Unified theory of the semi-collisional tearing mode and internal kink mode in a hot tokamak: implications for sawtooth modelling

Abstract

In large hot tokamaks like JET, the width of the reconnecting layer for resistive modes is determined by semi-collisional electron dynamics and is much less than the ion Larmor radius. Firstly a dispersion relation valid in this regime is derived which provides a unified description of drift-tearing modes, kinetic Alfvén waves and the internal kink mode at low beta. Tearing mode stability is investigated analytically recovering the stabilising ion orbit effect, obtained previously by Cowley et al. (Phys. Fluids 29 3230 1986), which implies large values of the tearing mode stability parameter ∆ ′ are required for instability. Secondly, at high beta it is shown that the tearing mode interacts with the kinetic Alfvén wave and that there is an absolute stabilisation for all ∆ ′ due to the shielding effects of the electron temperature gradients, extending the result of Drake et. al (Phys. Fluids 26 2509 1983) to large ion orbits. The nature of the transition between these two limits at finite values of beta is then elucidated. The low beta formalism is also relevant to the m = n = 1 tearing mode and the dissipative internal kink mode, thus extending the work of Pegoraro et al. (Phys. Fluids B 1 364 1989) to a more realistic electron model incorporating temperature perturbations, but then the smallness of the dissipative internal kink mode frequency is exploited to obtain a new dispersion relation valid at arbitrary beta. A diagram describing the stability of both the tearing mode and dissipative internal kink mode, in the space of ∆ ′ and beta, is obtained. The trajectory of the evolution of the current profile during a sawtooth period can be plotted in this diagram, providing a model for the triggering of a sawtooth crash.

Kinetic microtearing modes and reconnecting modes in strongly magnetised slab plasmas

“…is the familiar parameter of semicollisional theory [34,35]. In the following, for ease of comparison of our results with previous results, we will sometimes leave explicit the combination ρ e /L T ; however that should always be regarded as…”

Improved criterion for sawtooth trigger and modelling

“…Here, we establish the stable window in an operating diagram for the m = n = 1 drift-tearing mode and the resistive internal kink mode, defined in terms of the plasma beta, β = 8πp/B 2 , and the instability drive, represented by a quantity ∆ ′ which is inversely related to the potential energy, δW , for the internal kink mode. This diagram results from an earlier study [5] of the stability of these two modes based on a plasma model with semicollisional electrons and ions whose Larmor orbit exceeds the semi-collisional layer width where reconnection occurs. The trajectory of the core plasma state in this diagram as the profiles of q and plasma pressure, p, change during the sawtooth ramp can be monitored and the triggering of the sawtooth crash identified as the point at which the linear stability threshold is crossed.…”

“…The role of the drift-tearing and kink modes In Ref. [5] we developed a unified theory of the drift-tearing mode and internal kink mode relevant to the m = 1, n = 1 mode resonant at q = 1 in large hot tokamaks such as ITER. Specifically, we adopted a plasma model with semi-collisional electrons for which:…”

“…[5] with approximate value ∼ −η 1/2 e , τ = T e /T i , δ 0 = √ ω * e ν e r 1 L s /v the , and r 1 is the position at which q = 1. This result is valid at small ∆ ′ ρ i ∼β.…”

“…the sawtooth trigger conditions) employed in Ref. [4] by analytic marginal stability conditions derived in [5]. This has the effect of replacing unknown coefficients in Ref.…”