Occurrence of transport

As presented by GM98, the occurrence of transport is deduced in two steps:

1. 1. A “snow-particle” mobility index (hereafter called M_i, and varying between –1 and 1) is computed from the crystal characteristics of the different considered layers. A crust or a wet layer stops the vertical analysis.

2. 2. A snow-driftability index, Mt, is deduced for each layer by comparison of the previous index and the wind velocity Wt. This index attains a positive value if snow-drifting is possible.

Rate of transport: saltation and creep

During many winters, field experiments have shown that, at our site, the main part of the drifted snow was transported by saltation (observations made on poles equipped with several bags at different low vertical levels). Moreover, our experiments did not allow the creep and the saltation transport modes to be well separated. Many authors quoted by LS98 note the predominance of snow saltation over creeping This leads us to globally simulate the two effects through a unique empirical formulation inspired by the field measurements which are, as previously mentioned, representative of gentle slopes. We assume that a considerable fraction of the total amount of drifted snow is removed in this way. We call this ratio “r” and keep it at the crude constant value of 0.8, but this point needs further investigation.

As quoted by PG95, the saltation efficiency depends on the moved crystals. We shall also consider the shape of the blown crystal in guessing that an “old” snow grain is less driftable than a new one. The mobility index M_i is well suited for summarizing these effects and we simply set:

where e is the transport efficiency, varying between 1 and 0. Our field experiments corroborate the affirmations of PG95 relative to certain linearity of the saltation transport rate with the wind speed for speeds greater than a minimal threshold value. Unlike LS98, we shall thus keep this convenient relationship without higher-level assumptions and we shall write:

δsal (m) is the amount of blown snow at one precise location, and d is a convenient coefficient, with the dimension of time, integrating the different relevant length scales of the phenomena. It has been fixed experimentally at 0.075 s by examining local snow-depth differences between poles during saltation events. The “critical” threshold velocity, Wteri, is set to 4 m s–¹. This value, fitted on the instrumented site, is larger than those proposed by PG95 for their threshold shear velocity, but is consistent with those published by Reference Li and PomeroyLi and Pomeroy (1997).

Rate of transport: suspension and sublimation

Our measurement site is not very well suited to the experimental study of turbulent diffusion. Hence, we chose a formulation suggested by PG95:

As quoted by PG95, we consider here, especially through the threshold velocity, that the source of the turbulent suspension is the saltation. δh_sns represents the snow mass transported by turbulent suspension, which does not sublimate and will deposit afterwards. We assume that the sublimation occurs only in suspended snow and at higher wind speeds; this leads to the following expression:

Wt is limited to 20 m s–¹. The functions F and G represent, respectively, the separated effects of the air temperature (T) and relative humidity (Hu) in order to crudely take into account the different energy exchanges between the snow particle and its environment. These functions have been expressed as:

which gives, for a fixed snow layer “i”, a maximum depth of eroded snow of:

Operational mode

The purpose of the parameterization is to transfer every hour a small amount of snow, comprising different surface or near-surface layers, from one exposure to the opposite one at a given elevation. This snow mass is accounted for in the Crocus profiles and stems from a previous snowfall or wind-transport event. The phenomenon occurs only if the normal component of the transport velocity W_t, according to the windward exposure, is strong enough compared to the considered snow surface conditions and the threshold velocity. The blown snow is then aggregated (in terms of crystal characteristics and density) on the opposite aspect with the surface layer. A horizontal diffusion is also applied and can limit the total amount of transported snow by assuming transport from and to the bounds of the experimental area. As we want to make the most of having a full stratigraphy, provided by Crocus, the computation is done for every layer, with a specific transport efficiency based on the mobility index Mⁱ of the layer. Different layers on the windward slope can thus be eroded within the same hourly time-step. We assume a small densification of transported snow when it is deposited. Each eroded layer, with an initial density ρ_ier, will undergo density changes by compaction during the transport and have an equivalent deposited density of ρiac, , computed in the following way:

The maximum density ρmax was set to 0.350 kg m–³ and Mt is the snow-driftability index. The hourly total eroded depth ΔHer is then computed as the sum of n successive layers of respective depth δhi, from the top of the profile. The deepest layer n can be only partially eroded if its own depth δhⁿ is greater than the corresponding δhnma x:

The evaluation of the lowest hourly eroded layer n, for a given profile, is an important problem. Its determination is based on the following two criteria and is obtained when at least one of them is verified. Equation (12b) corresponds to the “partially eroded layer” case, while Equation (12c) is particularly suited for profiles which exhibit numerous thin layers:

Only the non-sublimated part will be deposited afterwards:

At every hourly time-step, we aggregate all the eroded and non-sublimated snow quantities in one unique leeward deposited layer with averaged characteristics. This constraint is mainly due to the Crocus modifications. Moreover, due to the compaction, the total deposited depth is slightly different from ΔHac. All these averaging operators are based on an approximation of a mean eroded density and of a mean accumulated density that can be expressed as follows:

The effective deposited snow depth on the lee aspect will be:

which will be aggregated in terms of depth, density and crystal shapes, with the already present surface structure of the lee slope.

Fluxes from and to outside

A complementary horizontal diffusion is also applied for reducing both the raw eroded and the accumulated snow mass in order to simulate the unknown effect of the surrounding. In fact, it is not unusual to observe a decrease of the snow depth on the lee side, especially in the case of strong wind on small slopes, as well as small snow-depth variations on the windward side when the drifting snow is immediately replaced by new snow (see, e.g., Reference Fierz and GauerFierz and Gauer, 1998). This kind of transport is not treated by the previous formulation, which considers only two fictitious locations on two opposite aspects, but these “combined”transport effects must be taken into account if we want to avoid locally a too intense transport of snow mass.

To achieve a crude estimation of the snow mass accumulated on the windward side, we use the previous formulation with the same crystal stratigraphy, but with a velocity limited to βW_t. This very simplified approach does not take into account a large number of effects, especially the three-dimensional ones; it aims only at representing the effects of a horizontal variation of the transport velocity along one slope of the one-dimensional channel. The best results, compared to the field data, are obtained for β ≈ 0.9. The corresponding results are presented in the following paragraphs. This new β coefficient is the only one used in these different estimations, in order not to “over-parameterize” the complex process of transport in this first attempt. The result is thus a partial replacement of the initial snow in the eroded profile by drifted snow. The same procedure is performed on the lee side and aims at removing a part of the snow that could have been deposited there. On this side we use the same (5 coefficient, but applied to the velocity analyzed by SAFRAN on this aspect.

These formulations have an implicit self-limitation: the transported snow is less subject to a new drifting effect during the following hours because it underwent some crystal changes which decrease the corresponding snow-driftability index and thus the transport ability.

Precipitation

The SAFRAN hourly precipitation rate is fully applied on the windward aspect, and the new snow (dendricity = 1, sphericity = 0.5) is aggregated with the snow surface layer. On the lee exposition, this rate is reduced in proportion to the wind velocity in order to take into account both the wind-deflection effect and the slope influence.

Morphological transformations of the drifted particles

The algorithm used is based on observation of the drifted crystals, done at the instrumented site for many years. It tends to make a new drifted crystal which is closer to small rounded grains than the previous shape; but such transformations are not obtained in one time-step and can take many hours.

The snow-crystal characteristics are described by using the parameters of Crocus (Reference Brun, David, Sudul and BrunotBrun and others, 1992). A recent snow crystal is thus defined in terms of dendricity (1–0) and sphericity (0–1). When drifted, its dendricity decreases and its sphericity increases according to the wind velocity. When the crystal has undergone some metamorphism, it is mainly described in terms of sphericity and size (>0.3mm). Such drifted crystal will thus decrease in size and increase in sphericity, always in function of the wind velocity.

Aggregations

The density of the new surface layer of the lee slope is simply obtained by linear combination of its own previous density and weighted with their respective depths.

Another merging is done on the crystal characteristics. It is performed twice: firstly for computing only one drifted averaged crystal based on the different eroded layers, and secondly for merging this “drifted” crystal with the surface lee layer. The numerical aggregation of two types of fresh snow or of transformed snow is obvious and is based only on linear transformations. For the mixing of new and transformed “snows”, an equivalent size for the new snow is computed. A linear combination is then performed; if the compound size is > 0.3 mm the resulting snow will be assumed transformed, otherwise it will stay new. In either case the same rules are thus applied as for homogeneous snow types.

Limitations of the model

The limitations of this model are numerous, but, as previously stated, it must be evaluated in terms only of parameterization inside a larger-scale numerical modelling. We can principally criticize the fact that we are in a one-dimensional channel with only two computation points, which prevents a real estimation of the horizontal divergence of the implied fluxes and so implies a sometimes wrong estimation of fluxes crossing our numerical boundaries. The main reason for this is that we do not have valid wind-field estimation both in time, with the implicit smoothing due to SAFRAN, and in space where we do not have a full field. These velocity-estimation discrepancies can cause severe gaps in the transported snow quantities because, as quoted by Reference GauerGauer (1998a) and also visible in our expression of δh_sus, the drifted-snow amount depends on a power greater than 1 of the wind speed. These facts are illustrated in the following section.

Full winter season 1998/99

Figure 2 shows different simulations of snow depth on the two main aspects of the site. The bold lines represent the SYTRON simulation (green = southern aspect, and red = northern aspect) and are labelled “Drift”. The corresponding coloured thin lines represent the current operational SCM runs without any wind redistribution, and are labelled “No-Drift”. The plotted squares are pole snow-depth measurements at two locations (north and south) of the instrumented site. These locations were chosen for their representativeness at the scale of the model, according to the homogeneous characteristics (size, slope, precise exposure) of the surrounding topography.

Fig. 2.

Different simulations of snow depths (SD) representative of the Lac Blanc site (2700 m a.s.l; Grandes Rousses massif, northern and southern aspects) compared to some pole observations during winter 1998/99 (see dates on bottom). The bold lines correspond to the SYTRON run (labelled “Drift”), while the thin lines represent the operational SCM suite where the wind-transport effects are not taken into account (labelled “NoDrift”). The red colour corresponds to the northern aspect, and green to the south. The plotted squares represent observed measurements on poles with the same colour code.

We can see that an intensive wind transport is simulated on 7 December 1998, with a northerly wind of 6–9 m s–1. During this period, the southern snow depth became deeper than on the northern aspect, contrary to the prediction of the operational suite. The pole measurements are difficult to interpret during this period, due to many small-scale phenomena, but still reveal some discrepancies in the southern aspect where the snow depth remains too deep after the drifting event.

Fortunately, the rest of the season is better represented, and the “drifted” snow depths are not too far from the observations which exhibit well the accumulation effects on the southern slope, especially during February. We also note the numerical stability of the “Drift” simulations, which can run for several months without observation of the site and without any re-initialization.

Beginning of winter 1999/2000

Figure 3a and b illustrate the beginning of the current season with the same colour code as previously. Figure 3a displays the same elements as Figure 2, with a supplementary plot of two new averaged snow-depth measurements, denoted by “Pole 2” (coloured circles). These new values represent about 20 manual measurements around a precise location. They have been added this season in order to better visualize the spatial heterogeneity of the site. Figure 3b mainly shows the difference between the “Drift” and “NoDrift” snow depths on two aspects.

Fig. 3.

Beginning of winter 1999/2000 on the same site, (a) Same elements as Figure 2 for this new season, but with the addition of a new set of poles which are plotted as circles. The meaning of the colours and lines remains the same. PB1, PB2 and PB3 refer to discrepancies commented on in the text, (b) Snow-depth differences between the two simulations (“Driff and” “No Drift”) on two aspects; the observed wind on the site (m s–¹) is represented by black bars, and the velocity from the SAFE AM analyses by thin blue lines; both are referenced on the right vertical axis. Vertical dashed lines indicate the previous problem dates.

In Figure 3a, we can readily see the same effects as during the previous year, namely, accumulation on the southern slope and erosion on the northern slope, which are typical of these climatic conditions. Considering the global mass balance in the “Drift” simulation, we notice, especially in Figure 3b, an important loss of snow mass on the northern slope, but less accumulation on the southern aspect, which gives a negative global mass balance compared to the “NoDrift” experiment. This effect is partially due to the extremely strong winds observed in France at the end of 1999. For this reason, we have also plotted the observed wind at the site in Figure 3b (black bars) and the SAFRAN transport velocity (thin blue lines) previously denoted by Wt.

Many gaps are still visible. The main points are labelled “Pbx” in Figure 3a and plotted as a vertical dashed line in Figure 3b; they are discussed below in inverse chronological order. They cover different kinds of errors, such as a wrong wind-velocity evaluation, a lack of drifting on the lee aspect or an erroneous parameterization of the outside fluxes.

PB3 occurred on 20 December 1999 during the first tempest event. Erosion was correctly simulated on the northern aspect, as observed on the poles, but the accumulated snow remained on the opposite aspect. The main reason for this is an underestimation of the northerly transport wind by SAFRAN, which calculated 8 m s–¹ when 13 ms–¹ was observed. From the pole values, we see that the lee aspect (south) was also completely drifted and has less snow depth than the northern aspect, which is not simulated at all. The second tempest event, on 25 December, with 13 ms–¹ observed but 15 m s–¹ simulated, was better estimated in terms of transport, precipitation rates and resulting snow depths.

The PB2 problem, again on the southern aspect, is clearly seen in the pole observations of 17 December, which exhibit much less snow depth than the numerical simulation. The identified cause is the event of 6 December 1999 where we missed the southern erosion and thus overestimated the snow depth on this aspect. At that date, the simulated surface layers on both sides were mainly composed of mixed transformed particles (rounded and faceted) with a density of about 185 kg m–³. Although somewhat underestimated (7–9 ms–¹ instead of 6–13 ms–¹ observed), the simulated velocity was strong enough to drive the erosion of such crystals on the northern aspects, but the induced changes in the shapes and densities of the transported grains made it impossible for deposited snow to drift again from the southern aspects. We thus kept an erroneous and too high snow depth on this aspect.

PB1 is, in some aspects, similar to PB2 and occurred on 23 November but was observed on 25 November. On the first date, the two experiments (”Driff” and “NoDrift”) were very close and the surface layers were mainly composed of recent light snow (fallen on 19 November) with a density of about 95 kg m–³. The main error comes from the fact that the transport occurred during a period of small wind speeds, not too badly simulated by SAFRAN as shown in Figure 3b. In reality, nearly all the snow on the northern slope was driftable and was effectively removed. In the simulation, the relatively small velocity induced only a partial transport on this aspect due to our crude formulation directly linked to the wind speed for both the amount and the end criterion. The second level of the trap is also due to this medium wind speed whose value is used for the evaluation of the outside fluxes. The β coefficient (< 1) of this formulation implied no new drifting of the previously deposited snow on the southern aspect.