## The Siple Dome model

Siple Dome is a well studied divide. As it is the site of a recent deep-drilling project by the United States Antarctic Program, Pettit (2003) uses available ice-core data and deformation measurements to solve an inverse problem for flow-law parameters that best describe the flow at Siple Dome utilizing an ice-flow model developed by H.P. Jacobson and T. Thorsteinsson. I focus on four flow-law parameters: the crossover stress (k), which I assume is spatially constant, and a three-layer isotropic enhancement parameterization, describing Holocene ice (Ej, ice-age ice

(E2), and the deep recrystallized ice (E3), where E. = —— is derived

' dpn from Equation (3) assuming A0nl is the suggested value from Paterson (1994, p. 91, based on several empirical studies of ice flow). These enhancement factors, E., are for isotropic enhancement only. The model accounts for anisotropic flow explicitly using an analytical formulation for anisotropic deformation developed by Thorsteinsson (2001). Pettit (2003) describes the k=o.oibar Qcl iooo 9oo soo "j? 7oo w eoo £ 5oo f4oo

Figure 58.1 Modelled steady-state isochrons for an idealized divide with a thickness of 1000 m and accumulation rate of 0.1cmyr-1, resembling Siple Dome, West Antarctica. The crossover stress used in the model increases from left to right. The importance of the linear term also increases from left to right, as reflected in the diminishing size of the isochron arch. W,,, is the non-dimensional ratio Tchlr/fc. (From Pettit & Waddington, 2003, reprinted from the Journal of Glaciology with permission of the International Glaciological Society.)

k=o.4bar Q char<1

Linear

Figure 58.1 Modelled steady-state isochrons for an idealized divide with a thickness of 1000 m and accumulation rate of 0.1cmyr-1, resembling Siple Dome, West Antarctica. The crossover stress used in the model increases from left to right. The importance of the linear term also increases from left to right, as reflected in the diminishing size of the isochron arch. W,,, is the non-dimensional ratio Tchlr/fc. (From Pettit & Waddington, 2003, reprinted from the Journal of Glaciology with permission of the International Glaciological Society.)

Figure 58.2 Horizontal (top panel) and vertical (bottom panel) velocity fields from the best-fitting model assuming steady state. For both panels, Bindschadler Ice Stream is to the right and Kamb Ice Stream is to the left. The dashed lines are velocity contours.

Distance from divide (km)

Figure 58.2 Horizontal (top panel) and vertical (bottom panel) velocity fields from the best-fitting model assuming steady state. For both panels, Bindschadler Ice Stream is to the right and Kamb Ice Stream is to the left. The dashed lines are velocity contours.

implementation of Thorsteinsson's formulation in this ice-flow model. The inputs to the model are the surface and bed geometry (from GPS and radar, respectively, Nereson et al., 1998b), the temperature (from hot-water drilled borehole logging; H. Engelhardt, personal communication), and the crystal fabric profile (from borehole sonic log; G. Lamorey, personal communication).

In order to invert for the four unknown flow-law parameters, I compare the model results to measurements of the vertical strain-rate profile at the divide and on the flank (e.g. Zumberge et al., 2002; Elsberg et al., 2004). The vertical strain-rate profile at the divide is sensitive to the relative magnitude of the linear term, whereas that at the flank is not. I chose to compare modelled and measured instantaneous deformation rates rather than internal layer shapes, as the internal layer shapes are a function of the deformational history of the divide.

Our model fits the data best when the crossover stress, k, is 0.22 bar. As the characteristic stress at Siple Dome is about 0.2 bar, I consider it to be a transitional divide, which means that the linear and non-linear terms contribute roughly equally to the overall deformation-rate pattern. The best-fitting enhancement factors are 1.3 for Holocene ice, 0.06 for ice-age ice, and 0.2 for the recrys-tallized ice near the bed. As the enhancement factor is defined relative to the suggested softness parameter from Paterson (1994), our result of 1.3 for Holocene ice at Siple Dome suggests that our model produces deformation rates close to those expected for clean Holocene ice under similar conditions. The enhancement factors for the two deeper layers are much lower than I expected.

This remains a puzzle, although Thorsteinsson et al. (1999), who also attempted to separate the softness due to anisotropy from that due to other ice properties, found a similar result at Dye 3 in Greenland. Possibly a poorly understood rate-limiting process such as grain-boundary migration increases the stiffness of the ice.

Using these best-fitting flow-law parameters, Fig. 58.2 shows the horizontal- and vertical-velocity fields for Siple Dome. The horizontal-velocity field shows a sharp change in gradient about 300 m above the bed. This shift corresponds in depth to a shift towards tighter crystal fabric alignment and has been dated to near the transition from ice-age to Holocene ice (Taylor et al., 2004). The tight vertically oriented crystal fabric of the ice-age ice concentrates the shear strain, as anisotropic ice is easy to deform along the basal planes.

Our goal in this case study was to determine the relative importance of linear deformation mechanisms in low-stress regions, particularly near ice divides. Toward this goal, I suggested a two-term flow law where the crossover stress, k, is a rheological property of ice. We found that k is about 0.2 bar for the temperature and grain size of the ice within Siple Dome. Divides worldwide range in characteristic stress from 0.1 bar (Valkyrie Dome, site of Dome Fuji Station) to near 0.4 bar (Quelccaya Ice Cap, Peru). When modelling flow at many divides, therefore, including a linear term in the flow law is likely to be important. Our model of Siple Dome also implies that other processes, in particular crystal-fabric anisotropy, are equally important to successfully capturing the characteristic behaviour of an ice divide.

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