Theory of the Drift-Wave Instability at Arbitrary Collisionality

Abstract

A numerically efficient framework that takes into account the effect of the Coulomb collision operator at arbitrary collisionalities is introduced. Such model is based on the expansion of the distribution function on a Hermite-Laguerre polynomial basis, to study the effects of collisions on magnetized plasma instabilities at arbitrary mean-free path. Focusing on the drift-wave instability, we show that our framework allows retrieving established collisional and collisionless limits. At the intermediate collisionalities relevant for present and future magnetic nuclear fusion devices, deviations with respect to collision operators used in state-of-the-art turbulence simulation codes show the need for retaining the full Coulomb operator in order to obtain both the correct instability growth rate and eigenmode spectrum, which, for example, may significantly impact quantitative predictions of transport. The exponential convergence of the spectral representation that we propose makes the representation of the velocity space dependence, including the full collision operator, more efficient than standard finite difference methods.

Linear theory of electron-plasma waves at arbitrary collisionality

“…4for 0.075 < α D < 0.2 and 0.015 < ν < 0.7 using the Coulomb collision operator, where a truncation at (P, J) = (18, 2) is used. Such values of (P, J) are in line with the estimate found inJorge et al (2018) which, by underestimating the effect of the collisional term C pj in the moment-hierarchy equation leads to P ∼ 4/ √ 2ν and J ∼ 2. For ν ∼ 0.1, this yields (P, J) ∼…”

“…We conclude therefore that the presence of additional momentum and energy conserving terms in the Dougherty operator with respect to the Lenard-Bernstein operator does not yield a damping rate closer to the Coulomb one. This was also pointed out in Jorge et al (2018), where a similar framework was used to derive the growth rate of the drift-wave instability.…”

“…Following Jorge et al. (2017, 2018), we solve the linearized kinetic equation, equation (2.10), at arbitrary collisionalities by expanding the perturbed distribution function into an orthogonal Hermite–Laguerre polynomial basis of the form where are physicists’ Hermite polynomials of order , defined by the Rodrigues’ formula and normalized via the orthogonality condition based on the scalar product with weight factor and the Laguerre polynomials of order , defined by the corresponding Rodrigues’ formula and orthonormal with respect to the weight Due to the orthogonality of the Hermite–Laguerre basis, the coefficients of the expansion in (2.13) can be computed via the expression With respect to a grid treatment using collocation points based on orthogonal polynomials (Belli & Candy 2008; Landreman & Ernst 2013), we note that the Hermite–Laguerre decomposition used in this work allows us to find the Rosenbluth potentials analytically as linear combinations of the moments of the distribution function, to find linear and nonlinear closures at an arbitrary number of moments and mean-free path, and to make much more evident the role of the parallel and perpendicular phase-mixing processes.…”

“…2016), but also to ion-acoustic waves (Epperlein, Short & Simon 1992; Tracy et al. 1993; Zheng & Yu 2000), and drift waves (Jorge, Ricci & Loureiro 2018). The difficulty associated with an accurate estimate of the collisional damping in a plasma at arbitrary collisionalities is related to the integro-differential character of the Coulomb collision operator (Helander & Sigmar 2005).…”

The collisional drift wave instability in steep density gradient regimes

“…After nearly 60 years density gradient driven DW instabilities remain an active theoretical research field. The latest efforts focus on a unified description of the collisional and collisionless DW instability [28,29] or proof instability for collisionless DWs in sheared magnetic fields after decades of misconception [19].…”