Effect of the Shafranov shift and the gradient ofβon intrinsic momentum transport in up–down asymmetric tokamaks

Abstract

Abstract.Tokamaks with up-down asymmetric poloidal cross-sections spontaneously rotate due to turbulent transport of momentum. In this work, we investigate the effect of the Shafranov shift on this intrinsic rotation, primarily by analyzing tokamaks with tilted elliptical flux surfaces. By expanding the Grad-Shafranov equation in the large aspect ratio limit we calculate the magnitude and direction of the Shafranov shift in tilted elliptical tokamaks. The results show that, while the Shafranov shift becomes up-down asymmetric and depends strongly on the tilt angle of the flux surfaces, it is insensitive to the shape of the current and pressure profiles (when the geometry, total plasma current, and average pressure gradient are kept fixed). Next, local nonlinear gyrokinetic simulations of these MHD equilibria are performed with GS2, which reveal that the Shafranov shift can significantly enhance the momentum transport. However, to be consistent, the effect of β (i.e. the radial gradient of β) on the magnetic equilibrium was also included, which was found to significantly reduce momentum transport. Including these two competing effects broadens the rotation profile, but leaves the on-axis value of the rotation roughly unchanged. Consequently, the shape of the β profile has a significant effect on the rotation profile of an up-down asymmetric tokamak.

Optimized up–down asymmetry to drive fast intrinsic rotation in tokamaks

“…We increased the aspect ratio of the flux surface of interest by setting major radius to R 0 = 3 and ρ = 0.54. This was done because reference [22] indicates that the momentum transport is sensitive to the aspect ratio (specifically, it increases with aspect ratio). Additionally, the magnitude of the shaping was increased somewhat to C N 2 = 0.5 and C N 3 = 0.4.…”

“…on every flux surface [14]. However, we can calculate a simple estimate of the level of rotation using local values of Π ζ and Q i [22]. First, we assume that the energy flux is dominated by the diffusion of a temperature gradient [43],…”

Turbulent momentum transport due to the beating between different tokamak flux surface shaping effects

“…This change was found to modify the ratio of the momentum flux to the heat flux by less than 5%. We are interested in this ratio because we expect it to be roughly proportional to the level of intrinsic rotation [34]. Figure 4 shows the time-averaged radial flux of energy carried by the ions Q i t , calculated by GS2 for the three scans.…”

“…This change was found to modify the ratio of the momentum flux to the heat flux by less than 5%. We are interested in this ratio because we expect it to be roughly proportional to the level of intrinsic rotation [34].…”

Intrinsic rotation driven by turbulent acceleration

“…A comprehensive theory including all of these symmetry-breaking mechanisms is given in [16,17,18,15,19]. There have also been a number of studies dedicated to individual mechanisms, including the effect of diamagnetic flows [20,21,22,23,24], up-down asymmetry of flux surfaces [25,26,27,28,29,30,31,32,33,34], slow poloidal variation of fluctuations [35], and 'global' effects [36,37,38,39], which include radial profile variation mingled with the other effects mentioned. Here we consider the effect of turbulent particle acceleration along the mean magnetic field.…”