The nonlinear normal mode dynamics is likely to be modified due to nonlinear, dissipative, and dispersive mechanisms in solar plasma system. Here we apply a plasma-based gravitoelectrostatic sheath (GES) model for the steady-state description of the nonlinear normal mode behavior of the gravitoacoustic wave in field-free quasineutral solar plasma. The plasma-boundary wall interaction process is considered in global hydrodynamical homogeneous equilibrium under spherical geometry approximation idealistically. Accordingly, a unique form of KdV-Burger (KdV-B) equation in the lowest-order perturbed GES potential is methodologically obtained by standard perturbation technique. This equation is both analytically and numerically found to yield the GES nonlinear eigenmodes in the form of shock-like structures. The shock amplitudes are determined (~0.01 V) at the solar surface and beyond at 1 AU as well. Analytical and numerical calculations are in good agreement. The obtained results are compared with those of others. Possible results, discussions, and main conclusions relevant to astrophysical context are presented.

The Sun, like stars and ambient atmospheres, has exploratively been an interesting area of study for different authors by applying different physical model approaches and observational techniques for years [

We are here going to propose a nonlinear stability analysis on the Sun on the basis of the plasma-based gravito-electrostatic sheath (GES) model [

The main motivation of this paper is to examine whether the solar plasma system, as a natural plasma laboratory, can support any nonlinear characteristic eigenmode through the GES model with plasma-boundary interaction taken into account. Spacecraft probes and Earth-orbiting satellites have also technically detected many wide-scale nonlinear mode features [

A distinct set of nonautonomous self-consistently coupled nonlinear dynamical eigenvalue equations in the defined astrophysical scales of space and time configuration is accordingly developed. In view of that, a unique form of KdV-Burger (KdV-B) equation [

A very simplified ideal solar plasma fluid model is adopted to study the GES model stability under a global hydrodynamic type of homogeneous equilibrium configuration. Gravitationally bounded quasineutral field-free plasma by a spherically symmetric surface boundary of nonrigid and nonphysical nature is considered. An estimated typical value ~10^{−20} of the ratio of the solar plasma Debye length and Jeans length of the total solar mass justifies the quasineutral behavior of the solar plasma on both the bounded and unbounded scales. A bulk nonisothermal uniform flow of solar plasma is assumed to preexist. For minimalism, we consider spherical symmetry of the self-gravitationally confined SIP mass distribution, because this helps to reduce the three-dimensional problem of describing the GES into a simplified one-dimensional problem in the radial direction since curvature effects are ignorable for small scale size of the fluctuations. Thus only a single radial degree of freedom is sufficient for describing the three-dimensional SIP and, hence, the SWP in radial symmetry approximation. This is to elucidate that our plasma-based theory of the GES stability is quite simplified in the sense that it does not include any complicacy like the magnetic forces, nonlinear thermal forces and the role of interplanetary medium or any other difficulties like collisional, viscous processes, and so forth.

Applying the spherical capacitor charging model [^{−1}, thereby showing that

The solar plasma is assumed to consist of a single component of Hydrogen ions and electrons. The thermal electrons are assumed to obey Maxwellian velocity distribution in an idealized situation. In reality, of course, deviations exist, and hence different kinds of exospheric models have already been proposed with special velocity distribution functions kinetically [

The basic normalized (with all standard astrophysical quantities) autonomous set of nonlinear differential evolution equations with all the usual notations [

In order to get a quantitative flavor for a typical value of

Applying the usual standard methodology of reductive perturbation technique [

The nonlinearities in our might have directly come from the large-scale dynamics (in space and time) through the harmonic generation involving fluid plasma convection, advection, dissipation, and so forth. These nonlinearities may contribute to the localization of waves and fluctuations leading to the formation of different types of nonlinear coherent structures like solitons, shocks, vortices, and so forth which have both theoretical as well as experimental importance [

In order to study the GES stability of the present concern, the relevant solar physical variables

We are involved in the dynamical study of the lowest-order GES potential fluctuation associated with the SIP system. Equation (

Now coupling (

This is clear from (

Equation (

Equation (

Equation (

A theoretical model analysis is carried out to study the GES fluctuation in a simplified field-free quasineutral solar plasma model in quasihydrostatic type of homogeneous equilibrium configuration. A distinct set of nonautonomous self-consistently coupled nonlinear dynamical eigenvalue equations in a defined astrophysical space and time configuration are developed. Applying the standard methodology of reductive perturbation technique over the defined GES equilibrium [

Let us now estimate the physical values of the main shock characterization parameters represented by (

The reductive perturbation method is, however, not very popular as a mathematically rigorous perturbation method, even conditionally. It, nevertheless, is a convenient, approximate, and easy way to produce certain mathematically interesting 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 solutions will subsequently show that there are many other shock-like structures that do not satisfy the “required ordering”. In fact, although shocks are frequently found experimentally, so far a shock 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 future interest to derive analytical results with greater accuracy with newer mathematical techniques so as to get more detailed picture of the self-gravitational fluctuations like in the Sun.

Our theoretical GES model analysis shows that the solar plasma system supports shock formation governed by KdV-B (

Profile of the lowest order GES potential fluctuation

Again Figure

Profile of the lowest order GES potential fluctuation

Lastly, Figure

Profile of the lowest order GES potential fluctuation

This has been recognized years ago that the compressional plasma in the solar atmosphere is a perfect medium for magnetohydrodynamic (MHD) waves. Applying the MHD model analyses [

GES versus MHD stability analyses.

S. no. | Items | GES stability analysis | MHD stability analyses |
---|---|---|---|

1 | Model | Ideal hydrodynamic | MHD |

2 | Plasma-boundary wall interaction and sheath formation mechanism | Included | Neglected |

3 | Effect of charge separation | Considered | Not considered |

4 | Floating surface (at which no net electric current) | Involved | Not involved |

5 | Magnetic field | Not considered | Considered |

6 | Description | Two-scale (SIP and SWP) | One scale (SWP) |

7 | Sonic range | Subsonic (SIP) and supersonic (SWP) | Supersonic (SWP) |

8 | Self-gravity (SIP) and external gravity (SWP) | Considered | Not considered |

9 | Transonic transition (subsonic to supersonic) | Involved (through SIP and SSB interaction process and thus transformed into SWP) | Not involved |

10 | Analytical solution | Bounded (SIP) | Unbounded (SWP) |

11 | Thermal species | Maxwellian | Single fluid (MHD) |

12 | Surface description and specification | Yes (at | Not precisely, but the diffused surface is electrically uncharged and unbiased |

13 | Source of nonlinearity | Plasma fluidity | Large-scale dynamics |

14 | Source of dispersion | Deviation from quasineutrality and self-gravity | Geometrical effect (also, some part of physical effect) |

15 | Source of dissipation | Weak collisional effects | Viscosity and magnetic diffusion |

16 | Sun and SWP coupling | Considered | Not considered |

17 | Nature of solutions | Mainly shocklike structures in the lowest ordered perturbed GES potential | Soliton and shocklike structures in the lowest ordered perturbed density and velocity |

18 | Solar atmosphere | Not stratified | Stratified (into a number of heliocentric layers) |

19 | Adopted technique | Standard reductive perturbation technique (about the GES equilibrium) | Standard multiple scaling technique (about the MHD equilibrium) |

20 | Convection and circulation dynamics | Not treated (for idealized simplicity) | Treated |

21 | Leakage process | Taken into account | Not taken into account |

22 | Main application | Surface origin of the subsonic SWP and its transonic flow dynamics | Solar chromospheric and coronal heating |

We scientifically admit that the neglect of collisional dissipations and deviation from Maxwellian velocity distributions of the plasma particles is not quite realistic. But our GES stability analyses even under some simplified and idealized approximations may provide quite interesting results for the solar physics community. The main points based on our analyses are summarized as follows.

The GES fluctuations appear in the form of various nonlinear structures (of shock-family eigenmodes) governed by a new analytic form of KdV-Burger (KdV-B) equation. Here the terminology “new analytic form” refers to the appearance of the new type of characteristic coefficients in the dispersive and dissipative terms in it.

The influence and presence of such eigenmodes are also experienced asymptotically even in the SWP scale. The structural modification here is due to the background initial conditions under which being excited. Such blast wave structures arise mainly due to violent disturbances of self-gravitational type. Their front thickness may, however, be a consequence of the homogeneous balance between self-gravitating solar plasma nonlinear compressibility and dissipative mechanisms like viscosity, heat conduction, and so forth. Similar observations in the Sun have also been reported by MHD-community under Hall-MHD approximation [

Self-gravity of the SIP mass distribution is normally found to have a tendency to depress (due to dissipation and dispersion) the nonlinear structures in the interior (subsonic SIP flow) and steepens (due to nonlinearity) them in the exterior (supersonic SWP flow) in our two-scale GES stability analyses.

Last but not least, the

This analysis, moreover, is carried out in a homogeneous kind of field-free quasihydrostatic equilibrium configuration under quasineutral plasma approximation. However, even in spite of these limitations, it may perhaps be useful for further investigation of dynamical stability on a nonlinearly coupled system of the SIP and SWP as an interplayed flow dynamics of heliocentric origin in presence of all the possible realistic agencies [

The dynamical stability of the GES model, although simplified through idealistic approximations, is analyzed in both analytical and numerical forms with standard perturbation formalism. It provides an idea into the interconnection between the SIP (Sun-) stability in terms of the lowest-order GES fluctuation appearing as various nonlinear structures (shock like) and their asymptotic propagation in the SWP scale as an integrated model approach. This is conjectured that the fluctuations are jointly governed by a new form of KdV-Burger type of nonlinear evolution equation having some characteristic model coefficients. Both analytical and numerical solutions are in qualitative and quantitative agreement. The main conclusions of scientific interest drawn from our present contribution are summarized as follows.

Nonlinear fluctuations of the GES in the SIP scale are governed by a KdV-Burger (KdV-B) type of equation with characteristic coefficients dependent on the solar plasma GES model.

Different forms of nonlinear eigenmodes exist in the SIP scale in different situations. Their presence, pretriggered strongly due to self-gravity on the SIP scale origin, is also experienced at asymptotically large distances beyond the SSB. It goes in qualitative conformity with those reported with different methodologies [

The structures are contributed mainly due to gravitoelectrostatically coupled self-gravity fluctuation of the solar plasma inertial ions under an integrated interplay of diverse nonlinear (hydrodynamic origin) and dispersive (self-gravitational origin) effects in the solar plasma system in presence of some internal dissipation.

The SIP is found to be more unstable (more fluctuation gradient) than the SWP (less fluctuation gradient) asymptotically. This is because of the GES fluctuation in presence of strong self-gravity in the bounded SIP scale and weak external gravity in the unbounded SWP scale.

Our two-scale theory of the GES is found to give two-scale dynamical variation of the GES stability as a gravito-electrostatically coupled system of the SIP (subsonic flow) and the SWP (supersonic flow) through the interfacial SSB.

Finally and additionally, the modified GES mode kinetics as a self-gravitationally triggered instability in an intermixed state of the gaseous phase of plasma and solid phase of dust grain-like impurity ions (DGIIs), by using a dissipative multi-fluid colloidal or dusty plasma model with dust scale size distribution power law taken into account, may be another interesting investigation to study DGII-behavior in an SWP-like realistic situation on a global scale. This is because the interplay between gravitational and electrostatic forces in the dynamics of such grains is responsible for many interesting phenomena in the terrestrial and solar environment (like rings of Saturn and Jupiter, satellites’ spoke formation, etc.). It eventually may have some useful characteristic implications of acoustic spectroscopy ([

The valuable comments, specific remarks, and precise suggestions by an anonymous referee, to refine the prerevised original paper into the present postrevised improved form, are very gratefully acknowledged. Moreover, the financial support received from the University Grants Commission of New Delhi (India), through the research project with Grant F. no. 34-503/2008 (SR), is also thankfully recognized for carrying out this work.