Macroscopic electrostatic stability properties of nonrelativistic non-neutral electron flow in a cylindrical diode with applied magnetic field

Electrostatic stability properties of nonrelativistic non-neutral electron flow in a cylindrical diode with applied magnetic field $B_0{\hat{e}}_z$ are investigated within the framework of the macroscopic cold-fluid-Poisson equations. The electrostatic eigenvalue equation is derived for perturbations about the general class of slow rotational equilibria with angular velocity profile ${\omega }_b^{-}\mathrm{}\left(r\right)=\frac{V_{\theta b}^0\mathrm{}\left(r\right)\mathrm{}}{r}=\left(\frac{{\omega }_c}{2}\right)\{1-{[1-(\frac{4}{{\omega }_c^2{r}^2}\left)\int a^{r}d{r}^{\prime }{r}^{\prime }{\omega }_{\mathrm{pb}}^2(r^{\prime }\right)]}^{\frac{1}{2}}\}$. Here, ${\omega }_c=\frac{e{B}_0}{\mathrm{mc}}$, ${\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)=\frac{4\pi n_b^0\mathrm{}\left(r\right)\mathrm{}{e}^2}{m}$ $n_b^0\mathrm{}\left(r\right)\mathrm{}$ is the equilibrium electron density profile, and the cathode is located at $r=a$ and the anode at $r=b$. Space-charge-limited flow is assumed with $E_r^0\mathrm{}(r=a)=0\mathrm{}$ and ${\phi }^0\mathrm{}(r=a)=0\mathrm{}$. The exact eigenvalue equation is simplified for low-frequency flute perturbations with $k_z=0\mathrm{}$ and ${|\omega -l{\omega }_b^{-}\mathrm{}(r\left)\right|}^2\ll {\omega }_c^2-{\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)\mathrm{}$, assuming ${\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)<{\omega }_c^2$ and a moderate-aspect-ratio diode $(R_0\gg b-a)$. In this regime, it is shown that $\frac{\partial n_b^0}{\partial r}\le 0$ over the interval $a\le r\le b$ is a sufficient condition for stability, and specific examples of stable oscillations (rectangular density profile) and weak resonant diocotron instability (gentle density bump) are analyzed in detail. Finally, the exact eigenvalue equation is solved numerically for a wide range of density profiles $n_0^b\mathrm{}\left(r\right)\mathrm{}$ and values of $\frac{{\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)\mathrm{}}{{\omega }_c^2}$ leading to weak and strong instability driven by velocity shear with $\frac{\partial {\omega }_b^{-}\mathrm{}\left(r\right)\mathrm{}}{\partial r}\ne 0$.