Use of Euler potentials for describing magnetosphere-ionosphere coupling

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1 Click Here for Full Article JOURAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005ja011558, 2006 Use of Euler potentials for describing magnetosphere-ionosphere coupling R. A. Wolf, 1 R. W. Spiro, 1 S. Sazykin, 1 F. R. Toffoletto, 1 P. Le Sager, 2,3 and T.-S. Huang 2 Received 30 ovember 2005; revised 16 March 2006; accepted 19 April 2006; published 26 July [1] We present a general formulation of the basic equations of large-scale magnetosphereionosphere coupling in terms of Euler potentials and describe a specific numerical implementation of this formulation in the context of the Rice Convection Model (RCM). When written in terms of Euler potentials, both the Vasyliunas magnetosphere-ionosphere coupling equation and the expression for bounce-averaged adiabatic drift assume particularly elegant forms, while the equation for the conservation of ionospheric current is only slightly more complicated to solve than the corresponding formula for a dipole case. For simplicity, large-scale models of convection in the inner magnetosphere have typically assumed strict symmetry between the northern and southern hemispheres, explicitly assuming the internal planetary magnetic field to be a dipole aligned with Earth s rotation axis and oriented perpendicular to the solar wind flow velocity. These approximations have precluded the realistic treatment of ionospheric longitude and seasonal effects as well as dipole-tilt and IMF- y -penetration effects in the magnetosphere. We present a scheme for constructing an Euler-potential-based computational mesh, in which the Euler potential a is set to zero at the dip equator for a reference altitude of 90 km, and b lines in the northern ionosphere follow lines of constant centered dipole magnetic longitude but are spaced equally in terms of total latitude-integrated magnetic flux. Properties of the Euler-potential-based grid are illustrated using an IGRF model for the Earth s internal field. Our procedure yiel an Euler-potential-based grid that covers the entire ionosphere, except for the southern polar cap and cusp. Citation: Wolf, R. A., R. W. Spiro, S. Sazykin, F. R. Toffoletto, P. Le Sager, and T.-S. Huang (2006), Use of Euler potentials for describing magnetosphere-ionosphere coupling, J. Geophys. Res., 111,, doi: /2005ja Introduction [2] Vasyliunas [1970] first presented a computational scheme for self consistently calculating the electrodynamic and plasma processes coupling the inner magnetosphere and ionosphere. Expressed in general form, this scheme consists of three sets of relations: [3] 1. The first relation is an expression relating plasma pressure gradients in Earth s magnetic field to field-aligned currents linking the magnetosphere and ionosphere. eglecting inertial drift currents compared to gradient/ curvature drift currents and assuming an isotropic plasma pitch angle distribution within a magnetic flux tube, this relation (termed the Vasyliunas equation) can be written as J kn J ks ¼ ^b rv rp n s 1 Department of Physics and Astronomy, Rice University, Houston, Texas, USA. 2 Department of Physics, Prairie View A&M University, Prairie View, Texas, USA. 3 ow at Institute for Environmental Research and Sustainable Development, ational Observatory of Athens, Athens, Greece. Copyright 2006 by the American Geophysical Union /06/2005JA011558$09.00 ð1þ where J kn and J ks are the density of irkeland current along the magnetic field direction just above the northern and southern ionospheres, respectively, ^b is a unit vector along the magnetic field, V = R / is the volume of a tube of unit magnetic flux, and P is magnetospheric particle pressure [e.g., Wolf, 1983; Heinemann and Pontius, 1990]. The right side of (1) can be calculated anywhere on the field line. [4] 2. The second set of relations consists of expressions for ionospheric current continuity. If one makes the conventional assumption that the induction electric field in the ionosphere is negligible, so that the electric field there is simply E = rf, then the standard expression for ionospheric conduction current is J ¼ s P ðrf þ v Þs H ðrf þ v Þ^b ð2þ where ^b is a unit vector in the direction of the magnetic field, and v is the neutral wind velocity. Setting the divergence in J equal to the computed field-aligned current yiel an elliptic partial differential equation that can be solved for the electric potential distribution in the ionosphere, given the electrical conductance and neutral wind fiel. Integrating (2) over the conducting region of each field line and requiring that the divergence of horizontal ionospheric conduction current be balanced by 1of7

2 WOLF ET AL.: RIEF REPORT the field-aligned current from the magnetosphere lea to the relation r h S $ ðrfþ þ w ¼ J k sin I ð3þ where S $ is the conductance tensor representing singlehemisphere field-line-integrated conductivities, w represents the field line integrals of products of conductivity and v (see, e.g., Forbes and Harel [1989] for explicit approximate forms) and I is the magnetic dip angle. Equation (1) is usually substituted in (3) using the assumption that J k / is antisymmetric between the two hemispheres. [5] 3. The third set of relations consists of expressions describing the bounce-averaged, energy-dependent adiabatic drift of magnetospheric plasma. In convection and ring current models, the bounce-averaged drift equation is usually written in a form like v D ¼ E ind 2 þ rh q 2 Here E ind is the induction electric field, the total electric field is E = E ind rf, and q is the charge. The Hamiltonian H is given by H ¼ qf þ W K where W K is the kinetic energy of a particle written as a function of position, time, and the appropriate adiabatic invariants. Specifically, when the particles are assumed to be undergoing strong, elastic pitch angle scattering, as in the RCM, we set W K ¼ lv 2=3 where l is the isotropic energy invariant, and V = R / is the volume occupied by a flux tube carrying one unit of magnetic flux, between the southern ionosphere and the northern. For the less restrictive case where the model particle distribution is organized in terms of the first two adiabatic invariants, as in most ring current-particle models, including specifically the Comprehensive Ring Current Model (CRCM) [Fok et al., 2001], W K is taken to be a function of the first and second adiabatic invariants m and J, as well as position and time. [6] The plasma distribution is advanced in time using t þ v D r f ¼ S L where S and L represent source and loss terms for the magnetospheric flux tube (e.g., upflowing particles, precipitation, and charge exchange), and f is a function of appropriate adiabatic invariants, position, and time. We note that the derivation of (1) requires neglect of inertial currents compared to gradient/curvature drift currents, an assumption that is valid only for timescales that are long compared to Alfvén wave travel times along field lines (typically a few minutes). The time dependence introduced in (7) occurs on a drift timescale, which is assumed to be much longer. ð4þ ð5þ ð6þ ð7þ [7] Early implementations of the Vasyliunas [1970] magnetosphere-ionosphere coupling scheme made many simplifying assumptions. Over the years many of the assumptions and limitations originally built into the Rice Convection Model [Harel et al., 1981] and other early convection models [e.g., Senior and lanc, 1984] have been eliminated [e.g., Toffoletto et al., 2003]. In this paper we formulate the magnetosphere-ionosphere coupling relations (1), (3), (4), and (7) in terms of Euler potentials in order to remove two important restrictions remaining from the original convection model implementations, namely, (1) the assumption that Earth s intrinsic magnetic field is a dipole with axis along the rotation axis, perpendicular to the solar wind flow, and (2) the assumption of simple conjugacy between northern and southern hemispheres: a field line that crosses the northern ionosphere reference altitude at magnetic latitude L and longitude f crosses the southern ionosphere reference altitude at magnetic latitude L and longitude f. [8] Even while allowing for the explication of the essential features of magnetosphere-ionosphere coupling, these assumptions have prevented the RCM and RCM-like convection models of the inner magnetosphere (e.g., the CRCM and the Magnetospheric Specification Model [e.g., Wolf et al., 1997]) from realistically treating both seasonal, and longitude variations in the ionosphere, and the effects of dipole tilt and solar wind y penetration in the magnetosphere. These simplifying assumptions have also proved to be a serious limitation on efforts [De eeuw et al., 2004; Toffoletto et al., 2004] to couple inner magnetosphere physics to global MHD codes, which have had to be dumbed down by the assumption of a magnetic dipole aligned perpendicular to the solar wind flow. Similarly, when coupling RCM-like models to ionosphere codes [Maruyama et al., 2005; Huba et al., 2005], the ionosphere codes have also had to be dumbed down to the case of equinox conditions and a rotation-axis-aligned magnetic dipole. [9] Elimination of restrictions 1 and 2 requires a basic change in computational coordinates, which, in the past, has always been a fixed, spherical-polar grid on an ionospheric reference sphere that does not rotate with the Earth. Our new nonorthogonal mesh is fixed in the northern ionosphere, rotates with the Earth, and is based on Euler potentials, which are constant along field lines. The southern ionosphere grid is the mapping along magnetic field lines of the northern one, and, thus, necessarily varies in time as the planet rotates through nonsymmetric and possibly time-varying magnetic field configurations. [10] The use of the Euler-potential-based computational mesh is discussed in detail here, in the hope that the same formulation might be useful for others seeking to model electrodynamic processes in the coupled magnetosphereionosphere system with realistic geometry. Another coordinate system constructed for a similar purpose is the magnetic apex system of Richmond [1995], in which the ionospheric latitude parameter is essentially the apex altitude that a field line would reach, if the magnetic field were everywhere equal to the intrinsic field of the Earth; the magnetic apex coordinate is similar to, but not identical with, the familiar McIlwain L. Though a bit abstract, Euler potentials allow the basic equations describing particle 2of7

3 WOLF ET AL.: RIEF REPORT motion in the inner magnetosphere and coupling to the ionosphere to take particularly elegant forms. 2. Expressing Particle Drift and Ionosphere- Magnetosphere Coupling in Terms of Euler Potentials [11] Substituting the definition of Euler potentials in terms of the magnetic field [Stern, 1970], ¼ra rb into equation (1), and using rv =(V/)ra +(V/)rb and a similar expression for rp lea to the particularly elegant form J kn n J ks s ¼ V P P V where the subscript s refers to values evaluated just above the southern ionosphere, and n refers to the same height above the northern ionosphere. The right side of (9) can be evaluated anywhere along the field line. [12] When the particle position is expressed in terms of Euler potentials (a(t), b(t)), equation (4), which describes charged particle drift, takes the form _a ¼ _b ¼ þ E ind ra t x 2 1 H q ¼1 H q t x þ E ind rb 2 þ 1 H q ¼ 1 H q ð8þ ð9þ ð10þ ð11þ where the subscript x indicates that the derivative is taken at constant position x. The second equality in each equation follows if we choose our gauge so that a particle drifting at velocity E ind / 2 maintains constant a and b. The drift equations of motion thus correspond to those of an ideal Hamiltonian fluid. [13] Some comment is needed concerning our treatment of induction electric fiel in (10) and (11). We assume that our (a, b) grid is fixed in the ionosphere and correspondingly neglect induction electric fiel there, setting E = rf n in the northern ionosphere, for example. The motion of the ionospheric footprint of the guiding center of a magnetospheric particle moves at _a = (1/q)H/, _ b = (1/q)H/. The equatorial map of a particle guiding center moves, because of the gradient of H but also because the equatorial crossing point of field line (a, b) moves in time, at a velocity E ind / 2. Thus we are not neglecting induction-electric field effects in the magnetosphere. Their effect is included in the time-dependent magnetic field mapping. We use a gauge in which the potential is constant along each magnetic field line, except for effects of fieldaligned potential drops, which may be estimated from the Knight [1973] or similar algorithm. [14] Written in terms of Euler potentials, equation (7) takes the form t þ _a þ b _ f ¼ S L ð12þ In (10), (11), and (12), H and f can be regarded as functions of either [l, a, b, t] for the isotropic distribution case where l is the isotropic energy invariant, or [m, J, a, b, t] for the more general case where m and J are the first two adiabatic invariants. [15] Our next task is to work out the generalization of equation (3), which the RCM solves for the ionospheric potential distribution, to the case where there is no symmetry between the two hemispheres and no assumption is made about constancy of the magnetic field along the conducting ionosphere regions of field lines. Using (8) to rewrite (2) in terms of Euler potentials yiel the expressions J ra ¼ F s PðraÞ 2 F ð s H þ s P ra rbþ þ ðv raþs H þ v rb ðrbþ 2 s P 2 þ s H ðra rbþ J rb ¼ F ð s H þ s P ra rbþ F s PðraÞ 2 þ ðv rbþs H þ v ra ðraþ 2 s P 2 þ s H ðra rbþ ð13þ Consider a flux tube that is bounded by Euler potentials a and a + Da and also by b and b + Db, so that it carries magnetic flux Da Db. Use of Euler potentials facilitates expression of current conservation in a flux tube. The conduction current flowing perpendicular to out of the northern ionosphere part of that flux tube is given by 2 DaDb 6 4 J ra þ J rb That current must be balanced by the irkeland current down into the northern ionosphere from the magnetosphere, which is given by J kn DaDb/ n. Here an under an integral sign indicates integration over the northern ionosphere part of a field line. The condition for current balance in the northern ionosphere is thus given by J ra þ J rb ¼ J kn n ð14þ and the corresponding condition for the southern ionosphere is S J ra þ S J rb ¼ J ks s ð15þ 3of7

4 WOLF ET AL.: RIEF REPORT The sign difference between equations (14) and (15) results from the fact that current parallel to is down into the northern ionosphere but up from the ionosphere in the south. [16] We assume that each field line is an equipotential through the conducting region of the northern and southern ionospheres, but allow a potential drop in the auroral acceleration region above each of them. Since the fieldaligned currents may have different densities in the two hemispheres, we allow the potential drops to differ, letting F n = F + F kn, F s = F + F ks, and DF k = F ks F kn, where F is now the potential on the magnetospheric part of the field line, above both the northern and southern hemisphere auroral acceleration regions. (This is consistent with the use of the potential F in equations (4), (10), and (11) to move magnetospheric particles.) Adding (14) and (15) and using (13) gives the master expression that must be solved for the potential F n : F n J kn J S Paa ks ¼ n s F n S Pbb where S Paa ¼ S Pbb ¼ F n F n S Pab S Pba þ S H s PðraÞ 2 ; S Pab ¼ S Pba ¼ s PðrbÞ 2 ; S H ¼ s H F n S H F n þ I Fk þ W ð16þ s Pra rb ; ð17þ and the integrals now include both northern and southern ionosphere en of the field line. Equation (17) represents a generalization of equation (3) in the old formulation. The term in (16) that results from any difference between the field-aligned potential drops above the northern and southern ionospheres is given by DF k S PaaS I Fk ¼ DF k S PbbS DF k S PabS þ S HS DF k S HS DF k DF k S PbaS ð18þ where the index S refers to the southern hemisphere, and S Paa = S PaaS + S Paa, etc. The wind term is given by W ¼ X i¼s J wai þ J wbi ð19þ where the index i refers to either the southern (S) or northern () ionosphere, and J wai ¼ J wbi ¼ i i " # ð v s H ra þ s P þ s H ra rbþrb ðrbþ 2 " # ð v s H rb þ s P þ s H rb raþra ðraþ 2 ð20þ Equations (16) (20) represent the generalization of equation (3) to the case where neutral win and field-aligned potential drops are included, no symmetry is assumed between hemispheres, and no assumption is made about the magnetic field being constant along the conducting region of each field line. Equations (16) (20) are physically equivalent to equations (2.6) (2.9) of Richmond [1995], except that we have included the effects of different fieldaligned potential drops on the northern and southern hemisphere en of the same field line; the coordinate systems used in the two cases are, of course, different (apex coordinates versus Euler potentials). [17] For our equatorial boundary condition, we follow Richmond [1995] and set the vertical current density equal to zero at the apex of the a = 0 field line, which just grazes the lower boundary of the conducting region. In earlier versions of the RCM, the grid has generally terminated about from the equator, and the equatorial electrojet has been approximately represented as a conducting band. The new procedure should allow us to resolve the equatorial electrojet. [18] The field-aligned potential drops can be estimated using the Knight [1973] relation, for example, and the values of J kn and J ks that were computed the previous time step. The wind velocities must come from an appropriate neutral wind model. Once the elliptic equation (16) is solved for the potential F n, then equations (14) and (15) can be solved (using (13)) for updated values of J kn and J ks.if there are serious differences between the field-aligned potential drops above the northern and southern ionospheres, one could compute new values of DF k and solve (16) again for more accurate values of F n, J kn and J ks. [19] ote that this approach allows calculation of J kn and J ks deep in the inner magnetosphere, where the currents generated by magnetospheric pressure gradients are negligible and J kn / n = J ks / s. That is, the formulation allows consideration of currents that flow along field lines between the northern and southern ionospheres, driven not by magnetospheric pressure gradients but by differences between win and conductances in the two hemispheres. 3. umerical Calculation of Euler Potentials [20] We begin by constructing the Euler potential grid in the northern ionosphere. Consistent with the assumption that E = rf at ionospheric height, we assume that the radial component of the magnetic field at ionospheric altitude is independent of time, and we set it equal to the radial component of the internal magnetic field. Magnetospherically driven currents can cause variations 1% in that component, but we neglect those effects in the present work, 4of7

5 WOLF ET AL.: RIEF REPORT where L is magnetic dipole latitude, L dip is the dipole latitude of the dip equator, and R i = R E + h ref. (ote that ir < 0 north of the dip equator.) [25] 4. Set the longitudinal spacing of the b lines by df db ðfþ ¼ 2p 0 df 0 p=2 2p L dip ðf 0 Þ p=2 ir ðl; f 0 Þcos L dl ir ðl; fþcos L dl ð22þ L dip ðfþ where the numerator is the total magnetic flux through the reference height divided by R i 2. Equations (21) and (22) together imply that the total magnetic flux per unit b is independent of b (and f). Substituting (22) in (21), we find that the value of a at the dipole north pole, given by a max ¼ R2 i 2p 2p 0 df p=2 L dip ðfþ ir ðl; fþcos L dl ð23þ which is also independent of f (and b). [26] With our definition of the zero point, the Euler potential a is the magnetic flux per unit b between the dip equator and the point in question. Applied to a dipole field, it follows a/a max = sin 2 L, so that we could define a ¼ a max sin 2 L eq ð24þ Figure 1. Sample RCM grid, plotted in terms of X = L gg cos f gg, Y = L gg sin f gg, where L gg and f gg are geographic latitude (in degrees) and longitude, respectively. (top) Entire northern hemisphere grid, but only every third latitudinal grid curve is plotted. (bottom) Limited to the polar region but shows all grid points in the region. because their inclusion would greatly complicate the analysis. [21] There is considerable flexibility in the definitions of Euler potentials for a given magnetic field. For convenience, we start with the northern ionosphere and make the following choices: [22] 1. ase the grid on a hemispherical shell at reference height h ref, which, for the example presented here, will be set to 90 km. [23] 2. Choose the b = constant curves to be contours of constant dipole longitude f, so that f = f(b). We let b go from 0 (at dipole longitude 0) to 2p. [24] 3. Set a = 0 at the dip equator and, using the fact that D(magnetic flux) = DaDb, calculate a along the line b = constant by the condition al; ð fþ ¼ R 2 df i db L L dip ir ðl 0 ; fþcos L 0 dl 0 ð21þ where L eq is an equivalent dipole latitude. [27] In the RCM implementation, we calculate the integrals in (21) and (22) on a very dense grid, much finer than the one we use for the full RCM, and then calculate the locations of the actual RCM grid points using (22) and (21) by interpolation. A sample RCM grid is shown in Figure 1, computed assuming an IGRF-2000 magnetic field. Figure 2 shows several aspects of our choice for the latitudinal grid spacing, specifically showing a, latitude, and apex altitude versus latitude grid index i. The latter two quantities actually depend on b: the single curves in Figures 2b and 2c were computed assuming a dipole field and are therefore only approximate. [28] ote the following: [29] 1. In Figure 1, the outer boundary of the grid is not a circle, because the distance from the magnetic pole to the dip equator varies with longitude. [30] 2. The constant-a grid curves in Figure 1 are smooth near the pole, as a result of our use of (22) to define the spacing between adjacent constant-b curves. [31] 3. We chose to make the latitudinal grid spacing relatively tight both at low latitudes and in the auroral zone. The low-latitude grid spacing was chosen to so that the apex heights at adjacent grid points are close enough to resolve the equatorial E and F regions (right side of Figure 2). The spacing was made tight in the auroral zone (approximately 0.4 to 1 ) to resolve that crucial region. Of course, the details of the distribution are easily adjusted, and the number of grid points can be increased if necessary. 5of7

6 WOLF ET AL.: RIEF REPORT [32] Once the northern ionosphere grid is determined, the southern ionosphere grid is calculated by numerically tracing field lines from the northern hemisphere at the reference altitude to the southern ionosphere at the same altitude. In the northern ionosphere, the b = constant lines were arbitrarily assumed to be curves of constant dipole longitude. The same is generally not true of the corresponding grid in the southern ionosphere. [33] Though it is reasonable to consider just the Earth s internal magnetic field to trace field lines through either the northern or southern ionosphere, that approximation is inadequate for the purpose of tracing field lines through the magnetosphere between northern and southern ionospheres. A magnetospheric magnetic field model is required for that purpose. In recent event simulations with the RCM [e.g., Garner et al., 2004; Sazykin et al., 2005], we have normally let the magnetic field vary in time, using the Hilmer and Voigt [1995] or Tsyganenko [2000] and Figure 3. Southern ionosphere grid computed for a Tsyganenko and Stern [1996] magnetic field model. Every third latitudinal grid point is shown. The orientation of the Earth is the same as in Figure 1, and the Sun is to the right. Figure 2. Details on latitudinal grid spacing. (a) a/a max as a function of grid index i. (b) Approximate latitudinal spacing and (c) approximate magnetic apex heights of the first 25 grid points. Tsyganenko et al. [2003] models to precompute magnetic field-mapping information at a series of mark times through the event (e.g., every 10 min); to let the magnetic field vary continuously in time, we reinterpolate every time step, between the nearest two mark times. For some recent runs in which the RCM has been coupled to codes that compute magnetic fiel that are in pressure balance [e.g., Lemon et al., 2004; De eeuw et al., 2004], magnetic fiel have been recomputed self-consistently at frequent intervals through events. In the new RCM, the values of the conductance and wind integrals ((17) and (20)) are recomputed by numerical integration every magnetic field mark time. Depending on the situation, the values that are used in (16) are either held constant between mark times, or linearly interpolated in time. [34] It should be noted that our procedure for constructing a southern ionosphere grid based on Euler potentials does not include the open field line region of the southern ionosphere. This is illustrated in Figure 3, which shows the result of using a T96_01 magnetic field model [Tsyganenko and Stern, 1996] to map the orthern hemisphere grid of Figure 1 to the Southern hemisphere. The central region where no grid lines are shown correspon to the region of open field lines in the T96 model. It is also clear from Figure 3 that the southern hemisphere Euler potential grid becomes irregular near the open-closed boundary, because of the near-singular mapping in that region. This problem occurs poleward of the modeling region of the RCM and other inner-magnetosphere models, but it does represent a problem for constructing a global ionospheric grid based on Euler potentials. Our procedure does not produce a useful grid in and near the southern polar cap. [35] One might argue that global-mhd codes treat the coupling of the two hemispheres in a much simpler and more natural way. They deal with coupling of the magnetosphere to the southern ionosphere independently of the coupling to the northern ionosphere, relying on the equa- 6of7

7 WOLF ET AL.: RIEF REPORT tions of ideal MHD to enforce equipotentiality of the magnetic field lines. However, that approach is subject to numerical error and sometimes allows substantially different potential drops between the equatorial magnetosphere and the northern and southern ionospheres. Our procedure allows specification of these potential drops directly from an explicit physical algorithm (e.g., Knight relation), rather than allowing them to be governed by numerical error. 4. Summary and Discussion [36] This paper presents a new formulation of the basic equations of magnetosphere-ionosphere coupling, a formulation that allows inner-magnetosphere convection models to be generalized beyond the earlier simplifying assumption of an aligned dipole, zero-tilt representation of Earth s internal magnetic field. The new formulation facilitates proper consideration of seasonal and IMF- y effects in the magnetosphere as well as seasonal and longitude effects in the ionosphere. The new formulation utilizes a computational mesh based on Euler potentials, allowing the adiabatic drift and irkeland current equations to be written in an elegant form. The expressions for conservation of ionospheric currents also are most naturally derived and expressed in terms of Euler potentials. Our procedure has the disadvantage that it does not produce a grid in the polar cap region of the southern ionosphere, but that does not limit its usefulness for models of inner-magnetospheric convection and the ring current. [37] Acknowledgments. This research has been supported by the Upper Atmospheric Sciences Section of the ational Science Foundation, under grant ATM to Rice and grant ATM to Prairie View A&M. The work has also received partial support from the Center for Integrated Space Weather Modeling (CISM), which is funded by the STC Program of the ational Science Foundation under agreement ATM The authors are grateful to the two referees for helpful comments on the manuscript. [38] uyin Pu thanks George Siscoe and Paul Song for their assistance in evaluating this paper. References De eeuw, D. L., S. Sazykin, R. A. Wolf, T. I. Gombosi, A. J. Ridley, and G. Toth (2004), Coupling of a global MHD code and an inner magnetospheric model: Initial results, J. Geophys. Res., 109, A12219, doi: /2003ja Fok, M.-C., et al. (2001), Comprehensive computational model of Earth s ring current, J. Geophys. Res., 106, Forbes, J. M., and M. Harel (1989), Magnetosphere-thermosphere coupling: An experiment in interactive modeling, J. Geophys. Res., 94, Garner,T.W.,R.A.Wolf,R.W.Spiro,W.J.urke,.G.Fejer, S. Sazykin, J. L. Roeder, and M. R. Hairston (2004), Magnetospheric electric fiel and plasma sheet injection to low L-shells during the 4 5 June 1991 magnetic storm: Comparison between the Rice Convection Model and observations, J. Geophys. Res., 109, A02214, doi: / 2003JA Harel, M., R. A. Wolf, P. H. Reiff, R. W. Spiro, W. J. urke, F. J. Rich, and M. 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McCormac, pp , Springer, ew York. Wolf, R. A. (1983), The quasi-static (slow-flow) region of the magnetosphere, in Solar Terrestrial Physics, edited by R. L. Carovillano and J. M. Forbes, pp , Springer, ew York. Wolf, R. A., et al. (1997), Modeling convection effects in magnetic storms, in Magnetic Storms, Geophys. Monogr. Ser., vol. 98, edited by. T. Tsurutani et al., pp , AGU, Washington, D. C. T.-S. Huang, Department of Physics, Prairie View A&M University, Prairie View, TX 77446, USA. P. Le Sager, Institute for Environmental Research and Sustainable Development, ational Observatory of Athens, I. Metaxa and Vas. Pavlou, Lofos Koufou, GR-15236, P. Pentili, Athens, Greece. S. Sazykin, R. W. Spiro, F. R. Toffoletto, and R. A. Wolf, Department of Physics and Astronomy, Rice University, Houston, TX 77251, USA. (rawolfrice.edu) 7of7