Magnetohydrodynamic waves Again starting with simple Jiří Krtička Masaryk University Langmuir waves Langmuir waves (plasma oscillations) In ionized plasma, lightweight electrons move around heavy nuclei. There is a new type of oscillations connected with this motion. Within the limit of plasma oscillations (Langmuir waves) we can consider nuclei as static and assume that electrons oscilate in the field of nuclei. These oscillations can be described by hydrodynamic equations describing free electrons ^+V-(pe^e) = 0, dve ^ _ at where E is the electric field induced by free electrons and A7e is electron density. i Langmuir waves: perturbations As always, we shall introduce perturbations of electron density and velocity in the comoving frame Pe = POe + $Pe, ve = 5ve, for which the hydrodynamic equations are (neglecting higher order terms) dSpe + p0eO\yOVe = 0, dt E. dt m e The electric field comes from the inhomogeneous distribution of density for which the Poisson equation gives div c =-ope. Langmuir waves: the wave equation All these equations of perturbations shall be combined. Calculating divergence of momentum equation and taking d'wSv from the continuity equation and E from the Poisson equation we derive the wave equation d2Pe w H--^rPOeOpe = 0, dt2 which describes Langmuir waves. This gives the plasma frequency 2 47re2 uj | =-ne. p me The waves with uj < up\ interact with free electrons and force the electrons to follow the electron motion. Therefore, the waves for frequencies uj < cjp\ are damped in plasma. On the contrary, the electrons are not able to react on higher frequency waves uj > ujp\ and such waves move freely in plasma. Alfven waves Alfven waves: the ansatz To start with truly MHD waves, we shall study perturbations in incompressible gas (div v = 0) in the comoving frame v = dv. B = B0 + 5B. We shall start with momentum equation d\/ 1 P^T +pv-Vv = -Vp + f + — (rot B) x 6, at 47r which in absence of external forces and considering the above mentioned perturbations ^ = ^(roUe)xe°' where we have assumed that derivatives of perturbations dominate. Alfven waves: perturbing the magnetic field Similarly, the induction equation dB ~dt transforms into = rot (v x B) = vot{Sv x B0). Using the corresponding vector identity rot(Sv x B0) = 5vd\vB0 - B0d'wSv + (B0V)5u - (5i/V)B0. Most terms on the right hand side of this equation can be dropped. The first one from the Maxwell equations, the second one from the incompressibility of the gas, and the last one due to the dominance of wave perturbvations. As a result, 85 B dt (BoV)Sv. Alfven waves: the wave equation Combining the resulting momentum and induction equations, we arrive at the wave equation for Sv P d2Sv 47T rot [(BqW)Sv] x 60 We can assume that the magnetic field Bo is parallel with z-axis. In such case the wave equation takes the form dz5vy dt2 1 d25vy dt2 d25vz i ( V dt2 I d2Svx d2Svz dz2 d25v, y dz2 dxdz d25vz dydz \ \ o / The z-component gives d25vz/dt2 = 0, which means that there are no oscillations in the direction of magnetic field. With initial conditions <5Vz|t=o = 0 and d5vz/dt\t=o = 0 this gives 8vz = 0. Alfven velocity With zero perturbation velocity in the z-axis direction, the wave equation simplifies to / d2Svx \ / d2Svx \ I Qz2 \ I \ d25v, y V dz2 0 dt2 d25v, y dt2 0 This corresponds to transversal waves (Alfven waves) moving in the direction of magnetic field (z-axis) with so-called Alfven velocity ß va 0 7 Alfven waves: the induction equation Similarly, from the time derivative of the induction equation d2SB ,„ „.dSv = (B0V) dt2 dt 4np (B0V) [(rot 56) x B0]. Again, assuming that the magnetic field Bo is parallel with z-axis, the wave equation takes the form of d25B> dt2 d25B y dt2 \ dt2 / ( B2o d2SBx d2SBz dz2 d2SB y dxdz d25Bz dydz \ \ o / The z-component gives d25Bz/dt2 = 0, which means that there are no oscillations in the direction of magnetic field. With initial conditions SBz\t=0 = 0 and dSBz/dt\t=0 = 0 this gives 8BZ = 0. 8 Alfven velocity With zero magnetic field perturbation in the z-axis direction, the wave equation simplifies to Airp ( d25B> dz2 d25B y \ dz1 0 / d25B> dt2 d2SB \ y \ dt2 0 / This also corresponds to transversal Alfven waves moving in the direction of magnetic field (z-axis) with Alfven velocity 6 0 9 Magnetoacoustic waves Magnetoacoustic waves: the ansatz We shall study a general form of MHD waves allowing for compression of gas. As always, we shall study perturbations in the comoving frame p = p + 5p, v = Sv. B = B0 + 5B. The MHD equations for perturbations are dSp dt + p0V -Sv = 0, 9Sv 1 „ 1 , _ _ 2r —- =--VSp H--(rot SB) x B0, Sp = azSp. ot po 47vp0 dSB dt rot (Sv x 60) 10 Magnetoacoustic waves: velocity perturbations Taking derivative of momentum equation with respect of time and insterting from the remaining equations d25v . 1 r „ m „ —= azVd\vSv H--rot rot (Sv x B0) x B0, or, using the vector identity d2Sv 1 —r = a2Vdiv Sv H--[Vdiv (Sv x B0) - A (5 v x B0)l x B0. Because div (Sv x Bo) = Bo • rotSv — Sv • rot Bo ~ Bo • rotSv, we have V(B0 • rot5v) = (B0 • V) rotSv + (rotSv • V) B0 + B0 x rot rot 5u + rot5u x rot B0 « (B0 • V) rot Sv + B0 x rot rot Sv = (Bo • V) rot Sv + Bo x (Vdiv Sv — ASv). The term with A cancels out and (B0 x Vd'wSv) x B0 = B$ Vd'wSv - (B0 • Vd'wSv) • B0. li Magnetoacoustic waves: the dispersion relation Finally, we arrive at the wave equation in the form of d2Sv ~dtr a2+ e° VdivJiH--{[(B0-V) rotSv]xB0 - (B0-Vd\vSv)-B0} We will seek the solution in the form of travelling waves Sv ~ exp [i(kr — ujt)\. In this case d\v Sv = ik • Sv, Vd'wSv = — (k • Sv) k, rotSv = ik x Sv, (Bo-V) rotSv = — (Bo • k) (k x and the dispersion relation is u2Sv = (a2+vl) (k • Sv) k+-^— [(B0 -k)(kx Sv) x B0 - (B0 ■ it) (it ■ 5u) B0] Inserting Bo = Bob with unity vector b, and using the vector identity (k x Sv) x Bq = (Bq • A) Sv — (Bo • 5v) k we derive the dispersion relation cu2Sv= [(a2 + ^) (k • (Jv) - ^ (k • 6) (5u • b)] k+v% \{k • 6)25u - (k - b) (Sv • 12 Sanity check: the Alfvén waves For incompressible flow k • Sv = 0 and the dispersion relation reads uj Sv = v, a (k • bf Sv - (k • b) (Sv-b)k Multiplying the wave equation by k and using the incompressible flow condition we have (k • b) (Sv • b) = 0. Consequently, either k • b = 0 or Sv • b = 0. However, the application of the former in the wave equation would preclude any oscillations, therefore Sv • b = 0, that is, the oscillations are perpendicular to the magnetic fields, as we have already seen. Consequently, the dispersion relation for the Alfven waves is uj = ±va (k • b). The group velocity is duj duj duj = ±v/\b. Therefore, the waves move in the direction of the magnetic field 13 General dispersion relation Multyplying the dispersion relation by k the last two right hand terms calcel out, and uj2 {k • Sv) = {a2 + vl) {k • Sv) k2 - v2k {k • b) (Sv • b) k2. The term Sv • b can be derived by multiplication of the dispersion relation by b, after which surprisingly nearly all term vanish and we simply get uj2 {b • Sv) = a2(k- Sv) (k-b). Inserting this into the former relation and calceling out k • Sv we finally arrive at the dispersion relation uj" - uj2{a2 + vi)kl + via1 (k • 6)z k 2\i .2 2 Jl 2 ,2 0. Denoting k • b = kcosO, where 9 is the angle between k and b, the wave speed is •2 i f.s /C2 a2 + vjj ± J(a2 + ^)2 - 4t/2a2 cos2 6 for fast and slow magnetoacoustic waves. 14 The dispersion relation of magnetoacoustic waves For v/\^> a the fast magnetoacoustic wave propagates nearly isotropically Vf ^ v/\, while the slow magnetoacoustic wave propagates at the velocity vs ^ a cos 9. The energy propagates with group velocity vg = V/cCj, which is \/g = v/\ for the fast wave and vg = ab for the slow wave. In a general case the dispersion relation depends just on a parameter 1 " 2 - 1 + P± + -4/9 cos2 6 CD 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 CD 1.5 1 0.5 0 -0.5 -1 -1.5 / \fast ■ / Alfven \ ■ b -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 vxl a -1.5 -1 -0.5 0 0.5 vxla 1.5 (3 = 2 (3 = 0.5 15 Suggested reading E. Battaner: Astrohysical fluid dynamics L. Mestel: Stellar magnetism L. C. Woods: Physics of Plasmas 16