Particle-laden gravity current

A particle-laden gravity current is driven by the excess density of suspended particles. Deposition decreases its reduced gravity, so a finite-volume current can approach a finite runout length.

A steady well-mixed dilute particle-laden gravity current above an absorbing porous bed loses carrier fluid at speed $W$ and particles at total downward speed $W+V$, where $V$ is their relative settling velocity magnitude. Volume conservation and the particle budget give $Q=Q_0-Wx$ and $\varphi =(Q/Q_0{)}^{V/W}$ for $W>0$. The momentum conservation equation is $(Q^2/h+g_0^{\prime }\varphi h^2/2{)}^{\prime }=-Wu$ if the withdrawn fluid carries its local horizontal velocity. Consequently $u{u}^{\prime }+g_0^{\prime }\varphi h^{\prime }+g_0^{\prime }h{\varphi }^{\prime }/2=0$. At $W=0$, $\varphi =e^{-Vx/Q_0}$ and the momentum flux is constant.

With reduced gravity $b$ and settling velocity $V_s$, a mixed rectangular-channel current obeys $Dh=-h{u}_x$, $Du=-b{h}_x-h{b}_x/2$, $Db=-V_{s}b/h$, where $D={\partial }_t+u{\partial }_x$. The characteristic speeds are $u,u\pm c$, with $c^2=bh$. Along the latter curves,The density differential matters: $u\pm 2c$ are Riemann invariants only when the reduced gravity is constant without deposition.

An axisymmetric gravity-current box model with no entrainment and negligible suspended-particle volume loss has constant bulk volume $\mathcal{V}$, uniform height $h$, and uniform particle volume fraction $c$. For an absorbing horizontal bed and downward settling velocity $w_s>0$, particle mass conservation gives $\dot{c}=-w_{s}c/h$. A constant front Froude number closes spreading by $\dot{R}=\mathrm{Fr}\sqrt{g_{0}ch}$, where $g_0=g({\rho }_p-{\rho }_a)/{\rho }_a$ is buoyancy per unit volume fraction. This model describes the mixed bulk and its head closure, not the initial inertial blast or a resolved head shape.

In a mixed particle-laden gravity current, a size class with settling speed $w$ has concentration $c(w,t)=c_0(w)e^{-wJ(t)}$. The shared reduced gravity is $g^{\prime }(J)=\int g_0(w)c_0(w)e^{-wJ}dw$, and $d{R}^4/dJ=4\mathrm{Fr}(\mathcal{V}/\pi {)}^{3/2}\sqrt{g^{\prime }(J)}$. Finite runout therefore requires ${\int }_0^{\infty }\sqrt{g^{\prime }(J)}dJ<\infty$. A positive minimum settling speed guarantees this; a distribution with arbitrarily small settling speeds need not. For $g_0{c}_0(w)\sim C{w}^p$ at zero, with $p>-1$, the integral converges precisely when $p>1$.

With $A=\mathcal{V}/\pi$, $g_0^{\prime }=g_0{c}_0$ and initial radius $R_0$, eliminating time from the axisymmetric settling box model gives $\sqrt{c/c_0}=(R_{\infty }^4-R^4)/(R_{\infty }^4-R_0^4)$. Put $s_0=R_0^2/R_{\infty }^2$ and $\tau =4A/(w_s{R}_{\infty }^2)$. Then $R^2/R_{\infty }^2=\tanh [t/\tau +\mathrm{artanh}{s}_0]$. Thus the finite limiting runout is approached at infinite time; treating it as an exactly attained finite-time stopping point contradicts these balances.

For the monodisperse circular ash runout, integrate the local particle deposition flux ${\rho }_p{w}_{s}c(t)$ from first arrival to infinite time. Here $s=\max (s_0,r^2/R_{\infty }^2)$ for $0\le r<R_{\infty }$ and $s_0=R_0^2/R_{\infty }^2$. The displayed $\Sigma$ is deposited particle mass per horizontal area, and vanishes outside the limiting current. It is constant under the initial disk, then decreases to zero; integrating $2\pi r\Sigma$ gives the initial ash mass ${\rho }_p{c}_0\mathcal{V}$. A deposit thickness additionally requires its packing fraction or bulk deposit density.

For fixed channel width $b(z)=\beta z$, cross-sectional area is $A=\beta h^2/2$. With reduced gravity $G\varphi$, uniform cross-sectional fields and downward settling velocity magnitude $w$, volume conservation, momentum and particle deposition flux giveThe hydrostatic pressure force contains ${\int }_0^h(h-z)\beta zdz=\beta h^3/6$. No streamwise widening term occurs in a prismatic channel. Deposition uses the projected boundary width $\beta h$ under the vertical-settling model.

For the prismatic triangular-channel shallow water equations, $c^2=G\varphi h/2$. The characteristic speeds are $u,u\pm c$. Along $dx/dt=u$, $d\varphi /dt=-2w\varphi /h$. Along $dx/dt=u\pm c$, the hyperbolic system has compatibility relationThe concentration differential cannot generally be discarded to obtain Riemann invariants $u\pm 4c$. Positive depth and concentration are required for this nondegenerate form.

The integrated budget is ${\partial }_t(g^{\prime }h)+B_x=S_B-v_s{g}^{\prime }$. A steady spatial increase needs a source $S_B$ exceeding deposition, for example sediment resuspension or lateral particle input. In unsteady flow, local depletion can produce $B_x>0$ even without particle addition.

For slowly depositing particles, $B_x=-v_{s}B/(uh)$ and local adjustment gives $u\sim {\beta }_{\ast }{B}^{1/3}$ with the constant-speed entraining gravity current on a slope coefficient. At leading order $h\sim h_0+E(x-x_0)$ and $B^{1/3}=B_o^{1/3}-[v_s/(3{\beta }_{\ast }E)]\ln [1+E(x-x_0)/h_0]$. Its validity requires settling slow compared with the current and, for small $E$, compared with entrainment.

Extrapolating the leading weak-settling attenuation of a sloping gravity current to zero flux gives $x_{\max }-x_0=(h_0/E)[\exp (3{\beta }_{\ast }E{B}_o^{1/3}/v_s)-1]$. Since speed tends to zero there, the weak-settling assumption fails in the terminal region; this is a formal leading-model estimate, not a uniformly accurate full-flow runout.

A dilute, vertically mixed particle-laden gravity current with no fluid entrainment, bed erosion or carrier-fluid leakage obeys volume conservation $q=uh=\text{constant}$. With downward settling velocity magnitude $v_s$, an absorbing bed gives $qd{g}^{\prime }/dx=-v_s{g}^{\prime }$. The variable-buoyancy shallow water equations conserve the kinematic momentum flux including hydrostatic pressure,These equations describe a steady interior supplied continuously from upstream, not the moving nose of a finite-volume release. The depth follows one of the depth branches of a steady depositing gravity current.

In a bidisperse particle suspension transported by a steady depositing gravity current, species with concentrations $c_i$ and downward settling velocity magnitudes $v_i$ satisfy $c_i(x)=c_{i0}{e}^{-v_{i}x/q}$. Their local particle deposition fluxes are $v_i{c}_i$. The local deposited particle-volume fraction belonging to species one isFor $v_1>v_2>0$ and both species present, $f_1$ decreases strictly downstream, since $f_1^{\prime }=(v_2-v_1)f_1(1-f_1)/q$. If $g_i^{\prime }=g_{0i}{c}_i$, use $c_i=g_i^{\prime }/g_{0i}$ rather than buoyancy ratios unless $g_{01}=g_{02}$. A number fraction additionally divides each species volume flux by its single-particle volume. Integrating deposition over $[0,x]$ instead gives cumulative weights $q{c}_{i0}(1-e^{-v_{i}x/q})$.

For positive $q,K,g^{\prime }$ in a steady depositing gravity current, the function $K(h)=q^2/h+g^{\prime }{h}^2/2$ has a unique minimum at $h_c=(q^2/g^{\prime }{)}^{1/3}$, where $\mathrm{Fr}=1$ and $K_c=3q^{4/3}(g^{\prime }{)}^{1/3}/2$. 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 $g_x^{\prime }=-v_s{g}^{\prime }/q$ givesThe supercritical depth decreases towards $q^2/K$ as $g^{\prime }\downarrow 0$, whereas the subcritical depth increases like $\sqrt{2K/g^{\prime }}$. 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.

A heated particle-laden layer released along a bed forms a gravity current while $g^{\prime }=g(\gamma \varphi -\theta )>0$. In a gravity-current box model, heating can make $g^{\prime }$ vanish in finite time, whereas settling velocity alone can give finite runout length of a gravity current approached asymptotically.

In a constant-volume gravity-current box model without settling, $\varphi ={\varphi }_0$ and $g^{\prime }=g{\varphi }_0(\gamma -\beta t)$. A constant Froude number then gives finite runout length of a gravity current at the neutral-buoyancy time $\gamma /\beta$. Further motion requires a different physical model once the stable density stratification is lost.

In a channel with width $2\lambda xz$ at height $z$, a well-mixed particle-laden gravity current of depth $h$ occupies area $\lambda x{h}^2$. Its volume conservation, momentum and particle deposition flux balances givewhere the reduced gravity is $g^{\prime }=d\varphi$. The $h/3$ coefficient is the cross-section-weighted mean of $h-z$ in the hydrostatic pressure force. The characteristic curves have speeds $u$ and $u\pm \sqrt{g^{\prime }h/2}$.

A finite-volume gravity-current box model in a widening triangular channel has volume proportional to $h^2{L}^2$, so $hL=K$ is constant. A gravity-current front condition and absorbing-bed particle deposition flux giveEliminating time makes $\sqrt{\varphi }$ affine in $L^{5/2}$, producing a finite limiting runout length of a gravity current despite an infinite idealized stopping time.

The runout length is the limiting horizontal distance reached by a gravity current after its driving buoyancy has been exhausted or balanced by resistance.