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Take inward fluid entrainment speed to be positive. The non-Boussinesq form of the Batchelor entrainment hypothesis iswhere $\alpha$ is the entrainment coefficient. Define physical, cross-section-integrated fluxes for the top-hat plume model byThus $Q$ is mass flux, $M$ is momentum flux, and $F$ is a density-weighted buoyancy flux. The kinematic buoyancy flux in the Boussinesq approximation is $F/{\rho }_0$. These definitions keep the factor $\pi$; conventions suppressing it simply rescale all three fluxes consistently.

For a homogeneous ambient, the sectional volume, mass and momentum balances areThe momentum source is the net upward buoyancy force; entrained ambient fluid initially supplies no vertical momentum. Subtracting the mass balance from ${\rho }_0$ times the volume balance gives conservation of the density deficit:Now $w=M/Q$, $\pi b^2\rho =Q^2/M$, and $\pi b^{2}g({\rho }_0-\rho )=QF/M$. Substitution produces the non-Boussinesq top-hat plume equationsFor reconstructing the fields without a Boussinesq approximation,The last expression uses the plume density in the acceleration denominator; it agrees with the ambient-density definition of reduced gravity to Boussinesq order.

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.

Let $V=4\pi R^3/3$, $w_p=w(Z)$ and ${\rho }_p=\rho (Z)$, with the subscript $p$ here referring to the plume fluid. The section at $Z(t)$ moves upward at $U=\dot{Z}$. Its incoming volume flux relative to that moving section is thereforeThe thermal's sectional model allows no direct ambient fluid entrainment, so its volume and mass balances areSubtracting these balances with $g_t=g({\rho }_0-{\rho }_t)/{\rho }_0$ and $g_p=g({\rho }_0-{\rho }_p)/{\rho }_0$ gives the buoyant thermal mass balanceFor the self-similar starting plume, $V\propto t^{9/4}$ and $g_t\propto t^{-5/4}$, so $g_{t}V\propto t$. HenceUsing $\pi B^2{w}_p$ instead of the relative inflow would miss the volume swept out by the rising matching section. The larger thermal reduced gravity reflects fluid accumulated from earlier, lower, more buoyant parts of the plume.

Let $f_0=F_0/(\pi {\rho }_0)$ as above. A self-similar starting plume has no source length or time scale; dimensional analysis therefore givesThe dimensionless constants depend on the entrainment coefficient and thermal closure. Thus $R/B$ is constant because the underlying Boussinesq point-source plume has $B=b(Z)=(6\alpha /5)Z$. The steadily increasing thermal volume scales as $t^{9/4}$, even though its reduced gravity decreases.

Define the concentration-dependent reduced gravityUnder the Boussinesq approximation, the three shallow water equations can be written in advective form asThe coefficient matrix of this quasilinear system has characteristic speedsThus this is a hyperbolic system when $g^{\prime }h>0$. Along the intermediate characteristic curve $dx/dt=u$, the concentration obeysIt decreases because ambient entrainment dilutes the chemical; detrainment does not change the concentration of a well-mixed parcel.

The final state is displacement ventilation: fresh air of density ${\rho }_0$ forms a cool lower layer, radiator-heated air forms a well-mixed upper layer, and the wall plume crosses their interface at height $h$. Steady volume conservation requires the plume volume flux there to equal the imposed ventilation flux,

Below the interface the ambient is uniform, so $B_0=F_0/{\rho }_0$ is constant. Put $J=M/{\rho }_0$. The triangular-profile wall line plume equations reduce toThe pure plume conditions $V,J\to 0$ at the radiator select the similarity solutionConsequently the displacement-ventilation interface height isThe upper-layer reduced gravity follows from its steady buoyancy balance as $g_u^{\prime }=B_0/A=F_0/({\rho }_{0}A)$. The two-layer solution applies when the calculated interface satisfies $0<h<H$.

Because $\hat{W}<0$, species 1 now settles more slowly and species 2 more rapidly than in the equal-speed case. The supplied inequality says that species 2 provides the larger initial particle-density contribution to the reduced gravity; with exact Boussinesq density contrasts, the corresponding condition is ${\tilde{\varphi }}_1({\rho }_1-{\rho }_a)<{\tilde{\varphi }}_2({\rho }_2-{\rho }_a)$. The dominant buoyancy contribution is therefore removed earlier, so $g^{\prime }(t)$ and the gravity-current front condition fall below their equal-settling values at first order in $ϵ$. The runout length decreases. The slower loss of the weaker species-1 contribution only partly compensates for this effect.

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.

Define the indoor-to-outdoor reduced gravityBecause the whole opening lies above the interface, its indoor side contains upper-layer fluid of density ${\rho }_H$. Under hydrostatic pressure, the pressure difference varies linearly about a neutral pressure level. Equal opening geometry and equal discharge coefficients for inflow and outflow put that level at the vent midpoint. At vertical distance $y$ from it, the ideal Bernoulli equation gives speed $\sqrt{2g_V^{\prime }|y|}$.

Writing $Q_V$ for either one-way volumetric flow rate, integration over one half of the opening givesThe total unsigned exchange is $2Q_V$. The ideal sharp-edged inviscid model has $C_d=1$; an empirical $C_d<1$ represents contraction and losses.

Define the reduced gravityand use the Boussinesq limit ${\rho }_1\simeq {\rho }_2$ except in buoyancy. The balances becomeThus, for $\mathbf{q}=(g^{\prime },h,u{)}^T$,which is the required entraining shallow-water layer system.

For a pure plume with $g_p^{\prime }=\gamma B^{2/3}{z}^{-5/3}$, a steady sharp interface at $z_i$ has $Q_p=B^{1/3}{z}_i^{5/3}/\gamma$ and warm-layer reduced gravity $g^{\prime }=B/Q_p$. A freely discharging upper doorway of height $D$ and width $W$ has available head $D-z_i$. Equating plume supply to the broad-crested weir flux gives the displayed implicit height relation, with $0<z_i<D$. Warm air above the doorway stores heat but adds no available head at its crest in this uniform-layer model.

Homogenizing the top depth $h$ of a constant-gradient stable stratification over horizontal area $\mathcal{A}$ creates mean density ${\rho }_t+{\rho }_0{N}^{2}h/(2g)$. Comparing gravitational potential energies before and after gives $\Delta P={\rho }_0\mathcal{A}{N}^2{h}^3/12$. The density jump at the base has reduced gravity $N^{2}h/2$. The cubic energy cost leads to a quadratic-in-depth coefficient multiplying the deepening rate.

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.

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.

For constant kinematic buoyancy flux $f_0$ from a point source, dimensional analysis gives head height and radius proportional to $f_0^{1/4}{t}^{3/4}$, head velocity proportional to $f_0^{1/4}{t}^{-1/4}$, and head reduced gravity proportional to $f_0^{1/4}{t}^{-5/4}$. Dimensionless coefficients require the chosen fluid entrainment and head closures.

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 $p>1$.

A source volume $Q$ makes roof outflow exceed floor inflow by $Q$. At zero floor inflow, an entraining plume cannot support a steady cold lower layer; the warm layer reaches the floor. With reduced gravity $g^{\prime }=B/Q$, the entire hydrostatic head drives roof discharge $Q=A_e\sqrt{g^{\prime }H}$. This gives the stated threshold. Equal splitting of pressure head is inappropriate at blocking. The same result in physical-area notation is $Q_c=(2C_d^2{A}_{\mathrm{geom}}^{2}BH{)}^{1/3}$.

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.