Transition to Turbulent Dynamo Saturation

Abstract

While the saturated magnetic energy is independent of viscosity in dynamo experiments, it remains viscosity-dependent in state-of-the-art 3D direct numerical simulations (DNS). Extrapolating such viscous scaling-laws to realistic parameter values leads to an underestimation of the magnetic energy by several orders of magnitude. The origin of this discrepancy is that fully 3D DNS cannot reach low enough values of the magnetic Prandtl number Pm. To bypass this limitation and investigate dynamo saturation at very low Pm, we focus on the vicinity of the dynamo threshold in a rapidly rotating flow: the velocity field then depends on two spatial coordinates only, while the magnetic field consists of a single Fourier mode in the third direction. We perform numerical simulations of the resulting set of reduced equations for Pm down to 2 · 10 −5 . This parameter regime is currently out of reach to fully 3D DNS. We show that the magnetic energy transitions from a high-Pm viscous scaling regime to a low-Pm turbulent scaling regime, the latter being independent of viscosity. The transition to the turbulent saturation regime occurs at a low value of the magnetic Prandtl number, Pm ≃ 10 −3 , which explains why it has been overlooked by numerical studies so far.The magnetic field of most astrophysical objects is believed to originate from the dynamo effect, an instability that converts part of the fluid kinetic energy into magnetic energy. The dynamo instability sets in when the flow is sufficiently vigorous to amplify magnetic field perturbations through electromagnetic induction and overcome Ohmic diffusion. In dimensionless form, this happens above a critical value Rm c of the magnetic Reynolds number Rm = U ℓ/η, where U and ℓ are the typical velocity and length scales of the flow and η = 1/µ 0 σ is the magnetic diffusivity, with σ the electrical conductivity of the fluid and µ 0 the magnetic permeability of vacuum. An immediate difficulty arises from the low value of the magnetic Prandtl number Pm = ν/η, where ν is kinematic viscosity: values of Pm of the order of 10 −5 are typical of liquid metals and solar system objects. As a consequence, when the flow reaches the O(1) threshold value Rm c , the kinetic Reynolds number Re = U ℓ/ν is in the range 10 5 − 10 6 and the flow is fully turbulent. This constitutes a challenge both experimentally and numerically: because of the high power needed to sustain a turbulent flow above Rm c , large experimental facilities are needed, and only three such experiments have succeeded in producing dynamo magnetic fields [1][2][3]. On the numerical side, direct numerical simulations (DNS) of the dynamo effect at realistic Pm values would require gigantic computational resources to accurately resolve the small scales of the fully turbulent flow. State-of-the-art dynamo DNS are therefore restricted to moderately low values of Pm, typically Pm ≥ 0.01 in triple periodic boxes [4] and Pm ≥ 0.05 in spherical geometry [5][6][7][8].Deriving and testing scaling laws that extrapolate numerical results...

Dynamo saturation down to vanishing viscosity: strong-field and inertial scaling regimes

“…We can summarize the findings of Seshasayanan et al (2017) and of the present study as follows: geostrophic base flows lead to the inertial scaling regime, with B 2 independent of the rotation rate, whereas base flows that are not in geostrophic balance can achieve the magneto-geostrophic scaling regime, with much larger magnetic energy, proportional to the global rotation rate. In spherical geodynamo simulations, which criterion should be retained to observe a ν-independent scaling regime -and whether this scaling-law involves the global rotation rate -may depend on the region of the sphere that contributes most to magnetic-field generation.…”

“…As opposed to standard weakly nonlinear methods (Nunez et al 2001;Seshasayanan et al 2017), our study is based on scale separation only and is not restricted to the immediate vicinity of the dynamo threshold. We therefore studied the scaling behavior of the magnetic energy at large distance from threshold: when both the Reynolds number and magnetic Reynolds number are large, the magnetic energy is independent of both viscosity and magnetic diffusivity.…”

“…It is interesting to notice that the criterion to achieve the strong-field regime in the present study is precisely that of a low-Ekman number (see section 1.5). Whether Ekman or Pm is the right parameter in fact depends very much on the geometry of the forcing: in a previous study (Seshasayanan et al 2017), we considered a forcing that is compatible with the Taylor-Proudman constraint -i.e., a forcing that is invariant along Ω (Gallet 2015)-and showed that the right criterion to neglect viscosity is low magnetic Prandtl number, Pm ≪ 1. The magnetic field then achieves the inertial or turbulent scaling regime: the magnetic energy is independent of viscosity but also of the global rotation rate.…”

“…As an example, we recently derived a reduced set of quasi-2D equations that is asymptotically valid in the limit of rapid rotation and in the vicinity of the dynamo threshold. We could then simulate these reduced equations down to very low values of the magnetic Prandtl number, thereby showing that the magnetic energy transitions to the turbulent scaling regime for low enough magnetic Prandtl number, Pm 10 −3 (Seshasayanan et al 2017).…”

Onset of three-dimensionality in rapidly rotating turbulent flows

“…Again, the determination of through fully 3-D DNS remains prohibitively expensive at large and low . Instead, the existence of FAIs, as well as whether and when a forward energy cascade develops, could be investigated through nonlinear extensions of this work: one could design a weakly nonlinear model by keeping only the first unstable vertically dependent mode and its feedback onto the 2-D base flow, in the spirit of Benavides & Alexakis (2017) and Seshasayanan, Gallet & Alexakis (2017). For physical systems that are amenable to 3-D DNS in extreme parameter regimes, models of this kind have predictive skills when compared to 3-D DNS, at least at a qualitative level (van Kan & Alexakis 2019).…”