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Take the current to be dilute, vertically well mixed and Boussinesq approximation, with constant channel width $w$, a horizontal absorbing bed, negligible ambient fluid entrainment, negligible bed friction, and no resuspension. The shallow-water approximation neglects vertical inertia and gives hydrostatic pressure. Particle loss alters reduced gravity but does not alter the leading-order carrier-fluid volume or inertia; otherwise additional mass and momentum terms would be necessary. The current occupies depth $h(x,t)$, has depth-uniform speed $u(x,t)$, and has particle volume fraction $c(x,t)$, with $g^{\prime }=g_{o}c$ and constant $g_o>0$.

Volume conservation of a short channel segment gives $h_t+(uh{)}_x=0$. Its suspended particle amount per unit bed area is $hc$, and the downward particle deposition flux is $v_{s}c$. Thus $(hc{)}_t+(uhc{)}_x=-v_{s}c$, which becomes $(h{g}^{\prime }{)}_t+(uh{g}^{\prime }{)}_x=-v_s{g}^{\prime }$ on multiplying by $g_o$.

At vertical coordinate $y$ above the bed, excess hydrostatic pressure divided by reference mass density is $g^{\prime }(h-y)$. Integrating across the depth gives the depth-integrated force flux $g^{\prime }{h}^2/2$. The horizontal momentum per unit bed area is $hu$ and the advected kinematic momentum flux is $h{u}^2$. The resulting variable-buoyancy shallow water equations areThe derivative of the pressure term acts on both $h$ and $g^{\prime }$; treating reduced gravity as horizontally constant while particles settle would lose part of the force.

For a steady interior, let $q=Q/w=uh>0$ denote volume flux per unit width. Then the particle balance is a first-order linear differential equation,Here $g_0^{\prime }=g^{\prime }(0)>0$, $v_s\ge 0$ is constant, and deposited particles are not returned to the current. In particular the decay length is $q/v_s=Q/(w{v}_s)$ for $v_s>0$, and the cancellation of depth follows from $uh=q$: dilution and transit time compensate in the deposition balance.

The steady momentum equation also givesThe inlet depth selects $K=q^2/h_0+g_0^{\prime }{h}_0^2/2$. Depth and speed at every downstream position are determined by the continuous positive root of a polynomial inThis implicit expression also makes clear why the inlet condition is needed: there are generally two possible positive depths.

The depth branches of a steady depositing gravity current are distinguished by the Froude number ${\mathrm{Fr}}^2=q^2/(g^{\prime }{h}^3)$. Differentiating the momentum invariant, and using the exponential decay of reduced gravity, givesFor $v_s>0$, supercritical flow has $\mathrm{Fr}>1$ and decreasing depth, with $h\to q^2/K$ and $u\to K/q$ as $x\to \infty$. Subcritical flow has $\mathrm{Fr}<1$ and increasing depth, with $h\sim \sqrt{2K/g^{\prime }(x)}\propto e^{v_{s}x/(2q)}$. This indefinitely deepening branch eventually violates the assumed shallow or confined-channel geometry. Hence the depth does not universally decrease: that conclusion needs a supercritical inlet.

For fixed $g^{\prime }>0$, $q^2/h+g^{\prime }{h}^2/2$ has its unique minimum at $h_c=(q^2/g^{\prime }{)}^{1/3}$, with $K_c=3q^{4/3}(g^{\prime }{)}^{1/3}/2$. The two positive roots when $K>K_c$ lie below and above $h_c$. Since $K_c$ decreases downstream while $K$ is constant, an initially strict branch remains separate from the other branch and cannot cross a critical point. A critical inlet with $v_s>0$ has no finite depth derivative: at $\mathrm{Fr}=1$ the first differentiated equation would equate zero to a positive value. It requires a singular entrance adjustment or physics beyond this steady model. If $v_s=0$, both reduced gravity and the continuous depth branch are constant. These steady solutions describe the supplied current interior, not a moving finite-release front.

For two particle populations, write $g^{\prime }=g_1^{\prime }+g_2^{\prime }$ and $g_i^{\prime }=g_{0i}{c}_i$, allowing different constant particle-density conversion factors. The bidisperse particle-laden gravity current equations areFor the steady case,Thus the depth is obtained from the same cubic polynomial invariant with the sum of the two exponentially decreasing reduced gravity contributions.

The local deposit consists of the two particle deposition fluxes $v_1{c}_1$ and $v_2{c}_2$. For a positive total deposition rate, the bidisperse gravity-current deposition fraction by particle volume isIf the particles have a common material density, this is also their local mass fraction. If they have common $g_o$, it can be written in the source's buoyancy notation asFor unequal $g_{0i}$ replace each $g_{i0}^{\prime }$ by $g_{i0}^{\prime }/g_{0i}$. The exponent sign matters: with both populations present and $v_1>v_2>0$, the faster-settling population has decreasing local deposit fraction downstream, since $f_1^{\prime }=(v_2-v_1)f_1(1-f_1)/q<0$. Equal positive settling speeds give a constant fraction. A nonsettling population contributes no deposit; if the entire local deposition rate is zero the deposit fraction is undefined.

If “fraction of particles” means number fraction rather than particle-volume or mass fraction, let $a_i$ be a single particle's volume in population $i$. The correct weights are then $v_i{c}_i/a_i$, soSettling speeds alone do not determine $a_i$ without particle properties and a drag law. Finally, if the intended deposit is cumulative over the bed from $0$ to $x$, rather than local at $x$, integrating particle deposition flux gives the species volumes per unit channel width $D_i(x)=q{c}_{i0}(1-e^{-v_{i}x/q})$. The cumulative particle-volume fraction isFor both $v_i>0$, its limit at infinity is the incoming particle-volume fraction, whereas its limit as $x\downarrow 0$ is the local deposition fraction at the inlet. Local sorting and the cumulative inventory are different observables.

Let $\mathcal{V}=H{L}_0$ be the layer volume per unit transverse width. The gravity-current box model imposes $hL=\mathcal{V}$; it approximates the front and scalar balances, rather than an exactly uniform solution of the local momentum equation. For a specified constant front Froude number $\mathrm{Fr}$, the heated particle-laden gravity current satisfiesThe initial data are $L=L_0$, $\varphi ={\varphi }_0$, and $\theta =0$. The gravity-current front condition applies only while $\gamma \varphi -\theta \ge 0$.

For heating without settling, $\varphi ={\varphi }_0$ and $\theta =\beta {\varphi }_{0}t$. Integration of the front equation givesHence the heated gravity-current runout isIt is reached at the neutral-buoyancy time $\gamma /\beta$ within this front model. A finite-momentum current could subsequently coast or lift off: the formula is the maximum predicted by the prescribed gravity-current front condition, which contains no independent front inertia.

For settling without heating, $\theta =0$. Set $K=\mathrm{Fr}\sqrt{g\gamma \mathcal{V}}$, so $\dot{L}=K\sqrt{\varphi }{L}^{-1/2}$. Eliminating time yieldsThus the runout length of a gravity current isThis length is approached as $t\to \infty$, because positive concentration cannot disappear at a finite time under the settling law. If distance means displacement from the removed barrier, the answer is $L_{\max }-L_0$ in either case.

Use the gravity-current front condition $\dot{L}=\mathrm{Fr}\sqrt{d\varphi h}$, with a specified order-one front Froude number. In a triangular-channel gravity-current box model, the depth and concentration are uniform over $0<x<L(t)$. The volume isThe horizontal projected deposition area is ${\int }_0^{L}Tdx=\lambda h{L}^2$. Consequently the integral model isThis front speed is a closure for the integral model, rather than the assertion that its uniform interior is an exact solution of the spatially varying shallow water equations.

Eliminating time and integrating givesThe limiting runout length of a gravity current is thereforeThe additional distance beyond the release gate is $L_{\infty }-L_0$. The limit is approached asymptotically as $\varphi \to 0$ and $\dot{L}\to 0$; the idealized box model does not predict a finite-time stop. Its numerical runout depends on the adopted front Froude number.

Use the kinematic fluxes $q=Q/(\pi {\rho }_0)$, $m=M/(\pi {\rho }_0)$ and $f_0=F_0/(\pi {\rho }_0)$. In the steady Boussinesq approximation, the non-Boussinesq top-hat plume equations reduce toEliminating $z$ gives $m^{5/2}=5f_0{q}^2/(8\alpha )$, where the integration constant vanishes for a pure plume from a point source. Write $b=az$ and $w=K{z}^{-1/3}$. The two balances give $a=6\alpha /5$ and $K^3=3f_0/(4a^2)=25f_0/(48{\alpha }^2)$. The Boussinesq point-source plume is thereforeHere $q=a^{2}K{z}^{5/3}$ and $m=a^2{K}^2{z}^{4/3}$ tend to zero at the source, while $f_0$ remains positive. The plume Froude number is independent of height:The ideal point source is a far-field similarity idealization. Since $g^{\prime }\propto z^{-5/3}$, the Boussinesq approximation fails near $z=0$ and is valid only when $g^{\prime }/g\ll 1$. In terms of $K$, this requires $z\gg [4K^2/(3g){]}^{3/5}$; the density formula must not be extrapolated into its unphysical negative-density region.

Put $a=6\alpha /5$ and $\lambda =R/B$, so that $B=aZ$ and $R=\lambda aZ$ with constant $\lambda \ge 1$. The volume balance givesConsequentlyFor the definition of $C_t$ printed in the question, using $g_p$ in its denominator, the starting-plume thermal Froude-number ratio isAll factors are constant in the self-similar starting plume. This expression decreases strictly with $\lambda >0$, givingThe stronger bound requested in the PDF does not follow from its stated definition and $R\ge B$. For example, $\lambda =1$ already contradicts that bound. If instead the thermal Froude number is normalized by its own reduced gravity, ${\tilde{C}}_t=U/\sqrt{g_{t}R}$, the preceding result $g_t=9g_p/4$ supplies the missing factor $2/3$:The inequality is strict for $R>B$, with equality allowed by the stated endpoint $R=B$. This alternative normalization recovers the numerical bound while keeping the distinction between the plume and thermal reduced gravities explicit.

The shallow-water equations cease to apply inside the narrow nose, so a gravity-current front condition is needed to relate its speed to the depth immediately behind it. Usewhere $Fr$ is the front Froude number; the ideal deep-ambient von Kármán condition gives $Fr=\sqrt{2}$.

In a uniform axisymmetric box model, conservation of contaminated volume givesThe total nondimensional heat content is $V\theta$, while cooling acts over area $\pi R^2$, soEliminating time and taking the initial release radius as negligible givesSpreading stops as $\theta \to 0$, atHence the maximum covered area is

Let the fixed current volume per unit channel width beA suitable high-Reynolds number deep-ambient gravity-current front condition iswhere the order-one Froude number $\mathrm{Fr}$ records the selected front closure. In the dilute limit the mixture's reduced gravity iswith ambient and carrier-fluid density ${\rho }_a$.

The well-mixed particle volume of species $i$ is $V_0{\varphi }_i$. Its deposition rate through the base of length $L$ is $L{\hat{W}}_i{\varphi }_i$, where ${\hat{W}}_i<0$. The resulting gravity-current box model is thereforeIt conserves fluid volume while suspended particle volume, and therefore the driving buoyancy, decreases by deposition.

For a steady rectangular-channel flow with discharge $Q=bhu$, the shallow-water specific energy relative to the bed is $E=h+Q^2/(2g{b}^2{h}^2)$. At fixed discharge and width, $E_h=1-F^2$, where the Froude number satisfies $F^2=Q^2/(g{b}^2{h}^3)$. The minimum is $E_c=3h_c/2$, with $h_c=(Q^2/(g{b}^2){)}^{1/3}$. Without dissipation, $E+H$ is the conserved Bernoulli equation head for bed elevation $H$.

For a spherical buoyant thermal fed only through the moving top of a Boussinesq point-source plume, put $\lambda =R/B$ and $B=(6\alpha /5)Z$. Relative inflow $\pi B^2(w_p-\dot{Z})$ and $\dot{V}=4\pi R^2\dot{R}$ implyThe ratio of Froude numbers is $5/[\sqrt{\lambda }(5+24\alpha {\lambda }^3)]$ when both use plume reduced gravity $g_p$. Normalizing the thermal with its own $g_t=9g_p/4$ multiplies this by $2/3$. These two definitions must not be interchanged when stating bounds.

A shallow gravity current is subcritical when its Froude number is less than one: a gravity disturbance can propagate upstream relative to the bed. This definition uses the characteristic speed $\sqrt{g^{\prime }h}$ appropriate to the stated model.

A shallow gravity current is supercritical when its Froude number exceeds one: both gravity-wave characteristic curves move downstream relative to the bed.