Destabilizing effects of edge infernal components on n = 1 resistive wall modes in CFETR 1GW steady-state operating scenario

The external kink instability, which occurs when plasma $\beta_{N}$ exceeds the no-wall beta limit $\beta_{N}^{no-wall}$ [1], can be fully suppressed by a close perfectly conducting wall up to the higher ideal-wall beta limit $\beta_{N}^{ideal-wall}$. Here, $\beta_{N}$ is the normalized plasma $\beta$ defined as in $\beta_{N}=\beta_{t}aB_{t}/I_{p}$, where $\beta_{t}=2\mu_{0}\langle p\rangle/B_{t}^{2}$, $\langle p\rangle$ the volume-averaged plasma pressure, $B_{t}$ the toroidal magnetic field, $a$ the plasma minor radius, and $I_{p}$ the toroidal plasma current. In reality, however, the finite wall resistivity can lead to the growth of resistive wall mode (RWM) and eventually the potential onset of plasma disruption, which is a major concern for tokamak operation and performance [2, 3].

The China Fusion Engineering Test Reactor (CFETR) project aims to bridge the gap between the experimental fusion reactor ITER [4] and the future fusion power plant DEMO [5]. During the past few years, significant progress has been made in the CFETR conceptual physics and engineering design [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Since 2018, a new design scenario of CFETR has been adopted with major radius $R_{0}=7.2m$ and minor radius $a=2.2m$, primarily targeting the long-pause self-sustainable burning plasmas with fusion power production up to 1GW in the steady-state operation (SSO) [19].

Such SSO scenarios, in which a high bootstrap current fraction is obtained together with good energy confinement and favourable MHD stability properties, are now considered to be the most promising for future reactor scale tokamaks [20]. However, the large fraction of bootstrap current, leads to significant reduction in the magnetic shear, and thus the associated stabilization against the MHD instabilities driven by the steep pressure gradients in the transport barrier regions [21]. In this case an pressure-driven MHD mode, the infernal mode, can appear if the zero-shear, i.e. the ‘flat-q’ region has a safety factor value close to a low-order rational value, which might then destroy the barriers [22, 23, 24]. For more common cases where the flat-q region is away from rational values, the infernal components become part of the RWM and lead to lower beta limits [25].

In earlier advanced tokamak studies, more attention is drawn to the influence of the central safety factor reversal. In DIII-D [26] and JET experiments [27], it is found that the no-wall beta limit for this kind of discharge is approximately $\beta_{N}^{no-wall}=2.5li$, where $li$ is the plasma inductance, whereas the ratio $\beta_{N}^{no-wall}/li$ usually scales as $4$ for the conventional scenarios  [28, 29, 30]. However, for future larger tokamaks, such as CFETR, the local bootstrap current density tends to be strong enough to form another flattened safety factor profile in the edge pedestal region. In this case the edge infernal components can develop and further destabilize the RWMs. Such a destabilizing effect of edge infernal components has been confirmed in this work, where the stability of the $n=1$ RWM has been analyzed using the AEGIS code [31, 32] for the CFETR 1GW SSO scenario. Consequently, stronger edge localized toroidal rotation is found to be required for the suppression of the $n=1$ RWM, which is above 1.5% the core Alfvénic speed for the 1GW design.

The rest of paper is organized as follows. In Sec. 2, the equilibria of the CFETR 1GW SSO scenario are briefly reviewed. In Sec. 3 the numerical scheme of AEGIS code is introduced. In Sec. 4 the stability analysis of the $n=1$ RWMs for the CFETR 1GW SSO scenario is reported, as well as the influence of edge infernal components on the $\beta_{N}$ limits. Sec. 5 evaluates the rotational stabilization effects, including those of rotation profiles. Conclusions and discussions are given in Sec. 6.

The CFETR 1GW SSO scenario is designed based on the integrated modelling in the OMFIT framework, where the EFIT code is employed to construct the MHD equilibrium used in this study [12, 13]. The main parameters are as follows: major radius $R_{0}=7.2m$, minor radius $a=2.2m$, toroidal magnetic field at magnetic axis $B_{0}=6.53T$, total plasma current $I_{p}=11MA$, plasma inductance $li=0.75$, the normalized beta $\beta_{N}=2.86$, and the Alfvénic speed at magnetic axis $v_{A}=1.004\times 10^{7}m/s$. The current density profile is locally peaked at $\psi=0.18$ in the core and $\psi=0.96$ at edge (figure 1). Correspondingly, the safety factor has a reversal in magnetic shear in the central region and a flat region in the edge pedestal ranging from $\psi=0.94$ to $\psi=0.99$. For comparison, a reference equilibrium is prepared by reducing the bootstrap current and pressure gradient in the edge pedestal region, so that the flat q profile at edge region is removed as shown in figure 1 with dashed lines. The total current and core plasma pressure are kept same as the SSO case.

The numerical studies in this work are carried out using the AEGIS code [31, 32]. The following linearized ideal MHD equation for toroidal plasma is solved:

where $\rho_{m}$ is the mass density, $\bm{\xi}$ the perpendicular fluid displacement, $\bm{J}$ the equilibrium current density, $\bm{B}$ the equilibrium magnetic field, $P$ the equilibrium pressure, $\mu_{0}\delta\bm{J}=\nabla\times\delta\bm{B}$, $\delta\bm{B}=\nabla\times(\bm{\xi}\times\bm{B})$, and $\delta P=-\bm{\xi}\cdot\nabla P$. The rotation effects are included mainly through the Doppler shift, $\hat{\omega}=\omega+n\Omega$ with $\Omega$ being the toroidal rotation frequency. For the subsonic rotation regime in most tokamak plasmas, the effects from centrifugal and Coriolis forces are negligible [33, 34]. In addition, since the RWM frequency in this study is much smaller than the ion acoustic wave frequency, the plasma compressibility results only in the so-called apparent mass effect through the coupling between parallel and perpendicular inertia motions [35]. Therefore, the contribution of plasma compressibility can be included by regarding $\rho_{m}$ as the apparent mass in the above momentum equation (1). To avoid the singularity in edge safety factor near the separatrix, the diverted equilibria of CFETR used in this study are all truncated at the poloidal flux surfaces where the safety factor value $q_{a}=9.1$ (figure 2). The conformal shaped wall is used in our AEGIS calculations.

In this section we analyze the linear stability of $n=1$ RWM for the CFETR 1GW SSO scenario, as well as the influence of flattened q-profile in the edge pedestal region. We compare the RWM stability of the SSO scenario and the reference equilibrium in absence of the flattened q-profile near the edge as described in section 2 (figure 3), where $\tau_{w}$ is the resistive diffusion time of the wall, typically a few milliseconds. For the designed wall location at $r_{w}=1.2a$, we find that the reference equilibrium without the edge flattened q-profile has higher beta limits ($[\beta_{N}^{no-wall},\beta_{N}^{ideal-wall}]=[3,3.17]$) than the SSO case ($[\beta_{N}^{no-wall},\beta_{N}^{ideal-wall}]=[2.76,3.03]$). This indicates that the edge flat q-profile is unfavourable for RWM stability. The target $\beta_{N}$ of CFETR 1GW SSO scenario is $2.86$, which falls within the unstable RWM regime when none of the passive or active stabilizing mechanisms is taken into account.

The presence of edge flat q region can significantly change the RWM mode structure. In the reference equilibrium case where the edge flattened q region is absent (figure 4), the $m=2$ harmonic peaks around the central flat q region. This is unusual since the minimum $q$ value is above 2 thus the $m=2$ harmonic is expected to be non-resonant. Meanwhile, the $m=3$ harmonic broadly domes over the region between the two $q=3$ rational surfaces with a greater amplitude. For the SSO case (figure 5), however, the edge infernal component with poloidal number $m=8$ becomes more dominant, even though the pressure gradient as well as the current density in the edge region is weaker than that in the central region, where the $m=2$ and $3$ harmonics are much reduced. In particular, unlike all other harmonics which peak around rational surfaces, the $m=8$ harmonic peaks in the edge flattened q region where $q_{flat}=7.38$ and $\psi=0.98$, which is also crucial to the rotation effects on the RWMs (see next section). Thus the edge flat q profile is more influential than the central flat q profile on the RWM mode structure and stability. This may be because that the RWM is an external MHD mode, so that the plasma current in the edge region has larger impact than that in the central region.

Next, the influence of safety factor value at the edge flat q region on the stability is studied. We slightly vary the total toroidal current of the SSO scenario equilibrium while keeping the shape of current density profile same. A series of equilibria with $q_{flat}$ ranging from $7.4-8.2$ are generated as shown in figure 6(a). The stability of $n=1$ modes for these equilibria are evaluated. Figure 6(b) shows the no-wall and ideal-wall $\beta_{N}$ limits as functions of the $q_{flat}$. One can see that both $\beta_{N}$ limits dramatically drop when $q_{flat}\simeq 8$, whereas apart from that they gradually decrease with $q_{flat}$. Moreover, both $\beta_{N}$ limits become nearly identical at $q_{flat}=8$, indicating the complete loss of wall stabilization. Note that in this case, the eigenfunction becomes delta-function-like and peaks at the resonance point for a single Fourier component, which can be identified as the infernal mode. Thus, the presence of the edge flat q region may subject the SSO scenario to the $n=1$ external instability even at a considerably low $\beta_{N}$.

In this section, we evaluate the stabilization effects on RWM due to toroidal plasma rotation with three types of radial profiles: the uniform rotation profile, the radially descent profile, and the Gaussian rotation profile.

In summary, we have studied the $n=1$ RWM stability in the CFETR 1GW SSO scenario using the AEGIS code. Due to the large bootstrap current contribution, such an SSO scenario has a deep safety factor reversal in the core region and a flat q profile in the edge region, corresponding to the internal and external transport barriers, respectively. As a consequence, the infernal mode components develop in both core and edge regions. The edge infernal component dominates the destabilizing effects for RWMs, which lower the $\beta_{N}$ limits and render the $n=1$ RWM unstable in the CFETR 1GW SSO scenario.

The $n=1$ RWMs can be fully suppressed by uniform rotation with frequency above $1.5\%\Omega_{A0}$. Due to the influence of the edge infernal components, it is the edge rotation that mainly contributes to the stabilization effect. Thus an edge localized toroidal rotation with a sufficiently broad profile and its peak rotation frequency maintained at above $1.5\%\Omega_{A0}$ can fully stabilize the $n=1$ RWMs in the CFETR 1GW SSO scenario. However, a robust control of RWM in CFETR SSO may require a full account of all other possible stabilizations from kinetic effects, as well as the active stabilizations from the feedback control coil system, which should be evaluated in other and future work.