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 $\unicode[STIX]{x03C0}$.

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.

Minimal perturbation flows that trigger mean field dynamos in shear flows

2018 ◽  
Vol 84 (3) ◽  
Author(s):  
W. Herreman
Keyword(s):  
Reynolds Number ◽  
Mean Field ◽  
Optimization Method ◽  
Magnetic Reynolds Number ◽  
Dynamo Model ◽  
Magnetic Fluctuations ◽  
Dynamo Action ◽  
Flow Perturbation ◽  
Mean Field Dynamo ◽  
Formula Type

Using a variational optimization method we find the smallest flow perturbations that can trigger kinematic dynamo action in Kolmogorov flow. In comparison to previous work, a second-order mean field dynamo model is used to track down the optimal dynamos in the high magnetic Reynolds number limit ($Rm$). The magnitude of minimal perturbation flows decays inversely proportional to the magnetic Reynolds number. We reveal the asymptotic high-$Rm$ structure of the optimal flow perturbation and the magnetic eigenmode. We identify the optimal dynamo as of $\unicode[STIX]{x1D6FC}{-}\unicode[STIX]{x1D6FA}$ type, with magnetic fluctuations that localize on a critical layer.

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 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 $Rm$ .

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.

Experimental investigation of dynamo effect in the secondary pumps of the fast breeder reactor Superphenix

2000 ◽  
Vol 403 ◽  
pp. 263-276 ◽  
Author(s):  
A. ALEMANY ◽  
Ph. MARTY ◽  
F. PLUNIAN ◽  
J. SOTO
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Magnetic Energy ◽  
Experimental Studies ◽  
Phenomenological Approach ◽  
Magnetic Reynolds Number ◽  
Experimental Investigations ◽  
Dynamo Action ◽  
Experimental Conditions ◽  
Dynamo Effect

The fast breeder reactors (FBR) BN600 (Russia) and Phenix (France) have been the subject of several experimental studies aimed at the observation of dynamo action. Though no dynamo effect has been identified, the possibility was raised for the FBR Superphenix (France) which has an electric power twice that of BN600 and five times larger than Phenix. We present the results of a series of experimental investigations on the secondary pumps of Superphenix. The helical sodium flow inside one pump corresponds to a maximum magnetic Reynolds number (Rm) of 25 in the experimental conditions (low temperature). The magnetic field was recorded in the vicinity of the pumps and no dynamo action has been identified. An estimate of the critical flow rate necessary to reach dynamo action has been found, showing that the pumps are far from producing dynamo action. The magnetic energy spectrum was also recorded and analysed. It is of the form k−11/3, suggesting the existence of a large-scale magnetic field. Following Moffatt (1978), this spectrum slope is also justified by a phenomenological approach.

A Kind of New Immune Genetic Algorithm and its Application

2011 ◽  
Vol 48-49 ◽  
pp. 25-28
Author(s):  
Wei Jian Ren ◽  
Yuan Jun Qi ◽  
Wei Lv ◽  
Cheng Da Li
Keyword(s):  
Genetic Algorithm ◽  
Large Scale ◽  
Optimization Problems ◽  
Logistic Map ◽  
Optimal Solution ◽  
Optimization Method ◽  
Local Optimum ◽  
Immune Genetic Algorithm ◽  
Chaos Optimization ◽  
Scale Optimization

According to the phenomenon of falling into local optimum during solving large-scale optimization problems and the shortcomings of poor convergence of Immune Genetic Algorithm, a new kind of probability selection method based on the concentration for the genetic operation is presented. Considering the features of chaos optimization method, such like not requiring the solved problems with continuity or differentiability, which is unlike the conventional method, and also with a solving process within a certain range traverse in order to find the global optimal solution, a kind of Chaos Immune Genetic Algorithm based on Logistic map and Hénon map is proposed. Through the application to TSP problem, the results have showed the superior to other algorithms.

A recursive implementation of the ALIENOR optimization method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Daniele Peri
Keyword(s):  
Design Methodology ◽  
Large Scale ◽  
Optimization Problem ◽  
Design Space ◽  
Optimization Method ◽  
Fine Tuning ◽  
Content Type ◽  
Design Variables ◽  
Scale Optimization ◽  
Similar Paper

PurposeA recursive scheme for the ALIENOR method is proposed as a remedy for the difficulties induced by the method. A progressive focusing on the most promising region, in combination with a variation of the density of the alpha-dense curve, is proposed.Design/methodology/approachALIENOR method is aimed at reducing the space dimensions of an optimization problem by spanning it by using a single alpha-dense curve: the curvilinear abscissa along the curve becomes the only design parameter for any design space. As a counterpart, the transformation of the objective function in the projected space is much more difficult to tackle.FindingsA fine tuning of the procedure has been performed in order to identity the correct balance between the different elements of the procedure. The proposed approach has been tested by using a set of algebraic functions with up to 1,024 design variables, demonstrating the ability of the method in solving large scale optimization problem. Also an industrial application is presented.Originality/valueIn the knowledge of the author there is not a similar paper in the current literature.

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