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How grounding lines work

The grounding line is where, in its forward path, the glacier switches from being grounded to being afloat. Glaciers that reach the sea have to have a calving front somewhere, because the ice can’t keep going forever, but there need not be a grounding line.

But suppose there is one. What is the problem? At one level, it is trivial. The glacier remains aground until it satisfies the condition for flotation: there must be just enough water to support the weight of the ice. Multiply ice thickness by ice density (about 900 kg m^-3), and water depth by seawater density (about 1028 kg m^-3). Unless other forces are at work, the ice ungrounds when the two products are just equal. Seven eighths (900 over 1028) of a column of floating ice is below the water level — which of course explains the phrase “the tip of the iceberg”.

Assume that the water depth doesn’t change. (It might, but we have enough to worry about without changes of sea level.) The position of the grounding line, then, depends on the thickness of the ice at the grounding line.

If you see something a bit peculiar in that assertion, you are getting a grip on the problem. Next, assume that we know where the grounding line is. What determines the ice thickness there? We can probably ignore the snow that falls on the grounding line itself, so it must be the imbalance, if any, between the delivery of ice from the thicker grounded ice sheet and the discharge of ice into the thinner floating ice shelf.

Suppose less ice arrives than leaves. The ice must get thinner. Now, instead of just meeting the condition for flotation, it more than meets it. So it starts to float. The grounding line retreats. This argument works the other way around: the grounding line advances if more ice arrives than leaves.

This is an excellent example of a nonlinear problem: the position of the grounding line depends on the position of the grounding line. It will stay put only if just as much ice arrives as leaves. That means that we have to consider the forces driving the ice towards the grounding line from the landward side and away from it on the seaward side.

This balance of forces was first stated accurately, but in an order-of-magnitude way, by Johannes Weertman in 1974. His equation has the grounding-line thickness on both sides of the equals sign: on one side, an expression appropriate for grounded ice, where the dominant force is shearing of the basal ice over the bed; and on the other an expression for flow due predominantly to along-flow stretching — acceleration of the now-floating ice towards the calving front.

Weertman graphed these two expressions. The rate of shearing in the grounded ice depends on the surface slope and the rate of change of the ice thickness, which means that it also depends on the slope of the bed. He found that, if the bed slopes upwards towards the grounding line, his two curves either fail to meet — the equation has no right answer at all — or they meet once. If the bed slopes the other way, the curves can meet twice — there can be two right answers.

It is not uncommon for nonlinear problems to have unexpected numbers of right answers. But there is a twist. The answers can be of two kinds, stable and unstable. In the two-right-answer case, the “small” answer is unstable in a reassuring way. If you decrease the snowfall, the ice sheet will go away. Increase the snowfall, and the ice sheet will grow until it reaches the “large” answer, which is stable.

The one-right-answer case is unstable. Knock it off its perch, for example by reducing the supply of snow below that with which it was in equilibrium, and the equation pushes the grounding line forward until it reaches the edge of the continental shelf, while if you increase the snowfall the equation makes the ice sheet dwindle and disappear.

It takes some getting used to, but all the evidence and analysis now point to the one-right-answer case being the “right” right answer. It seems that the West Antarctic Ice Sheet — an ice body grounded below sea level with its bed sloping upwards down-flow — can coexist peacefully with the rest of an unchanging universe only if its margin is at the edge of the continental shelf — or nowhere.

Posted by Graham Cogley on June 21, 2010 3:00 PM | Permalink