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Consider a steady, axisymmetric pure plume in a uniform ambient fluid, using the Boussinesq approximation and uniform top-hat cross-sectional profiles. If $r(z)$ is its radius, $U(z)>0$ its vertical velocity and $g^{\prime }(z)>0$ its reduced gravity, define the physical fluxesThey are respectively volumetric flow rate, momentum flux per reference mass density, and buoyancy flux. Only $B$ is conserved in this unstratified, source-free model. Fluid entrainment adds ambient volume to $Q$, and buoyancy increases $M$.

For constant entrainment coefficient $\alpha >0$, the Batchelor entrainment hypothesis gives an inward boundary speed $\alpha U$. Volume, vertical momentum and buoyancy balances therefore giveTo find a pure similarity solution, put $Q\propto z^p$ and $M\propto z^q$ relative to the plume virtual origin. The first balance gives $p-1=q/2$; the second gives $q-1=p-q$. Thus $p=5/3$, $q=4/3$, andIn particular $r=Q/\sqrt{\pi M}$ grows linearly while $U=M/Q$ decreases as $z^{-1/3}$. A finite source shifts the virtual-origin coordinate. Dividing all three fluxes by $\pi$ changes their balance coefficients, so pi-normalized and physical-flux formulas must not be mixed.

A buoyant plume is a localized rising or sinking flow driven by a density difference from its surroundings. Heating fluid can lower its mass density and produce upward buoyancy; a denser injected fluid can produce a descending counterpart. The sign of the density anomaly and the direction of gravity specify the motion being described.

As a plume travels, fluid entrainment incorporates ambient fluid, increasing its volume flux and diluting the source anomaly. A warm narrow stream can therefore carry a growing amount of air while its mean excess temperature decreases. The buoyancy flux has a different balance from the volume transport, especially when the ambient fluid is stratified or heat is lost. Dye or temperature measurements across successive heights reveal widening and dilution; their interpretation requires the cross-sectional velocity as well as anomaly size to determine a transported flux.

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 $g_t=9g_p/4$.

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 $W=F\sqrt{2a{g}^{\prime }}$.

An eddy diffusivity is a transport closure representing unresolved turbulence by a mean scalar flux proportional to a negative mean concentration gradient. A local estimate is $K\sim U\ell$, where $U$ and $\ell$ are turbulent speed and length scales. In a line plume with constant characteristic speed and width proportional to $z$, this gives $K\propto z$. Its coefficient is an independent closure parameter; fluid entrainment conservation alone does not determine it.

A filling box model describes a turbulent plume rising through a confined fluid and spreading beneath its ceiling. Fluid entrainment transfers initially unmodified ambient fluid into the plume; the discharged fluid forms a descending filling-box first front. A maintained heat source can have negligible source volume flux while maintaining nonzero buoyancy flux. A finite injected fluid flux requires an equal exhaust if the container volume is fixed.

Fluid entrainment is incorporation of surrounding fluid into a moving current, jet or buoyant plume. A downstream increase in the flow's volume flux can therefore occur without an additional localized source. The added fluid generally changes the mean velocity, temperature or constituent concentration of the moving region.

For a steady jet whose surroundings initially carry little axial momentum, entrainment increases transported volume while the conserved axial momentum flux constrains how the jet slows and spreads. In a warm buoyant plume, entrained ambient air similarly dilutes the temperature anomaly while increasing the total flow. Measuring only a centreline speed cannot determine the entrainment rate; one needs integrated transport across successive sections or a suitable boundary-flux estimate. Entrainment is different from molecular diffusion, although both can contribute to mixing and are often coupled in an observed flow.

Suppose $n$ independent axisymmetric pure plumes share total buoyancy flux $B$ equally, have negligible source volume flux, and are supplied with a separate ambient ventilation flux $F$ at floor level. Their combined flux is $P(h)=\lambda n^{2/3}{B}^{1/3}{h}^{5/3}$. Volume conservation at a steady displacement-ventilation interface height and buoyancy flux conservation giveA positive two-layer state needs $0<h_{\ast }<H$, sources small relative to $h_{\ast }$, and plumes sufficiently separated to remain independent. Splitting the sources increases total fluid entrainment and reduces the interface height by $n^{-2/5}$. A finite source flux cannot simultaneously be neglected and supply all the replacement air; no steady displacement layer without ambient supply explains this restriction.

Consider entraining plumes supplied with total source volume flux $S$, a separate lower-layer ambient inflow $L$, and ceiling exhaust $F=S+L$. Write $P(h)=S+E(h)$, where $E(h)$ is total fluid entrainment from the lower layer. Volume conservation gives $A\dot{h}=L-E(h)$. A steady positive lower layer therefore requires $L=E(h)>0$. If $L=0$ and $E(h)>0$ for every $h>0$, no positive steady displacement-ventilation interface height exists. In a shifted pure plume approximation $P_i(h)=\lambda B_i^{1/3}(h+z_i{)}^{5/3}$ matched to source flux $S_i$ at $h=0$, this follows immediately from $P_i(h)>S_i$ for $h>0$. This is a statement about the ideal two-layer model, not a prohibition of all possible stratification in a real ventilated room.

Let $\ell$ and $\mathcal{T}$ denote longitudinal length and time scales, and let $U$ be a typical streamwise velocity. The shallow water approximation requires $h/\ell \ll 1$, slow changes in the channel geometry $h|B^{\prime }/B|\ll 1$, and small longitudinal slopes. The vertical velocity is then of order $Uh/\ell$; its acceleration must be small compared with the restoring acceleration, for example $Uh/(\mathcal{T}\ell )+U^{2}h/{\ell }^2\ll g^{\prime }$ in a reduced-gravity layer. The leading pressure is hydrostatic pressure, and cross-sectional horizontal velocity is approximated by $u(x,t)$. High Reynolds number permits negligible interior viscous terms but does not itself guarantee the geometric or hydrostatic assumptions. We also require $B>0$ and $h>0$ in the wetted interior. The dimensionless $B$ need not be small: it measures the cross-section's opening angle, while the smallness condition concerns streamwise variation.

The cross-sectional area and volume transport areFor an impermeable bed and no fluid entrainment, volume conservation givesor, equivalently,The geometrical source term expresses the depth change needed when a moving layer encounters a wider or narrower triangular section.

Use a gravity-current box model: approximate the current by uniform depth $h(t)$ and length $L(t)$, with constant reduced gravity, no deposition and no fluid entrainment. The triangular cross-sectional area is $h^2/2$, so the fixed released volume isThe Benjamin deep-ambient front condition closes the integral model:For a finite initial length $L_0$, direct integration givesThe ideal point-release limit takes $L_0=0$ and has $L\propto t^{4/5}$, $h\propto t^{-2/5}$. Its divergent initial depth is outside the shallow water approximation, so a real finite reservoir regularizes the early stage. This is the finite-volume triangular-channel current integral law; it does not assert a spatially uniform exact solution of the local equations.

Use $G=g\beta (T-T_0(z))$ for the plume reduced gravity, and take $z$ upwards. For a symmetric line plume define fluxes per unit source length,The integrals span the plume; its edges can move horizontally with $z$. The fluid entrainment rate at one moving edge is the inward normal transport relative to that edge. For an edge $x=b(z)$ it is $E=w{b}^{\prime }-u$; symmetry supplies equal transport through the other edge. The Batchelor entrainment hypothesis closes its mean value as $E=\alpha w_c$, where $w_c$ is a representative vertical velocity and $\alpha$ is the entrainment coefficient. The integrated volume conservation equation is therefore $q^{\prime }=2\alpha w_c$.

Using volume conservation, the vertical acceleration equation has conservative form ${\partial }_x(uw)+{\partial }_z(w^2)=G$. Incoming ambient has zero vertical momentum; the integrated momentum conservation equation is consequently $m^{\prime }=\int Gdx$. For the temperature anomaly, $T_0$ depends on height andAmbient fluid enters with $G=0$, giving the buoyancy flux balance. The integral model isHere $N$ is the buoyancy frequency when $N^2>0$; the same algebra permits unstable temperature gradients with $N^2<0$.

These are mean integral balances. The advective equations printed in the question omit explicit turbulent stresses and heat fluxes; a turbulent plume interpretation requires the usual entrainment closure for that unresolved transverse transport. Assuming vanishing edge stresses and heat fluxes, and negligible additional axial turbulent-flux contributions, gives the displayed bulk model. It would be inconsistent to treat a discontinuous top-hat profile as a smooth exact pointwise solution of the inviscid temperature-advection equation.

In the top-hat plume model, vertical velocity $w(z)$ and reduced gravity $G(z)$ are uniform over $-b<x<b$. The ambient has zero vertical velocity, negligible ambient turbulence and a prescribed $T_0(z)$. Thenand the equations becomeEquivalently, the ordinary differential equations for the three unknowns areThe last equation displays dilution by fluid entrainment as well as loss of relative buoyancy while rising through a stably stratified ambient. The model assumes a steady slender two-dimensional turbulent plume, large Reynolds number, the Boussinesq approximation, constant $\beta$ and $\alpha$, negligible heat loss, negligible imposed crossflow, and negligible pressure departure from ambient hydrostatic pressure. The factors of two count both entraining edges; $b$ is a half-width. A finite source also requires its volume, momentum and buoyancy fluxes as boundary data.

Let the closed box have ceiling height $H$ above the source. The printed width $2R$ alone does not specify this height, so it must appear as an additional geometrical parameter. Take $t=0$ here when the plume has first reached the ceiling and its thin outflow has spread across the box. The leading filling box model assumes a negligible source volume, insulated boundaries, fast horizontal spreading, slow evolution compared with the plume transit time, negligible diffusion, and $b\ll R$. Below the interface the ambient still has its original temperature, so the local plume flux remains $q(z)=2\alpha w_{\ast }z$ from part (c).

The ceiling outflow creates a warm upper region. Fluid entrainment draws the original lower fluid into the plume, making that upper region deepen. The filling-box first front is the lower material boundary of fluid already modified by plume discharge, separating it from the initially unmodified ambient; it need not bound a uniformly mixed upper layer. With negligible occupied plume area, volume conservation at its height $h(t)$ givesThereforeIf source switch-on is the time origin instead, replace $t$ by elapsed time after the initial transit and spreading stage; the quasi-steady model does not resolve that stage.

The buoyancy jump requires distinguishing ambient fluid at the front from the plume crossing it. Just after the first ceiling outflow, the ambient buoyancy immediately above the nascent front isOutside the narrow plume, ambient temperature is transported without diffusion. If $G_a$ is its buoyancy relative to the original fluid, its equation isThe first front follows precisely these material trajectories, so the temperature of the fluid immediately above it is conserved. The initially unmodified fluid below has $G_a=0$. Thus the ambient jump isBy contrast, the contemporaneous plume immediately below the front has $G_p(h,t)=w_{\ast }^2/h(t)$, which grows as the front descends. That plume value is not the ambient jump across the material first front. Equating the two would inadvertently impose a uniformly mixed upper layer and discard the stable density stratification of the standard filling box model. Diffusion or extensive interfacial mixing can smooth and alter the ideal jump.

The filling box model requires the first outflow to spread across the enclosure and remain above unmodified fluid, while a narrow nearly vertical plume rises below it. Several mechanisms can defeat those assumptions. A tall narrow box makes the plume a substantial fraction of the width, so its compensating return velocity is no longer small and modifies both fluid entrainment and momentum flux. A fast ceiling current can descend the sidewalls and overturn the initially stable upper region. In both cases a horizontally uniform slowly descending filling-box first front is inappropriate. Strong plume displacement from the centre, imposed crossflow, large source volume, vents or substantial wall cooling likewise alter the volume and heat budgets.

A useful diagnostic is whether the ceiling outflow's inertial acceleration, of order $U_c^2/{\ell }_c$, overwhelms the restoring buoyancy of the nascent layer. If $U_c^2/(G_H{d}_c)$ is not small for its depth $d_c$, the outflow can penetrate or overturn the stratification rather than remain trapped above it. This is a diagnostic Froude number, not a universal threshold without specifying the wall-current geometry and mixing. Low Reynolds number invalidates a constant turbulent entrainment coefficient, and thermal diffusion or wall heat loss can erase the sharp first-front jump. Thus large aspect ratio, strong return circulation, overturning, ventilation or diffusion require a model beyond the standard filling-box approximation.

Normalize all fluxes by $\pi$, rather than including that factor in only some of them. The specific plume volume flux, momentum flux and buoyancy flux areThe ambient transport is $-Q$ and its specific momentum flux isPut $s=b^2/R^2=Q^2/(R^{2}M)$, with $0<s<1$. The total specific momentum flux isThe first differential equation follows immediately from relative-speed fluid entrainment:For the second, differentiate $J=M/(1-s)$ and use $s^{\prime }=2s{Q}^{\prime }/Q-s{M}^{\prime }/M$. Since $J^{\prime }=b^2{g}^{\prime }=B_{s}Q/M$, this yieldsThis is the requested two-equation system, with $B=B_s$ and zero total volume flux as the additional conservation conditions. Near the pure source, $Q,M\to 0$ and $s\to 0$. Eliminating height in that limit givesThe zero integration constant selects a pure plume; a nonzero imposed source momentum would select a different branch and generally a different turning height. The leading radius is $b\sim 6\alpha z/5$, which recovers the unconfined pure-plume limit near the source.

Put $G=g({\rho }_p-{\rho }_0)/{\rho }_0$ and $b=G\varphi >0$, the reduced gravity. Use the Boussinesq approximation, hydrostatic pressure, uniform vertical velocity and concentration profiles, negligible fluid entrainment and drag, and a flat bed. Integrating volume conservation, momentum conservation and the particle budget over the layer givesThe hydrostatic pressure resultant is ${\rho }_{0}b{h}^2/2$; the last right-hand side is the deposition flux. With $D={\partial }_t+u{\partial }_x$, these areThe quasilinear coefficient matrix in variables $(h,u,b)$ isIts eigenvalues are $u,u\pm c$, where $c=\sqrt{bh}$. Therefore the characteristic curves and their compatibility ordinary differential equations areFor the last equation the left eigenvectors are $(\pm c/h,1,\pm h/(2c))$, and multiplying the quasilinear equations by them gives the displayed source terms. This is the sedimenting rectangular-channel characteristic compatibility relation; it supplies one differential relation on each wave family, rather than three independently closed scalar equations there. The contact eigenvector has $du=0$ and $d(b{h}^2)=0$. For reference, an equivalent form isOnly for constant $b$ without settling do $u\pm 2c$ become genuine Riemann invariants. With settling, the spatial buoyancy gradient cannot be silently omitted.

Let $V=4\pi b^3/3$ and $B=Vg({\rho }_0-\rho )/{\rho }_{\mathrm{ref}}$. The four balances describe, respectively, fluid entrainment through the spherical surface at normal speed $\alpha u$, the increase of vertical momentum due to buoyancy, conservation of total buoyancy in a uniform ambient, and motion of the thermal centre. Entrained ambient fluid initially has no vertical momentum, so the momentum equation includes the inertia needed to accelerate it. This model neglects added mass and drag; these effects would alter the numerical similarity coefficients.

With $|\rho -{\rho }_0|\ll {\rho }_0$, the Boussinesq approximation replaces inertial density by ${\rho }_0$ while retaining the density difference in buoyancy. Take ${\rho }_{\mathrm{ref}}={\rho }_0$ to this accuracy. ThenFor a point source, $b(0)=z(0)=0$ and the initial integrated impulse is zero. Therefore $b=\alpha z$ and $Vu=B_{0}t$. Integrating $b^3\dot{b}=3\alpha B_{0}t/(4\pi )$ yieldsThe mass density follows from conserved buoyancy:Thus the radius and rise height grow as $t^{1/2}$, the rise speed decays as $t^{-1/2}$, and the density deficit decays as $t^{-3/2}$. In height coordinates, put $K=4\pi {\alpha }^3/3$ and $C=\sqrt{B_0/(2K)}$; then $V=K{z}^3$ and $u=C/z$.

If a reference density different from the inertial ambient density is retained, $d(Vu)/dt=({\rho }_{\mathrm{ref}}/{\rho }_0)B_0$: replace $B_0$ in the kinematic formulas by $({\rho }_{\mathrm{ref}}/{\rho }_0)B_0$, while retaining the original $B_0$ in the density-deficit formula. This keeps reference-density normalization separate from the physical acceleration.

The point-source spherical thermal similarity is singular at $t=0$ and cannot satisfy the small-density-difference condition arbitrarily close to its formal origin. A finite-radius, zero-impulse regularization has $b=b_s+\alpha z$ and $b^4=b_s^4+3\alpha B_0{t}^2/(2\pi )$, with $u=B_{0}t/V$. This displays the required initial data rather than attributing the singularity to a physical infinite initial speed.

In stable density stratification, ambient density decreases with height. A rising buoyant thermal encounters progressively lighter surroundings and entrains denser fluid from below, so its net buoyancy decreases. It reaches neutral buoyancy, overshoots through its remaining momentum, and eventually stops rising, spreads as a buoyant intrusion or excites internal gravity waves. The finite rise is a consequence of this restoring effect, not just drag.

Total buoyancy has dimensions $[B_0]=L^4{T}^{-2}$ and buoyancy frequency has $[N]=T^{-1}$. For an ideal point release with no other dimensional source length or time, dimensional analysis therefore givesThe dimensionless coefficient depends on entrainment coefficient, shape and the momentum model. A finite source radius or impulse supplies additional dimensionless parameters; dimensional analysis cannot determine the coefficient by itself.

Within precisely the spherical model, $b=\alpha z$ and the evolving thermal buoyancy satisfies $dB/dz=-N^{2}V=-K{N}^2{z}^3$, so $B(z)=B_0-K{N}^2{z}^4/4$. Neutrality occurs at $z_n^4=4B_0/(K{N}^2)$. For $P=Vu$, the momentum equation and $\dot{z}=P/V$ giveThe first nonzero-height stopping point is $z_{\max }^4=8B_0/(K{N}^2)$, so $z_{\max }=2^{1/4}{z}_n$. This illustrates both the dimensional scale and the distinction between neutral height and maximum rise; the precise coefficients rely on the specified ideal fluid entrainment and impulse model.

Differentiate $b=Q/\sqrt{M}$ and use both balances:Substituting $Q^2=8\alpha M^{5/2}/(5B_0)+c{Q}_0^2$ givesFor $c<0$, the radius grows faster than the pure-plume slope $6\alpha /5$ and its growth rate decreases toward that slope as $M$ increases. For $c=0$, it grows at that slope exactly. For $0<c<1$, it grows more slowly, possibly initially contracting, and its slope increases toward $6\alpha /5$. Since $M^{\prime }=B_{0}Q/M>0$ and $M\to \infty$, all branches satisfy $db/dz\to 6\alpha /5$ as $z\to \infty$.

Initial necking requiresThus a sufficiently lazy plume initially narrows as buoyancy accelerates its fluid faster than fluid entrainment supplies additional volume. It still has increasing volume and momentum fluxes; shrinking radius does not imply volume loss. At equality the initial slope is zero, and for $c>3/5$ the slope later becomes positive once $M^{5/2}=5c{B}_0{Q}_0^2/(12\alpha )$. This gives a finite minimum radius before the eventual pure plume regime.