108 MALKUS [chap. 4 



several properties in the atmosphere have been extensively measured (Mont- 

 gomery, 1940; Deacon, Sheppard and Webb, 1956) and are indeed logarithmic 

 when the stability is neutral, a wind is blowing, and the other physical assump- 

 tions are fulfilled. Furthermore, recent and more fundamental work on turbu- 

 lent shear flow (W. V. R. Malkus, 1956) has deduced theoretically the 

 logarithmic law, relationship (9), the constancy of t, and the linear ratio 

 {duldz)JT near the boundary in a simplified situation where the assumptions 

 outlined are clearly fulfilled. It should be cautioned, however, that other 

 theoretical work by the same author (W. V. R. Malkus, 1954), equally sup- 

 ported by controlled laboratory experiment, predicts that when heat flow alone 

 occurs in the absence of a shearing current, neither the logarithmic law, nor 

 linear increase of Kp is realized. Therefore, the range of validity of the transfer 

 formulas to be deduced should be restricted to conditions of a significant wind 

 blowing over the ocean, where turbulent shear flow is the major process govern- 

 ing the transports of all properties. Happily this condition is almost always 

 realized over the oceans where large sea-air fluxes are occurring, particularly 

 in the trades. Obviously, since the exchange formulas give fluxes linearly 

 proportional to wind speed u, they will give incorrect (zero) results as the wind 

 speed vanishes. Therefore, it may be said that the use of the exchange 

 formulas to obtain fluxes from the tropical oceans to the overlying air is 

 probably as valid a use of simple physically deduced laws as occurs anywhere 

 in large-scale meteorology or oceanography. In the lowest few tens of meters 

 there the air is well-stirred, shear-turbulence dominated (relatively free of 

 pronounced "selective buoyancy" forces), neutrally stable and barotropic. 



The useful forms of the exchange formulas are readily derived using the 

 above assumptions from equations of the type (9) for each property of interest, 

 namely, 



jr du 

 az 



^ J. Tde dT ..., 



Q, ^ -CpKs-^-^x -CpRs-^ (12) 



and 



Qe ^ LE = -LKe^^ (13) 



dz 



where equation (12) for the dependence of the sensible heat flux, Qs (which 

 strictly speaking should be written proportional to {TI6)d9ldz), has been 

 approximated as proportional to the actual temperature gradient. This is clearly 

 valid in surface layers only a few meters in depth (see Chapter 3, p. 66). 



It will now be assumed, partly for simplicity in derivation, that, firstly, the 

 height above the sea at which all three atmospheric properties are measured 

 is identical, and, secondly, that 



Km = Ks = Ke = K (14) 



