A new model for runaway electron transport based on chaotic Hamiltonian systems

Technical Summary: A New Model for Runaway Electron Transport Based on Chaotic Hamiltonian Systems

Problem Statement
The transport of runaway electrons (REs) in ergodic magnetic geometries, particularly during tokamak disruptions, is a critical challenge for controlled nuclear fusion. While the generation of RE beams via avalanche multiplication is well-understood, the survival of the initial seed population during the quench phase remains uncertain. Current modeling often relies on the Rechester-Rosenbluth (RR) diffusion approximation to describe transport in stochastic fields. However, the RR model assumes a fully enclosed ergodic volume and purely diffusive (exponential) decay. Simulation studies indicate that in open ergodic volumes or complex magnetic topologies, the RR approximation is insufficient. Specifically, it fails to account for "sticky regions"—trapping layers around magnetic islands where particle escape follows a power-law decay rather than an exponential one. This discrepancy limits the accuracy of reduced simulations used to interpret experimental data and validate RE models.

Methodology
The authors propose a semi-analytical transport model derived from the theory of chaotic Hamiltonian systems. The methodology relies on the correspondence between magnetic field line topology and the phase space of dynamical systems.

1. Theoretical Framework: The model addresses mixed phase spaces containing both chaotic seas and magnetic islands. It posits that particle survival probability, $P(t)$, is a superposition of two distinct decay mechanisms occurring simultaneously:
   • Exponential Decay: Dominated by particles in the chaotic sea, described by $e^{-\kappa t}$, where $\kappa$ is the escape rate.
   • Power-Law Decay: Dominated by particles trapped in "sticky regions" (remnants of broken islands), described by a term proportional to $(1 + (t/\tau)^\alpha)^{-1}$.
2. Model Formulation: The total survival probability is formulated as:
   $$P(t) = A e^{-\kappa t} + \frac{1-A}{1 + (t/\tau)^\alpha}$$
   where $A$ represents the initial fraction of particles in the chaotic regime, $\tau$ is the distinct time-scale for the power-law decay, and $\alpha$ is the power-law exponent. Unlike previous models that assign specific time intervals to these regimes, this model assumes both terms are present at all times.
3. Validation Cases: The model is tested against two distinct scenarios:
   • Ullmann-Caldas Map: A simplified 2D mapping representing magnetic field lines in a TBR-1 tokamak-like geometry.
   • JET Disruption Simulation: A complex 3D non-linear magnetohydrodynamic (MHD) simulation of a JET disruption (discharge #95135) using the JOREK code, followed by relativistic particle tracing.

Key Results

• Ullmann-Caldas Map: The model successfully fitted the numerical survival probability of 300,000 particles. The fit revealed an initial exponential loss phase dominated by the chaotic sea, followed by a prolonged tail governed by the power-law decay. The crossover time ($t_x$) between these regimes was calculated to be approximately 158 $\mu$s. The distinct time-scales were confirmed: the exponential decay time ($1/\kappa \approx 45.45$ $\mu$s) was notably different from the power-law time-scale ($\tau \approx 22.38$ $\mu$s).
• JET Disruption Scenario: Applied to a JOREK simulation of a JET disruption with 1.8 million tracer particles, the model achieved a "remarkably good fit" to the transport curve. The simulation showed a clear chaotic sea with prominent sticky regions around island chains. The model accurately captured the transition from rapid initial loss to the slower power-law tail, with a crossover time of approximately 125 $\mu$s.
• Parameter Consistency: In both cases, the fitted parameters ($A$, $\kappa$, $\tau$, $\alpha$) were physically consistent, with $\alpha$ values close to 1, supporting the theoretical derivation.

Significance and Claims
The paper claims that the proposed model offers a significant improvement over the Rechester-Rosenbluth approximation by explicitly incorporating the effects of sticky regions in mixed phase spaces. The primary contribution is the functional form that simultaneously accounts for exponential and power-law decays with distinct time-scales, rather than treating them as sequential or mutually exclusive regimes.

The authors demonstrate that this simple transport model can accurately describe the temporal evolution of particle confinement probabilities in both idealized mapping systems and complex, realistic 3D MHD simulations. By providing a better fit to direct particle trajectory simulations, the model offers a more robust tool for incorporating transport effects into reduced simulations of runaway electron dynamics. This enhanced accuracy is presented as crucial for the interpretation of present-day RE experiments and for the validation of models intended for future devices like ITER, where understanding the transport of the seed population is vital for predicting RE beam formation and mitigation.