boundary conditions

In reality, a thinning ice shelf will break off at its seaward rnargin once a critical rninirnurn thickness is reached (usually around 250 m). In the model, however, calving physics are not considered explicitly, and the ice shelf is actually rnade to extend all the way to the edge of the nurnerical grid. Testing indicates that this does not present a serious problern. Model runs, in which the ice shelf was 'cut off closer to the coast (but still outside the rnain ernbayrnents) did not have appreciably different ice thickness distributions or grounding line positions (which is of final interest).

Boundary conditions at the seaward margin are then applied for unconfined, freely floating and uniforrnly spreading (cornplete stress and strain-rate syrnrnetry in the ^X- and y-directions) ice shelves. This is equivalent to setting

= 7lYy =

%

; and = 0. Velocity components then follow frorn the flow law (eq. 4.5):

where an expression for the deviatoric Stresses can be found frorn the condition that the net total force on the ice shelf front rnust be balanced by the horizontal force exerted by the sea water:

From hydrostatic equilibrium (eq. 4.23), an expression for the stress deviator

=

-

⁰⁾ and T,, = -pg(H+h-z), this yields:

At the grounding zone, boundary conditions are not velocity gradients, but follow from the vertically integrated velocity components.

4.4.4. Stress transition Zone at the grounding line

In view of the basically different nature of the stress balance in grounded and floating ice, there must be a stress transition Zone located somewhere in between near the grounding line. Here, a change should take place frorn shear-dominated flow to a situation where longitudinal Stresses prevail, so that all stress components could be potentially important. As discussed before, the thickness at the grounding line results from a subtle balance between a whole number of competing mechanisms and feedback-loops, even so that by excluding one stress component almost any result can be obtained. Modelling the transition Zone is thus a delicate matter and requires careful consideration of the various terms in the force balance. Of interest in this respect are studies by Van der Veen (1985, 1987) and Herterich (1987), who addressed the flow at the grounding line in a 2-D (vertical plane) approach. In order to deal with the ice sheet

-

ice shelf interaction at the grounding line in a proper way, Van der Veen (1987) advocated to include the longitudinal deviatoric stress in the effective stress term of the flow law. The present 3-D approach incorporates this fundamental idea.

Two assumptions are made as follows. First, it is taken for granted that the width of the stress transition Zone is smaller than typical gridsizes in large scale numerical models. The validity of this assumption appears to be corroborated by the two-dimensional studies of Herterich (1987) and Van der

Veen (1987), although the length of the Zone over which longitudinal stresses are important may be larger where basal sliding is the dominant flow mechanisrn. Consequently, the grounding Zone in the model has a width of one gridcell and includes all grounded grid points that border the floating ice region. The second sirnplification assumes that deviatoric stress gradients can be neglected as a contribution to the shear stresses in the stress equilibrium (eqs. 4.8-4.9). This may represent a somewhat more dubious approxirnation, in particular in fast-flowing outlet glaciers and ice streams, where longitudinal stretching may Support part of the weight (e.g. McMeeking and Johnson, 1985). In spite of this, however, one could sirnply argue that this is probably the only option, because to do otherwise would irnply complicated and time- consurning stress integrations along the x- and y-axes respectively. On the other hand, in Van der Veen (1987), this assurnption did not appear to be crucial to the model outcome and as far as the ice sheet profile was concerned, the only thing that really mattered was the longitudinal deviatoric stress (and much less its gradient) at the grounding line only. A scale analysis of eqs. 4.8-4.9 for typical grid spacings in numerical models seems to support this. Moreover, in a recent detailed calculation of the stress distribution near the grounding Zone of Byrd Glacier (an outlet of the East Antarctic ice sheet), Whillans et al.(1989) found a close correlation between the driving stress and basal drag. These points seem to indicate that, as far as large-scale nurnerical models are concerned, the ice sheet approximation may still be valid, so that the shear stresses follow from the usual equations (4.10-4.11). Other assumptions in Van der Veen's analysis are also used here, such as replacing the deviatoric stresses by their vertical mean, because inforrnation On their vertical profile is lacking.

In view of this, the vertically integrated mass flux at the grounding line (that is of final interest) can also be found by integrating expressions for the shear strain rates

ixz

and

cyz

twice along the vertical, however incorporating all stress components in the effective stress terrn of the flow law:

- - -

where T ' ~ ~ , T ' ~ ~ , xVxy in the transition Zone cannot be specified directly, but have to be obtained by invoking the constitutive relation (4.5). The procedure relies On the fact that the Stress field can be calculated from the flow law, given the ice sheet geornetry, strain rates and the rate factor. Integrating expressions for

. - .

e X x , e Y y , eXy along the vertical and dividing by ice thickness to obtain vertical mean values then forrnally yields:

. .

for Exx , E _YY'

e

_{X Y}

After rnaking use of Leibnitz' rule to change the order of integration and some algebraic rnanipulations (details of which are ornitted here in order to save space), this leads to the following Set of coupled equations:

Since -tqxz, Yyz follow from eqs. 4.10-4.1 1 and all strain rates and gradients can be calculated from the velocities and ice thicknesses that result On the grid, this represents a non-linear system of three equations in the three

- - -

unknowns ( T ' ~ ~ , T ' ~ ~ , T ' ~ ~ ) that can be solved by an iterative method (see further

The main effect of treating the transition Zone in this way is that the ice is apparently softened there (the depth-averaged deviatoric longitudinal stresses may be up to 3 tirnes larger than the shear stresses), so that lower shear stresses are needed to produce the Same ice rnass flux. Also, this treatrnent fully couples ice shelf flow to the inland flow regirne. This is because longitudinal stresses reflect the dynamic state of the ice shelf as well. The deformational velocity at the grounding line (and indirectly also the basal sliding velocity through changes in the ice sheet geometry ) is therefore able to react to changes in ice shelf geornetry. For instance, restraining forces in the ice shelf which inhibit free spreading will reduce the velocity at the grounding line, whereafter the flux divergence decreases, in turn leading to thicker ice and smaller surface slopes. Of Course, the shear stresses may react to this and increase, causing a thinning, and so forth ... The important thing is that this approach is able to represent a number of potentially effective feedback mechanisms in the flow. The adopted coupling scheme seems to work well, because it allows the grounding line to move in both directions. As such, this certainly represents an improvement over previous Antarctic models.

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