nonlinear kinetic turbulence theory

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

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.)

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

(bi) kappa model

f (v) f (v|| )

v v||

Maksimovic et al. JGR, 110, A09104 (2005)

fMaxwellBoltzmannGauss ex 2

fkappa 1

(1 x 2 /)or

1(1 x 2 /) 1

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)

Nonlinear Plasma Interaction

e+

e–

E, B

E, BM M

M M

e+

e–

E, B

E, B

Klimontovich function

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

(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' |)

Phenomenological approach:Non‐Gaussian PDFMulti‐fractal scaling

Fluctuation Equation

Particle kinetic equation

Field equation

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:

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) ],

...

Particle Kinetic Equation

Spontaneous drag (discrete particle effect)

Velocity space diffusion

Wave Kinetic Equation

Spontaneous emission

Induced emission(Landau damping/Quasi‐linear growth/damping rate)

Linear wave‐particleresonance

Wave Particle

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

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)

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)

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)

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)


Maxwellian Core

Halo

Superhalo



Quasi‐thermal noise [by Stuart Bale]

I(k)LangmuirTurbulenceSpectrum

Turbulent Equilibrium

Particles and wave constantly exchange momentum and energy but are in dynamical steady state.

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)

Self‐consistent, steady‐state electron velocity and Langmuir spectrum

distributions

Electron VDF (cont’d)

e.g., thermal equilibrium

Self‐Consistent Solution: Turbulent Equilibrium

• Self‐consistent steady‐state particle and wave distributions leads to kappa distribution, but is undetermined:

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)

Balance of nonlinear terms within wave kinetic equation

Balance of spontaneous and induced emissions

Balance of spontaneous and induced scatterings

=0

=0

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.

• 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

• Turbulent equilibrium:

94 2.25.


Maxwellian Core

Halo

Superhalo



fobservation v , f theory v2 2 ~ v 6.5,

avgobservation 6.69 theory 6.5

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.

Part II. Electromagnetic Problem

e+

e–

E, B

E, B

Radiations in Space

Type III

Type II

BowshockRadiation

EscapingTerrestrial Continuum

Reiner et al.

X Ray

Type III radio

Electrons

Lin 1981

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

Culgoora Radio Spectrograph

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

Beam‐generated Langmuir turbulence

e

e

E, pe

Langmuir oscillation Ion‐sound wave

ee

E, kcSi

Debye cloud

k

pe (1 3k2D

2 /2)

kcS

– 2kL 2kL

Vb

kkL– kL

T

L

S

Bump‐in‐tailinstability

k||

k

S

L

Vb

Backscattered L wave

e iL

L’

S

k

S

L

T

pe

pe V0L’

k||

k

S

L

T

F Emission

eiL

F

S

Decay of L mode into S and T at Fundamental

F Emission

k

S

L

T

pe

pe V0

F

k||

k

S

L

T

H Emission

L

H

L’

Merging of Primary L and Backscattered L’ modes into T mode at Harmonic

k

S

L

T

pe

pe V0L’

H

H Emission

k||

k

S

L

T

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]

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

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

Ion‐acoustic wave kinetic equation

Spontaneous &induced emission

(S ‐> L+L) three wave decay

Particle kinetic equation

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

PIC simulation[Rhee et al., 2009]

PIC simulation [Rhee et al., 2009]


(L+I ‐> T) Fundamentalemission by scattering

(L+L ‐> T) Harmonic emission

(L+S ‐> T) Fundamental emission by decay


Dulk et al. 1984

Background Radiation

Blackbody Radiation

In the presence of plasma

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

Spectrum of solar radiation

Theory of background radiation

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

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.

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.

nonlinear kinetic turbulence theory

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