To calculate the dissipation rate at a specific range bin and a specific time ensemble:
# Extract or compute the along-beam bin center separation [r0] based on the instrument geometry
[[File:ADCPschematic SF.png|thumb|Schematic showing along-beam distance <math> r </math> and radial velocities. ]]
# Calculate the along-beam velocity fluctuation time-series in each bin, [v’(n, t)] from the Level 1 along-beam velocity data that has met the QC criteria
## If using burst sampling, calculations are done over the length of the burst or some sub-period over which the turbulent flow statistics can assumed to be stationary
## If using continuous sampling, calculations are dome over segments with a duration over which the turbulent flow statistics can assumed to be stationary
## For each data segment consisting of N profiles, the turbulent fluctuations are calculated separately for each beam and bin around either:
##* ''The mean over the data segment''
##* ''A linear detrend of the segment''
##* ''A low pass filtered signal''
# Select the maximum distance over which to compute the structure function based on conditions of the flow (e.g., expected max overturn) [r<sub>max</sub>] in bin separation distances [n<sub>max</sub> = r<sub>max</sub> / r<sub>0</sub>]
# The structure function for a data segment can be calculated using either a '''bin-centred difference''' or a '''forward-difference''' scheme
# For a '''bin-centred difference''' scheme
## start at bin n = (n<sub>max</sub> / 2) + 1
### start with <math>\delta</math> = 1
### if <math>\delta</math> is '''''even''''' compute the second order structure function D(n,<math>\delta</math>) as the segment mean of the square of the velocity difference between the bins separated by distance <math>\delta</math>r<sub>0</sub> centered around bin n: <br /><br />D(n, <math>\delta</math>) = <math>\langle</math> [v’(n+(<math>\delta</math>/2), t) - v’(n-(<math>\delta</math>/2), t)]<sup>2</sup> <math>\rangle</math> <br/><br /> where the angled brackets indicate the mean across all t for the data segment yielding a velocity difference after the application of the Level 1 QC criteria
### if <math>\delta</math> is '''''odd''''' compute the second order structure function D(n,<math>\delta</math>) as the segment mean of the mean of the square of the velocity difference between the bins separated by distance <math>\delta</math>r<sub>0</sub> centered on the upper and lower extent of bin n: <br/><br /> dv'<sub>lo</sub>(n, <math>\delta</math>, t) = v’(n+floor(<math>\delta</math>/2), t) - v’(n-ceil(<math>\delta</math>/2), t) <br/> dv'<sub>hi</sub>(n, <math>\delta</math>, t) = v’(n+ceil(<math>\delta</math>/2), t) - v’(n-floor(<math>\delta</math>/2), t) <br/><br /> where ''ceil'' and ''floor'' indicate the upper and lower integer value respectively, then <br/><br /> D(n, <math>\delta</math>) = <math>\langle</math> [dv'<sub>lo</sub>(n, <math>\delta</math>, t)<sup>2</sup> + dv'<sub>hi</sub>(n, <math>\delta</math>, t)<sup>2</sup>] / 2 <math>\rangle</math> <br/><br /> the angled brackets again indicating the mean across all t in the data segment yielding a velocity difference after the application of the Level 1 QC criteria
### increment <math>\delta</math> and repeat steps until <math>\delta</math> = n<sub>max</sub>
## increment n and repeat steps until n + (n<sub>max</sub> / 2) exceeds the bin number for which valid v’ are available
# For a '''forward-difference''' scheme
## start with n being the lowest number bin of the range over which the structure function is to be evaluated (number of bins in range must exceed n<sub>max</sub>
### start with <math>\delta</math> = 1
### compute the second order structure function D(n,<math>\delta</math>) as the segment mean of the square of the velocity difference between the bin n and bin n + <math>\delta</math>: <br/><br /> D(n, <math>\delta</math>) = <math>\langle</math> [v’(n, t) - v’(n+<math>\delta</math>, t)]<sup>2</sup> <math>\rangle</math> <br/><br /> where the angled brackets indicate the mean across all t for the data segment yielding a velocity difference after the application of the Level 1 QC criteria
### increment <math>\delta</math> and repeat steps until <math>\delta</math> = n<sub>max</sub> or n + <math>\delta</math> exceeds the last bin of the range over which the structure function is to be evaluated
## increment n and repeat steps until n + 1 is the last bin of the range over which the structure function is to be evaluated
# Including D(n, <math>\delta</math>) for <math>\delta</math> = 1 may be inappropriate since the velocity estimates from adjacent bins are not wholly independent, therefore the impact of its inclusion should be evaluated
# The number of instances when the squared velocity difference is evaluated for each bin n and separation distance <math>\delta</math>r<sub>0</sub> and their distribution are potential quality control metrics
# The impact of additional quality criteria can also be tested e.g. valid data requirements for all intermediate separation distances, so for a forward-difference scheme with n = 2 and <math>\delta</math> = 5, require all data in bins 2 to 7 to meet Level 1 QC requirements for the profile to be included when averaging to calculate D(n, <math>\delta</math>)
# Having calculated the segment D(n, <math>\delta</math>) for the appropriate range of bins and <math>\delta</math>r<sub>0</sub> separation distances, a regression of D(n, <math>\delta</math>) against (<math>\delta</math>r<sub>0</sub>)<sup>2/3</sup> is then undertaken.
## If D(n, <math>\delta</math>) was evaluated using a forward-difference scheme, the regression is done for the combined data from all bins in the selected range, hence the maximum number of D(n, <math>\delta</math>) values for each separation distance will be the number of bins in the range less 1 for <math>\delta</math> = 1, reducing by 1 for each increment in <math>\delta</math>, with the regression ultimately yielding a single <math>\varepsilon</math> value for the data segment
## If D(n, <math>\delta</math>) was evaluated using a bin-centred difference scheme, the regression can either be done:
### for each bin individually, with a single D(n, <math>\delta</math>) for each separation distance, ultimately yielding an <math>\varepsilon</math> for each bin; or
### by combining the data for all of the bins, with each separation distance having a D(n, <math>\delta</math>) value for each bin, with the regression again ultimately yielding a single <math>\varepsilon</math> value for the data segment
## The regression is typically done as a least-squares fit, either as: <br /><br /> D = a<sub>0</sub> + a<sub>1</sub> (<math>\delta</math>r<sub>0</sub>)<sup>2/3</sup>; or as <br /> D = a<sub>0</sub> + a<sub>1</sub> (<math>\delta</math>r<sub>0</sub>)<sup>2/3</sup> + a<sub>3</sub> ((<math>\delta</math>r<sub>0</sub>)<sup>2/3</sup>)<sup>3</sup> <br /><br /> the former being the canonical approach assuming no non-turbulent velocity difference between bins in v’, whilst the latter seeks to isolate the turbulent contribution from velocity differences between bins due to the orbital velocity forced by surface waves or any residual velocity shear retained due to the oscillatory motion of the ADCP on a mooring, both of which result in a velocity difference between bins which varies linearly with separation distance and hence contributes to D(n, <math>\delta</math>) as (<math>\delta</math>r<sub>0</sub>)<sup>2</sup>
## The coefficient a<sub>0</sub> (the intercept of the regression) is a function of with the noise of the velocity observations
## The coefficient a<sub>1</sub> is then used to calculate <math>\varepsilon</math> as <br /><br /> <math>\varepsilon</math> = (a<sub>1</sub> / C<sub>2</sub>)<sup>2/3</sup> <br /><br /> where C<sub>2</sub> is an empirical constant, typically taken as 2.0 or 2.1
# Extract or compute the [[along-beam bin center separation]] [<math>\delta r_0</math>] based on the instrument geometry
# Calculate the [[along-beam velocity fluctuation]] time-series in each bin <math>n</math>, where [<math>b’(n, t_s)</math>] from the along-beam velocity data that has met the QC criteria (i.e. the data in Level 2 of the netcdf file). Note <math> t_s </math> is the timeseries index within a segment.
# Select the maximum distance (<math>r_{max}</math>) over which to compute the structure function based on conditions of the flow (e.g., expected max overturn, spectral range corresponding to <math> k^{-5/3} </math>). The corresponding number of bins is [<math>n_{\text{rmax}} = r_{max} / \delta r_0</math>]
# Calculate the structure function <math>D_{ll}</math> for all possible bin separations <math>\delta</math> within <math>r_{max}</math> using either a [[bin-centred difference scheme]] or a [[forward-difference]] scheme.
# Perform a regression of <math>D_{ll}(n,\delta)</math> against <math>(\delta r)^{2/3}</math> for the appropriate range of bins and <math>\delta</math>r</sub> separation distances. Be aware of [[Regressing structure function against bin separation | special considerations for forward-difference, center-difference schemes]] in setting up the regression calculation. The regression is typically done as a least-squares fit, either as: <br /><br /> <math>D_{ll} = a_0 + a_1 (\delta r)^{2/3}</math>;
:: or as
:: <math>D_{ll} = a_0 + a_1 (\delta r)^{2/3}+a_3((\delta r)^{2/3})^3 </math> <br /><br /> the former being the [[canonical structure function method | canonical method]] that excludes non-turbulent velocity differences between bins, whereas the latter is a [[modified structure function method | modified method]] that includes non-turbulent velocity differences between bins due to any oscillatory signal (e.g. surface waves, motion of the ADCP on a mooring).
<ol type="1" start=6>
<li> Use the coefficient <math>a_1</math> to calculate <math>\varepsilon</math> as <br /><br /> <math>\varepsilon = \left(\frac{a_1}{C_2}\right)^{2/3}</math> <br /><br /> where <math>C_2</math> is an [[ Structure function empirical constant | empirical constant]], typically taken as 2.0 or 2.1.
</ol>
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Next step: [[Final data review (QA2) | Apply quality-control on dissipation rates (QA2)]] <br></br>
Previous step:[[Raw data review (QA1) | Apply quality-control on velocity time series data (QA1)]]<br></br>
Return to [[ADCP structure function flow chart| ADCP Flow Chart front page]]
Return to [[ADCP structure function flow chart| ADCP Flow Chart front page]]
[[Category:Velocity profilers]]
Latest revision as of 15:51, 30 May 2022
To calculate the dissipation rate at a specific range bin and a specific time ensemble:
Schematic showing along-beam distance [math]\displaystyle{ r }[/math] and radial velocities.
Extract or compute the along-beam bin center separation [[math]\displaystyle{ \delta r_0 }[/math]] based on the instrument geometry
Calculate the along-beam velocity fluctuation time-series in each bin [math]\displaystyle{ n }[/math], where [[math]\displaystyle{ b’(n, t_s) }[/math]] from the along-beam velocity data that has met the QC criteria (i.e. the data in Level 2 of the netcdf file). Note [math]\displaystyle{ t_s }[/math] is the timeseries index within a segment.
Select the maximum distance ([math]\displaystyle{ r_{max} }[/math]) over which to compute the structure function based on conditions of the flow (e.g., expected max overturn, spectral range corresponding to [math]\displaystyle{ k^{-5/3} }[/math]). The corresponding number of bins is [[math]\displaystyle{ n_{\text{rmax}} = r_{max} / \delta r_0 }[/math]]
Calculate the structure function [math]\displaystyle{ D_{ll} }[/math] for all possible bin separations [math]\displaystyle{ \delta }[/math] within [math]\displaystyle{ r_{max} }[/math] using either a bin-centred difference scheme or a forward-difference scheme.
Perform a regression of [math]\displaystyle{ D_{ll}(n,\delta) }[/math] against [math]\displaystyle{ (\delta r)^{2/3} }[/math] for the appropriate range of bins and [math]\displaystyle{ \delta }[/math]r separation distances. Be aware of special considerations for forward-difference, center-difference schemes in setting up the regression calculation. The regression is typically done as a least-squares fit, either as:
the former being the canonical method that excludes non-turbulent velocity differences between bins, whereas the latter is a modified method that includes non-turbulent velocity differences between bins due to any oscillatory signal (e.g. surface waves, motion of the ADCP on a mooring).
Use the coefficient [math]\displaystyle{ a_1 }[/math] to calculate [math]\displaystyle{ \varepsilon }[/math] as
