Peeling–Ballooning Modes in Tokamak Pedestals

• Peeling–Ballooning Modes (KPBMs) are global, intermediate-n MHD instabilities that couple pressure-gradient-driven ballooning and current-driven peeling to set the pedestal limits in tokamak plasmas.
• They are characterized by sensitivity to kinetic effects, local q-profiles, and magnetic shear, with stability thresholds derived from an energy balance between destabilizing drives and field-line bending forces.
• KPBMs significantly impact edge localized mode dynamics, metallic wall erosion, and overall fusion performance, guiding experimental design and numerical modeling for effective ELM control.

Peeling–Ballooning Modes (KPBMs) are global, intermediate-n magnetohydrodynamic (MHD) instabilities that define the pressure and current density limits in the edge pedestal of H-mode tokamak plasmas. Their physics is rooted in the coupling of the pressure-gradient-driven ballooning drive and the edge-current-driven peeling (kink) drive. KPBMs are considered the principal MHD limit triggering edge localized modes (ELMs), thereby controlling the pedestal height and influencing metallic wall erosion, core fueling, confinement, and fusion performance.

1. Theoretical Foundations: Coupled Pressure and Current Drives

KPBMs arise in regimes where both the edge pressure gradient and the edge (often bootstrap-generated) current are strongly enhanced. The instability is characterized by simultaneous excitation of ballooning (pressure-driven, high-n) and peeling (current-driven, low-n) branches, resulting in an MHD mode with finite, intermediate toroidal mode numbers ($3 \lesssim n \lesssim 30$) (Saarelma et al., 2013, Li et al., 2022). The stability of such modes relies on the balance between destabilizing drives (pressure gradient, edge current) and field-line bending energy, with the marginal threshold set by solving for zero total potential energy variation ($\delta W$) in the energy principle: $$\delta W = \int \left[ |\nabla_\perp\xi|^2 + ... - \alpha|\xi_\perp|^2 + \frac{\mu_0 J'}{B}\xi_\parallel\xi_\perp \right] dV,$$ where $\xi$ is the displacement, $\alpha$ the normalized pressure gradient, and $J'$ the local current density gradient (Saarelma et al., 2013, Zheng et al., 2014).

Kinetic Peeling–Ballooning Modes (KPBMs, Editor's term) generalize these physics by incorporating kinetic effects (finite Larmor radius, wave-particle resonances, trapped particle dynamics), revealed via gyrokinetic simulations in the extreme-gradient pedestal regime, yielding non-trivial structure, drive, and threshold behavior (Wan et al., 2012).

2. Mathematical Stability Criteria and Scaling Laws

KPBM stability is commonly assessed using a combination of normalized parameters:

• Normalized pressure gradient $\alpha$:

$$\alpha = \frac{-2\,\partial V/\partial \psi}{(2\pi)^2} \left(\frac{V}{2\pi^2 R_0}\right)^{1/2} \mu_0 \frac{\partial p}{\partial \psi}$$

• Magnetic shear $s$:

$$s = \frac{r}{q}\frac{dq}{dr} \approx \frac{\psi}{q}\frac{dq}{d\psi}$$

• Edge toroidal current $\langle j_\phi \rangle_{max}$

The critical threshold $\alpha_c(s, J_{bs}, n)$ as a function of $s$, edge current, and $n$ exhibits a window of instability for intermediate-$n$; it rises for both $n \to 0$ (peeling) and $n \to \infty$ (ballooning). The local growth rate can be expressed as: $$\gamma^2(n) = \omega_A^2 \left[ \alpha - \alpha_c(s, J_{bs}, n) \right] - \Lambda n^{-2},$$ where $\omega_A$ is the Alfvén frequency (Li et al., 2022).

Gyrokinetic modeling extends this local picture, finding near-threshold scaling for the KPBM branch: $$\gamma_{KPBM}(\hat s) \approx \gamma_0 (\hat s / 0.7)^{0.6},$$ with $\hat s$ the local shear. Collisionality further modifies thresholds and growth rates, especially for the high-$n$ KBM branch (Wan et al., 2012).

3. Numerical Modeling, Equilibrium Reconstruction, and Role of Bootstrap Current

Equilibrium input for KPBM analysis requires full knowledge of the density, temperature, and current (including the bootstrap current) profiles across the edge pedestal. Tools such as the HELENA code (for current/self-consistent $q$), MISHKA-1 (ideal MHD stability), and NIMROD (two-fluid extended MHD) are routinely used (Saarelma et al., 2013, Lin et al., 2019). For kinetic analysis, global electromagnetic gyrokinetic codes such as GEM are employed (Wan et al., 2012). The bootstrap current, computed via Sauter formulae, is key; in JET, peak edge bootstrap current densities reach 30–40 MA m$^{-2}$, flattening $q(\psi)$ and lowering the magnetic shear in the steep-gradient region, thus stabilizing high-$n$ ballooning and KBM, but allowing finite-$n$ KPBMs to persist (Saarelma et al., 2013).

Reducing the bootstrap current fraction causes the infinite-$n$ critical ballooning index to drop below unity, re-enabling KBM/ballooning instability throughout the pedestal. Experimentally, KPBM onset corresponds closely to the last $\sim$10% of the ELM cycle in low-fueling cases, with transitions tracked in the $(\alpha, \langle j \rangle)$ plane (Saarelma et al., 2013).

4. Kinetic Peeling–Ballooning Modes: Gyrokinetic Results and Pedestal Physics

Global gyrokinetic simulations have established that KPBMs form the intermediate-$n$ branch with strong curvature drive and finite but positive local shear, peaking in the steepest-gradient region ($\rho_N\approx0.95$–0.98) with electron-direction propagation (Wan et al., 2012). Stability is highly sensitive to local $q$-profile flattening: modest reductions in $\hat s$ sharply stabilize KPBM, while weakly destabilizing KBM, causing a modal transition in pedestal transport dynamics.

Mode structure analysis shows KPBM eigenfunctions have even parity and are radially localized, with $\Delta r/a\sim0.01$–0.02. Collisions (included as linearized operators) act to reduce KPBM growth and frequency (by 10–30%), and lower the KBM critical $\beta$, increasing its growth rate above threshold, altering the operational stability landscape (Wan et al., 2012).

5. Edge Topology, X-point Effects, and Peeling-Off Modes

The edge geometry, in particular the presence of X-points, fundamentally modifies the MHD spectrum. A dual-poloidal-region $q$-coordinate system reveals that KPBMs primarily sample a finite local $q$ away from the X-point; near the small poloidal wedge $\Theta_X$ around the X-point, field-line bending contributes a positive energy ($\delta W_X$) to the stability integral, thus raising the threshold for KPBM onset (Zheng et al., 31 Oct 2024). Neglecting the X-point region artificially lowers stability boundaries.

Furthermore, current jumps between the closed-flux edge and the scrape-off layer (SOL) can convert KPBMs into "peeling-off" modes: external kink or peeling–ballooning displacements convect equilibrium edge current across the LCFS, producing a convective current sheet with a discontinuity in parallel Ohm’s law. The resulting convective drive in the modified Rutherford equation ($A_c$ term) lowers the tearing threshold and leads to rapid magnetic reconnection, peeling off the pedestal plasma to the SOL and providing a mechanism for ELM ejection distinct from the conventional kink picture (Zheng et al., 2014). Peeling–off is associated with sharp ELM onsets and may reduce ELM size while raising repetition rates.

6. Dependence on Collisionality, Density Gradient, and Resistivity

The growth and saturation of KPBMs, and their role in ELM size regulation, depend strongly on pedestal collisionality, density profile, and resistivity. BOUT++ turbulence simulations (EAST) show that rising separatrix density or flattening the pedestal gradient weakens KPBM drive, decreases the most-unstable $n$, reduces linear growth rates and ELM size, and can eliminate avalanche spreading (Li et al., 2022). Table: summary of observed dependency.

Parameter	Effect on KPBM/ELM size	Regime
Separatrix density	↑ $n_{sep}$ → ↓ $n_{0}$, γ	From high-$n$ to P–B to stable
Collisionality	↑ $\nu_*$ → ↓ γ, ELM size	$\gamma_{max} \propto \nu_*^{-0.3 \ldots -0.5}$
Resistivity	↑ $Z_{eff}$ → ↓ γ, partial stabilization	EAST, NSTX (Lin et al., 2019)

Resistive stabilization via elevated $Z_{eff}$ and edge impurity content damps KPBM through $\eta J$ (current diffusion) and $\eta J^2$ (pressure-energy dissipation), but full suppression occurs only in specific devices and parameter regimes (Lin et al., 2019). Kinetic modifications to resistivity stabilization remain an active area; inclusion of non-ideal terms such as Hall, FLR, and Landau damping is required for completeness.

7. Phenomenological Implications and ELM Control

KPBMs serve as the ultimate hard limit for pedestal gradients in most diverted tokamaks. Their onset, location, and nonlinear outcome dictate the timing, size, and character of type-I ELMs. Edge topology (single vs. double null, X-point geometry), pedestal tailoring (via fueling, shaping, or impurity seeding), and resonant magnetic perturbations (RMPs) offer pathways for controlling or mitigating KPBM-triggered ELMs by altering local $q$-profiles, edge shear, or free-energy sources (Zheng et al., 31 Oct 2024, Li et al., 2022).

RMP strategies informed by dual-region $q$ analysis suggest improved ELM control can be achieved by aligning coil spectra to the actual local $q_c(\chi) h(\chi,\theta_c)$ in the X-point region. Experimental diagnostics—edge current profile measurements, detection of magnetic islands, and pedestal stability mapping—are essential to validate KPBM and related models (Saarelma et al., 2013, Zheng et al., 2014).

In summary, KPBMs and their kinetic and topology-modified generalizations determine the edge pedestal stability landscape in high-performance tokamak regimes. Quantitative modeling and experimental validation, especially of edge $q$-profiles, are prerequisites for predictive plasma scenario development and robust ELM control in future fusion reactors.