An examination of Vlasov theory of a plasma led to the hypothesis that a plasma may reside in a state of minimal change of the uniform distribution. This statement was made definite by determining that the change in the whole distribution can be minimized if the damping rate were maximized. A preliminary test of the theory shows that one would expect a plasma well fit by a kappa distribution to have a low kappa value

Pangia, Michael J.

English

NASA Technical Reports Server

1991
N92-15883
NASA/ASEE SUMMER FACULTY FELLOWSHIP PROGRAM
MARSHALL SPACE FLIGHT CENTER
THE UNIVERSITY OF ALABAMA
A THEORETICAL STUDY OF THE STEADY STATE OF A SPACE PLASMA
Prepared By:
Academic Rank:
Institution:
Michael J. Pangia, Ph.D.
Assistant Professor
LaGrange College
Department of Physics
NASA/MSFC:
Office:
Division:
Branch:
MSFC Colleagues:
Contract No.:
Space Science Laboratory
Solar-Terrestrial Physics
Magnetospheric Physics
Thomas E. Moore, Ph.D.
Craig J. Pollock, Ph.D.
Dennis L. Gallagher, Ph.D.
NGT-01-008-021
The University of AIabama
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https://ntrs.nasa.gov/search.jsp?R=19920006665 2020-03-17T14:20:07+00:00Z
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vAn important and outstanding problem in Space Plasma Physics is accounting for the velocity distributions
observed in the solar wind and the Earth's magnetosphere. This translates to seeking the steady-state
configuration of the plasma, the properties of which are strongly related to the wave activity in the plasma. In
many regions, particle distributions are well approximated by a single hump distribution known as the 1¢(Kappa)
distribution (see for example, Lui and Krimigis, 1981). Hasegawa et al. (1985) have derived the 1¢distribution as
a steady-state plasma distribution using a Fokker-Planck description of the plasma. However, from the study, it
is difficult to actually determine the value for 1( for comparing to fits of the data. This study restricts itself to
the situation of electrostatic wave activity in a plasma with zero magnetic field and where the (positive) ion
motion can be neglected. The end result of the study is a simple criterion for determining the shape of the dis-
tribution, or, more specifically, lc.
The fundamental equations for a plasma are the Vlasov equation for the dynamics of the electron distribu-
tion function f(t,x,v) and Gauss's law for the electrostatic electric field E(t,x). Written in Gaussian units, they
are, respectively,
_tf+v_xf- me_F_z3vf= 0 (1)
m
_E = 4roeno- 47reI dv f (2)
The independent variables are time (t), space (x) and particle velocity (v) in the x direction. A partial derivative
with respect to one of the variables is denoted by subscripting the partial derivative symbol 3. The electric
field vector, and the positive x and v axes all point in the same direction. The electron has mass m and charge
-e. The ions have charge e and are uniformly distributed over space with number density no. The normalization
for the electron distribution is that the number of electrons between (x,v) and (x+dx,v+dv) is f(t,x,v)dx dv.
Observed particle distributions represent a type of spatial average. For comparison with observations and
to get a simpler equation to analyze, the electron distribution is written as the sum of two distributions,
f(t,x,v) = f(t,v)+ f'(t,x,v). As seen from the arguments of functions, f(t,v) is the spatially uniform part of the dis-
tribution and f'(t,x,v) is the part which can vary with x. In practice, f(t,v) would be the observable part of the
distribution. It will be assumed that the plasma is infinite in extent. This means that one way to determine
f(t,v) is to take the limit of f(t,x,v) as Ixl _, and extract the uniform part of the distribution. Another way to
determine f(t,v), which will be applied here, is to define the following averaging process
Ldx
<A> = lim f :--, A(x) (3)
L_._% ZI.,
At infinity f" will at most oscillate with finite amplitude (i.e., wave amplitude is bounded), so that <f'> = 0.
Applying (3) to the total distribution function gives
L
Applying (3) to (2) gives
u
lim E(t,L)-- E(t,-L) = 4rLeno- 4_ I dv f(t,v) (5)
L_ 2L _..
It is unrealistic to have a large scale, uniform electric field, therefore, the condition <E> = 0 is imposed. This
amounts to using the boundary condition that E at most oscillates with a finite amplitude at infinite distances.
Equation (5) then says that the quasi-neutrality condition is enforced; namely, that
m
I dv f(t,v) = no (6)
In place of (2), one has
N
_xE = - 4r_e _ dv f' (7)
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Applying(3)to (1)gives
3tf(t,v)- e<E3_f'> = 0 (8)
m
As a check of consistency, it will now be verified that (6) is true for all time, along with determining
some other important properties of the first and second velocity moments of fit,v). Integrating (8) over all velo-
city values gives that the zeroth velocity moment of fit, v) is constant in time, which is consistent with (6). The
first velocity moment of (8) involves the zeroth moment of f', which is eliminated using (7). This gives that the
first velocity moment of fit, v) is also constant, and its value is taken to be zero by choice of reference frame.
Recalling that no large scale electric field is allowed, there can be no uniform current flowing in the plasma.
Therefore, the Maxwell-Ampere equation for electrostatic waves simplifies to
3rE = 4rue f dv v f" (9)
Making use of (9), the second moment of (8) is a statement of energy conservation, In summary, the first three
moments of (8) give
d[dvf(t,v)= O" d_dvvf(t,v)=O" "-7- -=m[dvv f(t,v)+ <E 2 =0 (10)
at _ ' at _ ' at [. z
Equation (8) is just one of the equations required to solve for fit,v). Since fit, v) depends on f" and E, both
(1) and (7) are also needed. Substituting (8) into (1) gives
The full set of equations for the present study are (7), (8) and (1 I). Since observations show that the _ distribu-
tion is a persistent feature in many regions of the space plasma environment, an analysis of the steady-state pro-
perties of the plasma is desired. The existing set of equations require that fit,v) and f' be specified at some time,
call it the "initial" time t=0 (E at each time is uniquely specified by (7) subject to the boundary condition that
the spatial dependence of E at most oscillates with a finite amplitude at infinity).
Consider the following initial condition
f'(0,x,v) = A g(v)sin (kx) and E(0,x) = Eocos (kx) (12)
where A, Eo and k are related constants. The averaging process encountered in evaluating the dynamics in (8)
and (11) can be performed using the trigonometric identities
1 1
2 sin (kx) cos (kx) = sin (2kx) and cos2(kx) = _-+_-cos (2kx) (13)
Application of (3) to (13) gives
<sin 0ax) cos (kx)> = 0 and <cos2(kx)> = 1 (14)
2
Substitution of (12) into (8) gives that the first partial time derivative of f(t,v) is initially zero. Evaluation of the
second partial derivative requires using (9) and (11) to determine the initial values of the first partial time
derivatives of E and f', respectively. The result is that second partial time derivative of fit,v) is not zero for any
f(0,v), which means that the uniform part of the dislribution will always change as a result of the initial condi-
tion. Therefore, at least for initial conditions of the same type as (12), there can be no steady-state of the
plasma, for after the plasma has evolved to a final state, the system can be caused to change again by externally
reinitiating a spatial perturbation. In respect to the Vlasov description of the plasma, external means of causing
spatial variation would be due to particle effects, such as discreteness and collisions. For later reference, both of
the terms in (11) enclosed within [ • • • } give zero contribution to the second derivative of f(t,v).
A second possibility to a steady-state equilibrium is that the plasma eventually resides in a state of
minimal change. In other words, as a result of the plasma response to perturbations, it eventually attains a state
for which the net change in fit,v) is a minimum, bu_!:not zero. This will be referred to as a quasi-steady state,
since the state could still be changed if perturbed. To proceed, the following asstirnpfio_ _are made:
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h
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v1) Waves are initiated by something external to the Vlasov theory of a plasma.
2) Waves are repeatedly being initiated.
3) The rate at which energy is delivered to the plasma through repeated wave initiations is much slower than
the rate at which the quasi-steady state is attained.
It should be emphasized that, in the present context, "external" means external to the Vlasov theory. Ultimately,
the source of energy comes from the plasma particles either from kinetic (.particle) or potential (electrical)
energy, but through a mechanism unaccounted for by the present theory. However, any more encompassing
theory would be correspondingly more difficult to handle. Assumptions (1) and (2) provide an operational
means of attaining quasi-steady state. Together they imply that it is insufficient to simply solve the initial-value
problem one time through, as the final state obtained could readily be change by repeating the perturbation.
Assumption (3) is made in light of energy conservation. As seen from the second moment equation in (10), the
energy brought into the system from a given perturbation goes into plasma thermal energy. Consistency of
Assumptions (1) and (2) together with the fact that plasma temperatures are finite, leads one to the assumption
that the energy from the perturbations is slowly supplied to the plasma.
Further work is necessary to specify a precise meaning of the statement that the net change in fit, v) must
be minimized in a quasi-steady state. Clearly, the statement should in some way apply to the whole distribution,
rather than some particular velocity or even some particular velocity moment. Consider the r distribution,
A,c (15)
f_ _ v____L_2)_
(1+ 2_:v_
where A,c is a normalization constant and v, is the thermal speed. The only parameter that should be affected by
minimizing the change in the whole distribution is _:, since it determines the shape of the function. The other
parameters, A,c and %, are subject to the constraints on the moments of the distribution, the number density and
thermal energy of the plasma, respectively.
An analytical expression for the change in f(t,v) is now sought subject to simplifying assumptions. Firstly,
as the system approaches quasi-steady state, it is reasonable to assume that f" is small compared to f(t,v). Con-
sistency with (7) implies that the electric field is also small. Consequently, the terms within { .-" } in (11),
which contain products of sma/l quantities, will be neglected. The resulting equation is the linearized Vlasov
equation with a time varying uniform distribution. Solving this equation for f'(t,x,v) with f'(0_t,v) as its initial
condition gives
l
f'(t,x,v) = _Idt'E(t',x-v(t-t'))8,f(t',v)+ f'(0,x-vt, v) (16)
(This expression can be verified by direct substitution into the linearized form of (11).) Substitution of (16) into
(8) gives
2 t
3tf(t,v) = e--_-_[dt'<E(t,x) E(t',x-v(t-t'))> + e---<E(t,x) f'(0_-vt,v)> (17)
m_t_ m
The right-hand side of this equation is quadratic in small quantities, which means that f(t,v) changes slowly.
These terms must be included to account for the leading order changes in f(t,v). Even though (17) is an approx-
imation, the main conclusions of the analysis thus far still apply. In particular, (10) still holds as well as the eon-
chsion that f(t,v) does change subject to the perturbation in (12), since the negligible terms in (11) do not con-
tribute to the initial value of the second partial time derivative of f(t,v).
Consistent with quasi-steady state being defined as a state of minimal change in fit, v) is the assumption
that the change in f(t,v) is small for this state. Therefore, in determining the change in f(t,v) at or near quasi-
steady state, f(t,v) can be treated as constant in (16) and on the right-hand side of (17), where the function f(0,v)
will be used in its place. Note that the starting time t = 0 here refers to the start time of one of the perturbations
that occurs when the system is close to quasi-steady state. Under this assumption, f" and E obey the standard
linearized Vlasov and Gauss equations. In particular, the temporal behavior of the electrostatic field can in many
situations be approximated by a product of a monotonic and an oscillatory function corresponding to damping
with rate _ (damping when 7> 0) and a oscillation of angular frequency COo,respectively (see for example,
Nicholson, 1983). The rates y and COoare determined from the root of the dielectric function, which itself
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dependson f(0,v). Therefore, the following explicit form for the electrostatic field will be used to calculate the
change in fit,v) near quasi-steady state
E(t,x) = Eoe"q'tcos (0_ot)cos (kx) (18)
The same initial condition of (12) is in use. Substitution of (12) and (18) into (17) enables one to calculate the
net change in fit, v) due to this perturbation. As an intermediate step, the time integral in (17) is evaluated to
give
t E 2 r e-'ft {--_cos (COot)+ (COo--Okv)sin (COo0}+ycos (kvt)--o (¢_0--okv)sin (kvt)"
dt'<E(t,x) E(t',x-v(t-t'))> = _ --_-e--rtcosI ,,: ±t z (COo _ + (COo__)_
The final expression for the net change in f(t,v) is
" e2 Eo2_vfi0,v) f-1
f(,o,v)-f(O,v) = Idt3tf(t,v)= Z Z
 +(COo kV) 1 +-o=+l tt = :1:14m2
v
t"
+ _-7--Ae Eo_v_g
COo--O kv
o=±1 4m [ "l,2+(COo-okv)2J
-_+_o--O kv "_-O I.t(coo--Okv)(coo-l.t kv) _
2(?+co ) + ? +(COo--- tk,,) J
(19)
From this expression, the damping rate y is the only factor that can be used to minimize the change in fit,v)
without singling out any one of the terms in (19). Therefore, it is concluded that the criterion for attaining
quasi-steady-state is that f((t,v) will be such that it maximizes the damping rate, or, in reference to (15), !¢
attains a value such that y is maximized. Physically, greater damping means that the waves dissipate faster and
exist for shorter times. That this leads to a smaller change in fit,v) is reasonable on the basis that it is the
waves that change fit,v) and waves which exist for shorter times with smaller amplitudes will produce less
overall effecL
It is necessary to discuss the special case of y=0. If such a mode exists, (19) shows that fit, v) changes
infinitely fast at all velocities. This is a clear breakdown of the approximations that lead to (19), chiefly the
assumption that f(t,v) changes slowly. In such a situation, one expects that the energy oscillates back and forth
between the particles and field without any loss. The criterion of minimal change in fit,v) in some sense still
applies, but now in reference to the time average, with there being no net change on the average. However, in
reference to the 1<distribution, a y= 0 mode exists only for infinite wavelength waves (k--g)); namely, a uniform
electric field that varies in time. For a uniform field, fit,v) would have to be changing as fast as the oscillation
to produce the required charge separation at infinity. Such a possibility was ruled out from the onset of this
study on the grounds that a uniform electric field infinite in extent is physically unlikely to exist.
As a preliminary evaluation of the theory, compare the damping rates y_ and y, for the _= 1 and K---)**
distributions, respectively, the latter being a Maxwellian distribution.
YI . y. 2 e_3/2 e= k _,i) , = (20)
COo COo k3_
The parameter )_Dis the Dcbye length for the plasma. The expression for y. applies only for k_ _ 1. Of these
two values, _c= l would be the quasi-steady state distribution, since it gives a larger damping rate_at least for
long wavelength perturbations. This is consistent with fits to observed particle distributions, which show that K
is not very large. The complete range of allowable _: must be analyzed for a better comparison with observa-
In summary, examination of the Vlasov theo_ of-a plasma lead to-the hypothesis---_at- a-phsma may
reside in a state of minimal change of the uniform_ di_stri_bution. Th__is_state_ment',yas madedefinite b_ydetermin-
ing that the change in the whole distribution can be minimized if the damping rate were maximized. A prelim-
inar,/test of the theory shows that one would expect a plasma well fit by a _: distribution to have a low g value.
..... References
Hasegawa A., K. Mima, and M. Duong-van, "Plasma d_stnbut_on function in a superthermal radmtmn field,
Phys. Rev. Letl 54,-2_8-i 1985. ........
Lui, A. T. Y., and S. M. Krimigis, "Energetic ion beam in the earth's magnetic lobe," Geophys. Res. Lett.
10, i3, 1983.
Nicholson, D. R., Introduction to Plasma Theory, John Wiley & Sons, New York, 1983.
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A theoretical study of the steady state of a space plasma

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