Numerical Modelling of the Antarctic Ice Sheet

Martin J. Siegert*

School of GeoSciences, University of Edinburgh, Grant Institute, West Mains Road, Edinburgh EH9 3JW, UK

ABSTRACT

Studies on Antarctic climate evolution have benefited increasingly over the last 20 years from numerical ice-sheet modelling. Such activities have led to the testing of geological hypotheses concerning past ice-sheet changes and a better understanding of macro-scale glaciological processes through space and time. Here, numerical ice-sheet models are reviewed, and their use in understanding former changes in Antarctica is outlined.

6.1. Introduction

Analyses of glacial geological data allow hypotheses concerning Antarctic Ice Sheet history to be formed. Testing these hypotheses can be undertaken in two ways. First, additional geological data can be acquired and, second, numerical ice-sheet modelling experiments can be performed. There are several advantages in taking the latter option over the former. Ice-sheet modelling allows time-dependent assessments of ice-sheet form and flow across the entire continent. Furthermore, model results can be analysed to determine where the ice sheet is most sensitive to changes and, in doing so, where the best sedimentary records of such changes may occur. Finally, the cost of undertaking numerical ice-sheet modelling is cheap relative to Antarctic data gathering. Hence, the use of ice-sheet modelling to test geologically based hypotheses has increased in recent years, and indeed is the

Corresponding author. Tel.: +44(0)131 650 7543; Fax: +44(0)131 668 3184; E-mail: [email protected] (M.J. Siegert).

focus of the Antarctic Climate Evolution programme. Ice-sheet models have limitations, however, concerning the necessary simplification of ice flow processes, and incomplete model input data (such as bed topography and climate forcing). Researchers need to comprehend these limitations when interpreting results. In this chapter, generic numerical ice-sheet modelling is reviewed, and ways in which Antarctic models may be validated are assessed. The chapter provides necessary background information to later chapters that discuss results from ice-sheet modelling.

6.2. Ice-Sheet Processes 6.2.1. Flow of Ice

Direct observations of glacier motion and laboratory studies of ice rheology (the flow and deformation of ice) have identified three main mechanisms by which ice masses move. These are by internal deformation of the ice itself, and through two processes which take place at the base of the glacier; basal sliding and the deformation of water-saturated weak sediments (Fig. 6.1). Flow by internal ice deformation takes place in all ice masses, and generally accounts for motion of a few metres per year. However, basal motion occurs only where the bed is at the pressure melting point such that water is present. Where basal motion takes place, glaciers may move at tens to hundreds, and sometimes a few thousand, metres per year.

The flow of ice sheets can be organized in general terms as follows. At the centre of an ice sheet, the flow speed is very low (of the order of metres per year), and controlled by internal deformation. A particle of ice on the ice-sheet surface will be buried by subsequent snowfall and so will possess a relatively significant vertical velocity component downwards into the ice. The flow of ice radiates from the ice divide, where there is no lateral flow. Ice-sheet interiors are characterized by a series of divides that define the margins of ice drainage basins. Ice sheets are effectively ''drained'' by fast-flowing rivers of ice, known as ice streams, transporting ice from the interior to the ice margin (Bennett, 2003). The velocity of ice streams is often several hundred metres per year. Ice streams flow quickly because water at their bases causes a reduction in subglacial friction allowing them to effectively slide across the subglacial topography, with ice deformation contributing only a small amount to the total velocity. The transition between the slow-moving ice sheet and the fast-flowing ice streams has been shown recently to occur in ''tributaries'' several hundred kilometres inland from the margin (Bamber et al., 2000).

(a) Warm-based glacier resting on bedrock

(b) Cold--based glacier resting on bedrock

(c) Warm-based glacier resting on deformable sediment

(a) Warm-based glacier resting on bedrock

(b) Cold--based glacier resting on bedrock

(c) Warm-based glacier resting on deformable sediment

Figure 6.1: Processes controlling the flow of large ice sheets. Ice will flow through internal deformation in all situations. If the ice rests on bedrock and is warm based (a) then basal sliding can occur. If the base is frozen (b) then only ice deformation will take place. If the base is warm and loose unconsolidated sediments are present (c) then their own deformation can add to ice deformation and basal sliding as a contribution to ice flow. Each of these processes can be accounted for in ice-sheet models. Adapted from

Siegert (2001).

Figure 6.1: Processes controlling the flow of large ice sheets. Ice will flow through internal deformation in all situations. If the ice rests on bedrock and is warm based (a) then basal sliding can occur. If the base is frozen (b) then only ice deformation will take place. If the base is warm and loose unconsolidated sediments are present (c) then their own deformation can add to ice deformation and basal sliding as a contribution to ice flow. Each of these processes can be accounted for in ice-sheet models. Adapted from

Siegert (2001).

6.2.2. Mass Balance

Antarctica gains mass through accumulation of snow in two ways. Direct precipitation of snow occurs close to the ice-sheet margin, where accumulation rates can reach ~1m per year. This is in contrast to accumulation rates in the interior, which are far lower (centimetres per year) and are associated with solid precipitation direct from water vapour. Broadly speaking, ice loss in Antarctica is through two main processes: iceberg calving and sub-ice-shelf melting (Jacobs et al., 1992). Consequently, ice-shelf processes are critical to the overall mass balance of the Antarctic Ice Sheet. On grounded ice, surface sublimation, which forms blue ice zones, and basal melting, responsible for at least 145 subglacial lakes (Siegert et al., 2005a), are not thought to contribute significantly to the overall ice-sheet mass balance.

6.2.3. Isostasy

When a glacial load is placed on the Earth's crust, the weight of ice will act to displace the crust into the viscous asthenosphere (beneath the lithosphere), which adjusts towards isostatic equilibrium according to Archimedes Principle, on time scales of a few thousand years. The process by which this is done involves flow in the asthenosphere and elastic deflection of the lithosphere. When the ice load is removed (deglaciation), the asthenosphere and lithosphere will relax back to their original state (i.e. isostatic uplift or recovery). There are several ways in which glacial isostasy can be accounted for in ice-sheet models, and a thorough description of the problem is provided in Le Meur and Huybrechts (1996).

6.3. Ice-Sheet Models

Numerical ice-sheet models have been used extensively since the mid-1980s to aid the reconstruction of former ice sheets, to comprehend existing ice sheets and to forecast future behaviour. The principle behind numerical icesheet modelling is that an ice sheet can be divided into a number of ''ice columns''. Each of these columns represents a ''cell'' in the model's 2D horizontal ''grid''. Ice-sheet models are usually arranged in a ''loop'' that begins by applying a series of algorithms, determining the flow of ice, mass balance and interaction with the Earth, in each cell. The loop is completed by a final equation (widely known as a ''continuity'' equation), used on the full grid, to calculate the interaction and flow of ice between cells. Each iteration of the loop advances the model through one time step. The accuracy of the model depends in part on the width of the grid cells, and the time step length. For continental-scale ice-sheet models, where the time-dependent change in ice-sheet behaviour needs to be calculated over several thousand years, the grid cell width is usually between five and twenty kilometres, and the loop time step is between 1 and 10 years. Because ice-sheet modelling is a valuable tool for palaeo-glaciologists, it is appropriate here to outline briefly how icesheet models are organized.

Most ice-sheet models are centred on the continuity equation for ice (Mahaffy, 1976), where the time-dependent change in ice thickness is associated with the specific net mass budget as follows:

where F(u, H) is the net flux of ice (m2 per year) (the flux of ice being the product of ice velocity, u, and ice thickness, H), V the divergence operator and bs the specific mass budget term (involving surface mass balance, iceberg calving and ice-shelf basal melting where necessary). In the simplest models, the depth-averaged ice velocity, u (ms-1), is calculated by the sum of depth-averaged internal ice deformation and basal motion (Fig. 6.1). Such models make the so-called ''shallow-ice'' approximation and assume that ice flow can be described by a single horizontal vector. More complex models involving the full three-dimensional flow of ice require a sophisticated solution to the stresses and strains which govern the flow of ice (named ''high-order stress models'').

Numerical ice-sheet models calculate glacial processes in a discrete manner (regardless of their sophistication). These processes are linked, however, such that feedback mechanisms can exist between the calculations. The relationship between distinct ice-sheet processes can be summarized by a flow diagram (Fig. 6.2). A good example of a feedback mechanism involves glacial isostasy, which acts to adjust bedrock elevation due to ice loading and, hence, the ice-sheet surface elevation. Surface mass balance and ice velocity will be modified by surface elevation changes, which in turn feed back on ice thickness and, hence, ice loading. Another feedback loop acts to change thermal conditions at the base of ice sheets. This is important because the ice velocity due to rapid basal motion (e.g. sliding) is influenced by the thermal regime. In a cold-based ice-sheet, ice velocities are generally low and so, theoretically, the ice sheet is allowed to build up with a minimum of basal sliding. However, as the ice sheet thickens, the basal temperature is likely to increase (because both surface temperatures and vertical temperature gradients remain relatively constant). Once the base of the ice sheet starts to melt, rapid basal motion becomes possible. Fast ice flow leads to more ice being advected to the ice-sheet margins and, so to ice-sheet thinning. Thinner ice then causes a reduction in the temperature of the icesheet base and possibly to the curtailment of rapid basal motion. This feedback process is complicated because the thermal regime is influenced by

Figure 6.2: Flow diagram of the operation of an ice-sheet model. Adapted from Siegert (2001).

Figure 6.2: Flow diagram of the operation of an ice-sheet model. Adapted from Siegert (2001).

the basal heat gradient, which is in part related to the heat due to basal sliding. Thus, rapid basal motion may maintain warm-based conditions even if the ice sheet thins.

One might consider when it is appropriate to employ simple flow models versus complex models. Both have a role to play in understanding the behaviour of large ice sheets. The benefit of simple models, such as those using the ''shallow-ice approximation'', is twofold. First, in central regions of the ice sheet, the approximation works well at a horizontal scale greater than five ice thicknesses. In other words, there is little change in output at this scale between a simple and a complex model. Second, they can be run quickly, which means multiple runs and sensitivity experiments can be undertaken with minimum computing costs. More complex models are required to solve particular ice flow issues, such as the dynamic change between floating and grounded ice that occurs at the ice-sheet/shelf transition (Schoof, 2007) and over subglacial lakes (e.g. Pattyn, 2003). Given that recent changes to the Antarctic Ice Sheet have been observed at the marine ice-sheet margin (for example, in the recent IPCC report in 2007), understanding ice-sheet response to future warming scenarios is likely to require complex models. High-order models are also able to run at greater spatial accuracy, meaning that important features such as ice streams are resolved. However, given the uncertainty in model inputs regarding palaeo-ice sheets (see below), it is arguable whether the complexity accounted for in high-order stress models adds any value to the output of simpler models.

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