The electrostatic stability of electron-positron plasmas is investigated inthe point-dipole and Z-pinch limits of dipole geometry. The kinetic dispersionrelation for sub-bounce-frequency instabilities is derived and solved. For thezero-Debye-length case, the stability diagram is found to exhibit singularbehavior. However, when the Debye length is non-zero, a fluid mode appears,which resolves the observed singularity, and also demonstrates that both thetemperature and density gradients can drive instability. It is concluded that afinite Debye length is necessary to determine the stability boundaries inparameter space. Landau damping is investigated at scales sufficiently smallerthan the Debye length, where instability is absent.
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