scholarly journals A trio of simple optimized axisymmetric kinematic dynamos in a sphere

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190308 ◽  
Author(s):  
D. Holdenried-Chernoff ◽  
L. Chen ◽  
A. Jackson
Keyword(s):  
Magnetic Energy ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Complex Flows ◽  
Dynamo Action ◽  
Link Type ◽  
Planetary Magnetic Fields ◽  
Spectral Coefficients ◽  
Simple Flows ◽  
Lagrangian Optimization

Planetary magnetic fields are generated by the motion of conductive fluid in the planet's interior. Complex flows are not required for dynamo action; simple flows have been shown to act as efficient kinematic dynamos, whose physical characteristics are more straightforward to study. Recently, Chen et al . (2018, J. Fluid Mech. 839 , 1–32. ( doi:10.1017/jfm.2017.924 )) found the optimal, unconstrained kinematic dynamo in a sphere, which, despite being of theoretical importance, is of limited practical use. We extend their work by restricting the optimization to three simple two-mode axisymmetric flows based on the kinematic dynamos of Dudley & James (1989, Proc. R. Soc. Lond. A 425 , 407–429. ( doi:10.1098/rspa.1989.0112 )). Using a Lagrangian optimization, we find the smallest critical magnetic Reynolds number for each flow type, measured using an enstrophy-based norm. A Galerkin method is used, in which the spectral coefficients of the fluid flow and magnetic field are updated in order to maximize the final magnetic energy. We consider the t 0 1 s 0 1 , t 0 1 s 0 2 and t 0 2 s 0 2 flows and find enstrophy-based critical magnetic Reynolds numbers of 107.7, 142.4 and 125.5 (13.7, 19.6 and 16.4, respectively, with the energy-based definition). These are up to four times smaller than the original flows. These simple and efficient flows may be used as benchmarks in future studies.

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.

Fully developed MHD turbulence near critical magnetic Reynolds number

1981 ◽  
Vol 104 ◽  
pp. 419-443 ◽  
Author(s):  
J. Léorat ◽  
A. Pouquet ◽  
U. Frisch
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Mhd Equations ◽  
Liquid Sodium ◽  
Turbulent Heat ◽  
Mhd Turbulence

Liquid-sodium-cooled breeder reactors may soon be operating at magnetic Reynolds numbers RM where magnetic fields can be self-excited by a dynamo mechanism (as first suggested by Bevir 1973). Such flows have kinetic Reynolds numbers RV of the order of 107 and are therefore highly turbulent.This leads us to investigate the behaviour of MHD turbulence with high RV and low magnetic Prandtl numbers. We use the eddy-damped quasi-normal Markovian closure applied to the MHD equations. For simplicity we restrict ourselves to homogeneous and isotropic turbulence, but we do include helicity.We obtain a critical magnetic Reynolds number RMc of the order of a few tens (non-helical case) above which magnetic energy is present. RMc is practically independent of RV (in the range 40 to 106). RMc can be considerably decreased by the presence of helicity: when the overall size of the flow L is much larger than the integral scale l0, RMc can drop below unity as suggested by an α-effect argument. When L ≈ l0 the drop can still be substantial (factor of 6) when helicity is a maximum. We examine how the turbulence is modified when RM crosses RMc: presence of magnetic energy, decreased kinetic energy, steepening of kinetic-energy spectrum, etc.We make no attempt to obtain quantitative estimates for a breeder reactor, but discuss some of the possible consequences of exceeding RMc, such as decreased turbulent heat transport. More precise information may be obtained from numerical simulations and experiments (including some in the subcritical regime).

scholarly journals The influence of stratification upon small-scale convectively-driven dynamos

2010 ◽  
Vol 6 (S271) ◽  
pp. 197-204 ◽  
Author(s):  
Paul J. Bushby ◽  
Michael R. E. Proctor ◽  
Nigel O. Weiss
Keyword(s):  
Magnetic Energy ◽  
Magnetic Reynolds Number ◽  
Small Scale ◽  
Initial Growth ◽  
Dynamo Action ◽  
Quiet Sun ◽  
Electrically Conducting ◽  
Solar Observations ◽  
Electrically Conducting Fluid ◽  
Convective Motions

AbstractIn the quiet Sun, convective motions form a characteristic granular pattern, with broad upflows enclosed by a network of narrow downflows. Magnetic fields tend to accumulate in the intergranular lanes, forming localised flux concentrations. One of the most plausible explanations for the appearance of these quiet Sun magnetic features is that they are generated and maintained by dynamo action resulting from the local convective motions at the surface of the Sun. Motivated by this idea, we describe high resolution numerical simulations of nonlinear dynamo action in a (fully) compressible, non-rotating layer of electrically-conducting fluid. The dynamo properties depend crucially upon various aspects of the fluid. For example, the magnetic Reynolds number (Rm) determines the initial growth rate of the magnetic energy, as well as the final saturation level of the dynamo in the nonlinear regime. We focus particularly upon the ways in which the Rm-dependence of the dynamo is influenced by the level of stratification within the domain. Our results can be related, in a qualitative sense, to solar observations.

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.

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 Kinematic dynamos in triaxial ellipsoids

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210252
Author(s):  
Jérémie Vidal ◽  
David Cébron
Keyword(s):  
Magnetic Fields ◽  
Boundary Conditions ◽  
Reynolds Numbers ◽  
Dynamo Action ◽  
Local Boundary ◽  
Electrically Conducting ◽  
Free Decay ◽  
Planetary Magnetic Fields ◽  
Induction Equation ◽  
The Magnetic Field

Planetary magnetic fields are generated by motions of electrically conducting fluids in their interiors. The dynamo problem has thus received much attention in spherical geometries, even though planetary bodies are non-spherical. To go beyond the spherical assumption, we develop an algorithm that exploits a fully spectral description of the magnetic field in triaxial ellipsoids to solve the induction equation with local boundary conditions (i.e. pseudo-vacuum or perfectly conducting boundaries). We use the method to compute the free-decay magnetic modes and to solve the kinematic dynamo problem for prescribed flows. The new method is thoroughly compared with analytical solutions and standard finite-element computations, which are also used to model an insulating exterior. We obtain dynamo magnetic fields at low magnetic Reynolds numbers in ellipsoids, which could be used as simple benchmarks for future dynamo studies in such geometries. We finally discuss how the magnetic boundary conditions can modify the dynamo onset, showing that a perfectly conducting boundary can strongly weaken dynamo action, whereas pseudo-vacuum and insulating boundaries often give similar results.

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.

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 Optimal dynamo action by steady flows confined to a cube

2015 ◽  
Vol 783 ◽  
pp. 23-45 ◽  
Author(s):  
L. Chen ◽  
W. Herreman ◽  
A. Jackson
Keyword(s):  
Incompressible Flows ◽  
Magnetic Energy ◽  
Magnetic Reynolds Number ◽  
Steady Flows ◽  
Dynamo Action ◽  
Electrically Conducting ◽  
Energy Growth ◽  
Optimal Flow ◽  
Complementary Set ◽  
The Magnetic Field

Many flows of electrically conducting fluids can spontaneously generate magnetic fields through the process of dynamo action, but when does a flow produce a better dynamo than another one or when is it simply the most efficient dynamo? Using a variational approach close to that of Willis (Phys. Rev. Lett., vol. 109, 2012, 251101), we find optimal kinematic dynamos within a huge class of stationary and incompressible flows that are confined in a cube. We demand that the magnetic field satisfies either superconducting (T) or pseudovacuum (N) boundary conditions on opposite pairs of walls of the cube, which results in four different combinations. For each of these set-ups, we find the optimal flow and its corresponding magnetic eigenmodes. Numerically, it is observed that swapping the magnetic boundary from T to N leaves the magnetic energy growth nearly unchanged, and both $+\boldsymbol{U}$ and $-\boldsymbol{U}$ are optimal flows for these different but complementary set-ups. This can be related to work by Favier & Proctor (Phys. Rev. E, vol. 88, 2013, 031001). We provide minimal lower bounds for dynamo action and find that no dynamo is possible below an enstrophy (or shear) based magnetic Reynolds number $Rm_{c,min}=7.52{\rm\pi}^{2}$, which is a factor of $16$ above the Proctor/Backus bound.

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