A Numerical Characterization of the Gravito-Electrostatic Sheath Equilibrium Structure in Solar Plasma ()
This article describes the equilibrium structure of the solar interior plasma (SIP) and solar wind plasma (SWP) in detail under the framework of the gravito-electrostatic sheath (GES) model. This model gives a precise definition of the solar surface boundary (SSB), surface origin mechanism of the subsonic SWP, and its supersonic acceleration. Equilibrium parameters like plasma potential, self-gravity, population density, flow, their gradients, and all the relevant inhomogeneity scale lengths are numerically calculated and analyzed as an initial value problem. Physical significance of the structure condition for the SSB is discussed. The plasma oscillation and Jeans time scales are also plotted and compared. In addition, different coupling parameters, and electric current profiles are also numerically studied. The current profiles exhibit an important behavior of directional reversibility, i.e., an electrodynamical transition from negative to positive value. It occurs beyond a few Jeans lengths away from the SSB. The virtual spherical surface lying at the current reversal point, where the net current becomes zero, has the property of a floating surface behavior of the real physical wall. Our investigation indicates that the SWP behaves as an ion current-carrying plasma system. The basic mechanism behind the GES formation and its distinctions from conventional plasma sheath are discussed. The electromagnetic properties of the Sun derived from our model with the most accurate available inputs are compared with those of others. These results are useful as an input element to study the properties of the linear and nonlinear dynamics of various solar plasma waves, oscillations and instabilities.
1. Introduction
The well known standard solar model (SSM) basically assumes a single neutral gas approximation under hydrostatic equilibrium description of the Sun [1-6]. This model ignores the Coulomb property of the solar plasma constituents and the role of coulomb interactions on binary and collective scales. This can be physically justified as because the quasi-neutral property of the solar plasma is well satisfied on the Jeans scale lengths. Consequently, the possible role of space charge electrical action due to plasma wall interactions is also absent. Nevertheless, a few curious minds [7-9] took an interest to investigate the possible effect and semi-empirical estimation of the electrical forces on the pressure in a star shown to have a net electrical charge on the surface. In an attempt made to determine the suspected electric field by the measurements of the Stark effect [10], no field could be detected below an upper limit of 100 V/cm. Gunn endeavored to indicate the importance of the electric fields and calculated its value using Pannekoek model for charge separation [10]. It amounts to 0.015 V/cm, a value which is too small to be measured by Stark effect. Afterwards the idea of electric field in the solar/stellar interior and exterior regions could not gather much momentum for further research. In these electrical models the problem of the origin mechanism and maintenance of the electric field still remains as an open question. Moreover, these models do not find out exact solutions of the basic structure equations, but provide empirical and simple theoretical estimations of the solar and stellar specific values.
Dwivedi, Karmakar and Tripathy in 2007 proposed a simple model of gravito-electrostatic sheath (GES) formation [11] around the solar surface boundary (SSB). This GES ranging from the bounded to unbounded scales is similar to the pre-sheath region of the laboratory scale plasma bounded by physical wall. It is normally believed that the terms plasma and sheaths were introduced in 1928-29 by Irving Langmuir in the study of electrical discharges in gases [12,13]. The continuous emission of the subsonic solar interior plasma (SIP) and its acceleration to the supersonic speed seems to be a necessity for sustaining the dynamically bounded GES-type of equilibrium of the Sun as a coupled system of two different plasmas. We wish to provoke that the gravitational squeezeing causes surface charge polarization due to excessive flux emissions of the electrons at successive spherical surfaces lying at variable radial points relative to the heliocentre. We can thus easily think that the successive surfaces will develop negative charges varying with radial position, being the highest at the SSB. Finally, a global scale charge polarization in the bulk SIP will be reflected on the SSB and beyond on Jeans scale length order. Thus one can understand and explain the interior origin of the solar wind plasma (SWP) and its supersonic acceleration due to space charge electrical field action in terms of the basic principles of plasma-wall interaction process. The existence of a spherical floating surface is an outcome of the GES theory. To say much more about its physical significance, we need more and more thinking and investigation.
This model provides an integrated and unique view of a self-consistently coupled system of the solar interior and exterior plasmas. Consideration of a two-fluid (one for the SIP, and the other, SWP) solar plasma model offers a new physical insight for describing the equilibrium hydrodynamic structure and space charge electrical state of the Sun and its atmosphere. In fact, the concept of a single neutral fluid model suffers deficiency of hiding the role of the Coulomb interactions on the binary and collective scales both. Hence it lacks in the complete description of the equilibrium properties of the solar plasma on both the bounded and unbounded scales. The bounded scale solar plasma is termed as the SIP, whereas the unbounded scale solar plasma is called the SWP. Both are of the same origin, but different on the basis of the dynamical behaviors. They are mutually interconnected and are sustained self-consistently as a steady state hydrodynamic equilibrium of a single unit of the self-gravity confined solar plasma and its own atmosphere. The GES concept allows the role of plasma-wall interaction physics to cause the origin of space charge electric field effect to play an important role in the selfgravitational confinement of the solar (stellar) plasma. It is sustained by the surface emission mechanism of the solar (stellar) wind plasma in general in a steady state (time-stationary) hydrodynamic configuration.
From the basic knowledge of a bounded plasma behavior on laboratory scale [14-16], one knows that the quasi-neutral property of the plasma is violated near the boundary wall. A non-zero and nonlinear space charge electric field (called plasma sheath or Debye sheath) develops near the wall surface and extends its impact over a few Debye lengths inside the confined plasma. This localized electrostatic field, nonlinear in space, confines the bulk quasi-neutral plasma enclosed within a solid boundary wall. In the case of a completely absorbing physical wall the loss of more flux of thermal electrons to the wall than that of the ions causes the origin of space electric field in the bulk plasma. The space charge electric field thus arising due to plasma-wall interaction process evolves and gets localized within a region of a few Debye lengths width [14-16].
In the case of solar plasma, there is no solid physical boundary wall located at some specified radial position as such, but the solar self-gravity itself acts as a gravitational potential wall having variable strengths in radial direction with the maximum strength at the SSB. The self-gravitational wall strength at any radial point is measured by the escape velocity of the plasma fluid at that point. This means that ion fluid requires certain minimum threshold velocity in supersonic range to cross over the gravitational barrier created and maintained by the entire solar plasma mass distribution itself.
This is to furthermore note that the wall strength in laboratory-produced plasma is a constant and is located at some fixed position with respect to our reference point in the plasma. In the case of the solar self-gravity confined plasma, on the other hand, it is continuous with variable strength at different radial positions from the heliocentre. The radial distribution of the solar selfgravity wall strength having the maximum value on the SSB is describable by the gravitational Poisson equation. Applying the basic conceptual knowledge of plasma-wall interaction physics in the GES model, we predict the existence of a concentric virtual spherical wall with floating surface potential away from the SSB. Our model offers a precise definition of the SSB and its associated electromagnetic properties. This also offers an alternate approach to understand the basic physics of the solar surface origin of the SWP and its supersonic transition through the transonic equilibrium. This physical model illustrates the idea of an internal origin of the SWP by plasma-boundary interaction processes and cross-border effects. This is in contrast to all the other proposed models reported so far as discussed above in earlier part of the introduction.
Many other authors [17-20] also discuss about the Sun and SWP physics. Moreover, the thermodynamics of the Sun and its non-ideal interior are well understood with a proper equation of state for the composing ionized matter based on the free energy minimization principle [26]. This is to acknowledge that the ideas of electric field, magnetic field, solar surface charge, etc. were known earlier [7-10]. But this is to comment that the exact solutions of the basic model equations were missing. The degree of accuracy on theoretical results requires accuracy in problem formulation, mathematical strategy, methodological calculations, and mathematical-physical consistency. For example, in our model calculation the inclusion of electrostatic Poisson term in further calculation of the space charge electric field arising at Jeans scale length order becomes redundant. This is well justified due to very wide range difference of the solar plasma Debye length and Jeans length scales. This simply implies that the quasi-neutral property holds well on the bounded and unbounded scales of the solar plasmas associated with fields and electric currents. As a result, consideration of electrostatic Poisson term in charge density to mass density ratio estimation as done earlier [8] leads to zero values of ~10–36 on a normalized scale length on Jeans order. So, to describe the property of the space charge electric field on Jeans scale length order, it seems inconsistent to include the electrostatic Poisson term in the theoretical analysis and quantitative estimation of the solar or stellar plasma electric field.
Our model calculation questions the earlier idea of the solar surface origin of the SWP and suggests its origin from deep inside the Sun i.e. in the core. One of the most important results of this model calculation is the flow of electric current inside the Sun and beyond the SSB in the solar atmosphere with current reversal property. In this research contribution, we derive and discuss the basic physics of the GES formation in detail, its existennce condition compared to other models [7-10], all the relevant characteristic lengths as well as the electrical and magnetic state of the Sun and its atmosphere.
Apart from the “Introduction” part described in Section 1 above, this paper is structurally organized in a usual simple format as follows. Section 2 includes the basic ideas and approximations of the model along with the mathematical formulations of the problem. In Section 3, we present the basic physical concept of the solar self-gravitational wall and existence condition of the GES formation. Section 4 describes the numerical results of the different physical parameters associated with the GES equilibrium structure in three added subsections. Subsections 4.1, 4.2 and 4.3 present the numerical results for the bounded SIP, unbounded SWP, and relative comments, respectively. Lastly, in Section 5, possible conclusions are drawn briefly and tentative future scopes with astrophysical importance are discussed to extend the problem further in more realistic astrophysical situations.
2. Physical Model and Formulations
A simplified ideal two-fluid plasma model is adopted to study the solar plasma equilibrium under the GES model framework on the bounded and unbounded scales in a field-free hydrodynamic equilibrium configuration. For mathematical simplicity, no collisions of any type are included and no magnetic field effect is considered. Applying the spherical capacitor charging model, the coulomb charge on the SSB comes out to be 
C. For rotation frequency of the solar surface corresponding to the mean angular frequency about the centre of the system 
Hz [9], the mean value of the strength of the solar magnetic field at the SSB in our model analysis is estimated as
T. This is negligibly small for producing any significant effects on the dynamics of the solar plasma particles. Thus the effects of the magnetic field are not realized by the particles due to the weak Lorentz force, which is now estimated to be 
N corresponding to a subsonic flow speed 
cm/s. Thus the Lorentz force has a very weak effect on the plasma particles and hence, neglected. It justifies the convective and circulation dynamics being neglected as well. Therefore our unmagnetized plasma approximation is well justified in our GES model configuration. All types of kinetic effects are also ignored to avoid complications of mathematics and physics both. An estimated value 
of the ratio of the solar plasma Debye length and the Jeans length of the total solar mass justifies the quasi-neutral behavior of the solar plasma on both the bounded and unbounded scales. The confining wall of the solar plasma looks like a spherically symmetric surface boundary of non-rigid and non-physical nature. The solar plasma is assumed to consist of a single component of Hydrogen ions and electrons. The electrons are assumed to have Maxwellian density distribution with gravitational potential term ignored due to zero mass approximation of the electrons. Ions follow the full inertial dynamics in one dimension of simplified radial degree of freedom.
The basic sets of dynamical evolution equations relevant for the bounded and unbounded solar plasma description and characterization are given and discussed in separate subsections as follows.
2.1. Basic Equations for SIP Scale Equilibrium
The basic autonomous coupled set of mathematical equations for the investigation of the SIP equilibrium properties is given below. All the equations are normalized and the normalizations are defined in our previous publication [11].
Solar self-gravity Poisson equation:
(1)
Ion continuity equation:
(2)
Ion momentum equation:
(3)
where
, 
is the thermal electron temperature and 
is the inertial ion temperature for the bounded solar plasma on the SIP scale (each in eV). Equations (2) and (3) are simplified by using the Maxwellian population density distribution for the electrons of the solar plasma system. In fact Equation (1) defines and describes the physical nature (strength and its radial variation) of the solar self-gravitational wall. The gravitational potential energy corresponding to the solar self-gravity at any given radial position quantifies the physical strength of the gravitational wall at that radial position. This is also expressible in the form of the escape velocity of the solar plasma ions as shown in Figures 9(a)-(c).
The mathematical notations
, 
, and 
as usual represent the equilibrium solar self-gravity, GES associated electrostatic potential and Mach number in normalized forms, respectively. Let us mention that the solar self-gravity is normalized by the solar free-fall (heliocentric) gravitational strength
. The GESassociated electrostatic potential is normalized by the electron thermal potential
. The ion flow velocity is normalized by solar plasma sound speed
. Moreover, the independent variables like time 
and space 
are normalized with Jeans time 
and Jeans length 
scales, respectively. Appendix may be useful to offer an instant and quick reference of the standard physical values [6-21] purposeful for solar plasma calculations for any reader of the paper conveniently. The equilibrium values of the relevant normalization parameters (/constants) useful for our work along with plasma parameters are estimated and enlisted in Appendices 2 and 3, respectively. Here we take 
and 
so that 
for both the SIP and SWP. This is found to be the best choice for which the hydrodynamic condition is fulfilled. Other formulae, constants and mathematical expressions of the relevant plasma parameters are directly adopted from NRL Plasma Formulary [21].
Astrophysical inhomogeneity scale lengths in the case of accretion disks have been derived and discussed in detail [22,23]. Applying the same methodologies in the Sun, the normalized forms of inhomogeneity scale lengths for self-gravity
, electrostatic potential
, Mach number
, population density 
and electric current density 
are respectively defined as,
(4)
(5)
(6)
(7)
(8)
Furthermore, the normalized forms of the gravitothermal coupling coefficient for electrons
, gravitothermal coupling coefficient for ions
, and gravitoacoustic coupling coefficient for ions 
are, respectively, expressed as follows
(9)
(10)
and
(11)
The expression for the SIP electric current density with ion thermal contribution included is written as follows in the form of Equation (12),
(12)
where 
defines the equilibrium ion Bohm current density for the SIP and 
specifies the mean SIP equilibrium population density.
Additional interesting properties of the SIP are electron Debye length
, Jeans frequency
, Jeans length
, electron plasma oscillation time scale
, ion plasma oscillation time scale
, SIP ion escape velocity
, etc. Their expressions are derived and given as follows
where
, (13)
where
, (14)
where
, (15)
where
, (16)
where
, (17)
and
(18)
Again when these time scales are normalized with Jeans time scales, the same expressions will read as
(19)
where 
and
(20)
where 
Again the SIP ions get energized under the influence of the electric field associated with the GES. The electric field-induced source velocity of the ions without thermal correction 
(for cold ions) and electric field-induced velocity with thermal correction 
(for relatively hot ions) associated with the SIP flow dynamics are respectively derived and given as follows
, (21)
and
. (22)
2.2. Basic Equations for SWP Scale Equilibrium
While exploring the SWP properties on an unbounded scale, this should be kept in mind that the self-solar gravity is switched off by electrical screening of the solar self-gravitational field. Now the Sun as a whole acts as a source of an external gravity, and it controls and monitors the dynamics of the SWP. The basic autonomous coupled set of the governing equations for the SWP equilibrium properties, as described on the SIP scale, is cast as follows
(23)
and
(24)
where 
is a normalization coefficient and
, 
is the solar plasma thermal electron temperature and 
is the inertial ion temperature as defined before. The normalized forms of the inhomogeneity scale lengths for electrostatic potential
, Mach number
, population density 
and electric current density 
are, respectively, defined on the unbounded SWP scale as follows,
(25)
(26)
(27)
and
(28)
The gravito-thermal coupling coefficient of the electrons
, gravito-thermal coupling coefficient of the ions
, and gravito-acoustic coupling coefficient of the ions
, respectively, are derived on the SWP scale and presented as follows
(29)
(30)
and
(31)
Expression for the SWP carrying electric current with the ion thermal contribution taken into account is written as follows
(32)
where 
is defined as the usual equilibrium Bohm current density on the SWP scale and 
specifies the SWP equilibrium population density.
Moreover, some additional interesting physical properties of the SWP are electron Debye length
, Jeans frequency
, Jeans length
, electron plasma oscillation time scale
, ion plasma oscillation time scale
, SWP ion escape velocity
, etc. Normalized expressions for escape velocity of the SWP ions
, electric filed-induced velocity without thermal correction 
(for cold ions) and electric filed-induced velocity with thermal correction 
(for relatively hot ions) associated with the SWP flow dynamics are respectively given as follows
, (33)
, (34)
and
. (35)
3. Conditions for the GES Formation
In the case of laboratory plasma the Bohm criterion [14] must be satisfied for the formation of Debye sheath near the wall boundary. This implies that the inertial ions must enter the non-neutral space charge layer known as the Debye sheath or simply plasma sheath with velocity exceeding the sonic velocity. Now, a presheath region must exist to accelerate the ions to acquire the requisite velocity as dictated by the Bohm criterion. In the case of completely absorbing wall the potential of the wall is raised to some maximum negative value so as to equalize the electron and ion particle flux densities received by the wall surface. Thus for such condition of the wall, no net electric current is drawn by the wall and hence, it is called the floating wall. To understand the basic physical process of plasma sheath formation in laboratory plasma, the following arguments are advanced. In an initial stage of the physically confined plasma the wall receives more thermal electron particle flux density than that of the inertial ions. This occurs due to the assumption of the zero electron-inertia with respect to heavier inertial ions. As a result, the wall acquires excess negative charge leaving an ion excess charge region inside the whole of the bulk plasma volume.
Now the wall-induced space charge polarized electric field comes into action that accelerates the bulk plasma ions towards the wall. Hence a dynamical process of space charge electric field evolution sets in and continues till the electron and ion fluxes are equalized at the wall. This is termed as the floating condition of the wall confining any simple two-component plasma. Initially produced space charge electrostatic potential extending over the entire plasma volume shrinks and localizes near the wall over a distance on the order of few Debye lengths. This distance is known as plasma sheath width. This is how the formation mechanism of plasma sheath in laboratory confinement of plasma is understood. This nonneutral space charge layer acts as an electrostatic fencing which confines quasi-neutral bulk plasma and protects it from any external influence. There are some recent theoretical reportings [24,25] about the existence of subsonic plasma sheaths too in the case of a predominant electron current flowing through the wall. Kinetic description of the plasma is used to arrive at this conclusion where the conventional Bohm criterion [14-16] is not able to describe the sheath edge transition.
Let us now discuss the basic physics of the GES formation in a self-gravitational confinement of solar plasma. The solar plasma creates its own confining boundary wall of gravitational potential barrier by virtue of Jeans collapse process of an interstellar dust cloud where the Sun is born. In the process of Jeans collapse of dust cloud, an excessive heating of the self-gravitationally collapsing matter converts it into a plasma state of matter. Now the plasma interacts with the solar self-gravitational field that acts as a squeezing agent to squeezing out the electrons from the gravitational wall surfaces producing thereby surface space charge polarization. The structure and strength of the gravitational wall is defined and described by the gravitational Poisson equation as expressed in Equation (1). The maximization of the solar self-gravity dictates the condition for the creation of a boundary wall confining the SIP. Any boundary, of course, may be defined as the one where physical field variables become extremum. This is to note that the gravitational wall has a variable structure and strength with the maximum value defining the wall boundary. This is important to note from Equation (1) that the maximization of the solar self-gravity does not occur in a plane geometry approximation.
In fact the lighter electrons are squeezed out of the successive spherical surfaces due to solar self-gravity acting on the bulk interior plasma leaving the heavier ions to compress radially inward. This leads to an enhancement of ions population inside the compressed volume with an enhancement of thermal electron flux emitted out of the successive spherical surfaces at different radial positions. As a result space charge polarization on the Jeans scale length order comes into action to accelerate the SIP ions against the solar self-gravity. The ions are then continuously accelerated radially outwards, but the self-gravity suppresses the bulk plasma flow. As a net effect, the bulk plasma leaks through the SSB with some minimum possible velocity of a few cm/sec. In fact, at the SSB the solar self-gravity effect is cancelled out by the space charge electric field. Consequently, the SIP leak velocity is controlled and monitored by the compressibility of the ion fluid flow and curvature effect acting at equal level. Thus the formation of the GES could be well understood in terms of physical phenomenon occurring on the laboratory scale plasma due to plasma-wall interaction process in the vicinity of some physical wall.
In the case of laboratory plasma the rigidity of the wall prevents the physical movement of the ions and electrons both across the wall. In the case of gravitationally confined plasma like solar plasma with solar self-gravity confinement, the electrons and ions are not equally hindered to prevent the motion across any spherical surface in the SIP. The electrostatic field strength is decided by the solar self-gravitational wall strength of the bounded SIP system. With these analytical arguments in mind, we denote the maximum value 
of solar self-gravity at some radial position 
where
. Applying the necessary condition for 
being the maximum at some radial position 
as 
in Equation (1), one yields
. However, it is not sufficient to justify the occurrence of the maximum value of 
until and unless the second derivative of 
is shown to possess some negative value at this specific radial point. To derive the sufficient condition for the maximum value of 
at
, let us once spatially differentiate Equation (1) to yield the following
. (36)
Now let us derive mathematical condition for sufficiency of 
being the maximum at
. Using the exact hydrostatic equilibrium approximation given by 
for the gravito-electrostatic force balancing near the SSB, following inequality is obtained,
(37)
The necessary condition for the maximum 
-value, which is the requirement for any self-gravitating bounded plasma system, from Equation (1) can analytically be expressed by the following relation,
. (38)
Again these two above Conditions (37) and (38) can be combined together to derive a single simplified condition for a bounded solution of the SIP to exist as
(39)
If we define escape velocity as
, the above Inequality (39) could be rewritten in the form of gravitational wall strength defined in terms of the SIP escape velocity as,
. (40)
This means that the strength of the solar self-gravitational wall must be such that the ions could not overcome the barrier and the wall could bear with the ram pressure of the supersonic bulk plasma flow. This is now to comment that like the usual Bohm condition, there exists a similar criterion for a bounded GES solution to exist in case of self-gravitating solar plasma in spherical geometry. Here the escape velocity given by Equation (18) measures the physical strength of the gravitational wall at the SSB. Figure 6 depicts the validity of the existential condition of the GES. This is also to note that the solar self-gravity wall is bounded, whereas the electrostatic potential field is unbounded and extends over many hundreds of Jeans length beyond the SSB (Figures 1 and 10). The virtual floating surface wall is found to exist at a distance on the order of seven times of Jeans length beyond the SSB (Figure 12(c)). This is again interestingly noted that the major electrostatic potential drop occurs beyond the SSB (Figures 10(a) and (b)).
This may be worth-mentioning that the 1st term on R.H.S. of the Inequation (37) arises due to the spherical geometry of the gravitational wall and represents the rate of spatial change (decrease) in curvature effect (in selfgravity). The 2nd term on R.H.S. of inequation (37) represents the rate of spatial change (decrease) in solar plasma density due to plasma-wall interaction process with wall acquiring negative potential. The location of the maximum solar self-gravity defines the SSB. As the strength of the gravitational wall increases with increase in radial position from heliocentre outward, the electrostatic potential as well as the electrostatic electric field too increases in magnitude. For a bounded solution of the wall to exist, the rate of spatial change (decrease) in solar plasma density (2nd term) must exceed the rate of spatial change (decrease) in curvature effect (1st term) on R.H.S. of Equation (37).
Defining the boundary as a point where an exact balancing of the gravito-electrostatic force occurs, Equation (3) reduces into a simplified form written as follows
. (41)
This is clear from Equation (41) that the unbounded supersonic SWP outflow is the transformed outcome of the bounded subsonic SIP in presence of curvature (geometrical) effect. Moreover, for a plane-geometry approximation
, one gets
. Thus there will be no acceleration of the subsonic SIP into SWP by gravito-acoustic coupling processes. It, therefore, implies that the subsonic GES formation is not possible under a plane-geometry approximation even in presence of self-gravity. This indeed is an astrophysical reality in a self-gravitating plasma system under nonplaner geometry applicable for the theoretical description of the fundamental issues of the self-gravitationally confined solar plasma flow dynamics.
4. Discussions of Numerical Results
In order to get a detailed picture of the hydrodynamic GES equilibrium features, we have used the well-known fourth order Runge-Kutta method (RK-IV method) for numerical analyses of the solar plasma system. The two scale equilibrium structures of the coupled solar plasma system are governed by Equations (1)-(3) for the bounded SIP, and Equations (23)-(24) for the unbounded SWP. Two different sets of realistic initial values of the relevant solar physical variables are specified for numerical solutions of the basic governing equations. The first set of realistic initial values for the SIP description is analytically specified by nonlinear stability analysis, which falls within the solar core region. The values of these physical variables at the SSB are the natural outcomes of the nonlinear dynamical solutions of the SIP Equations (1)-(3) as an initial value problem. Now the numerically calculated values of these physical variables at the SSB forms the second set of realistic initial values for the SWP description as carried out in our earlier work [11]. In fact, this is the way we have solved the nonlinearly coupled governing dynamical equations of the two-layer GES model description. These two sets of realistic initial values are listed in Table 1 as follows.
4.1. Numerical Results for Bounded SIP
The coupled structure Equations (1)-(22) for the GES characterization are numerically simulated with the initial values as tabulated in Table 1 on the SIP scale. Figures 1-9 describe the profile structures of the SIP equilibrium in terms of the GES physical parameters. As given in Table 3 the values of
, 
and 
are kept fixed as already specified in our earlier paper [11] for all the numerical plots for the SIP scale description. The other fixed initial values obtained by nonlinear stability analyses for the concerned characterization here are 
and
. Figure 1 contains the plots of the normalized solar selfgravity, its gradient and associated inhomogenity scale length (scaled down by division with 20) at various radial positions from the initial location. It is clear from Figure 1 that 
at
. Near the maximum value of the solar self-gravity associated with the SIP, i.e., near the SSB, the gradient indeed becomes zero which justifies the necessary condition for the maximization of the solar self-gravity. Moreover, near the SSB, the scale length becomes infinitely larger. The rate of spatial change of the solar self-gravity increases initially and then decreases to zero value as one approach the SSB. This means that the gravitational wall strength increases initially up to
. These profiles are physically quite consistent to each others.
Figures 2(a) and (b) depict the plots of the GES-associated normalized electrostatic potential, its gradient and associated inhomogeneity scale length. The scale length profile is non-monotonous in nature and exhibits a sudden decrease in the vicinity of the initial position and then increases rapidly. There is a minimum normalized scale length of the electrostatic potential variations of the order of 0.1. The basic features of Figure 2 near the initial location can obviously be understood from Figure 2(b), which is the enlarged view of Figure 2(a). This is interestingly noted from Figure 2(b) that 
at 
beyond which 
increases at a faster rate (more negative value). This is found from Figures 1 and 2 that 
and 
intersect approximately within the region defined by
. This means that 
and 
behave dynamically as exponentially varying functions of 
as 
and 
in this region particularly, respectively. This, in turn, helps the bounded solar plasma to leak through the self-gravitational potential barrier without overflowing through it.
When the self-gravitational potential barrier height increases, the initial bulk flow of the SIP is drastically reduced to the minimum possible values at the SSB as shown in Figures 3(a) and (b). Figures 3(a) and (b) depict the plots of the ion Mach number, gradient and scale length (scaled down by division with
). These plots exhibit monotonous behaviors having two scales of faster and slower regions of Mach number variations, respectively. Similarly, Figures 4(a) and (b) portray the plots
Table 1. Initial and boundary GES values.
Figure 1. Variation of the normalized values of solar selfgravity 
(solid blue line), self-gravity gradient 
(red dashed line) and self-gravity scale length
(black dotted line) associated with the SIP flow dynamics with normalized position 
from the heliocentre
. The initial values
, 
and 
are kept fixed. The other fixed initial values by nonlinear stability analyses are 
and
.
of solar plasma current density, its divergence and scale length. The gradient profile clearly shows the non-divergent behavior of the electron dominated current flowing through the gravitational wall of the SIP. From conventional viewpoint, the direction of current flow is towards the helio-centre. The magnitude of the current decreases as one moves from heliocentre to the SSB. The enlarged description of Figure 4(a) is given in Figure 4(b) showing the electrodynamics of the solar electric current near the heliocentre. This is found from Figure 4(b) that 
at both 
and
. But at the position
, it is seen that
, approximately.
Figure 5 exhibits the plots of the normalized solar plasma (population) density, its gradient and scale length. Figure 6 depicts the numerical plot of the second order differential derivative of the solar self-gravity and exhibits the validity of the condition for sufficiency of the maximization of the solar self-gravity at the SSB. Figure 7 exhibits the profiles of the normalized time scales for electron oscillations (scaled up by division with
), SIP ion oscillation (scaled up by division with
) and Jeans collapse. It is clear to note that the Jeans time scale is quite larger by several orders of magnitude relative to each of the rest. Figure 8 depicts the plots of the