Vertical structure of convective heating and the three-dimensional structure of the forced circulation on an equatorial beta plane

In this paper, the three-dimensional structure of the thermally forced atmo sphere on an equatorial beta plane is investigated. Special emphasis is pla ced on the relations between the vertical structure of hearing and the hori zontal structure of the forced response. By solving the vertical eigenvalue-eigenfunction problem in a vertically se mi-infinite domain, the authors obtain a complete set of vertical eigenfunc tions that includes a single barotropic (external) mode and a continuous sp ectrum of baroclinic (internal) modes. These eigenfunctions are used to dec ompose vertical healing profiles for two types of tropical heating: 1) deep heating representing the convective plume (CP) heating and 2) shallow heat ing representing mature cloud (MC) cluster heating. By examining the spectr al energy density of the heating profile, the contributions of each vertica l mode (spectral interval) to the overall structure are explored for each c ase, and the difference between the responses to these two profiles of heat ing is discussed. A dry spectral primitive equation model of the atmosphere is employed to verify the analytical results. The results from both the analytical approach and the numerical simulations are consistent in showing that the vertical structure of the heating is fu ndamental to the structure of the forced response. The CP is deep relative to the MC. Thus, the CP projects onto the vertical eigenfunctions of relati vely larger equivalent depth more so than does the MC. As a result, the CP- forced signals propagate away from the heat source much faster than those f orced by the MC. Hence, when the atmosphere is subjected to the same linear dampings (Rayleigh friction and Newtonain cooling), the spatial (mainly in the horizontal) decay rate of the CP-forced signals is significantly small er than that of the MC-forced signals, and the CP-forced signals extend far ther. To what extent a shallow-water system of a specified vertical mode (e.g., t he Gill model) can approximate the three-dimensional response is also exami ned. Results show that the effective gravity wave speed of the multimode sy stem varies greatly with location. Hence, it is extremely difficult to sele ct a globally suitable equivalent depth so that a one-mode shallow-water sy stem can approximate the spatially three-dimensional structure of the respo nse to a given heating.