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On the Evolution and Stability of Finite-Amplitude Mountain Waves

T. L. ClarkAtmospheric Environment Service, Downsview, Ontario, Canada M3H 5T4

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W. R. PeltierDepartment of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7

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Abstract

We describe a series of fixed Froude number numerical simulations of the generation of internal gravity waves by the flow of stably stratified fluid over an isolated obstacle. Upstream of the obstacle the parallel flow is shear free and the Brunt-Väisälä frequency is independent of height. Under these conditions the nonhydrostatic model which we employ does not support resonance modes. In this model the nonlinear lower boundary condition is treated via a general tensor transformation which maps the domain with an irregular lower boundary into a rectangle. We explore the characteristics of the wave field as a function of the aspect ratio of the topography and show that there exists a critical aspect ratio which, if exceeded, results in the generation of internal waves which are subject to a local convective instability. In the long time limit we compare the numerically determined wave drag, the vertical profile of Reynolds stress and the downslope wind amplification to the corresponding predictions of linear steady-state theory. In the limit of small aspect ratio the analytic and numerical results coincide; in particular the Eliassen-Palm theorem is recovered. In the unstable regime the drag on the obstacle increases drastically, the strength of the downslope flow is enhanced and the vertical profile of Reynolds stress is strongly divergent. We discuss the implications of these results to the understanding of certain characteristics of mountain waves in the atmosphere.

Abstract

We describe a series of fixed Froude number numerical simulations of the generation of internal gravity waves by the flow of stably stratified fluid over an isolated obstacle. Upstream of the obstacle the parallel flow is shear free and the Brunt-Väisälä frequency is independent of height. Under these conditions the nonhydrostatic model which we employ does not support resonance modes. In this model the nonlinear lower boundary condition is treated via a general tensor transformation which maps the domain with an irregular lower boundary into a rectangle. We explore the characteristics of the wave field as a function of the aspect ratio of the topography and show that there exists a critical aspect ratio which, if exceeded, results in the generation of internal waves which are subject to a local convective instability. In the long time limit we compare the numerically determined wave drag, the vertical profile of Reynolds stress and the downslope wind amplification to the corresponding predictions of linear steady-state theory. In the limit of small aspect ratio the analytic and numerical results coincide; in particular the Eliassen-Palm theorem is recovered. In the unstable regime the drag on the obstacle increases drastically, the strength of the downslope flow is enhanced and the vertical profile of Reynolds stress is strongly divergent. We discuss the implications of these results to the understanding of certain characteristics of mountain waves in the atmosphere.

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