## Governing Equations for Glacier Dynamics

Similar to models of atmosphere, ocean, and sea-ice dynamics, the flow of glaciers and ice sheets is mathematically described from the equations for the conservation of mass, momentum, and energy. For a point on the glacier with ice thickness H, the vertically integrated form of the conservation of mass is z = s z = s Figure 6.3. Schematic of glacier flow mechanisms. The surface velocity us = ub + ud (s). Basal velocity is the sum of deformation of underlying sediments and decoupled sliding at the ice-bed interface. Where ice is moving at the bed, these two processes can operate together (lower image), or only one of them may be active. High water pressure facilitates both processes.

Figure 6.3. Schematic of glacier flow mechanisms. The surface velocity us = ub + ud (s). Basal velocity is the sum of deformation of underlying sediments and decoupled sliding at the ice-bed interface. Where ice is moving at the bed, these two processes can operate together (lower image), or only one of them may be active. High water pressure facilitates both processes.

where u is the average horizontal velocity in the vertically integrated ice column, and bs is the specific mass balance rate at a point. The first term on the right-hand side describes the horizontal divergence of ice flux, and the second term describes the net local source or sink of mass associated with accumulation and ablation. The vertically averaged velocity includes ice flow due to both internal deformation and basal flow: u = ud + ub. Glacial ice moves slowly, so a year is typically adopted as the most convenient unit of time; hence, ice velocities are reported in units meters per year, and bs in (6.5) is expressed as meters per year of ice-equivalent gain or loss of mass.

The main challenge in modeling glaciers and ice sheets is evaluation of the velocity field. Acceleration and inertial terms are negligible in glacier flow, so the Navier-Stokes equations that describe conservation of momentum reduce to a case of Stokes flow, where gravitational stress is balanced by internal deformation in the ice:

Here, o is the ice stress tensor, pi is ice density, and g is the gravitational acceleration. Several texts present detailed derivation of this full system of equations and their solution.

As for sea ice, a constitutive relation is needed to express internal stresses in terms of strain rates in the ice. V-o can then be rewritten as a function of the three-dimensional (3D) ice velocity field, providing a framework to solve for u and integrate Eq. (6.5) to model the evolution of glacier geometry in response to variations in ice dynamics or climate. Because the timescales and stress regimes in glaciers and ice sheets are different from those in sea ice, the mechanical properties and modes of deformation of interest are distinct from those discussed in chapter 5. The next section describes the constitutive relation that is most commonly used in glacier modeling.

Ice rheology is strongly temperature dependent, so an additional equation is needed to solve for the 3D temperature distribution. The local energy balance gives the governing equation for temperature evolution in the ice sheet, pcwr 2Z (kf) + * (6.7)

Here, v is the 3D velocity vector, ci and ki are the heat capacity and thermal conductivity of ice, and J represents strain heat production due to deformational work, # = £ ¡J, where e is the strain rate tensor. This is the same form as the general equation governing the internal temperature evolution of snow or sea ice (Eq. 3.5), with the addition of an advection term that accounts for heat transfer due to movement of the glacier ice. The ad-vection term includes both horizontal and vertical flow, which are of comparable magnitude for heat advection in glaciers. Only the vertical component of diffusive heat transport is retained in (6.7), because vertical gradients in temperature are much larger than horizontal temperature gradients in glaciers and ice sheets. The solution to (6.7) is subject to prescription of air temperature on the upper boundary (the glacier surface) and geothermal or ocean heat flux at the base of the ice.

Glaciers in mild environments are isothermal or temperate; summer temperatures and the latent heat release from refrozen meltwater are high enough to give a mean annual surface temperature of 0°C. A winter cold layer may penetrate the upper ~10 m of the glacier, but the underlying ice is at the pressure melting point, Tpmp =

-bH, where b = 8.7 x 104 K m-1 in glacier ice (see chapter 2). Impurities can further lower the melting-point temperature. Unless a glacier is very thin, the bed is insulated from the air temperatures, and geothermal heat flux provides a source of heat energy that warms the base of the ice.

Cold or polythermal glaciers are found in sub-Arctic, Arctic, or high-elevation environments where the mean annual air temperature is far below freezing. These ice masses may be frozen to the bed or they may be warm-based (at the melting point). Many polar icefields, including the Greenland and Antarctic ice sheets, have mixed conditions: regions where ice is frozen to the bed and regions where ice is warm-based, with cold ice above (e.g., figure 6.4). Much of Antarctica is warm-based because the ice is extremely thick and has been there a long time, which has allowed the slow trickle of geothermal heat hundreds of millennia to warm things up. Once warmed to the melting point, additional heat input generates basal meltwater, which has ponded in parts of Antarctica to give large subglacial lake basins.

Peclet numbers in polythermal glaciers and ice sheets are of order 1; advection and vertical diffusion are similar in magnitude. This gives an interesting thermal structure where the upper half of a glacier or ice sheet is typically cold, strongly influenced by diffusion and vertical advec-tion of surface conditions (mean annual air temperature). At the ice divide in ice caps and ice sheets, advection toward the bed—"downwelling"—gives colder temperatures throughout the ice column; central Greenland is glaciers and ice sheets a. b.  3000 - 1 1 1 1 i 2500 - - 2000 - \ 1500 - #» % t \ \ 1000 500 0 i 1 1

Figure 6.4. Variation of (a) internal temperature and (b) effective viscosity with depth at points along (c) an E-W transect through the Greenland ice sheet at 72.6° N. The heavy solid lines are at the ice divide [Greenland Ice Core Project (GRIP) ice core location, 38.6° W], thin solid lines are along the western flank (45° W), heavy dashed lines are near the western margin (52.2° W), and thin dotted lines are near the eastern margin (28.2° W). Locations along the icesheet transect are indicated in (c).

cold-based for this reason (figure 6.4). Horizontal ad-vection of this "cold plume" is evident on the flanks and margins of the ice sheet, where a cold tongue of ice can be sandwiched between warmer ice both above (from warmer atmospheric conditions) and below. Near the ice sheet margins, high rates of shear deformation produce strain heating that supplements geothermal fluxes. This is effective in warming the ice near the bed. Rates of ice flow and the thermal diffusivity of ice make heat transfer in glaciers and ice sheets a slow process. It takes decades for a change in temperature in the atmosphere to penetrate to the base of a glacier, and tens of thousands of years in an ice sheet.

Given a 3D temperature distribution through the ice sheet, the effective rheology of the ice can be evaluated and the velocity field can be numerically determined. Knowledge of the temperature field is also essential to assessing whether the base of an ice mass is at the pressure melting point or not; if so, liquid water can be present at the bed, and the glacier or ice sheet is subject to basal flow.