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Research Papers

# Time Scale for Rapid Draining of a Surficial Lake Into the Greenland Ice Sheet

[+] Author and Article Information
James R. Rice

School of Engineering and Applied Sciences,
Department of Earth and Planetary Sciences,
Harvard University,
Cambridge, MA 02138
e-mail: rice@seas.harvard.edu

Victor C. Tsai

Seismological Laboratory,
California Institute of Technology,
e-mail: tsai@caltech.edu

Matheus C. Fernandes

School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138
e-mail: fernandes@seas.harvard.edu

John D. Platt

Department of Terrestrial Magnetism,
Carnegie Institution of Science,
Washington, DC 20015
e-mail: jplatt@dtm.ciw.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 2, 2015; final manuscript received March 11, 2015; published online June 3, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(7), 071001 (Jul 01, 2015) (8 pages) Paper No: JAM-15-1063; doi: 10.1115/1.4030325 History: Received February 02, 2015; Revised March 11, 2015; Online June 03, 2015

## Abstract

A 2008 report by Das et al. documented the rapid drainage during summer 2006 of a supraglacial lake, of approximately $44×106$ m3, into the Greenland ice sheet over a time scale moderately longer than 1 hr. The lake had been instrumented to record the time-dependent fall of water level and the uplift of the ice nearby. Liquid water, denser than ice, was presumed to have descended through the sheet along a crevasse system and spread along the bed as a hydraulic facture. The event led two of the present authors to initiate modeling studies on such natural hydraulic fractures. Building on results of those studies, we attempt to better explain the time evolution of such a drainage event. We find that the estimated time has a strong dependence on how much a pre-existing crack/crevasse system, acting as a feeder channel to the bed, has opened by slow creep prior to the time at which a basal hydraulic fracture nucleates. We quantify the process and identify appropriate parameter ranges, particularly of the average temperature of the ice beneath the lake (important for the slow creep opening of the crevasse). We show that average ice temperatures 5–7  °C below melting allow such rapid drainage on a time scale which agrees well with the 2006 observations.

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## References

Das, S. B., Joughin, I., Behn, M. D., Howat, I. M., King, M. A., Lizarralde, D., and Bhatia, M. P., 2008, “Fracture Propagation to the Base of the Greenland Ice Sheet During Supraglacial Lake Drainage,” Science, 320(5877), pp. 778–781. [PubMed]
Tsai, V. C., and Rice, J. R., 2010, “A Model for Turbulent Hydraulic Fracture and Application to Crack Propagation at Glacier Beds,” J. Geophys. Res., 115(F3), p. F03007.
Tsai, V. C., and Rice, J. R., 2012, “Modeling Turbulent Hydraulic Fracture Near a Free Surface,” ASME J. Appl. Mech., 79(3), p. 031003.
Adhikari, S., and Tsai, V. C., 2014, “A Model for Subglacial Flooding Along a Pre-Existing Hydrological Network During the Rapid Drainage of Supraglacial Lakes,” American Geophysical Union (AGU) Fall Meeting, San Francisco, Dec. 15–19, Abstract No. C33A-0355.
Weertman, J., 1973, “Can a Water-Filled Crevasse Reach the Bottom Surface of a Glacier?” International Association of Hydrologic Sciences, Wallingford, UK, IAHS Paper No. 095 0139, pp. 139–145.
Tinkler, K., and Marsh, J., 2006, “Niagara River,” The Canadian Encyclopedia (epub), available at:
Cuffey, K., and Patterson, W., 2010, The Physics of Glaciers, 4th ed., Butterworth-Heinemann, Burlington, MA.
Needleman, A., and Rice, J. R., 1980, “Plastic Creep Flow Effects in the Diffusive Cavitation of Grain Boundaries,” Acta Metall., 28(10), pp. 1315–1332.
Bower, A. F., Fleck, N. A., Needleman, A., and Ogbanna, N., 1993, “Indentation of a Power Law Creeping Solid,” Proc. R. Soc. London, Ser. A, 441(911), pp. 97–124.
Erdogan, F., Gupta, G. D., and Cook, T. S., 1973, “Numerical Solution of Singular Integral Equations,” Methods of Analysis and Solutions of Crack Problems, Recent Developments in Fracture Mechanics, G. C.Sih, ed., Noordhoff International Publications, Leyden, The Netherlands, pp. 368–425.
Viesca, R. C., and Rice, J. R., 2011, “Elastic Reciprocity and Symmetry Constraints on the Stress Field Due to a Surface-Parallel Distribution of Dislocations,” J. Mech. Phys. Solids, 59(4), pp. 753–757.
Garagash, D. I., and Detournay, E., 2005, “Plane-Strain Propagation of a Fluid-Driven Fracture: Small Toughness Solution,” ASME J. Appl. Mech., 72(6), pp. 916–928.
Gioia, G., and Chakraborty, P., 2006, “Turbulent Friction in Rough Pipes and the Energy Spectrum of the Phenomenological Theory,” Phys. Rev. Lett., 96, p. 044502. [PubMed]
Rubin, H., and Atkinson, J., 2001, Environmental Fluid Mechanics, Marcel Dekker, New York.
Nye, J. F., 1953, “The Flow Law of Ice From Measurements in Glacier Tunnels, Laboratory Experiments and the Jungfraufirn Borehole Experiment,” Proc. R. Soc. London, Ser. A, 219(1139), pp. 477–489.
Hutchinson, J., 1968, “Singular Behaviour at the End of a Tensile Crack in a Hardening Material,” J. Mech. Phys. Solids, 16(1), pp. 13–31.
Rice, J. R., and Rosengren, G., 1968, “Plane Strain Deformation Near a Crack Tip in a Power-Law Hardening Material,” J. Mech. Phys. Solids, 16(1), pp. 1–12.
Adhikari, S., and Tsai, V. C., 2015, “A Model for Subglacial Flooding Through a Preexisting Hydrological Network During the Rapid Drainage of Supraglacial Lakes,” J. Geophys. Res.: Earth Surface, 120(epub).

## Figures

Fig. 1

Sketch illustrating early October 2006 synthetic aperture radar image overlaid with the NASA moderate resolution imaging spectroradiometer (MODIS) image, showing the lake extent (blue) on July 29, 2006. This figure was redrawn here to approximately duplicate the features in Das et al. [1]. A GPS station measuring the ice displacement was located ~1/4 km from the lake shore. Hobo instruments located on the lakebed measured fluid pressure, from which lake depth versus time was inferred.

Fig. 2

Data from the 2006 lake drainage event, reproduced from Das et al. [1] with additional labeling by the authors: Falling lake level versus time, inferred from water-pressure loggers Hobo1 and Hobo2 (Fig. 1) is shown by the square symbols and by the curve starting along the upper left vertical axis and initially passing through those symbols, with values of the change in lake level being marked along the right vertical axis. The dashed continuation of that curve is a linear fit to the last two lake-level measurements before both loggers were left dry, and it suggests that the lake drained completely prior to ∼17:40 hours. Uplift Zrel at the GPS site (Fig. 1), acquired with 5-min temporal resolution, is shown by the curve starting low along the left vertical axis, and the uplift rate Zrel/dt, peaking around 17:00 hours, is also shown; values for both are given along the left vertical axis.

Fig. 3

Schematic of subglacial drainage system showing the vertical influx Qvert(t) from the lake through the crack–crevasse system feeder channel, and the resulting water injection along the ice–bed interface. The feeder channel horizontal opening Δu¯ includes contributions from elastic opening, Δu¯el, linearly proportional to the current fluid pressure, and prior creep opening, Δu¯cr, which accumulated over an extended time before the rapid drainage. The ice sheet height H is much larger than the lake depth and the basal fracture opening h(x, t). Additionally, the lake diameter is also significantly smaller than the ultimate horizontal spread of 2L(t) of the basal hydraulic fracture.

Fig. 4

Dependence of flow rate on inlet pressure for several values of the creep parameter C. In the model, the flow begins at pinlet=ρgH (hydrostatic pressure), but then the pressure drops as flow rate develops, until the pressure pinlet falls to the ice overburden pressure ρicegH.

Fig. 5

A plot showing how the basal fracture length, inlet pressure, conduit opening, and water flux into the basal fracture evolve for a range of values of C between 0 and 2.0. Dots at the ends of the curves mark complete drainage of the lake. These results were produced using the parameters given in Table 1. We see that the flux into the fracture increases with C, leading to more rapid growth of the basal fracture. The average flux of 8700 m3/s inferred in Ref. [1] is plotted as a dashed line. Our results also show that inlet pressures quickly fall from the initial hydrostatic value to close to σo and this is accompanied by elastic closing of the conduit opening.

Fig. 6

Data from the 2006 lake drainage event reproduced from Das et al. [1] with an additional curve showing our model prediction for the parameters are given in Table 1 and C = 2.0, meaning that the creep opening is twice what would be the initial elastic opening of the crack–crevasse system if subjected to hydrostatic water pressure

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