Drying front in a rainfall-driven Manning kinematic wave ($h_t+(h^{5/3}{)}_s=E(t),h\ge 0$)

Let $H(t)={\int }_0^{t}E(r)dr$, increasing until $t^{\ast }$ and decreasing afterwards. For dry initial data and zero upstream depth, a characteristic emitted at time $\tau$ has $h=H(t)-H(\tau )$ and position $s=\frac{5}{3}{\int }_{\tau }^t[H(r)-H(\tau ){]}^{2/3}dr$, for as long as its depth stays nonnegative. The height follows by integrating the source along a characteristic, and the position follows from the derivative of the Manning flux.

When $E<0$ and $H(t)>0$, define the earlier emission time ${\tau }_d(t)\le t^{\ast }$ by $H({\tau }_d)=H(t)$. The front where this water is just exhausted hasIt moves downstream because $s_d^{\prime }=-(10/9)E(t){\int }_{{\tau }_d}^t[H(r)-H(t){]}^{-1/3}dr>0$. Behind it depth is zero; ahead, unexhausted characteristic labels determine the wet profile. This construction presumes removal limited by water availability: continuing a negative source unchanged at zero depth would violate nonnegativity. If $H$ reaches zero, the remaining initially generated water has also vanished.