Thermally driven tropical circulations under Rayleigh friction and Newtonian cooling: Analytic solutions

In this paper, the atmospheric circulations on an equatorial beta plane in response to steady tropical heating are investigated by analytically solvin g a set of linear equations. Special emphasis is placed on the horizontal s tructure of forced response under the different combinations of momentum da mping and thermal damping, as well as the effect of the zonal domain on the forced responses. Two zonal domains are considered: a zonally cyclic domai n 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 exis tence of a continuous spectrum corresponding to energy propagation out the top of the troposphere. The resulting shallow-water equations are then solv ed using a method similar to that of Gill. Since the zonal decay scale is proportional to the inverse of the square ro ot of the product of the Rayleigh friction rate and the Newtonian cooling r ate, the solutions in a zonally unbounded domain can be good approximations for the solutions in a zonally cyclic domain only when both Rayleigh frict ion and Newtonian cooling are large enough. When either Rayleigh friction o r Newtonian cooling is very weak, the solutions are essentially zonally uni form regardless of the longitudinal location of the heat source in a zonall y cyclic domain except in a very narrow zone along the equator. The characteristic meridional scale of the shallow-water system is the equa torial radius of deformation of the shallow-water system multiplied by the fourth root of the ratio between the Rayleigh friction rate and the Newtoni an cooling rate. Therefore, the characteristic meridional scale is very lar ge for the Rayleigh friction-dominant case, and the forced response can ext end far outside the heating latitude. In contrast, in the Newtonian cooling -dominant case the characteristic meridional scale is very small and the fo rced response is confined to the heating latitudes. The implications of these solutions for both the thermally driven surface w inds and the zonally uniform low-frequency variation in pressure and temper ature in the upper half of the tropical troposphere are also discussed.