Numerical Study Of The Nonlinear Barotropic Instability Of Rossby Wave Motion

Abstract

Using β-plane geometry the non-linear barotropic, non-divergent vorticity equation is solved numerically over a long time period to test for stability Rossby’s original wave solution of the equation. This equation represents a simple model of the dynamics of the atmosphere. The numerical results show that the initial growth rate of the solution (when unstable) is correctly predicted by the linearized analysis of the problem conducted by Lorenz (1972) but that eventually the solution settles into a bounded oscillating state (which contrasts with the unbounded growth predicted by linear theory). This eventual bounded oscillating behaviour is shown to occur for all values of the basic wave amplitude, A which extends the result of the weakly nonlinear analysis of the problem by Loesch (1978) and Deininger and Loesch (1982) who showed similar behaviour for values of A slightly in access of A_C (the critical amplitude for instability determined by linear theory). Significantly, the numerical results show that A_C is not an accurate indicator of stability/instability for the nonlinear solution.