Diocotron mode instability for an electron plasma confined in the magnetic field of a long, straight, current carrying wire

As a large aspect ratio approximation to a pure electron plasma confined by the magnetic field of a current hoop [1], consider a pure electron plasma that is confined by the magnetic field of  a long, straight, current carrying wire.  This limit has the theoretical advantage that equilibrium states are cylindrically symmetric about the wire, facilitating stability analysis of diocotron modes.  For the case where the radial density profile multiplied by the radius squared [i.e.,n₀(r)r² ] is a top hat extending from r₁ to r₂ and where only E×B drifts are retained, analytic theory yields a mode growth rate of order (4π e²n₀)/(I k_z)  when k_z r₁is substantially larger than unity and k_z(r₂-r₁) is about one.  Here, I  is the current strength and k_z is the axial wave number of the diocotron mode.  Numerical results for similarly hollow, but rounded,  density profiles show similar growth  rates.  The modes cannot be stabilized by giving the wire a uniform negative charge  since the drift velocity v_d = c E_r /B_θ from the charged wire   is independent of r, and simply adds a shift  to the real part of the mode  frequency.  Numerical solutions show that curvature and grad-B drifts become important and do stabilize the mode when k_z λ_D approaches unity, where λ_D  is the Debye length.  

1.  H. Saitoh, et al., Phys. Plasmas 17,  112111; Stoneking, et al., J. Plasma Physics 86, 155860601