The first large-scale simulations of continuously driven, two-dimensional electromagnetic strong plasma turbulence are performed, for electron thermal speeds 0.01c⩽v⩽0.57c, by integrating the Zakharov equations for coupled Langmuir and transverse (T) waves near the plasma frequency. Turbulence scalings and wave number spectra are calculated, a transition is found from a mix of trapped and free T eigenstates for v⩾0.1c to just free eigenstates for v⩽0.1c, and wave energy densities are observed to undergo slow quasiperiodic oscillations

Jenet, F. A.

Melatos, A.

Robinson, P. A.

English

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Scholarworks@UTRGV Univ. of Texas RioGrande Valley

University of Texas Rio Grande Valley ScholarWorks @ UTRGV Physics and Astronomy Faculty Publications and Presentations College of Sciences 2007 Electromagnetic strong plasma turbulence A. Melatos University of Melbourne Frederick A. Jenet The University of Texas Rio Grande Valley P. A. Robinson University of Sydney Follow this and additional works at: https://scholarworks.utrgv.edu/pa_fac  Part of the Astrophysics and Astronomy Commons, and the Physics Commons Recommended Citation Melatos, A.; Jenet, Frederick A.; and Robinson, P. A., "Electromagnetic strong plasma turbulence" (2007). Physics and Astronomy Faculty Publications and Presentations. 335. https://scholarworks.utrgv.edu/pa_fac/335 This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Physics and Astronomy Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact justin.white@utrgv.edu, william.flores01@utrgv.edu. Electromagnetic strong plasma turbulenceA. MelatosSchool of Physics, University of Melbourne, Parkville, VIC 3010, AustraliaF. A. JenetDepartment of Physics and Astronomy, University of Texas at Brownsville, Brownsville, Texas 78520P. A. RobinsonSchool of Physics, University of Sydney, Sydney, NSW 2006, AustraliaReceived 17 October 2006; accepted 11 January 2007; published online 28 February 2007The first large-scale simulations of continuously driven, two-dimensional electromagnetic strongplasma turbulence are performed, for electron thermal speeds 0.01cv0.57c, by integrating theZakharov equations for coupled Langmuir and transverse T waves near the plasma frequency.Turbulence scalings and wave number spectra are calculated, a transition is found from a mix oftrapped and free T eigenstates for v0.1c to just free eigenstates for v0.1c, and wave energydensities are observed to undergo slow quasiperiodic oscillations. © 2007 American Institute ofPhysics. DOI: 10.1063/1.2472293Bursts of electromagnetic EM radiation at the plasmafrequency p and harmonics, coexisting with electrostaticES noise and quasineutral density wells, are often observedin beam-plasma1–3 and ionospheric-heating4,5 experiments inthe strong-plasma-turbulence SPT regime W=0E2 /4n0kBTe kD2, where E21/2 is the rms elec-tric field strength, n0 is the mean electron density, Te is theelectron temperature, k is the characteristic ES wave number,and D is the Debye length. One application of EMSPT is toexperiments where an ultrashort laser pulse is fired into anunderdense plasma, generating slow-moving, long-lived EMwave packets.6–8 It is also relevant to inertial confinementfusion research, e.g., at the National Ignition Facility andLaser Megajoule facility, where Te	80 MK.9 Considerableingenuity is needed to suppress hydrodynamic instabilities atthese facilities, and EMSPT can disrupt symmetry of implod-ing fuel capsules by converting some laser power to free EMwaves near p, and fast electrons via transit-timeinteractions,10,11 which redistribute energy unevenly in laserhot spots.9Theoretical investigations of ESSPT have yielded de-tailed understanding of its nonlinear dynamics e.g., pon-deromotive self-focusing and collapse, wave-energy scal-ings e.g., versus driver parameters, and wave spectra.10,12,13EMSPT has been studied less.4–8,14–22 Many of its propertiesare poorly understood, including the spatial structure of theturbulence, how ES waves trapped in density wells are con-verted to free EM waves, and EM back reaction on the dy-namics. Simulations of ionospheric heating are also incom-plete, because turbulence is evolved electrostaticallyalthough an EM driver is used, and EM radiation is calcu-lated perturbatively without including trapped radiation andits back reaction.4,5In this letter, we report simulations of two-dimensional2D EMSPT in which coupled ES Langmuir L modes andEM transverse T modes evolve self-consistently. The simu-lations, on a 2562 grid, are the first to produce a turbulentstate with many 102 wave packets simultaneously presenthence good statistics, cover a wide range of electron ther-mal speeds 0.01cv= kBTe /me1/20.57c, be continuouslydriven to a statistically steady state, explore dependencies ondamping and driver parameters, and reveal low-temperatureT-wave delocalization. Previous simulations were limited toa 642 grid containing 3 packets at a time, and to an un-driven case with v=0.08c.21 Analytic work has been re-stricted to vc, 	2p where a wave packet radiates as anantenna and the back reaction of T waves is slight,14–16 or tothe weak-turbulence regime W kD2,20 and always for asingle packet rather than a turbulent ensemble.14–16,18,19In the absence of a zeroth-order magnetic field, EMSPTis approximately described by the EM Zakharovequations10,18,19,22 for the slowly varying L-T electric-fieldenvelope Ex , t,iEt=  vT22p  	 	 − vL22p  · +pn2n0− iˆLE , 1defined by E=Ee−ipt, and for the slowly varying, ion-soundlike S, quasineutral density perturbation nx , t, 2t2+ 2ˆSt− vS22nn0=04n0mi2E2 , 2with vS2= 1+3Ti /Teme /miv2, vL2=3v21+ /5 / 1+5+2, vT2=c21+6+62 /5 / 1+5+2, p2= n0e2 /me02+ / 2+6+32, and =v2 /c2. We approximate relativisticexpressions for vL and vT here by Padé formulas,23 but ne-glect the relativistic ponderomotive terms in 2, which tendto be significantly smaller than the ones retained the resultsshould be viewed as semiquantitative due to this neglect andthe approximations inherent in the Zakharov equations,which neglect second-harmonic effects and thus also requirek2vT2,k2vL2p2, the operators ˆLx , t and ˆSx , t describe Land S damping and growth, and T waves are not damped/driven directly. We study the regime of weak ion-sounddamping Ti /Te=0.01, mi /me=183.6 to avoid suppressingS-T coupling.24,25 Previous EMSPT simulations that did notPHYSICS OF PLASMAS 14, 020703 20071070-664X/2007/142/020703/4/$23.00 © 2007 American Institute of Physics14, 020703-1average over fast oscillations in 1 and 2 showed that Tpower at 2p and 3p has only weak or negligible backreaction on the p dynamics,21 so we ignore T waves atharmonics of p. Neglect of harmonics is also implicit in thederivation of the Zakharov equations themselves. We solve1 and 2 using a spectral code with periodic boundaryconditions, verifying that it correctly simulates linear wavepropagation, ES parametric decays, and ESSPT Refs. 10,12, and 13 when the T waves are artificially suppressed.Figure 1 depicts steady-state EMSPT for a v=0.57cand b v=0.01c, via contours of WL,T=0EL,T2 /4n0kBTe. Inboth cases, the two-component spatial structure typical ofSPT is evident: a sea of low-level, incoherent, delocalizedwaves free modes is punctuated by intense, coherent wavepackets trapped modes, which nucleate in pre-existing den-sity wells and collapse to short scales and high fields viaponderomotive self-focusing.10 The L waves are trapped forall v, but T waves delocalize for v0.1c, forming long-wavelength clumps that are larger than, and do not coincidewith, individual density wells, except possibly the sites of themost intense collapses. This new phenomenon is discussedfurther below, and does not occur in ESSPT. In Fig. 1b, forv0.1c, overlapping, expanding, ring-like density depres-sions exist throughout the box. Each ring is a relic of a cir-cular S wave launched by a wave packet in the final stage ofcollapse. Similar rings were seen in simulations of clamp-driven ESSPT, for TiTe but were peanut-shaped.24,25Trapped and free modes in Figs 1a and 1b approxi-mately satisfy the Schrödinger equation it=Zˆ , where  denotes the vector electric field and Zˆ is the operator in pa-rentheses in 1 with n fixed and ˆL=0. As Zˆ does notdepend explicitly on t,   can be expanded as a sum =mcmme−imt of time-independent eigenmodes with Zˆ m=mm. By numerically decomposing trapped modes in Figs1a and 1b into 2D multipole harmonics meim, wefind they are chiefly dipolar, with L ,m=−m, T ,m= zˆ	m, and L,T±1 1+r2 /aL,T2 −1re±i, having peanut-shaped WL and WT contours with perpendicular major axes.The half-widths of the autocorrelation functions of the L andT fields, satisfy aL	a and aT	1.0c /v0.7a, where a is thelength scale of the density well. Within uncertainties over therange investigated, these results are consistent with aT	caL /v3, as expected if the 		 and · terms in 1are comparable and L	T. The density well is roughlycircular, with a fit nr 1+r2 /a2−2, as expected undersubsonic conditions t2vS22, n E2.10Trapped T states cannot exist for low v because T dis-persion overcomes refraction into density wells dug bytrapped L waves. The degenerate eigenvalue of a trapped,dipole m= ±1 T mode is T,±1 /p=4c2D2 /v2aT2+n0 /2n0. Using the empirical result n0 /n0	−31D2 /aL2, with aT as above, we find T,±10 detrap-ping for v0.13c. The L eigenvalue L,±1 /p=12D2 /aL2+n0 /2n0, is negative for all v. This explains the delocal-ized structure of T modes in Fig. 1b; WT contours meanderthrough the box, slightly favoring centers of density wells.Numerically decomposing the free T modes into 2D multi-poles, we find the m2 components have amplitudes com-parable to the noise; they approximate plane waves T,k=c2k2 /2p.Figure 2a displays WL and WT versus t for v=0.1c. Weinitialize Ex ,0 with curl-free noise with a k−2 spectrum tomimic the k	0 condensate of a parametric cascade, and setnx ,0=0.10 Initial rises in WL and WT mark a plane-wavemodulational instability, in which the condensate convertsinto a single, large wave packet that fills the simulation thencollapses. The e-folding time of WL is twice that of WTsecond-order process; growth takes place on the pondero-motive time scale WLp−1 and even faster if n0 initiallymore k-matched decay channels available.1–3 Similar be-havior is seen in undriven simulations for similar parametersv=0.079c but in different initial conditions e.g., singlepacket.21FIG. 1. Color EMSPT plots from the central part of the simulation box fora v=0.57 and b v=0.01, showing n /n0 blue scale; dark negative, lightpositive, with a typical range from circa −0.4 at burnout to 0.1 W betweenpackets, as in Ref. 13, WL=2,4 ,6 ,8 ,10	10−3 yellow contours, and WT=2,4 ,6 ,8 ,10	10−3 red.020703-2 Melatos, Jenet, and Robinson Phys. Plasmas 14, 020703 2007Once the initial modulational instability saturates, at t	400p−1 in Fig. 2a, memory of the condensate is lost andsteady-state turbulence is maintained by a negative dampingdriver at b ,kb= p ,0, of width kb=2kmax/N and peakgrowth given by L=−10−3p, where kmax is the maximum kand N is the number of grid units along each axis.10 PreviousEMSPT simulations were undriven and did not attain asteady state.17,21 We choose kmax=0.31D−1 and time step t2.1p−1v /c2 to follow the fast 		 term in 1. Damp-ing occurs at a rate L=0k+ak /kc2−1, with a=0.04p and kc=0.8kmax; the first term is linear Landaudamping, the second a proxy for coherent transit-time damp-ing, which is set to zero for kkc.10A prominent feature of Fig. 2a, absent from ESSPTdriven at k	0, is the existence of quasiperiodic, long-periodcycles in WL and WT, with period TLT= 1.2±0.2	104p−1,and similar fractional amplitudes factors of up to 2 and4, peak to trough, for L and T, respectively, with this 2:1ratio consistent with nonlinear generation of T waves. Theyintermittently die out and re-emerge, persisting for 50TLT.These global cycles are not the same as the nucleationcycles seen at individual collapse sites,10,12,24 which are alsoobserved here. On cross-correlating the time series, onefinds that WL lags WT by TLT= 7.7±1.1	102p−1, compa-rable to the ponderomotive i.e., modulational time scaleWLp−1. The period TLT varies by less than 20% over therange 0.01cv0.57c i.e., vT	vL at the upper bound,where c /3	0.57c is the sound speed in a relativistic gas;the lower bound is set by the point where k2vT2 becomeslarge, as seen in Fig. 3a, does not depend on box size, andFIG. 2. a Wave-energy density: total W dashed, WL top solid curve, andWT lower solid curve. Left panel shows initial modulational instability;right shows steady state EMSPT. b WT /WL squares and aT /aL triangles.FIG. 3. Angle-averaged spectra arbitrary units for v /c=0.01,0.03,0.1,0.3,0.57. a ELk2, b nk2, and c ETk2.Spectra are averaged over a time 5	104p−1.020703-3 Electromagnetic strong plasma turbulence Phys. Plasmas 14, 020703 2007is 	20 times longer than the growth times of i ES three-wave decay,10 ii EM modulational instabilityL+L→T+T,26,27 and iii sloshing between the mixed L-Tnormal modes of an inhomogeneous plasma. As these cyclesare absent from ESSPT, they likely involve 3- or 4-wavecoupling of L and T modes.Figure 2b shows the mean steady-state value of WT /WLversus v excluding the k=0 mode, which contains 	2% ofthe total energy, with aT /aL also plotted for reference.Least-squares fits yield WT /WLv2.4±0.1 v0.1c andWT /WLv1.2±0.1 v0.1c; the proportionality constants arefunctions of the driver and damping. However, when under-estimates of the low-k part of the spectrum see Fig. 3c,where the lower bound to each curve is determined by thebox size are taken into account, the results are consistentwith WT /WLv1.8 over the whole range investigated. Wealso obtain WLv−0.67±0.03 kb0.67, in accord with Eq.6.20 of Ref. 10 for 2D ESSPT.Wave number spectra WLTk= ELTk2 and nk2= nk2, averaged over time and angle, are shown in Fig.3. For v0.1c, we find power-law inertial ranges at knkkc, with kn= /aL=0.66W1/2D−1, WLkk−3.9±0.4, nk2k−2.3±0.5, and WTkk−5.8±0.8. For our parameters, theminimum nonzero k at the left end of each curve in Fig. 3ais at kminc /p	0.003c /v, the beginning of the inertial rangeat the right end of the shallowly sloped part of each curve isat knc /p	0.04c /v, the upper bound of each curve is atkmaxc /p	0.35c /v	1.25kcc /p, where kc is the upperbound of the Langmuir inertial range. The latter scaling isnew, while the first two agree with the two-component theoryof 2D ESSPT, which predicts WLkk−4 andnk2k−2.10,12,13 Although we have WTWL overall, wefind WTkWLk for kbkkn, where T waves are gen-erated by weak turbulence processes that partition energyroughly equally between modes.For v0.1c, a slight bump interrupts the inertial rangeof WLk at kT= /aT	0.2v /c−0.33p /c. Most T waves aregenerated near when collapse arrests at aL	15D. We findknkTkc for v0.1c. The bump also exists for v0.1c,but falls outside the inertial range kTkn. It may markenhanced focusing of L waves into density wells, deepenedby the additional ponderomotive force of the T waves, but itis unclear why there is no bump in WTk. T waves are notdamped directly, but are converted to damped L and S wavesvia 3- and 4-wave processes. Indirect T damping explains thei absence of a sharp cutoff in WTk at kc, because thedamping operates at all k, and ii sensitivity of WT, but notWL, to ˆS, which affects the steady-state level of S waves.The gradual decline of WTk above kc is due to the shortageof k-matched L waves converting to T waves.Several potential applications are flagged above. One isto interpret particle-in-cell PIC simulations of ultrashort,relativistic laser pulses propagating into an underdenseplasma, where 30%–40% of the incident energy is trans-formed into “solitons” behind the pulse.6–8 The group veloc-ity of most solitons is negligible, consistent with wave pack-ets in Fig. 1 and ESSPT Ref. 10 although in some PICsimulations7 solitons drift backwards at 0.3–0.8c. Solitonsnucleate in laser-generated channels, then collapse. After col-lapse is arrested, the fields relax to fill slowly expanding,ring-like density cavities which trap T modes.9,28 This behav-ior closely matches that of the density depressions inFig. 1b.Good prospects also exist to test our predictions in otherlaboratory situations; e.g., when packets containing coupledL and T modes see Fig. 1a approach a plasma-vacuuminterface, they radiate a burst of EM waves, whose spectrumreflects the field structure of the packet and hence WTk.7 Asthis structure changes at v=0.1c, it is possible in principle todetect the delocalization transition with this diagnostic. Inaddition, we only expect to see ring-like postsolitons for v0.1c. This ability to infer some parameters of SPT e.g., v,WL from remote measurements is also potentially useful inastrophysics14–17 and laboratory experiments demandingnoninvasive diagnostics.ACKNOWLEDGMENTSThe NSF, NASA, Miller Institute for Basic Research inScience, Australian Research Council, and Caltech Centerfor Advanced Computing and Research supported this work.1D. A. Whelan and R. L. Stenzel, Phys. Rev. Lett. 47, 95 1981.2P. Y. Cheung, A. Y. Wong, C. B. Darrow, and S. J. Qian, Phys. Rev. Lett.48, 1348 1982.3M. Schneider and M. Q. Tran, Phys. Lett. 91A, 25 1982.4E. Mjølhus, A. Hanssen, and D. F. DuBois, J. Geophys. Res. 100, 175271995.5P. Y. Cheung, E. Mjølhus, D. F. DuBois, J. Pau, H. Zwi, and A. Y. Wong,Phys. Rev. Lett. 79, 1273 1997.6S. V. Bulanov, T. Zh. Esirkepov, N. M. Naumova, F. Pegoraro, and V. A.Vshivkov, Phys. Rev. Lett. 82, 3440 1999.7Y. Sentoku, T. Zh. Esirkepov, K. Mima, K. Nishihara, F. Califano, F.Pegoraro, H. Sakagami, Y. Kitagawa, N. M. Naumova, and S. V. Bulanov,Phys. Rev. Lett. 83, 3434 1999.8K. Mima, M. S. Jovanović, Y. Sentoku, Z.-M. Sheng, M. M. Škorić, and T.Sato, Phys. Plasmas 8, 2349 2001.9D. Umstadter, J. Phys. D 36, R151 2003.10P. A. Robinson, Rev. Mod. Phys. 69, 507 1997, and references therein.11A. Melatos and P. A. Robinson, Phys. Fluids B 5, 1045 1993.12P. A. Robinson and D. L. Newman, Phys. Fluids B 2, 2999 1990.13P. A. Robinson and D. L. Newman, Phys. Fluids B 2, 3017 1990.14H. P. Freund and K. Papadopoulos, Phys. Fluids 23, 732 1980.15M. V. Goldman, G. F. Reiter, and D. R. Nicholson, Phys. Fluids 23, 3881980.16B. Hafizi and M. V. Goldman, Phys. Fluids 24, 145 1981.17J. Weatherall, Astrophys. J. 483, 402 1997.18J. Gibbons, S. G. Thornhill, M. J. Wardrop, and D. ter Haar, J. PlasmaPhys. 17, 153 1977.19M. Kono, M. M. Škorić, and D. ter Haar, J. Plasma Phys. 26, 123 1981.20P. L. Pritchett and J. M. Dawson, Phys. Fluids 26, 1114 1983.21K. Akimoto, H. L. Rowland, and K. Papadopoulos, Phys. Fluids 31, 21851988.22L. H. Li, Phys. Fluids B 5, 350 1993.23A. Magneville, J. Plasma Phys. 44, 191 1990.24G. D. Doolen, D. F. DuBois, and H. A. Rose, Phys. Rev. Lett. 54, 8041985.25D. Russell, D. F. DuBois, and H. A. Rose, Phys. Rev. Lett. 60, 5811988.26K. Akimoto, Phys. Fluids 31, 538 1988.27F. B. Rizzato and A. C.-L. Chian, J. Plasma Phys. 48, 71 1992.28N. M. Naumova, S. V. Bulanov, T. Zh. Esirkepov, D. Farina, K. Nishihara,F. Pegoraro, H. Ruhl, and A. S. Sakharov, Phys. Rev. Lett. 87, 1850042001.020703-4 Melatos, Jenet, and Robinson Phys. Plasmas 14, 020703 2007

Electromagnetic Strong Plasma Turbulence

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Electromagnetic Strong Plasma Turbulence

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