The Weakly Nonlinear Dynamics of a Planetary Green Mode and Atmospheric Vacillation

Abstract

Cold season atmospheric observations of vacillation point to a wave-mean flow interaction of baroclinic, planetary waves with their mean flow, and the observational data show that wave 2 is the largest contributor to the energetics and the heat flux. To verify this hypothesis we present a weakly nonlinear analysis of the evolution of a single, most unstable Green mode interacting with mean zonal flow in the presence of internal and Ekman layer dissipations, the former being larger than the latter. The derived amplitude equations for the wave and the mean fields transform into a Lorenz set of equations that admits stable, finite amplitude wave gates. No stable limit cycle or aperiodic solutions were found in the realistic parameter ranges that typify atmospheric winter conditions. When the system is disturbed away from these gable states, there is a monotonic or vacillators approach to equilibrium. Damped vacillation occurs when the internal dissipative time scale is longer than the efolding time scale of the inviscid, Green mode, a condition realized in the winter atmosphere. During the vacillation, due to the presence of the internal dissipation the tilt of the constant phase line may remain westward, and the horizontal he-at flux may be poleward throughout most of the cycle.