Abstract
 Top of page
 Abstract
 1. Introduction
 2. Ice Rheology and the Raymond Effect
 3. Experimental Design
 4. Numerical Modelling and Fitting Procedure
 5. Discussions and Conclusions
 Acknowledgments
 References
 Supporting Information
[1] The Glen exponent ncharacterizes the stressdependence of ice deformation, directly influencing the rate at which ice masses respond to external forcing. The slow deformation in large icesheets makes laboratory rheometry at representative strainrates difficult. We develop a new technique to estimaten insitu, deploying a phasesensitive radar to measure vertical strain rates of around 10^{−4} yr^{−1}within the top 1000 m of ice across ice divides at Summit and NEEM, Greenland. A fluiddynamical feature, the Raymond Effect, predicts strong vertical strainrate variation across divides over distances of a few icethicknesses. We achieve sufficient resolution to show this pattern, enabling us to estimaten= 4.5 by inverting our observations with flow modelling. This is higher than values previously used but consistent with other indirect measurements, implying laboratory measurements do not explore the full range of ice rheology and the consequent possibility of a greater sensitivity and responsiveness in icesheet dynamics.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Ice Rheology and the Raymond Effect
 3. Experimental Design
 4. Numerical Modelling and Fitting Procedure
 5. Discussions and Conclusions
 Acknowledgments
 References
 Supporting Information
[2] In recent years, the quality and quantity of data describing icesheet geometry and motion has increased considerably, so that our ability to describe the current state of the icesheets is no longer the limiting factor in making predictions. Despite this progress, theinsitu rheology of ice, which determines the rate at which the ice can flow and respond to external changes is still poorly understood and constrained. Uncertainties in the processes that govern the deformation of ice arise from the fact that insitustrainrates are not reproducible in laboratories, while field measurements are difficult to devise owing to the difficulties in completely characterising the strainrate and stress tensors.
[3] For ice flowing according to a nonlinear Glen rheology,Raymond [1983]predicted the presence of a nearlystagnant plug of stiff ice under ice divides, inducing horizontal variations of the vertical strainrate patterns, the Raymond Effect. So far the occurrence of this effectinsituhas principally been determined indirectly through the analysis of radar layers. By precisely measuring the differential displacement of internal radar reflectors, the phasesensitive radar (pRES) [Corr et al., 2002; Jenkins et al., 2006] allows a direct measurement of the Raymond Effect with sufficient resolution and accuracy to characterise the ice rheology at natural strainrates.
[4] Here, we present pRES data obtained along four transects crossing ice ridges on the Greenland ice sheet. Surface vertical strainrates are fitted using the results of a fullStokes numerical model of the Raymond Effect, where the flow along the ridge is taken into account. Finally, we discuss the implications of our estimate ofn for the operation of the Raymond Effect, for the ice rheology insitu and for ice dynamics in general.
2. Ice Rheology and the Raymond Effect
 Top of page
 Abstract
 1. Introduction
 2. Ice Rheology and the Raymond Effect
 3. Experimental Design
 4. Numerical Modelling and Fitting Procedure
 5. Discussions and Conclusions
 Acknowledgments
 References
 Supporting Information
[5] The flow law of ice is commonly characterised by Glen's flow law [Cuffey and Paterson, 2010], which relates the strainrate tensore to the deviatoric stress tensor τ by
where B is the temperature dependent stiffness factor, eis the second invariant of the strainrate tensor andnis the Glen index. Laboratory determination of the rheological parameters of ice have predominantly been at strainrates higher than those prevailing in natural conditions. Several physical mechanisms for ice deformation compete depending on stress, temperature, crystal size and impurity concentration [Schulson and Duval, 2009]. Consequently, a wide range of values for the rheological parameters of ice in ice sheets can be found in the literature. Several attempts have been made to measure the strainrates and characterise the flow law of ice insitu: from borehole inclination [e.g.,DahlJensen and Gundestrup, 1987], bubblyice densification [e.g.,Lipenkov et al., 1997], or strain sensors in boreholes [e.g., Elsberg et al., 2004; Pettit et al., 2011]. However, no agreement on the rheological parameters can be found, the main difficulty in the field being to properly characterise the stress state. In particular, a value of n = 3 is now commonly used in ice flow models, but published values range from 1 to 5.
[6] Because ice viscosity is nonlinearly dependent on the deviatoric stresses, the presence of a highly viscous plug of nearlystagnant ice is expected just under ice divides or ridges, leading to variation of the vertical strainrate profile from the ridge to the flank. Owing to this effect [Raymond, 1983], isochronic surfaces are expected to drape over the plug producing ‘Raymond arches’, which have been frequently observed in radargrams [e.g., Conway et al., 1999; Vaughan et al., 1999; Hindmarsh et al., 2011]. The size and the shape of these arches depend not only on the ice rheology [Pettit and Waddington, 2003; Martín et al., 2006], but also on the iceflow history [Nereson et al., 1998], and thus provide only indirect measurements of the Raymond Effect. Some surface strainrate measurements [Hvidberg et al., 2001] and vertical strainrate measurements from strainsensors in boreholes [Pettit et al., 2011] indicate differences in strainrates between divides and flanks that are consistent with the Raymond Effect, but do not have sufficient accuracy or resolution to determinen.
[7] Provided we can measure variations of the vertical strainrate across an ice divide predicted by the Raymond Effect with sufficient resolution, we can confirm that this is a sufficiently strong influence to cause the formation of Raymond arches. We can further expect to be able to infer the Glenexponentn insitu by comparing with the stress state obtained from ice flow modelling.
3. Experimental Design
 Top of page
 Abstract
 1. Introduction
 2. Ice Rheology and the Raymond Effect
 3. Experimental Design
 4. Numerical Modelling and Fitting Procedure
 5. Discussions and Conclusions
 Acknowledgments
 References
 Supporting Information
[8] Our phasesensitive radar has been deployed in the area of the topographic dome of Greenland and near the new deep ice core project NEEM to measure theinsitustrainrates (Figure 1). Two lines centered on the estimated divide position were surveyed at each site. Each line consist of 33 points of measurement, is 18 km long and is centered on the estimated divide position. For more details see section 1 of Text S1 in the auxiliary material.
[9] Our equipment consisted of a network analyzer (PNA E8356A) configured as a step frequency radar housed in a temperaturecontrolled case sitting on a sledge. Identical broadband aerials were positioned on the snow and separated by 8 m. They were operated at a center frequency of 314 MHz and a bandwidth of 142 MHz, using 7095 uniformly spaced frequency steps. The instrument was calibrated before each measurement by connecting the transmit and receive aerials. Since both phase and amplitude of the signal were recorded, changes in the reflector position can be measured to a small fraction of the wavelength [Jenkins et al., 2006]. Measurements were repeated after intervals of one year using the same setup and taking care to reposition precisely the equipment with respect to the bamboo markers left in place.
[10] The frequencydomain signal is converted to the complex timedomain [Corr et al., 2002], and then to an equivalent depth in ice, taking the speed of radar waves in solid ice to be 168 m μs^{−1}. The firn is left out of the analysis. Figure 2a shows a typical radar record of the amplitude against depth from NEEM Line 2. Beneath the direct breakthrough from the transmitter, amplitude falls to the noise floor at around 1000 m in this case. Zooms of amplitude and phase from two measurements taken one year apart are given in Figures 2b and 2c.
[11] Good reflectors exhibit a relatively high amplitude and a constant phase within them [Jenkins et al., 2006]. Stable reflectors exhibiting a good cross correlation of the amplitude and phase between the two measurements are selected and the mean phase difference is computed over the reflector width. Assuming the reflectors to be material surfaces, the phase difference between the two sets of measurements is proportional to the motion of ice, having accounted for the possibility of uncertainty in the integer number of wavelengths. As we only require differences to compute the strainrates, the displacement was set to a datum of zero for a good reflector at around 100 m depth, beneath the firn layer. The displacement is converted to an equivalent velocity in ice using the time between the two measurements and the velocity of the waves in solid ice. This procedure typically leads to ten vertical velocity measurements per 100 m of ice.
[12] The velocity profile measured at this point are plotted on Figure 2d. The large velocity gradient between 0 and 100 m is due to snow compaction. Below, from 100 m to 180 m, where the returned power decreases rapidly (Figure 2a), the amplitude and phase of the two sets of measurements are less stable, leading to some noise in the velocity profile. Below this (Figures 2b and 2c) the shape of both amplitude and phase are very similar, leading to a smooth derived velocity profile down to about 800 m depth, with increased variation lower down arising from the disappearance of the reflectors within the noise.
[13] The instrument error for each reflector is estimated from the amplitude of the reflector compared with the noise level. The phase error for each reflector takes into account a constant error of 3° for the procedure of selecting the reflectors and computing their mean phase difference. We obtain a relatively small error in the vertical velocity in the upper part of the icesheet, which increases with depth as reflector strength fades (Figure 2d).
[14] The velocity profile between 200 m and the depth where the reflectors disappear in the noise is fitted by a linear function of depth using a weighted leastsquares method, allowing us to estimate the error in the strainrate measurement. Additional quadratic terms are not found to be statistically significant, following the methodology ofJenkins et al. [2006]. This is in agreement with model results that show that the vertical strainrates are nearly uniform in the upper parts of the ice sheet [e.g.,Martín et al., 2006].
[15] Since velocity error increases with depth, the linear fit is largely determined by the upper points. The uniform vertical strainrate, the gradient of the velocity profile, reflects the vertical strainrate at the surface. The smooth velocity profile and the relatively small velocity errors on these upper points leads to a small uncertainty for the derived surface strainrates, between 1 and 5 × 10^{−5} yr^{−1}. Surface vertical strainrates and their uncertainty obtained at Summit and NEEM are given onFigures 3a–3c.
[16] For the two lines at NEEM, vertical strainrates do not show significant variation across the divide. In the Summit area, vertical surface strainrates are greater in magnitude beneath the divide and they are larger under the South divide. The strainrates also exhibit an asymmetry with respect to the divide.
4. Numerical Modelling and Fitting Procedure
 Top of page
 Abstract
 1. Introduction
 2. Ice Rheology and the Raymond Effect
 3. Experimental Design
 4. Numerical Modelling and Fitting Procedure
 5. Discussions and Conclusions
 Acknowledgments
 References
 Supporting Information
[17] We use Elmer/Ice to model the flow of ice. As true 3D fullStokes modelling is still too expensive to be easily used in an inverse procedure to fit the data [GilletChaulet and Hindmarsh, 2011], we use a simpler 2.5D model where we assume that the flow along the divide is nonzero but periodic and parameterized by the alongridge slopeγ (section 2 of Text S1). The stiffness parameter B in equation (1) varies with depth according to the GRIP temperature profile [Gundestrup et al., 1993]. Results obtained with this model with n = 3 for three different values of γ are given in Figures 3d–3f. As in the 2.5D model of Martín et al. [2009b], when γincreases, the velocity along the divide increases. As a consequence, the Raymond Effect, which implies higher absolute vertical strainrates at the divide compared with the flank, is muted by the alongridge flow and the maximum thinning rate under the divide decreases asγ increases. In consequence, the surface slope perpendicular to the divide shows stronger variations in the divide area when γ is low. The variation of the flow velocity perpendicular to the divide is identical for the three experiments. In Figure 3d, the slopes are compared with the slopes along the radar Lines 1 and 2 at Summit given by a 5km DEM [Bamber et al., 2001]. The slope along the divide where Line 1 crosses is 4 × 10^{−4} and for Line 2 it is 6 × 10^{−4}. There is good agreement between the surface slopes given by the model and surface slopes along the survey lines derived from the DEM.
[18] To fit the strainrates data obtained at Summit, we write the objective cost function
where _{zz} are the observed strain rates; C_{ee}is the error covariance matrix of the strainrate data, it is diagonal and derived directly from the strainrate error estimates;e_{zz}are the strain rates estimated at the measurement points; the strainratesf(x − x_{d}; n, a_{ℓ}, a_{r}, γ) are based on the 2.5D fullStokes model results and have four unknown parameters: the Glen indexn, the accumulation rates on either side of the divide a_{ℓ}, a_{r}, and the divide offset x_{d}, i.e., the offset between the estimated topographical divide position and the location of the minimum of the vertical strainrate. The Lagrange multiplierλconstrains the estimated strainratese_{zz} to being solutions of the flow model f.
[19] Best estimate of the four unknowns parameters that minimize J in equation (2) are given in Table 1and best fits to the Summit strainrate data are plotted inFigure 3. More details on the choice of f and the minimization procedure can be found in section 3 of Text S1.
Table 1. Best Fit Parameters Investigated for Summit Lines 1 and 2^{a}  n  a_{l} (m/yr)  a_{r} (m/yr)  x_{d} (km) 


Summit Line 1  4.5  0.22  0.26  0.62 
Summit Line 2  4.9  0.27  0.21  1.74 
[20] The fitting procedure used here takes into account only the measurement errors, leading to rather small errors for the best fit parameters that do not reflect the true sources of uncertainty. The errors on the stress state inferred from ice flow modelling are more difficult to estimate; the main model assumptions are discussed in details in section 4 of Text S1. Plausible errors in the alongridge slope give rise to an estimate ofnlying between 4.3 and 4.8, while 3D effect and uncertainties in the ice activation energy have a small effect. Numerical results with an anisotropic flow law show a reduction of the Raymond effect with a nonzero alongridge slope but confirm the influence of the ice rheology towards the base.
5. Discussions and Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Ice Rheology and the Raymond Effect
 3. Experimental Design
 4. Numerical Modelling and Fitting Procedure
 5. Discussions and Conclusions
 Acknowledgments
 References
 Supporting Information
[21] The uniform pattern of surface vertical strainrates in the NEEM area (Figure 3a) shows that the Raymond Effect is not operating, being muted by the flow along the divide. This is predicted by numerical simulations and also shows that there is no special accumulation pattern immediately at the divide. The results obtained at Summit show horizontal variations of the strainrate associated with an operating Raymond Effect. This is the firstdirect confirmation of the operation of the Raymond Effect from surface measurements, rather than inferring its operation from radar layer geometry.
[22] Our results confirm that the fact that Raymond arches are not seen in the radargrams in an area where the Raymond effect is operating means that the divide has not been in a stable position long enough for the arches to form over the appropriate advection timescale [Marshall and Cuffey, 2000]. Not only can we exclude nonoperation of the Raymond Effect arising from a (quasi)linear ice flow law, but we can also state that the higher absolute strainrates we measured over the divide indicate that wind scouring at the divide (where absolute strainrates would be lower) is not an explanation for the formation of Raymond arches. The values of accumulation inferred from optimal fitting are for a steady state divide. The different values deduced from best fit on each side of the divide (a_{l} and a_{r} in Table 1) either mean that there is a different pattern on each side of the divide, possibly associated with wind deposition, or that divide migration is giving rise to different strainrates.
[23] We have used our results to characterise the ice rheology. Assuming ice follows the Glen flow law implies a high value of n around 4.5. Such high values have also been reported from the study of Raymond arch amplitude [Martín et al., 2006]. Anisotropy does not seem to be an explanation as, in our 2.5D model calculations, it reduces the Raymond Effect by enhancing the shearing horizontal plane component along the ridge (see section 4 of Text S1). Nevertheless, in agreement with Pettit et al. [2007] and Martín et al. [2009a], the Raymond Effect is sensitive to the fabric profile in the lowest part of the icesheet. Below 2800 m at GRIP, the temperature is above −10°C and the microstuctures are characteristic of dynamic recrystallisation [De La Chapelle et al., 1998]. Dynamic recrystallisation is usually associated with a nonsteady or a tertiary creep regime for which values ofn = 4 have been reported [Schulson and Duval, 2009]. Thus, our value could be characteristic of this regime as, for grounded ice, most of the deformation is concentrated near the bed.
[24] Regarding the variations of temperature, microstructure, impurities content and strainrates with depth, several physical mechanisms are certainly competing to accommodate the deformation and our results cannot be held to contradict the low value of the stress exponent expected for cold ice at low deviatoric stresses. This could explain why the usualn= 3 value gives better results when investigations aim at matching the surface profile of the icesheet as in that ofGilletChaulet and Hindmarsh [2011]. Owing to the low strainrates measured, over the one year timeinterval used, the signal to noise ratio is relatively low, giving only a measurement of the vertical strainrates in the upper 30% of the icesheet.
[25] Our results challenge the isotropic Glen flow law with n = 3 used in most ice sheet models. Laboratory studies of ice rheology evidently do not provide a full description of insitu ice rheology. It is worth noting that pure metals have stress exponents of 4.5 [Weertman, 1999], but we offer no explanation as to why this should be appropriate for ice flow at divides. Our results have direct implications for precise flow modelling around cores at divide or dome positions. They support the nonlinear velocity profiles used for dating Dome C and Dome Fuji ice cores [Parrenin et al., 2007]. Furthermore, we need to establish their general applicability, since a high value of n can have significant consequences for grounding line retreat [Schoof, 2007] and transient response of icestreams to downstream perturbation [Hindmarsh, 2006]. Thus, more efforts are needed to constrain insituice rheology in polar ice sheets. Systems such as pRES, which can take closelyspaced measurements of vertical strainrates, are useful tools to achieve this goal.