Dynamics Days, January 2008 1
Phase-space methods in plasmawave theory
E. R. Tracy, William & MaryA. N. Kaufman, UC Berkeley and LBNL
A. J. Brizard, St. Michael's College
Supported by DOE, NSF
Dynamics Days, January 2008 2
Collaborators/contributors• Dan Cook• Tor Fla• Lazar Friedland• Andre Jaun• Yuri Krasniak• Yi-Ming Liang
• Robert Littlejohn• Steve McDonald• Jim Morehead• Steve Richardson• Yanli Xiao• Huanchun Ye• Nahum Zobin
Dynamics Days, January 2008 3
Outline:• A little history and background• Vector WKB: a good approximation• When good approximations good bad: caustics• When good approximations good bad: modeconversion• Applications• The future
Dynamics Days, January 2008 4
In the beginningthere wasHamilton…
And Hamilton said"Let there berays."
www.hamilton.tcd.ie/events/gt/gt.htm
Dynamics Days, January 2008 5
Themes• Geometrical pictures give insight.• Invariant formulations and symmetries are
important.• Modern mathematics and physics have
much to learn from one another.• General formulations will have wide
applicability.
Dynamics Days, January 2008 6
., 04 =!"#$=!" BE
.ctc
,tc
jE
BB
E!
="
"#$%=
"
"+$%
410
1
.t
0=!"+#
$#j
!
j(x, t) = dt'"(t # t ') d2x$(x,x',t # t')%
#&
+&
% 'E(x',t'), sim. ( E[ ]
Plasma preliminaries…
Maxwell’s eqs.
Model for the matter: e.g. current carrying fluid,kinetic model (Vlasov), etc.
Dynamics Days, January 2008 7
., 04 =!"#$=!" BE
.ctc
,tc
jE
BB
E!
="
"#$%=
"
"+$%
410
1
.t
0=!"+#
$#j
!
j(x, t) = dt'"(t # t ') d2x$(x,x',t # t')%
#&
+&
% 'E(x',t'), sim. ( E[ ]
Plasma preliminaries…
Maxwell’s eqs.
Model for the matter: e.g. current carrying fluid,kinetic model (Vlasov), etc.
In the following, we assume linear responses
Dynamics Days, January 2008 8
!
" #" #E +1
c2
$2E
$t 2+4%
c2
&(t ' t ')($)(x,x',t ' t')
$t*E(x',t') = 0.
Now use the identity:
.'
)','()'()'()'(''
),(2
2
2211
2
2
2
t
txxxxttxddt
t
t
!
!"""=
!
!#
xx $%%%
$
(sim. for gradients) to cast into standard form:
.)'t,'()'tt,',('dt'xd 02
=!"# xExxD
Dynamics Days, January 2008 9
Vector WKB (time-stationary, conservative)
2 ' ' ( , ', ') ( ', ') 0.d x dt t t t! " =# D x x E x
( , , ) ( , ) 0.t
i i t! " # $ =D x E x
( )( , , ) , ;2 2
i
mn mnD d d e D
!"! " "# $ % &' + $( )
* +,
k s s sx k s x x
Dispersion matrix(Weyl symbol)A hermitian matrixfor all real(x,k,ω)
Dynamics Days, January 2008 10
2 ' ' ( , ', ') ( ', ') 0.d x dt t t t! " =# D x x E x
( , , ) ( , ) 0.t
i i t! " # $ =D x E x
The dispersionfunction(Weyl symbol) willusually be smooth inits arguments.
If this is non-local
This will be infinite order(pseudo-differential)
( , , )!D x k
Dynamics Days, January 2008 11
A theme from mathematics: symbols ofoperators (H. Weyl, 1930's)
)',('ˆ xxxx AA =
A Invariants,geometry,harmonicanalysis (i.e.fourier analysison groups…
),( kxA
!
Dynamics Days, January 2008 12
A theme from mathematics: symbols ofoperators (H. Weyl, 1930's)
)',('ˆ xxxx AA =
The symbol of the operator (afunction on phase space)
AAn abstract operator acting insome Hilbert space
A matrix representation of theoperator in some particularbasis
),( kxA
!
Dynamics Days, January 2008 13
A theme from mathematics: symbols ofoperators (H. Weyl, 1930's)
)',('ˆ xxxx AA =
The symbol of the operator (afunction on phase space)
AAn abstract operator acting insome Hilbert space
A matrix representation of theoperator in some particularbasis
),( kxA
!
Dynamics Days, January 2008 14
Symbols of operators
),( kxA
!
A
This mapping is a type of fouriertransform. (N. Zobin)
Ordinary fourier transforms areassociated with the (commutative)translation group in x.
This new fourier transform isassociated with (non-commutative)shifts on phase space, (x,k) (theHeisenberg-Weyl group)
Dynamics Days, January 2008 15
Example of a symbol: the Wigner function
)'()(*'ˆ xxxx !!" =
),( kxW
!!" =ˆ
!"
#$%
&'!
"
#$%
&+=( )
* 22*),()(
''
sx
sxesdW in
spacephaseondensitya
+++ skkxx43421
Dynamics Days, January 2008 16
Symbols of products of operators
!
"AB(x,k) = "
A(x,k)# "
B(x,k)
= "A(x,k)"
B(x,k) $
i
2"A(x,k),"
B(x,k){ } +K
BA ˆˆ
A convolution on the Heisenberg-Weyl group: the 'Moyal product'
Poisson bracket
Moyal - 1930's (physics)Algebra deformations - current mathematics
Dynamics Days, January 2008 17
( , , ) ( , ) 0.t
i i t! " # $ =D x E x
Eikonal ansatz( )( ) ˆ( , , ) ( ) ( ) 0.i t
ti i e E
! "#$ %# & ' ( =) *x
D x x e x
( ) [ ]( ) ˆ( , , ) ( ) ( ) 0.i t
e i E! "
! "#
$ # $ % =x
D x x e x
( ) [ ]( )
!
ˆ( , , ) ( ) ( ) 0.i t
k
symmetrized
e i E! " ! "#
$ %& '( # ( )( + ) =& '* +
x
D x D x e xK14243
Back to vector WKB…
Dynamics Days, January 2008 18
ˆ( , , ) ( ) 0.! "# $ =D x e x
[ ]!
ˆ( , , ) ( ) ( ) 0.k
symmetrized
i E! "# $% &' ( ' )' + ) =% &* +
D x D x e xK14243
Leading order implies
1. must have a null eigenvalue at x.2. The polarization must be the associated null eigenvector.
( , ( ), )! "#D x x
Dynamics Days, January 2008 19
det ( , , ) ( , , ) 0.D! " ! "# $ # =D x x
The null eigenvalue condition implies
“Hamilton-Jacobi”
This condition is a nonlinear PDE forthe unknown θ(x).How to solve it? Hamilton showed theway…
Dynamics Days, January 2008 20
.0),,( =!= "#kxD
Given θ(x), this holds along any curve
!
D(x("),k("),#) = 0.
If we choose the particular curve satisfying Dd
dk
!"=#
x
Then:
!
dk
d"=#
xD
!
x(").
Dynamics Days, January 2008 21
!
dx
d"= #
$D(x,k)
$k,
dk
d"=$D(x,k)
$x.
'd'd
)'(d)'()( !
!
!"!+#$!# %
!x
k
0
0
Define the phase along any ray launched from(x0,k0) on the boundary:
Gives for a family of rays. The naturalplace to view this is in ray phase space.
!
"(x)
Dynamics Days, January 2008 22
Geometrical pictures and asymptotics
"Lagrangemanifold"
Dynamics Days, January 2008 23
Trouble in paradise: causticsMaslov theory (Keller, Heller, Berry, Kaufman, Friedland,Littlejohn, Delos, Weinstein, and many others…)
• In 1-dimension, caustics are just 'turning points'.• (A) Incoming wave: use WKB in x-space.• (B) Change from x- to k-space near x*. There is nosingluarity in the k-representation, so WKB works.• (B') Transform back to x-space: gives an Airy-typefunction• (C) Continue using WKB in x-space• Maslov argued this could be done in arbitrarydimensions.
x*
k A B
B'C
).(~)(x
x iniin
e !"
).(~)(x
x outiout
e !"
)(ˆ)(' kk!"i
e
Dynamics Days, January 2008 24
Scattering from aweakly attractivepotential with a hardcore (Fig. by A.Richardson, based ona paper by Delos)
See Littlejohn, Delos,and others for moreon caustics.
Multidimensional caustics are far more interesting!
Dynamics Days, January 2008 25
[ ]!
ˆ( ) ( ) 0.k
symmetrize
E! "# $% &% & =# $' (
D x e x14243
Back to amplitude transport (away from caustics)…
A little algebra
( )2
,
0, , kD
J J D ED
!
! "=#
##$ = = = %
x k
v v
Wave action flux conservation
Dynamics Days, January 2008 26
The importance of action principles and symmetries(Dewar, Holms, Kaufman, Morrison etc.)
[ ] ).,(),,(),(* tiiDtdAt
nxxxx !!! "#$= %
rr
),(~),( teti
xx !! "
=[ ] .),(~),,(,~'2
tDdAt
nxxx !"""! #$= %
.0),,(0~'
=!"#=t
DA
$$%&
&x
!= 0'
"#
"AAction flux conservation is due to a Noether symmetry!
!"" +# ),(),( tt xx
Dynamics Days, January 2008 27
Further developments: (McDonald/Kaufman)• Waves in cavities (early 'quantum chaos')• Covariant derivation of the wave-kinetic equation forvector fields
).()'()',(' xxxxxdn
jED =!"txk
ktx
!
!
"#=#
==
xk
kx ),,(),,(
!
dnx '" D(x,x ') #E(x')( ) ˜ E *(x' ') = j(x) ˜ E *(x ' ').
Dynamics Days, January 2008 28
Covariant derivation of the wave-kinetic equation(McDonald/Kaufman)
).''(~)()''(
~)'()',('
''''ˆ'
**xxxxxxxd
xxxxxd
n
n
EjEED
EED
=!= "
"
.ˆ EjEED =
( , )* ( , ) ( , ).jEx k x k x k=D W S
Abstract version
Weyl symbol calculus
Dynamics Days, January 2008 29
Covariant derivation of the wave-kineticequation (McDonald/Kaufman)
{ } .),(,2
),(),(),(*),(
)1()1()0(43421
K32144 344 21
O
jE
OO
kxi
kxkxkxkx SWDWDWD =+!=
.0),(),(:)0( =kxkxO WDCan simultaneously diagonalizeusing e-vectors of D.
0),(0),(0),(),(:)0( ==!= kxWorkxDkxWkxDO """"
{ } ).,(,:)1( kxSd
dWWDO
!
!
!!!
"== Wave-kinetic equation (almost)
Dynamics Days, January 2008 30
Mode conversion
The previous WKB derivation assumes thatthere is only one null eigenvalue (this is hidingin the ordering assumptions).
What happens when that assumption breaksdown? Mode conversion!
Dynamics Days, January 2008 31
What is mode conversion?
Plasmas, fluids and other media can support a wide variety oflinear wave types, with different dispersion characteristics andpolarizations.• In a weakly non-uniform medium, the dispersion characteristicsand polarizations are local objects.• For fixed frequency, , near a point x*, wave types ‘a’ and ‘b’(with different group velocities and polarizations) can have nearlyequal wavenumbers.• Mode conversion is due to a local phase resonance.
!
"
Dynamics Days, January 2008 32
Distinguish two cases:
CASE I) The two waves undergoing conversion have differentpolarizations. This can be reduced locally to a 2-componentvector wave problem.
CASE II) The two waves undergoing conversion have the samepolarization. This can be reduced locally to a scalar problem.(“Landau-Zener”, “avoided crossings”.)
Our work focuses on CASE I.
Dynamics Days, January 2008 33
, ( )
, , ,ˆ( , ) ( ) ( )a b
i i t
a b a b a bt e E
! "#=
x
E x x e x
Thought experiment: turn off coupling between the two modes
Dynamics Days, January 2008 34
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Linear mode conversion occurs inmany areas of physics• RF heating in plasmas• Ionospheric physics• Atomic, molecular and nuclear physics(Landau-Zener crossings, spin-orbit resonance)• Geophysics (e.g. equatorial waves)• Neutrino physics (‘MSW effect’)• Black hole theory• Solid mechanics• Magnetohelioseismology
Dynamics Days, January 2008 36
.0)','()',',('' =!"# tttdtd xExxDx
0),(),,( =!"#$ tiit
xExD
[ ])(ˆ)(ˆ),( xexexExk
bbaa
tiiEEet += !" # New ansatz
Congruent reduction procedure identifiesuncoupled polarizations:
Friedland and Kaufman (PRL, Phys. Fl. 1987)
baee ˆ,ˆ
Dynamics Days, January 2008 37
Friedland and Kaufman (PRL, Phys Fl. 1987)
.ˆ),,(ˆˆ *
a
t
aaaiD ekxDe !"#!$ %
.ˆˆ),,(ˆˆ ** t
abb
t
aabDiD =!"#!$ ekxDe %
*
ˆ ˆ
0.ˆ ˆ
aa ab a
tbab bb
D D E
ED D
! "! "=# $# $# $% &% &
Galerkin reductionfrom NXN to 2X2
Dynamics Days, January 2008 38
Friedland and Kaufman (PRL, Phys Fl. 1987)
( )( )
.0)(
)(
* *
*=!!
"
#$$%
&!!"
#$$%
&
'(+)'(
'(+)'(
x
x
RxxV
RxxV
b
a
bxb
axa
E
E
i
i
*
*
Multidimensionalx* is a point on the 'conversion surface'..
( )2| |exp | | /
| |
out
a
in
a
EB
E! " #= $ =
!
B = Va"R
b#V
b"R
a= D
aa,D
bb{ }
Poisson bracket.!!!
Dynamics Days, January 2008 39
Basic insight: Poisson brackets = phase space
!
B = Daa,D
bb{ }
=dD
aa
d"b
=dD
bb
d"a
Conversion occurs 'ray by ray', even thoughWKB is not valid in conversion regions!
Tracy and Kaufman, PRE (1991)Multidimensional conversion, computation ofWigner functions
Dynamics Days, January 2008 40
Using a generalization of the fourier transform (metaplectictransforms), we can replace
( )( )
.0)(
)(
* *
*=!!
"
#$$%
&!!"
#$$%
&
'(+)'(
'(+)'(
x
x
RxxV
RxxV
b
a
bxb
axa
E
E
i
i
*
*
(coupled PDEs with non-constant coefficients), with
.0);(
);(
* 21
21
1
1 =!!"
#$$%
&!!"
#$$%
& '
q
i
b
aq
(
(
)
) a 1st order ODE!Friedland and Kaufman, Phys Lett ATracy and Kaufman, PRE
Solution
Dynamics Days, January 2008 41
( ).)||(
2)(),||2exp()(
2
2/1
2
!!
"#!$!"!#
i%&=%'
Solution gives the WKB connectioncoefficients (Tracy, Kaufman, PRE 1993):
‘transmission’ ‘conversion’
These are applied ray-by-ray.
Dynamics Days, January 2008 42
Applications: for a practical algorithm
• Using ray-based methods for conversion requires:– 1. Detection/verification of conversion– 2. Finding the transmitted rays and local
uncoupled polarizations– 3. Galerkin reduction to local 2X2 form and
the local coupling const.– 4. Local field structure and fitting to
incoming/outgoing fields
Dynamics Days, January 2008 43
Sample applications:
• RF heating in fusion devices: ray tracing algorithms(Jaun, et al.)• Kinetic effects (wave absorption and emission byparticles) (Cook, et al.)• Upwelling in the Gulf of Guinea (Morehead, et al.)• Neutrino physics (MSW effect) (Brizard, et al.)• Magnetohelioseismology (Cally, et al.)
Dynamics Days, January 2008 44
Plasma Phys. Contr.Fusion 49 (2007) 43
Phys. Plasmas 8 (2007)
Dynamics Days, January 2008 45
Dynamics Days, January 2008 46
This is a ‘leaky’ dynamical system, but theleak is in k-space, not x-space.
Dynamics Days, January 2008 47
Why use ray tracing when there are ‘full-wave’ codes?
• Ray tracing provides valuable physical insight. Forexample, easier to see where the energy is flowing andhow small-scale structures develop (e.g. caustics).• Ray tracing should permit a wider range ofparameter studies than full-wave methods (ODEs vs.PDEs).• Complements and helps verify full-wave results.
Dynamics Days, January 2008 48
Dynamics Days, January 2008 49
Resonance crossing, and emission for minority ions ina non-uniform magnetized plasma
Friedland/Kaufman: resonance crossing is a form of mode conversion froma collective wave to a gyroballistic wave
!
D(x,"i#x,i#
t) $E(x, t) =
#t+ v#
x%
% #t+ i&(x)
'
( )
*
+ , Ea(x, t)
Eb(x, t)
'
( )
*
+ , = 0
!
D(x,"i#x,$) %E(x) =
"i#x" k
a&
& $ "'(x)
(
) *
+
, - Ea(x)
Eb(x)
(
) *
+
, - = 0, E(x,t) = e
" i$tE(x)
Dynamics Days, January 2008 50
( ) 0a aD k k k= ! =
!
D(x,k,") =k # k
a$
$ " #%(x)
&
' (
)
* + .
x
k
!
Db(x) =" #$(x)
Dynamics Days, January 2008 51
Gyroballistic waves propagate in k-space
!
D(x,"i#) $E(x) ="i# " k
a%
% & "'(x)
(
) *
+
, - Ea(x)
Eb(x)
(
) *
+
, - = 0!
˙ x = "#D
b
#k= 0,
˙ k =#D
b
#x= "$'(x)
x
k
!
Db(x) =" #$(x)
Dynamics Days, January 2008 52
Gyroballistic waves: what are they?
!
"B
t =0
Dynamics Days, January 2008 53
Gyroballistic waves: what are they?
!
"B
t >0
Dynamics Days, January 2008 54
Gyroballistic waves: what are they?
!
"B
t >0
!
"(x) = sin(kx),
with k growing linearly in t!
!
"
Dynamics Days, January 2008 55
Doppler effects due to ion motion along B means weget a continuum of resonances!
x
k
!
Db(x) =" # k
av #$(x) = 0
Dynamics Days, January 2008 56
1] Interactions via the electric field of Ea leads tophase synchronization!2] The synchronization pattern is a wave(like Smokey Mountain Fireflies?)(see, e.g. Strogatz’s work using Vlasov models,“From Kuramoto to Crawford…”, Physica D 2000)
(1993)Cooke et al.
Dynamics Days, January 2008 57
(1996)
WKB+Mode conversion+Case-van Kampen+Bateman-Kruskal+Crawford-Hislop=Analytic theory for excitation of the Bernstein wave
Dynamics Days, January 2008 58
Littlejohn and Flynn showed that, in multidimensions,we must keep the linear terms in the off-diagonal, too!
( )( )
*
*
. . . ( )0.
* . . .* ( )
a x a a
b x b b
i h o t E
h o t i E
!
!
" #$ + " # +% &% &=' (' (
+ " #$ + " # ) *) *
V x x R x
V x x R x
Multidimensional normal form
See also Colin de Verdiere; Braam and Duistermaat;Emmrich and Weinstein
Dynamics Days, January 2008 59
!
D(z') =q1
" + q2
+ i# p2
"*+q2$ i# p
2p1
%
& '
(
) * + h.o.t.
We can perform a combination of congruenceand canonical transformations to find thenormal form (Tracy and Kaufman, PRL):
!
"# =
Notice that the diagonals Poisson-commute with theoff-diagonals. Conjecture: we can take this as thedefinition of the normal form and extend to higherorder.
Dynamics Days, January 2008 60
The related 2X2 wave equation
!
ˆ q 1
" + ˆ q 2
+ i# ˆ p 2
"*+ ˆ q 2$ i# ˆ p
2ˆ p
1
%
& '
(
) * +
1
+2
%
& '
(
) * = 0
can be solved by separation of variables and ageneralization of the Fourier transformation.
Dynamics Days, January 2008 61
Conversion in the Gulf of Guinea (JFM, 1999)
• Shallow water theory• Non-uniform thermocline• Seasonal upwelling
Dynamics Days, January 2008 62
Neutrinos (MSW effect)
R. Bingham, R.A. Cairns, J.M.Dawson, et al., Phys. Lett. A232 (1997)257.
Brizard, unpublished
Dynamics Days, January 2008 63
Magnetohelioseismology (Cally)
Dynamics Days, January 2008 64
Work in progress:• Effects of higher order terms (Richardson poster)• Application of the ray-based algorithm to realistic tokamakscenarios (Jaun talk)• Direct comparisons of full-wave and ray-based approaches toconversion (Xiao, Richardson)• Path integral methods for non-standard conversions? (Zobin,Richardson PhD thesis, 2007)• Constructing 2X2 normal form following a ray, rather thanextending order by order in Taylor expansion: ‘semi-holonomic’dynamics• Nonlinear effects (auotresonance, Friedland)