= Forced stratified mixing with square-root power input
{title2=$\dot K=\sqrt K(A-K/L_v)$}
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 $L_v$.
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