A. Ishizawa

K. Uzawa

N. Nakajima

National Institute for Fusion Science (NIFS-Repository)

Propagation of magnetic island due to self-induced zonal ﬂowUzawa, K.(JAEA), Ishizawa, A, Nakajima, N.We have investigated the propagation of magneticisland caused by the drift-tearing mode by numericallysolving a reduced set of two-ﬂuid equations1). It is foundthat the island propagates into the ion diamagnetic di-rection when the island growth is saturated, and thepropagation velocity becomes small as the viscosity in-creases. We have found that the island phase velocityis approximately same as the E × B ﬂow averaged in-side the island at the saturation state. In Figure 1 thetotal E × B ﬂow velocity and each Fourier component�v(m)E � = �∂xφmei2πmy� from m = 0 to m = 2 at satura-tion state are shown as a function of the viscosity. Whenthe viscosity is around 5 × 10−5, excitation of m = 1mode is comparable to that of the m = 0 mode, andboth modes aﬀect the direction of the averaged E × Bﬂow. Since the signs of zonal ﬂow and other modes areopposite, the propagation directinon of the island is de-termined by the competition between them. The zonalﬂow component is slightly larger than other modes, andthus the total E×B ﬂow directs toward the ion diamag-netic direction. It is observed that the averaged E × Bﬂow velocity monotonically decreases as the viscosity re-duces. The contribution of the zonal ﬂow to the averagedE × B ﬂow becomes dominant to those of other modeswhen the viscosity is smaller than 10−5. This indicatesthat the self-induced zonal ﬂow plays an important rolein determining the propagation direction when the vis-cosity is small.-0.02-0.0100.01m = 0m = 1m = 2ν<          >10-510-6 10-4(m)vEtotalFig. 1: The total E × B ﬂow velocity �vE� and eachFourier component �v(m)E � from m = 0 to m = 2.We have examined the generation mechanism of thezonal ﬂow and found that the contribution of each stressdepends on the viscosity. Figure 2 plots contributions ofeach stress to zonal ﬂow generation at saturation state asa function of the viscosity. The sum of contributions fromthe Reynolds stress �v(R)0 �, the Maxwell stress �v(M)0 � andthe 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 zonalﬂow into the equilibrium electron diamagnetic directionin all viscoisty regimes. On the contrary, the direction ofthe ﬂow due to the Maxwell stress shows monotonicallydecreasing function with respect to the viscosity. Whenthe viscosity is small, the Reynolds stress generates thezonal ﬂow in the electron diamagnetic direction and theMaxwell stress almost cancels out the ﬂow generation bythe Reynolds stress. The small diﬀerence between themdrives the ﬂow in the electron diamagnetic direction. Theﬂow velocity driven by the ion diamagnetic stress is neg-ative so that it directs in ion diamagnetic direction. Thedirection of the ﬂow due to the Maxwell stress is negativefor small viscosity cases and is positive for large viscos-ity cases. The ﬂow velocity driven by viscous stress isnegative 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 theMaxwell stress almost cancel out each other, and thesmall diﬀerence between them �v(R)0 �+ �v(M)0 �+ �v(V )0 � ispositive and drives the ﬂow in the electron diamagneticdirection. The ﬂow due to the ion diamagnetic stressis larger than this ﬂow, as a consequence the zonal ﬂowdirects toward the ion diamagnetic direction when theviscosity is small. On the other hand, when the viscosityis large, the viscous stress is large, and it is counteractedby the Maxwell and the Reynolds stresses. The diﬀerencebetween them�v(R)0 �+ �v(M)0 �+ �v(V )0 � drives the ﬂow inthe electron diamagnetic direction. The ion diamagneticstress almost cancels out the ﬂow but is slightly largerthan the ﬂow, and thus the zonal ﬂow is very small anddirects toward the ion diamagnetic direction when theviscosity is large.-1.5-1-0.500.511.5<       >0vν0v (M)<          >0v (R)<          >0v (I)<          >10-6 10-5 10-40v (V)<          >0v (R)<          > 0v (M)<          >+ 0v (V)<          >+Fig. 2: Contributions of each stress to zonal ﬂow gener-ation in saturation regime as a function of the viscosity.Note that all contributions are averaged inside the island.1) Uzawa, K. et al. : Phys. Plasmas 17 (2010) 042508392§8. Propagatio   M ti  Island Due to Self-induced Zonal FlowUzawa, K. (JAEA), Ishizawa, A., Nakajima, N.

Propagation of Magnetic Island Due to Self-induced Zonal Flow

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