Exercise 7

The Gletzer-Okhitani-Yamada (GOY) model of turbulence is a shell model in which only a very reduced number of modes are considered. The model contains N 'shells' with a complex velocity, u_n(t), representing some sort of average of the Fourier components of the velocity field with $k_{n-1}<\vert{\bf k}\vert < k_n$. The momenta are defined as $k_n=k_0 \lambda^n$, where $\lambda$ is the shell spacing, ( $\lambda =2$ is the usual choice). The dynamical equation is,

$$\dot{u}_n = i k_n (u_{n+1}u_{n+2}-\frac{\delta}{\lambda} u_{n... ...\frac{1-\delta}{\lambda^2}u_{n-1}u_{n-2})^*-\nu k_n^2u_n +f_n,$$ (1)

where $\delta$ is a free parameter ( $\delta =1/2$ is the usual choice). This equation has structually the same form as the spectral Navier-Stokes equation.

Show that the energy defined as $E = (1/2)\sum \vert u_n\vert^2$ is an invariant when $\nu=f=0$.

What will be the equivalent of the K41 theory for the GOY model ? Hint: Express $\langle u_n^3 \rangle$ dimensionally in terms of k_n and the energy dissipation $\epsilon$.