MHD Waves Robertus vfs Robertus@sheffield.ac.uk SP RC, School of Mathematics & Statistics, The (UK)
What are MHD waves? How do we communicate in MHD? MHD is kind! MHD waves are propagating perturbations of magnetic field, plasma velocity and plasma mass density, described by the MHD (single fluid approximation) set of equations, which connects the magnetic field B, plasma velocity v, kinetic pressure p and density ρ. Non-relativistic approximation
Do we expect solar MHD waves? Corona has frozen in field condition Magnetic field rooted into turbulent photosphere Generates waves that dump energy in corona Alfvén/(Slow/Fast) Magneto-acoustic waves
Do we see MHD waves? SOHO/TRACE examples (mainly TR and higher)
Do we see MHD waves? - Moreton waves
Do we see MHD waves? Sun quakes Solar quakes: high energy electrons slam the solar surface
Do we see MHD waves? Surfing magnetic loops Example: Rapid (every 15 s) TRACE 171 Angstrom image Track changes in brightness Wave travels outwards from B to T
Do we see MHD waves? Surfing magnetic loops Difference image in brightness out from the loop base
Linear theory of MHD waves Static/steady stationary background Superimpose linear motions on this background Write physical quantities as f(r,t)=f (r)+f 1 (r,t); f 1 / f <<1 Reduce full set of nonlin PDEs of MHD to a set of ODEs Choice: initial value problem, boundary value problem, eigenvalue problem Eigenvalue problem of linear waves/oscillations: exp(iωt)
Linear theory of MHD waves Linearised ideal MHD equations ρ t 1 + ρ = V1 V t B B µ ( B ) B + µ ρ 1 1 1 = p1 + B t p = ( V ) 1 1 B 1 = cs ρ1 cs = γp /, ρ
Linear MHD waves in uniform plasma No characteristic length scale defined by the equilibrium Constant equilibrium magnetic field, e.g. B Superposition of linear waves = B sinαx + B cosαy exp(ik x x + ik y y + ik z z), k=(k x, k y, k z )=wave vector
Linear MHD waves in uniform plasma z B α k z x Characteristic speeds: Alfvén speed sound speed v A = B πρ 1/ (4 ) c s = ( γp ρ 1/ / )
Linear MHD waves in uniform plasma Consider dynamics of perturbations of this stationary state. In the linear limit, the set of MHD equation splits into two uncoupled subsets: (i) for the variables V y and B y (Alfvén wave) (ii) and for ρ, p, V x, V z and B x (magnetoacoustic waves)
Linear MHD waves in uniform plasma Consider dynamics of perturbations of this stationary state. In the linear limit, the set of MHD equation splits into two uncoupled subsets: (i) for the variables V y and B y (Alfvén wave) (ii) and for ρ, p, V x, V z and B x (magnetoacoustic waves)
Linear MHD waves in uniform plasma Alfvén waves t Properties: v z Az V y = v Az = B 1/ cosα /(4πρ) (i) Transverse oscillation driven by magnetic tension forces (ii) Does not perturb density è incompressible (in linear limit) (iii) Can t propagate across field lines (iv) Group velocity (δω/δk) is along B
Linear MHD waves in uniform plasma Alfvén waves t Properties: v z Az V y = v Az = B 1/ cosα /(4πρ) (i) Transverse oscillation driven by magnetic tension forces (ii) Does not perturb density è incompressible (in linear limit) (iii) Can t propagate across field lines (iv) Group velocity (δω/δk) is along B
Linear MHD waves in uniform plasma Alfvén waves t Properties: v z Az V y = v Az = B 1/ cosα /(4πρ) (i) Transverse oscillation driven by magnetic tension forces (ii) Does not perturb density è incompressible (in linear limit) (iii) Can t propagate across field lines (iv) Group velocity (δω/δk) is along B
Linear MHD waves in uniform plasma Alfvén waves t Properties: v z Az V y = v Az = B 1/ cosα /(4πρ) (i) Transverse oscillation driven by magnetic tension forces (ii) Does not perturb density è incompressible (in linear limit) (iii) Can t propagate across field lines (iv) Group velocity (δω/δk) is along B
Linear MHD waves in uniform plasma Alfvén waves When B z there can be two linearly polarized plane Alfvén waves, one perturbing V y, B y and the other V x, B x. For harmonic perturbations [exp(iωt -kz)] combination of two linearly polarized waves gives us elliptically polarized Alfvén waves: B y B x = Acos( ωt kz), = Bsin( ωt kz), A, B = const
Linear MHD waves in uniform plasma Alfvén waves The vector of the magnetic field perturbation rotates along an ellipse at the x,y-plane. When A = B, the wave is circularly polarized, with B =const. Circularly polarized Alfvén waves (even of finite amplitude) are an exact solution of the ideal MHD equations for a uniform plasma
Linear MHD waves in uniform plasma Magnetoacoustic waves t v Az z t s Harmonic perturbations: v z v V z Ax 4 t z V z =, ~ exp[i( ωt -kz)] v Ax = B 1/ sinα /(4πρ) Dispersion relation for MAW: ( ω v A cos αk )( ω v s k ) v A sin αω k = DR bi-quadraticè slow and fast magnetoacoustic waves
Linear MHD waves in uniform plasma Magnetoacoustic waves The polar plot for phase speeds (ω/k) for case β<1
Linear MHD waves in uniform plasma Magnetoacoustic waves ~ ~ ~ β =.36 <.5 β =.81 >. 5 β = 1 Here C = C + V C = min( C, V ); C = max( C, V ); β = C / C f S A ; m S A M S A m M ~
Linear MHD waves in uniform plasma Group velocity Initial perturbation corresponding either Alfven or slow or fast magnetosonic wave in the form of wave packet: f 1 = R{ Φ (r)exp( ik r) }, Lk <<1 (.13) Fourier expansion: Φ ˆ ( r)exp( ik r) = Φ( k)exp[ i( k + k) r] dk (.14) Contribution from k k << k :
Look for solution in the form f ( r, t) = R{ F( r, )} Each Fourier component has frequency 1 t ω(k) F ( r, t) = Φ ˆ ( k)exp ω + [ i( k + k ) r i ( k k ) t] d (.15) k k << k ω ω + Vg k, V g ω = ω( k ), V g = ω k k= k is group velocity (.16) F( r, t) = exp ˆ g ) [ i( k r ω t) ] Φ ( k)exp[ ik ( r V t ] dk [ i( k ω )] (.17) = Φ( r Vg t)exp r t Envelope propagates with group velocity Vg
VgA A / B For Alfven waves: = V B = V zˆ (.18) A For magnetosonic waves: V g± = 4 C ˆ ± k CSVA kz cosϕ C k[c ( C + V )] ± ± S A (.19) Curve ( ϕ ) ( ϕ < π ) V g is Fridrichs diagram Direct calculation shows that V g k = ϕ rule for finding V g (k)
Linear MHD waves in uniform plasma Magnetoacoustic waves The polar plot for group speeds (dω/dk)
Linear MHD waves in uniform plasma Slow waves Properties: (i) Anisotropic wave propagation largely confined to magnetic field (ii) Driven by magnetic pressure and tension forces (iii) Does perturb density/pressure (iv) Can t propagate across field lines
Linear MHD waves in uniform plasma Slow waves Properties: (i) Anisotropic wave propagation largely confined to magnetic field (ii) Driven by magnetic pressure and tension forces (iii) Does perturb density/pressure (iv) Can t propagate across field lines
Linear MHD waves in uniform plasma Slow waves Properties: (i) Anisotropic wave propagation largely confined to magnetic field (ii) Driven by magnetic pressure and tension forces (iii) Does perturb density/pressure (iv) Can t propagate across field lines
Linear MHD waves in uniform plasma Fast waves Properties: (i) Roughly isotropic wave propagation (ii) Driven by magnetic pressure and tension forces (iii) Does perturb density/pressure (iv) Propagates fastest perpendicular to B
Linear MHD waves in non-uniform plasma Characteristic length scale defined by the inhomogeneity Equilibrium quantities are functions of position Continuum of resonant Alfvén and slow waves Discrete slow and fast modes; discrete Alfvén modes Efficient damping in non-ideal MHD MHD waves with mixed character and wave transformation
Linear theory of MHD waves Original MHD theory by e.g., Edwin & Roberts (1983) modelled a flux tube as a magnetic cylinder. It was found that there are many different types of MHD waves, e.g., fast/slow magneto-acoustic modes in magnetically twisted cylinder (Erdélyi & Fedun 1). Courtesy Verwichte
Linear MHD waves in non-uniform plasma Properties of MHD waves depend upon the angle between the wave vector and the magnetic field è waves "feel" the direction of the field. When the magnetic field is not straight, Alfvén and slow waves should follow the field, because they are confined to the field. Even when the field is straight, inhomogeneities in the field absolute value, density and pressure affect the characteristic speeds of the waves (the Alfvén and the sound speeds) and, consequently, affect the waves. è Guided propagation of MHD waves, linear coupling of different MHD modes, phase mixing of Alfvén waves, resonant absorption, appearance of wave dispersion, etc.
Linear MHD waves in non-uniform plasma z v A (x) B v s (x) ρ (x) k x z
Linear MHD waves in non-uniform plasma Total pressure balance: Characteristic speeds: Alfvén speed Sound speed v v B ( x) ptotal ( x) = p( x) + = 8π A s B ( x) ( x) = 1/ [4πρ ( x)] ( x) = [ γp, ( x) / ρ ( x)] 1/, const. Tube (cusp) speed vsva v T ( x) = < 1/ [ v + v ] s A v s, v A.
Linear MHD waves in non-uniform plasma Fourier transform in homogeneous directions (y, z) f ( x)exp[i( ωt k y k z)], y z Boundary conditions at fixed x è Dispersion Relation D( ω, k y, kz,[ B ( x), ρ( x) and p( x)]) =.
Linear MHD waves in non-uniform plasma Consider: δ/δy=, though V y, B y (i.e..5d) Alfvén modes [perturbing V y, B y ] Magnetoacoustic modes [perturbing V x, V z, B x, B z, ρ ]
Linear MHD waves in non-uniform plasma Magnetoacoustic modes are governed by: d dx Λ( x) m ( x) dv dx x Λ( x) V x =, Λ( x) = ρ( x)[ ω kz va( x)], m ( kz vs ω )( kz va ω ) ( x) =, ( v + v )( k v ω ) A s z T + B.C.s=eigenvalue problem. Eigenfunctions define transversal (x) structure of waves; eigenvalues define dispersion for waves. Singularities: Alfvén ω / k ( ) { z = va x resonances! Cusp ω / k z = vt ( x)
Linear MHD waves in non-uniform plasma Magnetoacoustic modes Evanescent solutions: modes or trapped or guided (or ducted) waves; Dispersion is determined by the ratio of the longitudinal wavelength to the characteristic spatial scale of inhomogeneity. The modes can have different structures in x direction (inhomogeneity), which allows us to classify them: kink and sausage modes (perturbing or not perturbing the structure axis, respectively) body and surface modes (oscillating or evanescent inside the structure, respectively, and both evanescent outside the structure)
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Magnetosheath magnetopause - magnetosphere Heliopause interstellar wind Slow/fast solar wind boundary layer Sunspots/Coronal plums Helioseismology
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Magnetosheath magnetopause magnetosphere Generation of Pc5 waves, energisation of magnetosphere
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Magnetosheath magnetopause magnetosphere Heliopause interstellar wind Slow/fast solar wind boundary layer Sunspots/Coronal plums Helioseismology
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Heliopause interstellar wind
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Magnetosheath magnetopause magnetosphere Heliopause interstellar wind Slow/fast solar wind boundary layer Sunspots/Coronal plums Helioseismology
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Slow/fast solar wind
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Magnetosheath magnetopause magnetosphere Heliopause interstellar wind Slow/fast solar wind boundary layer Sunspots/Coronal plums Helioseismology
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Running penumbral wave generation by RFI
Kelvin-Helmholtz (KH) and Resonant Flow (RF) instability Magnetosheath magnetopause magnetosphere Heliopause interstellar wind Slow/fast solar wind boundary layer Sunspots/Coronal plums Helioseismology???
Kelvin-Helmholtz Instability (KHI) y ρ B U x Unperturbed state: for y < : ρ = ρ, B = B xˆ, V xˆ 1 = 1 1 U ρ 1 B 1 U 1 for y > : ρ = ρ, B = B xˆ, V xˆ = U C S >> V A Alfvenic perturbations can be considered as incompressible: ρ = const V =
Linearized equations: ρ & ( ' V t + (V ) V Solar Physics & Space Plasma ~ f ( t, x, y) = f + f ( t, x, y) V = (1) ) & + = p + B B ) ( * ' µ * + + (B ) B µ B t = V B + V B Boundary conditions: continuity of total pressure ( ) (3) p 1 + B 1 B 1 /µ = p + B B /µ at y = (4) Equation of perturbed interface y = η( t, x) Kinematic boundary condition ( ~ ): v 1, = η V = ( u, v,) t +U η 1, x at y = (5) ()
f exp[ i( k x ω )] 1 t Solar Physics & Space Plasma dv dy + iku = (6) ρ (ω ku )u = k p (7) ρ (ω ku )v = i & y p + B ( ' µ B x ) + kb B * µ y (8) ( ) (9) ωb x = i y U B y B v (ω ku ) B y = kb v (1)
Kinematic boundary condition at y = : v v 1, = i(ω ku 1, )η 1 v = (11) ω ku 1 ω ku Eliminate u and B ~ y from (6)-(1) d v dy k v = (1) p + B B x µ = iρ [(ω ku ) V A k ] k (ω ku ) v as y # v = A 1e k y, y < $ % A e k y, y > dv dy (14) (13)
Substitute (13) and (14) in boundary conditions (4) and (11) A 1 ω ku 1 A ω ku = (15) ρ 1 [(ω ku 1 ) V A1 k ]A 1 k (ω ku 1 ) + ρ [(ω ku ) V A k ]A k (ω ku ) = (16) Linear system with respect to A1 and A. There is non-trivial solution determinant is zero dispersion equation: ( 1 1 1 U ρ + ρ ) ω k ( ρ U + ρ ) ω +k ρ 1 (U 1 [ V ) + ρ (U V A1 )] = (17) A
ω = k ρ 1 U 1 + ρ U ± ρ 1ρ [U KH (U 1 U ) ] (18) ρ 1 + ρ U KH = (ρ + ρ )(ρ V 1 1 A1 + ρ V A1 ) (19) ρ 1 ρ U U < U I( ) = 1 KH ω Stability U U > U I( ) Instability, increment k 1 KH ω Particular case: ρ 1 = ρ, 1 = B VA 1 = VA = VA UKH = B V A
Model equilibrium state Inhomogen region Homogen I Z Homogen II L Steady equilibrium state
Kelvin-Helmholtz instability (KHI) Forwards Forwards Backwards Backwards KHI KHI Fast modes (a, θ=5 ; b, θ=75 ); β= (cold plasma); L= Forwards & backwards propagation, KHI