Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers

Saturation of Turbulent Helical Dynamos

The presence of large scale magnetic fields in nature is often attributed to the inverse cascade of magnetic helicity driven by turbulent helical dynamos. In this Letter, we show that in turbulent helical dynamos, the inverse flux of magnetic helicity toward the large scales Π_{H} is bounded by |Π_{H}|≤cεk_{η}^{-1}, where ε is the energy injection rate, k_{η} is the Kolmogorov magnetic dissipation wave number, and c an order one constant. Assuming the classical isotropic turbulence scaling, the inverse flux of magnetic helicity Π_{H} decreases at least as a -3/4 power law with the magnetic Reynolds number Rm: |Π_{H}|≤cεℓ_{f}Rm^{-3/4}max[Pm,1]^{1/4}, where Pm is the magnetic Prandtl number and ℓ_{f} the forcing length scale. We demonstrate this scaling with Rm using direct numerical simulations of turbulent dynamos forced at intermediate scales. The results further indicate that nonlinear saturation is achieved by a balance between the inverse cascade and dissipation at domain size scales L for which the saturation value of the magnetic energy is bounded by E_{m}≤cL(εℓ_{f})^{2/3}Rm^{1/4}max[1,Pm]^{1/4}. Numerical simulations also demonstrate this bound. These results are independent of the dynamo mechanism considered. In our setup, they imply that inviscid mechanisms cannot explain large scale magnetic fields and have critical implications for the modeling of astrophysical dynamos.

Magnetic field amplification in accretion discs around the first stars: implications for the primordial IMF

Magnetic fields play an important role in the dynamics of present-day molecular clouds. Recent work has shown that magnetic fields are equally important for primordial clouds, which form the first stars in the Universe. While the primordial magnetic field strength on cosmic scales is largely unconstrained, theoretical models strongly suggest that a weak seed field existed in the early Universe. We study how the amplification of such a weak field can influence the evolution of accretion discs around first stars, and thus affect the primordial initial mass function (IMF). We perform a suite of 3D ideal magneto-hydrodynamic simulations with different initial field strengths and numerical resolutions. We find that, in simulations with sufficient spatial resolution to resolve the Jeans scale during the collapse, even initially weak magnetic fields grow exponentially to become dynamically important due to both the so-called small-scale turbulent dynamo and the large-scale mean-field dynamo. Capturing the small-scale dynamo action depends primarily on how well we resolve the Jeans length, while capturing the large-scale dynamo depends on the Jeans resolution as well as the maximum absolute resolution. Provided enough resolution, we find that fragmentation does not depend strongly on the initial field strength, because even weak fields grow to become strong. However, fragmentation in runs with magnetic fields differs significantly from those without magnetic fields. We conclude that the development of dynamically strong magnetic fields during the formation of the first stars is likely inevitable, and that these fields had a significant impact on the primordial IMF.

The Turbulent Dynamo

The generation of magnetic field in an electrically conducting fluid generally involves the complicated nonlinear interaction of flow turbulence, rotation and field. This dynamo process is of great importance in geophysics, planetary science and astrophysics, since magnetic fields are known to play a key role in the dynamics of these systems. This paper gives an introduction to dynamo theory for the fluid dynamicist. It proceeds by laying the groundwork, introducing the equations and techniques that are at the heart of dynamo theory, before presenting some simple dynamo solutions. The problems currently exercising dynamo theorists are then introduced, along with the attempts to make progress. The paper concludes with the argument that progress in dynamo theory will be made in the future by utilising and advancing some of the current breakthroughs in neutral fluid turbulence such as those in transition, self-sustaining processes, turbulence/mean-flow interaction, statistical and data-driven methods and maintenance and loss of balance.

Magnetism, dynamo action and the solar-stellar connection

The Sun and other stars are magnetic: magnetism pervades their interiors and affects their evolution in a variety of ways. In the Sun, both the fields themselves and their influence on other phenomena can be uncovered in exquisite detail, but these observations sample only a moment in a single star's life. By turning to observations of other stars, and to theory and simulation, we may infer other aspects of the magnetism-e.g., its dependence on stellar age, mass, or rotation rate-that would be invisible from close study of the Sun alone. Here, we review observations and theory of magnetism in the Sun and other stars, with a partial focus on the "Solar-stellar connection": i.e., ways in which studies of other stars have influenced our understanding of the Sun and vice versa. We briefly review techniques by which magnetic fields can be measured (or their presence otherwise inferred) in stars, and then highlight some key observational findings uncovered by such measurements, focusing (in many cases) on those that offer particularly direct constraints on theories of how the fields are built and maintained. We turn then to a discussion of how the fields arise in different objects: first, we summarize some essential elements of convection and dynamo theory, including a very brief discussion of mean-field theory and related concepts. Next we turn to simulations of convection and magnetism in stellar interiors, highlighting both some peculiarities of field generation in different types of stars and some unifying physical processes that likely influence dynamo action in general. We conclude with a brief summary of what we have learned, and a sampling of issues that remain uncertain or unsolved.

Fate of Alpha Dynamos at Large Rm

At the heart of today's solar magnetic field evolution models lies the alpha dynamo description. In this work, we investigate the fate of alpha dynamos as the magnetic Reynolds number Rm is increased. Using Floquet theory, we are able to precisely quantify mean-field effects like the alpha and beta effect (i) by rigorously distinguishing dynamo modes that involve large-scale components from the ones that only involve small scales, and by (ii) providing a way to investigate arbitrary large-scale separations with minimal computational cost. We apply this framework to helical and nonhelical flows as well as to random flows with short correlation time. Our results determine that the alpha description is valid for Rm smaller than a critical value Rm_{c} at which small-scale dynamo instability starts. When Rm is above Rm_{c}, the dynamo ceases to follow the mean-field description and the growth rate of the large-scale modes becomes independent of the scale separation, while the energy in the large-scale modes is inversely proportional to the square of the scale separation. The results in this second regime do not depend on the presence of helicity. Thus, alpha-type modeling for solar and stellar models needs to be reevaluated and new directions for mean-field modeling are proposed.

Generation of Large-Scale Magnetic Fields by Small-Scale Dynamo in Shear Flows

We propose a new mechanism for a turbulent mean-field dynamo in which the magnetic fluctuations resulting from a small-scale dynamo drive the generation of large-scale magnetic fields. This is in stark contrast to the common idea that small-scale magnetic fields should be harmful to large-scale dynamo action. These dynamos occur in the presence of a large-scale velocity shear and do not require net helicity, resulting from off-diagonal components of the turbulent resistivity tensor as the magnetic analogue of the "shear-current" effect. Given the inevitable existence of nonhelical small-scale magnetic fields in turbulent plasmas, as well as the generic nature of velocity shear, the suggested mechanism may help explain the generation of large-scale magnetic fields across a wide range of astrophysical objects.

Kinetic helicity needed to drive large-scale dynamos

Magnetic field generation on scales that are large compared with the scale of the turbulent eddies is known to be possible via the so-called α effect when the turbulence is helical and if the domain is large enough for the α effect to dominate over turbulent diffusion. Using three-dimensional turbulence simulations, we show that the energy of the resulting mean magnetic field of the saturated state increases linearly with the product of normalized helicity and the ratio of domain scale to eddy scale, provided this product exceeds a critical value of around unity. This implies that large-scale dynamo action commences when the normalized helicity is larger than the inverse scale ratio. Our results show that the emergence of small-scale dynamo action does not have any noticeable effect on the large-scale dynamo. Recent findings by Pietarila Graham et al. [Phys. Rev. E 85, 066406 (2012)] of a smaller minimal helicity may be an artifact due to the onset of small-scale dynamo action at large magnetic Reynolds numbers. However, the onset of large-scale dynamo action is difficult to establish when the kinetic helicity is small. Instead of random forcing, they used an ABC flow with time-dependent phases. We show that such dynamos saturate prematurely in a way that is reminiscent of inhomogeneous dynamos with internal magnetic helicity fluxes. Furthermore, even for very low fractional helicities, such dynamos display large-scale fields that change direction, which is uncharacteristic of turbulent dynamos.

Turbulent magnetic diffusivity tensor for time-dependent mean fields

We reexamine the nature of the turbulent magnetic diffusivity tensor of mean field electrodynamics and show that the predicted growth rate of the mean field is, in general, incorrect if the tensor is calculated via consideration of time-independent mean magnetic fields. We describe how the traditional expansion procedure for the mean electromotive force should be extended to include time derivatives of the mean magnetic field, and illustrate the consistency of this approach by means of a perturbation analysis for a mean field varying on long spatial scales. Finally, we examine the magnitude of this new contribution to the magnetic diffusion for a particular flow.

Self-consistent mean-field magnetohydrodynamics

We consider the linear stability of two-dimensional nonlinear magnetohydrodynamic basic states to long-wavelength three-dimensional perturbations. Following Hughes & Proctor (Hughes & Proctor 2009 Proc. R. Soc. A 465 , 1599–1616 ( doi:10.1098/rspa.2008.0493 )), the two-dimensional basic states are obtained from a specific forcing function in the presence of an initially uniform mean field of strength . By extending to the nonlinear regime the kinematic analysis of Roberts (Roberts 1970 Phil. Trans. R. Soc. Lond. A 266 , 535–558 ( doi:10.1098/rsta.1970.0011 )), we show that it is possible to predict the growth rate of these perturbations by applying mean-field theory to both the momentum and the induction equations. If , these equations decouple and large-scale magnetic and velocity perturbations may grow via the kinematic α -effect and the anisotropic kinetic alpha instability, respectively. However, if , the momentum and induction equations are coupled by the Lorentz force; in this case, we show that four transport tensors are now necessary to determine the growth rate of the perturbations. We illustrate these situations by numerical examples; in particular, we show that a mean-field description of the nonlinear regime based solely on a quenched α coefficient is incorrect.