In this case, the temperature is out of phase with the heating, and represents a time average of the fluctuating heating. The peak temperature occurs later than the peak solar heating, since it takes time for the mixed layer to respond to the accumulating heating. Further, in this case, the seasonal temperature fluctuation becomes small as the mixed layer depth is made large, since the mixed layer becomes more and more efficient at averaging out the seasonal fluctuations of solar flux.
The variations in solar radiation over the course of a year are not sinusoidal, but we can nonetheless gain some further insight into the seasonal cycle by writing S' = S\ cos(wt). For this form of forcing, Eq. 7.15 can be solved most easily by using complex exponentials. Since S' = S\Real(exp(—iu>t)), the solution may be written T' = Real(Aexp(—iwt)). Substituting this form of solution into Eq. 7.15 we find si 1/t +i pcpH 1/t 2 + w2 where the phase and amplitude are
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