COURSE BLOG
The lecturer should give the audience full
reason
to believe that all
his powers have been exerted for their pleasure and instruction.
Michael Faraday.
Here I will post some information on the material
we have
covered in the past lectures, plans for the upcoming lectures,
suggestions
for additional reading, original references, example sheets, scheduling
notices
etc.
Note that my presentation will not necessarily be based on the
reading
suggestions below. These are not obligatory, they are given simply so
that
you know where to look for an alternative (and in many cases much more
extensive)
account of the material discussed in class.
Lecture 1 (9.10.06)
Preview of the course (pdf),
suggested
reading (pdf).
Introduction: magnetic fields and turbulence in astrophysics, physics
from
large to small scales, universality, Richardson cascade.
Here is a
wonderful
illustration of turbulence as multiscale disorder: this is a paper by
Yokokawa
et al. describing the biggest to date direct numerical
simulation of
turbulence done on the Earth Simulator machine in Japan. If you look
carefully
at the pictures, you should start having some reservations about the
qualitative
picture I described in my lecture. Do ask me about these reservations.
There will be no lecture on Wednesday
11.10.06 (we'll schedule a make-up lecture later). The next lecture is
on Monday 16.10.06.
Lecture 2 (16.10.05)
& Lecture 3 (18.10.05)
Statistical description of turbulence.
Correlation functions.
Symmetries (homogeneity, isotropy, parity). Incompressibility.
Spectrum.
The closure problem.
Reading: Davidson-Turbulence, §6.2.1 (x space), §8.1 (k
space), 8.2 (closure models)
Batchelor,
Chapters
II-III
Monin &
Yaglom, Chapter 6 --- the definitive account of
correlation
functions
McComb (on
closures)
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).
Lecture 4 (23.10.05) & Lecture
5 (25.10.05)
Kolmogorov's 1941 dimensional theory of turbulence.
Reading: Landau & Lifshitz §33 --- read this!
Davidson-MHD
§7.1.3
Davidson-Turbulence,
Chapter 5
Frisch, Chapter
7
Batchelor,
Chapter
VI
Monin &
Yaglom
§21, (see §24 on particle diffusion)
Kolmogorov's original paper: A. N. Kolmogorov, Dokl. Akad.
Nauk
SSSR 30, 299 (1941) [reprinted Proc.
Roy. Soc. A 434, 9 (1991)].
Here are two interesting historical papers:
In my lectures this year, I will not cover Kolmogorov's 4/5 law, but
you should read about it (obviously, this material is not examinable).
The reading suggestions are
Landau & Lifshitz §34
Frisch Chapter 6
Davidson-Turbulence
§§6.2, 6.3 (the latter section treats the decay laws), 8.2
(dynamics in k space)
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
The original von Karman-Howarth paper: T.
de Karman & L. Howarth, Proc. Roy. Soc. A 164, 192
(1938).
Kolmogorov's original paper on the 4/5 law: A.
N. Kolmogorov, Dokl. Akad. Nauk SSSR 32, 19 (1941)
[reprinted
Proc.
Roy. Soc. A 434, 15 (1991)].
If you
find it difficult to work out the 4/5 law etc. from the above, you may
ask me for some notes from last-year's version of the course (there was
more time, so I covered this material in class).
Further material on
turbulence, also omitted in this year's version of the course, concerns
intermittency models.
Here are some reading suggestions on this subject:
Frisch
Chapter 8; see §6.4 for detailed discussion
of
Landau's objection to Kolmogorov's theory
Davidson-Turbulence
§§6.5 (intro to intermittency), 7.3 (overview of numerical
results)
Biskamp-MHD
Turbulence
§§7.1,7.2,7.4
Kolmogorov's original paper on the refined-similarity
hypothesis:
A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962)
[not on
the web, alas, but all back volumes of JFM can be found on the ground
floor
of Pavilion G].
Here are some key recent papers on the
She-Lévêque
model of intermittency (not terribly clear except, perhaps, the last
one):
Discovery of extended self-similarity: R. Benzi et
al.,
Phys. Rev. E 48, R29 (1993)
Again, on intermittency, I can give
you some lecture notes.
Here is EXAMPLE SHEET I (pdf) --- you know enough to do it after Lecture 5.
I will discuss these examples in Ex. Class I. Note that Questions 2-3
take you through the dimensional theory of scalar turbulence, so even
if you are not planning to take the exam, I urge you to come to the ex.
class. If you have any trouble at all with the questions, do come to
see me before the ex. class --- I'll be happy to help you.
Lecture 6 (27.10.05) --- MAKE-UP
LECTURE on Friday 27 October @ 11:00-12:00 in
MR14
MHD equations: magnetic forces, the induction equation.
Reading: Davidson-MHD §§1.1-1.4, 2.1-2.6,
3.8-3.9
(equations; also 3.1-3.7 if you want to brush up on your fluid
mechanics)
Goedbloed &
Poedts
§4.1 (equations)
Kulsrud
§§3.1
(equations), 4.2 (forces)
Maxwell's
poetry (extracurricular)
If you would like to learn how to derive the MHD equations
properly
from the kinetic plasma theory, see Sturrock §§11.1-11.8,12.1
or
Goedbloed & Poedts §§2.4.1, 3.
Three more references on kinetic theory are
Yu. L. Klimontovich, The Statistical Theory of
Non-Equilibrium Processes
in a Plasma (MIT Press 1967) --- the mathematical construction of
the
kinetic theory
S. I. Braginskii, Reviews of Plasma Physics 1,
205 (1965) --- original calculation of collisional transport terms
(viscosity,
thermal diffusivity, magnetic diffusivity)
P. Helander & D. J. Sigmar, Collisional
Transport in
Magnetized Plasmas (CUP 2002) --- an excellent recent monograph on
collisional
transport, contains everything you need to know and more!
Lecture 7 (30.10.05)
Magnetic diffusion. Magnetic Reynolds number.
Flux freezing.
Zeldovich rope dynamo.
Reading: Davidson-MHD §§2.7 (diffusion), 4.1-4.3 (flux
freezing)
Sturrock
§§12.2
(flux freezing), 12.3 (diffusion)
Kulsrud
§§3.2-3.3
(flux freezing and its astrophysical applications)
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
Lecture 8 (1.11.05)
Lagrangian MHD. Cauchy solution of the induction equation. Action
principle.
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)
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)
Reading: Sturrock §§16.1-16.4 (action principle)
Kulsrud
§§4.8
(Cauchy solution), 4.7 (action principle)
Example Class I: Thursday 2.11.05 @ 14:00-16:00 in MR4
Lecture 9 (3.11.05) --- MAKE-UP
LECTURE on Friday 3 November @ 11:00-12:00 in
MR14
Conservation laws: mass, momentum, energy, helicity,
cross-helicity.
Reading: Kulsrud
§§4.3-4.5
Goedbloed &
Poedts
§§4.3 (conservation laws), 4.4 (same with dissipative terms)
Davidson-MHD §4.4 (helicity)
Sturrock
§13.8
(helicity)
If you wish to read something about the energy principle, MHD
equilibria, MHD
stability
etc., here are some pointers:
Energy principle: Kulsrud §7.2, Sturrock
§§16.1-16.4, Davidson-MHD §6.4,
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.
London
A244, 17 (1958)
MHD equilibrium: Kulsrud
§4.9 (cylindrical equilibria), Sturrock
§§13.1-13.7, 13.10 (force-free
fields), 13.9 (Woltjer theorem)
Instabilities: 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)
If you would like some notes on these things, ask me and I will
give
you a copy of 2005 lecture notes/example-sheet solutions where I
work
out the energy principle, instabilities of the cylindrical equilibria,
and
the instabilities in the presence of gravity (interchange
instabilities)
systematically using the Lagrangian approach.
Lecture 10 (6.11.05) &
Lecture 11 (13.11.05)
Linear MHD waves.
Finite-amplitude Alfvén waves. Elsässer variables.
Reading: Sturrock §14.1
Kulsrud
§§5.1-5.4
Goedbloed &
Poedts
§§5.1-5.2
Davidson-MHD
§6.1
There will be no lectures on
Wednesdays
8.11.06 and 15.11.06
Lecture 12 (20.11.05)
Alfvénic (anisotropic MHD) turbulence.
Reading: You may find this review
(§§1-2)
and references therein useful.
Here are the some papers on MHD turbulence (IK and GS):
P. S. Iroshnikov, Sov. Astron. 7, 566 (1964) --- English
translation of Iroshnikov's original paper
R.
H. Kraichnan, Phys. Fluids 8, 1385 (1965) --- Kraichnan's
original paper
M.
Dobrowolny, A. Mangeney & P. Veltri, Phys. Rev. Lett. 45, 144 (1980) --- IK theory
with imbalanced cascades (more + than - waves)
J.
C. Higdon, Astrophys. J. 285, 109 (1984) --- early
precursor of the GS theory
P.
Goldreich & S. Sridhar, Astrophys.
J. 438, 763 (1995)
--- original GS paper
Y.
Lithwick, P. Goldreich & S. Sridhar, astro-ph/0607243 --- GS
theory with imbalanced cascades
S.
Boldyrev, Phys. Rev. Lett. 96, 115002 (2006) --- a
modification of GS theory that has anisotropy but a -3/2
spectrum
On weak turbulence,
there is a lot of literature one could consult.
The main points of the formal weak turbulence scheme can be learned
from Zakharov, Lvov, Falkovich §§2.1.1-2.1.5 (obviously, to
understand
everything
properly, you need to read the whole book!)
Main papers on weak turbulence of Alfvén waves are (this is the
order
in which the main contributions have appeared):
S.
Sridhar & P. Golreich, Astrophys. J. 432, 612 (1994)
--- 4-wave theory (3-wave interactions argued empty)
D.
Montgomery & W. H. Matthaeus, Astrophys. J. 447,
706 (1995)
--- 3-wave interations defended
C.
S. Ng & A. Bhattacharjee, Astrophys. J. 465, 845
(1996)
--- 3-wave interactions demonstrated
C. S. Ng & A.
Bhattacharjee,
Phys. Plasmas 4, 605 (1997) --- more of the above
P.
Goldreich & S. Sridhar, Astrophys. J. 485, 680
(1997)
--- 3-wave interactions acknowledged and further analysed
S. Galtier et
al.,
J. Plasma Phys. 63, 447 (2000) --- a careful
calculation
A.
Bhattachrjee & C. S. Ng, Astrophys. J. 548, 318
(2001)
--- a numerical study
S.
Galtier et al., Astrophys. J. 564, L49 (2002)
---
a simpler version of their calculation (closest to what I did in class)
Y.
Lithwick & P. Goldreich, Astrophys. J. 582, 1220
(2003)
--- another version of the weak-turbulence calculation (plus imbalance
between
+ and - waves), previous work reexamined
I also have some lecture notes on weak turbulence that I can copy for
you if there is interest.
Lecture 13 (22.11.05) & Lecture
14 (24.11.05) --- MAKE-UP
LECTURE on Friday 24 November @ 11:00-12:00 in
MR14
Reduced MHD.
Decoupling of the 5 cascades.
Reading: This review
(§2)
and references therein.
Here is EXAMPLE
SHEET II (pdf)
Lecture 15 (27.11.05) &
Lecture 16 (29.11.05)
Small-scale dynamo in a linear velocity field.
Saturation of small-scale dynamo. Isotropic MHD turbulence.
Reading: This review
(§3)
and references therein.
There is a lot you can do analytically on the
small-scale dynamo if you consider the velocity field to be a random
Gaussian white noise --- this is called the Kazantsev model.
I can give you some lecture notes on this, but here are also some
extracurricular reading suggestions:
Statistical methods for dealing with multiplicative noise are
described
very thoroghly in van Kampen's book (on your reading list)
The specific method of averaging that I prefer,
as
well as extensions to small but finite correlation times, is 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
of small-scale dynamo is studied in many different ways:
A. P. Kazantsev, Soviet Phys. --- JETP 26, 1031 (1968)
See references and review of subsequent work in 1980s in Chapter 9 of
the
Zeldovich et al. book (on your reading list) --- they do
everything
in x space.
R.
M. Kulsrud & S. W. Anderson, Astrophys. J. 396, 606
(1992)
--- a very thorough study of the spectra of magnetic energy
A. Gruzinov, S. Cowley & R. Sudan, Phys. Rev. Lett. 77,
4342 (1996)
--- calculation of the spectrum in a different way
I.
Rogachevskii
& N. Kleeorin, Phys. Rev. E 56, 417 (1997)
---
the case of Pm<<1
K. Subramanian,
astro-ph/9708216
--- a WKB solution in x space
M. Chertkov
et
al., Phys. Rev. Lett. 83, 4065 (1999) ---
direct generalisation
of the linear-velocity calculation (as in my lectures) to the case of
random
FTLEs, higher moments of B
S. A.
Boldyrev &
A. A. Schekochihin, Phys. Rev. E 62, 545 (2000) ---
a systematic
development in terms of metric tensors
A.
A. Schekochihin, S. A. Boldyrev & R. M. Kulsrud, Astrophys. J. 567,
828 (2002) --- another calculation both in k and x
spaces
D.
Vincenzi, J. Stat. Phys. 106, 1073 (2002) --- a
numerical
solution, a range of Pm (from small to large) modelled
N.
Kleeorin, I. Rogachevskii & D. Sokoloff, Phys. Rev. E 65,
036303 (2002) --- x-space calculation with small but finite
correlation
time
A.
Schekochihin et al., Phys. Rev. E 65, 016305 (2002)
--- calculation of field structure in terms of field-line
curvature
etc.
S.
Nazarenko, R. J. West & O. Zaboronski, Phys. Rev. E 68,
026311 (2003) --- higher moments in k space
R.
J. West et al., Astron. Astrophys. 414, 807 (2004)
---
more
of the above
S.
A. Boldyrev & F. Cattaneo, Phys. Rev. Lett. 92,
144501
(2004) --- the case of Pm<<1 revisited
H. Arponen & P.
Horvai, nlin.CD/0610023 --- analytical extension of the Vincenzi
paper
Here are some recent theoretical papers on the
saturation of
small-scale
dynamo (representing several different views of what happens)
K.
Subramanian, Phys. Rev. Lett. 83, 2957 (1999)
E.
Kim,
Phys. Lett. A 259, 232 (1999)
E.
Kim, Phys. Plasmas 7, 1746 (2000)
S. V.
Nazarenko,
G. E. Falkovich, & Galtier, S., Phys. Rev. E 63,
016408
(2001)
A. A.
Schekochihin
et al., New J. Phys. 4, 84 (2002)
K. Subramanian, Phys. Rev. Lett. 90, 245003 (2003)
A. A. Schekochihin et al., Phys. Rev. Lett. 92,
084504
(2004)
A.
A. Schekochihin et al., Astrophys. J. 612, 276
(2004)
Example Class II: Tuesday 28.11.05 @ 14:30-16:30 in MR4
Once
in a lifetime opportunity: Part III
essay on Gyrokinetics (pdf)
If you are interested in exploring
other Part III essay or future Ph. D. research opportunities in plasma
physics (both astrophysical and fusion-oriented), turbulence, MHD
turbulence, dynamo theory etc., you are welcome to come to talk to me
about that.