Embed Size (px)
Citation preview
Instructions for use
Title Polythermal three-dimensional modelling of the Greenland ice sheet with varied geothermal heat flux
Author(s) Greve, Ralf; Hutter, Kolumban
Citation Annals of Glaciology, 21: 8-12
Issue Date 1995
Doc URL http://hdl.handle.net/2115/47803
Rights © 1995 International Glaciological Society
Type article
File Information Greve_Hutter_1995_AnnGlac.pdf
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
. flll/als !if C'1([ciologJ' 2 1 1995 (- fnt ern a ti o na l Glac io log ica l Soc ic l\'
Polythertnal three-ditnensional tnodelling of the Greenland ice sheet with varied geothertnal heat flux
R. GRE\' E .\.\'0 K, H lJTTER
Il/ slillllJiir .I/ echanik. T edllli.lche Hoc/mIll/le Darlllslar/I , D-6J28.9 D armJlaril, G'erl71a /~J'
ABSTRACT, C omputa tions o \'er 50000 yea rs into s tead y Sla te w ilh Gre\'e's pol y th erm a l ice-shee l m od el a nd ilS num e ri ca l cod e a re pe rfo rm ed fo r th e Greenl a nd ice shee t with tod ay's clima tological input (sur/ilce tempera ture and acc uml.!lati o n fun c tion ) a nd th rec \'al ues o f th e geo th erm al hea t flu x: (42, 54,6, 29 ,+) m \\' m 2 Ll is sh o\\'n th a t thro ug h th e th ermom ec ha ni ca l couplin g th e geo m etry as \\'e ll as th e th erm a l regim e. in p a rti cul a r th a t close to th e bed , res pond surpri , ing ly strong ly to th e basa l th erm a l hea t input. Th e m os t se nsili \'(' \ 'a ri able is th e basa l tempera ture fi e ld , but th e m ax imum he ig ht of th e summit a lso \'a ri es by more th a n ± 100 m, Furth e rm o re, so m e im('I'compa ri so n of th e m od el o utputs with th e real ice shee t is caHied o ut, sho\\'ing that th e mod el prO\' id es reaso na bl e res ults 1'0 1' th e ice-shee t geo m etry as we ll as fo r th e englacia l tempera tures ,
NOTATION 1. INTRODUCTION
Ai.I)
ALl>
A(T' ) A,(w) a~1 c D(w) E
9 h hll1 ;1'
fI t , 11111'
L
Qt,'otli T T' T~ I 7()
Tce TD,I'(':3
Tcmp V; Ol 11r"lllP ,1' , !J z (3
p a w
8
Ice-co\Tred basa l a rea Basa I a rea coyered b y tem pera te ice
R a te fac to r fo r co ld ice
R a te fac to r fo r tempera te ice \ 'o lum e flux through th e CTS S pec ifi c hea t or ice \Vater dra inage fun c ti o n
Enh a ncem ent facto r
Grcl\'ity accelera ti on
z coordina te of th e free surface .\l a ximum h or th e emire ice shee t .\laximum thi ckn ess or tempera te ice laye r La tent hea t o f ice
G eo th erm a l hea t flux
T em p era tu re
H omologo us tempera ture Pressure-melting po int ,\felting point a t ze ro press ure Basa l tempera ture a t C amp C entury ice co rc
Basal rem pe ra tu re a t D ye 3 icc co re
Basal tempera ture a t GRIP ice core T o ta l ice \ 'o lume \' 01 u me o f' tem pera te ice H o ri zo nta l ca rtes ia n coo rdin a tes \ ' erti cal ca rt es ia n coo rdi na te (heig h t a ,s, l. )
C la usius C la p ey ro n g radi ent
H eat co nduc ti\ 'it y o f ice
D ensi ty of ice Effec ti vc sh ea r stress \V a ter content (m ass fracti on o r wa te r in
tem pera tc ice )
Th res ho ld water content in th e dra in age
fun c ti on
I n thi s pa per ,,'e p resent a summ a r\, o r computat ions th a t \IT IT pe rfo rm ed ", ith th e poly th crma l ice-shee t model or
Gre\'e (disse rta ti on in prepa ra ti on ) using th e num eri ca l
cod e SICOPOLIS th a t \I 'as cb 'e loped fo r it. The ice \I'ithin th e Gree nla nd ice shee t is a llO\\'ed to beco me tempera te; in th e regions ",here it is cold the ene rgy equ a ti o n is used as a n e\ 'o luti o n equ a ti on 1'0 1' th e
tempera ture lield ; in th e tempera te region it se n TS as a n
equation gO\'erning th e produc ti on a nd ackec ti on of th e
\I'a ter content o f th e ice , The t\\'o regions are sepa ra ted b\' the so-ca ll ed co ld tempera te tra nsiti on surfacc (CTS ) II' hose m uti on is d esc ribed by th e kin ema ti c eq ua ti on a nd th e d ynamic co nditi ons pec uli a r to this typ e or S tera n
problem. The th eory is esse nti a ll y knml"ll \[ o\l"ler a nd
La rso n, 1978; Hut te r, 1982; Huller a nd o th ers, 1988; Bl a ttcr. 199 1; Bl a tt er and Hutter. 199 1; Huttcr, 1993 ) a nd is usecl here in th e fc) rm g i\'en by Gre\'e (disserta ti on in preparati on ), I ts implementa ti on to a numeri ca l cod e using finite-difference techniqucs has bee n perfo rm ed by
GIT \T; it ma kes efIi cient use o f' th e a-tra nsformati o n (sce
model-fea ture compil at ion in sec ti on 3) and stretched
coo rdin a tes in th e ve rti ca l, to <lchi e\'e th e utm os t acc uracy
in Ih e \'icinity or th e base \I'here th e ice may become tempera te in ra th er thin layers, \\'e sha ll no t prese nt a ny deta ils, but in til e roll oll' ing sect ion " ,ill se t out bri e fl \ ' th e bounda ry \"a lu e probl em th a t desc ribes th e problem ,
C omputa tions \I"ill be perfo rm ed for th e clim a te
d ri vi ng fo rces as follo\\ 's , Th e su rface tem pcra tu re di stributi on lI'ill be th a t 0 1' Ohmura ( 1987) as p a ra m e terized by CalO\' ( 199-1·), Simil a rl y, th e sno\\' bala nce a nd
m elting rat e a t th e surrace \\'ill be implem cnted acco rdin g
to Ohmura a nd R eeh 's ( 199 1) a nd R ee h ' s ( 199 1) sugges ti o ns as implem ent cd computational'" by Ca lm '
_ JIll/fill (!! (;I(lciolog J' 21 1995 r Interna[ional C\aciolog-ical Soc-ict\
Polythertnal three-ditnensional tnodelling of the Greenland ice sheet with varied geothertnal heat flux
R. GIU~ \ · I ~ .\:\\) K . H I·TT ER
III I/i/lfl jilr . \ I {'(" /iol/i/,. 7 i'dlll/v/lI' Ilor!IIr!IIIII' Dtlllill/adl. /J-64289 /Jal"lll j /nr/I. (,'l'r ll/flIO'
.\BSTR,\ CT. Com»u[alio", o,·cr .;OOOOye,m illlO sleach- \laic ,,";[10 Crnc·s poiythrTmal icc-:.. hcct model and it !'oo IlUllll'rical code ;n·t· p,.:rli)l"lIled roJ' Ill(' Cn'cnialld icC' ... hn·1 \\ ilh loda~· ... dimawlo t!;ical input sur!;\ce temperature: and ae<.: urnulalion function I and thrcl' \ ',-due, ur,lil' gl'oliwl'llwllwnl flu:--: ·12.51-.6. 29. ~ m\\ ' 111 2. It is sho" II thal through the thermoJllcchanical roupling- the g-eolll('tr~ as \\ell as tlic therlllal regilllt'. ill panicular that c1o ... r 10 th(' hrd , rf'spond s urpri~in~l~ ... trongly 10 the blt !o.al thnlllal heat input. The Ill()~ t stmiti\ L' \ariable is the ba'ia l tl'll1per(ltlllT lidd. bllt the maxilllulll height or the 'iUllllllil al .... o \ 'aries b~ lIlorl' tball ± 100 Ill. FUrlill'1'l1HJI"l', ~onH: illtl·ITOlllpilri .... oll or [ilt' nwdd OUlpUb \\ ilh the real icc ~ hCCI is c"!Tied out. showing that till' IlIOtlt-1 pnl\·idc·s 1'l':I ..... OII"hl,· rC':.c.JiI .... fiJI' [ht." in·- .... hn·t
gC'oIlH·tr> ;)S \\cll a:; fi)r [hl' eng-Iucial tl'mperatu res.
NOTATION 1. INTRODUCTION
AI _II
Au . A(T' ) A, (", ) a;/ ( .
JJ (uJ) E
9 II h l w \:\
/1, .11 111'0; L
Q~'()!h T T' 1\ [ I ()
Tee TD.n':~ Tcn[r \ ~O!
\I; "JUP ,1'. U
" (J
rr
uJ
..ulJLax
ICl'-('t}\t'!'C 'c\ ila.sal ,In';)
Basal arl'a co\'crc:d b> 1l'lllpCr;)t c ice
Ratc factor fin ('old irt'
Ratl' f<-ll"tol' for 14:lllperate iCt,
\"uiLlIlw flu:-- dll'ou~h the CT S Specific heal or in.'
\\ 'atn dr;lillage rUIIl"liOIi
EllhalltTllltllt factor
(; ra\'j t ~ accelera tion
Z ('oordinaH' of the free SUr!;HT
~ l a~illtlllll II of' [ile entin' icf' '''WPt ~ I a;o.:illlllln lhil'knc'ss of' lempcra((' icc laycr 1,:H{,111 hra[ or icc
Gcuthcrrnal ileat flu:--' l'clll pcra I lilT
H omologoll " l('mpcratlilT
l'n .. ·"'lIJ't'-rlwlli llg p(li!11
" \c lling point al /.l' !"l) prl' .... ..... u r(· Basal tt'lllpl'l"atur(' ,l{ Camp C(,l1Iltr~ IC(' COI"('
Basal ItnlptT~llure at D~e 3 icc core
Ba <.,ai t(,lllpl'I"alUre al GR IP ice corc
Total icc \ 'olumc , 'oIUl11l' or tl'Jllpl'J':tle icc' Il orizollt;11 l ·;II·tl· .... i;111 COOI'(Jill;l[t· ....
"cnilal (,:;lrte .... i;1Il (,'oordill(ltc ( ltt'ig-ht a ...... . 1. ) CI<lI I<.,iu s C:lapcyron grad ient Heltt conclucti\'iIY or icc Dl'nsit~ of'icc Efle-etin' shear strl'S:-I
\\':tln content (mass fractiun Of''';lttT III
tl'Il1P('I'Clt(' icc
Thrt'~ho l cl \\ 'aler con lent in Ihe drainage
fUllction
III thi '> paper \\r present a .... lInllllar~ OrnHllpllt;ltioll .... Ihal
\\TIT perlo nned \\ ilh Ihe pol) thermal in'-'1hlTt lIlodd or
l;re\T di ~'iert;lli on in prcpar:lIiOIl lI'iin~ Ihe nUlllerical
code SICOPOL IS thaI \\as clndo pl'd for it. The icC' \\'ithill till' Grel;' ldalld in' ..... 111'1 ,1 i ..... aIlO\\I·d 10 hecome !t·lI1pniltt·: in 1111: r('~iolh \\ here it i .... {'(Jld lht· rl1crg~
('qllation is Ll sed as all (..'\·olutioll equalion IIJr lire
U'IllIJlTalli IT fidd: in the telllperate rc~ioll it 'i tT\'('S a:; an
equation gO\erning thc production and <lch-cnioll of th(' \\ aIel' content or Ihe icf'. The t\\O regions a rc separated b>
[he <;o-caJled cold temperate trallsilioll SUI'EHT CT S \\ hO:-l(' motion is desnilwri hy lilt' killc·lllillit · npl<clioll ,cncJ lilt' d) 1I;lrllic cOll ri ilioll:-l peculia r ttl thi'i type ur Stclan
problcm, The tlll:ory is esselliiall) kllo\\ 11 Fo\\ I ...... IInel
Linson, 1978; H uttcr, 19H2: lI ul1(,(" and others. 1988:
Blatter. 1991: Blatter:llld H utt(T. 1991: H llth,(", 1993 and i .... lI '>ed here in the form ,!!;in'lI b~ (;n'\T di .... :.(·l'IaliOIl ill pn.' par<lticHi I h impl"llIl'lIlalioli to a 1I111lH'rical code u ..... ing finitl' -difft' ITIllT ledlltiqul''> iriiS becll pcrrorllwd b)
G IT\'(': it make ..... ('Hil'icllt use or the a-translurmatioll 'ict'
lllodc l-fpltlUIT compilatioll ill sl,(·tiOIl 3 .1IJ<.1 s tretched
coordinatc ..... in the \ l'nicaL to achi('\'c the ulmost acc uracy in Ihe \ · icinit~ of' the b;hl' wirl'rc tire ict' 111;1) bt'l'ollle
H'l11lJl'ratc in ratiler thin Ia~tr .... . \\'t· ~ h;tli not prt.'''cllt ;III~
dl' laib. but in thc follo\\ ing '1C1..'t;OIl \\'ill 'it' [ out brief!) the
bOlllldar~ \·;l1l1c problem Ihal de ..... crihes til{' problclll.
COlnpuwtion~ \\ ill be pcri(II'nwd for Ihe ctima[('d r i\'inl-{ 10rl.T\ '-I S rol lo\\· ...... T ire "'lIrf~IC(, tcm p eralure
distr ibutioll \\ill be that o rOhlllura 19137 ) as paralllt'tt'l'iZ('d b~ C"tc)\' 199-~ 1 . Similarl~. Ihe .... 110\\ balancc and
Illeilillg- rate at tire .... urC.cc \\ ill 11t' implt'mcnlcd accnrdin~
to Oilltlllra and R l'th' 'i 1991 and Rech's 1991
!o. ug-!4'l' !'oo liom a ... implellll' lI tt'd .... olllplltatiollal1~ b~ Calu\'
(199+ ). Bo th th ese input qua ntities \\·ill be held consta nt.
The basal su d~lce \I 'ill be held fi xed (using the database 0["
Letreguill y, and others ( 1990 )) , as o nl y steady-sta te conditi ons a re studi ed, and a co nsta m \'ertical geo th e r
m a l hea t Dux \\ill be app li ed. This hea t nu x \\ill be
\'aried, i. e. , three computatio ns will be perfo rm ed for
Q;" oth = (42 . 54.6.29 .4) m \ \' m 2 \I' ith I'Cl ri a tion s o f
±30o/u about th e mean: \\'e sha ll d em ons tra te th a t the thermal statc of the ice
c lose to th e base depends stro ng h ' o n thi s \ 'ariation o r th e
gco th erm a l h eat. Due (0 th e th ermom ec ha ni ca l co upling,
howe\Tr , both th e surface topograph\' a nd th e total \ 'o lume
of the Greenland ice shee t are eq ua ll y a ffected. Su ch res ults a re impo rtant findin gs , in part icula r in re lat ion to th eir impa ct o n future ant hl'Opogeni c m ass-ba la nce predictions.
2. THEORETICAL BACKGROUND
Th e th eo ry o f poly th e rmal ice presented he re has been fo rmul a ted by Grel 'e (disse rta ti o n in prepa ra tio n ) o n th e
bas is o r earl ie r a pproach es prO\'ided b y th e a uth o rs c ited
in the introductory pa ragraph. \\'e do not \I' ish to g ilT th e
d eri\'ation here, but m e nti o n th a t it is esse nti a ll y based o n
th e shall ow-i ce approxima ti o n (H utter, 1983; i\ r o rl a nd, 1984 ), i. e ., th e mode l equati o ns a rc scakd \I'ith res pec t to th e as pect ra tio ( t )'pical heig h t to leng th ) of th e ice shee t,
and on ly zero-ord er terms arc kept. Since we carn ' o ut
o nl y stead y-s ta te expe rim e nts in this paper, neither
bedrock sinking no r thermal inerti a oC th e lith ospherc is
taken in to acco unt. As fo r the stress-m'e tc hin g rela ti on . Glen's nO\l' law with a ll ex po nent n = 3 is used (sce e.g. P ate rson , 198 1) as is common in ice-shee t m ode lling.
Below, liT list th ose mode l eq uat ions th a t d esc ribe the
temperate regions within th e ice shee t. The complete se t \\·ill
be co m pil ed by Gr'C\'e (di sse rt at ion in preparation) . Fo r the
meaning of the diITere nt qu a ntiti es see the notation list. \\' a te r conte nt in th e te mperate regions:
Dw Dw Dw Dw 2 4 ~ + v.l'-;=;-+ vy-;=;-+ Vo !'.) = - LEAt(w)(T u t u X . u y uz P
(32 0'" c(3 (Oh oh oh ) 1 +--+- -+v.,-+v,--uz -- D(w).(2. 1) pL DT L 8t ox .J Dy (i
with
(T = pg(h - z) (Oh) 2 + (Oh) 2
OX oy (2.2)
Transition conditi o ns a t th e CTS:
(2 .3)
(i) a;;' > 0 (melting conditi on ) :
Wtemp, ice = O. OTco1d icc
Dz = p, (2.4)
.-\m az ingh ', the numerical \'alue 42 o f' the stan d a rd
Q~'Olh rep resents exac tl y th e Answer to th e Ultimate
Q~l es ti o n of Life, th e Uni\'erse a nd E\'erything (J\dams,
1979 ) . furth er stud y will be requ ired to figure o ut whether this corres pondence ca n lead to a m o re profound und ersta ndin g of a ll these thin gs o r is just an acci d ent.
Grae al/d Hulln: Po0'I/ierllZa! modelling of Green!al/d ice sileel
( ii ) a~ < 0 (freez in g conditi o n ) :
(
DTcOld it'(, ) 1. I{ 8z - (3 = Lpa",wt l'IIIP. icl" (2 .5)
3. THE THREE-DIMENSIONAL POLYTHERMAL ICE-SHEET MODEL, SICOPOLIS
SICOPOLlS (Simulation Code for Po ly th erm a l I ce
Sheets ) is a three-dim ensio na l ice-shee t m od el developed
by Gre\e (disse rtati o n in prepa rat io n ). It a ll ows time
dependen tin tegra ti on of the thermomechanical model fo r
poly therm a l ice shee ts as described in sec ti o n 2. H e reb\', th e ice-shee t (l Oll' is o btained as the respo nse to a g il 'C n climatic input. name ly surface te mpera ture, acc umula ti o n and a bl a ti o n. ,\ further boundary co nciiti o n is
pro\'ided by th e geo th e rmal hea t Dux, whose influe nce
on the res ults is im'estiga ted in thi s \I·o rk.
Th e m a in nell' f'eature o f thi s m od e l as compa red to prcI 'io us three-ciimensional ice-sheet m od els is th a t it acco ullts for polyth erl1l a l cond iti o ns within th e ice shee t,
i. e. th e possible prese nce of colel as \I 'ell as temperate
regio ns, in Cl physical'" adequate \I·ay . Th a t is to say th a t
th e \I 'ater conte nt in th e temperate regio ns is computed Iw so h-in g Equat ion (2.1 ); the dependence o f' the rate f~lc to r
A in the nO\l' la ll ' on th e \I ',Her co nt ent is co nside red: a nd th e pos iti o nin g o f' the CTS is ca rri ed ou t by fulfilling
Equati o ns 12.31 (2 .5 ). This e ntail s th e ., possibility 0 ['
discontinuities of' th e temperature g rad ient a nd th e water
conte nt a t th e CTS in case of' fi"eez in g conditions .
The model essc nti a ll y ass um es that \I'<Her in temperate regions traIT I, \I'ith the sa m c 1'Clocitl' as ice . H OlI'el'(' r , prelimina ry comp ut at io ns hal'e rel'ealcd that. \I' itho ut
a ll\' \I 'a te r-dra ina ge m ec han ism includ ed , the \I'a te r
co n ten t can exceed IOO (Yo fo r sCI'e ra I grid po i n ts, a
physically meaningless res ult , o f co urse . Th erefo re, it is
nccessa ry to prescri be a para meteriza tion for \I '<He r dra in age . On a hi g her Ic\TI o f' m od el complex itl·. thi s co uld be ac hi ClTd b y using some thin g like a n ex tend cd
D a rcy la 11' (H u ll e!'. 1993 ) for th e in te ract io n force
bet\l'een \I 'ater a nd ice tral'elling \I'ith diffe rent \'elo
citi es; hO\l'e\ 'e r, o ur fo rmulati o n \\'ith o ne single 111 0 m e ntum balancc for Ih e mixtulT ice plus lI'atcr d oes no t a llow thi s a pproac h. Thus. liT d ea l \lilh the pro bl e m bl introd ucing Cl dra inage function D(w) in th e lI'ater
con te nt Eq uat ion (2.1 ). Based o n m eas ure m e n ts or typica l
\I'a te r-conte n t \'al ues in g laciers, \IT c hoose
D(w) = { ~ w :::; Wllla , }
W > Wmax (with W llla ' = l o/c ). (3.6)
Thi s means th a t a 11\' water surplus exceed in g the
thres hold \'alue W IlJaX is ass um ed to be in sta nta neo usly
drained into the g ro und. \\'e a re all'a re of th e fact that
our water-drainage m ode l v io lates the loca l m ass balance oll' ing to \I 'ater a nnihil at io n o ut 0(' th e illle ri o r of the tempe ra te ice. H O\l·el·er. g lo ba l mass balance is satisfied
b y acco untin g fo r th e a m ount of dra ined \I'ater in the
com putat ion o f the e \'o lutio n oC th e ice s llrJ~lce.
SUD1.D1.a r y of further D1.ode l features
Threc-dim ensio nal modc l. based o n fin ite diITere nces
9
199-1- Both Ihese inpul qualltities \, 'ill be held constilll\.
Tlw basal slIrbcL' \\ ill be held lixed (ming the cialaba<;(' or LC Il'L'gllill~, and o the rs ( 1990 J, a s only "'(A;ld~-sl<-Hl'
conditioll 'i arc studied, a nd a CUlISlalit \Trlic;il g\'uthcr
III a I heal [lux will be applied, T his heat Ihl:\ niH be
\'ariccl, i,e" three COlllpurtll ions \\' ill he performed f()J'
Q~('\lll( = (-I2, 5-1.(-5, 29.-l). m\\'lll '2 \\' jlh \' ariations of ±30oAJ ahout til(:' mean,
\\'c shall dClllonstrate that thc thermal statc or the in.'
close to tlllA b;\se depends s trotlgl~ on thi'l \'a riatiOIl or Ih('
geot hermal heal. Due 10 tlte tlitTIIlOmecltallic111 coupling,
l!om'. .. \TL both the sllrbct' topograph~ anel the total \'olume
orthe Greenland ict' shcet ;In: equ(l1l~ ;)Ilt-'clt~d, Such r('sults
arc important lindings. ill p;l1',ic ll l;u ill relalioll 10 Ihcir illl]l;LCI Oil ['ulurt'· <lmhropo).,{l'nic nw..,s- bal '\I1 ec prcdi l'li on:-..
2. THEORETICAL BACKGROUND
Th e tlH'Ol'Y of' pol) Iht'rlll;d icc prc..,t·llIl'd 1lt'l"tA h:t'i been /ormulHtecl by Crc\'C d i:-;sl'rw tion in prcpa]';lIioll till tilt' ha<.,i:-; oj" earlicr approaches prU\'ided by thL' authur'i cited
ill the illtrodliclOr~ paragraph. \Ye d o not '\' ish to gi\'(' the
rkri\"alioll ht'IT . hut melltion that it is essC'ntialh b"..,('d un [he ,>ha llow-ice approxilTl:ltioll (H uller. 1983: \[orla11l1. 19U.+, i,t", Ille 1l1t1ci('1 ('qU;lliolis art' scaled \\i[h respect [(}
[he a~pc(t ratio t~pic;tl hcight to length nrllie in' ..,hel't.
and ()nl~ zero-ordn terlll~ arc kept. Sincc \\T carr~ oul
only steady-sta te experim ents in thi s papcr. ncithcr
bedrock 'iink ing nor thermal incrlia or lile litho:-.pll('rc i:-.
l<lk(An inlo accuunt. .\ <; for the stlTs<;-<;trnching rdation, Glen's flo\\' la\\' willl ;111 t'XPOll('1l1 n = ;1 is 1Is('d ~t't' l·.g", P;ltl'l"SOIl, 1981 as is ('oml11on in icc-!o.hcL'1 modelling.
Brlo\\·. \\'c lisl those model equaliolls thai (h.-scrihe the
lemperalc regions \\'ilhill thc ict' sheet. T ht completc set \\ ill
be cOlllpiled b~ Grc\'(' cl is.:.c rt<lti ol1 in prrparalinl1 I:or th('
IlH'aning- or [he dilkrent qualltitiC''1 :-.ee the notation li..,t. \\' a ter content in [he I C111])(']';1It' reg'ions:
uw Ow Dw Dw 2 " ,,+0."-8 +<'u-O +<"-0 =-Lt;A,(w)a o[ .r y Z f1
.3' 0" c.3 (Oil Oh Oh ) I +--.-+ - -+ U,.-+ c,,-,-- ", -- D (w).( 2.1 ) pL OT L at D.c . Du (I
a = pg(h - z) (22)
Transition conditioll.'1 ; 11 dll' CTS:
•
~:oltl ict' = 7; l'II![I, in' = 1:\1.
() ~ > 0 melting cundi lioll
DTc "Old in'
Dz
(2.3)
(2. I)
. \tnaz ill gl~, the nUll1crical \ 'alue ~2 of th e ..,tallrbrd
Q.L ,1 ITI11T.'1cn ts ('xanlv the .\nSWl'r to the Lltillltltt' )!,I'() I <
(,.2.uestion ol' l ,ifl', the Lni\'C'rs(' and E\"('rYlhing .\dams,
1979 Further stud~ will be required 10 figure out
\\ hctl1l'1' thi s correspondence ran lead to a more prololllld
Llllckrstandillg oj" all these thing-s ur i.., just all ;H:cidelll.
11 (/ 111 < 0 fn'C''l.ing condi lion :
(2.5)
3. THE THREE-DIMENSIONAL POLYTHERMAL ICE-SHEET MODEL, SICOPOLIS
SICO POL IS Sililulatiull Code for Polyt llennal Ice
ShC(,h is a thrcc-dimensional ice- I, hc(' t model d C'\Tlopcd
b~ Grc\ 'C' di ,,'>cnatiull in preparation It allo\\ .') timc
depenclent intt.-"~ ration orthe t-hermoll1('chanicalmocld I()r
p{)I~t1H"rmal icC' sh('('h a s dt":o.c riiwd in s('('! ion 2, H (-'rt'h~,
[ht' itT-:-.llt' t'1 flon i ~ obl;lint'd ;t.., 111(' n'spOIl'it' ]() " gi\ ('11
climatic inpul. namely sllri3cC' tCll1pcrrllure. acc1I 1l1uln [ion and ablalion. ,\ further boundan condition i:-.
pro\"idcd by till' geothermal heat flux. \dlOSC in!lu(,llcc Oil the ]'{'sult~ i .... imTstigcllt'd ill th i ~ \\'ork.
Thl"' main Ill'\\ I('alll],(' of l h i ~ modt,1 as compared 10
prn iuu<; Ihret·-dillll'n:-.ional icc-sheel Ilwdt'b i:-. [h~ll il ;IC("OIlIlI:-. (C/I ' polYlhel'1n;lI condilion:-. \\' itliin tilt' iet ..,Il('('!.
1.(.'. tht' [Jo~sihle presL'lIce of cold a'i wc ll as 1(,l1Iperatl'
r('ginn~. in (l physical ly adequate 'ray. That i.'l to sa~ thal
the \\ate r cOlHc nt ill the temperatC' regiurh i:'l computed b~
soki llg Eqll;tlioll 2,1 ; the dC]J('l1denlT elf' the rate ClclUr A in [hl' I1m\ b\\ 011 lhL' \\ <I[t'" culltell[ i'l t'oll...,idcl'ed: al1d tlil' posiliol1ing 0[' tlw CTS i~ (: ;l1Ti~'d {JU[ h~ rui!ilJing
EquatiulI;'} 2.3 2.j Thi.., t'\1\ilih the f" pos'libilil~ oC
discolltinuities or tile tcmperature gradiell t and the \\ 'ater
COil \('11 ( al ,ile eTS in cast' of IiTC/:ing condilion.."
TIl l' lll(Jd('Il':-;'il']lti~llly ~1'ISUIlll''' 111.1t water in temperate I'( 'giolls [I"<I\.t"! !o. \\'i[h [Ilt" sa llH ' \ ' t'lo("il~ ;1:-. in'. H o\\t,\,(,]"
prdimillar~ compulation.., ha\T rt..'\Talcd that. without
.[11\ \\ .Lln-drilillag'c llleChallislTI inclucled, Ihe ,,'ater
content can t'xcecd 100% lor s(', 'eral grid puillts . a
physicall~ meaninglcss re<.,tlh, of courst'. 'J' h('rd()t't:" il i ...
n('(T~'1ary 10 pre'ic ribe a para1llClerization for \\ ' ;11<.'1'
<l1":lil1;l g"(', On ;1 higlH'l' I(,H'I of 1110del ("mnplexit~, thi:-.
cc)uld he achin cd b~ u..,ing ,>o lll t'l hing lib, an t'xlended
Darcy 1<1\\' Hutt er , 1993 Cor tile interactiull i()ITl'
brt\\TCIl \\"alrr and icr tran'lli ng \\' jth dillcrcnl \ 'c!o
l.'iliL's: ho\\t'\'Cr. our formula Lion \\' itll (Jlle ~i llgk II 10111 ell
tum balance lOl" the mixture icc plus \\'atcr doL'~ not allo\\' thi.., ;l]Jproach. ·J'hu.., . we ekal \\ilh Ihe prubkll1 h~
introducing a drainage rlll1ction D(w) in the wa[cr·
contcnt Equ.lIioll 2.1 . Based 011 lllCa~UITITll:llts of typical
\\<ltcr-l'ontl'nt \ 'aILic'! in glac ier'l, \\c c hoose
D (",) = { ~ "",' :S """111;1".' }
w> "",'IIUI" (with L..:lln1 " = I ){) , (3.G)
T his llle;\lb Ih<ll any \\' ,11(']' surplm tx(('tx ling 1IH" thrc<;hold \'alue WlllllX i<; <l'iSUlllcd to be illst<lnt<llll'ousiy
drained into tile grou nd. \\\' arc aware of the rael that our \\;,ter-dr;,ill;tg'e lnt){lcl \'iol(1tl's Lite loc,d \11;\ <;S balallce
()\\illg 10 \\;llt'r ;llIlliltihllioll 0111 or Iht, illllTiol' or Iltl'
It'mperalt"' ice, H O\\("\tT, g lohal Ill,\-..., h"LIIHT i:-. s'lli.') Iil'd
l)~ accounting for Ihe amoll nt of drained \\',lIn in the
computatiull or the c\'olution ur thl' icc S Ur! ~IC(,.
Summary of further model features
9
Greve and /-fllller: Pol),tlzermal modellillg qf Greenland ice sheel
Poly th erm a l ealc ul a ti o n (as deseri bed a bm'e )
H ori zo nta l reso lutio n: 40 km
Tim e step: 10 a fo r th e calcula ti on of th e \'eloc iti es a nd th e
topogra ph y, 100 a fo r th e calcul a ti on of th e tempera ture a nd wa ter content
.I n th e \'e rti ca l: (T-coo rdina tes fo r each of th e three reg ions
(Iith os ph ere: II equidista nt g rid po ints; tempera te ice : II
equidi sta nt g rid points; co ld ice : 5 1 g rid points \\'ith
d ensifi ca ti o n towa rd s th e bo t to m ), i.e ., \ 'erti ca l columns a re m a pped o n [0 , 1] int en 'a ls
T em p e ra ture a nd wa te r- co n ten t eq ua ti o n: i m pli c i t
di se re ti za ti on of z d eri\ 'a ti\ 'es, exp li cit disc retization of :r
a nd y d eri\, a ti\ 'es (upwind sc hem e [or th e h o ri zo nta l
advec ti on te rms)
H eight (' \'o lution equ a ti on: ADI sc heme (i.e ., one of th e two direc ti ons is di sc reti zed impli citl y in a n a ltern a ting 11' (\ \ ')
Glen 's noli' la w with '11 = 3 a nd a n enha ncem ent fac to r
E = 3 (aeco un ting fo r th e red uced stiffn ess of \\'isco nsi
na n ice in th e near-basa l regions wh ere th e m a in shea rin g ta kes place ) used ; ra te facto r A d epending on homologous tempera tu re in case of co ld ice (common ex p onenti a l law
A(T') = Aa exp( R('il ~T' )) with two different ac ti\ 'a tio n enero' ies Q fo r T' < - 10 e a nd T' > - lODe res pec ti\'el,, '
b '.'
sce P ate rso n, 1981 ) a nd Oil wa ter content in case o f
temperate ice (A t(w) = A(T' = ODC) X (1 + 184w), accordin g to Liiboutl'y a nd Du n tl ( 1985 )
Pa ra m e teri za ti on of surface temperature, acc umu la ti on
a nd a bla ti on as used by C a lm ' ( 1994 ), based on Ohmu ra
( 198 7) , Ohmura a nd R eeh ( 199 1) a nd R eeh ( 199 1)
Basa l sliding in case o r a tempera te base (\\'eertm a n-t ype sliding la w as used b y C al m ' (1994)) ; no basal sliding in case o f a co ld base .
The va lues for th e ph ys ical qu a ntiti es occ urnng In th e
mod e l a re listed in T a bl e I.
4. THREE MODEL RUNS WITH VARIED GEOTHERMAL HEAT FLUX
In this sec ti o n we wa nt to ill\ 'es ti ga te th e influence of th e geo th e rm a l hea t flux on mod e ll ed stead y states o f th e Gree nl a nd ice shee t unde r present clim a te conditi ons. T o thi s end , three experiments ha lT bee n conduc ted with
S ICOPOLIS . In ex perim ent rcOOI th e sta ndard \'a lue
Q ;coth =42 m \\'m 2 is used . ex perim ent rc004 is carri ed
out with a 30% hi gher \ 'a lue (Q t coth = 54 .6 m \\· m 2), a nd expe rim e nt r c00 5 w ith a 30% lowe r \'a lu e (Q~~ol li =29.4 m \ \'m 2). Th ese \'a ri a ti ons cover ro ug hl y th e ra nge o f un certa inty o f Q~'otli (fo r a d e ta iled disc ussion of m easurem en ts of geothermal h eat fl uxes see
Lee ( 197 0 )) . Th e ite rati ons s ta rt with th e prese nt
topograph y as obta in ed from m eas urem ents by L e treg uill y a nd o th ers (1990 ) a nd a n iso th erm a l state w ith T = 10°C in th e en tire ice shee t; th ey a re cond uc ted fo r 50 000 mod el yea rs, bei ng su fTi cien t to reach a pprox i
m a te l y th e stead y sta te_
In Fi g ure l a-c, th e res ults o f the runs rcGOI , rc004
a nd rc005 [o r th e top ogra ph y o f th e free surface a re
10
T able J. r aLlies Ja r the distincl /)/~) '.ricaL qllall lities as llsed in Ihe model calculalions
OJ{{[ lIl i~J '
l ee d ensi t y p
H ea t conduc ti\'it y o f ice h:
S pec ifi c hea t of ice c L a tent hea t of ice L Enh a ncem ent fac tor E C la usius C la peyron g ra di ent f3 Geo th erma l hea t flu x Qi~Olli G ra \ ·i t)' accelera ti on g
r 'allle
9 10 kg m :; 2. I \ \ ' m I K I
2009 j kg I K I
335 kj kg I
3 8.7 x 10 I Km I
2 (+2,54.6 . 29 .4 ) m \\' m 9 .8 1m s 2
d e p ic ted. Fig u re 2 shows th e m odel o utput [o r th e ho m o logo us te mpe rature a t th e ice b ase a nd th e distribut io n of tempera te ice a bo\ 'C' th e base, a nd T a ble
2 gives the res ults fo r hill"", Vlot, VlCIllP l Ai .b , A t.h- Ht.tw, X 1
T CR lP, Tee a nd T D.Y0:3t (see no ta ti o n li st ) , supplem ented
by th e o bse rved n tlu es a t h a nd (to pogl'a ph y d a ta from Le trcg uill y a nd o th e rs ( 1990 ) ; ice-co re d a ta fro m \\'ee rlma n ( 1968 ) , Gund es trup and H a nse n ( 1984) ,
j ohnse n a nd o th ers ( 1992 ) a nd D a h I-j ensen (pe rso n a l
communi cati o n , 1994 )) .
Fig ure I a- c indi ca tes th a t th e free surfa ce geo m e try
d e pe nds surprising ly s tro ng ly o n th e amo unt oC th e geo th enn a l h eat flu x . N o t o nl y d o th e heig hts o rth e two d o m es difIe r (i-o m eac h o th e r b y ro ug hl y 100 m , th e loca ti o n a nd so m e d e ta il s in th e fo rm a re a lso di stin c t.
Thus, a n e rro r in th e es tima te o f the geotherma l hea t
flu x of 30 % (which is rea li s ti c) a ffec ts th e tota l m ass
ba la nce o f th e ice shee t in a n o n-negli g ibl e way . This is a lso e\'id ent fro m row 2 in T a bl e 2 w here th e to ta l ice \ 'o lum e V;ot is li s ted . Th ese res u lt s, in c id e nt a ll y, co rro bo rate th e opini o n o fte n expressed b y th eore ti
c ia ns th a t th e the rm o m ec ha ni ca l coupling is sig nifi cant
in qu a ntita ti ve ice-shee t d yn a m ics. Th ey show th a t th e
th e rma l e ffec ts feed co nsid e ra bl y o n the m echa ni cs, a nd th e res ults a re qua lita Live ly to be ex pec ted. In run rc005 ( Fig . l c ) the ice is co ld es t a nd thus the stiffes t o f th e three cases, implying tha t th e ice shee t must be th e
thi ckes t. In run re004 (Fig . I b ) th e h ea t input in to th e
ice from b elow is la rges t, th e bo ttom-mos t ice weakes t , h o ri zonta l a d\'ec ti o n la rgC"s t a nd he ig ht a nd to ta l \ 'o lume th e refore smal les t. 1\'o te that th e to ta l icecove red surface is nea rl y inse nsiti\ 'e to va ri a tions o f th e
geoth e rma l hea t fl ux .
Eve n m o r e affcc ted by th e va ri a tion o f th e
geo th erma l hea t nu x is th e th erm a l regime o f th e ice
shee t, in p a rti c ula r its res po nse c lose to th e b ase, as prese m ed in Fig ure 2a- e a nd in rows 3 a nd 5 of Table 2 . I n a ll three cases o f o ur m od el run , quite a broa d be lt o f
th e m a rg ina l basal ice is tempera te . This basal rin g oC tempera te ice is interrupted a t a few broad stripes in th e
cas t, no rth a nd west o f th e Gree n la nd ice shee t. [ n th e
These tempera tures a rc ca lcu la ted b y averaging th e
m od ell ed va lues a t th e four adj ace nt g rid POiIllS,
\V'e ig hed by th e il1l'erse dista nces o f th e g rid-point
positi o ns to th e ac tu a l bo reho le positi ons.
Grere and Hulfer: Po()'fhnl1la/ lIIodfiling of Greenland ire sheef
Polythcrrn ~d C:llclllatiull (as described abO\T )
Horizont<11 reso lution: 10 km
T itTl e step: 10 <l fur thc calculation orthe \T loeities a nd thc
IOpograph y, 100a lor Llw c-alclliatiOIi or thl' tl'll1]X'raturl' and \\al.<"1' COIll<"nl
I II l he H-rtical: (T-coordi l1 ates fur c.u.:h orthc thrce regions
l lithospherc: II equid istaut grid poi nts; tcmperat(' icc: I I eq u idistant gr id po il1ls: cold icc: 5 1 g rid po int s \\' ith
d Clisification lU\\'ard s thc bonom ). i,e .. \'Cnical columns
arc mapped on lO,IJ intcr\'al s
T f'lllpf'ra LUre an d \\ atcr-con lt: nl equation: irnpli('it
d iscretization or z d e l'i\'ati\'(::s. ex p licit discret ization or ."r
and !J dcri, 'ati\'('s lup\\'ind sc hC'mC' lor the hori:t,O Jl L.d
a(l\'cction If'rms )
H eight (,\'o lution equat io n : AD I sc hrll1t:' (i,r" one or LIlt' two directions is discretized impli citly in an alterna t ing \\ 'a)
Glen 's lIow la\\' ",ith 71 = :3 and (Ill enha ncement [ lc lOr
E = 3 (accou n titlg fur the red u cc.:d s t iITness of " 'iseons i
ll <lll icc in lIw Iwar-b<ls,,1 regio ll s wl1t.'l'e tilt: 1l1;lill shl';lrillg lake:.; p la('(" ) lIsf'd: I'al(, raclol' A dqH' lld illg on ho m o logolls Il" lll peratlll'e in case of' c\ld icc ( COIll H U)ll expolll' lI tialla\\'
A(T' ) = Ao eXP(lfClil?Tli) \I ' ith two ditlcrcn t <lni\'at ion
energies Q rar T' < - 10 C and T' > - lOoC. respecti"eh; sec Patcrson, 1981 ) a nd on \\'alel' content in C<l S(' of
tem peratc icc (A, (w) = A (T' = nOC) x (1 + 184 w) . according to I ,liboutry and l)u\'al 1985 )
Paramcterizat ion or SUr!ilCC tem perature, accumulalion
(lnd a b la t ion <I S used b) C(llo\' ( 199 1), based 0 11 Ohlll ura
( 1987 ). Ohml1ra anel Rcch ( 1991 ) and Reth ( 1991 )
Basa l ~ l i d ing in case of a temperate bas(' (\\'eenma n-I) pt~ sliding Ia\,' as ll scd by Calo\ ' 1994 )) : no basal sliding in case of a cold base,
The \'a ilies [01' the physical q ua ll tities occurri ng ill til(' model a r(' li stcd in T a h le I ,
4. T HREE MODE L R UNS WIT H VARIE D GEOTHERMAL H EAT FLUX
In this sl'ctio ll \\'C wallt to im 'cs t igate the influence ort he
geolht:'rlll(tl hcat fl u x Oil mod ellcd s tl'~ld y states of the Greenland icc sheet u nde r prestlll clima le c() lIdi liollS, T o this end. th rei.' experimen ts ha\'C been conducted w ith
S ICOPO Ll S, In experimen t rcOOI the sta ndard \'aluc
Q;'olb = 42 m \\'111 2 is used , experiment reDO-I is carried
Out with a 30 % h ig her , 'a iue ( Q ~('()th = 5+,6111 \\' rn '.! ) . a n d ('xpcr imf' ll t rc005 \\ i l h a ':> :10% In\\'C'r "<lIlli'
( Q~('C1t li = 29,4m \\ 'm 2) , These , 'a riations cover roug hly th e range or uncena inty uJ' Q ;'flt li (for a detailed discuss ion of' nlf'aSUIT m ents or gf'olhr l'm;:ll hem fluxf's see
Lee ( 1970 )) , ' I'he ite rations start wit h th e prese n t
topography as obta ined from m eaSUITI11Cn ts b y Lr-l rcgui ll y a nd others ( 1990 ) a nd an isothermal SI; 111:' willI T = 10 C in Ihf' f'ntir(' ic(' shf't"I: Ih ey £11'(' conduclt,d fol'
.10000 mo(h-l yea rs, bf'ing sufllcif'n t 10 reac-h appmx i
mal('l y the slcady stale.
I II F igu re 1<:1 c , the resu lts uf' the ru ns reOOI , rc004
and rc005 ro r the topop:r <lph y o f the Ii'ee surface a re
10
'Table I, , 'n/ul's/or lite di,I'liJlft J!/~J'.Ii{'(f1 qll(lIIlilil'J (lj 'on/ ill flu' lII od!'/ m/ru/a/ioll.\'
Icc density p H cat rond u(' ti \ 'ity or Ice Ii
S pccific Ilf'<lt of ice ('
I ,alent h ea t or ice L E nhancemcn t f~c t ur E CLtusi us Clapeyron gradient f3 Geothermal h ea t lIux Q~('(Jtll Cra" ity <lccekrmion 9
" allll'
910 kg III :l
2,1 '" II I 1 K 1
2009J kg I K I
335 kJ k" I . , 3 8,7 x 10 I .I'- m I (42_ .14,6 , 29.4,) ll1\\ ' m 2
9.81 n l s 2
depictcd, Figu re 2 sho,,'s the model OUlput for th (' \i clIl1olo,!.!;o us tem perature at lh e icc b a se and rhe c1i sLl'i but iO ll or tt'm perate icc abo\'{: the b ase . and T able
2 gin"s rhe resu lts lor h llllls.. \~Ol ' \l;.C'I!IP ' ALb , A Lh' I-It, lIla:\. '
Tbup, Te(, a nd Tn,\t, :~ t (sec notation list ). supplemented
b y the obsc l'\'ed \';.ilues at ha nd ( topography da ta (i-om Le l ri'g u ill y and other~ ( 1990); ic("-c-o r (~ dala J'roBl \\ 'L'cnman ( 1968 ), Gundcst r up and H ansen ( 19B-I,) ,
.J oh nse ll and ot h ('r~ ( 1992 ) and D a hl-J c IISt'1 1 (persUll(l1
cOllllllunicatioll , 1994 )) .
F igu re la (' indicates thm the rrcc s urb.ce geometry
d('p('nd ~ s llrpri s i ng l ~ st rongly un the a m ount of th e gco thf'l' ln a l llt'; !1 fl ll X , Not ollly do th e h eights of t i1c t\\'o dom cs di lkr li'orn eac h ol her by rough ly 100 m , tl u'
local i01l <l1ld sOl1'le deta ils in Ih e lorm arc a lso dis t inct.
T h ll s, an ('I'mr in the es t imate o f' the gco th cnnal h ea t
flux of 300/0 ( \I'h ie l'l is reali st ic ) afTects t he tolal mass
hal a nce 0(' the ice sheet ill a lIu ll -ncgligible way, 'This is also l'\'idt:' lll 11'om 1'0'" 2 in 'J'a bk 2 w h ere th e IOtal icc "olul1le \;;01 is li s ted, Th ese res u lts, in cic\f'lllally, l'o lTo l)oratl' thc opinion ortcn exprcsscd by !heo ret i
cia ns tha t lhe thermo l11 ech(lniea l coupling is sign ifi cant
in q uantitalive icc-sheet d YlJam ics, T hey sh ow Ih al the
tht' rmal eflcc(s fe ed considera b ly o n the m ec hanics, a nd l il t' res ul ts ;'ln~ qll~llil ;'lli\'t:'l y 10 h e c:"'pec (('ci, In run reOOS ( Fi g, I c ) th e icc is coldes t a nd th us th e st iflesl or lilt>
thrt'l' cases, implying lh a t the icC' shect mllS I be Ihe
thi ckes t. I n run rc004 ( Fig, Ib) th e heat input in to tile
irc from he lo\\' is larges l , lht' boltoll1-11l0 St ice weakes t ,
ho rizontal <lcl\'Cc tio ll l<lrges t and h e ig h I and IOlal vo lUlnl:' thelTi/ lIT ~ Ill ;dles t. Note that the tota l iccem 'cred s Llrla.ce is n f'<l rl~ ins("llsi t in' to \ 'a ria liuns of' lht'
geothnm a l heat n ux ,
E\' en mo r e afrected by Ihe \' aritlt ion of t h e
geot herm al h('al n ux is the therm al regime of' the icc
shl'l'L. ill particula r it s response c lose to the basc, as preSf' lI ll'd ill Fig H1T 2;1 I' ;'111 (\ in 1'() \\'S 3 and 5 ol' T a ble 2, In all three cases oroul' 1110d('1 1'1.111 , CJuitl.';1 b rO(l d 1)l.,It uf
the m a rgi ll al b asal icc is te rnpcratc, T h is basal ri ng or
te m p('l'at(' icc is inter ru pted <It a few broad st ripl's in the
caSI, north (l nd west of' th e G reenla nd ict' shect. In the
T hese tem pcratures arc calc u lated hy <1 \'f'raging the
modelled ",tlues al the fo u r a djaccnt g rid points,
\\ 'eighed b y the im'erse ciis t(lllces or the grid-point
positions to the a c tual bo reho le posi tio ns,
Grel'e alld Hillier: PO(J' thennal mode/ling l!/ Creel/land ia sheet
~ ".._B_O_·W_,__6",O·_W--' O_·W-,-_20_·W- --o,· __ ,-, ~
z o
8 '. ~
,
z l::
Z
'" '" ~
~
z ~
0 0 n ,
~ g <~ ~,
0
::. ,
o o
z
~
z ~
L
'" '" ~
!i
z
z o '"
0 0 n ,
E g ~~ ~,
0
~
o o o
o , L-___ ~ ____ ~~~~ __ n , , - 720 -J10 '00
x/km
-720 -310 5'0 920
'00 x/km
51 0 920 -120 -JlO '00 x/km
'''0 920
Fig. 1. Stead),-slate Fee slII/ace topograj)/o'I or the model run:, described ill the tnt with dijIering geotherll7al heat fill \ . Pallels (a,b.r) are Ior Q~eoth = (42.5/-. 6,29.4) 111 11 '177 2. all other rOllditiollJ beillg the same. Eqllidista ll re ~/ the laei lines is 200 m.
three cases o f Fig ure 2a- c th e te lllpenlle basa l surfaces a m o unt to 39'10,5 1 '10 a nd 33% o f' the e ntire ice-eO\ 'ereel
basa l surface a nd a re the re fo re surprising ly la rge . This
res ult qu a lit a ti\ 'Cly \ 'e rifies ea rli e r results b y Ca IO\· a nd
Hutle r ( 199-hl, b ) wh o d em o nstra ted using a much simpl er m odel th at th e th e rm a l conditi o ns o f' th e basa l ice d e pe nd criti call y o n th e th e rm a l input f'rom th e
surro unding bo undari es .
Fig ure 2 a lso ShOII'S th e positi ons in th e m od el runs
II'he re th e re occ urs a no n-\'<l ni shin g ba sa l laye r o f'
tempera te ice \I 'ith eith e r melting o r ii-eez ing CTS. It ca n be secn that a fi"ccz ing C:TS with th e' poss ibilit y o f'
di scontinuiti cs or th e tempera ture g radi ent a nd \\',H er
content a l\\'ays occ urs a t pos iti o ns near th e ice m argin . in
agreem ent II'ith \l'h a t is ex pected . Th e assoc ia ted \'o lul1l c o r th e tempera te ice is li sted in Table 2 a nd is o f' o rder 3000 km 3 (0.1 % of thc to tal icc I'o lumc) II'ith I'a ra ti o ns o f'
up to 10 % .
T a ble 2 g i\ 'es som c mo rc res ults o f' th e simul a ti ons as
a n O\·e r\'l C\,·. suppl em ent ed Iw o bse rn lli o ns. Ob\·ioush ·.
~ ".._B::..:O_'W_,_--:6.;.O'-W--'-o'-w ~20'-'W---o_' __ , ~ , 00
~ r-__ BO_'W~_6",O'.,..W...,.-'~O'_W ~2_0'_W _ _ 0.,,-' _~, ~ ~ r-"_B_O_'W~_6",O'_W...,.-'O_'W_,__20-'W---o,' _~, ~
0 0 n ,
E g ..;.~ ~I
0 0
N ,
0 0
7 -720
L L
~ § ~ 0 0 n , ,
k ~ E g E g ~~
-" 0 '-..N :.;, ~I
L z
16 ~ ~
u u
!i ~ 0 0
z
~ z 0
.; ... ~ ~ ,
0 temp . be se 0 tem p. be se 0 temp. bese
melt CIS • melt. CTS . melLCTS
freel . CTS z ffi !reel. CTS z ffi fre e l. CTS
0 0
'" 0 '" 0 0 0
n , ..L , '00 510 - J lO '00 5 10 920 -120 -3 10 100 510 920 -720 -.:510
x/km x/km x/km
Fig. 2. /so therms 0./ Ihe stea({l'-stale homologolls basal temjJeratllres ShOll.'1I as .Iolid lines . the homologolfj temjJeratllre being illdirated ill relltrigrade. O/JCII diall/olld s)'lIIbols illdiwte /)osilioll ;' ll'here the b(/jal ire is at the /nes;' lIre-melting /lII illt .),et lcilh 110 tel7l/Jerate la,.J'fr abOlle : Jull diamolld;, (JIIII circles) illdicate positiolls ll'here tilere i.1 {/ basal la. l'er q/ tem/Jelate ire /l' ith a meltillg (jiee::. i!)g) e T S. Pallels ( a.b ,c) are/or Q~ot h ( -12 . 5-1 .6 .29.4 ) m 11 '/11 2. all othn (OuditiollJ beillg the sall/e.
920
11
~
, "
~
~
~
0 0 0 ,
E g ~~ ~,
~
~ -120
, -J'O
(:Il'l'l' alld 1IIIIIr!" : Po(rlllf'!"lIIalllw(ltIlillg I!f (;II'm/alld ;((,llIl'cl
110'111' &0''01' 40'111' 2O"W
• " g
~
F g ~" ~,
< ~ ~ u " --- ~ --- , , ~ ~ ~
,,, '"
Fi,~. I. SI{'(u!r-I/alrlrr(' .11IIIa(r lolJOgrajJ/~I'Ior Ihe motlel HlI/.\ tle.llTibrd ill Ihr Inl ,I'illl difji-rillp, gevlflt'llll(J1 lI('al./lll\.
Pal/ell (a ,b.() aff.f;)r Q ;{,(Jlb = ( -12.5-1.6. 29..J) 111 11' 1/1 2 . all oiliel (vl/ tlilioll.1 bring Ille .l fllI/ i' . F:qllidil/al/cl' qjllll' 11'1'1,1 1i1ll'.1 il 20() III,
thn,'l' (";1' .. ('<" of Figurc 2a (' the tt'l11pt'ratf' ha,o.;a l o.;u rl ~ l('t · ... ;l l llULIlIt to 3~~1o , j J Uo and 3:)110 or th e ('mi r e iC{'-t 'O\Tr('d
basal SLlI'EICl: and are t1H_Tefu re surprisingl>' large . Tlli .') l"ew ll qualil:tli\ci ) \Trilic~ earlier rC!:I ults by Calm' and H Ull C: !" 199·Ja.1J \\110 demonstrated lI SlI1g ~ Illuch ~ ill1pJt 'J" Ilwdel that t he thermal condition.') or the ha~al iet' dq w llcl (" rili ('a ll ~ O il lilt' thenn;IJ in p ut li'om the
"' U1TOll lldill ,l{ houlldari t,.').
Figure 2 abo ~ h o\\') the positiol! ... in the mockl rum
\\·h('r!' therc occurs a non-\·alli .') hin ~ ba.')a l layt·), or
~ ~ 0
,
, "
, g E g ~~ ~" ~, ~ ,
~
~
~ ~ ~ ,
~ ~ i , , ,
-120 ". ,., ><, ". -/20 -JIQ
~/kfT1 '"'
It 'lI11wr;l tl' ict' \\ith e ith er Illclting o r f'n ·t'.l.i llg ( :T S. I t ('a ll 1Jt.' 'lct' ll that a II-cezing CT S \\ ilh 1111' ptl ~o.;i J,ilit~ or
di 'lCOll tilluilie ... or the tcmperature g radi ent and \\atcr
con tCI1\ al\\ ays occlIr~ at position-; 11I'ar th(' ict margin. in ;11{ITt' Ill t' liI \\ ilh wha t i'l expec tcd. TIlt' ao.;o.;t)ci<llt ·d H,lulI ll'
,,1' tIl(' It'lIllx,)";lIe in.' i ... li .. tcc\ in T able 2 and j ... of OnitT 3000 km ') O.I ~u ofllw tolal it ..... \'olullw \\ illl \'arat iolhof
lip 10 10° u.
T able 2 gi\t·~ ~OJ lI e lIlo re re:o. uil o.; of' th e 'limulat iollo.; a ...
a ll o\'CIY]('\". ",u pplementt'd b~ Oho.;l'IY:t tiolh, OI)\'ioll " I ~ .
~ 0 ,
~ F g ~" ~ ,
• ;,
~
~ ~
• ""'D. ~c .. • "'." C'~ • I,.e. CT:>
.. , ./km
Fig, 2. !.I vlhi'lIw q/lhe .!Jle(f(tr-Jlale /IOIIIOlo.t:Ollf /J(l.Iflllf'IIII}f'rallm'l litm/'II til wlid lilies. I"r /wlI/olognll' Il'lI/jJaolllll' beil/,~
illdiwled ill (t'IIlrigrade. O/){'I/ diamolld .~J' lIIhol.l illrii({lll' II/Jliliflll.1 Whl'tt' Ihe /J(wt/ iCt' il 01 lite /!I"i'.I.I/lII' -lIlI'llill,!!. jlflilll.)'t'lll'ilh
I/O 1t'llJjJl'I'olf' Ir~rl'r ohol't' : filII diomoll(h (fidl (ire/e.lj illdiwlt' jJ/J.\iliol/l l ,'liflf 11i,'/"{' i.1 f/ /Jal(lllr!J"a (!/ hmln,/,fI/t' ilf 1t'illt fI 1IIt/liIlP, r.frl"l'.:ill,l!, j (;T.\'. P(l1Irl.J ( fI.b.( ) are for Q~""l II (-12.5./.6.29 . ./ j III / r 11/ :' . {III IJllirl" (ondilim/I brill/!. 11i(' ifllIIl·.
II
(;rel 'e alld HI/lln : PO{I'lliemw/ lIlodelLillg (!f Green/and ice sheel
Table 2. ,\lode/ olll/JUI/or lOll' ( re005) . slandard ( re001 ) alld high ( re004 ) Q teoth ' res/Jeeliz'eL)' . (fud obserlw/l'a/I/es (obJ) , gil'clI/or h lll;L'" 1/;ot, 1/;Clltp' Ai.1)' At.I). H un"x'
TcnJP, Tee and TDYcJ as defined ill the lIotalion Lisl
re005 rIOO/ le004 obs
hlll ilX ( km ) 3 .429 3.308 3.207 3.247
1~0I ( 101; km:l ) 3.083 3 .016 2.945 2.827
11r,,",!, ( I 0 :1 k m3
) 2.694 3 .050 3 .1 34 G ') Au. ( 10 km - ) 1.7 17 1.720 1.710 1. 682 G ,)
At.b ( 10 km - ) 0.563 0.670 0.888
H t.nuL" (m ) +1.789 +2 .096 +2.790
TCRIP C 16. 149 - 11.550 7.4 13 - 9 .0
Tec C - 13 .185 11.1 20 - 9.580 - 13.0
TO.,",';l 'C 7. 334 5.702 4. 14·3 - 13.22
th e thermal reg ll11e a t the base, namely th e basal arca
cow reel by temperate ice. Al. iJ , anel th e basal tempe ra
tures at th e ice-core sites, T CBlP/ cC/ OyeJ' responel most
c riti ca lk to th e diffc re nt \ ',du es of the geot hermal hea t
flu" . H O\\"C \'er, there is a n o table se ns i(i\'ity of hmax and
Vrot as \\ 'e ll , i.e., the O\ 'e ra ll gcomctry of the ice sh eet is
allectcel by the dilIe re nt Q tcoth \ 'a lu es in a non-neg li g ible
way , as stateel above. The ice-cO\'c reel basa l area, Ai.b,
appears to b e a lmost in se nsiti\ ·e .
Th e ag reement betwee n L1w m o cl e ll ed ice s h ee ts and
th e real \\'o d d is re markabl v good. " ' ilh the e"ception of
Tn.\'d' th e o bSt'r\"Cd \ 'a lu es or the quantities li s ted in
T a blc 2 li e within or at le a s t \T ry close to th e rangc of
th e modelled ones; t h e rath e r big d e \ ' ia tion [or To,'c3 is
supposedly due to the rather small number of grid points
cO\ 'tTin g th e so uthern p a rt ort h e ice sh eet. H owe\,er, onc
mus t b ea r in mind that th e actua l ice sh eet is certainl y
n o t in s tead y s tate du e to its e"pe ri enced climatc hi s tory.
Thi s a ffec ts mu c h less the mass balance , whose response
time is rath e r small, than it does th e thermal regime. A tra ns ie nt S ICOPOLIS run driven with the sta nd ard
Q.l tl and a climate hi sLO t"\ as s lI!!ges ted bv th e GR I P g('U I ., 'J<' .I
ice ('ore prm'id es an ice sh ee t or co m para ble geometry
somc\\'h at thick e r ) , yet w ith notably lo\\'Cr englac ia l
te mpe ratures, and thus \\ 'ith a 1/;"'"[1 reduced by about
18 (10 and an Al.b redu ced by about 15°;;) compared to
the s teacl)-s tat c res ults (GrnC', di sse rtation in prepara
ti o n ) , clue to the memory of the \\'i sco nsina n ice age. [ n
ag re e ment wit h these cons iderat io n s the m ode ll ed
s tead y-sta te basa l ice-co re tcm p e ratu res tend to be
hi p;h e r than th e obsenTeI ones (except T CRIP from runs
rc005 a nel rcOO I ) . L'nl(:Jrlunately it is diffi c ult to c hec k the modcl output
fo r th c q ua n ti tics clescr i bi n g tcm pera te ice, na m e \\-
1;;"",1" A l." a nd !-ll.lmlx, against th e rea l \\'o rld beca use or
data sparsit y. Firs t, th e three d ee p ice co res mentioned
abO\'(' arc a ll situated in regions where no temperate ice is
fo und in th e elltire \ 'e rti ca l co lumn , in agreement with the
model rcsu lts 101' a ll three run s prese nt ed here . Secondl y,
boreh o les drilled a t Jako bsh a\ 'ns Isbra:' (W est Greenland ,
so uth o f" Di seo Is land ) ha\ 'e re \ 'ea lcd a temperate ice base
a nd sugges ted the probable e" is te nce of an O\ 'e rl y in g
te mperate laye r ( Ik e n a nd others , 1993 ) . Thi s is
confirm ed by lh e three m o d e l run s (see Fig. 2 ) , so thal
12
th e model outp ut in term s of" poly th erm a l features is at
leas t promising. e\'e n though quanti rati\'e c hecks ca nnot
be ca rri ed o ut at the moment. R e m ote-sc n sing techniques
b ased on the meas uremelll 0[" radi o -wa\'e \Tloeiti es
( i\l ac h e ret and o th crs, 1993 ) may remedy this lack o f"
data in the future.
ACKNOWLEDGEMENTS
While performing this work R. Gre"c was supported by th e
Stud ie ns tirtung des D eutsc h en \'o lk es a nd th e D epa rtm e nt
o r Geophysic al Sciences, U ni "e rsi t y of Chicago.
REFERENCES
.\dams, O. 1979. The hilrh hi!,,,'1 ,~liide 10 Ihe galal)', Londol1. Pan Books L.td. Hlallcr. 1-1 . 199 1. ElTC'ct of climate 0 11 the c ryospherl'. C limatc' co nditi o ns
and the po l ~ th ermal struCtllrr ()rglacicr~. :;'lirrlU'r Geogr. SrllI .. 41. 1 ~)8.
Blallcr. H. and K . Hu ttcr. 1991. Polnhcnnal condilions in .\ rnic glac iers. J. (;Io(iol ., 37 126 " 26 1 269.
Ca lm', R. 199+. Das thermonll"chani schc \ 'e rhalten des griinl ;indisehen Ei"c iJilclcs ul1lN der \\'irkung 'Tr,e hi cclcner Klimaszcna ri en ,\n twortc n c inl'~ thcorc li sch-numl'ri "ch cn \lodclls. Disse rt at io n , Techni ,c iJ c Hoch,citule, Dannstacll. '
Calm. R . and K . HU llcr. ln press a. Th c therlllomcchanica l responsc or thc Creenland ice sheet to "arious climate scenarios. Part 1: .\nalys is or the temperature regime. (;fillloll' D ),I/([lIIies.
Calm'. R. and K. Hurter. [n press iJ. The thermomeehanica l response or the Green land ice shee t to n trious climate scenarios. Part JI : ,\nalys is or now field ancl gco lll etl"\. C!iJlJole DI'Il(lJJlin.
Fo\\'le r .. \. C. ""d I) . . \ . Larson. 1978. On the 11 0\\ or poly thermal g lacier ... I . l\[ odr! alld p relimin a ry alla l y~ i ~. Prur . R . . )'oc. L ondun . Su. . 1. 363 , 1713t. 2172+2.
Gunde"rup. "J. S. and B. L. Hansen. 19B+. Bore-h ole sunT' at O)e ~ .
south Green land. J. G/ociol .. 30 106 1. 282 288. Hutter, 1'- . 1982 .. \ mathematica l model of poly thermal glac iers and ire
shce ts. (;" 0/);']'1 .. I,lro"Il)'I. Fluid 1~ )'Jl., 21 3-"' . 201 - 22+. Ilu tt<'l'. K . 1983. Theoreliw! /!.1{Jrioln.~r : ,"al",.ial Icimre oIiCf {Jllrilhl' )JJfr'h{JlIin
of g/tltier.1 al/d {Cl' Jli l'fiJ . Dordrccllt. ('tc. , D . R c id c l Pllbli ~ hin g; Co. 1 Toho. Terra Scicntific Publi shin g- Co.
HUller, K. 1993. ThertnO-lllcchan ica ll y coupl ed ice sheet rcsponse co ld , poly thermal , tcmpera te. J. (,'/lIcio l .. 39\ 131 ). 65- 86.
Hutter. K .. H. Blatler and \1. Funk. 1988. t\ mod el computation of moisture Contenl in pol),thermal glaciers. ]. Cl'o/)h ),; . R I'.I ., 93 BIO I. 12.205 12,21 L
Ike ll .. \. , K. Echelmcyr r. \\'. H"rri,on and ~1. Funk . 1993. ~j ech an i s ll1 ' orE"1 f1 0\\ in J acub:"J ha\'ll .... bhrat'. " 'est Gree nl and: Part I. \ieaslIrclll(,lllS of temperat ure and water InTI ill deep boreholcs.]. (;1(I(iol .. 39 131 , I:' 25.
J ohnsen. S . .) . and 9 olhm. 1992. I rregu lar glacial interstadia ls recorded in a nel\' Greenland ice core. , \ ·alure. 35916393 1. 3 11 - 313.
Lee. \\'. H. K. 1970. On the global \'ariations of terres trial heat n O\\.
Ph )', . I,arlh Pial/cl . Inll'l'., 215 ), 332 341. l.clr i·g- uilly . A. , :\. Rcch "lid P. I-Iuybrech". 1990. TO/JogrtlJ!hi((J1 dalajur
Gran/and. Re/Jorl. Brcmcril a\'(.'Il, j·\HJ '('c!- \\' egc ll e r-InsliLLIl fOr Polarlursc!lLin,q. Lliboutn·. L. and P. Du,·a!. 1985. Variou s iso tropic and ani so tropic icc"
lound in glac iers and polar ice caps and lheir corresponding rheologies .. lnJla/e; Gi'lJ/)!~)'li(("" 3(2). 207 22+.
:-'I ac herel. Yu . Y a .. :-'1. Yu . :-'Ioska!e"sh and E. \ '. \ ·asilenko. 1993. \ ' e locity o r radio \" ,1\ es in g lac iers a s an indicator of t hcir h,dro therlll al stall'. structure alld rco;inw. J. Glacial .. 39 132 . 373 38+.
i\l ori a lld. L.. \\'. 198+. ThlTmo-lllcchanica l )'" Jallces or ice , heel flo\\'.
(:,'o/J!!)'.I. . I,lro/AD. Fll/id I~ )'II .. 29. 237 266. Ohlllul'a .. \ . 1987. Ne\\ tempcralLtre distribution maps 10 1' Greenlalld. ';:.
(;/I'IJcherkd. Gla.::ia/geol., 23 11" I +5. Ohmura , :\. and N. Reeh. 199 1. :\e\\' precipitation and accumulation
maps for Glwnland.]' (;Iaciol" 37(125 ).140 1-1-8. l' ''IITSO I1 , \\ '. S. B. 1981. The /)/~ )'I ir.\ ,!/ ,~I{Jcicr.l . Secolld ediliall . Oxrord , etc ..
PCI 'g'<lI11()n P rL'~s.
R cc h. ~. I ~91. Para lll('l(' ri zali o n ur m c ll rate and s urf~l('l' tempe rature
O il the Greenland ice shecl. Pnlrllji,rsrhulIg, 59 3). 11 3 12B. \\·ce rtman . .). 1968. Comparison bet\\'cen measured and theoret ica l
tempera ture profiles of the Camp Cmtury. Greenland. borehole. J. (;('{Jf!/~)'I . Re.l .. 73 8 . 269 1 2700.
(; rlTl' alld !-Iulla: PO(l't/It'I"I/W/ //lor/filing of Crel.'lt/a lld iCI!\/ta(
T ab/I' 2 .. \ l odl'! olll/JIII jor !(In' (rc005 ) . J/(i I/f/rml (1"('00 I )
alld high (reOO-l) Q~~'(llh ' re.ljJl'rlil'f(l' . anrl o/Jsl'J"1'ef! l'a/lIl'J
(oll.I) . gil'l'I/Jor hulilx , \~Ul ' \!; e11lp , A UI ' AI .b . {fl.lllax ·
'1"(;B JP, Tee fllld T O,H'J (/I dPji1l1'f1 ill Ihl' lIolalioll IiJI
rc005 rrOO I rrOO-/ ob,1
h tll<lx (klll l 3.·f29 3.308 3.207 :12+7
\1; 01 l ob kJll ~ ) ~ . OB~ ~ . 0 1 6 2.9+5 2.827 . '~:~
\ / 1(,1IIP ( 10 k III 2 .69·~ :1 .050 3. 13+ Au. ( IOU krn ::! \ 1. 7 17 I. 720 1.710 1.682
A l.b I au kl1l ::! ) 0 .. 1G~ 0.670 0.888 II I .lIla."\; 111 ) 4 1.789 42.096 12.790
1(;[np C 16.1+9 - 11.550 iAI 3 9. 0
Tee c: - 1 ~ .185 11.1 20 9.580 13.0
Tl l.vC':1 C 7.33+ 5. 702 +.1+3 13 22
thl' tb l' rmal regim c a [ Ihe base, namely thl' basal arca
con ' red hy [(,lllpt:ratt' ice. A l. I)' a nd th e ba~,d tempera
tu re" :1t the icc-core '-j ites , Tc:n lf'/('('/ n.\"!':~, res pond 1110.>"\
c ri t ica ll y to tl1(' differc nt \ 'a lu c.., or the geo th ermal heal flu :-.:, H O\\"(:'\ '(' r , thc re is a ll o [<1blc scn" j[i\ 'it y or h lllflx a lld
1 ;Ul a s we lL i.e " lh e m'e rall geomC'lr) or th e ict' sh t' t't is
ani'Cl{,O 1)\ the dil le- rent Q~ I \ ',lIlIes in a non-lle rlio'ible . gt'ut! ;-, ;-,
\r a\" . as sta led a bo \·c . Th e in'-co\T ITd basal area. A i.h ,
appe<lt'!'> to be almus t illst'lls ili\"c. Til l' ; lg n -eml'lll b c twcen 1Il(' m od e lled ic e shc 'el s £Inri
the real \\'o rlel is l · t ' llI '1C · k;.lhl~ good. \\' jth the c:-.:ceplio n of" T j)Yf';! ' lilt" obsen T e! \ '<1 lues u l" the qu a ntilies li s tt'd in
T a hle 2 li{' w ithi1l o r at IeZi st \'e ry cl o~(' 10 th e range or the modelled o nes; til c ra ther hig dn'iation lo r T D.H>;l is
," lIppm.('dl~ · duc to the ("(1 th('r ~ma ll number o f g r icl ]Jo ints t"u\(-rin g th e sO llth c rn ]Ja rl o f"l ht' itT shet' t. HO\\,('\·L'r. o n c
111l1 'H 1ll' ; ll ' ill Illind lhal the a c tu a l icc s hc(' t i ~ ("cJ"l;lill ly
!lo t ill ~ It::td y stat e du e \0 it s experienced c1irnal t hi s tor~.
Th is all c.'c ls mu ch less the mass balanci.'. whuse respunse
lime i!'> rath er sm a lL th a n it cloes the Ihnlllal reg ime. A
Ira n",jf' llt S IC OPOLI S run dri \'t: n \1 ith th e s t tt llda rci
Q ;,..,tll ;llld a climate hi ~ t o r ~ a s suggested b~ tilt' C R I P icc cure pro \ 'id es ;111 in' sl 1l't" \ o f" co mpara ble gcom etry SOIll i.' \\· h at llrit"kt ·r l . ~(, l \I·itll no tabl y 100\"lT clIg b ci;.d
ll' lllpe ra turcs, <l nd thus " 'ilh a \I; ('mp reduced b ~ abo llt
18(% a nd a n A l.IJ lT rlU("('cl b~ abo llt 15 U;;t compared to
th e s tead ) -sla[ (" resillt s rGrr \ 'C. di ssertation ill p r t' lx lra
[ion l. due 10 tltt' nwmo r~ o Cth e \\'i sconsin a n ice ;I ge . 111
;lg l"(' t ' llll'l1I "ilh th e~e con s id t' ralioll ~ t ilt' 111 o (kllNl
s tea d ) -s tatc bas<1 1 i(' ('-co l"e t cmjJLl"a tuIT , tend to be
hig her Ihall the o bs('I"\'cd on es ( (' \:("(' ]11 T G BIP /i'u m runs
re005 <l nd rcOOI ). Cnro rtull ; lI e l) il is di!Ticul[ to c hec k tlll' lIIucie l o utput
for [h e qU ;lIltilit 's (It-scr ihin g- lcm pc ra tc icc . namely
\ ;~'l!IP' '--\ 1.10 alld [-/ t.lIJ:lX ' again"t th e )"ca l \\ () rlcl b CGIllSI.' o r data sp a rsil y. Fi rs t, Ih e t I1l"t'(' ekep icc CO lTS melltiolled a hO\T are all situatcd in regio ns " 'here no temperat e icc i<; [c.Hl11 c1 in Ihe {'IHire \ 't' rti cal column , ill ag rc('mcnt with th l'
mode l result , ]() r all thret' rullS presented here . Seco ndl y.
b o rcil o it-" dr ill ed at J a kobsha\' ns I sl l\"il' ( \V(,SI Greenland,
sOllllr o r Di :-<eo Islalld h :\\"C I"t'H';.i1t ' ri a 1f' 1l1pe ratc icc base
a nd suggest('cI th e p ro hahle exi 'l tcnce or ;1\1 oH' rl~ ' ing
tcmperate la ~e r Ike n a nd others. 1993 ) , Thi ~ is
conf i rm ed by Ih(' lI ln . .'e IllOciel rUI1 :-< ( ~ce Fig. 2 ), so that
12
th e m o d e l output in tc rms ur pol y thermal f"c ;ltlJrcS is £II
least promis ill g . e\" e ll tho ug h quan ti ta ti\'e ch('c ks ca nnot
h(' c arri{ 'd out at th e mo mellt. R emote-sensing tcchniques
based Oil the measurelllenl o f' ra ci io-wa\T \"eloc ities
(.\ Iacllne t ;tnd o tlt l,]"s, 1993 ) ma y \"c-rnedy th is lack or datil in th e fu ture,
ACKNOWLEDGEMENTS
\\'!tik perfo rming Ihis I\ 'o rk R . Greve \\'<1 S supported by tlte S ludi ens l inutlf!; d t's De l1l schen \ 'o lkes and [he De partment
of G COpll )'s ic ;ll Scien cc's . C ni\ rrs il Y o f" Chicago.
REFERENCES
.\ dartt'. D. 1979. Til, lIitrh h;~/,I·.lp.llid( 10 till' /!.{(I(/~l" London. Pan Bf)f)k~ Lid. Blal lt·r. H . 11)~1. EIl'T t ul"ditll:l t" fi ll til!' ,")}o~ph,' ! ". C li m:l ll' cond ili",h
,mel 1I 11' pul~ thl'l"m,t\ .,tl"lll t ll r{" ofg-Iaci(' t"" ;:'il"c/iu Grog', sdll' .. 41 . I 9R. HI.I\I ('l". II. ,tnd K . I ltt tu'l". 199 1. Polyt ht:rm a l {"Omlit jom i ll ,\ rqir
I{la{"i"l"~ . J. (;Iariol .. 37 126 . 261 269.
(:,l lm . R. 199-1-. D:n, dwrmonrt '(" lr a ni~ch l' \ 'LTlr <t!tl"1l dn Ari'Il!liilldi~chl'n
E i~,("hild(" Ull In tlc r \\' i rku ltg \~·N" loi, 'd, · Ilt' t' Klim av.(·nari~'n
,\ n(\\Ot"\ell eint" tltt'l)rt'(iwl' -lltl tll (" l'i , dJI ' JJ \ l fJf l(' lI s. Di" I'I"J:ltiOJ1 . T '"I"h JJi , d H" H ,wil ,t"i tuk . Ua rJ1h t,td1.
( :;] I'J\. R . ,l tlc! K. I l u lI I'r. In pre~~ a. The thermolllt'{"h ;t tlic.:a l rnpOl!' t' "f Ihl" ( ;rt'l"tl b nd 1('(' ~h ('t"l 11) \;lrio tl." , Iitllat(' ~("t' ll a ri os. Pa rt 1: , \ Il<lly~ i ..,
I)f lhe II.'mpcralUr~· t"('ginH". (;/illlfilf l~l'l/fllllicl.
C,tlm. R. and K. HU ller. In ])1"(.'\' h. Thc Ihc t'111()11l cc ha n ica I IT"pn IlS(' or Ihl' Grel"nl a ud itT , lnT t 10 \ ar i uu~ eli l11il tt' s(e n ario~ . Pitl"( 1[ : .\ llalr~i,
of noh' fidel and I-{eonl('u·~. Clilllille lJ, t'IIlIlIIi(). F{)\\ lfT .. \ , C, a Jld t) .. \ . 1.,JI ·.\,JJl . 197B. ()JJ lhl' !l1J\\ Ill' PIJ ! ~l h enn;tI
!.(I,t<"icr,. I. )\ lodl'l and prdi111i n ; l r~ an :Ji y, j,. Pm(. R. Sllr. 1.//IIIIi!/! . ,\f'I.
. 1. 363 1713 . 2172·12 . G llmk~tru p , :\ , S, a nd It L llatl,;("tl . 19H·L Bo rt' -h o lc ~UJ"\'c> at D)e 3.
~Qulh G recn la nd ,}. ( ;Ioriol .. 30 106 . 2B2 2Bfl . li utter. K. 19B2 . . \ m;H h('m,l tica I 1l1ud d ol'pOI) llte rma l glacier, and ict'
,,111"('1". (;{'I)/II!II .. h lr(J/II!n . Fluid I~!'II .. 21 3- 1.20 1 22 1. I I Ltt Il·r . K . I ~H::l. '1 Ii(Off/i(1I1 g/fiooloJ!.!,: I/wlrrifll ,drurr oj irr ((/111111, m,r/IIJJlif'
oj !;.IIIUI'I" lIud il"f' ,hrrl\. DlJl"lln·d ,r. , '11' .. D. Rc id d PLtbli~hill).{ Co, T uk)o. 'I'err;1 Sciell tiii, P uhl i, h ing Co,
Il uI I!'r. K. 1993 . Th t'rrno-llIcchanica ll) cou pled iet: , ll lTI t 't'~P() lt ~t'
cold, pol~the r 111 a 1. !f'11l Iw r;t\t'.). ( ,Iflciol .. 39 13 1 . Gj tlG, HU ller. K,. H, BI ,tllt"J" :\nd \ 1. FU IlL 1988, :\ mode! comp ut ,lIiol1 of
111oi,tllrt' WlLtt' tlt in pol)l lt t:" rtl t;tl ~b t'i(,I",), (;{'o/II!j'I, H(I .. 93 BIO . I~n~o,') I ~.~I I.
Ikl"l! . . \ .. K . E,lol'ln\('~(')". \\" , 1-i:J J"l'j,,,n and \1. Flin k. 1993. \ltdlatli~tlh 01'1:,,1 film itl .J ,ltub,h,I\II" I,hr.w. \\ 'l"t Gr{'l'tl\;ltld: \' ;11"1 I. \1 ,'aSlJ1TlIlt"nt, or
ll'tt1I'l'r; tlttre ;tml \Iittl'r In..! ill d,',"!, bor('hl)lt-~.]. (;1111;01 .. 39 lJi Ij ~5.
J o bmt'll. S . .J . fll/d Y olhm. 1992. I rrq.:-u lar g lat'j "J iJt tt'J">tad ia l, rcct)t'drd in
n 1\1"1 \ C ref'llbnr! ice rore . . \ ·lIllIl"r. 359 6393 1. 3 11 313,
I.("c. \\" H , K . 1970, O n d H' glnh:rl \'ari:lIiom of It: rrt'~t ria l !real 11011,
I'I!),L ,,"(lrlli 1'lal/rI, I I/Ia., 2 5 . 332 -3 11. 1.. ~'lrt'·!4"tti l l~. \ .. '\ . R,'I'h " tid P. H I!~I lJ't 'd l1., . 1990. 'TflIJO,l!,IrI/III/((Ilr/allljul
(,I{rlllllllt!. [{f/IOII. ll rrml'rh,l\ ~'l1 .. \ Ift·cd · \ \ ' tgellcr·1 tmitttl fUr l'ularli)]"«cit tlll~. LJ ibott t r~ . I .. ,1J1d 1" , Du\:t!, 19H5. \ 'a riou, iS~Jln)pk " tid 'l lli ,otJ"opic ien
IOllnd in gl:i ~'icr, a nd p ,)I, tr irt' C ,t p~ <lnd t heir \'u ITe~ pomlit lg'
rhc(l1(Jgir" . l lll/flll' ) (;I'()/I/~l'\if'(ll' . 3 2 1. 207 224 . \I.trlll'rr l. Yu. Y;r.. \1. Yu. \I o~klrk\'sk~ ami E. \ '. \ ',hilenko, 199:-1,
\ "' I f)("il~ IJI" ]",J<l i,) I\;t\,·~ in .~ I a(' i ( ' r, ,IS a n indicator of llll·ir h~dtUl ht' rtl1ai ,t;I1t:. ~tn,{'1tlft' .. ud ]"eg'itlH',]. (;Iariol .. 39 132 .37:1 :m-t.
\I orb nd. L. \\ ', 1981. Tlt rrtno· JlII"d t: J11i,·;oI ha 1aJt ("("~ o f in' ~h ,'('t 1101\.
(<rf)/d~)'I . . 1,11I111/!1"I. Flllid I~!II .. 29. :.1:n :.166. Oh tlHll"a .. \. I ~)H7 . ;\ll' " tClllPCL.tt Ut"t' di~trib ll ti on mapl> for (; r{'cnla ttc!. ~.
Ufl'll(hl'l/,d. ("{/:i(lI.~(()I .. 23 I I 45,
O hmura, .\ , ,In<l ;,\ , Rn·h . 19~11. :\n \ pl"('cip ila lio ll a nd ,tC("lI Ill Uia tiulI
Illap~ ii:J]" (; r("("Il la nd. J. ( ;Ifllil" .. 37 J25 . I ·W 1-I-1l. l';lh·nOIl. \\ '. S. B, I!JH I, Tile /IIU',in oJ,~/lIri(n. S/,f(lI/d ,dilioll, (h li"Jt·d . ,·1r. .
1\'J"g:a tllfHl l} tT\'. Reek \ -. 199 1, P;Ira JlH 'l,")"iF.aliou 01" nidi ra le a nd .~ u !"l;t(t' temper,Hure
f, n Il lI" (; n','u la tJd in' , lwei . 1)()/mJondlllll.f!,. 59 ~l 11 3 1~1.l,
\\ \,t'J"lltt;lJl . .J . 19G8. CUlt lp.tli'lul! bt't 1\ ,'I'tl nlCit, ttt 't'Ci and tlwore l ical
It'lIlpCrature profiles of tin- C a m p C(·t1 ttt 1"~. CI"t'r n !anr!. hore hole, ].
( ;/'UI)/~I' I. Nrl., 73 B . 209 1 ~700,
