Ice flows – but how, exactly?

by Steph Cornford

In response to some discussion generated on the cryolist email list about the ice flow formulation, we have put together a brief commentary on the ice flow formulation.

You can play the Alpha Test version at:

‘Ice flows’ is an educational game aimed at school-aged children, which means that we had to think about some potentially conflicting objectives. We wanted children to learn some lessons (perhaps guided by their teachers) about ice flow, and Antarctic ice flow  in particular, that reflected contemporary scientific understanding, so we had to make some decisions about the appropriate lessons to learn. At the same time, the game had to be playable,  so that any calculations needed to be carried out quickly enough that the animation appeared smooth, and changing any of the parameters (for example, the accumulation rate) had to lead to a new steady state within seconds, to make the link between cause and effect clear.

We want children to understand that ice sheets are dynamic environments, and that processes like iceberg calving and melting near the margins are always taking place. We want them to understand that if the accumulation of ice in the interior matches losses at the margin then an ice sheet will hold its shape, and that a change in either of these processes (and perhaps others, such as bed slipperiness) results in a change of shape.  We need to show ice contacting submarine beds where it is thick enough, and floating on water where it is not. We’ll be delighted if older children can see that flow speeds up toward the margin and that mass is conserved –  that downstream acceleration results in both horizontal stretching and vertical thinning. And, because it is an idea that has been so important in recent thinking, we want to include marine ice sheet instability.

There are a few things that we were confident could be left for university or later. Glen’s flow law, the Stokes equation, numerical solution of non-linear boundary value problems – these are all things that are important to glaciologists, but they would slow down the calculations, and omitting them does not preclude the broad understanding outlined above. They would also have made for a longer and more expensive period of  development  –  they are not the kind of calculations that game development tools are designed to support, or indeed, that game developers are necessarily familiar with.

In the end, we constructed a much simplified ice sheet model that relied only on the simplest kind of calculations. The ice sheet is built up from a queue of rectangular ice blocks, which move from left to right on the screen. We’ll turn to the velocity calculation below, but for now we need only remember that it is always positive and increases from left to right smoothly. At every time-step the left and right edges of each ice block  are moved according to the velocity field, which results in each block stretching (as the right edge is moving faster than the left).  The height of the ice block is reduced to keep its area constant, and then reduced further if there is any ice-shelf melting to account for. After that, any ice blocks right of the screen edge are removed, and, if there is space at the left hand side of the screen, a new ice block, whose height is determined by the accumulation, is added. In other words, we implemented a simple Lagrangian scheme for mass transport.

The velocity field is re-computed at every time-step from the ice sheet shape to follow a simple curve, starting from a chosen speed at the left edge of the screen and accelerating to reach a value computed from something like the famous Schoof 2007 expression – depending on thickness, buttressing, and rheology – at the grounding line. The strain-rate also varies from left to right, from zero at the left edge, to values derived from the calving front boundary condition beyond the grounding line. Obviously, this is a very crude approximation, but it contains the ideas we want to get across: acceleration toward the margin, the importance of buttressing, and even marine ice sheet instability.

At which point, it’s over to the real game developers to make something that children actually enjoy.

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