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1992 Theoretical Calving Rates from Glaciers Along Ice Walls Grounded in Water of Variable Depths Terence J. Hughes University of Maine - Main, [email protected]
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Repository Citation Hughes, Terence J., "Theoretical Calving Rates from Glaciers Along Ice Walls Grounded in Water of Variable Depths" (1992). Earth Science Faculty Scholarship. 106. https://digitalcommons.library.umaine.edu/ers_facpub/106
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 Glaciology, Vo!. 38, No. 129, 1992
Theoretical calving rates from glaciers along ice walls grounded in water of variable depths
T. HUGHES Department of Geological Sciences and In stitute fo r Qu aternary Studies, University of Maine, Orono, Maine 04469, U. S.A.
Calving has been studied for glaciers ranging fr om slow polar ABSTRACT.glaciers that calve on dry land, such as on Deception Island (63.0° S, 60.6° W) in Antarctica, through temperate Alaskan tide-water glaciers, to fast outlet glaciers that float in fiords and calve in deep water, such as J akobshavns Isbr
(Echelmeyer and Harrison, 1990), J akobshavns Isbne INTRODUCTION (69.2° N, 49.9° W) is the fas test known ice stream, and its A largely neglected field of glaciological research is the calving rate equals its velocity (Pelto and others, 1989). calving dynamics of glaciers and ice sheets along the Rapid calving rates are part of the J akobshavns effect, a fronts of grounded ice walls and floating ice shelves. Yet, series of positive feed-backs for rapid deglaciation of calving into the sea is the dominant ablation mechanism Jakobshavns Isbr 282 Hughes: Ca lving rates fr om glaciers grounded in water tensile bending stresses about one ice thickness behind the This mechanism was later observed to be dominant fo r calving front, which was compatible with the observation glaciers entering water in the Chilean Andes and along that tabular icebergs were of about these proportions. the Antarctic Peninsula, even when the ice wall was Fastook and Schmidt (1982) developed a finite-element undercut along the water line by wave action (Hughes model that duplicated these results and extended them to and Nakagawa, 1989). A theoretical fo rmula derived fo r include enhanced calving along water-filled crevasses. this mechanism on Deception Island had a summer Scattered crevasses on Jakobshavns Isbne are filled with calving rate proportional to the inverse sq uare of the ice water, but none reach the calving fr ont in that condition. wall height. A summer calving rate inversely proportional Holdsworth (1969, 1971, 1973, 1974, 1977, 1985) has to the square of ice height above ice height supported by made detailed theoretical and observational analyses of water buoyancy was reported fo r the ice wall of Columbia flexure along the grounding line of floating glaciers, Glacier by Brown and others (1982), who also observed caused by tidal and wave action, as a mechanism fo r an annual calving rate proportional to water depth fo r 12 releasing very large tabular icebergs. Other studies of this Alaskan tide-water glaciers, including Columbia Glacier. mechanism were by Robin (1958, 1979), Hughes (1977), These calving relationships were also discussed by Meier Lingle and others (1981), Holdsworth and Glynn (1978, and others (1980), Sikonia (1982) and by Bindschadler 1981) and Vinogradov and Holdsworth (1985). Hughes and Rasmussen (1983). (1983) has argued that large tabular icebergs would be Studies of calving dynamics, whether slab calving released from floating ice shelves along lines of weakness fr om ice walls or tabular calving fr om ice shelves, are a that formed fr om rifts alongside ice streams that supplied neglected part of glaciology largely because of dangers the ice shelf and fr om gashes through the ice shelf in the inherent in calving environments, especially calving into lee of ice rises, in addition to flexure along ice-shelf water. The mobility of field workers is severely limited grounding lines, and that these lines of weakness divided along marine and lacustrine calving ice margins, and an ice shelf into plates that were potential tabular icebergs calving from ice cliffs up to 100 m or more above water long before the plates reached the calving fr ont. Wordie level is unpredictable and can be fr equent. Our detailed Ice Shelf in the Antarctic Peninsula seems to be studies of calving dynamics on Deception Island were disintegrating along these lines of weakness at the present possible only because the calving ice wall was standing on time (Doake and Vaughan, 1991) . dry land and solifluctionof a thick ash layer on the glacier Iken (1977) made the first theoretical study of slab surface had produced a series of ash ramps that gave us calving from an ice wall undercut along the shoreline of a safe access to various heights along the ice wall (Brecher beach. She assumed homogeneous deformation based on and others, 1974; Hughes and others, 1974; Hughes, continuum mechanics to conclude that slabs were about 1989; Hughes and Nakagawa, 1989) . as thick as the height of the ice wall. Hughes (1989) reported non-homogeneous deformation in an ice wall on Deception Island (63.0° S, 60.6° W), fr om which slabs THEORY having much less thickness calved along shear bands caused by a bending moment at the base of the ice wall. Direct observations of the calving ice wall on Deception Fig. 1. Shear bands on the Deception Island (63.0° S, 60.tJ> W) ice wall revealed by vertical offs ets of ash layers in the ice. Access to the ice wall was by ash ramps of various heights, one of which is in the background. 283 Journal ofGlaciology , ( Fig. 2. A slab about to calve fr om the ice wall of an Antarctic glacier (photograph by D. Allan). Island, including tunneling into the wall at various stress is high. An overhanging ice wall is therefore heights, revealed numerous shear bands that rose necessary fo r fo rward calving but not for downward vertically from the bed behind the ice wall and curved calving. In Figure 2, the ice wall looks like it could calve increasingly fo rward with increasing distance above the either way. The slab bends fo rward at � 180 fr om the e bed, so that shear displacement increased from zero at the vertical and the crevasse formed at c :::::: O.4h behind the bed to a maximum at the surface (Hughes, 1989). Some wall, where h is the wall height. This combination of e of these shear bands are shown in Figure l. Non and clh, therefore, may be close to the transition fr om homogeneous bending creep produces the shear bands, downward calving to fo rward calving. There is no special and shear in them is similar to the slip between pages of a significance in this transition. book when the book is bent about its binding. The mechanics of slab calving used here will be based Shear rupture across a shear band leads to the kind of on the theory fo r bending beams, as presented by Popov slab calving shown in Figure 2. Shear rupture in a shear (1952) with appropriate modifications. Popov (1952, band at calving distance c behind the ice wall opens a p. 89) listed the eight steps in beam analysis needed to crevasse that removes coupling of the slab to the glacier. determine the shear fo rce, the axial fo rce, and the As the crevasse deepens, the entire weight of the slab must bending moment at any cross-section of a beam. An be borne by ice at the base of the slab, and by ice below initially vertical beam of ice resting on the bed will be the crevasse tip that still connects the slab to the glacier. If considered. The beam has length c, width wand mean the basal ice is crushed by the ice load above it, the slab height In general, an ice wall stands in water, so ri. will crash vertically downward. If the crevasse widens as distributed loads of ice and water act on opposite sides of it deepens, the slab will continue to bend until it topples the beam, causing the beam to bend fo rward. As shown in fo rward. Let be the angle of fo rward bending fo r the ice Figure 3, when water of depth d rises against a wall of e slab. The slab will collapse downward when and care height h, increasing buoyancy decreases basal shear stress e small, because the crushing stress is high, and topple TO, thereby causing a reduction of surface slope IX, such fo rward when and c are larger, because the bending that TO and ex approach 0 when the ice wall begins to float e ice / water / / \ / \ / \ fcter : / . \ . " . ':' ' ', ...... :. : . : : :; . :. : .: : .. .. Fig. 3. Th e effect ofwater depth on an ice wall. Arrows show the difference between the lithostatic pressure in ice and the hydrostatic pressure in water (sloping dashed lines). This pressure difference, basal shear stress TO, surface slope a, and perhaps slab-size decrease as water deepens, with TO = a = 0 at flotation depth, when tabular calving replaces slab calving. 284 Hughes: Calving rates fr om glaciers grounded in water z (6) where vertically averaged bending shear stress T8 induced by Fs acts parallel to calving area whe, and (7) where vertically averaged axial stress O'xx acts normal to ROCK calving area whe. Step 3 consists of measuring moment lever arms Ly, Lw, LB, LN, LQ, Ls and Le parallel to axes x, � fo r forces Fy, Fw, FB, FN, FQ, Fs and Fe, respectively, fo r bending Fig. 4. Free-body diagrams fo r a slab at an ice wall before moment where: shear rupture detaches the slab. Left: boundary conditions; My, right: fo rces and moments. Lr = �he (8) Lw =�d (9) LB LN = �c (10) at d = (py/Pw)h, where py and Pw are ice and water = densities, respectively, and are assumed to be constant. Lo = Ls = 0 (11) Questions to be examined are whether and c depend on e = �he. d and h, and how calving rate Uc depends on c, d and h. Le (12) e, Specifically, an approximate calving rate that depends on These are all shown in Figure 4. Equation (10) is exact these variables will be derived. only if the surface and bed are parallel. Step 1 consists of producing a fr ee-body diagram in Step 4 consists of applying the equations of statics to which all forces acting on the beam, taken as the slab that compute unknown reaction fo rces, assuming no inertial will calve, are referenced with respect to orthogonal forces. Force equilibrium along x requires that: coordinates. Figure 4 presents the diagram. The origin of Cartesian coordinates is on the bed at calving distance c Fr -Fw - Fa - Fe = 0 (13) behind the ice wall, with x horizontal and positive from which, up until a tensile crack opens the shear band: fo rward, y horizontal and parallel to the wall, and � vertical and positive upward. Action forces are the distributed horizontal lithostatic fo rce of ice Fy on the O'xx = �Prghe _�Pwgd2/he left side of the slab, the distributed horizontal hydrostatic = �prghe(1- Pwd2 / prh;) -To(c/he). (14) fo rce of water Fw on the right side of the slab, and the vertical body fo rce FB of the slab itself. These are all Force equilibrium along � requires that: gravitational fo rces, with g being gravity acceleration. s = Good approximations fo r dynamic equilibrium are: FB-F - FN 0 (15) from which: (1) O'zz = Prgh -Pwgd - 7she/c where � prghe is the mean lithostatic pressure pushing on area whe for ice thickness he at calving distance c, = prgh(l - Pwd/ prh) - Ts(he/c). (16) Moment equilibrium about the y-axis requires that, up until a tensile crack opens the shear band: where � Pwgd is the mean hydrostatic pressure pushing on area wd fo r ice thickness h at the ice wall and +Fe Le + FNLN +FwLw - FBLB - FrLr = 0 (17) My FB wchprg - wcdPwg = prgwch(l - Pwd/ prh) from which: = (3) My =�WChe(TO +78) -�prgwh;(1- Pwd2/prh;) where h � (h + he) is the average slab height, wch is the + prgh;(l - pwd3 /prh;) (18) = i slab volume and wed is the water volume it displaces. where bending moment causes bending creep at cross Step 2 consists of adding reaction fo rces to the fr ee My w body diagram. As located in Figure 4, these are a vertical section he that leads to shear rupture and crevassing in c basal normal fo rce FN, a horizontal basal traction fo rce the shear band at calving distance fr om the ice wall. Fo, a vertical shear fo rce Fs and a horizontal tensile fo rce Step 5 consists of specifying the fo rces at cross-section whe. Axial fo rce is obtained from Equations (7) and Fe that opens the crevasse at c: Fe (14) : (4) Fe = � Prgwh;(1 - Pwd2/ prh;) - TOWC. (19) where horizontally averaged basal axial stress O'zz acts through the substrate normal to basal area wc, Shear fo rce Fs is obtained fr om Equations (6) and (16): Fo TOWC (5) Fs = prgwch(l -Pwd/ prh) -O'zzwc. (20) = where horizontally averaged basal shear stress TO acts Force Fs causes shear rupture in the shear band and force parallel to basal area wc, Fe causes the shear band to open into a crevasse. 285 Journal of Glaciology ----�-+----�x ROCK Fig. 5. Free-body diagrams fo r a slab at an ice wall aft er shear rupture detaches the slab. Lift: boundary conditions; right: fo rces and moments. I I I ice Step 6 consists of isolating the slab after the crevasse I b opens and showing the fo rces and bending moments I s I a acting on it immediately prior to calving. As seen in I Figure 2, the crevasse can separate most of the slab before I it actually calves. The fr ee-body diagram fo r the ice slab water I at this stage of bending creep is shown in Figure 5. z Step 7 consists of major changes in the action and I reaction fo rces if some assumptions are made. First, I assume a thawed bed allows water to fill the crevasse to J depth d, so that: c------�.. 1/01,----- (21) Secondly, downward penetration of the crevasse stops x when at the crevasse tip, so that: axx = 0 Fig. 6. A bending ice slab after Ts has caused shear rupture (22) in a shear band and a xx has opened a crevasse along the ruptured shear band. The relationship between the angle () Thirdly, bending creep in numerous shear bands through in the crevasses, point x, z on the slab side of the crevasse length c has inclined the ice slab an angle from the 0 and velocities ux, Uz at point x, z is shown geometrically vertical, with increasing along z. Figure 6 shows similar 0 and is given by Equation (23). Note that Ux = Uc at the triangles fo r which 0 is the angle formed by the z-axis and top of the crevasse. the distance vector to point x,z on the bending curve, and by velocity vectors Ux and Uz of point x, z. Therefore: (23) entering My into the flexure fo rmula (Popov, 1952, p. 98- 102), and then deriving the differential equation for where fo rward displacement x of the slab increases along bending (Popov, 1952, p. 269-73) . z, with point x, z on the slab having fo rward velocity Ux In deriving the flexure fo rmula, bending stresses and downward velocity Uz due to bending creep. normal to the beam cross-section, tensile on the convex Step 8 consists of solving the equations of static side and compressive on the concave side, create an equilibrium fo r the fr ee body in Figure 5. No net internal bending moment that exactly resists the external horizontal fo rces are present. Summing vertical fo rces: bending moment (Popov, 1952). The important concepts (24) used to derive the flexurefo rmula are: (I) beam geometry which establishes the variation of strain in the beam cross Summing bending moments: section, (2) beam rheology which relates strain to stress, and (3) the laws of statics which locate the neutral axis My + FN (�c) - FB(�C + s) = 0 (25) and resisting moment at the beam cross-section. The where s is horizontal bending displacement x at height � h flexure fo rmula usually assumes elastic bending only, so of the centroid of the slab. Therefore, fr om Equation (23): that Hooke's law can be used to relate stress to strain. In this case, the flexure formula can be used to derive the (26) differential equation for the elastic bending curve for a Entering Equations (3) and (4) into Equation (24) gives: beam in static equilibrium (Popov, 1952, p. 269-73). When beam height h along z is large compared to beam azz FB/WC = prgh(1 - (lwd/prh). length in bending direction x and beam width a good = (27) C w, approximation is: Combining Equations (24) through (27) gives: - 1 L2 (29) My = FB8 � -2prgwch 0(1- (lwd/pih). (28) A calving criterion can be developed fo r the slab by where IS the elastic modulus and I = wc3 is the E f2 286 Hughes: Calving rates fr om glaciers grounded in water rectangular moment of inertial fo r basal cross-sectional n = 1 TI d2ux _ My (30) n=2 dz2 TJvl where TJy is the viscoplastic viscosity of creep. Instead of relating stress to strain using Hooke's law, as was done in T = _ ( d) av n - 1 3PIgh2() _ pw- 2 Ux - 1 z-. (33) (36) TJvC 2 prh ao n Viscoplastic viscosi ty at the moment of fr acture is half The slab calves when Ux = Ue measured at z = he ::::;h: TJv the slope of the tangent line at fa fo r a given value ofn (Hughes, 1983): 1 d d· 1 n . n-l TJv = 2( Tm / C) = 2aO /n cTm _ 1 n n-l _ 1 . _ where Ue is the calving velocity and c/h is the calving -2 A /n Tm -2Tmn/ crn-TJmn/ (37) ratio. The smaller the calving ratio, the better Equation (34) gives the calving rate. where 2TJm = T m/fm and TJm is the N ewtonian viscosi ty fo r Glen's law (Glen, 1955), used to compute TJv, is: purely viscous flow. In the glaciological literature, TJrn is often called the "effective viscosity" at effective stress T. It (35) is the slope of a straight line fr om the origin, f = T = 0, where f is the effective strain rate, T is the effective stress, that intersects a creep curve in Figure 7 at T fo r a given n. fa is the strain rate at plastic yield stress aa, n is a A tangent line to the creep curve is a better measure of the viscoplastic creep exponent and A is an ice-hardness physical property called viscosity. As seen in Figure 7, the parameter that depends on ice temperature and crystal tangent line at fa gives both the viscoplastic yield stress by fa bric. its intercept with the stress axis and the viscoplastic The calving criterion to be applied to the ice slab viscosity by half its slope. states that the slab calves when shear rupture occurs in the shear band at calving distance c behind the ice wall. A fr acture criterion fo r this situation requires that shear RESULTS rupture occurs when maximum shear stress Tm = � (al - ( ) reaches the viscoplastic yield stress av, where In applying Equation (34) to calving ice walls, such as the 2 al and a2 are maximum and minimum principal stresses. one in Figure 2, a question arises. Should () be measured This is the maximum shear-stress yield criterion. Setting when the crevasse first opens, in which case () is about 6° f = fm and T = Tm, Equation (35) is plotted as fm/fa (0.1 rad), as measured by the fo rward tilt of the left side of versus Tm/aO in Figure 7 to show how ay and TJv are the crevasse; or should () be measured when the slab obtained. Curves fo r all values ofn fr omn = 1 fo r viscous actually calves, in which case () is about 18° (0.3 rad) as creep ton 00 fo r plastic creep constitute the viscoplastic measured by the fo rward tilt of the right side of the = creep spectrum, and they all intersect at strain rate fa. In crevasse? As seen in Figure 8, other crevasses open behind 287 Journal of Glaciology Table 1. Average ice thickness h, ice height above water hw, water depth d and mean annual calving rate ue at the ice walls of12 Alaskan tide-water glaciers (Brown and others, 1982) Glacier h hw d ue I - m m m ma McCarty 42 30 12 600 Harvard 104 68 36 1080 Yale 222 69 153 3500 Meares 90 59 31 1010 Columbia 161 86 75 2185 Tyndall 113 49 64 1 740 Hubbard 172 92 80 2630 Fig. 8. Transverse crevasses behind the ice wall of a tide Grand Pacific 62 44 18 220 water outlet glacier in Greenland. Margerie 75 60 IS 463 J ohns Hopkins 126 70 56 2290 Muir 160 60 100 3700 South Sawyer 234 48 186 3200 the primary crevasse before calving, so () should apply to the calving event itself. A second question concerns the calving ratio cl h. For setting Ue = ue in Equation (34) and solving fo r c/h: convenience, let h = = he. Is clh a fu nction of total ice fi thickness or just ice thickness above water? Both (�) ( ) h, hw _ 2 = 3pr9h2() Pwd possibilities exist in Figure 2. Also, does depend on 1 (42) c/h h "lvue Prh . water depth d through flotation ratio (Pw/Pr)dlh for the ice wall? Figure 2 is fo r d = 0, so it provides no answers to Figures 9 and 10 show the respective linear and these questions. logarithmic plots of elh obtained from Equation (42), A third question concerns the value of viscoplastic versus (Pw/pr)d/h in Equations (40) and (41). The linear viscosity 7Jv to be used in Equation (34). If creep in the plot is preferred because it can include c/h = 0.4 at bending slab is primarily in shear bands, not between d = 0, from Figure 2. shear bands, then creep by easly glide, not hard glide, Table 3 shows results fr om entering Equations (38) dominates. Paterson (1981, p. 41) gave 7Jm = through (41) into Equation (34). Ideally, ue/ue = 1.00. 1 8 X 1013 kg m-1 8- fo r the minimum creep rate (hard Reasonable approximations of this ideal are attained for glide) and (To = 1 bar= as the plastic yield stress, for which (Tv= 0.67 bar for n 3 in Equation (36). Hughes and Nakagawa (1989) reported that the creep rate increases ten-fold in shear bands (easy glide), fo r which "lm = 1 8 X 1012 kg m-1 8- . These questions can be addressed using data from 12 0.4 calving tide-water glaciers in Alaska (Brown and others, 1982). Table I lists the glaciers and values of h, hw, d and ue for each one, where ue is the mean annual calving rate 0.3 that can be compared with Ue in Equation (34). Figure 2 c/h • 7 is consistent with fo ur expressions for c in Equation (34); a • linear dependence on h: • 4 5 12 0.2 .. elh = 0.4; (38) 10 6 3 • a linear dependence on hw: 0.1 elhw = c/(h - d) = 0.4; (39) a linear dependence on (Pw I pr)dlh: (40) °O���--�����--��--�� elh = 0.4 - 0.35(pw/Pr)d/h; 0.2 0.4 0.6 . 0.8 1.0 (Pw1p,)d/h and a parabolic dependence on (Pw / Pr)d/h: Fig. 9. A linear dependence of calving ratio c/h on (cl hi = (0.22)/[(PwlPr)d/hj. (41) buoyancy ratio (PwI pr)dlh fo r the ice wall in Figure 2 Values ofclh in Equations (40) and (41) are obtained by and the 12 tide-water glaciers in Tables 1 and 2. 288 Hughes: Calving rates fr om glaciers grounded in water most of the 12 glaciers if () = 0.3 rad and bending creep of 1 1 the ice wall is controled by 8 kg m- 8- for T/m= X 2 101 13 1 easy glide within shear bands, = 8 X 10 kg m- 8- T/m 1 fo r hard glide between shear bands, or fo r TJrn = 1 1 4 X kgm- S- for a combination of easy and hard 1013 glide within and between shear bands. In fa ct, ue/ue = c/h 1.03 ± .01 averaged for all the Ue values in Table 2. 0 These results are summarized in Table 3. The same results are obtained for () = 0.1 rad and TJv = �TJm for n = 3 in Equation (37). However, Equation (30), and therefore Equation (34), are exact only fo r n = in 1 Equation (37) (Popov, 1952, p. 109-13). ___ --'--__ _ -'=- --'-::-:' O.l':- _ --'- -='"OA:---:-'-:----' _ ...L...-O.B 0·1 0.2 0.3 0.5 0.6 07 0.9 1.0 (Pw1p,ld/h DISCUSSION Fig. 10. A logarithmic dependence of calving ratio c/ h on buoyanry ratio (Pw/ P I)d/ h fo r the ice wall in Figure 2 Results in Table 2 are consistent with both easy glide and the 12 tide-water glaciers in Tables 1 and 2. within shear bands and hard glide between shear bands Ta ble 2. A comparison of theoretical calving rates Ue and observed mean annual calving rates Ue fr om the ice walls of Alaskan tide-water glaciers Glacier Equation (38) Equation (39) Equation (40) Equation (41) Ue ue/ue Ue ue/ue Ue uclue Ue ue/ue ma- 1 ma- 1 ma-1 ma- 1 McCarty 241 0.40 94 0.02 137 0.23 120 0.20 Harvard 1333 1.23 624 0.58 895 0.83 808 0.74 Yale 2316 0.66 480 1.38 6274 1.79 2800 0.80 Meares 1001 0.99 468 0.46 670 0.66 604 0.60 Columbia 2504 1.15 1753 0.80 2242 1.03 2046 0.94 Tyndall 947 0.64 1018 0.58 1386 0.94 942 0.64 Hubbard 2858 1.09 2000 0.76 2822 1.07 2334 0.88 Grand Pacific 52 1 2.37 208 0.94 299 1.36 266 1.20 Margerie 875 1.89 273 0.60 400 .86 308 0.66 0 J ohns Hopkins 1610 0.70 1043 0.46 1479 .65 1254 0.54 0 Muir 1567 0.42 2234 0.60 3007 0.81 1718 0.46 South Sawyer 2245 0.70 10690 3.34 7190 2.25 2820 0.88 Ta ble 3. Average calving velocity ratios ue/ue computedfor combinations of wall bending angles ()and ice viscosities T/m and T/v Equation (38) Equation (39) Equation (40) Equation (41) Average u lue 1.02 1.04 1.04 0.71 () = ( c ) 18° = 0.3 rad I - -I 2 13 (kg m S ) 8 x 10 4 X 8 X 10 4 X TJm 1 1013 1013 () = 6° = 0.1 rad -1 -1 13 13 (kg m s 8/3 X 4/3 X 10 8/3 X 10 X TJv ) 1012 4/3 1013 289 Journal of Glaciology contributing to bending creep in the ice slab. This was buoyancy ratio (pw I PI)d/h depicted in Figures 9 and 10 confirmed by relating bending creep to ice fabrics in and is correct, then calving slabs get relatively taller and between shear bands behind the calving ice wall on thinner as water gets deeper prior to flotation of the slabs. Deception Island (Hughes and Nakagawa, 1989). The If backward calving exceeds the fo rward velocity of a calving ratio was clh � 0.3, but the ends of crevasses at tide-water glacier, its calving ice wall retreats, often into calving distance c behind the ice wall experienced deeper water if the glacier occupies a fo redeepened fiord. compression from converging ice flow. This violated the Hence, clh will be reduced and, if the section of the assumption that end effects did not exist or could be glacier weakened by transverse crevasses, as depicted in ignored. The assumption is valid for the ice wall in Figure Figure 8, becomes afloat, tabular calving for which = 2, where clh 0.4. Field studies on tide-water glaciers clh � 1 becomes possible in addition to slab calving for are needed to determine the dependence, if any, of clh on which clh However, tabular icebergs released from « 1. water buoyancy at the ice wall, as measured by this floating section of the glacier will be weakened at (PwI pr)dl h . intervals where shear rupture in vertically bending shear As illustrated by the Greenland tide-water glacier in bands allowed transverse crevasses to open. Flexural Figure 8, a series of transverse crevasses, about equally arching can therefore be accommodated by shear in these spaced, open and widen toward the calving ice wall. If shear bands, as depicted in Figure 12. Since ice slabs are this crevassed section of the glacier became afloat, only weakly held together across these shear bands, the columnar slabs would calve if clh < and tabular tabular iceberg may disintegrate by slab calving even 1 icebergs would calve if elh > 1. Water would fill the while it is being released fr om the floating ice front. crevasses up to sea level. When water in crevasses between Epprecht (1987) described this disintegration for blocks slabs reaches the flotation depth, the crevassed section becomes afloat if water continues to deepen, and the fo rward bending creep of a grounded ice wall also includes the upward flexural creep of a floating ice shelf. Reeh (1968) analyzed upward flexural creep of this kind and found that calving would occur at clh � 0.5 where Ts = Tm = !(O"l - 0"2 ) is maximized, but that calving could also occur at � 1.0 where O" = clh xx O"m was a maximum tensile stress at the top of the flexural arch. In finite-element modeling of these stresses, however, Fastook and Schmidt (1982) determined that 1 I water must at least partly fill the crevasses if they were to migrate downward through the ice and result in slab or tabular calving. The similarities and differences between forward Roe K bending of an ice wall and upward arching of an ice - - - shelf are compared in Figure 11. The main similarity 1-- 1 --- ., between the two is that both are caused by an .::..:::..::...f!.- asymmetrical imbalance between lithostatic and hydro I static horizontal gravity forces in ice and water at the I C E calving ice cliff. This allows upwardly curving shear I I I� J i 1 1 l 1 I bands to develop in both cases. The main difference J r � � 1 � I � I between the two is that removal of basal traction when ice I I floats allows forward bending at the ice cliff also to I � I include downward bending of basal ice, bending below I the buoyancy depth of ice, so that isostatic compensation , - -- ,I requires upward arching of ice behind the ice cliff. ==--=.".. =-=--- If the dependence between the calving ratio clh and w A T E R Fig. 12. A comparison between forward bending of an ice wall and upward arching of an ice shelf due to shear displacements between ice slabs. Top: behind the ice wall, = Ts = 0"xx 0 across crevasses. Widening of the crevasses I is due to increasing fo rward bending of ice slabs between ICE h H crevasses, so that extending flow at the ice surface is caused by bending stress Ts within each slab, not by within l�� 0"xx slabs, and this produces an increasing surface slope ROCK WATER (dashed line) . Bottom: in the ice shelf, upward arching is caused by shear within shear bands that close aft er the Fig. 11. A comparison between fo rward bending of an ice glacier becomes afloat. Th is shear relievesforward bending wall and upward arching of an ice shelfdue to asymmetry (dashed line) caused by 0"xx that is tensile on the top between lithostatic pressure in ice and hydrostatic pressure surface and compressive on the bottom surface, as shown in in water at the calving front. Figure 11. 290 Hughes: Calving rates fr om glaciers grounded in water H, J and K, tabular icebergs released from the north side ofJakobshavns Isbra: (69°IO'N, 5001O'W) in Greenland. Since clh :::::: 1 fo r these blocks, they began to rotate as they detached fr om the ice fr ont: "During the rotation of block H, more crevasses opened in the newly separated section of the glacier and, as a result, blocks J and K became more fr ee (Fig. 4). These then fell over, one after the other, like falling dominoes ... . Within 7 min, not only did the part which had become separated from the glacier by the main fissure disintegrate into blocks, but these blocks had also toppled over. Afterwards, they all lay as large white slabs, several hundred meters across, closely packed together in the fjord (Fig. 2). Their surfaces were relatively smooth and very clean." The calving section of Jakobshavns Isbra: is shown in Figure 13. It consists of ice walls at the north and south sides that become an ice shelf in the center as water deepens along the calving fr ont. CONCLUSIONS A possible decrease in claving ratio clh with increasing water depth (Figs 9 and 10), fo rward bending and upward arching both related to shear bands (Fig. 12) and the calving description by Epprecht (1987) fo r Jakobs havns Isbra: (Fig. 13) raise the possibility that Equation (34) may be a general calving law that has broad applications to all calving glaciers. In a private letter, A. Post (21 June 1990) stated that he recognized fo ur kinds Fig. 13. Calving fr om Jakobshavns Is brdJ, Greenland of calving from Alaskan tide-water glaciers. All fo ur may (orthophotograph by KUCERA International, Inc., with be represented in USGS photograph number 87V2-058 of elevation contours ill meters based on photogrammetric Miles Glacier, supplied by Post and reproduced here as measurements by H. Brecher) . Figure 14: I. Normal slab calving. The ice wall stands on dry land 10 m thick calve at irregular intervals. Normal slab (d = 0) or in shallow water (d « h) and may be fr ozen calving seems to be active on the photograph left side of to the bed. In this case, on Deception Island, a :::::: 0.3 near the calving front of Miles Glacier. Post called this the ice wall due to bending shear and a :::::: 0.06 fu rther "steady-state" calving. upslope due to bed traction (Hughes, 1989). Slabs the height of the ice wall, some tens of meters wide, and under 2. Unstable recessional calving. The calving rate exceeds the Fig. 14. Calving fr om Miles Glacier, Alaska (U.S.G.S. photograph 87V2-058, supplied by A. Post) . 291 Journal of Glaciology ice velocity, causing an ice wall standing in water to the ice wall does not lean forward as much as it bends retreat on a bed that slopes downward inland, thereby fo rward and may even lean backward. Ablation reduces s reducing the lithostatic overburden of ice on the bed in Equation (25) , and in Equation (26) is the fo rward e (d -+ hpr/Pw ) . This increases h and reduces a near the bending angle of the opening crevasse, not of the ice wall ice wall in positive feed-back, so that normal slab calving (see Fig. 6). Secondly, the ice wall is on a flat bed, so is replaced by accelerated calving of thinner slabs that extending flow due to ice riding up on to a terminal may create a calving bay if the calved slabs can be moraine is not considered. Instead, extending flow is evacuated fas ter than the calving rate. Unstable reces caused by C1xx in Equation (14) prior to shear bands sional calving seems to be active on the photograph right becoming transverse crevasses, and by bending creep side of the calving front of Miles Glacier. Post said a causing transverse crevasses to widen toward the calving "retracted stable position" would re-establish normal slab ice wall (see Fig. 12, top). Thirdly, steepening of surface calving at or near the head of tide water if the slope a toward the calving ice wall is primarily due to the downsloping bed sloped upward fu rther inland. widening of transverse crevasses by increasing bending creep (see Fig. 12, top), and this increase of a increases 3. Sudden tabular calving. A large tabular section of the the fo rward ice velocity and the backward calving rate. glacier suddenly and uniformly becomes uncoupled from Fourthly, downward calving of ice slabs requires either the bed, probably due to a build-up of basal hydrostatic crushing or ablating ice at the base of the ice wall. pressure to the flotation pressure (d = hpI/pw). The large However, these mechanisms do not control the calving transverse rift at the rear of the calving section on the rate; see Hughes and Nakagawa (1989). Fifthly, the photograph right side of Miles Glacier could have calving rate given by Equation (34) is most reliable when released this section as a single tabular iceberg if Miles the calving ratio c/h and viscoplastic exponent n are both Glacier had been floating. Post described these calving small; ideally, c/h « 1 and n = 1. Since 0.1 < c/h < 0.4 events as being virtually identical to the calving of tabular is observed fo r ten tide-water glaciers and n = 3 fo r ice, icebergs fr om the floating front of ]akobshavns Isbne. If Equation (34) only gives approximate calving rates. Use the planar dimensions of the tabular section exceed its of Equation (34) is even more problematic, given the thickness, a tabular iceberg moves away fr om the ice front ambiguities in specifying TJv and Even so, it has a e. intact. Otherwise, it rears up on one side, rolls over and fo undation in theory and in observation. disintegrates completely within minutes. A less obvious constraint on the bending-creep calving mechanism is the time needed to fo rm a shear band. In 4. Frequent buoyancy calving. Large sections of the ice front laboratory creep experiments, about 25 d were needed calve fr equently when they become fu lly buoyant when Ta was about I bar (Hughes and Nakagawa, 1989). (d > hpI/pw), while other grounded sections remain This time constraint can be illustrated by calving fr om I intact (d < hpI/pw). These two conditions seem to apply Columbia Glacier, fo r which Uc = 7 m d- , h = 160 m and to the photograph right side and the photograph left side, c/h = 0.22, so that an ice slab having c = 35 m calves respectively, of Miles Glacier. For other tide-water every 5 d. This means that bending creep must begin glaciers, Post observed this calving process when the about 210 m behind the calving fr ont if a shear band is to calving fr ont is retreating into water up to 400 m or more fr acture by shear rupture 35 m behind the calving fro nt. in depth, and it seems to be the dominant fo rm of calving fr om Columbia Glacier during the current rapid retreat. However, water in front of Miles Glacier is not this deep. ACKNOWLEDGEMENTS All of these fo ur kinds of calving might be accom modated by the bending-shear mechanism quantified in I thank A. Post fo r sharing his insights on calving fr om tide-water glaciers, R. Hooke fo r comments on an earlier Equation (34). The variables in this equation, h, d, c and version of this manuscript and P. Prescott fo r sharing his with a � at the calving fr ont, determine the calving e, e unpublished photogrammetric measurements of surface velocity uc. They have been observed by Post to be elevations, slopes, velocities and strain rates along the important, and they seem to have at least the correct calving front ofJakobshavns Isbra:. This work was fu nded qualitative relationships to observed calving rates. Data by U .S. National Science Foundation grant DPP- are insufficient to determine whether Equation (34) is a reliable quantitative expression fo r calving rates along the 8400886 and Battelle, PNL, subcontract 0 17112-A-B I. great variety of calving fr onts provided by Jakobshavns Isbne. However, it is clear that calving events are associated with new crevasses along the calving front REFERENCES and within one or two ice thicknesses behind the calving front that have no apparent relationship to older crevasses Andersen, B. G. 1981. Late Weichselian ice sheets in formed fa r upstream and carried passively to the calving Eurasia and Greenland. In Denton, G. H. and T.]. front (unpublished photogrammetric measurements by P. 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