Potential energy of mixed-layer deepening with an initial buoyancy jump ($\Delta \mathcal{P}/{\rho }_0=\Delta bad/2+N^{2}a{d}^2/4+N^2{d}^3/12$)

Let an initial upper mixed layer have finite depth $a$, buoyancy $b_1$, and fixed upper boundary; beneath it let $b=b_1-\Delta b+N^{2}z$ with the initial interface at $z=0$. Homogenization down to $z=-d$ conserves buoyancy and gives mixed buoyancy $b_1-[\Delta bd+N^2{d}^2/2]/(a+d)$. Integrating gravitational potential energy over the initial and final profiles gives the displayed increase per unit area. Its derivative is ${\rho }_0[\Delta ba/2+N^{2}ad/2+N^2{d}^2/4]$. Specifying absorbed mixing power is not enough to determine deepening unless the initial depth or an alternative upper-reservoir model is supplied.