Department of Geography University of Edinburgh Edinburgh EH8 9XP UK

This case study is concerned with the application of a three-dimensional ice flow model to Haut Glacier d'Arolla, a small temperate glacier in the Swiss Alps. The study illustrates both the potential and certain limitations of using high-resolution, three-dimensional flow modelling constrained by observations to aid our understanding of alpine glacier systems. In particular, the combination of flow modelling with detailed field measurements at Haut Glacier d'Arolla provides a powerful tool to investigate its rheological properties, the interaction between basal hydrology and ice dynamics, the occurrence and formation of surface structures, its past and present response and, ultimately, its future trajectory to climate change.

The model is based on Blatter's (1995) solution of the mass and force balance equations using a non-linear rheology (with the flow law exponent: n = 3) applied to the surface and bed topography of Haut Glacier d'Arolla at 70 m horizontal resolution with 40 vertical layers scaled to thickness. The model is steady-state; it computes the instantaneous stress and strain distribution based on Glen's (1958a) flow law:

Tij where eij is the strain rate tensor, A is the rate factor reflecting ice hardness, t¡¡ is the imposed stress tensor and Te is the effective stress given by

The model computes first-order terms, that is longitudinal and transverse deviatoric stresses (t' xx and t' yy) which result in compression and tension throughout the ice mass and which, in extreme circumstances, lead to observed structural failure such as overthrusting, shearing and crevassing. The resulting solution is highly dependent on the basal boundary condition, which can be specified as either a velocity or drag distribution or a combination of both, which enables the spatial interaction of slip/stick patchiness resulting from heterogeneity in the bulk subglacial properties, such as roughness, hydrology, effective pressure and sediment strength, to be modelled. This is achieved through the prescription of zero or reduced basal shear traction to replicate decoupled, low drag zones, whereas zero velocity or increased shear traction can be prescribed to simulate 'sticky' zones. For the purposes of initial model optimization though, the model is first specified with zero sliding across the whole of the bed.

By holding the flow-law exponent (n) constant then the only parameter that requires 'tuning' is the rate factor (A) related to ice viscosity. Assuming Haut Glacier d'Arolla to be temperate with negligible basal motion during winter, then comparison of modelled with observed winter surface velocities provides an effective means for calibrating A. Observations that winter surface velocities are consistently low and, furthermore, that there is virtually no supraglacial melt-water available to drive ice-bed separation, lend tentative support for this assumption. Tuning of the model against surface velocities measured over 10 days in January 1995 yields an optimum value of A = 0.063 yr-1 bar-3, corresponding to an R2 = 0.74 on a bivariate plot of modelled against measured surface velocities (Fig. 69.1). Although this value is half that expected (Paterson, 1994) it does lie within a narrow range of 0.07 ± 0.01yr-1bar-3 reported for other temperate glaciers modelled using higher order solutions (e.g. Gudmundsson, 1999). Such consistency lends confidence in the predictive quality of the internal strain component of these models but also suggests that they may be applied to temperate ice masses without significant tinkering with the rate factor.

Computation of the full stress and strain field enables model comparison with additional observables such as the occurrence and orientation of crevassing and measured principal strains. Comparison of zones of maximum computed surface tensile strain and their direction with the actual distribution and orientation of crevassing reveals a good general correspondence (Fig. 69.2a & b). Furthermore, the orientation of modelled principal strain faithfully reproduces the pattern measured from repeat survey of a dense network of strain diamonds from 1994 to 1995 (Fig. 69.3a & b). However, it is apparent that even though their

Figure 69.1 Modelled horizontal surface velocity across Haut Glacier d'Arolla at 70 m resolution for n = 3 and A = 0.063 yr-1 bar-3. The contours are plotted at 2.5myr-1 intervals. Overlain within circles are the winter 1995 velocity vectors and a bivariate analysis of modelled against observed velocities.

Figure 69.1 Modelled horizontal surface velocity across Haut Glacier d'Arolla at 70 m resolution for n = 3 and A = 0.063 yr-1 bar-3. The contours are plotted at 2.5myr-1 intervals. Overlain within circles are the winter 1995 velocity vectors and a bivariate analysis of modelled against observed velocities.

Figure 69.2 (a) The magnitude and orientation of modelled surface-parallel principal stresses at 140 m resolution; inward arrows indicate compression, outward arrows indicate extension. The shaded areas represent zones of maximum surface stress and indicate areas of potential ice failure. (b) The distribution of surface crevasses across Haut Glacier d'Arolla as observed from aerial photography and ground mapping.

Figure 69.2 (a) The magnitude and orientation of modelled surface-parallel principal stresses at 140 m resolution; inward arrows indicate compression, outward arrows indicate extension. The shaded areas represent zones of maximum surface stress and indicate areas of potential ice failure. (b) The distribution of surface crevasses across Haut Glacier d'Arolla as observed from aerial photography and ground mapping.

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Figure 69.3 (a) The relative magnitude and orientation of surface-parallel modelled and (b) measurement derived principal stresses and strains, respectively, in the region of the high-density strain network indicated in (a).

Figure 69.4 (a) Distribution of measured annually averaged (August 1995 to August 1996), horizontal velocity within a half cross-section at Northing 91700. (b) The modelled composite velocity distribution within the glacier cross-section composed of time-weight averages of 20/52 'winter' no sliding, 31/52 'normal summer' sliding and 1/52 enhanced 'spring event' sliding.

Figure 69.4 (a) Distribution of measured annually averaged (August 1995 to August 1996), horizontal velocity within a half cross-section at Northing 91700. (b) The modelled composite velocity distribution within the glacier cross-section composed of time-weight averages of 20/52 'winter' no sliding, 31/52 'normal summer' sliding and 1/52 enhanced 'spring event' sliding.

orientations correspond, their absolute magnitudes do show some discrepancy away from the centre-line, probably reflecting the effect of large lateral moraines on ice-fabric heterogeneity, combined with the fact that there is a significant component of basal motion influencing the annual flow regime. Both of these effects will significantly alter the relative magnitude of principal strains, but not their orientation.

The model may also be applied to contrasting basal boundary conditions in order to investigate an anomalous pattern of internal strain measured by repeat inclinometry of boreholes between

1995 and 1996 (Harbor et al., 1997) (Fig. 69.4a). The distinctive feature of this cross-sectional strain distribution is the 80-m-wide zone of rapid sliding, low strain 'plug-flow' which coincides with a subglacial pathway predicted on the basis of hydraulic potential analysis (Sharp et al., 1993) and characterized by highly variable water-pressures (often exceeding overburden) during the summer melt-season (Hubbard et al., 1995). To simulate this anomalous velocity distribution, annual flow is modelled as a time-weighted composite of three scenarios reflecting observed seasonal contrasts: (i) non-sliding during winter, (ii) 'normal sliding' during

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Figure 69.5 Comparison of modelled and observed annual surface velocity distribution from 1995 to 1996 overlain on the subglacial channel network reconstructed from hydraulic potential analysis (Sharp et al., 1993).

Figure 69.5 Comparison of modelled and observed annual surface velocity distribution from 1995 to 1996 overlain on the subglacial channel network reconstructed from hydraulic potential analysis (Sharp et al., 1993).

the summer melt-season and (iii) 'enhanced' sliding during a week-long spring-event, when observed surface velocities across much of the glacier were an order of magnitude higher than the annual mean. The pattern and magnitude of the modelled basal perturbation are defined with respect to the hydraulic potential analysis. The basal area covered by the main subglacial pathways was decoupled with zero-traction for the 'normal summer' scenario, and an extended area including and adjacent to the two main pathways was decoupled to match the pattern of the measured surface velocity anomaly during the spring-event. Application of these generalized basal scenarios provides a first step towards realistic modelling of melt-season dynamics and enables the reproduction of the key features of both the annual surface velocity distribution (Fig. 69.5) as well as that of the two-dimensional cross-section (Fig. 69.4b). Given the limitations imposed by the 70-m operating resolution, the modelled and observed annual velocity distributions compare well (R2 = 0.83) and confirm that variations in basal decoupling have a profound impact on the dynamics of Haut Glacier d'Arolla. These variations, however, are spatially and temporally complex, principally reflecting patterns of change in the glacier's basal drainage system.

Finally, in order to investigate the potential long-term response of Haut Glacier d'Arolla to climate change, this steady-state flow model can be coupled to a time-dependent model to yield the evolution of glacier thickness (H) through time (t) determined by the continuity equation:

— = b-V(HÜ ) dt where b is the net mass balance and V -(Hu) is the two-dimensional flux divergence operator. Using mass-balance

Figure 69.6 Modelled temporal-evolution of Haut Glacier d'Arolla surface centre-line in 10yr intervals from the mapped 1880 (maximum) extent through to the glacier's predicted demise in 2070. Observed 1940 and 1992 long-profiles are overlain for reference.

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Figure 69.6 Modelled temporal-evolution of Haut Glacier d'Arolla surface centre-line in 10yr intervals from the mapped 1880 (maximum) extent through to the glacier's predicted demise in 2070. Observed 1940 and 1992 long-profiles are overlain for reference.

measurements from 1989 to 1996 as a benchmark, the model is forced to match the maximum historical position mapped by the Swiss Survey in 1880. The model is then integrated forward in time to correspond to known positions in 1920, 1940 and 1992, and, assuming a constant 1989 to 1996 mass-balance distribution, is used to predict the evolving glacier geometry through to 2100 (Fig. 69.6). This modelling reveals that Haut Glacier d'Arolla has been undergoing accelerating retreat since the Little Ice Age and even under current conditions will not survive the next 70 yr. Such a response is no surprise given that the mean 1989 to 1996 equilibrium line altitude (ELA) was over 3000ma.s.l., reducing the effective accumulation area to ca. 15% of that required to maintain the present geometry.

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