Abstract
Microturbulence close to marginality with inclusion of electron dynamics and in the electrostatic limit [A. Weikl et al., Phys. Plasmas 25, 072305 (2018)] is revisited. In such states the E × B shearing rate ωE×B, i.e., the second radial derivative of the zonal electrostatic potential, a quantity often applied to study zonal flow structure formation, has been found to be dominated by radial fine scale features. Those features are significantly different from the mesoscale E × B staircase structures [G. Dif-Pradalier et al., Phys. Rev. E 82, 025401(R) (2010)] normally occurring close to the threshold. Instead of the E × B shearing rate, here, zonal flow structure formation is studied through the zonal flow shear induced tilt of turbulent structures, which is measured by director field methods. In contrast to dominant fine scale features in ωE×B, mesoscale zonal flow pattern formation on two disparate scales is identified: (i) A zonal flow with radial scale of the boxsize develops, (ii) superposed by zonal flow corrugations in form of shear layers emerging in the vicinity of lowest order rational layers. This mesoscale zonal flow pattern exhibits properties of E × B staircases: (i) A shearing rate of ∼10−1 vth,i/R0 (vth,i is the ion thermal velocity and R0 is the major radius), comparable to typical growth rates, can be attributed to both components of the mesoscale pattern. (ii) Avalanche-like turbulent transport events organize spatially on the same mesoscales. (iii) Shear stabilization by a background E × B shear flow requires values of the background shearing rate exceeding those connected to the mesoscale pattern. In conclusion, this work demonstrates that E × B staircases do develop, even when the E × B shearing rate ωE×B is dominated by radial fine scale features. The E × B shearing rate ωE×B, therefore, fails to estimate the shear provided by zonal flows when fine scale structures dominate its radial profile.
Highlights
A rich body of literature demonstrates the significance of radially sheared zonal flows (ZFs), i.e., toroidally symmetric plasma flows due to the E Â B-drift, for both the nonlinear saturation[1,2,3,4] as well as the nonlinear stabilization[5,6,7,8] of microturbulence in tokamak plasmas
Instead of the E Â B shearing rate, here, zonal flow structure formation is studied through the zonal flow shear induced tilt of turbulent structures, which is measured by director field methods
In contrast to dominant fine scale features in xEÂB, mesoscale zonal flow pattern formation on two disparate scales is identified: (i) A zonal flow with radial scale of the boxsize develops, (ii) superposed by zonal flow corrugations in form of shear layers emerging in the vicinity of lowest order rational layers
Summary
A rich body of literature demonstrates the significance of radially sheared zonal flows (ZFs), i.e., toroidally symmetric plasma flows due to the E Â B-drift, for both the nonlinear saturation[1,2,3,4] as well as the nonlinear stabilization[5,6,7,8] of microturbulence in tokamak plasmas. A metric that characterizes the strength of this shearing process is the E Â B shearing rate xEÂB, i.e., the radial derivative of the advecting ZF velocity.[9,14] Shear stabilization of microturbulence is often expressed in the form of the empirical Waltz rule xEÂB $ c,7,14 where c denotes the maximum linear growth rate of the underlying microinstabilities Gyrokinetic studies support this condition.[15,16]. Director field techniques[25] are applied to study zonal flow structure formation; a technique that has found application in the microturbulence context in Ref. 26 This method is based on the ZF induced tilting of turbulent structures and, represents a more direct way of quantifying the ZF shearing compared to the E Â B shearing rate.
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