## 1. Introduction

Snow-albedo feedback (SAF) enhances Northern Hemisphere (NH) extratropical climate sensitivity in climate change simulations (Budyko 1969; Sellers 1969; Schneider and Dickinson 1974; Robock 1983; Cess et al. 1991; Randall et al. 1994; Hall 2004) because of two changes in the snowpack as surface air temperature (*T _{s}*) increases (Robock 1985). First snow cover shrinks, and where it does it generally reveals a land surface that is much less reflective of solar radiation. Second, the remaining snow generally has a lower albedo because its optical properties are sensitive to

*T*. For example, wet melting snow, more common in a warmer climate, has a lower surface albedo than dry frozen snow (Robock 1980). Also, in a warmer climate the frequency of storms where precipitation falls as snow rather than rain decreases, particularly during the transition seasons when midlatitude atmospheric temperatures are close to freezing. Since snow albedo decreases with snow age, the resulting increase in average snow age will lead to lower average snow albedos (Wiscombe and Warren 1980). These snowpack metamorphosis effects are typically parameterized either with an explicit dependence of snow albedo on surface temperature (e.g., Robock 1980) or with a dependence of snow albedo on snow age (e.g., Wiscombe and Warren 1980). The two feedback loops associated with shrinking snow cover and the metamorphosis of the remaining snowpack are illustrated schematically in Fig. 1. Their interactions with other climate feedbacks are included in a climate feedback diagram given in Robock (1985).

_{s}*T*[following Cess and Potter (1988)]:Here

_{s}*Q*and

*Q*

_{net}are the incoming and net shortwave radiation at the TOA (where

*Q*is taken to be a constant),

*α*is the surface albedo, and

_{s}*α*is the planetary albedo. Equation (1) indicates that the strength of SAF is determined by the product of two terms, a coefficient representing the variation in planetary albedo with surface albedo (∂

_{p}*α*/∂

_{p}*α*) and another representing the change in surface albedo associated with a 1°C change in the surface air temperature (Δ

_{s}*α*/Δ

_{s}*T*). The magnitude of the second term is determined by the strength of the snow cover and snowpack metamorphosis feedback loops of Fig. 1. SAF is strongest in springtime because both

_{s}*Q*and snow cover are large (Robock 1983). In fact, Hall (2004) reports that approximately half the total NH SAF to simulated global climate change occurs during NH spring. Thus we focus on this season in our assessment of the two feedback loops controlling SAF strength.

Recently a highly accurate technique was developed to calculate the coefficient ∂*α _{p}*/∂

*α*given standard model output (Qu and Hall 2006). It was used to calculate springtime ∂

_{s}*α*/∂

_{p}*α*values in NH land areas for transient climate change experiments of the Fourth Assessment (AR4) of the Intergovernmental Panel on Climate Change (IPCC). The intermodel variation in ∂

_{s}*α*/∂

_{p}*α*is small, with most models agreeing to within 10% that a given

_{s}*α*anomaly results in an

_{s}*α*anomaly one-half as large (Fig. 2a).

_{p}This agreement occurs because the main factor controlling ∂*α _{p}*/∂

*α*is the transmissivity of the clear-sky atmosphere, where the models’ radiative transfer schemes converge in their handling of the atmosphere’s interaction with solar photons.

_{s}While there is consistency from model to model in ∂*α _{p}*/∂

*α*values, the second term in Eq. (1), Δ

_{s}*α*/Δ

_{s}*T*diverges widely. We calculate this term in the climate change context based on springtime values of

_{s}*α*and

_{s}*T*averaged over NH land areas from the beginning and end of the AR4 experiments (Fig. 2b). Not only is there almost a threefold spread in Δ

_{s}*α*/Δ

_{s}*T*but there is no clear preference for a central value. An understanding of the controls on SAF strength therefore amounts to a determination of what causes Δ

_{s}*α*/Δ

_{s}*T*to vary so much from model to model. This in turn requires an assessment of the relative contributions of snow cover reduction and snowpack metamorphosis to the surface-albedo reduction. In this study, we develop methods to separate and quantify contributions to Δ

_{s}*α*/Δ

_{s}*T*from these two feedback loops.

_{s}The causes of the spread of Δ*α _{s}*/Δ

*T*are particularly important in light of recent work (Hall and Qu 2006), demonstrating that many models have an unrealistic SAF strength in the context of the present-day seasonal cycle of

_{s}*T*. Because of a very strong correlation between the SAF strength in seasonal cycle and climate change contexts, addressing this bias in the seasonal cycle would significantly reduce the spread of SAF strength in climate change. In this study we develop a very simple picture of the reason for the spread in Δ

_{s}*α*/Δ

_{s}*T*in the current generation of climate simulations and relate SAF strength to specific choices about surface-albedo parameterizations and tunable parameters within them. This provides the climate modeling community with guidance in creating a more realistic SAF in individual models and ultimately reducing the spread in SAF.

_{s}Our study is presented as follows. The AR4 climate simulations and the models’ surface-albedo parameterizations are briefly described in section 2, followed in section 3 by a presentation of an idealized mathematical framework relating the two feedback loops controlling Δ*α _{s}*/Δ

*T*to quantities typically provided in model output. This framework is used in section 4 to separate and quantify the components of Δ

_{s}*α*/Δ

_{s}*T*in each model and to determine which components are responsible for its intermodel spread. In section 5, we discuss how SAF strength is related to surface-albedo schemes and tunable parameters within them. A summary and implications are given in section 6, followed by an appendix giving a description of the technique used to convert snow mass values outputted by many models to snow cover values necessary for our calculations.

_{s}## 2. Climate simulations and snow-albedo parameterizations

The simulated climates of the twentieth and twenty-second centuries were extracted from the “720-ppm stabilization experiment” with 18 AR4 models (see Table 1). In these experiments, historical twentieth-century forcing was imposed, followed by the IPCC’s AlB emission scenario for the twenty-first century. At year 2100, anthropogenic forcings were fixed for the remainder of the experiments, which end at year 2200. Though this forcing scenario was imposed on 23 models for the AR4, only the 18 listed in Table 1 had a complete time series of the necessary variables when our analyses were performed.

The schemes taking into account effects of snow on surface albedo can be grouped into four categories according to the way the masking effect of vegetation is parameterized. We list them here in order of decreasing complexity. The schemes of type 1 (including 5 models) employ a full-blown canopy radiative transfer model. In type 2 schemes (including 7 models), canopy albedo is prescribed according to vegetation type and then modified depending on whether snow is sticking to the canopy, and the overall surface albedo is a weighted mean of canopy albedo and ground albedo, with weights given by the fraction of vegetation coverage. In both types 1 and 2, ground albedo is given as a weighted mean of snow albedo and soil albedo, with weights determined by snow cover. And in both types, canopy reflectivity is increased when snow is allowed to stick to the canopy. In the simpler schemes of types 3 and 4 (each including 3 models), the canopy and ground albedos are not treated separately, and the overall surface albedo is simply a weighted mean of snow-free surface albedos and snow albedo, with weights determined by snow cover. In the type 3 schemes, the snow albedo depends on vegetation type, while the type 4 schemes are even simpler, with snow albedos independent of vegetation type.

An important difference between the first two and last two types is apparent where the ground is completely covered by snow. In this case, the surface albedo in types 3 and 4 is equivalent to the prescribed snow-albedo values while the surface albedo in types 1 and 2 is probably smaller than the prescribed snow albedo, since the canopy influences the surface albedo and is likely darker than snow. We emphasize this distinction throughout this paper by reserving the term “snow albedo” for the albedo of the snow itself, and referring to the albedo of surfaces with 100% snow cover as the “effective snow albedo.” They are equivalent and both tunable in type 3 and 4 models, while in type 1 and 2 models the effective snow albedo is determined by snow albedo and canopy effects. Table 1 shows the category each model belongs to. Links between the characteristics of snow-albedo parameterizations discussed here and the strength of SAF are examined in section 5.

## 3. An idealized model to understand Δ*α*_{s}/Δ*T*_{s}

_{s}

_{s}

In this section, we introduce a framework to separate and understand the two feedback loops determining the magnitude of Δ*α _{s}*/Δ

*T*in climate change simulations. In Fig. 1, snow cover, surface albedo, and

_{s}*T*are connected by arrows representing effects of one variable on another. There are four main effects, with

_{s}*κ*

_{1},

*κ*

_{2},

*κ*

_{3}, and

*κ*

_{4}representing the constant of proportionality translating the magnitude of an anomaly in one variable into the magnitude of an anomaly in the affected variable. The reduction in snow cover induced by a unit increase in

*T*is given by

_{s}*κ*

_{1}, and the change in surface albedo induced by a unit change in snow cover is then given by

*κ*

_{2}. Meanwhile, the

*T*increase also generates a surface-albedo reduction of the snowpack itself because of snowpack metamorphosis. The associated constant of proportionality is given by

_{s}*κ*

_{3}. Finally,

*κ*

_{4}gives the change in

*T*arising from a unit change in surface albedo through heat exchange processes between the surface and atmosphere.

_{s}*T*

_{s}_{0}, arising from an external forcing that sets in motion the interactions of Fig. 1: a decrease in snow cover and hence surface albedo occurs, as well as a decrease in albedo of the existing snowpack, leading finally to an additional increase in

*T*This incremental

_{s}.*T*increase and subsequent increments, represented by Δ

_{s}*T*

_{s}_{1}, Δ

*T*

_{s}_{2}, . . . , Δ

*T*, can be expressed in terms of

_{sn}*κ*

_{1},

*κ*

_{2},

*κ*

_{3}, and

*κ*

_{4}:Here, we assumed that

*κ*

_{1},

*κ*

_{2},

*κ*

_{3}, and

*κ*

_{4}are constant with each increment. Then, the total change in surface air temperature, Δ

*T*, can be obtained by summing up the initial perturbation and all the incremental changes:We assume the series in Eq. (5) converges when

_{s}*n*goes to infinity and thus we obtain

*T*perturbation, Δ

_{s}*α*

_{s}_{0}, Δ

*α*

_{s}_{1}, . . . , Δ

*α*, can be similarly derived:The total change in surface albedo, Δ

_{sn}*α*, is the sum of all the perturbationsUsing Eq. (6), we haveLikewise, we can obtain the total change in snow cover (

_{s}*S*):Based on Eqs. (6), (10), and (11), the ratios Δ

_{c}*α*/Δ

_{s}*T*and Δ

_{s}*S*/Δ

_{c}*T*are written asandPlugging Eq. (13) into (12), we obtain an expression for the SAF parameter in terms of the constants

_{s}*κ*

_{2}and

*κ*

_{3}:Equation (14) is our framework for understanding contributions to Δ

*α*/Δ

_{s}*T*. It implies that Δ

_{s}*α*/Δ

_{s}*T*is controlled by two components, one arising from the metamorphosis of the snowpack in a warmer climate (

_{s}*κ*

_{3}) and another stemming from the reduction of snow cover (

*κ*

_{2}Δ

*S*/Δ

_{c}*T*).

_{s}*κ*

_{2}and

*κ*

_{3}, we introduce the following approximation for surface albedo in snow-covered land areas for present and future climates:andwhere the superscripts “

*p*” and “

*f*” represent the present and future climate. Here

*α*

_{snow}represents effective snow albedo (the albedo of a completely snow-covered surface after taking into account possible canopy effects) and

*α*

_{land}is snow-free land albedo. Note that

*α*

_{land}is a constant, so that we are not considering changes in snow-free land albedo resulting from climate change here. Subtracting Eq. (15) from (16) and then manipulating, we obtain an expression for Δ

*α*that is the sum of two terms, one relating to the change in effective snow albedo and the other relating to a change in snow cover:Dividing Eq. (17) by the change in

_{s}*T*between the two climates, we haveComparing Eqs. (14) and (18), we obtain the following expressions for

_{s}*κ*

_{2}and

*κ*

_{3}:andEquations (19) and (20) indicate that

*κ*

_{2}may be interpreted as the mean contrast between effective snow albedo (averaged over the present and future climates) and snow-free land albedo. Meanwhile,

*κ*

_{3}represents changes in effective snow albedo with respect to changes in

*T*, modulated by mean snow cover.

_{s}## 4. Separating snow metamorphosis and snow cover components

To quantify and compare the contributions of snow metamorphosis and snow cover components to the reduction in surface albedo (Δ*α _{s}*), represented by the two terms on the right-hand side of Eq. (17), we need to estimate

*S*,

^{f}_{c}*S*,

^{p}_{c}*α*

^{f}_{snow},

*α*

^{p}_{snow}, and

*α*

_{land}for each model. The

*S*and

^{f}_{c}*S*can be readily derived from snow mass values outputted by many of the models (see the appendix for details on how this is done), and thus the only unknowns are

^{p}_{c}*α*

^{f}_{snow},

*α*

^{p}_{snow}, and

*α*

_{land}. As demonstrated in section 3, these three variables are constrained by Eqs. (15) and (16). To solve for them, we need a third constraining expression to complement the two equations. In this section, we construct such an expression by estimating values of (

*α*

^{f}_{snow}−

*α*

^{p}_{snow})/Δ

*T*. This term represents the variation of effective snow albedo with temperature, and as demonstrated in Eq. (18), this can be weighted by mean snow cover to give the contribution of snow metamorphosis to Δ

_{s}*α*/Δ

_{s}*T*.

_{s}For simplicity, we denote (*α ^{f}*

_{snow}−

*α*

^{p}_{snow})/Δ

*T*by

_{s}*F*(

*T*). Here, the dependence of

_{s}*F*on

*T*accounts for the fact that the snow metamorphosis effect varies significantly with the mean temperature of a snow-covered area undergoing climate change. In general, both the snow melting and aging effects increase as the mean temperature approaches freezing. To quantify

_{s}*F*(

*T*), we focus on the snow interior, defined as the regions where the surface is 100% snow-covered in both the current and future climate. In these regions any change in

_{s}*α*stems entirely from the change in the effective albedo of the snowpack itself. Here

_{s}*S*=

^{p}_{c}*S*= 1 by definition and thus according to Eqs. (14) and (20), Δ

^{f}_{c}*α*=

_{s}*F*(

*T*)Δ

_{s}*T*. Since Δ

_{s}*α*and Δ

_{s}*T*at every 100% snow-covered grid point are known from the model output, we can group them by intervals of

_{s}*T*and then average them for each

_{s}*T*interval. Denoting the resultant functions by

_{s}*T*) and

_{s}*T*), we can evaluate

_{s}*F*(

*T*) by dividing

_{s}*T*) by

_{s}*T*). We perform the calculation based on climatological monthly mean

_{s}*α*and

_{s}*T*data. This provides samples large enough to achieve stable statistics for most of the 18 simulations listed in Table 1. [The Parallel Climate Model (PCM), ECHAM5/Max Planck Institute Ocean Model (MPI-OM), ECHAM and the global Hamburg Ocean Primitive Equation (ECHO-G), and the Institute of Numerical Mathematics Coupled Model version 3.0 (INM-CM3.0) were not used in this calculation because the PCM did not have either snow cover or snow mass available when our analyses were performed, and ECHAM5/MPI-OM, ECHO-G, and INM-CM3.0 do not have enough 100% snow-covered grid points to produce stable statistics, particularly at temperatures close to freezing.]

_{s}To reveal general characteristics of *F*, we show its ensemble-mean values at different *T _{s}* in Fig. 3. For

*T*,

_{s}*F*is consistently negative, indicating that effective snow albedo decreases as

*T*increases. The

_{s}*F*increases from 250 to 270 K, where it reaches its maximum, signifying the largest snow metamorphosis effect there, and then decays rapidly above this point. The details of each model’s

*F*(not shown) vary somewhat, mostly because of differing representations of the snow metamorphosis effect in the simulations. However, in general they closely resemble the curve shown in Fig. 3.

Once *F* is known, we have a third expression constraining *α ^{f}*

_{snow}and

*α*

^{p}_{snow}in addition to Eqs. (15) and (16). They form a complete set of equations that we can solve for

*α*

^{f}_{snow},

*α*

^{p}_{snow}, and

*α*

_{land}. (The ensemble-mean

*F*values were used in the calculations for ECHAM5/MPI-OM, ECHO-G, and INM-CM3.0.) Then we can evaluate the change in

*α*attributable to snow metamorphosis and that attributable to snow cover at each snow-covered grid point using Eq. (17). We then average the 2 contributions to the

_{s}*α*reduction over the NH land areas poleward of 30°N and divide both of them by the

_{s}*T*increase averaged over the same area, giving the 2 components of Δ

_{s}*α*/Δ

_{s}*T*for each simulation. The results are shown in Fig. 4. The metamorphosis component is generally small compared to the snow cover component. This reveals that snow-albedo feedback results mostly from the reduction in snow cover rather than the reduction in albedo of the snowpack itself. Not only is the metamorphosis component generally small compared to the snow cover component, but it barely contributes to the spread of Δ

_{s}*α*/Δ

_{s}*T*seen in Fig. 2. As shown in Fig. 5, the intermodel variations of Δ

_{s}*α*/Δ

_{s}*T*are highly correlated with the intermodel variations of the snow cover component. There are a few cases where the snow cover component does not dominate. The metamorphosis component in the Meteorological Research Institute Coupled General Circulation Model version 2.3.2a (MRI CGCM2.3.2; model 1) is twice as large as the snow cover component, while the two components are comparable in the Centre National de Recherches Météorologiques Coupled Global Climate Model version 3 (CNRM-CM3; model 2) and Goddard Institute for Space Studies Model E-R (GISS-ER; model 4).

_{s}## 5. Controls on the snow cover component

Because of its dominance of SAF in most models as well as the intermodel spread of the feedback, we wish to decompose the snow cover component further and determine what controls it. According to Eq. (14), two factors govern its magnitude, one being the mean contrast between effective snow albedo and snow-free land albedo (*κ*_{2}) and the other being the sensitivity of snow cover to *T _{s}* (Δ

*S*/Δ

_{s}*T*). The second factor is straightforward to calculate from model output and is shown in Fig. 6a as an average over NH landmasses for each model. To quantify

_{s}*κ*

_{2}we simply divide the snow cover component (Fig. 4) by the Δ

*S*/Δ

_{c}*T*values of Fig. 6a. The

_{s}*κ*

_{2}values are shown in Fig. 6b. Though

*κ*

_{2}and Δ

*S*/Δ

_{c}*T*both exhibit variability from model to model,

_{s}*κ*

_{2}exhibits significantly more intermodel variation. Moreover, the variations of

*κ*

_{2}are more consistent with the intermodel variations of the total snow cover component than those of Δ

*S*/Δ

_{c}*T*. This is confirmed in the scatterplot of

_{s}*κ*

_{2}Δ

*S*/Δ

_{c}*T*vs

_{s}*κ*

_{2}(Fig. 7a). The square of the correlation between

*κ*

_{2}Δ

*S*/Δ

_{c}*T*and

_{s}*κ*

_{2}is 0.83. This implies that the contrast between effective snow albedo and snow-free land albedo accounts for about 80% of the intermodel variance in the snow cover component of SAF, with the relationship between snow cover and

*T*accounting for the remaining 20%.

_{s}To determine what factors are responsible for the intermodel variation in the contrast between effective snow albedo and snow-free land albedo, we average effective snow albedo for the current climate (*α ^{p}*

_{snow}) and snow-free land albedo (

*α*

_{land}) over the entire snow-covered region of each model. It turns out that the intermodel variability in

*α*

^{p}_{snow}is much larger than the intermodel variability in

*α*

_{land}, so that the intermodel variability in the mean contrast between these two quantities is overwhelmingly dominated by the

*α*

^{p}_{snow}contribution. This is confirmed by a scatterplot of

*κ*

_{2}against

*α*

^{p}_{snow}(Fig. 7b). The correlation coefficient implies that more than 80% of the intermodel variance in

*κ*

_{2}can be accounted for by the variation in the mean albedo of 100% snow-covered surfaces.

The successive links between the magnitudes of the total SAF (Δ*α _{s}*/Δ

*T*), the snow cover component of the feedback (

_{s}*κ*

_{2}Δ

*S*/Δ

_{c}*T*), the contrast between effective snow albedo and snow-free land albedo (

_{s}*κ*

_{2}), and finally the mean effective snow albedo itself (

*α*

^{p}_{snow}) are each strong enough that the mean effective snow albedo is a reasonably good predictor of the total SAF strength. This is clear in the scatterplot of Δ

*α*/Δ

_{s}*T*against

_{s}*α*

^{p}_{snow}(Fig. 8). The 2 variables are clearly related, and the correlation coefficient implies that the intermodel variation in mean effective snow albedo accounts for more than 2/3 of the intermodel variance in SAF strength.

There is a further relationship between mean effective snow albedo and SAF strength, and surface-albedo parameterization, as revealed by the color coding of the model numbers in Fig. 8. Models with the weakest SAF (models 1–5) all use either type 1 or 2 parameterizations (where ground albedo is given as a weighted mean of snow albedo and soil albedo, with weights determined by snow cover, and then overall surface albedo is a weighted mean of canopy albedo and ground albedo with weights given by the fraction of vegetation coverage), and they all have very low mean effective snow albedos. Models with very strong SAF (models 16–18) all use the type 4 parameterization (where overall surface albedo is a weighted mean of snow-free land albedos and snow albedo, with weights determined by snow cover), and they have the highest mean effective snow albedos. The models with type 3 parameterization (models 10, 11, and 15), as well as a few models with type 1 and 2 parameterizations (models 7–9 and 12–14), have intermediate mean effective snow albedos, and thus have intermediate SAF.

It might be assumed that the more complex models with explicit treatment of the vegetation canopy have more realistic SAF. However this may not be the case. In a previous study (Hall and Qu 2006), we showed that the simulated strength of SAF in the seasonal cycle context is an excellent predictor of SAF strength in the context of transient climate change. Thus models with the weakest SAF in the climate change context (models 1–5) all have unrealistically weak SAF in the seasonal cycle context by approximately a factor of 2. These models also all employ type 1 or 2 land surface parameterizations and have low albedos for snow-covered surfaces. We speculate that in these models, either 1) prescribed albedos of snow-covered ground or canopy when snow is present are too low or 2) that vegetation coverage determining the weighting of canopy albedo and ground albedo is too large, placing unrealistic emphasis on the vegetation canopy in the determination of overall effective surface albedo when the ground is at least partly snow-covered. At the same time, the models with the strongest SAF in the climate change context (models 16–18) all have unrealistically strong SAF in the seasonal cycle context by about 10%–20%, employ type 4 land surface parameterizations, and have very high albedos for snow-covered surfaces. We speculate that in these models snow albedos are assigned based on surface-albedo measurements of snow-covered surfaces with little protruding vegetation, so that completely snow-covered surfaces where vegetation does protrude in the real world are assigned unrealistically large albedos. This interpretation is supported by the fact that type 3 models, distinguished from type 4 models by the fact that snow albedos are prescribed according to vegetation type, have a somewhat lower mean effective snow albedo and a consistently weaker SAF.

## 6. Summary and implications

The strength of SAF in the current generation of transient climate change simulations is determined primarily by the surface-albedo decrease associated with loss of snow cover rather than the reduction in snow albedo due to snow metamorphosis in a warming climate. The large intermodel spread in SAF strength is likewise attributable mostly to the snow cover component. The spread in the strength of this component is in turn mostly attributable to a correspondingly large spread in mean effective snow albedo. Models with large effective snow albedos have a large surface-albedo contrast between snow-covered and snow-free regions and exhibit a correspondingly large surface-albedo decrease when snow cover decreases.

Effective snow albedo corresponds roughly with the type of surface-albedo parameterization used. The models without explicit treatment of the vegetation canopy typically have high effective snow albedos and generally have strong SAF. In models with explicit canopy treatment, surfaces that are completely snow-covered typically have lower albedos, and hence the simulations generally have weaker SAF. However, the greater complexity of the latter models does not guarantee greater realism. It is true that the models with the simplest surface-albedo parameterizations have unrealistically strong SAF in the seasonal cycle context, likely attributable to prescribed snow albedos that are too high. However, SAF in models with the most complex parameterizations is significantly weaker than observed. We speculate that in these models either snow albedos or canopy albedos when snow is present are too low, or perhaps more likely, vegetation shields snow-covered surfaces excessively. In any event, detailed, high quality observations of surface albedo in a representative sampling of snow-covered surfaces would be extremely useful in constraining these parameterizations and reducing SAF spread in the next generation of models. This need is particularly emphasized in the recent work of Zhang et al. (2006), who demonstrated the presence of significant discrepancies among the current available surface-albedo datasets in snow-covered areas.

Finally, we note that the relationship between snow cover and *T _{s}* makes a secondary though nonnegligible contribution to the spread in SAF, accounting for approximately 20% of the intermodel variability in SAF strength. In this study, Δ

*S*/Δ

_{c}*T*is the NH average change in snow cover divided by the average change in

_{s}*T*over the same region. The size of the snow cover anomaly associated with a 1°C

_{s}*T*anomaly in this case may be somewhat sensitive to the geographical structure of the Δ

_{s}*T*anomaly, which may in turn be affected by other feedbacks such as cloud- and sea ice–albedo feedback.

_{s}This research was supported by NSF Grant ATM-0135136. Opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect NSF views. We thank the international modeling groups for providing data, PCMDI for collecting and archiving this data, JSC/CLIVAR WGCM and their CMIP and Climate Simulation Panel for organizing the model analysis, and the IPCC WG1 TSU for technical support. The IPCC data archive at the Lawrence Livermore National Laboratory is supported by the U.S. Department of Energy. We also thank Alan Robock and one anonymous reviewer for their constructive criticism of this manuscript.

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# APPENDIX

## Converting Snow Mass to Snow Cover

Though snow mass per unit area is included in the output of nearly all 18 simulations listed in Table 1, *S _{c}* is given in only 5 of them. Here we use the following technique to convert snow mass to

*S*. At locations with snow mass greater than 60 kg m

_{c}^{−2}, we assume

*S*to be 1. At locations with snow mass smaller than this threshold, we assume

_{c}*S*to be proportional to snow mass. The coefficient of proportionality is chosen to be the inverse of the threshold so that

_{c}*S*is 1 at locations with snow mass equal to the threshold. This technique is similar to that used by Frei et al. (2003).

_{c}We can validate *S _{c}* obtained from snow mass by this technique in the five simulations providing both

*S*and snow mass. First, the climatological springtime mean values of predicted and outputted

_{c}*S*values agree with one another to within 1%–2% in both current and future climates in all 5 simulations (Fig. A1a). Second, there is nearly a perfect correlation between predicted and outputted time series of springtime mean

_{c}*S*values in both the current and future climate (Fig. A1b). And finally, the difference in

_{c}*S*between the two climates calculated from snow mass is almost exactly the same as the Δ

_{c}*S*values calculated directly from model output (Fig. A1c). The nearly perfect agreement in predicted and outputted climatological mean

_{c}*S*, interannual variations in

_{c}*S*, and anthropogenic change in

_{c}*S*in all five simulations gives a high degree of confidence in the technique we use to convert snow mass to

_{c}*S*in the other simulations.

_{c}Eighteen models used for AR4 climate change experiments and the description of surface-albedo parameterizations incorporated in the models. The “snow albedo” column describes how snow albedo is calculated, including the way metamorphosis effects are parameterized, either with an explicit temperature dependence or with a dependence on snow age. They are represented in the table by “temperature” and “snow age,” respectively. The “vegetation masking effect” column describes how snow albedo is used to calculate surface albedo. In type 1, surface albedo is calculated based on a radiation transfer scheme between canopy top and ground. Ground albedo is a weighted mean of snow albedo and soil albedo (weights determined by snow cover). In type 2, surface albedo is a weighted mean of vegetation albedo and ground albedo (weights determined by vegetation cover). In type 3, surface albedo is a weighted mean of snow-free land albedo and snow albedo (weights determined by snow cover). Snow albedo depends on vegetation type. In type 4, surface albedo is a weighted mean of snow-free land albedo and snow albedo (weights determined by snow cover). Snow albedo is independent of vegetation type. Note that models were ordered according to their values of Δ*α _{s}*/Δ

*T*.

_{s}