a. Data

We analyze simulations submitted to phase 6 of the Coupled Model Intercomparison Project (CMIP6; Eyring et al. 2016). We take the first 30 years (1850–79) in the historical experiment as the control (base) climate state and years 121–150 in the abrupt 4 × CO₂ (4 × CO₂ for short) experiment as the perturbed climate state. Climate changes in variables are characterized by subtracting the base state from the perturbed climate state. We use the outputs from nine models that have reported daily values of near-surface (2 m) air temperature (T), specific humidity (q), RH, surface soil moisture (SM, moisture in the top 10-cm soil layer), surface LH flux, surface SH flux, and upwelling and downwelling short- and longwave radiative fluxes from which we calculate the net radiation at the surface (R_n, downward positive), for both the historical and 4 × CO₂ experiments. The nine models are CanESM5, CMCC-ESM2, GFDL-CM4, MIROC6, IPSL-CM6A-LR, INM-CM5.0, MPI-ESM1-2-HAM, MPI-ESM1-2-HR, and MRI-ESM2-0. The limitation of these nine specific models mainly comes from the availability of surface soil moisture and surface flux outputs at daily frequency in the 4 × CO₂ experiment (e.g., daily SM is not available for CESM2 and daily surface and radiative fluxes are not available for the ACCESS model and EC-EARTH). Besides the limitation due to the data availability, we only include one model from each modeling institution even if outputs from the different versions of that model are available. In this case, we consider the more recent version or the higher-resolution version that has the more complete set of variables available. NorESM2-MM, FGOALS-g3, and Taiwan ESM (TaiESM1) have also reported daily surface soil moisture, but a large amount of the soil moisture values are invalid, and therefore, they are not included. In addition to the variables listed above, precipitation P in the historical experiment is used in order to calculate the climatological AI. AI is defined as AI = 0.8R_n/(L_υP). Arid regions have larger AI under this definition, and the constant 0.8 is set empirically to account for the fact that not all available energy goes into evapotranspiration even if soil moisture is abundant (Milly and Dunne 2016; Koster and Mahanama 2012). To calculate the near-surface air MSE (MSE = c_pT + L_υq + gz, where c_p is the specific heat of dry air, T is the surface air temperature, L_υ is the latent heat of vaporization, q is the surface air specific humidity, g is the gravitational acceleration, and z is the height above sea level), we also use the orographic height variable. Since the orographic height will not change with climate change, we only need this gz term when calculating MSE in one climate state. Temperature and geopotential height on pressure levels (variable ta and zg) during 1850–79 in the historical experiment are used to calculate the temperature lapse rate between 700 hPa and the surface. Since MRI-ESM2 only reports daily 3D variable after 1950 and daily 3D zg from CMCC-ESM2 is not available, we use only the remaining seven models for temperature lapse rates and profiles. We use the gridcell land area fraction variable to select land grid cells as those with a land area fraction ≥99%. Since we are interested in situations where we expect a strong land surface control on temperature, we focus on the warm season, which we define as the 150 days centered on 15 and 16 July for the Northern Hemisphere and the 150 days centered on 15 and 16 January for the Southern Hemisphere.

b. The daily SM/climatological AI phase space

The land surface is highly heterogeneous: it ranges from very arid regions such as the Sahara Desert to very moist regions such as the Amazon rainforest. For a given location, temporal variability of dryness/wetness can substantially affect the surface latent heat flux: LH that is more than 150 W m⁻² during average days can reduce to only a few tens of watts per square meter when the soil is dry (see Fig. A1g). Furthermore, when considering changes in these hydroclimatic variables under climate change, more complexity comes into play due to the nonlinear relationship between LH and soil moisture. In particular, for a fixed amount of net radiation at the surface, LH increases with SM in the transitional regime but is not sensitive to SM in the dry and wet regimes (e.g., Seneviratne et al. 2010) and decreases with SM in the active-rain regime (Duan et al. 2023). As a result, when soil dries under climate change, LH decreases in the transitional regime, while it increases in the wet regime due to increases in net radiation (Duan et al. 2023). These different changes in LH can lead to different responses in surface air temperature: warming will be magnified in the transitional regime due to reductions in latent cooling, in contrast to the behavior in the wet regime. Since places with different climatological dryness have different percentages of days in each of the LH–SM regimes, averaging over time that contains both local wet and dry days or averaging over locations with different local soil moisture distributions can lead to substantial canceling of signals and result in an averaged behavior that is hard to interpret. Moreover, models represent the spatiotemporal distribution of soil moisture and the evapotranspiration–soil moisture functions differently, resulting in further uncertainty.

Duan et al. (2023) presented a process-based phase space constructed using daily soil moisture and the climatological aridity index and showed that when organizing the highly heterogeneous land hydroclimatic variables in this phase space, one can acquire coherent patterns of changes across models and across variables. The coherent patterns in this SM/AI phase space are in contrast to the highly uncertain results in the geographical map view [Fig. S3of Duan et al. (2023)]. Here, we adopt this method and process the model fields that have dimensions of calendar days and geographical locations into phase-space grids consisting of 50 temporal bins and 50 spatial bins. The horizontal axis of the phase space is formed by the temporal bins and calculated by sorting the original data according to the daily SM and placing 2% of the 4500 days in each bin. These temporal bins represent local dry days to wet days. The vertical axis of the phase space is formed by the spatial bins and calculated by sorting the original grid cells (for both the base and the 4 × CO₂ climate states) according to the base-climate warm-season mean AI and placing 2% of the tropical land area (30°S–30°N) in each bin. These spatial bins represent the climatologically moist regions to climatologically arid regions. We then take the area-weighted average of data in each bin and plot the bin-averaged values as shadings. Figures A1 and 3 show a number of variables and their changes displayed in this SM/AI phase space. We refer to the average in each of the 50 × 50 bins as data “under a certain spatial–temporal condition.” We normalize changes in variables in the phase space by the tropical mean ocean warming ${[\overline{T}]}_o$ in each model before averaging over models to account for the different climate sensitivities across models.

c. The diagnostic linearized model for the magnitude of warming

Both perspectives consider a partitioning of total energy (MSE) or energy flux (R_n) between a temperature component (c_pT or SH) and a humidity component (L_υq or LH). Here, we reduce the variables in the balance relations and rearrange them to relate the temperature component to the total energy/energy flux by a partition factor. When considering climate change, we decompose changes in the total energy/energy flux into a component with an unchanged base-state partition factor and a component representing the contribution from changes in this partition factor. This method of decomposition helps us compare the partitioning in the two perspectives and diagnose the relative importance of the base-climate dryness versus the change in dryness under climate change in contributing to the amplified warming over tropical land. Distinguishing between the role of the base climate state and changes to that state is important because information about the base climate state is also available from observations and can be used to make predictions and constrain climate change projections. We derive the relations below.

While the surface flux perspective is based on the partitioning of the net radiative flux, the atmospheric dynamics perspective is based on the partitioning of the total energy MSE, which is assumed to be uniform across the tropics in QE–WTG theory. Leaving aside the gz term in MSE that will not change under warming, the moist enthalpy (ME; ME = c_pT + L_υq) consists of a temperature component c_pT (the dry specific enthalpy) and a moisture component L_υq. By a similar approach to that used above for surface fluxes, we can use the ratio between these two components as b = (c_pT)/(L_υq) to write ME as

$\text{ME}=\text{MSE}-gz=c_{p}T+L_{\upsilon }q=\frac{b+1}{b}{c}_{p}T=\psi c_{p}T,$(2)

where ψ = (b +1)/b = ME/(c_pT). In this way, we associate the total energy to surface air temperature, through a ratio ψ. The inverse, 1/ψ, is the fractional contribution of the temperature component in the total ME. Thus, the surface energy and atmospheric dynamic views can be written in somewhat comparable forms (see Table 1), and we will compare these parameters and their changes in the remainder of the paper.

Table 1.

A summary of the equations and terms considered for the base-climate physics and climate change perturbations in the atmospheric dynamics and the surface flux perspectives.

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The terms (c_pψ)⁻¹ in Eq. (6) and (κΨ)⁻¹ in Eq. (7) can be understood as the sensitivity of the temperature response to the forcing analog (the total energy/energy flux). The specific heat of dry air, c_p, is a physical constant. For simplicity, we also consider κ to be a constant as discussed above. Therefore, in the following sections, we take ψ⁻¹ and Ψ⁻¹ as the sensitivities in each framework, which are solely determined by the base-climate state.

The terms −c_pTΔψ and −SHΔΨ are the contributions of the change in the partition factor (ψ and Ψ, the inverse of which is the sensitivity discussed above) to the magnitude of warming, and we refer to them as the “repartition” terms. Specifically, the terms summarize the changes in the partitioning of MSE into T and q for the atmospheric framework or the partitioning of R_n into SH and LH for the surface energy balance framework. The sign of the repartition terms is consistent with their effect on the temperature response: positive values indicate that the balance is repartitioning in a way that will increase the temperature response and decrease the moisture response.

A summary of the balance relations, their linearized perturbation equations, and relevant terms for the two perspectives are listed in Table 1.

a. The atmospheric perspective

In the atmospheric perspective, the change in the total energy (ΔMSE) is split, by definition, between warming (c_pΔT) and moistening (L_υΔq). Each term in the phase space in CMIP6 models is shown in Figs. 2a–c. The ΔMSE (Fig. 2a) shows an increase across all 50 × 50 temporal and spatial bins with a value ranging from 3.3 to more than 4.2 kJ kg⁻¹ per degree of mean tropical ocean warming. The warming magnitude (ΔT; Fig. 2b) is smaller in moist conditions (lower right), while it is larger in dry conditions (upper left), maximizing over the desert regions. Warming around the critical soil moisture contours in moist regions is also strong, and this amplified warming center around SM_crit is more evident in changes in the daily maximum temperature (not shown). Specific humidity q increases under all conditions, and the magnitude is generally contrary to that of ΔT, i.e., wetter conditions moisten more and warm less, while drier conditions moisten less and warm more.

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Changes in surface air (a) MSE; (b) temperature converted to energy units (c_pT); (c) specific humidity converted to energy units (L_υq), between the 4 × CO₂ climate state and the base climate state displayed in the phase space of daily SM percentiles (horizontal axis) and climatological AI percentiles (vertical axis). (d) Deviation of the MSE over land from the corresponding (corresponding day and latitude) zonal-mean values over the ocean ([MSE]_o) in the base climate; (e) deviation of the MSE over land from the corresponding precipitation-weighted zonal-mean values over the ocean (${[\text{MSE}]}_o^P$) in the base climate; and (f) changes in the deviation of the land MSE from ${[\text{MSE}]}_o^P$ between the 4 × CO₂ climate state and the base climate state. Changes in variables are normalized by the mean tropical ocean warming ${[\mathrm{\Delta }\overline{T}]}_o$ in each model before averaging over the models. The black contours (solid for the base climate and dashed for the 4 × CO₂ climate) are the isolines of the surface SM value of 20 kg m⁻² which roughly marks the conditions in the transitional regime of the Budyko curve. The green contour is the isoline of the daily mean precipitation rate of 0.5 mm day⁻¹. See text for more details.

Citation: Journal of Climate 37, 18; 10.1175/JCLI-D-22-0955.1

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In the QE–WTG framework, ΔMSE is generally assumed to be uniform across various divisions of the tropics. For example, Byrne and O’Gorman (2013, 2018) assumed uniform annual-mean changes across land and ocean for a given latitude, and Byrne (2021) showed uniform changes in CMIP6 models for the upper 50 MSE percentiles across time and locations over land and ocean. With this assumption, amplified warming is then predicted in conditions with less moistening. The general “less moistening–more warming” correspondence we see in Figs. 2b and 2c seems to be consistent with the theory of this QE–WTG framework. However, it is interesting to note that we also see the less moistening–more warming relationship over the desert despite the fact that there is almost no moist convection over the desert, so the region is not expected to be in convective quasi-equilibrium as required by the theory. Indeed, as we can see in Fig. 2a, the change in MSE is noticeably smaller in deserts than in other regions, since there is a lack of coupling between the desert boundary layer and other regions that are dominated by moist convection. Thus, the assumption of equal change in MSE that has been applied previously to other groupings of regions in the tropics appears to break down in the phase space we consider for CMIP6 models, particularly in the drier days and regions that warm the most.

To further examine the QE–WTG framework in CMIP6 models regarding the assumption about the uniform MSE in the AI/SM phase space, we compare the MSE at a certain spatiotemporal condition over land to the corresponding day and latitude zonal-mean values over the ocean in the base climate in Fig. 2d. If all conditions satisfy QE and WTG (especially QE, since WTG holds relatively well across locations in the tropics; see Fig. 6c), the values in Fig. 2d should be approximately 0, indicating that MSE over land roughly equals the corresponding value over the ocean. However, Fig. 2d shows >0 values for wet conditions and <0 values for dry conditions. Zhang and Fueglistaler (2020) showed that this equal MSE assumption holds better when conditioned on precipitating situations (we refer to this as the revised equal MSE assumption below). Following their approach, Fig. 2e shows the MSE at each location over land with the corresponding precipitation-weighted zonal-mean values over the ocean (${[\text{MSE}]}_o^P$) subtracted. We can see that most of the moist conditions (to the right of the black line of the SM_crit or the green isoline of the precipitation rate 0.5 mm day⁻¹) approximately satisfy the revised equal MSE assumption. For dry conditions, land MSE still deviates from ${[\text{MSE}]}_o^P$ by more than 10 kJ kg⁻¹. This deviation is expected because, as mentioned above, these very dry conditions do not satisfy the QE assumption that relies on active moist convection.

QE–WTG suggests equal MSE over land and ocean in each of the base and the warm climate states, from which the equal change of MSE over land and ocean is derived. Figure 2f shows changes in the MSE deviation from the precipitation-weighted ocean zonal-mean value over land between the warm and the base climate states, i.e., $\mathrm{\Delta }(\text{MSE}-{[\text{MSE}]}_o^P)$, in CMIP6 models. Note that even in the moist conditions where the difference between land and ocean is close to 0 in the base climate (to the lower right of the black line in Fig. 2e), the MSE increase over land with climate change is still larger than the increase over the precipitating ocean. The magnitude of this deviation (difference between MSE increases over land and ocean) can be as large as 0.3–0.5 kJ kg⁻¹ (0.3–0.5 K in temperature units) for each degree of tropical mean ocean warming. Thus, while these assumptions are necessary to form the overarching theory of QE–WTG and have been used to successfully explain the behavior of climatological averages over large domains (e.g., land and ocean averages), their accuracy diminishes when considering the spatiotemporal distribution in this manner. In applying the QE–WTG framework to extreme temperatures at the daily time scale, Byrne (2021) added a term (Δh) to address the smaller changes in MSE during hot days over land that are evident in our phase space (Fig. 2f).

b. The surface perspective

For the surface flux perspective, changes in the total energy flux, i.e., the net radiation at the surface (R_n), are partitioned between changes in the surface sensible heat flux (ΔSH) and changes in the surface latent heat flux (ΔLH), with a negligible change in ground heat fluxes (not shown). Each term in the phase space in CMIP6 models is shown in Figs. 3a–c. The R_n increases under all spatiotemporal conditions of dryness and wetness, with the maximum increase in wet conditions (around 3 W m⁻² per degree of ocean warming) and the smallest increase over the desert (around 1 W m⁻² per degree of ocean warming). The pattern of ΔR_n across the spatiotemporal states is highly similar to the pattern of ΔMSE (Fig. 2a); their correlation across the 2500 bins in the phase space is 0.85. This close correspondence between the changes in the total energy in the surface air and changes in the total energy flux at the surface may be explained by the following two possibilities: one is that increases in R_n drive the increases in MSE; the other is that increases in q (Fig. 2c), which is the dominant component in ΔMSE (Fig. 2a), result in increases in R_n through the increased radiative emissivity of moister air.

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Changes in the surface (a) net radiation (R_n), (b) SH flux, and (c) LH flux, and (d) SM, (e) SH plus upwelling longwave radiation (SH + LW_up), and (f) RH between the 4 × CO₂ climate state and the base climate state, displayed in the phase space of daily SM percentiles (horizontal axis) and climatological AI percentiles (vertical axis). Changes in these variables are normalized by the mean tropical ocean warming ${[\mathrm{\Delta }\overline{T}]}_o$ in each model before averaging over the models. The black contours (solid for the base climate and dashed for the 4 × CO₂ climate) are the isolines of the surface SM value of 20 kg m⁻² which roughly marks the conditions in the transitional regime. See text for more details.

Citation: Journal of Climate 37, 18; 10.1175/JCLI-D-22-0955.1

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Except for the very dry conditions where changes in LH are near 0, and the transitional conditions around SM_crit where LH decreases and SH increases, changes in R_n are predominantly balanced by increases in LH. In particular, in the wet conditions (those to the right of SM_crit), LH increases by about 2–3 W m⁻² per degree of mean ocean warming, while SH changes much less (or even decreases), by ±1 W m⁻² per degree of mean ocean warming. Recall that in these same moist locations and times, increases in specific humidity also dominate the increases in MSE and the warming magnitude is similar to that over the ocean (Fig. 2b).

For the transitional conditions around SM_crit, decreases in LH in CMIP6 models are substantial (Fig. 3c). The decrease in LH is driven by the drying of SM (Fig. 3d; also indicated by shift of the SM_crit isoline to higher percentiles) and is compensated by strong increases in SH. The increase in SH and decrease in LH in the transitional conditions correspond to amplified warming in these conditions (Fig. 2b), reflecting the role of surface flux repartitioning in exacerbating the warming. It is interesting to note, however, that even though LH decreases in these transitional conditions, specific humidity increases (Fig. 2c). This reflects the role of other sources and sinks of surface air specific humidity besides the local land surface evapotranspiration. Meanwhile, RH decreases (Fig. 3f), suggesting that specific humidity has not increased enough to match the increase in the saturation specific humidity. RH is often taken as a reflection of atmospheric dryness; the decrease of RH along SM_crit despite the increase of the specific humidity indicates that soil moisture–temperature feedback is an important contributor to the decrease of RH over land.

In arid regions (upper quarter of the phase space), in the base climate state, LH is small (deserts are dry), and R_n is predominantly balanced by SH. With climate change, changes in LH will be small (deserts remain deserts), and we might expect large increases in SH, corresponding to the pronounced warming (Fig. 2a). However, we see here that changes in SH in these arid regions are also small. This is because both the surface and near-surface air warm substantially, and the increase of longwave radiation (which scales with T⁴) is faster than the increase of SH (which scales with T); i.e., upwelling longwave radiation (LW_up) is partly playing the role in the energy balance that is played by SH in other regions. Adding changes in LW_up, Fig. 3e reproduces the pronounced increases in the arid regions. Nevertheless, increases in R_n in these arid regions are small, and the surface energy balance (R_n ≈ SH + LH) is still obeyed. The comparisons in Figs. 3a–c and 3e are consistent with and supplement the discussion of surface flux changes under different regimes of the temperature distribution change in Duan et al. (2020).

c. A discussion on the moisture constraint

As mentioned in sections 1 and 3a, the stronger warming predicted over land in the QE–WTG framework emerges from the moisture constraint that Δq^L = γΔq^O, which is derived based on the transport of atmospheric moisture from the ocean (Byrne and O’Gorman 2016; Chadwick et al. 2016). While the derivation of this moisture constraint emphasizes ocean control—specific humidity over land is assumed to follow specific humidity over the ocean, which increases with warming approximately at the rate of the Clausius–Claperon scaling—land specific humidity is also affected by land–atmosphere exchanges (e.g., van der Ent et al. 2010). A drier land surface will tend to result in lower evapotranspiration and, for a given amount of moisture convergence (although the two do not operate independently), a smaller γ. When the moisture constraint is applied to climate model simulations, the value of γ is calculated as the local climatological q^L at each land grid cell divided by the corresponding zonal mean q^O (Byrne and O’Gorman 2016, 2018) and is spatially variable. Therefore, the moisture constraint (atmospheric perspective) contains information on the spatially variable surface dryness (surface perspective). Note that the results in Byrne and O’Gorman (2016, 2018) were presented as averages over latitudinal bands or the entire 40°S–40°N domain, which is an average across the underlying variability of γ.

We use the same method as in Byrne and O’Gorman (2016, 2018) to calculate γ on each day and present its spatiotemporal distribution in the SM/AI phase space in Fig. 4a. Examining across the spatiotemporal distribution, γ ranges from <0.2 in the very arid regions to about 1.2 in the wettest days (high local SM percentiles) in semiarid and semimoist regions. The lowest row (the lowest AI percentile bin) contains many grid cells in the upwind slope of the Tibetan Plateau where it rains heavily from the orographic lifting; therefore, AI is small (AI ∝ R_n/P), but the specific humidity (and γ) is not necessarily high (since temperature is low at high orographic altitude).

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(a) The land–ocean specific humidity ratio [the γ in Byrne and O’Gorman (2013, 2018), calculated as the specific humidity q at each land grid cell divided by the corresponding ocean zonal-mean specific humidity [q]_o]. (b) Fractional changes in γ normalized by the mean tropical ocean warming ${[\mathrm{\Delta }\overline{T}]}_o$, displayed in the phase space of daily SM percentiles (horizontal axis), and climatological AI percentiles (vertical axis). The black contour is the isoline of the surface SM value of 20 kg m⁻². (c) RH and the Bowen ratio B. Each dot represents one bin among the 50 × 50 spatial–temporal bins in the AI/SM phase space as in Figs. 3 and 5 and is color-coded by its SM value in the base climate. The blue/red dots in (c) show the theoretical scaling by the SFE, red using RH to calculate B and blue using B to calculate RH. See text for more details.

Citation: Journal of Climate 37, 18; 10.1175/JCLI-D-22-0955.1

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Under climate change, the moisture constraint assumes that the ratio γ remains constant, but this may not hold if evapotranspiration changes are substantial and not compensated by changes in moisture convergence. The change in γ is shown in Fig. 4b. We first draw the readers’ eye to the decrease of γ aligning along the critical soil moisture lines where changes in surface conditions are the strongest as emphasized by the surface perspective (see section 3b). In Fig. 4b, for each degree of warming, γ decreases by about 1% in the water-limited transitional regime for the CMIP6 multimodel mean. This suggests that when q^O increases by 7% K⁻¹, q^L in these conditions, due to the decrease of γ, will increase by 6% K⁻¹ (taking the log and then taking the climate change difference on both sides of q^L = γq^O yields Δq^L/q^L = Δq^O/q^O + Δγ/γ). For q^L ≈ 16 g kg⁻¹ (L_υq^L ≈ 40 kJ kg⁻¹; see Fig. A1c), the 1% change in γ will produce a deviation of about 0.4 kJ kg⁻¹ in L_υΔq^L (which, assuming an accurate constraint of ΔMSE by QE and WTG, leads to a 0.4-K deviation in the prediction of ΔT^L), for each degree of ocean mean warming. In applying the QE–WTG framework to predict the magnitude of land warming, this deviation brought by the moisture constraint assumption will partly compensate for the deviation brought by the equal MSE assumption (the MSE increase in these conditions is smaller; see Figs. 2a,f). Note that in these transitional conditions where γ decreases, LH also decreases (Fig. 3c). While we do not perform a moisture budget analysis here, this correspondence suggests the role of changing land-sourced moisture in controlling changes in γ. In Byrne (2021), where the QE–WTG framework is applied to discuss warming of the upper temperature percentiles, both the drier base-climate q and a decrease of RH on those high temperature percentile days contribute to the amplified warming, and both of these have connections to the land surface.

Another domain in the phase space in Fig. 4b that shows perceptible changes in γ is over the arid land regions: γ increases by a much larger fractional rate. The driving mechanism for this increase is not yet clear: at the top soil moisture percentiles in these arid regions, both rainfall and LH (see Fig. 3c) increase; the overall increase in γ and the moistening at the top soil moisture percentiles in these arid regions can result from both changes in the transport of moisture and local evapotranspiration.

Boundary layer RH in the base climate can be effectively regarded as a key indicator for the warming/moistening partition in the atmospheric perspective, similar to γ: while γ is the key parameter in the mathematical form of the QE–WTG framework, the physical intuition for the importance of base climate moisture in controlling warming magnitude draws upon the cloud-base height above which the temperature lapse rate shifts to follow a moist adiabat from a dry adiabat (see Fig. 1b). The cloud-base height is tightly related to the boundary layer RH (Betts 2009), and therefore, the base climate state of both can be regarded as the key control for the warming magnitude under the atmospheric perspective. In fact, when Byrne and O’Gorman (2013) first formed the QE–WTG framework based on equal equivalent potential temperature θ_e over land and ocean, the decomposition for the land/ocean warming ratio was based on RH. Altogether, the SFE relation in Eq. (9) that directly associates RH with the Bowen ratio is directly relating the key control in the atmospheric perspective to the key control in the surface perspective.

Figure 4c shows the relationship between RH and the Bowen ratio B for each box in the SM/AI phase space, with the color indicating the soil moisture. RH shows an inverse relationship with B, as expected. This inverse relationship suggests a strong land surface origin for the boundary layer RH: the surface influences RH by both controlling the source of water vapor through evapotranspiration and affecting temperature. The red and blue dots in Fig. 4c show the results when assuming SFE [Eq. (9); red dots use RH to calculate B and blue dots use B to calculate RH]. SFE qualitatively captures the inverse relationship between RH and B, but there are notable deviations from what is simulated by the climate models. For example, in wet conditions (green dots in Fig. 4c), SFE-based RH is higher than the simulated RH, while in dry conditions (brown dots in Fig. 4c), SFE-based RH is lower than the simulated RH. These deviations from the climate model may be expected with the assumptions being made by SFE, where atmospheric heat and moisture convergence are not making a net contribution. While these differences between SFE and CMIP6 make sense given the assumptions made in SFE, we have not compared with observations here, and it is possible that the real world may behave differently. For example, Chen et al. (2021) showed that the SFE-based annual-mean evapotranspiration calculated with CMIP6 model surface air temperature, specific humidity, and net radiation compares better with catchment observations than the evapotranspiration simulated by CMIP6 models. While some of these correspondences may seem intuitive, we make them explicit because soil moisture and its changes do not appear directly in the QE–WTG framework, while studies based on the surface flux perspective do not typically rely on the behavior of the large-scale transport of moisture.