Fluid equations are derived for nearly rigid electromagnetic displacements in two-dimensional equilibria described by distribution functions of the form F(H, Pz). Solutions in closed form of the Maxwell-Vlasov equations are given for the m=1 mode in z pinches, as well as for the corresponding free boundary mode in Extrap. If feedback stabilization is excluded, an instability of modes with large axial wavelengths is predicted, regardless of the shape of the equilibrium pressure profile. In z pinches, stability of modes with mod kza mod <0.5 requires that the radius of a conducting cylinder that surrounds the plasma is not more than twice the plasma radius. It is stressed that finite orbit effects at the magnetic O-point in z pinches are rigorously treated, since the formulas are essentially exact in the limit of negligible wall stabilization and small axial wavelength of the perturbation. Even chaotic particle orbits are adequately described in some cases.
Approximate equations for plasmas in mirror machines