Finite Amplitude Interactions between Unstable Baroclinic Waves and Resonant Topographic Waves

Abstract

Weakly nonlinear interactions between an unstable baroclinic wave and resonant topographic wave are investigated using asymptotic methods in a two-layer, quasi-geostrophic channel model on a midlatitude beta plane in the presence of sinusoidal topography and dissipation. The asymptotic analysis pivots about slightly supercritical, vertically sheared zonal flows for which the baroclinic wave is weakly unstable and the topographic wave nearly resonant. Two long time scales are required to describe the evolution of the baroclinic wave and zonal flow connections, while the topographic wave evolves only on the longest time scale. To facilitate the numerical analysis, the method of reconstitution is used to form amplitude and zonal flow equations on a combined time scale. Examination of the analytically derived amplitude evolution equations shows that the phase of the topographic wave relative to the mountain explicitly affects the nonlinear evolution of the baroclinic wave. In contrast, phase changes in the baroclinic wave have no direct effect on the evolution of the topographic wave. Numerical integrations of the reconstituted evolution equations reveal two distinct asymptotic states of the system: 1) a single (stationary) topographic wave state where the wave trough is upstream of the mountain ridge, or 2) a mixed wave state where the baroclinic wave propagates with fixed amplitude, while the topographic wave remains stationary with its trough upstream of the mountain ridge. Single wave states, or mixed wave states dominated by the topographic wave, are, relatively speaking, favored for large zonal scales, large topographic heights, small beta and weak dissipation. However, for sufficiently small zonal scales only mixed wave states exist which are dominated by the baroclinic wave.