

Article
A twodimensional toy model for geophysical turbulence
AbstractA toy model for large scale geophysical turbulence is constructed by making two modifications of the shallow water model. Unlike the shallow water model the toy model has a quadratic expression for total energy, which is the sum of Available Potential Energy (APE) and Kinetic Energy (KE). More importantly, in contrast to the shallow water model the toy model does not produce any shocks. Two numerical simulations are presented and compared with the simulation of a full General Circulation Model (GCM). The two simulations are forced in APE at a small and a medium wave number, respectively. The energy which is injected cascades in a similar way as in the GCM. First, some of the energy is converted from APE to KE at large scales. The wave field is then undergoing a forward energy cascade displaying spectra that scales approximately as $ k^{5/3} $ for both APE and KE, while the vortical field displays an approximate $ k^{3} $spectrum. At large wave numbers the wave field reaches a state in which KE and APE are equipartitioned. In the simulations with medium forcing wave number, some of the energy which is converted from APE to KE is undergoing an inverse energy cascade which is produced by nonlinear interactions only involving the rotational component of the velocity field. The inverse energy cascade builds up a vortical field at larger scales than the forcing scale. At these scales coherent vortices emerge with a strong dominance of anticyclonic vortices. The relevance of the simulation results to the dynamics of the atmosphere is discussed as is possible continuations of the investigation.

