J ournalorCLacioLoy, Vol. 44, No. 148, 1998

Glaciohydraulic supercooling: a freeze-on nlechanisnl to create stratified, debris-rich basal ice: 11. Theory

RIC HARD B. ALLEY,! D ANIEL E . L AW SON,2 EDWA RD B. E VEN SON,o J E FFREY C. STRASSER ,!

GRA H AME J. LARSON5

! Earth System Science Center and De/Jartment qfCeosciences, The PennsyLvania State Universiry, Universit.)' Park. Pennsylvania J6802. USA. 2 US Am!)' CoLd Regions Research and Engineering Laborat01~V, Anchorage, ALaska 99505, US.A.

~ DejJartment qf Earth and EnvironmentaL Sciences, Lehigh Univmiry, Bethlehem, Pennsylvania 18015, USA. "D epartment qfCeoLogy, Augllstana Coffege, Rock i sLand, ILlinois 6i201, CS.A.

5 DejJartment qfCeofogicaL Sciences, M ichigan State Universio', East LatlSing, 1I1ichigan 48824, USA .

ABSTRACT. Simple th eo ry supports fi e ld obse rvations (L a lVsoll and others, 1998) that subg lacia l water now o ut of overdeep en i ngs can cause accre ti on of laye red , debr is­bearing ice to the bases of glaciers. The large meltwater nu x into a temperate g lacier a t the onse t of summer melting can cause rapid water now throug h expanded basal cav ities or other now paths. If tha t now asce nds a suffi cie ntly steep slope o ut of an overdeepening, the water w ill supercool as the pressure-melting point ri ses, a nd basal-ice acc re tio n will occur. Diunla l, occas ional or a nnual flu ctuati o n s in water discharge will cause var ia ti o ns in acc re ti o n rate, debris content of acc reted ice or subsequent d iage nesis, produci ng laye rs. Under appropria te conditio ns, net acc reti o n o f debris-bea ring basa l ice will a llow debris nuxes tha t a re sign ificant in the glacier sediment budget.

1. INTRODUCTION

It has long been e\·ident , fro m theo ry and obsen 'ation, that water fl owing in conduits thro ugh or beneath a g lac ier will become supercooled if it ascends a sul1icientl y steep slope (e.g. Rothlisberger. 1968; R o thli sberge r a nd L a ng, 1987; H ooke a nd others, 1988). For example, ice has been obse rved to grow a ro und meltwater strca m s emerging from beneath g laciers whe n air temperatures were above freezing (Roth­li sberger a nd La ng, 1987; Strasser and others, 1992, 1996; L awson a nd others, 1998). I nst r umcntal observa ti ons have re\'ealed supercooling in such streams a nd associa ted la kes. Fraz il ice has been obse rved in such stream s (L awson, 1986; Strasser and others, 1992; Flcisher a nd others, 1993; Lawson a nd others, 1998), and in water rising in bo reholes (Hooke a nd Po l~ o l a, 1994).

Ordin a ril y, one would ex p ect water to Oow to a nd along the base of a temperate g lacier (Shreve, 1972). Hmveve r, basa l water fl owing in a cha nnel up a sufficiently steep bed slope from a n o\'e rdeepening will supercool a nd g row ice on the cha nnel wall s, tending to plug the cha nnel (Rbthli s­berger, 1968; Rbthlisberge r a nd La ng, 1987; H ooke a nd Po l~ o l a, 1994). A logical "extrapo la ti on" of the Ro thlisberger (1972) theory would allow fr eez ing to be ba la nced by cha n­nel expa nsion caused by water pressure in excess of ice pres­sure, but wa ter under super-fl o tat ion press ures wo uld be forced o ut o r the condui t a lo ng the ice- bed interface (per­sona l commu nication from R. L eE. Hooke, 1997). Much of the water !l ow through overdeepenings thus is di verted fro m central basa l channels into eng laeia l condui ts lac king the steep ri se of the bedrock (H ooke a nd others, 1988; Hooke a nd Pohj ola, 1994), into basa l cha nnels that run close enough to the sides of the glacier to avoid the overdeep ening (H antz

a nd L1iboutry, 1983; L1ibo ut ry, 1983), o r imo ca\·ities o r film s (Ro thlisberge r, 1968).

Lawson a nd o thers (1998; a lso L awson a nd Kull a, 1978; St rasser and oth ers, 1992, 1996) present e\·idence tha t thick sequences or la minated ice arc acc reting to the base of Mata nuska G lac ier, Alas ka, U.S.A., from a water system !lowing ou t of a n ove rdeepening. Th e intricate laye ring of the acc reted ice ( typica lly com prising layers much longer a nd wider th a n they are thick, tho ugh with occas ional cha nncl-form structures with small er aspec t ratios typically 1.5- 2) sugges ts acc re ti on not p rim a ril y from cha nnels but from linked caviti es, "canals", or other d istributed el ements o f the subglac ia l water system.

The data o f H ooke a nd Po l~ o la (1994) from the ove r­deepening of Storglacia ren in Sweden show tha t a t least some basal water fl ow does occ ur in a high-p ressure, di str i­buted system betwee n the ice a nd till. Such now rising from a n overdeepening can produce supercooling and ice grow th . T he high water pressure can m a inta in open ca\'iti es, canals or film s between the ice and its bed despite ice growing in to it (e.g. Iken and others, 1983; Iken a nd Bindschadler, 1986; A lley, 1996). \Ve suggest that within the range of natu ral glaciers, and specifica ll y beneath ?-.Ia ta nuska Glac ier, basa l freeze-on from di st ributed water systems occurs, entra ins debris and may contri bute signifi cantl y to sediment budgets.

2. T HEORY

2.1. Fundarrlen tal s

'Wc rev iew the simplest version of the problem here; for m ore-complete a nd more-elegam treatments of th e phys ics a nd thermod yna mics of subglac ia l water Oow, the reader is

563

J ournal cifG{aciology

referred to \ Veertman (1972), Nye (1976), Hooke (1984), R othlisberge r a nd Lang (1987) or Lawson (1993), among other excellent sources. \Ve restrict our calculati ons to tem­p erate glaciers, tbose at the pressure-melting temperature throughout; a brief discussion of this mechanism benea tb co ld glac iers is given by Alley and others (1997). Readers ,,·ho a lready are convinced tha t widespread waters 110wing up a sloping bed can supercool, even after allowance for geothermal nu xes and heat of sliding, should skip directly to sec tion 2.2. \ Vc fi rst state the m odel in text, and then using equations.

The melting point of ice increases with decreasing pres­sure. If we assume that subglacia l wa ter is in thermal equi­librium wilh a temperate glacier, then the water must wa rm as it movcs from a region of high pressure to one of lower pressure. Based on H ooke a nd o thers (1988) and Hooke and Pohjola (1994), the basa l wa ter pressure is likely to be close to the sta tic ice-overburden pressure in an overdeepen­ing, so water must warm as it 110ws up the adverse slope of a n O\u·deepening. Possible heat sources include the viscous di ssipa tion of the water 110w (the work done to move the water, ass uming no acceleration ), the geothermall1ux, the h eat of sliding (the work done to m ove the ice, ass uming no acceleration ) and latent heat if som e of the water freezes.

Depending on the air content of the water, the heat needed to wa rm the waler as its press ure decreases equa ls the heat di ssipa ted yiscously by the water 110w if the bed slope is about 20- 70 % steeper th an the ice-surface slope and in the oppos ite di recti on. Fo r a steeper bed (up to about 11 timcs thc magnitude of the ice-surface slope, beyond which the llow di rection of the water reverses ), additional geothermal, sliding, or latent hea t is needed to m a intain the water at the pressure-melting p oint.

For a ll but the smallest water l1uxes up the adverse slope of a su!Ticiently steep overdeep ening, the geothermal hea l a nd heat from sliding will not be enough to maintain the water at the pressure-melting p oint. Then the wa ter will begin to supercool. Because the wa ter is in intim ate contact with ice, supercooling should b e minimal and ice g rowth should occur, releasing the latent heat to maintain the water close to the melting temperature.

For a numerica l statement of this, refer to the simplified, two-dimensiona l case shown in Figure I. The x axis is taken as hori zonta l a nd increas ing in the direc tion of ice and water now. The Z ax is is vertical and increasing upwards. The ice surface is z = Zs and the bed is z = Zb . \Ve consider only those cases with (Bzs)/(ox) == et, < 0 and (OZb)/ (OX) == etb > 0, as shown in Figure I. Thus, we con­sider fl ow out of an overdeepening only.

Suppose tha t the average wa ter flu x per unit of glacier width in a subglacia l system is Q m 3 s Im 1 (ignoring all deta il s of how this is distributed ). Suppose further that the water pressure, Pw , falls below the ice-o\'e rburden pressure, Pi = Pighi. by a constant amount 6.P, in which g =

9.8 III s 2 is the g rav itati ona l acceleration, hi == Z, - Zb is the ice thickness, and Pi is the density of ice. We thus consider sta tic loads only, and not dynamic pressure variations asso­ciated with fl ow over riegels or with other fl ow p erturba­ti ons. T he p otenti a l of sub glacia l water, <D , then is

in which <Do is some a rbitra ry reference potentia l a nd Pw is the density of water. (Alternati vely, wc could lump 6.P into

564

Ice

w~

x

Fig. 1. Coordinate system used.

the reference potenti a l.) \ Vater 110ws down th e potenti a l gra­dient, w hich in our simplified, one-dimensional glacier is

0<D -0 = Pi g ets + (PI\" - Pi) gOb '

x (2)

The heat genera tion from water fl ow, R "" is the work done p er unit time per uni t area to move the water, given by

0<D R w = - Q ­

ox (3)

in which Q again is the water nu x and we have made the usua l ass umption tha t the water flow is not accelerating (e.g. WeerLman, 1972; R othlisberger and L ang, 1987). Addi­tiona l heat sources are the geotherm a l flux, H g. and the heat of sliding, Hs , which a re discussed in m ore detail below.

These heats are available to warm the water and ma in­tain it a t the pressure-melting point, and to melt basal ice or, if the m elt rate is negative, to allow freeze-on. The heat p er unit a rea, R pmI" needed to maintain water at the pressure­melting point with a wa ter l1ux Q is

oPw ( ) R pmp = Q ox (3Cs 4

in which Cs is the volumetric sp ecific heat of wa ter, (3 == (BT.,,) / (oPw) is the pressure dep endence of the melting p oint T m, and

(5)

Som e uncertainty is a ttached to (3. Rothli sberger (1972) a nd Rothli sberge r and L a ng (1987), among oth ers, favored using the value for pure wa ter, f3p .. whereas L1ibou try (1983), H ooke (1991) and others have favored th e la rger-magnitude value fo r air- saturated water, f3a . We carry out calculations for bo th .

Balancing these heats, a layer of basal ice thickens at the freez ing rate, j , given by

j = (R pmI' - R w - Hg - R s) cos etb (6) L

in wh ieh L is the volumetric heat of fu sion of ice, Hg is the

geothermal heat and Hs is the heat of sliding. The factor cos ab is needed in Equation (6) because j is measured nor­m a l to the sloping bed of the glac ier but thc other calcul a­tions are for the ho rizontal- vertica l x- z coordina te system defined in Figure I; however, for small ab typica l o[ g laciers, thi s [actor is nearly onc and can be omitted. Note that wc continue to consider averages over the glacier sole. Below, wc di scuss the effects of spati a l concentration of water Oow.

Of the work done in sliding, perhaps onc-third may bc used to produce new mineral surface during abras io n of sub­g lac ial or englac ia l ti ll or rock ( ~Ie tca lf; 1979), and a much sm a ll cr amo unt m ay be used to producc defec ts during dc­fo rmati on o f any new ice accretcd to the glac ier bccl. Tlw remainder is d iss i pa t cd as heat if no accelerations occu r. Thc work do ne per unit area in sliding is ThU" in which 11,

is the sliding \'Clocity and Tb is the basal drag. For simplicity, wc ignorc abras io n a nd defcct crea tion, a nd lVrite

(7)

which should ove res timate the heat of slid ing and thus undcrestimate the rate of basa l ice accretion. Accreted ice may be melted intern a ll y by the heat of dcforma tio n; the consen 'ati\'e way to deal with th is is to follow Robin (1955) . and ca ll U s the surface \ ·eloc ity.

We now can I-ewrite Equatio n (6) by substituting for H pIIlp from Equat ions (4) and (5), H I\' from Equations (2) and (3), a ncl H, from Equation (7). This yields

j = { Qg[pJ( a, + (PII' - pJ\')abl - Hg - ThUS} cos exb/ L:

l\" == 1 + (3Cs . (8)

Using the \'a lues o f consta l1lS li sted in 1able I, thi s becomes

j = 0.103 cos a" [Q (5282as + 45180.,,)

- Hp, - 3.17 x 1O-8Thus] : f3 = /3" (9a)

j = 0.103 cos 0." [Q(6184a-, +36160.,,)

- H I', - 3.17 x 1O -8 T"u,] : (3 = /31' (9b)

in wh ich Q is in m :1 s 1 m I, Hg in \Y m 2, Th in Pa, a nd.! a nd U, III m a 1

:\otc that the water-Oow mechan ism desc ribed here does not produce ne t accre ti on to the g lacier as a whole. The descent of water into the overdeepen ing mclls more ice than is frozcn bac k on during watcr ascc nl from thc o\'e rdeepen­ing. Howc\'C r, mllch o rlhe ice melted may be rela ti\'(:i y elean (es pecia lly that a ro und moulins descending thro ugh the g lac ier to the bed ), whereas d irtv basa l ice is aec re ted (Strasser and olhers, 1996), so ne t addition of debris to the g lac ier is likel y. Further, me lting a nd freez ing a rc spatially separated a t \ 'a r io us sca les. so lha t some regio ns of the glac ier wi ll exper ie nce a net acc re tion of basa l ice.

2.2. Example

rf liT se l H r; = H, = O. an appropriate approx im ~tion [or rapid water Oow in cha nnels, Equat ions (9) yield f = 0 for

0." = - l.20.s · (3 = (3". a nd o.b = - l.70." f3 = (31" the well­known result from previous wo rkers (e.g. Rothli sberger and La ng, 1987; H ooke, 1991). Rapid water 0 011' in a ny channel with an upwa rd slope> 1.2- 1.7 times the downwa rd surface slope should cause supercooling a nd ice acc re ti o n in the channel. Such a cha nnel r ising ou t o[ an o\'e rdeepening wou ld be ex pec ted to cease to fun ction effi cieml y as it is pa rtiall y or completely clogged with ice, so water now in

AII~)' and others: Glaciolf),draulic sll/Jercoolillg: If. T!t eo~J'

cha nnels across overdeepen ings is likely lo occ ur englac ia ll y with a gradi ent that d oes not lead to freez ing (Hooke a nd others, 1988; Hooke, 1991).

If wc ass ume that the onset of summcr melt causes a large, if non-steady, water Ou x up the adverse slope o r an O\'erdeepen ing, we can substitute likely \ 'a lues of H g . Th 'U, ,

Q, exs a nd ab into Equations (9) and ca lcul a te net acc re ti o n o[ basal ice. 1)'pieal sUllllllertime wate r Ou xes per unit width, a\'Craged ac ross the glacicr width, are 10 2 10 :1

m 3 s 1 III 1 on Findeleng lctschcr, S\\' itze rl and (Iken and Bindsehadl er, 1986), which apparently routcs much of its meltwater through basa l cavities at the o nse l o[ sumlller melting. \ Yater Ouxes approach 10 Im:l s Im Ion Colulllbi a Glacie r, Alaska, altho ug h the water paths are less we ll known there (persona l cOlllmunication from R . A. \\'a lters and others. 1986). D a ta from t>.Ia ta nuska G lac ier sugges t tha t its summertime dra inage may be 2: 10 Im:1s Im I.

\ ,Ye ha\'e calcula ted basa l freeze-on fo r a possibl e over­deep ened glacicr with cha rac ter istics li sted in Table I. \ Yater Oux through a basa l ca\ 'it y system \\'i1l \'ar)' on annual a nd shorter time-sca les; for simplicit y, I\'C ass ume that wate r Oows through basa l ca\·iti es from the o\'Crdeepening (or a frac tion of a yea r, till' and t hat no water Il ows out of the o\'er­deepen ing for the rcs t or the yea r, 1 - till ' The geotherm a l nux a nd hea t of sliding \\'ill tend to ca use basa l melting a ll year, but frceze-on frolll supercooling o r ri sing waleI' will occ ur o nl y duri ng till'

Our example glac ier has net li'eeze-on a t a rate 0('0.11 3 III

Table I. J "ariables used, with units . . \ illllerical mllles qf lJ/~Jlsical COlIstall/.' follolt, ROihfisberger alld Lang (1987) . . \ 11lJ1erica{ l'allles q/' .Iite-.s/Je('ijic l'ariables lIsed ill Ollr "slandard"calculatioll also are gi1'el7

Onlllanl

C D

I .'J h, H"

Ilplllp

H"

rn Q R

.1'

X z

.\/ullling

\\ 'a ll' l' sp('(' il i(' h ('a l

Chcl'd(, l'pl' ning m" x i111U111 deplh

In- frecz(,-oll n tH'

Grm'iUlliollfli ;'H.T4..' il'ralion

Rl'gl'iatiotl-ia \,(, l'thi('km'"

Gl'ollll'1'111aIIl1l .'

I-k at to kel'p "at('1' Illl~ a t PIT"UI'l'­

Illeltillg poillt He(lt gCIll'ra lcd f"rolll water \'iscous

di ss ipat ion 1+ ,iC, lel' fU'i iol1 hea l Basal melt I'atc

\\'atel' flux D ebri s/in' \ '0 1 u nl<' rat io

l1'anion or-year w ith Ca\'ily flo\\' Sliding \'c loc it \.

Horizonw l coordinatl' O\'Cl'ckl'pen ing ach-el'se-slopc Ienglh

\ (.- ni('al coordin ate Brei e! c\·ati o n

Ic l'-~u rfacc el c\ '<ltio ll

Brei slopc

SlIl'fitCC slop c

,\ ir-saluratcci 1l1elli llg-- po illl press ure

dependcnce

Pure melting-po int press ure ciept.'llcll'llcl' " 'ater pressure ice pr(,~s llrl'

Ice dellsit )'

\ \ 'atrl' densit )'

Basa l shear Slress

\\'a tcr poclll iai

Rcfl'l'l't1('C \\,atcr pot l' ntial

I it//le or /llIil.1

I. ~ x 10"j m I K I

111

III a 9.8 m s ;1

III

0.06 \\. III 1

\\'Ill ~

I\' m 2

:l.OG X 10"J 111 < I

m<l

IWI m < s I III I

0. 1 10 m a I

m

m

m

nI

III

O.Il') Il.OI

9.B x III H K p" I

7. I· x 10 H K Pa 1

p" 916 kgm <

1000 kg m l

10" p" Pa Pa

565

JOllmal qfClaciology

a I or 0.169 m a I (depending on (3) during t m , a nd melting at a rate of9.43 x 10-1 m a Iduring l - tm.Fortrn = 0.1. this gi\"Cs net accre tion of ice averaged over the base of the glac ier across the o\'erdeepening of 2.8 or 8.4 mm a I, depending on {3. The sensitivity of thi s calculation to varia­ti ons in the important pa ra meters is shown in Figure 2.

\ Vc expec t that ;'/Iatanuska Glac ier is among those glaciers w ith prominent, rapid basa l accretion. H owever, we belicve that basa l accretion is importa nt on many glac iers, albeit often at slowcr rates. We thus chose model parameters for our examplc g lac ier to produce somewhat slower acc re tion than for M atanuska Glacier, although Fig­ure 2 shows va lues more appropriate for ~Iatanu s ka Glacier.

Wc used a bed slope ab Icss than appropriate for ~ latanus ka Glacier (Lawson a nd others, 1998). ' t\le used only abo ut 10 % of the estimated summer water flux from be­neath M atanuska Glacier (L awson and others, 1998), based on the a rg uments of Hooke a nd others (1988) and Hooke (1991) that not all of the water now would occur subglaeially across the ove rdeepening. \ Ve assumed tm = 0.1 , a low \'alue consideri ng that frazil is obsen'ed in discharge from beneath ~Iatanus ka Glacier throughout the months-long

'I co

E-

'I co E-

'I co

E-

0.2

0.1 tm

0.0 Us

-0.1 0 50 100

tm(%), us(m a-')

0.2

0.1

0.0

-0.1 0.0 0.1 0.2

a", la,l

0.2

0.1

0.0 Hg

-0.1 0.0 0.1 0.2

Hg (Wm-\ Q (m3s-'m-')

Fig. 2. R eLa lion belween average nel Jree::.e -on ta the glacier sole, j , and other variables. Each plot shows how varying one jJarameter ji"Oln the "standard" calculation of Table I aJJects the amount of ice accreted during one year. In each case, the u/J/Jer curve isJor {3 = {3aJor air-saturated water, and the lower curve isfor {3 = (3pJor pure wate?: Nega tive j values indicate melting. The value of each parameter in the standard calculation is indicated by the arrow dosest to its label. The cOll7jJulation is especially sensitive to the cumulative water flux ( Qt lll ) and to the bed slope relative to Ihe swjace slope.

566

summer, and that some water discharge continues year­round as shown by aufeis growth in the winter. rf a ll of the estimated water flux of Matanuska Glacier were routed up the bed from the model overdeepening, ice would accrete at 1.69 m a- I during tn" leading to 0.16m a I net accretion if till = 0.1. For comparison, meters of ice have accreted to the base of Matanuska Glacier over decades (accretion 2: 0.1 m a I), as shown by the presence of atomic-bomb tritium in 4- 6 m thick sec tions of dirty ice in which evidence of sign ificant tectonic thickening has not been found (Strasser and others, 1996; Lawso n and others, 1998).

If an ice-contact water system a scending an overdeepen­ing leaks downward to recha rge a subglaeial aquifer (cr. Boulton and Hindma rsh, 1987), some of the geothermal heat may be advected away by the groundwater flow and not reach the glacier bed. Al so, if the adverse slope of the over­deepening is arm.ored by a deforming till, some of the heat of sliding may be removed in the groundwater flow. R educ­tion of Hg and H s by these mecha nisms wou ld increase ice accretion. Freeze-on also may occur slightly faste r than we have calculated above, because the basal ice a nd its entra ined debris will warm during now from the over­deepening, absorbing some heaL.

Aeereted ice is likely to conta in debri s, as discussed below. If the deb.ris/iee vo lume ratio is R with a net freeze­on rate of ice of f , then the debris flux into the ice is fR per unit area, and the steady debri 0 ux out of an overdeepening is j RX ~ j RD / ab per unit width, where X is the di sta nce from the deepest point to the down-glacier end of the over­deepening and D is the max imum depth. Returning to our example, suppose that {3 = {3a, g iving a net basal freeze-on of about 8.4· mm a- I, and suppose that the debris ratio R = 0.1 , a low \'a lue for ;,i(a ta nuska Glacier basal ice (Lawson and others, 1998). Then the g lacier will be entrain­ing all of the erosion products from the area if the eros ion rate isjust less than I mm a I, a "typical"glacier erosion rate (e.g. Boulton, 1979). For a long or deep overdeepening, thi s may become a sign ificant term in the sediment budget of the glacier. The order-of-magnitude la rger basal accre tion rate of M atanuska Glacier, together with its higher debris concen­trat ion, would allow its freeze-on to incorporate a ll of the erosion products from well beyond its O\'erdeepening or else would support quite rapid erosion of the overdeepening.

Hooke (1991) demonstrated the likelihood that over­deepenings develop, in part, because sediment protects the adverse slope from eros ion whi le the headwall is eroded rapidly. Our results arc not in conflict with this model. The adverse slope likely will red uce the sediment transport capacity of stream s, causing depositi on equal to or in excess of removal in basal ice (see review by Alley and others, 1997).

3. APPEARANCE OF BASALLY ACCRETED ICE

The calculations abO\'e are for spatial a\'erages across a g lacier. These cannot be accurate in detail; at any time there must be Oow paths carrying more water than the spati al average, separated by regions with little or no water Oow in which the heat of sliding is di ss ipated . In the water Oow pa ths, ice will grow more rapid ly than the averages cal­cul ated above, while melting m ay occur in the regions between the flow paths. \Ve now consider how the geometry of the water system wou ld a f.fect the appearance of the acc reted ice.

"Ve m odel ice growth as occurring prim a ril y from a di s­tributed water system, a term wc use to indicate water occu­pying much more of the bed tha n is typical fo r Ro thlisberger ch an nel s. \ Ve ass ume (c r. Iken alld othel-5, 1983; Iken a nd Bind schadl er, 1986) tha t the distributed water systcm spreads a nd thickens when water pressures a re ra ised by increased water suppl y to the glacier, so tha t the dra inage paths a re n ot plugged by ice g rowth. Sufficient di stributed fl ow to a llow significant ice accre ti on is likely in linked cayiti es ove r bedrock (Wa lder, 1986; Kamb, 1987), with fl ow paths of typical dimensions 1- 10 m laterally a nd 0. 1 m verticall y (Walder and Hall et, 1979). Caviti es a re also known to develop down-glac ier of la rge c1asts proj ecting from till under at least some circumsta nces (Boul to n, 1976), a ltho ug h soft till is unlikely to produce caviti es as la rge as those typical of bed rock beds. Dra inage over till may be do mina ted by fl ow in broad, shall ow, hi gh-pressure, poss ibly bra ided cha nnels in which water pressure increases with flu x ( the "cana ls" of Walder a nd Fowler, 1994). Transl'erse a nd yertical dimensions o f these fl ow pa ths a re likely to be o f the same m agni tudes as for bedrock-fl oo red cal'iti es, a nd longitudina l di mensio ns controll ed by bra iding might be simil a r to those for caviti es.

Ice g rowth in a Iinked-cal'ity or o ther di stributed system wo uld probably includc epitax ia l g rowth on the roo f; nucleation there wo uld be easy, and ver y little supercooling wo uld occur. However, form ation of true fraz il ice within the cavity o r of a nchor ice on or in the cavity fl oo r is also likely. Anchor ice that fl oated to the top of a cavity and bccam e incorpo ra ted in the cavity roof might a dd some sediment to the glacier. In situ a nchor ice could be added to the glacier by cav it y closure during fa lling water pressures. Fraz il ice inco r­pora ted in the roof of a cavit y, or dendritic epitax ial growth, might fo rm a sort of m es hwork that wo uld trap water-borne sediment (Hubba rd , 1991; St rasser a nd others, 1992; Lawso n a nd others, 1998). Thus, we II'ould ex p ec t acc reted ice to conta in debris (Lawso n a nd others, 1998).

If m ost of the m e ltwater from lVlata nuska Gl ac ier is routed through subg lacia l cav iti es, then acc retion ra tes could be rapid eno ug h that ice acc re ted during diurna l o r storm-induced high-di scha rge peri od s wo uld form a unit visibl e to the naked eye. Va ri ati ons in debris content mig ht occ ur, rel a ted to cha nges in suspended-sediment concentra­ti on in the water o r to cha nges in sediment-trapping effi­ciency in response to evolution of the dendriti c nature of the g rowing interface, producing layering a t a sub-a nnual sca le.

Basall y acc reted ice from a cal'ity "vould probabl y have its debris content a ltered , and usua lly increased, during rege la tion in a region o f intimate ice- bed contact between cav iti es. R ege la ti on ice is know n to incorpo rate sel'e ral p er cent of debris by I'o lume (Sugden andJohn, 1976, p. 61), in a laye r typicall y ~ IO mm thick (Kamb a nd L aChapelle, 1964; \ Veertma n, 1964). Rege la ti on ac ross the bed fo ll owing accre­ti on o f dirt y basal ice might increase o r dec rease the debri s concentrati on of tha t ice, depending o n de ta ils of debris en­tra inment a nd release. Net melting of ice will tend to co ncentrate the conta ined debri s if c1 as ts in contact with the bed regelate upwa rd as the ice a r o und them melts (H alle t, 1979). Rege la ti o n of ice into subg lac ia l sediments between cav iti es also wo uld a ll ow debris entrainme nt (1I'e rson, 1993; h 'erson a nd Semmens, 1995), and wo uld produce c1 as t-supporLed debris a nd thus quite high debris concentrations if active.

If regela ti on a ffec ts debris concentra tions and the ice

ALLey and others: Cla(io/~ydrallli( supercooling: 11. Theol)'

acereted during passage ac ross a cal'ity is thicker than the

regela ti on layer formed between caviti es, th en repeated pas­

sage of ice fro m cal'ity to inter-cal' ity region to cm' ity wo ul d

form layers o f ice with a lterna ting higher a nd lower debris

concentra ti o ns, with dimensio ns of the laye ring controll ed

by cm'ity sizes a nd spacings, If the accreted ice is thinner

tha n the regcla tion layer, the n a ll of the acc reted ice should

be altered a nd perhaps ho m ogeni zed by regela ti on during

passage ac ross the inter-cavity regions, Such p rocesses probably a re not espec ia ll y im porta nt

beneath lVlata nuska Glaciel-, because ice acc re ti on appea rs

to be so r apid that post-depos itional modifi cati on wo uld not

affect much o f the acc reted ice; sub-a nnua l features asso­

ciated with , 'arying accre ti o n conditions proba bly dominate

the appearance of the basa l ice of ~ Iatanuska G lacier. Post­

accretio n m odificati on wo uld be more importa nt if accre­

tion were slower or occurred fa rther up-g lac ie r, as is likely

beneath som e g laciers, \Ve hm'C constructed a simple model fo r basal-ice accre­

tion, to simul a te how acc re ted ice laye rs might appea r be­

neath a g lac ie r with sig nifi cant post-accre ti o n di age nes is.

We follow a length of ice fo r m any yea rs as it m Ol'es ac ross

a bed co nta ining cal' iti es o f fi xed leng th a t random

locati ons, Fo r plotting conve nience, we ig no re the straining

of this ice, based on the ass umption th at melting a nd freez­

ing a re ra pid compared lo cha nges in geo me tr y caused by

strain ra tes, During each a nnua l cycle, ice is acc reted to the

glacier bed o n caI'ity (or ca na l) roofs a t a spec i fi ed ra te / fo r

fract ion o f the yea r till ' a nd melti ng occurs a t a spec i fi ed ra te

rh for time 1 - till at cal·it y locations and for t he entire yea r

bet ween the C<l\·iti es, \Ve keep track or three ice typ es:

glac ie r ice, the ice present a t the sta rt of the expe riment;

acc reti o n ice, the ice tha t grel\' on a cav it y roof a nd never has been in a rege la ti o n layer; a nd

regela ti o n ice, any ice th a t has been within dista nce hr of' the g lac ier bed betwee n caviti es, with hr, the thickn ess or the rege la ti on layer, spec ifi ed.

The aCl il 'C rege lalion laye r m a intains a consta nt thickn ess

a nd mig ra tes upwards into g lac ier ice, acc re ti o n ice or older

regela ti o n ice a t thc melting ra te 'Ih due to geo th erma l heat

a nd heat fro m sliding, Results of o nc simul ati o n a re shown in Fig ure 3, II'hich

bea rs a t lea ' t a qualita ti, 'C resemblance to basal ice of some

glaciers (e.g. L awso n a nd o thers, 1998). Va rying the model

inputs produces differing appeara nces, but a w ide range of

simul ati o ns yields layers a few miilimeters thick with latera l

dimensio ns rel a ted to the cavit y size. In the non-physical

limit of caviti es occupying the entire bed for t ill of each year

and disappearing during the res t of the yea r, the laye ring

becomes la te ra ll y continuo us a nd a n inve rse ly I'an 'cd

sequence ca n be produced. Clea rly, ice accreted to a g lacier will be a ltered by tec­

tonic or other processes ove r time, p rod ucing cha nges in

fabric, continuity of laye rs, a nd so forth (Strasser a nd

others, 1994), Heat dissipa ted from defo rm a tio n in tem­

perate basal ice would be ex pected to produce meltwater

that would escape along g ra in boundari es, increas ing the

concentra ti o n of debri s in the remaining ice (\ Veertman,

1972; H a ll e t, 1979),

567

Journal qfGlacioLog)'

-0.06

~O.1

I.(m)

Fig. 3, Longitudinal section through basal stmtigraph), gflZer­ateel with the model described in the te.\'t, Glacier ice ( to/J ) and bed material (bottom ) are se/Jamted ~y regelation ice (dark) alld accretion ice not ajfecled ~Y regelation (white). The base qfthe glacier began at z = 0 at the onset riftheJirst melt season, and moved down (to /Jositive numbers) with net accretion during the 10),ear simulation; note thalthe coordi­nate ,fjlstem is Ji,led with resl)ect to the original glacier ice, which in reahO' moves ujJ dUTing caviIJI o/Jening. Calculations were conducted attm equal(J1 spaced grid/Joints; the eleventh gridpoint IJlotted is taken to be the same as theJirst, Qjwntities used include: j = 500 mm a 1]01' till = 0,1 over a 5 m long cavity placed IClndom[)I, with one cavi~y occurring beneath the 10111 Long seclion each ),ear, jUeLting OCCl/ rs at a mte rh rif 10 717117 a 1 wherever ({wities are absent, The cavi{J' /JoTtion qf the model domain n/Jeriences a net addition qf .jJ mm a 1

(ji-ee:::e -on rif 50 I17m in 0.1 aJoLlowed b)1 lI1eLt -qjJ rif 9 111 m during tlte olher 0,9 a), and Ihe non -cavi~y IJortion rifthe model domain e,l/Jeriellces net meltillg rif 10 mm Cl I; the average ac­cretion over the whole model domain is 31 mm Cl I The regeLa­tion -h~yer thickness, hr' is 10 mm,

4. OTHER MECHANISMS OF GLACIO-HYDRAULIC SUPERCOOLING

As noted by Lal\'son and others (1998), the complex sub­glacia l em' ironment may produce regions of decreas ing water pressure and net freeze-on e\'en though the a\'erage beha\'ior over the en ti re g lacier bed produces melting, Any pressure drop can lead to supercooling if other heat sources (geothermal, sliding, \'iscous dissipation in water) arc not sufficiently large, For rapid waler [l ows, \' iscous dissipation in the water is the dom inant heat source. It a lone will prevent supercooling if water pressure drops because of potential-gradient-dri\'en [l ow and if the work done on the water produces heat through \'iscous di ss ipation. However, supercooling is possible if some of the pressure drop occ urs because of flow up a sloping bed or because of o\'erburden rcmoval, or if some of the work on the water produces accel­eralion rather lhan viscous di ssipation.

Situations that might produce pressure drops with supercooling include g lac ier cah'ing events that remove some grounded ice (Lawson and others, 1998), or basal pres­sure changes related to g laciohydrau lie jacking (Murray and Clarke, 1995) or ice [low across bumpy beds (Robin, 1976; Lliboutry, 1993). H owe\'Cr, these arc generall y not ex­pected to produce significant basal layers. The Llibou try (1993) mechanism, which im'okes regelation water [low through ice rather than between ice and rock, produces a steady thickness of on ly 0.02 0.04 m of basal ice if ac tive.

568

Caking is largely a grounding-line, onc-time ewnt for any piece of ice. A calving-induced pressu re reduction of 106 Pa on a O.lm thick ea\'ity wou ld cause freeze-on of only about O.lmm of ice. \Vater-pressure reduction at one site on lhe glacier bed owing lO stress redistribution in response to water-pressure increase nearby (Robin, 1976; lVlurray and Clarke, 1995) could create similar freeze-on, but the process is reversed during subsequent water-pressure increase. Ice flow from such a site might temporarily preserve a littl e ac­crelion ice, but is unlikely to lead to significant net accre­tion, Similarly, the Robin (1976) mechanism (sec also Pate rson, 1981, p.121) is one of loca l freeze-on in a bed that is, on a\'erage, melting, and thus is not expected to produce net accretion. \Vate r fl owing up a \'a ll ey-side slope from a moulin-fcd channel may grow basal ice, but presen 'ation of this ice would require that th e water not subsequentl y [low back down the same slope.

5. DISCUSSION AND CONCLUSIONS

The subglacial em'ironment is complex, A variety ofrarLors, including di sequilibrium, g round wate r £low, impurities, water mixing, grad ients in shear stress affecting waler po­tential (Robin a nd Weenman, 1973) and others, could be invoked to complicate the si mple analysis here. R owel'er, it is quile clear that a large enough [low of subglacial or englac ia l water rising at a sufficientl y steep angle wil l super­cool and g row ice,

Overdeepenings arc common in the glac ia l el1\' ironment (Hooke, 1991). Channels carried inlO an overdeepen ing by ice fl ow are likely to be narrowed or clogged by rapid ice accretion. Broader areas of the bed on the adverse slope of an overdeepening may be affec ted by accretion from a dis­tributed water system such as subglacial cavities or wide, low "canals" (Walder and Fowler, 1994-). Some subglae ia l water [J ow has been documented across overdeepenings (Hooke and Pol~ ol a, 1994). If such [l ow occurs at sufTicienLly high volumes for enough of the year, lhen our calcu lations show that at least some glaciers will ha\'e net basal accretion of ice from this mechanism in overdeepened region.

Accreted ice entrai ns debris by \'arious mechanisms (Strasser a nd others, 1996; Lawson and others, 1998), Layer­ing defined by changes in debris COl1lent is likely, owing to time and space variations in accretion rates as well as to post-accretion changes. The typica l spatial scale of variation in the water system (order of meters laterally) is likely to occur in layering of the accreted ice. A simple model based on these ideas produces basal stratigraphy that is qualita­ti\'e!y simil ar to that observed beneath som e glac iers. If regelat ion affects accreled ice, isolOpic differences between accretion ice and regelat ion ice m ay occur if there is loss or addition of water (Souchez a nd Lorrain, 1991), a nd sen 'e to help test our model.

The general mechan isms a nd rates of debris entra in­ment by g laciers arc not full y understood. This hampers our interpretation of glacia l geology and glacier dynamics (cr. Beget, 1986; Alley, 1991; H ooke, 1991 ). The ev idence that basal accre tion from rising supercooled water occurs beneath at least onc glacier (Strasser and others, 1992, 1996; Lawson and others, 1998), and the probabi I i ty lhat th is obsen'ation is not unique, m ay a llow glae io logica l models to estim ate one term in the g lacial debris flux with greater acc uracy than was previously possibl e.

In summary, wc a re faced with strong observati ona l e\ ·i­dence of basal ice accre tion to a temperate glacier (Lawson and others, 1998). '10 exp la in thi s, wc have been led to com­bine two class ic res ults from glacier hydrology: that ri sing water can become supe rcooled, and tha t large inputs of surface-derived water to the glacier plumbing system cause water to spread ac ross the bed, pro\ 'id ing space in which water can now up the a dverse slope of a n overcleepen ing and g row ice. For pla usible va lucs of physical \'ariablcs, these mechani sms lead to predictions of n et basa l ice accre­tion beneath Matanuska G lacier and other glaciers. Signifi­calll debris entrainment by the soles of some glaciers is likely by this mechani sm. Spa ti a l and tempora l \'a ri ations in accretion and post-accretion modification wi ll produce stratifi cation in this acc reted ice.

ACKNOWLEDGEMENTS

Wc th ank J. S. Walder, R. L eE. Hooke, B. H allet and M. Sharp for helpfu l sugges tions. This work was supported in pa rt by thc U.S. Nati ona l Science Founda tion (grants OPP 9223007 and 9318677), the U.S. Army Cold Regio ns Research and Engineering L aboratory in th e Cold Regions Water R esources Progl'am (CWIS32689), and the D. a nd L. Packa rd Foundati on.

REFERENCES

All ey, R . E. 1991. Deforming-bed o ri g in lor southe rn L a urenti de till shec ts? ] Glacial., 37 (125 ). 67 76.

All ey. R . B. 1996. ' lowa rcb a hydrologic mode l fo r com pu teri zed ice· shee t sim ul a ti o ns. !-l),drol. Processes, 10 (+),6+9 660.

:\ lI ey. R . B. . K. i\ 1. c urre),. E. B. E"cnso n, J C. S. r assC'r, D. E. Lawsnn a n d

G.J L a rson. 1997. How g lac ie rs ellt rain a nd trans po rt sedi me nt at th e ir beds: phys ica l constra ints. 0{{{1. Sci. R,,,, .. 16. 1017 1038.

Begc t, J. E. 1986. Id ode ling th e influencc or til l rheo logy on the fl ow a nd

pro fil e of the La ke i\ l ic higa n lobe. so ut he rn L a urclltidc ice sh c(' l.

US A. ] Glarioi .. 32 (11 11. 235 2·1 I. Boult o n . G. S. 1976. The o ri g ill of g lac ia ll y flut cd su r laces obse rya ti o ns

a nd t heo ry..7 Glaciol.. 17 (76), 287 309.

Bou lw n , G . S. 1979. Processes o f' g lac icr eros ion o n d ille ren t subst rata . .7 Clariol., 23 (89), 15 38.

Bo ul tO n, G. S, am! R . C. A. I-lind marsh. 1987. Scd imclll deforma ti on bc­

nea th g lac i{' rs: rheo logy a nd geologica l consequc nces . .7 GeoJ)/~l'J. R es., 92 (B9), 9059 9082.

Fl eishcr, PJ . J t- 1. l~'a n z a n d J A. G a rd ncr. 1993. Ba th Yllletry a nd sed i­

mCllla r y ('1l\·irnnnWl1ls in proglacia l la kes '11 I he eastern B(Ting Picd m o nl G lac ier of ,\ la ska . ] (.'1'01. Edllc.. 41 (3 ), 267 27-1-.

H a llct. B. 1979 .. \ theo ret ica l m ockl of glacia l abras io n. ]. (;Iariol .. 23 (89), 39 50.

Ha lllz, D. a nd L. Lli bo lltr y. 1983. \\ 'a lnwa),s, ice pnl11cab ilil y at dep th, a n d wate r pressures at G lac ie r d Argc llli erc, I~-cnc h Al ps. ] (;Iaciol. . 29 (102), 227 239.

Hookc, R. Le B.198+. O n Ih (' role of mechanica l energy in m a in la iningsllbg la·

cia l wa lcr condu its at a tmosp heric pressu re.]' Glaciol. . 30 (105 ), 180 187.

Hooke, R . L eE. 1991. Pos iti"e feed bac ks assoc ia tcd w ilh erosion of g lac ia l cirques a nd o\'CTdce pcnings. (.'eol. Soc . . 1111. Bull., 103 (8), I I O·~ 11 08.

H ookc. R . L e B. a nd \'. A. Po l~o l a . 1994. H ydro logy ora segnw nl ofa g lac ier situa ted in a n OI'Crde('pening, S to rglac i,ire n, S\Ved e n.]. Glariol., 40 (l3·~) ,

11 0- 1+8.

Hooke, R . L cE .. S. B. i\ l il lc r a nd J Ka h le r. 1988. C h a l'ac te r afthe cnglac ia l

a nd subg lacia l dra inage system in the upper pa rt of the abla ti on area or S to rg lac ia ren, Sweden . .7 Glaciol., 34 (117.228 231.

Hubha rd , B. 1991. H-ccz ing·raLe e ileets on the p hys ica l charact eristi cs o f basa l icc fo rm cd by net ad freez ing. ]. (;Iaciol .. 37 (127),339- 3+7.

lke n. A. ;\\ld R . A. Bindsc had lcr. 1986. Combined l11 easuremelllS ofsubg la ­cia l wa ter pressu rc a nd surface \'c locil y of' Findclc ngletschcr, Switze r­

la nd: concl usions a bo ut dra inage systelll and sliding lllf'c ha nislll . .7 Cia(iol .. 32 11 0). 101 119.

[ken, A .. H . Rii thli sbcrge r. A. FI Olron and \\'. H aebe rli. 1983. The upli ft o f

Alley and olhers: Claciohydrall/ic .flljJPr(Ooling.· 'f. T hem)!

Untcraa rgle tscher at the begin ning orthe melt seaso ll a cnl1srquC'llcC' of wate r stO rage at the bed? ] (;Iarial., 29 1101 ,28 1-7.

h-('I'50n, 'J. R. 1993. Regela ti on o f ice t h roug h clebr is a t g lac ier becls: im pli­

ra ti ons fo r sed iment t ra nsport. Geolog,)', 21,(j " 559 562. h-crson, \I. R. a nd nJ Sem mcn s. 1995. Illt rusion ol' ic(' illt o porous media

by rege la ti o n : a mccha nisnl of sed ime nt e11l rainlll(,l1 t by g·lacicrs . .1 Gropl!)',. Res .. lOO ! B71, 10,219 10,230.

Kamb, B. 1987. G lac ier surge mecha ni sm based on linked (',l\· it,· conlig·ura ·

tion of th e basa l wa ter conduit s)'stem .]. Gl'o/Jhl'S. R"s.. 92 ( B9, 90R3 9100.

K amb, E. a nd E. L aC hapcl le.196+. Direct ohscn·at ion oft lw m edJan ism of

glac ier sliding O\w bed rock . .7 (;Io(iol., 5(38,. 159 172.

Lawson, D. E. 1986. Obsen 'at ions o n hydrau li c a nd Il", rma l cond iti ons at

the bed of r.Ia ta nuska G lac ie r . . \ laska. Eirl~. 1frh. lIor/m/llllf, ::iiric/i. lerslIchsollsl. 1 I {Isserball, I {!'rlml. (;Io::iol . . 1 fill. '10. 69 71.

Lawson. D. E. 1993. Glaciohydrolog ic a n d glarioh\'Cl rau li c dTcns on runolr

and sed ime lll ,·icld in glacier izecl bas ins. U,REL .Iiollogl: 93 · 02. Lawson, nE. a n d j. B. I-\.u ll a. 1978. All oxygen isolope imTSligation of lh ('

origin of t h e basa l zone of the i\ la ta n usk" Glacicr. ,\I aska.]. (;1'01 .. 86 '() ,. 673-68,~.

Lawson. nE., J c:. Sl rasser, E. B. E,'Cnson, R. B. .\I Ie), C. J Larso n and

S. A. Arcone. 1998. G lac io hydra uli c supercooling: a I;'('('z('-on

mecha ni sm tn create st rati fied , d e bri s-rich basa l ire: I. Fie ld c\·idencc.

]. Glacial., 44 (H 8), 5+7 562.

L1 ibo utry, 1.. 1983. i\ lodi lica tions to th e Iheor)'orilltraglac ia l wa term.ys for

th e case of's uhg lacia l oncs,.1 ( ;Ioriol .. 29 1021,216 226.

Ll ibo Lltr r, L. 1993. Illl erna ll11 elt ing a ll d ice accrct ion at th c bOllom oftem·

pcra tc g lac ie rs.]' Clariol.. 39 1 l3l , 50 (i.1.

\ietca lf. R . C. 1979. Encr.'!;y d iss ipat ion d uring subglacia l "br,\SiDn at "is·

qua il ), G lac ie r, Was hingto n, L'.S .. \ . ] (;Iariol .. 23 H'li. 2:3:; 2+6.

\ lurray,T a nd C. K . C. C la rk,·. 1~J9.1. Black·box modeli ng 0 1' 11 1(' subglacia l

wa ter system.] CeoJlhrs, Rn .. lOO t Bh 10,231 10,2+5.

">'yr. j. F 1976. Wa ter 110\\' in glacins:jii ktll h laups, tunnels a nci ,Tilt,,] (,'Ia­(jol .. 17 (76).1 81 207.

Patcrsnn, \ \ '. S. B. 1981. The /rlmil I' q! gla(im. Snolld edilioll. Oxford, ele. i'ergamon P ress.

Robin. G. d e Q 1955. Ice mO\'elllent a nd lC'mper<tltl re d istribut ion in

glac iers and icc sheets.]' Clacial .. 2 (18 '. 523 .'i:12.

Robi n. G. de Q 1976. Is the basa l ice of' a teml"Tate glac ier 8l t he pressure

melt ing po int" ] (;Iariol .. 16 7 1 . 1H3 196.

Robin , G. d e Q a ndJ \ \ 'eCrlllla n. 1<)7:1. Cycl ic surging orglac icr,.]' (;irlliol .. 12 (6+),3 18.

Riil hlisbergeL H. 1968. i-: ro,in' p ro'Ts,,'S whic h are li ke ly to acel'ntua le or

reci uce th e bottolll rel icf or \'alley g laciers. IlIlemaliollal .h.lIJ(ialioll I!/

Sril'lIli/ir Il rdrologl' Pllbiimlioll 7~) I (;Cll eml . I 1\I'lIIbir IIr Hfm 1967 .1'1101/ '

alld hI' ), 87 <)7.

Riit hli sberger, H . 1972. \ \ 'a!e r pressure in illt ra· and suhglac ia l cha n n('Is.].

(;Iaciol., 11 (62 1, 177- 203.

Riit hli sbe rgc r, I-I. a nd H . L a ltg. 1987. G lacial h>drnlogy. III C urnel l. .\ . i\ 1.

a nd \ I. j. C la rk, Nil. (;Ia(io:jlll l'ial IN/illlflll lralls/rr.· all aljJim /rn.I/lnli/ '('.

Chic hes ler, e tc .. J ohn \ \ 'i lc \, a nd SO il S. 207 28-1. Shre\T. R . L. 1972. i\! m'cmcll\ or\\'all'l' ill g lac iers . .7 (;I(II·iol.. 11 (62 ,. 205 21+.

Souchez, R . /\ . and R . D. Lorra ill. I')'JI. hI' (o1II/rolilioll alld girl(ifr ({rllalllin.

;\CW \ o rk . rlC .. Spring;cr-\ 'c ri ag. Springer Series in Ph "s ica l [m·iron·

me ll l 8.)

Slrass(' r. E. B. . D. E. L;\I \'solt, E. B. I':,·c nson. J c. G osse and R. B. c\li ey.

1992. Frazi l icC' g rowt h at til e ttTlll in lls or thf' \hll all u:-.ka G lacier.

A las ka . a lld iLs im pli catio ns for sed iment C'11l ra il1l l1 t'n l in g lac iers and

ice shee ls. [i\bst racl.i Geol. SOl' . . 1111 .. Ihslr Pro,~mllls. 24 31,78. Strasser, j. G. nE, Lawsnn, E. B. EH'",on and G. j. Lar,ol1. 1<)9 f. (:n·>I"I·

lograp hic a nd nl('~()Scale a na lyst's o r hasa l ZOIle' ice rro ll l the t l'rlllinl1~ or

thc i\ la tan llsk" G lacier. .\I as k<! : c" idr l1rt'lor basa l li ... ,'ze-on in an ope n

hyd rolog ic s),stCI11 . [Absl racl. i Ceol. Sot. . 1111 .. Ib.llr PmgmlllJ. 26 17 , ,\ 177. St rasser, J c., D. E. LawsolI , G. J L a rsn n. E. B. E"e nson a ncl R . B. ,\11 \,\'.

1996. Pre liminary rc's ults of tri t iulll a na lyses in hasa l ice. f\ latall uska

G la cie r, Alaska . L'.S.," .: c\'id c ll ce fo r subg-Iaria l ice accrcl io ll . . luII .

(;Iariol.. 22. 126- 133.

SII ,[(dcn. D. E. a n d B. S. J ohn. 1976. ( ,'Ia(im alld lalldlmjll': a geuIllOijJhologi((d a/I/m}()rit. L o ndon, Edward Arnolc!.

Wa lder.J S. 1986. H ydrau lics of su b g lac ia l (",·ities.] (;/(J(iol., 32 112 .+3') ++~.

\\ ·a lclcr. j. S. a n cl A. Fowler. 199+, C h a nll cl izecl sub,g; lacia l drai llage O\T r a

deformable bed .]. (;Iariol .. 4 0 (13+), 3 15.

\ \ 'a lcle r. j. a nd B. Ha lln 1'J79. Geom etry of' lorme r subglacia l water

cha nn e ls a nd C<l\· iti es.}. Clari"l .. 23 (89), 33:i :HG. \ \'ce rtl11 a n. J 196+. The Iheory oi'glac icr sli rl ing.]' (;I(II·inl., S (:~9 1 . 287 303.

\ \'c('rtman. J. 1 ~)72. G enera l I henr y of· wale I' Il ow at lh(' base or a g' lar ir r or

ice shee t. R ev. Ceol"!)'I. Spar/' Phi's .. 10 (1 ). 2R7 331

illS received 21 Oclober 1996 and acce/Jled il7 revised /01'1715 Mqv 1998

569