-
Hindawi Publishing CorporationPhysics Research
InternationalVolume 2011, Article ID 103893, 10
pagesdoi:10.1155/2011/103893
Research Article
Nonlinear Gravitoelectrostatic SheathFluctuation in Solar
Plasma
P. K. Karmakar
Department of Physics, Tezpur University, Napaam, Assam, Tezpur
784028, India
Correspondence should be addressed to P. K. Karmakar,
[email protected]
Received 4 November 2010; Revised 18 September 2011; Accepted 19
September 2011
Academic Editor: Eric G. Blackman
Copyright © 2011 P. K. Karmakar. This is an open access article
distributed under the Creative Commons Attribution License,which
permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
The nonlinear normal mode dynamics is likely to be modified due
to nonlinear, dissipative, and dispersive mechanisms in solarplasma
system. Here we apply a plasma-based gravitoelectrostatic sheath
(GES) model for the steady-state description of thenonlinear normal
mode behavior of the gravitoacoustic wave in field-free
quasineutral solar plasma. The plasma-boundary wallinteraction
process is considered in global hydrodynamical homogeneous
equilibrium under spherical geometry approximationidealistically.
Accordingly, a unique form of KdV-Burger (KdV-B) equation in the
lowest-order perturbed GES potential ismethodologically obtained by
standard perturbation technique. This equation is both analytically
and numerically found toyield the GES nonlinear eigenmodes in the
form of shock-like structures. The shock amplitudes are determined
(∼0.01 V) atthe solar surface and beyond at 1 AU as well.
Analytical and numerical calculations are in good agreement. The
obtained results arecompared with those of others. Possible
results, discussions, and main conclusions relevant to
astrophysical context are presented.
1. Introduction
The Sun, like stars and ambient atmospheres, has explo-ratively
been an interesting area of study for differentauthors by applying
different physical model approaches andobservational techniques for
years [1–13]. Some of the basicelectromagnetic properties of such
stars and stellar atmo-spheres have been reported in hydrostatic
equilibrium of theconstituent ionized gas [1, 2]. The steady
supersonic radialoutflow of the ionized gas from the Sun (or star),
called solarwind (or stellar wind), has been found to support
variousnonlinear eigenmodes [4]. Such nonlinear eigenmodes
areusually solitons, shocks, and so forth [4, 9–17] found to
existalmost everywhere in space including dust-contaminatedspace
plasma [14, 15]. Nonlinear stability analyses of the Sunand its
atmosphere have, however, been boldly carried outby many authors
applying magnetohydrodynamic (MHD)equilibrium configurations [9–13]
by means of standardmultiple scaling techniques. Nevertheless the
effects ofspace charge, plasma-boundary wall interaction, and
sheathformation mechanism have hardly been addressed in such
model stability analyses on the Sun and its atmospherereported
so far.
We are here going to propose a nonlinear stabilityanalysis on
the Sun on the basis of the plasma-based gravito-electrostatic
sheath (GES) model [3]. According to this GESmodel analysis, the
solar plasma system divides into twoparts: the Sun which is the
subsonic solar interior plasma(SIP) on bounded scale and the
supersonic or hypersonicsolar wind plasma (SWP) on unbounded scale.
The solarsurface boundary (SSB) couples the SIP (Sun) with theSWP
through plasma-boundary wall interaction processesin a
self-gravitating equilibrium configuration of hydro-dynamic type.
The SSB behaves like a spherical electricalgrid negatively biased
with the equilibrium GES potentialθΘ(ξΘ) ∼ −1 (= −1.00 kV) through
the process of the selfgravito-electrostatic interaction.
Henceforth, for coupledstructural information, the terms “GES
fluctuation”, “SIPfluctuation”, “GES perturbation,” and “SIP
perturbation”will be synonymously used to describe the “nonlinear
GESstability on the bounded SIP scale” in this investigation
ofsolar plasma fluctuation dynamics.
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2 Physics Research International
The main motivation of this paper is to examine whetherthe solar
plasma system, as a natural plasma laboratory, cansupport any
nonlinear characteristic eigenmode through theGES model with
plasma-boundary interaction taken intoaccount. Spacecraft probes
and Earth-orbiting satellites havealso technically detected many
wide-scale nonlinear modefeatures [4, 12, 13] like nonpropagating
pressure-balancestructures, collisionless shocks, turbulence-driven
instability,soliton, and so forth. These have particularly been
appliedto probe plasma kinetic effects in the form of collective
waveactivities in some important parameter regimes [4]
exper-imentally inaccessible to laboratories due to the
complexnature of the dynamics of the solar wind particles. Thus
atheoretical model analysis is highly needed in support of
thedescription of these experimental observations.
A distinct set of nonautonomous self-consistently cou-pled
nonlinear dynamical eigenvalue equations in thedefined
astrophysical scales of space and time configurationis accordingly
developed. In view of that, a unique form ofKdV-Burger (KdV-B)
equation [9–11, 14, 15] is methodolog-ically derived on the SIP
scale in terms of the lowest orderGES potential fluctuation. It is
then studied analytically aswell as numerically as an initial value
problem. The gravito-electrostatic features are asymptotically
examined even onthe SWP to explore some new observations on the
nonlineareigenmodes of the GES. Apart from the “Introduction”
partdescribed in Section 1 above, this paper is structurally
orga-nized in a usual simple format as follows. Section 2, as
usual,contains physical model of the solar plasma system
underinvestigation. Section 3 contains mathematical formulationand
required derived analytical equations and expressions.Section 4
shows the obtained results and discussions in threesubsections.
Sections 4.1, 4.2, and 4.3 give the analytical,numerical, and
comparative results, respectively. Lastly andmost importantly,
Section 5 depicts the main conclusions ofscientific interest and
astrophysical applicability.
2. Solar Plasma Model
A very simplified ideal solar plasma fluid model is adoptedto
study the GES model stability under a global hydro-dynamic type of
homogeneous equilibrium configuration.Gravitationally bounded
quasineutral field-free plasma bya spherically symmetric surface
boundary of nonrigid andnonphysical nature is considered. An
estimated typical value∼10−20 of the ratio of the solar plasma
Debye length andJeans length of the total solar mass justifies the
quasineutralbehavior of the solar plasma on both the bounded
andunbounded scales. A bulk nonisothermal uniform flowof solar
plasma is assumed to preexist. For minimalism,we consider spherical
symmetry of the self-gravitationallyconfined SIP mass distribution,
because this helps to reducethe three-dimensional problem of
describing the GES into asimplified one-dimensional problem in the
radial directionsince curvature effects are ignorable for small
scale size ofthe fluctuations. Thus only a single radial degree of
freedomis sufficient for describing the three-dimensional SIP
and,hence, the SWP in radial symmetry approximation. This is
toelucidate that our plasma-based theory of the GES stability
is quite simplified in the sense that it does not includeany
complicacy like the magnetic forces, nonlinear thermalforces and
the role of interplanetary medium or any otherdifficulties like
collisional, viscous processes, and so forth.
Applying the spherical capacitor charging model [3], thecoulomb
charge on the SSB comes out to be QSSB ∼ 120 C.More on the basic
electromagnetic properties of the Sun andits atmosphere could be
understood from the electrical stellarmodels [1, 2]. Let us
approximately take mean rotationalfrequency of the SSB about the
centre of the SIP system tobe fSSB ∼ 1.59 × 10−12 Hz. Applying the
electrical model[2] of the Sun, the mean value of the strength of
the solarmagnetic field at the SSB in our model analysis is
estimatedas 〈|BSSB|〉 = 4π2 QSSB fSSB ∼ 7.53 × 10−11 T, which
isnegligibly small for producing any significant effects onthe
dynamics of the SIP particles. Thus the effects of themagnetic
field are not realized by the solar plasma particlesdue to the weak
Lorentz force, which is now estimated to beFSIPL = e(vSIP ×
〈|BSSB|〉) ≈ 3.61 × 10−33 N correspondingto an average subsonic flow
speed vSIP ∼ 3.00 cm s−1, andhence, neglected. It equally justifies
the convective andcirculation dynamics being neglected in our SIP
model.Therefore our unmagnetized plasma approximation is
welljustified in our model configuration on the bounded SIPscale.
The same, however, may not apply for the unboundedSWP scale
description. This is because the Lorentz force forthe SWP comes out
to be FSWPL = e(vSWP × 〈|BSWP|〉) ≈1.64 × 10−2 N to a mean
supersonic flow speed vSWP ∼340.00 km s−1, thereby showing that
FSIPL /S
SWPL ∼ 10−31. ln
addition, the effects of solar rotation, viscosity,
nonthermalenergy transport, and trapping of plasma particles are
aswell, for mathematical simplicity, neglected in our
idealizedplasma-based model approach for the SIP description.
The solar plasma is assumed to consist of a single compo-nent of
Hydrogen ions and electrons. The thermal electronsare assumed to
obey Maxwellian velocity distribution inan idealized situation. In
reality, of course, deviations exist,and hence different kinds of
exospheric models have alreadybeen proposed with special velocity
distribution functionskinetically [4, 7, 8]. Again inertial ions
are assumed toexhibit their full inertial response of dynamical
evolutiongoverned by fluid equations of quasihydrostatic
equilibrium.This includes the ion fluid momentum equation as well
asthe ion continuity equation. The first describes the change inion
momentum under the action of the heliocentric gravito-electrostatic
field due to self-gravitational potential gradientand forces
induced by thermal gas pressure gradient. Thelatter is considered
as a gas dynamic analog of the solarplasma self-similarly flowing
through a spherical chamber ofradially varying cross-sectional area
with macroscopic bulkuniformity in accordance with the basic rule
of idealistic fluidflux conservation.
3. Mathematical Formulation
The basic normalized (with all standard astrophysical
quan-tities) autonomous set of nonlinear differential
evolutionequations with all the usual notations [1] constituting
aclosed hydrodynamical structure of the solar plasma system
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has already been developed in time-stationary form. Thesame set
of the basic structure equations defined in thegravitational scale
of space and time is developed in nonau-tonomous form and enlisted
as follows:
dM
dτ+ M
dM
dξ= −αdθ
dξ− gs, (1)
dθ
dτ+ M
dθ
dξ+dM
dξ+
2ξM = 0, (2)
dgsdξ
+2ξgs = eθ , (3)
d2θ
dξ2+
2ξ
dθ
dξ= 0. (4)
Here α = 1 + ∈T = 1 + (Ti/Te), Te is the thermal
electrontemperature, and Ti is the inertial ion temperature on
thebounded SIP scale (each in eV). This should be mentionedthat (4)
is a nonplaner geometrical outcome of the gener-alized
electrostatic Poisson’s equation with global quasineu-trality
approximation in a spherical symmetric distributionof the solar
plasma with all the usual notations [1] given as
(λDeλJ
)2[d2θ
dξ2+
2ξ
dθ
dξ
]= Ne −Ni. (5)
Here λDe =√Te/4πn0e2 denotes the SIP electron Debye
length. For instant information, the solar parametersM(ξ),
gs(ξ), and θ(ξ) represent the equilibrium Mach num-ber, solar
self-gravity, and electrostatic potential, respectively.They are
respectively normalized by plasma sound phasespeed (cs), solar
free-fall (heliocentric) self-gravity strength(c2s /λJ), and
electron thermal potential (Te/e). Moreover, theindependent
variables like time (τ) and position (ξ) are nor-malized with Jeans
time (ω−1J ) and Jeans length (λJ) scales,respectively, as already
carried out in our earlier publication[3] too.
In order to get a quantitative flavor for a typical valueof Te =
106 K, for example, one can estimate the value ofλDe/λJ ≈ 10−20
[3]. This implies that the size of the Debyescale length is quite
smaller than that of the Jeans scalelength of the solar plasma
mass. Thus on the typicalgravitational scale length of the
inertially bounded solarplasma system, the limit λDe/λJ → 0 becomes
a realistic(physical) approximation. By virtue of this limiting
scalecondition the entire SIP extended up to the SSB and
beyondobeys the plasma approximation of global quasineutrality,
asNe ≈ Ni from (5), in our defined self-gravitating solar
plasmasystem justifiably.
Applying the usual standard methodology of reductiveperturbation
technique [14, 15] over the coupled set of(1)–(4), we want to
derive a nonlinear dynamical equationin the lowest-order perturbed
GES potential in the SIPscale. Methodologically, the independent
variables are thusstretched directly as ξ = ∈1/2(x − λt) and τ =
∈3/2t.Thus in the newly defined space of stretched variables,
thelinear differential operators transform as ∂/∂ξ ≡
∈1/2∂/∂x,∂2/∂ξ2 ≡ ∈ ∂2/∂x2, and ∂/∂τ ≡ (∈3/2∂/∂t − λ∈1/2∂/∂x),
where λ is the phase speed of the GES perturbation and ∈is a
smallness parameter characterizing the balanced strengthof
dispersion and nonlinearity. The dimensionless amplitudeof the
lowest-order fluctuations is usually given by thisparameter ∈ [10,
11].
The nonlinearities in our might have directly comefrom the
large-scale dynamics (in space and time) throughthe harmonic
generation involving fluid plasma convection,advection,
dissipation, and so forth. These nonlinearitiesmay contribute to
the localization of waves and fluctuationsleading to the formation
of different types of nonlinearcoherent structures like solitons,
shocks, vortices, and soforth which have both theoretical as well
as experimentalimportance [14]. The scale size of all the nonlinear
fluctu-ations of current interest is assumed to be much shorter
thanall the characteristic mean free paths. These scaling
analyseshave, systematically, been derived under the conditions
thatthe normalized fluctuations in the dependent solar
plasmavariables are of the same order within an order of
magnitudein the astrophysical scale of space and time.
In order to study the GES stability of the presentconcern, the
relevant solar physical variables (M, gs, θ) arenow perturbatively
expanded around the respective well-defined GES equilibrium values
(M0, gs0, θ0) as follows:
⎛⎜⎜⎝M
gs
θ
⎞⎟⎟⎠ =
⎛⎜⎜⎝M0
gs0
θ0
⎞⎟⎟⎠+ ∈
⎛⎜⎜⎝M1
gs1
θ1
⎞⎟⎟⎠ + ∈2
⎛⎜⎜⎝M2
gs2
θ2
⎞⎟⎟⎠ + · · · . (6)
We now substitute (6) into the basic governing (1)–(4).Equating
the terms in various powers of ∈ from both sidesof (1), one
gets
∈1/2 : −λ∂M0∂x
+ M0∂M0∂x
= − α ∂θ0∂x
, (7)
∈3/2 : ∂M0∂t
− λ∂M1∂x
+ M0∂M1∂x
+ M1∂M0∂x
= − α∂θ1∂x
,
(8)
∈5/2 : ∂M1∂t
− λ∂M2∂x
+ M0∂M2∂x
+ M1∂M1∂x
+ M2∂M0∂x
= − α ∂θ2∂x
, etc.
(9)
Similarly, equating the terms in various powers in ∈ from(2),
one gets
∈1/2 : (M0 − λ) ∂θ0∂x
+∂M0∂x
= 0,
∈3/2 : ∂θ0∂t− λ∂θ1
∂x+ M0
∂θ1∂x
+ M1∂θ0∂x
+∂M1∂x
= 0, etc.(10)
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The order-by-order analysis in various powers of ∈ from
(3)similarly yields
∈0 :(
2ξ
)gs0 = 1 + θ0,
∈1/2 : ∂gs0∂x
= 0,
∈1 :(
2ξ
)gs1 = θ1,
∈3/2 : ∂gs1∂x
= 0,
∈2 :(
2ξ
)gs2 = θ2, etc.
(11)
The same order-by-order analysis in various powers in ∈from (4)
yields
∈1 : ∂2θ0∂x2
+(
2x
)∂θ0∂x
= 0, (12)
∈2 : ∂2θ1∂x2
+(
2x
)∂θ1∂x
= 0, (13)
∈3 : ∂2θ2∂x2
+(
2x
)∂θ2∂x
= 0, etc. (14)
We are involved in the dynamical study of the lowest-order GES
potential fluctuation associated with the SIP sys-tem. Equation
(8), therefore, is now approximately simplifiedinto the following
form:
M1 = −(
α
M0 − λ)θ1. (15)
A little exercise with the substitution of (15) in (9) (under
anapproximation of equal rate of harmonic covariation)
jointlygives
∂θ1∂x
= − 1(M0 − λ)
∂θ1∂t
+α
(M0 − λ)2θ1∂θ1∂x
. (16)
Again spatially differentiating (13), one gets
∂3θ1∂x3
+(
2x
)∂2θ1∂x2
−(
2x2
)∂θ1∂x
= 0. (17)
Now coupling (16) and (17) dynamically with no vari-ation of the
equilibrium parameters, one gets easily thefollowing modified form
of KdV-Burger (KdV-B) equation[9–11, 14, 15] for the description of
the nonlinear GESfluctuations in terms of θ1 as follows:
(2
M0 − λ)∂θ1∂t−[
2α(M0 − λ) 2
]θ1∂θ1∂x
+x2∂3θ1∂x3
+ 2x∂2θ1∂x2
= 0.(18)
This is clear from (18) that the temporal part (1st term)and
convective part (2nd term) have constant coefficients ina given
plasma configuration bounded quasihydrostatically.The dispersive
part arising due to the deviation fromglobal plasma quasineutrality
(3rd term) and dissipative partarising due to various internal loss
processes (4th term),however, have variable coefficients. We are
interested in time-stationary structures of dynamical fluctuations,
and hence,(18) is transformed into an ordinary differential
equation(ODE) with the transformation ξ ≡ (x − λt) so that
theoperational equivalence ∂/∂t ≡ −λ∂/∂ξ and ∂/∂x ≡ ∂/∂ξhold good.
Equation (18), therefore, with A = −2λ/(M0− λ)and B = −2α/(M0 − λ)2
gets transformed into a stationaryform as
A∂Θ
∂ξ+ BΘ
∂Θ
∂ξ+ ξ2
∂3Θ
∂ξ3+ 2ξ
∂2Θ
∂ξ2= 0, (19)
where Θ = θ1 (ξ) denotes the lowest-order fluctuation in theGES
potential.
Equation (19) clearly shows the possibility for theexistence of
some shock-like structures (due to energy dis-sipation) in addition
to soliton-like structures (due to energydispersion). The first
class of structures realistically ariseswhen the effect of
dissipation is significant in comparisonwith the joint effect of
the nonlinearity and dispersion,whereas for the second class, the
effect of dissipation isinsignificant in comparison with that
produced jointly bythe nonlinearity and dispersion [14]. Being of
nonlineartype, the exact solution of (19) is difficult without
anyasymptotic approximation. The approximate solutions areobtained
analytically by the method of integration with someboundary
conditions like Θ → 0, ∂Θ/∂ξ → 0, ∂2Θ/∂ξ2 → 0at ξ → ∞ as done by
others [14]. The explicit form of theanalytical solution (traveling
wave) of (19) with all the usualnotations is derived and presented
as follows:
Θ(x, t) = λ (λ−M0)α
[1 + tanh
{λ
2x(M0 − λ) (x − λt)}]
.
(20)
Equation (20), in fact, represents the asymptotic formof a
monotonic shock structure (laminar type) with shockspeed Ush = λ,
shock amplitude Ash = λ(λ − M0)/α,and shock front thickness Γsh =
2x(M0 − λ)/λ. Thefundamental difference of the solution (20) with
thoseobtained analytically by others [14] is that the
presentsolution represent shocks in the perturbed GES potentialin
the self-gravitating hydrodynamic SIP with plasma-boundary
interaction taken into account. The other reportedsolutions [14],
on the other hand, represents shocks inthe perturbed electrostatic
potential and density in dustyplasma in hydrostatic equilibrium,
but in the absence ofplasma-boundary wall interaction processes,
self-gravity,and gravito-electrostatic coupling effects. This,
interestingly,is noticed here that the shock width of (20) here is
a functionof the independent position coordinate x alone in the
definedself-gravitating solar plasma configuration. Eventually,
thisimpulsive character of the SIP blast wave is realistically
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justifiable due to infinite thermal pressure at the core of
theSun [3] generated by the effect of the strong
self-gravitationalcollapse leading to thermonuclear fusion
responsible fortremendous amount of energy production.
Equation (19) is furthermore numerically solved (byRunge-Kutta
IV method) to get a detailed picture of the basicfeatures of the
GES fluctuations on astrophysical scale underdifferent realistic
initial values of the relevant solar physicalvariables. These
realistic initial values are analytically arrivedat as a natural
outcome of the nonlinear dynamical stabilityanalyses around fixed
points, as carried out in our earlierwork [3], over the coupled
dynamical equations of the two-layer GES model description.
4. Results and Discussions
4.1. Analytical Results. A theoretical model analysis is
carriedout to study the GES fluctuation in a simplified field-free
quasineutral solar plasma model in quasihydrostatictype of
homogeneous equilibrium configuration. A distinctset of
nonautonomous self-consistently coupled nonlineardynamical
eigenvalue equations in a defined astrophysicalspace and time
configuration are developed. Applying thestandard methodology of
reductive perturbation techniqueover the defined GES equilibrium
[3], a modified form ofKdV-Burger (KdV-B) equation in terms of the
lowest-orderperturbed GES potential is obtained. An explicit form
ofthe approximate analytical solution (shock family) only isderived
with the help of conventional method of integration[14] by imposing
asymptotic boundary conditions. Similaranalytical results exist in
the literature [14, 15], but in termsof shocks in the perturbed
electrostatic potential and densityin dust-contaminated plasma in
hydrostatic equilibriumin absence of the effects of all
plasma-boundary wallinteraction, self-gravity, and
gravito-electrostatic couplingmechanisms as already mentioned above
in the previoussection.
Let us now estimate the physical values of the main
shockcharacterization parameters represented by (20) at the SSB.At
the SSB, (x − λt) = ξ ∼ 3.5,MSSB ∼ 10−7, and ∈T ∼ 0.4[3].
Typically, the smallness parameter is ∈ ≈ 10−2 [10]for solar plasma
atmosphere. Now for λ = USSBsh = 0.1and αSSB = (1 + ∈T) = 1.4, we
can analytically estimatethe physical value of the GES potential
shock amplitudeASSBsh, phys = 7.1 × 10−5 (=7.10 × 10−2 volts, or
2.36 × 10−4statvolts) and shock front thickness ΓSSBsh, phys = 7.0
(∼ 7.00 ×108 m, or 7.0 × 1010 cm). Thus gravito-electrostatic
shocksin the SIP system carry relatively lower energy of the
GES-induced mode.
The reductive perturbation method is, however, not verypopular
as a mathematically rigorous perturbation method,even
conditionally. It, nevertheless, is a convenient, approxi-mate, and
easy way to produce certain mathematically inter-esting paradigm of
nonlinear equations. By the free ordering,we may get almost any
explicit form of results as expected.In particular, the ordering
required to get, for example,a shock-like solution is now known.
Numerical solutionswill subsequently show that there are many other
shock-likestructures that do not satisfy the “required ordering”.
In fact,
although shocks are frequently found experimentally, so far
ashock that satisfies the KdV-B ordering has never been
found,whether in neutral fluid, lattice, or plasma. Thus,
analytically,it provides a new mathematical stimulus scope for
futureinterest to derive analytical results with greater accuracy
withnewer mathematical techniques so as to get more detailedpicture
of the self-gravitational fluctuations like in the Sun.
4.2. Numerical Results. Our theoretical GES model analysisshows
that the solar plasma system supports shock formationgoverned by
KdV-B (19). This is again integrated numericallytoo so as to get
some numerical profiles for differentinitial values on a more
detailed grip. The main features ofour observations based on our
numerical analyses may bediscussed as follows. Figure 1(a) shows
the Θ(ξ)-profile onthe SIP scale with ∈T = 0.4, λ = 0.8,M0 = 10−8,
andΘi = −0.0001. The various lines correspond to Case (1)ξi = 0.01,
Case (2) ξi = 0.03, Case (3) ξi = 0.05, and Case (4)ξi = 0.07,
respectively. Figure 1(b) similarly depicts the same,but on the SWP
scale. It is clear that the GES perturbationexcited on the SIP
scale gets propagated even at and upto an asymptotically large
distance due to space chargepolarization effects boosted up by the
supersonic SWP flow.The fluctuation assumes various nonlinear forms
of shockfamily sensitive to input initial position values relative
tothe heliocentric origin. The Θ(ξ) amplitude is found to varyfrom
−7.5 × 10−3 to +1.5 × 10−3 at the SSB (ξ = 3.5 λJ),the SWP base. It
is again on the order of 10−3 at 1 AU (ξ =750 λJ) approximately.
All these observations are relative tothe unperturbed GES potential
value at the SSB. The SSB isalready reported to act as a
spherically symmetric electricalgrid which is
gravitoelectrostatically negatively biased with anormalized value
of the equilibrium GES potential θΘ(ξΘ) ∼−1 (= −1.00 kV) [3]. The
SWP flow dynamics is the naturaloutcome of the solar plasma leakage
process through thiselectrical grid.
Again Figure 2(a) similarly depicts the Θ(ξ) profile onthe SIP
scale with ∈T = 0.4, ξi = 0.01,M0 = 10−8, and Θi =−0.0001. The
various lines are for Case (1) λ = 0.1, Case (2)λ = 0.2, Case (3) λ
= 0.8, and Case (4) λ = 5.1, respectively.Figure 2(b) gives the
same, but on the SWP scale. It is clearthat, as in Figure 1,
different shocklike structures arise fromdifferent flow velocities
with Θ(ξ) amplitude lying on theorder of 10−3 on both the SIP and
SWP scales.
Lastly, Figure 3(a) shows the Θ(ξ)-profile on the SIP scalewith
∈T = 0.4, ξi = 0.01,M0 = 10−8, and λ = 0.8.The various lines
specify Case 1: Θi = −0.0001, Case 2:Θi = −0.003, Case 3: Θi =
−0.01, and Case 4: Θi =−0.02, respectively. Likewise, Figure 3(b)
gives the same,but on the SWP scale. It is found that the
Θ(ξ)-amplitudevaries from −5.0 × 10−3 to +15 × 10−3 in the SIP
scale.Its order is the same even at 1 AU. This is interestinglyin
totality found numerically (Figures 1–3) that the GESpotential
fluctuation propagates as shock-like structures withamplitude Θ(ξ)
∼ 10−3 at both the SSB and at 1 AU,approximately. Thus for a
typical value of the smallnessparameter ∈≈ 10−2 [10] for solar
plasma configuration,the physical value of the GES potential shock
amplitudecan throughout the numerical analyses be calculated as
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0 1 2 3 4 5 6−8
−6
−4
−2
0
2
4×10−3
Distance
Pote
nti
al
Potential profile in SIP scale
Case 1Case 2
Case 3Case 4
(a)
0 100 200 300 400 500 600 700 800−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
Distance
Pote
nti
al
Potential profile in SWP scale
Case 1Case 2
Case 3Case 4
(b)
Figure 1: Profile of the lowest order GES potential fluctuation
Θ(ξ) on the (a) SIP scale and (b) SWP scale with∈T = 0.4, λ =
0.8,M0 = 10−8,and Θi = −0.0001. The various lines correspond to
Case 1: ξi = 0.01, Case 2: ξi = 0.03, Case 3: ξi = 0.05, and Case
4: ξi = 0.07, respectively.
0 1 2 3 4 5 6−1
−0.5
0
0.5
1
1.5
2×10−3
Distance
Pote
nti
al
Potential profile in SIP scale
Case 1Case 2
Case 3Case 4
(a)
0 100 200 300 400 500 600 700 800−0.02
0
0.02
0.04
0.06
0.08
0.1
Distance
Pote
nti
al
Potential profile in SWP scale
Case 1Case 2
Case 3Case 4
(b)
Figure 2: Profile of the lowest order GES potential fluctuation
Θ(ξ) on the (a) SIP scale and (b) SWP scale with ∈T = 0.4, ξi =
0.01,M0 =10−8, and Θi = −0.0001. The various lines correspond to
Case 1: λ = 0.1, Case 2: λ = 0.2, Case 3: λ = 0.8, and Case 4: λ =
5.1, respectively.
ΘPhys(ξ) =∈ ΘΘ(ξ) ≈ 10−5 = 10−2 volts. In all the cases,the Θ(ξ)
amplitude is usually found to go more andmore negative near the
heliocentre (ξ ∼ 0.5 λJ) due tostrong self-gravitational effects,
acting even on the plasmathermal electrons and thereby, tending to
prevent themto escape through solar self-gravitational potential
barrier.The reversibility of the magnitudes of the GES
shock-likestructures from negative to positive values is ascribed
due tothe solar plasma leakage process through the SSB grid.
4.3. Comparative Results. This has been recognized years agothat
the compressional plasma in the solar atmosphere is aperfect medium
for magnetohydrodynamic (MHD) waves.Applying the MHD model analyses
[9–11, 16, 17], severalauthors have investigated on the
corresponding nonlineareigenmodes in the compressional solar plasma
in presence ofmagnetic field. Some of the important distinctions
betweenour GES stability analysis and MHD stability analyses onthe
solar plasma fluctuation dynamics reported so far in
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Physics Research International 7
0 1 2 3 4 5 6−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
Distance
Pote
nti
al
Potential profile in SIP scale
Case 1Case 2
Case 3Case 4
(a)
0 100 200 300 400 500 600 700 800−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Distance
Pote
nti
al
Potential profile in SWP scale
Case 1
Case 2Case 3Case 4
(b)
Figure 3: Profile of the lowest order GES potential fluctuation
Θ(ξ) on the (a) SIP scale and (b) SWP scale with ∈T = 0.4, ξi =
0.01,M0 =10−8, and λ = 0.8. The various lines correspond to Case 1:
Θi = −0.0001, Case 2: Θi = −0.003, Case 3: Θi = −0.01, and Case 4:
Θi = −0.02,respectively.
the literature are worth mentioning here. The
followingtabulation, Table 1, shows the main distinctions
betweenthem.
We scientifically admit that the neglect of
collisionaldissipations and deviation from Maxwellian velocity
dis-tributions of the plasma particles is not quite realistic.But
our GES stability analyses even under some simplifiedand idealized
approximations may provide quite interestingresults for the solar
physics community. The main pointsbased on our analyses are
summarized as follows.
(1) The GES fluctuations appear in the form of variousnonlinear
structures (of shock-family eigenmodes)governed by a new analytic
form of KdV-Burger(KdV-B) equation. Here the terminology “new
ana-lytic form” refers to the appearance of the new typeof
characteristic coefficients in the dispersive anddissipative terms
in it.
(2) The influence and presence of such eigenmodes arealso
experienced asymptotically even in the SWPscale. The structural
modification here is due tothe background initial conditions under
which beingexcited. Such blast wave structures arise mainly dueto
violent disturbances of self-gravitational type.Their front
thickness may, however, be a conse-quence of the homogeneous
balance between self-gravitating solar plasma nonlinear
compressibilityand dissipative mechanisms like viscosity, heat
con-duction, and so forth. Similar observations in theSun have also
been reported by MHD-communityunder Hall-MHD approximation [9–11]
in terms
of the lowest-order perturbed solar density fluctu-ation. In
addition, such structures have also beenexperimentally observed and
reported in a dust-contaminated plasma system [14, 15] with
differentgrain population density in both oscillatory as well
aslaminar forms, but in absence of
gravito-electrostaticeffects.
(3) Self-gravity of the SIP mass distribution is normallyfound
to have a tendency to depress (due to dissi-pation and dispersion)
the nonlinear structures inthe interior (subsonic SIP flow) and
steepens (due tononlinearity) them in the exterior (supersonic
SWPflow) in our two-scale GES stability analyses.
(4) Last but not least, the Θ(ξ) fluctuations becomealmost
uniform at an asymptotically large distancerelative to the
heliocentre. This physically meansthat the SWP flow is a uniform
one asymptotically,and net electric current, contributed jointly by
solarthermal electrons and inertial ions, will remain con-served
(divergence-free current density in a steady-state description)
which is in good agreement withthe already reported results
[18].
This analysis, moreover, is carried out in a homogeneouskind of
field-free quasihydrostatic equilibrium configurationunder
quasineutral plasma approximation. However, evenin spite of these
limitations, it may perhaps be useful forfurther investigation of
dynamical stability on a nonlinearlycoupled system of the SIP and
SWP as an interplayed flowdynamics of heliocentric origin in
presence of all the possiblerealistic agencies [16, 17] like
collision, viscosity, and soforth. This is speculated that the
normal mode behaviors
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8 Physics Research International
Table 1: GES versus MHD stability analyses.
S. no. Items GES stability analysis MHD stability analyses
1 Model Ideal hydrodynamic MHD
2Plasma-boundary wall interaction andsheath formation
mechanism
Included Neglected
3 Effect of charge separation Considered Not considered
4Floating surface (at which no netelectric current)
Involved Not involved
5 Magnetic fieldNot considered(〈|BSSB|〉 ∼ 7.53 × 10−11 T)
Considered(〈|BΘ|〉 ∼ 1.30 × 10−6 T)
6 Description Two-scale (SIP and SWP) One scale (SWP)
7 Sonic range Subsonic (SIP) and supersonic (SWP) Supersonic
(SWP)
8Self-gravity (SIP) and external gravity(SWP)
Considered Not considered
9Transonic transition (subsonic tosupersonic)
Involved (through SIP and SSB interaction processand thus
transformed into SWP)
Not involved
10 Analytical solution Bounded (SIP) Unbounded (SWP)
11 Thermal species Maxwellian Single fluid (MHD)
12 Surface description and specificationYes (at ξ = 3.5 λJ ) and
it is negatively biased (withθs ∼ −1.00 kV) at the cost of thermal
loss of SIPelectrons
Not precisely, but thediffused surface iselectrically uncharged
andunbiased
13 Source of nonlinearity Plasma fluidity Large-scale
dynamics
14 Source of dispersion Deviation from quasineutrality and
self-gravityGeometrical effect (also,some part of
physicaleffect)
15 Source of dissipation Weak collisional effectsViscosity and
magneticdiffusion
16 Sun and SWP coupling Considered Not considered
17 Nature of solutionsMainly shocklike structures in the lowest
orderedperturbed GES potential
Soliton and shocklikestructures in the lowestordered perturbed
densityand velocity
18 Solar atmosphere Not stratifiedStratified (into a number
ofheliocentric layers)
19 Adopted techniqueStandard reductive perturbation technique
(aboutthe GES equilibrium)
Standard multiple scalingtechnique (about the
MHDequilibrium)
20 Convection and circulation dynamics Not treated (for
idealized simplicity) Treated
21 Leakage process Taken into account Not taken into account
22 Main applicationSurface origin of the subsonic SWP and its
transonicflow dynamics
Solar chromospheric andcoronal heating
of the global SSB oscillations could also be analyzed interms of
both the local as well as global gravito-electrostaticplasma
sheath-induced oscillations with such techniques.Additionally,
gaseous phase of the solar plasma is reportedto contain solid phase
of dust matter [19]. In the SWPscale of uniform flow, application
of the inertia-inducedacoustic excitation mechanism [20] may
further be carriedout for further stability analyses. The basic
principles ofthe nonlinear pulsational mode [21] of the
self-gravitationalcollapse model of charged dust clouds by applying
thepresented methodology may be another important futureapplication
in the self-gravitating solar plasma system.
5. Conclusions
The dynamical stability of the GES model, although simpli-fied
through idealistic approximations, is analyzed in bothanalytical
and numerical forms with standard perturbationformalism. It
provides an idea into the interconnectionbetween the SIP (Sun-)
stability in terms of the lowest-orderGES fluctuation appearing as
various nonlinear structures(shock like) and their asymptotic
propagation in the SWPscale as an integrated model approach. This
is conjecturedthat the fluctuations are jointly governed by a new
formof KdV-Burger type of nonlinear evolution equation having
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Physics Research International 9
some characteristic model coefficients. Both analytical
andnumerical solutions are in qualitative and
quantitativeagreement. The main conclusions of scientific interest
drawnfrom our present contribution are summarized as follows.
(1) Nonlinear fluctuations of the GES in the SIP scale
aregoverned by a KdV-Burger (KdV-B) type of equationwith
characteristic coefficients dependent on thesolar plasma GES
model.
(2) Different forms of nonlinear eigenmodes exist in theSIP
scale in different situations. Their presence, pre-triggered
strongly due to self-gravity on the SIPscale origin, is also
experienced at asymptoticallylarge distances beyond the SSB. It
goes in qualita-tive conformity with those reported with
differentmethodologies [2, 12, 13].
(3) The structures are contributed mainly due to
gravi-toelectrostatically coupled self-gravity fluctuation ofthe
solar plasma inertial ions under an integratedinterplay of diverse
nonlinear (hydrodynamic origin)and dispersive (self-gravitational
origin) effects in thesolar plasma system in presence of some
internaldissipation.
(4) The SIP is found to be more unstable (more fluc-tuation
gradient) than the SWP (less fluctuationgradient) asymptotically.
This is because of the GESfluctuation in presence of strong
self-gravity in thebounded SIP scale and weak external gravity in
theunbounded SWP scale.
(5) Our two-scale theory of the GES is found to givetwo-scale
dynamical variation of the GES stabilityas a
gravito-electrostatically coupled system of theSIP (subsonic flow)
and the SWP (supersonic flow)through the interfacial SSB.
Finally and additionally, the modified GES mode kineticsas a
self-gravitationally triggered instability in an intermixedstate of
the gaseous phase of plasma and solid phase ofdust grain-like
impurity ions (DGIIs), by using a dissipativemulti-fluid colloidal
or dusty plasma model with dust scalesize distribution power law
taken into account, may beanother interesting investigation to
study DGII-behavior inan SWP-like realistic situation on a global
scale. This isbecause the interplay between gravitational and
electrostaticforces in the dynamics of such grains is responsible
formany interesting phenomena in the terrestrial and
solarenvironment (like rings of Saturn and Jupiter,
satellites’spoke formation, etc.). It eventually may have some
usefulcharacteristic implications of acoustic spectroscopy ([20]and
references therein) as well on the basis of dispersion waveanalyses
to be characterized with different scale-sized inertialspecies
(DGIIs) in different realistic astrophysical conditions.The
mathematical methodology adopted may also be exten-sively
applicable to other types of nonlinear waves, whereverall being
considered as derivatives of shocks in presence ofnonlinearity,
dispersion, and dissipation, by applying kineticexospheric model
approaches [7, 8] with the more realisticSWP exobases taken into
concern. These mathematical anal-yses may be extended for further
investigation of fluctuation
and stability with more realistic assumptions like
grainrotations, spatial inhomogeneities, different gradient
forces,and so forth, taken into account in other astrophysical
andspace environments. These calculations, although tentativefor
any concrete application to any sharply specified stellarformation
mechanism, may be widely useful in the studyof fluctuation-induced
dynamics with electrostatic chargefluctuation of dust grains in
astrophysical environment ofdusty plasmas in the complex form of
self-gravitationallycollapsing dust cloud [21].
Acknowledgments
The valuable comments, specific remarks, and precise
sug-gestions by an anonymous referee, to refine the
prerevisedoriginal paper into the present postrevised improved
form,are very gratefully acknowledged. Moreover, the
financialsupport received from the University Grants Commission
ofNew Delhi (India), through the research project with GrantF. no.
34-503/2008 (SR), is also thankfully recognized forcarrying out
this work.
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