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Auxiliary Material
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“Break-up of the Larsen B Ice Shelf Triggered by Chain-Reaction Drainage of
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Supraglacial Lakes”
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Alison F. Banwell, Douglas R. MacAyeal and Olga V. Sergienko
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Methods
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1. Exact analytic solution for elastic plate subject to disk-shaped load
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We use an azimuthally symmetric solution valid for r>0 to the thin-elastic-plate flexure
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equation (also known as the Kirchoff-Love equation, but modified to account for
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buoyancy associated with ocean water below the thin plate) in which meltwater loads, or
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anti-loads associated with dolines (both of uniform load area density), are confined
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within a region r ≤ R using polar coordinates r, , and where R is the radius of the lake or
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doline. The vertical displacement of the elastic plate, (r), for 0 ≤ r ≤ ∞, is expressed in
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terms of Kelvin-Bessel functions (as derived by Lambeck and Nakiboglu [1980]), and is
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displayed in MacAyeal and Sergienko [2013].
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To illustrate the basic features of the solution, we plot in Figure S1 the vertical deflection
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resulting from the drainage of a lake 1 m in depth and 500 m in radius, (r), radial stress,
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Trr, and azimuthal stress, T, and the von-Mises stress (TvM), all evaluated at the upper
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surface of the ice shelf, TvM = (Trr2 + T2 – TrrT1/2, as functions of any given distance, r,
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from the lake center (given both in km, and expressed in units of the intrinsic flexural
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length scale L = {D/(sw g)}1/4 ~ 923 m, where D = EH3/[12(1-2)] is the flexural rigidity,
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sw = 1028 kg m-3 is the density of seawater, g = 9.81 m s-2 is the acceleration of gravity,
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E = 10 GPa is Young’s modulus,  = 0.3 is the Poisson ratio, and H = 200 m is ice
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thickness). The black line in the left panel of Figure S1 depicts the uplift associated with
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hydrostatic rebound of the missing load [MacAyeal and Sergienko, 2013]. A photograph
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of an uplifted doline on the George VI Ice Shelf surrounded by a down-warped moat is
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shown in Figure 1 of MacAyeal and Sergienko [2013].
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Figure S1. Analytic solution for axisymmetric (disk shaped) lake drainage-induced
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unloading. a) Radial stress, Trr, azimuthal stress, T and vertical deflection, ,
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introduced by drainage of a circular lake of radius 500 m and depth 1 m (note separate
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vertical scale, in cm, for ) as evaluated at the upper surface of the ice shelf. The two
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stress components vary linearly through the vertical dimension of the ice shelf, are zero at
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the neutral surface (central plane of the ice shelf half way between surface and base) and
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are equal but opposite in sign at the upper and lower surfaces of the ice shelf. Arrow and
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dot signify location of maximum compressive stress at the surface of the ice shelf, and
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maximum tensile stress at the base of the ice shelf, within the moat area. b) Von-Mises
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stress, TvM, associated with the stress conditions at the surface of the ice shelf. Shaded
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regions in both (a, b) denote the footprint of the lake.
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2. Replacement of exact shapes of observed lakes with circular disk loads
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Here, we justify the replacement of exact shapes of observed lakes (with equally
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distributed loads) with circular disk loads. In Figure S2, we display the geometry of the
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lakes and dolines observed in February 2000 (see Glasser and Scambos [2008] and
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Banwell et al. [in press]). Comparison of the observed lakes with their circular
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representations suggests that relatively few lakes are not well represented by circles.
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These few lakes tend to be the larger ones that are also long and narrow, with long
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dimensions aligned with the direction of ice flow, likely indicating the influence of a
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suture zone within the ice-thickness field of the ice shelf.
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To display the performance impacts of representing the lakes as circles, we use
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COMSOL® to numerically solve the ice-shelf elastic flexure equations for an idealized
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ice-shelf domain that is rectangular in which we arbitrarily place a given lake (or its
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circular representation) in the center. The resulting solutions for displacement, , and
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von-Mises stress, TvM, under the assumptions of either the observed lake geometry or
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circular lake geometry are shown in Figures S3 and S4. Figure S3 depicts the
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comparison between results of the exact observed geometry with the results of the
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circular representation for a typical lake. We regard the comparison to be sufficiently
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similar as to affirm our decision to simplify the lakes on the LBIS as circles so that the
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analytic solution could be used. For comparison, and to motivate future investigation, we
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show a worst-case performance of the circular representation of an observed lake that is
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long and narrow in Figure S4. Despite the relatively poor comparison of the two
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solutions in Figure S4, the relative rarity of large, long lakes on the LBIS motivated our
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decision to stick with circular lake representations.
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A comparison of the numerical and the exact analytic solutions for a circular lake is
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provided in MacAyeal and Sergienko [2013].
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Figure S2. Observed lakes (as observed in February 2000, see Glasser and Scambos
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[2008]) and their representation using circles of equal area. a) 2758 lakes and dolines on
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the LBIS. b) Representation of the lakes in a by circles. c) Close-up of lakes represented
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as circles in region outlined by box in (b). d) Close-up of lakes and dolines in region
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outlined by box in (a).
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Figure S3. Numerical demonstration of the effect of representing arbitrary lake shapes
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as circles—a typical lake showing acceptable performance of the circular representation.
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a) The deflection, , for a typical lake on the LBIS assuming a uniform lake depth of 5
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m. b) The von-Mises stress, TvM, evaluated at the ice-shelf surface associated with the
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solution shown in (a). c) As in (b), but for a circular representation of the lake. d) As in
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(a), but for a circular representation of the lake. Gray shading indicates the footprint of
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the lake or circular representation of the lake. Color bar for (a) also applies to (d), color
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bar for (b) also applies to (c). Only areas where TvM >70 kPa, are colored in panels (b, c).
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Figure S4. Numerical demonstration of the effect of representing arbitrary lake shapes as
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circles—an atypical lake showing worst-case performance of the circular representation.
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a) The deflection, , for an atypical lake (long and narrow) on the LBIS assuming a
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uniform lake depth of 5 m. b) The von-Mises stress, TvM, evaluated at the ice shelf
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surface associated with the solution shown in (a). c) As in (b), but for a circular
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representation of the lake. d) As in (a), but for a circular representation of the lake. Grey
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shading indicates the footprint of the lake or circular representation of the lake. Color bar
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for (a) also applies to (d). Color bar for (b) also applies to (c). Only areas where TvM >70
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kPa, are colored in panels (b, c).