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.

Ice streams are fast-moving rivers of ice embedded in the more sluggish-moving main body of ice sheets, and are responsible for the bulk of drainage from the interior in West Antarctica. Most ice streams start well upstream from the coast, some extending several hundreds of kilometers into the interior, and drain into floating ice shelves or ice tongues and are believed to represent the transition from inland-style “sheet flow” to ice-shelf spreading. The nature of this transition remains under debate, however.

In a long series of papers, T. Hughes presents the geometric approach to the balance of forces acting on ice shelves, ice streams, and interior ice (Hughes, 1986, 1992, 1998, 2003, 2009a, b, 2012; Hughes et al., 2011, 2016). Rather than working his way through the basic equations, as done by most other investigators, including Van der Veen and Whillans (1989) and Van der Veen (2013), he presents derivations based on graphical interpretation of triangles representing forces acting on an ice column. In essence, the transition in flow regime is achieved by introducing a basal buoyancy factor that describes the gradual ice–bed decoupling towards the grounding line.

The idea of basal buoyancy has been invoked many times before in glaciology, in particular in the context of formulating a sliding relation. In many models, the sliding speed is assumed to be inversely proportional to the “effective basal pressure” defined as the difference between the weight of the overlying ice and the pressure in the subglacial drainage system. Intuitively, this approach may seem to make sense: as the subglacial water pressure increases, the normal force on the bed should be reduced, thus allowing the glacier to move faster. However, this does not affect the balance of forces in the horizontal direction, as suggested by Hughes (2008, 2012).

The objective of this brief note is to evaluate the implications of Hughes' geometric approach to force balance by applying the results to Byrd Glacier, East Antarctica.

Analytic treatments of glacier force balance are numerous and derivations
of the depth-integrated force-balance equations are now standard fare in most
glaciology textbooks. In most cases, this balance of forces is discussed in
terms of stress deviators, defined as the full stress minus the hydrostatic
pressure. This is done because the flow law for glacier ice relates strain
rates to stress deviators. That is

It may be noted that the term “resistive” stress is an unfortunate choice,
perhaps, because these stresses do not necessarily always offer resistance to
flow. For example, gradients in longitudinal stress can act in cooperation
with the driving stress in pulling the ice forward. The more appropriate
terminology would perhaps be

Van der Veen (2013, Sect. 3.1) presents a derivation of the column-average
balance equations by integrating the momentum balance equations over the full
ice thickness. Van der Veen and Payne (2004) and Van der Veen (2013,
Sect. 3.2) present a discussion of force balance based on geometric arguments
and, not surprisingly, arrive at the same result. Without loss of generality,
flow in one horizontal direction may be considered. That is, the horizontal

The balance Eq. (2) is exact. No approximations are involved in deriving this expression from the basic equations describing the balance of forces on a segment of ice (Van der Veen and Whillans, 1989; Van der Veen, 2013, Sect. 3.1). Consequently, this equation applies to free-floating ice shelves where the gravitational driving stress is balanced entirely by gradients in longitudinal stress, yielding the classic Weertman (1957) solution (Van der Veen, 2013, Sect. 4.5), as well as laminar flow with basal drag providing sole resistance to flow (Van der Veen, 2013, Sect. 4.2). Except for these two end-member solutions, Eq. (2) does not permit analytic solutions without making additional assumptions. Nevertheless, because no approximations were made in its derivation, balance Eq. (2) applies equally well to transitory flow regimes such as ice streams and outlet glaciers.

Integrating the balance equation over the width of the flow band simplifies
the resistive term associated with drag at the lateral margins. Denoting the
lateral shear stress at the margins by

The geometric approach developed by Hughes arrives at a similar balance
equation, namely

Discussing force balance for stream flow, Hughes (2008, Sect. 11) equates

Hughes (2008) takes the geometric approach to another level and relates all
resistance to flow on ice streams to the basal buoyancy factor,

Balance of forces on Byrd Glacier, East Antarctica, was first discussed by Whillans et al. (1989), who used measurements of surface velocity and surface topography derived from repeat aerial photogrammetry, to evaluate the relative roles of lateral drag, gradients in longitudinal stress, and basal drag in resisting the gravitational driving stress. Van der Veen et al. (2014) reconsidered these calculations and also investigated the effect of drainage of two sub-glacial lakes in the catchment region. Both studies employed the analytic force-balance approach.

Reusch and Hughes (2003), Hughes (2009a), and Hughes et al. (2011, 2016)
discuss force balance on Byrd Glacier from the
geometric perspective and take
issue with the analytic approach of Whillans et al. (1989). None of these
studies explicitly shows how the various resistive forces vary along the
glacier and, instead, largely base their discussion on how the basal
buoyancy,

The geometry is shown in Fig. 1 (Van der Veen et al., 2014, Fig. 6). Only
the lower 30 km stretch upstream of the grounding line (at

The average driving stress is

Geometry of the lower part of Byrd Glacier, East Antarctica. The dashed line in the lower panel shows the buoyancy factor, calculated from Eq. (12).

Force-balance terms on Byrd Glacier according to geometric force balance, Eqs. (13)–(14).

To understand the limitation in the geometric approach to force balance, consider the forces along an ice stream flow line as discussed in Hughes (2008, p. 53 ff.) (see also Fig. 1 in Hughes (2003), and Hughes, 2012; Sect. 11). The geometry is shown in Fig. 3. While Hughes (2008, p. 53; 2012, p. 66) erroneously states that resistance from lateral drag vanishes at the centerline of an ice stream and therefore does not include this source of resistance in his discussion, this has no significant impact on the following discussion – lateral drag can be readily added to the basal drag term without altering the general tenets of the analysis.

Geometric force balance according to Hughes (2008).

According to Hughes (2008, 2012), the gravitational driving force at

The problem with this reasoning is that

It is not possible to relate resistive forces at any location to point values
such as basal water pressure or weight of the ice at location

While the geometric force-balance approach is severely limited, it is worth exploring the central premise of Hughes' ideas, namely that the transition from sheet flow to shelf flow is achieved through basal buoyancy, with interior ice firmly grounded on bedrock and ice shelves floating in sea water. It should be noted that for both these end-member solutions, at any location the weight of an ice column is fully supported from directly below: terra firma in the case of grounded ice, and sea water for ice shelves.

While not immediately obvious, the role of varying subglacial water pressure
is included in the force-balance Eq. (2), namely though bridging effects
(Van der Veen, 2013, Sect. 3.4). To clarify this, consider that resistive
stresses are linked to strain rates, or velocity gradients, by invoking
Glen's flow law for glacier ice (Van der Veen and Whillans, 1989; Van der
Veen, 2013, Sect. 3.3):

For brevity of notation, the along-flow resistive stress is written as the
sum of a contribution associated with along-flow gradients in velocity (first
term on the right-hand side of Eq. 17) and the vertical resistive stress:

Basal buoyancy may be important on ice streams and outlet glaciers according to the commonly adopted sliding relation in which sliding speed is inversely proportional to the effective basal pressure. Pfeffer (2007) suggests that this proportionality may explain rapid velocity increases on tidewater glaciers and Greenland outlet glaciers: as these glaciers thinned and thickness approached flotation, the effective basal pressure approached zero, resulting in a large increase in sliding velocity. Another possibility is that increased basal buoyancy reduces basal drag, thereby allowing glaciers to move faster. The importance of these effects can be evaluated from analysis of time series of surface speed and glacier geometry, or using numerical models based on the balance Eq. (7).

The primary difference between shelf flow and stream flow is that ice shelves are floating in water and basal drag is zero, whereas for sheet flow, basal drag provides most resistance to flow. Thus, it would seem reasonable to propose that the transition from sheet to shelf flow involves a gradual reduction in basal resistance, perhaps associated with the presence within deforming sediments, or gradual drowning of bed obstacles. As basal drag becomes less important, longitudinal stress gradients and lateral drag must increase and provide most or all resistance to the flow of ice streams.

The geometric approach to ice sheet modeling links ice–bed coupling directly to the stresses that resist horizontal gravitational motion (Hughes, 2008, p. 34). This basal buoyancy supposedly translates into a major component of gravitational forcing by which ice sheets discharge ice into the sea (Hughes, 2003). As shown in this contribution, the geometric force balance as presented by Hughes in a series of publications cannot be successfully applied to ice streams and outlet glaciers. This is not to say that a geometric approach is inherently flawed – if implemented correctly it should produce consistent and correct results, but this has yet to be achieved.

The claim that the analytic force-budget approach fails to account for basal buoyancy and excludes a “water buttressing force” on ice streams is incorrect. Equation (7) describing the depth-integrated balance of horizontal forces is derived without making any simplifying assumptions and applies equally well to floating ice shelves and firmly grounded interior ice. If some force is missing from this equation, this force must also be missing from the momentum balance equations that form the starting point for deriving Eq. (7).

Hughes is correct that ice streams and outlet glaciers represent the transition from sheet flow and shelf flow and that much remains to be understood about the nature of this transition. Advantageously, ongoing rapid changes on many of the outlet glaciers have been well documented through time series of surface elevation and surface velocity. The latter, in particular, are powerful indicators of the distribution of stresses on glaciers because strain rates (velocity gradients) are directly linked to stresses through the flow law for glacier ice. Improved understanding of the dynamics of rapidly changing ice-sheet components will come from interpretation of strain rates and temporal changes therein.

I am indebted to Ken Jezek for his continued support and careful reading of this paper, and to Leigh Stearns for additional comments. This research was supported by NASA grant nos. FED0066542 and UNI0072622.Edited by: F. Pattyn