Asymptotic scaling theory of electrostatic turbulent transport in magnetised fusion plasmas

TL;DR

The paper develops a simple, first-principles asymptotic scaling theory for electrostatic turbulence in magnetised fusion plasmas, unifying ETG- and ITG-driven transport by balancing linear growth, nonlinear decorrelation, and parallel propagation. It shows the heat flux depends on two geometry-sensitive quantities, the parallel system scale $L_\parallel$ and the outer-scale aspect ratio $\mathcal{A}^{\mathrm{o}}$, yielding ETG cubic scaling with the electron temperature gradient and ITG linear scaling, with extensive nonlinear gyrokinetic simulations across slab, tokamak CBC, and stellarator NCSX/W7-X confirming the predictions. The theory provides a physics-based foundation for fast, geometry-aware transport models and highlights how $L_\parallel$ and $\mathcal{A}^{\mathrm{o}}$ encode the essential saturation physics across diverse geometries. A key finding is that adiabatic responses govern $\mathcal{A}^{\mathrm{o}}$, making ETG largely $\mathcal{A}^{\mathrm{o}}\sim1$ while ITG exhibits geometry-dependent anisotropy, which drives their distinct heat-flux scalings. The framework thus offers a tractable pathway for reactor optimisation in both tokamaks and stellarators by linking turbulence to magnetic geometry through two physically transparent parameters.

Abstract

Turbulent transport remains one of the principal obstacles to achieving efficient magnetic confinement in fusion devices. Two of the dominant drivers of the turbulence are microscale instabilities fuelled by electron- and ion-temperature gradients (ETG and ITG), whose nonlinear saturation determines the cross-field transport of particles and energy. Despite decades of study, predictive modelling of this turbulence has been limited either to expensive gyrokinetic simulations or to reduced models calibrated by fitting to numerical or experimental data, restricting their utility for reactor design. Here we present a simple asymptotic scaling theory that unifies ETG- and ITG-driven turbulence within a common framework. By balancing the fundamental time scales of linear growth, nonlinear decorrelation, and parallel propagation, the theory isolates the dependence of the heat flux on equilibrium parameters to two key quantities: the parallel system scale and the outer-scale aspect ratio. We show that these quantities encapsulate the essential physics of saturation, leading to distinct predictions for ETG and ITG transport: a cubic scaling with the temperature gradient in the electron channel, and a linear scaling in the ion channel. Extensive nonlinear gyrokinetic simulations confirm that these theoretical predictions hold irrespective of the magnetic geometry (slab, tokamak, or stellarator), including the first numerical confirmation of the cubic ETG scaling anticipated by earlier theory. By isolating the dependence on just the parallel system scale and the outer-scale aspect ratio, our framework provides a physics-based foundation for fast, geometry-aware transport models, offering a pathway toward reactor optimisation in both tokamaks and stellarators.

Figures (17)

• Figure 1: Scaling results from the sETG simulations. (a) Time-averaged gyro-Bohm-normalised heat flux $Q_e/Q_{\mathrm{gB}e}$ [here $Q_{\mathrm{gB}e}= n_{e} T_{e} v_{\text{th}{e}} (\rho_e/L_{T_{e}})^2$] as a function of $L_\parallel/L_{T_{e}}$ (black circles), with the error bars showing the standard deviation over time. The dotted line shows the expected scaling [first expression in \ref{['eq:predictions_setg']}]. (b) Contour plot of the unstable solutions sETG dispersion relation \ref{['eq:setg_dispersion_relation']} in the $(k_y, k_\parallel)$ plane for $k_x=0$, $Z_i=1$, and $T_{i} = T_{e}$, with the dashed line showing the associated stability boundary \ref{['eq:setg_stability_boundary']}. The vertical dotted lines indicate to the minimum and maximum $k_y \rho_e$ in the simulation domain, with the horizontal dotted lines bounding the range of parallel wavenumbers corresponding to the different values of $L_\parallel/L_{T_{e}}$ in panel (a), each indicated by one of the horizontal coloured lines. The coloured diamonds in the inset denote the location of the intersection of these lines with the stability boundary. The maximum value of $k_\parallel L_{T_{e}} = 0.6$ for all simulations corresponds to the top edge of the plot. (c) Time-averaged value of the binormal outer scale $k_y^{\mathrm{o}} \rho_e$ (black circles), with the error bars showing the standard deviation over time. The coloured diamonds are values of $k_y \rho_s$ from those in panel (b) multiplied by an order-unity constant [see the discussion following \ref{['eq:spectrum_binormal']}]. The dotted line shows the expected scaling [the second expression in \ref{['eq:predictions_setg']}]. (d) Time-averaged outer-scale aspect ratio $\mathcal{A}^{\mathrm{o}}$ (black circles), with the error bars showing the standard deviation over time. The dotted line shows the expected scaling [the third expression in \ref{['eq:predictions_setg']}].
• Figure 2: Gyro-Bohm-normalised heat-flux time traces and spectra from the sETG simulations, with the colours indicating the value of $L_\parallel/L_{T_{e}}$ for a given simulation: (a) heat flux as a function of time; (b) time-averaged binormal spectrum of the heat flux \ref{['eq:spectrum_binormal']}; (c) time-averaged radial spectrum of the heat flux \ref{['eq:spectrum_radial']}.
• Figure 3: Heat-flux scalings from the CBC simulations. In all panels, the circles correspond to ETG cases and squares to ITG ones. The error bars show the standard deviation over time. (a) Heat flux as a function of the normalised temperature gradient $a/L_{T_{s}}$, with the dotted lines showing the predicted scalings [first expressions in \ref{['eq:predictions_etg']} and \ref{['eq:predictions_itg']}]. (b) Electron heat flux rescaled by $(a/L_{T_{e}})^3$ for three different values of the normalised temperature gradient, indicated by the different colours. The dotted line shows the predicted scaling [first expressions in \ref{['eq:predictions_etg']} and \ref{['eq:predictions_itg']}]. (c) The same as in (b) except the ion heat flux is rescaled by $(a/L_{T_{i}})^1$.
• Figure 4: Outer-scale binormal wavenumber scalings from the same CBC simulations as in \ref{['fig:CBC_heatflux_data']}. In all panels, the time-averaged values of $k_y^{\mathrm{o}}\rho_s$ are indicated by the points, with circles corresponding to ETG cases and squares to ITG ones. The error bars show the standard deviation over time, while the dotted lines show the predicted scalings [second expressions in \ref{['eq:predictions_etg']} and \ref{['eq:predictions_itg']}] at large $a/L_{T_{s}}$. (a) and (c): $k_y^{\mathrm{o}} \rho_s$ rescaled by the safety factor $q$ as a function of the normalised temperature gradient $a/L_{T_{s}}$ for ETG and ITG, respectively. (b) and (d): $k_y^{\mathrm{o}} \rho_s$ rescaled by $a/L_{T_{s}}$ as a function of $q$ at three different values of $a/L_{T_{s}}$, indicated by the different colours, for ETG and ITG, respectively.
• Figure 5: Outer-scale aspect ratio scalings from the same CBC simulations as in \ref{['fig:CBC_heatflux_data']}. In all panels, the time-averaged values of $\mathcal{A}^{\mathrm{o}}$ are indicated by the points, with circles corresponding to ETG cases and squares to ITG ones. The error bars show the standard deviation over time, while the dotted lines show the predicted scalings [third expressions in \ref{['eq:predictions_etg']} and \ref{['eq:predictions_itg']}]. (a) and (c): $\mathcal{A}^{\mathrm{o}}$ as a function of the normalised temperature gradient $a/L_{T_{s}}$ for ETG and ITG, respectively. (b) and (d): $\mathcal{A}^{\mathrm{o}}$ rescaled by $a/L_{T_{s}}$ as a function of $q$ at three different values of $a/L_{T_{s}}$, indicated by the different colours, for ETG and ITG, respectively.