Prediction of the future retreat of Columbia Glacier, Alaska, required a calving law for the boundary condition at the terminus. Qualitative observations on the variations of all major iceberg-calving glaciers of Alaska suggest that calving is high whenever glaciers terminate in deep water, and greatly reduced whenever they terminate in shallow water. Calving relations were investigated based on calculations of calving speed, defined as the volume rate of iceberg discharge from the terminus divided by the cross-sectional area of the terminus. The calving speed was determined for 12 glaciers for which measurements of glacier speed, advance and retreat rates, and other variables were obtained. To extend the range of data, four additional periods of rapid retreat were examined. Values for the terminus characteristics of water depth, cliff height, and thickness of the terminus, averaged over the width of the glacier and over a year, were then examined in relation to the calculated speeds of calving. A statistical analysis to determine the form and coefficients of an empirical calving relation that approximates the data shows that calving speed is best fitted by a simple proportionality to average water depth at the terminus:
c is the calving speed and h
w the water depth, both averaged over the width and over a year, and c a constant of proportionality. This gives a variance reduction fraction (similar to the coefficient of determination r2) of 0.90.
To investigate seasonal changes in calving, data based on shorter time intervals were obtained at the head of embayments from Columbia Glacier. At intervals of approximately two months, the following expression fits intra-yearly calving at Columbia Glacier:
where D is the meltwater discharge from the glacier, hj is the height of the ice column unsupported by water buoyancy, a, b, c are constants, and vc and hu are evaluated at the embayment head. D was determined by correlation with a nearby glacial stream, and hu = h _ hw PW/PJ, where h is glacier thickness and pi and pw the densities of ice and water. Best-fit values of b and c are approximately 0.5 and -2, respectively. This yields a variance reduction fraction r2 of 0.83.
Equation (2) does not fit data averaged over a year and over the width of the glacier and Equation (1) does not fit data obtained over shorter periods at the head of the embayment. Although the two equations are different in form, for similar or average values of D and h - hw (ice-cliff height), they give approximately similar results over the present range of the geometry of the terminus of Columbia Glacier. Whether this will be true after rapid retreat begins remains to be seen.