phase-space methods in plasma wave theorysymbols of operators!a(x,k) aö this mapping is a type of...

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


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


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


In the beginningthere wasHamilton…

And Hamilton said"Let there berays."

www.hamilton.tcd.ie/events/gt/gt.htm


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.


., 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.


., 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


!

" #" #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


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)


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


A theme from mathematics: symbols ofoperators (H. Weyl, 1930's)

)',('ˆ xxxx AA =

A Invariants,geometry,harmonicanalysis (i.e.fourier analysison groups…

),( kxA

!



)',('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

!


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)


Example of a symbol: the Wigner function

)'()(*'ˆ xxxx !!" =

),( kxW

!!" =ˆ

!"

#$%

&'!

"

#$%

&+=( )

* 22*),()(

''

sx

sxesdW in

spacephaseondensitya

+++ skkxx43421


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


( , , ) ( , ) 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…


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


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…


.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(").


!

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)


Geometrical pictures and asymptotics

"Lagrangemanifold"


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


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!


[ ]!

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


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


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 ' ').


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


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)


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!


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.

!

"


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.


, ( )

, , ,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


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


.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


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


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.!!!


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


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 =!!"

#$$%

&!!"

#$$%

& '

qq

qq

q

i

b

aq

(

(

)

) a 1st order ODE!Friedland and Kaufman, Phys Lett ATracy and Kaufman, PRE

Solution


( ).)||(

2)(),||2exp()(

2

2/1

2

!!

"#!$!"!#

i%&=%'

Solution gives the WKB connectioncoefficients (Tracy, Kaufman, PRE 1993):

‘transmission’ ‘conversion’

These are applied ray-by-ray.


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


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.)


Plasma Phys. Contr.Fusion 49 (2007) 43

Phys. Plasmas 8 (2007)


This is a ‘leaky’ dynamical system, but theleak is in k-space, not x-space.


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.


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)


( ) 0a aD k k k= ! =

!

D(x,k,") =k # k

a$

$ " #%(x)

&

' (

)

* + .

x

k

!

Db(x) =" #$(x)


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)


Gyroballistic waves: what are they?

!

"B

t =0



!

"B

t >0



!

"B

t >0

!

"(x) = sin(kx),

with k growing linearly in t!

!

"


Doppler effects due to ion motion along B means weget a continuum of resonances!

x

k

!

Db(x) =" # k

av #$(x) = 0


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.


(1996)

WKB+Mode conversion+Case-van Kampen+Bateman-Kruskal+Crawford-Hislop=Analytic theory for excitation of the Bernstein wave


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


!

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.


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.


Conversion in the Gulf of Guinea (JFM, 1999)

• Shallow water theory• Non-uniform thermocline• Seasonal upwelling


Neutrinos (MSW effect)

R. Bingham, R.A. Cairns, J.M.Dawson, et al., Phys. Lett. A232 (1997)257.

Brizard, unpublished


Magnetohelioseismology (Cally)


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)

phase-space methods in plasma wave theorysymbols of operators!a(x,k) aö this mapping is a type of...

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