The poleward heat flux can be calculated as follows. The poleward heat flux is
f = Cp J PvTdz = -a\J VmPmdz + J Vi Pidzj = a 4aH 2 h (H - h)2
We assume that the density balance in the oceanic interior can be approximately treated in terms of a one-dimensional balance between the vertical advection and vertical diffusion:
Using this equation, the scale for the mixing coefficient is k = WD, where W and D are the scales of vertical velocity and thickness of the stratified water. Therefore, the mechanical energy required for sustaining mixing in the ocean interior is estimated as cgAp2 3
e; = k(po + Ap - pm) = W(H - h)(Ap - Apy) = -^-f-h(H - h)3 (5.426)
The sensitivity of the MOC and poleward heat flux to changes in mixed layer properties in the model ocean can be explored through three cases with different linear mixed layer depth profiles (Fig. 5.196a). As stated in the previous section, the mixed layer depth in our simple model should be equal to the depth H of the ocean at the northern boundary, so that the differences between these three cases appear at lower latitudes. It can readily be seen that for these cases, if the mixed layer depth is larger at low and mid latitudes, the circulation rate is also larger there but remains unchanged at the northern boundary (Fig. 5.196b). At the same time, poleward heat flux is enhanced, and the position of maximum poleward heat a Upwelling rate 1.51-■-
Mixing energy a Upwelling rate 1.51-■-
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