Climate: Past, Present & Future


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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.

Corals, the thermometers of the past!

Corals, the thermometers of the past!
Name of proxy


Type of record

Oceanic variability


Fringing reefs, barrier reefs, or atoll

Period of time investigated

Mainly the last 200 years

How it works

What we usually picture as a coral is actually a colony of tiny living animals called coral polyps, which are closely related to jellyfish or anemones. They live in symbiosis with photosynthetic algae called Zooxanthellae (Figure 1).

Figure 1: Schematic of a coral with its individual parts (modified from Veron, 1986).

Each polyp secretes a skeleton made of aragonite -a form of calcium carbonate- whose chemical composition depends on ambient oceanic and climatic conditions. Coral skeletons can therefore serve as monitors of the past oceanic and climatic variability through time (Figure 2).

Figure 2: X-radiographs and coral images (modified from DeLong et al., 2011).

Corals are distributed in the tropical belt mostly in the central and western Pacific, the Indian Ocean, and the Caribbean. These areas are also the most affected by climate variability such as the El Niño Southern Oscillation (ENSO) phenomenon. At interannual time scale, this phenomenon influences worldwide patterns of sea surface temperature (SST). Our present understanding of ENSO variability is limited by the short duration of instrumental records. In the current context of climate change, we need to understand the past variability of this phenomenon to be able to predict its future evolution. A proxy for past SST changes in the tropical oceans is therefore highly desirable to extend the length of the instrumental record.

Key Findings

Coral skeletal Sr/Ca have been shown to be an accurate tracer (“proxy”) of SST at many sites (Corrège, 2006). There is an inverse relationship between coral Sr/Ca values and SST conditions, with low Sr/Ca values corresponding to high SST environments and vice versa. Regression of coral Sr/Ca to instrumental SST (Figure 3) leads to a calibration equation that allows reconstruction of SST variability further back in time. SST records that span at least the last 200 years allow to differentiate the contributions of natural climate variability from those that are anthropogenically forced (Solomon et al., 2011). These results place coral as a perfect tool to reconstruct past oceanic variability which leads to a better understanding of past climate variability and a tremendously useful record to help predict future changes.

Figure 3: Time series of Sr/Ca from a living coral from New Caledonia and local SST (left). Calibration of Sr/Ca vs. SST. Sr/Ca appears to be a robust SST tracer (right).

Further readings

Corrège, T. (2006), Sea surface temperature and salinity reconstruction from coral geochemical tracers, Palaeogeography, Palaeoclimatology, Palaeoecology, 232(2-4), 408-428, doi:10.1016/j.palaeo.2005.10.014.

DeLong, K. L., J. A. Flannery, C. R. Maupin, R. Z. Poore, and T. M. Quinn (2011), A coral Sr/Ca calibration and replication study of two massive corals from the Gulf of Mexico, Palaeogeography, Palaeoclimatology, Palaeoecology, 307, 117–128, doi:10.1016/j.palaeo.2011.05.005.

Solomon A, et al. (2011), Distinguishing the roles of natural and anthropogenically forced decadal climate variability: Implications for prediction. Bull Am Meteorol Soc, 92:141–156.

Veron, J.E.N. (1986), Corals of Australia and the Indo-Pacific. Angus and Robertson:London/Sidney.


Edited by Caroline Jacques and Célia Sapart

Ostracods, the sentinels of past oceanic circulation

Ostracods, the sentinels of past oceanic circulation
Name of the proxy


Type of proxy

Paleoenvironment proxy


All types of aquatic environments but here we will focus on marine waters

Period of time investigated


How does it work?

Ostracoda are crustacean of millimetre size which have inhabited all types of marine environments from the Ordovician to today (e.g. Salas et al. 2007) and colonized continental water bodies during the Carboniferous (Bennett et al. 2012). They are characterised by their bivalve calcified carapace articulated dorsally which encloses and protects the soft parts and appendages of the animal (Figure 1). The majority of Ostracoda live on or in the sediments: they are consequently highly sensitive to their environment.

What are the key findings that have been done using this type of proxy?

Throughout their history, marine Ostracoda inhabiting deep seas had very different morphologies from the contemporary shallow water species: thin shells, long, hollow and delicate spines and no eye spots (although this point is discussed; Figure 2). Based on the study of sediments, associated organisms and analogies with modern-days Ostracoda, ostracodologists concluded that those animals developed in low energy environments ranging from 500 to 5000 m depth in connection with global ocean cold water supplied by ice-caps (Lethiers & Feist 1991). This discovery provided a unique window into the oceanic circulation through geological times and the existence of a cold deep-water layer. The presence and characteristics of these Ostracoda have been cornerstones in understanding that the thermohaline circulation has not been constant through the Phanerozoic but rather existed only during the Late Ordovician, the Carboniferous-Permian interval and from the Eocene to today (Benson 1975).

Figure 2. Simplified geological time scale with Eras and Periods of the Phanerozoic. On the right are reported some archetypal deep-sea Ostracoda from the literature (for all photos, scale bar is 100 µm). A: Processobairdia spinanterocerata Bless & Michel, 1987; B: Cristanaria katyae Crasquin-Soleau, 2008; C: Gencella taurensis Forel, work in progress; D: Pedicythere klothopetasi Yasuhara et al., 2009.

Today, this field of research is very active as Ostracoda are the only metazoans regularly fossilized in deep-sea sediments over an extremely long period of the history of Earth. Their long fossil record spanning 5 mass extinctions and periods of extreme climatic changes make them precious tools to unravel the response of deep-water ecosystems to past climatic changes and the rhythms of their recovery. The extreme sensitivity and history of these peculiar animals make them sentinels of deep-sea ecosystems facing ongoing global temperature increase and acidification of marine waters.

  • Bennett, C.E., Siveter, D.J., Davies, S.J., Williams, M., Wilkinson, I.P., Browne, M., Miller, C.G. 2012. Ostracods from freshwater and brackish environments of the Carboniferous of the Midland Valley of Scotland: the early colonisation of terrestrial water bodies. Geological Magazine, 149, 366-396.
  • Benson, R.H. 1975. The origin of the psychrosphere as recorded in change of deep sea Ostracode assemblages. Lethaia, 8, 69-83.
  • Bless, M.J.M., Michel, M.P. 1967. An ostracode fauna from the Upper Devonian of the Gildar-Monto region (NW Spain). Leidse Geologische Mededelingen, 39, 269-271.
  • Crasquin-Soleau, S., Carcione, L., Martini, R., 2008. Permian ostracods from the Lercara Formation (Middle Triassic to Carnian?, Sicily, Italy). Palaeontology, 51, 537-560.
  • Lethiers, F., Feist, R. 1991. Ostracodes, stratigraphie et bathymétrie du passage Dévonien–Carbonifère au Viséen Inférieur en Montagne Noire (France). Geobios, 24, 71-104.
  • Salas, M.J., Vannier, J., Williams, M. 2007. Early Ordovician Ostracods from Argentina: their bearing on the origin of Binodicope and Palaeocope clades. Journal of Paleontology, 81, 1384-1395.
  • Yasuhara, M., Okahashi, H., Cronin, T.M. 2009. Taxonomy of Quaternary Deep-Sea Ostracods from the Western North Atlantic Ocean. Palaeontology, 52, 879-931.

Written by Marie-Béatrice Forel

Edited by Célia Sapart and Caroline Jacques

Hot towns, summer in the city!

Hot towns, summer in the city!

Cities obviously experience a different climate than natural landscapes. Already in 1810 the British meteorologist Luke Howard documented that the air temperature in the city of London was several degrees higher than in its surroundings. This so called urban heat island has several causes. In general the relatively dark surfaces of asphalt and roofs absorb solar radiation very efficiently and this heat is stored in building material during the day. At night this heat is released to the atmosphere, which keeps the city warm. Moreover, heat produced by air conditioning, traffic and industry contributes substantially to a city’s heat load. With a decreased amount of vegetation, cities also lose the shade and cooling effect of trees (Figure 1).

Currently already over 50% of the world’s population is residing in urban areas and this number is foreseen to increase even further in the future. Moreover, climate model projections indicate that heat waves will occur more often in the future. Together these developments will make many citizens potentially vulnerable to  urban heat. Although a slight temperature increase might look appreciable at first glance, elevated temperatures affect human health, since hospital visits and mortality are enhanced in warm conditions (about 2% per degree Celsius, e.g. Hajat et al., 2002). In addition, labour productivity in warm periods is reduced, resulting in economic losses. E.g. for Australia this was estimated to be about $650 per capita (Zander et al, 2015) which is a substantial contribution to the national income. Also, the urban energy demand needed for heating purposes in winter and cooling in summer is governed by urban weather. Finally cities are vulnerable to flooding in case of extreme precipitation by peak showers, when the sewage system capacity is hampered. Hence how do cities manage urban heat and keep dry feet?

Behind the general picture on the urban heat island, several scientific questions do remain. E.g. what is the temperature variability within a city, and how can we monitor temperatures? Also, can we make special weather forecasts for cities? Monitoring urban weather and climate is challenging since traditional weather stations are not suitable for urban areas since they require undisturbed terrain. Crowdsourcing, i.e. the collection of weather data by citizens has now become popular. Many hobby meteorologists have installed weather stations at home, and distribute their data directly via several websites as,, and These crowdsourced observations were crucial in estimation of the urban heat island effect in the Netherlands, a west-European country with a mild maritime climate were little attention was paid to urban climate. On hot summer days the urban heat island was over 6 degrees (Steeneveld et al., 2011, Heusinkveld et al., 2014)!

The urban climate can be efficiently monitored by tricycle traverse measurements (Figure 2). Bikes are excellent modules to measure the urban climate, since a wide variety of urban morphology (vegetation cover, building design and material) can be explored, especially outside the main roads. And it is carbon free, all driven by electricity generated by solar panels. Moreover, the bike is appealing for the general public.

Figure 2: A cargo tricycle equipped with a weather station to measure temperature, humidity, wind speed, solar and thermal radiation (source: Wageningen University; design Bert Heusinkveld).

What do we learn from such bike traverses? Figure 3 shows the variety of temperatures observed during a heat wave in a mid-size town in the Netherlands. Obviously, local temperature differences at the end of the afternoon may reach up to 3.5 ºC in this case. In general the town centre is relatively warm, though more surprisingly the relatively young neighbourhoods at the city edges appear to be warm too. In these neighbourhoods the vegetation is young, resulting in limited shadowing and therefore efficient heat absorption in roads and building walls. Local cool spots appear in parks and at small lakes. Luckily, enough room to escape from the heat!

Figure 3: Air temperature observations in the mid-size town Wageningen (the Netherlands, August 2nd 2013, 17.00 local time) obtained from two bike traverses. Source land cover maps:

No weather station in your garden? We still catch you via your smartphone! Smartphone users with the OpenSignal App that is intended to monitor wireless network capacity provide as a by product the smartphone battery temperature. Recently it was discovered that the temperature of your smartphone battery follows the air temperature outdoor (Overeem et al., 2013). Inversely this means that if we know the smartphone battery temperature, we can estimated the outdoor temperature. This insight offers a high potential for recording urban temperatures in areas where observations are rather scarce. The open question remains whether spatial and temporal scales these observations are applicable. Is it possible to get a reliable temperature record for your neighbourhood via available smartphones. Also, will these modern Big Data techniques change the paradigm on performing traditional measurements?

Weather forecasts on TV and radio never focus on the detailed weather for cities, which is somewhat surprising since many human activities, human health and critical infrastructures depend on the city temperature. Since computer power has rapidly grown the last decades, and still is, weather forecast models have refined their grid spacing. With this refinement urban areas “become visible” for these models. On one hand this offers a great potential for city specific forecasts. On the other hand information about the urban morphology is needed to feed these weather forecast models. For example they need to know whether urban districts contain skyscrapers or just three-story residential areas, and for example how much vegetation is present. Thereto the World Urban Database and Portal Tool is set up in which local experts document their city. You are welcome to join to describe your city and inform our weather forecast models how your city looks like!

  • Heusinkveld, B.G., G.J. Steeneveld, L.W.A. van Hove, C.M.J. Jacobs, and A.A.M. Holtslag 2014: Spatial variability of the Rotterdam urban heat island as influenced by urban land use, J. Geophys. Res, 119, 677–692.
  • Overeem, A.,  J. C. R. Robinson,  H. Leijnse,  G. J. Steeneveld,  B. K. P. Horn, and  R. Uijlenhoet (2013), Crowdsourcing urban air temperatures from smartphone battery temperatures, Geophys. Res. Lett., 40, 4081–4085, doi:10.1002/grl.50786
  • Steeneveld, G.J., S. Koopmans, B.G. Heusinkveld, L.W.A. van Hove, and A.A.M. Holtslag, 2011: Quantifying urban heat island effects and human comfort for cities of variable size and urban morphology in The Netherlands, J. Geophys. Res., 116, D20129, doi:10.1029/2011JD015988.
  • Zander, K.K., W.J.W. Botzen, E. Oppermann, T. Kjellstrom, S.T. Garnett, 2015: Heat stress causes substantial labour productivity loss in Australia, Nature Climate Change  5, 647–651.

Written by Gert-Jan Steeneveld, associate Professor at Wageningen University, The Netherlands

Edited by Célia Sapart and Caroline Jacques


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