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Giving glaciers the full Stokes treatment

The full Stokes equation is a precise description of the flow of a deformable continuum. It says that, in an ice sheet, the pressure gradient force and the force of gravity, resisted by the temperature-dependent stiffness of the ice, are balanced by the motion of the ice. Stated that way, it is simple arithmetic – but there is a devil of a lot of arithmetic to do.

At each point in the ice sheet, the pressures, or more accurately the shear stresses and the normal (compressive or extensional) stresses, are directed along each of three coordinate axes (pointing either way), and they can change from point to point along each of those axes. You want the best spatial resolution you can afford. But the resolution you need, if you are to maintain accuracy, is simply not affordable even on today’s fastest computers. In a recent study, Gaël Durand and others remarked laconically that in going from a grid spacing of 20 km to 2.5 km one of their simulations ballooned from two days of supercomputer time to two weeks.

Various simplifications of the full-Stokes treatment have been developed. If you ignore along-flow velocity gradients, tending to stretch or squeeze the ice in the horizontal direction, you get the so-called shallow-ice approximation, oddly named because it works better the thicker the ice. Ice sheets tend to feed floating ice shelves. Here the underlying water cannot support horizontal shearing, so the ice flows at the same speed throughout the thickness; the vertical gradients of the horizontal velocities are negligible. This is the shallow-shelf approximation. Both approximations are valuable time-savers.

Unfortunately they both break down near the grounding line that separates the ice sheet from the ice shelf. Faced with the impracticality of the full-Stokes treatment and of any one-size-fits-all approximation, the dynamicists have been working hard to make the problem tractable.

Rectangular arrays of grid cells are definitely old hat. Nowadays the favoured approach to the numerics is the finite-element method, in which you describe your ice sheet with cells of variable size and shape. This is laborious but reduces the computational burden later. You give the ice sheet the full-Stokes treatment everywhere, but spend little time where full Stokes isn’t really necessary.

There is an obvious snag. The grounding line is the focus of interest because it might migrate unstably towards the interior of the ice sheet. But if it migrates away from where you have laboriously set up lots of little cells, you are sunk. Instead of migrating in little steps it gains the computational freedom to take great big ones. It can, and may well, end up in some entirely unrealistic location.

So Durand and co-authors created an adaptive grid consisting of cells that were small near the grounding line, growing larger progressively with distance from it. But they re-centred the grid on the grounding line after each model time step, such that the little cells kept company with the grounding line. More purely preliminary labour, but with the smallest cells only 200 m in size they were able to obtain consistent numerical behaviour and to confirm Christian Schoof’s finding, from a different theoretical angle of attack, that grounding lines are indeed unstable when the bed slopes upwards.

There is irony in this greed for number-crunching power. Long before ice sheets became objects of scientific scrutiny, Stokes laid all of the conceptual groundwork with a pencil (or maybe a quill – did they have pencils in the 1840s?). Much of our understanding of how ice sheets work was developed on computers to which you would not give desk room (even if they would fit). Now, the glacier dynamicists are right up there with the astrophysicists, climate modellers and the like, baying for time on unimaginably fast computers that have trouble satisfying the demand.

The glaciological demand, though, is real and pressing. The full-Stokes treatment is getting attention because of the socioeconomic risks of grounding-line instability, which was identified in the Fourth Assessment by the Intergovernmental Panel on Climate Change as one of our biggest gaps in understanding of how the Earth works. My dynamicist colleagues have to have something to say about it in time for the IPCC’s next assessment, due in 2014. They have made tremendous progress by working overtime, but if yet more time is what it takes to crack the problem then I hope they will resist this pressure to deliver.


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