Exact solution of two fluid plasma equations for the creation of jet-like flows and seed magnetic fields in cylindrical geometry
EExact solution of two fluid plasma equations for the creation ofjet-like flows and seed magnetic fields in cylindrical geometry
H. Saleem , Department of Space Science, Institute of SpaceTechnology (IST), Islamabad, Pakistan Theoretical Research Institute Pakistan Academy of Sciences (TRIPAS),Islamabad, PakistanEmail: [email protected] (Dated: May 14, 2019)
Abstract
An exact solution of two fluid ideal classical plasma equations is presented which shows that thejet-like outflow and magnetic field are generated simultaneously by the density and temperaturegradients of both electrons and ions. Particular profiles of density function ψ = ln ¯ n (where ¯ n isnormalized by some constant density N ) and temperatures T j (for j = e, i ) are chosen which reducethe set of nonlinear partial differential equations to two simple linear equations generating longi-tudinally uniform axial outflow and magnetic field in cylindrical geometry in several astrophysicalobjects. This mechanism also seems to be operative for producing short scale plasma jets in thesolar atmosphere in the form of spicules and flares. The presented solution requires particular pro-files of density and temperatures, but it is a natural solution of the two fluid ideal classical plasmaequations. Similar jet-like outflows can be generated by the density and temperature gradients inneutral fluids as well.Key words: astrophysical jets, two fluid plasma outflows, neutral fluid jets, magnetic field gen-eration, baro-clinic vectors a r X i v : . [ phy s i c s . p l a s m - ph ] M a y . 1. INTRODUCTION First observation of astrophysical jets was made long ago [1] and later observations showedthat jets are collimated outflows of gases and plasmas from a variety of sources ranging fromyoung stellar objects (YSOs) to active galactic nuclei (AGNs) [2–5]. Jets emerging fromYSOs have speeds v smaller than speed of light c ( v ≤ − c ) and hence can be treatedusing classical plasma models while the jets emerging from AGNs have speeds approachingspeed of light and are highly relativistic . AGN jets have sizes of the order of 10 pc and YSOjets have sizes of the order of (10 − − pc . In between these two extremes are the outflowsof gases, electron ion plasmas and electron positron plasmas from normal stars, neutronstars, massive X-ray binary systems, galactic stellar mass black holes (or microquasars).Magnetic fields are considered to be an important constituent of astrophysical jets emerg-ing from accretion disks, compact stars and black holes [6–8]. Jets of active galactic nuclei(AGN) are extended up to kilo or mega parsecs with helical magnetic fields [9–11]. Themagnetic fields of these objects are generated by large scale currents and are amplified bydynamo effects. Gamma ray bursts are also associated with relativistic plasma outflow.Solar corona has millisions of plasma jets (spicules) and also flares [12–15]. The formationof short scale solar spicules and flares seems to be very similar to large scale non-relativisticgalactic jets. Cylindrical structures of plasma also emerge in solar corona along with loopsand play a role in solar coronal heating.Theoretical models and numerical studies have been performed to understand the dy-namics of relativistic and non-relativistic jets [16–19]. Non-relativistic simulations for theoutflows of plasma from accretion disks have been performed using magnetohydrodynamic(MHD) equations [20, 21]. Simulations using neutral fluid equations have been performed tocompare dynamics of relativistic and non-relativistic jets [22]. Kelvin-Helmholtz instabili-ties have also been investigated in supersonic jets in cartesian and cylinrical geometries [23].Plasma jets are also produced in laboratories to understand the physics of jet formation[24, 25].Despite a large difference in velocities, scale sizes, luminosities and magnetic energies ofthe jets emerging from different sources, the physical origin is believed to be very similar.Many physical mechanisms take part in the formation, stability and acceleration of astro-physical jets including rotation, Lorentz force, magnetic field topology, dissipative forces,2ravitation and pressure gradients. Extremely large gravitational effects in case of mate-rial outflow from massive black holes require general relativity to be invoked. Flows of hotionized material create currents and Lorentz force.Observations reveal that in YSOs, some jets rotate in the opposite direction to the ro-tation of the source i.e. star or disk. Theoretical model has been presented to explainthe counter-rotation of astrophysical jets and winds using classical magnetohydrodynamics(MHD) [26]. They have shown that the counter rotation is induced by variation of velocityalong the flow, or by shocks, or by inhomogeneous magnetic field. They have verified theobserved counter-rotation speed both by analytical approach and by computer simulation.The fundamental source of energy is the thermal energy which is converted into flow andelectromagnetic energies in these systems particularly in jets moving with velocities muchsmaller than speed of light. Our aim is to find out a simple physical mechanism whichgenerates perpendicular jet-like outflows in non-relativistic plasmas and neutral fluids. Thismeans the fundamental source for astrophysical jets is based on thermodynamics coupledwith fluid dynamics.Biermann effect produced by electron baroclinic vector is believed to be the source of seedmagnetic fields in stars [27] and galaxies [28]. It has been pointed out that it is not onlythe electron baro clinic vector which produces seed magnetic field; rather the ion dynamicsis also coupled with the electron dynamics in this mechanism. Hence ions should not betreated as a background of stationary positive charge [29, 30]. In these studies, cartesiangeometry was used and main focus was on the generation of seed magnetic field, therefore anadditional assumption was made that magnetic field B is related with the vorticity ( ∇ × v )of plasma as B = α ( ∇ × v ) where α was a constant. A theoretical model for the generationof magnetic field in relativistic plasma of early universe has also been presented [31].The ideal classical MHD equations were solved by decomposing the velocity vector intocomponents in cylindrical geometry to explain counter rotation of jets [26]. Then theseauthors showed that the rotation velocity component can change sign if the velocity of flowalong the flux tube becomes smaller than a threshold value. On the other hand, here itis demonstrated that if plasma density and temperatures have particular profiles, then theaxial longitudinally uniform flow v and magnetic field B are created simultaneously. Theflow becomes function of density and increases with time. Dissipative terms and gravity areignored to highlight the physical mechanism and to find out an exact analytical solution of3onlinear partial differential equations. Similarly, it is shown that jet-like flows can also becreated in neutral fluids by the density and temperature gradients. II. TWO FLUID PLASMA JETS
Here it is shown that thermodynamic forces produced by density and temperature gradi-ents of electron and ion fluids create jet-like plasma flows along z-axis in cylindrical geometry.Only electron dynamics is not sufficient to produce such flows; rather it is found that dy-namics of ions is coupled with the electrons and hence both positive and negative fluidsgenerate seed magnetic field and plasma flow simultaneously. For the investigation of thislong time behavior, electron inertia is ignored ( m e →
0) and hence momentum conservationyields, 0 (cid:39) − en e (cid:18) E + 1 c v e × B (cid:19) − ∇ p e (1)In the limit | ∂ t | << ω pe , Ω e , | c ∇ | where ω pe = ( πn e e m e ) is the electron plasma oscillationfrequency and Ω e = ( eBm e c ) is the electron gyro frequency, Maxwell’s equation reduces toAmperes’ law ∇ × B = 4 πc j (2)and quasi-neutrality ( n e (cid:39) n i = n ) holds. Electron velocity can be defined as, v e = v i − c πe (cid:18) ∇ × B n (cid:19) (3)Curl of Eq. (1) yields − c ∂ t B = − c ∇ × ( v e × B ) − ∇ × ( ∇ p e en ) (4)Using (3) and (4), we obtain ∂ t B = ∇ × ( v i × B ) − ( c πne ) {∇ × [( ∇ × B ) × B ] } (5)+ c πne {∇ ψ × [( ∇ × B ) × B ] } − ce ( ∇ ψ × ∇ T e )where ψ = ln ¯ n , ¯ n = nN and N is arbitrary constant density. Curl of ion equation of motion( ∂ t + v i · ∇ ) v i = en (cid:18) E + 1 c v i × B (cid:19) − ∇ p i (6)yields, 4 ∂ t B + ∂ t ( ∇ × v i ) = ∇ × [ v i × ( ∇ × v i )] + a ∇ × ( v i × B ) + 1 m i ( ∇ ψ × ∇ T i ) (7)where a = em i c . Ideal gas law p j = n j T j has been used for both electron and ion fluids where j = e, i . Let us assume longitudinally uniform flow by imposing the condition ∇ · v j = 0and hence density does not vary with time i.e. ∂ t n = 0. The continuity equations ∂ t n + n ∇ · v j + ∇ n · v j = 0 (8)demand ∇ ψ · v j = 0 (9)which due to (3) also requires, ∇ ψ · ( ∇ × B ) = 0 (10)Let the plasma density vary in ( r, θ )-plane and temperatures be functions of z coordinateonly, viz, ψ = ψ ( r, θ ) (11)and T j = − T (cid:48) j z = − ( T j δ j ) z (12)where T (cid:48) j is constant and it denotes derivative of T j with respect to z and δ j is the scalelength of the temperature gradient. Equations (11) and (12) yield, ∇ ψ = ( ∂ z ψ ) e r + ( 1 r ∂ θ ψ ) e θ (13)and ∇ T j = − T j δ j e z (14)Here e r , e z , and e θ are unit vectors along radial, axial and angular directions, respectively.Note that when ψ = 0, we have n = N . Using equations (11-14), we find( ∇ ψ × ∇ T e ) = ( − T e r ∂ θ ψ ) e r + ( T e δ e ∂ r ψ ) e θ (15)and ( ∇ ψ × ∇ T i ) = ( − T i r ∂ θ ψ ) e r + ( T i δ i ∂ r ψ ) e θ (16)5radient of ψ in ( r, θ )-plane along with the condition (10) forces B to have the followingform of spatial dependence, B = ( ∇ χ ( r, θ ) × e z ) = 1 r ( ∂ θ χ ) e r + ( − ∂ r χ ) e θ (17)which yields, ∇ × B = −∇ χ ( r, θ ) e z (18)Note that the vector potential A has only z -component non-zero i.e. A = A z e z = χ e z andhence the magnetic helicity H is zero, viz, H = A · B = 0 (19)Our aim is to show that time-independent forms of density and temperature gradientsare able to create time-dependent jet-like flows in the axial direction perpendicular to ( r, θ )-plane. Now we prove that following form of plasma axial flow is created by the abovementioned profiles of ∇ ψ and ∇ T j , v i = u ( r, θ ) f ( t ) e z (20)It will be seen later that plasma flow in axial direction becomes a function of density andtemperature. The nonlinear terms of electron and ion fluid equations can be expressed interms of scalars defined above as follows, ∇ × ( v i × B ) = { u, χ } f ( t ) e z (21) ∇ × [( ∇ × B ) × B ] = { χ, ∇ χ } e z (22)and ∇ ψ × [( ∇ × B ) × B ] = [ { χ, ψ }∇ χ ] e z (23)where { f, g } = 1 r ( ∂ r f ∂ θ g − ∂ θ f, ∂ r g ) (24)Note that ∇ × [ v i × ( ∇ × v i )] = 0 (25)6et us impose the following conditions on density, magnetic field and flow, { χ, ψ } = { u, χ } = 0 = { χ, ∇ χ } = 0 (26)Interestingly under these conditions all the nonlinear terms of Eqs.(5) and (7) vanish andthey reduce, respectively, to ∂ t B = − ce ( ∇ ψ × ∇ T e ) (27)and a∂ t B + ∂ t ( ∇ × v i ) = 1 m i ( ∇ ψ × ∇ T i ) (28)Due to (20), we have ∇ × v i = { ( 1 r ∂ θ u ) e r − ( ∂ r u ) e θ } f ( t ) (29)therefore ion vorticity (29), baroclinic vectors (15,16) and magnetic field (17) all becomeparallel to each other. Using (20) and (13) in equations (27) and (28), we obtain ∂ t χ = cT (cid:48) e e ψ (30)and u∂ t f = − ( T (cid:48) e + T (cid:48) i m i ) ψ (31)Let χ = h ( r, θ ) f ( t ) and integrate (30) and (31) for t : 0 → τ to get, respectively, χ ( r, θ, τ ) = h ( r, θ ) f ( τ ) = cT (cid:48) e e ψ ( r, θ ) τ (32)and v i ( r, θ, τ ) = u ( r, θ ) f ( τ ) = − ( T (cid:48) e + T (cid:48) i m i ) ψ ( r, θ ) τ (33)Since u = u ( ψ ) and χ = χ ( ψ ), therefore { χ, ψ } = 0 and { u, χ } = 0 hold. Note that f ( t ) = ( constant ) t has been assumed.Now we try to find out an exact solution of the nonlinear two fluid plasma equations bychoosing an appropriate form of ψ which fulfills all the above assumptions. For this, weimpose a condition on ψ , such that ∇ ψ = − λ ψ (34)where λ is a constant. Following relations of Bessel functions hold, dJ ( λr ) dr = λJ ( λr ) − r J ( λr ) (35)7nd d J ( λr ) dr = − ( 2 λ r ) J ( λr ) − λr J ( λr ) (36)where J and J are Bessel functions of order zero and one, respectively. The magnetic field χ and plasma flow u have been expressed in terms of density function ψ in equations (30)and (31). Let us define, ψ ( r, θ ) = ψ J ( λr ) cos( θ ) (37)where ψ is the magnitude of ψ and λ is the scale length of density variation. With thisdefinition of ψ the condition { χ, ∇ χ } = 0 mentioned in (26) also holds. Density function ψ approaches zero for large λr . The form of this function is illustrated in Fig. (1). Flow andmagnetic field have also similar forms but the direction of flow is axial and it increases withtime. Since all physical quantities vary like J therefore this solution is valid in the limitedregion of the source where ψ has the given form. At large λr the density becomes equal tothe background density, say n and phenomenon disappears.FIG. 1: Density function ψ = ln ¯ n is plotted vs λr : 0 →
10 and θ : 0 → π . Similaraxial flow v out of plasma disc is created and it increases with time because it is directlyproportional to ψ and t .It is very interesting to note that the form of density function given in (37) fulfills all theabove conditions. Thus an exact solution of two fluid plasma equations (5) and (7) exists incylindrical geometry which predicts that both magnetic field and ion vorticity are generatedsimultaneously by the baro clinic vectors of electrons and ions. Plasma is ejected in thez-direction like a jet perpendicular to ( r, θ ) plane. Velocity u becomes zero where J = 0or cos θ = 0. Strictly speaking these are the points where flow disappears. This means the8ow under this theoretical model is broken in pieces. At these points the density becomesequal to background density. III. NEUTRAL FLUID JETS
In this section, we show that neutral fluid is also ejected out of a two dimensional materialspread in ( r, θ ) plane in the form of a jet along z -axis in the cylindrical geometry. Let usconsider an ideal neutral fluid with density ρ , velocity v and pressure p . Its dynamics isgoverned by the following two simple equations, ρ ( ∂ t + v · ∇ ) v = −∇ p (38)and ∂ t n + ∇ · ( n v ) = 0 (39)where ρ = mn and m is mass of the fluid particle. We again assume longitudinally uniformflow ∇ · v = 0 with time independent density i.e. ∂ t ρ = 0. Then the continuity equationdemands ∇ n · v = 0 which can be expressed as ∇ ψ · v = 0 (40)where ψ = ln ¯ n and ¯ n = nN and N is constant density at axis of the cylinder ( r = 0). Curlof equation (38) gives, ∂ t ( ∇ × v ) = ∇ × [ v × ( ∇ × v )] + 1 m ( ∇ ψ × ∇ T ) (41)If nonlinear term vanishes ∇ × { v × ( ∇ × v ) } = 0 (42)then we obtain a simpler equation for the vorticity generation ∂ t ( ∇ × v ) = 1 m ( ∇ ψ × ∇ T ) (43)This shows that vorticity can be generated in the neutral fluid by the baro clinic vector.If this is true, then we need to find out a suitable solution which satisfies the conditions(40) and (42) along with equation (43). Let us consider density and temperature profiles asfollows, ψ = ψ ( r, θ ) (44)9nd T = − T (cid:48) z = − Tδ z (45)which give, ∇ ψ = ( ∂ t ψ ) e r + ( 1 r ∂ θ ψ ) e θ (46)and ∇ T = − T (cid:48) e z = − Tδ e z (47)where T (cid:48) = T δ is a constant, δ is temperature gradient scale length and ˆ z is unit vectoralong axis of the cylinder. Equations (46) and (47) yield,( ∇ ψ × ∇ T ) = − T (cid:48) (cid:20) ( 1 r ∂ θ ψ ) e r + ( − ∂ r ψ ) e θ (cid:21) (48)Equation (43) shows that vorticity is parallel to ( ∇ ψ × ∇ T ), hence we choose the form offlow as, v = u ( r, θ ) f ( t ) e z (49)Then (43) can be expressed as, { ( 1 r ∂ θ u ) e r + ( − ∂ r u ) e θ } ∂ t f = − T (cid:48) m { ( 1 r ∂ θ ψ ) e r + ( − ∂ r ψ ) e θ } (50)Integrating for t = 0 → τ , it yields, { ( 1 r ∂ θ u ) e r + ( − ∂ r u ) e θ } ∂ t f = − T (cid:48) m { ( 1 r ∂ θ ψ ) e r + ( − ∂ r ψ ) e θ } τ (51)Equation (50) gives a relation between flow u and density ψ as u ( r, θ ) = − T (cid:48) m ψ ( r, θ ) (52)where f ( t ) = ( constant ) t (53)Now we try to choose a particular form of ψ which should satisfy all the above mentionedassumptions. Let ψ = ψ J ( λr ) cos θ (54)The density at r = 0 is constant N and hence v = 0 at the axis. However, vorticity ( ∇ × v )will have several zeros in ( r, θ )-plane because this plane is expanded along r . Then we find u ( r, θ ) = − T (cid:48) m ψ J ( µr ) cos θ (55)10quation (52) shows that neutral material is ejected in the form of jet along z -axis out oftwo dimensional density structure while the temperature gradient is along the direction offlow. IV. DISCUSSION
Keeping in view the long time plasma behavior, a formulism has been presented in whichall the physical quantities turn out to be the functions of density ψ chosen in Eq. (37).Interestingly, all the nonlinear terms vanish naturally and hence all the assumptions usedto derive the simple linear equations (27) and (28) are satisfied. Solution of these equationsunder the condition (34) is given by Eq.(37). Here ψ is the amplitude and ψ varies alongradial direction like Bessel function of order one ( J ) and in θ -direction the variation ischosen to be like cos θ . This form of density function reduces the set of nonlinear partialdifferential equations to two linear equations. It seems important to mention here that thepresented solution requires particular profiles of density and temperatures, but it is a naturalsolution of the two fluid classical plasma equations which predicts the creation of jet-likeplasma outflow v ( r,θ ) and the magnetic field B ( r,θ ) simultaneously. Unlike self-gravitatingsystems [32], we just focus attention on the investigation of outflows from a plasma and afluid source (star or a disc) and do not include additional forces like gravity.First we have considered a two fluid classical ideal plasma to show that the driving forcefor the observed astrophysical and laboratory jets is based on the structures of density andtemperature gradients. This theoretical model suggests that magnetic field and plasmaoutflow are produced simultaneously in astrophysical objects and they evolve with timefrom a non-equilibrium ( T e (cid:54) = T i ) two fluid ideal classical plasma. Important point to noteis that the highly nonlinear system of two fluid plasma equations can create an orderedstructure in the form of a jet. The driving force of such flows and associated magneticfields is produced by the thermodynamic free energy available in the form of density andtemperature gradients. Similar mechanism in classical plasma in the Sun’s atmosphere seemsto be operative for producing short scale plasma jets in millions (the spicules) and solar flaresas well as cylindrical plasma structures in solar corona.Second, it has also been shown that jet-like outflows are also created in neutral fluids bythe same physical mechanism. Here it has also been found that the baro clinic vector of a11eutral fluid produces jet like structures expelling the material in the axial direction. Thusdensity and temperature gradients seem to be the fundamental source of jet formation inboth neutral fluids and classical plasma systems.The Fig. (1) illustrates the profile of ψ for the range of λr : 0 →
10 and θ : 0 → π .Equation (33) gives the form of axial flow as a function of ψ but increasing with time becauseflow is directly proportional to time t . Similar flow can be created in neutral fluids as well.Thus axial flows can be created by density and temperature gradients in both neutral fluidsand plasmas. Acknowledgments
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