Islands in the stream: The effect of plasma flow on tearing stability

Abstract

Reducing plasma flow clearly decreases the stability of tearing modes in multiple regimes (sawtooth, hybrid) in both high- and low-aspect-ratio tokamaks (DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)], Joint European Torus [M. Keilhacker and the JET Team, Plasma Phys. Controlled Fusion 41, 301 (1999)], National Spherical Torus Experiment [M. Ono, S. M. Kaye, Y.-K. M. Peng, Nucl. Fusion 40, 557 (2000)], each with distinct means of lessening rotation). Further, reducing flow makes pre-existing “saturated” islands larger at the same beta (β). Thus lower plasma flow impairs high-beta operation owing both to the destabilization and to the impact of tearing-mode islands. Experimental results suggest that flow shear (not flow) at the tearing rational surface is classically stabilizing, making the effective tearing stability index Δ′ of the total current density profile more negative (more stable). In this picture, with profiles and all else the same, the minimum metastable beta at which neoclassical tearing modes (NTMs) can destabilize is proportional to −Δ′ and hence lower flow and flow shear lead to possible destabilization (depending on seeding) at lower beta. Similarly, if destabilized, the saturated NTM island width is proportional to −β/Δ′ and thus increases as flow and flow shear are reduced. A working model gives a significant level of stabilizing shear if the plasma toroidal angular flow shear −dΩϕ/dr at a given rational surface is of order of the inverse of the product of the local values of the parallel magnetic shear length Ls and the Alfvén time τA. Experimental data are fitted for the effect of this normalization of flow shear in a simple empirical model for both onset and saturation of tearing modes. Most theoretical literature is on the consequence of flow shear on tearing stability at zero beta; tokamaks at high beta have large magnetic Prandtl number (an issue for the sign of the flow effect) and very large Lundquist number. It is in this regime that theory will be compared with experimentally based empirical models. The consequence for future tokamaks with low rotation may be lower tearing stability than now expected.

Influence of rotation on the (m,n) = (3, 2) neoclassical tearing mode threshold in the ASDEX Upgrade

“…This disagrees with the experiments at DIII-D, where a further decrease of the onset threshold with counter rotation was found. As already mentioned in previous works it was suggested that an offset of the threshold minimum exists which was not reached at DIII-D [1]. According to [23] the correct parameter to investigate the dependence of the NTM threshold on rotation is the velocity of the island in the laboratory frame f NTM,lab , where the local radial electric field E r is zero.…”

“…From this discrepancy one can infer that there exists an influence of plasma rotation on the underlying tearing stability. This means that at lower plasma rotation the ∆ ′ term changes such that the plasma is less stable against NTMs as already proposed in [7] and [1]. Further investigations have been made to disentangle the influence of rotation on the island stability itself as opposed to the trigger mechanism.…”

“…This island rotation frequency in the laboratory frame is defined as f NTM,lab = nf plasma,CXRS + f ⋆ i , where f ⋆ i is the flux surface averaged ion diamagnetic drift frequency, defined as nk B T i (dp i /dψ)/ep i (n is the toroidal mode number) and f plasma,CXRS is the plasma toroidal rotation frequency at the location of the island. As discussed in [1] this expression leads to a finite f NTM,lab even if f plasma,CXRS is zero and in consequence for f NTM,lab = 0 an offset in the counter direction at f plasma,CXRS ≈ f ⋆ i exists. In figure 5 the mode frequency f NTM at the onset is plotted against the toroidal plasma rotation frequency f plasma,CXRS at the resonant surface at the time of the NTM onset.…”

“…There exist several possibilities how plasma rotation can influence the NTM behaviour. It has been proposed that changes in rotation or rotation shear can reduce the normally stabilising effect of the classical tearing stability index ∆ ′ [1]. Rotation can also influence the stability of NTMs by means of small island effects.…”

Linear and nonlinear effect of sheared plasma flow on resistive tearing modes

“…In addition, flow shear at the rational surface is shown to make the effective tearing stability index ∆ more negative. 6,7 In the theoretical aspect, both analytical and numerical studies have been carried out. In the linear phase and in the slab geometry, plasma flow shear is found to either increase or decrease the linear growth rate, depending on the plasma viscosity, the magnetic shear and the strength of flow shear.…”