196 MALKUS [chap. 4 



difficulty was surmounted in the theoretical study, let us first omit this term 

 to illuminate some general properties of bottoin-heated, two-dimensional flows. 

 The heating function (Fig. 31b) is fairly well apjiroximated by an exponential 

 decay with height, so that the frictionless equation may be written 



'iJ^ + B^w' = Pe-'/d, (42a) 



where B-^ = g^lu^; P^gHolcpU^T. 



The decay -height of the heat source, d, is chosen from Fig. 31b to give the 

 correct integrated heating for the layer ; T is its mean temperature. For a deep, 

 uniformly stable air layer, bounded below by the ground, the general solution 

 to this equation is 



The divergence field is prescribed as 



dw' du' P 



Since the sine function is zero at the ground, the divergence must always be 

 jjositive there. This means a downstream acceleration of the surface wind and 

 subsidence in the low levels, regardless of the value of B, just so long as the 

 heat source is positive and decays with height. With weak stability, the di- 

 vergence extends through a 1-2 km deep layer and fades out approximately 

 with the heat source. Equations (33b) and (44) together show that the surface 

 pressure must drop downstream. However, without friction the magnitudes are 

 unrealistically related. The observed heating function gives much too great 

 a divergence or, conversely, the observed divergence requires a pressure drop 

 nearly two orders of magnitude lower than observed. Furthermore, from (44) 

 we see that the downstream divergence is maximum at the ground and decreases 

 monotonically upward, giving a similar-shaped profile of u + u'. Nevertheless, 

 we see the important relation between heating and subsidence in a statically 

 stable flow which is required to be two dimensional. A larger heating function 

 gives greater subsidence by means of enhancing the downstream pressure drop. 

 Thus an ageostrophic mass flow may be thermally driven and, contrary to 

 popular preconception, does not require friction for its existence ; the latter 

 only magnifies the pressure head and heat source required to maintain it. 



Realistic quantitative relations between the parameters are derived by the 



theory when frictional stress is introduced and (42) is solved for a two-layer 



model. The lower sub-cloud layer is adiabatic {6^ — 0) and the upper (cloud) 



layer is slightly stable, with a top boundary {w' = 0) at the inversion. In the 



original work, a trick was devised to deduce the total forcing function from the 



observations, approximating it by the difference in two exponentials in z, 



namely, 



d 2?// 



-^-^ + BW = -F{z) = Pe-'fd-Ge-'/^, (42b) 



