On dynamo action in a steady flow at large magnetic reynolds number

1989 ◽  
Vol 49 (1-4) ◽  
pp. 3-22 ◽  
Author(s):  
A. M. Soward
Keyword(s):  
Reynolds Number ◽  
Steady Flow ◽  
Magnetic Reynolds Number ◽  
Dynamo Action

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 Astrophysical dynamos: the limit of vanishing diffusivity

2015 ◽  
Vol 11 (A29B) ◽  
pp. 727-729
Author(s):  
Emmanuel Dormy ◽  
Ismaël Bouya
Keyword(s):  
Reynolds Number ◽  
High Performance ◽  
Extreme Values ◽  
Numerical Models ◽  
Magnetic Reynolds Number ◽  
Dynamo Action ◽  
Astrophysical Objects ◽  
Physics Journals ◽  
Turn Over ◽  
Performance Computing

AbstractAstrophysical dynamos are usually characterised by huge values of the magnetic Reynolds number (Rm). This reflects the short turn-over time compared to the resistive time. The extreme values of Rm relevant to astrophysical objects cannot be tackled with today's numerical resources and this number is always under-estimated by several orders of magnitudes in numerical models.Here we chose to focus on an extremely simplified problem (dynamo action from a periodic steady flow) and take advantage of this simplicity to numerically investigate the limit of very large magnetic Reynolds number. We present results recently published in physics journals, which highlight the difficulty of approaching the limit in which dynamo action is independent of the value of the ohmic resistivity (measured by 1/Rm), known as the “fast dynamo” limit. Using state of the art high performance computing, we present high resolution simulations (up to (40963) and extend the value of (Rm) up to (5⋅105).

scholarly journals Coronal influence on dynamos

2013 ◽  
Vol 9 (S302) ◽  
pp. 134-137 ◽  
Author(s):  
Jörn Warnecke ◽  
Axel Brandenburg
Keyword(s):  
Reynolds Number ◽  
Convection Zone ◽  
Magnetic Energy ◽  
Lower Layer ◽  
Solar Radius ◽  
Magnetic Reynolds Number ◽  
Layer Model ◽  
Dynamo Action ◽  
Turbulent Dynamo ◽  
Two Layer Model

AbstractWe report on turbulent dynamo simulations in a spherical wedge with an outer coronal layer. We apply a two-layer model where the lower layer represents the convection zone and the upper layer the solar corona. This setup is used to study the coronal influence on the dynamo action beneath the surface. Increasing the radial coronal extent gradually to three times the solar radius and changing the magnetic Reynolds number, we find that dynamo action benefits from the additional coronal extent in terms of higher magnetic energy in the saturated stage. The flux of magnetic helicity can play an important role in this context.

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 $\unicode[STIX]{x03C0}$.

scholarly journals Small-scale dynamo action in rotating compressible convection

2011 ◽  
Vol 690 ◽  
pp. 262-287 ◽  
Author(s):  
B. Favier ◽  
P. J. Bushby
Keyword(s):  
Reynolds Number ◽  
Energy Density ◽  
Thermal Stratification ◽  
Magnetic Energy ◽  
Critical Value ◽  
Magnetic Reynolds Number ◽  
Small Scale ◽  
Dynamo Action ◽  
Lower Bounding ◽  
The Mean

AbstractWe study dynamo action in a convective layer of electrically conducting, compressible fluid, rotating about the vertical axis. At the upper and lower bounding surfaces, perfectly conducting boundary conditions are adopted for the magnetic field. Two different levels of thermal stratification are considered. If the magnetic diffusivity is sufficiently small, the convection acts as a small-scale dynamo. Using a definition for the magnetic Reynolds number ${R}_{M} $ that is based upon the horizontal integral scale and the horizontally averaged velocity at the mid-layer of the domain, we find that rotation tends to reduce the critical value of ${R}_{M} $ above which dynamo action is observed. Increasing the level of thermal stratification within the layer does not significantly alter the critical value of ${R}_{M} $ in the rotating calculations, but it does lead to a reduction in this critical value in the non-rotating cases. At the highest computationally accessible values of the magnetic Reynolds number, the saturation levels of the dynamo are similar in all cases, with the mean magnetic energy density somewhere between 4 and 9 % of the mean kinetic energy density. To gain further insights into the differences between rotating and non-rotating convection, we quantify the stretching properties of each flow by measuring Lyapunov exponents. Away from the boundaries, the rate of stretching due to the flow is much less dependent upon depth in the rotating cases than it is in the corresponding non-rotating calculations. It is also shown that the effects of rotation significantly reduce the magnetic energy dissipation in the lower part of the layer. We also investigate certain aspects of the saturation mechanism of the dynamo.

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 Optimal kinematic dynamos in a sphere

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190675
Author(s):  
Jiawen Luo ◽  
Long Chen ◽  
Kuan Li ◽  
Andrew Jackson
Keyword(s):  
Reynolds Number ◽  
Flow Field ◽  
Small Region ◽  
Magnetic Reynolds Number ◽  
Optimization Approach ◽  
Optimal Solutions ◽  
Dynamo Action ◽  
Common Features ◽  
Slip Boundary ◽  
Large Velocity

A variational optimization approach is used to optimize kinematic dynamos in a unit sphere and locate the enstrophy-based critical magnetic Reynolds number for dynamo action. The magnetic boundary condition is chosen to be either pseudo-vacuum or perfectly conducting. Spectra of the optimal flows corresponding to these two magnetic boundary conditions are identical since theory shows that they are relatable by reversing the flow field (Favier & Proctor 2013 Phys. Rev. E 88 , 031001 ( doi:10.1103/physreve.88.031001 )). A no-slip boundary for the flow field gives a critical magnetic Reynolds number of 62.06, while a free-slip boundary reduces this number to 57.07. Optimal solutions are found to possess certain rotation symmetries (or anti-symmetries) and optimal flows share certain common features. The flows localize in a small region near the sphere’s centre and spiral upwards with very large velocity and vorticity, so that they are locally nearly Beltrami. We also derive a new lower bound on the magnetic Reynolds number for dynamo action, which, for the case of enstrophy normalization, is five times larger than the previous best bound.

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.

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