A Tale of Two Polarization Paradoxes I: the Diamagnetic Polarization Paradox

Abstract

An accurate calculation of the total polarization charge density in a plasma is essential for a self-consistent determination of the electric field. Yet, for a magnetized plasma, there are two different polarization paradoxes that are finally resolved in this work. The "diamagnetic polarization paradox'' refers to the fact that there is a paradoxical factor of 1/2 difference between the pressure-driven "diamagnetic polarization'' density calculated using the real space drift theory versus the action-angle space guiding center and gyrokinetic theory that has not been explained before. In this work, it is shown that the results of both approaches can be made consistent with one another. Half of the diamagnetic polarization is due to the transformation from the guiding center density to the real space density. The other half is due to the fact that, within the drift kinetic ordering assumptions, the guiding center density should be expressed as the gyroaverage of the density in the limit of vanishing Larmor radius. Expressions for the diamagnetic polarization density are given that are accurate to first order in amplitude and all orders in gyroradius within gyrokinetic theory for a constant magnetic field. Applications to anisotropic Maxwell-Boltzmann particle distribution functions are presented. Because the total energy and toroidal momentum are local invariants, they do not generate net polarization effects: the electric and thermodynamic polarizations must precisely cancel. In contrast, anisotropic dependance on the magnetic moment generates a net polarization proportional to the temperature anisotropy.

Figures (4)

• Figure 1: The finite size of the gyroradius, ${\rho}=v_\perp/\Omega$, is responsible for diamagnetic polarization effects.
• Figure 2: At constant particle position (black dot), an increase in gyroradius, illustrated by a change in color from blue to green to red, causes a shift in the guiding center position (colored dots) indicated by arrows.
• Figure 3: For drift theory, the two stages of FLR effects double the thermodynamic polarization, ${ \boldsymbol{\pi}_{K}}=q { \mathbf{d}_{K}}$, driven by gradients of the PDF. In the figure, the pressure gradient increases to the left. (1) The first stage displaces the zeroth order real space particle position (black circle) by the amount $-\boldsymbol {\rho}$ (black arrow) to the guiding center position (black). The pressure gradient shifts the polarization center further out by ${ \mathbf{d}_{K}}={\rho}_T^2/2L_p$ (red arrow) to the (red disk), where $1/L_p=\left|\nabla_\perp \ln{p}\right|$. (2) The second stage displaces the polarization center (red disk) back by the amount $\boldsymbol {\rho}$, but the shift including the pressure gradient is reduced to $\boldsymbol {\rho}-{ \mathbf{d}_{K}}$ (blue arrow), which yields the final polarization center (purple disk) in real space. This is equivalent to displacing the zeroth order particle position by $2{ \mathbf{d}_{K}}={\rho}_T^2/L_p$ (purple arrow, purple circle). Guiding center theory only has one stage, so the total pressure-driven displacement is ${ \mathbf{d}_{K}}={\rho}_T^2/2L_p$.
• Figure 4: For GC/GK/GF theory, the density in real space depends on the velocity space average of the guiding center distribution, ${ { \left< f \right>}_ z }={ { \left< F \right>}_ z }$, determined by the integral over the black circle. For DK/DF theory, there are two stages of finite orbit width effects. First transform $f_0(z)$ to guiding center space, then gyroaverage to obtain $F(Z)={ { \left< f_0 \right>}_Z }$. Second, transform $F(Z)$ back to real space, and, finally, gyroaverage to obtain the density. The two stage process implies that each point on the black circle is in turn defined by a gyroaverage over one of the blue circles. While the outer radius of the blue region is twice as large, the orbit integral over the blue region is determined by the root mean-square radius and yields twice the area of the black circle.