Long Nonlinear Waves in the Lower Atmosphere

D. R. Christie Research School of Earth Sciences, The Australian National University, Canberra, Australia

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Abstract

This paper is concerned with the theoretical description of long, finite-amplitude waves in the stably stratified lower atmosphere. The time evolution of these waves is governed to first order by the Benjamin-Davis-Ono (BDO) equation when frictional processes are negligible or by the BDO-Burgers equation when turbulent dissipation is significant. Numerical solutions of both of these model equations are presented for a wide variety of initial conditions ranging from long waves of finite volume to internal deep-fluid bore waves of infinite spatial extent. It is shown that initially smooth long wave disturbances evolve rapidly under ideal homogeneous waveguide conditions into solitary waves of exceptionally large amplitude. The BDO-Burgers equation is found to have highly stable, time-independent, deep-fluid internal bore wave solutions which may be either oscillatory or monotonic depending upon the degree of frictional dissipation. A number of specific models for the time evolution of long nonlinear atmospheric waves are proposed and discussed in detail. Explicit formulae are given for the wave propagation parameters, surface perturbation pressure, and wind components and these are illustrated for a simple, but realistic, boundary-layer waveguide model. A study has also been made of the influence on nonlinear wave propagation of either spatial or temporal variations in the degree of turbulent dissipation. It is shown that an increase or decrease in the frictional damping coefficient, such as might be encountered at a land-sea boundary, can induce a significant variation in the speed of propagation and a substantial change in the morphology of finite-amplitude boundary layer wave disturbances. Finally, it is shown that wave induced turbulence plays an important role in the evolution of long nonlinear atmospheric waves.

Abstract

This paper is concerned with the theoretical description of long, finite-amplitude waves in the stably stratified lower atmosphere. The time evolution of these waves is governed to first order by the Benjamin-Davis-Ono (BDO) equation when frictional processes are negligible or by the BDO-Burgers equation when turbulent dissipation is significant. Numerical solutions of both of these model equations are presented for a wide variety of initial conditions ranging from long waves of finite volume to internal deep-fluid bore waves of infinite spatial extent. It is shown that initially smooth long wave disturbances evolve rapidly under ideal homogeneous waveguide conditions into solitary waves of exceptionally large amplitude. The BDO-Burgers equation is found to have highly stable, time-independent, deep-fluid internal bore wave solutions which may be either oscillatory or monotonic depending upon the degree of frictional dissipation. A number of specific models for the time evolution of long nonlinear atmospheric waves are proposed and discussed in detail. Explicit formulae are given for the wave propagation parameters, surface perturbation pressure, and wind components and these are illustrated for a simple, but realistic, boundary-layer waveguide model. A study has also been made of the influence on nonlinear wave propagation of either spatial or temporal variations in the degree of turbulent dissipation. It is shown that an increase or decrease in the frictional damping coefficient, such as might be encountered at a land-sea boundary, can induce a significant variation in the speed of propagation and a substantial change in the morphology of finite-amplitude boundary layer wave disturbances. Finally, it is shown that wave induced turbulence plays an important role in the evolution of long nonlinear atmospheric waves.

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