O1 (WP1) – The theory of transport and machine learning architectures
Deliverable: A comprehensive turbulent transport model
Within this work package, we established the theoretical foundations for describing the dynamics of charged particles in fusion plasmas characterized by turbulence (Task 1.1), for the correct description of turbulent random fields (Task 1.2), and for the main machine-learning tools relevant to the purpose of this project (Task 1.3).
We consider a population of charged particles, with mass $m$ and charge $q$, confined in a tokamak configuration dominated by a strong magnetic field $\mathbf{B}$. The usual particle coordinates in phase space are $(\mathbf{x},\mathbf{v})$. When the magnetic field is sufficiently strong compared to other electromagnetic components, the motion can be viewed as a superposition of a smooth large-scale motion and a fast Larmor gyration. This scale separation is the basis of Lie perturbation theory [1], which provides a rigorous mathematical framework for charged-particle dynamics in magnetized plasmas.
In this approach, the true 6D phase space $(\mathbf{x},\mathbf{v})$ is replaced by the reduced gyrocentre coordinates $(\mathbf{X},v_{\parallel},\mu)$, which suppress the fast gyromotion through a perturbative expansion based on gyrokinetic ordering [2]. In this project we adopt the high-flow ordering [3], explicitly retaining the polarization drift [4].
The gyrocentre dynamics satisfy
$$ \frac{d\mathbf{X}}{dt} = v_{\parallel}\frac{\mathbf{B}^\star}{B_{\parallel}^\star} + \frac{\mathbf{E}^\star \times \mathbf{b}}{B_{\parallel}^\star} \qquad\text{(1)} $$
$$ \frac{dv_{\parallel}}{dt} = \frac{q}{m}\frac{\mathbf{E}^\star\cdot\mathbf{B}^\star}{B_{\parallel}^\star} \qquad\text{(2)} $$
$$ \frac{d\mu}{dt}=0 \qquad\text{(3)} $$
where $$\mathbf{E}^\star = -\nabla\Phi^\star - \frac{\partial \mathbf{A}^\star}{\partial t}, \qquad \mathbf{B}^\star = \nabla\times\mathbf{A}^\star,$$ and
$$ q\Phi^\star = \frac{m v_\parallel^2}{2} - \frac{m u^2}{2} + \mu B + q\phi_1^{\text{neo}} + q\phi_1^{\text{gc}} \qquad\text{(4)} $$
$$ \mathbf{A}^\star = \mathbf{A}_0 + \frac{m}{q}(v_{\parallel}\mathbf{b}+u+v_E) + \mathbf{A}_1^{\text{gc}} \qquad\text{(5)} $$
The zeroth-order magnetic potential $\mathbf{A}_0$ satisfies $\mathbf{B} = \nabla\times\mathbf{A}_0$. For axisymmetric equilibria, the field can be written as: $$ \mathbf{B} = F(\psi)\nabla\phi+\nabla\phi\times\nabla\psi, $$ where $\phi$ is the toroidal angle in cylindrical $(R,Z,\phi)$ or toroidal $(r,\theta,\phi)$ coordinates (COCOS = 2) [5].
The local safety factor $q_\theta(\psi,\theta)$, its flux-surface average $\bar{q}(\psi)$, and the generalized poloidal angle $\chi(\psi,\theta)$ are defined by:
$$ q_\theta(\psi,\theta) = \frac{\mathbf{B}\cdot\nabla\phi}{\mathbf{B}\cdot\nabla\theta} \qquad\text{(6)} $$
$$ \bar{q}(\psi) = \frac{1}{2\pi}\int_0^{2\pi} q_\theta(\psi,\theta)\,d\theta \qquad\text{(7)} $$
$$ \left.\frac{\partial\chi}{\partial\theta}\right|_{\psi} = \frac{q_\theta(\psi,\theta)}{\bar{q}(\psi)} \qquad\text{(8)} $$
Zeroth-order MHD equilibrium is toroidally invariant and constant on magnetic surfaces, so $n=n(\psi)$, $P=P(\psi)$, $T_i=T_i(\psi)$, $T_e=T_e(\psi)$.
A macroscopic radial electric field $E_0$ produces a toroidal rotation $$ \mathbf{u} = R^2\Omega_t(\psi)\,\nabla\phi, $$ where $\Omega_t(\psi)$ may have a gradient. The equations of motion are expressed in a frame rotating with $\mathbf{u}$ [3,7]. Poloidal flows are neglected [5], but a neoclassical poloidal potential $\phi_1^{\text{neo}}(\psi,\theta)$ appears:
$$ \phi_1^{\text{neo}} = \frac{m\Omega_t(\psi)^2}{2|e|\,(1+T_i/T_e)} \left(R^2 - \langle R^2\rangle_\theta\right) \qquad\text{(9)} $$
The first-order gyrocentre potentials $\phi_1^{\text{gc}}$ and $\mathbf{A}_1^{\text{gc}}$ include turbulent fluctuations and small external perturbations (RMPs). Gyro-averaging gives:
$$ \tilde{\phi}_1^{\text{gc}}(k,t) = \tilde{\phi}_1(k,t)\, J_0(k_\perp \rho_L), $$ with $\rho_L=\sqrt{2m\mu}/(qB)$.
The drift-wave turbulence relevant for ions is dominated by ITG/TEM modes [8].
Circular geometry
In circular approximation, $$ \mathbf{B} = B_0 R_0\,\nabla\phi + \frac{r b_\theta(r)}{R}\,\nabla\theta \qquad\text{(10)} $$ $$ F(\psi)=B_0 R_0 \qquad\text{(11)} $$ $$ \psi(r)=B_0R_0\int_0^r b_\theta(r')dr' \qquad\text{(12)} $$ $$ b_\theta(r)=\frac{r\,\bar{q}(r)}{R_0^2-r^2} \qquad\text{(13)} $$ $$ \chi(r,\theta)=2\tan^{-1} \left[\frac{R_0-r}{R_0+r}\tan\left(\frac{\theta}{2}\right)\right] \qquad\text{(14)} $$
Solov’ev equilibria
With linear profiles: $$ \mu_0\frac{dP}{d\psi}=A, \qquad \frac{dF^2}{d\psi}=\frac{2 A R_0^2 \gamma}{1+\alpha^2} \qquad\text{(15)} $$ The poloidal flux is $$ \psi=\frac{A}{2(1+\alpha^2)} \left[R^2 - \gamma R_0^2 - Z^2 + \frac{\alpha^2}{4}(R^2-R_0^2)^2\right] \qquad\text{(16)} $$ Minor radius: $$ a=\frac{R_0}{2\sqrt{2-\gamma-\sqrt{\gamma}}} \qquad\text{(17)} $$ Elongation: $$ \kappa=\frac{\alpha}{\sqrt{1-\gamma/(2-\gamma)}} \qquad\text{(18)} $$ Current: $$ F(\psi)=B_0R_0\left[1+\frac{\psi}{2A\gamma B_0(1+\alpha^2)}\right] \qquad\text{(19)} $$
Experimentally reconstructed equilibria
Experimental magnetic equilibria (G-EQDSK format) provide $\psi(R,Z)$ and profiles $F(\psi)$, $\bar{q}(\psi)$ on numerical grids. They are generated by reconstruction tools such as CHEASE [9], EFIT [10], PLEQUE [11], NICE [12], and EQUINOX [13]. T3ST uses these equilibria when analyzing specific discharges.
Given the definitions of the magnetic equilibrium quantities $q_\theta(\psi,\theta)$, $\bar{q}(\psi)$, and $\chi(\psi,\theta)$ (Eqs. (6–8)), one can show that the Clebsch form of the magnetic field is valid:
$$ \mathbf{B} = \nabla(\phi - \bar{q}\,\chi)\times\nabla\psi. $$
For representing small-scale turbulent fluctuations, most gyrokinetic (GK) codes adopt field-aligned coordinates $(x,y,z)$ [14], which simplify the description of fluctuations elongated along magnetic field lines. These coordinates correspond to the radial, binormal, and parallel directions ($\mathbf{B}\propto\nabla y\times\nabla x\propto\partial \mathbf{r}/\partial z$):
$$ x = C_x f(\psi) \qquad\text{(20)} $$
$$ y = C_y\,( \phi - \bar{q}\,\chi ) \qquad\text{(21)} $$
$$ z = C_z\,\chi \qquad\text{(22)} $$
In practice, T3ST uses constant values $C_x=a$ (the tokamak minor radius), $C_y=r_0/\bar{q}(r_0)$ (with $r_0$ a reference radius), $C_z=1$, and defines $f(\psi)=\rho_t = \Phi_t(\psi)/\Phi_t(\psi_{\text{edge}})$, where $\rho_t\in[0,1]$ is the normalized effective radius. The toroidal magnetic flux is
$$ \Phi_t(\psi)=\int_{\text{axis}}^{\psi} \mathbf{B}\cdot\mathbf{e}_\phi\, dS(\psi'), $$
thus $\rho_t \approx r/a$ (in circular geometry).
Instead of computing turbulent fields self-consistently, T3ST models them statistically as an ensemble of random fields $\{\phi_1, A_1\}$ with prescribed statistical properties, consistent with experimental ITG/TEM observations. Several moderate assumptions are used:
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Gaussianity — Fluctuations are assumed to follow a normal distribution $P[\phi_1]\sim\exp(-\phi_1^2/2\langle\phi_1^2\rangle)$. Small experimental deviations from Gaussianity have negligible impact on transport.
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Homogeneity — Turbulence is taken as homogeneous. Thus the autocorrelation function $$ E(\mathbf{r},t;\mathbf{r}',t') = \langle \phi(\mathbf{r},t)\phi(\mathbf{r}',t')\rangle = E(\mathbf{r}-\mathbf{r}',\,t-t') $$ contains all statistical information and can be evaluated from the power spectrum $S(\mathbf{k},\omega)=\langle|\tilde{\phi}_1(\mathbf{k},\omega)|^2\rangle$.
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Drift-type dispersion relation — ITG/TEM fluctuations originate from drift-wave instabilities described approximately even in the saturated state by a dispersion relation $\omega^\star(\mathbf{k})$. In the code: $$ \omega^\star(\mathbf{k}) = \frac{\mathbf{k}\cdot\mathbf{V}^\star}{1+\rho_s^2\,|\mathbf{k}_\perp|^2}, $$ where $\mathbf{V}^\star = -(\nabla p_s\times\mathbf{b})/(n_s q_s B)$ is the diamagnetic velocity of species $s$, and $\rho_s$ is its Larmor radius.
Based on these assumptions, T3ST constructs an ensemble of random fields $\phi_1(x,t)$ from white-noise fields in Fourier space $\eta(\mathbf{k},\omega)$ satisfying $$ \langle \eta(\mathbf{k},\omega)\,\eta(\mathbf{k}',\omega')\rangle = \delta(\mathbf{k}+\mathbf{k}')\,\delta(\omega+\omega'). $$ The Fourier components of the potential are
$$ \tilde{\phi}_1(\mathbf{k},\omega) = \sqrt{S(\mathbf{k},\omega)}\,\eta(\mathbf{k},\omega), $$
ensuring Gaussianity and correct correlations.
In toroidal geometry, the fields can be expressed through a double Fourier series:
$$ \phi(x,t)=\sum_{n,m}\phi_{nm}(r,t)\, e^{i(n\phi+m\theta)} \qquad\text{(23)} $$
which preserves natural periodicity. However, due to the anisotropic nature of turbulence (long correlation lengths along the field, short perpendicular), field-aligned representation is preferred. Thus T3ST evaluates $\phi_1$ in field-aligned coordinates, even though magnetic drifts are computed in cylindrical coordinates.
The random field is represented as:
$$ \phi_1(x,y,z,t) =\int dk\, d\omega\; \tilde{\phi}_1(\mathbf{k},\omega)\; e^{\,i(k_x x + k_y y + k_z z - [\omega^\star(\mathbf{k})+\omega] t)} \qquad\text{(24)} $$
For $k_\parallel\ll k_\perp$, the correspondence with toroidal mode numbers is approximately:
$$ k_y \approx \frac{n}{C_y},\qquad m = -\big[\bar{q}(x_0)\,n\big] + \Delta m,\qquad k_z = \frac{\Delta m + \{\bar{q}(x_0)n\}}{C_z}, \qquad\text{(25)} $$
where $[\cdot]$ and $\{\cdot\}$ denote the integer and fractional parts, and $n,\Delta m\in\mathbb{Z}$.
The spectrum $S(\mathbf{k},\omega)$ is modeled analytically to reproduce saturated drift-wave turbulence characteristics:
$$ S = A_\phi^2 \, \frac{\tau_c\,\lambda_x\lambda_y\lambda_z}{(2\pi)^{5/2}}\, \frac{ \exp\!\left[-(k_x^2\lambda_x^2 + k_z^2\lambda_z^2)/2\right] }{ 1+\tau_c^2\omega^2 }\, \Big[ e^{-\frac{(k_y-k_0)^2\lambda_y^2}{2}} \;-\; e^{-\frac{(k_y+k_0)^2\lambda_y^2}{2}} \Big] \qquad\text{(26)} $$
The parameters $\lambda_x,\lambda_y,\lambda_z$ are the correlation lengths in the field-aligned directions; $k_0$ selects the most unstable mode $$ k_{y,\text{max}} \approx \frac{k_0}{2} + \sqrt{\frac{k_0^2}{4} + \frac{2}{\lambda_y^2}}. $$ The correlation time $\tau_c$ measures deviations from the dispersion relation, and $A_\phi$ sets the turbulence amplitude. Typical parameters:
$$ \lambda_x \approx 5\rho_i,\quad k_0 \approx 0.1/\rho_i,\quad \lambda_y \approx 5\rho_i,\quad \lambda_z \approx 1,\quad \tau_c \approx R_0/v_{\text{th}},\quad |e|A_\phi \approx 1\%\,T_i. $$
Since $\lambda_z\approx 1$, we obtain $\lambda_\parallel\approx\bar{q}R_0$, consistent with experimental observations.
Drift-wave turbulence in tokamaks is generally a superposition of ITG and TEM modes, weighted by coefficients $A_i$ and $A_e$:
$$ \phi_1 = A_i\,\phi_1^{\text{ITG}} + A_e\,\phi_1^{\text{TEM}}, \qquad A_i + A_e = 1. $$
It is well known that magnetic curvature, shear, internal transport barriers, ELM activity, and blob structures can induce non-Gaussian behavior and spatial variations of turbulence. Thus the assumptions of Gaussianity and homogeneity are only approximate. Nevertheless, numerical results (Sec. 4) show that particle transport is predominantly local, with particles moving only a few Larmor radii from their initial surface. Therefore, although not globally exact, these assumptions remain valid locally for the purposes of the present analysis.
A possible extension recently implemented in the T3ST code is the ability to include inhomogeneities along the direction parallel to the magnetic field (“z”) through a simple prefactor:
$$ \phi_1 \rightarrow \phi_1 \, g(z) $$
$$ g(z) = e^{-\, z^2 / (2 \Lambda_z^2)} $$
where \( \Lambda_z \) is a measure of the parallel correlation and of the “ballooning” character specific to electrostatic turbulence.
O2 (WP2) – Developing and testing numerical codes
Deliverable: 1 code for test particle simulations, 2 codes for machine learning
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Within this work package, we implemented new technical improvements to the code (T3ST) for test-particle dynamics in turbulent fields (Task 2.1), developed scripts for implementing machine-learning methods (Task 2.2), and tested these two computational components (Task 2.3).
- T2.1: Improvements to the test-particle numerical code
- extract_contour_zero — finds all $(R,Z)$ points where $\phi=0$;
- order_points_by_angle — sorts contour points by polar angle to form a closed ordered loop;
- interp_point_along_contour — interpolates a point at a chosen position along the ordered contour.
- T2.2: Machine learning tool development
- T2.3: Testing
During this stage of the project, the development of the T3ST code has been completed. T3ST is a numerical FORTRAN code implementing the mathematical description of charged-particle dynamics in tokamak configurations, as presented earlier. Moreover, the code can include axisymmetric equilibrium magnetic fields at different levels of description (circular model, Solov’ev model, or equilibria reconstructed from experimental data). On top of this equilibrium field, Coulomb collisions and electrostatic turbulent fields are superimposed. The latter are modelled statistically using the principle of statistical representability, through random fields with predefined statistical characteristics.
Charged particles are dynamically followed in this electromagnetic environment using a numerical solver for the equations of motion, based on a 4th-order Runge–Kutta scheme. Once the particle trajectories are obtained, radial transport is estimated at the level of the convection and radial diffusion coefficients, computed in the usual way from the average position and dispersion of the trajectories:
$$ V_r(t\,|\,x)=\frac{d\langle X(t\,|\,x)\rangle}{dt}, \qquad D_r(t\,|\,x)=\frac12\frac{d}{dt}\left(\langle X(t\,|\,x)^2\rangle - \langle X(t\,|\,x)\rangle^2\right). $$
At the time of the funding proposal, the T3ST code (although unnamed at the time) was already about 90% complete and had been used in several theoretical investigations. As of this report, T3ST has undergone several improvements, detailed below.
Refinement of collision routines
Fokker–Planck operators for Coulomb collisions, $$ C[f,f_s] = -\partial_z\left(K_z f - D_{zz'}\,\partial_{z'} f\right), $$ have been previously derived in the literature for gyrokinetic coordinates [15].
Since T3ST follows particle trajectories $\{X(t), v_\parallel(t), \mu(t)\}$, we must describe collisions at this level. Collisions modify the gyroscopic dynamics, turning the deterministic ODE system into a set of stochastic differential equations (SDEs):
$$ d\mathbf{X} = \sqrt{2D_c\,(I-\mathbf{b}\otimes\mathbf{b})}\,d\mathbf{W}_X, $$
$$ dv_\parallel = v_\parallel\!\left[ -\nu + \left(\frac{2D_\parallel - D_\perp}{v^2} + \frac{\partial D_\parallel}{\partial v}\right) \right] dt + \Sigma_{v_\parallel,v_\parallel} dW_{v_\parallel} + \Sigma_{v_\parallel,\mu} dW_\mu, $$
$$ d\mu = \mu\!\left[ -2\nu + mE\!\left( v_\parallel \frac{\partial D_\parallel}{\partial v} + \frac{3(D_\parallel-D_\perp)(\mu B + 2E D_\perp)}{\mu B} \right) \right] dt + \Sigma_{\mu,v_\parallel} dW_{v_\parallel} + \Sigma_{\mu,\mu} dW_\mu, $$
where $dW_i$ are independent Wiener processes with zero mean.
Besides the full Coulomb operator, T3ST also implements a simplified Lorentz operator, useful especially for electron transport and providing a reasonable approximation for ions in coronal equilibrium with the background plasma. This formulation describes the evolution of the pitch angle $\lambda = v_\parallel/v$, where $v=\sqrt{v_\parallel^2 + v_\perp^2}$, and the kinetic energy $E_k$:
$$ \lambda(t+\Delta t) = \lambda(t) - \nu_d \lambda(t)\Delta t + \sigma\,(1-\lambda(t)^2)\nu_d\Delta t, \qquad (?) $$
$$ E_k(t+\Delta t) = E_k(t) - 2\nu_e\Delta t\, E_k(t) T \left(\frac{1}{T} - \frac{3}{2}\frac{E_k}{T} - \frac{d\ln\nu_e}{dE_k}\right) + 2\sigma\sqrt{\nu_e \Delta t\, E_k(t) T }. \qquad (?) $$
Here $\sigma=\pm1$ is a random sign chosen independently for each particle at each step. $T$ is the background plasma temperature, and $\nu_d$ and $\nu_e$ are the deflection and energy-transfer collision frequencies:
$$ \nu_d(v)= \frac{2\ln\Lambda\, n_i (Z_{\text{eff}} Z e^2)^2}{\pi\epsilon_0^2 m^2 v_{\text{th}}^3} \frac{\Phi(v)-\Psi(v)}{v^3}, \qquad (?) $$
$$ \nu_e(v)= \frac{\ln\Lambda\, n_i (Z_{\text{eff}} Z e^2)^2}{4\pi\epsilon_0^2 m^2 v_{\text{th}}^3} \frac{\Psi(v)}{v^3}. \qquad (28) $$
with $\ln\Lambda = 17$ the Coulomb logarithm, and the functions:
$$ \Phi(v) = \operatorname{erf}(v), \qquad (?) $$
$$ \Psi(v) = \frac{1}{2v^2}\big[\Phi(v) - v\,\Phi'(v)\big]. \qquad (?) $$
Equations (??) preserve both the isotropy of the pitch-angle distribution and the Maxwell–Boltzmann energy distribution: $$ P(\lambda, t+\Delta t) = P(\lambda). $$
Refinement of additional routines
Using real magnetic equilibria from preprocessed G-EQDSK files requires interpolation procedures to evaluate magnetic quantities along particle trajectories, which evolve in a Lagrangian manner, not on a fixed grid. Moreover, for initializing particle positions at $t=0$, most scenarios in T3ST require a uniform distribution of particles along a chosen magnetic flux surface. The previous implementation contained minor technical issues that introduced small spurious effects in the transport coefficients.
Within task T2.1, we identified, corrected, and improved the Fortran routine responsible for initializing particles on magnetic flux surfaces in experimentally reconstructed equilibria. The new routines are: psisurf_efit3, extract_contour_zero, order_points_by_angle, and interp_point_along_contour.
The routine psisurf_efit3 reconstructs magnetic flux surfaces from EFIT data and returns a representative point on each surface for a given parallel velocity $v_\parallel$. It reconstructs the poloidal flux $\psi(R,Z)$ and associated quantities $(F,\partial_R\psi,\partial_Z\psi)$ on a regular $(R,Z)$ grid, computes the magnetic field magnitude, and forms a modified scalar field:
$$ \phi(R,Z) = \psi - \psi_0 - K\,\frac{F}{|B|}\,V_p(k). $$
For each $v_\parallel$, the routine identifies the zero contour of $\phi(R,Z)$ — corresponding to the desired flux surface — using the auxiliary procedure extract_contour_zero, which locates sign changes of $\phi$ along cell edges and interpolates zero-crossing positions. A random point from this contour is selected as the particle position $(X(k), Y(k))$. If no valid contour is found, a fallback central value is used. The loop over all particles is parallelized with OpenMP.
Aggressive parallelization strategies
The most important improvement in T3ST in 2025 concerns program structure, leading to a roughly order-of-magnitude speed-up. This allows numerical simulations with satisfactory resolution (e.g., $N_p=10^6$, $N_c=200$, $N_t=1000$) on available hardware (~96 threads) in about one minute.
In previous versions, the most expensive computation was the evaluation of turbulent fields (and their derivatives), because it scaled as $\mathcal{O}(N_p N_c)$, where $N_p$ is the number of particles and $N_c$ the number of Fourier modes. By comparison, all other components scale as $\mathcal{O}(N_p)$. Thus, the earlier paradigm computed all quantities in series with vectorized operations, while only the turbulent components were parallelized.
Within WP2, we explored a complementary configuration: parallelizing all operations over the particles ($N_p$). Surprisingly, this approach led to a substantial performance improvement. A representative plot of computation time vs. particle number is shown in Figure 1 (log scale), where comparison between the old and new parallelization models reveals a speed-up of approximately a factor $\sim 11$.
The development of the T3ST code took place over several years and culminated in the present project, during which the last refinements and improvements to the numerical tool were made. Consequently, the code has gone through an extensive sequence of testing stages, targeting both specific implementations and large-scale results.
In this report we restrict ourselves to presenting three examples relevant to transport aspects. The first two refer to purely neoclassical dynamics (in the absence of turbulence) at the microscopic level, while the third test integrates almost all components of the code simultaneously.
We begin by checking whether the neoclassical trajectories (without turbulence) satisfy the minimal requirements regarding the correctness of their topology. For this purpose we analyse (Figure 2) two randomly chosen trajectories, which illustrate both the characteristic geometry of passing particles and that of “banana” orbits, both properly closed in the poloidal projection, in a circular configuration.
The second test targets the quantitative aspect of the trajectories. To this end, we generate a set of particles with a Maxwell–Boltzmann energy distribution, with pitch angles uniformly distributed in the interval \((0, 2\pi]\), and their initial positions placed on a magnetic surface corresponding to a circular equilibrium (thus at a fixed radius, but uniformly distributed in the toroidal and poloidal angles).
Using the circular model (see equation ??), it can be shown that a very good approximation of the radial velocity is provided by the analytical expression
In Figure 3 we present the distribution of radial velocities obtained in the analysed scenario, both from the analytical formula mentioned above and from the simulations carried out with the T3ST code. The two distributions are seen to coincide almost perfectly, which constitutes a solid validation of the way in which the guiding-centre drifts are calculated (at least in the non-turbulent regime).
The last and probably most important test is illustrated in Figure 4. The scenario considered is the following: hydrogen ions are distributed according to a Maxwell–Boltzmann distribution in energy and uniformly in pitch angle. The spatial distribution is chosen such that all particles possess approximately the same toroidal momentum, in direct connection with the new routine developed during stage T1.2:
This initial distribution is very close to a gyrokinetic equilibrium (as opposed to a distribution uniform on a magnetic surface). We therefore expect that particles distributed in this way will exhibit only a short transient phase, after which they will rapidly settle into a genuine equilibrium state, characterised by vanishing transport coefficients.
In this scenario, after reaching the quasi-equilibrium state, turbulence is initialised and the particle dynamics are then followed further. It can be shown—although the analysis is beyond the scope of this paper—that, under the given conditions, the quasilinear approximation of the diffusion (valid for short times immediately after the onset of turbulence) can be written as
The complexity of this test is also visible in the analytical formula inspired by quasilinear theory. It includes the turbulence intensity, the geometric effects associated with toroidicity, the spectrum of poloidal wavenumbers, as well as finite Larmor radius effects, which enter directly into the statistical average through the Bessel function \(J_0\).
In Figure 5 the results of this complex test are presented. As expected, in the absence of turbulence, the transient dynamics are short and the system rapidly converges towards a state without transport. This state is then “excited” by turbulence (around time \(t \approx 9\)), and the initial slope of the evolution is predicted with very high accuracy by the quasilinear result at short times.
O3 (WP3) – The construction of the database: extensive simulations
Deliverable: Database, trained and tested machine learning tools
...O4 (WP4) – Predictions and physical mechanisms
Deliverable: Characterization of transport mechanisms
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