요 약

올해 노벨 물리학상은 기후 시스템과 인위적 온실가스 배출에 대한 대응과 관련된 우리의 이해에 근본적 기여를 한 두 명의 기후 과학자에게 공동 수여되었다. 그들의 연구는 인간이 유발한 기후 변화를 이해, 시뮬레이션 및 감지하는 데 이정표가 되었다. 이 두 분 중 클라우스 하셀만(Klaus Ferdinand Hasselmann) 교수는 1976년에 아인슈타인에 의해서 제안된 브라우니안 운동(1905년)을 기후모델에 처음으로 적용하는 연구를 제안하였다. 그 후 확률적으로 교란된 에너지 균형 모델에서 파생된 하셀만 교수의 확률적 기후모델의 비선형 버전은 빙하기, 전자 회로, 중력파 간섭계, 수중 음향, 생물학적 시스템 및 급격한 기후 변화 이벤트에 이르는 동적 시스템 분석 및 응용 분야의 핵심 개념 즉 “확률적 공명(Stochastic Resonance)”으로 이어졌다. 흥미롭게도 1980년대 발간된 다양한 확률적 공명 관련 논문들에는 1976년에 발표된 하셀만 교수의 이론을 명확하게 언급하고 있으며, 올해의 또 다른 노벨 물리학상 수상자인 조르조 파리시(Giorgio Parisi) 교수가 공저자로 참여한 논문도 그 중 하나이다. 하셀만 교수의 2번째 기념비적인 업적은 기후 CSI (Crime Scene Investigation)로 알려진 “최적지문법(optimal fingerprinting)”의 개발이다. 이 방법은 시공간에서 데이터를 필터링하는 독창적인 방법으로 20세기 온도 관측에서 인간이 지구 온난화에 책임이 있다는 것을 명백하게 입증하는 핵심 요소였다는 것을 설명할 수 있었다. 결론적으로 하셀만 교수의 독창적인 기여 덕분에 “탐지와 원인분석(detection과 attribution)” 분야는 통계적 기후학의 가장 진보된 분야 중 하나가 되었다.

This year’s Nobel prize in physics is awarded - in part - to two climate scientists, who made fundamental contributions to our understanding of the climate system and its response to anthropogenic greenhouse gas emissions. Their research marks a milestone in understanding, simulating, and detecting human-induced climate change.

Klaus Hasselmann

Prof. Klaus Hasselmann, emeritus Director of the Max Planck Institute of Meteorology in Hamburg, Germany is recognized for his ground-breaking contributions in understanding how weather and climate interact and how global warming can be unequivocally distinguished in temperature observations from the internally generated variability in the climate system.

His 1976 paper “Stochastic Climate models” has become a milestone in climate physics - a must-read for every graduate student, ever since it was published. It adopts the concept of Brownian motion, which was first mathematically described by Einstein,1) to the climate system. Variations in the climate system are generated by the continuous accumulation of fast weather fluctuations, combined with amplifying, or damping feedbacks. This fundamental concept essentially explains 95% of all naturally occurring climate fluctuations. Climate scientists also call it the “Null hypothesis model” of climate.

The following paragraphs will give a brief and simplified introduction to Hasselmann’s stochastic climate model. In the 1960 and 70s climate researchers were on the one hand well equipped to understand the drivers of weather fluctuations, which occur on all timescales, but which have a preferential e-folding timescale of their autocorrelation (“decorrelation timescale”) of a few days up to 2 weeks due to the nonlinear, chaotic dynamics of the underlying equations of motion.2) On the other hand, climate researchers developed simple theories of forced climate variability, e.g. associated with Milanković forcing3) or solar variability. What was missing in the 1970s was a general understanding of how climate variability emerges in an unforced climate system. Klaus Hasselmann introduced a simple mathematical framework, which describes climate variability in terms of weather-generated and climate-modulated red noise.

According to Hasselmann, the climate system and its relationship to fast (decorrelated) weather fluctuations can be compared to Brownian Motion.1) According to Einstein’s theory, the erratic movement of pollen grains in a glass of water can be explained by the continuous random bombardment of individual excited water molecules. This can be described in terms of a stochastic differential equation.

To better understand the analogue between Climate/Weather and Brownian motion, let us first introduce a climate state vector \(\small z_i\) which contains a collection of climatic variables (e.g. temperature, pressure, etc.) at different grid points. The general evolution equation for this climate state can be described by a differential equation of the following type \(\small \frac{dz_i}{dt}=w_i (z)\). We split up the fluctuations into fast (short decorrelation timescale) weather fluctuations \(\small x_i\) and slower (longer decorrelation timescale) climate variables \(\small y_i\), which can be expressed by the following conditions:

\( \left [ x _{i} \left ( \dfrac{dx _{i}}{dt} \right) ^{-1} \right] = \tau_{x} \ll \tau _{y} = \) \( \left[ y_i \left( \dfrac{dy_i}{dt} \right)^{-1} \right] . \)

The evolution equations for weather and climate can be written in their most general form as \(\small \frac{dx_i}{dt} = u_i (x, y)\), \(\small \frac{dy_i}{dt} = \nu_i (x, y)\). According to these general equations, weather depends on climate and climate fluctuations depend on weather. But we have not fully implemented the timescale separation idea yet, except for the introduction of the variables \(\small x_i , y_i\). Let us look at the gradual evolution of a deviation \(\small \delta y_i (t)\) away from an initial climate state \(\small y_o\) for a period, which is still short in comparison to the climate decorrelation timescale \(\small \tau_y\). The deviation can be written as \(\small \delta y_i (t) = y_i (t) - y_{i,o}\). We need to acknowledge, that every weather realization will generate a slightly different climate trajectory. To account for this factor, we introduce stochasticity in the system, hence the mathematical model is referred to as stochastic climate model. Our climate evolution \(\small \delta y_i (t)\) can then also be expressed as the sum of an ensemble deviation \(\small y_i^\prime (t)\) and the ensemble mean \(\small \left< y_i \right>\) (averaged over all possible x-(weather) realizations but fixed \(\small y_o\)). We obtain: \(\small \delta y_i (t) = y_i^\prime (t) + \left< y_i \right>\). The ensemble means evolution of the disturbance is then governed by \(\small \left< \delta y_i (t) \right> = \left< \nu_i \right>t\), because there is no explicit \(\small y_i\) dependence on \(\small \nu_i\) for time periods smaller than \(\small \tau_y\), only a remaining \(\small x_i\) dependence. Therefore, we can now write the time evolution equation of the climate perturbation around the ensemble mean as: \(\small \frac{dy_i^\prime}{dt} = \nu_i (x, y) - \left<\nu_i\right> = \nu_i^\prime (x^\prime )\). The variable \(\small \nu_i^\prime\) is characterized by fast fluctuations because it only depends only the anomalous (deviation from ensemble mean) weather state. Using Fourier integrals we can then write:

\[ \frac{dy_i^\prime}{dt} = \nu_i^\prime (x^\prime ) = \int_{-\infty}^{\infty} V_i (\omega) e^{ i \omega t}d \omega \tag{1}\]

where \(\small V_i\) represents the Fourier transform of \(\small \nu_i^\prime\). We also use

\[y_i^\prime (t) = \int_{-\infty}^{\infty} Y_i (\omega) e^{ i \omega t}d \omega , \tag{2}\]

which after time differentiation and comparison with Eq. (1) leads the relation

\[ Y_i (\omega) = \frac{V_i (\omega)}{i \omega} . \tag{3}\]

We further assume that \(\small \nu_i^\prime\) represents a stationary stochastic process, which implies that all Fourier components are orthogonal to each other, i.e.

\[ \left\langle V_i (\omega) \tilde{V_j} (\omega^\prime) \right\rangle = \delta (\omega - \omega^\prime )F_{ij} (\omega), \tag{4}\]

where \(\small F_{ij}(\omega)\), represents the spectral co-variance tensor. From these assumptions, we can now calculate the spectrum of the process \(\small y_i^\prime (t)\), which characterizes climate fluctuations. According to the Wiener-Khinchin theorem, the power spectrum \(\small P(\omega)\) can be obtained from the Fourier transform of the auto-correlation function. Therefore, the correlation tensor of the climate anomaly \(\small y_i^\prime (t)\) (deviation from ensemble mean) can be expressed as:

\[ \langle y_i^\prime (t) y_j (t^\prime) \rangle = \int_{-\infty}^{\infty} P(\omega) e^{i \omega (t-t^\prime)} d\omega = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} Y_i (\omega) \tilde{Y_{ij}} (\omega) e^{i \omega t} e^{i \omega^\prime t^\prime} d\omega d\omega^\prime .\]

Using Eqs. (3) and (4), we can then write the power spectrum (variance as a function of frequency) of our climate anomalies as

\[ P(\omega) = \frac{F(\omega)}{\omega^2} . \tag{5}\]

It is fair to say that this equation is one of the most fundamental equations in climate dynamics because it links the spectral input of the weather \(\small F(\omega)\) with the power spectrum of naturally occurring climate fluctuations (Fig. 1). The power for low frequencies is increased relative to high frequencies. This effect is referred to as reddening of the climate spectrum and it can be observed for a wide range of climatic phenomena. The singularity for zero frequency emerges from the lack of explicit negative feedbacks in the stochastic model, which could limit energy growth. Equation (1) shows that climate fluctuations emerge essentially as a time-integration of rapidly fluctuating weather functions, which can be considered by themselves as a characterization of weather noise.

Hasselmann’s 1976 stochastic climate model has not been used very much in its most abstract version (Eq. (5)), except for several stochastic dynamical systems’ theory considerations, where Hasselmann’s ensemble approach and two-timescale separation has been adopted enthusiastically.4)5) However, simplifications of the 1976 model, such as the linear feedback version,6) have become very popular, because they can explain for instance observed sea surface temperature spectra with great accuracy. Nonlinear versions of Hasselmann’s stochastic climate model, derived from stochastically perturbed energy balance models, have led to the discovery of “Stochastic Resonance”7)8)9) - a key concept in dynamical systems analysis and applications that range from ice ages, electronic circuits, gravitational wave interferometers, underwater acoustics, biological systems and abrupt climate change events. The original Stochastic Resonance papers9)10) in the early 1980s explicitly acknowledge Hasselmann’s 1976 paper. Interestingly Giorgio Parisi, the 3rd Physics Nobel laureate in 2021 was a co-author of the Benzi et al. 1982 paper, thereby highlighting early connections in the science of Klaus Hasselmann and Giorgio Parisi. It is fair to say that Hasselmann’s stochastic climate greatly inspired the early work on stochastic resonance, which eventually developed into an extremely active and independent research field on its own with wide-ranging applications. Other conceptual extensions of the stochastic climate model include multiplicative noise,11) higher order noise terms,12) and spatial advection.13)14) In these cases, the dynamical models can generate noise-induced transitions, noise-induced instability, and spatial resonance, respectively. 

Here we briefly review the most famous linear version of the stochastic climate model.6)15)16) Let’s assume that an ocean layer with depth \(\small h\) is well mixed in temperature \(\small T\) and that it is forced by net surface heat fluxes \(\small Q\). We ignore lateral heat advection into the ocean slab and phase transitions of the water and can write down the temperature tendency equation for temperature \(\small \frac{dT}{dt} = \frac{Q}{c_p \rho_w h}\) where \(\small \rho_w\) and \(\small c_p\) are the density and heat capacity for seawater, respectively. The heatflux \(\small Q\) depends on many different factors, among them the sea surface temperature itself. Additionally, we introduce the fluctuations around a long-term mean state \(\small T = T^\prime + \bar{T}\). We further focus only on the temperature sensitivity of the heat fluxes and apply a Taylor expansion to the heat equation. Under certain assumptions, which I will not further elaborate on here, the temperature equation can then be approximated as

\[\frac{dT^\prime}{dt} = -\lambda T^\prime + \xi (t) ,\tag{6}\]

where we assumed a linear damping feedback between heat flux and ocean temperature. This situation can be found in wide swaths of the extratropical oceans. \(\small \xi (t) \) represents a temperature-independent heat flux contribution caused by the fast weather contributions, which can be represented as a white noise term with variance \(\small \sigma^2\). The autocorrelation function of a Gaussian white noise process can be expressed as \(\small \langle \xi (t) \xi (t^\prime ) \rangle = \delta (t - t^\prime )\sigma^2\). The power spectrum of temperature of this specific linear stochastic climate model Eq. (6) can be readily calculated by writing down Eq.(6) with the respective Fourier transforms \(\small \int_{-\infty}^{\infty} i \omega \mathit{\Theta}( \omega) e^{i \omega t} d\omega\)\(\small = \)\(\small -\int_{-\infty}^{\infty} \lambda \mathit{\Theta} (\omega) e^{i \omega t} d\omega\) \(\small +\) \(\small \int_{-\infty}^{\infty} V (\omega) e^{i \omega t} d\omega\). From this, it follows that \(\small \mathit{\Theta}(\omega) = \frac{V}{\lambda + i \omega}\) and after further manipulation: \(\small P(\omega) = \frac{\sigma^2}{\lambda^2 + \omega^2}\). This power spectrum is also known as the Lorentz spectrum of an Ornstein-Uhlenbeck process. The power spectrum of upper mixed layer ocean temperature fluctuations is completely determined by the variance of the white weather noise and the linear damping feedback between surface temperature and heat flux damping. The analogy with Brownian motion is as follows: the climate fluctuations (mixed layer ocean temperature variability) correspond to the macroscopic movement of a bigger particle (pollen on fluid surface) within the rapidly fluctuating environment of random weather events (microscopic molecular bouncing in Brownian motion case). Figure 1 (lower panel) illustrates the situation more clearly: white weather noise has constant variance on all timescales, although it has an infinitely short decorrelation timescale (note: the Fourier transform of a delta function is constant). The linear damping mechanism suppresses climate variations with \(\small \lambda < \omega\) and in this regime we obtain a power-law behaviour of the spectrum \(\small P(\omega) \sim \omega^{-2}\). For this situation, the model acts as a low-pass filter, which also becomes evident when we study the explicit solution of Eq. (6): \(\small T^\prime (t)\) \(\small =\) \(\small T^\prime (t_o )\) \(\small +\) \(\small \int_{t_0}^{t} e^{-\lambda(t-t^\prime)} \xi (t^\prime ) dt^\prime\), where we can see the damping kernel \(\small e^{-\lambda(t-t^\prime)} \), which serves as a weighting factor of the noise which is then further integrated. The larger \(\small \lambda \) (i.e. the smaller the damping timescale and stronger the damping), the fewer past noise components are considered in the integration. For small \(\small \lambda\) and weak damping, more noise contributions from further back in time are considered. In the second region, \(\small \lambda > \omega\), the power spectrum flattens \(\small P(\omega)\) \(\small \sim \) const. and the low frequency ocean dynamics is strongly determined by the low frequency atmospheric dynamics.

The original 1976 paper includes various other interesting titbits, such as a Fokker Planck formulation of climate variability and its application in the context of climate predictability. We leave this to the interested reader to explore further. The Hasselmann 1976 stochastic climate model paper can be downloaded from the website of the Max Planck society under: https://pure.mpg.de/rest/items/item_2514711/component/file_3220884/content.

It is worth looking back to understand why Hasselmann’s 1976 Tellus paper created such a paradigm shift in climate research. Prior to its publication, climate was mostly regarded as something that changes only very slowly – on timescales of \(\small > \) 30 years. Ice-ages were understood as the response of the complex earth system to astronomical forcing, associated with the dominant Milanković cycles of precession (earth’s wobble combined with precession of equinoxes), obliquity (earth axis tilt) and eccentricity (changes in orbital ellipticity) and its dominant periodicities of 21,000, 41,000 and 80,000-120,000 and 400,000 years. Moreover, already since the middle of the 19^th century, climate scientists were interested in how quasi-periodic changes in solar radiation, related to the major sunspot cycles with timescales of 11, 22, 80, and 300 years, could change the climate on our planet. Most of this research was statistical in nature, speculative and inconclusive. In the 1960s, partly due to pioneering research of Jacob Bjerknes, Jerry Namias, Klaus Wyrtki, it became clear that large-scale temperature variations in the Pacific do not require any external forcing. This became very apparent when researchers developed a better physical understanding of the El Niño-Southern Oscillation (ENSO) phenomenon, which characterizes the occasional warming (El Niño) and cooling (La Niña) of a \(\small \sim\)12,000\(\small \times\)4,000 km large area in the equatorial Pacific. In the late 1960s it was observationally demonstrated that a combination of extreme weather phenomena (so-called westerly wind bursts) can trigger an El Niño event.17) According to this view the accumulated effect of weather serves as a driver for major natural climate anomalies, which resembles the Hasselmann concept. But still in the late 1960s and early 1970s, a general theory that described how unpredictable weather can generate climate fluctuation on longer timescales was missing. Hasselmann’s 1976 paper filled this vacuum and provided an elegant conceptual framework (Einstein’s Brownian motion) that allowed climate researchers for the first time to understand the mechanisms of natural climate variability. Even nowadays extended Hasselmann-like models, also referred to as recharge oscillator models18) are used to capture the dynamics and predictability of ENSO.19)

It certainly is a peculiar coincidence that Klaus Hasselmann receives the Nobel Prize in Physics 100 years after Albert Einstein...

Prof. Klaus Hasselmann was also interested in applications of his stochastic climate model. To this end he developed an ingenious empirical method20) to derive stochastic climate model equations directly from multi-variate climate data sets. PIPs, principal interaction patterns are nonlinear parametric dynamical models that fit the complex spatio-temporal time-evolution of climate data in an optimal way. The derivation of these PIPs, using nonlinear optimization theory is cumbersome and unfortunately, only a few studies have applied this powerful method to climatic or hydrodynamic data.21)22)23)

However, the linear version of Hasselmann’s multivariate empirical model derivation method,20) which is referred to as Principal Oscillation Patterns (POPs), became widely popular in the context of climate variability and predictability research.38) Given a spatially discretized data field \(\small z_i (t)\), we can ask the question which matrix \(\small A_{ij}\) best matches the following vector tendency equation \(\small \frac{dz_i}{dt} = A_{ij}z_j + \xi_i (t)\), where \(\small \xi_i\) represents a Gaussian noise component. Essentially the task is to fit a multivariate linear stochastic climate model (similar to Eq.(6)) to empirical data and study the eigenmode properties of this empirical surrogate model. The task can be accomplished easily when the data are discretized in time. Then \(\small A_{ij}\) can be linked to the lag-1 auto covariance matrix. The eigenvectors, which in general are complex, along with the time-evolution coefficients (eigenvalues) describe the temporal evolution of the dominant multi-variate modes of climate variability captured in the climate data. This extremely powerful method, which has now become one of the leading contenders of operational seasonal El Niño predictions24) can be regarded as a direct application of Hasselmann 1976 Tellus paper to real world data.

The second breakthrough contribution recognized by the Nobel committee is Prof. Hasselmann’s 1993 optimal fingerprinting method, sometimes also nick-named climate-CSI,39) which has made it possible to detect the human effect in 20^th century temperature observations by an ingenious way of filtering the data in space and time. The application of the optimal fingerprint to temperature observations has been a key component in unequivocally demonstrating that humans are responsible for global warming.25)

The projections of the observations and the natural variability data (estimated, e.g., from an unforced natural climate model simulation conducted with the Coupled General Circulation Model) on the optimal fingerprint \(\small \sigma_i\)— the so-called detection variables \(\small \kappa (t)=\sigma^i e_i (t)\) — are then subject to further statistical tests. Detection at a given significance level is achieved if the hypothesis that the observed detection variable can be explained by natural climate variability alone can be rejected at that statistical level. As an example, we can identify when the detection variable \(\small \kappa (t)\) leaves the 2 standard deviation variability spanned by the natural variability \(\small \kappa^n (t) = \sigma^i e_i^n (t)\) and refer to this point as the time of detection of the anthropogenic signal.

The calculation of the fingerprint pattern, which bears some similarity to the mathematical operation of a “principal axis transformation”, can be elegantly performed in a truncated empirical orthogonal functions (EOF) space.40) We express the natural climate variability \(\small e_i^n\) in terms of a superposition of stationary basis functions, called EOFs \(\small g_j^k\), which are orthogonal to each other and which account for as much of the temporal variance as possible (first mode \(\small g_j^1\), explains most variance of the original observational data \(\small e_i^n\), second one optimizes the residual variance under the orthogonality conditions) (Fig. 2). Together with the corresponding projection coefficients \(\small \gamma^k (t) = e_j^n (t)g_j^k \) (referred to as principal components), which can be obtained from the scalar product between EOF pattern and observational data, we obtain \(\small e_i^n = \sum_k \gamma^k (t) g_i^k\). It can be shown that the EOFs are the eigenvectors of the covariance tensor \(\small C_{ij} = \langle e_i^n (t) e_j^n (t) \rangle\), which means that the calculation of the optimal fingerprint in the EOF transformed variables is particularly straightforward, because in EOF space \(\small C_{ij}^{-1}\) turns into a diagonal matrix with the inverse of the ranked eigenvalues as diagonal elements.

The mathematical formalism was originally developed by Klaus Hasselmann in 1979 and subsequent papers by Thomas L. Bell from the Goddard Space Flight Center, Greenbelt, USA in 1986 and by Hasselmann in 199327) refined the optimal fingerprint detection method. It was applied by Gabriele Hegerl (now at the University of Edinburgh, UK) and Hans von Storch (retired; formerly, GKSS, Germany) and Klaus Hasselmann in 1996 to a gridded compilation of land surface temperature data.25) The time evolution of the optimal detection variable demonstrated that the 20-year temperature trends ending in 1994 already left the envelope of natural climate variability, which was estimated from unforced natural climate computer model simulations. This served as a statistical “proof” that the human effect on surface temperature patterns (not only global mean value) was already visible. The key sentence from the Hegerl paper,25) which was widely discussed in the world’s media at that time is: “It is concluded that a statistically significant externally induced warming has been observed, …”. This key result was included in assessment report of the Intergovernmental Panel on Climate Change (IPCC) and a “climate change detection and attribution” chapter has become one of the center pieces of the 2^nd‒5^th IPCC assessment reports of Working Group 1. The Max Planck Institute landmark papers on climate change detection from the mid to late 1990s triggered an explosion of subsequent studies,28)29)30)31)32)33) which essentially used variants of Hasselmann’s original optimal fingerprinting method. Later on, it was further demonstrated that the optimal fingerprint method is equivalent to a regression with respect to generalized least squares.34)

The original optimal fingerprint method was further extended by Prof. Klaus Hasselmann and his collaborators to account for the existence of multiple forced patterns, e.g. due to solar forcing, aerosol forcing, and greenhouse gas forcing.35)36)37) This extension allowed scientists to “attribute” the observed temperature pattern to different types of physical forcing mechanisms as well as natural climate variability. A plethora of applications has been developed by an entire generation of climate scientists for various climate variables, including surface temperature, vertical atmospheric warming profiles, precipitation, marine biogeochemical variables, and extreme events. It is fair to say that thanks to the original contributions of Prof. Klaus Hasselmann, the field “detection and attribution”, has become one of the most advanced disciplines of statistical climatology.

Suki Manabe

(more details are provided in the manuscript of Prof. Ha, Kyung-ja)

The other “quarter” of the Nobel prize in physics is awarded to the founder of modern climate modeling: Dr. Syukuro (Suki) Manabe. Already in the 1960s Dr. Manabe began building the first computer models that solved the complex physical equations of the climate system on the first generation of supercomputers at the Geophysical Fluid Dynamics Laboratory in Princeton, USA. Initially the models described only the atmospheric motion, but later, thanks to a long-lasting collaboration with oceanographer Dr. Kirk Bryan, also the ocean circulation.

These so-called Coupled General Circulation models demonstrated for the first time, that a doubling of atmospheric CO₂ concentrations will increase the temperatures of our planet by about 2‒4 degrees Celsius and that warming on land and in polar regions will be even higher than the global mean. Dr. Manabe’s pioneering work, which also included the first ice age simulations, led to the development of climate models in other climate research centers worldwide, including Prof. Hasselmann’s Max Planck Institute in Hamburg. Tested repeatedly by different research teams and later paraphrased by the Intergovernmental Panel on Climate Change, the conclusion from thousands of independent climate model simulations is simple: anthropogenic greenhouse gas emissions cause global warming. If mankind does not stop emitting carbon into the atmosphere, our planet will continue to warm, sea-level will rise and large-scale rainfall patterns will shift.

I have had the pleasure and privilege to interact with both Nobel laureates in climate physics; with Prof. Hasselmann as my official PhD advisor at the Max Planck Institute of Meteorology and with Dr. Syukuro Manabe as a colleague, with whom I shared a common interest in North Atlantic Ocean circulation variability and ice ages. Both impressed and inspired me deeply by their intellectual generosity, their inexhaustible curiosity, and their motivation to make ground-breaking contributions to a research field with far reaching implications for humanity.

Acknowledgements

This study was supported by the Institute for Basic Science (project code IBS-R028-D1). A.T. acknowledges discussions with Prof. Christian Franzke from the IBS Center for Climate Physics.