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} \Psi(k) dk \end{equation} }[/math]

where [math]\displaystyle{ \Psi(k) }[/math] is the dimensional shear spectrum. The lower limit of spectral integration is often set to [math]\displaystyle{ k_0=0\ \mathrm{cpm} }[/math] although it can also be set to the lowest non-zero wavenumber of a spectrum. The spectrum at zero wavenumber, [math]\displaystyle{ \Psi(0) }[/math], is usually set to zero and it should be small if the spectrum is estimated properly.

The upper limit of integration, [math]\displaystyle{ k_u }[/math], is set by the smallest of a number of criteria that are listed next.

(1) The wavenumber range that is dominated by electronic noise can usually be determined from a minimum in the spectrum. Real shear is at wavenumbers smaller than this minimum while electronic noise is usually at higher wavenumbers. The spectral minimum may be found by fitting a low-order polynomial to the spectrum in log-log space. Third order is often sufficient. The wavenumber of the spectral minimum, [math]\displaystyle{ k_{\mathrm{min}} }[/math], sets one of the limits on [math]\displaystyle{ k_u }[/math].

(2) Another upper limit is [math]\displaystyle{ k_{150} = 150\ \mathrm{cpm} }[/math] that is imposed by the spatial resolution of a commonly used shear probe. You may use a different value if your shear probe has a spatial resolution different from that reported by Macoun and Lueck, 2004 At the wavenumber of [math]\displaystyle{ 150\ \mathrm{cpm} }[/math] the spectrum derived from the commonly used shear probe is boosted by a factor of 10. At higher wavenumbers the spectral correction is more than a actor of 10 and such large corrections are not recommended.

(3) 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_A \leq f_A/U }[/math]. Because most filters have a transition range from passing to attenuating a signals, it is wise to set this limit to value slightly smaller than the cut-off frequency. For example, [math]\displaystyle{ k_A \leq 0.9\, f_A/U }[/math].

(4) The user may impose an upper limit to exclude wavenumbers that have contaminations that are not correctable, [math]\displaystyle{ f_{\mathrm{lim}} }[/math]. For most instruments this limit is usually set to [math]\displaystyle{ \infty }[/math], but it may be prudent to set this limit to a finite value in some cases.

(5) The final wavenumber limit, [math]\displaystyle{ k_{95} }[/math], is the wavenumber at which the variance of shear is resolved to 95%. There is not incentive to integrate the spectrum beyond this limit because the correction that must be applied amounts to only 5%. This wavenumber is [math]\displaystyle{ k_{95} = 0.12\, (\varepsilon/\nu^4)^{1/4} }[/math] and the factor of [math]\displaystyle{ 0.12 }[/math], is nearly identical for all of the common approximations to the shear spectrum, such as the approximations to the Nasmyth , the Panchev-Kesich , and the Lueck non-dimensional universal spectra.

Thus, the upper limit of spectral integration is

[math]\displaystyle{ k_u = \mathrm{min}(k_{\mathrm{min}},\ k_{150},\ 0.9k_A,\ k_{\mathrm{lim}},\ k_{95}) }[/math].

The last of these upper limits, [math]\displaystyle{ k_{95} }[/math], presents us with a conundrum because it requires the rate of dissipation which is what we are trying to estimate by way of the integration of the shear spectrum. Clearly, we need to bootstrap this process by starting with a reasonable (but certainly a rough) estimate of the rate of dissipation.

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] is 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 provides an approximation to the Nasmyth spectrum and Lueck (2022) who provides an improved approximation to the Nasmyth spectrum, an approximation to the Panchev-Kesich (1975) spectrum, and a new approximation based on 14,600 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}\ \mathrm{W\, kg^{-1}} }[/math], this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, [math]\displaystyle{ \varepsilon }[/math], to the rate derived from integration to 10 cpm, [math]\displaystyle{ \varepsilon_{10} }[/math], is given by

[math]\displaystyle{ \frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 }[/math]

where

[math]\displaystyle{ \begin{split} a &= 1.25\times 10^{-9}\, \nu^{-3} \\ b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\ \varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\ \varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k \ \ . \end{split} }[/math]

Thus, integrating the spectrum, [math]\displaystyle{ \Psi }[/math], to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate?

The ratio of the true dissipation rate to the rate determined by integrating the spectrum to only 10 cpm, as a function of the 10-cpm rate for a range of kinematic viscosities, if the measured spectrum follows the Nasmyth spectrum.

The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of [math]\displaystyle{ \varepsilon }[/math] This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by [math]\displaystyle{ \varepsilon^{1/4} }[/math]. If the fraction of the variance resolved by a particular [math]\displaystyle{ k_u }[/math] exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations.

For very high rates of dissipation, such as [math]\displaystyle{ \varepsilon=1\times 10^{-5}\ \mathrm{W\, kg^{-1}} }[/math], the shear-probe cannot fully resolve the spectrum of shear. We recommend to estimate the rate by fitting to the spectrum in the inertial subrange.

In this range, the spectrum rises in proportion to [math]\displaystyle{ \varepsilon^{2/3}k^{1/3} }[/math] and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than [math]\displaystyle{ k(\nu^3/\varepsilon)^{1/4}=0.02 }[/math], and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear.

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References

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