Comparison of forcing functions in magnetohydrodynamics

Abstract

Results are presented of direct numerical simulations of incompressible, homogeneous magnetohydrodynamic turbulence without a mean magnetic field, subject to different mechanical forcing functions commonly used in the literature. Specifically, the forces are negative damping (which uses the large-scale velocity field as a forcing function), a nonhelical random force, and a nonhelical static sinusoidal force (analogous to helical ABC forcing). The time evolution of the three ideal invariants (energy, magnetic helicity and cross helicity), the time-averaged energy spectra, the energy ratios and the dissipation ratios are examined. All three forcing functions produce qualitatively similar steady states with regards to the time evolution of the energy and magnetic helicity. However, differences in the cross helicity evolution are observed, particularly in the case of the static sinusoidal method of energy injection. Indeed, an ensemble of sinusoidally-forced simulations with identical parameters shows significant variations in the cross helicity over long time periods, casting some doubt on the validity of the principle of ergodicity in systems in which the injection of helicity cannot be controlled. Cross helicity can unexpectedly enter the system through the forcing function and must be carefully monitored.

Fully resolved array of simulations investigating the influence of the magnetic Prandtl number on magnetohydrodynamic turbulence

“…where e 1 · e * 2 = e 1 · k = e 2 · k = 0 and e 1 and e 2 are unit vectors statisfying ik × e 1 = ke 1 and ik × e 2 = −ke 2 [28][29][30]. A(k) and B(k) are variable parameters that allow the injection of helicity to be adjusted; we chose to force nonhelically.…”

“…In our forced simulations we set k 0 = 5 and forced the velocity field at the largest scales, 1 ≤ k ≤ 2.5. The nature of the forcing function and the forcing length scale do not greatly affect the dynamics [28,32]. We also ran decaying simulations, where we were less interested in the inertial range energy spectra and more interested in RST, so we set the peak at k 0 = 40.…”

Cross helicity sign reversals in the dissipative scales of magnetohydrodynamic turbulence

“…In the dissipation range, we observe a significant level of the relative cross helicity, as well as reversal of cross helicity as we traverse from large scales to small scales. This feature has been reported earlier in [16], but it was not analysed in detail. The presence of relatively large levels of relative cross helicity in the dissipation range is in marked contrast to the vanishing kinetic helicity in dissipative range of hydrodynamic turbulence.…”

A priori study of the subgrid energy transfers for small-scale dynamo in kinematic and saturation regimes

“…Finally, E b is approaching a statistically stationary state. That is, it enters the saturated stage (III) which at unity magnetic Prandtl number and sufficiently large Re is characterized by the ratios of the dissipation rates ε b /(ε u +ε b ) 0.7 and energies E b /(E u +E b ) 0.25 [41][42][43][44]. Our data in stage (III) is consistent with these ratios, as can be seen from the values listed in table II, where a summary of the dynamo runs in the kinematic (I), nonlinear (II) and saturated (III) stages is provided.…”