Pangia postulated that the preferred state of a single component infinite plasma is the one that will change the least when perturbed. The plasma distribution of such a state would maximize the plasma wave damping rate, or minimizing gamma where negative gamma is the damping rate. The present study continues from the idea developed by Pangia to determine what distribution function maximizes the damping rate

Pangia, Michael J.

English

NASA Technical Reports Server

1992
NASMASEE SUMMER FACULTY FELLOWSHIP PROGRAM
MARSHALL SPACE FLIGHT CENTER
THE UNIVERSITY OF ALABAMA
THE KAPPA DISTRIBUTION AS A VARIATIONAL SOLUTION FOR AN
INFINITE PLASMA
Prepared By:
Academic Rank:
Institution and Department:
NASA/MSFC:
Office:
Division:
Branch:
MSFC Colleagues:
Michael J. Pangia, Ph.D.
Assistant Professor
Georgia Southwestern College
Department of Geology and Physics
Space Science Laboratory
Solar-Terrestrial Physics
Magnetospheric Physics
Dennis L. GaUagher, Ph.D.
Craig J. PoUoek, Ph.D.
Thomas E. Moore, Ph.D.
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https://ntrs.nasa.gov/search.jsp?R=19930008130 2020-03-17T09:10:39+00:00Z

The basis for this research is the 1991 Summer Faculty Fellowship work by the present author. In
review, Pangia [1992] postulated that the preferred state of a single component infinite plasma is the one that
will change the least when penufl_. The plasma distribution of such a state would maximize the plasma wave
damping rate, or, minimizing y where negative y is the damping rate. This explained the tendency for low
values of g:, when fitting plasma distributions to a K-distribution, which is
AK
(1 + v2/(2t:v 2) )r
[1]
where A,_ and v_ are the overall normalization factor and thermal speed parameter, respectively. A more
compelling question is why a _:-distribution works so well in fitting both electron and ion distributions.
Hasegawa et al. [1985] derived that the electron distribution subject to a superthermal radiation field is given by
[1], but that the ion distribution is Maxwellian for speeds less than the electron thermal speed. It is also
desirable to calculate K, which is not readily doable from the result ofHasegawa et al. [1985]. The present study
continues from the ideas developed by Pangia [1991] to determine what distribution function maximizes the
damping rate. A K-distribution will be the outcome for either electrons or ions, with !¢ tending toward 3/2.
It is necessary to extend the work of the author to include the general case of a multi-component
plasma. For the sake of completeness, this presentation will reproduce also the arguments of the previous work.
For a 1-dimensional, collisionless plasma with electrostatic field, E(t,x), the Vlasov and Maxwell equations are
3fs _fs qs E Ofs
-if+ V_x + _=o [2]
ms
oct
_ = E4rcqs j'dvf s [3]
S
_ = - E4r_ls Idv v fs [4]
S -oo
where fs(t,x,v) is the reduced distribution function for particles of type s. In use is the normalization that the
number of particles of type s that are between x and x+dx with velocities ranging from v to v+dv is fsdx dr.
Particles of type s have charge qs and mass m s.
In the absence of any external field, the steady state solution to the Vlasov-Maxwell equations is fs =
Fs(v) and E = 0, where Fs is any positive function normalized to the number density. One might then infer that,
in equilibrium, a plasma distribution is virtually likely to be anything. However, the response of the plasma to
a perturbation from equilibrium will greatly depend on the function Fs(v). Since plasmas evolve to equilibrium
under the influence of plasma waves, a particular steady state could be favored over all the others based on its
response to a perturbation. In an actual plasma, waves are usually present, indicating some sort of evolution of
the distribution function. Although equilibrium may never be fully obtained, the plasma should be at least
tending toward it. Changes in the background distribution function should diminish as equilibrium is
approached. Change can possibly be used as the criterion for identifying the preferred steady state of a plasma,
because a steady state that appreciably changes after having been perturbed is expected to evolve to a different
steady state which is more stationary. Therefore, it will be postulated that the plasma will tend toward the steady
state which changes the least when perturbed from equilibrium.
Since the steady state distribution in a plasma where external forces are absent is spatially uniform,
each distn"oution function will be expressed as a sum of a spatially uniform park fs(t,v), and the deviation from
uniformity, f's(t,x,v), with a similar expression for the electric field
fs(t,x,v) = fs(t,v) + f's(t,x,v)
E(t'x) = E(t) + E'(t,x)
[5]
[6]
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The unif_n electric field, E(t), may be zero, but, in general, it will exist if a zero wave number mode is part of
the perturbation or if it develops in time. A spatial average over all space will be used to formally define the
uniform part of any function Y(x); namely,
L
lim /--_ Y(x)<Y>x = L---_
[71
The subscript x on the angular bracket is used to identify it as a spatial average. In regard to [5] and [6], fs(t,v) =
<fs(t,x,v)>x and E(t) = <E(t,x)> x. Taking the spatial average, [2] through [4] become
_ fs(t,v) qs t3+ _s _- { E(t) fs(t,v)+<_E' f's_'x} =0
oo
0 = E4_qs _dv fs(t,v)
S
(ltd-'E(t)=" E 4r'xlsfdv v fs(t,v)
$ ,..oo
[8]
[9]
[10]
Subtracting [8] through [10] from [2] through [4], respectively, gives
Of's _f's qs
-_-+ v'_--+ _ss_ {E(t)f's + E' fs(t,v)+ E' f's" <E' f's>x } =0
oo
aE' fdv
_x'x = E4rU:ls f's
S .,oo
oo
_E' _dv
_t - "E 4r_ls v f's
$ -oo
[11]
[12]
[13]
Equations [8] through [13] are basically the starting equations for quasi-linear theory [see for example, Swanson,
1989], just extended to include a uniform electric field. Derivable from [8] through [13] are the conservation
laws for the system, which are written
(Itd-"_dv fs(t,v)= 0
S _
d v 2 fs(t,v) + E2(t)
ms v
\s "_
[14]
[15]
+ 4_ < E'2>x) = 0 [16]
Due to the quadratic non-linearity present in the Vlasov equation, an additional simplifying assumption
will be made in order to derive a result. Having perturbed a system from equilibrium, it will be assumed that,
eventually in its evolution, the predominant wave modes that remain are k--0 and k nearly zero, where k is the
wave number, all other modes having been depleted. This assumption is made even if an instability exists for
some particular range of modes. The rationale is that non-linear effects will channel wave energy to other modes,
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mostof whichdampaway.Tounderstandthebasiceffectof thequadraticnon-linearity,considerthe spatial
dependence of any mode k at some time L It will have the basic form eikx + e "ikx. The product of two modes,
kl and k2, will result in modes kl + k2, and k 1 - k2. In particular, if k 1 = k2, there will be mode conversion to
2kl and 0, and ff k1 and k2 are nearly equal, the nearly zero wave mode will resulL If kl is unstable, the process
will convert it to stable modes. Modes that find their way to the instability will have been damped beforehand,
amplified during the instability, and, finally, converted back to stable modes. Therefore, all wave activity will
experience some damping, either directly or indirectly, during some time in the system's evolution except for the
k--0 mode. Due to the initial perturbation having a small amount of total wave energy distributed over all the
modes, it is being assumed that even an unstable mode would be depleted of its field intensity through the
eventual conversion of all modes to k=0. There comes a stage in the plasma evolution when only the undamped
mode, k=0, and the least damped mode, k nearly zero, have survived.
When the plasma reaches the stage of predominantly k=0 and nearly zero wave modes, further
amplification of the k=0 mode will be negligible, because only the nearly zero mode, which is decaying, is
available to convert to k=0. At this point, E(t), which is the k=0 mode, is well-approximated as an oscillation at
the plasma frequency. From [10], fs(t,v) will oscillate with opposite phase to E(t). Therefore, on a time average,
the effects of the k=0 mode on changing the distribution function will be zero. Defining a time average for any
function Y(t) by
t+T
<Y>t = f-_ Y(t') [17]
where T is the plasma period, the time average of [8] is
0_<_fs(t,v)_,t = qs 0 ,
"ms 0-v <_E f's_>x,t
[18]
where the combined subscript x,t on the angular bracket indicates a double average over space and time.
The nearly zero mode, which is given by E'(t,x), will be in a regime where its evolution and effects are
describable by quasi-linear theory. In the quasi-linear approximation, products of E' and f's are retained in [8],
but neglected in [11]. In search of the distribution that changes the least, a restriction will be made to
distributions that change only slightly. Such a subclass of distributions can be insured to exist by making the
perturbation small enough. Specifically, based on [16], the initial wave field must satisfy the condition
E2(t) + _E'2_>x _<4_r_ms fdv v2 fs(t,v) [19]
at t=0. Then E(t) will be small, and the term E(t) f's will be negligible in [ 11 ], reducing it to
_f's Of's (:Is E' dFs
0=='_-+ v-_- + ms _ = 0
[20]
where fs(t,v) was replaced by Fs(V) since it only slightly varies. Equations [12] and [20] are the common
linearized equations for electrostatic waves. The solution is that, for a stable mode of wave number k, E'
oscillates with frequency 0) and exponential decays with damping rate ITI (negative y corresponds to damping)
given by [Nicholson, 1983]
e(o_+i3',k) = I- _ 4nq2/(k2ms) Idv dFs/dvC v - _-0 [21]
s k
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where the integration is in the complex v-plane along the Landau contour, and ¢ is the dielectric function. With
E' exponentially decaying at a damping of _, [18] says that, on the average, the change in fs(t,v) decays at least
as fast. Consequently, the distribution that changes the least will be the one that damps the nearly zero wave
mode the most. From [21], one seeks the Fs that maximizes ITl,or minimizes ?. Mathematically, the problem
at hand is under the classification of Calculus of Variations, where Fs is varied slightly and a minimum in _/is
sought.
The functions about which one varies have to be physical. To insure this, constraints must be
imposed. The conservation laws, [14] through [16], require that
fdvFs=ns; Ems[dvvFs--nP; Emsfdvv2Fs=nT
-oo S _ S ..oo
[22]
where, ns, n, P, and T are, respectively, the number density of particle type s, the total number density of all
particle types, the total plasma momentum, and the total plasma temperature,all of which are constants. The
existence of these velocity moments imply that the high speed dependence of Fs must be less than 1/Ivl3. This
puts a limit on damping rate, because, for small k, ? is proportional to the dFs/dv evaluated at high velocity
co/k [Nicholson, 1983]. Therefore, the damping is maximized for the function that has the steepest descending
slope, which is when the high speed dependence of Fs tends to I/Ivt3. In regard to [1], _: tending to 3/2 would
maximize damping.
To find the function for all velocity, additional constraints must be imposed to keep Fs physical. One
condition is that Fs must be positive. Another condition arises from the fact that each distribution function Fs
will consist of a definite number of plasma components, As, which is determined by the origin of the plasma.
Therefore, it is a fixed property of the system. Defining Fs,c as the distribution function for component c of
particle type s, the total distribution for particle type s is
A$
Fs = E Fs,c [23]
The number of components, As, is determined by counting the humps in Is. By constraining Fs,c to be
positive with only one hump, Fs is insured to be physical. This is done by maximizing the following integral
oo
Idv (Fs,c)r
,=oo
[24]
where the exponent r has to be determined, where r_l, so that maximizing [24] is independent from the first
equation in [22]. The problem is now well defined, whereby ? is minimized in [21] subject to the constraints in
[22] while maximizing [24]. A solution for arbitrary and small variations in Fs,c only exists if k approaches
zero. The answer is that each component is given by a r-disuibution with appropriate flow speed, where _:=l/(1-
r) with 3/'2 < _: < 0o, and _ tending to 3/2 gives the absolute minimum in T.
References
Hasegawa, A., K. Mima, and M. Duong-van, "Plasma distribution function in a superthermal radiation field,"
Phys. Rev. Lett. 54, 2608, 1985.
Nicholson, D. R., Introduction to Plasma Theory, John Wiley & Sons, New York, 1983.
Pangia, M. J., "A theoretical study of the steady state of a space plasma, "1991 NASA/ASEE Summer Faculty
Fellowship Program, N92-15850, 1991.
Swanson, D. G., Plasma Waves, Academic Press Inc., London, 1989.
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The kappa distribution as a variational solution for an infinite plasma

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