Bers, A.

Ram, A.K.

Strozzi, D.J.

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PSFC/JA-04-24   Electrostatic Fields in Density Cavities and Nonlinear Energization of Ions  A. K. Ram, D. J. Strozzi, and A. Bers   July 2004      Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge, MA 02139  U.S.A.    This work was supported by the National Science Foundation, Grant No. ATM-98-06328.  Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.   To appear in Proceedings of the 31st European Physical Society (EPS) Conference on Plasma Physics, London, England, June 28–July 2, 2004. Electrostatic Fields in Density Cavities and NonlinearEnergization of IonsA. K. Ram, D. J. Strozzi, and A. BersPlasma Science & Fusion Center, M.I.T, Cambridge, MA 02139, U.S.A.IntroductionThere are two events that are particularly ubiquitous in space plasmas. One is theexistence of short scale-length (of the order of a few ion Larmor radii) density gradientsand the other is transverse (to the geomagnetic field) energization of ionospheric ionswhich, eventually, make their way into the outer reaches of the terrestrial magneto-sphere. Rocket observations have shown the existence of density depleted structuresin the auroral ionosphere which are associated with enhanced electric fields [1, 2].These structures are a few tens of meters in width transverse to the geomagnetic fieldand are believed to be of the order of 100 km along the geomagnetic field. Associatedwith these enhanced fields, transversely accelerated ions with energies in the rangeof tens of eV are observed [1, 2]. These energies should be compared to the ambientthermal energies of approximately 0.3 eV.Resonant Electrostatic Fields in Density GradientsIf we assume a cylindrical density depression with its radial density variation beingacross the geomagnetic field, then, in the cold plasma approximation, the spatialevolution of the electrostatic potential is given by:d2φdr2+{1r+1K⊥dK⊥dr}dφdr+{mr1K⊥dK×dr− m2r2− k2zK‖K⊥}φ = 0 (1)where we have assumed that the azimuthal, longitudinal, and temporal variations ofthe fields are of the form exp(imθ + ikzz − iωt),K⊥ = 1−ω2peω2 − ω2ce−∑iω2piω2 − ω2ci(2)K× = − ωωceω2peω2 − ω2ce+∑iωciωω2piω2 − ω2ci(3)K‖ = 1−ω2peω2−∑iω2piω2(4)1where ωpe and ωpi are the electron and ion plasma frequencies, respectively, and ωceand ωci are the electron and ion cyclotron frequencies, respectively. The plasma andcyclotron frequencies are assumed to have a radial dependence due to the radial varia-tion of the plasma density and the geomagnetic field. Generally the radial scalelengthof the density depression is much smaller than the radial scalelength of the geomag-netic field variation, so that the geomagnetic field can be considered constant overthe transverse dimension of the density depression. From (1), it is clear that thedifferential equation for φ has a regular singularity at K⊥ = 0 corresponding to eitherthe lower hybrid resonance frequency ωLH or the upper hybrid resonance frequencyωUH . In the vicinity of either of these resonances, φ has a logarithmic singularity.For space plasmas of interest, we have considered a linear electron density profile:n =n0, for r ≤ r1n0 + (n1 − n0)(r − r1r2 − r1), for r1 < r < r2n1, for r ≥ r2where n0 and n1 are constants. In the regions r < r1 and r > r2, the solution to(1) can be readily expressed in terms of ordinary Bessel functions or modified Besselfunctions, depending on whetherK‖/K⊥ is less than or greater than zero, respectively.The solutions have to further satisfy that φ remains finite as r → 0 and as r → ∞.The complete solution to (1) over the entire density gradient is obtained by requiringthat φ match onto these Bessel function solutions at r = r1 and r = r2. This thenbecomes an eigenvalue problem. For fixed frequency ω the eigenvalue is kz while forfixed kz the eigenvalue is ω.For illustrative purposes, let us assume the following parameters which correspondto the plasma conditions in the auroral ionosphere around an altitude of 1000 km:n0 = 3.4× 109 m−3, n1 = 4.3× 109 m−3, singly charged oxygen O+ and hydrogen H+ions with the density ratio of 9:1, respectively, and B0 = 0.36 Gauss. For a fixed ωcorresponding to K⊥(r0) = 0 for r1 < r0 < r2 we solve, numerically, (1) for the eigen-value kz. It is worth noting that kz will depend on ω and the width of the gradientregion ∆r = r2− r1 and not on the specific values of r2 and r1. We find, numerically,that for a given ω and ∆r there are two distinct kz’s which satisfy the boundaryconditions. For ∆r = 1 m and a frequency of fLH = ω/2pi = 4.55 kHz (correspondingto the lower hybrid resonance being halfway up the density gradient), the two eigen-values are kz1 = (0.022 + 0.006i) m−1 and kz2 = (0.011 − 0.008i) m−1. If we choose∆r = 10 m and the same frequency, then the two eigenvalues are kz1 = (0.029+0.3i)m−1 and kz2 = (0.023 − 0.02i) m−1. Thus, the lower hybrid fields propagate along2the geomagnetic field for sharp density gradients and are essentially evanescent formore gradual density gradients. It is worth noting that even when the imaginarypart of kz is small compared to the real part, the fields are not “propagating” overlong distances before damping out spatially. The spatial extents are typically of theorder of 100 m. These distances are relatively short compared to what is inferredfrom observations. Hence such fields would have to be maintained by a source of freeenergy (e.g., an independent streaming instability) inside the density depleted region.Interaction of Ions With Localized FieldsThe interaction of ions with localized field structures is different from that witha plane wave or a set of plane waves, which we have studied in the past [3, 4]. Theprimary difference is that since these structures are smaller than the ion Larmorradius, the interaction of the ions with the fields occurs over only a small fraction oftheir orbit. Let us consider the situation in which the wave fields are independent ofthe coordinate along the geomagnetic field. We can then consider just the motion ofthe ions in a plane perpendicular to the geomagnetic field. For an ion in a uniformmagnetic field ( ~B = B0zˆ) interacting with electrostatic waves propagating in theradial direction across ~B, the equations of motion are:drdt= vr,dθdt=vθr(5)dvrdt= vθ(Ω +vθr)+QMEr,dvθdt= −vr(Ω +vθr)(6)where Ω is the angular ion cyclotron frequency, and Q andM are the charge and massof the ion, respectively. Consider the following form of a spatially localized electricfield that models the resonant electric fields in density gradients:Er = E0e−β (r − a)2 ∑nsin (knr +mnθ − ωnt+ φn)where a is the radius of the cavity, E0 is the electric field amplitude, β determinesthe radial width of the field, and kn, mn, ωn and φn are the radial wave vector,azimuthal mode number, frequency, and phase, respectively, of the n-th component.The equations of motion can be integrated numerically to determine the effect of thelocalized fields on the ion orbits. Our analysis shows that β À kn is required for ionenergization to occur. We set kn = 0 in the following results. For the ionosphericparameters discussed earlier, we consider the interaction of O+ ions with a single wavecomponent (n = 1, ω1 = ω). We assume that a = 10 m and ω/ωcO+ = 145.74. For an3ambient temperature of 0.33 eV, the initial Larmor radii of O+ is ρO+ ≈ 6.5 m. Thetop figure in Figure 1 shows the normalized Larmor radius ρ = ρO+/a as a functionof the normalized time for three values of the normalized electric field amplitude² = QE0/(MΩ2a). (For E0 = 200 mV/m ² = 2.6.) It is clear that there is athreshold in the amplitude of the electric field before ion energization takes place.The energization of low energy ions is chaotic from the very beginning. This is unlikethe case of ion interaction with coherent plane waves where the low energy ions are,initially, coherently energized before their motion becomes chaotic [3, 4]. The bottomfigure in Fig. 1 shows the change in Larmor radius due to varying widths of theinteraction region. Again there is a threshold in width, for a given amplitude, beyondwhich ion energization does not take place.In the case when the ratio of the wave frequencies to the ion cyclotron frequencyis a low number (corresponding to the interaction of H+ ions with lower hybrid struc-tures), we find that when two wave frequencies are separated by an integer multipleof the ion cyclotron frequency, the energy of the ions can increase monotonically as afunction of time. The monotonic increases in energy are akin to Le´vy flights [5]. Animportant observable signature of Le´vy flights is that the ion distribution functionwill have long tails. Such ion distribution functions are indeed observed [2].Work supported by NSF Grant number ATM-98-06328.References[1] P. M. Kintner, J. Vago, S. Chesney, R. L. Arnoldy, K. A. Lynch, C. J. Pollock,and T. E. Moore, Phys. Rev. Lett. 68, 2448 (1992).[2] K. A. Lynch, R. L. Arnoldy, P. M. Kintner, P. W. Schuck, J. W. Bonnell, andV. Coffey, J. Geophys. Res. 104, 28515 (1999)[3] D. J. Strozzi, A. K. Ram, and A. Bers, Phys. Plasmas 10, 2722 (2003).[4] A. K. Ram, A. Bers, and D. Be´nisti, J. Geophys. Res. 103, 9431 (1998);D. Be´nisti, A. K. Ram, and A. Bers, Phys. Plasmas 5, 3244 (1998).[5] E. W. Montroll and B. J. West, in Fluctuation Phenomena, eds. E. W. Montrolland J. L. Lebowitz (New York: North-Holland), 1979, Chapter 2.40 500 1000 1500 2000 2500 300000.511.522.5Ω tρ0 500 1000 1500 2000 2500 300000.511.522.5Ω tρFigure 1: Normalized Larmor radius fora O+ ion versus normalized time. Top:√βa2 = 6.25 × 103 with ² = 1.3 (blue),2.6 (green), and 5.2 (red). Bottom: ² = 5.2with√βa2 = 1.56×103 (green), 6.25×103(red), and 2.5× 104 (blue).5

Electrostatic Fields in Density Cavities and Nonlinear Energization of Ions

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Electrostatic Fields in Density Cavities and Nonlinear Energization of Ions

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