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Sorge’s law: the conversion of snow to ice

All of us who live near enough to a pole, or high enough up a mountain, have done experimental work on the densification of snow. When I was six I once, to test a hypothesis I have now forgotten, posted a snowball through the letterbox outside my school. My hypothesis was unsound, and in the aftermath I was taught my first, painful lesson about the complexities of densification.

When I grew up, I learned a little more. To be specific, I learned about Sorge’s Law. Sorge’s name (pronounced zor-guh, with the g hard) is well known to glaciologists because in 1954 Henri Bader suggested that it be commemorated by assigning it to a surprising but very important physical relationship: when snow accumulates steadily and there is no melting, the density is always the same at any one depth beneath the surface. The name has stuck.

Density is mass per unit volume. New snow can have a density as low as 50 kilograms per cubic metre if it is particularly fluffy, but the snowflakes collapse rapidly and as time passes the new snow, densifying under its own weight, takes up less and less space. We call it ice once it reaches about 830 kg m^-3, at which point the air spaces have pretty much closed off and turned into bubbles. Pure ice has a density of 917 kg m^-3, but most glacier ice is a bit less dense because of the remaining bubbles and other voids.

Ernst Sorge’s observations were made in Greenland in the winter of 1930–1931, in a hand-dug pit 16 metres deep. His Law actually has rather meagre observational support but, as Bader showed, it follows from some rather simple algebra. It means that, as new mass is added by snowfall, an equal mass of old snow turns into ice (that is, crosses the 830 kg m^-3 borderline) at depth.

That gives us a valuable payoff. Provided that nearly the whole thickness is ice, not snow, a change in that thickness, or equivalently in the surface elevation, can be converted to a change of mass by multiplying by the density of the ice. We rely on this extensively when trying to convert volume balances into mass balances. The volume balance, in cubic metres, is thickness change (m) times area (m²). There is nothing wrong with it, except that the mass balance, in kilograms (volume change times density), is more useful for purposes like estimating sea-level change. But a volume balance is considerably easier to measure. You can do it remotely, with laser altimeters, and you don’t have to measure the density in the field, expensively.

The savings from invoking Sorge’s Law are extremely attractive. For example, we can extend its range of validity by assuming that the density profile also remains unchanged when the rates of melting at the surface and refreezing at depth are constant and equal. But what if this assumption, or the more basic assumption of steady accumulation, does not hold? We don’t mind much if the snowfall rate varies from year to year, or with the seasons. (In interior Greenland, for example, more snow falls in summer than in winter.) We can handle that kind of flickering behaviour, as long as we know about it.

A more subtle problem is that the temperature may vary over the years, and that alters the compaction rate (slower when it is colder). A less subtle problem is that on many glaciers nowadays the density profile is being eaten away by continuing mass loss. In their middle reaches, the glaciers are losing stuff that would still be there, suffering compaction, if Sorge’s Law applied. Here, Sorge gives too great a density for the lost mass.

What this adds up to is a pressing need to know more about how often Sorge’s Law really holds on glaciers, and how to make accurate corrections where it doesn’t. Remote sensing can deliver much more knowledge than laboriously digging holes. What would be really nice would be a way to measure not just the glacier surface elevation changes from space, but also the density profile. That is not around the corner, and until we get there we all need to remember that volume change is not the same as mass change. The density we adopt for the conversion is a source of uncertainty.

Posted by Graham Cogley on December 7, 2009 3:00 PM | Permalink