JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 92, NO. B9, PAGES 9083-9100, AUGUST
10, 1987
Glacier SurgeMechanismBasedon Linked Cavity Configuration
of the Basal Water Conduit System
BARCLAY
KAMB
Division of Geologicaland PlanetarySciences,CaliforniaInstituteof Technology,Pasadena
Based on observationsof the 1982-1983 surgeof Variegated Glacier, Alaska, a model of the
surgemechanismis developedin termsof a transitionfrom the normal tunnelconfigurationof the
basalwater conduitsystemto a linked cavity configurationthat tendsto restrictthe flow of water,
resulting
in increased
basalwaterpressures
thatcause
rapidbasalsliding.Thelinkedcavitysystem
consistsof basalcavitiesformed by ice-bedrockseparation(cavitation),~1 m high and ~10 m in
horizontaldimensions,widely scatteredover the glacierbed, and hydraulicallylinked by narrow
connections
whereseparationis minimal (separationgap •<0.1 m). The narrowconnections,
called
orifices,controlthe waterflow throughthe conduitsystem;by throttlingthe flow throughthe large
cavities,the orificeskeep the water flux transmittedby the basalwater systemat normal levels
even thoughthe total cavity cross-sectionalarea (-200 m2) is much larger than that of a tunnel
system(-10 m2). A physicalmodel of the linked cavity systemis formulatedin terms of the
dimensionsof the "typical"cavity and orifice and the numbersof theseacrossthe glacierwidth.
The modelconcentrates
on the detailedconfigurationof the typicalorifice andits responseto basal
water pressureand basal sliding, which determinesthe water flux carriedby the systemunder
given conditions.Configurations
are workedout for two idealizedorifice types,steporificesthat
form in the lee of downglacier-facing
bedrocksteps,andwaveorificesthatform on the lee slopes
of quasisinusoidal
bedrock waves and are similar to transverse"N channels." The orifice
configurations
are obtainedfrom the resultsof solutionsof the basal-sliding-with-separation
problemfor an ice massconstitutinga near half-spaceof linear rheology,with nonlinearity
introduced
by makingtheviscosity
stress-dependent
on an intuitivebasis. Modification
of the
orifice shapesby melting of the ice roof due to viscousheat dissipationin the flow of water
throughthe orificesis treatedin detailunderthe assumption
of localheattransfer,whichguaranteesthat the heatingeffectsare not underestimated.This treatmentbringsto light a meltingstabilityparameterE that providesa measureof the influenceof viscousheatingon orifice
cavitation,similarbut distinctfor stepandwave orifices. Orifice shapesandthe amountsof roof
meltback
aredetermined
byE. WhenE •>1,sothatthesystem
is"viscous-heating-dominated,"
the
orificesareunstableagainstrapidgrowthin response
to a modestincreasein waterpressure
or in
orifice size over their steadystatevalues. This growthinstabilityis somewhatsimilarto the
j6kulhlaup-type
instability
of tunnels,
whicharelikewiseheating-dominated.
WhenE <•1, the
orificesarestableagainst
perturbations
of modestto evenlargesize. Stabilization
is promoted
by
high slidingvelocityv, expressed
in termsof a v 4/2 andv-• dependence
of E for stepandwave
cavities. The relationships
betweenbasalwaterpressureand water flux transmitted
by linked
cavitymodelsof stepandwaveorificetype are calculatedfor an empiricalrelationbetweenwater
pressureand slidingvelocityand for a particular,reasonable
choiceof systemparameters.In all
casesthe flux is an increasingfunctionof the waterpressure,in contrastto the inverseflux-versuspressure
relationfor tunnels.In consequence,
a linkedcavitysystemcanexiststablyasa systemof
manyinterconnected
conduitsdistributed
acrossthe glacierbed, in contrastto a tunnelsystem,
whichmustcondense
to oneor at mosta few maintunnels.The linkedcavitymodelgivesbasal
waterpressures
muchhigherthanthe tunnelmodelat waterfluxes•>1m3/sif the bed roughness
featuresthatgeneratethe orificeshavestepheightsor waveamplitudes
lessthanabout0.1 m. The
calculatedbasalwaterpressureof the particularlinkedcavitymodelsevaluatedis about2 to 5 bars
below ice overburdenpressurefor water fluxes in the range from about2 to 20 m3/s,which
matchesreasonablythe observedconditionsin VariegatedGlacier in surge;in contrast,the
calculated
waterpressure
for a single-tunnel
modelis about14 to 17 barsbelowoverburden
over
the sameflux range. The contrast
in waterpressures
for the two typesof basalconduitsystem
furnishes
thebasisfor a surgemechanism
involvingtransition
froma tunnelsystemat lowpressure
to a linkedcavitysystemat highpressure.The parameter
E is about0.2 for the linkedcavity
modelsevaluated,meaningthat they are stablebut that a modestchangein systemparameters
couldproduceinstability.Unstableorificegrowthresultsin the generation
of tunnelsegments,
Copyright1987 by the AmericanGeophysicalUnion.
Papernumber6B6328.
0148-0227/87/006B-632505.00
9083
9084
KAMB:
GLACIER SUROE MECHANISM
which may connectup in a cooperativefashion, leading to conversionof the linked cavity system
to a tunnel system, with large decreasein water pressureand sliding velocity. This is what
probablyhappensin surgetermination. Glaciersfor which E •<1 can go into surge,while thosefor
which E •>1 cannot. BecauseE variesas c•3/z(where c• is surfaceslope),low valuesof E are more
probable for glaciers of low slope, and becauseslope correlatesinversely with glacier length in
general,the model predictsa direct correlationbetweenglacier length and probabilityof surging;
sucha correlationis observed(Clarke et al., 1986). BecauseE variesinverselywith the basalshear
stressx, the increaseof x that takesplace in the reservoirareain the buildupbetweensurgescauses
a decreasein E there, which, by reducingE below the critical value ~1, can allow surgeinitiation
and the start of a new surgecycle. Transitionto a linked cavity systemwithout tunnelsshould
occur spontaneouslyat low enough water flux, in agreementwith observedsurge initiation in
winter.
1. INTRODUCTION
than in the nonsurgingstate [Brugman, 1986]. After surge
terminationthe meanwater transportspeedalongthe lengthof the
The fastest known glacier flow motions occur in glacier glacier (lower half) was 0.7 m/s, typical of basal water flow
surges. There has been much theorizingas to the cause(s)of the speedsfound in nonsurge-typeglaciers;during surge,in contrast,
fastflow (summarizedby Paterson[1981, p. 288]), but few if any the meanwatertransportspeedwasonly 0.025 m/s.
firm conclusionshave emerged becauseobservationsof the
5. In the dye-tracingexperimentduring surgethe dye was
processes
in action,which couldguidephysicalreasoning,have dispersed
acrossthe widthof the glacier,appearingin all outflow
beeninadequate.Studiesof the 1982-1983surgeof Variegated streams,whereasaftersurgeterminationthe dye appearedin one
Glacier, Alaska, provide a new, enlargedbody of observations streamonly.
[Kambet al., 1985]. I present
herea physical
modelof thesurge
6. Theoutflowstream
waterduringthesurgewasextremely
mechanism
developed
on thebasisof theseobservations.
A brief turbid(suspended
sedimentcontentat concentration
~100 kg/m3
sketchof themodelhasbeengivenby Kambet al. [1985,p. 478]. for particlesizes_<10[tm),muchmoreturbidthanaftersurgeor in
While the treatmentis basedon a "hardbed" model of the surge normal, nonsurgingglaciers (sediment content ~1-10 kg/m3)
mechanism,it seemslikely, as explained in section 10, that a [Brugman,1986, p. 88].
number of the important results are also applicable at least
3. FORMULATION
OF A MODEL OF THE SURGE MECHANISM
qualitativelyto "soft bed" models, in which sliding is over
deformable
basal
till.
The
discussion
concentrates
on
the
mechanismof surgingin springandsummerwhenrelativelylarge
amountsof water are availableto the basalwater conduitsystem.
The surge mechanismin wintertime can be consideredby
extendingthe conceptsdevelopedhere to conditionsof low water
flow; thiswill be donein a subsequent
paper.
2. OBSERVATIONAL
BASIS
The foregoing observationsthrow a sharp focus on what
needsto be explainedby a physical model of the surge mecha-
nism. The high slidingspeedsare explainedby the high basal
water pressures.A detailedexplanationrequiresa detailedmodel
of the relation betweenbasal water pressureand sliding speed.
Sucha modelcanbe developedwithin the frameworkof the basal
sliding mechanism discussedhere (or see Fowler [1986]), but
because,as will be argued, the detailed relationshipbetween
water pressureand slidingspeedis not the essentialelementof the
model needed to explain surging, I will here pass over these
details and insteadassumea simple empiricalrelation between
water pressureand sliding.
The essentialingredientof the surgemodel is what causesthe
high basalwaterpressures
in surge. How is the high basalwater
pressuremaintainedand indeed enhancedin springand early
summer,when, accordingto the standardmodelof the basalwater
conduitsystem[Rtthlisberger,1972], the pressureshoulddropas
an increasingflux of wateris carriedby the system?Sincehigh
basal water pressureand high basal sliding promote basal
cavitation,openingup holes(cavities)throughwhich water could
move at the baseof the glacier,and thusincreasingthe hydraulic
conductivityof the basal water conduit system, why does the
water not drain out from under the glacier more rapidly than in
nonsurge and thereby reduce the water pressureto subnormal
values? Why, on the contrary,doesthe glacierin surgeshowan
unusuallyhigh "retentivity"for water, as shownby the abnormally low water transportspeedrevealedby dye tracing? These
questionsgo to the heart of what is in my view the essential
physicaldifferencebetweenthe surgingandnonsurgingstatesof
the glacier. The surgemodel concentrateson explainingthis
The following observationsfrom the surge of Variegated
Glacier [Kamb et al., 1985; Raymond,this issue]form the direct
basisof the surgemodel:
1. The fast flow motionduringthe surgeis dueto rapidbasal
sliding.
2. During surge,the pressureof water in the basalconduit
systemis high,within 2-5 barsof the ice overburden
pressure,and
occasionallyreaching overburden;in the nonsurgingstate it is
distinctly lower, generally 4-16 bars below overburden,but with
occasionalpeaksto higherlevels. Peaksin pressure,particularly
those in which the water pressure rises to near overburden,
correspondto peaks in sliding motion, both in surge and out.
Thesefactsaretakenasindicationthat the directcauseof the high
slidingspeedin surgeis high basalwaterpressure.
3. Major slowdownsin surgemotion,andparticularlysurge
termination,are accompaniedby large flood peaksin the terminus
outflowstreamsandby a dropof theglaciersurfaceby 0.1-0.7 m.
This, in conjunctionwith observations
of uplift followedby drop
of the glacier surface in minisurges[Kamb and Engelhardt,
1987], is interpretedas an indicationthat the high slidingspeeds
and high basalwater pressuresin surgeand in minisurgesare
coupledwith extensivebasalcavitation,as expectedtheoretically
[Lliboutry,1968;Kamb, 1970,p. 720; Iken, 1981;Fowler, 1987]. difference and is therefore in the first instance a model of the
4. The flow of waterthroughthe basalwaterconduitsystem, basalwaterconduitsystemin surge.
as indicatedby dye-tracingexperiments,
is muchslowerin surge
The conduitsystemthat dominatesthe transportof water in
KAMB: GLACIER SURGEMECHANISM
the nonsurgingstateis a basaltunnelsystemof the kind discussed
in theoreticalterms by R(Sthlisberger
[1972], Weertman [1972],
Nye [1976], Spring and Hutter [1981, 1982], and Lliboutry
[1983]. It consistsof one or two main tunnels,of order 1 or a few
metersin diameter,runningalongthe lengthof the glacierat the
bed, usuallynear the centeror deepestpart, and probablyfed by
smallersidetunnelsheadingin glaciermoulins.
The water conduit system in the surging state is very
different. This is shownby the dye tracerexperimentsin termsof
the slowtransportspeedof waterthroughthe systemand the high
dispersionof the injected dye pulse [Brugman, 1986]. The
systemcannotconsistof the normal tunnelsof the nonsurging
statewith additionof conduitsformed by basal cavitationunder
the high basal water pressureand rapid basal sliding becausein
this case the water transportspeedwould remain high and the
total water transport(flux) would be increasedso that, as noted
above,the high water pressurecouldnot be maintained,at least
without an abnormallylarge throughputof water, which is not
observed. It follows that the normal tunnel systemmust not be
present.
In order to transportabout 5 m3/s of water at an average
longitudinalspeedof 0.025 m/s, the conduitsystemmusthave a
total cross-sectional
area of about200 m2 in the transverseplane
of theglacier. (An estimated5 m¾sis the averageflux at the time
of the tracerexperiment,taking into accountthe distributedinput
of meltwaterfrom the tracerinjectionpoint to the terminus,where
the dischargewas 7 m3/s.) If the conduitsare basalcavities,their
averageheight, areally averagedacrossthe 1-km width of the
glacier includingareasof ice-bedcontact(where the height is 0),
is 0.2 m. This is compatible with the average height 0.1 m
inferred from the drop in elevationof the ice surfaceon surge
termination
and also with the excess amounts of water released
from the glacier at surge termination and during the 4 days
thereafter,which correspond
to a dropin averagecavity heightof
0.1 and0.3 m, respectively. (Theseare upperlimits becausesome
of the surfacedrop may have beendue to ice strainand someof
the releasedwater may have been storedin intraglacialporosity;
seeKamb et al. [1985, p. 477].) Basalcavitiesof height~1-2 m,
distributedwidely overtheglacierbedandoccupying~ 10-20% of
the bed area, would provide passageways
for water flow of the
dimensionsneeded and would involve the widespreadcontact
between basal water and the ice-bed interface
that could account
for the extremelyhigh turbidity of the Outflow water in surge
(section 2, observation 6). The pattern of basal cavitation
visualized seems reasonable in relation to cavities that have been
actually observedunder glaciers [Carol, 1947; Kamb and La
Chapelle, 1964; Vivian and Bocquet,1973] or that can be inferred
from detailedobservationsof the abrasionmarkingson former
glacierbeds.
If the water conduitsystemin surgeconsistedof an openly
interconnected
network
of
basal
cavities
of
the
dimensions
suggested,
the water transportspeedthroughit would be ~ 1 m/s,
as it is throughnormal tunnelsystems,the lateral dimensionsof
tunnelsbeing of the sameorder. This is in strongcontradiction
with the observedtransportspeed,0.025 m/s. It follows that the
water flow throughthe conduitsof 200 m2._
cross-sectional
area
mustbe throttledin someway. Thereis a naturalreasonwhy this
should happen. In the sliding of ice over an irregularly undulatory bed, the distribution of normal stress across the
ice-bedrockinterface, which controlsice-bed separation,is such
that the large cavities that form tend to be isolatedfrom one
another, so that the water in them tends not to communicate
hydraulically. (A long steplike or wavelike roughnessfeature
9085
running transverseto the ice flow would develop a long,
connected cavity, but the transversedirection of hydraulic
communicationin it would not aid longitudinal transportof
water.) When the basalwater pressurebecomeshigh enoughand
the sliding velocity rapid enough, cavitation in the bed areas
intervening between the large cavities develops sufficiently to
provide hydraulic connectionsbetween the cavities, but the
connections are small features, much smaller than the cavities
theyconnect.It is thissystemof hydraulicallylinkedcavitiesthat
I here considerin a model of the surge mechanism. From the
evidencepreviouslydiscussed,
it appearsthatmostof the pressure
drop or potentialdrop in the water flow throughthe linked cavity
systemoccursin the narrowconnections,
or orifices,as I will call
them, and as a consequence,
theseorificesthrottleand controlthe
flow. On the other hand, for a parcel of water travelingthrough
the system,most of its time is spentmoving slowly throughthe
largecavities,so that the overalltransittime is tied to the 200 m2
total cross-sectional
area of the cavities. The water flow through
the orifices is probably fast and turbulent, as will be seen later,
and the water from eachorifice probablyemergesas a jet into the
cavity downstream,which helps to explain how the water is able
to pick up and carry a large amount of fine sediment in
suspension.
Once the basictopologyof the water conduitsystemin surge
is ascertained,it becomespossibleto formulatea physicalmodel
of it in sufficient detail to permit its hydraulicpropertiesto be
determined, analogouslyto what has been done for the normal
tunnel system by Riithlisberger [1972]. The results provide a
basis for deciding whether the underlying picture of the surge
mechanismis satisfactoryand for identifying the physical
conditionsthat distinguishthe surging and nonsurgingstatesof
glacier motion, from which one can reasonabout what causesa
glacierto be in the nonsurgingor surgingstate.
4. THE LINKED
CAVITY
MODEL
The patternof basalcavitationvisualizedin the linked cavity
model is shownschematicallyin Figures1 and 2. Figure 2 shows
schematically the longitudinal cross-sectionalshapes of the
leesideice-bed separationcavitieswhoseplan view shapesare
shownin Figure 1. The hachuredlines in Figure 1 represent
upstreamcavity boundarieswhere the ice separatesfrom bedrock,
often at the edgeof a topographicstepor sharpbreakin slopeof
the bedrocksurface,as shownin Figure2; the nonhachuredlines
in Figure 1 are wherethe ice recontactsthebed downstream.The
cavitationpatternin Figure 1 is rathersimilarto naturalexamples
of basalcavitationmappedby WalderandHallet [ 1979, Figure7]
and by Hallet and Anderson [1980, p. 174], except for the
presence of solution-etchedflow channels in the limestone
bedrockof thesenatural examples. Under high sliding velocity,
the cavitiesprobablyelongategreatly downglacier,but the basic
topologyof the hydrauliclinkageof cavitiesremainssimilar.
In Figure 1, the larger areasof the ice-bedrockseparation
representthe cavitiesof the linked cavity model, and the narrow
connectionsbetweenthese are the separation-gap
orifices. In
cross section these features appear as shown in Figure 2:
Figure 2a is a section through two separation cavities in
successionalong the flow line, while Figure 2b is a section
throughtwo orifices,somewhatexaggerated
in sizefor clarity.
The patternof water flow throughthe linkedcavity systemis
shownby the smallarrowsin Figure1. Becauseof the geometry
of cavitation, the flow water tends to be in the lateral direction,
particularlyin the orifices. This lateral flow is reflectedin the
9086
KAMB: GLACIER SURGE MECHANISM
(O)actual
I•nked-cawty
pattern
(schemahc)
(b) •deal•zed
hnked-cawt¾
model
:.:-:-:
iiiiiiiiiili
::::::::::::
he,cjht
of.... tyce,hng
=gc i!iiiiiiiiii
:::::2::::
::::::::::::
i:i ORIFICE
ii::'
CAVITY
/ce
wa/er
.........
flow
B'
'i:i:i:!:!:i
Fig.3. Localdetailanddimensions
in thelinkedcavitysystem
as seenin
mapview,withconventions
of Figure1. (a) A schematic
representation
of a realistic
linkagepattern
between
twocavities.(b)The idealized
geometryof the linkageassumed
in thelinkedcavitymodel.The length,
breadth,andheightparameters
L, l, andg arediscussed
in thetext.
. .......
......
i:
.....
"
/ .:i••
..
'
.
• 10m
l is the averagelateral dimension(breadth)of the orifice in
the direction transverseto water flow. This is always in the
direction
of
sliding
and
represents
the
length
ofthe
separation
gapbasal
between
ice and
bedrock,
which
forms
theorifice.
to is the average dimension(breadth) of the cavity in the
Fig. 1. Conception
of the linkedcavitybasalwaterconduitsystem,in direction transverseto water flow. In the cavitationpattern of
mapview,portraying
schematically
a smallareaof theglacier
bed,of Figure1 thisdimension
is generally
parallelto basalsliding,but
lateral
dimensions
~20m. Areas
oficecontact
with
thebed
areshaded,
in downglacier-elongated
cavities
it tends
to betransverse
to
areas
ofice-bed
separation
(cavitation)
are
blank.
Vertical
cross
sections
sliding.
alonglinesAA' andBB' are shownin Figure2. The largearrowindicates
thedirection
ofbasal
sliding.
Iceseparates
from
thebedalong
the
Loisthelength
oftheorifice
inthedirection
parallel
towater
hachured
linesandrecontacts
the bedalongtheplainlines. Onelarge flow. Becauseof the curving,sinuouspatternof the ice-bedrock
cavityandtwosmall"orifices"
in thelinkedcavitypattern
areidentified.separation
linesandrecontacting
lines(Figure1), the dimension
Directions
ofwater
flowthrough
thesystem
areshown
with
small
arrows.
Loisnotsharply
defined,
asFigure
3asuggests,
butinthemodel
Approximatescaleis indicatedby the 10-m bar.
the distinction between cavity and orifice is sharpenedby
wide lateral dispersionof dye in the tracer experimentduring idealizingtheirgeometryin the way shownin Figure3b.
Lo is the dimensionof the cavity in the directionparallel to
surge(section2, observation5).
water
flow, or, more precisely,the distancebetweensuccessive
In developinga quantitativemodelof the linkedcavity system
orifices
alongthe water-flowpath.
I will considerthe systemin termsof a typicalcavity anda typical
A = Lo/Lo is the "head gradient concentrationfactor"
linking orifice, illustratedin plan view in Figure 3. The dimensionsassociated
with the cavity andorifice,markedin Figure3, (section5).
are as follows:
gois the averageheightof the cavity.
g is the local height (measuredperpendicularto the bedrock
surface)of the separation-gap
orifice. It is a functionof position
::::::::::::::::::::::::
C :::::::::::::::::::::::::::::::::
.....
.........
'"""/C :::::::::::::::::::::::::::
acrossthe orifice, from the point of ice-bedrockseparationto the
point of recontact.
No is the number of independentorifices in a transverse
sectionacrossthe glacier. The averagelateral spacingbetween
independentorificesis W/No, where W is the glacier width. By
independentorifices I mean orifices that are not in succession
along water flow paths,so that their water fluxes add to make the
total water flux carriedby the system.
An actuallinked cavity systemis an ensembleof cavitiesand
orificesof differentshapesandsizes,resultingfrom cavitationby
basalslidingover the diverseand sundryroughnessfeaturesof the
~ lorn
bed, under a spatial distribution of basal shear stress, water
Fig. 2. Vertical crosssectionsthroughthe schematiclinked cavity system pressure,and ice overburdenpressure. The complexity of the
of Figure 1, alonglinesAA' and BB'. Heavy shadingindicatesbedrock, actual system is suppressedin the model developedhere by
light shadingice. The arrows showthe directionof basal sliding. The replacingthe spectrumof cavity and orifice dimensionswith a
blank areas are water-filled volumes formed by ice-bed separation
single set of "typical" dimensionsas indicated above and by
(cavitation)in the sliding process. Section AA' showstwo large
consideringhow thesedimensionsarecontrolledby a singlesetof
separation
cavities,while BB' showstwo separation
gaporifices,whose
gapheightis exaggeratedfor visibility in the drawing. Approximatescale values of basal shear stress '•, water pressure Pw, and ice
is indicatedby the 10-m bar.
overburdenpressurePI, when basal sliding takes place over
KAMB: GLACIERSURGEMECHANISM
9087
roughness
featuresof a singletype,with prescribed
dimensions.equivalent
to thestatement
thatall dropin hydraulic
headis taken
This idealization,whichis asgreata simplificationof the system acrossthe orifices;hencethe headgradientin the orificesis the
as can be made without losingwhat I regard as its essential averagegradientmultipliedby LdLo= A (section4). The orifice
physicalcharacteristics,
is chosenhereasthefirst approximationgapsareassumed
to be thincompared
to theirbreadth(g <<l), as
to the behaviorof the complicated
naturalsystem.It is madein seemsappropriate
for narrowice-bedrock
separation
gapsthatare
thesamespiritastheanalysis
of basalslidingovera bedwith a onlymarginally
open. For simplicity,
thetypicalorificeis taken
singletypeof roughness
feature[e.g.,Weertman,
1957]).
tohavea cross-sectional
shape
thatis constant
alongitslengthLo.
Becausethe flow of waterthroughthe linkedcavitysystemis The flow of water throughthe orifice is thendeterminedlocally
controlled
by the orifices,asdiscussed
in section3, andbecause by thelocalgapheightg(x), whichdepends
on a spatialcoordithe orificesare the ice-bedseparationgapsthat are mostsensitive nate x acrossthe breadth l of the orifice. Since the flow is
to theconditions
controlling
separation,
thehydraulic
behavior
of turbulent
(aswill be shown),
it canbeobtained
fromtheManning
thesystem
is muchmoresensitive
to thedetailed
geometry
of the formula [RSthlisberger,
1972, equation(9)] by putting the
orificesthanit is to the cavities.The cavitiesdoubtless
vary hydraulic
radiusequaltog(x)/2:
somewhatas physicalconditionschange,but the variationof the
Consequently,
inanalyzing
the
linked
cavity
model
I the
will
simply
u-w
(x)-M-1(_•)z
• (__•)
••
orifices
has
amuch
larger
effect
on
the
behavior
of
system.
(1)
assignreasonablevaluesto lG,L•, andg• and will concentratethe
effort
onhowtheorifice
dimensions
l andg aredetermined
bythe Here•w isthemean
water
flowvelocity
(averaged
across
thegap
sliding
process
andbywaterpressure
andflow. Thisis anotherheightg), M is theManning
roughness,
andetA/co
is thelocal
simplification
thatcanreasonably
bemadein a firsttreatment
of hydraulic
gradient
in theorificeasdiscussed
above.The total
flux of waterQw carriedby the linked cavity systemis obtained
the system.
There are three distinct physical processesthat, acting by multiplying(1) by the local gap width g(x), integratingover
together,determinethe hydraulicbehaviorof the linked cavity the breadthl of the orifice, and summingthe contributionsfrom
model: (1) for given roughnesscharacteristics
of the bed, given theNo independent
orifices(section4):
Pw andPi andgivenslidingvelocityv, thereis a particularorifice
5
geometry determinedby the cavitationprocessin basal sliding;
(2) for given orifice and cavity geometry and given hydraulic
[g(x)]
5d•
(2)
gradient,there will be a certainflow of water throughthe linked
cavity system;and (3) the water flow will result in generationof
heat by viscousdissipation,which will causeenlargementof the
The local rate of heat generationby viscousheatingis the
Qw22/3
M(•
orifices
bymelting
of theiceroof,resulting
in a modification
of localwaterflux •wg timesthepotential
gradient
pwgrctA/co
theresults
ofprocess
1 anda consequent
increase
in theflow (where
grisgravity
andPwisthedensity
ofwater).
Expressed
in
given by process2. Sincethe effectof heat dissipationis crucial
in the functioningof a tunnel systemand sincefor a given total
water flux down a given hydraulicgradientthe dissipationof heat
will be the samein a tunnelsystemand a linked cavity system,it
is essentialto takeit into accountin thelinkedcavity model.
Although there is a considerableparallelism between the
treatmentof the linked cavity model here and the treatmentof the
tunnel model by R6thlisberger [1972], the two models differ
termsof an equivalentvolumerate of meltingof ice by dividing
by piH, whereH is the latentheat of meltingand Pi is the ice
density,the local heatgenerationrate, per unit areaof the orifice
roof, is thus
(etA/co)
3/2 5/3
th- 22/3
DMg
(3)
substantially
because
thewaterflowandiceflowgeometries
are where
D = piH/pwg•
is a constant
withdimensions
of length(D =
verydifferent
andbecause
thetunnel
model
lacksa component
31km).
process
of type1 above.Thelinkedcavityconduits
havean
Therateof melting
caused
by theheatgeneration
in (3) is
essential
feature
of the"N channels"
introduced
byNye[1973]governed
byheattransfer
fromthewater
totheiceroofs
of the
andconsidered
byWeertman
[1972,
p.306],namely,
thattheyare orifices
andcavities.A fullydetailed
treatment
based
onthe
tiedtospecific
topographic
features
ofthebed.Theydiffer
inthis principles
ofheattransfer,
analogous
tothatcarried
outfortunnel
respectfundamentallyfrom tunnels("R channels"of Weertman systemsby Spring and Hutter [1981, 1982], is rather complicated
[1972]),
which,
inthemodel
concept
ofR•thlisberger
[1972],
are andwillherebeavoided
bythesimple
assumption
thattheheat
nottiedtoanytopographic
features
ofthebed.The"Nchannels"
generated
istransferred
locally
totheiceroof,aswasassumed
in
werespecifically
assumed
tofollow
bedrock
channels
incised
by theoriginal
treatments
of tunnel
systems
[R•thlisberger,
1972;
erosion
intothebed,whereas
thelinked
cavity
conduits
considNye,1976].Thisassumption
guarantees
thattheeffects
of heat
ered
here
aretiedtobedrock
roughness
features
ofdiverse
kinds. generation
onthelinked
cavity
model
willnotbeunderestimated,
becausea heattransferdistributedover the cavity roofs as well as
the orifices would reduce the melting in the orifices and thus
reduceits effect on the hydraulicbehaviorof the linked cavity
Given the geometryof the linkedcavity modelas specifiedin system. Under this assumptionand neglectingthe effect of the
section4, the flow of water throughthe systemcan be calculated pressuredependenceof the melting point [R6thlisberger,1972,
5. WATER FLOW AND VISCOUSHEATING
asfollows.
Calletw
theoverall
longitudinal
hydraulic
gradient
p.179],
which
averages
tozero
if etw= et,equation
(3)gives
the
(slopeof thehydraulic
gradeline)alongthelengthof theglacier. meltback
rateof theorificeroof.
In the evaluationhere, etwis taken to be equal to et, the surface
slopeof theglacier,but thisis not a necessary
assumption.If the
6. BASALCAVITATION
linked cavity systemhas an averagetortuosityco, the average
hydraulicgradientalongthe waterflow pathsis thenct/co.The
The sizesand shapesof basalcavitiesand separation
gap
model conditionthat the orifices control or throttle the flow is orificesof the linked cavity model (Figure 2) are determinedby
9088
KAMB: GLACIERSURGEMECHANISM
separationcharacteristicsand need to be distinguishedin the
linked cavity model. The first (Figure4a) is a downglacier-facing
stepin the bedrocksurface.The separationgap associated
with it
will be called a stepcavity or steporifice. Natural examplesare
stepcavity
shownby Hallet and Anderson[1980, Figure3] and by Karnband
La Chapelle[ 1964, Figure7]. The secondtypeof bedrockfeature
is a wave, idealizedas a sinusoid,with crestrunningtransverseto
ice flow. The separationgap that formson the lee slopeof sucha
wave form (Figure 4b) will be called a wave cavity or wave
orifice. A natural exampleis given by Karnb and La Chapelle
[1964, Figure2]. The main distinctionsbetweenstepcavitiesand
wave cavities(or the corresponding
orifices) are the following:
(1) in a stepcavitythe ice separates
from the bed at a sharpbreak
in slopeof the bedrocksurface,whereasa wave cavity has no
suchpredetermined
point of ice separation,and (2) a wave cavity
is inhibited from extending into the stoss slope downstream,
whereasno suchinhibitionis presentedto a stepcavity. Because
of the stressconcentrationat a sharp slope break, a step cavity
will remainopenat lower basalwater pressureand lower sliding
velocity than a wave cavity. Actual wavelike topographic
featuresof glacierbeds(rochesmoutonn•es,more or less) often
have a facetedlee slopethatis a hybridof the characteristics
of a
step and the roundedlee slope of a wave (Figure 4c); it will
generatea leesidecavity that is rather like a stepcavity at small
I
amountsof separationand becomeslike a wave cavity at large
Fig. 4. Idealizedgeometry(schematic)of ice-bedseparation
gapsformed amountsof separation. Natural examplesare shown by Carol
by cavitation
in basalsliding,seenin verticalcrosssection
parallelto the [1947], Vivian and Bocquet [1973, Figures 2-4], and Vivian
/•////////////,;,//,;),•
'•.,z•..-.-,-- -..x'.L'.'.z";
..
'."'-7-•'"-
.' .
slidingdirection
(arrows).Ice is stippled,
bedrock
cross-hatched;
[1980,Figure4]. Onecanalsopicturea rounded
step(Figure4d)
separation
gaps
areblank.
The
dashed
lines
show
theconfiguration
ofthe thatisadifferent
type
ofhybrid;
thecavity
thatforms
behind
such
cavity roofs at increasedsliding velocity and/or basal water pressure.
(a)A step
cavity
(orstep
orifice)
formed
intheleeofadownglacierarounded
step
willhave
thecharacter
ofawave
cavity
atsmall
facing
bedrock
step.(b)A wavecavity
ontheleeslope
of a bedrock
amounts
of separation
but will assume
moreandmorethe
wave. (c) Cavitation
develops
in theleeof a faceted
wavesuchasisoften characterof a step cavity as the amountof separationbecomes
seen
onglaciated
bedrock
surfaces;
it hascharacteristics
thatarea hybrid large. For maximumclarityanddistinctiveness,
in the linked
ofFigures
4aand
4b.(d)Adifferent
type
ofhybrid
cavity,
developed
in cavity
model
I willfocus
onthebehavior
ofthepureorextreme
the lee of a roundedbedrock step. (e) Cavitationin a transverse"N
channel"
ofthekind
visualized
byWeertman
[1972].
Thescale
ofthesetypes,
thestep
cavity
and
thewave
cavity.
Thetransverse
type
of
features
isarbitrary,
butfortheseparation
gaporifices
inthelinked
cavity"N channel"considered
qualitatively
by Weertman[1972,
systemassociated
with surgingas considered
herethe stepor wave Figure 10] and shownin Figure 4e is in fact very similar to a
heights
are~0.1m.
wavecavity.
The idealizedstep of the model (Figure 5) is a rectangular
the mechanics
of ice-bedrock
separation.Beforethe effectsof stepof heighth, runningtransverse
to theslidingdirection.The
heatdissipation
by waterflow throughthe linkedcavitysystem resultingice flow andseparation
problemis two-dimensional.
A
are brought into consideration,separationcan be treated as a
purelymechanicalconsequence
of the slidingof ice overbedrock
topographyunder the actionof basal water pressure[Lliboutry,
1968; Kamb, 1970, section19; Fowler, 1986]. If at any point on
the ice-bedrockinterfacethe normal pressureacrossthe interface
dropsbelow the basalwater pressure,then an ice-bedseparation
cavity will start to open at this point, providedthat water under
the given pressurecan actuallygain accessto the cavity. The
extent and shape of the cavity that develops is in principle
determinedby the solutionof a mixed-boundary-value
problemin
ice flow mechanics,the boundaryconditionbeing a constrainton
coordinatesystemx,z is chosenwith origin directlybelow the lip
of the step,with x runningdownstreamparallelto the bed and z
perpendicular
to the bed andupward(Figure5a). The heightof
the separationgap is g(x). The sliding-with-separation
problem
can be solvedin closedform under the assumptions
that the ice
rheologyis linearandthat the slopeof the sole,g'(x), is smallso
that the sliding ice constitutesa half-space, to an adequate
approximation. (Theseare the same assumptions
for which an
exact solutionof the basalslidingproblemwithout separationis
obtainable[Nye, 1969;Kamb, 1970, section6].) Derivationof the
solutionis too lengthy to include here and will be given in a
the stressover the separatedparts of the ice sole and a constraint separate
paper. The resultsneededfor thelinkedcavitymodelare
on the ice flow velocity over the nonseparated
parts. The problem as follows. The gap heightis
is tricky becausethe areasof basalseparation,
whichare the areas
where the stressboundary condition is to be applied, are not
prescribedin advancebut must be determinedas part of the
solution.
From the solution one obtains the breadth dimension
l
g(x)=h •1- •1sinl-I
12x
•i2 -x))(4)
2(2x
-l)•fx(l
for the orifices of the model, and the distributionof gap height
g(x) over the breadthof the orifices. The dimensionslc andgcfor where 0 < x < l. The gap shapegiven by (4) is shownby the
the cavities are determinedphysically in the same way, but the curvefor _• = 0 in Figure8. The gaplength(orificebreadth)is
model is not concernedin detail with them, as explained in
section 4.
Two types of bedrock topographicfeatures, illustrated in
idealized form in Figures 4a and 4b, have very different basal
l=4•rlv
(h
+m)
(5)
KAMB: GLACIERSUROEMECHANISM
(O) WITHOUT
ROOFMELTING
i'PZ" .....
'.sliding
velocity
•' •. ß....
//////////////-//,,Y///•,x
,i-•.'
/ •!!;'•'-.
t ,' •-•<-*•----'v..-.':::....
, •',"k
•'3'
i
•
(•) w•
• ••'.'.'"'
cavity
length
Z
.'.'.'.'.'.'.'
.....
"-'
-'
•
re-contactpoint
•oo• ••
'"'•"::;'"'•"
"[•v••o•'•
ß r
9089
oR. These two quantitiesare in principle related by the ice flow
law and the geometry of ice deformation in the sliding-withcavitationproblem;n = 3 will be assumed.The reasonableness
of
assumption(7) is shown by the fact that the formula giving the
closurerate of a circulartunnelfor linear rheologyis convertedby
(7) to the exact result for nonlinearrheology [Nye, 1953] except
for a numericalfactor, which can be absorbedinto TIRor oR.
The shapeof wave orifices will be consideredin terms of
separationin sliding over a "quasisinusoidal"bed with wave
crestsperpendicularto the sliding direction. It is basedon a sine
wave of wavelength)• and half amplitudea. The separation
geometryis shownin Figure6a. The lengthof the separationgap
h '•
'- •
".-."''.-..............is l, andthe gap heightis g(x), the origin for x beingat the headof
the gap (point of ice-bedrockseparation),the positionof which is
virtual
particle
trajectory•otal meltback
m
not prescribedin advance. The two-dimensionalsliding-with"virtuol
bed"
•'' • -• • -- separationproblem for linear rheology can be solved in closed
form if a <<•, asis appropriate
for waveorificeswhicharenarrow
separationgaps. The normal stressacrossthe ice-bedrockcontact
Fig. 5. Detailedgeometryof ideal stepcavityor steporifice (a) without before onset of separationis assumedto vary as (x - Xo)2, the
and (b) with ice roof melting by heating of basal water throughviscous amplitudeof the variationbeingmatchedto that for the sinusoidal
dissipation.The cavity is shownin verticalcrosssectionparallelto the
bedrockwave form in the vicinity of its inflection point, located
basal sliding direction. Ice is stippled,bedrock cross-hatched,
cavity
blank. The stepis of heighth. An x,z coordinatesystemis takenwith at Xo. The actual bedrock wave form that gives this parabolic
origin at the baseof the stepas shown. The gap heightg(x) is a function normal stressvariation in the x interval over which separation
of longitudinalcoordinatex. The cavity is filled with water at pressure later occursis here called a "quasisinusoidal"
waveform. For a
•
• •
' .ß. .'... .
Pw,andtheiceoverburden
pressure
isPI. Insliding
inthex direction
at strictly
sinusoidal
waveform
a restriction
l <<X onthevalidity
of
velocity
v,theiceseparates
from
thebedrock
atthelipofthestep,
x - 0, the sliding-with-cavitation
solution
wouldapply,but for a
and
recontacts
thebed
atx = l. Here
w(x)
isthezcomponent
ofice quasisinusoidal
waveform
asjustdefined,
it does
not. The
motion at the cavity roof. In Figure 5b the roof is meltedback a total z
distance
mastheicetraverses
thelength
ofthecavity.
Thepath
thatan method
of solution
isfundamentally
thesameasforstepcavities
iceparticle
leavingthesteplip wouldfollow,movingwiththeicemass,if
but is more involvedbecauseof the complicationcausedby the
it werenotremoved
bymelting
is shown
in Figure
5basa dashed
line initiallyunknownlocationof the separation
point. The lengthy
labeled
"virtual
particle
trajectory";
it hitsthe"virtual
bed,"
lowered
by derivation
willbegiven
in a separate
paper,
where
itsrelation
to
the distance m, at x = l.
whererl is the ice viscosity,v is the ice slidingvelocity,o = Pi- (0) WITHOUTROOFMELTING
Pw is the excessof ice overburdenpressurePi over basal water
pressure
Pw (i.e., the effectiveconfiningpressure),
andm = 0 if
there is no melting of the gap roof. The rate at which ice How
tendsto closethegapis givenin termsof the z component
of flow
velocityof theice of thegaproof at eachpointandis
w(x)
=vg'(x)
=-:r••lx(•
--x)
separation
sliding
velocity
point
'"":':'"""'":":'::"!'•(:':i
:.'::•ii:•:!3'?.!:5'...':.::':....i,.:
:..
......
2a
"'"'""-':'.:!"[":'!:
?.:.::
C.:.:..'.:.
....
........... '.':
i('..
(6)
II
•
X/2
-q
po,nt
This closurerate is balancedby the effect of slidingin tendingto
openthegap, sothatthegap is in a steadystate.
Accordingto (5), the gap lengthl tendsto 0 as o --> o%but l
remains nonzero for any finite confining pressureo, however
(b) WITH ROOFMELTING
large. Thus in the model, step orifices remain open to some
extentevenat very largePi or very low Pw. Under theseextreme ::-'.:
::::i':.:5:..::.:•
v.•.:...
t
conditions,when l s h, the approximationg'(x)<<1breaksdown,
so that the relationshipsin (1)-(6) are not strictly valid, but it is
assumedthat they describethe model system to an adequate
approximation
up to the largestvaluesof o thatoccurin practice.
Becauseactualice rheologyis nonlinear,(4)-(6) are only an
approximation
evenwheng'(x) is small. To dealwith this,we use
the simplestpossibleapproach,whichis to_makeTI shear-stressdependentin (5) and (6). Sinceall stresses
would be hydrostatic
if o = 0, it is logical to take o as the measureof shearstresslevel
in the sliding-with-separation
problem and hence in conformity Fig. 6. Detailed geometryof ideal wave cavity or wave orifice (a)
:.:.
.:"":'?:.::W
...!"?.
....
..:..':.-..'..:...-,
.:.••(
':'?:•":
......i(•.•.•::•:•:•.•.•
:..........
•
.zx• virtuo•
'""'2'••--_
g(x)
melt
b•ck
'•/,'•
porticle
trojectory----.-",*--•"*•
withoutand (b) with ice roof melting. Conventionsandsymbolsare as in
with the standard nonlinear flow law to take
n =q•
(7)
Figure5. Thebedrockwavehaswavelength
), andhalf amplitude
a. The
originof thex,z coordinatesystemis takendirectlybelowthe ice-bedrock
separationpoint;the x axis parallelsthe meanbed slopeof the wave.
Positions
of the separation
pointandrecantactpointaredeterminedin the
sliding-with-separation
solution.Bothpointsmoveto differentpositions
whereTIRis the effectiveviscosityat a referenceshearstresslevel whenthereisroofmelting(Figure6b).
9090
KAMB: GLACIER SURGEMECHANISM
7. EFFECT OF HEAT DISSIPATION
the different solutionsby Lliboutry [1968] andFowler [1986] for
the sameproblemwill be pointedout. The resultsneededhereare
as follows. The gap lengthis
AND WATER
ON CAVITATION
FLUX
When there is meltback of the cavity roof, amountingto a
total meltback
distance
m as the ice of the roof
traverses
the
(8) cavity
frompointof separation
topointof recontact,
it canbe,
taken into accountin the solutionof the sliding-with-separation
problemdiscussedin section6 by the simpleartifice of requiring
the ice to recontactthe bed at a point lower by a distancern than if
therewere no meltback. This is indicatedin Figures5b and 6b,
where, as before, o = Pi- Pw and where
12
= 8re
2•lav/)•
2
(9) where
thecavity
roofwithmeltback
isdrawn
witha solid
line,
while the dashedline showsthe imaginary ("virtual") trajectory
12 is called the wave cavitation parameterß
It is thelimitingthataniceparticle
leaving
theseparation
pointwouldfollow,
effective
confining
pressure
forcavitation:
foro > 12there
isno moving
withtheicemass,
if thesolewerenotmelted
backandif
separation.Thus,in contradistinction
to stepcavities,wave thebeddownstream
fromthe separation
pointwerelowereda
cavitiesform only for a sufficientlyhigh water pressurePw, such moving with the ice mass,if the solewere not melted back and if
the bed downstreamfrom the separationpoint were lowered a
that.Pw> Pi- 12.The gapheightis givenby
distancem. For the stepcavity, the heightof the "virtual" stepin
this caseis h + m, which explainsthe appearance
of this quantity
in
(5),
giving
the
gap
length
l
for
meltback
m.
3 7r
To obtain the steadystategap profile g(x) for a stepcavity
for 0 < x < l. The separationpoint lies a distance(3/8)/upstream with meltback,the meltbackrate (3) is put in oppositionto the
wideningor narrowing
from the inflectionpoint on the lee slopeof the sine wave. The gap closurerate (6) to give the progressive
of
the
gap
as
seen
moving
with
the
ice:
shapeof the separationgap is shownby the curve for E' = 0 in
g(x)
- 2rc-•4
a x5/2
(l- x)3/2
(10)
Figure 10.
Nonlinearrheologycan be introducedin the way done by
Kamb [1970, p. 693]. The strainrate dependence
of •1 is written
•1=N•
ß-1+ 1
'•
(11)
vg'(x) = rh(x)+ w(x)
(14)
When (3) and(6) areintroducedinto (14), afterfirst eliminatingo
from (6) with the use of (5), and when the variablesare nondimensionalizedby defining
y=g/h
where N and n are constantsand where • is the secondstrainrate
• = x/l
g = m/h
invariantßWe will take n = 3 andN = 0.94 bar yr1/3.For sliding
oneobtainsthe dimensionless
differentialequation
over a single sine wave of wavelengthX, the value of • at a
height Z/2rc above the sliding interface,which is assumedto
dy
T5/38
govern the effective viscosity as discussedby Kamb [1970,
p. 694], is
(15)
ß
•-•=2ax/(1
+g) - •(1+g)•/•(1
- •)
(16)
where
• = 4re2e-1 av )•-2
21/3
(ctA/cø)3/2
•' hg
*
E=tc•-V2DM
as can be obtainedfrom equations(47) and (49) of Kamb [1970]
with the assumption
)• <<)•o (the transitionwavelength)so that
(17)
regelationis negligible. Hencefrom (11), with n = 3,
e),2
) -32
•1=N 4•2av
,
The dimensionlessquantityE will be called the orifice meltingstability parameter,for reasonsthat appearlater (section8). It
providesa measureof the importanceof roof melting by viscous
dissipationin the linked cavity system.
(12)
Thegapprofiley(•) is obtained
by integrating
(16), startingat
Introducingthisinto (9), we obtain
I;-2N
(4rc2
e2
av.)
3I-•,2
•0) = +1. For a given valueof E, therewill be a valueof g such
thatwhen(16) is integrated
from• = 0 to 1, oneobtainsT(1)= 0.
This is the conditionfor theice to recontactthe bedat • = 1, as
(13)
showing that the wave cavitation parameterincreaseswith the
slidingvelocityas vm. The fact that the quantityo doesnot enter
into what determines•1 in (12) indicatesthat the treatmentis an
approximationin which the effect of the additional stresses
associatedwith the cavitationis neglectedßThis can be done as a
reasonableapproximationhere becauseo < 12,whereas for the
stepcavity, in (7), •1 mustdependon o, as in tunnelclosure,there
beingno ambientstresslevel analogousto 12.
required;henceit gives the nondimensionalized
meltbackg that
corresponds
to the given value of E. The steadystategap profile
•(•) corresponding
to E is givenby the integralof (16) for the
value of g so determined. The resultsof numericalintegrationof
(16) for a successionof values of E from 0 to 1.1 are shown in
Figures 7 and 8. Figure 7 gives the meltback quantity g as a
function of -Z, and Figure 8 gives the nondimensionalized
gap
profiles•(•) for severalvaluesof E. The amountof meltbackm
is directly seenfrom the curveof g = m/h in Figure7. For E ~ 1,
m becomesratherlarge, considerablylarger than the stepheighth.
KAMB: GLACIER SURGE MECHANISM
9091
is obtained from (13) in the same basic way, as suggestedby
Figure6b, with the appropriatew(x), relatedto (10). The form of
w(x) is complicatedby the fact that the positionsof both the
separationpoint and the recontact point are affected by the
meltback. I will pass over the somewhatinvolved details and
simplygive the results. If the nondimensionalization
is done by
t/I o
• = x/l andT = g/go,where
61
16
;5_
andif we define a "meltbackparameter"
8
andthe dimensionless
quantity
_
B •
(22)
(5 - 2v + v2)
1
thenthe differentialequationfor the gap profile can be written
I
dT=
I
O0 0.2 0.4 0.6 0.8 1.0 1.2
0
q/3_B2•3/2 -
+ - 5)
(23)
The melting-stabilityparameterE' that arisesin (23) is
•,,_
Fig.7. Steporificemodel:dependence
of dimensionless
parameters
on
the melting-stability
parameterE, whosedefinitionis givenin (!7). Here
• is the meltbackparameterm/h, determinedfrom (16) as describedin the
text,l / lo is the orificegapelongation
relativeto thegaplengthlofor no
roof melting (E = 0), and • is the flux factor definedin (19) and
calculatedfrom the resultsof integrating(16), given in Figure 8. Note
thatthereis a scalechangefor the I.t(E)curveat • -- 4 andfor the •(E)
27
,., - 32/357/3
(ctAto))3/2«a•- 1-
(24)
DM
curve at (I) -- 2.5.
9.31
Figure
8shows
how
the
shape
ofthe
gap
profile
responds
tothe
meltback. The roof is raised,and the peak of the roof is shifted
downstreamas meltingincreases.The biggesteffect, however,is
the lengtheningof the gap, which is not shown in Figure 8
becauseof the scalingof •. It canbe seenin termsof the ratio
l/lo, given in Figure 7; lo is the gap length for no meltback
\
\
• t0
(E = 0, Ix= 0), givenby(5)form = 0; thusl / lo= •/1+ Ix. The
Once
the
gap
profile
is
deterned,
the
water
flux
ca•ied
by
lengtheni.ng
of thegapwhenE -- 1 is by a factorof about3.
•e systemcanbe foundfrom (3). •e resultcanbe expressed
as
05
follows:
24/3
Nø
•
• h6.
(18)
where
0
0=•/1
+g;T5/3d•
o
(19)
02
0.4
0.•
0.8
Fig. 8. Steadystateconfiguration
of thesteporificerooffor valuesof the
melting-stability
parameter
E from0 to 1.0. Foreachvalueof E, thegap
height g(x) is shown in terms of g/h as a functionof dimensionless
longitudinal
coordinate
x/l, wherel is thegaplength. (Thel varieswith E
as indicatedby the l/lo curvein Figure7.) The curvesareobtainedby
The "flux factor" •, obtainedby numericalintegrationof the integration
of (16) with the appropriate
• valuesasexplainedin thetext.
resultsin Figure8, is givenasa functionof E in Figure7.
At E -- 1, integration
of (16) for thetwo indicated
valuesof • givestwo
The steadystategap profile for a wave orifice with meltback widelydivergingcurvesg(x) asshownby thedashedcurveswith arrows.
9092
KAMB: GLACIER SURGEMECHANISM
of which(8) is thespecialcasefor v = 0 (no meltback).Thus
2
m=(5- 2v+v2)2 17
(27)
The meltbackratio m/a dependson E' via v and in additionon
o/•; via the last factor in (27). Figure 9 gives a curve of the
quantity
(m/a)(1-(•/E) -22-2, whichdepends
ono onlyviathe
dependence
of E' on o and is equalto m/a for o/•; = 1/2. The
large indicatedamountof meltbackfor E'~ 1 is striking,as it is
alsofor steporifices.
Figure 10 showswave-orificegap profiles •)
for a
succession
of E' values. The gap height,plottedas the dimensionlessratio g/go in Figure 10, is scaledby the dimensiongo
givenin (20), whichis themidpoint
heightthatthegapwould
0.8
have if therewere no melting(value of g in (10) at x = l/2). The
0.8
gap heightin Figure 10 is of coursegreatlyexaggerated
in
relationto the gap lengthbecauseof the way the plot is made.
Thegaplengthis obtained
froml/lo = •/•, based
on(26). As
0.6
the curveof l/lo versusE' in Figure9 indicates,the gap length
changeslittle with increasingmeltback, contrary to the large
effectfor steporificesshownby thel / locurvein Figure7.
A surprisingfeature of the gap profiles is the effect of
meltbackin causinga saggingof theroof in the upstreampart of
0.6
-
0.4
1/
- 0.4
the slopecavity, seenby comparingthe E'= 0 and E'= 1 curves
in Figure10, for x/l < 0.5. The sagging
appears
to be linkedto
0.2
the fact thatthe pointof separation
shiftsdownstream
asmeltback
increases.The shift,whichis not depictedin Figure10 because
the originis at the separation
point,is indicatedby the following
relationbetweenmeltbackparameterv and the distanceXofrom
the separation
pointto theinflectionpoint:
- 0.2
0•/ • • • • • • -0
0
0.2
0.4
0.6
•,•/
0.8
1.0
1.2
1.4
3-V
Xo-
Fig. 9. Wave orificemodel: dependence
of dimensionless
parameters
on
themelting-stability
parameter
E', defined
in (24)or(25). Herev isthe
(28)
8
2w carried
c• L½ by the linkedcavitymodel
The waterdischarge
Qw
meltbackparameterdefinedin terms of m/a in (21) and calculatedfrom
(23) as explainedin the text, and l/lo is the relative orifice gap
elongation,
from l/lo -- B1/2,whereB is definedin (22). The quantity
] I ' I ' '1 ' I '
(m/a)/4(1 - (5/E)2givesa measureof the roof meltbackfrom (27). •F is the
flux factor definedin (30) and calculatedfrom the resultsof integrating
(23), givenin Figure 10.
.1.4--X•4
ß ,
As given in (24), it has a form similar to E in (17), differing
mainly in the way (5 and Z enter. However,Z/rl in (24) can be
eliminatedby substitutionfrom (9), and the parameterE' then
1.o
.2
,•
_
assumes a less similar form:
w'=
-
1T
211/2
(otA/fo)3/2•.a2/3(
•_)11
DM v
32/3
57/3
•
05
in which,notably,theeffectiveviscosityrl doesnot appear.
The integrationof (23) is donein the samebasicway as (16):
for a given_,
w' a valueof theparameterv is soughtsuchthatwhen
(23)isintegrated
from
• = 0 to1,starting
at•0) = 0,onearrives
at •1) = 0. Results
of thisprocedure
aregivenin Figures
9
775
/
02
0.4
0.s
0.8
t0
and 10.
Figure9 shows
[hemeltback
parameter
v asa function
of E'. Fig.10. Steady
state
configuration
ofthewave
orifice
gapforvalues
of
parameterE' from 0 to 1.4. For eachvalueof E', the
The actualmeltbackamountm is obtainedby solving(21) for m the melting-stability
and
introducing
vfrom
Figure
9and
thecavity
length
l from
l_4.(•ff_)«
(1-•)«
J[- •
gap
height
g(x)
isshown
interms
ofthe
ratio
g/go,
where
go
isthe
height
midwayalongthelengthof thegap(x -- l/2) in theabsence
of roofmelting
(26)
lengthl,
(E'
=the
0),
given
given
by
(20).
(26).
The
The
longitudinal
are
coordinate
obtained
by
xis
integration
scaled
by
the
of
(23)
gap
with
appropriate
values
of
vcurves
asexplained
inthe
text.
KAMB: GLACIER SUROE MECHANISM
9093
with wave cavity type orifices, from (2), (20), and (26), can be
instantof the pressureperturbation,the orifice gap immediately
written
beginsto lengthen
at a rate [(0)=v(1-O/Oo),and(2) the
5
maximum
rateof gaplengthening
at anytimeis v, thatis, • < v.
23
Becausewe are concernedwith the possibility of an unstable
orifice enlargementconsequentupon increasein water pressure
and because maximum stabilization of the orifice against
enlargementis provided by maximum closure rate and therefore
maximum length l, as indicatedby (6), conclusion2 makes the
following simpleprocedurevalid as a test for the existenceof an
Qw
=755/3
rcM
where
Wemaketheapproximation
that• isconstant,
• = fv,
(30) instability.
so that
is a "flux factor"analogousto (I) in (19). A curve of •P versusE',
obtainedfromtheresultsin Figure10, is givenin Figure9.
8. ORIFICE
STABILITY
In the integrationsleading to the curves in Figure 8, it is
found that as E increasesto near 1, the curves become very
sensitiveto the value of g. This is illustratedby the two dashed
curves(with arrows)for E = 1 in Figure8, obtainedby integrating
with g = 9.31 and g = 9.32, respectively. Under these circumstancesit becomesdifficult to find the value g that permits the
conditionT(1)= 0 to be satisfied,and for this reasonthe E = 1
curve in Figure 8 is showndashedbeyond where the calculated
curves diverge. A similar thing happensin the integrations
leadingto the curvesin Figure 10 when E' increasesto about 1.5.
¾(•) = 1 +f•
(32)
For anyparticularvalueof Eo, we try to choosethe proportionality constantf such that when (31) is integratedforward from
• = 0, T(0)= 1, thereference
pointhitsthebed(thatis,T(•) goes
to zero)at a "touchdown"
point•, compatible
withthei value
chosen,
whichrequires
T(•T) = •T, or, from(32),
f= 1- •T-1
(33)
If a valuef < 1 can be found suchthat theseconditionsare met,
then a stabletransientresponseof the orifice is possible. If, on
the otherhand,with the choicef= 1 in (32) the integrationof (31)
leadsto T(•) increasing
withoutbound,thenit is impossible
for
f = 1 gives
Thus,for E'= 1.5 the curveT(•) calculated
by integrating
with the orifice to havea stabletransientresponsebecause
the
maximum
possible
closure
rate
for
the
orifice
at
all stages
v = 0.679 divergesto T> 5.4 for • > 0.8, while the curvewith
(conclusion2 above).
v = 0.680dropsto T = -0.25 at • = 1.0. Thisbehaviorappears
to
be the manifestationof an orifice instability that commencesat
criticalparametervaluesE ~ 1 or E' ~ 1.5.
To shed light on this instability,it is helpful to analyze the
transientbehaviorof a steporifice subjectedto a perturbationin
water pressure.We startwith a steporifice in steadystate,under
effectiveconfiningpressureOo,with meltbackparametersEo and
go and initial gap length lo. The effective confiningpressureis
then abruptlydecreasedto o < Oo, so that the orifice begins to
Resultsof the aboveprocedureare shownin Figure 11 in
termsof trajectories
T(•) of reference
pointsfor an assortment
of
perturbationsat several values of Eo. The magnitudeof the
perturbationfor eachcurve is measuredby the value of 1- O/Oo
andis indicatedby the O/Oovaluesin Figure 11; thef valueused
in calculatingeachcurveis alsogiven. Figure 11 showsthat for
perturbations
as large as 1- O/Oo= 0.2, the transientresponseof
the orificecanbe stablefor Eo aslargeas0.7, but for Eo = 0.8, the
enlarge.We followthetransient
response
in termsof what perturbation
1- O/Oo
= 0.2 leadsto an unstable
response,
the
happens
toa reference
pointthatstarts
atthelipof thestepatthe orificeenlarging
without
bound.ForEo= 0.9,a perturbation
of
momenttheperturbation
is applied.The gaplengthl will now be
a functionof time or, equivalently,a functionof the position
• = x / loof thereference
pointasit movesforwardwith the ice.
Fromthe way equation(16) wasconstructed,
one can seethat
only 10% (O/Oo= 0.9) is sufficientto causean unstableresponse.
Theseresultsindicatethatsteporificeshavea runawayinstability
for Eo -> 1. This stabilitylimit is probablysomewhatoverestimated
because
nearthelimit, asf--> 1, (32) overestimates
¾,
withtwomodifications
it canbeused
todetermine
thegapheight atleastnear• = 0, where
theassumed
motion
• -->islargerthan
T(•)= g(•)/hthatdevelops
attheposition
• of thereference
point allowed
byconclusion
1 stated
intheprevious
paragraph.
asit movesalong:
It seemsvery likely thatthe sametypeof instabilityarisesfor
wave orificesfor E' >•1.5, as suggested
by the parallelismin the
behavior
of
the
integration
for
determining
the steady state
(31)
profilesof wave orificesandsteporifices,discussed
earlier.
The causeof the instabilitycan be linked to the increasein
The ratio O/Ooappearsin the right-handterm of (31) becausethe
melting rate that occurswhen the gap height increases. The
gap closure rates are proportionalto o, as (6) indicates. The instabilityis demonstrated
in Figure 11 underperturbations
of o,
d•- 2Eo
q1+•o -•
•oo
appearance
of¾(•)=l/lointhesame
term
derives
from
thel in butthere
is littledoubt
thatit would
appear
equally
under
(6),which
nowvaries
with• asjustexplained.
Tofindouthow
? perturbations
inT(•)itself.A closely
related
type
ofinstability
is
atthereference
pointvaries
withposition
• asit moves
along,
we afeature
oficetunnels,
asdiscussed
insection
11.
want to integrate(31), startingat T(0) = 1, ¾(0)= 1. To do this,
we needto knowthefunctionT(•), describing
themotionof the
recontact
pointwithtimet (or equivalent
• -- vt/lo).
The rigorousdetermination
of T(•) is a complicated
problem,
beyondthe scopeof thispaper. A detailedtreatment,to be given
elsewhere,yields two conclusions
of importancehere: (1) at the
The foregoingcalculation,in combinationwith conclusion1
statedabove,indicatesthattheresponse
of the steporificesystem
to pressureperturbations
is definitelystableif E and the pressure
perturbation
aresmallenough.
If thegaplengthening
ratei =fv
used in the calculationof a given transientresponsecurve in
Figure11 (as formulatedin (32)) is lessthanthe initial lengthen-
9094
KAMB' GLACIER SURGE MECHANISM
'
I•:
_
Figure 12 in terms of water level depth in an ice mass400 m
thick. Separatecurvesare given for a linked cavity systemwith
steporificesandfor onewith wave orifices.
The Qw versus o curve for the step orifice linked cavity
model in Figure 12 is calculatedfrom (18) with the following
parameter values and with nonlinear ice rheology taken into
accountby (7):
I
TRANSIENT
OF CAVITY
0.9
TRAJECTORIES
ROOF POINTS
%'0:0.9
/:1
I•= 0.8
_
%o:0.8 '•----uns/ab/e
/ra/gc/or/bs
f--1
_
_
s/able/ra/bc/or/bs
•%o:o••
•
D = 31 km
ct = 0.1
A = 10
(o = 4
piH/Pwgr
longitudinalhydraulicgradient
headgradientconcentration
factor(LG/ Lo)
conduitsystemtortuosity
No = 50
orifice number
Manning roughness
referenceice viscosity
referenceeffectivepressure
stepheight
M = 0.1 m-ms
fir- 0.1 bar yr
oR = 6 bars
h = 5 cm
Fig. 11. Transienttrajectoriesof a "referencepoint"thatfollowsthe step
The systemparametersA, (o,No, and h are chosenarbitrarily
orifice roof upon perturbationof the systemfrom the steadystateby a
of what one might
suddenincreasein basalwater pressurefor variousinitial and perturbed but with the intent that they be representative
conditions.
Thecurves
arecalculated
from(31)-(33)
asexplained
inthe expect
in a linkedcavitysystem
atthebedof Variegated
Glacier
text;the parameters
usedin calculating
eachcurveare givenalongside.in surge,with surfaceslopect--0.1. No = 50 meansone orifice
Here(50istheinitialeffective
confining
pressure
PI - Pwpriortothe every20m, on average,
across
the 1-kmwidthof theglacier.
perturbation,
(5theperturbed
value
((5<(50);
Eoistheinitial
value,
prior
to A = 10means
thatmost
ofthelength
ofthewater
flowpaths
isin
perturbation,of the melting-stabilityparameterE in (17); andf is the
constantin (32) and (33), which is requiredto be < 1 (see text). The
coordinatex is the longitudinalpositionof the referencepoint, which
starts
atthestep
lip(x--0,where
g--h)atthemoment
ofperturbation
and 50
L INKED - CAVITY SYSTEM
increaseswith time at the sliding speedv; it is scaledby lo, the initial E
steadystatecavity length. For stabletrajectories,in which the reference o
/WAVE-ORIFICEMODEL
point
ultimately
makes
contact
withthebed(g = 0),thedimensionless
•r
coordinatex/lo of the recontactpoint is •T in (33). Trajectories
that •
divergeupwardwith f = 1 representunstableresponses
to the imposed •
pressure
perturbation.
-'=
'-
::EP-ORIFIœE
•
MODEL
}/
lOO
/
/
o
ingratei(0) = v(1- o/(50)
required
byconclusion
1, thenit is •
quitelikelythatin theresponse
of theactualphysicalsystem
T(•) ,,,
will alwaysbe largerthantheT(•) usedin thecalculation,
which •
is based
ontheassumption
of constant
•. Thelarger¾ correspondsto a faster roof collapserate (from (6)) and thereforea
stronger assurancethat the gap does not go into runaway
enlargement. Thus a stable responseis definitely indicatedif
f < 1 - c•/Oo.This conditionis satisfiedby the curvesfor Eo = 0.1
and 0.3 in Figure 11. Clearly a stable responsewill also be
obtainedfor any perturbationsmallerthanthe one (1 - o/C•o= 0.5)
usedin calculatingthesecurves. The curvefor Eo = 0.7 appears
alsoto indicatea stableresponse,
but thisis not firmly established
because the above condition
is not satisfied
in this case.
The
150-
TUNNEL
_
_
/
_
/
200
/
_
/
/,
]
1
behavior of the steady-state-profileintegrations gives the
0.01
0.1
1
10
100
impressionthat both the step-orifice and wave-orifice systems
DISCHARGE
Ow
(m3/s)
havean appreciabledegreeof stabilityfor E or E' up to about0.8.
Stability will be increasedin an actuallinked cavity systemby Fig. 12. Resultsof the linked cavity model and comparisonwith the
transfer
of someof theheatof viscous
dissipation
to theroofsof tunnel
modelfor parameter
values
givenin section
9. Theeffective
thelargecavities,
withrelease
of theassumption
of localheat confining
pressure
(5= PI - Pw(iceoverburden
pressure
minus
basal
transfer
(section
5).
water
pressure)
isplotted
against
discharge
Qwcarried
bythebasal
water
9. QUANTITATIVE EVALUATION OF MODEL
conduitsystemfor two typesof linked cavity model(with steporifices
and wave orifices) and for the tunnel model. On the left, Pw is
represented
in termsof boreholewater level in ice 400 m thick, for ready
The hydraulic
performance
of the linkedcavitymodelis comparison
withdatafromVariegated
Glacier
inandoutofsurge.
The
shown
in Figure
12in terms
of curves
of water
discharge
Qw the
step
orifice
curve
isfrom
(38b),
thewave
orifice
curve
from
(42b),
and
tunnelcurvefrom (43). The melting-stability
parameterfor the step
versus
basalwaterpressure
Pwor effective
confining
pressure
orifice
curve
isE = 0.18,from(37),andforthewave
orifice
curve,
C•= PI - Pw. Forreadycomparison
withtheborehole
waterlevel E' < 0.20,from(4lb). Dimensional
features
associated
withthese
curves
observationsfrom Variegated Glacier, Pw is representedin arediscussed
in section
9.
KAMB: GLACIER SUROœMECHANISM
9095
cavities,only a tenthof the lengthbeingthroughorifices. The with VR= 1 rn/dandoR= 6 bars(corresponding
to a waterlevel of
ratio 10 is thoughtreasonable,
perhapsminimal,for a linked
cavitysystem
whenthereis veryextensive
cavitation
overthe
bed, with small orificesthrottlingthe flow, as visualizedin
Figure1. The Manningroughness
valueis onethathasbeen
100m depthin ice 400 m thick). The evaluation
of thelinked
cavitymodelin thissection
isbased
on(36),withx takenequalto
xRbecause
we consider
herethehydraulic
behavior
in thesurging
stateandchoose
thereference
valuexRtobethebasalshearstress
thought
appropriate
in application
to glaciertunnels
[R6thlis-in thesurging
state;
thisrestriction
isrelaxed
in section
12. An
berger,
1972,p. 181;Clarke
etal.,1984,pp.292and295].The evaluation
based
on(35)gives
overall
results
notgreatly
different
tortuosity
ischosen
somewhat
larger
thanthevalueo•= 3 thoughtfromthose
givenhereonthebasis
of (36),although
thereare
appropriate
forglaciertunnels
byBrugman
[1986,p.219]. The someinteresting
andilluminating
differences
in detail;thesewill
*IRvalueis whatis calculated
fromtheflowlaw (11)if we bediscussed
inasubsequent
paper.
assumethatthe secondstressinvariantis equalto C•R= 6 bars.
When (36) and (7) and the parametervaluesstatedabove are
In orderto calculatethe Qw-versus-ocurveit is necessaryto introducedinto (17) and (18), thereresults
specifythe sliding velocity v as a functionof o, sinceit enters
T1R
3
(18) bothdirectlyand throughthe flux factor • via its dependenceon E, which varieswith v as shownin (17). Althoughthe
• = rc1-•77DM
relationbetweenv and• (dependence
of slidingvelocityon basal
water pressure)can in principlebe obtainedfrom the mechanical and
theoryof slidingwith cavitationon whichthe foregoingtreatment
is based(or by the methodof Fowler [1986]), for the evaluation
24/3No
13 'r
here of the linked cavitymodelI usea simpleempiricalrelation
of the type
,-,21/3
(c•A/•o)
3/2
(VROR)«h•
(•_•R)2
=0.18
(37)
(38a)
v= VR
=p •PlJ--q
(34)
Qw= 0.63
m3/s
(38b)
wherex is the basalshearstressand vR is the slidingvelocity
where• is in bars;ß in (38) is 0.47, from Figure 7, for the value
of-Z foundin (37). The dependences
on • assumedfor '1 andv in
pressurec•R;the independent
constants
in (34) arep, q, and the (7) and(36) leadto theresult,in (37), thatE doesnot vary with •
under reference conditions of shear stress xR and effective
product
pkq= VRXR
-pO:Rq.
Theinclusion
of a basalshear
stressand has the constant value 0.18.
dependence
in (34)is notneeded
fortheimmediate
purpose
at
The Qw-versus-o
curvefor the step-orifice
systemin
hand,for whichwe will takex=xR,butis needed
later Figure
12isaplotof(38b).Theorifice
breadth
lRatthereference
(section12). On the basis of laboratoryexperimentson the conditiono = ORis lR = 1.4 m, from (5). If we introduce(7) and
slidingof ice over rock surfaces,Budd et al. [1979, p. 164] (36) into (5) and recognize that rn is constantbecauseE is
proposed(34) withp = 3 andq = 1'
constant,
we obtainl/lR = (C•R/C•)3,
SOthat the gap lengthl
increasesrapidly with the water pressure. At o = 3 bars it has
lengthenedto l = 10.8 m. It is thislengtheningthatmainly causes
the increasein Qw with Pw givenby (38) and shownby the step
orifice curvein Figure 12.
Bindschadler[1983] got an approximatefit to flow data for
Quantitativeevaluationof the wave orifice linked cavity
VariegatedGlacierin the nonsurging
stateby taking(35) with model, giving the correspondingcurve of Qw versus o in
3kl = 0.2md4 bar-2;
thefit wasnotimproved
bytaking
q• 1. Figure 12, is carriedout in a similar way. It is done with the
The laboratorydata for slidingover roughgranite[Buddet al., parameters
v=vR
•- =3kl •-
(35)
1979,Figure7] give3k• in therange0.4-1.1m d-• bar-2;the
nonconstancy
of 3k• canbe corrected
by takingq = 1.5, with
3•= 2 m
3kl.5= 2.5md-1 bar
4'5. Fromfitting(34) to ice sheetdata,
a = 7.8 cm sinewave amplitude
I5o= 12 bars waveorificecavitationlimit (seebelow)
bedrocksurfacewavelength
Buddet al. [1984]chosep = 1, q = 2, whereas
LingleandBrown
[1986]chose(35) buttook3k• to varyovera widerange(a factor
of ~100), sothatthechoicep = 3, q = 1 is mootin thiscase. The and with other parametersas listed above. The viscosity'1 is
slidingconditions
at the baseof Variegated
Glacierin surgeare given by (12); it doesnot appearexplicitly,but it affectsthe value
probablyratherdifferentfrom thosepresentin the laboratory of the wave cavitationparameter15 given by (13). For the
experimentsand the field situationscited; hence one does not parametersstated, orifice cavitation startsat • = 15o= 12 bars.
necessarily
expecta slidingrelationin agreement
with theseother This is calculatedby substitutingv from (36) into (13), putting
studies,which are moreovernot fully consistent
amongthem- • = 15= 15o,andsolvingfor 15o,with theresult
selves,asjust noted. To representby meansof (34) the observed
rangeof surgevelocities~1-10rn/dfor waterlevel depthsin the
(39)
range100-70m (exclusiveof transitorypeaksto higherlevels)
[Kambet al., 1985, Figures5 and 9] one finds that q = 3 is
appropriate.Hence I will use
v =VR • ß
(36)
15o
=(2N(:SR)
«(-•)31(aVR)
•(• •«
15o
is thereforenot an independent
parameter,beinggiven in rexres
of the other parametersby (39). I call 15othe wave cavitation
limit, asdistinctfrom the wave cavitationparameterZ. From (13)
and (36) it alsofollowsthatfor any o,
9096
KAMB: GLACIER
SURGE
MECHANISM
Z = Zo2/o
(40a) with the samevaluesof el,M, andN usedaboveandwith co= 3,
whichis compatible
with theresultsof dye tracingaftersurge
termination(Qw = 40 m3/s,•w/co= 0.7 m/s). Equation(43) is
closelyrelatedto equation
(11) of R6thlisberger
[1972],differing
tXw= tx(hydraulic
grade
(40b) onlyin thatit is derivedforthecondition
andhencetheratio (5/2;thatappearsin (24)-(29) is
(5/2;
= ((5/2;
0)2
line parallelto the glaciersurface),ratherthan for a horizontal
arefor an assumed
circularcylindrical
When (36) and (40b) are introducedinto (25) and (29) and conduit.Theseequations
tunnelshape.
evaluatedwith theparameters
given,we obtain
3
DM
VR
•,=0.162
({xA/CO)3/2
Xa
2/3
(_••(5)
E'=0.21
1-
(52 -6
10. INTERPRETATION: THE WATER FLUX/PRESSURE
RELATIO•OMPARISON
OF LINKED CAVITY
(41a)
AND TUNNEL
MODELS
Whereas
in a tunnelsystem
thewaterflux at steadystateis a
decreasing
(inverse)
function
of waterpressure,
in a linkedcavity
(41b) system
it is an increasing
(direct)function,as Figure12 shows.
Thisholdsforlinkingorifices
of eitherstepcavityorwavecavity
1- 12-•
type, althoughthe form of the functionis rather different for the
where (5 is in bars, and
two. A linkedcavitysystem
containing
orificesof bothtypeshas
a flux-versus-pressure
relationthat is a combinationof the two
Qw= 1.38•
(52 6
•
1- •
curvesandagainhasQw increasing
withPw. Qualitatively
this
resultis independent
of thedetailsof thesliding-versus-pressure
relation(36), aslongasv is anincreasing
functionof Pw.
q• (42a)
(5223
An
important
consequence
ofthe
direct
dependence
ofQw
on
(1-•/:)6tI• m3/s(42b)
Pw
isthat
linked
cavity
system,
unlike
atunnel
system,
exist
Qw
=9.8
stably
as
aasystem
ofmany
interconnected
conduits.
Incan
asystem
of interconnecting
tunnels,the smallertunnelsare unstablewith
with
(5again
inbars.
The
wave
orifice
curve
inFigure
12isaplotrespect
tothe
larger
ones,
which
enlarge,
capturing
the
drainage,
of(42b).
The
flux
factor
tpisfrom
Figure
9onthe
basis
ofE' while
thesmaller
tunnels
close
up[R6thlisberger,
1972,
p.180;
values
calculated
from
(4lb).For
the
wave
curve,
unlike
fortheShreve,
1972,
p.209].The
system
tends
toevolve
toasingle
step,
themelting-stability
parameter
E'varies
with
(5,as(41)trunk
tunnel.
Ifthe
linked
cavity
system
were
subject
tothe
same
indicates,
from
0toamaximum
ofE'=0.17,
at(5=8bars.
typeof instability,
thewidespread
network
of interconnected
Thewave
orifice
gaplength
at(5=6bars,
ascalculated
fromcavities
visualized
inFigure
1 could
notpersist.
Bythesame
(26)withtheparameters
stated,
isl = 1.43m. Twothirds
ofthe reasoning
thatshows
thatthistypeofinstability
isa feature
of
gaplengthening
takes
place
between
(5= 12bars
and9 bars.At conduit
systems
in which
thesteady
state
flux-versus-pressure
(5= 6 bars
theheight
ofthegapatitswidest
point
is9.2cm,from relation
isinverse,
it canbeshown
thattheinstability
does
not
(20)andFigure10.
ariseif theflux-versus-pressure
relation
isdirect.Thisshows
that
From(1)it isreadily
calculated
thattheReynolds
number
for a multiple
conduit
system
of linkedcavities,
widelydispersed
waterflowin orifices
of thelinkedcavitymodel
evaluated
above across
theglacierbedandcarrying
theglacialwaterflux,can
isgreater
thantheturbulent
flowthreshold
value
ofabout
2 x 103 reallyexistandpersist,
contrary
towhatwould
beexpected
from
providedthatthe gapheightis greaterthan3 mm. The step thepreviously
knownproperties
of tunnelconduits.
orificeandwaveorificegapsdiscussed
above
aremostly
higher A secondimportant
featureof the flux-versus-pressure
thanthis,so thatit is appropriate
to use(1) as thebasisfor relations
in Figure12isthatfora waterfluxgreater
than~1m3/s
calculating
thewaterflowthrough
them.
the linkedcavity modelrequiresmuchhigherbasalwater
At a gapheight
of 5 cm,typical
of theorifices
considered,
the pressures
thanthetunnel
model
does.Thisprovides
anexplanameanwaterflowvelocity
from(1) is0.4m/s.Thisismuchlarger tionforwhythebasalwaterpressure
ishighin thesurging
state
than the observedmean water transportvelocityin surge, of glaciermotion:a tunnelsystem
is absent,
andthelinkedcavity
0.025m/s,whichreflectsthefact,discussed
in section3, thatthe
conduitsystemmustconsistof largecavitieslinkedby narrow
orifices. As longas the cavitydimensions
lc (breadth)
andgc
(height)andthenumberof cavitiesN, in anycrosssectionof the
systemthatis presentrequireshighwaterpressures
in orderto
openup theorificecavitation
sufficiently
to carrythebasalwater
fluxfurnished
bywaterinputupstream.
Thedistinction
in Figure
12between
waterleveldepths
of ~60-90m in thelinkedcavity
glacier are large enoughto providethe neededcross-sectionalmodeland >180 m in the tunnelmodel,for waterfluxesin the
areaof 200m2(section
3), thatis,aslongasN,l•g,= 200m2,the range2-20m3/s,parallels
thedistinction
between
observed
water
cavitygeometry
isnotfurtherconstrained.
Theconstraint
seemslevelsin thesurging
andnonsurging
states[Kambet al., 1985,
readilysatisfied:
if N, = No= 50,thecavities
couldbe0.5-1m Figure9]. (In fact,thewaterlevelscalculated
for thetunnel
high and4-8 m broad,whichseemsreasonable
for heavy system
aresomewhat
lowerthantheaverage
of thoseactually
cavitation
overa roughbedat highwaterpressure
andhigh observed
aftersurge
termination,
corresponding
better
withthe
slidingvelocity.
lowestlevelsobserved.
A somewhat
similarsituation
wasnoted
Forcomparison
withthelinkedcavitymodel,
thehydraulicbyIkenandBindschadler
[1986,pp.113-114]
incalculated
water
performance
of anicetunnel
isshown
in Figure12in terms
of the levelsforbasaltunnels
in Findelen
Glacier,
Switzerland.
A lower
steady
stateflux-versus-water-pressure
curve,calculated
from
valueof N in (43), suchasthevalue0.50 baryrm chosen
by
R6thlisberger
[1972,p. 194]or thevalue0.81baryrm advocated
rCD4M3
(•)½(3_•)
12
Qw
=•-g
byLliboutry[1983,
p.
220],•iveswaterlevel
(43)level
(depth
110-130
m)
inVariegated
Glacier
after
surge.)
K^MB:
GLACIER SUROEMECHANISM
9097
The conclusionthat basalwaterpressuresare muchhigherin steady state under fixed hydraulic gradient is unstableagainst
a linked cavity systemthan in a tunnel system at water flux perturbationsin size or water pressure: if the size, or the water
levels.>1m•/s holdsfor stepheightsh and wave amplitudesa of pressure,is increased,the tunnelwill grow, at an ever accelerating
order10 cm or less,suchas thoseusedin the modelevaluations rate,because
the wall meltingrateincreases
fasterthanthe tunnel
in section
9. Because
Qwis anincreasing
function
of h, or of a closure
rateas thesizeincreases;
conversely,
if the sizeor the
and•., asshown
by(18)and(29)orby(38)and(42),withlarge pressure
is reduced,
thetunnelwill contract
andwill closeup
enough
values
ofthese
dimensions
itispossible
toincrease
Qwat completely,
forthesame
reason
withchange
ofsign.(Thegrowth
fixed
o byarbitrarily
large
amounts
andtherefore
todecrease
instability
manifests
itself
inthej6kulhlaup
phenomenon
[Nye,
correspondingly
the water pressureat a given flux to values as
low
as or lower
than for
a tunnel
with
the same flux.
For
1976; Spring and Hutter, 1981; Clarke et al., 1984].) In contrast,
the orificesof a linked cavity systemin steadystateunderfixed
hydraulicgradientare stableagainstinfinitesimalperturbations
of
example,
byincreasing
hinthestep
model
from
5to22cm,
thethis
type
according
tothemodel
developed
here:
if thesize
is
water
level
atQw= 10m•/s
islowered
from
64to190m. perturbed,
the
response
isareturn
toward
the
original
steady
state
Changing
other
parameters
in(38)and
(42)could
also
contribute
size,
and
if thepressure
isperturbed,
theresponse
isameasured
tolowering
thewater
pressure.
Itfollows
that
thesurging
state
of adjustment
toward
anewsteady
state
size
dependent
onthenew
glacier
motion,
made
possible
bythepresence
ofalinked
cavitypressure.
basal
water
conduit
system
athigh
water
pressure,
isachievableThefeature
that
provides
thebasis
fortheprimary
stability
of
only on bedsof certainroughnesscharacteristics,
that is, certain
the linkedcavity systemis the flow capacityof the systemin the
dimensions
andspatial
arrangements
of roughness
features,
absence
ofmelting
byviscous
dissipation,
when
theconduits
are
equivalent statisticallyto what is achieved in the step orifice
model in idealized form by arrangingsteps of height ~5 cm
formedentirelyby cavitationunderthe availablewater pressure.
No correspondingfeature is presentin the model conceptof a
among
larger
roughness
forms
insuch
awayastoprovide
orifices
tunnel
system
[R•thlisberger,
1972].Oneof thereasons
for
linking
theleeside
cavities
behind
thelarger
features.
developing
themodel
ofthelinked
cavity
system
insections
4-8
Thepresence
ofthelarge
cavities
inthebasal
water
conduit
istofindoutwhether
ornotthestabilizing
feature
provided
by
system
isnota feature
necessary
fortheforegoing
conclusion:
cavitation
persists
when
viscous
heat
dissipation
isintroduced
into
putting
A equal
to1in(38a)
and(42a),
which
would
mean
no thelinked
cavity
system.
Theresults
insection
8 show
thatit
large
cavities,
decreases
Qwonlybya factor
of about
3, whichdoes,
within
limits.
could
becompensated
byachange
inthedimensional
parameters Thelimitsaresetbythemelting-stability
parameter
E (for
orNo.Thenecessity
ofthecavities
inthelinked
cavity
system
step
orifices)
orE'(forwave
orifices),
asdiscussed
insection
8.
comes
notfromtheireffect
ontheQw-versus-Pw
curves
but These
parameters
provide
a measure
of themagnitude
of the
rather
fromtheobservational
evidence
ondyetransport
throughviscous
dissipation,
asshown
by thedirectconnection
between
the system
andalsofromthe logicalexpectation
thatthe themandtheamount
oforifice
roofmeltback
m,tiedtoparamecavitation
gapheights
andbreadths
overanactual
bedsurface
will tersg andv inFigures
7 and9. Thestabilizing
effect
ofthebasal
behighly
variable
from
place
toplace,
creating
naturally
asystem
cavitation
phenomenon
predominates
aslong
asthemagnitude
of
oflarge
cavities
interspersed
withnarrow
orifices
(section
3)ß
theviscous
heating,
measured
byEor_,
•' is smallenough,thatis,
A linked
cavity
conduit
system
can
form
over
a"soft"
bedaslong
asE•<1and
E'.<1.5.Theinstability
that
develops
ina
consisting
largelyof unconsolidated
rock debrisprovidedthe bed steporifice for E • 1 or in a wave orifice for E' >.1.5 manifests
hasasufficient
number
ofroughness
features
that
donotmoveitself
when
viscous
heating
results
inanaccelerating,
uncontrolled
with
the
ice,
such
as
bedrock
protuberances
or
large
protruding
orifice
growth
that
is
essentially
the
same
(and
with
thesame
bouldersthat are stuckat deeperlevels so that they do not move
cause)as the acceleratinggrowth of a tunnel, describedabove.
withdeformation
of the"soft"
parts
of thebed.ThelocalHowever,
this
growth
instability
oforifices
differs
from
thatof
conditions
forcavitation,
and
theshape
ofthecavities
and
orifices
tunnels
inthat
orifices
areunstable
only
under
finite
perturbations
thatform,
arenotfundamentally
different
fromthose
overa from
thesteady
state.Therequired
sizeofthedestabilizing
•'
and becomes
"hard"
(bedrock)
bed.
Consequently,
one
can
expect
that
a
linked
perturbation
decreases
with
increase
of
E or_,
cavitysystemon a softbedat highspeedsof ice motionwill have
a flux-versus-pressure
relation that is similar, at least qualita-
small for E = 1 and E'--1,5, as seen for the step orifice in
tively,
tothe
curves
forthe
hard
beds
modeled
above
and
also
thatFigure
11.Fortunnels
there
isnostability
parameter
analogous
it will be subjectto a melting-instability
parametersimilarto
E'insection
8.Such
aconduit
system
must
bepresent
whether
or This
perturbations
under
allconditions
offixed
hydraulic
gradient.
is becausethem existsno steadystatefor tunnelsof finite
not soft bed deformationis involvedin the surgemotionbecause
the water flux in spring or summer is much too large to be
transmittedby a layer of subglacialtill or any granularmedium
exceptvery coarsegravel [Iken and Bindschadler,1986, p. 104;
Shoemaker,J986]. It followsthat the surgemechanismmodeled
hereon the basisof hardbedsis alsoapplicablein a generalway
to surgingwhen soft bed deformationcontributesto the high
speedsof ice motion.
11. STABILITY/INSTABILITY
LINKED
CAVITY
FEATURES
AND TUNNEL
OF
SYSTEMS
diameterin the absenceof viscousheating. Finally, whereasthe
tunnelinstabilityis bilateral(i.e., it appliesfor eitherpositiveor
negativeperturbations),
the orifice instabilityoccursonly for an
increasein size or waterpressurefrom the steadystatecondition.
We see here a fundamentaldistinctionbetweenthe hydraulic
behaviorof the orificesof a linked cavitysystemandthe tunnels
of a normaltunnelsystem.
Figure13 showsa visualization
of whathappenswhena step
orifice goesinto unstablegrowth. It evolvesinto a tunnel, which
is carried downstream
with
the movement
of the ice.
The
upstreampartof theenlargingtunnelbeginsto collapseandcloses
Thestability
orinstability
of a multiple
conduit
system
andits off,leaving
a steporificebehind,
whilethetunneliscarried
away.
relation to direct or inverseflux-versus-pressure
functions,Whathappens
to it thereafter
is a complicated
matter,dependent
discussed
in section10, is a secondary
manifestation
of what I on the three-dimensional
complexitiesof the bed geometry.
will call primary stability/instability
of conduits. A tunnel at Withoutenteringinto the numerouspossibilitiesand complica-
9098
KAMB: GLACIER SUROEMECHANISM
STEP
ORIFICE
?,::.?.i•i•::?,i?."•::::?).;i?.•i•:i?:..•:•i
=o (•o=o5).............
..............
// x///2 '/ ////•'/////[ ///////////////////////////////////
ß
•'•...................-•':":::':..
'•'"/•'""
"':/.??)')'
?.;':'2
•':;:'::•'<*:'""::•'.....•
•'•':'.;"•
i:.:?,:
:..:::>:
:•.,.•.;.•i:;.::::;?i.:?;;???•
............. ..<.z.•,•.:..111.i::..
;,;/"A;;2;Z;,•'
.'.....
'-':'•'•:.""'"
'''........ :.:...:.•,:;..:,:,:::,:.•
ß.......
////////
............ .,,..
,:.•..:.:,•..:..
>:
....
surge termination, by which the basal water conduit system
convertsfrom a linked cavity systemto a tunnel system,with
consequentdrop in basal water pressureand slowdownof the
slidingspeed. It is noteworthythat the observedsurgetermination was immediatelyprecededby a high peak in water pressure
[Kamb et al., 1985, Figure 9], which could have providedthe
initiatingperturbationthatput the orificesinto unstablegrowth.
The valuesof the melting-instabilityparametersderivedfor
the linked cavity modelsin section9, E = 0.18 for the steporifice
model and E'<_ 0.17 for the wave orifice model, from (37) and
(41), are enoughsmallerthan 1 to assurethat thesemodellinked
cavity systemscouldexist stablyat the glacierbed. At the same
time E and E' are within "shootingdistance"of 1, so that for
reasonablechangesin themodelinputparametersin section9 it is
possibleto reacha conditionwhereE > 1 or E' > 1.5, for which
the linked cavity systemswould be unstableand would convertto
tunnelsystems,asdiscussed
above.
12. SURGE MECHANISM
The above considerations
imply that it is possiblefor some
glaciersto be in a state of rapid sliding motion with a linked
cavity basalwater conduitsystemat high basalwater pressure,
.........
•';;/;)'2
........ "::::.:':.•'.':.•:.....,.
......-?.:.¾
';'.V:::.
,7..
:'.?::!.
!.:.
5:33):.:
.3:.:.'•
t:.:::i.:.:.: while for others the developmentof a tunnel systemwith low
waterpressureprecludessucha state. Moreover,it is possiblefor
a glacierto make a transitionto this stateandback again. This is
Fig. 13. Schematicpicture of what happenswhen a steporifice goes
the essenceof a surgemechanism.
unstablefor E ). 1, shown in a time sequenceof diagramswith time
The possibility of a surging state of glacier motion is
increasingfrom top to bottom. The bed is markedwith crosshatchingand
'"":'"'
'•,;;'/•')):
;;•;;•,;•/
-///////////////7//////
theiceisshown
stippled,
inthestyle
ofFigures
4 and5. Atthetopthe predicated
upontwofeatures
of thelinkedcavitysystem:
(1)the
steporificeis shownin steadystateundersomeeffectiveconfining high basal water pressure(and consequenthigh basal sliding
pressure
c•o, andwith E approximately
0.5; the recontact
pointis not velocity)that the linkedcavityconfiguration
canforceuponthe
moving,thatis, ! = 0, wherel is theorificegaplength.Uponincrease
of basal water system at typical water transportfluxes of order
E to a valuegreaterthanabout1, andupondecrease
in effectivepressure 1-10 m3/s or larger, as discussedin section 10, and (2) the
to c• < (•oasin Figure11,theorificestarts
to expand
asshown
by the possiblestabilityof themultipleconduitlinkedcavityconfiguraarrows
in thesecond
diagram;
initiallythegaplength
increases
at a rate tionagainstconversion
to a single-conduit
tunnelsystemthrough
/(0)--(1- c5•/C5o)V,
where
visthesliding
velocity
(see
section
8).Asthe theaction
of viscous
heatdissipation,
discussed
in sections
10
expansion
continues,
thegaplengthening
rateapproaches
themaximum
and11. Forthistypeof surging
stateto bepossible
for a given
possible
value
! -->vasnoted
inthethird
diagram.
When
thegaplengthglacier,
it isnecessary
thatthewater-flux-versus-water-pressure
passes
thesteady
state
length
l• that
corresponds
toc5•
inaccordance
with curvefor its basalcavitation
system
be suchas to require
Figure
7 fortheperturbed
parameter
E•,thehead
ofthecavity
begins
to appropriately
highbasalwaterpressures
(~2-5barsbelowice
collapse,
asindicated
in thefourth
snapshot,
buttheenlarged
orificeoverburden
pressure
in thecaseof Variegated
Glacier)
at the
downstreamcontinuesto grow rapidlyby meltingundergreatlyincreased
water flow and becomeseffectively a tunnel segmentconnectingthe two currentwater flux levels imposedby input of water to the basal
cavities
thatwereinitially
connected
bythestep
orifice.
Ultimately
the watersystem
fromall sources,
andit is necessary
thatthe
headof thecavitycloses
off,asshown
in thelastsnapshot,
andthetunnel melting-stability
parameter(E or E' in sections7 and 8) for its
segment
is advected
downstream
at speed
v. If it doesnotencounter
a linkedcavitysystem
underthesecurrentconditions
belessthana
large
stoss
slope
downstream
andgetpinched
offandif it canswingcertain
criticalvaluethatseparates
stability
(lowE) from
around into a longitudinal orientation,it can become a more or less
permanently
established
tunnel
segment
inthebasal
water
conduit
system,
instability
(high
E) against
conversion
toa tunnel
system.
These
provided
thatthelocalwaterfluxandlocalwaterpressure
adjust
to the conditionscan be satisfiedonly if the systemparameters
lie
levelsneeded
for its stable
maintenance
asa tunnelin accordance
with withincertainranges,andthisis what,in principle,distinguishes
(43).
glaciers
thatcansurge
fromonesthatcannot
surge.Thelinked
cavity modelin sections7-11 providesa meansof identifyingin
tions, I surmisethat if viscousheatingin the systemis large simplifiedform what the relevantsystemparameters
are, and of
enough(E or E' largeenough)andif the initiatingperturbation
is showingthat for reasonable
valuesof theseparameters
(suchas
big enoughthat a large enoughnumberof orificesgo into thoselistedin section9) theconditions
for the surgingstatecan
unstablegrowth simultaneously,
a networkof interconnectedbe realized,while for otherreasonable
valuesthe conditions
are
tunnelsmay startto developout of cavitieslinkedby unstably notsatisfied
(sections
10 and11). Sincethe"orifices"
(section
3)
growingorificesor by thetunnelsegments
sheddownstream
by of the linkedcavitysystemcontrolthe waterflow, theirdimenthem. If thisdevelopment
is ableto continue,whichprobably sionalparameters
are particularlyimportant;in general,the
requiresthata largeenoughpartof the bedbe affectedto allow roughness
featuresthatgeneratethe orificesby basalcavitation
the process
to developcollectivelyandcooperatively,
the linked musthavesmallamplitude(.<0.1m) in orderthatthe conditions
cavity systemcan therebyconvertto a multiple-tunnel
system, for surgingcan be satisfied. Also importantis the sliding
whichthenproceeds
to degenerate
into a single-tunnel
systemby velocity, becauserapid sliding stabilizesthe orifices, by
multiple-conduitinstability. In my view, this is the processof decreasing
E in (17) and E' in (25). Slidingof ~1 rn/d undera
KAMB: GLACIER SUROEMECHANISM
9099
basal water pressureabout 6 bars below overburdenis about offsetby an increaseof o• in (37) and (41a), but in fact ct did not
sufficient
to conferstabilityuponorificesgenerated
by roughnessincreasein the area wherethe surgeactuallystarted(C. F.
featuresof amplitude~0.1 m, but if the slidingvelocitywere a Raymond,personalcommunication,
1986;C. F. Raymondet al.,
fifth this great,the orificeswouldbe unstable,accordingto the Variegated Glacier studiesm1979, unpublishedmanuscript,
modelresultsin section9. Sincetheslidingvelocityat anygiven Figure3).
basal shearstressand water pressurewill increaseas the bed
The proposedsurgemechanism,as describedabove,involves
roughness
is decreased,low roughnessagainfavorssurging,but entry into the surgingstatein winter. This is in agreementwith
the roughnessthat is relevanthere is probablynot that associated the fact that the surgeof VariegatedGlacier did start in winterwith the orifices but rather the larger-amplituderoughness time. However,the attainmentof a surgespeedof 2 m/d or more
associated
with wavecavitiesof thelinkedcavitysystem.
in midwinter,when the availabilityof water for the basalwater
Although the distinctionbetween "surgeable"and "non- systemis minimal, indicatesthat them is more to the surge
surgeable"glaciersthusseemsin principleclear, thereis a large initiationmechanismthanthe foregoingdiscussion
suggests.The
gap betweentheseprinciplesand theirpracticalapplicationto the conceptsof the linked cavity modelproveusefulin consideringin
problemof explainingwhy certainglacierssurgeand othersdo greaterdetail the causeof wintertimesurgeinitiation,which will
not. For one thing, the relationshipbetweenthe simple model be takenup in a subsequent
paper.
parametersand the complexactualfeaturesof real glacierbedsis
problematical,
andalso
there
isingeneral
nowaytomeasure
such Acknowledgments.
Thisworkwassupported
by grants
parameters
atthebottom
ofactual
glaciers.
Foranother
thing,
the DPP-8209824
andDPP-8519083
fromtheU.S.National
Science
connection
between
thesystem
parameters
andthebasal
slidingFoundation.
I thank
Paul
Hawley
forthetextprocessing.
Caltech
relation
((36b)
or(36c)orother)
isa complicated
matter.
This Division
ofGeological
andPlanetary
Sciences
contribution
4428.
latter problemcan be dealt with to someextent for exampleslike
Variegated
Glacier
andFindelen
Glacier
forwhichthereexist
REFERENCES
observational
datafor an empirical
relation
between
waterBindschadler,
R.,Theimportance
ofpressurized
subglacial
water
pressure
andsliding
uptothehigh
values
relevant
forsurging.inseparation
and
sliding
attheglacier
bed,
J.Glaciol.,
29,3-19,
Forglaciers
thatdonotexperience
highbasal
water
pressures,
the
needed
empirical
relation
would
beunobtainable.
Despite these difficulties, a test of the ability of the linked
1983.
Brugman,
M.M.,Water
flowatthebase
ofa surging
glacier,
Ph.D. thesis,Calif. Inst. of Technol., Pasadena,1986.
cavity
model
to distinguish
predictively
in a statistical
way Budd,
W.F.,P.L.Keage
and
N.A.Blundy,
Empirical
studies
of
betweensurge-typeand nonsurge-type
glaciersis providedby the
observedcorrelationbetweenglacier length and probability of
surgingin a large sampleof Yukon glaciers[Clarke et al., 1986,
Figure2b]. Glacier lengthdoesnot directlyenter as a parameter
in the model, but since there is a strong inverse correlation
betweenlengthand surfaceslopein the samplestudied[Clarke et
al., 1986, Figure 4a], the observedcorrelationcan be considered
an inverse correlationbetween surge probability and slope ct,
which is a parameter in the model. (Clarke et al. [1986]
concludedthat the correlationwas really with length alone and
that slopehad no effect, but a compellingstatisticalargumentthat
would requirethis conclusionwas not given.) The parameters2
in (37) and 2' in (41a) vary as o•/2; henceother thingsbeing
ice sliding,J. Glacœol.,
23, 157-170,1979.
Budd, W. F., D. Jensen and I. N. Smith, A three-dimensional
time-dependentmodel of the West Antarctic ice sheet,Ann.
Glaciol., 5, 29-36, 1984.
Carol, H., The formation of roches moutonn6es,J. Glaciol., 1,
57-59, 1947.
Clarke, G. K. C., W. H. Mathews and R. T. Pack, Outburstfloods
from glacialLake Missoula,Quat.Res.,22,289-299, 1984.
Clarke, G. K. C., J.P. Schmok,S. L. OmmaneyandS. G. Collins,
Characteristics
of surge-typeglaciers,J. Geophys.Res., 91,
7165-7180, 1986.
Engelhardt,
H. F., W. D. HarrisonandB. Kamb,Basalslidingand
conditionsat the glacier bed as revealedby boreholephotogra-
equal,
themodel
indicates
lower
values
of2 and2' forlonger phy,
J.Glaciol.,
20,469-508,
1978.
glaciers,
which
means
a higher
probability
ofvalues
lowenoughFowler,
A. C.,A sliding
lawforglaciers
ofconstant
viscosity
in
to stabilize
a linkedcavitysystem
andallowsurging.
Thus thepresence
of subglacial
cavitation,
Proc.R. Soc.London,
qualitatively,
atleast,
themodel
passes
thistest.
Ser.A,407,147-170,
1986.
Thequestion
ofhowandwhya surge-type
glacier
makes
the Hallet,
B.,andR. S.Anderson,
Detailed
glacial
geomorphology
transition
fromthenonsurging
state
tothesurging
state
andback of a proglacial
bedrock
areaat Castleguard
Glacier,
Alberta,
again
iscentral
tothemechanism
of surging.
It issuggested
in Canada,
Z.Gletscherk.
Glazialgeol.,
16,171-184,
1980.
section
11thatsurge
termination
occurs
whenthelinked
cavityIken,A., Theeffectof subglacial
waterpressure
onthesliding
system
goesunstable
andconverts
to a tunnel
system.
The velocity
of a glacier
in an idealized
numerical
model,
J.
simplest
mechanism
of surgeinitiation,
withintheframework Glaciol.,
27,407-421,
1981.
provided
by the linkedcavitymodel,is thatthe basalwater Iken,A., andR. A. Bindscradler,
Combined
measurements
of
conduit
system
firstbecomes
a linkedcavitysystem
without subglacial
waterpressure
andsurface
velocity
of Findelentunnels,
whichhappens
in winter,
whenthewaterfluxis low gletscher,
Switzerland:
Conclusions
about
drainage
system
and
enough,
andthen,
whenthefluxincreases
inspring,
2 happens
to sliding
mechanism,
J.Glaciol.,
32,101-119,
1986.
belowenough
thatthelinkedcavitysystem
doesnotdegenerate
Kamb,B., Slidingmotionof glaciers:Theoryandobservation,
to a tunnelsystem.Thelowering
of 2 to thepointwherethe Rev.Geophys.,
8, 673-728,1970.
linkedcavitysystemis stabilizedcan be causedin the reservoir Kamb,B., andH. Engelhardt,
Wavesof accelerated
motionin a
areaof theglacierby theincrease
in basalshearstress
thereasthe glacier approaching
surge: The mini-surgesof Variegated
ice thicknessbuildsup priorto surge:(37) and(4la) showthat an
Glacier,Alaska,J. Glaciol.,33, 27-46, 1987.
increase
in x decreases
2 and2'. In agreement
with thisconcept Kamb, B., and E. R. La Crapelie, Direct observation
of the
for surgeinitiation,the surgeof VariegatedGlacier startedin the
mechanismof glacier sliding over bedrock, J. Glaciol., 5,
reservoirarea. The effect of increasedx would to someextent be
159-172, 1964.
9100
KAMB: GLACIER SURGEMECHANISM
Kamb, B., C. F. Raymond,W. D. Harrison,H. Engelhardt,K.A. Shoemaker,E. M., Subglacialhydrologyfor an ice sheetresting
Echelmeyer,N. Humphrey,M. M. Brugmanand T. Pfeffer, on a deformableaquifer,J. Glaciol.,32, 20-30, 1986.
Glacier surge mechanism: 1982-1983 surge of Variegated Shreve,R. L., Movementof water in glaciers,J. Glaciol., 11,
Glacier, Alaska, Science,227, 469-479, 1985.
205-214 1972.
Lingle, C. S., and T. J. Brown, A subglacialaquiferbed model Spring,U., andK. Hutter,Numericalstudiesof j6kulhlaups,Cold
and water-pressure-dependent
basal-slidingrelationshipfor a
Reg. Sci. Technol.,4,227-244, 1981.
west-Antarctic
ice stream,in The Dynamicsof the West Spring,U., andK. Hutter,Conduitflow of a fluid throughits
Antarctic Ice Sheet (ed. C. J. van der Veen and J. Oerlemans),
solidphaseandits application
to intraglacialchannelflow, Int.
pp. 249-285, D. Reidel,Dordrecht,1986.
J. Eng. Sci., 20, 327-363, 1982.
Lliboutry,L., Generaltheoryof subglacialcavitationand sliding Vivian, R., The nature of the ice-rock interface: The results of
of temperateglaciers,J. Glaciol., 7, 21-58, 1968.
investigationon 20,000 m2 of the rock bed of temperate
Lliboutry,L., Modificationto the theoryof intraglacialwaterways glaciers,J. Glaciol., 25, 267-277, 1980.
for the caseof subglacialones,J. Glaciol., 29, 216-226, 1983.
Vivian, R., and G. Bocquet, Subglacialcavitationphenomena
Nye, J. F., The flow law of ice from measurements
in glacier underthe Glacier d'Argenti•re, Mont Blanc, France,J. Glaciol.,
tunnels,laboratoryexperimentsand the Jungfraufimborehole 12,439-451, 1973.
experiment,Proc. R. Soc.London,Ser. A, 219, 477-489, 1953.
Walder, J., and B. Hallet, Geometryof former subglacialwater
Nye, J. F., A calculationof the slidingof ice over a wavy surface channelsand cavities,J. Glaciol., 23, 335-346, 1979.
usinga Newtonian
viscous
approximation,
Proc.R. Soc. Weertman,
J., Onthesliding
of glaciers,
J. Glaciol.,
3, 33-38,
London,Ser.A, 311,445-467, 1969.
1957.
Nye,J.F.,Wateratthebedofaglacier,
IAHSPubl.,95,189-194,Weertman,
J., General
theory
of waterflowat thebaseof a
1973.
glacier
oricesheet,
Rev.Geophys.,
10,287-333,
1972.
Nye, J. F., Water flow in glaciers: j6kulhlaups,tunnels,and
veins,
J.Glaciol.,17,
181-207,
1976.
B. Kamb,
Division
of Geological
andPlanetary
Sciences,
Paterson,
W, S. B.,ThePhysics
ofGlaciers,
Pergamon,
NewCalifomia
lnstituteofTechnology,
Pasadena,
CA91125.
York, 1981.
Raymond,C. F., How do glacierssurge?A review,J. Geophys.
Res.,thisissue.
R6thlisberger,H., Water pressure in intra- and subglacial
channels,
J. Glaciol.,11,177-203, 1972.
(ReceivedNovember17, 1986;
revisedMarch31,1987;
accepted
May 7, 1987.)