Dissipation rate estimation from shear probes

The following items break down the derivation of the turbulent dissipation rate of kinetic energy ([math]\displaystyle{ \varepsilon }[/math]). Explanations for each step can be found after.

Briefly

1. Extract the section defined in step 2.
2. High-pass filter the shear-probe and (optionally) the vibration data.
3. Identify each diss-length segment in the profile.
4. De-spike the shear-probe data, and track the fraction of data affected by de-spiking within each diss-length segment. This will become a quality-control metric.
5. Calculate the frequency spectra and cross-spectra of shear and vibrations for each diss-length segment using the method described here.
6. Extract the original and the vibration-coherent clean shear-probe frequency spectra.
7. Correct shear and vibration frequency spectra for the high-pass filter.
8. Correct the cleaned frequency spectra for the bias induced by the Goodman algorithm.
9. Convert the frequency spectra into wavenumber spectra using the mean speed for each diss-length segment. That is, make the wavenumber [math]\displaystyle{ \begin{equation}k=f/U\end{equation} }[/math] and the wavenumber kinetic energy spectrum [math]\displaystyle{ \begin{equation}E(\kappa)=UE(f)\end{equation} }[/math] .
10. Correct the spectra of shear for the wavenumber response of the shear probe.
11. Apply an iterative spectral integration algorithm to estimate the variance of shear, which is described here.
12. Calculate the rate of dissipation by multiplying the shear variance by [math]\displaystyle{ 15/2 \nu }[/math] where [math]\displaystyle{ \nu }[/math] is the temperature-dependent kinematic viscosity.

Description

• De-spiking – There is currently no standard method of de-spiking shear-probe data. One method consists of calculating the absolute value of the (high-pass filtered) shear. A copy of this absolute shear signal is smoothed with a low-pass filter that has a cut-off frequency that is approximately the expected minimum duration of turbulence patches (usually about 1 meter divided by the speed of profiling). When the ratio of the instantaneous absolute shear divided by the smoothed absolute shear exceeds a threshold, the data is deemed to be a spike. A typical threshold is 8. A spike usually consists of a number of contiguous samples. A region surrounding the spike is then replaced by a local mean calculated using data from both sides of the spike (but excluding the spike itself). Because the response of the shear probe to a collision with plankton -- the ringing of anomalously large amplitude – is a temporal response, the amount of data replaced by a local mean is usually of fixed duration and not fixed in length. Typically, the amount of data replaced is 20ms before a spike and 40ms after a spike. This algorithm is applied iteratively until no more spikes are detected. The iterative application permits longer anomalies to be removed, such as those that might occur because of collisions with jelly fish and seaweed. The fraction of the data that is altered by a de-spiking routine must be noted for each diss-length segment because this is a quality-control metric. Dissipation estimates should be treated with caution if the fraction of altered data exceeds a few percent. But, there is currently no standard for what is an acceptable fraction.

• Auto- and cross-spectra should be calculated using the windowed and overlapped periodogram method. The window should be a periodic cosine window. The overlap should be 50%. Although this is not the optimal overlap, it is a convenient one and it is one that provides 90% of the maximum degrees of freedom attainable with the cosine window (Nuttall, 1971). The number of fft-segments within a diss-length segment of data is [math]\displaystyle{ N_f = 2 \frac{L_D}{L_f} }[/math] where [math]\displaystyle{ L_D }[/math] is the length of a dissipation estimate (diss-length) and [math]\displaystyle{ L_f }[/math] is the length of an fft-segment (fft-length).

• The high-pass filter correction is obtained by multiplying the spectrum by [math]\displaystyle{ \begin{equation} 1 + \left(\frac{f_c}{f}\right)^2 \end{equation} }[/math]