It is shown that nonlinear three-wave interaction, described by
vector-product type nonlinearities, in pair plasmas implies much more
restrictive conditions for a double energy transfer, as compared to
electron-ion plasmas

J. Vranjes

J. Weiland

S. Poedts

English

arXiv

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Nonlinear three-wave interaction in pair plasmas

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0 Nonlinear three wave interaction in pair plasmas

It is shown that nonlinear three-wave interaction, described by vector-product-type nonlinearities, in pair plasmas implies much more restrictive conditions for a double energy transfer, as compared to electron-ion plasmas.sponsorship: The results presented here are obtained in the framework of Projects No. G.0304.07 (FWO-Vlaanderen), No. C 90347 (Prodex), and No. GOA/2009-009 (K.U. Leuven). Financial support by the European Commission through the SOLAIRE Network (Contract No. MTRN-CT-2006-035484) is gratefully acknowledged. (FWO-Vlaanderen|G.0304.07, Prodex|C 90347, K.U. Leuven|GOA/2009-009, European Commission through the SOLAIRE Network|MTRN-CT-2006-035484)status: Publishe

Vranjes, Jovo

Poedts, Stefaan

Lirias

Nonlinear three-wave interaction in pair plasmasJ. Vranjes* and S. Poedts†Centre for Plasma Astrophysics, and Leuven Mathematical Modeling and Computational Science Research Centre (LMCC),Celestijnenlaan 200B, 3001 Leuven, BelgiumReceived 26 February 2010; revised manuscript received 29 March 2010; published 11 June 2010It is shown that nonlinear three-wave interaction, described by vector-product-type nonlinearities, in pairplasmas implies much more restrictive conditions for a double energy transfer, as compared to electron-ionplasmas.DOI: 10.1103/PhysRevE.81.067401 PACS numbers: 52.35.Mw, 52.30.Ex, 52.27.EpAn electron-ion e-i plasma, with a density gradient per-pendicular to the ambient magnetic field vector, supportsdrift waves driven by the density gradient. In the nonlinearregime, the vector-product-type nonlinearity leads to a three-wave interaction which allows for both a direct and an in-verse energy transfer 1–3, i.e., the transport of energy to-ward both shorter and longer wavelengths. A similarbehavior is obtained also in rotating self-gravitating astro-physical clouds 4, where the inverse energy transfer im-plies the formation of large-scale structures on time scalesmuch shorter than the gravitational contraction.In the case of pair plasmas electron-positron 5,6 orpair-ion plasmas, instead of the drift mode, one obtains con-vective cells 7. In the past, the physics of electron-positronplasmas has been investigated mainly related to problems ofactive galactic nuclei and neutron stars. Nevertheless, experi-mental techniques have been developed for collecting andkeeping positrons, and thus the antimatter plasma has beenthe subject of experimental investigations as well. However,the problem of particle annihilation that is inherent inelectron-positron plasmas is absent in the recently experi-mentally produced pair-ion plasmas 8–11. This experimen-tal success has triggered an increased activity in the field inthe past few years 12–17.In the present work, we shall study the nonlinear waveinteraction in pair plasmas. Hence, we assume a plasma withtwo components of equal mass and opposite chargeelectron-positron or pair-ion, and we use the continuity andmomentum equations for any of the two species in the formtn1n0 +   · v1 +vz1z+ v1 ·  n0 +n1n0 · v1 + v1 n1n0= 0, 1 t+ v1 ·  v1 =qm−  1 + v1  B 0 . 2The indices 0 and 1 here denote equilibrium and perturbedquantities, respectively. We study low-frequency perturba-tions exp−it+ ikr+ ikzz,  /t = qB0 /m, propa-gating nearly perpendicularly with respect to the magneticfield vector B 0=B0ez, i.e., k kz, and we allow for thepresence of the equilibrium perpendicular density gradient n0. It will be shown that the density gradient effects playno role in the present problem.From Eq. 2 we obtain approximatelyv1 =1B0ez   1 −1B0 t+1B0ez   1 ·   1,3 t+1B0ez   1 ·  vz = −qm1z. 4Using these two in Eq. 1, for any of the two species onemay writeLn1 =n0B0L2 1 − n0vz1z−n0B0ez   1 ·   log n0− n1  · v1 − n1vz1z− vz1n1z. 5Here,L =t+1B0ez   1 ·  .Now, writing two Eqs. 5 for the two species and equatingthe corresponding expressions in view of the assumedquasineutrality, it is seen that the terms with the equilibriumdensity gradients exactly cancel each other out. The resultingcombined equation for the species a ,b consequently reads 1B0a−1B0bL2 1 +zvbz1 − vaz1 = 0. 6Applying the operator L onto Eq. 6 once more, and afterusing Eq. 4, one obtains t+1B0ez   1 ·  	 t+1B0ez   1 ·  2 1 = − 221z2 .7Linearizing Eq. 7 one then obtains the dispersion equationfor the electrostatic convective cells as normal modes in this*jovo.vranjes@wis.kuleuven.be†stefaan.poedts@wis.kuleuven.bePHYSICAL REVIEW E 81, 067401 20101539-3755/2010/816/0674014 ©2010 The American Physical Society067401-1pair plasma, propagating almost perpendicularly to the mag-netic field vector,2 = 2 kz2k2 ,  = qB0/m . 8Note that using the Poisson equation instead of the quasineu-trality will only modify the constant on the right-hand sidesof Eqs. 7 and 8, 2→2 / 1+2 / 2p2, p2=q2n0 / 0m. This modification is not essential for our workand will be neglected, implying that 22p2.In order to study the three-wave interaction, we use thenonlinear Eq. 7 assuming the perturbed potential afteromitting the previously used index 1 in the formt = j=1j=3 jtexp− i jt + ik jr +  jtexpi jt − ik jr .9Here,  denotes the complex-conjugate quantity, and weshall use also tj→ − i j +		t, i j  	/	t ,where 	 /	t is the time derivative on a slow time scale, as aconsequence of the nonlinear three-wave interaction. In thiscase, using Eq. 8, from Eq. 6 one obtains the followingexpression for time variation of the jth amplitude:	 jt	t=i2 jkj2 1B0ez   t·  2 +2B0ez    ·  2 t+1B02 ez    ·  ez    ·  2  . 10On the right-hand side in Eq. 10,  should be taken as thesummation 9. In view of the approximative calculation ofthe time-varying mode amplitude, the remaining time andspace derivatives on the right-hand side imply  /t ,  im-plies il , kl l=1,2 ,3, where  appears due tocomplex-conjugate expressions, and out of all terms intro-duced by the summation 9 one should keep only the reso-nant ones, corresponding to the −i jt+ ik j ·r on the left-handside.Without any loss of generality we further assume the fol-lowing resonant conditions:1 = 2 + 3, k1 = k2 + k3. 11The remaining task of calculating the nonlinear terms in Eq.10 is presented below. For the mode 1 ,k1, from the firstterm on the right-hand side in Eq. 10 we have1,1 1B0ez   t·  2 ⇒iB0ez · k3  k22k32 − 3k2223.Note that without the time derivative this nonlinear termwould correspond to its counterpart in the electron-ionplasma 3.Similarly, the second nonlinear term yields1,2 2B0ez    ·  2 t⇒2iB0ez · k3  k23k32 − 2k2223.Consequently, Eq. 10 for the mode 1 ,k1 becomes	1t	t− 13 = −ez · k3  k22B01k12 2k32 − 3k22+ 23k32 − 2k2223. 12The meaning of the term 13 is obvious from Eq. 10; it isthe third nonlinear term there.In the same manner from the time evolution 10 for themodes 2 ,k2 and 3 ,k3, one obtains, respectively,	2t	t− 23 = −ez · k1  k32B02k22 1k32 + 3k12− 23k32 + 1k1213, 13	3t	t− 33 = −ez · k1  k22B03k32 1k22 + 2k12− 22k22 + 1k1212. 14Equations 12–14 describe time evolution of the threemodes due to their nonlinear interaction. For an arbitrary jthmode on the left-hand sides in Eqs. 12–14, the first andsecond nonlinear terms on the right-hand sides must includeterms k and l, where k l including the combination withthe complex-conjugate counterparts, too.On the other hand, the third nonlinear term  j3 includesproducts of all three amplitudes  j ,k ,l and, conse-quently, it cannot contain resonant exponential terms thatwould follow from the interaction between different modes.In other words, for the jth mode on the left-hand sides inEqs. 12–14, the third nonlinear term  j3 can only containthe self-interacting terms of the type cj · j, where at thesame time the interaction coefficient can only include termsof the form kk, ll, where k j and l j. Hence, al-though it describes the variation of the mode amplitude dueto three-wave interaction, it does not directly contribute to apossible energy transfer toward larger and shorter wave-lengths. It is a cubic nonlinearity yielding only a frequencyshifts due to the modulational interaction. As such it in prin-ciple introduces a mismatch in the perfect resonant condition11 for the frequencies, yet these effects will not be dis-cussed here.BRIEF REPORTS PHYSICAL REVIEW E 81, 067401 2010067401-2The double energy transfer can follow only from the firstand second nonlinear terms in Eq. 10; therefore, only thecontribution of those terms will be checked below. For thatpurpose, using the resonant conditions 11 we rewrite Eqs.12–14 in a more symmetric form	1t	t− 13 =ez · k2  k32B01k12 2k32 − 3k22+ 23k32 − 2k2223, 15	2t	t− 23 =ez · k2  k32B02k22 − 1k32 − 3k12+ 23k32 + 1k1213, 16	3t	t− 33 =ez · k2  k32B03k32 1k22 + 2k12− 22k22 + 1k1212. 17It is interesting to compare these equations with drift-waveequations in electron-ion plasma 3, where the terms  jk areabsent and the interaction coefficients for the three modeswithout some unimportant common terms are given, re-spectively, by1 = ez · k2  k3k32 − k2223, 182 = ez · k2  k3k12 − k3213, 193 = ez · k2  k3k22 − k1212. 20The double energy transfer implies an energy transfer towardboth shorter the direct one and longer the inverse transferscales. Taking as an examplek22  k32  k12, 21for an electron-ion plasma, from Eqs. 18–20 it turns outthat 30, implying the double energy transfer and the pre-cipitation of energy from the intermediate mode k3 to theother two modes. In the opposite case k22k32k12, we have30, 1,20 and the intermediate mode receives the en-ergy from the two others. Similarly, fork32  k22  k12, 22from Eqs. 18–20 it is seen that the intermediate mode k2loses the energy, 20, and the double energy transfer takesplace.However, in our case dealing with pair plasmas, for posi-tive frequencies and after using Eq. 11 to eliminate 3,from Eqs. 15–17 after disregarding the obvious positiveand common terms the signs of the coefficients of interac-tion are determined, respectively, by1 = k3221 − 2 − k221 + 2 , 232 = k321 − 22 + k121 + 2 , 243 = k221 − 22 + k122 − 21 . 25As compared with the coefficients  j from an e-i plasma, thecoefficients  j appear to be more complex, with a depen-dence on frequencies, too.For a more detailed comparison with the e-i plasma, wetake the case 21 given above. In order to have the doubleenergy transfer with the intermediate mode yielding the en-ergy, 30, 1,20, from Eqs. 23–25 we obtain the fol-lowing additional conditions for the frequencies and wavenumbers:1  22, k12  k221 − 2221 − 2, k32  k22 1 + 221 − 2.26Hence, the same double energy transfer in the pair-plasmaimposes three more conditions. Yet, it is still possible and asan example we take the following set of numbers: 1=3,2=1, 3=2 and k12=3, k22=1, k32=2. These indeed satisfy allconditions 11, 21, and 26 and allow for the double en-ergy transfer because 1=6, 2=14, 3=−14. From the deri-vations, and in particular from Eq. 10, it is clear that the kjhere imply the perpendicular components of the wave num-bers. On the other hand, because both  and the ratio kz /kremain free except for the assumption that the latter is muchless than unity, the dispersion relation 8 will easily besatisfied, too.Making one more comparison with the e-i plasma, wenow use the condition 22 and analyze the coefficients23–25. The requirement 20, in view of Eq. 11, nowyields22  1  2, k32  k121 + 2/22 − 1 . 27The condition 10 yields k32k221+2 / 21−2 and12 /2, with the latter being always satisfied in view ofthe condition 11. However, the obtained condition for thefrequencies 27 in fact makes the third required condition30 impossible. Hence, contrary to the e-i plasma case,which gives the double energy transfer, in the pair plasma forthis case it is absent. In general, the main reason for thesedifferences is the presence of frequencies in Eqs. 23–25,originating from the necessarily different decoupling ascompared to the e-i plasma used in the derivations of Eq.7.To conclude, the pair plasma introduces a lot of interest-ing physical phenomena, with the nonlinear three-wave in-teraction discussed in this Brief Report being yet anotherexample of it. We stress that pair properties imply that theresults presented here are valid for both homogeneous andinhomogeneous environments.The results presented here are obtained in the frameworkof Projects No. G.0304.07 FWO-Vlaanderen, No. C 90347Prodex, and No. GOA/2009-009 K.U. Leuven. Financialsupport by the European Commission through the SOLAIRENetwork Contract No. MTRN-CT-2006-035484 is grate-fully acknowledged.BRIEF REPORTS PHYSICAL REVIEW E 81, 067401 2010067401-31 A. Hasegawa and K. Mima, Phys. Fluids 21, 87 1978.2 M. Kono and E. Miyashita, Phys. Fluids 31, 326 1988.3 J. Weiland, Collective Modes in Inhomogeneous Plasmas In-stitute of Physics Publishing, Bristol, 2000.4 J. Vranjes and S. Poedts, Phys. Rev. Lett. 89, 131102 2002.5 N. Iwamoto, Phys. Rev. E 47, 604 1993.6 G. P. Zank and R. G. Greaves, Phys. Rev. E 51, 6079 1995.7 J. Vranjes and S. Poedts, Phys. Plasmas 15, 044501 2008.8 W. Oohara and R. Hatakeyama, Phys. Rev. Lett. 91, 2050052003.9 W. Oohara, D. Date, and R. Hatakeyama, Phys. Rev. Lett. 95,175003 2005.10 W. Oohara, Y. Kuwabara, and R. Hatakeyama, Phys. Rev. E75, 056403 2007.11 W. Oohara and R. Hatakeyama, Phys. Plasmas 14, 0557042007.12 J. Vranjes and S. Poedts, Plasma Sources Sci. Technol. 14, 4852005.13 I. Kourakis, A. Esfandyari-Kalejahi, M. Medhipoor, and P. K.Shukla, Phys. Plasmas 13, 052117 2006.14 H. Saleem, J. Vranjes, and S. Poedts, Phys. Lett. A 350, 3752006.15 B. Zhao and J. Zheng, Phys. Plasmas 14, 062106 2007.16 J. Vranjes, D. Petrovic, B. P. Pandey, and S. Poedts, Phys.Plasmas 15, 072104 2008.17 R. Gill and J. S. 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Nonlinear three wave interaction in pair plasmas

Nonlinear three wave interaction in pair plasmas

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