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.

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.

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.

scholarly journals Determination of the instantaneous geostrophic flow within the three-dimensional magnetostrophic regime

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180412 ◽  
Author(s):  
Colin M. Hardy ◽  
Philip W. Livermore ◽  
Jitse Niesen ◽  
Jiawen Luo ◽  
Kuan Li
Keyword(s):  
Rotation Axis ◽  
Logarithmic Singularity ◽  
Three Dimensional ◽  
Outer Core ◽  
Necessary Condition ◽  
Small Subset ◽  
Dynamo Action ◽  
Geostrophic Flow ◽  
Geostrophic Flows

In his seminal work, Taylor (1963 Proc. R. Soc. Lond. A 274 , 274–283. ( doi:10.1098/rspa.1963.0130 ).) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second-order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalization of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any three-dimensional Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully three-dimensional case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular.

scholarly journals Torsional waves driven by convection and jets in Earth’s liquid core

2018 ◽  
Vol 216 (1) ◽  
pp. 123-129 ◽  
Author(s):  
R J Teed ◽  
C A Jones ◽  
S M Tobias
Keyword(s):  
Magnetic Field ◽  
Density Gradient ◽  
Inner Core ◽  
Liquid Core ◽  
Outer Core ◽  
Dynamo Action ◽  
Radial Magnetic Field ◽  
Torsional Waves ◽  
Plausible Mechanism ◽  
Rich Liquid

SUMMARY Turbulence and waves in Earth’s iron-rich liquid outer core are believed to be responsible for the generation of the geomagnetic field via dynamo action. When waves break upon the mantle they cause a shift in the rotation rate of Earth’s solid exterior and contribute to variations in the length-of-day on a ∼6-yr timescale. Though the outer core cannot be probed by direct observation, such torsional waves are believed to propagate along Earth’s radial magnetic field, but as yet no self-consistent mechanism for their generation has been determined. Here we provide evidence of a realistic physical excitation mechanism for torsional waves observed in numerical simulations. We find that inefficient convection above and below the solid inner core traps buoyant fluid forming a density gradient between pole and equator, similar to that observed in Earth’s atmosphere. Consequently, a shearing jet stream—a ‘thermal wind’—is formed near the inner core; evidence of such a jet has recently been found. Owing to the sharp density gradient and influence of magnetic field, convection at this location is able to operate with the turnover frequency required to generate waves. Amplified by the jet it then triggers a train of oscillations. Our results demonstrate a plausible mechanism for generating torsional waves under Earth-like conditions and thus further cement their importance for Earth’s core dynamics.

scholarly journals The magnetorotational instability prefers three dimensions

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190622
Author(s):  
Jeffrey S. Oishi ◽  
Geoffrey M. Vasil ◽  
Morgan Baxter ◽  
Andrew Swan ◽  
Keaton J. Burns ◽  
...  
Keyword(s):  
Shear Rate ◽  
Angular Velocity ◽  
Laboratory Experiments ◽  
Three Dimensional ◽  
Linear Instability ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Magnetorotational Instability ◽  
Dynamo Action ◽  
Wide Range

The magnetorotational instability (MRI) occurs when a weak magnetic field destabilizes a rotating, electrically conducting fluid with inwardly increasing angular velocity. The MRI is essential to astrophysical disc theory where the shear is typically Keplerian. Internal shear layers in stars may also be MRI-unstable, and they take a wide range of profiles, including near-critical. We show that the fastest growing modes of an ideal magnetofluid are three-dimensional provided the shear rate, S , is near the two-dimensional onset value, S c . For a Keplerian shear, three-dimensional modes are unstable above S  ≈ 0.10 S c , and dominate the two-dimensional modes until S  ≈ 2.05 S c . These three-dimensional modes dominate for shear profiles relevant to stars and at magnetic Prandtl numbers relevant to liquid-metal laboratory experiments. Significant numbers of rapidly growing three-dimensional modes remainy well past 2.05 S c . These finding are significant in three ways. First, weakly nonlinear theory suggests that the MRI saturates by pushing the shear rate to its critical value. This can happen for systems, such as stars and laboratory experiments, that can rearrange their angular velocity profiles. Second, the non-normal character and large transient growth of MRI modes should be important whenever three-dimensionality exists. Finally, three-dimensional growth suggests direct dynamo action driven from the linear instability.

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.

Numerical Solution of Melting in Side-Heated Rectangular Enclosure Under Electromagnetically Simulated Low Gravity

2003 ◽  
Author(s):  
Eduardo Gonc¸alves ◽  
M. Faghri ◽  
Y. Asako ◽  
M. Charmchi
Keyword(s):  
Magnetic Field ◽  
Flow Field ◽  
Lorentz Force ◽  
Control Volume ◽  
Outer Space ◽  
Electromagnetic Simulation ◽  
Low Gravity ◽  
Conducting Phase ◽  
Electrically Conducting ◽  
Rectangular Enclosure

Electromagnetic simulation of low-gravity environment has been numerically investigated to study the transport phenomena associated with melting of an electrically conducting phase change material, (Gallium), inside a rectangular enclosure. Electromagnetic fields are configured such that the resulting Lorentz force can be used to damp and/or counteract the natural convection and thereby simulating the low gravity environment of outer space. The governing equations are discretized using a control-volume-based finite difference scheme. The solutions are obtained for true lowgravity environment as well as for the simulated-low-gravity. The results show that when the Lorentz force is due to the presence of magnetic field alone, the low-gravity condition is simulated by the damping effect, which is shown to have a profound effect on the flow field. On the other hand, it was shown that under electromagnetic field simulation, where the Lorentz force is caused by the transverse electric and magnetic fields, it is possible to minimize the flow field distortion caused by the high magnetic field and therefore achieve a much better simulation of low-gravity. Furthermore, it was found that under electromagnetic simulation of low gravity the flow field can be reduced or even reversed but never completely halted.

Spatial variations of magnetic permeability as a source of dynamo action

2013 ◽  
Vol 727 ◽  
pp. 161-190 ◽  
Author(s):  
B. Gallet ◽  
F. Pétrélis ◽  
S. Fauve
Keyword(s):  
Electrical Conductivity ◽  
Asymptotic Expansion ◽  
Magnetic Permeability ◽  
Spatial Variations ◽  
Parallel Flow ◽  
Conducting Fluid ◽  
Dynamo Action ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Spatially Varying

AbstractWe investigate dynamo action for a parallel flow of an electrically conducting fluid located over a boundary with spatially varying magnetic permeability. We first compute the dynamo threshold numerically. Then we perform an asymptotic expansion in the limit of small permeability modulation, which gives accurate results even for moderate modulation. We present in detail the mechanism at work for this dynamo. It is an interplay between shear (an $\omega $-effect) and a new conversion mechanism that originates from the non-uniform magnetic boundary. We illustrate how a similar mechanism leads to dynamo action in the case of spatially modulated electrical conductivity, a problem studied by Busse & Wicht (Geophys. Astrophys. Fluid Dyn., vol. 64, 1992, pp. 135–144). Finally, we discuss the relevance of this effect to experimental dynamos and present ways to increase the dynamo efficiency and reduce the instability threshold.

Collision Detection of Cylindrical Rigid Bodies Using Line Geometry

2005 ◽  
Author(s):  
John S. Ketchel ◽  
Pierre M. Larochelle
Keyword(s):  
Collision Detection ◽  
Nuclear Physics ◽  
Detection Algorithm ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Necessary Condition ◽  
Infinite Length ◽  
Line Geometry ◽  
Well Drilling ◽  
Necessary And Sufficient

This paper presents a novel methodology for detecting collisions of cylindrically shaped rigid bodies moving in three dimensions. This algorithm uses line geometry and dual number algebra to exploit the geometry of right circular cylindrical objects to facilitate the detection of collisions. First, the rigid bodies are modelled with infinite length cylinders and a necessary condition for collision is evaluated. If the necessary condition is not satisfied then the two bodies are not capable of collision. If the necessary condition is satisfied then a collision between the bodies may occur and we proceed to the next stage of the algorithm. In the second stage the bodies are modelled with finite length cylinders and a definitive necessary and sufficient collision detection algorithm is employed. The result is a straight-forward and efficient means of detecting collisions of cylindrically shaped bodies moving in three dimensions. This methodology has applications in spatial mechanism design, robot motion planning, workspace analysis of parallel kinematic machines such as Stewart-Gough platforms, nuclear physics, medical research, computer graphics and well drilling. A case study examining a spatial 4C robotic mechanism for self collisions is included.

Sign in / Sign up

Export Citation Format

Share Document