.1n/lal.r oJ GlaciologJ' 23 1996 (' Internatio na l G laciological Society

NUDlerical Dlodelling of a fast-flowing outlet glacier: experiDlents with different basal conditions

FR:-\;'-JK P ,\TTYi\ De/Jar/lllen / of G eograjJ/~) I . "rije ['"il'eni/eil Brum/, B-1050 Bumel, BelgiulII

A BSTRACT, R ecent o iJse J'\'ati o ns in Shirase Dra in age Basin, Enderb\' L a nd, Anta rcti ca, sholl' tha t the ice sheet is thinning a t the consid(: ra ble ra te of 0,5 (,0 m ai, S urface \'clocities in the stream a rea read; m ore tha n 2000 m ai, making S hirasc G lacier one of the fas test-fl o\l'ing glaciers in East Anta rcti ca, .\ numeri cal im'esti ga ti on of the pITSe nt stress fi eld in S hirase G lacier sholl's the existence of a large tra nsition zone 200 km in leng th w here bOlh shea ring a nd stretching a rc of equal im porta nce, fo ll owed b y a slI Tam zone 0(' a pproxima tely 50 km , \I'here stretchill g is the m ajor deform a ti o n process, In o rder to imprO\ 'e insig ht into the preselll tra nsient beha\'iour of the ice-sheet sys tem , a t\\'o-dimensiona ltime-dependent fl o\l'line model has been d e\'e loped, taking into account the ice-stream m echa ni cs, Bo th bedrock adj ustment a n cl ice tempera ture a re ta ken into ac('oulll a nd the templ'I'a lU re field is full y coupled to the ice-shcet \'C locit)' fiel d, Experiments were carried o ut \\'ith dilTerent basal m o ti o n conditio ns in order to understa nd their influen ce o n the cl\'na mic beha\'io ur of th e ice sheet a nd the stream a rea in pa rticul a r. R es ults revcaled' that \I,hen basa l mo ti o n becom es the d omina nt d eforma ti o n p rocess, (partia l) disintegra ti o n of the ice sheet is counteracted by cold er basal-ice tem pera tures due to hig her ad\'Cc ti on ra tes , This giyes rise to a cycli c behayi o ur in ice-sheet res ponse a nd la rge cha nges in local imba la nce \'a lues,

LIST OF SYMBOLS T' T rela ti\'C to p ress ure m elting (K v, H ori zonta l \'elocity (m a I) a Arrhenius consta nt (Pa lIa I) Basal-sliding \'C locit)' ( 111 a I )

A l ee fl ow-l a H ' pa rame ter (Pa 11 m 2" a I ) Vertical m ean \'eloeity (m a I) A, Sliding-l a w para m eter W \ ' crtical \'cloci t y (m a I ) b FI O\\'-ba nd \\' id th (m ) ,1' H o ri zonta l coordina te (m ) cl' S pecifi c heat capacity U kg 1 K I ) Z \'ertieal coordina tc (m ) 2 D DilTusio n coeffi cien t (m a I) G eom etric consta nt (;:::02 ,0 ) g G ra \'ita ti o n consta nt (9,8 Im s 2) W a ter-film d epth (111 ) G Geotherma l heat flu x (-54 ,6 m\\' 2) C ritical pa rticle size ( 111 ) h Bedrock ele\'a ti on a bo\'e present sea Ic\'(' 1 (m ) Kro neckcr d elta H Ice thickn ess (m ) H ori zonta l g radient a t d epth (

k i Therma l cond uctiyi t)' of ice (j m 1 K 1 a I ) Eflccti\'C shear-stra in ra te (a I) L Spec ifi c la tent heat of fu sion (3,35 x 10'; ] 111 1 K 1 a I) Shea r-stra in rate (a I) m Flo\\'-enha ncem ent factor G/ ki (geothe rl11 a l hea t) (Km I ) /lJ S urface m ass ba la nce (m a 1 ice equi\'a lent ) Bo unda ry n due (Q (surface) or I (bo llo m ) N ElTec ti \'e norm a l pressure (Pa ) Lithos ta ti c stress compo nent (Pa ) 11 Glen fl O\\'-l aw exponent (3) Viscosit\, of\l'a ler ( 1. 8 x 10 :l Pas) " 'l p Sliding-l a \\' coeffi cient (0 3) Ice d enSity (9 10kg m ') Pg Press ure g rad ient (P a 111 I) Sea-wa ter d ensit y ( 1028 kg 111 3) q Sliding-l a w coeffi cient (0 2) K orma l stress a t the base (P a ) Q Acti n llio n energy fo r creep U m oll ) Ice-shell' back-pressurc (Pa ) Q\\' W a ter flu x (m:ls I ) T cmpera ture-corrcctio n facto r (0 ,0 0, I Cl ) R U ni\'ersal gas consta nt (8,3 14J m ol I K I ) Basal shear stress (Pa ) R i) R es isti w' stress (Pa ) Effec ti \'(:' slress ( Pa ) S Basal-melting ra te (m a 1 ice equi\'a lent ) Dri\'ing stress Pa l t Time (a rycar j) Stress IPa ) T Ice tempera ture (K ) Stress d C'\, ia tor (Pa) 7() Absolute temperature (2 73 ,1 5 K ) V ertical dimensio n\css coordina te

237 Pattyn: Numerical modelling of a fast-flowing outlet gLacier

INTRODUCTION

The d ynamic behaviour of a large polar ice sheet is well understood; grounded ice flow is basically governed by shearing close to th e bed. In its most simplisti c form, the basal shear stress equals ice density times gravity times overburden depth times surface slope, whi ch Hutter (1993) defined as the glaciologist's first commandment. H owever, likc a ny physical sys tem, a n ice sheet is not onl y controlled by its internal d ynamics but a lso by its bounda ry conditions, which are less well understood. The environmental processes such as surfacc tempera ture a nd mass balance act as the upper bounda ry a nd a re in ma ny ice-shee t models regarded as the ex ternal sys tem forcing. The present environmenta l conditions th ereby se rve as a reference. The two remaining boundaries, on which we will further concentra te in this pa per, are th e basal motion as a lower and the margin d yna mics as th e outer boundary condition. The basal vclocity depends on the interna l ice dynamics, the bedrock conditions and the prese nce of water a t the ice-shee t base. Basa l velocity will thus play a role when the tempera ture of the basal ice is a t the melting point and will gain more importance cl ose to the ma rgin where it interacts with the ma rgin ice d ynamics . \Vhen the ice-sheet ma rgin termin a tes at th e sea in th e form of a flo a ting ice shelf, a tra nsition zone can be d efin ed where the ice-shee t dynami cs gradua ll y e\'olve towards ice-shelf d ynamics. E labora te studi es on this Fig. 1. Location mal) of Shirase Drainage Basin . The grounding zone have revealed th a t for small basal mo ti on dashed line represen ts the celltral.J7owline, as Inimm)' in/)lIt the width of the transition zone is of the same order of Jar the model/ing experiments. The.J7owLille starts at Dome magnitude as the ice thickness, so tha t the grounding zone F, folLo ws more or less t/ie 40° E meridian towards is reduced to a grounding line a nd the shallow-ice Llil:;,ow-Holmbukta , and ends at Sizirase GLacier in a approximation still holds, provided that there is no floating ice tongue. passive grounding of th e ice shelf. A full deri vati on of the Stokes ba la nce equa ti ons is not necessary; it suffices to calculate the longitudina l stress deviator along the grounding line and prescribe a longitudinal stress near Dome F (77 °22' S, 39°37' E ) to LLitzow-Holmbukta. gradient (Van der Veen, 1987). Grounding-lin e mO\'e­ Shirase Glacier is one of the fastes t-flowing Antarctic ment is thus only governed by ice-sheet mecha ni cs glaciers, with " elocities up to 2700 m a- I in the ice-stream (Hindma rsh, 1993). This situa ti on is the commones t region (Fujii, 1981 ). The ice flow in the central drainage outer boundary a t the edge of the East Anta rcti c ice shee t basin is furthermore characteri zed by large submergence where a steep surface slope, cha racterizing grounded ice, \'eloci ti es (Naruse, 1979). Consecutiye measurements in a bruptly changes to a sma ll surface gradient where the ice this area (N aruse, 1979; Nishio a nd others, 1989; T oh and begins floating. Shibuya, 1992) have pointed to a large thinning rate However, la rge amoun ts of An ta rctic ice are not ra nging from 0.5 to 2.0 m a I in the region where the drained along th ese ma rgins but through fas t outlet surface elevation is lower than 2800 m. K ameda and glaciers or ice streams, which a re beli eved to be more others (1990) concluded from a study of the tota l gas sensiti\'e to environmenta l changes. These ice streams a re contents of ice cores in this a rea that the thinning is a characterized by la rge basal \'elocities, rela tivel y small ra th er recent phenomenon and that the ice-sheet surface surface gradients and a low basal traction. The internal decreased by a bout 350 m during the la:t 2000 years. ice dynamics involve not only shearing but also long­ In a pre\'ious paper, Pattyn and Declei r ( 1995 ) itudinal stretching, rhus viola ting the first commandment. simulated the d yna mical beha\'iour of the Sh irase In \ Ves t Anta rctica, where ice streams a re a common Dra inage Bas in during the course of the last glacial­ drainage feature, they a re able to d es tabilize th e ice shee t. interglacia l cycle. Except for th e ice-stream area, th e The East Antarctic ice shee t is a lso in some pl aces model was able to desc ribe the main features of the drained by la rge ice streams, such as Lambert Glacier, dra inage basin. The present paper fo cuses on the jutulstraumen, Shirase G lacier or Slessor Glacier. In this processes in the ice-stream area and near th e basal paper we will fo cus on Shirase Glacier (70° S, 39° E ) in boundary, which we think a re important in controlling Enderby Land (Fig. I ). This important continental ice th e enha nced thinning of the ice sheet. First, the present stream drains a bout 90% of the total ice disc harge of the force bala nce of the ice-stream a rea is calcula ted Shirase Drainage Basin covering approximately 200000 progressively from the surface of the preselll ice sheet km 2 This S-sha ped basin extends from the polar plateau (as displayed in Figure 2a; using the observed surface

238 Pat£JII1: },,'lImerica/ modelling of a Jasl-Jlowing ollllel glacier

4000 a distribution, a nd assuming a consta nt ice d ensity, the 3500 continuity eq ua tion is used to form a n evolution equa ti on 3000 for the ice thickn ess H where 2500 oH 1 D 2000 -=---(UHb )+M-S (1) 1500 ot box 1000 with U the depth-averaged horizo nta l vcl ocity. H th e ic e 500 thi ckn ess. !I f the surf"ace mass balance and S the melting rate at th e base of the ice sheet. Divergence a nd 1000 -500 cO I1\ "C rgence of the ice Oow a rc ta ken into accou nt in

-1000 th e calcula ti on of the varia tion of" the ice flux alo ng th e Distance from ice divde [km] fl o\\'lin e throug h the parameter b, indicating the ,,'idth of th e fl o\l' band and ta ken perpendicula r to the fl owline. Bounda ry conditions to the ice-sheet sys tem arc zero 2500 b surface g radients a t the ice di vid e and a Ooating ict' shelf 2000 a t the seaward margin. For the d eri\'ation of the t"'o-dimensiona l stress fi eld 1500 in the ice-shee t sys tem, the method described by V a n der

1000 " ee n and \\'hill a ns ( 1989 ) a nd \ 'an der \ 'een ( 1989 ) \\'as followed as th e m a in reasoning. For numerical eOIl\'en­ 500 ience, a dimensio nless ,'Crtical coordinate is introduced to account for ice-thickn ess "ariations a long the OO\l'line, 100 200 300 400 \,"hich is d efin ed as ( = (H + h - z)/ H, so th a t ( = 0 at -500 th e upper surrace and (= 1 a t the base o r the ice shee t. -1000 The major assumption made in th e model formulation is Distance [km] to consid er plane strain, so that all stress compo nents Fig. 2 (.'eollletric dwmcteri.ltics oJ the Shiuoe Glacier ilwo h 'ing y arc neglected . In this way, th e Euler jloll'/ine (a) alld the sfea(/.)'-stafe reference ice sheet Jar fhe continuum in t\\'o dimensions is \\'ritten as /IIodelling e.\perimeJ/ls ( b).

(2. 1)

"elocity as a n upper boundary) to th e bottom, according to the method proposed by Va n d er " een a nd \\'hilla ns (1989). This experiment is desc ribed in the secti on (2.2) "Diagnostic model run". Secondly, by inverting this methocl, th e total stress-fi eld derivation is incorporated in " 'here a n ice-sheet sys tem model, where th e calculations start a t DH the bo ttom (basal "c1ocity) a nd progress towards the iee­ ~ =o(H + h)_ r (3) ( ox '> ox . sheet surface, as expla ined by Van (ler Veen (1989 ). In the la tter case, se nsitivity experiments a re carri ed out Foll olI'ing Whilla ns (1987 ) a nd Van d er V een a nd with difTcrent descriptions of the basal motio n in o rder to Whillans ( 1989 ), the full stress components from Equa­ understand the ice-sheet response \I'ith time. Therefore, ti o ns (2) are partitioned into a resistive (Rij ) a nd and a we took a simple ice-sheet conli g uration as given in Fig. lithos ta ti c (A) component, so th a t 2b, in order to reduce local effects imposed by the Shirase Glacier bedrock topogra phy. An S-shaped basin form \I'as Tij (4) kept to take into account the large cO I1\ 'C rgence in the ice­ = Rij + DijA: A = -pgH( stream a rea. with Dij the Kronecker delta, equa lling I \\"hen i = j a nd o o therwise. In trod uci ng the res isti" e-stress componen t into Equation (2. 1) a nd integrating from th e bo unda ry K VELOCITY AND STRESS-FIELD CALCULATION to a genera l depth ( in the ice shee t res ults in a n expression for the resisti,'e-shear stress. The boundary

The numerical ice-sheet system model used in this study is \'a lue /-i, is introduced so that calculations can start fr om a dynamic flowlin e model that predi cts the ice-thickness th e surface ( /-i, = 0 ) as well as from the bottom (K = 1) of distribution along a flowline in space a nd time as well as th e ice sheet. The sta rting conditio n K = 0 is applied in the t\l'o-dimensional Oo\\" regime ("elocity, stra in ra te and the diagnosti c run, (see la ter) \I'hile the starting condition stress field s) and temperature distribution in res ponse to K = 1 is used in the mod elling ex periments. R es isti,'C environmenta l conditions. stresses can then be replaced by shear-strain ra tes throug h A Cartesia n coordina te system (x. z) with the x a xis the constituti" e rela ti on a nd the relation betll'een the rull parallel to the geoid and the z axis pointing verti call y stress tensor a nd the de"iatoric stress . Glen's 00\1' la\\" is upwards (z = 0 at sea level) is delined. Sta rting from th e used as a constitutive relation (Pa terson, 1994). The known bedrock to pog raphy and surface mass-balance res ulting equa ti on then becomes (Van der Veen a nd

239 Pa/~) l n: .1\ "1I III erical modelling of a fast-flml'ing oll llel glacier

Whilla ns, 1989; Va n der Veen, 1989) expla ined in \ 'an der \T een ( 1989) a nd brie fl y hereafter. Once th e two-d imensio na l vel ocity fi eld is calcula ted, ho ri zonta l ice llu xes, whi ch are rela ted to the rate of ice­ thickn ess cha nge thro ug h Equation ( I) a re ro und by integrating the horizonta l \'eloci ty fi'om the bo ttom or the ice sheet to the surface, hence

UH = H t u(()d( + us H (9) + 6.( R zo + :L [l (H R oo d('] (5 .1) In a(H + h) t..:=0 => To = R.,z(O) - R.f.,(O) a = 0 (5 .2) \I· here Us is the basal- sli d ing \'elocit y, whi ch \I'ill he ,I' ah d efi ll C'd la ter. t..:= 1 => To = R,d1) - R,,(l) -a = Tb· (5.3) .L where TeI is the d ri\'ing stress (TeI = -pgHV' (H + h) ). ICE TEMPERATURE AND ISOST A TIC BEDROCK Eq ua ti on (5.1 ) can be integrated nu mericall y from the ADJUSTMENT surface ( K, = 0) or from the bo ttom (t..: = 1) of th e ice sheet ",ith bounda ry conditions (5.2) a nd (5.3), res pec­ T he te m pera ture d istribution in a ll ice sheet is gO\Trned li \·ely. The now-la w pa rameler A is a functio n of both by d ill'usio n a nd ach- ection, a nd is th erefo re tempera ture T * (corrected for th e dependence of th e d ependent no t onl y on bo undary conditions such as m elting point on press ure (T' = 1;) - (8 .7 x lO --I) H )) surface tem pera ture a nd geothermal heat nux but a lso on a nd obeys a n A rrhenius rela ti onship (Paterson, 1994): ice vel ocity. The th ermod yna mi c equa ti on can thus be m 'itten as (Huybrec h ts a nd O erl emans, 1988; Ritz, 1989)

A = 7TWCXP ( -Q ) (6) 2 RT ' aT I-.:. j a T aT at pCpH2 a(2 - u(() ax I( l where a= 1.14 x 10 5 Pa !la I a nd Q= 60 kjmol for 10 _ ~ aT { a( H + h) + u(() a( H + h) T * < 263.15 K , a= 5.74 X 10 Pa!la I and Q= 139 kj H a( at a,1: mol I for T * 2: 263. 15 K. An ex press io n for the \'C rtical rcs isti \'e stress is _( [aH +lI(()aH] -w(() } __l_auT.l z(() obtained by vertical! y in tegrati ng Eq ua ti on (2.2 ) from at a.I: pCp H a( the surface to a depth in th e ice sheet (. After a lgebraic ma nipula ti on simila r to the deri vati on of Equation (5), (10) the vertical resisti ve stress can be wri Uen as:

with T tem pera ture a nd k j a nd cl' the therma l conducti vity R zz (() = U/i" + 6 cA-lj IlEe (l j ll )- I E.rz(() a nd the specifi c heat capacity, res pecti vel y. The heat tra nsfer is thus a res ult of \'e nical d iffusion, hori zontal a nd II + :X [l(HA-'/Ee(I/II)-lEr:(()d('] \"e rtical ad\'ection a nd internal deforma ti on caused by hori zonta l shear-s tra in ra tes. Bounda ry conditions foll ow (7.1) fr om the annua l mean air tempera ture a t the surface a nd the geotherma l heat flu x (en ha nced with heat fr om sli ding) t..: = 0 => (7.2) a t th e base (Ritz, 1987) so th a t

K, = 1 => (7.3) aT] _ H [ H a(H + h) ( ) _ ] (11) [ !Ol - k pg a u 1 'Yg . u( ( = 1 " j X where Eq ua ti ons (7.2 ) a nd (7.3 ) a pply to the bounda ry condilions a t the surface or a l the bottom , res pecli vely. where 'Yg is the geoth er mal hea t entering the ice a t the Slra in ra tes a rc by definiti on rela ted to \"e loc ity gradients, base a nd ta ken as a consta nt fo r the line of experiments such tha t disc ussed here, The basal tempera ture in th e ice sheet is kept a t press ure-melting point whenever it is reached a nd . _ ~_ [aUi aUJ ] _ A (n- l) , the surplus energy is used fo r m elting . This may, in some E,) - !Ol + a - Te T;). . (8) 2 u Xj Xi . cases, lead to the fo rma ti on of a tempera te ice layer between (= 1 a nd (= (nwlt, so th a t the basal-melt ra te S Introducing the scaled \'erti ca l coordina le sys lem a nd is defin ed as a pply ing the incompressibi lity cond iti o n (au/ax+ aw/az = 0 ), a n expressio n can be fo und fo r the norma l S -_ ~HL [(aTor ) _ ( aT or )] + ~L 1 (""'1< aorlL T.I-z(()d( stra in ra te in the hori zonta l direc tio n. Bounda ry condi­ p '> b '> c PI '> ti ons to this sys tem of equa tions a re a stress-free surface (12) a nd known ho ri zonta l a nd \'C rtical ice veloci ti es a t the surface o r a t the basal layer. The whole \Tlocity fi eld can where incl ex b d eno tes the basal tempera ture gradien t as be calcu la ted fr om th e ice-shee t geometry, ta king into given in Equa tion ( 11 ) a nd incl ex c d enotes the basal account the proper bounda ry conditions. The numeri cal tempera ture g radient after correcti o n 1'0 1' press ure melt­ solutio n to the a bove-d escribed equations h as been ing. I n the ice shelL the \'ertical tempera ture profil e is

240 Pa!{ J'II : . \ 'lIm ericallllodellill/!, qf a.fas!-jlowing oll tlet glacier ta ken linea rl y \\'ith the surface tempera ture as the upper bo unda ry condition a nd a consta nt tempera ture of _2 u C a t the bo ttOm o f the ice shelf, :'1elting or refr eezing processes a t the ice- sea-water contac t a re no t ta ken into accoun t. 0,5 Bedroc k adjustment to the ice load is calcula ted as d escri bed by O eri em a ns a nd \' a n der V een ( 1984, cha pter 7), The elasti c properties o f the lithosphere d etermine the ultima te bedrock d epress ion, while the \'iscosity o f the underl ying astenosphere d etermines the relaxati on time, The time-dependent res po nse is mod ell ed Di stan ce from ice divid e (km l b\' a diffusion equa tion for bed rock e1 e\'a ti on, FI~[I, . 3. T u.' o-dilJlellsiol/al II/a/) o.f !/ie ifJec/il'e stress .field ill !/ie dolt'lIs/ream area o.f Shirase GlacifJ'.

NUMERICAL SOLUTION

The m a in pa rt of the numerical sc heme consists of soki ng topogra phic ma ps of the region. S urface \'elociti es arc the ba la nce Equa tion (5) for the shea r-stra in ra te a t known a t se\'eral sites 0 11 l\ [iz uho Plateau a nd a t the genera l d epth (, Therefo re, \'elociti es a t that d epth layer g lacier m a rg in. l\Iore d eta iled reference to the compil­ a nd the \"C locities of the layer underneath or a bO\'e need a ti on of the fl owline d a ta, prese nt mass-ba la nce a nd to be known, Fo r modelling purposes, integra ti on sta rts at surface-tempera ture distribution can be fo und in Pa ttyn the bed rock (K = 1 ), conditioned by a basa l velocity a nd a nd Decl eir (1995 ). T empera tu re in the ice sheet was \\'hich is go\'Crned b y bo th basal sliding mecha nisms. ice­ calcula ted on the basis of a stead y-sta te tempera ture stream a nd ice-shelf fl ow, furthermo re, the basal yelocit y distributio n a nd coupled to the \'clocity fi eld by m eans of is inlluenced by the t\\'o-dimensio na l stress fi eld a nd vice the Arrhenius rela ti onship. \'ersa, so tha t linking of the two mec ha nisms should be Adopting this ice-shee t confi g ura ti o n. it was possible perfo rmed d yna micall y. V a n d er V ee n ( 1989) o ITered a to obta in the two-dimensiona l stress a nd stra in fi eld in the solution fo r the integra tion oCth e fl ow field sta rting [i-om ice sheet, fi 'om the ice divide to the ed ge oCthe floating ice the bed . The numeri cal computa ti on is, however, ra ther shclC simpl y sta rting from the present hori zollla l a nd time-consuming a nd increases when eventua ll y a basal \'C rtical sUI'face \·e locitics. The present surface mass bo unda ry condition is linked to the Il ow field . H O\\T\'er, ba la nce was used as the upper bo unda ry to the \T rtical by choosing good initia l es tima tes to the basal drag a nd \'C locit y. The res ulting e fTccti \'e stress d istributio n (in­ no rma l stress, cO I1\T rgence can be a tta ined rather rapidly, corpora ting bo th shearing a nd stretching ) in the glacier e\'en when basal \'elocit y is influenced b y the basal drag. area (from a dista nce of 600 km from the ice di\·id e d O\\'I1 The lin a l fl o\\'-field d etermina ti on is stopped when the to the gro unding line) is displ ayed in Fig ure 3. In the stress-free surfaee condition is fulfill ed , i. e. Equa tions (5.3) u pstream a rea, the ice fl o\\' is prima ril y gO\'C rncd b y shear a nd (7.3). stresses (d own to 710 km fr om the di\'id e), because o f the The two-dimensional ice-sheet system model is imple­ rela ti\'C ly hi gh surface slopes impos ing a la rge dri\'ing mented on a fl owline which ex tends from the ice di\'id e to stress. The ice-stream a rea itse lf (fi'om 800 km furth er the ed ge o f the continenta l ice shelf. Because of the non­ d ownstream ) is a lmos t so lely dri\'C n by longitudina l linear na ture o f the \'elocity in the ice sheet. the continuity stre tching a nd shear stresses a re a lmos t negli g ible. Equa ti on ( I ) is reformula ted as a non-linear pa rtia l Bet\\'Ce ll these two a rcas both shear stress a nd long­ difTi:-rentia l equa ti on a nd \\Tiuen as itudina l stress a re of the same o rd er of magnitude a nd fo rm some kind of tra nsiti o n zonc. Des pite the poor oH 1 0 [ , O( H + h)] --;:::;-=-()!1 b(x) D(x) 0 + M -S (13) knowl ed ge of the d eta il ed surface a nd bedrock topogra­ u t b .1' u J' X ph y a nd surface \'e1 ociti es in the ice-stream a rea (700- 850 km fr o m the divid e ), Fig ure 3 d epicts the o\Tra ll with D(.r) a diffusion coefft cient which, in the grounded cha rac teri sti cs o f a n ice stream , i. e. a la rgc ho ri zonta l ice shee t, equals the mass flu x (U H ) di\'id ed by the \'C locit )' g radient a nd a \'C ry sma ll vcrtical gradient o f' the surface gradient. A solution to this equa ti o n can be hori zontal \'elocity (ou/ oz ~ 0 ). These features mo re obta in ed by \\Titing Equa ti on ( 13) in the conser\'a tion resemble ice-shelf beha\'iour tha n g rounded ice-shee t form (i\[itchell a nd Grifftths, 1980 ). Fo r the numerical d yna mi cs. computa ti on, a semi-implicit sc heme was a pplied a nd \ 'ertical resisti\'e stresses pl ay a role in those a reas ne\\' ice thickncsses on the Il ex t time step a re fo und by the where the fl o\\' regimc cha nges. i. e. fro m shearing to solution of a tri-diagona l system of equa ti ons. An implicit stre tching a nd \'ice \'('rsa. They a rc of the o rder or I numerical sc heme was ad opted [o r the ice-tempera ture 5 kPa , a lmos t (\\'0 orders of magnitude sma ll er tha n the fi eld , res ulting in a simila r set o r tri-diagona l equa tio ns. effecti ve' stress.

DIAGNOSTIC MODEL RUN BASAL VELOCITY

The ice-sheet geometry o[Shirase Dra inage Basin (surface Fo r mod elling purposes, the rull stress field in an ice-sheet a nd bedrock e1 e\'ati on ) was compiled from J a panese sys tem , which consists of a g ro unded ice sheet. its

2+ 1 Paltyn: Numerical modellillg oJ a Jast-flowing outlet glacier

Table 1: Constants and parametersJor the different basal velocity laws. HAB = ifJective normal jJressure calculatedfiom the height abo ve buoyancy (see text). The numbers between brackets rifer to the equation in the text

Type Author Us law N jJ q v

1 4 Budd and J enssen (l8) 1.5xlO- o 1.0 x l0 ( 1987) (ma I) (m) 2 Budd and J e nssen (17) HAB 5.0 x 109 2 0.1 ( 1987) (Nm la 1) (D C 1) 5 3 Alley a nd others ( 17) (20 ) 0.8 X 10 1.33 1. 8 0.0 (1989) (P a0 47 m a- I) 4 Alley and o thers (l 8) 2 x 10 11 2 1.0 x 10-" ( 1989) (P a 2 m- 1 a- I) (m) 5 V a n d er V een ( 17) H AB 2.0 x 10 7 3 5 1 ( 1987) (N 2 m a- )

grounding zone o r ice-stream a rea and a coupled ice shelf, the ice-overburden press ure. Sliding is thus only expected is d erived starting from a basal bounda ry condition. The where the basal ice is at the m elting point (Paterson, calculation of the two-dimension a l ice-velocity field is 1994) . In polar ice streams, the origin o[the basal wa ter is thus reduced to a one-dimensiona l problem , in whic h the most likely due to basal m elting primarily caused by basal-sliding mechanism in the ice sheet, the ice-stream strain heating. Percola tion of surface m eltwater a nd the Oow m echanism and the ice-shelf Oow need s to be forma tion of intra glacia l c hannels andmoulins might thus determined. Since the b asal bounda ry condition is b e n eglected. In most sliding laws, derived fr o m iterativel y updated with the d e rivation of the two­ laboratory experiments and data fitting fi-om observed dimensional now field , the final solution is a two­ values, basal sliding is a [unction of the shear stress a t the dimensiona l one. base, bed roug hness a nd the effective normal press ure a t The ice shelf is simply ta ken as a freely Ooating ice the base, the la tter equa lling the ice-overburden pressure shelf bounded in a parallel-sided bay, so tha t a ll stresses in minus the wa ter press ure a t the base of the ice sheet. A the y direction can be set to zero. Ve rtical shearing IS genera l sliding law h as the form of (Bindschadler, 1983; omitted, so tha t the force ba la nce equatio n reduces to Budd a nd others, 1984; Budd and J ensse l1 , 1987; V a n d er V een, 1987 ) (14) Us = AsTf:N - q exp [v(T* - To)] (17) In order to calculate the longitudina l devia toric stress [rom this equilibrium, its value a t the seaward ed ge of the where p a nd q a re positive numbers ra nging between 0 ice-sh elf must be known. H e re, the total force of the ice and 3 a nd the exponen ti a l term a correction for the basal must be ba la nced by the sea-water pressure (Thomas, tempe rature, introduced by Buclcl a nd J enssen ( 1987) . 1973), so tha t For g laciers that te rmina te in the sea, subglacia l water pressure at the grounding line may be calculated from the Ootation c riterion, so tha t the effective normal press ure N is proportional to the heig ht of the glacier surface a bove buoyan cy. Tn som e now m od els of the An tarc ti c ice sheet Using this expression to calcul a te the longitudinal stress (Budd a nd o the rs, 1984; Huy brechts, 1990), the effecti ve deviato r a t the edge o[the ice shell', Equa ti o n ( IS) is then press ure is calculated in the sam e fa shion over the whole integrated upstream to yield the longitudinal d eviatoric model doma in and not only a t the grounding line. stress in the w hole ice shelf. Once the vertical m ean H owever, the physical res triction is that the beclrock ho rizonta l velocity a t the g rounding line is known, the should li e below sea level and that the water press ure vertical m ean horizonta l ice-shelf velocity (which equa ls o ri gina tes from sea wa te r infiltrating a t the g rounding line the basal velocity) is calcul ated l'rom the d efinition of the and moving in the upstream directi o n. Since this type o f no rmal stra in rates a nd the constitutive rela tion: p a ra m e teri zation is wiclely used in ice-sheet models, it is adopted for the calcula tion of sliding type 2 a nd 5 in . OU A I n T a ble I (d e noted b y HAB). f.n = -;:;- = T · · (16) v X 1 .r Subglacia l water pressure or the pressure g radient The tempe ra ture-dependent now-Ia \-\." parameter A is unde rneath the ice shee t can be d eriyed fr o m the taken as a consta nt in the whole ice she lf, hased o n the existence oCsubglacial cha nnels a nd/or cavities . Bindsc ha­ m ean ice-shelf tempe ra ture. dler (1983 ) a pplied the so-called R oth li sbe rger-c ha nnels Basal sli d ing of the g ro unded ice sheet can orig ina te (R oth li sberger, 1972) fo r the d eriya tion of effecti\"C fi 'om sliding of the ice over its bed a nd d efo rma tion of the norma l press ure in Antarctic ice stream s. The theory bed itsclf, the latter if the bed consists of sediments and ph ys ical background of the fo rma tion of c ha nnels saturated with water a t a press ure tha t closely m a tc hes and caviti es is very complicated a nd is hi g hl y d epe ndent

242 Paltyll: ,lI,"wnericaL modeLLing oJ a Jast-flowing outlet gLacier on the d etailed bedrock topog raphy (bumps). Since basal the top, Starring from the initia l bedrock condition, we conditions at Shirase Glacier a re ha rdly known, the use of developed a n ice sheet bounded by the surface mass such phys ical principles rem a ins hig hl y specula ti\·e. balance a nd tempera ture confi g ura tion as o btained from A m ore stra ig htforward treatmen t o f the basal water d a ta of Shirase Dra inage Basin (Pa ttyn a nd D ecl eir, fl ow is to consider a water film underneath the ice sheet. 1995). T a king this simple bedrock topography, local According to W eertman and Birchfi eld ( 1982), the sliding extra neous eITects interacting with the basal motion laws velocity is d ependent on the depth of the water layer 0 a re excluded, Only an S-shaped basin form a nd the upper a nd the cri ti cal particle size Dc so tha t bounda ry (surface m ass ba la nce and temperature) are in accord wi th the presen t observa ti ons. After 40 000 years of calcula ti o n a full -grown stead y-state ice sh eet was obta ined with a bedrock lying approximately 400 m below sea le\'el (due to bed rock adjustment) a nd a (18) temperature fi eld coupled to the velocity fi eld. Basal and velocity was excl uded from this sta ndard run. The 1 12 g ro unding line was situa ted a t a dista nce of 470 km from 6 = ( /LQ w)3 (19) the ice di\'ide and d etermined by the no tation conditio n P g (Fig. 2a) , Using this stead y-state si ru a tion as a reference, where Qw is the water nux per unit width, calculated the m odel ran forward again 20000 years constra ined by thro ug h downstream integra tion of the basal m elting rate one of the fiv e selected basal m o tion laws, According to a nd Pg the press ure gradient (Alley, 1989). Fig ure 4, the proportion of basal m ovem ent in the total In the analysis a bQ\'e, the g lacier m oves over a n ba lance \'elocity is hig hl y d ependelll on the basal motion undeforma ble bed, a nd the water acts as a lubrication type, Basal-sliding types 2, 4 and 5 show a grad ua l layer between the bedrock a nd the ice. H owever, in the increase in the proportion of basal motion in the total presence of sub glacial till, it is very likely tha t, whene\'er velocity toward s the grounding line. These a re the Budd this till layer becom es saturated , the dra inage is a long the a nd V a n der Veen type based on Equation (17 ), with the till a nd/or bed (W a lder a nd Fowler, 1994). When the till eITecti\'e pressure independent ('rom the water production layer is sufTiciently thick, d eformation of the till layer (bed a t the ice-sheet base, and the Alley type for a water film d eformati on ) can occur (Alley, 1989). Combina tion of all (Equation ( 18 )) , These three types are a function of the these processes (cavities in ice and till, su bg lacial aq uifers) basal shea r stress, which increases towards the g rounding o bvio usly makes the b asa l sliding mechanism very line, H owever, the till-deformation law (type 3) a lso complicated. Nevertheless, Alley a nd o thers ( 1989) d epends on the basal shear stress, as can be seen in Table incorporated in a simplified way the bed d eforma ti on. I but, looking closely a t the governing equations for this Neglecting the sh ear streng th of the till a nd setting the type, one notices that the basal shear stress a lso enters the effective pressure (N ) independent of depth, the bed­ eITective press ure (N ) through Equation (20), so tha t the d eforma ti on velocity then takes th e form of the sliding basal motion in fact becom es in versely prop ortiona l to the veloci ty Equation ( 17 ) wi thout the exponen ti a l term. The basa l-shear stress due to the power constants p a nd q ( 1,33 effective press ure (N ) is calcul a ted as (Alley a nd others, a nd 1,8, respecti vely) , The rem a ining types I a nd 3 1989) (sliding Q\'er a water fi lm and till d eforma tion) a re cha racteri zed by a sudden increase in basal velocity N = fJTb (20) where thc basal-i ce tem pera ture reaches the press ure­ 1 + O.lloglo 6 melting point, and the proportion of the basal motion in the tota l \'elocity d ecreases in the downstream directi o n, where the denominator represents the fraction of the bed Bo th experiments create a n ice sheet with a disturba nce in occupied by the water film with depth 0, a nd fJ a the surface-topography profile a t the place where the geom etric constant rela ted to the bed roughness (fJ ~ 2.0 basal-i ce temperature reach cs thc press ure-melting point, in the a na lysis of Alley and o thers ( 1989)) . thus creating a kind of enha nced "block-flow" moyem ent In tota l, we consid ered fi\'e different basal velocity of the do\\'nstream pa rt of the ice sheet. At the cold ! laws, based on three differcnt basal m ovement concepts: tem pcra te intcrface, the ice is "sucked " o ut of the (i) basal sliding related to basal shear stress a nd effecti\'e upstream ice sheet, resulting in a sm a ll er surface gra di ent normal pressure a t the bed, (ii ) sliding due to the presence a nd a non-neglectable impo rLa nce 0(' the longitudinal of a watcr film a nd (iii ) till d eformatio n a t the ice- bed strcss , interface, The coefTi cients for the different basal-motion :\ questi on tha t arises when the ice sheet is subjectcd laws a rc g i\'en in Table 1. to a change in basal boundary conditions is whether o r not the ice mass \I'ill rctricve a nc\\' steady state. The answer is gi\'C n b y Figure 5, displaying thc mean SENSITIVITY EXPERIMENTS g round ed-i ce thickness during the CO LlI"e of 20000 years of integration for the fi\ 'C basal m o ti on experiments. For thc sta ndard model run, wc considcrcd a linC'ar \Vhcne\'er the basalmo\'cmcnt ca nno t be balanced by thc slo ping bcdrock, with a n elevati on of 500 m a.s, l. a t the ice input upstream (a nd surface m ass bala nce), the ice di\'idc, away from it gradua ll y lowering by 150 m per sheet \\'ill continuo usly shrink, g i\'cn that the em 'iron­ 100 km, a nd a horizon ta l grid size spacing of 10 km. In mental condition remains thc sa me or favours thc the \"Crti cal domain, 20 layers wcrc defined with the lower enha nccd basal mO\'Cment (such as a warmer clima te). layer spacing of ~ (= 0,015, g rad ua ll y in creasing to\\ a rds H O\l'c\'cr, the ice-shect temperature plays a n importa nt

243 Pal tyn: N umerical modelling oJ a Jast-jlowing outlet glacier

1.2

--type 1 ,- ,-; - -.-",-- / I "' / / - -- - type2 I : / - -- - type3 / / 0.8 ---. -. type 4 / / / I / - -- type 5 ----(... / :::l

VI 0.6 :::l ~- -! -: / / ./ I 0.4 I I / I I / I / I I : 0.2 , ; : ,. I .. , !:' ..- ./ , ,: . . ,.. .. - . . - . , - -- ~ .. - .. _. - - - 0 I 0 50 100 150 200 250 300 350 400 450 500 Distance from divide [km]

Fig. 4. Ralio of the basaL velocity 10 the lolal balance veloci!)' ill thf ice sheel acco rding 10 theIiz'e £vjJes oI basal veLoeil:), laws after 20 000years oIilltegralioll ( see text Ior eX/llanaliou).

role in controll ing the basal ice-shee t d yna mics. Introdu­ produc ti on of basal meltwater caused by strain heating. cing basal movement causes the ice sheet to movc more The a mounr of basal meltwa ter in(]uences d irec tly the ra pi d ly, hence increasing ho ri zol1la l a ncl vertical a cl vec­ basal-wa ter Jlu x a nd the sli d ing \·c1 oc ity. tion ra tes . Colcl er tempera tures reach thc base of the icc sheet, thus decreasing the tota l surface subjected to melting. Basal velociti es d ecrease, sta bilizing the ice-sheet DISCUSSION m o ti on . The whole process gives ri se to a cyclic behaviour; the slower ice sheet will tencl to grow, Payne ( 1995 ) showed the existence of cycli c behaviour in a cl veeti on ra tes cl ecl-ease, bottom melting increases, hence ice sheets with a thermomec ha ni cal ice-sheet modcl using res ulting in la rger basal velocities . This can be seen in a basal sli d ing mecha nism simi la r to the Budd sliding type Figure 6, where both mean ice th ickn ess a nd tempera ture 2 in this pa per. H e found tha t the cou pling of the ice-fl ow cha nge during the course of 20000 years a re d isplayed , fi eld a nd the tempera ture evolution gives rise to limit according to th e basal-sliding type I . In particul a r, the cycles in the basal therma l regime of the ice sh eet. These sliclin g la ws which are clirectly dependent on the cha nging cycl es a rc caused by the sw itching on a nd off of sliding as basal cha racteri sti cs such as the basal-melting ra te wi ll basal ice reaches the press ure-melting point. R es ults or show this a ppa rent behaviour. Furthermore, when basal­ our work do not show this cycli c behaviour for a sliding veloci t y g rad ien ts increase, stretc hi ng becom es the la w in whi ch the eITec t of basal wa ter is not explicitly d omina nt process, so tha t shearing in the basal layers included . This is proba bly due to a lower tuning \'a lue in becomes less effecti ve (uujuz ~ 0), thus reducing the our ex periments compa red to th e sliding law or Payne

2400

2300 ,­ typ \ - .. - typej e1 2 - -- - type3 1 I 2200 ., "...... ' ...... type 4 1 III Cl) c: --- type5 1 u '" 2100 E £i .!:!'" ..c: Cl) 2000 ~

1900 '. ~ i .;

1800 0 4000 8000 12000 16000 20000 Time la) Fig. 5. E volution of the mean grounded ice Ihickness when a/JjJ!J'illg the Ii/le dlfferfll t basal /'e/oci{J' laws.

244 Pal [l'II: . \ '/1111 erica I lIIodellillg 0/ a ./asl:jloll'illg ou llel glacier

o

--mea n ice thickness ,0,5 , , , , , . mean basal temperature -1 LJ 0

-1,5 ~ .'l ~ Q) -2 Q. E ~ ;;; en -2.5 co .Q '. c: co -3 Q) ::!:

-3.5

1-- ,4 8000 12000 16000 20000 Time tal Fig. 6, El'olu lioll of Ihe //lUll! groullded ice lliirkllfss alld /Ilea!! basal lelll/Jeralure u'lien a/)/J~J ' il/g brlJal .lfidillg [l'/Je J,

( 1995 ), By compa ring different basal-motion la ws, it ice sheets is \'ery complicated a nd its effect can o nl y be se('ms that those \\'hi ch d epend on basal \\'

245 Pattyn : N umerical modelling oJ a J ast flowing outlet gla cier

Budd, W. F. and D. J enssen. 1987. ~um e ri ca l modelling of the large­ Pattyn, F. a nd H . Decl eir. 1995 . Numeri ca l simula ti on of Shirase scale basal water flu x under the \Ves t Antarctic ice shee t. 111 Van der Glacier, Enderb), Land , Eas t Queen 1\1a ud La nd, Anta rcti ca. Veen. C.J. and .1 . O eri emans, eds. DJ>!"I/llics cif Ihe lI'esl Anlarc/ic ice Proceedings cif Ihe N IP!? SymposiunI on Polar lvl efeo r%g)' and Claciology sheel. Dordrec ht, etc., Kluwer Academic Publishers, 293 320. 9, 87- 109. Budd, W. F., D. J ensse n a nd I. N . Smith. 1984. A three-dimensional Payne, A.J . 1995. Limit cycles in th e basa l th ermal regime of' ice shee ts. time-dependent model of the Wes t Antarcti c ice sheet. Ann. Clacial., J. G'eoplrys . Res., 100(B3), 4249- 4263. 5, 29-36. Ritz, C. 1987. Time dependent boundary co nditions for calcula tion of fuj ii , Y. 198 1. Aerophotographic interpretati on of surface fea tures and tempera ture Geld s in ice sheets. i n/ernationlll Associa lion oJ I-I,)'drological es tima tion of ice disc harge a t the outlet of th e Shirase dra in age basin , Sciences Publication 170 (Symposium a t Vancouve r 1987 - p/~]lsical Anta rctica. An/arc/. Rec . 72 , 1- 1.'i . Basis oJ Ice Slteel M odelling ), 207-2 16. Hind marsh, R. C. A. 1993. Qua li tati\'c dynami cs of mari ne ice sheets. 111 Rilz, C. 1989. Inlerpretati on of' the tempera ture profil e measured a t

Pciti er, \ V. R ., er!. lee ill the climate ~)lstem. Berlin , etc. 1 Spri ngcr- \ ' eri ag, Vos lok, East Anta rcti ca. Ann. Glacial., 12, 138- 144. 67 99. (.'\' ATO AS] Series I : G lobal Enviro nmenta l Change 12. ) R ii lhlisbergc r, H. 1972. Wa ter press ure in intra- a nd subglacial Hlmer, K . 1993. Thermo-mec hani ca ll y coupled ice-shee t response channrls. J. Claciol., 11 (62 ), 177- 203. cold , poly thermal, temperate. J. Claciol., 39( 13 1).65- 86. Thomas, R . H. 1973. The cree p of ice sheh- es: theo ry. J. Glacial. , 12 (64 ). Huybrechts, P. 1990. A 3-D model fo r the Antarcti c ice sheet: a 45 53. se nsiti\'ity study on the glacial- interglacial contras t. Climate Dj'n. , T oh, H . a nd K . Shibuya. 1992. Thinning rate of ice shee t on 1\[izu ho

5 (2), 79- 92 . Pl a teau 1 East Antarc ti ca, d e termined by G PS diffe re n tial positi onin g. Huybrec hts, P. a nd J . O erl emans. 1988. Evo lulion of lhe Eas t Antarctic III Yos hi da, Y., fd. Recelll jJrogrrss ill Anlarctic farlh sciellce. Tokyo. Terra ice sheet: a llullle ri cai study of lhermo -1l1 cchani cal res po nse patle rns Sc ientific Publishing Company, 579 583. with changing cl im a te. Ann. (aaciol., 11 , 52- .19. \ 'a n der " ee n, C.J. 1987. Longitudinal stresses a nd basa l sliding: a Kameda, T., M . Na kawo, S. Mae, O . Wata na be and R . Naruse. 1990. co mparative study. III Van cl er \'ee n, C.J and J. O erl emans, eels . Thinning ofthc ice shee t es tima ted from total gas content of ice cores D)'IlIImics oJ Ihe lI"esl Alllarctic ice sheet. Dordrcc ht, etc., Kluwe r in Mizuho Pla teau, Eas t Antarctica. AIIII . Claciol .. 14, 13 1- 135. Academic Publishers, 223 248. Mitchell , A. R. a nd D . R . Griniths. 1980. The fillite difJerence melhod in Van der Vee n, C. ,} . 1989. A numeri ca l scheme for calcula ti ng Slresses jJarlial differenlial equalions. Chi ches ter, ete, J ohn \ Vile y ond Sons. a nd strai n rates in glaciers. illalh. Ceal., 21 (3), 363- 377. Naruse, R . 1979. Thinning of the ice shee l in Mizuho Pl a tea u, East Va n (ler \ 'ce n, C. J . a nd 1.1\ 1. Whillans. 1989. Force budge t: I. Theory Antarctica. J. Glaciol., 24(90 ), 45 52. a nd numeri cal methods. ]. Glacial., 35( 11 9), 53- 60. Ni shio. F., S. 1\1ae, H . Ohmae, S. T a kabas hi , M . Nakawo and K. \V ald er, ]. S. a nd A. Fowler. 1994. Cha nn eli zed subglacia l drain age over K awada. 1989. D ynami cal behaviour of the ice shee t in Mizuho a deformable bed. ]. Glacial., 40(134), 3- 15. Pl atea u, East Antarctica. Proceedings DJ tlte .'rIPR Sj'IIZ/JosiulIl all Polar Weertman,J a nd C. E. Birchficlcl . 1982 . Subglacial wa ter fl ow under ice lvleteorolog), and Clacialog)! 2, 97- 104. strea ms and \Vest Anta rctic ice sheel stabi lit y . .'11111 . Glacial. , 3, 3 16 O erl emans, J . a nd C. J . vo n der Veen. 1984. Ice sheels al/d climale. 320 Dordrcc ht, etc., D. Reidel Publishing Company. \\'hilla ns, 1. M . 1987. Force budget of ice shee ts. III Van del' Veen, C ..J . Palerso ll , W. S. B. 1994. The jJ /~) 's ics of glaciers. Third edilion. O xlord, etc., and J . O eri emans. eds . DYI/amics qf the lI 'esl Alltarclic i(f sheel. Elsevier Science Ltd . Dord rec ht, etc., Kluwe r Academi c Publishers, 17-36.

246