The Impact of the Explicit Representation of Convection on the Climate of a Tidally Locked Planet in Global Stretched-mesh Simulations

Fine-scale atmospheric phenomena, such as moist convection, are typically unresolved in exoplanet climate models because the grid in these models is too coarse. Therefore, most general circulation models (GCMs) rely on convection parameterizations (Arakawa 2004). While convection parameterizations are physically motivated, they are necessarily a simplification of the real process (Kendon et al. 2021; Rios-Berrios et al. 2022). Recent studies showed that resolving convection explicitly may impact the estimate of the global climate of terrestrial tidally locked exoplanets (Sergeev et al. 2020; Lefèvre et al. 2021; Yang et al. 2023), because of the cloud stabilizing feedback (Yang et al. 2013). However, global convection-resolving simulations are computationally challenging, especially for climate-scale runs.

Locally refined global grids offer an elegant solution. They allow one to focus on a specific area of the planet, while keeping the global coverage and allowing for interactions at different scales, e.g., between localized convection and planetary-scale water distribution. Additionally, locally refined grids avoid numerical artifacts associated with using limited-area, or regional, models (Fox-Rabinovitz et al. 2008; Uchida et al. 2016). Namely, there are no boundaries or sharp changes in resolutions between the high-resolution region and the global model. These grids also allow for the two-way interaction between the region of interest and the rest of the planet, unlike a typical regional setup as in, e.g., Sergeev et al. (2020). This is especially important for the climate of tidally locked exoplanets, for which the localized stellar forcing is the key driving mechanism (e.g., Wordsworth & Kreidberg 2022).

A form of localized resolution increase is to stretch or deform the global mesh (e.g., Fox-Rabinovitz 2000; Harris et al. 2016; Uchida et al. 2016; Bindle et al. 2021). The advantages of this method are that the mesh remains topologically the same and does not require modifications to the model transport schemes. Grid stretching is also performed gradually, resulting in fewer grid artifacts compared to other methods. On the other hand, grid stretching typically results in a lower resolution on the opposite side of the planet, though it was not found to noticeably degrade the solution (Harris et al. 2016).

Stretched grids have been successfully applied for modeling Earth's atmosphere, bringing improvements in reproducing phenomena such as tropical precipitation (Harris et al. 2016) and localized greenhouse gas emissions (Bindle et al. 2021). The topological simplicity of stretched grids has been shown to be particularly useful for simulating extreme precipitation and tropical cyclones in aquaplanet Earth simulations (Harris et al. 2016). Despite a substantial degree of stretching, there was no degradation of the global climate in these simulations and no spurious numerical noise contaminating the solution. An intercomparison of several stretched-mesh GCMs was shown to successfully reproduce the key aspects of regional climate (Fox-Rabinovitz et al. 2006), in which the major positive impact was due to a better representation of model dynamics and orography (Fox-Rabinovitz et al. 2008).

Tidally locked exoplanets offer an excellent use case for stretched-mesh GCMs because convective processes are hypothesized to occur mostly on the hot dayside, on which the local refinement can be centered. Our work is the first application of a stretched-mesh GCM for the case of a tidally locked terrestrial exoplanet. We perform four experiments, in which we move from a configuration with a quasi-uniform mesh with parameterized convection to a stretched mesh with explicit convection. This includes an experiment with a reduced parameterization, serving as an intermediate step between fully parameterized and explicit convection. We demonstrate that it is computationally feasible to maintain the simulation fidelity of the global climate and at the same time resolve mesoscale circulations associated with the moist convection. Our stretched-mesh simulations resolve fine-scale cloud patterns on the dayside to a similar degree as was done previously in limited-area (Zhang et al. 2017; Sergeev et al. 2020; Lefèvre et al. 2021) and pseudo-global (Yang et al. 2023) high-resolution simulations. Using an explicit convection configuration on a stretched mesh leads to a colder, lower relative humidity climate, and an increase in cloud water content and the rate of the most intense precipitation. At the same time, the structure of the large-scale atmospheric circulation typical for tidally locked terrestrial exoplanets is preserved, and mesh stretching does not produce any numerical artifacts.

We use LFRic-Atmosphere, the new 3D GCM of the Met Office (Adams et al. 2019), based on a novel dynamical core GungHo and a suite of well-tested physical parameterizations inherited from its forerunner, the Unified Model (UM). GungHo solves the fully compressible non-hydrostatic Euler equations on a quasi-uniform cubed-sphere mesh using a mimetic finite-element discretization and a mass-conserving finite-volume transport scheme (Melvin et al. 2019; Bendall et al. 2020, 2023; Kent et al. 2023). The radiative transfer is parameterized using a two-stream correlated-k scheme SOCRATES based on Edwards & Slingo (1996), for which we use the same setup as in Sergeev et al. (2022b). LFRic-Atmosphere reproduces a variety of atmospheric flows (Melvin et al. 2019, 2024; Brown et al. 2023; Kent et al. 2023), including climate benchmarks for terrestrial exoplanets (Sergeev et al. 2023).

Our simulation parameters are based on the TRAPPIST-1 Habitable Atmosphere Intercomparison (THAI; Fauchez et al. 2020), a set of steady-state climate scenarios for a tidally locked TRAPPIST-1e assuming a 1 bar atmosphere. Sergeev et al. (2023) have shown that LFRic-Atmosphere reproduces the THAI scenarios close to that obtained from other models (e.g., Paradise et al. 2022; Turbet et al. 2022; Wolf et al. 2022). Here, we use the nitrogen-dominated aquaplanet case, Hab 1 (Sergeev et al. 2022b), to highlight the impact of mesh stretching on cloud regimes and moist convection.

Importantly, we set the rotation period of the planet to be 12.2 days, i.e., twice as long as the one used in the THAI scenarios, while still assuming the planet is tidally locked. This is to avoid the climate bistability that often appears in 3D GCM simulations of the Hab 1 case (Sergeev et al. 2022a) and would complicate our analysis, which is focused on the impact of the grid resolution. However, for any given case of a tidally locked exoplanet within a similar region of parameter space, our findings are still pertinent. Additional simulations with the original rotation rate of 6.1 days indeed result in two distinct climate regimes, as we discuss in Appendix C.

Throughout this study, we use the C192 cubed-sphere mesh, in a stretched and non-stretched configuration (see Appendix A for more details on mesh stretching). We performed four experiments: three with the mesh stretched by a factor of 10, and one with a non-stretched, quasi-uniform mesh (Figure 1, Appendix A). In the UniformParam and StretchParam configurations, we use the same parameterizations as those used in Sergeev et al. (2023), including the standard mass-flux parameterization of convection. The StretchParam experiment differs from UniformParam only in the mesh configuration, while the StretchReduced experiment uses a reduced convective parameterization (for technical details, see Appendix B). The StretchReduced configuration is designed to allow small, unresolved convection to be handled by the parameterization, while allowing deeper convection to be explicitly resolved. Note that the transition between resolved and parameterized convection in StretchReduced is not defined by a specific threshold. Instead, it dynamically changes with the cell size (and is therefore particularly useful in a stretched-mesh setup): when the scale of convection is smaller than the grid scale, convection is handled by the parameterization, while when the scale of convection is greater than the grid scale, parameterization effectively switches off and convection is handled explicitly. The StretchReduced experiment thus serves as an intermediate step between fully parameterized and fully explicit convection. The StretchExplicit configuration handles convection explicitly, i.e., with the parameterization switched off (see discussion in Sergeev et al. 2020).

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All four experiments have 63 levels with a model top at about 41 km. This corresponds to the same vertical resolution below 41 km as that used by the Met Office's UM in global climate simulations (Walters et al. 2019; Tomassini et al. 2023). While the vertical resolution may impact the degree of convective aggregation (Jenney et al. 2023), the simulations for tidally locked terrestrial planets have not been sensitive to it (Wei et al. 2020; Yang et al. 2023). The impact of the vertical resolution in a stretched-mesh configuration will be investigated in a future study.

All simulations start from a dry isothermal (300 K) atmosphere at rest (Fauchez et al. 2020). We integrate the model for 1000 Earth days, which is sufficient, given the relatively thin atmosphere and the shallow (1 m) slab ocean at the bottom boundary (judging by the surface temperature and top-of-atmosphere energy balance reaching a steady state). Discarding the spin-up period, in Section 3, we present climate diagnostics for the final 500 days.

Adopting a refined mesh allows us to simulate convective clouds covering the substellar area in great detail. As the snapshots in Figures 2(a)–(d) show, the stretched-mesh simulations produce spatial patterns of total column cloud condensate and precipitation rate similar to those reported in a convection-permitting limited-area model (Sergeev et al. 2020). The coarse, quasi-uniform mesh, on the other hand, produces a smoother pattern with a smaller variation of cloudiness and less extreme rainfall.

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On the periphery of the substellar region, the stretched-mesh simulation without a convection parameterization (StretchExplicit), produces cloud bands reminiscent of cloud streets (Figure 2(a)). These parallel elongated bands of clouds with cloud-free conditions between them were reported by Yang et al. (2023) in quasi-global convection-permitting simulations for TRAPPIST-1e. They form as a result of the persistent advection of cold air from the nightside onto a warmer ocean surface of the dayside, resulting in the formation of convective boundary layers. This is similar to the roll convection occurring in cold air outbreaks on Earth (e.g., Gryschka & Raasch 2005). Cloud streets can have a cooling effect on the global climate because the outgoing longwave radiation is larger in the clear-sky regions than that in the cloudy regions (Yang et al. 2023). Our work supports the results from Yang et al. (2023) and suggests that cloud streets can be reproduced in a stretched-mesh global model at a reduced computational cost. However, they may also be an artifact of under-resolving turbulence because of the increasing grid spacing and the 1D boundary layer parameterization in our setup (for the relevant discussion, see Boutle et al. 2014).

The small-scale variability enabled by the mesh refinement and explicit convection is also reflected in a marked increase in the precipitation rate in the substellar region. As Figure 2(h) shows, the maximum instantaneous precipitation reaches 100 mm day⁻¹ in the UniformParam simulation. Moving from a quasi-uniform to a stretched mesh has little effect (Figure 2(g)), while reducing or disabling the convection scheme results in almost doubling the maximum precipitation rate (Figures 2(e), (f)). The histogram in Figure 2(i) distills this change in the precipitation maxima. It shows that the mesh stretching barely changes precipitation rates (after interpolating the data to the same mesh). However, reducing the impact of the convection scheme or even entirely switching it off results in substantially more intense precipitation, reaching 175 mm day⁻¹ at least twice within our analysis period of 500 days (0.001% of 221,184 mesh cells). This is because in the StretchExplicit and StretchReduced cases, much of the convection is handled by the grid-scale dynamics, permitting the formation of small but concentrated storms. Similarly high precipitation rate was reported for the stretched-mesh simulations of Earth as an aquaplanet (Harris et al. 2016) and for the convection-permitting limited-area simulations of the climate of TRAPPIST-1e and Proxima Centauri b (Sergeev et al. 2020).

The main effects of explicit convection on the global climate are a reduction in atmospheric water vapor content and an increase in condensed cloud water. While the UniformParam and StretchParam cases have a similar amount of water vapor (5.4–5.6 kg m⁻²), in the StretchReduced and StretchExplicit cases, it decreases by about 25%, with the latter simulation being the driest overall (Figure 3(a)). The overall drying of the atmosphere in the StretchExplicit and StretchReduced simulations is evident in the vertical profiles of absolute humidity (Figure 4(b)) and relative humidity (Figure 4(c)). The relative humidity is noticeably lower in the troposphere in the StretchExplicit and StretchReduced simulations, so the water vapor decrease cannot be explained only by the lower temperatures in these two simulations (Figure 4(a)). Indeed, since the stratospheric temperature is broadly the same in all four cases, it is the parameterization of turbulent mixing above the boundary layer included in the StretchReduced configuration that reduces the mixing across the tropopause and results in a less efficient upward transport of water vapor. This leads to a lower humidity in the upper layers of the nightside and terminator regions (not shown), which may have a small effect on the water features in the transmission spectrum as discussed, e.g., in Fauchez et al. (2022) and Sergeev et al. (2022a).

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At the same time, the global amount of cloud water increases from 7.9 × 10⁻² in the UniformParam experiment to 9 × 10⁻² and 8.7 × 10⁻² in the StretchReduced and StretchExplicit experiments, respectively (Figure 3(b)). The largest increase of cloud condensate for the StretchExplicit case is seen in the liquid phase (Figure 3(c)). It happens predominantly in the ring around the substellar region, i.e., where the grid resolution becomes relatively coarse due to the nature of the mesh stretching. The grid spacing is too large to fully resolve the shallow convection, and the representation of convective updrafts suffers from the lack of parameterization that can remove the instability. Explicit convection overcompensates for this, and as a result, convective updrafts remove moisture out of the boundary layer too efficiently, creating more cloud condensate (evident especially in the liquid cloud content in Figure 3(c)). In the StretchReduced experiment, the parameterizations are set up to represent shallow convection better. As a result, grid-aliased convection is not as intense, and the cloud content around the substellar region is lower than that in the StretchExplicit case (Figures 3(c), (d)). The liquid cloud increase in the reduced and explicit convection simulations contributes to the changes in the cloud fraction. As Figure 2(j) reveals, moving from parameterized convection (StretchParam and UniformParam) to reduced/explicit convection (StretchReduced and StretchExplicit), the global mean low cloud fraction increases by up to 6%. This change happens predominantly in the substellar region, and is indeed due to the liquid phase increase, as evident in the vertical profiles of the mixing ratio and bulk cloud fraction (Figures 4(d), (f)).

Compared to previous high-resolution studies of, e.g., Sergeev et al. (2020), our results broadly agree that the high-resolution model with explicit convection produces more low-level clouds than a coarse-resolution GCM with parameterized convection. This is unsurprising because most subgrid-scale physics parameterizations in LFRic-Atmosphere are inherited from the UM (Sergeev et al. 2023). The differences between our study and that of Lefèvre et al. (2021) are greater mostly because of the inter-model differences in physical parameterizations. Moreover, the high-resolution simulation in Lefèvre et al. (2021) is forced by a constant and spatially uniform heating rate, while in the present study, the impact of the global circulation on the substellar region is direct due to the gradual stretching of the mesh and no boundaries between nested and driving models. Finally, the planetary parameters are different between the aforementioned studies and the present study. As a result, the substellar atmosphere is warmer in the simulations in Lefèvre et al. (2021) and Sergeev et al. (2020) than in the present study. This likely leads to more water being in the vapor form than in the condensed cloud form in those two studies. Another consequence of a colder climate in our simulations is that the substellar convection is likely to be slightly weaker, resulting in a weaker transport of cloud condensate in the upper atmosphere and hence fewer high clouds.

The cloud cover differences between the parameterized and explicit convection experiments lead to changes in the radiation balance and, consequently, the surface temperature. In the StretchExplicit case, the top-of-atmosphere shortwave cloud radiative effect is −58.8 W m⁻², more than 10 W m⁻² greater in magnitude than that in the UniformParam case. A similar brightening of clouds was reported in Tomassini et al. (2023) for the reduced convection experiments. Expressed in terms of the global mean albedo, the amount of reflected shortwave radiation ranges from 21.9% in the UniformParam case to 25.2% in the StretchExplicit case. Together with the reduction in the greenhouse effect due to less water vapor, this leads to a 5 K drop in the global mean surface temperature in the StretchExplicit case (Figures 5(a)–(d)). The global mean surface temperature is 230, 231, 235, and 235 K in the StretchExplicit, StretchReduced, StretchParam, and UniformParam, respectively. Note that the StretchReduced climate is not colder than StretchExplicit despite having a substantially lower stratospheric humidity (Figure 4(b)). This is because the greenhouse effect of the stratospheric water vapor is small due to its very low absolute amount.

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The thermal maps in our simulations exhibit a day–night dichotomy typical for tidally locked planets, both in terms of surface temperature (Figures 5(a)–(d)) and top-of-atmosphere outgoing longwave radiation (Figures 5(e)–(h)). The substellar area reaches temperatures of 288 K–290 K, while the nightside coldest areas (stationary gyres) reach temperatures of 159 K–168 K. The latter is substantially lower than the value predicted by a coarse-mesh LFRic-Atmosphere simulations of the THAI Hab 1 case (Sergeev et al. 2023), which is likely due to the slower rotation rate of the planet in the present study. A similar effect of slowing the rotation rate on the climate was found in GCM simulations by, e.g., Edson et al. (2011) and Komacek & Abbot (2019). Likewise, the outgoing longwave radiation is at the lower end of the THAI inter-model spread (Sergeev et al. 2022b): its values range from 167.8 W m⁻² in the UniformParam experiment to 156.8 W m⁻² in StretchExplicit (Figures 5(e)–(h)). Clouds prevent even more heat from being lost to space, so their net effect on the top-of-atmosphere longwave radiation flux is positive. It is broadly the same in our four experiments (≈13 W m⁻²) and is comparable to that predicted by the models in Sergeev et al. (2022b) and Yang et al. (2023).

In agreement with the global temperature distribution, the large-scale circulation is broadly the same in all four simulations (see streamlines in Figures 5(a)–(d)). The main pattern of the wind field in the troposphere is a superrotating (prograde) jet at the equator and stationary cyclonic gyres in high latitudes on the nightside, as is typically expected for synchronously rotating terrestrial planets (e.g., Haqq-Misra et al. 2018; Komacek & Abbot 2019; Hammond et al. 2020). The vertical structure of the wind field reveals that the atmosphere has two regions of eastward flow: tropospheric (below ≈15 km) and stratospheric (above ≈25 km), with a quiescent region in between (Figures 5(i)–(l)). The tropospheric flow is characterized by intense upward motions in the substellar region (red contours in Figures 5(i)–(l)), associated with convergence near the surface and divergence aloft. The zonal mean vertical velocity is marginally stronger in the StretchReduced simulation, which is due to the changes in the convection parameterization. Mesh stretching alone brings a relatively small change in the time-averaged vertical velocity (Figure 5(k)). In the stratosphere, the eastward flow has two maxima in high latitudes, which become somewhat weaker moving from UniformParam to StretchExplicit. Hence, the 3D wind field in our simulations is broadly similar to and only slightly weaker than that in the UM output for THAI Hab 1 (Sergeev et al. 2022b).

Despite better resolving small-scale convective patterns (Figure 2), on average our stretched-mesh simulations produce a similar total precipitation rate as compared to that in the non-stretched-mesh simulations with parameterized convection (Figure 6). The precipitation rate has a broad peak over the substellar region, reaching ≈30 mm day⁻¹ at the substellar point—close to the highest values within the inter-tropical convergence zone in Harris et al. (2016). Averaged meridionally (across all latitudes), the peak of precipitation is about 10 mm day⁻¹ (not shown) and agrees well with the values in Yang et al. (2023). However, the way the model generates precipitation is different. Grid-scale precipitation, generated by the microphysics scheme, is the largest in the StretchExplicit and StretchReduced cases, accounting for almost all of the precipitation in these cases (Figures 6(e), (f)). Convective precipitation, on the other hand, the largest in the UniformParam case, and second-largest in StretchParam: ≈0.4 mm day⁻¹ (Figures 6(k), (l)). In the StretchReduced simulation, the reduced convection parameterization yields about two orders of magnitude less precipitation than the full parameterization. This is because the StretchReduced simulation has a larger convective available potential energy (CAPE) timescale (Appendix B), which is a key parameter controlling the amount of parameterized convective mass flux relative to the resolved vertical motion and associated rainfall in the model (Tomassini et al. 2023). In summary, stretching the mesh while keeping the convection parameterization the same results in relatively more grid-scale precipitation; while reducing or disabling this parameterization offloads the precipitation to the grid scale.

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Our results show that LFRic-Atmosphere is capable of simulating small-scale atmospheric processes using a stretched mesh while also reproducing the key characteristics of the global climate of a tidally locked terrestrial exoplanet. Using a locally refined mesh allows us to capture small-scale cloud patterns in the substellar region. In agreement with previous studies based on localized or global high-resolution grids (e.g., Harris et al. 2016; Kajikawa et al. 2016; Uchida et al. 2016; Rios-Berrios et al. 2022), resolving deep convection leads to more intense and localized precipitation. Thus, at a reduced computational cost (Table 1), LFRic-Atmosphere allows us to examine a regional phenomenon such as convection in great detail, and at the same time study, the interaction between local and global atmospheric circulation.

Tidally locked exoplanets offer a particularly suitable use case but a locally refined mesh has also shown potential for Mars climate modeling, e.g., at the scale of individual craters (Lian & Richardson 2023). Stretched-mesh GCMs may also be useful in studying Titan weather, in particular, the interaction between localized convective storms and large-scale Rossby waves (Battalio & Lora 2021). For Venus and slowly rotating Venus-like exoplanets, stretched-mesh GCMs may shed light on the convectively generated gravity waves and their interaction with the superrotating flow (Lefèvre et al. 2018). For non-tidally locked exoplanets, the mesh can be stretched in a different way, e.g., focusing the high resolution on the equatorial band, allowing for a better simulation of deep convection in the tropics (Rios-Berrios et al. 2022).

Our study is the first exoplanet application of LFRic-Atmosphere in a global configuration with explicit convection, allowed for by the stretched mesh. Our key finding is that the simulation with explicit convection (StretchExplicit) predicts a climate that is substantially drier (as seen in both specific and relative humidity) and colder (by ≈5 K), with about 10% more cloud condensate compared to the simulations with fully parameterized convection. The main reason for the colder climate is the higher cloud albedo, i.e., stronger reflection of the shortwave radiation by clouds (by ≈10 W m⁻²). Additionally, there is less water vapor and thus a weaker greenhouse effect. The increase in cloud condensate, especially in the low-level cloud liquid, can be partly explained by the fact that the grid resolution on the dayside's periphery may be too coarse to fully resolve the shallow convection. However, the experiment with reduced parameterization (StretchReduced) produces a similar cloud structure (between that in the StretchExplicit run and those with fully parameterized convection), supporting this trend. Note that the StretchReduced simulation serves as an intermediate solution between the parameterized and explicit convection because it is designed to activate convection depending not only on the grid cell size but the size of the convection relative to it.

We believe that overall our experiment with explicit convection simulates a climate close to reality because it truly represents the underlying physics of the atmosphere, without relying on parameterizations tuned for Earth. The main caveat is the grid coarsening on the dayside periphery where convection becomes under-resolved. A possible solution could be to use a base mesh with more grid points and a smaller stretching factor. In the absence of observational constraints, these simulations may serve as benchmarks for coarse-resolution GCMs with parameterized convection. At the same time, as noted in Tomassini et al. (2023), cloud parameterizations will likely require tuning for convection-permitting models. This may reduce the amount of shortwave radiation reflected by the clouds, resulting in a warmer climate. Future work should explore the representation of cloud microphysics and their impact on the climate in stretched-mesh simulations.

Another caveat of the present study is that the convection scheme in our model is not tuned for a stretched-mesh configuration. Indeed, in most kilometer-scale model convection is only partially resolved (Tomassini et al. 2023). This introduces the challenge of representing convection in a balanced way: partially explicitly and partially as a parameterized subgrid process. Scale-aware convection schemes are a promising solution (Kendon et al. 2021), which may also re-ignite interest in using stretched-mesh GCMs (Uchida et al. 2016). One example of a scale-aware scheme is CoMorph, a new convection parameterization under development at the Met Office (Daleu et al. 2023; Lavender et al. 2024). It is designed for use within both the UM and LFRic-Atmosphere and has already been shown to be close to cloud-resolving models in representing many aspects of convection (Lavender et al. 2024). The benefits of applying CoMorph in a stretched-mesh configuration will be explored for a tidally locked exoplanet in a future study.

Despite the differences discussed above, the key climate diagnostics and spatial patterns simulated in the stretched and non-stretched runs are relatively similar and the fidelity of the large-scale atmospheric circulation is preserved. We have thus shown that mesh stretching allows for a gradual transition into high resolution and does not produce numerical artifacts, confirming previous work on Earth climate (Fox-Rabinovitz et al. 2000; Harris et al. 2016; Uchida et al. 2016) and opening new avenues for studying 3D mixing in planetary atmospheres.

We thank Lorenzo Tomassini for his help with the reduced convection setup. We thank Thomas Melvin and Nigel Wood for helpful feedback on this work. We thank Thomas Melvin for his help with the cubed-sphere mesh visualization. Material produced using Met Office Software. We acknowledge the use of the Monsoon2 system, a collaborative facility supplied under the Joint Weather and Climate Research Program, a strategic partnership between the Met Office and the Natural Environment Research Council. Additionally, some of this work was performed using the DiRAC Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility. ⁵ The equipment was funded by BEIS capital funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1. DiRAC is part of the National e-Infrastructure. This work was supported by a UKRI Future Leaders Fellowship MR/T040866/1. This work also was partly funded by the Leverhulme Trust through a research project grant RPG-2020-82.

Software: The LFRic-Atmosphere source code and configuration files are freely available from the Met Office Science Repository Service ⁶ upon registration and completion of a software license. The UM and JULES codes used in the publication have been committed to the UM and JULES code trunks, having passed both science and code reviews according to the UM and JULES working practices; in the UM/JULES versions stated in the paper (vn13.3). Scripts to post-process and visualize model output are available on GitHub ⁷ and depend on the following open-source Python libraries: Aeolus (Sergeev & Zamyatina 2024), GeoVista (Little 2023), iris-esmf-regrid (Worsley 2023), Iris (Met Office 2023), Matplotlib (Hunter 2007), and NumPy (Harris et al. 2020).

The stretching of the mesh is achieved using the Schmidt (1977) transformation (see also Harris et al. 2016; Bindle et al. 2021). It preserves the mesh topology, so the number of grid points and their connectivity are the same as in the non-stretched mesh (Figure 1(d)). The stretching is smooth and does not have abrupt transitions in resolution such as those associated with grid nesting (Sergeev et al. 2020). The resulting grid has a refined resolution at the target location compensated by a coarser resolution on the opposite side of the sphere. Finally, the refined domain (the target face) diminishes in size proportional to the stretching factor.

The mesh stretching procedure involves two steps. First, the grid points of the original cubed sphere are attracted to the South Pole by the Schmidt (1977) transform

$$\begin{eqnarray*}&&\phi ^{\prime} (\phi )=\arcsin \left(\displaystyle \frac{D+\sin \phi }{1+D\sin \phi }\right)\,\mathrm{with}\,D=\displaystyle \frac{1-{S}^{2}}{1+{S}^{2}},\end{eqnarray*}$$

where S is the stretch factor (S > 1 causes stretching), ϕ is the original latitude, and $\phi ^{\prime}$ is the transformed latitude. The second step is the rotation of the stretched mesh so that the refined region is centered at the desired location. Thus, four parameters uniquely define a stretched mesh: the size of the cubed sphere Cn, the stretch factor S, and the target latitude ϕ_T and longitude λ_T . The larger the S factor, the finer and more localized the refinement is. The grid spacing at the target location is approximately S times finer compared to the non-stretched mesh, while at the antipode location, it is approximately S times coarser. Stretch factors used in existing literature are typically between 2.5 and 10 (e.g., Fox-Rabinovitz et al. 2006; Harris et al. 2016; Uchida et al. 2016; Bindle et al. 2021).

As shown in Figure 1, we use a stretched mesh with s = 10, which gives a resolution ≈10 × finer at the target location. The target location is the substellar point ϕ_T = 0°, λ_T = 0°. While the factor of 10 is high, it has been used for Earth applications, e.g., to better resolve nitrogen dioxide emissions in California (Bindle et al. 2021). By using a highly stretched mesh, we also test the boundaries of the computational stability of LFRic-Atmosphere. In future studies, we will probe other stretch factors, depending on the planet and problem in question.

The resulting stretched mesh has a cell width size of ≈4.7 km in the substellar region (Figures 1(a)–(c)), reaching the convection-permitting resolution. To achieve a similar resolution with a quasi-uniform mesh, one would have to use 100 times as many grid points, which is extremely computationally expensive for climate simulations. Similar 4 km grid spacing has been used for simulating convection explicitly in regional climate predictions for Earth (e.g., Stratton et al. 2018) and, more recently, for exoplanets with an Earth-like atmosphere, both on a global (Yang et al. 2023) and regional (Sergeev et al. 2020) scale. By the nature of grid stretching, the nightside of the planet has a relatively coarse-mesh spacing of up to ≈470 km. Nevertheless, even this resolution is close to those applied globally in previous studies for exoplanetary climates (e.g., Yang et al. 2013; Turbet et al. 2016; Paradise et al. 2022).

As Table 1 shows, the cost of performing stretched-mesh simulations is only about 2× higher than that of the uniform mesh simulation (mostly due to a smaller time step in the stretched-mesh setup), while allowing us to reach a convection-permitting resolution for the substellar point. Using a uniform mesh with the same high resolution for the whole planet would require a relative computational cost of ∼1000.

In the StretchReduced experiment, we follow a kilometer-scale configuration used for the UM in Tomassini et al. (2023), namely, their MidLevShConv15RAturb configuration. This configuration combines a reduced convection parameterization with a modified representation boundary layer scheme. As a result, convection is allowed to be partly explicit, accounting for the fact that in the focal region of our stretched mesh, the grid step reaches 4.7 km.

Briefly, the StretchReduced parameterization settings are different from those used in StretchParam and UniformParam in the following. Scaling of the shallow convective mass flux is set to 0.015 (default: 0.03)—the value found to produce more realistic low-level clouds in global kilometer-scale numerical weather prediction runs. The formulation of the mid-level convection scheme is also modified. The key modification that reduces the subgrid convective mass flux is a longer timescale for the CAPE closure: 2700 s instead of the default 1800 s. This effectively decreases the strength of the convection parameterization and allows convection to be partly explicit, i.e., represented by grid-scale vertical flow. The configuration also includes the turbulence blending scheme (Boutle et al. 2014), which allows for a transition between the 1D turbulence parameterization (used in coarse-grid GCMs) with a 3D Smagorinsky–Lilly turbulence scheme (used in cloud-resolving models). As mentioned in Section 3, this parameterization leads to a drier stratosphere compared to the other three simulations.

In additional experiments, we change the rotation rate of TRAPPIST-1e to show the climate bistability. We set it to its observed orbital period of 6.1 days, i.e., the value prescribed by the THAI protocol (Fauchez et al. 2020) and used in the majority of modeling studies for this planet (e.g., Wolf 2017; Turbet et al. 2018, 2022; Fauchez et al. 2019; Eager-Nash et al. 2020; May et al. 2021; Yang et al. 2023). Broadly, our conclusions hold for these simulations. The key difference is that in this case, the GCM is prone to a climate bistability: the atmospheric circulation settles on a double-jet regime in the StretchExplicit experiment, and on a single-jet regime in the other three experiments (for definitions, see Sergeev et al. 2022a). This corresponds to a distinct difference in climate characteristics, including the global cloud cover (Figure 7). In the StretchExplicit experiment, the cloud cover associated with the deep convection is narrower and more zonally elongated than that in the other three experiments.

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The fact that the circulation regime changes not when the mesh is stretched (comparing UniformParam to StretchParam) but when the convection scheme is disabled (comparing StretchParam to StretchExplicit), confirms our key finding that for the global climate mesh stretching is less important than changes in model parameterizations (see Section 3). Furthermore, our StretchExplicit experiment is qualitatively similar to the convection-permitting simulation of TRAPPIST-1e in Yang et al. (2023), in terms of the spatial distribution of temperature, wind, cloud cover, and precipitation. This implies that at least some of the differences between the coarse-grid GCMs and the convection-permitting model reported by Yang et al. are due to the circulation regime change and not due to resolving convection per se. And as Sergeev et al. (2022a) demonstrated, the circulation regime change can happen even in a coarse-grid GCM if the convection parameterization is modified or switched off.

Despite the circulation regime change in these simulations, our model predicts a climate within the spread of THAI GCM results (Sergeev et al. 2022b). Taking the experiment with the uniform mesh (UniformParam) as an example, the global mean surface temperature is 237 K (Figure 7(h)), which is lower than that predicted by ExoCAM and LMD-G but higher than that predicted by the UM for the Hab 1 case. Likewise, the maximum surface temperature (292 K) falls within the inter-model spread. The lowest surface temperature is 184 K, simulated in the regions of nightside stationary gyres. It is on the colder end of the spectrum of the THAI GCMs but still about 10 K higher than that in the UM simulation. Comparing the StretchExplicit case with the ROCKE-3D simulation in Sergeev et al. (2022b), we see the same trend in the surface temperature in both models: the minimum temperatures rise to about 200 K due to the shift of the stationary waves, while the maximum temperatures fall to 285–290 K. Thus, in the THAI Hab 1 case, LFRic-Atmosphere tends to produce a colder climate compared to other GCMs, though not as cold as that in the UM.