Role of Flow Shear in the Ballooning Stability of Tokamak Transport Barriers

A tokamak’s confinement time is greatly increased by a transport barrier (TB), a region having a high pressure gradient and usually also a strongly sheared plasma flow. The pressure gradient in a TB can be limited by ideal magnetohydrodynamic instabilities with a high toroidal mode number $n$ (“ballooning modes”). Previous studies in the limit $n\to \infty$ showed that arbitrarily small (but nonzero) flow shears have a large stabilizing influence. In contrast, the more realistic finite $n$ ballooning modes studied here are found to be insensitive to sub-Alfvénic flow shears, provided the magnetic shear $s\sim 1$ (typical for TBs near the plasma’s edge). However, for the lower magnetic shears that are associated with internal transport barriers, significantly lower flow shears will influence ballooning mode stability, and flow shear should be retained in the analysis of their stability.

Figure 1

Fourier mode amplitudes as a function of $x=n(q_0-q)$, for low magnetic shear $s=0.1$ and $\alpha =0.5$.

Figure 2

Growth rates as a function of flow shear $d\Omega /dq$, for $\Lambda =\infty$ (gray triangles) and $\Lambda =1.35$ (black squares, upper and lower curves). The continuous line shows the $n=\infty$ growth rates calculated by Miller et al. [9] ($s=1.0$, $\alpha =2.0$).

Figure 3

Fourier mode amplitudes as a function of $x=n(q_0-q)$ (a)–(c) and the mode structures in poloidal cross section (d)–(f). (c) contains the mode structures of the pair of solutions with complex conjugate growth rates ($s=1$, $\alpha =2$).