Part 3, Chapter 4

MHD Waves

Alfvén waves and magnetosonic modes

4.1 Linearized MHD

To study waves, we linearize about a uniform equilibrium with B₀, ρ₀, p₀, and u₀ = 0:

$$\rho = \rho_0 + \rho_1, \quad \mathbf{B} = \mathbf{B}_0 + \mathbf{B}_1, \quad p = p_0 + p_1, \quad \mathbf{u} = \mathbf{u}_1$$

Perturbations ≪ equilibrium quantities

Assuming plane wave solutions ∝ exp(ik·r - iωt):

$$-i\omega\rho_0\mathbf{u}_1 = -i\mathbf{k}p_1 + \frac{1}{\mu_0}(i\mathbf{k} \times \mathbf{B}_1) \times \mathbf{B}_0$$

4.2 Characteristic Speeds

Alfvén Speed

$$v_A = \frac{B_0}{\sqrt{\mu_0 \rho_0}}$$

Speed of magnetic tension wave propagation

Sound Speed

$$c_s = \sqrt{\frac{\gamma p_0}{\rho_0}}$$

Adiabatic sound wave speed

4.3 Alfvén Waves

Incompressible waves with velocity perpendicular to both k and B₀:

$$\omega = k_\parallel v_A = k v_A \cos\theta$$

Shear Alfvén Wave Dispersion

Properties

• Incompressible: ∇·u₁ = 0
• Transverse oscillation
• No density/pressure change
• Propagates along B₀

Physical Picture

Field lines act like taut strings under tension B²/μ₀. Perturbations propagate at the Alfvén speed along the field.

4.4 Magnetosonic Waves

Compressible waves involving both pressure and magnetic forces. The dispersion relation is:

$$\left(\frac{\omega}{k}\right)^2 = \frac{1}{2}\left[(c_s^2 + v_A^2) \pm \sqrt{(c_s^2 + v_A^2)^2 - 4c_s^2 v_A^2 \cos^2\theta}\right]$$

Fast (+) and Slow (−) Magnetosonic Waves

Fast Magnetosonic Wave

• Compression of both gas and magnetic field in phase
• Phase velocity: v_f ≥ max(c_s, v_A)
• Propagates in all directions (isotropic for θ = π/2)

Slow Magnetosonic Wave

• Compression of gas and field out of phase
• Phase velocity: v_s ≤ min(c_s, v_A)
• Cannot propagate perpendicular to B₀

4.5 Special Cases

Parallel Propagation (θ = 0)

Alfvén

$$\omega = k v_A$$

Fast

$$\omega = k \cdot \max(c_s, v_A)$$

Slow

$$\omega = k \cdot \min(c_s, v_A)$$

Perpendicular Propagation (θ = π/2)

Fast (Compressional Alfvén)

$$\omega = k\sqrt{c_s^2 + v_A^2}$$

Slow & Alfvén

ω = 0 (no propagation)

4.6 Wave Energy and Group Velocity

Alfvén Wave Energy

$$\mathcal{E} = \frac{1}{2}\rho_0 u_1^2 + \frac{B_1^2}{2\mu_0}$$

Equal kinetic and magnetic energy

Group Velocity

$$\mathbf{v}_g = \frac{\partial \omega}{\partial \mathbf{k}}$$

Alfvén waves: v_g = v_A along B₀

Key Takeaways

✓ Three MHD wave modes: Alfvén, fast magnetosonic, slow magnetosonic
✓ Alfvén waves: incompressible, transverse, ω = k∥v_A
✓ Fast waves: v_f ≥ max(c_s, v_A), propagate all directions
✓ Slow waves: v_s ≤ min(c_s, v_A), guided by B₀
✓ v_A = B/√(μ₀ρ) is fundamental Alfvén speed

Interactive Simulations

MHD Wave Speeds: Alfven, Fast, and Slow Modes

Python

script.py57 lines

import numpy as np

# Three MHD wave modes in a magnetized plasma:
# Alfven: v_A = B / sqrt(mu0 * rho)
# Sound: c_s = sqrt(gamma * P / rho)
# Fast/Slow magnetosonic: v_f,s^2 = (v_A^2+c_s^2)/2 +/- sqrt(((v_A^2+c_s^2)/2)^2 - v_A^2*c_s^2*cos^2(theta))

mu0 = 4 * np.pi * 1e-7
m_p = 1.673e-27
k_B = 1.381e-23
e = 1.602e-19
gamma = 5/3

# Tokamak-like parameters
B = 3.0           # T
n = 5e19          # m^-3
T_eV = 2000       # eV
T = T_eV * e / k_B

rho = n * m_p
P = n * k_B * T * 2  # electron + ion

v_A = B / np.sqrt(mu0 * rho)
c_s = np.sqrt(gamma * P / rho)
beta = 2 * mu0 * P / B**2

print("MHD Wave Phase Speeds vs Propagation Angle")
print("=" * 60)
print(f"B = {B} T, n = {n:.0e} m^-3, T = {T_eV} eV")
print(f"v_A = {v_A:.3e} m/s ({v_A/1e6:.2f} Mm/s)")
print(f"c_s = {c_s:.3e} m/s ({c_s/1e6:.2f} Mm/s)")
print(f"beta = {beta:.3f}")
print()

angles = np.linspace(0, 90, 19)

print(f"{'theta(deg)':>10} {'v_fast (Mm/s)':>14} {'v_Alfven':>14} {'v_slow':>14}")
print("-" * 54)

for theta_deg in angles:
    theta = np.radians(theta_deg)
    cos_th = np.cos(theta)

# Fast and slow magnetosonic
    sum_sq = v_A**2 + c_s**2
    disc = sum_sq**2 - 4 * v_A**2 * c_s**2 * cos_th**2
    disc = max(disc, 0)

v_fast = np.sqrt(0.5 * (sum_sq + np.sqrt(disc)))
    v_slow = np.sqrt(0.5 * (sum_sq - np.sqrt(disc)))
    v_alfven = v_A * abs(cos_th)

print(f"{theta_deg:10.0f} {v_fast/1e6:14.4f} {v_alfven/1e6:14.4f} {v_slow/1e6:14.4f}")

print(f"\nAt theta=0: fast=max(v_A,c_s), slow=min(v_A,c_s)")
print(f"At theta=90: fast=sqrt(v_A^2+c_s^2), slow=0, Alfven=0")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Alfven Wave Propagation Time in Astrophysical Systems

Fortran

program.f9067 lines

program alfven_times
  implicit none
  ! Alfven crossing time: tau_A = L / v_A
  ! Relevant for MHD equilibrium, instability growth
  real(8) :: mu0, pi, m_p, k_B, e_charge
  real(8) :: B, n, rho, v_A, L, tau_A, T_eV
  integer :: i

mu0 = 4.0d0 * 3.14159265d0 * 1.0d-7
  pi = 4.0d0 * atan(1.0d0)
  m_p = 1.673d-27
  k_B = 1.381d-23
  e_charge = 1.602d-19

write(*,'(A)') "Alfven Speed and Crossing Times"
  write(*,'(A)') "================================"
  write(*,*)
  write(*,'(A18,A10,A10,A10,A14,A14)') &
    "System", "B(T)", "n(m^-3)", "L(m)", "v_A(m/s)", "tau_A"
  write(*,'(A)') repeat("-", 78)

! Tokamak
  B = 5.0d0; n = 1.0d20; L = 6.0d0
  rho = n * m_p; v_A = B / sqrt(mu0 * rho)
  tau_A = L / v_A
  write(*,'(A18,ES10.1,ES10.1,ES10.1,ES14.3,A4,ES10.3,A2)') &
    "Tokamak", B, n, L, v_A, "  ", tau_A, " s"

! Solar corona (active region)
  B = 0.01d0; n = 1.0d15; L = 1.0d8
  rho = n * m_p; v_A = B / sqrt(mu0 * rho)
  tau_A = L / v_A
  write(*,'(A18,ES10.1,ES10.1,ES10.1,ES14.3,A4,ES10.3,A2)') &
    "Solar corona", B, n, L, v_A, "  ", tau_A, " s"

! Solar wind at 1 AU
  B = 5.0d-9; n = 5.0d6; L = 1.5d11
  rho = n * m_p; v_A = B / sqrt(mu0 * rho)
  tau_A = L / v_A
  write(*,'(A18,ES10.1,ES10.1,ES10.1,ES14.3,A4,ES10.3,A2)') &
    "Solar wind", B, n, L, v_A, "  ", tau_A, " s"

! Earth magnetosphere
  B = 3.0d-5; n = 1.0d7; L = 6.4d7
  rho = n * m_p; v_A = B / sqrt(mu0 * rho)
  tau_A = L / v_A
  write(*,'(A18,ES10.1,ES10.1,ES10.1,ES14.3,A4,ES10.3,A2)') &
    "Magnetosphere", B, n, L, v_A, "  ", tau_A, " s"

! Interstellar medium
  B = 3.0d-10; n = 1.0d6; L = 3.0d16
  rho = n * m_p; v_A = B / sqrt(mu0 * rho)
  tau_A = L / v_A
  write(*,'(A18,ES10.1,ES10.1,ES10.1,ES14.3,A4,ES10.3,A2)') &
    "Interstellar", B, n, L, v_A, "  ", tau_A, " s"

! Accretion disk
  B = 1.0d-2; n = 1.0d18; L = 1.0d9
  rho = n * m_p; v_A = B / sqrt(mu0 * rho)
  tau_A = L / v_A
  write(*,'(A18,ES10.1,ES10.1,ES10.1,ES14.3,A4,ES10.3,A2)') &
    "Accretion disk", B, n, L, v_A, "  ", tau_A, " s"

write(*,*)
  write(*,'(A)') "MHD instabilities grow on Alfven timescale"
end program alfven_times

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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