Plasma physics by Dr. imran aziz
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Plasma Physics DR.MOHAMMAD IMRAN AZIZ Assistant Professor(Sr.) PHYSICS DEPARTMENT PHYSICS DEPARTMENT SHIBLI NATIONAL COLLEGE, AZAMGARH (India). 1 [email protected]
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Plasma physics by Dr. imran aziz
Transcript of Plasma physics by Dr. imran aziz
Plasma Physics
DR.MOHAMMAD IMRAN AZIZAssistant Professor(Sr.)
PHYSICS DEPARTMENTPHYSICS DEPARTMENTSHIBLI NATIONAL COLLEGE, AZAMGARH (India).
• An ionized gasis characterized, in general, by a mixture of neutrals, (positive) ions and electrons.
• For a gas in thermal equilibriumthe Saha equationgives the expected amount of ionization:
Ionized Gases
gives the expected amount of ionization:
• The Saha equationdescribes an equilibrium situation between ionization and (ion-electron) recombination rates.
3/ 2/212.4 10 i BU k Ti
n i
n Te
n n−⋅�
• SolvingSaha equation
Example: Saha Equation
3/ 2/212.4 10 i BU k Ti
n i
n Te
n n−⋅�
/2 21 3/ 22.4 10 i BU k Ti nn n T e−⋅�
Backup: The Boltzmann Equation
The ratio of the number density (in atoms per m^3) of atoms in energy state B to those in energy state A is given by
NB / NA = ( gB / gA ) exp[ -(EB-EA)/kT ]
where the g's are the statistical weights of each level (the where the g's are the statistical weights of each level (the number of states of that energy). Note for the energy levels of hydrogen
gn = 2 n2
which is just the number of different spin and angular momentum states that have energy En.
From Ionized Gas to Plasma
• An ionized gasis not necessarily a plasma
• An ionized gas can exhibit a “collective behavior” in the interaction among charged particles when when long-range forces prevailover short-range forceslong-range forces prevailover short-range forces
• An ionized gas could appear quasineutralif the charge density fluctuations are contained in a limited region of space
• A plasmais an ionized gas that presents a collective behavior andis quasineutral
The “Fourth State” of the Matter
• The matter in “ordinary” conditions presents itself in three fundamental states of aggregation: solid, liquid and gas.
• These different states are characterized by different levels of bondingamong the molecules.levels of bondingamong the molecules.
• In general, by increasing the temperature(=average molecular kinetic energy) a phase transitionoccurs, from solid, to liquid, to gas.
• A further increase of temperatureincreases the collisional rateand then the degree of ionization of the gas.
The “Fourth State” of the Matter (II)
• The ionized gas could then become a plasma if the proper conditions for density, temperature and characteristic length are met (quasineutrality, collective behavior).
• The plasma state does not exhibit a different state of aggregationbut it is characterized by a different behavior when subject to electromagnetic fields.
1 Unmagnetized Plasmas
1.1 Charge in an Electric Field
1.2 Collisions between Charged Particles
1.1 Charge in an Electric Field
• Electric force:
F=qE
Dimensional analysis:Dimensional analysis:
N=C V/m
• A positive isolated chargeq will produce a positive electric field at a point distance r given by
• The force on another positive charge will be repulsivesince F=qE is directed asr
304
q
rπε= r
E 2
1/
V C
m F mm =
1.2 Collisions between Charged Particles
r0
v
• Interaction time T=r0/v
• Change in momentum:
01 2 1 22
0 0 00
1 1( )
4 4
rq q q qmv mv FT
v r vrπε πε∆ = = =�
• Impact parameter:
1 20 2
0
14q q
rmvπε
=
• Collisional cross section:
( )22 1 2
0 2 2 40
1
16
q qr
m vσ π
πε= =
Charge in an Electric Field
• Electric force:
F=qE
Dimensional analysis:Dimensional analysis:
N=C V/m
• A positive isolated chargeq will produce a positive electric field at a point distance r given by
• The force on another positive charge will be repulsivesince F=qE is directed asr
304
q
rπε= r
E 2
1/
V C
m F mm =
1.1 Charge in an an Uniform Magnetic Field
• Magnetic force:
Dimensional analysis:
N=C T m/s
• Equation of the motion for a positive isolated chargeq
m q= = ×F v v B&
• Equation of the motion for a positive isolated chargeqin a magnetic field B:
x y z
x y z
m q q v v v
B B B
= = × =
i j k
F v v B&
Charge in an an Uniform Magnetic Field (II)
• Case of a magnetic field B directed along z:
( ) ( ) ( )x y z y z z y x z z x x y y x
x y z
v v v v B v B v B v B v B v B
B B B
= − − − + −
i j k
i j k
• Case of a magnetic field B directed along z:
0zmv =&
x y zmv qv B=&
y x zmv qv B= −&
Charge in an an Uniform Magnetic Field (III)
• By taking the derivative of x y zmv qv B=&
• Then replacing : /v v qB m= −&
x y zmv qv B=&& &
/y x zv v qB m= −&
( )2/x x zv v qB m= −&&
• Analogously:
( )2/y y zv v qB m= −&&
Charge in an an Uniform Magnetic Field (III)
• The equations for vx and vy are harmonic oscillator equations.
• The oscillation frequency, called cyclotron frequencyis defined as:
/c zq B mω =
Charge in an an Uniform Magnetic Field (IV)
• The solution of the harmonic oscillator equation is
( ) ( )exp expx c cv A i t B i tω ω= + −
The Kinetic Theory
1 The Distribution Function
2 The Kinetic Equations
3 Relation to Macroscopic Quantities3 Relation to Macroscopic Quantities
The Distribution Function
1 The Concept of Distribution Function
2 The Maxwellian Distribution
1.1 The Concept of Distribution Function
• General distribution function: f=f(r,v,t)
• Meaning: the number of particles per m3 at the position r, time t and velocity between v and v+dvis f(r,v,t) dv, where dv= dvx dvy dvz
• The densityis then found as
• If the distribution is normalizedas
ˆ ( , , ) 1f t d∞
−∞=∫ r v v
then f^ represents a probability distribution
ˆ( , , ) ( , ) ( , , )f t n t f t=r v r r v
3( , ) ( , , ) ( , , )x y zn t dv dv dv f t f t d v∞ ∞ ∞ ∞
−∞ −∞ −∞ −∞= =∫ ∫ ∫ ∫r r v r v
The Maxwellian Distribution
• The maxwellian distribution is defined as:
3/ 2 2
2ˆ exp
2mB th
m vf
k T vπ −=
where
2 2 2x y zv v v v= + + 2 /th Bv k T m=
• The known result
ˆ ( ) 1mf d∞
−∞=∫ v v
2exp( )x dx π∞
−∞− =∫
yields
The Maxwellian Distribution (II)
• The root mean square velocity for a maxwellian is:
2 3 /Bv k T m=
• The average of the velocity magnitudev=|v| is:
recall 212 3 BW mv k T= =
• In one direction:
3 2ˆ ( ) 2 2 /thm B
vv vf dv k T mπ
π
∞
−∞= = =∫ v
• The average of the velocity magnitudev=|v| is:
0xv = ˆ ( ) / 2 /x m th Bv vf d v k T mπ π∞
−∞= = =∫ v v
The Maxwellian Distribution (III)
• For a Maxwellian
• The distribution w.r.t. the magnitude of v
0
( ) ( )g d f d∞ ∞
−∞=∫ ∫v v v v
• For a Maxwellian
3/ 2 22
24 exp
2mB th
m vg n v
k T vπ
π −=
The Kinetic Equations
1 The Boltzmann Equation
2 The Vlasov Equation
3 The Collisional Effects3 The Collisional Effects
1. The Boltzmann Equation
• A distribution function: f=f(r,v,t) satisfies the Boltzmann equation
• The r.h.s. of the Boltzmann equation is simply the
c
f f ff
t m t
∂ ∂ ∂ + ⋅∇ + ⋅ = ∂ ∂ ∂
Fv
v
• The r.h.s. of the Boltzmann equation is simply the expansion of d f(r,v,t)/dt
• The Boltzmann equation states that in absence of collisions df/dt=0
x
vx
tt+∆∆∆∆tMotion of a group of
particles with constant densityin the phase space: [email protected]
2. The Vlasov Equation
• In general, for sufficiently hot plasmas, the effect of collisions are less and less important
• For electromagnetic forces acting on the particles and no collisions the Boltzmann equation becomes
( )f q f∂ ∂( ) 0f q f
ft m
∂ ∂+ ⋅∇ + + ⋅ ⋅ =∂ ∂
v E v Bv
that is called the Vlasov equation
3. The Collisional Effects
• The Vlasov equation does not account for collisions
0c
f
t
∂ = ∂
• Short-range collisions like charged particles with neutrals can be described by a Boltzmann collision neutrals can be described by a Boltzmann collision
operatorin the Boltzmann equation
• For long-range collisions, like Coulomb collisions, a statistical approach yields the Fokker-Planck
collision term
• The Boltzmann equation with the Fokker-Planck collision term is simply named the Fokker-Planck
equation. [email protected]
4. Relation to Macroscopic Quantities
1 The Moments of the Distribution Function
2 Derivation of the Fluid Equations
1. The Moments of the Distribution Function
• If A=A(v) the averageof the function A for a distribution function f=f(r,v,t) is defined as
3x y zdv dv dv d v
∞ ∞ ∞ ∞
−∞ −∞ −∞ −∞=∫ ∫ ∫ ∫
• Notation: define
∞3
3
3
( , , ) ( , , )
( , )
( , , )
1( , , ) ( , , )
( , )
v
A t f t d v
A t
f t d v
A t f t d vn t
∞
−∞∞
−∞∞
−∞
= =
=
∫
∫
∫
r v r v
r
r v
r v r vr
The Moments of the Distribution Function (II)
• General distribution function: f=f(r,v,t)
• The densityis defined as the 0th order momentand was found to be
3( , ) ( , , ) ( , , )x y zn t dv dv dv f t f t d v∞ ∞ ∞ ∞
−∞ −∞ −∞ −∞= =∫ ∫ ∫ ∫r r v r v
−∞ −∞ −∞ −∞
• The mass densitycan be then defined as
3( , ) ( , ) ( , , )t mn t m f t d vρ∞
−∞= = ∫r r r v
The Moments of the Distribution Function (III)
• The 1st order momentis the average velocity or fluid velocity is defined as
31( , ) ( , , )
( , )t f t d v
n t
∞= ∫u r v r v
r
• The momentum densitycan be then defined as
3( , ) ( , ) ( , , )n t m t m f t d v∞
−∞= ∫r u r v r v
( , ) ( , , )( , )
t f t d vn t −∞
= ∫u r v r vr
The Moments of the Distribution Function (IV)
• Higher moments are found by diadicproducts with v
• The 2nd order momentgives the stress tensor (tensor of second order)
∞3( , ) ( , , )t m f t d v
∞
−∞= ∫Π r vv r v
• In the frame of the moving fluidthe velocity is w=v-u. In this case the stress tensor becomes the
pressure tensor3( , ) ( , , )t m f t d v
∞
−∞= ∫P r ww r v
2 Derivation of the Fluid Equations
• Boltzmann equation written for the Lorentz force
( )c
f q f ff
t m t
∂ ∂ ∂ + ⋅∇ + + × ⋅ = ∂ ∂ ∂ v E v B
v
• Integrate in velocity space:• Integrate in velocity space:
( )3 3 3 3
c
f q f fd v f d v d v d v
t m t
∂ ∂ ∂ + ⋅ ∇ + + × ⋅ = ∂ ∂ ∂ ∫ ∫ ∫ ∫v E v B
v
• From the definition of density
3 3f nd v fd v
t t t
∂ ∂ ∂= =∂ ∂ ∂∫ ∫
Derivation of the Fluid Equations (II)
• Since the gradient operator is independent from v:
( )3 3f d v f d v n⋅∇ = ∇ ⋅ = ∇ ⋅∫ ∫v v u
• Through integration by parts it can be shown that
( ) 3 0q f
d vm
∂+ × ⋅ =∂∫ E v Bv
( )m ∂∫ v
3 0c
fd v
t
∂ = ∂ ∫
• If there are no ionizations or recombination the collisional term will not cause any change in the number of particles (no particle sources or sinks)
therefore
Derivation of the Fluid Equations (III)
• The integrated Boltzmann equation then becomes
( ) 0n
nt
∂ + ∇ ⋅ =∂
u
• In general moments of the Boltzmann equationare taken by multiplying the equation by a vector
that is known as equation of continuity
taken by multiplying the equation by a vector function g=g(v) and then integrating in the
velocity space
• In the case of the continuity equation g=1
• For g=mv the fluid equation of motion, or momentum equationcan be obtained
Derivation of the Fluid Equations (IV)
• Integrate the Boltzmann equation in velocity space with g=mv
( )3 3 3
3
f fm d v m f d v q d v
tf
m d vt
∂ ∂+ ⋅∇ + + × ⋅ =∂ ∂
∂ = ∂
∫ ∫ ∫
∫
v vv v E v Bv
vc
m d vt
= ∂ ∫ v
• The first term is
( )3
3 3 33
fd vfm d v m fd v m fd v m n
t t t tfd v
∂ ∂ ∂ ∂= = = ∂ ∂ ∂ ∂
∫∫ ∫ ∫
∫
vv v u
Derivation of the Fluid Equations (V)
• Further simplifications yield the final fluid equation of motion
( ) ( ) collmn qnt
∂ + ⋅ ∇ = + × − ∇ ⋅ + ∂
uu u E u B P P
where u is the fluid average velocity, P is the stress tensor and Pcoll is the rate of momentum change
due to collisions
• Integrating the Boltzmann equation in velocity space with g=½mvv the energy equationis
The Kinetic Theory
1 The Distribution Function
2 The Kinetic Equations
3 Relation to Macroscopic Quantities3 Relation to Macroscopic Quantities
4 Landau Damping
4 Landau Damping
1 Electromagnetic Wave Refresher
2 The Physical Meaning of Landau Damping
3 Analysis of Landau Damping3 Analysis of Landau Damping
Electromagnetic Wave Refresher (II)
• The field directions are constant with time, indicating that the wave is linearly polarized
(plane waves).
• Since the propagation direction is also constant, this disturbance may be written as a scalar wave:
E = Emsin(kz-ωt) B = Bmsin(kz-ωt)E = Emsin(kz-ωt) B = Bmsin(kz-ωt)k is the wave number, z is the propagation
direction, ω is the angular frequency, Em and Bm
are the amplitudes of the E and B fields respectively.
• The phase constants of the two waves are equal (since they are in phase with one another) and
have been arbitrarily set to 0. [email protected]
The Physical Meaning of Landau Damping
• An e.m. waveis traveling through a plasma with phase velocity vφ
• Given a certain plasma distribution function (e.g.a maxwellian), in general there will be some particles with velocity close to that of the wave.
• The particles with velocity equal to vφ are called resonant particles
The Physical Meaning of Landau Damping (II)
• For a plasma with maxwellian distribution, for any given wave phase velocity, there will be more “near resonant” slower particles than “near resonant” fast particles
• On average then the wave will loose energy (damping) and the particles will gain energy(damping) and the particles will gain energy
• The wave damping will create in general a local distortionof the plasma distribution function
• Conversely, if a plasma has a distribution function with positive slope, a wave with phase velocity within that positive slope will gain energy
The Physical Meaning of Landau Damping (III)
• Whether the speed of a resonant particle increases or decreases depends on the phase of the waveat its initial position
• Not all particles moving slightly faster than the wave lose energy, norall particles moving slightly slower than the wave gain energy.slower than the wave gain energy.
• However, those particles which start off with velocities slightly above the phase velocity of the wave, if they gain energy they move away from the resonant velocity, if they lose energy they approach the resonant velocity.
The Physical Meaning of Landau Damping (IV)
• Then the particles which lose energy interact more effectivelywith the wave
• On average, there is a transfer of energyfrom the particles to the electric field.
• Exactly the opposite is true for particles with initial velocities lying just below the phase initial velocities lying just below the phase velocity of the wave.
The Physical Meaning of Landau Damping (V)
• The damping of a wave due to its transfer of energy to “near resonant particles” is called Landau damping
• Landau damping is independent of collisional or dissipative phenomena: it is a mere transfer of energy from an electromagnetic field to a particle energy from an electromagnetic field to a particle kinetic energy (collisionless damping)
Analysis of Landau Damping
• A plane wave travelling through a plasma will cause a perturbation in the particle velocity distribution: f(r,v,t) =f0(r,v,t) + f1(r,v,t)
• If the wave is traveling in the x direction the perturbation will be of the form
( )[ ]expf i kx tω∝ −
• For a non-collisional plasma analysis the Vlasov equation applies. For the electron species it will be
( )[ ]1 expf i kx tω∝ −
( ) 0f e f
ft m
∂ ∂+ ⋅ ∇ − + × ⋅ =∂ ∂
v E v Bv
Analysis of Landau Damping (II)
• A linearization of the Vlasov equation considering
0 1
;
0; 0
0
f f f= += + = += =
× =
0 1 0 1
0 0
E E E B B B ;
E B
v B (since only contributions along v are studied)
yields
0× =v B
011 1 0
ff ef E
t m
∂∂ + ⋅∇ − ⋅ =∂ ∂
vv
or, considering the wave along the dimension x,0
1 1 1x xx
fei f ikv f E
m vω ∂+ = −
∂
(since only contributions along v are studied)
Analysis of Landau Damping (III)
• The electric field E1 along x is not due to the wave but to charge density fluctuations
• E1 be expressed in function of the density through the Gauss theorem (first Maxwell equation)
or, in this case, considering a perturbed density n0en ε∇ ⋅ = −1E
or, in this case, considering a perturbed density n1
equivalent to the perturbed distribution f1
0xikE enε= −
• Finally the density can be expressed in terms of the distribution function as
31 1( , ) ( , , )n t f t d v
∞
−∞= ∫r r v [email protected]
Analysis of Landau Damping (IV)
• The linearized Vlasov equation for the wave perturbation
can be rewritten, after few manipulations as a
01 1 1x x
x
fei f ikv f E
m vω ∂+ = −
∂
can be rewritten, after few manipulations as a relation between ω, k and know quantities:
0 0 0ˆ /f f n=
where
( )
20
2
ˆ ( )1 p x x
xx
f v vdv
v kk
ωω
∞
−∞
∂ ∂=−∫
Analysis of Landau Damping (V)
• For a wave propagation problem a relation between ω and k is called dispersion relation
• The integral in the dispersion relation
( )
20
2
ˆ ( )1 p x x
xx
f v vdv
v kk
ωω
∞
−∞
∂ ∂=−∫
can be computed in an approximate fashion for a maxwellian distribution yielding
x−∞
20
2/
ˆ ( )1
2p x
px v k
f vi
vk ω
ωπω ω=
∂= + ∂
Analysis of Landau Damping (VI)
• For a one-dimensional maxwellian along the xdirection
20
1 2 3 2
ˆ ( ) 2expx x x
x th th
f v v v
v v vπ ∂ = − − ∂
• This will cause the imaginary part of the • This will cause the imaginary part of the expression
20
2/
ˆ ( )1
2p x
px v k
f vi
vk ω
ωπω ω=
∂= + ∂
to be negative (for a positive wave propagation direction)
Analysis of Landau Damping (VII)
• For a wave is traveling in the x direction the of the form
( )[ ] ( ) ( )[ ]( ) ( )[ ]( ) ( ) ( )
1 exp exp exp
exp exp
exp exp exp
R I
R I
R I
f i kx t ikx i i t
ikx i t
ikx i t t
ω ω ωω ω
ω ω
∝ − = − + =
= − + =
= −
a negative imaginary part of ω will produce an attenuation, or damping, of the wave.
( ) ( ) ( )exp exp expR Iikx i t tω ω= −
Plasmas as Fluids: Introduction
• The particle description of a plasma was based ontrajectories forgivenelectric and magnetic fields
• Computationalparticle modelsallow in principleto obtain a microscopic description of the plasmawith its self-consistentelectric and magnetic fields
• The kinetic theory yields also a microscopic,self-consistentdescription of the plasma based on theevolution of a “continuum” distribution function
• Most practical applications of the kinetic theoryrely also on numerical implementationof thekinetic equations
Plasmas as Fluids: Introduction (II)
• The analysis of several important plasma phenomena does notrequire the resolution of a
microscopic approach
• The plasma behavior can be often well represented by a macroscopic descriptionas in a fluid modelby a macroscopic descriptionas in a fluid model
• Unlike neutral fluids, plasmas respond to electric and magnetic fields
• The fluidodynamics of plasmasis then expected to show additional phenomena than ordinary hydro,
or gasdynamics
Plasmas as Fluids: Introduction (III)
• The “continuum” or “fluid-like” character of ordinary fluids is essentially due to the frequent
(short-range) collisionsamong the neutral particles that neutralize most of the microscopic
patterns
• Plasmas are, in general, less subject to short-range collisions and properties like collective effectsand quasi-neutralityare responsible for the fluid-like
behavior
Plasmas as Fluids: Introduction (IV)
• Plasmas can be considered as composed of interpenetratingfluids (one for each particle
species)
• A typical case is a two-fluid model: an electron and an ion fluids interacting with each other and and an ion fluids interacting with each other and
subject to e.m. forces
• A neutral fluidcomponent can also be added, as well as other ion fluids (for different ion species or
ionization levels)
The Fluid Description of Plasmas
1 The Fluid Equations for a Plasma
2 Plasma Diffusion
3 Fluid Model of Fully Ionized Plasmas3 Fluid Model of Fully Ionized Plasmas
Fluid Model of Fully Ionized Plasmas
. The Magnetohydrodynamic Equations
.Diffusion in Fully Ionized Plasmas
. Hydromagnetic Equilibrium
. Diffusion of Magnetic Field in a Plasma. Diffusion of Magnetic Field in a Plasma
Magnetohydrodynamic Equations
• Goal: to derive a single fluiddescription for a fully ionized plasma
• Single-fluid quantities: define mass density, fluid velocity and current density from the same quantities referred to electrons and ions:
( )m i i e e i em n m n n m mρ = + ≈ +
( ) ( )1( )i e
i i e em i e
m mm n m n
m mρ+
= + ≈+
i ei e
u uu u u
( ) ( )i i e e i ee n n ne= − ≈ −j u u u [email protected]
Magnetohydrodynamic Equations (II)
• Equation of motion for electron and ions with Coulomb collisions, ne=ni and a gravitational term (that can be used to represent any additional non
e.m. force):
( ) ( )ii i i i i i ie inm q n p m n
t
∂ + ⋅ ∇ = + × − ∇ + + ∂
uu u E u B P gi i i i i i ie it ∂
( ) ( )ee e e e e e ei enm q n p m n
t
∂ + ⋅∇ = + × − ∇ + + ∂
uu u E u B P g
• Approximation 1: the viscosity tensorhas been neglected, acceptable for Larmor radius small w.r.t. the scale length of variations of the fluid
quantities. [email protected]
Magnetohydrodynamic Equations (III)
( ) ( )ii i i i ii e iinm q n p m n
t+∂ = + × − ∇ + + ∂
⋅∇uE u B Pu u g
• Approximation 2: neglect the convective term, acceptable when the changes produced by the fluid convective motion are relatively small
( ) ( )ee e e e ee i eenm q n p m n
t+∂ = + × − ∇ + + ∂
⋅∇uE u B Pu u g
• These equation can be addedand by setting p=pe+pi, -qi=qe=e and Pei=-Pie obtaining:
( ) ( ) ( )i i e e i e i en m m en p n m mt
∂ + = − × − ∇ + +∂
u u u u B [email protected]
Magnetohydrodynamic Equations (IV)
• By substituting the definition of the single fluid variables r, u and j the equation
can be written as
( ) ( ) ( )i i e e i e i en m m en p n m mt
∂ + = − × − ∇ + +∂
u u u u B g
m mpt
ρ ρ∂ = × − ∇ +∂u
j B g
that is the single fluid equation of motionfor the mass flow. There is no electric force because the
fluid is globally neutral (ne=ni).
can be written as
Magnetohydrodynamic Equations (V)
• To characterize the electrical propertiesof the single-fluid it is necessary to derive an equation
that retains the electric field
• By multiplying the ion eq. of motion by me, the electron one by mi, by subtracting them and taking
the limit m m=>0, d/dt=>0 it is obtainedthe limit me/ mi=>0, d/dt=>0 it is obtained
( )1ep
enη+ × = + × − ∇E u B j j B
that is the generalized Ohm’s lawthat includes the Hall term(jxB) and the pressure effects
Magnetohydrodynamic Equations (VI)
• Analogous procedures applied to the ion and electron continuity equations (multiplying by the masses, adding or subtracting the equations) lead to the continuity for the mass density rm or for the
charge density r:
ρ∂ ( ) 0mmt
ρ ρ∂ + ∇ ⋅ =∂
u
0t
ρ∂ + ∇ ⋅ =∂
j
• The single-fluid equations of continuity and motion and the Ohm’s law constitute the set of
magnetohydrodynamic(MHD) equations. [email protected]
Diffusion in Fully Ionized Plasmas
• The MHD equations, in absence of gravity and for steady-state conditions, with a simplified version
of the Ohm’s law, are
0 p= × − ∇j B
η+ × =E u B j
• The parallel(to B) component of the last equation reduce simply to the ordinary Ohm’s law:
η+ × =E u B j
η=E j� � �
Diffusion in Fully Ionized Plasmas (II)
( ) η⊥ ⊥× + × × = ×E B u B B j B
• The component perpendicularto B is found by taking the the cross product with B
that is2B pη η⊥ ⊥ ⊥× − = × = ∇E B u j B
• The first term is the usual ExB drift (for both species together), the second is a diffusiondriven
by the gradient of the pressure
2B pη η⊥ ⊥ ⊥× − = × = ∇E B u j Band finally
2 2p
B B
η⊥⊥
×= − ∇E Bu
Diffusion in Fully Ionized Plasmas (III)
• The diffusion in the direction of -grad pproduces a flux
• For isothermal, ideal gas-type plasma the perpendicular flux can be written as
2n n p
B
η⊥⊥ ⊥Γ = = − ∇u
that is a Fick’s lawwith diffusion coefficient
perpendicular flux can be written as
2
( )B i B en k T k Tn
B
η⊥⊥
+Γ = − ∇
2
( )B i B en k T k TD
B
η⊥⊥
+=
named classical diffusion coefficient [email protected]
Diffusion in Fully Ionized Plasmas (IV)
• The classical diffusion coefficient is proportionalto 1/B2 as in the case of weakly ionized plasmas: it is typical of a random-walk type of process with
characteristic step length equal to the Larmor radius
• The classical diffusion coefficient is proportional to n, not constant, because does not describe the
scattering with a fixed neutral background
• Because the resistivity decreases with T3/2 so does the classical diffusion coefficient (the opposite of
a partially ionized plasma)
Diffusion in Fully Ionized Plasmas (IV)
• The classical diffusion is automatically ambipolar, as it was derived for a single fluid (both species
are diffusing at the same rate)
• Since the equation for the perpendicular velocity does not contain any term along E that depend on
E E itself, it can be concluded that there is no perpendicular mobility: an electric field
perpendicular to B produces just a ExB drift.
Diffusion in Fully Ionized Plasmas (V)
• Experimentswith magnetically confined plasmas showed a diffusion rate much higher than the one
predicted by the classical diffusion
• A semiempirical formula was devised: this is the Bohm diffusion coefficientthat goes like 1/Band
increases with the temperature:
116
B eBohm
k TD
eB⊥ =
• Bohm diffusion ultimately makes more difficult to reach fusion conditions in magnetically confined
Hydromagnetic Equilibrium
• The MHD momentum equation, in absence of gravity and for steady-state conditions is
considered to describe an equilibrium condition for a plasma in a magnetic field.
p∇ = ×j B
• The momentum equation expresses the force balancebetween the pressure gradient and the
Lorentz force
• In force balance both j and B must be perpendicular to grad p: j and B must then lie on
constant p [email protected]
Hydromagnetic Equilibrium (II)
B
grad p
j
• For an axial magnetic field in a cylindrical configuration with radial pressure gradient, the
currentmust be azimuthal
• The momentum equation in the perpendicular plane (w.r.t. B) will then give an expression for j
Hydromagnetic Equilibrium (II)
• The cross product of the momentum with B yields2p B× ∇ = × × =B B j B j
and, in the usual approximations, solving for j yield again the expression for the diamagnetic
( )2 2B i B ep n
k T k TB B
× ∇ × ∇= = +B Bj
current
• From the MHD point of viewthe diamagnetic current is generated by the grad pforce that
interacts (via a cross product) with [email protected]
Hydromagnetic Equilibrium (IV)
• The connection between the fluid and the particle point of viewwas previously discussed: the
diamagnetic current arises from an unbalanceof the Larmor gyration velocities in a fluid element
• From a strict particle point of viewthe confinement of the plasma with a gradient of pressure occurs because each particle guiding
centeris tight to a line of force and diffusion is not permitted (in absence of collisions)
Hydromagnetic Equilibrium (V)
• For the equilibrium case under consideration, the momentum equation in the direction parallelto B
will be simply0
pp
s
∂∇ = =∂
where s is a generalized coordinatealong the lines of force. of force.
• For isothermal plasmait will be 0n
s
∂ =∂
then the density is constant along the lines of force
• This condition is valid only for a static case(u=0).
• For example in a magnetic mirrorthere are more particles trapped at the midplane (lower line of force density) than at the mirror end [email protected]
Waves inPlasmas
1 Electrostatic Waves in Non-Magnetized
Plasmas
2 Electrostatic Waves in Magnetized Plasmas2 Electrostatic Waves in Magnetized Plasmas
E.S. Waves in Non-Magnetized Plasmas
1. Wave fundamentals2. Electron Plasma Waves3. Sound waves4. Ion Acoustic Waves4. Ion Acoustic Waves
Wave Fundamentals
• Any periodic motion of a fluid can be decomposed, through Fourier analysis, in a superposition of sinusoidal components, at different frequencies
• Complex exponential notationis a convenient way to represent mathematically oscillating quantities: the physical quantity will be obtained by taking the real physical quantity will be obtained by taking the real
part
• A sinusoidal plane wavecan be represented as
( )0( , ) expf t f i tω = ⋅ − r k r
where f0 is the maximum amplitude, k is the propagation constant, or wave vector (k is the wavenumber) and w the angular [email protected]
Wave Fundamentals (II)
• If f0 is real then the wave amplitude is maximum (equal to f0) in r=0, t=0, therefore the phase angleof
the wave is zero
• A complex f0 can be used to represent a non zero phase angle:
( ) ( ) ( )0 0exp exp expf i t f i i tω δ δ ω ⋅ − + = ⋅ − k r k r
• A point of constant phaseon the wave will travel along with the wave front
• A constant phase on the wave implies
( ) 0d
tdt
ω⋅ − =k [email protected]
Wave Fundamentals (III)
• In one dimension it will be
( ) 0d dx
kx t vdt dt k ϕ
ωω− = ⇒ = �
where vf is defined as the wave phase velocity
• The wave can be then also expressed by• The wave can be then also expressed by
( )0( , ) expf x t f ik x v tϕ = − • The phase velocity in a plasma can exceedthe velocity of the light c, however an infinitely long
wave train that maintains a constant velocity does not carry any information, so the relativity is not violated.
Wave Fundamentals (IV)
• A wave carries informationonly with some kind of modulation
• An amplitude modulation is obtained for example by adding to waves of different frequencies (wave
“beating”)
• If a wave with phase velocity v is formed by two • If a wave with phase velocity vf is formed by two waveswith frequency separation 2Dw , both the two
components must also travel at vf
• The two components of the wave must then also have a difference in their propagation constant k equal to
2Dk
Wave Fundamentals (V)
• For the case of two wave beatingit can be written
( ) ( )0( , ) cosAf x t f k k x tω ω = + ∆ − + ∆
( ) ( )0( , ) cosBf x t f k k x tω ω = − ∆ − − ∆
• By summing the two waves and expanding with • By summing the two waves and expanding with trigonometric identities it is found
( ) ( ) [ ]0( , ) ( , ) 2 cos cosA Bf x t f x t f k x t kx tω ω + = ∆ − ∆ ⋅ −
• The first term of the r.h.s. is the modulating component(that does carry information)
• The second term of the r.h.s. is just the “carrier” component of the wave (that does not carry
information)[email protected]
Wave Fundamentals (VI)
• The modulating component travels at the group velocitydefined as
• Thegroupvelocitycanneverexceedc
0g g
dv v
k dkω
ω ω∆ →
∆= ⇒ =∆
• Thegroupvelocitycanneverexceedc
Electron Plasma Waves
• Thermal motions cause electron plasma oscillations to propagate: then they can be properly called
(electrostatic ) electron plasma waves
• By linearizing the fluid electron equation of motion with respect equilibrium quantities according to
0 1e e en n n= + 0 1e e eu u u= + 0 1E E E= +
the frequency of the oscillationscan be found as
2 2 2 232pe thk vω ω= +
where2 2th B e ev k T m= [email protected]
Electron Plasma Waves (II)
• Electron plasma waves have a group velocityequal to
• In general a relation linking w and k for a wave is called dispersion relation
2 23 32 2th th
d k kv v
dk vϕ
ωω
= =
• The slopeof the dispersion relation on a w-k diagram gives the phase velocityof the wave
Sound Waves
• For a neutral fluidlike air, in absence of viscosity, the Navier-Stokes equation is
• From the equation of state
( )m pt
ρ ∂ + ⋅ ∇ = −∇ ∂
uu u
pp
γ∇ =• From the equation of statem
pρ
∇ =
then
( )mm
p
t
γρρ
∂ + ⋅ ∇ = − ∂
uu u
• Continuityequation yields
( ) 0mmt
ρ ρ∂ + ∇ ⋅ =∂
Sound Waves (II)
• Linearizationof the momentum and continuity equations for stationary equilibrium yield
where m is the neutral atom mass and c is the sound
1 2 1 20
0
Bs
m N
p k Tc
k m
γ γωρ
= = =
• For a neutral gas the sound wavesare pressure waves propagating from one layer of particles to another one
• The propagation of sound waves requires collisionsamong the neutrals
where mN is the neutral atom mass and cs is the sound speed.
Electromagnetic Waves in Plasmas
1E.M. Waves in a Non-Magnetized Plasma2 E.M. Waves in a Magnetized Plasma3Hydromagnetic (Alfven) Waves3Hydromagnetic (Alfven) Waves4Magnetosonic Waves
Electromagnetic Waves in a Plasma
• In a plasma there will be current carriers, therefore the curl of Ampere’s lawis
t
∂∇ × = +∂D
H j
• By taking the curl of Faraday’s law
( ) ( )20 t
µ ∂∇ × ∇ × = ∇ ∇ ⋅ − ∇ = − ∇ ×∂
E E E H
and eliminating the curl of H
( )2
20 2t t
µ ∂ ∂∇ ∇ ⋅ − ∇ = − + ∂ ∂
DE E j
Electromagnetic Waves in a Plasma (II)
• If a wave solutionof the form exp(k·r-wt) is assumed it can be written (D=e0E)
( ) 2 20 0 0i i k iωµ ω µ ε⋅ + = +k k E E j E
• By recalling that an e.m. must be transverse(k·E =0) and that c2=1/(m e ) it followsand that c2=1/(m0e0) it follows
( )2 2 20/c k iω ω ε− = −E j
• In order to estimate the current the ionsare considered fixed (good approximation for high
frequencies) and the current is carried by electrons with density n0 and velocity u:
0 en e= −j u [email protected]
Electromagnetic Waves in a Plasma (III)
• The electron equation of motionis
em e et
∂ = − − ×∂u
E u B
• The motion of the electrons here is the self-consistent solutionof u, E, B (E and B are not external imposed solutionof u, E, B (E and B are not external imposed
field like in the particle trajectory calculations)
• A first-orderform of the equation of motion is then
em et
∂ = −∂u
E
then 20
e e
n ee
i m i mω ω−= ⇒ =
−EE
Electromagnetic Waves in a Plasma (IV)
• Finally, substituting the expression of j in
it is found
( )2 2 20/c k iω ω ε− = −E j
( )2
2 2 2 2 2 2 20p
n ec k c k
mω ω ω
ε− = ⇒ = +E E( )
0pc k c k
mω ω ω
ε− = ⇒ = +E E
that is the dispersion relation for e.m. waves in a plasma(without external magnetic field)
• The phase velocityis always greater than c while the group velocityis always less than c:
222 2
2 2
pv ck kϕ
ωω= = +2
g
d cv
dk vϕ
Electromagnetic Waves in a Plasma (V)
• For a given frequencyw the dispersion relation
gives a particular k or wavelength (k=2p/l) for the wave propagation
• If the frequency is raised up to w=w then it must be
2 2 2 2p c kω ω= +
• If the frequency is raised up to w=wp then it must be k=0. This is the cutoff frequency(conversely, cutoff densitywill be the value that makes wp equal to w)
• For even larger densities, or simply w<wp there is no real k that satisfies the dispersion relation and the
wave cannot propagatethrough the plasma
Electromagnetic Waves in a Plasma (VI)
• When k becomes imaginary the wave is attenuated
• The spatial part of the wave can be written as
where d is the skin depthdefined as
( ) ( ) ( )exp exp exp /ikx k x x δ= − −�
c−
( )1
1/ 22 2p
ckδ
ω ω−= =
−
E.M. Waves in a Magnetized Plasma
• The case of an e.m. wave perpendicularto an external magnetic fieldB0 is considered
• If the wave electric field is parallel to B0 the same derivation as for non magnetized plasma can be
applied (essentially because the first-order electron equation of motion is not affected by B )equation of motion is not affected by B0)
• The the wave is called ordinary waveand the dispersion relation in this case is still
2 2 2 2p c kω ω= +
y
x
z
B0
k
E
E.M. Waves in a Magnetized Plasma (II)
• The case of the wave electric field perpendicular to B0 requires both x and y components of E since the
wave becomes elliptically polarized
y
z
B0
k
E
y
x
k
• A linearized (first-order)form of the equation electron equation of motion is then
0 0e em e e i m e et
ω∂ = − − × ⇒ − = − − ×∂u
E u B u E u [email protected]
E.M. Waves in a Magnetized Plasma (III)
• The wave equation now must keep the longitudinal electric fieldk·E=kEx
( ) 2 20 0 0i i k iωµ ω µ ε⋅ + = +k k E E j E
( )2 2 2 2 / /c k c kE i in eω ω ε ω ε− + = − = −E k j u
or
• By solving for the separate x and y components a dispersion relation for the extraordinary waveis
found as
( )2 2 2 20 0 0/ /xc k c kE i in eω ω ε ω ε− + = − = −E k j u
( )2 2 22 2
2 2 2 2 21 p p
p c
c k ω ω ωω ω ω ω ω
−= −
E.M. Waves in a Magnetized Plasma (IV)
• The case of the wave vector parallelto B0 also requires both x and y components of E
y
z
B0
kE
y
x
• The same derivation as for the extraordinary wave can be used by simply by changing the direction of k
E.M. Waves in a Magnetized Plasma (V)
• The resulting dispersion relation is
( )2 22 2
21
1p
c
c k ω ωω ω ω
= −m
or the choice of sign distinguish between a right-hand circular polarization (R-wave) and a left hand circular circular polarization (R-wave) and a left hand circular
polarization (L-wave)
• The R-wave has a resonancecorresponding to the electron Larmor frequency: in this case the wave
looses energy by accelerating the electrons along the Larmor orbit
• It can be shown that the L-wave has a resonancein correspondence to the ion Larmor [email protected]
Hydromagnetic (Alfven) Waves
• This case considers still the wave vector parallelto B0
but includes both electrons and ion motions and current j and electric field E perpendicular to B0
z
B0
k
E,jy
x
E,j
• The solution neglects the electron Larmor orbits, leaving only the ExB drift and considers propagation
frequenciesmuch smaller than the ion cyclotron frequency
Hydromagnetic (Alfven) Waves (II)
• The dispersion relation for the hydromagnetic (Alfven) wavescan be derived as
( )( ) ( )2 2 2
2 2 220 00 0
11
c c
k c BB
ωρµρ ε
= =++
where r is the mass densitywhere r is the mass density
• It can be shown that the denominator is the relative dielectric constantfor low-frequency perpendicular
motion in the plasma
• The dispersion relation for Alfven waves gives the phase velocity of e.m. waves in the plasma
considered as a dielectric [email protected]
Hydromagnetic (Alfven) Waves (III)
• In most laboratory plasmas the dielectric constant is much larger than unity, therefore, for hydromagnetic
waves,
( )2 20
1/ 2
0
A
B cv
k
ωρµ
≈ ≡( )0ρµ
where vA is the Alfven velocity
• The Alfven velocity can be considered the velocity of the perturbations of the magnetic linesof force due to
the wave magnetic field in the plasma
• Under the approximations made the fluid and the field lines oscillate as they were “glued” together
Magnetosonic Waves
• This case considers the wave vector perpendicularto B0 and includes both electrons and ion motions (low-
frequency waves) with E perpendicular to B0
z
B0
k
Ey
x
E
• The solution includes the pressure gradientin the (fluid) equation of motion since the oscillating ExB0
drifts will cause compressions in the direction of the wave
Magnetosonic Waves (II)
• For frequencies much smaller than the ion cyclotron frequency the dispersion relation for magnetosonic
wavescan be derived as 2 22
22 2 2
s A
A
v vc
k c v
ω +=
+ Ak c v+where vs is the sound speed in the plasma
• The magnetosonic wave is an ion-acousticwave that travels perpendicular to the magnetic field
• Compressions and rarefactions are due to the ExB0
drifts
Magnetosonic Waves (III)
• In the limit of zero magnetic fieldthe ion-acousticdispersion relation is recovered
• In the limit of zero temperaturethe sound speed goes to zero and the wave becomes similar to an Alfven
wave
APPLICATION OF PLASMA PHYSICSPHYSICS
1. Magnetohydrodynamic Generator
2. Thermonuclear fusion reactor
• MHD power generation uses the interaction of an electrically conducting fluid with a magnetic field to convert part of the energy of the fluid directly into electricity
1.Magneto hydrodynamic Generator
• Converts thermal or kinetic energy into electricity
Where• F is the force of the acting particle (vector)• V is the velocity of the particle (vector)• Q is the charge of the particle (scalar)• B is the magnetic field (vector)
Lorentz Force Law:Lorentz Force Law:F = F = QvBQvB
Conversion Efficiency
• MHD generator alone: 10-20%
• Steam plant alone: ≈ 40%• Steam plant alone: ≈ 40%
• MHD generator coupled with a steam plant: up to 60%
Advantages of Fusion
• Inexhaustible Supply of Fuel
• Relatively Safe and Clean
• Possibility of Direct Conversion
Requirements for Fusion
• High Temperatures
• Adequate Densities• Adequate Densities
• Adequate Confinement
• Lawson Criterion: nτ >
1020 s/[email protected]
Two Approaches
• Inertial Confinement:– n ≈ 1030 / m3
� τ ≈ 10-10 s� τ ≈ 10-10 s
• Magnetic Confinement:
– n ≈ 1020 / m3
� τ ≈ 1 [email protected]
Magnetic Confinement• Magnetic Field Limit: B < 5 T
• Pressure Balance: nkT≈ 0.1B2/2µ0
• ==> n ≈ 1020 / m3 @ T = 108 K
• Atmospheric density is 2 x 1025 / m3
• Good vacuum is required
• Pressure: nkT≈ 1 atmosphere
• Confinement: τ ≈ 1 s
• A 10 keV electron travels 30,000 miles in 1 [email protected]
