scholarly journals On magnetic helicity generation and transport in a nonlinear dynamo driven by a helical flow

2020 ◽  
Vol 86 (4) ◽  
Author(s):  
F. Cattaneo ◽  
G. Bodo ◽  
S. M. Tobias
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Travelling Waves ◽  
Magnetic Reynolds Number ◽  
Magnetic Helicity ◽  
Dynamo Action ◽  
Strong Constraint ◽  
Sheared Flow ◽  
The Relationship ◽  
The Magnetic Field

The relationship between nonlinear large-scale dynamo action and the generation and transport of magnetic helicity is investigated at moderate values of the magnetic Reynolds number ( $Rm$ ). The model consists of a helically forced, sheared flow in a Cartesian domain. The boundary conditions are periodic in the horizontal and impenetrable for the vertical. The magnetic field is required to be vertical at the upper and lower boundaries. There are two consequences of this choice; one is that the magnetic helicity is not gauge invariant, the second is that fluxes of magnetic helicity are allowed in and out of the domain. We select the winding gauge, define all the contributions to the evolution of the helicity in this gauge and measure these contributions for various solutions of the dynamo equations. We vary $Rm$ and the shear strength, and find a rich landscape of dynamo solutions including travelling waves, pulsating waves and non-wave-like solutions. We find that, at the $Rm$ considered, the main contribution to the growth of magnetic helicity comes from processes throughout the volume of the fluid and that boundary terms respond by limiting the growth. We find that, in this magnetic Reynolds number regime, helicity conservation is not a strong constraint on large-scale dynamo action. We speculate on what may happen at higher $Rm$ .

Turbulent dynamo action at low magnetic Reynolds number

1970 ◽  
Vol 41 (2) ◽  
pp. 435-452 ◽  
Author(s):  
H. K. Moffatt
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Magnetic Reynolds Number ◽  
Dynamo Action ◽  
Ohmic Dissipation ◽  
Magnetic Energy Density ◽  
Magnetic Diffusivity

The effect of turbulence on a magnetic field whose length-scale L is initially large compared with the scale l of the turbulence is considered. There are no external sources for the field, and in the absence of turbulence it decays by ohmic dissipation. It is assumed that the magnetic Reynolds number Rm = u0l/λ (where u0 is the root-mean-square velocity and λ the magnetic diffusivity) is small. It is shown that to lowest order in the small quantities l/L and Rm, isotropic turbulence has no effect on the large-scale field; but that turbulence that lacks reflexional symmetry is capable of amplifying Fourier components of the field on length scales of order Rm−2l and greater. In the case of turbulence whose statistical properties are invariant under rotation of the axes of reference, but not under reflexions in a point, it is shown that the magnetic energy density of a magnetic field which is initially a homogeneous random function of position with a particularly simple spectrum ultimately increases as t−½exp (α2t/2λ3) where α(= O(u02l)) is a certain linear functional of the spectrum tensor of the turbulence. An analogous result is obtained for an initially localized field.

scholarly journals The effect of velocity shear on dynamo action due to rotating convection

2013 ◽  
Vol 717 ◽  
pp. 395-416 ◽  
Author(s):  
D. W. Hughes ◽  
M. R. E. Proctor
Keyword(s):  
Magnetic Field ◽  
Velocity Field ◽  
Large Scale ◽  
Mean Field ◽  
Magnetic Reynolds Number ◽  
Velocity Shear ◽  
Magnetic Field Generation ◽  
Dynamo Action ◽  
Rotating Convection ◽  
The Magnetic Field

AbstractRecent numerical simulations of dynamo action resulting from rotating convection have revealed some serious problems in applying the standard picture of mean field electrodynamics at high values of the magnetic Reynolds number, and have thereby underlined the difficulties in large-scale magnetic field generation in this regime. Here we consider kinematic dynamo processes in a rotating convective layer of Boussinesq fluid with the additional influence of a large-scale horizontal velocity shear. Incorporating the shear flow enhances the dynamo growth rate and also leads to the generation of significant magnetic fields on large scales. By the technique of spectral filtering, we analyse the modes in the velocity that are principally responsible for dynamo action, and show that the magnetic field resulting from the full flow relies crucially on a range of scales in the velocity field. Filtering the flow to provide a true separation of scales between the shear and the convective flow also leads to dynamo action; however, the magnetic field in this case has a very different structure from that generated by the full velocity field. We also show that the nature of the dynamo action is broadly similar irrespective of whether the flow in the absence of shear can support dynamo action.

scholarly journals Synthesis of Small and Large Scale Dynamos

2002 ◽  
Vol 12 ◽  
pp. 739-741
Author(s):  
Kandaswamy Subramanian
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Magnetic Reynolds Number ◽  
Mhd Equations ◽  
Magnetic Helicity ◽  
Small Scale ◽  
Quantum Mechanical ◽  
Closure Model ◽  
Scale Fluctuation ◽  
Small Scale Fluctuation

AbstractUsing a closure model for the evolution of magnetic correlations, we uncover an interesting plausible saturated state of the small-scale fluctuation dynamo (SSD) and a novel anology between quantum mechanical tunnelling and the generation of large-scale fields. Large scale fields develop via the α-effect, but as magnetic helicity can only change on a resistive timescale, the time it takes to organize the field into large scales increases with magnetic Reynolds number. This is very similar to the results which obtain from simulations using the full MHD equations.

scholarly journals The optimal kinematic dynamo driven by steady flows in a sphere

2018 ◽  
Vol 839 ◽  
pp. 1-32 ◽  
Author(s):  
L. Chen ◽  
W. Herreman ◽  
K. Li ◽  
P. W. Livermore ◽  
J. W. Luo ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Helical Structure ◽  
Optimization Method ◽  
Global Optimum ◽  
Magnetic Reynolds Number ◽  
Steady Flows ◽  
Dynamo Action ◽  
Kinematic Dynamo ◽  
Scale Optimization

We present a variational optimization method that can identify the most efficient kinematic dynamo in a sphere, where efficiency is based on the value of a magnetic Reynolds number that uses enstrophy to characterize the inductive effects of the fluid flow. In this large-scale optimization, we restrict the flow to be steady and incompressible, and the boundary of the sphere to be no-slip and electrically insulating. We impose these boundary conditions using a Galerkin method in terms of specifically designed vector field bases. We solve iteratively for the flow field and the accompanying magnetic eigenfunction in order to find the minimal critical magnetic Reynolds number $Rm_{c,min}$ for the onset of a dynamo. Although nonlinear, this iteration procedure converges to a single solution and there is no evidence that this is not a global optimum. We find that $Rm_{c,min}=64.45$ is at least three times lower than that of any published example of a spherical kinematic dynamo generated by steady flows, and our optimal dynamo clearly operates above the theoretical lower bounds for dynamo action. The corresponding optimal flow has a spatially localized helical structure in the centre of the sphere, and the dominant components are invariant under rotation by $\unicode[STIX]{x03C0}$.

A kinematic theory of large magnetic Reynolds number dynamos

1972 ◽  
Vol 272 (1227) ◽  
pp. 431-462 ◽  
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Hybrid Approach ◽  
Magnetic Reynolds Number ◽  
Additional Contribution ◽  
Dynamo Action ◽  
Principal Mechanism ◽  
Azimuthal Magnetic Field ◽  
Field Note ◽  
The Magnetic Field

An asymptotic analysis is made of the magnetic induction equation for certain flows characterized by a large magnetic Reynolds number R . A novel feature is the hybrid approach given to the problem. Advantage is taken of a combination of Eulerian and Lagrange coordinates. Under certain conditions the problem can be reduced to solving a pair of coupled partial differential equations dependent on only two space coordinates (cf. Braginskii 1964 a ). Two main cases are considered. First the case is examined, in which the production of azimuthal magnetic field from the meridional magnetic field by a shear in the aximuthal flow is negligible. It is shown that a term J (analogous to electric current) is related linearly to the vector B which determines the magnetic field. (Note that B is not the magnetic field vector: see (1.33) and (2.35 b ).) The current J is likely to sustain dynamo action. Secondly, the case is considered, in which shearing of meridional magnetic field is the principal mechanism for creating the azimuthal magnetic field and the effect described above is one mechanism for creating meridional magnetic field from the azimuthal magnetic field. It is shown that the term J is not only linearly related to B , but has an additional contribution P x (V x B ), where P is characterized by the flow (see (4.15)). Both these effects have been predicted previously in theories of dynamo action produced by turbulent motions. Under certain restrictive conditions the resulting equations in the second case reduce to Braginskil’s (1964 a , b ) formulation for nearly symmetric dynamos. The words azimuthal and meridional are not used here in the usual sense. The difference in terminology is a consequence of a coordinate transformation.

scholarly journals The elemental shear dynamo

2012 ◽  
Vol 699 ◽  
pp. 414-452 ◽  
Author(s):  
James C. McWilliams
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Magnetic Energy ◽  
Orientation Angle ◽  
Domain Size ◽  
Random Force ◽  
Magnetic Reynolds Number ◽  
Dynamo Action ◽  
Alpha Omega ◽  
The Mean

AbstractA quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, ${\mathit{Re}}_{\eta } \ll 1$. Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $\gamma $ can occur for arbitrary values of ${\mathit{Re}}_{\eta } $, viscous Reynolds number ${\mathit{Re}}_{\nu } $, and random-force correlation time ${t}_{f} $ and orientation angle ${\theta }_{f} $ in the shearing plane. The value of $\gamma $ is independent of the domain size. The shear dynamo is ‘fast’, with finite $\gamma \gt 0$ in the limit of ${\mathit{Re}}_{\eta } \gg 1$. Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small ${\mathit{Re}}_{\eta } $ and ${\mathit{Re}}_{\nu } $, the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying ($\gamma {t}_{f} \gg 1$) and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.

scholarly journals Large-scale dynamo action of magnetized Taylor–Couette flows

2020 ◽  
Vol 493 (1) ◽  
pp. 1249-1260
Author(s):  
G Rüdiger ◽  
M Schultz
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Mean Field ◽  
Single Mode ◽  
Eddy Diffusivity ◽  
Magnetic Reynolds Number ◽  
Dynamo Model ◽  
Turbulence Energy ◽  
Dynamo Action ◽  
Taylor Couette

ABSTRACT A conducting Taylor–Couette flow with quasi-Keplerian rotation law containing a toroidal magnetic field serves as a mean-field dynamo model of the Tayler–Spruit type. The flows are unstable against non-axisymmetric perturbations which form electromotive forces defining α effect and eddy diffusivity. If both degenerated modes with m = ±1 are excited with the same power then the global α effect vanishes and a dynamo cannot work. It is shown, however, that the Tayler instability produces finite α effects if only an isolated mode is considered but this intrinsic helicity of the single-mode is too low for an α2 dynamo. Moreover, an αΩ dynamo model with quasi-Keplerian rotation requires a minimum magnetic Reynolds number of rotation of Rm ≃ 2000 to work. Whether it really works depends on assumptions about the turbulence energy. For a steeper-than-quadratic dependence of the turbulence intensity on the magnetic field, however, dynamos are only excited if the resulting magnetic eddy diffusivity approximates its microscopic value, ηT ≃ η. By basically lower or larger eddy diffusivities the dynamo instability is suppressed.

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