1 Turbulent Generation of Large Scale Magnetic Fields in Unmagnetized Plasma Vladimir P.Pavlenko...
-
Upload
kelley-oconnor -
Category
Documents
-
view
217 -
download
2
Transcript of 1 Turbulent Generation of Large Scale Magnetic Fields in Unmagnetized Plasma Vladimir P.Pavlenko...
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 33
OutlineOutline
Some notes from Fluid MechanicsSome notes from Fluid Mechanics MotivationMotivationModelingModelingSelf-consistent descriptionSelf-consistent descriptionNon-linear dynamicsNon-linear dynamicsConclusionsConclusions
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 44
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 55
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 66
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 77
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 88
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 99
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1010
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1111
Large scale structures: DefinitionsLarge scale structures: Definitions
Zonal Zonal
magnetic magnetic
fieldsfields
Magnetic Magnetic
streamersstreamers
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1212
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1313
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1414
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1515
Non-linear dynamicsNon-linear dynamics
Large scale Large scale Drift-type wave Drift-type wave
StructuresStructures turbulenceturbulence
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1616
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1717
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1818
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 1919
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 2020
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 2121
Non-linearNon-linear dynamics dynamics
Large scaleLarge scale Drift-type wave Drift-type wave
Structures Structures turbulance turbulance
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 2222
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 2323
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 2424
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
2007, May 292007, May 29 Workshop ’Reconnection and Turbulence’ UppsalaWorkshop ’Reconnection and Turbulence’ Uppsala 2525
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