nonlineargravitoelectrostaticsheath...

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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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)


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


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


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


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


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


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


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


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


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


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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