= Solution
Use a <gravity-current box model>: approximate the current by uniform depth $h(t)$ and length $L(t)$, with constant <reduced gravity>, no deposition and no <fluid entrainment>. The triangular cross-sectional area is $h^2/2$, so the fixed released volume is
$$
V_0=\frac12Lh^2,\qquad h=\left(\frac{2V_0}{L}\right)^{1/2}.
$$
The <Benjamin deep-ambient front condition> closes the integral model:
$$
\dot L=\sqrt{2g'h}=F\sqrt{g'}(2V_0)^{1/4}L^{-1/4},\qquad F=\sqrt2.
$$
For a finite initial length $L_0$, direct <integration> gives
$$
\boxed{L(t)=\left[L_0^{5/4}+\frac54F\sqrt{g'}(2V_0)^{1/4}t\right]^{4/5},\qquad h(t)=\sqrt{2V_0/L(t)}.}
$$
The ideal point-release limit takes $L_0=0$ and has $L\propto t^{4/5}$, $h\propto t^{-2/5}$. Its divergent initial depth is outside the <shallow water> approximation, so a real finite reservoir regularizes the early stage. This is the <finite-volume triangular-channel current> integral law; it does not assert a spatially uniform exact solution of the local equations.
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