DRAFT VERSION MAY 25, 2019 Preprint typeset using LATEX style emulateapj v. 08/22/09 REVERSED DYNAMO AT SMALL SCALES AND LARGE MAGNETIC PRANDTL NUMBER 1 2 3 4 5 AXEL BRANDENBURG , , , MATTHIAS REMPEL 1Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden 2Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden 3JILA and Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80303, USA 4McWilliams Center for Cosmology & Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 5High Altitude Observatory, NCAR, P.O. Box 3000, Boulder, CO 80307, USA (Revision: 1.88) Draft version May 25, 2019 ABSTRACT We show that at large magnetic Prandtl numbers, the Lorentz force does work on the flow at small scales and drives fluid motions, whose energy is dissipated viscously. This situation is opposite to that in a normal dynamo, where the flow does work against the Lorentz force. We compute the spectral conversion rates between kinetic and magnetic energies for several magnetic Prandtl numbers and show that normal (forward) dynamo action occurs on large scales over a progressively narrower range of wavenumbers as the magnetic Prandtl number is increased. At higher wavenumbers, reversed dynamo action occurs, i.e., magnetic energy is converted back into kinetic energy at small scales. We demonstrate this in both direct numerical simulations forced by volume stirring and in large eddy simulations of solar convectively driven small-scale dynamos. Low density plasmas such as stellar coronae tend to have large magnetic Prandtl numbers, i.e., the viscosity is large compared with the magnetic diffusivity. The regime in which viscous dissipation dominates over resistive dissipation for large magnetic Prandtl numbers was also previously found in large eddy simulations of the solar corona, i.e., our findings are a more fundamental property of MHD that is not just restricted to dynamos. Viscous energy dis- sipation is a consequence of positive Lorentz force work, which may partly correspond to particle acceleration in close-to-collisionless plasmas. This is, however, not modeled in the MHD approximation employed. By contrast, resistive energy dissipation on current sheets is expected to be unimportant in stellar coronae. Subject headings: dynamo — hydrodynamics — MHD — turbulence — Sun: corona, dynamo 1. INTRODUCTION should be an inefficient one. The magnetic fields of planets, stars, accretion discs, and A large magnetic Prandtl number implies that the magnetic galaxies are all produced and maintained by a turbulent dy- diffusivity is small, so one would have expected the dynamo namo (Zeldovich et al. 1983). Dynamos work through the to be efficient, because it suffers less dissipation. This imme- conversion of kinetic into magnetic energy. This energy con- diately leads to a puzzle. How can a dynamo be efficient in version is characterized by the flow field doing work against the sense of experiencing low energy dissipation, but at the the Lorentz force. It has been known for some time that this same time inefficient in the sense of having small energy con- energy conversion also depends on the microphysical value version? Here is where our suggestion of a reversed dynamo comes of the magnetic Prandtl number, PrM ≡ ν/η, the ratio of kinematic viscosity ν to magnetic diffusivity η (Brandenburg in. A reversed dynamo is one that does work by the Lorentz force—and not against it, as in a usual dynamo. Thus, it cor- 2009, 2011). The larger the value of PrM , the larger is also the ratio of kinetic to magnetic energy dissipation (Brandenburg responds to driving velocity by the Lorentz force and hence 2014). This is plausible, because large viscosity means large to a conversion of magnetic to kinetic energy. Therefore, the idea is that the flow is indeed an inefficient dynamo, but only viscous dissipation (ǫK ), and large magnetic diffusivity or re- at large scales (LS), where kinetic energy is converted to mag- sistivity means large resistive dissipation (ǫM ). Large values netic energy. At small scales (SS), however, magnetic energy of PrM are generally expected to occur at low densities, for 10 begins to dominate over kinetic energy, leading therefore to example in the solar corona (PrM ≈ 10 ; see Rempel 2017) 11 an efficient conversion of magnetic into kinetic energy. This and in galaxies (PrM ≈ 10 ; see Brandenburg and Subrama- nian 2005). means we have a reversed dynamo, as sketched in Figure 1(b), where we show the flow of energy separately for LS and SS. In the steady state, ǫM must be equal to the rate of kinetic to magnetic energy conversion. This becomes clear when look- To test this idea, we analyze solar convection simulations and ing at an energy flow diagram; see Figure 1(a). It shows perform idealized simulations of isotropically forced homo- that magnetic energy can only be supplied through work done geneous nonhelical turbulence over a range of different mag- against the Lorentz force, J ×B, where J = ∇×B/µ is the netic Prandtl numbers and calculate the spectrum of magnetic 0 to kinetic energy transfer. current density, B is the magnetic field, and µ0 is the vacuum permeability. Exactly the same amount of energy must even- Mahajan et al. (2005) introduced the concept of a reversed tually also be dissipated resistively. This implies that at large dynamo in the context of large-scale dynamos leading to the magnetic Prandtl numbers, not only must most of the energy formation of large-scale flows that are driven simultaneously be dissipated viscously, but also the magnetic energy dissipa- with the large-scale field by microscopic fields and flows. In tion must be small. Therefore, also the work done against the our investigation, we focus on small-scale dynamos and show Lorentz force must be small, which suggests that the dynamo that small-scale flows are driven by the Lorentz force when 2 FIG. 1.— Energy flow diagrams for (a) a standard dynamo and (b) a dynamo at LS with a reversed dynamo at SS. Dashed arrows indicate relatively weak flows of energy. PrM ≫ 1. The spectral range over which this microscopic than unity. Here, however, we analyze the solar convection reverse dynamo is operational is found to be PrM dependent setups that were described in Rempel (2018) with effective and is largest in the high PrM regimes. numerical (or pseudo) PrM on the order of 0.088, 1.77, and 54.6. It is important to note that the pseudo PrM = 0.088 and 2. DYNAMO SIMULATIONS AND ANALYSIS 1.77 cases have approximately the same Reynolds number, We consider two types of dynamo simulations. On the one Re, but different magnetic Reynolds number, ReM , whereas hand, we perform direct numerical simulations (DNS; as in the PrM = 1.77 and 54.6 cases have the same ReM , but dif- Brandenburg 2014), where viscous and magnetic dissipation ferent Re. Note that PrM = ReM /Re. are solved for explicitly, and large eddy simulations (LES), We should note that, owing to the strong vertical stratifi- where these terms are modeled. In both cases, we vary the cation of the density ρ in the LES, we have modified Equa- ratio of kinetic to magnetic energy dissipation to examine our tion (1) by computing Tu(k) with the Fourier transforms ideas about reversed dynamo action. as ρ1/2u and J × B/ρ1/2, where overbars denote horizon- Our main analysis tool is the spectrum of energy conver- tal averaging. However, the choice of a ρ1/2 was some- sion, defined as (Rempel 2014) what arbitrary and one could have used instead ρ1/3, because 3 ^ ∗ ρ(z)urms(z) is known to be an approximation to the convec- Tu(k)= ℜ uk · (J × B)k, (1) tive flux which, in turn, is expected to be approximately con- k−<X|k|≤k+ stant through the convection zone. Note that the multiplica- e tion and division by the same factor does not affect the di- where k± = k ± δk/2, δk = 2π/L is the wavenumber in- 1/3 mension of Tu(k). Incidently, a ρ factor has also been crement and also the smallest wavenumber k1 ≡ δk in the advocated by Kritsuk et al. (2007) in the context of super- domain of side length L, tildes denote Fourier transformation sonic interstellar turbulence. In the present simulations, how- either in all three directions in the homogeneous DNS or in 1/3 ever, ρ urms is seen to increase slightly with height, while the two horizontal directions in the inhomogeneous LES, and 1/2 ℜ denotes the real part. ρ urms decreases slightly, so the latter choice is equally The LES of Rempel (2014, 2018) are designed to model the well justified. solar convection zone and use realistic physics such as multi- Figure 2 shows the corresponding magnetic and kinetic en- frequency radiation transport and a realistic equation of state ergy spectra, EM(k) and EK(k), respectively, together with allowing for partial ionization effects. Here, the flow is driven the spectral transfer functions for the three cases with dif- by the convection resulting from radiative surface cooling at ferent pseudo PrM . Since the domain is periodic horizon- a rate high enough so that radiative diffusion leads to a su- tally, but stratified vertically, we consider here only the hori- peradiabatic stratification, which is Schwarzschild unstable. zontal Fourier transforms when computing power spectra and This model is periodic in the two horizontal directions, but transfer functions. In addition, quantities are averaged over not in the vertical. The employed LES approach allows for a height range of 800 km ranging from 700 to 1500 km be- different diffusivity settings in the momentum and induction neath the solar photosphere.