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 alsoChapter 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 withplasma 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

I The 1D wave equationI Algebraic solutionI Eigenmode solution

I Sound wavesI Linearization of equations of hydrodynamicsI Derivation of dispersion relationship

I MHD wavesI Linearization of MHD equationsI Introduce displacement vector ξ and MHD force operator F(ξ)I Derivation of dispersion relationshipI Shear Alfven, fast magnetosonic, and slow magnetosonic waves

I Observations of MHD wavesI Solar coronaI Space plasmasI Laboratory experiments

Why do we care about waves?

I Waves are ubiquitous in magnetized plasmasI Just as sound waves are ubiquitous in air

I Waves are the simplest way that a system responds todisturbances and applied forces

I Waves propagate information and energy through a system

I Waves are closely related to shocks, instabilities, andturbulence

I Plasmas display a rich variety of waves within and beyondMHD

Applications of waves in plasma astrophysics

I Space physicsI Earth’s ionosphere, magnetosphere, and solar wind

environment

I Solar and stellar physicsI Coronal heatingI Acceleration of solar and stellar winds

I Molecular clouds and star formation

I Interstellar medium

I Cosmic ray acceleration and transport

I Accretion disks and jets

I Pulsar magnetospheres

Whenever a plasma is disturbed, there will be waves!

Example: the 1D wave equation

I The wave equation for u in one dimension is

∂2u

∂t2= c2∂

2u

∂x2(1)

where c is a real constant that represents the wave speed

I The solutions are waves traveling at velocities of ±cI The wave equation is a hyperbolic partial differential equation

I Connection to conservation laws

The algebraic solution to the 1D wave equation

I Define two new variables

ξ(x , t) = x − ct η(x , t) = x + ct (2)

I Rewrite the wave equation as

∂2u

∂ξ ∂η= 0 (3)

I The solutions are then

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

I Use separation of variables and look for solutions of the form

uω(x , t) = e−iωt f (x) (6)

I Plug this solution into the wave equation

∂2

∂t2

[e−iωt f (x)

]= c2 ∂

2

∂x2

[e−iωt f (x)

](7)

−ω2e−iωt f (x) = c2e−iωtd2

dx2f (x) (8)

−k2f (x) =d2

dx2f (x) (9)

where k = ω/c. This is an eigenvalue equation for f (x).

I Next: identify eigenfunctions of the differential operator d2

dx2

with corresponding eigenvalue −k2.

Eigenmode decomposition of the 1D wave equation

I Look for solutions of the form

f (x) = Ae±ikx (10)

I The solution to the wave equation for this eigenmode is

uω(x , t) = Ae−ikx−iωt + Be ikx−iωt (11)

I Recall Euler’s formula

e ix = cos x + i sin x (12)

I Take the real part of Eq. 11 to get

uω(x , t) = A cos (kx + ωt) + B cos (kx − ωt) (13)

The solutions are waves propagating in the ±x directions. UseFourier techniques to find the full solution.

Definitions

I The lines in the u-x plane on which x − ct or x + ct areconstant are called characteristics

I The wave vector k points in the direction of wavepropagation and has a magnitude of k = 2π/λ where λ is thewavelength

I The phase velocity is the rate at which the phase of a wavepropagates through space

Vp =ω

k(14)

I The group velocity is the rate at which the overall shape ofthe waves’ amplitudes propagates through space

Vg =∂ω

∂k(15)

Finding the dispersion relationship for sound waves

I Represent variables as the sum of a background component(denoted ‘0’) and a small perturbed component (denoted ‘1’)

ρ(r, t) = ρ0 + ρ1(r, t) (16)

p(r, t) = p0 + p1(r, t) (17)

V(r, t) = V1(r, t) (18)

I Assume the background is homogeneous, time-independent,and static (V0 = 0)

I Look for solutions proportional to e i(k·r−ωt)

I Solve for a dispersion relationship that connects the wavevector k with the angular frequency ω

Linearizing the equations of hydrodynamics

I The equations of hydrodynamics are

∂ρ

∂t+∇ · (ρV) = 0 (19)

ρ

(∂

∂t+ V · ∇

)V +∇p = 0 (20)

∂p

∂t+ V · ∇p + γp∇ · V = 0 (21)

I Linearize the equations. Drop higher order terms. Use thatthe background is constant.

∂ρ1

∂t+ ρ0∇ · V1 = 0 (22)

ρ0∂V1

∂t+∇p1 = 0 (23)

∂p1

∂t+ γp0∇ · V1 = 0 (24)

Linearizing our first equation

I We start out with the continuity equation

∂ρ

∂t+∇ · (ρV) = 0 (25)

Substitute in ρ(r, t) = ρ0 + ρ1(r, t) and V(r, t) = V1(r, t).

∂ρ0

∂t︸︷︷︸=0

+∂ρ1

∂t+∇ · (ρ0V1) + ∇ · (ρ1V1)︸ ︷︷ ︸

second order

= 0 (26)

∂ρ1

∂t+∇ · (ρ0V1) = 0 (27)

I We dropped ∂ρ0∂t because the background is time-independent

I We dropped ∇ · (ρ1V1) because ρ1 and V1 are both small, sothe product resulting from this second order term will benegligibly small.

Deriving a wave equation for hydrodynamics

I Take the time derivative of Eq. 23

ρ0∂2V1

∂t+∇∂p1

∂t= 0 (28)

I Then substitute ∂p1∂t = −γp0∇ · V1 from Eq. 24 to get a wave

equation∂2V1

∂t2− c2

s∇ (∇ · V1) = 0 (29)

where the sound speed is

cs ≡√γp0

ρ0(30)

Assume that the solution is a superposition of plane waves

I Assume plane wave solutions of the form

V1(r, t) =∑

k

Vkei(k·r−ωt) (31)

I Differential operators turn into multiplications with algebraicfactors

∇ → ik,∂

∂t→ −iω (32)

I The problem is linear and homogeneous, so we consider eachcomponent separately.

I The wave equation then becomes

∂2V1

∂t2− c2

s∇ (∇ · V1) = 0

(−iω)2 V1 − c2s (ik) (ik · V1) = 0

ω2V1 − c2s k (k · V1) = 0 (33)

The dispersion relationship for sound waves

I Choose coordinates so that k = kz z, which then implies thatV1 = V1z z. Eq. 33 becomes(

ω2 − k2z c

2s

)Vz1 = 0 (34)

I The non-trivial solutions are

ω = ±kzcs (35)

Find the phase velocity and group velocity

I The dispersion relationship is

ω = ±kzcs (36)

I The phase velocity and group velocity are

Vp ≡ω

kz= ±cs (37)

Vg ≡∂ω

∂k= ±cs (38)

I Sound waves are compressional because ∇ · V1 6= 0

I Sound waves are longitudinal because V1 and k are parallel

How do we derive the dispersion relation for MHD waves?1

I Linearize the equations of ideal MHD.I Take a Lagrangian approach

I Partially integrate the equations with respect to timeI Write equations in terms of the displacement from equilibrium

I Assume solutions proportional to e−i(k·r−ωt)

I Derive a dispersion relationship that relates k and ω

I Investigate the properties of the three resulting wave modes

1Here we follow Boyd & Sanderson §4.5 and §4.8.

Begin with the equations of ideal MHD

I The continuity, momentum, induction, and adiabatic energyequations are

∂ρ

∂t+∇ · (ρV) = 0 (39)

ρ

(∂

∂t+ V · ∇

)V =

J× B

c−∇p (40)

∂B

∂t= ∇× (V × B) (41)(

∂

∂t+ V · ∇

)p = −γρ∇ · V (42)

The linearized equations of ideal MHD

I The continuity, momentum, induction, and adiabatic energyequations are linearized to become

∂ρ1

∂t= −V1 · ∇ρ0 − ρ0∇ · V1 (43)

ρ0∂V1

∂t=

(∇× B1)× B0

4π−∇p1 (44)

∂B1

∂t= ∇× (V1 × B0) (45)

∂p1

∂t= −V1 · ∇p0 − γp0∇ · V1 (46)

Here we ignored second and higher order terms and usedAmpere’s law.

I The terms −V1 · ∇ρ0 and −V1 · ∇p0 vanish if we assume thebackground is uniform

The displacement vector, ξ, describes how much theplasma is displaced from the equilibrium state2

I If ξ(r, t = 0) = 0, then the displacement vector is

ξ(r, t) ≡∫ t

0V1(r, t ′)dt ′ (47)

I Its time derivative is the perturbed velocity,

∂ξ

∂t= V1(r, t) (48)

2A side benefit of using slides is that I do not have to try writing ξ on thechalkboard.

Integrate the continuity equation with respect to time

I Put the linearized continuity equation with a uniformbackground in terms of ξ

∂ρ1

∂t= −V1 · ∇ρ0 − ρ0∇ · V1 (49)

= −∂ξ∂t· ∇ρ0 − ρ0∇ ·

∂ξ

∂t(50)

I Integrate this with respect to time∫ t

0

∂ρ1

∂t ′dt ′ =

∫ t

0

[− ∂ξ∂t ′· ∇ρ0 − ρ0∇ ·

∂ξ

∂t ′

]dt ′ (51)

which leads to a solution for ρ1 in terms of just ξ

ρ1(r, t) = −ξ(r, t) · ∇ρ0 − ρ0∇ · ξ(r, t) (52)

We can similarly put the linearized induction and energyequations in terms of ξ

I Integrating the linearized equations with respect to time yieldssolutions for the perturbed density, magnetic field, and plasmapressure:

ρ1(r, t) = −ξ(r, t) · ∇ρ0 − ρ0∇ · ξ(r, t) (53)

B1(r, t) = ∇×[ξ(r, t)× B0(r)

c

](54)

p1(r, t) = −ξ(r, t) · ∇p0(r)− γp0(r)∇ · ξ(r, t) (55)

The perturbed density ρ1 doesn’t appear in the otherequations, which form a closed set

I However, we still have the momentum equation to worryabout!

The linearized momentum equation in terms of ξ and F[ξ]

I Using the solutions for ρ1, B1, and p1 we arrive at

ρ0∂2ξ

∂t2= F[ξ(r, t)] (56)

which is reminiscent of Newton’s second law

I The ideal MHD force operator is

F(ξ) = ∇(ξ · ∇p0 + γp0∇ · ξ)

+1

4π(∇× B0)× [∇× (ξ × B0]

+1

4π{[∇×∇× (ξ × B0)]× B0} (57)

which is a function of the displacement vector ξ andequilibrium fields, but not of V1 = ∂ξ

∂t .

Building up intuition for the displacement vector ξ andforce operator F(ξ)

I The displacement vector ξ gives the direction and distancea parcel of plasma is displaced from the equilibrium state

I The force operator F(ξ) gives the direction and magnitudeof 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

I Assume that the plasma is uniform and infinite

I Perform a Fourier analysis by assuming solutions of the form

ξ (r, t) =∑k,ω

ξ (k, ω) e−i(k·r−ωt) (58)

I The linearized momentum equation,

ρ0∂2ξ

∂t2= F (ξ (r, t)) , (59)

then becomes

ρ0ω2ξ = kγp0 (k · ξ) +

{k× [k× (ξ × B0)]} × B0

4π(60)

Deriving the dispersion relation for MHD waves

I Choose Cartesian axes such that

k = k⊥y + k‖z (61)

I Expanding the vector products yields(ω2 − k2

‖V2A

)ξx = 0 (62)(

ω2 − k2⊥c

2s − k2V 2

A

)ξy − k⊥k‖c

2s ξz = 0 (63)

−k⊥k‖c2s ξy +

(ω2 − k‖c

2s

)ξz = 0 (64)

where cs is the sound speed

I The Alfven speed is defined as

VA ≡

√B2

0

4πρ0(65)

The dispersion relation for MHD waves

I To get a non-trivial solution (ξ 6= 0), we need

det

ω2 − k2‖V

2A 0 0

0 ω2 − k2⊥c

2s − k2V 2

A −k⊥k‖c2s

0 −k⊥k‖c2s ω2 − k2

‖c2s

= 0

(66)

I Eq. 66 reduces to the dispersion relation for MHD waves(ω2 − k2

‖V2A

) [ω4 − k2

(c2s + V 2

A

)ω2 + k2k2

‖c2s V

2A

]= 0

(67)

Non-trivial solutions of the dispersion relation for MHDwaves

I The solution corresponding to shear Alfven waves is

ω2 = k2‖V

2A (68)

I The solution corresponding to slow and fast magnetosonicwaves is

ω2 =1

2k2(c2s + V 2

A

) [1±√

1− δ]

(69)

where δ is

δ ≡4k2‖c

2s V

2A

k2(c2s + V 2

A

)2, 0 ≤ δ ≤ 1 (70)

I All three solutions are realI No growth or decayI No dissipation or free energy

Shear Alfven and magnetosonic waves

I Left: Shear Alfven waves propagating parallel to B0

I The displacement ξ is orthogonal to B0 and kI These are transverse waves

I Right: A magnetosonic wave propagating orthogonal to B0

I The displacement ξ is parallel to k but orthogonal to B0

I These are longitudinal waves

Properties of the shear Alfven wave

I The dispersion relationship is ω2 = k2‖V

2A

I The wave is transverse

I The restoring force is magnetic tension

I No propagation orthogonal to B0

I The displacement vector ξ = ξx x is orthogonal to bothB0 = B0z and k = k⊥y + k‖z

I Shear Alfven waves are incompressibleI Since k · ξ = 0, the linearized continuity and energy equations

show that both ρ1 and p1 are 0

Properties of slow and fast magnetosonic waves

I Magnetosonic waves are analogous to sound waves modifiedby the presence of a magnetic field

I Magnetosonic waves are longitudinal and compressible

I The restoring force includes contributions from magneticpressure and plasma pressure

I These are also known as ‘magnetoacoustic waves’ and‘slow/fast mode waves’

What is the difference between slow and fast magnetosonicwaves?

I Obvious differencesI Fast waves are faster (or the same phase velocity)I Slow waves are slower (or the same phase velocity)

I Plasma pressure and magnetic pressure perturbations maywork together or in opposition

I In the slow wave, these two effects are out of phase

I In the fast wave, these two effects are in phase

I The phase velocity depends on the angle of propagation withrespect to the magnetic field and plasma β

I Slow mode waves cannot propagate orthogonal to B0

I Fast mode waves propagate quasi-isotropically

Phase velocity and energetics

I Friedrichs diagrams plot the phase speed of waves as distancefrom the origin as a function of angle with respect to B0

I The wave energy includes contributions from kinetic,magnetic, and thermal energy

I Half of wave energy is kinetic energy for all three wavesI Half of the shear Alfven wave’s energy is magneticI The energetics of the slow and fast waves depend on the type

of wave, the angle of propagation, and plasma β

kx

kzkzkzkzkz

kx

β = 10β = 2β = 1β = 0.5β = 0.1

“Friedrichs diagrams” for MHD waves: Phase speed plotted as radial distance, with the angle between k and B0 shown as the angle awayfrom the y–axis. Here, β = (cs/VA)

2. Blue point: Alfven speed. Black point: sound speed. Curve color-codes shown below.

RED: FAST-MODE

BLUE:ALFVEN

GREEN: SLOW-MODE

Illustration of how MHD waves partition their total fluctuation energy into kinetic, magnetic, and thermal energy in various regimes: wavevectorsparallel to B0 (top row), an isotropic distribution of wavevectors (middle row), wavevectors perpendicular to B0 (bottom row); columns denote plasma βregimes. 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 thecontour closest to the origin, and the (red) fast mode is always the contourfurthest from the origin.

Limitations of this analysis

I We linearized the equations of ideal MHD and combined themto derive the dispersion relationship for shear Alfven waves,fast magnetosonic waves, and slow magnetosonic waves for auniform, static, and infinite background

Discussion questions: In what ways do our assump-tions limit the applicability of these results? What aresome situations where these assumptions are invalid?

In situ measurements of waves in space plasmas

I Spacecraft observations provide highly detailed localizedinformation

I Anticorrelations between δB and δV in Wind data are due toAlfven waves in the solar wind near 1 AU (Shi et al. 2015)

Observations of plasma waves in the solar corona

I Alfven waves are a leading mechanism for heating solar &stellar coronae and accelerating solar & stellar winds

I Power spectra of Doppler velocity observations showcounter-propagating waves, which are necessary for thedevelopment of turbulence (Morton et al. 2015)

Laboratory experiments on plasma waves

I Laboratory experiments offer an opportunity to study plasmawaves in detail

I Left: The Large Plasma Device at UCLA which is used tostudy Alfven waves, interacting magnetic flux ropes, and otherphenomena

I Right: Polarized shear Alfven waves detected in theexperiment (shown are isosurfaces of field-aligned current andperturbed magnetic field vectors)

X-ray stripes in Tycho’s supernova remnant are interpretedas cosmic ray acceleration sites

I Accelerated particles around supernova remnant shock wavesgenerate Alfven waves

I Laming (2015) proposed that the interaction between theseAlfven waves and the shock may result in these stripes

Summary

I Waves are ubiquitous in astrophysical, laboratory, space, andheliospheric plasmas

I The three principal wave modes for ideal MHD are the shearAlfven wave, the slow magnetosonic wave, and the fastmagnetosonic wave

I The shear Alfven wave is a transverse wave that propagatesalong the magnetic field

I Slow and fast magnetosonic waves are longitudinal waves thatmay propagate obliquely

I Plasma waves are well-studied in solar, space, and laboratoryplasmas and play important roles in a variety of astrophysicalplasmas