Interchange instability in the presence of the field-aligned current: Application to the auroral arc formation

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A3, 1106, doi: /2002ja009505, 2003 Interchange instability in the presence of the field-aligned current: Application to the auroral arc formation I. V. Golovchanskaya and Y. P. Maltsev Polar Geophysical Institute, Apatity, Russia Received 29 May 2002; revised 23 September 2002; accepted 22 October 2002; published 11 March [1] We have revisited the ideal MHD interchange instability for the case of the nonaligned gradients of the plasma pressure p and unit magnetic flux tube volume V, i.e., the background field-aligned current (FAC) included. Several possibilities of a directional difference between rp and rv in the Earth s plasma sheet are considered. The growth of an interchange perturbation leading to the pressure-gradient-driven FAC striation is illustrated through numerical simulation in a simple model of the nightside region 1 FAC by Karty et al. [1984] and Yang et al. [1994]. It is shown that the FAC is being striated by the interchange process on a timescale from minutes to tens of minutes. The application to the quiet time discrete arc formation is discussed. INDEX TERMS: 2752 Magnetospheric Physics: MHD waves and instabilities; 2731 Magnetospheric Physics: Magnetosphere outer; 2753 Magnetospheric Physics: Numerical modeling; 2764 Magnetospheric Physics: Plasma sheet; KEYWORDS: magnetosphere, field-aligned current, instability, aurora Citation: Golovchanskaya, I. V., and Y. P. Maltsev, Interchange instability in the presence of the field-aligned current: Application to the auroral arc formation, J. Geophys. Res., 108(A3), 1106, doi: /2002ja009505, Introduction Copyright 2003 by the American Geophysical Union /03/2002JA [2] The stability of the near-earth and midtail plasma sheet against the interchange mode, which is a special case of the ballooning-type instabilities, has been examined in a number of studies [Gold, 1959; Swift, 1967; Cheng, 1985; Southwood and Kivelson, 1987; Lee and Wolf, 1992; Ohtani and Tamao, 1993]. Most of these treatments were based on the assumption of the longitudinal symmetry, i.e., of perfectly aligned radial gradients of the plasma pressure and unit magnetic flux tube volume (further, the interchange instability considered under this restriction will be referred to as the standard interchange instability ). In this case, in spite of some deal of controversy (for an exhaustive review on the issues see Liu [1996]), it was generally accepted that average magnetospheric configurations consistent with the statistical magnetic field observations are interchange stable. This, of course, does not exclude occasional formation of very steep earthward pressure gradients leading to the interchange destabilizing, and interpretation of the resulting ripple structures, undulations, w bands, etc. in terms of the standard interchange instability [Yamamoto et al., 1997; Sazykin et al., 2002]. [3] On the other hand, as was demonstrated from the wave approach by Volkov and Maltsev [1986], Ivanov and Pokhotelov [1987], and Maltsev and Mingalev [2000] and from the energy principle by Liu [1996], any longitudinal plasma asymmetry causes destabilizing the interchange motions. Two points are noteworthy regarding this asymmetry: First, its maintenance requires an external source, and, second, its existence implies a background fieldaligned current [Vasyliunas, 1970], which has been typically excluded from stability analysis. At the same time, there is a class of magnetospheric phenomena, such as the auroral arcs, rapid magnetospheric streams, breakup related fingerlike structures, etc., which are known to develop against the background of the field-aligned currents (FAC), either the large-scale region 1/region 2 FACs or the FACs associated with the substorm current wedge. The striation process leading to these manifestations is still of a primary interest. This makes reasonable an attempt to interpret them in terms of the interchange instability revisited for the case of the background field-aligned current included (further it is referred to as the modified interchange instability ). [4] In section 2, we shall verify the stability of the radial plasma distribution using one of the latest magnetospheric magnetic field model T96 [Tsyganenko, 1996]. It will be demonstrated that except for the quiet conditions radial pressure profiles consistent with the T96 are interchange stable even with respect to the Gold [1959] criterion which is an excessively strict condition for stability [Liu, 1996]. Then, in section 3 we shall suggest a directional difference between rp and rv, where p is the isotropic plasma pressure and V is the unit magnetic flux tube volume, and simulate numerically the growth of an interchange perturbation, meaning the application to the discrete auroral arc formation. In section 4 the principal results will be discussed. 2. Interchange Stability of the Radial Gradients in the Magnetosphere [5] For typical magnetospheric configurations in which V increases with increasing radial distance r, Gold s criterion of the interchange stability has the form SMP 5-1

2 SMP 5-2 GOLOVCHANSKAYA AND MALTSEV: INTERCHANGE INSTABILITY AND AURORA Table 1. Values of the Model Parameters Adopted in Calculating pv g Under T96 Model a Conditions B z IMF, nt Dst, nt Quiet 5 10 Average 0 16 Moderately disturbed 4 50 Strongly disturbed a T96 is the magnetospheric magnetic field model of Tsyganenko [1996]. a d dr pv g > 0; ð1þ where g is the adiabatic politropic index equal to 5/3. We have checked if equation (1) holds in the magnetosphere under different geomagnetic conditions. In doing that we calculated V in two manners: straightforward from the Fairfield et al. [1994] database and from the T96 magnetic field model as Z ds V ¼ B ; ð2þ b ds being the element of a magnetic field line. Then under the assumption that the plasma pressure in the equatorial plane of the plasma sheet roughly balances the tail-lobe magnetic pressure, tailward profiles of p were restored. Since the results obtained in these two ways were very similar, we shall further illustrate only those obtained with the use of the T96. [6] The calculations were performed for four sets of the model parameters (in T96 these are B z and B y IMF components, solar wind dynamic pressure P sw and Dst index) corresponding to quiet, average, moderately disturbed, and strongly disturbed conditions (see Table 1). Only B z IMF and Dst index were chosen to vary, whereas average values of P sw = 2.1 npa and B y IMF = 0 were adopted. [7] Figures 1a 1c show the calculated distributions of V, p, and pv g along the x GSM coordinate for the midnight magnetosphere under different level of geomagnetic activity. One can see from Figure 1c that pv g always grows tailward except for the region from x GSM = 30 to x GSM = 40 R E, where under quiet conditions this quantity decreases with increasing distance from the Earth. The peculiar behavior of pv g in this region results from nonmonotonous decline down the tail of the B z component in the neutral sheet, a feature that has been revealed both in the Fairfield et al. [1994] database and T96 model under quiet conditions and not reliably interpreted so far. With this exception it can be concluded that the radial distribution of p and V in the average magnetosphere generally implies interchange stability. 3. Modeling the Instability for Nonaligned rv and rp [8] As was pointed out by Liu [1996], if the magnetosphere represents an open system, for example, there exists an external electric field or any other source maintaining a directional difference of the p and V gradients, such a configuration can be interchange unstable. In fact, nonalignment of rv and rp follows from the existence of region c Figure 1. Midnight magnetosphere profiles of the (a) unit flux tube volume, (b) plasma pressure, and (c) pv g from the T96 model for different levels of geomagnetic activity (see Table 1).

3 GOLOVCHANSKAYA AND MALTSEV: INTERCHANGE INSTABILITY AND AURORA SMP 5-3 1/region 2 field-aligned currents j k if they are driven by the pressure asymmetry mechanism and determined by the Vasyliunas [1970] formula j k ¼ e k ½rV rpš; ð3þ a where e k is the unit vector along the magnetic field line. Linear analysis of the ideal MHD equations, including equation (3) for the ground state, was performed by Volkov and Maltsev [1986]. The following dispersion equation was derived in the case of low-b plasma configuration g ½k rðpv ÞŠ½k rvš w ¼ kv i k P 2 P V g ; ð4þ B i where w and k are the frequency and wave vector of a perturbation, respectively, v is the plasma velocity, P is the height-integrated Pedersen conductivity of the ionosphere, and B i is the magnetic field at the ionospheric level. Expression (4) was generalized by Maltsev and Mingalev [2000] for the case of arbitrary b plasma as b g ½k rðpv ÞŠ½k FŠ w ¼ kv i k P 2 P V g ; ð5aþ B i e where F ¼rV m 0 Z ds B 3 rp; Z p ds e ¼ 1 þ gm 0 V B 3; ð5bþ ð5cþ m 0 =4p 10 7 H/m, and integration in equations (5b), (5c) are performed along a magnetic field line from the equatorial plane of the magnetosphere to the ionosphere. Relations (4) and (5a) imply that when the radial profiles of p and V are interchange stable, and the results of section 2 prove that this is the most typical case for the magnetosphere, the modified interchange instability can, nonetheless, develop. Specifically, the perturbations with the wave vectors lying between r( pv g ) and rvor, in a more general case, between r( pv g ) and F defined by (5b) will grow. [9] Figures 2a 2d illustrate schematically different possible orientations of the r( pv g ) and rv vectors in the c Figure 2. (opposite) Different possible orientations of the r(pv g ) and rv (the bold arrows) in the Earth s plasma sheet: (a) Interchange stable configuration, in which these vectors are parallel. The field-aligned currents (FACs) do not originate. (b) Nonalignment of r(pv g ) and rv caused by the sunward convection. The background FACs have the sense of the region 2 FACs [Vasyliunas, 1970]. (c) Nonalignment of r(pv g ) and rv suggested by Karty et al. [1984] and [Yang et al., 1994] to explain the nightside region 1 FACs. (d) The case of the standard interchange instability for perfectly aligned r(pv g ) and rv of opposite directions. The FACs do not originate. The wave vectors of the most unstable perturbations (the thin arrows) as well as topology of the developing interchange structures (the shadowed strips) are also shown. d

4 SMP 5-4 GOLOVCHANSKAYA AND MALTSEV: INTERCHANGE INSTABILITY AND AURORA Earth s plasma sheet as well as the topology of the most unstable interchange motions developing in the each case. Figure 2a is related to the configuration with the vectors r( pv g ) and rv being parallel. This is the only case when neither standard nor modified interchange instability works. The other three panels correspond to successive strengthening of the instability due to increasing angle between the r( pv g ) and rv under the given absolute values of these vectors. As was shown by Vasyliunas [1970], Jaggi and Wolf [1973], Lyatsky et al. [1974], and Yamamoto et al. [1996], the sunward magnetospheric convection leads to the directional difference between r( pv g ) and rv, which is depicted in Figure 2b. In this case the modified interchange instability develops against the large-scale field-aligned currents having the sense of the region 2 FACs. Figure 2c sketches the model of the nightside region 1 FACs by Karty et al. [1984] and Yang et al. [1994] adopted below in the numerical simulations. We shall argue in the discussion that in this case the modified interchange instability generates structures topologically consistent with the discrete arcs observed during periods of the steady magnetospheric convection. Finally, Figure 2d refers to the standard interchange instability in the case of a very steep plasma pressure gradient pointing toward the Earth. The resulting interchange structures can be relevant to the north-south arcs sometimes observed during breakups [Zesta et al., 2000]. [10] In this section we present a simulation of the interchange perturbation growth in the model with a small background gradient of pv g pointing to the flank (the y axis) and the vector of rv pointing away from the Earth (the x axis), meaning the pressure-gradient-driven mechanism of region 1 FACs generation proposed by Karty et al. [1984] and Yang et al. [1994]. Though we do not reject other possible sources of these currents, the chosen model was considered as the simplest and most appropriate to illustrate the very idea of the modified interchange instability development. This model is known to be quite adequate to the periods of steady magnetospheric convection [Sergeev et al., 1990], for which we perform comparison with observations in section 4. [11] We treat the case of low-b plasma; that is, the perturbations of the flux tube volume V caused by the interchange motions are neglected. In the case of arbitrary b plasma, one more equation interrelating pressure and volume perturbations should be included in the treatment. Though the simulation is performed for the dimensionless variables, the estimations of the typical quantities are made for the near-flank region of the dawn-midnight quadrant, with the coordinate center at (x GSM = 10 R E, y GSM = 10 R E ), R E being the Earth radius. The background convection velocities are sunward and determined as v ¼ a e k r pv g ; ð6þ where the dimensionless constant a In equation (6) and everywhere further we normalize distance over 10 R E ; p, V, and pv g are normalized over their values at x GSM = 10 R E and y GSM = 10 R E, which under average geomagnetic conditions are 0.7 npa, 0.7 R E /nt, and 0.7 npa (0.7 R E /nt) 5/3, respectively (see section 2). We set the x dependence of the volume in the form V =1+0.1x and y dependence of pv g as pv g =1+0.01y. This suggests a twofold increase down the tail in the volume and 10 per cent increase of pv g toward the flank over the region of simulation. According to the Vasyliunas formula (3) the background field-aligned current in our model is flowing down into the ionosphere, which, considering the dawnmidnight MLT sector, corresponds to the region 1 FAC. Its density can be estimated as A/m 2 in the magnetosphere or A/m 2 when reduced to the ionospheric level. Having taken P = 10 S, we have the characteristic potential drop j 0 over y =10R E equal to 30 kv, with the electric field being 0.5 mv/m and convection velocities of about v 0 = 50 km/s in the magnetosphere. The corresponding timescale t s, changing from 10 2 sto s for other possible values of the model parameters. Everywhere below we shall normalize the potential over j , i.e., 30 V, and time over t 0. The problem is reduced to the step-by-step solution of the Poisson-type equation for the disturbed electric potential (the disturbed values are denoted by the subscript 1 ), which presents the current continuity requirement and has the form j 1 ¼ 1 P V g with the boundary condition g rv rðpv Þ 1 j 1 j ¼ 0 and the equation of the adiabatic transport ð7þ ðpv g 1 þðv þ v 1 ÞrðpV g Þ 1 þv 1 rðpv g Þ ¼ 0; ð9þ where the disturbed velocities v 1 are calculated from the disturbed potential j 1 at each step. We point out that the considered problem is two-dimensional. The field-aligned direction is not involved in the problem, since the case of the interchange instability implies the movement of the magnetic field tube as a whole, with no quantities changing along the tube. The disturbed potential j 1 is found through numerical integration of the Green function as j 1 ðx; yþ ¼ 1 ZZ 4p h sðx 0 ; y 0 Þln ðx x 0 Þ 2 þ ðy y 0 Þ 2i dx 0 dy 0 ; ð10þ where s(x, y) stands for the right-hand side of (7). The initial perturbation of pv g was set negative as a two-dimensional Gaussoid elongated perpendicularly to a bisectrix of the angle between rvand r( pv g ) (see Figure 3, top left panel). As it is seen from formula (4) of linear analysis, such an orientation is the most favorable for the instability growth. Setting the initial perturbation in a special form, we mean that any arbitrary perturbation can be expanded into Fourier integral over strip-like harmonics. In the course of the instability development, only the strips of a certain orientation will grow, while all the other harmonics will be suppressed. The maximum depletion of pv g in the center of the perturbation was taken to be We note that, according to equation (9), the results would not change if at

5 GOLOVCHANSKAYA AND MALTSEV: INTERCHANGE INSTABILITY AND AURORA SMP 5-5 Figure 3. Perturbations of pv g and of electric potential j (top) at the initial moment and (bottom) in the course of the modified interchange instability growth, t =0.7t 0. The dimensionless quantities are shown (for their normalization see the text). the initial time a velocity perturbation v 1 were given instead of ( pv g ) 1. The perturbation of the electric potential at t =0is shown on the top right panel of Figure 3 in 30 V units. One can see that the disturbed electric field will move the strip of depleted pv g into the region of greater pv g, thus leading to the growth of the instability. Indeed, as the bottom panels of Figure 3 illustrate, in less than t 0 the perturbation prompted noticeably into the region where its contrast against the background is larger, with the disturbed electric potential increasing by an order of magnitude (we note that all the results presented in Figure 3 refer to the reference frame moving sunward with the background convection velocity v 0 ). Unfortunately, our grid does not allow us to simulate evolution of the perturbations on the transverse scales of discrete auroral arcs. Being reduced to the ionosphere, the transverse scale size of the perturbation depicted in Figure 3 is 200 km, that is, refers rather to the inverted-v structures. The maximum quantity of the perturbed electric field at t =0.7t 0 is 20% in excess of the unperturbed value. For the most prominent auroral arcs the adjacent perpendicular electric field can twice exceed the background level [Opgenoorth et al., 1990; Golovchanskaya et al., 2002]. Under the interchange instability mechanism such quantities can be reached in 3 4 t 0, i.e., on the timescales from minutes to tens of minutes. 4. Discussion and Summary [12] In the present study we state that in the Earth s magnetosphere, which generally represents an open system, the interchange process can develop in a more sophisticated manner than it was previously thought. The modified interchange instability considered in the previous sections works whenever the vectors rv and r( pv g ) are nonaligned. This instability seems energetically less restrictive than the standard one. The reason is that destabilizing of the standard interchange instability generally implies large plasma pressure gradients, which are the only source of the free energy required. Such large pressure gradients get inconsistent with the force balance condition in the observed magnetic field configurations, especially for large b [Ohtani and Tamao, 1993]. In the case of the modified interchange instability the energy is supplied by the external source through maintenance of the directional difference between rv and r( pv g ), and formation of unrealistically large radial pressure gradients is not needed.

6 SMP 5-6 GOLOVCHANSKAYA AND MALTSEV: INTERCHANGE INSTABILITY AND AURORA and illustrated in Figure 5a taken from Yahnin et al. [1994, Figure 6], in which the auroral structures seen by all-sky cameras at the first minute of the every UT hour during the SMC period are plotted. In order to compare the topology of the observed auroral structures with that of the implied interchange filaments, we have mapped the individual auroral arcs numbered from 1 to 14 in Figure 5a onto the plane of the magnetospheric neutral sheet (Figure 5b). We excluded from consideration the folded arcs presented in Figure 5a, for along with the interchange instability some other process is clearly involved in their physics. It should be noted that Figure 4. Sketch illustrating polarization of a strip-like and a bubble-like negative perturbation of pv g, leading to the growth of the interchange instability in the case of a strip. The arrows indicate directions of the perturbation movement. [13] We have also tried to demonstrate the idea that any pressure-gradient-driven FAC will be striated by the interchange process leading to the arc-like structures formation. The model by Karty et al. [1984] and Yang et al. [1994] of region 1 FAC generation due to the cross-tail pressure gradients directed toward the flanks has been used. The process supplying and restoring high pv g flux tubes on the flanks is not specified. Some other than reconnection mechanism may be involved which draws closed magnetic field tubes enriched with hot plasma from the cusp region to the night side along the flanks. There have been known other manifestations of rp and rv nonalignment in the magnetosphere, for example, the dawn-dusk pressure asymmetry [Liu and Rostoker, 1991], etc. In the light of the present consideration, the associated Birkeland currents should be tested with respect to the modified interchange instability, with discrete auroral structures being a very likely result. [14] Figure 4 clears up why the mechanism being considered generates just arcs and not, for example, bubbles. The reason is that the polarization electric field moves an appropriately oriented strip with depleted pv g into the region of higher pv g (see also section 3), thus amplifying the perturbation, while a bubble simply drifts along the convection lines pv g = const. From Figure 4 one can also see that there is always a component of the background electric field along the strip that will move it perpendicular to its elongation. This feature is often observed in the dynamics of discrete auroral arcs [e.g., Feldstein and Starkov, 1967]. [15] Sergeev et al. [1990] argue that the Karty et al. [1984] mechanism of the region 1 FAC generation is especially relevant to the periods of steady magnetospheric convection (SMC) and claim, with referring to the observations, 50% pressure increase on the flanks as compared to the central tail region. This is 5 times greater than we adopt in the simulation in section 3. Thus we can expect an extremely strong interchange striation accompanied by bright auroral displays during SMC periods. Observations of such a display are available for the SMC event on 24 November 1981 Figure 5. (a) Auroral display in CGL-MLT coordinates observed during the period of steady magnetospheric convection of 24 November 1981 according to [Yahnin et al., 1994]. The arcs numbered from 1 to 14 have been mapped onto the plane of the magnetospheric neutral sheet. (b) Magnetospheric images of the chosen individual auroral arcs.

7 GOLOVCHANSKAYA AND MALTSEV: INTERCHANGE INSTABILITY AND AURORA SMP 5-7 periods of SMC give us a good opportunity for mapping due to unique invariance of the solar wind and geomagnetic conditions for many hours during SMC. In performing the mapping we used the standard T96 model rather than its modification proposed by Sergeev et al. [1994] for SMC periods. It is justified, since the arc projections mostly go to the magnetospheric regions where the effect of the thin current sheet suggested in the midnight near-earth tail by Sergeev et al. [1994] is not strong. The topology of the magnetospheric arc images (see Figure 5b) is rather consistent with what is expected from the above consideration. Namely, the images are elongated intermediately between the convection streamlines and contours V = const. [16] In conclusion, we summarize the well-known morphological features of the quiet auroral arcs which can be reasonably interpreted under the modified interchange instability scenario: (1) strong elongation in the close to the latitudinal direction [Akasofu and Kan, 1981]; (2) equatorward movement transverse to their stretching [Feldstein and Starkov, 1967; Yau et al., 1981]; (3) embedding into the background FACs [Akasofu and Kan, 1981]; (4) attachment to closed magnetic field lines [Feldstein et al., 1974]; and (5) associated electric field distribution suggesting a magnetospheric origin [Lyons, 1981]. [17] Acknowledgments. Lou-Chuang Lee and Chin S. Lin thank Takashi Yamamoto and Igor Voronkov for their assistance in evaluating this paper. References Akasofu, S.-I., and J. R. Kan (Eds.), Physics of Auroral Arc Formation, Geophys. Monogr. Ser., vol. 25, AGU, Washington, D. C., Cheng, A. F., Magnetospheric interchange instability, J. Geophys. Res., 90, , Fairfield, D. H., N. A. 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Maltsev, Polar Geophysical Institute, Apatity, Murmansk region, , Russia. (golovchan@pgi.kolasc.net.ru; maltsev@pgi.kolasc.net.ru)