Basal buoyancy and fast moving glaciers : in defense of 1 analytic force balance 2

Abstract. The geometric approach to force balance advocated by T. Hughes in a series of publications has challenged the analytic approach by implying that the latter does not adequately account for basal buoyancy on ice streams, thereby neglecting the contribution to the gravitational driving force associated with this basal buoyancy. Application of the geometric approach to Byrd Glacier, Antarctica, yields physically unrealistic results, and it is argued that this is because of a key limiting assumption in the geometric approach. A more traditional analytic treatment of force balance shows that basal buoyancy does not affect the balance of forces on ice streams, except locally perhaps, through bridging effects.


Introduction 15
Ice streams are fast-moving rivers of ice embedded in the more sluggish-moving main body of ice sheets, 16 and are responsible for the bulk of drainage from the interior in West Antarctica. Most ice streams start 17 well upstream from the coast, some extending several hundreds of km into the interior, and drain into 18 floating ice shelves or ice tongues and are believed to represent the transition from inland-style "sheet 19 flow" to ice-shelf spreading. The nature of this transition remains under debate, however. 20 In a long series of papers, T. Hughes presents the geometric approach to the balance of forces acting on ice 21 shelves, ice streams, and interior ice [Hughes, 1986[Hughes, , 1992[Hughes, , 1998[Hughes, , 2003[Hughes, , 2009a[Hughes, , 2009b[Hughes, , 2012Hughes et al., 22 2011Hughes et al., 22 , 2016. Rather than working his way through the basic equations, as done by most other 23 investigators, including Van der Veen and Whillans [1989] and Van der Veen [2013], he presents 24 derivations based on graphical interpretation of triangles representing forces acting on an ice column. In 25 essence, the transition in flow regime is achieved by introducing a basal buoyancy factor that describes the 26 gradual ice-bed decoupling towards the grounding line. 27 The idea of basal buoyancy has been invoked many times before in glaciology, in particular in the context 28 of formulating a sliding relation. In many models, the sliding speed is assumed to be inversely proportional 29 to the "effective basal pressure" defined as the difference between the weight of the overlying ice and the 30 pressure in the subglacial drainage system. Intuitively, this approach may seem to make sense: as the 31 subglacial water pressure increases, the normal force on the bed should be reduced, thus allowing the 32 glacier to move faster. An analogy may be drawn with pushing a shopping cart across a sandy beach: the 33 less groceries are in the cart, the easier it is to push the cart forward. The difference is, of course, that the 34 weight of the groceries is pre-determined (by the ice thickness), so the only way to facilitate the forward 35 motion is through some force acting to lift the cart upward. Hughes [2008,2012] suggests that basal 36 decoupling provides this upward force. 37 The objective of this brief note is to evaluate the implications of Hughes' geometric approach to force 38 balance by applying the results to Byrd Glacier, East Antarctica. 39

Force balance: analytical approach 40
Analytical treatments of glacier force balance are numerous and derivations of the depth-integrated force-41 balance equations are now standard fare in most glaciology textbooks. In most cases, this balance of forces 42 is discussed in terms of stress deviators, defined as the full stress minus the hydrostatic pressure. This is 43 done because the flow law for glacier ice relates strain rates to stress deviators. That is 44 where the prime denotes the stress deviator and unprimed stresses are full stresses, and ij δ = 1 for i = j and 46 ij δ = 0 for i ≠ j. Deviatoric stresses are called for in the flow law for glacier ice because the rate of 47 deformation is in good approximation independent of the hydrostatic pressure. However, the use of 48 deviatoric stresses in discussing the balance of forces unnecessarily complicates the interpretation because 49 the longitudinal deviatoric stress in one direction depends on the full normal stresses in all three directions 50 of a Cartesian coordinate system. It is more convenient to consider stresses in a glacier as the sum of the 51 stress due to the weight of the ice (lithostatic stress) and stresses, ij R , due to the flow (resistive stresses).

52
This partitioning makes a clearer distinction between action and reaction in glacier dynamics [Whillans, 53 1987] and follows common practice in geophysics [Engelder, 1993, p In this expression, dx τ denotes the gravitational driving stress, defined as where ρ represents the density of ice, g the gravitational acceleration, H the ice thickness, and h the 75 elevation of the upper ice surface. The terms on the right-hand side of equation (2) represent the resistance 76 to flow associated with, respectively, drag at the glacier base, gradients in longitudinal stress ("pulling 77 power") and lateral drag arising from shear between the faster-moving ice stream and the near-stagnant 78 where ij σ represents the full stress, and -ρg(h -z) the lithostatic stress (weight of the ice above) at depth z.

83
The balance equation (2) In these balance equations, F σ is related to the deviatoric tensile stress; its exact interpretation has evolved 109 over the years. To avoid unnecessary confusion, a consistent notation is used in the following discussion, 110 based on Hughes [2008,2012]. Comparison of equations (7) and (9) shows that F xx R σ = ɶ . It is the way 111 this stress is calculated that sets Hughes' geometric approach apart from the analytical approach. In 112 essence, this stress is linked to basal buoyancy and, in later versions, downglacier-integrated resistance 113 from basal and lateral drag. While the force balance equation (7) 139 while the longitudinal stress gradient term is given by 140 The achievement here is that these equations are derived without consideration of ice velocity or physical 142 properties of the ice (temperature, stiffness, fabric development, etc.), or, for that matter, basal water 143 availability and balance. Presumably, all these factors are somehow reflected in the ice-stream geometry 144 and the inferred basal buoyancy. The geometry is shown in Figure 1 [Van der Veen et al., 2014, fig. 6]. Only the lower 30 km stretch 159 upstream of the grounding line (at x = -10 km) is considered here because that is the region laterally 160 bounded by near-parallel ford walls. Also shown in Figure 1 is the basal buoyancy factor calculated from 161 eq. (12); φ increases from around 0.7 a little more than 30 km upstream of the grounding line, to 1 where 162 the ice starts to float. While there is nothing in particular wrong or disturbing about this basal buoyancy 163 factor, the situation becomes more problematic when the actual forces are considered. 164 The average driving stress is ~160 kPa, but shows large spatial variations that appear to be temporally fixed 165 x, it would be more appropriate to refer to these quantities as areal averages. If the surface slope is 215 calculated over a distance 2∆x, the associated driving stress is the average over the interval (x -∆x, x + 216 ∆x), and similarly for basal drag. Nuancing common parlance to reflect this subtlety would render many 217 discussions of glacier dynamics unnecessarily cumbersome and should be superfluous for most readers 218 understanding the fundamentals of glacier dynamics. 219

Discussion 220
While the geometric force balance approach is severely limited, it is worth exploring the central premise of 221 Hughes' ideas, namely that the transition from sheet flow to shelf flow is achieved through basal buoyancy, 222 with interior ice firmly grounded on bedrock and ice shelves floating in sea water. It should be noted that 223 for both these end member solutions, at any location the weight of an ice column is fully supported from 224 directly below: terra firma in the case of grounded ice, and sea water for ice shelves. 225 While not immediately obvious, the role of varying subglacial water pressure is included in the force-226 balance equation (7) Here, B represents the temperature-dependent rate factor, and n = 3 the flow-law exponent; e ε ɺ is the 232 effective strain rate defined as the second invariant of the strain-rate tensor. The last term on the right-hand 233 side of equation (17) is the vertical resistive stress defined as 234 zz zz For brevity of notation, the along-flow resistive stress is written as the sum of a contribution associated 236 with along-flow gradients in velocity (first term on the right-hand side of equation (17)) and the vertical 237 resistive stress: 238 Force-balance in the horizontal direction can then also be written as 240 Where the weight of the ice is fully supported by the substrate below, the vertical resistive stress is zero. 242 This is the assumption usually made when considering the budget of forces acting on glaciers [e.g. Van der 243 Veen and Whillans, 1989]. Locally, however, bridging effects may be important, for example where a 244 water-filled cavity exists at the ice-bed interface [Van der Veen, 2013, sect. 7.2]. Where cavitation occurs 245 and basal ice becomes separated from the bed, the cavity cannot support the weight of the ice leading to 246 shear-stress gradients that effectively transfer the weight to surrounding areas where the ice is in contact 247 with the bed, such that the areal average of the vertical resistive stress is zero. Thus, on a large scale, such 248 as the length of ice streams and outlet glaciers, basal buoyancy is a non-issue where horizontal force 249 balance is concerned. Indeed, Hughes [1998, eq. (3.5)] does not include bridging effects in his discussions 250 and equates the total vertical stress at depth to the lithostatic stress. 251 Basal buoyancy may be important on ice streams and outlet glaciers according to the commonly-adopted 252 sliding relation in which sliding speed is inversely proportional to the effective basal pressure. Pfeffer 253 [2007] suggests that this proportionality may explain rapid velocity increases on tidewater glaciers and 254 Greenland outlet glaciers: as these glaciers thinned and thickness approached flotation, the effective basal 255 pressure approached zero, resulting in a large increase in sliding velocity. Another possibility is that 256 increased basal buoyancy reduces basal drag, thereby allowing glaciers to move faster. The importance of 257 these effects can be evaluated from analysis of time series of surface speed and glacier geometry, or using 258 numerical models based on the balance equation (7). 259 The primary difference between shelf flow and stream flow is not that on ice shelves the ice weight is 260 supported by water and on grounded interior ice this weight is supported by the bed below. The main 261 difference is that, because ice shelves float in water, basal drag is zero and resistance to flow must be 262 partitioned between gradients in longitudinal stress and lateral drag, whereas for sheet flow, basal drag 263 provides most resistance to flow. Thus, it would seem reasonable to propose that the transition from sheet 264 to shelf flow involves a gradual reduction in basal resistance, perhaps associated with the presence within 265 longitudinal stress gradients and lateral drag must increase and provide most or all resistance to the flow of 267 ice streams. 268

Concluding remarks 269
The geometrical approach to ice sheet modeling links ice-bed coupling directly to the stresses that resist 270 horizontal gravitational motion [Hughes, 2008, p. 34]. This basal buoyancy supposedly translates into a 271 major component of gravitational forcing by which ice sheets discharge ice into the sea [Hughes, 2003]. 272 The concept as presented by Hughes in a series of publications spanning the last 30 years has yet to come 273 up with a solution that can be successfully applied to ice streams and outlet glaciers. This is not to say that 274 a geometric approach is inherently flawed -if implemented correctly it should produce consistent and 275 correct results but this has yet to be achieved. 276 The charge that the analytical force-budget approach fails to account for basal buoyancy and excludes a 277 "water buttressing force" on ice streams is incorrect. Equation (7) describing the depth-integrated balance 278 of horizontal forces is derived without making any simplifying assumptions and applies equally well to 279 floating ice shelves and firmly grounded interior ice. If some phantom force is missing from this equation, 280 this force must also be missing from the momentum balance equations that form the starting point for 281 deriving equation (7). 282 Hughes is correct that ice streams and outlet glaciers represent the transition from sheet flow and shelf flow 283 and that much remains to be understood about the nature of this transition. Advantageously, ongoing rapid 284 changes on many of the outlet glaciers have been well documented through time series of surface elevation 285 and surface velocity. The latter, in particular, are powerful indicators of the distribution of stresses on 286 glaciers because strain rates (velocity gradients) are directly linked to stresses through the flow law for 287 glacier ice. Improved understanding of the dynamics of rapidly-changing ice-sheet components will come 288 from interpretation of strain rates and temporal changes therein. 289