The linear acoustic and gravity wave response to localized heating in a compressible atmosphere

The role of acoustic and gravity waves in the adjustment of a compressible atmosphere to an instantaneous, localized heat source is examined. An analytic solution to the 3-D, time-dependent, linear hydrostatic and geostrophic adjustment problem is obtained using a separation of variables technique. The horizontal structure is solved via the Fourier transform. The vertical structure is described by a Sturm-Liouville equation, whose solution is a linear combination of vertically orthogonal eigenfunctions. Each of these eigenfunctions has a unique time-dependence implied by the initial conditions and the dispersion relation. In this manner, we represent the solution as an explicit sum of acoustic and gravity waves, a Lamb mode, and a potential vorticity conserving steady-state. The time-dependent energetics have a similar orthogonal representation. The wave energetics are thus partitioned between acoustic and gravity waves, illuminating the contrasting roles of these waves in the hydrostatic and geostrophic adjustment process.