 Depth branches of a steady depositing gravity current

For positive $q,K,g^{\prime }$ in a steady depositing gravity current, the function $K(h)=q^2/h+g^{\prime }{h}^2/2$ has a unique minimum at $h_c=(q^2/g^{\prime }{)}^{1/3}$, where $\mathrm{Fr}=1$ and $K_c=3q^{4/3}(g^{\prime }{)}^{1/3}/2$. When $K>K_c$ there are two positive depth roots: $h_{-}<h_c$ is supercritical flow and $h_{+}>h_c$ is subcritical flow. Differentiating the momentum invariant and using $g_x^{\prime }=-v_s{g}^{\prime }/q$ givesThe supercritical depth decreases towards $q^2/K$ as $g^{\prime }\downarrow 0$, whereas the subcritical depth increases like $\sqrt{2K/g^{\prime }}$. Because $K_c$ decreases downstream, an initially strict branch never reaches a finite-distance critical point. A critical inlet instead gives a singular depth derivative, so a smooth critical continuation is not justified by this model.