TY - JOUR

T1 - Nonlinear hydrostatic adjustment

AU - Bannon, Peter R.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1996/12/1

Y1 - 1996/12/1

N2 - The final equilibrium state of Lamb's hydrostatic adjustment problem is found for finite amplitude heating. Lamb's problem consists of the response of a compressible atmosphere to an instantaneous, horizontally homogeneous heating. Results are presented for both isothermal and nonisothermal atmospheres. As in the linear problem, the fluid displacements are confined to the heated layer and to the region aloft with no displacement of the fluid below the heating. The region above the heating is displaced uniformly upward for heating and downward for cooling. The amplitudes of the displacements are larger for cooling than for warming. Examination of the energetics reveals that the fraction of the heat deposited into the acoustic modes increases linearly with the amplitude of the heating. This fraction is typically small (e.g., 0.06% for a uniform warming of 1 K) and is essentially independent of the lapse rate of the base-state atmosphere. In contrast a fixed fraction of the available energy generated by the heating goes into the acoustic modes. This fraction (e.g., 12% for a standard tropospheric lapse rate) agrees with the linear result and increases with increasing stability of the basestate atmosphere. The compressible results are compared to solutions using various forms of the soundproof equations. None of the soundproof equations predict the finite amplitude solutions accurately. However, in the small amplitude limit, only the equations for deep convection advanced by Dutton and Fichtl predict the thermodynamic state variables accurately for a nonisothermal base-state atmosphere.

AB - The final equilibrium state of Lamb's hydrostatic adjustment problem is found for finite amplitude heating. Lamb's problem consists of the response of a compressible atmosphere to an instantaneous, horizontally homogeneous heating. Results are presented for both isothermal and nonisothermal atmospheres. As in the linear problem, the fluid displacements are confined to the heated layer and to the region aloft with no displacement of the fluid below the heating. The region above the heating is displaced uniformly upward for heating and downward for cooling. The amplitudes of the displacements are larger for cooling than for warming. Examination of the energetics reveals that the fraction of the heat deposited into the acoustic modes increases linearly with the amplitude of the heating. This fraction is typically small (e.g., 0.06% for a uniform warming of 1 K) and is essentially independent of the lapse rate of the base-state atmosphere. In contrast a fixed fraction of the available energy generated by the heating goes into the acoustic modes. This fraction (e.g., 12% for a standard tropospheric lapse rate) agrees with the linear result and increases with increasing stability of the basestate atmosphere. The compressible results are compared to solutions using various forms of the soundproof equations. None of the soundproof equations predict the finite amplitude solutions accurately. However, in the small amplitude limit, only the equations for deep convection advanced by Dutton and Fichtl predict the thermodynamic state variables accurately for a nonisothermal base-state atmosphere.

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U2 - 10.1175/1520-0469(1996)053<3606:NHA>2.0.CO;2

DO - 10.1175/1520-0469(1996)053<3606:NHA>2.0.CO;2

M3 - Article

AN - SCOPUS:0030324362

SN - 0022-4928

VL - 53

SP - 3606

EP - 3617

JO - Journal of the Atmospheric Sciences

JF - Journal of the Atmospheric Sciences

IS - 23

ER -