j.t. mendonça instituto superior técnico, lisboa, portugal. recent advances in wave kinetics...
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J.T. Mendonça
Instituto Superior Técnico, Lisboa, Portugal.
Recent advances in wave kinetics
Collaborators:
R. Bingham, L.O. Silva, P.K. Shukla, N. Tsintsadze, R. Trines
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Outline
• Kinetic equations for the photon gas;• Wigner representation and Wigner-Moyal equations;• Modulational instabilities of quasi-particle beams;• Photon acceleration in a laser wakefield;• Plasmon driven ion acoustic instability;• Drifton excitation of zonal flows;• Resonant interaction between short and large scale perturbations;• Towards a new view of plasma turbulence.
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Wigner approach
Schroedinger eq.
€
h2
2m∇ 2 − ih
∂
∂t
⎡
⎣ ⎢
⎤
⎦ ⎥ψ = −Vψ
Wigner-Moyal eq.
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Quasi-classical approximation (sin ~ , h —> 0)
Conservation of the quasi-probability(one-particle Liouville equation)
Of little use in Quantum Physics
(W can be directly determined from Schroedinger eq.)
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Wigner-Moyal equation for the electromagnetic field
Field equation (Maxwell)
Kinetic equation
[Mendonca+Tsintsadze, PRE (2001)]
€
∇2 −1
c 2
∂ 2
∂t 2
⎛
⎝ ⎜
⎞
⎠ ⎟r E =
1
c 2
∂ 2
∂t 2(χ
r E )
€
Fk (r r , t) =
r E (
r r +
r s /2, t) ⋅∫
r E *(
r r −
r s /2, t)e−i
r k .
r s d
r s
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Photon number density
For the simple case of plane waves:
(R=0 is the dispersion relation in the medium)
Slowly varying medium
(photon number conservation)
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Dispersion relation of electron plasma waves in a photon background
€
χ ph (ω,k) = −ωph 0
2
2
k 2ωpe02
γ 02meω
2
dk
2π( )3
hk ⋅∂ ˆ N 0∂k
ωk2 ω − k ⋅vg (k)( )
∫
€
1+ χ e (ω,k) + χ ph (ω,k) = 0
Electron susceptibility Photon susceptibility
Resonant wave-photon interaction,
Landau damping is possible
[Bingham+Mendonca+Dawson, PRL (1997)]
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Physical meaning of the Landau resonance
Non-linear three wave interactions
€
ω +ω'= ω"r k +
r k '=
r k "
Energy and momentum conservation relations
Limit of low frequency and long wavelength
€
ω =ω"−ω'= Δω',with | Δω' |<<ω',ω"r k =
r k "−
r k '= Δ
r k ',with | Δ
r k ' |<<|
r k ' |,|
r k "|
€
ω rk
=Δω'
Δr k '⇒
dω'
dr k '
=r v g
(hints for a quantum description of adiabatic processes)
€
ω − r
k .r v g = 0
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Spectral features: (a) split peak, (b) bigger split, (c) peak and shoulder, (d) re-split peak
Simulations: R. Trines
Experiments: C. Murphy
(work performed at RAL)
Photon dynamics in a laser wakefield
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Experimental Numerical 1D
Also appears in classical particle-in-cell simulations
Can be used to estimate wakefield amplitude
Split peak
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Plasma Plasma
surfacesurface
LASER
ignition
DTcore
channel
Anomalous resistivity for Fast Ignition
LASER Fast electron beamElectron plasma wavesTransverse magnetic fields
Ultimate goal: ion heating
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Theoretical model:
Wave kinetic description of electron plasma turbulence
• Electron plasma waves described as a plasmon gas;
• Resonant excitation of ion acoustic waves
Dispersion relation of electrostatic waves
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Electron two stream instability
Maximum growth rate
Total plasma current
Dispersion relation
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Kinetic equation for plasmons
Plasmon occupation number
Plasmon velocity Force acting on the plasmons
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Ion acoutic wave resonantly excited by the plasmon beam
Maximum growth rate
Effective plasmon frequency
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Two-stream instability(interaction between the fast beam and the return current)
Freturn
Ffast
Unstable region:
Plasmon phase velocity vph ~ c
Electron distribution functions
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Plasmon distribution
Low group velocity plasmons: vph .vg = vthe2
Vg ~ vthe2 / c
Ion distribution (ion acoustic waves are destabilized by the plasmon beam)
Npl
Vph/ionac ~ vg
Fion
Npl
[Mendonça et al., PRL (2005)]
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Laser intensity threshold
For typical laser target experiments, n0e~1023 cm-3:
I > 1020 W cm-2
Varies as I-5/4
0 , u0e~ I1/2
Preferential ion heating regime
(laser absorption factor)
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Experimental evidence
Plastic targets with deuterated layers using Vulcan (RAL)
I = 3 1020 Watt cm-2
Not observed at lower intensities (good agreement with theoretical model)
[P. Norreys et al, PPCF (2005)]
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We adapt the 1-D photon code to drift waves:
Two spatial dimensions, cylindrical geometry,
Homogeneous, broadband drifton distribution,
A Gaussian plasma density distribution around the origin.
We obtain:
Modulational instability of drift modes,
Excitation of a zonal flow,
Solitary wave structures drifting outwards.
Coupling of drift waves with zonal flows
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Fluid model for the plasma (electrostatic potential Φ(r)):
Particle model for the “driftons”:
Drifton number conservation;
Hamiltonian:
Equations of motion: from the Hamiltonian
( ) kdNkk
kk
t k
r
r 22221
∫++
=∂
Φ∂
ϑ
ϑ
( ),1 22*
ϑ
ϑϑω
kk
Vk
rk
ri ++
+∂
Φ∂=
r
n
nV
∂∂
−= 0
0*
1
[R. Trines et al, PRL (2005)]
Quasi-particle description of drift waves
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Excitation of a zonal flow for small r, i.e. small background density gradients; Propagation of “zonal” solitons towards larger r.
Simulations
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Plasma Physics processes described by wave kinetics
Short scales Large scales Physical relevance
Photons Ionization fronts Photon acceleration
Photons Electron plasma waves
Beam instabilities; photon Landau damping
Plasmons Ion acoustic waves Anomalous heating
Driftons
(drift waves)
Zonal flows Anomalous transport
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Other Physical Processes
Short scales Large scales Physical relevance
Photons Iaser pulse envelope
Self-phase modulation
Cross-phase modulation
Photons polaritons Tera-Hertz radiation in polar crystals
Photons Gravitational waves Gamma-ray bursts
neutrinos Electron plasma waves
Supernova explosions
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Conclusions
• Photon kinetic equations can be derived using the Wigner approach;• The wave kinetic approach is useful in the quasi-classical limit;• A simple view of the turbulent plasma processes can be established;• Resonant interaction from small to large scale fluctuations; • Successful applications to laser accelerators (wakefield diagnostics); inertial fusion (ion heating) and magnetic fusion (turbulent transport).