The transition to topographic normal modes

The present note proposes an explanation of topographic normal modes that g row in a flow with a zonally symmetric mountain. An issue of practical impo rtance is whether one expects the maximum amplitude to occur poleward or eq uatorward of a mountain localized in the cross-stream direction. Using a qu asigeostrophic 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 i n the presence of an Eady normal mode is discussed first. The topographic c orrection to the Eady mode quickly acquires a negative meridional tilt at t he surface. This tilted structure penetrates upward to the lid and advects basic-state temperature. Because the phase tilt of the Eady mode varies wit h zonal wavelength, the phase of the topographic correction at the lid rela tive to the initial wave will vary similarly. For shorter zonal scales the primary cancellation of the initial wave is equatorward of the channel cent er, thus biasing the total disturbance amplitude poleward. The opposite occ urs for longer waves. At long times, topographic modes emerge. These modes involve the interactio n of gravest (symmetric) and first asymmetric modes meridionally through th e 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 r ate. The phase relation of the symmetric and asymmetric modes varies with z onal wavelength and determines the meridional bias in amplitude of the tota l solution.