Turbulence spectrum

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Turbulence energy cascade and its expected spectral representation

This is the common definition for Turbulence spectrum, but other definitions maybe discussed within the wiki.

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Spectra in the frequency domain are converted into the spatial domain via Taylor's Frozen Turbulence hypothesis. Convert time derivatives to spatial gradients along the direction of profiling using

<math> \frac{\partial}{\partial x} = \frac{1}{U_P} \frac{\partial}{\partial t} </math> .

Convert frequency spectra into wavenumber spectra using

<math> k = f/U_P </math> and <math> \Psi(k) = U_P \Psi(f) </math> .

If a sampled quantity, say <math>\zeta</math>, has a spectrum, <math>\Psi(k)</math>, then this spectrum provides the wavenumber distribution of the variance of <math>\zeta</math>. For example,

<math>\overline{\zeta^2}=\int_0^{k_N} \Psi(k)\, \mathrm{d}k </math>

provides the total variance of <math>\zeta</math>, where <math>k_N = \frac{1}{2} f_s / U_P</math> is the Nyquist wavenumber and <math>f_s</math> is the sampling rate <math>\zeta</math>. The variance located in the wavenumber band of <math>k_1</math> to <math>k_2</math> is

<math>\int_{k_1}^{k_2} \Psi(k)\, \mathrm{d}k \ \ .</math>

Thus, a spectrum has units of variance per wavenumber.

• Missing the y-axi variables
• Lowest frequency and wavenumber resolvable