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Register a new userUniform derivation of Coulomb collisional transport thanks to Debye shielding
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Dominique F. Escande
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The effective potential acting on particles in plasmas being essentially the Debye-shielded Coulomb potential, the particles collisional transport in thermal equilibrium is calculated for all impact parameters $b$, with a convergent expression reducing to Rutherford scattering for small $b$. No cutoff at the Debye length scale is needed, and the Coulomb logarithm is only slightly modified.
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This paper brings further insight into the recently published N-body description of Debye shielding and Landau damping [Escande D F, Elskens Y and Doveil F 2014 Plasma Phys. Control. Fusion 57 025017]. Its fundamental equation for the electrostatic potential is derived in a simpler and more rigorous way. Various physical consequences of the new approach are discussed, and this approach is compared with the seminal one by Pines and Bohm [Pines D and Bohm D 1952 Phys. Rev. 85 338--353].
The derivation of Debye shielding and Landau damping from the $N$-body description of plasmas is performed directly by using Newtons second law for the $N$-body system. This is done in a few steps with elementary calculations using standard tools of calculus, and no probabilistic setting. Unexpectedly, Debye shielding is encountered together with Landau damping. This approach is shown to be justified in the one-dimensional case when the number of particles in a Debye sphere becomes large. The theory is extended to accommodate a correct description of trapping and chaos due to Langmuir waves. Shielding and collisional transport are found to be two related aspects of the repulsive deflections of electrons, in such a way that each particle is shielded by all other ones while keeping in uninterrupted motion.
A bounded plasma where the electrons impacting the walls produce more than one secondary on average is studied via particle-in-cell simulation. It is found that no classical Debye sheath or space-charge limited sheath exists. Ions are not drawn to the walls and electrons are not repelled. Hence the plasma electrons travel unobstructed to the walls, causing extreme particle and energy fluxes. Each wall has a positive charge, forming a small potential barrier or inverse sheath that pulls some secondaries back to the wall to maintain the zero current condition.
Avoiding impurity accumulation is a requirement for steady-state stellarator operation. The accumulation of impurities can be heavily affected by variations in their density on the flux-surface. Using recently derived semi-analytic expressions for the transport of a collisional impurity species with high-$Z$ and flux-surface density-variation in the presence of a low-collisionality bulk ion species, we numerically optimize the impurity density-variation on the flux-surface to minimize the radial peaking factor of the impurities. These optimized density-variations can reduce the core impurity density by $0.75^Z$ (with $Z$ the impurity charge number) in the Large Helical Device case considered here, and by $0.89^Z$ in a Wendelstein 7-X standard configuration case. On the other hand, when the same procedure is used to find density-variations that maximize the peaking factor, it is notably increased compared to the case with no density-variation. This highlights the potential importance of measuring and controlling these variations in experiments.
High-Z impurities in magnetic confinement devices are prone to develop density variations on the flux-surface, which can significantly affect their transport. In this paper, we generalize earlier analytic stellarator calculations of the neoclassical radial impurity flux in the mixed-collisionality regime (collisional impurity and low-collisionality bulk ions) to include the effect of such flux-surface variations. We find that only in the homogeneous density case is the transport of highly collisional impurities (in the Pfirsch-Schl{u}ter regime) independent of the radial electric field. We study these effects for a Wendelstein 7-X (W7-X) vacuum field, with simple analytic models for the potential perturbation, under the assumption that the impurity density is given by a Boltzmann response to a perturbed potential. In the W7-X case studied, we find that larger amplitude potential perturbations cause the radial electric field to dominate the transport of the impurities. In addition, we find that classical impurity transport can be larger than the neoclassical transport in W7-X.
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