Department of Physics Seminar 1b Magnetohydrodynamic waves in a plasma Author: Janez Kokalj Advisor: prof. dr. Tomaž Gyergyek Petelinje, April 2016 Abstract Plasma can sustain different wave phenomena. In MHD theory of plasma three different wave modes can be observed (slow, fast and pure Alfven waves). In seminar, at first each mode is briefly described. Fluid velocity and propagation of them is presented in second part of the seminar. In case of non-perfect conducting plasma also dumping is present as described at the end of the seminar.
Contents 1. INTRODUCTION... 1 2. DESCRIPTION OF PLASMA... 2 3. MAGNETOHYDRODYNAMIC WAVES... 3 3.1. Sound waves... 3 3.2. Alfven waves... 3 3.3. Magnetosonic waves... 4 4. FLUID VELOCITY AND PROPAGATION OF WAVES... 5 4.1. Waves perpendicular to the magnetic field... 5 4.2. Waves parallel to the magnetic field... 6 4.3. Arbitrary direction... 7 4.4. Phase velocity... 8 5. DAMPING OF MHD WAVES... 10 6. CONCLUSION... 11 7. REFERENCES:... 11 1. INTRODUCTION Plasma is a mixture of ions, free electrons and also neutral atoms or molecules. It is the fourth state of the matter. Major difference between plasma and other matter is that plasma contain charged particles. Between them, there is Coulomb force that has long-range in a contrast of other interactions between particles. Because of the high electron mobility, plasma is very good electrical conductor. The consequence is that particles are trying to shade the electric field and the Debye length tell us the distance over which the interactions of the individual charged particles is felt. An important characteristic of plasmas is their ability to sustain a great variety of wave phenomena. [1] Some examples are longitudinal electrostatic waves, high-frequency transverse electromagnetic waves, low-frequency Alfven waves and magnetosonic waves. The study of plasma waves provides important information on plasma properties. The wave modes derived from MHD plasma theory are called magnetohydrodynamic waves and can be divided in three modes: - pure Alfven waves - slow MHD waves - fast MHD waves
They can be observed in Earth s ionosphere and are thought to be the process of heating the sun s corona, which has a temperature of million degree, whereas the temperature of the surface is few thousand degree. Understanding of the wave phenomena is a key to provide the stability of the fusion devices. 2. DESCRIPTION OF PLASMA Plasma consists of very large number of electrons and ions and consequently it is hard to observe each individual particle. It can be described with a help of statistical physics since we are not interested in individual particles but in the macroscopic properties of a plasma [2]. Particles with the same position and velocity can be combined in a classes and their distribution can be described with the distribution function. Time and space development of this function is defined with Boltzmann equation. The other possibility to describe plasma is hydrodynamic model. For different types of particles a set of fluid equations, very similar to Navier-Stokes equations, can be derived from Boltzmann equation. The next level in plasma description is magnetohydrodynamic model. Sometimes it is easier to describe a plasma as a conducting fluid with magnetic fields and electric current. Differences between different types of particles are neglected and plasma is described as a single conducting fluid, some kind of mixture of different fluid with kind of averaged features [3]. This model is MHD model. In practice usually several approximations are made with neglecting some of the terms. For steady-state situations or slowly varying problems, the MHD equations lead to results, which would not be easily obtained form the individual equations for the each particle species [4]. Set of simplified magnetohydrodynamic equations: Mass and energy equilibrium equation ρ t + (ρu ) = 0 (1) ρ u t + ρ(u )u = p + J B Adiabatic equation of energy conservation (2) Maxwell s equations p = c S 2 ρ (3) B = μ 0 J (4) E = B t (5) E + u B = 0 (6)
Where ρ is density, u velocity of a particles, J current density, B magnetic field, p pressure and E electric field. 3. MAGNETOHYDRODYNAMIC WAVES 3.1. Sound waves The most fundamental type of waves in compressible fluid is sound, which is longitudinal wave motion. In fluid are in that case changes in density - compressions and rarefactions. [1] Figure 1: Sound waves, compressions and rarefactions in media. [5] In thermodynamic the fluid can be described with the adiabatic energy equation pρ κ = const (7) where p stands for pressure, ρ for density and κ for the ratio of specific heats at constant pressure and at constant volume. With differentiating, we obtain the adiabatic speed of sound c for ideal gas as c = κp ρ = κkt m where k is Boltzmann constant, T is temperature and m is mass. In contrast with the solid matter, particles in plasma do not collide but the Coulomb force interacts between them in a similar way, just the distances are greater. 3.2. Alfven waves If fluid is compressible and conducting, it is possible to have other types of waves in a magnetic field. In a magnetic field of intensity B, the magnetic stresses are equivalent to a tension B 2 /μ 0 along the field line and an isotropic hydrostatic pressure B 2 /2μ 0 [1]. As that can be superposed on the fluid pressure, the magnetic field lines act like elastic cords under a tension B 2 /μ 0. The line of force in perfectly conducting plasma acts like string with mass, because plasma particles act as if they are tied to the magnetic field lines. In case of disturbance from equilibrium, the magnetic field lines act like elastic strings and transverse vibration can be observed. The velocity of propagation of those transverse vibrations is expected to be square root of ratio of tension and density and is it in that case (8)
c A = B2 μ 0 ρ (9) where c A is known as the Alfven velocity. In case of this type of wave, there is not any fluctuations in density or fluid pressure. Figure 2: Transverse Alfven waves. The propagation is along the field lines, whereas the fluid motion and magnetic field perturabtions B 1 are perpendicular to the field lines. [6] 3.3. Magnetosonic waves Anyway longitudinal oscillations are also expected to occur in compressible conducting fluid in a magnetic field. In this direction particles are free to move and waves are ordinary sound waves. Their propagation velocity is c (equation 8). That is because with the motion of particles in this direction, there is no magnetic field perturbation. Figure 3: Longitudinal sound waves. Their propagation is in the direction of the field lines. However, for the direction, perpendicular to the magnetic field, there is possibility for a new type of longitudinal waves. Besides the fluid pressure p there is also the magnetic pressure B 2 /2μ, which is normal to B. Subsequently the total pressure is p + B 2 /2μ. Velocity of propagation for those magnetosonic waves is (p + B2 2μ 0 ) = c M 2 ρ (10)
where c M is velocity of magnetosonic or magnetoacoustic waves. Since the lines of force are frozen in the conducting fluid, the magnetic flux BdS and the mass of a unit length ρds are both conserved during the oscillation, it can be written c M 2 = c 2 + c A 2. (11) Figure 4: Compressions and rarefactions of fluid and the lines of force in conducting fluid for a case of the magnetosonic waves. 4. FLUID VELOCITY AND PROPAGATION OF WAVES From MHD equations can be for small-amplitude waves with neglecting second-order terms and linearization produced equation for u speed of fluid. Small fluctuations from equilibrium for magnetic field B and density ρ can be described as x = x 0 + x 1 (12) whereas velocity u has only fluctuation (u 0 = 0). It leads to simple solution for wave propagation in the direction parallel or perpendicular to the magnetic field. [1] ω 2 u 1 + (c 2 S + c 2 A )(k u 1 )k + (k c A ) ((k c A )u 1 (c A u 1 )k (k u 1 )c A ) = 0 (13) 4.1. Waves perpendicular to the magnetic field For propagation perpendicular to the magnetic field the wave vector k is perpendicular to the magnetic induction B 0. Consequently k c A = 0 and from equation (13) can be observed u 1 = 1 ω 2 (c S 2 + c 2 A )(k u 1 )k. (14) u 1 is parallel to k and solution for u 1 is a longitudinal wave with the phase velocity c 2 S + c 2 A. If magnetic field is considered in form of
B 1 (r, t) = B 1 exp (ik r iωt) (15) Where index 1 means small perturbation from equilibrium (index 0). For that longitudinal wave the relate magnetic field, using vector identity and observing that k B 0 = 0 is B 1 = u 1 ( k ω ) B 0 (16) Relate electric field with this wave is E = u 1 B 0. (17) That wave is in some way similar to an electromagnetic wave. Time dependent magnetic field is perpendicular to the direction of propagation and parallel to the static magnetic field. Time dependent electric field is perpendicular to the static magnetic field and to the direction of the propagation. Because velocity of mass flow and mass density fluctuations associated with the wave motion are both in the direction of the wave propagation, this wave is called the magnetosonic wave, also known as fast Alfven wave. Since the phase velocity is independent of frequency this wave does not disperse. The magnetosonic waves cause compressions and rarefactions in the magnetic field lines but does not change their direction. Lines of force move with fluid since the fluid is perfectly conducting. The restoring forces (forces that are directed toward the equilibrium) in the magnetosonic waves are the fluid pressure gradient and the gradient of the compressional stresses between the magnetic field lines. [1] If the fluid pressure is much bigger than the magnetic pressure, the contribution of the magnetic field can be neglected and the magnetosonic wave become an acoustic wave ( ω k = c S). In opposite case, when magnetic field is much greater than the fluid pressure, the wave become the Alfven wave with phase velocity c A. 4.2. Waves parallel to the magnetic field When wave vector k is parallel to the magnetic field, it is k c A = k c A. From equation (13) we obtain (k 2 c 2 A ω 2 )u 1 + ( c S 2 2 c 1) k2 (u 1 c A )c A = 0. A (18) There are two possibilities for wave motion. When u 1 is parallel to B 0 and k, a longitudinal mode is possible with the phase velocity c S.
In that case, there is no electric field, electric current density or magnetic field associated with this wave, which is ordinary longitudinal sound wave. On the other side, there is transverse wave, perpendicular to magnetic field and wave vector. Phase velocity is c A. This wave is known as Alfven wave. There is no dispersion, because phase velocity is independent of frequency. The magnetic field associated with the Alfven wave is B 1 = B 0k ω u 1. (19) Since the magnetic field disturbance B 1 is normal to the static magnetic induction B 0, that gives the line of force a sinusoidal ripple. The Alfven wave is associated with no fluctuations in the fluid density and pressure. The fluid and the magnetic field lines oscillate in plane normal to B 0. The magnetic energy density is equal to the kinetic energy density of the fluid motion b 1 2 2μ 0 = 1 2 ρu 1 2. (20) The Alfven wave mode is also known as the slow Alfven wave. 4.3. Arbitrary direction In case of wave propagation in arbitrary direction, we introduce Cartesian coordinates. Let the z be the direction of the magnetic induction B 0 and the y direction is normal to the plane defined by wave vector k and B 0 and the angle θ is between them. For components, it applies next equations [1]. For the x component u 1x ( ω 2 + k 2 c A 2 + k 2 c S 2 sin 2 θ) + u 1z (k 2 c S 2 sin θ cos θ) = 0 (21) For the y component u 1y ( ω 2 + k 2 c A 2 cos 2 θ) = 0 (22) And for the z component u 1x (k 2 c S 2 sin θ cos θ) + u 1z ( ω 2 + k 2 c S 2 cos 2 θ) = 0. (23)
From equation (22) it can be observed that there exists linearly polarized wave with oscillations perpendicular to the k and B 0. These oscillations are B 1y and u 1y, so it is a transverse Alfven wave, known as pure Alfven wave. For propagation parallel to magnetostatic field (θ = 0) phase velocity is ω k = c A, but if propagation is perpendicular to the magnetostatic field this wave does not exist. 4.4. Phase velocity From equation (25) and (27) we got two nonzero solutions for a phase velocity ( ω k ) 2 = 1 2 (c S 2 + c A 2 ) ± 1 2 ((c S 2 + c A 2 ) 2 4c S 2 c A 2 cos 2 θ) 1 2 (24) Solution with plus is called fast MHD wave mode and solution with minus slow MHD wave mode. Because of ( ω k )2 we get solutions for wave s propagation also in the opposite direction. Phase velocity is a function of the angle θ between k and B 0. Figure 5: Phase velocity of the fast, slow MHD waves and Alfven waves as a function of θ. Graph (a) stands for c A > c S and (b) stands for c A < c S. [1] If c S < c A the phase velocity of the fast MHD waves increases form c A at θ = 0 to c S 2 + c A 2 at θ = 90. The slow MHD waves travel at the velocity c S at θ = 0 and at 0 when θ = 90. The fast MHD waves becomes the Alfven waves at θ = 0 and the magnetosonic for θ = 90. The slow MHD waves are the sound waves at θ = 0 and do not exist for θ = 90. If c S > c A the phase velocity of the fast MHD waves increases from c S at θ = 0 to c S 2 + c A 2 at θ = 90. The slow MHD waves travel at the velocity c A at θ = 0 and at 0 when θ = 90.
The fast MHD waves becomes the sound waves at θ = 0 and the magnetosonic for θ = 90. The slow MHD waves are the Alfven waves at θ = 0 and do not exist for θ = 90. Figure 6: Wave normal diagram for the fast, slow MHD waves and Alfven. Diagram (a) stands for c A > c S and (b) stands for c S > c A. [1] In magnetohydrodyamics the displacement current term ε 0 E / t from Maxwell equation is usually neglected. That is valid only for highly conducting fluids for low frequencies below the ion cyclotron frequency. [1] If it is attended the equations for wave propagation are slightly modified. Anyway, these results are valid only at frequencies where the effect of charge separation is not important. In case of propagation of waves across the magnetic field equation is similar to equation (14), but the square of the frequency is multiplied by (1 + c A 2 c2). Consequently the phase velocity of the longitunial magnetosonic wave become ω k = c 2 2 A + c S 1 + c A 2. (25) c For propagation along the magnetic field, there is no change in propagation of the longitudinal sound wave. However, for the transverse Alfven wave the square of the frequency is multiplied by (1 + c A 2 2). The phase velocity of the Alfven wave become c ω k = c A 1 + c. A 2 c 2 (26)
5. DAMPING OF MHD WAVES Fluids are in general not perfectly conducting. If it have finite conductivity or because of viscous effect (η k and η m, kinematic viscosity and magnetic viscosity respectively), the MHD waves will be damped. For the Alfven waves square of frequency is multiplied by two factors (1 + iη kk 2 ) and (1 + iη mk 2 ). If it is assumed, that viscosities are small and second order is ω ω neglected, equation for k can be simplified to k = ω c A + i(η k + η m )ω 2 2c A 3. (27) The imaginary part present the damping. If real part of k is written as k r and the imaginary as k i, the plane wave propagation in direction z can be written e ikz = e ik rz e k iz. (28) It has an exponential decreasing amplitude, which decrease to 1/e at the distance of 1/k i. Attenuation increases with frequency and decreasing with magnetic field intensity. As expected, it increases also with fluid viscosity. For longitudinal sound waves propagating parallel to B 0 it can be written k = ω c S + iη kω 2 2c S 3. (29) The attenuation increases with frequency and fluid viscosity, but decreases with sound velocity. For the longitudinal magnetosonic waves, propagating across the B 0, the terms for the wave vector can be simplified to k = ω c S 2 + c A 2 + iω2 2(c S 2 + c A 2 ) 3 2 ( η k + η m 1 + c S 2 c A 2 ). (30) The attenuation also increases with frequency and with viscosity, bur decreases with increasing magnetic field.
6. CONCLUSION Seminar tries to enlighten the phenomena of magnetohydrodynamic waves in a plasma. Using MHD theory to describe the plasma took away some individual particle feature. Nevertheless, it allow us also to simplify equation enough to solve them for general information of the events in the plasma, such as waves. They are small fluctuations around the equilibrium and some of them can be imagined as waves in other media, but some of them are unique for plasma. In seminar are presented some more important features of sound waves, Alfven waves and magnetosonic waves in plasma. 7. REFERENCES: 1 Bittencourt J. A.: Fundamentals of plasma physics 3 rd edition, Cambridge University Press, 2004 2 Ichimaru S.: Basic principles of plasma physics, A statistical approach, W. A. Benjamin; Revised ed. Edition, 1973 3 Filipič G.: Uvodni pojmi iz fizike plazme ter nekaj zgledov iz magnetohidrodinamike, seminar, Fakulteta za matematiko in fiziko, Univerza v Ljubljani 4 Tenenbaum B. S.: Plasma physics, McGraw-Hill, 1967 5 http://www.ducksters.com/science/physics/wave_sound.jpg, dostopano 2. 4. 2016 6 https://farside.ph.utexas.edu/teaching/plasma/lectures1/img1222.png, dostopano 30. 3. 2016