Variational Methods in Modeling Plasma Waves: Basic ... · Variational Methods in Modeling Plasma Waves: Basic Physics and Applications I ... 12/19/2014Plasma wave theory needs

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


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



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



x

Q

E2 B2E

Bx2 2α

Q1pxqQpxq

BEBx QpxqE 0

E9 1

Qpxqα14exp

i

» xQpxq12 dx


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.



x

Q

E2 B2E

Bx2 2α

Q1pxqQpxq

BEBx QpxqE 0

E9 1

Qpxqα14exp

i

» xQpxq12 dx


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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Variational Methods in Modeling Plasma Waves: Basic ... · Variational Methods in Modeling Plasma Waves: Basic Physics and Applications I ... 12/19/2014Plasma wave theory needs