= Solution
Far downslope let the typical height be $H(X)$ and the cross-slope half-width be $W(X)$. Slenderness gives $W\ll X$ and slowly varying longitudinal thickness. In a steady <porous gravity current>, the longitudinal spreading term is smaller than <advection> by the ratio $H/X$, while longitudinal <diffusion> relative to transverse <diffusion> is of order $(W/X)^2$. Thus the leading balance is
$$
\partial_Xh^\beta=\partial_Y(h^\beta h_Y).
$$
Integrating across the finite current, with zero flux through its dry side edges, makes the leading downslope <volume flux> constant:
$$
\int h^\beta\,dY=\text{constant},\qquad H^\beta W\sim\text{constant}.
$$
Balancing the two differential terms gives
$$
\frac{H^\beta}{X}\sim\frac{H^{\beta+1}}{W^2},\qquad W^2\sim HX.
$$
Eliminate $H\propto W^{-1/\beta}$ to find the <far-downslope width of a constant-flux porous current>:
$$
\boxed{Y_N(X)\propto X^\gamma,\qquad
\gamma=\frac{\beta}{2\beta+1}.}
$$
The typical height is correspondingly $H\propto X^{-1/(2\beta+1)}$. For every $\beta>0$, the width exponent is less than one half, so $W/X\to0$ and $H/X\to0$, consistent with the discarded longitudinal terms. The <porosity> exponent $\alpha$ drops out because it enters storage, which has zero time derivative in this steady problem.
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