Bauer2012_06_20DerivationPlasmaFluidEquations

8/11/2019 Bauer2012_06_20DerivationPlasmaFluidEquations

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Derivation of Plasma Fluid EquationsPlasma Physics Summer School

Bruno Bauer

UNR Physics Department

Center for Nonlinear Studies, LANLJune 20, 2012

BSB 6/15/12


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Abstract

Plasma physics is of great importance for science and technology.All plasma follows a common set of principles, whether it is thetenuous plasma of interstellar space or the ultradense plasmacreated in inertial confinement fusion experiments; or whether it is

the cool, chemical plasma used in the processing of semiconductorsor the hot, thermonuclear plasma of stars and fusion devices. Thissecond lecture of the Plasma Physics Summer School continues abroad outline of plasma physics. Starting with particle motion, webuild distribution functions and derive kinetic equations. Takingvelocity moments of the Boltzmann kinetic equation we reduce thedimensionality of the plasma description but increase the number of

partial differential equations. Through suitable approximations, weobtain and close f luid equations, including those of magnetohydrodynamics. At every stage in this cascade ofderivations, we should pause to solve our equations and admire anew perspective on the wondrous world of plasma equilibria, waves,and instabilities.

BSB 6/06/12


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Derivation of Plasma Fluid Eqns1. The challenge of plasma physics

2. Kinetic equations

Klimontovich, Boltzmann, Vlasov Distribution functions

3. Particle motion

4. Velocity moments of Boltzmann equation

+ approximations=> fluid equations &magnetohydrodynamics

5. Plasma equilibria, waves, instabilities,self-organization

BSB 6/15/12


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Fusion Core150 g/cc, 1.57!107K

Photosphere2!10-7 g/cc, 5777 K

Corona: e.g., 10-15

g/cc, 2!106

K

EUV(He-II304A)Image:ESA&NAS

A,

SOHO/EIT,um

bra.nascom.nasa.gov/

images/eit_19

970914_

0121_

304.jpg

BSB 6/20/12


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TPI, BSB 6/15/12

Solar wind drivesEarth magnetosphere

Fluid-like flow of the solar windImage:Artist'sConceptionoftheSun-EarthSystem,

SteveMercer,SpaceScienceInstitute

pwg.gsfc.nasa.gov/

istp/news/0005/MercerMural40.jpg


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! Explain macrostructure from microphysics

! Understand & control large collectionsof unbound charged particles & photons

Classical plasma physics simplifyingassumption:

" Assume plasma parameters such thatparticles can be described with classical physics

"

This is a complete, but intractable, set of equations

The challenge of plasma physics

{ }

( )

For 1,2,3, , : , , , ,

with Lorentz force

where spiky fields are determined by

the particles via Maxwell's equations

i i i i i

ii i i i

i

i N q m r v a

qa E v B

m

=

= +

! ! !"

! !! !

BSB 6/15/12


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Maxwell

s Equations

! ="o

QAdE

#

!!

RelatesEto charges.(Coulombs law)

=" 0AdB!!

Bis continuous.

No magnetic monopoles.

Gauss

Laws

! "

#=$dt

dldE

B!!

ChangingB

producesE

Faraday

sLaw

! "+=#

dt

d

IldB Eooo $

!! Ampre-

MaxwellLaw

Ior Changing

EproducesB

These equations describe all electric & magnetic phenomena(in the absence of dielectric or magnetic materials).

RAP, BSB 6/20/12


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Maxwell

s equations for spiky fields

0

E !

"# =

! !" ( )i ext

1

( , ) ( ) ( , )N

i

i

r t q r r t r t ! " !=

= # +$! ! ! !

( ) ext1( , ) ( ) ( ) ( , )

N

i i i

ij r t q v t r r t j r t!

=

= " +#! !! ! ! ! !

0B! ="

BE

t

!"# =$

!

! !

0 0

EB j

t !

"#$ = +

"

! ! !

where

BSB 6/15/12


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Delta functions: (x)dx= 1,0)=#, x) = 0 forx "0

Cartoon:S

.B.Cahn&B.E.Nadgorny,

AGuideto

PhysicsProblems,Pa

rt1,

PlenumPress,NY1994

BSB 6/20/12


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Derivation of Plasma Fluid Eqns1. The challenge of plasma physics

2. Kinetic equations Klimontovich, Boltzmann, Vlasov Distribution functions

3. Particle motion

4. Velocity moments of Boltzmann equation

+ approximations=> fluid equations &magnetohydrodynamics


BSB 6/15/12


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Cartoon: Astrophysics

Cartoon:NickD.Kim,strang

e-matter.net

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For each species of particle, $,define the spiky microscopic phase-space

distribution function of Klimontovich

Here N$

= # of particles of species $%

Take a statistical approach to trackingthe large number of plasma particles

K

1

( , , ) ( ( )) ( ( ))N

i i

i

f r v t r r t v v t!

! " "

=

= # #$! ! ! ! ! !

BSB 6/15/12


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DifferentiatingKlimontovich distribution=> Klimontovichequation

K

1

( , , ) ( ( )) ( ( ))N

i i

i


! " "

=

= # #$! ! ! ! ! !

K

K K

i iv

f dr dvf ft dt dt

!

! !

"= # $ # $

"

! !

" "

BSB 6/15/12

( )K! K K 0vf q

v f E v B f t m

! !

"+ # + + $ # =

"

! ! ! !! !" "

K

K Kv

fv f a f

t

!

! !

"=# $ # $

"

! !! !" "

Let the delta functions do the localizing to particles:

Inserting Lorentz force => Klimontovich equation:


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TheKlimontovich equation conservesparticle number and spikiness

K

1

( , , ) ( ( )) ( ( ))N

i i

i


! " "

=

= # #$! ! ! ! ! !

BSB 6/15/12

K!d

0d

f

t=Recognizing convective derivative:

( )K! K K 0vf q

v f E v B f t m

! !

"+ # + + $ # =

"

! ! ! !! !" "


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Ensemble average to obtain smoothfunctions and the Boltzmannequation

K

1

( , , ) ( ( )) ( ( ))N

i i

i


! " "

=

= # #$! ! ! ! ! !

BSB 6/15/12

K ,

,

, etc.

K

K

f f

f f f

E E

! !

! ! !"

# < >

# $

# < >

! !

( )! ! !c

d

d v

f f fqv f E v B f

t t m t ! !

" "# $= + % + + & % =' (" ") *

! ! ! !! !" "

Here the right-hand-side is short hand for a

complex collision term.

=> the Boltzmannequation:

( )K! K K 0vf q

v f E v B f

t m

! !

"+ # + + $ # =

"

! ! ! !! !" "


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A distribution function f(x,y,z,vx,vy,vz)is a density in phase space

Figure: F.F. Chen, Introduction to Plasma Physics and

Controlled Fusion, Vol. 1, 2nded., Plenum Press, NY, 1984.

This fvariesin space

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Collisions drive the plasmatowardmaximum entropy

0 M

( )( , ) | exp ( )

q rf r v n f v

k T

!! ! "

# !

"=

$ %&= ' (

' () *

!

! ! !

2 2- /

M 3 3/ 2

21( ) ,v a

k Tf v e a

a m

! "

"#

= $

!

Maxwellian distribution function

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Plasma distribution functions areoften non-Maxwellian

Figure: F.F. Chen, Introduction to Plasma Physics andControlled Fusion, Vol. 1, 2nded., Plenum Press, NY, 1984.

BSB 6/20/12


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Plasma distribution functions areoften anisotropic


Controlled Fusion, Vol. 1, 2nd

ed., Plenum Press, NY, 1984. BSB 6/20/12


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Phase-space evolution can becomplex and beautiful

Figure: Electron 2-stream instability,

H.L. Berk, C.E. Nielson, and K.V.Roberts, Phys. Fluids 13, 986 (1970).In F.F. Chen, Introduction to PlasmaPhysics and Controlled Fusion, Vol. 1,2nded., Plenum Press, NY, 1984.

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To better understand Boltzmann equation, followparticles in phase space & calculate collisions

BSB 6/16/12

One Boltzmann equation per species of particle

! Describes the evolution of the distributionof particles in phase-space

How do particles flow in phase space?

! What is the effect of the collision term (on rhs)?

How would fevolve without collisions?

Neglect collisions => Vlasov equation

( )! ! !c

d

d v

f f fqv f E v B f

t t m t ! !

" "# $= + % + + & % =' (" ") *

! ! ! !! !" "

( )! !d

0d

v

f f qv f E v B f

t t m

! !

"= + # + + $ # =

"

! ! ! !! !" "


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Derivation of Plasma Fluid Eqns

1. The challenge of plasma physics

2. Kinetic equations Klimontovich, Boltzmann, Vlasov Distribution functions

3. Particle motion

4. Velocity moments of Boltzmann equation+ approximations=> fluid equations &magnetohydrodynamics


BSB 6/15/12


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Charged Particles Curve in a

Magnetic Field

!A magnetic field can change the velocity of amoving charged particle, but not its speed.

!A moving particle

s direction will bechanged by a magnetic field if it has avelocity component perpendicular to thefield.

sin

F qv B

F qvB !

= "

=

!

!Magnetic force on a

moving charged particle:

RAP, BSB 6/19/12


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Cyclotron Motion

Orbital Period T:

2 2 2mv!R m

T v v qB qB

! !"

" "

= = =

Orbital Frequencyfc:1

2c

qBf

T m!= =

Cyclotron

Frequency

. . . . . .. . . . . .

. . . . . .

. . . . . .

. . . . . .

out of plane

R

vIf a moving charged particles

trajectory is entirely within

a uniform transverse

magnetic field, its

orbit is closed and circular.

B

!

RAP, BSB 6/19/12

2mv

F qv BR

!

! != =


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Motion of a charged particle in an arbitrarydirection relative to a magnetic field

Such motion of positiveions from the solar wind

ionizes the atmosphereand is responsible for thephenomenon of theAurora Borealis(northern lights)

v

!

B

!

B

!

If is neither perpendicular

nor parallel to , a helical

(spiral) motion about the

direction of results.

RAP, BSB 6/20/12

Fig.:H.D.Yo

ungandR.A.Freedman

,

Sears

and

Zemanskys

University

Physics,11the

d.,Pearson/Addison

Wesley,San

Francisco,CA,2004.

Figure:D.C.Giancoli,Physics

for

Scientists

&

Engineers

withModernPhysics,

3rde

d.,

PrenticeHall,NJ,2000.


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Solar storms make the news

BSB 6/20/12

The Daily Show with Jon StewartComedy Central, 2004


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A Magnetic Bottlefor Charged Particles

Magnetic fields are used to confine thecharged particles in hot ionized plasmas

(e.g., in fusion energy research).RAP, BSB 6/20/12

Fig.:H.D.YoungandR.A.Freedman,

SearsandZemanskysUnivers

ity

Physics,11the

d.,Pe

arson/AddisonWesley,SanFrancisco,CA,2004.


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Theoretical Physicists Design

Magnetic

Confinement

Device

Cartoon:ZT-H, LANL

BSB 6/20/12


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1953: Perhapsatron, LANL


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Cartoon: State-of-art

wheel

Cartoon:S.

Harris,

EinsteinSimplified:CartoonsonScience

,

RutgersUniversityPress,NewBrunswick,NJ,1989.

BSB 6/20/12


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Achieving a long &E~1srequires a large (10 m) plasma

InternationalThermonuclearExperimentalReactor (ITER)

500 MW400 seconds

Gain Q>10

$21 billion(July 2010)

RES, BSB 6/20/12Figure: ITER, www.iter.org


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E perpendicular to B makes acharged particle drift sideways

Figure: D.R. Nicholson, Introduction to Plasma Theory, Wiley, NY, 1983.

2, the " " drift

E

E Bv E BB

!= !

! !!

+

BSB 6/20/12


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Bonus Quiz

An electron in vacuum is released from rest at alocation near the Earth

s surface where E= 0,B= 0.5 G north, and g= 9.8 m/s2down. Describe andsketch the motion of the electron.

BSB 6/15/12

gWest East

B


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The electron makes a cusp motion,falling then rising, drifting west

!"! $%$&%&'

g

West East

B

This image is not quitecorrect: because v(0)=0,the tops of the orbit

have radius r = 0,i.e., they are points


Controlled Fusion, Vol. 1, 2nd

ed., Plenum Press, NY, 1984.


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2. Kinetic equations Klimontovich, Boltzmann, Vlasov

Distribution functions

3. Particle motion

4. Velocity moments of Boltzmann equation+ approximations=> fluid equations & magnetohydrodynamics


BSB 6/15/12


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Integrate f(x,y,z,vx,vy,vz) over 3D velocity spaceto obtain density & other fluid quantities

BSB 6/20/12

( , , ) ( , , , , , )x y z x y zn x y z dv dv dv f x y z v v v! !

" " "

#" #" #"

$ % % %

( , , , , , )x y z x y zn dv dv dv v f x y z v v v! ! !

" " "

#" #" #"

$ % % %u! !

( , , ) ( , , )x y z q n x y z! !!

" =#

( , , ) ( , , ) ( , , )x y z q n x y z x y z! ! !!

=

"j u!


Controlled Fusion, Vol. 1, 2nded., Plenum Press, NY, 1984.


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Integrate Boltzmann eqn to obtain fluid eqns thatcapture the main plasma collective properties

BSB 2/20/09

( )n

n St

!

! ! !

"+#=

"u

! !"

For each species $:

Mass conservation

Momentumconservation

Energy conservation: Heat equation#simplest to use Equation of State

dm n n q m S dt

!! ! ! ! ! ! ! ! ! !"

"

# $= + % & ' & +( ) *u

E u B P u R ! ! ! !! !

"

( )m n!" ! ! !" ! " "! #= $ $ = $R u u R ! !

d

dt t !

"= + #"

u

!!"! Convective derivative:

!

Resistive drag force density:

p

p nT Cn!

=

= =

P I


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With further approximations, obtain single fluidmagnetohydrodynamics (MHD), describing an electricallyconducting fluid in the presence of magnetic field

collision

1,L

C!

"

>>

e in n Z!

0 0B j j! " = # !="

, 0B

E Bt

!" # = $ "=

!

! ! ! !"

Slow

Quasi-neutral Thermal equilibrium

Equations:

Big mean freepath De L, ,L r! !>>

e iT T!

Mass Continuity

Equation of state

Ampere

s Law

Faraday Law

Ohm

s Law E u B j!+ " =!

!

d0, ( 1)

d

p Tp Z

t M

!

!

" #= = +$ %

& '

d ( ) 0d

u

t

!!+"=

! !"

Momentum Equationd

d

uj B p g

t! != " # $ +

! !! !

BSB 6/16/12


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2. Kinetic equations Klimontovich, Boltzmann, Vlasov

Distribution functions

3. Particle motion

4. Velocity moments of Boltzmann equation+ approximations=> fluid equations &magnetohydrodynamics


BSB 6/15/12


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Plasmas are complex systems

!Plasmas can exhibit self-organizedbehavior of high complexity

!Self-organization occurs in many areas"Space & astrophysics, biosystems,

micro- and nano- components, protein folding

"Selective decayprocesses, thermodynamics

Dissipation of some quantity on small scales(e.g., energy in eddies, turbulence)

Persistence of other quantities on larger spatialscales (e.g., helicity)

TPI, BSB 6/15/12


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Example of self-organization (numericalsimulation): reorganization of plasmapressure into stable profile

Plasma

Inner conductor

Vacuum

Hard core currentPlasma current

Pressure

Radius

Wall

Stableconcaveregion

Stable if'p and (are within

Kadomtsevbounds

%

RS, SF, VM, BSB 6/20/12

Outer conductor (implodes for MTF)


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Movie of dynamic formation phase

RES, VM, BSB 5/26/03


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Movie of equilibration phase

RES, VM, BSB 5/26/03


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Simulated plasma finds Kadomtsevmarginally stable profile

Point marked with xiswhere beta for simulated

profile is matched to

Kadomtsev beta

2 4 6 8 10 r, cm0

1

2

3

p, MPa

marginallystable

pressure

profile

!max"0.401

RS, VM, BSB 5/26/03

Through m=0 turbulence,the plasma organizes itselfinto the marginally stable

Kadomtsev state


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Field Reversed Configuration

high-(self-organized plasma ~ 1

compact torus

like spheromak

Can translateinto liner

The LANL FRC has parameters orders ofmagnitude different than previous FRCs.How will FRC behave under compression?How will liner interact with FRC?

RES , BSB 8/24/08


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Experiment on 1-MA/100-ns Zebra (UNR) studiesplasma formed by multi-MG field on aluminum

BSB, NLG 4/22/08


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Zebra Megagauss Experiment

Movie: Stephan Fuelling, UNR, Oct 2006

SF, BSB 6/20/12


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Some good plasma references

# P.M. Bellan, Fundamentals of Plasma Physics, CambridgeUniversity Press, Cambridge, United Kingdom, 2006.

# J.A. Bittencourt, Fundamentals of Plasma Physics, 3rded.,Springer Science, New York, NY, 2004.

# F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,Vol. 1: Plasma Physics, 2nded., Plenum Press, New York, NY, 1984.

# R.P. Drake, High-Energy-Density Physics: Fundamentals, InertialFusion and Experimental Astrophysics, Springer Verlag, New York,NY, 2006.

#

R.J. Goldston and P.H. Rutherford , Introduction to PlasmaPhysics, Institute of Physics Publishing, Philadelphia, PA, 1995.

# D.R. Nicholson, Introduction to Plasma Theory, John Wiley & Sons,New York, NY, 1983.

BSB 6/20/12


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Thanks!

BSB 6/20/12Photo: University of Nevada Reno www unr edu

Bauer2012_06_20DerivationPlasmaFluidEquations

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