6.1. Linearized Wave Equations in a Uniform Isotropic MHD Plasma. = 0 into Ohm s law yields E 0

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1 Chapter 6. Linear Waves in the MHD Plasma 85 Chapter 6. Linear Waves in the MHD Plasma Topics or concepts to learn in Chapter 6:. Linearize the MHD equations. The eigen-mode solutions of the MHD waves (a) The characteristics of the entropy mode (b) The characteristics of the intermediate mode (or Alfvén mode or shear Alfvén mode) (c) The characteristics of the fast mode (or compressional Alfvén mode) (d) The characteristics of the slow mode 3. The Friedrichs diagrams of the phase velocity and the group velocity of the MHD waves. Suggested Readings: () Chapter 7 in Nicholson (983) () Chapter 4 in Krall and Trivelpiece (973) (3) Chapter 4 in F. F. Chen (984) 6.. Linearized Wave Equations in a Uniform Isotropic MHD Plasma Table 6. Column () lists governing equations of magnetohydrodynamic (MHD) plasma with isotropic pressure and zero heat flux. Derivation of these equations has been introduced in Chapter 3. Appendix C shows that the MHD Ohm s law can lead to frozen-in flux, which is an important characteristic of MHD plasma. In addition to frozen-in flux, MHD linear wave modes are also important characteristics of MHD plasma. Substituting V 0 = 0 into Ohm s law yields E 0 Far from the source region, perturbations can be assumed in plane-wave format. A perturbation A (x,t) can be written as A (x,t) = A (k,ω)cos(k x ωt + φ A ) = Re{ A (k,ω)exp[i(k x ωt)]} where A (k,ω) = A (k,ω)e iφ A is a complex number. The wave amplitude A (k,ω) satisfies O(A ) = O(ε)O(A 0 ). Following procedures described in Sections 5. and 5., a set of linearized MHD equations in ( ω, k ) domain are obtained and listed in Table 6. Column() for V 0 = 0, E 0 = 0, and A 0 = 0, where A 0 denotes a background variable.

2 86 Chapter 6. Linear Waves in the MHD Plasma Table 6.. Governing equations of MHD plasma with isotropic pressure and zero heat flux () MHD equations in ( t, x ) domain () linearized MHD equations in (ω, k) domain Mass continuity equation ( t + V )ρ = ρ V Mass continuity equation MHD momentum equation ρ( t + V )V = p + J B MHD momentum equation ( iω) ρ = (ik) V (6.) ( iω) V = (i k) p + J (6.) MHD energy equation 3 [( t + V )ln(pρ 5/3 )] = 0 MHD charge continuity equation MHD Ohm s law J = 0 E + V B = 0 Maxwell s equations: E 0 B = 0 E = B t B = J MHD energy equation MHD charge continuity equation MHD Ohm s law Maxwell s equations: ( iω) p = γ p 0 ( iω) ρ (6.3) (ik) J = 0 (6.4) E + V = 0 (6.5) (i k) E 0 (6.6) (ik) B = 0 (6.7) (i k) E = iω B (6.8) (i k) B = J (6.9) Our goal is to reduce the system equations listed in Table 6. Column () into a set of equations for plasma flow velocity V. We shall focus on the momentum equation (6.). In order to eliminate p in Eq. (6.), we substitute Eq. (6.) into Eq. (6.3) to eliminate ρ, then substitute the resulting equation into Eq. (6.) to eliminate p. Likewise, to eliminate J in Eq. (6.), we can substitute Eq. (6.5) into Eq. (6.8) to eliminate E, then substitute the resulting equation into Eq. (6.9) to eliminate B, and then substitute the resulting equation into Eq. (6.) to eliminate J.

3 Chapter 6. Linear Waves in the MHD Plasma 87 Substituting Eq. (6.) into Eq. (6.3) yields!p = γ p 0!ρ ρ = C ρ S 0!ρ = C 0 k!v S 0 0 ω (6.3') Substituting Eq. (6.5) into Eq. (6.8) to eliminate! E, then substituting the resulting equation into Eq. (6.9) to eliminate! B, it yields!j = i k!b i k k!e i k k (!V ) = ω = ω = i k [k (!V )] ω Substituting Eqs. (6.3') and (6.9') into Eq. (6.) yields ( iω ) V! ρ = i kc 0 k V! S0 ω Multiplying Eq. (6.') by iω / k ω k (6.9') + i k [k (B! 0 V )] (6.') ω yields V! = C S0 ˆk ˆk V! + C A0 ˆB0 { ˆk [ ˆk ( ˆ V! )} where C A0 / is called Alfvén speed, and C S 0 γ p 0 / is called sound speed. as As a result, we can obtain a set of equations for flow velocity!v, which can be written D!V = 0 (6.0) where D = [ ω k C ( ˆB A0 0 ˆk) ] (C A0 + C S 0 ) ˆk ˆk + C A0 ( ˆ ˆk)( ˆ ˆk + ˆk ˆB0 ) (6.) For convenience, we can choose a coordinate system such that background magnetic field is along the ẑ -axis, and wave number k lies on x - z plane. Namely, = ẑ (6.) and k = k(ẑcosθ + ˆxsinθ) (6.3) where θ is the angle between k and. Substituting Eqs. (6.) and (6.3) into Eqs. (6.0) and (6.) yields (ω / k ) α 0 δ where 0 (ω / k ) C A0 cos θ 0 δ 0 (ω / k ) β! V! V y! V z = 0 (6.4)

4 88 Chapter 6. Linear Waves in the MHD Plasma α = C A0 cos θ + (C A0 + C S 0 )sin θ = C A0 + C S 0 sin θ (6.5) β = C A0 cos θ + (C A0 + C S 0 )cos θ C A0 cos θ = C S 0 cos θ (6.6) δ = C S 0 cosθ sinθ (6.7) Note that solutions of ω / k values of the following matrix α 0 δ 0 C A0 cos θ 0 δ 0 β for different wave modes can be considered as eigen Characteristics of different wave modes can be obtained from the corresponding eigen vectors. Exercise 6.. Review eigen values and eigen vectors of a symmetric matrix. Determine eigen values λ, λ, λ 3, and the corresponding normalized eigen vectors ê, ê, ê 3, of the following symmetric matrix 0 M = 0 Show that these eigen vectors of the symmetric matrix form an orthonormal basis and after coordinate transformation, the representation of matrix B = {ê, ê, ê 3 } becomes λ 0 0 M = 0 λ λ 3 B M in this new basis 6.. Linear Wave Modes in the MHD Plasma Number of linearized equations with time derivative term can lead to the same number of linear wave modes. There are seven equations in Table 6. that consist of a time derivative term. It will be shown in this section that, for θ 0 and θ π /, seven linear wave modes can be found in MHD plasma. Three of them are forward propagating waves.

5 Chapter 6. Linear Waves in the MHD Plasma 89 Based on their wave speeds, these three wave modes are called fast-mode wave, intermediate-mode wave, and slow-mode wave. The intermediate mode wave is also called Alfvén-mode wave or shear-alfvén wave. The other four wave modes are backward propagating fast-mode wave, intermediate-mode wave, slow-mode wave, and non-propagating entropy-mode wave. The fast mode, Alfvén mode, and slow mode are eigen modes of Eq. (6.4). The entropy mode is an additional wave mode, which can be obtained from equation of ρ (i.e., continuity equation) Entropy Mode Entropy mode in MHD plasma is characterized by ρ 0, but V = V y = V z = 0 and ω For ω = 0, the phase speed also vanishes. Thus, entropy mode is frozen in the plasma flow. In general, if V = V y = V z = 0, but ρ 0 and/or B 0, then ω must be zero (ω = 0 ), and (i k)!p + J! Proof: For V = V y = V z = 0, Eq. (6.0) or (6.4) is automatically fulfilled. Substituting V = 0 into Eq. (6.5) yields E Substituting V = 0 into Eq. (6.) yields ω!ρ Substituting E = 0 into Eq. (6.8) yields ω B! Thus, if ρ 0 and/or B 0, then we must have ω Substituting V = 0 into Eq. (6.) yields (i k)!p +! J = 0 (6.a) Substituting Eq. (6.9) into Eq. (6.a) yields k!p k(! B ) + (k )! B = 0 (6.b) Eq. (6.7) implies B k, thus Eq. (6.b) can be decomposed into two components. One of them is in k direction. The other is in B direction. That is k(!p +! B ) = 0 (6.c) and (k )! B = 0 (6.d)

6 90 Chapter 6. Linear Waves in the MHD Plasma Eq. (6.d) implies if B 0 then k Likewise, if k 0 then B Thus, solutions of ω = 0 can be classified into the following types: If B 0, ρ = p = 0, and k = 0, then wave mode with ω = 0 can be considered as perpendicular-propagated Alfvén-mode wave. Eq. (6.c) yields B = 0 in this case. If B 0, p 0, and k = 0, then wave mode with ω = 0 can be considered as perpendicular-propagated slow-mode wave. Eq. (6.c) yields B 0 in this case. If ω = 0, ρ 0 and k 0, then Eq. (6.d) and (6.c) yield p = 0 and B This wave mode is called entropy mode. Note that for ω = 0, Eq. (6.3) is automatically fulfilled. It can be shown that solutions of nonlinear MHD equilibrium states consist of Contact Discontinuity (CD), Tangential Discontinuity (TD), Rotational Discontinuity (RD), and Shock Waves. (e.g., Kantrowitz and Petschek, 966; and Chao, 970. Or see Chapter in my lecture notes of Nonlinear Space Plasma Physics.) It can be shown that Tangential Discontinuity (TD) can be considered as a nonlinear version of perpendicularly propagated Alfvén-mode wave or slow-mode wave. Contact Discontinuity (CD) can be considered as a nonlinear version of entropy-mode wave in MHD plasma Alfvén Mode (or Intermediate Mode) 6... Phase Velocity of the Alfvén-mode Wave Alfvén mode in MHD plasma is characterized by! V =! V z = 0 but! V y 0. For V! = V! z = 0 but V! y 0, Eq. (6.4) yields ω k = C cos θ (6.8) A0 Eq. (6.8) is the wave dispersion relation of Alfvén-mode wave. Since the phase speed of Alfvén mode is in between fast-mode and slow-mode wave speed, the Alfvén mode is also

7 Chapter 6. Linear Waves in the MHD Plasma 9 called intermediate mode. It can be shown that Rotational Discontinuity (RD) can be considered as a nonlinear version of Alfvén-mode wave in MHD plasma Group Velocity of the Alfvén-mode Wave From Alfvén-mode wave dispersion relation ω = ±k C A0 cosθ, we can determine group velocity of Alfvén mode to be v g = dω ω = ˆx + ẑ ω = ±ẑc d k k x k A0 = ± ˆ C A0 z Characteristics of the Wave Fields in the Alfvén-mode Wave Exercise 6.. () Show that for Alfvén wave ρ = 0, p = 0, and B Show that B can be determined from B = B ˆ. () Determine perturbation directions of V, E, B, and J for Alfvén-mode wave. (3) Determine relationship between B and V in Alfvén-mode wave. Show that variations of B and V are in phase if π / < θ < π, but out-off phase if 0 < θ < π / Fast Mode and Mode Phase Velocity of the Fast-mode and -mode Waves V! For V! y = 0 but V! z 0 0 Eq. (6.4) yields (ω / k ) α δ det δ (ω / k ) β = ω k ω k (α + β) + αβ δ = 0 where α, β, and δ are given in Eqs. (6.5)~(6.7), which yields α + β = C A0 and αβ δ = (C A0 Thus, we have ω k ω k (C A0 + C S 0 sin θ + C S 0 cos θ = C A0 + C S 0 + C S 0 sin θ)c S 0 cos θ C 4 S 0 cos θ sin θ = C A0 C S 0 cos θ + C S 0 ) + C A0 C S 0 cos θ = 0 (6.0)

8 9 Chapter 6. Linear Waves in the MHD Plasma Eq. (6.0) has two roots of ω / k. They are the fast-mode (+) and slow-mode ( ) dispersion relation ω = (v k ph = {(C + C A0 S 0 ) ± (C A0 + C S 0 ) 4C A0 C S 0 cos θ} (6.) Fast Group Velocity of the Fast-mode and -mode Waves The group velocity of a wave is defined by v g = dω ω = ˆk d k k + ˆθ ω k θ (6.) Since, for MHD waves, the phase velocity is a wavelength-independent function. i.e., ω k = v ph (θ). (6.3) Thus, the k -component of the group velocity can be obtained by ω k = k [kv (θ)] = v ph ph (θ). (6.4) Whereas, the θ -component of the group velocity can be obtained by ω k θ = θ (ω k ) = θ [v ph(θ)] = v ph (θ) θ [v ph(θ)] (6.5) From MHD dispersion relation, we can determine phase speed of the Fast-mode and -mode waves. Namely, we have θ [v ph(θ)] Fast = ± θ (C A0 + C S0 ) 4C A0 C S0 cos θ (6.6) Substituting (6.6) into (6.5), the θ -component of the group velocity can be rewritten as ( k ω θ = ± [v ph (θ)] Fast (C A0 C A0 C S0 cosθ sinθ + C S0 ) 4C A0 C S0 cos θ (6.7) Substituting the equations (6.4) and (6.7) into equation (6.), we can obtain the group velocity of the fast-mode and slow-mode waves (v g = ˆk(v ph ± ˆθ [v ph (θ)] Fast (C A0 C A0 C S0 cosθ sinθ + C S0 ) 4C A0 C S0 cos θ (6.8) where (v ph is given in Eq. (6.). Note that the denominator of the θ -component group velocity in equation (6.7) vanishes at θ = 90 for the slow-mode wave. It also vanishes at θ = 0 or 80 when C A0 = C S0 for both slow-mode and fast-mode waves. We shall use the L'Hôpital's rule to

9 Chapter 6. Linear Waves in the MHD Plasma 93 determine the θ -component group velocity in these special cases. Namely, when the denominator in equation (6.6) vanishes, the θ -component of the group velocity can be obtained by ( k = ± ω θ [ v (θ) ph ] θ Fast = [ v (θ) ph ] Fast θ (C A0 θ (C A0C S0 cosθ sinθ) + C S0 ) 4C A0 C S0 cos θ + [v ph (θ)] Fast θ (C A0 + C S0 ) 4C A0 C S0 cos θ = ± [ v ph(θ) ] θ Fast (C A0 C A0 C S0 cosθ + C S0 ) 4C A0 C S0 cos θ ± 4[v ph (θ)] Fast [ v ph(θ) ] θ Fast where (6.5) and (6.6) have been used to obtained Eq. (6.9). Eq. (6.9) yields [ v ph (θ) θ ] Fast = C A0 C S0 cosθ 4[v ph (θ)] Fast ± (C A0 + C S0 ) 4C A0 C S0 cos θ (6.9) (6.30) Special Case A: The slow-mode θ -component group velocity at θ = 90. Equation (6.30) yields lim ( ω θ 90 ε k θ ) = lim θ 90 C A0 C S0 cosθ 4[v ph (θ)] (C A0 + C S0 ) 4C A0 C S0 C S0 cos θ = C A0 (C A0 + C S0 ) (6.3) where the minus sign and the plus sign are chosen to match the group velocity at θ 90 ε and at θ 90 + ε, respectively, with ε 0 +. For C A0 = C S0 and θ 90 ε, the slow-mode group velocity is equal to (v g ) = ˆθ lim ( ω θ 90 ε k θ ) = ˆθ C A0 (6.3) Special Case B: The slow-mode and fast-mode group velocity at C A0 = C S0 and θ = 0. Equation (6.30) yields lim ( θ 0 C A 0 =C S 0 k ω θ = ± C A0 (6.33) where the sign is chosen to match the group velocity at θ 0 + ε with ε > 0.

10 94 Chapter 6. Linear Waves in the MHD Plasma Characteristics of the Wave Fields in the Fast-mode and -mode Waves Exercise 6.3. () Determine phase relationship of ρ and B, for fast-mode and slow-mode waves. () Determine perturbation directions of V, E, B, and J for fast-mode and slow-mode waves. (3) Show that VFast! V! Answer of Exercise 6.3(3) Proof! VFast! V = 0 Eq. (6.4) yields ( V! [(ω / k α] ( V! z δ = 0 and ( V! ) [(ω / k ) α] ( V! z ) δ = 0 Substituting the above two equations into!v Fast!V, it yields!v Fast!V = ( V! ( V! ) + ( V! z ( V! z ) = ( V! ( V! ) +{( V! [(ω / k α] / δ}{( V! ) [(ω / k ) α] / δ} = (! V (! V ) {+ [(ω / k α][(ω / k ) α] / δ } = (! V (! V ) {δ + (ω / k (ω / k ) α[(ω / k + (ω / k ) ]+ α }/ δ = (! V (! V ) {δ + α + 4 [b (b 4c)] αb}/ δ = (! V (! V ) {δ + α + c αb}/ δ where b = (α + β) c = αβ δ Thus!V Fast!V = (! V (! V ) {δ + α + (αβ δ ) α(α + β)}/ δ = 0 Answer of Exercise 6.3() (6.): ( iω )!ρ = (ik)! V (6.3') :!p = γ p 0!ρ = C S0!ρ (6.) : ( iω )!V = (ik)!p +!J

11 Chapter 6. Linear Waves in the MHD Plasma 95 (6.9) : (ik)! B =! J Substituting (6.9) into (6.), it yields ( iω )! V = (ik)!p + (ik! B ) / (6.') Substituting (6.3') into k (6.') to eliminate! J, then substituting (6.) into the resulting equation to eliminate! V and substituting (6.3') into the resulting equation to eliminate!p, it yields ( iω )k! V = k (i k)!p + k [(i k! B ) ] / ( iω )!ρ = ik C S0!ρ + i k B k! 0 B B ik 0 B! (6.") where k! B It can be shown that B = B = B ( / ) Thus, the above equation (6.") can be rewritten as (ω )! ρ = k!ρ C B S0 + k 0! B!ρ = k C S0 + k C A0!B ( ω k C S0)! ρ = C A0!B (6.''') Thus, for ω / k > C S 0, variations of ρ and B are in phase. For ω / k < C S 0, variations of ρ and B are out-off phase. It can be shown that, for fast-mode wave, we have (ω / k C S 0. Thus, for fast-mode wave, variations of ρ and B are in phase. For slow-mode wave, we have (ω / k ) C S 0. Thus, for slow-mode wave, variations of ρ and B are out-off phase. Note that (6.3') yields variations of ρ and p are in phase. Substituting equation (6.3') into equation (6.''') to eliminate!ρ it yields ( ω k C S0 )!p!b = C C A0 S0 or ( ω k C S0 )!p = C A0 γ p 0!B Thus, for ω / k > C S 0, variations of p and B are in phase. For ω / k < C S 0, variations of p and B are out-off phase.

12 96 Chapter 6. Linear Waves in the MHD Plasma Friedrichs Diagrams of the Phase Velocity and Group Velocity of the MHD Waves Exercise 6.4. () Ignoring the entropy mode, plot the phase velocities of the three MHD wave modes: fast-, Alfvén-, and slow-modes, on the Friedrichs diagram, where the polar coordinate (r,θ) = (ω / k,θ k,b0 ). () Ignoring the entropy mode, plot the group velocities of the three MHD wave modes: fast-, Alfvén-, and slow-modes, on the Friedrichs diagram, where the polar coordinate (r,θ) = (v g,θ vg ). Students are encouraged to read the classical paper written by Kantrowitz and Petschek (966) for detail discussion on the MHD wave modes. The application of the group-velocity Friedrichs diagram on wave expansion near the source region can be found in the two papers by Lai and Lyu (006, 008). References Chao, J. K. (970), Interplanetary Collisionless Shock Waves, Rep. CSR TR-70-3, MIT Center for Space Research, Cambridge, Mass.. Chen, F. F. (984), Introduction to Plasma Physics and Controlled Fusion, Volume : Plasma Physics, nd edition, Plenum Press, New York. Kantrowitz, A., and H. E. Petschek (966), MHD characteristics and shock waves, in Plasma Physics in Theory and Application, edited by W. B. Kunkel, p. 48, McGraw-Hill Inc., New York. Krall, N. A., and A. W. Trivelpiece (973), Principles of Plasma Physics, McGraw-Hill Book Company, New York. Lai, S. H., and L. H. Lyu (006), Nonlinear evolution of the MHD Kelvin-Helmholtz instability in a compressible plasma, J. Geophys. Res.,, A00, doi:0.09/004ja0074. Lai, S. H., and L. H. Lyu (008), Nonlinear evolution of the jet-flow-associated Kelvin-Helmholtz instability in MHD plasmas and the formation of Mach-cone-like plane waves, J. Geophys. Res., 3, A067, doi:0.09/007ja0790. Nicholson, D. R. (983), Introduction to Plasma Theory, John Wiley & Sons, New York.