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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.

For positive dense-fluid reduced gravity $G$ and upward coordinate $z$, an unstable gradient $G_z>0$ supplies buoyant work of order $d^2{G}_z$ over mixing length $d$. velocity $u\sim d\sqrt{G_z}$ and eddy diffusivity $K\sim ud$ follow. Order-one constants depend on the turbulence model. This closure is for an unstable gradient; its square root must not be applied to a stable negative gradient.

A buoyant thermal is a localized rising volume of fluid whose mass density is lower than its surroundings. It can form the head of a starting plume. A spherical mixed thermal of radius $R$ has volume $V=4\pi R^3/3$ and reduced gravity $g_t=g({\rho }_0-{\rho }_t)/{\rho }_0$ in the Boussinesq approximation.

If a buoyant thermal receives volume inflow $J$ with reduced gravity $g_p$ and receives no other fluid entrainment, its mass and volume balances imply $\dot{V}=J$ and $d(g_{t}V)/dt=g_{p}J$. If $V\propto t^{9/4}$ and $g_t\propto t^{-5/4}$, then $g_t=9g_p/4$.

For positive depth and reduced gravity, the characteristic speeds are $u,u\pm c$. Along the material characteristic, $D{g}^{\prime }=-2g^{\prime }{w}_e/h$. Along $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$, the left-eigenvector compatibility equations areIf $w_e=0$ and $g^{\prime }$ is spatially constant, these reduce to the Riemann invariants $u\pm 4c$. Variable buoyancy or entrainment prevents that simplification.

The volumetric coefficient of thermal expansion is the fractional volume increase per temperature increase at fixed pressure and composition:The mass density form requires fixed mass $m$. Its units are ${\mathrm{K}}^{-1}$; ${\alpha }_T$ is another notation for this coefficient. For small temperature changes $\theta$ at fixed pressure,For a finite change in a single thermodynamic phase, integrate the possibly varying coefficient: $\ln (V_2/V_1)={\int }_{T_1}^{T_2}\alpha (T,P,N)dT$. A constant-coefficient approximation gives an exponential, while the displayed linear formula additionally requires $|\alpha \theta |\ll 1$.

For an ideal gas at fixed particle number,Integrating $1/T$ recovers the exact fixed-pressure thermal expansion ratio $V_2/V_1=T_2/T_1$. Combining expansion with isothermal compressibility gives the local response of any smooth simple compressible system of fixed composition,The ratio has units $\mathrm{Pa}{\mathrm{K}}^{-1}$. For the one-mole ideal gas state in isothermal compressibility, it is $P_0/T_0=333.33\mathrm{Pa}{\mathrm{K}}^{-1}$: a $1\mathrm{K}$ increase at fixed volume raises the pressure by about $333.33\mathrm{Pa}$. Maxwell relations identify this same ratio as the entropy slope with respect to volume and give $C_P-C_V=TV{\alpha }^2/{\kappa }_T$.

Under the Boussinesq approximation, the small fractional mass density change produces buoyancy while mass density is otherwise treated as constant. A positive temperature excess $\theta$ gives reduced gravity $g^{\prime }=g{\alpha }_T\theta$ in the convention used in natural ventilation. The coefficient therefore converts a heat budget into a buoyancy budget. The same measurable expansion response supplies entropy changes and heat capacity differences through Maxwell relations, connecting fluid motion to equilibrium thermodynamics.

A horizontal scale comparing gravitational or buoyancy restoring forces with rotation. For a shallow-water gravity mode with propagation speed $c$, the deformation radius is $c/|f|$. The barotropic deformation radius uses $c=\sqrt{gH}$; internal layered modes use a speed set by reduced gravity and layer depths.

If a finite-volume gravity current loses or dilutes the scalar that supplies its reduced gravity, its front can approach a finite runout length of a gravity current. A box model couples the scalar balance to a gravity-current front condition; eliminating time then gives the front position directly as a function of the remaining scalar concentration.

For a prismatic channel of width $b(z)=z$, a Boussinesq current of depth $h$ occupies area $A=h^2/2$. Ambient fluid at rest enters across the interface at speed $w_e$, supplying volume but no horizontal momentum or excess density. The volume conservation, momentum and excess-density balances areHere $g^{\prime }$ is the reduced gravity. With $D={\partial }_t+u{\partial }_x$, these give $Dh+(h/2)u_x=w_e$, $Du+g^{\prime }{h}_x+(h/3)g_x^{\prime }=-2u{w}_e/h$, and $D{g}^{\prime }=-2g^{\prime }{w}_e/h$. The momentum source in primitive form describes acceleration of newly entrained fluid, even when bed drag vanishes.

A semi-infinite resting layer of depth $H$ and constant reduced gravity $b$ released into a deep ambient develops a rarefaction wave satisfying $u+2\sqrt{bh}=2\sqrt{bH}$. The head closure $u_f=F\sqrt{b{h}_f}$ gives $h_f=4H/(F+2{)}^2$ and $u_f=2F\sqrt{bH}/(F+2)$. A constant-depth shelf lies between the rarefaction tail speed $u_f-\sqrt{b{h}_f}$ and the front speed $u_f$; its width is positive for every finite positive $F$.

A gravity-current box model in a prismatic channel with width equal to height has volume $V_0=L{h}^2/2$. Closing the front speed by $\dot{L}=F\sqrt{g^{\prime }h}$ gives the displayed law and long-time spreading $L\propto t^{4/5}$. This is an integral approximation with constant volume and reduced gravity, not an exact uniform-depth solution of the local shallow water equations.

For a merged pair held at the original velocity and reduced gravity, pure plume balance requires the original individual radius, not the larger area-preserving radius. The required pure area is half the combined area. Shrinking to that area at unchanged velocity and reduced gravity would halve volume, momentum and buoyancy fluxes, contradicting conservation of the merged fluxes. A conservative merger must change its properties or begin away from pure balance.

A thin layer of very viscous liquid floating on an effectively inviscid denser liquid has negligible shear traction on both faces. Its leading horizontal velocity is a plug, and incompressibility gives vertical strain $w_z=-u_x$. Normal-stress balance shifts the pressure by $-2\mu u_x$, so the longitudinal excess stress is $4\mu u_x$. Combining depth-integrated momentum balance with hydrostatic reduced gravity gives the displayed extensional equations. This is distinct from a no-slip, shear-driven thin-film gravity current.

In a prismatic triangular channel with constant reduced gravity, the wave speed is $c=\sqrt{g^{\prime }h/2}$ and the Riemann invariants are $u\pm 4c$. A dam release into a deep ambient produces a simple-wave rarefaction with $u+4c=4c_0$. The imposed gravity-current front condition $u_f=F\sqrt{g^{\prime }{h}_f}$ gives the displayed uniform nose-adjacent state. The fan has $c=(4c_0-x/t)/5$ and $u=4(c_0+x/t)/5$, extending from $x/t=-c_0$ to $u_f-c_f$; a uniform region then reaches the front $x/t=u_f$. The nose closure supplies physics absent from the one-layer interior equations.

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

A dilute particle suspension can supply both excess mass density and heat. To first order its equation of state is $\rho /{\rho }_0=1+\gamma \varphi -\theta$, where $\varphi$ is the particle volume fraction and $\theta$ is fractional thermal expansion. Particle deposition flux reduces $\varphi$, while particle heating increases $\theta$. The competition can reverse the reduced gravity and create unstable density stratification.

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 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.

A local ratio of stabilizing buoyancy to shear across an interface of thickness $\delta$, using reduced gravity jump $g^{\prime }$ and characteristic velocity $u$. The dimensionless value is ${\mathrm{Ri}}_i=g^{\prime }\delta /u^2$. It is distinct from a whole-layer bulk Richardson number based on the total mixed depth. Local entrainment laws use this ratio when the transition thickness and velocity characterize the turbulence-producing region.