Nonlinear kinetic turbulence theory
Transcript of Nonlinear kinetic turbulence theory
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Nonlinear kinetic turbulence theory
Peter H. YoonUniversity of Maryland
6th East‐Asia School and Workshop on Laboratory, Space, and Astrophysical PlasmasTsukuba, Japan, 11 ‐ 16 July, 2016
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FLUID THEORIES KINETIC THEORIES
• MHD (Ideal,Resistive, Hall, etc).Well‐developedmature field.• Drift‐wave turb.(Hasegawa‐Mima)
• Strong turbulence(à la Zakharov)
• Weak turbulence forunmagnetized plasmas(Tsytovich, Melrose,Sitenko, Davidson,etc.). Most maturekinetic plasmaturbulence theory• Renormalized (strongturbulence à laKadomtsev, Dupree,etc.)
REDUCEDKINETIC THEORIES
• Drift Kinetic• Gyrokinetic (applicable for low‐frequency regime andfor stronglymagnetizedplasmas)
• Weak turbulence formagnetized plasmas(incomplete, but somepreliminary worksinclude those byPustovalov & Silin,Porkolab, etc.)
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Part 1. Electrostatic Problem
Wang et al., ApJ Lett. (2012)
Maxwellian Core
Halo
Superhalo
Interplanetary& galactic cosmicray electrons
SOLAR WIND ELECTRON DISTRIBUTION AT 1 AU
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(bi) kappa model
f (v) f (v|| )
v v||
Maksimovic et al. JGR, 110, A09104 (2005)
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fMaxwellBoltzmannGauss ex 2
fkappa 1
(1 x 2 /)or
1(1 x 2 /) 1
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fMaxwellBoltzmannGauss ex 2
fkappa 1
(1 x 2 /)or
1(1 x 2 /) 1
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fMaxwellBoltzmannGauss ex 2
fkappa 1
(1 x 2 /)or
1(1 x 2 /) 1
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Theories of “Kappa” Distribution
• 1. Collisional or Stochastic Transport Model (Scudder, 1990s; Schwandron, 2010)
• 2. Non‐Extensive Thermodynamics (Tsallis, 1980s; Treumann, 2000s; Leubner, 2000s; Livadiotis, 2000s)
• 3. Steady‐State Plasma Turbulence (Yoon, 2014)
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Nonlinear Plasma Interaction
e+
e–
E, B
E, BM M
M M
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e+
e–
E, B
E, B
Klimontovich function
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e+
e–
E, B
E, BElectrostatic approximation
Na (r,v,t) fa (r,v, t)Na (r,v,t) fa (r,v,t) Na (r,v,t)E(r, t) E(r, t)
Separation into average and fluctuation
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(r,t) (r’,t’)(r,t) (r’,t’)
(r,t) (r’,t’)(r,t) (r’,t’)
g2ab(| r r ' |, | t t ' |) Na (r,v, t)Nb(r ',v ', t ')
g2ab (r, t,|r r' |,| t t' |)
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Phenomenological approach:Non‐Gaussian PDFMulti‐fractal scaling
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Fluctuation Equation
Particle kinetic equation
Field equation
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Near absence of linear term in NS turbulence:<< 1
Robust linear physics in plasma turbulence:
Linear, QL, and weak turbulence theories are valid for plasmas
“Renormalization” is necessary at the outset
Navier‐Stokes turbulence:
Plasma turbulence:
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Perturbative plasma turbulence theory
Nk Nka(1) Nk
a(2) Nka(3)
Nka(1) Nk
0 ea
ma
1 k v
kk fa
v,
Nka(2)
ea
ma
1 k v
dk 'k'v[k ' 'Nkk ', '
a(1) k ' 'Nkk ', 'a(1) ],
Nka(3)
ea
ma
1 k v
dk 'k'v[k ' 'Nkk ', '
a(2) k ' 'Nkk ', 'a(2) ],
...
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Particle Kinetic Equation
Spontaneous drag (discrete particle effect)
Velocity space diffusion
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Wave Kinetic Equation
Spontaneous emission
Induced emission(Landau damping/Quasi‐linear growth/damping rate)
Linear wave‐particleresonance
Wave Particle
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ik k ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Spontaneous decay Induced decay
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ik k ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Nonlinear wave‐wave resonance
Wave (kk)
Wave(k–kʼk – kʼ)
Wave(kʼkʼ)
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ik k ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Spontaneous scattering
Induced scattering (NL Landau damping) (scattering off thermal ions)
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ik k ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Nonlinear wave‐particle resonance
Particle
Wave(kk)
Wave(kʼkʼ)
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Weak turbulence theory‐ L. M. Gorbunov, V. V. Pustovalov, and V. P. Silin, Sov. Phys. JETP
20, 967 (1965)‐ L. M. Al’tshul’ and V. I. Karpman, Sov Phys. JETP 20, 1043 (1965)‐ L. M. Kovrizhnykh, Sov. Phys. JETP 21, 744 (1965)‐ B. B. Kadomtsev, Plasma Turbulence (Academic Press, 1965)‐ V. N. Tsytovich, Sov. Phys. USPEKHI 9, 805 (1967)‐ V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum Press,
1970)‐ R. C. Davidson, Methods in Nonlinear Plasma Theory (Academic
Press, 1972)‐ V. N. Tsytovich, Theory of Turbulent Plasma (Consultants
Bureau, 1977)‐ D. B. Melrose, Plasma Astrophysics (Gordon and Breach, 1980)‐ A. G. Sitenko, Fluctuations and Non‐Linear Wave Interactions in
Plasmas (Pergamon, 1982)
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Wang et al., ApJ Lett. (2012)
Maxwellian Core
Halo
Superhalo
Interplanetary& galactic cosmicray electrons
SOLAR WIND ELECTRON DISTRIBUTION AT 1 AU
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Quasi‐thermal noise [by Stuart Bale]
I(k)LangmuirTurbulenceSpectrum
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Turbulent Equilibrium
Particles and wave constantly exchange momentum and energy but are in dynamical steady state.
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ikk ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Wave‐wave
Linear wave‐particle
Nonlinearwave‐particle
Asymptotic solution (∂/∂t = 0)
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Self‐consistent, steady‐state electron velocity and Langmuir spectrum
distributions
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Electron VDF (cont’d)
e.g., thermal equilibrium
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Self‐Consistent Solution: Turbulent Equilibrium
• Self‐consistent steady‐state particle and wave distributions leads to kappa distribution, but is undetermined:
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ikk ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Wave‐wave
Linear wave‐particle
Nonlinearwave‐particle
Asymptotic solution (∂/∂t = 0)
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Balance of nonlinear terms within wave kinetic equation
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Balance of spontaneous and induced emissions
Balance of spontaneous and induced scatterings
=0
=0
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Solving the above equation one obtains the steady‐state turbulence intensity that balances the spontaneous and induced scattering processes:
kdI(k)dk
4 2
Ti
d k
dk[I(k)]2
d k
dkI(k) 0.
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• We now have two alternative expressions for the steady‐state turbulence intensity:
Te 3/2 1
Ti, and 3234
• Balance of spontaneous and induced emissions
• Balance of spontaneous and induced scatterings
• The only way the two expressions can be reconciled is when
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• Turbulent equilibrium:
94 2.25.
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Wang et al., ApJ Lett. (2012)
Maxwellian Core
Halo
Superhalo
Interplanetary& galactic cosmicray electrons
SOLAR WIND ELECTRON DISTRIBUTION AT 1 AU
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fobservation v , f theory v2 2 ~ v 6.5,
avgobservation 6.69 theory 6.5
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Part 1. Conclusion• Perturbative nonlinear kinetic theory of Langmuir turbulence is well known [e.g., Tytovich, 1970; Davidson 1972; Melrose, 1980; Sitenko, 1982, etc.]
• Application of the theory to explain quiet time solar wind electron kappa distribution [Yoon, 2014] is, however, new. Comparison between theory ( = 6.5) and observation (avg = 6.69) is favorable.
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Part II. Electromagnetic Problem
e+
e–
E, B
E, B
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Radiations in Space
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Flare
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Type III
Type II
BowshockRadiation
EscapingTerrestrial Continuum
Reiner et al.
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X Ray
Type III radio
Electrons
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Lin 1981
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Most type III bursts do not have the iconic F‐H pair emission structure
Type II emissions are somewhat more clearly identifiable with the beam‐induced F‐H pair emissions
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Culgoora Radio Spectrograph
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Plasma emission scenario(Ginzburg & Zeleznyakov, 1958; Melrose, 1970s;
Robinson, Cairns, 1980 & 1990s)
• Electron beam produced during flare• Generation of Langmuir waves• Backscattered Langmuir waves by nonlinear processes
• Harmonic emission by merging of Langmuir waves
• Fundamental emission by Langmuir wave decay
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Beam‐generated Langmuir turbulence
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e
e
E, pe
Langmuir oscillation Ion‐sound wave
ee
E, kcSi
Debye cloud
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k
pe (1 3k2D
2 /2)
kcS
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– 2kL 2kL
Vb
kkL– kL
T
L
S
Bump‐in‐tailinstability
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k||
k
S
L
Vb
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Backscattered L wave
e iL
L’
S
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k
S
L
T
pe
pe V0L’
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k||
k
S
L
T
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F Emission
eiL
F
S
Decay of L mode into S and T at Fundamental
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F Emission
k
S
L
T
pe
pe V0
F
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k||
k
S
L
T
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H Emission
L
H
L’
Merging of Primary L and Backscattered L’ modes into T mode at Harmonic
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k
S
L
T
pe
pe V0L’
H
H Emission
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k||
k
S
L
T
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Plasma emission scenario
• Many advances have been made over six decades of research.
• We have recently carried out EM weak turbulence calculation of plasma emission from beam‐plasma instability to radiation generation [Ziebell, Yoon, Gaelzer, Pavan, ApJL, 2014; ApJ, 2015]
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ik k ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Langmuir wave kinetic equation
Spontaneous & induced emission
(L ‐> L+S) three wave decay
(L ‐> L+e) spontaneous &induced scattering
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2 ', ''1 k
L dk' Vk,k 'L ( k
L 'k 'L ' ' kk '
S )
kLIk '
'LIk k ' ''S 'k '
L Ik k ' ''S Ik
L ' ' k k 'L Ik '
'LIkL
Langmuir wave kinetic equation
Spontaneous & induced emission
(L ‐> L+S) three wave decay
(L ‐> L+e) spontaneous &induced scattering
e
e
E, pe
e iL
L’
S
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Ion‐acoustic wave kinetic equation
Spontaneous &induced emission
(S ‐> L+L) three wave decay
Particle kinetic equation
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Transverse EM wave kinetic equation
(L+I ‐> T) Fundamentalemission by scattering
(L+L ‐> T) Harmonic emission
(L+S ‐> T) Fundamental emission by decay
(L+T ‐> T) Higher‐harmonic emission
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Transverse EM wave kinetic equation
(L+T ‐> T) Higher‐harmonic emission
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PIC simulation[Rhee et al., 2009]
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PIC simulation [Rhee et al., 2009]
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Transverse EM wave kinetic equation
(L+I ‐> T) Fundamentalemission by scattering
(L+L ‐> T) Harmonic emission
(L+S ‐> T) Fundamental emission by decay
(L+T ‐> T) Higher‐harmonic emission
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Dulk et al. 1984
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Background Radiation
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Blackbody Radiation
In the presence of plasma
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10-8 10-7 10-6 10-5 10-4 10-310-20
10-18
10-16
10-14
10-12
10-10
10-8
wavelength
Spectrum
With plasma
vacuum
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Spectrum of solar radiation
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Theory of background radiation
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t
IkT
2 0
'1 dk ' dv Uk,k '
T [kT 'k '
L (kk ') v]
ne2
pe2 k
T Ik ' 'L 'k '
T IkT
2
( fe fi )
me
mi
Ik ' 'L Ik
L
2(kk ') fi
v
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Part 2. Conclusions and Discussion
‐ Despite 6 decades of research, the plasma emission was not understood in complete quantitative terms until now.
‐ We solved EM weak turbulence theory to demonstrate plasma emission for the first time and compared against few available 2D EM PIC simulation results.
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General Conclusion
• Perturbative nonlinear kinetic theory called “weak turbulence theory” successfully explains many facets of space plasma phenomena – electron kappa VDF, plasma emission, …
• Basic methodology can be extended for other applications, most notably, turbulent phenomena in magnetized plasmas.