Using meteorological sign conventions for turbulent fluxes, the energy budget of a non-melting snow surface (_sfc) of height (h) under non-precipitating conditions with volumetric ice-fraction vi is given by: 1ρivih⋅cpi∂Tsfc∂t︸Snow Temp. Change=-λ∂T∂z|z=h︸Conduction-Hs-Hl︸Turbulent Fluxes+LW↓-ϵσTsfc4+SW↓(1-α)︸Radiation where cpi is the specific heat capacity of ice, Tsfc is the temperature of the topmost snow layer of height h, ρi is the density of ice, λ∂T∂z is the conductive heat flux from/out of the top snow layer to lower layers (z is positive upward from the snow ground interface), λ is thermal conductivity of the snow layer interface, Hs is the sensible heat flux, Hl is the latent heat flux, LW↓ is the incoming longwave radiation from the atmosphere (3.5–50 µm), ϵσTsfc4 is the outgoing longwave emission (LW↑) from the snow surface where ϵ is emissivity and σ is the Stefan-Boltzmann constant (5.67×10-8 W m⁻² K⁻⁴), SW↓(1-α) is the net absorbed shortwave (solar) radiation where α is the snow surface albedo . For non-melting conditions, Hl is the energy associated with vapor sublimating (+Hl) or depositing (-Hl) onto the snowpack. At nighttime and in the absence of precipitation, Hl is balanced by Hs, the LWnet budget, cooling of the uppermost snow layer (cpss∂Tsfc∂t), and the conduction of heat into/out of lower layers of the snowpack (-λ∂T∂z). From the perspective of observing LW↑ emitted by the snowpack, the depth h for which Eq. () is defined is small since LW↑ is primarily emitted from the snow crystals very near the surface of the snow .

According to most prevailing theories, surface hoar forms via the deposition of water vapor from the atmosphere onto the snow surface . The sensible, latent, and momentum fluxes between the snow surface and atmosphere are given by: 2ls-1Hl=ρw′q′‾=ρCquΔq3cp-1Hs=ρw′T′‾=ρChuΔT4τρ-1=u⋆2=Cdu2 where the rightmost term represents the bulk approximation for the flux , Δq and ΔT are stand-ins for the air-snow specific humidity (q) and temperature differences (qsfc-qair and Tsfc-Tair), w′ is the deviation from the mean vertical wind velocity, u is the horizontal wind speed (wspd) at the height z, Cq and Ch are the bulk transfer coefficients terms for q and T respectively, Cd is the bulk transfer coefficient for momentum, ρ is the air density, and τρ-1 is the surface shear stress. ls and cp are the latent heat of sublimation (2838 kJ kg⁻¹) and specific heat capacity of dry air (1.006 kJ kg⁻¹ K) which are treated as constants. The small temperature dependency of cp is not considered in this study. Since we are ultimately concerned with surface hoar mass, we let Fq=ls-1Hl as a stand-in for the left hand term of Eq. (), which expresses water vapor flux (units of mass per unit area per time). While not a primary focus of the paper, the friction velocity u⋆ is important for understanding turbulent exchange during surface hoar events.

Most land and snow models use some form of the bulk approximation to estimate fluxes (rightmost terms of Eqs. –). Eddy-covariance (EC) instruments on the other hand can estimate fluxes by measuring qair, Tair, and wind velocity components at a high temporal frequency (middle terms of Eqs. –). Monin-Obukov (MO) theory is typically invoked to determine bulk transfer coefficients, which depend on both the characteristics of the surface and the thermodynamic stability of the near-surface air which can enhance (unstable) or suppress (stable) turbulent eddies through buoyancy forces . In the SUMMA model, bulk transfer coefficients for momentum, moisture, and heat are expressed as, 5Cq=k2[ln(z/z0)][ln(z/zq)]⋅f(Ri),6Ch=k2[ln(z/z0)][ln(z/zh)]⋅f(Ri),7Cd=k2[ln(z/z0)]2⋅f(Ri), where z is the measurement height, z0, zq, and zh are the roughness lengths for momentum, moisture, and heat respectively, k is the von-Karman constant (0.4), and f(Ri) is the stability correction function. The bulk Richardson number (Ri) is given by: 8Ri=z9.81(Tair-Tsfc)0.5(Tair+Tsfc)u2. Positive Ri values indicate stable conditions and negative values indicate unstable conditions. For simplicity and in keeping with many land models, zq, zh, z0 are assumed to be equivalent, but zq may in principle be significantly less than z0 . Given that we are concerned with nighttime winter processes, we test several stability correction functions implemented in the SUMMA model (described in Sect. ) for stable conditions (Table ). The a, ϵ and Ric parameters in Table  are tunable coefficients. For each function, f(Ri) approaches unity at neutral conditions (Ri∼0). The Ric parameter in the standard model is commonly assumed to be 0.2, beyond which turbulent exchange is effectively zero.

The humidity of the snow surface (qsfc) is saturated with respect to ice, and thus depends solely on Tsfc and Pbaro. The radiative surface temperature is commonly used to represent Tsfc in Eq. () , and can be found by rearranging the Stefan-Boltzmann law expressed in Eq. () .

We assume that snow emissivity (ϵss) is 0.98 . For air and the snow surface, q is computed by: 9q=0.622ePbaro-0.378e where e is the vapor pressure (in hPa) and Pbaro is the barometric air pressure. For snow, e is computed using the saturation vapor pressure formula for ice using the radiative Tsfc retrieved from LW↑ measured by down-looking pyrgeometers.

Together, the snow energy budget (Eq. ), flux formula (Eq. ), and qsfc relationship to Tsfc provide the governing equations that explain the sign and magnitude of Fq, which is sensitive to meteorological conditions and snowpack properties. In this study, we examine instantaneous 30 min fluxes as well as nightly and yearly averages of the integrated nighttime Fq between the snow and atmosphere. The overnight amount of modeled or observed integrated water vapor flux is simply the cumulative sum multiplied by the timestep, here denoted by ΣFq. A night with a negative ΣFq is considered a deposition (or, surface hoar event) and a night where ΣFq is positive is considered a sublimation event. To express the ramifications of climate change on the changing character of the overnight integrated water vapor flux, we define: 10Fq,tot‾=1Kj⋅ΣFq+‾︸avg. nightly subl.+n⋅ΣFq-‾︸avg. nightly depo. where Fq,tot is the annual average amount of the integrated nightly vapor flux taking into account both sublimation and deposition, K is the number of eligible winter nights per year, n is the number of winter deposition events, j is the number of winter sublimation events, and ΣFq+/-‾ is the average nightly amount of all deposition and sublimation events, respectively. To account for a potentially reduced snowpack with warming, we only consider nights where snow height (Snow_h) is greater than 0.1 m. For a non-leap year when this condition is always met, there are K=121 DJFM winter nights considered in the annual averages in Eq. ().

In addition to low frequency measurements of Tair and RH, the SOS tower at KP hosted pairs of sonic anemometers and gas analyzers to measure EC fluxes. Data from the 3 m heights are analyzed in this study. provides 5 min flux estimates and higher-order moments computed from the high frequency data after applying a planar fit tilt-correction . The 5 min fluxes were recomputed for a 30 min averaging timescale, and additional Webb and sonic temperature corrections were applied. EC measurements can be problematic in stable surface layers with weak turbulence, so additional tests to ensure data quality were employed . Flux data were discarded if any of the gas analyzer or anemometer flags were signaled. The commonly applied stationarity tests from were also calculated. Between December and April of the SOS observation period, 62 % of the 30 min timesteps contained valid observations based on sensor flags for the sensors at 3 m, 35 % pass the least stringent stationarity criteria, and 16 % pass the most stringent stationarity criteria for heat flux. These criteria likely reject periods when overnight frost forms on the infrared hygrometers that interfere with flux measurements, but a full accounting of the prevalence of this phenomenon is beyond the scope of this article.

The measurement height between the snow and sensor is important for interpreting observed fluxes and bulk calculations. A lidar system measured the height of the snowpack in the vicinity of the towers and is used to compute the absolute height difference between the snow surface and tower measurement height. Snow_h reached a peak of approximately 2 m above the ground level .

Several considerations are important for comparing SUMMA to EC flux observations. For fair comparison against the model, EC fluxes are only considered when the second-most stringent stationarity criteria are met and the absolute value of Hs is greater than 1 W m⁻². Countergradient fluxes are also ignored, since these conditions cannot be simulated by SUMMA.

The SOS campaign manually measured surface hoar during 2023 using weighing boxes similar to and referred to as “Stössel boxes” by the SOS campaign. On nine evenings in February 2023 when surface hoar was deemed likely, two snow samples were removed, weighed, and replaced in the snowpack at a location approximately a half kilometer up-valley from the M1 site. The samples were removed and reweighed the following morning. On a few occasions, a thin layer of frost was found on the underside of the box and removed prior to weighing the boxes. In the absence of disturbance, the mass difference of the snow sample between the morning and preceding evening is a measure of the cumulative deposited and/or sublimated mass of water vapor between the snow and atmosphere. Dividing the mass difference by the area of the sample yields the integrated flux (ΣFq) in units of mass per area.

Six observations showed appreciable water vapor deposition where the redundant observations agreed within 40 %, with values ranging between 40–180 gm⁻². The morning of the 12 showed that both observations agreed on a near-zero amount of surface hoar. It is worth noting that even the largest deposition amounts represent a small component of the mass addition to the snowpack. We caution that the Stössel box measurements were not directly co-located with instrumentation at M1 or KP, but nevertheless provide confirmation that surface hoar occurred for each evening, and a plausible order-of-magnitude expectation for surface hoar mass. Thirty-three additional Stössel box measurements were collected after the end of the S3 campaigns during the winter of 2024 (Billy Barr, Rocky Mountain Biological Laboratory, personal communication). The twenty-nine measurements showed overnight deposition with an average mass of 140 gm⁻² and three measurements exceeding 200 gm⁻². Since these measurements did not overlap with S3 instrumentation, we primarily use the data as an additional check on the modelled distributions of surface hoar mass in Sect. . In addition to data from S3 campaigns, we also use data digitized from to evaluate Cq coefficients and Ri, hereafter referred to as H97.

Examining Eq. () suggests that surface hoar formation may be sensitive to the thermal conductivity (λ) value and therefore the density of the snow layer, since the conductivity is strongly influenced by it . To isolate the effect of snow density via thermal conductivity on surface hoar, we conduct sensitivity experiments by prescribing a constant value of λ for all layer interfaces of the snowpack representative of a low (λ=0.1), medium (λ=0.3), and high (λ=0.5) density snowpack . This is achieved by using the “smnv2000” snow thermal conductivity option in SUMMA for these experiments. The model is then run for a single 36 h period encompassing a single surface hoar event and the effects on fluxes, surface energy balance, and surface hoar accumulation are quantified.

We use data from the WUS-D3 dataset to represent the effects of changing climate under the SSP3-7.0 emission scenario in the ERW. WUS-D3 uses the Weather Research and Forecasting model to dynamically downscale coarse CMIP6 model data to a 9 km horizontal grid spacing across the western United States. WUS-D3 downscaled an ensemble of CMIP6 models. We select 9 of the downscaled models to force SUMMA. Specifically these are access-cm2_r5i1p1f1, cesm2_r11i1p1f1, ec-earth3_r1i1p1f1, fgoals-g3_r1i1p1f1, ukesm1-0-ll_r2i1p1f2, canesm5_r1i1p2f1, cnrm-esm2-1_r1i1p1f2, ec-earth3-veg_r1i1p1f1m, and mpi-esm1-2-lr_r7i1p1f1 model configurations (where the text following the underscore is the model realization description).

Dynamical downscaling by WUS-D3 preserves much of the warming and wetting/drying trends from the parent GCMs but accounts for local heterogeneity related to orography and land-atmosphere feedbacks . The SSP3-7.0 scenario is a moderately high radiative forcing scenario driven by “business as usual” emissions . WUS-D3 is extensively vetted in other snow climate studies and in addition, various configurations of the WRF model have been tested in the ERW demonstrating skill simulating the regional snow climate.

Hourly surface meteorological data (wind speed, downwelling shortwave radiation, downwelling longwave radiation, barometric pressure, and precipitation) from the 9 models are used to force SUMMA, yielding a total of 9 SUMMA runs from 1980 to 2100 with hourly output of surface hoar relevant quantities. Prior to doing so, two additional bias corrections to the WUS-D3 data were applied. The wspd output from the model was bias-corrected using CDF-matching onto observed wspd at the KP location. This method was chosen since, given the 9 km grid spacing, nocturnal drainage flows may not be well represented as the model topography is relatively coarse. In addition, a (∼1 °C) small cold bias of the average early century Tair (relative to S3) observations was found and removed. Such biases are common in dynamically downscaled output over mountain regions . Other meteorological fields were not adjusted.

In order to evaluate climate change effects on surface hoar, the ensemble of SUMMA GCM-forced experiments are evaluated using Eq. () for cold-season (DJFM) nocturnal conditions only, which, for simplicity are defined as between 08:00 pm to 08:00 am local time. To account for natural year-to-year variability, surface hoar relevant quantities are grouped by 20-year blocks and averaged across time. For fair comparison and to avoid including snow-free conditions into the averages, only timesteps where the Snow_h is >0.1 m are considered. Given the cold baseline climate and significant snowfall in this location, these criteria reject relatively few timesteps among the models, even at the end-of-century. To report changes, we define the beginning-of-century as 1980–2020 (BOC), the middle-of-century as 2020–2080 (MOC), and the end-of-century as 2080–2100 (EOC). In addition, data are also grouped by the average annual warming for each model year relative to the GCM's average beginning-of-century temperature (e.g., the “warming level”) in order to more clearly isolate the effects of warming from interannual variability. The statistical significance of the change in ensemble mean between the EOC and BOC periods for different surface hoar relevant variables is quantified using the Welch's two-sided t-test.

To a large extent, Tair is correlated with many other features of the snow surface energy balance and rising Tair is the most basic and robust prediction for climate change. The extent to which models and observations agree with the proposed wspd thresholds for surface hoar formation also merits consideration, and the failure by models to do so would decrease confidence in projected changes in surface hoar. Therefore, we also evaluate model and observed sensitivities for surface hoar relevant quantities to wspd and Tair prior to evaluating the GCM-forced SUMMA experiments.

As we have shown, favorable conditions for surface hoar tend to occur on clear, cold nights with weak synoptic forcing (Fig. ). These conditions naturally lead to downslope katabatic flows in mountain terrain . We already see evidence of this, given the reversal in wdir from generally up-valley (from the southwest) during daytime hours to generally down-valley (from the northwest) at sunset (Fig. c; see also ).

Using the SOS tower data from the winter of 2023 and separating nights again into clear and cloudy periods shows that clear nights are typified by stronger stability as measured by Ri (Fig. a), wdir almost exclusively from the down-valley direction, and shallow katabatic flows. Normalizing the vertical profile of wspd by the tower averaged wspd highlights the shape of the katabatic flow regime (Fig. d), which peaks at around 5 m above the ground level (approximately 4 m above the snow surface in the mid-winter). The strength of the jet is relatively weak, and on average is roughly 10 % higher than the mean wspd across the entire 20 m tower. While the average maxima is found at 5 m, in many cases on clear-sky nights, the wspd maxima is found at the lowest available measurement height. Disturbed, cloudy periods, however, have a higher average wspd and both up and down-valley directions and a more commonly unstable or weakly stable surface layer. In these conditions, wspd versus height follows the standard log-linear profile of increasing wspd with height.

Previous sections have investigated Δq and ΔT, but quantitative estimates of turbulent exchange require taking the buoyant suppression of turbulence and surface roughness into account. Finally, f(Ri) functions from SUMMA (Table ) are compared against the bin-averages (Fig. ). Generally speaking, the assumption of a z0=2×10-4 m surface roughness for momentum, heat, and moisture leads to a reasonable approximation of bulk transfer coefficients. This is found by noting the intersection of the f(Ri) functions at near-neutral (Ri=0) conditions with the observed bin-averages (Fig. .) The effect of a higher or lower z0 value can be visualized by recomputing (Eqs. –) which has the effect of translating the f(Ri) curves up/down along the y-axis. Cq is perhaps slightly less than theoretical estimates, which is consistent with evidence suggesting that zq≪z0 during certain condition over ice .

Cd deviates strongly and consistently from theoretical values at Ri greater than 0.1 (Fig. a). Closer examination shows that such conditions are almost always characterized by katabatic flows during clear skies. This is not surprising, given that the observed wspd profile during such conditions does not follow the logarithmic profile required by MO theory. Cd is essentially an expression of turbulent intensity over the mean wspd, so a higher than expected value indicates higher turbulent intensity during katabatic flows than non-katabatic flows for the same wspd. CD is well characterized at low Ri values and when wspd increases approximately logarithmically with height (Fig. c).

A crucial finding emerges with relevance for subsequent modeling – surface hoar events occur at high Ri values (>0.2) beyond which the “standard” stability correction predicts a near-zero flux (Table ). Recomputed data from H97 also demonstrate that surface hoar measured by their study occurred at a high Ri number of 0.1. The “Louis” stability correction with a b parameter set to one better matches observations of both Cq and Ch at high Ri values, and these parameters are used in subsequent sections (as well as in (Fig. )). However, it is important to note that Cq values computed using data from Stössel box and H97 data implies that, for at least five cases, Cq values are as much as 3–4× higher than the Cq implied by EC observations and model parameterizations using a zq equivalent to z0.

To test these sensitivities, 30 min estimates of Δq and Fq are plotted against Tair and wspd and the same criteria for EC fluxes from the previous section are applied. Data are binned by 4 °C for Tair and 1 m s⁻¹ for wspd, and the mean, 25th, and 75th quantile are plotted for each bin (Fig. a–d).

The bin-averaged Δq magnitude increases slightly beyond -24 °C but then declines rapidly (becomes closer to zero) as Tair warms (Fig. a). Sublimation becomes favored rather than deposition above -8 °C (Fig. a). Encouragingly, the SUMMA model likewise shows a transition at the same Tair. Fq shows a very similar pattern (Fig. b; note that the y-axis uses a symmetric logarithmic scale, with a linear region between the blue lines), though the data are noisier and have a higher inter quartile range than the model values.

The relationship with wspd is similar (Fig. c, d). Fq and Δq are slightly stronger for wspd between 1–2 m s⁻¹ compared to 0–1 m s⁻¹. Similar to other work, we find a threshold wspd value of 2–3 m s⁻¹ that matches both observations and the SUMMA model, above which sublimation becomes favored over deposition. The bin mean of Δq at high wspd is lesser than that at warm Tair values, indicating that wspd alone may be a less robust predictor for discriminating water vapor deposition from sublimation than Tair.

Ultimately Tsfc, qsfc and overnight ΣFq are highly sensitive to the prescribed λ value (Fig. a–d). During daytime hours, Tsfc reaches 0 °C for all simulations. Models diverge overnight, and the low-density simulation reaches the coldest Tsfc value (and, the most similar to the observations; (Fig. a)). The overnight average ΔT values are -6.5 °C for the low-density simulation compared to -5.5 °C for the high-density simulation (Fig. c). This translates into an average overnight Δq of -0.4 and -0.15 g kg⁻¹ respectively (Fig. b) Water vapor deposition occurs in all model scenarios, but a combination of the reduced deltas and stability (not shown) leads to a 54 % reduction in the overnight ΣFq for the high-density scenario relative to the low-density scenario (Fig. d). The open-run model (Fig. ) yielded an amount similar to the medium-density scenario.

The fraction of winter nights with surface hoar (nK in Eq. ) declines linearly as a function of both time and annual warming at a rate of 2.5 nights °C⁻¹ of warming (p<0.001, R2=0.22; Fig. a). The EOC experiences 14 % fewer surface hoar events per winter compared to the BOC (p<0.001). The average deposition event, ΣFq-‾, averages between 120–130 gm⁻² across all models during the BOC, remarkably similar to observations (Sect. ).

Fq,tot.‾ increases (becomes less negative) annually at a rate of 6.1 gm⁻² °C ⁻¹ (p<0.001, R2=0.12). Starting in the MOC and at a warming level of 3 °C, the sign of Fq,tot.‾ switches for some models, indicating that sublimation contributes more to the total flux than the deposition of overnight water vapor. Averaged across all models, the annual magnitude of Fq,tot.‾ decreases by 81 % by the EOC compared to the BOC (Fig. b) (p<0.001).

The reduction in the magnitude of Fq,tot.‾ is caused by the following factors. While deposition events increase in size slightly up until 2 °C of annual warming (Fig. c), the frequency of those events declines. Meanwhile, the size of the average sublimation event (ΣFq+‾) increases with year and rising Tair and are 15 % larger by the EOC (Fig. c.) (p<0.001). At the same time, the frequency of sublimation events also increases at the complement of the decreasing rate of deposition.