The Section describes the formulation of the basal boundary condition. An interface for the upper boundary condition (atmospheric BC) is easily defined by the surface temperature and mass balance. Similarly, the basal boundary consists of mechanical and thermal boundary conditions. The complications arise because the thermal and mechanical boundary conditions depend on each other. The interface of the basal boundary can be described with the following fields (see also Fig.3.8):
Additionally, the ice sheet model calculates a melt/freeze rate based on the temperature gradient and basal water depth. This is handled by GLIDE.
If the ice is not frozen to the bed, basal décollement may occur. This can be parameterised by a traction factor, tb. Within the ice sheet model tb is used to either calculate basal sliding velocities, b, in the case of zeroth order physics, i.e.
where τb is the basal shear stress. Alternatively, tb can be used as part of the stress–balance calculations when the model is used with higher order physics. In simple models tb may be uniform or prescribed as a spatial variable. More complex models may wish to make tb dependant on other variables, e.g. basal melt rate. Typically tb will depend on the presence of basal water.
The second mechanical boundary condition, basal melting/freeze–on Ḃ, is handled within the ice sheet model. The details are described in Section 3.3.2.
The thermal boundary condition at the ice base is more complicated than the mechanical BC. The ice is heated from below by the geothermal heat flux. Heat is generated by friction with the bed. Furthermore, the ice temperature is constrained to be smaller or equal to the pressure melting point of ice. The thermal boundary is set to the basal heat flux if there is no water present. If there is water, the thermal boundary condition is set to the pressure melting temperature1 .
At the ice base, z = h, we can define outgoing and incoming heat fluxes, Ho and Hi:
|Hi||= -krockz=h- + b ⋅b +||(3.65b)|
Freeze–on occurs if Ḃ is negative, basal melting occurs if Ḃ is positive.
The heat flux accross the basal boundary depends on past temperature variations since temperature perturbations penetrate the bed rock if the ice is frozen to the ground (Ritz, 1987). The heat equation for the bed rock layer is given by the diffusion equation
where krock is the thermal conductivity, ρrock the density and crock the specific heat capacity of the bed rock layer.
Initial conditions for the temperature field T are found by applying the geothermal heat flux, G to an arbitrary surface temperature T0:
This ensures that initially the geothermal heat flux experienced by the ice sheet is equal to the regional heat flux. The basal boundary condition of the bedrock layer is kept constant, i.e.
Lateral boundary conditions are given by
At the upper boundary, the heat flux of the rock layer has to be matched with the heat flux in the basal ice layer when the ice is frozen to the bed, i.e.
Otherwise the temperature of the top bedrock layer is set to the surface temperature (if the cell has been occupied by ice, but there is no ice present) or the basal ice temperature (if there is ice). Equation (3.71) is automatically fulfilled if we set the top bedrock temperature to the basal ice temperature everywhere and then calculate the geothermal heat flux to be used as boundary condition for Equation (3.38).
The horizontal grid is described in Section 3.1.1. The vertical grid is irregular like the vertical grid of the ice sheet model. However, it is not scaled. Also for now, I have ignored topography or isostatic adjustment, i.e. the bedrock layer is assumed to be flat and constant.
The horizontal second derivative in Equation (3.67) becomes using finite–differences
and similarly for ∂2T∕∂y2. The vertical second derivative ∂2T∕∂z2 is similar to Equation (3.43):
Using the Crank-Nicholson scheme, Equation (3.67) becomes
with D = krock∕(ρrockcrock). Equation (3.74) is solved by gathering all Tt+1 terms on the LHS and all other terms on the RHS. The index (i,j,k) is linearised using ι = i + (j - 1)N + (k - 1)NM. The resulting matrix system is solved using the same bi–conjugate gradient solver as for the ice thickness evolution.
It is clear from the discussion above that the presence of basal water plays a crucial role in specifying both the mechanical and thermal boundary conditions. However, the treatment of basal water can vary greatly. Basal water is, therefore, left as an unspecified interface. GLIDE does provide a simple local water balance model which can be run in the absence of more complex models.
The basal boundary consists of the individual components described in the previous sections. All components are tightly linked with each other. Figure 3.9 illustrates how the modules are linked and in what order they are resolved.
The order of executions is then:
Clearly, this scheme has the problem that heat is lost if the basal heat flux is such that more water could be frozen than is available. This might be avoided by iterating the process. On the other hand if time steps are fairly small this might no matter to much.