= Symmetric line-plume filling-box front
{title2=$h(t)=H e^{-\alpha w_*t/R}$}
An ideal two-sided <line plume> has half-width $b=\alpha z$, constant <velocity> $w_*=(\mathcal B/(2\alpha))^{1/3}$ and <volume flux> $q=2\alpha w_*z$. In a closed box of width $2R$, neglecting the plume's occupied area gives $2R\dot h=-q(h)$ for the <filling-box first front>, initially at ceiling height $H$. With negligible diffusion the front is a material interface; its ambient <buoyancy> jump retains the initial ceiling-discharge value $\mathcal B/q(H)=w_*^2/H$. The contemporaneous plume <reduced gravity> $w_*^2/h(t)$ is a different quantity. The upper ambient generally becomes stratified rather than remaining one uniformly mixed layer.
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