The Transition to Topographic Normal Modes

Abstract

The present note proposes an explanation of topographic normal modes that grow in a flow with a zonally symmetric mountain. An issue of practical importance is whether one expects the maximum amplitude to occur poleward or equatorward of a mountain localized in the cross-stream direction. Using a quasigeostrophic model, it is shown that the occurrence of maximum amplitude to the south of the mountain occurs only for longer waves. For a given wave of medium scale, the amplitude is more likely to maximize to the south of the mountain for steeper slopes. The transient behavior resulting from the sudden appearance of a mountain in the presence of an Eady normal mode is discussed first. The topographic correction to the Eady mode quickly acquires a negative meridional tilt at the surface. This tilted structure penetrates upward to the lid and advects basic-state temperature. Because the phase tilt of the Eady mode varies with zonal wavelength, the phase of the topographic correction at the lid relative to the initial wave will vary similarly. For shorter zonal scales the primary cancellation of the initial wave is equatorward of the channel center, thus biasing the total disturbance amplitude poleward. The opposite occurs for longer waves. At long times, topographic modes emerge. These modes involve the interaction of gravest (symmetric) and first asymmetric modes meridionally through the topographic term in the lower boundary condition. The phase displacement of the symmetric and asymmetric modes is such as to damp the symmetric mode and amplify the asymmetric mode so that the two maintain the same growth rate. The phase relation of the symmetric and asymmetric modes varies with zonal wavelength and determines the meridional bias in amplitude of the total solution.