 Fixed-energy turbulent mixed layer

In a slowly deepening mixed layer driven by a rotating boundary, a phenomenological balance between energy input, turbulent loss and acceleration of entrained fluid can keep total kinetic energy approximately fixed. For a tank of radius $R$ this gives $u^{2}h=A{\Omega }^2{R}^3$, so velocity decreases as $h^{-1/2}$. The closure does not make the speed constant and is an additional physical assumption rather than a direct consequence of mass conservation.