Clarifications to the limitations of the s-alpha equilibrium model for gyrokinetic computations of turbulence

In the context of gyrokinetic flux-tube simulations of microturbulence in magnetized toroidal plasmas, different treatments of the magnetic equilibrium are examined. Considering the Cyclone DIII-D base case parameter set [Dimits et al., Phys. Plasmas 7, 969 (2000)], significant differences in the linear growth rates, the linear and nonlinear critical temperature gradients, and the nonlinear ion heat diffusivities are observed between results obtained using either an $s-\alpha$ or an MHD equilibrium. Similar disagreements have been reported previously [Redd et al., Phys. Plasmas 6, 1162 (1999)]. In this paper it is shown that these differences result primarily from the approximation made in the standard implementation of the $s-\alpha$ model, in which the straight field line angle is identified to the poloidal angle, leading to inconsistencies of order $\varepsilon$ ($\varepsilon=a/R$ is the inverse aspect ratio, $a$ the minor radius and $R$ the major radius). An equilibrium model with concentric, circular flux surfaces and a correct treatment of the straight field line angle gives results very close to those using a finite $\varepsilon$, low $\beta$ MHD equilibrium. Such detailed investigation of the equilibrium implementation is of particular interest when comparing flux tube and global codes. It is indeed shown here that previously reported agreements between local and global simulations in fact result from the order $\varepsilon$ inconsistencies in the $s-\alpha$ model, coincidentally compensating finite $\rho^*$ effects in the global calculations, where $\rho^*=\rho _s / a$ with $\rho_s$ the ion sound Larmor radius. True convergence between local and global simulations is finally obtained by correct treatment of the geometry in both cases, and considering the appropriate $\rho^* \rightarrow 0$ limit in the latter case.