Theory of the Drift-Wave Instability at Arbitrary Collisionality

A numerically efficient framework that takes into account the effect of the Coulomb collision operator at arbitrary collisionalities is introduced. Such a 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.

Figure 1

Growth rate of the DW instability obtained from the moment hierarchy, Eq. (4), as a function of ($k_{\parallel }$, $k_{\perp }$) and, from left to right, ${\nu }_{ei}=0.05$, 1, 10, and 500, in the cold-ion limit with ${\sigma }_e=0.023$.

Figure 2

Comparison of the growth rate $\gamma$ maximized over $k_{\parallel }$ and $k_{\perp }$, as a function of collisionality ${\nu }_{ei}$, between the solution of the drift-reduced Braginskii model, the collisionless model, the linearized moment hierarchy using a simplified Lenard-Bernstein, Dougherty, and the Coulomb collision operator. For the Coulomb case, a truncation at different ($p_{\max }$, $j_{\max }$) is considered.

Figure 3

Eigenvalue spectra obtained with the collisionless model, the linearized moment hierarchy equation using the Coulomb and the Dougherty collision operator at the wave number ($k_{\parallel }$, $k_{\perp }$) corresponding to the fastest growing mode in the cold-ion limit for ${\nu }_{ei}=0.4$ and ${\sigma }_e=0.023$. The analysis is carried out with $p_{\max }=15$ and $j_{\max }=2$.