A study of multiple equilibria in a β-plane and a hemispheric model of a barotropic atmosphere

We examine the equilibrium solutions of two topographically forced barotropic models, one having a β-plane geometry and the other being formulated in spherical coordinates. In both models the topography is represented by a single harmonic to facilitate the study of bent resonance, and the flow is forced through a mean zonal wind driving term. The horizontal scale of the topography and the dissipation time scales are selected to yield forced waves of comparable amplitudes in the limit of linear flows. For both models, a steady state version is used to obtain the equilibrium solutions and to derive the linear stability properties of the equilibria. A second, time-dependent nonlinear version of both models is also used to investigate the time evolution of the flow when the equilibrium solutions are perturbed. Our results for the β-plane model indicate that while multiple equilibria can be found for sufficiently low dissipation rates, only one equilibrium is stable and observable over a significant period of time in a time-dependent model. In regions of parameter space where multiple equilibria exist, the unstable equilibrium solutions are not surrounded by limit cycles. Initial states chosen close to these unstable equilibria therefore evolve towards the one stable solution. The spherical geometry model also possesses multiple equilibria for sufficiently low dissipation rates in association with the presence of a bent resonance. Two of the equilibria are stable, one corresponding to a low zonal index flow and the other to a high zonal index flow. The multiple equilibria exist, however, only over a rather narrow range of u^*, the mean zonal wind driving parameter.