Chapter 2 Basic Plasma Properties - University of Alaska...

Chapter 2

Basic Plasma Properties

2.1 First Principles

2.1.1 Maxwell’s Equations

In general magnetic and electric fields are determined by Maxwell’s equations, corresponding bound-ary conditions and the source (charges and currents) distributions.

∇ · E =1

ε0ρc (2.1)

∇×B− 1

c2∂E

∂t= µ0j (2.2)

∇× E +∂B

∂t= 0 (2.3)

∇ ·B = 0. (2.4)

where E and B are electric and magnetic fields.

c = 3 · 108ms−1, ε0 = 8.85 · 10−12Fm−1, and µ0 = 4π · 10−7Hm−1. Sometimes it is convenientto express the electromagnetic fields in terms of an electric potential Φ and a vector potential A suchthat

E(r, t) = −∇Φ(r, t)− ∂A(r, t)

∂tB(r, t) = ∇×A(r, t)

which requires to solve the electromagnetic field equations for the potentials for instance in the form

1

c2∂2Φ

∂t2−∇2Φ =

1

ε0ρc(r, t) (2.5)

1

c2∂2A

∂t2−∇2A = µ0j(r, t) (2.6)

where Φ and A satisfy the Lorentz gauge ∂Φ/∂t+ c2∇ ·A = 0.

12

CHAPTER 2. BASIC PLASMA PROPERTIES 13

2.1.2 Lorentz Equations of Motion

In electromagnetic fields the motion of charged particles is determined by the fields through theequations of motion

dridt

= vi (2.7)

dvidt

=qim

(E + v ×B)ri,t. (2.8)

If the forces are external, the corresponding equations of motion are complete and the particle motionis often called test particle motion. However, if the other particles contribute to the force on a partic-ular particle the forces have to be evaluated. The electromagnetic forces in a plasma depends on thecharge and current densities which are determined by the collective particle interaction:

ρcs = qsns = qs

∫ ∞−∞

d3vfs(x,v, t)

js = qsnsus = qs

∫ ∞−∞

d3vvfs(x,v, t)

with ρc =∑s ρs and j =

∑s js.

In a plasma the number of particles in a physical system is usually rather large. In addition the over-whelming majority of problems deal with the collective particle behavior rather than the individualone. Discrete particle dynamics can be important in some areas of plasma physics for sufficientlysmall (‘microscopic’) length or time scales. The collective behavior is usually well described in afluid approximation.

2.1.3 Plasma Properties and Parameters

A plasma is a gas of charged particles, which consists of free positive and negative charge carriers. Toallow free motion of the particles the typical potential energy must be much smaller then the typicalkinetic energy

〈eφ〉 �⟨m

2v2⟩

= kBT

• More than 99% of all know matter in the universe is in the plasma state.

• Magnetosphere provides an ideal laboratory to examine complex processes in a natural plasmaby in situ observations.

Debye Shielding: Coulomb potential of a charge q:

φ = q/(4πε0r). (2.9)

Test charge shielded by other charges:


φD =q

4πε0rexp

(− r

λD

)(2.10)

with the Debye length defined by

λ−2D =n0e

2

ε0kB

(1

Te+

1

Ti

)(2.11)

or λD =(ε0kBTenee2

)1/2for temperatures Te ' Ti, number density ne, Boltzmann constant kB = 1.38 ·

10−23JK−1 and electron charge e = 1.6 · 10−19C. This potential is sometimes called the Yukawapotential.

Exercise: Show that the Yukawa potential satisfies the Poisson equation

∇2Φ = − 1ε0qtδ(r) + 1

λ2DΦ(r)

Exercise: Derive the Poisson equation for the case of a test charge q = e in a plasma (Hint: Firstdemonstrate that the density of species s is ns = n0 exp(−qΦ/kBTs); see section 2.2; thenexpand the exponential under the assumption qΦ� kBTs).

Exercise: Compute the net charge of the shielding cloud.

Quasi-neutrality for any physical length L� λD otherwise binary interaction should be considered.

qt

-e

-e

+e

+e

qt

λD

E∼0

Figure 2.1: Sketch of the particle trajectories and of the Debye shielding charge distribution in thevicinity of a test charge.

Plasma Parameter: The number of of particles in a Debye sphere is N = 4π3nλ3D. The condition

that the potential energy is much smaller than the kinetic energy yields

Λ = nλ3D � 1 (2.12)

with Λ being the so-called plasma parameter.

Exercise: Calculate the electron thermal speed, Debye length, and the plasma parameter for(a) the tail magnetosphere with Te = 107 K, n0 =106 m−3

(b) the ionosphere with Te = 103 K, n0 =1012 m−3


Exercise: Can a plasma be maintained at temperatures of Te = 100 K (Hint: Calculate the densitylimit using the plasma parameter and explain your result).

Exercise: Λ ∝ n−1/20 T 3/2 While the dependence on temperature seems intuitively clear the density

dependence appears odd because lower densities mean less particles and less shielding. Whydoes the plasma parameter improve (increase) with decreasing density?

Plasma Frequency: Typical oscillations in a plasma are electron plasma oscillations with thefrequency

ωps =

(neq

2s

msε0

)1/2

(2.13)

Collisions: In partially ionized plasma the collision time between electrons and neutrals must bemuch larger then the inverse electron plasma frequency

ωpeτn � 1.

The collision frequency in a plasma is given by

νc =

√π

2

1

32π

ωpeΛ

ln [12πΛ]

Figure 2.2: Magnetospheric plasma parameters.

2.2 Kinetic Equations

To describe a plasma one can solve the coupled system of Maxwell’s equations and the particle equa-tions of motion. However, there are more efficient methods to solve the plasma dynamics using theabove approximations.


The first step toward a fluid description is the introduction of a phase fluid with a phase space con-sisting of the set of ordinary coordinates and velocities (r,u). The entire ensemble of n particles has3n spatial and 3n velocity degrees of freedom. By integrating the corresponding Liouville equation(see statistical mechanics) over 3(n− 1) spatial coordinate and 3(n− 1) coordinates one obtains theBoltzmann equation for the so-called single particle distribution function f(r,u, t)

∂f

∂t+ v · ∇f +

q

m(E + v ×B) · ∇vf =

∂f

∂t

∣∣∣∣∣coll

(2.14)

In the case of thermal equilibrium f assumes locally a Maxwell distribution in velocity space andthe collision term on the rhs of 2.14 vanishes. Equation 2.14is clearly a fluid (advection) equationthough in a 6 dimensional space. The lhs can be interpreted as the total derivative of f(r,u, t) alonga trajectory given by the 6 dimensional velocity v(6) = (v, F

m) where F is the Lorentz force.

In the magnetosphere collisions can be neglected almost everywhere except for the ionosphere (andon rather long time scale in the plasma sphere). In the absence of collisions the total derivative alongthe path determined by v(6) is

df

dt= 0

in analogy of the advection equation ∂f/∂t + v · ∇f = df/dt. This implies that the value of f isconserved along this trajectory in the six-dimensional phase space. This particularly means that anymaximum of the distribution remains exactly the same. If in addition the force is conservative, the6 dimensional flow is incompressible, that means that the phase space volume of a contour with anyconstant value of f is conserved during the evolution as illustrated in Figure 2.3.

xi

vi

xi

vi

t=t1

t=t2

f>fc

f>fc

Figure 2.3: Illustration of the evolution of a collisionless distribution conserving the area inside con-tours of constant f .

The Vlasov equations consist of the collisionless Boltzmann equation 2.14 complemented by Maxwell’sequations. The collision term on the rhs can consider many different physical or chemical processes.Chemical reactions, ionization or recombination, friction, diffusion, and energy exchange collisionsare contained in the collision term. Details depend on the corresponding processes.

The description with the single particle distribution function 2.14 and the actual equations of mo-tion 2.7 and 2.8 are in many respects equivalent. In mathematical terms the Lorentz equations ofmotion represent the characteristics of the collisionless Boltzmann equation. Each description has itsparticular advantages and disadvantages.

In the presence of constants of motion (steady state ∂/∂t = 0, or ignorable coordinate directions, e.g.,∂/∂y = 0) any function of the constants of motion Fs(Hs, Psy) for species s with the Hamiltonian


Hs = msv2/2+qsΦ and the generalized momentum Psy = msvy+qsAy solves the collisionless Boltz-

mann equation. However, it still remains to solve Maxwell’s equation which will be demonstrated insection 7. The solution to plasma waves can be found by linearizing the Vlasov equations.

2.3 Derivation of the Fluid Plasma Equations

Fluid equations are probably the most widely used equations for the description of inhomogeneousplasmas. While the phase fluid which is governed by the Boltzmann equation represents a first exam-ple, many applications do not require the precise velocity distribution at any point in space. Ordinaryfluid equations for gases and plasmas can be obtained from the Boltzmann equation or can be derivedusing properties like the conservation of mass, momentum, and energy of the fluid.

2.3.1 Definitions

The equations of ordinary fluids and gases as well as those for magnetofluids (plasmas) can be ob-tained from equation 2.14 in a systematic manner. Defining the 0th, 1st, and 2nd moment of theintegral over the distribution function f as mass density ρ, fluid bulk velocity u, and pressure tensorΠ

n(x, t) =∫ ∞∞

d3vf(x,v, t) (2.15)

u(x, t) =1

n

∫ ∞∞

d3vvf(x,v, t) (2.16)

Π(x, t) = m∫ ∞∞

d3v(v − u)(v − u)f(x,v, t). (2.17)

Note that all quantities should carry an index for the particle species which is omitted here for con-venience. With these definitions one also obtains number density ρ(x, t) = mn, charge densityρc(x, t) = qn, momentum density p(x, t) = ρu, current density j(x, t) = qu, and scalar pressure(the isotropic portion of the pressure) p(x, t) = 1

3Tr(Π) where the individual particle mass m and

charge q are used. The fluid equations are then determined by the moments of the Boltzmann equa-tion, i.e., ∫ ∞

∞d3v vi(BoltzmannEqu.)

To account for the collision term in (2.14) we define

Qρ(x, t) = m∫ ∞∞

d3v∂f

∂t

∣∣∣∣∣c

(2.18)

Qp(x, t) = m∫ ∞∞

d3v(v − u)∂f

∂t

∣∣∣∣∣c

(2.19)

QE(x, t) =1

2m∫ ∞∞

d3v(v − u)2∂f

∂t

∣∣∣∣∣c

(2.20)

The precise form of these terms depends on the particular properties of the systems and will not bespecified at this point.


2.3.2 Fluid Moments

To provide an example for the evaluation of the moments of the Boltzmann equations let us evaluatethe 0th moment of the integral.

∫ ∞∞

d3v

(∂f

∂t+ v · ∇f +

F

m· ∇vf

)=∫ ∞∞

d3v∂f

∂t

∣∣∣∣∣coll

The first term of the equation becomes∫ ∞∞

d3v∂f

∂t=

∂

∂t

∫ ∞∞

d3vf(x,v, t) =∂n(x, t)

∂t.

The second term is∫ ∞∞

d3vv · ∇f =∫ ∞∞

d3v∇ · (vf(x,v, t)) = ∇ ·∫ ∞∞

d3vvf(x,v, t) = ∇ · (u(x, t)n(x, t))

and the third term is∫ ∞∞

d3vF

m· ∇vf =

∑i

∫ ∞∞

d3vFim

∂f

∂vi=∑i

∫ ∞∞

d2v[Fimf]vi=∞vi=−∞

−∑i

∫ ∞∞

d3vf

m

∂Fi∂vi

The terms on the rhs. in the above equation are 0 because the f = 0 for vi =

{+∞−∞ for each

component vi and in because ∂Fi/∂vi = 0 (see homework for the Lorentz force). Therefore∫ ∞∞

d3vF

m· ∇vf = 0

The collision term of the Boltzmann equation reduces to Qρ (see equation 2.18) such that in summarythe 0th moment of the Boltzmann equation reduces to

∂n(x, t)

∂t+∇ · (nu) =

1

mQρ. (2.21)

This is the usual continuity equation for the particle number density with a source term on the rightside. Multiplying (2.21) with the particle mass yields the continuity equation for mass density

∂ρ(x, t)

∂t+∇ · (ρu) = Qρ (2.22)

The source term describes production or annihilation of mass for instance through chemical reactionsor ionization or recombination. It is noted that (2.22) is for one species only. In the case of severalneutral constituents or ion species a corresponding continuity equation is obtained for each species.The total production rate of mass has to be zero.


Similar to the 0th moment the 1st moment of the Boltzmann equation [m∫∞∞ d3v v (Boltzmann equation)]

and 2nd moment [32m∫∞∞ d3v v2 (Boltzmann equation)] yield the equations for the fluid momentum

(or velocity) and energy

∂ρu

∂t= −∇ · (ρuu)−∇ · Π + qn(E + u×B) + uQρ + Qp (2.23)

∂

∂t

(1

γ − 1p+

1

2ρu2

)= −∇ ·

(1

2ρu2u +

1

γ − 1pu + u · Π + L

)

+qnu · E +1

2u2Qρ + u ·Qp +QE (2.24)

with the heat flux L(x, t) = 12m∫∞∞ d3v(v− u)(v− u)2f(x,v, t) and γ is the ratio of specific heats,

i.e., γ = 5/3 if a gas has 3 degrees of freedom for motion. Note that

• As before the “fluid” Lorentz force E+u×B requires to solve the corresponding field equations(2.1) - (2.4) .

• The Lorentz force (or any velocity dependent force now depends on the bulk velocity andvelocity is a dependent variable in the fluid equations rather than an independent variable as inthe Boltzmann equation.

• The source terms for mass Qρ, momentum Qp, and energy QE depend on system proper-ties and need to be specified through these or through a systematic collision operator andthe corresponding velocity integrals. The terms reflect mass generation and annihilation Qρ,momentum exchange through friction Qp, and energy exchange collisions QE . The frictionterm Qp

s for species s can be expressed in terms of an effective collision frequency Qps =∑

t6=s νstmsn(vt − vs)

• The pressure tensor is often split into a scalar pressure and a viscous tensor Π = p1 + w, withp = 1

3Tr(Π) and the viscous tensor w.

• Often a kinematic viscosity σik = a(∂ui∂xk

+ ∂uk∂xi

)+b∂ul

∂xlδik is used on the right of the momentum

equation yielding a term∇ · σ = η∆u +(ζ + η

3

)∇(∇ · u).

• Elimination of ∂∂t

(12ρu2

)in the energy equation (with the aid of (2.22) and (2.23)) yields

1

γ − 1

(∂

∂tp+∇ · pu

)= −(Π · ∇) · u−∇ · L +QE (2.25)

The fluid equations without the source terms imply the conservation of the corresponding property(mass, momentum, and energy). Consider the mass in a given volume defined by

MV =∫Vρd3x

The change of mass in the volume is

dMV

dt=∫V

∂ρ

∂td3x = −

∫V∇ · vρd3x = −

∮SVρv · ds


where SV is the surface of V . In other words the mass in the volume V changes only if there is anonzero density flux (velocity) across the surface of the volume. Similarly momentum and energy areconserved. However, for the energy one has to include momentum and energy which is contained inthe fields as well.

Exercise: Determine the integral of the 1st order moment for the first two terms in the Boltzmannequation.

Exercise: Derive the 1st order moment force term for a gravitational force and the Lorentz force(velocity dependent).

Exercise: Do the same for the energy equation (i.e., multiply the Boltzmann equation (2.14) with12mv2 and integrate).

2.3.3 Typical Fluid Approximations

Equations (2.22) - (2.24) establish the typical set of fluid equations which are used in many simula-tions of fluids and gases like weather simulations, air flow around aircraft or cars, water flow in pipesor round boats, and many other research and technical applications. Using the set of equations (2.22),(2.23), and (2.25) we can derive most equations commonly used in fluid simulations:

• For a known velocity profile u and no sources Qρ = 0 it is sufficient to model the continuityequation for instance to derive the evolution of density of a gas or the concentration of dust,aerosols, etc. in any medium like air water etc.:

∂ρ

∂t+∇ · (ρu) = 0

• With the total derivative along the fluid path defined as ddt

= ∂∂t

+ u · ∇ the advection equationis

dρ

dt= −ρ∇ · u

• A system is called incompressible if the density of a fluid parcel along the convection path doesnot change, i.e., dρ/dt = 0 which is equivalent of the condition∇ · u = 0 for the convection.

• For no sources Qρ, Qp = 0, no viscosity σ = 0 (scalar pressure) and gravitational accelerationfor the force term one obtains Euler’s equation:

∂u

∂t+ u · ∇u = −1

ρ∇p (+ g)

• With a kinematic viscosity included the momentum equation is known as the Navier-Stokesequation:

ρ

(∂u

∂t+ u · ∇u

)= −∇p+ η∆u +

(ζ +

η

3

)∇(∇ · u) (+ ρg + ..)


• Neglecting pressure and external force terms and assuming a simplified viscosity one obtainsBurger’s equation:

∂u

∂t+ u · ∇u− ν∆u = 0

• Diffusion and heat conduction: A diffusion equation can be obtained from the continuity equa-tion and a redefinition of the bulk velocity in the presence of several particle species. However,the more straightforward equation for diffusion is obtained for heat conduction (i.e., diffusionof temperature). Equation (2.25) can be re-written in different forms. Defining p

γ−1 = ρεyields an equation for the internal energy ε of a gas. More commonly used is the ideal gas lawp = nkT to re-write the energy equation into an equation for temperature. Assuming scalarpressure, constant density, and a heat flux driven by a temperature gradient L = −κ∇T onobtains

∂T

∂t= −µ∆T

• Steady state equation are generated from the above sets by assuming ∂/∂t = 0. For steadycondition the velocity is often determined from a potential which can be scalar if the flowis assumed irrotational (u = ∇Φ) or incompressible flow is modeled sometimes by a vectorpotential (u = ∇×V).

Exercise: Using the continuity equation and the stated assumptions derive Euler’s equation.

Exercise: Derive the equation for heat conduction with the stated assumptions.

Exercise: Derive the heat conduction equation for nonzero velocity u.

Exercise: Derive the continuity equation and momentum equation for irrotational flow.

Exercise: Assume a scalar pressure, L = 0, and QE = 0 in the pressure equation (2.25). Considera function g = paρb and determine a and b such that the resulting equation for g assumes aconservative form, i.e., ∂g/∂t +∇ · gu = 0.

Exercise: Assume a scalar pressure, L = 0, and QE = 0 in the pressure equation (2.25). Consider afunction h = paρb and determine a and b such that the resulting equation for h assumes a totalderivative, i.e., ∂h/∂t +u · ∇h = 0. For γ = 5/3 this becomes the equation for an entropyfunction because entropy is conserved for adiabatic changes.

2.4 Plasma (Two-) Fluid Equations

In the absence of ionization and energy exchange collisions, considering a simple two component(ion and electron) plasma, and assuming a quasi-neutral plasma (implying ne = ni) one obtains theso-called two fluid equations. With quasi-neutrality the continuity equation is

∂n(x, t)

∂t+∇ · (nus) = 0. (2.26)


Momentum and energy equations are unchanged except for neglecting the collision terms.

∂ρsus∂t

= −∇ · (ρsusus)−∇ · Πs + qsn(E + us ×B) + Qps (2.27)

1

γ − 1

∂ps∂t

= − 1

γ − 1∇ · psus − (Πs · ∇) · us −∇ · Ls +QE

s (2.28)

where Qps describes friction between the two components and effectively describes an electric resis-

tance. Effectively one can express the friction terms as Qps =

∑t6=s νstmsn(vt − vs) where νst is the

collision frequency for particles of species s to collide with particles of species t. Momentum andenergy conservation (for a closed system) further imply

∑s

Qps = 0∑

s

QEs = −

∑s

us ·Qps

Specifically for a neutral plasma consisting of electrons and one ion species these relations imply

νeiνie

=mi

me

QEe +QE

i = (ui − ue)Qpe =

1

enj ·Qp

e

Often it is also assumed that the pressure is scalar or gyrotropic, i.e., is different parallel and perpen-dicular to the magnetic field. Also the heat conduction term is often neglected. Further use of theseequations will be made in sections 3 and 4.

2.5 Single Fluid or MHD Equations

While considerably much simpler the two fluid equation contain still considerable complexity whichis not needed for many plasma systems. Thus it is desirable to formulate a more appropriate set ofequations which is applicable for large scale systems. This set of equations are the so-called MHDequations. With qi = −qe = e the total current density is

j = en(ui − ue)

We can also define the total mass density as ρ = n(mi +me), effective mass M = mi +me, and bulkvelocity or total mass density flux as

ρu = n(miui +meue)

Exercise: Show that these definitions yield

∇ · j = 0∂ρ

∂t+∇ · ρu = 0 (2.29)


With the above definitions we can now uniquely express ui and ue in terms of u and j the goal beingto derive equations which substitute the two-fluid equations for momentum (2.27) and energy (2.28)density. It is also assumed that the pressure is scalar for both the electron and the ion componentseven though this is not necessary. By taking the sum of the momentum equations and substituting uiand ue one obtains:

∂ρu

∂t+∇ · (ρuu) +

memi

e2∇ · (1

ρjj) = −∇p+ j×B (2.30)

with p = pi + pe. Note that the total momentum has to be conserved such that Qpi + Qp

e = 0. Asecond equation is required for uniqueness (there are two momentum equations for the two fluids).This is obtained by multiplying the ion equation with qi/mi and the electron equation with qe/me andthe sum of the modified equations:

E + u×B =me

e2n

[∂j

∂t+∇ · (uj + ju)

]− 1

en∇pe +

1

enj×B + ηj (2.31)

with the resistivity η = meνc/ne2 where νc is the collision frequency between electrons and ions (or

neutrals). This equation is usually termed generalized Ohm’s law. In the above equation the first termon the rhs is often called the inertia term because it represent the electron inertia in this equation. Thesecond term is the electron pressure force and the third term is the Hall term.

Note that thus far there has been no approximation in our derivation (except for the pressure isotropywhich is not really required) such that the above form of Ohm’s law is fully equivalent to the two-fluidequations.

Finally one can take the sum of the electron and ion pressure equations and keep the electron pressureequation unmodified to obtain

1

γ − 1

(∂

∂tp+∇ · pu

)= −p∇ · u−∇ · L + ηj2 + ... (2.32)

1

γ − 1

(∂

∂tpe +∇ · peue

)= −pe∇ · ue −∇ · Le + ηj2

Note that the ohmic heating term is present both in the sum of the pressure equations as well as in theelectron equation. Here it is assumed that the frictional heating occurs almost entirely for the electronsbecause of their small mass compared to ions. For the MHD equations we will now neglect the termsassociated with the electron pressure, and the first, second, and third term on the rhs of generalizedOhm’s law. Summarizing the equations and complementing them with the non-relativistic Maxwellequations one obtains the following set of equations termed the resistive MHD equations:

∂ρ

∂t+∇ · ρu = 0 (2.33)

∂ρu

∂t+∇ · (ρuu) = −∇p+ j×B (2.34)

E + u×B = ηj (2.35)1

γ − 1

(∂

∂tp+∇ · pu

)= −p∇ · u + ηj2 (2.36)

∇×B = µ0j (2.37)

∇× E +∂B

∂t= 0 (2.38)


In Maxwell’s equations the displacement current has been neglected by assuming that there are noelectromagnetic waves propagating at the speed of light. Divergence B is satisfied through the initialcondition (show this) and quasi-neutrality satisfies Coulomb’s equation. The continuity and momen-tum equations are exact (except for the pressure isotropy) and the main assumptions (simplifications)went into Ohm’s law and to some degree into the total pressure equation (and the elimination of theelectron pressure equation). These assumptions can be better understood by examining Ohm’s law(2.31) through a normalization or dimensional analysis. The ideal MHD equations are give by theabove set with η = 0.

The normalization is applied by measuring all quantities in typical units, e.g., the magnetic inductionB in units of a typical magnetic field B0 such that B = B0B where B is now of order unity, densityin units of a typical density ρ0 = m0n0, Length in units of a typical length scales L0, etc. Thusvelocities should be measured in units of the Alfvén speed u0 = B0/

√µ0ρ0 and time in units of

Alfvén travel time t0 = L0/u0. Now we can examine the coefficients of the different terms in Ohm’slaw. Applying this scaling yields coefficients c2/ω2

peL20 for the first term on the rhs and c/ωpiL0. The

terms c/ωpe = (ε0mec2/n0e

2)1/2 and c/ωpi = (mi/me)

1/2c/ωpe are called electron and ion inertiascales (or electron and ion skin depth because of the extinction length of waves in a medium). Thus itis justified to neglect these terms if the gradients in the system are on a much larger length scale thenthese inertia scales. Using this normalization general Ohms law becomes

E + u× B =

(c

ωpeL0

)2 [∂ j

∂t+∇ ·

(uj + ju

)]− c

2ωpiL0

∇pe +c

ωpiL0

j× B + ηj (2.39)

with η =ηt0µ0L2

0

=η

µ0u0L0

=1

R(2.40)

where a dimensional analysis of magnetic diffusion (section 7.2) yields R as the so-called Lundquistof magnetic Reynoldsnumber. Note that the factor of 2 in the electron pressure gradient term arisesfrom a normalization of thermal pressure with p0 = B2

0/ (2µ0). In this normalization the plasmaβ = 2µ0p/B

2 (ratio of thermal to magnetic pressure or energy density) becomes simply β = p/B2.Typical magnetospheric values of c/ωpi are a few 100 km and of c/ωpe about 10 km demonstratingthat these coefficients are small if L0 ∼ 1RE or larger. This justifies to neglect the first 3 terms on theright of general Ohm’s law for many applications. Particularly the first term involving the derivative ofthe current (electron inertial term) is small for almost all applications unless structure evolves on thetiny scale of about 10 km. The resistive term is 0 based on Coulomb collisions. However, resistivitycan be caused by plasma turbulence in strong electric current or strong density or pressure gradients.In such cases resistivity can be important at thin boundaries (see also section 7).

Note, that equation (2.40) demonstrates that the effective resistivity η (or inverse Lundquist number)in systems with fixed collision frequency and fixed typical velocity is proportional to the inverselength scale L0.

As a second application of the normalization, consider a plasma interacting with a neutral background.In this case the normalized momentum equation is

∂ρu

∂t+ ∇ · (ρuu) = −1

2∇p+ j× B− t0νinρ (u− un)

Thus the effective plasma (ion) neutral collision frequency is measured in units of 1/t0 or νin =t0νin = L0νin/u0 = L0νin

√µ0ρ0/B0. Considering a fixed magnetic field and density (and therefore


density), we can identify the importance of the length scale. For L0 ∼ u0/νin, the coefficient forthe friction term is of order unity. For the case L0 � u0/νin the momentum equation is stronglydominated by the plasma-neutral friction. This is for instance used in the ionosphere where Ohm’slaw is expressed as j = σ · E with the conductivity σ, and is derived from the electron and ionmomentum equations by neglecting the inertial and pressure gradient terms. Note, however, thatthis is justified only on sufficiently large scales, and that small scale and high frequency phenomenarequire to consider the complete form of Ohm’s law (2.31).

Exercise: Assume a plasma density of 1 cm−3, temperature equivalent to 1 keV, and a magnetic fieldof 20 nT which are typical for the near Earth magnetotail. Determine electron and ion inertiascales. Assume that quasi-neutrality is violated in a sphere with the radius of the electroninertia length by 1 % (e.g. 1% of the ion charge is not compensated by electrons. If outsidewere a vacuum what is the electric field outside the sphere? What velocity perpendicular to themagnetic field is required by Ohm’s law to generate an electric field magnitude equal to that onthe surface of the sphere?

Exercise: For the plasma in the prior exercise, determine the temperature in degrees Kelvin. Deter-mine the energy density in kW hours/m3 and kW hours /R3

e (1 RE = 6370 km). For the sake ofsimplicity assume that the plasma sheet is represented by a cylinder with 10 RE radius and 100RE length. How long could a power plant with an output of 1000 MW operate on the energystored in the plasma?

2.6 Properties of the Two-Fluid and MHD equations:

The MHD equations are a very commonly used plasma approximation. They conserve mass, momen-tum, and energy. As mentioned above they are valid on scales larger than the ion inertia scale. It isimportant to note that the ideal MHD equation do not have any intrinsic physical length scale. Thisimplies for instance a self-similarity in the sense that the dynamics on small physical length scalesis exactly the same as for large scale systems with the only difference that the larger system evolveslower. This can be illustrated by normalizing the equations to a particular length L0 which impliesthat the typical time scale is τ0 = L0/uA (with uA = B0/

√µ0ρ0). For a system which is identical

except that is is 10 times larger the length scale is 10L0 and the time scale is 10τ0. Thus a simplere-normalization yields exactly the same dynamics.

B

u

C(t0)

C(t0+dt)

Figure 2.4: Illustration of the frozen-in condition.

Ohm’s law in the ideal MHD equations E + u×B = 0 implies that the magnetic flux is frozen intothe plasma motion. This can be seen from the following arguments. The magnetic flux through the


surface C is the surface integralΦC =

∫CB · ds

with dsC being the surface element of the contour C. The contour elements move with the fluidvelocity u. The change of the magnetic flux from time t0 do t0 + dt is

ΦC(t0 + dt)− ΦC(t0) =∫ C(t0+dt)

C(t0)B · ds + dt

∫C(t0)

∂B

∂t· ds

= dt∮∂C(t0)

B · (uC × dl)− dt∫C(t0)

(∇× E) · ds

= dt∮∂C(t0)

(B× uC) · dl− dt∮∂C(t0)

(−u×B) · dl

= dt∮∂C(t0)

[(u− uC)×B] · dl

where the first term on the rhs represents the contribution from the change of the shape of C and thesecond term the contribution from the change of B. It follows that

dΦC

dt= 0

if the surface is moving with the fluid uC = u. In other words the amount of magnetic flux throughany given cross sectional area of the MHD fluid does not change in time if this area which moves withthe fluid. The frozen-in condition can also be understood in the following way. Two fluid elementsare always connected by a magnetic field line if they were connected at one time by a field line (aline defined by the direction of the magnetic field at any moment in time). In other words a field linecan be identified by fluid plasma elements. This property is sometimes called line conservation. Thisrequires that ideal Ohm’s law applies, i.e., electrical resistivity is zero.

t0

t1

f1

f2f1

f2

Figure 2.5: Illustration of line conservation

A more complete form of Ohm’s law should be considered if gradients on smaller scales exist in aplasma. Since the ion inertia scale is by a factor of

√mi/me larger than electron inertia effects the

first terms to consider are the Hall term and the electron pressure term (note that the electron pressureis typically an order of magnitude smaller than the ion pressure such that contributions from this termare small. It is interesting to note that

ue = u− mi

eρj (2.41)

Comparing this with generalized Ohm’s law we can re-write this neglecting the electron inertia scaleas

E + ue ×B = −Meρ∇pe (2.42)


In other words the addition of the Hall term transforms Ohm’s law from E+u×B = 0 in MHD intoE + ue ×B = 0. With this form it is clear that the frozen-in condition for the magnetic flux appliesnow to the electron fluid (note that one can also include a scalar electron pressure term in Ohm’s lawif the density does not vary strongly).

There will be various applications using the fluid and the kinetic equations. Typical applications con-sider waves, discontinuities and shocks, instabilities, steady state solutions, and equilibrium solutions.Particularly for the last topic it is important to note the following terminology.

Steady state assumes time stationary solutions with nonzero velocity, ∂/∂t = 0 and u 6= 0.

Equilibrium solutions assume ∂/∂t = 0 and u = 0. Note that for kinetic systems the velocity inphase space is always nonzero for physical systems. Also the electron velocity is nonzero incurrent regions.

Electrostatic solutions assume ∂B/∂t = 0. This implies ∇ × E = 0 or E = −∇φ. In this caseOhm’s law must be replaced by the Poisson equation (∇ · E = 0).

2.7 Tables of Plasma Parameters

The first table gives some fundamental constants:

Property Symbol SI cgsSpeed of light c 3×108 m s−1 3×1010 m s−1

Boltzmann constant k 1.38×10−23 J K−1 1.38× 10−16 erg K−1

Electron mass me 9.1 ×10−31 kg 9.1 ×10−28 gProton mass mi 1836 me 1836 me

Elementary charge e 1.6×10−19C 4.8×10−10 statcoulombDielectric constant ε0 8.85×10−12 F m−1 -

Permeability of free space µ0 4π×10−7 H m−1 -

Other relations:

ε0µ0 =1

c2

1eV = 1.16× 104K

The following table presents several basic plasma frequencies, length scales and velocities:


Property Symbol SI cgs

Plasma frequency ωpe(nee2

ε0me

)1/2 (4πnee2

me

)1/2Electron gyro frequency ωge

eBme

eBmec

Coulomb collision frequency νeiωpe

4πnλ3Dln Λ ωpe

4πnλ3Dln Λ

Debye length λD(ε0kTne2

)1/2 (kT

4πne2

)1/2Skin depth (electron inertia) λe

cωpe

cωpe

Electron Gyroradius rgevtheωge

vtheωge

Electron Thermal velocity vthe(kTme

)1/2 (kTme

)1/2Alfvén speed vA

B

(µ0nmi)1/2

B

(4πnmi)1/2

Number of particles in a Debye sphere ND = 1g

4π3nλ3D

4π3nλ3D

The corresponding ion properties are

ωpi =√me/miωpe

ωgi = (me/mi)ωge

νie = (me/mi)νei

rgi =√mi/merge

vthi =√mi/mevthe

The following list provides numerical values for the various plasma parameters in a convenient form.Because it is more common in the field of magnetospheric physics everything is measured in cgs unitsin this table, i.e., n in cm−3, B in Gauss, T in eV, and ln Λ ≈ 20 (ions are protons). Note that 1 T =104 Gauss and 1 nT =10−5 Gauss.

ωpe = 5.64× 104 n1/2 [rad/sec]

ωge = 1.76× 107B [rad/sec]

ωgi = 9.58× 103B [rad/sec]

νei = 1.1× 10−5 ln(Λ)nT−3/2e

[sec−1

]λDe = 7.43× 102 n−1/2T 1/2

e [cm]

λe = 5.31× 105 n−1/2 [cm]

λi = 2.28× 107 n−1/2 [cm]

rge = 2.38× 100 T 1/2e B−1 [cm]

rgi = 1.02× 102 T1/2i B−1 [cm]

vthe = 4.19× 107 T 1/2e [cm/sec]

vthi = 9.79× 105 T1/2i [cm/sec]

vA = 2.18× 1011Bn−1/2 [cm/sec]

ND = 1.72× 109T 3/2e n−1/2

The last table is an overview of typical plasma properties in the magnetosheath/mantle, the outermagnetosphere/tail, and the inner magnetosphere(acceleration) regions.


Mantle/Sheath Outer M’Sphere/Tail Inner M’Sphere/Accel. Rn [cm−3] 10 [1-100] 1 [0.01-5] 102 [0.1-104]B [nT] 20 [5-100] 40 [10-100] 104 [103 -105]Te [eV] 50 [10-103] 500 [102 − 103] 103 [1-104]Ti [eV] 100 [10-103] 2×103 [103 − 104] 103 [1-104]ωpe 1.8× 105 5.7× 104 5.6× 105

ωge 3.5× 103 7.0× 103 1.8× 106

ωgi 1.9× 100 3.6× 100 9.6× 102

νei 6.2× 10−6 2.0× 10−8 7.0× 10−7

λDe [m] 1.7× 101 1.7× 102 2.3× 101

λe [m] 1.7× 103 5.3× 103 5.3× 102

λi [m] 7.3× 104 2.3× 105 2.3× 104

rge [m] 8.4× 102 1.3× 103 7.5× 100

rgi [m] 5.0× 104 1.1× 105 3.2× 102

vthe [m/sec] 3.0× 106 9.4× 106 1.3× 107

vthi [m/sec] 9.8× 104 4.4× 105 3.1× 105

vA [m/sec] 1.4× 105 8.7× 105 2.18× 107

ND 1.9× 1011 1.9× 1013 5.4× 1012

Chapter 2 Basic Plasma Properties - University of Alaska...

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