Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2007 / A / 2

The first Saint-Venant equation conserves water volume. The second balances material acceleration against downslope gravity, bed friction and the free-surface pressure gradient. With length scale $L$, depth scale $H$, slope $S$ and velocity scale $U$, the conventional gravity-friction scaling has $\epsilon =H/(SL)$ and Froude number $F=U/\sqrt{gH}$. The pressure-gradient/downslope-gravity ratio is $\epsilon$, and the inertia/downslope-gravity ratio is $\epsilon F^2$.

The uniform steady solution obeys $u^2=h$ and $hu=Q$, hence $\boxed{u_0=Q^{1/3},h_0=Q^{2/3}}$. In the constant-flux river momentum relaxation reduction, substitute $h=Q/u$ and $X=1-s$. Since $u_s=-u_X$ and $h_s=Q{u}_X/u^2$, the resulting quasilinear partial differential equation isA downstream boundary at $X=0$ supplies data into $X>0$ precisely when $c(u_1)>0$, equivalent to $\boxed{u_1<Q^{1/3}/F^{2/3}}$. Although water flows out there, the relevant characteristic curves carry information upstream in this subcritical regime.

Along a characteristic curve, $dX/dt=c(u)$ and $du/dt=(1-u^3/Q)/(\epsilon F^2)$. Positive $u$ relaxes toward $u_0$; near $u_0$, its displacement decays at rate $3/(\epsilon F^2{u}_0)$. Boundary-generated deviations therefore become small after a time of order $\epsilon$ and, while the characteristic speed stays bounded away from zero, within a distance of order $\epsilon$. The outer state is the initial uniform solution. A jump between initial and boundary values at their corner can produce a weak solution or a thin initial adjustment; a globally smooth corner solution requires compatible data.

For the moving boundary put $Y=X-vt$. The characteristic speed relative to it is $c(u)-v$, while the outer speed isFor the intended tidal slowing $0<u_1\le u_0$, $c$ is decreasing in $u$ and hence $c(u)\ge v_b$ throughout the relaxation. Thus $v<v_b$ leaves an incoming family and an attached thin boundary layer. The diagram shows numerical characteristics of this scalar reduction, with a slower boundary velocity. When $v>v_b$, the tidal boundary overtakes the unperturbed characteristic family; an attached smooth relaxation layer is no longer appropriate, and a tidal bore or shock wave can arise in a fuller river model.

There are two qualifications to the printed reduction. First, without the physical slowing assumption, $v<v_b$ alone need not make the boundary incoming: for $u_1>u_0$, also require $v<c(u_1)$. Second, imposing $hu=Q$ exactly everywhere in the original continuity equation gives $h_t=0$, and consequently $u_t=0$. The time-dependent characteristics above solve the requested momentum-only approximation, rather than the complete Saint-Venant equation system with an exactly constant flux.