Multiscale geometrical Lagrangian statistics of heavy impurities in drift-wave turbulence

TL;DR

This work addresses heavy impurity transport in edge drift-wave turbulence by performing direct numerical simulations of the 2D Hasegawa–Wakatani system and tracking inertial tungsten impurities. A Coulomb-scattering-based derivation yields an exact impurity relaxation time $\tau_p$, linking drag to the impurity flow via $d\boldsymbol{v}_p/dt = (\boldsymbol{u}-\boldsymbol{v}_p)/\tau_p + (Ze/m_p)(\boldsymbol{E}+\boldsymbol{v}_p\times\boldsymbol{B})$, with inertia quantified by the Stokes number $St=\tau_p/\tau_\eta$. The study reveals clustering at intermediate $St$ due to vortex centrifugal effects, while high-$St$ particles follow more ballistic, less structured trajectories; a multiscale geometrical Lagrangian framework based on the scale-dependent curvature angle and curvature $\kappa$ uncovers how inertia acts as a low-pass filter on trajectory changes across scales. These insights provide a quantitative basis for predicting impurity spatial distributions and transport in tokamak edge plasmas, with potential implications for tungsten accumulation and confinement in devices such as ITER.

Abstract

We investigate the behavior of heavy impurities in edge plasma turbulence by analyzing their trajectories using the Hasegawa-Wakatani model. Through direct numerical simulations, we track ensembles of charged impurity particles over hundreds of eddy turnover times within statistically steady turbulent flows. Assuming that heavy impurities lag behind the flow, a novel derivation of relaxation time of heavy impurities is proposed. Our results reveal that heavy impurities can cluster within turbulence. We provide multiscale geometrical Lagrangian statistics of heavy impurities trajectories. To quantify directional changes, we analyze the scale-dependent curvature angle, along with the influence of the Stokes number on the mean curvature angles and the probability distribution function of curvature angles.

Figures (7)

• Figure 1: Illustration of the curvature angle $\Theta$ and the curvature $\kappa$. $\bm{x}_p(t-\tau)$, $\bm{x}_p(t)$ and $\bm{x}_p(t+\tau)$ marked by red dots are the positions of the impurity particle $p$ at time instants $t - \tau$, $t$, and $t + \tau$, respectively. The inverse curvature $\kappa^{-1}$ corresponds to the radius of the unique circle passing through these three points, providing a discrete approximation at scale $\tau$ of the local trajectory curvature.
• Figure 2: Vorticity fields with $10^4$ impurity particles (sampled from a total of $10^6$) superimposed for $W^{60+} (St = 0.03)$, $W^{20+} (St = 0.20)$, $W^{10+} (St = 0.83)$, and $W^{3+} (St = 9.14)$ in the statistically steady state.
• Figure 3: Four plots are randomly selected for showing the trajectories of particles with different Stokes numbers, with their common initial position marked by a green cross. The periodic domain $[0, 64] \times [0, 64]$ has been extended when necessary to ensure continuity in trajectory visualization. For clarity, only the relevant portion of the domain is displayed.
• Figure 4: Left: Mean curvature angle $\theta(\tau)$ versus $\tau$ for different Stokes numbers $St$. Right: A magnified view for large $\tau$.
• Figure 5: PDF of the normalized curvature angle $x = \Theta/\pi$ for different Stokes numbers $St$ and different time lags $\tau$.