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Use the planar geometry indicated by the supplied $x,z$ equation and streamfunction; the source heat input is then measured per unit out-of-plane span. Let $\Theta =T-T_0$ and defineThis is the buoyancy part of Darcy law. Let $C_h$ denote the constant heat capacity per volume consistent with the given transport equation; for the fluid heat-flux convention $C_h={\rho }_0{c}_p$. Porosity or matrix heat-storage factors, if retained, must be used consistently in this coefficient and the effective transport parameters.

For plume width $b\ll z$, continuity gives $u\sim wb/z$. Both $u{\Theta }_x$ and $w{\Theta }_z$ are consequently of order $w\Theta /z$: lateral advection must not be discarded. Lateral thermal diffusion is of order $\kappa \Theta /b^2$, while vertical diffusion is smaller by $(b/z{)}^2$. The slender steady porous thermal plume model is thereforeWriting it in conservative form,and integrating across the plume, where $\Theta$ and lateral diffusive/advective heat transport vanish at large $|x|$, provesIn steady state with no lateral heat loss and negligible vertical conductive flux, this equals the heat supplied at the source. Define $Q=\beta F_h/C_h=\int w^{2}dx>0$.

For characteristic velocity $W$ and width $b$, flux conservation gives $W^{2}b\sim Q$ and the advection-diffusion balance gives $W{b}^2\sim \kappa z$. ConsequentlyThe dimensional scales are $b\sim ({\kappa }^2{z}^2/Q{)}^{1/3}$ and $W\sim (Q^2/(\kappa z){)}^{1/3}$. The plume becomes relatively more slender, $b/z\propto z^{-1/3}$, away from the source.

To find the profile, put $\eta =x/b(z)$ and $\psi =-W(z)b(z)f(\eta )$, so $w=W{f}^{\prime }$ andInserting into $u{w}_x+w{w}_z=\kappa w_{xx}$ givesIts coefficient is constant under the derived scaling. Choose the width normalization $W{b}^2=6\kappa z$. Integrating once with decaying velocity derivatives gives $f^{\prime \prime }+2f{f}^{\prime }=0$. A second integration gives $f^{\prime }+f^2=1$ after normalizing the centreline velocity to $W$; symmetry has $f(0)=0$. The decaying, positive solution is $f=\tanh \eta$. Hence the hyperbolic-secant porous plume profile isSince ${\int }_{-\infty }^{\infty }{\mathrm{sech}}^4\eta d\eta =4/3$, the precise amplitudes areThey satisfy both $W{b}^2=6\kappa z$ and $(4/3)W^{2}b=Q$. The derived $u$ supplies lateral entrainment; setting $u=0$ would not reproduce this profile or satisfy continuity.

A heat-weighted head speed of a porous plume estimate follows by filling the column below a head height $Z_h(t)$ with this steady profile. Its excess heat per unit height isConservation of the total injected heat gives $F_{h}t\simeq {\int }_0^{Z_h}\mathcal{H}(z)dz$. Differentiating,Thus a heat-weighted head advances at a speed of order the local centreline velocity and decelerates as $Z_h^{-1/3}$. With $K=(3Q^2/(32\kappa ){)}^{1/3}$,The factor $2/3$ is the energy-conserving estimate for a truncated steady column; the foremost centreline parcels have characteristic speed $W(Z_h)$ instead. The steady calculation does not resolve the transient nose or define an exact sharp temperature front. A truly axisymmetric point-source plume would require different geometry and cannot use this planar profile unchanged.