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3653 results in ebooks in fluid mechanics

Page 10 of 183

  • 1 - Introduction

  • 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 3-19

  • Summary

    Dynamo theory is introduced as the theory of the spontaneous generation of magnetic fields by internal inductive motions in planets, stars and galaxies. The historical background to dynamo theory is described, focussing mainly on the magnetic fields of the Earth and the Sun. The simplest self-exciting dynamo model (the homopolar disc dynamo) is described, and its limitations are indicated. Cowling’s anti-dynamo theorem is touched on, and the resulting need for departures from axisymmetry in the internal fluid motions is discussed. For flows that are turbulent (whether strongly nonlinear or weak, as in a field of random waves), the `mean-field’ approach is described, and the need for a lack of reflectional symmetry (or ‘chirality’) in the flow is stressed. Other highlights in the historical development since the 1950s are described: the analogy with vorticity in turbulence, the first rigorous examples of dynamo action in a simply connected fluid domain (the two-sphere model of Herzenberg and the stasis model of Backus), the early computational attack on the geodynamo problem by Bullard and Gellman, the `cyclonic events’ theory of E. N. Parker, and the incorporation of dynamic constraints imposed by the Navier–Stokes equation in a strongly rotating fluid.

  • 4 - The Magnetic Field of the Earth and Planets

  • 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 99-120

  • Summary

    The gross properties of the magnetic fields of the Earth and of Mercury, Venus, Mars, Jupiter, Saturn, Uranus and Neptune are described. Strong rotation and a liquid conducting interior, at least in part, are indicated as necessary requirements for the existence of a significant magnetic field. The magnetic fields of planetary satellites are briefly described. The interior structure of the Earth is described, conditions in the outer liquid core being of particular relevance for dynamo theory. The time variation of the dipole moment of the Earth over the last few thousand years, as inferred from archaeomagnetic studies, is discussed, and the random reversals of the dipole moment over periods of the order of millions of years, that are inferred from palaeomagnetic studies of rock magnetism, are described. The need for a dynamo theory for the Earth to explain the persistence and time-variation of its magnetic field over past millennia is established. Conditions at the core-mantle boundary are discussed, and the possible relevance of precession of the Earth’s angular velocity vector as the source of energy for the core motions responsible for dynamo action is briefly considered.

  • 12 - Dynamic Equilibration

  • 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 315-355

  • Summary

    Dynamic effects are introduced via the theory of Alfvén waves in a perfectly conducting fluid, the associated invariants and their relation with ‘cross-helicity’. Lehnert waves in a rotating fluid with an ambient magnetic field are then considered, and the dispersion relation is obtained. The important helicity-related concepts of `up-down symmetry breaking’ and the magnetostrophic limit are introduced. Transitory dynamo action associated with decaying Lehnert waves leading to a ‘fossil magnetic field’ is explained. Quenching of the α-effect due to the back-reaction of the Lorentz force and resulting magnetic equilibration are treated, first for waves generated by helical forcing, then for forced Lehnert waves, for which a resonant response in the presence of a growing magnetic field can occur. Boundary forcing is also considered. The generation of helicity is analysed, first due to the interaction of buoyancy and Coriolis forces, then due to magnetostrophic waves in an unstably stratified medium, then through instability due to ‘magnetic buoyancy’, and finally due to flow over the core–mantle interface, for which up-down symmetry is clearly broken.

  • 3 - Advection, Distortion and Diffusion

  • 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 59-98

  • Summary

    Alfvén’s theorem, the analogue of Kelvin’s circulation theorem, is proved. The concept of a ‘frozen-in’ field in a perfectly conducting fluid is described, and the associated conservation of magnetic helicity is proved. The analogy with vorticity in an ideal fluid under Euler evolution is presented. The evolution of a magnetic field subjected to uniform strain is considered, and the possibility of accelerated ohmic diffusion is described. Flux expulsion is introduced through the example of uniform shearing of a space-periodic field; magnetic instability associated with oscillating shear flow and with steadily rotating shear flow is described. Flux expulsion associated with differential rotation is then analysed, with focus on the initial distortion, the intermediate phase and the ultimate steady state. The law of isorotation and the generation of toroidal field by differential rotation are also analysed, with emphasis again on the initial phase and the ultimate steady state. The concept of topological pumping resulting from asymmetry between upward and downward convective flow is introduced, and the behaviour as a function of the magnetic Reynolds number is described.

  • 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.

Page 10 of 183