Statistical descriptions

Fourier transforms

We define the continuous Fourier transform on a periodic domain \([0, L]\) as:

\[\hat u(k) = \int_0^L u(x) e^{-ikx} \frac{dx}{L},\]

where \(k = 2\pi / \lambda\). The inverse Fourier transform is:

\[u(x) = \sum \hat u(k) e^{-ikx}.\]

Spectra

Spectra are defined by equations like this one:

\[{\langle {\boldsymbol{u}}^2/2 \rangle} = \int_0^\infty dk E(k)\]

Richardson cascade

\(U^3/L \sim u(l)^3/l \sim {\varepsilon}\) is constant in the inertial range (for scales \(l\) between the injection scale and the dissipation scale).

This gives the scaling law for the velocity at scale \(l\):

\[u(l) \propto ({\varepsilon}l)^{1/3}\]

In terms of spectra, this gives (\(k \sim 1/l\)):

\[E(k) \propto u(l)^2 k^{-1} \propto {\varepsilon}^{2/3} k^{-5/3}.\]

Kolmogorov spectrum.