Atomix - User contributions [en]

File:Rolf Lueck.jpeg

2024-06-09T17:40:02Z

Rolf: Picture of Rolf Lueck

== Summary ==
Picture of Rolf Lueck

Benchmark datasets for shear probes

2023-11-21T22:26:53Z

Rolf:

These datasets can be accessed from a [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAgL55HqB50DQd2J11m7kl9a?dl=0|read-only dropbox folder], and will eventually be housed at the BODC center with a DOI.

We provide a number of matlab routines to read in and compare data from grouped NetCDF files that follow our recommended structure on our [https://github.com/SCOR-ATOMIX/shear-probes GitHub repository]. The Matlab function ATOMIX_load.m there can be used to load a benchmark data file..

{| class="wikitable sortable"
|-
! Filename Prefix
! Platform
! Instrument
! Region
! PI (ATOMIX)
! Comment
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAajUI9HIodADTYRJSfYb3ba/VMP250_TidalChannel/VMP250_TidalChannel_024.nc?dl=0 VMP250_TidalChannel_024]
| Ship
| VMP-250
| Haro Strait
| Lueck
| Intense turbulence. Large up/down drafts. Most estimates require fitting to the inertial subrange.
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAD3PcHlwHjyrMAA7er6jRG_a/EPSIFISH_BLT_NORTHATL/epsifish_epsilometer_blt_north_atl.nc?dl=0 EPSILOMETER_RockallTrough]
| Ship
| Epsilometer
| Rockall Trough
| Le Boyer
| Strong turbulence in a canyon
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AABKhtIRZSuuZSfwTJ3xNNaNa/VMP2000_FaroeBankChannel/VMP2000_FaroeBankChannel.nc?dl=0 VMP2000_FaroeBankChannel]
| Ship
| VMP-2000
| Faroe Bank Channel (North Atlantic)
| Fer
| Ranging from quiescent mid-water to turbulent, deep gravity current
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAniGJlY-bwRplaiAV8AxC2a/MSS_BalticSea/MSS_Baltic_0330.nc?dl=0 MSS_BalticSea]
| Ship
| MSS
| Baltic Sea
| Holtermann
|
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 Nemo_MR1000_Minas_Passage_InStream]
| Mooring
| MicroRider
| Minas Passage (Bay of Fundy, NS)
| Lueck
| A swift tidal channel. All dissipation estimated from the inertial subrange
|-
|}

[[Category: Shear probes]]

-------------------------
return to [[Shear probes]] <br><br>
ATOMIX members: see also [[Tentative benchmarks for shear probes| tentative benchmarks for testers]]

Benchmark datasets for shear probes

2023-11-21T22:08:03Z

Rolf:

These datasets can be accessed from a [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAgL55HqB50DQd2J11m7kl9a?dl=0|read-only dropbox folder], and will eventually be housed at the BODC center with a DOI.

We provide a number of matlab routines to read in and compare data from grouped NetCDF files that follow our recommended structure on our [https://github.com/SCOR-ATOMIX/shear-probes GitHub repository]. The Matlab function ATOMIX_load.m there can be used to load a benchmark data file..

{| class="wikitable sortable"
|-
! Filename Prefix
! Platform
! Instrument
! Region
! PI (ATOMIX)
! Comment
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAajUI9HIodADTYRJSfYb3ba/VMP250_TidalChannel/VMP250_TidalChannel_024.nc?dl=0 VMP250_TidalChannel_024]
| Ship
| VMP-250
| Haro Strait
| Lueck
| Intense turbulence. Large up/down drafts. Most estimates require fitting to the inertial subrange.
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAD3PcHlwHjyrMAA7er6jRG_a/EPSIFISH_BLT_NORTHATL/epsifish_epsilometer_blt_north_atl.nc?dl=0 EPSILOMETER_RockallTrough]
| Ship
| Epsilometer
| Rockall Trough
| Le Boyer
| Strong turbulence in a canyon
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AABKhtIRZSuuZSfwTJ3xNNaNa/VMP2000_FaroeBankChannel/VMP2000_FaroeBankChannel.nc?dl=0 VMP2000_FaroeBankChannel]
| Ship
| VMP-2000
| Faroe Bank Channel (North Atlantic)
| Fer
| Ranging from quiescent mid-water to turbulent, deep gravity current
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAniGJlY-bwRplaiAV8AxC2a/MSS_BalticSea/MSS_Baltic_0330.nc?dl=0 MSS_BalticSea]
| Ship
| MSS
| Baltic Sea
| Holtermann
|
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 Nemo_MR1000_Minas_Passage_InStream]
| Mooring
| MicroRider
| Minas Passage (Bay of Fundy, NS)
| Lueck
| A swift tidal channel. All dissipation estimated from the inertial subrange
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 VMP_SA_Sodwana_Bay_KZN_048]
| Dive boat
| VMP-250-TE
| Agulhas Current(Sodwana Bay, SA)
| Lueck
| Western boundary current. A good and bad profile
|-
|}

[[Category: Shear probes]]

-------------------------
return to [[Shear probes]] <br><br>
ATOMIX members: see also [[Tentative benchmarks for shear probes| tentative benchmarks for testers]]

Benchmark datasets for shear probes

2023-11-21T22:06:36Z

Rolf:

These datasets can be accessed from a [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAgL55HqB50DQd2J11m7kl9a?dl=0|read-only dropbox folder], and will eventually be housed at the BODC center with a DOI.

We provide a number of matlab routines to read in and compare data from grouped NetCDF files that follow our recommended structure on our [https://github.com/SCOR-ATOMIX/shear-probes GitHub repository]. The Matlab function ATOMIX_load.m there can be used to load a benchmark data file..

{| class="wikitable sortable"
|-
! Filename Prefix
! Platform
! Instrument
! Region
! PI (ATOMIX)
! Comment
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAajUI9HIodADTYRJSfYb3ba/VMP250_TidalChannel/VMP250_TidalChannel_024.nc?dl=0 VMP250_TidalChannel_024]
| Ship
| VMP-250
| Haro Strait
| Lueck
| Intense turbulence. Large up/down drafts. Most estimates require fitting to the inertial subrange.
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAD3PcHlwHjyrMAA7er6jRG_a/EPSIFISH_BLT_NORTHATL/epsifish_epsilometer_blt_north_atl.nc?dl=0 EPSILOMETER_RockallTrough]
| Ship
| Epsilometer
| Rockall Trough
| Le Boyer
| Strong turbulence in a canyon
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AABKhtIRZSuuZSfwTJ3xNNaNa/VMP2000_FaroeBankChannel/VMP2000_FaroeBankChannel.nc?dl=0 VMP2000_FaroeBankChannel]
| Ship
| VMP-2000
| Faroe Bank Channel (North Atlantic)
| Fer
| ranging from quiescent mid-water to turbulent, deep gravity current
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAniGJlY-bwRplaiAV8AxC2a/MSS_BalticSea/MSS_Baltic_0330.nc?dl=0 MSS_BalticSea]
| Ship
| MSS
| Baltic Sea
| Holtermann
|
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 Nemo_MR1000_Minas_Passage_InStream]
| Mooring
| MicroRider
| Minas Passage (Bay of Fundy, NS)
| Lueck
| a swift tidal channel. Dissipation estimated from the inertial subrange
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 VMP_SA_Sodwana_Bay_KZN_048]
| Dive boat
| VMP-250-TE
| Agulhas Current(Sodwana Bay, SA)
| Lueck
| Western boundary current. A good and bad profile
|-
|}

[[Category: Shear probes]]

-------------------------
return to [[Shear probes]] <br><br>
ATOMIX members: see also [[Tentative benchmarks for shear probes| tentative benchmarks for testers]]

Benchmark datasets for shear probes

2023-11-21T22:05:44Z

Rolf:

These datasets can be accessed from a [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAgL55HqB50DQd2J11m7kl9a?dl=0|read-only dropbox folder], and will eventually be housed at the BODC center with a DOI.

We provide a number of matlab routines to read in and compare data from grouped NetCDF files that follow our recommended structure on our [https://github.com/SCOR-ATOMIX/shear-probes GitHub repository]. The Matlab function ATOMIX_load.m there can be used to load a benchmark data file..

{| class="wikitable sortable"
|-
! Filename Prefix
! Platform
! Instrument
! Region
! PI (ATOMIX)
! Comment
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAajUI9HIodADTYRJSfYb3ba/VMP250_TidalChannel/VMP250_TidalChannel_024.nc?dl=0 VMP250_TidalChannel_024]
| Ship
| VMP-250
| Haro Strait
| Lueck
| Intense turbulence. Large up/down drafts. Most estimates require fitting to the inertial subrange.
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAD3PcHlwHjyrMAA7er6jRG_a/EPSIFISH_BLT_NORTHATL/epsifish_epsilometer_blt_north_atl.nc?dl=0 EPSILOMETER_RockallTrough]
| Ship
| Epsilometer
| Rockall Trough
| Le Boyer
| Strong turbulence in a canyon
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AABKhtIRZSuuZSfwTJ3xNNaNa/VMP2000_FaroeBankChannel/VMP2000_FaroeBankChannel.nc?dl=0 VMP2000_FaroeBankChannel]
| Ship
| VMP-2000
| Faroe Bank Channel (North Atlantic)
| Fer
| ranging from quiescent mid-water to turbulent, deep gravity current
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAniGJlY-bwRplaiAV8AxC2a/MSS_BalticSea/MSS_Baltic_0330.nc?dl=0 MSS_BalticSea]
| Ship
| MSS
| Baltic Sea
| Holtermann
|
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 Nemo_MR1000_Minas_Passage_InStream]
| Mooring
| MicroRider
| Minas Passage (Bay of Fundy, NS)
| Lueck
| a swift tidal channel. Dissipation estimated from the inertial subrange
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 VMP_SA_Sodwana_Bay_KZN_048.nc]
| Dive boat
| VMP-250-TE
| Agulhas Current(Sodwana Bay, SA)
| Lueck
| Western boundary current. A good and bad profile
|-
|}

[[Category: Shear probes]]

-------------------------
return to [[Shear probes]] <br><br>
ATOMIX members: see also [[Tentative benchmarks for shear probes| tentative benchmarks for testers]]

Benchmark datasets for shear probes

2023-11-21T22:03:48Z

Rolf:

These datasets can be accessed from a [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAgL55HqB50DQd2J11m7kl9a?dl=0|read-only dropbox folder], and will eventually be housed at the BODC center with a DOI.

We provide a number of matlab routines to read in and compare data from grouped NetCDF files that follow our recommended structure on our [https://github.com/SCOR-ATOMIX/shear-probes GitHub repository]. The Matlab function ATOMIX_load.m there can be used to load a benchmark data file..

{| class="wikitable sortable"
|-
! Filename Prefix
! Platform
! Instrument
! Region
! PI (ATOMIX)
! Comment
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAajUI9HIodADTYRJSfYb3ba/VMP250_TidalChannel/VMP250_TidalChannel_024.nc?dl=0 VMP250_TidalChannel_024]
| Ship
| VMP-250
| Haro Strait
| Lueck
| Intense turbulence. Large up/down drafts. Most estimates require fitting to the inertial subrange.
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAD3PcHlwHjyrMAA7er6jRG_a/EPSIFISH_BLT_NORTHATL/epsifish_epsilometer_blt_north_atl.nc?dl=0 EPSILOMETER_RockallTrough]
| Ship
| Epsilometer
| Rockall Trough
| Le Boyer
| Strong turbulence in a canyon
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AABKhtIRZSuuZSfwTJ3xNNaNa/VMP2000_FaroeBankChannel/VMP2000_FaroeBankChannel.nc?dl=0 VMP2000_FaroeBankChannel]
| Ship
| VMP-2000
| Faroe Bank Channel (North Atlantic)
| Fer
| ranging from quiescent mid-water to turbulent, deep gravity current
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AAAniGJlY-bwRplaiAV8AxC2a/MSS_BalticSea/MSS_Baltic_0330.nc?dl=0 MSS_BalticSea]
| Ship
| MSS
| Baltic Sea
| Holtermann
|
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 Nemo_MR1000_Minas_Passage_InStream]
| Mooring
| MicroRider
| Minas Passage (Bay of Fundy, NS)
| Lueck
| a swift tidal channel. Dissipation estimated from the inertial subrange
|-
| [https://www.dropbox.com/sh/ybbpauv5e2n8xyp/AADqZw8zwf93HpZMhF8MWAd-a/Nemo_MR1000_Minas_Passage/Nemo_MR1000_Minas_Passage_InStream.nc?dl=0 Nemo_MR1000_Minas_Passage_InStream]
| Dive boat
| VMP-250-TE
| Agulhas Current(Sodwana Bay, SA)
| Lueck
| Western boundary current. A good and bad profile
|-
|}

[[Category: Shear probes]]

-------------------------
return to [[Shear probes]] <br><br>
ATOMIX members: see also [[Tentative benchmarks for shear probes| tentative benchmarks for testers]]

Iterative spectral integration algorithm

2022-12-09T20:20:20Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1
</math>

where

<math>
\begin{split}
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k \ \ .
\end{split}
</math>

Thus, integrating the spectrum, <math>\Psi</math>, to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

[[Figure_381.jpg]]
<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

File:Figure 381.jpg

2022-12-09T20:18:11Z

Rolf: The ratio of the true dissipation rate to the rate determined by integrating the spectrum to only 10 cpm, as a function of the 10-cpm rate for a range of kinematic viscosities. if the measured spectrum follows the Nasmyth spectrum.

== Summary ==
The ratio of the true dissipation rate to the rate determined by integrating the spectrum to only 10 cpm, as a function of the 10-cpm rate for a range of kinematic viscosities. if the measured spectrum follows the Nasmyth spectrum.

Iterative spectral integration algorithm

2022-12-09T20:15:24Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1
</math>

where

<math>
\begin{split}
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k \ \ .
\end{split}
</math>

Thus, integrating the spectrum, <math>\Psi</math>, to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

[[File:epsilon_10_to_epsilon_ratio.pdf]]
<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T20:12:16Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1
</math>

where

<math>
\begin{split}
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k \ \ .
\end{split}
</math>

Thus, integrating the spectrum, <math>\Psi</math>, to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

[[File:epsilon\_10\_to\_epsilon\_ratio.pdf]]
<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T20:10:36Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1
</math>

where

<math>
\begin{split}
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k \ \ .
\end{split}
</math>

Thus, integrating the spectrum, <math>\Psi</math>, to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

[[File:epsilon_10_to_epsilon_ratio.pdf]]
<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

File:Epsilon 10 to epsilon ratio.pdf

2022-12-09T20:07:16Z

Rolf: The ratio of the true dissipation rate to the dissipation rate determined by only integrating the spectrum of shear to 10 cpm, as a function of the 10cpm-rate for a range of kinematic viscosities.

== Summary ==
The ratio of the true dissipation rate to the dissipation rate determined by only integrating the spectrum of shear to 10 cpm, as a function of the 10cpm-rate for a range of kinematic viscosities.

Iterative spectral integration algorithm

2022-12-09T20:01:10Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1
</math>

where

<math>
\begin{split}
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k \ \ .
\end{split}
</math>

Thus, integrating the spectrum, <math>\Psi</math>, to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T19:58:41Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1
</math>

where

<math>
\begin{split}
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k \ \ .
\end{split}
</math>

Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T19:57:18Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1
</math>
where
<math>
\begin{split}
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k
\end{split}
</math>

Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T19:55:26Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\begin{split}
\frac{\varepsilon}{\varepsilon_{10}} &= \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 \\
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\
\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k
\end{split}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T19:54:06Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\begin{split}
\frac{\varepsilon}{\varepsilon_{10}} &= \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 \\
a &= 1.25\times 10^{-9}\, \nu^{-3} \\
b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\
\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k
\end{split}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T19:51:36Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\begin{split}
\frac{\varepsilon}{\varepsilon_{10}} &= \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 \\
a &= 1.25\times 10^{-9} \nu^{-3} \\
b &= 5.5\times 10^{-8} \nu^{-5/2}
\end{split}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T19:50:15Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1\\
a = 1.25\times 10^{-9} \nu^{-3} \\
b = 5.5\times 10^{-8} \nu^{-5/2}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-09T19:49:03Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}}= \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\rught) - 1\\
a = 1.25\times 10^{-9} \nu^{-3} \\
b = 5.5\times 10^{-8} \nu^{-5/2}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-07T18:54:11Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}}= \sqrt{1+a\varepsilon_{10}}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-07T18:52:46Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presets us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}}= \sqrt{1+a\varepsilon_{10}}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Iterative spectral integration algorithm

2022-12-07T18:49:14Z

Rolf:

Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman
<ref name="goodmanetal2006">{{Cite journal
|authors= L. Goodman, E. Levine, and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= On measuring the terms of the turbulent kinetic energy budget from an AUV
|year= 2006
|doi= 10.1175/JTECH1889.1
}}</ref>
algorithm, and on the wavenumber resolution of the shear probe.
The rate of dissipation is estimated using

<math>
\begin{equation}
\varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} \Psi(k) dk
\end{equation}
</math>

where <math>\Psi(k)</math> is the dimensional shear spectrum.
The lower limit of spectral integration is often set to <math>k_0=0\ \mathrm{cpm}</math> although it can also be set to the lowest non-zero wavenumber of a spectrum.
The spectrum at zero wavenumber, <math>\Psi(0)</math>, is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, <math>k_u</math>, is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, <math>k_{\mathrm{min}}</math>, sets one of the limits on <math>k_u</math>.

(2) Another upper limit is <math>k_{150} = 150\ \mathrm{cpm}</math> that is imposed by the spatial resolution of a commonly used shear probe.
You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004<ref name="macounlueck2004">{{Cite journal
|authors= P. Macoun and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= Modelling the spatial response of the airfoil shear probe using different sized probes
|year= 2004
|doi= 10.1175/1520-0426(2004)021
}}</ref>
At the wavenumber of <math>150\ \mathrm{cpm}</math> the spectrum derived from the commonly used shear probe is boosted by a factor of 10.
At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) The cut-off frequency, <math>f_A</math>, of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely <math>k_A \leq f_A/U</math>.
Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency.
For example, <math>k_A \leq 0.9\, f_A/U</math>.

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, <math>f_{\mathrm{lim}}</math>.
For most instruments this limit is usually set to <math>\infty</math>, but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, <math>k_{95}</math>, is the wavenumber at which the variance of shear is resolved to 95%.
There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%.
This wavenumber is <math> k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} </math> and the factor of <math>0.12</math>, is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= 10.1175/JTECH-D-21-0050.1
}}</ref>
, the Panchev-Kesich
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
, and the Lueck
<ref name="lueck2022b"></ref>
non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

<math>k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) </math>.

The last of these upper limits, <math>k_{95}</math>, presets us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum.
Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

The wavenumber range of the spectrum of shear depends on the rate of dissipation.
The spectrum broadens in proportion to <math>\epsilon^{1/4}</math> and the spectrum rises in proportion to <math>\epsilon^{3/4}</math>. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.

However, the non-dimensional spectrum

<math>G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k)</math>

where <math>\hat k=kL_k</math> and <math>L_k=(\nu^3/\varepsilon)^{1/4}</math> is the Kolmogorov length, is expected to be universal and independent of the rate of dissipation.
There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002)
<ref name="wolketal2002">{{Cite journal
|authors= F. Wolk, H. Yamazaki, L. Seuront, L., and R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= A new free-fall profiler for measuring biophysical microstructure
|year= 2002
|doi= 10.1175/1520-0426(2002)019
}}</ref>
who provides an approximation to the Nasmyth spectrum
<ref name="oakey1982">{{Cite journal
|authors= N. Oakey
|journal_or_publisher= J. Phys. Oceanogr.
|paper_or_booktitle= Determination of the Rate of Dissipation of Turbulent Kinetic Energy from Simultaneous Temperature and Velocity Shear Microstructure Measurements
|year= 1982
|doi= 10.1175/1520-0485(1982)012
}}</ref>
and Lueck (2022)
<ref name="lueck2022b">{{Cite journal
|authors= R. Lueck
|journal_or_publisher= J. Atmos. Oceanic Technol.
|paper_or_booktitle= The statistics of turbulence measurements. Part 2: Shear spectra and a new spectral model Shear spectra and a new spectral model
|year= 2022
|doi= TBD
}}</ref>
who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975)
<ref name="panchevkesich1969">{{Cite journal
|authors= S. Panchev and D. Kesich
|journal_or_publisher= Comptes rendus de lacademie Bulgare des sciences
|paper_or_booktitle= Energy spectrum of isotropic turbulence at large wavenumbers
|year= 1969
|doi= unknown
}}</ref>
spectrum, and a new approximation based on 14,600 dimensional spectra.

These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.

The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by

<math>
\frac{\varepsilon}{\varepsilon_{10}}= \sqrt{1+a\varepsilon_{10}}
</math>

where <math> a= 1.08 \times 10^9</math> Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as <math>\varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}}</math>, the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

<math></math>

==References==
<references />

----------------------------
return to [[Flow chart for shear probes]]

[[Category:Shear probes]]

Tentative benchmarks for shear probes

2022-06-13T16:15:20Z

Rolf:

Please follow the [[Filename convention for testing]] if changing the filename prefix supplied in this table.

==File name when testing==
FilenamePrefix_II.nc
where II is your initials (use as many letters as needed).

'''The table should include only prefix, not the full testing filename'''
{| class="wikitable sortable"
|-
! Filename Prefix
! Platform
! Instrument
! Region
! PI (ATOMIX)
! Comment
! Volunteer testers
|-
| VMP250_TidalChannel_024_cs
| Ship
| VMP-250
| Haro Strait
| Lueck
| Intense turbulence. | Large up/down drafts affecting fall rate; this one processed using a constant speed of <math>0.75\, \mathrm{m\,s^{-1}}</math>; some estimates require fitting to the inertial subrange.
| Dengler
|-
| VMP250_TidalChannel_024
| Ship
| VMP-250
| Haro Strait
| Lueck
| Intense turbulence. | Large up/down drafts affecting fall rate; this one processed using a speed equal to <math>\mathrm{d}P/\mathrm{d}t</math>; some estimates require fitting to the inertial subrange.
| All testers who have their own code.
|-
| FLOAT_MR_KY1310_Kuroshio
| Float
| Micro-Rider 1000
| Kuroshio
| Inoue
| Low epsilon with slow speeds past the sensor, NAVIS float.
| Bluteau, Fer
|-
| VMP_SA_Sodwana_Bay_KZN_048
| Ship
| VMP-250
| Agulhas Current
| Lueck
| Moderately intense turbulence. Two sections: one good, the other has descent glitches
| Holtermann, George
|-
| CPF_MR1000_Monterey_Bay_19_8
| Uprising float
| MR-1000
| Monterey Bay
| Lueck
| Example of a float rising at <math>0.20\,\mathrm{m\,s^{-1}}</math>; some wave orbital interference.
| Fer
|-
| mCTD_VL_StJohns_River_070
| Small boat
| mCTD
| St. Jons River, Florida
| Lueck
| Example of multiple shallow profiles in an estuary.
| Le Boyer
|-
| MicroCTD_UQAM_Malaysia_047_10
| Small boat
| mCTD
| Freshwater reservoir in Malaysia
| Lueck
| Example of poor data that looks good in a vertical profile of shear.
| Le Boyer, George
|-
| VMP2000_Test_HaroStrait
| Small ship
| VMP-2000
| Haro Strait, tidal channel
| Lueck
| Example of profiler with several narrow banded vibrations.
| Le Boyer
|-
| EPSILOMETER_RockallTrough
| Ship
| Epsilometer
| Rockall Trough
| Le Boyer
| Strong turbulence in a canyon
| Lueck, Bluteau, Ilker
|-
| AUV_MicroRider_BarentsSea
| AUV
| MicroRider
| Barents Sea (Arctic)
| Fer
| Propelled vehicle
| Dengler,Le Boyer
|-
| Glider_MicroRider_BarentsSea
| Glider
| MicroRider
| Barents Sea (Arctic)
| Fer
|
| Dengler
|-
| VMP6000_OgasawaraRidge
| Ship
| VMP-6000
| Ogasawara Ridge (western Pacific)
| Hibiya
| Untethered; >3000 m profile
| Lueck
|-
| Glider_MicroRider_Shelf_Peru
| Glider
| MicroRider<br />
| Continental slope off Peru, eastern tropical Pacific
| Dengler
| Electromagnetic current meter; strong tidal currents
| Le Boyer
|-
| MSS_BalticSea
| Ship
| MSS
| Baltic Sea
| Holtermann
| One good profile, one corrupted by jellyfish
| Bluteau, Fer
|-
| MSS_ArcticOcean
| Ice
| MSS
| Arctic Ocean
| Meyer
|
| Lueck
|-
| VMP6000_NorwegianSea
| Ship
| VMP-6000
| Norwegian Sea
| Fer
| In a strong & deep anticyclone, full depth (>3000 m)
| Bluteau, Inoue
|-
| Ice_VMP250upriser_ArcticOcean
| Ice
| VMP-250 upriser
| Arctic Ocean
| Fer
| One good and bad profile to highlight challenges with an upriser
| Bluteau
|-
| Nemo_MR1000_Minas_Passage_InStream
| Mooring
| MicroRider
| Minas Passage (Bay of Fundy, NS)
| Lueck
| a swift tidal channel. Dissipation estimated from the inertial subrange
| Fer, Bluteau
|}

[[Category: Shear probes]]

-------------------------
return to [[Shear probes]]

Level 4 data (shear probes)

2022-06-05T18:43:51Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name !! description
|-
| TIME_SPECTRA || length of the record of average times of spectral segments
|-
|N_SHEAR_SENSORS ||number of shear channels (shear sensors)
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || - || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| METHOD <math>^b</math>|| method used to make dissipation estimate || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate.

<math>^b</math> METHOD=0 for spectral integration, METHOD=1 for fitting to the inertial subrange.
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| KMIN || minimum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic_ energy_dissipation_estimate || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-06-05T18:42:52Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name !! description
|-
| TIME_SPECTRA || length of the record of average times of spectral segments
|-
|N_SHEAR_SENSORS ||number of shear channels (shear sensors)
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || - || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| METHOD <math>^b</math>|| method used to make dissipation estimate || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate.

<math>^b</math> 0 for spectral integration, 1 for fitting to the inertial subrange.
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| KMIN || minimum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic_ energy_dissipation_estimate || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-06-05T18:42:24Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name !! description
|-
| TIME_SPECTRA || length of the record of average times of spectral segments
|-
|N_SHEAR_SENSORS ||number of shear channels (shear sensors)
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || - || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| METHOD <math>^b</math>|| method used to make dissipation estimate || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate.
<math>^b</math> 0 for spectral integration, 1 for fitting to the inertial subrange.
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| KMIN || minimum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic_ energy_dissipation_estimate || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-06-05T17:36:01Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name !! description
|-
| TIME_SPECTRA || length of the record of average times of spectral segments
|-
|N_SHEAR_SENSORS ||number of shear channels (shear sensors)
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || - || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| KMIN || minimum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic_ energy_dissipation_estimate || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-06-05T17:35:07Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name !! description
|-
| TIME_SPECTRA || length of the record of average times of spectral segments
|-
|N_SHEAR_SENSORS ||number of shear channels (shear sensors)
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || - || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| <nowiki>PSPD_REL</nowiki> || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || - || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| KMIN || minimum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic_ energy_dissipation_estimate || - ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 3 data (shear probes)

2022-06-05T17:24:04Z

Rolf:

Level 3 data contains the raw (and optionally cleaned) [[Spectrum|spectra]] derived from level 2 time series. Level 3 parameters are defined along a new TIME [[Netcdf dimensions (shear probes)|dimension]], which is the average time of the individual spectral segments. The length of the TIME dimension equals the number of spectral segments. Parameter for the calculation of the spectra, e.g. segment length, are provided in the [[Netcdf meta data (shear probes)|meta data]].

=Dimensions=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>

{| class="wikitable sortable"
! Dimension name !! description
|-
| TIME_SPECTRA || length of the record of average times of spectral segments. This also equals time of dissipation estimates.
|-
| WAVENUMBER || length of the wavenumber array
|-
|N_SHEAR_SENSORS || number of shear channels (shear sensors)
|-
|N_***_SENSORS || number of *** channels (such as ACC and VIB)
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || - || TIME_SPECTRA
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| SH_SPEC || shear_probe_spectrum || s-2 cpm-1 || [TIME_SPECTRA, WAVENUMBER, N_SHEAR_SENSORS]
|-
| KCYC || Cyclic wavenumber || cpm || [TIME_SPECTRA, WAVENUMBER]
|-
| DOF || degrees_of_freedom_of_spectrum || - || 1
|}
</div>

=Optional Level 3 Variables=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| SH_SPEC_CLEAN || shear_probe_spectrum_clean || s-2 cpm-1 || [TIME_SPECTRA, WAVENUMBER, N_SHEAR_SENSORS]
|-
| ACC_SPEC || acceleration_sensor_spectrum || m2 s-4 cpm-1 || [TIME_SPECTRA, WAVENUMBER, N_ACCEL_SENSORS]
|-
| VIB_SPEC || vibration_sensor_spectrum || - || [TIME_SPECTRA, WAVENUMBER, N_VIB_SENSORS]
|-
| SH_VIB_SPEC ||shear_and_vibration_cross-spectral_matrix || - || [TIME_SPECTRA, WAVENUMBER, N_VIB_SENSORS*N_SHEAR_SENSORS]
|-
| SH_ACC_SPEC ||shear_and_acceleration_cross-spectral_matrix || - || [TIME_SPECTRA, WAVENUMBER, N_ACCEL_SENSORS*N_SHEAR_SENSORS]
|}
</div>

[[Category: Shear probes]]
-----
Return [[Dataset requirements for shear probes]]<br>
go to next: [[Level 4 data (shear probes)| Level 4 data]].

Netcdf meta data (shear probes)

2022-06-05T17:16:41Z

Rolf:

== Meta data (shear probes) ==

'''In progress''' <br>

=Required Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
|title || e.g., Vertical micostructure profiler data from cruise XXX
|-
|summary || required by ACDD
|-
|comment|| required by CF
|-
|platform_type ||research vessel, sub-surface glider etc
|-
|creation_time || yyyy-mm-ddTHH:MM:SSZ
|-
|date_created || yyyy-mm-ddTHH:MM:SSZ
|-
|date_update || yyyy-mm-ddTHH:MM:SSZ
|-
|time_reference_year || year for time reference
|-
| fs_fast || sampling frequency for fast channels
|-
| fs_slow <math>^a</math> || sampling frequency for slow channels
|-
| profiling_direction || horizontal, vertical, glide
|-
| fft_length || as data points (note, fft_lengths_sec in seconds is optional)
|-
| diss_length || as data points
|-
| overlap || as data points
|-
| goodman || 0=not applied; 1=applied
|-
| HP_cut || the high-pass filter cutoff frequency in Hz. Can be zero for no filtering.
|-
| conventions || CF-1.6, ACDD-1.3, ATOMIX
|-
| history || Version 1
|-
| data_mode || D #(D)elayed
|}
<math>^a</math> or fast_cdt or similar if such records are included
</div>

=Optional Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
| fft_length_sec || seconds
|-
| diss_length_sec || seconds
|-
| overlap_sec || seconds
|-
| f_AA || Hz, the anti-aliasing frequency.
|-
| FOM_limit || non-dimensional, if absent the FOM QC-flag is not set. Typically, 1.1.
|-
| diss_ratio_limit || non-dimensional, if absent the dissipation ration QC-flag is not set. Typically, 2.77.
|-
| despike_shear_limit || non-dimensional, if absent the de-spike fraction QC-flag is not set. Typically, 0.05.
|-
| despike_shear_pass_limit || non-dimensional, if absent the de-spike passes QC-flag is not set. Typically 8.
|-
|area || e.g., Arctic Ocean, Barents Sea
|-
|geospatial_lat_min ||
|-
|geospatial_lat_max ||
|-
|geospatial_lon_min ||
|-
|geospatial_lon_max ||
|-
|geospatial_vertical_min|| 0
|-
|geospatial_vertical_max|| in m
|-
|geospatial_vertical_positive|| down, up
|-
|time_coverage_start || yyyy-mm-ddTHH:MM:SSZ
|-
|time_coverage_end || yyyy-mm-ddTHH:MM:SSZ
|-
|institution ||
|-
|principal_investigator ||
|-
|authors ||
|-
|contact ||
|-
|project_name ||
|-
|cruise ||
|-
|vessel ||
|-
|source || From the SeaVoX Platform Categories vocabulary (L06) list, e.g. “subsurface mooring”, ”ship”, ""sub-surface gliders"", ""autonomous underwater vehicle"" (CF)
|-
|references ||
|-
|keywords ||
|-
|creator_name ||
|-
|creator_email ||
|-
|creator_url ||
|-
|acknowledgement||
|-
|station_name ||
|}

</div>

return to [[Dataset requirements for shear probes]]

[[Category:Shear probes]]

Netcdf meta data (shear probes)

2022-06-04T19:24:40Z

Rolf:

== Meta data (shear probes) ==

'''In progress''' <br>

=Required Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
|title || e.g., Vertical micostructure profiler data from cruise XXX
|-
|summary || required by ACDD
|-
|comment|| required by CF
|-
|platform_type ||research vessel, sub-surface glider etc
|-
|creation_time || yyyy-mm-ddTHH:MM:SSZ
|-
|date_created || yyyy-mm-ddTHH:MM:SSZ
|-
|date_update || yyyy-mm-ddTHH:MM:SSZ
|-
|time_reference_year || year for time reference
|-
| fs_fast || sampling frequency for fast channels
|-
| fs_slow <math>^a</math> || sampling frequency for slow channels
|-
| profiling_direction || horizontal, vertical, glide
|-
| fft_length || as data points (note, fft_lengths_sec in seconds is optional)
|-
| diss_length || as data points
|-
| overlap || as data points
|-
| goodman || 0=not applied; 1=applied
|-
| HP_cut || the high-pass filter cutoff frequency in Hz. Can be zero for no filtering.
|-
| conventions || CF-1.6, ACDD-1.3, ATOMIX
|-
| history || Version 1
|-
| data_mode || D #(D)elayed
|}
<math>^a</math> or fast_cdt or similar if such records are included
</div>

=Optional Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
| fft_length_sec || seconds
|-
| diss_length_sec || seconds
|-
| overlap_sec || seconds
|-
| FOM_limit || non-dimensional, if absent the FOM QC-flag is not set. Typically, 1.1.
|-
| diss_ratio_limit || non-dimensional, if absent the dissipation ration QC-flag is not set. Typically, 2.77.
|-
| despike_shear_limit || non-dimensional, if absent the de-spike fraction QC-flag is not set. Typically, 0.05.
|-
| despike_shear_pass_limit || non-dimensional, if absent the de-spike passes QC-flag is not set. Typically 8.
|-
|area || e.g., Arctic Ocean, Barents Sea
|-
|geospatial_lat_min ||
|-
|geospatial_lat_max ||
|-
|geospatial_lon_min ||
|-
|geospatial_lon_max ||
|-
|geospatial_vertical_min|| 0
|-
|geospatial_vertical_max|| in m
|-
|geospatial_vertical_positive|| down, up
|-
|time_coverage_start || yyyy-mm-ddTHH:MM:SSZ
|-
|time_coverage_end || yyyy-mm-ddTHH:MM:SSZ
|-
|institution ||
|-
|principal_investigator ||
|-
|authors ||
|-
|contact ||
|-
|project_name ||
|-
|cruise ||
|-
|vessel ||
|-
|source || From the SeaVoX Platform Categories vocabulary (L06) list, e.g. “subsurface mooring”, ”ship”, ""sub-surface gliders"", ""autonomous underwater vehicle"" (CF)
|-
|references ||
|-
|keywords ||
|-
|creator_name ||
|-
|creator_email ||
|-
|creator_url ||
|-
|acknowledgement||
|-
|station_name ||
|}

</div>

return to [[Dataset requirements for shear probes]]

[[Category:Shear probes]]

Iterations

2022-06-04T19:21:34Z

Rolf:

De-spiking is usually done in iterations.
Shear-probe data are replaced if they meet certain criterion for threshold and other properties.
The range that is replaced usually includes a small neighbourhood surrounding an anomaly.
However, the range that is replaced may not include all anomalies and consequently a subsequent attempt is usualy made to identify and remove the remaining anomalies.
These attempts are repeated until no more anomalies (or data extrema) are identified.

The number of passes or attempts made to clean the shear-probe data is a quality-control metric.
If many attempts are required to clean the data then the anomalies are extremely long and may be caused by collisions with objects large than the typical size of zooplankton such as, for example, jellyfish.

There is no objective criterion for the maximum number of passes that should be tolerated.
However, experience indicates that more than about 8 passes indicates that the data are very unusual and should not be used for the estimation of the rate of dissipation.

The number of de-spiking passes used to clean the shear data should be noted for every dissipation estimate and from every shear probe.
Similarly, the maximum number of passes that causes the estimate to be flagged for exclusion must also be noted.

Iterations

2022-06-04T19:20:00Z

Rolf:

De-spiking is usually done in iterations.
Shear-probe data are replaced if they meet certain criterion for threshold and other properties.
The range that is replaced usually includes a small neighbourhood surrounding an anomaly.
However, the range that is replaced may not include all anomalies and consequently a subsequent attempt is usualy made to identify and remove anomalies.
These attempts are repeated until no more anomalies (or data extrema) are identified.

The number of passes or attempts made to clean the shear-probe data is a quality-control metric.
If many attempts are required to clean the data then the anomalies are extremely long and may be caused by collisions with objects large than the typical size of zooplankton such as, for example, jellyfish.

There is no objective criterion for the maximum number of passes that should be tolerated.
However, experience indicates that more than about 8 passes indicates that the data are very unusual and should not be used for the estimation of the rate of dissipation.

The number of de-spiking passes used to clean the shear data should be noted for every dissipation estimate and from every shear probe.
Similarly, the maximum number of passes that causes the estimate to be flagged for exclusion must also be noted.

Netcdf meta data (shear probes)

2022-06-04T18:47:27Z

Rolf:

== Meta data (shear probes) ==

'''In progress''' <br>

=Required Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
|title || e.g., Vertical micostructure profiler data from cruise XXX
|-
|summary || required by ACDD
|-
|comment|| required by CF
|-
|platform_type ||research vessel, sub-surface glider etc
|-
|creation_time || yyyy-mm-ddTHH:MM:SSZ
|-
|date_created || yyyy-mm-ddTHH:MM:SSZ
|-
|date_update || yyyy-mm-ddTHH:MM:SSZ
|-
|time_reference_year || year for time reference
|-
| fs_fast || sampling frequency for fast channels
|-
| fs_slow <math>^a</math> || sampling frequency for slow channels
|-
| profiling_direction || horizontal, vertical, glide
|-
| fft_length || as data points (note, fft_lengths_sec in seconds is optional)
|-
| diss_length || as data points
|-
| overlap || as data points
|-
| goodman || 0=not applied; 1=applied
|-
| HP_cut || the high-pass filter cutoff frequency in Hz. Can be zero for no filtering.
|-
| conventions || CF-1.6, ACDD-1.3, ATOMIX
|-
| history || Version 1
|-
| data_mode || D #(D)elayed
|}
<math>^a</math> or fast_cdt or similar if such records are included
</div>

=Optional Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
| fft_length_sec || seconds
|-
| diss_length_sec || seconds
|-
| overlap_sec || seconds
|-
| FOM_limit || non-dimensional, if absent the FOM QC-flag is not set. Typically, 1.05.
|-
| diss_ratio_limit || non-dimensional, if absent the dissipation ration QC-flag is not set. Typically, 2.77.
|-
| despike_shear_limit || non-dimensional, if absent the de-spike fraction QC-flag is not set. Typically, 0.05.
|-
| despike_shear_pass_limit || non-dimensional, if absent the de-spike passes QC-flag is not set. Typically 10.
|-
|area || e.g., Arctic Ocean, Barents Sea
|-
|geospatial_lat_min ||
|-
|geospatial_lat_max ||
|-
|geospatial_lon_min ||
|-
|geospatial_lon_max ||
|-
|geospatial_vertical_min|| 0
|-
|geospatial_vertical_max|| in m
|-
|geospatial_vertical_positive|| down, up
|-
|time_coverage_start || yyyy-mm-ddTHH:MM:SSZ
|-
|time_coverage_end || yyyy-mm-ddTHH:MM:SSZ
|-
|institution ||
|-
|principal_investigator ||
|-
|authors ||
|-
|contact ||
|-
|project_name ||
|-
|cruise ||
|-
|vessel ||
|-
|source || From the SeaVoX Platform Categories vocabulary (L06) list, e.g. “subsurface mooring”, ”ship”, ""sub-surface gliders"", ""autonomous underwater vehicle"" (CF)
|-
|references ||
|-
|keywords ||
|-
|creator_name ||
|-
|creator_email ||
|-
|creator_url ||
|-
|acknowledgement||
|-
|station_name ||
|}

</div>

return to [[Dataset requirements for shear probes]]

[[Category:Shear probes]]

Netcdf meta data (shear probes)

2022-06-04T18:42:58Z

Rolf:

== Meta data (shear probes) ==

'''In progress''' <br>

=Required Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
|title || e.g., Vertical micostructure profiler data from cruise XXX
|-
|summary || required by ACDD
|-
|comment|| required by CF
|-
|platform_type ||research vessel, sub-surface glider etc
|-
|creation_time || yyyy-mm-ddTHH:MM:SSZ
|-
|date_created || yyyy-mm-ddTHH:MM:SSZ
|-
|date_update || yyyy-mm-ddTHH:MM:SSZ
|-
|time_reference_year || year for time reference
|-
| fs_fast || sampling frequency for fast channels
|-
| fs_slow <math>^a</math> || sampling frequency for slow channels
|-
| profiling_direction || horizontal, vertical, glide
|-
| fft_length || as data points (note, fft_lengths_sec in seconds is optional)
|-
| diss_length || as data points
|-
| overlap || as data points
|-
| goodman || 0=not applied; 1=applied
|-
| HP_cut || the high-pass filter cutoff frequency in Hz. Can be zero for no filtering.
|-
| conventions || CF-1.6, ACDD-1.3, ATOMIX
|-
| history || Version 1
|-
| data_mode || D #(D)elayed
|}
<math>^a</math> or fast_cdt or similar if such records are included
</div>

=Optional Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
| fft_length_sec || seconds
|-
| diss_length_sec || seconds
|-
| overlap_sec || seconds
|-
| FOM_limit || non-dimensional, if absent the FOM QC-flag is not set
|-
| diss_ratio_limit || non-dimensional, if absent the dissipation ration QC-flag is not set
|-
| despike_shear_limit || non-dimensional, if absent the despike fraction QC-flag is not set
|-
|area || e.g., Arctic Ocean, Barents Sea
|-
|geospatial_lat_min ||
|-
|geospatial_lat_max ||
|-
|geospatial_lon_min ||
|-
|geospatial_lon_max ||
|-
|geospatial_vertical_min|| 0
|-
|geospatial_vertical_max|| in m
|-
|geospatial_vertical_positive|| down, up
|-
|time_coverage_start || yyyy-mm-ddTHH:MM:SSZ
|-
|time_coverage_end || yyyy-mm-ddTHH:MM:SSZ
|-
|institution ||
|-
|principal_investigator ||
|-
|authors ||
|-
|contact ||
|-
|project_name ||
|-
|cruise ||
|-
|vessel ||
|-
|source || From the SeaVoX Platform Categories vocabulary (L06) list, e.g. “subsurface mooring”, ”ship”, ""sub-surface gliders"", ""autonomous underwater vehicle"" (CF)
|-
|references ||
|-
|keywords ||
|-
|creator_name ||
|-
|creator_email ||
|-
|creator_url ||
|-
|acknowledgement||
|-
|station_name ||
|}

</div>

return to [[Dataset requirements for shear probes]]

[[Category:Shear probes]]

Netcdf meta data (shear probes)

2022-06-04T18:36:17Z

Rolf:

== Meta data (shear probes) ==

'''In progress''' <br>

=Required Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
|title || e.g., Vertical micostructure profiler data from cruise XXX
|-
|summary || required by ACDD
|-
|comment|| required by CF
|-
|platform_type ||research vessel, sub-surface glider etc
|-
|creation_time || yyyy-mm-ddTHH:MM:SSZ
|-
|date_created || yyyy-mm-ddTHH:MM:SSZ
|-
|date_update || yyyy-mm-ddTHH:MM:SSZ
|-
|time_reference_year || year for time reference
|-
| fs_fast || sampling frequency for fast channels
|-
| fs_slow <math>^a</math> || sampling frequency for slow channels
|-
| profiling_direction || horizontal, vertical, glide
|-
| fft_length || as data points (note, fft_lengths_sec in seconds is optional)
|-
| diss_length || as data points
|-
| overlap || as data points
|-
| goodman || 0=not applied; 1=applied
|-
| conventions || CF-1.6, ACDD-1.3, ATOMIX
|-
| history || Version 1
|-
| data_mode || D #(D)elayed
|}
<math>^a</math> or fast_cdt or similar if such records are included
</div>

=Optional Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
| fft_length_sec || seconds
|-
| diss_length_sec || seconds
|-
| overlap_sec || seconds
|-
| FOM_limit || non-dimensional, If absent the FOM flag is not set
|-
| overlap_sec || seconds
|-
| overlap_sec || seconds
|-
|area || e.g., Arctic Ocean, Barents Sea
|-
|geospatial_lat_min ||
|-
|geospatial_lat_max ||
|-
|geospatial_lon_min ||
|-
|geospatial_lon_max ||
|-
|geospatial_vertical_min|| 0
|-
|geospatial_vertical_max|| in m
|-
|geospatial_vertical_positive|| down, up
|-
|time_coverage_start || yyyy-mm-ddTHH:MM:SSZ
|-
|time_coverage_end || yyyy-mm-ddTHH:MM:SSZ
|-
|institution ||
|-
|principal_investigator ||
|-
|authors ||
|-
|contact ||
|-
|project_name ||
|-
|cruise ||
|-
|vessel ||
|-
|source || From the SeaVoX Platform Categories vocabulary (L06) list, e.g. “subsurface mooring”, ”ship”, ""sub-surface gliders"", ""autonomous underwater vehicle"" (CF)
|-
|references ||
|-
|keywords ||
|-
|creator_name ||
|-
|creator_email ||
|-
|creator_url ||
|-
|acknowledgement||
|-
|station_name ||
|}

</div>

return to [[Dataset requirements for shear probes]]

[[Category:Shear probes]]

Netcdf meta data (shear probes)

2022-06-04T18:35:08Z

Rolf:

== Meta data (shear probes) ==

'''In progress''' <br>

=Required Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
|title || e.g., Vertical micostructure profiler data from cruise XXX
|-
|summary || required by ACDD
|-
|comment|| required by CF
|-
|platform_type ||research vessel, sub-surface glider etc
|-
|creation_time || yyyy-mm-ddTHH:MM:SSZ
|-
|date_created || yyyy-mm-ddTHH:MM:SSZ
|-
|date_update || yyyy-mm-ddTHH:MM:SSZ
|-
|time_reference_year || year for time reference
|-
| fs_fast || sampling frequency for fast channels
|-
| fs_slow <math>^a</math> || sampling frequency for slow channels
|-
| profiling_direction || horizontal, vertical, glide
|-
| fft_length || as data points (note, fft_lengths_sec in seconds is optional)
|-
| diss_length || as data points
|-
| overlap || as data points
|-
| goodman || 0=not applied; 1=applied
|-
| conventions || CF-1.6, ACDD-1.3, ATOMIX
|-
| history || Version 1
|-
| data_mode || D #(D)elayed
|}
<math>^a</math> or fast_cdt or similar if such records are included
</div>

=Optional Metadata=
<div class="mw-collapsible mw-expand" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Description
|-
| fft_length_sec || seconds
|-
| diss_length_sec || seconds
|-
| overlap_sec || seconds
|-
| FOM_limit || non-dimensional
|If absent the FOM flag is not set
| overlap_sec || seconds
|-
| overlap_sec || seconds
|-
|area || e.g., Arctic Ocean, Barents Sea
|-
|geospatial_lat_min ||
|-
|geospatial_lat_max ||
|-
|geospatial_lon_min ||
|-
|geospatial_lon_max ||
|-
|geospatial_vertical_min|| 0
|-
|geospatial_vertical_max|| in m
|-
|geospatial_vertical_positive|| down, up
|-
|time_coverage_start || yyyy-mm-ddTHH:MM:SSZ
|-
|time_coverage_end || yyyy-mm-ddTHH:MM:SSZ
|-
|institution ||
|-
|principal_investigator ||
|-
|authors ||
|-
|contact ||
|-
|project_name ||
|-
|cruise ||
|-
|vessel ||
|-
|source || From the SeaVoX Platform Categories vocabulary (L06) list, e.g. “subsurface mooring”, ”ship”, ""sub-surface gliders"", ""autonomous underwater vehicle"" (CF)
|-
|references ||
|-
|keywords ||
|-
|creator_name ||
|-
|creator_email ||
|-
|creator_url ||
|-
|acknowledgement||
|-
|station_name ||
|}

</div>

return to [[Dataset requirements for shear probes]]

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-05-24T22:04:40Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name
! description
|-
| TIME_SPECTRA
| Time axis for the spectral estimates
|-
|N_SHEAR_SENSORS
|number of shear channels
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || dimensionless || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || dimensionaless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || dimensionless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || dimensionless ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || dimensionless ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic_ energy_dissipation_estimate || dimensionless ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-05-24T22:03:14Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name
! description
|-
| TIME_SPECTRA
| Time axis for the spectral estimates
|-
|N_SHEAR_SENSORS
|number of shear channels
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || dimensionless || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || dimensionaless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || dimensionless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic_ energy_dissipation_estimate || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-05-24T22:02:40Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name
! description
|-
| TIME_SPECTRA
| Time axis for the spectral estimates
|-
|N_SHEAR_SENSORS
|number of shear channels
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || dimensionless || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_ turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || dimensionaless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || dimensionless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic energy_dissipation_estimate || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-05-24T22:01:29Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name
! description
|-
| TIME_SPECTRA
| Time axis for the spectral estimates
|-
|N_SHEAR_SENSORS
|number of shear channels
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || dimensionless || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || dimensionaless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || dimensionless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic energy_dissipation_estimate || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-05-24T22:00:33Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name
! description
|-
| TIME_SPECTRA
| Time axis for the spectral estimates
|-
|N_SHEAR_SENSORS
|number of shear channels
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || dimensionless || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || dimensionaless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || <math>\mathrm{m^2\,s^{-1}</math>m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || dimensionless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || standard_deviation_of_kinetic energy_dissipation_estimate || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-05-24T21:56:20Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name
! description
|-
| TIME_SPECTRA
| Time axis for the spectral estimates
|-
|N_SHEAR_SENSORS
|number of shear channels
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || dimensionless || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_turbulent_kinetic_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || dimensionaless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || dimensionless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || FIX THIS standard_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Level 4 data (shear probes)

2022-05-24T21:53:55Z

Rolf:

Level 4 data include the final dissipation estimates as time series, as well as indicators for the quality and accuracy of the estimates and additional derived parameters. Each dissipation estimate in level 4 corresponds to a spectrum in the level 3 data. Consequently, the level 3 and 4 TIME dimensions are the same. Parameter EPSI_FINAL is the final dissipation rate estimate, averaged of the selected estimates (using the [[Quality_control_coding | QC flags]]) at that depth.

=Dimensions=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
! Dimension short name
! description
|-
| TIME_SPECTRA
| Time axis for the spectral estimates
|-
|N_SHEAR_SENSORS
|number of shear channels
|}
</div>

=Required Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimensions
|-
| TIME || time || Days since YYYY-MM-DDT00:00:00Z || TIME_SPECTRA
|-
| SECTION_NUMBER || || dimensionless || TIME_SPECTRA
|-
| PSPD_REL || flow_past_sensor || m s-1 || TIME_SPECTRA ||
|-
| EPSI || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FINAL || specific_turbulent_kinetic_energy_dissipation_in_water || W kg-1 || TIME_SPECTRA
|-
| KMAX || maximum_wavenumber_used_for_estimating_turbulent_kinetic
_energy_dissipation || cpm || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_FLAGS <math>^a</math> || dissipation_qc_flags || dimensionaless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|}
<math>^a</math> Quality control coding for the final dissipation estimate
</div>

=Optional Level 4 Variables=
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
<br>
{| class="wikitable sortable"
|-
! Parameter Name !! Standard Name !! Units of measurement !! dimension
|-
| PSPD_REL || platform_speed_wrt_sea_water || m s-1 || TIME_SPECTRA
|-
| PRES || water_pressure || dbar || TIME_SPECTRA
|-
| KVISC || kinematic_viscosity_of_water || m2 s-1 || TIME_SPECTRA
|-
| FOM || figure_of_merit || dimensionless || [TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| MAD || mean_absolute_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| VAR_RESOLVED || variance_resolved || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|-
| EPSI_STD || FIX THIS standard_deviation || ||[TIME_SPECTRA, N_SHEAR_SENSORS]
|}

</div>

-----
Return [[Dataset requirements for shear probes]].

[[Category:Shear probes]]

Nomenclature

2022-05-24T00:31:20Z

Rolf:

== Background (total) velocity ==
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">

{| class="wikitable sortable"
|-
! Symbol
! Description
! Units
|-
| <math>u</math>
| zonal or longitudinal component of velocity
| <math> \mathrm{m\, s^{-1}}</math>
|-
| <math>v</math>
| meridional or transverse component of velocity
| <math>\mathrm{m\, s^{-1}}</math>
|-
| <math>w</math>
| vertical component of velocity
| <math> \mathrm{m\, s^{-1}}</math>
|-
| <math>u_e</math>
| error velocity
| <math>\mathrm{m\, s^{-1}}</math>
|-
| V
| velocity perpendicular to mean flow
| <math>\mathrm{m\, s^{-1}}</math>
|-
| <math>W_d</math>
| Profiler fall speed
| <math>\mathrm{m\, s^{-1}}</math>
|-
| <math>U_P</math>
| Flow speed past sensor
| <math>\mathrm{m\, s^{-1}}</math>
|-
| b
| Along-beam velocity from acoustic Doppler sensor
| <math>\mathrm{m\, s^{-1}}</math>
|-
| <math> b^{\prime}</math>
| Along-beam velocity from acoustic Doppler sensor with background flow deducted
| <math>\mathrm{m\, s^{-1}}</math>
|-
| <math> \delta{z}</math>
| Vertical size of measurement bin for acoustic Doppler sensor
| <math>\mathrm{m}</math>
|-
| r
| Along-beam distance from acoustic Doppler sensor
| <math>\mathrm{m}</math>
|-
| <math> \delta{r}</math>
| Along-beam bin size for acoustic Doppler sensor
| <math>\mathrm{m}</math>
|-
| <math> \theta</math>
| Beam transmit and receive angle relative to instrument axis for acoustic Doppler sensor
| <math>^{\circ}</math>
|}
</div>

== Turbulence properties ==
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">

{| class="wikitable sortable"
|-
! Symbol
! Description
! Eqn
! Units
|-
| <math>\varepsilon</math>
| The rate of dissipation of turbulent kinetic energy per unit mass by viscosity
|
| <math>\mathrm{W\, kg^{-1}}</math>
|-
| <math>B</math>
| Buoyancy production -- the rate of production of potential energy by turbulence in a stratified flow through the vertical flux of buoyancy.
| <math>B= \frac{g}{\rho} \overline{\rho'w'} </math>
| <math>\mathrm{W\, kg^{-1}}</math>
|-
| <math>P</math>
| The production of turbulence kinetic energy. In a steady, spatially uniform and stratified shear flow, turbulence kinetic energy is produced by the product of the Reynolds stress and the shear, for example <math>P = -\overline{u'w'}\frac{\partial U}{\partial z} </math> . The production is balanced by the rate of dissipation turbulence kinetic energy, <math>\varepsilon</math>, and the production of potential energy by the buoyancy flux, <math>B</math>.
| <math>P = -\overline{u'w'}\frac{\partial U}{\partial z} = \varepsilon + B</math>
| <math>\mathrm{W\, kg^{-1}}</math>
|-
| <math>R_f</math>
| Flux Richardson number; the ratio of the buoyancy flux expended for the net change in potential energy (i.e., mixing) to the shear production of turbulent kinetic energy.
| <math>R_f = \frac{B}{P}</math>
|
|-
| <math>\Gamma</math>
| "Mixing coefficient"; The ratio of the rate of production of potential energy, <math>B</math>, to the rate of dissipation of kinetic energy, <math>\varepsilon</math>.
| <math>\Gamma = \frac{B}{\varepsilon} = \frac{R_f}{1-R_f}</math>
|
|-
| <math>R_i</math>
| (Gradient) Richardson number; the ratio of buoyancy freqency squared to velocity shear squared
| <math>R_i = \frac{N^2}{S^2} </math>
|
|-
| <math>\kappa_{\rho}</math>
| Turbulent eddy diffusivity via the Osborn (1980) model
| <math>\kappa_{\rho} = \Gamma \varepsilon N^{-2}</math>
| <math>\mathrm{m^2\, s^{-1}}</math>
|-
| <math>D_{ll}</math>
| Second-order longitudinal structure function
| <math>D_{ll} = \big\langle[b^{\prime}(r) - b^{\prime}(r+n\delta{r})]^2\big\rangle</math>
| <math>\mathrm{m^2\, s^{-2}}</math>
|}
</div>

== Fluid properties and background gradients for turbulence calculations ==
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">

{| class="wikitable sortable"
|-
! Symbol
! Description
! Eqn
! Units
|-
| <math>S_P</math>
| Practical salinity
|
| <math> - </math>
|-
| <math>T</math>
| Temperature
|
| <math> \mathrm{^{\circ}C } </math>
|-
| <math>P</math>
| Pressure
|
| <math>\mathrm{dbar} </math>
|-
| <math>\rho</math>
| Density of water
| <math> \rho = \rho\left(T,S_a,P \right)</math>
| <math>\mathrm{kg\, m^{-3}} </math>
|-
| <math>\alpha</math>
| Temperature coefficient of expansion
| <math> \alpha = \frac{1}{\rho} \frac{\partial\rho}{\partial T}</math>
| <math> \mathrm{K^{-1}}</math>
|-
| <math>\beta</math>
| Saline coefficient of contraction
| <math> \beta = \frac{1}{\rho} \frac{\partial\rho}{\partial S_P}</math>
|
|-
| <math>S</math>
| Background velocity shear
| <math> S = \left[ \left( \frac{\partial U}{\partial z}\right)^2 + \left( \frac{\partial V}{\partial z}\right)^2 \right]^{1/2} </math>
| <math> \mathrm{s^{-1}} </math>
|-
| <math> \nu_{35} </math>
| Temperature dependent kinematic viscosity of seawater at a practical salinity of 35
| <math> \sim 1\times 10^{-6} </math>
| <math> \mathrm{m^2\, s^{-1} } </math>
|-
| <math>\nu_{00}</math>
| Temperature dependent kinematic viscosity of freshwater
| <math>\sim 1\times 10^{-6} </math>
| <math>\mathrm{m^2\, s^{-1} } </math>
|-
| <math>\Gamma_a </math>
| Adiabatic temperature gradient -- salinity, temperature and pressure dependent
| <math>\sim 1\times 10^{-4}</math>
| <math>\mathrm{K\, dbar^{-1} } </math>
|-
| <math>N </math>
| Background stratification, i.e buoyancy frequency
| <math>N^2 = g\left[ \alpha\left(\Gamma_a + \frac{\partial T}{\partial z} \right) - \beta \frac{\partial S_P}{\partial z} \right] </math>
| <math>\mathrm{rad\, s^{-1} } </math>
|}
</div>

== Theoretical Length and Time Scales ==
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">

{| class="wikitable sortable"
|-
! Symbol
! Description
! Eqn
! Units
|-
| <math>\tau_N</math>
| Buoyancy timescale
| <math> \tau_N = \frac{1}{N}</math>
| <math> \mathrm{s} </math>
|-
| <math>T_N</math>
| Buoyancy period
| <math> T_N = \frac{2\pi}{N}</math>
| <math> \mathrm{s} </math>
|-
| <math>L_E</math>
| Ellison length scale (limit of vertical displacement without irreversible mixing)
| <math>L_E=\frac {\langle \rho'^2\rangle^{1/2}}{\partial \overline{\rho}/\partial z}</math>
| <math> \mathrm{m} </math>
|-
| <math> L_Z</math>
| Boundary (law of the wall) length scale
| <math> L_Z=0.39z_w </math> with 0.39 being von Kármán's constant
| <math> \mathrm{m} </math>
|-
| <math>L_S</math>
| Corssin length scale
| <math> L_S = \sqrt{\varepsilon/S^3} </math>
| <math> \mathrm{m} </math>
|-
| <math>L_K</math>
| Kolmogorov length scale (smallest overturns)
| <math>L_K=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4}</math>
| <math> \mathrm{m} </math>
|-
| <math>L_o</math>
| Ozmidov length scale, measure of largest overturns in a stratified fluid
| <math>L_o=\left(\frac{\varepsilon}{N^3}\right)^{1/2}</math>
| <math> \mathrm{m} </math>
|-
| <math>L_T</math>
| Thorpe length scale
| <math>L_T</math>
| <math> \mathrm{m} </math>
|-
| <math>z_w</math>
| Distance from a boundary
| <math>z_w</math>
| <math> \mathrm{m} </math>
|}
</div>

== Turbulence Spectrum ==
<div class="mw-collapsible mw-collapsed" data-collapsetext="Collapse" data-expandtext="Expand">
These variables are used to express the [[Turbulence spectrum]] expected shapes.

{| class="wikitable sortable"
|- Style="font-weight:bold; "
! Symbol
! Description
! Eqn
! Units
|-
| <math>\Delta t</math>
| Sampling interval
| <math> \frac{1}{f_s} </math>
| <math> \mathrm{s} </math>
|-
| <math>f_s</math>
| Sampling rate
| <math>f_s=\frac{1}{\Delta t} </math>
| <math> \mathrm{s^{-1}} </math>
|-
| <math>\Delta s</math>
| Sample spacing
| <math> \Delta s = U_P \Delta t </math>
| <math> \mathrm{m} </math>
|-
| <math>\Delta l</math>
| Linear dimension of sampling volume (instrument dependent)
|
| <math> \mathrm{m} </math>
|-
| <math>f</math>
| Cyclic frequency
| <math>f=\frac{\omega}{2\pi}</math>
| <math> \mathrm{Hz} </math>
|-
| <math>\omega</math>
| Angular frequency
| <math>\omega = 2\pi f</math>
| <math> \mathrm{rad\, s^{-1}} </math>
|-
| <math>f_N</math>
| Nyquist frequency
| <math>f_N=0.5f_s</math>
| <math> \mathrm{Hz} </math>
|-
| <math>k</math>
| Cyclic wavenumber
| <math>k=\frac{f}{U_P}</math>
| <math> \mathrm{cpm} </math>
|-
| <math>\hat{k}</math>
| Angular wavenumber
| <math>\hat{k}=\frac{\omega}{U_P} = 2\pi k</math>
| <math> \mathrm{rad\, m^{-1}} </math>
|-
| <math>\tilde{k}</math>
| Normalized wavenumber
| e.g., <math>\tilde{k}=k L_K, L_K = \left(\nu^3/\varepsilon \right)^{1/4}</math>
| -
|-
| <math>\tilde{\Phi}</math>
| Normalized velocity spectrum
| e.g., <math>\tilde{\Phi}_u(\tilde{k}) = \left(\epsilon \nu^5\right)^{-1/4} \Phi_u(k)</math>
| -
|-
| <math>\tilde{\Psi}</math>
| Normalized shear spectrum
| e.g., <math>\tilde{\Psi}(\tilde{k}) = L_K^2 \left(\epsilon \nu^5\right)^{-1/4} \Psi(k)</math>
| -
|-
| <math>k_\Delta</math>
| Nyquist wavenumber, based on sampling volume size <math>\Delta l</math>
| <math>k_\Delta=\frac{0.5}{\Delta l}</math>
| <math> \mathrm{cpm} </math>
|-
| <math>k_N</math>
| Nyquist wavenumber, via Taylor's hypothesis
| <math>k_N=\frac{f_N}{U_P}</math>
| <math> \mathrm{cpm} </math>
|-
| <math>\Psi(k)</math>
| Shear spectrum. Use <math>\Psi_1</math>, <math>\Psi_2</math> to distinguish the orthogonal components of the shear. Use <math>\Psi_N</math> for the Nasmyth spectrum, <math>\Psi_{PK}</math> for the Panchev-Kesich spectrum and <math>\Psi_L</math> for the Lueck spectrum.
|
| <math> \mathrm{s^{-2}\, cpm^{-1}}</math>
|-
| <math>\Phi(k)</math>
| Velocity spectrum. Use <math>\Phi_u</math>, <math>\Phi_v</math>, <math>\Phi_v</math>, or <math>\Phi_1</math>, <math>\Phi_2</math> , <math>\Phi_3</math> for the different orthogonal components of the velocity. Use <math>\Phi_K</math> for the Kolmogorov spectrum.
|
| <math> \mathrm{m^2\, s^{-2}\, cpm^{-1}} </math>
|}
</div>

[[Category:Glossary]]

Quality control coding

2022-05-24T00:22:51Z

Rolf:

'''Shear-probe quality-control flags'''

The Q (quality control) flags associated with shear-probe measurements are not compatible with the Ocean Sites [http://www.oceansites.org/ Ocean Sites] for quality control (QC) coding.

Every dissipation estimate from every probe must have Q flag.
The numerical values of the Q flags are as follows:

Q =
0, if all checks pass
1, if FOM > FOM_limit
2, if despike_fraction > despike_fraction_limit
4, if |log(e_max)-log(e_min)|> diss_ratio_limit X \sigma_{\ln\varepsilon}
8, if despike_iterations > despike_iterations_limit

The Q flags are combined by their addition.
For example a Q value of 3 means that the dissipation estimated failed both FOM_limit test and the despike_fraction test.
A value of 15 means that all tests failed.
A failure of any one test (<math>Q\ne0</math>) means that a dissipation test should not be trusted.
The reasons for a failure can be decoded by breaking the value of Q down to its powers of 2.

Another flag that can be used (in addition to the Quality flag) is GOOD PROBE:
GOOD_PROBE
0, all probes are good
1, sh1 only
2, sh2 only
3, sh1 and sh2
4, sh3 only
5, sh1 and sh3
6, sh2 and sh3
7, sh1, sh2 and sh3
8, sh4 only
999, all bad

However, this is already provided by the above recommended Q flags.

'''Ocean Sites'''
Providing quality-control flags according to Ocean Sites is encouraged.
These are described at [http://www.oceansites.org/ Ocean Sites] for quailty control (QC) coding.
This flagging scheme is mostly compatible with the primary level flagging recommended by [http://www.ioccp.org/images/D4standards/IOC-OceanDataStandards54-3-2013.pdf Intergovernmental Oceanographic Commission of UNESCO (2013)].
However, only the flags of 0, 1, and 4 make sense for dissipation estimates derived from shear-probe data.

{| class="wikitable"
|-
! Flag
! Meaning
! Comment
|-
| 0
| unknown
| No QC was performed.
|-
| 1
| good data
| All QC tests passed.
|-
| 2
| probably good data
| Data have failed one or more QC tests but detailed examination after processing (e.g. by visual examination) suggests data is good.
|-
| 3
| potentially correctable bad data
| These data are not to be used without scientific correction or re-calibration (e.g. uncertain shear sensor sensitivity).
|-
| 4
| bad data
| Data have failed one or more tests.
|-
| 5
| -
| Not used
|-
| 6
| -
| Not used
|-
| 7
| nominal value
| Data were not observed but reported (e.g. instrument target depth.).
|-
| 8
| interpolated value
| Missing data may be interpolated from neighboring data in space or time.
|-
| 9
| missing value
| This is a fill value
|}

<br />
[Rolf left the stuff below in place because I do not know what to do with it.]

Climate and Forecast Metadata Convention (CF) requires that QC flags carry attributes. In netCDF (Network Common Data Form) data files, the following information for quality control flagging should be provided for each data variable <PARAM>. <br />
<br />
<PARAM>_QC <br />
<PARAM>_QC:long_name = “quality flag of <PARAM>”; <br />
<PARAM>_QC:conventions = “OceanSITES QC Flags”; <br />
<PARAM>_QC:flag_values = 0, 1, 2, 3, 4, 7, 8, 9; <br />
<PARAM>_QC:flag_meanings = “0:unknown 1:good_data 2:probably_good_data 3:potentially_correctable_bad_data 4:bad_data 7:nominal_value 8:interpolated_value 9:missing_value” <br/>

[[Category: Shear probes]]

Quality control coding

2022-05-24T00:18:32Z

Rolf:

'''Shear-probe quality-control flags'''

The Q (quality control) flags associated with shear-probe measurements are not compatible with the Ocean Sites [http://www.oceansites.org/ Ocean Sites] for quality control (QC) coding.

Every dissipation estimate from every probe must have Q flag.
The numerical values of the Q flags are as follows:

Q =
0, if all checks pass
1, if FOM > FOM_limit
2, if despike_fraction > despike_fraction_limit
4, if |log(e_max)-log(e_min)|> diss_ratio_limit <math>\times \sigma_{\ln\varepsilon}</math>
8, if despike_iterations > despike_iterations_limit

The Q flags are combined by their addition.
For example a Q value of 3 means that the dissipation estimated failed both FOM_limit test and the despike_fraction test.
A value of 15 means that all tests failed.
A failure of any one test (<math>Q\ne0</math>) means that a dissipation test should not be trusted.
The reasons for a failure can be decoded by breaking the value of Q down to its powers of 2.

Another flag that can be used (in addition to the Quality flag) is GOOD PROBE:
GOOD_PROBE
0, all probes are good
1, sh1 only
2, sh2 only
3, sh1 and sh2
4, sh3 only
5, sh1 and sh3
6, sh2 and sh3
7, sh1, sh2 and sh3
8, sh4 only
999, all bad

However, this is already provided by the above recommended Q flags.

'''Ocean Sites'''
Providing quality-control flags according to Ocean Sites is encouraged.
These are described at [http://www.oceansites.org/ Ocean Sites] for quailty control (QC) coding.
This flagging scheme is mostly compatible with the primary level flagging recommended by [http://www.ioccp.org/images/D4standards/IOC-OceanDataStandards54-3-2013.pdf Intergovernmental Oceanographic Commission of UNESCO (2013)].
However, only the flags of 0, 1, and 4 make sense for dissipation estimates derived from shear-probe data.

{| class="wikitable"
|-
! Flag
! Meaning
! Comment
|-
| 0
| unknown
| No QC was performed.
|-
| 1
| good data
| All QC tests passed.
|-
| 2
| probably good data
| Data have failed one or more QC tests but detailed examination after processing (e.g. by visual examination) suggests data is good.
|-
| 3
| potentially correctable bad data
| These data are not to be used without scientific correction or re-calibration (e.g. uncertain shear sensor sensitivity).
|-
| 4
| bad data
| Data have failed one or more tests.
|-
| 5
| -
| Not used
|-
| 6
| -
| Not used
|-
| 7
| nominal value
| Data were not observed but reported (e.g. instrument target depth.).
|-
| 8
| interpolated value
| Missing data may be interpolated from neighboring data in space or time.
|-
| 9
| missing value
| This is a fill value
|}

<br />
[Rolf left the stuff below in place because I do not know what to do with it.]

Climate and Forecast Metadata Convention (CF) requires that QC flags carry attributes. In netCDF (Network Common Data Form) data files, the following information for quality control flagging should be provided for each data variable <PARAM>. <br />
<br />
<PARAM>_QC <br />
<PARAM>_QC:long_name = “quality flag of <PARAM>”; <br />
<PARAM>_QC:conventions = “OceanSITES QC Flags”; <br />
<PARAM>_QC:flag_values = 0, 1, 2, 3, 4, 7, 8, 9; <br />
<PARAM>_QC:flag_meanings = “0:unknown 1:good_data 2:probably_good_data 3:potentially_correctable_bad_data 4:bad_data 7:nominal_value 8:interpolated_value 9:missing_value” <br/>

[[Category: Shear probes]]

Quality control coding

2022-05-24T00:17:21Z

Rolf:

'''Shear-probe quality-control flags'''

The Q (quality control) flags associated with shear-probe measurements are not compatible with the Ocean Sites [http://www.oceansites.org/ Ocean Sites] for quality control (QC) coding.

Every dissipation estimate from every probe must have Q flag.
The numerical values of the Q flags are as follows:

Q =
0, if all checks pass
1, if FOM > FOM_limit
2, if despike_fraction > despike_fraction_limit
4, if |log(e_max)-log(e_min)|> diss_ratio_limit <math>\times \sigma_{\ln\varepsilon}</math>
8, if despike_iterations > despike_iterations_limit

The Q flags are combined by their addition.
For example a Q value of 3 means that the dissipation estimated failed both FOM_limit test and the despike_fraction test.
A value of 15 means that all tests failed.
A failure of any one test (<math>Q\ne0</math>) means that a dissipation test should not be trusted.
The reasons for a failure can be decoded by breaking the value of Q down to its powers of 2.

Another flag that can be used (in addition to the Quality flag) is GOOD PROBE:
GOOD_PROBE
0, all probes are good
1, sh1 only
2, sh2 only
3, sh1 and sh2
4, sh3 only
5, sh1 and sh3
6, sh2 and sh3
7, sh1, sh2 and sh3
8, sh4 only
999, all bad

However, this is already provided by the above recommended Q flags.

'''Ocean Sites'''
Providing quality-control flags according to Ocean Sites is encouraged.
These are described at [http://www.oceansites.org/ Ocean Sites] for quailty control (QC) coding.
This flagging scheme is mostly compatible with the primary level flagging recommended by [http://www.ioccp.org/images/D4standards/IOC-OceanDataStandards54-3-2013.pdf Intergovernmental Oceanographic Commission of UNESCO (2013)].
However, only the flags of 0, 1, and 4 make sense for dissipation estimates derived from shear-probe data.

{| class="wikitable"
|-
! Flag
! Meaning
! Comment
|-
| 0
| unknown
| No QC was performed.
|-
| 1
| good data
| All QC tests passed.
|-
| 2
| probably good data
| Data have failed one or more QC tests but detailed examination after processing (e.g. by visual examination) suggests data is good.
|-
| 3
| potentially correctable bad data
| These data are not to be used without scientific correction or re-calibration (e.g. uncertain shear sensor sensitivity).
|-
| 4
| bad data
| Data have failed one or more tests.
|-
| 5
| -
| Not used
|-
| 6
| -
| Not used
|-
| 7
| nominal value
| Data were not observed but reported (e.g. instrument target depth.).
|-
| 8
| interpolated value
| Missing data may be interpolated from neighboring data in space or time.
|-
| 9
| missing value
| This is a fill value
|}

<br />
[Rolf left this here because I do not know what to do with it.]

Climate and Forecast Metadata Convention (CF) requires that QC flags carry attributes. In netCDF (Network Common Data Form) data files, the following information for quality control flagging should be provided for each data variable <PARAM>. <br />
<br />
<PARAM>_QC <br />
<PARAM>_QC:long_name = “quality flag of <PARAM>”; <br />
<PARAM>_QC:conventions = “OceanSITES QC Flags”; <br />
<PARAM>_QC:flag_values = 0, 1, 2, 3, 4, 7, 8, 9; <br />
<PARAM>_QC:flag_meanings = “0:unknown 1:good_data 2:probably_good_data 3:potentially_correctable_bad_data 4:bad_data 7:nominal_value 8:interpolated_value 9:missing_value” <br/>

[[Category: Shear probes]]

Quality control coding

2022-05-24T00:12:47Z

Rolf:

'''Example: DEFINE A NAME'''

The Q (quality control) flags associated with shear-probe measurements are not compatible with the Ocean Sites [http://www.oceansites.org/ Ocean Sites] for quality control (QC) coding.

Every dissipation estimate from every probe must have Q flag.
The numerical values of the Q flags are as follows:

Q =
0, if all checks pass
1, if FOM > FOM_limit
2, if despike_fraction > despike_fraction_limit
4, if |log(e_max)-log(e_min)|> diss_ratio_limit <math>\times \sigma_{\ln\varepsilon}</math>
8, if despike_iterations > despike_iterations_limit

The Q flags are combined by their addition.
For example a Q value of 3 mens that the dissipation estimated failed both FOM_limit test and the despike_fraction test.
A value of 15 means that all tests failed.
A failure of any one test (<math>Q\ne0</math>) means that a dissipation test should not be trusted.
The reasons for a failure can be decoded by breaking the value of Q down to its powers of 2.

This allows one to identify the unique ....

Another flag that can be used (in addition to the Quality flag) is GOOD PROBE:
GOOD_PROBE
0, all probes are good
1, sh1 only
2, sh2 only
3, sh1 and sh2
4, sh3 only
5, sh1 and sh3
6, sh2 and sh3
7, sh1, sh2 and sh3
8, sh4 only
999, all bad

However, this is already provided by the above recommended Q flags.

'''Example: Ocean Sites'''
Providing quality-control flags according to Ocean Sites is encouraged.
These are described at [http://www.oceansites.org/ Ocean Sites] for quailty control (QC) coding.
This flagging scheme is mostly compatible with the primary level flagging recommended by [http://www.ioccp.org/images/D4standards/IOC-OceanDataStandards54-3-2013.pdf Intergovernmental Oceanographic Commission of UNESCO (2013)].
However, only the flags of 0, 1, and 4 make sense for dissipation estimates derived from shear-probe data.

{| class="wikitable"
|-
! Flag
! Meaning
! Comment
|-
| 0
| unknown
| No QC was performed.
|-
| 1
| good data
| All QC tests passed.
|-
| 2
| probably good data
| Data have failed one or more QC tests but detailed examination after processing (e.g. by visual examination) suggests data is good.
|-
| 3
| potentially correctable bad data
| These data are not to be used without scientific correction or re-calibration (e.g. uncertain shear sensor sensitivity).
|-
| 4
| bad data
| Data have failed one or more tests.
|-
| 5
| -
| Not used
|-
| 6
| -
| Not used
|-
| 7
| nominal value
| Data were not observed but reported (e.g. instrument target depth.).
|-
| 8
| interpolated value
| Missing data may be interpolated from neighboring data in space or time.
|-
| 9
| missing value
| This is a fill value
|}

<br />
Climate and Forecast Metadata Convention (CF) requires that QC flags carry attributes. In netCDF (Network Common Data Form) data files, the following information for quality control flagging should be provided for each data variable <PARAM>. <br />
<br />
<PARAM>_QC <br />
<PARAM>_QC:long_name = “quality flag of <PARAM>”; <br />
<PARAM>_QC:conventions = “OceanSITES QC Flags”; <br />
<PARAM>_QC:flag_values = 0, 1, 2, 3, 4, 7, 8, 9; <br />
<PARAM>_QC:flag_meanings = “0:unknown 1:good_data 2:probably_good_data 3:potentially_correctable_bad_data 4:bad_data 7:nominal_value 8:interpolated_value 9:missing_value” <br/>

[[Category: Shear probes]]