Dilute frazil polynya similarity solution ($b,u,\varphi \propto x^{1/3}$)

In a uniformly wind-forced polynya with normalized heat loss $f$, let $H=L+c_{p}m{C}_{\infty }$, $K=f/H$, $J=3S/(4\rho )$ and $r=\rho -{\rho }_s$. The dilute equilibrium model gives $b{u}^2=Jx$, $\varphi bu=Kx$ and $rg\varphi b{b}^{\prime }=S/4$. Hence $u=(rgKx/\rho {)}^{1/3}$, $b=Jx/u^2$ and $\varphi =(K/J)u$. This similarity solution is valid only while the frazil ice volume fraction is small and no exported crystal flux has been neglected improperly.