MHD waves. Elementary, entirely elementary A story about compression and vorticity

Transcription

1 MHD waves Elementary, entirely elementary A story about compression and vorticity AstroNet5 QUB August 2016 Belfast, NI Marcel GOOSSENS Centre mathematical Plasma Astrophysics, KU Leuven, Belgium IAP P7/08 Charm KU Leuven GOA/ FWO Vlaanderen

2 Focus on MHD wave theory Theory is the essence of facts. Without theory scientific knowledge would only be worthy of the mad house. Electromagnetic theory O. Heaviside. Observations: MHD waves are everywhere in the solar atmosphere!

3 Those were the days, my friend. We thought they d never end. For we were young and sure to have our way. Mary Hopkins

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10 Leuven: Fonske Sapientiae

11 Leuven: Stella Artois

12 What to expect from the forces at work Tell me and I forget. Show me and I will remember. Involve me and I will understand. Step back and I will act Chinese proverb Equation of motion ρ d v dt = p + 1 µ ( B) B. ρ = plasma density, p = plasma pressure, B = magnetic field, v = velocity. Gradient plasma pressure force p Isotropic, component along B, drives acoustic waves. Lorentz force (LF)

13 j B = 1 µ ( B) B = 1 µ ( B. ) B ( B2 2µ ) Anisotropic, no component along B First term in RHS of LF magnetic tension force 1 µ ( B. ) B = B µ d ds (B 1 t ) = d ds B 2 1 t + B2 2µ µ s = the arc length along the magnetic field line. 1 t = unit vector tangential to the magnetic field line 1 t = 1 B = B/ B Recall Frenet-Serret formulae for a space curve Use the first formula of Frenet d 1 t ds = 1 n R c. d 1 t ds. R c = the local radius of curvature, 1 n = the unit vector along the normal directed towards the local centre of curvature.

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15 Decompose as 1 µ ( B. ) B = d ds B 2 1 t + B2 2µ µ 1 n R c. ( B2 2µ ) = d ds (B2 2µ ) 1 t ( B2 2µ ) = operator in the planes normal to the magnetic field lines. j B = ( B2 2µ ) + B2 µr c 1 n. LF is confined to planes normal to the B lines. ( B2 ) = the magnetic pressure force 2µ B 2 µr c 1 n = T n = magnetic tension force Behaves as the tension force in a string.

16 Gradient plasma pressure force + gradient magnetic pressure force p ( B2 2µ ) B introduces anisotropy in magneto-acoustic waves Constructive interaction = Fast magneto-sonic waves Destructive interaction = Slow magneto-sonic waves What about magnetic tension? Analogy with elastic string and transverse waves. Transverse waves propagate along the B lines with the velocity v A B 2 1/2 µρ = B µρ = v A = Alfvén velocity. Discovered by Alfvén in 1942 and named the Alfvén wave. H. Alfvén: 1942, Nature, Vol.150, Issue 3805, pp , Existence of Electromagnetic-Hydrodynamic Waves.

17 Historical note New theory of sunspots based on Magneto-hydrodynamic waves. H. Alfvén: 1945a, M.N.R.A.S. 105, 3-16, Magneto-hydrodynamic waves and sunspots. H. Alfvén: 1945b, M.N.R.A.S. 105, , Sunspots and Magnetohydrodynamic waves II T.G. Cowling: 1946, M.N.R.A.S. 5, , Alfvén s theory of sunspots. Summary: Alfvén s theory of sunspots is reviewed critically. The theory is shown to be, at a number of points, internally inconsistent, or inconsistent with observation. It is concluded, that while certain details of the theory may have permanent value, the theory as a whole is not acceptable. Heating of solar corona by Magneto-hydrodynamic waves. H. Alfvén: 1947, M.N.R.A.S. 107, , Granulation, Magnetohydrodynamic Waves, and the Heating of the solar corona It is possible that the very high temperature found in the corona is produced through this magneto-hydrodynamic heating. Tension force in magneto-acoustic waves?

18 Linear MHD waves Linear motions superimposed on a static equilibrium state. Linearise the original MHD around this static equilibrium. Starting point: Non-linear time dependent MHD equations dρ t = ρ v, dp dt ρ d v dt B t = γp dρ ρ dt {+gain and loss terms}, = p + 1 µ ( B) B + {viscous forces, } = ( v B) + {η 2 B}.. B = 0, j = 1 µ ( B), E = v B. η = the coefficient of magnetic diffusivity; γ = the ratio of specific heats. d dt = t + v

19 Equilibrium: no change in time t = 0 Static equilibrium v = 0. Equations for a magnetostatic equilibrium p + 1 µ ( B) B = 0,. B = 0. No exact MHD equilibrium state in the real world, but? Go back to the original non-linear MHD equations Superimpose motions on the equilibrium state, displace the plasma element that is at the position r to the position r + ξ r r + ξ. ξ = Lagrangian displacement Changes in density, pressure and magnetic field. Measure changes at a fixed geometrical position (Eulerian description) or while following the motion of the plasma (Lagrangian description).

20 Denote the physical quantities in the static equilibrium as f 0 ( r) and in the time dependent plasma as f( r; t) f ( r; t) = f( r; t) f 0 ( r) = the change in the quantity f at a fixed geometrical position = Eulerian perturbation of f. f f 0 1. δf( r; t) = f( r + ξ; t) f 0 ( r) = change in f while we follow the motion of the fluid = the Lagrangian perturbation of f. In linear theory δf( r; t) = f ( r; t) + ξ. f 0 ( r). Linearise the ideal MHD equations.

21 ρ =.(ρ 0 ξ), p = ξ. p 0 γp 0. ξ, B = ( ξ B0 ), ρ 0 2 ξ t 2 = p + 1 µ ( B 0 ) B + 1 µ ( B ) B 0. RHS = p + 1 µ ( B 0 ) B + 1 µ ( B ) B 0 = F ( ξ) F ( ξ) = force operator. Write the linearised equation of motion as ρ 0 2 ξ t 2 = F ( ξ) 3 unknown scalar functions in linear ideal MHD:? 8 unknown scalar functions in non-linear ideal MHD:?

22 Linear MHD waves seen as differential equations in time Constant coefficients. Possible solutions of the form f( r; t) = f( r) exp( iωt) f( r) is the time independent part of the solution and f is any of the perturbed quantities. Normal modes of the system ω 2 ρ 0 ξ( r) = F ( ξ( r)) ω 2 ξ( r) = L( ξ( r)) L( ξ( r)) = 1 ρ 0 F ( ξ( r)). L and F = linear and time independent operators. The eigenvalues of the operator L = the squares of the eigenfrequencies of the plasma system and the corresponding eigenvectors = the eigenmodes.

23 Linear MHD waves of a uniform plasma of infinite extent The most simple plasma equilibrium: a uniform plasma of infinite extent. The MHD waves can be put in separate boxes! Recall the time dependence exp( iωt). Constant magnetic field B = (0, 0, B z ), p 0 = constant and ρ 0 = constant. The coefficients of the partial differential equations are constants. Solutions in the form of plane waves. f( r) = ˆf exp(i k. r) = ˆf exp(i(k x x + k y y + k z z)) Combine the temporal and spatial factors f( r; t) = f( r) exp( iωt) = ˆf exp(i( k. r ωt)) = ˆf exp(i(k x x + k y y + k z z ωt)) ˆf= constant amplitude of f, k = k x 1 x + k y 1 y + k z 1 z = the wave vector, ω = the frequency of the wave.

24 / t and only act on exp(i( k. r ωt)) Equilibriun plasma is uniform t iω i k. p 0 = 0, ρ 0 = 0, B 0 = 0,. B 0 = 0 Simplified equations ρ = iρ 0 ( k. ξ), p = iρ 0 v 2 S( k. ξ), B = i k ( ξ B0 ) = i ξ( k. B0 ) i B 0 ( k. ξ), ρ 0 ω 2 ξ = i kp + i µ ( k B ) B 0 = ρ 0 v 2 S k ( k. ξ) i k B 0. B µ + i k. B 0 µ B

25 B 0 = 0 Acoustic waves The equation of motion ρ 0 ω 2 ξ = ρ0 v 2 S k ( k. ξ). Take the scalar product with k, ρ 0 (ω 2 k 2 v 2 S)Y = 0, Y = k. ξ Recall. ξ = i k. ξ = iy, Y = compression A non-trivial solution with Y 0 exists only for ω 2 = k 2 v 2 S Acoustic waves. Completely isotropic waves driven by the gradient plasma pressure force. Phase velocity v ph and group velocity v gr are equal v ph = σ k 2 k = v S 1 k, v gr = k σ = v S 1 k.

26 Alfvén waves Recall the equation of motion and the induction equation: Compute the LF ( k B 0 ) 2 ξ + ( k. B 0 ) µ µ ( ξ. B 0 ) k B2 0 µ ( k. ξ) k + ( k. B 0 ) µ ( k. ξ) B 0 Rewrite the equation of motion ρ 0 ω 2 ξ = ρ0 v 2 S k( k. ξ) ( k B 0 ) 2 ξ + ( k. B 0 ) µ µ ( ξ. B 0 ) k B2 0 µ ( k. ξ) k + ( k. B 0 ) µ ( k. ξ) B 0 Two options. Option 1: Focus on displacement perpendicular to the magnetic field lines that does not cause any compression. k. ξ = 0, B0. ξ = 0. No gradient plasma pressure force; no gradient magnetic pressure force; the Lorentz force = the magnetic tension force

27 ( k. B 0 ) 2 ξ. µ The equation of motion ρ 0 ω 2 ( k. B 0 ) 2 µρ 0 ξ = 0 A non-trivial solution with ξ 0 exists if and only if = ωa 2 ωa 2 = ( k. B 0 ) 2 = k 2 B0 2 = k µρ 0 µρ v 2 A 2 = kzv 2 A 2 = k 2 va 2 cos 2 θ, va 2 = B2 0 0 µρ 0 ω 2 ω A = the Alfvén frequency, v A = the Alfvén velocity, k = the component of the wave vector k parallel to B 0, θ is the angle between B 0 Classic Alfvén waves. Purely magnetic waves driven solely by magnetic tension T n = ρ 0 ωa 2 ξ = ( k. B 0 ) 2 ξ. µ

28 No variations in density and plasma pressure: ρ = 0 p = 0. The displacement is transversal to B 0 ( B 0. ξ = 0) and to k ( k. ξ = 0). The eigenvalue ω 2 = ω 2 A = k 2 zv 2 A is (infinitely) degenerate. It is independent of k x, k y. and only depends on k = k z v ph = ω A k 2 k = v A cos θ 1 k, v gr = k ω A = v A 1 z v ph changes in magnitude from v A for parallel propagation to 0 for perpendicular propagation. v gr is always oriented along B 0 and in magnitude equal to v A. Extreme anisotropy of the Alfvén wave. Option 2 { k [Equation of motion] } 1 B ρ 0 (ω 2 ω 2 A)Z = 0, Z = ( k ξ) 1 B ξ 1 z = i k ξ 1 z = i Z = component of vorticity parallel to B 0 The Alfvén waves transport vorticity.

29 General case Choice of wave variables ξ z = component parallel to B 0 ξ = i k ξ = i Y = compression ξ 1 z = i k ξ 1 z = i Z = component of vorticity parallel to B 0 Y = k ξ, ξ z, Z = ( k ξ) 1 z ω 2 ξ z k z vs 2 Y = 0, k 2 vak 2 z ξ z + (ω 2 k 2 (va 2 + vs 2 ))Y = 0, (ω 2 ωa)z 2 = 0. Two uncoupled subsets of equations.

30 Two types of solutions. Classic Alfvén waves Y = 0, Z 0, ξ z = 0. ω 2 = ω 2 A = ( k B) 2 µρ = k 2 z v 2 A, v 2 A = B2 µ ρ ω A = local Alfvén frequency. Vorticity: AW are the only waves that propagate vorticity in a uniform plasma of infinite extent. No compression, no parallel displacement. Restoring force = Magnetic tension force! General k = (k x, k y, k z ) Popular choice: take k y = 0 so that ξ = ξ y 1 y. Popular, but 1 of many choices. Often this is not what we want! k z R << 1 = m k y >> k z. Nearly perpendicular propagation!

31 Magneto-sonic slow and fast waves Y 0, Z = 0, ξ z 0 Compression, parallel displacement. No vorticity. Solutions ω 2 = ω 2 sl,f = k2 (v 2 S + v 2 A) 2 1 ± 1 4ω 2 C k 2 (v 2 S + v 2 A) 1/2 ω C = the cusp frequency, v C = the cusp velocity ω 2 C = v 2 S vs 2 + va 2 ωa 2 = k v 2 C 2 = kzv 2 C, 2 vc 2 = v2 Sv 2 A vs 2 + va 2. sl (slow) = the minus sign, f (fast) = the plus sign. Cut off frequencies ω I and ω II : 0 ωsl 2 ωi 2 = min(kzv 2 S, 2 kzv 2 A), 2 ωii 2 = max(k 2 vs, 2 k 2 va) 2 ωf 2 Four characteristic frequencies ωc, 2 ωi, 2 ωa, 2 ωii, 2

32 Division of the spectrum into the slow subspectrum, the Alfvén subspectrum and the fast subspectrum ω 2 C ω 2 sl ω 2 I }{{} slow ω 2 A ω 2 II ω 2 f }{{} fast Fast / slow magnetosonic waves. Driven by tension and pressure forces. Density and pressure variations. Plasma pressure and magnetic pressure variations are in phase / antiphase. Flow of energy is fairly isotropic / highly an-isotropic. Non-uniform plasma: ω C, ω I, ω A, ω II define four intervals of frequencies Non-uniform plasmas: MHD waves with mixed properties and resonant absorption. Vorticity 0 & Vorticity = 0 & compression = 0 : Alfvén waves compression 0: Magneto-acoustic waves

33 Phase velocities 1/2 2 2 ( c + V A ) f s A c V A x

34 Group velocities. z 1/2 2 2 ( c + V A ) f s V C c V A x

35 MHD waves on 1-D cylindrical plasma columns Equilibrium model Flux tube in static equilibrium. Straight cylindrical plasma column of radius R. Confine inhomogeneity to interval [b = R l/2 r a = R + l/2] l = 2R corresponds to fully inhomogeneous loop. z l ρ i ρ e L R B B ϕ r

36 Cylindrical coordinates r, ϕ, z. i refers to 0 < r R, e refers to r R Equilibrium quantities B = (0, B ϕ (r), B z (r)), p(r) and ρ(r). Radial force balance equation d B2 (p + dr 2µ ) = B2 ϕ µr, B2 = (Bϕ 2 + Bz) 2 Temporal and spatial dependence of MHD waves exp( iωt) Eigenvalue problem of linear motions: ω =? / steady state of driven problem: ω prescribed. exp(i(mϕ + k z z)) m, k z = azimuthal and axial wave numbers. k = (0, m/r, k z ), m = integer number of nodes in radial eigenfunctions n R Eigenmodes: dispersion relation DR(ω, m, k z, n R ) = 0. Graphical representation? 4-D configuration? exp(i(mϕ + k z z ωt)) standing waves / axially propagating waves / spinning waves.

37 MHD Waves on flux tubes with piece-wise constant density. l = 0 0 r R : ρ 0 = ρ i = constant, R r < : ρ 0 = ρ e = constant Straight field, B ϕ = 0, B z = constant ρ e < ρ i, B z = constant = ωai 2 = kzv 2 Ai 2 < ωae 2 = kzv 2 Ae 2 ξ = Lagrangian displacement, ξ ϕ, ξ z components in magnetic surfaces perpendicular/parallel to the magnetic field lines. P = P = p + B B /µ = Eulerian perturbation of total pressure;. p = Eulerian perturbation of plasma pressure. Bessel functions. Wentzel 1979, Spruit 1982, Edwin and Roberts 1983 ER 1983: 2 uncoupled equations for = ξ, Γ = ( v) 1 z. Equation for is solved. Apparently all solutions with 0 have Γ = 0.

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39 There is life outside the box, see Spruit Vorticity is killed by the assumption of constant density. Example: Fundamental radial kink (m = 1) mode Fig 1b R/L = 0.01, 0.02, 0.1, R/L = 0.1 = (2R)/(L/2) = 0.4 ξ ϕ,i (R) = ξ ϕ,+ (R)

40 Vorticity is concentrated at the boundary ( ξ) 1 z = 2 ξ ϕ,i(r) R δ(x 1), x = r/r Same result as for surface Alfvén waves at a Cartesian discontinuity! Fundamental radial mode of kink wave behaves as surface Alfvén(ic) waves! Goossens et al. (GASVAT) 2012 Thin tube approximation k z R << 1 ω 2 = ω 2 k = ρ i ω 2 Ai + ρ e ω 2 Ae ρ i + ρ e. ω A,i < ω k < ω A,e Wentzel 1979 : P. Equation for P is solved for different cases. Case Ai corresponds to Ionson 1978 heating by surface Alfvén waves in Cartesian geometry.

41 Spruit 1982: = ξ. Equation for 0 is solved In addition to the undamped waves HS also determines MHD waves damped by MHD radiation; life outside the box MHD waves damped by MHD radition / Leaky MHD waves. Not widely known in solar physics community: Spruit 1982, Cally 1985, GH1993, Stenuit et al. 1998, 1999, Goossens et al (GTAAB) 2009, Vasheghani et al Spruit 1982 Solutions for = 0 Alfvén waves. See flow lines for m = 0 and m = 1 One Alfvén frequency only possible for equilibrium with constant density! Goossens et al. (GER) 2011, Goossens et al Equilibrium with non-constant Alfvén velocity (density): frequencies. continuum of

42 Vortex motions in the solar atmosphere, Giagkiozis et al. 2016

43 MHD waves with mixed properties General case B 0 = B z 1 z + B ϕ 1 ϕ General case l 0. ξ = Lagrangian displacement, ξ, ξ components in magnetic surfaces perpendicular/parallel to the magnetic field lines. ξ = (ξ ϕ B z ξ z B ϕ )/B, ξ = ξ B/B P = Eulerian perturbation of total pressure. ξ r = radial component of Lagrangian displacement. Hain and Lust 1958, Appert et al. 1974, SGH1991, GHS1992, GRH1995 ω A = local Alfvén frequency, ω C = local cusp frequency: ωa 2 = ( k B) 2, ωc 2 = v2 S µρ vs 2 + va 2 ωa, 2 vs 2 = γp ρ, v2 A = B2 µ ρ

44 D d(rξ r) dr D dp dr = C 1 rξ r C 2 rp, = C 3 ξ r C 1 P, ρ(ω 2 ω 2 A)ξ = i B C A, C A = g B P 2f B B ϕ B z µr ξ r, ρ(ω 2 ω 2 C)ξ = if B B v 2 S vs 2 + va 2 C S, ρω 2 ξ = if B B δp, C S = P 2B2 ϕ µr ξ r, ξ = ( ξ) 1 B = ω 2 C S ρ(v 2 S + v 2 A)(ω 2 ω 2 C), i d { ρ0 (ω 2 ωa) 2 } 2 dr { ρ0 (ω 2 ω 2 A) } C A B + M ϕ ρ 0 (ω 2 ω 2 A). δp = Lagrangian variation of plasma pressure. δp = p + dp dr ξ r (1) B ϕ = 0 = M ϕ = 0

45 D = ρ(v 2 S + v 2 A) (ω 2 ω 2 A) }{{} (ω2 ω 2 C), C 1, C 2, C 3 see e.g Appert et al. 1974, SGH1991a, GHS1992, GRH1995 Equations for ξ r, P, ξ, ξ are coupled: C A, C S C A = g B P 2f BB ϕ B z ξ r, C S = P 2B2 ϕ ξ r µr µr SGH1991a, GSH1992, GRH1995, TG1996 f B = k B = k z B z + mb ϕ r, g B = mb z r k z B ϕ ξ : Alfvén waves ξ r : Fast waves ξ : Slow waves No pure magneto-sonic waves and no pure Alfvén waves. have mixed properties. MHD waves MHD waves have mixed properties and have different appearances in different parts of the plasma because of the inhomogeneity of the plasma (GAD2002, G2008, GTAAB2009, GER2011, GASVAT2012)

46 C A 0, C S 0, d dr { ρ0 (ω 2 ω 2 A) } 0 Vorticity = ( ξ) 1 B 0 & compression = ξ 0. One exception: straight field ( B ϕ = 0 ) and m = 0 Straight field B 0 = B(r) 1 z, B ϕ = 0; ξ = ξ ϕ, ξ = ξ z C A = mb z r P, C S = P B ϕ = 0 and m = 0 : C A = 0, but C S 0 Equation 3 is decoupled ρ(ω 2 ω 2 A(r))ξ ϕ = 0 Pure Alfvén waves for m = 0 in a non-uniform cylindrical plasma with a straight field. Each magnetic surface oscillates with its own local Alfvén frequency The axi-symmetric MHD waves are decoupled in axi-symmetric Alfvén waves and sausage magneto-sonic waves.

47 Simplify equations 2-3, and 5-6 for a straight field. dp dr = ρ(ω 2 ω 2 A)ξ r, ρ(ω 2 ω 2 A)ξ ϕ = im r P, ξ = ω 2 P ρ(v 2 S + v 2 A)(ω 2 ω 2 C), ( ξ) 1 z = i(m/r)p d { ρ0 (ω 2 ωa) 2 } 2 dr { ρ0 (ω 2 ω 2 A) }. Equation of motion in horizontal planes (ideal MHD, straight field). ρ ω 2 ξh = P ρ ω 2 A ξ h ξ h = (ξ r, ξ ϕ, 0), = ( d dr, i im r, 0) T h = ρ ωa 2 ξ h, P = ρ (ω 2 ωa) 2 ξ h, Λ(ω 2 ) = ω2 ωa 2 ωa 2.

48 Λ(ω 2 ) measures the relative importance of the pressure force to the magnetic tension force. Non-uniform plasma: v S (r), v 2 A(r), ω 2 A(r), and ω 2 C(r) depend on position. Λ(ω 2 ), ξ, ( ξ) 1 z depend on position. The nature of the MHD wave changes according to the properties of the plasma it travels through. Uniform plasma Λ = constant and the nature of the wave does not change as it always sees the same environment. Λ(ω 2 ) < 1: Magnetic tension force dominates over the gradient pressure force, wave is mostly Alfvénic. This is why I love elementary school so much. The kids really believe everything you tell them. Principal Seymour Skinner to Mrs. Edna Crabapple The Simpsons.

49 Resonant Alfvén waves in 1-D cylindrical equilibrium models Singularity is almost invariably a clue. Sherlock Holmes The Boscombe Valley Mystery Sir Arthur Cannon Doyle ω 2 A(r) and ω 2 C(r) vary with position r D = ρ(v 2 S + v 2 A) (ω 2 ω 2 A) }{{} (ω2 ω 2 C) ω [min ω A (r), max ω A (r) ] = D = 0 at position r A where ω = ω A (r A ) Equations have regular singular points at the positions r = r A. Continuous spectrum of resonant Alfvén (slow) waves [min ω A (r), max ω A (r) ] [min ω C (r), max ω C (r) ] Appert et al. 1974; Chen and Hasegawa, 1974; Goedbloed, Dispersion relation for Alfvén waves is locally satisfied on each magnetic surface. Frobenius-Fuchs solutions around singular point r = r A where ω = ω A (r A ). Fundamental conservation law for resonant Alfvén waves C A (s) g B P 2f BB ϕ B z µr A ξ r = constant

50 SGH1991, GRH1995, TG1996 s = r r A. Valid in the interval [ s A, s A ] Fundamental conservation law for resonant slow waves C S (s) P 2B2 ϕ µr ξ r = constant Resonant Alfvén / slow waves are controlled by coupling functions C A /C S. C A 0 & ω AC = RA in Alfvén continuum C S 0 & ω SC = RA in slow continuum Popular misconception 1: No RA in AC for axisymmetric (m = 0) waves. Popular misconception 2: A variation in density is necessary for AC. Twisted field B ϕ 0, m = 0 : GP1992, Giagkozis et al What happens at r = r A where ω = ω A (r A ). Look back at the equations for ξ (= ξ ϕ ), ( ξ) 1 B, Λ r r A ω 2 ωa(r) 2 s 0, = d dr (ω2 ωa). 2 Remember C A constant

51 ξ 1/s ±, ( ξ) 1 B 1/s 2, Λ s 0 Infinite spatial solutions for ξ, ( ξ) 1 B. The gradient pressure force is zero. At r = r A the MHD wave is an Alfvén wave. The dominant dynamics is in the perpendicular motions ξ ( in the magnetic surfaces perpendicular to the magnetic field lines). The confinement of the resonant Alfvén wave is not absolute. The AW is linked to the outside world by the component normal to the magnetic surfaces and pressure variations (C A 0). Resonant Alfvén waves in dissipative MHD Infinite ideal solutions are replaced by large but finite solutions The Alfvenic nature in the vicinity of the resonant point is confirmed. The perpendicular motions dominate the radial and axial motions by far. ξ : strong counterstreaming shear flows = KH-instabilities e.g. Terradas et al. 2008, Antolin et al. 2015

52 Mixed properties in space and time Example: Fundamental radial mode of kink wave in non-uniform equilibrium Recall results for the fundamental radial mode of kink wave in a flux tube with a piece-wise constant density. Spatial eigenfunctions in the TT approximation ξ r,i (r) R = C, ξ ϕ,i (r) R = i C, P i(r) (B 2 /µ) = C (k zr) 2 }{{} ρ i ρ e ρ i + ρ e r R, ξ i = C (k z R) 2 }{{} ρ i ρ e ρ i + ρ e r R, ξ ϕ,e (R > ) = ξ ϕ,i (R < ) Eigenfrequency in the TT approximation ω 2 = ω 2 k = ρ i ω 2 Ai + ρ e ω 2 Ae ρ i + ρ e.

53 Non-uniform flux tube: replace the jump in ω A by a continuous variation of ω A. The local Alfvén frequency ω A (r) is function of position r and defines the Alfvén continuum.

54 Frequency of kink MHD wave is in the Alfvén continuum. C A 0 & ω k AC Mixed properties: ξ 0, ( ξ) 1 z 0 Mixed properties : Λ(ω 2 ), ξ, ( ξ) 1 z vary with position. Fundamental radial mode of kink waves undergoes resonant damping. GHS1992, GH1993, RR2002, GAA2002, GTAAB2009 The discontinuous behavior of ξ ϕ is replaced by singular behavior in ideal MHD and by large but finite values in non-ideal MHD. I ve done everything the Bible says. other stuff. What more can I do? Ned Flanders. The Simpsons Even the stuff that contradicts the

55 Transverse and rotational motions. Kink modes owe their popularity to the fact that they are transverse. Where is the transverse motion? First attempt: piece wise uniform density and thin tube approximation. Introduce azimuthal dependency. 0 r R : Dependency on z : exp(ik z z) k z = nπ/l, n = 1 : sin(πz/l) ξ h (r, ϕ) R = C (cos ϕ 1 r sin ϕ 1 ϕ ) Fundamental longitudinal mode n = 1 : Jump in plane z = z 0 and define a system of Cartesian coordinates x = r cos ϕ, y = r sin ϕ 1 r = 1 x cos ϕ + 1 y sin ϕ, 1 ϕ = 1 x sin ϕ + 1 y cos ϕ From polar to Cartesian coordinates ξ = ξ x ξ y = C 1 0 = ξ T R

56 Uniform motion of the entire internal plasma along the x-axis. How did we get the uniform motion along the x-axis ξ r and ξ ϕ have equal amplitudes and ξ ϕ is a quarter of period ahead of ξ r. Necessary conditions: uniform, thin tube, piece-wise constant density. 0 r R. What happens outside tube r R? Remember strong rotational motions in non-uniform tube. ξ r and ξ ϕ have unequal amplitudes. ξ h (r, ϕ) 0 r R : = C T R cos ϕ 1 r C AZ sin ϕ 1 ϕ, C T R C AZ R Again from polar to Cartesian coordinates ξ = ξ x ξ y = C T R C ROT sin ϕ sin ϕ cos ϕ C ROT = C AZ C T R

57 ξ x = constant part + variable part ϕ. ξ y = variable part ϕ. Motion = uniform translation + rotational motion ξ = C T R 1 x C ROT sin ϕ 1 ϕ Rotational motion contains x and y components. These components depend on the angle ϕ. Doppler displacements due to kink motions. Non-uniform plasma Close to resonant layer r A where ω = ω A (r A ) C AZ >> C T R Motion is a non-axisymmetric m = 1 azimuthal motion Goossens et al. (GSTVV) 2014.

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60 He was astonished that it had been so easy. All you had to do was tell people what they wanted to hear and they would believe you no matter how implausible your story might be. Wilt. T. Sharpe