SECT. 2] SMALL-SCALE INTERACTIONS 53 



number can be written in terms of wind and potential temperature differences 

 between two levels, a and b, i.e. 



T du/dz Ua — Ub 



As a sufficiently good approximation for use in the correction term of (18) we 

 may write duldz = u^j'kz so that it becomes 



du u^ 



8z k 



1 ^ akg{da-db) 



Z U^T{Ua — Ub) 



which yields on integration between two levels zi and Z2, and division through- 

 out by ln(z2/zi) : 



U2-U1 _ u^ ag{0a-6b){z2-zi) 

 ln(z2/2i) k T{ua-Ub)\n{z2lzi) 



If now the values of the L.H.S. of equation (19) are plotted against (22 — 2:1)/ 

 ln(22/2i) and a straight line of slope a{glT){6a — 6b)l{ua — Ub) drawn to fit the 

 points as well as possible, then the intercept, w^^/A:, gives the required value of m^. 

 If air temperatures at the requisite two heights are not available, then approxi- 

 mate allowance for stability may be made by taking level b to be the sea 

 surface. This approximation also leads to the following air-sea temperature 

 differences between which (18) and (19) may be applied when dealing with 

 profile data extending up to 10 m. 



Wind speed, m/sec 5 10 15 



Air-sea temperature 



differences, °C -0.4to+1.2 -1.5to+4.o - 3 to + 9 



If the greater height is 5 m instead of 10 m then the temperature difference 

 ranges are doubled. 



At large instability, the convective action sets the pattern to the air motion 

 and Ellison (1957) suggests that it might then be expected that Kh/Km should 

 approach a constant value. Should this be so, the form of the wind-profile 

 would be similar to that of potential temperature. Now dimensional analysis 

 (Priestley, 1954, 1959) shows that under these conditions of free convection 



ddldz oc 2-4/3 



and so this line of reasoning would indicate a similar inverse four-thirds power 

 law for the vertical wind gradient. To provide a formulation which would give 

 this result at large instability and approximate to (18) under near neutral 

 conditions, Ellison (loc. cit.) proposed a relationship in terms of the flux 

 Richardson number 



dujdz = {uJkz){\-yRf)-y* (20) 



