instructions for use - huscap...351. the concept of homogenization models, also called micro-macro...

Instructions for use

Title A Continuum-Mechanical Model for the Flow of Anisotropic Polar Ice

Author(s) Greve, Ralf; Placidi, Luca; Seddik, Hakime

Citation低温科学, 68(Supplement), 137-148Physics of Ice Core Records II : Papers collected after the 2nd International Workshop on Physics of Ice Core Records,held in Sapporo, Japan, 2-6 February 2007. Edited by Takeo Hondoh

Issue Date 2009-12

Doc URL http://hdl.handle.net/2115/45440

Type bulletin (article)

Note II. Ice-sheet flow model

File Information LTS68suppl_013.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

A Continuum-Mechanical Model for the Flow of Anisotropic Polar Ice

R,,11f Greve * , Luca Placidi .. , Hakimc Scddik •

, Institute oj Low Temperatllre Science, Hokkaido University. Sapporo. Japan: greJle@/oIVtem.hoklldai.ac.jp

" Department a/Structural alld Georeclmical Engineering, "Sapienw", UlliI'ersiry of Rome, Rome, Italy;

IlIca.placid;@III1;romo I. it

Abstract: In order \0 study the mechanical behaviour of polar ice masses, Ihe method of continuum me­chanics is used, The newly developed CAFFE model (Continuum-mechanical, Anisotropic Flow model, based on an anisotropic Flow Enhancement faclor) is described, which compriscs an anisotropic now law as well as a fab­riccvolution equation. The now law is an extension orthe isotropic Glen's now law, in which anisotropy enters via an enhancement factor that depends on the deformability of the polycrystal. The fabric evolution equation results from an orientationalmass balance and includes constitu­tive relations for grain rot,lIion and recrystallization. The CAFFE model fulfi lls all the fundamental principles of classical continuum mech,lIlics, is sufficiently simple 10

allow numerical implementations in icc- now models and contains only a limited number of free parameters. The applicability of the CAFre model is demonstrated by a case study for the site of the EPICA (European Project for Icc Coring in Antarctica) icc core in Oronning Maud Land, East Antarctica.

Key words: Ice, polycrystal, now, anisotropy, continuum mechanics, icc core

1 Introduction

Ice in natural land icc masses, such as polar ice sheets, ice caps or glaciers. consists of zillions of individual hexagonal crystals ("crystallites" or "grains") with a typ­ic;!1 di;!meter of millimeters to centimeters. This length scale stands in contrast with the size of the ice masses. whic h mngcs from 10<)"s of meters to ](:x:x)"s of kilome­ters. It has long been known that. while the distribu­tion of the crystallographic axes (also known as optical axes or c-axes) at the surface of an ice sheet is essentially at random. deeper down into the ice, ditTerent types of anisotropic fabrics with preferred orientations of the c­axes tend to develop.

Many models have been proposed to describe the anisotropy of polar icc. On the one end of the range in complexity. a simple now enhancement factor is in­troduced in an ad·hoc fashion as a multiplier of the

isotropic ice nuid ity in order to ;!ccount for anisotropy ;!ndlor impurities. This is done in most current icc-sheet models, oftcn without explicitly mentioning anisotropy [19, 25, 46}. In macroscopic, phenomenological mod­els, an an isotropic macroscopic fonnulation for the now law of the polycrystal is postulated. To be usable, the rheological parameters that enter this law must be eval­uated as functions of the anisotropic fabric 114, 15, 34, 351. The concept of homogenization models, also called micro-macro modcls, is to derive the polycrystalline be­haviour al the level of individual crystals and the fab­ric [I, 4. 5,17, 26,29,50, 51]. As for the "high-cnd" complexity. full-field models solve the Stokes equation for ice now properly by decomposing the polycrystal into many elements, which makes it possible to infer the stress and strain-rate heterogeneities at the microscopic scale [27, 28, 30, 31]. A very comprehensive, up-to-date overview of thcse different types of models is given by Gagl iardini et al. [131 (in th is volume). However, the more sophisticated models arc usually too complex and computationally time-consuming to be included readily in a model of macroscopic icc flow.

Here, the Continuum-mechanical. An isotropic Flow model, based on an anisotropic Flow Enhancement factor ("CAFre mode]"), shall be described. It belongs to the elass of macroscopic models, and is laid down in detail in the study by Placidi et al. [42} (based on previous works by Faria [10, IIJ, Fariaet al. [12}, Placidi [40, 41 [, Placidi and HUller L43J). The now enhancement factor is taken as a function of a newly introduced scalar quantity called deformabilitJ. which is essentially a non-dimensional in­variant related to the shear stress acting on the basal plane of a crystallite, weighted by the orientation-distribution function which describes the anisotropic fabric of the polycrysta1. Fabric evolution is modelled by an orienta­tion mass balance which accounts for grain rotation and recrystallization processes.

The CAFre model fulfills all the fundamental princi­ples of classical continuum mechanics (see also Placidi et al. [44J), and it is a good compromise between nec­essary simplicity on the one hand. and considemtion of the major effects of anisotropy on the other. In order to demonstrate its performance, the model is applied to the site of the EPICA (European Project for Ice Cor­ing in Antarctica) ice core in Dronning Maud Land, East

-137 -

Quantity Value

3 Stress exponent, n Pre-exponenti al constant, Ao

Activation energy, Q

3.985 X 1O- 13 S- 1 Pa - 3

1.916 x 103 s- 1 Pa - 3

60kJ mol - 1

(forT' ::; 263.15 K) (for T' > 263.15 K)

(forT' ::; 263.15 K) (for T' > 263.15 K) 139kJ mol - 1

Tablc I: Physical parameters for Glell's jfow law.

Antarctica. for which data o n the icc now as well as on the ani sotropic fabric arc available.

2 CAFFE model

2.1 Glen's flo w law Let us brie ny review the isotropic case, for which poly­

crystalline ice is treated as an incompressible, viscous nuid. The Cauchy stress tensor T is split up according 10

T = - pI +5, (I)

where p denotes the pressure, and 5 is the traceless stress deviator (tr S = 0). Due to the incompressibili ty, the Oow law only determ ines the stress dev iator 5 and reads

5 = 211D, (2)

where D = sy m grad v is the strain-rate tensor (sym met­ric part of the gradient o f the veloci ty v ), and the coeffi ­e ient 'I) is the shear Iliscosity (or simply the I' iscosity). Its inverse, thejfuidity. cun be factorized as

~ ~ 2EA(T' )J(a ), 'I

(3)

where

(4)

is thc effective stress (sq uare root o f the second invari­ant o f the stress deviator), and the creep fll llctioll / (0' ) is g iven by the power law

/ (0' ) = O'n- l (5)

(Ihe parameter n is ca lled "stress exponent"). Further, the rate faclOr A (T' ) depends on the temperature rela­tive to pressure mel ting T' = T - Till + To (T: ab­solute temperature, T,,, = To - {3p: pressure mehing point, To = 273.16 1(: melting point at zero pressure, (3 = 9.8 X 10- 2 K :\ IPa- 1: Clausius-Clapeyron constant) via the Arrhe nius law

A(T' ) = Ao e-Q / RT', (6)

where Ao is the pre-exponential constant, Q the acti va­tion energy and R = 8.3l4Jmol- 1 K- I the universal gas constan\. The jfow ellhallcemcl1I f aclOr E is equal to un ity for pure iee, and can be set to values deviating from unity in order 10 account roughl y for e ffects o f impuriti es and/or anisotropy (Paterson 137]).

The isotropic now law for ice is now obtained by in­serting Eq . (3) (with the spec ifications of Eqs. (5) and (6)J in the viscous Oow law (2). Th is yields

D ~ EA(T' )J(a) 5, (7)

wh ich is ca lled Nye's generalization of Glen·s jfow law, or Glell ·sjfow law fo r shon (e.g. , Greve and Bl aller [20], Hooke [23] , Paterson [38], van der Veen [52]). Suitable values for the several parameters arc listed in Table I.

2.2 Anisot ropic generalization of Glen's flow law

2.2.1 Deformation of a crystallite

In order to derive a gencralization o f Glen's now law (7) which accounts for general, anisotropic fabrics of the ice pol ycrystal, we first considcr the de fo mlat ion of a crys­tallite embedded in the polycryst<l li ine aggregate. Fo l­lowing Pl acidi et al.[421. onl y thc dominnnt deformatio n along the bnsal plane is accounted for, whereas deforma­tio ns along prismatic and pyramidal planes. which arc at least 60 times more difficult to activate, shall be neglccted (fig. I ).

<C' : !

." 'y" "'.

Basa! Prismatic

'"V , ,"". , .'. " ,'. a " :\ . '. .. :'~ : \ '. . , ..... 1'\, "' ... '

Pyramidal

Figure I: Basal. prismatic and pyramidal glide planes ill the hexagonal ice crystal. sketched as a riglit liexagollal prism (Faria /9J).

Let n be the normal unit vector of the basal plane (di ­rection of the c-axis), then Tn is the resolved stress vec­tor (Fig. 2). Note that the tensor T is interpreted as the macroscopic stress which docs not depend on the orienta­tio n n . It is reasonable to assume that onl y the stress com­ponent S l tangenti al to the basal plane (resolved shear stress) contributes to its shear deformation , while the component nomlal to the basal pl ane has no effect.

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basal plane

Figure 2: Decomposition of the stress vector inlO a parI normal and a parrlangen/ialro rhe basal plane.

According to Fig. 2, the decomposition of the stress vector reads

Tn = (Tn· n)n + Stt., (8)

where t denotes the tangentia l unit vector. Inscrting the decomposition ( I) of the stress tensor T readity elim i­nates the pressure p and leaves

5n = (5n · n)n + Stt. (9)

As mentioned above, deformation of the crystalli te in the polycrystalline aggregate is attributed 10 the tangential component St only. Since wc aim at a theory which de­scribes the effects of anisotropy by a scalar, anisotropic flow enhancement faclOr, we define the scalar invariant

S~ = 5n · 5n - (5n · n )2 . ( 10)

This quantity has the unit of a stress squared. and so a nat­ural way 10 non-dimensionalize it is by the square of the effective stress a [Eq. (4)], which is also a scalar invari­ant. T hus, we introduce the deformabilitJ of a crystallite in the polycrystalline aggregate. which is loaded by the stress T , as

(II)

The factor 5/2 has been introduced mcrely for reasons of convenicnce, as it will become clear below.

2.2.2 Flow la\\' for polycrystalline ice

In polyerystalline icc, the crystallites within a control volume (which is assumed to be large compared 10 the crystallite dimensions, but small compared 10 the macro­scopic scale of icc flow) show a certain fabric. Extreme cases arc on the one hand the single maximum fabric , for which all c-axes are perfectly aligned, and on the other hand the isotropic fabric with a completely random di stri­bution of the c-axes. A general fabric, which is usually in between these cascs, can be described by the orielllarion mass densit), (OM D) p· (n ). It is defined as the mass per volume and orientation, the latter being specified by the normal unit veclOr (direction of the c-axis) n E S2 (S2 is

the unit sphere). Evidently, when integrated over all ori­entations, the OMD must yield the nOnl1:l1 mass density p, which leads to the normalization condition

J p'(n)d2n = p.

S'

( 12)

Alternatively, an orielllatioll dislribuliollfilllctioll (ODF) f' (11) can be defined :IS

n u) ~ p'(U), p

(13)

which is normalized 10 unity when integrated over all ori­enl<ltions.

We use the ODF in order to define the deformabilifJ of polycrystall ine ice by weighting the deform ability of the crystallite (II),

A = J A' (n)f'(n ) d2n

S'

5 J S;(U)f' ( ) I' ~- -- n (n 2 u'

s'

= sJ 5n ·Sn - (5n . n )2f' (u)d2n tr (52) .

S'

( 14)

Note thai, for the isotropic case, the ODF is r(n) = 1/(47r), and from Eq. ( 14) we obtain a deformability of A = 1 (Pl ilcid i el al. [42]). For that reason, the factor 5/2 has been introduced in Eq. (II).

UC/SM !

+-W-+ i

SS/SM -rn +--

Figure 3: Uniaxial compression on single maximllm (UC/SM) al1(l simple sllear on single maximum (SS/SM) for a small sample of polycrystallille ice. Stresses are ill­dicated as black arrows, alld tile single maximllmfabric is marked by tile dark-grey arrows within 'he ice sample.

TIle envisaged flow law for anisotropic polar icc can now be formulated. Essentially, we keep the form of the Glen's flow taw (7), but with a scalar. anisotropic en­hancement factor E(A) instead of the parameter E,

D ~ E(A) A(T')f(u) S. (15)

The func tion E(A) is supposed to be strictly increasing

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with the dcformabil ity A, and has the fixed points

£ (0) = Em;n

(uniaxial compression on single maximum),

E (l ) ~ I

(arbitrary stress on isotropic fabric) , ( 16)

£(~) = Em"x

(simple shear on single maximum).

T he " hard" case ( 16h and the "soft" case (16h arc illus­trated in Fig. 3. Note also that the defonnability cannot lake values larger than A = 5/2 (Placidi et a1. [42]).

For the detailed form £ (A) of the anisotropic en­hancement factor, in addition to Eq. (16), we demand that the function is continuously differentiable, lhat is, E E Gt [O, !J. Moreover. Azuma 121 and Miyamoto [32] have experimentally verified that the enhancemcnt factor depends on the Schmid fac tor (resolved shear stress) to the fourth power. that is, on the square of the deformabi l­ity A. T his yields

E(A ) ~

E",;n + (1 - E", ;n )A ' ,

8 £ ",,,x - 1 t = - ,

21 1 - E m;n

o ~ A ~ 1,

4A2(E rnax - 1) + 25 - 4Ernax

2 1

(17)

(for details sec Plaeidi et a1. L42]). Several investigations (e.g. Budd and Jacka [3[, Pimienta et al. [39], Russell­Head and Budd [45]) indicate that the paramcter E max (maximum softening) is approximately equal to ten. The parameter Em;n (maximum hardening) can be realisti­cally chosen between zero and one tenth , a non-zero value serving mainly the purpose of avoiding numerical problems. The fUllction (17) is shown in Fig. 4.

10

j 9 E "" 10 8 m"

E . "" 0 7 moo C 6 • E 5 • 4 g • 3 ~ 2 c w 1 --------- ------------ - --

00 05 1 15 2 25 Deformabitity

Figure 4: Aniso/ropic enhancelllellf faclOr £ (A) as a fllll c/ion of the deformability A according to £q. (17).for Emax = 10 and Em;n = O.

2.2.3 Inversion of the flow la w

As long as the creep function / (0-) is given by the power law (5), the anisotropic fiow law (15) can be inverted an­alytically. We find

(18)

whcre

( 19)

is the ejjeclil"e slrain rale. 11le defomlability A also needs to be expressed by strain rates instead of stresses [sce Eq. ( 14)]. In <lnalogy to Eq. (9), we consider the resolved strain-rate vector On in a crystallite in the poly­crystalline <lggregate, and decompose it according 10

On = (On · n )n + Dtt , (20)

where D t is the resolved shear rate in the basal plane (see also Fig. 2). As in Eq. ( 10), wedefinc the scalar invariant

D? = On · On - (On · n )2 . (21 )

Owing to the collinearity of the tcnsors 5 and 0 [sce Eqs. ( 15) and (18)], the deformability of a crystallitc in the polycrystall inc aggrcgate [Eq. (1 I) I can be readily ex­pressed by D t and d,

A "( ) = ~ D~(n) = ~ On · On - (On · n )2 (22) n 2 J2 ;) tr ( D2)

and the de formability of polycrystalline ice [Eq. ( 14)[ yields

A = J A ' (n ) J' (n) d2n

s'

~ 5/ On · On - (On · n )2 J' (n) d2n. (23) tr (0 2)

S'

This completcs the inversion of the an isotropic now law.

2.3 Evolution of a nisotropy

2.3. 1 O ricn ta tionlllass ba lance

The anisotropic now law in thc form ( 15) or ( 18) needs to be complementcd by an evolution equation for the anisotropic fabric . This is done by formulating the ori­ellfarion lIIass balance for the OMO p' ( 11 ).

-140-

" ,

" "

-7-,

'" n

/ s'

with Ihe usc of the Gauss theorem and Ihe mass­conservation requirement

J p" (n) r" (n) d2n = O.

s'

(27)

In order 10 solve lhe orientation mass balance (25), const itutive relations for Ihe orientation transition rate u ' (n ). the orientation flu x q ' (n ) and the orientat ion pro­duction rale r" (n ) need 10 be provided as closure cond i­tions.

2.3.2 Constitutive relation for the orientation transi-Figure 5: Orientation transition rare u "(n ) 011 the IlIIil lion ra te ~phere 52.

We arc not going to enler Ihe detailed formalism of orientation balance equations here (sec Pl acidi et al. [42j. :lnd references therein), Instead, we rather motivate the form of the orientation mass balance by generalizing the ordinary mass balance. The d ifference is that, in addi­tion to the dependencies on the posi tion vector x and the time t, the density and velocity fields also depend on the orientation vector n E S2, which is indicated by the no­tation p' (n ) and v' (Il). The velocity, which describes motions in the physical space, is complemented by an ori­elltaliol1frallSifioll rale u - (n ), which describes motions on thc unit sphere, that is. changes of the oriental ion due 10 gra in rOlation (Fig. 5). Also, an oriel//aliol1 flux q ' (n ) is considered. wh ich allows redistributions of the OMD due to rotation recrystallization (polygon ization). Conse­quently, the orientat ion mass balance reads

&!:t' + div (p·v' )+ divs~(p" u· + q· )= pT ·. (24)

The first two lerms on the left-hand side arc straightfor­ward general izat ions of the terms in the ordinary mass balance. The third tcrm on the left-hand side is the equiv­alen t of the seeond tcrm for the orientat ion transition rate lI' (n) and the orientat ion nux q "(n ), where divs2 is the divergence operator on the unit sphere. On thc right-hand side, a source term appears which allows that certain ori­entations can be produced:ll the expense of others. TI1C

quantity [' (n) is thcrefore called the Orielllalioll plVdllc­lioll fale. Physica ll y. it describes migration recrystalliza­tion a nd all othcr proccsses in which Ihe transport of mass from one grain , having a certain orientat ion, to another grain , having a different orientation, cannot be neglected.

In the following, we will make the reasonable assump­tion lhat the spatial velocity docs not depend on the ori­clllalion, Ihal is, v' (n ) = v. Therefore, lhe orienUltion mass balance (24) sim plifies to

&:t' + div (p"v)+ di vs2(p' 1I' + q ") = pT". (25)

Intcgralion over S2 (all orientations) gives lhe classical m3SS balance

c;:: + div (pv ) = 0, (26)

As mentioned above, lhe orielllation transition rate corre­sponds physically to gra in rotation. Since grain rotation is induced by shear deformation in the bas31 plane, we argue that it is controlled by the resolved shcar ratc Dtt [Eq . (20)[, and use the relation

u" (n ) = -t Dtt+Wn

= t · [( On ' n ) n - Dn[ + Wn (28)

(sce, c.g. , Dafal ias [6]). The parameter L is assumed 10

be a posi tive constant. The <lddi tional term Wn with the spi n tensor W = skw grad v (skew-symmetric part of the grad ient of the veloc ity v) describes the contribution of local rigid-body rotations.

In the special case L = 1, the basal planes are m<lterial area elements, that is, they carry out an affine rotation. However, due to geometric incompatibilities o f the de­formation of individual crystallites in the polycrystalline aggregate, an affine rotation is not plausible, and we ex­pect realistic values of L 10 be less than unity.

2.3.3 Constitutive n!lation for the orientation flux

The orientation flux is supposed to deseribe rotation re­crystallization (polygonization). Following the argumen­tation by GOdcTl [16[, it is modelled as a diffusive pro­cess,

q' (n) ~ - .\gmds, [p' (n)H ' (n)[, (29)

where the parameter). > 0 is the orientation ditTusivity, grads~ is the gradient operalOr on the unit sphere, and the "hardness" 'H.' (n ) is a monotonically decreasing function of the crystallite deformability A' (n) [see Eq. (l1)J. A simple choice for the hardncss function would therefore be

H' (n) ~ , A" (n )+f

(30)

the offset f « 1 bei ng introduced in order to prevent a singularity for A' = O. However, recent results by Du­rand et a1. [8[ suggest that rotation rceryst<lll izat ion is an isotropic process not affected by the orientation. In this case, the choice

H' (n): ' (31 )

is indicated, which renders Eq. (29) equivalent to Fick's laws of diffusion on the unit sphere.

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2.3.4 Constitutive rela tion fo r the o rientation pro· duction ra te

The driv ing forcc for the oricntation production rate. which models essentially migration recrystallization. arc macroscopic dcformations of the polycrystal . which can be more easily followed on the microscopic scale by grains oriented favourably for the given defonnation. Therefore. it is reasonable to assume that the orientation production rate for a certain orient:llion 11 is related to the crystallite deformability A' (ll ) lEqs. (11). (22)]. In the CAFFE model, the linear relation

r' (n) ~ r [A' (n) - AJ (32)

is proposed. Subtrac tion of the polycrystal deformabi l ~

ity A is required in order to fulfill the mass-conservation condition (27). The parameter r is assumed to be pos­itive, which guarantees a positive mass production for favourably oriented grains. and a negative production for unfavourably oriented grains (Fig. 6).

r

0+---:;*"-----A A'

Figure 6: OrielllariOIl production rate according to Eq. (32).

The CAFFE model is now fomlul:lIed completely. Equat ion ( 15) is the actual now law, which replaces its isotropic counterpart (7). Anisotropy enters via the en­hancement factor E(A) [Eq. (17)!, which depends on the deformability A defi ned in Eq. (14). Computation of the deformability requires knowledge of the orientation mass density p' , which is governed by the evolution equation (25) and the constitutive relations (28). (29) and (32).

3 Application to the EDML ice core

3. 1 Methods We have developed a one-dimensional now model , in­

cluding the CAI-"""FE model, for the site of the EPICA icc core at Kohnen Station in Dronning Maud Land, East Antarctica ('"EDML core", 75°00'06/15, 00°04'04"8. 2892 meters above sea level; see hnp:llwww.awi­bremerhaven.de/Polar/Kohnenl). For th is core with an overall length of 2774 m, preliminary fabric data are available from 50 m until 2570 m depth (I. Hamann, pers. comm. 2006). The t":lbric is essentially isotropic down to 3pproxim3(ely 600 111 depth. and shows a gr3dual tr3n­sition to 3 broad girdle fabric between 600 Olnd I()(X) m depth. Further down, the girdle fabric narrows until ap­proximately 2000 m depth. The fabric then experiences

an abrupt change towards a single maximum, which pre­vails below 2040 III depth. Tendencies of secondary or multiple maximOl are observed at several depths. The complete data set and a detailed interpretation will be pre­sented elsewhere (Hamann ct a1. [22]).

The location of the EDML site on a nank (rather than a dome like most other ice cores) allows deriving a one­dirncnsionaillow model based on the shallow-icc approx­imation (Hutter l24J. Morland [33]), with which the per­formance of the CAFFE model can be tested. We define a local Cartesian coordinate system such that Kohnen Sta­tion is located at the origin. the x-axis points in the 260° (WSW) direction, the y-axis in the 170° (SSE) direction , 3nd z (depth) points vertic311y downward (Fig. 7).

.74.51' 11".

·74'54'

·74·57"

·75·00'

x (260°) -75·03'

, Figure 7: Local coordinate system for the EDML site. UI/derlaid topography map by Wesche el al./53/.

Accord ing to the topographical data by Wesche et ::II. [531, the x-ax is is approximately aligned with the down­hill direction, and the gradient of the free surface eleva­tion, II, is

8h ,I ""i.l = - 9 x 1 0~ ± 10%, vI

oh -= 0. oy (33)

Th us, in the shallow-ice approximation, the only non­vanishing bed-parallel shear-stress component is Txz (= S"!z ), given by

(34)

where 9 is the acceleration due to gravity. Combination with the x-z-componenl of the Glen 's fl ow law (7) yields the isotropic horizontal velocity,

/I

v"! = -2P9~~ J A(T')an~ l i dE (35) ,

(e.g .. Grcve 1181, Greve and BI31\er [20]), where H is the icc thickness, the rate fac tor A(T') rmd stress exponent n arc chosen as listed in Sect. 2.1. and the enhancement factor E has been sct to unity. Sim ilarly, for anisotropic

-142-

,.r---r------------,

,'"\'----_c,c----_o,----c_,c----e, Ve<lICai .,.81"....&11 (.-' I 10-')

' rr---------------,

I = 1500 §

,ooo~=__c~_c=c--=o--~ 230 240 :lSO 260 270

TemperMure (K)

Figure 8: Dansgaard-JollIIsen type distributions oj the vertical strain rate (left panel) Gild the tempera/ure al the EOML j';te (righl panel). The depth of the killk.\" is a/ Iwo-rhirds oitlle focal ice thickness. The strain rale at the surface has been chosen slIch that Ihe dowl!\rard vertical velocity equals the accumulation rate. and the .~lIrface alld basal temperatures match the ice-core data.

condi tions and the correspondi ng flow law (15), the hori­zontal ve locity is

fI

oh J - , Vx = - 2P9&x E (A ) A(T' ) (7n- i dE, (36)

,

with the enhancement faclor function E(A) of Eq. (17). Note that no-slip conditions have been assumed at the ice basco that is, v'" (z = H) = O.

The unknowns in Eq. (36) arc the normal dev iatoric stresses (Su . Sy y, S zz ) which arc required together with the shallow-icc shear stress (34) for computing the de­formability A by Eq. ( 14), and then the enhancement fac­tor E (A) by Eq. ( 17). The normal deviatoric stresses arc computed by application of the inverse anisotropic fl ow law (18) with the deformability in the limn (23). The latter is evaluated with the calculated shallow-icc defor­mations :md an assumed venieal strain rate D:: in the form of a Dansgaard-Johnsen type dis tribution l71. which consists of a constant val ue of Dz: from the free surface down to two thirds of the ice thickness, and a linearly decreasing value of Dz: below. A simil ar d istribution is empl oyed for thc temperature profile (sec Fig. 8). We also assume cxtension in the x-dircetion onl y, so that the onl y non-zero horizont:ll strain rate entering the evalu:ltion of Eq. (23) is D~~ = - Dz: . Thc vertical velocity V z results from integrating the prescribed vertical strain rate D n ,

which gives a linear/quadratic profile (e.g., Greve el al. [2 ID.

For the ODF, we usc the preliminary data of the EDML fabric described above. However, since during the drilling process the orientation orthe core is not fixed. the horizontal orientation of the non-circularly symmet­ric g irdle fabric (between approximately 600 and 2040 m

depth) relative to our coordinate system , i.e., the direc­tion of ice flow, is unknown. For this reason, we need (0

assign an orientat ion for the fabric whcn computing the enhancement factor. We consider two limiting cases by rotating the initial data such that the girdle fabric at all dcpths is al igned with the x-axis (casc "RI T') and with the y-axis (case "R23"). respectively. This is illustrated in Fig. 9.

:::r:: " : .. .... :. :;.:: ." . • ;,s •• ."::.. r " .....

Figure 9: Sketch of the rotation of the girdle fabrics in order to align wirh the x -axis (case "RI3") and lI'irh the y-axis (case "R23") in/he Schmidt projection.

At thc surface. we assume isotropic condit ions, so that As = 1, and for the maximum softcning and hardcning parameters. we usc the values Erna~ = 10 and Em;" = 0, respectively.

In a second step, we attempt at solving the fabric evo­lution equation (25). For the lack of better knowledge, we neglect recrystallization. that is. we sci). = 0 and r = ° in the constitutive relations (29) and (32), respectively. By allowing only a dcpcndency of the oricntation mass density p' on the vertical coordinate :: (one-dimensional steady-state problem) and on the orientat ion n , the orien-

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tatian mass balance (25) yields an equation which gov­erns the fabric evolution along the EDML icc core.

(37)

where the orientational gradient operator Oi and the ori­entation transition rate u; , respectively, read in index no-tat ion as

(38)

and, due to Eq. (28),

l t l = t DMnhn ",ni - t D i j1lj + W ;j ll j . (39)

With Eq. (38), and by inserting the constitutive relation Eq. (39) in Eq. (37), it follows

op' •••• oz 'Uz + u; 0iP + P o;,u i

op' • = oz Vz + (Hlij - ~D;j)n/)i P

+ 3LP' DI! ",n~.nh = O. (40)

We assume that the local now field consists of vertical compression with the compression ratc (ncgative vcrt ical strain rate) c = -{)u~/{)z according to Fig. 8, horizontal extension in x -directi on, and the horizontal, bed-parallel shear rate

'Y = fl8Vz = 2pg fl8

h t eA ) ACT' ) O'n- t Z

, .< (4 1)

that results from Eq. (36). The velocity gradient L = grad v then reads

(

€ 0 l ~ 0 0

o 0 (42)

Conscquently, we obtain for the strai n-rate tensor 0 and the spin tensor W

D ~ ( € 0

101

) 0 0

1 'Y 0 -€

(43)

and

W ~ ( 0 0 11 ) 0 0 o . -h 0 0

(44)

With these expressions and the introduction of spherical coordinates, Eq. (40) reduces to

8p' 4 {)z V z

+ 3tp' [c(2sln2 800s 2r.p - 1 - 3eos 28)

+ 2'Y sin 28 cos r.pj

8p' [ sill r.p] + 2-8

asin 2rp + ')'{ - 1 + t) --" rp tan Q

8p' [ 1 + 2 {)8 - '2l£sin28 (cos2rp+ 3)

+')'(1 -t eOs 28) eOS (fl] = 0, (45)

where 8 and (fI arc the polar angle (co-latitude) and az­imuth angle (longitude), respectively. Note that, due to Eq. (41), the shear rate ,), depends on the fabric via the deformability A.

The shear now at the EDML station leads to the transport of ice particles over significant horizontal dis­tances. Huybrechts et al. (251 estimate. based on three­dimensional now modelling, that particles at 89% depth of the core originate from ~ 184 km upstream. Thi s is not taken into account in our spatially one-dimensional model. However, the variation of the shear upstream of the drill site is likely small due to the small varia tion of the surface gradient (Fig. 7), so that the error resulting from the neglected horizontal inhomogeneity should be limited .

In this study, we restrict the solut ion of Eq. (45) to the simplified case of a transversely isotropic (circularly symmetric) fab ric, so that the OMD p' i~ only a func­tion of the depth z and the polar ang le 8. TIlen Eq. (45) becomes, after integration over the azimuth angle (fl.

op' op' 4 {)z 'UZ - {)8 31£ sin 2(}

- 3tp'c(1 + 3 cos 28) = O. (46)

Equation (46) is solved by using a finite-difference dis­cretizat ion with the parameter t = 0.6.

3.2 Results

Figure 10 shows the variation of the enhancement fac­tor, the ice nuidity and the horizontal velocity along the icc core, computed with the ODF based on the fabric data described above. For both limiting cases R 13 and R23. the enhancement factor is close to unity in the up­per 600 m. which reflects the nearly isotropic fabrics in that pari of the EDML core. Further down, in the girdle fabric regime, the case R 13 is characterized by a moder­atc increasc of the enhaneemcnt fac tor to an average valuc of about two, whereas the case R23 exh ibils a strong de­crease of the enhancement factor to values close to zero. This demonstrates clearly that the girdle fabrics produce a sign ificant ly different mechanical response depending on the orientation relative to the icc now. Case R23 is probably closer to reality. because in the girdle fabric regime above 2000 m depth the deformation is essentially pure shear (vertical compression , horizontal extension in x -direction only). For this situation, a simple "deck-of­cards" model illustrates that thc coaxes turn away from the :r-axis and towards the z-axis. whcrcas nothing hap­pens in y-dircction. so that in the Schmidt projection a concentration perpendicular to the x -axis (now direction) results.

Below 2(x)() m depth, where the fabric switches 10 ,I

single maximum , the difference between the cases R I3 and R23 essentially vanishes. The crystallite basal planes

-144-

'~~-----C~~~ I --+--- DDlaRI3 ___ o.t.R23

, , - - - Isotropy , , __ Oat. AU

1--e-- 1hl1.o RI3 I ___ DatoA23

, '"

, ___ [)ala A23

, .. , , , , I , , , , , ,

:; 1500 £ 150(

! !

, I ,

, , ,

I :; 1500

! , , , .. -~ , ,

U , , , ,~ ,~ ,~

, . . .

''',~--",----.~--".---c.~--!,., "',~-cc-~,c--o,---.~-cc-~. ... , 6.2 0.. 0.6

Hori.ont.1 velocity 1m ,-') , ..

Enhanc........"I"",,,, Fluidity (m'lNt _ 10-1~

Figure J 0: Varia/ioll of ",e enhancement faclOr (lcCl panel), 'he ice iluidif), (middle panel) and rhe horizonral velocity (right panel) al0118 (he £DML ice core . .. Data R 13" and "Dara R23" represent rhe soll,,;olls obtained wirh the measured girdle fabrics !"Otalet! fO align wirll 'he x - Gild y-direcliol1. respectively, alld " ISOTropy'- represellfs iSOTropic condiTions.

arc favourably oriented for the now prevailing simplc­shear deformation, which leads \0 large defonnabilitics. Consequently, the enhancement faclOr shows a sharp in­crease to a maximum value of about eight, which is close to the theoretical maximum of E lllnx = 10.

The variabi lit ies of the enhancement factor and the ef­fective stress. as well as the increase of the temperature with depth, contribute to the nuidity profiles shown in Fig. lOb. Since the nu idity is very sma ll above 2000 m depth and increases only funher down, the di fference be­tween the cases R 13 and R23 in absolute values is sur­prisingly small. At 2563 m depth , the fluidity is about 200 times higher than the fluidity at 1000 m depth for the case R23 due to the counteract ing contributions from the favourably oriented c-axes, the higher temperature and the smaller e ffective stress. The lattcr is somewhat sur­prising; it is caused by the normal deviatoric stresses Sx:e and S zz , wh ich decrease strongly below 2000 m depth and ou tweigh the innuence of the increasing shcar stress Sxz in the effective stress.

Owing to the large enhancement factors close to the bottom, the an isotropic now law predicts significa ntly larger horizontal vclocit ies compared to the isotropic now law for the ent ire depth of the icc core (Fig . 10e). Atthe surface, thc an isotropic horizontal veloci ties arc by ap­proximately a faclOr 3.5 larger than their isotropic coun­terpan s, and the absolute value of ~ 0.7 m a-I agrces very well with measurements (H. OeTter, pers. comm. 2005; Wesche et a!. [53]). The difference between the cases R I3 and R23 amounts to ~ 10%, the larger val­ues bei ng obta ined for the case R23 owing to the slightly larger e nhancement factors below 2000 m depth. Inter­esti ngly. these differences show that the fabrics are not perfectly transversely isotropic below 2000 m depth, even though they are very close 10 the sing le-maxi mum type.

Let us now turn to the simu l:lIion in wh ich the fabric evolut ion is computed by solving Eq. (46) for a trans­versely isotropic fabric. Although this assumption is not consistent with the observed gird le fabric between ap­proximately 600 and 2000 m depth and is therefore a gross simplification. it is interesting to study the mechan­ical response of such a simplified system and the di ffer­ences to the ice now resulting from applying the mea­sured fahrics.

Figure ll a shows the comparison between the en­hancement factors resulting from the computed, trans­versely isotropic fabric (which wi ll be referred to as "modelled enhancement factor" in the following) and fro m the fabric data. Evidently, the agreemcnt is good despite the assumption of transverse isotropy. Down to 1800 III depth, the modelled enhancement factor lies in between the cases R l 3 and R23, which are the limiting cases for the orientation of the measured gird le fabric with respect to the ice-flow direction. Between 1800 and 1900 m depth, the modelled enhancement factor is very close to the low values of the case R23, for which the girdle fabric is aligned perpendicular to the now direc­tion. Below 2000 m depth , the sharp increasc is :l1so well reproduced; however, the maximum of the modelled en­hancement factor is more pronounced :md lies closer to the boltom than for the cases R 13 and R23.

For that reason. the modelled enhancement t:1ctor leads to larger near-basal shear rales than the enhancemcnt fac­tor based on the cases R 13 and R23. Consequently, the horizontal velocity resulting from the modelled enhance­ment factor is larger by about a factor two than the ve­locities for the cases RI3 and R23 (Fig. I [b). At the sur­face, a value of ~ 1.5 ma- 1 is reached, which is twice the measured surface velocity. Thi s highlights the great sensitivity of the ice dynamics 10 the processes ncar the

-145 -

'-V~------r"=~ I J _._ , ___ Dal. R13

- . - Dal. A23

'000

,~ ,~

'OOO,~-",c--.c--".--c.c---!" , ""', Enhanc..-' !Klot

i --- flotropy , , , 1-· ... , , , ..... OIltllRI3 ,

• - - Oata R23 , ! , , , • , I ,

,

I , , , • , , , , , ,

ld-0.5 1 1.5

Horizontal velocity (m .-')

,

Figure 11 : Variarion of tile enhancement jaclor (left pane l) and the horizontal velocity (right panel) a/ollg tile EDM Lice core. "Model" represenlS 'he solutions based oil/he fabric evollllion equation (46)/or transverse isotropy. For "Data RJ3", "Data R23" and "Isorropy" see ,lie cap/ioll of Fig. 10.

bollOI1l, which arc mOSI ditTiculllo mood precisely. Be­side the assumpt ion of transverse isotropy. a weak point in [hal context is Ihe neg lect ion of recrystall izll tion pro­cesses, which arc expected 10 become important for the fabri c evolution in Ihe lower part of the ice core. This point requires further attention.

4 Conclusions

The new ly developed CAFFE mode l (Continuum­mech,lllical, Anisotropic Flow model, based on an anisotropic Flow Enhancement factor), which compri ses an an isotropic now law as well as a fabric evolution equa­tion. was presented in this study. It is a good compro­mise between physica l adequateness and simplicity, and is therefore well suited for being used in now models o f ice sheets and glaciers.

The CAFl-"E model was successfully applied to the site of the EDM L icc core in East Antarctica. Two dilTerent methods were employed, (i) computing the anisotropic enhancement factor and the horizontal now based on fab­rics data, and (i i) solving the fabric evolution eq uation under the simplifying assumption of transverse isotropy. Method (i) demonstrated clearly the importance of the anisotropic fabric in the icc column for the now veloc­ity, ,md beller agreement with the measured surfacc ve­locity was achieved compared 10 an isotropic computa­tion. The anisotropic enhancement fac tor produced with method (ii) agr(."Cd reasonably well with that of method (i), despite the fa ct that the measured fabric is not trans­versely isotropic in large parts of the icc core.

A solution of the fabric evolution cquation (45) for the EDM L ice core without the assumption of trans­verse isotropy has been presented elsewhere (Seddi k et al. (48]). Further, the CAFFE model has already been im-

plemented in the three-dimensional , full -S tokes icc- now model Elmer/Icc (Seddik [47l, Seddik et al. [49]) in or­der to simul ate the icc now in the vic inity within 100 km around the Dome Fuji drill site (MolOyama 136]) in cen­tral East Antarctica.

Acknowledgements

The authors wish to thank Dr. Sergio H. Faria (Un i­versity of G6l1inge n, Germany), Dr. Olivier Gagliar­dini (Laboratory of Glaciology and Environmental Geo­physics, Grenoble, France) and Professor Kolumban Hut­tcr (Swiss Federal Institute of Technology, Zurich) for their collaboration in developing the new mode l for anisotropic polar ice, Thanks arc further due 10 Ms. Ilk a Hamann and Dr. Sepp Kipfstuhl for kindl y providing the prel iminary fabric data of the EDM L ice core. to Dr. Hans Oerter for communic<l ting the mC<lsured surface veloc ity, to Ms. Christine Wesche (all at Alfred Wegener Institute for Polar and Marine Research , Bremerhaven, Germany) for allowing us to use the topographic map of the vicin­ity of the drill site, and to an anonymous reviewer whose comments helped improving the clarity of the paper.

Thi s work was supported by a Grant-in-Aid for Cre­ative Sciemific Research (No. 14GS0202) from the Japanese Ministry of Education, Culture, Sports, Science and Technol ogy, and by a Grant-in-A id ror Scientific Re­se<lrch (No. 18340 [35) from the J<lpan Soc iety for the Promot ion of Science. We would like to express our grat­itude for the effi cient manage ment of the Creative Re­search project by the leader, Professor Takeo Hondoh, and the project assistant, Ms. Kaori Kidahashi.

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