Introduction

Large-scale ice-shelf flow is resisted in part by small-scale obstructions occurring wherever the ice shelf grounds on the sea bed (Reference ThomasThomas, 1979[a]). These obstructions are termed ice rises if the surrounding ice-shelf flow is diverted around the perimeter of grounded ice, or ice rumples if the ice shelf flows over the grounded region. Field surveys and simple model treatments of the Ross Ice Shelf in West Antarctica suggest that ice rises play a major role in regulating ice discharge off the grounded continental ice sheet (Reference Thomas, Thomas, Sanderson and RoseThomas and others, 1979; Reference Thomas and MacAyealThomas and MacAyeal, 1982). Crary Ice Rise shown in Figure 1, for example, is thought to reduce horizontal spreading rates along the grounding lines of Ice Streams A and B at a rate several orders of magnitude below that which would occur without the ice rise (Reference Thomas and MacAyealThomas and MacAyeal, 1982). Given their small size, and that climatic conditions could erode their perimeters under appropriate circumstances, ice rises and rumples are considered critical features In determining the climatic sensitivity of the West Antarctic ice shelves and the marine-based ice sheet which feeds them (Reference ThomasThomas, 1979[a], Reference Thomas[b]).

Fig. 1. The Crary Ice Rise complex at the outlets of Ice Streams A and B (polar stereographic projection). Light shading represents grounded ice. Heavy shading in upper right represents the Transantarctic Mountains range and the East Antarctic ice sheet. Clear regions represent floating Ross Ice Shelf. Darkened stripes on left and broken lines represent tracer trajectories mapped by radio echo-sounding (Reference Shabtaie and BentleyShabtaie and Bentley, 1987). The up-stream origins of the darkened stripes are the heavily crevassed margins which delineate the ice streams (labeled A through D). The dark irregular patch just down-stream of the largest ice rise represents its rifted wake. Field-measurement stations (some labeled, see also Figure 2) are represented by (•), and most are connected together by radio echo-sounding flight lines to form contours ⌈ and ⌈*. We use our field data (and that reported by Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes and BentleyThomas and others (1984)) to compute the force, mass, and mechanical-energy budgets of the region enclosed by ⌈ (the Crary Ice Rise complex). To investigate the influence of the ice-rise complex on the surrounding ice streams, we compare these budgets with those calculated for the ice-stream gateway contour ⌈* Station E₄ (shown by star) is designated as the projection pole for projecting our study region stereographically on to the x, y-plane (all figures, however, are standard south-pole referenced stereographic projections). Coordinate axes x and y are defined in Table IV.

Fig. 2. The Crary Ice Rise complex in detail. The configuration of the complex was determined by Reference Shabtaie and BentleyShabtaie and Bentley (1987) using radar-mapping techniques. Ice velocities over most of component A are near zero, indicating that this largest part of the complex is a true ice rise. Components B–E are clutter-free regions in the radio echo-sounding records. Component F was detected by tilt-meter analysis of tidal flexure (personal communication from S. Stephenson) and is known to be an ice rumple. Selected field stations along contours ⌈ and ⌈ * are labeled.

To understand better the role of ice rises and ice rumples in the maintenance of large-scale ice-shelf flow, stress scales and other physical parameters governing ice-rise/ice-shelf coupling must be better understood. A physical basis for predicting ice-rise resistance in terms of such parameters as size, shape, and crevasse conditions would be useful, for example, in research concerning the time evolution of the West Antarctic ice sheet. An additional objective is to understand how ice rises typically originate, and to determine whether their formation or decay accompanies other time-dependent fluctuations of the surrounding ice shelf or ice streams. Such an understanding would aid the interpretation of present-day features in terms of past history of the West Antarctic ice sheet, and would help to predict possible consequences of present-day imbalance detected in the mass budgets of certain ice streams (see Reference Shabtaie and BentleyShabtaie and Bentley, 1987).

As a step towards these goals, we present an analysis of the stress, mass, and energy-dissipation budgets of Crary Ice Rise located on the Ross Ice Shelf (Fig. 1) using field data collected during the 1983–85 austral summers and in previous field programs. We additionally calculate the net back pressure and ice-discharge rate along the grounding lines of Ice Streams A and B (Fig. 1) to assess the effect of the ice rise on the surrounding flow. Comparison of the ice-rise budgets with the analysis of grounding-line data confirms the influence of the ice rise on ice-sheet stability, and suggests that Crary Ice Rise may have formed recently in response to an acceleration of Ice Stream B. We speculate that feed-back between ice-stream acceleration and ice-rise formation may control the future evolution of Ice Stream B and promote long-term grounding-line stability in the face of strong natural fluctuations.

Field Program

To determine the force, mass, and mechanical-energy dissipation budgets of Crary Ice Rise, it was necessary to measure its thickness, velocity, and strain-rate around the perimeter of the ice rise. We performed these measurements along an imaginary contour designated ⌈ in Figures 1 and 2. This contour encloses the ice rise as well as parts of the surrounding ice shelf, and lies entirely on floating ice shelf (except possibly for several kilometers constructed over ice-shelf rifts or ice rumples not identified prior to the field operation; Figure 2). ⌈ was used instead of the natural perimeter of the ice rise for the following reasons: (1) the natural perimeter is not clearly defined or accessible, (2) the effects of severe crevassing typically found along the perimeter on ice flow are not known, and (3) surface measurements of strain-rate and velocity taken on floating ice sufficiently far from the ice-rise boundary are naturally representative of flow at depth (see Reference Sanderson and DoakeSanderson and Doake, 1979; and Reference Morland, Veen and OerlemansMorland, 1987, for a discussion of lead-order ice-shelf flow characteristics including the absence of significant vertical shear).

The contour ⌈ was constructed by establishing a box-like pattern of surface stations surrounding the ice-rise complex, and by connecting these stations with airborne radar-sounding flight lines as shown in Figures 1 and 2. Velocity and strain-rate measured at these surface stations during our 1983–85 field operations are summarized in Table 1. Data from three surface stations along ⌈ (J_10’ J_11’ and I₁₁ in Figure 2) were acquired by Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes and BentleyThomas and others (1984) during the Ross Ice Shelf Geophysical and Glaciological Survey (RIGGS) (see also Reference Bentley, Hayes and BentleyBentley, 1984). Radio echo-sounding data are reported by Reference Shabtaie and BentleyShabtaie and Bentley (1987), and are presented in Figure 3. Additional data collected during the 1983–85 operations or during RIGGS are presented in some figures to improve the display of regional flow conditions (see paper in preparation by R.A. Bindschadler and others for a detailed listing of these regional data).

Fig. 3. Ice-thickness cross-sections (thickness scale on edge in meters) along contours ⌈ and ⌈* (Reference Shabtaie and BentleyShabtaie and Bentley, 1987).

As a result of the radio echo-sounding surveys conducted during our field operations (see Reference Shabtaie and BentleyShabtaie and Bentley (1987) for details concerning flight lines not shown in Figures 1, 2, or 3), the ice-rise perimeter was found to encompass a greater area than that shown on previous maps (e.g. Reference DrewryDrewry, 1983). Reference Shabtaie and BentleyShabtaie and Bentley (1987) described the ice rise as consisting of a main oblong island of grounded ice, here referred to as Crary Ice Rise A, surrounded by possible “outrigger" ice rises, rumples, or crevasse-free ice-shelf “rafts" here referred to as Crary Ice Rises B through E. Crary Ice Rise F was identified as an ice rumple by a tidal tilt-meter survey performed in concert with ground-based radio echo-sounding (Reference Bindschadler, Bindschadler, MacAyeal, Stephenson, Veen and OerlemansBindschadler and others, 1987).

Surface-elevation and velocity measurements confirm that Crary Ice Rise A is indeed grounded, and that large parts of it are motionless with respect to the general flow of the surrounding ice shelf (Reference Bindschadler, Bindschadler, Stephenson, MacAyeal and ShabtaieBindschadler and others, in press). Ice Rises B through E, however, are detected by their clutter-free radio-echo signatures denoting the marked absence of surface crevasses within heavily crevassed perimeters common to other, confirmed ice rises. Inasmuch as these outlying features cannot unambiguously be identified as ice rises, ice rumples, or simply regions of crevasse-free ice shelf, we refer to the entire collection of features as the Crary Ice Rise “complex”.

Two other contours, ⌈* and ⌈^c were constructed to examine the force, mass flux, and energy dissipation along the grounding line of Ice Streams A and B, and within a control region of the Ross Ice Shelf (to test our methods in an ice-rise-free environment), respectively. The ice-stream grounding-line contour (Fig. 1) was constructed from eight line segments joining surface stations extending between G₄ on the northern edge of Ice Stream B to the edge of the ice shelf along the Transantarctic Mountains. As with ⌈, ⌈* is constructed outside the natural perimeter of grounded ice to avoid uncertainties in mapping the grounding line and to exploit the advantage of performing surface measurements on floating ice. Six of the eight segments forming ⌈* correspond to radio-echo flight lines. Thickness profiles along two segments (C₂ to F₉ and F₉ to the edge) were extrapolated from RIGGS data (Reference Bentley, Bentley, Clough, Jezek and ShabtaieBentley and others, 1979). Surface strain-rates were not measured at the station defining the edge of the ice shelf (Fig. 1), so the strain-rates observed at F₉ were used along the entire segment connecting F₉ to the edge in the analysis of force and energy dissipation. The error induced by this extrapolation is treated in the estimates of uncertainty which accompany our analysis.

The contour Г^c encloses an ice-rise-free part of the Ross Ice Shelf for use in a control experiment designed to test our data analysis and interpretation of ice-rise effects. This contour was constructed by connecting six RIGGS stations (P_14’ Q_14’ R_14’ R_15’ Q_15’ and P₁₅ shown by the map of field stations in Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes and BentleyThomas and others (1984) and in Table 1) in a closed circuit of line segments Ice thickness along these segments was interpolated from surface radar measurements at the RIGGS stations. Strain-rates and velocities observed, or extrapolated from surrounding stations, at the six RIGGS stations used to construct ⌈^c are summarized in Table 1.

Form Drag and Dynamic Drag: Theory

We determine the reaction force exerted by the Crary Ice Rise complex on the surrounding ice shelf and the ice streams located up-stream by computing the net surface traction acting on ⌈. This surface traction is partitioned into a glaciostatic contribution and a contribution arising from viscous friction associated with ice deformation, termed form drag and dynamic drag, respectively (Reference MacAyeal, Veen and OerlemansMacAyeal, 1987; see also Reference Whillans, Veen and OerlemansWhillans, 1987). This partition separates the resistive forces which depend on the constitutive properties of ice (dynamic drag) from those that do not (form drag).

The net surface traction vector F acting on the imaginary material surface surrounding the Crary complex is given by (see Reference JaegerJaeger, 1969, p. 5)

where T ≡ the stress tensor, n ≡ the outward-pointing vector of unit magnitude on the boundary contour ⌈ (this vector lies in the horizontal plane and points perpendicularly away from the Crary complex at each point on ⌈), z is the vertical coordinate (zero at sea-level, positive upwards), z_s and z_b are the z-coordinates of the surface and base of the ice shelf, respectively, and dλ is the horizontal length element of ⌈. We adopt the convention that F denotes the force acting on the surroundings of the Crary complex due to the physical features enclosed within ⌈. This convention accounts for the minus sign appearing in (Equation 1).

The stress tensor T satisfies the stress-equilibrium equations (Reference PatersonPaterson, 1981, p. 84), which are considerably simplified for an ice shelf of small aspect ratio (ratio of thickness to horizontal extent) (Reference WeertraanWeertman, 1957; Reference Morland, Veen and OerlemansMorland, 1987): the vertical shear stresses, T_xz and T_yz , are negligible and the surface traction T . n across any differential area element of ⌈ lies in the horizontal plane. In this circumstance, the pressure P = -1/3(T_xx + T_yy + T_zz ), is

where ρ(z) = local ice/firn density, g = 9.81 m/s², ζ is a dummy variable of integration, T’_zz is the direct vertical component of the deviatoric stress T’ given by

I is the identity tensor and x and y are horizontal coordinates.

With T given by (Equation 3), and using (Equation 2), (Equation 1) becomes

The right-hand side of (Equation 4) is partitioned into the sum of the form drag and the dynamic drag (F^f + F^d ) by writing

and

where the incompressibility condition is used to substitute (T_xx’ + T_yy’) for –T_zz′.

In our calculation of the form drag, we assume that the density-depth profile of the firn layer is spatially uniform and use the observed profile from drill site J₉ (located near the ice-rise complex, Figure 1) as representative of the entire study region (Reference Kirchner, Kirchner, Bentley and RobertsonKirchner and others, 1979). The exponential profile which best fits the J₉ data is given by

where α = 608.0 kg/m³, β = -0.043 m⁻¹, and ρ_i = 917 kg/m³. The integrals over ζ and z in (Equation 5) may thus be evaluated exactly giving

where H = (z _s - z _b).

The second and third terms of the integrand on the right-hand side of (Equation 8) represent a correction for the excess thickness of the ice shelf over a solid-ice shelf of the same mass. This correction is 15.4 m, which is smaller than the 17.4 m average over the entire ice shelf (Reference Shabtaie and BentleyShabtaie and Bentley, 1982). The low snow accumulation rates in our study area may account for this difference (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes and BentleyThomas and others, 1984).

Flow Law

The distinction between form and dynamic drags is advantageous from the standpoint that, for measurement from field data, only dynamic drag requires an assumed flow law, Reference Lliboutry and DuvalLliboutry and Duval (1985) outlined the physical factors that affect ice flow, and emphasized that it is currently not possible to adopt a simple relationship defining ice response to stress that will account for all factors and apply to all flow regimes. Since few of the thermal and chemical characteristics affecting ice flow in our field area are known in detail, we adopt relatively simple flow laws that account for temperature and firn density only. Additionally, we present several calculations of F ^d using alternative flow laws to demonstrate possible uncertainties inherent in our result.

Flow laws determined from laboratory and field experiments have been reviewed by Reference PatersonPaterson (1981; see also Reference Paterson1985), Reference HookeHooke (1981), Reference WeertmanWeertman (1983), and Reference Doake and WolffDoake and Wolff (1985[a]; see also Reference Doake and Wolff1985[b]). Following the trend of these reviews, we write the flow law in the following format:

where é is the strain-rate tensor, (summation convention is used), and n is the flow-law exponent. The ice-stiffness parameter B(ρ,θ, n) accounts for the effect of temperature and density, and is given by

where θ is the temperature (°K). Q is the activation energy, R = 8.3143 J mol⁻¹ deg⁻¹ is the gas constant, B ₀ is a rate-determining parameter accounting for all the physical effects not explicitly treated, p(z_s) is the firn density at z = z _S, and ρ _i is the density of ice at depth (917 kg/m³) (see Reference Thomas and MacAyealThomas and MacAyeal (1982) for discussion of density treatment). For convenience in treating our strain-rate data, we invert (Equation 9) to express deviatoric stress in terms of strain-rate (T′ _ij =* 2vė _ij) using the effective viscosity

where .

To illustrate the effects of flow-law uncertainty, we adopt three alternative sets of values for B ₀, Q, and n shown in Table II. The first two alternatives represent non-Newtonian flow laws (with n = 3) commonly used in other to be approximately 0.2 × 10⁸ Pa s^1/3 for flow laws #1 and #2 (approximately 13% of the value of B ^Z ), and 1.5 × 10¹⁴ Pa s (approximately 23% of the value of B ^Z ) for flow law #3 (see Reference Thomas and MacAyealThomas and MacAyeal, 1982, fig. 7). We adopt a uniform value of B ^Z calculated from the observed temperature-depth profile at J₉ (Reference Clough and HansenClough and Hansen, 1979), and regard the effect of regional temperature variation as uncertainty.

Fig. 7. Surface strain-rates (principal axes ) in detail around the Crary Ice Rise complex.

The values of B ^Z calculated from J₉ data using the three alternative flow laws and an evaluation of their uncertainty are presented in Table II. The two values of B ^Z corresponding to flow laws #1 and #2 can be directly compared because n is the same in each flow law. B ^Z calculated following Reference PatersonPaterson’s (1981) assessment of natural ice flow (flow law #2) is smaller (yielding softer ice) than B ^Z calculated using the laboratory-derived flow law (flow law #1), but is within the estimated uncertainty interval surrounding the laboratory-derived value.

Form Drag and Dynamic Drag: Measurement

To determine F ^f and F ^d from field data, we performed the integrations expressed in (Equations 8) and applications; the third represents a controversial Newtonian flow law (with n = 1) recently proposed by Reference Doake and WolffDoake and Wolff (1985[a], Reference Doake and Wolff[b]). Flow law #l represents parameter values used in other studies of Ross lce Shelf field data (Reference Thomas and MacAyealThomas and MacAyeal, 1982; see also Reference MacAyeal, MacAyeal, Shabtaie, Bentley and KingMacAyeal and others, 1986), and is based on laboratory measurements of ice creep (Reference Barnes, Barnes, Tabor and WalkerBarnes and others, 1971). Flow law #2 is based on Reference PatersonPaterson’s (1981, p. 39, table 3.3) compilation of natural ice-deformation data (bore-hole tilting and ice-shelf spreading; but modified to account for firn density as shown in (Equation 10)). Flow law #3 is adopted from Reference Doake and WolffDoake and Wolff’s (1985[a], Reference Doake and Wolff[b]) analysis of Reference HoldsworthHoldsworth’s (1982) Erebus Glacier tongue data which suggests a Newtonian (n = 1) rather than non-Newtonian (n > 1) rheology.

Expressing the dynamic drag in terms of the measured depth-independent strain-rate tensor and flow-law parameters gives:

where H = (z _s – z _b) is the observed ice thickness,

and B ^Z is the depth-averaged value of B(n,θ, ρ). B ^Z is determined by the local temperature-depth profile, and is thus influenced by snow accumulation, basal melting, and boundary temperatures at z _s and z_b Variation of B ^Z throughout the study area due to regional changes in the temperature-depth profile and uncertainty in the basal melting rate (assumed to be 0 ±0.1 m/year) was estimated (12), respectively, using the analysis procedures summarized in the Appendix (Table A-I). In brief, we divided ⌈ into 14 line segments on a polar stereographic projection (with station E₄ at lat. 83^o09’20"S., long. 171˚ 36’38"W. designated as the projection pole to reduce map-distortion error, and denoted by the star in Figure 1) using 14 surface stations as segment end-points. Each of the 14 line segments was further divided into as many sub-segments as necessary to represent the ice-thickness profiles acquired from the radio echo-sounding records. Strain-rate components at points along the line segments were interpolated from the observed values at the end-points. The integrations required by (Equations 8) and (12) were then performed using the trapezoidal method (Table A-I).

Confidence limits on the derived results were calculated using the technique summarized by Reference BoasBoas (1983, p. 734). Uncertainties due to measurement, navigation, and interpolation errors are summarized in Table III. Accuracy of ice thickness is estimated to be 8 m for flight lines conducted during our operations, and 15 m for the one RIGGS flight line used to connect stations J₁₀ and J₁₁ (Table III and Fig. 2). Three segments of ⌈ (K₃ to J₁₀, J₁₁ to I₁₁, and I₁₁ to J₃ in Figure 2) that comprise the down-stream end of ⌈ have thicknesses measured at their end-points only. For these segments, we interpolated the thickness profiles using the ice-thickness contour map derived from RIGGS (Reference Bentley, Bentley, Clough, Jezek and ShabtaieBentley and others, 1979). Accuracy of these interpolated profiles is approximately 50 m (10% of the 520 m average ice thickness along ⌈). Most of ⌈ is located sufficiently far from the ice-rise boundaries to expect relatively smooth spatial variation in the flow field. Interpolation errors of the strain-rates are thus expected to be approximately 10% of the value of ė = (0.5ė _ijė _ij)½ averaged between the segment end-points (Table III).

The computed force budget is presented in Table IV and, along with segment-by-segment break-downs of contributions from individual segments of ⌈, in Figures 4 and 5. Form drag F ^f, which arises because of non-uniformity of ice thickness, is clearly the dominant term of the total force budget. If the ice thickness were constant, the form drag would be exactly zero (assuming a spatially homogeneous firn layer). As shown in Figure 3, the ice thickness is greater up-stream of the ice-rise complex than down-stream, and is greater on the southern flank than on the northern. The form drag F ^f is thus diverted approximately 45° from alignment with the regional flow of the ice shelf. This indicates that forces generated by the ice rise do not entirely depend on the direction of flow. In the present circumstance, the ice rise acts to maintain thicker ice along the Transantarctic Mountains in addition to resisting the glaciostatic push of the ice streams.

Fig. 4. Form drag plotted for each boundary segment. Net plotted at station E₄.

Fig. 5. Dynamic drag computed using flow law #1 (Table III) plotted for each boundary segment (note different scale than Figure 4). Net plotted at station E₄.

Dynamic drags F ^d (Table IV; Fig. 5) computed using the three alternative flow laws presented in Table II exhibit similar magnitude and direction. This similarity suggests that conclusions drawn from our analysis are relatively insensitive to flow-law uncertainty. The magnitudes of the three alternative dynamic drags are 8, 7, and 5%, for flow laws #1 to 3, respectively, of the magnitude of the form drag. Unlike the form drag, the dynamic drag is aligned against the regional flow (Fig. 5) as expected for a purely frictional resistance.

Studies of large-scale ice-shelf flow have suggested that shear faults, or boundary zones of substantial ice failure, may exist on shear margins where the ice shelf flows past stagnant, grounded ice (Reference MacAyeal, MacAyeal, Shabtaie, Bentley and KingMacAyeal and others, 1986). The strain-rate and velocity measurements reported here are not from surface stations sufficiently close to the ice-rise perimeter to verify this hypothesis (see, however, other measurements taken during 1983–85 reported by Bindschadler and others (paper in preparation)). Analysis of the dynamic drag acting on different parts of ⌈ may indicate, however, which orientations of the ice-rise boundary with respect to the surrounding flow yield the greatest friction.

Segment-by-segment break-down of F ^d computed using flow law #1 as a standard of comparison (shown in Figure 5) reveals that the up-stream and down-stream ends of ⌈(C₂ to G₂ and J₁₀ to I₁₁, respectively) contribute more to F^d than do the flanks (C₂ to J₁₀ and I₁₁ to C₂), whose individual contributions to the transverse component of F^d tend to cancel. The dominant influence of the up-stream and down-stream ends of the ice-rise complex can be explained in terms of the observed strain-rate patterns shown in Figures 6 and 7. Up-stream of the ice-rise complex, the most compressive principal axis of the horizontal strain-rate tensor is oriented perpendicular to ⌈, whereas down-stream of the ice-rise complex, the most tensile axis is perpendicular to ⌈ This combination of orientations on the up-stream and down-stream sides produces the majority of the net dynamic drag. Shear stress along the flanks of the ice-rise complex is relatively weak (outside of the perimeter of ⌈). Tangential contributions to F ^d thus tend to be small.

Fig. 6. Surface strain-rates (principal axes ) at selected stations (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes and BentleyThomas and others, 1984; Reference Bindschadler, Bindschadler, MacAyeal, Stephenson, Veen and OerlemansBindschadler and others, 1987. in press). See also Table I.

Form Drag and Dynamic Drag: Interpretation

The significance of the bulk-force budget calculated for the Crary Ice Rise complex is best illustrated by comparing the force computed above with the imaginary force that would exist if the region enclosed by ⌈ were replaced with a sea-water-filled cavity. In this circumstance, the net force transmitted through ⌈ is entirely due to sea-water pressure and is given by

where ρw = 1028 kg/m³ is the density of sea-water. Note that F ^w is also the net horizontal component of the force that would be exerted against the bottom of the ice within ⌈ if it were entirely afloat. We computed F ^w using the same procedure used to compute F ^f outlined in the Appendix, and present the result in Table IV.

The magnitude of F ^wamounts to approximately 90% of the form drag F ^f; thus the vector difference (F ^f + F ^d − F ^d) is considerably smaller in magnitude than the total force (F ^f + F ^d) transmitted through ⌈. The difference (F ^f + F ^d– F ^w), shown in Table IV and in Figure 8, is the part of the total force resulting from grounding within ⌈. We refer to (F ^f + F ^d − F ^w) as the effective resistance of the ice-rise complex because it constitutes the part of the total force that arises due to friction associated with ice/sea-bed contact.

Fig. 8. Net-force vectors representing the effective resistance of the Crary Ice Rise complex on its surroundings, and the extra back-pressure force transmitted through the ice-stream gateway (computed using flow law #1; Table III). (1) F^f on ⌈; (2) (F^f + F^d - F^w) on ⌈ (3) F^d on ⌈: (4) (F^f + F^d - F^w) on ⌈^*; (5) (F^f + F^d - F^w) on ⌈ (same as (2) above).

Detailed analysis of the stress continuity equations (see Reference MacAyeal, MacAyeal, Shabtaie, Bentley and KingMacAyeal and others, 1986; Reference Morland, Veen and OerlemansMorland, 1987) reveals that the effective resistance of an ice-rise-free part of the ice shelf must vanish. The difference between F ^f and F ^w produces the driving force which causes the ice shelf to spread. This difference leads directly to the longitudinal deviatoric stress and thus to F ^d, It follows that (F ^f + F ^d + F ^w) should equal 0 across any floating section of the ice shelf if there is no resistance to spreading except for the water. We confirm this interpretation in a control experiment described in the next section. The only physical change within the contour ⌈ which could reduce the effective resistance to zero is the elimination of grounded ice. The effective resistance thus represents the part of the total ice-rise force that is sensitive to changing basal-contact area. This resistance can also be identified as the part of the total force most sensitive to climatic change.

Significance of the effective resistance generated by the ice-rise complex may be judged in terms of the force (F ^f + F ^d — F ^w)* transmitted across the grounding line of Ice Streams A and B through the contour ⌈ (shown in Figure 1). We refer to (F ^f + F ^d — F ^w)* as the extra back-pressure force (Reference ThomasThomas, 1979[b]) because it represents resistance generated by the entire ice shelf above that provided by sea-water pressure acting alone on the ice front and sloping under side of the ice shelf.

We computed the extra back-pressure force acting across ⌈* using the same procedure as used to calculate the effective resistance of the ice-rise complex. The effective resistance produced by the Crary Ice Rise complex is found to vary between 45 and 51% of the extra back-pressure force, depending on the flow law used to calculate dynamic drag (Table IV and Fig. 8). The ice-rise complex thus provides a significant restraint on the flow of Ice Streams A and B, and this result is relatively insensitive to flow-law uncertainties.

Another interpretation of the effective resistance (F ^f + F ^d − F^w) may be made in terms of an equivalent basal shear stress τ_b defined as the magnitude of the effective resistance divided by the estimated area of grounded ice contained within ⌈ (2.0 x 10⁹m²). The value of τ_b ranges between 0.045 × 10⁵ Pa and 0.1 × 10⁵ Pa (0.045-0,1 bar), depending on the flow law. This range is comparable to regional averages of the longitudinal component of basal stress computed using the balance expression τ_b = ρigHγ surface slopes γ ranging between 0.9 × 10⁻³ and 2.0 × 10⁻³, and an area-average ice thickness H = 550 m. An idealization of ice-rise resistance in terms of an effective basal shear stress may prove useful in predictive models of ice-shelf dynamics which cannot spatially resolve the details of flow surrounding small-scale ice rises.

Control Experiment

To test our data-analysis procedure and to verify our interpretations, we computed the effective resistance (F ^f + F ^d − F ^w)^c for an ice-rise-free part of the Ross Ice Shelf enclosed by the control contour ⌈^c constructed by linking six RIGGS stations located near the ice front (Table I; Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes and BentleyThomas and others, 1984). If our procedure and interpretations are correct, and if ⌈^c encloses floating ice shelf alone, (F ^f + F ^d – F ^w)^c should be zero (or within the estimated uncertainty interval surrounding zero). As shown in Table IV, the effective resistance computed for ⌈^c is within estimated uncertainty of zero. This test confirms our data-analysis procedure and verifies that effective resistance is a valuable measure of ice-rise/ice-shelf interaction.

Mass Balance

Mass balance, M (kg m⁻³), of the region enclosed by ⌈was computed under the assumption that basal melting is zero. In this circumstance, M is the sum of advection, denoted Q, and snow accumulation Ȧ (kg m⁻² s⁻¹), and is given by

where u (m s⁻¹) is the horizontal ice velocity (independent of depth at our ice-shelf measurement stations; see Morland, in press), and S is the surface area enclosed by ⌈ (1.1065 × 10¹⁰m² on the stereographic projection using station E₄ as the projection pole). We computed M using the following data: (i) the regionally averaged snow-accumulation rate determined by beta-activity analysis of snow cores from ten RIGGS stations (F₇, G₈, G₁₀, H₈, H₉, H₁₀, H₁₁, I₉, I₁₀, and I₁₁) reported by Reference Clausen, Clausen, Dansgaard, Nielsen and CloughClausen and others (1979); (ii) linearly interpolated values of u using the observed values of u at surface stations along ⌈ (Fig. 9; Table I); and (iii) ice thickness H derived from radio echo-sounding flights (or interpolated from RIGGS measurements over segments K₃ to J₁₀, J₁₁ to I₁₁, and I₁₁ to J₃). The discretized version of the contour integral in (Equation 15) used to calculate M is presented in the Appendix. Confidence limits on M are calculated using estimated uncertainties of the velocity component perpendicular to ⌈ and of A shown in Table III.

Fig. 9. Measured surface velocities used to calculate mass balance and the work done against resistive forces of the Crary complex (see also Table 1). Data sources: Bindschadler and others (1984); Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes and BentleyThomas and others (1984); Reference Bindschadler, Bindschadler, MacAyeal, Stephenson, Veen and OerlemansBindschadler and others (1987); paper in preparation by R.A. Bindschadler and others. All measurements were conducted using “geoceiver” satellite position-fixing equipment.

The computed mass balance M indicates that mass is accumulating within ⌈ at a rate of 1.40 ± 0.20 × l0⁵kgs⁻¹. This accumulation could be balanced in part, or in whole, by basal melting (at an average rate of 0.44 m/year) within ⌈ (see Reference Pillsbury, Jacobs and JacobsPillsbury and Jacobs (1985) for discussion of oceanic evidence for basal melting below the Ross Ice Shelf). Given the lack of sub-ice-shelf ablation measurements available from the Ross Ice Shelf, it is not possible to judge the likelihood of such a balance. Reference MacAyealMacAyeal (1984) suggested, however, that tidal fronts may develop in the sea-water cavity surrounding Crary Ice Rise as a result of high tidal amplitudes and vertical mixing. These fronts could promote locally vigorous basal melting at rates between 0.05 and 0.5 m/year. Our radio echo-sounding data indicate that basal echoes are strong in the vicinity of the ice rise, indicating possible basal erosion (see also Reference NealNeal, 1979). Strongest basal-echo amplitudes occur along the southern flank of the ice-rise complex facing the Transantarctic Mountains. A sea-bed trough is situated adjacent to this flank, and may provide a path for warm, saline water to penetrate from sources in the open Ross Sea (Reference MacAyealMacAyeal, 1984; see Reference Greischar and BentleyGreischar and Bentley (1980) for a map of sub-ice-shelf sea-bed elevation). If tidally triggered basal melting is active in the vicinity of the ice-rise complex, it is unlikely to affect more than a limited part of the area enclosed within ⌈ (Reference MacAyealMacAyeal, 1984). We therefore suggest that M is largely unbalanced by basal ablation.

Assuming that basal melting does not balance M, mass accumulation within ⌈ would support an area-average thickening rate of 0.44 ± 0.06 m/year. This apparent thickening rate is consistent with previous estimates of mass balance (Reference ThomasThomas, 1976; Reference Thomas and BentleyThomas and Bentley, 1978; Reference Jezek and BentleyJezek and Bentley, 1984; Reference Shabtaie and BentleyShabtaie and Bentley, 1987), and suggests that the ice-rise complex is growing. To test this suggestion, the temperature-depth profile could be measured at a number of positions on the grounded ice rise to see whether deviations from steady state support a history of recent grounding. This procedure was used to date the formation of ice rises on Ward Hunt Ice Shelf in the Arctic Ocean (Reference Lyons, Lyons, Ragle and TamburiLyons and others, 1972; see also Reference MacAyeal and ThomasMacAyeal and Thomas, 1980).

For reference, we also computed the net mass discharge, Q*, of Ice Streams A and B into the Ross Ice Shelf across the contour ⌈*. The apparent mass accumulation within the ice-rise complex M (Table IV) amounts to 10% of the net ice-stream discharge (Table IV). Our computed value of 0*, in terms of ice volume, amounts to 46.9 ± 1.4km³/year. Given the uncertainty, this value is equivalent to 49.2 ± 2.5 km³/year reported by Reference Shabtaie and BentleyShabtaie and Bentley (1987) for a slightly different cross-section of the ice-stream grounding line (see fluxes A, B, and B2 in their table 2a).

Reference Shabtaie and BentleyShabtaie and Bentley (1987) suggest that the present mass flux of Ice Stream B exceeds snow catchment in the region drained by the ice stream by approximately 10–20 km³/year (see their table 2c). If their estimate is correct, then approximately 25–50% of the excess discharge could be accumulating within the ice-rise complex or balanced by basal melting there. The current vigorous flow of Ice Streams A and B may thus be building ice rises.

Mechanical-Energy Budget

Gravitational potential energy must be released in the ice-shelf/ice-stream system to balance energy loss associated with flow past the ice-rise complex. This energy loss results from: (i) work done against friction (dynamic drag), and (ii) redistribution of gravitational potential energy associated with ice-shelf or ice-rise thickening within ⌈. In computing the redistribution of potential energy, it is necessary to account for potential energy associated with displaced seawater. We compute the energy loss by evaluating the rate at which work is done along ⌈ by ice-shelf movement against the effective resistance:

where (f ^f+ f^d – f ^w) is the local equivalent of the effective resistance.

Work done against the local form drag f^f represents the rate at which gravitational potential energy possessed by ice is carried by advection across ⌈ This flux exists only where there is net mass influx across ⌈ (see previous section). Archimedes’ principle requires that, for a given ice-shelf mass entering ⌈, an equal sea-water mass must flow out of ⌈. This requirement assumes that sea-level changes negligibly as a result of mass convergence into ⌈, and that ice-shelf flux convergence leads to ice-shelf thickening rather than ice-rise thickening. Gravitational potential energy flux associated with this displaced sea-water mass is smaller than that of the equal mass of ice; and is equal in magnitude, but opposite in sign, to the work done by ice shelf against hydrostatic sea-water pressure f ^w along ⌈ Work done by the ice shelf against the difference f ^f −f ^w thus represents the net potential energy change within ⌈ due to mass exchange between water and ice. Work done against dynamic drag F ^d along the contour ⌈ accounts for frictional energy dissipation. Values of W computed using the data-analysis procedures presented in the Appendix and the three alternative flow laws are presented in Table IV.

Energy loss within the ice-rise complex may originate up-stream of the grounding line where ice-stream thinning releases gravitational potential energy. In this circumstance, energy must be transferred to the ice shelf by work done at the ice-stream grounding lines. We examine this possibility by computing the work W* done against the extra back-pressure force (f ^f+f^d *- f ^w) by Ice Streams A and B along the grounding-line contour ⌈*. This work accounts for energy losses associated with resistance provided by the entire ice shelf. As shown in Table IV, the energy loss associated with the ice-rise complex W is found to range between 15 and 49% of the grounding-line energy flux W* (for flow laws #3 and #l, respectively). This result suggests that the ice-rise complex provides a substantial sink of potential energy released by ice-stream flow.

Ice-Rise, Ice-Shelf, and Ice-Stream Interactions

Our interpretation of field data yields the following conclusions: (i) approximately 45–51% of the extra back-pressure force on the grounding line of Ice Streams A and B results from the effective resistance generated by the Crary Ice Rise complex; (ii) the fraction of this extra back-pressure force contributed by the ice-rise complex is relatively insensitive to the flow law used to convert measured strain-rate into stress; (iii) the effective ice-rise resistance varies by a factor of two between non-Newtonian flow laws (n = 3) commonly used in analysis of ice-shelf flow (such as derived from laboratory creep tests by Reference Barnes, Barnes, Tabor and WalkerBarnes and others (1971), or from bore-hole tilt and ice-shelf spreading observations by Reference PatersonPaterson (1981) and the Newtonian flow law (n = 1) described by Reference Doake and WolffDoake and Wolff (I985[a], Reference Doake and Wolff[b])); (iv) ice accumulates within the ice-rise complex, primarily due to ice-flow convergence, at a rate exceeding estimates of regional basal melting derived from Reference MacAyealMacAyeal (1984) and from Reference Pillsbury, Jacobs and JacobsPillsbury and Jacobs (1985); (v) if this accumulation is not balanced by basal melting, the ice-rise complex may be expanding through ice-shelf thickening and grounding; (vi) this possible expan-sion may constitute a response to high discharge through Ice Streams A and B; and (vii) energy dissipated by flow around the ice-rise complex is balanced by energy transmitted across the grounding line of Ice Streams A and B.

Considering the possible consequences of climatic change such as atmospheric warming, oceanic warming, or sea-level rise, our analysis supports the view expressed by Reference Thomas, Thomas, Sanderson and RoseThomas and others (1979) that ice rises and ice rumples are key sources of resistance that stabilize the West Antarctic ice sheet. Ice-rise influence is large compared with the physical size of the ice rise or rumple; thus, relatively local changes of oceanic or atmospheric conditions in the vicinity of an ice rise or rumple may produce widespread modifications of the ice-sheet/ice-shelf system. In terms of ice-stream dynamics, our results suggest that ice rises may form or expand in response to ice-stream accelerations and given time to develop significant resistance, may ultimately damp the ice-stream fluctuations that formed them.