Effective stress profiles and seepage flows beneath glaciers and ice sheets

Transcription

1 Journal o Glaciology, Vol. 55, No. 191, Eective stress proiles an seepage lows beneath glaciers an ice sheets Alan W. REMPEL Department o Geological Science, University o Oregon, Eugene, Oregon , USA rempel@uoregon.eu ABSTRACT. The resistance to sliing an the extent o till eormation beneath sot-bee glaciers epen on the spatially average level o eective stress N, which is controlle by the istribution o water pressure at the be. Major subglacial conuits that acilitate large-scale water transport are expecte to be preominantly aligne with the irection o maximum hyraulic graient, which is normally parallel to the slope o the glacier surace. When the basal heat low promotes net melting or reezing, seepage transport can enable water exchange between these conuits an the rest o the basal surace area. For a simple glacier geometry with subglacial conuits that are aligne parallel to a uniorm slope, the seepage transport is riven primarily by graients in eective stress. Balance equations etermine how N varies with conuit spacing an the heat-low regime. Consierations o thermoynamic equilibrium require that ice penetrates the pore space at high eective stress. Even when the glacier base experiences net melting, or a given heat-low regime there are limits on the conuit spacing that can be attaine beore a inite till layer becomes partially rozen throughout. During net reezing, the resistance to low through partially rozen seiments limits the steay-state conuit spacing. The partially rozen zone can actually be restricte to smaller thicknesses when the reezing rate is greater. LIST OF SYMBOLS b Location o till base Length scale D Distance to rainage ivie D Location o irst ice-iniltration g Acceleration ue to gravity h Fringe thickness H Glacier thickness k Permeability k 0 Permeability o water-saturate till K Ice liqui interacial curvature K e Eective thermal conuctivity l Location o glacier base L Latent heat o usion N Eective stress at sliing surace N C Near-conuit eective stress N D Eective stress at rainage ivie Eective stress scale p Pore pressure ajacent to warmest ice p Eective stress or ice iniltration Q b Heat lux into glacier base Q Heat prouce by issipation Geothermal heat lux Q g R p S i t T T T l T m Raius o pore throats Ice-saturation level Time Temperature Ice-iniltration temperature Temperature at glacier till interace Bulk melting temperature u V x y z il p i l s n T b Darcy transport velocity Freezing rate Sliing rate scale Sliing rate Cross-glacier coorinate Down-glacier coorinate Vertical coorinate Exponent in permeability relation Exponent in ice-saturation relation Ice liqui interacial energy Pore ilm pressure ierence Viscosity o water Friction coeicient Density o ice Density o liqui water Density o soli particles Normal stress Magnitue o briging stresses Basal shear stress Porosity 1. INTRODUCTION The most signiicant cause o uture sea-level rise is likely to be enhance ischarge through outlet glaciers an ice streams that are lubricate by subglacial meltwater (Solomon an others, 2007). The istribution o water pressure at the glacier be controls the eective stress N, which is expecte to inluence the basal shear resistance b to glacier sliing an till eormation (e.g. Paterson, 1994). However, the sliing law that escribes how b, the sliing rate an N are relate has remaine elusive. The

2 432 Rempel: Eective stress proiles preictive abilities o moels or glacier an ice-sheet low are limite by this eiciency in our unerstaning o the basal bounary conitions (Fountain an Waler, 1998; Marshall, 2005). The work escribe here examines the controls that seepage lows exert on the average eective stress N beneath sot-bee glaciers. The morphology o subglacial rainage is thought to play a key role in the ynamics o glacier low (e.g. Weertman, 1972; Kamb, 1987). For example, observations (Harper an others, 2007; Bartholomaus an others, 2008) an theory (Lliboutry, 1968; Fowler, 1987; Schoo, 2005) suggest that changes in the water volume store in cavities beneath har-bee glaciers alter the surace area over which signiicant shear resistance is imparte to the glacier base. Similar behavior may also occur beneath sot-bee glaciers, particularly when the be itsel is eorme by the overriing glacier low (e.g. Schoo, 2007). While connections between basal water supply an glacier motion have long been recognize, several recent evelopments have rawn urther attention to the ynamic nature o subglacial processes. These inclue reporte correlations between summer surace melting an accelerate low near the margins o the Greenlan ice sheet (Zwally an others, 2002; Price an others, 2008), recent satellite observations o lake rainage events beneath West Antarctica (Fricker an others, 2007), global positioning system (GPS) measurements o short-term low variations in ice-stream regions (Binschaler an others, 2003) an the intriguing iscovery o glacial earthquakes with surace-wave magnitues o 5 an larger in Greenlan an Alaska (Ekström an others, 2003). Subglacial rainage systems are ominate by channelize lows that eiciently transport water over long istances (e.g. Röthlisberger, 1972; Weertman, 1972; Nye, 1976; Shoemaker, 1986; Waler an Fowler, 1994; Ng, 1998; Clarke, 2005). Beneath sot-bee glaciers with relatively lat bes, water transport between major subglacial conuits an most o the basal surace area is expecte to occur through seepage lows. The hyraulic graients neee to rive this local transport imply eective stress variations that can be integrate to evaluate the average eective stress N or a given rainage coniguration an heat-low regime. The average basal shear stress is expecte to epen on N; or example, b N i till behaves as a Coulomb-plastic material with riction coeicient an negligible cohesion. In aition to storing an transporting vast quantities o resh water, glaciers are among the most powerul erosive agents on the planet (Alley an others, 1997). The current work is o relevance to the manner in which sot-bee glaciers entrain seiments. The treatment has parallels with that escribe in an earlier stuy by Iverson (2000). A series o laboratory experiments (Iverson, 1993; Iverson an Semmens, 1996) an iel stuies (Iverson an others, 2007) provie convincing evience or the ability o ice to regelate into porous seiments when the eective stress is high. Iverson (2000) moiie an extene the theory o Philip (1980) to preict the steay-state epth o ice penetration that enables a balance between the rates o melting an pressure-inuce regelation. Since the preicte rate o regelation increases with eective stress, the thickness o the regelation layer, ientiie here as a ringe, is expecte to increase towars major rainage conuits. A signiicant uncertainty note by Alley an others (1997) an Iverson (2000) concerne the role o the ice liqui surace energy in impeing ice iniltration through inegraine till. The themoynamic an mechanical equilibrium conitions at the ice liqui interace have recently been ormulate to take these eects into account an etermine a lower-boun p on the eective stress require or ice to iniltrate a given seiment (Rempel, 2008). For sot-bee glaciers, a one-imensional treatment o the vertical orce, mass an heat balance at the glacier be emonstrates links between the eective stress N, the rate o basal reeze-on V (or melting with V < 0), sliing spee an etails o the near-be temperature istribution an till properties. A major avance is the preiction or N > p o the steaystate ringe thickness h uring both net melting an net reezing at the glacier be. For convenience, the assumption mae here is that the ringe is attache to the glacier base an hence able to transport its seiment loa with the glacier at sliing spee. Many past iel stuies o subglacial hyrological networks have taken place on har-bee valley glaciers that experience substantial iurnal an seasonal variations in the lux o meltwater that reaches the be (Iken an Binschaler, 1986; Stone an Clarke, 1993; Fountain an others, 2005; Lappegar an others, 2006; Bartholomaus an others, 2008). These glaciers oten have much smaller or negligible melt inputs to the be uring winter months, an water storage at the be is limite to the volumes containe in cavity systems ownstream o berock obstacles. There have been a ew important iel stuies o the lui pressure istributions beneath ice sheets that are suiciently col an thick to be relatively unaecte by inputs o surace meltwater. They are generally expecte to be unerlain by hyrological networks that are not strongly aecte by seasonal orcings (e.g. Engelhart an Kamb, 1997). The moeling eorts escribe below are most irectly applicable to relatively stable situations such as these, but moel extensions can be mae to investigate the ynamics in more transient environments. For example, an interesting intermeiate case is suggeste by extensive stuies o the hyrological networks near the margins o Breiðamerkurjökul, Icelan (Boulton an others, 2007a,b). There, high geothermal heat input causes major melt conuits to persist over ecaal-long observation perios, with hyrological networks augmente by a series o more ephemeral conuits that accommoate summer meltwater inputs. This paper is organize as ollows. First, a escription is given o the conitions or thermoynamic an mechanical equilibrium at the glacier base, the transport o heat an lui mass an the constitutive behavior o till. These consierations are then implemente to preict the major characteristics o hyrological networks that unerlie an iealize glacier that is separate rom impermeable berock by a layer o permeable till. The moeling is ocuse on cases o steay rainage where seepage low through the till transports water between subglacial conuits an the glacier base. A broa range o potential steay-state behavior is examine using a simpliie treatment that approximates the seepage low by ocusing on the epth-integrate horizontal transport. The eects o vertical lui low within the unrozen till are not expecte to alter the gross qualitative behavior, as emonstrate in the Appenix by comparisons between the epth-integrate treatment an a more complete treatment that accounts or these complications. The

3 Rempel: Eective stress proiles 433 Fig. 1. Schematic iagrams o the region near a glacier base above water-saturate seiments. (a) A macroscopic conuit is present; the glacier base is at T l T m an the lui pressure immeiately ajacent to ice is p n. (b) Seiment particles support part o the glacier weight; p < n an T l < T m, but T l still excees the level T neee or ice to exten through pore throats o raius R p (i.e. K 2=R < 2=R p ). (c) A partially rozen ringe o thickness h an ice saturation S i extens beneath the glacier base; at z l, T l < T an the temperature rises to T only at z l h, where the lui pressure p < n p. paper closes with a iscussion o the implications o this work an irections or uture research. 2. MODEL FORMULATION The temperature an lui pressure beneath warm-base, sot-bee glaciers are etermine by the local equilibrium conitions. The local heat balance etermines whether water reezes onto or melts away rom the glacier base. Mass-balance consierations necessitate lateral water transport through the unrozen till to acilitate this phase change. Basal equilibrium conitions For liqui water an ice to coexist in equilibrium, the temperature must be close to the pressure-melting point T m ðpþ. Lateral pressure graients in the water immeiately ajacent to the ice rive lui transport when the local heat balance promotes net reezing or melting. Thermoynamic an mechanical equilibrium are assume at the ice liqui interace. Three ierent potential geometries or portions o the basal interace are shown schematically in Figure 1. In Figure 1a, the till an ice are separate by a layer o water that is o a thickness (e.g. O(1 mm)) such that intermolecular orces between the till an ice are o negligible strength. There are many subglacial environments where such macroscopic water layers are present, incluing lakes, channels, water sheets an cavities. In this paper, the term conuit is use to reer to any such region that may acilitate rapi liqui transport (Weertman, 1972). In Figure 1a, the ice liqui interace is epicte as being lat or simplicity although, in practice, its geometry may be more complicate. Mechanical equilibrium requires that the liqui pressure p on the ice water interace be equal to the normal stress n in the ice. As eine here, the eective stress N n p 0 in such regions. (A common alternative einition not use here reers to the eective stress as the ierence between the ice pressure, that is the average o the principal components o the stress tensor, an the lui pressure p. The einition o N given here is chosen to simpliy the presentation o the vertical orce balance an be consistent with the use o Terzaghi s eective stress principle (Terzaghi, 1943) in the escription o rictional resistance below.) Thermoynamic equilibrium is achieve when the interace temperature T l T m ðpþ. In Figure 1b, the ice conorms to the upper suraces o till particles rom which it is separate by thin liqui ilms. Thermoynamic equilibrium requires that the temperature o the ice liqui interace T l < T m so that some o the glacier weight can be supporte by interactions between the ice an the till particles across the liqui ilms (e.g. Shreve, 1984; Dash, 1989; Dash an others, 1995, 2006; Wettlauer, 1999). Where the ice liqui interace veers away rom the particles, the Gibbs Thomson eect escribes how surace energy il prevents it rom penetrating through the pore throats when T l > T. For pore throats o characteristic raius R p, T T m 1 2 il = i LR p where i is the ice ensity an L is the latent heat o usion. For physical intuition, T m T 0:006 C when R p 10 mm. The pore geometry is iicult to quantiy irectly, but T can be inerre rom measurements o the water content o porous meia at ierent sub-t m temperatures. For many ierent seiment types, the value o T can be extracte rom tables o ata compile by Anerslan an Laanyi (2004). In the pores immeiately beneath the ice, mechanical equilibrium requires that (Rempel, 2008) N il K þ p i L T m T l, ð1þ T m where K < 2=R p is the curvature o the ice liqui interace in the pore throats an p is the ierence in lui pressure between the pores an the ilms that separate the ice rom the particles. Because rates o melting an reezing are typically quite low (i.e. <100 mm a 1 ), p can be consiere negligible everywhere except where water must low long istances through ilms aroun larger clasts an boulers. The strengths o ice till interactions increase as the temperature cools an the liqui ilms get thinner. For this reason, i N increases at the ice base, T l must ecrease to

4 434 Rempel: Eective stress proiles enable ice till interactions to support an increase loa. As Equation (2) inicates, this causes the curvature o the ice liqui interace to increase. Ice irst penetrates through the pore throats once K 2=R p so that T l T an N p i LðT m T Þ=T m. Because the presence o ice in the pores reuces the permeability, the pressure p is an important natural scale or eective stress variations associate with seepage transport uner sot-bee glaciers. For the example given above with T m T 0:006 C, p 7 kpa. Larger values o p are expecte or seiments that are characterize by smaller pore apertures (e.g. Anerslan an Laanyi, 2004; Rempel, 2008). In Figure 1c, the ice extens ownwars into the till to orm a ringe o thickness h an porosity, with a partial ice saturation S i that ecreases with temperature (e.g. Cahn an others, 1992). The temperature at the base o the ringe is T, whereas the temperature at z l where the ice irst encounters till is T l < T. For this case, mechanical equilibrium requires that at z l h (Rempel, 2008) N Z l lh ð s i Þgð1 Þ z þ p il T m Z l Z Tl T ð1 S i Þ T 2 i V ð1 S i Þ 2 2 z, ð2þ l lh k where s an l are the ensities o the soil particles an liqui water, g is the acceleration ue to gravity, is the viscosity o water, V is the rate o reezing (V < 0 or melting) an k is the ice-saturation-epenent permeability to lui low. The irst term on the right-han sie o Equation (2) accounts or the weight o till within the ringe. The next two terms are the net vertical orce per unit area prouce by intermolecular interactions between the ice an till. The inal term accounts or the eviation o the lui pressure rom hyrostatic conitions that is require to move water vertically through the ringe. Derivations o Equations (1) an (2) an urther iscussion o minor correction terms that are neglecte here are given in Rempel (2008). A similar relationship to Equation (2) can be extracte rom iel-teste moels o rost heave (e.g. O Neill an Miller, 1985; Nixon, 1991; Fowler an Krantz, 1994) that have recently been upate within the context o a contemporary unerstaning o pre-melting behavior (e.g. Dash an others, 1995, 2006; Rempel an others, 2001, 2004; Rempel, 2007). Heat balance The low o heat into the glacier base at z l controls the rate o reezing or melting. The latent heat consume uring reezing is balance by the heat transport into an away rom the interace so that the reezing rate is V 1 i Qb Q g þ hs i : ð3þ In Equation (3), Q b is the heat lux into the glacier base, Q g is the geothermal heat lux, Q is the rate o work perorme at the basal interace an hs i Z l lh S i z is the ice content o the ringe. Spatial an temporal changes in Q b are controlle by the rate o mechanical issipation in the ice in aition to avective an iusive heat transport through the ice. Q g is expecte to be nearly constant over relatively large istances an long timescales. For ice that slies over till at velocity, the rate o mechanical issipation at the sliing interace Q b can vary over short istances because o heterogeneities in b. The inal term in Equation (3) accounts or changes in the ice content o the ringe (i present). For physical intuition, the steay reezing rate preicte by Equation (3) is approximately 0:1½Q b ðq g þ Q Þ (mm a 1 mw 1 m 2 ) so that or a temperate glacier with Q b 0, V 6mma 1 when Q g þ Q 60 mw m 2. Water transport Water lows to or rom the glacier base to acilitate the melting or reezing require by the heat balance. Transport is riven by graients in the lui potential, an the volume lux along each pathway epens on the resistance to low prouce by interactions with soli suraces. For a given potential rop, low through large conuits is most eicient an is expecte to transport the greatest lui volumes (e.g. Röthlisberger, 1972; Waler an Fowler, 1994; Fountain an Waler, 1998). When most o the basal cross-sectional area o a sotbee glacier is ajacent to water-saturate till, lui transers with conuits are accommoate by seepage lows that move water accoring to Darcy s law at transport rate u k 0 r ð p þ lgzþ, ð4þ where k 0 is the permeability o the unrozen seiment. The hypothesis explore here is that the requirements or seepage transport can exert a ominant control on the average eective stress at the glacier base. The lui pressure p n N, where N satisies the local basal equilibrium consierations outline above. I the layer o unrozen seiment is thin in comparison to the scale o horizontal transport then the rate o vertical lui motion is small enough that the vertical lui pressure graient is approximately hyrostatic an the seepage transport rate can be written: u k 0 r xy i gðh þ lþþð s i Þg1 ð Þh N þ T : ð5þ In Equation (5), r xy is the horizontal graient operator an T is eine as the amount by which briging stresses cause n to excee the glacier weight per unit area. Minor terms involving the ensity ierence between water an ice are neglecte. The seepage low satisies a mass conservation conition so that within the water-saturate ð l Þþrð l uþ 0: Equation (6) can be vertically integrate over the thickness o a till layer with its base at z b to in that changes in the horizontal seepage low rate are l ðl b hþ þr xy l ðl b hþu i V: ð7þ To arrive at Equation (7), the till layer has been assume to rest upon an impermeable substrate an reezing at the top o the till layer is treate as a sink on the horizontal transport.

5 Rempel: Eective stress proiles 435 Weertman (1972) showe that beneath har-bee glaciers the requirements o creep closure can cause the normal stress istribution near the bounaries o Röthlisberger channels to prevent water exchange with the rest o the glacier base. This is consistent with the preictions o Equation (5) when r xy ð T NÞ changes sign as a conuit is approache rom outsie. Waler an Fowler (1994) argue that water supply to canals cut into sot seiments woul not be impee to the same extent as or conuits on top o impermeable substrates. Ng (1998, 2000) calculate the normal stress istribution at the ice till interace outsie an iealize canal with a with that greatly excees its epth. It was oun that j T j is negligible at istances much larger than the canal with, but increases abruptly to become uneine at the conuit bounary. Regularization o this result probably requires a more etaile analysis that incorporates aitional physical interactions within the thin bounary layer where j T j becomes signiicant (Ng, 2000). This iicult problem is not consiere urther here; r T is in act neglecte in the calculations that ollow. Instea, the simpliying assumption is that water is transporte across a bounary layer near the conuit wall, an a speciie eective stress N C is applie as a bounary conition on the till sie o the conuit. The behavior o ice in till Certain properties o the till are important or etermining the behavior o the subglacial system. Tests on a range o unconsoliate materials suggest that the epenence o ice saturation on temperature is well represente by an empirical relationship o the orm (Anerslan an Laanyi, 2003; Rempel, 2007, 2008) S i 1 T m T, ð8þ T m T where T < T an the exponent is typically less than unity. The permeability in the ringe is moele using an empirical relationship o the orm (Nixon, 1991; Anerslan an Laanyi, 2004) T m T k k 0, ð9þ T m T where the exponent is typically greater than unity an k 0 is the permeability o water-saturate (i.e. ice-ree) till. For the calculations presente here, Equations (8) an (9) are evaluate using parameters reporte or Chena silt, a ine-graine seiment that has been well characterize in terms o its ice-saturation behavior an permeability variations with sub-zero temperatures (Anerslan an Laanyi, 2004). Goo correlations between empirical ice-saturation parameters an T rom Equation (8) an the measure speciic surace areas SSA o many ierent silts an clays suggest that (SSA) might be use to estimate T an hence p or ierent subglacial seiments (Rempel, 2008). Notably, the SSA o Chena silt is within the range reporte or seiments recovere rom the base o Kamb Ice Stream, West Antarctica (Christoersen an Tulaczyk, 2003). It can be argue that most tills are likely to be more poorly sorte than those erive rom marine seiments near the Siple Coast. Nevertheless, the onset o ice iniltration that eines the pressure scale p is expecte to be controlle primarily by the most ine-graine raction o heterogeneous particle mixtures. Clearly, glaciers are unerlain by a rich variety o seiments an the soil parameters use here are only meant Fig. 2. Schematic iagram o a glacier that sits upon a layer o till cut by subglacial conuits. to illustrate the types o behaviors that are expecte. Direct measurements o variations in S i an permeability with temperature are neee or speciic iel settings. 3. STEADY HYDROLOGICAL NETWORKS Figure 2 shows a schematic view o the rainage system beneath a sot-bee glacier that is lowing in the by irection. Variations in H, l an b are also assume to occur only in the by irection. Uner these simpliie conitions, when ownstream variations in T are small, the maximum hyraulic graient an hence the subglacial conuits are expecte to be aligne with the surace slope. For a given conuit spacing 2D, the moel ramework escribe above can be use to preict the average eective stress that persists over the glacier be. When the sliing velocity 0 an there is a uniorm rate o basal melting, the results below agree with those given previously by Shoemaker (1986) who provies an interesting analysis o the connections between subglacial rainage an the nearterminus proiles o iealize ice sheets. Cross-sections through two potential steay-state hyrological networks are shown schematically in Figure 3. These can be thought o as close-ups o part o the lower portion o Figure 2 uner ierent heat-low regimes. In Figure 3a, the basal heat lux Q b is insuicient to remove the combination o geothermal (Q g ) an issipative (Q ) heat, so V < 0 an net melting occurs at the glacier base accoring to Equation (3). A seepage low transports meltwater to the conuit, an the eective stress istribution require to rive this low has N ecreasing towars the rainage ivie. In Figure 3b, the basal heat lux Q b excees the sum o Q g an Q so V > 0. Equation (3) escribes the rate with which net reezing takes place at the glacier base. As shown, mass balance is satisie by low through a conuit on the let that supplies the seepage low tappe by the reeze-on process. The ecrease in lui pressure neee to rive this low increases the eective stress N towars the rainage ivie at larger x. I the seepage low path is suiciently long that N > p ar rom the supply conuit, a ringe with partial ice saturation will orm (as inicate on the right). No ringe present: h 0 The controls on average eective stress are illuminate by examining simple cases in which r T! 0. With the ringe

6 436 Rempel: Eective stress proiles Fig. 3. Possible hyrological networks: (a) net melting takes place at the glacier base an (b) net reezing takes place an ice is accrete to the glacier base. absent, h 0 an Equation (5) becomes u bx N igðh þ lþby, ð10þ where bx an by are unit vectors. Seepage lows have components in both the irection o glacier low an in the cross-low irection. The total seepage lux in the ownglacier by irection is assume to be relatively constant in comparison with that o the cross-glacier bx irection. This is expecte to be the case when variations in till thickness l b an surace slope in the irection o glacier low occur over a length scale that is much longer than the istance D rom a conuit to the rainage ivie. Experiments suggest that till can oten be approximate as a Coulomb-plastic material (e.g. Terzaghi, 1943), with the local shear resistance satisying b N or a constant riction coeicient (Kamb, 1970; Clarke, 1987; Iverson an others, 1998; Tulaczyk an others, 2000; Fowler, 2003). For sliing rate, substituting Q N into Equation (3) an eining the length scale piiiiiiiiiiiiiiii ðlbþ =, the pressure scale ðq g Q b Þ=ð Þ an the velocity scale l k 0 L=½ðl bþ, Equation (10) combines with the steay-state mass-balance conition rom Equation (8) to give 2 N x 2 ðl bþ 2 þ N 2 : ð11þ Equation (11) escribes how the eective stress changes over a characteristic istance. For example with Nð0Þ N C, NðDÞ N D an N=x 0atx D, the proile satisies W 0 NðxÞ N D þ cosh D x, ð12þ where > 0. When Q g > Q b an melting takes place, the mass balance requires that N C > N D. However, N C < N D < j j = when Q b > Q g an reezing occurs. The upper limit on the eective stress at the rainage ivie N D < j j = uring reezing arises because higher values o N generate enough rictional issipation to cause net melting. Proiles o the eective stress istribution or this an other cases are shown in Figure 4, an their Fig. 4. Proiles o (a) scale eective stress N=p an (b) scale ringe thickness h=ðl bþ when D 2ðl bþ, Q g 60 mw m 2 an or the values o Q b given in the legen. mathematical escriptions are summarize in Table 1 an in the Appenix. O more importance to the overall glacier behavior than the local value o NðxÞ is the average eective stress over the glacier base: N 1 Z D NðxÞ x : ð13þ D 0 Importantly, N an by extension the average basal shear stress b N is sensitive to the glacier sliing rate even when the till behaves locally as a Coulomb-plastic material. The sensitivity o N to is greatest when Q b an Q g are similar in magnitue an the conuit spacing is large. For example, uring reezing with N D approaching j j =, D an Equation (13) preicts that N / Ws 1, as note in Table 1 (section A). Such rateweakening behavior suggests the potential or instability, as iscusse below in section 4. In the special case where the issipative heat lux Q 0, or example when 0 an the glacier oes not slie, the average eective stress between the conuit an the ivie is (or etails see Table 1, section B) N 1 3 N C þ 2 3 N D: ð14þ

7 Rempel: Eective stress proiles 437 Table 1. Equations escribing the steay-state behavior with Nðx 0Þ N C an Nðx DÞ N D A. melting or reezing with rictional heating, no ringe B. melting or reezing without rictional heating, no ringe Q N > 0; maxðnþ p : NðxÞ N D þ cosh D cosh D x N C þ = N D þ N D N C þ tanh D Q 0; maxðnþ p : NðxÞ N C N Q ðl bþ 2 Dx x2 2 qiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii D ðl bþ 2ðN C N D Þ= N 1 3 N C þ 2 3 N D C. melting with rictional heating, ringe present D. melting without rictional heating, ringe present Q N > 0; N C > p > N D ; Q g þ N C > Q b : Q 0; N C > p > N D ; Q g > Q b : W 0 NðD > x > D Þ N D þ cosh D x cosh D D p þ = N D þ NðD > x > D Þp N Q ðl bþ 2 Dx ð D Þ x2 D 2 2 qiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii D D ðl bþ 2ðp N D Þ= Nðx < D Þ Equation (15), hðd Þ0; hðx < D Þ satisies: 1 h N h x l b h ð s i Þg1 ð Þ x ðl bþ 2 þ N 2 h x xd W0 p þ tanh½ðd D Ws Þ= p Qg ð Þ 1 þ p Ws þ inq l ðlbþ 1 þ p Ws NQ W0 Ke TmT Qg Nðx < D Þ Equation (17), hðd Þ0; hðx < D Þ satisies: 1 h N h x l b h ð s i Þg1 ð Þ x ðl bþ 2 h x xd D D l b = p Q g ðl bþ K e ðt m T Þ þ i l E. reezing with rictional heating, ringe present F. reezing without rictional heating, ringe present Q N > 0; N D > p > N C ; Q g þ N D < Q b : Q 0; N D > p > N C ; Q g < Q b : NðD > x > 0Þ W0 W0 N C þ sinh½ðd Ws xþ=þ p þ sinhðx=þ Ws sinhðd =Þ NðD > x > 0Þ N C þ ðp N C Þ x þ D x x 2 D 2 ðl bþ 2 Nðx > D Þ Equation (15), hðd Þ0; hðx > D Þ satisies: 1 h N h x l b h ð s i Þg1 ð Þ x ðl bþ 2 þ N 2 h x xd þ W0 p þ coth D Ws NC W0 þ csch D Ws p Qg ð Þ 1 þ p Ws þ inq l ðlbþ 1 þ p Ws NQ W0 Ke TmT Qg Nð> D Þ Equation (15), hðd Þ0; hðx > D Þ satisies: 1 h N h x l b h ð s i Þg1 ð Þ x ðl bþ 2 h x xd þ ðp N C Þ l b D þ 2 D = p Q g ðl bþ l b K e ðt m T Þ þ i l x D where h=x 0orN=h ð s i Þgð1 Þ x D where h=x 0orN=h ð s i Þgð1 Þ Since N D > N C when reezing takes place at the glacier base, whereas N C > N D uring melting, Equation (14) preicts that or the same overall magnitue o eective stress ierence jn C N D j, N is higher uring reezing than uring melting. The argument that N is greater uring reezing than melting has been mae beore base on the epenence o voi ratio on eective stress. Here, even though changes in voi ratio have not been accounte or, N is still shown to be higher uring reezing than melting because o the istribution o NðxÞ require to rive the seepage transport. The N=p proile or Q b 30 mw m 2 an the lower o the two proiles with Q b 90 mw m 2 in Figure 4a provie examples where N ranges between similar but opposite values at the conuit an ivie. However, it is clear that N is higher or the reezing case (e.g. Q b 90 mw m 2 > Q g ). Figure 5a an b show preicte values o N=p an D=ðl bþ respectively as a unction o Q b with Q g 60 mw m 2 in the limiting cases where p > N > 0. Soli curves are or issipative heating Q N with 0:6 an 10 m a 1. Nominal values or the other control parameters are summarize in Table 2. Dashe lines are or the special case when Q 0. As long as Q jq g Q b j, the role o issipation in proucing

8 438 Rempel: Eective stress proiles Fig. 6. Eective stress proiles or several values o N D =p in the transitional region with Q b Q g þ N D. For the calculations shown here, N=p ranges rom a minimum o approximately (bottom curve) to a maximum o (top curve). Fig. 5. (a) N=p as a unction o Q b with Q g 60 mw m 2.The soli line shows the preictions o Equation (13) (where 10 m a 1 ) an the ashe lines epict the preictions o Equation (14). Between the vertical otte lines, N N max an reezing takes place with D. (b) The rainage ivie istance D=ðl bþ as a unction o Q b or the same conitions as in (a). aitional melt is minor so the soli curves ten towars the ashe lines at large an small values o Q b. As shown in Figure 5a, N is twice as large uring reezing with N C 0 an N D p as it is uring melting with N C p an N D 0. When Q g an Q b are closely matche, issipative heating exerts an important control on the quantity o liqui to be transporte. With N C 0 uring reezing, D increases rapily as N D approaches the limiting value j j =. Values o D=ðl bþ plotte in Figure 5b are the largest possible, with p > N > 0 everywhere between the conuit an ivie. When N > 0 everywhere, or larger D a ringe is expecte to orm; the ringe is locate near the ivie when reezing takes place an near the conuit when melting takes place. As suggeste by Figure 5, the rainage behavior is particularly sensitive to small changes in parameter values when Q b excees Q g by only a small amount. Figure 6 illustrates this urther with several eective stress proiles calculate using Q b 63:3mWm 2, D 20ðl bþ an the closely space values o N D =p liste in the legen. Although the basal heat lux is suicient to remove the geothermal heat, rictional heating still leas to net basal melting when Q b < Q g þ N D. Hence, although Q b > Q g is the same in each o the calculations shown in Figure 5, only the smallest two values o N D =p are low enough to allow net reezing to the glacier base. The value o N D =p 0:5 was chosen so that Q b Q g þ N D an no net lui low is require. Slightly higher values o N D =p prouce rictional melting an require N to increase towars the rainage conuit at x 0. This suggests that when Q b an Q g are similar in size, small changes in the eective stress at a rainage ivie have the potential to prouce major reorganizations in the subglacial low that are accompanie by isproportionately large changes in N. Fringe present: h > 0 When N > p, the local basal equilibrium consierations iscusse above require that a partially rozen ringe orm, as escribe by Equation (2) an shown schematically in Figure 1c. Rempel (2008) showe that h can be approximate reasonably well when the temperature graient is treate as constant an the etails o the temperature proile are neglecte. Using Equation (3) to obtain the steay-state reezing rate, an estimate o the ringe thickness h is etermine as the solution to N ð s i ik e 2 l L Þgð1 Þh þ p il Q b 1 Q g þ Q Z T T l T m Z Tl T ð1 S i Þ T 2 ð1 S i Þ T, k ð15þ where T l T ðq g þ Q Þh=K e. Figure 7 epicts solutions to Equation (15) using the soil parameters or Chena silt given in Table 2 to obtain S i an k rom Equations (8) an (9). N p at h 0 an, as h increases, the eective stress at the ringe base initially increases to support the ae weight o till. When Q b < ðq g þ Q Þ so that V < 0 an net melting takes place, water is riven ownwars through the ringe. This implies that the lui pressure graient through the ringe must be

9 Rempel: Eective stress proiles 439 Fig. 7. Approximate steay-state ringe thickness h as a unction o eective stress N, with Q g 60 mw m 2 an the values o Q b note in the legen. Dissipative heating is moele with Q N, an the temperature graient is approximate as uniorm an equal to ðq g þ Q Þ=K e. Fig. 8. Variation in N=p with D=ðl bþ or Q g 60 mw m 2 with the values o Q b liste in the legen. The calculations involving net reezing at the glacier base (i.e. Q b 90 an 120 mw m 2 ) were mae with Nð0Þ N C 0. less than hyrostatic so the last term in Equation (15) is positive an N increases monotonically as h gets larger. When Q b > ðq g þ Q Þ, the lui pressure graient through the ringe must be elevate to rive low upwars an N begins to ecrease once h is suiciently large. The soli curves in Figure 7 show preictions or h when 10 m a 1 an issipation along the assume sliing surace at the base o the ringe enhances the local heat supply. A comparison with the ashe curves or 0 reveals that increase issipation is associate with higher N at any given value o h. In regions along the glacier base where a ringe is present, the steay-state mass-balance conition rom Equations (5) an (7) can be written as 1 h N h x l b h ð s i Þgð1 Þ x ðl bþ 2 þ N 2 : ð16þ Here, the seepage lux in the own-glacier irection is again assume to vary graually in comparison with that in the cross-glacier irection. Solutions to Equation (16) are require to match with solutions to Equation (11) at x D, where N p an h 0. Further etails o the preicte behavior are summarize in sections C F o Table 1. Figure 8 shows preicte values or the scale average eective stress N=p as a unction o the scale istance to the rainage ivie D=ðl bþ. The soli curves labele with their corresponing Q b values are or the eault case where Q N an 10 m a 1. For comparison, the ashe curves in Figure 8 show the preicte behavior when Q 0. For the case o a temperate glacier with a basal heat lux o Q b 0, N increases monotonically with D until it reaches 2p near D 3ðl bþ. The lower soli curve or Q b 30 mw m 2 preicts similar behavior, but with a slower monotonic rise in N with D. Note that Q b < Q g or this case as well, so net melting takes place. The curve terminates at N=p 1:5 an D=ðl bþ 4 because the ringe becomes thick enough at this point to encompass the entire nominal till thickness o l b 3 m. For the case where Q b 0, h l b 3 m when D 3ðl bþ an N 2p. Calculations or cases with Q b Q g were perorme or an assume eective stress at the ivie o N D 0. Higher choices o N D woul prouce higher values o N or a particular choice o D. These calculations emonstrate that when melting takes place, only a limite channel spacing is permitte beore the entire till layer is iniltrate with ice. Two sets o calculations are shown in Figure 8 with Q b > Q g so that net reezing occurs at the glacier base. In each case, the eective stress at the conuit is set to N C 0. With increases in D, N increases initially until the threshol rom Figure 5b is reache or that particular value o Q b,at which point a ringe irst begins to orm at x D D. Increases in D are possible or a steay-state system with a ringe that extens rom the ivie to ever smaller values o D. A maximum steay-state ivie istance is soon reache, however. Further ecreases in D are also accompanie by ecreases in D so that with the same value o N C (i.e. N C 0 here) there are two possible steaystate values o N or any particular D. The higher steay-state N is or the case in which the ringe extens over a larger portion o the glacier base. As shown here, the range o possible steay-state values o D an N is greater or cases o reezing with Q b closer to Q g. These results show that when net reezing occurs, the maximum steay-state channel spacing oes not necessarily correspon to the entire till layer being iniltrate with ice. Instea, or the cases examine here, the maximum D is connecte with the limit to the steay-state N that can be achieve (reer to the let-most curves in Fig. 7) as a result o the reuce permeability to vertical lui transport through the ringe. The horizontal soli line in Figure 8 with N 0 correspons to the case where Q b Q g with N D 0. With no imbalance between the backgroun geothermal heat low an the basal heat lux, since no issipative heating is

10 440 Rempel: Eective stress proiles Table 2. Nominal parameters use or the calculations presente here. Values o T m T, k 0, an are or Chena silt (Anerslan an Laanyi, 2004). These parameter values imply that the pressure scale p 35 kpa, the velocity scale 130ma 1 an the istance scale 11 m. With the nominal value o Q g, ranges rom 24kPa when Q b 0 to 24kPa when Q b 120 mw m 2 Parameter Nominal value Unit g 9.8 m s 2 k m 2 l b 3 m K e 2 W (m K) 1 L Jkg 1 Q g 60 mw m 2 T m T K T m 273 K 10 m a Pa s i 920 kg m 3 l 1000 kg m 3 s 2650 kg m 3 expecte when N 0, no net water transport is require, the eective stress is constant everywhere an N 0. Figure 4 shows proiles o N=p an h=ðl bþ or the parameters use to prouce Figure 8. For the cases where Q b < Q g so that melting takes place, with ixe values o D an N D (calculations here were mae with D 2ðl bþ an N D 0) the requirements or lui rainage amit a unique proile o N=p. As expecte rom the values o N=p shown in Figure 8, the eective stress increases towars the rainage conuit more rapily when Q b 0 than when Q b 30 mw m 2. This is because larger lui luxes accompany more vigorous melting an require corresponingly larger eective stress graients to rive low. The ringe epth near the rainage conuit is greater when N=p is higher there, an so a larger volume o till is expecte to be rozen to the glacier base when Q b is low. The presence o eep rozen regions near the rainage conuit tens to restrict lui access. Since the ringe epth is reuce or lower conuit spacings, this suggests a potential mechanism or setting the spacing o rainage conuits. Note that more closely space conuits not only have the avantage that lui access is not restricte by thick ringes, but also allow or lower N as shown in Figure 8. For the cases where Q b > Q g þ Q so that reezing takes place, two steay-state N=p proiles are possible or suiciently small values o D=ðl bþ with ixe N C (calculations shown in Fig. 4 were mae with N C 0). Only a restricte range o steay-state conuit spacings is possible or a given value o Q b > Q g. For example, the N=p proile with lower N or Q b 90 mw m 2 requires no ringe ormation. However, the case with lower N or Q b 120 mw m 2 requires a ringe that extens to nearly one-ith o the epth o the till layer. This ierence in behavior can be trace to the larger lui supply rate require o the more vigorous reezing in the latter case. The eective stress graient vanishes at the rainage ivie where low in thebx irection ceases. As inicate by Equation (16), this happens when either h=x 0 or N=h ð s i Þg1 ð Þ. Further analysis shows that the ormer conition always applies or the lower N=p proile at a particular Q b > Q g þ Q an N C. Graients in h at the rainage ivie also vanish or the upper N=p proiles when N=p is suiciently close to its maximum shown in Figure 8 or a particular Q b > Q g þ Q. However, as shown in Figure 4b by the lower curve or Q b 90 mw m 2, h=x 6 0 or the higher o the two N=p proiles at this basal heat lux. A linear stability analysis shows that those steay-state solutions with N=h ð s i Þg1 ð Þ are only marginally stable. Perturbations that lea to slight increases in h at the ivie are expecte to cause N to rop slightly. This will reuce the rictional heat input an allow more reezing to take place until, eventually, the entire seiment layer becomes rozen. In other wors, a steay state with h=x 6 0 at the rainage ivie is not expecte to persist. 4. DISCUSSION The simple moels presente here illustrate how subglacial rainage systems impose limits on the steay-state behavior o glaciers an ice sheets. For example, the average eective stress N is preicte to be limite to a small multiple o the pressure scale p that marks the local value o the eective stress N at which ice irst invaes the pore space. Even or tills with very small pore apertures, p is typically expecte to be less than 1 bar; coarse-graine substrates can be associate with values o p that are vanishingly small. There are many glaciological settings in which the inerre average basal shear stress b is low enough to be consistent with till operating as a Coulomb plastic material, with b N Oðp Þ. There are also circumstances in which the inerre b is suiciently large that it greatly excees expecte values o p. This coul imply that the subglacial seiments are particularly ine-graine or are perhaps mantle by ine-graine material so that p is unusually large; it coul also imply that thick ringe layers exten beneath the glacier. A urther possibility that has not been explore here, but oten seems likely to hol, is that the inerre average basal shear stress is ominate by the eects o basal topography, much as is generally accepte to be the case or glaciers that slie over har bes (e.g. Kamb, 1970; Paterson, 1994). The moeling ramework employe here coul be extene to explore the eects o be unulations on subglacial reezing behavior to quantiy their inluence on b. Strong iel evience or the importance o subglacial heterogeneity in b inclues the inerre sticky spots that are thought to be responsible or temporal variations in sliing rate along the Siple Coast ice plain (Binschaler an others, 2003). When no ringe is present, or a given conuit spacing the average eective stress N is preicte to ecrease with increase sliing spee. Such behavior is contrary to what has sometimes been inerre an moele or tills that appear to eorm in a viscous ashion. In many circumstances the eects o basal topography are expecte to

11 Rempel: Eective stress proiles 441 prouce enhance b with increase, an this may be partly responsible or the inerre behavior. Moreover, a comparison between the soli an ashe curves in Figure 8 inicates that when a ringe is present an reezing takes place, N is lower or steay sliing an issipation at inite than it is when 0. The opposite situation arises or the case where melting takes place with a ringe present an the steay N is higher with issipation than without. In circumstances where more rapi sliing prouces lower sliing resistance, the potential exists or a runaway eect that coul be responsible or proucing regions o localize ast glacier low. The steay-state preictions shown here suggest that very low values o N can be associate with cases o intermeiate basal heat low, when Q g < Q b < Q g þ N D an conuit spacings are large because very little meltwater is generate. It is commonly assume that ast-lowing regions such as ice streams require rapi melt generation, yet etaile stuies suggest that the situation may be more complicate. In some cases (e.g. beneath Whillans Ice Stream, West Antarctica), the heat-low regime curently supports relatively low melting rates an even areas with net reezing (Joughin an others, 2004). The calculations shown here an illustrate in Figure 6 are consistent with the theory that small changes can prouce signiicant isruptions to the steay behavior. It is interesting to speculate whether the ynamics o seepage lows may bear signiicant responsibility or the transient behavior o ice streams. The conuit spacing D has been treate here as given, whereas in practice this key characteristic o hyrological networks shoul be etermine ynamically. Waler an Fowler (1994) suggeste that conuit spacing is set by a piping conition, where seepage luxes are limite by erosive processes at conuit bounaries. Shoemaker (1986) explore the potential or piping at the ice-sheet terminus to regulate the conuit spacing by limiting the maximum size o hyraulic graients, but avore a mechanism in which spacing was set by a maximum in the allowable pressure rop rom conuit to ivie. The current moel also suggests that the pressure change rom conuit to ivie limits the maximum allowable D, an urther constrains what this maximum allowable pressure change shoul be. When Q b < Q g þ Q so that net melting occurs, the conuit spacing reaches its limiting value when N C rises suiciently or the ringe to penetrate the entire epth o the till layer (i.e. h l b; see Fig. 8). By contrast, uner reezing conitions with Q b > Q g þ Q, the steay conuit spacing attains a stable maximum at intermeiate D that is controlle by the permeability structure o the partially rozen seiments o the ringe once it reaches a inite thickness (that can still be much lower than l b). It is interesting to note that or a ixe conuit spacing, the resistance to lui low through the ringe can be suicient to cause N or reezing conitions with Q b > Q g þ Q to be signiicantly lower than N or melting conitions with Q b < Q g þ Q. This behavior is contrary to what is commonly assume base on the poroelastic properties o tills. Graients in voi raction are not expecte to be signiicant or the steay behavior examine here an so have been neglecte, whereas temporal changes in voi raction can have a more important inluence on transient events. For the calculations presente here, p was taken as that corresponing to Chena silt. Other seiments are expecte to have larger or smaller values o p, epening on the sizes o their pore apertures. For example, with pore apertures o characteristic raius R p, p 2 il =R p varies rom 70 kpa to 7 kpa to 0.7 kpa as R p varies rom 1 mm to10mm to 100 mm. The value o the unrozen permeability k 0 4: m 2 or Chena silt is towars the low en o values typically quote or tills. As note in the Appenix, the epthintegrate seepage low moels presente here are expecte to perorm better or higher values o k 0, which ten to promote larger aspect ratios D=ðl bþ. Topographic variations an heterogeneous basal properties in natural glacier systems prouce a ar richer variety o behavior than the elementary moels shown here coul ever hope to capture. Nevertheless, the basic consierations employe here can be use as a guie or uture stuies aiming to explore the essential eatures o rainage interactions in more realistic settings. 5. CONCLUSIONS The average eective stress N beneath sot-bee glaciers is constraine by subglacial rainage requirements. When low between most o the glacier base an major conuits occurs primarily through seepage lows, the conuit spacing an heat-low regime strongly inluence N. Beyon a critical level o N, the hyraulic graients require or seepage transport lea to local values o N > p so that ice penetrates the pore space to orm a ringe o partially rozen seiment with thickness h, extening beneath the glacier base. When net melting takes place, the ringe is thickest near rainage conuits an higher values o N require increases in h that can eventually lea to reezing o the entire till thickness. When net reezing takes place, the ringe is thickest near rainage ivies an there is a maximum conuit spacing that can support steay-state behavior. For any conuit spacing below this maximum, two steay-state values o N are possible, the lower o which is always stable. The higher steay-state N has a more extensive ringe that can be unstable to small perturbations in its thickness. This suggests the potential or transient reezing through the till layer that coul prouce ramatic changes in basal shear stress. REFERENCES Alley, R.B., K.M. Cuey, E.B. Evenson, J.C. Strasser, D.E. Lawson an G.J. Larson How glaciers entrain an transport basal seiment: physical constraints. Quat. Sci. Rev., 16(9), Anerslan, O.B. an B. Laanyi Frozen groun engineering. Secon eition. Chichester, John Wiley & Sons. Bartholomaus, T.C., R.S. Anerson an S.P. Anerson Response o glacier basal motion to transient water storage. Nature Geosci., 1(1), Binschaler, R.A., M.A. King, R.B. Alley, S. Ananakrishnan an L. Paman Tially controlle stick slip ischarge o a West Antarctic ice stream. Science, 301(5636), Boulton, G.S., R. Lunn, P. Vistran an S. Zatsepin. 2007a. Subglacial rainage by grounwater-channel coupling, an the origin o esker systems: part I glaciological observations. Quat. Sci. Rev., 26(7 8),