Model studies indicate that black carbon (BC) deposited on snow and sea ice produces climatically significant radiative forcing at both global and regional scales by reducing surface albedo (“BC albedo forcing”) (e.g., Warren and Wiscombe, 1980; Hansen and Nazarenko, 2004; Jacobson et al., 2004; Flanner et al., 2007). Global, annual average radiative forcing by BC in snow has been assessed as +0.04 Wm-2 using model estimates adjusted to observed snow concentrations (Bond et al., 2013; Boucher et al., 2013). BC snow albedo forcing has been cited in particular as a possible contributor to warming in the Arctic (e.g., Flanner et al., 2007; Koch et al., 2009), reduced springtime Eurasian snow cover (Flanner et al., 2009), melting of glaciers on the Tibetan Plateau and Himalayas (Xu et al., 2009; Kopacz et al., 2011), and changes in the Asian hydrological cycle (Qian et al., 2011). Estimates of this BC albedo forcing and the resulting climate impacts rely on modeling and therefore on accurate model representation of surface snow BC concentrations.

A critical difference between forcing by BC in the atmosphere and BC in snow is that forcing by BC in the atmosphere scales with the vertically resolved burden of BC (e.g., kilograms per square meter of air column), while forcing by BC in snow scales with the mixing ratio of BC (e.g., kilograms of BC per kilogram of snow) in the surface snow layer. This difference is because snow is a highly scattering medium so incident sunlight only penetrates to ∼ 10 cm depth, depending on the snow density, grain size and the mixing ratio of absorbing impurities. Therefore, BC deeper in the snowpack does not produce significant forcing. Surface snow BC mixing ratios are determined by the mixing ratio of BC in snowfall (wet deposition), the settling of atmospheric BC onto the snow surface (dry deposition) and in-snow processes that reduce the amount of snow (melting, sublimation) or that reduce the amount of BC (washout of BC with snow meltwater). It is perhaps unsurprising that sublimation is effective at raising surface snow BC mixing ratios. Empirical evidence has shown that when snow melts, the meltwater washes down through the snowpack more efficiently than do particulate impurities, also leading to enhanced BC concentrations at the snow surface (Conway et al., 1996; Xu et al., 2012; Doherty et al., 2013; Forsström et al., 2013). For models to accurately represent snow BC mixing ratios, they must simulate all of these processes with fidelity.

To date, the Community Earth System Model version 1 (CESM1) is the only global climate model that accounts for all of these processes, through the SNow, ICe, and Aerosol Radiative model (SNICAR; Flanner et al., 2007) in the land component (known as the Community Land Model version 4, CLM4; Lawrence et al., 2012), which accounts for snow on land, among other things. A more simplified treatment of BC in snow that is on sea ice and in the sea ice itself is also included in the most recent version of the CESM1 sea ice model component, CICE4 (Holland et al., 2012). In addition to treating processes that determine snow BC mixing ratios, SNICAR captures both fast and slow feedbacks that amplify the radiative forcing by BC in snow: surface snow warmed by BC absorption generally transforms to larger snow grain sizes, which further reduces snow albedo. In addition, the reduction in albedo for a given mixing ratio of BC is greater for larger-grained snow (Fig. 3 of Flanner et al., 2007). These feedbacks further accelerate warming and lead to earlier snowmelt, which in turn leads to higher BC mixing ratios in surface snow as described above. Eventually this also leads to earlier exposure of the underlying surface, further reducing surface albedo (i.e., the classic “snow albedo feedback”) (Flanner et al., 2007, 2009; Fig. 29 of Bond et al., 2013).

This comprehensive treatment in CESM1 made possible the recent Atmospheric Chemistry and Climate Model Intercomparison Project (ACCMIP) studies where BC albedo forcing was estimated for surface deposition fields derived from a suite of climate models (Lee et al., 2013). This forcing was included in an overall assessment of modeled radiative forcing under ACCMIP (Shindell et al., 2013). In the Lee et al. (2013) study, each participating ACCMIP model calculated BC atmospheric abundances and deposition rates using a common set of emissions. The resulting deposition fields (e.g., grams of BC deposited per square meter per second in each grid box/day) were then used in CESM1 to calculate snowpack BC mixing ratios. Estimated BC albedo forcing for the different models' aerosol fields covered a wide range, reflective of differences in BC transport and deposition rates. Comparisons of the modeled snow BC mixing ratios with observed mixing ratios across the Arctic and Canadian sub-Arctic showed significant positive model biases for Greenland (a factor of 4–8), a factor of 2–5 low biases over the Arctic Ocean, and agreement to within a factor of 2–3 elsewhere, though, with the exception of one model (CESM1-CAM5, which has version 5 of the Community Atmosphere Model), the BC mixing ratio biases in the remaining regions were more often positive than negative (see Lee et al., 2013; Table 6).

Goldenson et al. (2012) also used CESM1 with prescribed atmospheric aerosol concentrations and deposition fluxes to compute the climate impacts of BC in snow on both land and sea ice and BC in sea ice. They found significant impacts on surface warming and snowmelt timing due to changes in BC deposition in year 2000 versus year 1850. They also found that forcing by BC in snow on land surrounding the Arctic had a larger impact on Arctic surface temperatures and sea ice loss than did BC deposited on sea ice within the Arctic. On sea ice, Goldenson et al. (2012) found poor spatial correlation between modeled and observationally estimated BC concentrations (see their Fig. 3), though the range of concentration is similar; on land, the two are better correlated but the model concentrations tend to be higher, by roughly a factor of 2 (Goldenson et al., 2012; Fig. 4).

Jiao et al. (2014) applied CESM1 to simulate BC in snow on land and sea ice using deposition fields from the Aerosol Comparisons between Observations and Models (AeroCom) suite of global simulations. In comparison with measurements of BC in Arctic snow and sea ice (Doherty et al., 2011), they found that models generally simulate too little BC in northern Russia and Norway, while simulating too much BC in snow elsewhere in the Arctic. As with Goldenson et al. (2012), they found poor spatial correlation between modeled and measured BC-in-snow concentrations, though the multimodel means, subsampled over the measurement domain, were within 25 % of the observational mean.

Here we test whether the use of prescribed BC mass deposition rates in CESM1, as was done in the Goldenson et al. (2012), Holland et al. (2012), Lawrence et al. (2012), Lee et al. (2013) and Jiao et al. (2014) studies, produces a bias in surface snow BC mixing ratios, and therefore a bias in snow albedo. The bias being investigated would result from the fact that BC deposition fluxes in CESM1 prescribed-aerosol runs are decoupled from snow deposition rates, combined with the fact that the model's top snow layer has a fixed maximum thickness and is divided when it exceeds this thickness. Note that the bias being tested for here is independent of any biases due to errors in input emissions or in modeled transport and scavenging rates; it is purely a result of the mathematical approach taken in the model to estimate surface snow BC mixing ratios.

Prescribed-aerosol fields are derived from prognostic-aerosol model runs, where the resulting atmospheric concentrations and dry and wet mass deposition fluxes are saved as model output. This is used as input to the prescribed runs. In prognostic model runs, aerosols are emitted directly or formed from aerosol precursors in the atmosphere. Aerosols and their precursors are transported, dry-deposited to the surface, and scavenged in rain and snowfall according to the modeled meteorology. In prognostic-aerosol models, wet deposition of BC occurs only when there is rain or snowfall. The mass of wet-deposited BC depends on the amount of precipitation, the ambient BC concentration, and the hygroscopicity of the BC, with these dependencies varying from model to model.

When prescribed, atmospheric aerosol concentrations and deposition fluxes are typically independent of the meteorological fields in the model, as is the case in CESM1; the meteorological fields themselves in these runs may be either prescribed or prognostic. Furthermore, the input aerosol fields are often interpolated in time from monthly means. Therefore the episodic nature of aerosol deposition in reality (owing to wet deposition) is generally absent in prescribed-aerosol fields. This was the case for the prescribed-aerosol studies of Goldenson et al. (2012), Lawrence et al. (2012), and Holland et al. (2012), and for all integrations of CCSM4 (i.e., CESM1-CAM4) that were submitted to CMIP5 (Climate Model Intercomparison Project Project Phase 5) and used in the Lee et al. (2013) and Jiao et al. (2014) studies. In the Lee et al. (2013) and Jiao et al. (2014) studies, these BC deposition fields were then coupled with prescribed meteorology from the Climatic Research Unit (CRU)/National Center for Environmental Prediction (NCEP) reanalysis data for the 1996–2000 (Lee et al., 2013) or 2004–2009 period (Jiao et al., 2014) to calculate surface snow mixing ratios of BC. The CRU/NCEP data set is described at ftp://nacp.ornl.gov/synthesis/2009/frescati/model_driver/cru_ncep/analysis/readme.htm.

To test the effect of using decoupled BC mass and snow mass deposition rates on surface snow BC mixing ratios, we first compare ensembles of prescribed-aerosol and prognostic-aerosol runs of CESM1/CAM4. The prescribed-aerosol runs use the same monthly-resolved, year 2000 BC aerosol mass deposition rates that were used in the 20th century integrations of CCSM4 that were submitted to CMIP5. These deposition fluxes themselves come from a separate prognostic model simulation (Lamarque et al., 2010) and are interpolated from monthly-input fields (as shown in Fig. 1 for two model grid boxes in Greenland corresponding to research camps where BC in snow has been measured in snow pits and ice cores). CESM1/CAM4/CLM4 prescribed-aerosol runs were done for 10 years at 2∘ spatial resolution and at daily temporal resolution using repeating year 2000 prescribed aerosols and year 2000 greenhouse gases. The prognostic-aerosol runs are from the CESM1/CAM5/CLM4 Large Ensemble Community Project (Kay et al., 2014; www2.cesm.ucar.edu/models/experiments/LENS). Under this project, 30 realizations of CESM1 were run at 1∘ resolution from 1920–2100 with small initialization differences for each run (Kay et al., 2014). Aerosol and aerosol precursor emissions for year 2000 of these runs were the same as those used by Lamarque et al. (2010) to generate the aerosol deposition fields used in our prescribed-aerosol runs. In both the prescribed- and prognostic-aerosol runs, in-snow processes such as melting and sublimation also affect snowpack BC mixing ratios, and feedbacks amplify these effects. The output of aerosol and precipitation variables from the prognostic-aerosol runs is provided at monthly-average resolution only; so, for this comparison we use the monthly means for year 2000 from all 30 members and compare them with the monthly means of the prescribed-aerosol run.

Below we compare surface snow BC mixing ratios from CESM1 prescribed-aerosol and prognostic-aerosol runs to see if there is a systematic difference between the two, despite the fact that the aerosols are derived from the same emissions year and nearly the same emissions database. In the model, the mixing ratio of BC in the surface snow layer (MRBC) at each time step n is determined by the addition of BC through dry deposition (BCdepdry) and wet deposition (BCdepwet) and by the addition of new snowfall to the surface snow layer (SWEsnowfall). In the “real world”, wet-deposited BC is added only with new snowfall, in the form of the mixing ratio of BC in snowfall (MRBC,snowfall). The prognostic-aerosol runs are much like in the real world, while in the prescribed-aerosol run, BCdepwet is decoupled from SWEsnowfall. Since the sum of a series of ratios (MRBC,snowfall) does not equal the ratio of a series of sums (total BCdepwet and total SWEsnowfall), we expect this decoupling of deposition and snowfall will lead to errors in MRBC. In addition, if there is a large amount of new snowfall, MRBC,snowfall will be anomalously low, but much of this low-mixing-ratio snow will be buried in the snowpack where less (or no) sunlight interacts with it. In contrast, if there is only a small amount of new snowfall, MRBC,snowfall will be anomalously high, and this high-mixing-ratio snow will be near the snow surface and interact with sunlight. In a model with multiple snow layers that are divided with snow accumulation, the mixing ratio in the topmost model snow layer will thus be biased high. The magnitude of the high bias will depend on the model's top snow layer thickness. In this way, low snowfall/high MRBC,snowfall precipitation events will have a greater influence on time-averaged snow albedo than high snowfall/low MRBC,snowfall precipitation events.

In addition to differences deriving from coupled versus uncoupled BCdepwet and SWEsnowfall, the comparison of prescribed-aerosol and prognostic-aerosol runs will be affected by other model differences, such as the simulated geographic and temporal distribution of snow cover and BC transport and scavenging in CAM5 (prognostic-aerosol runs) vs. CAM4 (prescribed-aerosol runs). Positive feedbacks (e.g., consolidation of BC in surface snow during snowmelt) are included in both runs, so any resulting differences in surface snow BC mixing ratios will be amplified. Therefore, we also conducted a series of offline calculations to isolate the effect of BC deposition being decoupled from snowfall rates in the prescribed runs (Table 1).

In CESM1, at each time step, n, surface snow BC mixing ratios, [MRBCn]model (ng g-1) are determined by the dry- and wet-deposited masses of BC (BCdepdryn and BCdepwetn; ng m-2), the mass of snow in the surface snow layer (SWEsurfn;  g m-2), the mixing ratio of BC from the previous time step [MRBCn-1]model; ng g-1), the fraction of the surface snow layer that is replaced by new snowfall, fn, (once the surface snow layer has reached its maximum thickness), and the combined effects of melt and sublimation on BC and snow-water masses in the surface layer, which we will simply denote here as X (ng g-1): [MRBCn]model=BCdepdrynSWEsurfn+BCdepwetnSWEsurfn+1-fn×[MRBCn-1]model+X, where fn=SWEsnowfalln/SWEsurfn. In Eq. (1), the surface snow BC mixing ratio at time-step n equals the sum of, respectively, dry-deposited BC during time-step n, the addition of wet-deposited BC during time-step n, the mass of BC and snow water remaining in the surface layer at time-step n from time-step n-1, and the impact of melt and sublimation on BC and snow-water content. By definition, in prognostic-aerosol runs BCdepwetn is zero if there is no precipitation (fn=0), so the second term in Eq. (1) is zero. However, in prescribed-aerosol runs there is both dry- and wet-BC deposition at every time step (e.g., see Fig. 1), even when there is no precipitation. Effectively this means that in prescribed-aerosol runs the mixing ratio of BC in snowfall, MRBC,snowfalln, approaches infinity as snowfall approaches zero since MRBC,snowfalln=BCdepwetn/SWEsnowfalln. In our offline calculations we diagnose the BC mixing ratio both in snowfall (MRBC,snowfalln) and in our model's surface snow layer (MRBCn). In CLM4, the surface snow layer is of variable thickness but always between 1 and 3 cm and is 1–2 cm thick when snow depth exceeds 3 cm (Oleson et al., 2010). In our calculations we set the surface snow layer BC mixing ratio on day 1 to that from day 1 in the prescribed-aerosol CESM1/CAM4/CLM4 run. The surface snow layer BC mixing ratios for all subsequent days in the year are then calculated offline. Values of BCdepdryn, BCdepwetn, SWEsurfn and SWEsnowfalln for each time step and grid box are taken directly from the prescribed-aerosol run of CESM1/CAM4. In our first set of offline calculations, we calculate surface snow mixing ratios that are equivalent to those from the prescribed-aerosol run, minus the effects of melting and sublimation: [MRBCn]d=BCdepdrynSWEsurfn+BCdepwetnSWEsurfn+1-fn×[MRBCn-1]d. If fn is greater than 1.0, the surface snow layer from time-step n-1 will be buried to the second (or deeper) layer and will play no role in determining the surface snow layer BC mixing ratio. Thus, if fn is greater than 1.0 we simply set fn=1.0. All calculations are done at daily resolution. By not including the effects encompassed by X (Eq. 1) in our offline calculations we are isolating how dry and wet deposition only affect MRBC. While the focus here is on BC, the same conclusions would apply for deposition/surface snow mixing ratios of dust and organic aerosols.

While Eqs. (1) and (4) allow for wet deposition of BC even in the absence of snowfall, a more physically realistic calculation of surface snow BC mixing ratios (minus the influence of in-snow processes) is given by MRBCn=BCdepdrynSWEsurfn+fn×MRBC,snowfalln+1-fn×MRBCn-1. In this calculation, the contribution of wet deposition to MRBCn is through the mixing ratio of BC in snowfall (MRBC,snowfalln), and this contribution goes to zero when the snowfall (fn) goes to zero. However, we cannot use in Eq. (5) MRBC,snowfalln as calculated directly from BCdepwetn and SWEsnowfalln from the prescribed-aerosol run, since, as noted above, this sometimes yields infinite values of MRBC,snowfalln. Therefore, we recalculate MRBC,snowfalln by assuming that total BC mass deposition flux scales with total snowfall (in snow-water equivalent) within each month and grid box, yielding the smoothed values [MRBC,snowfall]m and [MRBC,snowfall]y, which are calculated as follows.

[MRBC,snowfall]m: within each month of the multiyear model run, SWEsnowfall and BCdepwet from the prescribed-aerosol model run are summed. Monthly values of MRBC,snowfall are calculated from the ratio of the monthly-total BCdepwet and monthly-total SWEsnowfall.

[MRBC,snowfall]y: a monthly climatology of monthly-total SWEsnowfall is computed. Monthly values of MRBC,snowfall are calculated from the ratio of the monthly-total BCdepwet and the monthly climatology of SWEsnowfall.

These smoothed snowfall BC mixing ratios are compared to those given by using the prescribed-aerosol model values directly.

[MRBC,snowfall]d: each day MRBC,snowfall is calculated as the ratio of the prescribed daily BCdepwet (e.g., Fig. 1) and daily SWEsnowfall.

The wet and dry BC mass deposition rates used to calculate all values of MRBC,snowfalln are exactly those used in the prescribed-aerosol runs. The total BC mass and total snow mass deposited to the surface within a given month and grid box, averaged across all years, is the same across all three sets of these calculations, so the only difference in how they affect surface snow BC mixing ratios is through changes in the relative timing of when BC is deposited to the surface versus when snow is deposited to the surface.

Surface snow BC mixing ratios [MRBC]d for each grid box/day are then calculated using Eq. (4), and corresponding values of [MRBC]m and [MRBC]yare calculated using Eq. (5) with [MRBC,snowfall]m and [MRBC,snowfall]y, respectively (Table 1). We again emphasize that the values [MRBC]d are analogous to those in CESM1 when aerosol deposition fluxes are prescribed, minus the effects of melt and sublimation; i.e., time-averaged, smoothed prescribed BCdepwet is paired with daily-varying SWEsnowfall, and wet deposition is present even when there is zero new snowfall. In contrast, [MRBC,snowfall]m and [MRBC,snowfall]y use SWEsnowfall values that have been time-averaged over increasing temporal scales, and so are more physically consistent with BCdepwet, which is the product of averaging across multiple years of prognostic model runs using the same BC emissions. Furthermore, [MRBC]m and [MRBC]y are only affected by wet deposition when there is new snowfall.

We conduct two full sets of offline calculations of [MRBC,snowfall]d, [MRBC,snowfall]m, [MRBC,snowfall]y and [MRBC]d, [MRBC]m, [MRBC]y (Table 1). In one set of offline calculations, MRBC,snowfalln and fn are calculated using SWEsnowfall taken directly from our prescribed-aerosol model runs; we will refer to these as the “CESMmet” (CESM meteorology) calculations. In a second set of calculations, model snowfall rates were replaced with CRU/NCEP reanalysis daily precipitation for the years 2004–2009 in order to mimic the runs reported by Jiao et al. (2014); we will refer to these as the “CRUNCEPmet” calculations. The CRU/NCEP data set specifies precipitation rates but not whether it is rain or snow, so we made the simple assumption that when the reported surface air temperature was 0 ∘C or lower the precipitation was snowfall. In both cases, snow cover – specifically, the snow-water equivalent in the surface snow layer for each day and grid box – is the average across the 10 model years of the year 2000 CESM1-CAM4 run. Calculations are done for all variables for either 10 years, using SWEsnowfall values from the model (CESMmet; repeating year 2000 meteorology), or 6 years, using SWEsnowfall from the CRU/NCEP reanalysis data set (CRUNCEPmet; years 2004–2009 meteorology).

Note that while averaged values of SWEsnowfall were used to calculate [MRBC,snowfall]m and [MRBC,snowfall]y, the fraction of surface snow replaced by new snowfall (fn) is always calculated using the daily-varying value of SWEsnowfall from either CESM1-CAM4 (CESMmet) or the CRU/NCEP reanalysis data set (CRUNCEPmet). In other words, the rate of snowfall varies daily according to the model (CESMmet) or reanalysis (CRUNCEPmet) meteorology in all offline calculations, but the BC mixing ratio in that snowfall is either [MRBC,snowfall]d, [MRBC,snowfall]m or [MRBC,snowfall]y. This allows for realistic evolution of the snowpack water mass while testing the effect of using different estimates of the mass mixing ratio of BC in snowfall.

We compare the results of the prognostic-aerosol runs versus the prescribed-aerosol runs and across our six sets of offline calculations (Table 1) for three geographic regions where forcing by BC in snow on land is climatically important: Greenland (60—85∘ N, 290–340∘ W), North America (50–80∘ N, 190–300∘ W) and Eurasia (60–75∘ N, 30–180∘ W). Only those grid boxes containing snow on land are included in the statistics presented below; snowfall on sea ice and BC in snow on sea ice are not considered here.

We argue that prescribing temporally and geographically smoothed surface BC deposition fluxes in a model where snowfall varies on typical meteorological timescales (i.e., daily or faster) will produce high biases in time-averaged surface snow BC mixing ratios. Using comparisons of prescribed-aerosol and prognostic-aerosol model runs and offline calculations, we have demonstrated that (a) prescribed-aerosol runs have higher surface snow BC mixing ratios than prognostic-aerosol runs, by a factor of about 1.6–3.0, despite being based on the same BC emissions and accounting to first order for differences in total BC and snow deposited to the surface; and that (b) decoupling of BC wet deposition fluxes and snowfall rates leads to surface snow BC mixing ratios of a factor of about 1.5–2.5 higher than if the same mass of BC was wet-deposited in proportion to the snowfall snow mass. Both of these biases are significant at daily, seasonal and annual timescales.

Black carbon mass deposition fluxes in snowfall depend on ambient BC concentrations, the scavenging efficiency of BC in snow, and snowfall rates. Thus, while BC deposition fluxes do not depend solely on precipitation rates, removing any dependence on snowfall leads to biases in the mixing ratio of BC in snowfall, MRBC,snowfall. If BC deposition rates and snowfall rates are fully decoupled, MRBC,snowfall will be biased high on days of lower snowfall, when the fractional contribution to surface snow (fn) is lower than average. Conversely, MRBC,snowfall will be biased low on days when fn is higher than average. As our offline calculations have shown, low and high biases in MRBC,snowfall do not have offsetting effects on surface snow BC mixing ratios (MRBC). This is because the cases of high-biased MRBC,snowfall remain near the snow surface and therefore have a strong influence on MRBC. Conversely, cases of low-biased MRBC,snowfall may contribute to snow deeper in the snowpack and so have less influence on the surface snow BC mixing ratio.

We estimate that prescribed-aerosol model runs of CESM1 have approximately a high-bias factor of 1.5–2.5 in surface snow BC mixing ratios due to the use of climatological/smoothed BC mass deposition fluxes coupled with modeled, daily-varying snowfall. In CESM1 (i.e., in the SNICAR component of CLM) the surface snow layer is 1–3 cm deep. Sunlight usually can penetrate > 10 cm into the snowpack, depending on snow density (Warren and Wiscombe, 1980), so mixing ratios over this full depth are relevant for albedo reduction and BC albedo forcing. SNICAR accounts for this, with albedo being determined by MRBC in as many snow layers as is reached by sunlight (typically the top 2 or 3 layers). We expect the bias in surface snow BC mixing ratios will decrease as the depth of the top snow layer increases, becoming zero as the depth of the surface layer approaches the total snowpack depth. When multiple layers are represented, the high biases in BC mixing ratios in the surface layer will be accompanied by low biases in BC mixing ratios in deeper snow layers. However, since the amount of sunlight drops off rapidly with snow depth, the MRBC in the top few centimeters of the snowpack has the strongest influence on albedo. Most absorption of sunlight by BC will occur in the top few centimeters of the snowpack, i.e., the surface snow layer in SNICAR. It is beyond the scope of this study to calculate the exact impact on modeled albedo for snow of different densities and therefore different sunlight penetration depths. It is sufficient to point out the following: a.

Using climatological, prescribed mass deposition fluxes coupled with daily-precipitation rates produces a large positive bias in surface snow MRBC that is significant across daily, seasonal and annual-average timescales and from a grid box to broad regional (and therefore also global) geographic scales.

We suggest that, for wet deposition, one option is that instead of prescribing mass deposition fluxes (e.g., kg m-2 s-1 BC deposition) the model could instead prescribe mass mixing ratios in snowfall (e.g., nanograms BC per gram snowfall SWE, or parts per billion BC per snowfall water). These prescribed mass mixing ratios could be a climatology from a multiyear integration of a prognostic-aerosol model. The appropriate number of model run years would need to be determined by testing how both the mean and variability in snow mixing ratios change with number of years averaged. Aerosol dry deposition will need to continue to be prescribed as a mass flux since it does not scale with snowfall. The value of MRBC at time step n could then be calculated directly as given in Eq. (5), as used here in our offline calculations of [MRBC]m and [MRBC]y. This approach will produce an inconsistency in the mass balance of BC within the prescribed-aerosol model runs in that the change in the mass of BC in the atmosphere between time steps will not equal the mass of BC deposited to the surface. However, both the atmospheric BC concentrations and surface snow BC mixing ratios in the model calculation will be physically more realistic. This is preferable to maintaining the mass balance within the prescribed-aerosol run since both the atmospheric concentrations and deposition rates are anyhow prescribed, and the climatically important variable in studies of albedo forcing is the surface snow BC mixing ratio.