Dating ice flow change near the flow divide at Roosevelt Island, Antarctica, by using a thermomechanical model to predict radar stratigraphy

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jf000326, 2006 Dating ice flow change near the flow divide at Roosevelt Island, Antarctica, by using a thermomechanical model to predict radar stratigraphy Carlos Martín, 1 Richard C. A. Hindmarsh, 2 and Francisco J. Navarro 1 Received 21 April 2005; revised 31 October 2005; accepted 4 November 2005; published 22 February [1] Radar-detected internal layering contained in some ice divides shows upwarped arches termed Raymond bumps. The distribution of their amplitude with height can date the onset of divide flow, reflecting changes in the basin structure of the ice sheet. The distribution depends on rheology, surface geometry, accumulation rate, and temperature. Conway et al. (1999) used an isothermal parameterized ice flow model to estimate a date of 3200 years B.P., with no error estimate, for the onset of divide flow in Roosevelt Island, Ross Ice Shelf, which they associated with grounding line retreat of the West Antarctic Ice Sheet. No other retreat dating exists for an area of the Ross Ice Shelf distant from geological exposures. Employing a full thermomechanically coupled transient model, we use a direct search to determine which ice rheology best fits the observed bump distribution and surface profile, estimating the sensitivity of the dating to model parameters and exploring possible reasons for the observed asymmetry of the surface profile. Our main results are as follows: (1) A best estimate of the date of grounding line retreat near Roosevelt Island is 3000 years B.P., bounded by 2300 and 4200 years B.P. (2) Standard rheology (power n = 3) can only match the observed bump distribution for unrealistic isothermal models. (3) For thermomechanically coupled models a high power (n = 4) in Glen s law is required. (4) Low-power rheologies and wind scouring cannot produce correct bump amplitude distributions. (5) Asymmetric accumulation cannot explain the observed asymmetry of the surface profile. (6) Modeling isochrones in flanking regions also indicates changes in flow around 3000 years B.P. Citation: Martín, C., R. C. A. Hindmarsh, and F. J. Navarro (2006), Dating ice flow change near the flow divide at Roosevelt Island, Antarctica, by using a thermomechanical model to predict radar stratigraphy, J. Geophys. Res., 111,, doi: /2005jf Introduction [2] The annual throughput of water in Antarctica corresponds to a 5 mm layer of the ocean, substantially greater than the current sea level rise of 1 2 mm. The West Antarctic ice sheet (WAIS) contains sufficient water to raise sea level by 5 m. These are just two examples showing that changes in the imbalance of the Antarctic ice sheet have global significance, but their underlying cause is not yet known, and their predictability is therefore poor. Knowledge of the history of the WAIS over timescales of thousands of years is vital for the calibration of models, yet the Holocene retreat of the WAIS is unknown in sufficient detail to reliably ascribe mechanism. [3] In this paper, we focus on the formerly grounded portion of the WAIS known as the Ross Ice Sheet flowing 1 Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, Madrid, Spain. 2 British Antarctic Survey, Cambridge, UK. Copyright 2006 by the American Geophysical Union /06/2005JF toward the Ross Sea (Figure 1). Most of the evidence of its Holocene retreat is geological, located at the edges of the ice sheet: exposure age dating, trimlines and moraine limits near the Transantarctic Mountains and in Marie Byrd Land [e.g., Stuiver et al., 1981; Clayton-Greene et al., 1988; Denton et al., 1989; Bockheim et al., 1989; Dochat et al., 2000; Hall et al., 2000, 2002; Stone et al., 2003]. The presence of the Ross Ice Shelf prevents the use of traditional geologic methods and numerical ice flow modeling becomes a crucial tool for constraining the WAIS retreat [Anderson et al., 2004]. The most recent date specific to the retreat in the Ross Sea area puts the grounding line north of Roosevelt Island (RI in Figure 1) as recently as 3200 years B.P. [Conway et al., 1999], providing evidence that the Holocene retreat of the WAIS occurred relatively late in the Holocene and by implication is continuing. This is the only evidence for grounding line retreat in an area far removed from geological exposures. [4] The dating by Conway et al. [1999] is based on ice flow modeling of Roosevelt Island. It puts a date on a change in the ice flow regime which is believed to be associated with the passage of the grounding line of the 1of15

2 Figure 1. Study area showing the location of Roosevelt Island (RI) and T line. SDM is Siple Dome. Data kindly provided by H. Conway. main Ross Ice Sheet past Roosevelt Island. The change in flow regime was due to the formation of an ice divide which caused flow lateral to the previous flow toward the Ross Sea. The initiation of the divide flow resulted in the formation of Raymond bumps. These are convex arches in the layers seen in radar profiles obtained across some ice divides [e.g., Nereson et al., 1998b; Vaughan et al., 1999; Conway et al., 1999]. These layers are believed to be syndepositional and are therefore termed isochrones. The formation of isochrone arches under ice divides can be explained by the operation of a nonlinear flow phenomenon first predicted by Raymond [1983] and known as the Raymond effect. Unlike the constant viscosity of a linear (Newtonian) rheology, Glen s nonlinear rheology predicts high viscosities in a region near the bed at the divide as a consequence of the low deviatoric stresses. The stiff ice in this region impedes downward flow, so that a particular isochrone moves down more quickly on either flank that at the divide, producing a local arch in the isochrone. [5] Flow models can be used to predict the growth of these arches, and once started, the process of arch growth takes a predictable course, with arch amplitude dependent on depth and time since the initiation of the Raymond effect. However, the process of arch growth depends upon the accumulation rate [Nereson and Raymond, 2000; Nereson and Waddington, 2002], the evolving geometry of the ice mass [Hindmarsh, 1996; Nereson et al., 1998a, 1998b; Nereson and Raymond, 2001; Nereson and Waddington, 2002], the rheology of the ice [Pettit and Waddington, 2003]; the surface temperature and geothermal heat flux [Hvidberg, 1996; Nereson and Waddington, 2002]; and basal sliding [Pettit et al., 2003], all of which are rather poorly constrained. The combined effect of some of these processes is to strongly distort or eliminate bumps; e.g., divide migration [Nereson and Waddington, 2002], stochastic variations of divide position [Hindmarsh, 1996], strong basal sliding [Pettit et al., 2003] or nearlinear rheologies at low deviatoric stresses [Pettit and Waddington, 2003]. [6] The Raymond effect occurs under ice divide areas, where the slope magnitude is very small, and the shear stress is consequently also small. Before the Raymond effect started to operate in Roosevelt Island, northwesterly flow from the main body of the WAIS, approximately parallel to the present divide ridge [Anderson and Shipp, 2001] would have caused the slope of ice in the direction of the present divide ridge to have been large enough to suppress the Raymond effect. Exactly what configuration of Roosevelt Island and the main WAIS would have been necessary to cause the Raymond effect to operate is unclear. For example, Siple Dome (SDM in Figure 1) has a pronounced ridge, but is upstream of the grounding line. Thus, if ice streams flanked Roosevelt Island, the initiation date of the Raymond effect would be an indicator of a grounding line to the northwest of Roosevelt Island. In contrast, if there were no flanking ice streams, then onset of divide flow might not have occurred until after the main grounding line had retreated upstream of Roosevelt Island. One purpose of this paper is essentially to put error bars on this date of 3200 years B.P. by Conway et al. [1999], using a suite of flow modeling results that considers the effect of parameter uncertainty on the computed date. [7] A related use of divide modeling is given in a paper by Waddington et al. [2005], who infer ice sheet thickness changes in the central Ross Sea Embayment by using a transient ice flow model to find combinations of accumulation rate and ice sheet thickness histories that match the depth-age relationship, influenced by the Raymond effect, and the measured layer thickness pattern in the Siple Dome ice core. Their results show that Siple Dome was not much thicker at the Last Glacial Maximum than it is now; only by between 200 and 400 m. While one s first reaction might be skeptical (are ice parameters well enough known?), actually the argument is essentially based upon kinematics and incompressibility and is strongly constrained by data. The implications of this idea are quite far reaching: the geometry of the Ross Ice Sheet was very different to anything we have supposed it to be, and may not have any modern analogues. This impacts directly on our understanding of the Holocene retreat of the WAIS, which is essential information for validating any model of the WAIS used in forecasting its future behavior and its effect on sea level. [8] In summary, the modeling results from Siple Dome [Waddington et al., 2005], Roosevelt Island [Conway et al., 2of15

3 a flow model embedded in a control method. This allows us to examine asymmetry in the forcing as well as examine a hypothesis that Roosevelt Island has experienced recent disturbance. Figure 2. Radargram of the Roosevelt T line. Ice flow is into the page. Note the basal asymmetry, with a small basal bump to the east of the divide. Data kindly provided by H. Conway and A. Gades. 1999], and recent exposure age dating of nunataks in Marie Byrd Land [Stone et al., 2003] paint a complex picture of WAIS retreat. A particular advantage of Raymond effect dating, as compared to traditional geological techniques, is the proximity of sites to ice streams, whose variability played a strong role in WAIS retreat [e.g., Bindschadler, 1997]. [9] Both these models by Conway et al. [1999] and by Waddington et al. [2005] used parameterized descriptions of isothermal ice flow, where the ice rheology was not specified precisely, nor was the surface profile matched. Our modeling uses a self-consistent thermomechanical finite element ice flow model where rheology has to be specified. The dynamical part of our model solves the full Stokes system of differential equations, i.e., takes into account all stress components. By using different rheologies and comparing the model results for bump amplitude and surface profile to their observed values we found that matching the Roosevelt Island Raymond bumps and surface profile provided strong constraints on the ice rheology. Knowledge of ice rheology is key to matching any type of ice model to data and to making predictions of the way glaciers will behave in the future. It is known that wind scouring at the divide could enhance the Raymond effect [e.g., Nereson et al., 1998b; Nereson and Waddington, 2002], so we also consider its effects. Matching of the observed asymmetry of Roosevelt Island surface profile posed special difficulties, and we will test asymmetric accumulation rate as a possible reason for this surface profile asymmetry. Finally, we consider the isochrone geometry in the flanks, fitting it with 2. Data From the Roosevelt Island T Line [10] Roosevelt Island is an ice rise in the Ross Sea, almost completely surrounded by the Ross Ice Shelf (Figure 1). It is the second largest ice rise in Antarctica. It shows typical ice rise features, being lemon-shaped with a strong ice divide running down the middle. The ice is at most around 700 m thick. [11] Our modeling will focus on Roosevelt Island T line (see location in the inset of Figure 1 and radargram in Figure 2), transverse to the ice divide near its summit. We used field data on accumulation rate and estimates of thinning rate by Conway et al. [1999]. The accumulation rate is around 0.18 m yr 1 ice equivalent. Ice cap thinning results from an imbalance between accumulation and the downward velocity of ice at the surface, and is calculated as the difference between the ice equivalent accumulation rate and the horizontal flux divergence. The latter comes from measurements of ice thickness and the divergence of the horizontal surface velocity, which is, in turn, calculated from changes in measured pole positions. The uncertainties involved imply thinning rate estimates between 0.06 and 0.11 m yr 1 ice equivalent. We have taken 0.09 m yr 1 as the reference value and considered its uncertainty in the sensitivity analysis. The present surface temperature is about 20 C and we use a geothermal heat flux of G = 60 mw m 2 which is consistent with previous studies [Thomas et al., 1980]. [12] We aim to compute the age field, and plot its related distribution of bump amplitude versus bump base elevation above the bed, in order to compare these model results with the corresponding ones retrieved from the radar profile. From now on, we will refer to these curves as BAED (bump amplitude elevation distribution). We will also compare computed and observed surface profiles. In order to quantify the differences between model predictions and observations, we quantitatively define mismatch functions (see section 4.3) and seek to minimize them using a direct search method. The model equations are given below and the numerical solution is discussed in Appendix A. 3. Governing Equations and Numerical Model 3.1. Basic Equations [13] The setup of flow is illustrated in Figure 3. The coordinates are (x, y, z), where x is taken in the direction of divide flow, y in the direction of divide ridge and z direction is the vertical. Denoting r =(x, y), the thickness of the ice is given by z = H(r, t), while the surface and bed are given by z = s(r, t), z = b(r, t) respectively, and t represents time. The operators r H, r H, represent the horizontal gradient and divergence respectively. [14] The three-dimensional velocity field is conveniently represented by the vertical velocity w and the horizontal velocity vector u =(u x, u y ), and we also use v =(u x, u y, w). The volume flux Q = R s budz is used frequently throughout the paper. The continuity, kinematic and momentum balance 3of15

4 Figure 3. Illustration of the problem setup and notation. equations are, for t > x (starting time of the Raymond effect), r H u z w ¼ t s þ u rs ¼ w þ a; rs þ rg ¼ 0; S ðþ s n ðþ s ¼ 0; v ðþ b ¼ 0; the heat equations are, again for t > t q þ v rq ¼ kr 2 q þ 1 rc D; bðr; tþ z sðr; tþ; ð1aþ z ¼ sðr; tþ; ð1bþ bðr; tþ z sðr; tþ; ð1cþ z ¼ sðr; tþ; ð1dþ z ¼ bðr; tþ; ð1eþ bðr; t Þ z s ð r; t Þ; ð1fþ [15] In the above equations, superscripts (s), (b) indicate evaluation at the surface or base. (1a) expresses conservation of mass in the ice; (1b) is the free surface kinematic condition, where a is accumulation rate of ice, expressed as a volume rate per unit area; (1c), (1d) and (1e) describe conservation of momentum in the ice, where S is the stress tensor, r is the density of ice, g = g(0, 0, 1) is the gravitational acceleration vector and n is the normal vector at the indicated surface; (1f) (1h) represent conservation of heat in the ice, where q is the temperature in the ice, q s is the prescribed surface temperature, k is the thermal diffusivity of ice, c is the specific heat capacity, D = 1 2trace(T e) is the dissipation, T is the deviatoric stress, t = 1 2 trace(t2 ) is the deviator stress invariant, e is the strain rate, K i is the thermal conductivity of ice, and Q G is the geothermal heat flux; (1i) is the advection equation for age as a function of space and time, while (1j) expresses that the age at surface is zero, as the modeled area is located in the accumulation zone of the ice sheet. [16] We solve the above dynamic and thermal equations using finite element methods, while semi-lagrangian methods are used to solve the free surface evolution and age equations; details are given in Appendix A. [17] Our first simplification is to consider plane flow. The justification for this is that the T line runs perpendicular to the divide and follows a line of steepest descent in this strongly elliptic feature, so 3-D flow effects are not expected to influence our results. From this point on, we will use the two-dimensional (x, z) forms of the model equations, assuming that no flow occurs parallel to the divide ridge line (y direction). [18] Natural or physically intuitive boundary conditions corresponding to real ice sheet margins are so spatially remote that the computational domain required would be unfeasibly large. The solution domain we have used for the model computations is a region which extends, at each side of the divide, to a distance equal to 15 times the ice thickness at the divide H d. A sensible choice for boundary conditions in the sides, which has been adopted by, for example, Raymond [1983], Waddington et al. [2005], and Hvidberg [1996] is to impose shallow ice velocity fields, which export ice at a rate which conserves global mass balance. The 15H d distance between the divide and the boundary at either side ensures that boundary effects are minimized in the divide area [Raymond, 1983; Hvidberg, 1996]. In particular we will restrict our analysis of the Raymond effect to a region of 5H d to each side of the divide. Thus, at the boundary x = X, the momentum equations boundary conditions are q ðþ s q s ¼ 0; z ¼ sðr; tþ; ð1gþ ux; ð z Þ ¼ ux; ð sþ 1 z nþ1 ; ð2aþ K i rq ðbþ n ðbþ ¼ Q G ; z ¼ bðr; tþ; ð1hþ ux; ð sþ ¼ n þ 2 R X x d ax ðþdx n þ 1 HðXÞ ð2bþ and the age equation [e.g., Hindmarsh, 2001] t þ u r H þ w@ z ¼ 1; bðr; tþ z sðr; tþ; ð1iþ ðþ s ¼ 0; z ¼ sðr; tþ: ð1jþ wx; ð zþ ¼ ða cþ 1 z n þ 2 n þ 1 þ znþ1 þ ux; þ ð1 z n þ 1 ; ð2cþ 4of15

5 where z =(s z)/h, c is the thinning rate and x d represents the divide location. The heat equation boundary condition is chosen to be where n b ¼ 2:4; n g ¼ 1:8; n d ¼ ¼ 0: Far from the ice divide, horizontal gradients in temperature induced by advection are small, which makes this choice of boundary condition a reasonable one. [19] The hyperbolic age equation does not require boundary conditions at the margin outlet. Initial conditions are generated by solving the vertically integrated one-dimensional equation ð3þ Z s dz 0 ðx; zþ ¼ z wx; ð z 0 Þ : ð4þ 3.2. Constitutive Relationships [20] We consider flow by internal deformation only, and test several different rheological relationships. Our computations of basal ice temperature clearly show that at present, Roosevelt Island ice is frozen to its bed. However, basal melting and therefore sliding conditions, could have been attained in the past, through the insulating effect of a thicker WAIS [e.g., Steig et al., 2001]. Our preliminary tests including thinning suggested only a limited effect from basal sliding, restricted to cases of extreme values of model parameters and extremely high initial ice thickness. In such cases, we attained melting temperatures at the bottom only during the earlier stages of the simulation (always <1000 years, and usually much shorter). Consequently, we have not considered basal sliding in our model computations. The rheologies we have considered are as follows. [21] 1. The Glen constitutive law for isothermal ice, e ¼ At n 1 T; with rheological index n = 3, where the rate factor A (also known as the softness parameter) is a free parameter. [22] 2. The Glen constitutive relation with A(q) according to Dahl-Jensen [1989] (DJ), e ¼ At n 1 T; n ¼ 3; A ¼ 0:2071 e 0:5978q þ 0:09833 e 0:14747q Pa 3 yr 1 : ð7þ [23] 3. The Goldsby and Kohlstedt [2001] constitutive relation (GK), excluding diffusional flow (a creep mechanism usually occurring at very low stresses), e ¼ ð5þ ð6þ 1 A b t nb 1 þ 1! 1 A g t ng 1 þ A d t nd 1 T; ð8þ A b ¼ A 0 b exp Q b ; A 0 b Rq ¼ 5:5 107 MPa 2:4 s 1 ; Q b ¼ 60kJ mol 1 A g ¼ A0 g d p exp Q g ; Rq A 0 g MPa 1:8 m 1:4 s ¼ 3: Q g ; 1 3: kj mol 1 ¼ 49; q < 255K 192; q > 255K A d ¼ A 0 d exp Q d ; Rq A 0 d 4:0 105 Q d MPa 4 ¼ ; s 1 6: kj mol 1 ¼ 60; q < 258K 180; q > 258K ð10þ ð11þ ð12þ where d is crystal size and p = 1.4. In continuum mechanics literature, a viscous process is conventionally represented by a dashpot, and complex viscous models can be represented by putting dashpots in series and in parallel. Usually one would set each dashpot to represent a different flow law. The GK (Goldsby-Kohlstedt) measurements indicated a fourterm rheological relationship, with a set of three dashpots in series with strain rates depending on the stress to the powers 1, 1.8 and 4. This set of three dashpots is in parallel with another dashpot where the strain rate depends upon the stress raised to the power 2.4. There is also a dependence on crystal size. [24] 4. The Glen constitutive relation with A given by an Arrhenius relationship with a single activation energy (1Q rheology), e ¼ At n 1 T; A ¼ A 0 exp Q ; ð13þ Rq where n = {3,4}, and Q and A 0 are free parameters. [25] 5. The Glen constitutive relation with A given by an Arrhenius relationship with two different activation energies (2Q rheology), e ¼ A 40 t n 1 þ A 180 t n 1 T; ð14þ A 40 ¼ A 0 40kJ mol 1 40 exp ; ð15þ Rq A 180 ¼ A 0 180kJ mol exp ; ð16þ Rq where n = 4, and A , A 180 are free parameters. 5of15

6 Figure 4. Contours of dynamical fields and convergence study: (a) vertical velocity (m yr 1 ), (b) internal temperature ( C), (c) bump amplitude elevation (m), and (d) the age in years. Note the upwarping of the isolines characteristic of the Raymond effect in the divide area. Figures 4b and 4d are for node grids; Figures 4a and 4c show convergence studies for grid (solid line), grid (dotted line) and grid (dash-dotted line). Contour fill in Figure 4a is for grid. Contours for the two finer grids are coincident at this resolution. [26] We will search in the space of the free parameters in these rheologies to obtain best fits for the bump amplitudes and surface profiles. [27] The above rheologies do not include the polynomial rheology recently proposed by Pettit and Waddington [2003], which has terms in the stress to the first, third and fifth powers. We have already included Goldsby-Kohlstedt rheology, which has a laboratory basis and considers several deformation mechanisms with powers (n = 1.8, 2.4, 4.0) similar to those used by Pettit and Waddington (n = 1, 3, 5). The latter has the added difficulty of having many free parameters: enhancement factor, rate factor, crystal size and activation energy for each of the three terms Steady and Time-Dependent Models [28] We have considered two different sets of models, which we will refer to as steady state and transient. In the former, the glacier mass is considered to be constant in time. Starting from the present Roosevelt Island surface geometry, the model is run until it acquires a steady surface geometry. The corresponding steady velocity and temperature fields are then used to solve the age equation. [29] Measurements by Conway et al. [1999] of the surface strain rate show that Roosevelt Island is lowering by a considerable amount (0.09 m yr 1 ). This has a significant effect on the dependence of bump amplitude on elevation. In the transient model, which takes into account this thinning, the initial conditions are modeled as follows. [30] Suppose we wish to consider the case where the Raymond effect began operation x years ago, with a thinning rate c. We start with a flat profile with elevation cx above the present elevation of Roosevelt divide. We select a flat profile on the basis that prior to the onset of divide flow, the ice flow over Roosevelt Island was driven by the main flow from the WAIS, perpendicular to the flow line we are considering now. Velocity boundary conditions are prescribed on the flanks so that the total rate of mass loss will result in a profile x years later with approximately the same mass as is observed in Roosevelt today. We implement this in the model by equating the mass lost through the flanks in a given time step, to the difference between the mass deposited by snow accumulation and that lost due to the global thinning with rate c. The upper surface evolution is described by the appropriate kinematical equation (1b). 6of15

7 Figure 5. Plots of (left) match to surface data and (right) match to Raymond bump amplitude data. Cases are all steady state, with isothermal, DJ, and GK with two crystal sizes. This creates some spurious velocity oscillations near the margin, as the velocities are prescribed to follow there the vertical distribution corresponding to the SIA. This region of spuriously varying velocity extends to several ice thicknesses from the margin [e.g., Hvidberg, 1996]. However, in the center of the ice rise, the solution is smooth and the Raymond effect clearly operates. (Time steps for the kinematical equation are five years.) By ensuring that the appropriate amount of mass is lost at each time step, the mean profile after x years of evolution is approximately that observed now on the T line of Roosevelt Island. [31] The age distribution at the start of the evolution is set to be steady state using SIA distributions. This is incorrect if the ice had been lowering prior to the commencement of the operation of the Raymond effect, but since we do not know what was happening then, other distributions would be speculative. [32] The temperature distribution is a steady temperature distribution, updated every 25 years. Again, since we do not know what was happening prior to the commencement of operation of the Raymond effect, the choice of a steady initial condition seems reasonable. However, this is an initial condition appropriate to the initial geometry (a flat upper surface) and the effects of this would propagate over the evolution. For reasons of computational speed, we could not compute the temperature in the bedrock as well. The thermal inertia of the bedrock affects the time-dependent evolution of the temperature in the ice. [33] In view of these uncertainties we simply computed the steady temperature corresponding to the velocity field at each time step. Note that this does include the increased downward velocity due to thinning, which is the major nonsteady influence on the temperature (the Péclet number is around 6). [34] Some contour plots of steady fields are given in Figure 4. These are chosen to illustrate the operation of the Raymond effect, the reduced vertical velocity due to the high-viscosity stagnant plug, and its consequent effects upon the isochrones and the temperature and age fields. [35] Also shown is a convergence study of the effect of grid size on the computations. Plots for 31 31, and node grids are shown for the vertical velocity and the BAED. It can be seen that the BAED is slightly affected by the grid size, and the vertical velocity is slightly smoother for finer grids. We used the grid in our calculations, as it permitted a far larger number of computations in our direct search optimization routines. 4. Results and Discussion 4.1. Steady State Models [36] Firstly, we attempted matching using steady ice sheet geometries. These profiles had constant geometry through time. We did this for (1) isothermal n = 3 rheology, with tuning of the rate factor to produce an optimum match to both profile and bump amplitude elevation distribution (BAED), (2) Dahl-Jensen n = 3 rheology, and (3) Goldbsy-Kohlstedt rheologies with crystal sizes d = 0.5 and 1 mm. We chose these crystal sizes because they allowed bracketing of the surface profile (Figure 5, left). [37] The best profile and BAED fits are shown in Figure 5. We should emphasize here that the surface profile is a weak function of the internal deformation pattern, while the bump amplitude distribution is a strong function of it, so that a higher weight should be given to the latter. The BAEDs are all steady state. The GK and DJ cases (all with temperature-dependent viscosity) do not attain a sufficiently large bump amplitude. The isothermal case has a bump 7of15

8 Figure 6. Plots of (left) match to surface data and (right) match to Raymond bump amplitude data. Cases are all transient, with isothermal, DJ, and GK with two crystal sizes. amplitude that is sufficiently large to match the observed data, but the elevation of the maximum is nearly 200 m below the observed elevation. [38] Profile fits vary in quality. One GK case and the isothermal case are considerably better than the other two. A feature of all the model results is that it is not possible to capture the asymmetry of the Roosevelt Island surface profile Transient Models [39] Thinning is known to be important in obtaining a match to the BAED. When the ice is thicker than at present, Figure 7. Plots of (left) match to surface data and (right) match to Raymond bump amplitude data. Cases are all transient. Three 1Q models and one 2Q model are plotted, with indicated activation energies. 8of15

9 Figure 8. Plots of surface and bump amplitude mismatches E s, E b as a function of rate factor A for the indicated activation energies, 1Q model. Note that the surface data gives a strong well. the Raymond effect is stronger at a greater elevation compared with the steady case discussed above. Thinning therefore has the effect of raising the elevation of maximum bump amplitude. [40] We used the same rheological models as for steady state case, but prescribed a thinning rate of 0.09 m yr 1. Results are shown in Figure 6. The BAED is time-dependent, and the best fit in time is used in the presentation of the results and in the inference of the date of initiation of the Raymond effect. [41] The isothermal case produces a near perfect fit to the BAED as evidenced by the correct amplitude and elevation of maximum amplitude, while the DJ curve is too low and too small. The elevation problem could probably be corrected by using an increased thinning rate. The GK rheologies do not produce sufficiently large bumps, essentially because the dominant flow process is not sufficiently nonlinear (recall that the Raymond effect does not operate for Newtonian rheologies). GK rheology does not work for Roosevelt Island, because the stress levels are so low that lower-index rheologies dominate. [42] Profile fits remain variable in quality. The results are better than for the steady case, and the DJ and isothermal fits are quite good Rheological Index n = 4, One Activation Energy (1Q Rheology), and Two Activation Energies (2Q Rheology) [43] Experiments not reported here, as well as the results with the DJ rheology, show that a temperaturedependent viscosity with n = 3 cannot match the BAED. According to our model runs shown in Figure 6, successful fitting of the BAED will require a more strongly nonlinear flow law. There are good theoretical and observational reasons [Goldsby and Kohlstedt, 2001] to indicate that at high stresses n = 4, which is the limiting case for the GK rheology. It has been suggested [Pettit and Waddington, 2003] that in low deviatoric stress regions near ice divides a near-linear constitutive relationship for ice flow may be appropriate. Though the stress regime in Roosevelt Island divide is low, there is no indication of a reduction in n at low stresses, and the simplest interpretation of field data gives a rather high value of n =4[Thomas et al., 1980]. [44] Using the 1Q rheology (n = 4), we have been able to obtain a good match to the BAED and some quite good matches to the surface profile by altering the rate factor and the activation energy simultaneously. Results for different activation energies are shown in Figure 7. We define the mismatch between Roosevelt and simulated surface profiles (E s ) as the difference in area between both profiles normalized with respect to the flow plane area. Similarly, we define the mismatch between observed and computed BAED (E b ) as the difference in area between both curves normalized with respect to the area under the observed BAED curve. Some mismatches as a function of rate factor for different activation energies are shown in Figure 8. The rate factor affects the surface profile match strongly, but only has a weak effect on the BAED match. [45] The 2Q rheology is more or less the same as the 1Q rheology, except that the activation energy is changed to 180 kj mol 1 for higher temperatures, by analogy with the GK rheology. Our computations show that this has virtually no effect on the results, presumably because the Raymond effect is at its strongest where the ice is colder. The introduction of two activation energies therefore does not 9of15

10 Table 1. Sensitivity of the Date of Commencement of the Raymond Effect a a, myr 1 c,myr 1 q s (C) A 0,Pa 4 yr 1 x, kyr E s E b E E E E E E E E E E E E E E E E E E E E E E E E E E E a Read 1.0E-11 as improve the fit. One result for this rheological model is also shown in Figure Sensitivity of the Date of Commencement of the Operation of the Raymond Effect [46] Two fundamental goals of this paper were to check, using a more complete model, the estimate of 3200 years B.P. given by Conway et al. [1999] for the onset of divide flow in Roosevelt Island, and to understand how sensitive this dating was to uncertainties in model parameters. To accomplish it, we performed a large set of model runs for different choices of model parameters. We then calculated the mismatches between computed and observed values for bump amplitude (E b ) and surface profile (E s ). The sensitivity parameters that we analyzed were the accumulation rate a, the thinning rate c and the surface temperature q s. Preliminary calculations showed that the results were rather insensitive to geothermal heat flux, so we did not treat it as a sensitivity parameter. We allowed the accumulation and thinning rates to vary ±33% from those measured or estimated by Conway et al. [1999], while allowance for surface temperature was ±25% of its present value. For each choice of model parameters, we had to set a different starting time of divide flow in order to match the observed data for bumps and surface profile. We did all calculations in this section using the 1Q model with rheological index n = 4 and activation energy Q = 50 kj mol 1, which was the model providing the best fit for the reference values of the model parameters, as shown in earlier sections. The results of the model runs are shown in Table 1. [47] The perturbations in model parameters have in general little effect on the surface profile fit, while their effect on the bump amplitude fit is appreciable, because the bump amplitude distribution is a strong function of the flow, while the surface profile is a weak function of it. [48] There is little sensitivity to q s, except for high upper surface temperatures combined with low accumulation and thinning rates, for which melting temperatures at the glacier bottom beneath the divide are reached. [49] The best fit to the bump amplitude distribution is obtained for a starting date of 3000 years B.P., very close to the value estimated by Conway et al. [1999], using the same model parameters. Quite good fits are also obtained for the extreme cases in which both the accumulation and thinning rates are either low or high. The analysis of sensitivity graphs not reported here shows that for a given starting time, higher accumulation or thinning rates result in larger bump amplitudes, and consequently low values for both a and c require longer evolution times, while high values require shorter times. [50] Regarding our goal of giving an error bound to the onset of divide flow, dates between 2200 and 4300 years B.P. are shown in Table 1. This could be understood as a rough estimate of such bound, with the caution that it corresponds to extreme values, perhaps unrealistic, of the model parameters Exploring Wind Scouring Effects [51] We have been able to obtain good matches to the observed Raymond bumps by using a nonstandard rheology with a high index n = 4 in the nonlinear constitutive relation. An alternative approach is to suppose that the standard glaciological models (Glen s law with n = 3) are correct. Since these produce bump amplitudes that are too small, we need to seek a meteorological mechanism that increases the amplitude. Scouring at the divide could be such a mechanism, as was pointed out by Fisher et al. [1983] and later analyzed by, for example, Nereson et al. [1998b] and Nereson and Waddington [2002]. [52] We have investigated this effect by performing an experiment using a standard DJ rheology and a scouring of 15% the accumulation rate over an area extending to one ice thickness at each side of the divide. This produced fair matches of maximum bump size and elevation, though poorer in predicted elevation for the upper levels (see Figure 9, right), and a rather poor match to the upper surface profile (see Figure 9, left). Moreover, scouring pushes back the date of commencement of the Raymond effect to 5 kyr B.P. In view of the fact that the measurements by Conway et al. [1999] of accumulation at Roosevelt Island and their analysis of the spatial pattern of internal layering show no evidence of scouring at the divide, and the poor match that we have obtained, we do not believe that scouring has played a role in the development of the Raymond bumps at Roosevelt Island Exploring the Surface Profile Asymmetry [53] All of our results discussed earlier were unable to explain the observed asymmetry of the surface profile. Differential accumulation can cause asymmetry around the divide. This has been shown for a shallow ice approximation solution [e.g., Hindmarsh, 1996]: quite substantial differences in accumulation rate across the divide are needed to produce large effects in the surface profile. 10 of 15

11 Figure 9. Plots of (left) match to surface data and (right) match to Raymond bump amplitude data. Results for asymmetric accumulation and for scouring at the divide, for 1Q rheology, 1Q rheology with differential accumulation, and a DJ rheology with scouring. The curve shown for the latter corresponds to an onset time of 5 kyr B.P. Thomas et al. [1980] report an accumulation trend from 0.12 m yr 1 on the western grounding line to 0.24 m yr 1 on the eastern grounding line. Using the 1Q rheology (n = 4) with the activation energy and rate factor providing the best fit to bumps and surface for the reference values of the model parameters (Q = 50 kj mol 1 and A 0 =10 11 Pa 4 yr 1 ), we have tested the effects of asymmetric accumulation when the accumulation rate to the west of the divide is 33% higher than that to the east. The results are shown in Figure 9. Good fits of bump amplitude are retained, but the surface profile does not match appreciably better. [54] Some further conclusions regarding the asymmetry can be obtained by modeling of the areas away from the divide, using the shallow ice approximation. Figure 10 shows matches to the layer geometry obtained away from the divide, in a region where the shallow ice approximation is valid, using the methodology described by Siegert et al. [2003]. This method solves the age equation (1i) using steady state balance velocities deduced by integrating the accumulation rate along the length of the flow path. The vertical distribution of horizontal velocity is distributed according to a shape function [e.g., Reeh and Paterson, 1988] which assumes that the appropriate mechanical model is the shallow ice approximation. Consequently, this model is not applicable to zones where the Raymond effect operates. Vertical velocity is deduced from continuity. [55] The modeling presented here has the added refinement of being a true inverse methodology. The layer elevations are defined by a vector of functions Z(x) and the modeled predicted layers by H. A set of model parameters, in this case defining the accumulation rate variation, are defined by ;, and a set of a priori estimates of these parameters are denoted ;*. An objective function defined by J ¼ 1 2 ðh ZðÞ x ÞT C 1 Z ðh ZðÞ x Þþ 1 2 ð ; ;* ÞT C; 1 ð; ;* Þ; ð17þ is minimized subject to the constraint (imposed by Lagrangian multipliers) of the age equation being solved (note that the first term on the right-hand side is the fit of the model to the layers, and the second term on the righthand side is the fit to the parameters). Here the matrix C Z is error covariance of the layer data, and the matrix C g is the covariance of the errors in the a priori estimates of the linear trends. It can also be set to be so large compared with C Z that prior information does not affect the results. [56] Specifically, we set the covariance error of the layers to be around 10 m 2 away from the divide, and 10 5 m 2 in the region 3<x/km < 2, the point being to ensure that the SIA model underpinning the inversion did not attempt to match the layers to the mechanics describing the Raymond effect. We used two configurations. In the first configuration, we specified separate linear trends in accumulation rate for either side of the divide. This gives two sets of optimal layer fits, on either side of the divide. In the second configuration, one linear/quadratic trend covering both sides was permitted, which meant that each model layer was fitted across both sides. These trends were sufficient to obtain good fits 11 of 15

12 in the upper part of the ice sheet. The coefficients of the linear and quadratic trends were assigned a prior estimate zero and error covariance of the prior estimate the unit identity matrix. Following, for example, Siegert et al. [2004] and F. Parrenin et al. (Analytical solutions for the effect of topography, accumulation rate and lateral flow divergence on isochrone layer geometry, submitted to Journal of Glaciology, 2005) since the layers are undated it is not necessary to specify the absolute accumulation rate, and the accumulation rate linear trend is normalized by the divide accumulation rate. Minimization of (17) yields the best fitting layers and the best estimate of the spatial variation in accumulation rate. For the case of two separate linear trends, the trends are and m yr 1 km 1 on the left and right side respectively, approximately consistent with the trends measured by Thomas et al. [1980], while for the quadratic case the coefficients are 0.01 and m yr 1 km 1. [57] The method does assume a steady geometry. However, as will be seen, this does not prevent us from deriving useful conclusions. The results of the optimization are shown in Figure 10a. In the upper part of the ice sheet, the three lines corresponding to left, right and combined parts are almost coincident. It is only in the lower part that the lines are not coincident, implying that in earlier periods some sort of asymmetry occurred. The time at which symmetry was established can be estimated by plotting the ages of the left and right isochrones against each other (Figure 10b). Assuming a downward velocity of 0.27 m yr 1 shows that the coincident ages started around 3 or 4 kyr B.P. and continue until present. This strongly suggests that something happened about this age, exactly the conclusion obtained by the independent dating using the Raymond effect. [58] Another insight into the asymmetry could be gained by estimating the spatial variation in the rate factor A necessary to produce the flux, given the present geometry. Here we are examining hypotheses of whether nonuniformity in ice properties could account for the observed asymmetry. In plane flow under the SIA, the flux is related to the rate factor and geometry by the formula Q ¼ 2 n þ 2 AH nþ2 j@ x sj n x s; ð18þ Figure 10. (a) Fits of SIA model to radargram. Dotted lines with circles are some of the lines picked from the radargram (Figure 2). The solid lines are the modeled best fit radar lines, with dashed lines representing separate consideration of basins left and right of the divide as well as an inversion considering the combined basins. (b) Ages of right and left basin isochrones plotted against each other (solid line). Dotted line has unit slope and zero intercept. Noncoincidence of two lines indicates different flow conditions on either side of divide. (c) Rate factor necessary to produce the observed steady profile, assuming that SIA applies. Cases are n = 3 (solid line) and smoothed line (dash-dotted line) and n = 4 (dotted line) and smoothed line (dashed line). and the only unknown is the vertically averaged rate factor A. This is plotted in Figure 10c for cases of n = 3, 4. The raw estimates are noisy as the slope is a differentiated quantity. Also note that the model used is the SIA which does not describe the Raymond effect, so the anomalous behavior around the divide should not be regarded as real. Given this, the variations in the rate factor (5 times higher on the right-hand side for both cases) correspond to variations in basal temperature of around 5 at the modeled temperatures of between 10 C and 5 C. The latter, however, were obtained using a constant geothermal flux of 60 mw m 2 which could show lateral variation. The general trend observed in Figure 10c (excluding the divide area) is consistent with the increasing trend in geothermal heat flux (based on similar reasoning) suggested by Thomas et al. 12 of 15

13 [1980], from 50 mw m 2 at the northeast side of Roosevelt Island to 70 mw m 2 on the southwest side. 5. Conclusions and Further Work [59] Fundamentally, the argument of Conway et al. [1999] about the dating of divide formation, or at least the commencement of the start of the Raymond effect, is based on continuity. The detailed operation of the Raymond effect is represented through the introduction of a Dansgaard- Johnsen parameterization [Dansgaard and Johnsen, 1969]. [60] In this paper, we have attempted to match some quite detailed data obtained by Conway et al. [1999], using a selfconsistent, time-dependent, thermomechanically coupled model. We have tried to fit both the profile of the glacier (something not attempted by Conway et al. [1999]), and the distribution of bump amplitude with depth. We have been unable to obtain a match using standard glaciological models. We sought to obtain matches with steady state isothermal and thermocoupled models, and with transient thermocoupled models using standard dependence on the temperature. The requirement of matching the profile and the bump amplitude curve sometimes leads to conflicting requirements of the optimal model parameters. [61] We also attempted to obtain matches with the new results of Goldsby and Kohlstedt [2001]. This did not work, because the stress regimes in Roosevelt Island corresponded essentially to low-index flows (n = 1.8) in GK rheology, where it was impossible to get a good match to the BAED. We did manage to bracket the profile with different choices of crystals size, but an accurate match was not attempted. [62] We have obtained good matches to the BAED using n 3. In general, thermomechanical coupling reduces the strength of the Raymond effect. Obtaining a BAED match with n = 3 requires a very low activation energy, much lower than most laboratory determinations, in fact so low that the ice is almost isothermal. If the rheological index is raised to n = 4, the activation energy that produces a match is higher, and quite close to the value that Goldsby and Kohlstedt [2001] cite for ice below 258K. This could be viewed as a GK rheology where the rate factors for the lowindex terms in the GK rheology were lower in the Roosevelt Island ice than those obtained by Goldsby and Kohlstedt. Using the nonstandard 1Q rheology (n = 4) we have obtained excellent matches to the BAED. [63] An alternative approach is to suppose that the standard glaciological rheological models are correct. Since these produce bump amplitudes that are too small, one needs to seek a meteorological mechanism which increases the amplitude. We show that one such method is scouring at the divide, and matches can be obtained if an earlier onset of divide flow (5 kyr B.P.) is assumed. However, these matches are not quite as good as those obtained by invoking nonstandard rheological models. Moreover, scouring is not consistent with modern field measurements [Conway et al., 1999]. [64] The match with surface profile is more perplexing. We can obtain an excellent match with one side of the Roosevelt Island upper surface profile, but not with the other. We have seen that the observed surface asymmetry of Roosevelt Island profile cannot be explained by an asymmetry in the forcing via the bedrock topography or a difference in the accumulation rate, nor by lateral variations in the temperature distribution within the glacier if a constant geothermal heat flux is assumed. Further possible reasons of asymmetry to be explored include lateral variations in geothermal heat flux, already suggested for Roosevelt Island [Thomas et al., 1980], the effect of different boundary elevations [Nereson et al., 1998a, 1998b; Nereson and Raymond, 2001] or the influence of 3-D geometry. [65] Our best matches give very similar timings for the commencement of the operation of the Raymond effect to those obtained by Conway et al. [1999]: 3000 versus 3200 years B.P., respectively. Our sensitivity test brackets this timing somewhere between 2200 and 4300 years B.P. An analysis of the effects of accumulation and thinning rates variable in time has not been done yet. [66] Our model contains what are believed to be nearly all the relevant glacier physics for ice below the melting point, anisotropy being the most prominent material property not considered. Several questions are raised here: are we concerned that standard rheological models do not appear to work, and what should we do next? Ice rheology is affected by the existence of anisotropic fabrics, so one possibility is to attempt a fit with one or more models of anisotropic ice flow. At the moment, however, incorporating this into models of specific geophysical situations is somehow premature, in particular because radar sensing of fabric has only just started [e.g., Fujita et al., 2003]. A second possibility is attempting to match further ice divides, to see whether the same rheological model emerges, and use the results to pose questions of anisotropic models. With this aim, a second section from Roosevelt Island could be used, and sections from other divides, for example the Siple Dome, which has the added significant constraint of an age-depth relationship, but also the significant added complication of transverse motion of the divide. [67] There remain some issues of ice physics. Why does thermomechanical coupling reduce the magnitude of the Raymond effect? Other authors [e.g., Hvidberg, 1996; Nereson and Waddington, 2002] have discussed temperature effects, noting that in the thermomechanical case, ice is warmer near the base, so the contrast between viscosities at the top and bottom of the glacier must be smaller. This is a contributory factor; however, it is not a totally satisfactory answer, because the increased shear near the base in the flank areas causes isochrones to be lower here compared with the isothermal case. [68] The possible contribution of basal sliding should also be investigated further. Roosevelt Island ice is presently frozen to its bed. However, this could have been different in the past, when the ice was thicker. Our preliminary testing suggested only a limited effect from basal sliding, which was restricted to cases of extreme values of model parameters and extremely high initial ice thickness. Nevertheless, the alternative interpretation by Pettit et al. [2003] of Conway et al. [1999] dating that prior to 3200 years B.P. Roosevelt Island could have supported and ice divide over a wet bed that allowed sliding, is sufficient to imply that this subject deserves some further attention. [69] Summarizing, our main conclusions are as follows: (1) Our best estimate for the retreat of the grounding line of the WAIS near Roosevelt Island is 3000 years B.P., bounded by 2300 and 4200 years B.P. (2) The use of standard 13 of 15