= Potential energy of mixed-layer deepening with an initial buoyancy jump
{title2=$\Delta\mathcal P/\rho_0=\Delta b\,a d/2+N^2a d^2/4+N^2d^3/12$}
Let an initial upper mixed layer have finite depth $a$, buoyancy $b_1$, and fixed upper boundary; beneath it let $b=b_1-\Delta b+N^2z$ with the initial interface at $z=0$. Homogenization down to $z=-d$ conserves <buoyancy> and gives mixed buoyancy $b_1-[\Delta b\,d+N^2d^2/2]/(a+d)$. Integrating gravitational <potential energy> over the initial and final profiles gives the displayed increase per unit area. Its derivative is $\rho_0[\Delta b\,a/2+N^2a d/2+N^2d^2/4]$. Specifying absorbed mixing power is not enough to determine deepening unless the initial depth or an alternative upper-reservoir model is supplied.
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