Waves in magnetized plasma Introduction – some definitions Self-consistency, linear theory Transverse waves in unmagnetized plasma Longitudinal waves, Landau damping Waves in magnetized plasma Beam-plasma instabilities, plasma masers

Dispersion relation Dielectric permittivity tensor Linearization Consider cold plasma Equation of motion Linearization Linearization

Cyclotron (Larmor) frequency Omit for simplicity (for a moment) index 

Current density

Dielectric permittivity tensor

Dispersion relation isotropic medium Now, consider waves Particular cases: isotropic medium With regard to the field perpendicular to the external magnetic field, fully magnetized plasma behaves as vacuum Now, consider waves Dispersion relation General case is rather complicated. Let us consider waves propagating parallel to and perpendicular to the external magnetic field

Waves propagating along the magnetic field 2 kinds of solution correspond to different orientations of the wave electric field with respect to the external magnetic field Plasma oscillations along the external magnetic field are the same as in the unmagnetized plasma

What is the physical sense of different signs in the dispersion relations? Waves are circularly polarized. Signs correspond to different directions of the electric field vector rotation Ordinary wave Dispersion depends on the direction of rotation Extraordinary wave

Faraday effect Consider linearly polarized wave propagating in plasma along the external magnetic field The wave remains linearly polarized but the polarization plane rotates The rotation angle linearly depends on the distance

Rotation of the polarization plane is known in optics Rotation of the polarization plane is known in optics. Media, in which the rotation occurs, are called optically active media. An example is a solution of chiral molecules (e.g., sugar). There is, however, a fundamental difference that concerns the direction of rotation. The direction of rotation with respect to the wave propagation in the magnetized plasma depends on the direction of the wave propagation with respect to the magnetic field Direction of external magnetic field is a separated direction When the wave passes through a slab of optically active medium, reflects from a mirror, and comes back, the polarization plane rotation on the opposite path compensates that on the direct path. In the magnetized plasma, the rotation angles on the direct and opposite paths add together.

Faraday rotation is an important tool in astrophysics Faraday rotation is an important tool in astrophysics. In particular, Faraday rotation measurements of polarized radio signals from extragalactic radio sources shadowed by the solar corona allows for estimating both the electron density distribution and the direction and strength of the magnetic field in the coronal plasma. Similarly, Faraday rotation measurements of the polarized light beam are used for the diagnostics of the dense plasma in experiments on plasma implosion (Z-machine, etc.)

Return to the dispersion relation Analytic solutions are possible only at some assumptions Slightly modified dispersion of the transverse wave in isotropic plasma Cutoff frequency This solution is valid if

Very low-frequency waves If one assumes RHS is zero, the difference between the ordinary and extraordinary waves is lost Quasineutrality Alfven velocity Alfven wave

Alfven wave is a type of magnetohydrodynamic wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines. It is like a wave travelling along a stretched string – the magnetic field line tension is analogous to string tension. Alfven velocity is not necessarily the velocity of an Alfven wave Account correctly for the RHS in the dispersion relation Lower sign – ordinary wave – Alfven wave Upper sign – extraordinary wave – fast magnetosonic wave

Intermediate frequency It is possible to omit the terms relating to the ion component Suppose now that Solution exists only for the upper sign Extraordinary wave non-contradictory assumption

This wave is called whistler or helicon This wave is called whistler or helicon. Terrestrial whistlers are electromagnetic waves occurring at audio frequencies. They are produced by lightning strokes and travel along the Earth’s magnetic field lines. Dispersion of group velocity Perceived as a descending tone  due to the slower velocity of the lower frequencies 

Example of spectrogram

Summary: 5 branches of oscillations propagating along the magnetic field Electromagnetic waves Langmuir oscillations There are asymptotic solutions: Whistler wave Magnetohydrodynamic waves

Waves propagating across the magnetic field Again, 2 kinds of solution correspond to different orientations of the wave electric field with respect to the external magnetic field

Transverse waves. Magnetic field does not influence Waves are neither transverse, nor longitudinal purely Dispersion relation

Again, solutions are possible under some assumptions Magnetohydrodynamic wave

Summary: 4 branches of oscillations propagating across the magnetic field Asymptotic solutions: Upper hybrid Lower hybrid