The University of Maine DigitalCommons@UMaine
Earth Science Faculty Scholarship Earth Sciences
1992 On the Pulling Power of Ice Streams Terence J. Hughes University of Maine - Main, [email protected]
Follow this and additional works at: https://digitalcommons.library.umaine.edu/ers_facpub Part of the Earth Sciences Commons
Repository Citation Hughes, Terence J., "On the Pulling Power of Ice Streams" (1992). Earth Science Faculty Scholarship. 64. https://digitalcommons.library.umaine.edu/ers_facpub/64
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, Vol. 38, No. 128, 1992
On the pulling power of ice streams
T. HUGHES Department of Geological Sciences and Institute fo r Quaternary Studies, University ofMaine, Orono, Maine U.S.A. 04469-0110,
ABSTRACT. Gravity wants to pull an ice sheet to the center of the Earth, but cannot because the Earth's crust is in the way, so ice is pushed out sideways instead. Or is it? The ice sheet "sees" nothing preventing it from spreading out except air, which is much less massive than ice. Therefore, does not ice rush forward to fill this relative vacuum; does not the relative vacuum suck ice into it, because Nature abhors a vacuum? If so, the ice sheet is not only pulled downward by gravity, it is also pulled outward by the relative vacuum. This pulling outward will be most rapid where the ice sheet encounters least resistance. The least resistance exists along the bed of ice streams, where ice-bed coupling is reduced by a basal water layer, especially if the ice stream becomes afloat and the floating part is relatively unconfined around its perimeter and unpinned to the sea floor. Ice streams are therefore fast currents of ice that develop near the margins of an ice sheet where these conditions exist. Because of these conditions, ice streams pull ice out of ice sheets and have pulling power equal to the longitudinal gravitational pulling force multiplied by the ice-stream velocity. These boundary conditions beneath
and beyond ice streams can be quantified by a basal buoyancy factor that provides a life-cycle classification of ice streams into inception, growth, mature, declining and terminal stages, during which ice streams disintegrate the ice sheet. Surface profiles of ice streams are diagnostic of the stage in a life cycle and, hence, of the vitality of the ice sheet.
INTRODUCTION of the pulling fo rce F.", and ice velocity at that point, both measured horizontally along directifixon of flow: The best definition of an ice stream still may be the one Henri Bader gave three decades ago: an "ice stream" x (1) is something akin to a mountain glacier consisting of a broad accumulation basin and a narrower valley glacier; where is averaged through an ice column of height but a mountain glacier is laterally hemmed in by rock h in a flowfix band of width w. For steady-state flow, is slopes, while the ice stream is contained by slower the equilibrium mass-balance velocity u. Ux moving surrounding ice. The edges of the ice stream Analyzing the pulling power of ice streams involved are often crevassed, and the surface tends to be concave as the ice is "funnelled" down. Many of- the large out both old and new concepts. These concepts will now be identified. let glaciers in Greenland, particularly in the south, are the narrow outlets of large ice streams which reach back Gravitational force, F z = The gravitational force many scores of miles into the ice sheet. The largest ones Pg. is the product of the mass of ice (a scalar) multiplied by flowing into Disko Bay have corresponding depressions in gravity acceleration (a vector directed toward Earth's the glacier floor. An interesting question is whether the center) , so is a vector vertically downward. ice stream makes the depression or vice versa; and an in F:
teresting hypothesis is that the ice stream, once started, Pulling force, P,r; = Fp. The pulling force is the product is self-perpetuating because its ice mass, warmed up by of lithostatic pressure in ice (a scalar) multiplied by the heat of internal friction, has a lower viscosity than the vertical area across which the pressure is relieved (a vec surrounding ice (Bader, 1961). tor whose length is proportional to the area and whose Insights into these ideas may be found in the pulling direction is away from the ice mass) , so Fp is a vector power of ice streams. As defined here, pulling power is directed horizontally downstream. postulated as the ability of an ice stream to reach deep
into an ice sheet and pull out the ice. Formally, pulling Pushing force, Fx = Fp. The pushing force is the prod power P,r; at some position on an ice stream is the product uct of lithostatic pressure in ice (a scalar) multiplied by
125 Journal of Glaciology the vertical area across which the pressure is exerted (a ter (cPc 0), or both. The pulling fo rce is maximized vector whose length is proportional to the area and whose for full basal= buoyancy and is zero for no basalbu oyancy. direction is toward the ice mass), so is a vector dir Ice-bed coupling produces traction forces and ice-shelf ected horizontally downstream. Fp buttressing produces kinematic forces, both of which constitute the braking fo rce that balances the pulling Braking force, Fx = FB. Braking fo rces are either hor force. izontal traction forces or horizontal kinematic forces. A traction braking force is the product of shear stresses Background for the concept of pulling power will be axy and axz acting against side and basal surfaces of an presented, followed by an elaboration on the distinction ice stream, where shear traction exists, and the areas between pushing and pulling, and an elaboration upon of these traction surfaces. A kinematic braking fo rce ice-bed coupling and ice-shelf buttressing. Then the hor is the product of longitudinal deviator stress a�x that izontal fo rce balance is presented for sheet flow, shelf causes kinematic straining, and the cross-sectional area flow and stream flow. Pulling power is then related to affected by kinematic straining. The analyses of sheet four ice-stream surface profiles, (1) with minimal basal flow by Orowan (British Glaciological Society, 1949) and buoyancy, (2) with basal buoyancy continuously decreas of shelf flowby Weertman (1957a) were classic examples ing upstream, (3) with equilibrium basal buoyancy, and of using traction and kinematic horizontal braking forces, (4) with maximum basal buoyancy. These conditions respectively. of basal buoyancy become the basis for postulating a life-cycle classification for ice streams consisting of in HorizontaL force baLance, Fp - FB The horizon ception, growth, mature, declining and terminal stages, tal force balance requires dynamic equi= libriumO. between with the possiblity for rejuvenation at any stage after horizontal force Fp, whether pulling or pushing, both inception. A way to model the ice-stream life cycle is of which require gravity acceleration (which appears in presented, and driving the life cycle by pulling power is the lithostatic pressure), and braking forces FB, whether discussed. Next, pulling power is postulated as the mech traction forces or kinematic forces, both of which require anism for disintegrating marine ice sheets by its role in horizontal ice velocity (which causes straining). down-drawing ice sheets, discerping ice ridges between ice streams, controling the erosive power of ice streams, Pulling stress, a�x = �(axx - ad. The pulling stress regulating ice-shelf buttressing, widening ice streams and is an axial deviator stress in the direction of horizontal causing ice streams to surge. Finally, ways to initiate ice flow. It represents a deficiency of lithostatic pressure stream flow are discussed, and results for marine ice in the horizontal x-direction of ice flow, creating a relat streams are generalized to include terrestrial ice streams. ive vacuum in that direction that is filled by inrushing ice. Hence, strain rate Exx is tensile and proportional Background to pulling stress a�x' where a�x is the average difference Work related to the pulling power of ice streams dates at between the reduced lithostatic stress axx and the full least from the old dispute between Lliboutry (1958) and lithostatic stress azz, both of which are compressive, so Nye (1958) as to why transverse crevasses open when a that a�x is tensile. glacier enters a narrow valley. Lliboutry (1958, fig. 4) maintained "To open a crevasse we need a strain, not a Pulling power, Px = F'.-r;i1x. Pulling power is the product stress." His analog was a plastic slab being stretched bet of the horizontal pulling force Fp and mean ice velocity ux' Therefore, pulling power requires both gravity ac ween rolls, such that transverse compression by the rolls celeration (which causes the pulling force) and ice veloc causes longitudinal extension that opens pre-existing ity (which causes the braking force). Pulling power is a transverse cracks. Nye (1958, fig. 1) countered, "I say measure of the vitality of an ice stream, which changes a stress." His analog was a plastic slab cut into blocks during the life cycle of an ice stream as gravitational pot and compressed between frictionless parallel plates, the ential energy (represented by ice elevation) is converted cuts being the pre-existing transverse cracks. Transverse into kinetic energy (represented by ice veloci ty) as stream compressive stress exerted by the plates causes longitud flow moves the ice mass horizontally to lower elevations. inal extensional strain in the slab, but does not open the cracks. "If, however, we apply a (longitudinal) ten Basal buoyancy factor, 4>= (pwd/PIh)4>G' The basal sile stress ... the whole thing comes apart." Nye (1958) buoyancy factor quantifiesice-bed coupling and ice-shelf had proposed pulling power as a disintegration mech buttressing, as the two sources of traction and kinematic anism. His analog was for a valley glacier, but it has braking forces that resist the pulling force. Ice-bed coup significance for ice streams in view of the observation by ling is caused by the fraction of basal lithostatic pressure Bader (1961) that "An ice stream is something akin to in ice of height h and density PI that is supported by a mountain glacier (that enters) a narrow valley ..." . the bed, instead of by the basal hydrostatic pressure for Another indication that ice streams pull ice out of ice sheets was the analysis by Weertman (1963) of outlet water of depth d and density Full ice-bed coup = = (lW. glaciers draining a.n ice sheet fringed by mountain ranges. ling (4) 0) occurs when d No ice-bed coupling = = O. Although his analysis was for convex surface profiles, if (4) 4>G) occurs when d (pJ/ )h. Full basal buoy ancy (4) = 1) occurs when an icePw stream is not coupled it is applied to the concave surface of a marine ice stream to the bed and not buttressed by an ice shelf (cPc = 1). on a horizontal bed, the longitudinal strain rate is a max
No basal buoyancy (4) = 0) occurs when an ice stream imum across the surface-inflection line at the head of the is either fully coupled to the bed (d = 0), or fully but ice stream and a minimum across the basal grounding tressed by an ice shelf grounded around its entire perime- line at the foot of the ice stream, so that basal frictional
126 Hughes: On the pulling power of ice streams heat is conducted to the surface faster at the head than line at the heads of ice streams to retreat, ultimately at the foot of the ice stream. Chances for a frozen bed collapsing the West Antarctic ice sheet (Hughes, 1973). are therefore greater at the head, with stream-flow slid Necking has been observed directly by Whillans and oth ing on a thawed bed pulling sheet flow anchored to a ers (198 7) in longitudinal velocity gradients at the head frozen bed. of Ice Stream B (Fig. 1). The conclusion that necking is The ice-stream analysis by Weertman (1974) involves caused by pulling also emerges in using slip-line theory to pulling. He considered an ice stream as the transition represent the transition from sheet flow to stream flow zone between an ice sheet and an ice shelf in which to shelf flow as analogous to plastic material (ice) be the down-slope increase in a longitudinal tension stress ing successively drawn through a die, compressed bet allows a corresponding reduction of basal shear stress, ween plates and indented by a punch. Representing the and thereby produces a concave profile. Hence, concave sheet flow to stream-flow transition as being analogous "necking" of the ice stream is produced by a longitudinal to pushing ice through a die (Hughes, 1977, fig. 17) did tensile stress, just as are the transverse crevasses anal not explain the great transverse crevasses arcing around yzed by Nye (1958). A longitudinal tensile stress requires the head of Byrd Glacier (Fig. 1) in the zone of converg a longitudinal pulling force. ing flow. These crevasses would fo rm if Byrd Glacier The concave surface profile of West Antarctic ice was pulling ice out of East Antarctica, and the analog streams can be interpreted as a "necking" phenomenon was to ice being Ilulled through the die (Stuiver and oth that was causing the maximum-slope surface-inflection ers, 1981, fig. 8-7). This observation was the basis for
6S
I , , , I I I 'le I
.. . r?, \ '.
WEST ANTARCTICA EAST ANTAR CTICA
Fig. 1. Location map for selected Antarctic ice streams.
127 Journal of Glaciology an early attempt theoretically to quantify the concave distance Note that F.T is directed toward the remain ing ice mass,x. so that by definition is a pushing profile of ice streams (Hughes, 1981a), and to employ Fx Fp retreating ice streams as the vehicle for collapse of the fo rce. If TO is constant, equating the pushin= g force with West Antarctic ice sheet in a computer simulation (Stu the traction force gives (PI9/2To)h2. This is the con iver and others, 1981, fig. 8-7). It can be argued that vex surface profile of anx ice = sheet predicted by plasticity stream flowalso pulls flankingice into the ice stream, so theory, in which TO is the yield stress. that the proper analog with a plastic material between Horizontal pulling can be induced by removing the ice
x, �PI9Wh2 parallel plates is that extending plastic flow actively pulls sheet over distance so that Fx becomes a the plates together, rather than extending flow being a pulling force resisted by basal traction= force Tow(L ) passive response to the plates being pushed together. over distance L where L is the length of the - flow x band from the ice- xdivide to the margin, as shown in Pushing versus pulling Figure 2c. Note that F.T is directed away from the re Figure 2 illustrates the distinction between pushing and maining ice mass, so that by definition = is a pulling. Figure 2a shows horizontal spreading of an ice Fx Fp pulling force. Pulling extends only to the ice divide be sheet as resulting from vertical lowering caused by a cause TO reverses sign across the ice divide. Equating the downward pulling force equal to the mass of the ice F= fo rces for constant TO gives L - x = (PIg/2To)h2 as the sheet multiplied by gravity acceleration g. When the ice distance over which F.T is exerted for a given h and TO sheet is in steady-state equilibrium, the surface profile in Figure 2c. With F.T a pulling fo rce instead of a push is unchanged because accumulation rates match the low ing force, the ice surface over distance L - is quickly ering rate and ablation rates match the spreading rate. pulled down by as ice is pulled fo rward byx A sim Vertical pulling by gravity causes horizontal pushing. As F; Fr;. ilar result is obtained by not removing the ice sheet over seen in Figure 2b, if the ice sheet beyond distance from distance but, instead, having TO be large over - the ice margin is removed, it can be replaced by xa hor and smallx over x, which produces the concave surfaceL x izontal pushing force = (�PIgh)wh, Fx which is resisted of stream flow in Figure 2d. An ice stream can end as by basal traction force TOWX over distance in a flow a floating ice shelf, beneath which TO is zero (shown as band of constant width w, where PI is ice density,x h is terminal region 1), or as a grounded ice lobe, beneath ice height at position and TO is basalshear stress over which TO increases to the ice margin (shown as terminal x region 2). A marine ice stream supplies an ice shelf. A terrestrial ice stream supplies an ice lobe. Marine ice streams are the subject of this investigation because pre dictions can be tested by field studies on the numerous present-day marine ice streams. In Figure 2d, pulling fo rce Fp at distance from the ice-stream grounding line or the ice-lobe terminusx will be net horizontal force !:J.F.T = �PIgwh2 - �pwgwcf2, where pw is water density and d is the depth of water that would produce the hyd b - rostatic pressure of basal meltwater at As presented in Figure 2, horizontal fox.rce F.T is a push ing fo rce for sheet flow and a pulling force for stream flow, both produced by gravity, and both inducing traction fo rces that resist gravitational motion. Hence, F.T will be used to represent gravitational fo rcing, with F.T = Fp causing pushing or pulling, and Fa will represent brak ing fo rces caused by stresses that resist pushing or pulling motion.
Ice-bed coupling and ice-shelf buttressing As presented in Figure 2, stream flow is generated from sheet flowby reducing ice-bed coupling toward the term inus of a flowband. If that is the case, marine ice streams are the major dynamic links connecting a marine ice Fig. 2. Producing stream flowfrom sheet flow. sheet with its floating ice shelves, and stream flow is (a) In sheet flow, gravitationaL puLLing force Fz transitional between sheet flow and shelf flow. However, deforms the ice sheet from the soLid profiLe to stream flow cannot develop from sheet flow if reduced the dashed profiLe. (b) Inner dashed part of the ice-bed coupling is completely offset by increased ice ice sheet can be repLaced by horizontaL pushing shelf buttressing. To illustrate this, consider an ice cube force Fx . (c) Outer dashed part of the ice sheet in a pan. If water is added to the pan, an ice cube of h can be repLaced by horizontaL puLLing force Fx. height will begin to floatin water of depth d when the (d) HorizontaL puLLing converts the convex sur lithostatic pressure of ice and the hydrostatic pressure face of sheet flow into the concave surface of of water give the same overburden pressure ao on the stream flow, with stream flow becoming a float bottom of the pan: ing ice shelf in terminus 1 or a grounded ice Lobe in terminus 2. (2)
128 Hughes: On the pulling power ofice streams where is ice density, is water density and is grav ity accelerPI ation. Ice-bedP\V coupling is nil when gEquation (2) is satisfied. Now imagine that the ice cube is a large tabular iceberg and the pan is a larger lake. Elimination of ice-bed coupling allows pulling force Fp to spread the iceberg radially, in the manner analyzed by Weertman (1957a), until the iceberg grounds along the shoreline of the lake, thereby producing a braking force FB that com pletely cancels Fp. Hence, ice-bed coupling is removed in the center of the lake by a gain of basal buoyancy, and ice-shelf buttressing is attained around the perimeter of the lake by a loss of basal buoyancy. The above considerations suggest that the pulling power of a marine ice stream is controled by basal buoy ancy. Pulling power is increased when ice-bed coupling
"- beneath an ice stream is decreased by basal buoyancy. " " Pulling power is decreased when ice-shelf buttressing be h x_ yond an ice stream is increased by a loss of basal buoy ancy around the grounded parts of the ice shelf. Both .-If-·�(---ls� ice-bed coupling and ice-shelf buttressing can therefore �------l be quantified by a basal buoyancy factor 129 Journal of Glaciology and Robin, 1973), and the "pseudo ice shelves" of Ice band. Assumptions are (1) dynamic equilibrium exists, Stream C (Robin and others, 1970); examples of cl>a 1 (2) longitudinal gradients in stretching stress are small, are the floating ice tongues of Thwaites, David, Ninnis= and (3) the T-term can be neglected (Paterson, 1981, and Mertz Glaciers; and examples of 0 < cl>a < 1 are the p. 164). Assumption (1) is justified if pulling force Fp Ross and Ronne Ice Shelves. is added to braking force FB such that Fp FB = Refer to Figure 3, for which d changes along an ice Assumptions (2) and (3) are reasonable for a+ flow bandO. stream of length Ls and cl>a = 1 for an ice shelf of length of constant width on a horizontal bed (Muszynski, 1987; LF. Over length L - Ls, sheet flow with cl> = 0 exists Muszynski and Birchfield, 1987; Van der Ve en, 1987). when the bed is frozen, or thawed, but with only a basal The pulling fo rce in all cases is merely the expression water film. Over length Ls, stream flow with cl> = 0 of Newton's second law, Fp being gravity acceleration exists when basal water is only thick enough to drown· acting on the mass of the ice column, with Fp in the bedrock projections up to the controling obstacle size in horizontal x-direction being an average differential pres the Weertman (1975b) theory of basal sliding, and cl> sure exerted on faces of the ice column that are normal 1 when basal water saturates basal till or sediments so= to the x-direction. The braking force in all cases arises they are unable to support a basal shear stress; otherwise from motion or deformation of the ice column, with FB o < cl> < 1 in the ice stream, depending on how d varies consisting of the sum of stresses resisting the motion or along Ls, where d is statistically averaged for basal water deformation multiplied by the area of the ice column conditions across a given width w of the ice stream. upon which these stresses act. None of this is new, but Ice-bed coupling and ice-shelf buttressing are quan it will now be reviewed for sheet flow and shelf flow, be tified by cl> to provide a means for assessing the pulling fore analyzing stream flow as being transitional between power of ice streams using Equation (1). The plan to be these two. followed consists of: Force balance for sheet flow 1. Presenting a force balance for stream flowth at reduces Pushing force Fp for an ice sheet on a horizontal bed is obtained from the horizontal lithostatic forces shown to force balances for sheet flow when = 0 and shelf cl> in Figure 4 (right). A vertical ice column has down flow when cl> = l. slope height hand upslope height h + flh along incre 2. Deriving ice-stream profiles for a range of cl> values and computing pulling power along these profiles for mental length L1x over which flow-band width w is con steady-state equilibrium. stant. The pushing force per unit width is the difference 3. Classifying ice streams in a hypothetical scheme in between the areas of force triangles 1 and 2, defined as which the pulling-power curve is diagnostic of the pos the mean lithostatic pressures acting on the upslope and ition of an ice stream in its life cycle. downslope sides of the ice column, respectively. There 4. Applying the concept of pulling power to disinteg fore, for an ice-sheet flow band of width w: ration of marine ice sheets, notably in West Antarc tica. Fp - [�Plg(h + flh)] [w(h flh)J = + [!Plgh] [whJ � -Plgwhflh (4) The goal of this plan is to provide numerical models of + ice sheets with a parameter cl> that quantifies ice-bed coupling and ice-shelf buttressing for the ice streams that drain the ice sheet. Rapid collapse of marine ice sheets can then be simulated by allowing cl> to represent increases in ice-stream pulling power resulting from un h+llh" coupling and unbuttressing. h\." FORCE BALANCES ALONG A FLOW BAND Figure 3 shows an ice sheet flowband of constant width on a horizontal bed. The flow band begins as sheet flow (right) and ends as shelf flow (left), with stream flow (center), being a transition from sheet flow to shelf flow. Constant width and a horizontal bed are specifiedso the transitional nature of stream flow can be analyzed in Fig. 4. Horizontal forces on ice columns hav its simplest form, without complications introduced by ing constant width and no side traction. Hor laterally converging or diverging flow and variable bed izontal gravitational forces per unit width are topography. The origin of rectilinear coordinates is the the areas of triangles 1, 2 and 4, and rectangle ice-shelf grounding line, with x horizontal and positive 3. Traction force is basal shear stress TO times toward the ice divide and z vertical and positive upward. the basal area of the ice column. PuLLing force The flow band hasfloating length LF, and an ice stream is tensile deviator stress 2a�x times the trans of length Ls over grounded length L, with hG being ice verse cross-sectional area on the landward side height at the grounding line separating shelf flow from of the ice column. FoLLowing WhiLLans (1987), stream flow and hs being ice height at the inflection line the role of basal water pressure can be emphas separating stream flow from sheet flow. ized by distinguishing lithostatic pressure in Force balances are computed for ice columns in the triangles 1 and 2, and rectangle 3 from hydro sheet-flow, stream-flow and shelf-flow parts of the flow static pressure in triangle 4. 130 Hughes: On the pulling power of ice streams where the term containing can be ignored. Note (t:.h)2 is constant over length !1x. Braking force FB therefore that Fp is a force gradient, because it increases with results from a�x, which produces kinematic strain rate ice-thickness change £1h. Lithostatic pressure PIg(h + Exx as follows: pushes against area to produce !£1h)>=:::; PIgh w£1h Fp. Pushing gravitational motion induces creep controled by deviator shear stress axz, where = axz at z = 0 is the basal shear stress. Since resistanceTO to creep is provided by basal shear stress acting on basal area (9) w£1x,and ax: = To(l-z/h), theTO braking force is: where a� is expressed in terms of the strain rate Exx of = Tow£1x= axxhw£1x/(h -z) = Ahwt:.x/(h -z) x FB E��n (5) ice in longitudinal tension for the creep law: where ax: is expressed in terms of the strain rate Ex: of ice in simple shear fo r the creep law (10) in which Exx is constant through h over length t:.xand (6) T a�x for constant w. =When the ice column is in dynamic equilibrium, = in which T is the effective shear stress, is a viscoplastic n and (Weertman, 1957a): Fp exponent, A is an ice-hardness parameter, A averaged -FB through h is A and = ax: for simple shear. Exx = [prgh(l- pr/pw)/4Ar = [ao(1- pr/pw)/4Ar When the ice columnT is in dynamic equilibrium, = Fp (11) and (Nye, 1957): - -Fn where Exx = Ezz for constant w and volume conservat ion, and ao = Prgh is the basal lithostatic and hydrostatic = (h z)£1h £1x = (1-z/h)/ (7) Ex: [prg - / A r h Ar pressure. where = Prgh!1h/!1xis often called the "driving Force balance for stream flow stress" TOfor an ice sheet. Actually, is a braking stress Pulling fo rce Fp for an ice stream having longitudinal ex that resists ice motion. It takes onTO the role of a driving tension and both basal and side traction is obtained from the horizontal lithostatic and hydrostatic fo rces shown in stress only when Fp is equated with FB, as only Fp ex presses the product of ice mass and gravity acceleration. Figure 4 (center) . A vertical ice column has downslope height hand upslope height h + t:.halong incremental Force balance fo r shelf flow length £1x,over which basal water would rise in tem Pulling force for an ice shelf having only longitudinal perate bore holes to an average depth d above the bed. extension andFp no side traction is obtained from the hor The average d is based upon the fractions of the bed in izontal lithostatic and hydrostatic fo rces shown in Fig basal area w having no buoyancy ( 131 Journal of Glaciology low this height to produce the horizontal pushing force at the foot of a marine ice stream. Stream flow be per unit width given by the area of rectangle 3 in Fig comes laminar sheet flow when TS = O'�x = !:l(J'�xj /).x = ure 4. The fourth term is the mean hydrostatic pressure = 2TSh/).x ToW!:lX 2(0'�x Fo !:l(J'�x)w(h !:lh) ity, which pulls the glacier fo rward, and resistive forces, + + + + which hold it back." His table I compares the conven = (2Tsh Ts!:lh TOW)/).X tional division of stresses into spherical and deviatoric + + (13) components, on the one hand, with his lithostatic and resistive components, on the other hand, where his litho static force is gravitational and gradients in it produce where !:l (J'�x is the change in O'�x along !:lx and the term containing /)'O'�x!:lh can be ignored. his driving stress for glacial ice. In his figure2, Whillans (1987) showed a nearly exponential decrease of his driv When the ice column is in dynamic equilibrium, Fp = ing stress from the ice divide of West Antarctica to the -FB and the fo llowing expression for !:lhj/).x is ob tained: calving front of the Ross Ice Shelf, along a flowline con taining Ice Stream B. This same depiction of stream !:lh 2(hjw)TS TO flow as being transitional from sheet flow to shelf flow is shown in Figure 3 as resulting from a general increase !:lx h - (hjW)T+ - 20'�x PIg S in basal buoyancy from 'the surface-inflection line to the 2 � !:l �x 2 ) 132 Hughes: On the pulling power Dj ice streams tal length Llx along the ice stream, such that this term vanishes as Llxapproaches zero, leaving only the Ls where pulling stress 2(J"�x' given by Equation (15), acts on cross-sectional area who The G term in the vert ically integrated force balance (Paterson, 1981, p. 164) j is approximately the gradient of Fx. Hence, the pulling fo rce can be understood in relation to vertically integ rated momentum conservation. Mean ice velocity u can .-w-. .-w_ �w_ �w_ be computed to a good approximation by the fo llowing e expression fo r conservation of ice flux and steady-state ice ice ice j c equili bri um: V'J\rvr c o k rock rock rock hu = ushs ( a - b)(Ls - x) (17) + Fig. 6. Four possible cases of basal. water con where x = 0 at the grounding line of a marine ice stream, figurations shown as average d variations in x = Ls at the head of the ice stream, where Us is the Figure 3 (bottom) and average 133 Journal oJ Glaciology Case III shows wide water channels separated by nar row ridge crests along the entire length Ls of the ice stream, such that cP � (he/h)cPe averaged across w along Ls. This condition exists when basal water depths nearly match bedrock topographic variations in the direction of ice flow, and no net current exists in the water. It dif fers from case I only in that more basal water is present all along Ls, so that hydrostatic equilibrium is main tained in the water channels (channel water pressure where h > he is the same as at h = he , so there is no pressure-gradient current). Case IV shows a pad of water-soaked till or sediments between basal ice and bedrock, with deformation and permeability of the pad being such that et> et>e across w = III { and along Le (local basal water pressure is that required � � " � /- I � for the buoyancy of local ice thickness h, whereas in case 0, 0 A "A " \ " •I _ �,,,...... ;-, ,• ., , \.• _ / , • " "� ._"\ ,. _f ..- � - .. .:: : III it is that required for the buoyancy of ice thickness " .... ::.. " he at the grounding line). This situation can exist when t permafrost having a high ice fraction becomes thawed, w t and it requires thawing at x = Ls to produce new melt water as fast as pad deformation and permeability allow basal water to be discharged at Case IV would then require the surface-inflectionx line = O. of the ice stream �.�------Ls------'� to retreat into the ice sheet, as it is a thermal boundary x (the melting isotherm), and this retreat collapses the ice �4�------(; e sheet. i f low The amount of basal water along Ls depends on basal Fig. 7. A basal water configuration in plan melting and freezing rates in cases I through IV, and in view and longitudinal cross-section for the et> other cases that could be postulated. In cases I, 11 and variation of case I through case IV in Figure Ill, the basic assumption is that the ice overburden is 5 that is different from the basal water con mostly supported by basal water pressure above water figuration for these cases shown in Figure 6. channels and by bedrock between water channels (even The bed consists of hills and hollows instead of though the water film between channels has the over ridges and valleys, with et> = 0 in white areas, burden ice pressure). These cases require ice-stream ap cP = (hG/h)et>G in connected black areas, and plications of subglacial hydrology theories by Weertman cP = cPe in disconnected black areas, for which (1986), Kamb (1987) and Lliboutry (1987), among oth cPe = In case IV, basal water is trapped in ers. Case IV can be tested by theories of ice streams on a glaco.iall y eroded trough dammed by glacially deformable and permeable beds developed by Alley and deposited sediment till (dotted) at the ground others (1987), Lingle and Brown (1987), Alley (1989a, b) ing line, situations found in the inter-island and MacAyeal (1989), for example. channels and fiords of deglaciated landscapes. Figure 7 shows an alternative to Figure 6 for pres The vertical is exaggerated. enting et> variations in terms of basal water distribution. Bedrock hills and hollows determine the distribution in stead of bedrock ridges and troughs aligned in the dir these cases, et>G = 0 for ice above isolated bedrock lakes, ection of the ice stream. In case I, lakes exist only in but cPc > 0 if the till sill is permeable. the bottoms of isolated hollows, which act as slippery A water film over hilltops distributed like islands in a spots beneath the ice stream and a thin water filmex subglacial lake allows the hills to be local "sticky spots" ists elsewhere, so that et> = 0 along Ls. In case basal 11, where bedrock bears the ice overburden, and therefore water begins as lakes in isolated hollows and ends sur retards ice flow. Applying the Weertman (1986) anal rounding isolated hilltops along Ls, with et> = cPc for ice ysis of basal water pressure to ice streams, the sticky above isolated lakes, et> (hejh)cPe for ice surround = spots in Figure 7 would be the upstream side of bedrock ing isolated hills, and et> = 0 for ice in remaining areas hills, where a thin water film allows bedrock to resist where a thin water film allows bedrock to support the stream flow. Whether this situation exists in ice streams ice overburden. In case Ill, the thin water film exists is unknown, but Whillans and Johnsen (1983) observed only on isolated hilltops, which act as sticky spots where something like it, possible frozen patches on a thawed cP 0, but for the ice stream in general, et> = (hcjh)et>G = bed, in sheet flow near Byrd Station in West Antarc along Ls. In case IV, sediments beneath the ice stream tica (see Fig. 1). Vornberger and Whillans (1986) have have been eroded away and redeposited as till at the postulated sticky spots in Ice Stream B (see Fig. 1) as grounding line, leaving basal water in a bedrock trough producing bands of surface crevasses. dammed by a till sill, so et> = cPe along Ls for ice above the trough. Case IV produces fore-deepened troughs in Case Stream flow with minimal basal the inter-island channels, straits and fiordsof continental buoyancy1. shelves that had been glaciated by marine ice sheets. In Stream flow with cP = 0 was used to produce ice-stream 134 Hughes: On the pulling power Dj ice streams profiles in the CLIMAP reconstructions of global ice drown bedrock projections that control the basal sliding sheets at the last glacial maximum (H ughes, 1981a). velocity, but not the large projections, which therefore Ice streams without basal buoyancy may exist as outlet continue to support nearly all of the ice overburden. For glaciers through fiordsin coastal mountains, where ice is smaller projections, the ice overburden is supported by often thick and has a high surface slope for ice streams, the water layer. Hence, the fast sliding velocity of an ice implying some traction from a bedrock floorthat is rough stream can be attained without ubiquitous basal buoy and may be partly frozen. ancy, provided there is no ice-shelf buttressing (CPc= 1) Considerable attention has been fo cused on long and the basal water film drowns ratc-controling bedrock itudinal deviator stress (J�x and its downslope gradient projections (Weertman, 1986). o(J�x/ox in ice streams, notably by Weertman (1974), In ice streams reconstructed for CLIMAP, the bed Bindschadler and Gore (1982), Alley (1984), Alley and was assumed to be rigid, impermeable and rough in the Whillans (1984), McInnes and Budd (1984), Van der manner specified in the basal sliding theory for sheet flow Veen (1985, 1987), Muszynski (1987), Muszynski and proposed by We ertman (1957b). Basal sliding fo r stream Birchfield (1987) and Whillans (1987). These studies, flow was attained by having the bedrock projections of notably those of Van der Veen (1987) and Muszynski height that control the sliding velocity fo r sheet flow, (1987), show that o(h(J�x)ox is relatively unimportant be progressA, ively drowned in the downslope direction by in an ice stream. That o(h(J�x)/ox could be ignored in a basal water layer of thickness >- that increased from ice streams fu lly buttressed by an ice shelf was assumed � >- 0 at x = Ls to >- = at x = 0 along the ice stream. by Hughes (1981a) in the CLIMAP ice-sheet reconstruc The variation of >- with Ax over length Ls was unknown, tions. This assumption was based on the proposition = but on the assumption that o>-/fJx 0 at both x = Ls that (J�x at the ice-shelf grounding line is the buoyancy and x = 0, where x = 0 was the grounding line, the pulling stress minus a braking back-stress imposed by fo llowing cosine variation was employed: the ice shelf: 2 = �A [1 + cos (7rx/ Ls)] = cos (7rx/2Ls). (22) (19) >- A The undrowned height of these projections was - where he and (Jeare the ice thickness and the back and the bed-roughness factor for stream flow wouldA thenA, stress at the grounding line. For full ice-shelf buttressing, be: Hughes (1981a) assumed that: - >- A 1 - cos 2 7rX )] (20) ---;:;-A = A' [" ( 2Ls A . ? = -Sln- 7r�; (23) Taking d = 0 in Equation (3) maximizes ice-bed coup A' 2Ls ling beneath an ice stream and (Je given by Equation (20) ( -) maximizes ice-shelf buttressing beyond an ice stream. where was the distance between projections of height Equation (20) holds exactly only when grounding lines and A' N was a constant bed-roughness factor for completely enclose an ice shelf, making it a "pseudo ice / sheetA flowA in the Weertman (1957b) theory. shelf" above a subglacial lake (Robin and others, 1970; Basal sliding velocity Uo for sheet flow in the Weert Oswald and Robin, 1973). The ice stream has no pulling man (1957b) theory was: power in this case. Equation (19) can be related to the pulling force, given by Equation (16), by requiring that cp = where m = Hn + 1), = Bo(A/ N)2 and Ba was a constant that probably Bdep ended on basal water pres sure (Budd and others, 1979). In CLIMAP ice-sheet A reliable expression for (Je is a major problem in glaciol reconstructions, Equations (6) and (24) were used to ogy, because ice shelves have complex shapes and dynam compute convex surface flowline profiles characteristic of ics that impose both form drag and dynamic drag at the sheet flow from a formula proposed by Nye (1952). For grounding line (MacAyeal, 1987). However, in the ideal a horizontal bed, the case considered here, his formula ized case of an ice shelf having width w and thickness h was: identical to Wc and he at the grounding line, a balance of forces along floating length LF from the grounding line to i1h (25) the calving front of the ice shelf gives (Je i1x = 2Ts(LF/we), where side shear stress is assumed to be constant along LF (Thomas, 1977). ValuesTS of (Jefo r the Ross Ice Shelf The CLIMAP reconstructions assumed that basal shear in Antarctica have been computed from field data by stress TO had two characteristic variations for sheet flow Thomas and MacAyeal (1982) and by Jezek (1984). from the ice divide at x = L to the ice margin at x = 0, The concave surface profile that is characteristic of one being TF for a frozen bed computed from Equation stream flow can be produced even if cP = CPe= As (6) and the other being for a melted bed computed seen in Equation (3), cP -t 0 exists when d -t 0, a O.ph ys from Equation (24). If thTJl1e bed consisted of mixed frozen ical condition in which the basal water is thick enough to and thawed patches, as may exist in transition zones bet- 135 Journal of Glaciology ween zones where the bed is either completely frozen or Since A/A.'is raised to the 4m/(2m+l) power in Equat completely thawed, then: ion (28) for sheet flow in which bedrock projections are not drowned, progressive drowning for stream flowA = ITN! (1 - f)TF (26) according to Equation (22) requires that A - ). replaces TO + A as the undrowned height. Then, from Equations (23) where I was the thawed fraction in a given L1xstep of and (28): Equation (25). 4m/(2m+l) )] = sm2 - (29) Ablation in the CLIMAP ice sheets was confined to [. (27l' LXs the firststep in from the ice-sheet margin, with accumul TO 7}..1 ation being constant over all subsequent steps to the ice Substituting this expression for into Equation (25) divide. This constraint was replaced by Hughes (1981b), gives: TO and Table 1 lists the and expression for the mass balance expression: TM TF �m/(2m+l) LlAh TM . )] - = -- sm2 -- (30) a(L - E) + bE N (27) L1x PIgh [(2Ls7l'X = where the equilibrium line is at x = E, accumulation Equation (30) converts the convex sheet-flow profileover rate a is constant over flow-band length L - E, ablation length L given by in Equation (28) into a concave rate b is constant over flow-band length E and net mass stream-flow profile TMover length Ls. This is the concave balance N is zero for a steady-state ice sheet, positive profile for 4> = 0, so that PT 0 in Equation ( 1 8). for an advancing ice sheet and negative for a retreating = ice sheet. Stream flow develops from sheet flow over a thawed Case H. Stream flow with decreasing basal bed, and E hE b 0 in ice streams buttressed by buoyancy ice shelves, which= =was =the case for CLIMAP ice sheets. An expression for L1h/ L1x can be derived by using the Applying these limitations to in Table 1 gives: continuity condition to specify a�x in Equation (15). For TM constant w, the continuity equation is: = TM ( 11l+1) _ 1/ 2 8(hu)/8x - a 0 (31) (m + I)PI9a2 B'5m(A/ A.')4m(L x)(2m+l)/m ] = (2m + 1) [L(m+l)/m - (L - x)(m+l)/m] where u is the average velocity of the ice column fo r [ (28) mass-balance equilibrium. Integrating Equation (31) fo r Table 1. Variations of basal shear stress with distance along fiow bands of length L and constant width for sheet fiow over fro zen TO( = ) and thax wed ( = ) beds for constant surface accumulation and ablation rates a andTO b TFseparated by an equiliTO briumTM line at distance x = E from the margin of the ice sheet (Hug hes, 1981b)* For the ablation zone (0 ::::; x ::::; E): A [(1 + n/2)(N - bx)]I/n TF ------+ ) ------�-���--�------�} I/�(n�+�l) = { h�n 2 /n (2A/ PIgb)(1 n/2)1/n [(N - bx)(n+l)/n - (N - bE)(n+l)/n] + + B(N - bx)l/m = ------�--�------}�I�/(�2m-+��I) { m+I)/m ( ) b] (N - bx)(m+l)/m - (N - bE)(m+l)/m] ™ h� + [(2m + I)B/ m + I PIg [ For the accumulation zone (E ::::; x ::::; L): I/n ------A (1[- + n/2)(L- - x)] ------TF l) ) � ��� �� }��� 1/(n+ = { h�n+2 /n (2A/ Plg)(a an/2)I/n [(L - E)(71+1)/n - (L - x)(n+l)/n] + + - l/m ------B [a(L x)] ------} 1/(2m+l) = m+l)/m (m+ )/ (m+l)/m] 7}..1 --{ h� + [(2m- I)-Ba1/m/(m I)��Plg] [(L-=---- - E) I m - (L - x) ���� + + • The sign after the hE term was incorrectly a minus in Hughes (1981b). 136 Hughes: On the pulling power of ice streams constant a over length Ls and noting that u is negative: stream from the grounding line as follows: x = aJ3wh= 2fshx TOW (40) adx = ax = hu d(hu) = hu - heue (32) FJ3 X l + o where TS, and TO are the average values of TS, h and TO la hc:uc: li where he and Ue are values of h and u at the grounding over distance x. Equation (39) can then be replaced by: line. Equation (31) also yields: L'lh ha - h2 [(a�x - O'I3)/Aj" a = hou/ ox + uoh/ox = hExx + u/J.h/ .1 x (33) fj,x heue ax ? " ha - h- [-a�x +- 2fs(h:1;fhw) To(x/h) ] /An where Exx = ou ox is the longitudinal strain-rate and - / hCl1C + ax is given by Equation (10), in which T is the effective ( 41) shear stress and a�x is the longitudinal deviator stress. For constant width, the transverse deviator stress is a�y There is no mathematical way to derive Equation (41) zero: from Equation (39), but physically it means that the braking effect of shear stresses O'xy and ax= in restraining stream flow is mimicked if O'�:J:is reduced by Since TO results from basal traction caused0'13 . by ice where = Haxx ayy + O'=Jis the depth-averaged hyd bed coupling due to the weight of ice above buoyancy rostaticit press ure.+ Solving Equation (34) for ayy yields: height being supported by the bed, the effect of TO in Equation (41) is included if O'�x is given by Equation (35) (15). Equation (41) then becomes: Important stress components in T are O'xy, O'x=, axx, O'yy fj,h and 0'==. Using Equation (35): fj,x ha - h2 [�Plgh(l - PI/pw)q/ - 2Ts(h.x/hw)cp2]" /An T 2 = O'x 2 y ay= 2 O'= 2x hcuc + ax + + (ax+ )2 + (a k [ x - O'yy yy - 0'=0)2 + (0'==- axx)2] ( 42) = O'�y + a�= + HO'xx -0'=J2. (36) where cpexpresses the degree of ice-bed coupling and ice shelf buttressing, and is included in the TS term because Using Equation (35) to express O'�x in terms of axx and the braking force cannot exceed the pulling fo rce. 0'==: The variation of TS along x can be represented as a smooth decrease from a maximum at the head of the O'�x = axx - = O'xx -±(axx O'yy a=z) ice stream, where converging flow has produced strain it + + = �(axx - a=J. (37) hardened ice having an upper viscoplastic yield stress (Tv)s at x = Ls, to a minimum at the fo ot of the ice Equation (10) now becomes: stream, where laminar flowhas produced strain-softened ice having a lower viscoplasticyield stress (Tv)e at x = An empirical expression that represents this transition is:O. (38) (43) Equations (32) and (33) can be solved for .111,/ .1x, and If strain hardening remains all along Ls, then (Tv)s = Equation (38) can be substituted for Exx in the resulting (Tv)e = Tv and TS = Tv. If strain softening ends as expression to give: shear rupture at the grounding line, (Tv le = 0 and TS = Tv sin2(7l'x/2Ls). If shear rupture occurs all along /J.h a - hExx h(a - hExx) Ls, then (Tvls = (Tv)e = 0 and TS = This spectrum /J.x u heue ax of TS behavior obtained from EquationO. (43) is illustrated ha - h2 [(O'xx - a=J/+ 2A"] by the flow curves in Figure 8, in which increasing strain heue + ax softening allows a given side shear strain Exy to be pro duced by decreasing side shear stresses TS· [a� + � + (axx -0=0)2 (n-I)/2 y O' = ! ' ] (39) The average value of TS is provided by the relationship: heue + ax Equation (39) is difficult to evaluate further. TSX = ro TS dx. (44) A major simplification of Equation (39) is possible if J the braking effect of shear stresses O'xy and O'x=can be Substituting TS from Equation (43) for TS in Equat represented by a longitudinal braking stress aJ3th at red ion (44) gives: uces pulling by the gravity-driven longitudinal deviator = = TS (Tv)e + [(Tv)S - Tv)e] [�- HLs/7rx)] sin( x/2Ls) stress O'�x �(axx - 0'=0) given by Equation (15). This 7l' ( 45) implies a longitudinal braking force FB that arises from side and basal traction and increases with distance x up- fo r which fs = 0 at x = 0 and fs = �(T\' )s at x = S. 137 Journal of Glaciology Case II is a situation in which basal permafrost thaws • �------lS to produce a layer of water-soaked sediments through which bedrock crops out. An ice stream develops, with some of the ice overburden supported by these bedrock "sticky spots" and the remainder supported by basal pore-water pressure. Hence, TO exists and its variation along x can be represented by modifyingEquat ion (28), in which TO = TM for a thawed bed, to include the ef o �------�- fect of basal buoyancy that arises when basal permafrost thaws. Since the basal permafrost consists of unconsol idated sediments with a high ice fraction, thawing al Fig. 8. Possible flow curves for' lateral shear lows the basal water pressure to support the entire ice in an ice stream. Strain hardening in the zone overburden except at those sticky spots where bedrock of converging flow at the head of an ice stream projects through the sediments and penetrates the over causes an increase in side-shear stress TS and lying ice. If A is the height of these bedrock projections transverse shear strain until viscoplastic and A' is their average separation, and if ). is the thick yield stress Tv is reached.Exy Over l.ength Ls of ness of thawed water-soaked sediments between these stream flow, TS is constant if strain harden projections, they can be related to basal buoyancy factor ing and softening rates are in balance (curve cfJ as follows: 1), TS decreases if strain soft ening dominates (curves 2 and 3) and TS drops to zero if strain ( 47) soft ening leads to shear rupture (curve 3). Equation (47) requires that ). 0 when ifJ = 0 for no buoyancy and ), A when ifJ = cfJe for full buoyancy, Strain softening along x is not necessarily described with a linear variation= in between. Equation (28) can be by Equation (45). It can be related to the minimum modified to include this situation so that the following value of TS if 11 is replaced by mean height � (h he) and expression for TO results: side-strain softening controls the side-shear fo+ rce such that: TO = 1) m [(A A') 2fsh = (Tv)e(h he)x. ( 46) PIga2 Bg I + ] + [(m 1/{2m+l) Figure 9 shows the effect of Equation (46) as x in . (1 - cfJNc)rm(L - x){2m+I)/m creases in successive llxsteps. The mean side area is 1/{2m+l) _ _ l (h he)tlxfor the first step and (hs hc)Ls for the (2 1) L{m+I)/m (L x){m+ )/m I + I m + ]] last +step . The side area along x is overestimated more [ [ 1/{2m+l) with each step if the ice-stream surface is highly concave. [ 1)prga2 B2m(L _ x){2m+l)/m ] Progressive overestimation of side area offsets keeping TS (2m(m + + 1) [L{m+l)/m - (L - x){m+I)/m] at its minimum in a way that links strain softening to the m/{2m+l) concavity of the ice stream. The sooner strain softening . 1 _ develops in its side-shear zones, the more concave the ice [ ! r stream becomes. This is reasonable, because the ultim = TM (l - cfJNc)4m/{2m+l) (48) ate strain softening is shear rupture, for which TS = 0 and Equation (42) has the most concave profile. where B = Bo(AI A')2 in the Weertman (1957b) theory of basal sliding. This assumes the large "sticky-spot" bedrock projections that control stream flow have a height A - ). penetrating ice above thawed permafrost that is comparable to the "controling obstacle height" for sheet flow that Weertman (1957b) proposed. An expression for tlhl llxin which stream flowbegin s as sheet flow and ends as shelf flowis the following: tlh tlx ha- h2 [�PI9h(l-PII pw) - (Tv)C (1 +helh)xlwr cfJ2nlAn x heue ax TM (l - cfJNc)4m/{2m+l+) --����------( 49) Fig. 9. A method for approximating the side + PIgh area of an ice stream that uses successive straight lines to represent the concave top sur where Equation (46) is substituted into Equation (42), face, with one line for each additional tlx in Equation (48) is substituted into Equation (25) and the crement added to x. The less concave the sur two are added. Since braking vanishes when pulling van face, the better the approximation. ishes, (Tv)e 0 when ifJ = In Equation (49), sheet = O. 138 Hughes: On the pulling power of ice streams flow exists when cP = 0 and the second righthand term This is the case of maximum basal water pressure, for dominates, whereas shelf flow exists when cP = CPe and which Equation (49) reduces to: that term vanishes. Since braking stops when pulling stops, the limiting ice-stream length is: !:lh !:lx Prghw(l -Pr/ pw) { 2 ) - (Tv)e(l + he/h)x/wj" Ls = x = (50) ha - h [�PIgh(l - Pr! pw 4(Tv 1 + . )e( he/h) . cp�'/A" } /{heue ax}. (56) + If (Tv)e = 100 kPa, he = 1000 m, h = 2000 m and w = 30 km, Ls = 100 km. PULLING POWE R AND A LIFE-CYCLE Many ways to decrease cP as x increases along length CLASSIFICATION FOR ICE STREAMS of stream flow are possible, but one for which = Ls d Pulling power provides an analytical foundation for clas (pr!PW )he cos2(7rx/2Ls) so that in Equation (3) both cP sifying marine ice streams in a hierarchy in which loss of and dcp/dx are smooth functions is: gravitational potential energy over time produces an ice stream history characterized by inception, growth, mat (51 ) ure, declining and terminal stages of a life cycle, but al lowing for rej uvenation as well. Pulling power decreases Basal water pressure decreases gradually from eTa as an ice stream ages, and quantifies the vitality of the Prghe at x = 0 to eTa = 0 at x = Ls. Equation (49=) ice stream at a given stage in its life cycle. The life cycle becomes: of an ice stream must be inferred from present-day ice streams and from the glacial geological record of former !:lh 1 = { ha - h 2 [;jprgh(l -P /pw) - (Tv)e(1 he/h)x/w ] !:lx r " ice streams. These observations support an hypothesis + in which the pulling power of an ice stream is closely + ax} . [cp�cos4 (7rx/2Ls)j" /An} /{hcug related to the degree of ice-bed coupling beneath the ice 4mJ( m+ . 7rX 2 stream and the degree of ice-shelf buttressing beyond the sm2 -- 1) (52) 7i-1 [ ( ) ] ice stream. Letting numbers 2, 3, 4 and 5 denote no, + --Prgh 2Ls weak, moderate, strong and I,full ice-bed coupling, and Case Ill. Stream flow with equilibrium basal letters A, B, C, D and E denote no, weak, moderate, buoyancy strong and full ice-shelf buttressing. Table 2 presents a life-cycle classification of ice streams in which various If bedrock "sticky spots" cropping out in the water combinations of ice-bed coupling and ice-shelf buttress soaked sediments beneath an ice stream are small and ing are quantified as values of widely separated, or are not even present, and if the per rp. In this classification of an ice-stream life cycle, 1 for the inception stage, meability of the sediments allows the pore-water pressure cp = for growth stages, =rp for mature stages, to be in hydrostatic equilibrium with basal water pres rp 3/4 1/2 rp = 1/4 for declining stages =and rp = 0 fo r terminal sure at the grounding line, then d = (PI/ Pw )he all along Ls and Equation (3) becomes: stages. The fu ndamental assumption underlying the ice stream classificationin Ta ble 2 is that marine ice streams cP = (53) CPc(he/h). begin in inter-island channels on the outer edge of broad polar continental shelves or in the straits where large This is the case of constant basal water pressure, for continental embayments open on to polar oceans. These which Equation (49) becomes: continental shelves and embayments are sedimentary !:lh basins floored by permafrost at the beginning of a glaci ation cycle, when sea ice thickens and grounds on the ice !:lx cemented sediments, and continued thickening produces {ha - h2 [�PI9h(1 - pr!pw) - (Tv)c (1 he/h)x/wj " marine ice domes whose terrestrial margins transgress + . [cpc (he/h)j " /A" } /{heue ax} on to land and whose marine margins encroach on to 7i-1 (1 - he/h)4mJ(2m+l) + the outer continental shelf (Denton and Hughes, 1981; (54) Hughes, 1982, 1987a; Lindstrom and MacAyeal, 1987, P gh + -�-���---r 1989; Lindstrom, 1990). The life cycle of marine ice streams fo llows from this assumption. Case IV. Stream flow with maximum basal Since the mass balance of an ice stream changes dur buoyancy ing its life cycle, pulling power PT should be computed If bedrock does not crop out in the water-soaked sed for the actual mean ice velocity ux, not the velocity u for iments beneath an ice stream, and a till or bedrock sill mass-balance equilibrium given by Equation (17). An at its grounding line is too impermeable to pore water in equation for Ux that is based on the basal buoyancy fac the sediments to establish hydrostatic equilibrium under tor rp and combines mean sliding velocity ua with mean the ice stream, then pore-water pressure supports all of creep velocity ue along length Ls of stream flow is: the ice overburden. In this case, d = (pr! pw)h all along Ls, so no ice-bed coupling exists and in Equation (3) cp = cpu (1 - rp)ue reduces to its value for ice-shelf buttressing: Ux a + r" = d 1-rp) {Exz/h) dzdz cp ( Us lLSx Exx x) ( l lr rp= rpe . (55) + + a a 139 Journal of Glaciology Ta ble life-cycle classificationfo r ice streams* 2. A Stages in life cycle Stage cp during life cycle bO .S P...;::l lA lB lC lE Inception 3/4 1/2 1/4 0 0 u ID "0 0 2A 2B 2C 2D 2E rowth �3/4 3/4 1/2 1/4 0 .D G b 3A 3B 3C 3D 3E Mature 1/2 1/2 1/2 1/4 0 .�bO . '"S '" 0 4A 4B 4C 4D 4E Declining 1/4 1/4 1/4 1/4 0 ... u s:: ...... 5A 5B 5C 5D 5E Terminal 0 0 0 0 0 ! Increasing ice-shelf buttressing -+ *Pulling power decreases fr om 1 A to 5E defined as: 1. Full basal buoyancy along entire length. 2. Basal buoyancy slowly decreasing upstream. 3. Basal buoyancy steadily decreasing upstream. 4. Basal buoyancy rapidly decreasing upstream. 5. No basal buoyancy along entire length. A. No ice shelf or a freely floating ice shelf. B. Weak buttressing by a confined and pinned ice shelf. C. Moderate buttressing by a confined and pinned ice shelf. D. Strong buttressing by a confined and pinned ice shelf. E. Full buttressing by a confined and pinned ice shelf. East Antarctic ice sheet has the most likely candidates = us T is a complex function of basal buoyancy factor Pine Island Glacier because ice-bed coupling is nil and only sea ice exists and Thwaites Glacier are respective 1B, and 2A or 2B ice beyond the ice streams. The Wilkes Land margin of the streams in the Pine Island Bay polynya of the Amundsen 140 Hughes: On the pulling power of ice streams Sea embayment of the marine West Antarctic ice sheet west-central Greenland ice sheet into Jakobshavns Isfjord (Hughes, 1981b; Lindstrom and Hughes, 1984). and is the fastest ice stream known (Hughes, 1986). Mature stage of ice streams Modeling ice-stream life cycles Pulling power of marine ice streams increases rapidly About 90% of the ice in present-day ice sheets cover as grounding lines retreat downslope into the isostat ing Antarctica and Greenland is discharged by marine ically depressed marine sedimentary basins and decreases ice streams. If this was true of the marine margins rapidly as grounding lines retreat upslope out of these of former ice sheets, and if terrestrial ice streams were basins, since the pulling force is greatest when the prominent along their terrestrial margins, then computer grounding line is fu rthest below sea level. Hence, the models that simulate the advance and retreat history of mature stage of marine ice streams exists when a floating ice sheets should include any proposed life cycle for ice ice shelf occupies the seaward half of a marine sediment streams. The life cycle for ice streams proposed here ary basin and a grounded ice dome occupies the land would apply to deglaciation episodes in the history of a ward half, with ice streams drawing down the marine ice marine ice sheet, if the marine ice-transgression hypo dome and eroding thawed marine sediments beneath the thesis explains its glaciation episodes (Hughes, 1986). A dome. This situation produces 1C, 2C, 3C, 3B and 3A marine ice sheet is postulated to form when sea ice thick ice streams, depending on the degree of ice-bed coupling ens and grounds in shallow marine embayments or on and ice-shelf buttressing for ice streams at various sites shallow continental shelves fr inged by islands. Examples in the embayment. The Ross Sea and Wed dell Sea em are Hudson Bay, Foxe Basin and inter-island channels bayments of the marine West Antarctic ice sheet contain of the Queen Elizabeth Islands in North America, the mature ice streams (Hughes, 1977). Baltic, Barents and Kara Seas in Eurasia, Byrd Sub glacial Basin and the Ross and Weddell Seas in Antarc Declining stage of ice streams tica. Grounding occurs mainly on sediments in the per Retreating grounding lines of marine ice streams even mafrost condition, so that the marine ice sheet grows tually stabilize when they lie in water too shallow to on a frozen bed that precludes formation of marine ice sustain a significant pulling force or lie against the head streams. When a marine ice sheet becomes thick enough walls of fiords through coastal mountains, while a float to thaw the basal permafrost, both by depressing the ing ice shelf fills much of the marine sedimentary basin basal melting point and increasing basal-friction heat beyond the grounding line. Retreating inflectionlines at ing, ice streams will develop in outer inter-island chan the heads of ice streams can continue to retreat, pulling nels where most marine ice is discharged. This begins more ice into the marine basins and drawing down ter the deglaciation history of the marine ice sheet and the restrial ice domes. Marine ice streams become terres life cycle of its ice streams. trial ice streams at this stage, and they persist so long Computer models of ice sheets that include the life as accumulation over terrestrial ice domes replaces ice cycle of marine ice streams need to incorporate three pulled out by stream flow. This loss of marine character boundary conditions that constrain stream flow: (1) ice is typical of ID, 2D, 3D, 4D, 4C, 4B and 4A ice streams, shelf buttressing, (2) ice-bed coupling, and (3) side-shear and therefore represents the declining stage of marine traction. As definedby Equation (3), the basal buoyancy ice streams. Outlet glaciers through the Transantarc factor has potential for parameterizing the first two of tic Mountains are probably 3D, 4D and 4C ice streams these constraints, and the third constraint is made to that drain the largely terrestrial East Antarctic ice sheet depend on by way of Equation (46). The resulting and are buttressed by ice shelves floating in the mar expression fo r f1hlf1x is Equation (49) and it includes ine sedimentary basinsof West Antarctica (Swithinbank, the effects of all boundary conditions for specified values 1963). of . An empirical expression for that produces a spect Te rminal stage of ice streams rum close enough to the theoretical curves in Figure S When an ice stream is no longer able to deliver ice to to justify using it to simulate the life cycle of ice streams the calving front of its ice shelf fast enough to offset the in computer models of ice-sheet dynamics is: iceberg-calving rate, the ice stream enters its terminal stage. Ice streams are terminal in stages lE, 2E, 3E, 4E, = c (1 - xl Ls)C (S8) SE, SD, SC, 5B and SA. A calving bay carves away the ice shelf until its calving front coincides with its ground where c is a constant ranging from zero to infinity. Fig ing line to produce a calving ice wall. A 5E ice stream ure 10 is a plot of Equation (S8) for comparison with ideally exists in a fiord floored by rugged bedrock, so Figure S, showing that c reproduces case I and that ice-bed coupling is optimized; and supplies an ice 0 reproduces case IV, the= 00two cnd members of the shelf grounded on all sides, so that ice-shelf buttressing is spectrc = um. Case Il is approximated by c 1.0 and case optimized. Lambert Glacier, which pulls terrestrial East III is approximated by c = 0.1. During the= life cycle of Antarctic ice into the Amery Ice Shelf, may be a rejuv an ice stream, c increases from zero to infinity and c enated SE ice stream (Allison, 1979; Hughes, 1987b). decreases from one to zero as time progresses, with c rep Rej uvenation of a SE ice stream occurs when a calving resenting ice-bed coupling and c representing ice-shelf bay reduces ice-shelf buttressing, converting it into an buttressing along the various pathways depicted in Table ice stream in stages moving from SE to SA, thereby caus 2. ing a great increase of ice-stream velocity. This may be Equation (49) can be integrated numerically to pro the casewith J akobshavns Isbn!! that drains much of the duce the changing profilesof ice streams during an ice- 141 Journal of Glaciology 1500 0.8 1000 {m} h 0.6 500 0.5 4>G= 0.4 0 0 100 200 300 400 x (km) 0.2 Fig. 11. Evolution of an ice fiowline pro file from convex sheet flow to concave stream fiow as ice-bed coupling exponent c in Equat 0.6 ion (58) decreases from infinity to zero for in termediate ice-shelf buttressing given by CPc = x 0.5. Ils Fig. 10. The variation of ice-bed coupling N CPc with distance x upstream from the grounding line of a marine ice stream of length face profiles are plotted for c values of 0, 1/10, 1/5, 1 /2, Ls for c values ranging from zero to infinity in 1, 2, 5 and 10, following the c spectrum in Figure 10. Equation (58). The ice-stream surface elevation does not begin to climb along distance x upstream from the grounding line un til c 1/5, and the surface-inflection point separating lower= con cave parts from upper convex parts begins to stream life cycle. If flowline length L is divided into migrate toward the grounding line when c > 1, being at steps of constant length 11x over which ice elevation ini 240 km for c = 2, 120 km for c = 5, 80 km for c = 10 creases by variable height I1h= hi+1 - hi, where is and at the grounding line for c = 00 . Hence, the pulling an integer and stream flow occurs over length ofi L, Ls power of the ice stream, as represented by a down-draw Equation 49 can then be written: ( ) that creates a lowered concave surface reaches further and more strongly into the ice sheet as ice-bed coupling, h = i+1 represented by c, decreases. Figure 12 shows an instability in the spectrum when 1 1 c hi + hia-h - - { { � [P��1i( - ;�) (T��X( '�) r 0.4 7 0.49 m _ ( m+l)/m (m 1)plga2B2 (L x) 2 + 7/��::'-��--,=,,...... - o.so 2m 1 (m+ /m - (m+l)/m + ( ) [L l) -( L x) ] {[ + 4m 1/(2m+l) (m) ( } h 1 - 1 - ) / P[ghi I1x. 00 ( ;s ) ] } 5 (59) 4>G =l ° �----1�0�0-----2�0-0-----3�0�0----4�00 Equation (59) can be numerically integrated directly 0 using the Euler method or more accurately using the x (km) Runge-Kutta method. Ice-bed coupling is specified for Fig. 12. The lengthening reach of a mar o :S c :S 00 and ice-shelf buttressing is specified for 0 :S ine ice stream into an ice sheet during the � � = 10kPa, = 2kPa a m- (Hughes, 1981a) and A ide, with substantial down-draw of sheet fiow J B 470 a kPa (A-n = 3.16 x 1O-16 s-1 kPa-3) for a mean occurring after a surface-infiection instability temperature of -15°C through the ice thickness (Pater at 0.47 < c < 0.48. Stream fiow steadily rev son, 1981, table 3.3). erts to sheet fiowfor c > 1, causing the fiow Figure 11 shows the effect of ice-bed coupling for the line elevation to increase and its profile to be midpoint value of ice-shelf buttressing, CPc = k. Sur- come increasingly convex. 142 Hughes: On the pulling power of ice streams 1500 r------r------.------.-----, 1500 r------.------.------.------, 1000 (m) (m) h h 500 j,..::::::'-O:::::::=--'" O.B 0.20 100 77 200 300 400 100 200 300 400 (k m) x x (k m) Fig. 13. Surface profiles of marine ice streams Fig. 14. Surface pTofil.es of maTine ice streams for ice-shelf buttressing increasing fTom no for ice-shelf buttressing increasing fTom no buttressing ( 143 Journal ofGlaciology is illustrated in Figure 15, for which c = 0.1 for minimal ice-bed coupling and ,pc decreases from unity to zero 75 �----�------r------�----� as ice-shelf buttressing increases. Inception of stream flowbegins without ice-shelf buttressing, and when basal meltwater is produced from thawing permafrost at the head of the ice stream faster than it is discharged across the grounding line, so that the ice stream migrates into the ice sheet. Ideally, this is case IV with ,pc 1. Ret reat of the head and lengthening of the body =of the ice stream are too rapid for hydrostatic equilibrium to be established beneath the ice stream and fo r sheet flow to be down-drawn into the ice stream. Figure 15 shows 0.)( 2000 r-----�------�----�------� c = 0.1 100 200 300 400 1500 (km) o x Fig. 16. Pulling power during the life cycle of (m) 1000 h an ice stream emphasizing the growth stage, with c = 0.5 and ,pc decreasing from unity to zero. Pulling power decreases slowly up stream fro m the grounding line, especiul.l.y for the higher ,pc va'/'ues. O �----_4�--��----n_+_----� concave stream-flow surface profiles as retreating and 105 lengthening over the range 1.00 ::; rPG ::; 0.90 (top), during which pulling power increases with distance upstream from the groundingPx line (bottom). The sheet 90 flow surface profile, for which ,pc = 0, is undisturbed beyond the ice stream. However, down-draw of sheet flow is dramatic over the range fr om 0.90 ::; ,pc ::; 0.85 75 (top), during which PT changes dramatically from in III creasing to decreasing upstream from the grounding line � (bottom). The conclusion to be drawn from this is that c 0.E 60 a little ice-shelf buttressing goes a long way in regulating - both ice-stream retreat and ice-sheet down-draw when h -0- 0 is formulated by Equation (59) and PT is formulated by 0.)( 45 Equation (60). In this formulation, the inception stage becomes the growth stage when ,pc decreases from 0.90 to 0.85 and c is nil. 30 An example of the growth stage in the ice-stream life cycle when c = 0.5 is illustrated by Figure 16. Pulling power is strongest at the grounding line and re 15 mains strong farther up the ice stream than for the other combinations of c and rPGin Figures 16 through 18. Com pared to the inception stage, pulling power reaches fur 0 0 1000 200 0 300 400 ther into the ice sheet but pulls ice out with weaker grip (km) during the growth stage. This is because hydrostatic x equilibrium becomes established beneath the ice stream Fig. 15. Pulling power during the life cycle of during the growth stage. an ice stream emphasizing the inception stage, The mature stage in the ice -stream life cycle is repres with c = 0.1 and ,pc decreasing from unity ented by the combination c 1, ,pc = 0.75 in Figure 17. to zero. As the ice stream retreats without Pulling power remains strong= at the grounding line be concomitant down-draw of the ice sheet (top), cause an ice shelf is providing only moderate buttressing. pulling power is concentrated at the head of the Pulling power decreases rapidly as rPG decreases. ice stream (bottom). After down-draw begins Declining stages in the ice-stream life cycle are shown for ,pc < 0.90 (top), pulling power becomes when c = 0.5, 1, 2 are combined with 0.75 > ,pc > 0.25 greatest toward the foot of the ice stream (bot in Figures 16 through 18. The ice shelf has become some tom). what confined in the embayment and pinned to islands or 144 Hughes: On the pulling power oJ ice streams 75 �-----r------�----�------' 75 �----�------�-----r-----' 60 60 '"-;;;- 45 ...... = 2 c c..E .c 30 0 c..)( 15 100 200 300 400 100 200 300 400 x (km) x (km) Fig. 17. Put/.ing powe1' during the liJe cy de oJ Fig. 18. Pulling power during the life cycle of an ice stream entphasizing the mcLtur-e stage, an ice stream emphasizing thc declining stage, with c = 1.0 and CPc decreasing J1'Om unit,!) with c = 2 and CPC decreasing from unity to zer-o . Putting po'wer decreases steadil:!) up to zero . Pulling power decreascs rapidly up str-eam J1'Om the gr-ounding line, beginning at str'co,m from the grounding line for la,rg er CPc lower- values as cPc decreas es. values and is low e'ven at the grounding linc for- small cPc values. The tCTminal stages i'll, the life cycle of an icc sir'eam aTe Teprcscnted by CPc = 0 in Fig11TCS 15 througlt fOT which p.� =O. 18, shoals, so pulling power at the grounding line has been reduced. It reaches up the ice stream with decreasing strength as ice-bed coupling increases. Te rminal stages in the ice-stream life cycle are ap proached most closely in Figures 16 through 18 when sixth mechanism is ice-stream surges caused by inter c = 0.5, 1, 2 are combined with cPc < 0.25. The ice mittent increases in pulling power. These mechanisms shelf is now both confined and pinned in an embayment will now be examined. studded with islands and shoals. Pulling power at the Pulling power down-draws marine ice sheets grounding line is so weak that the degree of ice-bed coup Ice streams have surface profiles that reflect how far and ling upstream is relatively unimportant. how strong pulling power can reach up an ice stream and down-draw the ice sheet. The length of reach is controled PULLING POWER DISINTEGRATES by ice-bed coupling. The strength of reach is controled MARINE ICE SHEETS by ice-shelf buttressing. The concept of a life cycle for marine ice streams in which The longest reach is provided by c = 0 and the great pulling power is initially strong and remains strong up est strength is provided by cPc 1 in Equations (59) and stream to the inflectionline, as in ice streams, and be (60). Pulling power reaches full= strength up the entire comes weaker and more concentratedlA near the grounding length Ls of the ice stream. This occurs during the in line, to terminate finallyas a stagnant 5E ice stream, pro ception stage, and produces stage ice streams. How vides a blueprint for disintegration of marine ice sheets. ever, as shown in Figure 12, the concavelA part of stream Several mechanisms linked to pulling power contribute flow is only about 93 km long and it is too early in the to disintegration. life cycle for interior ice to be down-drawn into the ice The first mechanism is down-draw, which measures stream. Hence, the unlowered convex surface of sheet how far longitudinal pulling power can reach up an ice flow extends from there to the ice divide. There is no stream and into the heart of a marine ice sheet. The ice-bed coupling beneath the ice stream and no ice-shelf second mechanism is discerpation of ice ridges between buttressing beyond the ice stream. ice streams by the transverse pulling power of these ice Down-draw begins during the growth stage of ice streams. The third mechanism is basal erosion caused streams, when some sticky spots appear beneath ice by the pulling power of ice streams, so that stream flow streams and a confined or pinned ice shelf begins to fo rm migrates toward the ice divide of marine ice sheets. The seaward of retreating ice-stream grounding lines. These fourth mechanism is creation and destruction of ice rises conditions exist fo r 0 < c < 0.5 and 0.75 < CPC < 1.00 by pulling power, thereby controling the ability of an ice in Equations (59) and (60). The reach and strength of shelf to buttress marine ice streams. The fifth mechan pulling power are reduced, but the ice sheet is begin ism is widening ice streams by lateral pulling power, ning to lower as interior ice is down-drawn into the ice thereby enlarging the ice-stream drainage basin. The streams. Surface profiles illustrating this situation are 145 Journal of Glaciology shown in Figure 14. Down-draw is greatest during the ing line is included (Thomas, 1977), it becomes: initial reduction of Pulling power discerps ice ridges Ice ridges lie between ice streams. Since transverse pro where TS is the side-shear stress between an ice stream files are often steeper that longitudinal profiles on ice and its flanking ice ridge. Equation (66) is, of course, ridges, flow down the flanks into ice streams is usually purely arbitrary. However, it has the virtue of illustrat faster than flow along the crest into an ice shelf. The ing that the pulling force F." defined by Equation (16) longitudinal profileof an ice ridge can be computed from controls tlh/ tlxat x = 0, and that the contribution to solutions of Equation (39) at the head (x = Ls) and the tlh/tlx by F.T decreases to zero at x Ls in the same foot (x = 0) of its flanking ice streams, since these are way that Weertman (1957a) derived an expression for the long itudinal strain rate in freely floating ice having constant width, and if the ice-shelf back-stress aG at the ground- Since TO is constant along x in Equat�on (68), Equat- 146 Hughes: On the pulling power oJ ice streams ion (69) can be integrated to give the parabolic-surface ing to pulling power provides a framework for answering profile that is predicted by plasticity theory: the question posed by Bader (1961) as to whether an ice stream is likely to create its own subglacial depres h= [ h2 +(2To/PI9)X] !2 . (70) sion. If stream flowdevelops independently of bed topo c graphy and if pulling power can migrate upstream, the Equation (70) gives a parabolic ice-ridge profile when TO ice stream will erode a linear basal depression. Flow con is given by Equation (68). verging downslope from saddles on the ice divide should Comparing Equation (67) for TS alongside an ice be able to produce an ice stream without a trough in stream with Equation (68) for TO beneath an ice ridge the bed. However, pulling power at the head of an ice having no basal buoyancy shows that side traction ex stream co-exists with a peak in basal shear stress so ceeds basal traction for unit traction area of an ice ridge, the bed should be strongly eroded beneath the surf�ce because: inflection line. Retreat of the inflection line will be at a TO 2(hs/W)TS (71) velocity that equals the erosion rate, and will produce a = trough beneath the ice stream. The trough should form most rapidly for a lA ice stream, because basal erosion where w hs. The ice ridge will become part of its is merely a consequence of thawing ice-cemented per flanking ice» streams when total side traction equals total basal traction for an ice ridge, so that ToLs 2TsLsH, mafrost, and most slowly for a 4D ice stream, because basal erosion requires quarrying bedrock. In short, the where H is the mean side height of the iceW ridge = above greater its pulling power, the more easily an ice stream its bed and is the width of the ice ridge. Hence, is can erode a trough. the minimumW spacing between ice streams and, for equalW Pulling power also allows an assessment of the hypo basal and side traction and H � 4(h hs): c + thesis by Bader (1961) that an ice stream is self perpetuating once it forms, because ice viscosity is low = 2H(TS/TO ) = (w/2hs)(h + hs) (72) c . ered by the heat of internal fr iction. Frictional heat per W Examples of how ice-stream pulling power discerps in unit area is generated by basal traction, and is greatest tervening ice ridges are found along the West Antarctic for a 5A ice stream, which has minimal basal buoyancy grounding line of the Ross Ice Shelf, particularly the Siple and ice-shelf buttressing. Frictional heat per unit vol Coast, where Whillans and others (1987) have meas ume is generated by side traction, which is greatest for a ured surface velocities and deformation, and Shabtaie lA ice stream, which has maximum basal buoyancy and and others (1987) have mapped surface and basal topo minimum ice-shelf buttressing. Basal and side traction combine not only to reduce effective viscosity along these graphy (see Fig. 1). Ice ridge AB has a contorted sur face similar to that of its flanking Ice Streams A and boundaries by generating easy-glide ice fabrics, but also whereas ice ridge BC has a smooth surface similar to to produce fiord-like channels by basal and side erosion. Both processes tend to stabilize the ice stream, making it Ice13, Stream C, which flanks it on the north. All three elf-perpetuating, and thereby allowing long-term eros ice streams have surfaces not far above buoyancy along � much of their length, with Ice Stream B underlain by de lOn. forming sediments Blankenship and others, 1966), and An early stage in channel formation might be taking ( place at the head of Ice Stream B on the Siple Coast of much basal water along Ice Stream C (Robin and oth ers, 1970). Ice Streams A and move rapidly, inland ice West Antarctica. Whillans and others (1987) reported that the head of Ice Stream B is a region where "rafts" of augments local accumulation to13 give ice ridge AB a mod inland ice are being pulled into the ice stream. This is a erate velocity, inland ice is diverted into Ice Streams B and C by a local ice dome on ice ridge BC, giving it a slow region of converging flow, where the bed may be a mosaic of frozen and thawed patches that constitutes a trans velocity from local accumulation only, and Ice Stream C ition from sheet flow over frozen permafrost to stream scarcely moves at all (Whillans and others, 1987). From flow over thawed permafrost. If the "rafts" of relatively these observations, Ice Streams A and B are type 1C or undeformed ice exist over frozen patches of permafrost, 2C, whereas Ice Stream C is type lE or 2E, being al disintegration of the converging flow zone may proceed most fully buttressed by the Ross Ice Shelf. Hence, Ice because pulling power is a maximum and can tear out Streams and exert strong transverse pulling fo rces these ice rafts. Peak pulling power plucks! but Ice StAream C13 does not. Moreover, transverse ice vel ocity entering these ice streams is much higher from ice Pulling power may regulate ice-shelf buttressing ridge AB than from ice ridge BC. Ice ridge AB is there Ice shelves are able to buttress ice streams because they fore discerped by pulling power, whereas ice ridge BC is exert fo rm drag and dynamic drag MacAyeal, 1987), not, so ice ridge AB has the contorted pulled-apart sur ( largely because of ice rises, where ice shelves are loc face of an ice stream, whereas ice ridge BC has a smooth ally grounded. Crary Ice Rise is a complex of local ice surface. domes on the Ross Ice Shelf that has been variously Pulling power measures the erosive power of ice described as becoming grounded or ungrounded from a streams ridge on the Ross Sea floor (MacAyeal and others, 1987). Subglacial erosion requires a pulling force to loosen basal This bedrock ridge seems to continue ice ridge AB sep material and an ice velocity to transport the loosened ar·ating West Antarctic Ice Streams A and 13 (Bentley, material. Pulling power, being the product of this force 1984). It may therefore represent an advanced stage of and velocity, is a direct measure of the erosive capacity of disintegration of ice ridge AB. However, it also lies im an ice stream. Therefore, classifying ice streams accord- mediately downstream from Ice Stream 13, which is car- 147 Journal of Glaciology rying several "rafts" of relatively thick and undeformed ice streams in Wilkes Land, East Antarctica, may have ice on to the Ross Ice Shelf. Whillans and others (1987) formed when these ice streams surged, and Thwaites traced these "rafts" to high-stress regions at the head of Glacier in Pine Island Bay, West Antarctica, may now Ice Stream B, notably the "Unicorn" where Ice Stream be surging (see Fig. 1). Pulling power firstincreases and B forks into branches B1 and B2. This suggests that then decreases during a surge cycle, as both surface slope the Crary Ice Rise complex resulted from the pile-up of and ice velocity increase and decrease at the head of the ice rafts against the bedrock ridge. In that case, Crary mountain glacier or ice stream, where the surge begins, Ice Rise was created by pulling power disintegrating the probably in phase with increases and decreases of basal head of Ice Stream B. Moreover, pulling power may be buoyancy (Kamb and others, 1985). disintegrating Crary Ice Rise by plucking rafts of ice from If basal buoyancy, and therefore pulling power, con its sides and lee end. Hence, pulling power may both cre trols surges of marine ice streams, surges can be triggered ate and destroy certain kinds of ice rises. These ice rises by rising sea level when the grounding line retreats faster are temporary docking sites for ice rafts discharged on than the inflection line. Rising sea level that moves the to ice shelves by ice streams. Pulling power regulates grounding line closer to the inflection line, thereby in ice-shelf buttressing for these kinds of ice rises. creasing surface slope, also lifts the ice shelf offits pin Pulling power widens ice streams ning points, thereby decreasing buttressing. Both pro cesses increase the pulling power of the ice stream. As Ice Stream A supplies the southernmost corner of the more ice is pulled out, the inflection line must retreat Ross Ice Shelf, and it fo rms from the confluence of East and pulling power extends its reach into the ice sheet. Antarctic ice from Reedy Glacier and West Antarctic ice A 5E ice stream is most sensitive to changing sea level from Horlick Ice Stream. Some East Antarctic ice also because basal buoyancy and ice-shelf buttressing change enters Horlick Ice Stream from Shimizu Ice Stream and most dramatically at the grounding line. In contrast, a local ice from the Wisconsin Range supplies the head of lA ice stream is least sensitive to changing sea level be Ice Stream A, in addition to its branches into East and cause no ice shelf buttresses its grounding line and full West Antarctica. buoyancy extends back to its inflection line. There may Radio-echo sounding by Shabtaie and others (1987) be a natural evolution from a lA to a 5E ice stream, pro showed that the lateral shear zones defining the sides vided that a lA ice stream can erode its thawed basal of Ice Stream A lie about midway between two chan sediments down to bedrock and its actively retreating nels, a deep narrow channel along the center of a shal inflectionline can drag along its grounding line, to leave low wide channel. The deep narrow channel is presum behind a buttressing ice shelf. ably a fiord that turns southward beneath Reedy Glacier Table 3 compares the surface-slope expressions for into East Antarctica, where it ends at a headwall in the sheet flow and ridge flow with those for stream flow Transantarctic Mountains. The wide shallow channel having ice�bed coupling conditions specified by basal may have been eroded in ice-cemented sediments that buoyancy factor 148 Hughes: On the pulling power of ice streams Ta ble Flowline surface slopes above a horizontal bed 3. Sheet flow: Ridge flow: 11h _ hs he (rPGhG)n )n] 11x h [�'UG _ Ue ( 1-pP wi Stream flow: T h p ,/..2 _ ( v)C e) ,/..2 .]n ha h2 [( rgh) 1 - � 4m/(2m+l) 11h w 1 + ------rP - - p -- -- ( ) _ --''--'- ----'------'--( ..:....:.:�) �( h + 11x 4 A �� +'P � wA 'P X Tlvl 1 !....-�!-. �h --� Minimum basal buoyancy: rP= o. 2 Decreasing basal buoyancy: = cos rr ) rP 149 Journal of Glaciology ACKNOWLEDGEMENTS Alley. 1986. Seismic measurements reveal a saturated porous layer beneath an active Antarctic ice stream. For me, Charles Swithinbank is the Father of Ice Nature, 322 6074 , 54-57. Streams, for his pioneering work on Antarctic out ( ) let glaciers supplying the Ross Ice Shelf (Swithinbank, British Glaciological Society. 1949. Joint meeting of the 1963) . On one of them, Byrd Glacier, he revealed Swith British Glaciological Society, the British Rheologists' Glaciol., inbank's law to me: "All knowledge is obvious - once Club and the Institute of Metals. J. 1 (5), you have found it out." This work is dedicated to him 231-240. and the pursuit of his law. In my pursuit reported here, Budd, W. F., P. L. Keage and N. A. Blundy. 1979. Em I became aware of another law: if something is obvious pirical studies of ice sliding. J. GlacioL, 23(89) , 157- to me, but I cannot make it obvious to my readers, only 170. one thing is obvious, it isn't obvious to me either. Denton, G. H. and T. J. Hughes, eds. 1981. The last I presented the germ of the pulling-power concept great ice sheets. New York, etc., John Wiley and at the 1986 AGU Chapman Conference on Fast Glacier Sons. Flow. Since then I have become indebted to glaciologists Doake, C. S. M., R. M. Frolich, D. R. Mantripp, A. M. on four continents who helped me to clarify and quantify Smith and D. G. Vaughan. 1987. Glaciological stud the concept. In particular, I thank J. Fastook, who pro ies on Rutford Ice Stream, Antarctica. J. Geophys. duced and plotted the numerical computations of surface Res., 92(B9) , 8951-8960. profiles and pulling power for ice streams presented here, Hughes, T. J. 1973. Is the West Antarctic ice sheet and R. Franzosa, for a mathematical analysis of the as disintegrating? J. Geophys. Res., 78(33), 7884-7910. sumptions upon which these plots are based. These gen Hughes, T. J. 1976. The theory of thermal convection in tlemen will publish their own original work separately. polar ice sheets. J. Glaciol., 16(74), 41-7l. Among editors and referees, G. Clarke, K. Echclmeyer, Hughes, T. J. 1977. West Antarctic ice streams. Rev. D. MacAyeal and C. Raymond have been particularly Geophys. Space Phys., 15( 1 ) , 1-46. helpful. Most importantly of all, I thank B. Hughes fo r Hughes, T. J. 1981a. Numerical reconstruction of paleo typing and processing seven versions of this manuscript ice sheets. In Denton, G. H. and T. J. Hughes, eds. for two journals. This work was funded by the U.S. Nat The last great ice sheets. New York, etc., John Wiley ional Science Fondation (grant DPP-8400886) and the and Sons, 221-261. U.S. Department of Energy Battelle, PNL, subcontract ( Hughes, T. J. 1981b. Correspondence. The weak under 07112-A-B1 ). belly of the West Antarctic ice sheet. J. GlacioL, 27(97), 518-525. REFERENCES Hughes, T. J. 1982. Did the West Antarctic ice sheet cre Alley, R. B. 1984. A non-steady ice sheet model incorpor ate the East Antarctic ice sheet? Ann. Glaciol., 3, ating longitudinal stresses. Ohio State Univ. Inst. 138-145. Polar Stud. Rep. 84. Hughes, T. J. 1985. Thermal convection in ice sheets: we Alley, R. B. 1989a. Water-pressure coupling of sliding look but do not see. J. Glaciol., 31(107), 39-48. and bed deformation: 1. Water system. J. GlacioL, Hughes, T. J. 1986. The Jakobshavns effect. Geophys. 35(119) , 108-118. Res. Lett., 13( 1) ,46-48. Alley, R. B. 1989b. Water-pressure coupling of sliding Hughes, T. J. 1987a. Deluge n and the continent of and bed deformation: n. Velocity-depth profiles. J. doom: rising sea level and collapsing Antarctic ice. Glaciol., 35(119), 119-129. Boreas, 16(2 ) , 89-100. Alley, R. B. and 1. M. Whillans. 1984. Response of the Hughes, T. J. 1987b. The marine ice transgression hypo EastAnta rctic ice sheet to sea-level rise. J. Geophys. thesis. Geogr. Ann., 69A(2 ) , 237-250. Res., 89(C4), 6487-6493. Jezek, K. C. 1984. A modified theory of bottom crevasses Alley, R. B., D. D. Blankenship, S. T. Rooney and C. R. used as a means for measuring the buttressing effect Bentley. 1987. Till beneath Ice Stream B. 4. A coupled of ice shelves on inland ice sheets. J. Geophys. Res., ice-till flow model. J. Geophys. Res., 92(B9), 8931- 89(B3), 1925-193l. 8940. Kamb, W. B. 1987. Glacier surge mechanism based on Allison, 1. 1979. The mass budget of the Lambert Glacier linked cavity configuration of the basal water conduit Geophys. Res., drainage basin, Antarctica. J. Glaciol., 22(87), 223- system. J. 92(B9), 9083-9100. 235. Kamb, W. B. and 7 others. 1985. Glacier surge mechan Bader, H. 1961. The Greenland ice sheet. Hanover, NH, ism: 1982-1983 surge of Variegated Glacier, Alaska. U.S. Army Cold Regions Research and Engineering Science, 227(4686), 469-479. Laboratory. Lindstrom, D. R. 1990. The Eurasian ice sheet: formation Bentley, C. R. 1984. The Ross Ice Shelf Geophysical and collapse resulting from natural CO2 concentration and Glaciological Survey (RIGGS): introduction and variations. Paleoceanography, 5 (2), 207-228. summary of measurements performed. Antarct. Res. Lindstrom, D. R. and T. J. Hughes. 1984. Downdraw Ser., 42, 1-20. of the Pine Island Bay drainage basins of the West Bindschilodler, R. A. and R. Gore. 1982. A time Antarctic ice sheet. Antarct. J. V.S., 19(5) , 56-58. dependent ice sheet model: preliminary results. J. Lindstrom, D. R. and D. R. MacAyeal. 1987. Environ Geophys. Res., 87(C12) , 9675-9685. mental constraints on West Antarctic ice-sheet fo r Blankenship, D. D., C. R. Bentley, S. T. Rooney and R. B. mation. J. GlacioL, 33(115) , 346-356. 150 Hughes: On the pulling power of ice streams Lindstrom, D. R and D. R MacAyeal. 1989. Scandin Shabtaie, S., I. M. Whillans and C. R Bentley. 1987. The avian, Siberian, and Arctic Ocean glaciation: effect morphology of ice streams A, 13,and C, West Antarc Science, of Holocene atmospheric CO2 variations. tica, and their environs. J. Geophys. Res., 92(139), 245( 4918), 628-631. 8865-8883. Lindstrom, D. R and D. Tyler. 1984. Preliminary results Stuiver, M., G. H. Denton, T. J. Hughes and J. 1. Fas of Pine Island and Thwaites glaciers study. Antarct. took. 1981. History of the marine ice sheet in West V.S., 19(5), 53-55. Antarctica during the last glaciation: a working hypo J. thesis. In Denton, G. H. and T. J. Hughes, eds. The Lingle, C. S. and T. J. Brown. 1987. A subglacial aquifer last great ice sheet. New York, John Wiley and Sons, bed model and water pressure dependent basal sliding 319-436. In relationship for a West Antarctic ice stream. Van Swithinbank, C. W. M. 1963. Ice movement of valley der Veen, C. J. and J. Oerlemans, eds. Dynamics of glaciers flowing into the Ross Ice Shelf, Antarctica. the West Antarctic ice sheet. Dordrecht, etc., D. Science, 141(3580), 523-524. Reidel Publishing Company, 249-285. Thomas, R. H. 1977. Calving bay dynamics and ice sheet Lliboutry, 1. 1958. Glacier mechanics in the perfect plas retreat up the St Lawrence valley systems. Geogr. ticity theory. J. Glaciol., 3(23), 162-169. Phys. Quat., 31(3-4), 347-356. Lliboutry, 1. 1987. Realistic, yet simple bottom bound Thomas, R. H. and D. R. MacAyeal. 1982. Derived ary conditions fo r glaciers and ice sheets. Geophys. J. characteristics of the Ross Ice Shelf, Antarctica. J. Res., 92(B9), 9101-9109. GlacioL, 28(100), 397-412. MacAyeal, D. R 1987. Ice-shelf backpressure: form drag Van der Veen, C. J. 1987. Longitudinal stresses and versus dynamic drag. In Van der Veen, C. J. and J. basal sliding: a comparative study. In Van der Veen, Oerlemans, eds. Dynamics of the West Antarctic C. J. and J. Oerlemans, eds . Dynctmics of the West ice slteet. Donlrecht, etc., D. Reidel Publishing Com AntaTctic ice sheet. Dordrecht, etc., D. Reidel Pub pany, 141-160. lishing Company, 223-248. MacAyeal, D. R 1989. Large-scale ice flow over a viscous Van der Veen, C. J. and M. Whillans. 1989. Force bud basal sediment: theory and application to Ice Stream get: 1. Theory and numer1. ical methods. J. GlCLciol., B, Antarctica. J. Geophys. Res., 94(134), 4071-4087. 35(119), 53-60. MacAyeal, D. R, R. A. 13indschadler, S. Shabtaie, S. Vornberger, P. 1. and M. Whillans. 1986. Surface feat Stephenson and C. Bentley. 1987. Force, mass, and R. ures on Ice Stream 1. Marie 13yrdLand, West Antarc energy budgets of the Crary Ice Rise complex, Antarc tica. Ann. Gl.acio!13,. , 8, 168-170. tica. Gl.acioL, 33(114), 218-230. J. Weertman, J. 1957a. Deformation of floating ice shelves. McInnes, 13. J. and W. F. Budd. 1984. A cross-sectional Glaciol., 3(21), 38-42. model for West Antarctica. Ann. Gl.a.cio!.., 95-99. J. Weertman, J. 1957b. On the sliding of glaciers. J. Muszynski, I. 1987. The dynamics of coupled marine5, ice GlacioL, 3(21), 33-38. stream-ice shelf systems. (Ph.D. dissertation, North Weertman, J. 1963. Profile and heat balance at · the western University, Evanston.) bottom surface of an ice sheet fi'inged by moun Muszynski, I. and G. E. Birchfield. 1987. A coupled mar tain ranges. Internationa.l AssoC'iation of Scientific GLaciol., 33(113), ine ice-stream-ice shelf model. J. Hydrology Publication 61 (General Assembly of 3-15. BeTkeley 19G3 Snow and Ice), 245-252. 1952. Nye, J. F. A method of calculating the thicknesses Weertman, J. 1974. Stability of the junction of an ice Nature, 169(4300),529-530. of the ice-sheets. sheet and an ice shelf. Glaciol., 13(67), 3-11. 1957. J. Nye, J. F. The distribution of stress and velocity in Weertman, J. 1976. Glaciology's grand unsolved prob Proc. Soc. London, Ser. glaciers and ice-sheets. R. lem. Nature, 260(5549), 284-286. A, 239(1216), 113-133. Weertman, J. 1986. Basal water and high-pressure basal Nye, J. F. 1958. Comments on Professor Lliboutry's ice. Glaciol., 32(112), 455-463. paper. Glaciol., 3(23), 170-172. J. J. Whillans, M. 1987. Force budget of ice sheets. In Van Oswald, G. K. A. and G. de Q. Robin. 1973. Lakes be der Veen,1. C. J. and J. Oerlemans, eds. Dynamics oJ Nature, 245(5423), neath the Antarctic ice sheet. the West Anta.T·ctic ice sheet. Dordrecht, etc., D. 251-254. Reidel Publishing Company, 17-36. Paterson, W. S. B. 1981. The l)hysics of glaciers. Sec Whillans, I. M. and S. J. Johnsen. 1983. Longitudinal ond edition. Oxford, etc., Pergamon Press. variations in glacial flow: theory and test using data Robin, G. de Q. 1967. Surface topography of ice sheets. from the Byrd Station strain network, Antarctica. J. Nature, 215(5105), 1029-1032. Glaciol., 29(10 1), 78-97. Robin, G. de Q., C. W. M. Swithinbank and B. M. E. Whillans, M., J. Bolzan and S. Shabtaie. 1987. Velocity Smith. 1970. Radio echo exploration of the Antarctic 1. of ice streams B and C, Antarctica. J. Geophys. Res .. ice sheet. Intern(Ltiona.l Association of Scientific 32(B9), 8895-8902. Hydrolo9'!J Publication 86 (ISAGE), 97-115. Shabtaie, S. and C. R. Bentley. 1988. Ice-thickness map The accuracy of the refeTences in the text and in of the West Antarctic ice streams by radar sounding. this list is the responsibility of the author, to 'lli,wm Ann. Glaciol., 11, 126-136. queries should be addressed. MS received May 1988 and in 1'evised fomt 14 September' 1990 G 151