1. Introduction

Changes in the severity and duration of extreme heat stress events are potentially one of the most harmful impacts of climate change, affecting human health and productivity, and also damaging crops and ecosystems, among many other negative impacts [see Carleton and Hsiang (2016) for discussions of the negative social and economic impacts of extreme heat stress]. For large-enough global-mean warming, increases in heat stress may even make large parts of the tropics uninhabitable by humans (Sherwood and Huber 2010).

Heat stress is a result of elevated temperature and moisture levels: high temperatures cause more heat to be input into the human body, while high levels of moisture limit the ability of the human body to cool through evaporation, the primary method by which it dissipates excess heat in warm climates (Koppe et al. 2004). Understanding changes in heat stress in warmer climates thus requires understanding how local temperature and moisture extremes change, and the relative contributions each of these makes to the total change in heat stress.

A warmer climate will have hotter warm-temperature extremes, but it is less clear how changes in moisture will affect heat stress. Simple conceptual models suggest that near-surface relative humidity decreases over land with warming (Byrne and O’Gorman 2016), and this is robustly seen in observations and climate model simulations (Simmons et al. 2010; Byrne and O’Gorman 2013, 2018). In terms of specific humidity (q_υ), the Clausius–Clapeyron relation implies that q_υ will increase by roughly 7% °C⁻¹ over oceans (where relative humidity changes are small), but the expected relative humidity decreases over land mean that q_υ will likely increase more slowly than 7% °C⁻¹. Instead, changes in q_υ over land can be well approximated by assuming the same fractional changes in specific humidity over land as over the ocean source for the land moisture (Chadwick et al. 2016). But these conceptual models of how specific and relative humidity change over land have yet to be connected to changes in heat stress over land, and it is also unclear whether relative or specific humidity is more relevant for quantifying heat stress changes.

Uncertainty in the drivers of heat stress changes is partly a result of the variety of different heat stress metrics, which place differing emphases on the role of moisture (Buzan et al. 2015; Mora et al. 2017; Sherwood 2018). In the present-day climate, some metrics, such as the wet-bulb temperature (T_w), suggest that low-latitude heat stress extremes are primarily caused by moisture, while other metrics, such as the United States National Weather Service’s heat index, suggest that tropical and subtropical heat stress extremes are mostly due to elevated temperature (Buzan et al. 2015; Zhao et al. 2015). Still other metrics, such as the simplified wet-bulb globe temperature, show roughly equal contributions from temperature and moisture (Buzan et al. 2015). At a regional scale, Raymond et al. (2017), using T_w as their metric for heat stress, found that moisture extremes tend to dominate heat stress extremes over North America in the present climate, while Wang et al. (2019) showed that the relative contributions of temperature and moisture to T_w extremes over China varies region by region.

Changes in temperature and humidity covary in climate models, such that intermodel spread in the response of heat stress metrics such as T_w is smaller than if the intermodel spreads in the temperature and (relative or specific) humidity responses were independent (Fischer and Knutti 2012; Buzan and Huber 2020). The covariation of changes in temperature and moisture (conditioned on extreme heat stress events) is partly explained by the fact that extreme heat stress generally occurs in the summer, when the atmospheric state is largely determined by convection. Since most atmospheric profiles are close to moist convective neutrality in summer, this places bounds on the possible combinations of temperature, moisture, and pressure that can be expected at upper percentile heat stress levels for a given climate state (Buzan and Huber 2020; Zhang and Fueglistaler 2020). The limited set of possible temperature and moisture values means, for example, that the intensification of extreme warm events is projected to be associated with a reduction in the relative humidity associated with these events (Coffel et al. 2019). Although the allowable set changes with climate, constituting a “movable limit,” convective neutrality provides a useful first-order constraint on the possible combinations of temperature and moisture for a given climate.

A simple model of the response of heat stress extremes to warming was proposed by Willett and Sherwood (2012), who assumed a uniform shift of summertime simplified wet-bulb globe temperature (W) and fixed relative humidity to predict changes in regional W extremes. While this model was able to produce a reasonable match to observed W trends over many land regions, the assumption of fixed relative humidity during extreme W events is questionable over land, and the model does not provide an explicit separation of the relative contributions of temperature and moisture. So the relative contributions of temperature and moisture to heat stress changes have still to be separated and quantified.

In this study, conditions are derived on the baseline specific humidity for determining when changes in temperature or in specific humidity can be expected to dominate heat stress changes, with separate conditions over ocean and over land. A further condition is derived for when local relative humidity changes are the main contributor to heat stress changes, rather than temperature changes at fixed relative humidity. The arguments focus on equivalent potential temperature (θ_E), which is closely related to moist static energy and is conserved under moist pseudoadiabatic ascent, making it the most physically intuitive variable for theoretical investigations of heat stress. Also, θ_E is amenable to analysis, in contrast to some heat stress metrics that involve a large number of empirical constants, and several other heat stress metrics, such as T_w, scale with θ_E, or at least are strongly influenced by θ_E changes (see appendix A). Finally, note that using θ_E emphasizes specific humidity as the relevant moisture variable.

The theory is presented in the following section. Section 3 then investigates seasonal θ_E changes in 14 models participating in phase 6 of the Climate Model Intercomparison Project (CMIP6). Included in this section are investigations of the sources of uncertainty in θ_E changes, and of whether the baseline specific humidity can be used to develop emergent constraints on the response of seasonal-mean θ_E. Changes in extreme (98th-percentile) θ_E events are discussed in section 4, and the relative contributions of temperature and specific humidity to observed changes in θ_E are examined in section 5. The study ends with conclusions in section 6.

2. Theory

a. Over ocean

Equivalent potential temperature can be approximated as (Holton and Hakim 2013)

${\theta }_e\approx \theta \exp \left({\displaystyle \frac{L{q}_{\upsilon }}{c_p{T}_L}}\right),$(1)

where θ is potential temperature, L is the latent heat of warming, q_υ is the mixing ratio of water vapor (approximately equal to the specific humidity), c_p is the heat capacity of dry air, and T_L is the temperature at the lifting condensation level. Fractional changes in near-surface equivalent potential temperature can be further approximated as

${\displaystyle \frac{\text{Δ}{\theta }_e}{{\theta }_e}}\approx {\displaystyle \frac{\text{Δ}\theta }{\theta }}+{\displaystyle \frac{L}{c_{p}T}}\text{Δ}{q}_{\upsilon },$(2)

where the second-order T_L term is ignored, and T_L is approximated by the surface temperature T.

If near-surface relative humidity is assumed to stay fixed with warming (Held and Soden 2006), then (Δq_υ/q_υ) ≈ 0.07°C⁻¹ΔT, and substitution into Eq. (2) gives

${\displaystyle \frac{\text{Δ}{\theta }_e}{{\theta }_e}}\approx {\displaystyle \frac{\text{Δ}\theta }{\theta }}+0.07\mathrm{°}{\text{C}}^{-1}{q}_{\upsilon }{\displaystyle \frac{L}{c_p}}{\displaystyle \frac{\text{Δ}T}{T}}={\displaystyle \frac{\text{Δ}\theta }{\theta }}+174q_{\upsilon }{\displaystyle \frac{\text{Δ}T}{T}},$(3)

where q_υ now denotes the baseline specific humidity; L_υ is set to 2.5 × 10⁶ J kg⁻¹ and c_p to 1005 J kg⁻¹ °C⁻¹, so that 0.07 °C⁻¹ × L_υ/c_p ≈ 174. Assuming fractional changes in surface potential temperature are roughly equal to fractional changes in surface temperature (i.e., that surface pressure changes are small; see appendix B), the moisture term will be the primary driver of the fractional change in θ_e wherever

$q_{\upsilon }>{\displaystyle \frac{1}{174}}\approx 5.6{\text{gkg}}^{-1}=q_{\upsilon ,0}.$(4)

This is a low baseline specific humidity threshold, which is met throughout most of the tropics, subtropics, and midlatitudes in summer months (see Fig. 1d). As shown in appendix A, heat stress metrics that weight temperature more heavily, such as the simplified wet-bulb globe temperature, can have substantially higher threshold q_υ,₀ values; however, even these higher thresholds are met in most of the tropics.

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(a) Composite changes in JJA θ_E between years 71–80 and years 1–10 in transient warming simulations with 14 CMIP6 models. (b) Composite changes in JJA θ. (c) Composite changes in JJA q_υ, multiplied by L_υ/c_p. (d) Composite of JJA q_υ, averaged over years 1–10 of the simulations. The cyan contours show the 5.6 g kg⁻¹ isopleths.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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For a change in relative humidity of ΔRH, Eq. (3) is modified to

${\displaystyle \frac{\text{Δ}{\theta }_e}{{\theta }_e}}\approx {\displaystyle \frac{\text{Δ}\theta }{\theta }}+{\displaystyle \frac{L}{c_{p}T}}\left(0.07\mathrm{°}{\text{C}}^{-1}{q}_{\upsilon }\text{Δ}T+\text{Δ}\mathrm{RH}{q}_{\upsilon }^{\text{*}}\right)\approx {\displaystyle \frac{\text{Δ}\theta }{\theta }}+{\displaystyle \frac{q_{\upsilon }}{T}}\left(174\text{Δ}T+2490\mathrm{°}\text{C}{\displaystyle \frac{\text{Δ}\mathrm{RH}}{\mathrm{RH}}}\right),$(5)

where $q_{\upsilon }^{\text{*}}$ is the saturation specific humidity. This gives the new approximate condition for moisture to dominate θ_E changes:

$q_{\upsilon }>\left|{\displaystyle \frac{1}{174+{\displaystyle \frac{2490\mathrm{°}\text{C}}{\text{Δ}\theta }}{\displaystyle \frac{\text{Δ}\mathrm{RH}}{\mathrm{RH}}}}}\right|=q_{\upsilon ,0}.$(6)

For an initial relative humidity of 80%, a temperature increase of 2 K, and an increase in relative humidity of 1%,

$q_{\upsilon ,0}={\displaystyle \frac{1}{189}}\approx 5.3{\text{g kg}}^{-1}.$

Note that because ΔRH and Δθ may have opposite signs, the two terms in the denominator of Eq. (6) can cancel, causing q_υ,₀ to be undefined. This happens when a change in relative humidity exactly cancels the Clausius–Clapeyron change. The line of “critical” relative humidity and temperature changes is defined by

${\displaystyle \frac{\text{Δ}\mathrm{R}{\mathrm{H}}_c}{\text{Δ}{\theta }_c}}=-{\displaystyle \frac{174}{2490\mathrm{°}\text{C}}}\mathrm{RH}\approx -0.07\mathrm{°}{\text{C}}^{-1}\mathrm{RH}.$(7)

Figures 2a and 2b show q_υ,₀ for changes in relative humidity of −10% to +10% and temperature changes from −2° to +10°C, at baseline relative humidities of 60% (Fig. 2a) and 80% (Fig. 2b). The value of q_υ,₀ is very large in a band that stretches from the upper left quadrant of the figure down to the lower-right quadrant, for which Δθ ≈ Δθ_c and ΔRH ≈ ΔRH_c. q_υ,₀ decreases moving away from this band, with the largest decreases when increasing ΔRH at fixed Δθ, and q_υ,₀ is small for temperature changes close to 0 and lower for a higher baseline relative humidity.

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(a) The baseline specific humidity q_υ,0 above which moisture changes dominate changes in θ_E over ocean as a function of Δθ and ΔRH, for a baseline relative humidity of 60%. The values of q_υ,0 are calculated using Eq. (6). (b) As in (a), but assuming a baseline relative humidity of 80%. (c) As in (a), but showing the baseline specific humidity q_υ,0 over land [Eq. (11)], assuming a land warming amplification factor A of 1.5. (d) As in (c), but assuming a baseline relative humidity of 80%. In all panels, the gray shading denotes q_υ,0 values outside the color bar scale.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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3. Seasonal θ_E changes

To investigate the relative importance of changes in temperature and in specific humidity for θ_E changes, data were taken from simulations with 14 climate models participating in CMIP6 in which CO₂ concentrations were increased at 1% yr⁻¹ (see Table 1 for list of models).¹ For each simulation, Δθ_E, Δθ, and Δq_υ were calculated by taking the difference between averages over years 1–10 and over years 70–80. The value of θ_E was estimated using Eq. (1), with temperature at the lifting condensation level calculated using Eq. (21) of Bolton (1980), and multimodel composites were generated by linearly interpolating all of the model responses onto the same 2.5° × 2.5° grid.

Table 1.

r² values for correlations between JJA Δθ_E and JJA Δθ, between JJA Δθ_E and JJA Δq_υ, and between JJA Δθ_E and JJA q_υ, as well as r² values for the same correlations taken over land regions only. Boldface values have a p value less than 0.025, which gives an estimate of the statistical significance of the correlations.

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Note that the derivations in section 2 started by considering fractional changes in θ_E, but in the analysis of the CMIP6 data I focus on the absolute changes in θ_E to emphasize tropical regions, where the greatest risks of heat stress are found. (Results are shown at all latitudes, rather than in the tropics only, to illustrate the different regimes identified by the theory.) The conditions on q_υ for where moisture is the main driver θ_E changes [Eqs. (4), (6), (10), and (11)] can also be derived by starting from the equation for the absolute changes in θ_E, although the derivation is clearer when starting from the fractional change, and the final conditions are independent of whether fractional or absolute θ_E changes are considered.

a. Boreal summer

The boreal summer [June–August (JJA)] multimodel composite clearly shows that changes in moisture are the primary driver of the pattern of changes in equivalent potential temperature (cf. Figs. 1a and 1c). For example, there are large increases in θ_E over equatorial Africa, particularly along the coastline of the Bay of Guinea, and smaller increases over the Sahara, which match the pattern of specific humidity changes. By contrast, the potential temperature changes over Africa are much more uniform (Fig. 1b). Another notable example is in southwest North America, where there is a region of small q_υ and θ_E changes stretching southwest–northeast from Baja California into Arizona and New Mexico. This feature is not seen in the potential temperature field. Also, Δθ_E and Δq_υ are highly correlated throughout the tropical and midlatitude oceans.

To quantify the correlations, Table 1 gives r² values for pattern correlations between Δθ_E and Δq_υ, and between Δθ_E and Δθ. Whereas Δθ_E and Δq_υ are very highly correlated in the multimodel composite (r² = 0.79), and the average r² value across the individual models is 0.76, the correlation between Δθ_E and Δθ is weak (r² = 0.07) in the multimodel composite, although the correlation with Δθ tends to be higher in individual models (average r² = 0.31). Similar results are obtained when the correlations are taken over land areas only (columns 5 and 6 of Table 1), but the correlations with Δq_υ,L are generally higher and the correlations with Δθ_L generally lower. Taking correlations over tropical regions only (30°S to 30°N) further increases the correlations with Δq_υ and reduces the correlations with Δθ (not shown).

Figure 1d shows the multimodel composite of q_υ averaged over years 1–10 of the simulations, which is used as the baseline specific humidity. This is well correlated with Δθ_E in the multimodel composite (r² = 0.62), but the correlation is lower in individual models, roughly similar to the correlation with Δθ (average r² = 0.33). Considering land areas only improves these correlations (r² = 0.73 in the multimodel composite and r² averaged over all models = 0.38). Chadwick et al. (2016) showed that the approximation of Eq. (8) is less accurate in individual models than in the ensemble composite, because model disagreements in circulation changes and land surface feedbacks are averaged out in the composite, which likely explains why the correlation between q_υ and Δθ_E is higher in the multimodel composite.

The cyan contours in Fig. 1d indicate the 5.6 g kg⁻¹ isopleth, for which moisture changes will dominate θ_E changes over ocean if relative humidity is fixed. The areas with baseline specific humidities below this threshold include high-latitude oceans and desert regions (the Sahara, Arabia, the Kalahari, etc.). For example, the strong warming seen in the Southern Ocean leads to large θ_E changes there, despite small changes in q_υ (Fig. 1). Over land the temperature-dominated areas will be larger than the area enclosed by the cyan contours because the specific humidity threshold is higher.

Figure 3a quantifies the relative contributions of temperature and moisture by showing the ratio Q = (L_υΔq_υ)/(c_pΔθ) for the multimodel JJA composite. Over oceans there is close agreement with the theory, as the red contours in Fig. 3a, which denote where Q = 1, closely match the green contours, which show the 5.6 g kg⁻¹ isopleth again. Over land, Q < 1 over desert regions, with a larger extent than predicted by the green contours, and is also less than one over much of Europe and central Asia, central Brazil, and central India. Experimenting with other contour levels suggests that q_υ,0 varies over land between 5 and 10 g kg⁻¹, with substantial regional variations. For example, the North Atlantic experiences the slowest warming of any region, while Europe warms at a similar rate to other land regions at the same latitude (Fig. 1b). This results in a large amplification factor over Europe, so that temperature is the main driver of the θ_E response even though the baseline specific humidity is relatively high, and q_υ,₀ ≈ 9 g kg⁻¹ over Europe. By contrast, over Australia, southern Africa, and the southern part of South America the Q = 1 contours closely follow the q_υ,₀ = 5.6 g kg⁻¹ contours. These variations in q_υ,₀ highlight the role of regional factors in determining the relative contributions of humidity and temperature to heat stress extremes over land.

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(a) The ratio Q = (L_υΔq_υ)/(c_pΔθ) for the multimodel composite response of JJA θ_E in 14 CMIP6 models. (b) As in (a), but for DJF. In both panels, the red contours show the Q = 1 level and the green contours show the 5.6 g kg⁻¹ isopleths of q_υ, averaged over years 1–10.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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In summary, although temperature changes dominate the local changes in JJA θ_E over certain land regions, particularly over Eurasia, moisture changes are still the primary driver of the pattern of Δθ_E. This is because of the much larger regional variation of Δq_υ (cf. Fig. 2b and 2c), so that changes in θ_E can be approximated as coming from a spatially homogeneous distribution of potential temperature changes and a spatially heterogeneous pattern of specific humidity changes:

$\text{Δ}{\theta }_E\left(x,y\right)\approx {\theta }_E\left(x,y\right)\left[{\displaystyle \frac{\text{Δ}\theta }{\theta }}+{\displaystyle \frac{L}{c_{p}T}}\text{Δ}{q}_{\upsilon }\left(x,y\right)\right].$(13)

The greater spatial variation of Δq_υ reflects the much larger range of fractional changes in q_υ compared to fractional changes in θ: at constant relative humidity a warming of 1°C leads to a 7% increase in specific humidity, but only a ~0.33% °C⁻¹ (=1/300 K) increase in temperature. When relative humidity changes are accounted for, fractional changes in specific humidity can vary from <0% °C⁻¹ to more than 7% °C⁻¹, whereas the largest fractional changes in temperature will always be less than 1% °C⁻¹. Even over the oceans, where relative humidity changes are small and temperature is the main driver of the θ_E response (see section 2c), changes in relative humidity are sufficient for the pattern of Δθ_E to be more similar to the pattern of Δq_υ than the pattern of Δθ. Only in cold and/or very dry climates are larger absolute changes in θ able to overcome the larger fractional changes in q_υ.

b. Austral summer

Similar results are obtained in other seasons (i.e., the pattern of Δθ_E is primarily determined by the pattern of Δq_υ), with the notable exception of austral summer [December–February (DJF); Fig. 4]. In this season the strong Arctic amplification of warming, combined with the dryness of high-latitude winter climates, means that Δθ_E is mostly determined by Δθ at high northern latitudes and over much of the Northern Hemisphere continents (North America and Eurasia). The pattern correlations between Δθ_E and Δθ are higher in DJF, while the correlation between Δθ_E and Δq_υ are lower (not shown). Heat stress extremes are very unlikely to occur in these regions during boreal winter, but this example illustrates that Δθ can play a more important role in determining the pattern of Δθ_E in cold, dry climates.

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(a) Composite changes in DJF θ_E between years 71–80 and years 1–10 in transient warming simulations with 14 CMIP6 models. (b) Composite changes in DJF θ. (c) Composite changes in DJF q_υ, multiplied by L_υ/c_p. (d) Composite of DJF q_υ, averaged over years 1–10 of the simulations. The cyan contours show the 5.6 g kg⁻¹ isopleths. In all panels, the gray shading denotes values outside the color bar scales.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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Farther south, the θ_E changes in sub-Saharan Africa and South America are primarily caused by moisture changes, and in general the 5.6 g kg⁻¹ threshold accurately separates regions dominated by temperature changes and regions dominated by moisture changes, even over land (cf. red and green contours in Fig. 3b).² So in most places that are likely to experience heat stress extremes in austral summer, changes in specific humidity are still the key factor in seasonal θ_E changes.

c. Sources of uncertainty in Δθ_E

Uncertainty (intermodel spread) in Δθ_E is due to uncertainties in Δθ and Δq_υ. To quantify the contributions of Δθ and Δq_υ to uncertainty in Δθ_E, Fig. 5 shows squared semipartial correlation coefficients for correlations taken across models between Δθ_E and Δθ at each grid point (left column) and between Δθ_E and Δq_υ at each grid point (right column). The squared semipartial correlation coefficient between Δθ_E and Δθ is calculated as

$r_{\left(\text{Δ}{\theta }_E,\text{Δ}\theta \right)|\text{Δ}{q}_{\upsilon }}^2={\displaystyle \frac{{\left[r_{\left(\text{Δ}{\theta }_E,\text{Δ}\theta \right)}-r_{\left(\text{Δ}{\theta }_E,\text{Δ}{q}_{\upsilon }\right)}{r}_{\left(\text{Δ}\theta ,\text{Δ}{q}_{\upsilon }\right)}\right]}^2}{1-r_{\left(\text{Δ}\theta ,\text{Δ}{q}_{\upsilon }\right)}^2}},$(14)

where r_A,B is the correlation coefficient between A and B (Abdi 2007), and an analogous expression is used for calculating $r_{\left(\text{Δ}{\theta }_E,\text{Δ}{q}_{\upsilon }\right)|\text{Δ}\theta }^2$. Statistical significance is estimated using an F test with degrees of freedom d1 = 1 and d2 = 14 − 2 = 12 (Abdi 2007).

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(a) Semipartial correlation coefficients for correlations taken across the CMIP6 models between JJA Δθ_E and JJA Δθ. (b) Semipartial correlation coefficients for correlations taken across the CMIP6 models between JJA Δθ_E and JJA Δq_υ. (c) As in (a), but for MAM. (d) As in (b), but for MAM. (e) As in (a), but for SON. (f) As in (b), but for SON. (g) As in (a), but for DJF. (h) As in (b), but for DJF. In all panels, red stippling shows semipartial correlations that are significant at the 95% level, as estimated using an F test (see main text).

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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$r_{\left(\text{Δ}{\theta }_E,\text{Δ}\theta \right)|\text{Δ}{q}_{\upsilon }}^2$ measures the correlation between variations in Δθ_E and variations in Δθ that are independent of Δq_υ. That is, $r_{\left(\text{Δ}{\theta }_E,\text{Δ}\theta \right)|\text{Δ}{q}_{\upsilon }}^2$ removes the component of the correlation between Δθ_E and Δθ coming from correlations across models between temperature and humidity. For example, $r_{\left(\text{Δ}{\theta }_E,\text{Δ}\theta \right)|\text{Δ}{q}_{\upsilon }}^2$ will be small at a grid point where relative humidity is approximately fixed in all the models. [Note that since Δθ_E is roughly equal to a linear combination of Δθ and Δq_υ (ignoring surface pressure changes), the partial correlations between Δθ_E and Δθ, and between Δθ_E and Δq_υ are both ~1. That is, the correlation between variations in Δθ_E that are uncorrelated with variations in Δq_υ and variations in Δθ_E that are uncorrelated with variations in Δq_υ is very high, and vice versa. For this reason, semipartial correlations are used here, rather than partial correlations.]

Comparing the left and right columns of Fig. 5 shows that at most tropical and subtropical land locations Δq_υ contributes to much more uncertainty in Δθ_E than does uncertainty in Δθ. This includes much of South America, sub-Saharan Africa, India, Southeast Asia, and Australia. Exceptions include the northern Amazon in DJF (the dry season; Fig. 5g), the Sahara outside of boreal summer, and southern Australia in September–November (SON).

At higher latitudes, the semipartial correlation coefficients for both Δθ and Δq_υ are generally low over much of North America and Eurasia, suggesting that temperature and humidity changes are well correlated across models in these regions. The primary exceptions are seen in JJA, when $r_{\left(\text{Δ}{\theta }_E,\text{Δ}{q}_{\upsilon }\right)|\text{Δ}\theta }^2$ is high over much of Europe, central Asia, and North America, while $r_{\left(\text{Δ}{\theta }_E,\text{Δ}\theta \right)|\text{Δ}{q}_{\upsilon }}^2$ is weaker, though still notable in these regions. Also, Δθ is well correlated with Δθ_E over the Tibetan Plateau in boreal winter (Fig. 5g). The semipartial correlation coefficients are small over most ocean regions, where relative humidity changes are small.

d. Potential for emergent constraints

The correlations between baseline specific humidity and θ_E changes seen in Fig. 2 and quantified in Table 1 hint at the potential for emergent constraints between present-day specific humidity and changes in seasonal-mean θ_E with warming. To investigate this, r² values were calculated for correlations across models between the baseline q_υ and Δθ_E (Fig. 6). Values are only shown over land for ease of presentation and because these regions are of most societal relevance.

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(a) r² values for correlations across the CMIP6 models between baseline JJA q_υ (i.e., averaged over years 1–10) and JJA Δθ_E. Only values over land, with r² > 0.1, are plotted. (b) As in (a), but for MAM. (c) As in (a), but for SON. (d) As in (a), but for DJF values. In all panels, red stippling shows regions where the p value of the correlation is less than 0.025.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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In JJA, the baseline specific humidity is poorly correlated with Δθ_E at most locations (Fig. 6a), with few regions where the correlation is statistically significant (here defined as having a p value <0.025). There are patches of high r² values in equatorial Africa, western South America, parts of the Amazon, and over Pakistan. The results of correlations for other seasons are shown in the rest of the figure, and are similarly patchy, with few large regions of high r² values. This is consistent with the lower correlation between baseline q_υ and Δθ_E in individual models compared to the multimodel composite, and the real world would probably be expected to behave more like a single model realization. South America has patches of high r² values in DJF and, interestingly, the warming over much of North America and Eurasia is also well correlated with the baseline q_υ in DJF. This suggests that the amplitude of polar amplification could be constrained by the present-day specific humidity, though much more work is needed to investigate this possibility further.

Similar results are obtained when Δθ_E is normalized by the global-mean surface warming ($\text{Δ}\overline{\theta }$ or $\text{Δ}\overline{T}$) in each model or by local warming [Δθ(x, y)]. Henc e the baseline specific humidity seems to be a poor predictor of future θ_E changes over land. Intermodel variations in the land warming amplification factor (A), in the ratio of land specific humidity to ocean specific humidity (γ), in ΔA and Δγ, and in relative humidity changes could all weaken the connection between baseline specific humidity and Δθ_E in models. At fixed relative humidity, the ratio of the fractional change in θ_E to the fractional change in θ is proportional to q_υ:

$\left({\displaystyle \frac{\text{Δ}{\theta }_E}{{\theta }_E}}\right)/\left({\displaystyle \frac{\text{Δ}\theta }{\theta }}\right)\approx 1+174q_{\upsilon }.$

This could be used to constrain Δθ_E over ocean regions, given local fractional temperature changes, but will not hold over land regions with substantial relative humidity changes.

4. Changes in extreme events

Changes in extreme-θ_E events may be as important as seasonal-mean changes, but the combination of factors driving changes in extreme-θ_E events is likely to be more complex. For example, the assumption that fractional changes in moisture over land are equal to the fractional changes in moisture over the relevant oceanic moisture sources may not hold on the synoptic time-scales of extreme heatwaves. Furthermore, soil moisture feedbacks, which are ignored in the theory of section 2, often play a key role in extreme heat stress events (e.g., Diffenbaugh et al. 2007; Donat et al. 2017). Over oceans, the relative humidities associated with high-θ_E events may also have much larger responses to warming than seasonal-mean relative humidities. Nevertheless, the rapid increase of specific humidity with temperature, particularly at warmer temperatures, suggests that specific humidity changes are also likely to be the main driver of extreme-θ_E changes.

To investigate the roles of temperature and moisture in changing extreme-θ_E events, the analysis of the previous section was repeated for changes in the 98th percentile³ of the annual distribution of daily θ_E (Δθ_E,98), with Δθ and Δq_υ conditioned on these extreme events (Δθ₉₈ and Δq_υ,₉₈, respectively).⁴ Comparing Figs. 1a and 7a, the magnitudes of Δθ_E,98 are comparable to the magnitudes of JJA Δθ_E, but Δθ_E,98 is more spatially uniform, with similar increases over most land locations, whereas JJA Δθ_E is more tropically amplified.

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(a) Composite changes in the 98th percentile of daily θ_E between years 71–80 and years 1–10 in transient warming simulations with 14 CMIP6 models. (b) Composite changes in θ, conditioned on the 98th percentile of θ_E. (c) Composite changes in q_υ, multiplied by L_υ/c_p and conditioned on the 98th percentile of θ_E. (d) Baseline q_υ, conditioned on the 98th percentile of θ_E, averaged over years 1–10 of the simulations. The cyan contours show the 5.6 g kg⁻¹ isopleth.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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Just as for the seasonal-mean changes, the pattern of Δθ_E,98 closely resembles the pattern of specific humidity changes (Fig. 7c). For example, the largest increases in Δθ_E,98 and in Δq_υ,₉₈ over North America are in the Hudson Bay region, with the smallest increases over the southwestern United States and northwestern Mexico. Also, Δθ₉₈ is more uniform across North America (Fig. 7b), and generally has a smaller magnitude than Δq_υ,₉₈. Table 2 confirms this qualitative picture, as Δθ_E,98 is very highly correlated with Δq_υ,₉₈ (r² = 0.94 in the multimodel composite, 0.90 in the multimodel mean) and less well correlated with Δθ₉₈ (r² = 0.30 in the composite, 0.37 in the multimodel mean). Correlations taken over land regions only are similar for Δq_υ,₉₈, but lower for Δθ₉₈.

Table 2.

r² values for correlations between Δθ_E,98 and Δθ₉₈, between Δθ_E,98 and Δq_υ,₉₈, and between Δθ_E,98 and q_υ,₉₈, as well as r² values for the same correlations taken over land regions only. Boldface values have a p value less than 0.025, which gives an estimate of the statistical significance of the correlations.

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Δθ_E,98 is also well correlated with the baseline q_υ,₉₈ (q_υ conditioned on θ_E,98 and averaged over years 1–10) in the multimodel composite (Fig. 7d), with an r² of 0.65. The correlations are lower for individual models (multimodel mean r² = 0.47), and are similar when taken over land regions only. As with the seasonal-mean θ_E changes, correlations across models between q_υ,₉₈ and Δθ_E,98 indicate that the conditional baseline specific humidity is a poor constraint on changes in extreme heat stress events at most land locations (not shown).

Even more than the changes in seasonal Δθ_E, moisture primarily determines the response of extreme-θ_E events, so that Q₉₈ = (L_υΔq_υ,98)/(c_pΔθ₉₈) > 1 at almost all locations in the tropics, subtropics, and midlatitudes (Fig. 8). Exceptions are the Iberian Peninsula, parts of North Africa, central Asia, and the southern tip of South America. Extreme-θ_E events in these regions are all associated with specific humidities <10 g kg⁻¹ in the baseline climate (Fig. 7d). Also, Q₉₈ is less than one at high latitudes, where the green contours in Fig. 8 separate regions of q_υ,₉₈ > 5.6 g kg⁻¹ from regions where q_υ,₉₈ < 5.6 g kg⁻¹, and closely match the red Q = 1 contours.

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The ratio Q₉₈ = (L_υΔq_υ,98)/(c_pΔθ₉₈) for the multimodel composite response of the 14 CMIP6 models. The red contours show the Q₉₈ = 1 level and the green contours show the 5.6 g kg⁻¹ isopleths of q_υ, averaged over years 1–10.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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As another way of demonstrating the importance of specific humidity changes for extreme events, Fig. 9 plots the conditional specific humidity and temperature changes for locations over land where the 98th percentile of θ_E in the control climate is above 308 K (≈35°C) in the 14 CMIP6 models.⁵ The spread in the conditional specific humidity changes is larger than the spread in the conditional temperature changes in almost all of the models, with the exception of some grid points in the CanESM5 model. Inspection of the maps of Δθ_E,98, Δθ₉₈, and Δq_υ,₉₈ for CanESM5 shows that these grid points lie over the Tibetan Plateau, which experiences large temperature increases during warm, dry days in this model. Otherwise, changes in the very warmest θ_E events are associated with large q_υ,₉₈ responses. For these extreme heat stress events, specific humidity is again the leading factor driving the response to climate change.

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Scatterplots for the 14 CMIP6 models of changes in specific humidity (L_υΔq_υ,98/c_p) vs changes in temperature (Δθ₉₈) associated with 98th-percentile θ_E events that are ≥308 K. The markers are colored by their associated θ_E,98 value in the baseline climate.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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Sources of uncertainty in Δθ_E,98

Specific humidity changes are the primary control on the pattern Δθ_E,98, suggesting that they also control the intermodel spread, or uncertainty, in Δθ_E,98. Figure 10 repeats the calculations of Fig. 5, but now shows semipartial correlations across models between Δθ_E,98 and Δθ_E,98 and between Δθ_E,98 and Δq_υ,₉₈. Once again, Δq_υ,₉₈ explains a majority of the intermodel spread in Δθ_E,98 over most land locations, with high $r_{\left(\text{Δ}{\theta }_{E,98},\text{Δ}{q}_{\upsilon ,98}\right)|\text{Δ}{\theta }_{98}}^2$ and low $r_{\left(\text{Δ}{\theta }_{E,98},\text{Δ}{\theta }_{98}\right)|\text{Δ}{q}_{\upsilon ,98}}^2$ values. Two exceptions are northeastern South America, where both semipartial correlation coefficients are small, suggesting that temperature and specific humidity changes associated with extreme θ_E events are correlated across models, and the Tibetan Plateau, where $r_{\left(\text{Δ}{\theta }_{E,98},\text{Δ}{\theta }_{98}\right)|\text{Δ}{q}_{\upsilon ,98}}^2$ is larger than $r_{\left(\text{Δ}{\theta }_{E,98},\text{Δ}{q}_{\upsilon ,98}\right)|\text{Δ}{\theta }_{98}}^2$, mostly due to the CanESM5 model. The semipartial correlation coefficients are generally small over the oceans, though there is evidence of notable relative humidity changes in parts of the tropical and subtropical oceans, with $r_{\left(\text{Δ}{\theta }_{E,98},\text{Δ}{q}_{\upsilon ,98}\right)|\text{Δ}{\theta }_{98}}^2$ approaching 0.4.

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(a) Semipartial correlation coefficients for correlations taken across the CMIP6 models between Δθ_E,98 and Δθ₉₈. (b) Semipartial correlation coefficients for correlations taken across the CMIP6 models between Δθ_E,98 and Δq_υ,98. In both panels, red stippling shows semipartial correlations that are significant at the 95% level, as estimated using an F test (see main text).

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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Uncertainty in extreme θ_E changes can also be related to uncertainty in seasonal-mean θ and q_υ. High semipartial correlations between Δθ_E,98 and JJA Δq_υ are seen throughout much of North America, Europe, India, and the Sahel (Fig. 11b), suggesting that progress can be made in constraining the uncertainty in extreme heat stress events in these regions by constraining the response of boreal summer specific humidity. Similarly, the semipartial correlation is high between Δθ_E,98 and DJF Δq_υ in South Africa, Australia, and the southern half of South America (Fig. 11h). The responses of MAM and SON q_υ are well correlated with Δθ_E,98 in equatorial Africa (Figs. 11d,f), and the responses of MAM and SON θ are well correlated with Δθ_E,98 in the northeastern Amazon (Figs. 11c,e).

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(a) Semipartial correlation coefficients for correlations taken across the CMIP6 models between Δθ_E,98 and JJA Δθ. (b) Semipartial correlation coefficients for correlations taken across the CMIP6 models between Δθ_E,98 and JJA Δq_υ. (c) As in (a), but for MAM Δθ. (d) As in (b), but for MAM Δq_υ. (e) As in (a), but for SON Δθ. (f) As in (b), but for SON Δq_υ. (g) As in (a), but for DJF Δθ. (h) As in (b), but for DJF Δq_υ. In all panels, red stippling shows semipartial correlations that are significant at the 95% level, as estimated using an F test (see main text).

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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These results should be interpreted with caution, as few of the relationships are statistically significant at the 95% level. Nevertheless, these calculations suggest that intermodel variations in seasonal-mean Δq_υ during the hottest and most humid parts of the year (boreal summer in North America, austral summer in South Africa, the rainy season in central Africa, etc.) explain much of the intermodel variations in the response of extreme θ_E. The exception is the northeastern Amazon, where changes in θ explain more intermodel variation in Δθ_E,98, as was also seen for seasonal-mean Δθ_E (section 3c). The possibility that intermodel spread in Δθ_E,98 could be explained by changes in seasonal θ and q_υ suggests that these relationships merit further investigation as more data become available for the CMIP6 models.

5. Observed changes in θ_E

The relative contributions of temperature and specific humidity changes to observed changes in θ_E can be difficult to untangle because of the limited observational record, internal variability (which can induce large changes in relative humidity even over oceans), and confounding factors such as aerosol emissions and land-use changes. However, analysis of data from the ERA5 dataset (Copernicus Climate Change Service 2017) confirms again that specific humidity has generally been the leading driver of θ_E changes. Figure 12a shows the changes in JJA θ_E between 1979–88 and 2009–18, while Fig. 13a shows the changes in the 98th percentile of daily mean θ_E between 1979–88 and 2009–18. In both cases, the patterns are closely related to the patterns of q_υ changes (Figs. 12c and 13c), with r² values of 0.7–0.8 (bottom rows of Tables 1 and 2). Notably, the correlations are slightly lower when averaging over land areas only. By comparison, the patterns of Δθ (Figs. 12b and 13b) are more spatially uniform, and are less well correlated with Δθ_E, with r² values of ~0.5–0.6.⁶

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(a) Difference in JJA θ_E between 1979–88 and 2009–18. (b) Difference in JJA θ between 1979–88 to 2009–18. (c) Difference in JJA q_υ between 1979–88 and 2009–18, multiplied by L_υ/c_p. (d) JJA q_υ, averaged over 1979–88. The cyan contours show the 5.6 g kg⁻¹ isopleths and data are taken from the ERA5 dataset.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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(a) Difference in the 98th percentile of θ_E between 1979–88 and 2009–18. (b) Difference in θ, conditioned on the 98th percentile of θ_E, between 1979–88 and 2009–18. (c) Difference in JJA q_υ, conditioned on the 98th percentile of θ_E, between 1979–88 and 2009–18, multiplied by L_υ/c_p. (d) JJA q_υ, conditioned on the 98th percentile of θ_E and averaged over 1979–88. The cyan contours show the 5.6 g kg⁻¹ isopleths and data are taken from the ERA5 dataset.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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By comparing Figs. 12 and 13 with the CMIP6 results (Figs. 1 and 7) it can be seen that the changes in θ_E over the past 40 years are much more spatially heterogeneous than the changes in the 1% yr⁻¹ runs, with some regions even experiencing reductions in θ_E. These reductions have been caused by large decreases in relative humidity, and associated decreases in specific humidity (see Figs. 12c and 13c). For example, JJA and 98th-percentile θ_E have both decreased over southwest North America, despite increases in θ over the past 40 years. The changes in θ_E are also entirely decorrelated from the baseline specific humidity (q_υ averaged over 1979–88; see Figs. 12d and 13d, and bottom rows of Tables 1 and 2), because of the large changes in relative humidity. Finally, the ratios Q and Q₉₈ are also more spatially heterogeneous for the reanalysis data than for the CMIP6 data, because in many regions the changes in θ, q_υ, and θ_E are small (Fig. 14). However, it is notable that Q and Q₉₈ are both < 1 over much of Eurasia, particularly for the change in seasonal-mean JJA θ_E (Fig. 14a). As with the CMIP6 data, changes in temperature seem to have played a large role in θ_E changes over Eurasia.

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(a) The ratio Q = (L_υΔq_υ)/(c_pΔθ) for the ERA5 data. (b) The ratio Q₉₈ = (L_υΔq_υ,98)/(c_pΔθ₉₈) for the ERA5 data. In both panels, the red contours show the Q or Q₉₈ = 1 level, and values outside the color bar scale are masked in gray.

Citation: Journal of Climate 34, 3; 10.1175/JCLI-D-20-0262.1

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6. Conclusions

There is growing recognition that changes in heat stress could be one of the most devastating consequences of future climate change. Predicting these changes requires climate models that can make accurate prediction of how the many factors involved in extreme heat stress events respond to warming, while also making predictions at the fine scales required to take preventative action. But improved conceptual understanding of the factors governing heat stress changes is also necessary to guide the improvement of models and to ensure trust in model results.

In this study, simple conditions on the baseline specific humidity have been derived for when specific humidity can be expected to dominate changes in equivalent potential temperature (Δθ_E), with different conditions over ocean and over land. A condition was also derived for when changes in relative humidity determine the response of θ_E, rather than the response to warming at fixed relative humidity. These conditions have guided an analysis of θ_E, changes in transient warming simulations with 14 CMIP6 models. Specific humidity changes are found to be the primary control on the pattern of θ_E, changes, whether considering seasonal-mean changes or changes in the 98th percentile of θ_E, so that the response of θ_E can generally be approximated as coming from a spatially uniform (i.e., global mean) potential temperature change and a spatially varying pattern of specific humidity changes. Specific humidity changes also tend to control the intermodel spread, or uncertainty, in Δθ_E over land, particularly for extreme events. Over the oceans, where relative humidity changes are small, the temperature response is the main control on the responses of q_υ and θ_E, though relative humidity changes are still large enough for Δq_υ to be more highly correlated with Δθ_E, particularly in the tropics and subtropics. These model results are corroborated by reanalysis data, in which the patterns of θ_E changes are more heterogeneous, reflecting forced changes and internal variability, but are again highly correlated with the patterns of Δq_υ. In summary, improving our understanding of the regional pattern of θ_E changes and reducing the intermodel spread in θ_E, especially over land, can both be largely achieved by understanding and constraining the response of specific humidity to warming.

The key reason for the importance of specific humidity in θ_E changes is its rapid increase with temperature. Whereas temperature increases by ~0.3% °C⁻¹ (≈1/300), specific humidity increases by ~7% °C⁻¹ at fixed relative humidity. Changes in relative humidity, driven by dynamics, soil moisture feedbacks or land-use changes, can cause the local response of specific humidity to be <0% °C⁻¹ or to increase faster than the Clausius–Clapeyron scaling. Only in cold, dry climates are the larger fractional increases of specific humidity, and the larger spatial variation in these increases, overwhelmed by temperature increases, so that the pattern of Δθ sets the pattern of Δθ_E. Raymond et al. (2017) used similar reasoning to hypothesize that moisture will play an ever-greater role in determining wet-bulb temperatures in the future, and also that the dominance of moisture in wet-bulb temperature extremes will spread poleward as higher latitudes warm up.

The scalings derived in section 2 imply that Δθ_E is partly determined by the baseline specific humidity, q_υ, particularly over oceans. Pattern correlations confirm that q_υ and Δθ_E are related, for both seasonal-mean changes and for extreme events, although the correlations tend to be worse in individual models than in the multimodel composite. The relationship between q_υ and Δθ_E hints at the potential for emergent constraints, in which present-day specific humidity values are used to constrain future changes in heat stress, but q_υ is generally found to be a poor predictor for changes in Δθ_E over land in the models analyzed here. Intermodel variations in relative humidity, in the land warming amplification factor, in the ratio of specific humidity over ocean to specific humidity over land, and in the responses of these to warming could obscure the connection between q_υ and Δθ_E across models.

More detailed analyses are required to fully understand and constrain the pattern of heat stress changes; to understand local relative humidity changes, how surface processes, such as soil moisture feedbacks, affect local moisture levels, and how the dynamics of synoptic-scale weather events responsible for heat stress extremes change with warming. Furthermore, using alternative heat stress metrics, especially those that weight temperature more heavily, may give a different view of the relative importance of humidity and temperature, although ultimately the rapid Clausius–Clapeyron scaling of q_υ will always win out for warm, humid climates. But the analysis presented here provides a starting point for choosing what to focus on in future investigations. Using θ_E, the wet-bulb temperature T_w, or related metrics suggests that at most locations over land, constraining how the specific humidity during extreme heat stress events respond to global-mean warming is the most important step toward constraining future heat stress changes. Especially for extreme events, the local temperature response plays a secondary role in heat stress changes, and can essentially be set to a single, global-mean value. To put this another way, in most places changes in heat stress will be determined by changes in the body’s ability to dissipate excess heat through evaporation, rather than by changes in the amount of heat input into the body.