T3ST Code: Turbulent Transport Analysis

• T3ST Code is a computational tool that uses synthetic turbulence fields and a Lagrangian test-particle approach to analyze turbulent transport in tokamak plasmas.
• It employs statistical methods to extract transport coefficients and validate key properties like ergodicity, stationarity, and time-symmetry.
• The decoupled methodology enhances computational efficiency, enabling rapid parameter scans for optimizing plasma confinement models.

The T3ST code (Turbulent Transport in Tokamaks via Stochastic Trajectories) is a computational tool developed to analyze turbulent transport phenomena in tokamak plasmas. Unlike gyrokinetic codes that solve self-consistently for plasma and turbulence, T3ST employs a statistical test-particle methodology combined with synthetic turbulence fields characterized by prescribed spectra. This approach enables rapid, high-fidelity computation of transport coefficients and detailed paper of fundamental statistical properties of turbulent transport, such as ergodicity, stationarity, and time-symmetry.

1. Methodological Foundations

T3ST is grounded in a Lagrangian test-particle paradigm. The system tracks the dynamics of charged particles—specifically their gyrocenter trajectories—within realistic axisymmetric magnetic equilibria. The governing equations remove the fast Larmor rotation and focus on slower drift dynamics, encompassing equilibrium fields, Coulomb collisions, and turbulent fluctuations. Turbulence enters the equations of motion as an externally prescribed, statistically generated field, constructed using a Gaussian ensemble with parameterized spectral properties $S(\mathbf{k}, \omega)$.

A typical turbulent potential $\phi_1(x, y, z, t)$ is synthesized as: $$\phi_{1}(x, y, z, t) = \int d\mathbf{k}\, d\omega\, \tilde{\phi}_{1}(\mathbf{k}, \omega) \exp\left\{i\left[k_x x + k_y y + k_z z - (\omega_*(\mathbf{k}) + \omega)t\right]\right\}$$ where

$$\tilde{\phi}_1(\mathbf{k}, \omega) = \sqrt{S(\mathbf{k}, \omega)}\,\eta(\mathbf{k}, \omega)$$

and $\eta$ is a white noise process. This synthetic turbulence effectively decouples field evolution from plasma response, allowing systematic exploration of transport dependence on turbulence attributes (e.g., spectral peak, correlation length, decorrelation time).

2. Lagrangian Transport Formulation

Transport coefficients are extracted as Lagrangian moments of the time-evolving radial displacement $X(t | x_0)$ for particles initialized at position $x_0$. Specifically,

$$V(t | x_0) = \frac{d}{dt}\langle X(t | x_0)\rangle, \qquad D(t | x_0) = \frac{1}{2}\frac{d}{dt} \left( \langle X(t | x_0)^2 \rangle - \langle X(t | x_0)\rangle^2 \right)$$

where $\langle \cdot \rangle$ denotes averaging over both particle and turbulence ensembles. This trajectory-based statistical approach circumvents long-time averaging requirements and convergence issues often present in Eulerian (fluid element) approaches.

The code also supports Green–Kubo-based computation of the running diffusion coefficient: $D(t) = \int_0^t L(\tau) d\tau$ with $L(\tau) = \langle v_r(0) v_r(\tau) \rangle$ the Lagrangian velocity autocorrelation.

3. Implementation: Synthetic Field Generation and Phase-Space Sampling

T3ST constructs synthetic turbulent fields by summing over Fourier modes, typically including gyro-averaging effects through Bessel functions: $$\phi_{gc}(X, t) = \sqrt{\frac{2}{N_c}} \sum_{i=1}^{N_c} J_0(k_i^\perp \rho_L) \sin\left(k_i \cdot X - \omega_i t + \alpha_i\right)$$ where $N_c$ is the number of modes, $J_0$ is the zeroth-order Bessel function, $k_i$ are sampled wavevectors, $\rho_L$ is the Larmor radius, and $\alpha_i$ are random phases. This allows parametric exploration of turbulence regimes (e.g., ITG vs. TEM) by varying input spectra and field statistics.

The initial test-particle ensemble is sampled across phase-space, typically from Maxwell–Boltzmann velocity distributions and uniform spatial distributions. The sampling ensures broad exploration and meaningful ensemble averages, substantiating investigation of ergodicity and transport properties.

4. Statistical Properties: Ergodicity, Stationarity, Time-Symmetry

Recent studies utilizing T3ST demonstrate that, despite the inhomogeneity and compressibility of Eulerian (fluid element) gyrocenter drifts, the ion transport induced by drift-type turbulence in tokamaks exhibits approximate ergodicity, stationarity, and time-symmetry (Palade, 4 Sep 2025). The code shows:

• Ergodicity: Ensemble averages over many realizations of broad initial phase-space distributions are equivalent to time averages along a single trajectory. When the phase-space is well-filled, computed transport coefficients (diffusion, pinch) converge to steady-state values.
• Stationarity: The Lagrangian autocorrelation function $L(t, t')$ is approximately invariant under time shifts—i.e., it is nearly a function of $|t - t'|$ only.
• Time-symmetry: Integrated transport coefficients, such as running diffusion $D(t)$, are nearly symmetric when computed for time-forward and time-backward trajectories.

Results indicate that only severe constraints on initial distributions (e.g., fixing particles to a single radial line) significantly affect transport coefficients; minor restrictions yield negligible changes, confirming robust ergodic mixing and insensitivity to initial conditions.

5. Computational Efficiency and Applicability

The methodology decouples turbulence from plasma response, providing substantial computational advantages. T3ST achieves calculation speeds (minutes or seconds for low-resolution studies) unattainable by full gyrokinetic codes, facilitating rapid scanning of turbulence parameters and integrated modeling for device optimization (e.g., in ITER or DEMO contexts).

Its flexibility allows the exploration of:

• Transport dependence on turbulence spectrum, correlation length, decorrelation time, equilibrium gradients, and collisionality.
• Impact of zonal flows, pinch effects, and interplay between neoclassical and turbulent transport mechanisms.
• Sensitivity of confinement and transport barriers to input turbulence statistics.

6. Significance for Tokamak Plasma Transport Theory

T3ST provides an efficient platform to test and validate central theoretical concepts in reduced plasma transport models:

• Empirical validation of assumptions such as ergodicity and stationarity required for the correctness of Green–Kubo and decorrelation trajectory methods.
• Assessment of the independence of transport coefficients from minor initial phase-space restrictions under turbulent mixing conditions.
• Support for statistical approaches leveraging ensemble-averaged quantities, facilitating reliable application of reduced turbulence theories.

A plausible implication is that the code’s ability to reliably reproduce these statistical properties in synthetic turbulence regimes reinforces confidence in their application to experimental or theoretical scenarios where direct validation is challenging.

7. Concluding Perspective

T3ST code leverages Lagrangian test-particle dynamics in synthetic turbulence fields, providing an efficient means of calculating turbulent and neoclassical transport in tokamak plasmas. Its framework—comprising synthetic field generation, broad phase-space sampling, and trajectory-based averaging—yields robust transport coefficients and enables examination of key statistical properties. The combination of computational efficiency and fidelity to plasma physics principles establishes T3ST as an essential tool for both theoretical studies and experimental planning in the domain of magnetic confinement fusion research.