Numerical approach to the parallel gradient operator in tokamak scrape-off layer turbulence simulations and application to the GBS code

Abstract

a b s t r a c tThis paper presents two discretisation schemes for the parallel gradient operator used in scrape-off layer plasma turbulence simulations. First, a simple model describing the propagation of electrostatic shearAlfvén waves, and retaining the key elements of the parallel dynamics, is used to test the accuracy of the different schemes against analytical predictions. The most promising scheme is then tested in simulations of limited scrape-off layer turbulence with the flux-driven 3D fluid code GBS : the new approach is successfully benchmarked against the original parallel gradient discretisation implemented in GBS. Finally, GBS simulations using a radially varying safety profile, which were inapplicable with the original scheme are carried out for the first time: the well-known stabilisation of resistive ballooning modes at negative magnetic shear is recovered. The main conclusion of this paper is that a simple approach to the parallel gradient, namely centred finite differences in the poloidal and toroidal direction, is able to simulate scrape-off layer turbulence provided that a higher resolution and higher convergence order are used.

Global fluid simulation of plasma turbulence in stellarators with the GBS code

“…The z-dependence of the equilibrium density and electron temperature is retained because the gradients depend on the poloidal angle (they are typically steeper on the HFS). These assumptions lead to system in equation (22).…”

“…Since the geometrical operators ( 9)-( 13) have the same structure as the ones implemented in the axisymmetric version of GBS [14], their implementation does not require major changes in the structure of the code thanks to the use of a non field-aligned spatial discretization algorithm [22,23]. In fact, as in the previous versions of GBS, an Arakawa scheme is applied to implement the Poisson brackets [23], and the same solver used to invert equation (7) can be considered, since the perpendicular Laplacian only acts on the RZ-plane (to lowest order in the expansion parameters δ, ∆ and σ).…”

The GBS code for tokamak scrape-off layer simulations

“…The parallel gradient operators have been described in detail in Ref. [20]. Field-aligned and non-field-aligned strategies are possible.…”

“…The differential operators have been implemented using a finite volume approach, as suggested in [58] for variable coefficient problems. In GBS, in addition to the scalar λ inside the operator, one must also treat the shaping coefficients [20]. We express the problem in weak form, and the differential operator is treated using the divergence theorem ∇ · (λ∇ ⊥ ξ ) dV = λ ∇ ⊥ ξ ·n dℓ, which gives a line integral (n is an outward pointing vector normal to the integration contour, see Fig.…”

“…Magnetic shear effects were implemented using the ŝ − α metric coefficients [18], and then updated to include finite aspect ratio effects [19]. The numerical approach for the parallel gradient was recently generalized to allow varying q [20].…”