A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

Astrophysical magnetic fields and nonlinear dynamo theory

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e., to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in nonequilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.

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Particles and fields in fluid turbulence

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

http://www.math.ecnu.edu.cn/~zmwang/teaching/Advanced_ODE-2014-2015/1-Textbook_and_Reference_book/2-Ordinary%20Differential%20Equations%20with%20Applications.pdf

Ordinary Differential Equations with Applications

Introduction: the fast dynamo problem fast dynamo action in flows fast dynamo in maps. Methods and their applications: dynamos and non-dynamos magnetic structure in steady integrable flows upper bounds magnetic structure in chaotic flows nearly integrable flows spectra and Eigenfunctions strongly chaotic systems random fast dynamos dynamics.

Stretch, Twist, Fold: The Fast Dynamo

Introduction 1. Fundamentals of dynamo theory P. H. Roberts 2. Solar and stellar dynamics N. O. Weiss 3. Convection and magnetoconvection in a rapidly rotating sphere M. R. E. Proctor 4. Solar dynamos: computational background A. Brandenburg 5. Energy sources for planetary dynamos W. V. R. Malkus 6. Fast dynamos A. M. Soward 7. Non-linear planetary dynamos D. R. Fearn 8. The chaotic solar cycle E. A. Spiegel 9. The non-linear dynamo and Model-Z S. I. Braginsky 10. Maps and dynamos B. J. Bayly 11. Bifurcations in rotating systems E. Knobloch Index.

Lectures on solar and planetary dynamos

Saffman argues that in decaying two-dimensional turbulence approximate dis-continuities of vorticity will form, and the energy spectrum will fall off as k −4 Saffman assumes that these discontinuities are well separated; in this paper, we examine how accumulation points of such discontinuities may give an energy spectrum of between k −4 and k −3 . In particular we examine the energy spectra of spiral structures which form round the coherent vortices that are observed in numerical simulations of decaying two-dimensional turbulence. If the filaments of the spiral are assumed to be passively advected, the instantaneous energy spectrum has a $k^{-11/3}$ range. Thus we come some way to reconciling the argument of Saffman and the k −3 energy spectrum predicted by models of quasi-equilibrium two-dimensional turbulence based on a cascade of enstrophy in Fourier space.

Spiral structures and spectra in two-dimensional turbulence

The spiral wind-up and diffusive decay of a passive scalar in circular streamlines is considered. An accelerated diffusion mechanism operates to destroy scalar fluctuations on a time scale of order P1/3 times the turn-over time, where P is a Peclet number. The mechanism relies on differential rotation, that is, a non-zero gradient of angular velocity. However if the flow is smooth, the gradient of angular velocity necessarily vanishes at the centre of the streamlines, and the time scale becomes greater. The behaviour at the centre is analysed and it is found that scalar there is only destroyed on a time scale of order P1/2. Related results are obtained for magnetic field and for weak vorticity, a scalar coupled to the stream function of the flow. Some exact solutions are presented.

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Accelerated diffusion in the centre of a vortex

An analysis of small-scale magnetic fields shows that the Ponomarenko dynamo is a fast dynamo; the maximum growth rate remains of order unity in the limit of large magnetic Reynolds number. Magnetic fields are regenerated by a “stretch-diffuse” mechanism. General smooth axisymmetric velocity fields are also analysed; these give slow dynamo action by the same mechanism.

Andrew D. Gilbert

Papers

Stretch, Twist, Fold: The Fast Dynamo

Lectures on solar and planetary dynamos

Spiral structures and spectra in two-dimensional turbulence

Accelerated diffusion in the centre of a vortex

Fast dynamo action in the Ponomarenko dynamo