On Effects of Electrostatic Waves on the Auroral Ionospheric
PlasmaSusumu Saito

Solar-Terrestrial Environment Laboratory
Nagoya University, Japan

Acknowledgments
Chapters 3–4 of this thesis are mostly a reproduction of published materials (Saito etal., Observation of isotropic electron temperature in the turbulent E region, Annales
Geophysicae, vol. 19, 1, 2001; Saito et al., Effects of a kappa distribution function ofelectrons on incoherent scatter spectra, Annales Geophysicae, vol. 18, 1124, 2000; Buchert
and Saito, On the Pedersen current which is carried by electrons, SUBSTORM-4, 485,
1998).
I wish to express my sincere gratitude to my supervisor Prof. R. Fujii, who helped
to conduct my study. I would also like to express my sincere gratitude to Prof. S. C.
Buchert, who also helped provide guidance for my study. His knowledge and advice havegreatly helped me. I am deeply grateful to Prof. Matuura who was my supervisor in thefirst year of my master’s study program.
I am deeply grateful to Dr. F. R. E. Forme for his discussion and suggestions, which
greatly helped me in the study of Chap. 4 in this thesis. Prof. T. Ogawa and Dr. S.
Nozawa are greatly appreciated for their important help, discussion, and encouragement.
Prof. T. Hagfors is sincerely acknowledged for his discerning insight into my thesis.
Appendix A is based on Hagfors (1984) and his personal communication to me.
I would like to acknowledge the contribution of the staff and students of STEL for theirsupport, concern, and friendship. This thesis was finally written at the Auroral Observatory, University of Tromsø, Norway with the financial support of the Research Council of
Norway. I wish to acknowledge the staff and students of the Auroral Observatory for theirsupport and friendship to me as I studied in a foreign country. We are indebted to thedirector and staff of EISCAT for operating the facility and supplying the data. EISCATi

is an International Association supported by Finland (SA), France (CNRS), the Federal
Republic of Germany (MPG), Japan (NIPR), Norway (NFR), Sweden (NFR) and theUnited Kingdom (PPARC).
Data originated from the Imaging Riometer for Ionospheric Studies (IRIS), operated
by the Department of Communication Systems at Lancaster University (UK), funded by
the Particle Physics and Astronomy Research Council (PPARC) in collaboration with the
Sodankylä Geophysical Observatory.
The STARE radars are operated by the Max-Planck-Institut für Aeronomie in cooperation with ELAB, the Technical University of Norway, and the Finnish Meteorological
Institute. I am grateful to Dr. P. Eglitis of the Swedish Institute of Space Physics Uppsala Division (IRFU), Sweden for providing the data of the STARE system and for his
mportant advice.
I would like to express my gratitude to the track and field club, Nagoya University,and especially to Prof. H. Kunieda and Prof. Y. Kanao, for their support and friendshipthroughout my life at Nagoya University.
Finally, I wish to thank my parents for their continuously long and steady support of
my study program.

ii

Contents
AcknowledgmentiContents

iii

Figure List

vi

Notationx

Abstractxi

1 General Introduction
1.1

1.2

1

The Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Temperatures in the Ionosphere . . . . . . . . . . . . . . . . . . . .

3

1.1.2

Electric Currents in the Ionosphere . . . . . . . . . . . . . . . . . .

7

Different Types of Electrostatic Waves . . . . . . . . . . . . . . . . . . . .

9

1.2.1

Electrostatic Waves in an Unmagnetized Plasma . . . . . . . . . . . 10
Langmuir Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Ion Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2

Unstable Electrostatic Waves in the E region . . . . . . . . . . . . . 13
Farley-Buneman Wave . . . . . . . . . . . . . . . . . . . . . . . . . 16
Gradient drift Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Unified dispersion relation of Farley-Buneman and Gradient drift
instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 18

iii

1.3

Scattering of Electromagnetic Waves by Electrostatic Waves . . . . . . . . 18
1.3.1

Coherent Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.2

Incoherent Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Instruments

27

2.1

The EISCAT radar system . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2

The STARE radar system . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3

The Kilpisjärvi IRIS System . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 E region
3.1

3.2

3.3

3.4

35

Interaction Between the Farley-Buneman Wave and Electrons . . . . . . . 36
3.1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.2

Tristatic Observation of Electron Temperatures by EISCAT . . . . 38

3.1.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Energy Balance of Electrons: Pedersen Current Carried by Electrons . . . 43
3.2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.3

Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Cosmic Noise Absorption: Comparison with observation by IRIS . . . . . . 57
3.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.2

Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 F region

66

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3

Results of calculation and discussion . . . . . . . . . . . . . . . . . . . . . 74
4.3.1

Incoherent scatter spectra with kappa distribution I: Ion lines . . . 74

iv

4.3.2Difference of plasma parameters as a results of fitting with
Maxwellian distribution . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.3
4.4

Incoherent scatter spectra with kappa distribution II: plasma lines . 83

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Concluding Remarks

89

A The Electromagnetic Wave Propagation in the Turbulent Plasma

93

A.1 The wave propagation problem . . . . . . . . . . . . . . . . . . . . . . . . 93
A.2 The riometer case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
B Plasma Disersion Equation: Kinetic Treatment

101

B.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B.3 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B.4 Plasma Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 104?
C Derivation of Zκ,−
(ξκ )

106

C.1 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.2 Cauchy Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Bibliography

111v

List of Figures
1.1

Typical height profile of the electron density . . . . . . . . . . . . . . . . .

1.2

Left panel: Typical height profiles of Tn , Ti , and Te for a quiet condition.
Right panel: The same as on the left, but for a disturbed condition. . . . .

4

6

1.3

Dispersion relation of a Langmuir wave . . . . . . . . . . . . . . . . . . . . 12

1.4

Dispersion relation of an ion acoustic wave . . . . . . . . . . . . . . . . . . 14

1.5

Geometry of the system considered in Sec. 1.2.2. . . . . . . . . . . . . . . . 21

1.6

Electron and ion distribution functions for a ue0 < Cs , and b ue0 > Cs . . . 22

1.7

Schematic drawing of the mechanism of the gradient drift instability. After
Kelley (1989), Fig. 4.31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.8

The four types of radar spectra that are observed in the auroral electrojet.
After Kelley (1989) Fig. 8.27. . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.9

Schematic drawing of IS spectrum. a kradar λD À 1, b kradar λD ¿ 1,

. . . 25

1.10 IS spectra with different parameters. When a ne , b Ti , c Te /Ti , d ui , e νin ,and f ion composition are changed. . . . . . . . . . . . . . . . . . . . . . . 26
2.1

Geographical locations of the EISCAT radars. Tx/Rx means that the radarboth transmits and receives. Rx means that the radar only receives. . . . . 29

2.2

The STARE field of view. The position of the EISCAT Tromsø beam at
field-aligned direction is also plotted. . . . . . . . . . . . . . . . . . . . . . 31

vi

2.3

The projection of Kilpisjärvi IRIS beams onto the ionosphere at 90 km altitude. The outline is the locus at which the sensitivity is 3 dB lower than at
the beam center. (From IRIS homepage http://www.dcs.lancs.ac.uk/iono/iris
http://www.dcs.lancs.ac.uk/iono/iris /) The position of the EISCAT
Tromsø beam at a field-aligned direction is also plotted.

3.1. . . . . . . . . . 33

Overview of a data set with long periods of elevated Te in the lower E
region. The top panel shows |E| with a time resolution of 5 minutes. The
approximate threshold of the MTSI is about 20 mVm−1 . The middle andbottom panels show the electron and ion temperatures, respectively.

3.2. . . 40a Scatter plot of the electron temperatures from different aspect angles
observed with the Tromsø and Sodankylä EISCAT sites. The errors infitting to the IS spectra are also plotted. b The same as a, but the ion
temperatures are compared. . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3

The same as Fig. 3.2, but Tromsø and Kiruna data are compared.

. . . . 41

3.4

The same format as Fig. 3.1. But the electric field (top panel) is plottedwith a 1 min resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5Height profile of (left) temperatures of ions, electrons and neutrals, and(right) Pedersen conductivity due to ions (blue) and electrons (red).

3.6. . . 50

Electron flow angles relative to the E × B directions at 108 km are plotted.
The flow angle is measured in a counter-clockwise direction (from E × B to−E). Flow angles are calculated both with and without taking the coolingby ions into account (solid line and dashed line, respectively). . . . . . . . 51

3.7

Two-dimensional map of echo intensities and flow velocity vectors measured
by the STARE system at 1149 UT on September 16, 1999 is shown. The
echo intensities are measured in dB units with respect to the backgroundnoise level. The flow vectors are derived from two velocity components
measured at Midstandan and Hankasalmi applying the cosine rule.

vii. . . . 53

3.8

Similar to Fig. 3.6. Angles of flow derived from STARE data by applying
the cosine rule are also plotted (dash-dotted line).

3.9. . . . . . . . . . . . . 54

Semi-empirically obtained relation between height-integrated current andcross field voltage for a specific electron density profile. A straight line
(linear Ohm’s law, dashed line) would result without anomalous heating.
However, above the threshold for the MTSI, the total height-integrated
current JP,total = JP,e + JP,i (solid line) is higher.

. . . . . . . . . . . . . . 56

3.10 Overview of a data set with long periods of elevated Te in the lower E
region with a time resolution of 1 min. The top panel shows the absorption
measured by beam 9 of the Kilpisjärvi IRIS. The last four panels show
observation by the EISCAT, |E| (the second panel), Te at 108 km (third),
ne at 108 km (fourth), and ne at 90 km (fifth). . . . . . . . . . . . . . . . . 60
3.11 Absorptions estimated by the EISCAT data (red: νe = νen + νe? , green:νe = νen ) and the IRIS measurements (blue line) are compared between
1400 and 1500 UT when |E| is generally strong. . . . . . . . . . . . . . . . 62
3.12 The same as Fig. 3.11. But the results are for the period when Te is notenhanced, but ne at 90 km is enhanced. . . . . . . . . . . . . . . . . . . . . 63
4.1

A comparison between Maxwellian and kappa distribution with κ =3, 5, 10, and 40.

Maxwellian distribution (dashed line) corresponds toκ = ∞. x-axes are indicated by velocities normalized by vth,M and vth,κ=3 . 72
4.2

IS spectral density functions of kappa distribution plasma for a different
spectral index, κ = 3, 5, 10, and 40, together with one from Maxwellian
plasma (κ = ∞) are shown. For this case, ne = 2 · 1011 m−3 , Ti = 2000 K,
Te = 3000 K, and mi = 16 a.m.u. x-axes are indicated by corresponding
velocities normalized by vth,M and vth,κ=3 as well as by a frequency in kHz

viii

76

4.3

a, b. Solutions of dispersion equation for the ion acoustic waves for κ = 3
and Maxwellian. a Frequency b Damping rate are plotted over electron
temperature. Ion temperature and electron density are fixed to 1000 K
and ne = 2 · 1011 m−3 , respectively . . . . . . . . . . . . . . . . . . . . . . 78

4.4

An example of fitting on an IS spectrum (ne S(k, ω)) with kappa distribution by an IS spectrum with Maxwellian distribution. The model spectrum(plotted by circles) assumes Ti = 2000 K, Te = 3000 K, ne = 2 · 1011 m−3 ,and κ = 3. Solid line shows fitted IS spectrum with Maxwellian distributionfunction with Ti = 2002 K, Te = 1797 K, ne = 1.996 · 1011 m−3

4.5. . . . . . 80

The difference of parameters when kappa distribution IS spectra are interpreted by assuming Maxwellian distribution for Ti , Te , and ne are shown
from the top to the bottom. They are plotted as relative errors to real
(kappa distribution) parameters . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6

The two spectra in Fig. 4.4are shown for wider range of frequency. y-axis
is shown in logalithmic scale.

4.7. . . . . . . . . . . . . . . . . . . . . . . . . 82

Plasma lines (upshifted) for various electron temperatures (2000 - 10 000 K)
for κ = 3 (solid line) and Maxwellian (dashed line) are shown. Electron
density is fixed to 2 · 1011 m−3 . x-axis ranges are all 3 - 7 MHz. But y-axis
ranges are arbitrary for each plot . . . . . . . . . . . . . . . . . . . . . . . 85

4.8

Plasma lines (upshifted) are plotted for a different spectral index, κ =3, 5, 10, and 40, together with plasma line of Maxwellian plasma (κ = ∞)

4.9

86

Plasma lines (upshifted) are plotted for the sets of parameters appeared inFig. 4.4, (ne , Ti , Te , κ) = (2 · 1011 m−3 , 2000 K, 3000 K, 3) (solid line) and(1.996 · 1011 m−3 , 2002 K, 1797 K, ∞(M axwellian)) (dashed line) . . . . . 87

C.1 Contours of integration for signs and special values of ξκ are shown. The
contour C1 must be above a pole ξκ , because the Laplace transform isdefined as Im[ξκ ] < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

ix

Notation
General NotationBmagnetic flux density²0

permittivity of the vacuumEelectric fieldλD

Debye lengtheelementary chargeµ0permeability of the vacuumjcurrent densityνcollision frequency

kwave vector

ρqcharge densitykB

Boltzmann constantω

wave angular frequency

m

mass

Ω

gyrofrequencynnumber densityppressureqchargeTtemperatureubulk velocityvvelocity

Subscripts
eelectronenelectron-neutral

i

ion

in

ion-neutralnneutralsspecies of particlesxAbstract
We have studied some effects of electrostatic waves and turbulences on the plasma in theauroral both E and F regions both experimentally and theoretically.
In the E region, electron heating by Farley-Buneman waves are studied using the
EISCAT CP-1 data. We find that electrons are strongly heated in the magnetic field-line
direction during high electric field events. The remote site data show that the electron
temperature increases almost the same amount as in the field-perpendicular direction,and that electron heating by E region plasma turbulence is isotropic. We discuss theimplementations of our observation for the “plasmon”-electron as well as the wave Joule
heating models of the anomalous electron heating in the E region. St.-Maurice (1987)’smodel which shows that Farley-Buneman waves have an electric field component both
perpendicular and parallel to B and effectively heat electrons can explain our results. We
also suggest that if Te enhancement is rather due to a direct scattering of electrons bywave fields not involving electron-neutral collisions, our observation implies that thesewaves should be isotropic.
Because the total energy is conserved, we argue that electrons must partly drift along
the background electric field when the electron temperature is elevated above the background neutral temperature. We parameterize the effect of the waves by the anomalous(effective) electron collision frequency νe? . We find that a rotation of the electron flow
direction from a E × B direction can be up to ∼ 8◦ . Although we could not confirm
the rotation with the STARE data, we point out the possibility of using an optical instrument to test our results. We also discuss the importance of electron heating on the
Magnetosphere-Ionosphere coupling.
xi

By using the Kilpisjärvi IRIS data, we examine the nature of the anomalous (effective)
electron collision frequency. We find that including νe? in the absorption model leads tooverestimation of the cosmic noise absorption when compared with observation by the
Kilpisjärvi IRIS. We find that νe? has a limited range of application.
In the F region, we have studied incoherent scatter spectra which are a scattering ofelectromagnetic waves by electrostatic ion acoustic waves and Langmuir waves. We havetheoretically calculated incoherent scatter spectra for a plasma that consists of electrons
with kappa distribution function and ions with Maxwellian neglecting the effects of themagnetic field and collisions.
The ion line spectra have a double-humped shape similar to those from a Maxwellian
plasma. The electron temperatures are underestimated, however, by up to 40% wheninterpreted assuming Maxwellian distribution. Ion temperatures and electron densities
are affected little. Accordingly, actual electron temperatures might be underestimatedwhen energy input maintaining a high energy tail exists. We have also calculated plasmalines with the kappa distribution function. They are enhanced in total strength, and thepeak frequencies appear to be slightly shifted to the transmitter frequency compared tothe peak frequencies for a Maxwellian distribution. The damping rate depends on theelectron temperature. For lower electron temperatures, plasma lines for electrons witha κ distribution function are more strongly damped than for a Maxwellian distribution.
For higher electron temperatures, however, they have relatively sharp peaks.

xiiChapter 1
General Introduction
Various waves can propagate in space and in plasmas which are important because they
carry information (energy and momentum) and can even modify a plasma by interacting with particles, and even with other waves. Electrostatic waves are a part of plasma
waves which propagate in a direction parallel to the wave’s electric field. (Consequently
They have no oscillating magnetic field component.) They can propagate only in plasmas
because they need a density fluctuation of electric charges. Such waves contain information on plasmas in the ionosphere. Ground-based instruments such as radars utilize thescattering of transmitted radio waves to probe the ionosphere. Therefore, it is very important to study the effects of electrostatic waves not only as a natural plasma laboratory
but also for the correct analysis of radar experiments. In this study, we examine certain
phenomena in which electrostatic waves play important roles in the ionospheric E andF region. Among many interesting phenomena related to electrostatic waves, we explore
some topics which have close relationships with the electron temperature which is one of
the most basic and important parameters in the ionospheric plasma. In other words, thekeywords of our study are ‘electrostatic waves’ and ‘electron temperature.’ With these
keywords, we investigate how electrostatic waves interact with and modify the ionosphericplasma through the E and F region. They will be presented separately in Chap. 3 andChap. 4, respectively.
For a general introduction, the earth’s ionosphere and electrostatic waves are briefly
1reviewed in this chapter. In Sec. 1.1, a brief summary of the earth’s ionosphere is provided. In Secs. 1.1.1–1.1.2 the temperature is defined, and the heat production and theionospheric current are briefly described. Some properties of electrostatic waves and abrief explanation of some important electrostatic waves are given in Sec. 1.2. Basic concepts of scattering of radio waves, both coherent and incoherent scattering, are dealt within Sec. 1.3.

1.1

The Ionosphere

The ionosphere is the region where the earth’s atmosphere is ionized. At the end of
the 19th century the existence of an electrically conducting layer in the earth’s upper
atmosphere had been predicted by observations of the geomagnetic field. To explain thediurnal variation of the geomagnetic field, it was more reasonable to assume that anelectric current system exists in the sky than to ascribe it to inside the solid earth. In
the early 20th century, Kennelly and Heaviside pointed out that there was an electricalconducting layer in the upper atmosphere to explain a radio communication experimentacross the Atlantic ocean by Marconi. The existence of this layer was confirmed in 1925
by Appleton and Barnett, and Breit and Tuve, by measuring the delay time between
transmitting the radio wave vertically upward until the reflected wave was received todetermine the height of the layer.
The earth’s ionosphere consists of partially ionized gases which is created by ionizing
neutral molecules and atoms of the atmosphere caused by certain ionizing sources. The
main ionizing source is solar EUV radiation. Solar X-ray radiation, and high energy
particles like cosmic rays, auroral particles, radiation belt particles, and solar protons canalso ionize. But generally their contribution is smaller than that of solar EUV radiation.
The ionosphere has several peaks in density at different altitudes. Fig 1.1 shows the
typical height profile of the electron density, ne . The ionosphere is divided into layers bydifferent electron densities and production mechanisms. Between about 90 ∼ 150 km, it
is called E region.
2

Above about 150 km is the F region, and below about 90 km is the D region. Because
the density of the neutral atmosphere exponentially changes with altitude and the composition also changes in altitude, the dominant physical processes and dominant ionizedspecies change in altitude as well. In the F region, the dominant ion is O+ , while in the
E region they are NO+ and O+
2 . The increasing neutral density increases the collision
frequency between ionized species and neutrals. The collision frequency is important for
the ionospheric electric current. Most of the currents flow in the E region. In the D region, negative ions and cluster ions become important. Large collision frequency between
electrons and neutrals enhances cosmic noise absorption in the D region. All through theionosphere, the ionization degree is very low. Even around the peak in the F region, it
is ∼ 10−3 , and it is ∼ 10−6 or less in the E region. Therefore, the ionosphere is weakly
ionized plasma, where the effects of the neutral particles can be important.
In the polar ionosphere, the auroral particle precipitation can contribute significantly
to the formation of the ionosphere. Both protons and electrons precipitate into the earth’s
ionosphere and ionize. The species and the energy of the precipitating particles determine
the altitude of maximum ionization. For example, electrons with energy less than 1 keVionize the F region. Electrons with energy of 10 keV ionize mainly the E region, about
110 km in altitude. Higher energy electrons can penetrate into the D region, and enhance
the cosmic noise absorption which is measured in riometers.

1.1.1

Temperatures in the IonosphereFirst of all, the temperature must be defined. There are several ways of defining thetemperature. Here we define the temperature T of one species with mass m by the second
order moment of a distribution function f (v), or by mean kinetic energy of particles,
Z +∞
1 2
3
1
2
mv f (v)dv ≡ kB T
(1.1)
< m|v| >=
2
2
−∞ 2

3

Altitude (km)

500

300

F region

E region100

D region
10

9

10

10

10

11

10

12

Electron Density (m-3)
Figure 1.1: Typical height profile of the electron density

4

where kB is the Boltzmann constant. For an anisotropic distribution function, we candefine the temperature along a certain direction Tk ,1
< mvk2 >=
2

Z +∞
−∞

1 2
1
mvk f (vk )dvk ≡ kB Tk .
2
2

(1.2)

In the ionosphere, ions, electrons, and neutrals can have different temperatures. Gen<erally the neutral temperature increases above the mesopause. In lower altitudes (∼110 km), very frequent collisions between charged and neutral particles force them intoa state of thermal equilibrium, i.e., Tn ' Ti ' Te , where Tn , Ti , and Te , are the temperatures of neutrals, ions, and electrons, respectively. Because of a huge difference of mass
between an electron and a neutral particle, electrons more easily depart from thermal
equilibrium with decreasing electron-neutral collision frequency than ions. In a geomag<<

netically quiet condition, the following relation is found in the ionosphere, Tn ∼ Ti ∼ Te .
In a geomagnetically disturbed condition, however, Ti can be higher than Te at around
120 km due to the heat produced by an electric current flowing in the ionosphere. In
Fig. 1.2, a schematic drawing of height profiles of temperatures for a disturbed condition
is compared with those of a quiet condition.
The temperatures are determined by a balance between a heating rate and a coolingrate. There are various heating sources in the ionosphere. Solar illumination heats every
species. Particle precipitation heats electrons efficiently in the F region. A strong electricfield enhances the electric currents and produces heat by Joule heating. In the lower
E region at around 110 km, it is known that electrons can be heated up to ∼ 3000 K byelectrostatic plasma waves excited when the relative drift velocity between electrons andions is higher than the local ion acoustic velocity (Schlegel and St.-Maurice, 1981; St.Maurice et al., 1981; St.-Maurice and Laher, 1985; Robinson, 1986; Igarashi and Schlegel,
1987; Williams et al., 1990; Jones et al., 1991; Davies and Robinson, 1997). Elastic andinelastic collisions between charged and neutral particles transfer the thermal energy fromheated charged particles to neutral particles. Hence, the neutral atmosphere acts as a heatsink in the ionosphere.
For ions, the energy balance of heating due to friction with neutrals and cooling by
5

Tn

Te

Ti Te

140
Altitude (km)140
Altitude (km)

Ti

Tn

120

Ti

120

Te
100

0

100

500

1000

1500

0

Temperature (K)

500

1000

1500

Temperature (K)Figure 1.2: Left panel: Typical height profiles of Tn , Ti , and Te for a quiet condition.
Right panel: The same as on the left, but for a disturbed condition.

6

collisions with neutrals is given by (e.g. St.-Maurice and Hanson, 1982; Kelley, 1989; Rees,
1989)
mi ni νinmi3
mn |ui − un |2 φin = 2niψin νin kB (Ti − Tn )mi + mnmi + mn2
mi3+2ni
νie kB (Ti − Te )mi + me 2(1.3)

and this leads tomn φin|ui − un |2
3kB ψin(1.4)(mi + mn )νie
(Te − Ti ) .
mi νin ψin(1.5)

Ti = Teq +where
Teq = Tn +The dimensionless parameter φin and ψin depend on the nature of the collision between
ions and neutrals. For collisions between O+ and O (above about 200 km), φin and ψinare approximately unity.
For electrons, the temperature is determined by the energy balance (e.g. Rees, 1989;
Jones et al., 1991)
Qe = Le,el + Le,rot + Le,vib + Le,f O + Le,ion

Watt m−3(1.6)

where Qe is the electron heating rate. Le,el is the cooling rate due to elastic collision withneutral species, Le,rot and Le,vib are the cooling rates due to excitation of rotational andvibrational levels of neutral molecules, respectively, and Le,f O is the cooling rate due toexcitation of the fine structure level of the O atom. These cooling rates are modeled andreviewed by Schunk and Nagy (1978). Le,ion is the cooling rate by interaction with ions.
These quantities are complicated functions of Te − Tn and Te − Ti . Therefore, numericalcalculation is necessary to solve the energy balance equation of electrons (Eq.( 1.6)).

1.1.2

Electric Currents in the Ionosphere

Joule heating is caused by a dissipative current (parallel to the electric field). The energy
source of the electron heating in the lower E region is a large electron–ion relative drift
velocity, in other words, a current. Neglecting the effects of waves on charged particle
7

motions for the moment, solving the ion and electron mobility equations derives the ion
and electron velocities (e.g. Kelley, 1989; Brekke, 1997),uiue

κi E0⊥
1 E0⊥ × B
e
=
+
+
E0k
2
2
1 + κi B
1 + κiBmi νin00
1 E⊥ × Beκe E ⊥
= −
+
−
E0
2
2
1 + κe B
1 + κeBme νen k

(1.7)
(1.8)

where κi,e are the ratio of the ion- and the electron-neutral collision frequencies ,νin,en , tothe ion- and the electron-gyrofrequencies, Ωi,e , respectively, and E0 is the electric field in
the moving frame with neutral atmosphere, and B is the geomagnetic field, respectively.
Here we define the gyrofrequency as a positive valueΩi,e =

eB
.
mi,e

(1.9)

where e is the elementary charge. Then, by the definition of the current
j = ne e(vi − ve ),(1.10)

the ionospheric current is written as
j = σP E0⊥ − σH

E0⊥ × B+ σk E0k
B(1.11)

where σP , σH and σk are Pedersen, Hall and parallel conductivities, respectively, given by
·
¸
κe
κi
ne e
σP =
+
(1.12)
B 1 + κ2e
1 + κ2i
·
¸
ne e
1
1
σH =
(1.13)−
B 1 + κ2e
1 + κ2i·¸·¸
1ne e 11
12+=+(1.14)
σ k = ne eme νen mi νin
B κe κi
The first term on the right-hand-side of Eq. (1.11) is called the Pedersen current, andthe second term is called the Hall current. Since κe ¿ 1, σk is much larger than σP and
σH . aligned current could be very large and would immediately cancel out the potential
difference. Therefore, usually it is a good approximation for the magnetic field line to bean equipotential line. Because the Hall current is perpendicular to E, only the Pedersen
8current is dissipative and produces heat. Because κe ¿ 1, the first term on the righthand-side of Eq. (1.12) vanishes. Therefore, the Pedersen current is mostly carried by ions,and heats ions selectively at around 120 km altitude. For electron heating, the effects ofthe plasma waves must be taken into account (St.-Maurice et al., 1981; St.-Maurice andLaher, 1985; Robinson, 1986). The microscopic mechanism of electron wave heating is
still under debate.

1.2

Different Types of Electrostatic Waves

Electrostatic waves are plasma waves without a fluctuating magnetic field. Maxwell’sequations governing electric and magnetic fields are given byρq²0(1.15)∇·B=0(1.16)∂B∂t(1.17)∇·E=∇×E=∇ × B = µ0 j + µ0 ²0∂E∂t(1.18)For electrostatic waves, the wave vector k is parallel to the wave electric field (longitudinalwave), and the right-hand-side of Eq. (1.17) is zero.
∇×E=0(1.19)

Then Eq. (1.15) is the only way to create E. Therefore, electrostatic waves must be
associated with charge density fluctuations.
If we adopt a two-fluid plasma model, equations of continuity and momentum become
as follows:∂ns+ ∇ · n s us = 0
∂t
∂us
qs
∇ps
F0
+ us · ∇us −(E + us × B) +
=
∂t
ms
ns ms
ns ms

9

(1.20)(1.21)

Here we ignored the production and loss in the equation of continuity. F0 includes allother external forces like friction with other fluids. For a friction with neutrals,
F0 = −ns ms νsn vs .

(1.22)

The current density j and the charge density ρq are defined byj=X

n s q s us

(1.23)

s

and
ρq =

X

qs n s .

(1.24)s

To obtain a closed set of equations, we add the polytropic law of the pressure and thedensityps ∝ nγs s(1.25)

where γs is the ratio of specific heats. In the following sections, several examples ofelectrostatic waves are derived by using linear approximation of these equations.

1.2.1

Electrostatic Waves in an Unmagnetized Plasma

Although plasmas in the ionosphere are magnetized, waves in an unmagnetized plasma
are still important in some cases where the effects of a magnetic field can be neglected. In
this section, we derive linear dispersion relations of electrostatic waves in an unmagnetizedplasma, where B = 0.
Langmuir Waves
First we consider high frequency waves where ions can be regarded as an immobile background. We assume that there is no background electric field, E0 = 0, and no backgroundelectron drift, ue0 = 0. In what follows, we denote perturbation terms by a subscript “1,”and background terms by a subscript “0.” Assuming further that perturbations are small,that all perturbations are proportional to exp (−ikx + iωt) and in one-direction only, and10

that the process is adiabatic (γe = 3 for this one-dimensional case), we obtain−iωne1 + ikn0 ue1 = 0(1.26)−iωme n0 ue1 + n0 eE1 + i3kkB Te ne1 = 0(1.27)

ikE1 +

ene1
= 0.
²0

(1.28)For non-trivial solutions of these equations, the next relation is obtained.
¡¢2ω 2 = ωpe
1 + 3k 2 λ2D
where

µ
ωpe =

is the plasma frequency and

µ
λD =

n0 e2
me ²0

¶ 12

²0 kB Te
n0 e2

¶ 12

(1.29)(1.30)(1.31)

is the Debye length. A plasma wave that obeys this dispersion relation is called a Langmuir
wave. This dispersion relation is schematically shown in Fig. 1.3. The frequencies ofLangmuir waves are higher than the plasma frequency which is, in the ionosphere, 1∼10
MHz. For cold plasma or for long wavelengths, i.e., kλD ¿ 1, Langmuir waves are foundto be close to the plasma frequency.
Ion Acoustic Waves
Next we consider low frequency waves where effects of ions must be included. By the sameprocedure as for Langmuir waves but assuming isothermal plasma for a slowly oscillating
system (γi,e = 1), we obtain−iωne1 + ikn0 ue1 = 0(1.32)−iωni1 + ikn0 ue1 = 0(1.33)−iωme n0 ue1 + n0 eE1 + ikkB Te ne1 = 0(1.34)

11

ω

+ωpe
k
−ωpe

Figure 1.3: Dispersion relation of a Langmuir wave

12−iωmi n0 ui1 + n0 eE1 + ikkB Ti ne1 = 0

ikE1 −

eni1 − ne1= 0.
²0(1.35)(1.36)

Using the fact me ¿ mi , we obtain the dispersion relation of an ion acoustic wave
ω2 =k 2 kB Ti k 2 kB Te /mi+.
mi
1 + k 2 λ2D(1.37)For long wavelengths where kλD ¿ 1, the dispersion relation is reduced toω 2 = k 2 Cs2
where

µ
Cs =

Ti + TekB
mi(1.38)¶ 21.

(1.39)

Fig. 1.4 schematically shows the dispersion relation of an ion acoustic wave.
Langmuir waves and ion acoustic waves are fundamental waves which always exist in aplasma as thermal fluctuations. A kinetic treatment of these waves is given in Appendix B.

1.2.2

Unstable Electrostatic Waves in the E region

When certain conditions are satisfied, thermal fluctuations of a plasma grow exponentially.
This is called instability. Many kinds of instabilities are known to exist in the ionosphere.
In this section, so called electrojet instabilities in the E region are described. They areobserved by radars in the E region electrojet as coherent echoes. It is known that there
are two mechanisms which are important for the electrojet instability. One is the modified
two-stream instability (MTSI). Farley (1963), using kinetic theory, and Buneman (1963),
using fluid theory, independently developed the theory of instability for the equatorial
electrojet. This instability is also called the Farley-Buneman instability after these twoscientists. The other important instability is the gradient drift instability which was firststudied by Simon (1963) and Hoh (1963) for laboratory plasmas, and by Maeda et al.
(1963) for the equatorial ionospheric E region. These instabilities are also important for13

(a) Ti = 0ωω = k Cs
ω = ω piω pik(b) Ti = 0ωω = k Csk
Figure 1.4: Dispersion relation of an ion acoustic wave

14the polar ionosphere where the auroal electrojet which is typically more intense than theequatorial electrojet exists. In what follows, we briefly describe the linear theory of these
instabilities following Kelley (1989).
In the following derivations, we assume that neutrals are at rest (or equivalently oursystem is moving with neutrals), that the magnetic field is in a -z-direction (downward),(B = −Bez ), the background electric field is in a y-direction, (E0 = E0 ey , northward),
and that all the perturbations are proportional to exp (−ikx + iωt), k = kex . Here ex,y,z
means a unit vector in an x, y, and z-direction, respectively. Electrons which rarely collidewith neutrals,Ωe À νen(1.40)

drift in a -x-direction (westward). However, ions can be assumed to be at rest in thezero-order because of frequent collisions with neutrals.
Ωi ¿ νin(1.41)<

Eqs. (1.40) and (1.41) are justified when we look at the lower E region (∼ 110 km). The
geometry is sketched in Fig 1.5.
Farley-Buneman Wave
The electron continuity equation which is in equilibrium in the zero-order becomesiωn1 − ikn1 ue0 − ikn0 ue1x = 0.

(1.42)

Neglecting the electron inertia which is much smaller than the ion, the electron momentum
equation yields
eB
ue1x − νen ue1yme

(1.43)eeB
kB Te n1E1 −ue1y + ik− νen ue1x
me
me
me n0

(1.44)

0=in a y-direction, and0=−in an x-direction. For ions, the continuity equation isiωn1 − ikui1x n0 = 015(1.45)

Because vi0 = 0, only the x-component of the ion momentum equation is relevant. For
ions, keeping the inertia term ∂ui1x /∂t,iωui1x =ekB Ti n1
E1 + ik− νin ui1x
mi
mi n0

(1.46)

Here we have used Ωi ¿ νin . From Eqs. (1.42)–(1.46), we obtain a dispersion relation
ω − kue0 = −¤Ψ0 £ω(iω + νi ) − ik 2 Cs2
νin(1.47)

where
Ψ=

νi νe
Ωi Ωe

(1.48)and Cs is the ion acoustic velocity given by Eq. (1.39). If we set ω = ωr − iγ and assumeγ ¿ ωr , we obtain for the real partkue0
1 + Ψ0(1.49)

Ψ0 ωr2 − k 2 Cs2
νin 1 + Ψ0

(1.50)

ωr =and for the imaginary partγ=

For γ > 0, the waves grow exponentially. So the condition for the instability is
ue0 > (1 + Ψ0 )Cs(1.51)

Hence, when electrons drift at a higher velocity than the approximate ion acoustic velocity,
the system is unstable. This condition can be understood approximately as follows:
Fig. 1.6 shows the electron and ion distribution functions in the frame of ions. Fig. 1.6a
is the case when the electron drift velocity ue0 is less than the ion acoustic velocity
Cs . Then the derivative of the electron distribution function at Cs is negative, and thesystem is stable. When ue0 exceeds Cs (Fig. 1.6b), however, the derivative of the electron
distribution function at Cs is positive. Then the wave gains more energy from electrons
than it loses to electrons, and the wave grows. From the equations above, electron density
fluctuations have the following relationship with the amplitude of wave electric fields.
When we set Ωe À νen ,
νi
E1
n1
=
E0
Ωi (1 + Ψ) n0
16(1.52)

This means that the relative amplitude of the wave electric field is smaller than the density
fluctuation by a factor νin /Ωi (1 + Ψ0 ) which is much less than unity. Therefore, the wave
electric field cannot be so large compared to the background electric field.
Gradient drift Wave
For the gradient drift instability, the zero-order gradient of the electron density plays animportant role. In Fig. 1.7, an unstable geometry is shown. In the region where theperturbation electron density ne1 is negative, the perturbation electron velocity ue1y =E1x /B is in the y-direction, and the lower density region is transported into a higher
density region. Then the density fluctuation is intensified and the perturbation electricfield is also intensified. This is clearly unstable. So the necessary condition of instabilityis
E0 · ∇n > 0(1.53)

Unified dispersion relation of Farley-Buneman and Gradient drift instabilities
Fejer et al. (1975) have proposed the complete linear theory including a finite zero-orderion drift ui0 and recombination which introduces a term −αn2 in the continuity equations,and allows k to have a component parallel to B.
k · (ue0 + Ψui0 )
1½
+Ψ
¤
Ψ £
1
γ =
(ωr − k · ue0 )2 − k 2 Cs2
1 + Ψ νin¾
1
νin
+ (ωr − k · ui0 ) kk − 2αn0
Lk
Ωi

ωr =

(1.54)(1.55)

where L is the gradient scale length
1 ∂n0
1
=
L
n0 ∂y
and

"µ
Ψ = Ψ0

k⊥
k

µ

¶2
+
17

Ωe
νen

(1.56)
¶2 µ

kk
k

¶2 #
(1.57)

It is important that the wave vector can have a component along the magnetic field line.
Even if the wave electric field is small, the current associated with the electric field could
be significant because of very high conductivity along the magnetic field line as mentioned
in Sec. 1.1.2, and the energy dissipation as heat could be large (St.-Maurice, 1987).

1.3

Scattering of Electromagnetic Waves by Electrostatic Waves

In the previous sections, we described electromagnetic waves in a plasma. When electromagnetic waves pass through a plasma, electron density fluctuations associated withthose electrostatic waves scatter the electromagnetic waves. The scatterings are classified
into two categories by their nature, coherent scattering and incoherent scattering. Bothscatterings are used to probe the ionosphere by sending radio waves and measuring scattered waves by radar. In this section, coherent and incoherent scatter are briefly explainedreferring to Hunsucker (1991).

1.3.1

Coherent Scattering

Coherent scattering echoes are usually detected in the disturbed ionosphere where electron
density fluctuations are amplified by instabilities. Radio waves are scattered by sharp
electron density gradients associated with unstable waves and the spectrum widths are
narrow, typically less than 1 kHz (sometimes less than 100 Hz). For example, STARE
radars in Scandinavia use VHF waves and measure echoes from the auroral E region.
SuperDARN radars use HF waves and measure echoes from both the E and F region.
The waves responsible for the scattering in the E region are the Farley-Buneman and
the Gradient drift waves. The scattered spectrum reflects the nature of the scatteringwaves. The Farley-Buneman waves are thought to be so called type 1, and when the
Farley-Buneman waves are strong enough to heat electrons, type 4 echoes of which thespectral widths are very narrow occur. type 2 echoes of which the spectral widths are18

rather wide are thought to be due to the Gradient drift waves. The types of coherent
radar spectra are summarized in Fig. 1.8.

1.3.2

Incoherent Scattering

In contrast to coherent echoes, incoherent scattering (IS) echoes can be detected at any
time and at any altitudes in the ionosphere. Each electron is moved by the electric fieldof incident radio waves and re-radiate radio waves. This mechanism is called Thomson
scattering. The cross section is very small,
σT ' 6.653 · 10−29 m2(1.58)The echo power is much weaker than the power of a coherent echo. When the wavelengthof the probing radio wave is smaller than the Debye length, kradar λD À 1, where kradar is
the radar wave number, a radar observes motions of each “free” electron. In this case, thespectrum has a shape just similar to the electron distribution function. When the radar
wavelength is larger than the Debye length, kradar λD ¿ 1, however, a collective motion ofelectrons becomes important. A radar observes a collection of waves scattered by thermal
fluctuations of electron density which is described as ion acoustic and the Langmuir waves
in unmagnetized plasma as described in Sec. 1.2.1. Therefore, the scattered spectrum
consists of two features, the low frequency part (the ion line) and the high frequency part(the plasma line) corresponding to ion acoustic and Langmuir waves, respectively. Fig. 1.9
shows schematically the typical IS spectrum (the ion and the plasma lines) together witha gaussian shape spectrum which corresponds to a spectrum when the radar wavelength
λradar is much shorter than the local Debye length λD . Because the ion line contains
information about ions as well as electrons, IS radars can measure such parameters as
the electron density, the line-of-sight ion drift velocity, the ion temperature, the electron
temperature (or the ratio of the electron temperature to the ion temperature), the ionneutral collision frequency, and ion composition (if the signal-to-noise ratio is very high).
Fig. 1.10 shows how IS spectra change with some of these parameters.
In actual measurements, very powerful echoes of which one of or both peaks of the19

ion line are strongly enhanced are sometimes observed (Foster, 1988; Collis et al., 1991;
Rietveld et al., 1991). Such spectra cannot be explained by thermal fluctuations. Those
spectra are rather coherent echoes and indicate that some instabilities intensify the ion
acoustic waves. These anomalous spectra are often observed associated with phenomena
where ions flow up and away from the earth. The mechanism of how such anomalous
echoes are created, and its relationship with ion upflow phenomena are still controversial
topics.

20

Ey (North)x (East)z (Up)

B

ue

Figure 1.5: Geometry of the system considered in Sec. 1.2.2.

21

(a)

Distribution Function

ue0 Csion

electron0velocity(b)

Distribution Function

Cs

ue0

ion

electron0velocity
Figure 1.6: Electron and ion distribution functions for a ue0 < Cs , and b ue0 > Cs .

22

Figure 1.7: Schematic drawing of the mechanism of the gradient drift instability. After
Kelley (1989), Fig. 4.31.

23

Figure 1.8: The four types of radar spectra that are observed in the auroral electrojet.
After Kelley (1989) Fig. 8.27.

24

(a)
1206

Spectral Density Function

Spectral Density Function

1208

1204

1202

1200

1198

1196
−900

−600

−300

0
Frequency (KHz)

300-600 KHz -300 KHz

600

300 KHz900600 KHz

Frequency(b) Downshifted Plasma Line
6

2.5

4

x 10

7

6

x 10

6
2

4

3

Spectral Density Function

5

Spectral Density Function

5
Spectral Density Function

Spectral Density Function

7

Upshifted Plasma Line

Ion Line

4

x 10

1.5

1

2

4

3

2
0.5

1

0
−3.4

1

−3.2

−3

−2.8

0
−30

Frequency (MHz)-3.2 MHz-2.8 MHz

−20

−10

0
Frequency (KHz)-10 KHz

10

20

30

0
2.8

3

3.2

3.4

Frequency (MHz)10 KHz

2.8 MHz

3.2 MHz

Frequency

Figure 1.9: Schematic drawing of IS spectrum. a kradar λD À 1, b kradar λD ¿ 1.

25

7

(a)

3

n =1⋅1011 m−3en =5⋅1011 m−3e
12 −3
ne=1⋅10 m

ne = 1012m-3

2.5

6

(b)

x 10

2.5

x 10

Ti=1000K

Spectral Density Function

Spectral Density Function

22

5*1011

1.5

1

0
−2.5

−2

−1.5

−1

−0.5

0
0.5
Frequency (Hz)

1.5

2000
1

3000
0.5

1011

0.5

1

1.5

2

0
−4

2.5

4

2.5

(d)

Te/Ti=0.5
T /T =1
e i
Te/Ti=2

3.5−10Frequency (Hz)

1

2

3

4
4

x 10

x 10

ui=0 km/s
ui=0.5 km/s
ui=1 km/s
2

Te/Ti=0.5
Spectral Density Function

Spectral Density Function

−2

6

x 10

3

−3

4

x 10

6

(c)

Ti=1000 K
Ti=2000 K
Ti=3000 K

2.5

1
2

1.5

ui = 0 m/s
1.5

500
1

1000

2

1

0.5

0.5

0
−25

−20

−15

−10

−5

0
5
Frequency (KHz)

10

15

20

0
−2.5

25

6

(e)

4.5

−1.5

−1

−0.5

0
0.5
Frequency (Hz)

1

1.5

2

2.5
4

x 10

6

x 10

4

−2

(f)

ν =0 s−1i
4 −1
νi=1⋅10 sνι = 5*104 sec-1

3.5

5⋅104 s−1

x 10

3

3.5

composition:
O2+,NO+[O+,NO+]:100%2[O+]:100%[O+]:90%,[H+]:10%

4Spectral Density Function

Spectral Density Function

2.5
3

10

2.5

2

0

1.5

O+
2

O+,H+
1.5

1
1
0.5
0.5

0
−2.5

−2

−1.5

−1

−0.5

0
0.5
Frequency (Hz)

1

1.5

2

0
−3

2.5
4

x 10

−2

−1

0
Frequency (Hz)

1

2

3
4

x 10

Figure 1.10: IS spectra with different parameters. When a ne , b Ti , c Te /Ti , d ui , e νin ,and f ion composition are changed.

26

Chapter 2
Instruments
In this study, data obtained by the European Incoherent SCATter (EISCAT) radar system
are mainly used. In addition, data from the Scandinavian Twin Auroral Radar Experiment (STARE) radars and from the Imaging Riometer for Ionospheric Studies (IRIS) at
Kilpisjärvi, Finland, are used. In this chapter, these facilities are briefly described.

2.1

The EISCAT radar system

The EISCAT radar system is a set of incoherent scatter radars operated by the EISCATScientific Association supported by Finland, France, Germany, Japan, Norway, Sweden,
and the United Kingdom. The EISCAT radar system consists of one tristatic radarsystem at UHF frequencies and one monostatic radar at VHF frequencies located innorthern Scandinavia in the auroral zone, and two monostatic UHF radars in the Svalbard
archipelago in the polar cap region. The transmitter of the tristatic radar system islocated in Tromsø (69.58◦ N, 19.21◦ E), Norway, and its receiving antennas are located in
Kiruna (67.86◦ N, 20.44◦ E), Sweden, and Sodankylä (67.37◦ N, 26.65◦ E), Finland, as wellas in Tromsø. They operate at 933 MHz. These radars are all parabolic dishes (32 m
diameter). This is the only IS radar system in the world which can determine the 3dimensional ion velocity vector without any assumptions. At Tromsø, there is another
VHF radar operating at 224 MHz. With its longer wavelength and large antenna aperture27

(40 × 120 m cylindrical parabola), the VHF radar can reach the topside ionosphere at analtitude of more than 2000 km. This radar is also suitable for the D region study. The
radars in Longyearbyen (78.15◦ N, 16.05◦ E), Svalbard, Norway, are two independent UHF
radars both operating at 500 MHz called the EISCAT Svalbard Radar (ESR). One ofthem is a steerable parabolic dish (32 m diameter). The other is the latest antenna which
has been in use since 2000, not steerable but fixed in the direction along the magnetic
field line. It is a parabolic dish (42 m diameter) with better antenna gain. Geographicalpositions of the EISCAT radars are shown in Fig. 2.1.
By fitting a theoretical spectrum to a received spectrum, the EISCAT radars can derive
ne , ui , Ti , Te /Ti , νin , and, if the signal-to-noise ratio is very high, the ion composition. ne ,ui , Ti can be derived at all the altitudes. The effect of Te /Ti and νin are so similar that itis practically impossible to measure those two parameters at the same time. Therefore,
above about 105 km where the electron-neutral collision is less important for IS spectra,
Te (or Te /Ti ) is fitted with a model νin . And below about 105 km where a large value of
νin allows us to assume Te = Ti , νin is derived. When the electric field is strong enoughto heat electrons in the lower E region (down to ∼100 km) as mentioned in Sec. 1.1.1,
the boundary of the altitude must be changed to a lower altitude.

2.2

The STARE radar system

The STARE facility is operated jointly by Max-Planck-Institut für Aeronomie (MPAe)in Germany, and the Finnish Meteorological Institute (FMI) in Finland in cooperation
with ELAB (the Technical University of Trondheim) in Norway. It consists of two coherent radars located at Midtsandan (63.6667◦ N, 10.73◦ E), Norway, and at Hankasalmi(62.3047◦ N, 26.6494◦ E), Finland. The radars operate at 140 MHz and 143.8 MHz, respectively, and measure the phase velocities of meter-scale field-aligned irregularities inthe E region which are created by the plasma instabilities as mentioned in Sec. 1.2.2.
Therefore, the disturbed conditions (a strong electric field) are suitable, and the echoes
vanish when the electric field is too weak. Each radar has 8 beams, and the field of
28

40 o
80 o WN

40 o
E

Longyearbyen (ESR)
500MHz, Tx/Rx (32m)
500MHz, Tx/Rx (42m)
75 oN

Sodankylä
933MHz, Rx
Tromsø
933MHz, Tx/Rx
224MHz, Tx/Rx

o

70 N

Kiruna933MHz, Rx

65 oN

oE

20oW

o

20

60 N 0oFigure 2.1: Geographical locations of the EISCAT radars. Tx/Rx means that the radarboth transmits and receives. Rx means that the radar only receives.

29

view commonly covered by the two radars is about 400 km×400 km ∼160000 km2 which
covers northern Scandinavia including the ionosphere over EISCAT Tromsø. (Fig. 2.2)
The beams are wide in the elevation, and they illuminates the ionosphere at almost all
the altitudes. The radars detect signals from the point where the radar wave vectors areparallel to the irregularity wave vectors which are perpendicular to the local magnetic
fields. By combining the velocities measured at common volumes, the two-dimensional
velocity field can be derived over the wide field of view. The phase velocities of the E region waves are thought to be approximately equal to the relative drift velocities between
electrons and ions, ud = ue − ui . Since ui is almost zero in the lower E region due tovery frequent collisions between ions and neutrals, ud is approximately equal to ue whichis thought to drift at the local E × B velocity. Thus it approximates the two-dimensional
map of the electric field that can be derived from the STARE radar system. When thecondition kST ARE ⊥ B is achieved at a slightly higher altitude than 110 km, however,contamination by the ion velocity causes the measured velocity to depart from the E × B
velocity. It is also known that the STARE underestimates the E × B velocity when theelectric field is strong (∼ 17 mV/m), which is known as the saturation at the ion acoustic
velocity (Nielsen and Schlegel, 1983, 1985; Robinson, 1993). These facts should be kept
in mind when the velocities measured by STARE are interpreted.

2.3

The Kilpisjärvi IRIS System

IRIS is a modern version of a very sophisticated riometer. In contrast to the EISCAT and
the STARE system described above, a riometer is a passive instrument which measures
radio noise from extraterrestrial sources. When cosmic noise passes through the ionosphere, it is absorbed by ionospheric plasma. If the ionosphere is modified, for example,
by an electron precipitation, the absorption is enhanced and, as a result, the observed
cosmic noise becomes weaker. By comparing the observed noise power with the quiet-day
curve (QDC) which represents the power without ionospheric absorption, the absorption
is obtained in dB. Cosmic noise is absorbed most effectively in the D region where elec30

Fri Jan 12 18:45:51 2001

STARE NF

STARE
Field
View
Sep−16 1999,
UTof
18:00:00
73.03
28.4
26
23.6

70

21.2

EISCAT
Tromsø UHF

18.8
16.4

Geogr.Lat.
lat
Geogr.

14
11.6
9.2
6.8
4.4
2

Midtsandan

−0.4
−2.8

Hankasalmi

−5.2
−7.6
−10

60
2.128
4.73

10
22.22

13.5
30

Geogr. long * 0.45

Geogr. Long.

Figure 2.2: The STARE field of view. The position of the EISCAT Tromsø beam at
field-aligned direction is also plotted.

31

trons collide very often with neutrals. IRIS consists of a square array of antennae. It
produces an array of antenna beams which covers a wide field of view and enables us toobtain a two-dimensional snapshot image of ionospheric absorption.
The Kilpisjärvi IRIS system is supervised by Lancaster University, UK, and operated
in conjunction with Sodankylä Geophysical Observatory (SGO), Finland. It is located in
Kilpisjärvi (69.05◦ N, 20.79◦ E) in northern Finland. It operates at 38.2 MHz and produces
an array of 49 beams. The projection of beams onto the ionosphere at 90 km is shown in
Fig. 2.3. Each beam is numbered from 1 to 49 from the northeast corner to the southwest
corner as shown in Table 2.1. The projection of the beam of the EISCAT Tromsø UHF
radar at a field-aligned position (Azimuth 183.2◦ , Elevation 77.3◦ ) at 90 km is in beam 9.

32

Outline of Beams When Projected to 90km

72 oN

71 oN

Kilpisjärvi
70 oN

EISCAT
Tromsø UHF
69 oN

68 oN

67 oN

66 oN
12 oE

o

28 E

16oE

o

20 E

o

24 E

Figure 2.3: The projection of Kilpisjärvi IRIS beams onto the ionosphere at 90 km
altitude.

The outline is the locus at which the sensitivity is 3 dB lower thanat the beam center.

(From IRIS homepage http://www.dcs.lancs.ac.uk/iono/iris

http://www.dcs.lancs.ac.uk/iono/iris /) The position of the EISCAT Tromsø beam ata field-aligned direction is also plotted.

33

North
1

2

3

4

5

6

7

8

9

10

11 12 13

14

15

16 17

18 19 20

21

West 2223 24

25 26 27

28 East29

30 31

32 33 34

35

36

37 38

39 40 41

42

43

44 45

46 47 48

49

South
Table 2.1: Number of each IRIS beam.

34

Chapter 3
E regionIn this chapter, we deal with interactions between the ionospheric plasma and electrostatic
waves in the E region. We take up an electron temperature enhancement phenomenon
which is often observed in the lower E region and is believed to be resulted in by plasma
waves excited by a strong electric filed. In Sec. 3.1, we show the results of tristatic electron
temperature measurements measurements in the lower E region. This section is mostly areproduction of Saito et al. (2001). Next, in Sec. 3.2, we consider the energy balance ofelectrons and parameterize the effects of waves by the anomalous (or effective) electron
collision frequency, νe? . We interpret νe? into a rotation angle of an electron velocity vector
from E × B direction, and discuss its relationship with a flow angle difference between
E × B direction measured by the EISCAT and plasma flow velocity vector observed by
the STARE system. This section mainly consists of Buchert and Saito (1997) and its
extension. Finally, in Sec. 3.3, we examine the nature of νe? . The cosmic noise absorption
is selected as one example. By referring to data from the Kilpisjärvi IRIS, we make one
aspect of νe? clear.

35

3.1

Interaction Between the Farley-Buneman Wave
and Electrons

3.1.1

Introduction

The knowledge and understanding of irregularities in the E region was recently reviewed (Sahr and Fejer, 1996). Usually dominant is the modified two-stream (or Farley–
Buneman) instability (MTSI). It causes echoes in the HF and VHF ranges at angles nearly
perpendicular to the geomagnetic field. Radars like the STARE and also SuperDARN
detect such echoes and measure geophysical parameters such as the electric field. Understanding of the non-linear plasma physics involved in the MTSI is not only essential
for the correct analysis of these radar data. The unstable E region is also an importantnatural example of plasma turbulence.
The plasma in the E region is strongly influenced by collisions between charged andneutral particles. This normally enforces thermal equilibrium below about 110 km, i.e.,
temperatures of neutrals, ions, and electrons, Tn , Ti , and Te , respectively, are equal. Onlyat higher altitudes selective heat sources, like frictional heating of the ions, precipitating
particles, or ionizing solar radiation, can easily raise Ti or Te above Tn . Therefore, strong
enhancements of Te higher than 1000 K below 110 km, first reported by Schlegel and St.Maurice (1981) using the Chatanika, were unexpected. Further observations were donewith the EISCAT radar (Igarashi and Schlegel, 1987; Williams et al., 1990; Jones et al.,1991). Davies and Robinson (1997) also showed the electron heating in the E region on a
statistical basis using 900 hours of EISCAT CP-1 and CP-2 data. Generally it was foundthat Te starts to increase when the electric field strength |E| exceeds the threshold for
excitation of the MTSI, Eth . Te increases approximately linearly with |E| − Eth .
Above ∼ 90 km, the Joule (or frictional) heating of electrons are very small because
νen ¿ Ωe . In the lower E region, precipitating particles can raise Te by typically only 20–
50 K (Schlegel and St.-Maurice, 1981) which is quite insufficient to explain the measured
Te enhancements of more than 1000 K. St.-Maurice et al. (1981) first realized that the36electron heating is due to electrostatic waves generated in the MTSI. Schlegel et al. (1983)
has shown that E region electron temperature profiles obtained with a rocket experiment in
Antarctica are very good agreement with theoretical electron temperatures estimated witha heating model by unstable plasma waves suggested by St.-Maurice and Schlegel (1982).
From in-situ observations with rockets Primdahl and Bahnsen (1985) argued that theheating can be explained by assuming an anomalous collision frequency ν ? applying to theelectrons drifting in the background electric field. ν ? parameterizes all macroscopic effects
due to waves. Also Robinson (1986) put forward a more self-consistent model where all
macroscopic effects can be parameterized by a ν ? representing collisions between electronsand “plasmons”. The model is supported by mean-field turbulence theory (Sudan, 1983)
and was further extended (Robinson and Honary, 1990). Ogawa et al. (1980) has shownthat when the electric field is strong, the effective collision frequency of electrons should
be enhanced by a factor of 6 to explain the observed aspect angle dependence of phasevelocities of E region irregularities. Because the wave fronts seen at VHF frequencies
are close to field-aligned, Farley (1985) has argued that the ν ? applies only to the fieldperpendicular direction. The E region turbulent transport coefficient should be strongly
anisotropic.
The in-situ observed VHF wave amplitudes and density fluctuation are often too weak
to explain the heating rates derived from radar observations if the electrons are driven in
the perpendicular direction only. St.-Maurice and Laher (1985) postulated therefore theexistence of a small wave field component parallel to the magnetic field, δEk . This would
greatly enhance the average Joule heating rate
We = < δj⊥ · δE⊥ > + < δjk · δEk >= σP < δE⊥2 > +σk < δEk2 >(3.1)

because the parallel conductivity σk is much bigger than the Pedersen conductivity σP ,σk À σP . The existence of δEk was noted as an open question in the text book by Kelley(1989).
Experimental studies of anisotropy so far concentrated on the waves by using radar37

backscatter from the ground. VHF radar echoes are observed at angles further away fromperpendicularity to the earth’s magnetic field than expected from linear theory. Also at
higher frequencies, 930 MHz, the backscattering is not only surprisingly strong, but occurs
at aspect angles even further away from perpendicular (Moorcroft et al., 1988; Eglitis etal., 1996). Obviously the E region plasma turbulence becomes more isotropic at shorter
wavelengths.
In this work we search for a directional dependence of the electron temperature during
heating event using tristatic EISCAT UHF radar. If we find anisotropic temperatures, theheating would have to be caused by perpendicular wave fields directly because electronneutral collisions heat isotropically irrespective of the direction of E0 and δE. On theother hand, the finding of isotropic temperatures would be consistent with St.-Maurice’s
model as well as with Robinson’s electron-plasmon scattering. However, for the latter
viewpoint, a finding of isotropic temperatures implies restrictions which will be discussed
later in this work. Thus our analysis can potentially help to decide between two hitherto
indistinguishable interpretations of experimental results.
In the next section, we present a comparison of electron temperatures measured withdifferent aspect angles to the geomagnetic field, B. In Sec. 3.1.3 we interpret the resultsof Sec. 3.1.2 and discuss the implementation on the E region plasma turbulence.

3.1.2

Tristatic Observation of Electron Temperatures by EISCATFig. 3.1 shows a summary of the data obtained by the EISCAT CP-1-I experiment from1200 to 1700 UT on June 12, 1990 when we found several distinctive Te enhancement
events. The antenna of the transmitting and receiving site in Tromsø, Norway, wasdirected along B. The remote antennas in Kiruna, Sweden, and Sodankylä, Finland,measured alternately a common volume at 278 km altitude for electric field determinationand six common volumes in the E region that can be used to detect any plasma anisotropy.
The dwell times of the remote antennas are 50 to 60 sec. For E region measurements, the38

Tromsø site uses a five-pulse code with a range resolution of approximately 3 km. The
remote sites receive a long pulse of 350 µsec. The height resolution of the remote data isdetermined by the antenna beam width which is 0.8◦ . In the E region, this corresponds
to approximately 2.9 km for the Kiruna antenna and 5.5 km for Sodankylä. Thus theheight resolutions of both Tromsø and remote data are comparable.
During a period shown in Fig. 3.1, the electric fields measured in the F region are
generally high, except after 1555 UT. Strong electron heating between 100 and 115 km
with amplifications at around 1240, 1450, and 1515 UT are associated with high |E|. In
the most intense event Te reaches about 1700 K between 1454 and 1455 UT. Ions areheated up by frictional (Joule) heating above 115 km. The electron temperatures in thelower E region below about 115 km are generally enhanced over both the neutral and ion
background temperatures.
Using the E region measurements from the remote EISCAT sites, we have checked thethermal isotropy of the electrons and also the ions. Figs. 3.2 and 3.3 show scatter plotsof temperatures obtained simultaneously from two different sites using all the data from0800 UT, June 12 to 1345 UT, June 13, 1990. At 108 km the aspect angle of the Sodankylä
data is 39◦ , while that of Tromsø is 0◦ . In Fig. 3.2a the electron temperatures from thesetwo directions are compared with each other. At the highest Te of 1700 K observedby Tromsø, no simultaneous observation by the remote sites is available at this altitude,
therefore the plotted Te reaches only about 1100 K. In Fig. 3.2b the ion temperatures fromthe two sites are compared. The comparisons of the electron and the ion temperatures at
108 km between Tromsø and Kiruna where the aspect angle is 35◦ are shown in Fig. 3.3aand 3.3b, respectively. Occasionally some preferentially transverse ion heating seems tooccur even at this low altitude, but to a much lesser degree than the electron heating.
The important point in this context is that the plots indicate clearly the isotropy of Te at
all temperatures. Although not shown here, comparisons at other altitudes are similar.

39

EISCAT CP1−I on 12/06/1990
E (mV/m)

150

100

E
50
20
0
12
150

13

14

15

16

17
2000

Altitude (km)

140

1500

130

1000 Ti

120
110

500

100
90
12
150

13

14

15

16

17

2000

Altitude (km)

140

1500

130

1000 Te

120
110

500

100
90
12

0

13

14

15

16

17

0

Figure 3.1: Overview of a data set with long periods of elevated Te in the lower E region.
The top panel shows |E| with a time resolution of 5 minutes. The approximate thresholdof the MTSI is about 20 mVm−1 . The middle and bottom panels show the electron andion temperatures, respectively.

40

(a)

(b)
1500

1500

CP1−I 12/06/1990

CP1−I 12/06/1990

T (K) at Sodankylä

108 km

1000

500

0

1000

i

e

T (K) at Sodankylä

108 km

0

500

1000

T (K) at Tromsø

500

0

1500

0

500

1000

T (K) at Tromsø

e

1500

i

Figure 3.2: a Scatter plot of the electron temperatures from different aspect angles observed with the Tromsø and Sodankylä EISCAT sites. The errors in fitting to the IS
spectra are also plotted. b The same as a, but the ion temperatures are compared.

(a)

(b)
1500

1500

CP1−I 12/06/1990

CP1−I 12/06/1990
108 kmT (K) at Kiruna

1000

1000

i

e

T (K) at Kiruna

108 km

500

0

500

0

500

1000

T (K) at Tromsø

0

1500

e

0

500

1000

T (K) at Tromsø

1500

i

Figure 3.3: The same as Fig. 3.2, but Tromsø and Kiruna data are compared.

413.1.3

Discussion

The E region plasma waves, which are assumed to be the cause of electron heating, arehighly anisotropic with respect to B. From coherent radar observations it is known that at
wavelengths above about a meter the back-scattered wave amplitudes fall off within aspect
angles of about 2◦ away from the magnetic field. From this, one might expect anisotropic
electron temperatures. Nevertheless, our results show that the Te enhancement is clearly
isotropic.
If electrostatic waves heat electrons directly, the wavelengths must be smaller thanthe electron gyroradius rg (∼ 1 cm). Otherwise the wave-particle interaction is ineffectivein giving electrons enough energy to heat them up. For long wave lengths (À rg ) and
low frequencies (¿ the electron gyrofrequency Ωce ∼ 9MHz) the electrons perform alternating, deterministic E × B drifts. In-situ observations show that the wave amplitudes at
shorter wavelengths decrease substantially (Pfaff et al., 1984). Extrapolating this trend
to (unobserved) cm wavelengths, we find that direct heating by such waves is an unlikely
explanation for our observed isotropic enhancements of Te .
When taking into account also electron-neutral collisions, the strong but slow andanisotropic Farley-Buneman waves can lead to isotropic electron heating. The electrons’
deterministic (E+δE)×B drift gets in each elastic collision with a neutral completely andisotropically randomized. Due to the huge mass difference between electrons and neutrals,
an electron loses practically no energy in elastic collisions, but the energy is converted
from flow energy to heat. About every 100th collision with a neutral is inelastic, exciting
vibrational and rotational modes of N2 and O2 molecules as well as fine structure levelsof O (Schunk and Nagy, 1978). In an inelastic collision, the electron transfer typicallya large fraction of its energy to the neutrals. The efficiency of this processes enforcesthat Te gets hardly elevated above neutral temperature in the lower E region even when
there are DC electric fields of the order of 100 mV/m (which is typical for the auroralzone). The amplitudes of Farley-Buneman waves δE do not become larger than the DC
background field even in the non-linear stage of the instability. Therefore purely per-42

pendicular waves cannot explain the large observed enhancements of Te . Consequently
St.-Maurice (1987) postulated a field-parallel component of δE, δEk , which would heatelectrons very efficiently due to high parallel transportability as discussed in the introduction. Our observation of isotropic Te enhancements is in accordance with St.-Maurice’s
model, although we cannot add any quantitative estimates of the size of δEk .

3.2

Energy Balance of Electrons: Pedersen Current
Carried by Electrons

3.2.1

Introduction

It is obvious that electron heating must be associated with dissipative currents. In theliterature, electrons are said to drift in a EtimesB direction which is dissipationless Halldrift. Therefore, there must be an additional dissipative current which is carried byelectrons. Robinson (1993) has also pointed out that due to the drag effect of the waves
the electron drift motion should depart from E × B drift. Although his approach is basedon a concept of a microscopic electron-plasmon collision, it is basically the same as ours
from the point of view that electromagnetic energy must be dissipated. In our hypothesis,
microscopic heating processes are in fact in a black box. Instead of going into the detailsof microscopic heating processes, we parameterize all the heating effects of waves by adissipative electron drift.
To accomplish this, a first task is to obtain an estimate of the true average azimuthal
direction of the electron drift. Haldoupis et al. (1993) compared the observed azimuthal
direction of VHF wave fronts with the E × B direction and found systematic differences
up to 20 deg. We would expect a similar result with the electron drift. Here, we firstestimate how much the deviation of electron drift velocity could be based on observed
electron temperatures. The isotropy of Te shown in Sec. 3.1 allows us to assume that
the Te measured by the EISCAT Tromsø radar in the field-aligned direction applies in all
directions. We then try to compare the electron drift with results obtained by STARE as
43

Haldoupis et al. (1993) has done.

3.2.2

Methods

Our new method is based on measurements of the electron temperature Te . Electron
heating is observed regularly over several minutes, and to a certain extent, it can be
described as a stationary process. Therefore, the dissipative currents must be rather
stationary and similar to classical Pedersen currents. The local heating rate of electrons
must be equal to j · E.
Now we start from the momentum equation of electrons and ions (Eq. (1.21)). Because
we are interested in low frequencies and large scales, we set the left-hand-side of Eq. (1.21)
to 0. We also neglect the pressure gradient force. Since the energy to heat electrons is
extracted from the relative drift between electrons and ions, or the DC electric field, thewaves should act as a kind of drag force against the electron drift. We then assume asexternal forces F0 , and we introduce
F0 = −ne me νen ue − ne me νe? (ue − ui )(3.2)

F0 = −ne mi νin ui − ne mi νi? (ui − ue )(3.3)

for electrons, andThe first term of each equation represents a frictional force with neutrals. For electrons,since νen ¿ Ωe , the effect is rather small. And we have assumed un = 0. The last termsof each equation represent the momentum exchange between ions and electrons which areanomalously enhanced due to waves and turbulence. Because the effects of electrostatic
waves mostly originate in forces between ions and electrons, they should appear as terms
which represent interaction between ions and electrons. νe? and νi? include both classicaland anomalous interactions between ions and electrons,?
νe? = νei + νei

(3.4)

?
νi? = νie + νie

(3.5)

44

where νei and νie are the classical electron-ion and ion-electron collision frequencies which?
?
are usually much smaller than νen and νin , respectively, and νei
and νieare the anoma-

lous (or effective) electron-ion and ion-electron collision frequencies which are assumed
to describe effectively the effects of waves and turbulences, respectively. Whatever the
microprocesses of waves are, contributions of waves and turbulences to the total momen?
tum balance in the system must be in these forms, if νei,ie
can effectively parameterize

them. These drag forces should be internal forces and should cancel each other out. Thisrequires
νi? =

me ?
ν
mi e

(3.6)

The anomalous collision frequency for ions is much less than that for electrons. But it
can be dominant over νin when νe? is strongly enhanced.
For a steady state, the electron heating rate is equal to the electron cooling rate.
According to Poynting’s theorem, the divergence of the electromagnetic energy is equal
to j · E. From Eq. (1.6), the energy balance of electrons becomes
j · E = Le,el + Le,in + Le,ion(3.7)

where Le,in represents all the cooling rates per unit volume due to inelastic collisions withneutrals. Jones et al. (1991) and also Buchert and Saito (1997) used only the first two
terms on the right-hand-side of this equation. As those authors above did, we use themodel for the first two terms which are given in Schunk and Nagy (1978) in which isotropic
electron temperatures are assumed. We can take this model, since we know from Sec. 3.1
that Te is isotropic even when Te is strongly enhanced. For the last term, the cooling rateby ions, based on the analogy of the cooling rate of ions by electrons (the second term ofEq. (1.3)), is given by
Le,ion = 2ne3meνe? kB (Te − Ti )me + mi 2(3.8)

It should be noted that the current density j which is effective in heating electrons isessentially carried by electrons, because the ultimate energy source is the DC electricfield which is applied to the ionosphere from the magnetosphere and drives electrons in a45

E × B direction at the zero-th order. At ∼ 108 km where observed electron temperatureenhancements are strongest, we can assume ui ∼ un ∼ 0 because νi n À Ωi . Thena simpler equation can be used for the electron drift motion by just replacing νen byνen + νe? in the classical equation. This is the same as replacing κe in Eq. (1.8) by ananomalously enhanced one
νen + νe?
Ωe

(3.9)

κ?e E⊥
1 E⊥ × B
e
+
−
E0k
?2
?2
?
1 + κe B
1 + κeBme (νen + νe )(3.10)

κ?e =and we obtainue = −

Now let us assume there is no DC electric field parallel to B. Since the second term of theright-hand-side of this equation is perpendicular to E⊥ , only the first term contributes
to electron heating. It carries the Pedersen currents which are, however, anomalously
enhanced by the effects of waves and turbulences. The enhanced Pedersen currents which
are carried by electrons are now given by
jP e = n e e

κ?e E⊥≡ σP? e E⊥
1 + κ?2Be(3.11)

where σP? e is the anomalously enhanced (or effective) Pedersen conductivity by electrons
which is partly drifting in a −E direction due to effects of waves and turbulences. Finally
we obtain the energy balance equation of the electrons
ne e

3
me
κ?e |E⊥ |2= Le,el + Le,in + 2ne
νe? kB (Te − Ti )?2
1 + κe B
me + mi 2(3.12)?
This is valid as far as the νei
is concerned which is included in νe? and can parameterize

well the effects of waves and turbulences. This equation is a third-order equation of κ.
Out of three possible solutions, one solution should be adopted which converges to νen /Ωewhen |E⊥ | is weaker than the threshold of the instability.
It should be noted that according to Poynting’s theorem, j·E is equal to the divergenceof the electromagnetic energy flux. Our argument in relation to the total energy balanceis equivalent to that of Magnetosphere-Ionosphere (MI) coupling in which the electromagnetic energy is supplied from the magnetosphere to the ionosphere by the Field-AlignedCurrents (FACs) and dissipated by the Pedersen currents which close the FACs in the46

ionosphere. The electron heating by plasma waves and the classical Joule heating can be
macroscopically understood in the same frame of MI-coupling although their microscopic
processes are quite different.

3.2.3

Observation

We have applied the method described above to data obtained by the EISCAT CP-1-K
experiment on September 16, 1999. In contrast to the EISCAT CP-1-I used in Sec. 3.1,
the remote antenna directions are fixed to the common volume in the F region whichenables continuous observation of the three-dimensional ion velocity vector which can be
regarded as the E × B plasma drift velocity vector, which enables us to determine theelectric field. Overview plots between 0900 and 1600 UT are shown in Fig. 3.4. Below
we use data obtained between 1100 and 1200 UT when the electric fields are strong andelectrons are heated up in the lower E region.
The cooling rates in Eq. (3.7) are essentially functions of temperature differences. If
we take the parameters of neutrals from the existing model, the MSIS-90 model (Hedin,
1991) in our study, the parameters to be supplied by measurements are Ti and Te . Since
we know that these temperatures are isotropic in the lower E region from the study inSec. 3.1, or by Saito et al. (2001), we can use Ti and Te measured by the EISCAT Tromsø
radar along the magnetic field line. For a representative integration period, Fig. 3.5 (leftpanel) shows the height profiles of the observed Te and Ti as well as Tn according to the
MSIS90 model. Below 105 km where significant ion frictional heating is not expected, Tn
is very close to Ti . This we take as an indicator of the accuracy of the MSIS-90 model forthis specific time and place. Between 100 and 115 km, Te is greatly enhanced with the
maximum enhancements at 108 km. Above 111 km, Ti is enhanced by frictional heating.
?
In Fig. 3.5(right panel), the profile of σP,eis shown together with σP,i . The latter ion?
Pedersen conductivity is computed classically and σP,eusing (3.7). The contribution of?
σP,eto the total height-integrated conductivity is only up to about 20 %. Relatively,?
however, σP,ebecomes significant below 110 km, and is comparable to σP,i . The flow47

Tromso CP1−I on 16/09/1999
E (mV/m)

80
60

E

40
20
0

9

10

11

12

13

14

15

16

150

1500

Altitude (km)

140
130

1000

Ti

120
110

500

100
90

9

10

11

12

13

14

15

16

150

0
1500

Altitude (km)

140
130

1000

Te

120
110

500

100
90

9

10

11

12

13

14

15

16

0

Figure 3.4: The same format as Fig. 3.1. But the electric field (top panel) is plotted witha 1 min resolution.

48

◦angle rotation decreases the Hall current by a factor of 1/(1 + κ?2e ). For the angle 7 , this?
factor is about 1.5 %, and the Hall current is almost unchanged by νei.

In Fig. 3.6, estimated electron flow directions with respect to the E × B direction byusing the procedure described in Sec. 3.2.2 are shown. Corresponding to the variation of
the electric field strength and of the electron temperature, the flow angles change theirvalue and become up to 7◦ which corresponds to νe? ∼ 30νen . In the same plot, we plot theflow angles estimated without the cooling effect of ions as Jones et al. (1991) and Buchert
and Saito (1997) did. Compared to the case without taking the cooling rate by ions into
account, the angles are markedly increased by factors of 1.5 ∼ 2. This can be understoodas follows: Typically, Le,el is 0.05 ∼ 0.1 of Le,in . Le,ion works basically in the same way
as Le,el . Therefore, without the waves and turbulences it should be very small because
νei is much smaller than νen . With the existence of the waves and turbulences, however,
the effective collision frequency between electrons and ions is anomalously enhanced up
to several tens times larger than νen . This leads to Le,ion ∼ Le,in , and dissipated energy,
and equivalently the flow angle with respect to the E × B direction is doubled comparedto the case without the cooling rate by ions.

3.2.4

Discussion

In the previous section, we have estimated the rotation of electron velocity from a E × B
direction to a −E direction based on the hypothesis that the effects of the waves and turbulences can be parameterized by anomalous (or effective) electron-ion collision frequency?
νei. It is necessary to confirm the electron flow angle rotation by other independent meth-ods. As Haldoupis et al. (1993) did, we compare the electron flow velocities estimated byour method with those derived from the STARE system.
As mentioned in Sec. 2.2, the STARE radars measure the line-of-sight velocities of
1 m scale irregularities. Assuming that the radars observe the line-of-sight projectionof the velocity vector of the irregularities which are also assumed to be equal to theelectron flow velocity (the so called cosine rule), the two-dimensional velocity vector can49

E= 50.7mV/m

140

135

130

Altitude (km)

125

Ti
T
e
Tn

120

115

110

105

100

0

500

1000

Ti,e (K)

1500

0

0.5

1

σPi,e

1.5

2
−4

x 10

Figure 3.5: Height profile of (left) temperatures of ions, electrons and neutrals, and (right)
Pedersen conductivity due to ions (blue) and electrons (red).

50Angle measured counter−clockwise from E × B direction
10
L=Li+Ln
L=Ln

9

8

7

Angle (deg.)

6

5

4

3

2

1

0
1100

1110

1120

1130
Universal Time

1140

1150

1200

Figure 3.6: Electron flow angles relative to the E × B directions at 108 km are plotted.
The flow angle is measured in a counter-clockwise direction (from E × B to −E). Flowangles are calculated both with and without taking the cooling by ions into account (solidline and dashed line, respectively).

51

be reconstructed. Fig.3.7 shows an example of the STARE measurements. Red colors
indicate strong echoes, and blue colors weak. The arrows show the direction and themagnitude of flow velocities. We select the data at the common volume of the STARE
beams and the EISCAT Tromsø beam (field-aligned) in the lower E region. Since theelectric field vector can be determined by the EISCAT measurements, the flow angle
measured by the STARE system with respect to a E × B direction can be derived. In
Fig. 3.8, the angles obtained from the STARE measurements are compared with those of
our estimation. Although the directions of flow rotation are systematically the same, in the−E direction, the angles measured with the STARE are much larger than our estimation.
This result can be understood by the following two reasons: First, the measured flow
velocities can be contaminated by the ion velocity when kST ARE ⊥ B is achieved at aslightly higher altitude than 108 km. The ion velocity rotates the relative drift velocity
between electrons and ions in a counter-clockwise direction. Although we have not checked
the altitude where kST ARE ⊥ B is achieved, this explanation is quite probable because theion velocity effect always gives a counter-clockwise flow angle rotation. Second, as pointed
out and discussed by previous works (Nielsen and Schlegel, 1983, 1985; Robinson, 1993),
the phase velocity of the MTSI waves is saturated at the local ion acoustic velocity Cs .
When one of or both of the velocities measured by the two STARE radars is saturated,the velocity vectors derived by applying the cosine rule are no longer true electron velocity
vectors. In this time period, the common volume is in the eastward electrojet (westward
electron drift). Due to the geometry of the STARE system, radar beams of the Midstandansite may be in the unstable cone where the line-of-site electron drift velocities are fasterthan Cs more often than those of the Hankasalmi site. This leads to the consequence thatthe velocity saturation at Cs tends to rotate the flow vectors derived with the cosine rulein a counter-clockwise direction with respect to a E × B to a −E direction. Therefore, it
is impossible to confirm our hypothesis by measurements using the STARE system.
One possible method of testing our estimation is to use a very high resolution camera
which can detect meter-scale irregularities. Trondsen and Cogger (1997) has made such acamera which has a ∼ 2 m resolution and is sensitive to emissions typically from around52

Fri Jan 12 16:45:01 2001

STARE NF
Sep−16 1999, UT 14:11:00
73.03
28.4
26
23.6

70

21.2
18.8
16.4

Geogr.Lat.
lat
Geogr.

14
11.6
9.2
6.8
4.4
2

Midtsandan

−0.4
−2.8

Hankasalmi

−5.2
−7.6
−10

60
2.128
4.73

10
22.22

13.5
30

Geogr. long * 0.45

Geogr. Long.

Figure 3.7: Two-dimensional map of echo intensities and flow velocity vectors measured
by the STARE system at 1149 UT on September 16, 1999 is shown. The echo intensities
are measured in dB units with respect to the background noise level. The flow vectors arederived from two velocity components measured at Midstandan and Hankasalmi applying
the cosine rule.
53

Angle measured counter−clockwise from E × B direction60
L=Li+Ln
L=Ln
STARE
50

Angle (deg.)

40

30

20

10

0
1100

1110

1120

1130
Universal Time

1140

1150

1200

Figure 3.8: Similar to Fig. 3.6. Angles of flow derived from STARE data by applying thecosine rule are also plotted (dash-dotted line).

54

105 km altitude. If the camera could detect such meter-scale irregularities, however, wave
propagation directions could be determined from its two dimensional images. Simultaneous observation by the camera and the EISCAT radars could be a good combination of
testing our estimation of electron flow directions.
Another implication of our results is the effects of the electron drift parallel to −E on
the magnetosphere-ionosphere coupling by modifying the ionospheric current system. The
dissipative component of height-integrated Ohm’s law for the ionosphere, JP = ΣP E⊥ , isslightly non-linear. Fig. 3.9 shows semi-empirically the relationship between the heightintegrated Pedersen current, ΣP , and |E| for a representative profile of ne . To obtain theplot, observations of Te at different altitudes were plotted over |E|. Straight lines werethen fitted to the increase of Te (see also Schlegel and St.-Maurice, 1981; Jones et al.,1991) Using (3.7) je was calculated for a range of electric fields. Height integration givesR
the total current JP,total = JP,e +JP,i , JP,(e,i) = jP,(e,i) dh, which is plotted in Fig. 3.9. We
can see that at the onset of the MTSI at about 20 mVm−1 , the current-voltage relation
has a small non-linear kink.
That ΣP depends (weakly) on |E| could potentially destabilize MI coupling. When
|E| > Eth , the E region draws more than the proportional current from the magnetosphere.
This ionospheric non-linearity potentially generates higher harmonics in geomagnetic pulsations, and in Alfvén waves. However, as already expressed, the deviation from the linear
Ohm’s law is rather small.
Finally, macroscopic consideration of the effects of the electrojet irregularities on theflow of the current through the ionospheric plasma could be done in the same way as
Hagfors (1984). Although we have not estimated the effect in detail, the effect of theirregularities on the mean current flow does seem to lead to a deviation in the direction of
the current flow. The detailed assessment requires numerical calculations with knowledgeof the power spectrum of the density irregularities and is left for our future study.

55

Non−linear Ohm’s law in the ionosphere1JP,total
JP,iHeight−integrated Current Density (A m−1)

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

20

40
60
−1
Electric Field (mV m )

80

100

Figure 3.9: Semi-empirically obtained relation between height-integrated current andcross field voltage for a specific electron density profile. A straight line (linear Ohm’slaw, dashed line) would result without anomalous heating. However, above the thresholdfor the MTSI, the total height-integrated current JP,total = JP,e + JP,i (solid line) is higher.

56

3.3

Cosmic Noise Absorption: Comparison with observation by IRIS

3.3.1

Introduction

In Sec. 3.2, we have introduced the anomalous electron collision frequency νe? which macroscopically parameterizes all the microscopic effects. It is an effective collision frequency
based on the analogy of the classical collision frequency to represent the amount of anelectron Pedersen drift (energy dissipation), or deflection angles of electron drift velocity
from a E × B direction. To know the nature of νe? , it is important to examine whether theνe? can act in the same way as the classical collisions in other phenomena where collisionsof electrons play an important role.
Cosmic Noise Absorption (CNA) is an effect of the ionosphere that absorbs a fractionof radio waves from extraterrestrial sources. As mentioned in Sec. 2.3, it is measured inriometers. In CNA, collisions of electrons are essential to dissipate a fraction of energy ofincident extraterrestrial radio waves. The absorption rate K in dB/km when radio waves
pass through the plasma is given by Hunsucker (1991)¶µne ν e−2(db/km)K ' 4.6 · 10
ω 2 + νe2
for wave propagation perpendicular to B or for unmagnetized plasma, andµ¶ne ν e−2
K ' 4.6 · 10
(db/km)(ω ± Ωe )2 + νe2(3.13)(3.14)

for propagation parallel to B. For extraordinary waves, the minus sign and the plus signfor ordinary waves must be used. For riometer frequencies of ∼ 40 MHz À νe , K is
proportional to νe as well as ne . The balance between decreasing ne and increasing νewith a decreasing altitude causes a peak of K in the D region. The absorption measured
by riometers is given by an integral along the line of sight, [Eq.] If the νe? acts as areal collision frequency, we expect enhancement of absorption associated with νe? in the
E region.
57

Using an IS radar and a riometer at Sondre Strømfjord, Greenland, however, Stauning(1984) and Stauning and Olesen (1989) have shown that the absorptions observed by
the riometer can be explained by the absorption model (Eq. (3.14)) with the classicaleffective electron-neutral collision frequency which can be slightly enhanced by elevatedTe , and that adopting anomalous collision frequencies given by Robinson (1986) lead tosignificant overestimation, for example, 15.9 dB when 0.5 dB is observed. A macroscopic
theory which is not concerned with the microphysics of the waves also predicts that theanomalous absorption associated with the presence of the turbulences is negligible. (SeeAppendix A) In this section, we do a similar analysis as Stauning (1984) and Stauning
and Olesen (1989) with higher time resolution data in good quality from the EISCAT
Tromsø UHF radar and the Kilpisjärvi IRIS.

3.3.2

Observation

Fig. 3.10 shows the summary of data obtained by the Kilpisjärvi IRIS and the EISCATUHF radar between 0900 and 1600 UT on September 16, 1999 when the electric fields
are enhanced several times and electrons are heated. The absorption data were measured
by beam No. 9 which covers the ionosphere over Tromsø (see Fig. 2.3 and Table 2.1)
and corrected to values which would be observed if the line of sight is vertical by taking
into account the obliqueness of the line of sight. Between 0900 and 1030 UT, prominent
absorptions are found. After 1030 UT, the absorptions are generally weak. Occasionallysome negative absorptions, which indicates an increase of radio wave power, are observed
around 1130, 1200, 1300, 1420, and 1530 UT. They are due to solar radio emission andnot relevant for this study. The EISCAT UHF radar operated in the CP-1-K mode
where the Tromsø radar was pointed along the geomagnetic field line and the remotesites (Kiruna and Sodankylä) looked continuously at a common volume in the F region
for the determination of E. Above 90 km, we have used the data obtained by using thealternating-code which enables us to derive parameters like ne and Te . Below 90 km where
it can be rather safely assumed that Te = Tn , data are obtained by power-profile which58

provides ne down to 62 km with an assumption that Te = Ti which can be safely justified
in this altitude range. At 108 km where electron heating is generally strongest, several
Te enhancements up to about 1200 K are seen at around 1127, 1133, 1206, 1220, 1419,
and 1448 UT associated with strong electric fields. In general, the trend of ne at 90 km
and the absorption are correlated well, and Te enhancements (or strong |E|) seem to havepoor correlation.

3.3.3

Discussion

To check whether the electron anomalous collision frequency νe? is effective in CNA, wehave estimated the ionospheric absorptions with EISCAT data based on two different
assumptions:1. νe = νen + νe?
2. νe = νenand compared them with the absorption measured by the IRIS. If the absorption estimated by the anomalous collision frequency (case 2) was the same order as the observed
absorption, the possibility for the anomalous collision frequency to act as the ’real’ collision frequency is not excluded, although it is still uncertain. However, if the estimatedabsorption with the anomalous collision frequency is much larger than the observations,and if the classical collision frequency without the anomalous one (case 1) instead accounts
for the observation, it is clear that the anomalous collision frequency is not completely
equivalent to the classical collision frequency. We have argued this approach for two reasons. First, in fact, it has been shown by Stauning and Olesen (1989) that the calculationof absorption with the anomalous collision frequency introduced by Robinson (1986) with
the quasi-linear theory overestimates the absorption. Therefore, we had expected that
our anomalous collision frequency would not be effective on the radio wave absorption.
Second, it is difficult to estimate the collision frequency profile required for the observed
absorption, since the observed absorption is integrated along the line of sight. To solve
59September 16, 1999, Kilpisjärvi, beam 9
Absorption (dB)

1.5
1
0.5
0
−0.5

Electron Temperature (K)Electric field (mV/m)

−1
0900

1000

1100

1200
1300
Universal Time

1400

1500

1600

1000

1100

1200

1300

1400

1500

1600

1000

1100

1200

1300

1400

1500

1600

1000

1100

1200

1300

1400

1500

1600

1000

1100

1400

1500

1600

80
60
40
20
0
0900

2000
1500
1000
500
0
0900
12

−3

Ne (m ) at 90km

10

10

10

8

10
0900
12

−3

Ne (m ) at 108km

10

11

10

0900

1200
1300
Universal Time (hour)

Figure 3.10: Overview of a data set with long periods of elevated Te in the lower E region
with a time resolution of 1 min. The top panel shows the absorption measured by beam
9 of the Kilpisjärvi IRIS. The last four panels show observation by the EISCAT, |E| (thesecond panel), Te at 108 km (third), ne at 108 km (fourth), and ne at 90 km (fifth).

60

the equation of absorption for the collision frequency, a certain collision frequency model
must be assumed. In the case of 2, Te contributes to absorption only through the weak
1/2

dependence of νen on Te , because νen is approximately proportional to Te

in the E re-gion where the absorption is weak. On the other hand, for the case 1, Te contributes
to absorption through both νen and νe? . In estimating absorption, we have neglected theeffect of the geomagnetic field even if the IRIS beam 9 measures radio waves propagating
obliquely to the geomagnetic field line. This is justified because ω À Ωe . This assumption
allows us to use Eq. (3.13) in our estimation. Then we have integrated K over an altitude
from 80 to 120 km.
ZA=

Kdh = 4.6 · 10

−2

Z 120 km
80 km

ne ν edh (dB)ω 2 + νe2

(3.15)

Fig. 3.11 shows the results of the comparison between 1400 and 1500 UT. It is clearthat the estimated absorption with νe = νen + νe? is too high while the estimation withνe = νen shows very good agreement with the observation by IRIS. To check the accuracyof our calculation, we have also done the same analysis for a period when the absorption
observed by IRIS is strong (Fig. 3.12). In this figure, red circles (νe = νen + νe? ) are almost
hidden by green ones (νe = νen ) because mostly νe? = 0 due to weak |E|. The estimatedabsorption with νe = νen is in excellent agreement with absorption measured by the IRIS.
Our results confirm the analysis by Stauning (1984) and Stauning and Olesen (1989),
although our νe? is introduced differently from the anomalous electron collision frequency
in Robinson (1986). Also our results agree well with the macroscopic consideration given
in Appendix A. Therefore, our analysis strongly suggests that νe? introduced to parameterize the effect of the Farley-Buneman waves in heating electrons is not effective in theabsorption of radio waves. The anomalous collision can represent the electron heating
rate, but it has nothing to do with CNA phenomena, and it is not necessarily equivalent
to the classical collision frequency, but it rather has a limited range of application, although it can represent the drag effect of the waves equivalent to collisions of νe? times
per second.

61

September 16, 1999, Kilpisjärvi, beam 9
1.2

1Absorption (dB)

0.8

0.6

0.4

0.2

0

−0.2
1400

IRIS
EISCAT(anomalous)
EISCAT(classical)
1410

1420

1430
Universal Time

1440

1450

1500

Figure 3.11: Absorptions estimated by the EISCAT data (red: νe = νen + νe? , green:νe = νen ) and the IRIS measurements (blue line) are compared between 1400 and 1500 UT
when |E| is generally strong.

62September 16, 1999, Kilpisjärvi, beam 9
1.4

1.2

Absorption (dB)

1

0.8

0.6

0.4

0.2

0
0900

IRIS
EISCAT(anomalous)
EISCAT(classical)
0910

0920

0930

0940
0950
Universal Time

1000

1010

1020

1030

Figure 3.12: The same as Fig. 3.11. But the results are for the period when Te is notenhanced, but ne at 90 km is enhanced.

63

3.4

Summary

We have studied electron heating by Farley-Buneman waves in the lower E region using
data obtained by the EISCAT radar system, the STARE system, and the Killpisjärvi
IRIS. The results obtained in this study are as follows.
The electron temperatures observed in the E region are clearly isotropic over aspect
angles from 0◦ to about 40◦ even when they are strongly enhanced. If the actual heatingprocess is through electron-neutral collisions which have no directional preferences, it isnatural to expect isotropic Te . The very high heating rates suggest that Farley-Buneman
waves have an electric field component both perpendicular and parallel to B as proposedby St.-Maurice (1987). If Te enhancement is instead due to a direct scattering of electrons
by wave fields not involving electron-neutral collisions, our observation implies that thesewaves should be isotropic.
Because the total energy is conserved, electrons must carry a dissipative current whichmeans that the electron drift velocity should deviate from the E × B direction to the −E
direction. We have found an angle between ue and E × B of about 7◦ (νe? ∼ 30νen ) during
events with high |E|. This flow angle rotation could not be confirmed by observations by
the STARE system probably due to the effect of the ion velocity and the saturation of
the irregularity phase velocity at Cs . We have examined the importance of the electron<

Pedersen currents and have found that at altitudes ∼ 110 km this current can become
the main Pedersen current. Consequently, the current-voltage relation in the Earth’s
ionosphere is slightly non-linear.
We have shown that the cosmic noise absorption observed by the Kilpisjärvi IRIS
can be well explained by the model in which only the classical electron-neutral collision
frequency is taken into account, even when the anomalous electron collision frequencies
greatly exceed the classical ones associated with strong electric fields and strong electron
temperature enhancements. From this result, we concluded that νe? has nothing to do withthe absorption of radio waves in the ionosphere. Therefore, νe? introduced to parameterize
the heating effect of the Farley-Buneman waves has a rather limited range of application.

64

We conclude that the waves should have the drag effect equivalent to the collisions of νe?
times per second on the electrons drift, but νe? is not necessarily equivalent to the classicalcollision frequency in the other phenomena.

65

Chapter 4
F regionIn this chapter, we move to the F region. Incoherent scattering which is an interaction between radio waves and electrostatic (the ion acoustic and the Langmuir) waves are dealt
with. We calculate IS spectra for plasmas with electrons with the kappa distributionfunction, and their differences from Maxwellian IS spectrum are shown. Possible misinterpretation of the ionospheric parameters are estimated. Finally we discuss the plasmaline with electrons with the kappa distribution function, and point out the possibility
to measure deviation of an electron distribution function from a Maxwellian one. This
chapter is mostly a reproduction of Saito et al. (2000).

4.1

Introduction

Incoherent scattering is a very powerful tool to investigate plasma in the Earth’s upper
atmosphere and in the laboratory. Using incoherent scatter (IS) spectra, one can derivevarious parameters such as ne , Ti , Te , ui . Incoherent scattering allows us to see directly
properties of the two basic types of electrostatic waves in a plasma, the ion acoustic waves
and the Langmuir waves, and these wave properties can tell us not only the plasma parameters such as ne , Ti , Te , and ui but also something about the plasma itself like theparticle distribution functions. It is well known that the velocity distribution functions ofions and electrons in a plasma are Maxwellian, if the ions and the electrons are in thermal
66equilibrium. Often, however, external energy input causes significant deviations of the
distribution functions from a Maxwellian. For example, in the Earth’s ionosphere a highelectric field in combination with ion-neutral interaction leads to a bi-Maxwellian or even a
toroidal distribution function of the ions (St.-Maurice and Schunk, 1979). The corresponding incoherent scatter (IS) ion line was explored in detail (Raman et al., 1981). Zheng
et al. (1997) have calculated the spectrum of another type of non-Maxwellian electron
distribution function, the super-Gaussian distribution (f (v) ∝ exp (−(v/vth )x ), x > 2).
In this work we investigate the effects of a kappa distribution function of electrons on the
IS spectrum.
Theoretically it has been shown that the velocity distribution function obeys a powerlaw at energies higher than critical energy when a superthermal radiation field is present(Hasegawa et al., 1985). Such a power-law functional dependence is an approximation to
the more general distribution, the generalized Lorentzian or kappa distribution (Baumjohann and Treumann, 1996). Measurements of electron energy spectra with space craft
have been successfully modeled with kappa distributions (Vasyliunas, 1968; Christon etal., 1988; Bryant, 1999). Also plasma turbulence can interact with both thermal andsuprathermal particles and generate suprathermal tails of the particle velocity distribution (Galeev, 1989). For plasmas in which both electrons and ions have kappa distribution,
Summers and Thorne (1991) have calculated the modified plasma dispersion function theoretically, and found general properties of the ion acoustic waves and the Langmuir waves(See also Summers et al., 1996). Since incoherent scatter spectra are closely related tothose waves, some qualitative expectation of the spectra can be obtained, although it
is actually assumed that only electrons have kappa distribution while ion distributionfunction remains Maxwellian.
An IS spectrum consists of two characteristic features, the ion line and the plasmaline. The ion line is an echo that corresponds to the ion acoustic waves found in the low
frequency range (for a transmitter frequency of several hundred MHz, in the range of a few
kHz). Since the ion line contains information about ions as well as electrons, it is the ionline that is used to derive the ionospheric parameters. Therefore, it is important to know
67the effects of the kappa distribution function on the ion line for the correct derivation of
the parameters. This knowledge would be very important, for example, for estimating the
momentum balance of the ionosphere or the threshold of plasma instabilities. The othercharacteristic feature is that plasma lines are received in the high frequency range. This1

feature corresponds to the Langmuir waves found around (ωpe /2π)(1+3k 2 λ2D ) 2 , where λD
is the electron Debye length. Usually the lines appear at several MHz up- and downshiftedfrom the transmitter frequency depending on electron density and temperature. Since
ions are too heavy to oscillate at the high frequency range, the contribution of ions to theLangmuir waves is negligible. The shape and position of the plasma lines are determinedby electrons (and also by the radar k-vector). Therefore, it is expected that a kappa
distribution of electrons has clearer effects on the plasma line than on the ion line. Thismay be useful in obtaining a correct analysis of the ion line by providing more information
about electrons.
In the Earth’s topside ionosphere the European Incoherent Scatter (EISCAT) radars
observe frequently considerably enhanced electron temperatures. Unlike the electron heating in the E region, it is caused by soft precipitation. Candidates for the heating mechanisms are direct collisions and indirectly via the Langmuir waves and turbulence generated
for example by a bump-in-tail instability. Both of these mechanisms could in principle
enhance velocity-space diffusion and distorts the electron distribution function and cause
a high-energy tail in the electron distribution function. Therefore, one cannot exclude
the possibility that the electron distribution develops a high energy tail. It is known
that the electron heating by particle precipitation is also associated with ion upflow anddownflow (Wahlund et al., 1992; Ogawa et al., 2000). This field-aligned ion flow could be
explained by ambipolar diffusion. The data analysis is usually carried out under the assumption of Maxwellian distribution functions. In order to get more reliable, quantitative
understanding from incoherent scatter data, we need to consider the possibility that theelectrons have a high energy tail which might influence the fits of ionospheric parametersin the data analysis. In this work, we try to model the electron distribution function
with a a kappa distribution. We calculate the IS ion and plasma lines for a plasma with68

electron kappa distributions. The effects and possible misinterpretations of a Maxwellian
analysis are investigated. In Sec. 4.2 the theoretical basis for calculating the IS spectra ispresented. In Sec. 4.3.1 the ion line is discussed, and the errors caused by a Maxwellian
analysis are examined in Sec. 4.3.2. Sec. 4.3.3 investigates the plasma lines resulting froman electron kappa distribution.

4.2

TheoryThe procedure of the theoretical calculation of an IS spectrum is described by Sheffield(1975). We follow Sheffield (1975) for the following derivation. Scattered power in thefrequency range dω and in the solid angle dΩ with wave vector k is given byPs dΩdω ∝ ne0 S(k, ω)
where ne0 is the electron density. S(k, ω) is the spectral density function given by¿À
2γ |ne (k, ω − iγ)|2S(k, ω) = lim
γ→0 V
ne0

(4.1)

(4.2)

where ne (k, ω − iγ) is the Fourier- in space and Laplace- in time transform of the electron
density ne (r, t),ne (k, ω − iγ)
Z +∞
Z +∞
ik·r
=
e
dr
ne (r, t)e−i(ω−iγ)t dt (γ > 0)
−∞

(4.3)

−∞

When no magnetic field exists, or k is parallel to the magnetic field, the spectral density
function is given by

¯
¯ ¯2
¯
¯ (1 + Ci ) ¯2
¯ ¯
¯ Be + 2Z ¯ Ce ¯ Bi
(4.4)
S(k, ω) = 2 ¯¯
¯
¯ ² ¯
²
where Z is the charge number of ions. Functions Ce,i , Be,i , and ² in Eq. (4.4) are defined
taking into account the effect of collisions through a BGK model as follows:Z +∞qj2 nj0 k · ∂fj0 (v)/∂v
1
Cj =
dv
1 + Dj −∞mj ²0 k 2 ω − k · v − iνjZ +∞νjfj0 (v)Bj =dv
2(ω − k · v)2 + νj2|1 + Dj | −∞−|Dj |2
νj |1 + Dj |2

(4.5)

(4.6)
69

whereZ +∞Dj = iνjdv−∞

fj0 (v)
ω − k · v − iνj(4.7)

νj is the collision frequency, fj0 (v) the distribution function, mj the mass, qj the electriccharge, nj0 the density, ²0 the electric permittivity of the vacuum, and ω the angularfrequency. Here j can be e or i to denote electrons or ions, respectively. ²(k, ω, νe , νi ) is
the plasma dielectric function,²(k, ω, νe , νi ) = 1 + Ci + Ce

(4.8)

In integrating Eqs. (4.5) and (4.6), the integrals of a form need to be calculated.
Z +∞
fj0 (v)dv
ω − k · v − iνj
−∞

(4.9)

The integrand has a pole at k · v = ω − iνj The integration path must be above the polebecause we have defined the Laplace transform with negative imaginary part of frequency
and taken the limit of γ → 0 even in the case of a collisionless plasma.
Eq.s (4.4)–(4.8) are valid for arbitrary distribution functions as long as a plasma isstable. It is well known that the particle distribution function becomes Maxwellian when
the particles are in thermal equilibrium. Therefore, in usual incoherent scatter analysis
one can assume that fj0 is Maxwellian,
Ã

2

v
1
exp − 2fM (v) = √3vth,M( πvth,M )!
(4.10)

where vth,M is the characteristic velocity (or thermal velocity) of the Maxwellian distribution function given byrvth,M =

2kB T
m

(4.11)

Later, the temperature of different distribution functions is discussed. Therefore, thetemperature must be clearly defined in a consistent way for any distribution functions.
Eq. (4.11) has been derived according to the definition of temperature, Eq. (1.2). However, as mentioned in Sec. 4.1, distribution functions with high energy tail have been often
observed in space plasmas when energy input maintaining the high energy tail exists. It is70known that such distribution functions can be well modeled by a generalized Lorentzianfunction with spectral index κ (kappa distribution). The one-dimensional kappa distribution function is given byfκ(1) (v) = √1
Γ(κ + 1)3/2πvth,κ κ Γ(κ − 1/2)

Ã

v2
1+ 2
κvth,κ

!−κ
(4.12)

where vth,κ is also the characteristic velocity (or thermal velocity) of the kappa distributionfunction given byr2κ − 3 kB T
(4.13)
κmwhich satisfies the relation (1.2) when κ > 32 . (Summers and Thorne, 1991) Kappa
vth,κ =

distribution functions for different values of κ are compared with Maxwellian in Fig. 4.1.
It should be noted that the smaller the index κ is, the more high energy electrons exist.
In the limit of κ → ∞, the kappa distribution converges to Maxwellian.
To evaluate the spectral density function, it is assumed that the ion distributionfunction is Maxwellian because it is unlikely to be influenced severely by plasma waves or
turbulence. Following Sheffield (1975), Eqs. (4.5)–(4.7) for ions become as follows:
Ci
Di
Bi
where

2
1 1 2ωpi[1 + ξM Z− (ξM )]=2
1 + Di Z k 2 vth,Miνi
Z− (ξM )= −kvth,M
1|Di |2
= −
Im[Z
(ξ
)]
−
− M
kvth,M |1 + Di |2
νi |1 + Di |2

Z +∞ −ξ21e M
Z− (ξM ) = √ds
π −∞ s − ξM√2)(erf(iξM ) − 1)
= i π exp (−ξM

(4.14)(4.15)(4.16)(4.17)(4.18)andω − iνi(4.19)
kvth,M‘erf’ is the complex error function. Here the ion plasma frequency defined is introduced
ξM =

by

µ
ωpi =

ni0 Z 2 e2
mi ²0
71

¶ 12

(4.20)

−5

10

−6

Normalized Distributions

10

−7

10

−8

10

−9

10

−10

κ=3
5
10
40

Maxwellian(κ=∞)

10

−11

10

−12

10

−3

−5

−4

−2

−1

0

1

2

3

Velocity normalized by vth (Maxwellian)
−3

−2

−1

0

1

2

Velocity normalized by v (κ=3)

3

4

5

th

Figure 4.1:

A comparison between Maxwellian and kappa distribution with κ =3, 5, 10, and 40. Maxwellian distribution (dashed line) corresponds to κ = ∞. x-axes
are indicated by velocities normalized by vth,M and vth,κ=372

The function Z− (ξM ) given by Eq. (4.17) is very similar to the plasma dispersion function(Fried and Conte, 1961) except for the sign of the second term of the right hand side.
When the electron distribution function along the line-of-sight is a generalized LorentzianR +∞(1)
distribution (kappa distribution), i.e. −∞ dv⊥ f (v) = fκ (vk ), where v⊥ and vk are thevelocities perpendicular to and parallel to the line-of-sight, respectively, we can write
·
¸
2
2ωpe
1
1
?
Ce =
1−
+ ξκ Zκ,− (ξκ )(4.21)
2
1 + De k 2 vth,κ
2κ

De

Be

where

¶µ
¶3/2
κ
κ−1
κ − 3/2
κ
Ãr
!
κ−1
?
×Zκ−1,−
ξκ
κ

iνe
= −
kvth,κ

µ

µ
¶µ
¶3/2
κ
κ−1
1
= −
kvth,κ |1 + De |2 κ − 3/2
κ
"
Ãr
!#
κ−1
|De |2?
×Im Zκ−1,−ξκ−κνe |1 + De |2Zdsκκ−1/2 Γ(κ + 1) +∞?
Zκ,− (ξκ ) = √πΓ(κ − 1/2) −∞ (s − ξκ )(s2 + κ)κ+1(4.22)(4.23)(4.24)andξκ =ω − iνekvth,κ

(4.25)?
Zκ,−(ξκ ) can be analytically expressed for positive integer κ as

κ − 1/2 κ!
2κ3/2 (2κ)!
¶κ+1−lµκX(κ + l)!
2
κ−l
√
×
.
(−i)
l!
(ξ
κ / κ) − i
l=0

?
Zκ,−
(ξκ ) = −(4.26)?
is the complex conjugate of the modified dispersion function Zκ? introduced by
Our Zκ,−Summers and Thorne (1991) because we use a different complex frequency definition in the?
Laplace transform. The derivation of Zk,−
from Eq. (4.24) to Eq. (4.26) is demonstrated

in Appendix. C.
73

4.3

Results of calculation and discussion

4.3.1

Incoherent scatter spectra with kappa distribution I: Ionlines

We have calculated IS spectra for a kappa distribution plasma for different values of κ andcompared them with those from a Maxwellian plasma with the same set of parameters,
ne , Ti , and Te . In our calculation, the following assumptions were made.
1. Electron distribution function along the magnetic field line is a kappa distribution
and that in the plane perpendicular to the line-of-sight is Maxwellian, while that of
the ions (in all directions) remains Maxwellian because of their heavy mass.
2. Effects of the magnetic field are not taken into account. This assumption should bejustified when calculating the IS spectra from a field-aligned measurement.
3. The probing radio wave frequency is 933 MHz (wave number k0 = 19.55 m−1 ) corresponding to the Tromsø UHF radar, one of the EISCAT radar systems. Assumingbackscattering (the transmitting and receiving site are the same), this means thatthe wave number k becomes 39.1m−1 (= 2k0 ).
4. Although we can take the effects of collisions into account, we set νi,e = 0 for
simplicity since we are interested in phenomena in the F region where the collisions
are too rare to deform an IS spectrum noticeably.
5. In the case of the F region, we assume that the dominant ion is O+ (mi = 16 a.m.u.).
6. The mean drift velocity of ions, vi , is zero.
Fig. 4.2 shows an example of our calculations of the IS spectra for various κ at the low
frequency range which corresponds to the ion acoustic fluctuations (ion lines). The spectra
show double-humped shapes similar to the case of Maxwellian plasma. For the same setof ne , Ti , and Te , the total scattered power becomes stronger, the spectral width remains
74

almost the same, the spectral peaks are slightly downshifted, and the depth of the valley
between two spectral peaks becomes shallower. With increasing κ, the spectrum converges
to a Maxwellian spectrum. Indeed, the IS spectra for a kappa distribution plasma arequite different from those for a Maxwellian plasma, although ion distribution function
which mainly contributes to the ion lines is assumed to remain Maxwellian.
The increase in the total scattered power can be understood in the following way. The
level of excitation of the ion line is, among other functions, a function of the local electron
density for velocities around the phase velocity of the corresponding the ion acoustic
waves. For the chosen set of parameters, vth,M ∼ 3 · 105 msec−1 , the ion acoustic waves atfrequencies around ±10 kHz correspond to v/vth,M ∼ ± 0.0053. Looking at Fig. 4.1, it iseasy to see that in this velocity range the kappa distribution function has higher values
than the Maxwellian distribution. Therefore, the excitation level of the ion line is higher.
To explain the differences of the IS spectra from the viewpoint of thermal fluctuations
in a plasma more quantitatively, we have numerically solved the following dispersion
equation,

·µ
¶
µ
¶¸
2
2ωpe
ω
ω
1
?
1−
Z
1+ 2
k vth,e2κ kvth,e κ,− kvth,e
µ
¶¸
2 ·
2ωpi
ω
ω
+ 2
Z−
1−
=0
k vth,ikvth,ikvth,i(4.27)

where ω is a complex frequency. For derivation of this equation, see Appendix B. For
the ion acoustic waves, we have obtained the results represented in Fig. 4.3. It can beseen that the frequencies of the ion acoustic waves for κ = 3 are downshifted compared to
the Maxwellian case. This corresponds to our result(Fig. 4.2) that the spectral peaks areslightly downshifted for kappa distribution. The damping rates of the ion acoustic waves
for kappa distribution are stronger than those for Maxwellian. This also corresponds tothe fact that the valley between the two spectral peaks becomes shallower.
If the spectra from a kappa distribution plasma are interpreted by assuming a
Maxwellian plasma, the derived parameters would be quite different. So it is worthknowing quantitatively how much the differences of interpreted plasma parameters, suchas ne , Ti , and Te , could be by using Maxwellian spectra in fitting. In the next section, we
75

11

1.8
1.6

Spectral Density Function

1.4
1.2

−3

Ti=2000K, Ne=2×10 m , Te=3000K

−5

x 10

kappa= 35
1040, Maxwellian

1
0.8
0.6
0.4
0.2
0
−40

−30

−0.02

−0.03

−20

−10
0
10
Frequency (kHz)

20

30

−0.01
0
0.01
Frequency normalized by vth(Maxwellian)
−0.02

−0.01
0
0.01
Frequency normalized by vth(κ=3)

40

0.02

0.02

0.03

Figure 4.2: IS spectral density functions of kappa distribution plasma for a different
spectral index, κ = 3, 5, 10, and 40, together with one from Maxwellian plasma (κ = ∞)are shown. For this case, ne = 2·1011 m−3 , Ti = 2000 K, Te = 3000 K, and mi = 16 a.m.u.
x-axes are indicated by corresponding velocities normalized by vth,M and vth,κ=3 as well
as by a frequency in kHz

76

Frequencies and Damping rates of ion acoustic waves(a)Ti=2000K, Ne=2⋅1011 m−3

4

10.5

x 10

κ=3
Maxwellian

10Real Frequency (rad/sec)

9.5

9

8.5

8

7.5

7

6.5

6
1000

2000

3000

4000

5000

6000

7000

8000

9000

7000

8000

9000

Electron Temperature (K)(b)

Ti=2000K, Ne=2⋅1011 m−3

4

6

x 10

κ=3
Maxwellian

5.5Damping rate (rad/sec)

5

4.5

4

3.5

3

2.5

2

1.5
1000

2000

3000

4000

5000

6000

Electron Temperature (K)Figure 4.3: a, b. Solutions of dispersion equation for the ion acoustic waves for κ = 3
and Maxwellian. a Frequency b Damping rate are plotted over electron temperature. Ion
temperature and electron density are fixed to 1000 K and ne = 2 · 1011 m−3 , respectively
77

will estimate how large the differences could become as a result of assuming a Maxwellian
distribution function if the actual distribution is a kappa distribution.

4.3.2

Difference of plasma parameters as a results of fitting with
Maxwellian distribution

In our analysis, IS spectra for a kappa distribution plasma with parameters (ne , Ti , Te ,and κ), are fitted by an IS spectra model for a Maxwellian plasma using the least square
method. ne , Ti , and Te derived by the standard analysis are determined. Fig. 4.4 is an
example of fitting a Maxwellian spectra to ne S(k, ω) calculated according to the procedurein the previous section. For this case, the ion temperature and electron density are almost
the same, but the electron temperature is significantly underestimated to only 1800 K
instead of 3000 K. The results for various κ with the same plasma parameters as inFig. 4.4 are shown in Fig. 4.5. Again ne and Ti change little, but the fitted Te is always
lower than assumed for the kappa distribution spectrum. The underestimation is up to
40% when κ is very small (more deviation from Maxwellian). The results with different
sets of ne , Ti , Te , and κ are similar(not shown here).
It is surprising that the estimated electron density remains almost the same eventhough the total scattered power is larger for the spectrum with the kappa distribution.
This can be understood by considering the relationship between the total scattered power
and the ionospheric parameters. For the Maxwellian distribution, the total scattered
power Ps is proportional to ne /(1 + Te /Ti ). Therefore, the estimated ne is proportionalto Ps (1 + Te /Ti ). By lowering the estimation of Te , the estimated ne remains almost thesame even though Ps for the kappa distribution spectrum is larger than that for the aMaxwellian for the same set of ne , Ti , and Te .
This underestimate of Te may also be surprising, but this result is rather reasonable.
Ion lines are sensitive to the derivative of the ion and electron distribution functions at thevelocity range corresponding to the phase velocity of the ion acoustic waves. For electrons,
this velocity range is the very narrow region of the center of their distribution function

78

6

4

x 10

Fitted (Maxwellian)
Model (κ−distribution)

Spectral intensity (arbitrary unit)

3.5

3

2.5

Ti=2000K
Te=3000K
11 −3
Ne=2⋅10 m↓
Ti=2002K
Te=1797K
11 −3
Ne=1.996⋅10 m

2

1.5

1

0.5

0
−40

−30

−20

−10

0

10

20

30

40

Frequency (kHz)

Figure 4.4: An example of fitting on an IS spectrum (ne S(k, ω)) with kappa distribution
by an IS spectrum with Maxwellian distribution. The model spectrum (plotted by circles)
assumes Ti = 2000 K, Te = 3000 K, ne = 2 · 1011 m−3 , and κ = 3. Solid line shows fitted
IS spectrum with Maxwellian distribution function with Ti = 2002 K, Te = 1797 K,
ne = 1.996 · 1011 m−379Difference of parameters (κ−distribution spectra are fitted by Maxwellian)
3000

Ti (K)

2500

2000

1500

1000

0

10

20

30

40

50

60

0

10

20

30

40

50

60

10

20

30

40

50

60

4000
3500

Te (K)

3000
2500
2000
1500
1000

11

x 10

Ne (m−3)

2.1

2

1.9

1.8

0

κ

Figure 4.5: The difference of parameters when kappa distribution IS spectra are interpreted by assuming Maxwellian distribution for Ti , Te , and ne are shown from the top to
the bottom. They are plotted as relative errors to real (kappa distribution) parameters
80

7

10

6

Spectral intensity (arbitrary unit)

10

5

10

4

10

3

κ=3
ne=2⋅1011 m−3
Ti=2000 K
T =3000 KeMaxwellian 11 −3
ne=1.996⋅10 m
Ti=2002 K
Te=1797 K

10

2

10

1

10
−100

−80

−60

−40

−20
0
20
Frequency (kHz)

40

60

80

100

Figure 4.6: The two spectra in Fig. 4.4 are shown for wider range of frequency. y-axis isshown in logalithmic scale.

81

as shown in Fig. 4.1. There, the width of a kappa distribution function is narrower thanthat of Maxwellian with the same temperature, and a similar width of a Maxwellian
distribution function for a lower temperature.
As seen from Fig. 4.4, those two ion line spectra with different Te for the kappa (κ = 3)
and the Maxwellian distribution function of electrons are so similar that we can hardly
distinguish one from the other. Mathematically speaking, those two ion line spectra haveactually slight differences in their shapes, and they could be distinguished. But as shownin Fig. 4.6, the difference is very small. Hence in practice, the differene is too small
to determine both κ and Te simultaneously to see how much electrons there are in thehigh energy tail of the distribution function. Furthermore,noises make it more difficult,practically impossible. So we cannot derive both κ and Te simultaneously only from ionlines.
The same analysis was performed for the EISCAT VHF radar (transmitted frequency
224 MHz) and the EISCAT Svalbard radar (500 MHz). The same results were obtained
for both the radars (not shown here) as the EISCAT UHF radar shown already.

4.3.3

Incoherent scatter spectra with kappa distribution II:
plasma lines

The kappa distribution contains many more particles in the high energy tail. That shouldmake a significant effect in the high frequency part of the IS spectrum. Bjørnå andTrulsen (1986) have calculated the effect of a photo electron flux whose energy spectrum
are modeled by power law on plasma lines. They find that the total cross section at the
EISCAT UHF radar frequency is increased but the position remains the same as it is in
the Maxwellian case. Summers and Thorne (1991) have shown that the Langmuir waves
of a kappa distribution plasma are more damped than those of Maxwellian plasma, andthat the frequency of the Langmuir waves are closer to ωpe . Hence, when kλD ¿ 1, it isexpected that plasma lines show stronger damping and have a smaller shifts in frequency.
We have calculated plasma lines for various Te cases with fixed κ, Ti and ne and82

compared them to Maxwellian lines as shown in Fig. 4.7. It can be seen that plasma lines<for κ distribution are more strongly damped than they are for Maxwellian for lower Te (∼>4000 K). For higher Te cases (∼ 6000 K), however, plasma lines for Maxwellian are morestrongly damped than they are for κ distribution. For Te = 10 000 K, a plasma line forMaxwellian becomes very flat while that for κ distribution is still relatively sharp. At thesehigh electron temperatures, kλD becomes comparable to unity. Thus according to Thorne
and Summers (1991) wave damping for Maxwellian exceeds that for κ distribution. Our
results for higher temperatures are still consistent with previous works. It should be notedthat our calculation has been done for ne = 2 · 1011 m−3 in such a way that plasma lines forboth Maxwellian and kappa distribution function have the same order of their intensities
to see clearly the effect of the electron temperature. At the velocity corresponding to thephase velocity of the Langmuir wave with k = 39.1m−1 , both distributions have similar
number of electrons. If we increase ne , ωpe is also increased and the phase velocity of theLangmuir waves becomes faster. Then the plasma lines for Maxwellian distribution will
be much weaker than the plasma lines for kappa distribution because of the big differenceof electron densities at the phase velocity of the Langmuir wave. Svenes et al. (1992)
have found that their electron distribution function measurements can be explained bya bi-Maxwellian distribution which is a sum of two Maxwellian distribution functions,one of which has a higher temperature than the other. Their model distribution function
is still Maxwellian, so it decreases much more rapidly with a velocity in the velocity
space than a kappa distribution. Therefore, we expect that the plasma lines for a kappa
distribution can be much stronger than those for a bi-Maxwellian distribution, when thephase velocities of the Langmuir waves are enough fast (↔ electron densities are higher) to
make significant difference of values of a kappa and a bi-Maxwellian distribution functions
at the Langmuir wave velocities. The plasma line measurements could distinguish thosetwo (kappa and bi-Maxwellian) distribution functions.
In Fig. 4.8, a calculation of plasma lines for a specific set of Te and ne and for various
values of κ is shown and compared with a Maxwellian calculation. The positions areslightly shifted toward the transmitter frequency. This becomes clearer for smaller κ.
83

−7

−7

x 10

1.4

4

1.2

SPDF (arbitrary unit)

3

2.5

2

1.5

x 10

9

κ=3
Maxwellian

Te=2000K

3.5

SPDF (arbitrary unit)

−8

x 10

κ=3
Maxwellian

κ=3
Maxwellian

8

Te=3000K

Te=4000K

7

1

SPDF (arbitrary unit)

4.5

0.8

0.6

0.4

1

6

5

4

3

2
0.2

0.5

0

1

3

3.5

4

4.5

5

5.5

6

6.5

0

7

3

3.5

4

Frequency (MHz)

5

5.5

6

6.5

0

7

3

3.5

4

Frequency (MHz)

−8

8

4.5

5.5

6

κ=3
Maxwellian
6Te=5000K

7

x 10

7

κ=3
Maxwellian
7

6.5

−8

x 10

7

5

Frequency (MHz)

−8

x 10

4.5

κ=3
Maxwellian
6

Te=6000K

Te=7000K

5

4

3

5

SPDF (arbitrary unit)

SPDF (arbitrary unit)

SPDF (arbitrary unit)

6

4

3

2

5

4

3

2

2

1

1

0

3

3.5

4

4.5

5

5.5

6

6.5

0

7

1

3

3.5

4

4.5

Frequency (MHz)
−8

7

5.5

6

6.5

0

7

3

3.5

4

5.5

κ=3
Maxwellian

5

5.5

6

5

7

x 10

κ=3
Maxwellian

κ=3
Maxwellian
4.5

Te=9000K

4.5

6.5

−8

x 10

5

Te=8000K

4.5

Frequency (MHz)

−8

x 10

6

5

Frequency (MHz)Te=10000K

4

3

2

4

SPDF (arbitrary unit)

SPDF (arbitrary unit)

SPDF (arbitrary unit)

4
5

3.5

3

2.5

2

3.5

3

2.5

2
1.5
1

1.5

1

0

3

3.5

4

4.5

5

5.5

Frequency (MHz)

6

6.5

7

0.5

3

3.5

4

4.5

5

5.5

Frequency (MHz)

6

6.5

7

1

3

3.5

4

4.5

5

5.5

6

6.5

7

Frequency (MHz)Figure 4.7: Plasma lines (upshifted) for various electron temperatures (2000 - 10 000 K)
for κ = 3 (solid line) and Maxwellian (dashed line) are shown. Electron density is fixed
to 2 · 1011 m−3 . x-axis ranges are all 3 - 7 MHz. But y-axis ranges are arbitrary for each
plot

84

−7

6

x 10

Upshifted plasma lines (Te=1000K, Ne=1011m−3)

Maxwellian(κ=∞)
5Spectral Density Function

40
4

10

3
κ =3

5

2

1

0
2.5

3

3.5

4

Frequency (MHz)Figure 4.8: Plasma lines (upshifted) are plotted for a different spectral index, κ =3, 5, 10, and 40, together with plasma line of Maxwellian plasma (κ = ∞)

85

4

14

x 10

Spectral intensity (arbitrary unit)

12

Maxwellian 11 −3
ne=1.996⋅10 m
T =2002Ki
T =1797 K

10

e

8

6

κ=3
n =2⋅1011 m−3e
Ti=2000 K
T =3000 K
e

4

2

0
3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

Frequency (MHz)Figure 4.9: Plasma lines (upshifted) are plotted for the sets of parameters appearedin Fig. 4.4, (ne , Ti , Te , κ) = (2 · 1011 m−3 , 2000 K, 3000 K, 3) (solid line) and (1.996 ·
1011 m−3 , 2002 K, 1797 K, ∞(M axwellian)) (dashed line)86This smaller Doppler frequency of the plasma line with smaller κ can be explainedin the following way. The Langmuir dispersion relation is more exactlyω ' ωpe (1 +2 1/2
3k 2 < v 2 > /ωpe
) , where < v 2 > is the mean square velocity of electrons which is thekB Tsecond order moment of the distribution function. Since < v 2 >= 2κ−3
for the kappa
κmdistribution, < v 2 > is smaller for the smaller κ, and consequently the frequency of theLangmuir wave becomes smaller. This means that the Doppler frequency of the plasmaline should be smaller for the smaller κ.
We have also calculated plasma lines for the sets of parameters appeared in Fig. 4.4,
both of which result in very similar ion line spectra. It is shown in Fig. 4.9. In contrast
to the ion lines, these two plasma lines look quite different. Simultaneous measurementsof ion lines and plasma lines could enable us to determine both κ and Te simultaneously,
although it is impossible from ion lines only.

4.4

Summary

We have derived incoherent scatter spectra for a plasma when the velocity distribution isa generalized Lorentzian or kappa distribution function.
We have calculated incoherent scatter spectra for a plasma that consists of electrons
with a kappa distribution function and ions with a Maxwellian, and compared the spectrawith those for a Maxwellian plasma. The ion lines have a double-humped shape similar
to those for a Maxwellian plasma. The electron temperatures are, however, underestimated by up to 40 % when interpreting them assuming Maxwellian distributions for bothions and electrons. Ion temperatures and electron densities are almost unchanged. Our
results suggest that electron temperatures derived with the assumption of a Maxwellian
distribution function might be underestimated when soft electron precipitation is inferredor when intense wave activity is observed. This can lead to the incorrect estimation offorces like ambipolar electric field through the electron temperature.
We have also calculated the plasma lines for the kappa distribution function for electrons with different κ and different temperatures. For ne = 2 · 1011 m−3 which is used87

in our calculation, plasma lines for κ distribution are more strongly damped than they<>

are for Maxwellian for lower Te (∼ 4000 K). For higher Te cases (∼ 6000 K), however,plasma lines for Maxwellian are more strongly damped than they are for κ distribution.
For Te = 10 000 K, a plasma line for Maxwellian becomes very flat while that for κ
distribution is still relatively sharp. For smaller κ, the spectral peak is shifted toward thetransmitter frequency.
Although practically it is still difficult to measure the shape of a plasma line as wehave calculated, simultaneous measurements of plasma lines with ion lines would provide
more precise information on the electron distribution function.

88

Chapter 5
Concluding Remarks
We have studied various effects of electrostatic waves and turbulences on the ionosphericplasma in the E and F regions both experimentally and theoretically.
In Chap. 3, electron heating by Farley-Buneman waves in the lower E region arestudied using the data obtained by the EISCAT radar system, the STARE system, andthe Killpisjärvi IRIS.
In Sec. 3.1, we have shown that the electron temperatures observed in the E region
are clearly isotropic over aspect angles from 0◦ to about 40◦ by using the tristatic measurements of the EISCAT CP-1 data even when the electron temperatures are strongly
enhanced. The very high heating rates that we derived from the observed Te suggest that
Farley-Buneman waves have an electric field component both perpendicular and parallelto B as proposed by St.-Maurice (1987). If Te enhancement is instead due to a direct
scattering of electrons by wave fields not involving electron-neutral collisions, our observation implies that these waves should be isotropic. We think that our result of isotropic
Te is remarkable in light of the known strong anisotropy of the Farley-Buneman waves,which is the cause of heating. We feel that further clarifications of the underlying heatingprocesses are needed in the future.
In Sec. 3.2, we have argued that because the total energy is conserved, the chain ofprocesses, namely, growth of electrostatic waves, acceleration and heating of the electrons, and their cooling in inelastic collisions with neutrals must be maintained by a89

quasi-stationary MI current system. We have parameterized the effects of the waves andturbulences by the anomalous (or effective) collision frequency νe? as a kind of drag effect.
We have found that the bulk electron velocity deviates by up to about 7◦ (νe? ∼ 30νen ) fromthe E × B drift direction resulting in the additional Pedersen current carried by electrons
during events with high |E|. We have examined the importance of the electron Peder<

sen currents, and found that at altitudes ∼ 110 km this current can become the main
Pedersen current. Consequently, the current-voltage relation in the Earth’s ionosphereis non-linear. Although this non-linearity appears to be rather innocent, consequences
could occur in the Magnetosphere-Ionosphere coupling. This should be examined morethoroughly in the future. Although the flow angle rotation of electrons could not be confirmed by observations by the STARE system due to the saturation of the irregularity
phase velocity at Cs , we have pointed out the possibility of testing our estimation by
optical instruments.
In Sec. 3.3, we have examined the nature of νe? using CNA measurements by the
Kilpisjärvi IRIS. We have shown that the cosmic noise absorption observed by the
Kilpisjärvi IRIS can be well explained by Eq. (3.13) (or Eq. (3.14)) in which only theclassical electron-neutral collision frequency is taken into account, even when νe? greatly
exceeds the classical frequencies associated with strong electric fields and strong electron
temperature enhancements. These results strongly suggest that the νe? has no effect inabsorption of radio waves in the ionosphere, although it can parameterize the dissipationof the electromagnetic energy to electrons. Therefore, this anomalous collision frequency
has a rather limited range of application. The waves should have a drag effect equivalent
to the collisions of νe? times per second on the electron drift, but νe? is not necessarilyequivalent to the classical collision frequency in the other phenomena. Our results agreewith both previous experimental results (Stauning, 1984; Stauning and Olesen, 1989) andthe theoretical prediction with macroscopic consideration in Appendix A.
In Chap. 4, we have studied incoherent scatter spectra which are scattering of electromagnetic waves by the two basic types of electrostatic waves, the ion acoustic waves andthe Langmuir waves. The motivation of our study was based on the anticipation that in90

the F region energy inputs such as soft electron precipitation could excite the Langmuir
turbulence and could modify the electron distribution function. We have derived incoherent scatter spectra for a plasma when the velocity distribution is a generalized Lorentzianor kappa distribution function.
We have calculated incoherent scatter spectra for a plasma that consists of electrons
with a kappa distribution function and ions with a Maxwellian, and compared the spectrawith those for a Maxwellian plasma. The ion lines have a double-humped shape similar to
those for a Maxwellian plasma. The electron temperatures are, however, underestimatedby up to 40 % when interpreting them assuming Maxwellian distributions for both ionsand electrons. Ion temperatures and electron densities are almost unchanged.
This kind of analysis could be used when soft electron precipitation is inferred or whenintense wave activity is observed. For example, electron temperature enhancements causedby soft electron precipitation which is also associated with ion outflow are often observedwith EISCAT radars. Our results suggest that in such cases electron temperatures derived
with the assumption of a Maxwellian distribution function might be underestimated. Thiscan lead to the incorrect estimation of forces such as an ambipolar electric field throughthe electron temperature. Correctly analyzed data would significantly contribute to theunderstanding of ionospheric phenomena such as ion upflows and enhanced ion acoustic
echoes.
Another application of our analysis is for a heating experiment in which the ionosphereis artificially modified and heated by powerful electromagnetic waves. In such an experiment, one would expect a strongly distorted electron distribution function. Although it is
not certain if the electron distribution function can be modeled by the kappa distribution,
we believe that it is a good example to test our analysis.
We have also calculated the plasma lines for the kappa distribution function for electrons. We have found a significant difference in the shape and a slight difference in thefrequency shift. Although practically it is still difficult to measure the shape of a plasmaline as we have calculated, simultaneous measurements of the shape of plasma lines with
ion lines would enable us to determine the spectral index κ, and would provide more91

precise information on the electron distribution function.

92

Appendix A
The Electromagnetic Wave
Propagation in the Turbulent Plasma
In this appendix, we examine the effect of the irregularities on an electromagnetic wave
propagation through the turbulent plasma. We shall follow a procedure which was usedby Hagfors (1984) to discuss the effect of heater-induced field aligned plasma density
irregularities in ionospheric RF modification experiments. The mathematical procedure
was originally developed by Karal and Keller (1964) and found application by Liu (1967)
for magnetoplasmas for the case of high frequencies and for isotropic density irregularities.
The theory is a macroscopic theory, and is not concerned with the detailed microscopic
nature of ion-neutral and electron-neutral reactions.

A.1

The wave propagation problem

Consider an electromagnetic wave of frequency ω and wave vector k. In a homogeneous plasma of density N the electric field of the wave is assumed to be of the formE(k, ω) exp(i(k · r − ωt)). If the medium is no longer homogeneous but contains density
irregularities the propagation of an electromagnetic wave is changed and the effect may be
represented by a modification of the complex refractive index governing a “mean” wave
with field amplitude E0 (k, ω) and with wave vector k. We shall search for the effect of93

the plasma irregularities in the properties of this ”mean” wave.
Following closely the procedure of Hagfors (1984), but modifying the procedure where
it is unclear, we first Fourier expand the electric field and the polarization associated withthe wave asP(r, t) µ 1 ¶4 Z Z Z Z
P(k, ω)=d(k)dω
ei(k·r−ωt)E(r, t)
E(k, ω)
2π

(A.1)

We shall assume that the relation between the polarization of the medium and the electricfield is local in space and governed by a susceptibility tensor χ̃(r, t) (we shall use tilde to
denote tensors in what follows):
Z
P(r, t) = ²0dτ χ̃(r, t − τ ; t)E(r, τ )(A.2)

We allow the medium to be time-varying in addition to being dispersive in time, as
one must expect of the electrojet plasma. Note that this relation is deceptively simple.
The medium we are considering here is not translationally invariant. Strictly speaking
one has the following relation between the polarization and the electric field:Z
Z t0P(r, t) = ²0d(r )dt0 χ̃(r; r0 , t; t0 )E(r0 , t0 )
V

(A.3)

−∞

The convolution relationship normally used between the field and the polarization
therefore also depends on the position and the time, and not only on position and timedifferences. We can think of (A.2) as dealing with quantities after averaging over the
microscopic spatial scale, but not over the scale of the turbulent irregularities.
The Fourier components of the polarization is given by:
P0 (k, ω) = ²0 χ̃0 (ω)E0 (k, ω) +µ ¶4 Z Z Z Z
1d(q)dω 0 < χ̃1 (k − q, ω − ω 0 ; ω)E1 (q, ω 0 ) >
²0
2π
(A.4)
We have added an extra argument in the susceptibility function to indicate that thisfunction not only depends on differences in frequencies, but also on the frequencies themselves.
94

We have split the susceptibility and the field quantities into a mean over the ensembleof realizations of the turbulent medium and a zero mean part:χ̃ = χ̃0 + χ̃1< χ̃ >= χ̃0E = E0 + E1 < E >= E0(A.5)

P = P0 + P1 < P >= P0
If we can express E1 (q, ω) in (A.4) in terms of the mean field E0 (k, ω) we can substituteinto the equation to obtain a dispersion equation for the mean field, and determine therefractive index and the absorption of the mean wave.
By means of Maxwell’s equations we relate the polarization perturbation to the electricfield perturbation:
1
P1 (k, ω) = Ñ (k, ω)E1 (k, ω)²0where the elements in the Ñ (k, ω) matrix are:
µ 2 2
¶
ck
c2
Nlm =
−
1
δ
−
kl km
lm
ω2
ω2

(A.6)

(A.7)

where c is the velocity of light in vacuum. Using the constitutive relation, on the other
hand, we have:1
P1 (k, ω) = χ̃0 (ω)E1 (k, ω) +²0
µ ¶4 Z Z Z Z
1d(q)dω 0 χ̃1 (k − q, ω − ω 0 ; ω)E0 (q, ω 0 )
2π

(A.8)

Equating (A.8) and (A.6) we obtain the desired relation between E0 (q, ω) and E1 (k, ω)
for substitution into (A.4):µ ¶41[Ñ (k, ω) − χ̃0 (ω)]−1
E1 (k, ω) =
2π
ZZZZ
×
d(q)dω 0 χ̃1 (k − q, ω − ω 0 ; ω)E0 (q, ω 0 )(A.9)

Because the statistical properties of the medium are assumed to be independent of
position the susceptibility deviations satisfy the following condition:< χ̃1 (k1 , ω1 )χ̃1 (k2 , ω2 ) >=(2π)4 < χ̃1 (k1 , ω1 )χ̃1 (−k1 , −ω1 ) > δ(k1 + k2 )δ(ω1 + ω2 )95

(A.10)

we end up with a dispersion equation for the mean wave, essentially the same as in Hagfors(1984), except that the time variation of the fluctuations are accounted for, possibly inthe form of a density wave:µ[Ñ (k, ω) − χ̃0 (ω)]E0 (k, ω) =1
2π

¶4 Z Z Z Z
d(q)dω 0< χ̃1 (k − q, ω − ω 0 )[Ñ (q, ω 0 ) − χ̃0 (ω 0 )]−1 χ̃1 (q − k, ω 0 − ω) > E0 (k, ω)(A.11)
where we have left out the possible explicit dependence of χ̃1 on ω.
The assumption about the susceptibility in (A.10) basically means that we consider
the susceptibility deviations as a superposition of statistically independent plane waves
corresponding to density waves of the form:
N (r, t) = N0 + n1 · eiq(r−t·Cs ) + n∗1 · e−iq(r−t·Cs )(A.12)

Here Cs is the phase velocity of the acoustic waves in the electrojet, N0 the mean electron
density and n1 the amplitude of the density perturbation.
Introducing
X0 =

N0 e2
m²0 ω 2

X1 =

n1 e2
m²0 ω 2X1∗ =n∗1 e2m²0 ω 2

(A.13)

we obtain for the susceptibilities:χ̃ = −M̃ X = −M̃ X0 − M̃ X1 · eiq(r−t·Cs ) − M̃ X1∗ · e−iq(r−t·Cs )with the matrix M̃ defined by:






1
+
iZ
−iY
0




1
M̃ =
iY
1 + iZ0
(1 + iZ)2 − Y 2 




 0
2
2
0
((1 + iZ) − Y )/(1 + iZ)(A.14)(A.15)

assuming that the coordinate system is chosen so that the geomagnetic field is directedalong the z-axis, and the x and the y axes are normal to z, with the x-axis in the magnetic

96

meridian plane. In addition to the definition of X above we have also introduced thestandard notations:Y= Ωe /ω

Z = νe /ωωp2 = N (r, t)e2 /m²0
ω = wave frequencyΩe = eB/m
νe = electron collision frequency
In this notation it is shown in Hagfors 1984 that the dispersion equation, for plane wave
density striations, takes the form:−1
Det = ||M ˜(ω) Ñ (k, ω) + X0 (ω − Csq)I˜ +no−1< |X−1 (ω − Cs q)|2 > M̃ (ω − Cs q)−1 Ñ (k − q; ω − Cs q) + X0 (ω − Cs q)I˜+no−1
< |X1 (ω + Cs q)|2 > M̃ (ω + Cs q)−1 Ñ (k + q; ω + Cs q) + X0 (ω + Cs q)I˜||

=0

(A.16)

With this dispersion equation we ought to be able to discuss the problem of propagation ofan electromagnetic wave through a turbulent plasma in a single scattering approximation.
This should be appropriate to the discussion of anomalous attenuation of the riometerwaves through the electrojet.

A.2

The riometer case.

Substitution into (A.16) leads to rather complicated expressions, as shown in Hagfors(1984) even for simplified cases. There simplification was achieved by assuming that thelength of the propagating wave was much larger than the scale of the irregularities, andeven so the dispersion equation was quite complex. For the cases we consider here we shallmake other approximations. In the riometer, it will be assumed that the wave frequency
97

used, often in the range 30-50 MHz, by far exceeds the gyro-frequency, i.e. that Y ¿ 1,
so that the basic wave propagation will be as if in a non-magnetized plasma, i.e.
χ̃ = −M̃ X = −X ˜I
1 + iZ(A.17)We shall furthermore assume, somewhat unrealistically, that the density waves travel in a
direction perpendicularly to the magnetic field. This direction is that of the electron drift.
In reality, of course the mean current must be directed horizontally, and the waves must
be inhomogeneous also traveling horizontally but with their wave normal perpendicular
to B. This means that qz , the component of q along the magnetic field is zero. For the
purpose of assessing their effect on the absorption we shall adopt this simpler geometry.
We assume that the electromagnetic wave travels vertically and that the angle between k
and the z-axis is θ. As we shall see presently even this simplification lead to intractable
expressions, and we shall end up setting θ equal to zero thus considering the magnetic
field to also be vertical.
Substitute into the dispersion equation (A.16), noting that ω − Cs q ≈ ω and thatM̃ −1 ≈ (1 + iZ)I˜ we obtain:
Det = ||(1 + iZ)Ñ (k, ω) + X0 (ω)I˜no−1+ < |X−1 (ω)|2 > (1 + iZ)Ñ (k − q; ω) + X0 (ω)I˜no−1+ < |X1 (ω)|2 > (1 + iZ)Ñ (k + q; ω) + X0 (ω)I˜|| = 0(A.18)

Simplification of the general expressions containing k ± q does not seem to be possible
here because the electromagnetic wave used and the density wave of the electrojet may
have comparable wavelengths.
In order to obtain a tractable dispersion equation we have to explicitly determine
the form of both Ñ − (X0 /(1 + iZ))I˜ and its inverse. With the abbreviations D =1 − X0 /(1 + iZ) and A = c2 /ω 2 we have:


2
2



A(ky + kz ) − D−Akx ky−Akx kz22Ñ − χ˜0 =−Aky kz−Akx kyA(kx + kz ) − D







2
2
−Akx kz−Aky kz
A(ky + kx ) − D98

(A.19)

with the inverse, which is also needed:


2



Ak
−
D
Ak
k
Ak
k
x y
x z 
x




1
−1
2
(Ñ − χ˜0 ) =Akx ky Aky − D Aky kzD(D − Ak 2 )  Ak k

2
Ak
k
Ak
−
D
x z
y z
z

(A.20)As a model which may show effects of the density waves, and which will simplify hemathematics even further, we shall consider the case where the electromagnetic wave
travels along the magnetic field, i.e. where θ = 0. In this case the only component of k is
kz = k, and of q is qx = q. The two matrices needed in the dispersion equation now are:


2



Ak − D
0∓Aqk 

22(A.21)Ñ − χ˜0 =0A(q + k ) − D
0






 ∓Aqk0
Aq 2 − D
and


2



Aq − D 0±Aqk 1−1(Ñ − χ˜0 ) =0−D0D(D − A(k 2 + q 2 ))  ±Aqk2
0 Ak − DWith these substitutions in (A.18) we obtain the following dispersion relation:µ¶
2 < |X1 |2 > (Aq 2 − D)2
Ak − D +×
D(D − A(k 2 + q 2 ))¶µ
2 < |X1 |2 >2×
Ak − D −(D − A(k 2 + q 2 ))µ¶
2 < |X1 |2 > (Ak 2 − D)−D +=0
D(D − A(k 2 + q 2 ))(A.22)(A.23)

Setting to zero the first term in this product gives the dispersion equation for the xcomponent of the field, the second term the y component and the third term the z, orthe longitudinal component. The x and the y components are the transverse componentsof the electromagnetic wave, which is of interest here. The two have slightly different
dispersion relations because of the different orientation of the electric field with respect tothe density wave. The approximate solutions for the two equations under the assumption99

made are, for Ex , the field aligned with the wave vector q of the density wave:Ãµ ¶2 !
2
<
|X
|
>
k
1
n2 ≈ n20 + 2
1−
2
n0
q

(A.24)

and for Ey , the component with the field normal to q the solution is:n2< |X1 |
≈ n20 − 2
n20

2

>

µ ¶2
k
q

(A.25)

Here we have introduced the refractive index of the mean wave, n, and the refractive
index of the wave without density perturbations, n0 determined by n20 = 1 − X0 /(1 + iZ).
Typically we have X0 in the range 0.01 to 0.04, and probably the density perturbation
less than 10%, we have < |X1 |2 > in the range 10−6 to 10−5 so that the correction for thepresence of the plasma turbulence must be very small. Only when the frequency is neara plasma resonance in the non-perturbed medium is there a chance for the turbulence toplay a decisive role. With the present assumptions that is not the case, and we conclude
that anomalous absorption is of negligible importance in riometer observations.

100

Appendix B
Plasma Dispersion Equation: Kinetic
Treatment
In this appendix, we demonstrate the derivation of an equation which describes propertiesof plasma wave dispersion by kinetic treatment. First, useful transforms for this purpose,
the Fourier transform and the Laplace transform are explained. Then starting from basic
kinetic equations, we finally derive the plasma dispersion equation (Eq. (4.27)).

B.1

Fourier Transforms

The Fourier transform is one of the most commonly used integral transforms and very
useful in solving differential equations. This transform of f (r) which is a function inthree-dimensional space and its inverse transform can be defined asZ +∞f (k) =
drf (r)e+ik·r
−∞
Z +∞
1
dkf (k)e−ik·rf (r) =
(2π)3 −∞
This transform can include a transform in time,Z +∞f (ω) =
dtf (t)e−iωt
−∞
Z +∞
1
f (t) =dωf (ω)e+iωt
2π −∞
101

(B.1)
(B.2)

(B.3)
(B.4)

In the Fourier transform, k and ω are real. For the kinetic treatment of a plasma, we are
required to use the Laplace transform in time which is described in the next section.

B.2

Laplace Transforms

The Laplace transform may be defined asZ ∞f (ω) =
dtf (t)e−(iω+γ)t

(B.5)

0

where γ > 0. The factor e−γt must be large enough for this integral to converge. The
inverse transform is
1f (t) =
2π

Z +∞If we set ω = ω 0 − iγ,
1f (t) =
2π0dω 0 f (ω 0 )e+(iω +γ)t

(B.6)

−∞

Z −iγ+∞dωf (ω)eiωt

(B.7)

−iγ−∞

Since we have defined γ > 0, the transform applies to the lower half of the ω plane, andthe contour of integration must be closed below Im(ω) < −γ. When the contour needs
to be on the upper half plane, the function f(ω) must be analytically continued.

B.3

Basic Equations

Here we derive the plasma dispersion equation with a kinetic treatment. We start from a
microscopic distribution function,
Fs (r, v, t) =NqX

δ(r − rj )δ(v − vj )

(B.8)

j=1

where δ(r) is the Dirac delta-function. This means that the j-th particle is exactly at a
position r with a velocity v at a time t. Nq is the total number of particles which satisfiesZ
Nq = drdvFs (r, v, t)(B.9)
and the number density in space is given byZ
ns (r, t) = dvFs (r, v, t)
102(B.10)

The distribution function satisfies the Klimontovic equation
∂Fs
∂Fs
∂Fs
+v·
+a·
=0
∂t
∂r
∂v

(B.11)In an unmagnetized plasma, it is reasonable to set the acceleration term
a=

qs
E
ms

(B.12)For small fluctuation, we set
Fs = Fs0 + Fs1 (r, v, t)(B.13)

E = 0 + E1

(B.14)

ns = ns0 + ns1 (r, v, t)(B.15)

andWith this expanded distribution function, we form a hierarchy of equations (For details,
see Sheffield, 1975, A.2.7). The hierarchy is terminated at the two-particle correlationlevel, and for a stationary and homogeneous plasma at a low temperature with no magneticfield, the system is described by the following equations.
∂Fs0
∂Fs0
+v·
= 0
∂t
∂r
µ
¶
∂Fs1
∂Fs1
∂Fs1
qs
∂Fs1
+v·
+
E1 ·
=
∂t
∂r
ms
∂v
∂t c
X qs Z∇ · E1 =F1s dv²0s(B.16)(B.17)(B.18)The right-hand-side of Eq. (B.17) represents collisions. To include the effects of collisions,
we adopt the BGK collision termµ¶∂Fs1
= −νs [Fs1 − ns1 fs0 ]
∂t c
where

(B.19)

Zns1 =dvFs1 (r, v, t)(B.20)

and fs0 is defined by
Fs0 = n0s fs0
103(B.21)

B.4

Plasma Dispersion Equation

Below we adopt the Fourier transform in space and the Laplace transform in time. By
this transform, for the distribution function and for the density, we have
Z +∞
Z +∞
ik·r
Fs1 (k, v, ω) =edk
Fs1 (r, v, t)e−i(ω−iγ)t dt
−∞

and

Z +∞
ne (k, ω − iγ) =

ik·r

e

(B.22)

−∞

Z +∞
dr

−∞

ne (r, t)e−i(ω−iγ)t dt (Eq. (4.3))−∞

From Poission’s equation B.18 becomes1ρq1 (r, t)
²0
Z +∞
1 1
ρq1 (k, t)eik·r dk=²0 (2π)3 −∞∇ · E1 =and this leads toi 1
E1 =²0 (2π)3(B.23)Z +∞k
ρq1 (k, t)eik·r 2 · dkk−∞

(B.24)

Laplace transform of Eq. (B.17) with the collision term (Eq. (B.19)) becomes−Fs1 (k, v, t = 0) + [iω + νs − ik · v] Fs1 (k, v, ω)
qs
∂Fs0
= −i
ρq1 (k, ω)k ·+ νs ns1 (k, ω)fs0 − ns1 (k, t = 0)2²0 ms k∂v

(B.25)

Solving this equation for ρ(k, ω) with the help of the relation
ρq1 (k, ω) =X

qs ns1 (k, ω)(B.26)g(k, v, t = 0)P
1 + s Cs (k, ω)(B.27)s

we obtainρq1 (k, ω) =where g(k, v, t = 0) is a function which is related to the initial state of a plasma, andCs (k, ω) is given by Eq. (4.5). (not the ion acoustic velocity) By the inverse Laplacetransform, we obtain1ρq1 (k, t) =2π

Z −iγ+∞−iγ−∞

g(k, v, t = 0) iωtPe dω
1 + s Cs (k, ω)

104(B.28)

where g(k, v, t = 0) is a function of Fs1 (k, v, t = 0) which represents an initial perturbation
at t = 0. Poles of the integrand of the right-hand-side of this equation which contributeto the value of the integral come from both the numerator and the denominator. The
contribution from the numerator, g(k, v, t = 0) is, however, rapidly approaching zero.
Hence, for non-trivial solutions, we must have for ions and electrons²(k, ω) ≡ 1 + Ci (k, ω) + Ce (k, ω) = 0(B.29)²(k, ω) is the plasma dielectric function which appeared in Chap. 4 as Eq. (4.8). Thisequation can describe both high frequency Langmuir waves and low frequency ion acoustic
waves that are treated with fluid theory in Sec. 1.2.1 Solving this equation for k and ω,
we can obtain a relationship like Fig. 4.3. As shown in Sec. 4.2, Ci and Ce can be written?
in terms of Z− (ξ) (4.17) or Zκ,−(ξκ ) (4.24). Z− is given in terms of the complex error?
function. Appendix. C will be devoted to the derivation of Zκ,−.

105

Appendix C? (ξ )
Derivation of Zκ,−κ?
(ξκ ) (Eq. (4.26))
In this appendix, we demonstrate the derivation of the function Zκ,−

which appears in the theory of Chap. 4 following Summers and Thorne (1991).

C.1

Gamma Function

In this derivation we use some properties of a Gamma function,
Γ(0) = 1
µ ¶
√
1
Γ
=
π
2
Γ(x + 1) = xΓ(x)22x−1
Γ(2x) = √ Γ(x)Γ(x + 1/2)
π

(C.1)
(C.2)
(C.3)
(C.4)

For an integer κ, Eqs. (C.1) and (C.3) lead toΓ(κ + 1) = κ!

106

(C.5)

C.2

Cauchy Residue Theorem?
In evaluating Zκ,−(ξκ ), integration with a complex variable is used. Cauchy residue the-orem is extremely useful in integration of complex functions. This is given byIX
f (z)dz = 2πi
Res(zi )

(C.6)

i
(n)

where Res(zi ) is a residue at the i-th pole of f (z), and the residue at zi

which is then-th order pole of f(n)Res(zi ) =

C.3

³
´n i
1
dn−1 h(n)f (z) z − zi(n)(n − 1)! dz n−1
z=zi

(C.7)

Derivation?
Zκ,−(ξκ ) is defined by Eq. (4.24) as an integral form ofZdsκκ−1/2 Γ(κ + 1) +∞?
Zκ,− (ξκ ) = √πΓ(κ − 1/2) −∞ (s − ξκ )(s2 + κ)κ+1
We take the integration along the contour on the complex plane as shown in Fig. C.1.
The integration along the contour C2 vanishes when R → ∞. The task is to evaluate an
integralZ +∞F (ξκ ) =−∞ds(s − ξκ )(s2 + κ)κ+1

(C.8)

?
Even though our Laplace transform is defined as Im[ξκ ] < 0, Zκ,−can be defined also on

the upper half plane and on the real axis by analytic continuation. Therefore, we take√the integration contour C1 + C2 so that poles ξκ and −i κ are always enclosed inside by
deforming C1 if necessary. For all signs of Im[ξκ ], from Cauchy residue theorem, we find(Eq. (C.6))£√ ¤F (ξκ ) = −2πi Res(ξκ ) + Res(−i κ)(C.9)

The minus sign on the right-hand-side of Eq.( C.9) is because the integration is in aclockwise direction (Fig. C.1). Using Eq. (C.7), the residues are given by√
Res(−i κ) = (ξκ2 + κ)−(κ+1)
107(C.10)

and
µ
¶¯
¯
1
1 dκ
¯
√
Res(ξκ ) =κ! dsκ (s − ξκ )(s − i κ)κ+1 ¯s=−i√κ
¸
κ ·
1 X
κ!
dl
1
dκ−l
1
√
=
κ! l=0 (κ − l)!l! dsl (s − i κ)κ+1 dsκ−l s − ξκ s=−i√κ=κX(κ + l)!
l=0

l!κ!

1
1
√
√ κ+l+1 (−1)κ
(2i κ)(ξκ + iκ)κ+1−l(C.11)√
Eqs. (C.10) and (C.11) are valid for all values of ξκ (except possibly for ξκ = ±i κ which?
will be considered later). Then Zκ,−(ξκ ) becomesκκ−1/2 Γ(κ + 1)√
F (ξκ )πΓ(κ − 1/2)¶−(κ+1)µ√2 πiκ!
ξκ2= −1+
Γ(κ − 1/2)κ3/2
κ#"µ¶κ+1 X¶lκ µiξκiξκ (κ + l)!
1 1
1+ √
1− √
× 1−
κ! 2κ+1
l!2l
κ
κ
l=0

?
(ξκ ) =
Zκ,−(C.12)√For ξκ = −i κ which is actually a removable poleµ¶1κκ−1/2 Γ(κ + 1)1dκ+1√κ) = √(−2πi)(κ + 1)! dsκ+1 (s − i κ)κ+1πΓ(κ − 1/2)
(κ − 1/2)(κ + 1/2)
(C.13)= −i(κ + 1)κ2/3√
Similarly for ξκ = +i κ
·
µ
¶
√
1
dκ+1
1
κκ−1/2 Γ(κ + 1)?
√Zκ,− (+i κ) = √(−2πi)(κ + 1)! dsκ+1 (s + i κ)κ+1πΓ(κ − 1/2)
µ
¶¸
1 dκ
1
√
+
κ! dsκ (s − i κ)κ
= 0(C.14)
√

?
Zκ,−
(−i

Now we set the terms in [ ] of Eq. (C.12) as f− (ξκ )µ¶κ+1 X
¶l
κ µ
iξκ
1 1
iξκ (κ + l)!
1+ √f− (ξκ ) = 1 −1− √κ! 2κ+1
l!2l
κ
κ
l=0

(C.15)and define another function g− byf− (ξκ ) ≡ 1 − g− (ξκ )108(C.16)

Furthermore, if we define f+ and g+ as
f+ (ξκ ) ≡ f− (−ξκ ) = 1 − g− (−ξκ ) ≡ 1 − g+ (ξκ )(C.17)

f− + f+ = 1(C.18)f− (ξκ ) = 1 − f+ = 1 − (1 − g+ ) = g+

(C.19)

it can be shown that

This leads to

From Eqs. (C.12) and (C.19), we recover Eq. (4.26)
?
Zκ,−
(ξκ )

µ
¶−(κ+1)√ξκ2
2 πiκ!
1+= −Γ(κ − 1/2)κ3/2
κ"#µ¶κ+1 X
¶l
κ µ
1 1
iξκ
iξκ (κ + l)!
×
1− √
1+ √
κ! 2κ+1
l!2l
κ
κ
l=0
κ − 1/2 κ!
2κ3/2 (2κ)!
µ¶κ+1−lκX(κ + l)!
2
κ−l
√
×
(−i)
l!
(ξ
κ / κ) − i
l=0

= −

109

(Eq. (4.26))(C.20)

Im( ξ κ) < 0

Im( ξκ) > 0

Im(s)

+i κ

+i κ

+R
C1

Re(s)

ξκ

Re(s)

-i κ

-i κ

C2

C2
Im(s)

Im( ξκ) = 0

Im( ξκ) = -i κ

+i κ
+R

C1

-R
Re(s)

-i κ

+R
C1

Re(s)

ξκ= -i κ
C2Im( ξ κ ) =+i κ

Im(s)

+i κ
ξκ

-R

ξ

-R

+R

-R
C1

Im(s)

C2

Im(s)

ξκ= +i κ

-R

+R
C1

Re(s)

-i κ
C2

Figure C.1: Contours of integration for signs and special values of ξκ are shown. The
contour C1 must be above a pole ξκ , because the Laplace transform is defined as Im[ξκ ]
< 0.

110

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