Download Chapter 3 The ionosphere of the Earth

Chapter 3
The ionosphere of the Earth
The ionosphere of the Earth consists of weakly ionised gas. No more than about a
fraction of the order of 10−3 of the atmospheric molecules are ionised. This means
that the mass of neutrals surpasses overwhelmingly that of ions and electrons, and
the result is that neutrals also have a major effect on the motion of the ionised
species, whereas the effect of the latter ones on neutrals is weak or slow.
As a result of ion chemistry, the ionosphere consists of ions of several species, and
relative abundances of different ion species varies with altitude. A simplified picture
of ionospheric plasma dynamics is obtained by considering that the atmosphere
consists of gas of three particle species occupying the same volume. The three
species consist of neutral molecules, ions and electrons.
3.1
Ionospheric regions
The atmospheric regions discussed in the previous chapter contain ionized species
as follows.
• Thermosphere: positive ions and electrons
• Mesosphere: positive and negative ions of small amount
• Stratosphere and below: no free electrons
The region of main ionization exist thus in thermosphere. Historically the ionosphere
has been divided into D, E and F layers, but nowadays it is more common to speak
about D, E and F regions (Fig. 3.1). The lower part of the D region is in the
mesosphere. The approximate altitude limits for these regions and typical daytime
electron densities are:
• D region: 60–90 km, ne = 108 –1010 m−3
• E region: 90–150 km, ne = 1010 –1011 m−3
• F region: 150–1000 km, ne = 1011 –1012 m−3 .
39
Ionospheric
variability
40
CHAPTER 3.
THE IONOSPHERE OF THE EARTH
riations
gh in upper F
der of
V variations
ation
rs of magnitude
and D regions
and space
s
from seconds to
[from Richmond, 1987]
Figure 3.1: Typical ionospheric electron density profiles.
21.2.2007
The ionospheric layers were originally observed from radio wave reflections at different altitudes. It was then considered that the reflections take place at separate
layers. Later, however, it was noticed that the reflections were from bulges in a
continuous electron density profile, and therefore we nowadays rather speak about
regions. The nomenclature D, E and F has turned out to be useful since the regions
are physically different because of their different ion chemistry. A bulge called the
F1 layer forms in the bottomside F region electron density profile during daytime in
solar maximum conditions. The D region disappears during night. At high latitudes
the E region electron densities may increase drastically in the nighttime exceeding
even the F region peak during auroral electron precipitation.
Ionosphere has great variability:
• Solar cycle variations: one order of magnitude in upper F region
• Day-night variation: several orders of magnitude in lower F, E and D regions
• Space weather effects based on short-term solar variability (from seconds to
weeks): even several orders of magnitude in lower F, E and D regions
3.2. BASICS OF PLASMA DYNAMICS
41
Figure 3.2: The most typical ions species and the corresponding electron density
profile.
The ion composition changes in accordance with the change of molecular constituents in the neutral atmosphere. As illustrated in Fig. 3.2, NO+ and O+
2 are
+
the dominant ions below 150 km. At greater heights O ions are more abundant
and the number of H+ ions starts to increase rapidly above 300 km. D-region (not
shown) contains negative and positive ions and ion clusters. The distribution of ions
depends on solar and magnetospheric activity.
The temperature of the ionosphere is controlled by the absorption of solar UV
radiation in thermosphere. Since this radiation is able to ionize molecules and
atoms, free electrons as well as ions are formed which carry some excess energy
with them. Electrons have a larger heat conductivity so the electron temperature
becomes usually higher than the ion temperature. Ions are heavier and interact by
collisions more strongly with the neutral gas. Therefore much of the ion energy is
transferred to the neutrals and the ion temperature is less increased. Thus, typically
Te > Ti > Tn . However, in the high-latitude E region the Joule heating of ions may
result in Ti > Te . Electrons may also be strongly heated by plasma waves. An
example of ion, electron and neutral temperature profiles is shown in Fig. 3.3
3.2
Basics of plasma dynamics
The density of neutral molecules is denoted by n, their mass by m, pressure by p,
temperature by T and velocity by u. Correspondingly, these quantities for ions and
electrons are ni , ne , mi , me , pi , pe , Ti , Te , vi and ve . When the plasma consists
of a single positive ion species with a charge e, the plasma neutrality implies that
ni = ne .
42
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Altitude / km
500
Tn
Ti
400
Te
300
200
100
1000
1800
Temperature / K
2200
Figure 3.3: An example of neutral, ion and electron temperature profiles.
3.2.1
Continuity equations
Separate continuity equations are valid for all three gas species. Since only a very
small fraction of neutrals is ionised, the continuity equation of the neutral molecules
is not altered by ionisation, and therefore eq. (2.5) is valid for them.
Since ions and electrons are continuously created by ionisation and they also
disappear in recombination producing neutral molecules, the continuity equation for
charged particles is
∂ni,e
+ ∇ · (ni,e vi,e ) = q − l,
(3.1)
∂t
where q is the production rate per unit volume and l the loss rate per unit volume.
The unit of these quantities is particles per second per unit volume, i.e. m−3 s−1 .
Note that eq. (3.1) actually contains two equations, one for ions and one for electrons. Since we assume that all ions are singly ionised, the production and loss
rates for ions and electrons are the same. The production term is determined by the
density and properties of the neutral molecules as well as the ionising radiation and
the detailed ionisation process. The loss rate depends on the electron/ion densities
and the rate coefficients of the recombination process.
In the true ionosphere, more than a single ion species are present. Then a separate continuity equation is valid for each species and production and loss terms have
contribution from the ionospheric ion chemistry. This is because some ion species
disappear in chemical reactions producing ions of other kind. The ion chemistry produces links between the different continuity equations, which greatly complicates the
solving of the ion densities.
3.2. BASICS OF PLASMA DYNAMICS
3.2.2
43
Momentum equations and mobility
For the neutral gas, the momentum equation (2.11) is valid. In most cases, however,
we do not solve it but simply assume a neutral wind velocity u, which remains
unaltered. This is a reasonable approximation, because collisions with the dilute
ion and electron gas can change the neutral velocity only very slowly.
Ions and electrons experience Lorentz force density ±eni,e (E + vi,e × B), which
adds a new term in the momentum equations
!
ni mi
ne me
∂
+ vi · ∇ vi = ni mi g + eni (E + vi × B) − ∇pi − ni mi νi (vi −u) (3.2)
∂t
!
∂
+ ve · ∇ ve = ne me g − ene (E + ve × B) − ∇pe − ne me νe (ve −u). (3.3)
∂t
Here E is electric field, B is magnetic induction, pi and pe are the pressures of the
ion and electron gas, and the ion-neutral and electron-neutral collision frequencies
are denoted by νi and νe , respectively. Because force densities due to collisions
between ions and electrons are necessarily much smaller than the force densities due
to collisions with neutrals, these terms are dropped out from the very beginning.
The momentum equations are greatly simplified, if the plasma is in a stationary
and homogeneous state and gravitational field together with ∇p can be neglected.
Then the momentum equation for the ion gas is
e(E + vi × B) − mi νi (vi − u) = 0.
(3.4)
To solve the ion velocity, we use a notation
F = eE + mi νi u.
(3.5)
Then eq. (3.4) can be written as
vi =
1
(evi × B + F).
mi νi
(3.6)
A cross product with B gives
vi × B =
1
[e(vi · B)B − eB 2 vi + F × B]
m i νi
(3.7)
and a dot product with B gives
vi · B =
F·B
.
mi νi
(3.8)
By inserting this in eq. (3.7) we first obtain
1
e
vi × B =
(F · B)B − eB 2 vi + F × B
mi νi mi νi
(3.9)
44
CHAPTER 3. THE IONOSPHERE OF THE EARTH
and then, inserting this in eq. (3.6),
1
vi =
mi νi
e
e
(F · B)B − eB 2 vi + F × B + F .
mi νi mi νi
(3.10)
This can now be solved for vi . If eB is a unit vector in the direction B, i.e. B = BeB ,
"
eB
1+
mi νi
2 #
1
vi =
mi νi
"
eB
mi νi
2
#
eB
(F · eB )eB +
F × eB + F .
mi νi
(3.11)
Here
ωi =
eB
mi
(3.12)
is the angular gyro frequency, also called angular cyclotron frequency, of the ion.
Using this notation and noticing that 1/(mi νi ) = 1/(eB) · ωi /νi , we get
1
ω i νi
· 2
vi =
eB ωi + νi2
"
ωi
νi
2
#
ωi
(F · eB )eB +
F × eB + F .
νi
(3.13)
By dividing F into two components F = Fk + F⊥ this can also be written as
(
"
1
ω i νi
ωi
ωi
vi =
· 2
F⊥ +
F⊥ × e B + 1 +
2
eB ωi + νi
νi
νi
2 #
)
Fk .
(3.14)
It is conventional to present this equation in matrix form



Fx
kiP kiH 0



vi =  −kiH kiP 0   Fy  ,
Fz
0
0 kik
(3.15)
where
1
eB
1
=
eB
1
=
eB
kiP =
kiH
kik
ωi νi
ωi2 + νi2
ω2
· 2 i 2
ωi + νi
ωi
1
·
=
νi
mi νi
·
(3.16)
(3.17)
(3.18)
and the z axis points in the direction of B. This result is obtained by writing eq.
(3.14) in component form with F⊥ = Fx ex + Fy ey and Fk = Fz ez .
Eq. (3.15) can be written briefly as
vi = ki · F = ki · (eE + mi νi u).
(3.19)
Here ki is the ion mobility tensor. The tensor presentation is a convenient mathematical description of the result in which the velocity points in a direction different
3.2. BASICS OF PLASMA DYNAMICS
45
Ion mobilities (blue) and electron mobilities (red)
500
450
400
height (km)
350
300
keP
kiP
kiH
IkeHI
kiII
keII
250
200
150
100
kiH
50 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
s/kg
Figure 3.4: Ion and electron mobility tensor components calculated for Tromso,
1.4.2002, 24 UT.
from the direction of the effective driving force F. The mobility tensor is composed
of three different mobilities, the Pedersen mobility kiP , the Hall mobility kiH and the
parallel mobility kik (Fig. 3.4). It can be shown that the mobility tensor is invariant
in rotation around the z axis. This means that the directions of the x and y axes
can be freely chosen.
In the frame of reference where the neutral air is at rest, the neutral wind velocity
0
u = 0, the ion velcity vi0 = vi − u and E0 = E + u × B (Galilean transformation).
Eq. (3.19) must be valid in all inertial frames of reference. Therefore
vi0 = ki e · E0
(3.20)
vi − u = ki · e(E + u × B)
(3.21)
vi = ki · e(E + u × B) + u.
(3.22)
so that
and
Eqs. (3.19) and (3.22) are alternative presentations of the ion velocity.
The electron velocity can be calculated in a similar manner and therefore the
result can be written directly. One must, however, notice that e must be replaced by
−e and the angular gyrofrequency for both electron and ion are taken as a positive
46
CHAPTER 3. THE IONOSPHERE OF THE EARTH
quantity. Thus the components of the electron mobility tensor are
ω e νe
1
· 2
eB ωe + νe2
1
ω2
= −
· 2 e 2
eB ωe + νe
1
1 ωe
·
=
,
=
eB νe
m e νe
keP =
(3.23)
keH
(3.24)
kek
(3.25)
where
ωe =
eB
,
me
(3.26)
and the two expressions for the electron velocity are
ve = ke · (−eE + me νe u)
(3.27)
ve = −ke · e(E + u × B) + u.
(3.28)
At altitudes where the electron collision frequency is much smaller than the
electron gyrofrequency, keP ≈ 0 and keH ≈ −1/(eB). Then electrons move at a
velocity ve⊥ = E × B/B 2 in a plane perpendicular to B. This happens practically
500
Ion (blue) and electron (red) collision and gyro frequencies
450
400
height
350
νin
νen
ωiB
ωeB
300
250
200
150
100
50 −2
10
0
10
2
10
4
Hz
10
6
10
8
10
Figure 3.5: Ion and electron collision and gyro frequencies calculated for Tromso,
1.4.2002, 24 UT. Here νin = νi , νen = νe , ωiB = ωi and ωeB = ωe .
3.2. BASICS OF PLASMA DYNAMICS
47
at all ionospheric heights above the D region (Fig. 3.5). A similar drift is also
experienced by ions, but the ion collision frequency becomes smaller than the ion
gyrofrequency only in the F region (Fig. 3.5). So, in the F region (at altitudes
above about 200 km) the ions and electrons move together in a direction, which
perpendicular both to the electric field E and magnetic field B, the so called Ecross-B drift
vi,e⊥ =
E×B
B2
(3.29)
In the E region, electrons and ions move in a different way under the action of the
electric field. This leads to electric currents, which will be discussed in Chapter 4.
The parallel electron mobility is very high in the ionosphere. This is also a
consequence of the small electron mass. Thus electrons can easily move along the
magnetic field and they try to short circuit field aligned electric fields, which leads
to the fact that the electric field tends to be perpendicular to the magnetic field.
When field-aligned electric field is zero, the field aligned ion velocity calculated
from eq. (3.15) is
1
· mi νi uz = uz .
(3.30)
viz = kik Fz =
mi νi
Thus the neutral wind drives ions in the direction of the magnetic field and the ion
velocity in this direction will be adjusted to the value of the neutral wind component.
3.2.3
Ambipolar diffusion
Diffusion of charged particles in weakly ionised plasma contains one new aspect
which is not encountered in neutral gas. Due to the different masses of ions and
electrons, these two particle species tend to diffuse at different rates. This, however,
would lead to charge separation and generation of large electric fields. As a matter
of fact, a weak electric field will develop, which will adjust itself to a value which
forces ions and electrons to the same diffusion rate.
Let us first consider unmagnetised plasma (B = 0). Due to their light mass,
electrons are very mobile and they can quickly short circuit any external electric
field. The small electron mass also implies that, in the momentum equation (3.3),
ne me νe (ve − u) as well as me g can be put to zero. When the convective derivative
is also put to zero (stationary state and weak spatial variation in electron velocity),
the momentum equation for electrons is simply
ED = −
kB Te
1
∇pe = −
∇ne .
ene
ene
(3.31)
Here it is assumed that the electron gas obeys the ideal gas law and the electron
temperature has no spatial variation. The result shows that a stationary state
implies the existence of an electric field, which depends on the gradient of electron
density.
48
CHAPTER 3. THE IONOSPHERE OF THE EARTH
When the neutral air is at rest, the ion momentum equation then reads
eED + mi g − mi νi vi −
1
∇pi = 0.
ni
(3.32)
Next we divide the ion velocity into two components vi = viC +viD in such a manner
that
mi g − mi νi viC = 0.
(3.33)
This gives the convection velocity
viC =
g
.
νi
(3.34)
The diffusion velocity viD then obeys the equation
eED − mi νi viD −
kB Ti
∇ni = 0.
ni
(3.35)
Here Ti is the ion temperature. By inserting the diffusion electric field from eq.
(3.31) we then obtain the diffusion flux density of ions
ni viD = −
kB (Ti + Te )
∇ni = −Da ∇ni
mi νi
(3.36)
Because the plasma must keep charge neutrality, the diffusion flux densities of electrons and ions must be identical.
Eq. (3.36) defines the ambipolar diffusion coefficient
Da =
kB (Ti + Te )
.
mi νi
(3.37)
We notice that, if electrons and ions have the same temperatures, the ambipolar
effect doubles the diffusion coefficient compared to the neutral gas.
In magnetised palsma diffusion becomes anisotropic. By neglecting collisions
and gravity the stationary electron momentum equation reads
−e(E + ve × B) −
kB Te
∇ne = 0.
ne
(3.38)
We divide the electric field in two components E = E0 + ED in such a manner
that
E0 = −ve × B.
(3.39)
Obviously E0 is perpendicular to B. Eq. (3.39) gives the relation between electric
field and electron velocity in a homogeneous plasma (∇ne = 0) with no electron collisions. Then the electron velocity, electric field and magnetic field are perpendicular
to each other. Then, we get from eq. (3.38)
−eED −
kB Te
∇ne = 0 .
ne
(3.40)
3.2. BASICS OF PLASMA DYNAMICS
49
This gives
ED = −
kB Te
∇ne .
ene
(3.41)
Eq. (3.41) indicates that inhomogeneities in the electron density create and additional electric field ED . This is the same diffusion electric field, which is generated in
unmagnetised plasma (eq. (3.31)). The difference is that, in unmagnetised plasma,
diffusion electric field is the only electric field, whereas in magnetised plasma an
electric field perpendicular to the magnetic field is possible.
The stationary ion momentum equation can now be written as
e(E0 + ED + vi × B) + mi g − mi νi (vi − u) −
kB Ti
∇ni = 0.
ni
(3.42)
As above, we use convection and diffusion velocities vi = viC +viD in such a manner
that
e(E0 + viC × B) + mi g − mi νi (viC − u) = 0.
(3.43)
This is mathematically similar to eq. (3.4) and therefore its solution can be
written directly. One only has to replace u by u + g/νi . Hence the convection
velocity is
viC = ki · (eE0 + mi νi u + mi g).
(3.44)
The diffusion velocity must obey the equation
e(ED + viD × B) − mi νi viD −
kB Ti
∇ni = 0.
ni
(3.45)
By inserting the diffusion electric field from eq. (3.41) we obtain
eviD × B − mi νi viD −
kB (Ti + Te )
∇ni = 0.
ni
(3.46)
Again, this equation is mathematically similar to eq. (3.4) and the solution can be
written when we replace eE + mi νi u by −kB (Ti + Te )∇ni /ni . Therefore, the ion
diffusion velocity is
kB (Ti + Te )
viD = −
ki · ∇ni
(3.47)
ni
and the diffusion flux density is
ni viD = −Da · ∇ni ,
(3.48)
Da = kB (Ti + Te )ki
(3.49)
where
is the ambipolar diffusion tensor. This results shows the close connection between
the ion mobility tensor and the ambipolar diffusion tensor.
The diffusion velocity can now be calculated from eq. (3.47). In the ionosphere
we can assume that ∇ni is vertical, pointing either upward or downward, and the
50
CHAPTER 3. THE IONOSPHERE OF THE EARTH
x
∆
ni
N
I
B
h
y
z
Figure 3.6: Vertical density gradient and inclined magnetic field. N points to geomagnetic north in the horizontal plane, h is upwards, B is along the z axis, x
axis is located in the magnetic meridional plane and y completes the right-handed
coordinate system.
geomagnetic field has an inclination angle I. When altitude is denoted by h (see
Fig. 3.6),
dni
dni
eh =
(cos Iex − sin Iez ),
(3.50)
∇ni =
dh
dh
and

viD
 

DaP DaH 0
cos I
1 dni 
1

0 
0 
= − Da · ∇ni = − ·
 −DaH DaP
·
.
ni
ni dh
0
0
Dak
− sin I
(3.51)
After matrix multiplication we get the diffusion velocity in the xyz coordinate system

viD

DaP cos I
1 dni 

=− ·
 −DaH cos I  .
ni dh
−Dak sin I
(3.52)
This indicates that, although vertical density gradient is assumed, the diffusion
velocity has a horizontal component. The upward vertical component of diffusion
velocity is given by
viDh = viDx cos I − viDz sin I = −
1 dni
·
(DaP cos2 I + Dak sin2 I).
ni dh
(3.53)
Hence diffusion velocity is opposite to the direction of the density gradient. Both
Pedersen and parallel diffusion coefficient affect vertical ambipolar diffusion. If the
magnetic field is vertical (i.e. at geomagnetic poles), cos I = 0 and sin I = 1, so that
no Pedersen effect affects the diffusion. Above the magnetic equator the opposite
is true; cos I = 1 and sin I = 0, and vertical diffusion is solely due to the Pedersen
effect.
3.3. ABSORPTION OF IONIZING RADIATION
51
z
z+dz
z
χ
dz
χ
I+dI
I
ds = -secχ ·dz
s
Figure 3.7: Decrease of radiation intensity I in a slab of thickness ds.
The ratio of parallel and Pedersen ambipolar diffusion coefficients is
Dak
kik
ωi ωi2 + νi2
ω2 + ν 2
ω2
=
=
·
= i 2 i = 2i + 1.
DaP
kiP
νi
ωi νi
νi
νi
(3.54)
This approaches infinity when νi approaches zero and unity when νi approaches
infinity. Hence, at intermediate inclination angles I, field-aligned ambipolar diffusion
dominates at low collision frequencies, but at high collision frequencies field-aligned
and Pedersen diffusion become equally important.
3.3
Absorption of ionizing radiation
Let the intensity of the solar photon radiation at wavelength λ be I(λ, z) in units of
photons m−2 s−1 at an altitude z. Assume a slab with an area A and thickness ds,
perpendicular to the incident radiation (Fig. 3.7). The number of neutral molecules
in the slab is nAds, where n is the number density of the neutral molecules (single
species assumed). If the absorption cross section of a single molecule in units of
m2 is σ, the the total absorption cross section of the slab is σnAds. The number
of photons incident to the slab per unit time is IA, and the number of absorbed
photons per unit time is IσnAds. The decrease of intensity is given by the number
of absorbed photons per unit time divided by the area A. Therefore dI = −nσIds
and
dI
= −nσI .
(3.55)
ds
Absorption cross section σ is a measure for the probability of an absorption process. Generally, the absorption cross section depends on the wavelength for a given
molecule or atom.
If the ionization enters the atmosphere at a zenith angle χ,
ds = −
dz
= − sec χdz.
cos χ
(3.56)
52
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Then eq. (3.55) gives
dI
= −σnds = σn sec χdz,
(3.57)
I
where sec χ = 1/ cos χ. Using the notation I∞ for the intensity at the top of the
ionosphere (z = ∞) we then obtain
ZI
I∞
z
Z
dI
= σ sec χ n(z)dz.
I
∞
(3.58)
The integration gives

I(z) = I∞ exp −σ sec χ
Z∞
z

n(z)dz  = I∞ exp(−σ sec χ NT ) = I∞ exp(−τ ), (3.59)
where NT is the total number of neutral particles from height z to infinity per unit
area and the optical depth τ is defined by
τ (z) = σ sec χNT (z) .
(3.60)
Each wavelength will have its own optical depth. Note also, that optical depth is not
a constant, but a parameter replacing altitude in eq. (3.59): it takes into account
absorption produced by all the electrons the radiation has passed when entering
altitude z.
In the case of an isothermal atmosphere (with a single molecular species as we
have assumed so far)

I(z) = I∞ exp −σ sec χ · n0
Z∞

exp(−z/H)dz 
z
= I∞ exp[−σ sec χ · Hn0 exp(−z/H)] .
(3.61)
The optical depth is now
τ (z) = σ sec χ n(z)H.
(3.62)
The real atmosphere is composed of different neutral particles, and each particle species has an own density and wavelength-dependent absorption cross section.
Therefore, in a multicomponent atmosphere, eq. (3.58) can be written as
ZI
I∞
z
Z
X
dI
= sec χ
σj nj (z)dz.
I
j
∞
(3.63)
Here nj and σj are the particle density and absorption cross section of gas species
j, respectively. This gives

I(z) = I∞ exp − sec χ
X
j
σj
Z∞
z

nj (z)dz  = I∞ exp(−τ ),
(3.64)
3.3. ABSORPTION OF IONIZING RADIATION
53
Figure 3.8: Absorption cross sections for N2 (top) and O (bottom) in the wavelength
region from 0 to 100 nm and in values from 10−20 to 10−15 cm2 .
54
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Figure 3.9: Absorption cross section for O2 in the wavelength region from 0 to 200
nm and in values from 10−23 to 10−16 cm2 .
where the optical depth is
τ (λ, z) = sec χ
X
j
σj (λ)
Z ∞
z
nj (z 0 )dz 0 .
(3.65)
Figures 3.8 and 3.9 display the absorption cross sections for typical neutral constituents as a function of wavelength. Absorption takes place at UV and X ray
wavelengths and is highly structured for molecular species.
3.4
Ionization rate
The classical theory of ionospheric ionization by means of photoionization was presented by Chapman. To obtain the classical Chapman formula for the ion production
rate q, this theory assumes that the atmosphere is isothermal and obeys the hydrostatic equation so that the scale height H is independent of altitude. We also assume
that the atmosphere is composed of a single neutral species and the absorption cross
section is constant. Constant cross section is equivalent to assuming monochromatic
radiation.
It was concluded above that the total number of photons absorbed per unit time
in a slab with an area A and thickness s is IσnAds. Hence the total number of
photons absorbed per unit volume per unit time is nσI. However, not all their
energy goes to the ionization process. The ionization efficiency η is the (statistical)
number of photoelectrons produced per photon absorbed. Then the number of photo
3.4. IONIZATION RATE
55
electrons produced per unit volume per unit time, the ionization rate q (called also
the ion production function) can be expressed as
q = −η
dI
= ηnσI = ηnσI∞ exp(−τ ) .
ds
(3.66)
According to eq. (2.59), we write the neutral density profile as
n = nm,0 exp −
z − zm,0
H
= nm,0 e−h ,
(3.67)
where nm,0 is the molecular density at the reference height zm,0 and h = (z −zm,0 )/H
is the distance from the reference height, scaled by H. Then, according to eq. (3.62),
τ = σ sec χ Hn(z) = σ sec χ Hnm,0 e−h
(3.68)
and the ionization rate is
q(χ, h) = ηn(h)σI∞ e−τ (h) = ηnm,0 σI∞ exp −h − sec χ · σHnm,0 e−h .
(3.69)
The expression is still simplified if we now fix the reference height so that
σHnm,0 = 1.
(3.70)
Then
ηI∞
exp −h − sec χe−h
H
ηI∞
−h
exp 1 − h − sec χe
=
He
−h
= q exp 1 − h − sec χe
,
q(χ, h) =
(3.71)
m,0
where
η I∞
I∞
= ησnm,0
.
H e
e
If we still choose a height variable h0 = h − ln sec χ, we obtain
(3.72)
qm,0 =
0
q(χ, h) = qm,0 cos χ · exp 1 − h − e−h
0
.
(3.73)
In terms of variable h0 , the ionization rate can be presented for all zenith angles χ
with a single shape, which only has a scaling factor cos χ for different zenith angles.
The expressions in eqs. (3.71) or (3.73) are called the Chapman production function.
Since n increases and I decreases as we go down in altitude, the ion production
rate q has necessarily a maximum at some altitude. From eq. (3.73) we can show
that the maximum value is encountered at an altitude
h0m = 0
⇒
hm = ln sec χ .
(3.74)
56
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Since on the other hand hm = (zm − zm,0 )/H, we get
zm = zm,0 + H ln sec χ .
(3.75)
This equation relates the altitude of maximum ionization rate zm to the reference
height zm,0 . For zero zenith angle, χ = 0, zm is the same as the selected reference
height zm,0 . The notation selected in eq. (3.67) becomes now clear: in subscript
zm,0 , m refers to the maximum ionization rate and 0 refers to the zero zenith angle.
Thus, the maximum value of ionization rate is
qm = qm,0 cos χ.
(3.76)
This shows that the maximum production is encountered at χ = 0, i.e. when the
sun is shining overhead. Then also the maximum is encountered at lowest altitudes.
When the zenith angle increases, the maximum rises to higher altitudes and the
maximum value decreases (Fig. 3.10). It is obvious that, when the zenith angle is
high enough, the curvature of the Earth and the atmosphere should be taken into
account and theory should be modified. This is because, if the atmosphere would be
flat, the ray path in the atmosphere would approach to infinity when the elevation
angle would approach zero.
At very large values of h (z zm,0 ) the profile in eq. (3.71) takes the form
z
q = qm,0 exp −
H
(3.77)
Figure 3.10: Chapman production profiles for different solar zenith angles.
3.4. IONIZATION RATE
57
Figure 3.11: The altitude of unit optical depth calculated for zero zenith angle.
Ionization threshold wavelengths for several molecules and atoms are marked by
arrows (as well as the Ly-α wavelength).
so that well above the maximum the ionization rate profile decays with height in
the same way as the atmospheric density. This holds for h > 2, i.e. at distances
more than two scale heights above zm,0 . This relation arises because at those heights
the intensity of the radiation is only weakly reduced, and the rate of production is
essentially proportional to the gas density.
Eq. (3.68) can be used to show that at the altitude of maximum ionization rate
zm , the optical depth τm = 1. At the very same height the intensity of radiation
has a value Im = I∞ /e according to eq. (3.59). Figure 3.11 shows the altitude of
unit optical depth at wavelengths below 300 nm. Some spectral wavelengths are
absorbed very strongly, but the hydrogen Lyman-α at 121.5 nm penetrates to the
D region, where it ionizes NO.
The calculation of ion production rate in a real atmosphere with an incident
solar illumination with its true spectrum is complicated. The upper atmosphere
is composed of different gases with different scale heights and the solar radiation
spectrum consists of a myriad of lines and bands with different intensities (which
vary with solar activity). The different constituents in the atmosphere have different
ionization and dissociation threshold energies. Table 3.1 displays these energies in
eV for some of the most common neutral particles and the corresponding wavelength
(Vp = hν = hc/λ) which fall mostly in the EUV regime. Photons with energies larger
than Vp or wavelengths shorter than λ can ionize or dissociate the given species.
Dissociative ionization threshold wavelengths for O2 and N2 are 662 Å and 510 Å,
respectively (not shown in Table 3.1).
58
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Table 3.1
The approach is to assume a neutral atmosphere model with height distribution
of some of the major gases and the for finite number of wavelength steps to derive
the individual production profiles. Two examples are shown
in Fig.
3.12ofand
Chemical
effects
high Fig.
energy particle precipita
3.13.
For atomic species, the ionization efficiency η is unity so that the whole energy
of the ionizing photon goes into producing ion-electron pair(s). For molecules, a
photon may produce dissociation or ionization, so the ionization efficiency η ≤ 1,
Normal ionisation profile
Daytime ionisation
due to daytime
solar photons,
photoelectrons,
andphotons,
GCR photoelecFigure 3.12:rates
Calculated
ionisation
rates due to solar
(from
during solar
maximum
at latitude
70 equinox,
N, SZA at
= 69.6
trons,
and GCRequinox,
during solar
maximum
latitude
70P.Verronen,
N, SZA =2006)
69.6 (P.
Verronen, 2006).
SGO, E.Turunen, NordAurOpt Workshop 21.2.2007
3.4. IONIZATION RATE
59
Figure 3.13: Calculated ionisation rates (Hinteregger et al., 1965).
and η depends on the wavelength of the photon.
There is still another complexity associated with photoionization, and it caused
by the energy of the photoelectron. In general, the energy of the ionizing radiation
hν is much greater than the threshold energy Vp , and the excess energy will then
be left partly to the kinetic energy of the photoelectron and partly to the energy
of the excited ion. The photoelectron may then in turn produce further ionization.
Laboratory experiments have shown that the mean energy ¯ lost by a photoelectron
per ionization is almost constant and greater than the ionization energy. Hence the
energy of a photon consumed is
hν = V̄p + (Ne − 1)¯,
(3.78)
where Ne is the approximate number of electrons produced in the cascade of ioniza-
60
CHAPTER 3. THE IONOSPHERE OF THE EARTH
tion processes. This gives
hν − V̄p
.
(3.79)
¯
Table 3.2 gives values of ¯ for an ion-electron pair production by electron impact
ionization for different neutral species. It is customary to set V̄p = 15 eV and ¯ = 34
eV in air.
Ne = 1 +
Table 3.2
3.5
Recombination
After ions and free electrons are produced in the ionization processes, they will take
part in chemical reactions in which they may disappear or produce other types of
ions. Some of the ionization will also be carried away by transport mechanisms such
as diffusion, neutral winds and electric fields. This is controlled by the continuity
eq. (3.1) containing both the transport, production and loss terms. In this section we
will discuss the loss term which, together with production, determines the electron
density in the absence of transport.
When the convection term in the continuity equation is neglected and electron
density is constant, production and loss are in balance, i.e. there is a photochemical
equilibrium,
qi = li .
(3.80)
Electrons may recombine directly with positive ions to make neutral atoms or
molecules in the reaction
X + + e −→ X + hν
(3.81)
which leads to a production of a photon. This is called radiative recombination.
The probability of this reaction is low because of the demand put by simultaneous
conservation of energy and momentum.
Another possibility is dissociative recombination of a diatomic ions according to
the reaction
XY + + e −→ X + Y .
(3.82)
The conservation laws are more easily satisfied by this reaction which produces two
neutral atoms.
3.5. RECOMBINATION
61
The speeds of reactions like those in eqs. (3.80) and (3.81) are expressed in terms
of reaction coefficients. The probability of a single electron to recombine with must
be proportional to the ion density, since recombination implies that the electron must
hit an ion. The number of electrons lost per unit volume and unit time must also be
proportional to the number of electrons which can recombine, that is proportional
to electron density. Since the electron and ion densities are identical, the loss term
must be
li = αn2e ,
(3.83)
where α is called the recombination coefficient or reaction coefficient. Magnitudes
of the reaction coefficients for dissociative recombination are of the order of 10−13
m3 s−1 and for the radiative recombination 10−18 m3 s−1 . These numbers indicate
that dissociative recombination is much faster than radiative recombination.
When the Chapman production function from eq. (3.71) is inserted in the photochemical equilibrium (3.80) and dissociative recombination is assumed, the result
is
−h
= αn2e .
(3.84)
qm,0 exp 1 − h − sec χ e
The electron density solved from this equation is
r
ne =
q
=
α
r
qm,0
1
exp (1 − h − sec χ e−h .
α
2
(3.85)
When neglecting height variations of α (actually α does not depend directly on
height but on temperature) we find that the electron density has a maximum when
e−h = cos χ.
(3.86)
showing that the electron density maximum is given by
r
nm (χ) =
qm,0 √
√
cos χ = nm,0 cos χ.
α
(3.87)
We find that the maximum electron density is proportional to the square root of the
solar zenith angle. The electron density profile in eq. (3.85) is called the Chapman
α profile. It is representative of the ionospheric E region where the most common
+
+
ions are O+
2 and NO . The presence of NO might be surprising at the first sight,
since the neutral atmosphere contains only minute amounts of neutral NO. It turns
out that NO+ is produced by the E region ion chemistry. This will be discussed
later in detail.
If electrons are lost by attachment to a molecule
M + e −→ M − ,
(3.88)
the electron loss rate is only proportional to ne and can be expressed as
le = βne ,
(3.89)
where β is the attachment coefficient proportional to the density of neutral molecules,
nM . In this case the plasma consists of electrons and positive and negative ions,
62
CHAPTER 3. THE IONOSPHERE OF THE EARTH
and the positive ion density is the sum of electron and negative ion densities. By
assuming the Chapman production function, in equilibrium the electron density
profile is given by
qm,0
−h
.
(3.90)
exp 1 − h − sec χ e
ne =
β
This is called the Chapman β profile. As we can see in the next section, any loss
process resulting in a coefficient directly proportional to ne can be characterized as
β type.
If we again neglect height variations in β (which is not a good assumption since
the density of neutral molecules decreases with height) we get for the maximum
electron density
qm
nm =
cos χ = nm,0 cos χ .
(3.91)
β
In the E region, dissociative recombination with molecular ions is the most important loss mechanism for electrons. Production of negative ions takes place in D
region, where ion chemistry is complicated. In the F2 region loss mechanisms associated with atomic oxygen ions dominate. Between the E and F2 regions lies the F1
region, where both dissociative recombination and a reaction containing atomic oxygen ions are active and one of them dominates, depending on the values of reaction
rate coefficients.
3.6
O dominant ionosphere
In the F-region (above 150 km) the dominant ions formed by solar irradiance are
the O+ ions. In the following we discuss a simplified model which assumes that
these ions are the only species resulting from solar radiation and the other ions are
produced as a result of chemical reactions.
The O+ ions can be lost by several reactions. One possibility would be radiative
recombination with electron
αr
O+ + e −→
O + hν,
(3.92)
where αr is the radiative recombination coefficient (Table 3.3). This process is slow
and is neglected in the following discussion. A more rapid loss occurs through two
reactions,
k1
O+ + N2 −→
N O+ + N
(3.93)
followed by
α1
N O+ + e −→
N + O,
(3.94)
where k1 is the rearrangement coefficient and α1 is the dissociative recombination
coefficient (Table 3.3). The rate of process (3.93) is k1 [O+ ][N2 ] and the rate of
process (3.94) is α1 [N O+ ]ne . Square brackets are used to indicate the concentration
(number density) of each species.
3.6. O DOMINANT IONOSPHERE
O+
2
63
Another loss process for O+ ions is through a similar chain of reactions with
ions. First O+ makes a charge exchange with an O2 molecule
k2
O+ + O2 −→
O + O2+
(3.95)
and then O+
2 recombines with an electron and dissociates according to
α2
O2+ + e −→
O+O .
(3.96)
The rate of process (3.95) is k2 [O+ ][O2 ] and the rate of process (3.96) is α2 [O2+ ]ne .
In equilibrium conditions, and when radiative recombination is neglected, the
production of atomic oxygen ions must be equal to their loss qiven in eqs.(3.93) and
(3.95)
q(O+ ) = (k1 [N2 ] + k2 [O2 ])[O+ ] = l(O+ ).
(3.97)
.
64
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Table 3.3: Typical reaction rates. Here Tr = (Ti + Tn )/2 (Brekke, 1997).
A similar equilibrium condition valid for molecular oxygen and nitric oxide ions.
When their production by solar radiation is neglected, the production of O+
2 given
by eq. (3.95) must be equal to loss given by eq. (3.96)
q(O2+ ) = k2 [O2 ][O+ ] = α2 [O2+ ]ne = l(O2+ ) .
(3.98)
Production of NO+ given by eq. (3.93) must be equal to loss given by eq. (3.94)
q(N O+ ) = k1 [N2 ][O+ ] = α1 [N O+ ]ne = l(N O+ ).
(3.99)
Since charge neutrality must hold
ne = [O+ ] + [N O+ ] + [O2+ ].
(3.100)
From eq. (3.99),
[N O+ ] =
k1 [N2 ] +
[O ]
α 1 ne
(3.101)
and from eq. (3.98)
[O2+ ] =
k2 [O2 ] +
[O ].
α2 ne
(3.102)
By inserting these in eq. (3.100) we obtain
ne = [O+ ] +
k1 [N2 ] +
k2 [O2 ] +
[O ] +
[O ].
α 1 ne
α2 ne
(3.103)
The concentration of atomic oxygen ions, solved from this equation, is
[O+ ] =
ne
.
1 + k1 [N2 ]/(α1 ne ) + k2 [O2 ]/(α2 ne )
(3.104)
Inserting this in eq. (3.97) gives
q(O+ ) =
k1 [N2 ] + k2 [O2 ]
ne .
1 + k1 [N2 ]/(α1 ne ) + k2 [O2 ]/(α2 ne )
(3.105)
Using a notation β 0 for the coefficient of ne , this can also be written as
q(O+ ) = β 0 ne .
(3.106)
The values of k1 and k2 are of the order of 2 · 10−18 m3 /s while α1 and α2 are of
the order of 10−13 m3 /s. At altitudes above 250 km [N2 ] < 1015 m−3 , [O2 ] ≈ 1014
m−3 and ne ≈ 1011 m−3 , so α1 ne and α2 ne will be much larger than k1 [N2 ] and
k2 [O2 ], respectively. Therefore, in the upper F region, β 0 can be approximated by
β, which is
β = k1 [N2 ] + k2 [O2 ].
(3.107)
3.6. O DOMINANT IONOSPHERE
65
Then, from eq. (3.106),
ne =
q(O+ )
q(O+ )
=
β
k1 [N2 ] + k2 [O2 ]
.
(3.108)
Since N2 and O2 have almost similar scale heights we have approximately
z − z0
β = β0 exp −
,
H
(3.109)
where β0 is the loss coefficient at reference height z0 and H is the scale height.
In the lower F region, the reactions involving charge rearrangement between O+
and O2 and N2 will become more abundant because of the increase of these neutral
species. Therefore β 0 must be used instead of β. According to eqs. (3.105) and
(3.106)
k1 [N2 ] + k2 [O2 ]
1 + k1 [N2 ]/(α1 ne ) + k2 [O2 ]/(α2 ne )
β
=
1 + k1 [N2 ]/(α1 ne ) + k2 [O2 ]/(α2 ne )
β0 =
(3.110)
so that
1
1
1
=
1 + (k1 [N2 ]/α1 + k2 [O2 ]/α2 )
0
β
β
ne
1
1
+
=
,
β αeff ne
where
αeff =
β
.
k1 [N2 ]/α1 + k2 [O2 ]/α2
(3.111)
(3.112)
Then eq. (3.106) gives
q(O+ )
1
1
+
ne =
=
q(O
)
+
β0
β αeff ne
!
.
(3.113)
By studying the corresponding reaction coefficients (Table 3.3) and neutral densities it can be seen that below, say, 150 km the condition β αeff ne is easily valid.
Then an approximate value for ne can be solved from eq. (3.108) and it is
s
ne =
q(O+ )
αeff
.
(3.114)
Since β decreses with height and αeff ne increases with height below the F region
maximum, there must be a transition height zt such that
βt = αeff net ,
(3.115)
66
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Figure 3.14: (a) Ionization rate q as a function of altitude, (b), (d)-(f) ne as a function of altitude assuming different recombination processes and (c) recombination
coefficient β as a function of altitude.
where net is the electron density at zt . Above
this altitude electron density varies
q
as q/β (beta-type region) and below it as (q/αef f (alpha-type region).
Fig. 3.14 displays schematically the formation of the F1 region. Fig. 3.14 (a)
shows the Chapman production function q. It has a maximum at an altitude zm .
Fig. 3.14 (b) shows the electron density profile which would be generated if the ions
were diatomic and would be recombined by means of dissociative recombination
with a constant recombination coefficient α. This would produce a peak electron
density at the same altitude zm as the production peak. Fig. 3.14 (c) shows the
height variation of loss rate β, which decreases with altitude due to the decrease of
neutral density. Fig. 3.14 (d) shows the electron density which would be produced if
the ions were monoatomic and would decay by means of charge exchange (and the
successive recombination would be quick).
In the bottomside F region both decay processes are active. There is some transition altitude zt where the two recombination mechanisms are equally important;
dissociative recombination dominates below this altitude and charge exchange dominates above. The shape of the resulting total electron density depends on whether
the transition altitude lies above or below the production peak. If the transition
altitude lies above the production peak (Fig. 3.14 (e)), a local maximum in the
electron density profile is produced at zm . In the opposite case (Fig. 3.14 (f)) only
a knee at zm will appear.
This is roughly the mechanism of the generation of the F1 region (see Fig. 3.1).
3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION
67
The condition in Fig. 3.14 (e), i.e. zt > zm can be fulfilled in the daytime under
sunspot maximum conditions. The observation of an F1 layer does not necessarily
imply the existence of a maximum in the electron density profile; the F1 layer is
usually rather seen as a bulge in the profile.
We have seen that, at distances larger than two scale heights above the ionization
maximum, the ion production profile obeys eq. (3.77). Therefore we can write
q(z) ∝ exp −
z
,
HO
(3.116)
where H0 is the scale height of oxygen atoms (which is assumed as the dominant
neutral species at high altitudes). Since N2 is more abundant than O2 , we can
approximate eq. (3.107) to give
z
β(z) ∝ exp −
HN2
!
,
(3.117)
where HN2 is the scale height of molecular nitrogen. According to eq. (3.108) we
then obtain
"
#
!
HO
z
q(z)
∝ exp
.
(3.118)
−1
ne (z) =
β(z)
HN2
HO
Since HO /HN2 = 7/4, this leads to
3z
ne (z) ∝ exp
.
4HO
(3.119)
This is an unreasonable result, because it would indicate that the electron density
increases with height at great altitudes. In the real ionosphere F region has a maximum at an altitude of several hundred kilometers after which the electron density
decreases. Obviously the assumption of photochemical equilibrium (photochemical
production is equal to chemical loss, eq. (3.79)) is not correct. This means that
the transport term in the continuity equation cannot be neglected. The physical
mechanism behind the non-negligible transport term is ambipolar diffusion in the
topside ionosphere.
3.7
3.7.1
Multicomponent topside ionosphere and diffusion
Diffusion without magnetic field
When diffusion cannot be neglected, the continuity equation for ions in a horizontally
stratified ionosphere is
∂ni ∂(ni viz )
+
= qi − li .
(3.120)
∂t
∂z
68
CHAPTER 3. THE IONOSPHERE OF THE EARTH
For simplicity, let us first neglect the geomagnetic field. Then, according to eqs.
(3.34), (3.36) and (3.37),
ni vi = ni viC + ni viD =
ni g
− Da ∇ni .
νi
(3.121)
This is valid in geomagnetic poles where the magnetic field does not affect the
vertical plasma motion. In a horizontally stratified ionosphere this equation can be
written as
!
∂ni
ni g
ni viz = −Da
+
.
(3.122)
∂z
νi Da
We define the plasma scale height
Hp =
kB (Ti + Te )
mi g
(3.123)
and note that Hp = νi Da /g where the ambipolar diffusion coefficient Da is given by
eq. (3.37). Then
#
"
ni
∂ni
.
(3.124)
ni viz = −Da
+
∂z
Hp
If production and loss approximately cancel in the continuity equation, a stationary
state is obtained when viz = 0, i.e. when the convection and diffusion velocities also
cancel so that the ions have no vertical motion. Hence
dni
ni
+
= 0.
dz
Hp
(3.125)
This can be easily integrated and the result is
z − z0
ni = ni0 exp −
Hp
!
(3.126)
where ni0 is the ion density at a reference height z0 . Hence the equilibrium ion
density profile is exponential. The decrease with altitude is determined by the
plasma scale height. Notice that, assuming equal ion and electron temperatures,
the scale height is twice as great as it would be without the ambipolar effect.
Actually several ion species like O+ , N+ , H+ and He+ are present in the topside
ionosphere. When this is taken into account, the basic theory of ambipolar diffusion
must be modified. The approximations of the electron momentum equation are still
the same as before, and therefore the diffusion electric field is given by eq. (3.41)
ED = −
kB Te
∇ne .
ene
(3.127)
When we have k ion species with densities nj , j = 1, 2, . . . k, charge neutrality
implies
ne =
k
X
j=1
nj .
(3.128)
3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION
69
The convection velocity of each ion species is still (see eq. (3.34))
vjC =
g
,
νj
(3.129)
and the diffusion velocity is obtained from the equation (see eq. (3.35))
eED − mj νj vjD −
kB T
∇nj = 0,
nj
(3.130)
where we have assumed that all ion species have the same temperature Ti . If the
total velocity for ion j is zero,
vj = vjC + vjD =
kB Ti
eED
g
−
∇nj +
= 0.
νj
mj νj nj
m j νj
(3.131)
By multiplying this by mj nj and calculating a sum over j we obtain
g
k
X
j=1
mj nj − kB Ti ∇ne + ene ED = 0,
(3.132)
where we have taken into account the fact that the sum of all ion densities is equal to
the electron density. When the diffusion electric field is inserted here, the equation
can be solved to give
where
k
m+ g
g
1 X
1
mj nj =
∇ne =
·
,
ne
kB (Ti + Te ) ne j=1
kB (Ti + Te )
(3.133)
k
1 X
mj nj
m+ =
ne j=1
(3.134)
is the mean ion mass. Because different ion species have different height distributions, these will affect the electron density via the height dependence of the mean
ion mass. In a horizontally stratified situation eq. (3.133) can be written as
1 dne
m+ g
=−
.
ne dz
kB (Ti + Te )
(3.135)
When inserting this in eq. (3.127) we get an alternative form for the diffusion electric
field Ez ,
m+ g
c
eEz = Te ·
= m+ g
,
(3.136)
(Ti + Te )
1+c
where
Te
c=
(3.137)
Ti
is the ratio of electron and ion temperatures. When eq. (3.135) is inserted in eq.
(3.131), one can solve (note ED = Ez k̂ and g = −g k̂)
1 ∂nj
mj g
m+ gc
1
=−
+
=−
nj ∂z
kB Ti (1 + c)kB Ti
Hj
c
m+
1−
·
1 + c mj
!
=−
1
.
δj
(3.138)
70
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Figure 3.15: Relative ion concentration profiles for typical topside ionosphere ions.
Difusive equilibrium for electrons and ions in temperature of 1200 K is assumed (solid
lines). Dashed lines correspond to photochemical equilibrium (Ratcliffe, 1972).
where Hj = kB Ti /(mj g). This describes the vertical distribution of the jth ion
species and we see that in diffusive equilibrium (vj = 0) the scale height δj is
different from Hj . If the mass of the jth ion is the same as the mean mass m+ and
if the temperature ratio c is unity, the scale height for this ion species is the same as
the plasma scale height given by eq. (3.123). The scale heights of heavier and lighter
ions are smaller and greater, respectively. Especially, when mj < c/(1 + c)m+ , the
scale height is negative. This means that the number density of the lightest ions
may increase with height. An example of the height distribution of some ions is
given in Fig. 3.15.
One should notice that exact results for electron and ion densities cannot be
calculated by integrating eqs. (3.135) and (3.138). This is because both equations
contain m+ which is height-dependent. As a matter of fact, the ion momentum
equations (3.130) are coupled by means of the diffusion electric field, which depends
on electron density, i.e. on all ion densities. In order to find all ion densities, the
whole set of coupled differential equations should be solved.
Another point worth noticing is that, in the above theory, it was assumed that the
vertical velocity of all ion species is zero. This is not exactly the case. In the topside
ionosphere the diffusion electric field points upwards. This field accelerates light
positive ions, which will have upward velocity. Inside the polar cap this phenomenon
is known as polar wind. The classical theory of polar wind is based on diffusion
electric field. The polar wind, however, also contains heavier ions, which means that
there must be other acceleration mechanisms present generating this phenomenon.
3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION
71
Finally, one should notice that ambipolar diffusion cannot be expressed in terms
of a single diffusion coefficient, when more than one ion species are present in the
ionosphere. Mathematically this is associated with the fact that the ion momentum
equations (3.130) are coupled, as discussed above.
3.7.2
Diffusion with magnetic field
When we take into account the geomagnetic field and assume a single ion species in
the topside F layer, we can start from eqs. (3.48), and (3.44), which give
viD
1
= Da · − ∇ni
ni
(3.139)
and
viC = ki · (mi g).
(3.140)
Here we have assumed that the electric field E0 and the neutral wind u in eq. (3.44)
are both zero.
When we assume identical electron and ion temperatures (Te = Ti = T ), the
ambipolar diffusion tensor Da = 2kB T ki , and therefore the convection velocity can
be written as
mi
g .
(3.141)
viC = Da ·
2kB T
According to eq. (3.53), the vertical component of the diffusion velocity is (here
we denote altitude by z instead of h)
!
viDz = (DaP
1 dni
.
cos I + Dak sin I) − ·
ni dz
2
2
(3.142)
Comaprison of eqs. (3.139) and (3.141) then shows that, obviously
viCz = (DaP
mi
cos I + Dak sin I) −
g
2kB T
2
2
(3.143)
so that the total vertical ion velocity is
!
viz = viDz + viCz = (DaP
1 dni
mi
cos2 I + Dak sin2 I) − ·
−
g .
ni dz
2kB T
(3.144)
In accordance with (3.54),
DaP
kiP
1
ωi νi
νi2
=
=
· 2
·
m
ν
=
.
i i
Dak
kik
eB ωi + νi2
ωi2 + νi2
(3.145)
When νi is much smaller than ωi , this ratio is small. This means that, at sufficiently high altitudes, field-aligned motion dominates and Pedersen motion can be
neglected. When this approximation is used, eq. (3.144) gives a vertical ion flux
density
!
2
ni viz = −Dak sin I
dni
ni mi
+
g = −Dak sin2 I
dz
2kB T
!
dni
ni
+
.
dz
Hp
(3.146)
72
CHAPTER 3. THE IONOSPHERE OF THE EARTH
This shows that, at geomagnetic poles, the flux density is equal to that with no
magnetic field and, at the geomagnetic equator, the flux density will drop to zero.
To be more precise, a small vertical flux density still exists at the geomagnetic
equator. This is due to the Pedersen component of the ambipolar diffusion tensor.
In a stationary situation ∂ni /∂t = 0. If the loss term is l = βni , the complete
continuity equation reads
d(ni viz )
= q − l = q − βni .
dz
(3.147)
In order to obtain the differential equation for ni , the ion flux density in eq. (3.146)
must be derivated and inserted here. For doing that, we have to know the hight
dependence of Dak . Since Dak is inversely proportional to the ion-neutral collision
frequency and the collision frequency is proportional to the neutral density, we can
write
z
Dak = Dak0 exp
,
(3.148)
H
where H is the scale heigth of the neutral density. When this is inserted in eq.
(3.146) and the derivative is calculated, the result is
"
1
d(ni viz )
= −Dak sin2 I
dz
H
dni
ni
+
dz
Hp
!
d2 ni
1 dni
+
+
.
2
dz
Hp dz
#
(3.149)
Then the continuity equation is
1
d2 ni
1
−Dak sin2 I
+
+
2
dz
H Hp
"
!
#
ni
dni
+
= q − βni .
dz
HHp
(3.150)
This is a linear differential equation of second order. For solving it, the height
dependence of the production should be known.
(i) At lower altitudes, where νi is large enough, we can use an approximation
Dak ≈ 0. Then also ni viz ≈ 0 and production and loss are in balance, i.e.
ni =
q
.
β
(3.151)
Since both q and β are proportional to neutral density and q is also affected by
intensity of solar radiation, q decreases more slowly with altitude than β does.
Then the ion density must grow with altitude at heights where recombination is
more important than diffusion (see Fig. 3.14).
(ii) At higher altitudes, both q and β are small and, in addition, even the small
production and loss terms tend to balance. In this case d(ni viz )/dz ≈ 0. Then eq.
(3.149) gives
!
d2 ni
1
1 dni
ni
+
+
+
= 0.
(3.152)
dz 2
H Hp dz
HHp
This is a linear homogenous differential equation of second order with constant
coefficients, and the standard method of solution is to use a trial function ni ∝
3.7. MULTICOMPONENT TOPSIDE IONOSPHERE AND DIFFUSION
73
exp(az). Then dni /dz = ani and d2 ni /dz 2 = a2 ni . This leads to a characteristic
equation
!
1
1
1
2
+
a+
= 0.
(3.153)
a +
H Hp
HHp
Using the basic properties of the equation of second order we immediately see that
a1 + a2 = −1/H − 1/Hp and a1 a2 = 1/(HHp ). Then, necessarily, a1 = −1/H and
a2 = −1/Hp , so that the general solution of eq. (3.152) is
z
ni = C1 exp −
H
!
z
+ C2 exp −
.
Hp
(3.154)
This shows that the ion density goes to zero at great heights. Then, according to
eq. (3.146), the ion flux density must do the same. This means that
!
z
dni
ni
Dak0 exp
+
= Dak0 ez/H ·
H
dz
Hp
!
C1 −z/H C2 −z/Hp C1 −z/H C2 −z/Hp
− e
−
e
+
e
+
e
=
H
Hp
Hp
Hp
!
1
1
−
Dak0 C1
Hp H
(3.155)
must approach zero. Since H and Hp are constants, this is possible only if C1 = 0.
Therefore
!
z
ni = C2 exp −
.
(3.156)
Hp
The result indicates that, at altitudes where production and loss are small or balance,
ion density must decrease with increasing altitude.
Since at lower altitudes ion density increases with altitudes according to eq.
(3.151) and at higher altitudes ion density decreases, a maximum of ion density
must be found at some altitude. At the density peak dni /dz = 0. Then, according
to eq. (3.146)
ni
ni viz = −Dak sin2 I ,
(3.157)
Hp
so that the ion velocity at the peak altitude is
viz = −
Dak sin2 I
.
Hp
(3.158)
In summary, we have found that by assuming β type loss process and by taking
into account ambipolar diffusion, we can explain formation of the F region peak and
the decrease in ion density above it. In general, the F region peak doesn’t form at
the altitude of maximum ionization rate zm .
74
3.8
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Ionization and recombination in the E region
The main ion species produced by solar radiation between 90 and 130 km are O+
2
and N+
.
We
shall
next
study
the
photochemical
equilibrium
for
these
ions.
2
The O+
2 ions will recombine with electrons by means of dissociative recombination
α2
O2+ + e −→
O+O
(3.159)
or they can carry out charge exchanges with neutral constituents such as NO and
N2 in reactions
k3
O2+ + N O −→
N O+ + O2
(3.160)
or
k4
N O+ + N O.
O2+ + N2 −→
(3.161)
k5
N2+ + O −→
N O+ + N.
(3.162)
with rate coefficients k3 and k4 , thus forming NO+ ions which are abundant in the
E region. NO+ ions can also be borne in reaction
Another loss mechanism for N+
2 is dissociative recombination
α3
N2+ + e −→
N + N.
(3.163)
The NO+ ions produced in reactions (3.159), (3.160) and (3.161) are also destroyed
by means of dissociative recombination
α1
N O+ + e −→
N + O.
(3.164)
These reactions are the most important ones which make the E region ion chemistry. The E region electron density is stationary, if production and loss of each ion
species are in balance.
In the above model of ion chemistry, O+
2 is produced only by means of direct
photoionization. The loss is by means of direct dissociative recombination and
charge exchange with neutral N2 and NO. Hence the balance equation between
production and loss is
q(O2+ ) = α2 [O2+ ]ne + k3 [O2+ ][N O] + k4 [O2+ ][N2 ].
(3.165)
Similarly, N+
2 is produced by means of direct photoionisation and lost by means
of direct dissociative recombination and in producing NO+ . Thus
q(N2+ ) = α3 [N2+ ]ne + k5 [N2+ ][O].
(3.166)
When direct photoionization of the minority neutral constituent NO is neglected,
NO+ is produced by the three reactions (3.159), (3.160) and (3.161). The production
is
q(N O+ ) = k3 [O2+ ][N O] + k4 [O2+ ][N2 ] + k5 [N2+ ][O].
(3.167)
3.8. IONIZATION AND RECOMBINATION IN THE E REGION
75
Figure 3.16: Daytime solar minimum ion profiles (Johnson, 1966).
The loss is only due to dissociative recombination, i.e.
l(N O+ ) = α1 [N O+ ]ne .
(3.168)
+
The total production of ions due to photoionization (O+
2 and N2 ) must be equal
to the production of electrons. Therefore
q(e) = q(O2+ ) + q(N2+ )
= (α2 [O2+ ] + α3 [N2+ ])ne + (k3 [N O] + k4 [N2 ])[O2+ ] + k5 [N2+ ][O]. (3.169)
From eqs. (3.167) and (3.168) we notice that the last terms in eq. (3.168) are
(k3 [N O] + k4 [N2 ])[O2+ ] + k5 [N2+ ][O] = α1 [N O+ ]ne .
(3.170)
q(e) = (α1 [N O+ ] + α2 [O2+ ] + α3 [N2+ ])ne .
(3.171)
Hence
Since charge neutrality must always hold,
[N O+ ] + [O2+ ] + [N2+ ] = ne
(3.172)
By introducing the relative abundance ratios of the respective ions
r1 = [N O+ ]/ne
r2 = [O2+ ]/ne
r3 = [N2+ ]/ne
(3.173)
76
CHAPTER 3. THE IONOSPHERE OF THE EARTH
we get
r1 + r2 + r3 = 1.
(3.174)
Then, from eq. (3.171) (and using ne instead of e)
q(ne ) = (α1 r1 + α2 r2 + α3 r3 )n2e .
(3.175)
By introducing the effective recombination coefficient αeff
αeff = α1 r1 + α2 r2 + α3 r3
(3.176)
we can write
q(ne ) = αef f n2e .
(3.177)
Formally, this suggests that the shape of the electron density profile in the E region
is an α function. One should, however, notice that the relative abundances are
height dependent and therefore the height dependence of the electron density profile
is not only due to the height dependence of the production.
Fig. 3.16 shows some rocket observations of various ion density profiles. It indicates that the O+ and N+
2 densities decrease very rapidly below 150 km. Hence, we
may neglect the N+
density
in E region and put r3 to zero so that
2
r1 + r2 = 1
(3.178)
m+ = r1 · mN O+ + r2 · mO2+ .
(3.179)
and the mean ion mass is
It is a common practice to use a mean mass of 30.5 u at these altitudes, which leads
to
30 r1 + 32 r2 = 30.5.
(3.180)
This corresponds to the assumption
r1 = 3/4
and r2 = 1/4
(3.181)
in eq. (3.180). Hence NO+ is assumed to be more abundant than O+
2 in the E region.
As seen in Fig. 3.16, this is not true at all altitudes. If the assumption by eq. (3.181)
is used, the effective recombination coefficient is
αeff = 0.75 α1 + 0.25 α2 ,
where α1 and α2 are given in Table 3.3.
(3.182)
3.9. RECOMBINATION TIME CONSTANT
3.9
3.9.1
77
Recombination time constant
F region
Let us study the continuity equation (3.23) and neglect the convection term as above,
but leave the time derivative of electron density. Then the continuity equation reads
dne
= qe − le ,
dt
(3.183)
where qe and le are the electron production and loss rates, respectively. In the F
region, where O+ dominates, the loss is proportional to electron density. Then
dne
= qe − βne .
dt
(3.184)
If qe is constant, this can be solved by separating the variables. Hence
Zne
ne0
t
Z
dne
= dt,
qe − βne
(3.185)
0
which gives
qe − βne
ln
qe − βne0
so that
ne =
!
= −βt,
qe qe − βne0
−
exp(−βt).
β
β
(3.186)
(3.187)
Assuming zero electron density at t = 0,
ne = ne∞ [1 − exp(−βt)],
where
ne∞ =
qe
.
β
(3.188)
(3.189)
If, on the other hand, production is switched off at t = 0, electron density will decay
exponentially according to
ne = ne0 exp(−βt).
(3.190)
In these cases the electron density will change with a time constant
τ=
1
β
.
(3.191)
Now, eq. (3.190) can be written as ne = ne0 exp(−t/τ ) and τ is the recombination
time constant in the F region. In the case of eq. (3.188), electron density will
approach to a value qe /β and in the case of eq. (3.190) to zero.
78
CHAPTER 3. THE IONOSPHERE OF THE EARTH
When qe (t) is not constant in eq. (3.185), we have a complete linear differential equation of first order. A standard way of solving it is to use the method of
integrating factors. For a linear differential equation
y 0 + p(x)y = r(x)
(3.192)
the integrating factor is
Z
p(x) dx .
exp
(3.193)
In the case of eq. (3.184) the integrating factor is
exp
Z
β dt = exp(βt)
(3.194)
By multiplying eq. (3.184) by this integrating factor we obtain
dne
exp(βt) + βne exp(βt) = qe (t) exp(βt).
dt
(3.195)
The equation can be written in a more simple form to give
d
[ne exp(βt)] = qe (t) exp(βt).
dt
(3.196)
This can now be integrated and the result is
0
ne exp(βt) + C =
Zt
qe (t) exp(βt) dt,
(3.197)
0
so that
ne =

 t
Z
exp(−βt)  qe (t) exp(βt) dt + C  .
(3.198)
0
By applying an initial condition ne = ne0 when t = 0 we see that C = ne0 , and
finally we obtain

ne = exp(−βt)  ne0 +
Zt

qe (t) exp(βt) dt .
(3.199)
0
Thus the time variation of ne depends on how the production varies with time.
Notice also that putting qe to a constant in eq. (3.199) (and ne0 as zero) leads to eq.
(3.188).
3.9.2
E region
In the E region the loss rate is αeff n2e , so that the continuity equation is
dne
= qe − αeff n2e .
dt
(3.200)
3.9. RECOMBINATION TIME CONSTANT
79
Again, if qe is constant, the variables can be separated and the resulting equation
can be integrated. The resulting function is hyperbolic tangent (this is left as an
exercise).
If we assume that there has been some production causing an electron density
ne0 and the production is switched off at t = 0, we obtain
Zne
ne0
t
Z
dne
= −αeff dt = −αeff t.
n2e
(3.201)
0
The integration on the left hand side gives
ne − ne0
= −αeff t,
ne ne0
(3.202)
which can be solved for ne . The result is
ne =
ne0
.
ne0 αeff t + 1
(3.203)
Hence there is no exponential decay in this case. The electron density is halved after
a time
1
t1/2 =
,
(3.204)
αeff ne0
or, equivalently, the decay time corresponding to eq. (3.191) (time when electron
density has decreased to 1/eth fraction)
τ=
e−1
ne0 αeff
(3.205)
Hence the speed of decay depends on the initial electron density.
If the changes in the production are small or if we only consider small time
intervals, we only have to consider small changes of electron density. Then we can
write
ne = ne0 + n0e
(3.206)
where ne0 is the background density and n0e a small variation. Thus the continuity
equation is
dne
dn0
= e = qe − αeff (ne0 + n0e )2 = qe − αeff n2e0 − 2αeff ne0 n0e ,
dt
dt
(3.207)
where the small second order term αeff n0e 2 is neglected. Hence we have a linear
equation for n0e , mathematically similar to eq. (3.184). The difference is that qe in
eq. (3.184) is replaced by qe − αeff n2e0 in eq. (3.207) and β by 2αeff ne0 . Therefore,
following eq. (3.191), the time constant for the variations n0e is
τ=
1
2αef f ne0
.
This is often called the E region recombination time constant.
(3.208)
80
CHAPTER 3. THE IONOSPHERE OF THE EARTH
3.10
Ionization and recombination in the D region
Figure 3.17: D region ionization by diferent sources (G.C.R:= galactic cosmic rays)
(Whitten and Popoff, 1971).
The lower part of the ionosphere below 90 km and down to about 60 km is called
the D region. In the D region the atmospheric density is about 106 times the typical
F region density and collisions are dominant. The photochemistry is very complex
and not fully understood. Five important ionization sources the D region can be
identified. Their contribution to the ion production is illustrated in Fig. 3.17.
1. The Lyman-α line at 121.5 nm penetrates down to the D region and ionizes
nitric oxcide NO, which is a minor constituent in the neutral atmosphere. This
happens because the ionization limit of NO is 134 nm.
2. EUV radiation between wavelengths 102.7 and 111.8 nm ionizes excited oxygen
molecules O2 .
3. Hard X rays between 0.2 and 0.8 nm ionizes all constituents, thereby acting
mainly on the abundant molecules N2 and O2 .
4. Galactic cosmic rays ionize all constituents.
5. Relativistic electrons (100 keV - 1 MeV) and protons (1 - 100 MeV) may
occasionally penetrate deep into the D region and produce ionization.
3.10. IONIZATION AND RECOMBINATION IN THE D REGION
81
Main ions are
• in the upper D-region
– NO+ , O+
2
• in the lower D-region
– cluster ions H+ (H2 O)n , (NO)+ (H2 O)n
– negative ions like O−
2 , negative cluster ions
Figure 3.18: Concentration of positive ions detected by a mass spectrometer in the
D region (Narcisi and Bailey, 1965).
+
The positive ions formed directly by solar radiation in D region are O+
2 , N2 and
+
+
NO . The N2 ions rapidly experience charge exchanges with O2 to produce an O+
2
ions in the reaction
k6
N2+ + O2 −→
N2 + O2+
(3.209)
The reaction coefficient k6 is given in Table 3.3. The resulting O+
2 will quickly
participate in charge transfer in reactions
and
k3
N O + + O2
O2+ + N O −→
(3.210)
k4
O2+ + N2 −→
N O+ + N O,
(3.211)
82
CHAPTER 3. THE IONOSPHERE OF THE EARTH
which both produce NO+ . The latter of these reactions produces neutral NO, which,
however, is not an abundant constituent. Still, NO+ is an important ion species in
D region, since it is produced in reaction (3.210). This production is large because
the production of the primary ions O+
2 is effective (due to the large concentration
of neutral O2 ) and the concentration of the target molecules N+
2 is also great.
The concentrations of a few important D region ion species are shown in Fig.
3.18, which displays results of rocket-borne mass spectrometer observations. The
figure indicates that NO+ (mass number 30) and O+
2 (mass number 32) are major
ion species above about 85 km. Below that, hydrated ions (molecular mass numbers
19 - 37) such as H+ H2 O, H3 O+ H2 O are found. These hydrates occur when water
vapour concentration exceeds about 1015 m−3 . Heavy ions with molecular masses
above 45 are observed below 75 km. These are believed to be metallic ions, probably
debris from meteor showers.
An important feature of the D region is the formation of negative ions such as
O−
.
2 These are produced by three-body reactions
β1 , β2
O2 + M + e −→ O2− + M,
(3.212)
where M is one of the abundant neutral molecules O2 or N2 . The reaction coefficients
β1 and β2 are given in Table 3.3. In such reactions the role of M is to remove excess
kinetic energy from the reactants.
The electron affinity of O2− is small, only 0.45 eV. Therefore the electron might
be rapidly detached again by a low-energy quantum, such as infrared or visible light.
This reaction is
ρ
O2− + hν −→ O2 + e
(3.213)
with a reaction coefficient ρ. Negative ions may also be destroyed in collisions with
neutrals. Such reactions are
and
γ1 , γ2
O2− + M −→ O2 + M + e
(3.214)
γ3
O2− + O −→ O3 + e .
(3.215)
The rate coefficients are given in Table 3.4; γ1 is valid if M is N2 and γ2 if M is O2 .
Other possible loss processes are ion-ion recombination in reactions
and
αi1
O2− + O2+ −→
O2 + O2
(3.216)
αi2
O2− + O2+ + M −→
O2 + O2 + M.
(3.217)
These equations illustrate the major chemical reactions taking part in the D
region. Observations of ions in the D region have shown that the amount of negative
ions increases by decreasing height (Fig. 3.19). While there is less than one negative
ion per 100 electrons at 90 km at daytime, there may be more than 100 negative
3.10. IONIZATION AND RECOMBINATION IN THE D REGION
83
Figure 3.19: Negative ion-electron concentration ratio λ below 90 km. The model
is based on molecular oxygen ions only (Risbeth and Garriot, 1969).
ions per electron at 50 km. At night even larger concentrations of negative ions are
possible.
Since the ion chemistry in D region is very complicated, we will make a simplified
model in order to understand the main features of D region. Let us denote the
densities of electrons, positive ions and negative ions by ne , n+ and n− , respectively.
Positive ions are destroyed in recombinations with electrons (effective recombination
coefficient αD ) and in recombinations with negative ions (effective recombination
coefficient αi ). In addition, negative ions are destroyed in collisions with neutrals
(detachment coefficient γ) and by photons (detachment coefficient ρ). Radiative
recombination is neglected.
When the convection term is neglected, the electron continuity equation reads
dne
= q − αD n+ ne − βne nn + γn− nn + ρn− ,
(3.218)
dt
where q is the electron production rate due to photoionization and nn is the neutral
number density. The continuity equation for positive ions is
dn+
= q − αD n+ ne − αi n− n+ .
(3.219)
dt
Since electrons and positive ions are created in the same processes, they have equal
production rates. Finally, the continuity equation for negative ions is
dn−
= −αi n− n+ + βne nn − γn− nn − ρn− .
dt
Because of charge neutrality in plasma, we have
n+ = ne + n− .
(3.220)
(3.221)
By introducing the ratio between negative ions and electrons λ
λ=
n−
ne
(3.222)
84
CHAPTER 3. THE IONOSPHERE OF THE EARTH
we have
n+ = (1 + λ)ne .
(3.223)
With these notations αD n+ ne = αD (1 + λ)n2e and αi n− n+ = αi λ(1 + λ)n2e . Then eq.
(3.219) can be written as
(1 + λ)
dne
dλ
+ ne
= q − (αD + αi λ)(1 + λ)n2e ,
dt
dt
(3.224)
so that
dne
q
ne dλ
=
− (αD + αi λ)n2e −
.
(3.225)
dt
1+λ
1 + λ dt
In eq. (3.218), γn− nn = γλnn ne , and ρn− = ρλne . Therefore eq. (3.217) gives
dne
= q − αD (1 + λ)n2e − βne nn + γλnn ne + ρλne .
dt
(3.226)
These two equations lead to
"
#
1 dλ
q
= βnn − λ ρ + γnn + (αi − αD )ne +
.
1 + λ dt
(1 + λ)ne
(3.227)
Under most conditions λ is fairly constant, one value at night and the other in the
daytime. The electron attachment and detachment processes often occur so rapidly
that they nearly balance each other. This means that recombination with positive
ions becomes unimportant. Then, also in stationary situation the production and
recombination of positive ions are in balance. In this approximation we can put
dλ/dt to zero in eq. (3.227) and neglect the two last terms inside the brackets. Then
βnn − λ(ρ + γnn ) = 0,
which leads to
λ=
βnn
.
ρ + γnn
(3.228)
(3.229)
At night the detachment rate due to chemical reactions dominates over detachment rate due to solar radiation so that γnn > ρ and
λ=
β
.
γ
(3.230)
In the daytime, however, the opposite is true and
λ=
β
nn .
ρ
(3.231)
We see from Table 3.3 that β is proportional to [O2 ] or [N2 ] and λ will therefore
increase by decreasing height in agreement with Fig. 3.19. This increase, however,
will be stronger in the daytime than at night.
3.10. IONIZATION AND RECOMBINATION IN THE D REGION
85
Figure 3.20: Schematic electron density profiles in the D region for quiet and active
Sun.
If we assume that attachment and detachment are not in balance but λ is still
constant, eq. (3.224) gives
q
dne
=
− (αD + αi λ)n2e .
dt
1+λ
(3.232)
This equation is mathematically similar to eq. (3.199), and therefore the electron
density obeys a similar profile with an effective recombination coefficient
0
αeff
= αD + λαi
(3.233)
and a production profile
q
.
1+λ
In a stationary case the electron density has a shape of an α profile
q0 =
s
ne =
q0
=
0
αeff
s
q
.
(1 + λ)(αD + λαi )
(3.234)
(3.235)
Fig. 3.20 illustrates some typical electron density profiles observed in the D–
region for active and quiet solar conditions. The electron density decreases usually
from 1010 m−3 at 90 km to less than 107 m−3 below 60 km. A ledge in the profile
is often observed between 80 and 90 km where the density can decrease by one
magnitude. This ledge is believed to occur due to the presence of hydrated ions in
this height region. Since the hydrated ions are very effective in recombining with
electrons, they will rapidly destroy the electrons as soon as they are created.
86
CHAPTER 3. THE IONOSPHERE OF THE EARTH
Figure 3.21: Schematic figure of electron concentration in the lower D region with
time. R and S represent sunrise and sunset, respectively (Ratcliffe, 1972).
From radio wave propagation studies it has been observed that the electron
density in the D-region is during quiet conditions fairly constant at day and night
but changes rather rapidly at sunset and sunrise (see Fig. 3.21).
It has been recently found that ion chemistry is coupled to the neutral chemistry
in the D region and may lead to ozone loss (Fig. 3.22). This is because when
protons or electrons precipitate into the atmosphere, ions and secondary electrons
are produced. Ions and electrons react chemically and produce odd hydrogen (e.g.
H or OH), odd nitrogen (e.g. N, NO and NO2 ) and negative ions. This trio then
affects ozone O3 via catalytic reaction chains like
N O + O3 −→ N O2 + O2
N O2 + O3 −→ N O + 2O2
(3.236)
(3.237)
OH + O −→ H + O2
H + O3 −→ OH + O2
(3.238)
(3.239)
and
3.11
Plasmasphere
In the lower ionosphere heavy ions such as NO+ and O+
2 dominate, but in the upper
+
+
+
ionosphere the lighter ions such as O , He and H become more abundant. In the
daytime, plasma is produced and lost by different processes in the E and F region.
Above the F layer electron density peak we would expect plasma to be in diffusive
equilibrium. This equilibrium is, in general, established between electrons and O+
ions which, together, create the diffusion electric field pointing upwards. The lighter
ions such as He+ and H+ experience an upward force in this electric field and are
therefore accelerated upwards.
3.11. PLASMASPHERE
87
Figure 3.22: Schematic figure of effects of particle precipitation in the D region
(Turunen, 2007).
Figure 3.23: Earth’s plasmaphere.
At some height the ionosphere dominated by O+ gives way to the protonosphere,
which is dominated by H+ . Protonosphere is also known as geocorona. Actually
the term ”geocorona” refers to the solar far ultraviolet light that is reflected off the
cloud of neutral hydrogen atoms that surrounds the Earth. At these altitudes charge
exchange processes
H + + O −→ H + O+
(3.240)
and
O+ + H −→ O + H +
(3.241)
are extremely rapid. They tend to establish a chemical equilibrium in which concentrations at the transition altitude are related by
9
[H + ][O] = [H][O+ ],
8
(3.242)
88
CHAPTER 3. THE IONOSPHERE OF THE EARTH
where the factor 9/8 is obtained from statistical theory.
The transition between oxygen-dominated plasma and hydrogen-dominated plasma takes place somewhere between 500 km and a few thousand kilometers above the
ground. Since hydrogen ions can escape the Earth’s gravity field, the simple steadystate diffusive equilibrium cannot be maintained and a net upward flow of plasma
is possible from the topside ionosphere. At high latitudes, where the magnetic field
lines are open, this plasma flow is known as polar wind.
At lower latitudes the magnetic field lines are closed. There the upward flowing
ions can reach only a restricted region is space, determined by the configuration of
the geomagnetic field. In the daytime the plasma produced by the photoionization
may fill this enclosure and, at night, the plasma may flow back to the ionosphere
along the magnetic field lines to compensate the plasma lost there by recombination.
The result is a complex interaction between the ionosphere and a region of hydrogen
plasma trapped by the dipole magnetic field, the plasmasphere (Fig. 3.23). Near the
solstices, the flow can be interhemispheric. It is upward and out of the topside
ionosphere throughout the day and night in the summer hemisphere. In the winter
hemisphere, the flow is upward during the day, but downward during the night, so
during the night plasma can flow from the opposite hemisphere.
The situation is complicated by the fact that neutral winds in the upper ionosphere as well as electric fields both in the ionosphere and in the magnetosphere
strongly modify the plasma flow. Geomagnetic storms and substorms also affect the
flows. During storms, plasma in the outer plasmasphere is convected away owing to
enhanced magnetospheric electric fields. The high-altitude depletitions may be very
substantial, and the consequent reductions in plasma pressure induce ionospheric
upflows. The upflows typically occur throughout day and night in both hemispheres
and they can last many days after the storm.