on the role of mechani cal energy in maintaining

JournaL of

GLacioLogy ,

Vol. 30, No. 105, 1984

ON THE ROLE OF MECHANI CAL ENERGY IN MAINTAINING SUBGLACIAL WATER CONDUITS AT ATMOSPHERIC PRESSURE By ROGER LEB. HOOKE* (Naturgeografiska lnstitutionen, Stockholms Universitet, Stockholm, Sweden)

ABSTRACT.

Recent theoretical studies of glacier hydrology have assumed

that subglacial conduits are completely filled with water under steady-state conditions. This, however, is not necessarily the case. Where discharges are larger than a few tens of liters per second and the down-glacier slope of the

l'energie newtonnienne disponible en exces par rapport aux besoins du strict bilan peut remplacer l'ecoulement general de la glace. Les chenaux peuvent alors tendre

a prendre

une position diagonale par rapport

a la direction

de

I'ecoulement de la glace. Si un accroissement d e I'angle qu'un tel chenal fait

bed is more than a few degrees, the potential energy released by water

avec la direction de l'ecoulement de la glace provoque une augmentation

conduit many times faster than the conduit can close by plastic flow of the

ce qui enrraine un processus de retroaction positive.

descending this slope may be capable of melting the walls of a subglacial

ice. As a result, the pressure in such tunnels may normally b e atmospheric,

or possibly even at the triple-point pressure if there is no open connection

de la pente d\, dit chenal, plus d'energie newtonnienne devient disponible Des

chenaux

sous-glaciaire

a

la

pression

atmospherique

peuvent

influencer l'origine el la morphologie de certaines formes glaciaires tels que

to the glacier surface. Simple calculations suggest that such pressures in

les eskers et les formations du type "moulage plastique".

subglacial conduits may be more common than hercLOfore anticipated. turbations in discharge or ice veloci ty. This is because the mechanical

ZUSAMMENFASSUNG. Ober die Rolle der Newlon'schen Energie bei der AuJ­ rechlerhallung subglazialer Wasse1iihrungen unler LuJldruck. Neuere theoretische

offset the general flow of the ice. Cond ui ts can thus trend diagonally across

Giinge unter stationaren Bedingungen ganz mit Wasser gefUllt sind. Dies

The positions of such "open" conduits may be unstable to small per­

energy available in excess of that needed to balance closure can instead the direction of ice Aow. If an increase in the angle which such a conduit

makes with the ice flow direction also results in an increase in slope of (he

conduit, more mechanical energy will become available, resulting in a

positive feedback process.

Subglacial channels at atmospheric pressure may inAuence the origin and

morphology of certain glacial landforms, such as eskers and "plastically­ molded" features.

RESUME. Sur le role de l'inergie newtonnienne dam le maintien du riseau aquiflrl

sous-glaciaire cl la pression almosphirique. De recentes etudes theoriques d'hydrologie glaciaire ont suppose q u e le reseau sous glaciaire clait corn­

pletement rempli d'eau en condition d'cquilibre stable. Ceci, neanmoins,

Untersuchungen der Gletscher-Hydrologie nehmen an, dass subglaziale

mussjedoch nicht unbedingt der Fall sein. W o der AbAuss grosser als einige zehn Liter pro Sekunde ist und die Neigung des Bettes gletscherabwarts

mehr als einige Grad betragt, kann die potentielle Energie des Wassers, das Uber dieses Bett abfliesst, die Wande einer subglazialen Fuhrung vidrnal

schneller aufschmelzen, als sich die Fuhrung durch plastisches Fliessen des

Eises schliess!. Als Folge durfte der Druck in solchen Tunnels gewiihnlich

dem Luftdruck gleich sein oder moglicherweise sogar dem Druck am Drei­ fachpunkt, wenn keine offene Verbindung zur GletscheroberAiiche besteh!.

Einfache Rechnungen lassen darauf schliessen, dass soIche Druckverhalt­

nisse in subglazialen Fuhrungen hiiufiger als bisher angenommen vor­ komrnen.

Die Lage soIcher "offener" Fuhrungen durfte instabil gegenuber kleinen

a quelques

Storungen im AbAuss oder in der Eisgeschwindigkeit sein. Dies ist der Fall,

dizaines de litres par seconde et q u e la pente dU lit dU glacier est superieure

weil die Newton'sche Energie, die als Oberschuss gegenuber jener vor­

I'eau descendant

handen ist, die der Schliessung das Gleichgewicht halt, statt dessen den

n'est pas necessairement le cas. Lorsque les debits sonl superieurs

a quelques degres, I'energie potentielle mise en oeuvre par

celle pente peut "Ire capable de fondre les parois d'un chenal sous-glaciaire

beaucoup plus vite que le chenal ne peut se refermer par le Auage plastique

Gletscherftuss ausgleichen kann. Fuhrungen kiinnen so diagonal zur Fliess­

richtung des Eises laufen. Wenn die Zunahme des Winkels, den eine solche

de la glace. Il en resuhe que la pression dans de tels chenaux peut nor­

Fiihrung mit cler Fliessrichtung des Eises einschliesst, mil einer Zunahme

triple s'il n'y a pas de connection o uveTte vers la surface du glacier. Des

E nergie frei, was zu einer positiven Riickkopplung fLihrt.

glaciaires peuvent etre plus frequentes qu'il n'crair admis jusqu'ici.

und GestallUng gewisser glazialer Landformen beeinflussen, z.B. von Es­

rnalement etre atrnospherique ou, peut etre mcme a la pression du point caIculs simples montrent que de telles pressions dans les chenaux sous­

Les positions de tels chenaux "ouverts" peuvent ctre instables aux petites

der Neigung der Fuhrung verbunden ist, wird noch mehr Newton'sche

Subglaziale Kanale unter atmosparischem Druck konnen die Entstehung

kern und "platisch-geformten" Erscheinungen.

variations dans le debit ou la vitesse de la glace. Ceci provient de ce que

INTRODUC TION T heoretical stud ies d uring the past 15 years hav e contributed significantly to our und erstand ing of the internal and subglacial parts of a temperate glacier's d rai nage system. Nye (1953) analyzed the cl osure of circular passages by plastic d eformation and Haefeli (1970, p. 207) seems to h av e been among the first to mention the role of flowing water in enlarging such passages by melting. Ro thlisberger (1972, p. 178) and Shrev e (1972, p. 206) assumed a balance between these two processes in the stead y state, and used this to d ev elop theoretical mod els for water flow in such tunnels. T heir mod els are v alid only for situations in which the tunnels are completely filled with water. The pressure in the water is then just s1 i ghtly 1 ess than that in the ice, the latter being proportional to the ice thickness. Shrev e (1972, p. 209-10) mentions briefly the possibility of unfilled passages, and Ro t hlisberger (1972, p. 186, p. 193) encountered a few situations in which his solutions pred icted the existence of such cond uits. Neither author pursued this result to inv estigate the range of cond itions und er which such *Pcrmancnt Address: Department of Geo)ogy and Geophysics, University ofMin­

nesora, Minneapolis, MN 55455, U .S.A.

180

cond uits might be expected . L liboutry (1983), howev er, has looked at this possibility more closely, and recognizes that many passages may not be filled with water much of the time, ev en und er stead y-state con­ d itions. In the present paper this alternativ e is analyzed in greater d etail. The equations d ev eloped apply only when the potential melting rate on passage walls just barely exceed s the closure rate. It is shown that this possibility may be more common than prev iously anticipated . This may hav e significant implications for glacier behav ior and for the d ev elopment of som e common glacial land forms. T HEORE TICA L DE V ELOPMENT From turbulent pipe flow theory we hav e (Hunsaker and R ightmire, 1947, equation 8.11)

P2 - PI

+

22

L

-

21 -!Dc

v2

2g

o

(1 )

where Pi is the water pressure at location i in a cyl­ ind rical cond uit of d iameter Dc and friction co­ efficient!, which is carrying water of d ensity P w mov ing at v elocity V, zi is the elev ation of point i,

Hooke:

SubgZac:iaZ uxrter conduits

the pressure-melting temperature T m rises. However in a conduit in which the water pressure is constant, the temperature will not vary with ice thickness. On the other hand, due to the overburden pressure, ice in the walls of such a conduit will be colder than the water, so some of the energy in the water wi11 be lost by conduction to the ice. The remainder is, how­ ever, available for melting ice at a rate rn, thus

z Glacier surface ......

�Pim1TDsHL __ ��____-L__�______-i__________

X

used in EqUltion (1). Fig. 1. Definition of symboLs

is the acceleration due to gravity, and L is the distance between two cross-sect ions, i 1, 2 (Fig. 1 ). If the melt rate is slightly larger than the closure rate, the passage will be somewhat larger than neces­ sary to carry the discharge provided. Under these conditions the pressure at the water surface in the tunnel will be atmospheric if there is a connection to the glacier surface. In the absence of such a con­ nection, the pressure will be determined by the vol­ ume of air entrained in the water and in bubbles in the ice; if there is no trapped air, the pressure could fall to its triple-point value. In any case P2 P1 and the first term in Equation (1) vanishes. We wi11 refer to condui ts in which this occurs as "open" whether or not there is a connection to the glacier surface. It should be emphasized that Equation (1) applies only when the conduit is essentially full. As the air space in the tunnel increases, f will decrease and the flow will accelerate, thus further decreasing the cross-sectional area of the conduit occupied by water. In the following development we are concerned prin­ cipally with defining the conditions under which the transition from a full to an open conduit may occur. No attempt is made to explore in detail the charac­ teristics of flow in the open conduit. From continuity relations we have Q = nVDc 2/4 where Q is the water discharge. Geometrically, (2 2-21 )/L sin S, where S is the local bed slope. In accord with the coordinate system adopted in Figure 1, S is positive if the glacier bed slopes downward in the direction of flow. Making these sub­ stitutions and solving Equation (1 ) for Dc' we obtain

g

=

=

=

=

pwQgh

-

K

1TDSL dPi dTm - -2 dR dpi

--

(4 )

where P i is the density of ice, His the heat of fusion, R is the distance into the wall of the tunnel K is the thermal conductivity of ice, and p' is the ' pressure in the ice. If z(x) is the glacier \ hickness, Nye's (1953) relations can be used to show that dP i/dR 8pWZ/9D s on the tunnel wall. Then, noting that sin S = h /L, we obta in

1

=

4 dT m Pw Q -- sin S- -KZ -9 dPi Pi n

(5 )

•

With K = 2.1 J /m s deg and dT m /dp· = 9.B x lO - Bdeg/Pa for alr-saturated water and ice (flarrison, 1972), the second term within the brackets becomes 9.1 x 10 -Bz. Paterson (1971) notes, however, that the thermal con­ ductivity of ice very near the melting point is be­ tween 1% and 10% of the cold-ice value used above. In addition, the above calculations do not take into con­ sideration the temperature rise across the tunnel wall that is necessary for a finite rate of heat transfer. In view of the very low temperature differences in­ volved, this could easily reduce heat loss by a fac­ tor of two. Thus the second term is probably of the order 5 x 10 -9z or 5 x 1O-10Z whereas the fi rst is of the order 1O-2 Q or higher for appreciable bed slopes. The second term can thus be neglected when the ice thickness is less than a fe� hundred meters and dis­ charges exceed about 10 - 3 m3/s. The consequences of this assumption are explored more fully later. Neglecting this term and eliminating Ds from Equations (3) and (5) yields 5 3 m = C2 Q /5 sin6/ S

(6 )

with P i = 916 kg/m3 • Pw 999.B kg/m 3 at Doe, and H = 3.34 x 10 5 J/kg, C 3.73 x 10-5 m -4/\- 2/5. 2 The closure rate;' of the tunnel is approximately (Nye, 1953)

(2 ) As we are primarily interested in conduits at the glacier bed, a circular cross-section is unreasonable. Rothlisberger (1972, p. 192) and Shreve (1972, p. 213) use a semicircular cross-section in their examples, and that form is adopted here. The hydraulic radius of this semicircular passage should presumably be the same as that of a circular passage capable of carry­ ing the same discharge, if differences in friction factor are ignored. Thus D ((n+2)/1T)Dc' where D s is the diameter of the semicircular pass­ age. Assuming this scaling, and taking f = 0.05 as a plausible value for the friction coefficient (Hunsaker and Rightmire, 1947, p. 127), we obtain =

(3) where the constant factor Cl has the value 0.55 s 2/5 m -1 /5. The potential energy released per unit time as the water fall s through a vertical distance h wi 11 be fJ.,fJgh. In a full conduit, part of this energy may be needed to warm the water as the glacier thins and

(7)

where B is the viscosity parameter in Glen's flow law for homogeneous isotropic ice, {. = (T/B)n. Here £: and T are the effect ive strain-rate and stress, respectively. Equation ( 7 ) is strictly valid only when P; gZ � T , and when there are no frictional forces resisting sliding over the bed in a direction normal to the tunnel axis. The first of these con­ ditions is approximately satisfied, as PigZ will be a few megapascals for typical glacier thicknesses, whereas other stresses contributing to T are on the order of 0.1 MPa. The second condition is also prob­ ably reasonable, except very close to the bed, for instantaneous deformation of an initially semicircular tunnel. El iminating D s from Equations (3) and (7) and taking n 3 in the flow law yields =

(B)

1B1

JournaL of GLacioLogy

..

with B = 1.6 x 105 Pa a1 / 3 (Hooke, 1981; Lliboutry, -14 m-16/5 s-3/5 . 1983), C 3 = 5.70 x 10 From Equations (6) and (8) the condltlon m I' > can then be written as

[

,::

•

•

r

(9) 7 /5 , where C4 6.55 x 10 8 m12/5 s1/5. When this condition obtains, the potential melt rate on the tunnel walls will exceed the closure rate and the tunnel wl11 be larger than necessary to carry the discharge provided. [Roth1isberger's (1972) limiting condition for applicability of his theory (his equation (22)) is equivalent to Equation (9) above. To show thlS, dp/ dx (= p�Ch/dx) in his Eq�ation (9) is s�t equal to PWJ sin Il , in accord wlth the assumptlons . . made in the present paper, and the resu1tlng re1atlon for Q is then used in Equation (2) to obtain the reQ >

c,

=

lation k = 45/3(g/8f)1/ 2 /Dc1/6 between his roughness parameter k and the corresponding parameter f in the present paper. In addition, adjustment must be made for the conversion between the circular channel cross­ section which he used and the semicircular shape assumed herein. Weertman's (1972) equation (63) can be reduced to Roth1isberger's equatiion (22), but for a small constant factor, and thus it too is essentially equivalent to Equation (9) of the present paper.] In Figure 2, Il has been plotted as a function of for various values of Z and for the case when Q equals the right-hand side of Equation (9). For anyQ given combination of ice thickness and bed slope, the minimum value of that should lead to an open con­ duit can be read Q from this graph. For example, for a typical ice thickness of 250 m, water discharges in excess of about 0.1 m 3/s on a 50 bed slope should re­ sult in open conduits. As discharges i prog1acia1 � streams are typically in excess of 1 m /s lt lS clear

that most of the larger subg1acial conduits in valley glaciers may be "open" in areas of down-glacier sloping bed topography. L1iboutry (1983) reaches a similar conclusion but from different reasoning. To the extent that water draining from the surface is above the melting point (Shreve, 1972), or that there is a significant contribution of such "warm" water from groundwater (L1iboutry, 1983) the area of the glacier bed over which thi s condition obtai ns wi 11 be larger. To explore further the effect of heat loss to the tunnel walls, the second term within the brackets in Equation (5) was set equal to 5 x 10-10z, and the region of Figure 2 in which this term exceeded 10% of the first term was identified as a region in which such heat loss "probably" reduces in appreciably. The region in which 5 x 10 -9Z exceeds 10% of the first term is 1ikewise i dent i fied as a region in which such heat loss "may" reduce m appreci ably. It is apparent that this heat loss significantly af­ fects the conditions under which open conduits may be expected only when the discharge is quite smalL [Roth1isberger (1972, p. 188) applied his theory to vertical conduits, but reached a conclusion differ­ ent from that implied by Figure 2. This is because he began with the assumption m = ;., so the resu 1 t i ng solution yielded conduit diameters and hence water velocities, that satisfied the energy balance implicit inm = r. The resulting velocities-0.5 m/ s, for example, for water desc�ding i � a vertical conduit in ice with B = 1.8 x HP Pa a I 3 -are very low, and imply some form of constriction in the outflow from the bottom of the conduit.] Uncertainties

Rothlisberger (1972, p. 182) has pointed out the substantial uncertainties in such calculations. The largest sources of error are in the values o f the friction coefficient f and the flow-law viscosity parameter B . To illustrate the effect of varying these parameters within possible, but probably extreme,

90 60

---

40

20 en w

1

w

a:: N

IS'35E

- - Ice- thickness contours

------ Contours on

glacier

bed. Contour inler­ val 50m above 1300 m and 40 m

below

1240m with supple­ mentary contour lo­ cally at

HSOm.

67'S4'30"N

Fig. 4. Map of Stol'gLaciaren shotJing ice thickness and bed contours. Back-pt"essure effects in the over­ deepened areas should result in fined conduits in the areas encLosed by the heavy dashed Lines. outside of those areas op3n conduits are eXp3cted.

tor of five error in the estimate of the discharges over the Riege�s would have a negligible effect on the locations of these lines. The analysis suggests that subglacial conduits should be open under about 75% of Storglaciaren. Filled conduits are probably present primarily in overdeepened areas where effective bed slopes are zero or negative. Filled conduits are expected in these latter locations largely because back-pressure effects "drown" the bed, and bed slopes in these areas thus play no role. As discharges scale with glacier area, and as changes in discharge have a relatively modest effect on � (Fig. 2), similar results may be expected from other valley glaciers. Of course, ; increases as z3 so somewhat more extensive areas of filled conduits might be ex­ pected under thicker valley glaciers. Eskel"s

Shreve (1972) was reasonably successful in ex­ plaining many features of eskers formed beneath con­ tinental ice sheets with the use of a fi1 led-conduit , in ;. model. The present analysis suggests, however, that such a model may not be valid in areas where ice thicknesses are modest, discharges high, and bed slopes ;:Josit ive. In two �reas in Norway which I have studied, esk=

186

ers formed near the margin of the Scandinavian ice sheet become anastomosing. At distances of a few kilo­ meters from the margin a single low esker ridge may split from the main ridge and then rejoin it a few hundred meters down-stream. The geometry of the junc­ tions is such as to suggest either that both branches of the esker were active simultaneously, or that the lower branch was later. Only rarely is there a clear suggestion that the higher esker overlies the lower one. Further down-glacier bifurcations become more common, and eventually the esker complex comes to re­ semble a kettled outwash plain rather than a braided esker. I call such features eskel" nets. A reasonable hypothesis for the origin of such esker nets is that in > ;, so condu its were open and the water, rather than remaining on top of the esker, melted its way down its flank and formed a parallel esker for a short distance. In the two areas in which I have studied such es­ ker nets, northern Atnedalen (lat. 62°03'N., long. 10001'E.) and Bj(oira (lat. 62°23'N., long. 1l006'E.), Norway (see also Gjessing, 1960, plates 3 and 10), the eskers slope gently (0.4° and 1.5° respectively) down-glacier. With discharges on the order of 10� to 10 3 m3/s, the conduits in which the eskers formed might have been open under 100 to 200 m of ice (Fig. 2), which is reasonable for locations within a kilo­ meter or two of the ice margin. The hypothesis that esker nets form in areas where m>; is thus plausible. Anastomosing esker patterns may well be found elsewhere in areas of steeper bed slope farther from the glacier margin. In such cases, Figure 2 may pro­ vide a basis for speculating on the formative dis­ charges and corresponding ice thicknesses. P-forms

Plastically molded forms, or p-forms as they are often called, are a glacier-bed landform that has excited the imagination of glacial geologists for decades. Although bearing obvious evidence of ice abrasion in the form of striations, they often closely resemble fluvially eroded forms. Only recently has this apparent dichotomy been resol ved with the realization that cavities at the glacier bed can open during the summer melt season and close during the winter, so water and ice erosion can alternate sea­ sonally (Sollid, 1975). However the problem of the initiation of p-forms remains. Barnes (1956, p. 503) has suggested that sub­ glacial cavitation erosion may form depressions that later develop into potholes, and also notes a certain similarity between some other subglacial erosional forms and shapes "expected" from cavitation erosion. An unanswered question in Barnes' hypothesis is whether pressures in a subglacial water conduit can locally be reduced to the vapor pressure, a necessary prerequisite for cavitation erosion. The analysis presented herein suggests that this may well occur if the glacier bed slopes downward in the down-glacier direction. In view of the heat-transfer considera­ tions discussed earlier, the downward-sloping reach of the glacier bed should be on the scale of hundreds of meters so that the mechanical energy released will have time to enlarge the conduit t� the point where it is no longer full. If, in addition, the bed geo­ metry is such that the conduit is full both up- and down-glacier from the reach in question, and if there are no open connections to the glacier surface, the vapor pressure (actually the triple-point pressure) may be reached in the conduit even without appealing to the very high velocities normally associated with cavitation. Unfortunately, most descriptions of p-forms do not provide much information on the regional topog­ raphy. In many instances one may infer, however, that the condition of a positive bed slope is met. For ex­ ample, p-forms are spectacularly developed in the Port(oir area on the south-eastern coast of Norway (Gjessiny, 1965). As the Scandinavian ice sheet was

Hooke :

f lowing seaward in th i s area, th e regi onal slop e is clearly d own ward i n th e d own-g laci er di recti on . Si m ilarly in north ern Norway near Narv ik, p-form s are well d ev eloped in an area wh ere i ce was m ov i ng d own i nto fjord s f rom upland areas (D ah l, 1965). Th us h ere too, th e reg ion al bed slope was prob ably positiv e in m any i nstances. Th ese observ ations lend support to th e cavi tati on -erosi on h ypoth esis for th e i n i ti ati on of at least som e p-f orm s. CONC L U SI ONS I n areas wh ere g laci er b ed s slope d ownward in th e d i recti on of i ce f low, th e m ech anical energ y d i ssi pa­ ted i n water i n subg laci al cond ui ts d escend in g such slopes m ay en larg e th e con d ui ts m ore rapi rl l y th an th ey are closed b y plasti c flow of th e i ce. U nd er such ci r­ cum stances, th e pressure in th e cond uit wi 11 b e at­ m osph eric i f th ere is a con ne ction to th e g lacier sur­ f ace, and in any case wi l l b e ind epend en t of posi ti on along th e con d uit. A relati v ely si m ple stead y-state analysi s of th is probl em sug g ests th at such "open" cond uits m ay b e m ore comm on th an h eretof ore expected . Wh en th e add i tional com plication of non-stead y state f lows is in clud ed i n th e analysi s, th e probabi li ty th at m any con d uits are open m uch of th e ti m e increases substantially. Gen­ eraliz ations, h owev er, are d if fi cult as each g laci er presents a un ique com bination of water d isch arg e, ice th l ckness, and b ed topog raph y. Furth erm ore, b ack­ pressure effects f rom areas of low or rev ersed b ed slope wi11 red uce th e total area of th e bed ov er wh i ch open cond uits are expected . I n a stead y- state m od el, fi lled condui ts are con­ strained to run essenti ally parallel to th e ice f low d i rection. Howev er, open con d uits m ay trend d i ag on­ ally down b ed rock slopes at an an g le to th e d i recti on of ice flow. Th i s i s possib l e b ecause th e energ y d i s­ sipated i n th e water i n excess of th at need ed to b al­ ance closure can i nstead be used to off set th e g eneral f low of i ce toward th e con d ui t f rom th e up-g l aci er di recti on. Because th e energ y d i ssipated i ncreases as th e slope of th e conduit i ncreases, th e o ri en tati on of such d i ag onal conduits i s unsta b le to sm all per­ turb ations in d i sch arg e. Th us such cond uits m ay ch ang e posl tl on seasJ nally, or ov er sh orter ti m e peri ods, i n response to ch ang es i n d isch arg e. Areas of open cond ui t m ay b e responsib le for brai di ng of eskers and f or i n i tiati on of som e p-f orm s. O th er appli cati ons to prob l em s of g lacial g eom orph o­ log y sh ould b e consi dered . A s noted , th i s paper explores only th e cond i tions req ui red for a transi ti on from full to open ch annels, not th e ch aracter of th e open ch annels th em selv es. Th is rath er m ore com plex prob lem needs to b e add ressed in th e future, pref erably with th e ai d of fi eld d ata. R . L . Sh rev e (wri tten comm un i cation in M ay 19 83), for exam ple, sug g ests th at such ch annels m ay h av e a f lattened sh ape as m elti n g will be concentrated on th e si des (and b ottom i f th e cond uit is eng l aci al), and th at th i s m ay increase th e closure rate. ACKNOW LE D GEME NT S I sh ould like to express m y i nd ebted ness to P . H olm lund and B. W old f or sti m ulati ng m y i nterest in subg lacial d rai nag e, an d to J . Gjessi n g for f i rst sh owing m e th e m ag ni f i cen t eskers of north ern Atnedalen, Norway. R . L . Sh rev e, J . W eertm an, an d an anonym ous rev i ewer contri b uted a numb er of v aluable cri ti cal comm ents on an early v ersi on of th i s paper. R E FEKENCE S

Subglacial wte r condui t s

D ah l, R . 1965. P lastically sculptured d etail f orm s on rock surfaces i n north ern Nord l an d , Norway. Geog mfiska Anna l e r, V ol. 47A, No. 2, p. 83-14 0. Gjessi ng , J . 1960 . Isav sm eltni ng stid ens d reneri ng . Ad Novas (O slo), No. 3. Gjessi ng , J . 1965. On "plasti c scouring " and "sub ­ g laci al erosi on" . No rsk Ceog mfi s k Tidsskrift, Bd . 20, H t. 1-2, p. 1-37 . Haefel i, R . 197 0. C h ang es in th e b eh av i our of th e U nteraarg letsch er i n th e last 125 years. Journa l of Glaciology, V ol. 9 , No. 56, p. 19 5 -212. Harrison, W .D . 197 2. T em perature of a tem perate g laci er. Jo u rna l of Claci o l ogy, V ol. 11, No. 61, p. 15-29 . Hooke, R . L . 19 81. Flow law f or pol y crystalline i ce i n g laci ers: com pari son of th eoreti cal predictions, lab oratory d ata, an d f ield m easurem en ts. Reviews of 19 , No. 4 , Geophysics and Space Physics, V o1 p. 664 -7 2. Hooke, R . L . and othe rs . In press. Near surface tem ­ peratures near and b elow th e equi lib ri um line on polar and subpol ar g laci ers, by R . L . Hook e, J . E . Gould , and J . Brz oz owski . Zei t sc h rift fur Gle tscherkunde und Glazialgeologi e , Bd . 19 , H t. 2, [ for 19 83J . Hunsaker, J .C ., and R i gh tmi re, B.G. 194 7 . Engine e ri ng appli cations of fl uid mechanics . New Y ork, M cGraw­ Hi ll Book C o. , I nc. L li b outry, L .A. 197 1. P erm eabili ty, b rin e content, and tem perature of tem perate i ce. Journal of Claci o­ l ogy, V ol. 10, No. 58, p. 15-29 . L li b outry, L .A. 19 83. M odi fi cati on s to th e th eory of intrag laci al waterways f or th e case of subg laci al ones. Jo urnal of Glaciology, V ol. 29 , No. 102 , p. 2 16-26. Ni l sson , J ., and Sun d blad , B. 197 5. Th e internal d rai nag e of Storg laci aren and I sfall sg laci aren d escribed by an autoreg ressi v e m od el. Ceogmfi ska A nna l e r, V ol. 57A, Nos. 1-2, p. 7 3-9 8. Nye, J .F. 19 53. Th e f low law of i ce f rom m easurem en ts i n g lacier tunnels, lab oratory experi m ents, and th e J ung fraufi rn h oreh ole experim ent. Pr'Oceedings of Ser. A, V ol. 219 , t he Royal Socie t y of London, No. 1139 , p. 4 7 7 -89 . Nye, J .F ., am Frank, F .C . 197 3. Hyd rolog y of th e l � te� g � anular v � ins in a tem perate g laci er. Union •

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