In the previous chapters we assumed, explicitly or tacitly, that turbulence is isotropic, in particular that there is no mean magnetic field, so that spectra depend only on the modulus of the wave vector k and structure functions only on the distance l between two points. We also assumed that the kinetic and magnetic energies have similar magnitudes, which implies that the magnetic field is distributed in a space-filling way. However, even if the turbulence is globally isotropic, the local dynamics is not, differing strongly between the directions parallel and perpendicular to the local magnetic field. As discussed in Section 5.3.3, the small-scale fluctuations are dominated by perpendicular modes, and Alfvén waves propagating parallelly are only weakly excited, which gives rise to the observed Kolmogorov energy spectrum k−5/3 instead of the IK spectrum k−3/2.

In nature magnetic turbulence often occurs about a mean magnetic field, just as hydrodynamic turbulence occurs about a mean flow. However, whereas the latter can be eliminated by transforming to a moving coordinate system, the presence of a mean magnetic field has a strong effect on the turbulent dynamics. If this field is much larger than the fluctuation amplitude, turbulence becomes essentially 2D in the plane perpendicular to the field, since the stiffness of field lines suppresses magnetic fluctuations, Alfvén waves, with short wavelengths along the field. Hence turbulent motions tend to simply displace field lines without bending them. To deal with this situation quantitatively we derive a set of equations for a plasma embedded in a strong magnetic field B0 = B0ez.

Turbulence is a ubiquitous phenomenon. Wherever fluids are set into motion turbulence tends to develop, as everyday experience shows us. When the fluid is electrically conducting, the turbulent motions are accompanied by magnetic-field fluctuations. However, conducting fluids are rare in our terrestrial world, where electrical conductors are usually solid. One of the rare examples of a fast-moving conducting fluid, which has been of some practical importance and concern and to which authors of theoretical studies sometimes referred, is, or better, was the flow of liquid sodium in the cooling ducts of a fast-breeder reactor. It is therefore not surprising that, in contrast to the broad scientific and technical literature on ordinary, i.e., hydrodynamic, turbulence, magnetic turbulence has not received much attention.

The most natural conducting fluid is an ionized gas, called a plasma. It is true that laboratory plasmas, which are confined by strong magnetic fields, notably in nuclear-fusion research, exhibit little dynamics, except in short disruptive pulses. Only the reversed-field pinch, a toroidal plasma discharge of relatively high plasma pressure, exhibits continuous magnetic activity, such that it is sometimes considered more as a convenient device for studying magnetic turbulence rather than as a particularly promising approach to controlled nuclear fusion.

The interstellar medium (ISM), which had formerly been better known for the allegoric shapes of its nebulae and dark clouds than for its physical properties, has developed into a fascinating area of astrophyscical research during the past few decades. An important aspect is the turbulent flows observed in many regions of the ISM, with velocities that, at least in the cooler parts, far exceed the speed of sound. These flows seem to play a decisive role in the cloud dynamics, slowing down gravitational contraction and star formation. In contrast to the objects studied in the two previous chapters, which had well-defined physical and geometrical properties and thus allowed a detailed analysis, the ISM is a rather diffuse system, whose modeling is far more uncertain and arbitrary. While the various atomic processes, such as transition probabilities and excitation rates, are well known and also the thermodynamic properties are fairly well understood, we have only a coarse picture of their hydrodynamics, including the effect of magnetic fields. Thus it is, for instance, difficult to apply hydrodynamic-stability theory without very special assumptions regarding geometry and flows. Hence the results we discuss in this chapter are mainly of qualitative nature, in which general arguments, such as equipartition and virialization, play an important role. The precise numerical factors, which are often found in the astrophysical literature, imply special choices of geometry and profiles and should not be taken too literally. In Section 12.1 we give a brief overview of the characteristic properties of the ISM.

Turbulence is usually associated with the idea of self-similarity, which means that the spatial distribution of the turbulent eddies looks the same on any scale level in the inertial range. This is a basic assumption in the Kolmogorov phenomenology K41 and, on the same lines, the IK phenomenology introduced in Section 5.3.2. It is, however, well known that this picture is not exactly true, since it ignores the existence of small-scale structures, which cannot be distributed in a uniform space-filling way. In fact, in a real turbulence field experiments as well as numerical simulations show that smaller eddies, or higher frequencies, become increasingly sparse, or intermittent, which apparently violates self-similarity. This chapter deals with the various aspects of intermittency.

Section 7.1 gives a brief introduction. We illustrate the concept of self-similarity by some simple examples and clarify the notion of intermittency, distinguishing between dissipation-range and inertial-range intermittency. Section 7.2 deals with structure functions, in particular the set of inertial-range scaling exponents, which are convenient parameters for a quantitative description of the statistical distribution of the turbulence scales. We discuss the important constraints on these exponents imposed by basic probabilistic requirements. Since experiments and, even more so, numerical simulations deal with turbulence of finite, often rather low, Reynolds number, the scaling range may be quite short, or even hardly discernable, especially for higher-order structure functions, which makes determination of the scaling exponents difficult. The scaling properties can, however, be substantially improved by making use of the extended self-similarity (ESS), which often provides surprisingly accurate values of the relative scaling exponents.

Until now we have viewed the turbulent motions and fields mainly in space, or configuration space. Though spatial structures are important in MHD turbulence, for instance as final states of selective decay processes, the most characteristic property of fully developed turbulence is the presence of a wide spectrum of different scales. Turbulence scales have already been used in a very loose way in Section 4.2.2 in discussing the mechanism of dynamic alignment. In this chapter these ideas will be given a more precise meaning. In Section 5.1 we introduce the concept of homogeneous turbulence, a very useful idealization of a turbulence field far away from boundary layers. Here the Fourier components of the field play the role of the amplitudes at a certain scale l ∼ k−1. Section 5.2 considers an approximation that, at first sight, has little resemblance to real turbulence, namely a nondissipative system of Fourier modes truncated at a finite wavenumber and its relaxed states, which are called absolute equilibrium distributions. In spite of their artificial character such states can provide valuable information about the tendencies of the spectral evolution in dissipative systems, in particular the direction of the spectral fluxes called cascades. In Section 5.3 we then switch on dissipation in order to study the spectral properties of MHD turbulence.

In the previous chapters turbulence was assumed incompressible. As discussed in Section 2.3, this assumption is valid if either the sonic Mach number of the flow is small, Ms = υ/cs « 1, or the Alfvén Mach number is small, MA = υ/υA « 1. The former condition applies to a weakly magnetized plasma, in which υA « cs, or to motions along the magnetic field, while the latter applies to motions perpendicular to the field. If the flow is turbulent, there is some arbitrariness in the definition of the Mach numbers, since one may choose (a) the mean flow velocity, (b) the r.m.s. velocity fluctuation υ = 〈ῦ2〉1/2 = (Ek)1/2, or (c) the local velocity. Following convention in turbulence theory, we refer to the Mach number in terms of the r.m.s. velocity, noting that local Mach numbers may be considerably higher.

Since laboratory plasmas are usually confined by a strong magnetic field, they can be considered incompressible, the dynamics consisting mainly of cross-field motions. Also the motions in the liquid core of the Earth, which drive the Earth's dynamo, are incompressible, since Ms « 1 (here inertial effects are often neglected altogether, which is called the magnetostrophic approximation). By constrast, most astrophysical plasmas are compressible, for instance the interstellar medium, which is rather cold, such that, in the turbulent motions observed, Ms, and possibly also MA, tend to be large (see Chapter 12), or the turbulence in the interplanetary plasma, which is riding on the supersonic and super-Alfvénic solar wind (Chapter 10).

In the preceding chapters we considered the dynamics of an individual system. Starting from a smooth state, fine structures develop, which, in general, become unstable at some point. After the onset of instability the structure of the flow is very complex and irregular and, most importantly, the further behavior is unpredictable in the sense that minimal changes would soon lead to a completely different state. Such a behavior is commonly called turbulent. Though a direct view of the continuously changing patterns is certainly most eyecatching and fascinating, a pictorial description of these structures is not very suitable for a quantitative analysis. On the other hand, it is just this chaotic behavior which makes turbulence accessible to a theoretical treatment involving statistical methods. While individual shapes and motions are intricate and volatile, the average properties of the turbulence described by the various correlation functions are, in general, smooth and follow rather simple laws. A well-known paradigm is the turbulent behavior in our atmosphere. We try to predict the short-term changes, called weather, in a deterministic way for as long as is feasible, which, as daily experience shows, is not very long, while predictions of the long-term behavior, called climate, can be made only on a statistical basis.

Dividing the fields into mean and fluctuating parts, we derive equations for the average quantities, the generalized Reynolds equations, which contain second-order moments of the fluctuating parts, the turbulent stresses.

Consciousness is a nonphysical property that cannot be defined in physical terms, and indeed does not exist in the physical universe. It is impossible to determine by any physical means if an object is conscious. When presented with miscellaneous objects, such as an orange, a chair, a clock, a human being, a candle flame, and a crystal, an experimenter cannot determine by means of experiments with physical equipment which of these objects is conscious. This normally would constitute sufficient proof that consciousness does not exist anywhere in any form. One of the objects, however, could be myself, and I know beyond all doubt that I am a conscious being. I am more certain of my consciousness than was Dr. Johnson of the concreteness of his stone. Consciousness beyond all doubt exists, yet demonstrably does not exist in the physical universe. Consciousness belongs to the Universe not the physical universe. No other conclusion seems possible.

Everything is spread out in time. Things stretch away into the recent past as recalled in our memories and newspapers, and into the distant past as recounted by historians and geologists. They also stretch away into the near future as anticipated in our plans and foretold by fortune tellers, and into the distant future as predicted by geologists and astronomers.

Say no more of time! If you want a peaceful mind go no farther. Every step in quest of understanding time leads to greater bewilderment. Much of the problem is that our languages inadequately express our experiences of time.

“What, then, is time?” asked Augustine of Hippo in the Confessions. “If no one asks me, I know what it is. If I wish to explain what it is to him who asks me, I do not know.” He viewed time as a continuous temporal sequence from the past to the future, from Creation in the beginning to Judgment in the end. Time thus displays a historiography ordained by either God, fate, or natural law. This is much the same as our present commonsense general view. It caused him much perplexity, some of which is expressed by Austin Dobson in The Paradox of Time:

Some of the problem is easily stated: Nothing displayed in time can change! If you think of time in terms of space, as an extension, a sort of one-dimensional space, with everything displayed in it, such as birthdays, anniversaries, holidays, then everything has its fixed moment in time and cannot possibly change.

We have a picture of a seamless spacetime projecting into the space and time of each observer's world line. Though elegant and economic, in one sense it differs little from the Newtonian picture. Space in the Newtonian scheme was just a sort of nothing (a sideless box) spanning everything, and time was a similar sort of nothing in which everything also had location. In the theory of special relativity both came together to form an expanse of spacetime containing everything that again was just a sort of nothing (just a bigger sideless box).

Then in 1916 Einstein advanced the theory of general relativity and the picture changed dramatically. (How dramatically was not fully realized for many years.) Spacetime lost its state of nothingness and acquired a tangible physical reality. Gravity ceased to be a mysterious astral force acting instantaneously at a distance and became a property of dynamic curved spacetime.

In the new scheme spacetime itself guides the heavenly bodies and the old astrological action at a distance turned out to be the curvature of space and time combined into spacetime. We now have a spacetime that pulls and pushes and transmits shivers and shakes at the speed of light. We cannot eat spacetime, but it can be hit, and can hit back, and can eat us if we stray too close to a black hole. Spacetime in general relativity springs to life and becomes an active participant in the physical universe.

In the preface to the first edition of Masks of Universe I wrote: “At first I thought this book would take me only a few months to write. After all, the basic idea was simple, and only a few words should suffice to make it clear and convincing. But soon this illusion was shattered. A few months grew into three years, and now I realize that thirty years would not suffice. But enough! Other work presses, and life is too short.” Here I am, not thirty years but almost two decades later writing the preface to the second edition and struggling again to make clear the “simple idea.”

The idea rests on the distinction between Universe and universes. The Universe by definition is everything and includes us experiencing and thinking about it. The universes are the models of the Universe that we construct to explain our observations and experiences. Beneath the deceptive simplicity of this idea lies a little-explored realm of thought.

No person can live in a society of intelligent members unless equipped with grand ideas of the world around. These grand ideas – or cosmic formulations – establish the universe in which that society lives. The universes that human beings devise and in which they live, or believe they live, organize and give meaning to their experiences.

The Universe is everything. It includes us and the rational universes we collectively devise. Each universe unifies a society and dictates the “true” facts. Individuals suppose with unfailing confidence that their particular universe is the Universe, and their confidence is not in the least shaken by the fact that our ancestors lived in very different universes and our descendants in the future will live also in totally different universes. In all universes things have their causes, often hidden from ordinary mortals. We depend on our wise men – emperors, kings, shamans, priests, sages, prophets, and professors – to put us right and tell us the “true” facts. As long as somebody reliable knows the truth, all is right with the universe.

All universes have their rules of containment that define what is included as fitting and what is excluded as unfitting. Thales said the Ionian universe consisted of water; Anaximenes said air; Heraclitus said fire; Xenophanes said earth; Empedocles said earth, water, air, and fire. Democritus said the Atomist universe consisted only of atoms and the void; all else was illusion and opinion. Plato said the Platonic universe consisted of the eternal verities of the Mind; all else was shadow and deception. Aristotle said the Aristotelian universe consisted of earth, water, air, fire, and ether in ascending order, animated by Ideas, and nothing existed beyond the sphere of the stars. Saint Augustine said the Christian universe consisted of the Word of God, and all else was heresy.

The telescope, microscope, thermometer, barometer, precision clock, air pump, and other seventeenth-century inventions preceded the Age of Reason in the eighteenth century. The age of enlightened reason commenced with prophets proclaiming visions of a new universe: “I feel engulfed in the infinite immensity of spaces whereof I know nothing and which know nothing of me, I am terrified. … The eternal silence of these infinite spaces alarms me,” said Blaise Pascal. And “behold a universe so immense that I am lost in it. I no longer know where I am. I am just nothing at all. Our world is terrifying in its insignificance,” said Bernard de Fontenelle.

The mechanistic universe of the eighteenth century more than fulfilled the promise of the prophets. Outfitted with laws uniting the Earth and the heavens, with self-running celestial mechanistic systems distributed throughout endless space, with time ticking away regularly as in Huygens's precision pendulum clock, the mechanistic universe opened up the prospect of the human mind able at last to solve all the riddles of nature.

Lofty thoughts that formerly soared among the towers of the Eternal City descended to street level in an exhilarating new Earthly City. Pious otherworldly preoccupations transformed into practical worldly occupations. The reborn world was bright and young, free of the late medieval conviction that all was senile and exhausted. The rejuvenated human sciences, led by “lapsed Christians,” surged forward, achieving reforms that nowadays we take for granted as characteristic of Western society.

We take space and time for granted. Normally they do not trouble us, yet whenever we think about them we become puzzled.

Space seems simple enough. Here it is, all around us, stretching away and spanning everything in the external world. We are surprised when told that people in other cultures have different ways of regarding space. What is there about it that can possibly be different? Edward Hall in The Hidden Dimension says, “there is no alternative to accepting the fact that people reared in different cultures live in different sensory worlds” – in other worlds of space. It seems that the Arabs, Japanese, Hopi, and the people of many other cultures have different modes of expression concerning arrangements and relations in space; they live in different mental worlds – in other worlds of space.

Time is much more puzzling. Here it is in our imagination, stretching away, spanning everything in the past, present, and future. But unlike space it is not all around us and directly accessible. We experience time within ourselves, it seems, and cannot perceive it directly in the external world. Those intervals of minutes and hours on the face of a clock are actually intervals of space. A second cannot be displayed directly in pure form in the external world in the same way as a centimeter.

We live in the Solar System on the planet Earth that revolves with other planets around a star called the Sun. Light from the Sun hurrying at great speed takes 500 seconds to reach the Earth and five hours to reach the far-flung planet Pluto. The Earth that to us seems large is dwarfed by the Solar System with its whirling planets.

Starlight from the nearest stars travels for years before reaching the Earth. If we imagine the Sun having the size of a grain of sand, the nearby stars on the same scale would be at a distance of one hour's drive on an interstate highway. Scattered out to enormous distances in all directions are a hundred billion stars that constitute the whirlpool system called our Galaxy. The Galaxy – a glittering carousel of stars across which light takes 100,000 years to travel and around which the Sun journeys once every 200 million years – seems incomprehensibly large compared with the solar system.

Much has been discovered about the Galaxy: its many kinds of stars, sunlike stars, blue, yellow, and red giants, binary stars, white dwarfs, and dense neutron stars; its great spiral disk seen by us as the Milky Way where clouds of glowing gas and obscuring dust give birth to new stars; its even greater halo of very old stars and globular clusters; and still much that remains to be discovered.

Newton's universe of uniformly distributed stars has become Wright's universe.

Etienne Tempier, Bishop of Paris, roundly condemned all who dared to trifle with the power of the supreme being. Scholars and divines were free to admit reason into matters of faith provided full acknowledgment was made to God as an all-powerful being free of self-contradiction. Here was the Trojan Horse, introduced by the well-intentioned bishop, from which sallied forth in years to come thoughts that would topple the towers of the medieval universe.

Professors at Oxford and Paris in the fourteenth century made great progress in clarifying the nature of space, time, and motion. William Heytesbury and his colleagues at Merton College defined velocity and acceleration and then succeeded in calculating by graphical methods the distance traveled in an interval of time by a body having constant acceleration. William of Ockham participated in these studies while fighting a battle against needless abstractions. His celebrated principle of theoretical parsimony – known as Ockham's razor – states that in the use of concepts “it is foolish to accomplish with a greater number what can be done with fewer.”

Jean Buridan, a professor at Paris and formerly Ockham's student, revived the notion of impetus that can be traced back to Hipparchus in the second century B.C. and is now referred to as momentum. According to Buridan, impetus is proportional to the velocity of a body and also its quantity of matter (now referred to as mass), and the impetus of a thrown body maintains the body in a state of motion.