Turbulent Generation of Large Scale Magnetic Fields in Unmagnetized Plasma

11

Turbulent Generation of Turbulent Generation of Large Scale Magnetic Large Scale Magnetic

Fields in Unmagnetized Fields in Unmagnetized Plasma Plasma

Vladimir P.PavlenkoVladimir P.Pavlenko

Uppsala University, Uppsala University, Uppsala, SwedenUppsala, Sweden

2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 22

CoworkersCoworkers

Zhanna N.AndrushchenkoZhanna N.AndrushchenkoMartin JuckerMartin Jucker


OutlineOutline

Some notes from Fluid MechanicsSome notes from Fluid Mechanics MotivationMotivationModelingModelingSelf-consistent descriptionSelf-consistent descriptionNon-linear dynamicsNon-linear dynamicsConclusionsConclusions


Description of flows in a turbulent mediaDescription of flows in a turbulent media

, 0i i i iv v v v

iik i k ik

k

vp v v

t x

ik

i i ik i k

k i k k k

v v vpv v v

t x x x x x

(turbulent) Reynolds stress


MotivationsMotivations

Strong magnetic fieldsStrong magnetic fields- Solar flares- Solar flares

Magnetic field diffusivityMagnetic field diffusivityReconnectionReconnection

- Laser produced plasma- Laser produced plasmaStrong fields produced in unmagnetized plasmaStrong fields produced in unmagnetized plasma

70’s: Laser fusion experiments: 70’s: Laser fusion experiments:

Strong magnetic field observed in unmagnetized plasmaStrong magnetic field observed in unmagnetized plasma

Magnetic electron drift modesMagnetic electron drift modes


Modeling: AssumptionsModeling: Assumptions

0n n

0 0,B 0 0 0 0ˆ ˆ , x xn n x T T x

pi pe unpolarized electron fluid and immobile ions

magnetic electron drift modes

1

pe

ck

n 0 01, ln , lnn TTn T

k k

,B x yB z 2e

Bmc

v z

perturbed magnetic field electron fluid velocity

2

04

cB

en v z

electron vorticity

unmagnetized inhomogeneous plasma


Modeling: Equations Modeling: Equations

4 Ampere's law pe c

B j v B

electron flow is directly related to the instantaneous magnetic field

2 2 4 20

0

, ,e B c

B B B n Tt mc t en

evolution equation is nonlilear in intrincically due to convective nonlinearity, B v v

baroclinic vector vorticity source

2

2 00

0

2, ,

3

eTT eB T n B

t mc mcn


Model equationsModel equations

triad interactions: LHS ; RHS , i i ie e e kr k r k r k k k

2 0 2

3 n T

eT

mc

nc

e

42 2 2

2

, ,

, ,

T eB B B B

t y m

B eT B T

t y m

pe

c

22 2 2E B B T dxdy const

2U BT B T dxdy const

double energy cascade


Linear approximationLinear approximation

- purely growing for - purely growing for - no linear instability for - no linear instability for

- largest increment for - largest increment for

2, i.e. 0

3T n

Dispersion relation

2 2 ; ,

1ki t

k yk B T ek

0yk

0xk


Comparison to electrostatic drift Comparison to electrostatic drift wave turbulencewave turbulence

Two-field vs. One-field modelTwo-field vs. One-field model Electron skin depthElectron skin depth vsvs. . Ion Larmor radius with electronIon Larmor radius with electron temperaturetemperature

Direct andDirect and inverseinverse cascadecascade vs.vs. Inverse cascadeInverse cascade

42 2

2

,

MEDM:

,

T eB B B B

t y m

B eT B T

t y m

2

2 2 2 DWT: ,ss v

t y B


Large scale structures: DefinitionsLarge scale structures: Definitions

Zonal Zonal

magnetic magnetic

fieldsfields

Magnetic Magnetic

streamersstreamers


Large scale magnetic fields generationLarge scale magnetic fields generation

B B B

, , ,t Tx X

, , ,B B t T x X ,B B T X

2 2 4 2,e

B B BT Y mc

turbulent Reynolds stress

2 ,B e

BT Y mc

YX Y

vv v

T X

scale separation:

y xv vT y x


Large scale structures:Large scale structures: scale separation:

, . . ; i ik q

Bt t e t e c c

T

kr qr

k

r q k

,00qq

2 2 2

2 21

0

qx y k k

q

B e qk k B B

t m q

T

t

k

zonal fields

turbulent Reynolds stress


Self-consistent descriptionSelf-consistent description Define ”action-like invariant” or wave spectrumDefine ”action-like invariant” or wave spectrum

Wave kinetic equation (WKE)Wave kinetic equation (WKE)

Doppler shifted frequencyDoppler shifted frequency

22 24 1k kN k B

2NL NL

k k k k kk k k

N N NN St N

t

k r r kReNL

k k

2 2

2 2 2 2

1 2 1

1 1

q qB T

k

k k

k v k v

2

4B

eB

m

v z

2

4T

eT

m

v z


Non-linear dynamicsNon-linear dynamics

Large scale Large scale Drift-type wave Drift-type wave

StructuresStructures turbulenceturbulence


Large scale fields generation - RecipeLarge scale fields generation - Recipe

Model equationsModel equations

Wave spectrumWave spectrum Quasi-linear analysis, Quasi-linear analysis,

linearized linearized WKEWKE

Response functionResponse function Dispersion relationDispersion relation

2 2 22

2 21

0

qx y k

q

B e qk k B d

t m q

T

t

k

2 22 24 1k k kN k B

0k kN N N 0 ,k f

NN R p

k v

r k

1,

g

R pi

p v

2 2

2 2 202 2

,1

yq x

x

k Ni K q k R p d

k k

k2 2 2 2

22 2 216 1q

e qK

m q


Large scale field generation – Large scale field generation – Hydrodynamic regimeHydrodynamic regime

Hydrodynamic regimeHydrodynamic regime

- Monochromatic wave packet - Monochromatic wave packet

- Instability criterion- Instability criterion

- Explicit frequency- Explicit frequency

2 2

2 2 202 2

,1

yq x

x

k Ni K q k R p d

k k

k

0 0 0kN N k k

0g

x

v

k

0 1 2 2 2 2 2

0 0 05 22 21 2

1

y

g q x y

kqv iK q N k k

k


Large scale field generation – Large scale field generation – Kinetic regimeKinetic regime

Kinetic regimeKinetic regime

- Resonance (purely growing)- Resonance (purely growing)

- Instability criterion- Instability criterion

Contrary to Langmuir turbulenceContrary to Langmuir turbulence

2 2

2 2 202 2

,1

yq x

x

k Ni K q k R p d

k k

k

gpv 1,R p

1,

g

R pi

p v

00 0zfx

x

Nk

k


Large scale field generation – Large scale field generation – Modulation instabilityModulation instability

Modulation instabilityModulation instability

- Restart from basic model equations- Restart from basic model equations

- - Pump wave and flowsPump wave and flows

- - Triad interactions sidebandsTriad interactions sidebands

- - Explicit frequencyExplicit frequency

- - Well known Lighthill criterionWell known Lighthill criterion

,k k , q

0 k k q

k k q k k k q

k k

10g g

k k

v v

k

222 2 2

, 2g g

g k q kk

v qvqv i K B q


Large scale field generation - Large scale field generation - SummarySummary

Explicit incrementsExplicit increments

--Hydrodynamic regimeHydrodynamic regime

- Modulational instability- Modulational instability

- Note- Note

Instability criteriaInstability criteria

- Hydrodynamic, modulational- Hydrodynamic, modulational

- Kinetic regime- Kinetic regime

2 202 2 2 2 2 2

0 0 052 21 2

1

yhq x y

kK q N k k

k

2222 2 2

, 2gm

k q kk

v q vK B q

m h

0g

x

v

k

0 0xx

Nk

k


Non-linearNon-linear dynamics dynamics

Large scaleLarge scale Drift-type wave Drift-type wave

Structures Structures turbulance turbulance


ShearingShearing

Quasi-linear analysisQuasi-linear analysis

Wave kinetic equationWave kinetic equation

Diffusion in Diffusion in kk-space, i.e. shearing-space, i.e. shearing

Large Large kk Small scales Dissipation Small scales Dissipation

0kN N N

0NL NL

k k k k kN N N

t

k r r k

,

0 0

, ,

0x yk

x y x y

N ND

t k k

,

2 42 22 4

, , 2, 0

16x yk y x x y

eD k q B T R p

m


Large scale pattern and predator-Large scale pattern and predator-prey phenomenaprey phenomena

Reduction of the basic equationsReduction of the basic equations

minimal dynamical model (zero-dimensional approach) with dynamical model (zero-dimensional approach) with two two principal components: principal components:

small-scale waves (small-scale waves (prey) + ”zonal” magnetic pattern () + ”zonal” magnetic pattern (predator) )

++

Lotka-Volterra system

2

kx B

,00qq

qy B

2dx rrx x axy

dt K

dyaexy dy

dt


Predator-prey dynamics of the system composed of Predator-prey dynamics of the system composed of zonal magnetic pattern and wave turbulencezonal magnetic pattern and wave turbulence

Results of the numerical analysisResults of the numerical analysis


ConclusionsConclusions

Magnetic electron drift mode turbulence – Magnetic electron drift mode turbulence – model equationsmodel equations

Separation of scales: waves + ”fields”Separation of scales: waves + ”fields” Self-consistent description – wave Self-consistent description – wave kinetic kinetic

equationequation Waves ”Fields”: GenerationWaves ”Fields”: Generation ” ”Fields” Waves: ShearingFields” Waves: Shearing Long term dynamics Predator-prey dynamicsLong term dynamics Predator-prey dynamics

Documents

Turbulent Generation of Large Scale Magnetic Fields in Unmagnetized Plasma