Zonal flows are recognized as a mechanism for the self-regulation of turbulence in both neutral fluids and plasmas. They can largely affect the saturation of instabilities and turbulent transport.

The seminal work by Rosenbluth and Hinton [Rosembluth PRL 1998] demonstrated that in tokamaks in the collisionless limit, a long-wavelength zonal flow is not completely damped linearly, and a finite residual zonal flow remains for long times. This residual level is reached after some characteristic Geodesic Acoustic Mode (GAM) oscillations. In a hot turbulent plasma, these undamped zonal flows can survive and affect (reduce) the anomalous transport.

In stellarator devices, the linear evolution of zonal flows exhibits characteristic properties different from those in tokamaks: the residual level in the long-wavelength limit is zero and a new oscillation, at a frequency lower than the GAM, appears. The linear relaxation of zonal flows in stellarators has been extensively studied in simulations with the code EUTERPE in [Sanchez et al PPCF 2013, Velasco et al. PPCF 2016, Monreal PPCF 2016, Monreal PPCF 2017, Alonso PRL 2017, Sánchez et al PPCF 2018]. Here we show some examples of this kind of simulations.

**Zonal flow relaxation in a tokamak**

In the next figures, the zonal component (Fourier component of index m,n= 0,0) of the electrostatic potential is shown versus radius in a set of simulations of linear collisionless zonal flow relaxation in a tokamak (Rosenbluth-Hinton relaxation). The simulations are initiated with a long-wavelength perturbation to the potential with different radial scales k_s=0.5π and 5π. As the simulation evolves, the potential oscillates and damps out, reaching a steady value at long times (residual level). The initial perturbation is shown in the left axis and the evolution of the potential with time in the right axis.

In Figure 1, an initial perturbation with k_s=1.5 π is used. A different scale is used in the vertical axis for the initial perturbation and the time evolution in order to emphasize the residual level reached. In Figure 2 a perturbation with smaller radial scale is used and the residual level is larger.

**Zonal flow relaxation in stellarators**

Contrary to tokamaks, which have a finite residual level for long-wavelength zonal flows, in stellarators the residual level for long-wavelength zonal perturbations is zero, in general. As a result of the presence of particles trapped in helical ripple wells, a low-frequency oscillation appears in these devices, and the zonal potential oscillates until it damps out and reaches a zero value at long times.

Figure 3 shows the electrostatic potential in a simulation of linear collisionless relaxation of a zonal flow in the standard configuration of W7-X. The initial, long-wavelength perturbation, with k_s=1.5 π, is shown in the left axis and the evolution of the potential with time in the right axis.

Figure 4 shows the electrostatic potential versus radius in a simulation of zonal flow relaxation in W7-X. The initial perturbation is shown in the left axis and the evolution of the potential with time in the right axis. While for long-wavelength perturbations the long-time residual level of zonal flows is zero, at large radial wavelengths the residual is not null. This not null residual level is appreciated in this figure, in which a finite value is reached.

Figure 5 shows the electrostatic potential perturbation in a period of the standard configuration of TJ-II obtained in a linear simulation of zonal flow relaxation. The initial perturbation, in this case, is long-wavelength and the global character of the potential structure is clearly observed in all the toroidal cuts shown in the image. No clear m=1 structure is observed in the potential, as in the case of the density (next figure).

Figure 6 shows density perturbation at toroidal angle phi=0 in a simulation of linear zonal flow relaxation carried out in the standard configuration of TJ-II. At the beginning of the simulation, an m=1 component is clearly observed in the density, corresponding to the Geodesic Acoustic Mode (GAM) oscillation, which is strongly damped in TJ-II. The amplitude of this structure reduces with time as the simulation evolves.