Transcript 5. Simplified Transport Equations

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:
 m 
f (r, t )  n(r, t ) 
 2 kT 
32
 mc 2 
2
2
exp 
;
c

v

u



 2kT 
 3.44 
The ionosphere can be treated as a weakly ionized plasma, meaning that the
effects of Coulomb collisions are small compared to electron-neutral and ionneutral collisions. The opposite is true for a fully ionized plasma (neutrals are
still there). We limit our discussion to weakly ionized plasmas.
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1-x direction, see Fig. 5.1a. We assume that the
mean free path length   1 dn  , and that the velocity distribution is given by a
non-drifting Maxwellian.  n dx 
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
dn
n( x  x)  n  x   x  ...
dx
The particles arriving from the right had their last collision at x+x, and
n( x  x)  n  x  
dn
x  ...
dx
5.1 Basic Transport Properties (2)
The net particle flux crossing the plane at x then becomes
1
1
   fromleft   fromright  c n  x  x   c n  x  x 
4
4
c 
c dn
dn
dn 

n  x   x  n  x   x   
x
 5.2 

4 
dx
dx 
2 dx
dn
Notice is positive for the density in Fig.5.1a.
dx
c
For x  c  collision 
, where is the average collision frequency,

2

c

4 kT dn
dn
 D
 m dx
dx
dn
2 dx
8kT
2
Since c 
(App.H.21)
m
Fick's Law
 5.3
 5.4 
5.1 Basic Transport Properties (3)
where
4 kT
kT
 1.3
is the diffusion constant. D is proportional to the temperature
 m
m
T, and inversely proportional to the mass and the collision frquency.
The net flux vector Γ is
Γ  nu
 nu x e x for the situation in Fig.5.1a. The continuity equation (3.57) is
D
n
   nu  for zero production or loss.
t
nu x
n

2n
   nu x e x   

D 2
t
x
dx
x
n
2n
D 2
t
x
parabolic partial differential eq.
 5.6 
5.1 Basic Transport Properties (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
x2

4 t
N
n  x, t  
e
 5.7 
4 Dt
This solution has a Gaussian peak at x=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.
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 τ is
 xx  xy  xz 


τ   yx  yy  yz 
   
 zx zy zz 
where  xx is the transfer of x-momentum in the x direction,  xy the transfer of
y-momentum in the x direction, etc. For the conditions in Fig. 5.1b, only  yx is non-zero.
The viscous stress yx is the net transfer in the y-direction of x-momentum per m 2 and s
across the plane at y:
1
 yx  n  x  m c u x  y  y   u x  y  y  
4
 5.8



u
u
1
n  x  m c u x  y   x y  u x  y   x y 
4
y
y



u x
1
  n xm c
y where y    c / 
2
y
1
2 u x

n xm c
2
y

 yx  
u x
y
  1.3
nkT

where

1
nm c
2
viscosity coefficient
2

n 8kT
2 
 5.11
 5.12 
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):
Du
M
nm
 p    τ  ne  E  u  B   nmG 
.
Dt
t
For a uniform isothermal gas, p = 0. Assuming   τ is the dominant force term,
Du


the steady state  i.e.
 0  solution is obtained from
Dt


  τ  0.
The divergence of a tensor (L.26) is
  xx  yx  zx 
  xy  yy  zy 
  xz  yz  zz 
τ  






 ex  
ey  
 ez .

x

y

z

x

y

z

x

y

z






5.1 Basic Transport Properties (7)
Example of Viscosity Effect (2)
For our example, u  u x  y  e x , and only  yx is nonzero. Therefore


 

 
 
 
 
 
  τ   xx  yx  zx  e x   xy  yy  zy  e y   xz  yz  zz 
y
z 
y
z 
y
z 
 x
 x
 x

 yx e x  0
y
 yx
y
0 
  u x 
 
  0 where I used (5.11).
y 
y 
Assuming  
nkT

 constant, we can write
 2u x
 0  u x  y   Ay  B
2
y
With the boundary conditions at y = 0 an y = a
u x  0  =0, u x  a  = V0 we get
u x  y   V0
y
a
 5.16 
The velocity increases linearly from u x = 0 to u x = V0 . This means that
viscosity acts to smooth velocity gradients.
5.1 Basic Transport Properties (8)
Energy Flow
Thermal energy is m c 2 / 2  3kT 2. The energy flux is
q T  
m c2 n c
2
3k 3 2 n

2 2 m
x 
c

4
x x
x x
3kT n 8kT
3k 3 2 n
T 3 2  x  x   T 3 2  x  x  


2 4  m 2 2 m
9k 3 2 n 1 2 T
 3 1 2 T 
 2 2 T x x    2 2 m T x x

8kT
m

9k 3 2 n 1 2 T
q T   
T
x
2 2 m
q  T   
8kT
2
 m   9k n T T

 m x
T
x
Here  is used for the thermal conductivity:
9k 2 n
=
T
 m
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:
Mj
njmj
 p j    τ j  n j e j E  u j  B   n j m j G 
Dt
t
Du j
where the momentum transfer collision integral is given in  4.129b  :

Mj
z jj '  jj ' 
j
  n j m j jj '  u j  u j '    jj '
q j' .
 q j 
t
kT jj ' 
 j ' 
j'
If we neglect the heat flow, the momentum equation for the j particles becomes
Du j
njmj
 p j    τ j  n j e j E  u j  B   n j m j G  n j m j jn  u j  u n   5.23
Dt
where j stands for e (electrons) or i (ions), and n for neutrals.
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.
term 2/term 3:
n j m j u 2j / L
pj / L

n j m j u 2j / L
n j kT j / L

u 2j
kT j / m j
M 2j
Here u j is the average (drift) speed, kT j / m j
 5.24 
thermal speed, and M j the
Mach number, i.e., the ratio between drift speed and thermal speed. This means
the 2nd term in  5.24  can be neglected if M j
1, i.e., subsonic flow.
Similarly term1/term 3:
L
n j m ju j /  '
u j L / '
L / '
 ' 
Mj
kT j
n j kT j / L
kT j kT j
kT j
mj
mj
mj
mj
uj
 5.25
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.
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.
p j  n j e j E0   n j m j jn u j
 5.26 
  n j kT j   n j e j E0   m j jn Γ j
For an isothermal plasma (Tj = const):
kT j   n j   n j e j E0   m j jn Γ j , or
Γj  
kT j
m j jn
n 
n je j
m j jn
E 0   D j n   j E 0
 5.27 
where
Dj 
j 
kT j
m j jn
n jej
m j jn
is the diffusion constant as defined 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  0, (5.27) reduces to Fick's law derived from simple mean-freepath arguments
Γ j   D j n
 5.30 
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
Ds u s
 Ms
 ps    τ s  ns es  E  u s  B   ns ms G 
 3.58
Dt
t
Ds  3  5
 Es
p

p


q

τ
:

u



 3.59 
s
s
s
s
s

Dt  2  2
t
With reasonable assumptions valid at least for the height regime 60-160 km
ns ms
Schunk has shown that one can simplify:
ei
 E  ui  B    in (ui  u n )
mi
 5.31
0  3k  (Tn  Ti   mi  ui  u n 
 5.32 
2
 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 E is perpendicular
to the external B  Bb field, E = E .
5.3
Transport in a weakly ionized plasma (7)
Then
ui  u n 
ei
mi
  in


E'  2 ci 2 E'  b 
 2
2
 in   ci
  in   ci

 5.35
with
E'  E  u n  B, and  ci =
ei B
is the ion cyclotron frequency (rotations per s).
mi
mi
2
 ui  u n   Ti  Tn .
 5.36 
3k
The relative ion-neutral drift has two components: the Pederson component, which
Ti  Tn 
is parallel to E' , and the Hall component which is perpendicular to E' (and to B).
There is no component parallel to B since we had assumed that E  0.
The ion drift is
  in


E'  2 ci 2 E'  b  , and the ion current becomes
 2
2
 in   ci
  in   ci

2

ne  

J i   ni ei u n  i i  2 in 2 E'  2 ci 2 E'  b 
 5.110 
mi   in   ci
 in   ci

ui   u n 
ei
mi
2

ni ei2   in
 ci in '
'
J i   ni ei u n  
E

E

b
 2



mi in   in   ci2
 in2   ci2

  in2


J i   ni ei u n    i  2
E'  2 ci in 2 E'  b 
2
  in   ci

 in   ci


 5.111
ni ei2
with  i 
the ion conductivity.
mi in
5.112 
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.,
u s t  0; the flow is subsonic, i.e.,  u s   u s  0; the plasma is neutral, i.e., n e =n i .
We neglect Coriolis and centrifugul acelleration and the heat flow terms. The momentum
equation  5.50  becomes
 // ps    τ s  ns es  E  u s  B   ns ms G   ns ms st  ut  u s 
 5.50 '
s
Ion and electron momentum equations along the magnetic field:
 // pi     τ i  //  ni ei E //  ni mi G //  ni mi ie  u e  ui  //  ni mi in  u n  ui  //  5.51
 // p e     τ e  //  ne ee E //  ne meG //  ne me ei  ui  u e  //  ne me en  u n  u e  //  5.52 
Note ei  e, ee  e, also ni mi ie  ne me ei from eq.  4.158  , no proof. Add
the two equations:
 //  pi  p e      τ i    τ e  //  ni  mi  me  G //   ni  mi in  me en  u n  ui  //  5.53
5.5 Major Ion Diffusion (2)
Neglect terms with m e , including τ e which is proportional to m e . Then:
 //  pi  p e      τ i  //  ni mi G //   ni mi in  u n  ui  // , or with ps =n s kTs :
ui //  u n //
ui //  u n //
 // n i kTi  // n i kTe G //    τ i  //




ni mi in
ni mi in  in
ni mi in
k Ti  Te   // n i k Ti  Te   // Ti  Te  G //    τ i  //




mi in
ni
mi in
Ti  Te
 in
ni mi in
We set Ti  Te  2Tp , then:
ui //  u n //
2kTp  // n i 2kTp  //Tp G //    τ i  //
2kTp




. With Da 
mi in ni
mi in Tp e
 in
ni mi in
mi in
ui //  u n //
  // n i  //Tp mi G //    τ i  // 
 Da 



 ambipolar diffusion  5.54 
Tp e
2kTp
2ni kTp 
 ni
5.5 Major Ion Diffusion (3)
If we call r the coordinate along B, then  5.54  becomes:
mi g // 1 Tp
1 ni
1  i // un  ui




ni r
2kTp Tp r 2ni kTp r
Da
Remember Da 
2kTp
mi in
, i.e., it becomes large with increasing altitude where  in  0.
We therefore 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 G   ge r , i.e.,
g // = -g. Neglecting the stress term, we get the classical diffusive equilibrium eq.:
mg
1 ni
1 Tp
  i // 
ni r
2kTp Tp r
2kTp
1 ni
1
1 Tp


, Hp 
Plasma scale height
ni r
H p Tp r
mi g //
If Tp  constant:
ni  r   ni ,o
r  r0 

exp 
Hp
5.11 Electric Currents and Conductivities (1)
Similarly the elctron current becomes with ee  e, and  ce 
  en2

 ce en '
'
J e   ne eu n    e  2
E

E

b

  en   ce2   en2   ce2 


ne e 2
e 
me en
eB
me
 5.113
 5.115
The total perpendicular current, J   J i   J e , is therefore (for ei  e)

 in2
 en2  ' 
 ci in
 ce en
J    ni  ne  eu n     i 2


E






 i 2
e

e
  in   ci2
 en2   ce2 
 in   ci2
 en2   ce2



'
 b  E

J    ni  ne  eu n    p E'   H b  E'
J    ni  ne  eu n    p  E  u n  B    H b   E   u n  B   5.116 
2
2
 p  i 2
e 2
 in   ci2
 en   ce2
in
en
 H   i
I have resubstituted E'  E  u n  B.
 ci in
 ce en


Pederson and Hall conductivity.
e
 in2   ci2
 en2   ce2
5.11 Electric Currents and Conductivities (2)
For heights above ~100 km
2
 p  i 2
 in   ci2
 H   i
since  ce
 ce 
in
 .
en
f ce 
 ci in

  e en
2
2
 in   ci
 ce
 5.120 
eB 1.6 1019  5 105

 9 106 s 1
31
me
9.1 10
 ce
 1.4 MHz
2
Notice from  5.120  that the electrons contribute only to the Hall current, not to the Pederson current.
Currents along B :
To calculate u e use  5.23 for the electrons which are much more mobile than the ions and neutrals:
Du e
  pe     τ e   ne ee E   u e  B    ne meG  ne me en  u e  u n
Dt
Neglecting the stress term and the terms containing m e on the left side, we get
ne me
 pe  ne ee E  ne me en  u e  u n

where I used the fact that  u  B   0, since u
j
When an E exists, u e
un . Therefore
 pe  ne ee E  ne me en u e
Since pe  ne kTe and ee  e
ne k  Te  kTe ne  ne eE  ne me en u e
j
 B is a vector  to Β.
  5.23
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
J   ne eu e 
ne ek
kT e
n e2
n e2
 Te  e  ne  e E  e
me en
me en
me en
me en

 n ek
kTe
 ne   e  Te
E 
ne e

 me en
Setting
ne e 2
  e parallel electric conductivity
me en
ne ek
  e current flow conductivity due to thermal gradients
me en


kT
J   e  E  e  ne    e Te
ne e


 5.124 
Recall:
J    p E    H b  E  (for u n   0)
 5.116 
2
2
 ci in
 ce en
 p  i 2


;






e
H
i
e
 in   ci2
 en2   ce2
 in2   ci2
 en2   ce2
in
en
ni e 2
i 
mi in
Notice  5.124  becomes Ohm's law
J   eE
when the E term dominates
6. Wave Phenomena
6.1 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.
1  E1  1c  0 , 1c   es ns1;charge density disturbance,  0  8.85 x1012 SI units, permittivity
s
 2   B1  0
 3 xE1  
B1
t
 4  xB1  0 J1   0 0
where
E1
, 0  4 x107 SI units, permeability.
t
J1   es ns1us1
s
6.1 General Wave Properties(2)
We solve Maxwell’s equations by taking the curl of (3):

  B1
t
But   xE1     E1    2 E1
  xE1  

J1
 2 E1
and using(4): -   B1   0
  0 0 2 results in
t
t
t
 2 E1
J
 E1     E1    0 0 2  0 1
 6.6 
t
t
This quation can be easily solved for vacuum where  c1 = 0, and J1 = 0. In this case
2
 E1 
 c1
 0 according to 1 , and
0
 2 E1
 E1   0 0 2  0
t
1
or by setting
c
2
 0 0
1  2 E1
 E1  2
 0 in vacuum.
c t 2
 6.7 
2

Solution: E1  r, t   Re E10 e 
i K r  t 

 6.9 
6.1 General Wave Properties(3)

E1  r, t   Re E10 e 
i K r  t 
E
10
cos  K  r   t 
where for simplicity I assumed E10 is real. Recall: eix  cos x  i sin x.
From Mawell equation  3
B1
t
we can get B1. Notice that when the exponential functions are used, application of the operators
xE1  
 and

simply means multiplication with iK and -i , respectively. Therefore:
t
iK  E1  -i B1  K  E1   B1
This means that B1 is  to K and to E1. With the help of Maxwell equation 1  E1  1c  0
it is easy to show that for 1c  0 the electric field is  to K :
iK  E1  0  K  E1 .
Further from
K  E1   B1  K  K  E1   K  B1
K  K  E1   E1 K 2   K  B1  E1 

k  B1 .
K
Summary: In vacuum E1 , B1 , K are orthogonal to each other following the right-hand rule.
6.1 General Wave Properties(4)
Poynting Vector
The flow of energy carried by an electromagnetic wave in the direction K is given by
the Poynting vector:
S  E1   B1 / 0   E1  H1
 6.13
Since E1  H1 are sinusoidal time varying functions, S is a function of  t. In general we are not interested
in the fast in and out energy fluxes, but want to know the time-averaged flux:
1 w
2
S   E1  H1dt where Tw 
is the wave period. It is easy to verify that when using the exponential
Tw 0

T
notation:
1
S  Re E1  H*1 time-avearged Poynting vector
2
 6.14 
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):
ns
   (ns u s )  0 continuity eq.
 6.21
t
u
ns ms [ s   u s   u s ]  ps  ns es [E  u s  B]  0 momentum eq.  6.22 
t
Ds ps
  ps    u s   0 energy eq.
 6.23
Dt
n
n
1
1 Ds ns
 6.21  s  ns  u s  u s ns  0    u s   s  u s ns  
t
ns t ns
ns Dt
Substitute in  6.23 :
Ds ps
p Dn
 s s s  0
Dt
ns Dt
 6.25
6.2 Plasma Dynamics (2)
This implies that
Ds ps
 0, since

Dt  s
 6.26a 
Ds ps
ps Ds
ps Ds 
1 Ds ps
1  Ds ps







s
s
Dt  s  s Dt
 s 1 Dt
 s  Dt
 s Dt 
 1  Ds ps
ps Ds
ps Ds 
1  Ds ps


n
m



ns 
s s
 
 s  Dt
ns ms Dt

Dt
n
Dt
s 
s


And  6.26a  implies that

ps

s
 6.26b 
 const.
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 psV   const.)
From  6.26b  :
ps    const   s 1 s   
ps

s
  s 1 s 
 ps
 n kT
 kTs
 s  s s  s 
 s
s
ns ms
ms
 6.27 
6.2 Plasma Dynamics (3)
Substitute in the momentum equation (6.22):
u s
  u s   u s ]   s kTs ns  ns es [E  u s  B]  0
 6.28
t
The continuity equation was
ns
   (ns u s )  0
 6.21
t
We must solve these equations together with Maxwell's equations to find
ns ms [
n s , us , E and B (10 unknowns).
Using Perturbation Technique :
1. Solve for equilibrium conditions finding n 0 , u 0 , B 0 , E0 ( I dropped index s on n and u)
that satisfy the differential equations.
2. Perturb the equilibrium state of the plasma and assume that this will cause small changes in B 0 and E0 .
n  r, t   n0  n1  r, t 
u  r, t   u 0  u1  r, t 
E  r, t   E0  E1  r, t 
B  r, t   B 0  B1  r, t 
 6.31a 
 6.31b 
 6.31c 
 6.31d 
6.2 Plasma Dynamics (4)
Substitute these perturbed functions into the continuity and momentum equations:
  n0  n1 
t
   ( n0  n1 u 0  u1 )  0
 u 0  u1 
 u 0  u1    u 0  u1 ]   s kTs  n0  n1    n0  n1  es [ E0  E1   u 0  u1    B 0  B1 ]  0
t
Carry out the differenciations remembering that all 0-index terms are constants:
 n0  n1  ms [
n1
n
 n0  u1  n1  u1  u 0n1  1  n0  u1  u0 n1  0  6.33
t
t
where only first order (linear) terms in 1-index functions were kept. The momentum equation becomes
u1
 n0 m  u 0   u1   s kTs n1  n0es E0  n0es E1  n1es E0  n0esu 0  B 0  n1esu 0  B 0  n0esu 0  B1  n0esu1  B 0  0.
t
where es =  e for ions/electrons. But
n0 m
n0 es  E0  u 0  B 0   0 (equilibrium condition). Therefore
 u

n0 m  1   u 0   u1    s kTs n1  n0es  E1  u1  B 0  u 0  B1   n1es  E0  u 0  B 0   0.
 t

But the last    0, therefore
 u

n0 m  1   u 0   u1    s kTs n1  n0 es  E1  u1  B 0  u 0  B1   0  6.35 
 t

6.2 Plasma Dynamics (5)
We try plane wave solutions for all functions
n1 , u1 , E1 , B1  ei (k r  t ) .

 i .
t
i n1  n0iK  u1  u 0  iKn1  0
Remember   iK ,
  u0  K  n1  n0K  u1
 6.37 
n0 m  i u1   u 0  iK  u1    s kTs iKn1  n0es  E1  u1  B 0  u 0  B1   0
i   u 0  K  u1  iK
 6.37 
 s kTs n1
n0 m

es
E1  u1  B0  u0  B1   0
m
 6.38
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
 n1 , u1 , E1 , B1; n1 , u1 can have different values for the different species, n s1 , u s1  .
From slide 6.1(2):
 2 E1     E1    0 0
 iK 
2
 2 E1
J
 0 1
2
t
t
 6.6 
E1  iK  iK E1    0 0  i  E1  i 0 J1 ;  0 0 
2
2
2
 2  K  E1  K  K E1   i 0 J1
c

These are 3 more algebraic equations. from 6.11 :
1
c2
 6.20 
1
1  E1  1c  0 , 1c   es ns1  iK  E1   es ns1. One more algebraic eq.

s
0
s
 2    B1  0  K B1  0. Only tells that B1  K.
 3   E1  
B1
 KxE1   B1. Three more algebraic eqs.
t
Electrostatic Waves: B1= 0
6.3 Electron Plasma Waves (1)
We start the discussion with high frequency electron plasma waves assuming that B1  0.
The wave frequency is high enough so that the ions cannot follow the motion, i.e., ui1  0.
To simplify the discussion we also assume ui0  u e0  0, and Eo  B 0  0, then the algebraic
electron transport equations  6.37  and  6.38  become with es  -e :
 6.39a 
 ne1  ne 0K  u e1
i u e1  iK
 e kTe ne1
ne 0 me

e
E1  0
me
 6.39b 
From Gauss's law (see slide 6.2(5)):
 6.39c 
iK  E1  ene1 /  0
Our immediate goal is to find the dispersion relation that relates K and  .
Muliply  6.39b  with K  and use  6.39a  and  6.39b  to substitute for K  u e1 and K  E1 :
i K  u e1  iK  K
 e kTe ne1
ne 0 me

e
K  E1  0
me
 6.40 
i ne1
 kT n
e
 iK 2 e e e1 
 ene1 /  0   0
ne 0
ne 0 me
ime

 kT e 2 ne 0 
ne1   2  K 2 s s 
0
me
me 0 

 6.41
6.3 Electron Plasma Waves (2) (B1=0)
This gives the dispersion relation
  K
2
2
 e kTe
me
e 2 ne 0

 0, or
me 0
e 2 ne 0  e kTe 2
 

K , or
me 0
me
2
 2   2p   eVe2 K 2 ; usually  e is set equal to 3.
e 2 ne 0
p 
me 0
plasma frequency; Ve 
kTe
me
 6.42 
electron thermal speed.
The dispersion relation  6.42  relates  with the wavelength  (=2 /K). Notice there is no
propagating wave in a cold plasma where Te  0. In the cold plasma
 2   2p
plasma oscillation
 6.45 
6.4 Ion-Acoustic Waves (1) (B1=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 es  e :
 6.46a 
 ni1  ni 0K  ui1
i ui1  iK
 i kTi ni1
ni 0 mi

e
E1  0
mi
 6.46b 
In Gauss's law (see slide 6.2(5)) we must include positive and negative charges:
iK  E1  e(ni1  ne1 ) /  0 .
 6.46c 
Looking back at slide 6.3 (1), equation (6.40) for the electron motion:

e
 kT  n
i   2  K 2 e e  e1  K  E1  0 or
 6.40 
m
n
m
e
e  e0

 6.48
 me 2  K 2 ekTe  ne1  enie0 K  E1  0
Similarly from the ion transport equations we get from  6.46b 
m
 K 2 i kTi  ni1 
eni 0
K  E1  0
 6.49 
i
Assuming ne 0 =ni 0 (neutral plasma), and adding  6.48  and  6.48  gives
i
2
(mni1i  me ne1 ) 2  K 2 ( i kTni1   e kTe ne1 )  0. Since mi
(mi 2  K 2 i kT )ni1  K 2 e kTe ne1  0
me :
 6.50 
6.4 Ion-Acoustic Waves (2) (B1=0)
How do ni1 and ne1 relate? we can combine Gauss's law and the momentum equation for the electrons
given above as
 6.46c 
iK  E1  e(ni1  ne1 ) /  0 .
m 
e
2
 K 2 e kTe  ne1 
e(ni1  ne1 ) /  0 
ene 0
K  E1  0
i
 6.48
1
 me 2  K 2 ekTe  ne1
ene 0
 6.51
If we neglect the electron inertia term me 2 again for low frequencies:

K 2 e 0 kTe 
ni1  ne1 1 
0
e 2 ne 0 

ne1 
 kT
ni1
with D  02 e , the Debye length
2 2
e ne 0
1   e K D
 6.52 
Substitute into  6.50 
(mi 2  K 2 i kT )ni1  K 2 e kTe ne1  0,
gives the dispersion relation for ion plasma waves:

  kT
 e kTe
i


2 2

 mi

K


1
m


D 
e
i

 6.53
2  K2 
 2D 
For very long waves K 2D2 = 

  
2
1. We get the dispersion relation for ion acoustic waves, or ion sound waves:
  i kT   e kTe 
2 2
  K Vs ion acoustic waves
m
i


2  K2 
Vs 
 i kT   e kTe
mi
ion acoustic speed
 6.55
 6.56 
6.5 Upper Hybrid Oscillations
Assume B 0  0, and E0  0.Upper hybrid oscillations are high frequency oscillations directed  B 0 .
We start again with the electron continuity  6.37  and momentum  6.38  equations:
 n e1  ne 0K  u e1
 657a 
i u e1 
e
 E1  ue1  B0   0
m
We also use two Maxwell equations:
 6.57b 
iK  E1  ene1 /  0 Gauss's Law
 6.57c 
 6.57d 
 6.57e 
iK  E1  i B1 where B1  0 Faraday's Law
K  E1  0  K E1
The dispersion relation for the upper hybrid oscillation becomes:
 6.62 
 2   2pe   ce2
2
 eB0 
n0e 2
2
where   
 ,  pe 
me 0
 me 
 and K are not related, so there is no wave velocity defined. There is no wave.
2
ce
6.6 Lower Hybrid Oscillations
Assume again B 0  0, and E0  0. Lower hybrid oscillations are low frequency electrostatic oscillations directed  B 0 .
We must use the electron and ion continuity  6.37  and momentum  6.38  equations together with the two
Maxwell equations. This gives the algebraic equation:
 6.68
 2  ceci
where
 ce 
eB0
,
me
 ci 
eB0
mi
are the electron-cyclotron and ion-cyclotron frequencies. Again, no waves.
6.7 Ion-Cyclotron Waves
Assume again B0  0, and E0  0.The ion-cyclotron waves are low frequency electrostatic waves
that propagate in a direction almost perpendicular to B0 . The algebraic dispersion relation becomes:
 2   ci2  K 2Vs2
where
Vs2 
 e kTe
mi
 6.74 
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:
2
2
 2  K  E1  K  K E1   i 0 J1

c
Let's look first for transverse waves, i.e., forsolutions for which K  E1  K E1  0 :
 6.20 
2
2
 6.75
 2  K  E1  i0 J1
c


In a two-component plasma (electrons and one ion species) the current density is given by
J  eniui  eneu e
 6.76 
Perturbation and linearization:
J  J 0  J1
ns  nso  ns1 s  i / e for ions/electrons. Charge neutrality: nio  neo  no
u s  u s 0  u s1
J  eniui  eneu e
J 0  en0  ui  u e 
J  J 0  en0  ui1  u e1   eni1u i 0  ene1u e 0 , i.e.,
J1  en0  u i1  u e1   eni1u i 0  ene1u e 0
 6.80 
6.8 Electromagnetic Waves in a Plasma (2)
To simplify the notation, we assume E0  B 0  u i 0  u e 0  Ti  Te  0.Then
J1  en0  u i1  u e1 
High frequency approximation: ui1  0
 6.81
J1  en0u e1
The electron momentum equation  6.38 
i   u 0  K  u1  iK
i u e1 
J1  i
 s kTs n1
n0 m

es
E1  u1  B 0  u 0  B1   0 becomes
m
e
E1  0, therefore
me
 6.82 
n0e 2
E1
 me
Substitute into (6.75):
2
n0e 2
2
E1
 2  K  E1  0
c

m


e
Since E1  0, this gives the dispersion relation
2
 6.83
n0e 2
, or
1 2  K   0
c
 me
2
 2  c 2 K 2   2pe
 6.84 
6.8 Electromagnetic Waves in a Plasma (3)
The phase velocity of high frequency EM waves is
v ph 

 c2 
 2pe
2
K
K
The group velocity is
 c 1

c
c
vg 

c
c  c.
K  / K
v ph
2
 pe
2
c K
2
 c.
6.9 Ordinary and Extraordinary Waves (1)
We now look for a high frequency EM wave solutions in a plasma with B0  0
and begin with waves perpendicular to B0 , i.e., K  B0. Again we assume
E0  Te  0, ne0  ni0 , and we neglect ion motion. We will get different solutions
depending on whether E1  B0 or E1 B0. The wave with E1 B0 is called the
ordinary wave since B0has no effect on the wave propagation. The wave with
E1  B0 is called the extraordinary wave.
Note : in ionospheric radio science the terms ordinary and extraordinary waves
are used for waves with left-hand and right hand elliptical polarizations.
6.9 Ordinary and Extraordinary Waves (2)
We can use the following equations:
2
2
 2  K  E1  K  K E1   i0 J1
c

 ne1  ne 0K  u e1
i u e1 
e
E1  u e1  B 0   0 from  6.38
me
J1  en0u e1
 6.20 
 6.39a 
 6.81
For the ordinary wave B 0 E1  K  E1 since K  B 0  K E1  0.
Since E1 accelarates the electron along B 0 , then u e1 B 0  u e1  B 0 =0
The equation are therefore the same as for the case , and the dispersion relation relation is
again given by  6.84  :
 2  c 2 K 2   2pe
For the extraordinary wave E1  B 0  K E1  0.
Set E1 =E1  E1 and start cranking. The result is
  c K 
2
2
2
2
pe
 2   2pe
 2   2pe   ce2 
 6.96 
6.10 L and R Waves
Consider high frequency transverse EM waves propagating parallel to B 0 . Using the equations
2
2
 2  K  E1  K  K E1   i0 J1
c

 ne1  ne 0K  u e1
i u e1 
e
E1  ue1  B0   0 from  6.38
me
J1  en0u e1
 6.20 
 6.39a 
 6.81
K E1  0.
This system of equations has two solutions. If we orient the coordinate system such that B 0 and K
point in the direction of the the z-axis, then
 
 x  iy  e
E1R  E10 x  iy ei ( Kz  t ) right-hand circularly polarized wave  R wave, also e-wave 
E1R  E10
i ( Kz  t )
left-hand circularly polarized wave (L wave, also i-wave)
And the respective dispersion relations are:
 2pe
 c K 
1   ce / 
 R wave 
 6.102a 
 2pe
 c K 
1   ce / 
 L wave 
 6.102b 
2
2
2
2
2
2
6.11 Alfvén and Magnetosonic Waves
Low frequency transverse (i.e. E1  K )electromagnetic waves are called:
Alfvén waves, if K B0
magnetosonic waves, if K  B0
The dispersion relations are, respectively:
 2  K 2VA2
 6.103
Vs2  VA2
 K
1  VA2 / c 2
 6.104 
2
2
where
VA2 
B0
0 nio mi
9. Ionization and Energy Exchange Processes
1.
2.
3.
4.
5.
6.
Solar extreme ultraviolet (EUV) is the major source of energy
input into the thermospheres/ionospheres in the solar system.
Electron precipitation contributes near the magnetic poles.
 > 90 nm causes dissociation (O2 O+O)
  90 nm causes ionization
Energy losses (sinks), from the ionosphere point of view, are
airglow and the heating of the neutral atmosphere (thermosphere)
Energy flow diagram in Fig. 9.1
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
dI  s    I  s  n  s  ds
dI  s    a I  s  n  s  ds
 9.1a 
 9.1b 
where  a , the constant of proportionality, is usually called the absorption cross section. Of course,  a
depends on the wavelength . To understand the principle of "layer formation" we assume monochromatic
radiation for our dicussion. The positive direction 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
ds   sec  dz
dI  z    a I  z  n  z  sec  dz
dI
  a n sec  dz, or after integration from z to :
I
I  a
ln
  n  z  sec  dz
I  z  z
  a

I  z   I  exp     n  z  sec  dz 
 z

 9.2 
 9.1c 
 9.3
9.1 Absorption of Solar Radiation in the Thermosphere (2)
To get an estimate for the variation of I with z, we assume an exponential decrease of n with height:
 z  z0 
n  z   n  z0  exp  
 9.4 

H


and assume  a , H, and  are constant. Then the optical depth or optical thickness  becomes:

    a n  z  sec  dz   a n  z  sec  H
z
I  z ,    I  exp   a n  z  sec  H 
 9.5
In general:
I  z ,  ,    I     exp    z ,  ,   
 9.9 
Here  is added to the variables because I (I at top of thermosphere) and  a are functions of .
The simple realtion  9.4  holds only for an isothermal atmosphere. If T is not constant we can write
T  z0 
 z dz 
 z dz 
T  z0 
n  z   n  z0 
exp   
exp     .
  n  z0 
T  z
kT
/
mg
T
z


 z0

 z0 H 
kT
H
is the neutral gas scale height.
mg
9.1 Absorption of Solar Radiation in the Thermosphere (3)

The vertical columnar content  n  z  dz can be calculated by setting d  dz / H . Then
z0



kT
 n  z  dz   n   H   d   n   mg d
z0
0
0

 kT
kT  z0  
  n  z0 
exp    d 
d  n  z 0 
exp      0   d
T  z
mg 0
  0  mg
0

 n  z0 
T  z0 

kT  z0 
mg

z n  z  dz  n  z0  H  z0 
0

 n  z  dz  n  z  H  z 
 9.16 
z
To derive this equation equations we only had assumed that g does not vary with height.
If we assume that  a is height independent, the optical depth  for plane geometry becomes

  z,  ,      n  z  sec  dz  sec  
a

a
z
For a gas with different constituents:
  z,  ,    sec   sa    ns  z  H s  z .
z n  z  dz  sec      n  z  H  z .
a
 9.17 
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 excite the ion. If the probability for an absorbed photon to produce an
electron-ion pair is , the production rate is
PC  z ,     I  z ,   n  z  a   I  n  z  a e 
 9.21
with
  z ,  ,    sec   a    n  z  H  z  .
Sidney Chapman (1932) was the first to derive this Chapman production function. The two
factors in the product n  z  e  of opposite trends: with decreasing height, n increases while e 
decreases because  increases. At what height will PC be a maximum?
Answer: at the height where dPC / dz  0.
dPC d
d
 sec   a n  z  H 
  I  n  z  a e     I  n  z  a  ne
0

dz dz
dz 
For an isothermal gas (H = const):
a
dn
 I  n  z  e  sec   n  z  H 1  n  z  sec   a H   0
dz
 1  n  zmax  sec   a H  0  n  zmax    sec   a H 
1
Notice at the height of max production    max  n  zmax  sec   a H  1
9.3 Photoionization (2)
But for constant H:
n  z max   n0e
1
sec   H
a

zmax  z0
H
 n0e

zmax  z0
H
 zmax  z0  H ln  n0 sec   a H 
 9.22 
From (9.21):
PC  zmax ,     I  n  zmax  e
a  max
PC  zmax ,   
 I  a e 1
  I  n  zmax  e 
sec   a H
a 1
 I  cos 
He 1
 9.23
The maximum production 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
11.4 Ionospheric Layers
Earth'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
becomes
dn e
 Pe  Le
dt
In equilibrium, i.e., during most of the daytime
Pe  Le
11.49 
The loss is caused by recombination of electrons with ions. If we assume one dominant ion, then
n e  ni , and
Le  kd ne2 .
11.51
Using the Chapman production function (9.21) for Pe :
Pe   I  n  z  a e 
with
  sec   a n  z  H
11.53
 z  z0 
n  z   n  z0  exp  
H 

Pe   I  n  z  a e 
11.54 