= Potential energy of two-layer entrainment
{title2=$\dot E_P=\tfrac12\rho_lS(g'h)\dot h$}
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 $B=g'h$ is constant. With depth $z$ increasing downward, gravitational <potential energy> is $-gS\int_0^H\rho(z)z\,dz=E_*+\rho_lSBh/2$, 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.
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