Climate: Past, Present & Future

Climate of the Present

God does not play DICE – but Bill Nordhaus does! What can models tell us about the economics of climate change?

Climate change has been described as “the biggest market failure in human history”[1]. Although fuel is costly, emitting the by-product CO2 is for free; yet it causes damages to society. In other words, those who benefit, by using the atmosphere as waste dump, do not pay the full costs, i.e. the adverse effects climate change has on societies on a global scale. Can this market failure be cured? Should humankind sacrifice some of its present welfare to prevent future climate damages? William Nordhaus was jointly awarded the Nobel Prize for economy for providing a framework to answer these questions.

DICE– the Dynamic Integrated model of Climate and the Economy [2] – combines a simple economic model with a simple climate model. The aim is not to fully cover all details of economic and climate processes, but rather to provide a model that is sufficiently simple to be used by non-specialists, including policy makers. Figure 1 shows a simplified structure of the DICE model.

Figure. 1: Schematic illustration of the DICE model. The dark blue arrows correspond to the purely economic component of the model. The yellow and green arrows indicate how the economy impacts climate and vice versa. The light blue arrows illustrate the effect of climate policy.

The economy of the DICE model

The heart of DICE is an economic growth model (dark blue arrows in fig. 1). Economic production occurs when labour and capital is available. Labour is proportional to the world population, which is homogenous and grows according to externally prescribed data. Part of the economic production is invested to create capital for the next time step, while the remaining part is consumed. It is assumed that the “happiness” (called utility in the jargon) of the population depends exclusively on consumption, in a sublinear fashion: The more you consume, the happier you are. However, if you are already rich, then one extra Euro will not increase your happiness as much as when you are poor.

In this purely economic model, the only decision the world population has to take is to determine the saving rate – the fraction of economic production to invest for the next period. If we invest too much, we reduce our current happiness; if we invest too little, we have too little to consume next period. Therefore, the aim is to find an optimal pathway to be reasonably happy now and in the future. However, there is a twist: observations suggest that we, humans, value the present more than the future. E.g. if we are offered 1 Euro either now or next year, we would prefer to be paid now, even in absence of inflation or increasing income. However, if offered 1 Euro now or 1.03 Euro next year, we might begin to prefer the delayed, but larger payment. The extra amount needed to make later payment acceptable is called “Rate of Pure Time Preference”; in our example, it is 3% [3, p.28]. A high Rate of Pure Time Preference basically means that we care much less about future welfare than about the present one. If there is an economic growth (which is the case in DICE), there is an additional reason to prefer being paid now rather than later: In the future, you will be richer, so one additional Euro will mean less to you than now while you are still relatively poor. This effect means that the total “discount rate”, defined as the extra payment needed to make delayed payment attractive, is even higher than the rate of pure time preference [3, chapter 1] .


The impact of climate change

To bring climate change into play, Nordhaus assumed that apart from labour and capital, economic production also requires energy. However, energy production causes CO2 emissions. Part of the CO2 ends up in the biosphere or in the ocean, but another part remains in the atmosphere, leading to global warming.

Practically, everyone agrees that substantial warming will have damaging effects on the economy. Although there may not be “good” or “bad” temperatures a priori, ecosystems and human societies are adapted to the current climate conditions, and any (rapid) change away from what we are accustomed to will cause severe stress. For example, there may not be an “ideal” sea level, but strong sea level rise – or fall – will cause severe strain on coastal communities who are adapted to the current level[4].

These damages are extremely hard to quantify. First, we obviously have no reliable empirical data – we simply have not yet experienced the economic damages associated with rapid warming by several degrees. Second, there could be “low chance, high impact” events [5], e.g. events that even under climate change are deemed unlikely to our current knowledge, but would have dramatic consequences if they occur – for example, a collapse of large parts of the Antarctic ice sheet. Third, there are damages, like the loss of a beautiful glacial landscape or the human suffering inflicted by famine, which cannot be quantified in terms of money.

When formulating his Nobel prize-winning DICE model, William Nordhaus tried to solve the first problem by performing an extensive review of the (scarce) existing studies on climate-induced damages and greatly extrapolating the results. E.g. if data was available on reduced wheat production in the Eastern US during a heat wave, Nordhaus might assume that damage for all food crops in Africa is, say, twice as big (as Africa is more dependent on agriculture than the US). This may still be quite ad-hoc, but one might argue that even rough data is better than no data at all. The second and third of the above points where largely circumvented with the “willingness-to-pay” approach [2]: people were asked how much they would pay to prevent the extinction of polar bears or the collapse of the Antarctic ice sheet, for example, and the price they names was used as substitute for damages associated to these events.

Finally, Nordhaus came up with an estimate for climate damage:

D=k1T + k2T2

where D is the damage in % of the GDP, T is the global mean temperature change, and k are constants (k1 = -0.0035 K-1 and k2=+0.0045 K-1) [2, p. 207]. Note that the k1<0 implies that for small T, global warming is actually beneficial. 2.5 degree and 5 degree warming yield damages of 1.1% and 6.5% of the GDP, respectively. Later versions of DICE have k1=0.


To reduce global warming, humanity can reduce their carbon emissions. In other words, part of the global economic production is sacrificed to pay for greener energy. This will leave less money to spend on consumption and/or investment in capital, but it also diminishes future climate damages. Therefore, in order to maximise the “happiness”, two control variables must now be chosen at each time step: the saving rate and the emission reduction fraction. We want to reduce carbon emissions enough to avoid very dangerous climate change, but also avoid unnecessary costs.



Figure. 2 Results of the DICE model. The optimal policy (i.e. maximising “happiness”) in the 2013 version of DICE. The blue lines indicate the optimal policy, while yellow lines indicate no climate policy (i.e. zero emission reduction). The first plot shows the emission reduction fraction or “abatement”, i.e. the fraction of carbon emissions that are prevented. 1 means that no CO2 is emitted. The second plot shows the atmospheric CO2 concentrations in ppmv. For the optimal policy, CO2 concentrations peak at 770ppmv, whereas in absence of a policy, they rise beyond 2000ppmv. The pre-industrial value is 280ppmv. The third plot shows the global mean temperature change. For the optimal policy, it peaks at about 3.2K, i.e. above the limit of 2K or even 1.5K agreed by the Paris agreement.


Results and Criticism

The results in fig. 2 show that under the “optimal” policy, i.e. the policy which maximises “happiness”, the Paris agreement will not be met. This result suggests that the costs required for keeping global warming below 1.5 or 2ºC warming are too high compared to the benefit, namely strong reduction in climate damages. However, some researchers
criticise that DICE severely underestimates the risks of climate change. For example, the damage function might be too low and does not explicitly take into account the risk of “low chance, high impact events”. Including such events, occurring at low probability but causing high damages if they occur, will lead to more stringent climate action [6].

The rate of pure time preference has given rise to even fiercer discussions [7,8,9]. As explained above, a society’s discount rate can be estimated from market interest rates [3]. Knowing the economic growth, we can infer the rate of pure time preference used in market decisions. Many economists argue that the rate of pure time preference in models like DICE should be chosen consistent with observations[8]. Nordhaus followed this approach. However, one can argue that even if individuals care less for the future than the present, this does not mean that such an approach is ethically defendable in the context of climate change. Individuals are mortal and may choose to consume their own savings before they die. But climate change is a global and intergenerational problem, and it has been argued [7,9] that we should care for future generations as much as for ourselves. Therefore the rate of pure time preference must be (nearly) zero. Note that this still allows for some discounting, arguing that if future generations are richer, they might be able to deal better with climate change.

Another reason for the relatively weak carbon emission reduction in DICE’s optimal policy may be that it is too pessimistic concerning future costs of emission reduction. For example, DICE does not include the learning-by-doing effect: The more we reduce emissions, the more efficient technologies we discover, and the cheaper it gets. In addition, the costs for green energy are partly one-time investments, e.g. restructuring the energy distribution grids, which are now adapted for few, central energy providers, to a more decentralised structure with smaller providers (e.g. households with solar panels). Once these (large) efforts have been made, the costs for green energy will decrease. But if DICE overestimates the costs of carbon emission reduction, it will be biased towards recommending low reductions.

Due to the above, and many more, issues some researchers criticise that models like DICE are “close to useless”, and even harmful, as they pretend to give precise instructions to policy makers while in fact they struggle with huge uncertainties [10]. In my opinion, models like DICE should not be used for precise policy recommendations like fixing the carbon tax, but are still useful for a somewhat qualitative scenario exploration. For example, it can be fruitful to add “low chance, high impact events” or the learning-by-doing effect and investigate the qualitative effect on the optimal abatement.

Many more economy-climate models have been written in the last decades, some of which are much more sophisticated than DICE. Moreover, there are many models focussing only on specific aspects of the problem, for example, the details of the energy sector. This is still a very active field of research. So, however limited DICE may be, it has laid the foundations for a highly relevant scientific and societal discussion. And even if one should take its precise output with a lump of salt, it is a valuable tool to help policy makers to qualitatively grasp the essence of climate economy.

This post has been edited by the editorial board.


[1] Nicholas Stern: “The Economics of Climate Change” ( RICHARD T. ELY LECTURE )

[2] A thorough description of the model is given by William Nordhaus and Jospeh Boyer, “Warming the World. Economic Models of global warming” ( There are newer model versions available, but the underlying concepts remain the same.

[3] A thorough introduction to discounting is given in this book: Christian Gollier, “Pricing the Future: The economics of discounting and sustainable development” (, especially chapter 1.

[4] see e.g. Wong, P.P., I.J. Losada, J.-P. Gattuso, J. Hinkel, A. Khattabi, K.L. McInnes, Y. Saito, and A. Sallenger, 2014: Coastal systems and low-lying areas. In: Climate Change 2014: Impacts, Adaptation, and Vulnerability. Part A:
Global and Sectoral Aspects. Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change

[5] e.g. Lenton et al. “Tipping elements in the Earth’s climate system”,

[6] Cai et al., “Risk of multiple interacting tipping points should encourage rapid CO2 emission reduction”,
[7] The Stern Review on the Economics of Climate Change (

[8] Peter Lilley: “What’s wrong with Stern?” (

[9] Frank Ackermann: “Debating Climate Economics: The Stern Review vs. Its Critics” (

[10] Robert Pindyck, “The Use and Misuse of Models for Climate Policy”,


What can artificial intelligence do for climate science?

What can artificial intelligence do for climate science?

What is machine learning?

Artificial Intelligence, and its subfield of machine learning, is a very trending topic as it plays an increasing role in our daily life. Examples are: translation programs, speech recognition software in mobile phones and automatic completion of search queries. However, what value do these new techniques have for climate science? And how complicated is it to use them?

The idea behind machine learning is simple: a computer is not explicitly programmed to perform a particular task, but rather learns to perform a task based on some input data. There are various ways to do this, and machine learning is usually separated into three different domains: supervised learning, unsupervised learning and reinforcement learning. Reinforcement learning is of less interest to climate science, and will therefore not be touched upon here.

In supervised learning, the computer is provided both with the data and some information about the data: i.e. the data is labeled. This means that each chunk of data (usually called one sample) has a label. This label can be a string (e.g. a piece of text), a number or, in principle, any other kind of data. The data samples could be for example images of animals, and the labels the names of the species. The machine learning program then learns to connect images with labels and, when successfully trained, can correctly label new images of animals that it has not seen yet. This principle idea is sketched in Figure 1.

Figure 1: A schematic of supervised machine learning in climate science.

In a climate context, the “image” might be a rough global representation of for example surface pressure, and the label some local phenomenon like strong rainfall in a small region. This is sketched in Figure 2. Some contemporary machine learning methods can decide which features of a picture are related to its label with very little or no prior information. This is well comparable to certain types of human learning. Imagine being taught how to distinguish between different tree species by being shown several images of trees, each labeled with a tree name. After seeing enough pictures, you will be able to identify the tree species shown in images you had not seen before. How you managed to learn this may not be clear to you, but your brain manages to translate the visual input reaching your retina into information you can use to interpret and categorize successive visual inputs. This is exactly the idea of supervised machine learning: one presents the computer with some data and a description of the data, and then lets the computer figure out how to connect the two.


FIgure 2: example of using machine learning for predicting local rainfall.


In unsupervised learning, on the other hand, the machine learning program is presented with some data, without any additional information on the data itself (such as labels). The idea is that the program searches autonomously for structure or connections in the data This might be for example certain weather phenomena that usually occur together (e.g. very humid conditions in one place A, and strong rainfall in place B). Another example are “typical” patterns of the surface temperature of the ocean. These temperature patterns look slightly different every day, but with machine learning we can find a small number of “typical” configurations – which then can help in understanding the climate.

How difficult is it to implement machine learning techniques?

Machine learning techniques often sound complicated and forbidding. However, due to the widespread use of many machine learning techniques both in research and in commercial applications, there are many publicly available user-ready implementations. A good example is the popular python library scikit-learn1. With this library, classification or regression models based on a wide range of techniques can be constructed with a few lines of code. It is not necessary to know how the algorithm works exactly. If one has a basic understanding of how to apply and evaluate machine learning models, the methods themselves can to a large extent be treated as black-boxes. One simply uses them as tools to address a specific problem, and checks whether they work.

What can machine learning do for climate science?

By now you are hopefully convinced that machine learning methods are: 1) widely used and 2) quite easy to apply in practice. This however still leaves the most important question open: can we actually use them in climate science? And even more importantly: can they help us in actually understanding the climate system? In most climate science applications, machine learning tools can be seen as engineering tools. Take for example statistical downscaling of precipitation. Machine learning algorithms are trained on rainfall data from reanalyses and in-situ observations, and thus learn how to connect large-scale fields and local precipitation. This “knowledge” can then be applied to low-resolution climate simulations, allowing to get an estimate of the local precipitation values that was not available in the original data. A similar engineering approach is “short-cutting” expensive computations in climate models, for example in the radiation schemes. If trained on a set of calculations performed with a complex radiation scheme, a machine learning algorithm can then provide approximate solutions for new climatic conditions and thus prevent the need to re-run the scheme at every time-step in the real model simulation, making it computationally very effective.

However, next to this engineering approach, there are also ways to use machine learning methods in order to actually gain new understanding. For example, in systematically changing the extent of input data, one can try to find out which part of the data is relevant for a specific task. For example “which parts of the atmosphere provide the information necessary to predict precipitation/windspeeds above a specific city and at a specific height?

As final point, artificial intelligence and machine learning techniques are widely used in research and industry, and will evolve in the future independent of their use in climate research. Therefore, they provide an opportunity of getting new techniques “for free”: an opportunity which can and should be used.



This article has been edited by Gabriele Messori and Célia Sapart.

What is in the (European) air?

What is in the (European) air?

You thought that Mauna Loa was the only observatory to provide continuous measurements of atmospheric carbon dioxide concentration and were disappointed because Hawaii is way too far from your study area or because you wanted to know how bad  the air is in your hometown? The US have been monitoring the composition of the atmosphere since 1972, but what about Europe? Since 2008, Europe has its own measurement network that is managed by a research infrastructure called ICOS (Integrated Carbon Observation System).


Since the beginning of the industrial era (around 1750), atmospheric concentrations of greenhouse gases such as carbon dioxide (CO2), methane (CH4) and nitrous oxide (N2O) have increased, mostly because of human activities. As a consequence, the climate is getting warmer, which could have dramatic impacts on our daily life. The evolution of the atmospheric composition should therefore be closely monitored.

To improve our understanding of the climate system and achieve good climate predictions, high-precision measurements of greenhouse gas sources and sinks are needed. A large amount of datasets already exists, but the problem is that these data are often too difficult to access, too scattered, not consistent or not reliable.

ICOS main objectives

This is why the main goal of ICOS is to provide scientists, citizens and decision makers with harmonized and high-quality measurements of greenhouse gases in Europe. But the scope of ICOS mission is wider because these data can further be used to:

  • quantify greenhouse gas budgets
  • improve climate predictions
  • check how well/badly European countries are doing in reducing their greenhouse gas emissions
  • adapt policies

ICOS also encompasses an educational dimension by training young scientists through summer schools, workshops and conferences and by spreading knowledge about the carbon cycle to the general public.


ICOS is subdivided in national networks managed by research institutes. Twelve countries are currently members of ICOS: Belgium, Czech Republic, Denmark, Finland, France, Germany, Italy, Netherlands, Norway, Sweden, United Kingdom and Switzerland. The regional dynamics of greenhouse gases is monitored thanks to a network of 126 measurement stations implemented across these countries. Among these stations, 71 are ecosystem stations, 34 are atmospheric stations and 21 are ocean stations (Figure 1). ICOS grows rapidly and 8 other countries are expected to become members soon: Poland, Ireland, Estonia, Portugal, Spain, Hungary, Greece and South Africa.

To be part of the ICOS standardized network, candidate sites have to follow strict specifications regarding equipment, measurement protocols and data processing in order to ensure a homogeneous dataset. Periodic measurements are also carried out across the network with independent instruments to limit systematic errors. Moreover, ICOS is planning to render its data products compatible with outputs from other international measurement networks by taking part in an intercomparison program.

Atmosphere stations (Figure 2)

Atmospheric CO2, CO and CH4 concentrations are continuously measured in atmosphere stations, together with a range of usual meteorological variables such as air temperature, atmospheric pressure, relative humidity, wind direction and speed.

Figure 2: Cabauw atmosphere station in the Netherlands (ICOS ERIC,

Ecosystem stations (Figure 3)

Flux towers measure the exchange of water vapour, greenhouse gases and energy between the different types of ecosystems and the atmosphere. The list of variables collected at ecosystem stations is available here.

Figure 3: Brasschaat ecosystem station in Belgium (ICOS ERIC,

Ocean stations (Figure 4)

Ocean stations include ships, fixed buoys and flux towers. Carbon fluxes are measured at the ocean-atmosphere interface together with other marine variables such as pH, temperature, or salinity. You can have a look at the exhaustive list of measurements here.

Figure 4: VLIZ data buoy ocean station (ICOS ERIC,

Data products

Data collected by national network stations are gathered, processed and stored by central facilities called Thematic Centers (TC): the Atmosphere Thematic Center (ATC), the Ocean Thematic Center (OTC) and the Ecosystem Thematic Center (ETC).

You can access all these precious data for free here on the carbon portal. Among many examples, you can find ecosystem fluxes time series, atmospheric methane observations or global carbon budget. It is easy to handle as you can apply filters to refine your search, click on the “eye” icon to preview the data, or just select the dataset to obtain its description.

These data are protected by a Creative Commons Attribution 4.0 international licence, which means you can share and even modify them provided that you document any change, mention the original data source and give a link to the licence text ( It is of course necessary to cite  ICOS when you use the data. To make this as easy as possible, the citation is provided when you download the data set.

On the website of the Atmosphere Thematic Center, you can also find near real time data that are computed from all ICOS atmospheric stations every day in the morning. For example, Figures 5 and 6 show time series of the fraction of CO2 (top plot) and CH4 (bottom plot) in the air mass coming from the European continent measured at Mace Head station (MHD). Depending on the wind direction, this atmospheric station, located on the west coast of Ireland, is exposed to either the North Atlantic Ocean air mass and or the European continental air mass, offering a unique way to study these very different air masses. Time series for the period 2011-2017 show a clear upward trend for both greenhouse gases in the continental air mass. These increases are mainly caused by growing emissions associated to human activities.

Figure 5: CO2 molar fraction in continental air mass between 2011 and 2017 at Mace Head atmospheric station (ICOS ERIC,

Figure 6: CH4 molar fraction in continental air mass between 2011 and 2017 at Mace Head atmospheric station (ICOS ERIC,


Hopefully, this post helped you to get to know ICOS better. Do not hesitate to use this great tool in the future!

Find out more about ICOS

For those interested, the 3rd ICOS Science Conference will take place between the 11th and the 13th of September 2018 in Prague, Czech Republic.

Edited by Gabriele Messori and Célia Sapart


Of butterflies and climate: how mathematics helps us to better understand the atmosphere

Applied mathematics is often seen as an obscure field, which the general public has no hope of ever understanding. In the context of climate science, this is far from the truth. In fact, many mathematical concepts and ideas applied to the study of the climate system stem from intuitive arguments. While their implementation can be very complex, understanding the basic ideas behind them does not require a PhD in Science.

The Lorenz 1963 attractor, often known as the “Lorenz Butterfly”. Author: Paul Bourke (

For example, this is the case of some recent developments in the field of dynamical systems analysis applied to atmospheric data. The atmosphere changes continuously and in many ways: for example, winds become stronger or die down, temperatures rise or fall and rain comes and goes. Understanding this evolution is important in many domains, from weather forecasting to air traffic management to catastrophe response services. The basic idea of the dynamical systems approach is to visualize the evolution of the atmosphere as a series of points connected with a line, which form a trajectory. The figure above shows a well-known example of such a trajectory: the so-called “Lorenz butterfly” (Lorenz, 1963). Now imagine focusing on a specific variable – for example daily surface temperature – and a specific region – let’s say Europe. We can build a trajectory, similar to the one shown above, describing the day-by-day properties of this two-dimensional (latitude by longitude, just as in a geographical map) temperature field. From day to day, the temperature varies therefore each day corresponds to a different point along the trajectory. In the case where two days are very similar to each other, they will correspond to two points very close together. On the contrary, if they show very different temperatures, the points will be further apart. If the similar days are well separated in time, for example occurring during different years, the trajectory representing surface temperature over our chosen region will therefore return close to a point it had previously visited, meaning that the closeness of the points and their distance in time do not always correlate. In the figure below, for example, the three turquoise dots are close to each other and also correspond to successive days along the atmospheric trajectory. The two red dots correspond to temperature configurations similar to those of the turquoise dots, but are separated from the latter by several days.

The continuous black line represents an idealized trajectory, while the circles correspond to successive days along the trajectory. The arrows indicate the direction the time goes.

This way of visualizing the atmosphere might seem bizarre, but it can give us some very powerful insights on how the climate system works. Consider, for example, summer heatwaves in Europe. The most severe ones can persist for several days and can have major impacts on human health, the environment and the economy. As can be intuitively understood, their persistence is due to the fact that the large-scale atmospheric conditions causing them are also persistent. If we return to our atmospheric trajectory, this will mean that we have a large number of points which are close to each other and successive in time – such as is the case for the three turquoise dots in the figure above. Namely, the trajectory moves very slowly and for several days the large-scale circulation only changes very slightly. In mathematical terms, this is a “sticky” state, and again the name is very intuitive! Analyzing the stickiness of the atmospheric states help us to predict how long a given circulation configuration is likely to last, thus providing useful information for weather forecasts.

The next natural step is to try to predict what the atmosphere will do once it has left a sticky state. Dynamical systems theory can again help us. It is in fact possible to define another quantity called “local dimension”, which tells us how complex the state of the atmosphere is. Once again the word “complex” here means exactly what you imagine: a complex temperature state will be one with lots of small, complicated spatial patterns. A simple state will be one with only a small number of large-scale features: for example, a day with high temperatures across the Mediterranean region and cold temperatures over most of Continental and Eastern Europe. Returning to our trajectory, these complex (or high-dimensional) and simple (or low-dimensional) states can be interpreted as follows. In the simple case, it is easy to predict the direction the trajectory will take in the future. This is the same as saying that all similar states evolve in a similar way. So if we want to forecast tomorrow’s temperature and we know that today is a “simple” state, we can look for states similar to today in the past years and we know the evolution of today’s state will be similar to that of these past states. In the complex case, on the contrary, it is very difficult to predict what the trajectory will do in the future. This means that similar atmospheric states will evolve in very different ways, and looking at past days with similar temperatures to today will not help us to forecast tomorrow’s temperature. A complex, high-dimensional state will therefore be more challenging for weather forecasters than a simple, low-dimensional one.

Now imagine looking at a very long climate dataset, for example covering the last century. If the climate system is always the same, one would expect the trajectories for the first and second half of the century to be indistinguishable. If, however, the climate is changing, then one would expect the trajectories representing it to also change. To make an analogy, imagine taking your heart rate. If you measure it on two different days while you are at rest, the number of heart beats per minute will probably be equal. In this case the system – which is here your body – is always in the same state. However, if one day you take your heart rate at rest and the following day you take it while you are running, the results will be very different. In this case something in the system has changed. In just the same way, if the climate system is changing, its “pulse” – namely the trajectory – will change with it. The trajectories of the two half-centuries in the dataset will therefore look different, and their local dimensions and stickiness will display different properties – for example a different mean value. The same two indicators that can help us improve weather predictions at daily to weekly timescales can therefore also help us to understand how climate varies across the centuries.

The dynamical systems approach can be applied to a wide range of scientific problems beyond the examples discussed above. These range from turbulence in fluids to the analysis of financial datasets. Picturing such a complex system as the atmosphere as a “spaghetti plot” is therefore an excellent example of an intuitive mathematical approach that can help us advance our knowledge of the world around us.

Edited by Célia J. Sapart.

Reference: Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci., 20(2), 130-141.