A Search for Low Frequency Electromagnetic Waves in Unmagnetized Plasmas

A Search for Low Frequency Electromagnetic Waves in Unmagnetized Plasmas Hamid Saleem National Centre for Physics (NCP) Quaid-i-Azam University Campus, Shahdara Valley Road, Islamabad. --------------------------------------------------------------------------------------------------------------------- International Scientific Spring, 1-6 March, 010, NCP. 1

Abstract: It is proposed that the electrostatic and thermal fluctuations are the source of magnetic fields in unmagnetized inhomogeneous plasmas. It is also pointed out that the ion acoustic wave can become electromagnetic in a nonuniform plasma under certain conditions. It is shown that the description of the so called magnetic electron drift vortex (MEDV) mode suffers from serious contradictions and it cannot exist as a pure transverse mode of an inhomogeneous electron plasma. Dispersion relation of a new low frequency electromagnetic wave is presented.

Plan of the Lecture: 1. Background of the Problem. Brief Introduction of Plasmas 3. Transverse Waves in Plasmas 4. LFEM Waves in Electron Plasmas 5. Ion Acoustic Wave (IAW) 6. Electromagnetic IAW 7. LFEM Ion Wave 8. Summary 3

1. Background of the Problem The first experiment on laser interaction with the solid target was performed in USA [Ref. [1]: Stamper et. al. Phys. Rev. Lett. 6, 101, 1971]. Later, many experiments were performed in USA and Russia in 1970s. Most of the results appeared in Phys. Rev. Lett., JETP [-4]. Details can be seen in Review Article [5]. In early experiments, Laser pulse duration 10 9 sec Intensity' 10 1 W cm In many such experiments, large magnetic fields were observed, B '(10 3 10 6 )Gauss 4

One of the explanations of this phenomenon was that, some unstable low frequency transverse perturbation is produced in the plasma. A pure transverse mode was theoretically discovered to explain magnetic fieldsgeneratedinlaser- induced plasmas. A great deal of work on linear and nonlinear studies of this mode including numerical simulations has appeared in literature during last many years. But, in our opinion, the physics of this mode is incorrect.. Brief Introduction of Plasmas.1 Definition: Quasi-neutral statistical ensemble of charged particles which exhibits collective behavior. F Long range electromagnetic forces dominate over mechanical collisions of particles which are important in neutral uids. F Density uctuations of negative and positive charges create E fields and the motion produces currents and hence the B fields. F Debye Shielding: Applied electric potentials are shielded in plasmas within a distance about λ De. 5

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T e 6=0, so shielding is not perfect. T λ De =( e 4πn 0 e ) 1 Debye length Debye sphere= 4 3 πλ3 De If there are neutrals in the plasma, ν - frequency of electron collisions with neutrals ω pe =( 4πn 0e m e ) 1 electron plasma oscillation frequency and L = System size Then this system is a plasma, if 1. ν << ω pe (E-field dominates). λ De << L (quasi-neutrality) 3. 1 << N D = 4 3 De πλ3 n (statistical ensemble) 7

. Plasma Approximation Poisson eq..e =4πe(n i n e ) Even if n i ' n e is assumed, the E is not zero. Quasi neutrality means λ Dek << 1 That is the wavelengths of perturbations are much longer than λ De. Since m e 6= m i, we have many time scales in plasmas, So ω pi = ³ 4πn0 e 1 m i ; ω pe = Similarlymanyspatialscalesexistinplasmas. c ³ 4πn0 e 1 m e ; Ω i = eb 0 m i c ; Ω e = eb 0 m e c λ j = ω pi, (collision-less skin depth of jth species, j = e, i) ³ ρ j = v 1 ³ 1 tj Tj Ω j ; v tj = m j ; ρ s = c s T Ω i where c s = e m i and λ De. 8

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.3 Kinetic Model: Boltzmann eq: f t + v. f + F m. f v = µ f t f(r,v,t) is the distrubution Function F=force on a particle Time rate of change of f due to collisions. ³ f t c f j t + v j. f j + q j m j (E + 1 c v j B). f j v j =0 (.) c (.1) In hot plasmas, the collisions can be neglected. Then for jth species, we have Vlasov eq. 10

.4 Fluid Model Physical phenomena become clearer through uid models. two- uid equations for an ideal plasma are, m j n j ( t + v j. )v j = q j n j (E + 1 c v j B) p j (.3) t n j +.(n j.v j )=0 (.4) B = 4π c J + 1 c tb (.5) ( t + v j. )+ j p j.v j =0 (.6) E = 1 c tb (.7) The simplest Here j is the ratio of specific heats at constant pressure and volume. Plasma is a highly nonlinear complex system, in general. 11

.5 Nuclear Fusion: Thermocuclear Fusion Reactors will be advantageous over Fission Reactors. 1. Nuclear waste problems are lesser.. Fuel is available for centuries in the oceans. Coal burning: Fission C + O CO +4ev 9U 35 + 0 n 1 9 U 36 56 Ba 141 + 36 K 9 r +3 0 n 1 +(00)Mev Fusion: 1D + 1 T 3 He 4 + 0 n 1 +17.6Mev F 1kgm of U 35 produces energy equal to 3 10 6 kgm of coal. F In fusion, the energy release per kgm of fuel is about 4-times larger than fission. 1

I. Magnetic Confinement In Tokomaks: Many devices are studied like, pinches, mirrors and tokamaks for magnetic confinement. Tokamak is a device in which magnetic fieldisintheformof nested surfaces. Roughly speaking, n 10 14 cm 3 T 1kev Two types of operations are possible: (i) Pulsed operation (ii) Steady operation However, so far energy output is lesser than the energy input in such experiments. 13

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II. Inertial Confinement: Laser beams are focused on a fuel pallet, ablation takes place and the pallet is compressed to high densities and temperatures. Shock waves also help compression. Main problem is that the compression is not uniform and hence required results are not achieved. In stellar cores, the fusion takes place, but the stars remain intact due to gravity. 15

.6 Earth s Magnetosphere: As the highly conducting solar wind interacts with the earth s magnetic field, it compresses the field on sunward side and ows around it at supersonic speeds. The boundary is called Magnetopause. The inner region, from which the solar wind is excluded and which contains the compressed Earth s magnetic field, is called the Magnetosphere. The particles cannot easily leave the solar wind due to frozen-in characteristic of a highly conducting plasma. Since the solar wind hits the obstacle (magnetized Earth) with supersonic speed, a Bow Shock Wave is generated where the plasma is slowed down and substantial fraction of the particle s kinetic energy is converted into thermal energy. The turbulent region between the shock wave (Bow Shock) and the Magnetopause is known as the Magnetosheath. Van Allen Belts between (-6) R E (earth radius R E =6, 371 km). Solar Wind: (near Earth): n e 5cm 3 T e 5ev, T i 1ev B 5 10 5 Gauss v 3 10 km/sec 16

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Ionosphere: The solar ultraviolet radiation ionizesa fraction of the neural atmosphere. At about 80km, a permanent ionized population exists called Ionosphere. At high altitudes plasma sheet electrons can precipitate along magnetic field lines down to ionospheric altitudes, where they collide with and ionize neutral atmosphere particles. As a by-product, photons emitted by this process create the polar lights (Aurora). These auroras are typically observed inside the Auroral Oval..7 Sun: Age 4.5 10 9 Yrs M 0 1.99 10 30 kgm R 0 6.96 10 5 km 1

F Interior (3Regions): i. Core: T 1.6 10 3 ev n 1.6 10 5 kgm 3 It is a huge thermonuclear reactor. ii. Radiative Zone iii. Convective Zone F Outer Atmosphere: Lower region: Photosphere (visible) [T (0.1 0.4)ev]. Above photosphere is Chromosphere (0.4 0.6)ev Corona: Beyond Chromosphere T (few million degree). In between Chromosphere and Corona is a thin Transition Region ( 500km) where the temperature rises about thousand times (to a few million degrees). Its an unsolved problem yet.

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.8 Brief Summary of Parameters of some Plasma Systems: Plasma System N e (cm 3 ) T e ( 0 k) H(Gauss) Lonosphere:D 70km 10 3 cm 3 10 0.3 E 100km 10 5 (day) 10 3 (night) -3 10 0.3 F 300km 10 6 (day), 10 5 (night) 10 3 0.3 Interplanetary Space 1 10 4 10-103 (10 6 10 5 ) Solar Corona 10 4 10 8 10 3 10 6 10 5 1 Solar Chromosphere 10 1 10 4 10 3 Stellar Interiors 10 7 10 7.5 10 3 Pulsars 10 4 (center), 10 1 (surface) - 10 1 Interstellar Space 10 3 10 (av 0.03) 10 10 6 Intergalactic Space. 10 5 10 5 10 6. 10 8 9

3. Transverse Waves in Plasmas Linear dispersion relation in vacuum: in plasmas. ω = c k (3.1) ω = c k + ω pe (3.) Magnetized plasmas support a pure transverse low frequency wave; Alfven wave. ω = ν Ak z (3.3) ν A = µ B 0 4πn 0 m i 30

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Eq. (3.3) is obtained through MHD. In case the density uctuations are also taken into account, then (3.3) is modified using two uid theory as, If β < m e m i, then (3.4) blueuces to ω = v A k z(1 + ρ sk ) (1 + λ ek ) (3.4) ω ' v A k z 1+λ ek (slow shear Alfven wave) (3.5) If m e m i < β < 1, (3.4)blueucesto ω = ν Ak z(1 + ρ sk )(Kinetic Alfven wave) (3.6) β = c s ν A = kinetic pressure magnetic pressure = 4πn 0T e B 0 (3.7) 3

F No transverse mode at ion time scale has been discovered in unmagnetized plasmas. F MEDV mode is believed to be a low frequency transverse-mode of inhomogeneous electron plasma. But it has been described incorrectly. We try to look for some low frequency electromagnetic mode in nonuniform plasmas. The uniform unmagnetized plasma cannot support a low frequency electromagnetic wave. 4. Low Frequency Electromagnetic Waves in Electron Plasmas 4.1 MEDV Mode: Stationary ions [Ref: [7] Jones, Phys. Rev. Lett. 51, 169 (1983)]. Linear eqs. m e n 0 t v e1 = en 0 E 1 p e1 (4.1) B 1 = 4π c J 1 (4.) J 1 = en 0 v e1 (4.3) E 1 = 1 c tb 1 (4.4) 33

Since p 1 = n 0 T e1 therefore energy equation becomes, Curl of (4.1) gives, 3 n 0 t T e1 + 3 n 0(v e1. )T e0 = p 0.v e1 (4.5) t ( v e1 )= e m e c tb 1 + 1 ( n 0 T e1 ) (4.6) m e n o Equations (4.) and (4.3) yield, and hence v e1 = c B 1 (4.7) 4πen 0 v e1 = c 4πen 0 B 1 (4.8) where n 0 ( B 1 )=0due to the assumption k n 0 B 1. 34

Equation (4.8) pblueicts E 1 = E 1ˆx while n 0 = ˆx dn 0 dx, =(0,ik y, 0) and B 1 = B 1 ẑ have been chosen. Note the vorticity v e1 has been obtained from eq. of motion (4.6) and from Ampere s law (4.7). Then, (4.6) and (4.8) yield, (1 + λ e k ) t B 1 = c en 0 ( n 0 T e1 ) (4.9) where λ e = c ω. Compressibility is obtained from Amperes law (4.7), pe.v e1 = c 4πn 0 e n 0 n 0.( B 1 ) (4.10) and therefore one obtains, T e1 = 3 c 4πn 0 e k yκ n B 1 (4.11) 35

Question: What will happen if we evaluate.v ei from equation of motion (4.1)? Now (4.9) and (4.11) give the linear dispersion relation of MEDV mode as, ω = 3 C 0( κ n k y ) v Tek y (4.1) where C 0 = λ e k 1+λ and v te =(T e /m e ) 1 e k. The geometry of MEDV mode in cartesian co-ordinates is shown in Fig. 4.1. If T e0 6=0is assumed, then (4.1) becomes, ω κ n ( 3 = C κ n κ T ) 0 v k y k Tek (4.13) y where κ T = 1 dt e0 T e0 dx and T e0 =+ dt e0 dx ˆx. If( 3 κ n < κ T ), then the mode becomes unstable [Ref[8]: Yu & Chijin, Phys. Fluids 30, 3631(1987)]. In deriving (4.13), 36

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.E 1 =0,.v e1 6=0and n e1 =0have been assumed and it does not seem to be very convincing. This wave geometry does not give a pure transverse wave [Ref.[9]: Saleem, Phys. Rev. E54, 4469 (1996)]. Let us write eq. of motion (4.1) inxandycomponents, and t ν ex1 = e m e E x1 v Te t ν ey1 = e m e E y1 v Te ½ κ n T e1 T 0 ½ ik y µ Te1 T 0 + κ T n e1 n 0 + n e1 n 0 ¾ ¾ (4.14) (4.15) It is obvious from (4.15), that.v e1 6=0butitgivesE y1 = y ϕ 1 6=0. Therefore, electric field has both the longitudinal (E 1y ) and transverse (E 1x ) components. Hence.E =0should not be used and it implies n e1 6=0. It is important to note that density perturbation cannot be neglected. 38

Then the curl of Amperes Law (4.) yields a relation between E 1x and E 1y for ω << ω pe as, E 1x = 1 κ n (ie 1y ) (4.16) a k y where a =(1+λ ek y). Using equation of motion (4.1) for.v e1, instead of Ampere s law (4.10), in equation (4.5), we find, W 0 T e1 T 0 = v te k y where W 0 = 3 ω v te k y µ 1 3 κ T ky ³ 1 3 Γ 0 κ T κ n k y ne1 n 0 e m e µik y E 1y + 3 κ T E 1x and Γ 0 = (κ T κ n )κ T k y. (4.17) 39

The continuity equation yields, Ã L n e1 0 = 1+ v te k y n 0 W0 where L 0 = Poisson eq. (! µ (1 κ n/ky) i e k y E 1y m e ) µ e κ n E 1x m e 1+ 3 κ T vtek y κ n W0 (1 κ n/ky) n ω vtek y(1 κ n/ky) v4 te k4 y (1 κ W n/ky) 0.E 1 = 4πe µ ne1 ³ 1 3 canbewrittenas, ik y E 1y L 0 W0 ωpe W 0 + vtek y(1 κ n/ky) ª n 0 κ Γ k y Γ 0 (4.18) o.then (4.19) =(κ n E 1x )ω pe W0 + κ T v κ teky(1 κ n/ky) n (4.0) 40

Equations (4.16) and (4.0) yield a linear dispersion relation in the limit ω << ω pe as, a L 0W 0 ω pe W 0 + v tek y = ω pe(κ n /k y ) W 0 + 3 κ T κ n v tek y 1 κ n /k y ª 1 κ n /ky (4.1) This equation can be simplified as, [Ref.[10]: Saleem Phys. Plasmas (009)], H 0 W = µ 3 v te κ n 1 3 µ κ T (1 + λ e κ ky ) 1 3 κ T (4.) n κ n where H 0 = ½ (1 + λ Deky)+ ¾ 3 λ Deky a κ n/ky 41

The quasi-neutrality is not allowed, otherwise we have n e1 =0. The second term on right hand side of (4.) will disappear if E 1y =0is assumed. The first term and the factor λ eky in second term are the contributions of transverse component E 1x. ThusonecannotobtainMEDV-modedispersionrelationfor E 1y = 0 from (4.). It is interesting to note that equation (4.) can be simplified to obtain, ω = λ e 3H k y (v te κ n) µ1 3 κ T /κ n (4.3) 0 Note (4.3) contains effects of compressibility and hence the density perturbation in H 0 with the condition m e m i < λ De k y. So this electromagnetic wave has shorter wavelength compablue to electrostatic IAW which exists even for λ De m e m i. 4

However, the instability condition for this electromagnetic mode is the same as was for MEDV mode that is 3 κ n < κ T (4.4) The low frequency mode (4.3) is partially transverse and partially longitudinal. If E 1x =0is assumed, then equation (4.1) yields a low frequency electrostatic wave in a non-uniform unmagnetized plasma with the dispersion relation, 3 κ T /κ n ω = v teκ n (1 + λ De k y)+ 3 λ De k y (4.5) 43

5. IAW Ion acoustic wave is a fundamental low frequency electrostatic mode of unmagnetized and magnetized plasmas. It has always been treated as an electrostatic mode E = ϕ as follows: 5.1 Linear Dispersion Relation of IAW: Ion dynamics: For B 0 =0; n 0 =0(simple picture) m i n io t v i1 = en i ϕ 1 i T i n i1 (5.1) t n i1 + n i0.v i1 =0 (5.) These two eqs. give, Electron dynamics: µ ω it i k ni1 m i n 0 = T e m i k eϕ 1 T e (5.3) ω << ν te k, ω pe 44

So electron inertia is ignoblue. Then for m e 0, In linear limit (5.4) implies, 0 ' en e E 1 T e n e (5.4) n e1 n 0 ' eϕ 1 T e (5.5) For n e1 ' n i1, (5.3) and (5.6) give the linear dispersion relation of electrostatic IAW, ω = ν s k (5.6) ³ where νs = Te + i T i m i ³ For T i << T e, (5.6) is written as ω = c s k where c s = Te m i. 45

6. IAW and Magnetostatic Mode Here we shall show that transverse magnetostatic mode [Ref.[9]: Chu, Chu & Ohkawa, Phys. Rev. Lett. 41, 653 (1978)] which is obtained in the limit ω << ωpe can couple with IAW in a nonuniform plasma [Ref.[10]: Saleem, Phys. Rev. E54, 4469 (1996)]. Let k = kŷ, n 0 = ˆx dn 0 dx and E =(E 1x,E 1y, 0). Cold ions, t v i1 = e E 1 (6.1) m i Continuity equation, n i1 n 0 = e m i ω (κ ne 1x + ik y E 1y ) (6.) 46

We assume λ De k y 6=0and use Poisson equation which in the limit ω << ω pe becomes, v te k y ω ω pi (ω v te k y ) ω pe ω ik y E 1y ' [ω pi (ω v te k y )+ω pe ω ]κ n E 1x (6.3) Then curl of Amperes Law (4.16) E 1x = 1 a κ n k y (ie 1y ) and (6.3) give a linear dispersion relation as, ω = c sk y(a κ n k y ) (ab κ n k y ) (6.4) where c s = T e m i,andb =(1+λ De k y). 47

Note λ De < λ e and if quasi neutrality is used due to Ampere s law, then (6.4) yields the basic electrostatic IAW dispersion relation ω = c sk y. If the displacement current is retained and Poisson equation is used without using ω, ω pi << ω pe, then one obtains a dispersion relation of coupled three waves; ion acoustic wave, electron plasma wave and high frequency transverse wave [Ref.[11]: Saleem, Watanabe & Sato, Phys. Rev. E6, 1155 (000)]. 7. Low Frequency Electromagnetic Ion Wave Now we present a simple but interesting theoretical model for low frequency electromagnetic waves assuming ions to be cold. The electrostatic waves will also be considered and it will be shown that magnetic field perturbation is coupled with the dominant electrostatic field. For inertialess electrons and E = ϕ, the fundamental low frequency mode is ion acoustic wave, ω s = c sk y (7.) 48

in the quasi-neutrality limit. If dispersion effects are included, one obtains, ω s = c s k y 1+λ De k y (7.3) In the limit 1 << λ De k y, equation (7.3) gives ion plasma oscillations ω = ωpi. It is important to note that in the presence of inhomogeneity, a new scale κ n ky is added to the system. ³ If m e κ, m i << n ky then longitudinal and transverse components of electric field can couple to generate a low frequency electromagnetic wave. The divergence and curl of (4.1) give, respectively, t.(n 0 v e1 )= e m e n 0.E 1 e m e n 0.E 1 1 m e (. p e1 ) (7.4) and t ( v e1 )+(κ n t v e1 )= e m e κ n E 1 e m e E 1 (7.5) 49

where κ n = 1 dn 0 n 0 dx and n 0 =+ˆx dn 0 dx has been assumed. If initially electric field was purely electrostatic i.e. E 1 = ϕ 1, then it will develop a rotating part as well if n 0 E 1 6=0, as is indicated by the right hand side (RHS) of (7.5). The Poisson equation in this case is,.e 1 =4πn 0 e µ ni1 n 0 n e1 n 0 (7.6) Using continuity eqs. Of electrons (4.18) and ions (6.), the above equation canbewrittenas, µ ik y E 1y L 0 W 0 ω L 0 W 0 ω pi ω pe W 0 + v te k y 1 κ n ky = κ n E 1x L 0W 0 ω pi + ω pe ½ W 0 + 3 κ T κ n v tek y µ ¾ 1 κ n ky (7.7) 50

Then (4.16) and (7.7) yield, a L 0W0 ωpe(w 0 + vtek y) 1 κ n/ky = κ n k y ½ ωpe W0 + 3 + L 0W 0 ω ω pi κ T v te 1 κ κ n /k ¾ y n µ a κ n ky (7.8) In the limit ω, ω pi << ω pe, (7.8) gives a linear dispersion relation [Ref.[1], Saleem, Phys. Plasmas 16, 0810 (009)] ω = 1 H 0 (λ ek y)v teκ n µ 3 κ T + a κ κ n/ky c s ky n ½ µ 5 κ 3 T ky + κ T κ n k y ¾ (7.9) If ion dynamics is ignored, (7.9) gives the electromagnetic mode discussed for pure electron plasma. 51

In the electrostatic limit (E 1x =0),(7.9)becomes, ω = 1 H 1 v teκ n µ 3 κ T /κ n + c sky ½ µ 5 κ 3 + T ky + κ T κ n k y ¾ (7.10) where H 1 = (1 + λ De k y)+ 3 λ De k yª. For stationary ions, (7.10) reduces to (4.5). In electron-ion plasma, the electromagnetic wave dispersion relation in the quasi-neutrality limit can be written as, ω = λ eky µ (a κ n/ky) v teκ n 3 κ T + 5 κ n c sky (7.11) ThewavegeometryisshowninFig.7.1. This shows that ion acoustic wave can have electromagnetic part even in the quasi-neutrality limit if temperature perturbation is also considered. 5

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The instability can occur if (4.4) is satisfied along with 5 c s k y < λ ek y (a κ n/k y) v te κ n (7.1) 54

8. Summary 1. Ion acoustic wave which is treated as an electrostatic wave can become electromagnetic due to density inhomogeneity under certain conditions.. Electrostatic and thermal uctuations can also become the source for electromagnetic waves in unmagnetized plasmas. 3. The description of the so-called Magnetic Electron Drift Vortex (MEDV) mode is incorrect. Such a pure transverse mode cannot exist. 4. However a new low frequency wave may exist in nonuniform electron plasma. But this wave is partially transverse an partially longitudinal. 5. Since this wave has frequency closer to ion acoustic wave, therefore it can couple with it. 6. The thermal uctuations also modify the dispersion relation of electrostatic ion acoustic wave in inhomogeneous unmagnetized plasmas. 7. The electromagnetic and electrostatic new modes discussed can become unstable if 3 κ n < κ T in an open system where the steady state is assumed to be given and both density and electron temperature gradients can be parallel. 8. If p e0 =0is used as a condition for steady state then the above modes become stable. However many linear and nonlinear mechanisms can make these perturbations unstable. 55

References J. A. Stamper, E. A. McLean and B. H. Ripin, Phys. Rev. Lett. 40, 1177 (1978). A. Raven, O. Willi, and R. T. Rumsby, Phys. Rev. Lett. 41, 554 (1978). B.A.Altercop,E.V.Mishin,andA.A.Rukhadze,Pis mazh. Eksp. Teor. Fiz. 19, 91 (1974) [JETP Lett. 19], 170 (1974). L. A. Bol shov, Yu, A. Dreizin, and A. M. Dykhne, Pis ma Zh. Eksp. Teor. Fiz. 19, 88 (1974) [JETP Lett. 19, 168 (1974)]. K. A. Brueckner and S. Jorna, Rev. Mod. Phys. 46, 35 (1974). F.F. Chen, Introduction to Plasma Physics and controlled Fusion (Plenum Press, NY (1984)). R. D. Jones, Phys. Rev. Lett. 51, 169 (1983). M. Y. Yu and Xiao Chijin, Phys. Fluids 30, 3631(1987). C. Chu, M. S. Chu and T. Ohkawa, Phys. Rev. Lett. 41, 653 (1978). H. Saleem, Phys. Rev. E54, 4469 (1996). A.Stamper,K.Papadapoulos,R.N.Sudan,S.O.Dean,E.A.McLean,andJ. M. Dawson, Phys. Rev. Lett. 6, 101(1971). H. Saleem, Phys. Plasma 16 0810 (009). 56