Forced stratified mixing with square-root power input ($\dot{K}=\sqrt{K}(A-K/L_v)$)

For $A=U_0^2(1/L_f-1/L_{\rho })>0$, define $K_{\infty }=A{L}_v$ and $y=\sqrt{K/K_{\infty }}$. The positive-energy solution is $y(t)=[y_0+\tanh (\sqrt{K_{\infty }}t/(2L_v))]/[1+y_0\tanh (\sqrt{K_{\infty }}t/(2L_v))]$. Energies above and below $K_{\infty }$ approach it from their respective sides. The zero-energy equation is not Lipschitz and admits both permanent rest and delayed startup, so a unique positive steady limit needs a positive seed or an additional startup prescription. With negligible available and internal energy exchange, the long-time instantaneous mixing efficiency fraction is $L_f/L_{\rho }$, independent of $L_v$.