Iterative spectral integration algorithm
Spectral integration is an iterative procedure because the bandwidth required to estimate the variance of shear depends on the rate of dissipation, the level of electronic noise in the shear-probe signal, on the wavenumber of spurious signals that were not removed by the Goodman algorithm, and on the wavenumber resolution of the shear probe. The rate of dissipation is estimated using
[math]\displaystyle{ \begin{equation} \varepsilon = \frac{15}{2} \nu \int^{k_u}_{k_0} S(k) dk \end{equation} }[/math]
where [math]\displaystyle{ S(k) }[/math] is the dimensional shear spectrum. The upper limit of integration, [math]\displaystyle{ k_u }[/math], is et by the smallest of a number of criteria.
Electronic noise usually manifests itself as a minimum in the spectrum with real shear at wavenumbers smaller than this minimum and electronic noise at higher wavenumbers. The minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. The wavenumber of the spectral minimum sets one of the limits on [math]\displaystyle{ k_u }[/math].
Another upper limit is [math]\displaystyle{ k_u = 150 cpm }[/math] that is imposed by the spatial resolution of the shear probe. The cut-off frequency, [math]\displaystyle{ f_A }[/math], of the anti-aliasing filter used by the shear-probe sampler sets another upper limit of spectral integration, namely [math]\displaystyle{ k_u \leq f_A/U }[/math]. The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable. All of these limits are fixed for a particular data set.
The wavenumber range of the spectrum of shear depends on the rate of dissipation. The spectrum broadens in proportion to [math]\displaystyle{ \epsilon^{1/4} }[/math] and the spectrum rises in proportion to [math]\displaystyle{ \epsilon^{3/4} }[/math]. Thus, the fraction of the shear variance that is resolved at any particular wavenumber depends on the rate of dissipation.
However, the non-dimensional spectrum [math]\displaystyle{ G(\hat k) = \frac{L^2_k}{(\varepsilon \nu^5)^{1/4}} S(k) }[/math]
where [math]\displaystyle{ \hat k=kL_k }[/math] and [math]\displaystyle{ L_k=(\nu^3/\varepsilon)^{1/4} }[/math] s the Kolmogorov length, is expected to be universal and independent of the rate of dissipation. There are several analytic models of the non-dimensional spectrum that are based on approximations of empirically derived spectra such as Wolk et al (2002) who provide an approximation to the Nasmyth spectrum (Oakey, 1980), and Lueck (2021b) who provides an improved approximation to the Nasmyth spectrum and to a non-dimensional spectrum derived from 14,000 dimensional spectra. These spectral approximations also provide the fraction of the shear variance that is resolved at any particular non-dimensional wavenumber. For all of these analytic approximations, 95% of the variance of shear is resolved at a non-dimensional wavenumber of approximately 0.12. There is little motivation for integrating the spectrum beyond this wavenumber. This sets another upper limit to spectral integration once the rate of dissipation is known, or at least known approximately.
The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when [math]\displaystyle{ \varepsilon=1\times 10^{-10} W kg^{-1} }[/math]
[math]\displaystyle{ }[/math]
