Processing your ADCP data using structure function techniques

To calculate the dissipation rate at a specific range bin and a specific time ensemble:

1. Extract or compute the along-beam bin center separation [[math]\displaystyle{ r_0 }[/math]] based on the instrument geometry
2. Calculate the along-beam velocity fluctuation time-series in each bin [math]\displaystyle{ n }[/math], where [[math]\displaystyle{ v’(n, t) }[/math]] from the along-beam velocity data that has met the QC criteria (i.e. the data in Level 2 of the netcdf file)
3. 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). The corresponding number of bins is [[math]\displaystyle{ n_{\text{rmax}} = r_{max} / r_0 }[/math]]
4. Calculate the structure function [math]\displaystyle{ D_{ll} }[/math] for all possible bin separations [math]\displaystyle{ \delta }[/math] using either a bin-centred difference scheme or a forward-difference scheme. Consider QA2 requirements when choosing differencing scheme.
5. Perform a regression of [math]\displaystyle{ D_{ll}(n,\delta) }[/math] against [math]\displaystyle{ (\delta r_0)^{2/3} }[/math] for the appropriate range of bins and [math]\displaystyle{ \delta }[/math]r₀ separation distances. [JMM: THE FOLLOWING ITEMS ARE CONFUSING. SINCE THIS IS BEST PRACTICE, CAN WE JUST RECOMMEND ONE METHOD?]
   1. If [math]\displaystyle{ D_{ll}(n,\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 [math]\displaystyle{ D_{ll}(n, \delta) }[/math] values for each separation distance will be the number of bins in the range less 1 for [math]\displaystyle{ \delta }[/math] = 1, reducing by 1 for each increment in [math]\displaystyle{ \delta }[/math], with the regression ultimately yielding a single [math]\displaystyle{ \varepsilon }[/math] value for the data segment
   2. If [math]\displaystyle{ D_{ll}(n,\delta) }[/math] was evaluated using a bin-centred difference scheme, the regression can either be done:
      • for each bin individually, with a single [math]\displaystyle{ D(n, \delta) }[/math] for each separation distance, ultimately yielding an [math]\displaystyle{ \varepsilon }[/math] for each bin; or
      • by combining the data for all of the bins, with each separation distance having a [math]\displaystyle{ D_{ll}(n, \delta) }[/math] value for each bin, with the regression again ultimately yielding a single [math]\displaystyle{ \varepsilon }[/math] value for the data segment
   3. The regression is typically done as a least-squares fit, either as:
      
      [math]\displaystyle{ D_{LL} = a_0 + a_1 (\delta r_0)^{2/3} }[/math]; or as
      [math]\displaystyle{ D_{ll} = a_0 + a_1 (\delta r_0)^{2/3}+a_3((\delta r_0)^{2/3})^3 }[/math]
      
      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).
6. Use the coefficient [math]\displaystyle{ a_1 }[/math] to calculate [math]\displaystyle{ \varepsilon }[/math] as
   
   [math]\displaystyle{ \varepsilon = \left(\frac{a_1}{C_2}\right)^{2/3} }[/math]
   
   where [math]\displaystyle{ C_2 }[/math] is an empirical constant, typically taken as 2.0 or 2.1.

PERHAPS WE CAN INCLUDE A FIGURE LIKE THIS TO HELP DEFINE VARIABLES.

Next step: Apply quality-control on velocity time series data (QA1)

Previous step: Apply quality-control on dissipation rates (QA2)

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