Two dimensional hybrid code simulation of electromagnetic ion cyclotron waves of multi ion plasmas in a dipole magnetic field

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009ja015158, 2010 Two dimensional hybrid code simulation of electromagnetic ion cyclotron waves of multi ion plasmas in a dipole magnetic field Y. Hu, 1 R. E. Denton, 1 and J. R. Johnson 2 Received 3 December 2009; revised 9 April 2010; accepted 27 May 2010; published 24 September [1] A two dimensional hybrid code (particle ions and fluid electrons) is used to simulate EMIC waves in a H + He + O + plasma in a dipole magnetic field. The waves are driven by energetic ring current protons with anisotropic temperature (T?p /T kp >1). The initial state of the plasma is derived from an anisotropic MHD code so that the system is in MHD equilibrium, J B r P = 0. The cold species (with temperature of ev) are assumed to be isotropic and have a spatially uniform density distribution. We choose our parameters so that the EMIC waves are generated near the magnetic equator with frequencies W O + < w < W He +. The presence of each heavy ion species introduces a new dispersion surface. When the waves grow near the equator, they are dominantly left handed polarized and have small wave normal angle. While propagating toward high latitudes, the waves become linearly or right handed polarized with a larger normal angle, and they encounter the second harmonic of the O + cyclotron frequency, the He + O + bi ion frequency, and possibly the first harmonic of the O + cyclotron frequency. In this process, some waves are absorbed by the wave particle interaction, some waves are reflected by the He + O + bi ion frequency, some are transmitted on the same dispersion surface, and some may tunnel through the so called stop band. The relative importance of these effects varies with the ion composition and especially with the concentration of O +, h O + = n O +/n e. For instance, for h O + 0.5%, essentially all the wave energy passes through the resonances to reach the ionospheric boundary. For h O+ = 0.5% (the case examined in most detail), the time averaged Poynting vector at high latitudes is almost always in the poleward direction, even though clear evidence of some reflection at the He + O + bi ion resonance is seen. Citation: Hu, Y., R. E. Denton, and J. R. Johnson (2010), Two dimensional hybrid code simulation of electromagnetic ion cyclotron waves of multi ion plasmas in a dipole magnetic field, J. Geophys. Res., 115,, doi: /2009ja Introduction [2] The Electromagnetic Ion Cyclotron (EMIC) instability is driven by a hot anisotropic ion population with an approximate threshold condition T q ffiffiffiffiffiffi?i 1 ki > c T ; ð1þ T ki where T?i (T ki ) is the ion temperature perpendicular (parallel) to the ambient magnetic field, b ki =8pnT ki /B 0 2 (CGS units) is the parallel ion plasma beta, and c T is the threshold constant whose value usually varies from 0.1 to 1, depending on the initial plasma condition [Anderson et al., 1994; Gary and Lee, 1994; Gary et al., 1994a, 1994b, 1995]. EMIC waves are believed to be excited near the equator in 1 Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire, USA. 2 Plasma Physics Laboratory, Princeton University, Princeton, New Jersey, USA. Copyright 2010 by the American Geophysical Union /10/2009JA the magnetosphere where the energetic ring current protons provide the free energy. [3] The EMIC instability has been studied using several techniques. For a proton electron plasma, the linear theory [Gary, 1993] shows the waves have frequencies below the proton cyclotron frequency W H + and the most unstable mode appears when the wave vector k is parallel to the magnetic field. These waves are left hand circularly (elliptically) polarized for parallel (oblique) propagation. Numerical simulations using hybrid codes (particle ions and fluid electrons) have been deployed by several researchers to study the EMIC instability. Both one dimensional [Winske and Omidi, 1992; Denton et al., 1993; Gary and Lee, 1994] and two dimensional [Winske and Omidi, 1992; McKean et al., 1994] simulations in straight geometry with a uniform magnetic field have provided good agreement with the linear theory. More recently, Hu and Denton [2009], hereafter H&D, performed a two dimensional hybrid simulation in a dipole magnetic field which revealed how the polarization changes from dominantly left hand to linear as the wavefronts turned oblique due to the inhomogeneity of the magnetic field, and they estimated the energy propagation rate in the radial direction and also the coherence length 1of13

2 Figure 1. Dispersion curves for a three ion plasma. The frequency is normalized to W p, and the wave number is normalized to c/w pp. The instability is driven by hot protons with T?p /T kp =2,b kp = 0.8, and density n p =5cm 3. The cold isotropic species include H +,He +,O +, and electrons, whose b k are , , , and 0, respectively; densities are 31 cm 3,7cm 3,7cm 3, and 50 cm 3, respectively; B 0 = 100 nt. The sold (dashed) dispersion curves indicate left handed (right handed) polarization for parallel propagation. The curves labeled with the same letter belong to the same surface. The bi ion frequency w H He 0.34 W p, w He O 0.09 W p. The top shows the normalized growth rate for curves H and He for parallel propagation. associated with the EMIC wave structure. This current paper is a further development of H&D s work; multiple cold ions, including hydrogen (H + ), singly charged helium (He + ), and singly charged oxygen (O + ), are now included. [4] The presence of cold heavy ions such as He + and O +, common on the dayside in the magnetosphere where EMIC waves are observed, dramatically modifies the wave behavior, and also lowers the threshold condition (1) for instability [Gary et al., 1994b, 1995]. Each new heavy species introduces a cyclotron resonance and a bi ion resonance. The dispersion curves for a three ion (H +,He +, and O + ) component plasma are shown in Figure 1. (See Figure 1 caption for the detailed parameters.) For simplicity, we only draw dispersion curves for near parallel and perpendicular propagation. The curves belonging to the same dispersion surface are labeled with the same letters, and the solid (dashed) curves for parallel propagation indicate lefthand (right hand) polarization. Note that the dispersion curves cross at the crossover frequency for strictly parallel propagation (k? = 0) but remain uncoupled. For any oblique propagation (k? > 0) the surfaces do not cross and are separated, but the polarization on the same surface may change sign at frequencies close to the crossover frequency. (A more complete view of wave surfaces for a H + He + plasma can be found in the work of André [1985].) Generally, surface H lies above surface He, which lies above O, and the frequency of surface O approaches zero for perpendicular propagation. For parallel propagation, there are three resonances: W H +, W He +, and W O +, meaning the cyclotron frequency of H +,He +, and O +, respectively; for perpendicular propagation, there are two resonances: the H + He + bi ion frequency w H He and the He + O + bi ion frequency w He O ; there are also two cutoff frequencies and two crossover frequencies. In the cold plasma approximation, the values of these frequencies typically depend on the concentration of each species h a n a /n e, where a indicates a particle species. Figure 1 (top) shows the growth rate for the two dispersion curves H and He for parallel propagation (the positive growth rate for all other curves is negligible and is not shown). It is clear that for these parameters, the 2of13

3 waves on the He surface are more unstable than those on the H surface. For each surface, the growth rate is largest for the parallel direction and becomes negative (waves are damped) when k? increases. The growth rate is approximately zero or negative for perpendicular propagation. [5] For a dipole magnetic field, EMIC waves can be generated near the magnetic equator and travel toward high latitudes where the magnetic field is larger; on the dispersion curves, this corresponds to the waves moving down to smaller normalized frequencies [Johnson et al., 1995]. At the same time, a perpendicular component of the wave vector k? develops (H&D). This means that the waves in Figure 1 move from curves with parallel k toward perpendicular k within the same surface, e.g., from H k to H? and from He k to He?. [6] Ray tracing has been one of the primary tools to study the EMIC wave propagation in a dipole field. Rauch and Roux [1982] investigated a two ion (H + and He + ) plasma and found that the waves between W H + and W He + are reflected at high magnetic latitudes when the wave frequency becomes equal to the local H + He + bi ion frequency, while waves with w < W He + are not reflected and are able to reach the ground. Similarly, ray tracing for a three ion (H +, He +, and O + ) plasma [Horne and Thorne, 1990, 1993] shows that the waves between W He + and W O + are reflected once the wave frequency w = w He O. Those with w < W O + reach the ground. The reflection can be seen from the dispersion curves (Figure 1): if the waves on He surface are refracted so k? gets large enough, they will encounter the He + O + frequency, at which point they may be reflected since k? = 0; if the waves are not refracted much, then they will be reflected at a lower frequency (w < w He O ). It should be clear that waves on surface O do not suffer reflection since the frequencies approach zero when k k 0. The polarization of the waves is dominantly left hand near the equator and becomes linear or right hand when they are reflected [Rauch and Roux, 1982]. [7] However, both observations [Young et al., 1981; Perraut et al., 1984; Anderson et al., 1996b] and theoretical calculations [Johnson et al., 1995; Lee et al., 2008] suggest that not all of the waves are reflected at the bi ion frequency and [Johnson and Cheng, 1999] provide a calculation that shows that wave energy can reach the ground and discusses the dependence on heavy ion concentration. Since mode conversion may happen when two wave surfaces become close, which may occur when the waves propagate toward the crossover frequency or the cutoff frequency [Fuchs et al., 1981; Johnson et al., 1995], the wave energy from one surface can be transferred to another surface during propagation. Some waves may tunnel through the stop band, especially when the heavy ion concentration is low [Johnson et al., 1989; Horne and Thorne, 1993], and some waves are absorbed by the wave particle interaction and can possibly heat the heavy ions [Thorne and Horne, 1993, 1994; Horne and Thorne, 1997; Kim and Lee, 2003; Kim et al., 2008]. [8] The growth of EMIC waves has also been modeled in global (equatorial plane with bounce averaged particle motion) ring current simulations. Jordanova et al. [2008] integrate the convective growth rate along ray tracing paths and calculate the amplitude of the waves using an empirical model [Jordanova et al., 2003]. Khazanov et al. [2007] and Gamayunov et al. [2009] use a bounce averaged fluid equation for EMIC wave convection and growth. Their formalism includes an approximation for reflection at the bi ion resonance [Khazanov et al., 2006, 2007]. [9] Qian and Hudson [1989] used a one dimensional hybrid Darwin code in a dipole field to simulate a two ion plasma (H + and He + ) and found the stop band suggested by the dispersion curves; since their code was one dimensional, only parallel propagation could be investigated and there was no coupling between different modes. We extend our last study (H&D) from a single ion to a multi ion species plasma in which EMIC wave behavior is substantially different. The absorbing boundary and MHD equilibrium have been improved since the previous work. Consequently, the code is now capable to run more realistic parameters. Our simulation is a first principles fully self consistent nonlinear calculation of the EMIC wave dynamics in dipole geometry including a complete description of the field aligned dynamics for a multi ion plasma. The wave generation, propagation, reflection, absorption and tunneling will all be self consistently investigated. We will briefly describe our hybrid code in section 2, present the simulations and results in sections 3 and 4, and finally give a discussion and summary in section D Hybrid Code Using Dipole Coordinates [10] Our hybrid code was introduced by H&D. We use dipole coordinates q, r, and s [H&D], with q varying along the magnetic field and r (the normalized L shell) varying across; s is the azimuthal coordinate. Equally spaced increments of q correspond in real space to changes in distance proportional to B, which means that the grid points are farther apart (in real space) near the ionospheric boundaries (see Figure 2). While vector components of fields for all 3 components are maintained, the two dimensional simulation models spatial variation only in the q and r directions. The radial boundaries are r lo = 0.87 and r hi = 1.13, where the length corresponding to the middle L shell where r =1,L 0 = 300 c/w pp0, and p c/w ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pp0 is the normalization length in the code, where w pp0 = 4n e0 e 2 =m p The magnetic latitude of the ionospheric boundaries (q = ±1) for r =1isl 0 = 45 ; q =0 is at the magnetic equator; and the central point of the simulation domain is located at (q =0,r = 1) Basic Equations [11] The governing equations are essentially the same as those in H&D except that there are now multiple ion species. The equations in CGS units can be written as dv m dt dx m dt ¼ v m ; ð2þ ¼ m E þ 1 q c v m B J ; ð3þ ion ¼ X X m n e e ¼ ion ; q ðx x m Þ; ð4þ ð5þ 3of13

4 Figure 2. Mapping between the (a) coordinates (q, r) and (b) Cartesian coordinates (x,z) on the meridional plane (s = 0). Here x and z are also the SM X and Z coordinates at local noon plane. The vertical lines in Figure 2a, which are the constant r lines, correspond to the dipole magnetic field lines in Figure 2b; the grid spacing along the field lines (Figure 2b) is proportional to B; q =1( 1) is mapped to the north (south) ionospheric boundary and q = 0 to the magnetic equator. Here r lo = 0.8, r hi = 1.2, and the ionospheric magnetic latitude l i is 45 for the r = 1 field line. J ion ¼ X X m q v m ðx x m Þ; ¼ re i ¼ cre; J ¼ c 4 rb; ð7þ ð8þ J i ¼rB i ; ð17þ u ei ¼ 1 J i J ion;i ; ð18þ n ei u e ¼ 1 ð n e e J J ionþ; ð9þ E ¼ 1 c u e B þ J; ð10þ where the subscript m (a) is the particle (species) index; ion (e) stands for ions (electrons); r ion (J ion ) is the charge (current) density of all ion species; J is the total current density (ions and electrons). Every quantity is normalized using the normalization listed in Table 1. The normalized equations are dv m dt dx m dt ¼ v m ; ¼ E m þ v m B m J m ; ion;i ¼ 1 V i X J ion;i ¼ 1 V i X X l m¼f n ei ¼ ion;i ; X l m¼f ð11þ ð12þ Q m Sðx i x m Þ; ð13þ ð14þ Q m v m Sðx i x m Þ; ð15þ E i ¼ u ei B i þ J i ; ð19þ where f a (l a ) is the first (last) particle index of the species a; Q m is the (normalized) charge of the superparticle. The subscript i indicates that the quantity is defined on the ith grid point, where i represents a two dimensional location; V i is the volume of the i cell. The delta function d(x x m ) has been replaced by the shape function S(x i x m ) that yields linear interpolation. Table 1. Normalization in the Hybrid Code a Normalization Quantity m 0 q 0 v A0 t 0 x 0 n 0 r 0 J 0 E 0 Definition m p E pffiffiffiffiffiffiffiffiffiffiffiffi B0 4mpne 0 1 W p0 c!pp 0 n e0 en e0 en e0 v A0 va 0 B0 c h 0 4v 2 A 0 c 2 Wp 0 a The quantity X is normalized to X 0 ; for example, mass m is normalized to m 0 ; electric field E is normalized to E 0, etc. Subscript 0 in the definition column indicates quantities atp the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi central position on the simulation domain; W p0 = eb 0 /m p c; w pp0 = 4n e0 e 2 =m p, in CGS units. 4of13

5 r b1 =r lo + w r, and r b2 =r hi w r. The resistive function with a width of w q = 0.2 is shown in Figure 3 (solid curve). [13] A masking function [Tajima and Lee, 1981; Umeda et al., 2001] is also applied in the same q boundary region defined by f M ðqþ ¼ 8 >< >: 1 q q 2 b 1 ; w q if 1 q < q ; b1 1; if q b1 q q b2 ; 1 q q 2 b 2 ; w q if q < q 1: b2 ð23þ Figure 3. The resistivity function h q (solid curve using left axis) and the masking function f M (q) (dashed curve using right axis), defined in (20) and (23), respectively, where h q1 = 0.2, q b1 = 0.8, q b2 = 0.8, and w q = 0.2 (not the same values as used in the simulations) Boundary Conditions [12] The reflection boundary conditions for particles [Naitou et al., 1979] and perfect conductor boundary for fields [Denton and Hu, 2009] are used, as were used by H&D. Realistically, the Alfvén waves approaching ground from the magnetosphere suffer some reflection by the ionosphere [Knudsen et al., 1992], and the reflection coefficient is significant (can be as high as 60%) for low frequency waves with a perpendicular wavelength longer than 2 km [Lessard and Knudsen, 2001]. Here, however, we want to focus on the wave behavior near the bi ion frequency which usually occurs at locations far from the ionosphere. Therefore we will attempt to absorb all the waves that are transmitted through the bi ion frequency. Since the perfect conductor boundary reflects all incident electromagnetic waves [Tajima, 1989], we implement several mechanisms to damp waves near the boundaries. First we introduce resistivity h within a boundary layer, 8 ¼ q þ r ; q0 1 þ cos q þ 1 2 w q >< q ¼ 0; if q b1 q q b2 ; >: ; if q b2 < q 1: 8 q0 2 1 cos q q b 2 w q ; if 1 q < q b1 ; r0 1 þ cos r r lo 2 w r >< r ¼ 0; if r b1 r r b2 ; >: ; if r b2 < q r hi : r0 2 1 cos r r b 2 w r ; if r lo r < r b1 ; ð20þ ð21þ ð22þ where h q0 (h r0 ) is the resistivity coefficient in the q (r) resistive layer, which starts from the q (r) boundary and has a width of w q (w r ) and where q b1 = 1+w q, q b2 =1 w q, [14] The masking function f M (q) (dashed curve in Figure 3) is equal to unity in the central region, and smoothly reduces to zero in the resistive layer. We mask both the electric field and the magnetic field. The normalized equations (16) and (19) are replaced ¼ f M ðqþðre i Þ; ð24þ E i ¼ f M ðqþðu ei B i ÞþJ i ; ð25þ respectively. [15] Additionally, fourth order diffusion is applied with spatial differencing in equation (24) to reduce numerical fluctuations on the grid scale. With all the above damping terms in the resistive layer, the energy reflected by the q boundary is less than 5%. [16] Notice that the resistive layer should generally lie outside the reflection point, which puts a constraint on the width of the resistive layer. This, in turn, limits the amount of the waves the resistive layer can absorb. In our simulations the waves start on the helium branch with frequencies w 0.2 W p0. If the waves are refracted enough, they are reflected at the He + O + frequency, which occurs when the local magnetic field is about 3 times larger than the field at the equator. If the waves are not refracted as much, then they may be reflected at a lower frequency, therefore requiring a more narrow resistive layer. We have chosen a resistive layer with a width w q = 0.05 which is generally beyond the location where the local magnetic field is 4 times bigger (shown in Figure 4). [17] Note that the resistive layer and other damping terms are only applied outside the region of wave growth and reflection due to the He + O + bi ion resonance and wave absorption where w = W O +. These dissipative effects are not needed to stabilize the code. They are included to eliminate reflection at the ionospheric boundary MHD Equilibrium and Initialization [18] In order to successfully simulate EMIC waves, the plasma needs to start from a state that is nearly in MHD equilibrium, defined by the normalized force equation J B ¼rP; ð26þ P ¼ p? I þ p k p? ^b^b; ð27þ 5of13

6 Figure 4. Line 1 is the equator; on lines 2, 3, and 4, the dipole magnetic field is 2, 3, and 4 times stronger, respectively, than its value with the same L shellattheequator; the resistive layer with a width w q = 0.05 is indicated by the gray region. The EMIC waves between W He + and W O + should be reflected roughly where the wave velocity matches the w He O frequency in between lines 3 and 4. where J is the total current defined in equation (17), P is the gyrotropic pressure tensor, I is the unit tensor, ^b = B/B is the unit vector in the direction of the magnetic field, and p? (p k ) is the pressure perpendicular (parallel) to the magnetic field. [19] We solve the equilibrium using an anisotropic MHD model [Denton and Lyon, 2000], whose normalized equations can be summarized as dv ¼ rpþj B; dt ð28þ E ¼ v B; ð29þ t and " 3 k. At the same time that the velocity field is set to zero, we normalize the magnetic field such that its magnitude is unity at the center of the simulation. This process is repeated until the MHD equilibrium (26) is satisfied. The pressure ratio (equivalent to the temperature ratio) from the MHD equilibrium is peaked at the center of the simulation, and decreases along the equator and the field lines (see Figure 5). This temperature distribution causes the plasma to be most unstable to EMIC waves near the central equatorial region. [21] The above simulation is carried on in exactly the same domain as that used by the hybrid code. After MHD equilibrium is achieved, p?, p k, and B are used as input to the hybrid code. Since the hybrid code has multiple species, the MHD pressures p? and p k have to be distributed among each ion species p a? and p ak (the electron pressure in the hybrid code is assumed to be zero). The pressure is first split along the equator by assuming that each species takes up the same fraction of the MHD pressure. The fraction does not depend on spatial position and has the value b ak0 /S a b ak0, where b ak0 (b k for each species at the center) is the known input. All the cold species are isotropic (T?i = T ki ), and according to parallel force balance [Chan et al., 1994], the pressure along the magnetic field line of the species is constant. Therefore the pressure for all the cold species can be determined by tracing the field lines from the equator. Finally, the pressure for the hot protons is computed by subtracting the pressure of all the cold species from the MHD pressure. Considering that the b k of the cold ions is usually much smaller than that of the hot protons, the pressure of the hot protons is nearly equal to the MHD pressure. Note that, although the coordinate system is dipolar, the magnetic field from the equilibrium is not exactly dipolar anymore. [22] The number density of the cold species is uniform on the entire domain; the number density of the hot protons is uniform along the equator and varies along field lines according to 1 eq n p ¼ n p0 ; ð37þ 1 eq B ¼ E; B ¼rA; J p k B 3 ¼ rs "; ð30þ ð31þ ð32þ ð33þ ð34þ " ¼ 1 2 v2 þ 1 2 B2 þ p? þ 1 2 p k; ð35þ S " ¼ v 2 v2 þ 4p? þ p k þ E B þ p? p k vk : ð36þ [20] In order to obtain the equilibrium, the velocity v is set to zero after the total kinetic energy reaches a local maximum, that is, for time t 1 < t 2 < t 3, " k t 2 is greater than both " k t 1 Figure 5. The temperature ratio T? /T k from the MHD equilibrium. 6of13

7 Table 2. Key Parameters for All the Runs a Run h H + (%) b kh + h He + (%) b khe + h O + (%) b ko + w He O /W p0 q bi2 q 2WO a a The hot proton concentration h p = 10% for all runs and the temperature ratio T?p /T kp and b kp of the hot protons at the center are about 2 and 0.8, respectively. All the cold species are isotropic (T? /T k =1).Thecolumnofq bi2 (q 2WO+ ) indicates the absolute q position of the He + O + bi ion frequency w He O (second harmonic of W O +). Other parameters that are common for all runs: r lo = 0.87, r hi = 1.13, L 0 = 300 c/w pp0 for r =1,the magnetic latitude l = ±45 for (q = ±1, r = 1), number of grid points along q(r) is 501 (101), and there are 96 (8) particles per cell for the hot (cold) species. where n p0 is the constant density along the equator, the subscript eq indicates a quantity at the equator, and d eq = 1 p keq /p?eq [Chan et al., 1994]. Subsequently, the temperature and thermal velocity of each ion species are computed through T? ¼ 1 2 m v 2 th? ¼ p? n ; T k ¼ 1 2 m v 2 th k ¼ p k n : ð38þ ð39þ [23] The parallel and perpendicular velocity of the mth individual particle is loaded according to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v?m ¼ v th? ln R 1m ; ð40þ v km ¼ v thk erf 1 ðr 2m Þ; ð41þ where R 1m and R 2m are bit reversed numbers distributed between0and1toachieveaquietstart[birdsall and Langdon, 1991], and erf 1 (x) is the inverse of the error function Erf (x) =2 R 0 x exp( t 2 p ffiffiffi )dt/. Note that there are two degrees of freedom for the perpendicular velocity, and we decompose v?m to v?m1 = v?m cos R and v?m2 = v?m sin R using a random phase R. Finally (v?m1, v?m2, v km ) are rotated to the velocities in the dipole frame (v qm, v rm, v sm ). 3. Simulations [24] We present 5 different runs with different concentration of cold species using the key parameters listed in Table 2. All the simulations are run on the same domain using dipole geometry (see section 2.2). A total number of 501 (101) grid points is used in the q (r) direction; 96 (8) computer particles per cell are used for the hot protons (cold ion species); the time step to push the particles is Dt W p0 = The system size and the number of grid points were chosen to assure that the grid resolution is fine enough to simulate the waves with an adequate number of unstable wavelengths in the system. The grid spacing in the r direction resolves the proton gyroradius of the hot particles. Tests with higher and lower radial resolution (not shown) indicate that the radial resolution is adequate. [25] The plasma is initialized with a state that is in MHD equilibrium (discussed in section 2.3). The parameters are chosen such that EMIC waves are destabilized near the equator in the vicinity of the middle L shell (r = 1) with frequencies w < W He+, where W He+ is the helium cyclotron frequency at the center. The source of free energy driving the instability is provided by the hot anisotropic protons which approximately have T? /T k 2 and b k 0.8 at the center. The cold ion species (typically with b k < 0.001) include H +,He +, and O +. [26] As the waves leave the source region where they gain energy, they propagate (mostly along field lines) toward high magnetic latitudes, and encounter the second harmonic O + cyclotron resonance 2W O +, the He + O + bi ion frequency w He O, and the first harmonic O + cyclotron resonance W O +, in that order. The EMIC waves usually suffer reflection at w He O and absorption at W O+,2W O + and w He O due to the wave particle interaction [Thorne and Horne, 1993, 1994]. This is a complex process that is sensitive to the O + concentration h O + = n O +/n e0, which influences the value of w He O : smaller h O + results in smaller w He O that is closer to W O +, meaning the bi ion frequency lies closer to the q boundary. A larger change in B is needed to bring the normalized wave frequency down to w He O. Different concentration of O + will be investigated in our simulations, which are listed in Table 2. Basically Runs 1, 2, 3, and 4 have O + concentration of 0%, 0.05%, 4%, and 14%, respectively, with 14% He +. Run 3a has the same O + concentration (4%) as Run 3, but lower He + concentration (3%). All the runs have 10% hot protons and the concentration of H + is adjusted accordingly in each case to maintain quasineutrality. [27] Using jwhamp (Dartmouth s version of WHAMP [Rönnmark, 1982]) with the plasma parameters from the center in the simulations, the maximum growth rate g m /W p0 ranges from to 0.02, and the frequencies of these modes w m /W p0 vary from 0.18 to The waves of the branch w > W He + have growth rate less than W p0, which means that for a simulation with 10 e folding wave growth times, the wave energy with w > W He+ is 150 times smaller than that with w < W He +. The waves with w < W O + have even lower growth rate, which means we need only focus on the waves with w < W He + and their effects. The bi ion frequency w He O can also be estimated with jwhamp. The ratio of these two frequencies w m /w He O roughly implies how far along the field line the waves travel before their frequencies fall below w He O. Therefore it indicates where the resistive layers (see section 2.2) may be placed in the q direction: higher w He O indicates that w He O is located closer to the q boundary, suggesting the resistive layer should have a more narrow width. The resistive width w q in equation (20) is chosen to be 0.2 for Run 1, and 0.04 for all other runs; the width w r is 0.06 for all runs. We also estimate 7of13

8 describes how the wave with an absolute frequency is polarized: 8 ¼ 1 circularly left-hand polarization; >< 1 <<0 elliptically left-hand polarization; ¼ 0 linear polarization; 0 <<1 elliptically right-hand polarization; >: ¼ 1 circularly right-hand polarization: Figure 6. The power spectrum summing over all the spatial grid points versus frequency for the time interval 200 t W p0 300 for Runs (a) 1, (b) 2, (c) 3, and (d) 4. The bi ion frequency w H He, W He +,2W O +,andw O + are indicated by the vertical dashed lines. The fact that the power nearzero frequency is small indicates that the plasma starts with a good MHD equilibrium. in Table 2 the q position where waves encounter the bi ion frequency w He O. [28] AsinH&D,weusethemethodbyKodera et al. [1977] to apply a Discrete Fourier Transform (DFT) on the complex signal C(q, r, t n )=db r (q, r, t n )+idb s (q, r, t n ), where t n is the sampling time, and db r (db s ) is the r (s) component of the fluctuating magnetic field. The power is defined as P ðq; r;! k Þ ¼ ~C ðq; r;! k Þ~C* ðq; r;! k Þ; ð42þ where w k is the discrete Fourier frequency, ~C(q, r, w k ) is the DFT of C(q, r, t n ), ~C*(q, r, w k ) is the complex conjugate, and we use + ( ) to indicate power for which w k >0(w k < 0). Power P + (P ) defined in (42) corresponds to circularly righthand (left hand) polarization. The ellipticity, calculated as pffiffiffiffiffiffi pffiffiffiffiffiffi P þ P ¼ pffiffiffiffiffiffi p ffiffiffiffiffiffi ; ð43þ P þ þ P 4. Simulation Results 4.1. EMIC Wave Generation and Propagation [29] As discussed in section 3, the waves should be destabilized in the vicinity of the center with frequencies w < W He + in all cases. Figure 6 shows the power spectrum for Runs 1, 2, 3, and 4 in the time interval tw p0 = The power shown is integrated over all the spatial grid points. The linear theory (using parameters at the center) predicts that the wave frequency of the most unstable mode for Runs 1, 2, and 3 is around w m 0.18W p0 with a growth rate g m 0.024W p0 (the wave should have an energy gain of 4e foldings in a time interval of 100W p0 ), and for Run 4, w m 0.2W p0 and g m 0.016W p0. The bulk of the wave power falls between 0.1W p0 w 0.3 W p0. The lower limit is close to the w He O frequency, while waves with w > W He + are probably generated on the He branch but off the equator where the local proton cyclotron frequency is greater than W p0 (see H&D for waves on the H surface with w > W p0 ). Similar plots for later times also show almost all of the power is within this range, so we hereafter assume that the EMIC waves in our simulations have frequencies 0.1W p0 w 0.3 W p0. The power is predominantly left hand polarized (most power has negative frequency) for Runs 1, 2, and 3, indicating that waves are still in the source region gaining energy. The power for Run 4 is also left hand polarized, although not as dominantly as for the first 3 runs, since it is less unstable and the simulation noise is comparatively larger. [30] We integrate the power for the frequencies 0.1W p0 w 0.3 W p0 and 0.3W p0 w 0.1 W p0 to get the right and left hand polarized wave power in the frequency range of the EMIC waves, P ðq; rþ ¼ X0:3 j! kj¼0:1 P ðq; r;! k Þ: ð44þ [31] The overall EMIC power is the sum of P ± (q, r): P EMIC ðq; r Þ ¼ P þ ðq; rþþp ðq; rþ; ð45þ and the ellipticity " is calculated using (43). The two quantities P EMIC and " are represented using a twodimensional color map (Figure 7, introduced by H&D), in which the variation of the hue along the horizontal axis indicates the ellipticity and the variation of saturation along the vertical axis indicates the power. Blue (red) color generally stands for negative (positive) ellipticity, meaning left hand (right hand) polarization, and green stands for linear polarization. (Color blind people should note that the 8of13

9 Figure 7. Two dimensional color map representing both ellipticity and power. The variation of the hue along the horizontal axis indicates the ellipticity " and the variation of the saturation (color versus white) along the vertical axis indicates the wave power P normalized to the maximum value. Blue (red) indicates negative (positive) ellipticity, and green indicates ellipticity near zero (linear polarization). More intense color indicates more power, and pure white indicates that the power is zero. ellipticities plotted in this paper are generally 0.2 [no red color].) [32] Figure 8 shows the spatial distribution of the EMIC power and Poynting vector S averaged in different time intervals (indicated at Figure 8, right) for Run 2 (which has 14% pffiffiffiffiffiffiffiffiffiffiffiffi He + and 0.5% O + ). The first column (leftmost) shows P EMIC versus pffiffiffiffiffiffiffiffiffiffiffiffi (q, r) in gray scale and the second column shows both P EMIC and " using the 2 D color map (Figure 7). The waves grow near the equator and are predominantly left hand polarized (blue color) for 200 t W p0 300; the standing wave pattern is the result of the superposition of the waves traveling in both directions in the source region (discussed in H&D). In our self consistent model, as the waves grow, the source of free energy decreases and the source region becomes stabilized, which generates less and less waves. This can be seen for the time 300 t W p0 400, which shows the waves travel toward high magnetic latitudes and the polarization becomes linear, suggesting that the wave normal angle is large (see discussion by H&D). The waves continue to propagate and some of them reach the q boundaries at 400 t W p Therefore, some of the waves have propagated beyond the position of the He + O + bi ion resonance (indicated by the dashed lines in the first column). Considering that h O + = 0.5% in this case, the fact that some waves are transmitted agrees with the conclusion that a h O + concentration of less than 1% generally favors wave tunneling through the stop band [Johnson et al., 1995; Horne and Thorne, 1993]. The standing wave pattern inside the position of the bi ion frequency suggests that some waves are reflected, which will be seen more clearly in Figure 9. By the time the waves are linearly polarized, the azimuthal component of the magnetic field is roughly twice as big as the r component, which is also about twice as big as the q component. [33] The field line (q) component of the Poynting vector S of waves is shown in the third column of Figure 8 and both q and r components of S are shown in the fourth column. The plots show that the EMIC waves usually propagate along the magnetic field and away from the equator at high latitudes while in regions close to the equator, the net Poynting vector is small, probably the result of superposition of oppositely traveling waves. (The Poynting vector at later times (t W p0 > 700) at the lower L shells (r < 1) has a somewhat random direction; this region has oscillations in the equilibrium, instead of EMIC waves.) These characteristics of the Poynting vector are roughly consistent with the observations of Poynting vector from CRRES [Loto aniu et al., 2005], which exhibit bidirectional wave packet propagation at low magnetic latitudes and unidirectional (away from the magnetic equator) propagation at higher latitudes. While our results show that the wave energy usually propagates poleward at high latitudes, at t W p0 = (third row from top), there is a region (r = ) with equatorward propagation of energy (blue color) for q > EMIC Wave Power Evolution [34] Figure 9 shows the power integrated over the q or r direction versus time (as in H&D). The power spectrum at each time is calculated using a DFT of time interval 100 W 1 p0. From left to right, the columns correspond to Runs 1, 2, 3, 3a, and 4, respectively. Figures 9a 9e are the wave power integrated over r, plotted in gray scale; Figures 9f 9j are the same power together with ellipticity plotted using the color map in Figure 7; Figures 9k 9o and 9p 9t are similar to Figures 9a 9e and 9f 9j except that the power is integrated along the q direction. Generally, Figures 9a 9e and 9f 9j (Figures 9k 9o and 9p 9t) show the wave energy evolution along field lines (across L shell). [35] Run 1 has no O +. The waves with frequency w < W He + should travel all the way down the field lines without reflection [Rauch and Roux, 1982], and this can be seen in Figure 9a. The wave energy propagates along the field lines and hits the simulation boundaries where it is absorbed. Figure 9b shows the energy for Run 2 which has 14% He + and 0.5% O +. As shown in Figure 9b (also seen in Figure 8e) the waves clearly travel past the position where w is equal to the He + O + bi ion frequency w He O W p0 (the dotted line), which is almost where w = W O + (the dashed line). The waves near the reflection location show mixed polarization (not shown). This suggests that there are waves on both the He and O surfaces, which are slightly right hand and lefthand polarized, respectively. Also from Figure 9b, the wave reflection at w He O can be seen by both the standing wave pattern near w He O and the relatively light band of energy reflected from the same place. The reflected waves travel back toward the equator and remain guided and linearly polarized. When crossing the equator, the waves experience a slight gain of energy, indicating that the source region is still capable of generating waves, although much less than during the early stage. Correspondingly the polarization becomes a little bit more left handed (seen by cyan in Figure 9g). [36] As we increase h O + to 4% (Run 3, Figure 9c), the bi ion frequency increases to w He O W p0 (the dotted line). The energy in between the dotted line and the dashed line (location where w W O +) suggests that there is also wave transmission in this case, but unlike Run 2 (Figure 9b), the transmitted waves do not seem to propagate beyond the 9of13

10 Figure 8. The EMIC wave power and Poynting vector for Run 2 at different time intervals indicated at the right side. The first column shows the square root of power in gray scale, and the second column shows the square root of power and the ellipticity using the two dimensional color map in Figure 7. The dashed lines are approximately where the waves encounter the He + O + bi ion frequency. pffiffiffiffiffiffiffi The third column shows the signed square root of the q component of the Poynting vector ( js q j Sq / S q ). The fourth column shows both q and r components of the Poynting vector (not square root and scaled separately in each time interval). position where w W O +. Also compared to Run 2, more energy is reflected at the bi ion frequency. The reflected energy again crosses the equator. In both Run 2 and Run 3, only one crossing of the equator by the waves is clearly occurring, and the waves lose most of their energy after one bounce between reflection points, suggesting that some absorption is occurring. [37] In fact, resonant absorption of energy by the cold O + is occurring; this will be examined in a later paper. We also lower h He + in Run 3a (keeping the same h O + as Run 3), and the results are shown in Figure 9d. The bi ion frequency becomes w He O 0.08 W p0, slightly higher than that of Run 3, which may reduce the wave energy transmission at the bi ion frequency (Figure 9d). The waves generated near the equator (Figure 9i) are not purely left hand polarized as they are in Run 2 and Run 3 (Figures 9g and 9h), which may be because the waves in Run 3a are generated with larger k?, leading to more linearly polarized waves [Anderson et al., 1996a] or significant refraction may occur very close to the magnetic equator. Other than this, there is no essential difference between the results for the two runs, since the waves with w < W He + are not as sensitive to h He + as to h O + (in contrast, the waves with W He + < w < W H + are strongly dependent on h He +). [38] Figure 9e (for Run 4) shows a different energy evolution than all the previous runs. The wave energy does not seem to be guided and the polarization turns more righthanded than that in Runs 2 and 3 as the waves propagate. This is consistent with the fact that more O + causes a higher He + O + bi ion frequency, and thus pushes up the curve He? in Figure 1, leading to a broadened region of the wave surface where the wave is unguided and right hand polarized. Most of the wave energy does not pass beyond the position where w is equal to the bi ion frequency (the dotted line), which is w He O 0.1 W p0, about halfway in between the locations where w = W O + (the upper dashed line) and 10 of 13

11 Figure 9. The EMIC wave power integrated along the (a j) q and (k t) r directions versus time. Each column corresponds to the run indicated at the top. The gray scale plots the normalized power, and the color plots the normalized power and ellipticity using the two dimensional color map in Figure 7. The dotted lines in Figures 9b, 9c, 9d, 9e, and 9j are roughly the position where w w He O ; the dashed lines in Figures 9b, 9c, 9d, and 9e are the position of W O +; the second (lower) dashed line in Figure 9e is the position of 2W O +. 2W O + (the lower dashed line). Near the locations where w 2W O +, the wave vector is estimated to be k k w pp0 /c 0.2, k? w pp0 /c 0.1, and the ellipticity (Figure 9j) is around 0.2. Since the waves reach the crossover frequency at around q = 0.2 for parallel propagation, the waves in the simulation are probably on the He surface below the crossover frequency (Figure 1). (The ellipticity on the O surface with the above k k and k? is dominantly left handed with " 0.7). Therefore it appears that in this case the waves do not appreciably tunnel through the stop band. Figure 9j shows some right handed waves beyond the location where w = w He O, and this indicates that some waves are reflected at frequencies lower than w He O since the wave normal angle is not big enough. [39] Radially, all the runs display an outward radial energy propagation (Figures 9k 9o and 9p 9t), see H&D. For Runs 1 and 2, most of the energy is lost when the waves hit the q boundaries (at about tw p0 = 500) before they reach the r boundary. Because waves are able to tunnel through the bi ion frequency in Run 2 (Figure 9b), the energy loss (Figure 9l) might be due to the absorption at the q boundaries or cyclotron resonance. In comparison to Figure 9l, Figure 9m (Run 3 with 4% O + ) shows that there is relatively more energy reflected and the energy loss is slower when there is a higher O + concentration. Figure 9o suggests that Run 4 has a slower radial propagation of wave energy than Runs 2 and 3, probably because of the higher concentration of heavy ions in Run 4 which causes a lower Alfvén speed. The energy loss in the radial power plots (Figures 9k 9o and 9p 9t) is suggestive that there is wave absorption in each run. 5. Discussion and Conclusions [40] We have investigated the propagation, polarization, absorption, and transmission of EMIC waves in a multi ion 11 of 13

12 plasma using a two dimensional hybrid simulation. The plasma is initialized with an MHD equilibrium state in which the pressure and the magnetic field are obtained from an anisotropic MHD code. The EMIC waves are driven by the ring current hot protons, and the most unstable waves are destabilized near the equator with frequencies w < W He +.We have presented 5 different runs with different concentration of the cold ions. The thermal effect of the electrons was neglected in our simulations, and therefore the Landau damping by the electrons was not considered; instead, we focused on damping by the ions. [41] The length in the simulation is normalized to c/w pp0, which can be written as c! pp0 ¼ c 0:036 qffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R E ; ð46þ n e0 =cm 3 4n e0 e 2 m p where R E is the Earth s radius. Our simulations have a p ffiffiffiffiffiffi middle L shell of L 0 = 300c/w pp0 = 10.8R E / n e0. We may estimate the L shell to be L 0 3R E by assuming n e0 =10cm 3. This is somewhat, but not greatly smaller than realistic for this value of n e0 (4 or 5 R E would be better). Note, however, that the specific value of the electron density does not enter the hybrid code; as a result, the simulation results do not depend on the specific size of the simulation. [42] We summarize our simulation results here: [43] 1. In each case, the waves initially grow in the region close to the equator, and then propagate toward high magnetic latitudes. During this process, left hand polarization shifts to linear polarization (with some right hand polarization) and the wave normal angle increases. [44] 2. When there is no O +, the waves with w < W He + encounter no bi ion frequency and they are able to reach the q (ionospheric) boundaries where they are absorbed. (A more realistic ionospheric boundary condition would allow for some reflection.) [45] 3. In the presence of O +, the waves encounter the He + O + bi ion frequency, leading to reflection. A significant amount of waves may tunnel through the bi ion frequency if h O + is small (0.5% in our case). As h O+ increases, the wave tunneling is reduced while more reflection takes place. [46] 4. The EMIC waves propagate both away from and toward the equator in the source region (very low latitudes), resulting in a small Poynting vector there. Outside the source region (high latitudes), the Poynting vector usually (though not always) points away from the equator even though there is evidence of reflected wave power. This suggests that continued production of equatorial wave power (though small) may contribute a larger component of the net Poynting vector than that of the reflected waves, perhaps explaining the observations of Loto aniu et al. [2005]. [47] Acknowledgments. We thank Mary Hudson and Robyn Millan for helpful discussions. Y.H. was supported by NSF/DOE grant ATM R.D. was supported by NSF grant ATM (Center for Integrated Space Weather Modeling, CISM, funded by the NSF Science and Technology Centers Programs) and by NASA grant NNX08AI36G (Heliophysics Theory Program). J.J. was supported by NSF grant ATM , NASA grants NNH09AM53I and NNH09AK63I, and DoE contract DE AC02 76CH References Anderson, B. J., S. A. Fuselier, S. P. Gary, and R. E. Denton (1994), Magnetic spectral signatures in the Earth s magnetosheath and plasma depletion layer, J. Geophys. Res., 99(A4), , doi: / 93JA Anderson, B. J., R. E. Denton, G. Ho, D. C. Hamilton, S. A. Fuselier, and R. J. Strangeway (1996a), Observational test of local proton cyclotron instability in the Earth s magnetosphere, J. Geophys. Res., 101(A10), 21,527 21,543, doi: /96ja Anderson, B. J., R. E. Erlandson, M. J. Engebretson, J. Alford, and R. L. Arnoldy (1996b), Source region of 0.2 to 1.0 Hz geomagnetic pulsation bursts, Geophys. Res. 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