Bingham R

Kellett BJ

Quest KB

Shapiro VD

Ucer D

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The modified two stream instability at nonmagneticplanets.R. Bingham,† V.D. Shapiro,‡ K.B. Quest,∗ D. Üçer,‡ and B.J. Kellett,†January 29, 2014†Rutherford Appleton Laboratory, Chilton,Oxon,OX11 0QX, UK, ‡Department ofPhysics, University of California, San Diego, La Jolla, CA 92091, ∗ECE Department,University of California, San Diego, La Jolla, CA 92091,AbstractWe describe the role the modified two stream instability plays in the interac-tion of the solar wind with non-magnetized planets. The instability leads to theproduction of energetic electrons that can be responsible for the observed x-rayemission.At planets without intrinsic or very weak magnetic fields (Venus, Mars) an ionosphereis the only, however weak, obstacle to the solar wind flow, and the planetary bow shocksare located quite close to the planet. The ionospheres of both Venus and Mars aretherefore directly exposed to the solar wind. Data from particle and wave experimentson board Pioneer-Venus Orbiter (PVO) and Phobos-2 spacecrafts revealed the presenceof a thin ∆ ∼ 100km turbulent transitional layer at the boundary of both ionospheresusually referred as the plasma mantle (Taylor et al., 1980). In the mantle two types ofplasmas – solar wind and the ionospheric plasma are present with comparable densitiesn ∼ 102cm−3, but however with very different temperatures. The solar wind proton(electron) temperature is equal to 100(30)eV , and the solar wind drift velocity is closeto the proton thermal velocity. For the planetary plasma with oxygen the main ionicspecies, the oxygen and electron temperatures are close to 1 eV . There is also experimentalevidence for the presence of suprathermal electrons with energies exceeding 100 eV (Shutteet al., 1989; Szego et al., 1997) and oxygen (Taylor et al., 1980) populations in the mantleof both Mars and Venus. Finally, the large tailward escape of the planetary ions, most12probably originating from the dayside mantle, has been observed at Mars (Rosenbaueret al., 1989). Since the mantle thickness is much less than the gyroradius of the pickedup planetary ion, the usual scheme of E × B pick up of the planetary ions by the solarwind flow is not applicable. Ion pickup of a collective nature organized by waves mustbe responsible for the observed tailward planetary ion escape. This has been proposedfirst for Mars (Sagdeev et al., 1990) and then for Venus (Szego et al., 1991). Intensewave activity at the dayside Venusian mantle was observed by the 100Hz channel of thePVO electric field detector with amplitudes of the order of around tens of mV/m (Scarfet al., 1980), and even stronger wave activity in the 5 − 50Hz frequency range has beenobserved by the Phobos-2 spacecraft at the boundary of the Martian ionosphere (Grard etal., 1989), both these wave frequency ranges contain the lower-hybrid resonance frequency.There are two types of instabilities that can be responsible for the observed waveactivity at the planets, the modified two stream instability MTSI (McBride et al., 1972;Sagdeev et al., 1990; Shapiro et al., 1995a) and current driven ion acoustic instability(Huba, 1993). Characteristics of the waves excited by these two instability mechanismsare quite different. The MTSI results in the excitation of sufficiently long wavelengthoscillations with wavelength of the order of the solar wind electron gyroradius and withfrequency close to the lower-hybrid frequency (30 − 40Hz for the mantle conditions). Atthe same time waves excited by the ion acoustic instability have much shorter wavelengthsof the order of electron Debye scale lengths and frequencies close to the ion plasma fre-quency (∼ 1kHz for the mantle conditions). It was proposed by Crawford et al. (1993)that background noise due to the rapid variations of the plasma parameters at the iono-spheric boundary, especially of Debye length may be generated contributing to the 100Hzsignal. At the same time the averaging procedure used in (Shapiro et al., 1995a) for thewave data of approximately 50 orbits demonstrated that the 100Hz signal is dominantand real. Finally, a full statistical analysis of the peak values of the wave data performedrecently by (Strangeway and Russell, 1996) demonstrated that the 100Hz signals are realand well correlated with ionopause crossings. It is generally believed now that the peakat 100Hz (unfortunately, measurements at the lower frequency have not been availableat PVO) is not excluded from being natural. Therefore, the experimental data demon-strate that the mantle turbulence is dominated by lower-hybrid waves, and a theoreticalmodel explaining this is needed. We proposed such a model based on the modified twostream instability (MTSI). The model is also applicable to the study of the solar windwith artificial releases of plasma, such as in the AMPTE experiments (Bingham et al.,1991).Analysis of the dispersion relation of MTSI in a 4 component plasma (2 ion and 2 elec-tron components, corresponding to the solar wind and the ionospheric plasmas) has beenperformed in (Shapiro et al., 1995a). It was concluded that two branches are unstable.3In the vicinity of the lower-hybrid frequency there is a fluid-type branch correspondingto the waves propagating almost transverse to the magnetic field. At the same time thewave data above the lower-hybrid frequency may be attributed to the kinetic branch ofthe MTSI, corresponding to waves propagating obliquely to the magnetic field and havingfrequencies several times above the lower-hybrid frequency. A new branch of the MTSIbetter matching the observational data has been proposed (Dobe et al., 1999). Thisbranch is generated by the interaction between the ionospheric oxygen ions and the coldelectron beam created by E × B solar wind pick up.Electron pick up takes place during several electron gyroperiods, a time interval muchshorter than the lower-hybrid wave period. Although the cold electron beam has littlekinetic energy of its own, the resonance between the beam and waves can be sustainedfor a longer time due to electron pick up by the solar wind under the combined action ofthe magnetic and the convective electric field in the solar wind. This makes it possibleto transfer a substantial amount of the energy from the proton flow to the waves. Theexcited waves are predominantly electrostatic ωpe ≪ kc. They propagate over a range ofangles θ ∼ 60 − 90o with respect to the ambient magnetic field.The dispersion relation describing wave excitation can be written asω2peω2ce(1 +ω2pek2c2)−ω2pe(ω − kue)2k2∥c2k2c2 + ω2pe+1 + wpZ (wp)k2D2p+1 + Z (wi)k2D2i= 0 (1)A similar dispersion relation for a two component plasma (the proton flow through elec-trons) has been derived by (Hsia, 1979) in the hydrodynamical approximation, the kineticeffects have been taken into account by (McBride and Ott, 1972). In Eq.(1) we made thesame assumptions as in the numerical simulations described below only one component ofthe cold fluid electrons is present, both protons and planetary ions are treated kinetically.The following notation is usedwp =ω − kupkvTp, wi =ωkvTi,and Z(z) =∫∞−∞ dξe−ξ2z−ξ is the plasma dispersion function. The notation vTp (vT i) forthermal velocities of the protons (ions) and ue, up for the drift velocities of electrons andprotons are used. It is assumed that the wave frequency satisfies conditions ωcp ≪ ω ≪ωce, correspondingly electrons are magnetized in these oscillations, while protons and ionsare not.4The most unstable mode in the solution of the dispersion relation (1) corresponds tothe case when the Doppler shifted wave frequency (ω − kue) coincides with the frequencyof the plasmas eigenmode, in our case the whistler wave. The instability is similar tothe well known Buneman instability (Buneman, 1959) of the unmagnetized plasma, whenthe Doppler shifted wave frequency coincides with the Langmuir frequency. As for theBuneman instability, ω ≪ ku, the resonance condition can be written askue = ωcek∥kc2k2c2 + ω2pe(2)The numerical solution of the dispersion relation is plotted in Fig. 1. The growth rateof the most unstable mode, satisfying condition (2) is shown as a function of the modefrequency. A parameter which changes along the curve is κ =k∥k√mpme, i.e. the angle ofwave propagation with respect to the ambient magnetic field. For κ > 1 the frequencyincreases almost linearly with κ, while the growth rate reaches its maximum at κ = 6 andthen slowly decreases with κ. Triangles in Figure 1 show the values of the frequency andthe growth rate of the most unstable mode obtained in the numerical simulations of (Dobeet al., 1999) for the different values of κ listed in the figure. As can be seen analytical andnumerical results are in a good agreement. Simulations of the MTSI have been carriedout by Quest et al. (1997) and Dobe et al. (1999) using a hybrid code with kineticions and finite mass electrons. These simulations follow the nonlinear development of theMTSI within the mantle. The linear and nonlinear stages of evolution of this branch of theMTSI has been investigated using the hybrid particle simulation code (Quest et al., 1997).The code treats two ion species (protons and oxygen ions) using particle in cell methods,the electrons are modelled as a fluid, in which inertial and electromagnetic effects areretained. It conserves the total energy and momentum with an accuracy better than onepercent. In the simulations of Dobe et al. (1999) waves are permitted to propagate at anarbitrary angle with respect to the magnetic field with the boundary conditions (periodicin x) and initialisation (a relative drift between Maxwellian protons and ions) the sameas in Quest et al. (1997). The parameters used in the simulations are the following: thesolar wind is initialized with the drift velocity up = 1.2vTp in the x direction, vTp =√Tpmpis the thermal velocity. The density of protons and oxygen ions are assumed to be equal,and the β for the different plasma components are βp = 0.8, βi = 0.0032 and βe = 0.016,where βj is the ratio of plasma thermal pressure to the magnetic pressure for the jthcomponent. If we assume that electron number density is 102cm−3, then the simulationcorrespond to the case of Tp = 10eV, Te = 1eV, Ti = 0.2eV, up = 120km/s and B = 50nT .Conditions of quasineutrality ne = np + ni and zero net current in the flow (x) directionue = npup/ne =12up are imposed on the simulations. The latter condition is a consequenceof the one dimensionality and the assumption about the absence of the magnetic field5shear, releasing the condition of zero net current in the analytic dispersion relation it ispossible to prove that this condition does not alter the evolution of instability, at leastin its linear stage. Results of the simulation are presented in Figures 2 and 3. Theseresults correspond to the case of κ > 1, smaller κ values have been considered (Quest etal., 1997), but have not been described in this section.A particle in cell code in which the kinetic evolution of the electron can also be followedhas been developed (Quest et al., 2001). In these 1–D particle in cell simulations, Questet al. (2001) demonstrated that significant electron acceleration and heating can occur inthe mantle region as a result of the MTSI.In January 2001, Venus was observed for the first time using an X-ray satellite (Dennerlet al., 2002). Venus is clearly detected as a half-lit crescent, the X-ray emission has beenexplained as fluorescence scattering of solar X-rays (Dennerl et al., 2002). The emissionis observed at discreet energies, mainly the oxygen Kα line at 0.53 keV , the carbon Kαline at 0.28 keV , and marginally the nitrogen Kα line at 0.39 keV . Fluorescent scatteringproduced by collision between the neutral atmosphere and the energetic electrons cannotbe ruled out as another explanation. In fact the lack of solar variability observed in theX-ray spectrum tends to favour the energetic electrons as the source of X-rays.It is possible to estimate the fluorescence intensity from energetic electron collisionsusing the following expression for the total X-ray luminosity (Bingham et al., 2008)LFℓ = 10−17 π Λ nonTe R2 ∆R1√εe(eV )ergs/secwhere the numerical factor Λ for oxygen (Z = 8) atoms is 0.1. Using typical planetaryand energetic particle values no = 105cm−3, nTe = 102cm−3, εe = 100eV and ∆R = 107cmwhere no is the neutral density at the mantle of width ∆R and distance from planetR = 109cm, nTe is the number of accelerated electrons and εe their typical energy wecalculate the fluorescence luminosity due to energetic electrons as LFℓ ≈ 1013 erg/sec.The electrons are accelerated by the spectrum of the observed lower-hybrid waves.Acknowledgments This Paper is dedicated to the memory of Professor Vitali D.Shapiro. This work was partially supported by NASA grant NAG5-11754 as well as theScience and Technology Facilities Council’s Centre for Fundamental Physics (CfFP).6References[Bingham et al.(1991)] Bingham R., Shapiro V. D., Gil’Man M., Tsytovich V. N., deAngelis U., 1991, PhFlB, 3, 1728[Bingham et al.(2008)] Bingham R., Quest K. B., Shapiro V. D., Kellett B. J., 2008,AnGeo, 26, 1829[Buneman(1959)] Buneman O., 1959, PhRv, 115, 503[Crawford T. et al.] in Plasma Environments at non Magnetic Planets, ed, by T. Gombosi(Pegamon Press), p.253 (1993).[Dennerl et al.(2002)] Dennerl K., Burwitz V., Englhauser J., Lisse C., Wolk S., 2002,A&A, 386, 319[Dobe et al.(1999)] Dobe Z., Quest K. B., Shapiro V. D., Szego K., Huba J. D., 1999,PhRvL, 83, 260[Grard et al.(1989)] Grard R., Pedersen A., Klimov S., Savin S., Skalskii A., 1989, Natur,341, 607[Hsia et al.(1979)] Hsia J. B., Chiu S. M., Hsia M. F., Chou R. L., Wu C. S., 1979, PhFl,22, 1737[Huba(1993)] Huba J. D., 1993, GeoRL, 20, 1751[McBride & Ott(1972)] McBride J. B., Ott E., 1972, PhLA, 39, 363[McBride et al.(1972)] McBride J. B., Ott E., Boris J. P., Orens J. H., 1972, PhFl, 15,2367[Quest et al.(1997)] Quest K. B., Shapiro V. D., Szegö K., Dobe Z., 1997, GeoRL, 24,301[Quest, Ucer, & Shapiro(2001)] Quest K. B., Ucer D., Shapiro V. D., 2001, APS..DPPB,1022P[Rosenbauer et al.(1989)] Rosenbauer H., Shutte N., Galeev A., Gringauz K., Apathy I.,1989, Natur, 341, 612[Sagdeev et al.(1990)] Sagdeev R. Z., Shapiro V. D., Shevchenko V. I., Zacharov A., Ki-raly P., Szego K., Nagy A. F., Grard R. J. L., 1990, GeoRL, 17, 8937[Scarf et al.(1980)] Scarf F. L., Taylor W. W. L., Russell C. T., Elphic R. C., 1980, JGR,85, 7599[Shapiro et al.(1995a)] Shapiro V. D., Szegö K., Ride S. K., Nagy A. F., Shevchenko V. I.,1995a, JGR, 100, 21289[Shutte et al.(1989)] Shutte N. M., Diachkov A. V., Kiraly P., Cravens T. E., GombosiT. I., 1989, Natur, 341, 614[Strangeway & Russell(1996)] Strangeway R. J., Russell C. T., 1996, JGR, 101, 17313[Szego et al.(1991)] Szego K., Shapiro V. S., Shevchenko V. I., Sagdeev R. Z., KasprzakW. T., Nagy A. F., 1991, GeoRL, 18, 2305[Szegő et al.(1997)] Szegő K., Dóbé Z., Knudsen W. C., Nagy A. F., Shapiro V. D., 1997,JGR, 102, 2175[Taylor et al.(1980)] Taylor H. A., Brinton H. C., Bauer S. J., Hartle R. E., Cloutier P. A.,Daniell R. E., 1980, JGR, 85, 776581 Figure captionsFIG. 1: Relationship between the maximum growth rate and wave frequency for thedifferent κ values (κ =k||k√mpme), solid line analytical calculations, triangles correspond tosimulation results of the dispersion relation (89) for the modified two stream instability.FIG. 2: Phase space plots of protons and oxygen ions at saturation of the linearinstability (left hand plots) and in the fully developed turbulent phase (right hand plots).FIG. 3: Evolution in time for different k of the bulk momentum for (a) solar windprotons and (b) oxygen ions. Both are normalised to the initial proton bulk momentum.9Figure 1:10Figure 2:11Figure 3:

The modified two stream instability at nonmagnetic planets

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