Between the coast and stake E40

Along these first 33 km, a set of annual accumulation values measured at 40 stakes is available for the period 1971 to 1983 (Table I). The mean values of the annual balances are plotted in Figure 4 as a function of the distance to the coast, together with the mean surface slope between successive stakes. Note that each irregularity in the topography shows a corresponding major change on the balance curve, mainly involving a deficit up-stream of the break in slope and an excess just down-stream. But the main characteristic of this set of balance values is the great inter-annual variability, in the order of 70% of the mean. This value is slightly lower than that deduced from accumulation variations between the two decades 1955‒65 and 1965‒75 for a large part of Antarctica (Reference Pourchet, Pourchet, Pinglot and LoriusPourchet and others, 1983).

Fig. 4 Upper curves: mean values of the annual balance between the coast and stake E40 deduced from stake measurements for the period 1971‒83. The 1σ precision is plotted. Lower curves: slope change from one stake to the next. (-----) three points running mean. Fig. 5. Principal component analysis of reduced centred fluctuations of annual accumulation. For 10 years and 16 stakes (located between the coast and stake 37).

The question that immediately arises when confronted by a set of data affected by such high variability is whether these variations are randomly distributed or whether they in fact express a spatio-temporal distribution mode. This is important in a set of measurements in order to distinguish what is characteristic of the particular location from what may be a feature of the year; for example, to take into account several sets of measurements made at several stations in Antarctica when making a comparison.

For this purpose, we tested two mass-balance distribution models using the values in Table I, only retaining complete annual series, i.e. 16 stakes over a 10 year period (D05, 05 bis, 07, 08, 09, 10, 11, 15, 19, 21 22, 23, 27, 36, and 37).

The first simple model tested was the one proposed by Reference LliboutryLliboutry (1974), which assumes an equal inter-annual variation around the mean value of the balance for the stations If b_jt is the balance measured at site j for year t, it may be broken down into three terms:

where α _j is the mean value characteristic of site j, independent of time with

β _t is the centred variation of the balance

which does not depend upon the site and is identical for all the stakes, and ε _jt is a centred random residual, the standard deviation of which expresses the adequacy of the model with respect to reality, In practice, in order to test this model on real values by variance analysis, a table of centred values b _jt – α _j , which are the terms of the expression β _t ₊ ε _jt , was deduced from the table of b _jt , (J stakes and T years), for each of T years; the value of β _t , is given by

and the real values for each stake are different by ε _jt

The overall variance of the sample

is the sum of the variance explained by an identical β _t each year at the J sites:

and the residual variance

The degree of explanation τ = S² _e ⁄S ² _t is 0.52, indicating that the linear model of balance variation takes into account 52% of the total variance of the sample.

The remaining 48% unexplained by this model is expressed by a residual standard deviation of 19 cm of water equivalent, i.e. about 40 cm of snow. Note that this corresponds to the mean height of the sastrugi observed in the area. This shows that it would be worthwhile to add a network of bamboos around each stake in order to obtain a mean value of the accumulation over a few square meters which would be more representative of the site than the single-point value.

In fact, the incomplete values determined at four bamboo networks available for one axis and the year 1972 alone (Table II), although insufficient to lead to significant numerical results, confirm the importance which should be attached to micro-relief.

Table II. Accumulation Measurements at the Bamboo Networks for the Year 1972

Note also that these observations are similar to those already formulated for Dome C station (lat. 74°39’S., long. 124°10’E.) where the surface micro-relief is responsible, at a fixed-point sample, for a background value of the same order of magnitude as the annual accumulation: σ= 2.6 g cm^–2 year^–1 for an accumulation of 3.6 g cm^–2 year^–1 (Reference Petit, Petit, Jouzel, Pourchet and MerlivatPetit and others, 1982).

However, we find that the linear model gives only a rough approximation, since it is based on identical standard deviations along the profile, while the table of measure-ments indicates that they are highly variable. A second variance analysis can be conducted by comparing the variations of β _t for a site reduced by the standard deviation, in order to analyse the relative variations.

This is the basis of principal component analysis, which compares reduced centred fluctuations. The results are given in Figure 5 which shows a common factor responsible for 64% of the variance and a second factor accounting for 10% of the variance, with 26% distributed on the other axes.

Fig. 5 Principal component analysis of reduced centred fluctuations of annual accumulation. For 10 years and 16 stakes (located between the coast and stake 37).

Axis 1 is interpreted as the consequence of the variation of homogeneity of the reduced centred balance, while axis 2 shows another trend differentiating the values in an apparently geographic manner, from the coast to the interior; this is somewhat unclear and opposite to other balance distributions such as those found in alpine-type mountain ranges (Reference Reynaud, Reynaud, Vallon, Martin and LetreguillyReynaud and others, 1954). Finally, no known structure is apparent for the 26% of the variance explained on the other axes.

This is, however, not surprising taking into account the conditions of snow redistribution on the surface by wind which forms sastrugi. Under these conditions, it can be said that the spatio-temporal distribution homogeneity is certainly even more marked than found, and that far better results could be obtained if we were able to eliminate the background noise induced by the sastrugi. Using both radioactivity measurements and stake values, we also computed the mean value of accumulation for all the stations in this area and for the widest time period available (Table III). In each case, we mention the methods used as well as the relevant time period and the number of years represented. In some cases, our observations are discontinuous and as a result the number of years represented may be less than the time period of observation. For each accumulation value, we calculated an uncertainty σ = s⁄(n‒1) ^1⁄2 where s is the standard deviation of the annual accumulation at each stake, during the period 1971‒83, and n is the number of years of observation.

Table III. Topographic Measurements and Accumulation Rate Determined between Stake Jo I And Stake E40

This degree of uncertainty is much greater than the measurement errors. As previously mentioned, it mainly reflects the temporal variation of accumulation and allows us to compare stations measured at different times.

Between the coast and stake D15 (Fig. 4), we noted a well-marked altitude effect. After D15, this effect becomes secondary relative to major deviations observed from one stake to another. These deviations cannot be directly attributed to variations in relief; between L25 and E40, accumulation increases in the “hollows” and decreases on the “hills”; however, this effect is reversed between stakes D15 and L25.

For this time period, ranging from 1971 to 1982, we computed the annual accumulation in the area from D10 to D21(Table IV). Comparing these values with mean annual temperatures at Dumont d’Urville (Fig. 6), we can deduce that the local mean annual temperature does not govern this variation in accumulation directly.

Fig. 6 Mean annual temperature at Dumont d’Urville (upper curve) compared with the mean annual accumulation between: D10 to D21 (middle curve); D01 to D40 (lower curve).

Table IV. Mean Annual Temperature at Dumont D’urville Compared to Mean Annual Accumulation between D10 to D21

From the total set of observations, we conclude that between the coast and stake E40, the spatial variations in accumulation are considerable and cannot readily be explained by the relief. On the other hand, the temporal variations are quite homogeneous from one stake to another, but these variations are not governed simply by temperatures.

References

1.Lorius and others, 1970.

2.Lambert and others, 1977.

3.Pourchet and others, 1983.

Beyond stake E40

Table V gives all the relevant data available for this area. For each accumulation figure, we give a mean standard deviation (σ) corresponding to the 90% annual variability we measured in the preceding area.

From stake E40 (Fig. 7), a continuous decrease in accumulation (A) is observed when plotted against the distance to the coast. This decrease can be expressed by

where x is the distance to E40, and γ and ω are two constants, When A is expressed in g cm^–2 year^–1 and x in km, we find

Fig. 7 Variations of three parameters along the Dumont d’Urville‒R80 axis. Upper curve: 10 m depth snow temperature. Middle curve: accumulation measurement for the longest period observed. _____ Individual measurement. Accumulation explained by temperature variations.---Accumulation explained by temperature variations and wave function. The la uncertainties associated with individual measurements are given by the number of years of observation. Lower curve: 1 km mean slope along the Dumont d’Urville‒R80 axis.

Between E40 and R60, large cyclic variations are superimposed on a decreasing mean trend. For this area, the accumulation (Y) can be expressed by the product of two factors: a decreasing (A) governed mainly by temperature and a deviation (S) associated with a wave function that can be expressed:

This correcting term (B) is given by

In our case we find

The complete expression of the accumulation between E40 and R60 becomes

These periodic oscillations in the accumulation revealed between E40 and R60 cannot be explained by a simple relationship with the relief as seems to be the case in the vicinity of Wilkes Station (Reference Black and BuddBlack and Budd, 1964). We suggest a meteorological explanation for this phenomenon.

Outline of the problem

The above results show that from the point of break in slope of the Antarctic plateau, near stake R60, the accumulation of snow (which is almost uniform on this plateau) changes in a periodic manner with distance, until reaching the coastal area, with a quasi-constant wavelength of about 40 km.

In this area there is no melting of snow. The accumulation of snow therefore represents roughly the annual precipitation. However, the spatial distribution can be non-uniform due to the combined effects of wind and relief, which will now be discussed.

Precipitation conditions

In order to establish the most frequently occurring conditions of precipitation in the Terre Adélie area, we use the data collected during the International Geophysical Year (1957-58). This information is quite old but it is more continuous and better documented than data collected from automatic weather stations installed since the I.G.Y. In par-ticular, the Charcot meteorological weather station (lat. 69°22’S., long. 139°01’E., 2400 m elevation) was operational during the I.G.Y. period, next to the break-in-slope location. This location is ideal for an appreciation of the meteorological conditions both on the plateau and at the summit of the slope line.

On the other hand, the climate at the permanent Dumont d’Urville meteorological station is largely influenced by the ocean and by the katabatic winds.

Table VI shows that in winter (April‒September) at Charcot, the number of days of blowing snow is always higher than the number of days of snowfall. This confirms the idea that wind plays a major role in spreading the snow accumulation. Furthermore, the number of days of blowing snow at Charcot is higher than at Dumont d’Urville. This strong contrast can be explained by the fact that Dumont d’Urville is influenced by oceanic conditions and that in the case of katabatic wind, the hydraulic jump, physically described as the “blowing-snow wall” (Reference ValtatValtat, 1960), which greatly reduces the strength of the surface wind, often occurs on the slope up-stream of Dumont d’Urville.

Snow transport by wind

The snow is transported by wind, either during precipitation or by uplift from the surface by turbulence. Figure 8 shows the probability of finding snow particles in suspension in the air as a function of wind speed at a height of 10 m (Reference RadokRadok, 1968). In order to define this curve, the author assumed that the mean snowfall speed, in the proximity of the surface, was 0.3 m⁄s. This value was determined from measurements conducted at Byrd Station (Antarctica) and corresponds well to the type of snow falling on the well studied area. Moreover, these measurements have shown that the average fall speed near the surface is weakly dependent on the horizontal wind strength.

Fig. 8 Cumulative frequency curves of suspension up-currents for silt, snow, and sand (after Reference RadokRadok, 1968).

At a wind speed of 5 m⁄s, 10% of upward currents already exceed the fall speed of the snow. At Charcot, curtains of blowing snow were observed in spite of wind speeds no greater than 5 m⁄s (Reference GarciaGarcia, 1960). If the wind is greater than 20 m⁄s, 70% of upward currents exceed the fall speed. Table VII shows the mean values of the main meteorological features at Charcot Station in 1958. Over the year, there is a total of 326 days of blowing snow. The monthly mean wind speed is never less than 6.9 m⁄s. The monthly maximum wind speeds are always greater than or equal to 20 m⁄s, except in February (15 m⁄s).

Table VI shows that in 1957 the number of days of blowing snow was less (261). But, even if 1958 was an exceptional year, we can still conclude that deposition of snow occurs mainly under strong wind conditions.

Generally, there are two modes of snow transport by wind (Reference Male and ColbeckMale, 1980):

Table V. Observations and Measurements between Stake E40 and Dome C

References

Lambert and others, 1977.

Pourchet and others, 1983.

Petit and others, 1982.

Saltation.

Within a 10 cm thick layer above the surface, the snow particles follow a curved trajectory under the influence of gravity and the drag force resulting from the relative speed between the wind and the particles (Fig. 9).

Fig. 9 Trajectory of a particle during saltation (after Reference Male and ColbeckMale, 1980).

Diffusion.

When the wind reaches a certain level, the particles are transported in suspension in the air by turbulent diffusion.

In either case, experience shows that, due to inhomogeneities of the medium and the surface beneath, the travel distance of the particles is low, around a few meters, and that they give rise to low-height undulations on the surface.

Dunes.

We assume that dunes are built up in the same way as undulations (Reference DallavalleDallavalle, 1948). The heavier particles build up at the top of undulations until they form ridges and then dunes when the wind frequently blows in the same direction. The undulation wavelength depends on the wind strength and increases together with their crest height.

Effect of relief.

The effect of relief on the accumulation of snow has not been studied for the relevant scale (about 50 km in horizontal distance). On a small scale (small hills or buildings), snow is known to deposit on the wind-protected side (Reference RadokRadok, 1977).

However, Figure 10 (Reference DallavalleDallavalle, 1948) demonstrates that in the dune travel mechanism, the particles accumulate on the exposed side instead of in the hollows between undulations.

Fig. 10 Schematic illustration of dune travel (after Reference Dallavalle Dallavalle. 1948).

Thus, hills do not fill in but move instead, very slowly, in the direction opposite to that of the prevailing wind.

Experimental remarks.

In the area of interest, there is a strong correlation between the isotope content of the snow and the mean annual temperature of the related location (Reference Lorius and MerlivatLorius and Merlivat, 1977). The isotope content varies linearly with the distance from the coast. These observations are incompatible with snow transport by wind, heterogeneous by nature, over long distances (exceeding 10 km).

Table VI. Number of Days of Snowfall and Snow-Blow at Dumont D’urville and Charcot during the International Geophysical Year 1957-58)(After Cornet and others, 1960)

A comparison of the curves in Figure 7 shows that snow accumulation and the slope of the surface, as a function of distance from the coast, are not correlated.

We observe a strong contrast between the plateau, where accumulation is relatively uniform and slopes are shallow, and the slope area, where accumulation takes on a cyclic mode and slopes are steep

Conclusion.

These last three observations allow us to conclude that the mechanisms described previously can explain undulation formation with a wavelength up to about 10 km.

However, the redistribution of snow by wind in such a way cannot occur on a sufficient scale to explain the cyclic mode of snow accumulation with a wavelength of 40 km.

We must therefore seek a large-scale atmospheric phenomenon causing the modification of snow deposition. Such a phenomenon could be an atmospheric wave inducing “lift-up” or “fall-down” areas. These two distinct areas could locally inhibit or enhance snowfall, which would then be deposited in a non-uniform manner on the surface.

However, each system must respect certain special conditions: the wind must have an almost uniform direction and be of sufficient strength; the atmospheric wavelength must be almost constant and the disturbance must be triggered by the “break in slope”.

Determination of the system of equations

The aim is to establish a system of equations relative to the given problem allowing us to Study the conditions for the formation of a gravity‒inertia wave.

Antarctic climatology shows that, due to temperature differences between the continent and the sea, an eastward circulation prevails in the low troposphere, from the surface up to 3000 m (Reference AltAlt, 1960).

Resulting from this, and in the area of interest, the winds mainly show a constant direction, satisfying one of the wind conditions and reducing the problem to two dimensions.

Observations show that katabatic winds are very frequent in this area (Reference Valtat, Valtat, Gilbert and MagniezValtat and others, 1960).

We can therefore assume that the second condition, strong winds, will generally be satisfied. A katabatic wind is induced by a cold (dense) air flow, caused by gravity in the direction of the slope. Over the cold air layer is warmer air separated from the former by a fluid interface, and generated in the atmosphere by a thin temperature inversion layer.

This type of flow can be studied using the two-layer model proposed by Reference BallBall (1956). However, because of the horizontal scale involved (in the order of 100 km) and the proximity of the South Pole, we cannot neglect Coriolis force effects. Once again, based on a two-layer model, the effects of the Coriolis force have been studied by Reference BallBall (1960) and by Reference Gutman and KhainGutman and Khain (1975).

In the following sections, we shall follow these works but restrict ourselves to the study of the conditions of formation of gravity‒inertia waves in the area of interest.

Equations.

We choose the case causing the flow of two fluids of different densities superimposed and separated by an interface (Fig, 11). ρ₀ is the density of the upper fluid, ρ₁ is that of the lower fluid, h is the height of the interface measured up from the surface, and δ is the height relative to the x‒y plane.

Fig. 11 Air flow beneath an inversion layer. The base of the inversion is idealized as an interface between two fluids of different densities ρ₀ and ρ₁

The Y-axis is perpendicular to the plane of Figure 11. We assume that the problem can be reduced to two dimen-sions by in variance perpendicular to the x‒z plane.

At elevation H in the upper fluid, the atmospheric pressure is P _H and we assume that geostrophic equilibrium is established, expressed as:

Table VII. Meteorological Data During the International Geophysical Year (1957-58) at Charcot (Lat. 69⁰22’S., long. 139°01’E., 2400 m altitude)

(1)

The hydrostatic equation allows us to express the pressure P ₁ in the lower layer, at a height Z, as

(2)

The standing-wave equations can be written

(3)

where f is the Coriolis parameter, u and v are the horizontal components of the speed respectively in directions x and y. We can write the expressed variables in a dimen-sionless form by introducing a critical speed U_c = (g’ Q) ^1/3 where Q = Uh = constant and

where Δθ is the potential temperature difference between the cold air layer of temperature θ which flows down the slope and the warmer air which is on top.

The dimensionless variables, with a bar over them, are expressed by

and

Then the system of Equations (3) is written in dimen-sionless form, omitting the bars:

(4)

where

is the dimertsionless parameter for the problem which depends on a Froude number. In fact,

The relief is expressed in a simple way in accordance with Figure 11.

In the system of Equations (4), h can be eliminated, and the system becomes, for x > 0:

(5)

This system in Equations (5) can be integrated:

(6)

where C is an integration constant.

We are looking for the periodic behavior of the vertical component of speed W. Based on the assumption made earlier, W can be written, for z = h, as

Because of the continuity equation of system (3), we can express W as

Thedu/dx expression we could obtain, using Equations (5) and (6), would be too complicated to demonstrate easily its periodic behavior along the x-axis. However, we know that if u shows a periodic behavior along the x-axis, then the same holds for du/dx and consequently for W.

We can study graphically the behavior of u and v deduced from Equation (6). Three cases are to be considered:

(Fig 12).

Fig. 12 Schematic representation of possible phase-plan variants.

The curves are drawn with k =0. Curves corres-ponding to different values of k may be obtained by a translation on the v-axis.

The arrows show the movement for increasing values of x, and are obtained by studying the sign of the following equations:

(7)

The case where α > 1 is the only one giving a closed curve with a continuous movement, i.e. with the arrows always in the same direction.

In this case, where α> 1, u and v vary periodically along the x-axis, around an equilibrium point u = 1 and v = 0. In the following sections, we only consider this case, excluding the others.

Wavelength.

Figure 13 shows the graphical solution of Equation (6) for the case of α >1 and k <0.

Fig. 13 Schematic representation of phase-plan integral curves used to construct the wave solution of the problem (α > 1).

When x > 0, there is an unique solution: u = 1 and V = 0, designated as point a, which corresponds to geo-strophic balance. For x = 0, as |k| increases, we obtain ellipses increasing in size and centred around the point v = k and u = 1, corresponding to different waves as long as the curve is closed.

The maximum value of the periodic solution is reached when the solution crosses a singular point: v = k _max and u = α^-1/3, which corresponds to a value of the integration constant:

If | k | is greater than | k_max | = C ₀ ^1/2 = (α^1/3 ‒ 1)^3/2, the curve is not closed and there is no periodic solution.

In this case, the maximum amplitude of the wave is obtained for v = k, u = α^-1/3. Then we can integrate Equations (7) between the two values where v = k, and we obtain the wavelength:

(8)

Otherwise, we can assume that the amplitude of oscillation is small and that we remain near geostrophic equilibrium. In this case, the graphical solution is a small ellipse centred at the point: v= 0, u = 1.

A simple analytical solution can be obtained by linear-ization of the system of Equations (7), which becomes

(9)

v can be removed from these two equations, giving

We write where A is a constant and dimension less wavelength

(10)

Return to dimensional values and discussion.

To explain the cyclic variation of the accumulation of snow along a slope by an atmospheric wave, the atmospheric wavelength λ ₀ must be of the same order of magnitude as the wavelength of accumulation observed at the surface, i.e. λ₀ = 4 – 10⁴ m. Reference BallBall (1960) has shown that the wavelengths for the low-amplitude solution (Equation (10)) and for the long-amplitude solution (Equation (8)) are very close to each other.

So we can write the dimensional wavelength

with f being the Coriolis parameter: f =1.4×10^-4s^-1

This gives

In the expression of U _c, g’ and h are well known.

Observations (Reference Valtat, Valtat, Gilbert and MagniezValtat and others, 1960) show that we can expect temperature gradients from 5K up to 10K between the cold air layer and the warmer air on top. In this case, for a mean cold-air temperature of 250K, g’ varies from 0.2 to 0.4 m s^-2. The thickness of the cold air layer usually varies from 500 to 1000 m, C7, however, is not known. It is the mean speed of the cold air which flows down the slope due to gravity in a uniform and steady manner (Reference BallBall, 1956). For U, let us take a very low value of 1 m s^-1 together with the lowest values for g’ (0.2) and for h (500 m). In this case, U _c = 4.64 m s^-1.

Considering these assumptions, we obtain a maximum value of α

α must satisfy the α > 1 condition, and nevertheless remains at a value around 1.

This result excludes the exact maximum-amplitude solution (Equation (8)). Indeed, we respect the condition:

but

Since the relief drops about 2000 m over 200 km, we have

Furthermore, since α must be about 1, U _c is required to be of the same order of magnitude as U _g . If we choose a high value of U _c (30 m s^-1) and a low value of g’ (0.2), so that |k| must be small, we find:

We conclude that the only possible solution is the one corresponding to a disturbance occurring near geostrophic equilibrium and satisfying the system of Equations (9).

However, we must verify that α can be equal to 1 and that it stays near this value.

We have seen that α саn be written as the product of two Froude numbers:

where Fg = u _g /(g’ h) ^1/2 is a Froude number induced by the geostrophic wind and Fu = u/(g’ h) ^1/2 is a Froude number induced by the uniform speed of the flow. Reference BallBall (1956) has shown that this Froude number depends only on the relationship Fu = k/C _D where k is the slope of the field and C_D is a viscosity coefficient, depending on the surface structure.

So, although U and h can vary along the slope (we generally admit that g’ variation is low), the Froude number Fu of uniform flow remains constant.

We know that k =10^-2 but C_D is unknown for the study location. However, Reference BallBall (1956) obtained C _D values ranging from 10^-2 down to 5 – 10^-3 so Fu therefore changes from 1 to 2 and the condition α 1 therefore implies that:

Fg values will range between Fg = 1 and Fg = 1.26. These values will easily be reached, since the selection of relatively large values for g’ (0.4 m s^-1) and h (1000 m) gives Fg = 1.26 if U _g = 25.2 m s^-1, which is very reasonable considering the mean wind speed observed in Terre Adélie.

Finally, we note that the order of magnitude of values given by observations for the variables giving the Froude number (geostrophic wind, modified gravity, and layer thickness) lead to Fg 1.

This Froude number represents flow occurring in an area showing marked regularity: uniform surface slope over a long distance; very homogeneous nature and temperature of the surface; atmospheric circulation induced by a slowly changing temperature gradient (continent to sea); stable cold air.

Unfortunately, we presently have insufficient data from lower-layer soundings to confirm this hypothesis.

It can be concluded that a gravity‒inertia wave dis-turbing the geostrophic equilibrium may be formed. Its amplitude is weak but the triggering occurs at the break-in-slope location and the wavelength is of the same order of magnitude as the accumulation wavelength. The conditions for formation of this wave are strictly local in character.

such an atmospheric wave ongmates the cyclic behavior of the snow accumulation along the slope in Terre Adelie, we must emphasize that it is a very special case which is unlikely to occur other than in Antarctica.