Effects of Dissipation on Parallel Shear Instability near the Ground

P. A. Davis Department of Physics, University of Toronto, Toronto, Oniario, M5S 1A7 Canada

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W. R. Peltier Department of Physics, University of Toronto, Toronto, Oniario, M5S 1A7 Canada

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Abstract

The model used by Davis and Peltier (1976) to study the linear stability of a compressible, stratified parallel shear flow underlain by a rigid boundary is extended to include the effects of turbulent dissipation. It is shown that the incorporation of both eddy viscosity and thermal diffusivity removes the critical level singularity that occurs in the inviscid compressible model. Both Kelvin-Helmholtz and resonant instabilities continue to exist in the presence of dissipation. The stability characteristics of both modal types' are investigated as functions of the parameters of the background flow, including Reynolds number. Dissipation is found to reduce the range of horizontal wavenumbers for which Kelvin-Helmholtz instability is possible, primarily by stabilizing the short-wavelength disturbances. The dissipative resonant modes are also found over a reduced range of parameter space, but the entire region of resonant instability is shifted to shorter horizontal wavelengths. This behavior is explained in terms of three interrelated factors: the effect of dissipation on the overreflection properties of the critical level, the effect of dissipation throughout the fluid as a whole, and the fact that the vertical wavelength of the instability in the region between the bottom of the shear layer and the ground is quantized. The long-wavelength propagating disturbances generated by a nearly isentropic shear layer in the inviscid model appear to be stabilized by dissipation. The growth rates of both Kelvin-Helmholtz and resonant modes are reduced by dissipation, but the reduction is more severe in the case of the resonant modes. The evolution of instabilities in a real shear layer is discussed in the light of this result.

Abstract

The model used by Davis and Peltier (1976) to study the linear stability of a compressible, stratified parallel shear flow underlain by a rigid boundary is extended to include the effects of turbulent dissipation. It is shown that the incorporation of both eddy viscosity and thermal diffusivity removes the critical level singularity that occurs in the inviscid compressible model. Both Kelvin-Helmholtz and resonant instabilities continue to exist in the presence of dissipation. The stability characteristics of both modal types' are investigated as functions of the parameters of the background flow, including Reynolds number. Dissipation is found to reduce the range of horizontal wavenumbers for which Kelvin-Helmholtz instability is possible, primarily by stabilizing the short-wavelength disturbances. The dissipative resonant modes are also found over a reduced range of parameter space, but the entire region of resonant instability is shifted to shorter horizontal wavelengths. This behavior is explained in terms of three interrelated factors: the effect of dissipation on the overreflection properties of the critical level, the effect of dissipation throughout the fluid as a whole, and the fact that the vertical wavelength of the instability in the region between the bottom of the shear layer and the ground is quantized. The long-wavelength propagating disturbances generated by a nearly isentropic shear layer in the inviscid model appear to be stabilized by dissipation. The growth rates of both Kelvin-Helmholtz and resonant modes are reduced by dissipation, but the reduction is more severe in the case of the resonant modes. The evolution of instabilities in a real shear layer is discussed in the light of this result.

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