Rayleigh-Taylor Instability in Magnetized Plasma

Transcription

1 World Journal o Mechanics, 4, 4, 6-7 Published Online August 4 in SciRes. Rayleigh-Taylor Instability in Magnetied Plasma G. A. Hoshoudy Department o Applied Mathematics and Computer Science, Faculty o Science, South Valley University, Kena, Egypt g_hoshoudy@yahoo.com Received April 4; revised 8 May 4; accepted 5 June 4 Copyright 4 by author and Scientiic Research Publishing Inc. This wor is licensed under the Creative Commons Attribution International License (CC BY). Abstract The Rayleigh-Taylor instability in stratiied plasma has been investigated in the presence o combined eect o horiontal and vertical magnetic ield. The linear growth rate has been derived or the case where plasma with eponential density distribution is conined between two rigid planes by solving the linear MHD equations into normal mode. Some special cases have been particularied to eplain the roles the variables o the problem play; numerical solutions have been made and some stability diagrams are plotted and discussed. The results show that, the growth rate depends on the horiontal and vertical components o magnetic ield and also depends on the parameter λ λl D ( λ is constant and L D is the density-scale length). The maimum instability happens at λ 5. and to get more stability model we select λ such that it is dierent than λ 5.. The vertical magnetic ield component have a greater eect than the horiontal magnetic ield component in the case o large wavelength, while in the case o short wavelength, the horiontal magnetic ield components have greater eect than the vertical magnetic ield component. Keywords Rayleigh-Taylor Instability, Plasma, Horiontal and Vertical Magnetic Field. Introduction It is well-nown that, plasma is a hot ionied gas consisting o approimately equal numbers o positively charged ions and negatively charged electrons. The characteristics o plasmas are signiicantly dierent rom those o ordinary neutral gases so that plasmas are considered a distinct ourth state o matter. There are numer- How to cite this paper: Hoshoudy, G.A. (4) Rayleigh-Taylor Instability in Magnetied Plasma. World Journal o Mechanics, 4,

2 ous everyday uses or plasmas, or eample, luorescent lights and neon signs wor because o plasma. Circuit eatures on micro-processor chips in computers also contain plasma. Even the ehaust that is emitted during rocet launches is actually plasma, not gas. Plasma is also used to manuacture industrial diamonds and superconducting ilms. Plasma is also a ey technology in the development o alternative energy sources. Nuclear usion, which is plasma based, is one o the most promising candidates or the energy needs o the uture when ossil uels inally run out. One o the important models rises in hydrodynamic plasma called the Rayleigh- Taylor instability (RTI) problem [] []. RTI can occur when dense plasma is supported against the gravity. Studies on RTI show that this instability is troublesome because it obstructs the realiation o ICF [3] [4]. In astrophysics, this instability is related to the stellar structure and evolution [5] [6]. Hence, it is important to understand the physical mechanisms that can aect such instability, especially its suppression as well as the details o this instability. Such nowledge will aid our understanding o the origin o white dwars and type-ia supernovas. Also, it is well-nown that, the plasmas are strongly inluenced by magnetic ields (where the plasma parameters may vary with the application o magnetic ield), where the behavior o plasmas in the presence o a magnetic ield is among the oldest problems in plasma physics. It has been central in plasma usion research since the early eperiments on plasma coninement by a magnetic ield in the 95s, and it has remained great interest in plasma usion studies that use contemporary sophisticated devices [7]. It is also an important problem or many plasma discharges used in processing semiconductor materials where the application o a magnetic ield results in enhancement o some desirable eatures o speciic plasma sources [8]. The linear growth rate o RTI has been obtained by Goldston and Rutherord [9] under ied boundary conditions. The hydromagnetic stability o a magnetied plasma o variable density is o considerable importance in several astrophysical situations, e.g. in theories o sunspot magnetic ields, heating o solar corona and the stability o stellar atmospheres in magnetic ields. Ariel [] investigated the stability o an inviscid compressible luid o variable density in the presence o a uniorm vertical magnetic ield and viscous plasma has been investigated by Bhatia []. The instability o stratiied plasmas in the presence o horiontal magnetic ield o compressible plasmas is studied by Bhimsen []. The RTI o a plasma layer in the presence o a horiontal magnetic ield is investigated, taing into account the eects o Hall-currents and an arbitrarily large density gradient by Ariel [3]. The eects o horiontal magnetic ield, Hall currents and viscosity have been studied on the RTI o an incompressible ininitely conducting stratiied plasma by Ahsan and Bhatia [4]. The eect o horiontal magnetic ield o Hall currents have been investigated on the RTI o a initely conducting stratiied partially ionied plasma by Aiyub and Bhatia [5]. The instability o stratiied plasmas in the presence o horiontal magnetic ield o incompressible plasmas with the eect o a transverse velocity shear was studied by Wu et al. [6]. The eect horiontal magnetic ield on RTI o a plasma layer in the presence o quantum mechanism was studied by Jintao et al. [7]. The eects o vertical magnetic ield in the presence o quantum mechanism on RTI inviscous and viscous plasma were studied by Hoshoudy [8] [9]. The eects o magnetic ield gradient on the Rayleigh-Taylor instability (RTI) with continuous magnetic ield and density proiles were analytically investigated by Yang et al. []. The analytically and numerically investigated stabiliation o the linear growth o the RTI rom density gradients, magnetic ields, and quantum eects, in an ideal incompressible magnetied plasma was studied by Wang et al. []. Because o the great scientiic interest in magnetied plasma problem, it is attempted to discuss the RTI problem o a stratiied plasma layer in the presence o an variable magnetic ield, where this problem corresponds physically (in astrophysics) to the RTI o an equatorial section o a planetary magnetosphere or o a stellar atmosphere where the magnetic ield are perpendicular or parallels to gravity. In all the above-mentioned studies the behaviour o growth rates is considered in the presence o a variable magnetic ield in direction only or in direction only. Here, the eect o magnetic ield in both and direction on RTI problem or a inite thicness layer o incompressible plasmas are studied. The dispersion relation is obtained analytically and numerically analyed.. Linearied Equations We consider the strata o incompressible and inviscous plasma as a luid o electrons and immobile ions. The plasma is immersed in a magnetic ield, where the relevant liner perturbation equations may be written as (Res. [9]-[7]) U ρ p+ ρg + ( B) B+ ( B) B t µ, () 6

3 Here U, p, p and U () B ( U B) (3) t ρ + ( U ) ρ (4) t ρ, are the perturbations in the velocity U, pressure p, magnetic ield B, and density ρ, respectively. While U ( u, uy, u), B ( B, By, B), g (,, g ) and the luid is arranged in horiontal strata, then ρ is a unction o the vertical coordinate only (i.e. ρ ρ ) and B B e + B e. Then the system o Equations ()-(4) become: u p B B B ρ + B + B t µ, (5) uy p By B B y B ρ + B + B, (6) t y µ y y u ρ B B B B B g, (7) p + + ρ t µ u u u y y + +, (8) u y B + ( B u B u), y u B y ( B, By, B) B ( B uy ),, t t u y ( B u B u ) B. y ρ t dρ + u d I we assume that the perturbation in any physical quantity taes the orm:. () ( yt,,, ) ep{ i ( y y t) } ψ ψ +, () where and y are horiontal components o the wave-number vector such that + y and (may be comple ( r + i) ) is the requency o perturbations or the rate at which the system departs rom equilibrium. Using the Epression () in the system o Equations (5) - (), we have: iρu i p + B B + B B i B µ (9), () iρu i p + ib B B + B B i B y y y y y y µ, (3) p B B iρ u ρg + B + B ib, (4) µ i u u + i yuy+ (5) 6

4 iyb uy + ( B u B u), i { B, By, B} ib uy ( B uy ),, i( B u B u ) iyb u y. (6) dρ iρ+ u (7) Eliminating some variables rom the system o Equations () - (7) we have: d B B 4 3 d u d d u d u + 4 4B ib B 3 { ρ A} µ d µ d d d dρ du dρ + B ρ C g u, d + + d d where, d B db db db A B + 3B + B + 3i B + B µ d d d d 3 db d B db d B db B + B + B 3 d d d d d B µ d B db db db d B + i B B B + + d d d d d 3 3 d B db d B db db c B + i B + B 3 3 B µ d d d d d (9) 3. A Continuously Stratiied Plasma Layer In this section we consider the case o incompressible continuously stratiied plasma layer o thicness h units conined between two rigid boundaries, in which the density and magnetic ield distribution are given, respectively, by ρ ρ( ) ep( ), B B( ) ep( ) and B B( ) ep( ) where ρ ( ), B ( ) and B ( ) and L D (the density-scale length) are constants, then Equation (8) taes the orm:. (8) where d u d u 5 3i υ υ d u υ { υ υ υ } υ υ i d d 4 d 5 d u + υ + υ + i L υ υ L D D 4 4 d g + υ i υ υ 3, u 4 ( ) ( ) B υ and υ µρ ( ) ( ) B are Alvén velocity. Now, i we choose u in the orm µρ () 63

5 nπ u sin ep( λ) and by substituting in Equation (6), we will have an equation in both h π and cos n h. Then coeicients both sin n π h and cos n π, respectively, are given by: h 4 4 nπ nπ λ nπ υ 6 { } 3 λ + λ + υ + i υ υ λ h h h 5 3 iυ υ nπ υ υ λ 4 h λ 5 + υ + υ + iυ υ 4 4L g + υ i, υ υ 3 4 D π sin n h () nπ 5 υ nπ 4λυ 3 λ + λ υ + υ + λ h 4 h 6λ nπ 5 + υ + υ 3 i + υ υ λ L D h. 4 Now, we deine the dimensionless quantities: υ υ,,, λ λ,, h L D h pe pe pe g ρe, g,. pe pe me Then Equations () and (), respectively, tae the orm: λ λ h nπ + { } + λ + λ + λ λ + λ + λ h h h 4 h nπ nπ nπ 5 nπ 6 3 nπ nπ 5 + λ λ 3 3 i. g h λ h λ () (3) (4) nπ λ + + λ + λ + 4 h { } nπ 5 + 6λ + 6λ i. + h 4 Now, we put r + i and or r (stable oscillations), then Equations (4) and (5) may be given by: (5) 64

6 λ λ h nπ + + { } + λ + λ + λ λ + λ + λ h h h 4 h nπ nπ nπ 5 nπ , h h 4 4 nπ π 5 + λ λ 3 3 λ n λ i g λ λ λ 4 h nπ { } { } nπ 5 + 6λ + 6λ i + h 4 To discuss the role o parameter s problem we consider the net special cases rom Equations (6) and (7). (i) In the case o. From Equation (7) we get λ.5, and substituting in Equation (6) we ind that the square normalied growth rate given by: g Goldston and Rutherord nπ h. (6) (7), (8) This case is considered by Goldston and Rutherord (see Re. [9]), which represents an eponentially growing perturbation (instability case). (ii) In the case o,. A second time, rom Equation (7) we get λ.5, and substituting in Equation (6), then the square normalied growth rate given by: horiont magnetic ield nπ h g. (9) This case studied in Res. ([6] [7]). It is clariied that, the horiontal magnetic ield has stabiliing eect on RTI problem. This inluence is obvious rom Equations (8) and (9), where, <. Also, one can see that the square normalied growth rate decreases as horiont magnetic ield Goldston and Rutherord increases and the system arrives to complete stability case at nπ + 4 h relation + y y under the condition C C C, C <. Then Equation (9) becomes: then At horiont magnetic ield C nπ + + horiont magnetic ield, the critical point becomes This implies that the complete stability happens at 4 h c g c g y < y y C g nπ +. C 4 h g <. n π C + h 4. Now, i we use the <, C <. (3) 65

7 Also, i the perturbation in the Equation () in the orm ( y t) i ( t) and the square normalied growth rate becomes horiont magnetic ield,,, ep( ). Then C g, and the criti- nπ h g nπ cal point o stability becomes c, + this results correspond with Re. [7]. 4 h (iii) In the case o,. Again, rom Equation (7) we get λ.5, and substituting in Equation (6), the square normalied growth rate given by: vertical magnetic ield nπ nπ + + g h 4 h nπ nπ h 4 h. (3) Now, comparing between Equations (8) and (3), someone can observe that, the stabiliing role or the vertical magnetic ield on the considerable system, where < and the sys- tem arrives to complete stability case at c nπ h g n π + h vertical magnetic ield 4 Goldston and Rutherord. This indicates that, in the presence o g the stability role stratiies under the condition <. nπ + h 4 (iv) For the general case (, ), i we can eliminate the term i and (7). Then the square normalied growth rate given in the rom nπ 5 nπ 6λ + 6λ + λ λ { + }, between Equations (6) nπ nπ 5 nπ nπ 5 λ + λ 6λ 6λ { λ } λ λ 3 3 λ λ h h 4 h h 4 4 h 4 h + λ + λ + λ λ + + λ h h h 4 4 h 4 4 nπ nπ nπ 5 nπ 6 6 g nπ 3 nπ nπ 5 { λ + } { } + λ + λ λ λ λ λ h h h 4 4 In the general case, i we wish to loo into the eect o both horiontal and vertical magnetic ield together on the instability o the considered system, Equation(3) is to be numerically solved, where is unction in the dimensionless quantities o horiontal ( ) and vertical ( ) components o the magnetic ield, the wave number and λ λ, where λ is constant and L D is the density-scale length. In these igures C, C < ( C.), h, n and g. The role o λ as a unction horiontal in the presence o both horiontal and vertical magnetic ield.5 is plotted in Figure and Figure, where the square normalied growth rate components ( ) (3) 66

8 (a) (b) Figure. The square normalied growth rate ( ) against the square normalied wave number at (.5) (a) at λ.5,,.5 (b) at λ.5,,.5. is plotted against the square normalied wave number ( ). Figure (a) shows the role o λ (.5), or eample λ.5,,.5. One can see that decrease with increasing o λ. While Figure (b) shows the role o λ (.5) or λ.5,,.5, where decrease with decreasing o λ. These implies that the maimum instability in the presence o both vertical and horiontal magnetic ield happens at λ.5. The same phenomenon hold in Figure (a) and Figure (b) at (.5). In this case, i we put λ.5 in Equation (3), then the maimum square normalied growth rate is: 4 nπ nπ nπ h 4 h 4 h g ma + nπ nπ h 4 h. (33) 67

9 (a) Figure. The same as Figure but at (.5) (b). From Equation (33) it can see that, the maimum square normalied growth rate (in the presence o both horiontal and vertical magnetic ield components) given as we add the second term in Equations (9) and (3) to Equation (8). Figure 3 shows the role o both horiontal and vertical components magnetic ield.6 on the considered system at λ.5, where the square normalied growth rate is plotted against the square normalied wave number. One can see that, the magnitudes o in the presence o these parameters (, ) or either them is less than their magnitudes in the Goldston and Rutherord case, which means that these actors have a stabiliing eect on the considered system. Also, it can seen that, no mode o maimum instability eists when ( or ) as the square normalied growth rate usually increases by increase with the square normalied wave number values. While, in the presence o quantum term (.6), there is a mode o maimum instability, where the square normalied growth rate increases with increases through the range < < ma (at ma the square normalied growth rate arrives to the maimum instability, and when > ma the square normalied growth rate starts to decreases as increases and then goes to the complete stable at c ( c is the critical value or stability, at this point the square normalied growth rate goes to ero). This means that the horiontal component o mag- 68

10 netic ield has a crucial capability to suppress the instability that satisies or the large wave number (short wavelength), while the vertical component o magnetic ield has this strength just or the very short wave number (large wavelength). Also, the considerable model is more stability in the presence o both horiontal and vertical components o the magnetic ield (.6). This case (, ) shows in Figure 4, where the magnitudes o decrease with increasing o both horiontal and vertical components o magnetic ield Figure 3. The square normalied growth rate ( ) against the square normalied wave number at λ.5 in the presence o.6,.6 and,.6. Figure 4. The square normalied growth rate ( ) against the square normalied wave number at λ.5 or dierent values o the magnetic ield,.4,.5,.6. 69

11 (.4,.5,.6) that are less than their counterpart at sults). Figure 5(a) and Figure 5(b) show the role o magnetic ield (.4,.5,.6) (Goldston and Rutherord re- when λ have (a) Figure 5. The square normalied growth rate ( ) against the square normalied wave number (b) or dierent values o the magnetic ield,.4,.5,.6. (a) at λ.5 (b) at λ.5. 7

12 dierent values than λ.5 (at λ.5,.5, respectively). In Figure 5(a), one can see that, the mai- mum instability (maimum square o growth rate ) at (.4,.5,.6) in Figure 5(a), respec-. Also, the, they are less than their counter- in Figure 5(a) at λ.5 are less than their counterpart in Figure 4 at λ.5. The same phenomenon holds in Figure 5(b) but at λ.5, where both the maimum instability and the critical point or stability in Figure 5(b) are less than their counterpart in Figure 4 at λ.5, which also indicates that, the magnitudes o at λ.5 are less than their counterpart at λ.5. tively, are ( 4.,.5,.5), they are less than their counterpart in Figure 4 ( 4.95, 3.4,.7) critical point or stability in Figure 5(a), respectively, are ( c 3,8, 5) part in Figure 4 ( c 45,3, 69). This implies that the magnitudes o 4. Conclusions Finally, we have investigated the eect o magnetic ield in both horiontal and vertical direction on the Rayleigh-Taylor instability o stratiied incompressible plasmas layer. We can summarie the results as ollows: (i) In the presence o horiontal magnetic ield only or vertical magnetic ield only the growth rate depends on h h λ.5. While, in the presence o both horiontal and L o plasma layer and λ taes one value h vertical magnetic ield, the square growth rate (Equation (3)) depends on the dimensionless thicness h L and also depends on the parameter λ λ. (ii) The constant λ λ plays an important stabiliing role in the growth rate, where the maimum instability happens at λ.5 and to get more stability model, the values λ must be dierent than λ.5 (i.e..5 < λ or λ <.5 ). In other words, in studying the combined eect o horiontal and vertical magnetic ield on RTI o stratiied plasma, the system will be more stable when we accept that λ is dierent than.5 (i.e. λ.5 ). (iii) Our model is more stable than those considered in previous study (horiontal magnetic ield only or vertical magnetic ield only). This divergence ascribes to the stabiliing role that the magnetic ield plays on RTI problem in the presence o both horiontal and vertical components, where the presence o both these parameters dissipates the energy o any disturbance and thereby the system becomes more stability. Reerences [] Rayleigh, L. (88) Investigation o the Character o the Equilibrium o an Incompressible Heavy Fluid o Variable Density. Proceedings o the London Mathematical Society, 4, [] Taylor, G.I. (95) The Instability o Liquid Suraces When Accelerated in a Direction Perpendicular to Their Planes. Proceedings o the Royal Society o London. Series A, Mathematical and Physical Sciences,, [3] San, J. (994) Sel-Consistent Analytical Model o the Rayleigh-Taylor Instability in Inertial Coninement Fusion. Physical Review Letters, 73, [4] Lindl, J.D. (995) Development o the Indirect-Drive Approach to Inertial Coninement Fusion and the Target Physics Basis or Ignition and Gain. Physics o Plasmas,, [5] Cabot, W.H. and Coo, A.W. (6) Reynolds Number Eects on Rayleigh-Taylor Instability with Possible Implications or Type-Ia Supernovae. Nature Physics,, [6] Timmes, F.X. and Woosley, S.E. (99) The Conductive Propagation o Nuclear Flames. I Degenerate C + O and O + NE + MG White Dwars. Astrophysical Journal, 396, [7] Blinniov, S. and Soroina, E. (4) Type Ia Supernova Models: Latest Developments. Astrophysics and Space Science, 9, [8] Chen, F.F. (974) Introduction to Plasma Physics. Plenum, New Yor. [9] Goldston, R.J. and Rutherord, P.H. (997) Introduction to Plasma Physics. Institute o Physics, London. [] Ariel, P.D. (97) Rayleigh-Taylor Instability o Compressible Fluids in the Presence o a Vertical Magnetic Field. Applied Scientiic Research, 4, [] Bhatia, P.K. (974) Rayleigh-Taylor Instability o a Viscous Compressible Plasma o Variable Density. Astrophysics and Space Science, 6,

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