Processing your ADCP data using structure function techniques

To calculate the dissipation rate at a specific range bin:

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{ 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 are [[math]\displaystyle{ n_{max} = r_{max} / \delta r_0 }[/math]], where [math]\displaystyle{ \delta r_{0} }[/math] is the radial separation between bins.
4. The structure function for a data segment can be calculated using either a bin-centred difference scheme or a forward-difference scheme
5. Including D(n, [math]\displaystyle{ \delta }[/math]) for [math]\displaystyle{ \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
6. The number of instances when the squared velocity difference is evaluated for each bin n and separation distance [math]\displaystyle{ \delta }[/math]r₀ and their distribution are potential quality control metrics
7. 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]\displaystyle{ \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]\displaystyle{ \delta }[/math])
8. Having calculated the segment D(n, [math]\displaystyle{ \delta }[/math]) for the appropriate range of bins and [math]\displaystyle{ \delta }[/math]r₀ separation distances, a regression of D(n, [math]\displaystyle{ \delta }[/math]) against ([math]\displaystyle{ \delta }[/math]r₀)^2/3 is then undertaken.
   1. If D(n, [math]\displaystyle{ \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]\displaystyle{ \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 D(n, [math]\displaystyle{ \delta }[/math]) was evaluated using a bin-centred difference scheme, the regression can either be done:
      1. for each bin individually, with a single D(n, [math]\displaystyle{ \delta }[/math]) for each separation distance, ultimately yielding an [math]\displaystyle{ \varepsilon }[/math] for each bin; or
      2. by combining the data for all of the bins, with each separation distance having a D(n, [math]\displaystyle{ \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:
      
      D = a₀ + a₁ ([math]\displaystyle{ \delta }[/math]r₀)^2/3; or as
      D = a₀ + a₁ ([math]\displaystyle{ \delta }[/math]r₀)^2/3 + a₃ (([math]\displaystyle{ \delta }[/math]r₀)^2/3)³
      
      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]\displaystyle{ \delta }[/math]) as ([math]\displaystyle{ \delta }[/math]r₀)²
   4. The coefficient a₀ (the intercept of the regression) is a function of with the noise of the velocity observations
   5. The coefficient a₁ is then used to calculate [math]\displaystyle{ \varepsilon }[/math] as
      
      [math]\displaystyle{ \varepsilon }[/math] = (a₁ / C₂)^2/3
      
      where C₂ is an empirical constant, typically taken as 2.0 or 2.1

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