PPPL Theory12/19/2014
Variational Methods in Modeling Plasma Waves:Basic Physics and Applications
I. Y. Dodin
PPPL, Princeton University
In collaboration with:
N. J. Fisch, D. E. Ruiz, C. Liu, I. Barth,
P. F. Schmit (Sandia), A. I. Zhmoginov (UC Berkeley)
Research & Review Seminar
PPPL, 2014
PPPL Theory12/19/2014Wave theory has been neglected but is still full of challenges
Suppose an RF wave in stationary homogeneous plasma with k2 Qpω,N q. Thensuppose slow N pxq. Replacing k Ñ iBx leads to a WKB solution.
x
Q
E2 B2E
Bx2QpxqE 0
E9 1
Qpxq14 exp
i
» xQpxq12 dx
PPPL Theory12/19/2014Wave theory has been neglected but is still full of challenges
1. The substitution k Ñ iBx can lead to wrong results. More severe and complexmanifestations: at mode conversion in tokamaks, waves in nonstationary plasmas.
x
Q
E2 B2E
Bx2 2α
Q1pxqQpxq
BEBx QpxqE 0
E9 1
Qpxqα14exp
i
» xQpxq12 dx
PPPL Theory12/19/2014Wave theory has been neglected but is still full of challenges
1. The substitution k Ñ iBx can lead to wrong results. More severe and complexmanifestations: at mode conversion in tokamaks, waves in nonstationary plasmas.
x
Q
E2 B2E
Bx2 2α
Q1pxqQpxq
BEBx QpxqE 0
E9 1
Qpxqα14exp
i
» xQpxq12 dx
2. Now consider nonlinear waves. Typically, the dispersion is calculated by assuminga stationary wave, Ept, xq Epωt kxq, and some f0pvq with V 0; that gives
ωpk, Eq ω0pkq δωNLpk, Eq
PPPL Theory12/19/2014Wave theory has been neglected but is still full of challenges
1. The substitution k Ñ iBx can lead to wrong results. More severe and complexmanifestations: at mode conversion in tokamaks, waves in nonstationary plasmas.
x
Q
E2 B2E
Bx2 2α
Q1pxqQpxq
BEBx QpxqE 0
E9 1
Qpxqα14exp
i
» xQpxq12 dx
2. Now consider nonlinear waves. Typically, the dispersion is calculated by assuminga stationary wave, Ept, xq Epωt kxq, and some f0pvq with V 0; that gives
ωpk, Eq ω0pkq δωNLpk, Eq kV pk, EqIncreasing amplitude accelerates plasma Ñ new f0pvq Ñ Doppler shift. To find it,a nonstationary wave must be considered Ñ much more complicated PDEs.
PPPL Theory12/19/2014Wave theory has been neglected but is still full of challenges
1. The substitution k Ñ iBx can lead to wrong results. More severe and complexmanifestations: at mode conversion in tokamaks, waves in nonstationary plasmas.
x
Q
E2 B2E
Bx2 2α
Q1pxqQpxq
BEBx QpxqE 0
E9 1
Qpxqα14exp
i
» xQpxq12 dx
2. Now consider nonlinear waves. Typically, the dispersion is calculated by assuminga stationary wave, Ept, xq Epωt kxq, and some f0pvq with V 0; that gives
ωpk, Eq ω0pkq δωNLpk, Eq kV pk, EqIncreasing amplitude accelerates plasma Ñ new f0pvq Ñ Doppler shift. To find it,a nonstationary wave must be considered Ñ much more complicated PDEs.
3. What is the wave energy-momentum? No (unambiguous) answer in Maxwell’s eqs.Do electrostatic waves have momentum?.. nonlinear forces on the medium?..
PPPL Theory12/19/2014Plasma wave theory needs qualitatively new approaches
Conventional theories of plasma waves ignore specialsymmetries (adiabatic invariants) that nondissipativelinear and nonlinear waves have as dynamic objects.
ë Dissipation can be added as a perturbation, but thelocal symmetries can, nevertheless, be critical.
ë Building on these symmetries can help modernize wavetheory and find new physics.
Investments in modernizing foundations today is anopportunity to lead wave theory in the future.
It also helps continue the tradition of innovation thatPPPL has had in wave physics and Hamiltonian dynamics:
- current drive for waves in all frequency regimes;
- alpha channeling for tokamaks and mirror machines;
- classic research by Stix, Dewar, Kruskal, Bernstein, Greene...
PPPL Theory12/19/2014The way to deal with symmetries is to develop a field theory
Dealing with given variational principles (VP) is understood, especially for linearnondissipative waves. Recent book: Tracy, Brizard, Richardson, & Kaufman (2014).
But, except in simplest cases, VPs are hard to guess ad hoc:.
- general linear waves in inhomogeneous nonstationary plasma; e.g., Langmuir ë
- nonlinear, dissipative, and non-eikonal waves are even worse. . .
Goal: replace ad hoc VPs with more general and unified first-principle VPs
: Dodin et al., PoP (2009); Dodin and Fisch, PRD (2010). . .
PPPL Theory12/19/2014Waves as oscillations on (unbounded) manifolds: Lagrangians
Any wave has a “photon wave function” ψ and a Hermitian H such that
L pi2qrψ:pBtψq pBtψ:qψs ψ:Hpt,x,i∇qψ parametric/nonlinear
iBtψ Hpt,x,i∇qψ, ψ:ψ action operator
Quantum mechanics emerges as a special case Ñ cantreat particles and waves on precisely the same footing.
Everything has a unified interface, L. What remains isto describe the coupling through this unified interface.
Sufficient semiclassical ( cold-fluid) approximation:
ψ eiθ?I GOÑ Lrθ, Is rBtθ Hpt,x,∇θqsI
Classical limit is redundant yet subsumed too: I δpx Xptqq,³L d3x P 9X Hpt,X,Pq, P
. ∇θpt,Xq
Dodin, PLA (2014); cf. Hayes (1973); Whitham (1974) . . .
PPPL Theory12/19/2014Test wave experiencing cross-phase modulation (XPM)
Scalar waves, geometrical-optics limit: Lrθ, Is rBtθ Hpt,x,∇θqsI- δθL 0 Ñ action conservation theorem: BtI ∇ pIvgq 0
- δIL 0 Ñ Hamilton-Jacobi/dispersion: Btθloomoonω
Hpt,x, ∇θloomoonk
q 0
Consider a test wave modulated weaklyat some Ω
. BtΘ and K. ∇Θ:
ω ω Re rωcpt,x,kq eiΘpt,xqs
Averaging over Θ gives a “dressed” Lagrangian, L rBtθ Hpt,x,∇θqs I.Instantaneous Kerr effect: now conservative & through the linear ωpkq.
H ω K
4 BBk
|ωc|2Ω K vg
dtx BkH, dtk BxH
Dodin and Fisch, PRL (2014)
PPPL Theory12/19/2014Ponderomotive forces on rays and particles. Asymmetric barriers
Photons/plasmons, ω0 pω2p k2c2q12
H ω0
1
ω2p
4ω20
n
n0
2 pΩ2 K2c2q
pΩ K vgq2loooooooooooooooomoooooooooooooooonKerr (ponderomotive) term, º0
Nonreciprocal propagation due to the “group
resonance” at Ω K vg.
Electrons, ~ω0 ~2k22m eϕ
H P 2
2m
e2|E|24m
pΩ K Vq2 p~K22mq
2looooooomooooooonbeyond GO
-30-20
+20+30
-100-200-300-400-1.0
-0.5
0.0
+0.5
+1.0
EE
max
k0x
-350 -300 -250 -20050
150
250
350
450
k0x
Κ02 Ω
p0t
1.0
0.5
0.0
NΚ
02
-1.0-0.50.0
+0.5+1.0
NΚ 0
2
-1.0-0.5
0.00.51.0
-400 -200 0 200 400
EE
max
HaL k0¤x
-400 -200 0 200 4000
200400600800
k0¤x
Κ02 Ω
p0t
HbL-400 -200 0 200 400
0200400600800
k0¤x
Κ02 Ω
p0t
HcL
Quantumlike derivations are often simpler, since equations are kept linear.
EM waves may be better modeled with vector rays. “Spin” effects at ray tracing?
Dodin and Fisch, PRL (2014); Ruiz and Dodin, in preparation
PPPL Theory12/19/2014Hamiltonian dynamics of multiple coherent waves. Mode conversion
Example: Langmuir waves coupled via low-frequency drive ηptq cospkNxq
i 9ψn °mHnmptqψm ωnψn ηptq pψnN ψnNq
Plasma mode conversion quantumLandau-Zener problem. The totalaction is conserved manifestly:
°n |ψ|2 const
Autoresonant trapping of plasmons at
Ω Kvg
Quantum-ladder-climbing emulationin a classical system. BGK wavesmade of plasmons/quantum particles
Barth, Dodin, and Fisch, in preparation
PPPL Theory12/19/2014Self-consistent dynamics and dispersion. Photons are polarizable
Low-frequency wave pΘ,J q interacts with many high-frequency waves pθn, Inq:
L rBtΘ Ω0sJ °nrBtθn pωn βnJ qlooooooomooooooon
ponderomotive effect
sIn
rBtΘ pΩ0 °n βnInqlooooooooomooooooooon
self-consistent frequency, Ω
sJ °n
pBtθnqInloooooomoooooonindependent of pΘ,J q
EM wave interacting with particles and HF waves:
L 1
16πE
p1 χparticlesloooooomoooooonstandard ε
χwaveslooomooon“anomalous”
q E |B|216π
Wave quanta contribute to linear ε just like electrons and ions. An improvedformula for a Langmuir wave in a photon bath: extra term + evolution allowed.
ΩpKq Ω0pKq K2ω4
p0
8men0Ω
» ∇k rKf0pkqs
pΩ K vgqω20
d3k
K2ω4p0
8men0Ω
»f0pkq
ω30
d3k.
cf. Tsytovich (1970); Bingham et al. (1997); Mendonca (2000); Dylov and Fleischer (2008) . . .
PPPL Theory12/19/2014Wave energy and momentum. Effect of kK on RF-driven rotation
The variational approach is the only unambiguous and reliable way to define thewave energy-momentum, both “canonical” and “kinetic”.
- Example: The effect of kK has been unclear, sincekK does not cause resonant acceleration along B0.
- Quasilinear calculation: [Lee, Parra, Parker, andBonoli, PPCF (2012)]. But detailed dynamics isunimportant. The symmetry of L already gives
∆PK kK∆Eω Ñ ∆rgc ckK∆EpeB0ωqÑ charge separation Ñ Poynting momentum
Guan, Dodin, Qin, Liu, and Fisch, PoP (2013)
PPPL Theory12/19/2014Same ideas apply to nonlinear waves, where H is more complicated
Klein-Gordon Hamiltonian Ñ Chen-Sudan Lagrangian, Akhiezer-Polovin waves
ω2 k2c2 ω2p r1 xpeAmc2q2ys12 (same in the Dirac model)
Ruiz and Dodin, in preparation
Advanced kinetic nonlinearities: waves with autoresonant trapped particles
L xE2y8π
np xεppJq ωJ mv2
ph2yloooooooooooooomoooooooooooooongeneralized ponderomotive potential
nt xεtpJqyloooomoooonbouncing
ntmv
2ph2loooomoooon
t.p. inertia
Dodin, Fusion Sci. Tech. (2014); Dodin and Fisch, PRL (2011), PoP (2012a,b,c); Schmit et al., PRL (2013)
PPPL Theory12/19/2014Example: BGK wave dispersion in the limit when E eikx
Lpa, ω, kq E2m16π
npxεpy((((
(((((((
((((
npxmv2ph2 ωJy ntxεty
ntmv2ph2
a keEmpmω2q, ε is the single-particle energy in the wave rest frame
0 BaL BaE2
m16π nxεy
ω2
2ω2p
a
» 8
0
gpJq F pJq dJloooomoooonensemble averaging
ë Deeply trapped particles: ω2 ω2L 2ω2
t a1
ë Flat-top beam: ω2 ω2L β a12
ë Smooth distribution: ωNL C1 a12 C2 ln a
cf. Goldman and Berk (1971); Manheimer and Flynn (1971); Dewar (1972);
Rose and Russell (2001); Khain and Friedland (2007); Krasovsky (2007) . . .
There are 3-4 communities working on this and largely ignoring each other.
PPPL Theory12/19/2014Anharmonic waves: clumps & holes, Maxwellian distribution
Bisinusoidal-wave model: φ a1 cos θ a2 cos 2θ, two independent amplitudes
ω2
ω2p
2
a1
» 8
0
GpJ, a1, a2qF pJq dJ 1
2a2
» 8
0
KpJ, a1, a2qF pJq dJ
Indistinguishable from Breizman’s exact solution for a clump. Automaticallyaccounts for both fluid and kinetic nonlinearities, unlike in ad hoc models.
Liu and Dodin (in preparation); cf. Winjum et al. (2007); Krasovsky (2007); Breizman (2010)
PPPL Theory12/19/2014Adiabatic dynamics of BGK-like waves
Wave action ( momentum) conservationin compressing plasma: e.g., k const
vph ωk ωpptqk anptq
inv ³BωL dx E2
m Bωεp16πq ntmvphk
Schmit et al., PRL (2013); Dodin and Fisch, PoP (2012a,b,c)
Self-action is not described by the NLSE
ipBtψ vg0 Bxψq 12 v
1g0 B
2xxψ ωNLψ
The modulational instability is revised:
γ vg0 ∆kb
eEmkmω2
p
S2 p2S 1q
cf. Dewar et al. (1972); Rose (2005) . . .
0 2000 4000 6000 8000-100
-50
0
50
100
ΩptHx-
Υg0
tLΛD
0 200 400 600 800-150
-100
-50
0
50
Ωpt
Hx-Υ
g0tLΛ
D
S = 14 S = 1
S trapped-e energy flux
passing-e energy flux
PPPL Theory12/19/2014Nonadiabatic dynamics. Negative-mass instability (NMI)
0246
1000 1500 2000 2500 3000
0246
W
W0
Ωp0t
0 0.2 0.4 0.6
0.5
1
ΩΩp0
È∆WΩ
2
1500 1700 1900 2100
1
0.1
0.01
Ωp0t
∆W
HaL
HbL HcL
cf. Smith and Pereira (1978); Adam et al. (1981);
Tennyson et al. (1994); Herrera et al. (2013) ...
New instability is identified for BGK waves:trapped particle bunching due to Ω1pJq 0cf. Kruer et al. (1969); Goldman and Berk (1971); Liu (1972) . . .
1 ω2t
εeff
» J0
JF 1tpJq
Ω2pJq pω kvphq2dJ
1εeff 1εpω ω, k kq 1εpω ω, k kq
Applies also to aperiodic waves. Akin to theNMI in accelerators, FELs, ion traps. . .Kolomenskii and Lebedev (1959); Strasser et al. (2002) . . .
Follow-up studies by other groupsBrunner et al. (2014); Hara et al, submitted (2014)
Connection with KEEN wavesDodin and Fisch, PoP (2014)
Dodin, Schmit, Rocks, and Fisch, PRL (2013)
PPPL Theory12/19/2014Summary
New “object-oriented” approach to linear and nonlinear plasma waves is beingdeveloped that blends variational methods of fluid and quantum theories.
- Every object is a wave, every interaction is ponderomotive, every equation roots in a VP.
- Lagrangians are unified and serve as building blocks to model multiple-wave coupling.
Immediate advantages:
- Unified and manifestly conservative equations, even beyond geometrical optics.
- Lowest-order wave-wave coupling gives linear dispersion, “photons are polarizable”.
- Energy-momentum is well defined, even dissipation enters naturally.
Applications to nonlinear waves (e.g., BGK-like) and quantum plasmas
PPPL Theory12/19/2014Research status and plans
Present: developing robust understanding of variational principles (VPs) for waves
- VPs for dissipative waves; photon polarization & Landau damping (with A. I. Zhmoginov, UCB)
- Origin and limitations of reduced VPs explored through quantum parallels (with D. E. Ruiz)
- Resonant ponderomotive effect on photons: BGK modes comprised of plasmons (with I. Barth)
- Nonlinear dispersion of anharmonic nonlinear waves, including BGK modes (with C. Liu)
- KEEN waves as the NMI-saturated modes (with A. H. Hakkim and U. Verma)
- Reduced analytic model for nonlinear dissipation of BGK-like waves through separatrix crossing...
Future: improving robustness of modeling linear RF waves in fusion and otherplasmas; synergistic nonlinear physics of energetic-particle modes and LPI
- Manifestly conservative analytical framework for modeling linear RF waves in realistic settings
- Symplectic algorithms for modeling linear RF waves (quantum methods?), test simulations
- Nonlinear waves: manifestly conservative reduced nonlinear models + trapped particles
- Working on synergistic applications in RF physics (as in the case of RF-driven plasma rotation)
Teaching: updating the “waves” course [cf. Tracy, Brizard, Richardson, & Kaufman,Ray tracing and beyond: phase space methods in plasma wave theory (2014)]
The work follows the recently approved 5-year plan for the BPPG effort in modernizing RF basic
theory. Synergies with campus grants and teaching provide 50% of funding.
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