Numerical simulations of transport processes in magnetohydrodynamic

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NIVERSITATEA DIN

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D OCTORAL T HESIS

Numerical simulations of transport processes in magnetohydrodynamic

turbulence

Author:

Bogdan T^EACA˘

Supervisor:

Daniele CARATI

Supervisor:

Dan G^RECU

21 July 2010

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Acknowledgments

The present doctoral thesis represents the culmination of a long and arduous road, during which I had the support of many wonderful people. I will always be grateful to them and I would like to thank all the people that help me to reach this stage.

I would like to thank professor Dan Grecu for accepting to be my co-adviser for the joint doctoral program.

I would also like to acknowledge professor Daniele Carati. As a co-adviser, it was due to him that I started this line of work. It was a pleasure to work with him on so many projects and learn from one of the best teachers in the field. He opened many doors for me and started may of the collaborations in which I took part. Re- lated to his support, I would like to acknowledge Universit´e Libre de Bruxelles and the Statistical and Plasma Physics group for their financial assistance, which allowed me to concentrate on my work. Daniele, thank you for all!

Most of all, I would like to thank my parents and my family for investing in my future and for their uninterrupted support.

Mai mult decat orice, as dori sa multumesc parintilor mei si familiei mele pentru ca au investi in viitorul meu si pentru sprijinul lor neintrerupt.

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Preface

The strong couplings between different scales in a flow represent one of the pri- mary attributes of turbulence. They are expressed mathematically by the non-linear terms that enter the fluid balance equations and, in the turbulent regime, dominate the dynamics of the flow. In three-dimensional fluid turbulence, these couplings are the channel used to transfer the kinetic energy from the large geometry dependent scales to the small scales associated with dissipative effects. This energy exchange between scales is known as the Richardson cascade. The universal properties of highly developed hydrodynamic turbulence proposed by Kolmogorov, relies im- plicitly on the Richardson cascade to be a local in scale process in the inertial range. That is to say, the energy from one scale is transferred to an adjacent scale by the interaction of comparable scales. By multiple, successive exchanges, the information related to larges scales will be destroyed and the classical assumptions of isotropy and universality of small scales will be valid. On the other hand, if the transfers would be nonlocal,i.e.transfers involving scales of highly different size compared to local ones, which couple directly large and small scales, the characteristics related to the large scales will not be destroyed fast enough for the small scales to exhibit universality properties and the classical picture of turbulence will be weakened or even lost. Without locality, turbulent flows would be strongly influ- enced by large-scale forcing or geometrical properties of the boundaries throughout the cascade, making a general theory of turbulence impossible.

In magnetohydrodynamics (MHD), the Lorentz force influences the momentum balance equation and the number of non-linear terms is four instead of one like in the case of nonconductive fluid turbulence. This physical phenomenon is described by the union between hydrodynamics and electrodynamics in the classical mechanics limit and is characterized by non-linear equations. The various channels generated by the nonlinear terms can then be used to transfer the energy from the large scales to the dissipative range. In the MHD case, the problem com- plicates itself even more as new ideal invariants exists, with implications in the dynamics of the evolution equations. A good description of the energy redistribu- tion among scales is crucial for the development of adequate turbulence models.

The energy transfers in both fluid and MHD turbulence are usually presented in spectral space by computing the energy exchanges between the Fourier modes. The energy transfers between modes in turbulence are completely characterized by triad interactions. However, for strong turbulence regimes, the total number of modes

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active in the system is too large to be represented systematically by the triad interactions. Since the majority of modes have similar properties as their wave-number neighbors and bring similar contribution to the energy exchange between scales, the analysis of energy transfers is usually simplified by partitioning the spectral space into sub-domains and look at the averaged energy transfers between these sub-domains. The partitioning of the spectral domain is arbitrary but several convenient geometrical structures are preferred. The spectral spherical symmetry present in the case of isotropic turbulence naturally suggests a decomposition of the spectral domain into wave-number shells. For this case the energy transfer is described in terms of shell-to-shell transfer functions and spherical energy fluxes that have been studied in detail. In the presence of a mean magnetic field, the flow develops a preferred direction and exhibits anisotropy. The degree of anisotropy depends on the strength of the mean magnetic field. The angular dependence with respect to the preferred direction then becomes as relevant as the wave vector amplitude in the spectral space partition, and a simple shell decomposition is not sufficient for getting a more detailed picture of energy transfers. Coaxial cylindrical domains aligned with the preferred direction and planar domains transverse to each direction have both been used in the past to partition the spectral space. In the current work, another partition that is based on a ring decomposition of shells is proposed.

This novel approach provides many details on the energy transfers in an anisotropic system. Moreover, it allows to recover easily the isotropic shell transfer functions which have been extensively studied in literature. The ring decomposition proposed here is ideal for any form of anisotropic system.

The main goal of this thesis is to investigate the energy transport between scales using direct numerical simulations (DNS) of MHD turbulence. The locality properties of energy transport among scales for isotropic and anisotropic turbulence, generated by the presence of a constant magnetic field, is emphasize. A secondary objective is to establish a framework for the study of charged test particles transport in a turbulent electromagnetic field,i.e.an electromagnetic field generated by the motion of a conductive fluid, which possess multiple scale structures.

The thesis consists of three parts totaling seven chapters. A preface and after- word accompanies the presentation without being regarded as chapters themselves.

The structure of the thesis is presented in detail below, synthesizing each chapter separately. In the first part, consisting of the first two chapters, the author introduces concepts of turbulence, both hydrodynamic and magneto-hydrodynamics.

The first two chapters are chapters of synthesis. The first chapter is meant as a way to familiarize the reader with the phenomenon of turbulence, without resort- ing to a mathematical formalism. Starting instead from the physical intuition of an average scientist and from the observations of turbulence in nature, the necessary concepts are presented. The second chapter presents the theoretical description of magneto-hydrodynamic turbulence. MHD equations are introduced in real space and the symmetry properties and conservation laws are listed. The spectral form of these equations is obtained and a theoretical analysis, using dimensional arguments, presents the few theoretical predictions available.

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The second part is the main source of original results. Chapter three introduces the numerical methods used in solving equations, namely the pseudo-spectral methods. A new type of force, which imposes the level of dissipation of all ideal invariants, is introduced. In this way, the dynamics of the evolution equations are partially controlled. This approach allows for a systematic study of energy transfer fluid between different scales. In chapter four, an integrated energy transfer analysis of different scales of MHD turbulence is made. The energy transfer formalism is introduced. To investigate the energy transfer, the spectral space is decomposed into a series of wavenumber shells. The average energy transfers between these shells are then investigated. For anisotropic systems, the novel idea of spectral space decomposition in ring structures is presented and the analysis of energy transfer between these structures is presented. For isotropic turbulence, in chapter five, the locality of energy fluxes is investigated using locality functions.

For the hydrodynamic case these functions give an asymptotic scaling exponent of 4/3 in according with predictions given by closure theories. MHD turbulence, is found to have a more pronounced nonlocal behavior.

In the last part, which consists of chapters six and seven, the formalism for tracking trajectories of charged particles evolving in turbulent electromagnetic field is introduced. Chapter six concentrates on a problem of numerical nature, namely the influence on the particle solver of the interpolation method used. The final chapter, chapter seven, presents concepts related to the particle transport and diffusion regimes. The adiabatic nature of the charged particles motion is discussed and the transport of charged particle in a turbulent magnetic field is shown as example.

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The physics of turbulence

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Phenomenology of turbulence

1.1 Intuitive reasoning

The study of turbulence, like all other research topics in the field of natural sci- ences, began with observations of the world around us. Columns of smoke rising from fires, swirls of water in rapid rivers, plumes of ash expelled by erupting vol- canoes and the clouds in the sky are only a few typical occurrences that show clear signs of turbulent behavior. But what is really turbulence as a physical phenomenon? How does it form and how do we describe it? In this section we will try to introduce the concept of turbulence starting purely from physical intuition without the use of any mathematical formulation.

To start, we note that when we think about turbulence we almost always have in mind pictures of smoke or clouds and wrongly think of them as being turbulent.

In fact, the medium itself is turbulent, in this case the surrounding air, while the smoke or cloud vapors are just moved along in an action known as advection.

In this situation, the smoke or vapors are known as tracers¹ as they permit us to see the turbulent nature of the medium, as they trace a complex path when being subjected to the advective motion². The rare instances in which we notice directly the turbulent nature of the medium, like in the case of rapid water flows, are due to changes in the light reflection on the external flow surface, which in this case separates the water from air.

To understand turbulence, we must first look at a flow in complete absence of turbulence. In such a state, a flow is called laminar. Typically reserved for relatively slow moving flows, highly viscous fluids or small size flows, this state is quite rare in nature when compared to its turbulent contra-part. If in a laminar flow at rest we introduce a tracer, we observe the slow, omnidirectional diffusion of the tracer in that medium. Even if we start with a strange shaped tracer blob,

1A more technical name for the tracers is passive scalars as they do not influence the turbulent behavior. If the scalars influence the turbulent motion, then they should be considered as being part of the medium itself or the entire system should be looked at in the multi-fluid approach.

2For the advective motion to be seen, the diffusion of the tracer in the medium should be extremely small. Turbulent scales smaller then the tracers diffusively scale, will not be perceived.

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the diffusion motion will eventually smooth out any structures until a uniform, homogenous distribution of the tracer in the medium is observed. The same type of motion occurs for the fluid medium itself. This is due to viscosity, a measure of the resistivity of the flow to uneven¹motion tendencies. Due to viscous effects, any small irregularity in the flow will be smoothen out. For higher level of viscosity this behavior will be more pronounced. To better visualize this process, one can think of this effect as happening mainly at a particular scale, a viscous scale. If we introduce a motion to the flow at a scale smaller than this, the motion will be immediately dampened. We see that for uneven motion to exist its size should be larger then the viscous scale. The larger the scale of motion, the less pronounced will the smoothing effect due to viscosity be and the initial motion perturbation will take longer to be attenuated. Of course, large initial perturbations will take longer to subside. Due to viscosity, all turbulent systems tend toward a laminar state.

In the above paragraph, we have introduced in our reasoning the idea of scales.

The physical condition necessary for turbulence to occur, but not sufficient in itself, is that all the scales of the flow are coupled together. The fact that all scales are coupled does not mean that all motion scales are active or excited². This property characterizes the physics of turbulence, but this is also the main cause for compli- cations regarding its study. In the absence of viscosity³, any initial perturbation of a given intensity will be transferred by the coupling between scales until all of them, even initially dormant ones, reach a state of equipartition. In this state, all scales possess the same piece of the perturbation intensity, or simply put, in an equipartition state the initial perturbation energy is equally split between all scales. In this situation the smallest scale is theoretically infinitely small, while the large scales are bounded by the system size,e.g.the size of the Earth bounds the size of the atmospheric turbulence potentially possible. This theoretical case represents the quintessential turbulent state, an everlasting exchange of energy between motion scales.

Now, let us look at the entire picture and consider the flow of a common fluid⁴, which possesses a great number of scales larger than the viscous one. We consider the flow to be initially in a laminar state. Due to numerous causes related to the flow geometry, temperature and density gradients or uneven body forces, instabilities tend to form in the flow. Without the initial instabilities or if the viscosity can damp quickly the induced motion, a laminar flow will tend to stay in a laminar state even though a large range of scales is available in the system. In the situation when the instabilities are generated at a sufficiently large scale, the scale coupling will advect their effect to the adjacent range of scales. The instabilities will grow

1Shear stresses due to velocity gradients.

2The intensity of the motion, measured by the kinetic energy, can be zero for a particular scale.

3This theoretical medium is known as a Euler flow and is physically different from a flow possessing even an infinitely small level of viscosity.

4Some special fluids, like corn syrup, poses a different response to sheer-stresses which can not be accounted just by viscosity. They are known as non-Newtonian fluids.

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in the system, populating more and more motion scales and the laminar state will be lost. For the laminar flow, this can be seen at first as an oscillatory, sinuous motion of the flow that amplifies until any form of large scale pattern is lost. This latter stage we call fully developed turbulence, where similar initial perturbation will give completely different results in the flow’s evolution.

To answer our initial questions, turbulence represents the phenomena of propagation of an motion between the different scales of a flow. It is formed by motion instabilities that propagate until they populate the entire range of scales available.

In the absence of instabilities due to strong viscous effects or if the instability source is turned off, the flow will tend towards the laminar state, a situation known as decaying turbulence. We are just left with the non-trivial task of describing turbulence. For this, we turn first to empirical observations of flows in nature, before embarking on the path of mathematical reasoning.

1.2 Turbulence in nature

Although the coupling between scales represents one the main property of turbulence, turbulent flows are not universal in appearance at large scales. The large scales of a flow tends to depend more on the geometry of the problem and the size of the system then anything else. In Figure1.1 we clearly see that the cloud vortices generated by the island obstacle are of the same diameter as the island itself and have little to do with the size of turbulent structures already present in the flow. In this example, the vortex structures part of the vortex street can be seen as precursory to turbulence and even though their propagation is related to the same advective motion responsible for turbulence, may people do not consider this motion behavior as being turbulent on its own. In the top right corner of Figure1.1 we observe a multi scaled cloud pattern, more generally associated with turbulence. The vortices can be seen as seeds of turbulence as they will brake down into smaller and smaller structures, cascading energy to ever smaller scales in a typical behavior of turbulence, until the dissipative effects due to viscosity can take over and attenuate the entire motion. However, for quasi two-dimensional flows generated by the effects of gravity or planetary axial rotation, there is a tendency for small vortices to join up in forming large vortex structures that tend to dominate the system.

Due to its appearance, sometimes, people misinterpret turbulence as having a chaotic, random behavior. This is far from being true. In fact, one of the defin- ing characteristics of turbulence consists in the coherent structures being present at all scale. This can be best seen while looking at Jupiter atmospheric turbulence, Figure 1.2. The large red eddy, known as Jupiters Great Red Spot, not only is correlated in space, being the size of Earth, but is also correlated in time. It has been suggested that the Great Red Spot will survive for thousand of years¹. In

1A flow structure that tends to exist in a region of space for long times compared to typical times of interest in the flow are called zonal flows.

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Figure 1.1: Landsat 7 satellite image of clouds off the Chilean coast near the Juan Fernandez Islands on 15 September 1999. A pattern usually present for flows around cylindrical obstacles called avon Karman vortex streetis observed. Here, the cylinder is replaced by Alejandro Selkirk Island. The island is about 1.5 km in diameter, and rises 1.6 km into a layer of marine stratocumulus clouds. Source:

NASA.

the same pictures, due to the shear stress induced by the differential rotation of the atmosphere, one can see rows of turbulent eddies that tend to interact among themselves. The Great Red Spot was formed by the merging of small vortexes, a typical action in the gravity stratified planetary atmosphere. We can now denote a characteristics for a flow to be considered turbulent. For fully developed turbulence we observe a large number of different flow structures which show signs of correlation between scales.

Until now we have looked only at turbulent flows that are electrically nonconductive even though nature abounds in electrically conductive flows in term of plasmas. A plasma is an ionized medium which easily conducts electric currents but, due to self screening of ions and electrons, is electro-neutral,i.e.does not possess local charges. In general, the forces that keep the ions and electrons together and enforce the electro-neutrality of a plasma are so strong that even emerged in an

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Figure 1.2: Left: Multi-frame mosaic of Jupiter’s Great Red Spot taken by Voy- ager I on 03 March 1979. Slightly smaller than Earths moon, Europa is visible to the right of in this. Source: NASA/JPL and Michael Benson, Kinetikon Pictures.

Right:True color mosaic of Jupiter constructed from images taken by the narrow angle camera onboard NASA’s Cassini spacecraft on 29 December 2000. Jupiter here looks the way that the human eye would see it and is the most detailed global color portrait of Jupiter ever produced to date. Source: NASA/JPL/SSI.

external electric field the plasma will not experience charge separation. However, the ease with which an electric current is propagated makes the plasma easily sus- ceptible to magnetic fields influences. For a plasma¹in the presence of a magnetic field, an extra body force in the form of the Lorentz force will act on the flow. Not only that, but the turbulent eddies generated current will induce a self-consistent magnetic field that will in turn influence the subsequent flow motion. Plasma turbulence now has to be studied in term of magneto-hydrodynamics (MHD) as the induced magnetic field structure has to be considered. This is the typical situation for galactic clouds, interstellar medium, solar physics and Earth ionosphere.

The best source of magneto-hydrodynamic turbulence can be seen by observing the Sun, Figure1.3. Turbulent structures are trapped by strong magnetic fields, eddies are stretched and elongated and magnetic field lines reconnect releasing huge amounts of energy in the process. In general, MHD turbulence is at the base of numerous important phenomena like the generation of the Earth magnetosphere by means of the dynamo effect in the planet’s core and, trough the process of particle acceleration by the turbulent magnetic field, the generation of the solar wind, itself a MHD turbulent medium. In astrophysics, MHD turbulence is responsible for galactic level magnetic fields, which evolve at characteristic time scales of mil- lennia and are obtained by preferential collapse of turbulent protogalactic plasma clouds.

Although we have presented only a few cases of turbulence in nature, its effects are present in one form or another all around us. From an anthropological view, understanding the phenomenon of turbulence is important for the advancement of technological knowhow in fields like aeronautics, metallurgy and energy genera-

1We only look at plasmas in the one-fluid approximation and neglect such effects as ambipolar diffusion and Hall effects.

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Figure 1.3: Left: Careful processed ultraviolet data of the Sun, obtained from space probes on 30 July 1999, gives this picture approaching true color. Source:

TRACE Project, Stanford-Lockheed Institute for Space Research, NASA, Michael Benson, Kinetikon Pictures. Right: This image shows in great detail a solar prominence taken from a 30 March 2010 eruption. The twisting motion of the plasma trapped in the magnetic field is the most noticeable feature. Source:

NASA/SDO/AIA.

tion and also in our ability to predict and control the weather, both atmospheric and solar¹. For the later, the understanding of magneto-hydrodynamic turbulence is crucial.

1.3 Historical highlights in turbulence study

In the following section we will list a few important highlights in the history of turbulence study. It is not intended as an exhaustive historical account or even a complete one, but rather as a way of presenting the ideas that changed the way people looked at the field of turbulence.

The phenomena of turbulence is so common in nature, that stands to reason that early man would off be fascinated by the intricate and sometimes beautiful patterns it produces. Neolithic spiral patterns, and similar motives from all over the world would indicate a fascination of early man with the sinuous, curved lines so present in the flow of water. Regardless, the first observations of turbulence, and not just

1With our increasing dependency on satellites for every days activities, like communication and navigation, protecting them from solar storms effects is becoming a growing interest.

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Figure 1.4: One of Leonardo da Vinci’s drawings of turbulent flows.

hydrostatic fluids, is credited to Leonardo da Vinci¹, who looked at the phenomena ofturbolenza. Regarding turbulence, Leonardo da Vinci states:

Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to random and reverse motion.

A few drawings from that time, shown in Figure1.4, depicts observation of water flowing around an obstacle, a preferred way of generating turbulence in en- gineering applications even today. These drawings represent the first systematic observation of the phenomena in an attempt to better understand it.

Off course, fluid turbulence can not be completely separated from the broader study of fluid dynamics and the mathematical framework of continuum media, so

1Lived: 15 April 1452 – 2 May 1519.

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people like Archimedes, Pascal, Newton, Bernoulli, d’Alembert, Euler, Navier and Stokes, to name just a few, brought contributions to the development of turbulence study. However, none of them set out as a goal to understand the underling phenomena behind it. The first attempt to do so, in modern science, was done by Osborne Reynolds¹. He studied the conditions in which a pipe flow transitioned from laminar flow to turbulent flow, introducing the dimensionless Reynolds number, the ratio of inertial forces to viscous forces, as the control parameter. Reynolds also introduced what is now known as Reynolds-averaging of turbulent flows, by expressing quantities like the velocity and pressure as the sum of mean and fluctuat- ing components. Applying the Reynolds-averaging to the Navier-Stokes equations, permits us to describe the movement of the bulk of a turbulent flow.

Another important step in the study of turbulence was done by Lewis Fry Richardson². As an meteorologist he introduced the idea of eddies in describing turbulence as the interaction of vortex structures of different sizes. More importantly he saw turbulence as the process where a large eddie brakes into multiple smaller eddies and passing them its entire energy. The subsequent eddies will themselves brake into even smaller eddies, again without the loss of energy, until a scale is reached where the dissipation due to viscosity transforms the motion kinetic energy into heat. The entire process is known now as the Richardson cascade.

Inspired by a poem of Johathan Swift, Richardson wrote, Big whorls have little whorls That feed on their velocity,

And little whorls have lesser whorls And so on to viscosity.

– Lewis F. Richardson

It was later that Andrey Nikolaevich Kolmogorov³formulated the classical picture of turbulence, by relating the energy cascade in the inertial range, a range where large and small scales effects can be neglected, to the fluid viscosity and the total energy dissipated by the turbulent flow. We will present and use the Kolmogorov’s scaling law in a subsequent section.

Seeing how turbulence is affected by a huge range of scales, many researchers tried to simplify the problem of turbulence by reducing the number of degrees of freedom for the system. We mention two attempts, representing two schools of thought. One was done by Werner Heisenberg⁴who argued that the motion of small scales affect the motion of large scales the same way molecular motion affects mean motion via viscosity. The assumption ofeddy viscositytries to incorporate all the small scales physics into an effective term. Another approached was introduced

1Lived: 23 August 1842 – 21 February 1912.

2Lived: 11 October 1881 – 30 September 1953.

3Lived: 25 April 1903 – 20 October 1987.

4Lived: 5 December 1901 – 1 February 1976.

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by Robert Kraichnan¹in form of the Direct Interaction Approximation, by mainly assuming that knowing how the turbulent energy is transferred is sufficient for describing turbulence, allowing to close the set of equations in a mathematical formulation.

Compared to the field of hydrodynamical turbulence, magneto-hydrodynamics is a relative recent field which came into existence with the discovery of magneto- fluid waves [6], known as Alfv´en waves by Hannes Olof Gsta Alfv´en². Referred sometimes in old literature as hydro-magnetic turbulence, MHD turbulence has established itself as a field of study on its own. The coupling of Maxwells electromagnetic equations with the Navier-Stokes hydrodynamic equation, has led to the beginning of understanding of a rich phenomenological problem.

Today, with the advancement of computer algorithms and the increase of their power, the multitude of physical phenomena involved in complex turbulent flows are studied with the help of numerical methods, in the detriment of analytical methods to some extent. Pioneered by S. Orszag for turbulence studies, pseudo-spectral methods represent the best of both worlds as the simplicity and accuracy of the the method allowed for analytical ideas to be tested directly from the evolution of the equations. The pseudo-spectral methods were quickly embraced by people like A.

Pouquet, Y. Zhou, K. Ohkitani, S. Kida, R.S. Rogallo, J.A. Domaradzki, F. Waleffe and many other members of the community, including the present author who used such methods for the original work presented in the current thesis.

1.4 Empirical laws of turbulence

Similar to other complex physical systems, experiments can provide answers about the nature of a phenomena despite the fact that mathematical theories can not be properly formulated. Pipe flows, channel flows or wind tunnels represent the most common form of turbulent experiments. Pipe flow experiments rely on the boundary generated instabilities to propagate in a laminar flow and represent a good method for studying the transition to turbulence. Channel experiments can be used in a similar way, but also in studying the type of turbulence generated by a simple obstacle which possess various symmetries or objects with complex, realistic shapes like ships or sea floors. For the generation of homogenous, fully developed turbulence, passing a flow trough a grid in a wind tunnel represents the best approach.

For all of these experiments, using an anemometer type device, the velocity of the flow can be measured in one point and various statistics can be then determined.

Typically, the information of at least two points located in the direction of the flow is used to compute thelongitudinal velocity increment,

δu_�(x,�)≡[u(x+�)−u(�)]·�

� , (1.1)

1Lived: 15 January 1928 – 26 February 2008.

2Lived: 30 May 1908 – 2 April 1995.

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For homogenous turbulence, we can drop the dependency ofx. Thus,δu_�(�)represents the velocity increment between two points separated by the distance�, pro- jected onto the line of separation. Using the longitudinal velocity increment an entire class of statistical moments can be defined in the form of structure functions,

S_p(�)≡ �[δu_�(�)]^p�. (1.2) The volume average can be replaced experimentally by an ensemble average. The work consists in relating theS_p(�) functions to different experimental conditions and more importantly to the Reynolds number defined as,

R≡ U L

ν , (1.3)

whereν is the viscosity of the fluid and U andL are the characteristic velocity and length of the turbulent system chosen either on the flow or on the experimental setup. Experimentally two empirical laws were determined, namely,

Two-thirds law: In a turbulent flow, at very high Reynolds number, the second order structure functionS2(�) behaves approximately as the two-thirds power of the distance�separating the two-points.

S₂(�)∼�^2/3. (1.4)

Finite energy dissipation law:For identical experimental circumstances, the energy dissipation per unit massε_ν tends towards a finite positive limit when the viscosity of the fluid is reduced as much as possible.

νlim→0ε_ν >0 or, more precisely, lim

ν→0ε_ν ∼ S₃(�)

� . (1.5)

The last relation can be determined theoretically and it implies that the two-point, third order statistical momentum for turbulence is non-vanishing for� > 0. The finite energy dissipation result is, to a degree, non-intuitive. Since the dissipation is proportional with the viscosity, one would assume that reducing the viscosity to zero would reduce the dissipation to zero as well. In fact, the dissipation is also proportional to the intensity of the nonuniform velocity strain, the shear stresses.

When the viscosity becomes small, these stresses start to become large, a fact that compensate the overall energy dissipation. The finite energy dissipation law ex- plains why a flow in the limitν → 0behaves in a different way compared to an Euler flow for whichν = 0.

Considering the equations (1.4) and (1.5), from dimensional arguments, we can now write a relation for high Reynolds number turbulence,

S₂(�)∼ε^2/3_ν �^2/3 . (1.6)

This expression relates the energy contained in a eddy, of size�, to the total energy dissipation rate in the range where viscous effects can be neglected.

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Chapter notes & Acknowledgments

� This chapter represents my attempt to familiarized the reader with the problem of turbulence, without the use of a mathematical formalism. Although the description is based on my own intuition, the historical dates are obtained from the internet, mainly: www.wikipedia.org. Since the dates accuracy are not critical for the current work, we allow ourselves the liberty of using an open source encyclopedia.

� As a first read, for any novice scientist that wants to obtain insight into the filed of turbulence, we recommend the book of U. Frish, Turbulence: The legacy of A. N. Kolmogorov, [48].

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Theory of turbulence

2.1 The evolution equations

Deriving the magnetohydrodynamic equations

The derivation of the magneto-hydrodynamic (MHD) equations can be performed using multiple approaches, based on the framework employed by each person.

However, regardless of the approach used, be it averaging kinetic equations [10, 49], constructing a covariant formalism for relativistic MHD (similar to the method employed by [107] and [81] for the hydrodynamic case) or simply obtaining them for a continuum medium [16, 49], the physical results are unique as long as the different frameworks overlap and the assumptions made in each case hold true. In our studies, we consider the evolution equations as being derived for an electrically conductive medium calledplasma, in the continuum, fluid limit. This approached represents possibly the oldest but also one of the strongest way of introducing the evolution equations.

Traditionally, the starting point consists in deriving the Navier-Stokes equation, representing the momentum conservation equation for a nonconductive fluid, from the second law of classical mechanics for a continuum media. An equation of state relates the hydrodynamic pressure to the mass densityρ and temperature, which are found by solving the density continuity equation and temperature equation, respectively. In our studies, we will employ the incompressibility limit. This approximation simplifies our work immensely, as the momentum conservation equation, which reduces to the flow’s velocity evolution, decouples from the temperature equation for a constant mass density. Since there are a lot of good books presenting this derivation [13,90,71] and continuum media mechanics is part of any good university curriculum, we will forgo tradition and just postulate the Navier-Stokes equation. However, we would like to mention that the following MHD derivation is independent of the incompressibility assumption for the flow.

In the incompressibility limit, the evolution of the velocity u = u(x, t) for a 27

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nonconductive flow is given by the Navier-Stokes equation,

∂u

∂t =−u·∇u+ν∇²u− ∇p , (2.1) where ν is the kinematic viscosity and p = p(x, t) is the hydrodynamic pressure divided by the constant mass density. Due to the incompressibility condition,

∇·u= 0, the pressure is not an independent variable and depends onu through its derivatives. This can be easily seen by taking the divergence of equation (2.1) which results in a Poisson equation for the pressure,

∇²p=−∇u:∇u. (2.2)

Depending on the complexity of the boundary conditions for the velocity field, obtaining a solution for the pressure reduces now to the insurmountable or trivial task of inverting the Laplacian operator,∇².

For an electrically conductive fluid, of electric permittivityεand magnetic per- meabilityµ, an additional body force acts on the flow, namely the Lorentz force, fL = σE+J×B. This force couples the momentum conservation given by the Navier-Stokes equation to electromagnetism given by Maxwell’s equations,

∇·B= 0 →Zero divergence law, (2.3)

∇ ×E=−∂B

∂t →Faraday’s law, (2.4)

∇·E= 1

εσ →Gauss law, (2.5)

∇ ×B=µJ+µε∂E

∂t →Amp`ere’s law. (2.6)

The first two equations relay consistency informations about the electric E = E(x, t) and magneticB = B(x, t) fields, while the last two equations in the set denote the influences of the sources on the fields, whereσ =σ(x, t)is the charge density andJ=J(x, t)is the electric current. The two sources are related through the continuity equation,

∂σ

∂t +∇·J= 0, (2.7)

which is not an independent relation and can be obtained from the divergence of the Amp`ere’s law and the Gauss law. The current generated by the evolution in time of the electric field is known as the displacement current and its usually quite small compared toJfor the majorities of plasmas.

We make now the most important assumption in the derivation of the MHD equations. By definition, the plasma medium is considered to be electrically conductive but not to have free electric charges. This is known as the electro-neutrality¹

1The electro-neutrality property can be seen as the self screening effect of the electrons and ions as they are not confined to a crystal lattice but mobile in the fluid. The forces enforcing the screening are high frequency and can be considered as instantly opposing any tendency of charge separation.

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property of a plasma¹. Because of this property, the electric currents J are non zero², even thoughσ is taken to be zero in the Maxwell’s equations. For this reason, the Lorentz force is now properly given for a plasma to be,

f_L=J×B, (2.8)

sinceσE = 0. Remembering that the electric permittivityεand magnetic perme- abilityµare constants related to the speed of light as c = (εµ)⁻^1/2, we rewrite the displacement current appearing in the Amp`ere law, given by equation (2.6), as c⁻²∂E/∂t. For a slow moving medium in rapport toc, we can neglect the various contributions brought by the displacement current. By doing so, extracting the electric current from the reduced form of the equation (2.6),

J= 1

µ∇ ×B, (2.9)

and introduce it into the Lorentz force expression, we obtain a form of the force depending only on the electromagnetic fields,

f_L= 1

µ(∇ ×B)×B. (2.10)

Using the vectorial identity(∇ ×B)×B= (B·∇)B− ∇(B·B)/2, we write the Navier-Stokes equation for a plasma medium to be,

∂u

∂t =−u·∇u+ν∇²u− ∇p+1

ρf_L, (2.11)

=−u·∇u+ν∇²u− ∇p+ 1

ρµB·∇B− ∇|B|²

2ρµ . (2.12) We see that a new variable has been introduced in form of the magnetic field B(x, t). To account for its evolution in time a new equation has to be solved, namely Faraday’s induction equation (2.4), which tells us that the evolution of the magnetic field is minus the curl of the electric field. Using Ohm’s law for a conductive fluid, which consists in assuming that the electric field depends linearly on the electric current in the fluid co-moving frame of referenceE^� =χJ, we obtain in the laboratory frame of referenceE^� =E+u×Ban expression forEas,

E=−u×B+χJ, (2.13)

whereχ is the electric permeability. Introducing this expression for the electric field in equation (2.4) and using the expression for the electric current obtained

1Remembering that we denote generically the medium as plasma, we mention that for a liquid metal plasma the electro-neutrality property is even stronger then for an ionized gas.

2This can be seen microscopically as�σu�^D = Jwhere the�. . .�^D denotes averaging over the smallest domain for which the electro-neutrality property is true,�σ�D= 0.

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from equation (2.9), we obtain the evolution equation for the magnetic field,

∂B

∂t =∇ ×(u×B)− χ

µ∇ ×(∇ ×B), (2.14)

=−u·∇B+B·∇u+ χ

µ∇²B, (2.15)

where we have used the appropriate vectorial identities,∇×(u×B) =u(∇·B)− B(∇·u) + (B·∇)u−(u·∇)Band∇ ×(∇ ×B) =∇(∇·B)− ∇²B, while keeping count of the zero-divergence nature of the velocity an magnetic fields.

Technically, we have derived the set of MHD equation, however we will use the MHD equations in the following form,

∂u

∂t =−u·∇u+b·∇b+ν∇²u− ∇p , (2.16)

∂b

∂t =−u·∇b+b·∇u+η∇²b, (2.17) whereη is the magnetic diffusivity (η = χ/µ), the pressure now incorporate the magnetic pressure term (∼ |B|²) and the magnetic field is expressed in Alfv´en velocity unitsb=B/√ρµ. In this form the nonlinear terms influences are easier to investigate. When needed the electric field and electric current are found from the algebraic relation,

e=−u×b+η∇ ×b, (2.18)

j=∇ ×b. (2.19)

To simplify our analysis even further we consider the magnetic Prandtl number Pr =ν/ηto be unity by takingν=ηthroughout our work.

Els¨asser representation

Since the MHD equations possess a certain symmetry inuandb, we can rewrite them in term of the Els¨asser variables, [41], defined as,

z^±=u±b. (2.20)

The MHD equations in Els¨asser representation have the form,

∂z^±

∂t =−z^∓·∇z^±+ν⁺∇²z^±+ν⁻∇²z^∓− ∇p , (2.21) where ν^± = (ν ± η)/2 and the total pressure p can be eliminated due to the divergent free condition,∇·z^± = 0. The advantage of this form of the MHD equations consists in the non-linear term which is responsible only for the cross- coupling ofz⁺andz⁻, without any self-coupling phenomena. The two nonlinear terms can be seen now as the scattering of contra-propagating waves.

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Addressing an inconstancy

We have introduced the MHD equation in a non-relativistic limit by coupling a non- relativistic flow, which possesses a Galilean invariance, with the electromagnetic field, which possesses a Lorentz invariance. This introduces an inconsistency in the theory that has to be addressed¹.

For a Lorentz boost in an arbitrary direction with velocityV, it is convenient to decompose the spatial vectorxinto components perpendicular and parallel to the velocityV, so that forx=x_⊥+x_�only the componentx_�will be affected by the transformation.

t^� =γ

�

t−x·V c²

�

, (2.22)

x^�_� =γ(x_�−Vt), (2.23)

x^�_⊥=x_⊥, (2.24)

whereγ ≡(1−V·V/c²)^−1/2. As presented in [52], the Lorentz transformation for the electric and magnetic fields are expressed²as,

E^�_� =E_� , (2.25)

E^�_⊥=γ[E_⊥+V×B], (2.26)

and

B^�_�=B_� , (2.27)

B^�_⊥=γ[B_⊥− 1

c²V×E]. (2.28)

We see that in the classical limit, whenc→ ∞, the factorγ tend towards unity. In this situation³we recover the Galilean transformation of the electric and magnetic fields,

E^� =E+V×B, (2.29)

B^� =B, (2.30)

1Of course, this problem would not appear if we would use a Lorentz invariant flow, expressed by a four-vector velocityu^µ=γdx^µ/dt, wherex^µ= (c t,x).

2The electromagnetic anti-symmetric field-strength tensor, defined in term ofEandBas,

F^µν=







0 −Ex/c −Ey/c −Ez/c Ex/c 0 −Bz By

Ey/c Bz 0 −Bx

Ez/c −By Bx 0





 ,

transforms under an arbitrary Lorentz transformation,Λ^µ_ν, asF^µν = Λ^µ_αΛ^ν_βF^αβ. The Lorentz transformations for the electric and magnetic fields are found accordingly, considering that the form ofΛ^µνcan be identified from equations (2.22-2.24) which represent a Lorentz boostx^µ=Λ^µνx^ν for the four-vector,x^µ= (c t,x).

3Since from dimensional arguments�E�=�u��B�, whereuis the velocity of the charge generating the magnetic field, the terms proportional with�V��u�/c²can be neglected for�V�and�u� much smaller thenc.

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for a Galilean boost,

t^� =t , (2.31)

x^� =x−Vt . (2.32)

From a physical point of view, this transformation is valid only for phenomenas that are much slower then the propagation of the electromagnetic waves¹. In the classical MHD approximation, the flow induced magnetic field and the consistent electric field are in fact Galilean invariant but do not account anymore for the electromagnetic coupling betweenE andB, as variations in time of the electric field will not generate magnetic fields any longer.

2.2 Symmetry groups

In physics, symmetries are an important aspect of any phenomena, as they depict constrains in the dynamics that can simplify the development of subsequent theories for the phenomena in question. More precisely, the term symmetry refers to any discrete or continuous invariance group of a dynamical theory. ForG, a group of transformations acting on the space-time functionsu(x, t)andb(x, t), it is said to be a symmetry group of the MHD equations if, for alluandbwhich are solution of the MHD equation, the functionsgu andgbare also solutions, where g ∈ G is an element of the group. By settingb = 0 we recover the symmetries for the Navier-Stokes equation, as listed by [48].

Since the MHD equations are dissipative and not fundamental in any way, some of the symmetries we will list can only be considered when appropriate limits are taken. We will mention these shortcomings when appropriate. Also, for high turbulence levels these symmetries will be broken in a point wise sense but they will be recovered in a statistical sense. This implies a constrained on the statistical states allowed rather then on the equation dynamics. For the symmetries to have maximal level of validity, the turbulent system should be unbounded, extending to infinity, or at east possess periodic boundary conditions.

Space translations

For anyr∈R³, we define the space translation transformationg^spacer :

t→t^� =t , (2.33)

x→x^� =x+r, (2.34)

u→u^�=u, (2.35)

b→b^�=b. (2.36)

1This is equivalent in our case with placing the displacement currentc⁻²∂E/∂tto zero, which results in∇²E= 0and∇²B= 0,i.e.the instantaneous propagation of an electromagnetic wave.

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which leaves the MHD equations invariant. This signifies that MHD turbulence is continuously homogenous in absence of boundaries for a laminar state. In the presence of boundaries, like for an object immerse in a plasma flow, this symmetry is broken by the onset of turbulence. A weaker symmetry of discrete homogeneity may be recovered forr^�of the dimension of the object for an appropriate magnetic boundary condition. For high Reynolds number the discrete homogeneity will be completely lost but the plasma will be homogenous in a statistical sense, as the turbulent structures will be uniformly distributed in space.

Time translations

For anyτ ∈R, we define the time translation transformationg^time_τ :

t→t^�=t+τ , (2.37)

x→x^�=x, (2.38)

u→u^� =u, (2.39)

b→b^� =b, (2.40)

which leave the MHD equations invariant. This symmetry signifies that the MHD equations are time homogenous or time stationary for a laminar state. In the case of fully developed turbulence, a statistical stationary state can be found if the energy introduced in the system is equal to the energy lost due to dissipative effects.

Galilean transformation

The Galilean transformationg_V^Galis defined as:

t→t^� =t , (2.41)

x→x^� =x+Vt , (2.42)

u→u^�=u+V, (2.43)

b→b^�=b, (2.44)

for any constant velocityV ∈ R³. We see that for a velocityu(x+Vt, t) +V and magnetic fieldb(x+Vt, t), the partial time derivative terms are generating an additional term,

∂

∂t[u(x+Vt, t)] = ∂u

∂t +V·∇u, (2.45)

∂

∂t[b(x+Vt, t)] = ∂b

∂t +V·∇b, (2.46)

which cancel with the respective terms generated by[u+V]·∇uand[u+V]·∇b in the right hand side of the MHD equation. The Galilean invariance allows us to remove any constant velocity field for the flow and recover isotropy for a turbulent velocity field imbedded in a constant velocity flow.

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Parity transformation

Under parity transformationg^parity:

t→t^� =t , (2.47)

x→x^� =−x, (2.48)

u→u^�=−u, (2.49)

b→b^�=−b, (2.50)

all the terms in the MHD equations change sign, in particular∇ → −∇. Under a parity transformations the electric current and the fluid vorticity, defined as the curl of the velocity fields, do not change sign. The evolution equations for these quantities are not invariant under a parity transformation.

Time inversion transformation

For ideal MHD, obtained asν = η = 0, or in a weaker sense in the limitν → 0 andη →0, a time inversion transformationg^inversion:

t→t^� =−t , (2.51)

x→x^� =x, (2.52)

u→u^�=−u, (2.53)

b→b^�=−b, (2.54)

leaves the MHD equations invariant. The braking of this symmetry by the dissipative process, shows the irreversibility of the MHD equations evolution.

Rotation transformation

For a rotation operatorS ∈SO(R³)a rotation transformationg_S^rot:

t→t^� =t , (2.55)

x→x^� =Sx, (2.56)

u→u^�=Su, (2.57)

b→b^�=Sb, (2.58)

will leave the MHD equation invariant. This symmetry is equivalent with the assumption of isotropy. As we will see later, a external constant magnetic field will break this symmetry, although a rotationS^� ∈SO(R²)in the plane perpendicular to the direction of the external magnetic field will still be possible as the spherical symmetry will be replace by a cylindrical one.

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Scaling translations

For ideal MHD, obtained asν = η = 0, or in a weaker sense in the limitν → 0 andη→0, a scaling transformation can be definedg^scal_λ :

t→t^� =λ¹⁻^ht , (2.59)

x→x^� =λx, (2.60)

u→u^� =λ^hu, (2.61)

b→b^� =λ^hb, (2.62)

whereλ ∈ R+ andh ∈ R. Under the scaling transformation¹, all the terms in the MHD equation will be multiply by aλ^2h−1 factor, except for the dissipative terms which are multiplied byλ^h⁻². In particular, for the dissipative situation only h =−1case is permitted. The scaling transformation is related to the similarity principle of turbulence but can also be seen as special case of conformal symmetry².

2.3 Conservation laws

In physics, identifying the system symmetries (e.g.spherical symmetry. . . ) is the first step, immediately followed by the definition of proper invariance groups (e.g.

. . . rotation invariance. . . ). These are important since an invariance leads to conservation of a physical quantities (e.g.. . . angular momentum) which represents a constrained on the dynamics of the phenomena in question. For Lagrangian systems, the Noether theorem provides a simple method for identifying the conserved quantities of a system. Unfortunately, the MHD equation, like the Navier-Stokes one, are dissipative. However, in the ideal case³, when the dissipative terms due to viscosity and magnetic diffusivity are put to zero, we can talk about conservation

1We assume here that the magnetic and velocity fields scale in the same way.

2The conformal transformations, together with the Poincar´e group generate the conformal symmetry group. A conformal symmetry inddimensions is equivalent to an AdS symmetry ind+ 1dimen- sion. This led to the AdS/CFT correspondence, which maps a fully nonlinear Navier-Stokes type equation inddimensions to an Einstein type gravity equation ind+ 1dimensions. This line of work may bring new results in the field of fully non-linear systems, [14,15,47,53].

3We should note that working in the ideal MHD case adds an additional problem, [42,55,25,46, 44]. Viscosity and magnetic diffusivity smooth out any irregularities of the fields, enforcing the smoothness of the fields in a natural way. The lack of singularities and discontinuities, permits the usually employed mathematical manipulations such as integration by parts and derivation of products in an Hilbert space. For the ideal limit, the fields smoothness must be considered in the sense of H¨older continuity,|f(x)−f(y)| ≤ C|x−y|^α, whereC and αare nonnegative real constants. Ifα= 1, then the function satisfies a Lipschitz condition which is a smoothness condition stronger than regular continuity and and ifα= 0, then the function simply is bounded.

For turbulence, from flux scaling arguments, it is expected thatα= 1/3. In this situation is more rigorous to work with field incrementsδf(r) =f(x+r)−f(x)in anL^pspace where the p-norm is defined as,�δf(r)�=�|δf(r)|^p�^1/p.

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laws of global quantities obtained by averaging over the volumeV,

�q�= 1 V

�

q(x)dx. (2.63)

These global quantities are known as ideal invariants. We will list the conservation of global quantities for MHD turbulence, but also the conservation laws in the limit ofb→0in which we recover the Navier-Stokes equation. We will systematically employ,

�∇q�= 0, (2.64)

�∇·q�= 0, (2.65)

for a scalar and vectorial fields that tend towards a constant at±∞ or fields that poses periodic boundary conditions.

The conservation laws presented below are for three-dimensional turbulence.

In the two-dimensional case, some of these laws are always true due to geometrical considerations and new relevant conserved quantities appear.

Conservation of energy

The conservation of energy is probably one of the most important aspect of any system. For MHD turbulence, we can define the kinetic energy and magnetic energy at each point in space as,

E^u(x, t) =1

2u(x, t)·u(x, t) = 1

2�u(x, t)�² , (2.66) E^b(x, t) =1

2b(x, t)·b(x, t) = 1

2�b(x, t)�² , (2.67) where we omit the constant mass density, which we consider to be unity in value, in the definition of the energy densities. The evolution equations for the global kinetic energy E^u = �E^u(x, t)� and global magnetic energy E^b = �E^b(x, t)� can be obtain form equations (2.16-2.17) by contracting them withuandb, accordingly and then taking the volume average,

1 2

d

dt�(u·u)�=−�(u·∇u)·u�+�(b·∇b)·u�+ν�(∇²u)·u� − �∇p·u�, (2.68) 1

2 d

dt�(b·b)�=−�(u·∇b)·b�+�(b·∇u)·b�+η�(∇²b)·b�, (2.69) First, we note that the pressure term does not enter in the evolution of the global kinetic energy,�∇p·u�=�∇·(pu)�= 0. Noting that we can rewrite the fist term in each equation as the average of a divergence,

�(u·∇u)·u�= 1

2�∇·[u(u·u)]�= 0, (2.70)

�(u·∇b)·b�= 1

2�∇·[b(b·b)]�= 0, (2.71)

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we rewrite the global energy evolution equations as, d

dtE^u =T_b^u+εν , (2.72)

d

dtE^b =T_u^b+ε_η , (2.73)

where εν represents the global kinetic energy dissipation and εη represents the global magnetic energy dissipation,

εν =ν�(∇²u)·u�, (2.74) ε_η =η�(∇²b)·b�, (2.75) and

T_bû =�(b·∇b)·u�, (2.76) T_u^b =�(b·∇u)·b�. (2.77) We see that the global energies do not conserve themselves independently in the ideal case (ν =η = 0), for whichε_ν =ε_η = 0. This is due to the transfer terms T_bû andT_u^b, that are responsible for an exchange of energy taking place between the kinetic and the magnetic energy. The two transfers are of the same net value but of different signs,T_bû =−T_u^b, a fact seen best by considering their sum,

T_b^u+T_u^b =�(b·∇b)·u�+�(b·∇u)·b�=�∇·[b(u·b)]�= 0. (2.78) This fact implies that the total energy, obtained as the sum of the kinetic and magnetic components, is an ideal invariant as it is a conserved quantaty in the ideal case,

d

dt(E^u+E^b) = dE

dt = 0. (2.79)

In general, the global evolution of the total energy equals the total energy dissipation rateε=εν +εη,

dE

dt =ε. (2.80)

We see that for dissipative magnetohydrodynamic turbulence, the total energy de- creases monotonically in absence of sources, even though global exchanges increase and decrease the kinetic and magnetic components, respectively.

Conservation of cross-helicity

For MHD turbulence, we can define cross-helicity at each point in space as, H^c(x, t) =u(x, t)·b(x, t), (2.81)

Numerical simulations of transport processes in magnetohydrodynamic

&

U

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D OCTORAL T HESIS

Numerical simulations of transport processes in magnetohydrodynamic

turbulence

Contents

Acknowledgments

Preface

The physics of turbulence

Phenomenology of turbulence

1.1 Intuitive reasoning

1.2 Turbulence in nature

1.3 Historical highlights in turbulence study

1.4 Empirical laws of turbulence

Chapter notes & Acknowledgments

Theory of turbulence

2.1 The evolution equations

2.2 Symmetry groups

2.3 Conservation laws