Outburst Flooding Under Ice Sheets as Turbulently Driven Hydraulic Fracture

The origin of unsteady glacier motions, an area to which Robert M. McMeeking contributed significantly [1,2], is of interest for assessing stability of major ice sheets (Greenland, Antarctica). Meltwater and its pressure p at the bed of glaciers are known to have major influence on flow. Our work [3,4] focuses on outburst under-flooding of an ice sheet, as a means of delivering highly pressurized water to its bed, transiently with p > o (= ice overburden pressure). The process is viewed as a turbulently driven hydraulic fracture along the ice/bed interface. Such can result [5] when the glacier dams a rising lake, or from geothermal heating of a sub-glacial lake. The particular scenario we address is, instead, rapid drainage into the ice of a large surficial meltwater lake, like recently documented in Greenland [6] during mid-summer. This first involves Weertman gravitationally driven hydraulic cracking from the lake to the bed, and then rapid spreading of water along the bed, initially as a high-p sheet flow. We compare modeling to results of the Greenland study [6], for which a 0.043 km 3 lake disappeared into the ice, mostly within 1.5 hr. The lake drainage rate in that case, and the ~3 km length, parallel to the glacier surface, of a crevasse/moulin system along which drainage apparently occurred, suggest that in the rapid early phases of the underflooding the Reynolds number (based on basal fracture opening h and thickness-averaged flow speed U along it) was of order 10 6 . Accordingly, we adopted a Manning-Strickler-Nikuradse description of wall shear stress in the rough turbulent range to relate U(x,t), h(x,t), p(x,t), and the wall roughness scale k, where x is the coordinate in the direction of fracture propagation. We further assumed linear elastic response of ice and bed, and (for the large fracture propagation lengths of interest) negligible KIc . By extending studies like in [7,8] to that rough turbulent flow range, we have thus solved approximately the plane strain version of the hydraulic fracture problem when the basal crack length 2L is modest compared to ice sheet thickness H (i.e., for a crack in an unbounded body, subject to crack face pressure p o ). We outline those results here and will, in the presentation, report some preliminary improvements on them in current work to account for the range L/H of order 1 and larger, often a practically interesting one [4-6], by explicitly accounting for a nearby free surface. That adopts the approach of [9] to numerically relate the h(x,t) and p(x, t ) o distributions. To fully solve the problem for the rate of fracture propagation, and volume storage of meltwater within the fracture, the distributions must also be constrained to satisfy the governing fluid equations (analogously to [10], but for turbulent rather than laminar flow). Those fluid equations, to be solved on – L(t) 0) (hU )