Alfvén waves: A journey between space and fusion …phoenix.ps.uci.edu/scidac09/viewgraphs/Chen_Alfven_Lecture_EPS08.… · Hersonissos 06/09/2008 Alfvén waves: a journey between

Hersonissos 06/09/2008 Alfvén waves: a journey between space and fusionplasmas

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35th EPS PPC 2008 Liu Chen

Alfvén waves:A journey between space and

fusion plasmas*

35.th EPS Plasma Physics Conference,Hersonissos, Crete, GR, June 9-13, 2008

Liu Chen†

Dept. of Physics and Astronomy, Univ. of California, Irvine CA

† Also at Inst. for Fusion Theory and Simulation, Zhejiang Univ. Hangzhou PRC* Supported by U.S. D.o.E. and N.S.F


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Outlines(I) Introduction(II) Geomagnetic pulsations, continuous spectrum

and kinetic Alfvén waves(III) Shear Alfvén waves in Tokamaks(IV) Kinetic Excitations of shear Alfvén waves(V) Nonlinear physics of shear Alfvén waves(VI) Concluding remarks and Acknowledgment


3


(I) Introductiono Hannes Alfvén [1942] discovered e&m waves can propagate in a

conducting fluid (plasma) in the presence of finite B0

⇒ Hydromagnetic Alfvén waves ⇒ prevalent in nature & laboratoryplasmas; e.g.

o Alfvén instabilities excited by energetic particles in fusion plasmaso Geomagnetic oscillations in solar-wind disturbed magnetosphere

o Significance of Alfvén waves:finite δE & δB ⇒ energy and momentum exchange with chargedparticles

⇒ Acceleration/heating of charged particlesSolar corona heating [Science, Dec. 7, 2007]

⇒ Possible rapid loss of energetic/alpha particles in burning fusionplasmas (ITER)


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Ideal MHD Plasma

k=k⊥+k|| ς

Compressional Alfvén Wave(CAW)

ω≈kVA

IsotropicCompressible in B, nDifficult to excite

Shear Alfvén Wave (SAW)

ω≈k||VA• Anisotropic• Vg ≈VA e||⇒ Wave energypropagates only along B0• Incompressible• Much easier to excite

⇒ SAW

Focus of this talk!

ς , B0

x

y

k

k⊥


5


(II) Geomagnetic pulsations, continuousspectrum and kinetic Alfvén waves

(i) Geomagnetic pulsations:o Oscillations in Earth’s B, periods - O(10-103) seconds, observed

both at ground and satellites.o Hydromagnetic Alfvén waves [Dungey]o Ground magnetometer measurements [Samson; Lanzerotti;

~1970]⇒ Puzzling features in wave intensity and polarization.

⇒ Theoretical explanation?

N

E

H

D

θ


6


[Samsong etal, 1971][Lanzerotti etal, 1972]


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(ii) Conceptual Breakthrough⇒ Continuous spectrum,

phase mixing,collisionless dissipation

Echoes, shocks and phase mixingDissipation appears in a time-reversibletheory in the guise of phase mixing;analytically, it is recognized as acontinuous spectrum. The basic point isthat any finite or infinite discrete sum, Σ anexp (iωnt), oscillates indefinitely; an integral,∫a(ω) exp (iωt) dω, can, however, decay.The most important qualitative feature of acontinuous spectrum is that it preservesmuch of the information fed into the systemby initial and boundary data and gives riseto much more complex phenomena than adiscrete normal mode, which is primarily aproperty of the medium.

Harold GradPhys. Today (Dec. 1969)

Phase mixing with collisionlessdamping is not restricted tokinetic models but also foundin macroscopic theory odAlfvén waves and in coldplasma and magnetoionictheory.


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(iii) Continuous spectrum (radial nonuniformity)o Slab model

o Ideal MHD equation, cold plasmao δξ⊥ = δζ⊥(x) exp [i (k|| z + ky y - ωt)]⇒

o DA(x) = ω2/VA2(x)- k||

2 : SAW dispersion function⇒

o Uniform plasma ⇒ dDA(x)/dx = 0⇒ DA(x) δζx = 0 , SAW ; (d2/dx2 - ky

2 + DA) δζx = 0 , CAW⇒ Nonuniform plasma ⇒ dDA(x)/dx ≠ 0

⇒ SAW and CAW coupled !!

LL

B

00 x

y

z

0

B0 = B0 z

n0 = n0(x) , Te,i = 0

VA(x) = B0/[4πn0(x)mi]1/2


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o DA(x) = 0 ⇒ ω2 = k||2 VA

2(x) ≡ ωA2 (x) ⇒ singular solutions

ωA2 (x) : continuous spectrum, SAW continuum

o Singular structures : ω = ω0 CAW external driving frequency ω0

2 = ωA2 (xr) ⇒ DA(xr) = 0 ; xr : Alfvén resonant layer

⇒

⇒

o Including dissipations ⇒ explains the observed features⇒ field-line-resonance theory [C&H ; Southwood ; 1974]

o Existence of SAW continuum in dipole B [with Cowley]

!

d

dx" D A(x

r)(x # x

r)[ ]

d

dx$%

x& 0

!

"# x $ C / % D A (xr )[ ]ln(x & xr )

'( )"#( $ 0*"# y $ i C /ky% D A (xr )[ ]/(x & xr)

C $ ky

2"B|| /B0


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o Alfvén resonant energy absorption: energy piling up at x=xr

o Collisionless absorption ⇒

o Time-asymptotic responses: (perturbations due to solar-wind pulses)o As t → ∞ , |∂x |2 >> |∂y|2 >>|DA | , (ω ⇒ i ∂/∂t) Broadband source

o SAW equation

⇒

⇒ SAW continuum

⇒o Earth’s magnetosphere phase mixing

δζx ⇔ δBR (radial) ; δζy ⇔ δBE (East-West)o AMPTEE/CCE satellite observed SAW continuum of δBE at the dayside

magnetosphereo |∂x | ≡ |kx| ⇒ ω’At ⇒ “singular” structure time asymptotically

!

"

"x

" 2

"t 2+#

A

2(x)

$

% &

'

( ) "

"x*+

x(x, t) = 0

!

dW

dt="0

8

#

$ %

&

' ( LyLz ky)B||

2

* D A (xr )

!

"# y (x, t) = i C /ky( )exp ±i$A (x)t[ ]

!

"

"x

#

$ %

&

' ( )* x (x,t) + ,iky)* y (x,t)

!

"#x(x, t) = mi

C

$ % A(x)

&

' (

)

* +

1

t

&

' ( )

* + exp ±i%

A(x)t[ ]


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AMPTE/CCE [Engebretson et al, 1987]

δBE

δBR

δB||


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(iv) Kinetic Alfvén Wave (KAW)o Singular structures ⇒significance of ⊥ B microscopic effects

⇒ finite ρi , ρs = cs/Ωi

o Kinetic SAW wave equation

o εAk = k||2ρk

2 ∇⊥2 + DA(x) , DA(x) = ω2/VA

2(x)- k||2 , ρk

2 = ρi2 (3/4+Te/Ti)

o εAk = 0 ⇒ Kinetic Alfvén wave dispersion relation⇒ ω2= ωA

2(x)[1+ k⊥2 ρk2] ⇒ finite Vg⊥ ⇒ singularity removed

o Mode conversion at x ≅ xr

o

⇒

o |δE||/δE⊥| ∼|k||k⊥ ρs2

| ⇒ acceleration, transporto Paradigm for Geodesic Acoustic Mode and mode conversion to

Kinetic GAM [with Zonca 2008] ⇔ GAM: continuous spectrum !!

!

"# $%Ak"#&' x= 0

!

d

dx"

k

2 d2

dx2

# D A(x

r)(x $ x

r)

%

& '

(

) *

d

dx+,

x- 0

ω2> ωA2 : ideal MHD + KAW (outward propagating)

ω2< ωA2 : ideal MHD


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[1976]

MHD

KineticAlfvénWave


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Tokamak Magnetic Field

o r = constant magnetic surface o Pitch of B

!

rB"

RB# (r)$ q(r)

!

dq(r)

dr" 0

0

2π

0 2π

θ

φ

B

q

r

B = Bφ φ + Bθ θ

|B|~1/R0(1+r/R0 cos θ)


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(III) Shear Alfvén waves in tokamakso Effects of nonuniformities.

o Radial nonuniformities ⇒ SAW continuumo Parallel periodic “lattice” ⇒ gaps in SAW continuumo Localized “defects” in “lattice” symmetry ⇒

⇒ discrete Alfvén Eigenmode (AE) inside the gapo Ballooning-mode formalism [~1980] ⇒ theoretical studies feasible!o SAW equation with large toroidal mode number

o ;

o

periodic “lattice” “defect” at |η|∼Ο(1)

o ; magnetic shear

with Cheng,Chance, 1985

(SAW vorticityEqn.)

!

d2d"2 +V (")[ ]#(") = 0

!

"# <$ <#

!

V (") =#21+ 2$0 cos"( ) % ˆ s f( )

2

!

f (") =1+ ˆ s 2"2

!

ˆ s = r " q /q


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o Two scale-length and asymptotic matching analysisoo Asymptotic matching ⇒ (B/A)outer = (B/A)inner

⇒

o δTf>0 ⇒ Ω2L < Ω2

TAE < Ω2U ⇒ TAE [1985]

o Variations in MHD equilibrium ⇒ variations in δTf (“defects”)⇒ Variations in TAE! ⇒ Generalized to toroidal symmetry

breaking (Stellarator)

!

i"T (#) $#2 %#L

2( )#U

2 %#2( )

&

' ( (

)

* + +

1/ 2

=,s

4$ -Tf!

"(#) = A(#)cos(# /2) + B(#)sin(# /2)

[Exp verification.~1991 Wong etalHeidbrink etal]!

"U ,L

2 =1

41± #

0( )

ω2

rm-2 rm+2rm+1rmrm-1

ω2Am-1

ω2Am-1

ωU

ωL

!

VA

2qR

"

# $

%

& '

2

!

"TAE

~VA

2qR

TAE


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o In magnetosphere plasmaso Geomagnetic “storms” ⇒ β ~ O(10-1 - 1), T⊥ > T||

o Geomagnetic pulsations with large azimuthal wave number,

o |m|>>1 ⇒ Alfvén - ballooning mode (ABM) [Chan, 1994]o Localized Pstorm,ion ⇒ “defects” ⇐ Ballooning-mode formalism

⇒ Discrete ABM Eigenmodes [Vetoulis, 1996]

~

|m|~O(10-102)>>1

⇒

|δB/B|~O(5-10%)δBR

δBE

δB||

e

i


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(IV) Kinetic excitation of SAWo Stimulation: “Fishbone” instability in PDX [McGuire; 1983]o Insights from PDX experiment:

o ω~ωdE : toroidal precession frequency of the beam ions⇒ mode freq not determined by background MHD normal mode !

“Beam” mode in a beam-plasma system;⇒ beam ion dynamics non-perturbative⇒ equal footing for background MHD and energetic particles

o ω2 = k||2(r) VA

2(r) ⇒ SAW continuum damping ⇒ Beam power threshold for “fishbone” (PDX observations)

o Advances in gyrokinetic theories [~1980] ⇒ theory feasible !

⇒ Fishbone theory! [with White, Rosenbluth; 1984]

_

Analogy


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Fishbone modes on PDX

McGuire et al PRL 50, 891 (1983)Mode Particle Pumping

White et al [1983]

m/n2/1

1/1

q=1 q=2


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Fishbone instability in PDX [McGuire et al, 1983]


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o High-n SAW Instabilitieso Wave eqn including kinetic excitation: generalized SAW vorticity eqn

!

d2d"2 +V

k(")[ ]#(") = 0

!

"# <$ <#

!

Vf (") =#21+ 2$

0cos"( ) %

& cos"

p(")%(s%& cos")2

p(")2'

( )

*

+ ,

2

!

(s," = #q2R $ % ) modelequilibrium

!

V (") =Vf (") +V#c (")

!

"2

=#(# $#*pi) #A

2

!

p(") =1+ s" #$ sin"( )2

(“defects”)

!

V"c (#)$(#) = %4&'q2R2

k(2c2p(#)1/ 2

)

* +

,

- . e'd/G E

B curvature couplingto plasma compression

⇒

!

"d

= k# $VdB

!

V||

qR

"

"#$ i(% $%d )

&

' (

)

* + ,GE = i

e

m

&

' (

)

* + E

%"

"-+

) % *

&

' (

)

* + F0E

1

p(#)

&

' (

)

* +

1/ 2

%d

%

&

' (

)

* + E

.(#)

• k⊥2ρi2 , β <<1 ⇒ δE|| , µ∂||δB|| neglected

wave-particle resonance driveVdbXδB⊥


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o 2-scale-length asymptotic matching ⇒

o Circulating energetic ions; e.g.,

• General wave-particle resonancesω - k||v|| - pωt = 0 , p = 0, ±1, … ; ωt = v||/qR

• TAE ⇒ k|| ~ 1/2qR , p=0 ⇒ ω = ωt/2 p=1 ⇒ ω = 3ωt/2

!

"Tf = s# 4( ) 1$% s2( )

!

i"T (#) $#2 %#L

2( )#U

2 %#2( )

&

' ( (

)

* + +

1/ 2

= ,Tf + ,Tk (-)ω~VA/2qR: Toroidal

Alfvén freq. Gap.

!

" d = (k#2/4)$L

2(1+ q

22)

!

"TkU #$ 2

4s

e2

mc2

%

& '

(

) * E

q2R2

k+2

) , d

2 ) , *F0

- d (1+ - d )3 / 2

,

, t

24 ., 2

+,

9, t

24 ., 2

/

0 1

2

3 4

E

wave-particle resonances

⇐

δTk : energetic ion contribution in the |η|~O(1) ideal region

ω~VA/2qR


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o TAE dispersion relation with energetic particles

⇒ Two types of discrete unstable eigenmodes(i) “Gap” Alfvén Eigenmode (AE)

• δTf + Re(δTk) > 0 ⇒ Re(ΛΤ2) < 0 ⇒ ΩL

2 < Ω2 < ΩU2 ⇒ TAE

• Im(δTk) > thermal-particle damping in ΛΤ2

(ii) “Continuum” Energetic Particle Mode (EPM)• Re(ΛΤ

2) > 0 ⇒ Ω2 inside the continuum: {Ω2 < ΩL2 , Ω2 > ΩU

2}• Im(δTk) > Re(ΛΤ) ⇒ instability drive > continuum damping

• Near marginal stability⇒ Re[δTk(ω)] ~ - δTf : δTf independent of ω⇒ ω tracks characteristic energetic particle frequency in δTk(ω)⇒ ω ∝ (ωdE , ωbE , ωtE) ⇒ EPM !!

!

i"T (#) = $Tf + $Tk (%)

_


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o Low-frequency SAW instabilityo |ω| ~ thermal ion freq. ⇒ thermal ion pressure compressiono Generalized fishbone dispersion relation:⇒ [Zonca et al.]

• ΛL(ω) : inertial “singular” layer ⇐• δWf : ideal MHD δW• δWk(ω) : energetic-particle compression in ideal “regular”

regiono Kinetic thermal-ion “gap” AEs : BAE/KBM/AITG, …o Energetic particle “continuum” mode (EPM) ⇒

“fishbone” first EPM !!o AEs and EPMs observed in tokamak experiments

!

i"L (#) = $Wf + $Wk (%)

thermal ioncompression +w-p resonances


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Continuum of EPM to TAE-like modesFr

eque

ncy

(kH

z)NSTX δB

Observations[Fredrickson etal 2006]

• On left, bursting, chirping EPM-like modes.• Evolutions to nearly coherent, TAE-like modes on right.• Large TAE/BAE gaps, large rotation shear results in mixed frequency ranges.


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Nazikian et al [2006]

o n=40 ⇒ kθρit~O(1) ⇒ SAW instability by thermal ions !!


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o Kinetic excitation of geomagnetic pulsationso Gyrokinetic theory ⇒ magnetospheric space plasmas (1991)o Alfvén-ballooning mode (ABM) via

ω − ωd = pωb , (p=−1) ⇒ drift-bounce resonance

⇒ ω = ωd − ωb ⇒ NSTX experiments [Fredrickson, 2006]

o β ~ O(1), T⊥ > T|| , magnetic trapped particles⇒ drift-mirror mode [Hasegawa]⇒ trapped-particle compressional mode

[Rosenbluth ; Crabtree et al]o Interesting issues:

Realistic B ⇒ finite κc = b⋅∇b curvature⇒ Mirror-compressional mode Coupled !

Alfvén-ballooning mode No comprehensive theory !

_

_

Significant dB|| !


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(V) Nonlinear physics of Alfvén waveso Broad scope of research activities –

o Nonlinear charged-particle dynamics ⇒ saturation of TAE [Berk,Breizman]; transport [Sigmar et al]; heating [White]

o Nonlinear mode-mode coulings ⇒ zonal flow/current [Chen et al;Diamond et al]; Nonlinear ion Landau damping of TAE [Hahm];nonlinear (n=0, m=±1) equilibria [Zonca et al]⇒ Limited discussions here !!

o Pure Alfvénic State (B0 = const)(i) Ideal MHD : δE + δuXB/c = 0(ii) Incompressibility : ∇⋅δu = 0(iii) Walén relation: δu/VA0 = ± δB/B0

⇒ Break (i), (ii) & (iii) ⇒ effective nonlinear SAW dynamics !

**

**

o SAW : self-consistent nonlinear stateo ω2 = k||

2VA2


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o Ponderomotive forces and Alfvénic Stateo ⇒ SAW vorticity equation

⇒

o β <<1 ⇒ δu ~ δu⊥ ⇒ Walén relation ⇒ <Fp⊥>surface-averaging

⇒ 0 ⇒ KAW and EPM (more than TAE) ⇒ Zonal flow/current

o Finite parallel plasma compressibility and ponderomotive forceo At slow-sound freq. ⇒ finite compressibility along B0

⇒ SAW ⇒ δns (ωs,ks)⇒ δns ⇒ modulate Jpol of SAW

!

Fp = "J#"B /c $ %m "u & '( )"u

!

" #Fp =" #1

4$%B & "( )%B ' (m %u & "( )%u

)

* + ,

- . / 00

Walénrelation

!

Fp || = "1

8#n0

b $ % &B'( )2[ ]


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o |ωs| ~ |ks|||vti ⇒ nonlinear ion Landau damping⇒ Multiple - TAE freq. Cascading ↓

o Single TAE ⇒ ωs ~ 0 , ks = (ns=0 , ms=±1) ⇒ δns : TAE freq. gapnarrowing ⇒ saturation of TAE

o Interesting issues:Fp|| ⇒ δns , δφs ⇒ finite thermal ion compressibility⇒ ? Zonal flows ?⇒ ? Cross-gap coupling to AEs in Kinetic-thermal-ion gap?

o Perpendicular compressibility [Zonca et al ‘95]o In tokamak : ∇⋅δu⊥ ~ - 2κc⋅δu⊥ κc ⇒ B curvature

⇒ (ns=0 , ms=±1) ⇒ nonlinear magnetic surface change⇒ TAE freq. gap narrowing ⇒ saturation of TAE

!

"

"t#B$

(2)= %#B$

(1)& '#u$

(1)[ ] = 2#B$

(1)kc' #u$

(1)[ ]

(e.g. BAE)

single-n

TAE

(n1,ω1)(n2,ω2)(n3,ω3)


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o KAW enhancement in Fp||

o Ideal MHD : |δJ⊥ |~ |δJpol | = n0e |ω/Ωi | | δuE|o KAW : |δJ⊥ | ~ O(k⊥2ρi

2) n0e | δuE|⇒ [Fp||]KAW/ [Fp||]MHD ~ O(k⊥2ρi

2 Ωi /ω) >> 1 !! [1976] [1982]⇒ Important to keep microscopic thermal ion physics !

o Interesting issues relevant to burning plasmas• Coupled EP and multiple-AE dynamics• EPM nonlinear physics• Long-time-scale coupling between meso-scale SAW physics &

microscale thermal-particle turbulences; etc. [2006, 2007]o Magnetospheric plasmas

o B ⇒ κc ⇒ mirror-compressional mode & Alfvén-ballooning modeo Nonlinear studies ⇒ Sparse and simple decoupled limits !o Realistic (gyrokinetic) simulation & theoretical analysis⇒ Advance understandings !


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(VI) Concluding remarks andAcknowledgment

o SAW in plasmas confined by realistic B ⇒ rich, interesting physics!o Radial nonuniformities ⇒ continuum, “singular” structures, KAW

⇒ Cross radial-scale-length coupling (macro/meso → micro)⇒ Acceleration, heating, ⊥ B transport

o Parallel nonuniformities ⇒ “lattice” symmetry ⇒ “gaps” in continuum.Equilibrium “defects” ⇒ AEs in the “gap”

o Kinetic excitations:• W-P resonance via finite-κc coupling to pressure compressions• Non-perturbative energetic-particle (EP) effects ⇒ (MHD & EP

equal footing!)⇒ Generalized fishbone dispersion relation⇒ “Gap” AE and “continuum” EPM (e.g., “fishbone”)


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o Nonlinear Physicso Wave-particle resonance ⇒ nonlinear phase-space dynamics of

charged particles ⇒ trapping, stochastic motion; etc. Nonlinear mode-coupling processes

⇒ Deviations from the pure Alfvénic state⇒ Finite δE||, finite plasma compressibility, non-perturbative EP⇒ Enhanced nonlinear effects due to µscopic structures

o Positive, fruitful feedbacks & cross-fertilization between space ⇔fusion research. Also,

Exps/Observations ⇔ Theory ⇔ Simulationo Exciting, challenging issues remain ⇒ nonlinear physics

• Realistic conditions (e.g. nonuniformities, B curvature, etc.) ⇒ play crucial dynamic roles

• Simulations ↔ Theory ↔ Exps/Observations ⇒ Advances inAlfvén wave physics ⇒ Intellectually exciting! Practicallyimportant!


34


‘‘ Learning and studying so very often,what a pleasure!

Having friends from far away places,what a happiness! ’’

– Confucius (Kongtze)

Summary of “Journey”


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Acknowledgment(i) Trainings: C.K. Birdsall and Akira Hasegawa(ii) Learning from giants: (Rosenbluth, Frieman,

Oberman, Greene)(iii) Colleagues and friends: notably White(iv) Talented graduate students, postdocs, junior

colleagues (Hahm, Zonca, Crabtree, Y. Lin, Z.Lin)

(v) Generous and steady support from U.S. D.o.E.and N.S.F. for many years!


36


‘‘ Not recognized by others, and yet not upset;What a noble person! ’’

– Confucius(Kongtze)

Documents

Alfvén waves: A journey between space and fusion …phoenix.ps.uci.edu/scidac09/viewgraphs/Chen_Alfven_Lecture_EPS08.… · Hersonissos 06/09/2008 Alfvén waves: a journey between