Seasonal to decadal scale variations in the surface velocity of Jakobshavn Isbrae, Greenland: Observation and model-based analysis

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jf002110, 2012 Seasonal to decadal scale variations in the surface velocity of Jakobshavn Isbrae, Greenland: Observation and model-based analysis Ian Joughin, 1 Ben E. Smith, 1 Ian M. Howat, 2 Dana Floricioiu, 3 Richard B. Alley, 4 Martin Truffer, 5 and Mark Fahnestock 6 Received 2 June 2011; revised 10 April 2012; accepted 16 April 2012; published 25 May [1] Using new data, we build upon the nearly two-decade long record of observations from Jakobshavn Isbrae to investigate the processes driving its dynamic evolution. While winter flow speed has not increased substantially over the last three winters, there remains a strong seasonal variation in flow speed that coincides with a cycle of summer thinning and winter thickening. We relate changes in glacier speed to geometry through variations in basal traction and horizontal stresses, using ice-flow models constrained by satellite and airborne observations. These results suggest that the bed provides little flow resistance along the main trough within about 20 km of the terminus. While the loss of buttressing from the retreat of grounded and floating ice likely contributed to the initial speedup, other processes are of comparable significance at seasonal to decadal time scales. From analysis of the models, we hypothesize that thinning-induced change in basal effective pressure is the dominant process influencing near-terminus behavior, while diffusive processes drive the upstream response. The apparent need for the terminus to thin to near flotation before it can calve may limit the rate at which retreat occurs. Our analysis of the processes controlling the speed suggests little potential for further large acceleration. Thinning and elevated speeds may continue at rates similar to present, however, putting the glacier on course to retreat to the head of its deep trough in about a century, at which point it likely would stabilize with a thinner terminus. Citation: Joughin, I., B. E. Smith, I. M. Howat, D. Floricioiu, R. B. Alley, M. Truffer, and M. Fahnestock (2012), Seasonal to decadal scale variations in the surface velocity of Jakobshavn Isbrae, Greenland: Observation and model-based analysis, J. Geophys. Res., 117,, doi: /2011jf Introduction [2] Over the last decade many of Greenland s marineterminating outlet glaciers sped up by 50% or more [Rignot and Kanagaratnam, 2006], including the ice sheet s three 1 Polar Science Center, Applied Physics Laboratory, University of Washington, Seattle, Washington, USA. 2 School of Earth Sciences and Byrd Polar Research Center, Ohio State University, Columbus, Ohio, USA. 3 Remote Sensing Technology Institute, German Aerospace Center (DLR), Wessling, Germany. 4 Department of Geosciences and Earth and Environmental Systems Institute, Pennsylvania State University, University Park, Pennsylvania, USA. 5 Geophysical Institute, University of Alaska Fairbanks, Fairbanks, Alaska, USA. 6 Institute for the Study of Earth, Oceans, and Space, University of New Hampshire, Durham, New Hampshire, USA. Corresponding author: I. Joughin, Polar Science Center, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA , USA. (ian@apl.washington.edu) Copyright 2012 by the American Geophysical Union /12/2011JF largest marine outlets in terms of ice discharge: Jakobshavn Isbrae, Helheim Gletscher, and Kangerdlugssuaq Gletscher [Joughin et al., 2004a; Howat et al., 2005; Luckman et al., 2006]. Many of these abrupt speedups occurred as the Intergovernmental Panel on Climate Change (IPCC) was convening for their 4th Assessment. The stark contrast between these observations of rapid increases in speed and the far more subdued changes predicted by large-scale ice sheet models prompted the IPCC to conclude that our knowledge of ice dynamics was insufficient for determining a meaningful upper bound on 21st Century sea level rise [IPCC, 2007]. [3] Following the IPCC 4th Assessment, Pfeffer et al. [2008] proposed an upper limit of 0.47 m on the Greenland Ice Sheet s 21st Century contribution to sea level by assuming that all of its glaciers collectively could speed up by no more than a factor of 10. Despite their dramatic nature, most recent speedups in Greenland have been by less than a factor of 2 [Joughin et al., 2010; Moon et al., 2012], increasing total ice discharge by roughly 50% above the level required to maintain equilibrium [Rignot and Kanagaratnam, 2006]. Thus, Greenland s future contribution to sea level via ice discharge hinges on whether its outlet glaciers are already 1of20

2 Figure 1. TerraSAR-X image showing Jakobshavn Isbrae, Aug 26, 2009 when the terminus was the farthest back as observed with our regular 11-day sampling. The white line indicates the location of the profile shown in subsequent plots. The solid-colored squares show the 2009 seasonal variation of the terminus. The locations of points plotted in Figures 3 (solid colored triangles), 5 (solid colored circles), and 6 (open red and black squares). TerraSAR-X image copyright DLR, flowing near their maximum rates or instead they can speed up much further, potentially reaching the hypothesized 10x speed limit. [4] Four processes have been identified that either solely or in combination control the recent speedups of Greenland s marine-based outlet glaciers. In the first of these processes, as a grounded or floating glacier terminus retreats, the glacier must speed up to increase strain rate dependent resistive stresses upstream to compensate for the loss of downstream contact with the bed and/or fjord walls. This is consistent with numerous observations that show speedup coincident with terminus retreat [e.g., Amundson et al., 2008; Howat et al., 2005; Nettles et al., 2008]. The second process depends on the commonly applied inverse relation between sliding speed and effective pressure [Bindschadler, 1983], which is the difference between the ice overburden and the basal water pressure. Thus, it is hypothesized that a thinning glacier reduces its overburden relative to what is assumed to be fixed basal water pressure, decreasing effective pressure and increasing sliding speed [Meier and Post, 1987]. This process is believed to be a major influence on Alaskan tidewater glaciers and may also affect glaciers in Greenland [Pfeffer et al., 2008]. In the third process, strong thinning on the main trunk induced by the other two processes steepens interior slopes, causing the speedup to diffuse farther inland [Joughin et al., 2008b; Payne et al., 2004; Howat et al., 2007]. Finally, the fourth process is related to enhanced lubrication by seasonal meltwater, which observations suggest generally has a relatively modest effect on fast moving outlets in Greenland [Joughin et al., 2008a]. The relative role of each of these processes in controlling the behavior of Greenland s outlet glaciers remains poorly understood. [5] The largest glacier on Greenland s west coast, Jakobshavn Isbrae (Sermeq Kujalleq), changed dramatically from 1998 to 2003, doubling its speed as its 15-km-long floating ice tongue disintegrated [Joughin et al., 2004a; Luckman and Murray, 2005; Thomas et al., 2003]. From 2004 to 2007 the glacier continued to accelerate over much of its trunk at a rate of 5 to 7% per year [Joughin et al., 2008b]. Although there was virtually no seasonal variation of the glacier s speed in the 1980s [Echelmeyer and Harrison, 1990], observations from 2004 onwards show that a strong seasonal variation in speed developed coincident with the annual advance and retreat of a transient winter ice tongue [Joughin et al., 2008b]. [6] The periods during which Jakobshavn Isbrae and many other outlet glaciers sped up were relatively brief (months to a few years), and they occurred during a time when there were few satellite or airborne observations to monitor the temporal evolution of speed and glacier geometry. Since Jakobshavn Isbrae s initial speedup, the Canadian RADAR- SAT and German TerraSAR-X satellites (e.g., Figure 1) have provided frequent observations for measuring the glacier s surface velocity. This glacier also has been monitored almost 2of20

3 Figure 2. (a) Surface and bed elevations for Jakobshavn Isbrae along the profile shown in Figure 1. (b) Force, F, due to terminus as function of terminus position (see equation (1)). Colored symbols show the terminus position in 2006 (circles) and 2009 (squares; see also Figure 1). In both Figures 2a and 2b, the insets show detail near the terminus with vertical lines indicating points of maximum retreat. annually with airborne lidar [Krabill et al., 2004] and icepenetrating radar [Li, 2010] for well over a decade. These data sets, along with the strong seasonal to decadal variation in speed, make Jakobshavn Isbrae an ideal site for studying the controls on fast ice flow as described below. 2. Data [7] We used a combination of airborne and spaceborne data sets collected over the last two decades in our modelbased analysis of the factors contributing to flow variability on Jakobshavn Isbrae. Here we summarize the relevant data and the methods used to derive them Surface Elevation [8] Surface elevation data were acquired by the Airborne Topographic Mapper (ATM), which flew several grids over Jakobshavn Isbrae at approximately 5-km spacing with additional profiles down the centerline of the glacier at several different times since 1993 [Krabill et al., 2004]. We interpolated these data to produce two different elevation data sets. [9] First we interpolated the data to produce elevations (Figure 2a) along a profile running down the approximate center of the glacier (Figure 1) at three different times (1990s, 2006, and 2009). For the 1990s profiles, we used data collected from 1993 to 1998, over which time changes in the surface elevation were relatively small. These data extend down the fjord onto the floating ice tongue. [10] Our 2-D numerical ice stream model experiments required a digital elevation model (DEM) covering the complete model domain. Thus, we combined ICESat (Version 531, GLA12 available through the National Snow and Ice Data Center), ATM [Krabill et al., 2004], and LVIS (Land 3of20

4 Figure 3. Relative elevation is shown as a function of time at several points along an often-repeated ATM flightline. The mean height (given as subscript in legend) has been subtracted from each profile and an arbitrary height assigned to each set of point for purposes of display. The locations of the points are shown with the same plot symbol in Figure 1. The black circles show the relative change in height between the highest (z 719 ) and lowest points (z 195 ), with decreasing values indicating a relative lowering of the near terminus point (z 195 ) with respect to the inland point (z 719 ). Vegetation, and Ice Sensor) data [Hofton et al., 2008]. We treated each data set as a cloud of elevation points in space and time, to which we fit a moving-surface model that was constrained to be smooth in space and time at each grid point, while fitting the data to within specified tolerances. This provided a DEM for June 1, 2000, posted at 200-m intervals, and a set of annual elevation-difference maps for the years between 1997 and 2010, posted at 800 m. Cubic interpolation of these maps provides elevations for any intermediate time. For our numerical model, we used this process to produce DEMs for 1997 and 2009, which we smoothed to a resolution of 2 km. [11] To evaluate thinning on the main trunk, we used an ATM profile that was repeatedly flown down the trunk of the glacier [Krabill et al., 2004]. Figure 3 shows relative elevations at several points (marked as triangles in Figure 1) along Jakobshavn Isbrae s main trunk. Using data from the four springtime flights, we evaluated the thinning trend at each point, finding a fairly consistent lowering of the springtime surface. Data also were acquired in September 2007, and the results for the point nearest the terminus (red) show lowering of 27 m relative to the multiyear trend, only a small fraction of which can be accounted for by summer melt [van de Wal et al., 2005], suggesting enhanced summer dynamic thinning. By the following July, the near-terminus surface had risen 3 m, although it was still 18 m below the trend. By late April 2009 the surface had increased in elevation by an additional 18 m. [12] Despite sparse temporal sampling, Figure 3 suggests a near-terminus pattern of spring-to-fall thinning (30 m) followed by fall-to-spring thickening (15 m), which results in a net annual thinning trend of 15 m yr 1. The amplitude of this seasonal variation diminishes with distance upstream. As a result, the near-terminus point (red) is lowering at an annual rate of 7 m yr 1 relative to the uppermost point (cyan), but there is a seasonal variation of the relative height between these two points of 25 m (black) Bed Elevation [13] The bed DEM was developed at the University of Kansas and was created from data collected by their airborne depth-sounding radar [Li, 2010; van der Veen et al., 2011], and it is distributed by the Center for the Remote Sensing of Ice Sheets ( Most of the radar and ATM data were collected concurrently, resulting in similar spatial sampling (5 km) for the interpolated surface and bed DEMs. Figure 2a shows the bed elevation along our nominal centerline profile along with flotation height h float, which is the threshold elevation that the glacier surface must stay above to remain grounded Terminus Position [14] Figure 2b shows the centerline terminus position derived from a time series of RADARSAT (2006) and TerraSAR-X (2009) (see also Figure 1) synthetic aperture radar (SAR) images. Comparison of the inland limit of the terminus position for these two years indicates that the position of maximum summer retreat increased by 1.8 km over the 3-year period. For both years, the extent of the maximum retreat corresponds to a location where the surface elevation is 20 m above h float Ice Flow Velocity [15] We applied standard speckle-tracking techniques [Joughin, 2002] to TerraSAR-X data to extend an existing record derived from ERS-1, ERS-2, RADARSAT, and LandSAT data (Figure 4) [Joughin et al., 2004a, 2008b]. Velocity uncertainty is dominated by the spatially varying errors of 2 3% associated with slope corrections applied to 4of20

5 Figure 4. Speed on Jakobshavn Isbrae over an 18-year period, updated from an earlier record [Joughin et al., 2008b]. The number in the marker name gives the nominal distance from the late summer terminus. The locations of the first 5 markers (M6-M20) are shown in Figure 1, with the other 2 markers (M26 and M43) located farther upstream along the main trunk. Dashed lines show speedup trends calculated with the 2004 to 2007 data and solid lines show the same trends calculated for the 2004 to 2010 record. the across-track (range) velocity component [Joughin, 2002; Joughin et al., 2008b]. When images are collected with the same imaging geometry, however, such slope errors cancel in estimates of flow speed change. Thus, remaining errors that influence velocity differences are generally small (<1%). [16] The 2004-to-2007 RADARSAT-derived data were used in an earlier study to estimate trends in speedup of 5.2 to 6.5% yr 1 (dashed lines in Figure 4) [Joughin et al., 2008b]. We augmented this record using TerraSAR-X data from 2008 onwards. While only a few such images were acquired in 2008, from 2009 onwards TerraSAR-X data were acquired almost every 11 days. Using these new data, we evaluated the longer time series, which yielded smaller speedup trends ranging from 2.6 to 4.4% yr 1 due to a diminished rate of speedup in the later years. [17] Figure 5 shows the speedup expressed as a percentage of the early January 2009 speed near both the upper and lower ends of the TerraSAR-X data set (marked as black/red squares in Figure 1). While the seasonal variation makes it difficult to evaluate short-term trends, the January 2009 speeds are almost identical to the minimum speeds for the subsequent two winters. This suggests that the speeds may be stabilizing, albeit with winter minimum speeds more than twice those of the 1990s. While the rate of winter speedup Figure 5. Speedup as percentage of the January 2009 speed at a location just above the terminus and at location near the inland edge of the TerraSAR-X data set. See corresponding plot symbols in Figure 1 for locations. The solid filled near-terminus symbols indicate the times shown in Figure 8. 5of20

6 Figure 6. (a) Speed profiles for the 1990s, 2000, and The 2009 velocity was sampled every 11 days and color is used to indicate the corresponding date. (b) Observed summer (Day 187) and winter speeds (average of Day 33 and Day 363). Also shown is the winter speed (black) scaled by, V = V0, as determined using equation (3) and the summer (Day 187) terminus position. appears to be declining, the seasonal amplitude in 2009 and 2010 was greater than in 2006 and [18] The 2009 TerraSAR-X acquisitions provided 31 estimates of speed computed every 11 days over the area shown in Figure 1, with corresponding centerline speeds shown in Figure 6a. These data indicate that at 10 km along the profile, 2009 speeds relative to the mid 1990s are elevated by factors of about 2.1 and 2.8 in mid-winter and mid-summer, respectively. We also used a similar set of velocity profiles (not shown) from RADARSAT data collected in 2006 [Joughin et al., 2008b]. 3. Model [19] We used two ice-flow models to examine Jakobshavn Isbrae s behavior. The first was a simple terminus-driven model that relates seasonal variability in speed to changes in terminus position and geometry. An advantage of this 6of20

7 terminus-driven model is that we can explicitly account for the dynamic boundary condition at the grounded terminus. The second was a numerical shallow-shelf approximation ice stream model [MacAyeal, 1989] to which we applied both forward and inverse methods [MacAyeal, 1993; Joughin et al., 2004b]. Our implementation of this model, however, does not allow a dynamic boundary condition for a grounded terminus. Instead, we implemented it to use a kinematic (specified velocity) boundary condition on the model domain s perimeter Seasonal Terminus-Driven Model [20] Force-balance techniques are commonly applied to study the resistive stresses that influence glacier flow [van der Veen, 1999; van der Veen et al., 2011]. These methods, however, involve a double differentiation of the velocity data, producing results that are noisy and often difficult to interpret. Instead, we focus on examining the driving stresses, which are less subject to noise. In addition to the usual slope-and-thickness-dependent driving stress, t d, there is a net force, which is due to the presence of a free calving face. The pressure boundary condition at this face is determined by the height above water of the calving front and the density of seawater. This pressure profile is different from that of hydrostatic ice, which is a good approximation away from the calving face. Following Howat et al. [2005], the difference of these two forces at the terminus can be expressed as F ¼ gr i 2 1 r w r i Hf 2 þ r w h f 2H f h f r i Nm 1 ; where g is the acceleration due to gravity, H f is the thickness at the terminus with a corresponding surface elevation, h f, and r i and r w are the densities of glacial ice (910 kg m 3 ) and seawater (1028 km m 3 ) respectively. For a grounded terminus, equation (1) accounts for any additional height above flotation, while for a floating terminus it reduces to the commonly used H 2 dependence [van der Veen, 1999; van der Veen et al., 2011]. For the surface profiles measured in the 1990s, 2006, and 2009, Figure 2b illustrates the dependence of F on terminus position; because F applies only at the terminus, which thus far has ranged over only the first 7.5 km of the profile, the inland part of the curve for F in Figure 2b is hypothetical (i.e., at any point it only applies if the terminus actually retreats to that point). [21] To achieve force balance for a grounded or floating terminus, the frontal force, F, must be resisted upstream by some spatial distribution of resistive stress. The observed changes in speed well upstream of the terminus (e.g., Figure 6a) indicate the region over which longitudinal stresses likely redistribute much of this force. While gradients in the column-integrated longitudinal stresses often are treated as resistive stresses in force budget methods [e.g., van der Veen, 1999], here we treat the portion of the longitudinal resistive stress that originates from F at the terminus, t F, as an addition to the driving stress since its sign is such that it acts as a pull on the ice over a stress-coupling range that extends upstream from the terminus [Kamb and Echelmeyer, 1986]. Over this coupling range, the enhanced driving stress (i.e., t d + t F ) either is resisted directly at the bed beneath, or is transmitted via lateral shear stress to the bed near the margins, both of which lead to ð1þ increases in strain rates and speed. While this description oversimplifies the complicated nature of the balance of forces [van der Veen, 1999], conceptually it characterizes the major forces associated with the terminus [Thomas, 2004; Howat et al., 2005]. [22] The velocity data in Figure 6a suggest that the seasonal component of t F is at a maximum at the terminus (0 to 7 km) and is nearly negligible at 29 km. Based on this observation, we make the simple ad hoc assumption that the influence of F falls off linearly with distance from the terminus, such that t F FL x1x 2 ðþ; x where L x1 x 2 is a simple weighting function (with units of inverse distance) that is zero outside the interval (x 1, x 2 ), declines linearly from its value at x 1 (terminus) to zero at x 2 (upstream limit of stress coupling), and is normalized such that the along-flow integral of t F equals F. Later, we use our numerical model to investigate the validity of this empirically determined length scale. [23] To assess the role of F in driving seasonal oscillations in speed, we express F as the sum of a seasonally varying component, F s, and a constant reference value, F o. Analogously, we state the additional stress associated with the terminus as the sum of a reference and a seasonally varying component, t s, such that t F = t o + t s. Treating these terms as additional driving stresses, the velocity, V(t), normalized by a corresponding reference velocity, V o, is given by: ð2þ Vðx; tþ ¼ t d þ t o þ t 3 s : ð3þ V o t d þ t o Here we assume V t d 3, which is applicable for several types of flow such as increased resistance either from greater internal ice deformation (e.g., laminar flow or lateral shearing) or basal sliding over a hard, rough bed (i.e., power law sliding with n = 3) [van der Veen, 1999]. With this equation, the terminus-driven variation in t s modulates the speed relative to the reference, V o. We equate V o to the average midwinter speed, since that is when speeds are most stable (see section 4.1). We note that this is a highly simplified model meant to examine the sensitivity of speed to changes in terminus geometry, and as such, it has limited prognostic ability Numerical Model [24] While our simple terminus-driven model is appropriate for investigating the effect of seasonal advance and retreat on ice-flow speed, we use a numerical model to examine a more complicated geometry and a broader range of processes related to outlet-glacier flow. For this purpose, we use a finite element implementation of a standard vertically integrated, shallow-shelf approximation ice streamflow model [MacAyeal, 1989] given by: x y 2nH 2nH 2 u x þ v y 2 v y þ u x þ y þ x nh nh u y þ v x u y þ v x t b;x r i gh z s x ¼ 0 t b;y r i gh z s y ¼ 0: ð4þ 7of20

8 In this equation the basal shear stress is represented by t b, the Cartesian horizontal coordinates by x and y, and the corresponding velocity components by u and v. The effective viscosity, n, is given by: n ¼ E 1 =n B 2 2 þ þ 1 2 u x v y 4 u y þ v x 2 þ u x n 1 v y ð Þ = 2n ; ð5þ with rate factor B, exponent n, and enhancement factor E. [25] We solve these equations with kinematic (i.e., velocity or Dirichlet) boundary conditions. As in earlier work [Joughin et al., 2009], t b is modeled as power law sliding with an exponent, m, that produces Coulomb-plastic behavior as m [Cuffey and Paterson, 2010]. The terms with velocity derivatives represent lateral and longitudinal stress gradients in the horizontal plane, which we collectively refer to as membrane stresses [e.g., Hindmarsh, 2009]. For our experiments, we use an m = 3 sliding law such that ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t b;x ¼ b 2 p 1=3 u 2 þ v 2 u pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ v 2 ð6þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t b;y ¼ b 2 p 1=3 u 2 þ v 2 v pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : u 2 þ v 2 This equation follows MacAyeal s [1992] convention of using a squared drag coefficient, b 2, to ensure a positive value from the inversion. [26] Despite being depth averaged, an important feature of the model is that it does include membrane stresses, which are significant for fast flow through a narrow (a few ice thicknesses), well-incised channel such as Jakobshavn Isbrae [Lüthi et al., 2003; van der Veen et al., 2011; Vieli and Nick, 2011]. The model does not account for vertical shear, which has two main effects. First, vertical shear is not included in the effective viscosity. With a weak temperate layer, most of this effect should be confined to the lower part of the ice column, and have little effect on viscosity in the colder, stiffer upper part of the column that has the greatest influence on membrane stresses [Lüthi et al., 2003; Truffer and Echelmeyer, 2003]. While we cannot entirely ignore this deficiency, it is likely of similar or lesser consequence than the uncertainties in the other factors that influence the model viscosity (e.g., temperature and fabric). Second, the model does not include motion due to vertical shearing, and performs best when motion is dominated by sliding [Hindmarsh, 2004; MacAyeal, 1989]. High ratios of sliding to deformation are expected if the bed beneath the main ice stream is weak (<50 kpa) with respect to the driving stress (300 kpa) [Thomas, 2004]. Furthermore, the presence of temperate basal ice [e.g., Funk et al., 1994] should concentrate any vertical shearing near the bed, producing results similar to sliding with m = 3 [Lüthi et al., 2002]. [27] We used inverse methods applied to the forward model given by equation (4) to infer the basal shear stress [Joughin et al., 2004b; MacAyeal, 1993, 1992]. Due to the limitations of the model, we focus more on a qualitative interpretation of the inferred patterns, rather than a quantitative interpretation of the inferred basal shear stress at any given point. In the case of Pine Island Glacier, Antarctica, comparison of inversion results from this type of model and a full-stokes model shows little qualitative difference [Morlighem et al., 2010]. 4. Model Results [28] We performed a series of model experiments with both the simple terminus-driven model and the numerical flow model. Each model has different strengths, and collectively they reveal much about the different factors contributing to the past and present speed of Jakobshavn Isbrae Terminus-Driven Model [29] An advantage of the simple terminus driven model is that, via equations (1) and (2), we can use the observed geometry and terminus variation to calculate values of F and t F. Using an appropriate value of F 0 (see next paragraph), we can decompose F in fixed and seasonal components and compute V = V0 using equation (3). Here we use winter for our reference epoch and assume that V 0 is equivalent to the minimum winter velocity (average of profiles corresponding to Days 33 and 363). Thus, we can scale this V 0 by the ratio calculated using equation (3) to estimate V at other times of the year. As an example of this approach, we estimated the speed for early July 2009 (Day 187) as a rescaled version of the winter speed. As Figure 6 illustrates, this approximation agrees well with the observed velocity along the full length of the profile. This agreement indicates that the additional force exerted by the change in terminus position can account for much of the summer speedup, when distributed linearly inland. Numerical model results described below indicate that similar agreement is maintained year-round near the terminus, but farther upstream other processes can be important. [30] To further explore the relationship between speed and terminus position, we used equations (1) through (3) to compute V(t)/V o from the 2006 and 2009 terminus positions (Figure 7). The transient winter ice tongue complicates this analysis. While the ice tongue did advance in early 2009, no tongue formed in the winter of 2009/10. The early 2009 tongue appears to have been weak, yielding little effect on flow speed. As a result, we assumed F o = 0.45 GN m 1 corresponding to the point where the surface reaches flotation at 5.5 km. Seaward of 5.5 km we set F = F o (Figure 2b), which is consistent with the lack of a tongue in late 2009 and roughly approximates an unconfined (i.e., providing no backstress) ice tongue earlier in the year. For 2006, we used F o = 0.3 GN m 1 (the approximate value when the terminus was fully advanced) and set F equal to the value determined from the flotation height seaward of 5 km. This is equivalent to assuming that the tongue was buttressed in 2006 with contact from the fjord walls providing backstress. Alternatively, in 2006 the advance of a thicker terminus may have maintained contact with the bed farther downstream than the point where we assume it reached flotation (i.e., seaward of 5 km). In either case, most of the change in F occurred between 4 and 5 km, at the downstream end of the overdeepened region (i.e., area from 4 to 11 km in Figure 2a). Consequently, the terminus retreat from 0 to 4 km had little effect on the modeled flow speed. 8of20

9 Figure 7. (a) Normalized observed speed for 2009 (blue) compared to corresponding estimates based on terminus position (green; equation (3)). Open circles show the response to the 2009 terminus position, but with the 2006 surface to illustrate the sensitivity to geometry. (b) Normalized observed speed (blue) compared to corresponding estimates (red) for Both plots are computed for a location corresponding to 11-km along the profiles shown in Figure 2. [31] Figure 7 shows the model results for a location several kilometers upstream from the terminus (11-km in Figure 2). For both years the terminus-driven model approximately reproduces the observed seasonal cycle of V/V o. There are two spurious points in the 2009 times series, which are discussed further below. The early stages of speedup in 2009 precede the model estimate by 11 days, indicating either quantization error (11-day sampling in 2009) or unmodeled backstress from the last 800 m of the ice tongue. Our model showed similarly good agreement for the early part of 2006, although observed speeds decline more slowly in the fall than predicted by the model. This difference largely occurs during the terminus advance from 5 to 4 km, and may result from other, uncharacterized processes associated with seasonal thinning as described below. [32] Figure 7 also illustrates the influence of thinning over time by showing the computed 2009 variation in speed using the 2006 geometry and the 2006 variation using the 2009 geometry (open circles). The results indicate the seasonal variation would have been far greater in 2009 if the glacier 9of20

10 had not thinned substantially from 2006 to Conversely, there would have been almost no seasonal variation in 2006 if the surface elevation had been similar to that of [33] The absolute value of F used in equation (1) is subject to errors in the assumed ice-column density, which is far less certain than seawater density. We assumed a uniform density (r i = 910 kg m 3 ) throughout the column [Lüthi et al., 2002], which does not account for the reduction in nearsurface density due to the heavy surface crevassing. In our calculation this mainly affects the value of F o so that the seasonal variation in F s about this value is relatively insensitive to the assumed density of ice. Thus, a better density estimate would change our value of F o, but this would have little effect on the estimated seasonal velocity variation. [34] An important assumption used in the terminus-driven model is that the spring surface remains constant throughout the year, even though Figure 3 shows that there are nonnegligible seasonal fluctuations in elevation. The Figure 2a inset, however, indicates that most of the region over which the terminus varied in 2009 was near flotation that spring and likely did not vary substantially in elevation. Substantial thinning may have occurred later in the summer as the terminus reached locations well above flotation (Figure 2a). This corresponds to the region where our model produced two estimates substantially larger than the observed speeds. Thus, by not accounting for summer thinning, our model may have overestimated the speedup in this region. Nonetheless, through the remainder of the summer the model provides relatively good agreement with observations Inverse Results for Basal Shear Stress [35] Our terminus-driven model was based on the assumption of a linearly varying longitudinal stress distribution upstream of the terminus, which likely represents an oversimplification. To further examine this assumption, we performed several inversions using the numerical model to determine the basal shear stress, t b, beneath Jakobshavn Isbrae. Figure 8a shows the speed used to constrain the 2009 inversions and a similar map (not shown) was used for the 1990s inversions. Figure 8b shows the driving stress, t d, for the 1990s, indicating areas with t d > 400 kpa consistent with earlier findings [Clarke and Echelmeyer, 1996]. In general, the substantial thinning from the 1990s to 2009 increased t d in areas upstream, while reducing it near the terminus (Figure 8c). [36] Ice rheology plays an important role in determining the influence of membrane stresses through the parameter B in the effective viscosity (see equation (5)). In general softer ice (smaller B) reduces the effect of membrane stresses so that t b supports a larger fraction of t d locally. In the limiting case as B goes to zero, the complete lack of membrane stresses would yield a fully local force balance (t d = t b ) as in the Shallow Ice Approximation. To examine the sensitivity of our estimates of t b to ice rheology, we performed inversions with several different values of B. The first of these was our reference model with a mean value of B ¼ 9: Pa s 1=3 over a domain that has negligible spatial variation of this parameter. This model is equivalent to an isothermal ice column with a temperature of 3.5 C using the flow law temperature dependence given by Cuffey and Paterson [2010]. For comparison, at a point about 10-km upstream from the area shown in Figure 1, measured borehole temperatures [Lüthi et al., 2002] provide an estimate of B equivalent to 15 C isothermal ice [Cuffey and Paterson, 2010], indicating our reference model has some enhancement (E = 6) relative to the borehole. We also used a soft model where our reference model B was halved, such that E = 8 relative to the reference model, or equivalently, E = 48 relative to the value derived from borehole temperatures [Lüthi et al., 2002]. In addition, we used a hard model where the reference B was doubled (E = 1/8). Finally, we produced a mixed model where we used the soft model B value along the margins and the hard model value in the central trunk. [37] We performed our first two inversions using the reference model applied to the 1990s and 2009 data (Figures 8d and 8e). The results for both periods show a weak (<40 kpa) bed extending along much of the main channel. While the patterns are similar, the weak bed is more prominent in the 2009 estimate and covers a broader area, largely by extending outward toward the margins relative to the 1990s result. Force balance estimates by van der Veen et al. [2011] suggest a considerably stronger bed while using a value of B ( 9: Pa s 1=3 ) similar to that used in our reference model. Their force balance results were averaged across the trough, and thus, may include both the weak center and strong margin shown in Figure 8. [38] Inverting the 2009 data with the softer model (Figure 8f) reduced the influence of membrane stresses, leading to a stronger bed estimate relative to the reference model. Nonetheless, even with this large degree of enhancement, the membrane stresses distribute much of the main channel s high driving stress to the margins or to isolated strong spots within the channel, so that areas of weak bed are present. Using the hard model expands the area where the bed is estimated to be weak (Figure 8g), producing a uniformly weak bed estimate along the main channel. The mixed model (weak margins/ strong trunk) also yields a weak bed over much of the channel, yielding a result similar in character to the strong-model result. [39] The 2009 models all estimate speeds with root mean square differences between the model and the observations of about 145 to 165 m yr 1, with most of the discrepancy confined to the regions of fast flow. Only the hard model is substantially different from the others, with larger modeldata differences because the stiffer margins resulted in slower speeds near the terminus where sufficient shear could not be generated across the margins. In the mixed model adding the weaker margins while retaining the strong core produce a residual misfit error that was not significantly different from that for the soft or reference models. Thus, the inversion results are insensitive to E and provide little information about the true ice rheology, except for largely eliminating the hard model for the fastest parts of the ice stream Forward Model Experiments [40] The process of inverting for t b yields a tuned forward model that we perturbed to further investigate the glacier s response to various forcings. We conducted two sets of experiments, with the first set designed to examine the role of terminus position on the seasonal flow variation, and the second set aimed at understanding the factors contributing to the longer-term speedup. 10 of 20

11 Figure 8. (a) Winter flow speed for 2009 as determined using TerraSAR-X data overplotted with unlabeled colored dots corresponding to markers M6-M20 shown in Figure 1. (b) Driving stress for the 1990s and (c) change in driving stress from the 1990s to 2009 computed using elevation and thickness data sets. Inversions for basal shear stress determined using the reference flow model for (d) the 1990s and (e) Similar inversions computed for 2009 using the differing rheologies of the (f) soft, (g) hard, and (h) mixed models (see text for description) Response to Seasonal Terminus Forcing Experiments [41] While the data-model misfits from the inversion results cannot distinguish between differing flow law parameterizations, the response to seasonal changes in terminus position provides additional information. In our terminusdriven model, we did not explicitly include the rheology, and instead, we empirically determined the form and distance over which stresses act. Here, we use the forward model to explicitly determine which rheological model is consistent with these assumptions and to examine the roles of other processes in contributing to seasonal flow variability. [42] Ideally, we would apply a dynamic (i.e., stress) boundary condition to our model that accounts for the forces associated with the changes in the terminus position. At present, however, our model is implemented to allow only a 11 of 20

12 dynamic boundary condition in the case of a floating terminus. Thus, we cannot apply this boundary condition to Jakobshavn Isbrae, where the terminus is often grounded above flotation. This deficiency provided much of the motivation for our development of the simple terminusdriven model. [43] Instead of applying a dynamic boundary condition at the terminus, we use kinematic boundary conditions based on the observed velocity. By definition, we impose the correct speed on the boundary, but this speed could result from changes that are either external or internal to the model domain. If there are no changes within the domain (e.g., all domain-internal parameters remain fixed), then changes in the kinematic boundary represent only forcing external to the model domain (e.g., terminus change). On the other hand, a change in speed at the terminus (boundary) could result entirely from changes internal to the domain (e.g., change in basal lubrication). [44] Since our terminus-driven model assumes all speedup is terminus driven, modeled changes in speed are in phase with the forcing along the entire profile. The data in Figure 5 clearly show that this not always the case. For example, the terminus began speeding up in May 2009 but the area upstream did not begin to speed up until late June, indicating much of this inland speedup is not directly in response to changes in force balance near the terminus. Instead this may represent an indirect response involving multiple processes as the glacier geometry subsequently evolves in response to changes at the terminus. [45] To investigate the delayed inland response, we forced the model with the June 9 20 terminus speed, which was the period just after the terminus retreated and sped up in 2009 but just before the upstream area sped up. Figure 9a shows the results of this experiment for the different ice rheologies. The reference model (red) produced good agreement with the observed speedup (solid black), yielding only small model-data differences (dashed black). The mixed-model response extended inland slightly farther, but still produced good agreement with the data. In contrast, the soft model under-predicted the inland extent of the speedup, while the hard model overestimated the upstream limit. From this experiment, we conclude that, despite considerable uncertainty, the reference and mixed models provide the closest approximation to the true ice rheology of the four models evaluated. [46] Next we conducted several experiments in which we forced the model by scaling the kinematic boundary condition at the modeled terminus to match the 2009 speeds at the observed terminus. Note that we used a fixed model domain that extended beyond the actual terminus for some times. In each of these experiments, we increased the model terminus speed to a level that matched the speed at the observed terminus. Figure 9b shows the result for July 12 23, when the glacier was near its peak summer speed. In this case, the modeled response provided a reasonable fit (within several percent) over the first several kilometers above the terminus, but accounted for less than about half of the speedup inland of 16-km. For September the terminus forcing was similar in magnitude to the June 9 20 period, but the observed upstream response is much larger than the modeled response. The final experiment (November 21 to December 2) shows that speeds remain slightly elevated (3%) in the upstream regions despite little (1%) terminus forcing relative to the winter reference Decadal Scale Forcing Experiments [47] Figure 2 shows that since the speedup began in the late 1990s [Luckman and Murray, 2005] much of the glacier s downstream region has thinned by more than 100 m so that the value of F at the present terminus is too small to directly explain the elevated speed. Furthermore, the direct response to terminus variation (e.g., Figure 9) does not extend far enough inland to account for the observed 13-year speedup, which is still nearly twofold at 28 km inland (Figure 6a). Thus, while the loss of the ice tongue likely initiated the speedup, other processes almost certainly have contributed to and sustained the speed up, which also is consistent with similar conclusions based on flow-band model experiments [Vieli and Nick, 2011]. We used the forward model to evaluate the relative contributions of such processes. To do this, we began with the basal shear-stress distribution determined from the inversion of the 1990s data with our reference rheology (e.g., Figure 8d). Figure 10a compares the observed speeds (black) and the model results from the inversions (magenta) along our standard profile. The goal of these experiments was to perturb the 1990s forward model to determine the changes in geometry and bed conditions necessary to match the 2009 speeds. [48] As a first step, we applied the 2009 kinematic boundary conditions to the entire perimeter of the 1990s model to obtain the result shown in Figure 10a (blue). The kinematic boundary condition imposes a speedup near the boundaries, but introduces little change near the profile center. The boundary condition alone accounts for about 26% of the observed mean change in speed from the 1990s, with speedup located mostly in the near-terminus regions. Since no changes were applied to the domain interior, this emulates an external forcing (e.g., ice shelf loss). In actuality, however, the observed changes on the boundary may have resulted from changes in the interior of the model domain rather than external forcing. While it is difficult to separate internal and external influences, the speedup imposed by the boundary condition does represent an upper limit on the direct response from the loss of the ice tongue due to external forcing. Thus, while external forcing could account for much of the near-terminus speedup, the change in force balance due to loss of the ice tongue likely cannot directly account for most of the speedup farther inland. Instead, loss of the ice tongue may have triggered the speedup through other indirect processes. [49] With thinning of more than 100 m over much of the area included in the model domain, the driving stress changed substantially over the long-term progression of the speedup. To evaluate this effect, we replaced the model s 1990s surface and thickness with that of As Figure 10a indicates (red), the evolution of the surface and resulting change in t d accounts for much (30% average increase relative to the blue curve in Figure 10) of the observed speedup, particularly in the upstream regions, consistent with earlier inferences [Joughin et al., 2008b]. [50] While about half of the speedup on the main trunk likely was due to the combined loss of the ice tongue and the subsequent steepening of the surface, change in t b is a likely cause for much of the remaining speedup, consistent with 12 of 20

13 Figure 9. Comparison of modeled (color) and observed speedup in 2009 (solid black) for (a) June 9 20, (b) July 12 13, (c) September 16 27, and (d) November 21 December 2. These four periods are indicated with solid circles in Figure of 20

14 Figure 10. (a) Observed speed (black curves) and forward model results (color) for standard profile. Magenta curves show the original inversion results V inv (1990s) and V inv (2009). The 1990s forward model was adjusted successively by adding to the 2009 boundary condition (blue), changing the 2009 surface geometry (red). On top of these changes we also ran the model with the two water pressure distributions (green), P1 and P2. The resulting scaling, N , N 1990 are shown for (b) P1 and (c) P2. the inversion indicating a weaker bed in As described above, if water pressure remains relatively constant, thinning will reduce the overburden relative to the water pressure to lower the effective pressure and increase sliding. We assumed sliding speed is inversely proportional to the effective pressure, N [e.g., Cuffey and Paterson, 2010]. As a result, for our m = 3 sliding law, we scaled the drag coefficient to account for a change in effective pressure such that. 3b b ¼ N ð7þ N 1990s 1990s : The actual water-pressure distribution is not known, so we postulated two pressure distributions to examine the model s sensitivity to thinning-induced changes in effective-pressure. First, following Pfeffer [2007], we assumed that basal water 14 of 20 was equal to equivalent oceanic pressure (i.e., r w gz b ) for the areas where the bed lies below sea level (P 1 ). Since most of the fast moving ice lies in the deep trough, we assumed no change in the areas above sea level. Using this pressure model, we used the ice thickness to compute effective pressures needed for the scaling given by equation (7) (Figure 10b). For the second pressure model (P 2 ), we set the water pressure to 72% of the 1990s overburden pressure, which yielded the scaling shown in Figure 10c. In this model, percentage of the overburden was selected to provide an approximate match to the observations. [51] With the 2009 surface and our P 1 distribution, the forward model produced a closer match to the 2009 data, but still underestimated the speed in the middle region of the profile. The speedup in this case (green-dashed line Figure 10) was relatively modest, accounting for about 20%

15 of the total. This was because the weakening was concentrated over a region with negligible shear strength near the terminus. [52] Using the second water-pressure model, the forward model produced a reasonable match along the entire length of the profile. In this case, the speedup was substantially greater because the weakening was concentrated nearer the margins where the bed is strong. Since we fixed the water pressure to be a fraction of the ice thickness, the greatest weakening occurred near the margins where the thinning as a fraction of the ice thickness is the greatest. While this model works well along the centerline, it tends to widen the fast-flowing trunk more than is observed because the bed is weakened over a relatively broad area near the margins. 5. Discussion [53] The data and model results described above reveal many aspects of the variation in flow on Jakobshavn Isbrae, which we discuss in further detail below Basal Conditions and Ice Rheology [54] Several investigations have attributed much of the fast flow of Jakobshavn Isbrae to vertical shearing of a weak temperate layer [Lüthi et al., 2002; Funk et al., 1994; Iken et al., 1993]. Although boreholes and models do indicate that such a warm layer exists, vertical shearing will contribute substantially to the surface motion only if the bed supports much of the main trunk s high (>300 kpa) driving stress. Thomas [2004] noted that the half-width of the glacier trough is comparable to the depth so that after accounting for the margins, this author estimated a basal shear stress of 50 KPa, which would allow vertical shearing to produce only a small contribution to the overall motion. Consistent with this earlier work, all of our inversions suggest the bed is weak over much of the trunk, which also agrees with recent independent indications derived from airborne gravity surveys of weak sediments in the main trough [Block and Bell, 2011]. It is important also to note that the temperate layer was determined at boreholes 10-km beyond the upstream limit of our model domain, so at that location our results do not rule out a much stronger bed with substantial vertical shear, as suggested by earlier analysis [Lüthi et al., 2002; Funk et al., 1994; Iken et al., 1993; Lüthi et al., 2003; van der Veen et al., 2011]. [55] Clarke and Echelmeyer [1996] analyzed reflection coefficients from near the margins of Jakobshavn Isbrae and concluded that the bed consists of lodged tills or compacted sediments. Their data, however, were acquired well upstream of where we find the weakest bed. In addition, their results also were near the shear margins, where the inversions indicate a relatively strong bed. Thus, these seismic results do not contradict our finding of large areas of weak bed in the downstream regions of the main trunk. [56] Our terminus forcing experiments (Figure 9a) suggest that our reference and mixed models both provide reasonable matches to the observations. There are several reasons that may explain why the mixed model works well. First, it is important to note that our model assumes isotropic ice. If shearing at the margins produces a crystal fabric favorable to lateral shear, then ice with this fabric should be strong with respect to longitudinal tension [Cuffey and Paterson, 2010]. We have not attempted to include anisotropic behavior in our numerical model, but our mixed model provides a crude approximation to such anisotropy with a soft B in the margins and a hard B in the central trunk. Thus, while softer margins would tend to diminish the lateral drag, the resulting fabric would enhance the longitudinal stresses, which would determine the extent to which near-terminus changes directly influence upstream flow. Thus, anisotropy may play an important, but rarely simulated, role in governing the stress distribution in narrow outlet glacier troughs. [57] Lateral shear heating could produce isotropic softening of the ice at the margins [van der Veen et al., 2011; Vieli and Nick, 2011; Thomas, 2004]. Several kilometers upstream of the area shown in Figure 1, however, borehole measurements indicate little warming at the margins or in the main trunk [Lüthi et al., 2002; Iken et al., 1993]. The center borehole speed of 1000 m yr 1 is relatively slow, and the rate of shear heating is greater for the much faster flow farther downstream. At these faster locations, however, ice also should transit through the high shear regions relatively quickly, allowing little time for warming. Even if lateral shearing does warm and soften the downstream margins, much of the central area of the trunk would be cold ( 15 to 22 C) ice advected from the upstream region near the boreholes [Lüthi et al., 2002], yielding a rheological variation qualitatively similar to that of our mixed model. [58] As described above, a deficiency of the numeric model is that, while it includes the depth-averaged membrane stresses, it excludes vertical shearing and depth-varying stresses. Nonetheless this should not prohibit our model from determining a hard bed since high basal shear stress can account for the motion through sliding, rather than vertical shearing. A weak bed is determined because the influence of the depth-averaged membrane stresses is large. Thus, even a relatively large enhancement factor (e.g., soft model) relative to the estimated borehole values [Lüthi et al., 2002] yields t b (basal shear stress) much lower than t d (driving stress). In contrast to the middle of the trunk, where the weak bed estimates suggest relatively little vertical shear, the area where the model deficiencies are likely to be the greatest are where stresses are redistributed to the bed on the outboard sides of the shear margins [Truffer and Echelmeyer, 2003]. For such reasons, we have not discussed the details of the stress distribution near the margins. Thus, consistent with Thomas s [2004] earlier analysis and more recent gravity estimates, our inversions provide a qualitatively robust result indicating that much of the bed is weak (a few kpa to a few 10s of KPa) at the downstream end of the central trunk, suggesting a fairly widespread distribution of weak sediments. Even in the upstream regions of our model domain,t b remains comparatively small relative to t d, possibly indicative of heterogeneous distribution of hard bed and weak sediments Thinning and Effective-Pressure Variations [59] We assumed two simple fixed water-pressure distributions to evaluate sensitivity to thinning-induced changes in effective pressure, neither of which is likely to faithfully reproduce the true water-pressure distribution. With changes in thickness and slope altering the hydrologic potential driving water flow [Cuffey and Paterson, 2010], it also is likely that there are commensurate changes in the basal 15 of 20