Marine ice-sheet dynamics. Part 1. The case of rapid sliding

Abstract

Marine ice sheets are continental ice masses resting on bedrock below sea level. Their dynamics are similar to those of land-based ice sheets except that they must couple with the surrounding floating ice shelves at the grounding line, where the ice reaches a critical flotation thickness. In order to predict the evolution of the grounding line as a free boundary, two boundary conditions are required for the diffusion equation describing the evolution of the grounded-ice thickness. By analogy with Stefan problems, one of these conditions imposes a prescribed ice thickness at the grounding line and arises from the fact that the ice becomes afloat. The other condition must be determined by coupling the ice sheet to the surrounding ice shelves. Here we employ matched asymptotic expansions to study the transition from ice-sheet to ice-shelf flow for the case of rapidly sliding ice sheets. Our principal results are that the ice flux at the grounding line in a two-dimensional ice sheet is an increasing function of the depth of the sea floor there, and that ice thicknesses at the grounding line must be small compared with ice thicknesses inland. These results indicate that marine ice sheets have a discrete set of steady surface profiles (if they have any at all) and that the stability of these steady profiles depends on the slope of the sea floor at the grounding line.

Similitude of ice dynamics against scaling of geometry and physical parameters

“…We thus do not neglect further terms in the SSA stress balance by comparing orders of magnitudes of acting stresses (as done in, e.g., Schoof, 2007a) but consider the general case in which both membrane and basal stresses balance the Figure 8. Scaling of response time τ , surface mass balance δ and ice softness ζ under the assumption of Vialov-type geometric scaling (β = α 1/2 ) and identical basal friction (γ = 1).…”

“…9 and 19). In other studies the scales for ice-sheet thickness, length and time are used to express the friction parameter C, resulting in a dimensionless SSA stress balance that is characterized by a single scaling parameter (often denoted as , Schoof, 2007a;Tsai et al, 2015). In the present study we consider C to be an independent parameter/scale and thus obtain two scaling parameters θ and φ in the SSA stress balance.…”

“…A value of m = 1/n = 1/3 is commonly assumed to represent sliding over rough bed (Schoof, 2007a;Joughin et al, 2009;Cuffey and Paterson, 2010). The evolution equation for the ice thickness, i.e., the ice thickness equation (ITE), which results out of mass conservation (Greve and Blatter, 2009) reads…”

“…(12) and (13) we obtain contour maps for the ratios τ , ζ and δ in the α-β phase space (Fig. 7a-c for the common choice of an exponent value of m = 1/n with n = 3; e.g., Schoof, 2007a;Greve and Blatter, 2009;Cuffey and Paterson, 2010). Therein the blue and red areas correspond to the regimes of an increasing and decreasing parameter value under geometric scaling, respectively, which are separated by a white curve along which the considered parameter remains the same.…”

“…The non-dimensionalized form of the SSA itself and the involved dimensionless coefficients that result from the introduction of typical scales for, e.g., ice-sheet thickness and velocity, have been used to consider asymptotic limits of SSA ice flow in previous work (Schoof, 2007a;Dupont and Alley, 2005;Tsai et al, 2015;Haseloff et al, 2015). In the present study we utilize these coefficients to derive ice-sheet scaling laws for the geometry, response time and other physical icesheet parameters, a step that, to our knowledge, has not been taken before.…”