## Introduction

Freshwater from snowmelt accounts for almost one-sixth of the global human consumption and agricultural use of water1, whereas the economic and societal impacts of snow-cover loss can surmount trillions of dollars2,3. Existing trends in the reduction of snowfall4,5 and snow coverage6,7,8,9 are expected to continue under the projected global warming10.

Potential evaporation or evapotranspiration (PET) refers to the maximum rate of water vaporization from a saturated (or unstressed) surface to an unsaturated atmosphere mainly driven by the atmospheric water vapor pressure deficit (D) and the available energy (AE; primarily net radiation) absorbed by the surface11. Operationally, it is estimated as the evaporation occurring from a free water surface11,12,13,14 or the evapotranspiration of a well-watered short grass12,14,15 with fixed surface albedo (0.08 for water and 0.23 for a reference crop) under existing atmospheric conditions. PET is widely used to assess continental drying through aridity indices11,16 and drought indices17, whereas it is also used to estimate the atmospheric demand of water in hydrologic models18,19,20 and offline assessments of climate change impacts on water resources21. However, the lack of a definition for PET over snow-covered surfaces has hampered progress towards understanding the fate of water resources for colder, snow-dominated regions, which are commonly excluded from global studies11,21. Under such conditions, the assumptions of a free water surface or the reference crop as the choice of evaporative surface may lead to unrealistic estimates of PET.

Here we define PET from a landscape partially covered by snow as the maximum water vaporization (evaporation and/or sublimation) from an unstressed water surface covered by an unstressed snow surface, of which the coverage is changing seasonally. The term “unstressed” refers to a zero surface resistance to molecular diffusion of water vapor through the roughness sublayer to the turbulence-dominated surface layer. To account for the dynamically changing surface albedo caused by seasonal snow accumulation/ablation, we used a physically based land surface model (LSM) driven by observed atmospheric forcing to estimate PET. We report the results from this alternative method for computing PET and compare these results to estimates of PET following a conventional definition that assumes the evaporation of a free water surface13,14. We use both methods to compute daily values of PET for 671 catchments from the Catchment Attributes and Meteorology for Large-Sample Studies (CAMELS)22 located within the conterminous United States (Supplementary Fig. 1a), over a period of 30 hydrologic years (1983–2013). We explore some immediate implications of the proposed changes by comparing 30-year climatology of both estimates as a function of average catchment temperature ($$\bar T$$) and also compute the aridity index (ϕ), a variable commonly utilized to assess global patterns of water availability accounting for both atmospheric water demand (PET) and supply (Precipitation). Following that, we offer an explanation of the existing paradigm stating that a reduction in snowfall fraction leads to a decrease in the mean streamflow at the catchment scale23, based on a framework24 that assumes water balance to be controlled by ϕ. Finally, we demonstrate how temperature increases might lead to much higher reductions in streamflow in colder regions as compared to warmer regions.

## Results

### Estimation of PET

Under prolonged sub-freezing temperatures, water will not exist in its liquid form but will be frozen or exist as accumulated snow over the land surface, leading to significant changes in the surface albedo of the evaporative surface. For most catchments, snow is the dominant form of frozen water and the presence of snow will greatly increase the amount of reflected solar radiation and therefore reduce the net radiation25. Although a variety of analytical equations exist for estimation of PET for liquid water surfaces14, there are no simple analytical schemes that account for the vaporization from a surface subject to varying snow cover. For this reason, we used the physically based Noah-MP LSM26 (“Methods”) to compute PET over two scenarios of unstressed surfaces as follows: (1) a liquid water surface (PETwater) and (2) liquid water surface with seasonal snow cover (PETsnow). We ran the model for 30 years from 1983 to 2013 driven by observed hourly near-surface atmospheric data. We then computed monthly PET climatology (the average of the monthly model outputs over the 31 years) and PET climatology (the average of all data over the 30 years). Figure 1a shows monthly PET values (in mm equivalents) for a catchment with significant amount of snowfall (fs = 0.55), located in the state of Montana, USA. The monthly climatology of PETwater (mm) is greater than that of PETsnow in snow seasons but converges in summer. Over all CAMELS catchments, the difference in the 31-year PET climatology becomes more apparent for colder catchments with $$\bar T$$ ranging from −5 °C to 10 °C (Fig. 1b). PETwater becomes much larger than PETsnow by up to 50% in colder catchments with $$\bar T$$ around 0 °C (Fig. 1c) and snow-dominated catchments with fs above 0.7 (see “Methods”) (Supplementary Fig. 2), suggesting that the conventional approach overestimates PET, progressively increasing towards colder, snow-dominated catchments.

### The sensitivity of PET to increasing temperatures

To answer the question of how PET will change over snow-dominated catchments under a warming climate, we performed additional model experiments to compute PETwater and PETsnow with +1.0, +2.0, and +3.0 °C increases in surface air temperature. To discern the physical mechanisms that may contribute to the warming-induced changes in PET, we used an attribution analysis21,27 based on the well-established Penman28 equation using monthly North American Land Data Assimilation System project phase-2 (NLDAS-2) atmospheric data and model outputs. Under the +3.0 °C warming in South Crow Creek, Montana (USA), warming-induced increase in the water vapor pressure deficit (ΔD) appears to be the most dominant contributor to the increases in PET estimated by the conventional method ($${\Delta} _{{\mathrm{PET}}_{{\mathrm{water}}}}$$) followed by changes in the slope of water vapor pressure vs temperature (ΔS) and available energy (ΔAE), which reduce the positive contributions by ΔD (Fig. 2a). The negative effect of ΔAE on $${\Delta} _{{\mathrm{PET}}_{{\mathrm{water}}}}$$ are caused by the energy loss through increases in the outgoing longwave radiation due to the surface warming (increases in the downward longwave radiation under a warming climate is not considered in the warming experiment). When the snow albedo effects are considered, ΔAE becomes a positive contributor to the increase in PET ($${\Delta} _{{\mathrm{PET}}_{{\mathrm{snow}}}}$$, Fig. 2b) due to warming-induced decreases in snow coverage and thus surface albedo. Between the two cases (Fig. 2a, b), the largest change in these contributors is ΔAE followed by changes in aerodynamic factor (Δfu). Warming-induced changes in fu (see Eq. (2) in “Methods”) are due mainly to the changes in the surface drag coefficient, CM, for both cases. For $${\Delta} _{{\mathrm{PET}}_{{\mathrm{water}}}}$$, Δfu is negligible, whereas for $${\Delta} _{{\mathrm{PET}}_{{\mathrm{snow}}}}$$, Δfu becomes a positive contributor, because the warming may induce increases in surface roughness length, z0, which are weighted by fractional snow cover (SCF), due to reduced snow cover (z0 for water is 0.01 m, whereas 0.002 for snow surface). Over all the CAMELS catchments, warming-induced changes in PET are generally greater for partially snow-covered surfaces than for water due mainly to the warming-induced increases in available energy as a result of reduced surface albedo or surface darkening (Fig. 2c). The warming effects progressively increase as the extent of warming increases from +1.0 °C to +3.0 °C. The catchments with $$\bar T$$ around 5 °C appears to be the most sensitive catchments, where the duration of snow coverage, changing more dynamically with seasons, is more sensitive to the warming.

A decrease in the fraction of precipitation falling as snow has recently been suggested to lead to a decrease in the mean streamflow at the catchment scale23 when long-term catchment fluxes are analyzed through the widely adopted Budyko24 framework. The Budyko24 framework assumes that, at sufficiently long timescales, the mean annual precipitation is partitioned into evapotranspiration (E) and streamflow (Q), and this partitioning is controlled by the aridity index (ϕ), which represents the competition between atmospheric demand and supply of water. ϕ is commonly estimated as the ratio between the mean annual values of PET and mean annual precipitation (P). The basis for studying catchment-scale mean annual water balance within the Budyko framework is the definition of PET and ϕ, as their estimation determines the location of catchments along the x-axis in the Budyko space (ϕ vs. E/P). The snowfall-streamflow paradigm (SSP)23 suggests that catchments with a greater snowfall fraction show a lower value of E/P than that predicted by the Budyko framework, suggesting that catchments with a greater snowfall fraction can produce more runoff, Q/P. Therefore, a warming-induced shift snow to rain would lead to a decrease in runoff. The PET estimates used in ref. 23 do not account for the presence of snow, as PET was estimated based on pan evaporation measurements29, from the months of May through October, whereas the estimates for the remaining months were extrapolated through a non-physically based procedure30. A physically based explanation for the SSP has been recently suggested for a study31 within the context of the Colorado River Basin, where the reductions of streamflow were associated to an increase in evapotranspiration due to reduction in snow cover. However, such conclusions were drawn through a more expensive Monte Carlo simulation with a radiation-aware hydrologic model. Here we attempt to generalize the previous findings of the decreasing streamflow under a warming climate23,31 through the lenses of our snow-corrected PET estimation.

We analyzed the positions of the catchments in the Budyko space to assess whether there are significant differences between the normalized streamflow (Q/P = 1 − E/P) of snow-dominated catchments and others with significantly less or no snowfall. For this analysis we have reduced the sample size to 214 catchments (Supplementary Fig. 1b), after a careful quality-control assessment (“Methods”). We first assessed whether fs is associated with anomalies in Q/P when using PETwater and PETsnow to compute ϕwater and ϕsnow, respectively (“Methods”). Figure 3a, b show the positions of the CAMELS catchments based on ϕwater and ϕsnow (“Methods”), in which catchments are differentiated with respect to snowfall fraction, where a value of fs = 0.5 was chosen to highlight the differences in the catchment location within the ϕ − E/P space. Although an outlier behavior is suggested for snow-dominated catchments when ϕwater is used (Fig. 3a), the same pattern cannot be seen in the case of ϕsnow (Fig. 3b). We then assessed how Q/P changes with varying degrees of fsQ/P, “Methods”). A significant positive relationship (r = 0.3, p = 10−6) between ΔQ/P and fs exists when ϕwater is used, in which a 20% increase in Q/P occurs for each incremental increase in fs (Fig. 3c), consistent with the SSP as shown in ref. 23. However, when ϕsnow is used, the trends are not significant (r = 0.02, p = 0.6) and negligible for ΔQ/P, suggesting the normalized runoff of snow-free and snow-dominated catchments to be approximately the same for catchments having similar values of aridity index. This also suggests the Budyko24 framework to be valid for both groups, allowing the SSP to be properly hypothesized through the Budyko24 framework as follows: In a warming climate, the increasing temperatures leading to a shift from snow to rain (i.e., decreasing fs) will increase the atmospheric demand for water (i.e., the increasing PET and ϕ), leading to a reduction in streamflow at the mean annual timescale.

To further explore the interpretation of streamflow changes with the extent of warming for catchments with different snowfall extents, we estimated the changes in the normalized streamflow $${\Delta} (\frac{Q}{P})$$ caused by changes in PET (and therefore ϕ) computed for the different warming scenarios (“Methods”) by assuming both PETwater and PETsnow leading to $${\Delta} (\frac{{{Q}}}{{{P}}})_{\mathrm{{water}}}$$ and $${\Delta} (\frac{{{Q}}}{{{P}}})_{{\mathrm{snow}}}$$. Under the same +3.0 °C warming scenario, colder catchments appear to lose more water and thus result in a larger drop in the normalized streamflow (∆$$(\frac{{\boldsymbol{Q}}}{{\boldsymbol{P}}})$$) when using PETsnow than using PETwater, with the difference being inversely proportional to catchment-averaged temperature (Fig. 4). Our results point to a much higher sensitivity of colder, snow-dominated regions than for warmer snow-free regions. For example, when comparing $${\Delta} (\frac{{\mathrm{Q}}}{{\mathrm{P}}})_{{\mathrm{snow}}}$$ of catchments within 0 ± 2 °C average temperature with catchments within $$\bar T$$ ≈ 20 ± 2 °C, this difference is of almost four times ($${\Delta} (\frac{{{Q}}}{{{P}}})_{{\mathrm{snow}}} \approx 0.11$$, $$\bar T$$ ≈ 0 °C vs. $${\Delta} (\frac{{{Q}}}{{{P}}})_{{\mathrm{snow}}}0.03$$, $$\bar T$$ ≈ 20 °C), with similar results for other warming scenario (Supplementary Fig. 3). This is explained by the higher positive changes in mean annual PET and therefore increased aridity experienced by colder regions due to reduction in snow-driven atmospheric demand as previously explained.

## Discussion

The modeling assumptions adopted here represent an attempt towards a conceptual and physically meaningful estimation of PET for snow conditions, and we believe that additional investigations might be useful for its proper advancement. Also, the warming experiments performed with Noah-MP in this study do not include potential changes in other climatic variables that may contribute to changes in PET10, for instance, increases in the downward longwave radiation and specific humidity under a warming climate as consistently projected by many earth system models. Nonetheless, the prescribed warming experiments performed in this study provide a first-order approach towards the understanding of how PET and ϕ may respond to future climates. Our results also carry some important uncertainties arising from the limitations embedded in our water-balance analysis. The Budyko framework24 is based on the simple climate-based reasoning on the long-term water balance, lacking detailed description of physical processes controlling the partitioning of precipitation into runoff and evapotranspiration. Moreover, the lack of a robust way for its application when predicting changes occurring in time vs. its ability to explain spatial patterns of water-balance partitioning32, as well potential issues arising from the water-balance closure assumption33 should be carefully considered if a similar analysis is intended for prediction purposes.

We conclude that the conventional approaches, e.g., the open water PET, overestimate PET and thus aridity index in snow-dominated catchments. We have shown that colder regions might experience a larger decrease in streamflow than warmer regions due mainly to increases in net radiation caused by decreases in surface albedo or surface “darkening.” This result is consistent with a recent study, suggesting “warming-driven loss of reflective snow energizes evaporation” through a Monte Carlo simulation with a radiation-aware hydrologic model31 and provides an explanation for the suggested catchment-scale reduction of streamflow occurring from a shift from snowfall to rainfall at the mean annual timescale23. Although current global-scale assessments of future changes in water availability performed through the lens of the Budyko framework do not include cold regions in their analysis11,21, we believe this study provide a promising step towards a generalized understanding of how water availability of colder regions is affected by increasing global temperatures. Finally, our results confirm recent concerns1,2,3,23,31 and reinforce the call for greater attention towards water supply policies in snow affected environments.

## Methods

### Estimation of PET and fs

To estimate PET over unstressed surfaces of free water and partially snow covered, we employed a widely used LSM, Noah with multi-physics (Noah-MP)26. Noah-MP is a widely used LSM with demonstrated ability in simulating land surface fluxes26,34,35. Noah-MP considers detailed snowpack physics (e.g., liquid water retention and refreezing, densification, heat released or absorbed due to phase changes, and surface albedo changes due to snow aging), and surface energy and mass exchanges with the atmosphere (e.g., sublimation/deposition, patchy snow effects on surface albedo, and surface turbulent heat and moisture exchanges, etc.). Noah-MP explicitly represents sublimation/deposition by converting the latent heat flux over sub-freezing surfaces by the latent of sublimation, which is the sum of fusion (334 kJ kg−1) and vaporization (2265 kJ kg−1). The surface albedo and roughness length of a water surface partially covered by snow is the areal average weighted by fractional snow cover, SCF, which is diagnosed from the model predicted snow depth using a snow depletion curve. Noah-MP was driven by the hourly North America Land Data Assimilation System Phase-2 near surface atmospheric forcing data over the contiguous US (CONUS) from 1980 to 2015 at 1/8th degree resolution36. We assumed two distinct scenarios, both of which assumed “unstressed” surfaces with a zero surface resistance to water diffusion in the roughness sublayer. Scenario 1 (water) was based on a liquid free water surface (i.e., assuming surface albedo is free water albedo = 0.08), whereas Scenario 2 (snow) allows snowpack to be built on “water” surface. For Scenario 1, we set up the snowfall fraction of the precipitation equal to 0, whereas no snow cover at the water surface was allowed by setting up SCF of the surface equal to 0. The water body temperature is solved at four layers with thicknesses of 0.1, 0.3, 0.6, and 1.0 m (totally 2 m) with a zero-heat-flux lower boundary condition at 8 m. The same as for soil. Noah-MP does not compute the frozen water depth but determines if the water surface is frozen from the resulting water surface temperature (Tg) (frozen if Tg ≤ 0 °C), where Tg is computed by iterating the surface energy balance equation26. Noah-MP assumes that snowpack can build up over water surfaces when the surface temperature is below the freezing point (Tfrz), while neglecting energy consumed/released during phase change. In addition, to ensure a conceptual agreement between Noah-MP evaporation and that of conventionally employed analytical solutions11,13,28, we excluded the stability correction to the surface drag coefficient, though the atmospheric warming may substantially increase the surface layer stability over snow and lake surfaces27 suppressing the surface turbulent exchanges of water and energy with the atmosphere. We estimated fs based on a new snow-rain partitioning scheme using a sigmoidal function of wet-bulb temperature, which was demonstrated to improve snow simulations in the CONUS35.

### Attribution analysis of the changes in PET

We adopted an “offline” attribution analysis21,27 to elucidate the mechanisms contributing to the changes in PET with increasing temperature for both PET scenarios. The temperature increments adopted here were uniformly applied to the historical timeseries. The attribution analysis consists of using the monthly NLDAS atmospheric forcing data36 and outputs from Noah-MP (net radiation and ground heat flux) to compute PET values through the commonly used Penman28 equation, in which PET (mm) can be written as:

$${\mathrm{{PET}}} = \frac{{\mathrm{s}}}{{{\mathrm{s}} + \gamma }}A_{\mathrm{E}} + \frac{\gamma }{{{\mathrm{s}} + \gamma }}f_{\mathrm{u}}{{DL}}_{\mathrm{V}}$$
(1)

where, s is the slope of saturated water vapor pressure vs. air temperature (Pa K−1), γ is the psychrometric constant (Pa K−1), AE is available energy (W m−2), a residual of net radiation (Rn, in w m−2) minus ground heat flux (G, in w m−2), fu is the wind function, D is the 2 m water vapor pressure deficit (Pa), and LV is the latent heat of vaporization (J kg−1). The wind function encapsulates the aerodynamic effects on evaporation and is defined as:

$$f_{\mathrm{u}} = \frac{\rho }{P}C_{\mathrm{M}}u_2$$
(2)

In which ρ is the air density (kg m−3), P is the 2 m air pressure (Pa), CM is the unitless surface drag coefficient (or surface exchange coefficient), and u2 is the 2 m windspeed (m s−1). CM controls the turbulence transport of momentum through the surface layer and is parameterized as a function of the surface roughness length, zero-displacement height (lifting up the evaporative surface), and surface layer stability through the Monin–Obukhov similarity theory26.

We then approximate the changes in PET (ΔPET) to the first order as:

$${\Delta} {\mathrm{PET}} \approx \frac{{\partial {\mathrm{PET}}}}{{\partial {{s}}}}{\Delta} {{s}} + \frac{{\partial {\mathrm{PET}}}}{{\partial \gamma }}{\Delta} \gamma + \frac{{\partial {\mathrm{PET}}}}{{\partial {{A}}_{\mathrm{E}}}}{\Delta} {{A}}_{\mathrm{E}} + \frac{{\partial {\mathrm{PET}}}}{{\partial {{D}}}}{\Delta} {{D}} + \frac{{\partial {\mathrm{PET}}}}{\partial {f_{\mathrm{u}}}}{\Delta} {f_{\mathrm{u}}} + \frac{{\partial {\mathrm{PET}}}}{{\partial {{L}}_{\mathrm{V}}}}{\Delta} {{L}}_{\mathrm{V}}$$
(3)

where the terms on the right-hand side represent the individual contribution to the total change ΔPET. These terms can be written as:

$$\frac{{\partial {\mathrm{PET}}}}{{\partial s}} = \frac{{\gamma \left( {{{A}}_{\mathrm{E}} - f_{\mathrm{u}}{{DL}}_{\mathrm{V}}} \right)}}{{\left( {s + \gamma } \right)^2}}{\Delta} s$$
(4)
$${\frac{{\partial {\mathrm{PET}}}}{{\partial \gamma }} = \frac{{\gamma \left( {{{f}}_{\mathrm{u}}{{DL}}_{\mathrm{V}} - A_{\mathrm{E}}} \right)}}{{\left( {s + \gamma } \right)^2}}{\Delta} \gamma }$$
(5)
$${\frac{{\partial {\mathrm{PET}}}}{{\partial A_{\mathrm{E}}}} = \frac{s}{{\left( {s + \gamma } \right)^2}}{\Delta} {{A}}_{\mathrm{E}}}$$
(6)
$${\frac{{\partial {\mathrm{PET}}}}{{\partial D}} = \frac{{\gamma f_{\mathrm{u}}L_{\mathrm{V}}}}{{s + \gamma }}{\Delta} D}$$
(7)
$${\frac{{\partial {\mathrm{PET}}}}{{\partial f_{\mathrm{u}}}} = \frac{{\gamma DL_{\mathrm{V}}}}{{s + \gamma }}{\Delta} {{f}}_{\mathrm{u}}}$$
(8)
$${\frac{{\partial {\mathrm{PET}}}}{{\partial L_{\mathrm{V}}}} = \frac{{\gamma Df_{\mathrm{u}}}}{{s + \gamma }}{\Delta} {{L}}_{\mathrm{V}}}$$
(9)

The offline ΔPET values obtained using Eqs. (3) through (9) showed a good agreement with ΔPET values from Noah-MP (Supplementary Fig. 4), which allows the use of a simpler analytical equation to assess how different drivers might be responsible for changes in PET.

### Catchment selection

We used daily streamflow from the CAMELS dataset22, which includes catchments from the contiguous United States (CONUS). We specifically looked at the hydrologic years 1 October through 30 September) within the span of 31 years (1983–2013). A few quality-assessment steps where undertaken for selection of catchments for the Budyko framework analysis (Figs. 3 and 4) as follows: (i) we eliminated all catchments with missing streamflow; (ii) the CAMELS dataset provides two different estimates of catchment area (km2). As all fluxes used in here are represented in mm, the estimation of catchment area plays an important role in the flux magnitudes. Therefore, we removed catchments based on the criteria of 1% difference between the two estimates. (iii) Catchments having negative evaporative fraction ($$\frac{E}{P}\ <\ 0$$), or evaporative fractions > 1 ($$\frac{E}{P}\ > \ 1$$), were eliminated, to accommodate the assumptions of long-term water balance comprising the Budyko framework and (iv) catchments with areas smaller than 200 km2 were removed to account for the inaccuracies arising from the transposition of NOAH-MP (pixel size ≈ 170 km2) results onto the basin masks. (v) Catchments with significant surface water bodies (fraction of water of a catchments area >5%) were also removed to reduce the effects of storage in the analysis. The resulting procedure returner in 214 catchments (Supplementary Fig. S1b) used for the analysis of Figs. 3 and 4.

### Aridity Index (ϕ) computation

ϕ is conventionally computed as the ratio between mean annual values of PET (water demand) and P (water supply), in mm equivalents per year. When computed as energy equivalents, ϕ can be regarded as a ratio between the mean annual energy associated with the evaporation at the potential rate (PETε, in MJ m−2 year−1) and the energy required to evaporate all the mean annual precipitation (Pε, in MJ m−2 year−1). In this way, Pε allows for the inclusion of the energy required for sublimation (or melt and evaporation) of the solid snow particles, which would not be possible when assuming mm equivalents. Therefore, ϕ was estimated as:

$$\phi _{{x}} = \frac{{{\mathrm{PET}}_{\varepsilon_{{x}}}}}{{P_\varepsilon }} = \frac{{{\mathrm{PET}}_{\varepsilon_{{x}}}}}{{P\rho _{\mathrm{w}}[f_{\mathrm{S}}\lambda _{\mathrm{S}} + \left( {1 - f_{\mathrm{s}}} \right)\lambda _{\mathrm{V}}]}},$$
(10)

where x denotes either scenario “water”, leading to $${\mathrm{PET}}_{\varepsilon _{{\mathrm{water}}}}$$ (MJ m−2 year−1) or scenario “snow” leading to $${\mathrm{PET}}_{\varepsilon _{{\mathrm{snow}}}}$$ (MJ m−2 year−1), P is the mean annual precipitation (m year−1), λv is the latent head of vaporization of water (λv = 2.501 MJ kg−1), ρw the density of water (103 kg m−3), and λs is the latent heat of sublimation (λs = 2.835 MJ m−2), being equal to the sum of λv and the latent heat of fusion (λf = 0.334 MJ m−2),

### Streamflow comparison

We tested whether there are significant differences between normalized streamflow (Q/P of catchments with and without significant amounts of snowfall across the whole range of observed ϕ for the CAMELS catchments according to the following: first, we fitted a Budyko-type curve (seen in Fig. 3) to the observed values of ϕ and $$\frac{E}{P}$$:

$${\frac{E}{P} = F\left( \phi \right) = \left( {\frac{1}{{1 + \left( {\frac{1}{\phi }} \right)^{1.9}}}} \right)^{0.85}}$$
(11)

Equation (2) was fitted based on a sub-sample with little influence of snowfall (fs < 0.2), which showed no significant difference between ϕ1 and ϕ2 (Supplementary Fig. 5c, d). The choice for a fitted curve rather than the original Budyko24 equation was motivated by the bias in estimated vs. observed Q/P obtained by using the latter equation (Supplementary Fig. 5a, b). Following that, we calculated the difference between the predicted and observed values of $$\frac{Q}{P}({\Delta} _{{{Q}}/{{P}}})$$ for all catchments and computed a linear regression (Fig. 3) between the deviation from the curve vs. fs. The results shown in Fig. 3 were also computed using the conventional formulation of the aridity index ($$\phi = \frac{{{\mathrm{PET}}\left( {{\mathrm{mm}}} \right)}}{{{\mathrm{P}}\left( {{\mathrm{mm}}} \right)}}$$, i.e., assuming no differentiation between snow and rain in the denominator) and did not show significant differences (Supplementary Fig. 6).

### Sensitivity of streamflow to increasing temperatures

Values of streamflow sensitivity to changes in temperature were computed as the changes in normalized streamflow (ΔQ/P) due to changes in temperature through the partial differentiation:

$${{\Delta} _{{{Q}}/{{P}}_{{x}}} = \frac{{\partial F}}{{\partial \bar T}}{\mathrm{d}}\bar T}$$
(12)
$$\hskip 3pc = \frac{{\partial F}}{{\partial \phi }}\frac{{\partial \phi }}{{\partial \bar T}}{\mathrm{d}}\bar T$$
(13)
$$= \frac{{\partial F}}{{\partial \phi }}\left[ {\frac{{\partial \phi }}{{\partial {\mathrm{PET}}_{{x}}}}{\Delta} {\mathrm{PET}}_{{x}} + \frac{{\partial \phi }}{{\partial P_\varepsilon }}{\Delta} P_\varepsilon } \right]$$
(14)

where the subscript x stands for scenarios of “water” or “snow,” and F represents Q/P obtained through Eq. (11). The term ΔPETx represents the change in PET, obtained from the differences between Noah-MP runs with increments of +1.0, +2.0, and +3.0 °C, and a baseline run without warming. The term ΔPε represents the change in the energy required to evaporate/sublimate the mean annual precipitation and was obtained as the difference in Pε [the denominator of Eq. (10)] between the baseline and warming scenarios.