Lecture notes on Plasma physics

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Lecture Notes in Physics Introduction to Plasma Physics Michael GedaliniiContents 1 Basic definitions and parameters 1 1.1 What is plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Debye shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Plasma parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Plasma oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ∗ ∗ 1.5 Ionization degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Plasma description 9 2.1 Hierarchy of descriptions . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Fluid description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Motion (Euler) equation . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 State equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.7 Order-of-magnitude estimates . . . . . . . . . . . . . . . . . . . . . . 14 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 MHD equilibria and waves 17 3.1 Magnetic field diffusion and dragging . . . . . . . . . . . . . . . . . . 17 3.2 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 MHD waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Alfven and magnetosonic modes . . . . . . . . . . . . . . . . . . . . . 22 3.5 Wave energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 iiiCONTENTS 4 MHD discontinuities 27 4.1 Stationary structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 Why shocks ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Two-fluid description 33 5.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Reduction to MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.3 Generalized Ohm’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6 Waves in dispersive media 37 6.1 Maxwell equations for waves . . . . . . . . . . . . . . . . . . . . . . . 37 6.2 Wave amplitude, velocity etc. . . . . . . . . . . . . . . . . . . . . . . . 38 6.3 Wave energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Waves in two-fluid hydrodynamics 47 7.1 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 Unmagnetized plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.3 Parallel propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.4 Perpendicular propagation . . . . . . . . . . . . . . . . . . . . . . . . 50 7.5 General properties of the dispersion relation . . . . . . . . . . . . . . . 51 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8 Kinetic theory 53 8.1 Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8.3 Relation to hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 55 8.4 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.5 Landau damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 9 Micro-instabilities 61 9.1 Beam (two-stream) instability . . . . . . . . . . . . . . . . . . . . . . 61 ∗ ∗ 10 Nonlinear phenomena 63 A Plasma parameters 65 ivPreface vCONTENTS viChapter 1 Basic definitions and parameters In this chapter we learn in what conditions a new state of matter - plasma - appears. 1.1 What is plasma Plasma is usually said to be a gas of charged particles. Taken as it is, this definition is not especially useful and, in many cases, proves to be wrong. Yet, two basic necessary (but not sufficient) properties of the plasma are: a) presence of freely moving charged particles, and b) large number of these particles. Plasma does not have to consists of charged particles only, neutrals may be present as well, and their relative number would affect the features of the system. For the time being, we, however, shall concentrate on the charged component only. Large number of charged particles means that we expect that statistical behavior of the system is essential to warrant assigning it a new name. How large should it be ? 19 −3 Typical concentrations of ideal gases at normal conditions are n∼ 10 cm . Typical −3 concentrations of protons in the near Earth space aren∼ 1−10cm . Thus, ionizing only a tiny fraction of the air we should get a charged particle gas, which is more dense than what we have in space (which is by every lab standard a perfect vacuum). Yet we say that the whole space in the solar system is filled with a plasma. So how come that so low density still justifies using a new name, which apparently implies new features ? A part of the answer is the properties of the interaction. Neutrals as well as charged particles interact by means of electromagnetic interactions. However, the forces be- tween neutrals are short-range force, so that in most cases we can consider two neutral atoms not affecting one another until they collide. On the other hand, each charged par- ticle produces a long-range field (like Coulomb field), which can affect many particles at a distance. In order to get a slightly deeper insight into the significance of the long-range fields, let us consider a gas of immobile (for simplicity) electrons, uniformly distributed inside an infinite cone, and try to answer the question: which electrons affect more the 1CHAPTER 1. BASIC DEFINITIONS AND PARAMETERS one which is in the cone vertex ? Roughly speaking, the Coulomb force acting on the chosen electron from another one which is at a distance r, is inversely proportional to 2 the distance squared, f ∼ 1/r . Since the number of electrons which are at this dis- r 2 0 tance, N ∝ r , the total force, N f ∼ r , is distance independent, which means that r r r that electrons which are very far away are of equal importance as the electrons which are very close. In other words, the chosen probe experiences influence from a large number of particles or the whole system. This brings us to the first hint: collective effects may be important for a charged particle gas to be able to be called plasma. 1.2 Debye shielding In order to proceed further we should remember that, in addition to the densityn, every gas has a temperatureT , which is the measure of the random motion of the gas particles. Consider a gas of identical charged particles, each with the charge q. In order that this gas not disperser immediately we have to compensate the charge density nq with charges of the opposite sign, thus making the system neutral. More precisely, we have to neutralize locally, so that the positive charge density should balance the negative charge density. Now let us add a test chargeQ which make slight imbalance. We are interested to know what would be the electric potential induced by this test charge. In the absence of plasma the answer is immediate: φ = Q/r, where r is the distance from the charge Q. Presence of a large number of charged particles, which can move freely, changes the situation drastically. Indeed, it is immediately clear that the charges of the sign opposite to Q, are attracted to the test charge, while the charges of the same sign are repelled, so that there will appear an opposite charge density in the near vicinity of the test charge, which tends to neutralize this charge in some way. If the plasma particles were not randomly moving due to the temperature, they would simply stick to the test particle thus making it "neutral". Thermal motion does not allow them to remain all the time nearQ, so that the neutralization cannot be expected to be complete. Nevertheless, some neutralization will occur, and we are going to study it quantitatively. Before we proceed further we have to explain what electric field is affected. If we measure the electric field in the nearest vicinity of any particle, we would recover the single particle electric field (Coulomb potential for an immobile particle or Lienard- Viechert potentials for a moving charge), since the influence of other particles is weak. Moreover, since all particles move randomly, the electric field in any point in space will vary very rapidly with time. Taking into account that the number of plasma charges producing the electric field is large, we come to the conclusion that we are interested in the electric field which is averaged over time interval large enough relative to the typical time scale of the microscopic field variations, and over volume large enough to include large number of particles. In other words, we are interested in the statistically averaged, or self-consistent electric potential. 2CHAPTER 1. BASIC DEFINITIONS AND PARAMETERS Establishing the statistical (or average) nature of the electric field around the test charge we are able now to use the Poisson equation Δφ =−4πρ−4πQδ(r), (1.1) where the last term describes the test point charge in the coordinate origin, while ρ is the charge density of the plasma particles, ρ =q(n−n ). (1.2) 0 Here n is the density of the freely moving charges in the presence of the test charge, while n is their density in the absence of this charge. Assuming that the plasma is 0 in thermodynamic equilibrium, we have to conclude that the charged particle are dis- tributed according to the Boltzmann law n =n exp(−U/T), (1.3) 0 whereU =qφ. Strictly speaking, the potential in the Boltzmann law should be the local (non-averaged) potential, and averaging hexp(−qφ/T)i = 6 exp(−qiφi/T). However, sufficiently far from the test charge, whereqφ/T  1 we may Taylor expand qφ hexp(−qφ/T)i = 1−h i T so that 2 n q 0 ρ =− φ, (1.4) T where nowφ is the self-consistent potential we are looking for. Substituting into (1.1), one gets 2 1 d d 4πn q 0 2 r φ = φ (1.5) 2 r dr dr T for r 0 and boundary conditions read φ → Q/r when r → 0, and φ → 0 when r→∞. The above equation can be rewritten as follows: 2 d 1 (rφ) = (rφ), (1.6) 2 2 dr r D where p 2 r = T/4πn q (1.7) D 0 is called Debye radius. The solution (with the boundary conditions taken into account) is Q φ = exp(−r/r ). (1.8) D r 3CHAPTER 1. BASIC DEFINITIONS AND PARAMETERS We see that for r  r the potential is almost not influenced by the plasma particles D and is the Coulomb potential φ ≈ Q/r. However, at r r the potential decreases D exponentially, that is, faster than any power. We say that the plasma charges effectively screen out the electric field of the test charge outside of the Debye sphere r = r . D The phenomenon is called Debye screening or shielding, and is our first encounter with the collective features of the plasma. Indeed, the plasma particles act together, in a coordinated way, to reduce the influence of the externally introduced charge. It is clear that this effect can be observed only if Debye radius is substantially smaller than the size of the system,r  L. This is one of the necessary conditions for a gas of charged D particle to become plasma. It is worth reminding that the found potential is the potential averaged over spatial scales much large than the mean distance between the particles, and over times much larger than the typical time of the microscopic field variations. These variations (called fluctuations) can be observed and are rather important for plasmas’ life. We won’t discuss them in our course. The two examples of the collective behavior of the plasma (Debye shielding and plasma oscillations) show one more important thing: the plasma particles are "con- nected" one to another via self-consistent electromagnetic forces. The self-consistent electromagnetic fields are the "glue" which makes the plasma particles behave in a co- ordinated way and this is what makes plasma different from other gases. 1.3 Plasma parameter Since the derived screened potential should be produced in a statistical way by many charges, we must require that the number of particles inside the Debye sphere be large, 3 N ∼ nr  1. The parameterg = 1/N is often called the plasm parameter. We see D D D that the condition g  1 is necessary to ensure that a gas of charged particles behave collectively, thus becoming plasma. We can arrive at the same parameter in a different way. The average potential energy 2 of the interaction between two charges of the plasma isU ∼ q /r¯, wherer¯ is the mean distance between the particles. The latter can be estimated from the condition that the 3 −1/3 there is exactly particle in the sphere with the radiusr¯: nr¯ ∼ 1, so thatr¯∼ n . The average kinetic energy of a plasma particle is nothing butT , so that 2 1/3 U q n 1 2/3 ∼ ∼ =g . (1.9) 2/3 2 K T n r D Ifg 1, as it should be for a plasma, then the average potential energy is substantially smaller than the average kinetic energy of a particle. In fact, we could expect that since in order that the charges be able to move freely, the interaction with other particles should not be too binding. If U/K  1, the plasma is said to be ideal, otherwise it is 4CHAPTER 1. BASIC DEFINITIONS AND PARAMETERS non-ideal. We see that only ideal plasmas are plasma indeed, otherwise the substance is more like a charged fluid with typical liquid properties. 1.4 Plasma oscillations In the analysis of the Debye screening the plasma was assumed to be in the equilibrium, that is, the plasma charges were not moving (except for the fast random motion which is averaged out). Thus, the screening is an example of the static collective behavior. Here we are going to study an example of the dynamic collective behavior. Let us assume that the plasma consists of freely moving electrons and an immobile neutralizing background. Let the charge of the electron beq, massm, and densityn. Let us assume that, for some reason, all electrons, which were in the half-spacex 0, move to the distanced to the right, leaving a layer of the non-neutralized background with the charge density ρ =−nq and width d. The electric field, produced by this layer on the electrons on both edges isE = 2πρd =−2πnqd (for the electrons at the right edge) and 2 E = 2πρd = 2πnqd (for the electrons at the left edge). The forceF =qE =−2πnq d accelerates the electrons at the right edge to the left, while the electrons at the left edge experience similar acceleration to the right. The relative acceleration of the electrons at 2 the right and left edges would be a = 2(qE/m) = −4πnq d/m. On the other hand, ¨ a =d, so that one has 2 2 2 ¨ d =−ω d, ω = 4πnq /m. (1.10) p p The derived equation describes oscillations with the plasma frequencyω . It should be p emphasized that the motion is caused by the coordinated movement of many particles together and is thus a purely collective effect. In order to be able to observe these oscillations their period should be much smaller than the typical life time of the system. ∗ ∗ 1.5 Ionization degree A plasma does not have to consist only of electrons, or only of electrons and protons. In other words, neutral particles may well be present. In fact, most laboratory plasmas are only partially ionized. They are obtained by breaking neutral atoms into positively charged ions and negatively charged electrons. The relative number of ions and atoms, n /n , is called the degree of ionization. In general, it depends very much on what is i a making ionization. However, in the simplest case of the thermodynamic equilibrium the ionization degree should depend only on the temperature. Indeed, the process if ionization-recombination,a↔ i+e, is a special case of a chemical reaction (from the point of view of thermodynamics and statistical mechanics). Let I be the ionization 5CHAPTER 1. BASIC DEFINITIONS AND PARAMETERS potential, that is, the energy needed to separate electron from an atom. Then n g i i+e = exp(−I/T), (1.11) n g a a where g = g g and g are the number of possible states for the ion+electron and i+e i e a atom, respectively. Usually g ∼ g ∼ 1. However, g is large. It can be calculated a i e precisely but we shall make a simple estimate to illustrate the methods, which are widely 3 3 used in plasma physics. The number of available states for an electrong ∼ ΔVΔ p/h , e 3 where ΔV is the volume available for one electron, Δ p is the volume in the momenta space, and h is the Planck constant. It is obvious that ΔV ∼ 1/n . The volume in the e momenta space can be estimated if we remember that the typical kinetic energy of a 2 3 3/2 thermal particle,p /m∼ T , from whichΔ p∼ (Tm ) . Eventually, e 3/2 (Tm ) e g ∼ , (1.12) e 3 n h e and (1.11) takes the following form: 3/2 n n (Tm ) i e e ∼ exp(−I/T). (1.13) 3 n h a Let us proceed further by assuming that the ions are singly ionized, which givesn =n , i e and introduce the total density n = n +n and the ionization degree z = n /n, then a i i one has 2 3/2 z (Tm ) e ∼ exp(−I/T). (1.14) 3 1−z nh When the density is low, the pre-exponential in (1.14) is large, and even forT I the ionization degreez may be close to unity,1−z 1. In this case we say that the plasma is fully ionized. The expression (1.14) in its precise form is called Saha formula. 1.6 Summary • Plasma is a gas of ionized particles. 2 1/2 • Debye length (single species): r = (T/4πnq ) . D • Debye screening of a test particle: φ =Qexp(−r/r )/r. D 3 • Plasma parameter: g = 1/nr  1. D 2 1/2 • Plasma frequency (single species): ω = (4πnq /m) . p 6CHAPTER 1. BASIC DEFINITIONS AND PARAMETERS 1.7 Problems PROBLEM 1.1. Calculate the Debye length for a multi-species plasma: n ,q ,T . The plasma s s s P is quasi-neutral: n q = 0. s s s PROBLEM 1.2. Calculate the plasma frequency for a multi-species plasma: n , q , m . The s s s P plasma is quasi-neutral: n q = 0. s s s PROBLEM 1.3. Calculater ,g andω for the plasmas in Table A. D p PROBLEM 1.4. A parallel plate capacitor charged to±σ is immersed into an electron plasma (immobile ions). What is the potential distribution inside the capacitor ? What is its capacity ? 7CHAPTER 1. BASIC DEFINITIONS AND PARAMETERS 8Chapter 2 Plasma description In this chapter we learn about possible methods of plasma description, and derive the powerful but limited MHD. 2.1 Hierarchy of descriptions In order to deal with plasma we have to choose some method of description. The most straightforward and most complete would seem to use the motion equations for all par- ticles together with the Maxwell equations for the electromagnetic fields. However, it is impossible as well as unreasonable for a many-particle system with a collective behav- ior. A less precise but much more efficient description would be to describe all particles of the same species as a fluid in the phase space (r,p). This would correspond to the assumption that on average behavior of each particle is the same and independent of other particles, following only the prescriptions of the self-consistent fields. In this ap- proach we forget about the possible influence of the deviations of the fields from the self-consistent values (fluctuations) and direct (albeit weak) dependence of a particle on its neighbors (correlations). This is the so-called kinetic description. The further step toward even greater simplification of the plasma description would be to average over momenta for each species, so that only average values remain. In this case each speciess is described by the local densityn , local fluid velocityv , local s s temperatureT or pressurep . This is the so-called multi-fluid description. s s Finally, we can even forget that there are different species and describe the plasma as one fluid with the mass density ρ , velocity V , and pressure p. It is clear that m electromagnetic field should be added in some way. The rest of the chapter devoted to the description of plasma as a single fluid. The description is known as magneto- hydrodynamics (MHD) for the reasons which become clear later. 9CHAPTER 2. PLASMA DESCRIPTION 2.2 Fluid description In order to describe a fluid we choose a physically infinitesimal volumedV surrounding the pointr in the momentt. The physically infinitesimal volume should be large enough to contain a large number or particles, so that statistical averaging is possible. On the other hand, it should be small enough to not make the averaging too coarse. With- out coming into details we shall assume that qualitative meaning of this "infinitesimal" volume is sufficiently clear and we can make such choice. The fluid mass ρ density is simply the sum of the masses of all particles inside m P this volume divided by the volume itself, ρ = m /dV . Since the result may be i different for volumes chosen in different places or at different times, the density can, in general, depend onr and t. The hydrodynamical velocity of this infinitesimal volume P is simply the velocity of its center of mass: V = mv /ρdV . Again,V =V (r,t). i i Pressure is produced by the random thermal motion of particles (relative to the center- of-mass) in the infinitesimal volume. In order to avoid unnecessary complications we shall assume that the pressure is isotropic, that is, described by a single scalar function p(r,t). In what follows we shall consider plasma as an ideal gas, that is, p = nT , where n(r,t) is the concentration and T(r,t) is the temperature. Thus, we have four fields: ρ (r,t),V (r,t),p(r,t), andT(r,t), for which we have to find the appropriate m evolution equations, connecting the spatial and temporal variations. For brevity we do not write the dependence(r,t) in what follows. 2.3 Continuity equation We start with the derivation of the continuity equation which is nothing but the mass conservation. Let us consider some volume. The total mass inside the volume is Z M = ρ dVρ (2.1) m m V This mass can change only due to the flow of particles into and out of the volume. If we consider a small surface element,dS =n ˆdS, then the mass flow across this surface during timedt will bedM =ρ Vdt·dS. The total flow across the surfaceS enclosing m the volumeV from inside to outside would be I Z dJ = ρ V ·dSdt = div(ρ V )dVdt (2.2) m m S V Since the flow outward results in the mass decrease, we write Z Z d ρ dV =− div(ρ V )dV ⇒ (2.3) m m dt V V 10CHAPTER 2. PLASMA DESCRIPTION Z   ∂ρ m + div(ρ V ) dV = 0⇒ (2.4) m ∂t V ∂ρ m + div(ρ V ) = 0. (2.5) m ∂t The last relation follows from the fact that the previous should be valid for any arbitrary (including infinitesimal) volume at any time. Equation (2.5) is the continuity equation. 2.4 Motion (Euler) equation The single particle motion equation is nothing but the equation for the change of its momentum. We shall derive the equation of motion for the fluid considering the change of the momentum of the fluid in some volume V . The total momentum at any time would be Z P = ρ VdV (2.6) m V The momentum changes due to the flow of the fluid across the boundary and due to the forces acting from the other fluid at the boundary. Let us start with the momentum flow. The fluid volume which flows across the surfacedS during time dt isVdt·dS. This flowing volume takes with it the momentumdP = (ρ V )(Vdt·dS). Thus, the total m flow of the momentum outward is I dP = (ρ V )(V ·dS)dt (2.7) m S The total force which acts on the boundaries of the volume from the outside fluid is I F =− pdS (2.8) S Combining (2.6)-(2.8) we get Z I I d ρ VdV =− (ρ V )(V ·dS)− pdS (2.9) m m dt V S S Further derivation is simpler if we write (2.9) in the component (index) representation: Z I I ∂ (ρ V )dV =− (ρ V V )dS − pδ dS (2.10) m i m i j j ij j ∂t V S S and use the vector analysis theorem: I Z ∂ A dS = A dV (2.11) ij j ij ∂x j S V 11CHAPTER 2. PLASMA DESCRIPTION Now we get the motion equation in the following form: ∂ ∂ ∂ (ρ V )+ (ρ V V ) =− p (2.12) m i m i j ∂t ∂x ∂x j j If we recall that the continuity equation can be written as ∂ ∂ ρ (ρ V ) = 0 (2.13) m m j ∂t ∂x j we can rewrite (2.12) in the following widely used form:   ∂ ρ V +(V ·∇)V =−gradp (2.14) m ∂t One has to be cautious with the form of the equation since (V ·∇)V is not a good vector form and cannot be easily written in curvilinear coordinates. Instead one has to use the proper vector representation   2 V (V ·∇)V = grad −V × rotV (2.15) 2 It is worth noting that the force−gradp the volume force, that is, the forth per unit volume. If other volume forces exist we should simply add them to the right hand side of (2.14). 2.5 State equation We have derived 4 equations (one for the scalar and three for the vector equation) for 5 variables: ρ , three components ofV , and p. Therefore, we need another equation m for the pressurep. Either we have to derive it from the first principles, as we did for the continuity equation and the motion equation, or to use some sort of approximate closure. For this course we just assume that the pressure is a function of density,p =p(ρ). 2.6 MHD So far we have been treating a single fluid, without any relation to plasma. What makes the fluid plasma is its ability to carry currents. If the current density in the plasma is j then it experiences the Ampere force (1/c)j×B, once the magnetic field B is present. In principle, electric volume force ρ E may be also present. However, in the q non-relativistic MHD approximation the plasma is quasi-neutral and this term is absent, and for the rest of the chapter we do not write the indexm forρ - it is always the mass 12CHAPTER 2. PLASMA DESCRIPTION density. (If you ever wish to learn relativistic MHD do not forget the electric force.) Thus, the motion equation takes the following form:   ∂ 1 ρ V +(V ·∇)V =−gradp+ j×B (2.16) ∂t c However, now we have two more vector variables: j and B. It is time to add the Maxwell equations: divB = 0, (2.17) 4π 1 ∂ rotB = j + E, (2.18) c c∂t 1 ∂ rotE =− B. (2.19) c∂t We do not need the divE = 4πρ equation since quasi-neutrality is assumed and this q equation does not add to the dynamic evolution equations, but rather allows to check the assumption in the end of calculations. Eq. (2.17) is a constraint, not an evolution equation since it does not include time derivative. It is also redundant since (2.19) shows that once (∂ divB/∂t) = 0, and once (2.17) is satisfied initially it will be satisfied forever. It can be shown (we shall see that later in the course) that non-relativistic MHD is the limit of slow motions and large scale spatial derivatives, so that the displacement current is always negligible, and (2.18) becomes a relation between the magnetic field and current density c j = rotB. (2.20) 4π The only evolution equation which remains is the induction equation (2.19). However, it includes now the new variable E which does not seem to be otherwise related to any other variable. Ohm’s law comes to help. The local Ohm’s law for a immobile conductor is written as j = σE. Plasma is a moving conductor and the Ohm’s law 0 0 should be written in the plasma rest frame,j = σE . For non-relativistic flows the rest 0 0 frame electric fieldE =E +V ×B/c, whilej =j because of the quasi-neutrality condition. Thus, the Ohm’s law should be written in our case as j =σ(E +V ×B/c). (2.21) This relation is used to express the electric field in terms of the magnetic field: 1 c E =− V ×B + rotB, (2.22) c 4πσ and substitute this in (2.19): 2 ∂ c B = rot(V ×B)+ ΔB, (2.23) ∂t 4πσ 13CHAPTER 2. PLASMA DESCRIPTION thus getting an equation containing onlyB andV . Now, substituting (2.20) into (2.16) we get the equation of motion free of the current:   ∂ 1 ρ V +(V ·∇)V =−gradp+ rotB×B. (2.24) ∂t 4π 2.7 Order-of-magnitude estimates LetL be the typical inhomogeneity length, which means that when we move byΔx,y,z∼ L the variable under consideration, say B changes by ΔB ∼ B. Now substitute (∂B/∂x) ∼ ΔB/Δx ∼ B/L, that is∇ ∼ 1/L. Similarly, if T is the typical vari- ation time, we have (∂/∂t) ∼ 1/T . Typical velocity then is estimated as V ∼ L/T . Using these definitions we can estimate from the induction equationE∼ (V/c)B. Re- spectively, the ratio of the displacement current to rotB term will be   2 1∂E LE V /rotB∼ ∼ c ∂t cTB c and is very small for nonrelativistic velocities. This is the reason, why it is usually neglected. For the charge density we have ρ = divE/4π ∼ E/4πL. Thus, the ratio of the q electric and magnetic forces   2 ρ E 1 V q ∼ (1/c)j×B 4π c and is also negligible. 2.8 Summary Let us write down again the complete set of the MHD equations: ∂ ρ+ div(ρV ) = 0, (2.25) ∂t d 1 ρ V =−gradp+ rotB×B, (2.26) dt 4π 2 ∂ c B = rot(V ×B)+ ΔB, (2.27) ∂t 4πσ where we introduced the substantial derivative d ∂ = +(V ·∇). (2.28) dt ∂t 14

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