Propagation of shock structures in a high density plasma
A theoretical investigation has been made to study the cylindrical
and spherical electron-acoustic shock waves (EASWs) in an
unmagnetized, collisionless degenerate quantum plasma system
containing two distinct groups of electrons (one inertial
non-relativistic cold electrons and other inertialess
ultra-relativistic hot electrons) and positively charged static
ions. By employing well known reductive perturbation method the
modified Burgers (mB) equation has been derived. It is seen that
only rarefactive shock waves can propagate in such a quantum
plasma system. The effects of degenerate plasma pressure and
number density of hot and cold electron fluids, nonplanar
geometry, and positively charged static ions are responsible to
modify the fundamental properties of EASWs. It is also observed
that the properties of planar mB shocks are quite different from
those of nonplanar mB shocks. The findings of the present
investigation should be useful in understanding the nonlinear
phenomena associated with nonplanar EAWs in both space and laboratory plasmas.
Keywords: Electron-acoustic waves, modified Burgers equation, Shock waves, Degenerate pressure, Relativity, Compact objects
The apparent interest for quantum plasmas has been retained their interest due to their existence both in laboratory and space plasma environments Haas2003 ; Shukla2008 ; Eliasson2006 ; Khan2007 ; Misra2007 ; Sah2009b ; Hossain2013 ; a1 ; a2 ; a3 ; a4 . Generally, electron-acoustic (EA) waves (EAWs) occur in a plasma environments (white dwarfs, neutron star, etc.) consisting of two distinct temperature electrons (as hot and cold electrons) Watanabe1977 ; Yu1983 ; Tokar1984 ; Mace1990 ; Kourakis2004 ; Mamun2004 . EAWs are nothing but high-frequency electrostatic mode, for which inertia is provided by the cold electron motion, while the restoring force comes from the hot electron thermal pressure. The positively charged ions may be safely assumed to be stationary, simply maintaining the quasineutrality condition of the plasma system. In such acoustic mode, the frequency lies in the range between the plasma frequency of the cold and hot electron fluids. Watanabe and Taniuti Watanabe1977 have first shown the existence of the electron-acoustic (EA) mode in a plasma of two-temperature (cold and hot) electrons. Some past decades, EA waves has received a great deal of renewed interest not only because of the two distinct group of electron plasma is very common in laboratory experiments Derfler1969 ; Henry1972 ; Kadomtsev1971 ; Armstrong1979 ; Sheridan1991 and in space Dubouloz1993 ; Pottelette1999 ; Singh2001 ; Ang2007 ; Barnes2003 ; Fedele2002a ; Fedele2002b but also because of the potential importance of the EA waves in interpreting electrostatic component of the broadband electrostatic noise (BEN) observed in the cusp of the terrestrial magnetosphere Tokar1984 , in the geomagnetic tail Schriver1989 , in white dwarfs and neutron stars Shapiro1983 , etc.
Now a days, researchers of plasma community gives great attention to study the nonlinear behavior of astrophysical compact objects e.g. white dwarfs, neutron stars, etc. The plasma particle number density for such compact objects is so high (in white dwarfs it can be of the order of , even more) Hossen2014 ; Hossen2014c ; Hossen2014d ; Hossen2014e that the electron Fermi energy is comparable to the electron mass energy and the electron speed is comparable to the speed of light in a vacuum. Chandrasekhar Chandrasekhar1931a ; Chandrasekhar1931b presented a general expression for the relativistic ion and electron pressures in his classical papers. The pressure for electron fluid can be given by the following equation
where is the electron number density and
for the non-relativistic limit (where , and is the Planck constant divided by ). And
A large number number of works on relativistic degenerate quantum plasma have been accomplished considering different acoustic waves in the recent years Shah2015a ; Shah2015b ; Shah2015c ; Shah2015d ; Hossen2014i ; Hossen2015c ; Hossen2014k ; Ema2015 ; H2014a ; H2014b ; H2014c ; H2014d ; H2014e ; H2015a ; H2016a ; H2016b . Han et al. Han2013 investigate the existence of electron-acoustic shock waves and their interactions in a non-Maxwellian plasma with q-nonextensive distributed electrons . Later on, Han et al. Han2014 theoretically investigated the nonlinear electron-acoustic solitary and shock waves in a dissipative, nonplanar space plasma with superthermal hot electrons. Sahu and Tribeche Sahu2013 considered electron acoustic shock waves (EASWs) in an unmagnetized plasma whose constituents are cold electrons, immobile ions and Boltzmann distributed hot electrons and studied the effects of several parameters and ion kinematic viscosity on the basic features of EA shock waves. By considering quantum plasma El-Labany et al. EL-Labany2013 investigated the effects of Bohm potential on the head on collision between two quantum electron-acoustic solitary waves using the extended Poincar -Lighthill-Kuo method. Mahmood and Masood Mahmood2008 illustrated that an increase in quantum diffraction parameter broadens the nonlinear structure. Recently, Sah Sah2009 demonstrated that the width, the amplitude, and the velocity of electron-acoustic double layers, in three component dense quantum plasmas consisting of stationary background ions and two electron populations: one cold and the other hot , are significantly affected by the ratio of unperturbed cold to hot electron densities. Again, the effect of static ions is very common in plasma physics literature Pakzad2011 ; Tribeche2010 ; Amour2012 . To the best of our knowledge, none of the authors did consider the combine effects of nonplanar geometry, effects of relativistic limits (i.e., both non-relativistic and ultra-relativistic) and degenerate plasma pressure which can significantly modify the propagation of solitary and shock waves.
Ii Governing Equations
We consider a cylindrical and spherical EA waves in an unmagnetized, collisionless plasma, which is composed of non-relativistic inertial cold electrons, both non-relativistic and ultra-relativistic degenerate hot electron fluids, and static positive ions. Thus at equilibrium, we have , where is the equilibrium number density of the species ( for cold electrons, hot electrons, positive ions, respectively). The nonlinear dynamics of the electrostatic waves propagating in such a degenerate quantum plasma system is governed by the following normalized equations
where for one dimensional planar geometry, (2) for nonplanar cylindrical (spherical) geometry, (s=c, h, i) is the the plasma species number density normalized by its equilibrium value , is the plasma fluid speed normalized by with () being the hot electron (cold electron) rest mass, is the speed of light in vacuum, is the electrostatic wave potential normalized by . Here is the ratio of ion-to-cold electron number density. The time variable () is normalized by , and the space variable () is normalized by . The coefficient of viscosity is a normalized quantity given by . We have defined and .
Iii Derivation of modified Burgers Equation
We derive a dynamical modified Burgers (mB) equation for the nonlinear propagation of the EA waves by using equations (5)-(9). To do so, we employ a reductive perturbation technique to examine electrostatic perturbations propagating in the relativistic degenerate dense plasma system due to the effect of dissipation, we first introduce the stretched coordinates Maxon1974 :
where is the wave phase speed ( with being the angular frequency and being the wave number), and is a smallness parameter measuring the weakness of the dissipation (). We expand the parameters , , , , and in power series of as:
Now, expressing equations (5)-(9) (using equation (10), in terms of and , and substituting equations (11)-(15), one can easily develop different sets of equations in various powers of . To the lowest order in , we have: , , , , where and . The relation represents the dispersion relation as well as the phase speed for the EA type electrostatic waves in the degenerate quantum plasma under consideration.
To the next higher order in , we obtain a sets of equations
Iv Discussion and Results
In this section, our first intention to numerically analyze the Burgers equation. However, for clear understanding, we first briefly discuss about the stationary shock wave solution for equation (20) with , though the solution is similar for both IA and EA waves (excluding the values of A and B). We should note that for a large value of , the term is negligible. So, in our numerical analysis, we start with a large value of (viz. ), and at this large (negative) value of , we choose the stationary shock wave solution of equation (23) [without the term ] as our initial pulse. The stationary shock wave solution of this standard Burgers equation is obtained by considering a frame (moving with speed which is the ion fluid speed at equilibrium) and the solution is H2016m ; H2016n ; H2016o
where and .
Now, we investigate the dynamical properties of EA shock waves in terms of the intrinsic parameters of our model, namely the ion-to-cold electron density ratio and the (cold electron) kinematic viscosity . It is important to not that the dissipation term only depends on the electron kinematic viscosity . The shock profile is nothing but sudden increase or decrease of the permanent profile of the waves H2016p ; H2016q ; H2016r ; H2016s . It is observed that EA waves are significantly modified when cold electron being non-relativistic degenerate () and hot electron being ultra-relativistic degenerate () than both cold and hot electron being non-relativistic degenerate (). It is important to mention that modified Burgers equation derived here is valid only for the limits , and . It is also important to note that for () rarefactive shock waves are found but no compressive shock waves exist at . We have considered for our numerical analysis of EASWs for the plasma system under investigation here.
Figure 1 shows the variation of phase speed () with ion to cold electron number density ratio . It is found that the phase speed decreases with the increasing values of . It is expected as the phase speed (derived from this considered plasma) is higher for lower values of (see the expression of ). The variation of the rarefactive amplitude of shock structures for both planar and nonplanar geometry is shown in Fig. 2 where cold electron and hot electron being nonrelativistic degenerate. Figure 3 shows the variation of the rarefactive amplitude of shock structures for both planar and nonplanar geometry where cold electron being nonrelativistic degenerate and hot electron being ultrarelativistic degenerate. The cylindrical and spherical variation of the amplitude of EASWs for both non-relativistic and ultra-relativistic limits is shown in Figs. 4-7. Finally, the results that we have found in this investigation can be summarized as follows:
The cylindrical and spherical plasma system under consideration supports only rarefactive shock waves with negative potential, but no compressive shock waves exist.
The fundamental properties of EASWs are found to be significantly modified by the relativistic parameters, nonplanar geometry and plasma particle number densities.
It is observed that the phase speed (Vp) of these EA shocks inversely proportional to the square root of ion to cold electron number densities ratio .
It is also found that the phase speed () of EA waves decreases with the increasing values of (see Fig. 1).
It is observed that the amplitude of the shock is maximum for the spherical geometry, intermediate for cylindrical geometry, while it is minimum for the planar geometry (see Figs 2-3).
The amplitude of shocks proportional to the fluid speed but inversely proportional to the constant A.
In conclusion, our simplified theoretical model represents a small yet steady step towards the rigorous understanding of the behavior of cylindrical and spherical EA shocks in degenerate plasma environments, which appear to be of fundamental importance in a wide range of astrophysical Dubouloz1993 ; Pottelette1999 ; Singh2001 ; Ang2007 ; Barnes2003 ; Fedele2002a ; Fedele2002b and laboratory scenarios Derfler1969 ; Henry1972 ; Kadomtsev1971 ; Armstrong1979 ; Sheridan1991 .
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