mechanics, but we shall nevertheless indicate what these
laws are, and what implications they have on the flow of the
ice caps. In the following, we shall only deal with the case of
ice assumed to be incompressible. This hypothesis does not
hold true for snow, whose density increases from the surface
down to a maximum depth of 100 m, but in most cases, the
mechanical effect of this layer of snow and firn is the same as
a layer of ice of the same weight would be, which amounts
to removing about twenty meters from the total thickness of
ice at the chosen spot (the thickness is of the order of a few
kilometers).
If we consider the balance of forces applied to an ice
particle, the only body force is gravity. The other forces are
surface forces arising from contact with other ice particles, or
with the base or water (when it is floating). The ice flows
sufficiently slowly so that we can ignore accelerations and
inertial forces (Coriolis), and the balance of forces applied to
ice sheets is often referred to as a quasi-static equilibrium. In
terms of mechanical behavior and at the scale of the strain
rate occurring in ice sheets, ice is considered as a viscous
fluid, i.e. the strain rate (deformation per unit of time, directly
expressed as function of the spatial derivative of the velocities) is connected to the stress. Water, for example, is also a
viscous fluid, but its viscosity (lower than that of ice) does not
depend on the stress (nor on the strain rate). This is called
linear viscosity (also known as Newtonian). Ice, on the other
hand, is characterized by non-linear viscosity which decreases with stress according to a power law (with an exponent of
approximately 2): the more the ice deforms, the easier it is to
deform. This type of behavioral law is not exceptional, it is
also found for lava, mud and even chocolate. As with most
viscous materials (maintaining the analogy with chocolate),
the viscosity of ice decreases as its temperature increases (in
an exponential relationship). Depending on the location of
the ice sheet, the temperature can vary from −50 °C at the
surface to melting point at the base, and this can influence the
viscosity by a factor of up to 500.
Combining the quasi-static equilibrium, the law of viscous behavior, incompressibility and the various boundary
conditions, we arrive at a system of equations which rigorously takes into account the mechanical equations. This
system (called ‘full stokes’) can be solved numerically, but
the cost in terms of computation time is such that it is not
conceivable at this time to apply this method to the entire ice
sheet, especially when the aim is to also study its temporal
evolution. This approach is therefore limited to localized
research. Fortunately, an approximation exists which takes a
‘thin layer’ approach making it possible to treat the ice sheet
overall. Indeed, a characteristic of ice sheets is their very
small aspect ratio, i.e. the ratio of thickness to expanse. For
Antarctica, for example, the thickness is around 3 km and
the expanse is 3000 km (ratio of 1/1 000). If it were made
into a scale model 3 m wide, it would only be 3 mm thick
and would look like a thin sheet of ice. Capitalizing on this
aspect ratio, two separate approximations have been proposed: one for the part of the ice sheet resting on land
(Shallow Ice Approximation, SIA), the other for the floating
part (Shallow Shelf Approximation, see later). These
approximations are used in ice sheet models (Ritz 2001).
Moreover, they allow a qualitative understanding of the
interaction between the geometry of an ice sheet and its flow.
For the resting part for example, the SIA shows that the
velocity of the ice (averaged over its thickness) is proportional to the thickness to the power of four and to the slope
of the surface to the power of three. Although the thickness
shows little variation over the entire ice sheet, the slope
varies from 10
−3 in the central regions to nearly 10
−2 at the
edges, thus implying a speed 1000 times greater (the size of
variation observed in reality). This also explains why the
thickness of an ice sheets is strongly related to their expanse
with the amount of snow accumulation being of only minor
importance. When an ice sheet grows, its slope at the surface
increases and its drainage increases greatly, creating a negative feedback which limits thickening. Another result of the
SIA, used in the interpretation of ice cores, concerns the fact
that most of the deformation is concentrated in the layers
near the bottom, and that higher up, horizontal speed changes little with depth and in a first approximation, the thinning of the ice layers decreases linearly with depth.
The ice flows through deformation but its velocity at the
interface with the bedrock (basal velocity) also contributes to
the flow. Two processes intervene, the sliding over the
bedrock and the deformation of the underlying sediment.
Based on Antarctic observations, it appears that it is the
latter mechanism which is the most effective, leading to
speeds of several hundred meters per year. In both cases, the
basal velocity is negligible as long as the temperature at the
interface is below melting point. On the other hand, at the
melting point, not only is sliding possible but water is produced. This brings about another mechanism, the higher the
water pressure, the greater the basal velocity, due to both a
lubrication effect (less friction) and the fact that the
water-saturated sediment is easier to deform.
The temperature field in the ice thus affects the flow in at
least three ways: through the viscosity, through the threshold
(melting point) from which basal movement is possible and
through the subglacial water pressure. The temperature in the
ice can be estimated reasonably well by solving the heat
equation and taking into account any changes in surface
temperature over time. In general, temperature increases
with depth. It is very cold at the surface. At around 10 m
depth, where seasonal variations are mitigated, it has the
value of the mean annual temperature. In Antarctica, for
example, the temperature at 10 m varies from about −20 °C
at the coast to −60 °C in the center. At the base of the ice,
the temperature is often close to melting point because
308
C. Ritz et al.
laws are, and what implications they have on the flow of the
ice caps. In the following, we shall only deal with the case of
ice assumed to be incompressible. This hypothesis does not
hold true for snow, whose density increases from the surface
down to a maximum depth of 100 m, but in most cases, the
mechanical effect of this layer of snow and firn is the same as
a layer of ice of the same weight would be, which amounts
to removing about twenty meters from the total thickness of
ice at the chosen spot (the thickness is of the order of a few
kilometers).
If we consider the balance of forces applied to an ice
particle, the only body force is gravity. The other forces are
surface forces arising from contact with other ice particles, or
with the base or water (when it is floating). The ice flows
sufficiently slowly so that we can ignore accelerations and
inertial forces (Coriolis), and the balance of forces applied to
ice sheets is often referred to as a quasi-static equilibrium. In
terms of mechanical behavior and at the scale of the strain
rate occurring in ice sheets, ice is considered as a viscous
fluid, i.e. the strain rate (deformation per unit of time, directly
expressed as function of the spatial derivative of the velocities) is connected to the stress. Water, for example, is also a
viscous fluid, but its viscosity (lower than that of ice) does not
depend on the stress (nor on the strain rate). This is called
linear viscosity (also known as Newtonian). Ice, on the other
hand, is characterized by non-linear viscosity which decreases with stress according to a power law (with an exponent of
approximately 2): the more the ice deforms, the easier it is to
deform. This type of behavioral law is not exceptional, it is
also found for lava, mud and even chocolate. As with most
viscous materials (maintaining the analogy with chocolate),
the viscosity of ice decreases as its temperature increases (in
an exponential relationship). Depending on the location of
the ice sheet, the temperature can vary from −50 °C at the
surface to melting point at the base, and this can influence the
viscosity by a factor of up to 500.
Combining the quasi-static equilibrium, the law of viscous behavior, incompressibility and the various boundary
conditions, we arrive at a system of equations which rigorously takes into account the mechanical equations. This
system (called ‘full stokes’) can be solved numerically, but
the cost in terms of computation time is such that it is not
conceivable at this time to apply this method to the entire ice
sheet, especially when the aim is to also study its temporal
evolution. This approach is therefore limited to localized
research. Fortunately, an approximation exists which takes a
‘thin layer’ approach making it possible to treat the ice sheet
overall. Indeed, a characteristic of ice sheets is their very
small aspect ratio, i.e. the ratio of thickness to expanse. For
Antarctica, for example, the thickness is around 3 km and
the expanse is 3000 km (ratio of 1/1 000). If it were made
into a scale model 3 m wide, it would only be 3 mm thick
and would look like a thin sheet of ice. Capitalizing on this
aspect ratio, two separate approximations have been proposed: one for the part of the ice sheet resting on land
(Shallow Ice Approximation, SIA), the other for the floating
part (Shallow Shelf Approximation, see later). These
approximations are used in ice sheet models (Ritz 2001).
Moreover, they allow a qualitative understanding of the
interaction between the geometry of an ice sheet and its flow.
For the resting part for example, the SIA shows that the
velocity of the ice (averaged over its thickness) is proportional to the thickness to the power of four and to the slope
of the surface to the power of three. Although the thickness
shows little variation over the entire ice sheet, the slope
varies from 10
−3 in the central regions to nearly 10
−2 at the
edges, thus implying a speed 1000 times greater (the size of
variation observed in reality). This also explains why the
thickness of an ice sheets is strongly related to their expanse
with the amount of snow accumulation being of only minor
importance. When an ice sheet grows, its slope at the surface
increases and its drainage increases greatly, creating a negative feedback which limits thickening. Another result of the
SIA, used in the interpretation of ice cores, concerns the fact
that most of the deformation is concentrated in the layers
near the bottom, and that higher up, horizontal speed changes little with depth and in a first approximation, the thinning of the ice layers decreases linearly with depth.
The ice flows through deformation but its velocity at the
interface with the bedrock (basal velocity) also contributes to
the flow. Two processes intervene, the sliding over the
bedrock and the deformation of the underlying sediment.
Based on Antarctic observations, it appears that it is the
latter mechanism which is the most effective, leading to
speeds of several hundred meters per year. In both cases, the
basal velocity is negligible as long as the temperature at the
interface is below melting point. On the other hand, at the
melting point, not only is sliding possible but water is produced. This brings about another mechanism, the higher the
water pressure, the greater the basal velocity, due to both a
lubrication effect (less friction) and the fact that the
water-saturated sediment is easier to deform.
The temperature field in the ice thus affects the flow in at
least three ways: through the viscosity, through the threshold
(melting point) from which basal movement is possible and
through the subglacial water pressure. The temperature in the
ice can be estimated reasonably well by solving the heat
equation and taking into account any changes in surface
temperature over time. In general, temperature increases
with depth. It is very cold at the surface. At around 10 m
depth, where seasonal variations are mitigated, it has the
value of the mean annual temperature. In Antarctica, for
example, the temperature at 10 m varies from about −20 °C
at the coast to −60 °C in the center. At the base of the ice,
the temperature is often close to melting point because
308
C. Ritz et al.