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 (7.10.05)
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.
Here
is a recent talk for the general
public which
contains some pretty pictures of magnetic fields, turbulence and
magnetic
turbulence.
Lecture 2 (10.10.05)
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
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:
A.
M. Yaglom, Ann. Rev. Fluid Mech. 26, 1 (1994) on A.
N.
Kolmogorov and the founding of the Russian school of turbulence.
H.
K.
Moffatt,Ann. Rev. Fluid Mech. 34, 19 (2002) on G. K.
Batchelor
and the Cambridge school of turbulence.
Lecture 3 (12.10.05)
Particle diffusion in turbulence: exponential separation, Richardson
law,
turbulent diffusion.
Correlation functions.
Reading: Davidson-Turbulence §6.2.1
Batchelor,
Chapters
II-III
Monin &
Yaglom
§24 (particle diffusion), Chapter 6 --- the definitive account of
correlation
functions
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).
If you were intrigued by the example I gave you at the end of
the lecture
(the calculation of magnetic-field spectrum from Faraday Rotation in
clusters),
here are three original papers where the method is explained and the
data
analysis performed:
T. A.
Enßlin
& C. Vogt, Astron. Astrophys. 401, 835 (2003)
C. Vogt
&
T. A. Enßlin, Astron.
Astrophys.
412, 373 (2003)
C.
Vogt
& T. A. Enßlin, Astron.
Astrophys.
434, 67 (2005)
Lecture 4 (14.10.05)
Correlation functions continued: spectra.
Infrared scaling (long-range correlations).
Kolmogorov's 4/5 law: start of the derivation.
Reading: Davidson-Turbulence §8.1 (k space corr.
functions)
See reading
suggestions
for the next lecture
NB: There will be a seminar at CMS
on
21 October, where Peter
Davidson
will talk about long-range correlations and decaying turbulence.
Lecture 5 (17.10.05)
The closure problem.
The 3d-order correlation function.
von Karman-Howarth equation.
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
McComb (on
closures)
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)].
Example Sheet I (pdf) --- Note that
Problems
2-4 take you step-by-step through the main results of the theory of
scalar
turbulence (passive scalar theory). Doing these problems is a good way
to
check if you understand Kolmogorov-style reasoning. Regardless of
whether
you planning to take the exam in the course, I urge you all to attend
Examples
Class I, when I will explain these results in detail.
NB: There will be a seminar
today in MR14 @16:00, where François Rincon
will talk about the 4/5 law in turbulent convection.
Lecture 6 (19.10.05)
Intermittency: intro, the refined similarity hypothesis.
Reading: Frisch Chapter 8; see §6.4 for detailed discussion
of
Landau's objection
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].
Lecture 7 (21.10.05)
Intermittency: the lognormal model, She-Lévêque theory.
Reading: see previous lecture.
Here are some key recent papers on the
She-Lévêque
model of intermittency (not terribly clear except, perhaps, the last
one):
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)
Lecture 8 (24.10.05)
Intermittency: summary and discussion of the She-Lévêque
theory.
Extended self-similarity.
Discovery of extended self-similarity: R. Benzi et
al.,
Phys. Rev. E 48, R29 (1993)
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!
No lectures on 26.10.05 and 28.10.05 (to be
rescheduled).
Next lecture on 31.10.05.
Lecture 9 (31.10.05)
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 10 (2.11.05)
Lagrangian MHD. Cauchy solution of the induction equation. Action
principle.
Conservation laws: mass, momentum, energy.
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)
Reading: Sturrock §§16.1-16.4 (action principle)
Kulsrud
§§4.8
(Cauchy solution), 4.3-4.5 (conservation laws), 4.7 (action principle)
Zeldovich et
al.
§9.1 (dynamo)
Goedbloed &
Poedts
§§4.3 (conservation laws), 4.4 (same with dissipative terms)
Lecture 11 (4.11.05)
Conservation laws: energy (completed), helicity, cross-helicity.
MHD Equilibrium. Force-free fields.
Reading: Davidson-MHD §4.4 (helicity)
Sturrock
§13.8
(helicity), 13.1-13.7,13.10 (force-free fields)
Kulsrud
§§4.9
(cylindrical equilibria)
If you wish to read something about the energy principle, 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)
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)
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)
If you would like some notes on these things, ask me and I will
give
you a copy of last year's 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.
Example Sheet II (pdf) ---
examples on MHD
Lecture 12 (7.11.05)
Force-free fields completed: Woltjer theorem
MHD waves
Reading: Sturrock §§13.9 (Woltjer theorem), 14.1
Kulsrud
§§5.1-5.4
Goedbloed &
Poedts
§§5.1-5.2
Davidson-MHD
§6.1
Lecture 13 (8.11.05 @ 12:00-13:00
in MR5 --- make-up lecture, note time and venue!)
MHD waves completed.
Finite-amplitude Alfvén waves. Elsässer variables.
Example Class I: 8.11.05@ 16:00-17:30 in MR4
Lecture 14 (9.11.05)
Alfvénic (anisotropic MHD) turbulence: an overview of
theoretical
uncertainties.
Reading: You may find this review
(§§1-2)
and references therein useful.
Lecture 15 (11.11.05)
Reduced MHD.
Decoupling of the Alfvén-wave cascade.
Lecture 16 (14.11.05)
Decoupling of the Alfvén-wave cascade: completed.
NB: 15.11.05 (today!) @ 13:00 in
MR14
Astrophysics
Lunchtime Seminar given by Eugene Parker, one of the creators of
Astrophysical
Fluid Dynamics
Example Class II: 15.11.05 @ 14:30-16:00 in MR15
(Note change of time!)
Lecture 17 (16.11.05)
Weak turbulence of Alfvén waves.
Reading: Zakharov, Lvov, Falkovich §§2.1.1-2.1.5 (the
main
points of the weak turbulence scheme; obviously, to understand
everything
properly, you need to read the whole book!)
Papers on weak turbulence of Alfvén waves (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
PARTY! --- For students
interested
in research opportunities with the Astrophysical Fluid Dynamics
Group/DAMTP
Come meet members of our research group informally 17 November @ 6pm
in
N2 Great Court/Trinity College (Prof. Proctor's rooms)
Lecture 18 (18.11.05)
Weak turbulence of Alfvén waves.
Discussion of difficulties and unresolved issues in the
Alfvén-wave
turbulence.
Lecture 19 (21.11.05)
Weak turbulence completed: Zakharov transformations.
Small-scale dynamo in a linear velocity field.
Reading: My
review
(§3.1) and references therein.
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, 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), where you will
also find
further references.
See also the Childress & Gilbert book.
NB: 22.11.05 (Tuesday) @ 13:00 in
MR14
--- I am giving an Astrophysics
Lunchtime Seminar on the Alfvén and slow-wave cascades in
MHD and
kinetic theory of astrophysical plasma turbulence. Roughly the first
half
of the seminar will be similar to my Lecture 15, the rest will be an
overview
and discussion of the kinetic generalisation of this theory. So both
those
wanting a recap and those wanting to know more are welcome to attend.
Lecture 20 (23.11.05)
Small-scale dynamo in a linear velocity field.
Lecture 21 (25.11.05)
Small-scale dynamo in a linear velocity field: discussion.
The Kazantsev--Kraichnan Model.
Lecture 22 (28.11.05)
The 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 specific 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
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
A. Gruzinov, S. Cowley & R. Sudan, Phys. Rev. Lett. 77,
4342 (1996)
--- calculation of spectrum similar to the one I did in class
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
Lecture 23 --- make-up lecture.
The magnetic-field spectrum in the Kazantsev model.
Lecture 24 (30.11.05)
Small-scale dynamo: saturation.
Isotropic MHD turbulence.
I will (hopefully) also have some time left to answer any questions
that
you care to ask.
Reading: My
review
(§3.2) and references therein.
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 Sheet III (pdf) ---
examples on MHD turbulence etc.
Example Class III will be
held
some time in the Lent Term.
In the meanwhile, you are welcome to come and see me if you have any difficulties/questions/thoughts/suggestions
and also if you want to discuss possible Ph. D. projects on MHD
turbulence,
plasma astrophysics, dynamo, and on the physics of galaxy clusters (the
DAMTP application deadline is 23 December). I am in town till 26
December
and then from 23 January onwards.
HAPPY NEW YEAR! --- I am sure it'll
mark the beginning of a brilliant
career in research or whatever else you decide to do with
yourself.