Top-hat plume model

A top-hat plume model takes velocity and reduced gravity to be uniform across the plume and zero outside it. Integral volume, momentum, and buoyancy fluxes then depend only on plume width and these uniform values.

In an impermeable cylinder, an axisymmetric turbulent plume with negligible source volume flux requires a downward return flow. With uniform upward plume velocity $w$ and ambient velocity $v$, volume conservation gives $b^{2}w+(R^2-b^2)v=0$. An entrainment coefficient $\alpha$ based on the relative speed gives $Q^{\prime }=2\alpha b(w-v)$. A specified pressure closure is essential for momentum conservation. If the departure from ambient hydrostatic pressure and wall force are neglected in the whole-section integral, the total specific momentum flux $J=b^2{w}^2+(R^2-b^2)v^2$ obeys $J^{\prime }=b^2{g}^{\prime }$. Homogeneous unmodified ambient and no buoyancy loss give $b^{2}w{g}^{\prime }=B_s$. These are leading integral-model closures; restoring the ambient dynamic pressure can change the singularity criterion.

For the pressure-neglecting confined plume with compensating return flow, define specific plume momentum flux $M=b^2{w}^2$ and area fraction $s=Q^2/(R^{2}M)$. Then $J=M/(1-s)$ andThe branch issuing from a pure plume reaches a fold at $s=1/2$, where plume and ambient have equal areas, opposite velocities and equal specific momentum fluxes. The numerator is nonzero on this branch, so $M^{\prime }$ diverges although $M,Q,w$ remain finite. This signals failure of the steady uniform-profile description, not infinite physical velocity.

Source volume flux is negligible only when entrained volume greatly exceeds it. In the jet limit of the top-hat plume model, this requires $z\gg Q_0/(c\sqrt{M_0})$. In a pure plume with $Q=C_Q{z}^{5/3}$, it requires $z\gg (Q_0/C_Q{)}^{3/5}$, where $C_Q=(3c/5)(9c{F}_0/20{)}^{1/3}$ for full-cross-section fluxes and $c=2{\alpha }_e\sqrt{\pi }$. These scales differ from the jet length which measures imposed momentum relative to buoyancy. Neglecting source volume does not automatically permit neglecting source momentum.

With zero source volume flux and zero buoyancy flux but positive source momentum flux, the axisymmetric top-hat plume model reduces to a turbulent round jet. For full-cross-section kinematic fluxes, put $c=2{\alpha }_e\sqrt{\pi }$. Entrainment gives $Q^{\prime }=c\sqrt{M}$, while momentum conservation gives $M=M_0$. Hence $Q=c\sqrt{M_0}z$, speed $w=\sqrt{M_0}/(cz)$ and radius $r=cz/\sqrt{\pi }$. A finite source flux replaces $z$ by $z+Q_0/(c\sqrt{M_0})$. The ideal zero-flux source has singular speed and is a far-field model, not a finite nozzle.

For plume area $A$ and reduced gravity $g^{\prime }$, the principal matrix in variables $(A,w,g^{\prime })$ is $\left(\begin{array}{ccc}w& A& 0\\ 0& w& 0\\ 0& 0& w\end{array}\right)$. At $A>0$ it has a triple real speed $w$ but only two independent eigenvectors. Thus the model is weakly hyperbolic, not strongly or strictly hyperbolic. Real speeds alone do not provide a complete set of characteristic fields or a uniform high-frequency estimate. Shape factors representing nonuniform radial profiles can alter this degeneracy.

Use half-width $b$, upward speed $w$, constant inertial reference density $R$, and positive buoyant reduced gravity $g^{\prime }$. Half-section fluxes are $Q=Rbw$, $M=Rb{w}^2$ and $F=R{g}^{\prime }bw$. With two-sided entrainment $u_e=\alpha w$, the Boussinesq balances areFull-width fluxes are twice these definitions. The shape coefficients differ from triangular transverse profiles, so their unsteady balances cannot be used unchanged here.

In a homogeneous ambient fluid, the displayed nonsteady power-law ansatz solves the unsteady top-hat line-plume balances for $\tau =t+t_{\ast }>0$. It gives $Q=R\alpha z^2/(3\tau )$, $M=2R\alpha z^3/(9{\tau }^2)$ and $F=2R\alpha z^3/(9{\tau }^3)$. The solution is singular at zero age and carries zero source flux at its unshifted ideal origin. It describes a separable branch away from a source; a finite-width boundary, virtual-origin shift or matching region is needed to supply nonzero physical source flux. It is distinct from the steady constant-flux line plume, whose top-hat half-width grows as $\alpha z$.

If two equal pure plumes merge while keeping their common velocity and reduced gravity, conservation of cross-sectional area gives radius $\sqrt{2}$ times the individual radius. All three integral fluxes double. The plume balance parameter increases by $2^3/2^{5/2}=\sqrt{2}$, so the immediate combined plume is a lazy plume. Similarity attraction may restore pure balance farther downstream, but instantaneous pure balance does not follow from adding two pure plumes.

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.

With common factors of $\pi$ removed, steady kinematic fluxes satisfy $Q_z=2\alpha \sqrt{M}$, $M_z=FQ/M$ and $F_z=-N^{2}Q$. The Batchelor entrainment hypothesis supplies volume, the integrated buoyancy force supplies momentum, and the environmental mass density gradient reduces buoyancy flux. This uses the Boussinesq approximation.

For $N^2=K{z}^{\beta }$ and positive pure-power fluxes, the similarity solution exponents are $q=3+\beta /2$, $m=4+\beta$ and $f=4+3\beta /2$. Increasing volume and momentum and decreasing positive buoyancy flux require $-4<\beta <-8/3$. At fixed nonzero $K$ the endpoint coefficient limits are singular.

For prescribed $N^2=-G^2<0$, the steady Boussinesq top-hat plume in a stratified ambient has kinematic fluxes $Q={\alpha }^{2}G{z}^3/9$, $M={\alpha }^2{G}^2{z}^4/36$, and $F={\alpha }^2{G}^3{z}^4/36$. Balancing powers in $Q_z=2\alpha \sqrt{M}$, $M_z=FQ/M$, and $F_z=G^{2}Q$ gives exponents $(3,4,4)$ and these positive amplitudes. The ambient's static instability is prescribed without allowing it to drive ambient motion; the Boussinesq approximation additionally requires small density differences over the modeled height range.

For the Boussinesq top-hat plume in a stratified ambient with $N^2=C{z}^p$, the exact pure-power solution has the displayed radius and velocity, and $g^{\prime }=-2C{z}^{p+1}/(3p+8)$. With positive entrainment, radius, upward speed and reduced gravity, stable $C>0$ requires $-4<p<-8/3$, whereas unstable $C<0$ requires $p>-8/3$. The stable branch has divergent buoyancy flux at zero height, so it is not a finite-flux point-source solution. The unstable branch has zero limiting source buoyancy flux and extracts buoyancy from its maintained unstable ambient. The cases $p=-6,-4,-8/3$ are singular at nonzero $C$; $C=0$ instead admits the ordinary Boussinesq point-source plume.

Normalize volume, momentum and buoyancy by a positive plume similarity in power-law stratification and use $\xi =\ln (z/z_{\ast })$. The normalized equations are ${\hat{Q}}_{\xi }=q(\sqrt{\hat{M}}-\hat{Q})$, ${\hat{M}}_{\xi }=m(\hat{F}\hat{Q}/\hat{M}-\hat{M})$ and ${\hat{F}}_{\xi }=f(\hat{Q}-\hat{F})$. They form an autonomous differential equation.

The linearized relative-flux matrix is $J=[(-q,q/2,0);(m,-2m,m);(f,0,-f)]$. Its characteristic polynomial has constant term $qmf<0$ for physical positive-buoyancy similarity, so at least one real eigenvalue is positive. The growing mode behaves as $z^{\lambda }$ and makes the similarity trajectory unstable with increasing height, rather than with physical time.

Dividing the mass flux, momentum flux, and density-weighted buoyancy flux by ambient mass density gives the kinematic fluxes $q=\pi b^{2}V$, $m=\pi b^2{V}^2$, and $f=\pi b^{2}Vg({\rho }_0-\rho )/{\rho }_0$. This convention makes dimensional similarity solutions independent of the arbitrary ambient density unit.

For a turbulent plume in a homogeneous ambient of density ${\rho }_0$, define actual sectional mass flux, momentum flux and density-weighted buoyancy flux by $Q=\pi b^2\rho w$, $M=\pi b^2\rho w^2$ and $F=\pi b^{2}wg({\rho }_0-\rho )$. With non-Boussinesq Batchelor entrainment $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, their balances areHere $Q^2/M$ is sectional mass, $QF/M$ is sectional density-deficit buoyancy, and the entrained ambient initially has zero vertical momentum. The last equation follows by subtracting mass conservation from ${\rho }_0$ times volume conservation. Scase, Caulfield, Dalziel and Hunt's time-dependent plume model derives the equivalent system with the common factor $\pi$ suppressed.

For an ideal-gas heated pure plume with entrainment proportional to $w\sqrt{\rho /{\rho }_0}$, the excess heat flux implies conserved kinematic buoyancy flux $B$. Put $q=Q/{\rho }_0$, $m=M/{\rho }_0$. Their balances are $q^{\prime }=2\alpha \sqrt{m}$ and $m^{\prime }=Bq/m$, identical to the pi-suppressed Boussinesq pure-plume flux equations. With $d=6\alpha /5$ and $s=[25B/(48{\alpha }^2){]}^{1/3}$, $q=d^{2}s{z}^{5/3}$, $m=d^2{s}^2{z}^{4/3}$ and $z_b^{5/3}=B/(g{d}^{2}s)$. The radius is $b=dz\sqrt{1+(z_b/z{)}^{5/3}}$ and reduced gravity is $g/[1+(z/z_b{)}^{5/3}]$. At $z=z_b$ density is half ambient; far above this height the Boussinesq approximation becomes accurate.

A pure plume is a similarity regime in which the source supplies buoyancy flux but no dynamically persistent source volume or momentum flux. Its finite source region is represented by a plume virtual origin.

For the constant-entrainment axisymmetric plume equations $Q^{\prime }=2\alpha \sqrt{M}$, $M^{\prime }=FQ/M$ with $F>0$, pure balance means $M^{5/2}=5F{Q}^2/(8\alpha )$ at every height. This is a similarity relation between fluxes, not a requirement that the physical source have zero radius. A finite source on this curve is represented by a plume virtual origin. For fluxes normalized by pi and top-hat fields, the same relation is $g^{\prime }b/w^2=8\alpha /5$.

When $F$ and $\alpha$ are constant, eliminating height between $Q^{\prime }=2\alpha \sqrt{M}$ and $M^{\prime }=FQ/M$ gives $d(M^{5/2})/dQ=5FQ/(4\alpha )$. Thus $M^{5/2}-5F{Q}^2/(8\alpha )$ is constant. Its sign classifies forced and lazy departures from pure plume balance, and its vanishing makes a finite-source plume exactly self-similar after a virtual-origin shift.

For an axisymmetric top-hat plume model in uniform ambient density, use full kinematic fluxes and $c=2{\alpha }_e\sqrt{\pi }$. The equations $Q^{\prime }=c\sqrt{M}$, $M^{\prime }=F_{0}Q/M$ conserve $M^{5/2}-5F_0{Q}^2/(4c)$. Solve this invariant for $M(Q)$, then integrate $dz=dQ/(c\sqrt{M})$ to obtain the displayed exact quadrature. It determines the transition from the near-source jet limit of the top-hat plume model to the distant pure plume. The two regimes cannot be added by linear superposition because the flux equations are nonlinear.

For positive source fluxes and positive constant buoyancy flux, the plume flux-balance invariant has a finite constant while $Q$ grows without bound. Dividing by $Q^2$ makes the source imbalance vanish relatively. The balances then give $Q\sim (6\alpha /5)(9\alpha F/10{)}^{1/3}{z}^{5/3}$ and $M\sim (9\alpha F/10{)}^{2/3}{z}^{4/3}$. This attraction concerns relative similarity and does not assert that different finite sources share the same virtual origin.

The dimensionless ratio $\Gamma =5F{Q}^2/(8\alpha M^{5/2})$ compares a plume to the constant-entrainment pure balance. Values one, below one and above one describe pure plume balance, a momentum-excess forced plume, and a momentum-deficient lazy plume, respectively. For fluxes normalized by pi and a top-hat profile, $\Gamma =5g^{\prime }b/(8\alpha w^2)$.

For a top-hat axisymmetric pure plume of radius $r$, speed $U$, and reduced gravity $g^{\prime }$, the conserved fluxes are $Q=\pi r^{2}U$, $M=\pi r^2{U}^2$, and $B=\pi r^{2}U{g}^{\prime }$. The Batchelor entrainment hypothesis gives $Q\propto B^{1/3}{z}^{5/3}$.

For pi-normalized top-hat fluxes, $b=Q/\sqrt{M}$. The pure similarity solution therefore has $b=(6\alpha /5)(z+z_0)$, so the radius grows linearly with slope $6\alpha /5$. The corresponding plume virtual origin is $-z_0$ and a finite source has $z_0=5b_s/(6\alpha )$. This law translates geometric plume-contact criteria into heights.

A constant surviving buoyancy flux sets the amplitudes of a pure plume: $Q\propto F^{1/3}$ and $M\propto F^{2/3}$. Relative-flux perturbations have the neutral ratio $(\delta Q/Q,\delta M/M,\delta F/F)\propto (1,2,3)$. This is the zero-eigenvalue limit of growing mode of power-law plume similarity; nonzero endpoint stratification can still create cumulative logarithmic flux loss.

A steady top-hat plume model supplied with kinematic buoyancy flux $f_0>0$ but zero source volume flux and momentum flux hasThe 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 line plume is invariant along one horizontal direction, so its integral fluxes are measured per unit span. A wall line plume entrains through one exposed edge and has width proportional to distance from its plume virtual origin.

A two-sided line plume whose axial velocity and density anomaly both have triangular transverse profiles. For half-width $b$, ${\int }_{-b}^{b}f(x/b{)}^{n}dx=2b/(n+1)$. These profile moments set the different coefficients of stored mass, momentum, and their fluxes. It differs from a wall plume because both edges entrain ambient fluid.

In the Boussinesq approximation, density-weighted fluxes are $Q={\rho }_{0}bW$, $M=(2/3){\rho }_{0}b{W}^2$ and $F=(2/3)gbW({\rho }_0-\rho )$. With two-sided Batchelor entrainment $e=\alpha W$, the integral balances areThe factors $4/3$ and $3$ depend on the triangular profile and the definition of the entrainment coefficient.

The unsteady Boussinesq triangular-profile line-plume balances admit a buoyant similarity solution $Q=\alpha {\rho }_0{z}^2/\tau$, $M=(2/3)\alpha {\rho }_0{z}^3/{\tau }^2$ and $F=(2/3)\alpha {\rho }_0{z}^3/{\tau }^3$, with $\tau =t+t_{\ast }>0$. It reconstructs $b=\alpha z$, $W=z/\tau$ and reduced gravity $z/{\tau }^2$. Origins can be shifted, but the zero-time unshifted solution is singular. Source and initial data are required to select the physical member and region of validity.

For a triangular-profile line plume with centreline density $\rho$, speed $W$, half-width $b$, ambient density ${\rho }_0$, and inward relative edge speed $e$, volume, mass and momentum conservation giveSubtracting mass conservation from ${\rho }_0$ times volume conservation also gives buoyancy conservation. Batchelor entrainment supplies a closure for $e$.

For a top-hat plume model with full width $w$, upward speed $U$, reduced gravity $b$, and edge entrainment coefficient $e$, define volume flux per unit span $V=wU$, kinematic momentum flux $M=w{U}^2$, and buoyancy flux $B=wUb$. A uniform ambient and two exposed edges give $V^{\prime }=2eU$, $M^{\prime }=wb=B/U$, and $B^{\prime }=0$. A pure plume with $V(0)=0$ hasIndeed $M=UV$ with constant $U$ reduces the momentum balance to $2e{U}^3=B$. If width means half-width, its growth coefficient is $e$, not $2e$; profile shape factors also alter the coefficients.

A wall line plume is a one-sided line plume adjoining a solid wall. Under a top-hat pure-plume model, $d(bW)/ds=\alpha W$ because only its outer edge entrains ambient fluid.

A glacier supplies fresh meltwater at speed $U>0$ to a rising wall line plume whose buoyancy is dominated by salinity. With one entraining edge, entrainment coefficient $\epsilon$, volume flux $Q=bw$ and negligible wall stress, $Q^{\prime }=\epsilon w+U$ and $(b{w}^2{)}^{\prime }=g{\beta }_C(C_{\infty }-C)b$. Enthalpy and salt balances give $[Q(T_{\infty }-T){]}^{\prime }=U[L/c_p+T_{\infty }-T_s]$ and $[Q(C_{\infty }-C){]}^{\prime }=U(C_{\infty }-C_s)$. If latent heat dominates and entrainment dominates the volume supply, integration from a virtual origin $Z=0$ and power-law balance give the displayed width and speed. The temperature and salinity deficits are $ULZ/(c_{p}Q)$ and $U(C_{\infty }-C_s)Z/Q$, both proportional to $Z^{-1/3}$. The approximation is a far-field plume solution, not a solution at the virtual origin.

An ice-bearing plume at local liquidus equilibrium conserves excess enthalpy $E$ and salt $S$ relative to the ambient liquidus water. Eliminating its temperature and brine composition gives $\varphi Q[L+c_{p}m{C}_0-L\varphi ]=-E-c_{p}mS+E\varphi$. To dilute-crystal order this fixes its advected ice volume, hence its leading buoyancy flux. A positive rising branch needs a positive source invariant.

For one exposed entraining edge and conserved positive dilute ice flux $I_0$, the line-plume equations give the displayed pure far-field solution. Its width grows linearly, speed is constant and ice fraction decays inversely with distance from a virtual origin. Two exposed edges replace $e$ by $2e$. The dilute solution is invalid near the formal origin, where its ice fraction diverges.

In a triangular-profile wall line plume, axial velocity and reduced gravity decrease linearly from their wall values to zero across the plume width $b$. If the wall speed is $W$, its volume and kinematic momentum fluxes per unit span are $V=bW/2$ and $J=b{W}^2/3$.

A line plume of kinematic buoyancy flux $B_0$ rising through uniform buoyancy frequency $N$ has the dimensional height scaleIt measures the height over which ambient stratification removes an order-one fraction of the plume buoyancy before the plume reaches neutral buoyancy and spreads as a buoyant intrusion.

A plume virtual origin is the extrapolated point at which a similarity plume's width and volume flux vanish. It absorbs finite source geometry into a shift of the axial coordinate.

For a horizontal forced plume, the plume virtual origin lies downstream and below the physical source on the far-field vertical asymptote. The origin shift in centreline arc length is distinct from its vertical height: if $z=s-D+o(1)$ and $\hat{s}=s+s_0$, the virtual height is $z_v=-D-s_0$. Both geometric shifts scale with the jet length at fixed entrainment coefficient.