Magnetohydrodynamic Waves Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 17, 2016 These slides are largely based off of 4.5 and 4.8 of The Physics of Plasmas by Boyd & Sanderson (see also Chapter 6), Wikipedia articles on the wave equation and eigenstuff, Chapter 5 of Principles of Magnetohydrodynamics by Goedbloed and Poedts, lecture notes by Steve Cranmer, and a discussion with plasma wave expert Mahboubeh Asgari-Targhi. Extensive discussion of waves beyond MHD is included in Plasma Waves by D. G. Swanson and Waves in Plasmas by T. Stix.
Outline The 1D wave equation Algebraic solution Eigenmode solution Sound waves Linearization of equations of hydrodynamics Derivation of dispersion relationship MHD waves Linearization of MHD equations Introduce displacement vector ξ and MHD force operator F(ξ) Derivation of dispersion relationship Shear Alfvén, fast magnetosonic, and slow magnetosonic waves Observations of MHD waves Solar corona Space plasmas Laboratory experiments
Why do we care about waves? Waves are ubiquitous in magnetized plasmas Just as sound waves are ubiquitous in air Waves are the simplest way that a system responds to disturbances and applied forces Waves propagate information and energy through a system Waves are closely related to shocks, instabilities, and turbulence Plasmas display a rich variety of waves within and beyond MHD
Applications of waves in plasma astrophysics Space physics Earth s ionosphere, magnetosphere, and solar wind environment Solar and stellar physics Coronal heating Acceleration of solar and stellar winds Molecular clouds and star formation Interstellar medium Cosmic ray acceleration and transport Accretion disks and jets Pulsar magnetospheres Whenever a plasma is disturbed, there will be waves!
Example: the 1D wave equation The wave equation for u in one dimension is 2 u t 2 = c2 2 u x 2 (1) where c is a real constant that represents the wave speed The solutions are waves traveling at velocities of ±c The wave equation is a hyperbolic partial differential equation Connection to conservation laws
The algebraic solution to the 1D wave equation Define two new variables ξ(x, t) = x ct η(x, t) = x + ct (2) Rewrite the wave equation as The solutions are then 2 u ξ η = 0 (3) u(ξ, η) = R(ξ) + L(η) (4) u(x, t) = R(x ct) + L(x + ct) (5) where R and L are arbitrary functions traveling at velocities ±c (to the right and to the left)
Eigenmode decomposition of the 1D wave equation Use separation of variables and look for solutions of the form Plug this solution into the wave equation u ω (x, t) = e iωt f (x) (6) 2 t 2 [ e iωt f (x) ] = c 2 2 x 2 [ e iωt f (x) ] (7) ω 2 e iωt f (x) = c 2 e iωt d2 f (x) (8) dx 2 k 2 f (x) = d2 f (x) (9) dx 2 where k = ω/c. This is an eigenvalue equation for f (x). Next: identify eigenfunctions of the differential operator d2 dx 2 with corresponding eigenvalue k 2.
Eigenmode decomposition of the 1D wave equation Look for solutions of the form f (x) = Ae ±ikx (10) The solution to the wave equation for this eigenmode is Recall Euler s formula u ω (x, t) = Ae ikx iωt + Be ikx iωt (11) Take the real part of Eq. 11 to get e ix = cos x + i sin x (12) u ω (x, t) = A cos (kx + ωt) + B cos (kx ωt) (13) The solutions are waves propagating in the ±x directions. Use Fourier techniques to find the full solution.
Definitions The lines in the u-x plane on which x ct or x + ct are constant are called characteristics The wave vector k points in the direction of wave propagation and has a magnitude of k = 2π/λ where λ is the wavelength The phase velocity is the rate at which the phase of a wave propagates through space V p = ω k (14) The group velocity is the rate at which the overall shape of the waves amplitudes propagates through space V g = ω k (15)
Finding the dispersion relationship for sound waves Represent variables as the sum of a background component (denoted 0 ) and a small perturbed component (denoted 1 ) ρ(r, t) = ρ 0 + ρ 1 (r, t) (16) p(r, t) = p 0 + p 1 (r, t) (17) V(r, t) = V 1 (r, t) (18) Assume the background is homogeneous, time-independent, and static (V 0 = 0) Look for solutions proportional to e i(k r ωt) Solve for a dispersion relationship that connects the wave vector k with the angular frequency ω
Linearizing the equations of hydrodynamics The equations of hydrodynamics are ( ρ p t ρ t + (ρv) = 0 (19) ) t + V V + p = 0 (20) + V p + γp V = 0 (21) Linearize the equations. Drop higher order terms. Use that the background is constant. ρ 1 t + ρ 0 V 1 = 0 (22) V 1 ρ 0 + p 1 t = 0 (23) p 1 t + γp 0 V 1 = 0 (24)
Linearizing our first equation We start out with the continuity equation ρ t + (ρv) = 0 (25) Substitute in ρ(r, t) = ρ 0 + ρ 1 (r, t) and V(r, t) = V 1 (r, t). ρ 0 t }{{} =0 + ρ 1 t + (ρ 0V 1 ) + (ρ 1 V 1 ) }{{} second order = 0 (26) ρ 1 t + (ρ 0V 1 ) = 0 (27) We dropped ρ 0 t because the background is time-independent We dropped (ρ 1 V 1 ) because ρ 1 and V 1 are both small, so the product resulting from this second order term will be negligibly small.
Deriving a wave equation for hydrodynamics Take the time derivative of Eq. 23 ρ 0 2 V 1 t + p 1 t = 0 (28) Then substitute p 1 t = γp 0 V 1 from Eq. 24 to get a wave equation 2 V 1 t 2 c2 s ( V 1 ) = 0 (29) where the sound speed is γp0 c s (30) ρ 0
Assume that the solution is a superposition of plane waves Assume plane wave solutions of the form V 1 (r, t) = k ˆV k e i(k r ωt) (31) Differential operators turn into multiplications with algebraic factors ik, iω (32) t The problem is linear and homogeneous, so we consider each component separately. The wave equation then becomes 2 V 1 t 2 c2 s ( V 1 ) = 0 ( iω) 2 V 1 cs 2 (ik) (ik V 1 ) = 0 ω 2 V 1 c 2 s k (k V 1 ) = 0 (33)
The dispersion relationship for sound waves Choose coordinates so that k = k z ẑ, which then implies that V 1 = V 1z ẑ. Eq. 33 becomes ( ω 2 kz 2 cs 2 ) Vz1 = 0 (34) The non-trivial solutions are ω = ±k z c s (35)
Find the phase velocity and group velocity The dispersion relationship is The phase velocity and group velocity are ω = ±k z c s (36) V p ω k z = ±c s (37) V g ω k = ±c s (38) Sound waves are compressional because V 1 0 Sound waves are longitudinal because V 1 and k are parallel
How do we derive the dispersion relation for MHD waves? 1 Linearize the equations of ideal MHD. Take a Lagrangian approach Partially integrate the equations with respect to time Write equations in terms of the displacement from equilibrium Assume solutions proportional to e i(k r ωt) Derive a dispersion relationship that relates k and ω Investigate the properties of the three resulting wave modes 1 Here we follow Boyd & Sanderson 4.5 and 4.8.
Begin with the equations of ideal MHD The continuity, momentum, induction, and adiabatic energy equations are ρ + (ρv) = 0 (39) ( t ) ρ t + V V = J B p (40) c B = (V B) (41) ( ) t t + V p = γρ V (42)
The linearized equations of ideal MHD The continuity, momentum, induction, and adiabatic energy equations are linearized to become ρ 1 t V 1 ρ 0 t B 1 t p 1 t = V 1 ρ 0 ρ 0 V 1 (43) = ( B 1) B 0 4π p 1 (44) = (V 1 B 0 ) (45) = V 1 p 0 γp 0 V 1 (46) Here we ignored second and higher order terms and used Ampere s law. The terms V 1 ρ 0 and V 1 p 0 vanish if we assume the background is uniform
The displacement vector, ξ, describes how much the plasma is displaced from the equilibrium state 2 If ξ(r, t = 0) = 0, then the displacement vector is ξ(r, t) t Its time derivative is the perturbed velocity, 0 V 1 (r, t ) dt (47) ξ t = V 1(r, t) (48) 2 A side benefit of using slides is that I do not have to try writing ξ on the chalkboard.
Integrate the continuity equation with respect to time Put the linearized continuity equation with a uniform background in terms of ξ ρ 1 t Integrate this with respect to time t 0 ρ 1 t dt = = V 1 ρ 0 ρ 0 V 1 (49) = ξ t ρ 0 ρ 0 ξ t t 0 (50) [ ξ t ρ 0 ρ 0 ξ ] t dt (51) which leads to a solution for ρ 1 in terms of just ξ ρ 1 (r, t) = ξ(r, t) ρ 0 ρ 0 ξ(r, t) (52)
We can similarly put the linearized induction and energy equations in terms of ξ Integrating the linearized equations with respect to time yields solutions for the perturbed density, magnetic field, and plasma pressure: ρ 1 (r, t) = ξ(r, t) ρ 0 ρ 0 ξ(r, t) (53) [ ] ξ(r, t) B0 (r) B 1 (r, t) = (54) c p 1 (r, t) = ξ(r, t) p 0 (r) γp 0 (r) ξ(r, t) (55) The perturbed density ρ 1 doesn t appear in the other equations, which form a closed set However, we still have the momentum equation to worry about!
The linearized momentum equation in terms of ξ and F[ξ] Using the solutions for ρ 1, B 1, and p 1 we arrive at 2 ξ ρ 0 = F[ξ(r, t)] (56) t2 which is reminiscent of Newton s second law The ideal MHD force operator is F(ξ) = (ξ p 0 + γp 0 ξ) + 1 4π ( B 0) [ (ξ B 0 ] + 1 4π {[ (ξ B 0)] B 0 } (57) which is a function of the displacement vector ξ and equilibrium fields, but not of V 1 = ξ t.
Building up intuition for the displacement vector ξ and force operator F(ξ) The displacement vector ξ gives the direction and distance a parcel of plasma is displaced from the equilibrium state The force operator F(ξ) gives the direction and magnitude of the force on a parcel of plasma when it is displaced by ξ Discussion question: What is the sign of ξ F(ξ) when the configuration is unstable? Why?
Deriving the dispersion relation for MHD waves Assume that the plasma is uniform and infinite Perform a Fourier analysis by assuming solutions of the form ξ (r, t) = ξ (k, ω) e i(k r ωt) (58) k,ω The linearized momentum equation, then becomes 2 ξ ρ 0 = F (ξ (r, t)), (59) t2 ρ 0 ω 2 ξ = kγp 0 (k ξ) + {k [k (ξ B 0)]} B 0 4π (60)
Deriving the dispersion relation for MHD waves Choose Cartesian axes such that k = k ŷ + k ẑ (61) Expanding the vector products yields ( ) ω 2 k 2 V A 2 ξ x = 0 (62) ( ω 2 k 2 c2 s k 2 VA 2 ) ξy k k cs 2 ξ z = 0 (63) k k cs 2 ξ y + ( ω 2 k cs 2 ) ξz = 0 (64) where c s is the sound speed The Alfvén speed is defined as V A B0 2 4πρ 0 (65)
The dispersion relation for MHD waves To get a non-trivial solution (ξ 0), we need det ω 2 k 2 V 2 A 0 0 0 ω 2 k 2 c2 s k 2 V 2 A k k c 2 s 0 k k c 2 s ω 2 k 2 c2 s = 0 (66) Eq. 66 reduces to the dispersion relation for MHD waves ( ) [ ω 2 k 2 V A 2 ω 4 k 2 ( ] cs 2 + VA) 2 ω 2 + k 2 k 2 c2 s VA 2 = 0 (67)
Non-trivial solutions of the dispersion relation for MHD waves The solution corresponding to shear Alfvén waves is ω 2 = k 2 V 2 A (68) The solution corresponding to slow and fast magnetosonic waves is ω 2 = 1 ( 2 k2 cs 2 + VA) [ 2 1 ± ] 1 δ (69) where δ is δ 4k2 c2 s VA 2 k ( ) 2 cs 2 + VA 2 2, 0 δ 1 (70) All three solutions are real No growth or decay No dissipation or free energy
Shear Alfvén and magnetosonic waves Left: Shear Alfvén waves propagating parallel to B 0 The displacement ξ is orthogonal to B0 and k These are transverse waves Right: A magnetosonic wave propagating orthogonal to B 0 The displacement ξ is parallel to k but orthogonal to B 0 These are longitudinal waves
Properties of the shear Alfvén wave The dispersion relationship is ω 2 = k 2 V 2 A The wave is transverse The restoring force is magnetic tension No propagation orthogonal to B 0 The displacement vector ξ = ξ xˆx is orthogonal to both B 0 = B 0 ẑ and k = k ŷ + k ẑ Shear Alfvén waves are incompressible Since k ξ = 0, the linearized continuity and energy equations show that both ρ 1 and p 1 are 0
Properties of slow and fast magnetosonic waves Magnetosonic waves are analogous to sound waves modified by the presence of a magnetic field Magnetosonic waves are longitudinal and compressible The restoring force includes contributions from magnetic pressure and plasma pressure These are also known as magnetoacoustic waves and slow/fast mode waves
What is the difference between slow and fast magnetosonic waves? Obvious differences Fast waves are faster (or the same phase velocity) Slow waves are slower (or the same phase velocity) Plasma pressure and magnetic pressure perturbations may work together or in opposition In the slow wave, these two effects are out of phase In the fast wave, these two effects are in phase The phase velocity depends on the angle of propagation with respect to the magnetic field and plasma β Slow mode waves cannot propagate orthogonal to B0 Fast mode waves propagate quasi-isotropically
Phase velocity and energetics Friedrichs diagrams plot the phase speed of waves as distance from the origin as a function of angle with respect to B 0 The wave energy includes contributions from kinetic, magnetic, and thermal energy Half of wave energy is kinetic energy for all three waves Half of the shear Alfvén wave s energy is magnetic The energetics of the slow and fast waves depend on the type of wave, the angle of propagation, and plasma β
kz kz kz kz kz kx kx β = 0.1 β = 0.5 β = 1 β = 2 β = 10 Friedrichs diagrams for MHD waves: Phase speed plotted as radial distance, with the angle between k and B0 shown as the angle away from the y axis. Here, β = (cs/va) 2. Blue point: Alfvén speed. Black point: sound speed. Curve color-codes shown below. GREEN: SLOW-MODE RED: FAST-MODE BLUE: ALFVÉN Illustration of how MHD waves partition their total fluctuation energy into kinetic, magnetic, and thermal energy in various regimes: wavevectors parallel to B0 (top row), an isotropic distribution of wavevectors (middle row), wavevectors perpendicular to B0 (bottom row); columns denote plasma β regimes. Kinetic energy fractions are denoted vi, magnetic energy fractions are denoted Bi, and the thermal energy fraction is denoted th. Accessibility note for the top row of plots: The (green) slow mode is always the contour closest to the origin, and the (red) fast mode is always the contour furthest from the origin.
Limitations of this analysis We linearized the equations of ideal MHD and combined them to derive the dispersion relationship for shear Alfvén waves, fast magnetosonic waves, and slow magnetosonic waves for a uniform, static, and infinite background Discussion questions: In what ways do our assumptions limit the applicability of these results? What are some situations where these assumptions are invalid?
In situ measurements of waves in space plasmas I Spacecraft observations provide highly detailed localized information I Anticorrelations between δb and δv in Wind data are due to Alfve n waves in the solar wind near 1 AU (Shi et al. 2015)
Observations of plasma waves in the solar corona Alfvén waves are a leading mechanism for heating solar & stellar coronae and accelerating solar & stellar winds Power spectra of Doppler velocity observations show counter-propagating waves, which are necessary for the development of turbulence (Morton et al. 2015)
Laboratory experiments on plasma waves Laboratory experiments offer an opportunity to study plasma waves in detail Left: The Large Plasma Device at UCLA which is used to study Alfvén waves, interacting magnetic flux ropes, and other phenomena Right: Polarized shear Alfvén waves detected in the experiment (shown are isosurfaces of field-aligned current and perturbed magnetic field vectors)
X-ray stripes in Tycho s supernova remnant are interpreted as cosmic ray acceleration sites I I Accelerated particles around supernova remnant shock waves generate Alfve n waves Laming (2015) proposed that the interaction between these Alfve n waves and the shock may result in these stripes
Summary Waves are ubiquitous in astrophysical, laboratory, space, and heliospheric plasmas The three principal wave modes for ideal MHD are the shear Alfvén wave, the slow magnetosonic wave, and the fast magnetosonic wave The shear Alfvén wave is a transverse wave that propagates along the magnetic field Slow and fast magnetosonic waves are longitudinal waves that may propagate obliquely Plasma waves are well-studied in solar, space, and laboratory plasmas and play important roles in a variety of astrophysical plasmas