Global Theory of Microtearing Modes in the Tokamak Pedestal

The pedestal of H-mode tokamaks displays strong magnetic fluctuations correlated with the evolution of the electron temperature. The microtearing mode (MTM), a temperature-gradient-driven instability that alters magnetic topology, is compatible with these observations. Here we extend the conventional theory of the MTM to include the global variation of the temperature and density profiles. The offset between the rational surface and the location of the pressure gradient maximum ($\mu$) emerges as a crucial parameter for MTM stability. The extended theory matches observations on the Joint European Torus tokamak.

Figure 1

Magnetic spectrogram showing fluctuations throughout several ELM cycles from JET-C pulse #78697. The vertical broadband lines are ELMs. The inter-ELM bands of fluctuations with toroidal mode numbers $n=-4$ and $n=-8$ are clearly visible. The negative sign indicates the mode is propagating in the electron diamagnetic direction. The low frequency bands with positive $n$ numbers are core modes and not relevant to our study. Figure courtesy of Ref. [7].

Figure 2

Growth rate and real frequency of the mode as $\mu$ is increased. The blue line indicates the growth rate computed using the full spatial dependence of ${\omega }_{*}$. The orange line (dotted) shows the growth rate if ${\omega }_{*}$ is evaluated at the rational surface and treated as uniform (i.e., the local approximation). (a) and (b) show the eigenvalue corresponding to $r=0.28$ and $\widehat{s}=0.006$. (c) and (d) correspond to $r=0.14$ and $\widehat{s}=0.012$. Other parameters are set to $x_{*}/{\rho }_s=10.0$, $\beta =0.002$, $\nu /{\omega }_{\mathrm{peak}}=1.0$, and $\eta =2.0$.

Figure 3

The eigenmode and the conductivity evaluated at $\omega$. (a) and (c) correspond to the rational surface and ${\omega }_{*}$ peak aligning, and (b) and (d) correspond to an offset of $\mu =3.0$. The eigenvalues were determined to be $\omega /{\omega }_{\mathrm{peak}}=1.084+0.045i$ and $\omega /{\omega }_{\mathrm{peak}}=1.009+0.009i$, respectively. Parameters were set to $x_{*}/{\rho }_s=10.0$, $\beta =0.002$, $\widehat{s}=0.012$, $\nu /{\omega }_{\mathrm{peak}}=1.0$, $\eta =2.0$, $Z_{\mathrm{eff}}=1.0$, and $m_i/m_e=1836$. Here, $V(x)=i\sigma$.

Figure 4

Here, we display the comparison of the slab model to global linear gyrokinetic electromagnetic numerical experiment (GENE). (a) Displays the Gaussian fit of the ${\omega }_{*}$ profile and the location of the rational surfaces determined from the $q$ profile. (b) Displays the growth rates of each toroidal mode number. (c) Displays the real frequencies of the unstable modes. The stable markers indicate a negative growth rate for both the slab model and GENE (excluding $n=7$ and $n=9$). $n=7$ (gray) is predicted to be unstable by GENE; however, due to the large jump in real frequency inter alia, it is not an MTM. $n=9$ is predicted to be unstable by GENE, but the growth rate is marginal.