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\begin{document}
\title{Auxiliary material for: \\Constraints on the lake volume required
for hydro-fracture through ice sheets}
\author{M J Krawczynski, M D Behn, S B Das, I Joughin
\date{}}
\maketitle
\section{Numerical Approach}
The first question we address when discussing supraglacial lake
drainage is whether there are favorable conditions for crack
formation. We use the stress intensity factor at the crack tip to
determine whether critical crack propagation will occur for a given
set of parameters, and calculate the crack length if there is
propagation. The stress intensity factor, $K{}_{tot}$, at the tip of
the crack is a linear combination of the individual stress
intensities:\begin{equation} K_{tot}=K_{T}+K_{I}+K_{W}\end{equation}
where the terms on the right hand side of the equation are expressed
as:\begin{equation} K_{T,I,W}=\sigma_{t,i,w}\sqrt{\pi\cdot
z}\end{equation} Thus, $K_{tot}$ is a function of the deviatoric
(longitudinal) stress $\sigma_{t}$ (assumed to be independent of z),
the ice overburden pressure $\sigma_{i}=-\rho_{i}gz$, as well as the
added stress of the water filling a crack
$\sigma_{w}=\rho_{w}g[z-d_{w}]$. Here, $z$ is the crack depth,
$\rho$is density, and $d_{w}$is the depth to the top of the water in
the crack. When $K_{tot}$is greater than the fracture toughness of
the ice, the crack will critically propagate. For simplicity we use
a single value for the fracture toughness of 0.1 MPa
[\textit{Hooke}, 2005], which strictly speaking is a function of the
temperature. However, we feel this is a reasonable approximation, as
the fracture toughness within the bulk of the ice sheet will not
vary greatly.
Because of the dependence of equation (1) on $z$, critical crack propagation
requires an initial (small) crack to be present in order for $K_{tot}$
to exceed the fracture toughness. We have calculated the depth an
initial crack must reach before it starts to critically propagate
for the water-filled condition as a function of the differential stress
which varies widely in the ice sheet from tensile to compressive.
We find this initial depth is typically less tha 1 m, but under cases
of neutral differential stress, or slight compression, initial cracks
of only 4-7 m are necessary (Figure S1). This initial flaw
length is frequently observed in many regions of the Greenland Ice
Sheet in the form of dry crevasses. These dry crevasses may be created
far from supraglacial lakes in areas of overall tension, and once
advected into the lake basin may serve as the initial cracks needed
for water-filled propagation.
The depth of a crack can then be determined as a function of the
fracture toughness, water content, and the longitudinal stress. This
crack length is unbounded in the case of water-filled cracks because
$K_{W}$ and $K_{I}$ are of opposite signs, and $K_{W}$ is always
greater than $K_{I}$ (Figure S2). The length of the crack ($z$) is
then used to calculate the opening geometry of an edge crack after
\textit{Weertman} [1996]:
\begin{multline}
D(y)=\frac{2\alpha\sigma}{\mu}\sqrt{z^{2}-y^{2}}+\frac{2\alpha\rho_{i}g}{\pi\mu}z\sqrt{z^{2}-y^{2}}-\frac{2\alpha\rho_{w}g}{\pi\mu}\sqrt{z^{2}-d_{w}^{2}}\sqrt{z^{2}-y^{2}}- \\
2\alpha\rho_{i}g\frac{y^{2}}{2\pi\mu}\ln\left(\frac{z+\sqrt{z^{2}-y^{2}}}{z-\sqrt{z^{2}-y^{2}}}\right)+2\alpha\rho_{w}g\frac{y^{2}-d_{w}^{2}}{2\pi\mu}\ln\left|\frac{\sqrt{z^{2}-d_{w}^{2}}+\sqrt{z^{2}-y^{2}}}{\sqrt{z^{2}-d_{w}^{2}}-\sqrt{z^{2}-y^{2}}}\right|-\\
2\alpha\rho_{w}g\frac{d_{w}y}{\pi\mu}\ln\left|\frac{d_{w}\sqrt{z^{2}-y^{2}}+y\sqrt{z^{2}-d_{w}^{2}}}{d_{w}\sqrt{z^{2}-y^{2}}-y\sqrt{z^{2}-d_{w}^{2}}}\right|+
2\alpha\rho_{w}g\frac{d_{w}^{2}}{\pi\mu}\ln\left|\frac{\sqrt{z^{2}-y^{2}}+\sqrt{z^{2}-d_{w}^{2}}}{\sqrt{z^{2}-y^{2}}-\sqrt{z^{2}-d_{w}^{2}}}\right|
\end{multline}
where stress, $\sigma$, is expressed as:
\begin{equation}
\sigma=\sigma_{t}-\frac{2\rho_{i}gz}{\pi}-\rho_{w}gd_{w}+\frac{2}{\pi}\rho_{w}gd_{w}\arcsin\left(\frac{d_{w}}{z}\right)+\frac{2\rho_{w}g}{\pi}\sqrt{z^{2}-d_{w}^{2}}
\end{equation}
The derivation of these equations is outlined by \textit{Weertman}
[1973, 1996]. $D(y)$ is the displacement of one side of the crack
with respect to the center line as a function of the depth, $y$.
$\alpha$ is $(1-\nu)$, where $\nu$ is Poisson\textquoteright{}s
ratio and a value of 0.3 is used for ice. The depth ($d_{w}$) to the
top of the water in the crack as measured from the surface can be
varied, and $d_{w}=0$ for water filled cracks. The shear modulus for
ice ($\mu$) has been found to vary based on loading rates, grain
size, and temperature, none of which are constrained by our model.
We use 3 different values across the known range for the shear
moduli (3.9, 1.5, and 0.32 GPa) to determine the sensitivity in our
model to this variable [\textit{Vaughn}, 1995]. The density of water
($\rho_{w}$) and ice ($\rho_{i}$) are 1000 and 920
kg/m$^{\text{3}}$, respectively. $g$ is the gravitational constant
taken here at 9.78 m/s$^{2}$. We ignore uncompacted snow or firn at
the surface of the ice sheet as it is likely a negligable component
for the lake bottom environment in the Greenland ablation zone. Two
example calculations are shown in Figure S3, one for the dry case
($z=d_{w}$) and one for a completely water filled case ($d_{w}=0$).
Because the general shape of a water-filled crack does not vary
significantly with depth, we approximated it as a channel with
parallel sides in the calculations of lake drainage time. The mean
opening of the crack is used for this calculation, and is determined
by dividing twice the integral of $D$ from $0$ to $z$ by $z$. The
Reynolds number for this system is calculated to be on the order of
$10^{6}$ and thus falls in the turbulent regime. We follow the
approach of \textit{White} [1974] to estimate the turbulent flux of
water through a channel with a mean opening determined as above
(Figure S4). Using the flux the drainage time can be calculated for
any lake of a known volume. If a crack is positioned underneath a
supraglacial lake and is to remain water-filled throughout its
propagation, then the cross sectional area of a conical lake must be
at least equal to the cross-sectional area of the crack. There is a
simple geometric relationship between the area of the crack and the
minimum mean lake diameter necessary to keep the crack water-filled:
\begin{equation} lake\: diameter\geq\sqrt{400\cdot A}\end{equation}
Where $A$ is the cross-sectional area of the crack, which can be
calculated by integrating equation (3) over the entire length of the
crack and multiplying it by two, to account for both sides of the
crack. Using the above equation a minimum lake size can be estimated
for a given ice sheet thickness (as shown in Figure 4). We also
calculated drainage times for turbulent pipe flow, used to similate
a moulin, which is a possible drainage mechanism for supraglacial
lakes. The moulin drainage (pipe flow) represents a volume per time
calculation, while the calculation for a crack drainage (channel
flow) is a 2-D flux. Thus, in this case we use lake volume rather
than lake area to estimate drainage time. We note that channel flow
is always faster, due to the larger frictional forces associated
with the pipe walls.
\section{Satellite Imagery}
Daily MODIS images collected throughout the 2006 melt season were
used to determine the maximum summer lake extent of 1300 lakes
across our study area. Lake boundaries were estimated by
thresholding the ratio of the blue to the other visible channels.
Lake locations were tracked through time and transient features were
discarded (e.g., single-day false detections caused by clouds).
Daily lake extent was then tabulated for each lake throughout each
summer allowing us to determine the maximum lake surface area.
Following our discussion in the main text, we prescribe an aspect
ratio (mean surface diameter:maximum depth) of 100:1 to then
calculate the water volume.
\section{Field Observations}
Although it is impossible to measure the shape and depth of many of
these crevasses at depth in the field, the observations we do have
support the validity of the Weertman model for opening geometry, as
well as our calculations of the crack volume and our conclusions
about modeling lake drainage through channel flow. Areas of the ice
sheet that had previously been submerged beneath a supraglacial lake
display a distinctive \textquoteleft{}egg-carton\textquoteright{}
like appearance on the surface, due to the melting processes that
occur at the lake bottom. We have observed many crevasses that are
now frozen closed (healed) running across the ice sheet through
areas displaying these egg-carton textures (Figure S5), indicating
that they were once at the bottom of a lake. It is easy to see these
crevasses because the ice filling them has a distinctly bluer color
than surrounding lake bottom material. It is common for these
crevasses to be longer than 1 km in their horizontal dimension, and
they maintain a constant opening width across their entire length.
Healed crevasses also typically have thin lines of bubbles
indicating that they re-froze from the outside in.
Ice canyons, which are erosional features formed from surface meltwater
flow, can be tens of meters deep, and provide 3-D observations of
these healed crevasses (Figure S6). In these cases we see
that the dilation of the crevasses remains remarkably constant in
the vertical as well as horizontal dimension, consistent with a model
in which the crack was water filled during its formation (Figure
S3). The \textquoteleft{}healing\textquoteright{} process for these
crevasses is ideal because it prevents them from being subjected to
viscous flow and deformation processes, and \textquoteleft{}fossilizes\textquoteright{}
the shape the crevasse has when it initially forms (i.e., parallel
sides, constant opening width, etc.).
\\
\\
\\
\textbf{References}
\\
Hooke, R. L. (2005), \textit{Principles of Glacier Mechanics}, 2nd
ed., Cambridge Univ. Press, New York.
Vaughan, D. G. (1995), Tidal flexure at ice shelf margins,
\textit{J. Geophys. Res.}, \textit{100}, 6213--6224.
Weertman, J. (1973), Can a water-filled crevasse reach the bottom
surface of a glacier? Symposium on the Hydrology of glaciers: Water
within glaciers II, \textit{Int. Assoc. Sci. Hydrol.}, \textit{95},
139--145.
Weertman, J. (1996), \textit{Dislocation Based Fracture Mechanics},
World Sci., River Edge, N. J.
White, F. M. (1974), \textit{Viscous Fluid Flow}, McGraw-Hill, New
York.
\end{document}