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5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system of equations, we resort to a simpler description by assuming collision dominance. For such gas the drifting Maxwellian is a good approximation for the velocity distribution function: The ionosphere can be treated as a weakly ionized plasma, meaning that the effects of Coulomb collisions are small compared to electron-neutral and ion-neutral collisions. The opposite is true for a fully ionized plasma (neutrals are still there). We limit our discussion to weakly ionized plasmas. 32 2 2 2 (,) (,) exp ; 3.44 2 2 m mc f t n t c kT kT r r v u
51

5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

Dec 22, 2015

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Page 1: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5. Simplified Transport Equations

We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system of equations, we resort to a simpler description by assuming collision dominance. For such gas the drifting Maxwellian is a good approximation for the velocity distribution function:

The ionosphere can be treated as a weakly ionized plasma, meaning that the effects of Coulomb collisions are small compared to electron-neutral and ion-neutral collisions. The opposite is true for a fully ionized plasma (neutrals are still there). We limit our discussion to weakly ionized plasmas.

3 2 2

22( , ) ( , ) exp ; 3.442 2

m mcf t n t c

kT kT

r r v u

Page 2: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (1)

To develop the general idea, we assume an isothermal gas with a density n(x) that has a constant gradient in the -x direction, see Fig. 5.1a. We assume that the mean free path length , and that the velocity distribution is given by a non-drifting Maxwellian.

We expect diffusion of particles from higher to lower density, i.e., from left to right in Fig. 5.1a. Let us calculate the net particle flux (# of particles per m2 and s) through the y-z plane at x. Appendix H.26 gives the thermal particle flux at x as [n(x) <c(x)>]/4 (HW#6: derive H.26). If the density n were constant, the net flux through the plane x = const would be zero because of the equal flux from the left and the right. The particles reaching the plane x had their last collision (in the average) in the plane x-x, where x ~ . The density at x-x is

The particles arriving from the right had their last collision at x+x, and

( ) ...dn

n x x n x xdx

( ) ...dn

n x x n x xdx

11 dn

n dx

Page 3: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (2)The net particle flux crossing the plane at x then becomes

1 1

4 4

5.24 2

Notice - is positive for the density in Fig.5.1a.

For , where is the average collision frequency,

fromleft fromright

collision

c n x x c n x x

c cdn dn dnn x x n x x x

dx dx dx

dn

dxc

x c

2

2

5.32

8Since (App.H.21)

4Fick's Law 5.4

c dn

dxkT

cm

kT dn dnD

m dx dx

Page 4: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (3)

where

41.3 is the diffusion constant. D is proportional to the temperature

T, and inversely proportional to the mass and the collision frquency.

The net flux vector is

for the six x

kT kTD

m m

n

nu

Γ

Γ u

e

2

2

2

2

tuation in Fig.5.1a. The continuity equation (3.57) is

n for zero production or loss.

t

n

t

nparabolic partial differentialeq. 5.6

t

xx x

n

nu nnu D

x dx x

nD

x

u

e

Page 5: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (4)

2

4

Solutions of the DE depend on the initial conditions, say the value n(x,0),

and two boundary conditions, say n ,t and n(- ,t). A solution is

, 5.74

This solution has a Gaussian peak at x

x

tNn x t e

Dt

=0, and goes to 0 for x . As

t , n 0. And for t 0, n 0 for all x (except for x=0 where

the function is not defined for t=0). See Fig.5.2.

Page 6: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (5)Viscosity

The viscosity of a gas determines the transport of momentum perpendicular to the flow

direction. Such transport occurs when there is a velocity gradient like the one in

Fig. 5.1b). In general, the stress tensor is

where is the transfer of x-momentum in the x direction, the transfer of

y-momentum in the x direction, etc. For the conditions in Fig. 5.

xx xy xz

yx yy yz

zx zy zz

xx xy

τ

τ

2yx

1b, only is non-zero.

The viscous stress is the net transfer in the y-direction of x-momentum per m and s

across the plane at y:

1n x c 5.8

4

1n x c

4

yx

yx x x

x xx x

m u y y u y y

u um u y y u y y

y y

2

2

1n x c where /

2

1n x c

2

1 8where c 5.11

2 2

1.3 viscositycoefficient 5.12

x

x

xyx

um y y c

y

um

y

u n kTnm

y

nkT

Page 7: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (6) Example of Viscosity Effect (1)

A gas flowing in the x direction between two infinite plates at y = 0 and y = a. Assume the plate at y = 0 is fixed and the plate at y = a moves in the x direction with velocity Vo. The particles close to y = a will tend to move with the plate in the x direction, while the particles at y = 0 will be at rest. What is the velocity at y? To find the answer we must solve the momentum equation (equation of motion, or force equation) (3.58):

.

For a uniform isothermal gas, = 0. Assuming is the dominant force term,

the steady state i.e. 0 solution is obtained from

0.

The divergence of a tenso

Dnm p ne nm

Dt tp

D

Dt

u Mτ E u B G

τ

u

τ

r (L.26) is

.yx xy yy zy yzxx zx xz zzx y zx y z x y z x y z

τ e e e

Page 8: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (7)Example of Viscosity Effect (2)

For our example, y , and only is nonzero. Therefore

0

0 0 where I used (

x x yx

yx xy yy zy yzxx zx xz zzx y

yxx

yx x

u

x y z x y z x y z

y

u

y y y

u e

τ e e

e

2

2

0

0

x x 0

5.11).

Assuming constant, we can write

0

With the boundary conditions at y = 0 an y = a

0 =0, a = V we get

5.16

The velocity increases linearly from u = 0 to u = V . Thi

xx

x x

x

nkT

uu y Ay B

y

u u

yu y V

a

s means that

viscosity acts to smooth velocity gradients.

Page 9: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.1 Basic Transport Properties (8)Energy Flow

2

2 3 23 2 3 2

3 2 3 21 2 1 2

3 2 21 2

Thermal energy is / 2 3 2. The energy flux is

3 8 3

2 4 2 4 2 2

3 3 92

22 2 2 2

8

89 9

2 2

x x

x x

m c kT

m c n c kT n kT k nq T T x x T x x

m m

k n T k n TT x T x

x xm m

kTc mx

kTk n T k nmq T T

xm

2

Here is used for the thermal conductivity:

9=

TT

m x

Tq T

x

k nT

m

Page 10: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.3 Transport in a weakly ionized plasma (1)

Weakly ionized means that Coulomb collisions are of negligible importance in comparison with collisions with neutral particles. We start again with the momentum equation:

' '' ' ' '

' ' '

where the momentum transfer collision integral is given in 4.129b :

.

If we neglect the heat flo

j jj j j j j j j j j

j jj jj jj j jj j j jj j j

j jj j

Dn m p n e n m

Dt t

zn m

t kT

u Mτ E u B G

Mu u q q

w, the momentum equation for the j particles becomes

5.23

where j stands for e (electrons) or i (ions), and n for neutrals.

jj j j j j j j j j j j jn j n

Dn m p n e n m n m

Dt

uτ E u B G u u

Page 11: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.3 Transport in a weakly ionized plasma (2)Scale Analysis

We start with the so-called diffusion approximation, for which the inertia terms can be neglected. To estimate the importance of the different terms look at the ratios.

2 2 2

2

j

term 2/term 3:

/ /5.24

/ / /

Here u is the average (drift) speed, / thermal speed, and the

Mach number, i.e., the ratio between drift speed and thermal speed.

j

j

j j j j j j j

j j j j j

j j

n m u L n m u L uM

p L n kT L kT m

kT m M

j

This means

the 2nd term in 5.24 can be neglected if M 1, i.e., flow.

Similarly term1/term 3:

/ ' / ' / '' 5.25/

jj j j j

jjj j j j j

j j j j

Lun m u u L L

MkTn kT L kT kT kTm m m m

subsonic

Page 12: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.3 Transport in a weakly ionized plasma (3)

This means the first term in (5.23) can be neglected if Mj << 1, and ’, the time constant for the plasma processes, is long. Neglecting the time derivative eliminates the description of plasma waves.

Summary:

The diffusion approximation is valid for a slowly varying subsonic flow.

Page 13: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.3 Transport in a weakly ionized plasma (4)

Consider a special case: no neutral wind (un = 0) and a dominant electric force due to an external field E0.

0

0

j

0

0 0

5.26

For an isothermal plasma (T = const):

, or

5.27

where

is the diffusion constant as de

j j j j j jn j

j j j j j jn j

j j j j j jn j

j j jj j j

j jn j jn

jj

j jn

p n e n m

n kT n e m

kT n n e m

kT n en D n

m m

kTD

m

E u

E Γ

E Γ

Γ E E

0

fined before, and

is called the mobility coefficient.

The +sign in 5.27 is valid for j = i, and the -sign for j = e. In the absence of the

electric field, E 0, (5.27) reduces to Fick's law

j jj

j jn

n e

m

derived from simple mean-free-

path arguments

5.30j jD n Γ

Page 14: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.3 Transport in a weakly ionized plasma (5)

To analyze the transport in a weakly ionized plasma (Coulomb collisions neglected) we start with the momentum and energy equations

3.58

3 5: 3.59

2 2

With reasonable assumptions valid at least for the height regime 60-160 km

Schunk has shown that one can simplify:

e

s s ss s s s s s s s s

s ss s s s s

Dn m p n e n m

Dt tD E

p pDt t

u Mτ E u B G

q τ u

i

i

2

( ) 5.31m

0 3 ( 5.32

5.31 states that the Lorence force is equal to the friction between ions and neutrals.

It is assumed that the external (superimposed) electric field is

i in i n

n i i i nk T T m

E u B u u

u u

E perpendicular

to the external field, = .B B b E E

Page 15: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.3 Transport in a weakly ionized plasma (7)

' '2 2 2 2

' i

2

5.35

e, and = is the ion cyclotron frequency (rotations per s).

. 5.363

The relative ion-neutral drift has t

i in cii n

i in ci in ci

n cii

ii n i n i n

Then

e

m

with

B

m

mT T T T

k

u u E E b

E E u B

u u

' '

wo components: the Pederson component, which

is parallel to , and the Hall component which is perpendicular to (and to ).

There is no component parallel to since we had assumed that 0.

The i

E E B

B E

' '2 2 2 2

2' '

2 2 2 2

22'

2 2

on drift is

, and the ion current becomes

5.110

in

i in cii n

i in ci in ci

i i in cii i i n

i in ci in ci

i i ci ini i i n

i in in ci

e

m

n en e

m

n en e

m

u u E E b

J u E E b

J u E

'2 2

2' '

2 2 2 2

2

5.111

with the ion conductivity. 5.112

in

in ci

ci ini i i n i

in ci in ci

i ii

i in

n e

n e

m

E b

J u E E b

Page 16: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.5 Major Ion Diffusion (1)+Consider a plasma with one major ion, like O in the F region of the ionosphere,

electrons, and one species of neutrals. Let us assume the plasma is stationary, i.e.,

0; the flow is subsonic, i.es t u

e i

// s

., 0; the plasma is neutral, i.e., n =n .

We neglect Coriolis and centrifugul acelleration and the heat flow terms. The momentum

equation 5.50 becomes

p 5.

s s

s s s s s s s s st t ss

n e n m n m

u u

τ E u B G u u

// i // //// // //

// e // //// // //

50 '

Ion and electron momentum equations along the magnetic field:

p 5.51

p 5.52

Note ,

i i i i i i i ie e i i i in n i

e e e e e e e ei i e e e en n e

i e

n e n m n m n m

n e n m n m n m

e e e

τ E G u u u u

τ E G u u u u

// i e //// //

, also from eq. 4.158 , no proof. Add

the two equations:

p p 5.53

i i ie e e ei

i e i i e i i in e en n i

e n m n m

n m m n m m

τ τ G u u

Page 17: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.5 Major Ion Diffusion (2)

e e

// i e // s s s// //

// i // i // //// //

// //

Neglect terms with m , including which is proportional to m . Then:

p p , or with p =n kT :

n n

e

i i i i i in n i

ii ei n

i i in i i in in i i in

i n

n m n m

kT kT

n m n m n m

τ

τ G u u

τGu u

u u

//// i // //

//// i // //// //

//// i// //

n

We set 2 , then:

2 2 2n. With

n

ii e i e i e

i in i i in i e in i i in

i e p

ip p p pi n a

i in i i in p in i i in i ine

pi n a

i pe

k T T k T T T T

m n m T T n m

T T T

kT kT T kTD

m n m T n m m

T mD

n T

τG

τGu u

u u // // ambipolar diffusion 5.54

2 2ii

p i pkT n kT

τG

Page 18: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.5 Major Ion Diffusion (3)

// //

If we call r the coordinate along , then 5.54 becomes:

1 1 1

2 2

2Remember , i.e., it becomes large with increasing altitude where 0.

We therefo

pi i i n i

i p p i p a

pa in

i in

Tn m g u u

n r kT T r n kT r D

kTD

m

B

//

re can neglect the last term.

If the magnetic field force is negligible, we can use this equation in any arbitrary

direction, for example the vertical direction. We recall that , i.e.,

g = -g. NergG e

//

//

0p ,

glecting the stress term, we get the classical diffusive equilibrium eq.:

1 1

2

21 1 1, Plasma scale height

If T constant: exp

pi i

i p p

p pip

i p p i

i i op

Tn m g

n r kT T r

T kTnH

n r H T r m g

r rn r n

H

Page 19: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.11 Electric Currents and Conductivities (1)

e

2' '

2 2 2 2

2

Similarly the elctron current becomes with e , and

5.113

5.115

The total perpendicular current, , is therefo

en

cee

ce ene e n e

en ce en ce

ee

e en

i e

eBe

m

n e

n e

m

J u E E b

J J J

i

2 2' '

2 2 2 2 2 2 2 2

' '

2

2 2

re (for e )

5.116

in en

in

ci in ce eni e n i e i e

in ci en ce in ci en ce

i e n p H

i e n p n H n

p iin ci

e

n n e

n n e

n n e

J u E b E

J u E b E

J u E u B b E u B

2

2 2 2 2 2 2

'

Pederson and Hall conductivity.

I have resubstituted .

en ci in ce ene H i e

en ce in ci en ce

n

E E u B

Page 20: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.11 Electric Currents and Conductivities (2)

2

2 2 2 2

19 56 1

31

For heights above ~100 km

5.120

1.6 10 5 10since . 9 10

9.1 10

1.42

Notice from 5.120 that the electrons contribute

in

en

ci in enp i H i e

in ci in ci ce

ce cee

cece

eBs

m

f MHz

e

only to the Hall current, not to the Pederson current.

Currents along :

To calculate use 5.23 for the electrons which are much more mobile than the ions and neutrals:

ee e e e e e e

Dn m p n e

Dt

B

u

uτ E u

e

5.23

Neglecting the stress term and the terms containing m on the left side, we get

where I used the fact that 0, since is a vector

e e e e en e n

e e e e e en e n

j j

n m n m

p n e n m

B G u u

E u u

u B u B

e

e

to .

When an E exists, u . Therefore

Since and e

n

e e e e e en e

e e e

e e e e e e e en e

u

p n e n m

p n kT e

n k T kT n n e n m

Β

E u

E u

Page 21: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

5.11 Electric Currents and Conductivities (3)

Since the electron mobility is much larger than the ion mobility, the current is carried by the electrons. The field-aligned current is therefore

2 2

2

Setting

parallel electric conductivity

current flow conductivity due to thermal

e e e e e ee e e e e e

e en e en e en e en e e en

ee

e en

ee

e en

n ek kT e n e n e kT n ekn e T n n T

m m m m n e m

n e

m

n ek

m

J u E E

n

2 2

2 2 2 2 2 2 2 2

2

gradients

5.124

Recall:

(for u 0) 5.116

;

Notice 5.124 becomes Ohm's law

in en

ee e e e

e

p H

ci in ce enp i e H i e

in ci en ce in ci en ce

ii

i in

kTn T

n e

n e

m

J E

J E b E

when the term dominateseJ E E

Page 22: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6. Wave Phenomena6.1 General Wave Properties(1)

Following Schunk’s notation, we use index 1 to indicate the electric and magnetic wave fields, E1 and B1, and the plasma variations, 1c, caused by the waves.The direction of the propagating wave is given by the propagation constant K.

To find the plasma waves we must solve Maxwell’s differential equations in the plasma environment.

121 1 0 1 s 1 0

s

1

11

711 0 1 0 0 0

1 s 1 1s

1 , e ;charge density disturbance, 8.85 10 ,

2 0

3

4 , 4 10 , .

where e

c c s

s s

n x SI units permittivity

xt

x x SI units permeabilityt

n u

E

B

BE

EB J

J

Page 23: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.1 General Wave Properties(2)We solve Maxwell’s equations by taking the curl of (3):

1 1

21 1 1

21 1

1 0 0 0 2

22 1 1

1 1 0 0 02

c1 1

But

and using(4): - results in

6.6

This quation can be easily solved for vacuum where = 0, and = 0. In th

xt

x

t t t

t t

E B

E E E

J EB

E JE E

J

c11

0

22 1

1 0 0 2

0 0

22 1

1 2 2

1 10

is case

0 according to 1 , and

0

1or by setting

10 in vacuum. 6.7

Solution: , Re 6.9i t

t

c

c t

t e

K r

E

EE

EE

E r E

Page 24: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.1 General Wave Properties(3)

1 10 10

10

11

1

, Re cos

where for simplicity I assumed is real. Recall: cos sin .

From Mawell equation 3

we can get . Notice that when the exponential functions are used,

i t

ix

t e t

e x i x

xt

K rE r E E K r

E

BE

B

1 1 1 1

1 1

application of the operators

and simply means multiplication with i and -i , respectively. Therefore:

-i

This means that is to and to . With the help of Maxwell equa

t

i

K

K E B K E B

B K E

1 1 0

1

1 1

1 1 1 1

21 1 1 1 1

1 1

tion 1

it is easy to show that for 0 the electric field is to :

0 .

Further from

.

Summary: In vacuum , , are orthogonal to

c

c

i

KK

E

K

K E K E

K E B K K E K B

K K E E K B E k B

E B K

each other following the right-hand rule.

Page 25: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.1 General Wave Properties(4)Poynting Vector

1 1 0 1 1

1 1

The flow of energy carried by an electromagnetic wave in the direction is given by

the Poynting vector:

/ 6.13

Since are sinusoidal time varying functions, is a function of t. In

K

S E B E H

E H S

1 1

0

general we are not interested

in the fast in and out energy fluxes, but want to know the time-averaged flux:

1 2dt where is the wave period. It is easy to verify that when using the e

wT

ww

TT

S E H

1

*1

xponential

notation:

1Re time-avearged Poynting vector 6.14

2 S E H

Page 26: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.2 Plasma Dynamics(1)

The propagation of waves in a plasma is governed by Maxwell’s equations and the transport equations. We assume that the 5-moment simplified continuity, momentum, and energy equations (5.22a-c) can describe the plasma dynamics in the presence of waves. If we neglect gravity and collisions these equations become (Euler equations):

( ) 0 continuity eq. 6.21

[ ] [ ] 0 momentum eq. 6.22

0 energy eq. 6.23

1 16.21 0

Substitute in 6.23 :

ss s

ss s s s s s s s

s ss s

s s s ss s s s s s s

s s s

nn

t

n m p n et

D pp

Dtn n D n

n n nt n t n n Dt

u

uu u E u B

u

u u u u

0 6.25s s s s s

s

D p p D n

Dt n Dt

Page 27: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.2 Plasma Dynamics (2)

1

This implies that

0, since 6.26a

1 1

1 1

And 6.26a implies that

.

s s

s

s s s s s s s s s ss s

s s s s s

s s s s s s s ss s s

s s s s s

s

s

D p

Dt

D p D p p D D p p D

Dt Dt Dt Dt Dt

D p p D D p p Dn m n

Dt n m Dt Dt n Dt

pconst

1

6.26

Notice that this is the equation of state of a gas. The value for is =3/5 for adiabatic flow,

and =1 for isothermal flow (Can also be written as .)

From 6.26 :s

s s s

b

p V const

b

pp const

1 6.27s s s s ss s s s s

s s s s s

p n kT kT

n m m

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6.2 Plasma Dynamics (3)

Substitute in the momentum equation (6.22):

[ ] [ ] 0 6.28

The continuity equation was

( ) 0 6.21

We must solve these equations together with Maxwell's equations

ss s s s s s s s s s

ss s

n m kT n n et

nn

t

uu u E u B

u

s s

0 0 0 0

to find

n , , and (10 unknowns).

:

1. Solve for equilibrium conditions finding n , , , ( I dropped index s on n and u)

that satisfy the differential equations

u E B

Using Perturbation Technique

u B E

0 0

0 1

0 1

0 1

0 1

.

2. Perturb the equilibrium state of the plasma and assume that this will cause small changes in and .

, , 6.31

, , 6.31

, , 6.31

, , 6.31

n t n n t a

t t b

t t c

t t d

B E

r r

u r u u r

E r E E r

B r B B r

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6.2 Plasma Dynamics (4)

0 10 1 0 1

0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Substitute these perturbed functions into the continuity and momentum equations:

( ) 0

[ ] [ ] 0

Carry out the differencia

s s s s

n nn n

t

n n m kT n n n n et

u u

u uu u u u E E u u B B

1 10 1 1 1 0 1 0 1 0 1

tions remembering that all 0-index terms are constants:

0 6.33

where only first order (linear) terms in 1-index functions were kept. The momentum equation be

n nn n n n n

t t

u u u u u

10 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0

s

0 0 0 0

10 0 1

comes

0.

where e = e for ions/electrons. But

0 (equilibrium condition). Therefore

s s s s s s s s s

s

n m n m kT n n e n e n e n e n e n e n et

n e

n mt

uu u E E E u B u B u B u B

E u B

uu u

1 0 1 1 0 0 1 1 0 0 0

10 0 1 1 0 1 1 0 0 1

0.

But the last 0, therefore

0 6.35

s s s s

s s s

kT n n e n e

n m kT n n et

E u B u B E u B

uu u E u B u B

Page 30: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.2 Plasma Dynamics (5)

( )1 1 1 1

1 0 1 0 1

0 1 0 1

0 1 0 1 1 0 1 1 0 0 1

0 1

We try plane wave solutions for all functions

, , , .

Remember , .t

0

6.37

0

i t

s s s

s

n e

i i

i n n i i n

n n

n m i i kT i n n e

kTi i

k ru E B

K

K u u K

u K K u

u u K u K E u B u B

u K u K

11 1 0 0 1

0

1 1 1

0 6.38

6.37 and 6.38 are 4 algebraic equations that must be satisfied for 6.36 to be solutions.

We also must make use of Maxwell's equations to solve for the unknowns

n , , ,

s sn e

n m m E u B u B

u E B

1 1 1 1 1

22 1 1

1 1 0 0 02

2 2

1 1 0 0 1 0 1 0 0 2

22

1 1 0 12

; n , can have different values for the different species, n , .

From slide 6.1(2):

6.6

1;

6.20

These

s s

t t

i i i i ic

K ic

u u

E JE E

K E K K E E J

E K K E J

1 1 0 1 s 1 1 s 1s 0

1 1 1

11 1 1

are 3 more algebraic equations. from 6.1 1 :

11 , e e . One more algebraic eq.

2 0 0. Only tells that .

3 . Three more algebraic eqs.

c c s ss

n i n

xt

E K E

B K B B K

BE K E B

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Electrostatic Waves: B1= 06.3 Electron Plasma Waves (1)

1

i1

We start the discussion with high frequency electron plasma waves assuming that B 0.

The wave frequency is high enough so that the ions cannot follow the motion, i.e., 0.

To simplify the discussion

u

i0 e0 0

1 0 1

11 1

0

we also assume 0, and 0, then the algebraic

electron transport equations 6.37 and 6.38 become with - :

6.39

0 6.39

From Gauss's law (see slide 6

o

s

e e e

e e ee

e e e

e e

n n a

kT n ei i b

n m m

u u E B

K u

u K E

1 1 0

1 1

11

.2(5)):

i / 6.39

Our immediate goal is to find the dispersion relation that relates K and .

Muliply 6.39 with and use 6.39 and 6.39 to substitute for and :

e

e

e e ee

en c

b a b

kT ni i

K E

K K u K E

K u K K

10

21 11 0

0 0

22 2 0

10

0 6.40

/ 0

0 6.41

e e e

e e e ee

e e e e

s s ee

e e

e

n m m

i n kT n eiK en

n n m im

kT e nn K

m m

K E

Page 32: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.3 Electron Plasma Waves (2) (B1=0)

22 2 0

0

22 20

0

2 2 2 2e

20

0

This gives the dispersion relation

0, or

, or

; usually is set equal to 3. 6.42

plasma frequency; electron thermal s

e e e

e e

e e e

e e

p e e

e ep e

e e

kT e nK

m m

e n kTK

m m

V K

e n kTV

m m

2 2

peed.

The dispersion relation 6.42 relates with the wavelength (=2 /K). Notice there is no

propagating wave in a cold plasma where 0. In the cold plasma

plasma oscillation 6.45

e

p

T

Page 33: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.4 Ion-Acoustic Waves (1) (B1=0)

1 0 1

11 1

0

We now consider the low frequency waves for which the ion motion must be included. The

ion transport equations are, similar to 6.39 , with :

6.46

0 6.46

In G

s

i i i

i i ii

i i i

e e

n n a

kT n ei i b

n m m

K u

u K E

1 1 1 0

2 2 1

0

auss's law (see slide 6.2(5)) we must include positive and negative charges:

i ( ) / . 6.46

Looking back at slide 6.3 (1), equation (6.40) for the electron motion:i e

e e e

e e

e n n c

kT n ei K

m n m

K E

1

2 2 01 1

2 2 01 1

0 0

0 or 6.40

0 6.48

Similarly from the ion transport equations we get from 6.46b

0 6.49

Assuming = (neutral plasma), and adding 6.48 and 6.48 gives

e

ee e e e

ii i i i

e i

enm K kT n

i

enm K kT n

in n

K E

K E

K E

2 21 1 1 1

2 2 21 1

( ) ( ) 0. Since :

( ) 0 6.50

i e e i i e e e i ei

i i i e e e

mn m n K kTn kT n m m

m K kT n K kT n

Page 34: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.4 Ion-Acoustic Waves (2) (B1=0)

1 1

1 1 1 0

2 2 01 1

2 21 1 0

0

How do and relate? we can combine Gauss's law and the momentum equation for the electrons

given above as

i ( ) / . 6.46

0 6.48

1( ) /

i e

i e

ee e e e

i e e e ee

n n

e n n c

enm K kT n

i

e n n m K kTen

K E

K E

1

2

20

1 1 20

1 01 D2 2 2

0

2 2

6.51

If we neglect the electron inertia term again for low frequencies:

1 0

with , the Debye length 6.521

Substitute into 6.50

( )

e

e

e ei e

e

i ee

e D e

i i

n

m

K kTn n

e n

n kTn

K e n

m K kT n

21 1

2 2

2 2

22 2 D

0,

gives the dispersion relation for ion plasma waves:

6.531

2For very long waves = 1. We get the dispersion relation for ion acoustic wa

i e e e

i e e

i i e D

D

K kT n

kT kTK

m m K

K

2 2 2 2

ves, or ion sound waves:

ion acoustic waves 6.55

ion acoustic speed 6.56

i e es

i

i e es

i

kT kTK K V

m

kT kTV

m

Page 35: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.5 Upper Hybrid Oscillations

0 0

e1 0 1

1 1

Assume 0, and 0.Upper hybrid oscillations are directed .

We start again with the electron continuity 6.37 and momentum 6.38 equations:

n 657e e

e e

n a

ei

m

0B E high frequency oscillations B

K u

u E u

1 0

1 1 0

1 1 1

1 1

0 6.57

We also use two Maxwell equations:

i / Gauss's Law 6.57

i where 0 Faraday's Law 6.57

0 6.57

The dispersion relation for the upper hybrid oscillation becomes

e

b

en c

i d

e

B

K E

K E B B

K E K E

2 2 2

2 22 20 0

0

:

6.62

,

and K are not related, so there is no wave velocity defined. .

pe ce

ce pee e

eB n ewhere

m m

There is no wave

Page 36: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.6 Lower Hybrid Oscillations

0 0Assume again 0, and 0. Lower hybrid oscillations are directed .

We must use the electron and ion continuity 6.37 and momentum 6.38 equations together with

0B E lowfrequency electrostatic oscillations B

0 0

2

the two

Maxwell equations. This gives the algebraic equation:

6.68

where

,

are the electron-cyclotron and ion-cyclotron frequencies. Again, .

ce cie i

ce ci

eB eB

m m

no waves

Page 37: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.7 Ion-Cyclotron Waves

0 0

0

2 2 2

Assume again 0, and 0.The ion-cyclotron waves are

that propagate in a direction to . The algebraic dispersion relation becomes:

ci K

B E low frequency electrostatic waves

almost perpendicular B

2

2

6.74

where

s

e es

i

V

kTV

m

Page 38: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.8 Electromagnetic Waves in a Plasma (1)

Now we consider the case where E1 and B1are non-zero. We start with the general wave equation (6.20) assuming again a plane wave solution:

22

1 1 0 12

1 1

22

1 0 12

6.20

Let's look first for transverse waves, i.e., forsolutions for which 0 :

6.75

In a two-component plasma (electrons and one ion species) th

K ic

K ic

E K K E J

K E K E

E J

0 1

1

0 1

0 0

0 0 1 1 1

e current density is given by

6.76

Perturbation and linearization:

/ for ions/electrons. Charge neutrality:

i i e e

s so s io eo o

s s s

i i e e

i e

i e i

en en

n n n s i e n n n

en en

en

en en

J u u

J J J

u u u

J u u

J u u

J J u u u

0 1 0

1 0 1 1 1 0 1 0

, . .,

6.80

i e e

i e i i e e

en i e

en en en

u

J u u u u

Page 39: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.8 Electromagnetic Waves in a Plasma (2)

0 0 0 0

1 0 1 1

1

1 0 1

10 1 1 1 0 0 1

0

To simplify the notation, we assume 0.Then

High frequency approximation: 0

6.81

The electron momentum equation 6.38

i e i e

i e

i

e

s s s

T T

en

en

kT n ei i

n m m

E B u u

J u u

u

J u

u K u K E u B u B

1 1

20

1 1

2 22 0

1 0 12

1

2 22 0

1 02

2 2 2 2

0 becomes

0, therefore 6.82

Substitute into (6.75):

6.83

Since 0, this gives the dispersion relation

, or

6.84

ee

e

e

e

pe

ei

m

n ei

m

n eK

c m

n eK

c m

c K

u E

J E

E E

E

Page 40: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.8 Electromagnetic Waves in a Plasma (3)

2 22

2 2 2

The phase velocity of high frequency EM waves is

1 .

The group velocity is

./

pe peph

gph

v c c cK K c K

c cv c c c

K K v

Page 41: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.9 Ordinary and Extraordinary Waves (1)

0

0 0

0 0 0

We now look for a high frequency EM wave solutions in a plasma with 0

and begin with waves perpendicular to , i.e., . Again we assume

E 0, , and we neglect ion motion. We will get die e iT n n

B

B K B

1 0 1 0 1 0

0

1 0

fferent solutions

depending on whether or . The wave with is called the

ordinary wave since has no effect on the wave propagation. The wave with

is called the extraordinary wa

E B E B E B

B

E B

ve.

: in ionospheric radio science the terms ordinary and extraordinary waves are used for waves with left-hand and right hand elliptical polarizations.Note

Page 42: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.9 Ordinary and Extraordinary Waves (2)

We can use the following equations:

22

1 1 0 12

1 0 1

1 1 1 0

1 0 1

0 1 1 0 1

1 0 1

6.20

6.39

0 from 6.38

6.81

For the ordinary wave since 0.

Since accelarates the electron along , then

e e e

e ee

e

e

K ic

n n a

ei

m

en

E K K E J

K u

u E u B

J u

B E K E K B K E

E B u B

0 1 0

2 2 2 2

1 0 1

1 1 1

=0

The equation are therefore the same as for the case , and the dispersion relation relation is

again given by 6.84 :

For the extraordinary wave 0.

Set = and start

e

pec K

u B

E B K E

E E E

2 2

2 2 2 2

2 2 2

cranking. The result is

6.96pepe

pe ce

c K

Page 43: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.10 L and R Waves

0

22

1 1 0 12

1 0 1

1 1 1 0

1 0 1

1

Consider high frequency transverse EM waves propagating parallel to . Using the equations

6.20

6.39

0 from 6.38

6.81

0.

This sy

e e e

e ee

e

K ic

n n a

ei

m

en

B

E K K E J

K u

u E u B

J u

K E

0

( )1 10

stem of equations has two solutions. If we orient the coordinate system such that and

point in the direction of the the z-axis, then

right-hand circularly polarized wave R wi Kz tR E x iy e

B K

E

( )1 10

22 2 2

22 2 2

ave, also e-wave

left-hand circularly polarized wave (L wave, also i-wave)

And the respective dispersion relations are:

R wave 6.1021 /

L wa1 /

i Kz tR

pe

ce

pe

ce

E x iy e

c K a

c K

E

ve 6.102b

Page 44: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

6.11 Alfvén and Magnetosonic Waves

Low frequency transverse (i.e. )electromagnetic waves are called:

Alfvén waves, if

magnetosonic waves, if

The dispersion relations are, respectively:

0K B

0K B

1 E K

2 2 2

2 22 2

2 2

2 0

0

6.103

6.1041 /

where

A

s A

A

Aio i

K V

V VK

V c

BV

n m

Page 45: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

9. Ionization and Energy Exchange Processes1. Solar extreme ultraviolet (EUV) is the major source of energy

input into the thermospheres/ionospheres in the solar system.

2. Electron precipitation contributes near the magnetic poles.

3. > 90 nm causes dissociation (O2 O+O)

4. nm causes ionization

5. Energy losses (sinks), from the ionosphere point of view, are airglow and the heating of the neutral atmosphere (thermosphere)

6. Energy flow diagram in Fig. 9.1

Page 46: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

9.1 Absorption of Solar Radiation in the Thermosphere (1)

The photon flux I(s) decreases along its path s through the atmosphere. The flux change dI due to path

element ds is

9.1

9.1

where , the constant of proportionality, is usual

a

a

dI s I s n s ds a

dI s I s n s ds b

ly called the absorption cross section. Of course,

depends on the wavelength . To understand the principle of "layer formation" we assume monochromatic

radiation for our dicussion. The positive di

a

rection for ds is from the sun towards the earth. If the solar rays

have an angle with the vertical (solar zenith angle, Fig. 9.2), then

sec 9.2

sec 9.1

sec , or after integr

a

a

ds dz

dI z I z n z dz c

dIn dz

I

ation from z to :

Iln sec 9.3

I z

I z exp sec

a

z

a

z

n z dz

I n z dz

Page 47: 5. Simplified Transport Equations We want to derive two fundamental transport properties, diffusion and viscosity. Unable to handle the 13-moment system.

9.1 Absorption of Solar Radiation in the Thermosphere (2)

00

To get an estimate for the variation of I with z, we assume an exponential decrease of n with height:

exp 9.4

and assume , H, and are constant. Then the or a

z zn z n z

H

optical depth optical thi

a

becomes:

sec sec

, exp sec 9.5

In general:

, , exp , , 9.9

Here is added to the variables because I (I at top of thermosphere) and are functions o

a a

z

a

ckness

n z dz n z H

I z I n z H

I z I z

0 0

0 00 0

f .

The simple realtion 9.4 holds only for an isothermal atmosphere. If T is not constant we can write

exp exp ./

is the neutral gas scale hei

z z

z z

T z T zdz dzn z n z n z

T z kT mg T z H

kTH

mg

ght.

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9.1 Absorption of Solar Radiation in the Thermosphere (3)

0

0 0 0

0 0 0

0

0 00 0 0

00

0

can be calculated by setting / . Then

exp exp

z

z

z

n z dz d dz H

kTn z dz n H d n d

mg

T z kT zkTn z d d n z d

T z mg mg

kT zn z

mg

n z dz n z

The vertical columnar content

a

0

9.16

To derive this equation equations we only had assumed that g does not vary with height.

If we assume that is height independent, the optical depth for plane geometry becomes

z

n z dz n z H z

z

H z

a a a

as

, , sec sec sec .

, , sec . 9.17

For a gas with different constituents:

s s

z z

z n z H z

n z dz n z dz n z H z

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9.3 Photoionization (1)Photons that exceed the ionization energy threshold can produce electron-ion pairs when

absorbed by neutral particles. Th excess energy is transfered to the kinetic energy of the

electron, or used to

a

excite the ion. If the probability for an absorbed photon to produce an

electron-ion pair is , the production rate is

, , 9.21

with

, , sec .

Sidney Chapman (1932) was th

a aCP z I z n z I n z e

z n z H z

e first to derive this . The two

factors in the product of opposite trends: with decreasing height, n increases while

decreases because increases. At what height

Chapman production function

n z e e

a

a

C

C

secC

sec a

amax

will P be a maximum?

Answer: at the height where dP / 0.

dP d d0

For an isothermal gas (H = const):

1 sec 0

1 sec 0

n z Ha a

n z H

dz

I n z e I n z nedz dz dz

dnI n z e n z H

dz

n z H n z

1amax

amax max

sec

Notice at the height of max production sec 1

H

n z H

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9.3 Photoionization (2)

max 0

max 0

max

max 0

a0 max 0 0a

11

max max max a

max 1

But for constant H:

n z

1ln sec 9.22

sec

From (9.21):

,sec

cos, 9.23

The maximum production

z z

H

z z

H

aa a

C

C

n e

n e z z H n HH

I eP z I n z e I n z e

H

IP z

He

rate occurs at local noon.

The height of max production increases with according to 9.22 . This has ben observed for the

E and F1 layers in Earth's ionosphere

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11.4 Ionospheric LayersEarth's ionosphere is usually stratified in D, E, F1, and F2 layers during the daytime. At night,

only the F2 layer survives. When transport processes can be neglected, the ion continuity equation

bec

e

e

2

omes

dn11.49

dtIn equilibrium, i.e., during most of the daytime

The loss is caused by recombination of electrons with ions. If we assume one dominant ion, then

n , and

. 11.51

Using t

e e

e e

i

e d e

P L

P L

n

L k n

e

a

00

he Chapman production function (9.21) for P :

with

sec 11.53

exp 11.54

ae

ae

P I n z e

n z H

z zn z n z

H

P I n z e