Pure plume

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, $\Gamma =5g^{\prime }b/(8\alpha w^2)$.

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.

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 pi-suppressed kinematic buoyancy flux $f_0=b^{2}w{g}^{\prime }>0$ but zero source volume flux and momentum flux hasThe full-cross-section buoyancy flux is $\pi f_0$. The 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.

Consider an equilibrium ice-water wall line plume with volume flux per unit span $Q$, ice volume fraction $\varphi$, liquid salinity $C$, and temperature $T=-mC$. The positive coefficient $m$ is the slope magnitude of a linear liquidus; ambient liquid has salinity $C_0$ and temperature $T_0=-m{C}_0$. Use equal reference densities and equal specific heat capacities $c_p$ in the thermodynamic bookkeeping, and let $L$ be the latent heat of fusion per unit mass.

When entrainment only adds that reference liquid and no crystals or heat escape, the excess enthalpy and salt fluxes are conserved:Combining them gives $E+c_{p}mS=-\varphi Q(L+c_{p}mC)$. Eliminating $C$ with the salt relation yieldsThus the leading dilute-crystal balance is $\varphi Q\simeq (-E-c_{p}mS)/(L+c_{p}m{C}_0)$. More precisely, this limit requires both $L\varphi \ll L+c_{p}m{C}_0$ and $|E|\varphi \ll |E+c_{p}mS|$; cancellation in the source combination can invalidate it even at small $\varphi$.

This conserved leading ice flux can drive a self-similar ice-bearing wall plume. Suspended frazil ice also changes buoyancy in ice-shelf plume models, whose additional crystal-growth, deposition and pressure effects need their own balances.^[ref]

In a dilute, well-mixed wall line plume, let $b$ be its width, $w$ its upward velocity, $\varphi \ll 1$ its ice volume fraction, and $e>0$ its one-edge entrainment coefficient. Define $g_i=g({\rho }_l-{\rho }_i)/{\rho }_l$ from the liquid and ice mass densities. The mixture's reduced gravity is approximately $g_i\varphi$, so $g_i$ itself is a material density-contrast scale, not the changing mixture reduced gravity.

Neglect crystal slip, deposition and other buoyancy contributions. If the ice-bearing wall-plume thermodynamic invariant supplies a constant positive ice flux $I_0=\varphi bw$, then volume flux $Q=bw$ and kinematic momentum flux $M=b{w}^2$ obeyA constant-velocity solution has $b=eZ$ with $Z$ measured from its plume virtual origin. Substitution givesIts width grows linearly and ice fraction falls inversely with height. Two exposed entraining edges replace $e$ by $2e$. The formal origin lies outside the dilute model because the ice fraction diverges there; finite-source conditions require matching away from that origin.

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.