Resistive Reduced MHD Modeling of Multi-Edge-Localized-Mode Cycles in Tokamak $X$-Point Plasmas

The full dynamics of a multi-edge-localized-mode (ELM) cycle is modeled for the first time in realistic tokamak $X$-point geometry with the nonlinear reduced MHD code jorek. The diamagnetic rotation is found to be instrumental to stabilize the plasma after an ELM crash and to model the cyclic reconstruction and collapse of the plasma pressure profile. ELM relaxations are cyclically initiated each time the pedestal gradient crosses a triggering threshold. Diamagnetic drifts are also found to yield a near-symmetric ELM power deposition on the inner and outer divertor target plates, consistent with experimental measurements.

Figure 1

Magnetic and kinetic energies of the $n=8$ mode without diamagnetic effects. The time in $x$ axis is normalized to the Alfvén time $t_A$.

Figure 2

Magnetic field topology in the $n=8$ ELM simulation without diamagnetic effect, given at the peak of the ELM crash (a)–(c) and after the ELM crash at $t\sim 7\times 10^3{t}_A$ (b)–(d). The Figs. 2 and 2 show the Poincaré plot of the magnetic topology in the poloidal plane, and the Fig. 2 [respectively, 2] is a zoom of the plasma edge of the Fig. 2 [respectively, 2] in ($\psi$, $\theta$) coordinates for a normalized flux $\psi \ge 0.8$. Note the full edge reconnection at the peak of the crash (destroyed magnetic surfaces) and the remaining partial reconnection after the crash (island structures with an ergodic very edge).

Figure 3

Kinetic energy of the modes $n=2–8$ in the multiharmonic $n=0:2:8$ simulation with diamagnetic effects.

Figure 4

Pedestal pressure gradient normalized over flux surfaces before (dash line), during (dash-dot line), and after (full line) the ELM crash.

Figure 5

Peeling-ballooning diagram: evolution of the maximum pressure gradient as a function of the mean pedestal current in quasiperiodic regime. The time between two points is $20t_A$, and each color represents one ELM.

Figure 6

Integrated power reaching the inner (dash line) and outer (full line) divertor plates.

Figure 7

Radial distribution of the heat flux (in $\mathrm{GW}/{\mathrm{m}}^2$) reaching the inner and outer divertor target plates: without flows included in simulation (full line) or with diamagnetic, neoclassical, and toroidal flows (dashed line).