Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 74 / 2

The Boussinesq approximation requires $\delta =(\rho -{\rho }_0)/{\rho }_0=\varphi ({\rho }_p/{\rho }_0-1)\ll 1$. The reference mass density ${\rho }_0$ can then be used in inertia, while the small density excess is retained in buoyancy. Dilute particle volume fraction $\varphi \ll 1$ alone does not suffice if ${\rho }_p/{\rho }_0$ is exceptionally large. Define $G=g({\rho }_p-{\rho }_0)/{\rho }_0$, so the reduced gravity is $g^{\prime }=G\varphi$.

The shallow water equations require depth small compared with the horizontal evolution scale, weak vertical acceleration and approximately hydrostatic pressure. Here the ambient is deep and quiescent, so its leading contribution is a hydrostatic reference pressure; the excess pressure in the layer is ${\rho }_{0}G\varphi (h-z)$. The model also needs nearly uniform streamwise velocity across the section and the stated absence of secondary circulation. The advancing gravity current nose is a separate region where these assumptions can fail.

A nearly uniform particle volume fraction can be maintained by turbulent mixing with eddy diffusivity $K_z$, provided $h^2/K_z\ll h/w$ and the mixing time is short compared with horizontal advection and bulk evolution. Equivalently, $wh/K_z\ll 1$ keeps the bulk settling-induced gradient small. The particles should respond rapidly enough to follow the mixing motions. A thin boundary layer adjacent to the absorbing bed or walls need not be well mixed.

Uniform bulk concentration does not imply zero sedimentation. The relative downward particle velocity still supplies a particle deposition flux of approximately $w\varphi$ to an absorbing boundary. Mixing redistributes the remaining particles and maintains the bulk concentration profile while its overall level decreases; it need not suspend every particle indefinitely. Deposition requires negligible resuspension in the model.

The channel is prismatic: its width depends on height, not on $x$. A layer of depth $h$ has cross-sectional area $A={\int }_0^h\beta zdz=\beta h^2/2$. Volume conservation gives $A_t+(Au{)}_x=0$. With $g^{\prime }=G\varphi$, the integrated excess hydrostatic pressure force isThus the streamwise momentum balance is $(Au{)}_t+(A{u}^2+G\varphi \beta h^3/6{)}_x=0$. Under the specified vertical-settling approximation, the horizontal projection of the depositional boundary has width $\beta h$, so the particle balance is $(A\varphi {)}_t+(Au\varphi {)}_x=-\beta hw\varphi$, where $w=w_{0}f(\varphi )>0$ is the downward speed magnitude. The prismatic triangular-channel shallow water equations are thereforeThe pressure coefficient $h/3$ is the section-weighted mean of $h-z$. The factor two in deposition comes from top width divided by area. There is no streamwise widening term, because $\beta$ is constant along the channel.

Define the positive characteristic speed scale $c=\sqrt{G\varphi h/2}$. In variables $(u,\varphi ,c)$ the equations becomeThe coefficient matrix of this hyperbolic system has eigenvalues $u,u+c,u-c$. Hence the three characteristic families areAlong $dx/dt=u$, the concentration equation is the ordinary differential equation $d\varphi /dt=-2w\varphi /h$. Along $dx/dt=u\pm c$, the left eigenvectors $(1,\mp 4c/(3\varphi ),\pm 4)$ give the sedimenting triangular-channel characteristic compatibility equationsHere every derivative in a given equation follows that characteristic family, and $h=2c^2/(G\varphi )$. These three compatibility equations form the characteristic description; they are not three independent closed equations for all fields on any one curve. In particular, $u\pm 4c$ are not conserved Riemann invariants when the concentration varies: its differential and the deposition source must both be retained. The description assumes $h,\varphi >0$; the dry or zero-buoyancy limit is degenerate.

At the gravity current nose, the depth and velocity change over a distance comparable with the depth, with appreciable vertical acceleration and mixing. The hydrostatic approximation and the depth-uniform interior model do not resolve that structure. A suitable gravity-current front condition iswith a positive, order-one Froude number determined by the nose closure and geometry. It is not fixed by the interior equations alone. This definition uses $\sqrt{g^{\prime }h}$; if the front condition is expressed using $c=\sqrt{g^{\prime }h/2}$, its numerical coefficient changes by $\sqrt{2}$.

For a late-time gravity-current box model, take the depth and particle volume fraction to be uniform over $0<x<X(t)$, and neglect entrainment and resuspension. The initial volume is $\mathcal{V}=\beta h_0^{2}L/2$, soThe total particle volume is $\mathcal{V}\varphi$. Deposition over the projected boundary width gives $d(\mathcal{V}\varphi )/dt=-\beta hXw\varphi$. With the specified hindered settling factor, the prismatic triangular-channel gravity-current box model isInitializing this approximate model with $X(0)=L$ and $\varphi (0)={\varphi }_0$ matches the original volume and concentration; it does not assert that the immediate dam-break flow is a uniform box.

Eliminate time to obtainSince the left primitive is $2\mathrm{artanh}\sqrt{\varphi }$, the hindered-settling runout invariant in a prismatic triangular channel isThis relation, the volume constraint and $\dot{X}$ determine the late-time evolution. If desired, the time is given by the quadrature $t={\int }_L^{X(t)}[\mathrm{Fr}\sqrt{G\varphi (s)h_0\sqrt{L/s}}{]}^{-1}ds$. The physical branch has nonnegative right-hand side in the boxed relation; it ends when that expression reaches zero.

Thus the limiting runout length of a gravity current, measured from the closed end, isThe advance beyond the dam is $X_{\infty }-L$. In the dilute limit, $\mathrm{artanh}\sqrt{{\varphi }_0}\simeq \sqrt{{\varphi }_0}$. The ideal model approaches its finite runout length of a gravity current only as $t\to \infty$: near the limit $\varphi$ decays exponentially with rate $2w_0/h_{\infty }$ and the front speed tends to zero. The exponent $7/4$ uses the prismatic triangular geometry; a channel widening with $x$ has a different volume constraint and runout law.