= Solution
Take $A>0$ for the repulsive molecular interaction, $p_0$ as the common reference pressure, and let $T_i=T_m-\delta T$ be the ice-film <temperature>. All ratios involving $T_m$ use its absolute value, approximately $273.15\,\mathrm K$, rather than the numerical Celsius value zero. The <premelting> film permits water to reach and freeze at the bottom of the <ice> while the upper surface is displaced upward without <freezing> there.
For equal phase <mass densities>, equality of the solid and liquid <chemical potentials> gives the <Clapeyron pressure relation for equal-density phases>. Indeed, $d\mu=-s\,dT+dp/\rho$ and $s_l-s_s=L/T_m$, so expansion around coexistence at $(T_m,p_0)$ yields
$$
0=\mu_s-\mu_l=\frac{L}{T_m}(T_i-T_m)+\frac{p_s-p_l}{\rho},
\qquad
p_s-p_l=\frac{\rho L}{T_m}\delta T.
$$
This is a pressure difference between phases, not a same-pressure Clausius-Clapeyron slope. The molecular <disjoining pressure> supports this difference:
$$
p_T=p_s-p_l=\frac{A}{6\pi a^3}.
$$
Treat the normal solid load as its effective pressure in this planar model. The gravitational load corresponding to <hydrostatic pressure> is $p_s=p_0+\rho gh$, so $p_l=p_0+\rho gh-p_T$. Refer the bath pressure to the sheet's upper gravitational datum, or neglect the sheet-scale hydrostatic head. The <Darcy flow law> then gives the upward supply and the <ice> growth rate
$$
\dot h=\frac{\Pi}{\mu d}(p_0-p_l)
=\frac{\Pi}{\mu d}(p_T-\rho gh).
$$
Here $\Pi$ is <permeability of a porous medium>, $\mu$ is <dynamic viscosity>, and equal densities identify supplied liquid volume with added <ice> volume to leading order in $a/h$.
To recover the printed law, use the usual <hydraulic control of frost heave> idealization: the sheet's top is at $T_m$, the liquid film and <ice> have a common <thermal conductivity> $K$, and the flow is slow enough that latent-heat production is small compared with the conductive heat passing through the layer. In addition to a quasi-steady <ice> <temperature> field, this needs
$$
\frac{\rho L|\dot h|h}{K\Delta T}\ll1.
$$
The leading <heat flux> is consequently continuous through the film and <ice>. Their thermal resistances give
$$
\frac{\delta T}{a}=\frac{\Delta T-\delta T}{h},
\qquad
\delta T=\frac{a}{h+a}\Delta T\simeq\frac ah\Delta T.
$$
Let $B_T=\rho L\Delta T/T_m$, a pressure scale. The <thermomolecular pressure> and the molecular law then give
$$
\frac{A}{6\pi a^3}\simeq B_T\frac ah,\qquad
a\simeq\left(\frac{Ah}{6\pi B_T}\right)^{1/4},\qquad
p_T\simeq B_T^{3/4}\left(\frac{A}{6\pi h^3}\right)^{1/4}.
$$
Substitute this pressure into the <Darcy flow law>:
$$
\boxed{\dot h=\frac{\Pi}{\mu d}
\left[\left(\frac{\rho L\Delta T}{T_m}\right)^{3/4}
\left(\frac{A}{6\pi h^3}\right)^{1/4}-\rho gh\right]}.
$$
The first term draws water toward the undercooled <ice>; the increasing gravitational load eventually cancels it.
Quasi-steady <temperature> alone is not sufficient to fix this exact prefactor. With film conductivity $K_l$, <ice> conductivity $K_s$, and nonnegligible latent heat, the appropriate additional balance is
$$
\rho L\dot h=\frac{K_s(\Delta T-\delta T)}h-\frac{K_l\delta T}{a},
\qquad
\delta T=\frac{T_mp_T}{\rho L},\qquad
a=\left(\frac{A}{6\pi p_T}\right)^{1/3}.
$$
Together with the Darcy equation, this is the implicit growth model. Even in the slow-flow limit, a conductivity ratio $K_s/K_l=r$ changes the leading driving pressure by $r^{3/4}$. Taking $r=2$ is a counterexample to obtaining the printed prefactor from quasi-steadiness alone. Thus the boxed equation is the intended equal-conductivity, thin-film, hydraulically limited model, rather than a consequence of only the stated quasi-steady assumption. This distinction is the <heat-balance correction to a premelted-film growth model>.
For that reduced equation define
$$
F=B_T^{3/4}\left(\frac{A}{6\pi}\right)^{1/4},\qquad
h_*=\left(\frac{F}{\rho g}\right)^{4/7},\qquad
t_*=\frac{\mu d}{\Pi\rho g},\qquad
\eta=\frac h{h_*},\quad\tau=\frac t{t_*}.
$$
The scale $h_*$ balances $Fh_*^{-3/4}$ against $\rho gh_*$; $F$ has units of pressure times length to the power $3/4$, so $h_*$ is a length and $t_*$ is a time. They yield
$$
\boxed{\frac{d\eta}{d\tau}=\eta^{-3/4}-\eta}.
$$
Set $u=\eta^{7/4}$. The equation becomes linear:
$$
\frac{du}{d\tau}=\frac74(1-u).
$$
For initial thickness $\eta(0)=\eta_0\ge0$, the <explicit solution of gravity-limited frost heave> is
$$
\boxed{\eta(\tau)=
\left[1+\bigl(\eta_0^{7/4}-1\bigr)e^{-7\tau/4}\right]^{4/7}}.
$$
The usual zero-initial-thickness sketch uses
$$
\boxed{\eta(\tau)=\left(1-e^{-7\tau/4}\right)^{4/7}}.
$$
Its early and late behaviour are
$$
\eta(\tau)\sim\left(\frac{7\tau}{4}\right)^{4/7}
\quad(\tau\downarrow0),\qquad
\eta(\tau)=1-\frac47e^{-7\tau/4}+O(e^{-7\tau/2})
\quad(\tau\to\infty).
$$
For $0<\eta<1$, the derivative is positive and $\eta''=(-\tfrac34\eta^{-7/4}-1)\eta'<0$, so the curve is increasing and concave downward. It approaches the stable thickness $h_*$ exponentially. Initial thickness above $h_*$ instead relaxes downward; at $h_*$ it is stationary.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-71-premelting-growth.png]
{title=Exact reduced premelting growth from zero thickness with the early power law and late exponential approach to equilibrium}
{height=540}
The singular slope at zero belongs to the formal reduced solution. Since $a/h\propto h^{-3/4}$, the thin-film assumption eventually fails as $h\to0$. The early power law therefore describes an intermediate continuum regime after any microscopic startup, not arbitrarily early physical times. A finite positive $\eta_0$ supplies a regular initial condition when that regime begins. \b[Gravity limits the ultimate thickness; molecular suction and the Darcy resistance set the growth toward it.]
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