Jou nal o Glaciology, Vol. 71, e123, 2025 (doi: 10.1017/jog.2025.10100) 1
On non-dimensional o ms o basal sliding laws and low
laws o ice-shee and glacie modelling
Ral GREVE1,2
1Ins i u e o Low Tempe a u e Science, Hokkaido Uni e si y, Sappo o, Japan
2A c ic Resea ch Cen e , Hokkaido Uni e si y, Sappo o, Japan
Co espondence: Ral G e e <g [email p o ec ed]>
ABSTRACT. Ice shee s and glacie s low h ough basal sliding and in e nal
de o ma ion, each go e ned by physical laws commonly exp essed as powe -
law ela ions. These o mula ions include coe icien s – he sliding coe icien
and a e ac o – whose alues and uni s depend on he espec i e exponen s.
This dependency complica es he sys ema ic explo a ion o pa ame e space,
especially in ensemble simula ions. To add ess his, we p opose dimensionless
o mula ions o bo h sliding and low laws, in which he coe icien s a e o o de
uni y and decoupled om he exponen s. This sepa a ion simpli ies sensi i i y
s udies and pa ame e a ia ions. The dimensionless laws a e s aigh o wa d
o implemen in exis ing models; we demons a e his wi h he SICOPOLIS
ice-shee model using h ee es simula ions in an idealized se -up. These
simula ions illus a e ha independen a ia ion o exponen s and coe icien s
is easible and p ac ical, suppo ing he use o dimensionless laws in e o s o
be e cons ain ice dynamics in pas and u u e clima e scena ios.
1 INTRODUCTION
Ice shee s and glacie s low due o wo di e en p ocesses, namely basal sliding and in e nal de o ma ion.
Basal sliding desc ibes he sliding o glacie ice on he unde lying subs a e, which can be ei he ha d
bed ock o a de o mable sedimen laye be ween ice and bed ock. In e nal de o ma ion is go e ned by he
– Au ho 's Ve sion –
G e e: Non-dimensional sliding and low laws 2
non-linea iscous p ope ies o “ho ” polyc ys alline ice ( ha is, wi h a homologous empe a u e T{Tm
nea uni y, whe e Tis he absolu e empe a u e and Tm he p essu e mel ing poin ).
In a dynamic/ he modynamic ice shee o glacie model, bo h p ocesses mus be included. Basal sliding,
which in eali y is a complex p ocess ha depends on a mul i ude o ac o s such as he basal empe a u e,
oughness o he bed ock, so ness o he subglacial sedimen laye (i exis ing) and hyd ological condi ions,
is usually pa ame e ized by a sliding law ha ela es he sliding eloci y o he basal s esses. In e nal
de o ma ion can be modelled by a non-linea iscous low law ha desc ibes he ela ion be ween he
mac oscopic de o ma ion (s ain a e) and in e nal s esses (e.g., Hooke, 2005; G e e and Bla e , 2009;
Cu ey and Pa e son, 2010).
Popula o ms o such ela ions a e he Wee man–Budd sliding law and he Nye–Glen low law (see
below o e e ences). They ha e in common ha hey a e exp essed as powe laws wi h some exponen s,
o which he op imal alues a e deba ed, and con ain a ac o o close he espec i e equa ion. This
ac o , he “sliding coe icien ” in case o he sliding law and he “ a e ac o ” in case o he low law,
may con ain emaining dependencies, such as on he empe a u e. In a dimensional o mula ion, he uni s
and nume ical alues o hese ac o s depend s ongly on he choice o he exponen s, which makes i
cumbe some o a y he exponen s o e hei po en ial ange o alues, o ins ance, wi hin an ensemble
o simula ions o a gi en scena io. To o e come his obs acle, we p opose ully o pa ly dimensionless
e sions o he sliding and low laws, which ha e in common ha he espec i e ac o is dimensionless
and gene ally o o de uni y. These o mula ions decouple he alue o he exponen s om he alue o he
ac o , so ha he ac o s and exponen s can be a ied independen ly. We demons a e his use ul ea u e
by some simple simula ions wi h he ice-shee model SICOPOLIS (SImula ion COde o POLy he mal Ice
Shee s; SICOPOLIS Au ho s, 2025).
2 BASAL SLIDING LAWS
Basal sliding laws (aka basal ic ion laws) ela e he shea s ess (d ag) a he base o an ice shee o
glacie , τb, o he basal no mal s ess, Nband he basal sliding eloci y, b. In a gene al, implici o m, a
basal sliding law can be exp essed as
p b, τb, Nbq “ 0,(1)
G e e: Non-dimensional sliding and low laws 3
whe e is a unc ion unspeci ied a his s age. The lis o a iables is no necessa ily exhaus i e as u he
dependencies, o ins ance on basal empe a u e o he p esence o basal wa e , may be included. No e
ha , in he p esence o subglacial wa e , he basal no mal s ess is o en unde s ood as he di e ence
be ween he ice o e bu den s ess, Nb,i, and he basal wa e p essu e, pb,w,
Nb“Nb,i´pb,w,(2)
and hen called he educed no mal s ess o , al e na i ely, he e ec i e p essu e.
A popula o m o a basal sliding law is he Wee man–Budd sliding law, which esul s when assuming
an explici o m o Eq. (1) sol ed o b, wi h powe -law dependencies on τband Nb:
b“Cb
τp
b
Nq
b
,(3)
whe e Cbis he sliding coe icien and pp, qqa e he non-nega i e sliding exponen s (Wee man, 1957; Budd
and o he s, 1979, 1984; Budd and Jenssen, 1987). Al e na i ely, Eq. (3) can be sol ed o he shea s ess,
τb“C‹
b 1{p
bNq{p
b,wi h C‹
b“C´1{p
b,(4)
whe e C‹
bis he basal ic ion coe icien . Fo he case q“0, ha is, igno ing he dependence on he
no mal s ess Nb, he abo e o ms a e o en e e ed o as he Wee man sliding law.
In p inciple, band τba e ec o quan i ies. Fo simplici y, we o mula e he sliding laws only wi h he
espec i e magni udes. Howe e , o in e p e he esul s co ec ly, i mus be kep in mind ha band τb
a e an i-pa allel o each o he due o he na u e o ic ion.
Le us now non-dimensionalize he sliding law (3) by in oducing scales ( ypical alues) o he ele an
quan i ies (e.g., Hu e and Jöhnk, 2004). We conside a si ua ion nea he edge o an ice shee whe e
G e e: Non-dimensional sliding and low laws 4
basal sliding is mos ele an :
Hs “ 1 km p ypical hicknessq,(5a)
ε“10´2p ypical su ace slopeq,(5b)
Nbs “ ρg Hs “ 107Pa p ypical no mal s essq,(5c)
τbs “ ε Nbs “ 105Pa p ypical shea s essq,(5d)
bs “ 100 m a´1p ypical sliding eloci yq,(5e)
whe e we ha e used app oxima e alues ρ«103kg m´3 o he ice densi y and g«10 m s´2 o he
accele a ion due o g a i y, which is su icien ly accu a e o he sake o a scaling analysis. This scaling is
consis en wi h he linea sliding law [Cb“10´3m a´1Pa´1,pp, qq“p1,0q] used o he EISMINT Phase 2
Simpli ied Geome y Expe imen s (Payne and o he s, 2000):
100 m a´1
loooomoooon
b
“10´3m a´1Pa´1
looooooooomooooooooon
Cb
ˆ105Pa
loomoon
τb
.(6)
An app op ia e choice o he scale o he sliding coe icien esul s om Eq. (3) as
Cbs “ bs Nbsq
τbsp.(7)
We now use he abo e scales o in oduce dimensionless quan i ies as ollows:
b“ bs˜ b,(8a)
τb“ τbs˜τb,(8b)
Nb“ Nbs˜
Nb,(8c)
Cb“ Cbs˜
Cb,(8d)
whe e he quan i ies ma ked by he ilde symbol a e he non-dimensional basal sliding eloci y, shea s ess,
no mal s ess and sliding coe icien , espec i ely. Inse ing Eqs. (8) in he sliding law (3) yields i s ully
non-dimensional o m,
˜ b“˜
Cb
˜τp
b
˜
Nq
b
,(9)
G e e: Non-dimensional sliding and low laws 5
in which all quan i ies a e supposed o be o o de uni y.
A dimensional o m o Eq. (3) can be kep by only making use o he scaling (8d) o he sliding coe icien ,
b“ Cbs˜
Cb
τp
b
Nq
b
,(10)
which has he ad an age ha i s implemen a ion in an exis ing model based on dimensional quan i ies
equi es only minimal adap a ions.
In o de o ob ain he ully o pa ly dimensionless coun e pa s o Eq. (4), we no e he scaling and
non-dimensionaliza ion o he ic ion coe icien C‹
b:
C‹
b“ C‹
bs˜
C‹
b,wi h C‹
bs “ Cbs´1{p“ τbs
bs1{p Nbsq{p.(11)
The ully non-dimensional o m o Eq. (4) esul s hen as
˜τb“˜
C‹
b˜ 1{p
b˜
Nq{p
b,(12)
and he dimensional o m in which only he scaling (11) o he ic ion coe icien is used eads
τb“ C‹
bs˜
C‹
b 1{p
bNq{p
b.(13)
Why do we p omo e using Eqs. (10) o (13) ins ead o Eqs. (3) o (4) in an ice shee o glacie
model? In Table 1 we ha e compiled some pa ame e combina ions ha we e used along wi h Wee man
o Wee man–Budd sliding laws in he li e a u e. The impossibili y o compa ing he a ious dimensional
sliding coe icien s Cb o di e en exponen s pp, qqbecomes immedia ely e iden . They do no e en ha e
a common uni , and he espec i e nume ical alue ells no hing abou he ac ual s eng h o basal sliding.
In he second case, pp, qq“p1,2q, he nume ical alue o Cbis g ea e han 109; howe e , he small
dimensionless alue means ha i p oduces only e y li le basal sliding ( ˜
Cb«0.04). By con as , in he
hi d case, pp, qq“p3,0q, he nume ical alue o Cbis me ely 10´12; howe e , i co esponds o p onounced
basal sliding ( ˜
Cb“10). The dimensionless sliding coe icien s ˜
Cbgi e a much be e idea abou wha he
espec i e alue means physically, and allow compa ing alues ac oss di e en sliding laws.
To u he s eng hen ou poin , suppose ha we wish o es a sliding law wi h a new se o exponen s,
o ins ance pp, qq “ p3,1.5q. Wo king wi h he dimensional sliding coe icien Cb, we would no ha e any
G e e: Non-dimensional sliding and low laws 6
pp, qqCb Cbs˜
CbRe e ence
p1,0q10´3m a´1Pa´110´3m a´1Pa´11Payne and o he s (2000)
p1,2q3.985 ˆ109m a´1Pa 1011 m a´1Pa 0.03985 Budd and o he s (1984):
p3,0q10´12 m a´1Pa´310´13 m a´1Pa´310 Co n o d and o he s (2020)
p3,1q1.607 ˆ10´6m a´1Pa´210´6m a´1Pa´21.607 Sai o and o he s (2016):
p3,2q6.72 m a´1Pa´110 m a´1Pa´10.672 Rückamp and o he s (2019)
Table 1. Sliding exponen s pp, qq, dimensional sliding coe icien s Cb, scales Cbsand dimensionless sliding coe i-
cien s ˜
Cb o se e al Wee man (q“0) o Wee man–Budd (qą0) sliding laws used in he li e a u e.
:: Ra he han using he no mal s ess Nb, hese sliding laws we e o mula ed wi h he p essu e head Z“Nb{pρgq.
We con e ed he sliding coe icien s gi en in hese s udies acco dingly, using ρ“910 kg m´3and g“9.81 m s´2.
idea which o de o magni ude may be sui ed o i s nume ical alue, and which ange o alues mean
s ong o weak sliding. Howe e , i he dimensionless sliding coe icien ˜
Cbis used, we can immedia ely
s a wi h an ini ial guess ˜
Cb“1and, om he e on, e ine he sliding law by, e.g., uning o obse ed low
speeds. Acco ding o he scaling (7), (8d), he dimensional equi alen o ˜
Cb“1would be Cb“ Cbs “
10´2.5m a´1Pa´1.5“3.162 ˆ10´3m a´1Pa´1.5.
We ha e only discussed cases wi h a cons an sliding pa ame e ; howe e , he non-dimensionaliza ion
me hod is o cou se no limi ed o his. I can also be applied o a spa ially a iable sliding coe icien , which
may a ise om an in e sion p ocedu e (e.g., Mo lighem and o he s, 2013). Al e na i e sliding laws, such as
he Coulomb-limi ed ules discussed by Co n o d and o he s (2020), allow simila non-dimensionaliza ion,
al hough we e ain om wo king ou he de ails he e.
3 FLOW LAWS
A simila p oblem o uni s and hugely a ying nume ical alues a ises o he low law o polyc ys alline
ice. I is a iscous low law ha ela es he s ain- a e (s e ching) enso dij o he s ess de ia o D
ij.
The s ain- a e enso is de ined as
dij “1
2ˆB i
Bxj
`B j
Bxi˙pi, j “1,2,3q,(14)
whe e xideno es he Ca esian coo dina es (x1“x,x2“y,x3“z), and iis he eloci y ec o . The
s ess de ia o is he aceless pa o he ull s ess enso ij,
ij “ ´p δij ` D
ij ,(15)
G e e: Non-dimensional sliding and low laws 7
whe e p“ ´ ii{3is he p essu e (we assume he Eins ein summa ion con en ion: summa ion o e he
wice-appea ing index iimplied, hus ii is he ace o he s ess enso ), and δij is he K onecke del a
symbol, in o he wo ds, he uni enso in index no a ion.
Fo he low law, usually collinea i y be ween he symme ic enso s dij and D
ij is assumed. We no e
he o m gi en by G e e and Bla e (2009),
dij “A pτeq D
ij ,(16)
whe e Ais he a e ac o , τe“ p D
ij D
ijq{2s1{2 he e ec i e s ess (summa ion o e bo h iand jimplied),
and pτeqis he c eep unc ion. The a e ac o depends on he empe a u e ela i e o he p essu e mel ing
poin , T1, ia an A henius law (e.g., Cu ey and Pa e son, 2010), bu i is some imes chosen as a cons an
pa ame e o simplici y. In he Nye–Glen low law (Glen, 1955; Nye, 1957), he c eep unc ion is exp essed
as a powe law,
pτeq “ τn´1
e,(17)
so ha
dij “Aτn´1
e D
ij ,(18)
whe e nis he s ess exponen . A alue o n“1would co espond o a New onian luid; howe e , he
de o mabili y o ice di e s ma kedly om ha beha iou , and he alue is equen ly chosen as n“3,
o wi hin he ange om 1.5 o 4.2 (Cu ey and Pa e son, 2010) (while ecen e idence om labo a o y
expe imen s ac ually suppo s n“1 o empe a e ice; Schohn and o he s, 2025).
The Nye–Glen low law (18) can also be in e ed o he s ess de ia o ,
D
ij “A‹d´p1´1{nq
edij ,wi h A‹“A´1{n,(19)
whe e A‹is he associa ed a e ac o and de“ pdijdijq{2s1{2 he e ec i e s ain a e (e.g., G e e and
Bla e , 2009).
Simila o he p ocedu e in Sec . 2, we now in oduce scales ( ypical alues) o he ele an quan i ies,
G e e: Non-dimensional sliding and low laws 8
conside ed sui able o a eas whe e a he la ge de o ma ions ake place:
τs “ 105Pa p ypical de ia o ic s essq,(20a)
ds “ 2.5ˆ10´2a´1p ypical s ain a eq,(20b)
whe e he scale τsis deemed app op ia e o bo h D
ij and τe. Using Eq. (18) en ails he choice o he
scale o he a e ac o :
As “ ds
τsn.(21)
We in oduce he dimensionless quan i ies
D
ij “ τs˜
D
ij ,(22a)
τe“ τs˜τe,(22b)
dij “ ds˜
dij ,(22c)
A“ As˜
A , (22d)
whe e he quan i ies ma ked by he ilde symbol a e he non-dimensional componen s o he s ess de ia o ,
e ec i e s ess, componen s o he s ain- a e enso and a e ac o , espec i ely.
Fo n“3, Eq. (21) yields As “ 2.5ˆ10´17 a´1Pa´3“7.922 ˆ10´25 s´1Pa´3. This alue is close o
he ecommenda ion by Cu ey and Pa e son (2010) o T1“ ´6˝C, which demons a es he alidi y o ou
scaling.
We ob ain he ully non-dimensional o m o he low law (18) as
˜
dij “˜
A˜τn´1
e˜
D
ij ,(23)
(all quan i ies supposed o be o o de uni y). A o m wi h only Ascaled esul s i we only apply he
scaling (22d):
dij “ As˜
A τn´1
e D
ij .(24)
Analogous o he sliding law (10), his o m can be implemen ed in a model based on dimensional quan i ies
wi h only minimal changes.
To ob ain he ully o pa ly dimensionless e sions o Eq. (19), we no e he scaling and non-dimensio-
G e e: Non-dimensional sliding and low laws 9
Fig. 1. Dimensionless a e ac o ˜
Aas a unc ion o he empe a u e ela i e o p essu e mel ing T1, ollowing
he ecommenda ion by Cu ey and Pa e son (2010): A henius law wi h ac i a ion ene gies Q“60 kJ mol´1 o
T1ď ´10˝C,Q“115 kJ mol´1 o T1ě ´10˝C,A“3.5ˆ10´25 s´1Pa´3 o T1“ ´10˝Cand n“3.
naliza ion o he associa ed a e ac o A‹,
A‹“ A‹s˜
A‹,wi h A‹s “ As´1{n“ τs
ds1{n,(25)
and o he e ec i e s ain a e, he scale (20b) is used,
de“ ds˜
de.(26)
The ully non-dimensional o m o Eq. (19) is hen
˜
D
ij “˜
A‹˜
d´p1´1{nq
e˜
dij ,(27)
and he o m wi h only A‹scaled eads
D
ij “ A‹s˜
A‹d´p1´1{nq
edij ,(28)
Figu e 1 shows he empe a u e-dependen a e ac o ollowing he ecommenda ion by Cu ey and
Pa e son (2010), non-dimensionalized wi h he scaling (21), (22d). Howe e , while he o iginal ecommen-
da ion in dimensional o m is alid only o n“3, his dimensionless o m can be used o any alue o he
s ess exponen n. I is he e o e much mo e lexible.
Conside he case n“4, which has ecen ly been discussed by, e.g., Bons and o he s (2018); Mills ein
G e e: Non-dimensional sliding and low laws 16
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