1. Introduction
The transport of snow by the wind plays a prominent part in many problems of applied glaciology. Two main problem regions may be distinguished. In mountains, drift-snow forms a major nourishment of glaciers and a prime ingredient of avalanches; drift-snowa-ccumulations also have potential or real value as a form of hydrological storage. On the featureless plains of northern Eurasia and America, drifting snow creates problems for road and rail traffic and, by eroding a heat-insulating surface cover, for agriculture. Furthermore, surface structures tend to create snow accumulations which can interfere with their use or even bury them completely. This is an acute problem on polar ice sheets where the supply of dry loose snow is virtually unlimited and snow drift an incessant phenomenon.
The greater simplicity of the air flow over a featureless surface has made it easier to gain an understanding of basic snow-drift processes there rather than in the mountains. As a result, most systematic work on snow drift has been carried out on the Siberian plains and on the polar ice sheets of Greenland and Antarctica. However these studies have emphasized different aspects. In Siberia, snow is a seasonal problem and has practical interest. On the polar ice sheets the persistent drift-snow forms a potentially significant part of the mass balance and perhaps of their very origin. Reference BergeronBergeron (1965) has argued in detail how a nonlinear increase in snow-drift flux with wind velocity, suggested by all observational and theoretical models (cf. Section 3 below), could account for the observed precipitation regions and the thick ice of central Greenland and Antarctica. But this has been questioned by Loewe, who first formulated the quantitative aspects of snow drift from his own observations during the Wegener expedition to Greenland (Reference LoeweLoewe, 1933), and for Antarctica from his own observations during the second expedition of Expeditions Polaires Françaises to Terre Adelie (Reference LoeweLoewe, 1956), before combining all the available evidence in the most authoritative estimates to date of the drift-snow losses from Greenland and Antarctica (Reference LoeweLoewe, 1970).
These results will not be discussed here since they are somewhat outside the framework of “applied” glaciology; nevertheless they have been the driving force behind the search for a theory of drifting snow that can do more than provide rough estimates for the quantity of snow likely to be collected by a snow fence in prescribed climatic conditions. Any review of snow drift as a problem in applied glaciology must take note of the somewhat esoteric polar studies because they have provided the keys to such more special questions as visibility, electrical phenomena, evaporation, etc.
The reviewer's task is complicated by the fact that the subject has developed in two largely independent streams, reported in different languages (English and Russian). Thus even a simple comparison of results becomes a laborious undertaking, and disputes over subtleties, intractable. Nevertheless, the merging of the two streams for the benefit of applied glaciology seems an urgent necessity, and the present review makes one step in that direction.
A starting point is provided by two monographs covering all its aspects (Reference DyuninDyunin, 1963; Melior, 1965). A further source of great value is a bibliography compiled by Reference GoldGold (1968). These publications have made it possible to devote this review mainly to more recent advances.
2. Drift processes and their measurements
Two distinctly different processes are involved in the drifting of snow. In weak winds the snow particles glide or skip along the surface and only here and there are concentrated into streams deflected upward by surface irregularities. This “saltation” mode of motion has been beautifully recorded on film by Reference KobayashiKobayashi (1972) at velocities of around 5–6 m/s at the 1 cm level (cf. Fig. 1). Quantitative analysis of such film records shows the snow concentration in the lowest millimetre above the surface to be around I particle per cm3 and to decrease, at these wind velocities, to one hundredth of that value at a height of 1 cm above the surface.
Giné pictures of snow saltation (Reference KobayashiKobayashi, 1972).
When the wind velocity increases above a threshold which depends on the nature of the snow surface, and perhaps other factors such as temperature, the snow drift begins to go into a state of suspension by turbulent diffusion. Threshold velocities have been quoted by various authors as generally falling within the range from 8–16 m/s. At larger velocities the layer of the atmosphere with drift-snow takes on the appearance of dense fog, and snow plumes can rise hundreds of metres even over level ground. Such snow drift typically is not uniform but pervaded by snow “trombes”, eddies with vertical axes, which were described already by Reference WegenerWegener (1911) and are an impressive sight under special illumination. The reviewer recalls a night near the coast of Terre Adelie in 1952 when from the ship the snow drift against the bright sky above the dark outline of the ice sheet, some 15 km away, resembled protuberances during a solar eclipse. An incessant sequence of snow trombes seemed to grow to heights of a hundred or more metres in a matter of seconds and remained visible for many tens of seconds while slowly moving down the ice slope.
More commonly observed are localized drift clouds associated with topographic flow disturbances or internal flow instabilities akin to hydraulic jumps. Their practical significance outside the polar setting is to underline the large height range in which appreciable snow quantities can be transported by the wind. This clearly makes the problem of measuring the drift flux much more difficult.
For the measurement of drift-snow flux a variety of mechanical traps have been used since Reference AndréeAndree (1886) made the first measurements on Spitsbergen during the first Polar Year. A systematic comparison of the collection efficiency of different types of mechanical drift traps was reported by Reference Budd, W. F., W. R. J., U. and RubinBudd and others (1966). The majority of existing drift measurements have been made with two traps: the B02 trap (cf., e.g. Reference DyuninDyunin, 1963, fig. 24) favoured by Soviet workers, and the rocket-shaped trap developed by Reference MellorMellor (1960). A continuously recording version of the Mellor trap has been described by Reference Tabler and JairellTabler and Jairell (1971). An operational comparison of the efficiency of these devices in collecting polar snow in the wind range 9–14 m/s at the 1 m level showed the Mellor trap to collect 37% more, on the average, than the BO2 trap. This difference must be kept in mind for any comparison of Siberian and Antarctic results.
More recently photoelectric methods have come into use for measurements of snow concentration. These do not interfere with the free air stream and provide the drift orientation directly; on the other hand, for drift flux determinations, care must be taken to combine them with suitable wind measurements (cf. Section 4).
The principle of photoelectric drift-snow gauges was discussed by Reference Landon-Smith and Wood berryLandon-Smith and Woodberry (1965) and Reference BelovBelov (1971); actual designs have been published by Reference WishartWishart (1965) and Reference Bird, I. G., J.J. and E. R.Bird and others (1971). A particularly promising photoelectric drift gauge developed by R. A. Reference Schmidt and SommerfeldSchmidt and Sommerfeld (1969) counts individual drift particles and measures their velocity, a quantity of basic importance for the understanding of snow-drift dynamics. According to a personal communication from Dr R. A. Schmidt, this gauge is now being used operationally in drift monitoring.
Another key factor, the particle size, has not yet been tackled with photoelectric means and presents special measurement problems in blizzards. When particles are caught directly on velvet or on slides coated with a form-var-polyvinylchloride solution the particles tend to cluster and sinter almost instantaneously, giving rise to inflated sizes. To overcome this difficulty, Dingle during the “Byrd” drift project (Reference Budd, W. F., W. R. J., U. and RubinBudd and others, 1966) constructed a special particle trap which provided him with some 3000 individual particles and detailed spectra for different heights. Some doubts have been raised about these in view of possible systematic differences between formvar replicas and the original particles (Reference DyuninDyunin, 1974), but a thorough study of replicas and their changes with time by Reference BattyeBattye (unpublished) has revealed no such effects. Nevertheless, direct size measurements of airborne particles represent a significant need in snow-drift research, in view of the decisive role played by the particle size spectrum in the rival theories of snow drift, which will now be reviewed.
3. Theories of snow drift
Two main theories of drifting-snow processes have developed independently in Siberia and in Australia. The principal Soviet worker, A. K. Dyunin, has named them the “dynamical” and “diffusion” theories; their basic ideas go back to the work of Reference BagnoldBagnold (1941) and Reference SchmidtW. Schmidt (1925), respectively. Their most telling distinction is in terms of dominant processes and vertical scales: the Siberian theory views snow drift as a near-surface phenomenon due to small eddies in the lowest 10 cm producing mainly saltation; the Australian theory (more properly described as relating to conditions on polar ice sheets) attaches the main importance to the larger eddies in the free air stream extending to tens or even hundreds of metres above the surface. The occurrence of drift particles at such heights is not in dispute, only the proportion of the total drift transport taking place in the different layers.
The only comparison to date of the two theories has recently been attempted by Reference DyuninDyunin (1974) almost exclusively in terms of estimates for the total transport Q as function of wind speed. Since these estimates and the underlying measurements were made with a variety of techniques, some discrepancies must be expected, and it seems essential, for a valid comparison of the theories, to go back to their basic arguments. This is especially important for the Siberian theory, which so far lacks a critical exposition in English. Such an exposition of Dyunin's theory is attempted here on the basis of the most recent account by Reference DyuninDyunin (1974), which has been translated in part for the purpose.
The basic equations of the two theories are of somewhat marginal relevance from the practical point of view, and their discussion has therefore been relegated to an Appendix. Each theory results in an expression for the drift-snow concentration involving a mean eddy flux term of the form 
where the variables have been assumed as sums a′ + a″ of a mean component a′ and a deviation a′, with 
and 
. In the diffusion theory that relationship is
where ω is the particle fall velocity (m s-1) and nz the drift-snow “density” (g/cm3) at level z (m). The corresponding relationship in the dynamic theory is
with i = ωgn, and I, W being pure air-flow components for which the Appendix discussion suggests simpler alternatives. This is not a major issue since for further progress the eddy terms in (A. 11) and (12′) are expressed by mean flow parameters.
For this parameterization the “diffusion theory” used a gradient-flux relationship of the form
and the working hypothesis that the eddy diffusivity for drift-snow K s does not differ significantly from that valid in neutral conditions for momentum as well as heat and moisture, viz.
where k is von Kármán's constant and 
is the friction velocity. The corresponding parameterization of the “dynamic” theory is not discussed in Dyunin's latest paper but can be found in the monograph (Reference DyuninDyunin, 1963). An eddy layer close to the surface is postulated where Bernoullian pressure reductions produce entrainment of snow particles. The pressure deficit has the form 
and vertical gradient ρv 1 (∂v 1/∂z). Since the horizontal velocity v, vanishes at the surface and its gradient becomes relatively small at some quite low level, the eddy stress in Equation (A. 12′) and hence the drift concentration j should, on this argument, reach a maximum very close to the surface and be proportional to v 12. Dyunin derived a quantitative representation from a simplified boundary-layer shear profile (Reference DyuninDyunin, 1963, equation (128)). Two of his deduced drift-density profiles are shown in Figure 2; for comparison, a rather striking verification of the vertical density profiles predicted by the “diffusion” theory has been added from Reference BuddBudd's (1966[b]) study. Reference KobayashiKobayashi's (1972) measurements at much lower wind velocities down to 1 mm above the surface do not show Dyunin's maximum which theoretically would be expected to occur at a fraction of the roughness length (Mellor and Radok, 1960).
Drift density profiles predicted by the " dynamic" (left) and "diffusion" (right) theories qf snow drift.
From Dyunin's drift-density expression above it follows immediately that the drift flux close to the surface must be proportional to v 13. Observations and a more refined argument suggest that a threshold pressure deficit or velocity must be exceeded to break the bonds between the surface snow grains, and in due course this leads to Dyunin's well-known drift transport relationship
By contrast, the diffusion theory of snow drift without further assumptions yields the quantitative details of the drift density as function of height or wind velocity; for constant particle fall velocity ω these are (Reference Dingle and RadokDingle and Radok, 1961)
and
where z 0 is the roughness length and Z a reference level. An interesting corollary is that measurements of nz(vz) over a restricted height range can be represented with practically the same precision by one of the popular power laws of the form
but the advantage of Equation (5) is that it predicts the values of the exponents α to be expected for different levels; thus Dyunin's value of α = 2 would be expected, in polar conditions, near the 10 cm level, and the value α = 5 deduced by Reference LiljequistLiljequist (1957) from visibility data, near the 2 m level (cf. Reference Dingle and RadokDingle and Radok, 1961, fig. 2).
The cost of all this information is a more involved transport formula which must be found by vertical integration as
Even with the assumption of a constant mean particle fall velocity ω this gives rise to rather complex expressions; but in reality both the composition of the drift and its bulk concentration per unit mass are functions of height and wind-speed. Tractable expressions for the change in the total drift transport with wind velocity of a single reference level have therefore frequently been constructed by fitting simple power laws to sums of partly measured and partly calculated layer transports. An example of such a formula is that derived by Reference Budd, W. F., W. R. J., U. and RubinBudd and others (1966) for the transport through a vertical surface I m wide at right angles to the wind from I mm to 300 m above the Antarctic ice sheet at “Byrd” station,
Such simplifications have proved useful in broad assessments of the role of snow drift in the Antarctic mass balance (Reference LoeweLoewe, 1970), and in explaining the snow accumulation up-wind of an elevated building on the Greenland ice sheet (Reference RadokRadok, 1968) but should not be used as evidence against the validity of the diffusion theory, as has been done by Reference DyuninDyunin (1974), By the same token, the diffusion theory does not invalidate the important results produced and reported by Dyunin, Kobayashi, and others from field measurements and wind-tunnel experiments on the drift flux close to the surface, in terms of net accumulation or snow collected in ditches and horizontal containers. However, the arguments here presented do show that unless one wishes to study the detailed and distinct behaviour of snow particles and of the surrounding air, the approach from the momentum equation is an unnecessary detour.
The real power of the diffusion theory comes from its detailed predictions about drift concentrations and from the verifications by direct measurements of that relatively simple quantity. For details it is necessary here to refer to Reference MellorMellor (1965) and to more recent specialized accounts by Reference Budd, W. F., W. R. J., U. and RubinBudd (1966[b]) and Reference RadokRadok (1970). Reference KobayashiKobayashi's (1972) measurements confirm the theoretically predicted order of magnitude of the surface-layer drift concentration, and measurements with the photoelectric particle counter have suggested comparable drift-particle and wind velocities (personal communication from R. A. Schmidt). This further supports the diffusion theory, and makes it the essential stepping stone to the understanding of various unsolved basic problems of snow drift. Some of these will be discussed in the next section.
4. Some basic problems of snow drift
The first unsolved problem to be mentioned concerns the vertical scale of snow transport. Transports found by extrapolation to layers above measurement levels are as large or larger than those found near the surface. Observational confirmations of the computed drift concentrations at high levels are needed to round off the otherwise satisfying verification of the diffusion theory. Such high-level measurements seem well within the capabilities of photoelectric devices. In using this type of system for estimates of drift flux during a time T at a given level
it will be necessary to multiply the outputs of anemometers and drift-density gauges with similar time resolutions, rather than use average winds and densities whose product neglects the potentially significant eddy contribution
As a second basic problem, electric phenomena associated with snow drift have been noted since the beginning of polar exploration (e.g. Reference SimpsonSimpson, 1919-23, Vol. I, 1921, p. 309; Reference SheppardSheppard, 1937) but present rather difficult measuring problems. The most advanced study to date was made by Reference WishartWishart (1970) who measured bulk as well as individual particle charges and currents created in wires strung across the wind. The wire currents were found to be substantially larger than the charge carried by the drift-snow, suggesting the additional presence of negative ions as well as space charges. Further measurements are needed to confirm these and to establish the charging mechanisms responsible for blizzard electricity; an elaborate programme has been drawn up by Reference WishartWishart (unpublished) for this work. No backer has as yet been found for this but the links between blizzard and cloud electricity, the role of blizzard discharges in the terrestrial ozone balance, and the practical implications of blizzard statics for radio communications, will all in due course demand greater attention for this field of study.
The final basic problem to be mentioned may also play a role in blizzard electricity but has great practical significance in its own right. This is the evaporation of drifting snow, a well-researched though as yet largely untranslated topic in the Russian literature, which has recently also caught the attention of American workers. To do it full justice would therefore require the comparative approach of Section 3, but this will have to be done in a separate paper. Here it must suffice to mention the most important recent results and their practical implications.
The majority of these stem from an ambitious attempt by Reference SchmidtSchmidt (1972) to model all the major processes contributing to the evaporation (termed “sublimation” by him) of single drift particles. The study finds that at drift densities occurring in natural blizzards interaction effects are negligible, except possibly in the lowest few millimetres above the snow surface. The evaporation rate appears to double for each IQ deg temperature rise in the range from –20°C to 0°C and more than double when the particle diameter is doubled. The percentage mass loss in unit time by evaporation, however, increases markedly with decreasing particle size. The evaporation rate is proportional to the saturation deficit when conduction and convection are considered alone, and doubled by allowing for solar radiation effects. Altitude and particle-shape effects appear to be of lesser importance, but further work by Reference LangLang (unpublished) has brought out the important role played by atmospheric turbulence (personal communication from R. A. Schmidt).
The theory will no doubt be extended in due course by more sophisticated treatments of the radiation transfer in snow drift and of diabatic stability conditions. Even without this, the inferred humidity and temperature profiles corresponding to steady-state steadily evaporating snow drift are of considerable importance for practical snow-control considerations (cf. Section 5 below). They also point the way towards an improved model of the katabatic wind systems of polar ice sheets which may be regulated by the interplay of adiabatic heating with radiational and evaporative cooling in the associated snow drift (Reference RadokRadok, 1973).
5. Control of snow-drift accumulation and erosion
Techniques and experiments in controlling snow-drift accumulation and erosion long preceded the more fundamental studies discussed so far and are well summarized in Reference MellorMellor's (1965) monograph. Reference DyuninDyunin (1963) provides a great deal of additional field and wind-tunnel information on the quantitative effects of snow fences and vegetation barriers on the nearby wind field; this needs to be translated and compared with the independent, more recent experimental results of Reference MartinelliMartinelli (1973[a]) and Reference Tabler and VealTabler and Veal (1971).
Reference MartinelliMartinelli (1973[b]) has summed up the features that have been found useful in practical tests. Reference TablerTabler (1974) has reported in more detail on the quantitative aspects of fence design and efficiency. According to Reference MartinelliMartinelli (1973[b]) the trapping efficiency of a fence appears to increase with fence height H up to about H = 3.4 m, close to the practical upper limit. The fence “density” (solid percentage cross-section area of the fence) should be between 40 and 50%; slightly higher values have been recommended by Reference DyuninDyunin (1963) who uses a “transparency” factor defined as (100—density). The main requirement is that the fence must remain permeable; this condition also suggests the size of the bottom gap, in the range 0.1 H to 0.15H. For a fence with bottom gap equal to 0.IH experience shows that the final lee drift will vary in length from 10H (100% density) to 15–17H (50% density). Height and density have similarly systematic effects on the maximum depth and its location along the drift. Fences should be at least 2H long and continuous; inclined and tandem fences are not significantly better.
In special contexts the relocation of a fence has been used to create drift accumulations of up to 10 m (cf. Mellor, 1965, p. 52). In principle it ought to be feasible, if perhaps not cost-effective, to design a fence which rises automatically as the snow settles behind it.
With reasonably general agreement on what makes an efficient trap for surface drift, the main current research is going into the problem of evaporative losses and their control. This topic also occupied a good deal of Reference DyuninDyunin's (1963) monograph, in addition to his treatment in an earlier work (Reference DyuninDyunin, 1961), but both await being adequately translated. In the English literature Reference SchmidtR. A. Schmidt's (1972) seminal modelling discussion was preceded by the more pragmatic approach of Reference TablerTabler (1971) which has not been accessible to this reviewer but appears from later summaries, closely to parallel that of Reference DyuninDyunin (1963).
The principal new concept is the “transport distance” R m (Reference TablerTabler, 1971), called by Dyunin “effective length of snow-collecting basis, L s”". This is defined as the distance over which the average drift particle is completely evaporated and differs in general from the “contributing distance” (R c in the English literature, L s in Reference DyuninDyunin (1963)). The amount of drift-snow collected by a fence is the fraction θ of the precipitation P which is blown off this surface over the distance R c, reduced by the evaporation loss. θ is called “relocation ratio” by Tabler and “blow-off coefficient” by Dyunin. In these terms the snow collected becomes (Reference TablerTabler, 1975[a])
or (Reference DyuninDyunin, 1963, equation (273))
where I s is the evaporation rate per unit saturation deficit Δe. The second term on the right-hand side equals the total evaporation loss, θPR c2/2R m.
Typical values of θ are quoted by Reference TablerTabler (1975[a]) in the range of 0.5–0.9 for Wyoming, and by Reference DyuninDyunin (1963, table 39, p. 274) from 0.37 to 0.88 for Siberia. The magnitude of R m is of the order of a few kilometres and similar to that of R c in natural condition. In order to reduce evaporation loss in areas where drift accumulation is desired, the natural contributing distance R c can be subdivided by additional fences, but this evidently must be judged in terms of total cost effectiveness.
The foregoing approach has recently been considerably refined by Reference TablerTabler (1975[a]) with Reference SchmidtSchmidt's (1972) drift evaporation model. The revised procedure is based on the particle-size distribution measured in Antarctica by Dingle and reduced to a handy mathematical form by Reference BuddBudd (1966[b]). Figure 3 shows schematically the amount of detail that can now be handled; q s and P r are the water-equivalents of snow stored and precipitated, respectively, and need to be measured or estimated. Comparison of calculated and observed drift accumulations show very satisfactory agreement, for all practical purposes.
The control of drift accumulation (Tabler, 1975); for detaiLs see text.
Progress has also been made with the corresponding problems of drift accumulation by natural topographical features and by substantial structures. A very general attack, by Reference Berg and CaineBerg and Caine (undated), on the first problem has not yet been completely successful in predicting the shape of natural drift accumulations, but points the way to a complete solution. For practical purposes a multiple regression formula involving terrain slopes appears to have achieved considerable success (Reference TablerTabler, 1975[b]). This empirical approach could be extended by means of sequences of aerial photographs taken during a melt season. Such photographs would show the most persistent drift accumulation which could then be systematically analysed in terms of mapped topographical parameters and weather records for the preceding accumulation season.
The problem of drift-snow accumulations on buildings and structures also has not yet found its definite solution. Following the progressive obliteration of several I.G.Y. stations in Antarctica, two different remedies were explored, viz. underground and elevated construction. The latter approach spawned a lot of wind-tunnel work and the exploration of the similarity laws of snow drift (Reference OdarOdar, 1965). A review of problems and possibilities in this area has recently been made by Reference NoremNorem (1975). However, it is not practicable to model basic snow processes such as sintering and cornice formation, and difficult in most wind tunnels to produce realistic boundary-layer depths of several times the height of the model building. Hence more recent wind-tunnel studies have mostly limited themselves to deducing the likely regions of snow deposition from the structure of the wind field and the surface stress (Reference Melbourne and StylesMelbourne and Styles, 1967).
Greater realism can be achieved in this context by gradually building up the tunnel surface in regions of Row stagnation (Reference CermakCermak, 1966). A useful alternative has been provided by observations on relatively large model buildings set up near, or exposed to similar field conditions as, the actual buildings (Reference Tobiasson and ReedTobiasson and Reed, 1966).
A major result of these studies was to throw doubt on the original reasoning behind the elevated-construction approach. In line with the central tenet of the “dynamical” drift theory, this was that the bulk of the drifting snow remains close to the surface and should be able therefore to pass through unhindered below a raised building. Against this, Reference RadokRadok (1968) showed that the drift accumulation up-wind of a radar building on the Greenland ice sheet incorporated broadly all the snow-drift flux divergence in the layer between the surface and the top of the building, almost 100 ft (30 m) above the surface.
The drift accumulations on this and another radar building on the Greenland ice sheet have remained under constant surveillance (Reference Tobiasson, W., G., K. and A.Tobiasson and others, 1972, Reference Tobiasson, W., R. and W.1975) and developed characteristic shapes at the two sites (Fig. 4). These may be due to the different surface slopes of the ice sheet; although small, such differences in slope and also in surface curvature can have very marked effects on the snow accumulation rate (Reference Budd and RubinBudd, 1966[a]) which might be further accentuated by an obstacle. Another example is provided by the elevated building at "Casey" station, Antarctica, where the drift accumulation each year tends to take quite different shapes along two wind flow lines only 400 ft (120 m) apart (Fig. 5). It would be interesting to apply the statistical relationship developed by Reference TablerTabler (1975[b]) to these topographical situations.
The accumulation of drift snow on two radar stations in Greenland (Reference Tobiasson, W., R. and W.Tobiasson and others, 1975).
The accumulation of drift snow along two windjlow lines past the station building at Casey, Antarctica (Reference Melbourne and StylesMelbourne and Styles, 1967, and Antarctic Divisioll data).
The main practical problem in the case of elevated structures is whether especially the up-wind drift accumulation can be allowed to build up to an equilibrium shape without blocking the space below the building. An experiment is currently being carried out at “Casey” station by P. Keage of the Australian National Antarctic Research Expeditions to measure the rates of build-up along lines Band F of Figure 5 during individual blizzards. His preliminary measurements have already shown that a substantial slowing-down of the flow occurs during the approach to the building even in the layer seemingly free to move through the gap. This is accounted for by the general pressure pattern up-wind of such structures, as established by Reference Melbourne and JoubertMelbourne and Joubert (1971) and schematically shown in Figure 6.
The jlow around a tail building with bottom gap (Reference Melbourne and JoubertMelbourne and Joubert, 1971).
6. Conclusion
It has not been possible here to do more than touch on the main current activities in the field of snow-drift research and operations, and in particular to mention the diversified related work going on in mountainous regions. But the advances made with quantifying the effects of natural flow around surface obstacles, and with the understanding of drift evaporation, provide an assurance that, as a problem in applied glaciology, drifting snow is well under control in the plains. The mountains, however, are a different matter.
Acknowledgements
A great deal of useful material for this review was provided by A. K. Dyunin and R. A. Schmidt; the necessary typing and drafting was carried out by Vonne Murphy, Kim Bay, and Russell Littleson. All this help is gratefully acknowledged.
Appendix Basic equations of two theories of snow drift
Reference DyuninDyunin (1974, p. 10–13), to explain the “dynamic theory”,
“proceeds from the dynamical equation for a two-phase solid flow. The vertical component of that equation for the particular case of a plane current moving a long a horizontal surface has the following form
where g is the acceleration of gravity, m/s2, i, q are the mass fluxes of drift-snow in the direction of the coordinate axes z (vertical) and x (horizontal), respectively, in kg/m2 s, t is the time, s, w is the average vertical velocity of the snow particles, m/s, s is the volume concentration of the solid particles, γs, γ are the specific weights of snow and air, kg/m3, ε is the total acceleration of other mass forces, e.g. the force of electrical attraction and repulsion between charged particles m/s2, p is the pressure on a horizontal surface kg/m2, τ is the shear stress along a vertical surface at the given point of the flow, kg/m2” [translated from Reference DyuninDyunin, 1974].
Although subsequently simplified out of recognition this equation needs a closer analysis as basis and starting point of the “dynamic theory”.
To arrive at (7′) it appears that the author has started from the vertical equation of two-dimensional (x, v horizontal; z, w vertical) motion of a “two-phase” snow-air mixture of density ρ = sρ s + (1–s) ρa, viz.
where p is the atmospheric pressure, τ the vertical shear stress, ε the sum of other body forces and g the acceleration of gravity. Multiplication by the total density gives
where the convective terms have been transformed according to
the last term on the right in Equation (A.3) vanishing by virtue of mass conservation when evaporation of the drift particles is disregarded.
The two “phases” of the flow can now be considered additively. Changing with Dyunin to weights instead of densities by writing ρ sg = γs; ρ ag = γ, and distinguishing between the vertical velocities of the air (w) and snow (ω), Equation (A.3), becomes
The two lines show the balances for the two phases, but only the first is really relevant to the discussion. Introducing finally Dyunin's notation
and dividing by g we obtain finally
The first line resembles Dyunin's basic equation (7′) except that there the factor s has been omitted in the pressure and stress gradient terms; moreover the introduction in the first line of an Archimedean force requires a correspondingly modified air density in the second line. Subsequent simplifications of the basic equation obliterate these differences.
Before simplifying Equation (A.7) the corresponding development in the “diffusion theory” should be noted. This can be written down directly as the continuity equation for the drift-snow mass per unit volume at level z, called “drift density” for short and denoted by nz:
For the only concrete case considered in practice, that of steady-state horizontally homogeneous drift, Equation (A.8) reduces to
whereas the first line of Equation (A.7) becomes approximately
if the small Archimedean and other mass forces are neglected.
As next step in both developments, the vertical drift fluxes are i = i′ + i″, w = w′ + w″, etc., and 
, 
, etc., where the bar denotes a time-space average. Applying the averaging process to Equation (A.9) then gives
Dyunin's manipulation is more involved and follows again in translation; the present discussion has however established that the terms ∂p/∂z and ∂τ/∂x should everywhere have s as factor:
“Equation (7′) is the equation of turbulent two-phase flow averaged over space and time. Let q, i, w be mean values, q′, i′, w′ their instantaneous values, q″, i″, w″ their fluctuations
where the bar indicates time-space averaging. The second terms on the right (the correlations between q, i, w) do not vanish; the fluctuation i″, w″ and i″, q″ can have the same or opposite signs.
It follows from this that
as is easily derived from (7′), where pM, τM are the mean stresses and the correlations represent additional turbulent stresses.
Stricter reasoning gives
where I″, G″ are the fluctuations of the vertical and horizontal air flows and W″ is the fluctuation of the transverse component of the air flow.
Clearly the stresses p, τ have complex structures due to the correlation components.
Let us consider the case of established snow drift realizing its full transport capability and maximum transport rate.
For such a drift the vertical mass flux is zero on the average. Otherwise the snow carried by the current would increase or decrease, contradicting the assumption of steadiness. With w zero on average irrespective of i the right-hand side of Equation (7′) becomes zero.
Let us introduce the quantity
which is the snow concentration in a unit mass of the snow–air mixture (allowing for Archimedean forces). In that case we find from Equation (7′)
It is natural to expect in a steady drift flow that
the values γ s/i ≈ γ/γ s ≈ 10-3, and the other force can be neglected in comparison with gravity, as shown in (8, §8). The normal pressure pM in Nature equals the barometric pressure and is equal over the entire thickness of the current. Therefore
Measurements of normal pressures in wind do not reveal appreciable deviations from zero in the gradient ∂ρ/∂x with increasing distances from the Earth's surface. Therefore deviation from zero in Equation (12′) would be expected in the immediate neighbourhood of the surfaces. This leads to the conclusion of the “surface nearness” of low drift. For very low hydraulic values ω, particles can be lifted by the wind through the diffusivity
to great heights; at first glance this confirms the correctness of the diffusion theory but only for very small snow particles, which exist only for a short time, as we shall see below” [translated from Reference DyuninDyunin, 1974].
Dyunin's equation (12′) contains a term relating to the air itself: this arises from his inclusion of the total pressure and shear-stress gradients in Equation (7′), instead of only a proportion relating to the drift phase as in the first line of Equation (A.7). The difference is however lost in the subsequent parameterization of the eddy-flux term of (12′). That parameterization, and the corresponding development for the right-hand side of Equation (A. 11), contains the real substance of the two theories and are dealt with in the review itself (Section 3).
Discussion
A. Dyunin: The large clouds showing on your slide down-wind of the rock outcrops could be visually misleading. The main flow of snow will be invisible in the shadows. What is your opinion about this? Also, what is the basis for your use of a constant diffusion coefficient? I would like to thank Professor Radok for his stimulating contributions; much could be gained in the future through the merging of our two lines of investigation.
U. Radok: Transport of snow below the 3 cm level is approximately equal in amount to that in the next 20 cm. Our highest measurements were made at 8 m, but we need data from much higher levels, even to 100 m. Drift extends much higher than is sometimes assumed.
H. Lister: The deflation approach has no measure of the surface other than that manifest in the velocity, while the diffusion approach has a surface-roughness coefficient expressed in units of height. Surface characteristics can be very different, even with the same roughness coefficient, e.g. in dull weather, one day and five days after a snowfall. Drift on each of these days can be very different. Can you therefore suggest modifications of the roughness coefficient to incorporate these cohesion differences?
Radok: The snow surface creates its own roughness.
P. M. B. Föhn: The diffusion theory seems to yield only ratios of snow-drift densities; thus to gain the absolute drift density for a given level it is necessary to work from data obtained from a reference level. Would it be possible to calculate the drift density at a higher level by first deriving the “reference” drift density close to the ground using the Siberian method (nz α v 12)?
Radok: The drift density is quite constant for different levels, so that this is a small problem.
