a. A simplified presentation of the conditional mean framework

Consider a case where temperature is only affected by two processes, call them A and B. The temperature budget equation is

$\frac{\partial T}{\partial t}=A+B.$(1)

This balance applies instantaneously, and it also applies to the average at a given percentile [see Linz et al. (2020)], so if we use the notation ${⟨*⟩}_{T^e}$ to represent the mean at a given percentile e of temperature, we can also write the following balance:

${⟨\frac{\partial T}{\partial t}⟩}_{T^e}={⟨A⟩}_{T^e}+{⟨B⟩}_{T^e}.$(2)

This can be rewritten to express the left-hand side as the rate of change of the temperature at a given percentile:

$\frac{\partial T^e}{\partial t}={⟨A⟩}_{T^e}+{⟨B⟩}_{T^e}.$(3)

Since we are considering the temperature distributions to be stationary in time, there is no time tendency at a particular percentile. At a given percentile, then, there is a balance between A and B:

$0={⟨A⟩}_{T^e}+{⟨B⟩}_{T^e}.$(4)

How can we make any inferences about causality in this case? Each percentile of temperature has a balance of terms—which of these is “driving” temperature anomalies toward more extreme values and which is “damping” them toward the median? To think about this, consider a particular event that falls into, say, the 75th percentile T⁷⁵, where it is getting warmer. That means that right now

$\frac{\partial T}{\partial t}=A+B>0.$(5)

A and B were balanced, but in order to get warmer, now A has to get larger or B does. Which one is causing the temperatures to get hotter? We return to the statistical perspective and see what the balance is, statistically, when it is warmer. If the conditional means of A and B look like the example shown in Fig. 1a, ${⟨A⟩}_{T^{75}}<{⟨A⟩}_{T^{76}}$, so on average the contribution of A is even more positive at the 76th percentile than the 75th. At the 76th percentile, B is more negative, so B cannot be doing the work of making the temperatures more extreme. Therefore A drives the temperature from the 75th to the 76th percentile. A more complicated example is shown in Fig. 1b. Now if at a particular time, the temperature is at the 75th percentile and it is getting warmer, ${⟨A⟩}_{T^{75}}>{⟨A⟩}_{T^{76}}$, but ${⟨B⟩}_{T^{75}}<{⟨B⟩}_{T^{76}}$, so now B is driving the warm extreme values (even though the tendency due to B is actually a negative tendency). The behavior changes over the distribution in this example also, so A drives cold extremes, but B drives warm ones.

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Two examples of the balance described in Eq. (4). (a) The two physical processes act consistently between hot and cold percentiles. (b) A less intuitive example where the process that always acts as a negative tendency (B) drives warmer extremes and the process that always acts as a positive tendency (A) drives cold extremes.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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The simplest way to understand the role of different processes in causing the shape of the temperature distribution heuristically is to remember that a term with a positive linear slope (in T^e versus tendency) for a certain range of percentiles is driving temperatures toward more extreme values in that range. One with a negative slope is acting to damp anomalies. With that in mind, we will consider temperature advection alone before moving on to other processes.

b. The role of horizontal temperature advection in shaping temperature distributions

We start by looking at the horizontal temperature advection (−v ⋅ ∇T), because this is the easiest term to understand. At each grid point, we first calculate the conditional mean temperature values T^e, where T^e represents temperature at the eth percentile. Then we calculate the conditional mean horizontal temperature advection ${⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}$ when temperature is at T^e. Notice that ${⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}$ represents the conditional mean horizontal temperature advection at the eth temperature percentile, rather than the conditional mean value at the advection percentile ${⟨-\mathbf{v}\cdot \nabla T⟩}_{{(-\mathbf{v}\cdot \nabla T)}^e}$, which in general are very different [see Fig. 1 in Zhang et al. (2022)]. The relationship between T^e and ${⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}$ shows how horizontal temperature advection shapes the temperature probability distribution at that grid point.

Following Zhang et al. (2022), in this article we calculate the conditional mean values on 49 different percentiles (e = 2, 4,…, 98), and each conditional mean value is the average over temperature percentile range [e − 0.5, e + 0.5]. All calculations are based on 6-hourly 850-hPa ERA5 reanalysis data in boreal summer (JJA). We only focus on land, though the approach is also applicable over ocean. Since we use data on the 850-hPa pressure level, regions where atmospheric pressure does not reach 850 hPa are ignored.

We then calculate the Pearson correlation between T^e and ${⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}$, $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$, to explore the relationship between horizontal temperature advection and the local temperature probability distribution. Figure 2a shows the global distribution of $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$. The implication of $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$ can be explained by idealized examples in Fig. 2b: if $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})=+1$ (the red line), there is warm (cold) advection when temperature is high (low). Therefore, horizontal temperature advection drives temperature to extreme values by making hot (cold) days hotter (colder). By contrast, the horizontal temperature advection drives temperature back to the median value by making hot (cold) days colder (hotter) in areas where $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})=-1$ (the blue line). Most places in Fig. 2a have a significant positive correlation, while a few small regions have a negative one. This is consistent with the result in Zhang et al. (2022) (also based on ERA5 reanalysis data), but slightly different from Linz et al. (2020) (based on data generated by an idealized aquaplanet model) due to the lack of land–sea contrast in the aquaplanet model.

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(a) The distribution of the Pearson correlation between 850-hPa JJA conditional mean horizontal temperature advection and temperature [$\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$] on land. Regions where the 850-hPa pressure level is under the surface are ignored. Grid points whose correlation value meets the 5% confidence level in a shuffling test are shaded. In this map we mark two black squares, six green squares, and two yellow squares. We will choose one representative region from each color to do a case study in section 3. (b) Idealized examples when conditional mean horizontal temperature advection and temperature have a perfect linear relationship $T^e=\tau {⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}+T_{\text{eq}}$. (c)–(e) The $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$ in three representative regions marked by black, green, and yellow squares, respectively. The dashed squares indicate subregions in which we will compute regional averages. We will do case studies in these three regions in section 3.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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Note that the sign of $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$ does not tell the entire story, especially in the tropics and subtropics. For example, at the Arabian Peninsula, the fact that $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})\approx +1$ in Fig. 2a tells us horizontal temperature advection drives temperature to extreme values, but we cannot say the hot (cold) extreme temperatures result from warm (cold) advections, because the Arabian Peninsula is among the hottest places on Earth in JJA and hot temperatures there do not advect from other hotter places. The conditional mean horizontal temperature advection ${⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}$ is negative at every T^e at the Arabian Peninsula, and more negative at low temperatures while close to zero at high temperatures. As shown in Fig. 1b, “horizontal temperature advection drives temperature to extreme values” means “horizontal temperature advection increases with temperature” but not necessarily means “warm (cold) advection occurs at hot (cold) temperatures.” Therefore, the correlation between conditional mean horizontal temperature advection and temperature percentiles is thus a quick way to check the general behavior in a region, but a more detailed examination is sometimes necessary.

Although τ and T_eq have been written this way in analogy to a Newtonian relaxation, they cannot be understood in the standard way. A positive τ gives a sense of the time scale of persistence of the temperature anomalies due to advection. Notice that $T_{\text{eq}}\ne T^{50}$, and T_eq is useful only in so much as the sign of $T_{\text{eq}}-T^{50}$ will give a sense of the role of other processes at T⁵⁰ [if T_eq is larger/smaller than T⁵⁰, “other processes” $-(T^{50}-T_{\text{eq}})/\tau$ will be positive/negative and have a warming/cooling effect at $T^{\prime 50}$]. The τ < 0 is physically uninterpretable as this simplified picture is not relevant when temperature advection is acting to damp anomalies (applying τ and T_eq fields derived here in a dynamical core would not be meaningful and would indeed be unstable). The way to use these values is as a sanity check and, where τ > 0, as a comparison of the strength of other processes relative to temperature advection. Alternatively, consider them to just be coefficients from a Taylor expansion that have been rewritten to have simpler units. They are determined by doing a linear regression, and we label them in Fig. 2b.

c. The roles of other processes in shaping temperature distributions

In the regions where $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})\approx -1$ (highlighted by blue color in Fig. 2a), processes other than the horizontal temperature advection drive temperature to extreme values, while the horizontal temperature advection drives temperature back to the median value. Most of these regions are coastal summer monsoon regions (such as the six green squares in Fig. 2a; please refer to appendix A for the definition of monsoon region and Fig. A1 for a map), so we expect humidity to play an important role in shaping their local temperature distributions.

Figure 3a shows the global distribution of $\text{corr}({\theta }_{\text{eq}}^e,{⟨-\mathbf{v}\cdot \nabla {\theta }_{\text{eq}}⟩}_{{\theta }_{\text{eq}}^e})$, which is calculated using the same data as Fig. 2a; therefore, a comparison of the correlations displayed in these figures can reveal the role of humidity in setting local temperature distributions. In particular, regions with positive correlations in Fig. 2a retain their positive correlation in Fig. 3a (like west Siberia in Fig. 2c), while many regions with negative correlations in Fig. 2a (especially coastal monsoon regions marked by green squares like Somalia and Kenya in Fig. 2d) are positive in Fig. 3a, indicating a switch from a damping role for temperature advection to amplifying one for θ_eq advection. In those regions horizontal temperature and moisture advection together cause θ_eq values to be more extreme.

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(a) The distribution of the Pearson correlation between 850-hPa JJA conditional mean horizontal θ_eq advection and θ_eq on land. Regions where the 850-hPa pressure level is under the surface are ignored. Grid points whose correlation value meets the 5% confidence level in a shuffling test are shaded. The squares are same as those in Fig. 2a. (b) Iidealized scatterplots when all terms have a perfect linear relationship with T^e. The sum of all terms is zero according to Eq. (13). (c)–(e) the $\text{corr}({\theta }_{\text{eq}}^e,{⟨-\mathbf{v}\cdot \nabla {\theta }_{\text{eq}}⟩}_{{\theta }_{\text{eq}}^e})$ in three representative regions. The dashed squares indicate subregions in which we will compute regional average.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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We can study the roles of these five terms by studying each one’s relationship with conditional mean temperature T^e. At grid points where the result behaves like the idealized example in Fig. 3b, the horizontal humidity advection and vertical temperature advection drive temperature to extreme values, while the horizontal temperature advection and the vertical humidity advection drive temperature back to the median value. The residual term has a negligible effect. Zhang et al. (2022) did a similar diagnostic analysis, but they mainly focused on vertical temperature advection ${⟨-\omega (\partial T/\partial p)+\omega (\kappa T/p)⟩}_{T^e}$ and did not take humidity into consideration.

So far, we have generalized the original conditional mean framework in Linz et al. (2020) and Zhang et al. (2022) to study how processes other than the horizontal temperature advection shape temperature probability distributions. Next, we will apply the method to the three representative regions shown in Figs. 2c–e as case studies.

a. The role of horizontal temperature advection

First, we will examine the role of horizontal temperature advection (−v ⋅ ∇T) in setting the temperature distribution, and explain its diverse roles in different regions by composite analysis.

The first example is west Siberia, an inland plain in Fig. 2c, which represents most regions where $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})\approx +1$. [See Zhang et al. (2022) for a comparison of regions.] The points with color gradient show horizontal temperature advection in Fig. 4a with the color indicating the percentile. This region has a cold (warm) horizontal temperature advection when temperature is low (high), so the horizontal temperature advection drives temperature to extreme values. To study the mechanism connecting horizontal temperature advection to local temperature distribution, we create and analyze composite maps at different temperature percentiles.

Composite maps in Fig. 5 show how the horizontal temperature advection (−v ⋅ ∇T) drives temperature to the extreme values: the direction of the temperature gradient ∇T does not depend on temperature percentile T^e (the south side is hotter than the north side in each of the four maps, though the magnitude of temperature gradient depends on temperature. See Fig B1 in appendix B for details), but the direction of wind vector v depends on T^e. When temperature is low (Fig. 5a, on cold days), the wind heads from north (cooler) to south (warmer), creating cold temperature advection in the dashed black square (−v ⋅ ∇T < 0). By contrast, when temperature is high at these points (Fig. 5d, on hot days), southerly winds drive warm temperature advection (−v ⋅ ∇T > 0). The other two composite maps (Figs. 5b,c) are simply transitional states between the extremes. Therefore, the horizontal temperature advection (−v ⋅ ∇T) tends to make hot days hotter and cold days colder, thereby driving temperature to extreme values. Figure 4a also shows that the vertical humidity advection has a positive slope for the upper half of the distribution, so it also contributes to warm extremes. This example is a simple one that nevertheless corresponds to many regions across the globe.

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The composite maps in west Siberia, Russia (Fig. 2c), at different temperature percentiles: (a) 2%, (b) 34%, (c) 66%, and (d) 98%. The white lines are elevation contours, and the dashed black line marks the subregion where we compute regional averages. The black arrows represent conditional mean wind vectors, and the color indicates temperature.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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Our second case study is Somalia and Kenya, on the east coast of equatorial Africa in Fig. 2d. This region and many other tropical regions where $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})\approx -1$ (marked by green squares in Fig. 2a) are typical coastal summer monsoon regions (Li and Zeng 2000; An et al. 2015), which suggests that the coastal summer monsoon influences the role of horizontal temperature advection in these regions. [We use a monsoon index (Li and Zeng 2000) to define monsoon regions; details are in appendix A]. This analysis is only done for boreal summer, so we cannot comment on $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$ in these regions during other nonmonsoonal seasons.

From the points with color gradient (which represents horizontal temperature advection) in Fig. 4b we know that this region has warm (cold) horizontal temperature advection when temperature is low (high). Composite maps in Fig. 6 show that the coastal summer monsoon contributes to the negative $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$ in Somalia and Kenya. The direction of the wind vector v does not depend on temperature percentile T^e (v has the same directions in four maps in Fig. 6), because the prevailing background monsoon wind is from the ocean to the land. Conversely, the direction of temperature gradient ∇T depends on T^e. When temperature is low (Fig. 6a, in cold days), the ocean is warmer than the land (because of the ocean’s higher heat capacity), so the sea breeze transports warm air from the ocean to the land, resulting in the warm advection in the dashed green square (−v ⋅ ∇T > 0). When temperature is high (Fig. 6d, in hot days), the ocean is cooler than the land and cold air is advected onto land by the prevailing winds (−v ⋅ ∇T < 0). The other two composite maps (Figs. 6b,c) are transitional states between the extremes. Therefore, the horizontal temperature advection (−v ⋅ ∇T) associated with the coastal summer monsoon tends to make hot days cooler and cold days warmer, thereby driving temperatures back to the median values.

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The composite maps in Somalia and Kenya, the east coast of equatorial Africa (Fig. 2d), at different temperature percentiles: (a) 2%, (b) 34%, (c) 66%, and (d) 98%. The solid black line is the coastline, the white lines are elevation contours, and the dashed green line marks the subregion where we compute regional averages. The black arrows represent conditional mean wind vectors, and the color indicates temperature.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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Our third case study is the south corner of Argentina (Patagonia) in Fig. 2e, where $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})\approx -1$ but there is not a coastal monsoon. The other region of this class is in central Africa (as denoted by a yellow square in Fig. 2a). Both of these are distinguished by substantial local topography, which should play an important role in setting the shape of temperature distributions (Lutsko et al. 2019). Unlike the coastal monsoon regions, the horizontal temperature advection ${⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}$ (points with color gradient) in Fig. 4c always has a negative value: when temperature is high (low), the cold advection is stronger (weaker). Therefore, the horizontal temperature advection (−v ⋅ ∇T) drives temperature back to median values, and it has a cooling effect at both high and low temperatures.

The composite maps at different temperature percentiles in Fig. 7 differ from those in Somalia and Kenya (Fig. 6). Here, the wind direction depends on temperature percentile T^e, and the temperature distribution does not have land–sea contrast. Figure 7 shows that the horizontal temperature advection is actually a negative contribution at every temperature percentile, with a stronger negative tendency at the highest percentiles. This example shows that the negative correlation does not show the entire story, as at the coldest temperatures, horizontal temperature advection still has a negative tendency, thereby has a cooling effect. As we mentioned earlier, the correlation between conditional mean horizontal temperature advection and temperature percentiles is a quick way to check the general behavior in a region, but a more detailed examination is sometimes necessary.

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The composite maps in the south corner of Argentina (Fig. 2e), at different temperature percentiles: (a) 2%, (b) 34%, (c) 66%, and (d) 98%. The solid black line is the coastline, the white lines are elevation contours, and the dashed yellow line marks the subregion where we compute regional averages. The black arrows represent conditional mean wind vectors, and the color indicates temperature.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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In Fig. 8 we explain the diverse roles of horizontal temperature advection in shaping local temperature probability distribution at different places. In most inland regions (e.g., west Siberia in the Northern Hemisphere Fig. 2c), the south is always warmer than the north. Northerly wind at low temperatures cools the black box in Fig. 8a to lower temperatures, while southerly wind at high temperatures warms the black box in Fig. 8b to higher temperatures, so horizontal temperature advection drives temperature to extreme values (see Fig. 5). By contrast, in most coastal monsoon regions (e.g., Somalia and Kenya at the east coast of equatoral Africa in Fig. 2d), there is always sea breeze (because we are looking at the conditional mean wind that filters out the local land–sea circulation) in boreal summer. When temperature is low (high), the ocean is warmer (cooler) than the land because of higher heat capacity, resulting in warm (cold) advection in the green box in Fig. 8c (Fig. 8d), so the horizontal temperature advection drives temperature back to median values (see Fig. 6). Nevertheless, there are also a few regions with special topography (like south Argentina in Fig. 2e) where the negative $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$ is not caused by summer monsoon. There are some regions where the slope of horizontal temperature advection changes sign for different percentiles, but generally, temperature advection is well behaved so that the sign of the correlation coefficient provides information over almost all of the distribution.

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The schematic figure explaining the diverse roles of horizontal temperature advection in shaping local temperature probability distribution at different places: (a) an inland region (in the Northern Hemisphere), when temperature is low; (b) an inland region, when temperature is high; (c) a monsoon region, when temperature is low; and (d) a coastal monsoon region, when temperature is high. The minus sign means cold advection, while the plus sign means warm advection.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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b. The roles of other processes

Now we look back to Figs. 3 and 4 to study the roles of other processes in shaping local temperature probability distribution.

Considering θ_eq instead of T, the correlation turns from negative to positive in Somalia and Kenya (Fig. 3d), but remains approximately the same in west Siberia and south Argentina (Figs. 3c,e). Such contrast shows that humidity plays an important role in determining the sign of the correlation in the monsoon (Somalia and Kenya) region, while it is not important in the inland (west Siberia) or nonmonsoon (Argentina) region. This is consistent with the results in Figs. 2a and 3a, where we see that most coastal monsoon regions with green squares have negative $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})$ but positive $\text{corr}({\theta }_{\text{eq}}^e,{⟨-\mathbf{v}\cdot \nabla {\theta }_{\text{eq}}⟩}_{{\theta }_{\text{eq}}^e})$.

We study the roles of different processes more explicitly by the budget analysis in Fig. 4. In west Siberia (Fig. 4a), horizontal temperature advection ${⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e}$ (points with color gradient), and, to a lesser extent, vertical humidity advection $(L_{\upsilon }/c_p){⟨-\omega (\partial q/\partial p)⟩}_{T^e}$ (red points) drive temperature to extreme values. In Somalia and Kenya (Fig. 4b), horizontal humidity advection $(L_{\upsilon }/c_p){⟨-\mathbf{v}\cdot \nabla q⟩}_{T^e}$ (green points), and, to a lesser extent, vertical temperature advection ${⟨-\omega (\partial T/\partial p)+\omega (\kappa T/p)⟩}_{T^e}$ (blue points) drive temperature to extreme values. In south corner of Argentina (Fig. 4c), vertical temperature advection ${(p_0/p)}^{R_d/c_p}{⟨-\omega (\partial T/\partial p)+\omega (\kappa T/p)⟩}_{T^e}$ (blue points), and, to a lesser extent, horizontal humidity advection $(L_{\upsilon }/c_p){⟨-\mathbf{v}\cdot \nabla q⟩}_{T^e}$ (green points) drive temperature to extreme values. A summary of the same diagnostic analysis in other regions (marked by squares in Fig. 2a) is shown in Table 1. Now we analyze the roles of other processes by case studies.

1) The role of horizontal humidity advection

We find that in lots of coastal monsoon regions (marked by green squares in Fig. 2a), horizontal humidity advection, instead of horizontal temperature advection, is the dominant term that drives temperature to extreme values. Therefore, we take Somalia and Kenya in Fig. 2d as an example to explain the role of horizontal humidity advection (it drives temperature to extreme values, see the green points in Fig. 4b) in detail by looking at humidity and wind composites at different temperature percentiles.

First, we focus on specific humidity and wind composites. Similar to Fig. 6, in Fig. 9 the direction of the wind vector v does not depend on temperature percentile T^e, because the prevailing background wind is always from the ocean to the land. Conversely, the direction of specific humidity gradient ∇q depends on T^e. When temperature is low, (Fig. 9a, in cold days), the sea breeze transports dry air near the coast to wetter inland regions, so $(L_{\upsilon }/c_p){⟨-\mathbf{v}\cdot \nabla q⟩}_{T^e}<0$ in the dashed green square; when temperature is high (Fig. 9d, in hot days), the sea breeze transports wet air near the coast to drier inland regions, so $(L_{\upsilon }/c_p){⟨-\mathbf{v}\cdot \nabla q⟩}_{T^e}>0$ in the dashed green square. The other two composite maps (Figs. 9b,c) are transitional states between the extremes. Therefore, horizontal humidity advection might drive temperature to extreme values by affecting latent heat release from condensation: At low temperatures dry advection reduces latent heat release and has a cooling effect, while at high temperatures moist advection increases latent heat release and has a warming effect.

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The specific humidity and wind composite maps in Somalia and Kenya, the east coast of equatorial Africa (Fig. 2d), at different temperature percentiles: (a) 2%, (b) 34%, (c) 66%, and (d) 98%. The solid black line is the coastline, the white lines are elevation contours, and the dashed green line marks the subregion where we compute regional averages. The black arrows represent conditional mean wind vectors, and the color indicates specific humidity.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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To see whether horizontal humidity advection drives temperature to extreme values via latent heat release from condensation, we not only look at specific humidity (Fig. 9 above), but also look at relative humidity (RH, Fig. 10 below). At low temperatures (Fig. 10a), the highest RH in the subregion enclosed by the green square is already 100%, so condensation releases latent heat. The dry advection (Figs. 9a and 10a) replaces the saturated moist air by dry air and reduces latent heat release, so it has a cooling effect. However, at high temperatures (Fig. 10d), RH in the green square or the source of the advection are both far from 100%. Although moist advection (Figs. 9d and 10d) makes the green square wetter, the air is not saturated, so the moist advection does not increase latent heat release. Therefore, horizontal humidity advection has a cooling effect at low temperatures by reducing latent heat release, but has a warming effect at high temperatures through other mechanisms at Somalia and Kenya. We are not able to explain the latter right now, and we leave this for future research.

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The relative humidity and wind composite maps in Somalia and Kenya, the east coast of equatorial Africa (Fig. 2d), at different temperature percentiles: (a) 2%, (b) 34%, (c) 66%, and (d) 98%. The solid black line is the coastline, the white lines are elevation contours, and the dashed green line marks the subregion where we compute regional averages. The black arrows represent conditional mean wind vectors, and the color indicates relative humidity.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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Similar humidity and wind composite analysis can be done in other representative regions. We find that in most coastal monsoon regions (green squares in Fig. 2a), horizontal humidity advection is the dominant factor driving temperature to extreme values (see Table 1 for details), although the underlying mechanisms might be different. That is why $\text{corr}(T^e,{⟨-\mathbf{v}\cdot \nabla T⟩}_{T^e})<0$ (Fig. 2d) but $\text{corr}({\theta }_{\text{eq}}^e,{⟨-\mathbf{v}\cdot \nabla {\theta }_{\text{eq}}⟩}_{{\theta }_{\text{eq}}^e})>0$ (Fig. 3d) in coastal monsoon regions.

2) The role of the residual term (diabatic heating)

Somalia and Kenya in Fig. 2d is also a good example to study the role of the residual term. In Fig. 4, the residual term $R_{T^e}$ (yellow points) is presumed to be from diabatic heating, which is dominated by radiation absorption at the 850-hPa pressure level. Figure 4b tells us that at Somalia and Kenya the residual term $R_{T^e}$ is large and positive at low temperatures, while close to zero at high temperatures. Therefore, numerically we would expect that radiation absorption related to cloud cover has a significant warming effect at low extremes but does not play an important role at high extremes.

Figure 11 shows the cloud cover fraction in Somalia and Kenya at different temperature percentiles. At low temperatures (Fig. 11a), the cloud cover in the green square is anomalously high. Low cloud at 850 hPa absorbs solar radiation and results in a positive heating rate, so radiation absorption related to cloud cover has a warming effect at low temperatures. At high temperatures (Fig. 11d), the cloud cover in the green square is close to zero, so the heating rate is very small and radiation absorption does not affect temperature. Results from Fig. 11 are consistent with our expectation above based on Fig. 4b.

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The cloud cover fraction at Somalia and Kenya, the east coast of equatorial Africa (Fig. 2d), at different temperature percentiles: (a) 2%, (b) 34%, (c) 66%, and (d) 98%. The solid black line is the coastline, the white lines are elevation contours, and the dashed green line marks the subregion where we compute regional averages. The color indicates cloud cover fraction.

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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3) The role of vertical temperature advection

Subsidence during summer is known to be important for warm extremes. Now we use south Argentina as the example to study the role of vertical temperature advection. Figure 4c tells us that vertical temperature advection (blue points) has a positive slope, so numerically it drives temperature to extreme values, and seems to have a large warming effect at high temperatures at the south corner of Argentina.

Figure 12 shows the horizontal and vertical velocity composites in south Argentina at different temperature percentiles. At low temperatures (Fig. 12a), the vertical velocity in the yellow square is relatively small, and close to zero on average. As a result, vertical temperature advection is small at low temperatures. At high temperatures (Fig. 12d), the yellow square is dominated by subsidence [vertical velocity $\omega =dP/dt>0$]. Given that $-(\partial T/\partial p)+(\kappa T/p)>0$ in this region (figure not shown), vertical temperature advection ${⟨-\omega (\partial T/\partial p)+\omega (\kappa T/p)⟩}_{T^e}$ is positive and has a significant warming effect. Because the horizontal advection comes from the Andes, we can conclude that adiabatic warming from downslope winds off the Andes is a major contributor to warm extremes in this region.

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The horizontal and vertical velocity $\omega =dP/dt$ composite maps in the south corner of Argentina (Fig. 2e), at different temperature percentiles: (a) 2%, (b) 34%, (c) 66%, and (d) 98%. The solid black line is the coastline, and the dashed yellow line marks the subregion where we compute regional averages. The black arrows represent horizontal wind vectors, and the color indicates vertical velocity $\omega =dP/dt$ (positive means subsidence).

Citation: Journal of Climate 36, 15; 10.1175/JCLI-D-22-0556.1

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