Spectra of velocity gradients

Short definition of Spectra of velocity gradients
Often called dissipation spectra, velocity spectra multiplied by ${\kappa }^2$ or ${\kappa }_1^2$

This is the common definition for Spectra of velocity gradients, but other definitions may be discussed within the wiki.

The spectra of the gradients of velocity are closely related to the rate of dissipation, $\epsilon$, and are often called dissipation spectra. These spectra are the velocity spectra multiplied by ${\kappa }^2$ or ${\kappa }_1^2$, whichever is appropriate. The rate of dissipation is related to the gradient of the three-dimensional velocity spectrum by

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \begin{split} \varepsilon &= 2\nu \int_0^{\infty} \kappa^2 E(\kappa)\, \mathrm{d} \kappa = 2\nu \left(\varepsilon\nu^5 \right)^{1/4} \int_0^{\infty} \kappa^2 F(\hat{\kappa})\, \mathrm{d} \kappa \\ &=2\nu \left(\varepsilon\nu^5 \right)^{1/4} L_K^{-3} \int_0^{\infty} \hat{\kappa}^2 F(\hat{\kappa})\, \mathrm{d} \hat{\kappa} \\ &= 2\varepsilon \int_0^{\infty} G(\hat{\kappa})\, \mathrm{d} \hat{\kappa} \end{split} \end{equation} }

Thus, the universal (non-dimensional) gradient spectrum is $G={\hat{\kappa }}^{2}F$ , and its integral over all wavenumbers must equal 1/2. The along-profile gradient of the along-profile velocity fluctuations often called the rate of strain (or, simply strain), is related to the rate of dissipation by

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \begin{split} \varepsilon &= 15\nu \int_0^{\infty} \kappa_1^2 E_{11}(\kappa_1)\, \mathrm{d}\kappa_1 = 15\nu \left(\varepsilon\nu^5 \right)^{1/4} \int_0^{\infty} \kappa_1^2 F_{11} (\hat{\kappa}_1)\, \mathrm{d} \kappa_1 \\ &= 15\varepsilon \int_0^{\infty} G_{11} (\hat{\kappa}_1)\, \mathrm{d} \hat{\kappa}_1 \end{split} \end{equation} }

where $G_{11}={\hat{\kappa }}_1^2{F}_{11}$ is the universal (and non-dimensional) rate of strain spectrum, which must integrate to 1/15. Similarly, the shear spectrum is related to the rate of dissipation by

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \begin{split} \varepsilon &= \frac{15}{2} \nu \int_0^{\infty} \kappa_1^2 E_{22}(\kappa_1)\, \mathrm{d}\kappa_1 = \frac{15}{2}\nu \left(\varepsilon\nu^5 \right)^{1/4} \int_0^{\infty} \kappa_1^2 F_{22}\, (\hat{\kappa}_1) \mathrm{d} \kappa_1 \\ &= \frac{15}{2}\varepsilon \int_0^{\infty} G_{22} (\hat{\kappa}_1)\, \mathrm{d} \hat{\kappa}_1 \end{split} \end{equation} }

where $G_{22}={\hat{\kappa }}_1^2{F}_{22}$ is the universal shear spectrum which must integrate to 2/15.