Cuspate Surfaces of Melting Ice and Firn

American Geographical Society

Cuspate Surfaces of Melting Ice and FirnAuthor(s): John LeighlySource: Geographical Review, Vol. 38, No. 2 (Apr., 1948), pp. 300-306Published by: American Geographical SocietyStable URL: http://www.jstor.org/stable/210861 .

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CUSPATE SURFACES OF MELTING ICE AND FIRN

JOHN LEIGHLY

IN HIS recent article on the Wolf Creek glaciers of the St. Elias Range, Robert P. Sharp describes and illustrates a type of surface of melting glacier ice that he calls "scalloped."' His term was evidently suggested

by the profile of a section normal to the surface. A more suitable name for a three-dimensional surface such as he describes might be "cuspate" or "nega- tively mammillate." According to Sharp, "scalloped" surfaces are associated with fluted ones, and always face downward. "Scallops" appear on a horizontal or nearly horizontal surface and pass into flutings as the inclination of the surface from the horizontal increases. Sharp is unable to suggest a "satisfactory explanation ... for the form and pattern of flutes and scallops."

I propose here a hypothesis of the origin of these surface forms of melting ice. As a supplement to the illustrations in Sharp's article, Figure i is in- cluded, reproduced from a photograph of part of the roof of a stream tunnel under a firn bank. I took the photograph on August 7, I940, in the canyon of Lowell Creek, which flows into Resurrection Bay at Seward, Alaska. In winter, snow accumulates to great depths in parts of this canyon, as a result of the piling up of avalanches that have slid down its steep tribu- tary valleys. As late in the summer as the date when the photograph was taken, three such accumulations had not yet melted. All the exposed snow had been changed into firn, and the stream flowed through tunnels under the remaining banks. Figure i was taken from the upper end of one of these tunnels, looking downstream. The avalanche snow is rather dirty, so that many of the ridges in the melting surface of the firn are lined with bits of rock and plant material. Most of this foreign matter has been carried to the ridges and cusps by the water that flows as a film along the melting surface toward the cusps and there drips off. In the upper right of the pic- ture, melting has exposed an ice crust containing debris; it has an appearance different from that of the true cuspate or scalloped surface.

MELTING ON OVERHANGING SURFACES

A part of the heat that melts downward-facing surfaces of ice and firn is probably derived from the absorption of reflected solar radiation and of

I R. P. Sharp: The Wolf Creek Glaciers, St. Elias Range, Yukon Territory, Geogr. Rev., Vol. 37, I947, pp. 26-52; reference on pp. 46 and 47 (Figs. 27 and 28).

>- MR. LEIGHLY, professor of geography in the University of California, Berkeley, has long been interested in phenomena associated with turbulence in bodies of water and ice.



SURFACES OF MELTING ICE 30I

long-wave radiation from neighboring ice-free surfaces and the atmosphere. Most of the heat, however, is communicated to the melting surface as sensible heat by convection and, ultimately, by conduction; or, if the dew point of the air is above freezing, as latent heat released when the water vapor of the air is condensed on the surface. This transfer of heat depends

FIG. i-Part of the roof of a stream tunnel under a bank of firn, canyon of Lowell Creek near Seward, Alaska.

on the intimate contact with the melting surface, the temperature of which remains at the freezing point, of air having a temperature above freezing, a contact that must be continuously renewed. The continuous supply of air to the melting surface and its equally continuous removal are effected by convection of air near the surface. The conditions under which such convection proceeds should be examined.

When the adjacent air has a temperature above freezing, an overhanging surface of ice provides one of the best conditions for free thermal convection to be found in nature. The air in contact with the ice is cooled and becomes denser than the air beneath it. Convection is induced in the unstable air, in the course of which the colder air sinks away from the surface and is replaced by warmer air from below. The intensity of this convection depends on the amount by which the temperature of the air exceeds the freezing point.! The convective process next to the cold overhanging surface is thus the

2 Such convection may also occur when both ice and air have temperatures below freezing and the air is unsaturated. Sublimation of ice at the surface would cool the air next to it in the same manner as melting cools it when the temperature of the air is above freezing. Since the present discussion is con- cerned only with melting surfaces, the question of forms produced by sublimation will not be pursued.



302 THE GEOGRAPHICAL REVIEW

mirror image of that which is set up over an upward-facing surface when it is heated to a temperature higher than that of the air above it.

CELLULAR CONVECTION

Movement of fluid away from a boundary surface and its replacement by fluid from a distance, however induced or maintained, always proceeds

A -B~~~~~ a a~ ~ ~~a ~

FIG. 2-Diagrammatic representation of the movements of fluids in cellular convection near a

boundary surface: A, in section normal to the surface; B, in plan. Key: a, centers of divergence at axes of

cells; b, intercellular zones of convergence and movement away from the boundary surface.

with some difficulty, since all movement normal to the limiting surface is impeded by the presence of the surface itself In an unstable body of air, as in all other fluids, only a negligible part of the exchange of air between the

layer next to the surface and the remoter layers proceeds by the diffusion of individual molecules. Except for this negligible molecular process, the elements of the fluid that effect the exchange are of larger dimensions. The exchange proceeds in a cellular fashion, such as is shown schematically in section normal to the boundary surface in Figure 2, A, and in plan in B of the same figure. The fluid approaches the surface relatively slowly, in polygonal columns, and withdraws from it more rapidly in sheets that intersect at intervals to enclose the columns or cells in which the fluid is

moving toward the surface. Movement away from the surface is most rapid at the places where the intercellular sheets of fluid intersect, where the cross section of the streams of fluid moving away from the boundary surface is largest. In Figure 2, A, the surface of the fluid is drawn bulged slightly outward in the cells, which are centered at points a, and depressed at the loci of return flow, b. This is the shape assumed by the free surface of a liquid in which active cellular convection is taking place. The plan of the cell represented in Figure 2, B, is drawn, perhaps with too great simpli- fication, as a regular hexagon; the intercellular zones in which the fluid

withdraws from the boundary surface are stippled. At and near the boundary



SURFACES OF MELTING ICE 303

surface the fluid in the cells diverges from their axes and converges toward the intercellular zones. The arrows in B show the directions of this divergent and convergent flow next to and parallel to the boundary surface. In accordance with the relative speed of motion in the cells and in the intercellular zones, the cells, in which the fluid approaches the boundary surface, occupy a good deal more area in a section parallel to the surface than do the intercellular zones, in which the fluid is moving away from it.

Cellular convection of the kind just described may be rendered visible at the free surface of a layer of viscous fluid heated at the bottom and may also be observed in numerous natural phenomena.3 It may be observed at the surface of a stream that carries a heavy suspended load, in which water rising from below with its heavy load of silt is easily distinguishable from the less turbid water sinking in the intercellular zones. Surface vortices, the most familiar evidence of turbulence in streams, occur in the intercellular sheets, most frequently at their intersections. Here the boundary surface is the free surface of the fluid itself In streams, however, the whole body of water is in motion downslope, and the convection is forced, not free as in a stationary unstable layer of fluid. The convection depends, in fact, on the flow and the consequent friction, both that between the stream and its bed and the internal friction of the water. When the fluid is in motion, the cells are distorted or may move with the general flow. Many clouds formed in unstable layers of air of limited thickness below temperature inversions that act as boundary surfaces display a cellular structure. One may sometimes see evidence of cellular convection in the air at the ground; for example, in pillars of steam fog formed in cold air moving over a warm water surface. Fog pillars are observed in the ascending elements of convection; that is, in intercellular sheets of air moving upward from the boundary surface, and in particular at their intersections. Their place in the system of convective movements is thus the same as that of surface vortices in turbulent streams- the place of the downward-moving elements marked b in Figure 2, A and denoted by stippling in B of the same figure. The air is usually in motion when steam fog is formed, and the cells and intercellular zones of convection are not stationary. I have seen stationary pillars of fog only once, but in a situation that may well occur frequently. This was in a rather thin "tule" fog in the San Joaquin Valley, California, in calm air on a winter day. The sun was fairly high in the sky, and its radiation, although weakened by the

3 Note, for example, the illustrations in Sobhag Mal: Forms of Stratified Clouds, Beitr4ge zur Physik derfreien Atmosphdre, Vol. I7, I930, pp. 40-66; and Sir G. T. Walker: Recent Work by S. Mal on the Forms of Stratified Clouds, Quart. Journ. Royal Meteorol. Soc., Vol. 57, I93 I, pp. 4I3-42I.



304 THE GEOGRAPHICAL REVIEW

fog, was warming the ground and inducing free convection in the lowest layer of air. Fields were thickly studded with slender pillars of fog. These pillars were nearly or quite stationary, but they differed in density and swayed irregularly with fluctuations in intensity of the convection.4

CELLULAR CONVECTION AT MELTING SURFACES OF ICE

Probably few overhanging surfaces of ice or firn are originally smooth. However formed, most such surfaces undoubtedly have initial irregularities. The salient elements of the initial irregularities must be the primary loci of movement of air away from the surface, just as positive elements of the relief of the land and such salient structures as sloping roofs are the primary loci of updraft of air when terrestrial surfaces are heated by sunshine. From the beginning of convection next to the melting surface, the re-entrant elements would thus be the places where air approaches the surface and diverges along it toward the salients. The melting effected by the air in its cycle of divergence from the centers of the convective cells and its return to the general body of the atmosphere would round out the depressions and sharpen the ridges and cusps between them, just as the licking tongues of cattle dissolve rounded depressions in the faces of blocks of salt set out for them. The central parts of the cells, where air approaches the surface, melt their way into the ice more rapidly than the peripheral parts because the air in contact with the ice is warmest there. Moreover, if condensation of water vapor from the air contributes to the melting process, it is the air at the axes of the cells, the air that is just approaching the surface, that has the highest content of water vapor. As the air diverges and moves along the surface of the ice toward the intercellular zones, from a toward b in Figure 2, A, it is progressively cooled; and if its original water-vapor content exceeds the saturation vapor content at freezing, it is progressively dried. It therefore yields progressively less heat to the melting surface as it approaches the margins of the cells, where it moves away from the surface. The ridges and cusps that separate the depressions mark the intercellular zones, the loci of movement of air away from the bounding surface, where the air in contact with the ice has least power to melt it.

The rounded depressions in the melting surfaces seem to approach uniformity in diameter. This phenomenon suggests that the initial irregularities cannot be held exclusively responsible for the features of cuspate

4 Some such phenomenon as this must have been in Christina Rossetti's mind when she wrote the line "Making the serried mist to stand afloat" in the third sonnet of the sequence "They Desire a Better Country."



SURFACES OF MELTING ICE 305

surfaces. The size of cells formed at a smooth boundary surface depends on the viscosity of the fluid and the intensity of convection. At melting surfaces of ice, and with convection induced by the difference in temperature between ice and air, these factors do not vary manyfold. It is to be expected, therefore, that when the surface comes into an approximate equilibrium with the convection of the adjacent air there should be no extreme differences in magnitude among the concavities in the surface that correspond to the cells of the convection. Such an approximate equality would be attained, first, by the breaking up of the circulations in initial re-entrants of the surface that are too large for the texture of the convection naturally set up in the air, so that these larger re-entrants would be succeeded, in the course of continued melting, by smaller ones. Second, those re-entrants that are too small for the texture of the convection would be eliminated by the expansion of adjacent ones closer to the size of the convectional cells appropriate to the difference in temperature between ice and air.5 Cellular convection thus seems competent to transform any initial surface, whether irregular or smooth, into one consisting of rounded concavities and cuspate salients of approximately uniform size.

Sharp describes and illustrates a transition from approximately spherical or paraboloid concavities, formed where the melting surface is nearly horizontal, to elongated ones that appear when the inclination of the surface increases. This transition is in strict accordance with the changes im shape of convectional cells that have been observed in experiments when a layer of fluid is set in motion. If a movement of translation into the plane of the page is imposed on the convective motion represented in Figure 2, A, the air near the bounding surface assumes helical paths. The cell centered at a, in which air approaches the surface and diverges, becomes elongated in the direction of flow. If the flow is slow, the polygonal cells may be merely stretched in the direction of flow, and still be terminated by transverse intercellular zones of convergence and motion away from the surface. With increasing speed the cells become strips continuous with the stream of fluid, separated by equally continuous lines of convergence and withdrawal from the surface, the lines into which the intercellular zones are drawn out. That is the pattern given by the superposition of flow into the plane of the page

5 It is to be presumed that the air temperatures at the warmest hours of the day would determine the predominant size of the concavities formed by the cells of the convection, since the ice melts most rapidly at these hours. But several days might be required to attain the equilibrium postulated in the text; where weather changes frequently during the summer, such equilibrium might never be attained. It is sufficient for the argument, however, that in a succession of days on which the midday temperature is approximately constant a condition of equilibrium is approached.



3o6 THE GEOGRAPHICAL REVIEW

on the motions normal to the boundary surface indicated in Figure 2, A. The melting resulting from a flow of air down the inclined surfaces of ice, superimposed on cellular convection, seems to be adequate to account for the transition from cuspate to fluted surfaces as the surfaces depart from a horizontal position.

FURTHER OBSERVATIONS NEEDED

It would not be profitable, without additional observations, to speculate further on the surface forms of ice to which Sharp has called attention. Questions arise, however, that are worth pursuing by anyone who has an opportunity to observe these forms. The following features seem worthy of further study:

i. The absolute relief of the cuspate surfaces. A priori, it would appear that the more completely cellular convection dominates the process of melting, the more deeply would the individual depressions in the surface be excavated. Forced convection produced on clean ice by wind, it seems, should produce less relief than free convection. Melting by absorption of radiation would also be fairly uniform over the surface. The greatest relief should therefore be found where the action of free cellular convection is most nearly exclusively responsible for melting; that is, in situations most sheltered from wind and radiation.

2. Diameter of the concavities. The size of the convective cells should be fairly constant, since it is primarily dependent on the viscosity of the air. But it is to be expected that some depressions grow more rapidly than, and at the expense of, others. Do the convective cells in large depressions then become unstable and break up into smaller ones? If a surface could be watched at close intervals from the time when it is first exposed, or from the time when melting begins on it, the evolution of the fully formed cuspate surface could be traced, and its relation to initial irregularities defined. Surfaces might be smoothed artificially and observed to determine whether, as would happen according to the hypothesis set forth here, cuspate surfaces appear spontaneously and independently of initial irregularities.

3. Possible differences in surface forms produced by differences in temperature and humidity of the air. The nature of the convection close to the melting surface could probably be determined by no more elaborate an apparatus than a lighted cigarette, the smoke from which would reveal the manner in which air moves in the layer close to the surface. But the intensity of convection would vary with the difference in temperature between the air and the surface, and the rate of melting with this difference and the absolute humidity of the air. The curious inquirer would therefore like to see the temperature and humidity of the air observed along with the forms of ice surfaces, since the rate of melting may affect the shapes and sizes of those forms.

Other questions will suggest themselves to the reader. These few are set forth only for the purpose of sharpening the eyes of observation, which still has much to record concerning the forms assumed by melting ice.



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Cuspate Surfaces of Melting Ice and Firn