Convergence of Roberts flow dynamos in Eulerian and Lagrangian codes

Abstract

Standard benchmark tests for numerical magnetohydrodynamics codes focus on one- and two-dimensional problems that ignore the possibility of dynamos, i.e., an exponential instability converting kinetic energy into magnetic energy. They also tend to ignore magnetic helicity conservation that can have dramatic effects in periodic domains. The existence of magnetic diffusion is crucial for dynamos, as was exemplified by the impossibility of finding dynamo solutions when using Euler potentials, which is a perfectly valid approach in the absence of magnetic diffusion. The Roberts flows~I, II, III, and IV are two-dimensional flows that support three-dimensional dynamos, all of which are also large-scale dynamos in the sense that their planar averages are of significant strengths, even though flow~II is pointwise non-helical. Here we demonstrate the numerical convergence properties of solutions obtained using discretisation schemes of second, sixth, and tenth order in the mesh width in the Pencil Code, compare the results with the SPH-based SWIFT code, and present a list of averaged quantities that facilitate their use in characterising the solutions as benchmarks. In addition, we present a procedure to obtain two-dimensional time-independent magnetic field visualisation for all Roberts flows, and perform per-pixel comparison between the Pencil and SWIFT codes. The rich combination of features makes Roberts flows an excellent dynamo benchmark for the direct numerical simulations of magnetohydrodynamics equations.