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2000 Glacier Calving: A Numerical Model of Forces in the Calving-Speed/-Depth Relation Brian Hanson

Roger Hooke University of Maine - Main, [email protected]

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Repository Citation Hanson, Brian and Hooke, Roger, "Glacier Calving: A Numerical Model of Forces in the Calving-Speed/Water-Depth Relation" (2000). Earth Science Faculty Scholarship. 40. https://digitalcommons.library.umaine.edu/ers_facpub/40

This Article is brought to you for free and open access by DigitalCommons@UMaine. It has been accepted for inclusion in Earth Science Faculty Scholarship by an authorized administrator of DigitalCommons@UMaine. For more information, please contact [email protected]. Journal of , Vo l. 46, No.153, 2000

Glacier calving: a numerical model of forces in the calving-speed/water-depth relation

Brian Hanson,1 Roger LeB. Hooke2 1Department of Geography, University of Delaware, Newark, Delaware 19716, U.S.A. 2Department of Geological Sciences, University of Maine, Orono, Maine 04469-5790, U.S.A.

ABSTRACT. Empirical data suggest that the rate of calving of grounded glaciers ter- minating in water is directly proportional to the water depth. Important controls on calv- ing may be the extent to which a calving face tends to become oversteepened by differential flow within the and the extent to which bending moments promote extru- sion and bottom crevassing at the base of a calving face. Numerical modelling suggests that the tendency to become oversteepened increases roughly linearly with water depth. In addition, extending longitudinal deviatoric stresses at the base of a calving face in- crease with water depth. These processes provide a possible physical explanation for the observed calving-rate/water-depth relation.

INTRODUCTION BACKGROUND

Glaciers ending in bodies of water lose ice by calving. Calv- In studies of Alaskan tidewater glaciers, Brown and others ing occurs both from floating ice shelves and from 01982) found a strong correlation between mean annual calv- glaciers that remain grounded essentially all the way to ing speed, uc, and the average depth of the water body along their termini. The blocks of ice thus lost range in size from the calving face, hw, but the physical reasons for this were un- individual ice separated from a calving face by clear. They also considered the possibility that uc might be along grain boundaries to tabular hun- related to accumulated strain, ice speed, ice thickness, effec- dreds of square kilometers in area that break off from ice tive , state of the tide, water and the shelves. Melting may influence the calving process by alter- volume of subglacial runoff 0Brown and others, 1982, p. C8). ing the shape of a calving face and is itself included impli- Intuitively, some dependence of calving speed on some of citly in most calculations of calving speed. The water into these parameters is likely, but they found ``no direct evidence which glaciers calve may be either salty or fresh and either that these variables need to be separated explicitly''. Meier stable or mixed by and tidal currents, all of which 01994) incorporated 14years of recent data from Columbia affect melt rates. Given these complexities, it is not surpris- Glacier into Brown and others' dataset and found that the ing that no known theoretical or empirical rule governs the new data were generally consistent with the earlier values, ex- cept for the last 2 years, when the calving rate may have been rate at which these processes, combined, result in ice loss affected by floating. from calving faces. Calving speeds of polar glaciers in West and In this paper, we limit our discussion to grounded valley and of temperate glaciers ending in fresh-water glaciers calving into tidal salt-water and inlets. For are also proportional to water depth. However, the this restricted calving environment, an empirical relation- calving speeds of the polar glaciers are about one-third of ship between calving speed and water depth has been ad- those of Alaskan glaciers in water of similar depth 0Pelto vanced, but the fundamental physical processes that and Warren, 1991), and those of temperate glaciers ending govern the calving rate are still poorly understood. Our ob- in are only about one-tenth of those of their jective herein is to propose some such processes that could Alaskan counterparts 0Funk and Ro« thlisberger,1989). Reeh account for the empirically observed approximately linear 0cited by Meier, 1993) compiled data for floating polyther- increase in calving rate with water depth. mal glaciers in Greenland and found that uc was propor- Calving of grounded glaciers is of considerable interest: tional to ice thickness, but calving rates were only about it can be responsible for icebergs that threaten shipping; it one-fifth of those of grounded temperate Alaskan glaciers. affects the way in which valley glaciers react when In contrast, Sikonia 01982) found that over time-spans of raise levels near their termini 0Funk and Ro« thlisberger, weeks to months, calving rates in an embayment in the calv- 1989; Hooke and others,1989); and if ice sheets in Greenland ing face of Columbia Glacier showed little relation to local and begin to disintegrate as a result of climatic water depth. However, he and Sikonia and Post 01979) found warming, calving will control the rate of collapse. Initially, that over these time-spans uc was directly related to water calving in Antarctica will be largely from ice shelves, but as discharge in the nearby Knik River and inversely related to these vanish, grounded ice , resembling tidewater the effective pressure at the bed.We will return to this later. glaciers, will play an increasingly important role. Hughes and Nakagawa 01989) have suggested that bend- 188 Hanson and Hooke: Glacier calving ing shear controls calving speed. Observations of a vertical uc is normally determined by measuring ui and subtracting ice face on dry on Deception showed that as the the rate of change in glacier length, dL=dt,thus: top of the face moved forward faster than the bottom, shear dL u ˆ u À : 1† zones developed along arcuate surfaces much as the pages of c i dt a book shear past one another if the binding is held fixed on 0Note that u thus encompasses all ice-loss processes at the a horizontal surface and the pages are bent to the side. Fail- c margin including, for example, melt and sublimation as well ure eventually occurs along surfaces that follow shear zones as failure of small and large ice blocks.) Because dL=dt is downward and then, near the base, veer outward toward small compared with u under most conditions, there is the face at about 45³, following a plane of maximum shear i bound to be a good correlation between u and u ,asVan stress. This type of failure is probably less important in the c i derVeen 01996) has noted. tongue of a pervasively fractured tidewater glacier. Because u is determined in this way, one may argue, as Hughes 01992), expanding on these ideas, developed a c Van der Veen 01996, p.381) appears to, that it is circular to physically based calving model, using beam theory. Inter- use the correlation between u and u as support for a physi- estingly, agreement with observation is reasonably good. i c cal relation between these two variables, or between one or However, in contrast to the situation with calving tidewater both of them and a third variable, such as water depth. If it glaciers: 01) stresses are assumed to be distributed linearly as were possible to measure u independently, however, circu- in an elastic beam; 02) calving blocks are assumed to extend c larity would no longer be of concern. In short, the fact that over the full height of the calving face, including the sub- the measurement of u is indirect and involves measuring u merged part; 03) the calving speed calculated is the speed c i does not detract from the real possibility that u is somehow of movement of the top of individual calving blocks at the c physically related to u . moment of failure; and 04) the stress field assumed is not ap- i Supporting Van der Veen's model are data spanning propriate where the water is deep and the glacier ap- about 50 years that suggest that at the terminus of Hasbreen proaches flotation. Consequently, Hughes' approach does in Svalbard, u and u have indeed varied roughly synchro- not appear to be appropriate for the present problem. c i nously, although the water depth has remained nearly con- stant 0Jania and Kaczmarska, 1997). Unfortunately, this dataset is small, with some values representing averages ROLE OF CREVASSING over as much as 20 years, so the result must be viewed as only suggestive. Jania and Kaczmarska also note that on an annual basis, u begins to increase inJune as u increases, Van der Veen 01996) proposed a quite different model of c i and both peak in mid-July. calving. He believes that when a tidewater glacier thins suf- Sikonia's 01982) observation that on short time-scales u ficiently, though remaining grounded, it becomes weak and c varies directly with water pressure may also support Van der fails 0Van der Veen, 1996, p.382). Hooke 01983; see Meier, Veen's model, as u also should vary in this way. Similarly, his 1994) suggested that such weakening might be due to perva- i finding that u varies directly with water discharge in the sive fracturing resulting from high longitudinal strain rates, c Knik River, which he assumes is a proxy measure of the sub- and indeed longitudinal strain rates in a tidewater glacier glacial discharge, supports Van der Veen's model because, on are likely to increase as the glacier approaches flotation. time-scales of days, higher subglacial water fluxes are likely to Hooke 01983), Meier 01994) and Van der Veen 01996, fig. 3) be associated with higher water and hence higher have all found weak correlations between strain rate and u .Van derVeen 01996, p.381), in addition, speculates that the calving speed. In the Van der Veen model, the continuum i higher subglacial water pressures weaken the ice, and that that is failing is rather like a pile of irregular, weakly bonded this causes the increase in u . Higher water pressures might blocks of being pushed from behind, as by a bulldozer: c weaken the ice by leading to higher stretching rates, as men- as the front of the pile becomes oversteepened, blocks col- tioned above. 0On longer time-scales, in contrast, high water lapse off of it. In this model, the calving speed depends upon fluxes may increase the sizes of conduits, and this may reduce the flux of ice to the calving face, much as the rate at which water pressure 0Ro« thlisberger,1972).) water comes out of a pitcher depends upon the rate at which the pitcher is tilted. Post 01997) argues that the absence of icebergs of signifi- TEMPERATURE cant size in the forebay of Columbia Glacier today is evi- dence for such pervasive fracturing at depth. Similarly, In an effort to explain why calving rates are lower on polar Venteris 01997) found that the divergence of the velocity glaciers and in winter, Van der Veen 01997) suggested that field at the surface of Columbia Glacier suggested a thin- temperature may influence calving speeds. , for ning rate well in excess of that measured. He attributed the example, may weaken ice 0Liu and Miller,1979) or, by fill- discrepancy to pervasive internal and bottom crevassing. ing , may increase stresses leading to propagation 0e.g. Van der Veen, 1998b). In addition, as dis- cussed next, melt rates on submerged parts of calving faces will be higher when the water body is warm. 0As noted ROLE OF GLACIER SPEED above, uc as defined by Equation 01) includes melting.)

If uc depends upon the flux of ice to the calving face, it will SUBMARINE MELTAND CALVING correlate closely with the mean ice speed at the terminus, ui, which, in turn, depends on factors such as geometry The highest in water bodies into which and depth 0Van derVeen,1996).Indeed, there is a strong cor- glaciers calve occur in the fall 0Matthews,1981,fig.6;Walters relation between uc and ui, but this may be largely because and others, 1988, fig. 5). This is of considerable interest 189 Journal of Glaciology because calving is generally more vigorous between THE MULTIVARIATE PROBLEM October and December 0Meier and others,1985). Funk and Ro« thlisberger 01989) attributed the low No physical mechanism has been proposed that would ex- calving speeds of glaciers terminating in fresh water to the why there should be a dependence of calving speed low buoyancy contrast between glacier meltwater and fresh on water depth, and available data seem to suggest that the water compared with the buoyancy contrast between constant of proportionality varies from temperate to polar glacier meltwater and water. Circulation near the settings, from salt to fresh water and even within a dataset calving face should thus be greater in sea water. Such circu- from a single retreating glacier. Consequently, some authors lation would be significant, however, only if melting played have speculated that the correlation between uc and hw may a major role in the process we call ``calving''. On the other be due to some correlation between these two parameters hand, the buoyancy of ice is also about 30% lower in fresh and a third 0unspecified) variable 0Pelto and Warren, 1991; Van derVeen,1996, p.381). water than in sea water, and this may affect the rates at However, it should be recognized that if a dependent which blocks of ice become detached from a calving face variable, u , is a function 0physically) of several indepen- below the water level in saline and fresh water. c dent variables, A set of observations of calving events made by Warren and others 01995) on Glaciar San Rafael, , may also be uc ˆ f x1;x2;x3; ...;xn† ; 2† cited in support of melting. They calculated calving fluxes then a change, duc,inuc may be a result of a change, dxi,in by noting the time and size of every visible calving event, any one or more of the n independent variables: both subaerial and submarine, occurring during daylight Xn @f hours for 31days.The flux from the subaerial part of the cliff du ˆ dx : 3† c @x i ^1 iˆ1 i corresponded to uc  10.7 m d ,whilethatfromthesub- ^1 marine part implied a speed of only 1.4 m d . Thus, about At one time of year, or in one situation, duc may be domin- 87% of the ice loss from the submerged part of the cliff went ated by dx1 whereas at another time or place it may be con- undetected.This may have been because melting dominated trolled largely by dx2 or dx3. Furthermore, the sensitivity of the ice loss there. duc to variable xi depends upon @f=@xi, which may, in Theoretical studies 0e.g. Gade,1979; Josberger and Mar- turn, be a function of other independent variables and/or tin, 1981) of melting on a vertical ice face submerged in sea time. water have concentrated on the circulation 0or free convec- Thus, when a strong correlation exists for some datasets tion) set up by the rising of fresh water released by melting at or some parts of a dataset, we should seek a physical cause the face. So far, these studies have failed to demonstrate the for this correlation. The fact that some observations do not likelihood of a large volume of ice loss by submarine melting. fit the correlation suggests that other factors are important, However, simple calculations suggest that the in the and the challenge is then to identify these factors. far-field water is often sufficient to melt ice at rates compar- In the case of calving, evidence presented above suggests able to observed ``calving'' speeds. The question is whether that three of the key variables are likely to be water depth, this far-field water can be brought to the calving face by con- longitudinal strain rate 0or accumulated longitudinal vection processes associated, for example, with rising plumes strain) and temperature. Our objective herein is to seek a of subglacial water or tidal action. physical explanation for the dependence on water depth. Calving of ice blocks from the submerged part of a face must be driven primarily by buoyancy forces. The stresses STRESSES IN A CALVING FACE involved are relatively small, however, compared with those on the subaerial part of a calving face, so forces become suf- Calving presumably occurs when the stresses applied to a ficient to dislodge blocks only when the blocks are large or block of ice with a near-vertical face exceed its strength. As the face is weakened in some manner, such as by wave-cut noted, these stresses have sometimes been analyzed using notches 0Hughes, 1987; Syvitski, 1989; Kirkbride and War- elastic-beam theory 0e.g. Reeh, 1968; Hughes, 1992). How- ren, 1997), pervasive fracturing or bottom crevassing. This ever, owing to the non-linearity of the flow law, stresses in a may be why most observed submarine calving events in- calving terminus are not distributed elastically, and further- volve large blocks. more a grounded glacier is supported along its base and thus Hughes and Nakagawa 01989) have suggested that the does not freely bend downward into its bed. Other analyses submerged part of the calving face might project a consider- have used the finite-element method. Examples are studies able distance out into the water from the subaerial part.This of a floating 0Fastook and Schmidt,1982), of an ice would increase the stresses on any failure plane projecting tongue resting on land but undercut by water 0Iken,1977), of more-or-less vertically downward from the subaerial cliff. a grounded glacier terminating in 0fresh) water 0Visher and In many observations of submarine calving events, it is others,1991) and of a tidewater glacier, Columbia Glacier, in noted that icebergs emerge vertically >100 m from the calv- a fjord 0Sikonia,1982).We, too, have modeled the stress dis- ing face. Warren and others 01995, p.278) comment that tribution using the finite-element method 0Fig. 1), and in- when icebergs emerge relatively far from the cliff, water deed a primary objective of this paper is to present the between the cliff and the emerging is not disturbed. results of this effort. These observations support the conclusion that, at least oc- casionally, toes project a significant distance seaward from a Model description calving face. On the other hand, submersible observations in Glacier suggest that the face is usually either nearly The model used for these simulations is a version of the two- vertical or slightly overhanging 0personal communication dimensional, vertical flow-plane model described in Han- from R. Powell,1990). son 01990) and Hanson and Hooke 01994). The basic equa- 190 Hanson and Hooke: Glacier calving

arise solely from choices of boundary conditions. For a given set of boundary conditions, we normally made runs with the five values of h0 noted above.

Boundary conditions

In all simulations, the upper surface of the glacier and the sub- aerial part of the calving face were assumed to be stress-free. Heights of subaerial parts of calving faces, hc, are re- markably uniform. In ten of Brown and others' 01982) seven- teen examples they are in the range 50^70 m, and in four others the uncertainties in the estimated heights allow the possibility of their being in this range. Consequently, in the modeling discussed herein we held hc constant at 60 m and Fig. 1. Finite-element simulation of velocities in a flow plane allowed the total height of the face, and hence water depth, near a calving face.The thickness at the calving face is 200 m. to vary. Vectors are shown, with 20 vertical exaggeration, for every At the up-glacier boundary of the model domain, a hori- fourth node in each dimension. zontal velocity was specified in order to create an inflow of ice to the domain. In most simulations, this inflow was calculated tions of the model, as used here, are steady conservation of from the Nye 01952) plane-strain relation in which u L; z† is momentum and conservation of mass for an incompressible determined by the surface slope and ice thickness, thus: medium: nno 1 g L† n‡1 n‡1 @ @p u L; z†ˆub L†‡ h L† À‰h L†ÀzŠ ; jk À ‡ g ˆ 0 4† 2 B @x @x j j j 7† @u where L† and h L† are the surface slope and ice thickness j ˆ 0 ; 5† at the up-glacier end of the domain, and u L† is the sliding @xj b velocity there, normally taken to be 1000 m a^1. where  is the deviatoric stress tensor, p is pressure, g is the jk j The submarine part of the calving face had an applied gravity vector, u is velocity, x is position and  is density. j j stress boundary condition to account for the hydrostatic The convention of summing repeated indices over their water pressure range is used. The model is two-dimensional, with coordin- pw ˆ wg hw À z† ; 8† ates x1 ˆ x horizontal and x2 ˆ z vertically upward, with ^3 corresponding velocities u1 ˆ u and u2 ˆ w. The stress^ where w is the density of sea water,1030 kg m , and hw is strain-rate relationship used is a Glen 01955)-type power law, the water depth at the calving face.

1À1 Boundary conditions at the base of the simulated do- jk ˆ B"_n "_jk ; 6† main were more complicated. There, a mixed velocity and where "_jk is the strain-rate tensor, "_jk ˆ @uj=@xk ‡ @uk=@xj†; stress boundary condition was applied.The vertical compo- 1 and "_ is the effective strain rate, "_ ˆ‰1=2 "_jk"_jk†Š2: Constants nent of the velocity was held fixed at w ˆ 0. Horizontal in all simulations reported herein are n ˆ 3, B ˆ 0.2 MPa a1/ n velocities were free, but subjected to a resistive stress. The 0appropriate for temperatures at the pressure-melting point), resistive stress, b, was derived from consideration of force ^3 ^2  ˆ 900 kg m , and g ˆÀg2 ˆ 9.8 m s . balances on the set of calving glaciers studied by Brown Equations 04) and 05) are solved on a finite-element grid and others 01982). Consider a section of a glacier of length consisting of four-node Taig quadrilaterals 0Irons and Ah- X and width W in a rectangular channel. The thickness at mad, 1980).Velocity components are defined at each nodal the calving face is h0. Define a mean slope  over the dis- point, whereas stresses, pressures and densities are constant tance X such that hX  h X†ˆh0 ‡ X . Balancing forces across the element. The numerical equations are con- on this block yields the traction, ,thus: b structed using Galerkin's method and two-point Gauss^Le- 1 1 1  ˆ gh2 À  gh2 gendre integrals, producing a sparse, banded-symmetric set b X 2 X 2 w w of linear equations that are solved via Gaussian elimination. Z 9† 2 hX h Grids for these models were constructed of elements in 0 À xx X; z† dz À A s columns that were 5 m wide everywhere. Elements were X 0 W 0 5 m high at the calving face, but increased back from the 0e.g. Thomas, 1973), where xx X; z† is the longitudinal calving face in order to span the increasing glacier thickness stress deviator at the upflow end of the block, h is the mean without increasing the number of elements in a column. All thickness of the block,s is the mean drag on fjord walls and of the grids for these experiments had perfectly bottoms, A is a shape factor that accommodates convergence or di- a calving face thickness, h0, of 100, 150, 200, 250 or 300 m, vergence of the valley walls in the down-glacier direction and a simulated length up-glacier from the calving face, L, 0A ˆ1ifW is constant). of 2000 m. Only the section within a few hundred meters of Let us now define another shape factor, Sf, to partition the calving face was analyzed; the rest of the simulation was the drag forces between the bed and the fjord walls. Let  primarily intended to buffer the calving face from effects of Ahs=W ˆ 1=Sf À 1†b.Thus,Sf ˆ 0.5 corresponds to the up-glacier boundary conditions. the situation in which half of the drag is supplied by the In all our simulations, the internal dynamics of the ice walls and half by the bed, whereas lower values of Sf repre- were considered to be known. Thus, the changes we discuss sent situations in which less of the resistive stress is supplied 191 Journal of Glaciology by the bed, as might occur when basal water pressures are t i o n 011), u s i n g T ˆ a ‡ bh0, provides a relation for the slope high.We now define a pressure-gradient forcing term, T: at the margin: Z a ‡ bh  h2 h hX 0 w w 0 b 2 0 0 ˆ ‡ À : 13† T ˆ ‡ xx dz: 10† gh0 2h0X 2X Sf X 0 Figure 2b shows the resulting surface profiles for the stan- Combining Equations 09) and 010) yields: dard set of ice thicknesses used in the simulations.  We used Sf T as an initial basal-stress boundary condi- 1 1 2 1 2 tion. 0Shape factors are introduced in the model through T ˆ ghX À wghw : 11† X 2 2 modification of the value of B in Equation 06).) Use of Sf T as a stress boundary condition actually incorrectly transfers Va l u e s o f T can now be calculated for the glaciers studied by the force from the vertically integrated longitudinal stress Brown and others 01982), using ice thicknesses and water deviator to the bed. As the model must iteratively solve the depths at the calving face from their report, obtaining sur- equations of motion, owing to the non-linear flow law, this face slopes from maps, assuming a horizontal bed and tak- problem was readily remedied by adjusting the basal-stress ing X ˆ 1000 m. The values thus obtained were well boundary condition during each iteration. In each iteration, correlated with h0 0Fig. 2a); a regression line of the form the model estimates a velocity field, uses the velocities to cal- ^1 T ˆ a ‡ bh0 yielded a ˆ 81 kPa and b ˆ 0.667 kPa m .If culate a strain-rate tensor field, uses those strain rates to re- we assume that the driving stress over the block, d, is con- estimate the effective viscosity and then repeats the estima- stant and equal to gh , the surface profile will be parabolic tion of the velocity.The longitudinal stress field is generated and described by: essentially as a by-product of each model iteration, so the in- q tegrated longitudinal stress deviator could be subtracted 2 from the basal boundary condition at each iteration. This h x†ˆ h ‡ 2 0h0x ; 12† 0 did not appreciably increase the number of iterations required for model convergence, because the integrated lon- where is the surface slope at the calving face. Solving 0 gitudinal stress deviator is small compared to  .Infact,it Equation 012) for and combining the result with Equa- b 0 would not have changed our results significantly had we neglected this stress adjustment entirely.

Simulations with varying thickness

Some results of our modelling are presented in Figures 3^7. A pervasive feature is a zone of high ui just below the water- line 0Figs 3 and 4). Such a high speed might be anticipated from Hughes' 01992, fig. 3) analysis. As a result of this flow, there is a zone of very slight compression at the surface, ex- tending about 50 m up-glacier from the calving face. This zone was present even in simulations, described below, that included crevasses. In Figure 4 it will be seen that both ui and the velocity gradient between the level of the velocity maximum and the bed increase with water depth, as one would expect. This velocity gradient is plotted as a function of h0 in Figure 5 for two different values of Sf. These velocity gradients are significant because they suggest that the calving face will tend to develop an overhang, and that this tendency will be greater on thicker glaciers. Such overhangs would promote fail-

Fig. 2. 0a) Relation between T 0Equation 011)) and ice thickness at the calving face for glaciers studied by Brown Fig. 3. Contours of horizontal velocity, ui,inthelast500mof and others 01982).0b) Longitudinal surface profiles for stan- a glacier 200 m thick at the calving face. Sf ˆ 0.5 for this dard set of ice thicknesses. simulation. Contour interval is 5 m a^1. 192 Hanson and Hooke: Glacier calving

Fig. 4.Variation with depth of horizontal velocity at the calv- ing face for five different values of h0 and two different values of Sf. Value of h0 appropriate for any curve can be inferred from ordinate value of bottom of curve.

0 Fig. 6. Contours of xx in the last 500 m of a glacier that is 200 m thick at the calving face. Contour interval is 20 kPa. 0a) Sf ˆ 0.5. 0b) Sf ˆ 0.25. There is a zone of high extending stresses along the bed near the calving face 0Fig. 7).We attribute these stresses to the high vertical stresses needed to balance the bending mo- ment applied to the terminus of the glacier by the unequal distribution of hydrostatic and cryostatic pressure on either side of the calving face 0see also Reeh, 1968; Hughes, 1992, fig. 4). Such stresses would lead to a tendency toward extru- sion flow near the base of the calving face, much as a col- umn of ice sitting on a rigid substrate would tend to bulge out at the base.This would be likely to generate bottom cre- 0 vasses. Van der Veen 01998a) estimates that xx  50 kPa is required to initiate bottom crevasses, and the stresses in Fig- Fig. 5.Variation in rate of overhang development with h0 for S ˆ 0.25 and 0.5. Lines are based on linear regressions. ure 7 are significantly larger than this. Such crevasses would f promote submarine calving. Of particular interest is the fact 0 ure 0or collapse) of the top of a calving face, particularly one that the average value of xx within 20 m of the calving face that was pervasively fractured. increases with water depth 0Fig. 8). This could be another Such oversteepenings are rarely noted, presumably cause of an increase in uc with increasing hw. because the ice collapses as fast as the overhang develops, and perhaps because the overhang is largely below . Sensitivity studies For example, Kirkbride andWarren 01997) say that they saw no evidence of gradual flow-induced overhang development Some of the boundary conditions and modeling assump- in a sequence of photographs of the calving face of Maud tions used in these simulations may be questioned. However, Glacier, . However, their photograph and we found that the main conclusions are robust under a vari- sketches suggest that outward-leaning calving faces were ety of different modeling scenarios. In what follows, we treat present on occasion. Similarly, Motyka 01997) reports that the run in which h0 ˆ 200 m and Sf ˆ 0.5 as a control ex- the calving face of LeConte Glacier, , has been periment and test the effects of changing some of the bound- observed to``lean outward ... at various times''. ary-condition parameters from those in the control run. In Figures 6 and 7 we show some spatial variations in the The overall length of the simulated domain, L, and the 0 longitudinal stress deviator, xx.There is a persistent high in basal inflow into the up-glacier end of the domain, ub L†, 0 xx just below the water surface and a few tens of meters both affected the overall speed of the ice near the calving 0 back from the calving face. Note that xx typically exceeds face without affecting the stresses. Simulations with ub ˆ 0 ^1 100 kPa.VanderVeen01998b, p.41)notes that stresses of only or 2000 m a at this face, compared with our standard ub ˆ about 30^80 kPa are needed for crevassing.Thus, the exten- 1000 m a^1, simply added or subtracted 1000 m a^1 from the sive crevassing observed up-glacier from the terminus of a velocities near the calving face. Increasing the overall calving glacier is readily understood. length of the domain to 3000 or 4000 m showed that speeds 193 Journal of Glaciology

0 Fig. 8.Variation with h0 of mean xx at the bed within 20 m of the calving face. Crevasses provided a more interesting change in the results. A realistic depiction of downward-propagating sur- face cracks, the width of which varies with depth, and the depth of which varies almost at random, is beyond the scope and capability of the model. However, a characteristic of a glacier surface that is heavily crevassed into a field is that stresses are much more readily relieved by slippage along crevasse faces than in uncracked ice. We can thus simulate some aspects of the heavily crevassed surface by softening the ice, by means of a substantial decrease in B where B is understood to apply to the bulk, heavily cracked ice rather than to small-scale samples. Realistically, such an effect should be anisotropic, as there are preferred direc- tions for stress relief, but that is also beyond reasonable use of a two-dimensional vertical model.The simulation also re- tains absolute mass conservation and incompressibility, which does not necessarily apply to an ice mass that is in- creasing its volume via crevassing. In two perturbation runs, B was reduced by approxi- mately a factor of 5, resulting in a 100-fold increase in "_jk for agivenjk, in elements in every third column within 500 m of the calving face. Thus, columns of softened ice 5 m wide were separated by columns of regular ice 10m wide. In the Fig.7.Longitudinal deviatoric stresses at the calving face and two runs, the top four and eight elements, respectively, were 20, 40, 60 and 80 m back from the calving face for 0a) S ˆ thus softened, so these ``crevasses'' extended to depths of 20 f and 40 m, respectively, at the margin. The depth increased 0.5, h0 ˆ 100 m; 0b) Sf ˆ 0.5, h0 ˆ 200 m; 0c) Sf ˆ 0.5, h ˆ 300 m; and 0d) S ˆ 0.25, h ˆ 200 m. slightly up-glacier as ice thickness increased. 0 f 0 One visible effect of crevassing is that it increases the over- increased as length increased.This demonstrates that we do all speed of the glacier, top to bottom, and that greater depths not have a comprehensive simulation of the stress going well of crevassing produce a greater speed increase 0Fig. 9a).This back from the calving face; rather, it seems likely that both may be true for a real glacier, but it should probably be con- basal and side drag increase up-glacier. However, neither sidered a side-effect of the model. Softening some 3000^ 2 the perturbations in ub L† nor those in L produced graphi- 6000 m of the flow plane is bound to cause a speed increase: cally visible changes in the longitudinal stress deviators or in the softening is numerically equivalent to reducing the side shear stresses near the calving face. drag. The extension of the speed increase to the glacier bed Varying the water level for a constant glacier thickness indicates that even for a change applied only to the top 20 m, 0thus varying the height of the subaerial cliff) had a substan- the 500 m of horizontal extent of change is sufficient to pro- tial effect on glacier speed, as would be expected: lower duce a response that is coupled 200 m downward to the bed. water levels decrease the back stress at the calving face, so The more interesting element of the speed increase is the glacier moves faster. None of the results discussed here, that its maximum is at the base of the crevassed zone, which however, were affected, as maxima in both speed and stress enhances the velocity maximum at the waterline. Longitu- still appeared in the same positions relative to the waterline. dinal stress deviators in the crevassed cases clearly show a 194 Hanson and Hooke: Glacier calving

Fig. 9. Effects of crevasses on simulated speeds 0a) and simulated longitudinal stress deviators 0b) in a glacier 200 m thick. decreased ice strength in the upper layers where softening Similarly, Sikonia inferred that on seasonal and sub-- has occurred, and a greater extension at the waterline 0Fig. onal time-scales uc varies directly with subglacial water dis- 9b).We conclude from both the horizontal velocity changes charge and inversely with effective pressure. By reducing and the longitudinal stress deviator changes that effective basal drag, both of these may increase the speed at the base softening of heavily crevassed near-surface ice increases the of the calving face 0see effect of Sf in Fig.4), thus facilitating likelihood of buckling outward at the waterline, and hence failure by propagation of bottom crevasses. enhances the effects seen in the uncrevassed case. In conclusion, we underscore the point made above 0Equations 02^3)), that in certain situations, it may be a change in one of the variables, xi, that results in a change CONCLUSIONS in uc, while in other situations it is another. Water depth is likely to be only one of the variables that affect calving Calving speed increases with water depth, other factors re- speed. maining constant 0Brown and others,1982; Funk and Ro« th- lisberger, 1989; Pelto and Warren, 1991). If it turns out that REFERENCES there is a physical connection between uc and hw , despite Van der Veen's 01996, 1997) reservations, our modeling sug- Brown, C. S., M. F. Meier and A. Post. 1982. Calving speed of Alaska tide- gests that this may be, in part, because the distribution of water glaciers, with application to Columbia Glacier. U. S. G e o l. S u r v. Prof. longitudinal stresses and velocities leads to oversteepening Pap. 1258-C. of calving faces, with the rate of development of the over- Fastook, J. L. and W.F. Schmidt. 1982. Finite element analysis of calving from ice fronts. Ann. Glaciol., 3,103^106. steepening increasing with water depth.When the ice is per- Funk, M. and H. Ro« thlisberger. 1989. Forecasting the effects of a planned 0 vasively fractured, as it must be when xx is so high, such reservoir which will partially the tongue of Unteraargletscher in oversteepening will destabilize the face, facilitating calving. . Ann. Glaciol., 13, 76^ 81. In addition, zones of high 0 at the base of the glacier near Gade, H. G.1979. Melting of ice in sea water: a primitive model with applica- xx tion to the Antarctic ice shelf and icebergs. J. Phys. Oc ea n og r. , 901), 18 9 ^ 19 8. the calving face may promote bottom crevassing, particu- Glen, J.W.1955. The creep of polycrystalline ice. Proc. R. Soc. London, Ser. A, larly in thicker glaciers. 22801175), 519 ^ 538. Calving rates are also known to increase in the fall 0Si- Hanson, B.1990. Thermal response of a small to climatic forcing. J. konia,1982; Meier and others,1985).This may also be partly Glaciol., 360122), 4 9 ^ 56. Hanson, B. and R. LeB. Hooke. 1994. Short-term velocity variations and due to oversteepening resulting from increased basal drag basal coupling near a , Storglacia« ren, Sweden. J. Glaciol., as subglacial water pressures decrease. In addition, the 400134), 67 ^ 74. observation that calving speeds are highest when mean Hooke, R. LeB.1983. Report on activities.Tacoma,WA, U.S. Geological Survey. water temperatures are highest suggests that increased sub- Hooke, R. LeB., T. Laumann and M. I. Kennett. 1989. Austdalsbreen, Nor- way: expected reaction to a 40 m increase in water level in the lake into marine melting, which would also lead to oversteepening, which the glacier calves. Reg. Sci.Technol., 1702), 113^126. may play a role. Hughes,T.1987. Ice dynamics and models when ice sheets col- We think that this model of calving is consistent with lapsed. In Ruddiman, W.F. and H. E. Wright, Jr, eds. North America and both the Brown and others 01982) and the Sikonia 01982) adjacent during the last deglaciation. Boulder, CO, Geological Society of America,183^220. 0The of North America K-3.) calving relations. As noted, in an annual time-frame Brown Hughes, T. 1992. Theoretical calving rates from glaciers along ice walls and others found that calving rates increased linearly with grounded in water of variable depths. J. Gl a c i ol. , 380129), 282 ^ 29 4. water depth, and we find that the velocity field in the tongue Hughes, T. and M. Nakagawa. 1989. Bending shear: the rate-controlling of a calving glacier results in tendencies toward oversteepen- mechanism for calving ice walls. J. Glaciol., 350120), 260^266. Iken, A. 1977. Movement of a large ice mass before breaking off. J. Gl a c i ol. , ing and bottom crevassing that also increase with water 19081), 595^605. depth; in the case of oversteepening, the increase is linear. Irons, B. M. and S. Ahmad. 1980.Techniques of finite elements. NewYork, etc., 195 Journal of Glaciology

JohnWiley and Sons. Reeh, N.1968. On the calving of ice from floating glaciers and ice shelves. J. Jania, J. and M. Kaczmarska. 1997. Hans Glacier ö a tidewater glacier in Glaciol., 7050),215^232. southern Spitzbergen: summary of some results. Byrd Polar Res. Cent. Rep. Ro« thlisberger, H. 1972. Water pressure in intra- and subglacial channels. J. 15, 95 ^ 10 4. Glaciol., 11062), 177^203. Josberger, E. G. and S. Martin. 1981. A laboratory and theoretical study of Sikonia, W.G. 1982. Finite-element glacier dynamics model applied to the boundary layer adjacent to a vertical melting ice wall in salt water. Columbia Glacier, Alaska. U.S. Geol. Surv. Prof. Pap.1258 -B. J. Fluid Mech., 111,439^473. Sikonia, W.G. and A. Post. 1979. Columbia Glacier, Alaska: recent ice loss Kirkbride, M. P. and C. R. Warren. 1997. Calving processes at a grounded and its relationship to seasonal terminal embayments, thinning, and ice cliff. Ann. Glaciol., 24,116^121. glacier flow. U.S. Geol. Surv. Open File Rep.79-1265. Liu, H.W. and K. J. Miller. 1979. Fracture toughness of fresh-water ice. J. Syvitski, J. P.M. 1989. On the deposition of within glacier-influenced Glaciol., 22086),135^143. fjords: oceanographic controls. Mar. Geol., 8502/4),301^329. Matthews, J. B. 1981. The seasonal circulation of the Glacier Bay, Alaska Thomas, R. H.1973.The creep of ice shelves: theory. J.Glaciol., 12064),45^53. fjord system. Estuarine Coastal Shelf Sci., 12, 679^700. Van derVeen, C. J.1996.Tidewater calving. J. Glaciol., 420141), 375 ^ 385. Meier, M. F.1994. Columbia Glacier during rapid retreat: interactions be- Van der Veen, C. J.1997. Controls on the position of iceberg-calving fronts. tween glacier flow and iceberg calving dynamics. In Reeh, N., ed. Report Byrd Polar Res. Cent. Rep.15,163^172. ofaWorkshop on ``The Calving Rate of theWest Greenland Glaciers in Response to Van der Veen, C. J. 1998a. Fracture mechanics approach to penetration of '', Copenhagen, 13^15 September 1993. Copenhagen, Danish bottom crevasses on glaciers. Cold Reg. Sci.Technol., 2703), 213 ^ 223. Polar Center, 63^83. Van der Veen, C. J. 1998b. Fracture mechanics approach to penetration of Meier, M. F., L. A. Rasmussen, R. M. Krimmel, R.W. Olsen and D. Frank. surface crevasses on glaciers. Cold Reg. Sci.Technol., 2701), 31 ^ 47. 1985. Photogrammetric determination of surface altitude, terminus Venteris, E. R. 1997. Evidence for bottom crevasse formation on Columbia position, and ice velocity of Columbia Glacier, Alaska. U. S. G e o l. S u r v. Glacier, Alaska, U.S.A. Byrd Polar Res. Cent. Rep.15,181^185. Prof. Pap.1258 -F. Vischer, D., M. Funk and D. Mu« ller. 1991. Interaction between a reservoir Motyka, R. J.1997. Deep-water calving at Le Conte Glacier, southeast Alaska. and a partially flooded glacier: problems during the design stage. In Byrd Polar Res. Cent. Rep.15, 115 ^ 118. Dix-septie©me Congre©s des Grands Barrages 0ICOLD),Vienne, 1991. Paris, Com- Nye, J. F.1952.The mechanics of glacier flow. J. Gl a c i ol. , 2012), 82 ^93. mission Internationale des Grands Barrages,113^135. 0Q.64, R.8.) Pelto, M. S. and C. R.Warren.1991. Relationship between tidewater glacier Walters, R. A., E. G. Josberger and C. L. Driedger. 1988. Columbia Bay, calving velocity and water depth at the calving front. Ann. Glaciol., 15, Alaska: an``upside down''. Estuarine Coastal Shelf Sci., 26, 607^617. 115 ^ 118. Warren, C. R., N. F. Glasser, S. Harrison,V.Winchester, A. R. Kerr and A. Post, A. 1997. Passive and active iceberg producing glaciers. Byrd Polar Res. Rivera. 1995. Characteristics of tide-water calving at Glaciar San Ra- Cent. Rep.15, 121 ^ 134. fael, Chile. J. Gl a c i ol. , 410138), 273 ^ 28 9.

MS received 26 January 1999 and accepted in revised form 20 October 1999

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