Evolution of Aligned States Within Nonlinear Dynamos (article)

This is the final version. Available on open access from Taylor & Francis via the DOI in this recordThe dataset associated with this article is located in ORE at: https://doi.org/10.24378/exe.1603The Archontis dynamo is a rare example of an MHD dynamo within which forcing drives a dynamo where the flow and magnetic fields are almost perfectly aligned and the energies are approximately equal. In this paper, I expand upon our knowledge of the dynamo by showing that the intermediate steady states of the kinetic and magnetic energies observed by Cameron and Galloway are not a necessary feature of aligned dynamos. Furthermore, I show that the steady state into which the flow and magnetic fields eventually evolve is remarkably robust to the addition of time dependence and asymmetry to the forcing.Engineering and Physical Sciences Research Council (EPSRC

Dissipative structures in a nonlinear dynamo

This paper gives new results concerning magnetic field generation leading to a steady equilibrated state, the so-called Archontis dynamo. A combination of numerical work, analysis of PDEs and functional analysis is used to derive information on scaling laws for dissipative structures in the dynamo.This is an Accepted Manuscript of an article published by Taylor & Francis in Geophysical and Astrophysical Fluid Dynamics, Volume 105, Issue 6, 2011, available online on 2 Dec 2010: http://wwww.tandfonline.com/10.1080/03091929.2010.513332.Author's accepted manuscriptThis paper considers magnetic field generation by a fluid flow in a system referred to as the Archontis dynamo: a steady nonlinear magnetohydrodynamic (MHD) state is driven by a prescribed body force. The field and flow become almost equal and dissipation is concentrated in cigar-like structures centred on straight-line separatrices. Numerical scaling laws for energy and dissipation are given that extend previous calculations to smaller diffusivities. The symmetries of the dynamo are set out, together with their implications for the structure of field and flow along the separatrices. The scaling of the cigar-like dissipative regions, as the square root of the diffusivities, is explained by approximations near the separatrices. Rigorous results on the existence and smoothness of solutions to the steady, forced MHD equations are given.Royal SocietyCNRSLeverhulme TrustAgence Nationale de la Recherche, FranceRussian foundation for basic researc

The role of the Yoshizawa effect in the Archontis dynamo

The generation of mean magnetic fields is studied for a simple non-helical flow where a net cross helicity of either sign can emerge. This flow, which is also known as the Archontis flow, is a generalization of the Arnold--Beltrami--Childress flow, but with the cosine terms omitted. The presence of cross helicity leads to a mean-field dynamo effect that is known as the Yoshizawa effect. Direct numerical simulations of such flows demonstrate the presence of magnetic fields on scales larger than the scale of the flow. Contrary to earlier expectations, the Yoshizawa effect is found to be proportional to the mean magnetic field and can therefore lead to its exponential instead of just linear amplification for magnetic Reynolds numbers that exceed a certain critical value. Unlike $\alpha$ effect dynamos, it is found that the Yoshizawa effect is not noticeably constrained by the presence of a conservation law. It is argued that this is due to the presence of a forcing term in the momentum equation which leads to a nonzero correlation with the magnetic field. Finally, the application to energy convergence in solar wind turbulence is discussed.Comment: Accepted for publication in MNRA

Alignment and Structure in MHD Dynamos

Magnetic fields are ubiquitous within astrophysical settings. There is strong evidence to suggest that some of these magnetic fields, for example the Sun’s, are maintained through a dynamo process whereby energy is exchanged between a flow and a magnetic field. Magnetohydrodynamics (MHD) is the branch of mathematics where this interaction is studied. The initial amplification of a weak seed field is often modelled using the kinematic dynamo approximation where the flow is not influenced by the magnetic field. This approximation to the early behaviour of a nonlinear dynamo problem, where the magnetic field grows exponentially during a kinematic phase and then saturates into a nonlinear regime, has the benefit of being far less computationally intensive.In this thesis, I examine three different topics within MHD dynamos. First, I examine how measuring alignment of the flow and magnetic field during a kinematic dynamo can reveal changes to the magnetic field structure. This I show to be useful both within individual simulations and when comparing magnetic fields within parameter studies. Secondly, I examine nonlinear dynamos where the flow and magnetic field are strongly aligned and have almost identical energies. I reproduce, and give an explanation for, a previously unexplained behaviour. Furthermore, I show that aligned flow and magnetic fields can exist for increasingly complex forcings and as such the aligned state is remark-ably robust. Finally, I consider a number of different nonlinear dynamos for a family of forcings with different magnetic field structures during their kinematic phase. Using Minkowski Functions to quantify the structures, I show that, where the magnetic field becomes sufficiently strong, the magnetic fields become (or remain) ribbon-like in the nonlinear regime. As such, the influence that stagnation points in the flow have on the magnetic field structure is less than in the kinematic dynamo equivalent

Yoshizawa's cross-helicity effect and its quenching

A central quantity in mean-field magnetohydrodynamics is the mean electromotive force EMF, which in general depends on the mean magnetic field. It may however have a part independent of the mean magnetic field. Here we study an example of a rotating conducting body of turbulent fluid with non-zero cross-helicity, in which a contribution to the EMF proportional to the angular velocity occurs (Yoshizawa 1990). If the forcing is helical, it also leads to an alpha effect, and large-scale magnetic fields can be generated. For not too rapid rotation, the field configuration is such that Yoshizawa's contribution to the EMF is considerably reduced compared to the case without alpha effect. In that case, large-scale flows are also found to be generated.Comment: 10 pages, 8 figures, compatible with published versio

Cross-helicity effects and turbulent transport in magnetohydrodynamic flow

In the presence of large-scale vortical motions and/or magnetic-field strains, the turbulent cross helicity (velocity--magnetic-field correlation in fluctuations) may contribute to the turbulent electromotive force and the Reynolds stress. These effects of cross helicity are considered to balance the primary effects of turbulence such as the turbulent magnetic diffusivity in magnetic-field evolution and the eddy viscosity in the momentum transport. The cross-helicity effects may suppress the enhanced transports due to turbulence. Physical interpretation of the effects is presented with special emphasis on the difference between the cross-helicity effect and the usual $\alpha$ or helicity effect in the dynamo action. The relative importance of the cross-helicity effect in dynamo action is validated with the aid of a direct numerical simulation (DNS) of the Kolmogorov flow with an imposed magnetic field. Several mechanisms that provide turbulence with the cross helicity are also discussed.Comment: 10 pages, 6 figures, Journal of Physics Conference Series: 13th European Turbulence Conference (ETC13

Mean electromotive force proportional to mean flow in mhd turbulence

In mean-field magnetohydrodynamics the mean electromotive force due to velocity and magnetic field fluctuations plays a crucial role. In general it consists of two parts, one independent of and another one proportional to the mean magnetic field. The first part may be nonzero only in the presence of mhd turbulence, maintained, e.g., by small-scale dynamo action. It corresponds to a battery, which lets a mean magnetic field grow from zero to a finite value. The second part, which covers, e.g., the alpha effect, is important for large-scale dynamos. Only a few examples of the aforementioned first part of mean electromotive force have been discussed so far. It is shown that a mean electromotive force proportional to the mean fluid velocity, but independent of the mean magnetic field, may occur in an originally homogeneous isotropic mhd turbulence if there are nonzero correlations of velocity and electric current fluctuations or, what is equivalent, of vorticity and magnetic field fluctuations. This goes beyond the Yoshizawa effect, which consists in the occurrence of mean electromotive forces proportional to the mean vorticity or to the angular velocity defining the Coriolis force in a rotating frame and depends on the cross-helicity defined by the velocity and magnetic field fluctuations. Contributions to the mean electromotive force due to inhomogeneity of the turbulence are also considered. Possible consequences of the above and related findings for the generation of magnetic fields in cosmic bodies are discussed.Comment: 7 pages, 1 figure, Astron. Nachr. (submitted

Nonlinear dynamos at infinite magnetic Prandtl number

The dynamo instability is investigated in the limit of infinite magnetic Prandtl number. In this limit the fluid is assumed to be very viscous so that the inertial terms can be neglected and the flow is slaved to the forcing. The forcing consist of an external forcing function that drives the dynamo flow and the resulting Lorentz force caused by the back reaction of the magnetic field. The flows under investigation are the Archontis flow, and the ABC flow forced at two different scales. The investigation covers roughly three orders of magnitude of the magnetic Reynolds number above onset. All flows show a weak increase of the averaged magnetic energy as the magnetic Reynolds number is increased. Most of the magnetic energy is concentrated in flat elongated structures that produce a Lorentz force with small solenoidal projection so that the resulting magnetic field configuration was almost force-free. Although the examined system has zero kinetic Reynolds number at sufficiently large magnetic Reynolds number the structures are unstable to small scale fluctuations that result in a chaotic temporal behavior

Applications of a finite-volume algorithm for incompressible MHD problems

We present the theory, algorithms and implementation of a parallel finite-volume algorithm for the solution of the incompressible magnetohydrodynamic (MHD) equations using unstructured grids that are applicable for a wide variety of geometries. Our method implements a mixed Adams-Bashforth/Crank-Nicolson scheme for the nonlinear terms in the MHD equations and we prove that it is stable independent of the time step. To ensure that the solenoidal condition is met for the magnetic field, we use a method whereby a pseudo-pressure is introduced into the induction equation; since we are concerned with incompressible flows, the resulting Poisson equation for the pseudo-pressure is solved alongside the equivalent Poisson problem for the velocity field. We validate our code in a variety of geometries including periodic boxes, spheres, spherical shells, spheroids and ellipsoids; for the finite geometries we implement the so-called ferromagnetic or pseudo-vacuum boundary conditions appropriate for a surrounding medium with infinite magnetic permeability. This implies that the magnetic field must be purely perpendicular to the boundary. We present a number of comparisons against previous results and against analytical solutions, which verify the code's accuracy. This documents the code's reliability as a prelude to its use in more difficult problems. We finally present a new simple drifting solution for thermal convection in a spherical shell that successfully sustains a magnetic field of simple geometry. By dint of its rapid stabilization from the given initial conditions, we deem it suitable as a benchmark against which other self-consistent dynamo codes can be tested