= Porous thermal plume
{title2=$w=\beta(T-T_0)$}
A slender planar plume in a saturated porous medium has buoyancy velocity $w=\beta(T-T_0)$, with $\beta=\rho_0g\alpha\Pi/\mu$. The leading heat equation retains both <advection> components and lateral <thermal conduction>: $uT_x+wT_z=\kappa T_{xx}$. Its convective heat input per span is conserved, $F_h=C_h\int w(T-T_0)dx$. Defining $Q=\beta F_h/C_h$ gives width proportional to $z^{2/3}$ and vertical speed/temperature excess proportional to $z^{-1/3}$.
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