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# Plasma Waves

This module calculates various properties (the index of refraction, dispersion relation, phase velocity, group velocity, ellipticity, and wavelength) below the electron cyclotron frequency for linear cold plasma waves in an infinite, homogeneous, and magnetized plasma.

The formulas used are those of T.H. Stix (The Theory of Plasma Waves, McGraw-Hill, 1962). The inertial effects of the ions and electrons are retained and all dissipative effects, including collisions, are neglected. This model contains neither sound waves, particle bunching, nor Landau damping, and it does not describe plasma shock waves.

We assume that the plasma consists of multiple ion species, which can be specified by the users, and electrons. The electron density, however, is calculated based on the assumption that the plasma is in a neutral state, i.e., N+ = N-. The magnetic field B is along the positive z-direction. The wave vector k is in the xz-plane (ky = 0) with an inclination to the magnetic field.

## Graph selection

This module offers the following graph types:

• Index of refraction
• This options calculates the index of refraction (n = c/Vph or ck/ω) of the plasma and plots it versus the frequency in the top right panel.

The bottom right panel displays the polar plot of the index of refraction at the frequency you choose. It shows how the index of refraction at the frequency depends on the propagation angle, the angle between k and B. In this plot, the magnetic field B is parallel to the vertical axis. Different colors represent the two different modes: Blue for left-handed and red for right-handed.

• Dispersion relation
• This option shows the dispersion relation of the plasma, i.e., a plot of the frequency of the wave versus its wave number. Note that the frequency scale is vertical.

The lower graph displays the polar plot of k (1/km) for waves at the frequency you choose. It shows how the value of k at this frequency depends on the propagation angle, the angle between k and B. In this plot, the magnetic field B is parallel to the vertical axis.

• Phase velocity
• This option calculates the phase velocity and plots it versus the frequency in the top right panel.

The lower graph displays the Clemmow-Mullaly-Allis (CMA) diagram for waves at the frequency you choose. It shows how the phase velocities of waves at this frequency depend on the propagation angle, the angle between k and B. In this plot, the magnetic field B is parallel to the vertical axis.

• Group velocity
• This option calculates the group velocity and plots it versus the frequency in the top right panel.

The lower graph displays the polar plot of the group velocity of the wave at the frequency you choose. It shows how the direction of group velocities is controlled by the magnetic field direction. In this plot, the magnetic field B is parallel to the vertical axis. The polar plot displays the group velocity as a function of the B-Vgr angle.

Note that this polar plot does not indicate the direction of wave vector k but the direction of the group velocity. In general, the group velocity is not parallel to the phase velocity.

• Parallel group velocity
• This option calculates the group velocity parallel to the wave vector k and plots it versus the frequency in the top right panel.

The lower graph displays the polar plot of parallel group velocity for waves at the frequency you choose. It shows how the parallel group velocity of the wave at this frequency depends on the propagation angle, the angle between k and B. In this plot, the magnetic field B is parallel to the vertical axis.

Note that the parallel group velocity refers to the component of group velocity along the wave vector k, or phase velocity, but not the magnetic field B.

• Perpendicular group velocity
• This option calculates the group velocities perpendicular to the wave vector k and plots it versus the frequency in the top right panel.

The lower graph displays the polar plot of the perpendicular group velocity of the wave in km at the frequency you choose. It shows how the perpendicular group velocity of the wave at this frequency depends on the propagation angle, the angle between k and B. In this plot, the magnetic field B is parallel to the vertical axis.

Note that the perpendicular group velocity refers to the component of group velocity transverse to the wave vector k or phase velocity, but not the magnetic field B.

• Ellipticity
• This option calculates the wave ellipticity and plots it versus the frequency in the top right panel.

The wave ellipticity is defined as iBy/Bxz, where By is the magnetic field perturbation in the y-direction and Bxz is the magnetic perturbation in the xz-plane. In other words, By is perpendicular to the background magnetic field B and thus is the purely transverse component. Bxz is in the plane containing both the k vector and the background magnetic field, and thus it contains both transverse and compressional components. Positive ellipticity represents right-handed polarized wave and negative ellipticity represents left-handed polarized wave.

The lower graph displays the average normal surface for waves at the frequency you choose. It shows how the phase velocities of waves at this frequency depend on the propagation angle, the angle between k and B. In this plot, the magnetic field B is parallel to the vertical axis.

• Wavelength
• This option calculates the wavelength and plots it versus the frequency in the top right panel.

The lower right graph displays the polar plot of the wavelength in km at the frequency you choose. It shows how the wavelength of the wave at this frequency depends on the propagation angle, the angle between k and B. In this plot, the magnetic field B is parallel to the vertical axis.

## Basic parameters

You can specify the field strength, wave propagation angle, and plasma properties (mass, charge states, and number density) of at most four ion species by entering those values in the text boxes.

## Normalization

The Normalize the quantities plotted check box allows you to graph normalized values, as explained below.

• Select this option to normalize the frequency f by the cyclotron frequency of the ion species in the first column (fci1); the wavenumber k is normalized by fci1/VA, where VA is the Alfvén velocity based on the total mass density; and the speed v is normalized by the Alfvén velocity VA, the velocity of the left-hand mode parallel to the magnetic field at very low frequencies.
• Clear this option to plot the original values. The frequency f is in Hz; the wavenumber k is in 1/km, the speed v is in km/s, and the index of refraction is dimensionless.

## Characteristic wave properties

If you want to have a better understanding of the wave at one particular frequency, you can select a frequency by either clicking in the top right panel or manually entering the frequency value under the Selected frequency text box to the left.

If you click in the top right panel, the chosen frequency will be marked as a vertical dashed line (except in the dispersion relation graph, in which case it will be marked as a horizontal dashed line).

The bottom left panel displays characteristic wave properties at the frequency you choose. The two columns give the properties for each of the two modes whenever either mode propagates at the chosen frequency. The lower right panel shows the corresponding wave properties.