Sir,
I can only admire the article of Dr. H.Wexler in the March 1959 number,1 and the novel application of his equations. But it seems to me that, in addition to the 31.6 cal. cm.−2 yr.−1 geothermal heat flux which he uses for F 0 in these equations, there should also be added the heat of friction if the glacier is in a steady state yet accumulating 30 cm. of ice per annum (if it were motionless, the steady state could not be). Nor is the heat of friction difficult to compute—30 cm. (25 g.) of ice added over the length of any glacier 3,000 m. thick is the equivalent of 25 g. falling 3,000 m. in one year over each square centimeter of surface. This gives about 7 × 109 ergs,or 150 cal. cm.−2 yr.−1, five times the geothermal heat flux! No appreciable amount of this work could have been syphoned off as kinetic energy of the moving glacier before conversion into heat; even at a velocity of 100 m. yr.−1 for the glacier, the kinetic energy of that ice column would be quite negligibly small.
Also as regards the value of K, the thermal conductivity of the glacier ice, which Wexler takes as 5.3 × 10−3, I believe that the coefficient for any Arctic-type ice of density 0.88 or so (compared with 0.92 g.cm.−3 for laboratory ice) will be substantially less, due to “resistance” to heat flow by included air bubbles (the extreme case, snow, is a great thermal insulator). For the same reason K, the thermal diffusivity, will also be less. So with a much larger supply of bottom heat, and a lower thermal conductivity and lower diffusivity, I believe that the family of curves in Wexler’s fig. 2—at least all curves for times over 10,000 years—will ground themselves at a temperature just below 0° C., implying that the basal region of any accumulating, thick glacier which is in a stable state will consist of isothermal ice, a conclusion of importance in considering the even thicker Pleistocene ice sheets.
In order to give some idea of the thermal conductivity of bubbly ice, I have had my guide, Armand Perron of Valtournanche, make a series of measurements which should approximate a comparison of the thermal conductivity of normal (isothermal) glacier ice of density 0.92 in the Gornergletscher (altitude 2,800 m.) with that of bubbly (isothermal) glacier ice, density 0.87, in the same glacier. Illustrations of small specimens of these two types of ice from the same area were reproduced in an earlier paper.2 These tests were made by inserting aluminum tubes containing refrigerated brine mixtures at about –9° C. into close-fitting bore holes in the two types of ice in situ, and by measuring the time interval for the respective tubes of cold brine to warm up to the temperature (0° C.) of the surrounding glacier ice. The tubes inserted in normal glacier ice required on the average 21 min.; those in the bubbly ice, 42 min. It would seem that there is a very substantially lower thermal conductivity for bubby ice compared with normal ice, of the order of one half.
Also, on this same subject, Birch and Clark3 report substantially lower thermal conductivity for limestone, marble, and even for gabbro and diabase compared with the conductivities, suitably weighted, of their constituent minerals—due, they show, to the minutest films and wedges of air or other gases between the mineral crystals. Such differences of conductivity run up to some 20 per cent for these rocks; if in rock the minute remnants of air persisting between mineral crystals after millions of years of exposure to thousands of tons pressure at high temperature cause such differences, it seems probable that substantially greater diminution of thermal conductivity occurs in cold firn, in which occluded air has only had a few thousand years to escape under only a few thousand pounds pressure. Both in rock and ice it is the breaking up of continuous paths by multiple minute air spaces, rather than the resulting slightly lesser density, which decreases thermal conductivity. I am planning measurements in the field of the thermal conductivity of this cold firn, probably at 4,000 m. on the Monte Rosa, this summer or next.
But the original method of attack by Dr. Wexler has my admiration! I thank him.
16 July 1959
