Plasma Physics
DR.MOHAMMAD IMRAN AZIZ
Assistant Professor(Sr.)
PHYSICS DEPARTMENT
SHIBLI NATIONAL COLLEGE, AZAMGARH (India).
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Ionized Gases
• An ionized gas is characterized, in general, by a
mixture of neutrals, (positive) ions and electrons.
• For a gas in thermal equilibrium the Saha equation
gives the expected amount of ionization:
ni T 3 / 2 −Ui / kBT
2.4 ⋅ 1021 e
nn ni
• The Saha equation describes an equilibrium situation
between ionization and (ion-electron) recombination
rates.
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Example: Saha Equation
• Solving Saha equation
ni T 3/ 2 −U i / kBT
2.4 ⋅1021 e
nn ni
ni2 2.4 ⋅ 1021 nnT 3 / 2e −Ui / kBT
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Example: Saha Equation (II)
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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
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.
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From Ionized Gas to Plasma
• An ionized gas is not necessarily a plasma
• An ionized gas can exhibit a “collective behavior” in
the interaction among charged particles when when
long-range forces prevail over short-range forces
• An ionized gas could appear quasineutral if the charge
density fluctuations are contained in a limited region
of space
• A plasma is an ionized gas that presents a collective
behavior and is quasineutral
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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 bonding among the molecules.
• In general, by increasing the temperature (=average
molecular kinetic energy) a phase transition occurs,
from solid, to liquid, to gas.
• A further increase of temperature increases the
collisional rate and then the degree of ionization of the
gas.
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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
aggregation but it is characterized by a different
behavior when subject to electromagnetic fields.
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The Particle Picture
1 Unmagnetized Plasmas
2 Magnetized Plasma
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1 Unmagnetized Plasmas
1.1 Charge in an Electric Field
1.2 Collisions between Charged Particles
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1.1 Charge in an Electric Field
• Electric force:
F=qE
Dimensional analysis:
N=C V/m
• A positive isolated charge q will produce a positive
electric field at a point distance r given by
q r V = C 1
E= m F / m m2
4πε 0 r 3
• The force on another positive charge will be repulsive
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since F=qE is directed as r
1.2 Collisions between Charged Particles
r0
v
• Interaction time T=r0/v
• Change in momentum:
q1q2 1 r0 q1q2 1
∆ (mv) mv = FT = =
4πε 0 r0 v 4πε 0 r0 v
2
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• Impact parameter:
q1q2 1
r0 =
4πε 0 mv 2
• Collisional cross section:
σ =π r02 =
( q1q2 )1
2
16πε 0 m v
2 2 4
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Charge in an Electric Field
• Electric force:
F=qE
Dimensional analysis:
N=C V/m
• A positive isolated charge q will produce a positive
electric field at a point distance r given by
q r V = C 1
E= m F / m m2
4πε 0 r 3
• The force on another positive charge will be repulsive
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since F=qE is directed as r
2 Magnetized Plasmas
2.1 Charge in an Uniform Magnetic Field
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1.1 Charge in an an Uniform Magnetic Field
• Magnetic force:
F = mv = qv × B
&
Dimensional analysis:
N=C T m/s
• Equation of the motion for a positive isolated charge q
in a magnetic field B:
i j k
F = mv = qv × B = q vx
& vy vz
Bx
By Bz
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Charge in an an Uniform Magnetic Field (II)
i j k
vx vy vz = i (v y Bz − vz By ) − j(vx Bz − vz Bx ) + k (vx By − v y Bx )
Bx
By Bz
• Case of a magnetic field B directed along z:
mvx = qv y Bz
&
mv y = −qvx Bz
&
mvz = 0
&
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Charge in an an Uniform Magnetic Field (III)
• By taking the derivative of mvx = qv y Bz
&
mvx = qv y Bz
&& &
• Then replacing :
v y = −vx qBz / m
&
vx = −vx ( qBz / m )
2
&&
• Analogously:
v y = −v y ( qBz / m )
2
&&
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Charge in an an Uniform Magnetic Field (III)
• The equations for vx and vy are harmonic oscillator
equations.
• The oscillation frequency, called cyclotron frequency
is defined as:
ω c = q Bz / m
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Charge in an an Uniform Magnetic Field (IV)
• The solution of the harmonic oscillator equation is
vx = A exp ( iω ct ) + B exp ( −iω ct )
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The Kinetic Theory
1 The Distribution Function
2 The Kinetic Equations
3 Relation to Macroscopic Quantities
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The Distribution Function
1 The Concept of Distribution Function
2 The Maxwellian Distribution
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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+dv
is f(r,v,t) dv, where dv= dvx dvy dvz
• The density is then found as
∞ ∞ ∞ ∞
n(r, t ) = ∫ dvx ∫ dv y ∫ dvz f (r, v, t ) = ∫
3
f (r, v, t )d v
−∞ −∞ −∞ −∞
• If the distribution is normalized as
∞
∫ f (r, v, t ) dv = 1
ˆ f (r, v, t ) = n(r, t ) f (r, v, t )
ˆ
−∞
then f^ represents a probability distribution
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The Maxwellian Distribution
• The maxwellian distribution is defined as:
3/ 2
m −v 2
fm =
ˆ
exp 2
2π k BT vth
where
v= 2
vx + vy
2
+ vz
2
vth = 2k BT / m
• The known result
∞
∫ exp(− x )dx = π
2
−∞ yields
∞
ˆ ( v ) dv = 1
∫ f maziz_muhd33@yahoo.co.in 32
−∞
The Maxwellian Distribution (II)
• The root mean square velocity for a maxwellian is:
v 2 = 3k BT / m
recall W = 1 mv 2 = 3k BT
2
• The average of the velocity magnitude v=|v| is:
∞
v = ˆm ( v )dv3 = 2vth = 2 2k BT / π m
∫ vf
π
−∞
• In one direction:
∞
vx = 0 vx = ∫ vf m ( v )dv = vth / π = 2k BT / π m
ˆ
−∞
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The Maxwellian Distribution (III)
• The distribution w.r.t. the magnitude of v
∞ ∞
∫ g ( v)dv = ∫ f ( v ) dv
0 −∞
• For a Maxwellian
3/ 2
m −v 2
g m = 4π n v 2 exp 2
2π k BT vth
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The Kinetic Equations
1 The Boltzmann Equation
2 The Vlasov Equation
3 The Collisional Effects
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1. The Boltzmann Equation
• A distribution function: f=f(r,v,t) satisfies the
Boltzmann equation
∂f F ∂f ∂f
+ v ⋅ ∇f + ⋅ =
∂t m ∂v ∂t c
• 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
vx
Motion of a group of t+∆t
particles with constant density t
in the phase space: aziz_muhd33@yahoo.co.in 36
x
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
+ v ⋅ ∇f + ( E + v ⋅ B ) ⋅ = 0
∂t m ∂v
that is called the Vlasov equation
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3. The Collisional Effects
• The Vlasov equation does not account for
collisions ∂f
=0
∂t c
• Short-range collisions like charged particles with
neutrals can be described by a Boltzmann collision
operator in 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
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equation.
4. Relation to Macroscopic Quantities
1 The Moments of the Distribution Function
2 Derivation of the Fluid Equations
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1. The Moments of the Distribution Function
• Notation: define
∞ ∞ ∞ ∞
∫ dvx ∫ dv y ∫ dvz = ∫ d 3v
−∞ −∞ −∞ −∞
• If A=A(v) the average of the function A for a
distribution function f=f(r,v,t) is defined as
∞
∫ A(r, v, t ) f (r, v, t )d 3v
A(r, t ) v
= −∞ ∞
=
3
∫ f (r, v, t )d v
−∞
1 ∞
= ∫ A(r, v, t ) f (r, v, t )d 3v
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The Moments of the Distribution Function (II)
• General distribution function: f=f(r,v,t)
• The density is defined as the 0th order moment
and was found to be
∞ ∞ ∞ ∞
n(r, t ) = ∫ dvx ∫ dv y ∫ dvz f (r, v, t ) = ∫ f (r, v, t )d 3v
−∞ −∞ −∞ −∞
• The mass density can be then defined as
∞
ρ (r, t ) = mn(r, t ) = m ∫ f (r, v, t )d 3v
−∞
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The Moments of the Distribution Function (III)
• The 1st order moment is the average velocity or
fluid velocity is defined as
1 ∞
u(r, t ) = 3
∫ vf (r, v, t )d v
n(r, t ) −∞
• The momentum density can be then defined as
∞
r , t ) = m ∫ vf
n(r, t )mu(aziz_muhd33@yahoo.co.in(r, v, t )d 3v 42
−∞
The Moments of the Distribution Function (IV)
• Higher moments are found by diadic products
with v
• The 2nd order moment gives the stress tensor
(tensor of second order)
∞
Π (r, t ) = m ∫ vvf (r, v, t )d 3v
−∞
• In the frame of the moving fluid the velocity is
w=v-u. In this case the stress tensor becomes the
pressure tensor
∞
P (r, t ) = m ∫ wwf (r, v, t )d 3v
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−∞
2 Derivation of the Fluid Equations
• Boltzmann equation written for the Lorentz force
∂f q ∂f ∂f
+ v ⋅ ∇f + ( E + v × B ) ⋅ =
∂t m ∂v ∂t c
• Integrate in velocity space:
∂f 3 q ∂f 3 ∂f d 3v
∫ ∂t d v + ∫ v ⋅ ∇f d v + m ∫ ( E + v × B ) ⋅ ∂vd v = ∫ ∂t
3
c
• From the definition of density
∂f 3 ∂ ∂n
∫ ∂t d v = ∂t ∫ fd v = ∂t
3
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Derivation of the Fluid Equations (II)
• Since the gradient operator is independent from v:
∫ v ⋅ ∇f d 3v = ∇ ⋅ ∫ vf d 3v = ∇ ⋅ ( nu )
• Through integration by parts it can be shown that
q ∂f 3
m ∫ ( E + v × B ) ⋅ ∂vd v = 0
• 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
∂f d 3v = 0
∫ ∂t
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c 45
Derivation of the Fluid Equations (III)
• The integrated Boltzmann equation then becomes
∂n
+ ∇ ⋅ ( nu ) = 0
∂t
that is known as equation of continuity
• In general moments of the Boltzmann equation are
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 equation can be obtained
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Derivation of the Fluid Equations (IV)
• Integrate the Boltzmann equation in velocity space
with g=mv
∂f 3 ∂f 3
m ∫ v d v + m ∫ vv ⋅ ∇f d v + q ∫ v ( E + v × B ) ⋅ d v =
3
∂t ∂v
∂f d 3v
= ∫ mv
∂t c
• The first term is
∂f 3 ∂ ∂ 3 ∫ vfd 3v ∂
m ∫ v d v = m ∫ vfd v = m ∫ fd v
3
3
= m ( nu )
∂t ∂t ∂t
∫ fd v ∂t
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Derivation of the Fluid Equations (V)
• Further simplifications yield the final fluid
equation of motion
∂u + u ⋅ ∇ u = qn E + u × B − ∇ ⋅ P + P
mn ( ) ( )
∂t
coll
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 equation is
obtained
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The Kinetic Theory
1 The Distribution Function
2 The Kinetic Equations
3 Relation to Macroscopic Quantities
4 Landau Damping
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4 Landau Damping
1 Electromagnetic Wave Refresher
2 The Physical Meaning of Landau Damping
3 Analysis of Landau Damping
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1 Electromagnetic Wave Refresher
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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)
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.
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The Physical Meaning of Landau Damping
• An e.m. wave is 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
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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
• The wave damping will create in general a local
distortion of 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
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The Physical Meaning of Landau Damping (III)
• Whether the speed of a resonant particle increases
or decreases depends on the phase of the wave at
its initial position
• Not all particles moving slightly faster than the
wave lose energy, nor all particles moving slightly
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.
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The Physical Meaning of Landau Damping (IV)
• Then the particles which lose energy interact more
effectively with the wave
• On average, there is a transfer of energy from the
particles to the electric field.
• Exactly the opposite is true for particles with
initial velocities lying just below the phase
velocity of the wave.
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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
kinetic energy (collisionless damping)
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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
f1 ∝ exp [i ( kx − ω t )]
• For a non-collisional plasma analysis the Vlasov
equation applies. For the electron species it will be
∂f e ∂f
+ v ⋅ ∇f − ( E + v × B ) ⋅ = 0
∂t m
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∂v 58
Analysis of Landau Damping (II)
• A linearization of the Vlasov equation considering
f = f 0 + f1
E = E0 + E1 ; B = B0 + B1 ;
E0 = 0; B 0 = 0
v × B = 0 (since only contributions along v are studied)
yields
∂f1 e ∂f 0
+ v ⋅ ∇f1 − E1 ⋅ =0
∂t m ∂v
or, considering the wave along the dimension x,
e ∂f 0
iω f1 + ikvx f1 = − E1x
m
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∂vx 59
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)
∇ ⋅ E1 = −en ε 0
or, in this case, considering a perturbed density n1
equivalent to the perturbed distribution f1
ikE x = −en ε 0
• Finally the density can be expressed in terms of
the distribution function as
∞
, t ) = ∫ f1 (r, v
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−∞
Analysis of Landau Damping (IV)
• The linearized Vlasov equation for the wave
perturbation
e ∂f 0
iω f1 + ikvx f1 = − E1x
m ∂vx
can be rewritten, after few manipulations as a
relation between ω, k and know quantities:
ω2
p
∞
∂f 0 (vx ) ∂vx
ˆ
1= 2 ∫ dvx
k −∞ vx − (ω k )
where
f 0 = f 0 / n0
ˆ
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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
ω 2 ∞ ∂fˆ0 (vx ) ∂vx
p
1= 2 ∫ dvx
k −∞ vx − (ω k )
can be computed in an approximate fashion for a
maxwellian distribution yielding
π ω p ∂fˆ0 (vx )
2
ω = ω p 1 + i
2k 2
∂vx v =ω / k
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Analysis of Landau Damping (VI)
• For a one-dimensional maxwellian along the x
direction
∂f 0 (vx )
ˆ 2v x vx
2
= − 1 2 3 exp − 2
∂vx π vth vth
• This will cause the imaginary part of the
expression
ω 2 ∂fˆ0 (vx )
π p
ω = ω p 1 + i
2k 2
∂vx v =ω / k
to be negative (for a positive wave propagation
direction)
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Analysis of Landau Damping (VII)
• For a wave is traveling in the x direction the of the
form
f1 ∝ exp [i ( kx − ω t )] = exp ( ikx ) exp [ −i (ω R + iω I ) t ] =
= exp ( ikx ) exp [( −iω R + ω I ) t ] =
= exp ( ikx ) exp ( −iω R t ) exp (ω I t )
a negative imaginary part of ω will produce an
attenuation, or damping, of the wave.
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The Fluid Description of Plasmas
The Fluid Equations for a Plasma
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Plasmas as Fluids: Introduction
• The particle description of a plasma was based on
trajectories for given electric and magnetic fields
• Computational particle models allow in principle
to obtain a microscopic description of the plasma
with its self-consistent electric and magnetic fields
• The kinetic theory yields also a microscopic, self-
consistent description of the plasma based on the
evolution of a “continuum” distribution function
• Most practical applications of the kinetic theory
rely also on numerical implementation of the
kinetic equations
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Plasmas as Fluids: Introduction (II)
• The analysis of several important plasma
phenomena does not require the resolution of a
microscopic approach
• The plasma behavior can be often well represented
by a macroscopic description as in a fluid model
• Unlike neutral fluids, plasmas respond to electric
and magnetic fields
• The fluidodynamics of plasmas is then expected to
show additional phenomena than ordinary hydro,
or gasdynamics
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Plasmas as Fluids: Introduction (III)
• The “continuum” or “fluid-like” character of
ordinary fluids is essentially due to the frequent
(short-range) collisions among the neutral
particles that neutralize most of the microscopic
patterns
• Plasmas are, in general, less subject to short-range
collisions and properties like collective effects and
quasi-neutrality are responsible for the fluid-like
behavior
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Plasmas as Fluids: Introduction (IV)
• Plasmas can be considered as composed of
interpenetrating fluids (one for each particle
species)
• A typical case is a two-fluid model: an electron
and an ion fluids interacting with each other and
subject to e.m. forces
• A neutral fluid component can also be added, as
well as other ion fluids (for different ion species or
ionization levels)
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The Fluid Description of Plasmas
1 The Fluid Equations for a Plasma
2 Plasma Diffusion
3 Fluid Model of Fully Ionized Plasmas
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Fluid Model of Fully Ionized Plasmas
. The Magnetohydrodynamic Equations
.Diffusion in Fully Ionized Plasmas
. Hydromagnetic Equilibrium
. Diffusion of Magnetic Field in a Plasma
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Magnetohydrodynamic Equations
• Goal: to derive a single fluid description 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:
ρ m = mi ni + me ne ≈ n( mi + me )
1 ( mi ui + meue )
u= ( mi ni ui + me neue ) ≈
ρm (mi + me )
j = e ( ni ui − ne u e ) ≈ ne ( ui − u e )
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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):
∂ui
nmi + ( ui ⋅ ∇ ) ui = qi n ( E + ui × B ) − ∇pi + Pie + mi ng
∂t
∂u e
nme + ( u e ⋅ ∇ ) u e = qe n ( E + u e × B ) − ∇pe + Pei + me ng
∂t
• Approximation 1: the viscosity tensor has been
neglected, acceptable for Larmor radius small
w.r.t. the scale length of variations of the fluid
quantities.
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Magnetohydrodynamic Equations (III)
• Approximation 2: neglect the convective term,
acceptable when the changes produced by the
fluid convective motion are relatively small
∂ui
nmi + ( ui ⋅ ∇ ) ui = qi n ( E + ui × B ) − ∇pi + Pie + mi ng
∂t
∂u e
nme + ( u e ⋅ ∇ ) u e = qe n ( E + u e × B ) − ∇pe + Pei + me ng
∂t
• These equation can be added and by setting
p=pe+pi, -qi=qe=e and Pei=-Pie obtaining:
∂
n ( mi ui + me u e ) = en ( ui − u e ) × B − ∇p + n ( mi + me ) g
∂t aziz_muhd33@yahoo.co.in 74
Magnetohydrodynamic Equations (IV)
• By substituting the definition of the single fluid
variables r, u and j the equation
∂
n ( mi ui + me u e ) = en ( ui − u e ) × B − ∇p + n ( mi + me ) g
∂t
can be written as
∂u
ρm = j × B − ∇p + ρ m g
∂t
that is the single fluid equation of motion for the
mass flow. There is no electric force because the
fluid is globally neutral (ne=ni).
aziz_muhd33@yahoo.co.in 75
Magnetohydrodynamic Equations (V)
• To characterize the electrical properties of 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 me/ mi=>0, d/dt=>0 it is obtained
1
E + u × B = η j + ( j × B ) − ∇pe
en
that is the generalized Ohm’s law that includes the
Hall term (jxB) and the pressure effects
aziz_muhd33@yahoo.co.in 76
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 m
+ ∇ ⋅ ( ρmu ) = 0
∂t
∂ρ
+∇⋅j= 0
∂t
• The single-fluid equations of continuity and
motion and the Ohm’s law constitute the set of
magnetohydrodynamic (MHD) equations.
aziz_muhd33@yahoo.co.in 77
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 = j × B − ∇p
E + u × B =ηj
• The parallel (to B) component of the last equation
reduce simply to the ordinary Ohm’s law:
E =η j
aziz_muhd33@yahoo.co.in 78
Diffusion in Fully Ionized Plasmas (II)
• The component perpendicular to B is found by
taking the the cross product with B
E × B + ( u ⊥ × B ) × B = η⊥ j × B
that is
E × B − u ⊥ B 2 = η ⊥ j × B = η ⊥ ∇p
and finally
E × B η⊥
u ⊥ = 2 − 2 ∇p
B B
• The first term is the usual ExB drift (for both
species together), the second is a diffusion driven
by the gradient of the pressure
aziz_muhd33@yahoo.co.in 79
Diffusion in Fully Ionized Plasmas (III)
• The diffusion in the direction of -grad p produces
a fluxη
Γ ⊥ = nu ⊥ = −n ⊥ ∇p
2
B
• For isothermal, ideal gas-type plasma the
perpendicular flux can be written as
η⊥ n(k BTi + k BTe )
Γ⊥ = − 2
∇n
B
that is a Fick’s law with diffusion coefficient
η⊥ n(k BTi + k BTe )
D⊥ =
B2
named classical diffusion coefficient
aziz_muhd33@yahoo.co.in 80
Diffusion in Fully Ionized Plasmas (IV)
• The classical diffusion coefficient is proportional
to 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)
aziz_muhd33@yahoo.co.in 81
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 itself, it can be concluded that there is no
perpendicular mobility: an electric field
perpendicular to B produces just a ExB drift.
aziz_muhd33@yahoo.co.in 82
Diffusion in Fully Ionized Plasmas (V)
• Experiments with 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 coefficient that goes like 1/B and
increases with the temperature:
1 k BTe
D⊥ Bohm =
16 eB
• Bohm diffusion ultimately makes more difficult to
reach fusion conditions in magnetically confined
plasma
aziz_muhd33@yahoo.co.in 83
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
balance between 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 surfaces
aziz_muhd33@yahoo.co.in 84
Hydromagnetic Equilibrium (II)
j
B
grad p
• For an axial magnetic field in a cylindrical
configuration with radial pressure gradient, the
current must be azimuthal
• The momentum equation in the perpendicular
plane (w.r.t. B) will then give an expression for j
aziz_muhd33@yahoo.co.in 85
Hydromagnetic Equilibrium (II)
• The cross product of the momentum with B yields
B × ∇p = B × j × B = jB 2
and, in the usual approximations, solving for j
yield again the expression for the diamagnetic
current
B × ∇p B × ∇n
j= 2
= ( k BTi + k BTe )
B B2
• From the MHD point of view the diamagnetic
current is generated by the grad p force that
interacts (via a cross product) with B
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Hydromagnetic Equilibrium (IV)
• The connection between the fluid and the particle
point of view was previously discussed: the
diamagnetic current arises from an unbalance of
the Larmor gyration velocities in a fluid element
• From a strict particle point of view the
confinement of the plasma with a gradient of
pressure occurs because each particle guiding
center is tight to a line of force and diffusion is not
permitted (in absence of collisions)
aziz_muhd33@yahoo.co.in 87
Hydromagnetic Equilibrium (V)
• For the equilibrium case under consideration, the
momentum equation in the direction parallel to B
will be simply
∂p
∇p = 0 =
∂s
where s is a generalized coordinate along the lines
of force.
∂n
• For isothermal plasma it will be = 0
∂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 mirror there are more
particles trapped at the midplane (lower line of
force density) than at the mirror end sections 88
aziz_muhd33@yahoo.co.in
Waves in Plasmas
1 Electrostatic Waves in Non-Magnetized
Plasmas
2 Electrostatic Waves in Magnetized Plasmas
aziz_muhd33@yahoo.co.in 89
E.S. Waves in Non-Magnetized Plasmas
1. Wave fundamentals
2. Electron Plasma Waves
3. Sound waves
4. Ion Acoustic Waves
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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 notation is a convenient way to
represent mathematically oscillating quantities: the
physical quantity will be obtained by taking the real
part
• A sinusoidal plane wave can be represented as
f (r, t ) = f 0 exp i ( k ⋅ r − ω t )
where f0 is the maximum amplitude, k is the
propagation constant, or wave vector (k is the
wavenumber) and w the angular frequency
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Wave Fundamentals (II)
• If f0 is real then the wave amplitude is maximum
(equal to f0) in r=0, t=0, therefore the phase angle of
the wave is zero
• A complex f0 can be used to represent a non zero
phase angle:
f 0 exp i ( k ⋅ r − ω t + δ ) = f 0 exp ( iδ ) exp i ( k ⋅ r − ω t )
• A point of constant phase on the wave will travel
along with the wave front
• A constant phase on the wave implies
d
(k ⋅ r − ωt ) = 0
dt aziz_muhd33@yahoo.co.in 92
Wave Fundamentals (III)
• In one dimension it will be
d dx ω
( kx − ω t ) = 0 ⇒ = vϕ
dt dt k
where vf is defined as the wave phase velocity
• The wave can be then also expressed by
f ( x, t ) = f 0 exp ik ( x − vϕ t )
• The phase velocity in a plasma can exceed the
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.
aziz_muhd33@yahoo.co.in 93
Wave Fundamentals (IV)
• A wave carries information only 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 vf is formed by two
waves with 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
aziz_muhd33@yahoo.co.in 94
Wave Fundamentals (V)
• For the case of two wave beating it can be written
f A ( x, t ) = f 0 cos ( k + ∆k ) x − (ω + ∆ω ) t
f B ( x, t ) = f 0 cos ( k − ∆k ) x − (ω − ∆ω ) t
• By summing the two waves and expanding with
trigonometric identities it is found
f A ( x, t ) + f B ( x, t ) = 2 f 0 cos ( ∆k ) x − ( ∆ω ) t ⋅ cos [ kx − ω t ]
• 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”
the wave (that
component of aziz_muhd33@yahoo.co.in does not carry 95
information)
Wave Fundamentals (VI)
• The modulating component travels at the group
velocity defined as
∆ω dω
vg = ⇒ vg =
∆k ∆ω →0 dk
• The group velocity can never exceed c
aziz_muhd33@yahoo.co.in 96
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
ne = ne 0 + ne1 ue = ue 0 + ue1 E = E0 + E1
the frequency of the oscillations can be found as
3 2 2
ω 2
= ω2
pe + k vth
2
where
vth = 2k BTe me
2
aziz_muhd33@yahoo.co.in 97
Electron Plasma Waves (II)
• Electron plasma waves have a group velocity equal to
dω 3 k 2 3 k 2
= vth = vth
dk 2 ω 2 vϕ
• In general a relation linking w and k for a wave is
called dispersion relation
• The slope of the dispersion relation on a w-k diagram
gives the phase velocity of the wave
aziz_muhd33@yahoo.co.in 98
Sound Waves
• For a neutral fluid like air, in absence of viscosity, the
Navier-Stokes equation is
∂u + u ⋅ ∇ u = −∇p
ρm ( )
∂t
γp
• From the equation of state ∇p =
ρm
then
∂u + u ⋅ ∇ u = − γ p
ρm ( )
∂t ρm
• Continuity equation yields
∂ρ m
+ ∇ ⋅ ( ρmu ) = 0
∂aziz_muhd33@yahoo.co.in
t 99
Sound Waves (II)
• Linearization of the momentum and continuity
equations for stationary equilibrium yield
12 12
ω γ p0 γ k BT
= = m = cs
k ρm0 N
where mN is the neutral atom mass and cs is the sound
speed.
• For a neutral gas the sound waves are pressure waves
propagating from one layer of particles to another one
• The propagation of sound waves requires collisions
among the neutrals
aziz_muhd33@yahoo.co.in 100
Electromagnetic Waves in Plasmas
1E.M. Waves in a Non-Magnetized Plasma
2 E.M. Waves in a Magnetized Plasma
3Hydromagnetic (Alfven) Waves
4Magnetosonic Waves
aziz_muhd33@yahoo.co.in 101
Electromagnetic Waves in a Plasma
• In a plasma there will be current carriers, therefore
the curl of Ampere’s law is
∂D
∇×H = j+
∂t
• By taking the curl of Faraday’s law
∂
∇ × ∇ × E = ∇ ( ∇ ⋅ E ) − ∇ E = −µ0 ( ∇ × H )
2
∂t
and eliminating the curl of H
∂ ∂2D
∇ ( ∇ ⋅ E ) − ∇2 E = −µ0 j + 2
∂t ∂t
aziz_muhd33@yahoo.co.in 102
Electromagnetic Waves in a Plasma (II)
• If a wave solution of the form exp(k·r-wt) is assumed
it can be written (D=e0E)
ik ( ik ⋅ E ) + k 2 E = iωµ 0 j + ω 2 µ 0ε 0 E
• By recalling that an e.m. must be transverse (k·E =0)
and that c2=1/(m0e0) it follows
( ω 2 − c 2 k 2 ) E = −iω j / ε 0
• In order to estimate the current the ions are
considered fixed (good approximation for high
frequencies) and the current is carried by electrons
with density n0 and velocity u:
j = − n0 eu e
aziz_muhd33@yahoo.co.in 103
Electromagnetic Waves in a Plasma (III)
• The electron equation of motion is
∂u
me = −eE − eu × B
∂t
• The motion of the electrons here is the self-consistent
solution of u, E, B (E and B are not external imposed
field like in the particle trajectory calculations)
• A first-order form of the equation of motion is then
∂u
me = −eE
∂t
then 2
−eE n0 e E
u= ⇒ j=
aziz_muhd33@yahoo.co.in 104
−iω me iω me
Electromagnetic Waves in a Plasma (IV)
• Finally, substituting the expression of j in
( ω 2 − c 2 k 2 ) E = −iω j / ε 0
it is found
n0 e 2
( ω 2 − c2 k 2 ) E =
ε0m
E ⇒ ω 2 = ω p + c2 k 2
2
that is the dispersion relation for e.m. waves in a
plasma (without external magnetic field)
• The phase velocity is always greater than c while the
group velocity is always less than c:
ω 2
ωp 2
dω c 2
vϕ = 2 = 2 + c aziz_muhd33@yahoo.co.in vg =
2 2 =
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Electromagnetic Waves in a Plasma (V)
• For a given frequency w the dispersion relation
ω 2 = ω p + c2 k 2
2
gives a particular k or wavelength (k=2p/l) for the
wave propagation
• 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
10 20 s/m3
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Two Approaches
• Inertial Confinement:
– n ≈ 1030 / m3
τ ≈ 10-10 s
• Magnetic Confinement:
– n ≈ 1020 / m3
τ≈1s
aziz_muhd33@yahoo.co.in 135
Magnetic Confinement
• Magnetic Field Limit: B 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 s
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