Dynamics and mass balance of Taylor Glacier, Antarctica: 2. Force balance and longitudinal coupling

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi:1.129/29jf1329, 29 Dynamics mass balance of Taylor Glacier, Antarctica: 2. Force balance longitudinal coupling J. L. Kavanaugh 1 K. M. Cuffey 2 Received 28 March 29; revised 16 June 29; accepted 1 July 29; published 3 November 29. [1] Taylor Glacier, Antarctica, exemplifies an ice sheet outlet that flows through a region of rugged topography dry climate. In contrast to other well-studied outlets, Taylor Glacier moves very slowly, despite a thickness of order 1 km driving stresses averaging 1.5 bars. Here we analye new measurements of glacier geometry surface velocity to elucidate flow dynamics of Taylor Glacier. Force balance basal temperatures are calculated at six locations along the glacier s length using an algorithm developed for this study. The effects of stress-gradient coupling on longitudinal flow variations are also examined; we ask whether Kamb Echelmeyer s (1986) linearied theory adequately describes the observed response of flow to large-amplitude variations in driving stress. The force balance calculations indicate that no basal motion is needed to explain the observed flow of Taylor Glacier. Inferred basal temperatures are within a few degrees of the melting point in regions of kilometer-thick ice well below the melting point elsewhere; deformation of subfreeing ice largely controls the flow of Taylor Glacier. Basal drags are mostly in the range.9 to 1.2 bars, lateral drags are in the range.2 to.5 bar. Stress-gradient coupling strongly reduces the variability of velocities along the glacier. The velocity variations can be described as the convolution of a forcing function with a spatial filter, as Kamb Echelmeyer suggested, but the form of the forcing function differs from the theoretical relation derived for small-amplitude perturbations (the power on driving stress is one, not three). Citation: Kavanaugh, J. L., K. M. Cuffey (29), Dynamics mass balance of Taylor Glacier, Antarctica: 2. Force balance longitudinal coupling, J. Geophys. Res., 114,, doi:1.129/29jf Introduction [2] A general description of Taylor Glacier, its special interest to studies of Antarctic geomorphology, is given in a companion paper [Kavanaugh et al., 29]. Here, we use the surface flow data topographic models discussed in that paper to examine three aspects of the glacier s flow dynamics: the force balance, the basal temperatures, the role of stress-gradient coupling in controlling longitudinal variations of flow Motivation [3] Taylor Glacier differs from other well-studied glaciers, ranging from polar ice sheet outlet glaciers to small alpine glaciers; Figure 1 illustrates the contrast. For any viscous gravitational flow with significant basal drag, the velocity (U) increases with the product of driving stress t d thickness H. The ratio U/(t d H) can be regarded as an index of how readily the flow occurs. Evidently, the tendency for Taylor Glacier to flow is exceptionally low for a glacier; it 1 Department of Earth Atmospheric Sciences, University of Alberta, Edmonton, Alberta, Canada. 2 Department of Geography, University of California, Berkeley, California, USA. Copyright 29 by the American Geophysical Union /9/29JF1329 moves sluggishly despite a large driving stress a thickness of order 1 km. Taylor Glacier, in addition, passes through a rugged mountainous lscape, so its flow regime changes rapidly as it flows over bedrock ridges basins, through valley narrows, around large bends. [4] Outlet glaciers of this sort do not contribute much to the overall transport of ice off the Antarctic continent. They are very interesting, nonetheless, because they are abundant features along some sectors of the ice sheet margin. Furthermore, their flow is a large-scale manifestation of the viscous creep deformation properties of cold polycrystalline ice, an important controversial subject [Paterson, 1994, pp ; Goldsby Kohlstedt, 21; Duval Montagnat, 22] Prior Work [5] In a companion paper [Kavanaugh et al., 29] we present extensive new measurements of Taylor Glacier s surface velocities, elevations, ice thicknesses discuss their main features. Prior to our investigations, the most comprehensive analysis of Taylor Glacier dynamics was that of Robinson [1984]. Using two separate analyses, one of measured ice flow rates one of modeled temperature-depth profiles, Robinson concluded that in the valley center, about half of the motion of Taylor Glacier must be due to basal sliding over a thawed bed. There are three main weaknesses of these analyses: (1) Ice thicknesses 1of11

2 Figure 1. Flow index U/(t d H) plotted as a function of ice thickness H, for typical locations on several well-studied glaciers. Shown are Taylor Glacier (TG; filled square) examples of alpine polar glaciers (circles: MC, McCall Glacier; ME, Meserve Glacier); alpine temperate glaciers (triangles: ST, Storglaciären; BL, Blue Glacier; WO, Worthington Glacier); one temperate tidewater glacier (pluses: C77, Columbia Glacier, year 1977; C95, Columbia Glacier, year 1995); strong-bedded polar ice streams (diamonds: J7, Jakobshavn Isbrae, 7 km from terminus; J4, Jakobshavn Isbrae, 4 km from terminus; JU, Jutulstraumen Ice Stream); weak-bedded polar ice streams (squares: WIS, Whillans Ice Stream; PIG, Pine Isl Glacier; NEG, Northeast Greenl Ice Stream). Data sources are Kavanaugh et al. [29], Rabus Echelmeyer [1997], Holdsworth [1974], Hanson Hooke [1994], Meier et al. [1974], Harper et al. [21], O Neel et al. [25], Clarke Echelmeyer [1996], Rolstad et al. [2], Engelhardt Kamb [1998], Thomas et al. [24], Joughin et al. [21]. were underestimated in some locations; (2) the role of lateral drag in reducing basal shear stresses was not considered; (3) the temperature analysis neglected horiontal heat advection. [6] Higgins et al. [2] revisited this issue. They modeled the ice thickness profile along the glacier, constrained by estimated ice fluxes, compared the results to available airborne radar measurements of ice thickness from Drewry [1982]. Higgins et al. concluded, in contrast to Robinson [1984], that the glacier bed is froen. This was an important conclusion for interpreting the geologic record created by past advances of Taylor Glacier; because a froen bed implies negligible basal slip, morainal debris here likely originates by entrainment during advance into lakes rather than by subglacial erosion on a thawed bed. We note that the basal elevation model used by Higgins et al. study was poorly constrained differs substantially from the basal topography inferred by Kavanaugh et al. [29], especially in the western (upper) half of the glacier. [7] Hubbard et al. [24] performed a rigorous analysis of the lowest 8 km of Taylor Glacier. They constructed a map of bed elevations using radar surveys, also estimated basal temperatures by linking a thermal model to a sophisticated ice flow model [Blatter, 1995]. They concluded that basal temperatures here, where ice thicknesses average only about 5 m, are well below melting point for pure ice. The maximum in their study region was estimated to be 7 to 8 C. Strong radio reflections from the bed indicated the presence of water in the deepest regions, most likely indicating the presence of hypersaline water. The water is not abundant enough, however, to show in airborne radar data [Holt et al., 26]. [8] Our study covers a much larger portion of the glacier than examined by Hubbard colleagues, but at lower spatial resolution. One of our sites, discussed in the present paper, sits within the region examined by Hubbard et al. [24], allowing a cross comparison of results Goals of This Paper Force Balance [9] Here we use measurements of surface velocities strain rates to estimate how driving stress is partitioned into different resisting forces at several locations along the glacier. Such force balance analyses, a type of inverse problem, are widely applied in glacier dynamics studies [e.g., Rolstad et al., 2; O Neel et al., 25; Maxwell et al., 28], but the methodology in any situation must be adapted to case-specific factors, including the availability of data constraints, the spatial scale of interest, the ice dynamical setting (e.g., isothermal versus nonisothermal conditions, significant versus insignificant basal drag). [1] This is necessary because the problem of inverting surface measurements for conditions at depth in a glacier is fundamentally ill-posed; a direct application of the momentum equations, as originally suggested by van der Veen Whillans [1989], cannot generate a unique or stable solution because errors grow exponentially with depth [Bahr et al., 1994]. Essentially nothing can be learned about distributions of stresses velocities near the bed at spatial scales shorter than about one ice thickness [Balise Raymond, 1985; Maxwell et al., 28]. For variations at larger scales, such as the place-to-place differences considered in the present paper, approximate solutions can be achieved by constraining the analysis with assumptions that are plausible considering the general properties of glacier flow (see the discussions by Truffer [24] Maxwell et al. [28]). Here we present a method, applicable to nonisothermal glaciers with significant basal drag, that relies on assumptions about how variables change with depth. Our method is numerically stable focuses only on obtaining quantities averaged through a vertical column; we do not try to resolve details of the deformation in layers near the bed. [11] We use the analysis to address three questions: How much of Taylor Glacier s motion can be accounted for by internal creep rather than basal slip? What are likely values for basal temperatures? How important are different deformations (basal shear, lateral shear, longitudinal stretching compression) in controlling the glacier s flow? Longitudinal Coupling [12] Stress-gradient coupling reduces the along-glacier velocity fluctuations arising from variations of driving stress ice thickness. Kamb Echelmeyer [1986] made a theoretical analysis of this problem for small-amplitude perturbations of stress flow. They concluded that longitudinal flow variations can be described by the convolution of a forcing function with a filter function, with the forcing function dependent on powers of driving stress 2of11

3 Figure 2. Map view of survey marker locations (blue dots), GPS base stations (blue triangles), flow lines used for analysis of longitudinal flow variations in section 3 (green curves). Force balance analyses were performed on the central flow line, the axis of fastest flow, where it intersects the across-glacier transects labeled T3 through T22. ice thickness. This process is equivalent to a longitudinal averaging. Acknowledging the many assumptions that underly this theory, Kamb Echelmeyer [1986, p. 267] referred to this approach as semi-quantitative. Nonetheless, it has been shown to provide reasonable estimates of velocity variations along two temperate valley glaciers [Kamb Echelmeyer, 1986] one polythermal polar alpine glacier with an ice thickness of about 15 m [Rabus Echelmeyer, 1997]. [13] Taylor Glacier provides an opportunity to examine longitudinal flow variations on a large polar glacier. Here we adopt a simple empirical approach; we identify the forcing filter functions that give a best match between observed calculated velocities. The relation between longitudinal variations of stress flow should be regarded as a fundamental property of the glacier, comparable to the balance of forces the basal conditions. (In fact, longitudinal coupling expresses one aspect of the balance of forces.) In addition, the question of how to best perform the longitudinal averaging has general interest. At Taylor Glacier, the variations of driving stress are large, whereas the theoretical relations were constructed using a lineariation technique appropriate for small perturbations. In addition, the theory strictly applies for a straight flow line. From Kamb Echelmeyer s analysis we would expect that longitudinal averaging must involve lengths as great as 24 km, substantially longer than any of the straight flow line segments. This suggests that the formulation might work poorly. 2. Force Balance Basal Temperature [14] We analyed the force balance for the six locations on the central flow line where our survey network included complete across-glacier profiles. Figure 2 shows the locations. We here summarie the method give results. Appendix A provides a more detailed description of the method Summary of Method [15] We choose the x axis as horiontal in the direction of flow, the y axis as horiontal across flow, the axis as vertical (with = at the bed in the center of the region, = H at the ice surface). The gravitational driving stress acting upon the glacier is t d = r i gah. (Here r i is the density of ice, g the gravitational acceleration, a the surface slope.) This is the horiontal force acting on a vertical column spanning the entire glacier thickness. For a subsection of this column with base at top at H, the driving stress ^t d () is h ^t d ðþ¼r i gha 1 i : ð1þ H Neglecting local deviations of vertical normal stresses from hydrostatic, the balance of forces on this column in direction x is [van der Veen Whillans, 1989] ^t d ðþ¼t x 2t xx þ t yy d t xy d : ð2þ x y fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflffl{fflfflfflfflfflfflfflfflffl} term 2 term 3 The three terms on the right are resisting forces. The first term represents drag from shear on horiontal planes; the second term, the net force from variations of stretching compression along the glacier ( longitudinal drag ); the third term, drag from across-glacier variations of shear on vertical planes ( lateral drag ). At the bed, the total resistance per unit area, acting up glacier, is the basal drag t b. This equals the value of t x at = times a small correction for the area of bedrock surface per horiontal area. We assume, following stard practice, that deforma- 3of11

4 tion rates deviatoric stresses obey the generalied Glen s law [Nye, 1957] _e ij ¼ AT ð Þt n 1 II t ij _e II ¼ AT ð Þt n II ; where A(T) is a temperature-dependent coefficient (taken from Paterson [1994, p. 97]), n = 3 is the creep exponent, t II the second invariant of the deviatoric stress tensor, _e II the second invariant of the strain rate tensor. Given that u x / 2_e x, the glacier surface moves at a rate U x ¼ U b þ 2 ð3þ A 1=n _e ðn 1Þ=n II t x d : ð4þ All three factors in the integr are functions of. U b denotes the basal velocity, initially assumed to be ero. [16] We solved equations (2), (3), (4) by an iterative method that adjusts the temperature at depth in order to match observed surface velocities, uses observed strain rates at the surface to specify strain rates at depth. The following discussion outlines the assumptions used to achieve a solution; the Appendix A provides details of the method. To justify the procedure, we note that it is (1) much better than the default assumption that t x () = ^t d (), (2) commensurate with the type of measurements available to us (we have no measurements of flow or strain rate at depth), (3) numerically stable, unlike some forms of force balance analysis. The assumptions quantitatively capture the dominant features of the glacier mechanics while making no attempt to represent the details. No method, including the most complete ice flow models, can represent accurately the details of flow deformation deep in the glacier because variations of the bed topography ice viscosity are not known at a high resolution. Moreover, as noted in the Introduction, inverting surface measurements for conditions at depth in a glacier is a fundamentally ill-posed problem; assumptions must be made to constrain solutions [Truffer, 24]. [17] We first calculated _e xx, _e yy, _e xy at the glacier surface from velocities measured at groups of survey poles. We eliminated _e from the equations by applying the incompressibility condition ( _e = _e xx + _e yy ), then assumed that some quantities vary with depth according to scaling relations: _e y ¼ U x U y _e x ; YðÞ¼ u xðþ Y S with Y 2 ¼ _e 2 xx U þ _e2 yy þ _e2 xy þ _e xx_e yy ; ð6þ x L x ¼ L ðþ L L x with S y ¼ S ðþ S S y L ðþ¼ with ð5þ 2t xx þ t yy d; ð7þ S ðþ¼ t xy d: ð8þ In the above relations, U x U y are the surface components of ice velocity in the along-flow acrossflow directions (defined by the local mean of the surveyed velocities), u x () u y () are the velocity components at depth. In the expressions for Y, L, S, subscript S denotes a surface value while signifies a value calculated for =. These scalings assume that the direction of flow does not change with depth, that the secondary deformations (_e xx, _e yy, _e xy ) decrease with depth at the same rate as the dominant velocity component u x. The second of these assumptions is unlikely to be valid deep in the glacier. Its violation should have little effect, however, because horiontal plane shears dominate the secondary deformations in the lower half of the ice thickness. [18] Another assumption concerns the horiontal gradients of depth-averaged deviatoric stresses (t ij ) within the small regions of the glacier used for the analyses. The regions are typically about 1 km 2, contain a cluster of about 1 survey poles. The gradients (2t xx + t yy )/x t xy /y were taken as constants in each such region. This is essentially equivalent to assuming that one can neglect the third higher derivatives of the velocity with respect to horiontal distance. Given that glacier flow varies smoothly with distance, this should be a very good assumption within these small regions; moreover, the surface velocity data indicate it is so [see Kavanaugh et al. [29, Figure 3a]. [19] A final major assumption concerns the variation of temperature with depth. We assume it increases linearly, from a given surface temperature to a value at the bed that is determined as part of the analysis. Preliminary modeling of the steady state temperature distribution, using the full three-dimensional energy equation, shows that a linear depth profile provides a good approximation in this glacier. Upward ice flow, as occurs in ablation ones, tends to make the temperature profile convex up. Down-valley advection of cold ice tends to make the temperature profile concave up. Together, these competing factors produce nearly linear temperature profiles. The slow flow of Taylor Glacier makes this possible; in rapidly flowing glaciers, such as the major ice streams, horiontal advection creates a minimum in temperature at midrange depths. Here, the velocity field for temperature modeling was constrained by measured surface values (for the horiontal components) by measured ablation rates (for vertical components). [2] In section 2.3, we discuss how our results would change if the primary assumptions are invalid. [21] The equations are solved by iteration. From an initial guess at the stress distribution, equation (4) is integrated upward to calculate a surface velocity. The basal temperature ( hence the temperature profile T()) is then adjusted until the calculated velocity matches the measured value. Integrating equation (2) down from the surface then gives new estimates for stresses. The velocity profile is recalculated, with adjustments of the basal temperature as before. Repetition continues until a solution is obtained. [22] As noted above, values used for the ice softness coefficient A(T) are those from Paterson [1994]; no adjustment has been made in these values for freeing point depression due to pressure or salinity effects or for the greater softness of ice age ice. Because both of these effects increase the rate of flow at a given temperature, their inclusion would result in a lower calculated basal temperature value. We note 4of11

5 Figure 3. Results of force balance calculations for transect T12. (a) Velocity profile u x (). (b) Ice temperature profile. (c) Vertical profiles of stress components t x (black), t xx (blue), t yy (red), t xy (green). (d) Vertical profiles of force balance terms t d (black), t x (blue), term 2 (red), term 3 (green). Terms 2 3 are as defined in equation (2). here that any basal temperature T B. C calculated by the method described above indicates a froen bed in this location, as ice creep alone can account for the observed rate of motion at the ice surface. Calculated basal temperatures exceeding. C imply that to explain the observed surface velocities, basal motion or enhanced ice shear must occur Results [23] Figure 3 shows one example of the calculated vertical distributions of stress flow. Table 1 summaries the primary results for all six locations on the central flow line. Basal stresses are estimated to be about 1 bar (the mean for the six sites is 1.1 bars), as is typical for slow flowing glaciers. Because the glacier is confined to a valley, lateral drags oppose the driving stress reduce the forces available for basal shear by 1 4%. As the glacier velocity nominally varies as the third power of the basal shear stresses, this implies that lateral drags reduce the velocity by about 3 to 8% (given a constant ice thickness). Longitudinal forces are important in some places. They restrain the flow at location T12, where the valley curves strongly. At T19, in contrast, a steep icefall located a few kilometers west of the site pushes the glacier forward. [24] Estimated basal temperatures are well below melting point for pure ice. Thus, no basal motion is needed to explain the observed flow anywhere on the glacier. [25] Our site T3 falls within the region examined by Hubbard et al. [24]. They modeled this section of the glacier using the scheme of Blatter [1995] for including longitudinal lateral stress effects. They concluded that the basal temperature near our site T3 was about 8.5 C. This compares very well with our estimate of 7.5 C can be taken as confirmation of our method s validity. [26] On the other h, the estimated temperature at T6 is probably too low. Here the glacier flows through a comparatively narrow canyon. At the spatial resolution of our measurements, some of the lateral drag provided by the sides of the canyon (near its bottom) are effectively ascribed to basal drag t b. (t b is the total resistance, normalied to horiontal area, on a section of the glacier base of sie 1 km 2.) 2.3. Sensitivity of Results to Assumptions [27] Two major glaciologic factors are poorly constrained: the shape of the temperature-depth profiles, the variations of flow strain rates deep in the glacier. We now discuss the sensitivity of our results to these uncertain quantities. We also illustrate the sensitivity to the assumption of uniform stress gradients. Because of the high precision of measurements with GPS at the site [Kavanaugh et al., 29], uncertainties in values for surface velocities strain rates are negligible in comparison. Elevations are also tightly constrained by GPS surveys. Thus ice thicknesses are the most poorly known factors in driving stresses. With the radar system used, the uncertainty on thickness is estimated as about ±1 m for an ice thickness of 7 m [Kavanaugh et al., 29], for a range of about 3%. Supposing that surface velocity varies as the third power of driving stress, this corresponds to a change of velocity by a factor of about 1%. The same change would result from a temperature change of about 1 C deep in the glacier Sensitivity to Shape of the Temperature Profile [28] If the temperature profile is convex up, as would be the case in an ablation one with negligible horiontal advection, our basal temperature estimates are too high. Such a situation strengthens our conclusion that no basal motion occurs. In contrast, a concave-up temperature profile, the typical form in regions of strong horiontal advection, implies that our basal temperature estimates are too low the bed may be thawed. Measured surface velocities provide a strong constraint, however, because any increase of temperatures near the bed will have a much larger effect on velocities than will a decrease of temperatures in the middle of the ice thickness. The larger sensitivity to basal temperatures reflects the higher stress, the dependence of the strain rate on the third power of the stress, the exponential decrease of ice viscosity as temperature increases. [29] Consider one demonstration of the magnitude of the effect, using a typical situation for Taylor Glacier (basal temperature of 5 C, ice thickness of 8 m). Suppose the Table 1. Results of Force Balance Calculation a Transect t d (bars) t b (bars) Term 2 (bars) Term 3 (bars) T() ( C) T T T T T T a The unit 1 bar = 1 5 Pa. Terms 2 3 represent longitudinal lateral coupling, respectively, T() is the basal temperature. Driving stress is defined as positive down glacier. All resistances (t b, term 2, term 3) are shown here as negative if they act up glacier. Terms 2 3 are as defined in equation (2). 5of11

6 Figure 4. Force balance model perturbation examples results for transect T12. (a) Sample of 1 romly perturbed velocity profiles u x () (blue curves); profile determined from force balance calculation shown for comparison (red curve). Across-glacier velocity profiles u y () are perturbed in a similar (but independent) manner. (b) Sample of 1 romly perturbed profiles of ice flow law prefactor A() (blue curves) with profile determined from force balance calculation assuming linear temperature profile shown for comparison (red curve). (c) Probability density for basal stress t b resulting from n = 1, perturbations of u x (), u y (), A() as shown in Figures 4a 4b. (d) Probability density for basal stress t b resulting from n = 1 rom variations of values C 1 C 2 across clusters U, R, L. profile changes from linear to one that is strongly concave, such that the temperature is uniform in the upper half of the glacier but varies linearly in the lower half. To maintain the same surface velocity, the basal temperature would need to increase to about 3 C, assuming no change in the stresses. (If basal temperature increased all the way to the pure ice melting point, the surface velocity would increase by a factor of 1.7, a very large change compared to the accuracy of measurements.) If the stresses t x are simultaneously reduced by 2%, to reflect an increase of resisting forces due to colder ice in the middle of the glacier, then the basal temperature would need to increase to about 2 C. [3] Thus, given the absence of any temperature measurements at depth, we cannot completely exclude the possibility that the bed reaches melting point at the locations with thickest ice (T12 T16, Table 1). It is clear, however, that if any basal motion occurs it must be a small fraction of the surface velocity inconsequential for the glacier dynamics Sensitivity to Deep Layer Dynamics [31] Velocity profiles deep in polar glaciers often do not conform to theoretical expectations [e.g., Paterson, 1994, pp ]. Viscosity variations, unrelated to temperatures, are one reason. Complicated flow patterns induced by bedrock topography are another. To assess how such variations add uncertainty to the stresses determined with our force balance analysis, we adopt a Monte Carlo approach. We suppose that the velocity-depth profiles at our analysis sites in the adjacent vertical columns do not match the calculated ones, but instead take a wide range of perturbed forms in the lower half of the ice thickness. Figure 4a illustrates some examples. (The velocity at the surface is fixed by the measured value.) We also assume that the A values (the coefficient in Glen s law) vary romly as arbitrary sinusoidal functions of depth, with a factor of 5 range about the A profile given by the original analysis. This fivefold range corresponds to typical variations of viscosity seen in borehole tilt experiments in polar ice [see, e.g., Paterson, 1994, pp ]. [32] Figure 4b again illustrates some examples. Because the velocity profiles are specified, the perturbed A values represent changes in the effective viscosity for the secondary deformations (_e xx, _e yy, _e xy ), not for the horiontal plane shears. This implies anisotropic ice, as expected if viscosity variations reflect different c axis fabrics. (Note that the A value for horiontal plane shears also varies romly, but implicitly, as a function of the stresses the velocity profile. Because we are specifying the velocity profiles, these variations of A should be regarded as arbitrary. This analysis shows the sensitivity of calculated stress terms but has no bearing on calculated basal temperatures.) [33] Using perturbed velocities A values, we abon the scaling assumption that the secondary deformations decrease with depth in proportion to the velocity. We calculate the force partitioning (equation (2)) using secondary strain rates determined by spatial gradients in the perturbed velocities at depth. Figure 4c shows, for location T12, the distribution of basal drag values given by a large number of such perturbation experiments. The likely range corresponds to about ±.8 bar, or, at this site, about ±15% of the difference between driving stress basal drag. Thus the longitudinal side drag terms are plausibly larger or smaller than the values given in Table 1 by at least 15% Sensitivity to Assumed Uniformity of Stress Gradients [34] Finally, in Figure 4d, we show the effect of aboning the assumption that the horiontal gradients of depthaveraged stresses are uniform within the region of analysis. The gradients on the edges of the study region are allowed to vary between.5 2 times the value in the center of the region, a new solution for both stresses temperatures found by iteration as before. (In other words, with reference to terms defined in the Appendix A, the constants C 1 C 2 in columns U, R, L vary between.5 2 times their values in the central column.) The likely range of basal drags corresponds to ±.5 bar, the basal temperatures vary by ±1. C. 3. Longitudinal Coupling 3.1. Definitions of Variables [35] Kamb Echelmeyer s [1986] analysis of stressgradient coupling derived the following relationships between longitudinal variations (assumed to be small) of 6of11

7 Figure 5. Longitudinal variations of flow stress along two flow lines. (a) Driving stress values t d for lower glacier flow line (location shown in Figure 3). (b) Driving stress values t d for upper glacier flow line (location shown in Figure 3). (c) Comparison of measured (black) calculated (blue) perturbations of velocity along the lower flow line. Velocity perturbation is defined as the deviations from unity of velocity normalied to its mean: U/U 1. (d) Comparison of measured (black) calculated (blue, red) velocity perturbations along the upper flow line. Unlike the blue curve, the red curve uses a constant value for the averaging length scale ; it fails to show some of the structures revealed by measurements. driving stress t d (x), basal drag t b (x), ice thickness H(x), velocity U(x) (see equation 39 of Kamb Echelmeyer): Or, equivalently, t b ðþh x p ðþ¼t x ½ d H p Š* J ðþ with ¼ lh ð9þ U ¼ K t n b H q : ð1þ U ¼ K t n d H q * J ðþ: ð11þ Here the * denotes convolution J signifies a filter function, which can be approximated as a symmetrical triangle with total base length of 4 (called the longitudinal averaging length ). The power p is 1/n for flow by internal creep but ero for basal slip. Likewise, q equals one or ero, respectively. At Taylor Glacier, the temperature increases with depth so concentrates shearing toward the bed (see, for example, Figure 3a). Thus, for this glacier, it is possible that p q are closer to ero than to 1/n 1, even if no basal slip occurs, because the warm basal layers might act as a shear one of roughly uniform thickness. [36] The averaging length scale,, is proportional to ice thickness. Kamb Echelmeyer [1986] estimated the proportionality l as one to three for temperate valley glaciers, but four to ten for ice sheets. The larger scale for ice sheets arises from two factors. First, increases if the effective viscosity for longitudinal deformation exceeds that for basal shear; this is the expected situation in polar glaciers because of the temperature contrast between the upper layers the basal layers. Second, is smaller in a confined valley than in a broad sheet, because, in a valley, lateral drags take up some of the driving stress; only the portion of driving stress that is resisted by basal longitudinal drags plays a role in the longitudinal coupling. A factor f, corresponding to the value of t b /t d in a longitudinally uniform section of glacier, can be inserted in both sides of equation (9) to account for this [Kamb Echelmeyer, 1986]. However, only the variations of f, not its mean, affect the variations of U Method [37] Here we examine how the Kamb Echelmeyer [1986] formulation applies to Taylor Glacier. We ask what form of equation (11) works best at this site, if any. From the theory we would expect that because temperature varies by 1 2 C between bed surface, the value for l would be 4 6. This implies a longitudinal averaging length (4lH) of as much as 24 km, longer than any of the straight flow line segments, suggests that the formulation might work poorly. On the other h, the mathematical form of relations like equations (9) (1) implies that the amplitude of calculated velocity fluctuations increases with the power n but decreases with the averaging length. This suggests that by reducing the value of n, a smaller averaging length can be used. [38] Such an approach retains the center weighting implied by the physics of longitudinal coupling, while suppressing the unrealistic behavior arising from application of a smallperturbation theory to finite amplitude perturbations. It is a compromise justified largely by its mathematical properties; the Kamb-Echelmeyer theory was not designed to work for finite amplitude perturbations. (Here we ask, empirically, whether such a compromise provides a good match to data.) A physical consideration of possible relevance is that longitudinal extending compressing deformations decrease the effective viscosity of ice, a consequence of ice s nonlinearity [Nye, 1957]. Softer ice, in turn, reduces the length scale of longitudinal coupling [Jóhannesson, 1992; Gudmundsson, 23]. [39] We examined two longitudinal transects on Taylor Glacier, one in the upper ablation one, one in the lower (the green curves shown in Figure 2). Along both, ice thickness is known well from the radar surveys reported by Kavanaugh et al. [29]. In the middle of both, the flow line changes direction by almost 9. Driving stresses vary by a factor of 5 on the upper one, by a factor of 3.5 on the lower one. Along both, we calculated the basal drag variation as t b = [t d H p ] * J( = lh) the velocity variation as t b m H q, for different values of m, p, q l. The results were compared to the velocities measured by interferometry [Kavanaugh et al., 29] Results Comments [4] Figure 5 shows that longitudinal velocity variations along Taylor Glacier can be simulated roughly by using the 7of11

8 simple relation U = t d * J, with the parameter l = 2 on the lower glacier l = 3.5 or 4 on the upper glacier. This relationship, for which m = 1 p, q =, works much better than the relation for small-amplitude variations suggested by Kamb Echelmeyer (m = 3). The latter predicts velocity variations substantially larger than the observed ones, by a factor of 2 3. If the length-scale is increased to correct this problem, then the prediction no longer accurately depicts the pattern of velocity maxima minima. All relations with m 2 p, q.5 work poorly for this reason. [41] The greater coupling length needed for the upper glacier confirms that lower surface temperatures strengthen the coupling. (An even better match can be obtained for the upper glacier by using a value = 3 for its lower half, which flows along a trough, a value = 4.5 for its upper half, where the flow is spread wide.) Figure 5d shows that some of the measured velocity variations only occur because the length scale varies in proportion to ice thickness; local regions of thin ice restrict the range of influence of longitudinal coupling. This is not a controversial statement, but we know of no better demonstration of this behavior expressed in longitudinal variations along a glacier. [42] Theoretical analyses show that the length scale of horiontal stress transmission along a glacier increases with the fraction of motion due to basal sliding rather than internal creep [Lliboutry Reynaud, 1981; Jóhannesson, 1992; Gudmundsson, 23]. On fast flowing ice streams moving mostly by basal slip, the one of influence of transverse longitudinal stresses can be many times the ice thickness. The comparatively short length scale found in our empirical analysis of Taylor Glacier is therefore consistent with flow dominantly by creep, as implied by our force balance analysis. 4. Conclusions [43] No basal motion is needed to explain the observed flow of Taylor Glacier, as Higgins et al. [2] first concluded, most of the bed is likely froen. The glacier might therefore be useful as a natural laboratory for studying the deformation of subfreeing ice at prevailing strain rates (of order.1 a 1 smaller). [44] In the regions of thickest ice (1 km), our inferred basal temperatures are within a few degrees of melting point for pure ice. Given uncertainties in ice flow parameters, we cannot exclude the possibility that some melt occurs. Saline water might also be present [Keys, 1979; Hubbard et al., 24], although not in sufficient quantities to cause sliding at a detectable rate. [45] The gravitational driving stress acting on Taylor Glacier averages about 1.5 bars, but varies widely as the glacier passes over ice falls subglacial basins. Basal drag generally amounts to bars, while lateral shears generate about.2.5 bars of drag. Such values are typical of many alpine glaciers. The glacier flows slowly, given its driving stress thickness, largely because of the low temperatures of the ice bed, partly because of lateral drag. [46] Longitudinal coupling strongly affects the alongglacier variations of flow on Taylor Glacier; large variations of driving stress produce substantially muted variations of velocity. Away from sharp bends, the effects of longitudinal coupling can be described roughly as a proportionality between ice velocity F * J, where F = t d is driving stress, * denotes convolution, J is a triangular filter of total base length =4lH. The scale factor l is approximately 2 on the lower part of Taylor Glacier but 3.5 to 4 on the upper glacier. These lengths, short compared to those on glaciers with slippery beds, are consistent with motion primarily by internal creep. [47] Allowing to vary with H captures features of the velocity pattern not seen if is assumed constant (compare the red blue curves in Figure 5d). This demonstrates that longitudinal coupling is limited by regions of thin ice. All of these results match expectations from the Kamb- Echelmeyer theory. However, the forcing function suggested by theory, F = t d 3 H (or t d * J raised to a power 3), works poorly on this glacier where driving stress varies over a large range. The best relation is F = t d. Appendix A: Force Balance [48] The deviatoric stress strain rate invariants in equations (3) (4) are, assuming incompressible flow, t II ¼ _e II ¼ t 2 xx þ t2 yy þ t xxt yy þ t 2 xy þ t2 x þ t2 y ða1þ 1=2 1=2: _e 2 xx þ _e2 yy þ _e xx_e yy þ _e 2 xy þ _e2 x þ _e2 y ða2þ Thus, the total state of deformation, or stress, needs to be known to calculate any one component of strain rate _e ij.as summaried in the text, we use scaling relations, in which values at depth are taken to be a fraction of those at the surface, to constrain some of these terms. This simplification is justified in the present context because most of the shear responsible for the glacier flow occurs in the bottom third of the glacier, where the horiontal plane shears (t x t y ) are dominant; here the other terms can be treated in an approximate fashion, because they contribute little to the magnitude of the stress invariant. At the same time, the force effects of the other terms are largely determined by deformation in the upper two thirds of the glacier, hence are strongly tied to surface values for strain rates, which we have measured. A1. Estimation of Surface Components [49] Surface strain rates were calculated from measured velocity values using groups of four poles, arranged on the glacier surface as approximate squares. First, velocity gradients were calculated using a least squares fit to the velocity differences between each of the six pairs of poles in the group of four. For example, the difference in U x between two poles ( 1) is U x1 ¼ U x þ U x x Dx þ U x y Dy; ða3þ 8of11

9 the system of six equations of this form was solved for U x /x U x /y. The same process was used for terms in in U y. Strain rates were then calculated according to the stard definitions: _e xx ðh Þ ¼ U x x ; _e xy ðhþ ¼ 1 2 _e yy ðhþ ¼ U y y : U x y þ U y x ; ða4þ A2. Estimation of Components at Depth [5] We first define a quantity y, such that y 2 = _e 2 xx + _e 2 yy + _e xx _e yy + _e 2 xy. We also assume that _e y =(U y /U x )_e x. This assumption affects the calculations very little, because U y is much smaller than U x ; x is chosen to parallel the flow in the center of the clusters of poles used for the analysis. Equation (A2) can be written where A is the value appropriate for 1 C given by Paterson [1994, p. 97] for n =3,Q is the activation energy for creep R = J K 1 mol 1 is the gas constant. A higher value for Q was used above 1 C than below. As described in the text, we assumed the temperature varies as a linear function of depth, from a fixed ice surface value. Ice surface temperatures were assumed to decline with elevation, by 6.45 C km 1, from a sea level value of 15.5 C. Results depend only weakly on surface temperature. A4. Relations for Stress [52] The balance of forces on a vertical column of ice, extending from height above bed to the surface at = H,is t x ¼ ^t d ðþþ x 2t xx þ t yy d þ y t xy d ða1þ with driving stress at depth being ^t d () =rga[h ]. For convenience, we define the longitudinal (L) side drag (S) contributions as " _e 2 II ¼ y2 þ 1 þ U # 2 y _e 2 x U : x ða5þ The value of y at the surface, y S, is calculated directly from the surface strain rates (equations (A4)). We then assume that at depth, y scales as Z H 2t xx þ t yy d ¼ x x L ðþ¼l ðþ y t xy d ¼ y S ðþ¼s ðþ: ða11þ ða12þ yðþ¼ u xðþ y U S : ða6þ x The individual components _e xx, _e yy, _e 2 xy are assumed to scale in the same fashion. Combining with equation (3) gives, for the x component of strain rate, _e x ¼ A 1 n ðþ ( u x ðþ " 2 y U þ 1 þ U # )n 1 2 y 2n S _e 2 x x U x t x : ða7þ Because u /x, the variation of velocity u x with height above the bed is found by integration from _e x ¼ 1 2 u x Z u x ðþ¼2 _e x d ; ða8þ into which equation (A7) is substituted for _e x. Once the velocity profile is calculated (which requires determining t x, as described below), all strain rate components are determined by the scaling relations, the corresponding deviatoric stress components can be calculated from the ice constitutive relation (equation (3)). A3. Temperature [51] The temperature profile of the glacier strongly controls the rate of flow. The flow law coefficient A() is taken to vary with temperature as A ðþ¼a expð Q=RTðÞ Þ; ða9þ Recasting the integrals in these three relations in terms of depth-averaged values of the stress components t xx, t yy t xy yields the following expressions (an overbar denotes a depth-averaged value): L ðþ¼ x H 2t xx þ t yy ¼ H x 2t xx þ t yy þ 2txx þ t yy x H S ðþ¼ y Ht xy ¼ H y t xy þ t xy y H: ða13þ ða14þ [53] We now consider a cluster of poles, consisting of four subclusters for which strain rates are known. Of primary interest are the gradients L () S () that apply to the central subcluster. In terms of values at the subclusters up glacier (U), to the right (R) left (L) sides, these can be approximated as L C ðþ ¼ L CðÞ L U ðþ x C x U ða15þ SC ðþ¼s RðÞ S L ðþ : ða16þ y R y L 9of11

10 Here x y are the mean horiontal coordinates of the given subcluster. [54] To find approximate values for L () S () in the subclusters surrounding the central one, we note first that by rearrangement, H x 2t xx þ t yy ¼ L ðþ 2t xx þ t yy x H H y t xy ¼ S ðþ t xy y H: ða17þ ða18þ We next approximate the horiontal gradients of depthaveraged stresses as constants in the region of analysis. (This should be a very good approximation, as stresses are transmitted through the ice over distances equal to a multiple of several ice thicknesses; velocities thus vary smoothly.) The constants, denoted C 1 C 2, are C 1 ¼ x 2t 1 xx þ t yy ¼ L ðþ 2t xx þ t yy x H H C 2 ¼ y t xy ¼ S 1 ðþ t xy y H H : ða19þ ða2þ Values for these constants are calculated using parameter values from the central subcluster. Then, for the surrounding subclusters (i = U, R, L), the gradient terms are evaluated using subcluster-specific values t xx(i), t yy(i), t xy(i), H i, H i / x H i /y in the relations: L iðþ¼ x L iðþ¼h i C 1 þ 2t xxðþ i þ t yy i ðþ x H i Si ðþ ¼ y S iðþ¼h i C 2 þ t xyðþ i y H i: ða21þ ða22þ [55] Finally, for all subclusters (i = C, U, R, L), the force gradient terms on the ice column with base at arbitrary height, top at the glacier surface, is estimated using the scaling relations Z H 2t xx þ t yy d ¼ L x i ðþ L iðþ L i ðþ Z H t xy d ¼ Si y ð Þ Si ðþ S i ðþ : ða23þ ða24þ The horiontal plane shear stress at height in subcluster i is then, by substitution into equation (A1), t x ðþ¼r i gh ð Þa x þ L L i ðþ i L i ðþ þ S i ðþ S i S i ðþ : ða25þ A5. Solution by Iteration [56] The goal is to solve for the profiles of stress, strain rate, velocity, temperature beneath the central subcluster of survey poles. To do so, it is necessary to evaluate the preceding relations for all four subclusters. [57] We begin by assuming a value for basal temperature a known stress (specifically, t x () =t d ()). We then integrate equations (A7) (A8) to calculate a surface velocity adjust the basal temperature to improve the match between calculated observed velocities. We repeat this process until the match is nearly exact. Using the corresponding velocity profile, the assumed stress, measured values for surface strain rates, equations (A25), (A15), (A16), (A5), (A6), we integrate equation (A1) downward to obtain a new estimate for t x (). We then iterate by calculating a new velocity profile from integration of equations (A7) (A8) (this time, with all strain rate components included in _e II ), adjusting the basal temperature until calculated observed velocities match, integrating equation (A1) to find an adjusted t x profile. This iteration is repeated until convergence is achieved. [58] Acknowledgments. This work was funded by the U.S. National Science Foundation, grant OPP to K.C. We gratefully acknowledge additional support from the Hellman Family Faculty Fund to University of California, Berkeley, the contributions of our collaborators on other parts of this project, especially Sarah Aciego, Andy Bliss, Howard Conway, Dave Morse, Eric Rignot. We thank the anonymous reviewers for their comments. The Associate Editor, M. Truffer, offered particularly helpful suggestions for improving the manuscript for improving the argumentation in an earlier draft. References Bahr, D., W. T. Pfeffer, M. F. Meier (1994), Theoretical limitations to englacial velocity calculations, J. Glaciol., 4, Balise, M. J., C. F. Raymond (1985), Transfer of basal sliding variations to the surface of a linearly viscous glacier, J. Glaciol., 31, Blatter, H. (1995), Velocity stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients, J. 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