thermal ratchet models for glacial motion

There appears to be two, thermally induced, ratchet processes that could be at work in the movement of glacial ice. Depending upon the nature of the glacial terrain either could be playing a major role in the flow, or more correctly the motion, of glacial ice. What they share is an energy source derived from the solar induced cyclical variations of temperatures. They differ only as to the effects these cyclic temperatures have upon the ice.


In this posting these thermal ratchet processes will be described for an idealised conceptual model based upon the assumptions that the temperature increases are uniform through the thickness of the ice and that the movement of the ice is resisted by an idealised bed shear failure taking the form of a, so called, fully mobilised, kinematic friction. Furthermore, vertical sections through the ice before the temperature change will be considered to remain vertical after the temperature change. These assumptions are invoked merely to allow the mechanisms at work to be more clearly established. Later, brief consideration will be given to the more realistic circumstances of thermal gradients existing through the ice thickness, giving rise to stress gradients both through the thickness and laterally in the case of alpine glaciers. The ways that these stress gradients would develop non-uniform elastic-plastic shear distortions through the ice, in addition to the shear failures at the glacier-rock interfaces will also be considered in a later posting. Furthermore, in real glacial situations discrete obstructions will occur, such as protruding rocks, which will present a localised constraint to the motions. How these thermally based mechanisms would be modified to account for the sudden fracture of such obstacles will also be briefly touched upon in a later blog.

Material Ratchet: This process is closest to that described for the development of pingos (Croll. 2004b). It is also most likely to be the major driving force in the motions of continental ice, including the well documented outward flows of our contemporary ice-caps and those of even greater extent that occurred during glacial periods of the ice ages. Because these continental glacial motions account for the major part of the earth’s continuing ice movements, this mechanism is suggested to be potentially the more important ratchet process affecting glacial behaviour.

Consider a length l of essentially horizontal ice sheet depicted in Figure 1(a). This would be typical of that occurring at the upper glacier basin (cirques), or on the continental ice shelves. When during the spring to summer warm-up period an average increase in temperature of T occurs, a section B would if unrestrained move out relative to a section A which is assumed to remain effectively fixed in position. For a continental ice shelf the section A might be thought of as a central location of the ice sheet. For the upper reaches of an alpine glacier the section A could be thought of as the rigid rock face on the upstream side of the glacier or cirque containing the upper ice lake – like the flat accumulation zones talked about in the earlier posting. However, it is likely that the bedrock and valley walls will present a rough surface with the bond between the rock and the ice constraining movement by the development of friction between the underside of the glacial ice and the bedrock. If the friction between sections A and B, as shown in Figure 1(b), prevents all relative movement then for a temperature increase of T a compression force per unit width of magnitude C=alfa.T.E.t will be generated, where alfa is the coefficient of thermal expansion, E the modulus of elasticity, and t the assumed thickness of the ice sheet. As simple scoping calculations will later show these forces C can for even quite modest rises in temperature become immense, partly as a result of the extremely high coefficient of thermal expansion of ice. It is likely therefore, that at an early stage during this warm-up period the induced bed shear will become great enough to cause shear failure. When this occurs the interface in the present idealised model is taken to develop an assumed uniform mobilised frictional resistance of q per unit length. In this mobilised state the maximum compression force at A will remain at the level C=q.l, with section B moving outwards from A to relieve the thermal energy that would otherwise have been developed. By the end of the warm period it is likely that there will have been a number of such sudden releases of thermal energy, at various locations over the length of the glacier. In this way the glacial ice will have moved forward over its entire length, with a maximum outward motion of alfa.T.l, as shown in the sketch of Figure 1(c).

Another consequence of this form of slip-stick, fracture, behaviour will have been the release of the energy and an associated loss of the compressive energy stored within the ice. This means that at the end of the warm period the compression force will be considerably lower than that which would have been present had no shear failures occurred. As a consequence, when the cold season starts it will take only a small decrease in temperature before this residual compression force is overcome by the tensile strains arising from the constraint to thermal contraction. Continuing decreases in temperature will induce tensions in the ice. If the contraction was to occur about the geometric centre of the ice sheet then major fissures and crevices would be expected at the rigid restraint of the ice wall at A. At the free end B the fissures would likely be less pronounced and instead some recovery of the former outward deformation would occur. With ice being so weak in tension the tensile stresses at an early stage of the cooling cycle will cause tensile cracking, both in the form of discrete cracks and fissures, or major crevices. These tension failures would be likely to occur at force levels considerably lower than those needed to induce reversed bed shear failure at the interfaces between the ice and the bedrock. Consequently there would be little recovery of the glacial advances that had occurred during the warm period and the relative deformation that had occurred between sections B and A will not be recovered during the succeeding cooling period. This is suggested in Figure 1(d). Into the major fissures will flow precipitation and surface melt water runoff which will be quickly turned to ice. By the start of the next warm season the glacial ice will to a great extent have recovered its continuity and present a new, relatively stress free, integral ice sheet for the compressive actions to be repeated during the next cycle of warm-up. Over long periods it might be envisaged that the depth and the outward movement of the ice would reach a mass equilibrium with the rates of precipitation falling over the body of the ice sheet.

With the above mechanism being repeated each year, and to a lesser extent during any shorter periods, even daily thermal cycles, it is possible to envisage a gradual outward movement u of the glacial ice sheet relative to the assumed stationary section A. This thermal cycle and its associated internal compression force C, together with the outward motion u, are depicted in Figure 2. Figure 2 is of course a gross simplification since there will be a lot of other complexity occurring over a typical thermal cycle. However, it should serve to illustrate the mechanics involved in the postulated thermally induced, material property dependent, ratchet process. During the early warm-up period there will be a fairly rapid build-up of compression stress associated with the force C, shown by the line ab in Figure 2(b), but little forward movement u. When the compressive energy reaches levels required to overcome the static bed shear friction resistance there could be a fairly sudden, surge, forward movement u and a possible relief of the compression; this is depicted by the line bc. A period of steady forward motion would then accompany relatively steady compressive stresses required to overcome the assumed constant kinematic friction during the warmest season, shown as line cd. This residual compression would be quickly overcome during the start of the cooling period, line de, with an associated halt in the forward advance of the ice. Continuing cooling would be expected to result in a period of maximum tension cracking and an associated unstable form of glacial motion with relatively little net advance, as suggested by the jagged section ef.

It is significant that this form of slick-slip, thermal ratchet, behaviour could occur without any gravity forces being present. For this reason the ratchet action made possible by the discontinuous material failure characteristics of ice, provides a mechanism that could be important in both alpine but especially ice-sheet and continental glacial ice motions. This phenomenon of material failure induced ratchet is closely related to the process observed to occur in the motions of lake ice (Frellson, 1963).

Gravity Ratchet: Where the ice sheet sits upon an inclined plane as shown in Figure 3(a), a second form of thermal ratchet process can be set in motion. This ratchet continues to have thermal energy as its driving force but no longer requires the material to display discontinuous material failure properties. The trigger for this ratchet effect is provided by the relatively small component of gravity force acting parallel to the slope.

Consider a section of glacial ice similar to that of Figure 1(a) to be now located on a plane having slope θ. When subject to a temperature increase T, sufficient to overcome the static friction, the downhill section B will move downwards while an uphill section A would move upwards, with these relative motions being resisted by the fully mobilised bed kinematic friction q. These opposing motions will occur about a stagnation point D that remains fixed in position, as shown in Figure 3(b). This stagnation point will be located at a position offset by a distance x from the geometric centre O of the sheet, in such a way that the net upward friction force 2.q.x will equilibrate the downhill component of gravity force 2.gamma.t.l.sin theta, where gamma is the weight of ice per unit volume. Ignoring the elastic shortening associated with the compressive forces in the ice, the point O will as a result of the expansion move downward relative to the stationary point D through a distance given by u = alfa.T.x, where x = gamma.t.l.sin theta / q, so that

u = alfa.T.gamma.t.l.sin theta / q (1)

If the temperature is now lowered by T the ice sheet will contract with kinematic friction forces being developed as shown in Figure 3(c). During this phase of the thermal cycle the relative motions will occur about a stagnation point E located a distance x downhill from O so that the uphill friction force will now provide the equilibrant to the downhill gravitation component. Relative to the point E that does not move, the geometric centre O of the ice sheet will again move downhill by amount given by eqn (1). Each increase or decrease in temperature will cause the geometric centre of the ice mass to move a little further down the slope.

This form of thermal ratchet is directly related to the magnitude of the downhill gravitational component 2.gamma.t.l.sin theta. It might be anticipated to occur whenever the overlying sheet has a thermal expansion coefficient that is greater than the underlying bed material, or where the sheet has an average temperature change higher than in the bed material. With ice having such a high coefficient of thermal expansion, this form of ratchet action might be expected to be of considerable significance in the observed motions of certain forms of alpine glaciers.

Just to stress the point that above all relates to fairly idealised models of how thermal ratchets could be at work in glacial motion. It is intended to illustrate mechanical phenomena rather than reality. Some more realistic and practical considerations will be briefly covered in a future posting.