introduction to plasma physics

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Introduction to Plasma Physics

Univ.-Prof. Mag. Dr. Helmut O. Rucker

Institute of Physics

Karl-Franzens-University Graz, Austria


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The lecture “Introduction to Plasma Physics” is intended as a first approach to thephysics of particles and fields in space with fundamental phenomena and correspondinglaws. The content is a selection out of the books of Chen [1], Kippenhahn and Möllenhoff [2], and Lyons and Williams [3] with specific typical emphasis to space physics as a base

for further lectures like “Planetary magnetospheres” and “Radio and plasma waves”.

It will be highly appreciated to send any comments, corrections, and suggestions for im-provements to [email protected].

Helmut O. Rucker

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http://localhost/var/www/apps/conversion/tmp/scratch_1/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/[email protected]


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Contents

1 Basics of Plasmaphysics 6

1.1 The Three Plasma-Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Different types of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Influence of a Magnetic Field on the State of Plasma 17

2.1 Gyration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Special Magnetospheric Processes . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Motion of Charged Particles in Magnetic and Electric Fields . . . . 192.2.2 Derivation of the dipole moment . . . . . . . . . . . . . . . . . . . 222.2.3 Derivation of the magnetic moment . . . . . . . . . . . . . . . . . . 242.2.4 Oscillation (“bounce motion”) . . . . . . . . . . . . . . . . . . . . . 252.2.5 E × B-Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Plasmas as Fluids 46

3.1 Plasmaphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 The Convective Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Consideration of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Consideration of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Equation of continuity . . . . . . . . . . . . . . . . . . . . . . . . . 503.5.2 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5.3 The full set of MHD equations . . . . . . . . . . . . . . . . . . . . 523.5.4 MHD-Drift perpendicular to B: . . . . . . . . . . . . . . . . . . . . 55

4 Plasma Oscillations and MHD Waves 59

4.1 Plasma oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Magnetohydrodynamic Waves . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.1 Linear perturbation theory: . . . . . . . . . . . . . . . . . . . . . . 634.3 MHD waves with arbitrary angle to the magnetic field . . . . . . . . . . . 71

4.3.1 Development of a hodograph: . . . . . . . . . . . . . . . . . . . . . 82

4.4 CMA-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4.1 Structure of the CMA-diagram . . . . . . . . . . . . . . . . . . . . 854.4.2 Wave-modes and transitions . . . . . . . . . . . . . . . . . . . . . . 86

5 Appendix 90

5.1 Physical Quantities in Plasma Physics . . . . . . . . . . . . . . . . . . . . 90Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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1 Basics of Plasmaphysics

1.1 The Three Plasma-Criteria

Matter can appear in four different states. The particles setting up matter have themore possibilities to move the higher the temperature is. In solid-state matter atomsand molecules have the least freedom to move. In liquid state the freedom to moveis a little more extensive. In gas, atoms and molecules can move independently butthe electrons are – obeying the laws of nuclear physics – bound to their atoms. In theplasmatic state the electrons are completely separate from the atoms and therefore have

entire freedom of movement. If atoms or molecules have lost one or more electrons theycarry positive charge outwardly, in this case they become positive ions. “Plasma” istherefore considered as gas showing collective behavior and consisting of particles whichcarry positive and negative charges – in the extent that the overall charge comes to zero.

Definition: Plasma is electrically neutral to the outside, if the number of positive and negative charges equals in a sufficiently large volume and for a sufficiently long interval of time. This balance is referred to as “quasi-neutrality”.

ne = Zni (1.1)

or Zni − ne

Zni 1 (1.2)

ne . . . electron-density (number of electrons per volume unit)ni . . . ion-density (number of electrons per volume unit)Z . . . Z -times charge

Space plasmas like the solar wind are quasi-neutral over given areas. In a given volume(in a defined span of time) there are as many positive as negative particles. The physicalexplanation is the good conductivity of plasma.

Positive charges lead to electric fields that attract negative charges, the latter are muchmore mobile and thus cause compensation of charges. Electric charge-effects have tobe dominant (effects must be at least of the same intensity) over effects of thermalmotion so that typical plasma properties occur or can be held up. Only in this case thefar-reaching electric forces can affect neighbouring particles. This results in collectivereactions which are determined by electric forces, this is how plasma and ordinary gascan be distinguished. One of the Maxwellian equations is used to determine the first

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plasma-criterion:

∇ · E = div E = ρε0

(1.3)

ε0 . . . dielectric constant (= 8.85 × 10−12A s/V m)ρ . . . charge-density (mass density and electric charge)E . . . electric field

The electrical potential Φ is directly connected to the electric field:

E = −∇ Φ (1.4)

Equivalent notation:

∇ Φ = grad Φ = ∂Φ∂x

i + ∂ Φ

∂y j +

∂ Φ

∂z k (1.5)

Relation between charge density and potential:

div E = −div grad Φ = −( ∂ 2

Φ∂x2

+ ∂ 2

Φ∂y2

+ ∂ 2

Φ∂z2

) (1.6)

Forming the divergence of the gradient is expressed by the symbol ∇2 (Nabla squared)or by the Laplacian operator ∆:

∇2 = ∆ = ∂ 2

∂x2 +

∂ 2

∂y2 +

∂ 2

∂z2 (1.7)

The Laplacian operator is the scalar product of the Nabla operator with itself.

∇ · ∇=

∂

∂x

i + ∂

∂y

j + ∂

∂z

k · ∂

∂x

i + ∂

∂y

j + ∂

∂z

k = ∇2 = ∆, (1.8)

where i · j = δ ij. The relation div E = ρ/ε0 thus leads to Poisson’s equation:

∆Φ = − ρε0

(1.9)

In spherical coordinates this relation can be written as:

∇2Φ = 1r2

∂

∂r

r2

∂Φ

∂r

+

1

r2 sin ϑ

∂

∂ϑ

sin ϑ

∂Φ

∂ϑ

+

1

r2 sin2 ϑ

∂ 2Φ

∂ϕ2

= − ρ

ε0(1.10)

The charge density ρ symbolizes the overall charge density of electrons and ions:

ρe = ne · (−e) and ρi = ni · (Ze) (1.11)

ne . . . electron density (number of electrons per volume)ni . . . ion density (number of ions per volume)Z . . . charge (Z -times)(to simplify matters Z = 1)e . . . elementary charge

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According to classical statistics the number of particles of a given overall number n0 stay-ing at the position x, y, z in a potential field Φ(x,y ,z) is given by Boltzmann’s equation:

n(x,y ,z) = n0 exp−eΦ(x,y ,z)kT (1.12)eΦ . . . potential electric energykT . . . thermal energy

k . . . Boltzmann’s constant (k = 1.38 × 10−23W s/K)Boltzmann’s equation for electrons is given by:

ne(x,y ,z) = ne,0 exp

+

eΦ(x,y ,z)

kT e

(1.13)

Physically this relation expresses that electrons are very mobile due to their low massand that they can be accelerated to high energies quickly by an appropriate force. Elec-trostatic forces emerge because electrons cannot leave a plasma region as a whole withoutgenerating a significant positive (ion-) space-charge, which balances electrons and ions.Because of the condition of quasi-neutrality the following relation is applied to the quan-tities with indices zero:

ne,0 = Zni,0 = n0 (1.14)

ne,0 = ni,0 = n0 for Z = 1

Applying ne, ni and n0 to Poisson’s equation (1.9) we get

∆Φ = − 1ε0

[−en0 exp( eΦkT e

) + en0] = en0

ε0[exp(

eΦ

kT e) − 1] (1.15)

The term [−en0 exp( eΦkT e )] is applied to electrons, the second term (en0) is applied topositive charges. In areas where ( eΦkT e ) 1, where thermal energy surpasses the potentialelectric energy substantially the exponential expression can be developed into a Taylor

series:

exp( eΦ

kT e) = 1 + (

eΦ

kT e) +

1

2(

eΦ

kT e)2 + . . .

The series is aborted after the second term. Applying these terms to eqn. 1.15, nowusing spherical coordinates, see eqn. 1.10, and considering the radial component solely,

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we get

∆Φ = en0

ε0

1 +

eΦ

kT e− 1

1

r2∂

∂r

r2

∂Φ

∂r

=

en0ε0

1 +

eΦ

kT e− 1

1

r2

∂

∂rr2

∂Φ

∂r + r2

∂ 2

∂r2Φ

=

e2n0kT eε0

Φ

1

r2

2r

∂Φ

∂r + r2

∂ 2Φ

∂r2

=

e2n0kT eε0

Φ

∂ 2Φ

∂r2 +

2

r

∂Φ

∂r − e

2n0kT eε0

Φ = 0

D2 + 2r

D − n0 e2ε0kT

Φ = 0 (1.16)using the differential operator D := ∂ ∂r . This leads to the characteristic equation

λ2 + 2

r λ − n0e

2

ε0kT = 0, (1.17)

which has the two following solutions

λ1,2 = −1

r ± 1r2 + n0e2ε0kT , (1.18)whereupon only the negative part provides a solution that is physically meaningful, aswill be shown below. The potential Φ is given by:

Φ(r) = C 1 eλ1r + C 2 e

λ2r (1.19)

= C 1 e(− 1

r−

1

r2+

n0e2

ε0kT )r

+ C 2 e(− 1

r+

1

r2+

n0e2

ε0kT )r

= C 1 e−1

C 1 e−

1+

n0e2r2

ε0kT + C 2 e−1

C 2 e

1+n0e

2r2

ε0kT .

This differential equation has to satisfy the following initial conditions:

• Φ(∞) = 0

• Φ(0) = Φ0

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Calculating the term n0e2r2

ε0kT at terrestrial orbit one obtains

107 m−3 · (1.602 × 10−19)2 A2 s2 · (1.5 × 1011)2 m28.85 × 10−12 A s /V m · 1.38 × 10−23 A V s/K · 4 × 104 K ∼ 10

21 1.

As a consequence the following approximation is evident: 1 +

n0e2r2

ε0kT

n0e2r2

ε0kT = e

n0ε0kT

r (1.20)

Φ(∞) = 0 ⇒ C 2 = 0Φ(0) = Φ0 ⇒ C 1 = Φ0

⇒ Φ(r) = Φ0 e−e

n0ε0 k T

r

= Φ0 e− rλD

λD =

ε0kT en0e2

(1.21)

Within the Debye shielding length λD the potential Φ(r) of an individual charge decreasesby a factor of 1/e due to space charge effects of the neighboring charges. λD decreases if plasma density increases and λD increases with increasing temperature T e. Electrontemperature is very important in this context because shielding mainly occurs on accountof electrons and their high mobility.

The Debye shielding length, eqn. 1.21 is of great importance: in areas smaller thanλD electric fields are too weak to take influence on the motion of particles. Thus quasi-neutrality is only given beyond a given volume. The lower limit in space for the occurrenceof quasi-neutrality is given by the Debye sphere, with Debye length λD as radius. Quasi-neutrality is only given in areas greater than this sphere, thermal motion of particles isdominant within it. Quasi-neutrality therefore is a good assumption for studying plasma

Figure 1.1: Description of the Debye length

phenomena with a scale length L, being substantially longer than the Debye length.

λD L 1st plasma criterion (1.22)

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Example Is quasi-neutrality of the solar wind given in some area of the solar system(orbit of Neptune: r ≈ 30 AU)?

To answer this question one has to investigate if the first plasma criterion can beapplied everywhere in interplanetary space. The variables in the relation for λD (eqn.

1.22) are temperature and density. Assuming approximately constant temperature of solar wind plasma the density of solar wind remains the only variable quantity. Over adistance of r ≈ 30 AU it decreases to 1/r2 of the density at earth orbit.How large is the Debye length λD:

(a) at orbit of Earth (r = 1 AU)?

(b) at orbit of Neptune (r = 30 AU)?

Putting in the parameters in eqn. (1.21):ε0 . . . dielectric constant 8.85 × 10−12 A s/V mne . . . electron density 10

7 m−3 (at 1 AU)np . . . proton density 10

7 m−3 (at 1 AU)k . . . Boltzmann’s constant 1.38 × 10−23 W s/Ke . . . elementary charge 1.602 × 10−19 CT e . . . electron temperature 1.5 × 105 KT p . . . proton temperature 4 × 104 K

ad (a) At earth orbit, calculation for protons:

λD =

8.85 × 10−12 A s/V m · 1.38 × 10−23 A V s/K · 4 × 104 K

107 m−3 · (1.602 × 10−19)2 A2 s2 12

= [8.8 · 1.38 · 42.56

· 10−35+4−7+38 m2] 12 = [18.975 × 100 m2] 12

λD,proton = 4.4 m λD,electron = 8.4 m

With a scale length of Ldensity ∼ 108 m λD = 4.4 m for protons and 8.4 m forelectrons, respectively. Thus the first plasma criterion is met.

ad (b) At Neptune-orbit n decreases by 1/r2 within the solar wind, that is

n(r = 30 AU) ≈ n(r = 1 AU)/302

If n decreases by about a factor of 900, λD increases by√

900 = 30. Ldensity ∼108 m λD ≈ 131 m for protons and 252 m for electrons, respectively. Here thefirst plasma criterion is also met.

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Figure 1.2: Scales of plasma, [source: M. Heyn: Vorlesungsunterlagen zur Plasmaphysik]

Although this is only a coarse estimation it can be said that in the entire known spaceof the solar wind plasma (with the exception of the direct vicinity of the Sun) it is inquasi neutral condition. Using magnetohydrodynamics therefore is fully guaranteed forthe description of large and small scaled (∼ 103 m) structures.The Debye sphere is defined by the volume V Debye:

V Debye = 4π

3 λ3D (1.23)

In a Debye sphere there are V Debyene electrons – N Debye charged particles in general. Fora magnetohydrodynamic approach the following relation has to be applied:

N Debye = ne4π

3 λ3D 1 (1.24)

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N Debye is the number of particles in the Debye sphere.

Λ = neλ3D 1 2nd plasma criterion (1.25)

If Λ, which is called plasma parameter, is much greater than 1, the term “plasma” isused. In other words: collective behavior of particles in a plasma requests N Debye 1.A charged particle can leave its neutral position up to a defined length, the Debye lengthλD. Then the particle oscillates around its former neutral position, attenuation can beneglected. The mathematical approach in neglect of attenuation is written as:

m∂ 2x

∂t2 + eE = 0 (1.26)

m∂ 2x

∂t2 deviating force

eE counterforce due to electric field

To simplify matters a one-dimensional approach is taken: only the x-direction is takeninto account. From the Maxwellian equation (eqn. 1.3), mentioned before, we get:

E = ne

ε0x (1.27)

applying (1.27) to (1.26):

m∂ 2x

∂t2 +

ne2

ε0x = 0

With the approach x = x0eiωt, the solution of this differential equation can be determined

immediately:

ẋ = iωx0eiωt

ẍ = −ω2x0eiωt

We obtain:

− mω2 = −ne2

ε0

ω =

ne2

mε0(1.28)

This leads to the plasma frequency or Langmuir frequency:

f p = ωp2π

= 1

2π

ne2

mε0(1.29)

The equation of motion leads to the fact that the deviated particle carries out a harmonicoscillation with the frequency ωp. This frequency is called Langmuir or plasma frequency .

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It is a characteristic frequency for a system of charged particles and it depends on theparticles’ mass and density. One can clearly see that plasma consisting of two componentshas a characteristic frequency for electrons as well as for ions. Inserting correspondingvalues in eqn. 1.29 results in plasma frequency for, e.g. heavy particles (ions), and for

lighter particles (electrons), leading to a lower plasma frequency for ions with respectto electrons. The Langmuir frequency f p is the characteristic oscillation frequency forelectrostatic disturbances in a plasma.

Example Estimating the plasma frequency in the solar wind

ε0 . . . dielectric constant (= 8.85 × 10−12 A s/V m)ne . . . electron density 10

7 m−3 (at 1 AU)np .. . proton density 10

7 m−3 (at 1 AU)me . . . electron mass 9.1 × 10−31 kgmp . . . proton mass 1.67 × 10−27 kge . . . elementary charge 1.602

×10−19 C

• Langmuir frequency for electrons: f p,e ≈ 28.4kHz• Langmuir frequency for ions: f p,p ≈ 663Hz• The ratio f p,e : f p,p =

mp/me ≈ 43.

In order to define a “plasma”, a further condition, the 3rd plasma criterion must be met:

ωτ > 1 3rd plasma criterion (1.30)

τ is the average time between collisions of charged and neutral particles, ω is the circularfrequency of a typical plasma oscillation (see e.g. eqn 1.29). If this condition is not given,in other words: if there are too many collisions with neutral particles, particle motioncan rather be described by hydrodynamic equations than by electromagnetic equations.Plasma physics in the classical sense can not be applied.

Summary of the plasma criteria:

1. λD L2. Λ = neλ

3D 1

3. ω · τ > 1

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Figure 1.3: Speed of the solar wind depending on direction (Geophys. Res. Lett., 25 , 1-4,1998)

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2 Influence of a Magnetic Field on theState of Plasma

Space plasma is in general a magnetoplasma. Electric charges describe circular orbitsaround homogeneous (invariant in space) and static (invariant in time) magnetic fields.

2.1 Gyration

mdv

dt = e(E + v × B) (2.1)

Assumption: E = 0

mdv

dt = e(v × B) (2.2)

Inner product with v

mdv

dt ·v = ev

·(v

×B)

v · (v × B) = 0

mdv

dt · v = d

dt(

mv2

2 ) = 0

Kinetic energy of the particles and thus |v| remain constant. The magnetic field is givenby B = (0, 0, B)

v × B =

i j k

vx vy vz0 0 B

= i(vyB) − j(vxB) + k(0)

mv̇x = eBvy (2.3)

mv̇y = −eBvx (2.4)mv̇z = 0 (2.5)

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2 Influence of a Magnetic Field on the State of Plasma

Second time derivative:

mv̈x = eBv̇y

mv̈y = −eBv̇x, inserting (2.3): v̇x = eB

m vy

mv̈y = −eB eBm

vy

v̈y = −

eB

m

2vy

v̈y = −ω2c vyv̈x = −ω2c vx,

the cyclotron frequency being

ωc = eB

m (2.6)

With rL = v⊥ωc

the Larmor radius becomes

rL = mv⊥

eB

rL = mv sin α

eB . (2.7)

rL . . . Larmor radius (also gyration radius rg, or cyclotron radius rc); [r] = m

m . . . particle mass; [m] = kgv⊥ . . . component of particle velocity perpendicular to B; [v] = m/sB . . . magnetic field; [B] = Te . . . elementary charge 1.602 × 10−19 CAccording to (2.7) electrons with equal v⊥ have a – by the factor

mpme

= 1836.1 – shorter

Figure 2.1: Components of the velocity vector regarding B

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2.2 Special Magnetospheric Processes

cyclotron radius than protons.

rL,e = mev⊥

eB (2.8)

rL,p = mpv⊥

eB (2.9)

⇒ rL,e = rL,p1836.1

Therefore the cyclotron radius can be written as:

rc = v⊥ωc

(2.10)


The magnetosphere provides a variety of different plasma populations. Particles movealong complicated paths which leads to specific current systems.

2.2.1 Motion of Charged Particles in Magnetic and Electric Fields

The motion of a charged particle can basically be regarded as a helical path around afield line. The helical path can be separated into

• gyration around the field line and• translation along the field line.

Figure 2.2: Motion of the electron around a magnetic field line

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If there is no or only an insignificant temporal variation within the magnetic field theparticle motion can be regarded as circular motion around a so-called guiding center ,which moves along the field line.

r = R + rc (2.11)

r . . . position vector of the path of the charged particleR . . . position vector of the guiding centerrc . . . radius vector of the cyclotron path

The cyclotron radius of a charged particle in a magnetic field B can be obtained by

Figure 2.3: Gyration of an electron around a magnetic field line where v and B are not parallel

setting equal Lorentz-force and the centrifugal force:

F = e[v × B] = m v2⊥

r (2.12)

ev⊥B = mv2⊥r

rc = mv⊥

eB

rc = mv sin α

eB

v⊥ . . . component of velocity perpendicular to magnetic fieldα . . .∠(v, B), i.e. angle between velocity vector of the particle and magnetic field;(sin α = v⊥/v)

In order to take into account relativistic effects it is necessary to introduce the Lorentz- factor (c . . . velocity of light):

γ R = 1

1 − v2c2

(2.13)

rc = mγ Rv⊥

eB (2.14)

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In the following, we do not assume relativistic conditions, thus we will not use this factorany more, γ R ≡ 1.

The gyration period τ c is given by the time a particle with velocity v needs to pass the

circular orbit 2rcπ.τ c = 2π

rcv⊥

(2.15)

rcv⊥

= m

eB

Thus τ c is given by:

τ c = 2π m

eB (2.16)

the cyclotron frequency is therefore:

f c = 1

τ c=

1

2π · eB

m (2.17)

A field can be considered as not or slightly variable in time or space if the followingconditions are met:

|∇ · B||B|

1

rcsmall spatial changes in B (2.18)dBdt

Bτ c small temporal changes in B (2.19)If these requirements are met the field can be seen as static , the particle is not suppliedwith energy. In this case the magnetic flux Φ through the circular area described by thecyclotron radius is constant:

Figure 2.4: The magnetic flux, ΨMF

dΦdt

= 0 (2.20)

Φ can be written as:

ΦMF = πr2c B (2.21)

ΦMF = m2v2⊥

e2B2 πB =

π

e2 · p

2⊥

B = const (2.22)

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p . . . momentum perpendicular to B.

If ΦMF is multiplied by a constant value of e2/2πm we get:

ΦMFe2

2πm

= π

e2

·

p2⊥

B ·

e2

2πm

= p2⊥

2mB

= const (2.23)

µ = p2⊥2mB

= const (2.24)

µ is the so-called first adiabatic invariant .

2.2.2 Derivation of the dipole moment

In the following sections we will need a relation between the magnetic moment and themagnetizing force of a dipole; in particular we will also require the vector componentsof the magnetic induction in spherical coordinates. For this reason, these physical quan-tities shall be derived right here. A fictitious magnetic monopole + p at P (x, 0, z) has a

Figure 2.5: Magnetic dipole with vector r to test point P, under the angel ϑ to the z -axis

potential of

Φ+ = 1

4π

p

r+. (2.25)

Therefore the entire potential of a dipole becomes

Φ+ + Φ− = 1

4π

+ p

r++

− pr−

. (2.26)

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The poles are placed at + p

0, 0, − l2

and − p 0, 0, + l2 and have a distance of r+ =x2 +

z + l2

21/2and r− =

x2 +

z − l2

21/2respectively from the test point P . Since

y = 0 we get r2 = x2 + z2. The distances may be written as

r+ =

x2 + z2 + z l + l2

4

1/2=

r2 + z l + l2

4

1/2

r− =

x2 + z2 − z l + l

2

4

1/2=

r2 − z l + l

2

4

1/2.

Is l r, quadratic terms of l may be omitted and the entire potential becomes

Φ = 1

4π

+ p

(r2 + z l)1/2 +

− p(r2 − z l)1/2

= 1

4 π p r2 + z l−1/2 − r2 − z l−1/2 . (2.27)

By performing the following development we may simplify these terms:

(r2 + z l)−1/2 =

r2

1 + z l

r2

−1/2=

1

r

1 +

z l

r2

−1/2=

1

r

1 − 1

2

z l

r2 + . . .

1

r 1 − z l2 r2So we get for Φ:

Φ = p

4π

1

r

1 − z l

2 r2

− 1

r

1 +

z l

2 r2

Φ = p

4π

1

r − z l

2 r3 − 1

r − z l

2 r3

= − plz

4πr3 = − 1

4π pl

cos ϑ

r2 , (2.28)

with cos ϑ = z/r. The dipole moment M being p · l, we now may equate the potential of the dipole

Φ = − 14π

M cos ϑ

r2 = − 1

4π

M · rr3

. (2.29)

Out of this we get the vector components of the magnetic field.

B = −µ0 grad Φ = −∇ µ0M · r4πr3

(2.30)

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with (M · r)/r3 = M r cos ϑ/r3. Since the potential Φ does not depend on the azimuthangle ϕ, the azimuth component of B, i.e. Bϕ equals zero.

Br = −

µ0∂Φ

∂r =

−µ0

∂

∂r − 14π M cos ϑr2 =

µ0M

4π cos ϑ

∂

∂r

1

r2

=

µ0M

4π cos ϑ

− 2

r3

= −µ0M

4π

2cos ϑ

r3 (2.31)

Bϑ = −µ0 1r

∂Φ

∂ϑ = −µ0 1

r

∂

∂ϑ

− 1

4π

M cos ϑ

r2

=

µ0M

4π

1

r3∂

∂ϑ cos ϑ =

µ0M

4π

1

r3 (−)sin ϑ

= −µ0M 4π

sin ϑ

r3 (2.32)

Bϕ = 0 (2.33)

The dipole is axially symmetric; every meridian layer offers the same structure of field.For − p and + p are placed at (0, 0, +l/2) and (0, 0, −l/2) respectively, the dipole momentis directed downwards (as is presently the situation at the earth).

Using the magnetic field components we may equate the absolute value of magneticinduction:

B =

B2r + B2ϑ + B

2ϕ =

−µ0M

4π

2cos ϑ

r3

2+

−µ0M

4π

sin ϑ

r3

21/2

= µ0M

4πr3

4cos2 ϑ + sin2 ϑ

B = µ0M

4πr3

1 + 3 cos2 ϑ (2.34)

B ∝ 1r3

(2.35)

2.2.3 Derivation of the magnetic moment

Under the conditions for a static field a gyrating particle can be considered as a circularcurrent.

i = ef c = ev⊥2πrc

, (2.36)

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f c being the gyration frequency already derived in equations (2.15) and (2.17). Thiscircular current flows round the area A = r2c π and generates a magnetic moment µB:

µB = iA = ev⊥

2πrcr2c π =

1

2

ev⊥rc (2.37)

µB = 1

2

ev2⊥m

eB =

mv2⊥2B

(2.38)

µB = E kin,⊥

B (2.39)

This is the 1st adiabatic invariant:

The ratio of kinetic energy of the particle and the magnetic induction is con-stant everywhere along the field line.

2.2.4 Oscillation (“bounce motion”)

Non-relativistic representation

The relation for the magnetic moment of a particle directly leads to the analysis of particlemotion in magnetic fields with converging field lines. In this case B is not constant inspace:

µ = E kin,⊥

B =

E kin sin2 α

B = const (2.40)

E kin,⊥ . . . kinetic energy of the particle

⊥ to B

µ = mv2⊥

2B =

mv2 sin2 α

2B = const (2.41)

For a particle moving in a converging B field, the angle α increases until it becomes π/2(sin π/2 = 1). In this point the guiding center of the particle inverts its direction andmoves back towards the direction it has come from. In a static field:

mv2 sin2 α12B1

= mv2 sin2 α2

2B2= const

sin2 α1B1

= sin2 α2

B2= const (2.42)

Principle of the magnetic mirror: The invariance of the dipole moment µ results in thereflection of gyrating particles by the so- called magnetic mirror. The effect of reflectionis also characterized by the direction of the force (v × B) which is always directed awayfrom the region with higher magnetic induction.

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“Magnetic bottle” in the terrestrial magnetic field

B0 . . . magnetic induction at the equatorα0 . . . “pitch angle” between velocity vector and direction of the magnetic field (see fig.2.7)

sin2 α

B =

sin2 α0B0

sin2 α = sin2 α0B

B0(2.43)

Equation for the magnetic induction of the dipole:

B = µ0M D

4πr3

1 + 3 cos2 ϑ (2.44)

B = µ

0M

D4πr3 1 + 3 sin2 λ

r = r0 cos2 λ (2.45)

B = µ0M D

4πr3

1 + 3 sin2 λ =

µ0M D4π

·

1 + 3 sin2 λ

r30 cos6 λ

(2.46)

Applying sin2 λ = 0 and cos2 λ = 1 for B0 at λ = 0:

B0 = µ0M D

4π · 1

r30(2.47)

B

B0=

µ0M D4π

· 1 + 3 sin2 λr30 cos

6 λ · 4π

µ0M D· r

30

1 =

1 + 3 sin2 λcos6 λ

sin2 α = sin2 α0

1 + 3 sin2 λ

cos6 λ (2.48)

α = 90◦ in the mirror point. With sin2 α = 1 the transformed equation leads to:

1 = sin2 α0B

B0

BM =

B0

sin2 α0 (2.49)

This equation gives the magnetic induction BM in the magnetic mirror point, with agiven B0 and α0, which is determined by

sin2 α0 = cos6 λM

1 + 3 sin2 λM(2.50)

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There is only gyration for a pitch angle of α0 = 90◦. The mirror points of a gyrating

particle on the equator (λ = 0◦) are located there too (λM = 0◦). Therefore B /B0 = 1.

If the pitch angle α = 0◦, the particle moves exactly parallel to the field lines, it can –theoretically – reach a latitude of λ = 90◦, but here B/B0 → ∞.

The relation between pitch angle α0, equation (2.49), and the magnetic latitude λMfollows the curve as shown in fig. 2.8. Intermediate values:

λM = 55◦ sin α0 = 0.1432 α0 = 8.2

◦

BMB0

= 48.7

λM = 30◦ sin α0 = 0.5647 α0 = 34.4

◦

BMB0

= 3.1

Perpendicular and parallel component of the velocity of a particle during oscillatingmotion can be given (in addition to the magnetic induction BM at the mirror point andthe relation between pitch angle and magnetic latitude λM of the mirror point):

v2⊥ = v2 sin2 α = v2 sin2 α0

B

B0

v2 = v2 cos2 α = v2(1 − sin2 α0 B

B0)

If the mirror point BM is located high enough above the atmosphere, a magnetic “reflec-tion” will occur without a problem. If the pitch angle is too small – if the particle canspiral along a field line longer and thus gets into deeper atmospheric layers then collisionand absorption by atmospheric particles is more probable than reflection. h = 100 kmcan be considered as effective height of the dense terrestrial atmosphere. The followingrelation can be determined:

sin2 αMBM

= sin2 α0

B0(2.51)

sin2 α0 = B0B100 km

α0,100 km = arcsin

B0

B100km(2.52)

All particles with pitch angle between

0◦ ≤ α ≤ α0,100 kmenter the atmosphere and are considered lost. The velocity vector of the particles lies inthe “atmospheric loss cone” with the aperture angle

α0,100 km = αloss (2.53)

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Figure 2.8: Relation between the pitch angle α0 and magnetic latitude λM

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Figure 2.9: The atmospheric loss cone with loss cone angle αLoss

If reflection comes to pass at location BM above the atmospheric loss zone, continuousoscillation occurs between the mirror points in the northern and southern hemisphere of the dipole: Within a full period of oscillation the distance between λ = 0◦ and λ = λM

Figure 2.10: Particle oscillation between the two mirror points

is passed four times. The oscillation period τ b (τ bounce) is given by:

τ b = 4 λM

0

dl

v= 4

λM

0

dl

dλ

dλ

v(2.54)

The infinitesimal path element along a field line is given by (curve length dl):

(dl)2 = (dr)2 + r2(dλ)2 (2.55)

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The second invariant is determined by integration along the field line between two con- jugate mirror points. The so-called integral invariant I is defined by:

I = J

2mv =

1

2mv 2

lM2

lM1

mvdl =

lM2lM1

vv

dl =

lM2lM1

v

1 − sin2 α0 BB0v

dl

The magnetic induction in the mirror point is known already:

I =

lM2lM1

1 − B

BMdl (2.61)

The advantage in using the integral invariant is that in a static magnetic field I onlydepends on the field configuration. If ∂B/∂t = 0 equation (2.60) has to be applied.

2.2.5 E×B-DriftStarting from the equation of motion

mdv

dt = e(E + v × B) (2.62)

the parallel component is examined first:

mv̇ = eE (2.63)

v × B has no component parallel to B. The electric field E cannot be maintainedbecause electrons parallel to B are extremely mobile. In the following the perpendicularcomponent will be investigated. In order to do so we assume: E⊥ = E xi

mv̇x = eE x + eBvy

v̇x = e

mE x +

eB

m vy

v̇x = e

mE x + ωcvy (2.64)

v̇y = 0 there is no E y

−ωcvx (2.65)

Second time derivative:

v̈x = ωcv̇y =

−ω2c vx (2.66)

v̈y = −ωcv̇x = −ωc( em

E x + ωcvy)

v̈y = −ω2c (vy + E x

B ) (2.67)

Substitution: vy = vy + E x

B (2.68)

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After this substitution the second derivative leads to the following expressions:

v̈x = −ω2c vx (2.69)v̈y =

−ω2c v

y (2.70)

This is a cyclotron motion. It is superimposed by a drift of the guiding center in y-direction. This drift is the so-called E × B-drift:

vD = E × B

B2 (2.71)

The physical fundamentals of the E ×B-drift are founded in the Lorentz transformation.

Figure 2.12: Scheme of the E × B-drift

In a moving system the following relation is given:

E = E + v×

B (2.72)

For an independent particle the following condition has to be met: E = 0. E = −v × B.The Lorentz transformation is independent from charge, therefore the E × B-drift ischarge-independent too.

If an external force F acts on a charged particle; it carries out a motion that can bedescribed by the following equation:

dp

dt = F + e(v × B) (2.73)

In a moving system, which translates with the velocity of the guiding center, the particlesconduct a circular motion around the field line. With these assumptions the force F⊥must be compensated by an induced electric field or by a force, which can be deducedfrom the induced electrical field: eE. In a moving system the fields are given by:

B ≈ B (2.74)E = E + (vD × B) (2.75)

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The B field is almost equal in both systems. The induced E field however consists of an E field that may be already existing and a part that occurs in the dashed system:(vD × B). vD is the transversal velocity of the moving system, and as well the velocityof the guiding center.

The sum of the appearing forces is zero as well in a static system as in a moving system:

eE + F⊥ = e(vD × B) + F⊥ = 0 (2.76)Cross product with B leads to:

(2.76) e [vD × B] E

×B + F⊥ × B = 0 (2.77)

F⊥ × B = eB × [vD × B]= e{(B · B)vD − (B · vD)B}

= eB2vD − eB(vD · B)

vD = F⊥ × B

eB2 =

F × BeB2

(2.78)

For the cross product only the perpendicular component of F is used. Due to this factthe cross product is made with F instead of F⊥. The drift velocity is perpendicular tothe external force F and to the magnetic induction. The external force F can be replacede.g. by an electrical field with the so-called electric force eE:

F = eE (2.79)

vDE = eE × B

eB2

= E × B

B2

, (2.80)

which, in addition, shows v = E/B . The following assumptions are applied: The driftdoes not depend on charge of a particle, mass or energy. It is only dependent on theconfiguration of the E and B fields. To give further details about particle motion in themagnetosphere, those external forces that stem from the dipole are especially relevant:

(a) the gradient of the magnetic field

(b) the curvature of the field lines

ad (a) Drift generated by a gradient in the magnetic field The force component parallelto the gradient of the B field is given by:

F L cos φ = −ev⊥ cos ϕ(B + |∇⊥B|rc cos ϕ) (2.81)The Lorentz force is always of inverse direction of the gradient

ϕ = 0◦ : B + |∇⊥B|rc = B2ϕ = 90◦ : B + 0 = B

ϕ = 180◦ : B − |∇⊥B|rc = B1

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Figure 2.13: Gradient of magnetic field strength

Figure 2.14: Drift caused by the magnetic field (GC . . . guiding center)

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Force averaged over full gyration:

|F| = 12π

2π

0ev⊥(B + |∇⊥B|rc cos ϕ)cos ϕ dϕ

= 1

2πev⊥

2π0

(B cos ϕ)dϕ 0

+

2π0

(|∇⊥B|rc cos2 φ) dφ

2π

0(cos2 ϕ) dϕ = (

1

2ϕ +

1

4 sin(2ϕ)) |2π0 = π +

1

4 sin(4π)

0

−(0 + 0)

(2.82)

Fav = − 12π

ev⊥∇⊥Brcπ

Fav = −12

ev⊥∇⊥Brc (2.83)This force is directed oppositely to the Lorentz force. It gives rise to a drift, theso-called gradient drift. With

rc = mv⊥

eBone obtains:

Fav = −12

ev⊥∇⊥B mv⊥eB

= −mv2⊥

2B ∇⊥B = −µB∇⊥B (2.84)

From equation (2.78) follows:

vDG = F × B

eB2 =

−µB∇⊥B × BeB2

(2.85)

B gradient drift is dependent on the energy of particles (the kinetic energy can befound in µB) and on the charge e. Again it is demanded that the variation of theexternal field B must be small compared to the cyclotron radius:

Postulation : rc|∇⊥B|

|B| 1 (2.86)

ad (b) The curvature drift occurs due to a force FC which results from the motion of a

mass along a curved field line, thus centrifugal forces emerge.

centrifugal force: FC =mv2RC

n (2.87)

vDC =mv2

RCeB2n × B =

mv2

eRC

n × BB2

(2.88)

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Figure 2.15: Parameters describing the curvature drift

Important: Neither the drift resulting from the gradient of a B field nor thecurvature drift appear isolated because curvature of the field line always ap-pears in a natural non-homogeneous B field. Thus both kinds of drift alwaysappear combined. If currents are negligible, the rotation can be applied forthe B field: rot B = 0.

The perpendicular gradient ∇⊥B can be juxtaposed with the curvature radius of thefield lines RC:

∇⊥B = − BRC

n (2.89)

The exact derivation can be conducted within the scope of MHD. The normal vector n

is given by:n = −RC∇⊥B

B (2.90)

Insertion into (2.88) yields:

vDC =mv2

eRC

(−RC )(∇⊥B) × BB3

vDC = −mv2(∇⊥B) × B

eB3 (2.91)

From eq.(2.85) follows:

vDG = −mv2

⊥∇⊥B

×B

2BeB2 (2.92)

Addition of both drifts leads to:

vD = −mv2⊥∇⊥B × B

2eB3 −

mv2∇⊥B × BeB3

vD = m

2eB3 (v2⊥ + 2v

2) B × ∇⊥B (2.93)

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If the relations for v⊥ and v are put in, the following can be obtained:

(v2⊥ + 2v2) = v

2 sin2 α + 2v2 cos2 α = v2(sin2 α + 2 cos2 α)

v2(1 + cos2 α) 1−sin2 α

= v2(2−

sin2 α)

v2(2 − sin2 α) = v2

2 − sin2 α0 BB0

= v2

2 − sin2 α0

1 + 3 sin2 λ

cos6 λ

Thus a drift velocity vD for a particle with a defined equatorial pitch angle α0 along agiven field line in geomagnetic latitude λ can be obtained.

vD = mv2

2eB3

2 − sin2 α0

1 + 3 sin2 λ

cos6 λ

B × ∇⊥B (2.94)

Including the following approximation: ∇⊥B = (−B/RC)n and with the restriction thatoscillation only takes place near the equator – in magnetic latitudes about ≈ ±20◦ thedipole field line can be approximated by the osculating circle with the curvature radius

RC = r0

3 , (2.95)

r0 is the apex distance of the field line.

∇⊥B = −B 3r0

n (2.96)

vD = mv

2

2eB3 2 − sin2 α0 1 + 3 sin2 λcos6 λ B(− 3r0 B)b̂ × n|vD| = 3

2

mv2

eBr0

2 − sin2 α0

1 + 3 sin2 λ

cos6 λ

B = µ0M D

4πr3

1 + 3 sin2 λ

B = µ0M D

4πr30

1 + 3 sin2 λ

cos6 λ

One finally obtains:

|vD| = 32

mv2

er0

4πr30 cos6 λ

µ0M D

1 + 3 sin2 λ

2 − sin2 α0

1 + 3 sin2 λ

cos6 λ

|vD| = 6πmv2r20

µ0M De

cos6 λ 1 + 3 sin2 λ

2 − sin2 α0

1 + 3 sin2 λ

cos6 λ

(2.97)

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This is the approximation for oscillation along the osculating circle (λ ≤ ±20◦). vD givesthe drift velocity of charged particles in a dipole field. To simplify matters the structureof the dipole field line is replaced by the osculating radius. Therefore equation (2.97) canonly be applied to a restricted range of latitude of about −20◦ ≤ λ ≤ +20◦.

Another estimation for the value of the combined drift velocity is made, supposing thatoscillation occurs only narrowly over and under the magnetic equator: pitch angle α0 ≈90◦, λ ≈ 0◦. So the expression in parentheses in equation (2.97) is reduced to:

≈1 cos6 λ

1 + 3 sin2 λ ≈1

2 − sin2 α0≈1

1 + 3 sin2 λ

cos6 λ ≈1

= 1

With the definition of the shell parameter

L = r0rp

(2.98)

r0 . . . apex distance of the field linerp . . . radius of the planet

In case of minimum oscillation around the magnetic equator:

vD = 6π

µ0· mv

2r2pM De

L2 (2.99)

The relation vD = r0∂ϕ

∂t (2.100)

( ∂ϕ∂t = ϕ̇), gives the angular velocity of a particle moving away of the guiding field line.This motion occurs transverse to the field structure.

The angular velocity can be estimated:

ϕ̇ = vD

r0=

6πmv2

µ0M De · r

2pL

2

rpL

ϕ̇ =

vD

r0 =

6π

µ0 · mv2rp

M De L (2.101)

Protons hardly ever reach velocities high enough for relativistic effects to occur, forelectrons the Lorentz factor is relevant. Another interesting effect is to be mentionedhere: ϕ̇ ∝ L, this means that particles with greater distance in the magnetosphere, witha given amount of energy move faster around the planet! These relations for the driftvelocity are only valid if the pitch angle α0 ≈ 90◦.

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Gyration Oscillation DriftElectrons ≈ µs tenth s ≈ minProtons ≈ ms ≈ s ≈ hrs

If charged particles get to their starting point after a drift of 2π, in other words after afull orbit around the planet, this motion starts from the beginning: Particles are trappedin a dipole field due to their complicated paths, they are referred to as “trapped particles” .The motion around the field line, the so-called gyration is crossed out by averaging, sothe gyration is not relevant for the motion between the mirror points. Motion betweenthe mirror points, the so-called oscillation is crossed out if drift motion is considered,oscillation is not relevant for motion around the Earth. oscillation and drift are importantif the surface on which the guiding center moves is viewed. Complete azimuthal motion

Figure 2.17: Part of a drift shell

yields a surface which is in three dimensional view arched: the so-called drift shell . The drift shell is the sum of all guiding field lines .

The 3rd adiabatic invariant In case temporal variations occur very slowly compared tothe drift period (if the following relation is given):

τ drift|∂ B∂t ||B| 1 (2.102)

The third theorem of conservation can be expressed: The magnetic flux Ψ, which isencircled by the drift shell of the particle is constant.

Ψ =

A0 · dx = const (2.103)

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r0 . . . zenith distance of a field line

We get the magnetic flux by using equation (2.103) and applying the integral theoremof Stokes:

Ψ = A0 · dx = Ω

(∇ × A0) B

·dS = Ω

B · dS (2.105)

It is identical with the value derived from integration over the part of the equatorial planeΩ which lies outside the intersection line of the shell with Ω . All particles reflecting onthe same magnetic field line (dipole field line) have the same value for the third adiabaticinvariant.

Figure 2.20: Illustration of the determination of the magnetic flux Ψ

Ψ =

π/2λ

2π0

Br · dS (2.106)

=

π/2

λ

− µ0M

4π

2

r3 sin λ

Br·r dλ

2π

02r cos λπ dϕ

For the radial component of B we take the result already obtained above, (eqn. 2.34)

Br = −µ0M 4π

2

r3 cos ϑ

sinλ

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Ψ = −µ0M 4π

· 2r3

· 2r2π π/2

λsin λ cos λ dλ

2π0

dϕ

Ψ =

−

µ0M

r π/2

λ

sin λ cos λ dλ 2π

0

dϕ

π/2λ

sin λ cos λ dλ = sin2 λ

2 |π/2λ =

1

2

1 − sin2 λ = 1

2 cos2 λ

Ψ = −µ0M 2r

cos2 λ · 2π = −µ0πM r

cos2 λ (2.107)

Ψ ∝ cos2 λ (2.108)

The magnetic flux therefore reaches a maximum at λ = 0◦ (defining the equator) and aminimum at λ = 90◦ (defining the pole, here Ψ = 0).

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3 Plasmas as Fluids

3.1 Plasmaphysics

In plasma the situation is substantially more complicated than the one-particle consider-ation that has been dealt with up to now. Electric and magnetic fields cannot been stated”fixed”, they are determined by positions and movements of the particles themselves.

One has to solve a self consistent problem: The globally dominant fields have to be

supplied with the fields that are generated or attenuated. If one tried to consider everysingle particle – even with a super computer – and to determine resulting paths andfields, this would turn out to be impossible. One thing to consider: change in position of one single particle causes a deviation of position of all the other (n − 1) particles (e.g. ina laboratory plasma there are n ∼ 1012 protons and electrons within 1 cm3), in additionthe particle causes a change of all prevailing three-dimensional fields.

In the MHD theory (magnetohydrodynamics) the individual particle is negligible, onlythe motion of a “fluid element”, this is an ensemble of particles is considered. Of coursethere are problems that can be poorly or not at all be dealt with. In such cases onehas to go back to the one-particle model. In this context the Monte Carlo method is

worth mentioning. Position, velocity and resulting fields of up to 104

or 105

particles canbe defined by means of this method. The main problem concerning this method is therequirement for a huge storage capacity within the computer. Another possibility if theMHD theory fails is the kinetic theory, but it is connected with a substantially higherlevel of difficulty.

In plasma-physics one uses the Maxwell equations for vacuum. They are written as:

div E = ∇ · E = ρε0

(3.1)

curl E = ∇ ×

E =−

∂ B

∂t (3.2)

div B = ∇ · B = 0 (3.3)

curl H = ∇ × H = j + ∂ D∂t

(3.4)

(B = µ0µH ) ∇ × B = µ0 j + 1c2

∂ E

∂t (3.5)

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3.2 The Convective Derivative

In the so-called “vacuum equations” E and B are used (instead of D and H) as well asthe relations ε = 1 and µ = 1.

Maxwell’s equations determine the conditions of E and B for a given state of plasma. In

order to solve this self consistent problem it is necessary to know the response of plasmato the E and B field, in other words the response has to be defined by an equation. Fora single particle the already known equation of motion can be applied:

mdv

dt = e(E + v × B) (3.6)

Supposing that collisions and thermal motion are not considered a general velocity u canbe defined for all particles in a fluid element:

mndu

dt = en(E + u × B) (3.7)

n . . . particle number density (particles per volume unit)

At this point it is necessary to define the so-called convective derivative :

3.2 The Convective Derivative

In the single particle model (equation with v) the derivative with respect to time is used– in a coordinate system moving with the particle. An equation for a liquid element mustbe defined in a fixed coordinate system. The following relation can be applied for anyquantity if the corresponding transformation is conducted:

G = G(x, t)

G is one-dimensional at the moment because it depends on x. In a coordinate systemmoving with the fluid temporal variation of G is given by the sum of two terms:

dG(x, t)

dt =

∂ G

∂t +

∂ G

∂x

∂x

∂t =

∂ G

∂t + ux

∂ G

∂x (3.8)

∂ G/∂t represents the variation of G in a fixed reference point, ux ∂ G/∂x represents thevariation of G that way that the observer moves with the fluid into a region with adifferent value in G.

The following relation can be given generally in three dimensions:

dG

dt =

∂ G

∂t + (u · ∇)G (3.9)

This equation is the convective derivative . Sometimes it is written as DG/Dt. Theexpression (u · ∇) (means “u in Nabla”) is a scalar differential operator . The following

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3.4 Consideration of collisions

different pressures emerge from this: p⊥ = nkT ⊥ and p = nkT . Pressure becomes apressure tensor which is symbolized by P̄ , e.g.:

P̄ = p⊥ 0 0

0 p⊥

00 0 p

P̄ =

p 0 00 p 00 0 p

Isotropic temperature distribution leads to the second of the tensors mentioned above.According to this the motion equation has to be generalized too:

mn

∂ u

∂t + (u · ∇)u

= en(E + u × B) − ∇P (3.13)

Remark: In this case an isotropic liquid element was assumed,this means that the char-acteristic features of the liquid element are constant in all directions. This is not justifiedfor in presence of a magnetic field B.

3.4 Consideration of collisions

If collisions between plasma particles are taken into account this has of course an effecton the equation of motion. To simplify matters only collisions with neutral particlesare considered. (Collisions among charged particles require a more detailed discussion of this subject, which is not necessary at this point). Any collision of plasmatic and neutralparticles leads to a change of momentum. This is a change in mn∆u where the change of momentum is proportional to the relative velocity u − u0. Here, u0 is the velocity of the

“neutral” fluid. If the time between two successive collisions can be considered constant– with τ as mean free path time – the resulting force resulting from such collisions isgiven by:

−mn(u − u0)τ

. . . Momentum per time

Thus the generalized equation of motion (it includes non-isotropic pressure and collisionswith neutral particles) is given by:

mn

∂ u

∂t + (u · ∇)u

= en(E + u × B) − ∇P̄ − mn(u − u0)

τ (3.14)

3.5 Hydrodynamics

Ordinary liquids obey the Navier-Stokes equation:

ρ

∂ u

∂t + (u · ∇)u

= −∇ p + ρν ∇2u (3.15)

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3.5 Hydrodynamics

The integrands have to be equal because this relation can be applied to any surface:

∂n

∂t = −∇ · (n u)

∂n

∂t + ∇ · (n u) = 0 (3.20)

This is the equation of continuity . On the right hand side any sources and sinks have tobe taken into account.

3.5.2 Equation of state

In order to complete the system of equations one more relation is necessary: the relationbetween p and ρ which is substantially determined by the behavior of temperature:

p = p(ρ, T )

Supposing an isothermal state the following relation is applied:

p = const · ρ ⇒ ddt

p

ρ

= 0

With p = nkT we can write:

∇ p = ∇(nkT ) = kT ∇n (3.21)

∇ p p

= γ ∇nn

(3.22)

(From Chen, p. 58)

The adiabatic relation is applied to a gas (plasma) when it cannot release energy:

p = const · ργ

Adiabatic change of state

d

dt p

ργ = 0

γ is the so-called adiabatic exponent, which is the ratio of the two specific heats: γ =cp/cV. In an ideal gas γ can be determined by the number of degrees of freedom:

γ = f + 2

f (3.23)

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3 Plasmas as Fluids

3.5.3 The full set of MHD equations

Now the complete set of MHD equations is given, where the plasma is to consist of twocomponents – electrons and ions:

charge density: σ = niei + neee (ei = −ee) (3.24)current density: j = nieiui + neeeue = e(niui − neue) (3.25)

At this point velocities of the fluid element are used instead of velocities for single par-ticles. ui is applied for ions and ue for electrons.

Maxwellian Equations: See eqns. (3.1) to (3.5) for the definitions of the four Maxwellianequations

div E = ρ

ε0(3.26)

div E = 0 (without particular space charges) (3.27)

curl E = −∂ B∂t

(3.28)

div B = 0 (3.29)

curl B = µ0 j + 1

c2∂ E

∂t (3.30)

Equation of motion: (without collisions and viscosity)

mknk ∂ uk∂t + (uk · ∇)uk = eknk(E + uk × B) − ∇ pk (3.31)k = i . . . ionsk = e . . . electrons

equation of continuity:

∂nk∂t

+ ∇ · (nkuk) = 0 (k = i, e) (3.32)

Equation of state:

pk = C (mknk)γ k (k = i, e) (3.33)

Normally, Ohm’s law is added to these equations; Ohm’s Law:

jk = λ(E + uk × B) (3.34)

λ . . . conductivity

The following unknown quantities appear:

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3 Plasmas as Fluids

If the processes in MHD approximation are considered to be sufficiently slow, the dis-placement current on the right side (second expression on the right) of the correspondingMaxwellian equation is negligible:

∇ × B = µ0 j + 1

c2∂ E

∂t

Inserting into equation above:∂ u

∂t + (u · ∇)u

= −1

ρ∇ p + 1

µ0ρ[(∇ × B) × B]

Vector Analysis provides the rules, the gradient operates on products of vector-fields. Inthe following (∇ × B) × B is going to be calculated:

∇(a · b) = (b · ∇)a + (a · ∇)b + b × (∇ × a) + a × (∇ × b)∇(a · b) − (b · ∇)a − (a · ∇)b − b × (∇ × a) = a × (∇ × b)

(∇ × b) × a = (b · ∇)a + (a · ∇)b + b × (∇ × a) − ∇(a · b)a = b:

(∇ × b) × b = (b · ∇)b + (b · ∇)b + b × (∇ × b) −(∇×b)×b

−∇(b · b)

2(∇ × b) × b = 2(b · ∇)b − ∇b2

(∇ × b) × b = (b · ∇)b − 12∇b2

Thus:

(∇ × B) × B = (B∇) B − 12∇B2 (3.35)

The following is obtained:∂ u

∂t + (u∇)u

= −1

ρ∇ p + 1

µ0ρ(B∇)B − 1

µ0ρ

1

2∇B2

The expression (B · ∇)B/(µ0ρ) becomes zero if the magnetic field does not change withprogression in direction of B. The following is obtained for the acceleration perpendicularwith respect to the B field:

du⊥dt = −1ρ∇ p − 1ρ ∇B2

2µ0

du⊥dt

= −1ρ∇

p + B2

2µ0

(3.36)

Plasma moves perpendicular to the magnetic field as if the magnetic pressure B2/2µ0would work besides the pressure p.

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3 Plasmas as Fluids

v⊥ ⊥ B ⇒ v⊥B is connected with the cosine cos π/2 = 0. Therefore:

0 = en[E

×B]

−env⊥B

2

− ∇ p

×B

en v⊥B2 = en[E × B] − [∇ p × B]

v⊥ = [E × B]

B2 − 1

enB2[∇ p × B]

= vE + vD

vE = E × B

B2 normal E × B-drift

vD = −∇ p × BenB2

(3.37)

vD is the so-called diamagnetic drift which only occurs if plasma is regarded as fluid, it isdescribed by the expression ∇ p. vD only appears perpendicular to ∇ p, this is why (v·∇)vcan be neglected; this expression takes into account variations of v in direction of v, butnot variations perpendicular to v. The plasma- and field configuration is illustrated infigure 3.5.4. The equation of state is given by:

Figure 3.1: Representation of B and the diamagnetic drifts of ions and electrons

∇ p p

= γ ∇n

n

⇒ ∇ p = γp ∇nn

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3.5 Hydrodynamics

Inserting ∇ p into the relation for diamagnetic drift (+/− signs for ions and electrons,respectively), we get:

vD =

−

∇ p × B

enB2

= − 1enB2

γ p∇n

n × B

= −nkT γ ∇n × ẑBenB2n

= +γkT ẑ × ∇n

eBn

electron: vDe = −γ kT

eBn ẑ × ∇n (3.38)

ion: vDi = +γ kT

eBn ẑ × ∇n (3.39)

What is the reason for this drift? The following field and density configuration shouldbe given: The gradient in density is symbolized by the number of gyration radii. In an

Figure 3.2: Schematics of the diamagnetic drift

arbitrary volume element a greater number of particles moves downward than upwardbecause the particles streaming downward come from a region of higher density. Thus a

“fluid drift” perpendicular to ∇n and B exists where the guiding centers stay stationary .Because of the opposite direction of gyration, the diamagnetic drift of electrons is directed

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4 Plasma Oscillations and MHD WavesAny periodic motion of a medium can – by Fourier analysis – be divided into a superposi-tion of sine or cosine waves with varying frequency ω and wavelength λ. If the oscillationfrequency is small, the waveform is sinusoidal in general. Only one component has to beused.

Example: Oscillation in density

n = n̄ exp[i(k · r − ωt)] (4.1)

n̄ . . . constant, defined by the amplitude of the oscillation in densityk . . . propagation vector

Written in cartesian coordinates:

k · r = kxx + kyy + kzzk only has an x-component for wave propagation in direction of x:

n = n̄ exp[i(kx − ωt)] = n̄ei(kx−ωt)

It is intended by convention that if exponential notation is applied, the real part describesthe quantity to be measured. Applying De-Moivre’s theorem:

eiϕ = cos ϕ + i sin ϕ

Re(n) = n̄ cos(kx − ωt)A wave of time-constant phase, d(phase)/dt = 0 shows that

d

dt(kx − ωt) = 0

kx − ωt = constx =

ω

kt + C

dx

dt =

ω

k = vphase (4.2)

Within a wave the E field oscillates.

E = E0 ei(kx−ωt) (4.3)

real part: Re(E) = E0 cos(kx − ωt)

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4 Plasma Oscillations and MHD Waves

Because it does not carry any information the phase velocity vphase can be greater thanthe velocity of light. An unmodulated wave of infinite length and constant amplitudedoes not carry information. The modulation itself – the information – propagates by thegroup velocity , which cannot exceed the velocity of light.

A modulation will be viewed in the following, to do so two waves with almost equalfrequency are added:

E 1 = E 0 cos[(k + ∆k)x − (ω + ∆ω)t]E 2 = E 0 cos[(k − ∆k)x − (ω − ∆ω)t]

(k − ∆k)x or (k + ∆k)x: Both waves must have the same phase velocity – both of thempropagate in the same medium – a difference of 2∆k has to be allowed in the spreadingfactor k. (ω − ∆ω)t or (ω + ∆ω)t: E 1 and E 2 vary about 2∆ω in frequency.

substitution: a = kx − ωtb = (∆k)x

−(∆ω)t

E 1 + E 2 = E 0 cos[kx − ωt + (∆k)x − (∆ω)t]+E 0 cos[kx − ωt − (∆k)x + (∆ω)t]

= E 0 cos(a + b) + E 0 cos(a − b)= E 0(cos a cos b − sin a sin b + cos a cos b + sin a sin b)= 2E 0 cos a cos b

E 1 + E 2 = 2E 0 cos[kx − ωt] cos[(∆k)x − (∆ω)t] (4.4)

This is a sinusoidally modulated wave. Information is carried by the envelope of the

Figure 4.1: Modulation of the wave

wave, it is given by cos[(∆k)x − (∆ω)t]. The corresponding velocity of information is∆ω/∆k. Group velocity can be determined in the transition ∆ω → 0:

vg = dω

dk (4.5)

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4.1 Plasma oscillations

This group velocity cannot exceed the velocity of light.

4.1 Plasma oscillations

An important form of plasma oscillation is the plasma frequency ωp:

for electrons: ωpe =

ne2

meε0

1/2(4.6)

for ions: ωpi =

ne2

miε0

1/2(4.7)

This plasma oscillation is only dependent on particle density n. It is important to

mention that it does not depend on k, i.e. the group velocity turns out to be zero:dω/dk = 0. The oscillation or disturbance does not expand! One process that canlead to expansion of plasma oscillations is the thermal motion: Electrons getting out of regions with such oscillations by thermal motion have impressed information about theoscillation. This case can be understood as plasma wave, to be exact, as electron plasma wave. A derivation which will not be conducted in detail here.

ω2plasmawave = ω2p +

3

2k2v2th (4.8)

(One dimensional consideration with v2th = (2kT e)/m, where k is Boltzmann’s constant.)

The frequency now depends on k; the group velocity can be derived by differentiation of the equation above:

2ωdω = 3

2v2th2kdk

vg = dω

dk =

3

2v2th

k

ω

vg = 3

2

v2thvphase

(4.9)

vg is the group velocity of the electron plasma wave.

4.2 Magnetohydrodynamic Waves

Starting point are the MHD equations.

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Maxwellian equations:

∇ × B = µ0 j

∇ × E = −∂ B

∂t

∇ · B = 0(∇ · E = ρ

ε0) will not be used here

Ohm’s Law:

j = λ(E + v × B)

Equation of motion:

ρ∂ v

∂t + ρ(v · ∇)v = j × B − ∇ pEquation of continutity:

∂ρ

∂t + ∇ · (ρv) = 0

Equation of state:

p = p(ρ, T )

These fundamental equations will be used to describe simple wave solutions in the follow-ing section. A linearization of the equations will be conducted: The equilibrium solution

is indexed “0”, the time dependent disturbance is indexed “1”. It is also important to notethat the disturbance must be small compared to the not-disturbed quantity. Lineariza-tion is conducted for: E → E = E0 + E1, B, v, j, ρ and p. The objective is to describe aresting plasma, a set of equations for stationary state will be compound. The followingrelation is given in stationary state: ∂ ∂t = 0. Thus the MHD zero-indexed equations aregiven by:

Maxwellian equations:

∇ × B0 = µ0 j0 (4.10)

∇ ×E0 = 0 (4.11)

∇ · B0 = 0 (4.12)

(∇ · E0 = ρε0

) not neccessary here (4.13)

Ohm’s law:

j0 = λ(E0 + v0 × B0) (4.14)

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Equation of motion:

ρ0(v0 · ∇)v0 = j0 × B0 − ∇ p0 (4.15)Equation of continuity:

∇ ·(ρ0v0) = 0 (4.16)

Equation of state:

p0 = p0(ρ0, T ) (4.17)

4.2.1 Linear perturbation theory:

Disturbance variables are now added to this – simplified – stationary case. Distortionsof quadratic order and higher are neglected. A set of MHD fundamental equations isobtained for perturbations. The set of equations will be simplified by the followingassumptions:

B0 = (B0, 0, 0), (B0 being constant), λ → ∞, v0 = 0 (4.18)

ρ0 und p0 are constant in space.

∇ × B1 = µ0 j1 (4.19)

∇ × E1 = −∂ B1∂t

(4.20)

∇ · B1 = 0 (4.21) j

λ = E1 + v0 × B1 + v1 × B0

for λ → ∞ and v0 = 0 we get:

0 = E1 + v1 × B0 (4.22)

ρ0∂ v1∂t

= −∇ p1 + j1 × B0 (4.23)

∂ρ1∂t

+ ∇ · (ρ0v1) = 0 (4.24) p1 p0

= γ ρ1ρ0

(4.25)

γ = 5/3 for a single-atom gas. This set of equations provides a homogeneous linearsystem for spatial and temporal behavior of the disturbance quantities E1, B1, j1, v1, p1

and ρ1. The disturbance quantities should show the same behavior in time as e.g. v1:∂ ∂t (ρ1) = −iωρ1. In order to solve the set of equations (4.19) – (4.25) v1 can be givenand a differential equation for v1 can be searched for; approach:All disturbance variables dependent on v1 are inserted into equation (4.23) subsequently.

From this a linear, homogeneous differential equation for v1 is derived. Any special casecan be investigated with this scheme, where a well defined assumption has to be madeat the beginning.

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1st step: v1 into equation (4.22) gives E1(v1)2nd step: E1 into equation (4.20) gives B1(v1)3rd step: B1 into equation (4.19) gives j1(v1)4th step: v1 into equation (4.24) gives ρ1(v1)

5th

step: ρ1 into equation (4.25) gives p1(v1)

1st special case: Alfvén waves

Assumptions:

(a) The disturbance variable is dependent only on x and t and is given in the y com-ponent: v1 = (0, v1(x, t), 0). Approach: v1y(x, t) = C e

ikx−iωt

(b) The B field is given by B0 = (B0, 0, 0)

Figure 4.2: Directions of B0 and v1,y in space

(c) Supposition of an incompressible plasma oscillation ∇ p1 = 0

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Inserting into the solution scheme:

• 1st step:

E1 = −v1 × B0 = − x̂ ŷ ẑ

0 v1y 0B0 0 0

= +(v1yB0)ẑE1 = (0, 0, v1yB0)

• 2nd step:

∇ × E1 = −∂ B1∂t

∇ × E1 =

x̂ ŷ ẑ∂/∂x ∂/∂y ∂/∂z

0 0 v1yB0

=

∂

∂yv1yB0

x̂ −

∂

∂xv1yB0

ŷ + 0ẑ

∇ × E1 = (0, −∂v1y∂x

B0, 0)

v1y depends on x and t. For v1y exclusively, a spatial derivation may be foundbecause B0 is homogeneous and constant.

B1 = −

(∇ × E1)dt + C 1

= −

−∂v1y∂x

B0

dt

C 1 = const, is supposed to be 0, B0 is stationary, therefore temporally constant.

B0

∂

∂x(v1y)dt = B0

∂

∂x(C eikx−iωt)dt =

1

−iω∂

∂x(v1y)B0

therefore: B1 = (0, − 1iω

∂v1y∂x

B0, 0)

• 3rd step:∇ × B1 = µ0 j1

∇ ×B1 =

x̂ ŷ ẑ∂/∂x ∂/∂y ∂/∂z

0 − 1iω ∂v1y∂x B0 0 = − ∂

∂z − 1

iω

∂v1y

∂x

B0 x̂− [0]ŷ +

∂

∂x

− 1

iω

∂vly∂x

B0

ẑ

j1 = − 1iωµ0

(0, 0, ∂ 2v1y

∂x2 B0)

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Now all disturbance variables (dependent on v1) are inserted. An incompressible, coldplasma is assumed “Cold Plasma Approximation” (CPA) (T = 0, therefore ∇ p1 = 0);

ρ0∂ v1

∂t

= j1

×B0

j1 × B0 =

x̂ ŷ ẑ

0 0 − 1iωµ0∂ 2v1y∂x2

B0B0 0 0

= (0, − 1

iωµ0

∂ 2v1y∂x2

B20 , 0)

ρ0∂ v1∂t

= −∇ p1 + j1 × B0 ∇ p1 = 0( because of CPA)

ρ0(−iωv1y) = − 1iωµ0

B20∂ 2v1y∂x2

ρ0ωv1y = − B20ωµ0

∂ 2v1y∂x2

ρ0ωv1y + B20ωµ0

∂ 2v1y∂x2

= 0

∂ 2v1y∂x2

+ ρ0ω

2µ0B20

v1y = 0 DE of an oscillating string

−k2v1y + µ0ρ0B20

ω2y1y = 0

By using the approach v1y = C eikx−iωt the following relations are obtained:

∂

∂xv1y = ikv1y and

∂ 2

∂x2v1y = −k2v1y

The characteristic equation is written as:

−k2v1y +

µ0ω2ρ0

B20

1y = 0

−k2 + µ0ω2ρ0

B20= 0

ω2µ0ρ0B20

= k2

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k2

ω2 =

µ0ρ0B20

ω2

k2 =

B20µ0ρ0

≡ v2A

vA = ω

k =

B0√ µ0ρ0

(4.26)

vA is the so-called Alfvén velocity

ω

k = vphase velocity of the distortion

Two modes are obtained by taking the root: one mode with +vA and one mode with−vA, both of them propagate along the field line B0. k = ω/vA is derived by insertingvA again:

v1y(x, t) = C e±ikx−iωt = C e±i ωvA x−iωt = C eiω(±

xvA−t)

The initial situation of this special case is to be repeated at this point:

• incompressible plasma• conductance λ → ∞• B0 = (B0, 0, 0) and• v1 = (0, v1(x, t), 0)

Figure 4.3: Propagation of the Alfvén wave

The distortion B1 – the excursion of the field line – impressed onto B0 has only a y-component, it propagates along the field line with velocity vA. This so-called Alfvén-wave

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is the most simple kind of MHD waves.

Thus the assumption of initial conditions leads to a special analytic solution, the Alfvénwaves. The Alfvén velocity only depends on the magnetic induction B and the plasma

mass density; it is therefore a characteristic quantity of the given plasma configura-tion. Alfvén waves are transversal waves because the direction of propagation (here:x-direction) is directed perpendicular to the changing amplitude (y-direction). If ρ → 0,the Alfvén velocity becomes higher, within the scope of this approximation vA → ∞.Detailed investigation shows that vA converges towards the velocity of light, in this con-nection Alfvén waves change into electromagnetic waves.

2nd special case: magnetohydrodynamic compression-waves

Assumptions:

• B0 = (B0, 0, 0)

• v1 = (0, v1y(y, t), 0), the disturbance variable only depends on y and t• v1y(y, t) = C e±iky−iωt – wave propagation is permitted in positive and negative

direction

The y-component of the distortion depends on y, thus the wave is a longitudinal wave(the direction of propagation coincides with the direction of the distortion amplitude), aso-called MHD compressional wave. For the plasma is compressed, the adiabatic relation

Figure 4.4: Propagation of MHD compressional waves

is needed:

p1 p0

= γ ρ1ρ0

p1 = γ p0

ρ0ρ1

vsonic =

γ

p0ρ0

1/2(4.27)

p1 = v2sonic ρ1 (4.28)

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A differential equation for the disturbance variable v1 is to be found, the same schemelike in the first special case is used. The only difference is the expression grad p1, it isnow part of the motion equation:

ρ0

∂ v1y

∂t = −∇ p1 + j1 × B0Approach:

• 1st step:E1 = −v1 × B0 = (0, 0, v1yB0)

• 2nd step:

∇ × E1 = −∂ B1∂t

∇ ×E1 =

x̂ ŷ ẑ∂/∂x ∂/∂y ∂/∂z

0 0 v1yB0 = ∂

∂y

(v1yB0) x̂ − ∂

∂x

(v1yB0) ŷ + [0] ẑ∇ × E1 = ( ∂v1y

∂y B0, 0, 0)

B1 is derived from integration over t:

B1 = ( + 1iω

∂v1y∂y B0, 0, 0 )

• 3rd step:∇ × B1 = µ0 j1

∇ × B1 = x̂ ŷ ẑ∂/∂x ∂/∂y ∂ /∂z1

iω∂v1y∂y B0 0 0

= [0] x̂ −−

− 1

iω

∂

∂z

∂v1y∂y

B0

ŷ +

− 1

iω

∂ 2v1y∂y2

B0

ẑ

∇ × B1 =

0, 0, − 1iω

∂ 2v1y∂y2

B0

= µ0 j1

j1 = ( 0, 0, − 1iωµ0

∂ 2v1y∂y2

B0 )

Inserted into the equation of motion:

j1 × B0 =

x̂ ŷ ẑ

0 0 − 1iωµ0

∂ 2v1y∂y2

B0

B0 0 0

j1 × B0 =

0, − 1iωµ0

∂ 2v1y∂y2

B20 , 0

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• 4th step: This step is necessary because of the use of the adiabatic relation. Inser-tion into the equation of motion:

∂ρ1∂t

+ ∇(ρ0v1) = 0

∂

∂t → −iω

∇ → ∂ ∂y

(The other derivations ∂/∂x and ∂/∂z are zero)

−iωρ1 + ρ0 ∂v1y∂y

= 0

ρ1 = 1

iω

ρ0∂v1y

∂y

• 5th step: Inserting ρ1 into the state equation p1/p0 = γρ1/ρ0: p1 p0

= γ

ρ0

1

iωρ0

∂v1y∂y

p1 = γp0

ρ0 v2s

1

iωρ0

∂v1y∂y

p1 = v2s

ρ0iω

∂v1y∂y

Putting into the equation of motion:

ρ0∂ v1∂t

= −∇ p1 + j1 × B0 = −∂p1∂y

+ j1 × B0

−iωρ0v1y = −v2sρ0iω

∂ 2v1y∂y2

− 1iωµ0

B20∂ 2v1y∂y2

This leads to an equation of motion in the y-component:

ω2v1y = −v2s∂ 2v1y∂y2

− B20

µ0ρ0 v2A

∂ 2v1y∂y2

= −(v2s + v2A)∂ 2v1y∂y2

−(v2s + v2A)∂ 2v1y∂y2

+ ω2v1y = 0

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4.3 MHD waves with arbitrary angle to the magnetic field

This is the required differential equation. Approach:

v1y = C eiky−iωt

∂v1y

∂y = ikv1y

∂ 2v1y∂y2

= −k2v1y

(v2s + v2A)(−k2v1y) + ω2v1y = 0 characteristic equation

ω2

k2 = v2s + v

2A ≡ u2MHD−compr. (4.29)

This is the velocity of phase of a magnetohydrodynamic compressional wave that propa-gates perpendicular to the field lines. The square of the phase velocity can be expressed

by the sum of the squares of sonic speed v2s and Alfvén velocity v2A. Which situation isgiven in the limit cases B0 → 0 an

introduction to plasma physics

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