SECT. 2] SMALL-SCALE INTERACTIONS 49 



where 6 denotes potential temperature and q specific humidity. In (6), Ca is 

 known as either the friction or drag coefficient. For heat transfer ha in (7) is 

 the Stanton number, while, for evaporation, it would be appropriate to call da 

 the Dalton number as Dalton first proposed a relationship of the form of 

 equation (8). 



In dealing with heat transfer the well-known Reynolds analogy treatment is 

 to take the Stanton number equal to the drag coefficient, a procedure which 

 has been found to give approximately correct results in some problems but to 

 be quite considerably in error in others. The corresponding treatment for 

 evaporation is to take the Dalton number equal to the drag coefficient. i 



It is readily seen from equations (4) and (5) that these treatments would only 

 be accurate if equation (4) applied not only to the fully turbulent layers, but 

 right down to the sea surface itself. Such extension ignores the fact that in the 

 transitional and laminar layers the various transfer coefficients are not equal. 

 Furthermore, in the layer between the troughs and crests of the waves, the 

 momentum-transfer mechanism must differ from that of the other entities 

 owing to the operation of the pressure forces which give rise to the form drag 

 of the waves. The various theoretical attempts that have been made to take 

 account of these effects are discussed later. 



2. Momentum Transfer and the Wind-Profile 



A. The Neutral Wind-Profile 



Studies both in the laboratory and in the field have established for steady 

 flow over rigid surfaces and neutral conditions of stability a simple relation- 

 ship between the vertical gradient of fluid speed and the shearing stress in the 

 fully turbulent layers. This is 



0'll> ^d; / f\\ 



dz kz 



in which u = mean speed at height z, u^ = {rlpy''', t = shearing stress, and 

 /) = fluid density. As the shearing stress is the vertical flux of momentum in the 

 direction of the mean flow : 



TJp = —u'w' 

 and so u^^ has the dimensions of a velocity and is called the friction velocity. 

 The Karman constant, k, has a value of around 0.4; 0.41 according to Clauser 

 (1956) is the best value fitting the various laboratory investigations and, as 

 the work over natural surfaces (Rider, 1954; Deacon, 1955) gives a closely 

 similar value, this is used hereafter. The eddy viscosity, Km, is U:^^l{duldz) and, 

 for neutral conditions, it follows from (9) that Km = ku^z in the fully turbulent 

 layers. 



1 In obtaining the evaporation from measurements of the heat available for the two 

 processes, evaporation and sensible-heat transfer together, the apportionment is custo- 

 marily made by assuming equality of Dalton and Stanton numbers. This is the Bowen 

 ratio method applied in such heat-balance analyses as those of Jacobs (1951, 1951a) and 

 Budyko et al. (1954). 



3- s. I 



