20. Alfven waves. ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma:

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1 Phys780: Plasma Physics Lecture 20. Alfven Waves Alfven waves ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma: 1. electrostatic Langmuir waves and ion-sound waves: oscillations of electrons and ions relative to each other 2. electromagnetic waves - high-frequency waves affected by motions of electrons in plasma in the oscillating electric field of the waves and in the stationary magnetic field (we assumed that ions are heavy and don t move) The third types of waves are MHD waves in which plasma oscillates as a single fluid. These waves are described by using single-fluid plasma equations and Maxwell equations.

2 Phys780: Plasma Physics Lecture 20. Alfven Waves. 2 Consider Maxwell equations: E = 1 c B t B = 1 E + 4π } c {{ t } c j displacement current B = 0 E = 4πq = 0 We derive equation for E by applying operator to the first equation and substituting B from the second equation: Using ( E) = 1 c B t = 1 c 2 2 E t 2 4π c 2 j t ( E) = ( ) E ( ) E = 2 E

3 Phys780: Plasma Physics Lecture 20. Alfven Waves. 3 we obtain 2 E = 1 c 2 2 E t 2 + 4π c 2 j t where j is determined from Ohm s law (lecture 13): E + v B = [ j c σ + 1 j ] B p e en c + m e e 2 n Previously, we considered only slow processes and neglected the last term. However, for high-frequency electromagnetic waves with frequencies close to plasma frequency this term significant. Plasma velocity v is determined from the equation of motion ρ d v dt = p+ 1 c ( j B), and pressure is determined from the continuity and energy equations: ρ t + (ρ v) = 0 j t

4 Phys780: Plasma Physics Lecture 20. Alfven Waves. 4 p/ρ γ = const The last equation is the adiabatic law meaning that the plasma entropy remains constant. In general, under any given set of circumstances there are four modes at a given frequencies (some of these may be imaginary). Their properties and described by the dispersion relations, ω = ω(k), where ω is the wave frequency, and k is the wavenumber. Without magnetic field there are: A) electrostatic (a) plasma wave (Langmuir waves) - oscillations of electrons relative to ions: ω 2 = ω 2 p +3k 2 v 2 T (b) ion-sound waves (electrons and ions move together, inertia is determined by ions, pressure is due to both electrons and ions): ω 2 = c 2 Sk 2

5 Phys780: Plasma Physics Lecture 20. Alfven Waves. 5 where c 2 S = T e +3T i M is the ion-acoustic speed. B) electromagnetic waves ( E k) of two polarization states. In general, the theory of plasma waves is complicated because of the complexity of the single fluid equations. We considered only simple cases. For instance, for high-frequency waves without magnetic field we can keep only two terms in the Ohm s law: Hence the equation for E is: E = m e 2 n j t 2 E = 1 c 2 2 E t 2 + 4πe2 n c 2 m E From this obtain the dispersion relation for electromagnetic waves in

6 Phys780: Plasma Physics Lecture 20. Alfven Waves. 6 plasma: ω 2 c 2 = k2 + ω2 p c 2. In general, a wave in a magnetic field involves both electric and magnetic forces. A high-frequency wave is a combination of electromagnetic wave with a longitudinal electrostatic wave. Density gradients may produce coupling between different types of waves. With magnetic field, for electromagnetic waves traveling along the field lines there are two wave modes: R-waves (with right circular polarization): c 2 k 2 ω 2 = 1 ω 2 p ω(ω ω e ) where ω e = eb mc is the electron cyclotron frequency.

7 Phys780: Plasma Physics Lecture 20. Alfven Waves. 7 L-waves (with left circular polarization): c 2 k 2 ω 2 = 1 ω 2 p ω(ω +ω e ) In the first case, the wave polarization vector ( E) rotates in the same direction is the gyration of electrons. This gives rise to a low-frequency whistler mode (electron-cyclotron wave) with the frequency below the electron cyclotron frequency (Lecture 17). Previously, we considered ions as stationary. When motion of ions is taken into account then the R- and L-wave modes are modified, and new type of hydromagnetic waves appear at low frequencies smaller than the ion cyclotron frequency ω < ω ci. In this case the dispersion relations for the R- and L-waves are the following:

8 Phys780: Plasma Physics Lecture 20. Alfven Waves. 8 R-waves (with right circular polarization): c 2 k 2 ω 2 = 1 ω 2 p (ω +ω i )(ω e ω) where ω i = eb Mc is the ion cyclotron frequency. L-waves (with left circular polarization): c 2 k 2 ω 2 = 1 ω 2 p (ω ω i )(ω +ω e ) The additional mode (ion cyclotron wave) appears in the L-wave case because the polarization electric field vector rotates in the same direction as the gyration of ions (Figure 1).

9 Phys780: Plasma Physics Lecture 20. Alfven Waves. 9 Figure 1: Dispersion diagram for L mode of waves propagating along the magnetic field (including the low-frequency branch of so-called Alfven shear waves). For the left circular polarized waves the polarization vector rotates as in the same direction as the direction of gyration of ions. These waves accelerate ions and cannot propagate with frequencies above the ion cyclotron frequency.

10 Phys780: Plasma Physics Lecture 20. Alfven Waves. 10 Figure 2: Dispersion diagram for R mode. For the right circular polarized waves the polarization vector rotates as in the same direction as the direction of gyration of electrons. At low frequencies, for waves traveling perpendicular to magnetic field lines a compressible Alfven wave mode (also called fast MHD wave) appears (we ll consider this wave in the next lecture).

11 Phys780: Plasma Physics Lecture 20. Alfven Waves. 11 For low ω the L-wave dispersion relation is: c 2 k 2 and if c 2 k 2 /ω 2 1 we get: where V 2 A = B2 4πnM = B2 4πρ ω 2 = 1+ ω2 p ω i ω e = 1+ 4πnMc2 B 2, ω 2 = B2 4πnM k2 = V 2 Ak 2 is the Alfven speed. In these waves (called shear Alfven waves), inertia is due to ions, and the restoring force is j B. These waves can be regarded as waves of the magnetic lines of force, which behave like strings loaded with plasma particles.

12 Phys780: Plasma Physics Lecture 20. Alfven Waves. 12 Alfven waves Let us consider now the low-frequency hydromagnetic waves. In we neglect the displacement current, Hall effect, pressure gradient, and compressibility. Then, the equations have the following form: If 2 E = 4π c 2 j t ρ v t = 1 c j B B = (0,0,B 0 ) v = (v x,0,0) E = (0,E y,0) j = (0,j y,0)

13 Phys780: Plasma Physics Lecture 20. Alfven Waves. 13 Figure 3: Geometry of an Alfven wave propagating along B 0.

14 Phys780: Plasma Physics Lecture 20. Alfven Waves. 14 then ρ v x t = 1 c j yb 0 E y 1 c v xb 0 = 0 2 E y z 2 = 4π j y c 2 t v x = ce y B 0 j y = cρ B 0 v x t = c2 ρ B E y z 2 = 4πρ B E y t 2 E y t Thus, the dispersion relation of these waves (Alfven waves) is ω 2 = B2 0 4πρ k2 = V 2 ak 2

15 Phys780: Plasma Physics Lecture 20. Alfven Waves. 15 where is the Alfven speed. V 2 A = B2 0 4πρ Consider basic properties of Alfven waves. If ( E y = E 0 sinω t z ) V A then j y = c2 ρe 0 ω B 2 0 cosω v x = ce 0 B 0 sinω ( t z ( t z V A We find the oscillating magnetic field of the wave from the Maxwell equation B = 4π c j V A ) )

16 Phys780: Plasma Physics Lecture 20. Alfven Waves. 16 the y-component of which has the following form Hence B x z = 4π c j y B x z = 4πcρ E y B0 2 t Substituting E y and integrating over z we get B x = ce 0 V A sinω ( t z V A We see that v x and B x oscillate in antiphase. We can calculate the kinetic and magnetic energy densities averaged over the wave period, taking into account that: ( sin 2 ω t z ) = 1 2 ρv 2 x 2 V A = ρc2 E 2 0 4B 2 0 )

17 Phys780: Plasma Physics Lecture 20. Alfven Waves. 17 B 2 x 8π = c2 E 2 0ρ 4B 2 0 Thus, the kinetic and magnetic energies of Alfven waves are equal. For the relative amplitudes of velocity and magnetic field oscillations we obtain v x = B x = ce 0 V A B 0 B 0 V A If v x is large then B x is also large. Hence Alfven waves can amplify the initial magnetic field and transport it to large distances. However, the condition of incompressibility requires that the Alfven speed is much smaller than the speed of sound V A c s. Consider now the equation for the magnetic lines of force: dx = dz B x B 0 dx dz = B x B 0 = ce 0 B 0 V A sinω(t z/v A )

18 Phys780: Plasma Physics Lecture 20. Alfven Waves. 18 The general solution for the lines of force displacement is: x = x 0 + ce 0 B 0 ω cosω(t z/v A) The corresponding velocity of the line of force is: dx dt = ce 0 B 0 sinω(t z/v A ) = v x Hence the magnetic field lines are frozen into the plasma. When the electrical resistivity of plasma is zero (σ = ) the waves are non-dissipative, otherwise the Alfven waves dissipate. We can calculate the averaged over the period Joule dissipation and the corresponding change of the wave energy: dw dt == 1 T T 0 j 2 σ dt

19 Phys780: Plasma Physics Lecture 20. Alfven Waves. 19 where W = 1 T T 0 ( ) 1 2 ρv2 x + B2 x dt 8π We obtain W = ρc2 E 2 0 2B 2 0 j 2 = c4 ρ 2 E 2 0ω 2 2B 4 0 = Wω2 ρc 2 B 2 0 dw dt = Wω2 ρc 2 σb 2 0 = Wω2 c 2 4πσV 2 A = W τ where Here τ = 2πσV 2 A c 2 ω 2 = V 2 A 2ω 2 ν m = L2 ν m ν m = c2 4πσ

20 Phys780: Plasma Physics Lecture 20. Alfven Waves. 20 is called magnetic viscosity, L = V A ω = λ 2π is a characteristic size of variations in plasma, λ = 2π/k = 2πV A /ω is the wavelength. The dissipation rate relative to the wave period is where τ/p = τω/2π = V 2 A 4πων m = 1 2π V A L ν m = 1 2π Re m Re m = V AL ν m is the magnetic Reynolds number. It determines the relative time scale of the Joule dissipation compared to the dynamic time scale.

21 Phys780: Plasma Physics Lecture 20. Alfven Waves. 21 Magnetic Reynolds number It plays a fundamental role in the plasma MHD theory. Consider the equation for the magnetic field evolution in the presence of Joule dissipation B = 4π c j ( j = σ E + 1 ) c v B E = 1 c B t E = j σ ( v B) c j = c 4π B

22 Phys780: Plasma Physics Lecture 20. Alfven Waves. 22 Finally, we obtain B t = ( v B) [ ] c 2 4πσ ( B) The magnetic Reynolds number determines the relative role of the two terms in the right-hand side: magnetic field advection and dissipation. The relative importance of these terms for a process of a characteristic scale L, velocity v is determined by the magnetic Reynolds number: R M = vb L c 2 4πσ = 4πσLv B c 2. L 2 For typical coronal conditions: T = 10 6 K, σ = s 1, L = 10 8 cm, v = 10 7 cm/s, R M 10 5 >> 1. For uniform σ the last term can be simplified: ( B) = ( B) 2 B = 2 B.

23 Phys780: Plasma Physics Lecture 20. Alfven Waves. 23 B t = ( v B)+ c2 4πσ 2 B Then, if v = 0 we get a diffusion equation: where B t = D 2 B, D = c2 4πσ is a diffusion coefficient for magnetic field. Exercises: 1. Estimate the characteristic scale of dissipation of magnetic field in solar flares. The duration of solar flares is 10 3 sec. c2 t L 4πσ 105 cm = 1km. This is smaller than the observed flare structure. What does that mean?

24 Phys780: Plasma Physics Lecture 20. Alfven Waves Estimate the decay time of sunspots (L 10 9 cm, T 10 4 K, σ 10 9 s 1 ). t 4πσL2 c sec 4 months. This is longer the observed lifetime of sunspots. Why?