J Ollrnaloj(;/(Lciology, Va /. 43, .\ 0. 144, 1997
The role of lateral drag in the dynamics of Ice Stream B,
Antarctica
I. M. WHILLANS, * Gj.
VAN D E R VEEN
Byrd Polar Resemdl Cmler. and ~ D epartment of Geological Sciences, T he Ohio State Unil'ersit.y, Collllnblls, Oh io 132/0, U S.A.
ABSTRACT.
The p a rtiti oning of r es isti ye force b e t wee n th e be d a nd sides of Ic e
Stream B, Anta rcti ca , is obta ined fo r three large a reas th a t have bee n m eas ured using
re p eat a eri a l photogramm etr y. Problems associated with d ata errors a nd loca l vari ations
in ice strength a nd velocity a re reduced by considering th e a reall y ave raged budge t o f
fo rces for each photo bl ock. Results indicate that the b ed under Ice Strea m B must be very
wea k a nd un able to prov ide mu ch r es ista nce. Mec h a ni ca l control o n thi s ice strea m
em a n ates a lm os t enti rely from the late ra l m a rgins.
IN TRODUCTION
Ice streams a re fas t-moving glac iers within the grounded
port ion of th e ''''est Anta rcti c ice sh ee t. Some tens of kilometers wide a nd sever a l hundred s of ki lometers long, th e
ice strea ms are the principa l dra in age r o utes fo r the inla nd
rese n 'o ir of ice. C h a nges in ice-stream Oow, such as th e
recent stoppage of Ice Strea m C, a lter the storage o f ice
a nd a ffect globa l sea level. The bas ic mechanism o f icestream moti on has not been understood . A necessa ry ste p
in achi eving such understa nding is to identify the sites o f
res ista nce to now.
Th e Om-v of a ll g laciers is dri ve n by g r a\'ity, as described
by th e dri\'ing stress. On most fas t glaciers, large velociti es
a rc a ttain ed because th e dri ving stress is la rge. For example,
Byrd Gl acier, a n o ut let g lac ier (I'om E as t Anta rctica, travel
at a bo ut 800 m a 1 a nd h as a dri\'ing stress 01' 250 kPa (Whilla ns a nd others, 1989). Th e grounded p o rti on ofJ a kobsh av ns
Isbra: in Greenla nd travel s at about 3 km a 1 under a driv i ng
stress of a kw hund red kPa ( Echelmeyer and H a rriso n,
1990; Echelmeye r a nd others, 1991). Such fa. t now du e to
large grav itationa l forces is understa nd a ble in terms o f
co nve nti ona l glac io logica l theory. H owever, fee Strea m B
reac hes speeds of as much as 824 m a 1, ye t its driving stress
is o nly a bout 15 kPa (Al ley a nd Whilla ns, 1991), less th a n
10% of the value fo r o ther fas t glaeiers. Ice Streams D a nd
E have somewh at la rge l- dri\'ing stresses a nd simil a r fas t
speed s (Bind sc had le r a nd Sca mbos, 1991; wlac Ayea l a nd
others, 1995). Thi s lack of obvious link age betwee n speed
a nd dri ving stress has bee n a m<uo r enig m a when addressing
q ues ti o ns of th e ice sh eet a nd its sta bility.
G lacier now is co ntroll ed by a ba la nce be tween dri ving
stress a nd stresses oppos ing the nm.v. Th e small driving
stresses on th e Wes t Anta rctic ice stream s indi cate th a t n et
res ista nce to now must be equa ll y sma ll. This resista nce
m ay ori ginate at th e g lacier bed (basa l drag ), at th e la te ra l
m a rg ins (latera l drag ) or from gradi ents in longitudin a l
stress (Va n der Vee n a nd Whi llans, 1989). M easured surface
\'e1 ocities ha\'e been used to show th a t longitudin a l stress
gradients arc unimporta nt to the large-sca le balance o f
forces o n Ic e Stream B (\ Vh il la ns and Va n d erVeen, 1993a ).
Thus, res ista nce on this ice stream must b e so me combin ation of drag from the b ed a nd lateral ma rg i n s.
Th e prese nt stud y addresses the iss ue o f th e relati\'e importance o f latera l a nd basa l drags for Ice Stream B. Th e
bas ic d a ta se t used deri vcs fro m repea t aeria l photogrammetr y, prov iding num ero us d eterminati o ns o f surface speed
and surface el evation. Fo r each of thrce sel ccted regio ns
(shown in Fig ure I), ma ny tra nsects of vel oc it y a re a\'aila ble,
allowin g the role of latera l drag to be e\"a lu a tedmore fu lly
than in Ix io r studies.
Prior stud ies have a ddressed the role o f la teral drag as
well. to r Icc Stream B, Bi nd sc had ler a nd o th ers (1987) a nd
Echelm eye r a nd others (1994) use single tra n. ec ts of\'elocity
near th e DN B and UpB ca mps, respec ti\·e1 y. For Ice Stream
E, i\.Iac Ayea l and oth ers (1995) appli ed inve rse control
methods to infer basa l d rag. T he limitati o n in th at wo rk is
that it has bee n clif1icult to o bta in sufficient ly acc urate va lues
of the reg io nal surface slo pe, as needed to ca lcu late th e
dri \' ing stress, and there m ay be importa nt limits on th e
spati al reso luti on of \'elocity determination in shear zones.
Th e present study has th e ad\'a ntage o f' good spat ia l
coverage o f velocit y determi na tion and prec ise a nd acc ura te
surface slo p e determin a ti o ns.
ESTIMATING LATERAL DRAG
The ory
Resista nce to fl ow from la tera l drag ca n b e estim ated by
consider ing how th e latera l shear stress, R J .y , va ri es across
the ice stream. The x ax is is chosen in the direction of fl ow
and th e y ax is perpendi c u la r to it; th e components of
vel ocity in these d irec ti o ns are represe nted by U a nd V ,
res pec ti vel y. Th e shea r stress is linked to the shea r stra in
rate, or tra nsverse gradien t in ice velocity, using the inverse
23 1
Joumal qfGlaciology
60
'loI
South \
S 840 29' 49"
W 1340 16 ' 49 "
--;"/,1
Shear Margin
A
Camp
Linear Feature
@
Station moving
less than 5 m/a
Crevasses
H-Mi
Basal (?) Feature
North A
S 820 53' 32"
W 1360 39' 37"
-
o
-
o
50 km
200
400
1
!
o
)
20
m/a
m/a
40
' ~
Fig. l. Locatioll maps/towing the main suljacefiaturesq/ Ice Stream B, based all aerial jJ/IOLogrammetlY. Numbers refer La groundbased stations where velociLies ( re/msented by the arrows; note the two different scales) were measured using Tiwzsit sateLLite
tracking ( r Vhillans and Van da Veen, /993a). Shading rejJTesents the aeLive ice stream. Th e three areas studied in this contribution
are indicaLed ~v the bores, and cover tributaries El and B2 and the narrow region downstream q/Lhe confluence a/these tributaries.
formul a tion of Glen's fl ow law for glacier ice, with exponent
3:
71,=
R ,u = B
~
Dy
("2l DU)
(1)
whi ch B represenrs the temperatu re-dependent rate
factor (H oo ke, 1981).
In a dopting this eq ua ti on, a number of simplifying
approxi m a tions a re introduced. The fi rst approxim ation is
that f.. ,y is the domina11l strain rate, so th at the effective
strain rate may be replaced by the shearing ra te. Whill ans
and ot hers (1993) show that longitudina l tension and lateral
compressio n are of seco nd a ry importance in the region near
the UpB camp, so that these strain rates can be neglected .
Inspectio n of the surface velocities in the other areas considered in this study ind icates that thi s result may be ex tended
to all photo blocks. The seco nd approxim ati on made here is
that flow I i ne turning, as described by the a lo ng-fl ow gradient in the tra nsverse co mpo nem of ice velocity, DV 18x, is
negligibly sm a ll. Thus, th e shear strain rate is mainly determin ed by th e trans\'erse g ra die11l in the a long-flow veloc ity,
8U I Dy. No ne of th ese approximations is critical. A furt her
approximati o n is that th e shear stra in ra te may be considered constant with depth, Th e large, peeds on Ice Stream
B suggest that most of the flow is associated with basal sliding and so th e shear strain ra te varies li tt le with depth, Thus
the res istive force from th e sides can be estimated from
observed surface speeds,
Two equivalent method s for estim atin g full-thickn ess
forces were considered. One approac h is to use a rate factor,
B , in the fl ow law (Equ ation (I)) appropri ate for near-surface
tempera tures, a nd th e depth of strong ic e, which is less tha n
111
232
the full ice thi ckn ess, The other a pproach is to use a depthaveraged a nd temperature-weighted rate factor toge ther
with the full thickn ess of the glacier, The second approach
is sim pler, a nd is used here. Thu s, the lateral force equals
HR"y per unit length in the ice-strea m direction, in which
H represe nts full thickn ess, and a temperature-we ighted
rate facto r is used in Equ ation (I).
The net r esistance from latera l drag per unit width of ice
stream is (e.g. Van der Veen and \ Vh ill a ns, 1989)
D
Fiat= Dy(HR".II )'
(2)
Th e ice thi ckness, H , is nea rl y co nsta nt across the ice stream
(Retzlaff and others, 1993). Taking H independent o f y and
substituting Equ a tion (I) for t he shea r stress gives
BH
(au)-~ 8
Fiat = 3:l j2 ay
2
8 U
y2 '
(3)
Applying this expression to measured rransects of velocity,
U(y) , all ows th e role of lateral drag to be assessed across
the width o f th e ice strea m,
Th e steps involved in the calcul ation a re sh own in Figure 2 for a single transect. The upper panel shows one transect of the a long-Il ow component of velocit y, U, derived from
repeat ae rial ph otogra mmetr y. The lack of data for the interval16 < y < 31 km is clue to the absence of visibl e surface
feat ures in the ce ntral part of th e ice stream. From these
velocities, the sh ea r strain ra te (second pa nel) is estimated,
Applying th e fl ow law (Equation (I )), the shear stress (th ird
panel) can be eva luated. The tra nsverse variation in shear
stress is a measure for resistance to fl ow associa ted with lateral drag (lower panel ). In this calcul ation, a constant rate
Wh d!all s andVall der reen: Role of/atera! drag il1
~Vllami(s qf l ee
Stream B
I
400
'(1)
-S
"0
Q)
Q)
300
(
••
200
Cl.
Cl)
100
(1)
(')
50
0
-50
Co
•,
The calCLtl a ted res ista nce fro m la tera l drag is propo rti o na l
to the scco nd d e riva ti w of ice sp eed as shown by Equ a tio n
(3). Conseque nt ly, sma ll meas urem elll erro rs m ay tra nsla te
into la rge unce rt a inti es in t he res ults. The uncerta inty in
velocity deter min ation, o"u, i a bo ut 4 m a 1 (\Vhill a ns a nd
o thers, 1993). Th e propaga ted e rror in shea r stra in ra te,
fr y = 1/ 2( f)U / fJy) , is 0"11 / (V2f). y) , in which ~ y r e presents
the d ista nce O\"('r which th e t ran s\'e rse \'eloeit y gra dient is
calc ulated . 10 a vo id un acce pta bl y la rge errors, a nd ye t still
desc ribe local p a t terns, \'eloc ity d e term inati o l1s a t leas t I km
a pa rt but no m o re tha n 2 km ap a rt a re used. Th is leads LO a
m ax imum e rro r in stra in ra te, 0"("" , of 3 x 10 ~ a 1. Shea r
stra in rates reac h " a lues th a t a rc a n order o f m ag nitude
la rger th a n t h is e rror estim a te (Fig. 2). Th e assoc ia ted er ro r
in shear stress ca n be estim a ted [1'0 111
•
\
.~
en
Error propagation
\
Q)
c
\,
•
\
,.. ,
100
";-
~
~
,•
0
b
factor is used (B = 5--1-0 kPa a "). Thi s " a lue is the te mpera ture-weig hted ra te fano r, based on the temperature profil e
meas ured in a bo rehole to th e bed nea r th e U pB ea mp
(Ellgdha rdt a nd o thers, 1990) a nd va lues o f B g iw l1 in
H ooke (1981).
,
500
Q)
.r:
Cl)
-100
0" H,"
co
0...
6
C/)
C/)
~
en
Co
Q)
.r:
Cl)
250
200
150
100
50
0
-50
-100
-150
-200
-250
nel), whil e t.h e shear stra in ra te is on the o rder o [ 0.05 a 1.
Thus, th e e rro r in the shea r stress a t th e m a rgins is a bo ut
4 kPa. Appl y ing thi s error in Equ a tion (2), th e uncerta int y
in resista nce fro m la tera l drag, 0"/'1,,, , is about 3 kPa fo r trans'Trse g rad ie J1ts calcul ated 0\"(' 1" di sta nces la rge r th a n I km
(th e ice thi ckn ess is a bout 1000 m ). The error m ay b e considerabl y la rge r to wa rd s th e ce nt er of the ice stream where
sma ll \'a lues o f DU/ ay tend to m ag nify the effec t o [ measurement unce rt a inti es. Co nsequ entl y th e error ba rs in th e
lower panel o f Fig ure 2 a rc la rge r near th e ice-stream
ce nter.
~, t
co
0...
6
Cl
~
0
~
Local fluctuations
~
Q)
+
Cil
-l
-50
0
20
10
30
Transverse distance (km)
40
Fig. 2. Stp/}S ill estimation qflateml dmgFom measurelllet/t q/
along a sillgle t ransert. The II/)/}f}" panel shows the ice
speed, [ ~ alld the second panel the shear straill rate or haif
tranSl'erse gradifllt, E.r!! = 1/ 2(DU / Dy). The third /)ane!
.lholf'S the shear streSJ. R nI'
ca/rulatedjivm
the sltear strain
.
I
rate lIsing a rate/artor B = 5+0 kPa (l "fiom Hooke (1981),
and temperatllre-weighted aaordillg to the temjJerature profile
measured in a boreho!e 10 tlt e bed lIear the ['/) B rall1/) ( Ellgelhardt and others, 1990). T he lower /)({I1e1 shows lateral dmg.
({[Im/a ted fiom F la t = (fJ / [)y) (If R.ry). T his /)(lItiw/a r
transecl CONTS tribll/{l7] B2, about haifu'a'y between station
J-j and lhe L/) 8 cam/). l ee.flow is to wards the rear/el:
l'floc£ ~JI
(4)
In th e shear m a rg ins, R.ry reach es a bout 200 kPa ( third pa-
50
"0
= -31R'''!I
-.- 0" C,"
E.,.!)
Th ere a rc loca l nUCluati ons in ice " eloc ityo n t he ice strea m.
Such \'ari a ti o ns haw bee n obse r\'ed on th e stra in g rid d c-ployed nea r th e U pB ca mp (\Vhill a ns a nd Va n d e r Veen,
1993 b; Hulbe a nd Whill a ns, 1994) a nd a rc a lso evide11l in
the \'eloc ities o bta ined fr om re pea t ae ri a l photog ra mm etr y
(Whill a ns and o th ers, 1993). fni tia ll y, it was be li eved tha t
these nuctu a ti o ns mig ht be associa ted with no w o\T r a nd
a round basal fea tures. Howe ' T r, th e pattern s in velocity
a nd surface slo p e do not co rrel a te as wo uld be expec ted
due LO a basa l di sturba nce. M o reO\"C r, th e surface to pog raphy mig rates with time (Hulbe a nd Whill a ns, 1994).
The a no m a lo us loca l patte rns in w loc ity m ay be du e to
hori zonta l co ntras ts in ice stre ng th. Deform a ti o n is co nce ntrated in wh a t see m to be na rrow (a bo ut 500 m w id e ) ba nd s
of so ft ice embedded in the m a in bod y of th e ice stream.
HOWe\Tr, th e we ig hted-mea n ra te facLO r ac ross the ice stream,
a llowing for soft ba nd s, is onl y a lit tlc small er tha n th e ,'a lue
used here. Th e e ffect of loca l \'a ri a ti ons in th e ra te factor is
ta ken LO be like no ise that ca n be mitiga ted by aver aging.
This loca l no ise is removed by acc umul a ting m a ny
tra nsects of , "(' loc it y. Instead o[ eo nsideri ng si ngle tra nsee ts,
as in Fig ure 2, multipl e transec ts o bta ined from re pea t ae ri a l
photogramm e tr y a rc used. Thi s results in the g ra ph s shown
in th e upper p a nel of Fig ure 3.10 furth er reduce th e e ffec t of
233
J ournalofClaciology
800
'ro
.s
Tributary B2
Tributary B1
,
. . . . . . . . . 4Oblook
600
400
::>
200
0
0 .15
~
0.10
Ql
0.05
~
Cl
c
0 .00
Ql
£
(j)
-0.05
A - --
.~
-«.;
.
\
• -
- 1 ...
-<fIIa....,
-0. 10
300
200
C?
a..
6
100
,0
>x
a:
-100
-200
-300
<l .0 0.2 0.4 0 .6 0 .8 1.0
0.0 0 .2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Dimensionless tran sverse distance
Fig. 3. Profiles of measured ice sjJeed and derived quantities across/he ice streamjor the three /J/1O[o blocks shown in Figure I.
D irection offlow is towards the viewer: T he shear stresses in the lower jJanels are calculated using Equation (IJ with mteiactor
I
= 540 kPa a 'l . T he solid line rejJresents the theoretical gradient if the sides OjJ/Jose aLL qfthe action qIthe ire stream.
loca l Ouctuations, th e calcul ations a re a lso conducted using
the tra nsverse g radient in shear stress eva luated over o nly
the central part of the ice stream in the disc ussion of force
budge t. Thi s reduces the effect of measurement error a nd
cffccti vely averages ove r local Ouctu ations.
DATA
Photogramme try
Velociti es a nd surface elevati ons used in thi s stud y d erive
from repeat ae rial photogramme tr y co nducted in J a nu a ry
1985 and J a nu a r y 1986. The positions of surface feat ures
(m a inly crevasses) recogni zabl e o n both se ts of photogr aphs
a re m easured to obta in di placem ents a nd thus velocities
(Whill ans a nd others, 1993). Scale a nd ori entation fo r th e
linked aeri al photographs are provided by photo targe ts se t
up at so me of the surface sta tions where velociti es were
m easured using Tra nsit satellite tracking (Whillans a nd
Va n der Veen, 1993a). The a rrows in Fig ure I represent
velocities at th e ground stations. The locations of th e three
photo blocks used in this stud y a re shown in Figure I; t he" 10
block" cove rs th e t'vvo tributari es BI a nd B2, and the "4·0
block" the narrow p o rti on of the combined ice stream.
Photogrammetric Yelocity determin ations are restric ted
to those parts of th e ice stream wh ere surface crevasses a re
present, which is prim a ri ly th e regio n o f the shear m a rg ins
a nd inwa rd . Near the center, th ere a re fewer, if a ny,
crevasses, res ulting in data gaps. Fo r tributary B2, additi on a l velocity de termin ations along the strain g rid nea r
the UpB camp (deployed in a region free from exp osed surface crevasses ) a r e ava il able and used in thi s study. These
234
measurem ents are represented by the cluster of points
ce ntered a round s = 0.6 (8 being th e transverse coord inate)
in the upper left panel o f Fig ure 3.
Transect s of veloc ity
A sing le accumul ated tra nsec t of velocity is prepared for
each of th e three photo blocks by stacking sep arate na rrow
transects. Velocity deter mina tions within each I km wide
transverse band are used to calcul ate th e sh ear stress, R. ry .
Th ese va lues are added acco rding to a dimensionless tra nsverse coo rd inate. This is necessar y beeause, in general, the
oppos ing sides of the ice stream a re not mutu a ll y parallel.
For example, over the leng th of the B2 photo bl ock (27 km ),
the width of the ice stream decreases by a bo ut 15%. Th e
dimensionlcss coord ina te is defin ed as the dista nce from
the so uthern margin, di vided by th e ice-stream width. Th e
width is the d istance betwee n sites of zero a long-Oow component o f velocity. Th ese sites are determined by visually
ex trap o la ting the meas ured speeds nea r both m argins. The
speed approaches zero asymptoticall y, a nd so the width is
not precisely determ ined . H owever, a m ore sophisticated
procedure to determin e th e local width would not significantl y a lter the res ults presented here. The acc umu lated
transec ts a re presented in Fig ure 3.
Thi s procedure introd uces an error in acc umulated shear
stress because of uncerta inty in separating tra nsverse velocit y
gradients from the a long-fl ow gradient in speed. For the
photo block covering tributa ry B2, th e along-Oow stretching
rate is typicall y about 5 x 10 3 a I (Whilla ns a nd oth ers,
1993), a nd the speed m ay change by as much as 5 m a lover
the width of each band. This constitutes a velocity error fo r
each I km wide band a nd, by itself, leads to a n error in lateral
r' "'ni/mLS and V{1/7 del' reen: R ole qflaleral drag in dynamics if fee St ream B
shearing rate of as much as 0.03 a \ which is additi o na l to th e
measurement errors disc ussed earlier. Th e net uncertain ty in
a single determin ation of shea r stress is then abo ut +0 kPa in
th e lateral m a rgins. For th e entire photo block, th ere are 27
transec ts, so the crror in shca r stress reduces to a bo ut 8 kPa.
An a ltern a tive procedure, th a t wo uld elimin a te b oth th e
error i11lroduced by coll ecti ng a ll veloc ity determ inations in
each band into a single tra nsec t, a nd the need for using a
dimensionlcss tranS\'e rse coo rdina te, is to interpo late the
irregu la rl y sp aced data to reg ul a r g rid nodes. Thi s was done
by Whill ans a nd oth ers (1993) for the B2 photo bl ock. Howe\'er, the tes ted g ridding ro utines p erform rath er poorl y nea r
rhe latera l shear margins where sha rp changes in the alongOow component of speed occur. \ Vhill ans and oth ers (1993)
tri ed to a llc\·iate this probl em by a dding synthetic data just
outboard of th e shear ma rg ins, thereby forcing the g ridded
va lues to become zero in the shear ma rgins. H owever, this
procedure alTects th e gridded result s even in th e shear margins which, fo r the present study, a re potenti a lly th e most
importa nt pa rts of the ice strea m. Therefore, a ttempts at
gridding the d a ta we re aband oned a nd the ac tu a l, irreg ul arl y
spaced \'elocity determinati ons a re used.
RESULTS
Shear stress
Shea r stress is ca lcul ated from th e tra nsec ts ofm eas Llred ice
speed. For eac h o f the three ph oto bl oc ks. the res ults of th e
maj or steps it1\"o h -cd in thi s calcula ti on arc shown in Figure
3. Th e upper p a nels of Fi g ure 3 di spl ay th e m easured
sprrcl s, a nd th e middle p anel s sh ow the calcul a ted shear
strain rate, o r ha lf th e tra nS\'erse g radient in ice sp eed.
Unce rt a int y in th e rate fac to r, B , due to experim enrallimitations (H ooke, 1981) allows for va lu es as much as 25%
different in the lower panels. Fo r this ca lcul ati on, th e middle
va lue for th e r ate fac tor as n:comm ended by H oo ke (1981) is
used, a nd no a ll owa nce is made for so ft ening of th e ice in the
m a rgms.
Comparison with driving stress
To assess wheth er latera l drag is important in eo ntrollinoth e moti on o f th e ice stream, th e g radient in shear stress is
compa red to rh e clri\·ing stress. Th e dri\'ing stress is calcul ated using th e fa mili a r fo rmul a invoking surface slope
a nd ice thi ckn ess,
Td.r
=
oh
- pgH " , '
u .1:
(5)
where h represe nts the ele\'ati o n of th e ice surface, a nd p th e
d ensit y of ice. Th e surface slo pe is th e a\'Crage ovc r th e full
length a nd width or eac h ph oto bloc k. If latera l drag pro\·ides a ll res ista nce to Oow, fo rce b a la nce requires
(6)
o r, using Equ a ti ons (2) a nd (5) a nd ta king th e ice thickn ess,
H , co nsta nt,
oh
OR.r.ll)
( -oy- = - pg -ox: .
(7)
Thi s theoreti ca l g radient is indicated by the so lid lin es in
the lower pa nel s of Figure 3. Tt is th e physicall y mos t extreme gradient in shea r stress that co uld occ ur. A la rge r
tra nS\'erse g ra di ent in R.ry wo uld indi ca te latera l resista nce
to fl ow exceeding the driving stress, which would be p oss iblc
o nl y under unusua l circumstances.
Th e values of the dri\'ing stress cl eri\'e from elevati o ns
a nd thi cknesses g ive n in Retzlaff a nd o thers (1993). Th e ele\'ati o ns are ultim a tel y ti ed to good-qu a lity surveys by Tra nsit sta tellite trac king a nd have a sta nd a rd error of 4-9 m.
Th e es tim ated sta nd a rd error in thi ckness is 30- 60 m. These
va lu es lcad to a calcul a ti on unce rta inty oflcss th an I kPa in
dri v ing stress.
The poims in th e lower panels o f Fi g ure 3 approx im a tel y
foll ow th e solid lin e b ased on thi s driving stress. Scatter is
g r eatest near th e cente r line, wh erc calcul ati on unce r ta inties a rc largest. Nea r er the margins, p oints tend to be m ore
dilTerent from ze ro tha n th e solid li n e, p erhaps due to ice
so ftening. At the extre me left and ri g ht limits, the calcul ati o ns show a trend of \'a lues decrea ing towa rd zero. Thi s is
a n elTect of increased b asal drag at th e o utboa rd edges o f the
ice stream. Neglecting va lues from th e m a rgins, the p a ttern
of ca lcul ated va lues o f R.rfl within th e ice stream acco rds
with the solid lin e. Thi s agreeme nt indicates that la tera l
drag is sufficient to oppose most or a ll o f th e dri ving stress.
Or, sta ted differentl y, res istance to now associated with la ter a l drag balances m os t of the d riving stress. By implicati o n,
b asa l drag must be nea r zero.
M"ost of the seaLte r a pparent in Fi g ure 3 can be a ttributed to loca l Ouetu a ti o ns in ice speed . Va riations in R J'!J a re
la rgest nea r th e ce nter of th e ice tream where th e sh ea r
stra in rate approach es ze ro. Because o f the non-linearit y of
th e n ow law, sma ll Ouct uations in \'elocity a nd shea rin g rate
le a clto large r Ouctu a ti o ns in calcul a ted shear stress he re.
Softening of the margins
Nea r so me of the la te ra l margins, th e calcul ated points in
th e lowe r panels o f Fig ure 3 tend to b e m ore different fi-om
zero th a n th e solid , th eo retical lin e. That is, the points te nd
to c url away from thi s lin e. The mos t straig htforwa rd interpre ta ti on is th at th e ice may be soft er a t those sites tha n supp osed in the ca lcul a ti o n. Strain heating associated w ith
inte n se lateral shea r ra ises th e tempe ra ture of th e m a rg in a l
ice, m a king thi s ice so ft er. In additi o n, t he strength of th e ice
in th e m a rgins may b e a ltered if a p ro no unced fabric d evelops. Fo r exa mple, if th e c axes a rc a l ig n ed perpendi cu la r to
th e shea r margin, the ice would b e a b o ut 40% soft er th a n
iso tro pic ice (Va n d e r Vee n and Whill a n s, 1994). An a lte rn ati\'e a nd bas ica ll y equiva lem interpre ta ti o n is th at a sm a ll er
effec li ve thickn ess sh o uld be used in the m argins, because
dee p ice is wa rmer and hence wea ke r th a n near th e ieestream center lin e a nd less of the thi ckn ess supports la rge
stresses. Softening o r reduced effecti\'e thickness co uld be
inco rpo rated in th e ca lcul ati ons by usi ng a smaller ra te facto r w here appropri a te. Thi s would reduce th e calcula ted
va lues o f the shear stress, R.1·fI, and bring the points in Fi g ure
3 cl oser to the solid lin e. rIo elimin ate the ex treme ca lcul a ted
va l ues of R.ry , a 20% so f't ening or effecti\'e thickness reducti o n in the shea r m a rg i ns would be suffic ient.
An independent estim ate of th e soft ening in the m a rg in
is no t avail able. Th e a m ount of softeni ng of the ma rg in a l ice
rel a ti ve to the ice in th e ce nter of th e ice stream depend s o n
the te mperature o f th e ice before entering the shear m a rg in s, o n possibl e cha nges in fabric, a nd p erhaps oth er factors. Much of thi s unce rta inty may be a\'oided by omitting
m a rg in a l ice from th e calcul ati on o f fo rce budget, a nd con235
J ournal cifGlacio/()gy
sidering onl y th e central pa rt o f the ice stream. It is done
both ways in the next subsection .
few kPa, with an uppe r limit of abo ut 4 kPa. For tributa r y
Bl, the upper estim ate of basa l drag is abo ut twice thi s va lue
(7 kPa ).
L a r ge-scal e force budget
A nother way to disc uss the res ul ts shown in Fi g u rc 3 is to
consider the bala nce of forces ac ting not on a n infinitesim a ll y na rrow sli ce of ice stream as in th e lower p a nel o f Figure 2, but on t hc full , or nearly full , width. T his is do ne by
integrating Eq ua ti on (2) over th e w idth, 2W. This g i,'Cs the
m ean resista nce from late ra l drag ac ti ng on the ice stream,
p,
1nl
_ J R,!!(W) - R .ry( -W)
- l:
2W
.
(8)
where y = ± W represe nts the p ositi ons of the la ter a l margins. If th e shea r m a rgins a re to be omitted, W represents a
sm a ll er dista nce from the dyn a mi c center line. Th e shear
stress is approx im atel y equ al ( but of opposite sig n ) a t both
m a rgins (Fi g. 3). D enoting th e va lue of IR.t!!1 at th e outer
limits of the calc ul ation by Ts, E q ua ti on (8) becom es
(9)
It may be noted that thi s expression for the ne t late ral
drag ac ting on th e ice strea m d oes nor im·oh-e a n y ass umpti on about the trans,'C rse ,·ari a ti o n in shear stress.
Th e wid rh-a,·e raged force-ba lance eq uati on can be written as
(10)
in which gra di ents in longitu d in a l stress a re neglec ted
because they are unimportal1l if sp atia l a,"Crages are considered (Whill a ns a nd Van der Vee n, 1993a; on th e local scale
longitudin a l stresses may be impor tant, as shown by Whillans a nd Va n de r Vee n (1993 b )). T hc overba r d enotes the
ar eally ave raged \"a lue for th e pho to bl ock. Res ults for the
th ree photo bloc ks a re give n in Ta bl e I.
According to these calc ul a ti o ns, late ra l d rag supports
m ost or a ll of th e dri,· ing stress in a ll th ree area s studied.
Fo r the region near the UpB ca mp (tri buta ry B2) a nd the
4·0 block farther dow n-glacier, inferred basal drag is onl y a
COMPARISON WITH OTHER STUDIES
Th e region near th e DNB camp, in th e vicinity of th e
g r o unding zone whe re Ice Stream B t: lllers th e Ross Ice
Shelf, has been studi ed by Bindsch a dle r a nd others (1987).
T he very small dri ving stress (2.58 ± 0.20 kPa) is ba la nced
prim arily by resista nce from later a l drag (2.14 ± 0.41 kPa ),
with a negligible cOl1lributi on fro m basal drag
(0.44 ± 0.46 kPa ). Gra diel1ls in lo ng itud inal stress a re unimportant in thi s r egio n. These resu lts a re simi la r to t hose
obta ined here.
E chel meyer a nd o thers (1994) m eas ured a tra nsect of
, ·eloc ity from th e ce nter of the ice stream near the UpB
camp, so uthwa rd thro ugh the shear m a rgin and on to th e
slow-m oving interstream ridge. Fo ll owing basically th e
sam e procedure as in Figure 2, the shear stress at the m a rg in s is calcul ated to be 170 kPa (simi la r to ,·a lues obta in ed
h ere ). This ,·a lue suggests that th e m a rg ins prO\·ide m ost or
a ll of the res istive fo rce on the whole sec ti on of the ice stream
con sidered (3.7 x lOBN per meter of ice-stream leng th in
t he fl ow direction, compa red to a to ta l g r avitationa l driving
8
fo rce of +.0 x 10 -:\I ). Howe ,·e l~ furth e r a na lysis a nd nl.odclling of the ,·e!ocit y profi le along the t ra nsec t leads Echelmeyer a nd others (1994) to conelude th a t resista nce to flow
must be partiti o ned equally be twee n latera l a nd b asal
d rags. Thi s second, co ntradictory, interp re tation a rises from
a n ass umption tha t local OUClu ations in ice vel oc ity a re due
to la rge-scale forces a nd not loca l strength a nd dr ivingst ress ,'ariations (as d o ne here).
E chelmc ye r a nd o thers (1994) also co nducted numerical
m o d elling. In a first a Llempt, the th eore tical velocity profil e
ob tained by ass uming zero basal d,-ag a nd constant ice stiffness agrees poorl y w ith the meas ured p rofile. To improve
ag r eement , Echelm cyer a nd others (1994) apply Eq uati o n
(3) to calc ul ate latera l drag along th e tra nsect. Ignoring lat-
Table 1. Force-budget ten nsJor the three jJ/wto blocks on Ice Slream B. T he two values I.IsedJor the T([lejacto1" in the non I
I
linemflow lawforglacier ice correspond la the average value cif540 kPa a ", and to the lower limit cif400 k Pa a ] ajJjJlicable to the temperature prcifile measured lIeaT UpB using the lemjJeratllre dependence cif the rateJactor as given in Hooke
(1981). Compared to the uncertainty in rate jactol; other sources cif error are negligibly small. T he center /Jo1"tioll cif each
photo block is dejinee! as the central 60% ojthe ice stream, extendingfrom s = 0. 2 to s = 0.8, wilh [; the dimensionless
transverse coordinate
B2
II' hole
Ccntcr
Wh ole
CCIllCl"
Wh o lc
CCnlcr
12
1000
16
12
1000
10
21
1.~00
14
21
1500
8.5
9
925
2+
9
925
15
225
165
120
90
200
150
110
80
250
185
120
90
1410
12
9
21
16
19
14
10
j
540k Paa j
400 kPa a l
2
2
0
3
0
.')
2
7
2
+
Td,· ( kP"
fI (m
l!' (k ill
T,
-10 blod,
81
(kPa )
540kPaa
400k Paa
H r,/lI" (kPa
540 kP" a
400 kPa a
I
'
j
',
5
1i, ( k Pa
236
2
+
r I'hillans and Van del' Tleen: R oLe ciflalera! drag in c£ynamics if Ice Slream B
eral flu ctu a tions in driving stress, th ey find th a t ca lcul ated
values of basal drag a re negati\·e. To eliminate these ul1lena ble res ults, large transverse \'a ri a ti ons in the ra te fa ctor arc
needed . Ho wever, short-distance flu ctuati ons in th e dri\'ing
stress ca n reach more tha n six times the a real ave rage a nd
should be included wh en aLLempting to model loca l velocity
variati ons a lon g a transect.
Another stud y th at addresses the partiti oning of flow resista nce on \ Vest Al1la rcti c ice streams is that of M a cAyeal
a nd oth ers (1995) for Tce Stream E. Th at stud y indi ca tes that
resista nce from la teral drag, and p erhaps gradients in longitudina l stress, a re important in the O\'era ll budge t o f forces,
with la rge a reas of th e ice stream s experi encing a lm os t zero
basal drag. It is co neluded that the a rea-a\'Craged b asal drag
supports a bo ut 29 % of the drivin g stress of Ice Stream E.
Thi s is compa ra ble to th e upper estimates of b asa l drag
g iven in Tabl e 1. ?vlos t of th e caleul ated basal res ista nce on
Ice Stream E is concemrated in small-scale sticky spots.
There is no e\·ide nce for stieky spots on Ice Strea m B (\ Vhilla ns a nd Va n d er Vee n, 1993a, b; \tVhill ans a nd othe rs, 1993),
but, as noted by M ac Ayeal and oth ers (1995), the a reas on
Ice Stream B co nsid ered in these sLUdies a re compa rable in
size to so me of the sticky-spot-free zones on Ice Stream E.
Furth ermore, o n m ost of Ice Stream E, the dri ving stress is
a bout 50 kPa, which is about fo ur times th e areally averaged
driving stress on Ice Stream B. Thi s indicates a m~ o r diflcrence between the two ice stream s, Ice Stream E being
steeper a nd thi cker, a nd so \'a ri o us resista nces to moti on
must be la rge r.
CONCLUSIONS
L a teral drag is th e prim a ry co ntrol on th e fluw of l ee
Strea m B, with a lm ost ze ro c011lribution from basal drag.
This findin g ag rees with res ults fro m seismic studi es co nducted in th e v ic inity of th e UpB ca mp suggesting that, at
leas t locall y, the b ed under Ice Strea m B consists o f a weak,
water-sa turated laye r of till (Bl a nkenship and others, 1986),
It also ag rees with streng th tes ts p erformed on samples retrieved from th e bed in the sa m e vicinit y th at indicate a
\'ery small yield strength, abo ut 2 kPa (Ka mb, 1991), The
prese nt wo rk shows th at th ese properti es a re no t restricted
to the \'icinit y of the UpB camp, bUl appl y to la rge p a rts of
the ice stream. The ice stream must be mainl y und erl a in by
a soft, weak laye r of ponded wa ter o r weak debri s.
The maj or limitations tu a ny stud y addressing th e budget of fo rces a rc uncerta inti es assoc iated with the \'a lue of
the rate fac tor, E , a nd, perhaps, the form of th e constitutive
rel ati on. In thi s stud y, as well as in those referred to a bm'e,
the ice is ta ken to defo rm according to Gl en's flow law. M ore
importa nt is th e a ppropri ate \'a lue to be used for th e rate
factor in thi s flow law. According to Hooke (1981), th e unce rta inty in B m ay be as large as 25%. H owever, eve n in the
ex treme, with the lower limit of the rate fac tor used , a nd
thr potellli all y so ft m a rgins exc lud ed, lateral drag provides
55% (40 bl ock) to 75% (tributa r y B2) of the fl ow res ista nce.
Th e model for ice-stream mec ha nics th at follows from
this study i · ve r y simple, but ra th er different from m odel s
a dvanced earli er. Prior models emphas ized fri cti on em a na ting from th e bed a nd disc ussed the importance of' \ 'isc ous
di ssipa tion in mobile drift, the thickn ess of th e subg lacia l
wa te r layer, a nd th e r o le of drain age conduits. \ Vhilc th ese
processes arc very impo rta lll to th e questi on of wea kening
of the bed, once the ice stream has sta rted, basal processes
d o not seem lo co ntro l the speed of th e ice stream. In stead ,
co ntrol is from th e sides where ra th e r la rge (up to a b o ut
200 kPa ) shea r stresses op erate,
ACKNOWLEDGEMENTS
\Ve th a nk R. Bindsc h a dlcr and a n a no nymous referee [o r
comments. Thi s r esearch was supporl.ed by th e U.S.
Na tio na l Science Fo und ati on (gra nt No. OPP- 9316509). Thi s
is Byrd Pola r Resea rch C enter contributi on No. 1031.
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237