Line plume

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.