Linearly driven flow on a rotating sphere

We investigate a generalized Navier–Stokes (GNS) equation as an analytically tractable minimal model for fluid flows driven by active stresses. The GNS dynamics couple an advective nonlinearity with a generic linear instability and have been shown to permit exact solutions in a stationary 2D spherical geometry. Here, we extend the analysis to actively driven flows on rotating spheres, motivated in part by the complex flow patterns observed in planetary atmospheres. The resulting model generalizes the widely studied barotropic vorticity equation by accounting for internal forcing effects that depend on the flow vorticity itself. We find exact solutions of the GNS equations corresponding to time-independent zonal jets and their superposition with westward propagating Rossby waves. Simulations for large rotation rates confirm that the statistically stationary state is close to these exact solutions. The phase speed of the nonlinear Rossby waves measured in the simulations agrees well with analytical predictions.