A four-field gyrofluid model with neoclassical effects for the study of the rotation velocity of magnetic islands in tokamaks

Abstract

At equilibrium, in a tokamak, magnetic field lines lie on surfaces forming a family of nested tori, named magnetic surfaces. This structure of nested magnetic surfaces can be affected by instabilities. One of the most important ones is the so-called tearing mode, which is an instability "tearing" and reconnecting magnetic field lines. Magnetic reconnection locally breaks the topology of magnetic surfaces leading to a more energeticallyfavorable configuration. Magnetic islands result from the nonlinear evolution of tearing modes and represent a serious obstacle for obtaining nuclear fusion in magnetic confinement devices. In fact, the breaking of magnetic surfaces causes an increase in the heat and particle fluxes. The uncontrolled growth of magnetic islands can also lead to major disruptions, causing serious damage to the device. Many efforts have been made in the past decades to develop a theory of magnetic island dynamics in tokamaks. The interest in this kind of studies is to understand the conditions for the onset of the islands in the tokamak experiments and to control them to prevent their growth to large amplitudes and the consequent negative effects on confinement. Magnetic islands arise from the nonlinear evolution of tearing modes [1]. In the presence of an equilibrium density and temperature gradient, the tearing mode acquires a propagation frequency and the instability is said a drift-tearing mode [2]. According to the linear drift-tearing dispersion relation, the propagation frequency of the instability should be close to the electron diamagnetic frequency, ω − ω E ≈ ω * e [3,2], where the frequency is related to the velocity through the wave vector k, ω = k · v. The tearing mode is an instability characterized by a long wavelength, which corresponds to a small wavevector. ω E is the E ∧ B-drift frequency, due to the equilibrium electric field. In fact, the plasma as a whole rotates with the E ∧ B velocity, so that this contribution must be subtracted from the island rotation velocity (Doppler shift). Experimental observations of magnetic islands in tokamaks, under specific conditions, show a rotation frequency closer to the ion diamagnetic frequency, ω − ω E ≈ ω * i [4,5,6]. This disagreement between the predictions of the linear theory and the experimental observations raises doubts on the validity of the most credited theoretical models describing magnetic island dynamics. According to recently developed models, in the presence of significant electron temperature gradients, the introduction of the so-called "mode inductivity" [7] in the Ohm's law permits the existence of modes propagating with the ion diamagnetic frequency. This effect arises naturally in the linear regime, but the experimental observations of the island rotation concern nonlinear islands, thus a direct check of the validity of this model is not currently possible. Another widely accepted interpretation of the observed rotation velocity is that, when the island width becomes larger than the ion-acoustic radius, the ion fluid canno...

Study of stability and rotation of a chain of saturated, freely-rotating magnetic islands in tokamaks

“…The full procedure of calculation is described in [12]; here only the main ideas and the results are reported. The gyrokinetic equation for the ions is solved, under appropriate hypotheses, by including a collision operator to keep the collisional effects into account.…”

“…(1) 0 and V (1) 0 are ordered with δ 0 ∆ 1 . The quantities A, B, C, E, F, G, K, H are complicated functions of the zero order fields, and are defined in the appendix of [12]. The quantities x 2 and x represent the fluctuating quantities: x 2 = x 2 − x 2 / ⟨1⟩, such that x 2 = 0.…”

“…Ion polarization current depends on the island rotation velocity through the coupling between the rotation of the island and the rotation of the ion fluid [7]. This system of equations includes non-ideal effects (viscosity, diffusivity), lowest order FLR effects and neoclassical effects (damping, rotation) [12]. The FLR effects were included by performing a series of perturbative expansions on a system of gyrofluid equations [13,14] and by calculating the lowest order FLR corrections to the neoclassical poloidal flow damping.…”