Drift kinetic theory of the NTM magnetic islands in a finite beta general geometry tokamak plasma

Abstract

In [K Imada et al. Nucl. Fusion 59 (2019) 046016] a new 4D drift kinetic nonlinear theory, valid in the limit of a low beta, small inverse aspect ratio, circular cross section, toroidal geometry, to describe the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is derived. In [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] this theory is reduced in a low collisionality limit, which allows a dimensionality reduction to a 3D problem to efficiently resolve the collisional dissipation layer in the vicinity of the trapped-passing boundary. [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] adopts an improved model for the magnetic drift frequency, which reduces the threshold magnetic island half-width from $8.73 \rho_{b i}$, where $\rho_{b i}$ is the trapped ion banana orbit width, to $1.46 \rho_{b i}$, making it in a closer agreement with experimental observations for the large aspect ratio tokamak equilibrium. In the present paper, the theory is extended to a high beta, arbitrary tokamak geometry to capture the plasma shaping effects on the NTM threshold physics with the focus on the non-zero triangularity discharges that are known to have a strong impact on the plasma MHD stability. First, it is found that the higher triangularity plasma is more prone to NTMs which is in agreement with the $2/1$ tearing mode onset relative frequency measurements in DIII-D. Second, the NTM threshold dependence on the tokamak inverse aspect ratio obtained in [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] is refined and extended to a finite aspect ratio limit. Third, the NTM threshold dependence on poloidal beta is obtained and benchmarked against the EAST threshold island width measurements.

Drift kinetic theory of neoclassical tearing modes in tokamak plasmas: polarisation current and its effect on magnetic island threshold physics

“…At w ≳ ρ ϑi , the layer around drift islands remains narrow, broadening the layers from both sides of the magnetic island. As was previously noted in [20,21], these separatrix layer effects are expected to be less significant for non-rotating magnetic islands but become crucial when the magnetic island propagation frequency is non-zero (or alternatively in the presence of the equilibrium radial electric field if one works in the magnetic island rest frame). This current version of RDK-NTM is being referred to as RDK-NTM (v.3) in table 1.…”

“…A prime denotes the derivative with respect to ψ. In [21], it was shown that the second term on the right hand side of equation ( 2) associated with the perpendicular component of the perturbed vector potential, A 1 ⊥ , results in the O δ * ∆ 2 Vt L f 0 corrections in the drift kinetic equation and thus is to be omitted. Here δ * = ρ/L is a small parameter associated with the drift kinetic ordering (ρ is the particle Larmor radius and L characterises the system size) [22], V t is the particle thermal velocity and f 0 is the equilibrium distribution function 6 .…”

“…Following [18][19][20][21], we work in the island rest frame. Therefore, the Φ ′ eqm (and hence ω E ≡ mΦ ′ eqm /q s ) variation in the island rest frame provides the island propagation frequency, ω, dependence in the reference frame in which the equilibrium electric field is zero far from the magnetic island (see section 2.5 of [23]).…”

“…There are two boundary layers: one is in pitch angle space in the vicinity of the boundary between passing and trapped particles, and one that surrounds the drift island separatrix and results from different topologies (and thus averages in helical angle) inside and outside the island. In [20,21] we have introduced a model further reduced in collisions (being referred to here as the RDK-NTM approach): being valid in the limit when the particle collision frequency is much less than the magnetic drift frequency and/or parallel streaming around the island flux contours, it allows one to better resolve boundary layers and thus learn more about the physics related to collisional dissipation, providing an extension to the DK-NTM formulation to low collisionality plasmas (see table 1 for the DK-and RDK-NTM comparison). In this low collision frequency limit, the collision operator can be treated perturbatively, and one then finds that all the spatial variation can be captured in a single stream function, S = S (p φ , ξ, λ, V; σ) so that g (0) = g (0) (p φ , ξ , λ, V) = g (0) (S, λ, V) and the problem reduces to 3D.…”

“…In the limit of finite beta, C also includes a contribution that results from the Vlasov operator (second term on the right hand side of equation ( 12)). Following [21], we require…”