Chapter I Introduction * q: This thesis is concerned mainly with the instability studies on ion cyclotron (IC) waves in space and fusion plasmas described by a generalired distribution function, namely the anti loss cone distribution function. In the last 30 years the field of Plasma Physics in general and plasma waves, instabilities and radiations, in particular, have grown tremendously. Two reasons can be put forward for this continuous growth. The first has been the active program of research pursued towards the achievement of thermonuclear fusion. 1 his has led to an enhanced understanding of plasmas under normal laboratory and near-reactor conditions. The second has been a large number of experiments launched with satellites to provide in situ data on the properties and nature of plasmas in the Earth and other planetary magnetosphere and the solar wind. Both these areas of research have yielded abundant evidence showing new types and characteristics af waves and instabilities and their interaction with plasmas. A number of texts have been authored that deal at least in part with the topic of plasma waves. This list has, unfortunately, grown very long; we, however, -list a few of those which are now considered as "classics" (Akhiezeret a1.,1967; Akhiezer et a1.,1975; Cap 1976; Davidson 1972; Hasegawa 1975; lchimaru 1973; Kadomstev 1965; ~rall' and , $ Trivelpiece 1973; Stix 1962; ?idman and Krall 1971; Tsytovich 1977; etc.,etc) and others which can be considered as representative of the more modern period (Bittencourt 1986; Chen. F.F 1974; Chen L 1987; Gary 1993; Kenell et al. 1979; Nicholson 1983; Sazhin 1993; Stacey 1981; etc.). In addition to these in the continuing series "Reviews of Plasma Physics" (ed. Leontovich) and "Advances in Plasma Physicsu- (ed. Simon and Thompson). We begin with a brief review of waves and instabilities in general. I Waves and Instabilities Any periodic motion can be considered as a wave; so too a variation in the distribution of energy (both kinetic and potential) in a medium. So a:ny disturbance in the distribution of energies creates waves. J I
Transcript
Chapter I
Introduction * q:
This thesis is concerned mainly with the instability studies on ion cyclotron (IC) waves in
space and fusion plasmas described by a generalired distribution function, namely the anti
loss cone distribution function.
In the last 30 years the field of Plasma Physics in general and plasma waves,
instabilities and radiations, in particular, have grown tremendously. Two reasons can be
put forward for this continuous growth. The first has been the active program of research
pursued towards the achievement of thermonuclear fusion. 1 his has led to an enhanced
understanding of plasmas under normal laboratory and near-reactor conditions. The second
has been a large number of experiments launched with satellites to provide in situ data
on the properties and nature of plasmas in the Earth and other planetary magnetosphere
and the solar wind. Both these areas of research have yielded abundant evidence showing
new types and characteristics af waves and instabilities and their interaction with plasmas.
A number of texts have been authored that deal at least in part with the topic of
plasma waves. This list has, unfortunately, grown very long; we, however, -list a few of
those which are now considered as "classics" (Akhiezeret a1.,1967; Akhiezer et a1.,1975;
Cap 1976; Davidson 1972; Hasegawa 1975; lchimaru 1973; Kadomstev 1965; ~ r a l l ' and , $
Trivelpiece 1973; Stix 1962; ?idman and Krall 1971; Tsytovich 1977; etc.,etc) and others
which can be considered as representative of the more modern period (Bittencourt 1986;
Chen. F.F 1974; Chen L 1987; Gary 1993; Kenell et al. 1979; Nicholson 1983; Sazhin
1993; Stacey 1981; etc.). In addition to these in the continuing series "Reviews of Plasma
Physics" (ed. Leontovich) and "Advances in Plasma Physicsu- (ed. Simon and Thompson).
We begin with a brief review of waves and instabilities in general.
I Waves and Instabilities
Any periodic motion can be considered as a wave; so too a variation in the distribution of
energy (both kinetic and potential) in a medium. So a:ny disturbance in the distribution
of energies creates waves.
J
I
An instability is any disturbance or variation in the quasi-equilibrium state of the
system that reduces the free energy and brings the system to a true thermodynamic state.
Plasma instabilities are normal modes of a system that grow in space or time. Thus
t he w r d 'instability' in plasma implies a well-defined relationship between the wave vector
and frequency w ; thus plasma fluctuations relates the fret energies of the system.
By 'free energy' we meah particle kinetic energy which can be transferred to fluc-
tuating fields (that is, the inverse Landau damping). The two forms of distributional free
energy are:
1. A non-Maxwellian energy distribution with an excess of high energy particles.
2. A velocity distribution that is anisotropic in space; for example, with the temperature
parallel to the field lines TI not equal to the temperature perpendicular to the field lines
T-. These free energy sources are not always separable. For example, the loss - cone distri-
bution (occurring in magnetic mirror machines) is a combination of both non-Maxwellian
energy distribution with an excess of high - energy particles and a velocity distribution that
is anisotropic in space.
During an instability this free energy of the distribution is converted to copious
radiation and changes in the distribution lead to enhanced particle losses.
Not all the deviations from a Maxwellian distribution lead to an instability. Con-
sider an infinite, horn-eneoup plasma with an isotropic velocity distribution in a uniform
magnetic field. If the distribution function f (Ek) decreases monotonically with energy Ek,
that is if < 0 for all Ek, then the plasma is stable (Gardner; 1%3), even if the distri-
bution is non-Maxwellian (for a Maxwellian distribution -v < 0 in general). Therefore
a necessary requirement for instability in a plasma with an anisotropic velocity distribution I
is > 0, for some value of Ek 4
I
If the fluctuations are relatively weak the linear theory is appropriate to describe the
physics of the instability. The traditional development of the linear theory of instabilities 4
in collisionless plasmas follows a well-esta blished procedure; the linear Vlasov equation is
subjected to a Fourier/Laplace analysis in space/tirne, yielding fluctuating particle densities
and particle flux densities that are inserted into Maxwell's equations to yield a dispersion
equation. The solution of this dispersion equation relates w and k and thereby determines
the normal modes of the plasma.
The dispersion equation may be solved either as a boundary value problem (w is
given as real, and one solves for a complex component of k) or as an initial value problem
( k is given as real, and one solves for a complex w). The latter approach is subject to fewer
mathematical ambiguities and it is the approach more often followed in the literature; we
follow it exclusively.
Thus throughout this thesis the complex frequency w = w,+ir , where w, is the real
frequency and y is the growth or damping rate. We regard as a heavily damped oscillation
any solution of the linear dispersion equation that satisfies y < -Iwrl < 27r. We use L - Iwl-1
the term "waves" to describe those weakly damped solutions that satisfy ,, < y < 0
and describe as "instabilities" growing solutions with 7 > 0. All fluctuations will denote
both stable waves and instabilities. The phase speed of a fluctuation, the speed a t which
a point of constant phase of a single mode propagates through the plasma, is 2; the
relative motion between the observer and the medium bearing the wave , the damping or
the growth rates 7 calculated from homogeneous plasma theory are all independent of the
frame in which the calculations Bre performed.
If the distribution function of each plasma species is Maxwellian and no external
electric fields are present, the dispersion relation typically yields non-growing roots. In
order to yield one or mpre plasma instabilities, the dispersion relation must be based on 1
distribution functions involving free energy; that is, having a non- Maxwellian property
corresponding, for example, to an anisotropy or an inhomogenity.
2 Fluid and Kinetic Instabilities:
As the free energy (say the relative drift speed between the two components) is increased,
the imaginary part of the frequency, 7, of a damped mode becomes less negative until y = 0
is reached at some wave vector. We term this condition the threshold of the associated
instability because any further increase of the free energy leads, a t some wave vector, to .,
1 > 0, that is wave growth. And some where above the threshold, it is often true that at "?
least one component (j) is resonant with the instability; that is, lCjl 5 1 where 6 is the
argument of the dispersion function Z(Cj) used in the linear dispersion relation. In this
regime, wave growth depends on velocity-space details of the 33h component distribution
function and the instability is termed "kinetic".
If the free energy is further increased (for example if the relative drift speeds of the
components become much greater than the component thermal speeds), the maximum
growth rate also continues to increase; all plasma components often become non-resonant
(1CI >> I), and the dispersion equation can be reduced to a cold plasma form. In this
regime, the growing mode is usually termed a "fluid" instability.
Given a particular source of free energy, a plasma may be unstable to several dif-, 4
ferent modes. So the classification of any micro-instability requires identification of both
the free energy and the dispersion properties.
3 Longitudinal and Transverse Instabilities
When discussing plasma wave and instabilities, it is convenient to separate the fluctuating
electric fields into two types: Longitudinal (k x E(') = 0) and transverse ( k . ~ ( ' ) = 0). ,
The complete solution of the general dispersion relation will typically have contributions
from both types of fields: We define these as = kk.% and = q, respectively.
Plasma fluctuations that have only a longitudinal electric field may be derived
through the use of a kinetic equation and a single Maxwell equation, the Poisson's equation. 4
Such waves and instabilities h h e B(') = 0 and are termed "electrostatic". In contrast,
waves and instabilities with fluctuating electric and magnetic fields perpendicular to the
wave vector and with no longitudinal electric field can be described through the use of an
appropriate kinetic equation and Faraday's equation and Ahpere-Maxwell equation. These
fluctuations are called If electromagneticff.
Most fluctuations in space plasmas of non-zero 9 have both the transverse and
longitudinal components. If
0 < ld$)12 + ( B ( ~ ) I ? < 1E(,')l2
we will term the wave primarily "electrostatic" and if
the mode is primarily "electromagnetic". Finally, the term "electromagnetic" will encom-
pass f\uctuations with arbitrary ratios of longitudinal and transverse fluctuating fields.
In addition to the above classification, instabilities may also be classified on the
basis of their frequency, the growing mode and the driving mechanism. Thus we have
1 .Classification on the basis of frequency:
i) Micro instabilities
ii) Macro instabilities
2.Classification on the basis of the growing mode:
i) Convective instability
ii) Absolute instability
and
3.Classification on the basis of the driving mechanism:
i) Streaming instability
ii) Rayleigh-Taylor instabilities
iii)lJniversal instabilities and
iv) Kinetic instabilities
- 4 1
3.1 Micro vs Macro
The most general classification of growing modes in a plasma divides them into two broad
categories:
Macro instabilities occur a t relatively long wavelengths and micro instabilities at
shorter wavelengths.
Macro instabilities depend on the configuration-space properties of the plasma and
are well described by the fluid equations. /
Micro instabilities are driven by the departure from thermodynamic equilibrium of .:
the plasma velocity distribution and therefore must be described by the Vlasov equation
or other kinetic equations.
In a magnetised plasma with the gyro-radius r~ of a characteristic ion. macro
instabilities generally grow most rapidly at krL << 1 , whereas the micro instabilities I
generally have maximum growth rates at k7bL 2 1 . Distinction between the tho categories is not clear cut; a macro mode may have
appreciable growths a t short wavelengths and a micro mode may persist even a t small
wavenumbers. The distinction is further blurred by the fact that the wavenumber a t
maximum growth rate of some instabilities depend on the plasma parameters and may
slide from the micro- to the macro - regime as the parameters change. Nevertheless, the
macro vs micro distinction is a convenient one to begin 'any general discussion of unstable
plasma modes.
Further, high frequency (or short wavelength) instabilities tend to have a more
"fine-grained" nature than the low frequency instabilities. This difference is sometimes
expressed by designating the high frequency instabilitie as "microinstabilities" incontrast
with the low frequency "macro instabilities" but again there is nosharp division between the
two types. High frequency instabilities manifest themselves through enhanced emission of
electromagnetic radiation, increased particle losses and thange. in the particle distribution
function in the direction of thermodynamic equilibrium. 4
The main driving mechanism for the high frequency instabilities is a departure of
the ion or the electronfdistr+ution from a Maxwellian velocity distribution.
With a Maxwellian distribution, the waves are damped (Landau damping); but a
wave can grow in some particle distributions that depart from Maxwellian distributions.
Landau or cyclotron damping occurs because of the transfer of energy from waves
to a group of particles whose velocities satisfy some resonance condition. The rate of
damping depends on the distribution of particles in this special region of velocity space. If
the distribution is not Maxwellian there is the possibility tha t the energy transfer may be
modified and goes the other way, so that the wave grows a t the expense of the particle
kinetic energy. Thus microinstabilities depend on the microscopic details of the velocity
distribution.
Microinstabilities also result in turbulence and .the rapid diffusion of the plasma t
across the confining magnetic field. These instabilities involve fluctuating electric or mag-
netic fields inside the plasma; compared with magnetohydrodynamic instabilities they are
often slow growing and not associated with any large scale motion which makes them more
difficult to detect. They however produce an enhanced level of fluctuation. Any anomalous
particle l o u which may result is fairly readily determined from density measurements, but
an identification of the instability is usually much more complicated. The diffusion in ve-
locity spaceLresulting from microinstabilities is much slower than the speed of sound but it
can nevertheless be too fast to allow confinement for thermonuclear times. Experimentally
fast anomalous diffusion, sometimes called Bohm diffusion, has been obse~ed in many
toroida l experiments.
Other contributing factors far high frequency instabilities are :
1. Density gradient
2. Temperature gradient and
3. Magnetic field gradient.
Low Frequency instabilities
lnsta bilities with frequencies less than the ion cyclotron frequency have been investigated
extensively because of the possible link between the instabilities and the diffusion of plasma
acros the magnetic field. Experiments by Hoh and Lehnert :(1960) showed the fim un-
ambiguous correlation between diffusion and instabilities (fluctuation).
Quite apart freh the qelation between plasma lass and instabilities, low frequency
oscillations have their own interest as manifestation of characteristics of plasmas such as
~ a n d a u damping, anisotropy introduced by magnetic field, etc.
3.2 Absolute and Convective Instabilities
Although the criterion discussed above can determine if an instability can occur, it does
not determine if the instability is a convective instability or an absolute instability.
For a convective instability the wave propagates along the system SO that the
disturbance a t a fixed point inspace may amplify or decay with time. On the other hand
an absolute instability increases with time a t every point in space. A convective, instability b
can be likened to a traveling (evanescent) wave whik an absolute instability should be
compared with a standing (amplifying) wave.
Whenever we have a dispersion relation
which has either complex k for real w, or complex w for real k, then the wave may be
described as either stable or unstable depending on whether the wave grow w d-ys in
space or time. Unfortunately, however, the distinction is not always that simple, for there
exist classes where a wave may grow in space but decay in time at a f d point. We shall
use the nomenclature that a wave is unstable if for some real k t with w = w, + ir, w , has
positive real imaginary part, or .y > 0. W e shall call a wave with complex k = k, + ik,. for
real w, an amplifying wave if the wave grows in space in the direction of energy flow and
evanescent if the wave decays in space in the direction of energy flow. It is not sufficient
to find only the sign of the imaginary part of kt since growth in the direction of the phase
velocity may lead to a decay in the direction of the group velocity, as for a backward wave.
A further distinction is that if a finite source (in space and time) leads to growth in time
a t every point in space. we call this as absolute instability. However, whenever a growing
disturbance propagates in space (convects) such that, a t a fixed paint in space, the wave
eventually decays in time, we call this a convective instability. This distinction is not o b 1
\ server independent; cleaily, since an observer moving along with the growing disturbance
would see it growing everywhere, so it would appear to him to be an absolute instability.
Examples of these two types of instabilities are illustrated in Figure 1 by showing snapshots
at successive times for some hypothetical unstable system with a pulsed local source. '
3.3 Classification on the basis of Free energy
On the basis of the type of free energy available to drive the instabilities, they are classified
as follows:
1.Streaming instabilities
In this case, either a beam of energetic particles travels through the plasma. or a current is
driven through the plasma so that the different species have a drift relative t o one another;
the drift energy is used to excite waves and oscillation energy is gai,ned at the expense of
drift energy in the unperturbed state.
2.Rayleigh-Taylor instabilities
In this case the plasma has a density gradient or a sharp boundary, so that it is not uniform.
In addition an external, non-electromagnetic force is applied to the plasma. It is this force
which drives the instability.
3.Universal instabilities
Even when there is no obvious driving force such as an electric or a gravitational field, a
plasma is not in perfect thermodynamic equilibrium as long as it is confined. The plasma
pressure tends to make the plasma expand and the expansion energy can drive an insta-
bility. This type of free energy is always present in any finite plasma and the resulting
instabilities ate called universal instabilities.
4.Kinetic instabilities
In fluid theory the velocity distributions are assumed to be Maxwellian. If the distributions
are, in fact, not Maxwellian there is a deviation from thermodynamic equilibrium; insta-
bilities can be driven by this anisotropy of the velocity distribution. Such instabilities are
called kinetic instabilities.
4 Ion Cyclotron Waves
- ",
As mentioned above, this the& is concerned mainly with the instability studies of ion
cyclotron waves in anti-loss cone plasmas. We therefore consider some aspects of this
- wave - its occurrence, generation and uses will be dealt with briefly in what follaws.
4.1 Ion Cyclotron waves in fusion plasmas
Theoretical considerations indicate that, in moderately dense plasmas, there occurs nat-
ural oscillations at frequencies slightly lower than the ion cyclotron frequency in a given
magnetic field. These oscillations, called ion cyclotron waves, are the short wavelength,
low density limit of transverse hydromagnetic waves (also called magnetosonic waves). If
an external magnetic field is applied to the plasma, the !individual ions move in circles
around the lines of force, just as at low densities. But the phases and amplitudes of I
- the ion velocities now vary sinusoidally both in space and time, because the motions of
the plasma particles are coupled electromagnetically to one another. The oscillations (or
waves) thus represent a cooperative organised motion of the plasma as a whole. The ion
flow is divergent, as in the single particle picture, and this would be expected to produce
large space charge and accompanying radial electric fields. l ow ever, since, as a result of
the wave motion, ion flow pattern in the axial directioniis periodic, the electrons are able
to flow along the lines of force and neutralize the space charge.
Thus, if a t some instant the ions at a given point have an inward motion, then at a
distance of half a Debye wavelength they will be moving'outward with the same amplitude.
The ion density will thus vary in the axial direction and so electrons can flow along the
lines of force from a region where the ion density is low to one where it is high. There is
experiments l evidence, that the flow of this neutralizing'electron current produces heating
of a plasma (Glasstone and Loveberg, 1960).
A much more important heating mode, is expected to arise from the damping of the
ion cyclotron waves, called ion cyclotron resonance heating (ICRH); a more detailed discus-
sion of which is beyond the scope of this review. However, the essence of the mechanism
is as follows: When the wave is allowed to propagate into a region of slowly decreasing
field. the decreasing field will cause the wavelength to become shorter and shorter and the
frequency will approach the local ion cyclotron frequen~y given by
_ L\ t
where B is the magnetic field; 2, the charge and M, the mass of the ion. '
Any ion entering the plasma wave will be subjected to an oscillation a t this resonant
frequency and so will pick up energy from the field. As a result, the plasma wave amplitude
will damp out; there is thus a decrease in the wave energy accompanied by an increase in
the particle energy. This energy is perpendicular to the field lines, but it soon becomes
random in direction; that is, it k thbrnalised. It is to be noted that the heating by cyclotron
damping is not a direct process, but involves the intermediary action of the ion cyclotron
wave.
Thus ion cyclotron waves and i ts stability plays an important role in device plasmas.
We give below the different types of ion cyclotrqn waves.
Name of,wave (or instability) Necessary conditions Excited wave frequency
O bl iquely propagating /$<< 1 klI << k , k l p i Ei l ion Bernstein wave lib > &/h1 W >_ nu,, (e.s.) (electron beam) (n=1.2,. . .)
Electromagnetic /ji > ?ri,/?ni kll < k ~ , klp i 2 1 ion cyclotron wave Vb > >r /k / l W (= 7 ~ ~ , i
(e.m.) (electron beam) (n=1,2.. . .)
VD > (W a:)/kll kll k ~ , k ~ ~ i I 1 Ion cyclotron drift wave (electron beam w > n w , (e.s.) +diamagnetic drift) (n=1,2 ...)
Dory-Guest-Harris instability ~r [ J , ( ~ ) ] ~ R ~ V ~ > 0 W
(e.s. ) (ion beam) (nz1.2 ...)
Harris instability
(e.s. )
r,
Atfven ion cyclotron wave
( e .m. )
Drift-cone instability fo(V' = 0) = 0 W g rtWci,
(ems.) \ (loss cone distribution) (n=1,2,. .) In the above ear. denotes an electrostatic wave and e.m. an electromagnetic
wave-U, is the ion cyclotron or gyrofrequency.
4.3 Ion cyclotron waves in space plasmas
The first definitive measurement of these waves were made by Frederick and Russell
(1973), though the presence of ULF electromagnetic energy yas noticed even earlier (Scarf
et al 1972). These ion cyclotron waves with a frequency of 0.67 - 0.87 fd. (f, is the ion
gyrofrequency) were of small amplitude and propagated a t an angle to the magnetic field.
Later observations by Taylor and Lyons (1976) revealed the presence of large am-
plitude (0.4 to 6 gammas) ICW's in the frequency range o f 1-30 Hz. While some of these
wave events were associated with enhanced ion fluxes others were not. A similar search by
the satellite Hawkeye 1, however, turned in only five wave events which occurred during the
recovery phase of geomagnetic storms near the plasma pause (Kitner and Gu rnett. 1977).
Perraut et al (1978) and Perraut (1982) detected electromagnetic waves in the
frequency range 0.2- l lHz on board the GEOS 1 and 2 satellites. These waves, identified
as ICW's with their magnetic field in a plane perpendicular t o the ambient magnetic field.
can under certain circumstances propagate along a line of force and reach the ground.
Mauk and McPherron (1980) too, using the ATS 6 geostationary satellite, observed ICW's J
near the afternoon and dusk region of the earth's magnetosphere. These waves with a
normalised frequency ranging between 0.05 and 0.5 were generated within the equatorial
or minimum (B( region. r,
More recent observatidns of EMlC waves, observed in the Earth's equatorial mag-
netosphere around ORE, have been due to Young e t al 1981; Fraser. 1985; Erlandson et
at 1990 and Anderson et al 1992). These EMlC waves appear to be generated near the
magnetic equator by anisotropic (TL > Til) protons with energies of 10 to 50 keV.
5 Alfven waves
Alfven waves were first predicted t o exist by Hannes Alfven, in 1942 and were subsequently
observed in laboratory plasmas in the late 1950's by Allen ct al(t959), Jephcott(l959) and
Wilcox et al(1960). Most of the laboratory experiments with Alfven waves are now con-
cerned with the physics and engineering issues of heating plasmas to thermonuclear con-
ditions. The most spectacular results have been obtained in 16rge tokamak devices where
ICRF (ion cyclotron range of frequencies) heatinghas been used + generate temperatures
exceeding 5keV.
5.1 Physical properties of Alfven Waves
There are two distinct Alfven wave types, known as the "torsional" and tfcornpressional" . .
Alfven waves. The torsional wave is also known as the shear or dow Alfven wave or simply
as the "Alfven wave". At frequencies near the ion cyclotron f v y , . .. . :,: c it is usually called
the ion cyclotron wave. When the kinetic effects are irn&ant: it k&lld kinetic Alfven
wave. The to i ional Alfven wave does not propagate a t frequenBb &t theion cyclotron
frequency, at least in a cold plasma. In a warm plasma, the ion cyclotron wave popapt=
a t frequencies above the ion cyclotron frequency.
The compressional wave is often referred to as the fast wave. At frequencier well
above the ion cyclotron frequency, the compressional wave is more commonly known as a
helicon or whistler wave, while at frequencies near the electron cyclotron frequency it is
called the electron cyclotron wave.
5.2 Theoretical Approximat ions
In most laboratory plasmas, the Alfven wavelengths are comparable with the dimensions
of the plasma. Since the plasma parameters vary significantly over a wavelength, most
theoretical treatments- 6f w a y propagation remain approximate. The simplest possible
approximations are made, assuming a t first that the plasma is homogeneous and of infinite
extent. This approximation may seem to be totally unrealistic but it is a useful first
approximation, especially for the torsional Alfven wave. This wave is strongly guided by
the steady magnetic field and does not, therefore, reflect off the container vessel walls. In
this respect, the plasma boundary is effectively a t infinity.
The basic equations are simplified by ignoring plasma kinetic pressure. The kinetic
pressure can usually be neglected when the acoustic speed, "af1, is much less than the
Alfven speed, VA, where u = [%I$ and VA = with T as the plasma temperature, (ponimi)'12
the ion mass, B the ambient magnetic field and n,, the ion number density and v =
513. In large tokarnaks and in host other magnetically confined laboratory plasma, the
condition a < < VA is usually well satisfied. However, kinetic effects can become important,
even when o << V4, if the ion gyro radius is a significant fraction of the perpendicular
wavelength.
5.3 Alfven waves in space plasmas
They have been observed not only in the Sun but also in the ionosphere and the mag-
netosphere and even a t the Earth's surface. Correlation between satellite and ground
observations provides direct evidence that the kinetic Alfven wave is guided through the
magnetosphere by the Earth's magnetic field. There is, of course, no plasma a t the Earth's
surface, but Alfven wave fields can still be detected in the insulating boundary adjacent to
the ionosphere.
Kinetic effect of Alfven waves are important in the magnetosphere if the perpen-
dicular wavelength is smaller than about 1000 Km. The low frequency Alfven waves in the
magnetosphere, or in the Sun, are waves with a period of a few minutes. The solar wind
and the magnetosphere, probed with spacecrafts also make a very good laboratory for the
study of Alfven waves.
Cesarsky (1980) and Wentzel (1974) have reviewed the effect o f Alfven wave gen-
eration by cosmic rays. The observed isotropy of cosmic rays arriving a t the Earth indicate
that either that there is a very homogeneous distribution source within and outside our
Galaxy, or that a mechanism must exist for scattering cosmic rays before they reached the 1
Earth. The mechanism most favoured is scattering by Alfven waves which are generated
by the cosmic rays themselves. Particles moving through a plasma can generate Alfven
waves when the particle velocity exceeds the Alfven velocity. This process is essentially the
inverse of Landau damping, which is the process whereby waves give up energy to particles
moving a t about the same speed as the wave. Cosmic rays travelling along the magnetic
field line are subsequently scattered by the wave, resulting in an isotropic distribution at
the Earth. This effect may also be important in laboratory plasmas since alpha particles
created in fusion reactions may be scattered by the Alfven waves generated by the alpha f
particles themselves.
In recent years, it has been recognized that Alfwn waves may be responsible for
. heating the solar corona. The surface temperature of the sun is about 6000K. Above the
surface, the temperature rises slowly and then jumps through a narrow region to about 1
or 2 * 106K in the solar corona.
Most coronal heating models involve transport of wave energy from below the Sun's
surface and into the corona. Measurements of flux of acoustic wave energy made in the
period 1976-1980 by Mein and Meon (1980) have shown that it is much too small to heat
the corona. Consequently, a strong effort is being made to identify a mechanism to damp
Alfven waves in the corona. Several possibilities have been identified by Davila (1987),
Hollenbach and Thronson (1986), Priest (1982,1985) which include Alfven wave resonance
heating, phase mixing of the Alfven wave, etc.
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