Thermally Driven Tropical Circulations under Rayleigh Friction and Newtonian Cooling: Analytic Solutions*

Abstract

In this paper, the atmospheric circulations on an equatorial beta plane in response to steady tropical heating are investigated by analytically solving a set of linear equations. Special emphasis is placed on the horizontal structure of forced response under the different combinations of momentum damping and thermal damping, as well as the effect of the zonal domain on the forced responses. Two zonal domains are considered: a zonally cyclic domain and a zonally unbounded domain. The linear model is decomposed in terms of the vertical eigenfunctions in a vertically semi-infinite domain. A new feature of the solution is the existence of a continuous spectrum corresponding to energy propagation out the top of the troposphere. The resulting shallow-water equations are then solved using a method similar to that of Gill. Since the zonal decay scale is proportional to the inverse of the square root of the product of the Rayleigh friction rate and the Newtonian cooling rate, the solutions in a zonally unbounded domain can be good approximations for the solutions in a zonally cyclic domain only when both Rayleigh friction and Newtonian cooling are large enough. When either Rayleigh friction or Newtonian cooling is very weak, the solutions are essentially zonally uniform regardless of the longitudinal location of the heat source in a zonally cyclic domain except in a very narrow zone along the equator. The characteristic meridional scale of the shallow-water system is the equatorial radius of deformation of the shallow-water system multiplied by the fourth root of the ratio between the Rayleigh friction rate and the Newtonian cooling rate. Therefore, the characteristic meridional scale is very large for the Rayleigh friction–dominant case, and the forced response can extend far outside the heating latitude. In contrast, in the Newtonian cooling–dominant case the characteristic meridional scale is very small and the forced response is confined to the heating latitudes. The implications of these solutions for both the thermally driven surface winds and the zonally uniform low-frequency variation in pressure and temperature in the upper half of the tropical troposphere are also discussed.