316
the free boundary problem, and in fact it is linearly stable. It then seems that thermal
runaway is unlikely to occur in practice.
A slightly different perspective may allow runaway, if we admit non-steady ice fluxes.
Formally, we can derive a suitable model if A = a/f3 = 0(1), f3 -+ 00. In this case, we
can expect T to tend rapidly to equilibrium of (2.33), and then C reacts more slowly via
mass conservation, thus
(2.40)
An x-independent version of (2.40), consistent with the p~evious discussion, is
ac at = s - q((),
(2.41)
and this will allow relaxation oscillations if q(() is multivalued as a function of C - which
will be the case. Surging in this sense is conceivable, but the limit f3 -+ 00 is clearly
unrealistic, and unlikely to be attained. The earlier conclusion is the more likely.
2.3 The sliding law
The sliding law relates the basal shear stress Tb to the basal sliding velocity 1J.b. The
classical theory, enunciated by Lliboutry, Weertman, Nye, Kamb, and others, considers
ice flowing at the base of a glacier over an irregular, bumpy bedrock. The ice is lubricated
at the actual interface by the mechanism of regelation, or melting-refreezing, which allows
a thin film (microns thick) to exist at the ice-rock interface, and allows the ice to slip. The
drag is then due to two processes; regelation itself, and the viscous flow of the ice over the
bedrock. Regelation is dominant for small wavelength roughness, while viscous drag is
dominant for large wavelengths, and early work emphasised the importance of a controlling
(intermediate) wavelength (of several centimetres). More recently, the emphasis has been
away from regelation and considered only the viscous flow.
A suitable model for discussion is that of a Newtonian fluid over a rough bedrock of
'wavelength' [x] and amplitude [y], given, in coordinates scaled with [x], by y = IIh(x),
where y is now the vertical coordinate, and
II = [y]/[x]
(2.42)
is a measure of corrugation. The governing equation for slow, two-dimensional flow is the
biharmonic equation
\J41jJ = 0
(2.43)
the free boundary problem, and in fact it is linearly stable. It then seems that thermal
runaway is unlikely to occur in practice.
A slightly different perspective may allow runaway, if we admit non-steady ice fluxes.
Formally, we can derive a suitable model if A = a/f3 = 0(1), f3 -+ 00. In this case, we
can expect T to tend rapidly to equilibrium of (2.33), and then C reacts more slowly via
mass conservation, thus
(2.40)
An x-independent version of (2.40), consistent with the p~evious discussion, is
ac at = s - q((),
(2.41)
and this will allow relaxation oscillations if q(() is multivalued as a function of C - which
will be the case. Surging in this sense is conceivable, but the limit f3 -+ 00 is clearly
unrealistic, and unlikely to be attained. The earlier conclusion is the more likely.
2.3 The sliding law
The sliding law relates the basal shear stress Tb to the basal sliding velocity 1J.b. The
classical theory, enunciated by Lliboutry, Weertman, Nye, Kamb, and others, considers
ice flowing at the base of a glacier over an irregular, bumpy bedrock. The ice is lubricated
at the actual interface by the mechanism of regelation, or melting-refreezing, which allows
a thin film (microns thick) to exist at the ice-rock interface, and allows the ice to slip. The
drag is then due to two processes; regelation itself, and the viscous flow of the ice over the
bedrock. Regelation is dominant for small wavelength roughness, while viscous drag is
dominant for large wavelengths, and early work emphasised the importance of a controlling
(intermediate) wavelength (of several centimetres). More recently, the emphasis has been
away from regelation and considered only the viscous flow.
A suitable model for discussion is that of a Newtonian fluid over a rough bedrock of
'wavelength' [x] and amplitude [y], given, in coordinates scaled with [x], by y = IIh(x),
where y is now the vertical coordinate, and
II = [y]/[x]
(2.42)
is a measure of corrugation. The governing equation for slow, two-dimensional flow is the
biharmonic equation
\J41jJ = 0
(2.43)