Froude number ($\mathrm{Fr}$)

The Froude number compares inertial speed with a gravity-wave speed. For a shallow gravity current of depth $h$ and reduced gravity $g^{\prime }$, $\mathrm{Fr}=U/\sqrt{g^{\prime }h}$.

For a steady slowly varying rectangular channel, mass conservation and the Bernoulli equation give $(W/w{)}^2=((F+2)/F)s^2-(2/F)s^3$, where $s=h/H$. The positive cubic has its maximum at $s=(F+2)/3$, with value $(F+2{)}^3/(27F)$. This gives the displayed minimum possible width for the prescribed upstream flux and energy. Smooth subcritical and supercritical branches connected to upstream data remain distinct unless the critical width is attained.

A hydraulic control is a critical section where a shallow water flow passes smoothly between subcritical flow and supercritical flow. One long-wave characteristic speed relative to the bed vanishes there. Regularity supplies an extra relation between discharge, depth and channel geometry, allowing the control to set the discharge or upstream state. A critical depth alone is not sufficient for a regular transcritical solution: the channel forcing must also be compatible there.

A weir is an obstruction across a channel that can regulate volume flux by creating a hydraulic control. In a shallow water model a sufficiently smooth crest has a critical section, with upstream flow subcritical and downstream flow supercritical under free discharge. Losses, submergence and geometry determine whether this ideal control applies.

For reservoir head $\delta$ above a broad crest, negligible approach velocity, constant reduced gravity, and no head loss, hydraulic control gives $\delta =3h_c/2$ and the displayed discharge. Reflecting the vertical coordinate gives the same result for a light layer escaping beneath a lowered roof. An orifice discharge coefficient is a separate closure and should not be silently inserted into this inviscid weir law.

A light shallow layer flows below a roof of height $H(x)$ and width $W(x)$ through a deeper approximately motionless ambient. Hydrostatic equilibrium and inviscid steady Bernoulli equation give the displayed specific energy for constant volume flux $Q$ and reduced gravity $g^{\prime }$. At fixed position, $E_h=g^{\prime }(1-{\mathrm{Fr}}^2)$ and $h_c=[Q^2/(g^{\prime }{W}^2){]}^{1/3}$. A roof depression raises the minimum admissible energy, just as a bed elevation does for a heavy lower layer.

For two same-height throats carrying the same light-layer volume flux and reduced gravity, the narrower throat has the larger critical minimum energy. A narrower downstream throat can back up and submerge an upstream throat. If the first throat is narrower, it can control discharge and feed a supercritical current through the second; a hydraulic jump followed by a second control is also possible when the available energy drop and intermediate geometry permit it. Equal throat widths are a marginal lossless case. Relative width alone does not fix jump position or mixing in the intervening reservoir.

For loss of fluid carrying its local axial velocity, the inviscid Bernoulli equation head stays constant while discharge decreases. On a flat bed a regular control requires a positive local maximum of $q=Q/b$. In the parabolic-width channel $b=b_0(1+x^2/L^2)$, this maximum occurs at $x_c=-S{L}^2/(Q_0+\sqrt{Q_0^2+S^2{L}^2})$, with $q_c=(Q_0+\sqrt{Q_0^2+S^2{L}^2})/(2b_0)$. Its critical depth is $(q_c^2/g{)}^{1/3}$. A control exists only when the prescribed total head equals the maximum critical-head envelope and the reach includes that point while retaining positive discharge.

For a prescribed positive head-loss rate $\gamma (x)$, the steady energy balance is $(E+H{)}_x=-\gamma$. The accumulated loss is added to the bed elevation when applying hydraulic control compatibility: $H_x+\gamma =h_c{b}_x/b$. A local loss law alone does not determine the accumulated loss between a remote inlet and the control; that integral must also be specified.

For a smooth rectangular channel, conservation of discharge and Bernoulli equation head gives $(1-F^2)h_x=F^{2}h{b}_x/b-H_x$. A regular hydraulic control has $F=1$ and the displayed geometric compatibility condition. At fixed discharge the critical-head envelope is $T(x)=H(x)+3h_c(x)/2$, with $h_c=(Q^2/(g{b}^2){)}^{1/3}$. A simple smooth control requires a local maximum of $T$: $T_x=0$ and $T_{xx}<0$. A local minimum cannot be crossed at the critical head because the real depth branches disappear nearby.

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

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 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 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.

The shallow water equations at a dry bed, with negligible ambient mass density, end in a vanishing-depth edge, rather than a finite-depth gravity current head. In a prismatic triangular channel the Riemann invariant $u+4\sqrt{g^{\prime }h/2}$ gives the speed $2\sqrt{2g^{\prime }{h}_0}$ after release of a resting semi-infinite layer of depth $h_0$. In a rectangular channel the corresponding speed is $2\sqrt{g^{\prime }{h}_0}$; the channel geometry matters.