Convert the shear probe data

copy&paste from V1 docx

Convert the shear-probe data samples into physical units using the standard equation,

$s=\frac{N_s}{2\sqrt{2}S{U}^{2}G\gamma }$

where $s$ is the shear signal in physical units of ${\mathrm{s}}^{-1}$ , $N_s$ are the raw numeric samples (the output of an analog-to-digital converter), $S$ is the calibrated sensitivity of the shear-probe in units of $\mathrm{V}/\mathrm{(}\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}{\mathrm{)}}^{\mathrm{2}}$, $U$ is the speed of profiling in $\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$, $G$ is the gain of the differentiator in the electronics of the shear probe in units of $\mathrm{s}$, and $\gamma$ is the gain of the analog-to-digital converter used to create the samples in units of $\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}{\mathrm{V}}^{\mathrm{-}\mathrm{1}}$. The above formula assumes that the electronics of the shear probe uses a differentiator to produce a signal that is proportional to the time rate of change of the cross-profile velocity fluctuations.

The sensitivity of the shear probe to shear is proportional to the square of the speed of profiling. Thus, one should set a minimum speed for the conversion of the shear-probe data into physical units. Otherwise, the conversion may produce enormously large and quite unrealistic values. Realistic minimum speeds for the conversion into physical units are $0.05$ to $0.1\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$ because the shear-probe signal is likely to be dominated by electronic noise and the angle of attack will be large ($>20^{\circ }$) even for low levels of dissipation (\citet{Lueck, R.G., D. Huang, D. Newman, and J. Box, 1997, Turbulence measurements with a moored instrument, J. Atmos. Oceanic Techno., 14, 143-161}).

If your instrument does not have a differentiator, then the recorded signal is proportional to the cross-profile velocity fluctuations, say $u$. You must convert this signal into the time derivative of cross-profile velocity using the first difference operator

$s=f_s[u(n)-u(n-1)]$

where $f_s$ is the sampling rate of your data and $u$ are the samples of the shear probe that have been converted into velocity fluctuations in units of $\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$ following the recommendation of the manufacturer of your instrument.

The above equation is an approximation of a time derivative that deviates from a true (continuous-domain) derivative with increasing frequency. The square of the magnitude of the transfer function of a first difference operation is

Failed to parse (syntax error): {\displaystyle \left|𝐻_{\Delta}(𝑓)\right|^2 = \left[4𝑓_𝑁\, \sin(\frac{\pi}{2} \frac{𝑓}{𝑓_𝑁}) \right]^2 }

where $f_N=f_s/2$ is the Nyquist frequency. The square of the magnitude of the transfer function of a differentiator in the continuous domain is

$|H_c(f)|={\left[2\pi f\right]}^2$.

Shear spectra computed using shear-probe data that has been processed by a first difference operator must be corrected by multiplying such spectra by the ratio of these two transfer functions. That is, the spectra are multiplied by

Failed to parse (syntax error): {\displaystyle \left| \frac{H_c}{H_{\Delta}} \right|^2 = \left[ \frac{2\pi f}{4𝑓_𝑁\, \sin(\frac{\pi}{2} \frac{𝑓}{𝑓_𝑁})} \right]^2 \ \ .}

This correction factor has its maximum at the Nyquist frequency where it equals ${\left(\frac{\pi }{2}\right)}^2$.