= Steady friction-dominated grounding-line thickness relation
{title2=$\frac{g'}{8g}\mathcal H^5=\mathcal Q(\mathcal Q-\alpha\mathcal H^2)$}
With constant <volume flux per unit width> $Hu=q_0>0$, define $\beta=l/\lambda$, $\mathcal H=H/\beta$, and $\mathcal Q=\nu q_0/(g\beta^3)$. The friction closure gives $H_x=\alpha-q_0\nu/(\beta gH^2)$. Differentiating constant flux and applying the <unbuttressed Newtonian grounding-line stress> condition gives $q_0H_x/H+g'H^2/(8\nu)=0$. Substitution proves the displayed relation. For $\alpha\ge0$, $g'>0$ and $\mathcal Q>0$, the polynomial $(g'/8g)\mathcal H^5+\alpha\mathcal Q\mathcal H^2-\mathcal Q^2$ increases strictly on the positive half-line, begins negative and tends to infinity; hence it has exactly one positive root. Its root satisfies $\mathcal Q>\alpha\mathcal H^2$, so the thickness decreases seaward at that steady grounding line.
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