AN Alternative Two-Fluid Formulation Of a Partially Ionized Plasma
IIOP
PublishingJournal Title Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX AN Alternative Two-Fluid Formulation Of a Partially Ionized Plasma
V. Krishan
Indian Institute of Astrophysics, Bangalore-560034, India E-mail: [email protected] Received xxxxxx Accepted for publication xxxxxx Published xxxxxx
Abstract
In a recent paper (Krishan 2021), a two - fluid description of a partially ionized plasma was present-ed in which electron fluid and the neutral fluid were combined appropriately into one fluid, christened as ENe fluid, and treat the ions as the second fluid. Some of the electrostatic modes of this two-fluid system were studied. Here, I discuss another possibility in which the ion fluid and the neutral fluid are combined into one fluid, christened as INe fluid, and treat the electrons as the second fluid. There can be a huge variation between the rela-tive masses of the neutrals and the ions. Thus both have to be treated as the inertia carry-ing species. After establishing the framework for the INe-electron fluids, some of the characteristic wave modes of this novel plasma are investigated.
Keywords: partially ionized plasma, waves, hydromagnetic fluid xxxx-xxxx/xx/xxxxxx © xxxx IOP Publishing Ltd . Introduction A multi-species particle system is hard to crack. Often, simplifications are sought which can reduce the effective degrees of freedom of the system. Statistical averaging is a powerful technique by which a collec-tion of a large number of particles can be transformed into a single fluid with its characteristic properties of mass density, flow velocity, pressure and viscosity. It is like getting water from water molecules. The simplest plasma is a fully ionized plasma consisting of electrons and ions. This plasma can be described by a single conducting fluid by appropriately combin-ing the dynamics of the electrons and the ions Alfve`n and Falthammer (1957). The magnetohydrodynamic (MHD) fluid de-scription so obtained has served well in accounting for a variety of phenomena in the area of plasma physics. The simplest partially ionized plasma consists of three species of particles viz. electrons, ions and the neutrals. Such a system can be modeled in diverse ways. One can describe it as a three-fluid system consisting of an electron fluid, an ion fluid and a neutral fluid. The three fluids can again be appropriately combined into a single conducting fluid in the manner of the MHD ( Krishan, V. 2014 and references therein). This system has been investigated variously, e.g. Krishan and Gangadhara, (2008); Hiryaki et al. (2010); Krishan and Varghese (2009), Ganadhara et al (2014). , Paradhkar et al (2019). If one desires more information about the system, the two-fluid description can be ac- cessed in three ways 1) one fluid by com-bining the electron and the ion fluids to ob-tain the MHD conducting fluid and the second the neutral fluid, 2) one fluid by combining the electron and the neutral flu-ids and the second the ion fluid and 3) one fluid by combining the ion and the neutral fluids and the second the electron fluid. While there is some work on the the first possibility, the second possibility, to the best of my knowledge, have been explored only recently (Krishan, 2021). In this paper I explore the third possibility. Here, in sec-tion 2, the dynamics of the ion and the neutral fluids will be combined to generate a single fluid which is christened as the INe fluid. In section 3, the dynamics of the electron fluid is described. The Poisson equation is established in section 4.The study of some of the normal modes in the two, INe and electron fluids is presented in sections 5-7. The paper is concluded in sec-tion 8.
The INe fluid is generated by combin-ing the ion and the neutral fluids ap-propriately in the manner of the MHD. We derive the mass, momentum , the electric charge and the energy conser-vation laws for the INe fluid. We begin with the mass conservation law + .( ) = 0 (1) of the ion fluid of mass density and velocity ∂ ρ i ∂ t ∇ ρ i V i ρ i V i . ournal XX (XXXX) XXXXXXAuthor et al The mass conservation law of the neutral fluid of mass density and velocity is + .( ) = 0 (2) Adding Eqs. (1) and (2) begets the mass conservation law of the INe fluid + .( ) = 0 (3) where is the mass density and is the center of mass velocity of the INe fluid. The momentum conservation law for the neutral fluid is = - - (4) where
The momentum conservation law for the ion fluid is = - (5) where
On defining the relative velocity to be = (6) we find = , = (7)
We can now substitute for and
In Eqs. (4) and (5) from Eq.(7) and add them up. Neglecting the nonlinear convective derivative terms as is done in the formulation of MHD, we find ln( ) = - (8) where = + , = (9)
Eq (8) describes the momentum con-servation law of the new INe fluid with number density , mass density , ve-locity , pressure and an effective charge density = ) . The INe fluid is thus seen to consists of parti-cles of mass m= , electric charge = ρ n V n ∂ ρ n ∂ t ∇ ρ n V n ∂ ρ ∂ t ∇ ρ V ρ = ρ i + ρ n V = ρ i V i + ρ n V n ρρ n D n − ∇ P n ρ n ν ne ( V n − V e ) ρ n ν ni ( V n − V i ) D n = [ ∂ V n ∂ t + ( V n . ∇ ) V n ] ρ i D i en i [ E + V i × Bc ] − ∇ P i ρ i ν ie ( V i − V e ) − ρ i ν in ( V i − V n ) D i = [ ∂ V i ∂ t + ( V i . ∇ ) V i ] u u ( V i − V n ) V i V + ρ n ρ uV n V − ρ i ρ u V i V n ρ ∂ V ∂ t + ρ i u ∂∂ t ρρ i qn E + ( V + ρ n ρ u ) × Bc − ∇ P −∇ ρ n ( ρ i ρ ) u ( ν ie ν ne ) P P i P n qn en i n ρ V P ( q n ) ( en i ρ / n q ournal XX (XXXX) XXXXXXAuthor et al /n ). The additional terms de-pending upon the ion and electron flu-id characteristics will be considered as the nonideal effects. The difference of Eqs.(4) and (5) gives the evolution of the relative velocity as - - + (10) in the limit We should also address the question of electric charge conservation. The ion fluid charge conservation is described as + .( ) = 0 (11) which can be written in terms of the charge density of the INe fluid as + .[ )] = 0 (12)
On using mass conservation, Eq.(3) and substituting for the velocity dif-ference ) from Eq. (7), we find +V. ] + .[ =0 (13) Or + .[ = 0 (14)
This along with Eq.(10) for deter-mines the charge density variation of the INe fluid. One can see that in the equilibrium with no relative flow be-tween the neutral and the ion fluids i.e. for = 0, = ). The energy conservation for the INe fluid is described by taking the equa-tion of state to be (15) where and are respectively the pressure, the density and the square of the sound speed of a fluid. Thus the mass, momentum, electric charge density and the energy conser-vation of the INe fluid are respectively contained in Eqs.(3), (8), (14) and (15).
The mass conservation for the electron fluid is + .( ) = 0 (16) The momentum conservation of the electron fluid is : ( en i ∂ u ∂ t = em i E + ( V + ρ n ρ u ) × Bc − ∇ P i ρ i ∇ P n ρ n u ( ν in ν ni ) ρ e → ∂ ( en i ) ∂ t ∇ en i V i ∂ ( qn ) ∂ t ∇ qn ( V + V i − V ( V i − V n [ ∂ q ∂ t ∇ q ∇ n q ρ n ρ u ] n d qdt ∇ n q (1 − ρ i ρ ) u ] uu q q = ( en i / n ∇ P = c ∇ ρ P , ρ c ∂ ρ e ∂ t ∇ ρ e V e ournal XX (XXXX) XXXXXXAuthor et al = - - (17) where and are the electron- neutral and the electron-ion collision frequencies. We can substitute for the neutral and the ion velocities in terms of the INe fluid velocity and the rela-tive velocity u to get = - - - (18) The charge conservation for the elec-tron fluid is + .( ) = 0 (19)
The equation of state for the electron fluid is = (20) where is the square of the sound speed in the electron fluid. Thus Thus the mass, momentum, elec-tric charge density and the energy conservation of the electron fluid are respectively contained in Eqs.(16), (18), (19) and (20).
The electric field is described through the Poisson equation as: . = 4 π ( - (21 ) The electromagnetic waves in magne-tized INe and electron fluids can be studied by using the wave equation ( . ) = + (22) where J , the current density, is de-fined as = e( - (23) which in terms of the INe fluid veloci-ties becomes = e[ (V + u) - (24) We now have the complete mathe-matical formulation of the dynamics of the two fluids, the INe and the elec-tron fluids along with the Poisson equation and the electromagnetic wave equation to study the existence of electrostatic and electromagnetic waves. 5 . The Linear Electrostatic Waves
The equilibria of the INe and the elec-tron fluids are taken to be uniform and static. After substituting the one di-mensional plane wave variation, exp( - , of all the perturbed quantities in the linearized form of Eqs. (3), (9), (10), in the absence of a magnetic field, one gets ρ e [ ∂ V e ∂ t + ( V e . ∇ ) V e ] − en e [ E + V e × Bc ] − ∇ P e ρ e ν en ( V e − V n ) ρ e ν ei ( V e − V i ) ν en ν ei V [ ∂ V e ∂ t + ( V e . ∇ ) V e ] en e ρ e [ E + V e × Bc ] − ∇ P e ν en ( V e − V + ρ i ρ u ) ν ei ( V e − V − ρ n ρ u ) ∂ ( en e ) ∂ t ∇ en e V e ∇ P e c e ∇ ρ e c e ∇ E en i en e ) ∇ E −∇ ∇ E π c ∂ J ∂ t c ∂ E ∂ t J n i V i n e V e ) J n i ρ n ρ n e V e ] ikx i ω t ) ournal XX (XXXX) XXXXXXAuthor et al + = 0 (3a) - + (8a) = (15a) ( + + - (10a) + = 0 (16a) ( + (18a) = (20a) = 4 π - (21a) where = - = - ) , for (25) and b = Substituting for the density perturba-tions from the continuity equations, the Poisson Equation becomes = ( ) - (26)
The dispersion relation of the electro-static waves can be determined using the well known procedure of substitut-ing for all the perturbed quantities in terms of one of them. This gives 1. for all pressure perturbations and collision frequencies = 0, = , = , = (27) and the dispersion relation is = + (28)
Thus the role of the neutrals dis-appears in the absence of colli-sions and pressure perturbations. − ωρ k ρ V ωρ V = iqn o E ia u kP P c ρ ω i ν i ) u = ieE m i k P i ρ i k P n ρ n − ωρ e k ρ e V e ω i ν e ) V e = − ieE m e + i ν e V + ibu + k P e ρ e P e c e ρ e ikE e ( n i n e ) a ρ n ( ρ i ρ )( ν ie ν ne )( ρ e ρ )( ν ei ρ n ν en ρ i → ρ e → a ρ e iE π en i ω V + ( ρ n ρ u V e ) V ien i E ωρ u ieE ω m i V e ieE ω m e ω ω ep ω ip ournal XX (XXXX) XXXXXXAuthor et al
2. retaining pressure perturbations in the INe fluid and in the electron fluid with all collision frequencies = 0, one gets = (1- , = , = (1- (29)
The dispersion relation is now found to be
In the limit and much less than , one recovers the dispersion relation , Eq. (28). In the opposite lim-it when and are both much larger than , one gets = ) [1+ ) + (31) which can also be written in terms of the plasma frequency of the INe fluid as = ) [1+ + (32) where = (33) is the plasma frequency of the INe fluid consisting of particles of mass equilibrium density and equilibrium electric charge ). The dispersion relation in Eq.(32) may be christened as the INe-acoustic mode since it is remi-niscent of the ion-acoustic wave dispersion in a fully ionized plasma. One may note that the INe-acoustic mode vanishes in a fully ionized plasma. 3. Retaining all collision frequen-cies and neglecting all pressure perturbations, from Eqs. (8a) and (10a), one gets, (34) = (35) P P e V ien i E ωρ k c ω ) − u ieE ω m i V e ieE ω m e k c e ω ) − ( ρ i ρ ω ip ω − k c ( ρ n ρ ω ip ω ω ep ω − k c e k c k c e ω k c k c e ω ω ( ρ n ρ ω ip ( ρ i ρ ω ip k c ω ep k c e ] − ω ( ρ n ρ i ω p ω p k c ω ep k c e ] − ω p π n ( n i en ) m m , n ( n i en V = i en io E ωρ u i eE m i ( ω + i ν i ) ournal XX (XXXX) XXXXXXAuthor et al From Eq. (18a), with the neglect of all pressure perturbations, the elec-tron fluid momentum equation gives ( + (36)
After substituting for and , one gets = [ - ] (37)
After substituting for the velocities in the Poisson, Eq.(21) , the dispersion relation of the electrostatic waves can be determined. In the low frequency limit such that << and we find = + + ] (38) which is a purely damped mode for > 0. Now from Eq.(22) can be shown to be - ) = -( + ) (39) (39) which could acquire a negative value under special conditions. Nevertheless the other two terms in Eq.(38) are larger than the b term and thus the mode remains damped. One could also consider the case for and the dispersion rela-tion turns out to be + - = 0 (40) where - has been used .
6. Electrostatic Waves in Magnetized INe and Electron Fluids
We determine the dispersion relation of electrostatic waves in the presence of a uniform magnetic field , say in the z direction. The electrostatic waves have their electric field and the wave vector k parallel to each other and perpendicular to the uni-form magnetic field. The linearized form of the momentum equation (8) of the INe fluid with the plane wave variation for the first or-der quantities reads = ( + )+ (8b) ω i ν e ) V e = − ieE m e + i ν e V + ibu V u V e − ieE m e ωρ − i ν e ρ e ωρ ( ω + i ν e ) i bm e m i ( ω + i ν e )( ω + i ν i ) ω ν e ν i ω − i [ ω ip ν i ω ep ν e b ω ip ν e ν i bb b = ( 1 ρ )( ν ei ρ n ν en ρ i ν ei ν ei ν en ρ i ρ ν e > > ω > > ν i ω i ω ep ων e ω p ρ ρ i b ≈ ν e ρ i ρ B E x V x i ω ec ρ e ωρ V y ρ n ρ u y ien i E x ωρ ournal XX (XXXX) XXXXXXAuthor et al = - ( + = 0 which furnish (41) where T= (1- (1- S = (1- (1- (42) = , =
The corresponding electron fluid equa-tion gives = - (1- (18b)
Equation (15) for the relative velocity between ions and neutrals gives = (43) where
R= (44)
Substitution in the Poisson equation ( ) - (45) begets the dispersion relation of the electrostatic waves in the presence of the uniform magnetic field
1- (1- [ ] (1- = 0 (46)
For =0 so that =1 and stationary ions, one can easily recover the dis-persion relation + (47) of the upper hybrid waves in an elec-tron-ion plasma. For = 0 and in the plasma approximation, amounting to the neglect of first term one recovers the dispersion relation (48) of the lower hybrid waves in an elec-tron-ion plasma. In order to determine V y i ω ec ρ e ωρ V x ρ n ρ u x ) V z V x = ieE x ω m T (1 − S ) n i n α i ω ic ω ) − α n ω ic ω ) − α i ω ic ω α i ω ic ω α i ω ic ω ) − α n ω ic ω ) − α n ρ n ρ α i ρ i ρ V ex ieE x ω m e ω ec ω ) − u x ieE x ω m − ρ n ρ R )( RT (1 − S ) + mm i ) ω ic ω π en i ω V x ` + ( ρ n ρ u x V e x ) ω ip ω α n ω ic ω ) − α n ω ic ω + α i (1 − α n ω ic ω )(1 − α i ω ic ω ) − α n α i ω ic ω − ω ep ω ω ec ω ) − α n α i ω = ω ep ω ec α n n e = n i , ω = ω ic ω ec ournal XX (XXXX) XXXXXXAuthor et al the effect of the degree of ionization on these modes, the dispersion rela-tion, Eq.(46) must be plotted as a function of or . the inclusion of collisional terms contribute to the damping of the wave modes.
7. Electromagnetic Waves in Magne-tized INe and Electron Fluids
The electromagnetic waves in magne-tized INe and Electron Fluids can be studied by using equations (22 and (24). The linearization gives = e [ + - (49) where in the equilibrium, = has been used. We can study the electro-magnetic waves propagating parallel and perpendicular to the ambient magnetic field. We first consider the perpendicular propagation such that , = and the electric field , , is par-allel to This characterizes the or-dinary wave. The linearized wave equation becomes - ) -4 [ + - (50) On examining the z components of the velocities, we find that the predomi-nant contribution is from the electron fluid. The other two terms contribute terms of the order or smaller than the ion plasma frequency. Substituting for fetches (1+ + (51)
It shows that the ordinary wave under-goes damping predominantly due to electron-neutral collisions.
This wave is characterized by = = ( , which is perpendicular to . The wave equation takes the form -4 [ + - - ) -4 [ + - (52) The extraordinary wave is also a high frequency wave. Therefore the elec-tron fluid velocity is the main contrib-utor to the current density. The INe fluid contributes terms of the order or less than the ion plasma frequency. One can determine the dispersion re-lation including the electron- neutral collision frequency. 8 . Conclusion
The new fluid, here, christened as the INe fluid is obtained by combining the ion and the neutral fluids appropriate-ly. A partially ionized plasma thus can α n α i J n i V α n u V e ] n i n e B = B z k k x E = E z B . ( ω k c E z = π ei ω n e V z α n u z V e z ] V e z ω = ω ep i ν e ω ) − k c k k x , E E x E y ) B ω E x = π ei ω n e V x α n u x V e x ]( ω k c E y = π ei ω n e V y α n u y V e y ] ournal XX (XXXX) XXXXXXAuthor et al be described by two fluids the INe flu-id and the electron fluid. Some of the normal modes in this system have been investigated. The role of the de-gree of ionization is obvious. A more detailed parametric study of the vari-ous modes is highly desirable and would bring out the novelties of the system. Acknowledgements
Discussions with Professor Abhijit Sen are grate-fully acknowledged.
References [1]
Krishan V,
A New Electrostatic mode in Two-Fluid (ENe-Ion) Formalism of a Partially Ionized Plasma,
DOI: [2]
Alfven, H. and Falthammer, C., 1963, Cosmi-cal Electrodynamics [3]
Krishan V., 2016, Physics of partially ionized plasmas. Cambridge University Press [4] Krishan V., Gangadhara R., 2008, Monthly Notices of the Royal Astronomical Society, 385, 849 [5] Gangadhara R., Krishan V., Bhowmick A.,Chitre S., 2014, The
Astrophysical Journal, 788, 135 [6] Hiraki Y., Krishan V., Masuda S., 2010, The Astrophysical Jour- nal, 720, 1311 [7]
B. S. Paradkar, ⋆ S. M. Chitre and V. Krishan , 2019, MNRAS,, 2019, MNRAS,