= Depth branches of a steady depositing gravity current
{title2=$h_-(x),h_+(x)$}
For positive $q,K,g'$ in a <steady depositing gravity current>, the function $K(h)=q^2/h+g'h^2/2$ has a unique minimum at $h_c=(q^2/g')^{1/3}$, where $\operatorname{Fr}=1$ and $K_c=3q^{4/3}(g')^{1/3}/2$. 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 $g'_x=-v_sg'/q$ gives
$$
\frac{dh}{dx}=\frac{v_sh}{2q(1-\operatorname{Fr}^2)}.
$$
The supercritical depth decreases towards $q^2/K$ as $g'\downarrow0$, whereas the subcritical depth increases like $\sqrt{2K/g'}$. 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.
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