Instability in a system

In nature, any nearly uniform (ie, not changing significantly with position in at least one of the three possible directions) field is subject to random forcing.

Here the term "field" is used to represent a property of the system which varies with position – for example till thickness, sand thickness, air moisture content.

Consider sand ripples which are created where sand is being transported by currents.

The current is never constant, but varies in time as a result of wave action and wind action.

This means that the rate of sediment transport varies in time. The variability of the rate of sediment transport is complicated and we can consider it to be some kind of a random process.

For subglacial sediments, the randomness is perhaps due to variations in the till lithology, subglacial water pressure etc.

In the somewhat restricted technical sense used here, a system is unstable when it acts to amplify small disturbances. This means that some manner of positive feedback is operating.

Positive feedback

If the system is unstable, then small natural variations (perturbations) which occur as a result of the forcing, become larger through the operation of positive feedback, eventually disrupting the near uniformity of the field.

For example, on an initially flat sand surface on a beach, a small protuberance (ie, variation in the sand thickness) encourages local sediment accretion and the protuberance consequently grows.

If a pattern or wave is to form, these disturbances must have a preferred horizontal scale, ie a wavelength.

Such an instability generally grows exponentially with time at least initially and in practice grows fastest at a particular preferred wavelength, called the wavelength of maximum growth rate.

This wavelength is determined by the physical operation of the system.

There is no absolute principle which says that there must be such a wavelength, but in practice there almost always is – otherwise we would not observe any patterns. 

For example, if the wavelength of maximum growth rate of a sand ripple were larger than the beach, we would see no ripple.

---

Instabilities in subglacially deforming till 

Several workers have now demonstrated mathematically that instabilities can be generated in subglacially deforming till [Hindmarsh, 1998a, 1998b, 1998c; Fowler, 2000; Schoof, 2007].

This theory joins a large volume of instability theories that have successfully explained patterns in nature [e.g., Philips, 1993; Ashton et al., 2001; Yamamoto et al., 2003] and is an appealing idea because it eliminates the requirement for specific localized conditions to be in place to generate subglacial bedforms.

A characteristic feature of instabilities is that they produce the same pattern repeatedly.

We would therefore expect an instability that operated in the subglacial zone to manifest itself repeatedly, and in a similar fashion beneath various ice sheets and places.

If the system is inherently unstable, then one would expect the instability to operate over large areas of that system.

Whilst we strongly suspect that an instability mechanism produced subglacial bedforms, the race is on to find the actual process interactions that generate the instability.

Most work has been done so far on the processes within a deforming layer of sediment at the base of the ice, which seems the most promising route.

However, the instability might arise from other process interactions such as from water and ice or solely within the ice itself.

Much of the above is adapted from our description of instabilities in Dunlop, Clark and Hindmarsh (2008) and which provides further explanation and detail on the concept.