D. Mean Temperature Distribution Equations 



Time- smoothed mean temperature distributions at the surface and 

 i+OO feet provide general information on distributions and their varia- 

 bility with latitude. Further exploitation of these distributions per- 

 mits derivation of empirical models similar to equation (3) for various 

 types of water in the North Atlantic. For this purpose smoothed curves 

 were expanded in Fourier series of the type: 



dii]- —^ + X (OnSin ncjt + bpCOS ncut) 



The constants Qq, a^ and bn v©re computed for four harmonics. The am- 

 plitude and phase angle for each harmonic were determined by: 



and a„ = - tan' 



'(^) 



Since an equation is desired for one level only (below the seasonal 

 thermocline at UOO feet), the parameters will not be functions of depth 

 z. Phase angles at AOO feet are designated as 0^, 400 and are obtained 

 directly from expansion of the 400-foot curves, because phase angles at 

 400 feet were not computed in terms of surface phase angles, the phase 

 lag factors may be discarded. With these simplifications equation (3) 

 may be rewritten: 



^400=^s,Q + ^ + A6'"'' cos (a,t- 0,^^00)+ (4) 



A^"'2cos(2ujt-a2^40o) + """ 



where 9^^ q is again the mean annual surface temperature, and A is the 

 annual temperature amplitude of the smoothed curve for the surface. This 

 general amplitude was substituted for individual amplitudes of each har- 

 monic of the surface curve, because individual exponents of each harmonic 

 at hOO feet were computed in terms of this general amplitude as well as in 

 terms of the individual amplitudes of each harmonic of the ^KXJ-foot curve. 



The parameter / is obtained from —^ values at the surface and 

 at 400 feet from corresponding expansions 



^ 400 *^ surf 



The term [-^) is less than {-^) \ therefore / is negative. 



2 400 2 surf 



The exponents for each harmonic are obtained by 



"l,400 ^2,400 



After the / and r parameters were computed for all ocean weather 

 stations being considered, variatioB of the parameters with latitude was 

 determined by grouping the stations with similar oceanographic regimes. 



10 



