Neoclassical and turbulent E × B flows in flux-driven gyrokinetic simulations of Ohmic tokamak plasmas

Experiments have been conducted for similar hydrogen and deuterium discharges in the low current regime at the FT-2 tokamak ($I_{\rm p} = 19.5$ kA, $B_{\rm t} = 2.25$ T) [25, 26]. The large aspect ratio tokamak (R₀  =  55 cm, a  =  7.9 cm) has a circular cross section and a limiter configuration. The plasmas studied here have similar temperature and density profiles (T_i  =  150 eV, T_e  =  440 eV, and $n_e = 2.5 \times 10^{19}$ m⁻³ at axis as seen in figure 3), while impurity fraction is larger for deuterium ($Z_{\mathrm{eff}} = 2.8$) than for hydrogen ($Z_{\mathrm{eff}} = 2.3$). Spectroscopic measurements and temperature profiles point to partially ionised oxygen O⁶⁺ being the main impurity. The radial $Z_{\mathrm{eff}}$ profile is assumed to be uniform. High collisionality and large ion orbit widths ($\nu_{\ast i} > 5.0$ and $\rho_p \approx 1.0$ cm) characterise the parameter regime of the FT-2 plasmas. More detailed description of the discharges is available in [25, 26].

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Experimental measurements have found anti-correlation between the turbulence level and GAM activity in the discharges [25]. The simultaneous measurements of plasma poloidal velocity oscillations associated with the GAM and turbulent density fluctuations were performed using a combination of Doppler enhanced scattering (ES) and standard equatorial probing O-mode reflectometry. ES is a microwave technique that utilises the backscattering off the small scale density fluctuations in the probing wave upper hybrid resonance [27, 28]. The diagnostics also provide mean $E \times B$ flow measurements. The discharges were divided into time periods of strong and weak GAM amplitude, and the comparison of turbulent spectra between these periods shows that the turbulent fluctuations decrease when GAM amplitude is larger. This indicates that the the flow oscillations are able to suppress the turbulence. The effect is stronger for the deuterium discharge compared to hydrogen. At the same time, particle source estimations based on the H_β and D_β line radiation measurements indicate an isotope effect on particle confinement for the discharges [26].

We apply the gyrokinetic code ELMFIRE to study the interplay of flows and turbulence in these discharges. The simulation grid allows the resolution of turbulence for $k_{\theta}\rho_i < 3.0$ inside r/a  =  0.63, after which resolution declines to $k_{\theta}\rho_i \approx 1$ at r/a  =  1.0. Unlike in [25], the computations now cover the plasma from magnetic axis to the wall and both poloidal and radial resolution vary radially. A total of $4.9 \times 10^5$ ($2.4 \times 10^5$) grid points and $3.6 \times 10^9$ ($1.9 \times 10^9$) particle markers are used to obtain grid size equal to ion Larmor radius $\rho_i$ in radial and poloidal direction up to r/a  =  0.63 and an average of 7400 (7800) markers per grid cell in the hydrogen (deuterium) case. The particle count is sufficient to keep the noise below 1%, while no noise accumulation occurs in full-f simulations [34]. The time step is $\Delta t = 30$ ns in this case, and it is mainly limited by the accuracy condition for the parallel electron streaming, i.e. $k_{\parallel} \Delta t < 1$, and the number of toroidal grid cells (N_z  =  8 in this case) [40].

Available computational resources and the balance between sources and sinks limit the total physical time of these flux-driven simulations, which is relatively short (180 μs $\approx 0.2\tau_E$). The energy confinement time of the discharges is approximately $\tau_E \approx 1$ ms, and the particle confinement time is understood to be longer. As a perfect match between sources and sinks is not achieved, the mean profiles slowly diverge from the experimental equilibrium that is used as an input. The effect is worse for the hydrogen parameters (figure 3). The deviation is within error bars for density and ion temperature, but the electron temperature relaxes strongly at the edge. The total energy of the plasma increases with a rate of 18.3 kW in the end of the hydrogen discharge simulation compared to the estimated Ohmic input power of 48.5 kW. The total particle number remains constant due to the source model employed by ELMFIRE but the preset recycling profile does not completely match the flux profile, and so the density profile relaxes mainly by accumulating ions and electrons at the edge. As the particle confinement time is longer than the energy confinement time, the density profiles are maintained closer to the input during the simulation.

ELMFIRE predictions agree better with the Hinton–Hazeltine analytical formula than with the Hirshman–Sigmar estimate for FT-2 parameters when turbulence is excluded from the calculations (figure 4). For this comparison, only the flux-surface-averaged charge separation is again solved (m  =  0, n  =  0) and three different physics settings are applied to simulations of the hydrogen discharge: adiabatic electrons, kinetic electrons, and kinetic electrons with impurities (O⁶⁺). The collisionality of the plasmas lies in the transition zone between Plateau and Pfirsch–Schlüter regimes depending on the radial position ($\nu_{\ast i} > 5$). The good agreement with the Hinton–Hazeltine formula is in contradiction with the collisionality scan presented earlier. Finite orbit width (FOW) effects are the most likely explanation for this change, because inverse aspect ratio values are similar to the parameters used in the collisionality scan and cannot thus explain the difference. The analytical predictions agree on the very edge as the plasma transitions completely to the Pfirsch–Schlüter regime.

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Agreement between ELMFIRE results and the Hinton–Hazeltine formula is very good in the core, where the plasma is more in the plateau regime. Towards the edge the agreement worsens as the plasma transits to the collisional regime and the temperature gradient term starts to have a larger influence on the total E_r (figure 5). This is to be expected based on the collisionality scan. Toroidal rotation has a negligible effect on the radial electric field for these parameters: it is largest at 0.7  <  r/a  <  0.8, where it corresponds to 6.5% of the total Hinton–Hazeltine analytical prediction in the hydrogen case and on average to just 3.0%. Inclusion of kinetic electrons also changes the behaviour of the boundary condition and introduces a clear orbit loss effect that results in a wide radial electric field well on the edge. The analytical formula captures a part of the orbit loss effect as the inclusion of kinetic electrons causes a depletion of density and temperature very close to the wall. Otherwise the effect of kinetic electrons and even impurities is small despite the finite effective charge values ($Z_{\mathrm{eff}} = 2.2$ in the hydrogen case of figure 4). The former result is in agreement with results from the collisionality scan.

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Inclusion of turbulence significantly alters the E_r and thus the $E \times B$ flow and shear profiles in ELMFIRE simulations (figures 6 and 8). This is caused by three main factors: pressure profile modifications due to transport of energy and particles, turbulence-driven flows (low frequency zonal flows and finite frequency geodesic acoustic mode), and the non-adiabatic response of passing electrons. The first factor is clearly seen by comparing the analytical prediction for the initial profiles and the profiles averaged over time. Heating of the plasma reduces temperature gradients and ion collisionality in r/a  =  0.6–1.0, which decrease the $\nabla T_i$ contribution to E_r in the analytical formula. The parallel flow term $\langle U_{i, \parallel} B \rangle$ in equation (5) increases to 6.1% of the total E_r prediction on average from the 3.0% in the neoclassical simulations. The prominent E_r well seen in the neoclassical simulations narrows down, as the turbulent particle flux dominates over the orbit loss mechanism near the boundary.

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Turbulence-driven zonal flows are the main reason for the deviation of ELMFIRE prediction from the analytical Hinton–Hazeltine estimate. This is seen by comparing the difference in the mean $E \times B$ flow predictions and the turbulent flow drive (figure 7). Unlike in the comparison to the experimental measurements in figure 6, there is no need to consider how close the plasma profiles are to the input in this case, so the values are averaged over the whole saturated phase of the simulation lasting 150 μs. The simulated $E \times B$ flow $V_\theta$ is again calculated directly from the flux-surface-averaged radial electric field as $V_\theta = -E_r/B_{\rm t}$, while the E_r prediction of equation (5) is used for $V_{\theta, \mathrm{HH}}$ based on the T_i, n_i, and $\langle U_{i, \parallel} B\rangle$ profiles from the same simulation and time interval. The turbulent drive is quantified by Reynolds stress

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R_s = \langle \delta v_r \delta v_\theta \rangle \nonumber \end{align} \tag{ 10 }$$

and its derivative, the Reynolds force

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F_{\rm R} = -\frac{\partial R_s}{\partial r} = - \frac{\partial}{\partial r} \langle \delta v_r \delta v_\theta \rangle. \label{freyn} \nonumber \end{align} \tag{ 11 }$$

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Here $\delta v_r$ and $\delta v_\theta$ are taken as the radial and poloidal $E \times B$ drifts, respectively, while $\langle \cdot\rangle$ denotes averaging over the flux surface. The time averaged Reynolds force correlates strongly spatially with the deviation of the simulated $E \times B$ flow from the analytical values. Additionally, there is a significant temporally varying E_r component due to the geodesic acoustic mode, which is discussed in the following subsections.

The sharp zig-zag-like pattern of the flow and its shear is associated with the rational surfaces and the non-adiabatic response of passing electrons at these surfaces (figure 8) [15, 16]. The often used adiabatic electron assumption breaks down around surfaces for which the safety factor has values q_s  =  −m/n, where m and n are small integers. The parallel wavenumber vanishes as $k_{\parallel}\approx (nq_s + m)/(Rq_s) = 0$ and the adiabatic condition $k_{\parallel}v_{t, e} \gg \left(\nu_e^2 + \omega^2 \right){}^{1/2}$ is violated at and around the surface. As a consequence, the passing electrons also participate in turbulent fluctuations, which increases electron temperature fluctuations and electron diffusivity around rational surfaces. Electron temperature and density thus flatten at e.g. q_s  =  2.0 and steepen around it to account for the pattern in diffusivity. The impact on the ion temperature profile is limited, as observed earlier by Waltz et al [14]. The flow shear is close to zero at the rational surface and minimum and maximum next to it where the pressure profiles are steepened. The shear pattern stays fixed at these magnetic surfaces unlike the meandering $E \times B$ staircase observed by Dif-Pradalier et al [13]. Comparison of simulation and analytical predictions indicates that the fine structures are larger in amplitude than the neoclassical theory would predict based on the morphing of the profiles (figure 7).

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The parameter range and model employed in this manuscript have some crucial differences relative to the other work where similar flow patters have been identified. The interplay of rational modes and radial electric field is complicated by neoclassical background, full-f model, impurities, collisions, and flux-driven nature of the simulations compared to the work of Dominski et al [15, 16], while inclusion of impurities and kinetic electrons were missing in the studies of Dif-Pradalier et al [13]. The inclusion of kinetic electrons is crucial for the development of the staircase-like pattern presented in this manuscript and work of Dominski et al, as it enables the simulation of resonant modes, the effect of which could be exacerbated by the the temperature ratio $T_e/T_i > 1$. Meanwhile, impurities effectively change the collisionality of the plasma and influence the neoclassical radial electric field profile, increase the damping of zonal flows, and inhibit the linear growth of turbulence. The local flattening of the profiles, changes in E_r and turbulent fluctuation profiles, and localisation on rational modes have similarities with magnetic islands associated with internal transport barriers (ITBs) [41], although as an electrostatic code ELMFIRE cannot simulate their formation. Waltz et al also note that the effect of rational surfaces was observed with electrostatic simulations and without the magnetic signature of islands in experiments. Due to the short simulation time, the spatio-temporal flow pattern of the geodesic acoustic mode also plays a role in defining the averaged flow profile as it propagates radially and its amplitude evolves in time.

The $E \times B$ flow $V_{\theta} = - E_r/B_{\rm t}$ predictions by ELMFIRE agree well with experimental measurements of poloidal flow on average in both discharges (figure 6). The experimental values are based on the Enhanced Scattering diagnostic which measures the velocity of plasma fluctuations as a sum of the $E \times B$ drift and the phase velocity of the fluctuations $V_\theta = V_{E \times B} + V_p$. In the comparison of figure 6, the term $V_p$ is effectively neglected by taking $V_{E \times B} = V_\theta$. The phase velocity contribution is generally smaller than the uncertainty of the measurements based on local linear gyrokinetic simulations with GENE [42]. As the measurements pick the phase velocity for known wavenumbers at each radial position, the contribution of the phase velocity to the measurements can be estimated via the linear gyrokinetic simulations. For r/a  =  0.63 in the hydrogen case, the $V_\theta$ value is based on poloidal wavenumber $k_\theta \rho_i \approx 1.36$ and the simulations predict $V_p = 42.3$ m s⁻¹ in the ion diamagnetic direction. Collisions and impurities are included in these linear calculations.

The small-scale corrugations of the flow pattern cannot be compared to experiments due to limited resolution of the measurements, but the differences between the analytical Hinton–Hazeltine predictions and measurements indicate that either there are significant turbulent contributions or that the actual plasma profiles deviate meaningfully from the ones used in the simulations and ASTRA analysis, or a mix of both effects. Given the uncertainty in the ion temperature and that the $V_{\theta}$ values based on the relaxed profiles are closer to the measurements, the ion temperature profile may be actually less steep at the edge than is used as an input for simulations. Uncertainties in e.g. impurity distribution and safety factor profiles and their impact on the $E \times B$ flows are more difficult to quantify and would require more extensive and specific studies.

The oscillating component of the flow exhibits the basic characteristics of the geodesic acoustic mode with axisymmetric (m  =  0, n  =  0) $E \times B$ poloidal rotation and the associated m  =  1 up-down asymmetric density fluctuation. The dominating frequency of the flow oscillations is within 20% of experimentally measured values and theoretical prediction for the geodesic acoustic mode frequency (figure 9). The radial slope of the frequency profile in experiments and simulations is consistent with continuum-type GAM, for which the frequency of the mode scales according to the local plasma properties. Considering the density and safety factor profiles of the discharges, this is in line with computational and experimental evidence from TCV [44, 45].

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The theoretical prediction of the GAM frequency has been calculated according to the dispersion relation derived by Guo et al [43], which accounts for impurities and finite orbit width effects. The O⁶⁺ impurities are included according to ELMFIRE density profiles that are consistent with the experimental effective charge $Z_{\mathrm{eff}}$ in the beginning of the simulation, while experimental value is used for the radial wavenumber k_r, for the experimental measurements and computational predictions of $\lambda_{\mathrm{GAM}}$ have previously been shown to be close to each other [28]. For the hydrogen case, the simulations predict $\lambda_{\mathrm{GAM}} \approx 2.5$ cm and measurements indicate $\lambda_{\mathrm{GAM}} \approx 3.0$ cm, which agree within uncertainties. Relaxation of the temperature and density profiles is a characteristic feature of the flux-driven full-f simulations. In this case it mainly increases temperatures and thus expected frequencies in the edge plasma, and accounting for the relaxation improves the agreement between the analytical estimate and ELMFIRE prediction.

The amplitude of the oscillating flows is large enough to momentarily even reverse the poloidal $E \times B$ rotation from electron diamagnetic to ion diamagnetic direction in experiments and simulations (figure 9). This is in agreement with L-mode experiments in other tokamaks [19, 46, 47]. Comparison of Fourier analysis and standard deviation of the E_r time series shows that the latter is a good substitute for the GAM amplitude in ELMFIRE simulations of FT-2 plasmas due to the prominence of the oscillating mode. In fact, the standard deviation provides a better match for the experimental values. For the radial electric field, it is approximately 4.0 kV m⁻¹ at r/a  =  0.5 in both cases, while the mean value is $E_r \approx -3.0$ kV m⁻¹. As a consequence, the oscillating component of flow shear rate dominates over mean flow shear in most of the plasma volume (figure 8). The peak GAM amplitude is larger by a factor of two in the deuterium case compared to hydrogen, although the difference in the maximum standard deviations of $V_{\theta}$ is only approximately 50% as in the experiments. Given the propagation of the GAM in radial direction, longer simulations with improved power balance and reduced profile relaxation could improve the agreement between simulations and experiments.

The geodesic acoustic mode is excited in the non-linear saturation phase of the simulations, and the frequency and propagation signature of the GAM are also visible in other quantities as avalanche-like bursts that propagate mostly inwards in radial direction. This is a manifestation of the non-linear coupling of turbulence and flows. Turbulence transfers part of its energy to the flow, which is then dissipated by the damping of the flow back into the bulk plasma. In the following, we analyse this interaction in ELMFIRE simulations. We use the frequency associated with the peak E_r component predicted by ELMFIRE as the characteristic GAM frequency $f_{\mathrm{GAM}}$. This definition breaks down close to the axis, where the GAM signatures disappear.

The linear growth rate of the trapped electron mode (TEM, driven by the density gradient in these discharges) turbulence is approximately $\gamma = 4.0 \times 10^5$ s⁻¹ at r/a  =  0.63 in both discharges [26]. The standard deviation of the shearing rate $\omega_s$ easily exceeds this, while the mean value does not. It is difficult to draw conclusions by comparing the mean flow shear directly to linear growth rates, because also the mean electron temperature and density profiles steepen around the rationals in flux-driven full-f simulations as discussed earlier. The linear analysis is conducted with the initial profiles and the local morphing of profiles by transport is not taken into account.

The combined effect of mean and oscillating shear can be strong enough to provide significant rotation shear for turbulence suppression once or twice per GAM period. According to the theory proposed by Hahm et al [21, 24, 25], a condition for the effective shearing rate can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega_{\mathrm{eff}} = \vert \tilde{\omega}_{s}H + \overline{\omega}_s \vert > \gamma, \label{eff:shear} \nonumber \end{align} \tag{ 12 }$$

where $\overline{\omega}_s$ is the mean and $\tilde{\omega}_{s}$ the fluctuating component of the shearing rate

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega_s = \frac{r}{q_s B}\frac{\partial}{\partial r} \left( \frac{q_s}{r} \frac{\partial \phi}{\partial r}\right). \label{eqn:shear} \nonumber \end{align} \tag{ 13 }$$

The reduction factor $H\, =\, \left[\left(1\,+\,3F\right){}^2\,+\,4F^3\right]^{1/4}\,/$$\left[(1+F)\sqrt{1+4F}\right]$ (with $F = (2\pi F_{\mathrm{GAM}}/\gamma){}^2$) discounts the effect of the oscillating flow shear, since it is not as effective in suppressing turbulence as a stationary shear. The suppression mechanism itself is not sensitive to the sign of the gradient, but $\overline{\omega}_s$ is on average large enough to guarantee that the condition in equation (12) is fulfilled only in one half of a GAM period, i.e. when the mean and fluctuating component have the same sign.

The effective shearing manifests itself as a modulation of particle flux at the GAM frequency, and the effect is stronger in deuterium due to larger effective shearing rate. The spatial average of absolute mean shearing rate is $\vert \overline{\omega}_s \vert = 101$ kHz (105 kHz) in r/a  =  0.4–0.9 of the hydrogen (deuterium) case, while the standard deviation is $\sigma(\omega_s) = 611$ kHz (717 kHz) and reduction factor H  =  0.56 (0.60) in the same interval. The clear difference in the GAM amplitude is diluted in terms of shearing rate due to larger wavelength $\lambda_{\rm GAM}$ of the heavier isotope since $\tilde{\omega}_s \sim A_{\mathrm{GAM}}/\lambda_{\mathrm{GAM}}$. Temporal phase shift between the flow shear and particle flux is close to $\pi/2$ at the GAM frequency in both cases (figure 10). The flow shear follows the particle flux, and this relationship holds for the wide radial range where the GAM exists. The phase shift α is calculated as a function of temporal frequency f as $\alpha(\,f) = \mathrm{arg}[\omega_s^\ast(\,f) \Gamma_e(\,f)]$ after applying a temporal Fourier transform on the flow shear $\omega_s$ and the particle flux $\Gamma_e$. Here ^* denotes the complex conjugate.

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As in the case of the mean flow, we quantify the turbulent flow drive via Reynolds stress $R_s = \langle \delta v_r \delta v_\theta \rangle$ and its derivative Reynolds force $F_{\rm R} = -\partial R_s / \partial r$. The Reynolds power density or the energy transfer from the turbulence to the flow per unit of time in unit volume is calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P_{\rm R} = F_{\rm R} V_{\theta}, \label{gam:drive} \nonumber \end{align} \tag{ 14 }$$

where $V_{\theta} = -E_r/B_{\rm t}$ stands for the poloidal $E \times B$ rotation based on the axisymmetric radial electric field in the simulations [48]. The dissipation of the GAM, meanwhile, is generally expected to be induced by collisionless and collisional damping. We focus on the latter mechanism due to high collisionality of the FT-2 plasmas ($\nu_{i*} \geqslant 5.0$ in the whole plasma volume for both discharges). The Landau-like collisionless damping has its maximum in the the core region because it decreases quickly with the safety factor, although finite orbit effects tend to strengthen its effectiveness [49]. Collisional dissipation depends on the ion collision frequency $\nu_{i}$ and should thus be especially significant in the edge plasma [50]. We use the basic expression

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \gamma_{\mathrm{GAM}}= \frac{4}{7} \nu_i \label{collidamp1} \nonumber \end{align} \tag{ 15 }$$

for the collisional damping with $\nu_i$ calculated according to equation (7). To account for the effect of impurities, we multiply $\gamma_{\mathrm{GAM}}$ with the effective charge $Z_{\mathrm{eff}}$ [51]. The dissipation rate of the flow energy density is then calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P_{\rm D} = 2 \gamma_{\rm GAM} V_{\theta}^2. \label{gam:damp} \nonumber \end{align} \tag{ 16 }$$

The total Reynolds power density is approximately equal in the hydrogen and deuterium plasmas (figure 11). Balance between the Reynolds power density and collisional dissipation of equations (14) and (16) is well maintained for deuterium, except on the very edge. A more notable gap between the two remains for hydrogen parameters. This is likely due to the crude collisional damping estimate and short simulation duration. For the comparison, we have integrated the driving and dissipating power over the volume inside the magnetic surface and averaged over time. The results suggest that the drive and dissipation of the geodesic acoustic mode can be explained within the framework of Reynolds force and collisional damping in the FT-2 plasmas. This would thus support the earlier conclusion that the higher GAM amplitude in the deuterium discharge would be due to decreased collisional damping for the heavier isotope [25].

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The Reynolds force F_R alternates its direction between electron and ion diamagnetic directions by oscillating in time in the region of largest GAM amplitude (r/a  =  0.4–0.8). In this radial interval, the mean value of the force is close to zero compared to the fluctuating components, while inside r/a  =  0.4 the mean component starts to be significant (figure 12). The total Reynolds force fluctuations are 45% larger in hydrogen compared to deuterium according to the temporal standard deviation. As the standard deviation of the Reynolds force follows roughly the radial distribution of radial $E \times B$ drift fluctuation amplitude, the difference between the isotopes is largely explained by the larger turbulent $E \times B$ fluctuations $\langle (\delta v_{E, r}){}^2 \rangle$ in the hydrogen case. Combining this with the equivalent Reynolds power density P_R, the estimated turbulence consumption rate $P/\langle (\delta v_{E, r}){}^2 \rangle$ (the ratio of Reynolds power density and turbulent $E \times B$ energy) is actually larger for deuterium by roughly 45%. This is not obvious when comparing just the density fluctuation levels, which are comparable for the two cases. How $\delta n$ translates into $\delta v_{E, r}$ depends on the poloidal spectra of the density and, by extension, potential fluctuations. The conclusion is nevertheless in agreement with the earlier effective shearing rate calculations.

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The Reynolds power density is comparable between the isotopes due to larger amplitude of the GAM and the stronger correlation between the Reynolds force and the flow oscillations in the deuterium case (figure 13), despite the difference in the Reynolds force amplitudes (figure 12). The GAM frequency dominates the temporal correlation function for deuterium in the GAM populated radial zone, while higher frequency components are clearly present in the hydrogen case. The correlation between Reynolds force and poloidal $E \times B$ velocity at GAM frequency becomes negligible outside the GAM dominated region inside r/a  <  0.4 in the deuterium case. The phase shift $\alpha(\,f) = \mathrm{arg}[V_\theta^\ast(\,f) F_{\rm R}(\,f)]$ between the Reynolds force and the flow oscillation changes sign close to $f \approx f_{\mathrm{GAM}}$, indicating a small or vanishing phase shift for that frequency (figure 14). The behavior is consistent for both hydrogen and deuterium discharges in the radial range where the flow oscillations are observable in the simulations. The energy transfer from turbulence to the flows thus depends on the spatio-temporal relationship between Reynolds force and flow in addition to their absolute magnitudes according to Reynolds power density in equation (14). The synchronised oscillations allow the positive transfer of energy from turbulence to flows.

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