Suppression of temperature-gradient-driven turbulence by sheared flows in fusion plasmas

TL;DR

The paper develops a phenomenological theory for how perpendicular flow shear suppresses temperature-gradient-driven turbulence in magnetised fusion plasmas, distinguishing weak-shear and strong-shear regimes controlled by the outer-scale energy-injection rate. It derives scaling laws for the outer-scale wavenumbers and fluctuation amplitudes, and predicts heat-transport suppression patterns that depend crucially on the outer-scale aspect ratio $\mathcal{A}^\text{o}(0)$, validated by both a 2D electrostatic fluid ETG model and gyrokinetic ITG flux-tube simulations. The theory captures the transition between regimes and explains observed phenomena such as eddy tilting, isotropisation of the inertial range, and potential transport bifurcations, with detailed predictions for momentum transport and the Prandtl number. These results have implications for understanding cross-scale interactions, turbulence suppression by ion-scale flows, and the design of shear-based control strategies in fusion devices.

Abstract

Starting from the assumption that saturation of plasma turbulence driven by temperature-gradient instabilities in fusion plasmas is achieved by a local energy cascade between a long-wavelength outer scale, where energy is injected into the fluctuations, and a small-wavelength dissipation scale, where fluctuation energy is thermalised by particle collisions, we formulate a detailed phenomenological theory for the influence of perpendicular flow shear on magnetised-plasma turbulence. Our theory introduces two distinct regimes, called the weak-shear and strong-shear regimes, each with its own set of scaling laws for the scale and amplitude of the fluctuations and for the level of turbulent heat transport. We discover that the ratio of the typical radial and poloidal wavenumbers of the fluctuations (i.e., their aspect ratio) at the outer scale plays a central role in determining the dependence of the turbulent transport on the imposed flow shear. Our theoretical predictions are found to be in excellent agreement with numerical simulations of two paradigmatic models of fusion-relevant plasma turbulence: (i) an electrostatic fluid model of slab electron-scale turbulence, and (ii) Cyclone-base-case gyrokinetic ion-scale turbulence. Additionally, our theory envisions a potential mechanism for the suppression of electron-scale turbulence by perpendicular ion-scale flows based on the role of the aforementioned aspect ratio of the electron-scale fluctuations.

Figures (10)

• Figure 4: (a) Time-averaged, saturated radial turbulent heat flux, normalised to its value at zero flow shear, as a function of normalised flow shear $\hat{\gamma}_{E}$ [normalised per \ref{['eq:gammanorm']}] for the sets of simulations detailed in \ref{['tab:sims']}. The data from all four sets overlays due to the scale invariance of \ref{['eq:phi']}--\ref{['eq:T']}. The black dashed and dash-dotted lines show the theoretical predictions \ref{['eq:heatflux_weakly_sheared']} and \ref{['eq:heatflux_strong_shear']}, respectively, where, for the former, the curve is plotted using $\hat{\gamma}_{\text{c}} \approx 39$, found by fitting to the data presented here. The vertical black dotted line marks the approximate shearing rate $\hat{\gamma}_{E}\approx 100$ where the system transitions from the weak- to the strong-shear regime. The values of $\gamma_{E} \approx \gamma_\text{max}$ are shown using vertical dotted lines of the same colour as the data points for each respective set of simulations. (b) The outer-scale wavenumbers $k_x^\text{o}(\gamma_{E})$ and $k_y^\text{o}(\gamma_{E})$, defined as those that maximise \ref{['eq:Q_poloidal_avg']} and \ref{['eq:Q_radial_avg']}, respectively, for the Sim1 set of simulations. The dashed line indicates a linear dependence on the flow shear, $k \propto \gamma_{E}$. The left vertical dotted line is the same as in panel (a) and marks the location $\gamma_{E} \approx 100$ where the system transitions from the weak- to the strong-shear regime. In the former, $k_y^\text{o}$ is (approximately) pinned to $k_{y}^\text{o}(0)$ but $k_x^\text{o}$ increases linearly with $\gamma_{E}$. In the strong-shear regime, $k_x^\text{o} \sim k_y^\text{o} \propto \gamma_{E}$. The right vertical dotted line indicates the value of flow shear that is equal to the largest growth rate $\gamma_\text{max}$, where the outer scale $k_y^\text{o}$ reaches, at least approximately, the scale of the most unstable mode $k_{y, \text{max}}\rho_\perp \approx 3.7$. Note that, at low $\gamma_{E}$, \ref{['eq:Q_poloidal_avg']} is sometimes maximised at $k_x = 0$. In those cases, represented by the hollow triangles, we have set $k_x^\text{o}\rho_\perp$ to the box scale $2\pi \rho_\perp/L_x \approx 0.063$.
• Figure 5: Snapshots of $\varphi$ (top row) and $\delta T_e/T_{e}$ (bottom row) in the $(x, y)$ plane for Sim1 simulations with four different values of $\gamma_{E}$, as specified above each column. For each snapshot, the amplitudes are normalised to lie in the range $[-1, 1]$, with the values in this interval corresponding to colours between dark blue and dark red, respectively. The second column corresponds to the weak-shear regime (i) from \ref{['fig:heatflux_anisotropic']}(a), where the flow shear is too weak to influence the saturated state significantly. The third column also corresponds to the weak-shear regime, with $k_y^\text{o}(\gamma_{E})$ pinned to $k_{y}^\text{o}(0)$ but with $k_x^\text{o}(\gamma_{E})$ increased by the influence of the flow shear, which here clearly manifests itself as the tilting of the eddies. In this case, the structures have a similar size in $y$ to those in the first- and second-row panels, but a shorter length scale in $x$ due to being sheared. The last column shows the saturated state in the strong-shear regime (ii) of \ref{['fig:heatflux_anisotropic']}(a), where the flow shear has manifestly pushed the outer scale to much shorter wavelengths.
• Figure 6: Radial localisation of turbulent perturbations at very large values of flow shear. Taken from a Sim4 simulation with $\hat{\gamma}_{E} = 540$, which is just over the largest growth rate $\hat{\gamma}_\text{max} \approx 493$. The simulation has achieved a steady state with time-averaged normalised heat flux $\hat{Q}(\gamma_{E})/\hat{Q}(0) \approx 4 \times 10^{-7}$, which is why it is not visible in \ref{['fig:Q_vs_gammaE']}.
• Figure 7: (a) The time-averaged, saturated-state, radial turbulent heat flux, normalised to its value at $\gamma_{E} = 0$, and (b) the outer-scale poloidal wavenumber $k_y^\text{o}\rho_i$ as functions of the flow shear for two different values of the ion-temperature gradient (see \ref{['sec:ITG']} for other relevant numerical parameters). The black dashed line corresponds to the trend $Q \propto \gamma_{E}^{-1}$, while the blue and red dashed lines are linear fits for $k_y^\text{o}$ as a function of $\gamma_{E}$ for each temperature gradient. The vertical dotted lines correspond to $1.5\gamma_\text{max}$ for each of the simulations. The flow shear is normalised to $a/c_s$, where $a$ is the minor radius and $c_s$ is the ion sound speed.
• Figure 8: Radial turbulent heat flux versus time for $R/L_{T_i} = 14$ and four different values of $\gamma_{E}$, as labelled in the title of each panel. The blue lines are time traces from simulations initialised with small-amplitude noise, while the red ones represent simulations restarted from a saturated $\gamma_{E}=0$ run.