Compressible f ‐plane solutions to body forces, heatings, and coolings, and application to the primary and secondary gravity waves generated by a deep convective plume

We derive the analytic, linear, f ‐plane compressible solutions to local, interval, 3‐D horizontal and vertical body forces, and heat/coolings in an isothermal, unsheared, and nondissipative atmosphere. These force/heat/coolings oscillate at the frequency â and turn on and off smoothly over a finite interval in time. The solutions include a mean response, gravity waves (GWs), and acoustic waves (AWs). The excited waves span a large range of horizontal/vertical scales and frequencies ω . We find that the compressible solutions are important for GWs with vertical wavelengths | λ z | > ( 1 to 2 ) × π ℋ if the depth of the force/heat/cooling is greater than the density scale height ℋ . We calculate the primary GWs excited by a deep convective plume, ray trace them into the thermosphere, and calculate the body force/heat/coolings which result where the GWs dissipate. We find that the force/heat/cooling amplitudes are up to ∼40% smaller using the compressible (as compared to the Boussinesq) GW spectra. For a typical plume, the force/heat/coolings are deeper than ℋ and have maximum amplitudes of ∼0.2 to 0.6 m/s 2 and ∼0.06 to 0.15 K/s for solar maximum to minimum, respectively. The heat/cooling consists of dipoles at z ∼150–200 km and a heating at z ∼240–260 km. We find that the compressible solutions are necessary for calculating the secondary GWs excited by these thermospheric force/heat/coolings.