We solve the quasigeostrophic potential vorticity equation,
\begin{equation}
\frac{\partial }{\partial t} \left( \nabla^2 \psi – \lambda^2 \psi \right) + J\left( \psi, \nabla^2 \psi \right) = F + \alpha \lambda^2 \psi -r \nabla^2 \psi + \nu Q(k) \nabla^4 \psi
\end{equation} where we have included thermal damping ( \( \alpha \lambda^2 \) ), frictional damping (\(r\)), and hyper diffusion (\(\nu\)). The variable \(F\) represents the forcing, \(J\) is the Jacobian and \(Q(k)\) is the spectral vanishing viscosity filter.

The two movies show relative vorticity, floats advected by the fluid (colored by their initial position) and the energy spectrum.

In this first video the forcing is relatively large, 64 times the grid size. The forcing injects vorticity directly into the fluid. It doesn’t take long before the small scale vorticity structures to emerge, and the enstrophy cascade, with slope \(k^{-3}\), becomes apparent in the energy spectrum.

The second video shows smaller scale forcing, only 16 times the grid size. In this case the small scale structures and enstrophy cascade form even more quickly, but the inverse energy cascade, with slope \(k^{-5/3}\), takes much longer. In fact, energy continues to build up at larger scales even as the movie ends. This is not as evident in the vorticity, but it apparent by looking at the size of the fluid structures advecting the particles.

The code is available on github and includes a matlab script for generating the movies seen above.