Alfvén Waves

Alfvén waves are low frequency magnetohydrodynamic plasma waves or oscillations. They were first theoretically predicted by Hannes Alfvén in the 1950's, and he earned the Nobel Prize for his work in plasma physics. Alfvén waves are of fundamental importance in the behavior of many laboratory and space plasmas. Basically, these waves can be thought of as waves on a magnetic string. The magnetic field acts like a string, and the plasma particles act like beads. Heavier beads - heavier plasma ions - means the waves are slower and the wavelength is longer.

Alfvén waves communicate information about changes in magnetic field topologies, and are especially important in the dynamics of magnetic reconnection. For example, changes in the auroral current magnitude and spatial configurations, or changes in the magnetospheric configuration, involve propagation by Alfvén waves. These waves are also present in fusion plasma experiments, and toroidal Alfvén eigenmodes may adversely affect the plasma confinement of burning fusion reactors. These waves are somewhat difficult to study in laboratory experiments. We do so at the University of New Mexico using helicon sources.

Alfvén Speed

The Alfvén speed is defined as

v_A² = B^2//m₀r

where r is the mass density. There are two types of Alfvén waves, Shear and Compressional. Shear waves bend the magnetic field lines, but the local plasma kinetic pressure remains constant. Compressional waves change the total plasma pressure.

Shear Waves

The figure at the right shows shear waves propagating shear along the z axis, ie. along the magnetic field. In any x plane the waves have the same phase. However, in adjacent planes the waves can have a different different phases. Thus, the wavevector k can be oriented at an arbitrary angle to the background magnetic field. The relationship between frequency and wavenumber, w and k, is the dispersion relation. For shear waves this becomes

w/k = v_A cos q

where q is the angle between k and the magnetic field. But, the wave energy still propagates only along the magnetic field. Shear waves are generated in response to magnetic field line bending, radiating away the energy.

Close to the ion cyclotron frequency the shear wave dispersion relation changes to account for the ion cyclotron resonance. It becomes:

where w_ci is the ion cyclotron frequency, qB/m_i. At the cyclotron frequency the Alfvén wave becomes resonant, and the wavelength goes to zero.

Shear waves can be further categorized as either inertial or kinetic depending on whether the Alfvén speed is less than or greater than the electron thermal speed.

Compressional Waves

Compressional waves, on the other hand can propagate at any angle to the magnetic field. As shown in the figure at the left, adjacent field lines in the same plane can have different phases and "compress" the plasma between them. The compressional wave propagates at the same speed regardless of the direction of k.

w/k = v_A

If we include thermal effects the compressional wave has two modes: the slow and fast mode. In the slow mode changes in magnetic pressure are out of phase with changes in kinetic pressure, and the total pressure remains constant. For the fast mode the total pressure fluctuates. Physically, the slow mode tends to reduce parallel pressure gradients, while the fast mode tends to reduce perpendicular pressure gradients. The dispersion relation for the two mode is:

c_s is the ion sound speed.

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