Porous thermal plume ($w=\beta (T-T_0)$)

A slender planar plume in a saturated porous medium has buoyancy velocity $w=\beta (T-T_0)$, with $\beta ={\rho }_{0}g\alpha \Pi /\mu$. The leading heat equation retains both advection components and lateral thermal conduction: $u{T}_x+w{T}_z=\kappa T_{xx}$. 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 $z^{-1/3}$.

For a slender porous thermal plume, Darcy law gives $w=\beta (T-T_0)$, with $\beta =Kg\alpha /\nu$. If $C_h={\rho }_0{c}_p$ and $F_h=C_h\int w(T-T_0)dx$ is the conserved excess convective heat flux per span, then the kinematic momentum flux is $M=\int w^{2}dx=\beta F_h/C_h$, and the buoyancy flux is $B=\int g\alpha (T-T_0)wdx=g\alpha F_h/C_h$. The hyperbolic-secant porous plume profile has $Q=(36\kappa Mz{)}^{1/3}$. Multiplication of $Q$ and $M$ by ${\rho }_0$ gives physical mass flux and the density-weighted Darcy second moment. Intrinsic pore-velocity momentum flux instead has an additional inverse-porosity factor. Momentum conservation here follows from the heat/velocity proportionality, not from an inertia balance: porous drag balances buoyancy locally.

The symmetric steady porous thermal plume has $w=W{\mathrm{sech}}^2(x/b)$, $T-T_0=w/\beta$ and $\psi =-Wb\tanh (x/b)$. Its exact normalizations are $W{b}^2=6\kappa z$ and $(4/3)W^{2}b=Q$, so $b=(48{\kappa }^2{z}^2/Q{)}^{1/3}$ and $W=(3Q^2/(32\kappa z){)}^{1/3}$. The streamfunction supplies lateral entrainment required by continuity.

Truncating the steady porous thermal plume profile below a head height and conserving supplied heat gives ${\dot{Z}}_h\simeq \int w^{2}dx/\int wdx=(2/3)W(Z_h)$. Thus $Z_h\propto t^{3/4}$. This is an integral bulk-head estimate, while fastest centreline parcels have characteristic speed $W$; the approximation does not resolve the transient nose.