Introduction

Accurate climate modelling requires a sound understanding of energy and hydrological budgets1. However, one of the key metrics, radiative forcing, still has large uncertainties in satellite observations and multi-model simulations2, primarily due to clouds and aerosols3,4. Clouds cover more than 60% of the Earth, and general circulation models (GCMs) systematically underestimate the cloud coverage despite overestimating their optical properties to compensate for bias5.

Radiative biases associated with clouds and aerosols are recognised as long-standing issues in GCMs6,7. Such cloud problems are ‘known unknowns’, but the potential radiative uncertainties associated with precipitation are ‘unknown unknowns’. Although little attention has been paid to the radiative effects of precipitation (REP) on climate, precipitating particles modify atmospheric radiation by perturbing incoming shortwave (SW) and outgoing longwave (LW) radiation. To date, no observational products are available that can provide global information on REP because it is difficult to distinguish between clouds and precipitation, and between liquids and solids, and to retrieve the size distributions and shapes of precipitating hydrometeors vertically, even with active sensors8. In terms of the modelling approach, computing radiative forcing due to precipitating hydrometeors (raindrops and snowflakes) requires a more sophisticated representation of precipitation. However, most GCMs traditionally treat precipitation diagnostically (DIAG), meaning that precipitating hydrometeors fall to the surface within a single model time step9. The assumptions of zero residence time of the precipitating hydrometeors and/or not passing of their information to the radiative transfer calculation are inaccurate; therefore, most GCMs participating in the Coupled Model Intercomparison Project Phase 6 (CMIP6) do not consider REP10.

To this end, prognostic parameterisation for precipitation (PROG) has been incorporated into some state-of-the-art GCMs11,12,13,14,15. Employing PROG improves representations of aerosol-cloud interactions16, cloud-to-rain conversion17, and cloud feedback18. A few recent studies that focused on REP suggest that the contribution from snowflakes is more significant10,19 than that from raindrops20, implying that REP modulates the global three-dimensional atmospheric structure from the higher troposphere to mid-latitudes and polar regions. Although systematic multi-model studies have shown snow radiative effects on biases in sea surface temperature (SST), surface wind, and ENSO variability over tropical regions21,22, extracting REP alone remains difficult because of the complicated atmosphere-ocean feedback and multi-model variabilities of schemes and performances. Consequently, a complete view of the REP on regional and global climates has not yet been achieved.

Motivated by these scientific needs, this study investigated how precipitating hydrometeors influence radiation budgets among the surface (SFC), atmosphere (ATM), and top of the atmosphere (TOA) on global and regional scales and discusses how REP alters temperature and precipitation. For this purpose, we adopted a single-model approach to exclude contamination from different treatments of model parameterisation and frameworks. We quantified the REP and its link to precipitation treatment (i.e., DIAG versus PROG) by employing three sub-versions of the same GCM, MIROC6 (see ‘Methods’ for details).

Here, using sets of simulations and multiple satellite observations, we show that the local thermodynamical modulation due to REP propagates remotely to tropical precipitation and polar temperature changes. This is clarified by the energy perspective, showing an increase of more than 1 K in the polar surface temperature during winter. This surprising impact of the REP can be explained by the distinct contrast between the ‘SW parasol effect’ and ‘LW warming effect’ on the surface. The strong seasonal and regional dependences of REP are in line with the CMIP6 multi-model evidence in the literature, which shows large diversities in winter SFC LW radiation and temperature23,24,25 and potentially faster sea ice retreat in stronger downward LW models26,27. Given that the systematic bias of slower sea ice retreat has not been eliminated in GCMs28, this study provides one of the key ‘unknown unknowns’ for accurate climate modelling.

Results

Global three-dimensional view of REP

The global annual mean radiation budgets under the all-sky condition from a suite of experiments (DIAG and PROG, with and without REP) are shown in Fig. 1. The CERES EBAF29,30 satellite estimates are also shown. The clear-sky radiation budget is shown in Supplementary Fig. 1, and there are slight differences among the three sub-models, indicating that the differences originated mainly from the cloudy scenes, though there are slight compensating errors between clear and cloudy sky conditions31,32.

Fig. 1: The global annual mean energy budget of Earth.
figure 1

Numbers in the figure denote fluxes in W m−2. Yellow and dark-red arrows represent shortwave (SW) and longwave (LW) radiation, respectively. Observation data was obtained from CERES EBAF Edition 4.2 (grey), and three models from DIAG, and PROG without and with radiative effects of precipitation (REP) are in black, blue, and red, respectively.

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For SW radiation, more sunlight was shown as reflected to space by updating from DIAG (103.5 W m−2) to PROG (106.4 W m−2) precipitation treatment. This was due to improved representations of cloud coverage in the PROG by more than 12% (Supplementary Table 1), primarily owing to high-thin cirrus33 (Supplementary Fig. 2) over the tropics. The impact with and without REP was comparable (~2.1 W m−2), although cloud coverage was almost the same between the PROG REP-ON and REP-OFF simulations (Supplementary Table 1 and Supplementary Fig. 3). Nevertheless, these updates worsened the TOA upward SW radiation against CERES, suggesting that there were compensating errors between cloud coverage and cloud radiative forcing. Consequently, the SFC downward SW radiation was decreased in the PROG simulations compared to the DIAG, and the so-called ‘parasol effect’ due to REP attained ~6 W m−2 at the SFC. Notably, the SW absorption in the ATM was increased in PROG simulations, which is in close agreement with the CERES observation. A comparison between DIAG and PROG REP-OFF suggests that increased cloud coverage increases ATM SW absorption by approximately 3 W m−2, and a comparison between PROG REP-OFF and REP-ON implies that SW multiple scattering due to precipitating hydrometeors effectively enhances ATM absorption.

Although the upward SFC LW radiation was similar among the simulations because they are determined by the SFC temperature (Fig. 2a) following the Stefan-Boltzmann law, the differences in the downward SFC LW radiation were larger among the sub-models. The PROG REP-ON model shows the strongest ‘warming effect’, which is more than 7 W m−2, compared to the DIAG model. This SFC warming caused by precipitating hydrometeors reduces the outgoing LW radiation at the TOA (i.e., OLR), whereas ATM LW cooling is relatively less sensitive to precipitation treatments.

Fig. 2: Global statistics of physical parameters from observations and models.
figure 2

Seasonal cycles of (a) 2m temperature, (b) top of the atmosphere (TOA) SW flux, (c) TOA LW flux, (d) precipitation rate, (e) atmospheric (ATM) net radiative cooling, (f) surface (SFC) downward SW, (g) SFC upward SW, (h) SFC upward LW, and (i) SFC downward LW. Observations are shown in shades of grey (CERES EBAF Edition 4.2 for 2001–2022; GPCP version 3.2 for 2000–2020), and three models from DIAG, and PROG without and with REP are in black, blue, and red, respectively. The error bars and shaded area represent ± 1 standard deviation of interannual variability.

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As a consequence of the combined effects of SW heating and LW cooling, the ATM net radiative cooling is significantly mitigated in the PROG REP-ON (4.41 W m−2), mainly due to the former process, whereas the PROG REP-OFF does not show this feature, as the increased SW heating and increased LW cooling cancel each other out. Finally, the mitigation of the ATM net cooling results in a decrease in sensible heat (SH) and latent heat (LH) in the PROG REP-ON against REP-OFF (4.41 W m−2), which must be balanced with the difference in the ATM cooling; thus, the latter is linked to the weakening of global mean precipitation (Supplementary Table 1). These results suggest that the REP stabilises the ATM, the geospatial responses of which are later discussed.

Although including REP should provide a more realistic physical representation, the model performance does not always show improvement when compared to observations (Fig. 2). The TOA radiation budgets (Fig. 2b, c), ATM net cooling (Fig. 2e), and SFC budgets (Fig. 2f–i) are generally well reproduced by the default MIROC6 DIAG; however, a more sophisticated modelling framework worsened their radiation budget performances. With regard to precipitation, PROG REP-ON performed the best among the sub-models (Fig. 2d). Given that most CMIP6 models perform well in simulating radiation budgets without REP34, this error compensation between precipitation and radiation budgets will be an ongoing bottleneck in the next generation of CMIP models.

Dependence of regional climate on REP

The regional precipitation responses with and without REP were examined from an energy perspective35,36, as shown in Equation (3). Figure 3 shows how the global precipitation change due to perturbations from the REP (Fig. 3a) is related to changes in the ATM diabatic cooling ΔQ (Fig. 3b) and the energy transport term ΔH (Fig. 3c). The global mean change in the latent heat released from precipitation is − 3.43 W m−2, which is equivalent to − 0.12 mm day−1 in mass flux. We further observed the main source of this precipitation change from convective or stratiform parameterisations (Supplementary Fig. 4) and found that the convective rainfall is dominant. This is attributable to the decreased SH due to the stabilisation of the atmosphere and is consistent with the slight increase in low cloud coverage (Supplementary Fig. 3d) due to less rain.

Fig. 3: Geographical distributions of the energy budget terms.
figure 3

Changes in (a) the atmospheric latent heating rate from precipitation (LΔP), (b) the atmospheric diabatic cooling rate (ΔQ), and (c) the dry static energy flux divergence (ΔH), between PROG with and without REP simulations (i.e., ON-OFF). The numbers at the top right of each panel are the global mean values, which are energetically balanced (see ‘Methods’ for details). The dotted regions indicate significant differences at the 95% confidence level.

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Notably, although the mitigation in the ATM diabatic cooling (ΔQ) mostly comes from mid-latitude precipitation (Fig. 3b), where the snow mixing ratio is abundant in the ATM, precipitation changes are found over the tropics due to the energy transport ΔH (Fig. 3c). This is in line with the aqua-planet theoretical framework, which shows that the weaker Coriolis force over the tropics effectively contributes to a larger ΔH relative to the extra-tropics37. This suggests that the REP can affect local precipitation by modifying atmospheric circulation. Statistically significant regions are widely found in the tropical and subtropical regions and mid-latitudes.

In addition to precipitation changes, we examined the impact of the REP on temperature and associated SFC radiation budgets (Fig. 4). Significant warming was observed in the polar regions, where the mean values reached 1.01 K and 1.51 K over the Arctic and Antarctic, respectively (Fig. 4a). In contrast, temperature changes over tropical and subtropical regions were relatively small, although AGCM simulations could vary the surface temperature over the continents. To understand the physical mechanisms of the two interesting features, ‘tropics-dominant precipitation change’ and ‘polar-dominant temperature change’, responses of the SFC radiation to REP are shown by decomposing SW and LW radiation, and the net changes (Fig. 4b–d). As explained in Fig. 1, the SFC net SW radiation decreases globally because of the parasol effect from precipitation but is less effective in polar regions (Fig. 4b) because of limited sunlight. In contrast, the increased SFC net LW radiation due to the warming effect from precipitation is observed in higher latitudes (Fig. 4c) because snowfall is abundant over polar regions, and the LW effect remains throughout the year. These synergistic contributions characterise the net radiation budget of the SFC (Fig. 4d), which shows warming over the polar regions against cooling over the tropical regions and mid-latitudes.

Fig. 4: Geographical distributions of the REP-induced surface change.
figure 4

Changes in (a) the 2 m temperature, (b) the surface SW flux, (c) the surface LW flux, and (d) the surface net flux between PROG with and without REP simulations. Note that fluxes are downward positive and red and blue indicate heating and cooling, respectively. The numbers at the top right of each panel are the global annual mean values. The dotted regions indicate significant differences at the 95% confidence level.

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Given that the SST is fixed, the cooling imbalance over the tropical and subtropical oceans should contribute to the mitigation of the SH and LH, which warms the SFC (Supplementary Fig. 5). This is consistent with decreased oceanic precipitation over lower latitudes (Fig. 3a). These energy analyses emphasises the important mechanisms of REP on global and regional climates to simultaneously induce the ‘tropics-dominant precipitation change’ and the ‘polar-dominant temperature change’.

Seasonality of REP in the Arctic climate

Because of the distinct occurrence of day and night cycles throughout the year in the polar regions, the seasonality of REP should be examined to understand how the implicit treatment of REP in most GCMs is linked to a systematic bias in relevant processes. Considering that most GCMs tend to underestimate sea ice retreat over the Arctic24,28, we further investigated REP seasonality to show potential uncertainties in the Arctic climate.

The increase in 2m temperature due to REP is pronounced during the winter season, which is more than twice as large as the summer warming (Fig. 5a). Despite the temperature increase during winter, which increased the upwelling LW radiation at the SFC, the TOA net radiation was almost unchanged between REP-ON and REP-OFF (Fig. 5b) because REP contributed to enhanced downwelling LW radiation, compensating for each other. Notably, the column water vapour during winter was almost the same between REP-ON and REP-OFF (Fig. 5c), indicating that winter warming is attributable to the treatment of precipitation rather than a change in water vapour. Because the ATM net cooling is increased (Fig. 5d) and SFC SW radiation is zero (Fig. 5e, f) in winter, the 2m temperature can be attributed to the increased downward LW by REP, although it is partly mitigated by the increased upward LW radiation (Figs. 5g, h, 6a).

Fig. 5: Arctic seasonal cycles of REP derived from the difference between PROG with and without REP simulations.
figure 5

Differences in (a) 2m temperature, (b) TOA net flux, (c) column water vapour, (d) ATM net cooling, (e) SFC downward SW, (f) SFC upward SW, (g) SFC upward LW, (h) SFC downward LW, and (i) lower tropospheric stability. The shaded area indicates ± 1 standard deviation of interannual variability. The Arctic is defined as the area north of 66°N.

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Fig. 6: The Arctic mean energy budget.
figure 6

Statistics for (a) the winter polar night season (December) and (b) the summer polar day season (June). Numbers in the figure are the fluxes in W m−2. Yellow and dark-red arrows represent SW and LW, respectively. Observation data were obtained from CERES EBAF Edition 4.2 product (grey), and three models from DIAG, and PROG without and with REP are in black, blue, and red, respectively. The Arctic is defined as the area north of 66°N.

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Interestingly, although the ATM is cooled during the winter season (Fig. 5d) and found in the temperature profile (Supplementary Fig. 6c), it is warmed in the Arctic during summer. This leads to an increased column water vapour during summer, which partly contributes to increased TOA net radiation (downward positive; i.e., decreased outgoing radiation) with synergistic contributions from the REP (Fig. 5b, c). The weakening of ATM cooling (Fig. 5d) increased the ATM temperature significantly in the summer (Supplementary Fig. 6e), although the change in the 2m temperature was small (Fig. 5a). These vertical temperature changes are consistent with the seasonal variation in lower tropospheric stability (LTS) with and without REP (Fig. 5i), showing destabilisation in winter due to SFC warming and stabilisation in summer due to ATM warming.

The seasonal characteristics of the radiation budget structure are shown in Fig. 6. The emphasis here is on the magnitude of ATM net cooling in REP-ON, which is stronger in winter (130.9 W m−2, compared to REP-OFF of 127.5 W m−2) but weaker in summer (73.3 W m−2, compared to REP-OFF of 79.0 W m−2). Stronger winter ATM cooling contributed to SFC warming (Fig. 6a), and summer ATM SW heating due to the REP contributed to ATM warming (Fig. 6b). It should be emphasised that these changes in radiation budgets due to REP improved the model bias against CERES observations in both seasons (i.e., ATM net cooling, SFC downward and upward SW in summer, and SFC upward and downward LW in winter). Given the significant Arctic warming in winter, including REP in GCMs will increase Arctic amplification and improve the systematic bias of underestimating sea ice retreat in most GCMs24,28. We obtained the signal indicating that REP will increase Arctic amplification from the CMIP6 multi-model analysis (Fig. 7, discussed later), which supports our results from the single-model analysis.

Fig. 7: Seasonality of the Arctic amplification index (AAI) for 1980–2014.
figure 7

The 34 CMIP6 models are classified into three groups (Supplementary Table 2) according to the treatment of precipitation and REP: (1) diagnostic precipitation without REP (black), (2) prognostic precipitation without REP (blue), and (3) prognostic precipitation with REP (red). The ERA5 statistic for 1980–2014 (matched with the CMIP6 historical record) is shown by the green squares.

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Discussion

The present study indicates a significant contribution of the REP on both global and regional scales. Three-dimensional radiation budget analysis revealed that in the tropics, precipitation change is the main effect, while in the polar regions, temperature change is the main effect of REP. To understand the fundamental importance of REP in GCMs, the following three questions need to be discussed.

The first question that arises is: How does the REP reduce GCM biases? In terms of precipitation, most CMIP models with an excessive upper-level LW cooling bias over the tropical regions result in excessive precipitation bias owing to atmospheric instability10. Double ITCZ bias and excessively large tropical precipitation are also longstanding issues in CMIP models38. Our results show that REP mitigates ATM LW cooling to stabilise the atmosphere, and thus reduces precipitation around the ITCZ (Fig. 3), implying that incorporating REP in GCMs could improve such systematic precipitation biases. In terms of temperature, there is a well-known cold temperature bias over continents39, particularly in arid regions40 in CMIP models. Including REP could improve the cold bias caused by the SFC warming via the modification of global radiation budgets (Fig. 4). Notably, CMIP6 models generally perform well in representing TOA radiation budgets, but there are still large diversities in the ATM and SFC budgets34. REP also tends to remove biases in underestimated ATM SW absorption and overestimated SFC SW absorption compared with CERES observations29,34. Considering this, including REP in GCMs with model tuning will help mitigate the compensating errors between TOA radiative fluxes and the hydrological cycle41. However, such model biases are closely related to cloud representation42, not only the REP.

The second question is: Can systematic biases of polar climate in CMIP models be attributed to the REP? Given that REP strengthens polar warming during the winter (Figs. 4a, 5a), the seasonal contrast of Arctic amplification43,44 should be more pronounced in REP-ON models. Here, we further examined the seasonal characteristics of the Arctic amplification among the 34 CMIP6 models (Fig. 7 and Supplementary Table 2). The results indicate that the mean of the ensemble model with REP performs best against ERA5, showing higher Arctic amplification indices (AAI), except in summer, compared to the models without REP. The means of the multi-model ensemble AAI with and without REP were 3.1 and 2.5, respectively, supporting our single-model experiments. However, there were lower AAI models with REP, and higher AAI models without REP (Supplementary Fig. 7). This large model diversity could be due to differences in the parameterisation framework among the models, rather than the treatment of precipitation10. For example, CMIP6 multi-model analysis showed that the representation of Arctic sea ice depends strongly on ocean models45 because poleward ocean heat transport primarily characterises the magnitude of the Arctic climate46,47. Furthermore, the seasonal cycle of clouds, cloud structures (i.e., amount, altitude, and liquid/ice phase partitioning), and consequent feedback are also significantly different among models48,49,50. Arctic amplification can respond not only to carbon dioxide but also to aerosol perturbations51. Given that the complex polar climate interacting with the atmosphere, ocean, and cryosphere52,53 could mask REP signals (Supplementary Fig. 7), the REP should first be evaluated within a single-model framework, as shown in the present study.

The Antarctic temperature is also highly sensitive to REP inclusion (Fig. 4a) Notably, although the seasonality is similar between Arctic and Antarctic (Supplementary Fig. 8), the responses of vertical profile in temperature and humidity are not symmetrical (Supplementary Fig. 6). This can be attributed to the different meteorology owing to the land-ocean asymmetry and elevation, as well as atmospheric flow pattern54,55, thereby diversifying Antarctic sea ice in CMIP6 models56. Considering this, future studies should investigate different impacts of REP on the Arctic and Antarctic climate for better insights into polar warming57,58.

The final question is: How will the REP contribute to the future climate? In tropical and subtropical climates, atmospheric stabilisation due to the REP can be linked to climate sensitivity1,59. Although the global mean differences in total, high, middle, and low cloud coverages with and without REP were negligible (Supplementary Fig. 3), low cloud coverage tended to increase over the tropical and subtropical oceans. The modification of the atmospheric stabilisation by the REP may influence how future warming influences cloud coverages60. In terms of mid-latitude and polar climates, future global warming will change the cloud phase from ice to liquid, contributing to negative feedback61. This is particularly important in the Arctic based on the phase change62, but it may potentially be outweighed by a positive surface feedback in the Arctic63. Prognostic treatment of precipitation can enhance this phase feedback through a snow-to-rain phase change64, and/or a change in the precipitation efficiency63,65 to perturb the climate. This so-called ‘precipitation feedback’ is excluded from most GCMs (i.e., DIAG and REP-OFF models) and its effect on climate remains controversial66,67. The changes in cloud types or regimes in different models (Supplementary Fig. 2) will also affect cloud lifetime and feedback63,68. Thus, the present study motivates further investigations on how the REP is linked to future climate change on global and regional scales.

Lessons learnt from the energy constraints on REP can be beneficial for other GCMs when considering the incorporation of REP in the upcoming multi-model intercomparison. This study provides valuable information on the possible reasons why the REP alters temperature and precipitation owing to the associated changes in radiation budgets. This is one of the biggest ‘unknown unknowns’, which must be unravelled to understand multi-model diversities at the fundamental process level.

Methods

Model description

The latest version of MIROC6 was used in this study. This version incorporates prognostic precipitation19,33 through the Cloud-Hydrometeors Interactive Module with Explicit Rain and Radiation (CHIMERRA), whereas the version used in CMIP6 treats precipitation diagnostically69. CHIMERRA is a two-moment, large-scale microphysics parameterisation that includes both the mass and number mixing ratios of rain (qr, Nr) and snow (qs, Ns). High-density ice hydrometeors, such as graupel and hail, have not yet been treated explicitly.

The prognostic equations for precipitating hydrometeors are provided as follows:

$$\frac{\partial {q}_{p}}{\partial t}=-\frac{1}{{\rho }_{a}}\nabla \cdot ({\rho }_{a}{{{\bf{u}}}}{q}_{p})-\frac{1}{{\rho }_{a}}\frac{\partial ({\rho }_{a}{q}_{p}{v}_{{q}_{p}})}{\partial z}+{S}_{{q}_{p}},$$
(1)
$$\frac{\partial {N}_{p}}{\partial t}=-\frac{1}{{\rho }_{a}}\nabla \cdot ({\rho }_{a}{{{\bf{u}}}}{N}_{p})-\frac{1}{{\rho }_{a}}\frac{\partial ({\rho }_{a}{N}_{p}{v}_{{N}_{p}})}{\partial z}+{S}_{{N}_{p}},$$
(2)

where p ∈ r and s (rain and snow); ρa is air density (kg m−3); u denotes the wind vector representing horizontal advection (m s−1); \({v}_{{q}_{p}}\) and \({v}_{{N}_{p}}\) are the mass- and number-weighted fall velocities (m s−1), respectively, representing vertical sedimentation; and \({S}_{{q}_{p}}\) and \({S}_{{N}_{p}}\) represent the source and sink tendencies (s−1), respectively. The microphysics was solved with 60 s iterations, but shorter time steps were used to satisfy the Courant-Friedrichs-Lewy (CFL) criteria during the vertical sedimentation of precipitation19.

Radiation calculation for precipitating hydrometeors

Precipitating hydrometeors are radiatively active in the PROG ON-REP. The prognosed mass mixing ratios of rain and snow from large-scale condensation and their effective radius are passed to the radiation calculation. The radiative transfer in CHIMERRA was resolved using the k-distribution-based two-stream approximation radiative transfer package mstrnX70, which included 29 spectral bands discretised into 15 SW and 14 LW bands19.

In the radiative transfer calculation, liquid droplets and raindrops are treated from 1–200 μm, and ice crystals and snowflakes are treated from 5–500 μm in radii. The radiative impact of raindrops is negligible as reported in the literature20, whereas that of snowflakes is significant33. The optical properties (scattering, absorption, and polarisation) of the solid hydrometeors were derived from Yang et al. (2013)71 by assuming hexagonal columns for cloud ice and dendritic crystals for snowflakes.

The difference between PROG with and without REP includes REP-induced changes in meteorology. Although the instantaneous REP forcing can be alternatively quantified within a single experiment by a ‘double call’ of the radiation code with and without precipitating hydrometeors in a time step (at fixed meteorology), we applied the former method since REP-induced feedback on meteorology was the main focus of this study to understand the CMIP model diversity.

Experimental design

We employed three sub-versions of MIROC with different precipitation and radiative calculation treatments: (1) DIAG without REP, (2) PROG without REP, and (3) PROG with REP. Apart from the precipitating hydrometeors, the radiation table used in the DIAG and PROG schemes remained consistent, and the tuning parameters, other than precipitation scheme, were the same.

Atmospheric GCM simulations with prescribed climatological SST and sea ice were performed for 20 years, and the last 15 years were used for subsequent analysis. The cyclic boundary conditions for the year 2000 were used, and the annual cycle was repeated for each simulation year. The model resolution was a T85 spectral truncation (1.4° × 1.4°) with 40 vertical levels (T85L40). The standard model time step was 12 min, except for microphysical processes in the PROG, where a 60 s iteration was applied. Herein, the values reported were derived from the “native” outputs of the model rather than the “simulator” outputs (except for cloud coverage) since this study primarily focuses on the differences between the three sub-models (Supplementary Table 1).

Statistics

The statistical significance of the differences between PROG with and without REP was determined at the 95% confidence level using a two-tailed Student’s t-test with their interannual variabilities. To ensure the validity of the simulation period (20 years) and spin-up period (5 years), the dependence of the statistics on ensemble members (i.e., simulation years annually repeated) was examined using a bootstrap method (Supplementary Fig. 9). The ensemble means of the annual Arctic 2 m temperature were calculated for ensemble sizes of 3, 5, 7, 9, 11, 13, 15, 17, and 19, which were randomly selected from 20 members and iterated 100 times. The mean and minimum/maximum values were calculated from the 100 different results.

Energy budget analysis

The target of this study was to determine how the energy flux components at the TOA, ATM, and SFC respond to precipitation treatment and its radiative effect. Given that global precipitation as latent heat is energetically controlled by atmospheric radiative cooling (ARC) and surface sensible heat flux (SH), the local energy budget in an equilibrium state can be described as35,36,

$$L\Delta P=\Delta Q+\Delta H$$
(3)

where Δ is the difference in REP, L is the latent heat of condensation (2.5 × 106 J kg−1), P is the precipitation mass flux (kg m−2 s−1), Q is the atmospheric diabatic cooling (W m−2), which is composed of ARC and downward SH, and H is the energy transport term (W m−2) which is globally conservative37,72. The ARC can be derived from the difference between the net TOA and SFC fluxes as follows:

$$ARC={F}_{Net,TOA}-{F}_{Net,SFC}=(L{W}_{TOA}+S{W}_{TOA})-(L{W}_{SFC}+S{W}_{SFC})$$
(4)

and is strongly related to the global precipitation rate35,73 whereas H characterises the change in the distribution of precipitation74.

CMIP6 multi-model and ERA5 reanalysis datasets

We used 34 climate models from the CMIP6 data archive. The models are listed in Supplementary Table 2. To understand the links between precipitation and radiation treatments for Arctic amplification, the variable named tas was obtained from historical experiments. The Arctic amplification index (AAI) was calculated as the ratio of linear trends for surface air temperature between the Arctic and the entire Earth, as a metric of how the Arctic warms faster44. The analysis period was from 1980 to 2014 and was compared to the reanalysis product, ERA575.