Introduction
It has long been known that local mass balance (net snow accumulation) variation is closely related to local topographic variation on inland ice sheets (Reference SchyttSchytt, 1955; Reference SwithinbankSwithinbank, 1959; Reference Black and BuddB’ack and Budd, 1964; Reference Gow, Gow, Blander de, Crozaz and Picciotto.Gow and others, 1972). Where topography is pronounced this may be due to orographic effects on precipitating air masses. However mass balance is related to slope even far from sources of precipitation and where slope changes are very small (Reference Seckel and StoberSeckel and Stober, [1969]), Mass balance is, moreover, greater where slope is less, and the reverse relationship would be expected from orographic interference on air masses. Mass redistribution by blowing snow and surface drift must be important in these places, but slope changes are so small that an adequate explanation for the balance pattern has been lacking.
Wind Response to Slope Change
Reference Mahrt and SchwerdtfegerMahrt and Schwerdtfeger (1970) have studied the effect on surface winds of a surface temperature inversion. Temperature inversions are very common on ice sheets and they cause steady inversion winds, a kind of katabatic wind, inland and the related gusty-katabatic winds near ice-sheet margins. The cold surface air is denser than air of the order of 100 m above the surface. Due to the slope of the snow surface, air is also denser than air at the same elevation, in the down-slope direction. This latter density contrast produces a thermal wind. Mahrt and Schwerdtfeger included friction effects and allowed for wind above the friction layer (the upper geostrophic wind), and have obtained an equation for inversion wind velocity under steady-state conditions.
Inversion winds are thus governed by the slope of the snow surface and will be stronger and the wind shear will be greater where the slope is steeper. Faster wind speeds result in more snow removal by drifting and blowing snow. In the case of undulating topography, annual snow thickness will be greater where the slope is less.
In order to investigate the applicability of this hypothesis we first analyze Mahrt and Schwerdtfeger’s inversion wind to determine its sensitivity to topographic variation and then, using an empirical snow-drift formula, show that calculated wind-speed changes can explain the observed mass balance variations in upper Marie Byrd Land, Antarctica.
Reference Mahrt and SchwerdtfegerMahrt and Schwerdtfeger (1970) were able to obtain an analytic solution for the surfacewind regime by approximating the temperature profile above the snow surface by an exponential relationship:
In which
and T is the temperature at height z above the snow surface, T0 is the temperature at the top of the inversion, ДT0 is the temperature difference between the top and the bottom of the inversion, K is the coefficient of eddy diffusivity, f is the Coriolis parameter; and sis an empirical dimensionless constant without a simple physical meaning which Mahrt and Schwerdtfeger set equal to unity on the basis of data from South Pole and Plateau Stations, Antarctica (Reference SchwerdtfegerSchwerdtfeger, 1968,1970). For generality, we will carry s through the computations.
The pressure gradient force due to a temperature inversion on a very gently sloping surface is
in which C = g▵T0/fT and g is the magnitude of the acceleration due to gravity, T is mean temperature, and α is the surface-slope vector, directed up the slope.
Reference Mahrt and SchwerdtfegerMahrt and Schwerdtfeger (1970) solved the equation for dynamic equilibrium. For the southern hemisphere, expressing vectors by complex numbers (i = (–1)½), their solution for air velocity is:
where J = (2s)−1 and u gh is the upper geostrophic wind velocity. This is essentially an Ekman spiral with the effect of the thermal wind due to the temperature inversion included.
The constants b 1, and b 2 are determined by the boundary conditions, u = o at z = o, and ∂2 u/∂z2 = 0 at the top of the inversion layer at z= h, (or z∗= h∗). From which:
The constant b 1, is negligibly small for most practical examples.
Consider the reaction of this wind to slope changes. A given wind u, would be in equilibrium with a slope α. And α may be found from Equations (1) and (2):
which is also usually negligibly small.
If the actual slope is α’, then there will be an unbalanced pressure gradient force:
causing an acceleration of the air mass:
where t represents time.
Substituting in this, one obtains a differential equation in u:
whose solution is of the form:
In which u 0 is the velocity vector that would be in equilibrium with α, and 1/τm and i/τd are the real and imaginary parts, respectively, of the coefficient of u in Equation (3).
For high latitudes, wind near the snow surface (z= 1 m), K= 10_1 m2s_1, and s in the range 0.5 to 10, which fits all the temperature profiles in Reference SchwerdtfegerSchwerdtfeger (1970), the decay constant for wind speed τm is of the order of 102 to 103 s, and for any example the wind direction decay τd is several times longer. The typical inversion wind of 10 ms–1 is thus sensitive to slope changes over distances of the order of 1 to 10km and responds first by a change in wind speed rather than in wind direction.
The response times depend chiefly upon the Coriolis parameter, the coefficient of eddy diffusivity, the parameter s, and height. Response is fastest near the snow surface. Neither the strength of the inversion nor the surface slope affects the response time.
Since ice sheets occur at high latitudes, the Coriolis parameter varies within narrow limits and the decay times and hence the size distribution of topographic features affecting inversion winds will be everywhere much the same.
Snow Transport
Reference Budd, Budd, Dingle, Radok and RubinBudd and others (1966), in their study of snow drifting at Byrd Station, Antarctica, found that more snow is carried by faster winds, according to:
where Q is the mass flux of snow passing through a section indefinitely high, perpendicular to the wind direction, and u10 is the wind speed at z = 10 m. On steeper slopes winds are faster and more snow can be transported. Where the slope decreases, by an amount –▵α, snow will be deposited at a rate:
In upper Marie Byrd Land, West Antarctica, slopes change by 0.001 in 3km. Such a change in slope changes the inversion wind at the 10 m height by 1 ms−1 (Equation (1)), and the mass of drift in transport by more than 10−2;kg m−1 sl. If this snow is deposited over that same distance 0f 3km, local mass balances should vary by 100kg m−2 a−1. Measured mass balances for a whole year vary by about 20kgm−2 a−1 (Fig. 1), which could be explained by this mechanism operating for 20% of the year.
Fig. 1. Surface balance, < ba>, and maximum surface slope αm (plotled upside down for convenience) against distances from the ice crest. Both are means for 3km by 3km squares. Data from a traverse north-east of “Byrd” station, Antarctica.
Discussion
Wind and temperature data for “Byrd” station are available, but have not been studied with respect to inversion-wind frequency, but it is certain that they occur during more than 20% of the year (Reference Rubin, Weyant and HathertonRubin and Weyant, 1965, fig. 103, p. 380). The mechanism described above is more efficient than the data indicate. Perhaps the complementary process of snow erosion and inclusion in drift transport has a much longer time constant than that for snow deposition. The explanation may be more likely that, in the absence of snow precipitation, snow erosion on the steeper slopes becomes more difficult with time, as older and more sintered snow is uncovered, so that snow redistribution is principally confined to occasions of inversion conditions with current or very recent snow precipitation.
The balance variations will, with time, affect the topography. Reference Black and BuddBlack and Budd (1964) treated this problem by assuming that balance variations are proportional to slope variations. Equation (5) supports this assumption because the derivative du10/dα obtained from Equations (1) and (2) does not depend on slope α. Their treatment applied to our data (Fig. 1) suggests a migration of the undulations up-strcam at about 20 ma−1, Against this tendency, the flow of the glacier carries the undulations down-stream at 5 to 10 ma−1 (Reference WhillansWhillans, 1973) in the region where the undulations are best developed. The significance of this result is difficult to evaluate because any phase differenceFootnote * between balance and slope variations reduce the speed of migration, and because variations in ice-flow character such as those suggested by Reference NyeNye (1959), Reference RobinsonRobinson (1966), Reference Robin.deRobin (1967), Reference Robin.de, Robin.de, Swithinbank and SmithRobin and others (1970), and Reference BuddBudd (1968,1970) are not considered in the theory.
Conclusion
(1) Inversion winds are sensitive to slopes measured on the scale of 1 km or longer and respond to changes in slope first in wind speed and second in wind direction change.
Short-length features such as snow drifts and sastrugi do not affect inversion winds. These features are well known to be transient both in size and position.
On longer scales changes in wind speed in response to slope change alter the snow-carrying ability of the wind and snow is preferentially deposited where slope decreases.
Over 20km and longer, the general shape of the ice sheet is affected by snow removal by inversion winds (Reference Mahrt and SchwerdtfegerMahrt and Schwerdtfeger, 1970) but is also governed by ice-flow and climatologic considerations (for example Reference NyeNye, 1959; Reference HughesHughes, 1973). Note that the balance is highest in the region of the ice divide where inversion winds are not so well developed due to smaller mean slope.
(2) On the scale of the reaction length of wind to slope change, about 3km, snow accumulation is greater on the gentler slopes of surface topographic undulations. In time, this can affect the topography (Reference Gow and RowlandGow and Rowland, 1965), and waves in the surface topography of wavelength of two times the reaction length, or about 6km, should form. We suggest that surface undulations of this wavelength (Fig. 1) are due to this mechanism. The positions of the undulations are however probably controlled by ice base irregularities and ice-flow variations,Footnote † but the form of the undulations is controlled by inversion winds.
Acknowledgements
I thank C. Bull, R. A. Hamilton, T. Hughes, M. McSaveney, and W, Schwerdtfeger for help and comment. This work was supported by National Science Foundation Grant GV-26137X awarded to The Ohio State University Research Foundation and the Institute of Polar Studies.
