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.

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 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 $Q\propto B^{1/3}{z}^{5/3}$.

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 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.

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.