Dynamo action and its temporal variation inside the tangent cylinder in MHD dynamo simulations

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.

Download Full-text

Three-dimensional kinematic dynamos dominated by strong differential rotation

Journal of Fluid Mechanics ◽

10.1017/s0022112096001292 ◽

1996 ◽

Vol 306 ◽

pp. 223-265 ◽

Cited By ~ 28

Author(s):

Graeme R. Sarson ◽

David Gubbins

Keyword(s):

Magnetic Fields ◽

Meridional Circulation ◽

Boundary Effect ◽

Fluid Motion ◽

Three Dimensional ◽

Thick Layer ◽

Stationary Solutions ◽

Dynamo Action ◽

Kinematic Dynamo ◽

Mhd Dynamo

In the kinematic dynamo problem a fluid motion is specified arbitrarily and the induction equation is solved for non-decaying magnetic fields; it forms part of the larger magnetohydrodynamic (MHD) dynamo problem in which the fluid flow is buoyancy-driven. Although somewhat restrictive, the kinematic problem is important for two reasons: first, it suffers from numerical difficulties that are holding up progress on the MHD problem; secondly, for the geodynamo, it is capable of reproducing details of the observable magnetic field. It is more efficient to study these two aspects for the kinematic dynamo than for the full MHD dynamo. We explore solutions for a family of fluid flows in a sphere, first studied by Kumar & Roberts (1975), that is heuristically representative of convection in a rotating sphere such as the Earth's core. We guard against numerical difficulties by comparing our results with well-understood solutions from the axisymmetric (αω) limit of Braginskii (1964a) and with solutions of the adjoint problem, which must yield identical eigenvalues in an adequate numerical treatment. Previous work has found a range of steady dipolar solutions; here we extend these results and find solutions of other symmetries, notably oscillatory and quadrupolar fields. The surface magnetic fields, important for comparison with observations, have magnetic flux concentrated by downwelling flow. Roberts (1972) found that meridional circulation promoted stationary solutions of the αω-equations, preferred solutions being oscillatory when no such circulation was present. We find analogous results for the full three-dimensional problem, but note that in the latter case the ‘effective’ meridional circulation arising from the non-axisymmetric convection (a concept made precise in the asymptotic limit of Braginskii 1964a) must be considered. Thus stationary solutions are obtained even in the absence of ‘true’ meridional circulation, and the time-dependence can be controlled by the strength of the convection as well as by the meridional circulation. The preference for fields of dipole or quadrupole parity is largely controlled by the sign of the velocity: a reversal of velocity from a case favouring a dipole will favour quadrupole parity, and vice versa. For the comparison problem of Proctor (1977b) this symmetry is exact; for the physical problem the boundary conditions make a difference. The boundary effect is first removed by surrounding the dynamo region with a thick layer of quiescent conducting fluid, and then studied numerically by progressively reducing the thickness of this layer to zero. The insulating boundary contributes to the difficulty of obtaining dynamo action, and to the numerical difficulties encountered. The effect of an inner boundary on dynamo action is also considered, but found to be slight.

Download Full-text

Chaos in geophysical fluids I. General introduction

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1994.0102 ◽

1994 ◽

Vol 348 (1688) ◽

pp. 431-443 ◽

Cited By ~ 2

Keyword(s):

Large Scale ◽

Classical Mechanics ◽

Theoretical Research ◽

Geological Time ◽

General Introduction ◽

Dynamo Action ◽

The Earth ◽

Mhd Dynamo ◽

Low Dimensional ◽

Geomagnetic Polarity

Irregular buoyancy-driven flows occur in the atmospheres and fluid interiors of the Earth and other planets, and of the Sun and other stars, where they influence and often control the transfer of heat. Their presence is manifest in or implied by a wide variety of observed phenomena, including external magnetic fields generated by self-exciting magnetohydrodynamic (MHD ) dynamo action. Based on the laws of classical mechanics, thermodynamics and, in the case of electrically conducting fluids, electrodynamics, the governing mathematical equations are well known, but they are generally intractable owing to their essential nonlinearity. Computers play a key role in modern theoretical research in geophysical and astrophysical fluid dynamics, where ideas based on chaos theory are being applied in the analysis of models and the assessment of predictability. The aim of this paper is to provide a largely qualitative survey for non-specialists. The survey comprises two parts, namely a general introduction (Part I) followed by a discussion of two representative areas of research, both concerned with phenomena attributable to symmetry-breaking bifurcations caused by gyroscopic (Coriolis) forces (Part II), namely ( a ) large-scale waves and eddies in the atmospheres of the Earth, Jupiter and other planets (where, exceptionally, laboratory experiments have been influential), and ( b ) MHD dynamos. Various combinations of Faraday disc dynamos have been studied numerically as low-dimensional nonlinear electromechanical analogues of MHD dynamos, particularly in efforts to elucidate the complex time series of geomagnetic polarity reversals over geological time. The ability of the intensively studied Rikitake coupled disc dynamo system to behave chaotically appears to be a consequence of the neglect of mechanical friction, the inclusion of which renders the system structurally unstable.

Download Full-text

Dynamos of the Sun and Stars, and Associated Convection Zone Dynamics

Symposium - International Astronomical Union ◽

10.1017/s0074180900029910 ◽

1983 ◽

Vol 102 ◽

pp. 247-270

Author(s):

Peter A. Gilman

Keyword(s):

Ad Hoc ◽

Convection Zone ◽

Dynamical Model ◽

Dynamo Action ◽

Kinematic Dynamo ◽

Medical Experiments ◽

Mhd Dynamo ◽

Self Consistent ◽

Fluid Dynamical Model ◽

Stellar Dynamo

I see my assignment in this talk as being to focus on the interaction between the magnetic fields produced by dynamo action and the dynamics of the fluid flow which drives this dynamo, and to make some connections to the solar and stellar dynamo problems. To do this really requires we start with a fluid dynamical model that satisfies relevant laws of fluid dynamics, and in which the flow can actually respond to the induced magnetic field. Thus the so-called kinematic dynamo models are not enough for our purposes, and we must address the full MHD dynamo problem in a self-consistent way. For example, we do not allow ourselves the license to vary independently the convection and differential rotation, as is commonly done in kinematic dynamo calculations, because the laws of physics do not allow that. We have been attempting to do self consistent MHD dynamo modeling at Boulder for the past several years, starting from a nonlinear fluid dynamical model for convection in a rotating spherical shell. This model we believe is physically complete in itself, with a minimum of ad hoc assumptions. It is much simpler than the real sun, but contains a lot of the physics we consider most relevant to the solar and stellar dynamo problem. I like to view the model more as an analog to the solar or a stellar convection zone, rather than as an approximation—much as a laboratory rat or monkey is used as an analog to a human being in many medical experiments. Each exists and is physically complete, and much, though not all, of their biochemistries are the same or are closely related.

Download Full-text

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

International Astronomical Union Colloquium ◽

10.1017/s0252921100064861 ◽

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.

Download Full-text