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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.

Keeping sediment acceleration while neglecting water storage and a small gravity-friction residual gives a reduced model for the bed elevation $s$, water speed $u$ and sediment speed $v$. A prescribed water flux $q$ gives $h=q/u$; integrated water momentum gives $s=C-F^2{u}^2/2-q/u$. Around the normalized uniform state $h=u=q=1$, let $(1-v_0{)}^2=\kappa v_0$, $0<v_0<1$. A Fourier mode of wavenumber $k$ has growth-rate real partThus nonuniform modes decay for subcritical Froude number and grow for supercritical Froude number. The spatially constant mode is neutral, and the critical value needs terms neglected in this reduction. In a general uniform state the sign change is at the actual mean-flow Froude number equal to one, rather than at a velocity-scale parameter unrelated to that state.

In an ideal Boussinesq gravity current advancing beneath a much deeper ambient, the local head Froude number is $u_f/\sqrt{g^{\prime }{h}_f}=\sqrt{2}$. The ambient must be displaced around the finite-depth head. This differs from the Saint-Venant dry-front condition, which neglects ambient inertia. The coefficient belongs to the ideal deep-ambient head model; confinement, mixing and drag can change it.

A steady top-hat plume model supplied with pi-suppressed kinematic buoyancy flux $f_0=b^{2}w{g}^{\prime }>0$ but zero source volume flux and momentum flux hasThe full-cross-section buoyancy flux is $\pi f_0$. The Batchelor entrainment hypothesis and momentum balance yield these constants. Its Froude number satisfies $w^2/(g^{\prime }b)=5/(8\alpha )$. The Boussinesq approximation requires $g^{\prime }/g\ll 1$ and therefore fails at the ideal point source itself.

A dilute bubbly thermal is a rising, entraining fluid volume whose gas bubbles provide part of its buoyancy. With negligible bubble slip and gas loss, Boyle's law gives gas volume proportional to $(1-z/H_p{)}^{-1}$, where $H_p=p_0/({\rho }_{0}g)$. Spherical fluid entrainment with normal entrainment velocity $\alpha W$ gives $da/dz=\alpha$. Bubble concentration follows by dividing the gas volume by the growing thermal volume. A diameter-based Froude number closure is $W=F\sqrt{2a{g}^{\prime }}$.

A common high-Reynolds number closure for a deep-ambient gravity current is $\dot{L}=\mathrm{Fr}\sqrt{g^{\prime }h}$, where $L$ is front position and the order-one Froude number depends on the chosen front model.

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.

For a constant-volume unit-width particle-laden gravity current with $M=Lh$, define the particle-fluid reference reduced gravity $g_0^{\prime }=g({\rho }_p-{\rho }_0)/{\rho }_0$, so the current has $g^{\prime }=g_0^{\prime }\varphi$. A constant front Froude number and absorbing-bed hindered settling give $\dot{L}=F\sqrt{g_0^{\prime }\varphi M/L}$ and $\dot{\varphi }=-(V_{s}L/M)\varphi (1-\varphi )$. Eliminating time and integrating gives the displayed runout length of a gravity current. For large runout and dilute initial concentration, $L_{\infty }^5\sim 25F^2{M}^3{g}_0^{\prime }({\varphi }_0+2{\varphi }_0^2/3+\cdots )/V_s^2$. This $g_0^{\prime }$ is buoyancy per unit particle fraction, not the initial current reduced gravity $g_0^{\prime }{\varphi }_0$.

A slowly varying, steady, inviscid channel flow has constant discharge $Q=uh$ per unit width and the displayed Bernoulli equation. The bed-height function of depth has its maximum at $h_c=(Q^2/g{)}^{1/3}$, where the Froude number is one. At a nondegenerate critical crest, a differentiable profile changes between the subcritical flow and supercritical flow branches. With subcritical upstream squared Froude number $F<1$ and upstream depth $h_1$, the shock-free supercritical downstream depth is $h_1(F+\sqrt{F^2+8F})/4$.

In a stationary rectangular-channel hydraulic jump, orient the discharge $q=hu>0$ from side 1 to side 2. Momentum conservation gives $q^2=g{h}_1{h}_2(h_1+h_2)/2$ for unequal depths. The specific mechanical energy is $B=u^2/2+gh$, soDividing by $g$ gives the displayed loss of head, measured as a length. The mechanical energy flux lost is $\rho q(B_1-B_2)$, so a dissipative jump must have $h_2>h_1$. With $R=h_2/h_1>1$, the squared Froude numbers are $R(1+R)/2>1$ upstream and $(1+R)/(2R^2)<1$ downstream. Thus the shallow water equations connect supercritical flow to subcritical flow through the admissible jump.

In the limiting model of infinitely strong stable density stratification, small Froude number and fixed vertical scales suppress vertical displacement. Each horizontal level evolves by two-dimensional vortex dynamics; $z$ labels independent copies of the horizontal Poisson equation. The leading normalized Ertel potential vorticity is $Q/N^2=\zeta$. This is a constrained asymptotic model: finite buoyancy frequency restores vertical motion and coupling, and arbitrarily short vertical scales invalidate the independent-layer limit. McIntyre's discussion of potential-vorticity inversion explains the limiting independent-layer picture.

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.

Bedload transport carries sediment by rolling, sliding and short hops near the bed. Suspended sediment is held in the water column by turbulent mixing. The competition between particle settling velocity and the turbulent vertical velocity scale chiefly determines which description applies; particle size and density influence that settling velocity.

Scale the depths and bed/free-surface elevations by $d$, both velocities by $U$, distance by $L$, and time by $T$. A choice producing exactly the requested coefficients isThe sediment continuity equation then has unit coefficients, while both material-acceleration time derivatives acquire the factor $L/(UT)=a/d$. Dividing water momentum by $gd/L$ and sediment momentum by $U^2/L$ givesFor example $d=1\mathrm{m}$, $a=10^{-3}\mathrm{m}$, $S=10^{-3}$, $f=10^{-2}$ and ${\rho }_s/{\rho }_w=2.65$ give $\epsilon =10^{-3}$ and $\delta =2.65\times 10^{-4}$. These are illustrative values: small effective mobile-layer thickness and small slope explain the two small parameters.

Set $\epsilon =\delta =0$ at leading order. Water continuity gives $hu=q$, with spatially constant flux; take a prescribed constant $q$. Water momentum integrates to $\eta +F^2{u}^2/2=C$. Since $s=\eta -h$,The bedload transport model with sediment inertia has remaining equations $s(u{)}_t+v_x=0$ and $v{v}_x=(u-v{)}^2-\kappa v$.

Choose a uniform state of this reduced model with $h_0=u_0=q=1$, constant $s_0=C-F^2/2-1$, andIt satisfies $(1-v_0{)}^2=\kappa v_0$. For small perturbations $U,V$ of the two velocities, put $a_1=2(1-v_0)>0$ and $b_1=a_1+\kappa >0$. Linearization givesFor a Fourier mode $e^{ikx+\lambda t}$ this yieldsEvery nonzero mode decays for $F<1$ and grows for $F>1$. The spatially uniform mode is neutral, and $F=1$ is a singular threshold where the neglected terms matter. In a general uniform state the controlling sign is $q/u_0^2-F^2{u}_0$: equivalently the actual mean-flow Froude number is less or greater than one. The displayed $F=1$ criterion uses the stated unit-depth, unit-velocity normalization of the reduced steady state.

For the supplied hillslope values,The Froude number is below one, but it is not extremely small numerically; the much smaller products actually multiplying inertia and the depth-gradient term justify their leading neglect. Positive flow then satisfies $u=h^{2/3}$ andIn overland flow the source is net rainfall input after infiltration into soil, with evaporation or other local losses included when important. Lateral inflow can also contribute. Negative net input must be limited by water availability once the surface dries.

Write $R=k_R\sqrt{A}$, $\bar{h}=k_H\sqrt{A}$, where the given geometry givesTake a Chézy drag law $\tau =\rho c_{d}u|u|$ and restrict to downstream flow $u>0$. Choose an area scale $A_0$, length scale $L$, and speed scale $U_0$ satisfyingThis makes gravitational driving and drag equal in the reference uniform steady flow. Dividing the printed momentum equation by $gS$ givesMass conservation retains $A_t+(Au{)}_x=0$. In this notation $F$ is the square of the velocity-to-gravity-wave-speed ratio based on the chosen depth scale; authors sometimes instead denote its square root by Froude number. If the actual triangular pressure head from the preceding solution is used, replace $k_H$ by $2k_H$ in both parameter definitions. The displayed dimensionless form remains the same, with consistently redefined parameters.

For the travelling wave $\xi =s+ct$, continuity gives $(ch+hu{)}^{\prime }=0$. The upstream state fixes the constant as $1+c$, henceSubstituting in momentum yieldssoA constant downstream state $H=h_{+}>1$ requires $H^3=(1+c-cH{)}^2$. The positive square-root choice would give $c<0$, so the upstream-moving wave uses $1+c-cH=-H^{3/2}$. ThusThe downstream flow reverses. Where the numerator is positive in $1<h<H$, a smooth increasing profile requires a positive denominator. Its minimum occurs at $h=1$, givingEquality does not give the prescribed asymptotic upstream state: cancellation of the two simple zeros at $h=1$ leaves a nonzero slope there. For the strict inequality, separation of variables constructs the monotone profile whenever the numerator remains positive between its endpoint zeros.

There is an additional restriction hidden in the printed positivity hint. Put $r=\sqrt{h}$ and $R=\sqrt{H}$. Since $c=R+1/(R-1)$, the potentially vanishing factor of the numerator isThe other factor, $h^{3/2}-(1+c-ch)$, is positive for $h>1$. Therefore the asserted positivity throughout $1<h<H$ holds precisely for $1<R\le 2$, that is $1<H\le 4$. For $H>4$, the zero at $h=[R/(R-1){]}^2$ intervenes before $H$, preventing the requested monotone connection. For example $H=9$, $c=3.5$, $h=4$ gives $h^3-(1+c-ch{)}^2=-26.25$. Thus the smooth upstream tidal-wave admissibility conditions areAt $h_{+}=4$ the downstream zero is double, giving an algebraic approach; below four the endpoint approaches are exponential. The displayed Froude number bound is the requested necessary condition, with the source's missing amplitude qualification made explicit.

Let $B$ be the constant canal width, large compared with water depth. Then $A\simeq Bh$ and $l\simeq B$, so friction is $f{u}^2/h$. Choose a reference uniform water depth $H$, normal-flow speed $U$ satisfying $f{U}^2/H=gS$, and scalesThe continuity equation becomes ${\hat{h}}_{\hat{t}}+(\hat{h}\hat{u}{)}_{\hat{s}}=0$. Dividing momentum by $gS$ and dropping hats givesThe Froude number compares velocity to long gravity-wave speed. At the reference state, $F<1$ is subcritical flow, $F=1$ critical flow, and $F>1$ supercritical flow. In dimensional variables the wave characteristics travel at $u\pm \sqrt{gh}$: one can propagate upstream in subcritical flow, while both move downstream in supercritical flow. Away from the reference state the local Froude number is $F|u|/\sqrt{h}$.

Assume nonzero upstream speed, so $F>0$. For steady constant-density flow, mass conservation gives the constant volume fluxThe Bernoulli equation states that $u^2/2+p/\rho +gz$ is constant along each streamline of a steady inviscid constant-density flow under gravity. It need not have the same value across unrelated streamlines in a general rotational flow. Here uniform upstream velocity supplies the same head, and the slowly varying, nearly horizontal channel permits hydrostatic pressure $p=p_{\mathrm{atm}}+\rho g(h-z)$. Neglecting losses, the Bernoulli equation therefore givesSubstitute $u=UWH/(wh)$ and set $s=h/H$, $F=U^2/(gH)$, the square of the upstream Froude number. Rearrangement givesFor positive depth, this cubic increases to its maximum at $s_{\ast }=(F+2)/3$ and then decreases until it reaches zero at $(F+2)/2$. Its maximum is $G(s_{\ast })=(F+2{)}^3/(27F)$. Consequently the critical width of a rectangular open-channel contraction isAt equality the two positive-depth branches coalesce and the local Froude number equals one.

When $w>w_c$ everywhere, a smooth solution connected to the upstream state cannot change branch without passing through that coalescence. At $w=W$ the upstream depth is $s=1$, with $G^{\prime }(1)=2(F-1)/F$. If $F<1$, this point is on the descending, subcritical branch. At a wider location $w>W$, the left-hand side falls below $1=G(1)$, so the depth on that branch must increase: $h>H$ for subcritical upstream flow $F<1$. If $F>1$, the upstream state is on the ascending, supercritical branch and decreasing the left-hand side decreases $s$: $h<H$ for supercritical upstream flow $F>1$.

For $F=1$, $w_c=W$ and the upstream state is critical. A literally constant-width upstream region then conflicts with the strict assumption $w>w_c$ everywhere. If the upstream state is interpreted only as an asymptotic critical limit, a widened channel has one solution with $h<H$ and one with $h>H$; a further branch condition is required. Merely solving the cubic at one location, without retaining the upstream-connected branch, would not determine the physically continued depth.

For a one-layer shallow water flow the Froude number is $F=u/\sqrt{g^{\prime }h}$. Supercritical flow has $F>1$, so both characteristic speeds $u\pm \sqrt{g^{\prime }h}$ point downstream. Subcritical flow has $F<1$, allowing information to travel upstream. At hydraulic control the flow passes through $F=1$ at a geometry-induced critical section; that condition fixes discharge in relation to upstream head when the exit is free.

For escaping light air, reflect the usual lower-layer picture vertically: a lowered doorway roof acts as an inverted broad-crested weir. The lower return flow can be ignored dynamically only while it has a sufficiently large available area and small velocity. If the warm interface lies far below the doorway top, the cool inflow occupies a substantial dynamically active fraction of the opening; its inertia and pressure cannot be replaced by a motionless deep ambient. A two-layer exchange model is then required. If the warm layer reaches below the doorway bottom, the assumed separate lower replacement path is lost entirely.

Outside a freely controlled doorway the outgoing current becomes supercritical and then rises rapidly. That upturn violates the small-vertical-acceleration shallow water approximation, but it does not alter the upstream discharge as long as it does not submerge or back-pressure the control. Information cannot return through the supercritical outflow to the critical section. A blocked conservatory or a drowned doorway is different and can move the control upstream.

Potential-vorticity inversion is the diagnostic recovery of the flow from its advected vortical scalar, using the balance relation and specified boundary conditions. Its usefulness is a separation of tasks: transport the scalar by the current velocity, invert to find the new velocity, and repeat. The inverse is spatially nonlocal even though the transport law is local. The four models differ chiefly in the operator being inverted and the information needed at boundaries.

In two-dimensional vortex dynamics, planar incompressibility permits $\mathbf{u}=(-{\psi }_y,{\psi }_x)$, and vertical vorticity is $\zeta ={\nabla }^2\psi$. The curl of inviscid momentum givesThere is no vortex stretching in strictly planar motion, so vorticity is materially conserved. The two-dimensional potential-vorticity inversion is the Poisson equation ${\nabla }^2\psi =\zeta$. On the plane, with suitable integrability and far-field behavior, its Green's function is $(2\pi {)}^{-1}\log |\mathbf{x}-{\mathbf{x}}^{\prime }|$, givingDifferentiation gives the planar Biot-Savart kernel. Here ${\psi }_h$ is a harmonic function fixed by boundary, circulation and far-field data. A rigid impermeable wall is a streamline, so $\psi$ is constant on it; constants or circulations around distinct holes require specification. The vorticity alone cannot distinguish two flows differing by a permitted harmonic contribution, such as a uniform current on an unbounded plane. A constant added to $\psi$ has no physical effect.

In nonrotating layerwise-two-dimensional vortex dynamics, strong stable density stratification constrains motion to almost horizontal layers. In the ideal independent-layer limit, vertical velocity vanishes at leading order, $(u,v)=(-{\psi }_y,{\psi }_x)$ andhold separately at each fixed $z$. Height is a parameter: the horizontal inverse Laplacian contains no vertical derivative. With background buoyancy $N^{2}z$, the leading Ertel potential vorticity is $Q=\mathbf{\omega }\cdot \nabla b\simeq N^2\zeta$, so the materially advected normalized scalar is $Q/N^2\simeq \zeta$. The same horizontal Poisson equation is inverted layer by layer, using each layer's boundary data. This is an asymptotic constrained model, not arbitrary three-dimensional Euler motion with vertical velocity set to zero: its validity requires small Froude number and controlled vertical scales. At finite buoyancy frequency, the weak vertical motion and tilted isopycnals restore coupling. The inversion may be independent by layer while the reconstructed fields vary with height. The horizontal Euler equations for an inviscid fluid determine the pressure gradient, while hydrostatic balance relates its vertical derivative to buoyancy.

In shallow-water quasi-geostrophic inversion, begin with the exact shallow-water potential vorticity $(f+\zeta )/h$. Linearizing its anomaly about depth $H$ gives, after multiplying by $H$, $\zeta -f_0\eta /H$. For small Rossby number, geostrophic balance gives $\psi =g\eta /f_0$, while $\zeta ={\nabla }_h^2\psi$. Keeping the beta plane background gradient givesThus $q-\beta y$ is inverted with the modified Helmholtz equation $({\nabla }_h^2-L_D^{-2})\psi =q-\beta y$, under the specified boundary and far-field conditions. The velocity follows from $\psi$ and surface elevation from $\eta =f_0\psi /g$. A horizontal Fourier mode of the anomaly satisfiesThe Rossby deformation radius therefore limits the reach of the balanced response. Layer stretching, represented by $-\psi /L_D^2$, is absent from ordinary planar vorticity inversion. Unlike the pure Poisson equation, the finite-deformation operator also responds to the constant part of $\psi$, which specifies mean elevation rather than an arbitrary streamfunction offset. Specified bottom topography adds a known source to $q$; it is subtracted along with the background before inversion.

In stratified quasi-geostrophic inversion, continuously varying vertical displacement replaces the shallow layer's single stretching term. For positive $N^2(z)$ and constant nonzero $f_0$, the three-dimensional quasi-geostrophic potential vorticity isGeostrophic balance and hydrostatic balance give velocity $(-{\psi }_y,{\psi }_x)$ and buoyancy perturbation $b=f_0{\psi }_z$. Invert the anisotropic elliptic boundary value problem using interior $q$, prescribed top and bottom buoyancy as Neumann boundary conditions, and the lateral, far-field and global-flow data. Boundary buoyancy evolves by its own material conservation when the boundary is impermeable. It cannot be discarded: the Eady edge wave in the preceding solution has zero interior potential vorticity anomaly but nonzero boundary buoyancy and a nonzero induced flow.

For constant $N,f_0$, the stretched coordinates for quasi-geostrophic inversion $Z=Nz/|f_0|$ turn the inversion operator into the three-dimensional Laplacian. Its infinite-space Green's function is $-1/(4\pi |\mathbf{X}-{\mathbf{X}}^{\prime }|)$ in $(x,y,Z)$ coordinates. Hence a horizontal disturbance of scale $L$ communicates vertically over scale $|f_0|L/N$, explicitly coupling layers. In a bounded domain with homogeneous difference data, multiplying the difference of two inversions by its streamfunction and applying integration by parts givesThis proves uniqueness up to a constant when the relevant boundary terms vanish; a mean or pressure reference fixes the remaining constant. Compatible boundary fluxes are needed for a purely Neumann boundary-value problem.

Across all four models, inversion provides the induced flow, while scalar transport supplies time evolution. The independent horizontal inverse in the first two models becomes a finite-deformation inverse in shallow water and a vertically coupled elliptic inverse in continuous stratification. The principle applies within the stated balance or constrained model. In the unrestricted Boussinesq equations or shallow water equations, gravity waves and divergent motion can share the same potential vorticity, so the vortical scalar alone does not reconstruct every possible flow.