Instabilities Produced in a Stationary Plasma by an "Almost Circular" Electron Beam

This investigation deals with an assembly of electrons which are moving along spiral trajectories in a cold and uniform ionized medium immersed in a static magnetic field ${\mathbf{B}}_0$. The electrons have velocity components $v_{\mathrm{II}}$ and $v_{\perp }$ which are, respectively, parallel and perpendicular to ${\mathbf{B}}_0$. It is assumed that $v_{\mathrm{II}}\ll v_{\perp }$ and, consequently, the assembly of electrons forms an "almost circular beam." The interaction of a low-density almost circular electron beam with the ionized medium is examined for $P$ (strong) and $B$ (weak) instabilities which produce growing waves aligned with ${\mathbf{B}}_0$. It is shown that the $P$ instabilities give rise to four excited waves—two with frequencies in the neighborhood of the ion gyrofrequency ${\Omega }_i$ and the electron gyrofrequency ${\Omega }_e$, and two with "intermediate" frequencies in the neighborhood of ${\Omega }_e{(1-{{\beta }_{\perp }}^2)}^{\frac{1}{2}}$ (where ${\beta }_{\perp }=\frac{v_{\perp }}{c}$, and $c$ is the velocity of light). The intermediate frequency waves have the largest rate of growth and represent, therefore, the dominant instabilities. These dominant instabilities are analyzed for various values of ${\beta }_{\perp }$ in several representative plasma-beam systems. The results of this investigation are applied to the study of certain natural very-low-frequency (VLF) radiations produced by streams of charged particles which are trapped by the earth's magnetic field and which interact with the outer regions of the ionosphere (exosphere). In this paper the term VLF radiation includes both audible and subaudible frequencies. It is shown that the excited waves in the VLF range have a continuous spectral distribution which includes both whistler and magnetodynamic waves. Both whistler and magnetodynamic waves occur for any ${\beta }_{\perp }$ whether in the relativistic or nonrelativistic range. (One should exclude, however, ${\beta }_{\perp }\approx 0$, i.e., when the beam is aligned in the direction of $B_0$, since in such case there is no excitation of whistler waves). It is significant that for values of ${\beta }_{\perp }$ less than approximately 0.97 all the excited waves within the VLF range, both whistler and magnetodynamic waves, have the same rate of growth $\left|\mathrm{Im}\delta \right|$ equal to $\left(\frac{\sqrt{2}}{2}\right){\sigma }^{\frac{1}{2}}{\omega }_e{(1-{\beta }^2)}^{\frac{1}{2}}{\beta }_{\perp }$, where ${\omega }_e$ is the plasma frequency of the ionized medium, $\beta$ is equal to $\frac{{({v_{\mathrm{II}}}^2+{v_{\perp }}^2)}^{\frac{1}{2}}}{c}$, and $\sigma$ is the ratio of the density of electrons in the beam to the density of the electrons in the plasma-beam medium. Thus the rate of growth $\left|\mathrm{Im}\delta \right|$ is independent of the frequency of the excited waves within the entire VLF range under consideration.