SECT. 2] LARGE-SCALE INTERACTIONS 195 



where equations (37-39) are the hydrostatic, continuity and gas laws respec- 

 tively. The small approximations and neglections therein have been detailed 

 in the original paper by Malkus (1956) and justified in similar meteorological 

 studies of flows under imposed heating (Malkus and Stern, 1953; Stern and 

 Malkus, 1953; Stern, 1954; Smith, 1955). 



One more equation is needed to complete the set and eliminate between the 

 five dependent variables u' , w' , p', p' and T'. The crucial step is the introduc- 

 tion of the first law of thermodynamics to achieve this ; by this means the heat 

 source is related to the motion and temperature fields. A standard meteoro- 

 logical form of the first law, namely, 



is expanded and linearized, using the hydrostatic law, to obtain 



H _dT' ,/dT g\ er ,,^ ^ 



where dTldz= —y, the vertical lapse rate of upstream-end temperature, T, and 

 — gjCp — F, the dry adiabatic lapse rate. The net heat source H (cal g~i sec~i) is 

 the same as that in equation (31) and is to be evaluated from Fig. 31b in terms 

 of ddjdt, the substantial time rate of change of potential temperature following 

 air trajectories. 



Elimination of all other dependent variables is now readily performed to 

 obtain a simple differential equation in w' , which is 



dz^ u^ CpU^T pu dz^ ' ^ ^ 



where 6^ ^{r—y)IT ^{Ijd) ddjdz, the given static stability at the upstream end. 

 The latter is, from the large-scale viewpoint, independent of the distance co- 

 ordinate, s, as is the heating function, H, so that writing (42) as an ordinary 

 differential equation with constant coefficients (within each vertical layer) is 

 justifiable. An important physical hypothesis underlying the formulation is the 

 separation of scales of motion. The dependent variables to be solved for, w' and 

 u' , are of the large, tropical-cell scale, while the heating function and stress 

 distribution are created by the far smaller turbulent-convective scale. The 

 latter may, therefore, to first order, be treated as "forcing function" dependent 

 upon the vertical co-ordinate only. Although left out of the steady-state 

 formalism, feed-back between variations in the large-scale trade and its forcing 

 function may clearly be significant ; it is discussed later when we analyze the 

 steadiness of the flow. Meanwhile we inquire what distribution of u' and w' are 

 brought about when we impose the average forcing function as evaluated from 

 the Pacific trade observations. 



To determine the vertical distribution of the second term on the right of (42), 

 which involves second derivatives of tsz, would clearly be pure speculation from 

 the kind of observations presently available. Before describing how this 



