Helical Core Dynamics in Fusion Plasmas

• Helical Core (HC) is a spatial region forming from the nonlinear saturation of m/n=1 instabilities, driving a helical displacement of the magnetic axis in fusion plasmas.
• It is analyzed using simulation tools like MEGA and VMEC to track magnetic axis displacement (δHC) and assess the influence of alpha particle pressure (βα) on plasma equilibrium.
• Secondary pressure-driven MHD instabilities can trigger magnetic chaos and confinement degradation, highlighting the need for precise profile control in ITER-scale devices.

A helical core (HC) refers to a spatial region within a plasma (typically in advanced fusion devices like ITER or JT-60U) that develops as a long-lived, saturated, non-axisymmetric equilibrium following the nonlinear evolution of an $m/n=1/1$ kink or quasi-interchange instability. The formation, evolution, and impact of HCs are of significant current interest due to their influence on confinement, magnetic topology, and stability in high-performance fusion plasmas. HC formation is also relevant in the design of advanced cooling channels, hard-core lattice and statistical models, and analysis of material and fluid structures with intrinsic helicity; however, in the context of ITER and burning plasma operations, the primary focus is on HC dynamics in the presence of fusion-born alpha particles.

1. Nonlinear Saturation and HC Formation

The HC emerges when ideal or quasi-interchange instabilities driven by pressure gradients and magnetic shear become nonlinearly saturated in low-shear, high-$q$ plasmas typical of hybrid scenario tokamaks. In such plasmas, if the central safety factor $q$ is slightly above unity but no $q=1$ surface exists (i.e., no resonant reconnection occurs), the instability does not evolve into a complete magnetic island, but rather generates a helical displacement of the magnetic axis. This displacement, $\delta_{\mathrm{HC}}$, quantifies the radial offset of the axis: $$\delta_{\mathrm{HC}} = \frac{1}{2\pi} \int_0^{2\pi} \sqrt{(R_a - R_{a0})^2 + (Z_a - Z_{a0})^2} \, d\phi$$ where $(R_a, Z_a)$ and $(R_{a0}, Z_{a0})$ denote the helical and axisymmetric axis positions respectively.

Key simulation tools such as MEGA (MHD-PIC hybrid) and VMEC (3D equilibrium solver) allow detailed tracking of $\delta_{\mathrm{HC}}$ for prescribed pressure and $q$ profiles. The magnitude of $\delta_{\mathrm{HC}}$ increases as central $q$ approaches unity and as the region of low shear (parameterized by $\rho_{q_{\min}}$) extends radially outward, revealing clear control of the HC structure via equilibrium configuration.

2. Role of Fusion-Born Alpha Particles

Fusion-born alpha particles introduce an additional pressure component $\beta_{\alpha}$, impacting both equilibrium and stability properties of the HC. For ITER-relevant regimes ($\beta_{\alpha}(0) \lesssim 1\%$), the alpha particle pressure can be treated as a flux function additive to the bulk MHD pressure: $$\beta_{\mathrm{MHD}}(\rho) = \beta_b(\rho) + \frac{1}{2\pi} \int_0^{2\pi} \beta_\alpha(\rho, \theta) \, d\theta$$ Within this regime, HC formation is enhanced ($\delta_{\mathrm{HC}}$ increases with $\beta_{\alpha}$) but omnigenity—good confinement on nested flux surfaces—is preserved, and the plasma exhibits only mild kinetic effects (e.g., a slight reduction in linear growth rate, typically $\sim 10\%$).

If $\beta_{\alpha}$ is increased beyond the nominal regime ($\beta_{\alpha}(0) \sim 3\%$), both bulk and alpha pressure profiles begin to flatten. This flattening signals reduced omnigenity and increased free energy expended in profile redistribution. Kinetic effects are more prominent: test-particle studies reveal modulation of orbit topology (development of "passing-trapped transitional" orbits) and enhanced radial excursions, especially for moderate HC displacement.

3. Secondary Pressure-Driven MHD Instability

After HC saturation, a resistive, pressure-driven secondary MHD mode can become unstable in the compressed flux region of the HC—where pressure gradients are maximum. Distinctive features of this mode include:

• Broad toroidal mode spectrum: Fourier components with $n = 5$ to $n = 16$ all grow at similar rates, forming a coherent eigenmode
• Growth rate scaling: $\gamma_{\mathrm{2nd}} \propto \hat{\eta}^{1/3}$, characteristic of resistive ballooning or kink-like instabilities
• Magnetic chaos: As the instability amplitude grows, flux surfaces around $q_{\min}$ are destroyed, and Poincaré plots demonstrate loss of nested surfaces and onset of radial mixing

Confinement degradation depends on the allowed toroidal spectrum: narrow ($n=1$ only) mode filtering is stabilizing, whereas a broad spectrum triggers abrupt collapse of $\delta_{\mathrm{HC}}$ and strong mixing.

4. Diagnostic Metrics and Simulation Results

A summary of the primary simulation metrics and observed behaviors is provided:

Parameter	ITER-Scale Limit	Effect on HC/Confinement
$\delta_{\mathrm{HC}}$	$\lesssim 0.8$ m	Increases with $\beta_\alpha$ and $q \to 1$
$\beta_\alpha(0)$	$\lesssim 1\%$	HC stable, omnigenity preserved
$\beta_\alpha(0)$	$\gtrsim 3\%$	Pressure flattening, secondary modes
$\gamma_{\mathrm{2nd}}$ scaling	$\hat{\eta}^{1/3}$	Magnetic chaos and confinement loss

Within the ITER operating window, the HC forms without significant loss of confinement. Exceeding $\beta_{\alpha} \sim 1\%$ increases susceptibility to pressure flattening and secondary instabilities.

5. Context, Implications, and Limitations

HC formation in burning plasma conditions is a generic saturation mechanism for low-$n$, low-shear plasmas and must be accounted for to optimize operational scenarios. The effect of fusion alpha particles is predominantly additive at moderate $\beta_{\alpha}$, but nonlinear and kinetic effects become important as $\beta_{\alpha}$ increases.

The presence of secondary resistive pressure-driven modes demonstrates a clear route to degraded confinement if pressure gradients steepen or collisionality increases. This secondary instability is localized, coherent, and induces magnetic chaos, in contrast to simple island overlap or stochasticity by high-$n$ Alfvenic modes.

The permanence and structure of the HC, along with sensitivity to equilibrium shaping, must be considered in future tokamak designs. Computational limitations remain for very broad mode spectra and kinetic effects at elevated $\beta_\alpha$.

6. Wider Connections: HC in Related Physical Systems

While the primary discussion is focused on burning plasma fusion and MHD, "helical core" phenomena (understood as the development of spatially non-axisymmetric, helical structures) also arise in:

• Magnet channel design for muon beam cooling, where novel solenoid–dipole–quadrupole arrangements yield six-dimensional emittance reduction via controlled continuous dispersion (Yonehara et al., 2012, Yonehara et al., 2012)
• Hard-core statistical and lattice models, where translation-invariant or period-two Gibbs measures define phase coexistence and symmetry breaking in countable-spin systems (Rozikov et al., 2022)
• Optical and condensed matter systems (e.g., vectorial helico-conical beams, helical antiferromagnetic ordering), where intrinsic helicity and spatial structure lead to polymorphic phase behavior, rigidity, and topologically distinct excitation manifolds (Medina-Segura et al., 2023, Sangeetha et al., 2019, Tomanek et al., 2017)

7. Outlook for ITER and Fusion Devices

The intricate interplay between core displacement ($\delta_{\mathrm{HC}}$), alpha pressure ($\beta_\alpha$), omnigenity, and secondary resistive instabilities governs confinement in ITER-scale plasmas. Future work must focus on:

• Developing mitigation schemes for secondary pressure-driven modes via equilibrium profile control and mode filtering
• Understanding kinetic–MHD coupling, especially at higher $\beta_\alpha$ approaching and exceeding 1%
• Refining diagnostics and simulation tools for measuring $\delta_{\mathrm{HC}}$, pressure flattening, and confinement degradation
• Quantifying the effect of HC dynamics on alpha particle transport and burning regime sustainment

A plausible implication is that maintaining $\beta_\alpha$ within the nominal operating window and actively controlling pressure profiles will be critical for avoiding secondary instabilities and ensuring stable, high-performance fusion operation. The HC paradigm thus represents both a useful diagnostic and a challenge for next-generation burning plasma devices.