A theoretical investigation of hydrostatic and geostrophic adjustment in a compressible atmosphere

by 1975- Chagnon, Jeffrey M.

iii The adjustment of a compressible, stably stratified atmosphere to sources of hydrostatic and geostrophic imbalance is investigated using a linear model. Imbalance is produced by prescribed, time-dependent injections of mass, heat, or momentum that model those processes considered “external” to the scales of motion on which the linearization and other model assumptions are justifiable. Solutions are demonstrated in response to localized warming characteristic of small isolated clouds, larger thunderstorms, and convective systems in order to determine how the spatial and temporal details of the injection affect the adjustment. The response to injections of different type (e.g. mass versus heat versus momentum) is also demonstrated in order to determine how the injection type affects the adjustment. For a semi-infinite atmosphere, solutions consist of a set of vertical modes of continuously varying wavenumber, each of which contains time dependencies classified as steady, acoustic-wave, and buoyancy-wave contributions. Additionally, a rigid lower boundary condition implies the existence of a discrete mode - the Lamb mode - containing only a steady and acoustic-wave contribution. The forced solutions are generalized in terms of a temporal Green’s function, which represents the response to an instantaneous injection. The partitioning of the energy among the acoustic, buoyancy, and Lamb waves and the steady state is examined. The energy associated with each of these classes is distinct and, after the external injection is shut off, constant in time. The characteristics of iv this partitioning depend on the spatial-temporal detail of the injection, as well as whether the imbalance is generated by injection of heat, mass, or momentum. Injections that generate identical potential vorticity distributions constitute an interesting set of cases for comparison. Although the asymptotic steady state is identical in these cases, the energy of such potential-vorticity-equivalent injections depends on the manner by which the potential vorticity is introduced. If the potential vorticity is introduced rapidly rather than slowly, then more high frequency waves will be generated. Unlike the steady-state response, the transient response to a given injection may be very different than that to its averaged injection. v