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Plasma Physics
Table of Contents
Unit 6 Overview: Magnetohydrodynamics
6.1 MHD equations and approximations
6.2 MHD equilibrium and stability
6.3 MHD waves and instabilities

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6.3 MHD waves and instabilities

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Magnetohydrodynamics (MHD) waves and instabilities are key to understanding plasma behavior in magnetic fields. These phenomena shape how energy moves through plasmas and impact their stability, crucial for fusion research and space physics.

MHD waves come in different flavors: Alfvén waves, fast magnetosonic waves, and slow magnetosonic waves. Each type has unique properties that affect how energy travels in plasmas. Meanwhile, MHD instabilities can disrupt plasma confinement, posing challenges for fusion reactors and astrophysical systems.

MHD Wave Modes

Alfvén and Magnetosonic Waves

  • Alfvén waves propagate along magnetic field lines, causing oscillations in the magnetic field and plasma velocity
  • Alfvén waves do not compress the plasma, maintaining constant density during propagation
  • Magnetosonic waves involve both magnetic field and plasma pressure perturbations
  • Fast magnetosonic waves propagate faster than the Alfvén speed, compressing both the magnetic field and plasma
  • Slow magnetosonic waves travel slower than the Alfvén speed, with magnetic and plasma pressure perturbations out of phase

Wave Characteristics and Dispersion Relation

  • Alfvén wave speed depends on magnetic field strength and plasma density: vA=Bμ0ρv_A = \frac{B}{\sqrt{\mu_0 \rho}}
  • Fast magnetosonic waves have speeds greater than both Alfvén and sound speeds
  • Slow magnetosonic waves have speeds less than both Alfvén and sound speeds
  • Dispersion relation describes the relationship between wave frequency and wavenumber
  • For MHD waves, the dispersion relation takes the form: ω2=k2vA2cos⁡2θ+12k2(vA2+cs2)±12k2(vA2+cs2)2−4vA2cs2cos⁡2θ\omega^2 = k^2 v_A^2 \cos^2 \theta + \frac{1}{2} k^2 (v_A^2 + c_s^2) \pm \frac{1}{2} k^2 \sqrt{(v_A^2 + c_s^2)^2 - 4v_A^2 c_s^2 \cos^2 \theta}

Wave Properties

Phase and Group Velocities

  • Phase velocity represents the speed at which wave crests move: vp=ωkv_p = \frac{\omega}{k}
  • Group velocity describes the speed at which wave energy propagates: vg=∂ω∂kv_g = \frac{\partial \omega}{\partial k}
  • For Alfvén waves, phase and group velocities are equal and constant
  • Fast and slow magnetosonic waves exhibit different phase and group velocities depending on propagation angle
  • Group velocity determines energy transport in plasma, crucial for understanding wave-particle interactions

Wave Propagation Characteristics

  • Alfvén waves propagate anisotropically, with maximum speed along magnetic field lines
  • Fast magnetosonic waves propagate nearly isotropically, with slight preference for perpendicular propagation
  • Slow magnetosonic waves propagate primarily along magnetic field lines
  • Wave damping occurs due to collisional and collisionless processes in plasma
  • Resonant absorption can transfer wave energy to particles, heating the plasma

MHD Instabilities

Current-Driven Instabilities

  • Kink instability occurs in cylindrical plasmas when current exceeds a critical value
  • Kink instability causes helical deformation of plasma column, potentially leading to disruption
  • Sausage instability results in periodic constrictions along a plasma column
  • Current-driven instabilities often limit the maximum current in fusion devices (tokamaks)
  • Stabilization techniques include external conducting shells and magnetic shear

Pressure-Driven Instabilities

  • Rayleigh-Taylor instability arises when a heavy fluid supports a lighter fluid against gravity
  • In MHD, magnetic fields can stabilize Rayleigh-Taylor instability up to a critical wavelength
  • Ballooning instability occurs in regions of adverse magnetic field curvature
  • Ballooning modes limit plasma pressure in toroidal fusion devices
  • Interchange instability involves the exchange of flux tubes in inhomogeneous plasmas

Tearing Mode Instability

  • Tearing modes lead to magnetic reconnection and formation of magnetic islands
  • Tearing instability occurs when magnetic field lines break and reconnect in a different topology
  • Growth rate of tearing modes depends on plasma resistivity and magnetic shear
  • Neoclassical tearing modes can degrade confinement in fusion plasmas
  • Stabilization methods include current profile control and localized heating/current drive