The nonlinear gyrokinetic equations are frequently used as a basis for simulations of small-scale turbulence in magnetized toroidal plasmas. In this context, field-aligned coordinates are usually employed in order to minimize the number of necessary grid points. The present work proposes a system of Clebsch-type coordinates which does not depend on the existence of flux surfaces. The construction and use of these coordinates is explained, and the corresponding formulation of the nonlinear gyrokinetic equations is accomplished. This setup paves the way toward the investigation of nonaxisymmetric toroidal geometries, also in the region of magnetic islands as well as inside the ergodic layer where flux surfaces cease to exist. For testing purposes, in the axisymmetric, large aspect ratio case, the well-known ŝ-α expressions are recovered for closed flux surfaces. Moreover, geometric data for a specific stellarator configuration are computed and discussed.

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