SOLAR MHD Lecture 2 Plan

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1 SOLAR MHD Lecture Plan Magnetostatic Equilibrium ü Structure of Magnetic Flux Tubes ü Force-free fields Waves in a homogenous magnetized medium ü Linearized wave equation ü Alfvén wave ü Magnetoacoustic waves ü Phase diagrams ü Poynting flux in MHD Waves in an inhomogenous magnetized medium

2 MHD Equations ρ dρ + (ρv) = 0 or + ρ v = 0 (Continuity Equation) t dt ρ dv dt = p + 1 4π ( B) B + ρg + ρν v (Momentum equation) d% p ( ' * = 0 (Energy equation for an adiabatic fluid), d& ρ γ ) (where d/dt = / t + v ). B = (v B) +η B, t In the limit of infinite conductivity (η=0), B/ t = (v B) (induction equation) = B v (v )B + (B )v (since B = 0)

3 Magnetostatic Equilibrium Structure of a cylindrically symmetric flux tube Consider a cylindrically symmetric magnetic field of the form, using cylindrical polars (R,φ,z) : B = [0, B φ (R), B z (R)] (3.1) Substituting the above in the magnetostatic equation, p + 1 4π ( B) B = 0. (3.) (assuming that gravity can be neglected), the equilibrium equation becomes: dp dr + d dr! B φ + B $ z # " 8π & + B φ = 0, (3.3) % 4πR where the second term represents the magnetic pressure and the third term the magnetic tension.

4 The equation of a field line can be determined simply from the equation Rdφ = dz. (3.4) Bφ Bz Consider a section of a flux tube of length L. Thus the twist over L is: Φ(R) = dφ = L 0 Bφ RBz dz = LBφ(R) RBz(R). (3.5) The term 4πL/Φ is referred to as the pitch P of the field and gives the length over which the field lines circle the axis once [since from Eq. (3.4), P = πrb z /B ϕ = 4πL/Φ]. Vertical Field (B ϕ =0) When the azimuthal component of the field is zero, Eq. (3.3) becomes d B (p + dr 8π ) = 0, which has the solution p+b /8π = constant i.e. total pressure is constant in the radial direction.

5 Azimuthal field (B z =0) When the vertical component of the field is zero, Eq. (3.3) becomes dp dr + d dr! B $ φ # " 8π & + B φ % 4πR = 0, (3.6) From Maxwell s relation, B ϕ is related to the current density through: jz = 1 4πR d (RBφ). (3.7) dr Let the total current I flow within a cylinder of radius a. Integrating Eq. (3.7), gives: B ϕ = IR/a, R < a and B ϕ = I/R, R > a. (3.8) The corresponding pressure is obtained by integrating Eq. (3.6): " I % p = p + π $ ' # πa & (a R ), R < a and p = p, R > a. Within the cylinder of radius a and B ϕ increase linearly with R, while the gas pressure p decreases.

6 Force-free fields (p=0) When p=0, Eq. (3.3) yields: d! B φ + B $ z dr # " 8π & + B φ % 4πR = 0, (3.9) Let us write: B = f (R), (3.10) Then Eq. (3.9), yields: B ϕ = -½ R df/dr, (3.11) and B z = B B ϕ. (3.1) Uniform twist or constant pitch force free field(gold and Hoyle 1957) Substituting B ϕ /B z = ΦR/L=πR/P [from Eq. (3.5)] in Eq. (3.9), we have d dr which can be integrated to yield Bz =! $ #[1+ (πr P ) ]Bz& = 0, (3.13) " % B0 1+ (πr / P ), πr Bφ = P B0 1+ (πr / P ), (3.14) where B 0 is the field (purely vertical) on the axis and P = 4πL/Φ is the pitch of the field that is assumed to be constant. Such a configuration was first proposed by Gold and Hoyle (1958) in connection with a flare model.

7 Effect of twisting a tube (Parker 1977) Consider the effect of twisting a force-free flux tube of radius a, that is confined by an external plasma pressure p e =B (a)/8π (follows from the condition of pressure balance R=a). During twisting, assume that the field remains cylindrically symmetric, but the radius varies from its initial value a (0) to a. Then the mean-square axial field <B z > is constant, since: < B z >= RB a z dr a 0 = a" Rf + 1 df % $ a # R ' dr from Eq. (3.1) 0 dr & = a d a dr (R f ) dr 0 = f (a) = B (a) using Eq. (3.10) Consider the uniform twist case, given by Eq. (3.14), with B 0 =B (0) (1+Φ a /4L ) ½, which varies with the twist Φ in such a way as to keep the total field equal to a constant B (0) at the tube edge. B (0) is the initial uniform axial field strength when the tube is untwisted. From flux conservation: π RBzdR = π[a (0) ] B (0) or (1+ Φ a / 4L ) 1/ log(1+ Φ a / 4L ) =[a (0) Φ] / 4L 0 a It transpires from the above equation, that as the twist Φ increases, so do B ϕ and a increase. At the same time B z (0) increases (since an outwards magnetic pressure is required to balance the inwards tension force produced by twisting), while B z (a) decreases. The mean longitudinal field is reduced, rather than enhanced by twisting the tube because <B z > = <B z > only if B z is uniform - any inhomogeneity causes <B z > < <B z >. But <B z > does not increase so the mean field is reduced.

8 Linear Force-Free Fields For a force-free field, the force balance equation reduces to j B = 0. (3.15) Since j curl B, Eq. (3.15) is satisfied for curl B = αb, (3.16) where α is in general a function of position. Taking the divergence of Eq. (3.16), we find: (B. )α = 0, (3.17) which implies that B lies on a surface of constant α (as well as j). When α is constant, taking the curl of Eq. (3.16), we have: ( +α )B = 0, (3.18) which holds for linear force-free magnetic fields.

9 Example of a constant α force-free field Consider a field of the form B = [0, B y (x),b z (x)], for which Eq. (3.15) i.e. j B = 0 becomes which integrates to: The solution therefore is: d dx (B +B y z ) = 0 B y +B z = B 0 (say) is constant. B = [0, B y, (B 0 - B y ) ½ ]. For the particular case of constant α, the z component of Eq. (3.16) (i.e. curl B = αb), is: dby dx = α(b 0 B y ) 1/, so that with the origin to be a zero of B y, the solution is: B y =B 0 sin αx, B z =B 0 cos αx (3.19)

10 MHD WAVES IN A HOMOGENEOUS MEDIUM Linearized Wave Equation Consider a uniform vertical magnetic field in a homogeneous medium in which the fluid is initially at rest everywhere (i.e. v = 0). The magnetic field vector defines a preferred direction, but the two directions perpendicular to B 0 are equivalent, so the wave problem is effectively two-dimensional. In Cartesian coordinates (x,y,z), take the initial magnetic field B 0 along the z-axis, of the form: where B 0 is constant. B 0 = [0, 0, B 0 ] (4.1) Consider small departures from the initial equilibrium state of the form: ρ = ρ 0 + δρ, p = p 0 + δp, v = δv 1, B = B 0 + δb, where δρ is a perturbation in the density and so on. Substituting these in the equations of motion and induction, we find: δv ρ 0 = δp + 1 t 4π ( δb) B. 0 (4.) δb = (δv B 0 ). t (4.3)

11 The continuity equation becomes δρ + ρ t 0 δv = 0, (4.4) where we have assumed that ρ 0 is constant. We can relate δρ and δp, from the adiabatic energy equation as follows: δp t γp 0 ρ 0 δρ t = 0, or δp t c S δρ t = 0, (4.5) where γ is the ratio of specific heats and c S is the sound speed defined as Taking the time derivative of Eq. (4.), we have c S = γp 0 ρ 0, (4.6) δv t = 1 % δp ( ' * + 1 ρ 0 & t ) 4πρ 0 % ' & δb t ( * B 0 ) = c S ( δv) +, - { (δv B 0 )}. / B 0, (4.7) 4πρ 0 where we have used Eq. (4.5) to eliminate δp, Eq. (4.4) to eliminate δρ Eq. (4.3) to eliminate δb.

12 We now look for plane wave solutions of the form: δv = v 1 e i (k.r ωt ), and similarly for the other perturbed variables, where k is the wavenumber vector and ω is the angular frequency of the wave. We can now replace / t in Eq. (4.7) by iω and by ik. Eq. (4.7) now becomes: ω v 1 = c S k(k v 1 ) + {k [k (v 1 B 0 )]} B 0 4πρ 0. (4.8) Sound Waves When B 0 = 0, so that the only force acting on the fluid is pressure, Eq. (4.8), becomes: ω v 1 = c s k(k.v 1 ) We take the scalar product of Eq. (4.9) with k and assuming that k.v 1 does not vanish, we obtain, ω =k c s This is the dispersion relation for acoustic waves, which propagate isotropically with a phase speed c s.

13 Magnetic Waves When the magnetic field is sufficiently large (i.e. β<<1), so that the gas pressure can be neglected, Eq. (4.8) becomes: ω v 1 = {k [k (v 1 ˆB 0 )]} v A ˆB0, (4.9) where v A = B 0 / 4πρ 0 is known as the Alfvén speed. Let e z be a unit vector in the z-direction i.e. directed along the field. Expanding the vector product in Eq. (4.9), yields: ω v 1 / v A = (k e z ) v 1 (k v 1 )(k e z )e z + " #(k v 1 ) (k e z )(e z v 1 ) $ % k, or in terms of the angle θ between the wave vector k and B 0, ω v 1 / v A = (k cos θ)v 1 (k v 1 )k cosθ e z + " #(k v 1 ) k cosθ(e z v 1 ) $ % k. (4.10)

14 From the condition div B = 0, we have k.b 1 =0, which means that the magnetic field perturbation is normal to the direction of propagation. Taking the scalar production of Eq. (4.10), with e z, we find: e z.v 1 = 0, (4.11) from which it follows that the perturbed velocity is normal to the background field B 0. Now if we take the scalar product of Eq. (4.10) with k, we get: Properties of Alfvén Waves Shear Alfvén Waves (ω k v A )(k.v 1 )=0, (4.1) If the perturbation is incompressible i.e. div v 1 = 0, so that Substituting Eq. (4.13) in Eq. (4.10) yields, k.v 1 =0. (4.13) ω = kv A cos θ, (4.14) which is the dispersion relation for Alfvén waves (sometimes called shear Alfvén waves) propagating in the positive z-direction (i.e. in the direction of the magnetic field). These waves travel with a phase speed v A cos θ. Eq. (4.14) can also be written as ω = k z v A. Differentiating with respect to k z, gives the group velocity as v g = v A e z.

15 A polar diagram for Alfvén waves (solid curve) and compressional waves (dashed curve). The length of the radius vector at an angle of inclination θ to B 0 is equal to the phase speed. The directions of v 1 and B 1 relative to B 0 and the k. For (a) the vectors v 1 and B 1 are normal to the plane of k whereas for (b) v 1 and B 1 lie in the same plane as k and B 0. The variation of the phase speed with θ can be seen in a polar diagram, which takes the form of two circles of diameter v A. The waves propagate fastest along the field, but not normal to it. The direction of wave-energy is along the magnetic field. Energy propagates along field lines, independent of direction, despite the fact that individual waves can travel at any inclination to the field (other than 90 ). From Eq. (4.13), we find that Alfvén waves are transverse the velocity perturbation is perpendicular to the propagation direction. We also find from Eqs. (4.4)-(4.5), that there are no density or pressure changes associated with these waves.

16 From the induction equation, for plane waves, we find: or -ωb 1 = k (v 1 B 0 ) -ωb 1 = (k.b 0 ) v 1 - (k.v 1 ) B 0. (4.15) Since k.v 1 =0 (from Eq. 4.11), Eqs. (4.14) & (4.15), give: v 1 = - B 1 /(4πρ 0 ) ½, (4.16) which implies that B 1 and v 1 are in the same direction, both lying in a plane parallel to the wave front. Using Eq. (4.11), i.e. e z.v 1 = 0 we find: B 0. B 1 =0, (4.17) so that the magnetic field perturbations are normal to the field. From the Lorentz force, j 1 B 0 /c = (k B 1 ) B 0 /4π, we find: = (k.b 0 ) B 1 /4π - k (B 0. B 1 )/4π. (4.18) The first term comes from the magnetic tension and second from the magnetic pressure. Using Eq. (4.17), we find that the driving force for Alfvén waves comes solely from the magnetic tension. From Eq. (4.16), we find: B 1 8π = 1 ρ v, (4.19) 0 1 which shows that for Alfvén waves, there is equipartition between magnetic and kinetic energy.

17 Torsional Alfvén Waves In cylindrical geometry with an axial field B 0 e z, there exist waves which possess only an azimuthal component in the field perturbation. Let at time t = 0, a circular disk perpendicular to the (initially uniform) magnetic field be put into rotation at a constant angular velocity Ω. A wave front moves up at the Alfvén speed along the rotating field bundle. Compressional Alfvén Waves The second solution of Eq. (4.1) is ω =k v A which represents compressional Alfvén waves. The phase speed is v A, regardless of the propagation angle. The group velocity is v A k, so that energy is propagated isotropically. Eqs. (4.10) and (4.11) imply that v 1 lies in the (k,b 0 ) plane and normal to B 0. It possesses components both along and transverse to k. From Eq. (4.15), B 1 is in the plane of v 1 and B 0, but is normal to k. For θ=π/, Eq. (4.10) shows that v 1 is parallel to k and hence the wave is longitudinal and from Eq. (4.18) only the magnetic pressure is the driver for the wave. For θ=0, the wave is transverse and identical to an ordinary Alfvén wave; it is driven solely by the magnetic tension and produces no compression (despite its name).

18 Magnetoacoustic Waves The generalization of Eq. (4.10) when pressure and magnetic forces are important is: ω v 1 / v A = k cos θ v 1 (k v 1 ) k cosθ e z + + " #(1+ c S / v A )(k v 1 ) k cosθ(e z v 1 ) $ % k. Since v 1 appears in the combinations k.v 1 and e z.v 1,let us take the scalar products with k and e z to get: and ( ω + k c S + k v A )(k v 1 ) = k 3 v A cosθ(e z v 1 ) k cosθ c S (k v 1 ) = ω (e z v 1 ). (4.19) If k.v 1 vanishes, one recovers the Alfvén wave solution. Otherwise, from the two equations above, the condition for a non-trivial solution yields the following dispersion relation: ω 4 ω k (c S +v A ) + c S v A k 4 cos θ = 0, (4.0) For outward propagating waves (ω/k >0), there are two distinct solutions: ω k = " 1 (c +v S A ) ± 1 c % $ 4 +v 4 S A c S v A cosθ ' # & 1/. (4.1) The higher frequency mode is known as the fast magnetoacoustic wave and the other one the slow magnetoacoustic wave. The Alfvén wave phase speed lies between the two.

19 Properties of Magnetoacoustic Waves The phase speed does not depend on frequency i.e. the waves are non-dispersive; They are anisotropic (i.e. have a dependence on θ) also the phase speed is not the same as the group speed. For θ = 0, the phase speed u(=ω/k) is either c S or v A, whereas for θ = π/, the phase speed is (c s +v A ) 1/ for the fast mode and 0 for the slow mode. Consider the limits v A >> c S or c S >> v A, when u f (c s +v A ) 1/, (v A >> c S or c S >> v A ) (4.) i.e. u f is the larger of c S or v A and is independent of θ i.e. the fast mode propagates isotropically. The slow mode speed becomes: u S c T cos θ, (v A >> c S or c S >> v A ) (4.3) c T = v A c s / (c s +v A ) 1/ (4.4) i.e. u S is the smaller of c S or v A and its angular dependence is the same as that of the Alfvén wave. Consider the limiting case cos θ << 1, that is for nearly perpendicular propagation to B 0. Expressions (4.) and (4.3) hold in this limit as well, but they now apply for arbitrary c S or v A. In this context, Eq. (4.) represents the fast magnetoacoustic speed, while the quantity given by Eq. (4.4) is referred to as the cusp speed. When c S >> v A, the fast mode is akin to a sound wave modified by the magnetic field. In the opposite limit c S << v A, it propagates at the Alfvén speed, with fluid displacements nearly perpendicular to B 0 as in an Alfvén wave.

20 Phase velocities (left) and group velocities (right) for fast & slow waves propagating at an angle θ to the magnetic field direction: u S and u f are the slower of the Alfvén speed v A and sound speed c S. Propagation diagrams of the fast mode (left) and slows mode (right) for v A =c S. Curves (u) show the phase speed, arrows the group speed v g as a function of the direction (dotted) of the phase speed. Directions of fluid displacement (red) in the fast mode (top) and slow mode (bottom) at low (left) and high (right) v A /c s. Grey: angular dependence of the phase speed

21 Poynting Flux in MHD The Poynting flux S of electromagnetic energy: S= c(e B) /4π, (4.5) is usually thought of in connection with electromagnetic waves in vacuum. It is in fact defined quite generally, and has a MHD-specific interpretation. With the MHD expression for the electric field, E = v B/c, we have S= B (v B) /4π. (4.6) S thus vanishes in flows parallel to B. Writing out the cross-products, and denoting by v the components of v in the plane perpendicular to B, we have: S = v B / 4π. (4.7) An MHD flow does not have to be a wave of some kind for the notion of a Poynting flux to apply. It also applies in other time dependent flows, and even in steady flows Also, the divergence of the Poynting flux S may be written as:. (E B) = -E. B + B. E and transformed using Maxwell s equations to: -c. (E B)/4π = E.j + / t (B /8π). (4.8) The physical interpretation is that an inflow of electromagnetic energy ((E B) produces electrical energy (E.j) for the plasma an an increase in the magnetic energy (B /8π). In turn the electrical energy given to the plasma by the electromagnetic field may be rewritten after substituting for E as: E.j = j /σ + v.j B/c. (4.9) The electrical energy appears as heat by Ohmic dissipation and work done by the Lorentz force.

22 MHD Equations ρ dρ + (ρv) = 0 or + ρ v = 0 (Continuity Equation) t dt ρ dv dt = p + 1 4π ( B) B + ρg + ρν v (Momentum equation) d % p ( ' * = 0 (Energy equation for an adiabatic fluid), dt & ρ γ ) (where d/dt = / t + v ). B = (v B) +η B, t In the limit of infinite conductivity (η=0), B/ t = (v B) (induction equation) = B v (v )B + (B )v (since B = 0)

23 Linearized Wave Equation in an inhomogeneous medium Consider a vertical magnetic field B 0 (x) e z in an unbounded medium in which the fluid is initially at rest everywhere (i.e. v=0). The x-dependence of the magnetic field leads to a similar dependence of the pressure p 0 and density ρ 0. The initial equilibrium (static) state is such that: d p 0 = p 0 (x), ρ 0 = ρ 0 (x), dx (p + B 0 ) = 0, (5.1) 0 8π The last equation in (5.1) follows from the x-component of the momentum equation for the initial equilibrium state. Consider small departures from the initial equilibrium state of the form: ρ 0 δv t δb t ρ = ρ 0 + δρ, p = p 0 + δp, v = δv, B = B 0 + δb, where δρ is a perturbation in the density and so on. Substituting these in the equations of motion and induction, we find: & = δp + 1 4π B δb ) ( ' * 4π (B 0 ) δb + 1 4π (δb ) B 0. (5.) = (δv B 0 ). (5.3)

24 The continuity equation becomes δρ + (ρ t 0 δv) = 0, (5.4) We can relate δρ and δp, from the adiabatic energy equation as follows: δp t! +δv p 0 = c S # " δρ t $ +δv ρ 0 &, (5.5) % where γ is the ratio of specific heats and c S = (γp 0 / ρ 0 ) ½ is the sound speed. Let us define the following variables: From Eqs. (5.1)-(5.5), we find and δp t Δ = δv, Γ = δv z z, p T = δp + B 0 4π δb z. $ = ρ 0 c S Δ dp ' 0 & )v % dx x, p T ( t = ρ 0 v A Δ ρ 0 (c S +v A )Δ, (5.6) δb x t δv = B x 0 z, δb y t δv = B y 0 z, δb z t = B 0 (Γ Δ) ( db 0 dx )δv x, (5.7) where v A = B 0 / 4πρ 0 is the Alfvén speed.

25 The components of the momentum equation (5.) give: # ρ 0 t v % A $ z # ρ 0 t v % A $ z δv z t Δ = c S z. & (δv x = ' x [ρ (c +v 0 S A )Δ ρ 0 v A Γ], (5.8a) & (δv y = ' y [ρ (c +v 0 S A )Δ ρ 0 v A Γ], (5.8b) (5.8c) We now look for plane wave solutions of the form: δv = v(x)e i (ωt k y y k z z), and similarly for the other perturbed variables, where k y and k z are the y and z components of the the wavenumber vector respectively and ω is the angular frequency of the wave. We can now replace / t in Eqs. (5.8) by iω, / z by -ik z and / y by -ik y. Eqs. (5.8) now can be reduced to a single equation for v x (x):

26 where d dx " $ ρ 0 (x)[k z v A (x) ω ] # %$ [m 0 (x) + k y ] dv x dx & $ ' ($ ρ (x)[k v 0 z A (x) ω ]v x = 0, (5.9a) m 0 (x) = [k c z S (x) ω ][k z v A (x) ω ] [c S (x) +v A (x)][k z c T (x) ω ], (5.9b) c T (x) = c (x)v S A (x) c S (x) +v A (x). (5.9c) For a uniform medium, when ρ 0, c S, v A, c T and m 0 are constants, Eq. (5.9) becomes: " (k z v A ω ) d v x dx (m % $ + k 0 y )v x ' = 0. (5.10) # & Eq. (5.10) has two possible solutions (for k y 0): either ω = k z v A, with v x (x) arbitrary; or ω k z v A, and v x (x) satisfies: d v x dx (m + k 0 y )v x = 0, (5.11)

27 In a uniform unbounded medium, Eq. (5.10) admits plane wave solutions, with an x- dependence satisfying: e ik x x (k z v A ω )(m 0 + k y + k x ) = 0, the roots of which give the well-known dispersion relations for Alfvén waves (ω =k z v A ) or magnetoacoustic waves discussed earlier. For a non-uniform medium (assuming that the motions are independent of the y- coordinate (i.e. k y =0 and the propagation is in the x-z plane), we see that v y is given by (see Eq. 5.8b) (k z v A ω )v y (x) = 0, and so unless v A is uniform (in which case Alfvén wave propagation is possible), we have v y = 0. With k y =0 and v y = 0, we see that for -D motions v=(v x,0, v z ), Eq. (5.9) becomes: d dx " $ ρ 0 (x)[c S (x) +v A (x)][k z c T (x) ω ] # %$ [k z c S (x) ω ] dv x dx & $ ' ($ ρ (x)[k v 0 z A (x) ω ]v x = 0, (5.1)

28 Equation (5.1) describes the propagation of -D waves in a compressible nonuniform medium (varying in the x-direction). The v z component of the perturbation is related to v x as follows: (k z c S ω )v z = ik z c S dv x dx, and the amplitude of the total pressure perturbation using Eq. (5.6) is given by : p T = i ω (c +v 3 S A )(k z c T ω )ρ 0 Δ = iρ 0 ω (c +v S A ) (k c z T ω ) (k z c S ω ) dv x dx. Eq. (5.1) has singular solutions at locations x=x s, where ω =k z c T. Eq. (5.9) is singular at ω =k z c T (x), k z v A (x). Such singularities have important physical repercussions such as continuous spectra.

29 Waves in a Sharply Structured Medium Consider the behaviour of modes in a medium where the non-uniformity takes the form of a single magnetic interface, the field being uniform (but of different magnitude) on either side of the inteface. Suppose that the basic state of the plasma is one in which the magnetic field changes discontinuously from B 0 to B e, so that! # B B 0 (x) = e, x > 0, " $# B 0, x < 0, where B 0 and B e are constants. From Eq. (5.1) which expresses pressure balance, we have at the interface x = 0, the following relation: p e + B e 8π = p + B 0 0 8π. (5.13) Now the medium on either side of the discontinuity at x = 0 is uniform and so Eq. (5.10) holds with constant coefficients.thus, we can have an Alfvén wave in x < 0 with ω =k z v A, v x (x) and v y (x) arbitrary, and v z, p, p T, and Δ are all zero. For example, an Alfvén wave can propagate in x < 0 without disturbing the interface at x = 0 or the fluid in x > 0.

30 In addition to the Alfvén wave there are magnetoacoustic modes which are governed by Eq. (5.11). Thus, in x < 0, we have: d v x dx (m + k 0 y )v x (x) = 0, x < 0, (5.14) where the constant m 0 maybe positive or negative (for real ω and k z ). Similar conditions apply in x > 0 with m e being defined analogously. Surface waves owe their existence to the presence of the discontinuity in B 0 (x) and may arise if (m 0 + k y ) and (m e + k y ) are both positive. Solving Eq. (5.14) and the corresponding equation in x > 0 for v x (x), gives: " $ v x (x) = # $ % α e e (m e +k y ) 1/ x, α 0 e (m 0 +k y ) 1/ x, x > 0, x < 0, (5.15) (m 0 + k y ) ½ > 0 and (m e + k y ) ½ > 0. In choosing the form of Eq. (5.15) we have imposed the condition that v x (x) tends to zero at x = ± (we exclude laterally propagating waves). Thus only surface modes arise.

31 Across the interface at x =0, v x (x) and p T (x) must be continuous. Thus, α e = α 0, while continuity of p T at x =0 yields: ρ 0 (k z v A ω )(m e + k y ) 1/ + ρ e (k z v Ae ω )(m 0 + k y ) 1/ = 0, (5.16) valid for both (m 0 + k y ) ½ and (m e + k y ) ½ positive. We have written m e (x) = (k c z Se ω )(k z v Ae ω ) k z c Se v Ae (c Se +v Ae )ω, where c Se = (γp e / ρ e ) ½ and v Ae = B e / 4πρ e are the sound and Alfvén speeds respectively in x > 0. Eq. (5.14) is the dispersion relation for surface waves at a single magnetic interface. Eq. (5.14) may also be rewritten as where ω k z = v R A R +1 (v v A Ae ) = v Ae + 1 R +1 (v v A Ae ), (5.16)' " R = (ρ e / ρ 0 ) m + k % 0 y $ m # e + k ' y & 1/ > 0 is a function of ω. From the above we clearly see that the longitudinal phase speed of a surface wave lies between v A and v Ae.

32 There are a number of special cases of the general dispersion relation. Consider the incompressible limit (c S ), we obtain m k z and from Eq. (5.16) we have: ω! = ρ v + ρ 0 A e v $ Ae k # z " ρ 0 + ρ & e % Waves in Magnetic Flux Tubes We consider the generalization of Eqs. (5.9) in cylindrical coordinates (r, θ, z), with B 0 =B 0 (r) and ρ 0 =ρ 0 (r) and replacing v x with v r where 1/ v r (r, θ, z,t) = v r (r) exp [i (ωt + nθ k z z)] (5.17) for mode number n = 0,1,,.Then Eq. (5.9a) in cylindrical co-ordinates becomes. d dr " $ ρ 0 (r )[k z v A (r ) ω ] # $ [m (r ) + n % $ r ] 1 r & d(rv r ) $ ' ρ dr 0 (r )[k z v A (r ) ω ]v r = 0, (5.18) $ ( $ where m is given by Eq. (5.9b).

33 Isolated Flux Tubes Consider an isolated flux tube of radius a, for which!# B 0 (r) = " $# B 0, r < a, 0, r > a,! #, ρ 0 (r) = " $# ρ 0, r < a, ρ e, r > a,, (5.19) To determine the modes, it is convenient to solve for p T instead of v r, which satisfies the following equation: ρ 0 (k z v A ω ) 1 r d dr " $ 1 ρ 0 (k z v A ω ) r dp & $ T # ' %$ dr ($ = (m + n r )p, (5.0) T For a medium in which ρ 0, v A and m are constants, this is simply Bessel s equation, d p T dr + 1 dp T r dr (n r + m )p T = 0, (5.1) The relevant solution of Eq. (5.1) for region inside the flux tube is: p T = A 0 I n (m 0 r), r < a, (5.) where I n denotes a modified Bessel function of order n. There is no restriction on the sign of m 0 ; modes with m 0 > 0 are evanescent whereas m 0 < 0 have an oscillatory character. Modes with m 0 > 0 are referred to as surface waves while those with m 0 < 0 are called body waves.

34 where m 0 is given by m 0 = [k c z S ω ][k z v A ω ], (5.9b) [c S +v A ][k z c T ω ] c T = c v S A c S +v. A (5.9c) Here c T is referred to as the cusp or tube speed.

35 The external atmosphere outside the flux tube is field free. We consider Eq. (5.1) with v A = 0 and p T is the perturbed gas pressure. Requiring p T 0 as r, we select the modified Bessel function K n as the solution of Eq. (5.1), which yields p T = A 0 K n (m e r), r > a, (5.3) n = 1 where m e = k z ω c. (5.4) Se n = 0 The requirement of a declining pressure function outside the tube means that we are imposing the constraint m e > 0 and so ω < k z c Se. The two solutions (5.) and (5.3) are matched by requiring that the radial velocity component v r and the pressure perturbation p T be continuous at the interface r = a. This results in the following dispersion relation: I! ρ e ω m n (m 0 a) 0 I n (m 0 a) + ρ (k K! v 0 z A ω )m n (m e a) e K n (m e a) = 0, (5.5) where the dash denotes the derivative of the Bessel function with respect to x.

36 Body modes have phase speeds ω/k z that that are between c T and c S and correspond to slow modes. The slow body modes may be viewed as waves that are constrained within the tube, bouncing from side to side of the tube as they propagate along its interior. Body modes have m 0 < 0 and so are more conveniently obtained from Eq. (5.5) if we rewrite as follows: v A < c Se v A > c Se J! ρ e ω n n (n 0 a) 0 J n (n 0 a) + ρ (k K! v 0 z A ω )m n (m e a) e K n (m e a) where n 0 = m 0 = (ω k z c S )(ω k z v A ) (c S +v A )(ω k z c T ) = 0, (5.6). (5.7) In addition there are also slow surface waves. These can be understood by taking the incompressible limit of Eq. (5.5) (c S and c Se ). The result is k z v A ω " = 1 ρ % I e n (( k z a)k n ( k z a) $ ' # & ρ 0 v A > c Se I n ( k z a) K n (( k z a), (5.8) The phase speed as a function of In the limit k z a 0 (thin tube limit), we find for the kink mode (n=1):! ω ~ k z c K 1+ 1 ( ρ $ # e )k ρ e + ρ z a K 0 ( k z a) & " 0 % where c K = ρ 0 /(ρ 0 +ρ e ) v A is referred to as the kink speed (c K < v A ). k z a for waves in an isolated magnetic flux tube. Solid curves are sausage modes (n = 0), dashed curves are kink modes (n = 1) and dotted curves are n =.

37 Embedded Flux Tubes Consider an equilibrium magnetic field and density of the form:! # B 0 (r ) = " $# B 0, r < 0, B e, r > 0,! #, ρ 0 (r ) = " $# ρ 0, r < 0, ρ e, r > 0,, (5.9) The dispersion relation for the case of an isolated flux tube can be generalized to the present case by replacing m e in Eq. (5.4) with: m e = (k c z Se ω )(k z v Ae ω ) (c Se +v Ae )(k z c Te ω ). (5.30) It is straightforward now to obtain the dispersion relation which is the generalization of Eq. (5.6) given by: Magnetic flux tube idealized as a cylinder of radius a with a uniform vertical magnetic field of strength B 0 embedded in a medium with a uniform vertical field of strength B e. The subscripts refer to equilibrium quantities inside and outside the flux tube respectively. J" ρ e (k z v Ae ω )n n (n 0 a) 0 J n (n 0 a) + ρ (k K" v 0 z A ω )m n (m e a) e = 0, (5.31) K n (m e a) with n 0 given by Eq. (5.7). When v Ae = 0, Eq. (5.31) reduces to Eq. (5.6).

38 Coronal Tubes In the solar corona, the Alfvén speed typically exceeds the sound speed. Then Eq. (5.31) possesses two sets of modes, namely fast and slow body modes that can be seen in the figure. There are no surface waves (n 0 < 0 ). Both sets of modes are dispersive, with the fast mode being strongly dispersive. The fast body waves occur only if v Ae > v A i.e. with v Ae,v A > c S, c Se, fast body waves occur only for flux tubes that satisfy the condition v Ae > v A, and so body waves occur in region of low Alfvén speeds. When B 0 = B e, regions of low Alfvén speed correspond to regions of high gas density. Coronal loops act as wave guides for fast magnetoacoustic waves. The phase speed ω/k z as a function of k z a for waves in a low β magnetic cylinder embedded in an external field (typical in the solar corona). Solid curves are sausage modes (n = 0) and dashed curves are kink modes (n = 1).

39 Summary MHD waves in magnetic tubes Dispersion relation ω(k) for tubes K ' ρ 0 (k z v A ω )m n (m e a) e K n (m e a) = ρ (k I ' e zv Ae ω )m n (m o a) 0 I n (m o a) m 0 > 0 à surface waves K ' ρ 0 (k z v A ω )m n (m e a) e K n (m e a) = ρ (k J ' v e z Ae ω )n n (n o a) 0 J n (n o a) m 0 = -n 0 < 0 à body waves. Note n=0 refers to sausage, n=1 to kink modes, etc. Main modes: Fast sausage ( B, ρ) Fast kink (almost incompressible) Alfvén torsional (incompressible) Slow (acoustic) type (ρ, v) n=0 n=1 Torsional waves

40 Summary contd MHD waves in magnetic tubes for typical conditions in the chromosphere Note n=0 refers to sausage and n=1 to kink modes. Main modes: Alfvén torsional (incompressible) Fast sausage ( B, ρ) Fast kink (almost incompressible) Slow (acoustic) type (ρ, v) Torsional waves n=0 n=1

41

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