Lamb's hydrostatic adjustment for heating of finite duration

Lamb's hydrostatic adjustment problem for the linear response of an infinite, isothermal atmosphere to an instantaneous heating of infinite horizontal extent is generalized to include the effects of heating of finite duration. Three different time sequences of the heating are considered: a top hat, a sine, and a sine-squared heating. The transient solution indicates that heating of finite duration generates broader but weaker acoustic wave fronts. However, it is shown that the final equilibrium is the same regardless of the heating sequence provided the net heating is the same. A Lagrangian formulation provides a simple interpretation of the adjustment. The heating generates an entropy anomaly that is initially realized completely as a pressure excess with no density perturbation. In the final state the entropy anomaly is realized as a density deficit with no pressure perturbation. Energetically the heating generates both available potential energy and available elastic energy. The former remains in the heated layer while the latter is carried off by the acoustic waves. The wave energy generation is compared for the various heating sequences. In the instantaneous case. 28.6% of the total energy generation is carried off by waves. This fraction is the ratio of the ideal gas constant R to the specific heat at constant pressure c_p. For the heatings of finite duration considered, the amount of wave energy decreases monotonically as the heating duration increases and as the heating thickness decreases. The wave energy generation approaches zero when (i) the duration of the heating is comparable to or larger than the acoustic cutoff period, 2π/N_A ∼ 300 s, and (ii) the thickness of the heated layer approaches zero. The maximum wave energy occurs for a thick layer of heating of small duration and is the same as that for the instantaneous case. The effect of a lower boundary is also considered.