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3
The Universe and the Earth

Contemporary of Newton, Leibniz, and French mathematicians René Descartes (1596–1650) and Pierre de Fermat (1607–1665), Blaise Pascal (Figure 3.1) embodied an ideal form of all of them: that of a knowledgeable man. A physicist, mathematician and philosopher, he evoked, in one of his most famous texts, the singular place of humanity on Earth and in the Universe:

“Let humanity therefore contemplate the whole of nature in its high and full majesty, let us keep our sight away from the low objects that surround us. Let us look at this bright light put like an eternal lamp to illuminate the Universe, let the earth appear to us as a point at the cost of the vast tower that this star describes and let us be surprised that this vast tower itself is only a very delicate point towards the one that these stars, which roll in the firmament, embrace” ([PAS 60], translated from French).

Are researchers in astrophysics and geophysics, in the 21st Century, the heirs of the 17th Century scientists? With numerical simulation, they nowadays carry out real thought experiments, supported by data, in fields where concrete experimentation is not easily accessible – or even simply possible.

Solving certain enigmas of the Universe, which extend Pascal’s philosophical observations, and contributing to the analysis of geophysical risks – in order not to reduce it to a probability calculation – are two areas in which numerical simulation is becoming increasingly important.

images — **Figure 3.1.** *Blaise Pascal (1623–1662)*

COMMENT ON FIGURE 3.1.– Early inventor of a computing machine (Chapter 4 of the first volume), Blaise Pascal was passionate about physics and mathematics. He was interested in the notion of vacuum, experimented with the laws of fluid hydrostatics and laid the foundations for the calculation of probabilities. His mystical experience turned him from science to theology: his ambition was to write a treatise on it, of which Les Pensées, found after his death, are the working notes (source: Blaise Pascal, anonymous, 17th Century, oil on canvas, Château de Versailles).

3.1. Astrophysics

An orchestral explosion punctuated by the percussion of timpani inaugurates the Representation of Chaos. It slowly fades into a long decrescendo ending in a marked silence. In a tiny pianissimo, an uncertain sound world is then born, whose tonality only gradually asserts itself. Through contrasting and apparently erratic episodes, ranging from the imperceptible murmur to the brutal explosion, the overture to The Creation, written by Austrian composer Joseph Haydn (1732–1809), then subsides into a transition giving Raphael the floor. The latter describes, in a short recitative, the world as “formless and empty” of the “dark abyss”. In a pianissimo breath, the choir entered in turn by evoking “the spirit of God” and proclaimed fortissimo “let there be light!” He is accompanied by the entire orchestra, supporting his song to a chord of D Major, powerful, radiant and luminous! Uriel concludes with a very short recitative of the introduction to this oratorio: “And God saw that the light was good… God separated the light from the darkness”. First performed in Vienna in 1799, this musical work begins with the first three minutes of the Universe. It seems to anticipate by more than a century the Big Bang theory, an idea supported in the 1930s by the observations of the American astronomer Edwin Hubble (1889–1953).

The quest for the origin of the Universe – as well as its possible future – has been built by improving the theoretical knowledge and sensitivity of observational instruments over time. Astrophysicists studying the Universe, its formation and evolution, nowadays have an additional tool at their disposal: computer simulations. Patrick Hennebelle, a researcher at the CEA, at the Institute of Fundamental Laws of the Universe, explains:

“For more than twenty years, astrophysics has been taking advantage of numerical simulations. They allow us to understand and predict the behavior of certain celestial bodies and are applied in many fields: star physics, galaxy dynamics, planet formation or the evolution of the gaseous interstellar medium, etc. Simulations make it possible to reproduce astronomical observations, which involve the interaction of systems with highly variable densities, covering a very wide range, typically from 1 to 10¹⁰ particles per cm³! The models take into account electromagnetic forces, gravitational force and current research aim to better account for radiation processes, which are still poorly described in some simulations”.

Scientific computation also offers researchers the possibility of reproducing evolutionary sequences that cannot be understood in their entirety by other means, because they take place over several billion years! It is also a means of optimizing the use of the most modern astronomical observation resources (Figure 3.2). The latter are shared within an international community and their access, subject to calls for projects examined by expert commissions, is very competitive. Simulation allows researchers to prepare an observation sequence, by testing different hypotheses in advance, and can also be used to interpret the results.

Numerical models are becoming widespread in all disciplines of astrophysics, particularly the most advanced ones, contributing to the description of black holes and the search for gravitational waves, whose recent discovery is also due to modeling techniques (Box 3.1).

3.1.1. Telling the story of the Universe

Our Universe was born some 13 billion years ago and the matter of which it is composed has been sculpted by the action of different fundamental physical forces. While the first few seconds of the Universe are still an enigma to cosmologists, its subsequent evolution is better known. Current physical models explain how the first particles (electrons, protons, neutrons – and beyond, elementary particles they are made of) and atoms (hydrogen and its isotopes, and then the range of known chemical elements) are formed under the influence of nuclear and electromagnetic forces (Figure 3.3). The history of the Universe is therefore that of a long aggregation of its matter that sees the formation of the first stars and galaxies and other celestial bodies (clusters and superclusters of galaxies). This organization is, among other things, the result of gravitational force.

The latter is described by Newton’s formula that some astrophysicists use to understand the structure of the Universe. In 2017, for example, researchers conducted one of the most accurate simulations to date of a piece of the Universe, using a so-called “N-body model” [POT 17]. The latter describes the gravitational interaction between a very large number of celestial bodies: the problem posed has no explicit mathematical solution and only a numerical method is able to give one. Romain Teyssier, one of the researchers behind this simulation, explains:

“This is a description of the gravitational forces acting on a large set of particles. They are of different sizes, and they represent celestial bodies: stars, star clusters, small or large galaxies (dwarfs or supernova), galaxy clusters, etc.”

COMMENT ON FIGURE 3.3.– This illustration summarizes the history of our Universe, which spans almost 14 billion years. It shows the main events that occurred in the first eras of cosmic life, when its physical properties are almost uniform and marked by very small fluctuations. The “modern history” of the Universe begins some 1 to 10 million years after the Big Bang, from the moment it is dense enough for the gravitational force to orchestrate its global organization. It is marked by a wide variety of celestial structures: stars, planets, galaxies and galaxy clusters. Long considered as the “start” of the history of the Universe, the Big Bang became for modern physicists the marker of a transition between two states of the Universe. The question of the origin of the Universe remains open [KLE 14] (source: ©European Space Agency/C. Carreau/www.esa.int).

The simulation figures are, strictly speaking, astronomical. In order to achieve it, scientists use a square digital box, one side of which measures 3 billion parsec¹.

The calculation box contains 2 trillion particles (2 million million or 2 thousand billion). It allows us to represent galaxies 10 times smaller than our Milky Way, which corresponds, for example, to the star cluster of the Large Magellanic Cloud. The simulation thus involves some 25 billion galaxies, whose evolution is observed over nearly 12.5 billion years and it is carried out in just under 100 h of computation.

On the day of the simulation, this resolution is considered very satisfactory by astronomers and astrophysicists. The calculation represents the places where matter has organized itself in the Universe under the influence of gravity (Figure 3.4) and provides data that are sufficiently accurate to be compared with observations made on the cosmos within a few years. Until now, the calculations did not have the accuracy required for this comparison to be relevant.

The simulation is made possible through two innovations:

– The use of particularly efficient algorithms to model a very large number of particles. Directly simulating the interactions between N particles requires N² operations: for the 2 trillion particles required for simulation, the amount of calculations to be performed is inaccessible to computers, even the most powerful ones! An appropriate method, known as the “fast multipolar method”, considerably reduces the number of calculation operations. It consists of representing the interactions of one particle with others by means of a tree, where the direct approach represents it by means of a network (Figure 3.5). In the tree, the interaction between the particles i and k is represented by the one they have with the particle j. By exploring the tree more or less deeply, we find all the interactions described by the network. Exploring the tree requires N operations, while the network one requires N² operations. Under these conditions, the calculation at 2 trillion particles becomes possible. The validation of the simulation is obtained by comparing a direct calculation and an optimized calculation on a sample of a few million particles. The simulation describing the 2 trillion particles is then performed with the optimized algorithm.

– The use of supercomputers that exploit the processing capacity offered by graphics cards (GPUs). Note that 5,000 graphics cards were simultaneously used on the supercomputer of the Swiss National Supercomputing Centre (Chapter 3 of the first volume), whose performance and architecture are currently unique in the world. The 100 hours of computing time required for simulation is entirely appropriate to the pace of the researchers’ work. By comparison, the duration of a parallel calculation on the same number of conventional processors (CPUs) is estimated by researchers at 20 years.

So, the calculation code is the result of a development work of about 20 years and its adaptation to the specificities of this simulation took about 3 years. Porting the algorithm to other supercomputers should allow for simulations with 10 or 100 times more particles, thus offering calculations a higher resolution.

Box 3.1. Newton’s equation

Legend has it that it was by watching an apple fall to the ground that the British physicist Isaac Newton (1643–1727) began to develop the theory of universal gravitation. It originated from a conversation that Newton’s doctor and confidant, William Stukeley (1687–1765), recounted in his Memoirs of Sir Isaac Newton’s Life (1752):

“As the weather got hot, we went into the garden and drank tea under the shade of a few apple trees, just him and me. During the conversation, he told me that he had found himself in the same situation, long before the notion of gravitation had suddenly occurred to him, while he was sitting in a contemplative mood. Why does this apple always fall perpendicular to the ground, he thought to himself. Why doesn’t it fall sideways or upwards, but constantly towards the center of the Earth? And if matter attracts matter in this way, it must be in proportion to its quantity; therefore, the apple attracts the Earth in the same way that the Earth attracts the apple” (quoted by [HER 62]).

Whether authentic or invented, the anecdote allows us to conceive Newton’s path of thought. Before the ultimate understanding, presented as sudden, the slow intellectual maturation of a theory has its roots in the observation of the world. For the physicist, it is a matter of expressing it in a general relationship that allows him to give a summarizing account of it, by means of a mathematical writing, the Newton equation being written as:

With this equation, Newton states that two bodies attract each other according to a force F proportional to the product of their masses (all the greater the greater the size of each one) and unlike the square of the distance that separates them (all the smaller the greater the distance).

This force is directed like an arrow u between the two bodies that interact through gravity. G refers to the gravitational constant: it is an invariant quantity, like the number π. Apples or planets, stars, galaxies, etc.; all bodies are subject to the effect of gravitational force, which Newton described as universal.

In order to conceive his theory, Newton relied on the laws of the German astronomer Johannes Kepler (1571–1630), who studied the heliocentric hypothesis – that the Earth revolves around the Sun, placed at the center of our planetary system – formulated by the polish astronomer and mathematician Nicolaus Copernicus (1473–1543). In his book Mysterium cosmographicum, which he published in 1596, Kepler proposed, among other things, a model of the Universe based on regular polyhedra (Figure 3.6).

Figure 3.6. The Universe model proposed by Kepler is based on five regular polyhedra and is close to the spherical shape symbolizing divine perfection

COMMENT ON FIGURE 3.6.– Tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron: Plato’s five solids allow Kepler to construct a model of the Universe by noting that they can be interposed between the orbits of the six planets known at that time [KEP 96, Tabula III: Orbium planetarum dimensiones, et distantias per quinque regularia corpora geometrica exhibens] (source: www.commons.wikimedia.org).

The law of orbits, the law of areas and the law of periods describe the main properties of the movement of planets around the Sun. Their orbit is an ellipse, of which one focus is the Sun, the Sun–planet ray sweeps equal areas for equal time intervals and the speed of a planet thus becomes greater as the planet approaches it: it is maximum in the vicinity of the shortest ray (perihelion) and minimal in the vicinity of the largest ray (aphelion). Finally, the square of the period of revolution is proportional to the cube of the half-larger axis of the orbit, according to the relationship a³/T² = C proposed by Kepler. Newton combined Kepler’s laws with the laws of motion set out in his Philosophiæ Naturalis Principia Mathematica and obtained the form of the equation of gravitation, the driving force behind the motion of the stars.

It should also be noted that Hypatia (360–415) can be credited, without certainty, with the discovery of the Earth’s elliptical orbit, nearly twelve centuries before Kepler. This is the starting point of Alexandro Amenabar’s film (Figure 3.7) – if not a historical reality.

Figure 3.7. Hypatia is played by Rachel Weisz in the film Agora [AME 09]

COMMENT ON FIGURE 3.7.– Hypatia, a Greek philosopher and mathematician, directed the Neoplatonist School of Alexandria in the 4th Century. In Amenabar’s film [AME 09], she is presented as a woman free of spirit, rejecting the social and religious conventions of her time and devoting her life to the search for truth. The movement of celestial bodies was explained by her contemporaries using the Ptolemy system, postulating circular trajectories. The film stages how Hypatia noted the inconsistencies between this description and her observations, and thus discovered that the Earth revolves around the Sun, by describing a circle with two centers, an ellipse. The film also incorporates historical elements, with anachronisms that the experts could discuss at length. It presents the contrasts between reason and superstition, between the power of thought and that of beliefs (source: poster of the film Agora, distributed in France by Mars Films/www.marsfilms.com).

With the laws he has laid down, Newton saw the movement of celestial bodies as falling motions. The French writer Paul Valéry (1871–1945) wrote about how Newton imagined movements: “You had to be Newton to see that the Moon is falling when everyone can see that it is not” ([VAL 41], translated from French). Valéry thus explained that Newton’s intuition was to understand and formalize that the falling motion of celestial bodies in the Universe is compensated by the force of attraction that connects them. He also praised Newton’s intelligence, who was able to see our world in a way which was different from common sense.

On the earth’s surface, we feel the gravity field, which combines the effects of gravitational force and centrifugal force due to the rotation of the Earth – the first being very clearly predominant. The gravity field shows variations due to the uneven distribution of masses on and within the Earth’s surface (Figure 3.8).

Figure 3.8. Geoid: representation of the equipotentials of the gravitational field at the Earth’s surface. For a color version of this figure, see www.iste.co.uk/sigrist/simulation2.zip

COMMENT ON FIGURE 3.8.– In 2011, ESA’s Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite enabled the high-precision measurement of changes in gravity at the Earth’s surface. The geoid, the surface obtained by grouping points of the same gravitational potential, is a reference for many topographic or oceanographic measurements. The accuracy of the data it provides is crucial for the various uses for which it is intended, for example to understand sea level changes, ocean circulation, iceberg drift, land layer dynamics, etc. GOCE is said to have collected an unprecedented amount of data, providing scientists with one of the most accurate geoid data to date (source: ©European Space Agency /www.esa.int).

While gravitation fascinates many artists, such as the filmmakers who direct it [CUA 13, VAN 07] or the dancers who experience it [HEY 16, WEN 11], we live with it forgetting its omnipresence, except perhaps in special circumstances. For example, when our buttered bread slice falls to the ground and (almost) always on the wrong side! When it falls from the edge of the table, it tends to slide and turn before being dragged to the ground. The parameters defining the initial state before the fall are the friction of the bread on the table, the height of the table and the size of the toast. The falling distance is not large enough to let gravity prevail over the rotation and friction of the bread on the table as it tilts, so that the toast first undergoes a half-turn rotation before crashing to the ground on the buttered side! With a table at least three meters high, the toast can make a full turn and its chances of falling on the buttered side or not are then equivalent to a coin toss: 50–50.

But 3-m tables are not part of our daily lives. Humans do not exceed this size: a fall from this height would be fatal to them. Nature does things well by limiting our growth (which it allows for more stable quadrupeds, such as giraffes). Evolving in a world where our tables are necessarily limited in height, our sandwiches (almost) always fall on the wrong side.

“We show that toast does indeed have an inherent tendency to land butter-side down for a wide range of conditions. Furthermore, we show that this outcome is ultimately ascribable to the values of fundamental constants. As such, the manifestation of Murphy’s Law appears to be an ineluctable feature of our Universe […]. We end by noting that, according to Einstein, God is subtle, but He is not malicious. That may be so, but His influence on falling toast clearly leaves much to be desired…” [MAT 95].

This is the conclusion of a British scientist, Robert Matthews, with both humor and rigor, who proposes a theoretical and experimental study that helps explain why our breakfasts can sometimes become catastrophic.

The theory of universal gravitation, based on a seemingly simple formula, makes it possible to calculate the date of eclipses, the passage of a comet in the sky or an asteroid near our planet. It is also because of Newton’s equation that the French astronomer Urbain Le Verrier (1811–1877) predicted in 1846 the existence and position of Neptune. This was the first time a planet had been discovered through the resolution of mathematical equations. Until then, only astronomical observations allowed it.

Having an equation is not enough to predict the orbit of the planets! However, a solution must be found, an explicit mathematical formula that can be calculated. Describing the gravitational interaction between any number of these bodies is a problem that has no analytical solution. The “three-body problem”, describing the interaction between the Earth, Moon and Sun, thus preoccupied, in addition to Newton, Leonhard Euler and Jospeh-Louis Lagrange. In 1772, Lagrange published Essais sur le problème des trois corps (Essay on the three-body problem), in which he calculated the position of points of particular space (Figure 3.9). The attraction forces exerted by two celestial bodies of large mass on a third, whose mass is negligible, compensate each other. This makes it possible to place satellites or observation probes, for example, because the relative positions of the three bodies are fixed. Two of them are “stable” – the objects placed there tend to keep their balance position, like how a marble in the hollow of a gutter remains at rest. The other three are “unstable” – the objects that occupy their position tend to move away from it, like how a marble at the top of a speed bump would roll to a lower point.

Figure 3.9. Lagrange points for the Sun and Earth. For a color version of this figure, see www.iste.co.uk/sigrist/simulation2.zip

COMMENT ON FIGURE 3.9.– The calculation of the position of the Lagrange points is done by considering the balance of a body of negligible mass under the effect of gravitational forces with two other bodies, for example the Earth and the Sun. The Lagrange L_I and L₂ points are “unstable “ balances. A neutral body cannot remain there naturally while a probe can be placed there at the cost of reasonable fuel consumption, as the gravitational field is weak in its vicinity. The L_I point of the Earth-Sun system makes it possible to observe the Sun without interference with the Earth or the Moon, and thus to detect solar disturbances before they reach the Earth’s atmosphere. It is also the ideal position for space weather missions, such as the one performed by SOHO (Solar and Heliospheric Observatory) of the European Space Agency. Located nearly I million and 0.5 kilometers from Earth, L₂ is the furthest from the Sun. It is close enough to the Earth that data collected by satellites can be transferred to it at a high rate. L₂ is used by major space astronomical observatories launched over the past two decades, such as the European Space Agency’s Planck satellite (source: © NASA/WMAP Science Team/www.nasa.gov).

In the solar system, the Lagrange points form a particular set, for example drawing an optimal trajectory to guide objects at a lower energy cost (Figure 3.10). The American probe, Genesis, launched in 2001, was placed in orbit at point Li and collected information on the Sun for two years. It was redirected to point L₂, before returning to Earth in 2004 with its precious cargo. The trajectory of this mission, defined by researchers at NASA’s Jet Propulsion Laboratory, passes near Lagrange points, used as “gravitational relays”. Unlike a terrestrial roadway, the shape and direction of this interplanetary network is constantly changing with the configuration of the planets in the solar system. A trip on one of the lanes of this “interplanetary highway” requires minimal energy, but is at the cost of a longer travel time than that allowed by a propelled trajectory.

Figure 3.10. The Interplanetary Transport Network (ITN) passes through the Lagrange points of different planets of the solar system (source: © NASA). For a color version of this figure, see www.iste.co.uk/sigrist/simulation2.zip

COMMENT ON FIGURE 3.10.– The figure is an artist’s representation of the interplanetary highway. The green ribbons correspond to one of the possible paths in a mathematically calculated set, represented by the dark-colored tubes. The places where the trajectories suddenly change direction correspond to Lagrange points.

A new look is taken at the Universe and gravitation with the work of Albert Einstein [BER 04, LOC 15]. In 1915, he presented the Theory of General Relativity in which he interpreted gravitation not like a force as Newton did, but as the curvature of space time. Einstein’s equations thus establish the relationship between the geometry and curvature of space time (represented by G_μv) and the momentum and energy of a system which evolves in it (represented by T_μv):

G_μv and T_μv are matrices (or arrays) that contain 10 unknowns. Solving these equations yields the geometry of space time and the dynamics of systems according to general relativity. For instance, the trajectories of the planets correspond to an inertial movement (in the sense of Newton’s firt law) passing through the shortest path between two points and following the curves of space time, distorted by the presence of very massive celestial bodies, such as the Sun (Figure 3.11). Einstein’s equations can be used to predict the existence of black holes, a topic to which British physicist Stephen Hawking (1942–2018) dedicated most of his life in order to piece together the parts of the puzzle [HAW 88]. One of the densest structures in the Universe, they attract mass, energy and light into the well they dig in space time: no light radiation emanates from it. Black holes fascinate us as much as the scientists who discovered their existence and began to understand their properties, which disrupt the conception of time, space, mass and gravitation [NOO 18].

Figure 3.11. Signals sent between the Earth and the Cassini probe (green wave) are delayed by the deformation of space and time due to the mass of the Sun (source: © NASA). For a color version of this figure, see www.iste.co.uk/sigrist/simulation2.zip

COMMENT ON FIGURE 3.11.– This artist’s representation represents the influence of the deformation of space time by very dense bodies. Here, the path of a signal transmitted from the Earth to a space exploration probe is distorted by the presence of the Sun. The propagation of the signal thus follows the deformation that it imposes on space time. In 1915, Einstein used his theory to explain Mercury’s singular trajectory: the ellipse described by this planet, the closest to the Sun (therefore more sensitive to gravitational effects), rotates slightly faster than it should. Newtonian mechanics does not allow us to fully understand this movement, while Einstein’s calculations explain them well. A second confirmation of the theory of general relativity comes four years later from observations made during solar eclipses. Carried out under the direction of the British astronomer Arthur Stanley Eddington (1882–1944), they highlight the deviation of light rays by the Sun, as shown by Einstein’s calculations.

Einstein’s theory further predicts that when two massive objects interact, they produce a “gravitational wave”, a deformation of space time itself, propagating in the Universe at the speed of light, as a wave in a fluid deforms in space and travels over a liquid expanse. If such a wave causes a deformation visible to the naked eye, attested by the movement of a floating body, detecting a gravitational wave is not so easy: the deformation it produces is in fact in the order of 10^–21 (1 for 1,000,000,000,000,000,000,000). Identifying it is equivalent to measuring the distance from the Earth to the Sun with an accuracy equivalent to the size of the hydrogen atom! Gravitational waves are somehow “hidden” in Einstein’s equations, as waves in a fluid are embedded in the equations of flows. The study of gravitational waves has benefited from researches undergone by mathematicians and theoretical physicists. Early studies in the 1980s involved complex (and tedious!) analytical calculations and, more recently, numerical simulation based on Einstein’s equations have allowed scientists to predict what the signature of gravitational waves would be (Figure 3.12).

Figure 3.12. Simulation of gravitational waves resulting from the interaction between two massive celestial bodies. For a color version of this figure, see www.iste.co.uk/sigrist/simulation2.zip

COMMENT ON FIGURE 3.12.– In 2012, a team of NASA astrophysicists was using models to calculate the properties of a gravitational wave and to understand the type of signal it generates. The scientists simulated the fusion of two small black holes using a computer calculation program. The emission of the gravitational wave is visualized in this image which represents the deformations of space time resulting from its passage. The yellow curves near the black holes indicate the regions in which strong interactions occur between the gravitational fields of the two bodies. The simulation is to date one of the most significant among those operated on NASA supercomputers (source: ©NASA/C. Henze).

Simulation techniques produce data useful for the development of gravitational wave detection instruments in support of instruments developed by scientists. The result of several decades of research, various laser interferometric observatories have been developed worldwide based on international scientific collaborations: Virgo in Europe and the LIGO (Laser Interferometer Gravitational-wave Observatory) in the United States are two examples².

Figure 3.13. Aerial view of the LIGO interferometer in the United States (Livingstone, Louisiana site)

COMMENT ON FIGURE 3.13.– The interference between two waves is characterized by regions in which vibrations are added or submerged, producing areas of high and low intensity, allowing them to be detected and characterized. This phenomenon is observed in the laboratory for certain electromagnetic waves using an interferometer. The detection of gravitational waves by means of interferometers poses many difficulties: for example, the signals can be disturbed by undesirable vibrations, such as those of seismic waves. Two remote sites in the United States host LIGO interferometers, which allows in particular the comparison of signals, freeing themselves from site effects. Another gravitational wave detector is under construction in Europe (Virgo) and two others, in Japan and India, will complement the observations of the three current detectors. With these tools, whose detection quality is constantly improving, scientists expect to discover the traces of many cosmic events and to understand more precisely the history of the Universe (source: www.ligo.org).

The two LIGO detectors, built at the initiative of three American astrophysicists, Kip Thorne, Rainer Weiss and Barry Barish, made it possible to detect a gravitational wave for the first time on September 14, 2015 [DAM 19]. The recorded event occurred very far from our Milky Way: nearly 1.3 billion light years away! This discovery earned Thorne, Weiss and Barish the 2017 Nobel Prize in Physics, awarded at the end of a scientific life mobilized by the quest for gravitational waves [BAR 16].

Figure 3.14. First detection of gravitational waves, named after the day of their discovery, September 15, 2014: GW150914 [ABO 16]. For a color version of this figure, see www.iste.co.uk/sigrist/simulation2.zip

COMMENT ON FIGURE 3.14.– The figure shows the vibration signals signing a gravitational wave and recorded at two LIGO observatory sites in the United States (Hanford, Washington and Livingstone, Louisiana). Raw signals of deformations are the signature of gravitational waves (top). They are reconstructed according to the prediction of the general relativity model (middle) and filtered in order to provide evidence for the variation of their frequency with respect to time (bottom). Data processing serves the scientist to thoroughly analyze the sequence of cosmic events that trigger the gravitational wave. The measured deformation is in the range of 10^–21: by comparison, the vibration of a guitar string, observable to the naked eye, is in the range of 10^–2! The event is analyzed here over half a second and involves vibrations whose frequency varies between 30 Hz and 500 Hz, a range corresponding to that of many musical instruments, such as the guitar. The discovery of gravitational waves is the fruit of one particular international collaboration: the historical publication from which the figure is extracted is signed by all scientists who worked to develop measurement systems and signal analysis methods from gravitational wave detectors (source: www.physics.aps.org).

Nowadays, simulations provide data for artificial intelligence algorithms. In 2018, American researchers used the results of several hundred simulations of the fusion of two black holes to predict the characteristics of the final black hole (its mass, rotation speed) and the shape of the gravitational wave that this collision should produce [VAR 19].

Gravitational waves offer astrophysicists the opportunity to observe one of the most impressive physical phenomena in the cosmos. They open up a new interpretation of the Universe and the physical laws that humans have built in order to understand it – starting with the fall of an apple?

3.1.2. Observing the formation of celestial bodies

The gravitational field plays an important role in the formation of black holes, and in order to understand the dynamics of these structures observed in the Universe and certain complex phenomena such as instabilities, researchers propose different physical and numerical models. The simulation then becomes a digital observatory to test many ideas or hypotheses explaining observations.

Some of the physics at play in the dynamics of celestial bodies (such as planets or black holes) is that of plasmas, a state of matter made up of charged particles. It may be described by the equations of magneto-hydro-dynamics (MHD), which express the conservation of the mass, momentum and energy of particles, on the one hand, and the propagation of electromagnetic waves within the plasma, on the other hand.

Two examples of simulations are proposed for the Rossby³ wave instability (Figure 3.15):

– the first (top) corresponds to the evolution of a black hole and uses a calculation code developed by astrophysicists to study an instability evolving very quickly, in a few seconds;
– the second (bottom) reproduces the formation mechanism of the heart of a planet, governed by similar physical mechanisms developing this time on longer time scales.

From the simulations, astrophysicists develop radiation spectra: a curve of light that indicates the presence of a black hole – or any other entity studied. Compared to spectra from celestial observations, they allow researchers to propose explanations of the observed phenomena and to test their validity.

To observe the formation of a planet’s core, the simulated physical time is a few million years and requires one month of calculation. The evolution of instabilities is significant over several hundred simulated years. The calculation must reproduce them and, in order to be reliable, not artificially generate them, which happens when digital schemas are not totally conservative. In such a case, the calculation may introduce numerical errors whose effect is to pollute the simulation. The simulated quantities are tainted with errors that produce oscillations, without any physical meaning.

In order to validate the calculation method – and to ensure the stability of the numerical schemas – a simulation is started in a neutral initial state, free of disturbance. It is a matter of ensuring that this state does not change. The energy signal emitted by the plasma is then zero over the entire duration of the simulation: in the absence of an external effect, absolutely nothing happens and this is what is desired. If the numerical scheme is conservative, then the calculation method can be used to see the plasma evolve under the effect of physical phenomena, only contained in the equations – and not induced by numerical calculation errors. This is an issue that is also found in many other industrial applications.

NOTE.– Conservative numerical schemes.

A numerical method generally makes it possible to transform partial differential equations, involving variations in time and space of different physical quantities, into differential equations, involving variation in time solely. These concern the evolutions over time of the unknowns of the problem: a step-by-step calculation algorithm can be used to calculate them at different times (Chapter 1 of the first volume).

Different numerical schemes may be implemented and they do not have the same digital properties. In some cases, there may be a loss of information during the simulation. Due to the time division, part of the mechanical energy of the simulated system is “filtered” by the calculation scheme. The calculated energy loss is more or less significant according to the diagrams and for most industrial simulations, it is known, controlled – and acceptable to engineers [BAT 82].

In other cases, such as astrophysics, meteorology or combustion, simulations extend over very long times, or with a large number of time steps. These require very robust digital schematics, not squandering the mechanical energy of the system they are calculating.

Let us illustrate the subject with the simplest mechanical model: the harmonic oscillator – a ball attached to a spring, a yo-yo! Its mechanical energy is due to the movement of the mass (kinetic energy) and the deformation of the spring (potential energy). Away from its original position as the guitar string tensioned by the musician’s fingerboard, and left to its own devices without friction, it will vibrate regularly and endlessly.

The mechanical energy of a harmonic oscillator depends on its position and speed. It is printed at the beginning of its movement and is preserved in the absence of losses of physical origin. All the initial conditions of the movement, the starting point, may be represented on two axes, position and speed (Figure 3.16).

Figure 3.16. Highlighting the properties of certain numerical schemes [CHH 08]

It has a given shape: an ellipse or a circle (shown in black in the figure). The surface of this figure represents the energy of the oscillator. It is plotted over time with position and velocity coordinates for different simulations (in white on the figure) based on different numerical schemes.

The energy conservation properties are thus highlighted:

– Scheme n°1 artificially produces digital energy for the oscillator whose movement it calculates: the area of the ellipse increases.
– Scheme n°2 digitally destroys the energy of the oscillator: the area of the circle decreases.
– Scheme n°3 preserves energy: the area of the ellipse, which can nevertheless deform, remains constant.

This theoretical experiment is used to test the properties of an energy-conserving simulation. The latter are essential to represent subtle physical phenomena, over long or even very long periods of time.

For some industrial simulations, such as the rotation of helicopter blades, it is imperative to represent the evolution of mechanical energy as accurately as possible. Otherwise, the calculation would produce a result without any physical reality, at the risk of not anticipating critical phenomena, such as instabilities, that could affect the reliability and safety of an aircraft in flight. Only calculations based on a numerical scheme with the adequate properties can be used in these cases.

The writing of robust methods digitally preserving physical properties refers to a current field of research based on Noether’s theorem⁴ [CHH 08]. It contributes to increasing the overall reliability of simulations: these theoretical advances will ultimately benefit engineers implementing simulations.

3.1.3. Predicting the mass of stars

Stars play a fundamental role in our Universe. They synthesize elements essential to certain life forms (carbon, oxygen, iron, etc.) and they contribute to the in-depth renewal of the composition of galaxies throughout their lives. They emit light energy necessary for certain chemical reactions to occur and, during explosions marking the end of activity of the largest ones, they emit a large quantity of matter into the Universe. The properties of stars vary with their mass:

– High-mass stars have a “hot” temperature, their lifespan is relatively short (on the scale of the Universe!): up to a few tens of millions of years. Marked by intense energy activity, they burn gases by injecting turbulence into the Universe and contributing to its radiation.
– Low-mass stars, like our Sun, have a “cold” temperature. Long-lasting, they contribute to the stability of planetary systems and generally host planets – potential life-bearers!

The distribution of the mass of the different stars makes it possible, among other things, to understand the history and functioning of the Universe. From observations of the cosmos (Figure 3.17), the distribution curve shows, for example, that the most frequent mass of stars is about one-third of the solar mass (denoted 0.3 Mo). It decreases quite quickly below (to the lower masses) and beyond (to the larger masses).

COMMENT ON FIGURE 3.17.– Created in 2018 by the European Space Agency using data from the “Gaia “ mission, the image represents a three-dimensional map of the stars observed in our galaxy, representing the most massive ones, which are also the warmest and brightest. The latter are mainly located near their training site, in the heart of the Milky Way. Regions of high stellar density are represented in pink and purple, intermediate density regions in violet and light blue, and low-density regions in dark blue. The map also shows areas of high concentrations of stardust, in green, and known clouds of ionized gas are identified by brown spheres. The map lists nearly 400,000 stars over a distance of 10,000 light-years from the Sun. Centered around the latter, the map represents the galactic disc observed from a point very far from our galaxy. With a database of the positions and trajectories of more than a billion celestial bodies, the “Gaia” mission is one of the most prolific to date, providing astronomers with detailed information (source:© European Space Agency/K. Jardin/www.esa.int).

This distribution seems to change very little from one part of the galaxy to another, and even from one galaxy to another, while physical conditions, such as density, can vary considerably. Patrick Hennebelle used numerical simulation (Figure 3.18) to understand the origin of this distribution, while studying the case of the stellar cloud:

“We have performed a series of calculations describing the dynamics of a collapsing stellar cloud of a thousand solar masses. By varying over large amplitudes, from 10 to 10,000 for example, different model parameters, such as density or turbulence intensity in the initial state of the cloud, or by modifying the laws of gas thermodynamics, we have studied numerous evolution scenarios and identified which physical parameters influence the dynamics of the stellar cloud”.

COMMENT ON FIGURE 3.18.– The figure represents the density of stars calculated in a collapsing cloud. It is represented in color by increasing values from blue (low density) to green (high density). The red dots mark stars of different masses whose formation is observed during the calculation (source: © CEA-DAp).

The result of the simulations first of all surprised the researcher, as the analysis of the calculations showed that thermodynamic phenomena explain the distribution observed on the masses of the stars. The gas state equation describes the radiation transfer processes that occur in the collapsing cloud; it controls the cooling of the gas, including its ability to remove excess gravitational energy. By varying the parameters of this equation, researchers have better understood the dynamics of the collapsing cloud.

“My initial idea was that thermodynamics played a minor role and we made this discovery almost by chance! The simulations then helped us in interpreting the results and developing a theory to explain them”.

The mechanism proposed after the analysis of the simulations predicts the correct value of the characteristic mass of the stars: it effectively leads to a stellar mass distribution in accordance with the observations (Figure 3.19).

COMMENT ON FIGURE 3.19.– The graph shows the distribution of stars produced in the collapse of a cloud according to their mass and under different assumptions introduced in the simulations. The distribution of the mass of stars has a maximum value around a mass of one-tenth of the solar mass, close to that observed in the cosmos (source: © CEA-DAp).

If simulation nowadays allows discoveries in astrophysics, it is difficult to attribute the sole merit to it:

“The calculations allow virtual experiments for which two questions are asked: the validation and understanding of the physical phenomena at work in the observed systems. A discovery involves a chain as a whole: observations, simulations and their interpretations”.

Simulation, which now plays an essential role in the development of knowledge in astrophysics, benefits from the advances of many digital disciplines: HPC computing techniques and AI algorithms are making their way into the astrophysicists’ toolbox [VAR 19]. In addition, some calculation methods originally developed by astrophysicists have more or less direct applications in the industry. This is the case with the so-called “SPH method”, for example, a particulate method initially invented to describe certain celestial dynamics. It is also used by engineers to solve flow equations under conditions where conventional methods encounter difficulties (Figures 2.1, 2.19 and 3.20).

COMMENT ON FIGURE 3.20.– This aquaplaning simulation helps to improve vehicle handling. The difficulty of the calculation lies in modeling the contact of the tyre on the road, which pinches a fluid blade. Most of the methods used for fluid mechanics simulations fail to reproduce aquaplaning. Based on the description of particles, the SPH method, initially developed to conduct astrophysical calculations, proves to be adapted to this situation.

3.2. Geophysics

Journey to the Center of the Earth is the third adventure novel published by Jules Verne [VER 64]. He recounts a scientific discovery – that of the depths of our planet, coupled with an initiatory adventure, that of the narrator of the story, the young researcher Axel Lidenbrock, who accompanies his uncle in his underground research – and emerges from this extraordinary experience. The novel was written at a time when geology was in full development. Extrapolating on the knowledge of his time, Verne’s imagination describes an Earth full of mineralogical and paleontological wonders (Figure 3.21): eternal diamonds, mushrooms or petrified trees, fantastic creatures – all species extinct from the surface of the globe (algae, fish or prehistoric monsters).

Advances in geophysics at the beginning of the 20th Century made it possible in particular to draw up a map of the Earth’s crust that was more in line with reality, highlighting its dynamic nature. The theory of plate tectonics was validated by the international scientific community in the 1960s. It explains the constant movements of the Earth’s mantle, which cause earthquakes, eruptions and tsunamis in various parts of the world. Responsible for terrible human losses, as well as very significant material damage, these events have left their mark on humans and their places of life for many years to come.

COMMENT ON FIGURE 3.22.– The colors on the map represent the age of the ocean lithosphere. The most recent zones, formed at the level of oceanic faults, are indicated in red. The age of the oldest areas (represented in purple) is estimated at 280 million years. The areas in green correspond to an average age of 130 million years. The black lines represent the boundaries of the tectonic plates. The movement speeds of the plates are in the order of a few centimeters per year. The fastest movements, about 20 cm/year, are recorded in some regions of Southeast Asia – such as Papua New Guinea – and the Pacific – such as the Tonga-Kermadec archipelago (source: https://www.ngdc.noaa.gov).

Over the past decade, digital simulation has become a widely used tool in geophysical sciences. Used by researchers in this discipline, it helps in the assessment of natural disasters risks. In the following, we will examine three fields of application: earthquakes, tsunamis and eruptions.

3.2.1. Earthquakes

Italy is located in an area of intense seismic activity: the life of the geological layers of this region is surrounded by the Alps to the northwest and by Mount Etna to the southeast of the country, which are evidence of tectonic activity. Characterized by violent ground movements, with acceleration levels far exceeding the acceleration of gravity, an earthquake is perceptible in all three spatial directions and produces strong mechanical stresses on buildings, often leading to their destruction. On Monday, April 6, 2009 at 3:52 am, the country experienced a major earthquake near L’Aquila, a town in the mountainous region of Abruzzo. With a magnitude of 6.2, its human toll was terrible: more than 300 dead and 1,500 wounded. The earthquake also destroyed many villages: about 40,000 buildings were demolished or severely damaged, affecting the lives of more than 70,000 people in the region. It is one of the most violent earthquakes recorded in Italy (Figure 3.23).

COMMENT ON FIGURE 3.23.– The map shows earthquake records in Italy, classified by magnitude. This magnitude, representing the logarithmic energy released by an earthquake, is related to the damage it causes. A magnitude increase of 1 corresponds to a 30-fold increase in seismic energy.

A chain of solidarity was set up after this disaster: providing relief, shelter, care and comfort. For the rescuers, a race against time is underway. The aim is to search for possible survivors in the rubble as soon as possible. Hours of work without sleep, in very difficult conditions – and let us not forget: this is a Western country, rich and organized, with extensive human and technical resources.

In the case of such disasters, an additional danger comes from very strong aftershocks that can occur in the following days, on the same geological fault or on other faults located in the region. These aftershocks may endanger the lives of rescuers and people working in the debris. On Thursday, April 9, 2009, a very strong aftershock, magnitude 5.1, occurred and completed the destruction or damage of buildings in the L’Aquila region. Fortunately, residents had been evacuated from the heavily affected area since the first earthquake and the aftershock only caused material damage.

In a discreet way, a geophysics researcher and numerical simulation expert helped to assess the risk of this eventuality by locating potentially dangerous ground vibrations. It was carried out using numerical simulations of possible aftershocks from the initial earthquake.

After the first seismic tremors, seismologists gained accurate data on the disaster – for example, the coordinates of the epicenter of the earthquake as well as its approximate depth, and the accelerations recorded at different locations. They also became aware of the other geological faults in the impacted region and can speculate on how the original fault, or even others in the region, could fail. Dimitri Komatitsch tells us:

“In the aftermath of the L’Aquila earthquake, I was contacted by seismologists from the National Institute of Geophysics and Volcanology (INGV) in Rome to build a numerical model of the region concerned. Using a calculation code that a community developed over the years to simulate ground vibrations in the event of an earthquake, I built such a model in less than 12 hours, feeding it with geological data from the L’Aquila region and INGV data on the first earthquake”.

The calculations, sent to the Italian authorities for analysis, made it possible to obtain in a few hours the results of several aftershock hypotheses in the region for the days following the initial earthquake (Figure 3.24). The aftershock shown on the image was calculated in 2009. On 30 October 2016, the region experienced a seismic event whose global characteristics (magnitude, location of the epicenter and nature of the geological fractures) were very close to this calculation.

Second, the partial erasure of the aftereffects of such a disaster is slow. Before reconstruction, an inventory of damaged buildings must be logged (Figure 3.25) and safe repair options must be sought. For some residents, it takes years to return to normal life.

In order to use simulation in emergency situations, it is crucial to perform fast and accurate seismic aftershock calculations and the code used by the researcher has two advantages:

– a particularly efficient calculation method solves the equation of seismic wave propagation in geological layers. These are described by means of a viscoelastic behavioral law [TRO 08]. This equation, also discussed in Chapter 1 of the first volume, is similar to that used for the vibration of musical instruments. It yields accurate results for large-scale seismic models (that of a given region);
– the computing power of a supercomputer, at the time a partition of nearly 500 computing cores of a machine, assigned urgently for these life-size calculations by GENCI in France. The development choices made on the seismic simulation code at the time of its development made it particularly suitable for porting to HPC computing means [KOM 11].

The improvement of this seismic calculation tool remains at the heart of geophysicists’ research today:

“One of the main areas of development is a more detailed characterization of the basements. It is accomplished through seismic imaging, by comparing data recorded in situ after an earthquake and calculations made in these situations. The difference between the seismicity data recorded and their simulation allows for the iterative correction of geological layer models. To do this, the calculation codes constantly perform simulations: there are sometimes thousands of them for correcting a particular problem!”.

Simulations obviously do not make it possible to predict the date of the seismic event, its location or its intensity. They contribute to the production of probable data for risk analysis. As a reminder, a few days before the major earthquake, a commission of Italian experts met in L’Aquila to analyze a series of earthquakes that had occurred in the region in the previous months. While they were obviously unable to predict the imminent arrival of a stronger earthquake, they made recommendations on the potential dangers facing the region at that time. For some Italian citizens, these were insufficient. After the earthquake, the seven scientists on this commission were found guilty of negligence in their seismic risk analysis. They were accused of giving too much reassuring information to the population, who could have taken measures to protect themselves. The experts were sentenced to prison terms and acquitted 2 years later on appeal. These trials have had a strong impact on the families of the victims of the earthquake and have also caused a great deal of misunderstanding and emotion in the international scientific community. These facts raise questions about the place and role of scientists in the prevention of natural risks, and about citizens’ expectations of experts and even the sciences. Legitimate in the obligation of means and in the search for proven errors, such as negligence or breaches of mandatory and legal procedures, these expectations can sometimes be disproportionate, even unrealistic, given the current capacity of science and scientists to predict with certainty certain events and their effects. This capacity often remains limited, as in the case of seismic risk:

“The only current earthquake prevention measure is to identify areas at risk (L’Aquila was one), build buildings seismically (designed to withstand earthquakes) and educate people to take shelter at the very beginning of seismic movements…”⁵

Let us conclude this section by noting that the Earth shakes under the effect of tectonic movements as it also can under the effect of crowds –with a vibration energy that is nevertheless much lower [SIM 18]. During the 2018 football world cup, the vibrations generated by the jumps of joy of Mexican fans were recorded by the seismographs of the Mexican Atmospheric and Geological Research Institute⁶.

3.2.2. Tsunamis

Hokusai, master of Japanese printmaking, was nearly 70 years old when he published his 36 Views of Mount Fuji in 1830. His sumptuous images are a true worship of the sacred mountain: it is present in an imposing way, or by becoming more discreet, in all the landscapes or scenes of life that the artist represents there. The first print in the series is probably Hokusai’s best-known print. The Great Wave [CLA 11] testifies to Japan’s ancestral vulnerability to tsunamis.

On March 11, 2011 at 2:45 am, an earthquake of magnitude 9.0 occurred 130 km (81 miles) off the coast of Sendaï on Hinsu, the main island of the Japanese archipelago. This earthquake was the most devastating in Japan after the Kobe earthquake in 1995 (5,500 deaths). It was the fourth most intense earthquake ever recorded since the first measuring devices were deployed in 1900. The tsunami caused was the deadliest since the Hokkaido tsunami in 1993 (200 deaths).

Images of the country’s devastation and the pain of its inhabitants left their mark on the entire world. The official report drawn up in February 2015 by the Japanese authorities indicated that 15,890 people were killed and 2,590 missing or presumed dead, as well as 6,152 injured in the country’s 12 prefectures. The earthquake and tsunami caused nearly $220 billion in damage in Japan and contributed to the disaster at the Fukushima I (Daiichi) nuclear power plant – with severe consequences for the health of the country’s population and economy.

COMMENT ON FIGURE 3.26.– A tsunami is a train of ocean waves generated by seabed movements, themselves resulting from an earthquake or eruption. At high depths, the tsunami wave has a very small amplitude in the order of a few centimeters. When it arrives near the coast, it can take on a much greater amplitude and break into a turbulent wall several meters high.

The effects of the tsunami were felt throughout the Pacific Rim. It caused damage estimated at nearly $30 million on the island of Hawaii, destroying U.S. Navy facilities, and caused more than $6 million in losses to the fishing industry in the city of Tongoy, Chile – more than 16,000 km from the epicenter of the earthquake. Satellite observations also revealed for the first time the effects of the tsunami on the collapse of the Antarctic ice cap.

The prevention of such disasters is based, among other things, on an observation of the state of the seas. In the Pacific Ocean, the DART (Deep-ocean Assessment and Reporting of Tsunamis) network consists of buoys that measure wave elevation and detect the formation of a tsunami. The data collected are used, for example, by various state agencies, such as NOAA* in the United States, to make predictions about the spread of a tsunami once it is detected. The models focus on the tsunami’s velocity (Figure 3.27) and the amplitude of the waves formed (Figure 3.28).

COMMENT ON FIGURE 3.27. – In the case of the Japanese tsunami in March 2011, the detection occurred about 25 min after the initiating earthquake and propagation calculations allowed the authorities of many countries bordering the Pacific Ocean to take measures to ensure the safety of the populations and facilities.

COMMENT ON FIGURE 3.28.– The image represents a simulation of the wave height generated in the Pacific Ocean by the tsunami of March 11, 2011. The model is established by NOAA’s Pacific Marine Environmental Laboratory, using a calculation method adapted to this type of event, and based on the use of data collected by a network of buoys at sea. A tsunami spreads like a “lonely wave of infinite wavelength”: the wave seems to move as a whole. The propagation speed of a wave varies globally like the square root of the product of gravity and depth: in deep water, the speed of a tsunami can reach nearly 800 km/h while not exceeding a few tens of centimeters in height. The largest waves are located in the area near the epicenter of the earthquake off the coast of Japan. Their amplitude decreases as they spread to the deepest regions of the Pacific Ocean. They break when they encounter shallow waters near the coast. The energy carried by the waves weakens as they move away from where they originated: the effect of the tsunami is less devastating off Hawaii than it is on the Japanese coast (source: https://www.tsunami.noaa.gov/).

Tsunamis can be caused by geophysical events other than earthquakes: underwater volcanic eruptions or island volcano collapses are other possible causes. While numerical simulations nowadays make it possible to properly account for the propagation of tsunamis, one of the major uncertainties in the calculation is that of the conditions under which they are generated. What energy is transported by the collapse of an island volcano slope? Is it enough to trigger a major tsunami? This risk is real in different parts of the world and for many volcanic islands that experience regular eruptive episodes (Canary Islands in Spain, Hawaii in the United States, Reunion Island in France, Java in Indonesia). The Cumbre Vieja volcano (Figure 3.29) on the island of La Palma in the Canary Islands attracts the attention of scientists because of its position and geology.

In 2001, two geophysical researchers, the American Steven Ward and the British Simon Day, published a study on the risk of a collapse of a part of the Cumbre Vieja during an eruption. According to their calculations, this event would create a tsunami whose effects would be felt across the Atlantic ocean, as far as the Florida coast, and would also affect Africa and Western Europe [WAR 01]. Their article had a certain impact on the general public and the risk of the volcano’s collapse was sometimes presented in a caricatured manner, as if by a sensationalist press invoking biblical predictions [BEI 17].

After its publication, the conclusions of Ward and Day’s study were widely discussed by the scientific community. The risk that the two researchers highlighted with their simulation proved overestimated, as the calculations are based on very conservative assumptions – they assume, for example, that half of the volcano collapses during an eruption, a hypothesis that some geologists dispute [DEL 06].

The scientific controversy has highlighted the need to better control the hypotheses of the simulations, in particular by refining models of volcanic debris flow. This is achieved through more accurate experimental data and calculation codes, allowing a more realistic assessment of the risks of a collapse and of a tsunami [ABA 12].

Sylvain Viroulet, a young researcher and author of a PhD thesis on the generation of tsunamis by geophysical flows [VIR 13], explains how a simulation based on experimental campaigns is constructed:

“I have worked on different landslide models to understand the dynamics of debris flow. My research consisted in developing an experimental device and a numerical model. Validated on the controlled configurations that the experiment allows, the simulation enables extrapolations to situations similar to those encountered in the field. Using numerical calculation, it is possible to study a wide variety of scenarios of a volcano’s collapse in the ocean, which allows for a better characterization of the wave thus created and the intensity of the tsunami it may generate”.

In the laboratory, the aim is to reproduce different configurations representative of landslides caused by the flow of a debris stream or by the impact of debris on a static cluster (Figure 3.30).

COMMENT ON FIGURE 3.30.– The figure shows photographs of debris flow experiments in two situations. On the left, a jet flow: the materials encountering a bump detach from the ground and follow a parabolic trajectory, before touching the ground again. Conditions of “stationary flow”, independent of time, then establish themselves. On the right, a debris flow is triggered by the impact of materials on a cluster upstream of a bump. After contact, there is a sudden decrease in the overall velocity of impacting particles and an increase in the thickness of the flow. These conditions propagate upstream of the resting area until the flow reaches an equilibrium position and then develops a stationary flow [VIR 17].

The experiments allow us to understand the physics of granular flows, involving different phenomena, among which erosion and deposition of matter, the presence of gas in the fluid bed or the segregation of particles according to their size – the more they are mixed the less they mix, so that large particles form the front of the flow and constrain it by retaining the smallest particles.

“With the computer, there are several options available to researchers and engineers. Particulate methods consist in solving Newton’s laws grain by grain, taking into account interactions with a surrounding fluid, the latter being represented by the Navier–Stokes equation. However, computation times are very long with this type of approach and researchers are developing methods from fluid mechanics to apply these simulations to geophysical cases. The challenge is then to represent the granular rheology using an equivalent friction”.

In Navier–Stokes equations, for example, this granular rheology can be represented by a viscosity that changes with the pressure and shear effects observed on moving grains [LAG 11]. The avalanche of a column of volcanic debris is, to some extent, accessible by simulation (Figure 3.31), but the entry into the water is not yet well represented: this physical ingredient is still missing in current models.

A forthcoming eruption of Cumbre Vieja, like other active volcanoes on the planet, is certain. The risk of a Great Wave cannot be scientifically ruled out in such circumstances. The use of simulations based on a physical analysis allowed by laboratory experiments thus contributes to evaluating the consequences of different collapse scenarios in a more realistic way than the apocalyptic predictions, which some may sensationally claim.

COMMENT ON FIGURE 3.31.– The figure shows a simulation of the debris avalanche that occurred in 2010 on the Scopoma volcano in Chile. It is carried out with the VolcFlow code⁷. This calculation tool can be used to determine the rheology of pyroclastic flow or debris avalanches. It allows for a visualizaton of the surface deformations of these flows and an interpretation of their physics. It can also be used to simulate the formation and propagation of mudslides or tsunamis. The code is made available by its developers to a wide scientific community, particularly in Asian and South American countries that do not have the research resources necessary for their development. Using this code, French and Indonesian researchers studied a scenario of a landslide of the Krakatoa volcano in Indonesia in 2012. Their calculations described the characteristics of the tsunami that would result from an awakening of the volcano [GIA 12]. Such a tsunami did indeed occur in December 2018 (source: Karim Kelfoun, University of Clermont-Ferrant).

3.2.3. Eruptions

The second print in Hokusai’s series, 36 Views of Mount Fuji, depicts the volcano in all its glory [BOU 07]: a red pyramid rising into the sky at a quivering dawn, coloring the sky a royal blue and dotted with cloud lines striating the atmosphere. The slopes of the volcano, zebra-striped at its peak with snowflows, soften at its base to a gradation of reds and oranges mixed with the greens of a sparse forest (Figure 3.32). There is no reason to believe that there is any danger behind this peaceful landscape: we contemplate Mount Fuji, forgetting that it is a volcano, still considered active today, although its latest eruption only dates back to the 18th Century.

With more than 110 active volcanoes, Japan is located in the “Pacific Ring of Fire”, a vast area that contains most of the world’s earthquakes and volcanic eruptions. In 2018, a Japanese government study indicated that an awakening of the volcano could cause the accumulation of 10 centimeters of ash in central Tokyo, 100 km from Fuji. These ashes would make roads impassable, blocking transport and thus the city’s food supply [ICH 18].

Volcanic eruptions have both local and global consequences, often dramatic for populations, as these two examples show:

– In 1991, the eruption of Mount Pinatubo in the Philippines was one of the most significant of the 20th Century. The surroundings of the volcano were profoundly disturbed: the mountain was losing a significant amount of altitude and the surrounding valleys were, over hundreds of meters, completely filled with materials resulting from the eruption. The forest on the mountain slopes was completely destroyed, the animal species that used to live there died. With a death toll of nearly 1,000, it made the country pay for a heavy human and economic loss.
– In 2010, the Icelandic volcano Eyjafjöll projected a large plume of water vapor, volcanic gases and ash. Driven by the prevailing winds that brought them down to continental Europe, it caused major disruptions in global air transport for several weeks.

Volcano activity is closely monitored by the security authorities of the many countries affected by their potential eruption. Karim Kelfoun is a researcher at the Magmas and Volcanoes Laboratory, Clermont Ferrand Observatory of Earth Physics, Clermont Auvergne University. He develops tools for simulating volcanic flows [KEL 05, KEL 17a, KEL 17b] used for research and risk analysis purposes:

“Volcanic eruptions are obviously rare and dangerous and laboratory experiments exist only for the study of the physics at stake. These are confronted with scale problems that do not make them relevant for field interpretation. Only numerical simulations may be used to study different eruption scenarios, such as changes in crater topography or slope topography, and they assess the consequences of assumptions made by volcanologists. Calculations help to refine volcanic risk maps, for example for predicting trajectories and potentially destroyed areas. Numerical tools are becoming more widespread in this field, particularly because of the increasing demands that populations at risk make on their country’s security authorities. The simulations are currently quite accurate, however, some parameters of the calculations are based on observations, not on physics. This limit can be problematic when it comes to predicting eruptions with rare characteristics. The forecast keeps an approximate aspect because the characteristics of the next eruptions are not known in advance”.

Numerical modeling is based on the conservation equations of the different physical quantities monitored over time. The simulation consists of solving these equations using a numerical method describing the mass and momentum fluxes and the forces exerted on the calculation cells of which the model is composed.

The main difficulty is to model the rheology of flows: classical models, describing friction within the material, are not adapted to volcanic materials. Most of the research work is carried out in this field by comparing different theoretical rheological models with field observations. In addition, the numerical methods used must be “stabilized” and “optimized”, thus the calculation code:

– continually checks that the elementary physical principles are respected, for example that flows do not go up the slopes: this effect is obviously impossible in reality, but a calculation can artificially produce it⁸;
– uses efficient algorithms to perform simulations in the shortest possible time. Depending on the case, the calculation times can take between a few minutes and 1 hour on a standard computer.

It is by looking into the past, i.e. by analyzing field data collected during known eruptions, that researchers try to predict the future through their simulations. Numerical models are built from the topographic and geological surveys of volcanoes, and it is crucial to know them beforehand in order to carry out simulation risk studies. “Geophysical monitoring, carried out by volcanological observatories, is also very important: it provides precise data, useful for calculations”, explains Karim Kelfoun.

An example of a simulation is presented below for the Indonesian Merapi volcano, which erupted several times in the 20th Century (Figure 3.33).

COMMENT ON FIGURE 3.33.– The photograph is an aerial view of the Indonesian Merapi volcano, taken during its eruption in 1930. A volcanic plume escapes from the crater at its top. The latter is partially blocked by a lava dome, visible as the dark mass under the crater (source: www.commons.wikimedia.org).

The simulation performed is that of the 2010 eruption (Figure 3.34) and according to its author, it makes it possible to properly reproduce the major phases of the eruption, and to understand qualitatively the phenomena involved. However, the code has serveral limitations of modelling, which researchers are currently working on.

“In reality, the role of temperature and atmospheric ingestion in flow, which would explain its very high fluidity, is not yet rigorously described in the simulations. In the laboratory, the calculation code reproduces experiments to study the effect of air on flowability. Our current research aims to better understand these mechanisms in order to quantify them at scale and under field conditions…”

As a conclusion, let us remark that the calculation code used for these simulations is developed in France by researchers from the Magmas and Volcanoes Laboratory, located in the heart of the Auvergne Volcanoes National Park: with mountains somehow as majestic as Mount Fuji – but less threatening, since they have long been plunged into a long volcanic sleep.

1 The parsec is a unit of measurement for long distances, as used in astronomy: 1 parsec corresponds to 3 light years, the distance traveled by light in 3 years, at a speed of 300,000 km/s. It is also the distance that separates the Alpha Sun from the Centaur, the closest star to our galaxy. The size of the simulation domain represents 3 billion times this distance, that is 9 billion light years.
2 See, for example, http://www.virgo-gw.eu/ and https://www.ligo.org/.
3 Carl-Gustaf Arvid Rossby (1898–1957) was a Swedish meteorologist. He was interested in the movements of large fluid scales in the atmosphere and the ocean. The wave instability that bears his name explains, for example, the characteristic shape of the Great Red Spot of Jupiter or that of cyclones. Found in the formation of black holes or planets, this mechanism, long known in meteorology and oceanography, is more recently studied in astrophysics.
4 Emmy Noether (1882–1953) was a German mathematician who contributed to major advances in mathematics and physics. The theorem which bears her name stipulates that the symmetry properties encountered in some equations of physics are related to the conservation of a given quantity. A “symmetry” refers to any mathematical transformation of a physical system that lets the transformed system be indistinguishable to the non-transformed one. This means that the equations that describe the system are the same before and after transformation. For instance, Noether’s theorem indicates that whenever a system is symmetric with respect to translation, there is a conserved quantity that can be recognized as the momentum. Emmy Noether demonstrated her theorem in 1915 at the time Albert Einstein conceived the General Relativity Theory – and Noether’s theorem was a decisive contribution to Einstein’s work. Noether’s theorem set up the groundwork for various fields in theoretical physics and can also be applied in engineering sciences [ROW 18]. As a woman of her time, Noether struggled to make her way in science – a man’s world – and she had to seek the support of eminent scientists, such as David Hilbert, to pursue her academic career. Still, she was not allowed to teach using her own name, let alone received wages from this… She also had to face the disapproval of some of her male colleagues, sometimes expressed in a violent form: “No women in amphitheaters, science is a man’s business!” [CHA 06]. Emmy Noether proved them wrong… and is considered one of the most brilliant minds of the 20th Century.
5 www.planet-terre.ens-lyon.fr.
6 Available at: http://iigea.com/sismo-artificial-por-celebracion-de-gol-en-mexico/.
7 Available at: http://lmv.uca.fr/volcflow/.
8 As we mentioned in Chpater 4 of the first volume, the question as to whether a simulation correctly renders physical phenomena applies to all kinds of calculations.

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Table of Contents for 3 The Universe and the Earth

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