Abstract
The stability of a cold ionring velocity distribution in a thermal plasma is analyzed. In particular, the effect of plasma temperature and density on the instability is considered. A high ring density (compared to the background plasma) neutralizes the stabilizing effect of the warm background plasma and the ring is unstable to the generation of waves below the lowerhybrid frequency even for a very high temperature plasma. For ring densities lower than the background plasma density, there is a slow instability where the growth rate is less than the backgroundion cyclotron frequency and, consequently, the backgroundion response is magnetized. This is in addition to the widely discussed fast instability where the wave growth rate exceeds the backgroundion cyclotron frequency and hence the background ions are effectively unmagnetized. Thus, even a low density ring is unstable to waves around the lowerhybrid frequency range for any ring speed. This implies that effectively there is no velocity threshold for a sufficiently cold ring.
The authors would like to thank Dr. Chris Crabtree, Dr. Sidney Ossakow, and Dr. Dan Winske for a careful reading of this article.
This work was supported by ONR and DARPA.
I. INTRODUCTION
II. DISPERSION RELATION OF AN IONRING DISTRIBUTION
III. FAST INSTABILITY WITH UNMAGNETIZED BACKGROUND IONS
A. Fast instability for a cold ring distribution in a warm plasma with
B. Fast instability for a cold ring distribution in a warm plasma with
IV. SLOW INSTABILITY WITH MAGNETIZED BACKGROUND IONS
A. Slow instability for a cold ionring distributionin a warm plasma with
B. Slow instability for a cold ring distribution in a warm plasma with
V. INSTABILITY FOR A HIGH DENSITY RING DISTRIBUTION
VI. FAST INSTABILITY FOR A WARM RING DISTRIBUTION
VII. DISCUSSION
VIII. CONCLUSION
Key Topics
 Plasma instabilities
 51.0
 Dispersion relations
 44.0
 Plasma temperature
 23.0
 Plasma waves
 22.0
 Magnetic susceptibilities
 13.0
Figures
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).
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).
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 Zfunction from Eq. (25). is normalized to the hydrogenelectron mass ratio . The maximum parallel wavenumber is approximately given by Eq. (26).
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 Zfunction from Eq. (25). is normalized to the hydrogenelectron mass ratio . The maximum parallel wavenumber is approximately given by Eq. (26).
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.
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.
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.
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.
Numerical solution of Eq. (39) when and . Shown in solid lines are the curves (36). The dashed line is the righthand side of Eq. (39). These solutions are the continuation of the modes from Eq. (32) for .
Numerical solution of Eq. (39) when and . Shown in solid lines are the curves (36). The dashed line is the righthand side of Eq. (39). These solutions are the continuation of the modes from Eq. (32) for .
The ratio from Eq. (30) for several values of the growth rate for . only from and .
The ratio from Eq. (30) for several values of the growth rate for . only from and .
(a) The real frequency for the high density instability when , , and . For small , . Shown in () is the frequency (11) of the lowerhybrid 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 .
(a) The real frequency for the high density instability when , , and . For small , . Shown in () is the frequency (11) of the lowerhybrid 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 .
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 backgroundion distribution . The solid black line is the total growth and is the sum of the growth and damping.
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 backgroundion distribution . The solid black line is the total growth and is the sum of the growth and damping.
The solution of the dispersion relation (55) when , indicating the regions of stability in {, , } space. The ratio of the ring thermal velocity to the backgroundion velocity is plotted as a function of for different values of the ring thermal velocity . Values of the parameters above the curve are unstable.
The solution of the dispersion relation (55) when , indicating the regions of stability in {, , } space. The ratio of the ring thermal velocity to the backgroundion velocity is plotted as a function of for different values of the ring thermal velocity . Values of the parameters above the curve are unstable.
Tables
Electrostatic wave modes from a cold ring distribution.
Electrostatic wave modes from a cold ring distribution.
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