SECT. 2] LARGE-SCALE INTERACTIONS 109 



SO that we may substitute from 



A = r/^^ (lla) 



into (12) and (13) in order to express the fluxes of heat and vapor in terms of 

 \^'hat we may deduce about stresses and wind profiles. 

 Furthermore, 



Tq = pCDUa^, (15) 



where the subscript refers to the surface and a to "anemometer height" or 

 the level of measurement. Equation (15) may either be deduced from observa- 

 tions or from the classical turbulence work (see Rossby and Montgomery, 

 1935), in which case the drag coefficient Cd is expressed in terms of surface 

 roughness, anemometer height and a universal constant (von Karman's). 



Finally we exjjress the derivatives in (11-13) in terms of finite differences for 

 the vertical interval Az = Za — 0, so that 



Au = Ua — = Ua 



AT = Ta-To = -{To-Ta) (16) 



Aq = qa-qo == -{qo-qa) 



and substitute relations (16), (14), (lla) and (15) into (11-13) to obtain the 

 three transfer equations, all similar in form, namely. 



To = pCDUa^, (17) 



Qs = pCpCD{To-Ta)Ua, (18) 



Qe = LF = pLcD{qo-qa)ua. (19) 



The foundations, shortcomings and range of validity of this simplified 

 derivation are examined with more care in the previous chapter, in relation to 

 the internal structure of the interface layer. Operationally, equations (17-19) 

 are the forms in which the exchange formulas are used, expressing the transfer 

 in terms of the low-level wind speed, Ua, the difference in property between sea- 

 level and the level a, a few meters above, and a parameter, usually taken as a 

 constant. Montgomery (1940) has shown that equations of this form are 

 derivable even if the measurement levels are not the same for the three proper- 

 ties (the leading parameter is very insensitive to small variations in measure- 

 ment height) and that the form of the equations is not dependent upon the 

 equality of coefficients as assumed in (14) provided their height dependence is 

 linear. He made detailed models of the lowest few centimeters, in which he 

 believed the transition from molecular to turbulent regime to occur, with quite 

 different modelling assumptions for the so-called "hydrodynamically smooth" 

 and "hydrodynamically rough" sea. He thus deduced a sharp increase, of a 

 factor of 2-3, in Cd as the sea went from smooth to rough. The details of his 

 transition-layer modelling are uncertain and extremely difficult to test. We 

 prefer the above formulation in view of usefulness, simplicity and in the light 



