Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Browse subjects
  3. Mathematics
  4. Fluid dynamics and solid mechanics
  5. Content listing

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

7997 results in Fluid dynamics and solid mechanics

Page 70 of 400

  • 17 - Magnetic Relaxation in a Low-β Plasma

  • from Part III - Dynamic Aspects of Dynamo Action
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 463-481

  • Summary

    Magnetic field relaxation in a plasma of very low density is considered, first in terms of a simple one-dimensional model in which the formation of current sheets can be explicitly realised. The buildup of fluid density at the location of the current sheets is very marked. Azimuthal field relaxation in a cylindrical annulus leading to the ‘pinch effect’ is then considered, and a similarity solution for current collapse in an unbounded fluid is obtained. Relaxation of a helical field and Taylor’s conjecture leading to a ‘Taylor state’ of prescribed global helicity is presented, with application to the reversed field pinch. The mechanism by which a reversed field can appear during the relaxation process is then considered; on the assumption that helical turbulence resulting from instabilities of the collapsing field is in some way responsible, an α-effect is incorporated in the relaxation process. It is shown that this can induce negative diffusivity and violent axial field instability near the cylinder boundary, with the tentative conclusion that an α-effect may indeed provoke axial field reversal near the boundary. Eruption and relaxation of twisted flux ropes in the solar corona are briefly considered.

  • 15 - Helical Turbulence

  • from Part III - Dynamic Aspects of Dynamo Action
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 417-440

  • Summary

    Kolmogorov’s theory of non-helical turbulence is reviewed, and the energy cascade is described, together with the effects of intermittency of the rate of energy dissipation; effects of helicity are then considered, taking account of the invariance of helicity as well as energy in the cascade process. Realisability conditions are obtained. Numerical results use the eddy-damped quasi-Markovian closure scheme (EDQNM). Nonlinear interaction of Alfvén waves leading to the Kraichnan–Iroshnikov spectrum is described. Batchelor’s early analogy between magnetic field and vorticity leading to a dynamo criterion in terms of the magnetic Prandtl number is reviewed. The Malkus–Proctor theory, whereby dynamo saturation is achieved through modification of the mean velocity field, is presented, in some cases leading to a Taylor state with superposed torsional oscillations. Magnetostrophic turbulence, i.e. turbulence in a strongly rotating fluid and in the presence of a strong dynamo-generated magnetic field, is described; here, the helicity that is responsible for dynamo action is generated by the rise of buoyant parcels of fluid driven by either thermal or compositional convection.

  • 14 - Astrophysical dynamic models

  • from Part III - Dynamic Aspects of Dynamo Action
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 396-416

  • Summary

    This chapter addresses the generation of magnetic field in astrophysical objects on a wide variety of scales. A range of models of increasing complexity are introduced, starting from low-order models, then using mean-field formalism, and finally using direct numerical simulations. The test-field approach, which establishes a connection between the last two approaches, is introduced. Dynamos in stars are our first concern in this chapter, in particular the existence of stars characterised by a dominant large-scale axial dipole, and others by a fluctuating smaller scale field, usually exhibiting cyclic behaviour. The dipole breakdown mechanism, by which excessive inertial effects prevent the generation of a large-scale dipolar field, is introduced. The concept of a kinematically unstable field is then presented as well as its application to the dipole breakdown mechanism. On a somewhat larger scale, mechanisms for generation of magnetic fields in galaxies are introduced. Finally, the theory of accretion discs and the magnetorotational instability to which they are subject is presented and discussed.

  • 8 - Nearly Axisymmetric Dynamos

  • from Part II - Foundations of Dynamo Theory
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 216-230

  • Summary

    Braginskii’s theory of nearly axisymmetric dynamo action, as reinterpreted by Soward, is presented. The departure from axisymmetry is what allows Cowling’s anti-dynamo theorem to be bypassed. A Lagrangian transformation of the induction equation leads first to a natural interpretation of Braginskii’s ‘effective variables’ and second to an axisymmetric mean-field equation in which a regenerative dynamo term appears that is wholly diffusive in origin, resulting from a diffusive phase shift between velocity and magnetic perturbation fields. This regenerative term is again associated with chirality in the perturbation velocity field. In the geomagnetic context, matching conditions to the external poloidal field are obtained, and the secular variation of the external field admits interpretation in terms of the fluctuating part of this external field. Soward’s theory involves hybrid Euler–Lagrange techniques, which also arise in the theory of the interaction between waves and mean flows on which they may be superposed.

  • 6 - Laminar Dynamo Theory

  • from Part II - Foundations of Dynamo Theory
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 145-184

  • Summary

    The kinematic dynamo problem is defined, and a necessary condition for dynamo action in terms of the maximum rate-of-strain of the velocity field is obtained. It is shown that, under frozen-field conditions, the dipole moment of the current distribution in a sphere is bounded but that diffusion across the spherical surface, in conjunction with interior convection, can lead to its sustained increase. Cowling’s anti-dynamo theorem and variants are proved and some consequences described. Rotor dynamos, including Herzenberg’s two-sphere dynamo and Gailitis’s ring-vortex dynamo, are analysed. Helical dynamo action is illustrated by the Ponomarenko dynamo. The Riga dynamo experiment, involving a propellor-driven helical flow of liquid sodium, is described; in this, dynamo action is observed as the propellor speed is increased beyond a critical threshold. The Bullard–Gellman formalism is described, and early convergence issues are noted. It is noted that axisymmetric flows in a sphere can generate non-axisymmetric fields, of either steady or oscillatory form. Finally, the ‘stasis’ dynamo of Backus, involving three phases well separated in time, is described; this was an early model through which dynamo action in a spherical domain could be rigorously established.

  • 11 - Low-Dimensional Models of the Geodynamo

  • from Part III - Dynamic Aspects of Dynamo Action
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 299-314

  • Summary

    Dynamical aspects of dynamo theory are introduced through consideration of a number of low-dimensional models: the segmented disc dynamo, which shows chaotic behaviour with random polarity reversals; the disc dynamo driven by thermal convection (coupling a Bullard disc and a Welander loop) which shows a primary thermal bifurcation, followed by a secondary dynamo bifurcation; and the Rikitake two-disc dynamical system, again showing random polarity reversals. A different mode coupling is introduced between a dipolar and a quadrupolar family. There are similarities with the problem of a particle in a bi-stable potential well in that random excitation can trigger a transition from one minimum to the other. Turbulent fluctuations provide the excitation, whereas the distance from a saddle-node bifurcation accounts for the barrier. Good correspondence is achieved with both VKS and paleomagnetic data.

  • 5 - Astrophysical Magnetic Fields

  • from Part I - Basic Theory and Observations
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 121-142

  • Summary

    The internal structure of the Sun, consisting mainly of ionised hydrogen and helium, is described. Dynamo action occurs in the ‘convection zone’ – the outer 30% (by radius) of the solar interior – bounded below by the ‘tachocline’, a layer of strong differential rotation. The velocity field in the Sun as observed on the surface and as inferred by helioseismology in the interior is described; sunspots and the 22-year solar cycle are also described, as also the toroidal magnetic field as inferred from sunspot polarities. The general poloidal magnetic field of the Sun is described, with reference also to the surface granulation pattern; the phase coupling between toroidal and poloidal ingredients is noted. The fields of magnetic stars are described in terms of the oblique-rotator model, in possible conjunction with the dynamo process. The dynamo process in fully convective M dwarf stars is also noted. The interaction between the Sun and its planetary system is briefly surveyed; diffusion of particles across the magnetosheath leading to auroral phenomena are noted. The magnetic fields of galaxies, including the Milky Way, are described. Finally, the situation in neutron stars is considered, for which Hall currents within the crust influence the magnetic field evolution.

  • 13 - The Geodynamo: Instabilities and Bifurcations

  • from Part III - Dynamic Aspects of Dynamo Action
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 356-395

  • Summary

    The problem of the origin of the Earth magnetic field is introduced and discussed in mathematical terms. The first instability relevant to this problem is the onset of thermal convection. The latest theoretical developments are presented. These involve the important distinction between local and global instabilities. The onset of dynamo action is then addressed. Particular attention is given to the concepts of weak and strong field and the relevant bifurcation sequence. Comparison of asymptotic and numerical models is performed, and mixed asymptotic and numerical models are introduced. The challenging limiting case of vanishing viscosity and inertia, known as the ‘Taylor state’, is then introduced. Deviations in terms of viscous ‘Ekman state’ or inertial ‘torsionnal oscillations’ are presented. Finally, scaling laws, which are sometimes used as a tool to relate numerical simulations to the parameter regime relevant to the Earth’s core, are presented and discussed.

  • 9 - Solution of the Mean-Field Equations

  • from Part II - Foundations of Dynamo Theory
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 231-278

  • Summary

    Mean-field equations are derived for axisymmetric systems, including both the α-effect and the ‘ω-effect’ of differential rotation. The distinction between α2-dynamos and αω-dynamos is explained. The α2-dynamo is treated, first as regards its free modes, then for the space-periodic ‘ABC-flow’, then for a spherical domain of fluid. Numerical results for α2-dynamos when α is antisymmetric about the equatorial plane are presented. The αω-dynamo is then treated, first as regards free modes, in this case taking the form of ‘dynamo waves’, then through a succession of models involving well-separated concentrated layers of α-effect and shear. A model for the galactic dynamo is analysed, showing dynamo modes of dipole or quadrupole symmetry, steady or oscillatory in each case. A ‘stasis’ dynamo of αω-type is described, and a range of numerical results for prescribed distributions of α and ω is presented. The Karlsruhe experiment modelling a space-periodic α2-dynamo and the `VKS’ experiment modelling an αω-dynamo with or without field reversals are described. Finally, the evolving Taylor–Green vortex and the manner in which concentrated flux tubes emerge from the dynamo action associated with this flow are presented.

  • 10 - The Fast Dynamo

  • from Part II - Foundations of Dynamo Theory
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 279-296

  • Summary

    The ‘fast dynamo’ is introduced through the ‘stretch-twist-fold’ mechanism whereby the magnetic field in a circular flux tube may be doubled in intensity. This apparently provides a dynamo whose growth rate is independent of diffusivity η in the limit η —> 0. However, the writhe generated by the stretch-twist-fold cycle is compensated by the appearance of twist within the tube, leading to the development of a pathological structure under infinite iteration of the cycle. A related mechanism for a helicity cascade to fine scales is proposed. The non-existence of smooth fast dynamos is proved, with the inference that fast-dynamo eigenfunctions have fine structure on scales O(η1/2) as η —> 0, non-differentiable in the limit. The homopolar disc dynamo is shown to be slow, rather than fast, when azimuthal current in the disc is taken into account. The Ponomarenko dynamo with discontinuous velocity is shown to be fast with the above pathological structure. Similarly, it is shown that certain space-periodic dynamos can be fast, the magnetic field being concentrated in narrow cigar-shaped regions. The manner in which the space-periodic Archontis dynamo evolves from slow to fast is described.

  • 7 - Mean-Field Electrodynamics

  • from Part II - Foundations of Dynamo Theory
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 185-215

  • Summary

    A two-scale approach is introduced, in which l0 is the dominant ‘energy-containing’ scale either of turbulence or of a random wave field, and L >> l0 is the scale of quantities averaged over the scale l0 . ‘ Mean-field electrodynamics’ obtains an equation for the mean magnetic field B0, by averaging the induction equation and by determining the ‘mean electromotive force’ < u x b > in terms of B0, so that the equation for B0 may be integrated. Here, u and b are the fluctuating parts of the velocity and magnetic fields, which are `cross-correlated’ only if the turbulence lacks reflection symmetry. The existence of the dominant ‘α-effect’ responsible for dynamo action is established for both isotropic and non-isotropic turbulence. At second order in l0/L, a generalised `eddy diffusivity’ is also established. Statistical properties of turbulence are defined, with emphasis on the energy and helicity spectrum functions. The α-effect is determined for a helical wave field and for turbulence under the `first-order smoothing approximation’ and is explicitly related to the helicity spectrum function. A Lagrangian approach to the weak-diffusion limit and a renormalisation approach based on successive averaging are also developed.

  • 2 - Magnetokinematic Preliminaries

  • from Part I - Basic Theory and Observations
  • Keith Moffatt, University of Cambridge, Emmanuel Dormy, Ecole Normale Supérieure, Paris
  • Book:
    Self-Exciting Fluid Dynamos
    Published online:
    13 May 2019
    Print publication:
    25 April 2019, pp 20-58

  • Summary

    The Biot–Savart law giving the vector potential A of a magnetic field B in terms of B is obtained. Structural properties of the general magnetic field are described, and magnetic helicity is introduced together with its topological interpretation in terms of flux-tube linkages. Simple mechanical examples of chiral behaviour are described. Toroidal/poloidal decomposition is defined and exemplified for axisymmetric and two-dimensional fields.The multipole expansion of the exterior field produced by a confined current distribution is obtained. Laws governing force-free fields are obtained, and particular force-free fields having field lines in the form of torus knots are described. The Lagrangian description of a magnetic field in a moving conductor is introduced, and the field evolution equation based on Faraday’s Law of Induction and Ohm’s Law is obtained. The free decay modes of a field in a spherical geometry are described, and the possibility of diffusive increase of the dipole moment is indicated. Fields exhibiting Lagrangian chaos are described. The notions of writhe and twist for knotted flux tubes are introduced, and the relation between helicity and writhe-plus-twist is explained. The behaviour when a flux tube passes through an inflexional configuration is explained.

Page 70 of 400