Formulation of ice shelf dynamic boundary conditions in terms of a Coulomb rheology

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 91, NO. B8, PAGES 8177-8191, JULY 10, 1986

Formulation of Ice Shelf Dynamic Boundary Conditions in Terms of a Coulomb Rheology

D. R. MACAYEAL

Department of the Geophysical Sciences, University of Chicago, Illinois

S. SHABTAIE AND C. R. BENTLEY

Geophysical and Polar Research Center, University of Wisconsin-Madison

S. D. KING

Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena

Coastal boundaries where fast flowing ice shelves shear past stagnant, grounded ice are typically riven with surface crevasses, seawater-filled basal crevasses, and tidal strand cracks. Here we formulate a boundary condition describing stress transmission through these fractured boundaries in terms of the Coulomb law. As a result of this formulation, agreement between finite element simulations of the Ross Ice Shelf flow and field observations is improved over agreement obtained with formulations which do not account for ice failure. Our results additionally suggest that shear stress transmitted through ice shelf boundaries is lower than previously thought.

INTRODUCTION

The boundary between fast flowing ice shelf and stagnant, land-fast ice is typically crevassed, is often broken into void- filled ice rubble, and, in some instances, displays the physical characteristics of strike-slip faults extending parallel to the flow [Thiel and Ostenso, 1961; Collins and Swithinbank, 1967; Barrett, 1975; Vornberger and Whillans, 1985]. With little else restraining the seaward flow of floating ice shelves, these boundaries provide a fundamental frictional control on ice shelf flow and evolution in response to climate [Thomas, 1979; Hughes, 1983]. Despite this important role, few observations of the rheological properties and stress regime of ice shelf boundaries have been conducted. Forecast studies of ice shelt flow have thus largely resorted to application of ad hoc boundary conditions not previously verified.

Thomas [1973a, hi, for example, developed a solution for ice shelf flow through a rectangular channel in which the depth-integrated stress at the sides is a function of the local ice shelf thickness and a plastic yield-stress parameter. The value of this parameter was chosen to provide a solution consistent with observations of the interior flow of the Brunt, Ross, and Amery ice shelves. The same boundary condition formulation was subsequently used in studies describing ice shelf response to climatic change [Thomas and Bentley, 1978; Thomas et al., 1979; Lingle, 1984]. This boundary condition is adequate for the studies referenced above but is not for studies that employ finite element methods to resolve the general two-dimensional stress regime governing the more complex flow geometries found in nature [MacAyeal and Thomas, 1982]. The inad- equacy arises because the Thomas formulation presupposes a condition of plastic yield at all boundaries without regard for the actual stress regime. For studies employing the finite element method it is preferable, and more dynamically consis-

Copyright 1986 by the American Geophysical Union.

Paper number 5B5831. 0148-0227/86/005 B-5831 $05.00

tent, to allow the flow solution itself to dictate conditions of plastic yield or failure.

In this study we address the dynamic inconsistency described above by formulating a new dynamic boundary condition applicable to ice shelf boundaries in which ice failure introduces a Coulomb rheology similar to that describing slip along faults in rock mechanics [Jaeger and Cook, 1976]. Be- cause our formulation embraces feedback between the interior

flow and the stress regime of the boundary, it provides (1) a criterion for determining where marginal strike-slip faults may be active, and (2) a relationship between normal and tangential forces acting across such active faults. In deriving the above relationship, we postulate that seawater-filled cracks or fissures extend upward .from the ice shelf base and rupture a significant portion of the total ice thickness, thereby reducing the level of tangential force transmitted across faulted boundaries. This idealization is motivated by observations of seawater-filled basal crevasses downstream of boundary contact regions [dezek et al., 1985, 1979; dezek, 1984; Clough, 1974] and by observations indicating basal rupture and strand cracking during tidal flexure [Lingle et al., 1981; Stephenson, 1984].

We test our formulation of ice shelf boundary friction by simulating the flow of the Ross Ice Shelf, Antarctica, using the finite element model developed and tested by MacAyeal and Thomas [1986, 1982]. Simulations in which our formulation is applied are found to agree more closely with observations than simulations which differ only in that the "no-slip" kin- ematic condition is applied at all ice shelf/grounded ice boundaries. This improvement suggests that ice failure and discontinuous flow are influential controls on ice shelf flow

and evolution in response to climate.

PHYSICAL CHARACTERISTICS OF ICE-SHELF MARGINS

Our formulation of ice shelf boundary friction stems from the following three physical characteristics of high-shear coastal margins, evident through aerial photography, radio

8177

8178 MACAYEAL ET AL.' ICE SHELF BOUNDARIES

......

...........

• ]..?:•......::,::: .....

.

•:•..•:,•....

•:.•ii':':'::•;'•:...-.•ii•i'i'11 ................... ii!i• ß ::::::::::::::::::::: .'?..

MACAYEAL ET AL.: ICE SHELF BOUNDARIES 8179

170E 180 ' I

ice front / I

I /

/

Roosevelt I.

_ _

/170W

/

] Crary Ice ] •! Rise Radio echo

/

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c

Fig. 1. (continued)

echo sounding, and tidal flexure measurement: (1) a dense system of surface and basal crevasses that fracture the ice shelf through much, if not all, of its thickness, (2) apparent strike- slip faults and suture zones extending parallel to the direction of flow, and (3) seawater-filled macroscopic voids or cavities downstream of ice shelf flow obstructions such as ice rises and

ice rumples. The aerial photograph (courtesy of the U.S. Navy) in Figure

l a displays an example of the first two characteristics evident along the edge of the Ross Ice Shelf where the effluent of ice stream B speeds past a complex terrain of grounded ice rises and stranded ice shelf [Bindschadler et al., 1985; S. Shabtaie and C. R. Bentley, unpublished manuscript, 1986]. This photo is representative of surface conditions that exist along an approximately 200-km section of the side of ice stream B and of the Ross Ice Shelf [Vornberger and Whillans, 1985]. The ice in the right side of the photo is part of the ice shelf (although within the grounding zone where there is intermittent contact with the seabed) and is moving toward the viewer at approximately 500 m/yr (R. A. Bindschadler, personal communication, 1985). The ice in the left side of the photo is stationary (R. A. Bindschadler, personal communication, 1985).

The band of intense surface crevassing that runs through the center of the photo defines the outer limb of the fast moving ice shelf. Crevasse superposition at different orienta- tions has caused this band to appear granular, with individual grains approximately 10-100 m in horizontal span. The depths of the cracks between the grains are not known, but glaciostatic pressure presumably closes these cracks at depths greater than • 50 m [Zumberge et al., 1960]. The stagnant ice to the left of the granular band is crevasse-free except for a narrow sideband of regularly spaced, half-crescent-shaped fractures abutting the wider granular band to the right.

The abrupt change of surface appearance between the two adjoining crevasse bands described above is suggestive of a dipping fault undergoing active slip. Direct confirmation of such slip is not available because a field survey has not been conducted. Indirect evidence in support of fault slip is derived from the surface crevasse patterns.

Crevasses initially open in directions perpendicular to the greatest principal tensile stress in the horizontal plane [Meier, 1958; Zurnberge et al., 1960; Hughes, 1977]. Pure shear at the sides of an ice stream or glacier thus requires that newly formed crevasses be oriented roughly 45 ø to the direction of flow and that their upstream tips should be closer to the center of the ice stream or glacier than their downstream tips [see Hughes, 1977, Figure 24]. As crevasses age, they slowly bend and rotate from their initial orientation and are subse-

quently overprinted with younger crevasses having the initial orientation described above. This rotation and overprinting eventually produces the chaotic ice surface depicted in the wide granular band of Figure 1.

To date, little research has been done to characterize the evolution of surface crevasse patterns in response to natural ice shelf flow regimes (see, however, Vornberger and Whillans [1985]). We propose that the patterns displayed in Figure 1 are unusual because of the abrupt boundary (suture zone or fault defined above) between the relatively mature pattern represented by the wide granular band and the relatively young pattern represented by the narrow sideband. The evidence for the characterization of the wide granular band on the right side of the photo as mature is that individual crevasses are no longer visible as a result of extreme distortion and overprinting. Curious lineations within the mature pattern are also visible (Figure 1) and are apparently structurally stable with respect to further distortion. The narrow sideband on the left displays little evidence of severe distortion or overprinting (close examination from low-flying aircraft does reveal, however, minor overprinting indicative of early terrain development). The juncture between the mature and young surfaces is so abrupt that it suggests that the ice on the two sides have undergone different evolutions. The existence of a strike-slip fault which allows the mature ice to slide past the young ice provides a simple explanation.

Aerial photos are useful indicators of surface conditions like those characteristic of a potentially faulted ice shelf boundary depicted in Figure 1. They do not, however, provide an indica- tion of how deep such faults, or the neighboring chaotic surface crevasses, penetrate nor of conditions where snow drift may mask the surface.

Radio echo sounding data overcome these difficulties. Figure 2 displays a radio echo cross section of a Ross Ice Shelf boundary taken across the south side of the Crary Ice Rise (see Figure 1). In this cross section the relatively clutter-free radio echo from the ice bottom in the stagnant ice to the right is abruptly masked by englacial radio scatter in the fast moving ice shelf to the left. This scatter is caused by surface crevasses, basal crevasses, and buried englacial voids or sea- ice-filled inhomogeneities interpreted as relict crevasse features (S. Shabtaie and C. R. Bentley, unpublished manuscript, 1986).

The abrupt transition between clean and heavily scattered echoes is consistent with a faulted boundary along the side of the Crary Ice Rise. Examination of radio echo cross sections from a variety of other locations reveals that such abrupt transitions are common to all the ice shelf/grounded ice boundaries (S. Shabtaie and C. R. Bentley, unpublished manuscript, 1986). This suggests that densely fractured ice conditions are more pervasive than would otherwise be suspected from casual visual inspection of the ice surface.

Another feature revealed by radio echo sounding is the presence of upwardly penetrating basal crevasses [Clough, 1974]. These crevasses differ from surface crevasses in that they are filled with seawater (or brine-laden sea ice, if they are

8180 MACAYEAL ET AL..' ICE SHELF BOUNDARIES

relict). This seawater provides hydrostatic pressure necessary to counterbalance glaciostatic stress tending to close off such crevasses. Basal crevasses are thus capable of penetrating far more of the total ice thickness than are dry surface crevasses [Weertman, 1980]. This suggests that basal crevasses in ice shelf boundary regions would have a greater ability to weaken mechanical coupling across the boundaries than would dry surface crevasses.

Radio-echo profiles of the Ross Ice Shelf (Figure 2b) reveal what appear to be basally crevassed wakes extending into the ice shelf interior from boundary contact regions [Jezek and Bentley, 1983; Jezek, 1984; Jezek et al., 1985]. These wakes suggest that boundaries are preferred sites for basal crevasse formation. Examination of radio echo profiles from the boundary contact regions do not, however, reveal high basal crevasse concentrations. This lack of direct verification may result from masking by surface clutter in the radio echo or from the possibility that newly formed basal crevasses are too thin to be detected by radio echo sounding [Weertman, 1980].

One possible mechanism for basal crevasse formation near ice shelf boundaries comes from the theory of tidal flexure [Holdsworth, 1977; Hughes, 1977]. Observations indicate that the ocean tide propagates freely in ice-shelf-covered waters [Williams and Robinson, 1980] and can cause significant flexure along ice shelf grounding lines [Stephenson, 1984]. Studies of such flexure suggest that a significant portion of the ice thickness is fractured as a result of tidally induced bending stresses [$withinbank, 1957; Lingle et al., 1981; Stephenson, 1984]. Stephenson [1984], for example, observed tidal bending at the grounding line of the Rutford Ice Stream on the Filch- ner/Ronne Ice Shelf. To reconcile his observed tidal-flexure profiles with theory, Stephenson [1984] proposed that basal cracks extend through approximately 1/4 of the total 1800 rn ice thickness. Such basal crevasses, or tidal strand cracks, would not be detected by radio echo sounding because of their thin profiles, but their effect would be significant in that repeated refracture and brine-laden ice trapped within the crevasses would reduce the local fracture strength of the ice shelf and resistance of the ice shelf to shear [Hughes, 1977, 1983; Weertman, 1980; Sanderson, 1984; Weeks and Assur, 1967].

EQUATIONS GOVERNING LARGE-SCALE ICE SHELF FLOW

As a first step in formulating a boundary condition address- ing the effects of ice failure at the sides of ice shelves, we derive the depth-integrated stress equilibrium equations governing large-scale ice shelf flow. This derivation is presented here to better identify scale assumptions and to delineate the circum- stances under which our treatment of ice shelf flow will be

valid. We restrict our consideration here to the limit of small

aspect ratio (ratio of thickness to horizontal span) and exclude possible effects of small-scale details of ice shelf flow on the basic large-scale flow. We accept this restriction in the present study because our numerical model and field observations address only the basic large-scale state of the Ross Ice Shelf. If our present analysis appears fruitful, an extension of our formulation without the small aspect ratio restriction, perhaps through boundary layer expansion techniques, should • be undertaken.

Steady creep of ice shelves is governed by the stress- equilibrium equations [Paterson, 1981, p. 84]:

c9--•- + '•y + • = 0 (1)

er•x er. er• ex + -•y + • = 0 (2)

•T•x •T. •T• •-;- + • + • = p(z)• (3)

where T is the stress tensor (in pascals), x and y are horizontal coordinates, z is the vertical coordinate (positive upward, zero at sea level), p(z) is the density depth profile (in kilograms per cubic meter), and g = 9.81 m s-2. The ice/atmosphere contact at z = z• is assumed to be stress-free. This assumption yields the following boundary conditions applicable at z = z•(x, y):

Oz• Oz•

Txx(z•) • + •x(z•) • - Tx•(z•) = 0 (4) Oz• Oz•

•x(z•) X + •(z•) •- •(z•) = 0 Oz• Oz•

T•x(z•) • + T.(z•) •- •,(z•) = 0 (6) The ice/ocean contact at z = zo(x, y) is assumed to transmit the hydrostatic pressure of seawater only:

Ozo Ozo Ozo (7) Txx½•) • + •x½•) • - Tx•½•) = -•aH •

Ozo Ozo Ozo (8) •x(Z•) X + •(z•) • - •(z•) = -•oH • Ozo Ozo

T•x½•) • + T.(z•) • - •(•) = •aH (9) where H = zs - z•.

Equations (1) and (2) are integrated over depth to yield

• (T•x' - P) dz + T•y' dz - 0 (1 O) • x •yy - p g m •xx

•x rx,' dz + •yy T•,' -- P) dz - pgH •yy = 0 (11) b b

where T' is the deviatoric stress, and P =-1/3(Tx,, + Tyy + T=) is the pressure (in pascals) (not necessarily the glaci-

ostatic stress). Observe that equations (10) and (11) are exact, and references to Txz' and Ty•' are eliminated through and application of the boundary conditions at z = z• and z = zb.

Equation (3) is integrated to obtain an expression for P:

V(z) = pg d• + r='(z)-•xx T•x' d•-•yy rzy d• (12)

where • is a dummy variable of integration. As with equations (10) and (11), equation (12) is exact and is acquired through application of the boundary conditions at z = z•. Equation

(12) indicates that vertical shear stress (Tzx' and T:y')affects the depth integrated horizontal stress equilibrium equations (equations (10) and (11)) only in so far as it affects the pressure. We show below that the contributions of the vertical shear

stress terms in equation (12) are negligible when the aspect ratio of the flow is small. Observe that this will not amount to

a "hydrostatic approximation" because the T::'(z) term of equation (12) will still be retained.

To assess the influence of vertical shear stress on ice shelf

flow, we adopt dimensionless variables and compare the scales

MACAYEAL ET AL.' ICE SHELF BOUNDARIES 8181

8182 MACAYEAL ET AL.: ICE SHELF BOUNDARIES

of the various terms in equation (12). Dimensionless variables (denoted by primes) are chosen as follows:

x,y=Lx',y'

z = Hoz'

P, Txx', Tyy', T•z', Txy'= HP', Txx", Tyy", Tzz", Txy"

T j. T.'= Tj. T."

P = PoP

where L _• 10 km and H o -• 500 m are typical horizontal and vertical length scales of ice shelf flow, H = pogHo, and ß = el-I, where e = Ho/L _• 5 x 10 -2 is the aspect ratio. The scale of vertical shear stress ß is reduced by a factor of e below the scale of the horizontal stress H by requiring that all terms of the dimensionless basal boundary conditions for horizontal force components be of order 1. Substitution of the dimensionless variables into equation (12) yields

e'(z') = p' d• + Tzz"(z' )

Lx' r•x" d• +•-fy, T•/' d• (13) The value of e2 is typically of order 2.5 x 10-3 for large-scale ice shelf flow. The contribution of the vertical shear stress

terms in the definition for pressure may thus safely be disregarded. In this circumstance, equation (13) may be approxi- mated by

v(d=f;s'p'dC+ Tzz"(z' ) (14) We adopt equation (14) henceforth in our study both because of its applicability to large-scale ice shelf dynamics and because removal of the vertical shear stress permits us to use a two-dimensional rather than three-dimensional finite element

model in our study. We remark that our justification of the neglect of vertical shear stress in ice shelf dynamics differs substantially from the treatment given by Sanderson and Doake [1979].

The validity of our assumption that e2 is much less than 1 can be challenged on the grounds that strong curvature of lateral margins, large-width crevasse geometry, strong ice thickness gradients, and strain heating will all tend to decrease the horizontal scale L near ice shelf boundaries (G. E. Birch- field and I. Muszinski, personal communication, 1986; G. Schubert, personal communication, 1986). We do not dispute these possibilities. In the present study, however, we strictly disregard such possibilities so that we can (1) use our two- dimensional ice shelf model and (2) formulate our parameterization of ice shelf boundary friction in terms of model variables. We remark, as further justification, that the horizontal scale of the few crevassed ice shelf margins that we have actually observed (as shown in Figures 1 and 2) do satisfy 8 2 << 1.

The constitutive relations applied to the interior ice shelf flow not subject to Coulomb failure are that (1) the ice is incompressible, and (2) an effective viscous theology based on laboratory data describes steady state creep [MacAyeal and Thomas, 1982; Hooke et al., 1979]:

T' = 2v(/•, 0)• (15)

where • is the symmetric strain rate tensor (in s-1) and v is the effective viscosity (in kg m-1 s-1) taken as a function of temperature 0 (in kelvins) and the second invariant of the strain rate tensor /•=(1' ß 1/2 2eijeij) . The form of v is derived from Glen's flow law [Glen, 1955; Hooke, 1981]:

B o exp n-• (16) V -- 2/•1-1In

where R = 8.3143 kJ/mol øK is the gas constant [Weast, 1968] and Bo and Q are temperature-dependent parameters acquired from laboratory creep tests to account for liquid water softening in grain boundaries at 0 > 260 ø [Thomas and MacAyeal, 1982; Barnes et al., 1971; Baker and Gerberich, 1979]:

Bo = 1.3 N m -2 S 1/3 0 >_ 260øK

Bo = 625 N m -2 S 1/3 0 < 260øK

Q = 120 kJ/mol 0 > 260øK

Q = 80 kJ/mol 0 < 260øK

The dimensionless stress is related to a dimensionless strain

rate by

/ you \ ,., T•j" = kpogHoL)2V e 0 ij va xz or yz

(17) / you \ ,.,

Tif'= e(pogHoL) 2v ½ij ij = xz or yz where U is the horizontal velocity scale (• 500 m/yr) and Vo is an effective viscosity scale (• 5 x 10 •4 kg m- • s- •). Note that the non-Newtonian ice rheology may not strictly permit v' to remain of order 1.

Before using equation (17) to replace T" in equations (10), (11), and (14) an additional approximation valid for e2 << 1 is applied: •xx'•yy' and •xy' are independent of depth. The ap- proximations are justified by showing that the vertical deriva- tive of •xx is of order e2 as follows (arguments corresponding to •yy and •xy follow similarly):

- - 2 X 2 (18) where u is the x velocity and w is the z velocity. Equation (17) states that •/is of order e(U/L). The incompressibility condition implies that w' • eu' and

The nondimensional form of equation (18) thus states that •d•y'/•z' is of order e 2. Variation of d•' with depth must thus be disregarded to maintain consistency with the small e 2 limits adopted previously.

The governing equations for ice shelf flow in the limit of small 82 may thus be written

• {2v'•n'(2•/+ e.)} + • •X •

_ pogH• •z•' • •' p' d• = A,H' •x' J - vo(U/L) p'H •x' •x' •, , (19)


horizon

grounded ice

cavitation

ice shelf

fault

level

sea

•ture

Fig. 3. Schematic diagram of idealized ice shelf margin describing our formulation of ice shelf boundary friction.

and,

• { 2v'=n'g '} + {2v'=H'(2•yy '+ c3X' -- -xy

vo(U/L) p'm' , p' A•H' (20)

where v '= is the vertical average of v'. The equalities expressed above in parentheses are valid when p'= 1 (constant ice shelf density) and show the relationship between the large-scale ice shelf flow equations and the equations derived by England and McKenzie [1983] for large-scale continental deformation. The nondimensional parameter

pogHoLF1 - (Po/Pw)] ,dr=

you

is the Argand number [England and McKenzie, 1982] and measures the contribution of local ice thickness gradients to the large-scale forcing of ice shelf motion.

To solve equations (19) and (20) for the ice shelf flow, boundary conditions are required on the contours in the horizontal plane defining the ice front, the ice stream mouths, and the shear margins where the ice shelf flows past grounded ice or mountains. On the seaward ice front the horizontal force of

the depth-integrated hydrostatic pressure is applied. On ice stream entry margins the horizontal velocity, assumed to be depth-independent, is applied. On margins where stagnant grounded ice, or mountains, abut fast moving ice shelf, the Coulomb law and the ideal fault geometry described above is specified as follows.

FAULT FRICTION PARAMETERIZATION

The physical characteristics of ice shelf margins discussed in previous sections suggest that the rheology of fractured, join- ted, or granular materials possessing possible faults is also applicable to ice shelf boundaries. This rheology, known in its

most simple form as the Coulomb law [Jaeger and Cook, 1976 p. 95], defines a linear relationship between the maximum tangential stress and the normal stress transmitted across any material plane. For bodies that exhibit well-defined faults such as possibly exist in ice shelves, a separate Coulomb law, usu- ally representing lack of cohesion and weaker material strength, applies to the preferred material plane defined by the fault. In our formulation we apply the Coulomb law to the imaginary vertical surface that separates grounded ice from ice shelf.

Our formulation, derived below, will be discussed in terms of an idealized boundary shear fault that is ruptured by seawater-filled, or briney-ice-filled, basal fissures of small width (much less than the ice thickness). The geometry of this idealized fault, presented in Figure 3, is likely to be far more simple than the geometry we would expect to find in nature. Formulations derived from more realistic distributions of

crevasses, joints, and ruptures, however, are not expected to differ substantially from ours because of the generality of the Coulomb law to such material characteristics [Jaeger and Cook, 1976, p. 405]. Additionally, we remark that our formulation is designed to be implemented easily in our numerical model.

Given the idealized fault configuration displayed in Figure 3, the magnitude of the tangential stress T acting to resist slip along this fault is related to the normal stress F (positive tensile' negative compressive) at any vertical level z within the plane of the fault by the Coulomb law modified to account for basal crevasses (we henceforth work in dimensional variables):

T = /•F z_> O (21)

T=0 z<0

where//is the coefficient of friction (the cohesion [daeger and Cook, 1976, p. 95] is assumed zero). Here T is set to zero for z < 0 under the assumption that a seawater-filled basal rupture completely eliminates mechanical coupling across the fault at all depths below sea level. By specifying rupture to sea


level, we assume the maximum extent of vertical penetration of brine-filled basal fissures resulting from cyclic tidal flexure. Less extensive rupture would be that associated with Weert- man's [1980] formulation of basal crevasse penetration. Weertman's formulation does not, however, account for tidal flexure, so it is not adopted here.

Integrating equation (21) over depth yields the relation between the total tangential force per unit length of the fault (T)z and the total normal force per unit length l • transmitted through the portion of the fault that lies above sea level'

(T)z = #F (22)

where ( )• is the vertical integration operator. l e differs from the vertically integrated normal force (F)• by the normal force due to hydrostatic pressure within the basal crevasse.

To derive œ, we consider the force balance of an imaginary ice shelf section of rectangular plan bounded by the fault plane (assumed vertical) and by an imaginary vertical plane parallel to the fault but displaced several ice thicknesses to- wards the ice shelf interior (Figure 3). Defining horizontal coordinates x (perpendicular to the fault) and y (parallel to the fault) as shown in Figure 3, the balance of forces acting in the x direction implies

ff =--«pwOzb 2 + T,,,,'-P) dz- poH-•x dX X=Xf b X=XO 0

+ [T•y'(y2)- Txy'(y•)] dz dx (23) o b

where «p,•g%21•,=•, s is the vertical integral of the x component of the hydrostatic stress within the basal rupture of the fault (p,• is assumed in this study to be 1027.9 kg m-3), Xo and x s are the x dimensions of the ice shelf section, and yl and Y2 are the y dimensions of the ice shelf section (Y2- Y• - 1 is assumed). This expression is exact and does not require the assumption of small vertical shear stresses. We simplify this expression by (1) disregarding terms involving Oz•/Ox (which will be of order e), (2) assuming that variation of T•' along the direction of the fault is small (this amounts to assuming that fault line curvature is small), and (3) adopting the small limit of T•'-P given by equation (14). After evaluating subject to the above simplifications, the expression for becomes

(T) z = # -- 2v ZH(2•,•, + •yy)--«pwgz• 2 + pg d• (24) b

TABLE 1. Specified Ice Stream Input

Input Velocity*, Ice Stream m/yr

A 400

B 700•- C 0•' D 450 E 450 Mulock Glacier 290

Byrd Glacier 740 Nimrod Glacier 150 Beardmore Glacier 330

*For independent estimates and observations of these velocities, see Brecher [1982], Hughes [1977], Lingle [1984], and Giovinetto et al. [1966]. Locations of ice input are displayed in Figures 1 and 4.

'•R. A. Bindschadler (personal communication, 1985).

This expression relates (T)• to the depth-average horizontal deviatoric stresses, T• '• = 20•,•, and Tyy '• = 20z•yy, in the ice shelf interior and to the ice shelf thickness. These variables are

given as output by the finite element model that we use in this study.

The dimensionless coefficient of internal friction # associated with our idealized formulation of ice shelf friction is

derived from laboratory studies of granular ice behavior and from studies of ice jams in rivers and sea ice arching in the ocean [Weiss et al., 1981; Hanes and lnman, 1985; Forland and Tatinclaux, 1985; Sodhi, 1977; Stewart and Daly, 1984]. The # values reported in these studies range from approximately 0.01 to approximately 1.0. Here, we choose # = 0.1 because it yields simulation results consistent with observations. An analysis of the effect of different values of # on our numerical simulations of the Ross Ice Shelf would be useful as

a future study.

NUMERICAL IMPLEMENTATION

The finite element model used to test the above formulation

of ice shelf boundary friction is derived by MacAyeal and Thomas [1986] (see also MacAyeal and Thomas [1982] and MacAyeal and Holdsworth [1986]). Although equations (19) and (20) describe the dynamics of ice shelf flow, our model is based on a different statement of the same dynamics based on the virtual work principal consistent with the finite element method I-MacAyeal and Thomas, 1982]. Input parameters associated with this model are described in the text and are

summarized in Table 1. Model output consists of grid point solutions for the horizontal velocity u and element solutions for the horizontal components of • and T and for O z. Grid resolution used in this study is 22.2 km (1/5 of a degree of latitude). A higher resolution was desirable but could not be implemented on the small desktop computer (HP 9000) used to run the model. A map of this grid is provided in Figure 4. Digitization of this numerical domain, and of observed values of H, the snow accumulation rate ,4, and the surface temperature O s , required as model input, was accomplished using the work by Drewry [1983] and bivariate interpolation of field data presented by Bentley et al. [1979] and by Thomas et al. [1984].

As stated previously, boundary conditions required by the model consist of either (1) specification of the velocity, (2) specification of the stress at the junction between ice shelf and land-fast ice, or (3) specification of hydrostatic pressure at seaward ice fronts. These boundaries are represented on the numerical grid by piecewise linear contours along the edge of the grid. At ice stream and glacier input boundaries we specify the observed input velocities given in Table 1. We remark that these velocities are uncertain by as much as 25% and that the coarse horizontal grid resolution used in this study (22.2 km) is not amenable to precise application of ice stream and glacier inputs (many small glaciers are accordingly not treated). At other boundaries we apply the condition of no normal flux (u ß n = 0, where n is the outward pointing normal) and specify either (1) no tangential flux (u x n = 0) or (2) (T)z given by equation (24). The choice of which condition, 1 or 2 above, to apply on a particular boundary depends on whether (T)• calculated by the model when condition 1 is applied exceeds the maximum value of (T) z given by equation (24). If so, then the model repeats its solution after switching from condition 1 to condition 2 at the particular boundary segment.

Fault slip is implemented in an iterative fashion to ensure


A B

Fig. 4. (a) The finite element grid (resolution 22.22 km). This grid was constructed from coastal boundary data presented in Figure lc. Points labeled with open circles represent where ice stream or glacier influx was specified (see Table 1). The boundary pbint labeled with a star represents the location of the boundary force analysis presented in Table 3. Elements labeled with solid circles represent where seawater-filled cavities are specified. (b) The basal melting rate /J (meters per year ice equivalent) specified in all simulations according to the oceanographic circulation patterns discussed by MacAyeal [1984]. Similar maps of specified snow accumulation rate ,,i and surface temperature O s are presented by Thomas et al. [1984].

/

/

that (1) all faulted boundary segments are correctly identified, (2) the values of (T}z specified are consistent with the values of •xx and •yy provided by the model solution, (3) stress redistribution due to identification of new faults is complete, and (4) the flow law given by equations (15)(16) is satisfied. The numerical procedure is outlined as follows:

1. First apply a no-slip boundary condition at all junctions between ice shelf and land-fast ice, and determine the normal and shear forces, J• and tentative values of the shear stress (T}z ø acting across those junctions.

2. Next test each boundary segment of the finite element grid (see Figure 4) to see if the tentative values (T}z ø exceed the envelope defining fault friction given by equation (24).

3. If any tentative values (T}• ø exceeds the envelope, the individual boundary segment is marked as faulted, and (T}z given in equation (24) is specified as a boundary condition on subsequent iterations of the model.

Steps 2 and 3 are repeated as many times as necessary to allow stress redistribution, to create faults at additional

boundary segments, and to ensure (T}z, I •, •z, •xx, and •yy satisfy equation (24). Typically, this requires 25 iterations. We do not allow faults to disappear as a result of stress redistribution during the iterative process.

Our convention of not allowing faults to disappear during the iterative implementation of our numerical treatment of ice shelf dynamics brings to question the issue of solution unique- ness. Physical intuition would suggest that stress redistribution could allow some faults activated early in the iterative scheme outlined above to potentially deactivate during subsequent iterations. Such "deactivation" would lead to a different but equally valid solution for the ice shelf flow. A criterion for fault deactivation consistent with our parameterization of fault friction would be whether the ice shelf stress distribution

acquired after "turning off" a particular fault did not exceed the failure envelope at the location of the deactivated fault. We did not test for such possibilities. Nevertheless, we suspect that the particularly low shear stress criterion used in our parameterization of fault activation would have led to the same distribution of faults regardless of the order in which the faults were activated in the iterative scheme. We thus suspect that relatively few faults would have disappeared in our simulations had we applied a testing scheme for fault deactivation.

Other details of the numerical solution procedure are the same as presented by MacAyeal and Thomas [1986, 1982] with the exception of the temperature depth profile. In this study, O(z) was not constructed numerically as was done by MacAyeal and Thomas [1986] but rather was specified using the analytic expression derived by Crary [1961]. This expression is written

O(z)=O s+ Cexp .4•- 2H •2 d• (25) s

where 0 s is the surface temperature specified from observations [Thomas et al., 1984]; .4 is the surface snow accumulation rate in meters of ice equivalent per second, also specified from observations [Thomas et al., 1984]; • is the basal melting rate (in meters per second) specified according to the oceanographic theory presented by MacAyeal [1984]; K = 1.4 x 10 -6 m2/s is the thermal diffusivity of ice at -16øC [Weller and Schwerd•eger, 1971]; and C is a constant given by

• z• (A + B) •2 d• (26) C = (0• - 0•) exp A•- 2H s

where 0• is the basal melting temperature (about -2.2øC) given as a function of ocean salinity and pressure at the depth


/

/

/

/

/

100% /

/

/

/ / /

Fig. 5. (a) Velocity magnitude (in meters per year) of the no-slip simulation. (b) Ice flow lines produced by the no-slip simulation. The flow lines produced by the fault-slip simulation are essentially the same. (c) Velocity difference between the no-slip simulation and the observations reported by Thomas et al. [1984]. Vector lengths are normalized by the magnitude of the observed velocity (see also Table 2). Vectors pointing upstream indicate that the simulated velocity is lower than the observed velocity by a percentage given by the length of the vector.

of the ice shelf base zb [Millero, 1978]. Contour maps of/i,/J, and 0s used as model input are presented in Figure 4 and by Thomas et al. [1984].

CAVITATION

A technical difficulty associated with our implementation of faulted boundaries is the problem of fault intersection at corners of the finite element mesh. Fault dynamics can not be

Fig. 5. (continued)

applied at these corners because the local curvature is singular and the no-normal-flux condition can not be applied without setting all velocity components to zero. Fault interruptions due to these computational asperities are unnatural and result in a no-slip condition over large portions of the boundary regardless of the shear stress.

Numerical simulation studies of faults in continental litho-

sphere overcome this difficulty by (1) adapting the finite element technique to permit curvilinear element boundaries and (2) by representing faults as special one-dimensional elements with zero area [Bird and Baumgardner, 1984]. Here we overcome the fault interruption difficulty by simpler means. We allow ice shelf flow cavitation to occur at selected positions where the intersection of two faults would cause the ice shelf

to separate from the boundary. Treatment of fault intersection by means of cavitation is preferable in ice shelf studies because open cavities (horizontal areas filled with seawater and sea ice) are often observed in ice shelves along shear margins.

We invoke ice shelf flow cavitation by allowing the ice shelf to separate from the coast on the leeside of a corner in the boundary to form a void having the size of one element. A pictorial description of the kinematics of this flow cavitation and a map showing the location of these cavities on our numerical grid are provided in Figures 3 and 4. We treat each void as filled with seawater and treat its boundaries in the

same manner as any other seaward ice front. Our choice of cavity location is designed primarily to correspond with observed cavity locations. Most notable are those downstream of the various ice rises [Barrett, 1975; Shabtaie and Bentley, 1985a, b] and at the side of the Byrd Glacier where it enters the Ross Ice Shelf [Hughes, 1977; Lucchitta et al., 1985].

It is desirable eventually to develop a specific criterion for the development of ice shelf cavitation and flow separation similar to the criterion proposed by Sanderson [1979]. In this study, however, we prefer a simple a priori specification so as not to confuse the evaluation of boundary friction viewed as our primary purpose.


/

/

lOO% /

/

cavity

Fig. 6. (a) Velocity magnitude (in meters per year) of the fault-slip simulation. (b) Network of faults produced by the fault-slip simulation plotted against a background displaying the outline of the numerical grid. (c) Velocity difference between the fault-slip simulation and the observations reported by Thomas et al. [1984]. Vector lengths are normalized by the magnitude of the observed velocity (see also Table 2). Vectors pointing upstream indicate that the simulated velocity is lower than the observed velocity by a percentage given by the length of the vector.

NUMERICAL RESULTS

We test our formulation of ice shelf boundary friction by comparison of two finite element simulations of the Ross Ice Shelf flow with observed velocities and strain rates reported by Thomas et al. [1984•. In the no-slip simulation, treatment of ice failure is suppressed, and both horizontal velocity components are specified zero at all ice shelf/grounded ice bound-

Fig. 6. (continued)

aries except those representing ice stream and glacier input (Figure 4 and Table 1). In the fault-slip simulation the boundary condition formulated in the previous sections is adopted. Many of the boundary nodes are thus free to adopt nonzero tangential velocities wherever the simulated stress regime dic- tates Coulomb failure. Both simulations are compared with the observed horizontal velocities and strain rates derived

from the Ross Ice Shelf Glaciological and Geophysical Survey (RIGGS) [Thomas et al., 1974]. These data represent results of position fixes using satellite navigation equipment and of stake network resurvey. Accuracy of these velocity and strain rate measurements are summarized by Thomas et al. I-1984].

The velocity magnitude (in meters per year) the flow lines, and the comparison with observed velocities for the no-slip simulation are presented in Figure 5. Agreement with the observed flow is good throughout much of the ice shelf (Figure 5c). In the large central region downstream of the various ice rises the simulated flow is within 5-10% of the observed flow.

In view of the uncertainties in basal melting rates, internal ice temperatures, and ice stream input velocities, better agreement than that achieved in the central region would most likely be fortuitous.

The no-slip simulation shows substantial disagreement with observations, however, in the region extending along the Transantarctic Mountains from the outlets of ice streams A

and B and in the region surrounding Roosevelt Island. Along the Transantarctic Mountains, the simulated flow falls approximately 100 m/yr below that observed. The velocity deficit of the simulation may be caused by excessive friction associated with the no-slip boundary condition. The fault-slip simulation, discussed below, is seen to improve the match with observations in this area by reducing the friction. In the channel east of Roosevelt Island, the no-slip simulation exceeds the observed flow by approximately 50-100 m/yr. This excess, which becomes more pronounced in the fault-slip simulation, is due to other inadequacies of the model besides the boundary condition. MacAyeal and Thomas 1-1986], for exam-


TABLE 2. Statistical Comparison Between Simulated (Us)and Observed (u0) Velocities

_

Simulation 1•8 145 luo - usl /(148) • luo - Usl/148

No-slip 0.255 96.4 Fault-slip 0.237 89.9

Values are in meters per year. Data given by Thomas et al. [1984]. Statistics compiled for 148 RIGGS stations (excluding N16).

ple, demonstrate using a similar finite element model with a better treatment of ice shelf heat flow that the simulated flow

in this region can be corrected (reduced by 75-100 m/yr) simply by increasing the specified basal melting rates to values more consistent with mass balance equilibrium. Other, more isolated instances of disagreement between model and observation, like those occurring at points south of the Crary Ice Rise and near the grounding line of ice stream C, are due largely to infidelity of the model boundaries to the real coast- lines and to the low-velocity magnitudes in these areas.

The results of the fault-slip simulation presented in Figure 6 show significant improvement over the no-slip simulation in all areas except the region east of Roosevelt Island. Along the Transantarctic Mountains disagreement with observation is generally reduced to approximately 10% from the roughly 30% level attained in the no-slip simulation. We remark that some isolated points of disagreement such as the point next to Crary Ice Rise are not effected by the fault-slip boundary condition because the no-slip condition still applies to corner nodes of the numerical grid. Higher numerical resolution would likely improve the fault-slip simulation along the Crary Ice Rise and at other locations of strong boundary curvature. A statistical summary of the comparisons between the two simulations and the observed flow is presented in Table 2.

Improvement of the fault-slip simulation over the no-slip simulation is attributed to the fault network displayed in Figure 6b and to the reduction of stress transmitted through the faulted boundaries. The low value (0.1) of tt chosen in this study resulted in faulting at virtually all straight boundaries where fault-slip could be implemented. This fault network does not, however, eliminate all instances of high ice shelf/- grounded ice coupling because corner nodes of the boundary are still constrained by the no-slip velocity condition wherever cavitation is not additionally specified.

Comparisons between the depth-averaged stress fields produced by the two simulations and between the stress acting through the boundary along the Transantarctic Mountains (Figure 4a), are presented in Figure 7 and Table 3. The analysis presented in Table 3 shows that the depth-averaged tangential stress transmitted through the boundary is reduced by 80% in the fault-slip simulation over that derived from the no-slip simulation. Furthermore, the fault-slip simulation ex- hibits a depth-averaged tangential stress at this boundary location significantly (65-80%) below that specified by previous formulations of ice shelf flow dynamics (0.5-1.0 x l0 s Pa) [Thomas, 1973a, b; Hughes, 1977; Paterson, 1985]. The low stress value derived from the fault-slip simulation suggests that tangential stresses transmitted through ice shelf shear margins are relatively insignificant in controlling ice shelf flow. Perhaps the previous formulations mentioned above require artificially high specified shear stress to overcome an inability to address complex ice shelf flow geometry or the associated

/

/ 2.0 x10•Pa/ / /

/

2.0 x 10•pa/ /

Fig. 7. (a) Depth-averaged horizontal deviatoric stresses (in pascals) of the no-slip simulation plotted at every second grid element. Crosses represent magnitude and orientation of principal axes. Arrow heads denote tension or compression. (b) Depth-averaged horizontal deviatoric stresses of the fault-slip simulation plotted at every second grid element.

drag provided by glaciostatic stress acting through curved boundaries. This point is discussed further by MacAyeal [1985].

The deviatoric stress regime displayed in Figure 7 also displays the shear stress reduction at faulted ice shelf boundaries in the fault-slip simulation. Note, however, that shear stress is still prevalent in the fault-slip simulation throughout most ice shelf boundary regions because of restrictions imposed by boundary curvature. These restrictions arise from geometrical aspects of the ice shelf flow rather than from tangential friction encountered along straight boundaries and accentuate the

MACAYEAL ET AL.: ICE SHELF BOUNDARIES 8189

TABLE 3. Boundary Stress Regime at Point Labeled With Star in Figure 4a

(T):/H*, r/H•', (F)../H,, Simulation 10 5 Pa 10 5 Pa 10 5 Pa

No-slip 0.902õ 2.834 30.099 Fault-slip 0.182E 2.675 29.940

All quantities are divided by H (679 m) to allow direct comparison with results of previous studies.

*Calculated from model output, and, for the fault-slip simulation, from equation (15). H at the boundary location analyzed is 679 m. Note that stress is defined in the finite element model at the element

centroid and not at the element boundary. •'Calculated according to equations (13)-(14) using model output. :l:Calculated according to equation (14)using model output. õCalculated for the element centroid located 7.86 km from the

boundary. ôCalculated by substitution of i e given at the element centroid into

equation (12).

importance of correct account of ice shelf flow geometry in ice shelf flow studies.

Direct comparison between the simulated stress field and observation is not possible because the observed stress is derived from observed strain rate (see also the method of Jezek et al. [1985]). Simulated and observed strain rates are presented for comparison in Figure 8. The displayed simulated strain rates are derived from the fault-slip simulation. Strain rates of the no-slip simulation do not differ substantially from those of the fault-slip simulation, except in certain boundary regions, and so are not shown. Complexity of both the observed and simulated strain rate patterns makes it difficult to compare the two quantitatively. Qualitative agreement is, however, fairly good. Notable features well produced by the fault-slip simulation are (1) the longitudinal compression and transverse extension upstream of the Crary Ice Risc, (2) the predominance of shear in the west central sections of the ice shelf, and (3) extension parallel to the seaward ice front within about 50 km of the ice front.

CONCLUSION

Our formulation of ice shelf boundary friction and fault slippage substantially improves agreement between finite element simulations of the Ross Ice Shelf flow and field observa-

tions over the agreement obtained by using a no-slip boundary condition. Improvement was achieved most notably in the area examined along the Transantarctic Mountains and south of the Crary Ice Rise. Substantial disagreement between simulation and observation east of Roosevelt Island could not be

corrected by our formulation. We attribute this disagreement, however, to inadequately specified basal-melting rates which are possibly higher in nature. The improvements that we have achieved highlight the importance of proper boundary condition specification in ice shelf studies and suggest that ice failure and bottom crevassing significantly affect the mechanical coupling between an ice shelf and its land-fast margins. Perhaps the most fundamental extension of our study is to observe directly the stress regime at an ice shelf shear margin. In particular, observations are required to determine whether shear faults described by a Coulomb rheology modified to account for basal rupture actually exist, or whether relatively narrow shear bands of continuous deformation define what we

treat in our study as faults. Our results are inconsistent with previous formulations

1.5 x 10 "ø • /

/

I 1.5x10 s-' / / • /

/

Fig. 8. (a) Horizontal strain rates (in s-•} of the fault-slip simulation plotted at every second grid element. Crosses represent magnitude and orientation of principal axes. Arrow heads denote extension or compression. (b) Observed horizontal strain rates reported by Thomas et al. [1984].

[Thomas, 1973a, b; Sanderson, 1979; Lingle, 1984] of ice shelf boundary friction that presuppose a plastic yield stress in the range of 0.5-1.0 x 105 Pa (0.5-1.0 bar), in that the shear stress acting across faulted ice shelf boundaries in our formulation (<0.20 x 10 • Pa) is considerably lower than the minimum presupposed stress. Furthermore, fi'ee slip si•nulations (not shown), in which the shear stress is set to zero, also produce remarkably good agreement with observations. The apparent agreement between observations and studies that use the plastic yield stress formulation may be reconciled with our results as follows:

Most previous studies of ice shelf flow, such as those conducted by Thomas [1973a], Lingle [1984]. Sanderson [! 979],


and Paterson [1985], do not consider glaciostatic pressures that act perpendicular to ice shelf margins. When coastal geometries are not rectangular and when the flow is not ideally aligned with parallel coasts, these glaciostatic pressures have a restrictive effect on ice shelf flow. MacAyeal [1985] terms the restrictive action of such glaciostatic stress form drag to distinguish it from dynamic drag, which accounts for shear stresses at ice shelf margins arising from ice motion. In studies of ideal ice shelves with simple, channel flow geometries, form drag can be disregarded (except in that it restricts ice shelf spreading to one direction only). We suggest that in studies where complex ice shelf flow geometry is analyzed with methods applicable to ideal geometries only, a high plastic yield stress at ice shelf margins must be specified to correct for the inability to account for form drag. Our numerical formulation of ice shelf dynamics does not require such specification because our methods treat the full two-dimensional geometry of ice shelf flow and can therefore properly distinguish form drag from dynamic drag.

Acknowledgments. This research was supported by National Sci- ence Foundation DPP 84-01016 awarded to the University of Chicago and by NSF DPP 81-20332 and NSF DPP 84-12404 awarded to the University of Wisconsin-Madison. Support for acquisition of map data was provided by the Charles A. Lindbergh Fund. We express our thanks to Michael Weaver, who digitized model input data; to Glenda York for editorial assistance; to Robert Bindschad- ler, Simon Stephenson, and Dean Lindstrom for fruitful discussions; and, finally, to G. Edward Birchfield and Isabelle Muszinski, who substantially improved the discussion of ice shelf dynamics through their thoughtful and careful review of this manuscript.

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Formulation of ice shelf dynamic boundary conditions in terms of a Coulomb rheology

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