Bootstrap Current Fractions in Fusion Plasmas

• Bootstrap Current Fractions are ratios measuring the neoclassically generated plasma current relative to the total current, essential for fusion device performance.
• Sophisticated kinetic models and full drift-kinetic solvers, like ZOW and PENTA, accurately compute these fractions by enforcing momentum conservation in both axisymmetric and non-axisymmetric systems.
• Accurate bootstrap fraction estimations guide experimental validations and design optimizations in magnetic confinement devices such as tokamaks and stellarators.

Bootstrap Current Fractions

The bootstrap current fraction quantifies the proportion of plasma current in a magnetically confined system (tokamak, stellarator, or helical device) generated by neoclassical transport processes, primarily through particle drifts and collisional dynamics, as opposed to externally driven currents. This fraction, typically denoted $f_{\rm bs}$, is critical for both device optimization and predictive modeling, affecting the equilibrium, stability, and confinement properties of fusion-grade plasmas. The accurate computation of bootstrap current fractions requires sophisticated kinetic models, careful treatment of drift- and collisional operators, and, in non-axisymmetric systems, enforcement of exact parallel momentum conservation. Bootstrap current fractions are also of interest in the conformal bootstrap of quantum field theory, where they measure the relative OPE weight carried by conserved current sectors.

1. Definitions and Formulation

Bootstrap current fraction is defined as the ratio of the bootstrap current to the total current on a flux surface:

$f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$

or, in an integrated form,

$f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$

where $j_{\rm bs}$ is the local bootstrap current density and $j_{\rm tot}$ is the total (typically toroidal) current density. The precise kinetic definition, as encoded in drift-kinetic models for non-axisymmetric systems, is

$$j_{\rm bs} = e\,n\,\bigl(U_{\parallel,i}-U_{\parallel,e}\bigr)\frac{\langle B\rangle}{\langle B^2\rangle}$$

with flux-surface-averaged parallel flows $U_{\parallel,a}$ for each species $a$ (Huang et al., 2016).

In conformal field theoretic bootstrap, the analogue, for mixed correlators involving a scalar and a conserved current $J_\mu$, is the OPE “current fraction”—i.e., the ratio of contributions from conserved spin-1 currents to the total conformal block sum (Reehorst et al., 2019, Kiaee et al., 23 Dec 2025):

$$\mathrm{fraction}_J(u,v) = \frac{ \sum_{\Delta,J=1}\lambda_{\phi\bar\phi J}^2\,G_{\Delta,1}(u,v) }{ \sum_{\rm all}\lambda^2_{\mathit{Q},\ell}\,G_{\Delta,\ell}(u,v) }$$

2. Theoretical Models and Calculation Methodologies

Reliable calculation of bootstrap fractions requires both kinetic theory and global plasma modeling.

• Local neoclassical theory: Formulas such as those of Sauter et al. (1999) express $f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$0 as a linear combination of normalized density and temperature gradients, weighted by dimensionless coefficients $f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$1 that interpolate across the banana, plateau, and Pfirsch-Schlüter regimes:

$f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$2

(Landreman et al., 2012).

• Full drift-kinetic solvers: For non-axisymmetric (stellarators/helical) systems, it is essential to solve the drift-kinetic equation with exact parallel momentum conservation:

$f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$3

and ensure the surface-averaged momentum balance is respected (Huang et al., 2016, López, 31 Oct 2025).

• Code benchmarks: Three principal computational approaches are compared:
  • DKES: Monoenergetic local drift-kinetic solver employing only pitch-angle scattering.
  • ZOW: δf Monte Carlo code, energy-dependent, includes tangential drift and momentum-conserving like-species operators.
  • PENTA: DKES transport coefficients with Sugama-Nishimura momentum correction.

ZOW and PENTA, which enforce exact momentum balance, return consistent results, while DKES (without correction) can overestimate the bootstrap-fraction by 600–800% (Huang et al., 2016).

• Strong-gradient/pedestal theory: In regions where the density/temperature scale length is comparable to the ion orbit width (e.g., tokamak pedestal), standard local neoclassical predictions must be amended by global Fokker-Planck or δf codes which include finite-orbit-width (FOW) and strong gradient modifications (Landreman et al., 2012, Landreman et al., 2012, Trinczek et al., 3 Apr 2025).

3. Parameter Dependence and Scaling Laws

The bootstrap fraction depends sensitively on collisionality, magnetic geometry, and plasma profiles:

• Collisionality ($f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$4): At low $f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$5 (banana regime), $f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$6 rises as trapped particle effects dominate, and falls off as $f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$7 in the Pfirsch-Schlüter regime. In the $f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$8 regime for standard stellarators/tokamaks, the bootstrap coefficient $f_{\rm bs} = \frac{j_{\rm bs}}{j_{\rm tot}}$9, but in omnigenous or optimized quasi-symmetric fields, $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$0 saturates or even decreases as $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$1 (López, 31 Oct 2025, Helander et al., 2017, Albert et al., 2024).
• Ripple amplitude and field symmetry: The presence of magnetic ripple increases the trapped particle fraction and alters the transport scaling. In omnigenous fields (including quasi-isodynamic or piecewise-omnigenous configurations), the average radial drift of all orbits vanishes, suppressing the bootstrap current entirely ($f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$2) (Calvo et al., 5 May 2025). For generic non-axisymmetric fields, alignment of field maxima and equivalent ripple conditions are necessary to minimize bootstrap offsets and attain the Shaing-Callen limit (Albert et al., 2024).
• Gradient strength: The magnitude and alignment of density and temperature gradients, as well as the flow profiles (e.g., strong radial electric fields $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$3), directly drive the bootstrap fraction. In strong-gradient regimes, FOW corrections can reduce or enhance $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$4 by up to 30%, particularly in the edge/pedestal region (Trinczek et al., 3 Apr 2025).
• Aspect ratio and global geometry: Lower aspect ratios and strong shaping (elongation, triangularity) reduce bootstrap drive compared to the large-aspect-ratio, circular case (Rekhviashvili et al., 2023).

4. Zero Bootstrap Current and Control Strategies

Recent advances have enabled the design of stellarator configurations with identically zero bootstrap current for all profiles—a property not present in generic designs.

• Piecewise-omnigenous fields: In such configurations, the geometric parameter $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$5 defined from the surface partition vanishes, yielding $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$6 independently of collisionality or plasma profiles. This condition is accessible to analytic design and has been numerically validated with high precision. These fields provide reference cases for code benchmarking and serve as a baseline for quantifying symmetry-breaking effects (Calvo et al., 5 May 2025).
• QA + piecewise-omnigenous ensembles (QA-pwO): By blending quasi-axisymmetry with piecewise-omnigenous perturbations, it is possible to tune the bootstrap current continuously from well above the axisymmetric value to zero or negative, which is critical for matching divertor operation constraints (such as in island-diverted reactors) or for enhancing bootstrap drive in steady-state tokamaks (Velasco et al., 20 Mar 2026).

5. Benchmarks, Code Validation, and Physical Limits

Robust predictive capability requires cross-verification between analytic models and numerical solvers:

Configuration	Model/Code	f_bs Values (sample)	Agreement/Deviation
FFHR-d1 helical reactor	DKES	0.35 (core)	Overestimates by ~10x
	ZOW	0.04 (core)	Consistent with PENTA
	PENTA	0.05 (core)	Consistent with ZOW
Tokamak (M3D-C1)	Sauter/Redl	—	Agreement with NEO, XGCa, SFINCS <3%
TJ-II stellarator	DKES/NEO-MC	F_bs ≈ 0.9–1.1	Consistent with experiment
Stellarator (piecewise-omnigenous)	MONKES	D_{31} crosses zero at Δ=0	Machine-precision zero

Numerical codes such as MONKES (Legendre-based solver), SFINCS (full 4D global kinetic), and M3D-C1 (extended-MHD with neoclassical closure) enable rapid and accurate estimation of $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$7, bootstrap fractions, and their optimization in design workflows (López, 31 Oct 2025, Saxena et al., 7 Jul 2025, Albert et al., 2024).

6. Experimental Evidence and Practical Implications

Experimental validation on devices such as TJ-II demonstrates the physical reality and diagnostic accessibility of bootstrap current fractions:

• TJ-II: For various density and temperature regimes, calculated and measured total bootstrap currents agree within 10–20%, with the bootstrap contribution often dominating the total plasma current ($f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$8), including sign reversals at high density (Velasco et al., 2011).
• Tokamak pedestals: In DIII-D and ASDEX Upgrade-like pedestals, local neoclassical formulae systematically overpredict $f_{\rm bs} = \frac{I_{\rm bs}}{I_{\rm tot}}$9 by 20–100% in the plateau or strong-gradient regime; global codes including FOW and E_r reduce this discrepancy (Landreman et al., 2012, Landreman et al., 2012, Trinczek et al., 3 Apr 2025).
• Optimization and self-consistency: In device optimization, particularly for quasi-symmetric stellarators, imposing a penalty function for mismatch between equilibrium and analytically predicted bootstrap current profiles yields self-consistent configurations with optimized confinement and controlled $j_{\rm bs}$0 (Landreman et al., 2022).

7. Generalization to Field Theory: Bootstrap "Current Fractions" in the CFT Context

In conformal field theory, current fractions formalize the share of OPE weight in mixed scalar-current correlation functions that is due to conserved current exchange, with direct connections to finite-temperature transport (e.g., conductivity in the 3d O(2) model).

• Numerical bounds (O(2) model): At the crossing-symmetric point $j_{\rm bs}$1, the current fraction is $j_{\rm bs}$2, while the stress tensor fraction is $j_{\rm bs}$3 (Reehorst et al., 2019).
• Large charge bootstrap: At large global charge, the current fraction in the OPE is determined by the number of Regge trajectories and related to EFT parameters, with closed analytic formulas for one- and two-trajectory solutions (Kiaee et al., 23 Dec 2025).
• Physics connection: These fractions directly relate to high-frequency conductivity in 2+1d CFTs via a Kubo formula, with bootstrap-predicted parameters yielding precise agreement with QMC data.

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• (Huang et al., 2016) Benchmark of the bootstrap current simulation in helical plasmas
• (Calvo et al., 5 May 2025) A new class of optimized stellarators with zero bootstrap current
• (López, 31 Oct 2025) Fast and accurate calculation of the bootstrap current and radial neoclassical transport in low collisionality stellarator plasmas
• (Reehorst et al., 2019) Mixed Scalar-Current bootstrap in three dimensions
• (Saxena et al., 7 Jul 2025) Bootstrap Current Modeling in M3D-C1
• (Trinczek et al., 3 Apr 2025) Strong gradient effects on neoclassical electron transport and the bootstrap current
• (Helander et al., 2017) Stellarator bootstrap current and plasma flow velocity at low collisionality
• (Rekhviashvili et al., 2023) Gyrokinetic simulations of neoclassical electron transport and bootstrap current generation in tokamak plasmas in the TRIMEG code
• (Velasco et al., 20 Mar 2026) Control of the bootstrap current in approximately quasi-axisymmetric magnetic fields
• (Velasco et al., 2011) Calculation of the bootstrap current profile for the TJ-II stellarator
• (Albert et al., 2024) On the convergence of bootstrap current to the Shaing-Callen limit in stellarators
• (Landreman et al., 2022) Optimization of quasisymmetric stellarators with self-consistent bootstrap current and energetic particle confinement
• (Landreman et al., 2012) Local and global Fokker-Planck neoclassical calculations showing flow and bootstrap current modification in a pedestal
• (Landreman et al., 2012) Changes to neoclassical flow and bootstrap current in a tokamak pedestal
• (Kiaee et al., 23 Dec 2025) Large charge bootstrap with U(1) current probes