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 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.

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 Global MHD phenomena and their importance for stellar surfaces

2010 ◽  
Vol 6 (S273) ◽  
pp. 141-147
Author(s):  
Rainer Arlt
Keyword(s):  
Magnetic Fields ◽  
Mean Field ◽  
Momentum Equation ◽  
Small Scale ◽  
Dynamo Action ◽  
Complex Activity ◽  
Mhd Instabilities ◽  
Stellar Magnetic Fields ◽  
Mean Field Dynamo

AbstractThis review is an attempt to elucidate MHD phenomena relevant for stellar magnetic fields. The full MHD treatment of a star is a problem which is numerically too demanding. Mean-field dynamo models use an approximation of the dynamo action from the small-scale motions and deliver global magnetic modes which can be cyclic, stationary, axisymmetric, and non-axisymmetric. Due to the lack of a momentum equation, MHD instabilities are not visible in this picture. However, magnetic instabilities must set in as a result of growing magnetic fields and/or buoyancy. Instabilities deliver new timescales, saturation limits and topologies to the system probably providing a key to the complex activity features observed on stars.

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 Grand Minima in a spherical non-kinematic α2Ω mean-field dynamo model

2020 ◽  
Vol 10 ◽  
pp. 9 ◽  
Author(s):  
Corinne Simard ◽  
Paul Charbonneau
Keyword(s):  
Differential Rotation ◽  
Large Scale ◽  
Mean Field ◽  
Cyclic Behavior ◽  
Dynamo Model ◽  
Cycle Period ◽  
Cycle Amplitude ◽  
Primary Cycle ◽  
Mean Field Dynamo ◽  
Grand Minima

We present a non-kinematic axisymetric α2Ω mean-field dynamo model in which the complete α-tensor and mean differential rotation profile are both extracted from a global magnetohydrodynamical simulation of solar convection producing cycling large-scale magnetic fields. The nonlinear backreaction of the Lorentz force on differential rotation is the only amplitude-limiting mechanism introduced in the mean-field model. We compare and contrast the amplitude modulation patterns characterizing this mean-field dynamo, to those already well-studied in the context of non-kinematic αΩ models using a scalar α-effect. As in the latter, we find that large quasi-periodic modulation of the primary cycle are produced at low magnetic Prandtl number (Pm), with the ratio of modulation period to the primary cycle period scaling inversely with Pm. The variations of differential rotation remain well within the bounds set by observed solar torsional oscillations. In this low-Pm regime, moderately supercritical solutions can also exhibit aperiodic Maunder Minimum-like periods of strongly reduced cycle amplitude. The inter-event waiting time distribution is approximately exponential, in agreement with solar activity reconstructions based on cosmogenic radioisotopes. Secular variations in low-latitude surface differential rotation during Grand Minima, as compared to epochs of normal cyclic behavior, are commensurate in amplitude with historical inferences based on sunspot drawings. Our modeling results suggest that the low levels of observed variations in the solar differential rotation in the course of the activity cycle may nonetheless contribute to, or perhaps even dominate, the regulation of the magnetic cycle amplitude.

scholarly journals Mean field dynamo action in shear flows. I: fixed kinetic helicity

2020 ◽  
Vol 495 (4) ◽  
pp. 4557-4569 ◽  
Author(s):  
Naveen Jingade ◽  
Nishant K Singh
Keyword(s):  
Magnetic Field ◽  
Mean Field ◽  
Magnetic Activity ◽  
Turnover Time ◽  
Cycle Period ◽  
Dynamo Action ◽  
Navier Stokes Equation ◽  
Axisymmetric Mode ◽  
Dynamo Wave ◽  
Mean Field Dynamo

ABSTRACT We study mean field dynamo action in a background linear shear flow by employing pulsed renewing flows with fixed kinetic helicity and non-zero correlation time (τ). We use plane shearing waves in terms of time-dependent exact solutions to the Navier–Stokes equation as derived by Singh & Sridhar (2017). This allows us to self-consistently include the anisotropic effects of shear on the stochastic flow. We determine the average response tensor governing the evolution of mean magnetic field, and study the properties of its eigenvalues that yield the growth rate (γ) and the cycle period (Pcyc) of the mean magnetic field. Both, γ and the wavenumber corresponding to the fastest growing axisymmetric mode vary non-monotonically with shear rate S when τ is comparable to the eddy turnover time T, in which case, we also find quenching of dynamo when shear becomes too strong. When $\tau /T\sim {\cal O}(1)$, the cycle period (Pcyc) of growing dynamo wave scales with shear as Pcyc ∝ |S|−1 at small shear, and it becomes nearly independent of shear as shear becomes too strong. This asymptotic behaviour at weak and strong shear has implications for magnetic activity cycles of stars in recent observations. Our study thus essentially generalizes the standard αΩ (or α2Ω) dynamo as also the α effect is affected by shear and the modelled random flow has a finite memory.

scholarly journals Flux-transport and mean-field dynamo theories of solar cycles

2012 ◽  
Vol 8 (S294) ◽  
pp. 37-47
Author(s):  
Arnab Rai Choudhuri
Keyword(s):  
Solar Cycle ◽  
Meridional Circulation ◽  
Mean Field ◽  
Dynamo Model ◽  
Solar Cycles ◽  
Poloidal Field ◽  
Flux Transport ◽  
Mean Field Model ◽  
Mean Field Models ◽  
Mean Field Dynamo

AbstractWe point out the difficulties in carrying out direct numerical simulation of the solar dynamo problem and argue that kinematic mean-field models are our best theoretical tools at present for explaining various aspects of the solar cycle in detail. The most promising kinematic mean-field model is the flux transport dynamo model, in which the toroidal field is produced by differential rotation in the tachocline, the poloidal field is produced by the Babcock–Leighton mechanism at the solar surface and the meridional circulation plays a crucial role. Depending on whether the diffusivity is high or low, either the diffusivity or the meridional circulation provides the main transport mechanism for the poloidal field to reach the bottom of the convection zone from the top. We point out that the high-diffusivity flux transport dynamo model is consistent with various aspects of observational data. The irregularities of the solar cycle are primarily produced by fluctuations in the Babcock–Leighton mechanism and in the meridional circulation. We summarize recent work on the fluctuations of meridional circulation in the flux transport dynamo, leading to explanations of such things as the Waldmeier effect.

scholarly journals Mean-field dynamo action in renovating shearing flows

2012 ◽  
Vol 86 (2) ◽  
Author(s):  
Sanved Kolekar ◽  
Kandaswamy Subramanian ◽  
S. Sridhar
Keyword(s):  
Mean Field ◽  
Dynamo Action ◽  
Mean Field Dynamo
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