II. THEORY 



A. Heat Transfer 



The vertical temperature conduction in a uniform field (no 

 advection) is expressed by the equation: 



^.^(^^)=0 (2) 



dt dz \ p dz I 



where 9 = temperature 

 t = time 

 fj. - eddy conductivity, and 



p - density. 



pi 

 Assuming -5- is constant, that average temperature is a linear 



function of depth, and that the annual march of surface temperature is 



closely approximated by equation (l), the solution of equation (2) with 



use of notations used in reference 1 is: 



^z,t = ^s,a+-^z + A,^"'''^°s("'^-°,-^^^+ (3) 



Agi?" 2 cos(2wt-a2-r22) + --- 

 where 9, ^- temperature at depth at time 



fig g= mean annual surface temperature 



^z = constant determining mean annual temperature at a depth 



z with respect to 



s,a 



A , A ,•"= annual surface temperature amplitudes for various harmonics 

 QoiQoi'''^ phase angles for various harmonics at the surface 



, = f^, , -_ /^... ,and cu =4^ 

 I V 2y^ 2 V M T 



where T = fundamental period , 



The assumptions introduced in integrating equation (2) do not 

 correspond to conditions in nature — eddy conductivity, /i , is variable 

 with depth and time. The average temperature is not a linear function of 

 depth, and the linear phase angle variation with depth can hardly be expected 

 to be of the same magnitude as the exponent of amplitude variation. In 

 addition, all these factors quite obviously vary with latitude. 



Parameters assiamed to be constant in equation (3) are certainly 

 not constant but are functions of depth, time, latitude, flow, and possible 

 several other factors. However, equation (3) can serve as a general frame, 

 because the present problem is limited to temperature distribution at one 

 level. At a given location these parameters would be constant at a given 



