*G0B28A* *0-1*

**Plasma Physics of the Sun**

**Stefaan Poedts**

*Centre for Plasma-Astrophysics, KU Leuven (Belgium)*

Course G0B28A, KU Leuven

Notes by S. Poedts (and R. Keppens& J.P. Goedbloed)
based on *PRINCIPLES OF MAGNETOHYDRODYNAMICS*

*G0B28A* *0-1*

**Aims**

### 1.

to introduce to the students a number of*specific applications of plasma*

*astro-physics in the closest star:*

**the Sun**

### 2.

to give the students the insight that the Sun plays a**key role**

*for our insight in the*

*physics of stars*and other astrophysical and laboratory plasmas

*G0B28A* *0-1*

**Aims**

### 1.

to introduce to the students a number of*specific applications of plasma*

*astro-physics in the closest star:*

**the Sun**

### 2.

to give the students the insight that the Sun plays a**key role**

*for our insight in the*

*physics of stars*and other astrophysical and laboratory plasmas

*G0B28A* *0-1*
**Nuclear forces**
*⇓*
quarks / leptons
nuclei (*+) / electrons (−)* 10*−15*m
**Electrostatic forces**
*⇓*
atoms / molecules 10*−9*m
(ordinary matter: electrically neutral)

**..**
**..**
**..**
**..**
**.**
**Gravity**
*⇓*

stars / solar system 109*/*1013m
galaxies / clusters 1020*/*1023m

*G0B28A* *0-1*

**Aims**

### 1.

to introduce to the students a number of*specific applications of plasma*

*astro-physics in the closest star:*

**the Sun**

### 2.

to give the students the insight that the Sun plays a**key role**

*for our insight in the*

*physics of stars*and other astrophysical and laboratory plasmas

### 3.

to use**magnetohydrodynamics**

*as a mathematical model*(equations, boundary conditions, estimates, approximations, etc.) to describe magnetic phenomena in the outer layers of the solar atmosphere

*G0B28A* *0-1*
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*G0B28A* *0-1*

**Aims**

### 1.

to introduce to the students a number of*specific applications of plasma*

*astro-physics in the closest star:*

**the Sun**

### 2.

to give the students the insight that the Sun plays a**key role**

*for our insight in the*

*physics of stars*and other astrophysical and laboratory plasmas

### 3.

to use**magnetohydrodynamics**

*as a mathematical model*(equations, boundary conditions, estimates, approximations, etc.) to describe magnetic phenomena in the outer layers of the solar atmosphere

### 4.

to offer the students the opportunity to**apply a number of mathematical**

**tech-niques in specific situations**

*: e.g. solving ordinary and partial hyperbolic*

*differen-tial equations (incl. numerically: shock waves), solving non-linear elliptic differendifferen-tial*
*equations, complex analysis (branch cuts in the complex plane, Riemann-surfaces,*
*etc.), perturbation analysis, initial value problems, solving hermitic and non-hermitic*
*eigenvalue problems, etc.*

*G0B28A* *0-2*

**Contents**

### 1. Introduction & elements of plasma physics

### [ book: Chaps. 1 & 2 ]

*plasma: definitions, occurrence, conditions, different descriptions*

### see also G0P71a: Introduction to plasma dynamics

### 2. MHD model

### [ book: Chap. 4 ]

*laboratory and astrophysical plasmas from one point of view*

### see also G0P71a: Introduction to plasma dynamics

### 3. MHD waves

### [ book: Chap. 5 & Sect. 7.4 ]

*waves in homogeneous and inhomogeneous plasmas, phase and group diagrams,*
*continuous spectra*

### 4. Magnetic Structures

### [ book: Chap. 8 ]

*sun, planetary magnetospheres, stellar winds, astrophysical jets*

### 5. Initial value problem and wave damping

### [ book: Chap. 10 ]

*wave damping, quasi-modes, leaky modes*

### 6. Resonant absorption and wave heating

### [ book: Chap. 11 ]

*ideal MHD theory, wave heating*

*G0B28A* *0-3*

**Literature**

**Introductory plasma physics:**

*•*

*J.A. Bittencourt, Fundamentals of Plasma Physics (1986).*

*•*

*A.R. Choudhuri, The Physics of Fluids and Plasmas. An introduction for Astrophysicists*(1998).

**Magnetohydrodynamics:**

*•*

*J.P. Freidberg, Ideal Magnetohydrodynamics (1987).*

*•*

*J.P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics (2004).*

http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521626072
www.rijnh.nl/users/goedbloed *(ErrataPrMHD.pdf)*

*•*

*M. Goossens, An introduction to plasma astrophysics and magnetohydrodynamics*(2003)

**Plasma astrophysics:**

*•*

*E.R. Priest, Solar Magnetohydrodynamics (1984).*

*G0B28A* *0-4*

**Plasma physics on www**

*•*

**Fusion energy**

### www.fusie-energie.nl

*(nuclear fusion and ITER, in Dutch)*

*•*

**Solar physics**

### dot.astro.uu.nl

*(Dutch Open Telescope)*

### www.spaceweathercenter.org

*(space weather)*

*•*

**Plasmas general**

### www.plasmas.org

*(basics, applications of plasmas)*

*•*

**Related notes**

*G0B28A* *0-5*

**Lectures**

*•*

Thursdays: 10.45–12.45, 200B
**‘Take-home’ assignments**

*•*

between consecutive lectures: ‘take-home’ assignments (obligatory)
*•*

compensated by less lectures
**Evaluation**

*•*

oral exam with written preparation (60 %)
*Introduction:* *Overview* *1-1*

**Chapter 1: Introduction and elements of plasma physics**

** **

### Overview

*•*

**Motivation:**plasma occurs nearly everywhere, magnetized plasma unifying theme for laboratory and astrophysical plasma physics; [ book: Sec. 1.1 ]

*•*

**Thermonuclear fusion:**fusion reactions, conditions for fusion, magnetic confinement

in tokamaks; [ book: Sec. 1.2 ]

*•*

**Astrophysical plasmas:**examples of astrophysical plasmas

*•*

**Definition of plasma:**usual microscopic definition (collective interactions), macro-scopic definition (the magnetic field enters). [ book: Sec. 1.4 ]

*Introduction:* *Overview* *1-1bis*

**Chapter 2: Elements of plasma physics**

** **

### Three theoretical models:

*•*

**Theory of motion of single charged particles**in given magnetic and electric fields;

[ book: Sec. 2.2 ]

*•*

**Kinetic theory of a collection of such particles,**

*describing plasmas microscopically*by means of particle distribution functions

_{f}

_{f}

_{e,i}_{(r, v, t)}

; [ book: Sec. 2.3 ]

_{(r, v, t)}*•*

**Fluid theory (magnetohydrodynamics),**describing plasmas in terms of averaged

*macroscopic functions of*

**r**

and _{t}

. [ book: Sec. 2.4 ]
_{t}

Within each of these descriptions, we will give an example illustrating the plasma property relevant for our subject, viz. plasma confinement by magnetic fields.

*Introduction:* *Motivation* *1-2*
** **

### Plasma

*•*

Most common (_{90%}

) state of matter in the universe.
*•*

On earth exceptional, but obtained in laboratory thermonuclear fusion experiments at
high temperatures (_{T ∼ 10}

8_{T ∼ 10}

_{K}

).
*•*

Crude definition: *Plasma is a completely ionised gas, consisting of freely moving*

*positively charged nuclei and negatively charged electrons.*

**
**

### Applications

*•*

Magnetic plasma confinement for (future) energy production by Controlled
Thermonu-clear Reactions.
*•*

Dynamics of astrophysical plasmas (solar corona, planetary magnetospheres, pulsars,
accretion disks, jets, etc.).
*Introduction:* *Nuclear fusion (1)* *1-3*

**
**

### Reactions of hydrogen isotopes

**n**

**+**

**n**

**+**

**n**

## +

**n**

**+**

**n**

**+**

## +

**n**

### D

### T

### He

### n

3.5 MeV 14.1 MeV

### Two products

*•*

Charged _{α}

particles:
_{α}

captured in plasma magnetic field

*⇒ α*

particle heating
*•*

Neutrons:
*Introduction:* *Nuclear fusion (2)* *1-4*
**
**

### Why plasma?

*•*

To overcome electrostatic repulsion
of nuclei need _{10 keV}

*⇒ T ∼ 10*

8 _{K}

_{(ionisation at}

_{14 eV}

_{).}

**+**

**n**D

**2**

*⇒*

*Plasma*

*≡*

*completely ionised gas consisting of freely moving positively charged*

*nuclei and negatively charged electrons.*

** **

### How to confine?

*•*

Magnetic fields:
### 1.

charged particles gyrate around field lines;### 2.

fluid and magnetic field move together (“**B**

frozen into the plasma”);
### 3.

thermal conductivity:_{κ}

_{κ}

_{}* κ*

*.*

_{⊥}*Introduction:* *Nuclear fusion (3)* *1-5*
** **

### Power balance:

*•*

Fusion power *⇔*

radiation _{+}

transport
losses (Lawson criterion).
*•*

At minimum (_{T ∼ 25 keV !}

):
_{T ∼ 25 keV !}

*nτ*

*E*

*∼ 0.6 × 10*

20 ### m

*−3*

### s ;

typically:

*n ∼ 10*

20 ### m

*−3*

*→ τ*

*E*

*∼ 0.6 s !*

*•*

upper curve: view that power losses
should be completely balanced by _{α}

-particle heating
_{α}

*⇒*

at _{T ∼ 30 keV}

:
_{T ∼ 30 keV}

*nτ*

*E*

*∼ 1.5 × 10*

20 ### m

*−3*

### s ;

*⇒*

*Magnetic fields provide the only way to confine matter of such high temperatures*

*during such long times.*

*Introduction:* *Nuclear fusion (4)* *1-6*

**
**

### Interaction of currents and magnetic fields

*•*

Schematic history of
fusion experiments:
**Tokamak:**

delicate balance between equilibrium & stability

**z - pinch:**

very unstable (remains so in a torus)

θ** - pinch:**

end-losses (in torus: no equilibrium)

**B**
**j**

**B**

**B**
**j**

*Introduction:* *Nuclear fusion (5)* *1-7*
** **

### Tokamak

*•*

Magnetic confinement:
poloidal coils producing toroidal magnetic field

transformer winding (primary circuit) resultant helical field plasma contained by magnetic field

iron transformer core

plasma current p p (secondary circuit) Bpol: poloidal magnetic field Btor: toroidal magnetic field

*Introduction:* *Nuclear fusion (6)* *1-8*
** **

### Tokamak

*Introduction:* *Astrophysical plasmas (1)* *1-9*

** **

### The Standard View of Nature

**Nuclear forces**

*⇓*quarks / leptons nuclei (

*+) / electrons (−)*10

*−15*m

**Electrostatic forces**

*⇓*atoms / molecules 10

*−9*m

(ordinary matter: electrically neutral)

**..**
**..**
**..**
**..**
**.**
**Gravity**
*⇓*

stars / solar system 109*/*1013m

galaxies / clusters 1020*/*1023m

universe 1026m

*Introduction:* *Astrophysical plasmas (2)* *1-10*

**
**

### The universe does not consist of ordinary matter

*•*

*> 90%*

*is plasma:*

electrically neutral, where the nuclei and electrons are not tied in atoms but

*freely move as fluids.*

*•*

The large scale result is *Magnetic fields*

(example: interaction solar wind – magnetosphere).

**
**

### Geometry

*•*

Spherical symmetry of atomic physics and gravity (central forces) not present on the
plasma scale:
**∇ · B = 0**

**∇ · B = 0**

*is not compatible with spherical symmetry*

*Introduction:* *Astrophysical plasmas (3)* *1-11*

**
**

### Example: The Sun

**a magnetized plasma!**

*Introduction:* *Astrophysical plasmas (5)* *1-12*

**
**

### Example: Coronal loops (cont’d)

*Introduction:* *Astrophysical plasmas (6)* *1-13*

**
**

### Example: Stellar wind outflow (simulation)

*•*

Axisymmetric magnetized
wind with a ‘wind’ and a
‘dead’ zone
[ Keppens & Goedbloed,
**Ap. J. 530, 1036 (2000) ]**

*Introduction:* *Astrophysical plasmas (8)* *1-14*

**
**

### Example: Polar lights

**Beauty of the polar lights** (a1smallweb.mov)

*Introduction:* *Astrophysical plasmas (9)* *1-15*

**
**

### Example: Accretion disk and jets (YSO)

Young stellar object (

_{M}

_{M}

*∗ ∼ 1M*

*):*

_{ }accretion disk ‘seen’ edge-on as dark strip, jets colored red.

*Introduction:* *Astrophysical plasmas (10)* *1-16*

**
**

### Example: Accretion disk and jets (AGN)

Active galactic nucleus (

_{M}

_{M}

*∗ ∼ 10*

8_{M}

_{M}

*): optical emission (blue) centered on disk, radio emission (red) shows the jets.*

_{ }*Introduction:* *Astrophysical plasmas (11)* *1-17*

**
**

### Example: Accretion disk and jets (simulation)

Stationary end state from the simulation of
a Magnetized Accretion Ejection Structure:
disk density surfaces (brown), jet magnetic
surface (grey), helical field lines (yellow),
accretion-ejection particle trajectory (red).
**[ Casse & Keppens, Ap. J. 601, 90 (2004) ]**

*Introduction:* *Definitions of plasma (1)* *1-18*
** **

### Crude definition:

*Plasma is an ionized gas.*

Rate of ionization:

_{n}

_{n}

*n*

*i*

*n*

### =

*2πm*

*e*

*k*

*h*

2

_{3/2}*T*

*3/2*

*n*

*i*

*e*

*−Ui/kT* _{(Saha equation)}

– air:

_{T = 300 K}

, _{T = 300 K}

_{n}

_{n}

_{n}_{= 3 × 10}

25 _{= 3 × 10}

_{m}

*−3*,

_{U}

_{U}

_{i}_{= 14.5 eV}

_{= 14.5 eV}

*⇒ n*

_{i}_{/n}

_{/n}

_{n}*≈ 2 × 10*

*−122*(!) – tokamak:

_{T = 10}

8 _{T = 10}

_{K}

, _{n}

_{n}

_{i}_{= 10}

20 _{m}

*−3*,

_{U}

_{U}

_{i}_{= 13.6 eV}

_{= 13.6 eV}

*⇒ n*

_{i}_{/n}

_{/n}

_{n}*≈ 2.4 × 10*

13

### Microscopic definition:

*Plasma is a quasi-neutral gas of charged and neutral particles which exhibits collective*
*behaviour (Chen).*

(a) Long-range collective interactions dominate over binary collisions with neutrals (b) Length scales large enough that quasi-neutrality (

_{n}

_{n}

_{e}*≈ Zn*

*) holds*

_{i}*Introduction:* *Definitions of plasma (2)* *1-19*
** **

### Collective behavior

**Conditions:**(a)

_{τ τ}

_{τ τ}

_{n}*≡*

### 1

*n*

_{n}*σv*

_{th}

*tokamak:*

*6*

_{τ 2.4 × 10}_{s}

*corona:*

*τ 2 × 10*20s ; (b)

_{λ λ}

_{λ λ}

_{D}*≡*

_{0}

*kT*

*e*

2*n*

*tokamak:*

*λ*

_{D}*= 7×10−5*m

*corona:*

_{λ}_{D}*(c)*

_{= 0.07 m ;}_{N}

_{N}

_{D}*≡*

4
3*πλ*

3
*D*

*n 1*

*tokamak:*

*N*

_{D}*= 1.4 × 10*8

*corona:*

_{N}_{D}*9*

_{= 1.4 × 10}*tokamak air T corona sun 1032 108 1 core sun 102 104 106 1010 n 108 1024 1016 λD = 10-8 m τn = 6 x 1017s λD = 10-4 m λD = 1m τn = 1s ND = 108 tokamak ND = 1016 ND = 1*

_{.}*Introduction:* *Definitions of plasma (3)* *1-20*

**So far, only the electric field appeared.** **(LOCAL)**

**
**

### Macroscopic definition:

*For a valid macroscopic model of magnetized plasma dynamical configurations, size,*
*duration, density, and magnetic field strength should be large enough to establish*
*fluid behavior and to average out the microscopic phenomena (i.e. collective plasma*
*oscillations and cyclotron motions of electrons and ions).*

**Now, the magnetic field enters:** **(GLOBAL !)**

(a)

_{τ Ω}

_{τ Ω}

*−1*

_{i}*∼ B*

*−1*(time scale longer than inverse cyclotron frequency);

(b)

_{λ R}

_{λ R}

_{i}*∼ B*

*−1*(length scale larger than cyclotron radius).

**
**

*Elements of plasma physics:* *Overview* *2-1*

**Chapter 2: Elements of plasma physics**

** **

### Three theoretical models:

*•*

**Theory of motion of single charged particles**in given magnetic and electric fields;

[ book: Sec. 2.2 ]

*•*

**Kinetic theory of a collection of such particles,**

*describing plasmas microscopically*by means of particle distribution functions

_{f}

_{f}

_{e,i}_{(r, v, t)}

; [ book: Sec. 2.3 ]

_{(r, v, t)}*•*

**Fluid theory (magnetohydrodynamics),**describing plasmas in terms of averaged

*macroscopic functions of*

**r**

and _{t}

. [ book: Sec. 2.4 ]
_{t}

Within each of these descriptions, we will give an example illustrating the plasma property relevant for our subject, viz. plasma confinement by magnetic fields.

*Elements of plasma physics:* *Single particle motion (1)* *2-2*
**
**

### Equation of motion

of charged particle in given electric and magnetic field,

**E(r, t)**

and **E(r, t)**

**B(r, t)**

:
**B(r, t)**

*m*

**dv**

**dv**

*dt*

**= q(E + v × B) .**

**= q(E + v × B) .**

### (1)

*•*

Apply to constant magnetic field **B = Be**

**B = Be**

*,*

_{z}**E = 0**

:
(a) projection on **B**

gives _{m}

_{m}

*dv*

*dt = 0*

*⇒ v*

### =

const*,*

(b) projection on

**v**

gives *d*

*dt (*

12*mv*

2*) = 0 ⇒ v*

*⊥*

### =

const.*•*

Systematic solution of Eq. (1) with **v = dr/dt = ( ˙x, ˙y, ˙z)**

gives two coupled differential
equations for motion in the perpendicular plane:
**v = dr/dt = ( ˙x, ˙y, ˙z)**

### ¨

*x − (qB/m) ˙y = 0 ,*

### (2)

### ¨

*y + (qB/m) ˙x = 0 .*

*Elements of plasma physics:* *Single particle motion (2)* *2-3*
**
**

### Cyclotron motion

This yields periodic motion in a magnetic
field, with *gyro- (cyclotron) frequency*

*Ω ≡*

*|q|B*

*m*

### (3)

and *cyclotron (gyro-)radius*

*R ≡*

*v*

*⊥*

### Ω

*≈*

*√*

*2mkT*

*|q|B*

*.*

### (4)

**B**R

_{i}+ R

_{e}

*--⇒*

Effectively, charged particles stick to the field lines.
Opposite motion of electrons and ions about guiding centres with quite different gyro-frequencies and radii, since

_{m}

_{m}

_{e}* m*

*:*

_{i}### Ω

*e*

*≡*

*eB*

*m*

_{e}* Ω*

*i*

*≡*

*ZeB*

*m*

_{i}*,*

*R*

*e*

*≈*

*√*

*2m*

*e*

*kT*

*eB*

* R*

*i*

*≈*

*√*

*2m*

*i*

*kT*

*ZeB*

*.*

### (5)

*Elements of plasma physics:* *Single particle motion (3)* *2-4*

**
**

### Cyclotron motion (cont’d)

**Orders of magnitude**

*•*

Typical gyro-frequencies, e.g. for tokamak plasma (_{B = 3 T}

):
_{B = 3 T}

### Ω

*e*

*= 5.3 × 10*

11 ### rad s

*−1*( frequency of

### 84 GHz

)*,*

### Ω

*i*

*= 2.9 × 10*

8 ### rad s

*−1*( frequency of

### 46 MHz

)*.*

*•*

Gyro-radii, with _{v}

_{v}

_{⊥}_{= v}

_{= v}

_{th}

*≡*

_{2kT /m}

for _{2kT /m}

_{T}

_{T}

_{e}_{= T}

_{= T}

_{i}_{= 1.16 × 10}

8 _{= 1.16 × 10}

_{K}

:
*v*

_{th,e}*= 5.9 × 10*

7### m s

*−1*

*⇒ R*

_{e}*= 1.1 × 10*

*−4*

*m ≈ 0.1 mm ,*

*v*

*th,i*

*= 1.4 × 10*

6### m s

*−1*

*⇒ R*

*i*

*= 4.9 × 10*

*−3*

*m ≈ 5 mm .*

*⇒*

Tokamak time scales _{(∼ 1 s)}

and dimensions _{(∼ 1 s)}

_{(∼ 1 m)}

_{(∼ 1 m)}

*justify averaging.*

Since the gyro-frequencies essentially depend on

_{B}

alone
_{B}

*Elements of plasma physics:* *Single particle motion (4)* *2-5*
** **

### Drifts

*•*

Single particle motion in constant **E (= Ee**

**E (= Ee**

_{y}_{) ⊥}

constant _{) ⊥}

**B (= Be**

**B (= Be**

_{z}_{)}

.
*•*

Transverse equations of motion:
### ¨

*x −*

*qB*

*m*

*˙y = 0 ,*

### (6)

### ¨

*y +*

*qB*

*m*

*( ˙x − E/B) = 0 ,*

replacing

_{˙x → ˙x − E/B ⇒}

gyration superposed with constant drift in _{˙x → ˙x − E/B ⇒}

_{x}

-direction.
_{x}

*•*

Hence, *⊥*

electric field gives **E × B**

drift :
**E × B**

**v**

d ### =

**E × B**

**E × B**

*B*

2 *,*

### (7)

independent of the charge, so that elec-trons and ions drift in same direction!

**E**
(y)
**B**
(z)
**v**d
(x)
+

*--Elements of plasma physics:* *Single particle motion (5)* *2-6*
**
**

### Drifts (cont’d)

*•*

Reason: periodic acceleration / deceleration of moving charge in electric field **E**

.
*•*

Lorentz transformation to a frame moving with **v**

_{d}yields:

**E**

**= γ(E + v × B) = 0**

**= γ(E + v × B) = 0**

*⇒*

particles move to ensure *vanishing of the electric field in the moving frame !*

*•*

Replace _{qE}

by any other force

_{qE}**F**

:
**v**

d ### =

**F × B**

**F × B**

*qB*

2
*⇒*

drift velocity now _{q}

-dependent: electron and ions drift in opposite directions
_{q}

*⇒*

electric current flow.
*•*

Other drifts (all due to periodic changes of the gyro-radius)
*⇒*

through gradients of the magnetic field: **B × ∇B**

drift ,
**B × ∇B**

*Elements of plasma physics:* *Single particle motion (6)* *2-7*
** **

### Mirror effect

*•*

Particles entering region
of higher *|B|*

are reflected
back into region of smaller
*|B|*

where gyro-radius is
larger and _{v}

_{v}

*smaller*

_{⊥}*⇒*

(a) mirror, (b) cusp.

coil
**B**
I
a b
coil
I
coil
I
coil
I

*•*

Both confinement schemes have been dropped in thermonuclear fusion research
(because of interchange instabilities and leakage through the ends), but the mirror
remains important concept to explain *trapping of particles*(e.g. van Allen belts).

*•*

Also, important for the systematic theory of fast periodic particle motion in the slow
variation of inhomogeneous magnetic fields *⇒*

*adiabatic invariants.*

For example, the reflection of charged particle spiraling into higher field regions of the mirror is described by an adiabatic invariant

*∼ v*

_{⊥}_{R}

, with _{R}

_{R ∼ v}

_{R ∼ v}

_{⊥}_{/B}

.
_{/B}

*Elements of plasma physics:* *Single particle motion (9)* *2-8*
**
**

### Application to magnetosphere

Example: Charged particles in
the magnetosphere.
J1: gyration
N
S
z
**B**
r
φ
J3: drift
W
E
J2: bouncing
i
e

(a) Electrons and ions rapidly gyrate about the magnetic field, conserving

_{J}

_{J}

_{1};

(b) The guiding centres bounce back and forth between the mirrors on the northern and southern hemisphere on a slower time scale, conserving

_{J}

_{J}

_{2};

(c) They drift in opposite longitudinal directions on a slower time scale yet, conserving

*J*

3 (magnetic flux inside the drift shell): This invariance is easily invalidated by the
*Elements of plasma physics:* *Kinetic theory (1)* *2-9*
** **

### Distribution functions

*•*

A plasma consists of a very large number of interacting charged particles *⇒*

*kinetic*

*plasma theory*derives the equations describing the

*collective behavior*of the many charged particles by applying the methods of statistical mechanics.

*•*

The physical information of a plasma consisting of electrons and ions is expressed in
terms of *distribution functions*

_{f}

_{f}

_{α}_{(r, v, t)}

, where

_{(r, v, t)}_{α}

= _{α}

_{e}

, _{i}

. They represent the density
of particles of type _{α}

in the _{α}

*phase space*of position and velocity coordinates. The probable number of particles

_{α}

in the 6D volume element centered at _{α}

_{(r, v)}

is given
by

_{(r, v)}_{f}

_{f}

_{α}_{(r, v, t) d}

3

_{(r, v, t) d}_{r d}

3_{r d}

_{v}

. The motion of the swarm of phase space points is described by
the total time derivative of _{v}

_{f}

_{f}

*:*

_{α}*df*

*α*

*dt*

*≡*

*∂f*

*α*

*∂t*

### +

*∂f*

*α*

**∂r**

**∂r**

*·*

**dr**

*dt*

### +

*∂f*

*α*

**∂v**

**∂v**

*·*

**dv**

*dt*

### =

*∂f*

*α*

*∂t*

**+ v ·**

**+ v ·**

*∂f*

_{α}**∂r**

**∂r**

### +

*q*

_{α}*m*

*α*

**(E + v × B) ·**

**(E + v × B) ·**

*∂f*

_{α}**∂v**

**∂v**

*.*

### (8)

*Elements of plasma physics:* *Kinetic theory (2)* *2-10*
**
**

### Boltzmann equation

*•*

Interactions (collisions) between the particles determine this time derivative:
*∂f*

*α*

*∂t*

**+ v ·**

**+ v ·**

*∂f*

*α*

**∂r**

**∂r**

### +

*q*

*α*

*m*

_{α}**(E + v × B) ·**

**(E + v × B) ·**

*∂f*

*α*

**∂v**

**∂v**

*= C*

*α*

*≡*

*∂f*

*α*

*∂t*

coll
*.*

### (9)

*•*

Here, **E(r, t)**

and **E(r, t)**

**B(r, t)**

**B(r, t)**

*are the sum of the external fields and the averaged*inter-nal fields due to the long-range inter-particle interactions.

_{C}

_{C}

*represents*

_{α}*the rate of*

*change of the distribution function due to the short-range inter-particle collisions*. In a plasma, these are the cumulative effect of many small-angle velocity changes ef-fectively resulting in large-angle scattering. The first task of kinetic theory is to justify this distinction between long-range interactions and binary collisions, and to derive expressions for the collision term.

*•*

One such expression is the *Landau collision integral*(1936). Neglect of the collisions (surprisingly often justified!) leads to the

*Vlasov equation*(1938).

*Elements of plasma physics:* *Kinetic theory (3)* *2-11*

**
**

### Completing the system

*•*

*Combine the Boltzmann equation*, determining

_{f}

_{f}

_{α}_{(r, v, t)}

,

_{(r, v, t)}*with Maxwell’s equations*, determining

**E(r, t)**

and **E(r, t)**

**B(r, t)**

. In the latter, charge density **B(r, t)**

_{τ (r, t)}

and current density

_{τ (r, t)}**j(r, t)**

appear as source terms. They are related to the particle densities **j(r, t)**

_{n}

_{n}

_{α}_{(r, t)}

and
the average velocities

_{(r, t)}**u**

_{α}_{(r, t)}

:

_{(r, t)}**τ (r, t) ≡**

**τ (r, t) ≡**

*q*

*α*

*n*

*α*

*,*

*n*

*α*

**(r, t) ≡**

**(r, t) ≡**

*f*

*α*

**(r, v, t) d**

3**(r, v, t) d**

*v ,*

### (10)

**j(r, t) ≡**

**j(r, t) ≡**

*q*

*α*

*n*

*α*

**u**

*α*

**, u**

**, u**

*α*

**(r, t) ≡**

**(r, t) ≡**

### 1

*n*

*α*

**(r, t)**

**(r, t)**

**vf**

**vf**

*α*

**(r, v, t) d**

3**(r, v, t) d**

*v.*

### (11)

This completes the microscopic equations.

*•*

Solving such kinetic equations in seven dimensions (with the details of the single
particle motions entering the collision integrals!) is a formidable problem
*Elements of plasma physics:* *Kinetic theory (4)* *2-12*
** **

### Moment reduction

*•*

Systematic procedure to obtain macroscopic equations, no longer involving velocity
space details, is to expand in *finite number of moments of the Boltzmann equation*, by multiplying with powers of

**v**

and integrating over velocity space:
*d*

3*v · · · ,*

*d*

3**v v · · · ,**

**v v · · · ,**

*d*

3*v v*

2 *· · · |*

truncate *.*

### (12)

*•*

E.g., the *zeroth moment*of the Boltzmann equation contains the terms:

*∂f*

_{α}*∂t*

*d*

3_{v =}

_{v =}

*∂n*

*α*

*∂t*

*,*

**v ·**

**v ·**

*∂f*

*α*

**∂r**

**∂r**

*d*

3_{v = ∇ · (n}

_{v = ∇ · (n}

*α*

**u**

*α*

*) ,*

*q*

_{α}*m*

*α*

**(E + v × B) ·**

**(E + v × B) ·**

*∂f*

*α*

**∂v**

**∂v**

*d*

3_{v = 0 ,}

_{v = 0 ,}

*C*

_{α}*d*

3*v = 0 .*

Adding them yields the *continuity equation* for particles of species

_{α}

:
_{α}

*∂n*

*α*

*Elements of plasma physics:* *Kinetic theory (5)* *2-13*

**
**

### Moment reduction (cont’d)

*•*

The *first moment*of the Boltzmann equation yields the

*momentum equation:*

*∂*

*∂t*

*(n*

*α*

*m*

*α*

**u**

*α*

*) + ∇ · (n*

*α*

*m*

*α*

**vv**

**vv**

*α*

*) − q*

*α*

*n*

*α*

**(E + u**

*α*

**× B) =**

**× B) =**

*C*

*αβ*

*m*

*α*

**v d**

3**v d**

*v . (14)*

*•*

The *scalar second moment*of Boltzmann Eq. yields the

*energy equation:*

*∂*

*∂t*

*(n*

*α*1 2

*m*

*α*

*v*

2_{}

_{}

*α*

*) + ∇ · (n*

*α*1

_{2}

*m*

*α*

*v*

2**v**

**v**

*α*

*) − q*

*α*

*n*

*α*

**E · u**

**E · u**

*α*

### =

*C*

*1*

_{αβ}_{2}

*m*

_{α}*v*

2*d*

3*v . (15)*

*•*

This chain of moment equations can be continued indefinitely. Each moment
intro-duces a new unknown whose temporal evolution is described by the next moment of
the Boltzmann equation. The infinite chain must be truncated to be useful. *In fluid*
*theories truncation is just after the above five moments:* continuity (scalar), momentum
(vector), and energy equation (scalar).

*Elements of plasma physics:* *Fluid description (1)* *2-14*

**
**

### From kinetic theory to fluid description

*•*

**(a) Collisionality:**Lowest moments of Boltzmann equation with transport closure

*gives system of two-fluid equations in terms of the ten variables*

_{n}

_{n}

*,*

_{e,i}**u**

*,*

_{e,i}_{T}

_{T}

*. To establish the two fluids, the electrons and ions must undergo*

_{e,i}*frequent collisions*:

*τ*

H * τ*

*i*

*[ τ*

*e*

*] .*

### (16)

*•*

**(b) Macroscopic scales:**Since the two-fluid equations still involve small length and

time scales (

_{λ}

_{λ}

_{D},

_{R}

_{R}

*,*

_{e,i}_{ω}

_{ω}

*,*

_{pe}−1_{Ω}

*−1*consider

_{e,i}), the essential step towards the MHD description is to*large length and time scales*:

*λ*

MHD *∼ a R*

*i*

*,*

*τ*

MHD *∼ a/v*

A * Ω*

*−1*

_{i}*.*

### (17)

The larger the magnetic field strength, the more easy these conditions are satisfied.
*On these scales, the plasma is considered as a single conducting fluid .*

*•*

**(c) Ideal fluids:**Third step is to consider plasma dynamics on time scales

*faster than*

*the slow dissipation* causing the resistive decay of the magnetic field:

*τ*

MHD * τ*

R *∼ a*

2*/η .*

### (18)

This condition is well satisfied for the small size of fusion machines, and very easily for the sizes of astrophysical plasmas

*⇒*

*model of ideal MHD.*

*Elements of plasma physics:* *Fluid description (2)* *2-15*

### In summary:

**Kinetic theory**

*⇓*

*frequent collisions*

*⇓*

**Two-fluid theory**

*⇓*

*large scales*

*⇓*

**Diss. MHD** *⇒ slow dissipation* *⇒* **Ideal MHD**

*Elements of plasma physics:* *Fluid description (3)* *2-16*
**
**

### Resistive MHD equations

*•*

Define one-fluid variables that are linear combinations of the two-fluid variables:
*ρ ≡ n*

*e*

*m*

*e*

*+ n*

*i*

*m*

*i*

*,*

*(total mass density)*

### (19)

*τ ≡ −e (n*

_{e}*− Zn*

_{i}*) ,*

*(charge density)*

### (20)

**v ≡ (n**

**v ≡ (n**

*e*

*m*

*e*

**u**

*e*

*+ n*

*i*

*m*

*i*

**u**

*i*

*)/ρ ,*

*(center of mass velocity)*

### (21)

**j ≡ −e (n**

**j ≡ −e (n**

*e*

**u**

*e*

*− Zn*

*i*

**u**

*i*

*) ,*

*(current density)*

### (22)

*p ≡ p*

*e*

*+ p*

*i*

*.*

*(pressure)*

### (23)

*•*

Operate on pairs of the two-fluid equations
*•*

Evolution expressions for _{τ}

and _{τ}

**j**

disappear by exploiting:
*|n*

*e*

*− Zn*

*i*

*| n*

*e*

*,*

*(quasi charge-neutrality)*

### (24)

**|u**

**|u**

_{i}**− u**

**− u**

_{e}*| v ,*

*(small relative velocity of ions & electrons)*

### (25)

*Elements of plasma physics:* *Fluid description (4)* *2-17*

**
**

### Resistive MHD equations (cont’d)

Combining one-fluid moment equations thus obtained with pre-Maxwell equations
(dropping displacement current and Poisson’s equation) results in *resistive MHD*

*equations:*

*∂ρ*

*∂t*

**+ ∇ · (ρv) = 0 ,**

**+ ∇ · (ρv) = 0 ,**

*(continuity)*

### (27)

*ρ (*

**∂v**

**∂v**

*∂t*

**+ v · ∇v) + ∇p − j × B = 0 ,**

**+ v · ∇v) + ∇p − j × B = 0 ,**

*(momentum)*

### (28)

*∂p*

*∂t*

**+ v · ∇p + γp∇ · v = (γ − 1)η|j|**

2 **+ v · ∇p + γp∇ · v = (γ − 1)η|j|**

_{,}

_{,}

_{(internal energy)}_{(29)}

**∂B**

**∂B**

*∂t*

**+ ∇ × E = 0 ,**

**+ ∇ × E = 0 ,**

*(Faraday)*

### (30)

where**j = μ**

**j = μ**

*−1*

_{0}

**∇ × B ,**

**∇ × B ,**

*(Amp `ere)*

### (31)

**E**

**≡ E + v × B = η j ,**

**≡ E + v × B = η j ,**

*(Ohm)*

### (32)

and**∇ · B = 0**

**∇ · B = 0**

*(no magnetic monopoles)*

### (33)

*Elements of plasma physics:* *Fluid description (5)* *2-18*
**
**

### Ideal MHD equations

*•*

Substitution of **j**

and **E**

*in Faraday’s law yields the induction equation:*

**∂B**

**∂B**

*∂t*

**= ∇ × (v × B) − μ**

**= ∇ × (v × B) − μ**

*−1*

0

**∇ × (η∇ × B) ,**

**∇ × (η∇ × B) ,**

### (34)

*where the resistive diffusion term is negligible when the magnetic Reynolds number*

*R*

*m*

*≡*

*μ*

0*l*

0*v*

0
*η*

* 1 .*

### (35)

*•*

Neglect of resistivity and substitution of **j**

and **E**

leads to *the ideal MHD equations:*

*∂ρ*

*∂t*

**+ ∇ · (ρv) = 0 ,**

**+ ∇ · (ρv) = 0 ,**

### (36)

*ρ (*

**∂v**

**∂v**

*∂t*

**+ v · ∇v) + ∇p − μ**

**+ v · ∇v) + ∇p − μ**

*−1*0

**(∇ × B) × B = 0 ,**

**(∇ × B) × B = 0 ,**

### (37)

*∂p*

*∂t*

**+ v · ∇p + γp∇ · v = 0 ,**

**+ v · ∇p + γp∇ · v = 0 ,**

### (38)

**∂B**

**∂B**

*∂t*

**− ∇ × (v × B) = 0 ,**

**− ∇ × (v × B) = 0 ,**

**∇ · B = 0 ,**

**∇ · B = 0 ,**

### (39)

*Elements of plasma physics:* *Conclusion* *2-19*
** **

### Plasma coherence

We introduced the three main theoretical approaches of plasmas (theory of single
particle motion, kinetic theory of collections of many particles, and theory of
mag-netohydrodynamics pertaining to global macroscopic plasma dynamics in complex
*magnetic fields). Three effects were encountered giving plasmas the coherence that*
is necessary for thermonuclear confinement of laboratory plasmas and which is also
characteristic for magnetized plasmas encountered in nature:

*•*

In the single particle picture, we found that *particles of either charge stick to the*

*magnetic field lines by their gyro-motion* which restrains the perpendicular motion.

*•*

In the kinetic description, we found that, because of the large electric fields that
occur when electrons and ions are separated, deviations from neutrality can occur
only in very small regions (of the size of a Debye length). Over larger regions, *ions*
*and electrons stay together to maintain approximate charge neutrality*.

*•*

In the fluid picture, it was found that*currents in the plasma create their own confining*

*magnetic field* and that *Alfv ´en waves act to restore magnetic field distortions*. We
also encountered the first destructive effect, viz. *the external kink instability*.