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Plasma Physics
By Satyam Ladva
August 2, 2016

1 Basic properties of Plasma
1
...
1
...


•

Plasmas - Collection of electrons and ions which are not highly co-ordinated (like atoms in a crystal) that move
around individually, like a gas
...
These can be time dependant
...


•

If the second term was dierent to the rst, an electric eld would be present which would be neutralised if the
electrons moved around
...
As

T → 0,

perfect neutrality impossible since ions and electrons recombine
...
Gases, however, only interact via nearest neighbour method e
...
Van der waals
...
Particles act together to form macroscopic eects e
...
waves,
shocks etc
...
E and B elds aect the bulk motion of the particles
...
1
...
1
...


F = q(E + v × B)

•

Gravity - Useful for very large stellar objects

•

Nuclear - Strong and weak force within stars and govern particles
...
2 Debye Shielding
1
r whereas far from the charge, other particles in the plasma screen out the potential
...


•

Close to the charge

φ(r) ∝

•

For this to work well, forget quasi-neutrality
...


1

•

The ion density remains constant and unchanging but the electrons move and, due to thermal motion, the screening
cloud expands so imperfect screening
...
2
...
Use Poisson's equations:

to relate potential to number density

eφ
E
ne (x) = ne0 exp(− kB Te ) = ne0 exp(+ kB Te )

2
...


where

ne0

is the number density at 0 potential
...


2

3
...
Assuming

=

ne0 e
0

|eφ|

[exp( eφ(r) )
k B Te

kB Te ,

φ(r) =

− 1]

the term can be expanded using exponential power series and this gives:


...

0 kB Te

This gives a 2nd order ODE
...


5
...
2
...


•

A slab of thickness d, containing electrons, is displaced a distance d from a slab containing positive ions
...


•

The density charge:

∆ne = ne − n0 ∝ φ

•

The length also sets the scale for the penetration depth of an externally applied E eld into the plasma body
...
3 The Plasma Parameter
4π 3
3 λD ne is the plasma parameter and it is the number of electrons in the Debye sphere
...


• ND =
• ND

1

gives an ideal plasma whilsts

ND < 1

ne = Zni

is the bulk

is a strongly coupled plasma (i
...
Much closer to liquid and solid

region if small enough)
...
3
...


−1/3

•

Electron ion separation:

•

KE
PE

•

Boltzmann statistics is good for a Debye sphere with many electrons and the small potential assumption (1
...
1)

=

k B Te
eφ(r=¯)
a

a = ne
¯

>1

holds true for distance

r≥a
¯

although breaks down any lower
...
4 Maxwellian velocity
1
...
1 1D Maxwellian
•

Classical particles in thermal equilibrium is valid for
...


2

1
...
2 Maxwellian speed distribution
• < v 2 >=

3kB T
and
m

kB T
m

vth =

1
...
3 Maxwell Boltzmann energy distribution
E
• fM B ∝ exp(− kB T )

where

E = qφ + 1 mv 2
2

gives the total particle energy
...
5 Plasma Oscillations
•

Assuming neutral plasma, a slab of electrons is displaced from xed ions and then released
...

However, the electrons overshoot and the slab oscillates about equilibrium position
...


•

Then consider equation of motion for electron in the slab under action of the E-eld where electron position given
by x(0) + X(t) where x(0) is the equilibrium position
...


•

This frequency gives the electron oscillation frequency, which is linked:

λD =

vth
ωp

1
...

Larmor frequency - The angular frequency of the circular motion
...


rL =

a = acent )

and

in order to get cyclotron frequency
...

Also, electron

frequency is much larger than ion frequency
...
Particles are tied to
B-eld lines
...


1
...
7
...


•

Pressures given from ideal gas law and total plasma pressure is the linear sum of the ion and electron pressures
separately
...


•

Electron uid motion:

me ne

dv e
dt

= − Pe − ne eE = 0

where

ve

is electron uid ow velocity,

− Pe

is the force

density
...


The electron density prole:
balance equation and

scale
...
7
...


•

ne (x)

is not a perfect step but over the Debye length, changes from

An external electric eld penetrates the plasma at a distance
the surface charge to spread the distance
...


Zni → 0
...
7
...


and

• ne = δne + < ne >

and

E

are turbulent, with structures of size

φ = δφ+ < φ >

λD

occuring at timescales

∼

1
ωp
...


φ

where the delta terms are results of the uctuations and the latter is the

averaged out wave - macroscopic scale
...
8 Plasma redenition
•

An ionised gas in which characteristic length scale satises the ideal plasma criterion:

−1/3

ne

λD

L

where the

LHS is the ionisation condition and the RHS is the QN condition
...
9 Non-Ideal Plasmas
1
...
1 Strongly Coupled
•

When PE>KE, Plasma becomes dense and cold
...


Γs =

=

[qφ]s
kB Ts where, for electrons,

[qφ]e =

e2
4π a ae and for ions,

1/3

[qφ]i =

Z 2 e 2 ni
4π a


...
Have Crystalline-like structure
...


1
...
2 Dusty Plasmas
•

Plasmas that contain pieces of solid (dust) or liquid (mist)
...


•

Electrons rapidly charge up dust grains then surround and shielded by ions (opposite of Debye shielding)

•

If Grain separation:

1
...


2
...
9
...


λDB =

ve = vth
2

2

2/3

2

•

Fermi energy:

•

Fermi degenerate pressure: Use Ideal gas P=nkT

•

Degeneracy aects pressure

Ef =

2me (3π

ne ) 3 ∝ ne

Pf

Pcl ,

resistivity, eqn of state, helps maintain ionisation when cold
...
1 Basic Physics of Fusion Reactions
2
...
1 Nuclear fusion
•

2 light nuclear are combined together in order to form a heavier nucleus and a second particle with a net release of
KE
...


2
...
2 Binding energy per nucleon
•

EB
c2 = Zmp +
constituents
...
1
...


•

D-D reaction

•

Conservation laws: Energy, linear and angular momentum, charge, baryon number (total number of nucleons), lepton
number
...


2
...
4 Coulomb barrier_0
•

In order for strong force to cause nuclei to react, they have to approach very close and, in order to do this, the
coulomb repulsion has to be overcome
...
2 The need for fusion
•

Global energy consumption is high and a lot of energy required
...


•

Fission would result in 476T of U needed per day
...


2
...

Reaction rate:

R = n1 n2 σv

where n represents the number densities of the beam (type 1) and target (type 2) nuclei
...


•

Derivation:

1
...
(Assume that the beam
does not spread)
...


2
...
total area presented by target - 2
...


A2

4
...
The number of beam particles passing through target in time:
6
...
x 5
...
The reaction rate is: 6
...


•

Cross section curves - It is a strong function of beam particle velocity, since it depends on the Coulomb barrier
...
Because

•

σD−e

σD−T

Thermonuclear fusion - The solution is to use hot plasma and, by using this, the electrons, D and T all have a
thermal energy distribution, of the same temperature
...
4 Ignition and Breakeven
•

Remember idealised fusion reactor diagram

U
τE where τE is the energy connement time
...
U is the internal
energy of the plasma
...


•

Condition to mantain the plasma at fusion temperature:

Pheat ≥ PL

where

Pheat > PL

would result in a temperature

increase
...
4
...


•

Pα ≥ PL where Pα
1
Rth = nD nT < σv >= 4 n2 < σv >
...


Ignition criterion:

nτE ≥

Pα = Rth Eα

where

is the

12kB T
<σv>Eα

2
...
2 Breakeven
• ηPout = PL

where

Pout = PL + Pth

where

•

Combine the two expressions and use the

•

Lawson's criterion:

Pth
PL

is the thermonuclear power
...
4
...
Good
for pulsed systems
...


2
...
5
...


•

Toroidal (B-phi) and poloidal (B-theta) eld components of the B-eld
...
Poloidal - unstable and plasma wriggles out
...


a
R0

minor radius
major radius and

Inverse aspect ratio:

•

Density, temperature and pressure peak at centre of tokamak and: particles and energy diuse out in time

=

=

B∝

1
R

•

particle collisions also relevant
...


•

Plasma beta:

•

Plasma heating: Ohmic heating (From current: not eective at high T, caused by plasma resistivity-

ηj 2

where

β=

resistivity ∝

Pohm
vol = E
...


2
...
2 Inertial connement fusion
•

Short connement time and high density approach
...


nτ ∝ ρR

where

τ≈

R
cs and

n=

ρ


•

Compress since energy too high at liquid density
...


•

Why use a shell? Pressure of required hot D-T >> pressure from the laser - Shell gradually accelerated inward by
laser and, due to spherical convergence, energy given to shell compressed in space-time
...
(See slides)
...


3
...


•

Dierentiate each component W
...
T time and substitute known expression
...


can obtain Larmor frequencies
...
r
...


•

Centre of the circular orbit is known as the guiding centre
...


•

The velocity components of particles follow a maxwellian distribution
...
2 Guiding centre drifts
3
...
1 E x B Drift
•

If a uniform E-eld is present in the plasma the guiding centre doesn't stay on a single eld line but drifts across
the eld lines
...

Remove
velocity:

qE⊥ ,

transform the co-ordinates by moving into a reference frame which moves in the x-direction with
E⊥
E
so that vx = vx − ⊥
B
B

•

Substitute into original EOM's and use the static elds approximation
...
This means that the cross product can move the whole plasma but cannot cause electric currents or
charge separations
...


where the velocity is that needed for the magnetic force on the guiding centre (gc) to balance the electric

force on it
...
2
...


vD =

3
...
3 Curvature drift
1
r3

•

B eld normally curved and inhomogeneous and falls o at

•

The eld has a gradient and curvature and each dipole causes a drift
...


•

Assume G
...
can be treated as a charged particle and

v gc = v c
⊥

ensures

q(v gc × B)
⊥

force provides centripetal

acceleration necessary for circular motion of the gc
...
2
...
(See diagram)
...


Then use unperturbed LO's:

vx = v⊥ cos(Ω0 t), vy = −v⊥ sin(Ω0 t)

and

y = y0 +

Fy = −qvx B(y)
v⊥
Ω0 cos(Ω0 t) and

Taylor expand
...


•

2π
Ω0 = τg ) gives:
obtained from the above substitution]
...
2
...


The average over 1 gyration (

< Fy >=

0

Fy dt =< 1 > + < 2 >[1

and 2 are components

3
...
5 Drifts in Tokamaks
•

See diagram on handout

•

Poloidal B eld stops plasma from drifting outwards by short-circuiting charge separation which leads to plasma
connement
...
3 Magnetic moment and Plasma diamagnetism
•

For a current loop magnetic moment:

•

Diamagnetic current:

•

Plasmas are diamagnetic and the eld from j opposes the B eld
...


8

3
...


•

Assume cylindrical symmetry
...


F = −µ

B

Assume E=0 and that particle KE is conserved for static B therefore magnetic moment is conserved
...

∂t
dt
dt

Note:

Fz = −µ ∂Bz and
∂z

dBz
dt

=

∂Bz
∂t (=

0)+vz ∂Bz
...


3
...
- see diagram
...
Particles can be reected

as a result (Trapping)
...


Mirror trapping condition:







Field minimum:

B = B0 ,

Particle reects at:

v⊥ = v⊥0 ,

B = BR

1
2
KE invariant:
2 mv⊥R

=

where

1
2
2 mv⊥0

Magnetic moment invariant:
Reection occurs when

+

2
mv⊥R

2BR

BR < Bm

v

v =v

R

= 0,

0

v⊥R = 0

1
2
2 mv 0

=

2
mv⊥0
2B0
...
5
...


•

The collisions




•

Scatter the particles
Cause diusion within the cone
Perpetuate loss of energy

See diagram: Edge of cone

v 0
v⊥0

Bin
B0

−1

3
...
2 Van Allen radiation belts
•

Energetic protons trapped in inside the belt and energetic electrons trapped on outer
...


•

Escaping particles stream down eld lines to the poles to create an aurorae
...


3
...
3 Banana orbits in Tokamaks
•

Caused due to electron drifts and mirroring

•

See diagram

•

Useful in connement of particles

9

4 Collisions
4
...
1
...


•

This results in momentum exchange whereby a large angle of deection causes a loss of previous motions
...


λmpf
the collision time, average momentum forgetting time
...
1
...


•

This results in many small angle deections, that cause a large cummulutive eect which leads to a loss in momentum
knowledge
...
1
...


4
...


• tan( χ ) = ( 4π
2

 b > b0 ,
 b ≤ b0 ,
•

ze2
1
2 )( b )
0 m e ve

χ < 90o
χ ≥ 90o

Mean free path




λ0

= ( bb0 )

where

b0

is the impact parameter at

χ = 90o

χ

by colliding with a stationary ion

- Landau Parameter
...


b0 and length l has a collisional CSA: σ0 = πb2
0
2
within b0 of electrons: Ni = ni πb0 l where, for Ni = 1, l = λ0

Giant cylinder diagram, radius
Average number of ions

τ0 =

 λ0 ∝

λ0
ve
4
ve

10

and the collision time:

4
...



•

Rutherford scattering:

tan( χ ) =
2

b0
b and small angle approximation:

χ(b) ≈

2b0
b via expansion of tan function
...


•
•

Mean squared angular deection:

< ∆θ2 >b = Nb χ2
b

Integrate expression over the shells and between the limits:

bmax = λD

(due to screening) and

bmin = b0

or

λDB

due to the uncertainty principle
...


Λ=

bmax
bmin

λe
ve

•

Temperature dependence:

•

Ion collisions:

τi ≈

√

ve = vrms = ( 3kTe )1/2
me

mi
2 z12 ( me )1/2 τe

where the rst term is due to the reduced mass, the second is due to Coulomb

force and the third is due to ions being bigger
...
4 Eects of Collisions
4
...
1 Resistivity
•

Apply an Electric eld to a plasma in order to create a current (See diagram)

•

Drude model - Balances momentum gain and loss (electron drift and, between collisions, electron reach the drift
velocity)

•

Drift equation:

−eE τe + −me vdrif t = 0 where the rst term is acceleration momentum and the second is a frictional
¯

term
...

in addition to substituting for

j

and

in order to get expression for the resistivity
...
2 out and it consists of averaging

•

Ohmic heating:

Power per unit volume =

E
...


and so tokamak diusion temperature cannot be

reached ohmically
...
4
...


where the energy connement time:

1 a2
2 D

τE ≈

radius2
D

= τcol ( ra )2
Li
∂f
∂t

2

= D∂ f
∂x2

whilst non-linear 3D diusion equation:

f (x, t) =

is a constant
...
(D f )
√

t

and the area

4
...
3 Bremstrahlung
•

See diagram on lecture notes
...


•

q 2 a2
2b
ze2
1 2 e2
2b
6π 0 c3 × ve ≈ ( 4π 0 b2 me ) 6π 0 c3 × ve where the rst term is
related to Larmas formula for power radiated by accelerating charge and the second term is the collision duration
...


•

The energy from one electron colliding with many ions in time

(ve ∆t)ni

∆t

is given by:

∆Etot =

´ bmax =∞
bmin =λDB

δE × 2πbdb ×

where the second term is the cross section of the shell, the third term is the cylinder length x number of

ions within the shell
...


5 Magnetohydrodynamics
•

Considers plasma as a single uid (macroscopic) where the individual particles are not dened
...
1 MHD equations
∂ρ
∂t

•

Mass continuity equation:

•

Equation of motion - momentum equation:

•

Equation of state (Energy equation):

•

Ohm's law:

•

Maxwell's equations:

+


...

No monopoles, Faraday's law and Ampere's law where neglect displacement current since

uid is slow moving
...


5
...
1 Continuity equation
•

Mass is conserved in the region, albeit not at a point due to mobility of particles
...
67-68) of xed cell in order to derive the continuity equation
...
1
...
]C

See Mathematical derivation
...
1
...


•

The force densities are a result of pressure gradients and

qv × B

ends up becoming

electrons and ions
...


•

Pressure forces - Newtons second law where only thermal pressure included:

 ma = ρ∆V dux = ρ∆x∆y∆z dux
dt
dt
 Force on left face: P (x)∆y∆z(ˆ)
x
12

j×B

when summing over




•

Add left and right face pressure and equate to N2 law expression in order to get:
Applying same to y and z components, combining, gives Euler's equations:

Force on right face:

P (x + ∆x)∆y∆z(−ˆ)
x
ρ dux = − ∂P
dt
∂x

ρ du = − P
dx

Magnetic Forces - Derived from Lorentz force law for electrons and ions

 F tot =

j (F e )j

+

k (F i )k where there are

Ne

electrons which follow:

F e = −e(E + v e × B)

and

Ni

ions

which follow similar force
...


Due to quasineutrality, charge density equals 0 and so Electric eld contribution vanishes however, this is only
for a large scale
...
)u] = − P + j × B + ρg
∂t

where Maxwell's equations and Ohm's law

determines the magnetic eld and electric current
...


•

Reynolds number =

|ρ(u
...
1
...


•

Due to heat ux in stationary matter which moves internal energy from hot to cold regions:







∂U
∂t

+(u
...

Divergent heat ow:
Ohmic heating:

ηj

−
...


•

Adiabatic equation of state - Use adiabatic gas law since heat ow is assumed negligible
...
1
...

 The − Pe terms and the more terms are ignored
...

 For ideal MHD: E = −u × B
∗
∗
∗
•

Dynamo like induction of the electric eld due to a conductor moving past a magnetic eld
...


LT of EM elds - Transforming from frame where: E=0 and B into

E = −u × B

where

B = γB

Due to Quasineutrality, Gauss' law can't be used to calculate E eld but, instead, Ohm's law can be used
...
u

5
...


• Te = Ti

and assume isotropic pressure
...
To avoid

this, rapid collisions to siphon

•

ρ, u etc
...
MHD only works well if:

the B-eld and

λmf p

LM HD

rl

LM HD

across

along it
...


Localisation - Particles must be tied to uid elements
...
This allows the Maxwellian to vary with
position otherwise Plasma would have uniform density and temperature
...


5
...
When the magnetic eld lines appear to get carried along with
the ow of a perfectly conducting uid
...




Proof: Consider a surface S, bound by a closed contour C which moves exactly with the uid ow
is not uniform, C will deform during subsequent motion
...
ds

u

and if ow

where two factors inuence

this ux:

∗
∗



B eld is time dependent
The area can deform and change

δl of the contour where small refers to u being approximated as uniform
δt, it sweeps out a parallelogram of area: δs = uδt × δl
...
In time

of the ux and substitute B-eld evolution with 0 resistivity (see above) and rearrange the second term, using
cyclic shift, in order to get:


•

dΦc
dt

=0

The ux doesn't change but the eld itself can, since the area is prone to deformation
...


It allows the magnetic eld to diuse through the

plasma
...

This process removes Magnetic eld energy and converts it into thermal energy via Ohmic heating
...


∆x → δc =
and ∆t → (¯c )e
τ

Step length
plasma)

•

∂B
∂t

c
ωp (collisionless skin depth - Depth that a low freq
...


Magnetic Reynold's number -

Rm =

convection
dif f usion

=

| ×(u×B)|
η
| ×( µ
×B)|

≈

0

this leads to turbulence, especially common at high temperatures

14

µ0 ul
η
...
4 Magnetic ux density
•

Using Ampere's law, j x B becomes:

1
µ0 (B
...
This pressure augments thermal pressure
...




3rd term is the magnetic tension
...
In order to do this, replace the del operator with
1/Radius of curvature
...
1 EM waves in Plasmas
•

EM waves are dispersionless in a vacuum and all frequencies propagate at the same velocity
...


This induces dispersion and introduces a

minimum wave frequency that can propagate through plasmas
...
1
...


This leaves only the

2

E

term from the above

procedure
...
Electrons drift with the E-eld where thermal motion ignored and then
linearised
...


This gives the x-component of the electron equation of

motion
...
Assume unperturbed plasma is uniform and stationary

∗
∗
∗
∗



ne = n0 + n1 (z, t) where |n1 |
uex = 0 + ux1 (z, t)
Ex = 0 + Ex1 (z, t)
By = 0 + Bx1 (z, t)

n0

Linearise the current by neglecting the products of small quantities and variables

∗ jx1 = −e(n0 + n1 )ux1 = −en0 ux1 − en1 ux1

where take the time derivative of the current and neglect the

small 2nd order term
...




For weak linear waves

d
∂
∗ dt → ∂t
∗ ne can be assumed constant
∗ v × B force is negligible compared



to the E-force unless the light is relativistic
...
1
...


•

Cut-o frequency - No propagation of low frequency waves below the cut-o frequency because the plasma electron
current cancels the displacement current in Ampere's law
...


•

Refractive index -

•

c
vp

≤1

Examples:



η=

Laser-Solid interaction: Due to EM wave dispersion relation, wavenumber changes in the plasma
...
The laser beam heats the surface rapidly and ionises it into a
plasma - See P
...




Critical density: At

ωp (x) = ωL
...


ne = ncrit =

2
0 me ωL
e2


...

Refraction: Total internal reection
...


Heat ow from critical

surface to ablation surface, strong enough to make headway against the supersonic ablation outow
...




Reection of radio and microwaves o the plasma (generated by intense frictional heating of the crafts underside
with the air) is a result of communication blackout during spacecraft reentry
...
2 Langmuir waves
•

Longitudinal waves due to electron-uid oscillations
...


•

Restoring forces = Electric elds (Space charge generated where the space charge is due to excess/decit of electrons
and ions are stationary) + Thermal pressure
...


•

With nite temperature, perturbing electrons in one direction results in a wave whilst those pushed at 0 temperature,
will oscillate back and forth
...


Equations and unknowns

∗
∗
∗
∗

δne = n1
δuex = ux1
pressure δPe = P1

From mass continuity equation, perturbed electron density
From the momentum equation, electron uid velocity
From the energy equation, perturbed electron
From Gauss' law, the electric eld

Ex = Ex1

16

T0

and ion density

ni =

n0
Z



Substitute the linearised expressions into the momentum equation and adiabatic law to relate perturbed pressure
and density
...


6
...
Motion of the entire plasma
...
3
...


•

For B=0, an ordinary sound wave:

•

With inclusion of thermal pressure:

•

For

B2
2µ0

P

vp =
vp =

restoring f orce
inertia
γP
ρ

= cs

get compressional Alfven waves
...
The number of degrees

of freedom is 2 and

γ=

d+2
2

=2

which gives:

vp =

B2
µ0 ρ

= vA

which is the Alfven speed
...
3
...


phenomena and shocks form
...
3
...
g
...


uy = By = 0
•

Unperturbed plasma is uniform, static and stationary:

•

Assume perturbations, along with the x and z change in velocity, are small
...
where the expression is a result of shear forces buckling the eld

lines
...

Combine the two expressions to get wave equation for a transverse B-eld perturbation travelling in z-direction:

2
2 2
∗ ∂t Bx1 = vA ∂z Bx1

17

7 Magnetic Connement
7
...
plasmas, that obey MHD
...


•

Current lines (lines following the current density vector eld) lie on ux surfaces
...





Proof: In order to be in MHD eqm
...

Equate MHD F=0 and dot product remaining expression with B to get 0
...
Same type of argument for j
...


•

B-eld strength can vary on the tube since internally stronger than external in tokamak surface
...
2 Z-pinches
•

Magnetic connement devices where there is no external eld (Plasma is conned due to its own azimuthal B-eld,

Bθ ,

generated by an axial current,

jz ,

driven through the plasma cylinder
...


•

For Cylindrical equilibria (calculate radial structure of the pinch)







Assume pinch in innitely long so can ignore end losses
...


There is a denite edge where density drops to 0
...

Take radial component of the force balance at equilibrium equation:

∂P
∂r

= jθ Bz − jz Bθ

where assume the rst

j and B term equates to 0
...
and mag
...


7
...
1 Skin Current
• η → 0:



•

All current ows along pinch surface in innitessimally thin layer:

Inside pinch

r < a:

r>a: No plasma so

Thermal pressure uniform and no B-eld so

B∝

P =0

− a)

for eqm
...


B-eld prole: B-eld prole obtained from Ampere's law (E+M rst year)
...
2
...
Inside, the amperian path model is used:

I(r) = πr2 jz
...
Current also diuses in until it becomes uniform:

See graphs
...
Integrate and determine

7
...
3 Bennett relation
•

Expression for the average temperature of a cylindrically symmetric plasma at MHD eqm
...


• N=
•

0

ni (r)2πrdr

and

¯
T =

1
N

´a
0

ni (r)T (r)2πrdr

For skin current plasma:




•

´a

Assume temperature is Isothermal and so density is uniform (P is constant within r=a)

P = (ne + ni )kB T = (Z + 1)ni kB T

Plasma ideal gas of electrons and ions thermal pressure:
Equate thermal pressure to

P =

where

Ti = Te = T

µ0 I 2
8π 2 a2

General proof:



Derive radial form of radial pressure, and don't expand:



Get both B factors inside derivative to obtain:





r2 ∂P =
∂r

∂P
1 ∂
∂r = − rµ0 ∂r (rBθ )Bθ
1 ∂
− 2µ0 ∂r [rBθ (r)]2

Integrate both sides from r=0 to r=a and evaluate B by using wire formula:

Bθ =

µ0 I
2πa

The LHS integrated by parts and use P(a)=0 BC
...
3 MHD instabilities
7
...
1 Sausage instabilities (m=0)
•

Perturbations to the pinch radius grow exponentially with time:
perturbation wavenumber and

•

γ

a(z, t) = a0 + δa0 cos(kz)exp(γt)

where k is the

is the perturbation growth rate
...




•

This causes

Bθ

to increase and the magnetic pressure at neck to increase faster than the pressure increase
...


Instability growth rate:

γ∼

1
τA and

τA =

a0
vA where

vA

is Alven wave speed
...
3
...


•

Due to curvature, density of B-eld lines greater inside the column edge which result in pressure acting outwards
and spiral distortion to grow
...
3
...


For m=0,

Bz

is frozen into the plasma and, if a neck tries to form, axial lines compress together raising

Bz

and, if this is >Bθ then instability stops
...


eld exceeds the magnetic force density due to

This leads to Kruskal-Shafranov equation:

λ < 2πa0 Bz
Bθ
19

where

Bθ ,

λmax

If curvature of restoring force

screw pinch is stable
...


7
...
4 Tokamaks and the Safety factor
•

For stabilizing at m=0, toroidal component of B-eld > poloidal component
...


Bφ >

2πR0
2πa Bθ

q(r) =

r Bφ (r)
R0 Bθ (r) can vary across the ux surface and, using krus
...


1

which changes the toroidal ux surfaces into cylindrical and, if open up ux surface,

poloidal direction
...


Bφ

Plasma beta is small since

Bθ
β=

where the toroidal B-eld controls stability and poloidal component connes

2
Bφ
2
Bθ

7
...


7
...
1 Neo-classical diusion
•

Takes into account inhomo and curved B-elds in Tokamaks
...
4
...


• DGM ≈

kB T
qB

8 Kinetic Theory
8
...
1
...


•

Distribution function -

f (r, v, t)

- 7D space
...


•

Independent variables - The 3 components in f are independent and, despite linkage due to N2 law, HUP works for
particles
...
1
...
(f v) + v
...
fv is the ux of f in conguration space while fa is
ˆ
the ux in the velocity dimensions of phase space
...
(f v) = v
...
v)
v
...
(

vf )

+ f(

where v is independent of r so the second term equates to 0
...
m ) where force is independent of velocity so the second term equates to 0
...


Collisions - Diuse the distribution function in velocity space
...
1
...
(v × B)

= 0 and,

while neglecting collisions,

∂fα
∂t

q
+ v
...

α

v fα

= 0 where alpha

applies to each species
...


8
...
4 Collision term
•

Krook term:

( ∂f )coll = − f −fM
∂t
τC

8
...


•

P
...


8
...
1 Continuity equation
•

Take Zeroth moment of Boltzmann kinetic equation where the rst term can be integrated directly, the second term
can be integrated directly (v

= v(r))

and the E-eld in the nal term can be integrated by using Gauss' law and

E = E(v)
•

The surface integral is over a sphere, in velocity space, where

•

The v x B term can be expanded using the product rule and the middle term vanishes due to same limit as third
integral above and the

•

v
...


limit
...


∂n
∂t

•

This gives:

•

MHD single uid continuity equation:

+


...
(me ne ue + mi ni ui ) =

∂ρ
∂t

+


...


8
...
2 Momentum equation
•

u is unknown but this equation determines its evolution
...
Multiply it by mv
and integrate which can be reduced for electrons, if thermal pressure is isotropic
...


•

MHD single uid momentum equation - Adding electron reduced equation and ion reduced form, assuming quasineutrality

mi ni

me ne

where

ρ du = − P + j × B
dt
21

8
...
3 Energy equation and closing the uid equations
•

Energy equation - Governs evolution of P can be obtained by taking second moment of boltzmann equation: mul-

1
2
2 mv and integrating but this introduces 3rd order uid quantity heat ow q
...

tiplying by

•

Closure - The above process is continuous and, to avoid, close set of uid equations by using transport coecients
...
g
...


•

q = −κ T

Localization - Assume particles interact only in close proximity (a larmor radius much smaller than the total length
of the plasma)
...
3 Langmuir Waves
•

Perturbing distribution function introduces a wave
...


8
...
1 The Dispersion relation
•

Real term is the Langmuir wave dispersion relation, as depicted in section 6
...


•

The imaginary term is the damping term caused by the singularity
...
4 Landau Damping
•

It is a collisionless phenomena
...


8
...
1 Resonant Particles and trapping
•

Longitudinal modulation - Electric eld and potential:



•

In lab frame, where waves propagate:

Ex = − ∂φ
∂x

n1 = δn0 sin(kx − ωt)

are modulated
...
2 and limits: In frame of a moving wave, electron velocity

vx = vx − vph

Passing particles - Particles moving much faster/slower than the wave and, as such, pass through wave's modulations
and experience a varying potential and so, wave doesn't eect their motion
...

If they are in the accelerating part of the wave, can lose KE
...
This means trapped particles oscillate back and forth in wave
...
4
...
4 prole, more particles are accelerated
...


•

Slow resonant electrons gain energy move backwards, wrt the wave
...
For slow electrons in negative regions, they move more backwards wrt the
E-eld and are decelerated less eectively so KE gain > KE loss
...
4
...


•

Short wavelengths -

kλD ∼ 1 which suggests wave should damp in less than 1 period
...


8
...
4 Wider implications
•

Instabilities - Landau damping acts in reverse and results in exponential wave growth
...
g
...
Micro-instabilities are non-maxwellian instabilities
that can only be decribed by kinetic theory
...


8
...


•

Ponderomotive force - Due to intensity gradient, laser exerts force on a plasma
...
This means the
phase velocity of the wakes that trail the plasma pulse close to c (under extreme density plasmas) can be captured
and increased in energy
...
Can use Gauss' law to determine
E-eld max

---

Title: Plasma Physics 26 Lecture note summary,
Description: Summarised from the taught module "Plasma Physics" at Imperial College London. Fourth Year Optional Module.

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