1. Introduction

Extreme temperature events such as heat waves and cold spells can present a serious threat to humans, livestock, and agricultural production (Bindoff et al. 2013; Field et al. 2012). Assessing the impact of a changing climate on temperature extremes remains a key challenge, and much work is still needed in order to provide reliable predictions of changes in their frequency, intensity, and persistence. While climate models robustly agree that the mean surface temperature is projected to increase due to anthropogenic warming, the higher-order temperature variability changes are less certain (e.g., Tamarin-Brodsky et al. 2020). The latter are especially important for temperature extremes, which lie in the tails of the probability density functions (PDFs).

Much effort in recent years has been directed to studying the mean temperature and temperature variance response to climate change (Schär et al. 2004; Screen 2014; Fischer and Schär 2009; Volodin and Yurova 2013; Parey et al. 2013; Kodra and Ganguly 2014; Schneider et al. 2015; Gao et al. 2015; Holmes et al. 2016; Bathiany et al. 2018; Dai and Deng 2021; Xu et al. 2020). It is now generally acknowledged that the temperature variance is projected to decrease in the Northern Hemisphere (NH) during winter under greenhouse gas warming, due to the excess warming of the Arctic region and associated decrease in the meridional (equator-to-pole) temperature gradient (Screen 2014; Schneider et al. 2015; Tamarin-Brodsky et al. 2020). The basic idea behind the variance decrease, proposed by Screen (2014) and then formulated by Schneider et al. (2015), can be understood by relating the temperature fluctuations, through linear meridional advection, to the meridional background temperature gradient. For example, if the meridional background temperature is uniform everywhere (zero gradient), then no temperature perturbation is generated by advection of air between the tropics and the pole. Similarly, a weakening of the meridional temperature gradient will be associated with a weakening of the temperature anomalies (and thus a decrease in the temperature variance), all else being equal.

Recently, there has been a growing interest in understanding the higher-order moments of the temperature variability, due to their importance for temperature extremes. Specifically, many recent studies have highlighted the importance of skewness [$S=\overline{T^{\prime 3}}/{\left(\overline{T^{\prime 2}}\right)}^{3/2}$, the third-order moment of the temperature PDF], which relates to the asymmetry between the cold and warm temperature anomalies, for capturing correctly the temperature distributions and their response to climate change (Petoukhov et al. 2008; Fischer and Schär 2009; Ruff and Neelin 2012; Perron and Sura 2013; Kodra and Ganguly 2014; Sardeshmukh et al. 2015; Garfinkel and Harnik 2017; Linz et al. 2018; Loikith and Neelin 2019; Tamarin-Brodsky et al. 2019; Linz et al. 2020; Tamarin-Brodsky et al. 2020). A zero skewness implies that the positive and negative tails are symmetric around the mean (e.g., Figs. 1a,b), while a positive skewness implies that the positive tail is longer than the negative tail (e.g., Figs. 1c,d) (with the opposite for negative skewness). Two main mechanisms have been proposed to understand how temperature skewness can be generated dynamically, through meridional advection. The first involves nonlinear meridional advection of temperature anomalies by anomalous cyclone–anticyclone pairs that are responsible for the equatorward (poleward) movement of the cold (warm) temperature anomalies (Garfinkel and Harnik 2017; Linz et al. 2018; Tamarin-Brodsky et al. 2019). This gives rise to the dipole skewness structure seen in the midlatitude storm track regions, which is most pronounced in the Southern Hemisphere (SH) storm tracks (see Fig. 2a). The second mechanism involves linear meridional advection of temperature anomalies generated by spatially asymmetric background temperature gradients (Tamarin-Brodsky et al. 2020) and is mostly important in the NH due to the abundance of continents that create strong temperature gradients. There is strong evidence that a positive skewness change is projected to occur over most of the NH during winter (Gao et al. 2015; Tamarin-Brodsky et al. 2020), indicating that the cold anomalies weaken more strongly than the warm anomalies (relative to the already warmer mean temperatures). This occurs because the largest gradient decreases, which are due to Arctic amplification, occur closer to the pole. Hence, cold anomalies advected from the Arctic weaken significantly more than warm anomalies advected from the tropics (Tamarin-Brodsky et al. 2020).

An illustration of PDFs with different kurtosis and skewness values. A PDF with (a) positive excess kurtosis (K − 3 > 0) and zero skewness (S = 0) (black), (b) negative excess kurtosis (K − 3 < 0) and zero skewness (S = 0) (black), (c) positive excess kurtosis (K − 3 > 0) and positive skewness (S > 0) (black), and (d) negative excess kurtosis (K − 3 < 0) and positive skewness (S > 0) (black). In all panels, a Gaussian PDF (S = 0 and K = 3) is shown in red for reference.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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An illustration of PDFs with different kurtosis and skewness values. A PDF with (a) positive excess kurtosis (K − 3 > 0) and zero skewness (S = 0) (black), (b) negative excess kurtosis (K − 3 < 0) and zero skewness (S = 0) (black), (c) positive excess kurtosis (K − 3 > 0) and positive skewness (S > 0) (black), and (d) negative excess kurtosis (K − 3 < 0) and positive skewness (S > 0) (black). In all panels, a Gaussian PDF (S = 0 and K = 3) is shown in red for reference.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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An illustration of PDFs with different kurtosis and skewness values. A PDF with (a) positive excess kurtosis (K − 3 > 0) and zero skewness (S = 0) (black), (b) negative excess kurtosis (K − 3 < 0) and zero skewness (S = 0) (black), (c) positive excess kurtosis (K − 3 > 0) and positive skewness (S > 0) (black), and (d) negative excess kurtosis (K − 3 < 0) and positive skewness (S > 0) (black). In all panels, a Gaussian PDF (S = 0 and K = 3) is shown in red for reference.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The 850-hPa temperature (a) skewness and (b) excess kurtosis, based on ERA-Interim data averaged over the years 1980–2015 during December–February (DJF). (c) The estimated excess kurtosis (see text for details). Regions where the skewness and excess kurtosis values are statistically significant are stippled.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The 850-hPa temperature (a) skewness and (b) excess kurtosis, based on ERA-Interim data averaged over the years 1980–2015 during December–February (DJF). (c) The estimated excess kurtosis (see text for details). Regions where the skewness and excess kurtosis values are statistically significant are stippled.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The 850-hPa temperature (a) skewness and (b) excess kurtosis, based on ERA-Interim data averaged over the years 1980–2015 during December–February (DJF). (c) The estimated excess kurtosis (see text for details). Regions where the skewness and excess kurtosis values are statistically significant are stippled.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Kurtosis, on the other hand, has been studied much less. Kurtosis [$K=\overline{T^{\prime 4}}/{\left(\overline{T^{\prime 2}}\right)}^2$, the fourth-order moment of the temperature PDF] is indicative of the “extremity” of the tails, and is therefore especially important for extremes. The kurtosis of a Gaussian distribution is exactly 3 (Fig. 1, red), which is why it is often convenient to consider the excess kurtosis (K − 3). Positive excess kurtosis (K > 3) generally implies that more of the PDF lies around the mean and at the extreme tails, while less of the PDF is in the midrange (Fig. 1a). In such a PDF, extreme events occur more frequently than in a Gaussian distribution. Similarly, negative excess kurtosis (K < 3) implies that extreme events occur less frequently than in a Gaussian distribution (Fig. 1b). Note however that this is not necessarily true for both tails if skewness is nonzero. Figure 1c shows an example where both skewness and excess kurtosis are positive but only the positive tail is longer than a Gaussian, while the negative tail is shorter than a Gaussian. The opposite would be true for a negative skewness and positive excess kurtosis. In Fig. 1d, skewness is positive and excess kurtosis is negative, and while both tails are shorter than a Gaussian in this case, it is clear that the negative tail is significantly shorter. Hence, care must be taken with regards to extremes and preconceptions about kurtosis, when skewness is nonzero.

While some studies have included an analysis of temperature kurtosis (Perron and Sura 2013; McKinnon et al. 2016), no deeper investigation of temperature kurtosis and its projected changes has been performed, to our knowledge. The observed kurtosis structure of near-surface temperature based on reanalysis data was presented in Perron and Sura (2013), who noted that regions of large skewness values tend to collocate with large kurtosis values, but did not examine it further. McKinnon et al. (2016) analyzed observed data from weather stations to estimate the temperature distribution changes during summer, and suggested that most of the PDF changes can be explained by a shift in the mean, while the changes in the remaining variability were small. Several previous studies have found an interesting parabolic relationship between kurtosis and skewness for variables such as sea surface temperature (SST) (Sura and Sardeshmukh 2008), sea surface height (Sura and Gille 2010), vorticity and sea level in ocean jets (Hughes et al. 2010; David et al. 2017), and plasma fluids (Krommes 2008; Guszejnov et al. 2013). This remarkable parabolic relation between kurtosis and skewness is not a fundamental statistical result (i.e., it does not follow directly from the definitions of skewness and kurtosis) but has been shown to hold in several complex systems (Schopflocher and Sullivan 2005; Sattin et al. 2009; Cristelli et al. 2012).

For SSTs, Sura and Sardeshmukh (2008) have shown how a linear mixed layer model for SST with a mixture of additive (SST-independent) and multiplicative (SST-dependent) noise can account for the skewness–kurtosis parabolic relation, by writing a Fokker–Planck type equation for the stationary PDF of the SST anomalies. Alternatively, Hughes et al. (2010) explored the statistics of vorticity and sea level height in ocean jets and developed a simple model that captures the squared relationship. They considered a sharp jet, which may be approximated as a step in vorticity. At the center of the jet, there is a flip from negative to positive vorticity anomalies, as the mixing barrier meanders due to passing Rossby waves. The model describes the statistics as the sum of two Gaussians, accounting for the positive and negative anomalies, and noise is then represented by the width of the Gaussians. Despite its simplicity, the model captures the key features such as the squared relationship between skewness and kurtosis, and explains why strong jets acting as mixing barriers tend to be associated with zero skewness and a low kurtosis.

In this paper we extend the model proposed by Hughes et al. (2010). We apply it to low-level atmospheric temperature to explain the squared relationship between temperature skewness and kurtosis, and further explore how it can be used to better understand the future temperature variability changes. The two-Gaussian model used in Hughes et al. (2010) is appropriate for jets in a staircase model of two distinct well-homogenized fluids with a strong mixing barrier. Hence, we cannot expect it to work well for temperature, whose horizontal distribution in the atmosphere is not generally well represented as a staircase of homogenized temperatures. In the proposed model, the temperature PDF is written as the sum of three (rather than two) Gaussians, with two Gaussians representing the cold and warm anomalies, and another near-zero Gaussian representing small departures from the mean temperature (i.e., a Gaussian mixture model, with the choice of three Gaussians motivated by a physical rather than a statistical perspective). Note that if one assumes Gaussianity of the temperature PDF (e.g., Schneider et al. 2015), then only one Gaussian is considered around the mean. Hence, our model can be thought of as an extension of the single Gaussian, to include the effect of long-range advection associated with coherent motions (represented by the warm and cold Gaussians), as opposed to random noise (represented by the Gaussian around the mean). A similar three-Gaussian model was used in David et al. (2017) to study the statistics of turbulent barotropic ocean jets, but the analytical solutions of the three-Gaussian PDF were not explicitly written or investigated further there.

The three-Gaussian model also extends our results from a previous study (Tamarin-Brodsky et al. 2020), where simple equations were derived for temperature variance and skewness in terms of the mean intensity of cold and warm anomalies, assuming a Bernoulli distribution. The oversimplification of the Bernoulli distribution used in that study did not allow for a proper representation of kurtosis. The essential inclusion of the near-zero anomalies, as well as the introduction of noise by allowing for deviations around the mean intensities, is enough to capture the essence of the temperature variability, yet keeping the simplicity in terms of interpretation. We show how the three-Gaussian model can be useful for better understanding temperature variability and its projected changes, by relating the PDF variance, skewness, and kurtosis changes to changes in the intensity and frequency of the cold, near-zero, and warm anomalies separately.

The paper is organized as follows. The methods and data used are first described in section 2. Section 3 gives an overview of the observed temperature skewness and kurtosis and the parabolic relationship between them. In section 4, we review previous studies and extend them to develop the three-Gaussian model. We present interesting limits of the solutions to build intuition of how the model works, and present idealized examples to demonstrate the role of the different model parameters. The model is then used in section 5 to investigate the latitudinal dependence of the parabolic relation, and its interpretation for temperature variability in the SH midlatitudes is discussed. In section 6, the usefulness of the model for interpreting future temperature variability changes is presented. A summary and discussion are given in section 7.

2. Data and methods

In this study we use the 6-hourly 850-hPa temperature field from the ECMWF interim reanalysis (herein ERAI) dataset (Dee et al. 2011) covering the period 1980–2014, where the background climatology is defined for every 6-hourly time period as its average over the 35 years in order to remove both the diurnal and the seasonal cycle. We concentrate on the NH winter season [December–February (DJF)], and on the 850-hPa level, which is mostly above the boundary layer but still highly correlated with the surface temperature (Tamarin-Brodsky et al. 2020), because the focus here is on the dynamical origin of the temperature anomalies and in winter we expect temperatures to be mostly governed by large-scale dynamics, whereas in summer other processes (such as local land surface feedbacks) may be more important (Fischer and Schär 2009).

For the projected temperature variability changes, we use the 6-hourly 850-hPa temperature field from 26 CMIP5 models (see the model list in Table S1 in the online supplemental material). We analyze the r1i1p1 ensemble member from the representative concentration pathway 8.5 (RCP8.5) emissions scenario (Taylor et al. 2012). The data cover a period of 19 years in the historical period (1981–99) and 19 years in the projected period (2081–99) during DJF. The historical simulations are forced by both the observed anthropogenic and natural atmospheric forcings, and in the projected simulations the radiative forcing increases by about 8.5 W m⁻² by year 2100. Similar to ERAI, for each model the background climatology is defined for every 6-hourly time period as its average over the 19 years. Perturbations are defined as deviations from the 6-hourly climatology (for the historical and projected simulations separately). The skewness and kurtosis are calculated first for each model separately, and then averaged together to produce the ensemble means.

3. The observed relationship between temperature skewness and kurtosis

The spatial structure of the low-level (850 hPa) temperature skewness from ERAI data (Fig. 2a) has been presented previously in several studies (e.g., Tamarin-Brodsky et al. 2019), but is presented here for completeness. Temperature skewness is generally negative (positive) on the equatorward (poleward) flank of the midlatitude storm tracks over the oceans in both hemispheres. This is mainly due to nonlinear meridional temperature advection by anomalous cyclone–anticyclone pairs, which advect the cold (warm) temperature anomalies equatorward (poleward) in regions of strong localized meridional temperature gradients (Garfinkel and Harnik 2017; Linz et al. 2018; Tamarin-Brodsky et al. 2019). Other processes, such as linear advection of asymmetric meridional temperature gradients (Tamarin-Brodsky et al. 2020) and regional land surface feedbacks or the vicinity to ocean and mountains (Lutsko et al. 2019; Loikith and Neelin 2019), can also influence temperature skewness.

The low-level (850 hPa) temperature excess kurtosis in ERAI (Fig. 2b) was shown in Perron and Sura (2013), but not investigated further to our knowledge. Temperature kurtosis is generally high (positive excess kurtosis) in the tropics, and a clear dipole structure can be seen in the SH midlatitude storm track region, but with a center that is farther to the south compared to the skewness dipole. Just by comparing Figs. 2a and 2b it is hard to see the parabolic relationship between kurtosis and skewness. This can be more easily seen by examining scatterplots of kurtosis versus skewness in different regions, for example, in the SH (Fig. 3a) and in the NH (Fig. 3b) midlatitudes. In the tropics (Fig. 3c), this relationship is less pronounced, as points are scattered more widely, but still apparent. Similar plots have been identified in previous studies for other variables (as discussed above), which indicates that this might be a fundamental aspect of advective fluid systems (Schopflocher and Sullivan 2005).

Scatterplots of temperature kurtosis vs skewness, based on ERA-Interim data covering the period 1980–2015 during DJF. (a)–(c) Kurtosis vs skewness is plotted for SH midlatitudes (30°–70°S), NH midlatitudes (30°–70°N), and the tropics (25°S–25°N), respectively. (d)–(f) As in (a)–(c), but for kurtosis vs skewness squared. The black lines in (a)–(c) are the best parabolic fit of the points, while in (d)–(f) a best linear fit is used.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Scatterplots of temperature kurtosis vs skewness, based on ERA-Interim data covering the period 1980–2015 during DJF. (a)–(c) Kurtosis vs skewness is plotted for SH midlatitudes (30°–70°S), NH midlatitudes (30°–70°N), and the tropics (25°S–25°N), respectively. (d)–(f) As in (a)–(c), but for kurtosis vs skewness squared. The black lines in (a)–(c) are the best parabolic fit of the points, while in (d)–(f) a best linear fit is used.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Scatterplots of temperature kurtosis vs skewness, based on ERA-Interim data covering the period 1980–2015 during DJF. (a)–(c) Kurtosis vs skewness is plotted for SH midlatitudes (30°–70°S), NH midlatitudes (30°–70°N), and the tropics (25°S–25°N), respectively. (d)–(f) As in (a)–(c), but for kurtosis vs skewness squared. The black lines in (a)–(c) are the best parabolic fit of the points, while in (d)–(f) a best linear fit is used.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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To get a rough idea of the coefficients of this relationship, in Figs. 3d–f the points are plotted for the same latitudinal bands, but for kurtosis (K) and skewness squared (S ²). We find that kurtosis can be estimated as K ≈ aS ² + b, with a = 1.6, 1.6, and 2.3 and b = 2.7, 2.8, and 3.5 for the SH midlatitudes, NH midlatitudes, and tropics, respectively. Hence, depending on the value of the skewness S, it is clear that the excess kurtosis K − 3 can be either negative or positive in the midlatitudes (since b is close to 3 and b – 3 < 0), while it is mostly positive in the tropics (since b − 3 > 0), as can indeed be seen from Fig. 2b.

The statistical significance of the observed skewness and excess kurtosis values can be determined using the standard errors of skewness and kurtosis, given by ${\sigma }_S=\sqrt{\left(6/N_i\right)}$ and ${\sigma }_K=\sqrt{\left(24/N_i\right)}=2{\sigma }_S$, respectively, where N _i is the number of independent degrees of freedom (Brooks and Carruthers 1953). Skewness and excess kurtosis values are then considered significant at the 95% level if they are larger in magnitude than 2σ _S and 2σ _K, respectively (Perron and Sura 2013). We can estimate N _i = 450 (90 days for a season multiplied by the 35 years and divided by a typical atmospheric decorrelation time scale of 7 days), which gives 2σ _S ≈ 0.23 and 2σ _K ≈ 0.46. Regions where skewness and excess kurtosis values are larger than these thresholds are stippled in Fig. 2 (and in Fig. 8 for the historical CMIP5 values) and denote regions where the PDFs deviate significantly from a Gaussian. It is interesting to note that temperatures over land are generally more Gaussian than temperatures over oceans, particularly over Eurasia (Fig. 2).

4. A simple model for temperature variability

While some of the high kurtosis regions seen in Fig. 2b are indeed captured by high values of S ² + 1, this relationship is oversimplified. The excess kurtosis (K − 3) estimated from (3) (see supplemental Fig. S1) is negative everywhere (because the skewness squared values are smaller than 2), unlike the actual excess kurtosis shown in Fig. 2b. Hence, while the simplified skewness from the Bernoulli distribution given by (2) recovers well the actual skewness (see Fig. 2 from Tamarin-Brodsky et al. 2020), the simplified kurtosis given by (3) fails to recover correctly the actual kurtosis (Fig. 2b). One of the main limitations of this simple approximation for studying kurtosis is the assumption that only warm or cold anomalies can occur, while ignoring the weight of the distribution around the mean.

For p ₀ = 0, one recovers the earlier Bernoulli results given in (3). For large p ₀ values (approaching one), Eq. (4) predicts high kurtosis (which can explain the high kurtosis values found in the tropics). In contrast to Eq. (3), the kurtosis estimated from Eq. (4) can now have a positive excess kurtosis (depending on the value of p ₀). However, the coefficient in front of S ² predicted from Eq. (4) is still one, while the observed slope of a best linear fit between K and S ² (Fig. 3) clearly shows that the coefficient should be larger than one.

a. The three-Gaussian model for temperature variability

We next extend our model to include also noise, which was done in Hughes et al. (2010) for the two-state Bernoulli problem. Instead of constant T _c, T ₀ = 0, and T _w, we take Gaussian distributions with mean values T _c, T ₀, and T _w, and with standard deviation $\widehat{\sigma }$ (see Fig. 4). This represents the random variability that can occur for the cold, near-zero, and warm temperature variations around their averages. The Gaussians around T _c and T _w represent the temperature anomalies associated with synoptic-scale features, while the Gaussian around T ₀ represents small departures from the mean temperature. The width of the three Gaussians $\widehat{\sigma }$ is chosen equal for simplicity, but in general the width can be different (and this assumption is probably a poor approximation in certain regions).

A schematic illustrating the three-Gaussian model. In this model, the total PDF is written in terms of three Gaussians, describing the cold, near-zero, and warm anomalies. The mean values of these Gaussians are −T _c, T ₀ = 0, and T _w; their amplitudes are p _c, p ₀, and p _w, respectively; and their widths are denoted as $\widehat{\sigma }$ (chosen equal for simplicity; see text). Note that the probabilities p _c, p ₀, and p _w are are not necessarily equal, and satisfy by construction p _c + p ₀ + p _w = 1.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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A schematic illustrating the three-Gaussian model. In this model, the total PDF is written in terms of three Gaussians, describing the cold, near-zero, and warm anomalies. The mean values of these Gaussians are −T _c, T ₀ = 0, and T _w; their amplitudes are p _c, p ₀, and p _w, respectively; and their widths are denoted as $\widehat{\sigma }$ (chosen equal for simplicity; see text). Note that the probabilities p _c, p ₀, and p _w are are not necessarily equal, and satisfy by construction p _c + p ₀ + p _w = 1.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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A schematic illustrating the three-Gaussian model. In this model, the total PDF is written in terms of three Gaussians, describing the cold, near-zero, and warm anomalies. The mean values of these Gaussians are −T _c, T ₀ = 0, and T _w; their amplitudes are p _c, p ₀, and p _w, respectively; and their widths are denoted as $\widehat{\sigma }$ (chosen equal for simplicity; see text). Note that the probabilities p _c, p ₀, and p _w are are not necessarily equal, and satisfy by construction p _c + p ₀ + p _w = 1.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Taking the PDF to be

$p=p_{c}N\left(-T_c,\widehat{\sigma }\right)+p_{0}N\left(0,\widehat{\sigma }\right)+p_{w}N\left(T_w,\widehat{\sigma }\right),$(5)

where p _w + p _c + p ₀ = 1 and $N\left(\mu ,\widehat{\sigma }\right)$ represents a Gaussian distribution with mean μ and standard deviation $\widehat{\sigma }$, we find (see section 4 in the supplemental material for further details)

${\sigma }^2={\widehat{\sigma }}^2+T_w{T}_c\left(1-p_0\right)$(6)

and

$S={\displaystyle \frac{T_w{T}_c\left(1-p_0\right)\left(T_w-T_c\right)}{{\sigma }^3}}.$(7)

We then further find that the kurtosis can be written as

$K=a{S}^2+b,$(8)

where

$a\equiv {\displaystyle \frac{{\widehat{\sigma }}^2}{T_w{T}_c\left(1-p_0\right)}}+1$(9)

and

$b\equiv {\displaystyle \frac{1}{\left(1-p_0\right)}}{\displaystyle \frac{1}{a^2}}-{\displaystyle \frac{3}{a^2}}+3.$(10)

The coefficient $a={\widehat{\sigma }}^2/\left[T_w{T}_c\left(1-p_0\right)\right]+1$ essentially measures the ratio between the noise (given by width of Gaussians $\widehat{\sigma }$) and the signal [related to the variance of the modified three-parameter Bernoulli system, T _w T _c(1 − p ₀)]. The parameter b, for a given a, is then inversely related to 1 − p ₀. In section 4b we explore interesting limits of the three-Gaussian model, and in section 4c we demonstrate how different a and b parameters give rise to different decompositions of Gaussians, which ultimately control the non-Gaussian shape of the total PDF.

Note that the three-Gaussian PDF written in (5) was used in David et al. (2017) to examine the kurtosis and skewness structure of idealized barotropic ocean jets (see Fig. 9 in David et al. 2017), but the analytic expressions were not written explicitly or investigated further there.

c. Interpreting the parameters a and b

Sattin et al. (2009) proposed that the parabolic relationship between kurtosis and skewness can be found in many physical systems obeying certain constraints. In the simplest case, assume that there is some parameter δ that controls the deviation from Gaussianity of the PDF. Hence, K and S are both functions of δ, and it is further assumed that S (δ) is smooth and invertible. This gives δ = δ(S), and therefore K = K(S). Expanding K in a Taylor series around small values of S (Sattin et al. 2009), one finds

$K\simeq K_0+{{\displaystyle \frac{S}{2}}}^2{\displaystyle \frac{{\partial }^{2}K}{\partial S^2}}.$(16)

Here it was also assumed that the system is invariant with respect to the sign inversion, so the odd derivatives are zero since S is an odd function and K an even function with respect to the inversion operation. Note that the higher-order terms in this expansion can be neglected to a reasonable extent, given that the observed temperature skewness values are typically smaller than one (e.g., Fig. 2a).

From (16), it can therefore be identified that b = K ₀ is the kurtosis that would exist in the absence of skewness. It is thus related to the relative frequency of the small (near-mean) and the extremely strong (in the far tails) anomalies, compared to anomalies in the middle range, regardless of the sign. Setting S = 0 gives K = b, which implies that the sign of b − 3 determines whether the symmetric PDF would be platykurtic (negative excess kurtosis) or leptokurtic (positive excess kurtosis).

For the three-Gaussian model, we find that $b-3=1/\left[\left(1-p_0\right)a^2\right]-3/a^2=\left(1/a^2\right)\left[\left(3p_0-2\right)/\left(1-p_0\right)\right]$. Hence, if $p_0>2/3$, the PDF with zero skewness has positive excess kurtosis, whereas if $p_0<2/3$ the PDF has a negative excess kurtosis. This is presented in Figs. 5a–d, which show examples of PDFs composed of the sum of three Gaussians, where the width of the total PDF is set to unity (σ = 1) and skewness to zero (S = 0 implying T _w = T _c), and we also arbitrarily set a = 1.8. This allows us to examine the effect of changing b, or effectively p ₀, through $b=1/\left[\left(1-p_0\right)a^2\right]-3/a^2+3$, on the PDF structure. For low values of b (and therefore p ₀) (Figs. 5a–c), the PDF is indeed characterized by a negative kurtosis, indicated by the shallower-than-Gaussian peak and shorter-than-Gaussian tails, whereas for high values of b (and thus p ₀) (Fig. 5d) the PDF is characterized by a positive kurtosis, indicated by the higher-than-Gaussian peak and longer-than-Gaussian tails. Note that the values of T _w, T _c, p _w, p _c, and $\widehat{\sigma }$ in these idealized examples are determined from b (or p ₀), a, σ, and S (see section 6 in the supplemental material).

Idealized examples of PDFs composed of the sum of three Gaussians, with varying a and b, to explore their influence on the shape of the total PDF. (a)–(d) a is held constant at a fixed value of a = 1.8, and skewness is set to zero (S = 0 implying T _w = T _c). The values of b—or effectively p ₀, through b = 1/[(1 − p ₀)a ²] − 3/a ² + 3—are (a) b = 2.5 (corresponding to p ₀ = 0.2), (b) b = 2.6 (corresponding to p ₀ = 0.4), (c) b = 2.8 (corresponding to p ₀ = 0.6), and (d) b = 3.6 (corresponding to p ₀ = 0.8). (e)–(h) b is held constant at a fixed value of b = 2.7, and skewness is set to S = 0.4. The values of a, which effectively control the signal-to-noise-ratio, are (e) a = 1.1, (f) a = 1.4, (g) a = 1.7, and (h) a = 2.3. In all cases, the width of the total PDF is set to unity (σ = 1), and a Gaussian distribution with a unit width is shown for reference with a gray line. The resulting kurtosis is given in the upper-left corner of each panel. In (e)–(h), the upper-right box is a zoom-in into the right positively skewed tail.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Idealized examples of PDFs composed of the sum of three Gaussians, with varying a and b, to explore their influence on the shape of the total PDF. (a)–(d) a is held constant at a fixed value of a = 1.8, and skewness is set to zero (S = 0 implying T _w = T _c). The values of b—or effectively p ₀, through b = 1/[(1 − p ₀)a ²] − 3/a ² + 3—are (a) b = 2.5 (corresponding to p ₀ = 0.2), (b) b = 2.6 (corresponding to p ₀ = 0.4), (c) b = 2.8 (corresponding to p ₀ = 0.6), and (d) b = 3.6 (corresponding to p ₀ = 0.8). (e)–(h) b is held constant at a fixed value of b = 2.7, and skewness is set to S = 0.4. The values of a, which effectively control the signal-to-noise-ratio, are (e) a = 1.1, (f) a = 1.4, (g) a = 1.7, and (h) a = 2.3. In all cases, the width of the total PDF is set to unity (σ = 1), and a Gaussian distribution with a unit width is shown for reference with a gray line. The resulting kurtosis is given in the upper-left corner of each panel. In (e)–(h), the upper-right box is a zoom-in into the right positively skewed tail.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Idealized examples of PDFs composed of the sum of three Gaussians, with varying a and b, to explore their influence on the shape of the total PDF. (a)–(d) a is held constant at a fixed value of a = 1.8, and skewness is set to zero (S = 0 implying T _w = T _c). The values of b—or effectively p ₀, through b = 1/[(1 − p ₀)a ²] − 3/a ² + 3—are (a) b = 2.5 (corresponding to p ₀ = 0.2), (b) b = 2.6 (corresponding to p ₀ = 0.4), (c) b = 2.8 (corresponding to p ₀ = 0.6), and (d) b = 3.6 (corresponding to p ₀ = 0.8). (e)–(h) b is held constant at a fixed value of b = 2.7, and skewness is set to S = 0.4. The values of a, which effectively control the signal-to-noise-ratio, are (e) a = 1.1, (f) a = 1.4, (g) a = 1.7, and (h) a = 2.3. In all cases, the width of the total PDF is set to unity (σ = 1), and a Gaussian distribution with a unit width is shown for reference with a gray line. The resulting kurtosis is given in the upper-left corner of each panel. In (e)–(h), the upper-right box is a zoom-in into the right positively skewed tail.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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As for the parameter a, from (16), it is clear that $a\equiv \left(1/2\right)\left({\partial }^{2}K/{\partial }^{2}S\right)$. This also gives (for constant a) $a\equiv \partial K/\partial \left(S^2\right)$. Hence, a measures the sensitivity of K to changes in the “intensity” of skewness (i.e., to changes in skewness squared). The larger a is, the larger kurtosis K can be found for the same S ². Moreover, for a given nonzero skewness S and a given b, a will determine if the PDF will have a positive or negative excess kurtosis. If b > 3 then K = aS ² + b will also be larger then 3, regardless of a (since aS ² > 0). However, if b < 3, the PDF can still have a positive excess kurtosis if a is large enough.

In the three-Gaussian model $a=\left[{\widehat{\sigma }}^2+T_w{T}_c\left(1-p_0\right)\right]/\left[T_w{T}_c\left(1-p_0\right)\right]={\sigma }^2/\left[T_w{T}_c\left(1-p_0\right)\right]$, which can therefore be interpreted as the enhancement of the total variance of the three-Gaussian distribution, σ ², compared to the variance of the modified Bernoulli distribution with three variables, T _w T _c(1 − p ₀). Alternatively, $a={\widehat{\sigma }}^2/\left[T_w{T}_c\left(1-p_0\right)\right]+1$ can be interpreted as the ratio (plus 1) between the noise in the system, given by the width ${\widehat{\sigma }}^2$ of the Gaussians, to the signal, defined here as the modified Bernoulli distribution with three variables. Hence, the signal-to-noise ratio is related to how well separated the Gaussians are compared to their width.

From its definition, it is clear that a ≥ 1 (with a = 1 only achieved for the modified Bernoulli system with three parameters, where $\widehat{\sigma }=0$). Figures 5e–h show PDFs composed of the sum of three Gaussians with varying a, where the total width of the PDF is again set to unity (σ = 1), and we also set skewness to S = 0.4 and b = 2.7 (hence b in the case of zero skewness would give a negative excess kurtosis). For low values of a (Fig. 5e) we find narrow PDFs that are more distant from each other, and the PDF becomes trimodal (which is not what is usually found for realistic low-level temperature distributions). In this case, the excess kurtosis is negative (K < 3) because the localization of the signals effectively make the Gaussian longer in the tails. As a increases (Figs. 5e–g), the PDFs become wider and closer to each other (the signal-to-noise ratio decreases) until eventually a is large enough (Fig. 5h) such that the overall PDF has a positive excess kurtosis (K > 3) and the right-skewed tail becomes longer than a Gaussian (see boxes in Figs. 5e–h, which show a zoom into the region of the right-skewed tail).

Hence, different a and b values can give very different PDF structures, from trimodal distributions to unimodal PDFs with a non-Gaussian shape. In the following sections, we estimate the parameters a and b from the linear relationship between K and S ², first in a longitudinally independent form and then for different local regions, and explore the resulting PDF decompositions, their interpretation, and their projected changes.

5. The latitudinal dependence of $\tilde{K}=a{S}^2+b$

For clarity, we henceforth denote the actual kurtosis as K, while the approximated kurtosis is denoted as $\tilde{K}=a{S}^2+b$. The coefficients a, b of the approximated kurtosis can be estimated for each latitude by taking all longitudinal points and finding a best linear fit between K and S ². The resulting dependence on latitude is shown in Fig. 6 (after applying a customary smoothing with respect to latitude using MATLAB’s “smooth” function). Also shown for completeness are the zonally averaged skewness S(y) (Fig. 6a) and the zonally averaged excess kurtosis K(y) − 3 (Fig. 6b). Note however that the zonal averages hide a lot of the regional structure in the NH. For example, the skewness is generally very small in the zonal mean, but Fig. 2a shows that this is a result of compensation between strong positive skewness in northern oceans and strong negative skewness, particularly over the west coast of North America. Nonetheless, the purpose here is to get the general latitudinal dependence of the parameters a and b, even if they might vary longitudinally. Moreover, since we are not averaging, but rather taking all longitudinal points in a given latitude, even if K and S vary significantly longitudinally they might still possess the same relationship in terms of the fit K = aS ² + b.

The latitudinal distribution of the 850-hPa (a) zonally averaged skewness S(y), (b) zonally averaged excess kurtosis, the estimated parameters (c) a(y), (d) b(y), and (e) p ₀(y), and (f) the recovered excess kurtosis $\tilde{K}\left(y\right)-3=a\left(y\right)S^2\left(y\right)+b\left(y\right)-3$, based on ERA-Interim data averaged for the period 1980–2015 during DJF. The parameters a(y) and b(y) are estimated by calculating, for each latitude separately, the coefficients a, b of $\tilde{K}=a{S}^2+b$, by fitting a linear relation between K and S ² in all the corresponding longitudinal points. The parameter p ₀(y) is then estimated from a(y) and b(y) using the relation $b\left(y\right)=\left\{1/\left[1-p_0\left(y\right)\right]\right\}\left[1/a{\left(y\right)}^2\right]-3/a{\left(y\right)}^2+3$.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The latitudinal distribution of the 850-hPa (a) zonally averaged skewness S(y), (b) zonally averaged excess kurtosis, the estimated parameters (c) a(y), (d) b(y), and (e) p ₀(y), and (f) the recovered excess kurtosis $\tilde{K}\left(y\right)-3=a\left(y\right)S^2\left(y\right)+b\left(y\right)-3$, based on ERA-Interim data averaged for the period 1980–2015 during DJF. The parameters a(y) and b(y) are estimated by calculating, for each latitude separately, the coefficients a, b of $\tilde{K}=a{S}^2+b$, by fitting a linear relation between K and S ² in all the corresponding longitudinal points. The parameter p ₀(y) is then estimated from a(y) and b(y) using the relation $b\left(y\right)=\left\{1/\left[1-p_0\left(y\right)\right]\right\}\left[1/a{\left(y\right)}^2\right]-3/a{\left(y\right)}^2+3$.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The latitudinal distribution of the 850-hPa (a) zonally averaged skewness S(y), (b) zonally averaged excess kurtosis, the estimated parameters (c) a(y), (d) b(y), and (e) p ₀(y), and (f) the recovered excess kurtosis $\tilde{K}\left(y\right)-3=a\left(y\right)S^2\left(y\right)+b\left(y\right)-3$, based on ERA-Interim data averaged for the period 1980–2015 during DJF. The parameters a(y) and b(y) are estimated by calculating, for each latitude separately, the coefficients a, b of $\tilde{K}=a{S}^2+b$, by fitting a linear relation between K and S ² in all the corresponding longitudinal points. The parameter p ₀(y) is then estimated from a(y) and b(y) using the relation $b\left(y\right)=\left\{1/\left[1-p_0\left(y\right)\right]\right\}\left[1/a{\left(y\right)}^2\right]-3/a{\left(y\right)}^2+3$.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The coefficient a(y) is generally high in the tropics and lower in the midlatitudes (Fig. 6c). Note that the 850-hPa level intersects topography, and we have thus limited our analysis here to 75°S–75°N. The parameter b(y) is also high in the tropics and achieves the lowest value in the middle of the SH jet (where skewness is zero) (Fig. 6d). The frequency p ₀ of the near-zero anomalies [calculated by plugging a and b in relation (10) and inverting it to find p ₀] is generally similar to b (Fig. 6e). The overall recovered excess kurtosis, $\tilde{K}-3=a{S}^2+b-3$ (Fig. 6f), is indeed similar to the zonally averaged kurtosis (Fig. 6b) and captures its main features.

The estimated coefficients a(y) and b(y) can also be used to plot a spatial map of the estimated excess kurtosis $\tilde{K}-3=a\left(y\right)S^2+b\left(y\right)$ (i.e., the same coefficients a and b are used for every longitude as a function of latitude only, but we keep the full spatial field of skewness; Fig. 2c). This recovers well the excess kurtosis structure (a spatial correlation coefficient of 0.85), albeit with somewhat lower values (see Fig. S2). For example, it captures the high kurtosis values in the tropics (which are not achieved for the simplified relation K = S² + 1; see Fig. S1), and also the high kurtosis values on the poleward flanks of the midlatitude storm track regions and on the west coast of North America.

b. The three-Gaussian interpretation for the SH midlatitudes

More intuitively, these results can be directly seen by inspecting the three-Gaussian model decomposition (Fig. 7). The seven unknowns in the model ($\widehat{\sigma }$, T _c, T ₀, T _w, p _c, p ₀, and p _w) can be found as follows. First, from the normalization of the PDF, we have p ₀ = 1 − p _w − p _c. Next, we set T ₀ = 0, and from the first-order moment of the PDF (the mean) we thus also have T _w p _w = T _c p _c. The second- and third-order moments [given in (6) and (7), respectively] give two more equations for the known variance and skewness. Last, from the fit between K and S ² we have two more equations for the estimated coefficients a and b [given in (9) and (10), respectively]. Hence, inverting these relations (see section 6 in the supplemental material for full derivation), we can find all the model parameters. Note that kurtosis is thus not used directly. Rather, we are using the fitted relationship between K and S ² to extract more information about the system.

Examples of the three-Gaussian vs the two-Gaussian decomposition for the SH midlatitudes. (a)–(c) The 850-hPa raw temperature PDFs (black line) for anomalies at the latitudinal bands (a) 30°–40°S, (b) 45°S, and (c) 55°–75°S, based on ERA-Interim data averaged over the years 1980–2015 during DJF. The red lines show the corresponding Gaussian distributions (the same variance but setting skewness to 0 and kurtosis to 3). (d)–(f) The approximated PDFs based on the three-Gaussian model for the same latitudinal bands, respectively. The thin blue and red lines are the Gaussians representing the cold and warm anomalies, respectively, while the green Gaussian represents the near-zero anomalies. The sum of the three Gaussians is shown by the thick black line, which recovers well the shape of the corresponding raw PDF, shown in dashed black for reference. (g)–(i) The corresponding decomposition but for the two-Gaussian model, calculated in the same manner but assuming p ₀ = 0. The correlation coefficients for the match between the model and the raw PDFs for each case is denoted in the upper-left corner of each panel. The parameters used for finding the three Gaussians are derived from the variance, skewness, kurtosis, a, and b, calculated for each region of interest separately (see section 6 in the supplemental material for more details).

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Examples of the three-Gaussian vs the two-Gaussian decomposition for the SH midlatitudes. (a)–(c) The 850-hPa raw temperature PDFs (black line) for anomalies at the latitudinal bands (a) 30°–40°S, (b) 45°S, and (c) 55°–75°S, based on ERA-Interim data averaged over the years 1980–2015 during DJF. The red lines show the corresponding Gaussian distributions (the same variance but setting skewness to 0 and kurtosis to 3). (d)–(f) The approximated PDFs based on the three-Gaussian model for the same latitudinal bands, respectively. The thin blue and red lines are the Gaussians representing the cold and warm anomalies, respectively, while the green Gaussian represents the near-zero anomalies. The sum of the three Gaussians is shown by the thick black line, which recovers well the shape of the corresponding raw PDF, shown in dashed black for reference. (g)–(i) The corresponding decomposition but for the two-Gaussian model, calculated in the same manner but assuming p ₀ = 0. The correlation coefficients for the match between the model and the raw PDFs for each case is denoted in the upper-left corner of each panel. The parameters used for finding the three Gaussians are derived from the variance, skewness, kurtosis, a, and b, calculated for each region of interest separately (see section 6 in the supplemental material for more details).

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Examples of the three-Gaussian vs the two-Gaussian decomposition for the SH midlatitudes. (a)–(c) The 850-hPa raw temperature PDFs (black line) for anomalies at the latitudinal bands (a) 30°–40°S, (b) 45°S, and (c) 55°–75°S, based on ERA-Interim data averaged over the years 1980–2015 during DJF. The red lines show the corresponding Gaussian distributions (the same variance but setting skewness to 0 and kurtosis to 3). (d)–(f) The approximated PDFs based on the three-Gaussian model for the same latitudinal bands, respectively. The thin blue and red lines are the Gaussians representing the cold and warm anomalies, respectively, while the green Gaussian represents the near-zero anomalies. The sum of the three Gaussians is shown by the thick black line, which recovers well the shape of the corresponding raw PDF, shown in dashed black for reference. (g)–(i) The corresponding decomposition but for the two-Gaussian model, calculated in the same manner but assuming p ₀ = 0. The correlation coefficients for the match between the model and the raw PDFs for each case is denoted in the upper-left corner of each panel. The parameters used for finding the three Gaussians are derived from the variance, skewness, kurtosis, a, and b, calculated for each region of interest separately (see section 6 in the supplemental material for more details).

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The three-Gaussian decomposition is illustrated for three latitudinal regions in the SH midlatitudes (30°–40°, 45°, and 55°–75°S). The actual PDFs (referred to as “raw” but presented with a kernel density smoother) are shown in black in the first row of Fig. 7. For comparison, their corresponding Gaussian distributions (i.e., same variance but with S = 0, K = 3) are shown in red. Equatorward of the SH jet (30°–40°S; Fig. 7a), the positive warm tail is shorter than a Gaussian, while the negative cold tail is longer than a Gaussian, consistent with the negative skewness there (S = −0.2). In addition, kurtosis is larger than that of a Gaussian (K = 3.3). The three-Gaussian model decomposition (Fig. 7c) gives T _w ≈ 3.3 and T _c ≈ 4.9. Hence, equatorward of the jet, the mean intensity of cold anomalies is larger than that of warm anomalies (T _c > T _w) (recall that anomalies are measured relative to the background field, which is warm in this region). Consistent with this, the decomposition also shows that averaged cold anomalies occur less frequently, p _c < p _w (p _w ≈ 0.25 and p _c ≈ 0.17). This is a simple consequence of the zero mean of the entire distribution (p _w T _w − p _c T _c = 0); that is, the stronger anomalies must be less frequent in order for the total time-mean anomaly to be zero. However, note that the very extreme cold anomalies (left tail in the Gaussian describing T _c; blue line in Fig. 7c) have a higher probability than very extreme warm anomalies (right tail in the Gaussian describing T _w, red line in Fig. 7c). Hence, while the averaged cold anomalies occur less frequently (to conserve zero time-mean anomaly in total), extremely strong cold anomalies have a higher probability than extremely strong warm anomalies. These results are exactly the characteristics of the negative skewness found in this region. The three-Gaussian model also reveals that the frequency p ₀ (p ₀ ≈ 0.58; that is, the frequency of small departures from the mean) is significantly higher than either p _w or p _c. The high p ₀ together with the nonzero skewness are what give rise to the positive excess kurtosis found in this region.

In the center of the SH jet (45°S; Fig. 7b), skewness is almost zero (S ≈ −0.02) and kurtosis is low (K ≈ 2.4). The three-Gaussian decomposition now gives (Fig. 7e) T _w ≈ 4 and T _c ≈ 4.2 (hence T _w ≈ T _c) and $p_w\approx p_c\approx p_0\approx 1/3$. Hence, T _w and T _c anomalies occur at similar magnitudes, and at similar frequencies as T ₀. This confirms our previous interpretation of the underlying dynamics at the center of the jet, discussed in section 5a.

Poleward of the jet (55°–75°S; Fig. 7c), we find the opposite result compared to the equatorward side. In this region the model predicts stronger warm anomalies, T _w > T _c (T _w ≈ 5.8 and T _c ≈ 2.5), which occur on average less frequently, p _w < p _c (p _w ≈ 0.1 and p _c ≈ 0.24), but with a much longer warm tail overall. In addition, we find higher frequency of near-zero anomalies (p ₀ ≈ 0.66). These results are consistent with the high positive skewness (S ≈ 0.6) and higher than Gaussian kurtosis (K ≈ 4) found in this region.

In each case shown in Figs. 7d–f, the overall sum of the three Gaussians (black line) recovers well the actual PDF in the region (shown in black in Figs. 7a–c and given by the dashed black line in Figs. 7d–f for ease of comparison). The corresponding correlation coefficient is reported in the upper left corner of each panel.

To summarize, the model allows us to truncate the full PDF (which has an infinite number of moments) into a sum of three Gaussians. These illustrate directly the main ingredients of the underlying PDF, namely the mean intensity of cold, near-zero, and warm anomalies, their frequencies, and width. These recover the variance, skewness, and the parabolic relation between kurtosis and skewness by construction, but also give a much simpler interpretation of the temperature variability characteristic of the region.

6. Interpreting future temperature variability changes

We next investigate the projected higher-order temperature variability changes using an ensemble of 26 CMIP5 models driven by the RCP8.5 emissions scenario. Specifically, we concentrate on the skewness and kurtosis changes (Fig. 8). The first row shows the historical values (Figs. 8a,b), which are very similar to the results found for the ERAI data (Figs. 2a,b), and the second row shows the corresponding projected changes of skewness (Fig. 8c) and kurtosis (Fig. 8d).

The historical (1981–99) ensemble-mean temperature skewness and kurtosis, and their projected changes (2081–99 minus historical), based on 26 CMIP5 RCP8.5 ensemble members. The 850-hPa temperature (a) skewness and (b) kurtosis during DJF, and (c),(d) the corresponding projected changes. Regions where the skewness and kurtosis values are statistically significant are stippled. The black boxes highlight regions of interest that are further explored in Fig. 10.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The historical (1981–99) ensemble-mean temperature skewness and kurtosis, and their projected changes (2081–99 minus historical), based on 26 CMIP5 RCP8.5 ensemble members. The 850-hPa temperature (a) skewness and (b) kurtosis during DJF, and (c),(d) the corresponding projected changes. Regions where the skewness and kurtosis values are statistically significant are stippled. The black boxes highlight regions of interest that are further explored in Fig. 10.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The historical (1981–99) ensemble-mean temperature skewness and kurtosis, and their projected changes (2081–99 minus historical), based on 26 CMIP5 RCP8.5 ensemble members. The 850-hPa temperature (a) skewness and (b) kurtosis during DJF, and (c),(d) the corresponding projected changes. Regions where the skewness and kurtosis values are statistically significant are stippled. The black boxes highlight regions of interest that are further explored in Fig. 10.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The model developed here can be used to better understand the future temperature variability changes in two different ways. First, from the approximate expression for kurtosis $\tilde{K}=a{S}^2+b$, we can inspect how each of the changes in a, b, and S ² influence changes in $\tilde{K}$. Second, using the three-Gaussian PDF given in (5), we can examine how each one of the changes in p _c, p ₀, p _w, T _c, T _w, and $\widehat{\sigma }$ contributes to the overall change in the temperature PDF. This enables easier visualization and understanding of the projected temperature changes in different regions over the globe.

a. Decomposition of kurtosis changes

From the relation $\tilde{K}=a{S}^2+b$, we can decompose the projected kurtosis changes in terms of changes in a, b, and S ² as

$\mathrm{\Delta }\tilde{K}=S^2\mathrm{\Delta }a+a\mathrm{\Delta }\left(S^2\right)+\mathrm{\Delta }b.$(17)

Each of the terms in (17) can be estimated separately from the historical and projected simulations. For simplicity, we consider here only the longitudinally independent estimations a(y) and b(y), but keep the full spatial field of skewness. The actual projected excess kurtosis change (shown again in Fig. 9a for ease of comparison) is captured well by the estimated projected excess kurtosis change $\mathrm{\Delta }\left(\tilde{K}-3\right)=\mathrm{\Delta }\tilde{K}=\mathrm{\Delta }\left(a{S}^2+b\right)$ (Fig. 9b), with a relatively small difference (and a spatial correlation coefficient of 0.76), showing the largest discrepancies in the tropics.

Projected kurtosis change and its decomposition, based on 26 CMIP5 RCP8.5 ensemble members at the 850-hPa level during DJF. The (a) projected excess kurtosis change ∆(K – 3), (b) total change in the estimated excess kurtosis $\mathrm{\Delta }\left(\tilde{K}-3\right)=\mathrm{\Delta }\left(a{S}^2+b\right)$, and (c) their difference, and the decomposition of the approximated kurtosis change into (d) ∆(a)S ², (e) a∆(S ²), and (f) ∆(b). Here the longitudinally independent estimations a(y) and b(y) are used, while for skewness the full S(x, y) field is used.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Projected kurtosis change and its decomposition, based on 26 CMIP5 RCP8.5 ensemble members at the 850-hPa level during DJF. The (a) projected excess kurtosis change ∆(K – 3), (b) total change in the estimated excess kurtosis $\mathrm{\Delta }\left(\tilde{K}-3\right)=\mathrm{\Delta }\left(a{S}^2+b\right)$, and (c) their difference, and the decomposition of the approximated kurtosis change into (d) ∆(a)S ², (e) a∆(S ²), and (f) ∆(b). Here the longitudinally independent estimations a(y) and b(y) are used, while for skewness the full S(x, y) field is used.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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Projected kurtosis change and its decomposition, based on 26 CMIP5 RCP8.5 ensemble members at the 850-hPa level during DJF. The (a) projected excess kurtosis change ∆(K – 3), (b) total change in the estimated excess kurtosis $\mathrm{\Delta }\left(\tilde{K}-3\right)=\mathrm{\Delta }\left(a{S}^2+b\right)$, and (c) their difference, and the decomposition of the approximated kurtosis change into (d) ∆(a)S ², (e) a∆(S ²), and (f) ∆(b). Here the longitudinally independent estimations a(y) and b(y) are used, while for skewness the full S(x, y) field is used.

Citation: Journal of Climate 35, 1; 10.1175/JCLI-D-21-0310.1

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The advantage of examining the estimated kurtosis change is that we can now decompose it into each of the terms in (17) for $\mathrm{\Delta }\tilde{K}$. The decomposition in Figs. 9d–f shows that the estimated kurtosis change is dominated by the skewness squared changes, through the term aΔ(S ²) (Fig. 9e), while the other terms are considerably smaller (Figs. 9d,f). Even though these are calculated for a(y) and b(y) only, it is clear from comparing Figs. 9a and 9b that these are reasonable approximations, and that indeed most of the change in kurtosis originates from the changes in the skewness squared. This implies that kurtosis changes can be predicted to first order by the skewness squared changes:

$\mathrm{\Delta }\tilde{K}\approx a\mathrm{\Delta }\left(S^2\right)=2aS\mathrm{\Delta }S.$(18)

It is easy to see from (18) that if skewness changes oppose the historical skewness [SΔ(S) < 0], then the kurtosis change will be negative, while if the skewness change reinforces the historical skewness [SΔ(S) > 0] then the kurtosis change will be positive. In other words, if the change in the asymmetry between cold and warm anomalies increases, then kurtosis increases too, since more of the PDF must lie in one of the tails (and vice versa if the asymmetry decreases).

In addition, (18) implies that the meridional advection arguments used in previous studies (Garfinkel and Harnik 2017; Linz et al. 2018; Tamarin-Brodsky et al. 2019; Linz et al. 2020; Tamarin-Brodsky et al. 2020) to explain skewness changes can also be used to understand kurtosis changes. For example, it was suggested that the skewness increase projected over most of the NH during winter (Fig. 8c) can be understood by linear advection arguments (Tamarin-Brodsky et al. 2020), since cold anomalies advected from the Arctic encounter a significantly reduced background temperature gradient compared to warm anomalies advected from the tropics (and the cold anomalies therefore weaken more than the warm anomalies). The positive skewness change, together with the spatial structure of the historical skewness (Fig. 8a), are enough to understand the sign of the projected kurtosis changes (Fig. 8d); these are generally positive where the sign of ∆S is the same as S, and negative where the sign of ∆S is opposite to that S, as predicted from (18).

We note however that in some regions, the contributions from changes in a, and especially b, could be important too (as seen from Figs. 9d,f) and could reflect changes in extremes that do not contribute directly to an asymmetry between cold and warm anomalies (e.g., through changes in $\widehat{\sigma }$ or p ₀).