Fixed-energy turbulent mixed layer ($u^{2}h=A{\Omega }^2{R}^3$)

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.

For potential energy of two-layer entrainment with constant $B=g^{\prime }h$, the fixed-energy turbulent mixed layer assumption gives $u_U^{2}h=C_K^2{\Omega }^2{R}^2{h}_0$. If $u_I=C_I{u}_U$ and the interfacial power model for entrainment applies with fixed coefficients, integration gives the displayed depth law with $A=5C_W{c}_D{C}_I^3{C}_K^3{\Omega }^2{R}^3/(B{h}_0)$. The relation applies only before the layer reaches the tank bottom. Its interfacial Richardson number $B{d}_I/(h{u}_I^2)$ stays constant for fixed interface thickness, whereas it decreases like $h^{-1}$ in the constant-speed entrainment model.

With constant interface thickness and the local entrainment exponent $3/2$, the fixed-energy turbulent mixed layer satisfies $\dot{h}=C{\Omega }^4{R}^6/(N^3{\delta }^{3/2}{h}^{7/2})$. Integration gives a linear growth law for $h^{9/2}$. Once an initial-depth offset is negligible, $h/R$ scales as $[{\Omega }^{2}R/(N^2\delta ){]}^{1/3}(\Omega t{)}^{2/9}$. The law applies before the mixed layer reaches the tank bottom and after the initial forcing transient.