NONLINEAR HYDROSTATIC ADJUSTMENT

The final equilibrium state of Lamb's hydrostatic adjustment problem i s found for finite amplitude heating. Lamb's problem consists of the r esponse of a compressible atmosphere to an instantaneous, horizontally homogeneous heating. Results are presented for both isothermal and no nisothermal atmospheres. As in the linear problem, the fluid displacem ents are confined to the heated layer and to the region aloft with no displacement of the fluid below the heating. The region above the heat ing 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 th e heat deposited into the acoustic modes increases linearly with the a mplitude 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 l apse rare of the base-state atmosphere. In contrast a fixed fraction o f the available energy generated by the heating goes into the acoustic modes. This fraction (e.g., 12% for a standard tropospheric lapse rat e) agrees with the linear result and increases with increasing stabili ty of the base-state 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 accur ately. However, in the small amplitude limit, only the equations for d eep convection advanced by Dutton and Fichtl predict the thermodynamic state variables accurately for a nonisothermal base-state atmosphere.