As introduced above, the training data used for our model aims to represent the harmonics in the atmosphere-terrain interaction of terrain-induced flows with large-scale forcing. Rather than using a set of real terrain, which would be subject to potential sampling bias, we build a data set of synthetic topographies that reflect fundamental spectral characteristics found in real terrain. For this, we first analyze spectral slope characteristics of real terrain and subsequently use these characteristics to build a set of new, synthetic topographies.

We start by analyzing terrain for spectral slope characteristics following the same method as introduced by with 2D-applications by and . We analyze the mountain regions depicted in Fig. a, focusing on glaciated mountain ranges in mid- to high latitudes ranging over the entire world with dominant large-scale atmospheric forcing. In a first step, 30 m resolution digital elevation models (DEMs) of the Copernicus DEM GLO-30 of the regions shown in Fig. a are resampled to 50 m horizontal resolution and cut to tiles of 256×256 points in order to fit the requirements in terms of domain size and resolution of the later model (Fig. b–c). To avoid large spectral amplitudes at wavenumber 0, the deviation from a linearly fitted plane is calculated. Additionally, a cosine filter is applied on the outermost 10 grid points at each border to taper out the terrain in order to avoid discontinuities at the tile edges. On these filtered and de-trended DEMs a 2D Fast Fourier Transform (FFT) is applied . We take the module of the complex values and normalize by the domain size in order to get the amplitude spectrum A. Subsequently, as indicated in Fig. d, the decay of spectral amplitude with increasing wavenumber k is described by the function 1A=akb, with a and b being the intercept and slope of the power-law scaled spectrum. Here, stronger negative values of b indicate less spectral amplitude contained in smaller wave lengths and thus more smooth terrain. This procedure is applied to in total 54 022 tiles.

Figure  shows the distribution of spectral slope characteristics for the analyzed mountain regions. With the values resembling earlier studies for different regions , and values only differing slightly between the regions analyzed here (not shown), this analysis indicates the transferability of these spectral slope characteristics.

Our method to create new, synthetic topographies (“Fourier Land”) builds on replicating the spectral slope characteristics described above. It follows the approach shown in for artificial surfaces in material sciences, the approach of to create artificial fields of snow height distribution, and methods used to create landscapes and water surfaces in movies and computer games. On a matrix of white noise a 2D FFT is applied, the amplitude is scaled following Eq. (). After that an inverse FFT is applied to create the topography. In total, a set of 72 topographies was created with values for a and b randomly drawn from values in the range of the ones observed in real terrain (Fig. ). A small subset of example topographies with different spectral slope settings is shown in Fig. S1 in the Supplement. These 72 synthetic topographies are later used as terrain input for the numerical simulations. In order to avoid steep slope angles (<40°) that would cause issues with numerical stability in the simulations, we omit the smallest negative values of b>-1.7 and very high values of a>18. Very low values of a<13 are also neglected, as these stem from tiles with almost flat topography. With this in mind, our topographies are slightly smoother than real topography and do not represent the extreme values in the variability of slope angles as well as of other commonly used terrain descriptors like topographic position index TPI, , maximum upwind slope angle or the terrain curvature (not shown).

In order to create the training data set, simulations with the Weather Research and Forecasting (WRF) model are conducted. We use the Advanced Research WRF version 4.3.1 with the coupled snow drift module of and online LES diagnostics WRFlux, version 1.3.2 of . Each model domain consists of 256×256 grid points with a horizontal grid spacing of Δx=50 m and 81 terrain-following vertical levels.

The lowest mass point is located at approximately 10 m above the ground, the model top is set to 12 000 m with Rayleigh damping activated for the upper 5000 m.

With the ideal-case setup of our simulations, only a reduced number of physical parameterizations are used. No parameterizations for radiative transfer or microphysics are employed. We use the Revised MM5 surface layer scheme and the scale-adaptive sub-grid scale turbulent closure scheme (SMS-3DTKE) of (km_opt = 5) blending between LES with the turbulence closure of and the PBL-scheme of at the meso-scale limit. No land-surface parameterization is employed which also means that no explicit treatment of snow processes on the ground is in place. Instead we define a “passive” snow layer with prescribed thickness and density that only experiences wind-driven erosion or deposition. All simulations employ the drifting snow scheme of , in which snow erosion depends on snow density and surface shear stress. Airborne snow is transported by the resolved three-dimensional wind and parameterized turbulent mixing with a super-imposed particle subsidence. In our simulations, sublimation from drifting snow particles and its cooling and moistening effect on the ambient atmosphere is represented; the surface particle radius is set to 2×10-4 m.

In total we run 720 individual simulations with unique atmospheric conditions. For each simulation the terrain height is defined by one of the artificial topographies described above, which means that ten simulations with different atmospheric conditions are conducted on each topography. All simulations are initialized with horizontally homogeneous profiles of potential temperature, specific humidity and the meridional and zonal wind component. These profiles are calculated from values of wind speed and direction, static stability, expressed as Brunt-Väisällä Frequency, and ground-level pressure, temperature and relative humidity. The values for ground-level pressure, temperature and humidity are taken randomly from within the ranges shown in Fig. c–e, reflecting the variability over winter-time mountain environments. For each simulation a unique combination of wind speed and stability (ranging from neutral to isothermal conditions) is drawn. From the arrays of wind speed and stability, each possible combination is used once as initial conditions, therefore, every stability class covers the entire range of wind velocities and vice versa (Fig. h). Every full degree of wind direction is represented twice throughout the simulations. Note that initial profiles of wind speed, direction, stability, and relative humidity are vertically constant, neglecting, e.g., conditions with vertical wind shear or variations in stability. In order to ensure that snow is continuously present on the ground for the entire simulation period, it is initialized with a thickness of 1 m. Snow density is drawn randomly from the range of plausible values (very low-density snow to ice density) with the distribution skewed towards fresh-snow densities. The roughness length is set constant over each domain with values representing snow-covered surfaces Fig. g, . In our simulations, the entire domain is assumed to be uniformly snow covered, and free of vegetation.

We run the simulations with a time step of 0.5 s until the near-surface turbulent fluxes have stabilized. We then use the stabilized, quasi-steady state fields with established turbulent structures and drifting snow fluxes as training data for the machine learning model. Our synthetic topographies are periodic by design, which allows us to employ periodic boundary conditions. These lead to a quick convergence of the turbulent fluxes throughout the entire domain without the need of buffer zones or perturbations at the domain boundaries e.g.,. Visual inspection showed that turbulent fluxes stabilized within the first three to four hours of internal model hours, in line with earlier ideal-case atmospheric simulations e.g.,. Thus, all simulations were run for six hours with the first four hours disregarded as spin-up. The fields of the last two hours were averaged and accumulated, and used for later training. The averaging over two hours is used to smooth out potential un-steady fluctuations especially in weak-wind situations and in wake regions, while larger, steady-state features remain.

In the past, convolutional neural networks (CNNs) have proven to successfully identify spatial structures especially in gridded data . Building on that, we use the U-Net as the basic architecture in our study. U-Nets are fully convolutional networks that were first introduced for image segmentation, but have also been successfully applied for model emulating tasks in atmospheric sciences e.g.,. Typically, U-Nets consist of an encoder path and a decoder path. In the encoder path, blocks of convolutional layers (in our case 3×3 convolution kernels with stride of 1 and circular padding, Fig. ) and non-linear activation functions (in our case leakyReLU), followed by pooling layers (in our case 2×2 Max Pooling with stride of 2) are used to encode spatial patterns with increasing levels of abstraction and decrease the spatial resolution of the data. After each pooling layer, the number of feature maps is increased to compensate for the loss in spatial information. In the decoder path, blocks of up-sampling operations, followed by convolutional layers and non-linear activation functions are used to reconstruct high-resolution relationships. Additionally, skip connections are employed, where information is directly passed from the encoder path to the decoder path in order to preserve high-resolution spatial information. In total, our U-Nets consist of each four encoder and decoder blocks and a bottleneck of dimensions 16×16×512 (Fig. ).

We train a separate U-Net for each of our four predictants (near-surface winds (wind_surf), snow mass change rate (ΔMsnow), vertically integrated sublimation rate (ΔMsubl), vertically integrated snow transport rate (fsnow, vertically integrated product of snow particle mass concentration and wind vector). As predictors, we use the high-resolution field of terrain height, and the low-resolution atmospheric fields and surface values (snow density ρsnow, roughness length μ, offset elevation z0) that the WRF simulations were driven by (Fig. ). The atmospheric fields include meridional and zonal wind components (u, v), static stability expressed as Brunt-Väisällä Frequency (N), and surface-level values of air pressure (p), temperature (T), and relative humidity (rH). In addition to the terrain height we provide extra terrain information as a predictor similar to previous work . Here we provide cos⁡a-d, the cosine of the difference angle between the ambient wind direction and the local slope aspect angle to indicate slopes exposed and sheltered from the ambient wind. Apart from that, the predictions of wind_surf are given as an additional predictor to the other U-Nets. This additional input and the transfer of information from the wind model to the snow-related models drastically improved the learning process of the subsequently executed models.

We perform a z-score normalization on all input and output fields. Additionally, we applied additional log-transformations for ΔMsnow and ΔMsubl as well as a square-root transformation on the terrain height and for fsnow, in order to reduce the positive skewness in the respective distributions and improve the training process of the model.

The most important hyperparameters for the model training are summarized in Table S1 in the Supplement. For all but fsnow the mean squared error (MSE) was used as the loss function. Here, the mean absolute error (MAE) proved more successful. During model development we tested to include penalty terms in the loss function to ensure mass conservation between ΔMsnow and ΔMsubl. However, dividing into separate U-Nets for ΔMsnow and ΔMsubl and using the MSE provided better results.

We split the data set in test (1/8, 90 samples: all respective simulations over 9 randomly drawn topographies), validation (1/8) and training (6/8). With the periodic nature of our data, we are able to employ data augmentation by shifting the fields in x or y direction by a random distance (see example in Fig. S2), which much improved model robustness . Other commonly used data augmentation techniques like rotating, flipping, linear rescaling or splitting of the data were deemed impractical for our applications. Thus, in each training epoch, the training set is once presented in unchanged form and once with each sample shifted by a random distance. The individual models are trained until convergence is reached and no indications of overfitting are present (Fig. S3).

To provide an outlook on potential applications of SNOWstorm, we revisit the case study of 8 February 2021 on Hintereisferner, a glacier in the Austrian Alps, studied in detail by , of which the main results are summarized in the following. Figure  provides an overview of the region and the locations of available weather stations. The event was characterized by a cold front passage during the night with fresh snowfall and a subsequent increase in wind speed and shift in wind direction, leading to large amounts of snow redistribution in the second half of the day. The amounts of snowfall and redistribution were observed by three terrestrial laser scans (TLS). The laser scanner is installed at the same location on the ridge above the glacier as the station IHE. Additionally, nested large-eddy simulations at Δx=48 m in WRF (HEF-LES) with the coupled snow drift scheme of were performed. Validation of wind speeds and direction against three automated weather stations (Fig.  for location) showed a high accuracy of the simulated flow field. Comparison to the TLS-observed snow height change revealed that the overall amounts of snow redistribution are underestimated by the HEF-LES by about 9 %. Slope-scale patterns of snow redistribution like the position of maximum erosion on summits and exposed ridges are captured well in the HEF-LES, while smaller-scale patterns such as dune formations and interaction with sub-grid scale topography are not represented. As the TLS signal is comprised of snow redistribution, compaction and avalanching, and is limited in its scanning geometry, we will not use the TLS for direct comparison with SNOWstorm, but assume the HEF-LES to be validated against the TLS, and use the HEF-LES for validation of SNOWstorm. For more details in the methods and results we refer to the original publication of .

To better understand the behavior of SNOWstorm in real-world applications, experiments with different coupling strategies as described below and summarized in Table  are run and validated against the results of the HEF-LES as well as against the observations from the weather stations. The coupling to the atmospheric input is done following the procedure described in Sect. . To explore the influence of meso-scale atmospheric information, we perform experiments driving SNOWstorm with input taken from ERA5 (experiments S_ERA_∗) and the two outer domains of the WRF simulations of D01 (Δx=6 km, experiments S_WD1_∗) and D02 (Δx=1.2 km, experiments S_WD2_∗). Additionally, we run experiments with the smoothed digital elevation model used in the HEF-LES (experiments S_∗_W), and with the un-smoothed DEM of GLO-30 (experiments S_∗_G), resampled to Δx=50 m (see Table  for overview of experiment settings).

For the experiments we do not couple SNOWstorm to any snow model but prescribe a snow density (200 kg m⁻¹) and roughness length (0.1 mm) consistent with the ones in the HEF-LES and representative for the fresh-snow conditions on the day of the case study. Given the short duration of the case study, we consider this reasonable. However, for future long-term investigations, coupling to a snow model will be necessary.

We will first focus on the second half of the day (after 14:00 UTC), the phase with highest wind speeds and snow redistribution taking place. Here, the flow fields predicted by SNOWstorm driven with input from the two meso-scale domains (S_WD1_W and S_WD2_W, Table ) overall agree with the HEF-LES (Figs. b–d, ). General flow features, such as the flow deflection and splitting on the windward side of Weißkugel summit (see Fig. ), the deflection on the ridge north of the glacier, as well as the acceleration of the flow in summit and ridge regions, are represented in the SNOWstorm predictions (Fig. b–d). The channeling effect in the valley is underrepresented by SNOWstorm, leading to a too strong westerly component here (Fig. b–d), which is also evident in the comparison to the two observation sites of Station Hintereis (STH, Fig. b) and the temporary station on the glacier (AWS28, Fig. f). With the ridgeline of Rofenberg south of the glacier (see Fig. ) being aligned almost parallel to the ambient westerly flow, both the HEF-LES and SNOWstorm fail to reproduce the local flow field here. HEF-LES and SNOWstorm simulate a south-westerly and westerly flow on this ridge, and thus fail to reproduce the southerly flow and therefore the main direction of overflow (Figs. b–d, d).

Predicted wind velocities at the three observation sites generally agree with measured overall velocities, though with absolute errors between 1.6–4.4 m s⁻¹ and a mean overestimation of 3–4 m s⁻¹ at the stations IHE and AWS28, while errors at STH are generally lower (Fig. a–b). Pointwise comparison to the HEF-LES shows an MAE of about 3.9 m s⁻¹, with negligible bias, however, localized velocity maxima in summit and ridge regions are underestimated by SNOWstorm (Fig. b–d). Errors in wind direction are in line with the aforementioned underestimation of the valley channeling and the misprediction of the local flow field around IHE described above and are in the range of 70 to 100° (Fig. c). These errors are higher than the overall errors seen in the cross validation experiments (Sect. ) and for other comparable modeling approaches: in real-world settings, report an MAE in wind speed of 1.1 to 1.4 m s⁻¹, a bias of -0.17 to -0.24 m s⁻¹, and an MAE in wind direction of 14 to 57°. find for wind speed an MAE of 1.0 to 1.5 m s⁻¹, and a bias of -0.04 to -0.16 m s⁻¹, as well as an MAE in wind direction of 32 to 58°. Keeping in mind the short duration of the case study and the resulting limited transferability, these errors have to be viewed in context. Observed and simulated velocities have not been corrected for the height difference between the model output of HEF-LES and SNOWstorm (10 m a.g.l.) and the measurement heights (STH: 3.3 m, IHE: 3 m, AWS28: 2 m). Correcting the predicted velocities to the height difference under the assumption of a neutral stratification reduces the errors to an MAE of 1.7 to 3.9 m s⁻¹ and a bias of -0.6 to 3.7 m s⁻¹ (Fig. S5). However, with lacking information on the near-surface stratification and slightly changing surface conditions throughout the day, these height corrections can only be viewed as an estimate. While unsurprising and in line with , that errors in the real-world application are larger than in the semi-idealized testing environment, absolute errors in the more comparable, higher velocity classes (wind speed higher 10 m s⁻¹) in the cross validation experiments are between 1.1 and 1.5 m s⁻¹ (Fig. a), thus closer to the errors found in the real-world experiment here. Errors of the HEF-LES benchmark to the observations are in a similar range as for SNOWstorm, providing an estimate of the overall predictability of this event. This also indicates a possible error due to an underrepresentation of microtopography, resulting in an underestimation of turbulence which is relatively common for numerical simulations in complex terrain and might have been inherited by the ML model. Errors of the coarse-scale input wind interpolated to the observation sites are in a similar range or even slightly lower than the SNOWstorm predictions and HEF-LES (see Table S2), though with the missing additional information on local flow features.

While SNOWstorm predictions in S_WD1_W and S_WD2_W show comparable results, the experiments with ERA5 input (S_ERA_W) capture the overall flow structure, though with generally too weak winds (Figs. a, , ). This indicates that the meso-scale flow accelerations, which is lacking in ERA5, is necessary for SNOWstorm to capture the strength of the local flow field. As S_WD1_W and S_WD2_W only differ slightly, the meso-scale flow structure seems to be already represented enough in the WRF D01, and no additional information is provided by the finer input data. Nevertheless, at specific locations with large positive bias in S_WD1_W and S_WD2_W (IHE and AWS28), S_ERA_W outperforms the experiments with meso-scale input (Fig. ). The experiments with SNOWstorm driven on the un-smoothed GLO-30 topography show similar structures (Fig. S6) and comparable errors (Table S3), which is remarkable, as the steepest slope angles here exceed all slope angles seen during training. With the finer structure in this topography also finer features in the wind field, e.g., around secondary ridge lines can be simulated.

In the first half of the day, SNOWstorm fails to capture the weak northerly flow caused by shallow cold-air inflow and overspill after the frontal passage (see Fig. S7). During the transition phase around 12:00 UTC SNOWstorm predicts the increase in wind speed about three hours too early (Fig. ) which consequently causes a too early onset of snow redistribution and increased errors when considering the full day (Fig. ). Both effects are explainable as SNOWstorm has the assumption that the local wind field adapts instantaneously to changes in the large-scale forcing and effects of shallow cold air advection and cold air pools have not been seen in training.

Consistent with the wind predictions, accumulated snow mass changes simulated by SNOWstorm with meso-scale input (S_WD1_W and S_ WD2_W) overall agree with the ones in the HEF-LES (Fig. f–h). Maximum snow erosion is predicted at the summit region of Weißkugel, and at the ridges north-west and south-east of Hintereisferner. Regions of maximum erosion are slightly more localized with higher amounts in the HEF-LES, consistent with the underestimation of maximum wind velocities in the summit regions by SNOWstorm. For example, the snow erosion in the summit region of Weißkugel is underestimated by roughly 4 kg m⁻² in SNOWstorm with meso-scale input. Apart from that, erosion zones are shifted from the ridge more towards the lee-side slopes at several places (Langtauferer Spitze, Rofenberg, see Fig.  for location). Due to the high overall wind speeds only very few regions of snow deposition are simulated in the HEF-LES as well as by SNOWstorm. These include especially the upper part of Langtauferer Ferner and the area around AWS28 at several instances in time in phases of stronger lee-side deceleration. The deposition zone in the lee of Rofenberg simulated in HEF-LES is not captured by SNOWstorm due to the high wind velocities here. Consistent with the too low wind velocities, SNOWstorm simulates a much smaller change in snow mass in the experiment with ERA5 input (S_ERA_W, Fig. e). Pointwise validation against the HEF-LES shows an MAE between 2.3 and 2.8 kg m⁻² and a correlation coefficient between 0.4 and 0.6 for the different experiments. As already seen for the wind field, the experiments with GLO-30 topography show similar results, though with smaller-scale features captured by the more detailed input topography (Fig. S6). Sublimation from drifting snow particles plays a negligible role in both modeling approaches (Fig. S8). Similar to the other variables, the overall amounts and the placement of sublimation zones predicted by SNOWstorm generally agree with the ones simulated in HEF-LES.

In summary, SNOWstorm manages to capture the general shape and strength of the flow field as well as the overall amounts and location of snow redistribution during this case study. Local details in the flow field and transition periods in the large-scale forcing remain challenging for SNOWstorm. The large advantage of the ML model, however, lies in its computational efficiency: the computations for the HEF-LES of required about 104 core hours. In contrast, the predictions with SNOWstorm can be run on a single CPU in 4 s for the entire day with ERA-5 input and 0.1 and 0.3 h, respectively, with WRF input due to the longer processing time during input data preparation. This means a speedup factor on the order of 105 to 107 and reasonable computational demands for simulations over entire accumulation seasons.