Spectral integration

Short definition of Spectral integration
Estimating [math]\displaystyle{ \varepsilon }[/math] by spectral integration.

This is the common definition for Spectral integration, but other definitions maybe discussed within the wiki.

The shear probe provides a measure of the turbulent shear, and this data can be used to estimate the spectrum of the shear. The variance of shear is often estimated by integrating the spectrum of shear, namely

[math]\displaystyle{ \varepsilon = \frac{15}{2}\nu \overline{\left(\frac{\partial u}{\partial z} \right)^2} \equiv \frac{15}{2}\nu \int_0^{\infty} \Phi(k) \,\mathrm{d}k \approx \frac{15}{2}\nu \int_0^{k_u} \Phi(k) \,\mathrm{d}k }[/math]

where [math]\displaystyle{ \Phi }[/math] is an estimate of the spectrum of shear, [math]\displaystyle{ \Psi }[/math], and the upper limit of spectral integration, [math]\displaystyle{ k_u }[/math], is imposed by practical considerations. Thus, only a fraction of the shear variance is resolved by this method. There is value in having a mathematical approximation for the spectrum of shear and for the fraction of the variance that is resolved by integration to a finite upper wavenumber. The model spectrum provides a gauge for judging the quality of the estimate of the spectrum. The model of its integral provides a means to correct (upwards) the estimate of [math]\displaystyle{ \varepsilon }[/math] provided by spectral integration up to a finite wavenumber. Details of spectral integration are discussed not a good link needs attention.