Lecture 1 (21.01.05)  

Introduction: physics from small to large scales.
Preview of the course (pdf), suggested reading (pdf).

Note on web sources: There are many sets of lecture notes related to MHD on the web. You can google for them if you want them.
Here are three good picks: here is a pdf file with recent lecture notes on MHD by Per Helander (Euratom/UKAEA);
                                                here are notes on MHD, Astrophysical Fluid Dynamics, and Accretion Discs for a related course by Gordon Ogilvie (DAMTP);
                                                this is a link to lecture notes on plasma physics by Richard Fitzpatrick (UT Austin).
NRL Plasma Formulary is a great place to look up vector identities, physical constants and much more (they will also send you a free paper copy courtesy of US Navy).

MHD equations.

Reading: Davidson-MHD §§1.1-1.4, 2.1-2.6, 3.8-3.9, also 3.1-3.7 if you want to brush up on your fluid mechanics
                Goedbloed & Poedts §4.1
                Kulsrud §3.1
                Maxwell's poetry (extracurricular)

Lecture 2 (24.01.05)

Overview of the derivation of MHD from kinetics; limits of MHD description; extensions of MHD.
Note: the details of the kinetic theory are non-examinable.

Reading: Sturrock §§11.1-11.8,12.1
                Goedbloed & Poedts §§2.4.1, 3

If you would like to learn the details of the coarse-graining procedure that led from the Klimontovich (exact) distribution function to the kinetic equation for the smoothed distribution, see
Yu. L. Klimontovich, The Statistical Theory of Non-Equilibrium Processes in a Plasma (MIT Press 1967)

Two additional advanced references on the calculation of collisional transport in plasma are
S. I. Braginskii, Reviews of Plasma Physics 1, 205 (1965) --- original calculation of everything
P. Helander & D. J. Sigmar, Collisional Transport in Magnetized Plasmas(CUP 2002) --- an excellent recent monograph

Lecture 3 (26.01.05)

Kinetic derivation of MHD completed.

Example Sheet 1: Problems 1-3 (pdf)

Lecture 4 (28.01.05)

Diffusion of magnetic field. The magnetic Reynolds number.
Flux freezing.

Reading: Davidson-MHD §§2.7, 4.1-4.3
                Sturrock §§12.2-12.3
                Kulsrud §§3.2-3.3, 4.8

Example Sheet 1: Problems 4-7 (pdf)

Lecture 5 (28.01.05)

Flux freezing completed (Cauchy solution).
Introduction to the dynamo problem.
Helicity.

Reading: Davidson-MHD §4.4
                Sturrock §13.8
                Zeldovich et al. §9.1 (dynamo)

The induction equation is extremely reach: books have been written just about solutions of this equation --- such studies often have to do with the dynamo problem. We will return to some aspects of this problem in the part of the course that deals with MHD turbulence. There will be more dynamo in Prof. Proctor's course next term. In the meanwhile, if you feel you must know more now, see books by Parker, Moffatt, Childress & Gilbert from your reading list. Here are some extra dynamo books for the insatiable:

M. R. E. Proctor & A. D. Gilbert, Lectures on Solar and Planetary Dynamos(CUP 1994) --- a widely used set of lecture notes from a Newton Institute workshop
A. A. Ruzmaikin, A. M. Shukurov & D. D. Sokoloff, Magnetic Fields of Galaxies (Kluwer 1988) --- everything you ever wanted to know about the mean-field dynamo theory for galaxies
F. Krause & K.-H. Rädler, Mean-Field Magnetohydrodynamics and Dynamo Theory (Pergamon 1980) --- a VERY meticulous exposition of mean-field theory by people who invented it
V. I. Arnold & B. A. Khesin, Topological Methods in Hydrodynamics (Springer 1998) ---- their chapter on kinematic dynamo tells you how the dynamo problem might appeal to a pure mathematician

This should take you through the weekend!

Lecture 6 (4.02.05)

Conservation laws in MHD.
Forces in MHD: magnetic curvature and pressure forces.
MHD equilibrium: cylindrical configurations.

Reading: Goedbloed & Poedts §§4.3 (conservation laws), 4.4 (same with dissipative terms)
                Kulsrud §§4.2 (forces), 4.3-4.5 (conservation laws), 4.9 (cylindrical equilibria)

Example Sheet 1: Problems 8-9 (pdf)

Lecture 7 (7.02.05)

Force-free solutions.
Woltjer theorem/Taylor relaxation.
Virial theorem.

Reading: Sturrock §§12.4-12.6 (virial theorem), 13.1-13.7,13.10 (force-free fields),13.9 (Woltjer theorem)
                Kulsrud §4.6 (virial theorem)

Example Sheet 1: Problem 10 (pdf)

Lecture 8 (9.02.05)

Lagrangian MHD.
Action principle for MHD.
MHD stability: the energy principle.

Reading: Kulsrud §§4.7 (action principle), 7.2 (energy principle)
                Sturrock §§15.2 (energy principle), 16.1-16.4 (action principle)
                Davidson-MHD §6.4 (energy principle)
                Goedbloed & Poedts §§6.1-6.6 (a very extensive account of the MHD stability theory)

The original famous paper on the MHD energy principle is I. B. Bernstein, E. A. Frieman, M. D. Kruskal & R. M. Kulsrud, Proc. Roy. Soc. LondonA244, 17 (1958)
Lagrangian formulation of MHD and the action principle are discussed in the excellent original paper by Newcomb:
W. A. Newcomb, Nucl. Fusion: 1962 Supplement, Part 2, p. 451 (distributed in class)
A more recent useful reference is D. Pfirsch & R. N. Sudan, Phys. Fluids B 5, 2052 (1993)

Examples Class 1 (9.02.05, 14:30 in MR5) Example Sheet 1.

Lecture 9 (11.02.05)

Energy principle completed.

Lecture 10 (14.02.05)

Instabilities (z-pinch instabilities as an example of energy-principle calculations).

Reading: Sturrock §§15.1-15.5 (z-pinch instabilities)
               Kulsrud §7.3 (interchange and Parker instabilities)
               Goedbloed & Poedts §§7.2, 7.5, 9.4 (various fairly advanced stability calculations)

Here is an example of a very sophisticated nonlinear instability calculation based on the Lagrangian MHD formalism: S. C. Cowley & M. Artun, Phys. Reports 283, 185 (1997)

Lecture 11 (16.02.05)

MHD waves.

Reading: Sturrock §14.1
               Kulsrud §§5.1-5.4
               Goedbloed & Poedts §§5.1-5.2
               Davidson-MHD §6.1

Lecture 12 (18.02.05)

MHD waves completed.
Finite-amplitude Alfvén waves. Elsässer variables.

Lecture 13 (21.02.05)

Magnetic reconnection.

Reading: Kulsrud §§14.1-14.11

Here is a downloadable earlier review by R. M. Kulsrud, Phys. PLasmas 5, 1599 (1998)
A very nice review of current sheets and reconnection is S. I. Syrovatskii, Ann. Rev. Astron. Astrophys. 19, 163 (1981)
The reference for X-point collapse in incompressible MHD is E. D. Chapman & P. C. Kendall, Proc. Roy. Soc. London A 271, 435 (1963)
See also Biskamp's books on your reading list. Another recent book on reconnection is
E. Priest & T. Forbes, Magnetic Reconnection (CUP 2000)

Lecture 14 (23.02.05)

Tearing mode.

Reading: Sturrock §§17.1-17.6

The original classic paper on the tearing mode is H. P. Furth, J. Killeen & M. N. Rosenbluth, Phys. Fluids 6, 459 (1963)
A further, nonlinear, stage of the tearing mode evolution (known as the Rutherford regime) was analysed by P. H. Rutherford, Phys. Fluids 16, 1903 (1973)
Saturation from this regime (in the small Delta' limit) was treated recently by
F. Militello & F. Porcelli, Phys. Plasmas 11, L13 (2004)
D. F. Escande & M. Ottaviani, Phys. Lett. A 323, 278 (2004)
Here are two classic reviews that cover tearing mode (both are more fusion-oriented than astrophysical):
R. B. White, Rev. Mod. Phys. 58, 183 (1986) --- more mathematical
B. B. Kadomtsev, Rep. Prog. Phys. 50, 115 (1987) --- more qualitative
See also Biskamp's books on your reading list.

Example Sheet 2 (pdf)

Lecture 15 (25.02.05)

Turbulence. Kolmogorov's 1941 dimensional theory.

Reading: Landau & Lifshitz §33 --- read this!
                Davidson-MHD §7.1.3
                Davidson-Turbulence Chapter 5
                Frisch Chapter 7
                Batchelor Chapter VI

Lecture 16 (28.02.05)

Correlation functions.

Reading: Davidson-Turbulence §§6.2.1 (x space), 8.1 (k space)
                Batchelor Chapters II-III

A downloadable account of correlation functions in d dimensions: Appendix A in A. A. Schekochihin, S. A. Boldyrev & R. M. Kulsrud, Astrophys. J. 567, 828 (2002) (not very pedagogically written, I am afraid).

Two events this week pertaining to MHD & Turbulence:

28.02.05 @14:00 in MR4 --- Meeting for Part III (Maths & Astro) students interested in PhD opportunities in astrophysics at DAMTP
                                             (there will be short presentations by Professor John Papaloizou, Dr Gordon Ogilvie, and myself)
1.03.05 @13:00 in MR14 --- I will give a lunchtime seminar "Turbulence in clusters of galaxies and the kinetic description of magnetised astrophysical plasmas"

Lecture 17 (2.03.05)

von Karman-Howarth Equation.
The closure problem in turbulence.
Kolmogorov's 4/5 Law.

Reading: Landau & Lifshitz §34
                Frisch Chapter 6
                Davidson-Turbulence §§6.2, 6.3 (the latter section treats the decay laws, which I did not cover, but you should read about them), 8.2 (dynamics in k space and closure models)
                Davidson-MHD §§7.1.4, 7.1.5 (these are more concise versions of §§6.2, 6.3 of his Turbulence book)
                Batchelor Chapter V

Examples Class 2 (2.03.05, 14:30 in MR5) Example Sheet 2.

Lecture 18 (4.03.05)

Intermittency.

Reading: Frisch Chapter 8
                Davidson-Turbulence §§6.5 (intro to intermittency), 7.3 (overview of numerical results)
                Biskamp-MHD Turbulence §§7.1,7.2,7.4

Here are some key recent papers on the She-Lévêque model of intermittency:
Z.-S. She and E. Lévêque, Phys. Rev. Lett. 72, 336 (1994)
B. Dubrulle, Phys. Rev. Lett. 73, 959 (1994)
Z.-S. She and E. C. Waymire, Phys. Rev. Lett. 74, 262 (1995)
S. Boldyrev, Astrophys. J. 569, 841 (2002)  

Example Sheet 3 (pdf) Note that Problems 2-4 take you step-by-step through the main results of the theory of scalar turbulence (passive scalar theory). I urge you all to attend the Examples Class 3, when I will explain these results in detail  --- this Examples Class will  be more like a supplementary lecture than a supervision.

Lecture 19 (7.03.05)

Intermittency completed.

Lecture 20 (9.03.05)

Anisotropic MHD turbulence: an overview of theoretical uncertainties.

Here (pdf) is my recent review talk on this from the 1st CMPD Plasma Physics Winter School at UCLA.

Examples Class 3 (9.03.05, 14:30 in MR5) We will be discussing Example Sheet 3. In particular, I will explain the basic theory of passive scalar turbulence in detail.

Lecture 21 (10.03.05) --- make-up lecture 12:00 in MR14

Small-scale dynamo in a linear velocity field.

Small-scale dynamo in a linear velocity field is analysed in Ya. B. Zeldovich et al., J. Fluid Mech. 144, 1 (1984) [all back volumes of JFM are available on the ground floor of Pavilion G].
The interpretation of their picture that I have given you is in A. A. Schekochihin et al., Astrophys. J. 612, 276 (2004).
The folded structure of the magnetic field is also discussed in the above paper and, on a more mathematical level, in A. Schekochihin et al., Phys. Rev. E 65, 016305 (2002).
A different (but complementary) formalism for understanding field growth and structure based on quantifying flux cancellation properties of the magnetic field was developed by Ott and coworkers in 1990s.
Their work is reviewed in E. Ott, Phys. Plasmas 5, 1636 (1998).

Lecture 22 (11.03.05)

Small-scale dynamo in a linear velocity field completed.
The Kazantsev-Kraichnan model.

Lecture 23 (14.03.05)

Small-scale dynamo: magnetic-field spectrum in the Kazantsev model.

Statistical methods for dealing with multiplicative noise are described very thoroghly in  van Kampen's book (on your reading list).
The method of averaging that I have given you, as well as extensions to small but finite correlation times, are described in A. A. Schekochihin & R. M. Kulsrud, Phys. Plasmas 8, 4937 (2001).

Here is a (very incomplete) list of papers where Kazantsev's model is studied in many different ways:
A. P. Kazantsev, Soviet Phys. --- JETP 26, 1031 (1968)
See review of subsequent work in 1980s in Chapter 9 of the Zeldovich et al. book (on your reading list)
R. M. Kulsrud & S. W. Anderson, Astrophys. J. 396, 606 (1992)
A. Gruzinov, S. Cowley & R. Sudan, Phys. Rev. Lett. 77, 4342 (1996)
I. Rogachevskii & N. Kleeorin, Phys. Rev. E 56, 417 (1997)
K. Subramanian, astro-ph/9708216
M. Chertkov et al., Phys. Rev. Lett. 83, 4065 (1999)
A. A. Schekochihin, S. A. Boldyrev & R. M. Kulsrud, Astrophys. J. 567, 828 (2002)
D. Vincenzi, J. Stat. Phys. 106, 1073 (2002)
N. Kleeorin, I. Rogachevskii & D. Sokoloff, Phys. Rev. E 65, 036303 (2002)
S. Nazarenko, R. J. West & O. Zaboronski, Phys. Rev. E 68, 026311 (2003)
R. J. West et al., Astron. Astrophys. 414, 807 (2004)
S. A. Boldyrev & F. Cattaneo, Phys. Rev. Lett. 92, 144501 (2004)

Example Sheet 4  (pdf)
Some references on scalar decay are
R. H. Kraichnan, Phys. Fluids 11, 945 (1968) --- equation for the spectrum (derived from closure theory)
R. H. Kraichnan, J. Fluid Mech. 64, 737 (1974) --- equation for the spectrum (derived from the joint PDF approach)
R. T. Pierrehumbert, Chaos Solitons Fractals 4, 1091 (1994) --- strange mode introduced
T. M. Antonsen et al., Phys. Fluids 8, 3094 (1996) --- theory based solely on the viscous-convective range physics
E. Balkovsky & A. Fouxon, Phys. Rev. E 60, 4164 (1999) --- theory based solely on the viscous-convective range physics
G. A. Voth et al., Phys. Fluids 15, 2560 (2003) --- experimental results
D. R. Fereday & P. H. Haynes, Phys. Fluids 16, 4359 (2004) --- extensive discussion of the strange mode
A. A. Schekochihin, P. H. Haynes & S. C. Cowley, Phys. Rev. E 70, 046304 (2004) --- strange mode spectrum

Lecture 24 (16.03.05)

Small-scale dynamo: overview of the nonlinear theory.
Isotropic MHD turbulence.

My discussion was based on the following three papers:
A. A. Schekochihin et al., New J. Phys. 4, 84 (2002)
A. A. Schekochihin et al., Phys. Rev. Lett. 92, 084504 (2004)
A. A. Schekochihin et al., Astrophys. J. 612, 276 (2004)
Here (pdf) is a recent talk that summarises all that.

Thank you all for staying with the course. I will be available to answer your questions etc. until the exam time, except  20 March -- 17 April and 22--28 May, when I will be out of the country. E-mail or ring me if you want help.

Examples Class 4 (9.05.05 at 12:00 in MR5 --- note change of venue) We will be discussing Example Sheet 4 and any questions you have in the run up to the exam.

Upcoming events of interest: