We have investigated the propagation of magnetic island caused by the drift-tearing mode by numerically solving a reduced set of two-fluid equations 1) . It is found that the island propagates into the ion diamagnetic direction when the island growth is saturated, and the propagation velocity becomes small as the viscosity increases. We have found that the island phase velocity is approximately same as the E × B flow averaged inside the island at the saturation state. In Figure 1 the total E × B flow velocity and each Fourier component �v (m) E � = �∂ x φ m e i2πmy � from m = 0 to m = 2 at saturation state are shown as a function of the viscosity. When the viscosity is around 5 × 10 −5 , excitation of m = 1 mode is comparable to that of the m = 0 mode, and both modes affect the direction of the averaged E × B flow. Since the signs of zonal flow and other modes are opposite, the propagation directinon of the island is determined by the competition between them. The zonal flow component is slightly larger than other modes, and thus the total E × B flow directs toward the ion diamagnetic direction. It is observed that the averaged E × B flow velocity monotonically decreases as the viscosity reduces. The contribution of the zonal flow to the averaged E × B flow becomes dominant to those of other modes when the viscosity is smaller than 10 −5 . This indicates that the self-induced zonal flow plays an important role in determining the propagation direction when the viscosity is small. We have examined the generation mechanism of the zonal flow and found that the contribution of each stress depends on the viscosity. Figure 2 plots contributions of each stress to zonal flow generation at saturation state as a function of the viscosity. The sum of contributions from the Reynolds stress �v (R) 0 �, the Maxwell stress �v (M ) 0 � and the viscous stress �v (V ) 0 � is also plotted as a reference. Note that all contributions are averaged inside the island. The Reynolds stress is positive so that it forces the zonal flow into the equilibrium electron diamagnetic direction in all viscoisty regimes. On the contrary, the direction of the flow due to the Maxwell stress shows monotonically decreasing function with respect to the viscosity. When the viscosity is small, the Reynolds stress generates the zonal flow in the electron diamagnetic direction and the Maxwell stress almost cancels out the flow generation by the Reynolds stress. The small difference between them drives the flow in the electron diamagnetic direction. The flow velocity driven by the ion diamagnetic stress is negative so that it directs in ion diamagnetic direction. The direction of the flow due to the Maxwell stress is negative for small viscosity cases and is positive for large viscosity cases. The flow velocity driven by viscous stress is negative and is very small when the viscosity is small. The balance of these stresses depends on the viscosity. When the viscosity is small, the Reynolds stress and the Maxwell stress almost cancel out each other, and ...

The effects of an external mean flow on the generation of zonal flow in drift wave turbulence are theoretically studied in terms of a modulational instability analysis. A dispersion relation for the zonal flow instability having complex frequency ω q = Ω q + iγ q is derived, which depends on the external mean flow's amplitude |φ f | and radial wave number k f . As an example, we chose an ion temperature gradient (ITG) turbulence-driven zonal flow as the mean flow acting on an electron temperature gradient (ETG) turbulence-zonal flow system. The growth rate of the zonal flow γ q is found to be suppressed, showing a relation, where γ q0 is the growth rate in the absence of mean flow and α is a positive numerical constant. This formula is applicable to a strong shearing regime where the zonal flow instability is stabilized at α|φ 2 f |k 2 f 1. Meanwhile, the suppression is accompanied by an increase of the real frequency |Ω q |. The underlying physical mechanism of the suppression is discussed.

The rotation direction of a magnetic island in the saturation regime and the underlying physical mechanism are numerically investigated based on a four-field model that includes the effects of both ion and electron diamagnetic drifts as well as parallel ion motion. It is found that diamagnetic effects vanish inside the island, and that the rotation direction is determined by nonlinearly generated zonal flow. The direction of zonal flow is sensitive to the viscosity and the finite Larmor radius (FLR) effect. The radial mode structure of zonal flow is found to be deformed by that of other modes as the viscosity increases. We have also shown that the FLR effect enhances island rotation toward the ion diamagnetic drift direction through energy transfer to the zonal flow by a nonlinear ion diamagnetic stress tensor.

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The spectral effect of the zonal flow (ZF) on its generation is investigated based on the Charney–Hasegawa–Mima turbulence model. It is found that the effect of finite ZF bandwidth qualitatively changes the characteristics of ZF instability. A spatially localized (namely, global) nonlinear ZF state with an enhanced, unique growth rate for all spectral components is created under a given turbulent fluctuation. It is identified that such state originates from the successive cross couplings among Fourier components of the ZF and turbulence spectra through the sideband modulation. Furthermore, it is observed that the growth rate of the global ZF is determined not only by the spectral distribution and amplitudes of turbulent pumps as usual, but also statistically by the turbulence structure, namely, their probabilistic initial phase factors. A ten-wave coupling model of the ZF modulation instability involving the essential effect of the ZF spectrum is developed to clarify the basic features of the global nonlinear ZF state.