Pressure-Driven and Ionosphere-Driven Modes Of

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provided by UTokyo Repository JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, A02224, doi:10.1029/2008JA013663, 2009 Click Here for Full Article

Pressure-driven and ionosphere-driven modes of magnetospheric interchange instability Akira Miura1 Received 3 August 2008; revised 9 December 2008; accepted 18 December 2008; published 28 February 2009.

[1] A general stability criterion for magnetospheric interchange instability, which includes an ionospheric destabilizing contribution, is derived for an arbitrary finite-b magnetospheric model satisfying the magnetohydrostatic force balance. The derivation is based on the magnetospheric energy principle. Unperturbed field-aligned currents in finite-b nonaxisymmetric magnetospheric models are assumed to close via diamagnetic currents in the magnetosphere or in the ionosphere. By exploiting the limit of a very large perpendicular wave number and the eikonal representation for the perpendicular plasma displacement, the magnetospheric interchange mode is shown to be compressible. In this limit the kink mode makes no contribution to the change in the magnetospheric potential energy. By using magnetospheric flux coordinates, the explicit form of the magnetospheric potential energy change is calculated for interchange perturbations, which do not bend magnetospheric magnetic fields. For a nonaxisymmetric finite-b magnetospheric model, a combined effect of the pressure gradient and field line curvature, not only in the meridional plane but also in the plane parallel to the longitudinal direction, is responsible for pressure-driven interchange instability. For an axisymmetric, north-south symmetric and low-b magnetospheric model, in which the magnetic field is approximated by a dipole field, the m =1orm = 2 ionosphere-driven mode, where m is the azimuthal mode number, has an upper critical equatorial b value for instability in the order of 1. Thus a substantial region of the inner magnetosphere or the near-Earth magnetosphere may be unstable against ionosphere-driven interchange instability caused by a horizontal plasma displacement on the spherical ionospheric surface. Citation: Miura, A. (2009), Pressure-driven and ionosphere-driven modes of magnetospheric interchange instability, J. Geophys. Res., 114, A02224, doi:10.1029/2008JA013663.

1. Introduction corresponds to the adiabatic gradient for the magnetohy- drostatics of a plasma in a dipole field with R being the [2] The possibility of spontaneous large-scale interchange distance from the earth’s center in the equatorial plane, the motion of magnetic flux tubes and the plasma contained in magnetospheric plasma is unstable against fast adiabatic them by pressure-driven interchange instability in the mag- convection. Gold [1959] suggested that a better understand- netosphere has been discussed by Gold [1959]. Such an ing of the processes of magnetic storms and auroras and of interchange perturbation does not bend magnetic field lines the Van Allen radiation belts would all require better in the magnetosphere and allows a class of motions to occur estimates of the interchange motions. The interchange without the need of overcoming magnetic tension force. instability has also been widely discussed as a potentially Since then, many studies have been devoted to clarify important mechanism for the redistribution of mass in characteristics of magnetospheric interchange instability planetary magnetospheres. Therefore, to know accurately [e.g., Sonnerup and Laird, 1963; Cheng, 1985; Rogers the conditions necessary for interchange instability is very and Sonnerup, 1986; Southwood and Kivelson, 1987; important in clarifying the dynamics of the magnetosphere. Ferrie`re et al., 2001]. In particular, stability criteria for [3] The primary objective of the present study is to derive magnetospheric interchange instability have been discussed a general stability criterion for interchange instability, which widely for several limited magnetospheric models. Gold includes an ionospheric destabilizing contribution, for an [1959] discussed the stability of low-b magnetospheric arbitrary finite-b and nonaxisymmetric three-dimensional configuration assuming a dipole magnetic field. He showed magnetospheric plasma, on the basis of a magnetospheric that for pressure gradients steeper than R20/3,which energy principle [Miura, 2007]. The unperturbed magnetospheric plasma is assumed to satisfy the magnetohydrostatic force balance J B = rp and is bounded by ideal 1Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan. ionospheric boundaries. The magnetospheric energy principle is an extension of the energy principle [Bernstein et al., Copyright 2009 by the American Geophysical Union. 1958] valid for any finite-b magnetohydrostatic equilibrium 0148-0227/09/2008JA013663$09.00

A02224 1of18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 configuration used in fusion plasmas to a magnetospheric system with ideal ionospheric boundary conditions, which satisfy the self-adjointness of the force operator. [4] A stability criterion for the interchange instability has been discussed by Bernstein et al. [1958] for a finite-b axisymmetric system with periodicity in the direction of the axis of symmetry. Hameiri et al. [1991] derived a stability criterion for finite-b axisymmetric magnetosphere with closed field lines, which was originally derived by Spies [1971]. They assumed that field lines are closed loops without any boundaries such as ionospheres and thus Figure 1. A three-dimensional view of the plasma volume avoided complicated problems arising from taking into P surrounded by an ideal ionospheric boundary and by the account ionospheric boundary conditions in the real mag- outer flux surface Sout and the inner flux surface Sin in the netospheric plasma. In these criteria the plasma pressure and magnetohydrostatic magnetospheric equilibrium, which are the specific volume of a magnetic flux tube U are consid- shown by the light grey and the dark grey surfaces, ered to be a function of only one variable representing the respectively. radial coordinate. Here, I d‘ U ¼ internal energy is conserved, a magnetospheric energy prin- B ciple is applicable. The present approach based on the variational magnetospheric energy principle is applicable and the integration is taken for one period or for a closed for any magnetospheric equilibium model satisfiying mag- field line loop. netohydrostatic force balance with and without axisymmetry. [5] These studies assume an axisymmetric three- [7] Since magnetospheric interchange instability is not dimensional system. Therefore, there is no unperturbed field- driven by the solar wind-magnetosphere interaction, one is aligned current. When the plasma pressure p or the specific interested in the stability of a plasma with a finite volume P flux tube volume U is nonaxisymmetric, however, an unper- in the magnetosphere. The plasma volume P is surrounded turbed field-aligned current apppears and stability is different by an ideal ionospheric boundary and two flux surfaces in a from that without a field-aligned current. When there is a field- magnetohydrostatic equilibrium. Shown in Figure 1 are two aligned current in the magnetosphere, the current must be flux surfaces Sout and Sin in the magnetohydrostatic mag- closed three-dimensionally either in the ionosphere or in the netospheric equilibrium, which are the light grey surface magnetosphere to satisfiy current continuity. When there is and the dark grey surface, respectively. Those two flux an unperturbed field-aligned current and when this is closed by surfaces Sout and Sin are virtual boundaries located far from Pedersen current in the ionosphere, Volkov and Mal’tsev disturbed field lines in the plasma volume P. The outer flux [1986] calculated the growth rate of the interchange instability surface Sout is taken inside the magnetopause. Therefore, for a finite-b plasma by using a perturbation stability energy conservation holds in the plasma volume P and one analysis in a local Cartesian coordinate system. can thus apply the magnetospheric energy principle to study [6] However, when the field-aligned current is closed by the magnetospheric interchange instability in the plasma ionospheric Pedersen current, there arises Joule dissipation volume P. in the ionosphere and the zeroth-order state assumed for [8] The energy conservation necessary for the magneto- instability is no longer a steady state but decays with a time spheric energy principle requires that the ideal MHD force constant larger than the Alfve´n transit time [Miura, 1996]. operator must be self-adjoint. Miura [2007] obtained four Hence, the sum of the magnetic energy, kinetic energy and ideal ionospheric boundary conditions, which are compat- the internal energy in ideal magnetohydrodynamics (MHD) ible with the self-adjoint property of the force operator and is not conserved. Therefore, any stability analysis assuming satisfying J B = rp in the magnetosphere. Since the a current closure by Pedersen current in the ionosphere interchange of flux tubes involves the motion of a whole cannot be a stability analysis of the steady state in the flux tube, the interchange of magnetospheric flux tubes strictest sense, because the zeroth-order state is not a steady requires horizontal displacement of magnetic field lines in state. In other words, when the field-aligned current is the ionosphere. According to the magnetospheric energy closed by ionospheric conduction currents, the electric field principle there are two ideal ionospheric boundary con- is set up for current closure and this electric field sets the ditions allowing the horizontal displacement of magnetic magnetospheric plasma in motion by E B drift. There- field lines in the ionosphere. One is the horizontally free fore, the magnetospheric plasma cannot be a static equilib- boundary condition for compressible perturbation and the rium satisfying J B = rp. In order to avoid this decaying other is the free boundary condition, which requires that nature of the zeroth-order state, which is incompatible with the perturbation is incompressible. Between these two the magnetohydrostatic force balance, it is assumed in the ionospheric boundary conditions, the horizontally free present study that the unperturbed field-aligned current is boundary condition with a compressible perturbation is closed by a diamagnetic current perpendicular to the unper- shown to be necessary for magnetospheric interchange turbed magnetic field. Therefore, the framework of ideal instability. MHD is retained strictly and the unperturbed state remains a [9] When the horizontal displacement x? is nonzero at steady state. Since in ideal MHD the sum of the kinetic the ionosphere and the ionosphere is a spherical surface, energy and potential energy consisting of magnetic and there arises a change in the potential energy in the

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ionopshere dWI according to the magnetospheric energy operator, but the existence of steady flows causes the principle [Miura, 2007]. This term is negative. Thus the appearance of a nonself-adjoint operator. Therefore, the plasma displacement on the spherical ionospheric surface stability of a system with steady flows cannot be studied gives a negative contribution to the change of the potential by a powerful minimizing principle, i.e., the energy principle. energy for interchange perturbation. The detailed calcula- [14] In order to represent a magnetospheric magnetohy- tion in the present study shows a possibility of ionosphere- drostatic equilibrium satisfying equation (1), some coordi- driven interchange instability due to a nonzero ionospheric nate system must be specified. The unperturbed magnetic horizontal plasma displacement on the spherical ionospheric field satisfying equation (1) can generally be written as surface. Such a possibility has never been pointed out previously and the instability criterion for an ionosphere- B ¼ry rc; ð2Þ driven interchange mode is discussed in detail in the present study on the basis of the magnetospheric energy principle. where y(r) and c(r) are scalar functions of position. They [10] In representing the magnetospheric equilibrium mod- are Clebsch potentials, which are also called Euler el, choice of the coordinate system is very important. By potentials [Stern, 1970]. In general, ry and rc are not choosing a proper coordinate system, the stability analysis orthogonal. In the magnetospheric plasma y is chosen as the based on the energy principle becomes particularly simple magnetic flux and represents a ‘‘radial-like’’ variable. The and tractable as has been demonstrated in fusion plasmas other, defined as c, describes a toroidal ‘‘angle-like’’ [see, e.g., Freidberg, 1987]. Therefore, a flux coordinate is variable. The third coordinate that needs to be defined is a introduced to simplify the stability analysis in the present ‘‘length-like’’ variable measuring distance along the mag- study of magnetospheric interchange instability. netic line. This coordinate is denoted by z. It is convenient [11] The organization of the present paper is as follows. to treat the general perturbation problems in these The definition of an unperturbed state and a flux coordinate coordinates and the use of y, c, and z makes the variational is given in section 2. Magnetospheric energy principle and stability analysis particularly simple. Figure 1 shows that y ionospheric boundary conditions used for the analysis of increases outwardly and c increases counterclockwise. magnetospheric interchange instability are reviewed in [15] For closed magnetic field lines (i.e., closed loops of section 3. Eikonal ansatz, compressibility of the magneto- lines of force), which are used in the axisymmetric magne- spheric interchange mode and an expression for the varia- tospheric model of Hameiri et al. [1991], the specific flux tional change in the potential energy in the magnetosphere tube volume is defined by dWF are explained in section 4. The change in the potential I energy dWF is further reduced for interchange perturbation d‘ UðÞ¼y ; ð3Þ in section 5. The total change of the potential energy dW = B dWF + dWI is calculated in section 6. Stability criteria for different magnetospheric models are presented in section 7. where the integration is taken for one period along the field Discussion is presented and a realistic evaluation of the line loop and ‘ is the distance along the field line. This is a criterion for ionosphere-driven interchange instability is function of the only y and U(y) is a flux label. However, in given in section 8. Summary and conclusion are presented nonaxisymmetric magnetospheric model, the specific vo- in section 9. The four ideal ionospheric boundary condi- lume of a magnetic tube must be defined by tions, which are compatible with the magnetospheric energy Z principle, are derived physically from the requirement of N d‘ energy conservation in Appendix A. Field line bending in UðÞ¼y; c ; ð4Þ B perturbations at the ionosphere is clarified in Appendix B. S where S is the footpoint of a field line at the southern 2. Unperturbed State, Current Closure, ionosphere and N is the foot point at the northern hemisphere. and Flux Coordinates For the nonaxisymmetric magnetospheric model, U is generally a function of y and c,i.e.,U = U(y, c). The unperturbed [12] In the present study, the interchange instability of a magnetohydrostatic magnetospheric configuration satisfying plasma pressure p is also generally a function of y and c, i.e., p = p(y, c). In such a nonaxisymmetric magnetospheric model, there is an unperturbed field-aligned current J . This J B ¼rp ð1Þ k can be seen from the following equation, which is derived from rÁJ =0[Schindler, 2007]: is investigated, where J, B, and p are an unperturbed current, the magnetic field, and the pressure, respectively. J J J Árs J Árs @p @U As long as a magnetospheric configuration is in a k k ¼ ? þ ? þ magnetohydrostatic equilibrium, any magnetospheric con- B N B S B N B S @c @y @p @U figuration, whether it is dipole-like or tail-like, can be used ; ð5Þ in the following analysis. @y @c [13] Although steady flows are often present in the magnetospheric regions of interest, a magnetohydrostatic where rs is the vector along the unperturbed magnetic field equilibrium must be assumed in the present analysis on the and J? is the current perpendicular to the unperturbed basis of the energy principle. This is because the energy magnetic field. Since J? is perpendicular to rs, the first two principle is based on the self-adjoint property of the force terms on the right hand side of equation (5) vanish.

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Therefore, only the third and fourth terms are left on the sign at different places in the ionosphere. Thus, in the present right hand side. Thus, it is obvious that any nonaxisym- ionosphere without conduction currents the field-aligned metry of p = p(y, c)orU = U(y, c) gives rise to a net current at the ionosphere can close via diamagnetic currents. unperturbed field-aligned current at the ionospheric height. In other words, the distribution of B in the magnetohydro- For an axisymmetric magnetospheric model characterized static model of the magnetosphere and the ionosphere is by p = p(y)orU = U(y), [Jk/B]N [Jk/B]S vanishes. The determined, so that the current closure or rÁJ =0is unperturbed field-aligned current affects the stability satisfied in the unperturbed configuration of the magneto- property of the magnetospheric interchange instability. sphere and the ionosphere. Notice that when the unperturbed magnetic field is incident [19] The following variational stability analysis based on obliquely on the ionospheric surface, the plane pependicular the magnetospheric energy principle is valid for any finite-b to the unperturbed magnetic field at N or S intersects the magnetohydrostatic equilibrium. Although no specific finite- neutral atmosphere. Therefore, J? at N or S in equation (5) b magnetohydrostatic magnetospheric equilibrium model is cannot flow in an arbitrary direction. This may restrict used in the present study, there are several numerical non- current closure in the system of the magnetosphere and the axisymmetric magnetospheric models satisfying equation (1). ionosphere and thus an unperturbed state in that system may For example, there are nonaxisymmetric, north-south sym- be somewhat restricted. However, if one assumes that the metric magnetospheric models characterized by p = p(y) unperturbed magnetic field is incident vertically on the and U = U(y, c)[Cheng, 1995], and by p = p(y, c) and U = ionospheric surface, J? at N or S in equation (5) can flow in U(y, c)[Zaharia et al., 2004]. any direction in the plane perpendicular to the unperturbed magnetic field and thus the above restriction caused by the 3. Magnetospheric Energy Principle and Ideal oblique incidence of the unperturbed magnetic field on the Ionospheric Boundary Conditions ionospheric surface is removed. [16] When there is a field-aligned current in the magne- [20] The magnetospheric energy principle states that a * tosphere, it must close in the magnetosphere or in the plasma equilibrium is stable if and only if dW(x , x)=dWF + ionosphere to satisfy rÁJ = 0. Since no unperturbed dWI 0 for all allowable displacements x. Here, dWF is electric field is allowed in the magnetospheric energy the variational change of the potential energy for the principle, this field-aligned current cannot be closed in the magnetospheric plasma, which is calculated for the unper- ionosphere via conduction currents in the present study. turbed plasma volume P. The volume P is surrounded by Therefore, one assumes in the present study that the field- two lateral boundaries Sout and Sin, which are taken to be aligned current is closed by diamagnetic currents flux surfaces in a magnetohydrostatic magnetospheric equilibrium (see Figure 1), and is also bounded by an unper- B rp turbed ionospheric surface. The boundaries Sout and Sin are J ¼ ð6Þ ? B2 located far enough from disturbed field lines in the three- dimensional magnetospheric model. Here, dWI is the iono- in the magnetosphere or in the ionosphere. spheric surface contribution to the variational change of the [17] From Ampe`re’s law rB = m0J, one obtains potential energy and is calculated for the unperturbed spherical ionospheric surface. The single assumption of this 1 1 magnetospheric energy principle is that the unperturbed J? ¼ m0 B k m0 b rB ð7Þ magnetic field at the ionosphere is perpendicular to the and spherical ionospheric surface [Miura, 2007]. [21] The specific form of dWF for the magnetospheric energy principle is the same as dW as given by Freidberg J ¼ m1Bb ÁrðÞb ; ð8Þ F k 0 [1987] and can be written as where b = B/jBj = B/B is the unit vector parallel to the Z h 1 1 2 1 2 2 unperturbed magnetic field and k =(b Ár)b. Since rB, k dWF ¼ dr m0 jQ?j þ m0 B jr Á x? þ 2x? Á kj 2 P and rb are all finite quantities at the ionosphere, both J? þ gpjr Á xj2 2ðÞx Árp ðÞk Á x* and Jk remain finite at the ionosphere. The perpendicular i? ? current (7) in the ionosphere, where there are no conduction J ðÞÁx* b Q ; ð10Þ currents, is provided by diamagnetic current (6), which is k ? ? obviously a finite quantity. [18] Since p is constant along the field line, the diamag- where g is the ratio of specific heats. This is the intuitive netic current given by equation (6) does not change direc- form of dWF originally suggested by Furth et al. [1965, tion along the field line. However, from equation (6) one p. 103] and Greene and Johnson [1968]. Here, obtains Q B ¼rðÞx B ; ð11Þ 1 2 1 rÁJ ¼ B Árp r : ð9Þ ? B B where the subscript ‘‘1’’ represents the perturbed quantity. [22] The last two terms in the integrand of equation (10) can be positive or negative and thus can drive instabilities. Therefore, rÁJ? can change sign at the ionosphere according 1 The first of these is proportional to rp J? B while the to the change of rÁB and therefore r(Jkb) can also change

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second is proportional to Jk. Thus, either perpendicular or plasma can be MHD unstable under proper conditions even parallel currents represent potential sources of instability. without potential sources of pressure-driven modes or The former type are sometimes referred to as pressure- current-driven modes. driven modes, which are subdivided into interchange and ballooning modes, and the latter as current-driven modes or 4. Eikonal Ansatz, Compressibility of the kink modes. Magnetospheric Interchange Mode, and [23] The plasma displacement vector x at the ionospheric boundary must satisfy one of the following boundary Expression for dWF conditions, i.e., [27] In this section an explicit form of dWF is derived for an arbitrary magnetospheric equilibrium by assuming an

xk ¼ 0 and ðÞb Ár x? ¼ 0; ð12Þ eikonal ansatz for x?. Retaining compressibility is shown to be essential for magnetospheric interchange instability. [28] Significant simplifications occur in the stability analysis of an arbitrary magnetospheric equilibrium by using rÁx ¼ 0 and ðÞb Ár x? ¼ 0; ð13Þ magnetospheric energy principle if one focuses attention on interchange modes. The most unstable modes in the interchange instability are usually characterized by a highly rÁx ¼ 0 and x? ¼ 0; ð14Þ localized k? !1perturbation, where k? is the perpendicular wave number. In the following, dW is reduced from its original three-dimensional form involving the three xk ¼ 0 and x? ¼ 0; ð15Þ components of x into a more tractable one-dimensional form involving only the normal component of x. In order where x = x? + xkb. For these combinations of the boundary conditions at the ideal ionosphere, the force to exploit the k? !1limit, an eikonal representation is operator becomes self-adjoint. Since the self-adjointness of used for x? [Freidberg, 1987] the force operator is equivalent to energy conservation of iS the system under consideration [Bernstein et al., 1958; x? ¼ h?e ; ð17Þ Freidberg, 1987; Miura, 2007], these four ideal ionospheric boundary conditions can also be derived from the require- where k? is defined using the eikonal S as ment of energy conservation in the system of the magnetosphere and the ideal ionosphere (see Appendix A for details). k? ¼rS ð18Þ [24] It is obvious that among the above four boundary conditions, the interchange mode requires nonzero x? at the ideal ionosphere. Therefore, only boundary conditions (12) B ÁrS ¼ 0: ð19Þ and (13) satisfy the requirement for the interchange mode.

The boundary conditions (12) and (13) are called the [29] The quantity h? is assumed to vary ‘‘slowly’’ on the horizontally free boundary condition and the free boundary equilibrium length scale a: jarh?j/jh?j1. In contrast, the condition, respectively. It is important to point out here that assumption k? !1implies that the variation of S is rapid: the horizontally free boundary condition (12) becomes an jarSj1. Notice that there is no assumption about the insulating boundary condition for a flat ionosphere, since parallel component xk. there is no finite B1? at the ionosphere and hence no [30] Substitution of (17) into (11) yields ionospheric surface current for a flat ionospheric surface [Miura, 2007]. Q ¼ eiS½rðÞh B : ð20Þ [25] The specific form of dWI for the three-dimensional ? ? ? configuration assuming a spherical ionospheric surface is Z Z !By substituting equations (17) and (20) into equation (10) 2 2 2 2 1 B jx?j B jx?j the exact form of dWF can be expressed as dWI ¼ dS þ dS ; ð16Þ 2m North RI South RI Z 0 1 2 2 dW ¼ dr ðÞrðÞh B þ B jik h þrh F 2m ? ? ? ? ? where RI is the sum of the Earth’s radius RE and the 0 ionospheric height h (R = R + h R ) and ‘‘North’’ and þ 2k h j2 þ m gpjr Á xj2 2m ðÞh Árp ðÞh* Á k I E E ? 0 0 ? ? ‘‘South’’ denote unperturbed ionospheric surfaces in the * Northern Hemisphere and the Southern Hemisphere, m0JkðÞÁrh? b ðÞðÞh? B ? : ð21Þ respectively. [26] The surface contribution dWI is negative for both horizontally free and free boundary conditions. Therefore, [31] In order to reduce dWF further, precise knowledge of this term is destabilizing for these boundary conditions. rÁx is necessary. The minimization condition of dWF with Since R appears in the denominator of equation (16), this respect to xk is obtained from equation (21) and can be I written as [Miura, 2007] destabilizing effect by dWI occurs for a spherical ionospheric surface for the three-dimensional configuration. The ðÞrÁB Ár x ¼ 0: ð22Þ existence of negative dWI suggests that a magnetospheric

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This means that rÁx is constant along the unperturbed order contribution to dWF, which is written as dW0, reduces magnetic field line. Therefore, in order to calculate rÁx in to the magnetosphere, one needs to specify the ionospheric Z boundary condition on rÁx. As was shown in the previous 1 2 2 dW0 ¼ drB jk? Á h?0j : ð25Þ section, there are two ideal ionospheric boundary conditions 2m0 allowing the horizontal displacement of the field line at the ionosphere. One is the horizontally free boundary condition Clearly, the perturbation which minimizes dW0 satisfies k? Á (12), which requires x =0at‘ = ‘ and ‘ = ‘ , where ‘ k S N N h?0 = 0 and therefore h?0 can be written as and ‘S are end points of a field line in the Northern Hemisphere and Southern Hemisphere, respectively. The h 0 ¼ Yb k?: ð26Þ other is the free boundary condition (13), which requires ? rÁx =0at‘ = ‘S and ‘ = ‘N. Here, Yis a scalar quantity, varying on the ‘‘slow’’ equilibrium- [32] For the free boundary condition (13), it is obvious scale length. that rÁx is zero everywhere in the magnetosphere owing to [36] The first nonvanishing contribution to dWF occurs the minimization condition (22). So, there is no need to take 2 in second-order proportional to (k? Á h?1) . That is, dWF = into account the m gpjrxj2 term in equation (21). For the 0 dW0 + dW2 + ÁÁÁ = dW2 + ÁÁÁ. In this expression, the only horizontally free boundary condition (12), the integration of appearance of the quantity h?1 is in the magnetic equation (22) along the field line from ‘ = ‘S to ‘ = ‘N yields compression term, which is written as R Z N 1 S B rÁx?d‘ 1 2 2 rÁx ¼ R : ð23Þ dW2ðÞcomp ¼ drB jik? Á h?1 þrÁh?0 þ 2k Á h?0j : N 1 2m S B d‘ 0 ð27Þ It follows that for both ionospheric boundary conditions (12) and (13), dWF can be written by using only the normal Obviously, the minimum of dW2(comp) occurs when component of h. Z þ 2h Á k ¼ 0; ð28Þ [33] Notice that in many fusion plasma applications, the ?0 plasma compressibility term rÁx can be neglected [e.g., where Freidberg, 1987]. Therefore, the general reduction of dWF proceeds without taking into account the m gp x 2 term 0 jr Á j Z ¼ ik? Á h?1 þrÁh?0: ð29Þ

in equation (21). The calculation of dWF for the incompressible case is described in detail by Freidberg [1987]. # [37] Now one can check aposteriori whether Á x term However, in the magnetospheric case, Gold [1959] showed can really be neglected for the free boundary condition (13) intuitively that, for a dipole field, the specific flux tube 4 in the magnetospheric interchange mode. In the lowest order volume is proportional to R , where R is the distance from the rÁx term in equation (21) can be written as the earth’s center to the flux tube in the equatorial plane. Therefore, for an interchange perturbation in the dipole xk field, the compressibility must be retained, since the specific rÁx ¼ b ÁrB þ b Árxk þrÁx?; ð30Þ flux tube volume changes with the change in R caused by B the interchange motion. Whether or not the compressibility where

is essential is not clear for the interchange perturbation for # iS iS iS an arbitrary finite-b magnetospheric model, since the spe- Á x? ¼ e Z ¼ e ðÞrÁh?0 þ ik? Á h?1 ¼ e ðÞ2h?0 Á k : cific flux tube volume cannot be calculated explicitly as a ð31Þ function of y and c for an arbitrary finite-b magnetospheric model. Therefore, in the following, one performs calcula- Since h?0 !1as k? !1from equation (26), rÁx? !1 tions for both the ideal ionospheric boundary conditions as k? !1from equation (31). Since xk is finite (bounded by (12) and (13), and then one checks aposteriori whether rÁx the field line length from the southern ionosphere to the is really important for magnetospheric interchange mode. northern ionosphere) and b Árxk is considered to be small for [34] An examination of equation (21) indicates that the the magnetospheric interchange mode, the first two terms on only explicit appearance of S (i.e., k?) occurs in the magnetic the right hand side of equation (30) cannot cancel out the rÁ 2 2 compression term, i.e., B jik? Á h? + rÁh? +2k Á h?j . x? term in the k? !1limit. Therefore, in the k? !1limit, Following Freidberg [1987], one is now motivated to rÁx cannot be zero. This contradicts the original assumption consider the limit k? !1(geometrical optics limit) since of the free boundary condition (13) that means rÁx =0 dWF can be systematically minimized by expanding everywhere in the magnetosphere. Therefore, the boundary condition (13) is discarded for the magnetospheric inter-

h? ¼ h?0 þ h?1 þÁÁÁ; ð24Þ change mode. [38] One needs, therefore, to use the horizontally free with jh?1j/jh?0j1/k?a. boundary condition (12) for the magnetospheric interchange [35] Let us first consider the case of the free boundary mode and to retain rÁx in equation (21). Since 2

condition (13). In this case there is no m0gpjr Á xj term in equation (21). Therefore, in the k? !1limit, the zeroth- # iS Á x? ¼ e ðÞrÁh? þ ik? Á h? ; ð32Þ

6of18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 one obtains from equation (23) [41] From equations (31) and (41), one obtains R ! N 1 2 2 j S B ðÞrÁh? þ ik? Á h? d‘j m gp 2hh Á ki jr Á xj ¼ R : ð33Þ rÁx ¼ eiSZ ¼ eiS 2h Á k þ 0 ?0 : ð43Þ N 2 ? ?0 2 1 1 B 1 þ gm0ph 2i S B d‘ B

Therefore, in the k !1limit, the zeroth-order contribution In the k? !1limit, rÁx? !1. Equation (30) also ? holds for the horizontally free ionospheric boundary to dWF reduces to condition (12). Since x is finite and b Árx is considered 2 3 k k Z R to be small for the magnetospheric interchange mode, rÁx 1 6 j N B1ik Á h d‘j27 cannot be zero in the k !1limit. Therefore, indeed, the 4 2 2 S ? ?0 5 ? dW0 ¼ dr B jk? Á h?0j þ gpm0 R : 2m N 2 rÁx term cannot be neglected for the horizontally free 0 B1d‘ S boundary condition (12). This is consistent with the original ð34Þ assumption of the ionospheric boundary condition (12). [42] In the evaluation of other terms in dW2 it is useful to simplify the quantity [ (h B)] as follows Again, the perturbation which minimizes dW0 satisfies k? Á r ? ? [Freidberg, 1987]: h?0 = 0 and therefore h?0 can be written as

½rðÞh B ¼rðÞX k? : ð44Þ h?0 ¼ Yb k?: ð35Þ ? ? ?

If one now writes rX = r?X + b(b ÁrX), then the per- [39] The first nonvanishing contribution to dWF occurs in second-order proportional to (k Á h )2. This second-order pendicular component of [r(h? B)]? can be expressed ? ?1 as contribution to dWF is denoted by dW2. In this expression 0 the only appearance of the quantity h?1 is in the dW2 (h?1) : term, which is the integral of the sum of the magnetic ½rðÞh? B ?¼ ðÞb ÁrX b k? ð45Þ 2 2 compression term, B jik? Á h?1 + rÁh?0 +2k Á h?0j , and 2 the plasma compression term m0gpjr Á xj . That is, [43] Using this relation one can show that the kink Z contribution to dW2, which is written as dW2(kink) is given 0 1 by [Freidberg, 1987] dW2ðÞ¼h?1 dydcW2; ð36Þ 2m0 Z 1 dW2ðÞ¼kink drJkðÞÁrh*? b ½ðÞh? B ? where 2 Z 1 ÂÃÀÁ Z ¼ dr J =B X *ðÞb ÁrX Á ½¼k Á ðÞb k 0: N hi2 k ? ? 2 2 2 d‘ W2 ¼ B jZ þ 2h?0 Á kj þ m0gpjhZij ð37Þ ð46Þ S B Thus, in the k? !1limit, the kink term makes no and contribution to stability. Therefore, by adding all the contributions to dW2 one obtains from equation (21) the full expression of the minimized form of dW2 h?0 ¼ ðÞX =B ðÞb k? ; ð38Þ Z 1 Z dW2 ¼ dydcW; ð47Þ 1 N Z 2m0 hZi¼ d‘; ð39Þ U B S where Z where X YB. Here, by setting z = ‘, where ‘ is the N 2m W ¼ k2 jb ÁrX j2 0 ½ðÞÁrb k p ½jðÞÁb k k X j2 length along the field line, one finds that dr = Jdydcdz = ? B2 ? ? dy dcd‘/B, since J is the Jacobian of the transformation S R N X 2 d‘ 4j 2 k Á ðÞb k? d‘j Á þ gpm RS B R : ð48Þ 1 1 1 1 0 N d‘ N d‘ J ¼ ¼ ¼ ¼ : ð40Þ B þ gpm 3 ðÞÁrry rc z B Árz B Ár‘ B S B 0 S B [44] Miura [2007] has shown in the magnetospheric [40] Therefore, in order to minimize W with respect to Z, 2 energy principle that for the rigid ionospheric boundary one has to obtain Z, which satisfies d(W ) = W (Z + dZ) 2 Z 2 condition (15), compressible ballooning modes occur in the W2(Z) = 0. The solution for this equation is shown to be magnetosphere. The potential energy change dW2 for the

2 compressible ballooning mode is given by the same dW2 as B ðÞ¼Z þ 2h?0 Á k m0gphZi: ð41Þ equation (47) for the horizontally free boundary condition, since both boundary conditions satisfy xk = 0 at the Therefore, one obtains ionosphere and rÁx is given by equation (23). For the two- dimensional cylindrical magnetospheric configuration, in 2hh?0 Á ki which the dawn-dusk direction is parallel to the cylindrical hZi¼ 1 : ð42Þ 1 þ m0gphB2i axis, a similar form of dW2 has been derived for compressible

7of18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224

[48] For the interchange trial function X^ without any z dependence, W in equation (48) reduces to " Z N ^ 2 1 d‘ WðÞ¼y; c 2m0jX j Á 2 ½ðÞÁrb k? p ½ðÞÁb k? k S B B 3 R 2 N 1 S B2 k Á ðÞb k? d‘ 7 ÀÁ5 þ 2gp 1 : ð50Þ U 1 þ gpm0hB2i

The requirement B ÁrS = 0 implies that

SðÞ¼y; c; z SðÞy; c : ð51Þ

Therefore, the wave vector k? immediately follows: Figure 2. The solid line shows an unperturbed field line in the magnetosphere. The short-dashed circle is the iono- @S @S k? ¼rS ¼ ry þ rc: ð52Þ spheric boundary. Shown schematically by a long-dashed @y @c line is a field line without line bending, which is perturbed Since B Árp =0,rp can be expressed as by an interchange mode. The amplitude of perturbation is exaggerated. Adapted from Miura [2007]. @p @p rp ¼ ry þ rc: ð53Þ @y @c ballooning modes using different methods [e.g., Lee and Wolf, Similarly, since the curvature vector, k =(b Ár)b, 1992; Bhattacharjee et al., 1998; Miura, 2000; Schindler and satisfies b Á k = 0, it is convinient to expand the curvature Birn, 2004]. vector k as [45] One observes in equation (48) that the derivative appearing on X involves only ‘. Hence, y and c enter W k ¼ kyry þ kcrc; ð54Þ solely as parameters. Stability can thus be tested one where magnetic line at a time (i.e., for fixed y and c). The ðÞÁb rc k three-dimensional stability problem has thus been reduced k ¼ ð55Þ to the solution of a sequence of one-dimensional problems, y B representing an enormous reduction in effort.

ðÞÁb ry k 5. Reduction of dW2 for Interchange kc ¼ : ð56Þ Perturbations B From equations (52) and (53), one obtains [46] In this section the variational magnetospheric potential energy change dW obtained in the previous section is further 2 @S @p @S @p reduced to a simple form for interchange perturbations. ðÞÁrb k p ¼ B : ð57Þ ? @y @c @c @y [47] Attention is now focused on the stability of interchange perturbations. This type of disturbance corresponds Substitution of equation (57) into equation (50) yields to a trial function of the form X(y, c, z)=X^(y y0, c ^ Z c0) where X is a localized function about y = y0, c = c0, N 1 WðÞ¼y; c 2m jX^j2 ðÞÁb k kd‘ the magnetic line under consideration. Since X is assumed 0 B2 ? to be independent of z, the first term in the integrand of W in " S R # N 1 k b k d‘ equation (48) vanishes. That is, @S @p @S @p S BÀÁ2 Á ðÞ ? Á þ2gp 1 : @y @c @c @y U 1 þ gpm0hB2i iS Q? ¼ B1? ¼ e ðÞb ÁrX b k? ¼ 0 ð49Þ ð58Þ From equations (52) and (54) one obtains in the magnetosphere. Thus, the interchange perturbation sets jQ j2 = 0, thereby eliminating the stabilizing effects of ? @S @S the line bending. Although there is no line bending in the b k? Á k ¼ B kc ky : ð59Þ magnetosphere as equation (49) shows, there is line bending @y @c in perturbations at the ionosphere (see Appendix B for details). Figure 2 is a schematic of the midnight meridian Therefore, one obtains plane and shows an unperturbed field line (solid line) and a Z Z Z N N N field line, which shows no line bending (dashed line), 1 @S kc @S ky 2 ðÞÁb k? kd‘ ¼ d‘ d‘: ð60Þ perturbed by an interchange mode. The amplitude of S B @y S B @c S B perturbation is exaggerated in Figure 2.

8of18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224

Therefore,R in order toR calculate W one is now left to Substitution of equation (71) into equation (69) yields N kc N ky calculate S B d‘ and S B d‘. Z @p Z [49] One obtains from the pressure balance equation N k 1 J J N k c d‘ ¼ kN kS þ @c y d‘: ð72Þ 2 @p @p 1 B b S B 2 BN BS S B k ¼ m rp þr ðÞB ÁrB : ð61Þ @y @y B2 0 2 B2 Substitution of equation (72) into equation (60) yields Therefore, one obtains Z y B 1 B2 y B N @S r r 1 @y JkN JkS kc ¼k Á ¼ m0rp þr Á : b k kd‘ B2 B2 2 B2 2 ðÞÁ ? ¼ @p S B 2 BN BS @ y ! ð62Þ @p Z @S @S N k þ @c y d‘: ð73Þ Using vector formulae, this is further reduced to @y @p @c B @y S m 1 ry B k ¼ 0 ry Á J þ rÁ : ð63Þ c 2B2 ? 2 B2 [53] On the other hand, Appendix B and equation (49) of Hameiri et al. [1991] give Using equation (53) one obtains I ky 1 dp 1 dU 1 @p 1 @p d‘ ¼ m0 U 2 ð74Þ J ¼ J J b ¼ ðÞB ry þ ðÞB rc : ð64Þ B 2 dy B dy ? k B2 @y B2 @c Taking the divergence of this equation yields for a closed field line loop used in their calculation. This equation is obtained by taking a volume integral of rÁ 2 ÀÁ ry B @p rc B @p (rc B/B ) in a volume surrounded by closed field lines rÁ J b ¼rÁ þrÁ : (see Appendix B of Hameiri et al. [1991] for details). If one k B2 @y B2 @c calculates a volume integral of rÁ(rc B/B2)ina ð65Þ volume surrounded by ionospheric surfaces and field lines from S to N, one obtains a similar equation [50] From the static pressure balance equation one has Z N rp ¼ ðÞrJ Árc y ðÞrJ Áry c: ð66Þ ky 1 @p 1 @U d‘ ¼ m0 U 2 ð75Þ S B 2 @y B @y The comparison of this with equation (53) yields @p @p by using a similar calculation as in Appendix B of Hameiri ¼ry Á J ; ¼rc Á J : ð67Þ @c ? @y ? et al. [1991]. The only difference of such a calculation from that in Appendix B of Hameiri et al. [1991] is that a surface 2 [51] Therefore, the use of (67) and substitution of equa- integral of rc B/B over the ionospheric surface arises. tion (65) into equation (63) yield However, this ionospheric contribution vanishes owing to "#the assumption that unperturbed field lines are incident vertically on the ionospheric surface. Therefore, a complete 1 ÀÁ m 1 rc B @p k ¼ rÁ J b 0 þ rÁ : ð68Þ similarity between equations (74) and (75) arises. Substitu- c @p k 2B2 @p B2 @c 2 @y 2 @y tion of equation (73) and (75) into equation (58) yields By dividing this equation by B and then integrating along m jX^j2 @S J J 0 kN kS the field line one obtains WðÞy; c ¼ ÀÁ2 " 1 @p @y BN BS Z U 1 þ gpm0hB2i @y N k 1 J J @p m 1 c kN kS 0 @S @p @S @p @p 1 @U d‘ ¼ @p þ U 2 S B 2 BN BS @c 2 B þ m0U @y @y @c @c @y @y B2 @y Z # N 1 1 rc B @S J N J S @S @p @S @p gp k k @p rÁ 2 d‘ : ð69Þ Á B B @y BN BS @y @c @c @y 2 @y S @p @U U þ gp : ð76Þ [52] Using equations (61) and (67), one obtains @y @y m @p 1 rc B k ¼ 0 rÁ : ð70Þ y 2B2 @y 2 B2 6. Calculation of dW = dWF + dWI By dividing this equation by B and then integrating along [54] The ionospheric contribution to the change of the the field line, one obtains potential energy dWI is given by equation (16). Here, Z Z N N 1 1 rc B @p 1 ky 2 2 ^ 2 2 rÁ d‘ ¼ m U 2 d‘: ð71Þ jx?j ¼jh?0j ¼ jX j k?; ð77Þ 2 0 2 B2 S B B @y B S B

9of18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 where become p(y) and U(y), respectively. Under this condition, W 0 in equation (85) can be written as @S 2 @S 2 @S @S k2 ¼ ðÞry 2 þ ðÞrc 2 þ 2 ry Árc: ð78Þ " ? @y @c @y @c m @S 2 W 0ðÞ¼jy; c X^j2 ÀÁ0 Notice that y and c are not orthogonal in general. Since 1 r r U 1 þ gpm0hB2i @c the unperturbed field lines are assumed to be incident # 2 vertically on the ionospheric surface, dr = dSd‘ at the dU 1 dp dU dp 2k? m0U 2 gp þ U : ionospheric surface. Therefore, one obtains dy B dy dy dy RI BI ð86Þ dydc dS ¼ : ð79Þ B It is obvious that W 0 0 is the stability condition for interchange perturbations. This stability criterion for inter- Therefore, equation (16) can be written as change perturbations is ! Z 2 Z 2 2 ^ 2 ^ 2 1 k?jX j k?jX j m @S dU 1 dp dWI ¼ dydc þ dydc : ÀÁ0 m U 2m0 North RI B South RI B U 1 þ gpm h 1 i @c dy 0 B2 dy 0 B2 ð80Þ dU dp 2k2 gp þ U ? 0: ð87Þ dy dy RI BI From equations (47) and (80) one obtains

Z [57] In the limit of planar ionospheric surface (R !1) 1 1 I dW ¼ dW þ dW ¼ dydcWðÞy; c or in the absence of ionospheric destabilizing contribution, 2 I 2m 2m Z 0 Z 0 !equation (87) can be written as k2 jX^j2 k2 jX^j2 ? dydc þ ? dydc : ð81Þ R B R B dU 1 dp dU dp North I South I m U gp þ U 0: ð88Þ dy 0 B2 dy dy dy

[55] When there is north-south symmetry, one obtains from equation (81) This stability criterion for interchange instability is the same as that derived by Spies [1971] and Hameiri et al. [1991] for Z 2 1 closed field line configurations, although their h1/B i is dW ¼ dydcW 0ðÞy; c ð82Þ defined by the integral along the closed field line instead of 2m 0 the integral from S to N. Although they assume that there are no boundaries along the field line and the field line is a closed loop without any boundary, the present stability k2 jX^j2 W 0 ¼ WðÞy; c 2 ? ; ð83Þ criterion (88) was derived for the realistic magnetospheric RI BI configuration with horizontally free ionospheric boundary condition. Since this criterion includes high-b terms where BN = BS = BI. Since equation (5) can be written as expressed by h1/B2idp/dy, this criterion is a fully general criterion for interchange stability in the finite-b axisym- J J @p @U @p @U kN kS ¼ ; ð84Þ metric and north-south symmetric magnetospheric model. BN BS @c @y @y @c 7.1.2. Low-b Plasma 2 [58] For low-b plasma, the h1/B idp/dy term in equation substitution of equation (84) into equation (83) yields (86) can be neglected and equation (86) becomes particularly simple as " 2 m0 @S 1 @p @U "# 0 ^ ÀÁ 2 W ðÞ¼jy; c X j 1 m0U 2 U 1 gpm @y B2 @c @c 0 2 m0 @S dU dU dp 2k? þ 0hB2i W ðÞ¼jy; c X^j gp þ U : U @c dy dy dy R B @S @U 1 @p @S @U @p I I þ m U gp þ U @c @y 0 B2 @y @c @y @y ð89Þ # @S @U @p 2k2 gp þ U ? : ð85Þ In the low-b limit, the unperturbed magnetic field may be @y @c @c RI BI represented by a dipole field for interchange perturbations. Therefore, 7. Stability Criteria for Different Magnetospheric pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m M 1 þ 3 sin2 F Models BRðÞ¼; F 0 ; ð90Þ 4pR3 cos6 F 7.1. Axisymmetric and North-South Symmetric Magnetospheric Models where 7.1.1. High-b Plasma [56] When the unperturbed magnetospheric states are axi- 2 R ¼ RE cos F0 ¼ LRE: ð91Þ symmetric and north-south symmetric, p(y, c)andU(y, c)

10 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224

Here, M is the magnetic moment of the earth, F is the latitude, Since the stream function y0 is proportional to the poloidal 0 and F0 is the latitude at which the fieldpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi line intersects the magnetic flux yp, one can choose y as the flux function y. earth’s surface. Since d‘ = RcosF 1 þ 3sin2 FdF,the That is, specific flux tube volume U(y) is written as m M 1 1 yðÞ¼R 0 : ð99Þ Z N 4p RE R d‘ 8p 4 U ¼ ¼ R f ðÞF0 ; ð92Þ S B m0M [61] In the dipole coordinate, the magnetic field in the equatorial plane is expressed as where m0M Z BðÞ¼R R^ ef; ð100Þ F0 3 7 4pR f ðÞ¼F0 fLðÞ¼ cos FdF 0 where R^ is the unit vector in the radial direction in the 16 1 3 5 equatorial plane and e is the unit vector in the azimuthal ¼ sin F0 1 þ þ 2 þ 3 f 35 2L 8L 16L direction. Since 16 35 7 ¼ 1 ÁÁÁ : ð93Þ 35 128L4 64L5 m M ry ¼ 0 R^; ð101Þ 4pR2 1 Therefore, neglecting terms of O(L4) and smaller, the specific flux tube volume U is proportional to R4. Notice that in an the comparison of equations (2) and (100) shows that intuitive argument of Gold [1959], he derived U / R4, since in the dipole field the magnetic field strength is proportional 1 3 rc ¼ e : ð102Þ to R and the cross section of the flux tube is proportional to R f R3, and he assumed that the flux tube length is proportional to R. The extra R dependence contained in f(L) in the exact Since c = c(f), one obtains relation (93) is due to the fact that the field line end points are accurately taken into account in equation (92). 1 dc rc ¼rcfðÞ¼ ef: ð103Þ [59] In the dipole field, it is more convenient to use the R df dipole coordinate (R, f, F), where f is the longitude, instead of the flux coordinate (y, c, z). Therefore, one Therefore, dc/df = 1 and one can take c = f. needs to transform (y, c)to(R, f). Here, y = y(R) and c = [62] For the axisymmetric equilibrium configuration, one c(f) in the dipole field. The flux function y should be can Fourier analyze with respect to f proportional to the poloidal flux yp, which is defined by Z xðÞ¼y; c; z xðÞy; z expðÞimf ; ð104Þ y ¼ B Á dA: ð94Þ p p with m being the azimuthal mode number. The f dependence is explicitly accounted for in the interchange mode analysis The stream function y0 can be defined in the equatorial by writing plane as SðÞy; c ¼ SRðÞ; f ¼mf þ S~ðÞy : ð105Þ 1 @y0 B ¼ : ð95Þ z R @R No other f dependence appears in the analysis. Therefore, X = X(y). In the dipole coordinate, ry and rc are or- [60] Since one has thogonal. Therefore, from equation (78) one obtains

2 m M m M 2 dS~ m2 B ðÞ¼R BRðÞ¼ 0 ; ð96Þ k2 ¼ 0 þ : ð106Þ z 4pR3 ? 4pR2 dy R2

Substitution of equation (106) into equation (89) yields one obtains from equations (95) and (96) " 0 2 2 m0 dU dU dp m0M 1 1 W ðÞ¼jy; c X^j m gp þ U y0ðÞ¼R ; ð97Þ U dy dy dy 4p RE R !# 2 2 m M 2 dS~ m2 0 0 where the integration constant was chosen so that y (R = 2 þ 2 : ð107Þ RI BI 4pR dy R RE) = 0. It is then straightforward to show that Z Therefore, for m = 0 mode, one obtains R y ¼ B 2pRdR ¼ 2py0: ð98Þ "# p z 2 R 2 ~ E 0 ^ 2 2 m0M dS W ðÞ¼jy; c X j 2 < 0: ð108Þ RI BI 4pR dy

11 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224

0 1 Since W < 0 is the condition for interchange instability, By neglecting terms of O(L4) and smaller, bcr is further this inequality means that the m = 0 mode is uncondi- reduced to tionally unstable. 0 [63] When m is nonzero, the stability criterion W 0 1 3 9 27 1 b ¼ 1 becomes cr 16g f ðÞ1 q=ðÞ4g L2 8L 128L2 1029L3 "# "#0 2 2 m dU dU dp 2 R2 dS~ R2 dS~ 0 Á 1 þ : ð116Þ gp þ U 2 1 þ 2 0: 2 U dy dy dy RI BI R m dR m dR ð109Þ Therefore, when the plasma b at the equator is larger than or After some calculation, this inequality is reduced to equal to some critical b (bcr), which is given by the right "#hand side of equation (115), the magnetospheric plasma is 2 d‘nU d‘np m M 2 1 2 dR R2 dS~ stable. g 0 1 0: þ 3 þ 2 [67] However, when b < b , the magnetosphere is dR dR 4pR m p RI BI dU m dR eq cr 0 unstable for interchange perturbations because of the desta- ð110Þ bilizing ionospheric contribution. Therefore, when 4g q > 0, beq < bcr is the interchange instability criterion. When From equations (92) and (93) one obtains by neglecting d‘np +4g > 0 and b < b , dW <0,dW > 0 and dW + 1 d‘nR eq cr I F F terms of O(L4) and smaller dWI < 0. Therefore, all the energy to destabilize the magnetospheric interchange perturbation satisfying X(y, d‘nU 4 ^ ’ : ð111Þ c, ‘)=X (y y0, c c0) comes from the ionosphere. dR R d‘np Therefore, the unstable interchange perturbation for d‘nR + 4g > 0 and b < b is different from the normal pressure- Substituting equation (111) and B from equation (90) into eq cr I driven interchange perturbation, in which dW <0. equation (110) yields F "#Therefore, to differentiate this mode, which is driven by 2 d‘np R cos6 F R2 dS~ ionospheric potential energy dWI < 0, from the normal þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 1 þ 4g; pressure-driven interchange mode, this mode is appropri- d‘nR 2R b 2 m2 dR E eq 1 þ 3 sin F0 f0 ately called ionosphere-driven interchange mode. Therefore, ð112Þ ionosphere-driven interchange instability occurs for low-b plasma. Notice that unlike pressure-driven interchange in- where beq is the plasma b in the equatorial plane at R = LRE stability for q > 0, the ionosphere-driven interchange insta- and f0 = 16/35, for stability. bility occurs even for q <0ordp/dR > 0 when beq < bcr is [64] The first term on the left hand side of equation (112) satisfied. comes from the pressure-driven destabilizing term and the second term represents the ionosphere-driven destabilizing 7.2. Nonaxisymmetric and North-South Symmetric term. The right hand side of equation (112) represents the Magnetospheric Model stabilizing contribution by compressibility. [68] The magnetospheric energy principle gives the gen- [65] When eral expression of W 0(y, c) for arbitrary finite-b and nonaxisymmetric magnetospheric models represented by p(y, c) d‘np þ 4g < 0; ð113Þ and U(y, c). Therefore, the stability criterion for this case d‘nR becomes the stability criterion (112) is not satisfied for any beq. m @S 1 @p @U Therefore, the system is unstable. When the plasma pressure 0 m U UðÞ1 þ gpm h1=B2i @y 0 B2 @c @c distribution has an inverse power law distribution such as 0 q @S @U 1 @p @S @U @p p / R , d‘np/d‘nR = q. Therefore, this instability criterion þ m U gp þ U @c @y 0 B2 @y @c @y @y means that the q >4g case is unstable. This is consistent with the intuitive argument of Gold [1959]. However, unlike @S @U @p 2k2 gp þ U ? 0: ð117Þ Gold [1959], the q =4g case could also be unstable, since @y @c @c RI BI the ionosphere gives additional destabilizing contribution. [66] When d‘np þ 4g ¼q þ 4g > 0; ð114Þ 8. Discussion d‘nR 8.1. Realistic Evaluation of the Criterion for the magnetosphere is not interchange unstable by pressure- Ionosphere-Driven Interchange Instability in an driven mechanism. For this case, the stability criterion (112) Axisymmetric, North-South Symmetric and Low-b becomes Magnetospheric Model "# 2 [69] In order to evaluate the ionosphere-driven inter- 1 L cos6 F R2 dS~ b b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 1 þ : change instability criterion beq < bcr for m 6¼ 0 and 4g eq cr d‘np 2 2 m2 dR 4g þ ‘ 1 þ 3 sin F0 f0 q > 0, where bcr is given by equation (116), one needs to d nR ~ 2 ~ ð115Þ calculate (dS/dR) . Since the variation of S with R is considered to be very rapid in the present k? !1ap-

12 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 proximation and for a realistic evaluation of the criterion interchange instability. Therefore, while pressure-driven one needs to calculate the order of magnitude of dS~/dR,let interchange instability [Gold, 1959] requires q >4g, which us simply assume is not easily satisfied in the inner magnetosphere, the instability condition for ionosphere-driven interchange in- 2p stability is easily satisfied in the inner magnetosphere. Thus, S~ ’ kRR ¼ R ð118Þ lR ionosphere-driven interchange instability may cause magnetohydrodynamic disturbances in the inner magnetosphere. instead of trying to obtain S~(R) accurately. [73] One notes that the m = 0 ionosphere-driven mode is [70] From the original assumption of jarSj1, lR must unconditionally unstable. Note that the m = 0 mode has k? be smaller than the perpendicular inhomogeneity scale parallel to ry, and hence h?0 is parallel to the direction of length a in the R direction. If one assumes lR 1 RE on rf. Since one has not calculated growth rates of different m the basis of the assumption of a being a few RE, substitution modes in the present magnetospheric energy principle, it is of equation (118) into beq < bcr yields difficult to determine which mode is dominant. However, if one considers that the real magnetosphere is nonaxisym- 1 2p 2 1 3 9 27 1 metric, the m = 0 mode would be strongly influenced by beq < 1 2 3 such a nonaxisymmetry. Therefore, one conjectures that m = 16g f0 m 1 q=ðÞ4g 8L 128L 1029L 1orm = 2 mode would survive in the presence of non- m 2 Á 1 þ : ð119Þ axisymmetry and m =1orm = 2 mode would be a dominant 2pL mode. [74] The ionosphere-driven interchange instability may be Notice that for L 1 and O(m) 1, the right hand side of viable even in a high-b region such as the near-Earth tail at equation (119) has only a small dependence on L. For the quiet times. However, a realistic evaluation of the upper adiabatic case (g = 5/3) and F0 =60° (L = 4), this critical b value for such a high-b region is difficult, because ionosphere-driven interchange instability criterion for low b an accurate high-b magnetospheric model must be used to magnetospheric plasma becomes obtain the criterion. Therefore, it is only suggested in this study that the ionosphere-driven interchange instability in 3:64 1 m 2 b < 1 þ : ð120Þ such a region would be important when the equatorial eq m2 1 3q=20 8p plasma b remains small and the magnetic field remains dipole-like such as before the growth phase or in the [71] For jqj4g = 20/3 and m =1,beq < 3.65. For jqj recovery phase of a substorm. In the near-Earth high-b 4g = 20/3 and m =2,beq < 0.916. For jqj4g = 20/3 and region, the ionosphere-driven interchange instability would m =3,beq < 0.410. Since the dipole field is assumed, the be quenched as the plasma b increases in the equatorial present ionosphere-driven interchange instability criterion is plane with progress of the growth phase. considered to be applicable to the inner magnetosphere 8.2. Relevance to Previous Stability Analyses Without covering the plasmasphere, radiation belts and the ring Ionospheric Destabilizing Contribution current region. Notice that for F0 =45°,60°,70°, L =2, 4, 8.5, respectively, and the angles Q between the unper- [75] Using a local Cartesian coordinate system, Volkov turbed magnetic field vector and the horizontal ionospheric and Mal’tsev [1986] performed a stability analysis of 1 pressure-driven interchange instability when there is an plane, which are given by tan(p/2 Q) = (2tanF0) ,are 63°,74°,80°, respectively. Therefore, for L >2the unperturbed field-aligned current in the z direction in a assumption of the normal incidence of the unperturbed finite-b plasma. They assumed a perturbation with the form magnetic field on the ionospheric surface in the magneto- exp[i(k Á r wt)], where k = kx^x + ky^y. They assumed an spheric energy principle may well be justified. unperturbed field-aligned current Jz [72] It is known that the quiet time radial profile of the pressure p is peaked around L 3. However, it is very Jz ¼ 2ez ÁrðÞVrp ; ð121Þ improbable that with increasing radius measured in the equatorial plane, the pressure diminishes more rapidly than where ez is the unit vector along the magnetic field and 20/3 R in the q >0ordp/dR < 0 region in the inner Z magnetosphere, which lies typically in L > 3. Therefore, ‘N dz V ¼ ; ð122Þ by assuming jqj4g = 20/3 in this region, one can apply 0 B the above criterion (120). In the inner magnetosphere in L > 3, beq < 3.65 is easily satisfied and beq < 0.916 may at times where the integration is from the equator (‘ =0)tothe be satisfied, but beq < 0.41 may not be satisfied, One ionosphere (‘ = ‘N) in the Northern Hemisphere. Since conjectures, therefore, that the ionosphere-driven inter- north-south symmetry is assumed in their model, V = U/2. change mode with m = 1 or 2 would be destabilized in Equation (121) becomes the inner magnetosphere even if the inverse power law index q is much smaller than the critical value of 20/3 for @p @V @p @V J ¼ 2 : ð123Þ pressure-driven interchange instability. Notice that even for z @y @x @x @y q <0ordp/dR > 0, ionosphere-driven interchange instability occurs when beq < bcr, although bcr is reduced from that They assumed that the field-aligned current is closed by for 0 < q 4g. This may suggest that the q <0ordp/dR > Pedersen current in the ionosphere. 0 region in L < 3 is also unstable against ionosphere-driven

13 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224

[76] Their growth rate wi (imaginary part of w)ofthe stability considered by Gold [1959], which is caused by the instability is given by combined effect of dp/dy and ky, is a special case of such a general pressure-driven interchange instability. hih i 2Vp b g wi ¼ k r‘np2 V Á k r‘npVðÞ; 8.3. Relevance to Magnetospheric Interchange S B k2ðÞ1 þ bg=2 p i Instability When the Unperturbed Field-Aligned ð124Þ Current is Closed Via Pedersen Current [79] In the real magnetosphere an unperturbed field-aligned where Bi is the magnetic field strength at the ionosphere and current generated in the magnetosphere is more likely to be Z closed via conduction currents such as Pedersen currents in the ‘N 1 2m0p dz 2m0p 1 ionosphere. Equation (84), which is derived from equation (5), b ¼ 2 ¼ 2 ¼ 2m0p 2 ð125Þ V 0 B B B B gives Jk at the ionosphere. This relation has also been derived previously [Vasyliunas, 1970]. When Jk at the is the average plasma b along the flux tube. It is straight- ionosphere is assumed to close via Pedersen current in the forward from equation (124) to show that ionosphere, it is driven by an unperturbed electric field. In such a case the static pressure balance equation J B = rp 2 @V 1 @p is no longer valid in the magnetosphere, since the electric w ¼ k m V i S B k2 1 m gp 1=B2 x @y 0 B2 @y field sets the magnetospheric plasma in motion. Therefore, pi ð þ 0h i the actual magnetospheric equilibrium state in such a case @V 1 @p @p 1 @V ky m0V 2 kx þ gpV must be determined by solving @x B @x @y @y @p @V k þ gpV 1 : ð126Þ rðÞV Ár V ¼ J B rp; ð128Þ y @x @x where V is the unperturbed flow velocity equal to the E B [77] On the other hand, when the ionospheric destabiliz- drift velocity. This equilibrium state is not a steady state, 0 0 ing contribution to W (y, c) is neglected, W (y, c)=W(y, because it decays with a decay time constant td, which is c) in equation (85) can be written as given by Miura [1996] by 2 m0 @S @U 1 @p 1 WðÞ¼jy; c X^j m U td 2m0SpVA 2 0 2 ¼ ÀÁ; ð129Þ 1 þ m0gph1=B i @y @c B @c 2 tA 1 þ m SpVA @S @U 1 @p @S @p @U 0 m U þ gpU 1 @c @y 0 B2 @y @y @c @c where tA =2‘/VA is the Alfveń transit time with 2‘ being the @S @p @U þ gpU 1 : ð127Þ field line length from the ionosphere in one hemisphere to @c @y @y the ionosphere in the opposite hemisphere and VA being the average Alfveń speed in the magnetosphere. By changing y ! x and c ! y and assuming the lowest- [80] The horizontally free ionospheric boundary condition order approximation @S/@y ! kx and @S/@c ! ky in necessary for magnetospheric interchange instability means equation (127), one sees that W(y, c) is proportional to wi. that the field line has a finite horizontal displacement on the The pressure-driven interchange instability criterion is W < spherical ionospheric surface. This means that Sp is very 0. This corresponds to wi > 0 and hence to an unstable small, since otherwise the field line is more or less tied to the perturbation in the stability analysis. Therefore, although the ionosphere and cannot move freely in the horizontal direc- magnitude of the growth rate is inversely proportional to Sp tion. In the small Sp limit, one obtains from equation (129) in equation (126), the instability criterion is the same in both equations (126) and (127). That is, there is agreement t 1 d ’ : ð130Þ concerning the instability condition between the perturba- tA 2m0SpVA tion stability analysis with the closure of unperturbed currents via Pedersen current and the criterion derived from [81] In order for the unperturbed state to remain a steady the magnetospheric energy principle. state, td tA is necessary. Therefore, m0VASp 1is [78] For a standard pressure-driven interchange instability necessary. Thus, the present analysis of the magnetospheric envisaged by Gold [1959], p = p(y) and U = U(y). Therefore, interchange instability based on the magnetospheric energy there is no unperturbed field-aligned current. From equation principle would be applicable to the real magnetosphere, (127) it is obvious that when there is an unperturbed field- when Sp is small or m0VASp 1. This means that the present aligned current and hence when p or U is also a function of c, analysis would be more applicable to the nightside magneto- @p/@c and @U/@c play the same roles in equation (127) as sphere, since S in the nightside is smaller than the dayside. @p/@y and @U/@y. Therefore, just as a combined effect of @p/ p [82] For a typical nightside quiet ionosphere in the high @ y and ky causes a pressure-driven interchange instability, a latitude one may have S ’ 0.5 mho and an average Alfveń @ @ p combined effect of p/ c and kc also causes a pressure- speed V ’ 500 km/s. Therefore, one has m V S ’ 0.31. driven interchange instability. Thus, a general pressure- A 0 A p This may validate m0VASp 1, which is necessary for the driven interchange instability involves field line curvature application of the present analysis. However, the ring both in the meridional plane and in the plane parallel to the current and the near-Earth plasma sheet in the nightside longitudinal direction. The pressure-driven interchange in- are more likely to be mapped onto the auroral oval, where

14 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 conductivity is greatly enhanced by energetic particle pre- magnetosphere, is derived for an arbitrary finite-b magne- cipitation. The present analysis is based on the magneto- tospheric model satisfying the magnetohydrostatic force spheric energy principle, which assumes no unperturbed balance. The derivation is based on the magnetospheric flow and hence no unperturbed electric field. Therefore, in energy principle [Miura, 2007], the only assumption of applying it to vaious regions of the real magnetosphere, the which is that the unperturbed magnetic field is incident realistic evaluation of m0VASp seems to be important. vertically on the spherical ionospheric surface. The criterion includes the effect of an unperturbed field-aligned current, 8.4. Closure of Current Perturbations which exists in finite-b nonaxisymmetric magnetospheric [83] Taking the perturbation of Ampe`re’s law rB1 = models, and the ionospheric destabilizing contribution m0J1, one obtains caused by a finite horizontal plasma displacement on the ÀÁ spherical ionospheric surface. The unperturbed field-aligned 1 1 1 J1? ¼ m0 ðÞþrB1? m0 B1kb k m0 b rB1k ð131Þ current is assumed to close via diamagnetic currents in the magnetosphere or in the ionosphere, so that the ideal MHD and the magnetospheric energy principle are applicable. 1 1 [87] By exploiting the k limit and thus using the J1k ¼ b Á J1 ¼ m0 b ÁrB1? þ m0 B1kb ÁrðÞb : ð132Þ ? !1 eikonal representation for x?,itisshownthatthe magnetospheric interchange mode is compressible. Using From rÁB1 = 0, one also obtains the horizontally free ionospheric boundary condition for compressible perturbations, the explicit form of dWF is rÁB1 B ðÞb Ár‘nB þ b ÁrB ¼ 0: ð133Þ ? 1k 1k calculated by using magnetospheric flux coordinates. By choosing X(y, c, z)=X^(y y0, c c0) and thus assuming In the magnetosphere, B1? is zero, but B1? is finite at the ionosphere (see Appendix B). Therefore, from equation (131) no line bending in the magnetosphere (Q? = B1? =0),dWF is further reduced for interchange perturbations. In the k? ! there is a nonzero J1? at the ionosphere. Since there is a jump 1 limit the kink mode makes no contribution to dWF. of B1? at the ionosphere, the first term of equation (131) [88] For arbitrary magnetospheric models, the general and hence J shows a d function-like behavior at the 1? stability condition for interchange instability becomes dW = ionosphere. This peculiar distribution of J1? seems to be unavoidable because of the assumption of an infinitely thin dWF + dWI 0, where dW is given by equation (81) and ionosphere and the normal incidence of the unperturbed dWI is the ionospheric contribution. Here, dWI is negative x magnetic field on the ionospheric surface in the magneto- when ? 6¼ 0 at the ionosphere and thus the ionosphere spheric energy principle [Miura, 2007]. gives a destabilizing contribution. Notice that when there is north-south symmetry, this stability can be tested one [84] From the perturbed equation of motion (equation (12) of Miura [2007]), one obtains in the ideal ionosphere magnetic line at a time and the stability condition can be written as W 0(y, c) 0, where W 0(y, c) is given by ÀÁ 1 2 2 equation (85). The general stability criterion is valid for J1? ¼B w rb x? B B1kJ? JkB1? arbitrary finite-b and nonaxisymmetric magnetospheric B1b rðÞx Árp þ gprÁPx ; ð134Þ models and is not restricted to any particular magnetospheric models. This general stability criterion shows that in where the first term represents the inertia current and the last the general pressure-driven interchange instability a com- term represents the perturbed diamagnetic current. Thus, the bined effect of @p/@y and ky or a combined effect of @p/@c ionospheric surface current J1? given by equation (131) can and kc destabilizes pressure-driven interchange instability. be provided by equation (134) in the ideal ionosphere. [89] If one specialises to an axisymmetric finite-b mag- [85] In the actual ionosphere, where there are also neutral netospheric model in the absence of ionospheric contribu- components, the perpendicular current perturbation for tion, the stability criterion tested for one magnetic field line constant plasma density and stationary neutral components becomes similar to the stability criterion derived by Spies is usually expressed by [1971] and Hameiri et al. [1991]. [90] If one further specialises to an axisymmetric, north- J1? ¼ sPE1 sH E1 b; ð135Þ south symmetric and low-b magnetospheric model, in where sP and sH are Pedersen and Hall conductivities, which the magnetic field is approximated by a dipole field, respectively. For the typical E-layer ionosphere in the a stability criterion by Gold [1959], i.e., q <4g is recovered 2 1 high latitude, one has sP ’ n0e /(minin) and sH ’ n0e/B, by neglecting terms of O(L4) and smaller and the iono- where nin is the ion neutral collision frequency and mi is the spheric destabilizing contribution. Furthermore, for this ion mass. When sP and sH are height integrated, they yield axisymmetric, north-south symmetric and low-b magneto- SP and SH. It is obvious that even in the ideal MHD spheric model, the existence of ionosphere-driven inter- ionosphere, where there are no conduction currents given change mode is shown, when the ionospheric destabilizing by equation (135), there are perpendicular currents given by contribution is included in the criterion for instability. The equation (134), which help close the current required by m = 0 ionosphere-driven interchange mode is uncondition- Ampe`re’s law. ally unstable. For jqj4g, thus for a stable case in the pressure-driven mechanism, the m =1orm = 2 ionosphere- driven mode has an upper ciritical equatorial b value for the 9. Summary and Conclusion instability in the order of 1 for a reasonble parameter set. [86] A general criterion for magnetospheric interchange Thus, such a mode with m =1orm = 2 would be a viable instability, which does not bend magnetic fields in the instability in the inner magnetosphere, where the magnetic

15 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 field remains dipole-like. Unlike the pressure-driven inter- real time domain and the perturbation is a function of change mode for q > 0, the ionosphere-driven interchange position r and time t. mode becomes unstable even for q <0ordp/dR > 0, which [95] One starts from a rigorous local energy conservation may occur in the quiet time in L < 3, when beq < bcr is equation of ideal MHD describing the time evolution of satisfied. total energy at any point inside the magnetosphere and the [91] When one specializes to a local Cartesian coordinate ideal ionosphere system and when the ionospheric destabilizing contribution " is neglected and there is no translational symmetry in a @ 1 p B2 1 p finite-b magnetospheric plasma, the expression for the rv2 þ þ ¼rÁ rv2 þ p þ v @t 2 g 1 2m 2 g 1 lowest-order approximation of W given by equation (127) 0 # becomes proportional to wi. Here, wi is derived from a 1 perturbation analysis in a local Cartesian coordinate system þ E B ; ðA1Þ m0 [Volkov and Mal’tsev, 1986], in which the unperturbed field- aligned current is assumed to close via Pedersen current. where, contrary to the notation used in the main text, r, p, B, Therefore, for such a case the stability criterion derived v, E, and B are all total quantities, which are functions of the from the magnetospheric energy principle becomes the position r and time t and not unperturbed quantities. same as the stability criterion derived from a perturbation [96] Taking the second-order perturbation of (A1) and analysis in a local Cartesian coordinate system with the then integrating the resultant equation over the unperturbed closure of unperturbed field-aligned currents via Pedersen plasma volume P, one obtains current. [92] The general stability criterion for magnetospheric Z Z @ 1 2 interchange instability derived in the present study provides r0v~1 þ w~2 dr ¼ u~2 Á ndS; ðA2Þ a framework for the stability analysis of interchange modes @t P 2 S in realistic arbitrary finite-b magnetospheric equilibria. It is very improbable that in the q >0ordp/dR < 0 region in the where S is the unperturbed surface surrounding the unper- inner magnetosphere the plasma pressure diminishes more turbed plasma volume P, w~2 is the sum of the second-order rapidly than R20/3 and thus pressure-driven interchange perturbations of internal energy and magnetic energy, i.e., instability may not be easily destabilized. However, a substantial region of the inner magnetosphere or the near- p~2 1 ~ ~2 w~2 ¼ þ 2B0 Á B2 þ B1 ðA3Þ Earth magnetosphere may be unstable against the iono- g 1 2m0 sphere-driven interchange instability caused by a horizontal plasma displacement on the spherical ionospheric surface. and u~2 is the second-order perturbation of the energy flux density, i.e.,

Appendix A: Physical Derivation of Ideal ÂÃÀÁ g 1 ~ ~ Ionospheric Boundary Conditions u~2 ¼ ðÞp~1v~1 v~1 B1 B0 þ ðÞv~1 B0 B1 : ðA4Þ g 1 m0 [93] The physical results in the present study are obtained ~ for a specific choice of boundary conditions given in Since x? =0onSout and Sin, u~2 Á n =0onSout and Sin,wheren equations (12) to (15). Therefore, the understanding of is the outward normal vector on Sout and Sin. Therefore, only how those ionospheric boundary conditions are derived is the integral over the ionospheric surface contributes to the right essential for understanding the physical results in the present hand side of equation (A2). Owing to the assumption of study. Although those ideal ionospheric boundary conditions normal incidence of the unperturbed magnetic field on the are derived from the requirement of the self-adjointness of ionospheric surface, i.e., n = b on the ionosphere of the the force operator by Miura [2007], the self-adjointness of Northern Hemisphere, one obtains from equation (A4) the force operator is equivalent to the energy conservation of the system under consideration. Therefore, in this appen- g ÀÁ u~ Á n ¼ x~ Árp þ gp rÁx~ ~v þ ~s Á n dix, those ideal ionospheric boundary conditions are derived 2 g 1 ? 0 0 1k 2 h directly from the requirement of energy conservation, that is, g ÀÁB2 ÀÁ ~ ~ ¼ x~ Árp þ gp rÁx~ ~v 0 v~ Á ðÞb Ár x~ the conservation of H(t)=K(v~1, v~1)+dW(x, x)=K + dWF + ? 0 0 1k 1? ? g 1 i m0 dWI in the system of the magnetosphereR and the ideal ÀÁÀÁ 2 ~ ~ v~ x~ b ; A5 ionosphere,R where K(v~1, v~1)=1/2 rv~1dr and dW(x, x)= 1? Á ? Ár ð Þ ~ ~ P 1/2 x Á F(x)dr. In the following, K, dW, dWF, and dWI P ~ ~ ~ ~ ~ ~ denote K(v~1, v~1), dW(x, x), dWF(x, x), and dWI(x?, x?), where ~s2 is the second-order Poynting vector (see Appendix B respectively. The mathematical details of the calculation are of Miura [2007] for details). ~ ~ described in Appendix B of Miura [2007]. [97]Since(x? Ár)b is equal to x?/RI on the iono- [94] Notice that, contrary to the notation in the main text, spheres of the Northern Hemisphere owing to the assump- in this appendix the subscript 0 is added explicitly to the tion of n = b (see Appendix A of Miura [2007]) unperturbed quantity in order to avoid confusion. Subscripts 2 2 1 and 2 denote the first-order linear perturbation and the B ÀÁ1 B @ 2 ~s Á n ¼ 0 v~ Á ðÞb Ár x~ 0 x~ : ðA6Þ second-order linear perturbation, respectively, and the tilde 2 m 1? ? 2m R @t ? on the perturbation means that the calculation is done in a 0 0 I

16 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224

Therefore, one obtains from equation (A2) Let us define Z @ ðÞ¼K þ dW þ dW u~0 Á ndS; ðA7Þ B0 ¼ BðÞb Ár x BðÞx Ár b ðB2Þ @t F I 2 1 ? ? R S where dWF P w~2dr and then by using vector formulae one obtains

ÀÁ2 ÂÃÀÁ b B0 Bx k: B3 0 g ~ ~ B0 ~ Á 1 ¼ ? Á ð Þ u~2 Á n ¼ x? Árp0 þ gp0rÁx ~v1k v~1? Á ðÞb Ár x? g 1 m0 0 2 1 B @ 2 Therefore, B1 contains a component parallel to b. One 0 ~ ðA8Þ ¼ u~2 Á n þ x?: obtains, therefore, from equations (B2) and (B3) 2m0 RI @t ÀÁ In order for energy conservation in the present system of the B ¼ B0 b Á B0 b ¼ BðÞb Ár x BðÞx Ár b þ BðÞx Á k b: magnetosphere and the ideal ionosphere to hold, H = K + 1? 1 1 ? ? ? ðB4Þ dWF + dWI must be conserved. For boundary conditions 0 (12), (14) and (15), u~2 Á n vanishes on the ionosphere and therefore H is conserved. For the boundary condition (13), Substitution of equation (35) into equation (20) yields one obtains iS Q? ¼ B1? ¼ e ðÞb ÁrX b k?: ðB5Þ 0 g ~ u~2 Á n ¼ ~v1kx? Árp0: ðA9Þ g 1 Since the interchange trial function X = X^(y y0, c c0) does not have z dependence, Therefore, a finite term is left on the right hand side of equation (A7) after integration of equation (A9). However, B1? ¼ 0 ðB6Þ the right hand side of equation (A7) in this case is shown to be much smaller than @dWI/@t on the left hand side for in the magnetosphere. Therefore, from equation (B4) low-b ionospheric plasma (see Appendix B of Miura [2007] for details). Therefore, energy conservation is also valid for ðÞb Ár x? ðÞx? Ár b þ ðÞx? Á k b ¼ 0 ðB7Þ the boundary condition (13), when the ionosphere is a low-b plasma. Thus, for all the boundary conditions (12) to (15), H must be satisfied in the magnetosphere. This gives a is conserved. There are no other combinations of boundary constraint on eikonal S. conditions, which validate the constancy of H. Therefore, one [100] Since the horizontally free boundary condition is finds that the four ideal ionospheric boundary conditions (12) adoptedforinterchangemode,(bÁr)x = 0 at the ionosphere. 0 ? to (15) are derived from the condition of u~2 Á n = 0 at the Therefore, at the ionosphere one obtains from equation (B4) ionosphere or the requirement of the conservation of H. 0 [98] Notice that for a flat ionosphere (RI !1) u~2 Á n = B1? ¼BðÞx? Ár b þ BðÞx? Á k b: ðB8Þ u~2 Á n. Since u~2 Á n at the ionosphere is the normal component of the total second-order energy flux density on the iono- 0 At the ionosphere in the Northern Hemisphere, Appendix A spheric surface, u~2 Á n = u~2 Á n = 0 at the ionosphere means of Miura [2007] gives that there is no second-order energy exchange between the magnetosphere and the neutral atmosphere across a flat x ðÞx Ár b ¼ ? : ðB9Þ ionospheric surface. One also notes that for all ionospheric ? R boundary conditions (12)–(15), first-order energy conserva- I tion is well satisfied for low-b ionospheric plasma in the Therefore, B1? 6¼ 0 at the ionosphere. This means that there is present system of the magnetosphere and the ideal ionosphere a field line bending in perturbations at the ionosphere. (see Appendix B of Miura [2007] for details). Thus, for the [101] Let us assume that ‘ = 0 at the ionosphere in the four ideal ionospheric boundary conditions (12) to (15), Northern Hemisphere and ‘ > 0 in the magnetosphere above the constancy of the total energy H in the system of the ionosphere, and ‘ < 0 below the ionosphere. From the magnetosphere and the ideal ionosphere is guaranteed equations (B8) and (B9) one obtains without the need to take into account the energy in the neutral atomosphere under the ionosphere. x? B1?ðÞ¼‘ ¼ 0 B : ðB10Þ RI Appendix B: Field Line Bending at the Ionosphere Thus, from equations (B5) and (B10) one obtains at ‘ =0

[99] In this appendix, it is shown that although there is no ÀÁ B iS ^ field line bending in the magnetosphere in interchange x? ¼ e b ÁrX b k?: ðB11Þ perturbations, at the ionosphere, however, there is indeed RI a line bending in interchange perturbations. From equation (84) of Miura [2007] one has Since

iS B1 ¼BbðÞþrÁx? BðÞb Ár x? BðÞx? Ár b bðÞx? Ár B: x? ¼ h?0e ; ðB12Þ ðB1Þ

17 of 18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 one obtains from equation (B11) Bhattacharjee, A., Z. W. Ma, and X. Wang (1998), Ballooning instability of a thin current sheet in the high-Lundquist-number magnetotail, Geophys. ÀÁ Res. Lett., 25(6), 861. B ^ h?0 ¼ b ÁrX b k?: ðB13Þ Cheng, A. F. (1985), Magnetospheric interchange instability, J. Geophys. RI Res., 90(A10), 9900. Cheng, C. Z. (1995), Three-dimensional magnetospheric equilibrium with Substituting isotropic pressure, Geophys. Res. Lett., 22(A17), 2401. Ferrie`re, K. M., C. Zimmer, and M. Blanc (2001), Quasi-interchange modes X^ h ¼ Yb k ¼ b k ðB14Þ and interchange instability in rotating magnetospheres, J. Geophys. Res., ?0 ? B ? 106(A1), 327. Freidberg, J. P. (1987), Ideal Magnetohydrodynamics,PlenumPress, into equation (B13) yields New York. Furth, H. P., J. Kileen, M. N. Rosenbluth, and B. Coppi (1965), Plasma 1 Physics and Controlled Nuclear Fusion Research: Proceedings of the ðÞb Ár ‘nX^ ¼ : ðB15Þ 2nd International Conference, Culham 1965,vol.I,Int.At.Energy RI Agency, Vienna. Gold, T. (1959), Motions in the magnetosphere of the Earth, J. Geophys. ^ Res., 64(9), 1219. Since X is not a function of ‘ in the magnetosphere (‘ >0), Greene, J. M., and J. L. Johnson (1968), Interchange instabilities in ideal equation (B15) is considered to define (b Ár)‘nX^ at ‘ =0.If hydromagnetic theory, Plasma Phys., 10, 729. one assumes ficticiously X^(‘ < 0), which satisfies Hameiri, E., P. Laurence, and M. Mond (1991), The ballooning instability in space plasmas, J. Geophys. Res., 96(A2), 1513. @ 1 Lee, D.-Y., and R. Wolf (1992), Is the Earth’s magnetotail balloon unstable?, ‘nX^ ¼ ðB16Þ @‘ J. Geophys. Res., 97(A12), 19,251. RI Miura, A. (1996), Stabilization of the Kelvin-Helmholtz instability by the transverse magnetic field in the magnetosphere-ionosphere coupling sys- at ‘ =0, which is just below the ionosphere, this equation is tem, Geophys. Res. Lett., 23(A7), 761. formally integrated to give Miura, A. (2000), Conditions for the validity of the incompressible assumption for the ballooning instability in the long-thin magnetospheric equili- ^ ^ ‘ brium, J. Geophys. Res., 105(A8), 18,793. X ðÞ¼y y0; c c0;‘ X ðÞy y0; c c0; 0 exp ðB17Þ RI Miura, A. (2007), A magnetospheric energy principle for hydromagnetic stability problems, J. Geophys. Res., 112, A06234, doi:10.1029/ ^ 2006JA011992. at ‘ =0. Therefore, although X is continuous at ‘ = 0 at the ^ ^ Rogers, B., and B. U. O¨ . Sonnerup (1986), On the interchange instability, ionosphere, [(b Ár)X ]‘¼0 6¼ [(b Ár)X ]‘¼0þ = 0. This means J. Geophys. Res., 91(A8), 8837. that although x? is continuous at the ionosphere, B1? is Schindler, K. (2007), Physics of Space Plasma Activity, Cambridge Univ. discontinuous at the ionosphere, if one defines [(b Ár)X^]‘=0 Press, Cambridge, U. K. [(b ÁrÞX^ . Therefore, there is a field line bending at the Schindler, K., and J. Birn (2004), MHD stability of magnetotail equilibria ‘¼0 including a background pressure, J. Geophys. Res., 109, A10208, ionosphere. doi:10.1029/2004JA010537. ¨ [102] Since (b Ár)X^ is nonzero only at ‘ = ‘S and ‘ = ‘N Sonnerup, B. U. O., and M. J. Laird (1963), On magnetospheric interchange in equation (48) and it is not infinite, the finite (b Ár)X^ instability, J. Geophys. Res., 68(1), 131. Southwood, D. J., and M. G. Kivelson (1987), Magnetospheric interchange term at ‘ = ‘S and ‘ = ‘N in the integrand of W in instability, J. Geophys. Res., 92(A1), 109. equation (48) does not give any finite contribution to W.In Spies, G. O. (1971), A necessary criterion for the magnetohydrodynamic other words, the field line bending in perturbations at the stability of configurations with closed field lines, Nucl. Fusion, 11, 552. Stern, D. P. (1970), Euler potentials, Am. J. Phys., 38, 494. ionosphere does not affect the value of the variational Vasyliunas, V. M. (1970), Mathematical models of magnetospheric convec- magnetospheric potential energy change dWF. tion and its coupling to the ionosphere, in Particles and Fields in the Magnetosphere, edited by B. M. McCormac, pp. 60 – 71, D. Reidel, Hingham, Mass. [103] Acknowledgments. This work was supported by Grant-in-Aid Volkov, M. A., and Y. P. Mal’tsev (1986), A slot instability of the inner for Scientific Research 20540436. boundary of the plasma layer, Geomagn. Aeron., 26, 671. [104] Amitava Bhattacharjee thanks the reviewers for their assistance in Zaharia, S., C. Z. Cheng, and K. Maezawa (2004), 3-D force-balanced evaluating this manuscript. magnetospheric configurations, Ann. Geophys., 22, 251.

References Bernstein, I. B., E. A. Frieman, M. D. Kruskal, and R. M. Kulsrud (1958), A. Miura, Department of Earth and Planetary Science, University of An energy principle for hydromagnetic stability problems, Proc. R. Soc. Tokyo, Hongo, 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan. (miura@eps. London Ser. A, 244, 17. s.u-tokyo.ac.jp)

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