Rotating-disc mixed-layer depth law ($h/R\propto [{\Omega }^{2}R/(N^2\delta ){]}^{1/3}(\Omega t{)}^{2/9}$)

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.