A mapping method for solving dynamo equations

1995 ◽  
Vol 448 (1932) ◽  
pp. 1-28 ◽  
Keyword(s):  
Magnetic Field ◽  
Decay Rate ◽  
Mapping Method ◽  
Three Dimensions ◽  
New Method ◽  
Dynamo Model ◽  
Dynamo Action ◽  
Electrically Conducting ◽  
Induction Equation ◽  
Conducting Fluids

A new method is presented for integration of the equations of magnetohydrody­namics in rapidly rotating, electrically conducting fluids, and in particular for studying dynamo action in such systems. Tests of the method are reported. The decay rate of magnetic field in a stationary sphere are recovered, as are the results of a number of α 2 - and αω -dynamos. These are solutions of the electrodynamic (induction) equation. An intermediate dynamo model, in which the dynamics are also partly allowed for, is also studied. This is due to Braginsky ( Geomag. Aero­naut . 18, 225 (1978)) and is of ‘model- Z type’. All models considered here are axisymmetric, but the possibility of generalization to three-dimensions is allowed for.

On kinematic dynamos

1970 ◽  
Vol 316 (1525) ◽  
pp. 153-167 ◽  
Keyword(s):  
Magnetic Field ◽  
Flow Pattern ◽  
Physical Characteristic ◽  
Major Change ◽  
Three Dimensions ◽  
Dynamo Action ◽  
Electrically Conducting ◽  
The Core ◽  
Electrically Conducting Fluid ◽  
Two Component

The method developed by Bullard & Gellman, to test flows of electrically conducting fluid in a sphere for dynamo action, is applied further to the two-component T 1 S 2c 2 flow pattern they proposed. In agreement with Gibson & Roberts, it is found that the results of the test are negative, which substantiates the indication from Braginskii’s work that the T 1 S 2c 2 flow pattern has too great a symmetry for it to act as a dynamo. However, the addition of a third component, S 2s 2 , to the flow pattern reduces the symmetry and produces results which indicate strongly that the three-component T 1 S 2c 2 S 2s 2 flow does act as a dynamo. Harmonics of magnetic field up to degree six have been taken into account, and this level of truncation appears to be justified. The streamlines of the T 1 S 2c 2 S 2s 2 flow form a distinctive whirling pattern in three dimensions, and this may be a physical characteristic necessary for dynamo action. The main magnetic fields of the T 1 S 2c 2 S 2s 2 dynamo are all toroidal, and the possibility is established that the geomagnetic dynamo is similar, with the dominant components of field being completely contained within the core. Variation of the subsidiary poloidal components of the field may then produce secular variation and even dipole reversals, without major change in the series of interactions between the toroidal components that form the basic dynamo.

Solar Internal Rotation and Dynamo Waves: A Two-Dimensional Asymptotic Solution in the Convection Zone

2000 ◽  
Vol 179 ◽  
pp. 379-380
Author(s):  
Gaetano Belvedere ◽  
Kirill Kuzanyan ◽  
Dmitry Sokoloff
Keyword(s):  
Magnetic Field ◽  
Asymptotic Solution ◽  
Internal Rotation ◽  
Convection Zone ◽  
General Idea ◽  
Dynamo Model ◽  
Axially Symmetric ◽  
Dynamo Action ◽  
Regeneration Rate ◽  
Dynamo Waves

Extended abstractHere we outline how asymptotic models may contribute to the investigation of mean field dynamos applied to the solar convective zone. We calculate here a spatial 2-D structure of the mean magnetic field, adopting real profiles of the solar internal rotation (the Ω-effect) and an extended prescription of the turbulent α-effect. In our model assumptions we do not prescribe any meridional flow that might seriously affect the resulting generated magnetic fields. We do not assume apriori any region or layer as a preferred site for the dynamo action (such as the overshoot zone), but the location of the α- and Ω-effects results in the propagation of dynamo waves deep in the convection zone. We consider an axially symmetric magnetic field dynamo model in a differentially rotating spherical shell. The main assumption, when using asymptotic WKB methods, is that the absolute value of the dynamo number (regeneration rate) |D| is large, i.e., the spatial scale of the solution is small. Following the general idea of an asymptotic solution for dynamo waves (e.g., Kuzanyan & Sokoloff 1995), we search for a solution in the form of a power series with respect to the small parameter |D|–1/3(short wavelength scale). This solution is of the order of magnitude of exp(i|D|1/3S), where S is a scalar function of position.

The structure of Taylor's constraint in three dimensions

2008 ◽  
Vol 464 (2100) ◽  
pp. 3149-3174 ◽  
Author(s):  
Philip W Livermore ◽  
Glenn Ierley ◽  
Andrew Jackson
Keyword(s):  
Lorentz Force ◽  
Inner Core ◽  
Rotation Axis ◽  
Analytic Form ◽  
Spectral Expansion ◽  
Three Dimensions ◽  
Necessary Condition ◽  
Dynamo Action ◽  
Azimuthal Component ◽  
Electrically Conducting

In a 1963 edition of Proc. R. Soc. A , J. B. Taylor (Taylor 1963 Proc. R. Soc. A 9 , 274–283) proved a necessary condition for dynamo action in a rapidly rotating electrically conducting fluid in which viscosity and inertia are negligible. He demonstrated that the azimuthal component of the Lorentz force must have zero average over any geostrophic contour (i.e. a fluid cylinder coaxial with the rotation axis). The resulting dynamical balance, termed a Taylor state, is believed to hold in the Earth's core, hence placing constraints on the class of permissible fields in the geodynamo. Such states have proven difficult to realize, apart from highly restricted examples. In particular, it has not yet been shown how to enforce the Taylor condition exactly in a general way, seeming to require an infinite number of constraints. In this work, we derive the analytic form for the averaged azimuthal component of the Lorentz force in three dimensions after expanding the magnetic field in a truncated spherical harmonic basis chosen to be regular at the origin. As the result is proportional to a polynomial of modest degree (simply related to the order of the spectral expansion), it can be made to vanish identically on every geostrophic contour by simply equating each of its coefficients to zero. We extend the discussion to allow for the presence of an inner core, which partitions the geostrophic contours into three distinct regions.

Planar velocity dynamos in a sphere

2006 ◽  
Vol 462 (2072) ◽  
pp. 2439-2456 ◽  
Author(s):  
A.A Bachtiar ◽  
D.J Ivers ◽  
R.W James
Keyword(s):  
Magnetic Field ◽  
Outer Core ◽  
Magnetic Reynolds Number ◽  
Main Magnetic Field ◽  
Dynamo Action ◽  
Field Growth ◽  
Planar Velocity ◽  
Induction Equation ◽  
Planar Flows ◽  
Magnetic Induction Equation

The Earth's main magnetic field is generally believed to be due to a self-exciting dynamo process in the Earth's fluid outer core. A variety of antidynamo theorems exist that set conditions under which a magnetic field cannot be indefinitely maintained by dynamo action against ohmic decay. One such theorem, the Planar Velocity Antidynamo Theorem , precludes field maintenance when the flow is everywhere parallel to some plane, e.g. the equatorial plane. This paper shows that the proof of the Planar Velocity Theorem fails when the flow is confined to a sphere, due to diffusive coupling at the boundary. Then, the theorem reverts to a conjecture. There is a need to either prove the conjecture, or find a functioning planar velocity dynamo. To the latter end, this paper formulates the toroidal–poloidal spectral form of the magnetic induction equation for planar flows, as a basis for a numerical investigation. We have thereby determined magnetic field growth rates associated with various planar flows in spheres. For most flows, the induced magnetic field decays with time, supporting a planar velocity antidynamo conjecture for a spherical conducting fluid. However, one flow is exceptional, indicating that magnetic field growth can occur. We also re-examine some classical kinematic dynamo models, converting the flows where possible to planar flows. For the flow of Pekeris et al . (Pekeris, C. L., Accad, Y. & Shkoller, B. 1973 Kinematic dynamos and the Earth's magnetic field. Phil. Trans. R. Soc. A 275 , 425–461), this conversion dramatically reduces the critical magnetic Reynolds number.

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.

Kinematic dynamo action in a network of screw motions; application to the core of a fast breeder reactor

1999 ◽  
Vol 382 ◽  
pp. 137-154 ◽  
Author(s):  
F. PLUNIAN ◽  
P. MARTY ◽  
A. ALEMANY
Keyword(s):  
Magnetic Field ◽  
Experimental Studies ◽  
Magnetic Reynolds Number ◽  
Liquid Sodium ◽  
Dynamo Action ◽  
Periodic Cell ◽  
The Core ◽  
Induction Equation ◽  
Breeder Reactors ◽  
Dynamo Effect

Most of the studies concerning the dynamo effect are motivated by astrophysical and geophysical applications. The dynamo effect is also the subject of some experimental studies in fast breeder reactors (FBR) for they contain liquid sodium in motion with magnetic Reynolds numbers larger than unity. In this paper, we are concerned with the flow of sodium inside the core of an FBR, characterized by a strong helicity. The sodium in the core flows through a network of vertical cylinders. In each cylinder assembly, the flow can be approximated by a smooth upwards helical motion with no-slip conditions at the boundary. As the core contains a large number of assemblies, the global flow is considered to be two-dimensionally periodic. We investigate the self-excitation of a two-dimensionally periodic magnetic field using an instability analysis of the induction equation which leads to an eigenvalue problem. Advantage is taken of the flow symmetries to reduce the size of the problem. The growth rate of the magnetic field is found as a function of the flow pitch, the magnetic Reynolds number (Rm) and the vertical magnetic wavenumber (k). An α-effect is shown to operate for moderate values of Rm, supporting a mean magnetic field. The large-Rm limit is investigated numerically. It is found that α=O(Rm−2/3), which can be explained through appropriate dynamo mechanisms. Either a smooth Ponomarenko or a Roberts type of dynamo is operating in each periodic cell, depending on k. The standard power regime of an industrial FPBR is found to be subcritical.

scholarly journals General relativistic magnetic perturbations and dynamo effects in extragalactic radiosources

2010 ◽  
Vol 6 (S274) ◽  
pp. 393-397
Author(s):  
L. C. Garcia de Andrade
Keyword(s):  
Magnetic Field ◽  
Magnetic Energy ◽  
Magnetic Fluctuations ◽  
Dynamo Action ◽  
Magnetic Contrast ◽  
Mhd Dynamo ◽  
Curvature Energy ◽  
Induction Equation ◽  
The Universe ◽  
The Magnetic Field

AbstractBy making use of the MHD self-induction equation in general relativity (GR), recently derived by Clarkson and Marklund (2005), it is shown that when Friedmann universe possesses a spatial section whose Riemannian curvature is negative, the magnetic energy bounds computed by Nuñez (2002) also bounds the growth rate of the magnetic field given by the strain matrix of dynamo flow. Since in GR-MHD dynamo equation, the Ricci tensor couples with the universe magnetic field, only through diffusion, and most ages are highly conductive the interest is more theoretical here, and only very specific plasma astrophysical problems can be address such as in laboratory plasmas. Magnetic fields and the negative curvature of some isotropic cosmologies, contribute to enhence the amplification of the magnetic field. Ricci curvature energy is shown to add to strain matrix of the flow, to enhance dynamo action in the universe. Magnetic fluctuations of the Clarkson-Marklund equations for a constant magnetic field seed in highly conductive flat universes, leads to a magnetic contrast of ≈ 2, which is well within observational limits from extragalactic radiosources of ≈ 1.7. In the magnetic helicity fluctuations the magnetic contrast shows that the dynamo effects can be driven by these fluctuations.

scholarly journals Unsteady two-layered fluid flow of conducting fluids in a channel between parallel porous plates under transverse magnetic field in a rotating system

2016 ◽  
Vol 21 (2) ◽  
pp. 423-446 ◽  
Author(s):  
T. Linga Raju ◽  
B. Neela Rao
Keyword(s):  
Magnetic Field ◽  
Fluid Flow ◽  
Differential Equations ◽  
Taylor Number ◽  
Rotating System ◽  
Electrically Conducting ◽  
Electrical Conductivities ◽  
Secondary Velocity ◽  
Conducting Fluids ◽  
Layered Fluid

AbstractAn unsteady MHD two-layered fluid flow of electrically conducting fluids in a horizontal channel bounded by two parallel porous plates under the influence of a transversely applied uniform strong magnetic field in a rotating system is analyzed. The flow is driven by a common constant pressure gradient in a channel bounded by two parallel porous plates, one being stationary and the other oscillatory. The two fluids are assumed to be incompressible, electrically conducting with different viscosities and electrical conductivities. The governing partial differential equations are reduced to the linear ordinary differential equations using two-term series. The resulting equations are solved analytically to obtain exact solutions for the velocity distributions (primary and secondary) in the two regions respectively, by assuming their solutions as a combination of both the steady state and time dependent components of the solutions. Numerical values of the velocity distributions are computed for different sets of values of the governing parameters involved in the study and their corresponding profiles are also plotted. The details of the flow characteristics and their dependence on the governing parameters involved, such as the Hartmann number, Taylor number, porous parameter, ratio of the viscosities, electrical conductivities and heights are discussed. Also an observation is made how the velocity distributions vary with the rotating hydromagnetic interaction in the case of steady and unsteady flow motions. The primary velocity distributions in the two regions are seen to decrease with an increase in the Taylor number, but an increase in the Taylor number causes a rise in secondary velocity distributions. It is found that an increase in the porous parameter decreases both the primary and secondary velocity distributions in the two regions.

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