Potential energy of two-layer entrainment (${\dot{E}}_P=\frac{1}{2}{\rho }_{l}S(g^{\prime }h)\dot{h}$)

In a closed tank of cross-sectional area $S$, entrainment into a uniform upper layer conserves the density deficit $({\rho }_l-{\rho }_u)h$. Thus $B=g^{\prime }h$ is constant. With depth $z$ increasing downward, gravitational potential energy is $-gS{\int }_0^H\rho (z)zdz=E_{\ast }+{\rho }_{l}SBh/2$, so its rate of increase is proportional to the rate of deepening. This includes homogenization throughout the upper layer, not merely lifting a thin parcel across the interface.

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.