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Stability of an ion-ring distribution in a multi-ion component plasma
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Image of FIG. 1.
FIG. 1.

The growth rate , solved using the dispersion relation (18) with , normalized to the growth rate in a cold background (16) as a function of the temperature. As decreases, the growth rate is diminished according to Eq. (20).

Image of FIG. 2.
FIG. 2.

The numerical solution of Eq. (21) for the growth rate vs parallel wavenumber for a range of electron thermal velocities with . The growth is inhibited at larger due to the imaginary part of the electron Z-function from Eq. (25). is normalized to the hydrogen-electron mass ratio . The maximum parallel wavenumber is approximately given by Eq. (26).

Image of FIG. 3.
FIG. 3.

A graphical solution of Eq. (32a), . The top figure shows as a function of in (---) when , representing a cold plasma. The black line is the curve of . There is a solution only when . The middle figure shows in (---) when , representing a warm plasma. The harmonic structure of is evident. In addition to the solution , there are solutions for , where . The bottom figure shows the derivative as a function of with . The solid curve is for the warm plasma and the (---) curve is for . For a warm plasma, the derivative has a nonzero minimum value between the harmonics.

Image of FIG. 4.
FIG. 4.

A graphical representation of from Eq. (33) as a function of the growth rate and the temperature of the plasma. When , for any plasma temperature, which means the solution to the dispersion relations (28), (29), (30a), and (30b) are valid and there is an instability with frequency and growth are as in Sec. III. However, for there exists a new possibility for instability at harmonics of the cyclotron frequency , with growth rate (34) when in a warm plasma.

Image of FIG. 5.
FIG. 5.

Numerical solution of Eq. (39) when and . Shown in solid lines are the curves (36). The dashed line is the right-hand side of Eq. (39). These solutions are the continuation of the modes from Eq. (32) for .

Image of FIG. 6.
FIG. 6.

The ratio from Eq. (30) for several values of the growth rate for . only from and .

Image of FIG. 7.
FIG. 7.

(a) The real frequency for the high density instability when , , and . For small , . Shown in (---) is the frequency (11) of the lower-hybrid branch corresponding to the fast instability in a cold background, where has been defined from the resonance condition. (b) The solution for the growth rate to the dispersion relation (45). For small , , but as increases Landau damping decreases the growth rate such that .

Image of FIG. 8.
FIG. 8.

An example growth rate (58) as a function of the phase velocity. The (---) curve is the growth due to the ring distribution function (1) and the curve is the damping due to the background-ion distribution . The solid black line is the total growth and is the sum of the growth and damping.

Image of FIG. 9.
FIG. 9.

The solution of the dispersion relation (55) when , indicating the regions of stability in {, , } space. The ratio of the ring thermal velocity to the background-ion velocity is plotted as a function of for different values of the ring thermal velocity . Values of the parameters above the curve are unstable.


Generic image for table
Table I.

Electrostatic wave modes from a cold ring distribution.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Stability of an ion-ring distribution in a multi-ion component plasma