A Tale of Two Paradoxes: The Diamagnetic Polarization and Spitzer Polarization Paradoxes

An accurate calculation of the total polarization charge density in a plasma is essential for a self-consistent determination of the electric field. Yet, there are two different polarization paradoxes for magnetized plasmas that are finally resolved in this work. The “diamagnetic polarization paradox” refers to the paradoxical factor of 1/2 difference between the pressure-driven “diamagnetic polarization” density calculated using the real space drift theory versus guiding center and gyrokinetic theory. We prove that, with the standard definitions used in the literature, the drift centers and guiding centers are not identical! Yet, the results of both approaches can be made consistent with one another: Half of the diamagnetic polarization is due to the pullback from the guiding center density to the real space density. The other half is due to the pushforward because, within the drift kinetic ordering assumptions, the guiding center density must be expressed as the gyroaverage of the density in the limit of vanishing Larmor radius. For arbitrary magnetic field geometry, one must also resolve the “Spitzer polarization paradox”: the inertial response to a magnetization current that changes in time induces a polarization current with a nontrivial polarization charge density. Finally, it is shown that a certain change of reference frame can be made that allows the guiding center and drift kinetic reference frames to agree at second order.