The Evolution of Mirror Mode Uctuations in the Terrestrial

Planetary and Space Science 50 (2002) 593–599 www.elsevier.com/locate/pss

The evolution ofmirror mode !uctuations in the terrestrial magnetosheath

M. TÃatrallyay ∗, G. Erd˝os

KFKI Research Institute for Particle and Nuclear Physics, P.O. Box 49, 1525 Budapest, Hungary

Abstract

Mirror type !uctuations were identiÿed in magnetic ÿeld data measured aboard the ISEE-1=2 spacecraft in di.erent regions of the magnetosheath. Minimum variance analysis was applied to ÿnd the mirror type !uctuations, and the amplitude ofthe !uctuations was determined individually. Assuming that the source ofthe mirror mode instability is at the bow shock, the growth rate ofthe ÿeld strength perturbations was determined for four selected magnetosheath passes when mirror type !uctuations were observed in the outer, middle, and inner region ofthe sheath. The obtained growth rate values are in the range 0 :002 s−1 ¡¡0:0035 s−1 being almost an order ofmagnitude smaller than the maximum growth rate values calculated by Gary et al. (J. Geophys. Res. 98 (1993) 1481) based on a numerical evaluation ofthe fullkinetic dispersion relation. The signiÿcant di.erence between the observed and calculated growth rate values can be explained ifthe mirror type !uctuations observed in di.erent regions ofthe magnetosheath do not always originate from the bow shock, but the source may be somewhere else (e.g. at the magnetopause, inside the magnetosheath, or in localized regions ofthe bow shock). Also, the linear approximation applied in the above-mentioned calculations may be inappropriate for describing the growth ofthe observed !uctuations. ? 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction The numerical evaluation ofthe fullkinetic dispersion relation by Gary (1992) and Gary et al. (1993) showed that In several space plasma regimes, the temperature ofthe the maximum growth ofthe proton cyclotron instability can proton component may be larger in the direction perpendic- reduce while the mirror instability stays una.ected when in- ular to the background magnetic ÿeld than it is in the paral- cluding a small percentage of particles (He++ ions) in lel direction. If Tp⊥ is larger than Tp in an electron–proton the electron–proton plasma with densities and temperatures plasma, several instabilities may arise. Ifthe electron tem- corresponding to those observed in the solar wind. These perature is not much higher than the ion temperature, then results are in good agreement with the systematic observa- two di.erent electromagnetic modes may be observed in a tions ofmirror mode !uctuations not only in the terrestrial high ÿ plasma below the proton cyclotron frequency: one is magnetosheath, but also in the magnetosheath ofother plan- the ion cyclotron instability and the other is the mirror in- ets. Mirror mode waves were observed at Jupiter and Sat- stability. Although theory and computational evidence sug- urn by Pioneer 11 (Tsurutani et al., 1982), in the subsolar gest that the ion cyclotron instability should be the dominant magnetosheath ofSaturn by Voyagers 1 and 2 (Violante mode, there have been several observations oflarge ampli- et al., 1995), in the dusk magnetosheath ofJupiter by Ulysses tude !uctuations ofboth low-frequencymodes in di.erent (Balogh et al., 1992), in the wake ofIo by Galileo (Rus- plasma regions where T⊥ ¿T and ÿ¿1, among others in sell et al., 1999), and also at Uranus by Voyager 2 (Russell the magnetosheath. et al., 1989). Mirror mode waves were present almost continuously from the bow shock to the magnetopause in the magnetic ∗ Corresponding author. ÿeld data measured by Voyager 2 when crossing Saturn’s E-mail address: [email protected] (M. TÃatrallyay). subsolar magnetosheath during the inbound pass lasting for

0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0032-0633(02)00038-7 594 M. Tatrallyay,Ã G. Erdos˝ / Planetary and Space Science 50 (2002) 593–599 almost 7 h. Bavassano Cattaneo et al. (1998) found that the ÿled data were used as measured by the !uxgate magne- amplitude and wavelength ofthese !uctuations tend to in- tometer aboard ISEE-1 and ISEE-2 (Russell et al., 1978). crease with increasing distance from the quasi-perpendicular Therefore, !uctuations shorter than 8 s had to be excluded bow shock, except close to the magnetopause in the plasma from this investigation. The proton gyrofrequency is typi- −1 depletion layer. cally p ≈ 1 − 3s in the magnetosheath. Ulysses also almost continuously observed mirror mode In order to select orbits when mirror waves were observed !uctuations in the magnetic ÿeld during its outbound in the magnetosheath for longer time intervals, minimum pass through the dusk magnetosheath ofJupiter formore variance analysis was applied. According to the results of than 1 day. Erd˝os and Balogh (1996) studied the statisti- the minimum variance analysis, mirror mode !uctuations oc- cal properties ofmirror mode depressions. Similar to the cur when the maximum variance direction is closely aligned above-mentioned observations in the magnetosheath ofSat- with the main ÿeld and the ratio ofthe maximum eigen- urn, the amplitude ofthe !uctuations was decreasing when value to the intermediate eigenvalue ofthe variance matrix approaching the bow shock. Close to the magnetopause, is relatively large. a saturation was observed in the minimum magnetic ÿeld The amplitude and period ofindividual !uctuations was level. The average duration ofthe depressions was about determined separately taking into account the change in the 40 s, no characteristic di.erence was found between the total ÿeld direction and the direction ofthe variance vector inner and outer region ofthe magnetosheath. in accordance with the above discussed minimum variance Low-frequency magnetohydrodynamic !uctuations were conditions. A dip was selected as mirror type !uctuation if: identiÿed as mirror mode waves also in the terrestrial magnetosheath in most cases in the inner region (cf. Tsurutani (1) the change in the ÿeld direction was less than et al., 1982; Hubert et al., 1998; Lucek et al., 1999; 20◦ between two successive maxima, i.e. when ◦ ◦ TÃatrallyay and Erd˝os, 2000). (Bmax1; Bmin) ¡ 20 , (Bmin; Bmax2)¡20 , and (Bmax1; ◦ In this paper the occurrence ofmirror mode waves and Bmax2) ¡ 20 where Bmax1 and Bmax2 is the ÿeld vec- the evolution ofthese !uctuations will be studied in di.erent tor at the time oftwo successive maxima and Bmin is regions ofthe magnetosheath fromthe bow shock to the the ÿeld vector at the time ofminimum between these magnetopause based on magnetic ÿeld data measured by maxima; ISEE-1 and ISEE-2. (2) the direction ofthe B variance vector was close to parallel to the ÿeld direction at maximum, i.e. ◦ ◦ (B1; Bmax1) ¡ 26 and (B2, Bmax2) ¡ 26 where 2. Data analysis B1 =(Bmax1 − Bmin) and B2 =(Bmax2 − Bmin).

The twin spacecraft of the International Sun-Earth Ex- In the ÿrst 30 orbits ofISEE-1, nine inbound and three plorer Program were launched together into the same highly outbound passes were found when mirror type !uctuations elliptical orbit in October 1977 (Ogilvie et al., 1977). Dur- were observed throughout the magnetosheath, i.e. mainly in ing most oftheir 10 year long lifetime,ISEE-1 and ISEE-2 the inner and middle region and some in the outer region. were separated by a relatively small distance. During the In about 70% ofall magnetosheath passes (even when there ÿrst 30 orbits data were collected in the dayside magne- were data gaps), mirror type !uctuations were found some- tosheath in the regions between 30◦ and 110◦ solar zenith where in the sheath, in most cases in the inner region. There angles (SZA). In these orbits the satellites passed through were only a few passes when no signature of mirror type the magnetosheath crossing the bow shock and the magne- !uctuations was detected. topause. When data were available from ISEE-2 (there were more Mirror mode waves are best identiÿed by the anticorre- gaps in this data set), mirror type !uctuations were observed lation between the magnetic and plasma pressure and by simultaneously with ISEE-1 since the distance between the the instability criterion 1 ¡ÿ⊥ (Ti⊥=Ti − 1) described by two spacecraft was only a few 100 km in the magnetosheath Hasegawa (1969) where Ti is the ion temperature. Magnetic during the ÿrst 30 orbits. The characteristic parameters of ÿeld depressions (dips) raised by the mirror mode instabil- mirror type !uctuations detected by ISEE-2 were very sim- ity typically last several tens ofseconds, thus the frequency ilar to those observed by ISEE-1 therefore they will not be ofthe associated waves is well below the gyrofrequencyof discussed separately. plasma particles. Sometimes mirror mode !uctuations ap- Figs. 1 and 2 show two representative cases from those 12 pear as peaks in a lower ÿeld region or they can look sinu- magnetosheath passes when mirror type !uctuations were soidal (Lucek et al., 1999). found in the outer, middle, and inner regions of the sheath. In this study the occurrence ofmirror mode !uctuations On 8 November 1977 (orbit 7 outbound), multiple magne- was determined from the characteristics of the magnetic ÿeld topause crossings were observed around 70◦ SZA (14:30 variations, i.e. large amplitude in the magnitude, almost lin- –14:40 UT), while there was a single quasi-perpendicular ◦ ◦ ear polarization parallel to the ÿeld resulting in a fairly sta- Bn =85 bow shock crossing at 65 SZA (18:52 UT). ble ÿeld direction. For the analysis 4 s averaged magnetic The upstream solar wind velocity was around 280 km=s. M. Tatrallyay,Ã G. Erdos˝ / Planetary and Space Science 50 (2002) 593–599 595

Fig. 1. The variation ofthe total magnetic ÿeld Bt, the invert ofthe sine ofthe angle between the direction ofthe main ÿeld and the maximum variance vector 1=sin(B0; Bmax), the ratio ofthe maximum to intermediate eigenvalue ofthe variance matrix max=int, and the normalized amplitudes NB=B of the individually selected mirror type dips forthe outbound leg oforbit 7 ofISEE-1.

Fig. 2. The same parameters as in Fig. 1 forthe inbound leg oforbit 24 ofISEE-1.

On 18 December 1977 (orbit 24 inbound) the bow shock ÿeld magnitude at the time oftwo successive maxima and ◦ ◦ was crossed several times between 75 (04:30 UT) and 70 Bmin is the ÿeld magnitude at the time ofminimum between ◦ SZA (12:17 UT) when Bn = 55–80 was observed. The these maxima. magnetopause was crossed three times around 60◦ SZA The length ofthe sliding window applied forthe mini- (16:10–16:32 UT). The upstream solar wind velocity was mum variance analysis was 240 s, therefore there are longer ∼ 350 km=s. data gaps in the middle two panels than in the upper and The uppermost panel ofFigs. 1 and 2 presents varia- bottom one. The coinciding relatively large values ofthe pa- tions ofthe total magnetic ÿeld Bt while passing through rameters 1=sin(B0; Bmax) and max=int indicate during both the magnetosheath. The second panel from the top shows magnetosheath crossings that the magnetic ÿeld variations the variations ofthe angle between the main magnetic ÿeld show the characteristic features of mirror type !uctuations B0 and the maximum variance vector Bmax as determined in most regions ofthe sheath. by minimum variance analysis. For better illustration ofthe The bottom panel ofFigs. 1 and 2 shows that the nor- small values ofthis angle, the invert ofthe sine ofthe an- malized amplitude NB=B ofthe individually selected mirror gle 1=sin(B0; Bmax) is shown. When this value is above type dips is signiÿcantly increasing with increasing distance 3, then the angle between B0 and Bmax is smaller than from the bow shock towards the magnetopause. A similar 20◦. The ratio ofthe maximum eigenvalue to the interme- tendency was observed in other magnetosheath passes when diate eigenvalue ofthe ÿeld variance matrix max=int is mirror type !uctuations were observed in di.erent regions of presented in the next panel. Mirror mode waves are char- the magnetosheath. Two cases were found, however, when acterized by large max=int values. The bottom panel shows the normalized amplitudes ofthe mirror type dips were not the normalized amplitudes NB=B ofthe individually selected larger in the inner magnetosheath than they were in the mid- mirror type dips where NB =(Bmax1 + Bmax2)=2 − Bmin and dle ofthe sheath. In both cases some disturbance occurred B=(Bmax1 +Bmax2)=4+Bmin=2 where Bmax1 and Bmax2 is the between the middle and the inner region. Also, the main 596 M. Tatrallyay,Ã G. Erdos˝ / Planetary and Space Science 50 (2002) 593–599 magnetic ÿeld B was signiÿcantly increasing in the inner mined using the model ofKobel and Fl uckigerQ (1994). This magnetosheath as the satellite was approaching the magne- model was originally introduced to deliver an analytical form topause which was closer to the Earth than its average loca- for the magnetic ÿeld components in the magnetosheath tion (upstream solar wind dynamic pressure was larger than based on the upstream ÿeld vector. Both the magnetopause 3 nPa). and the bow shock were assumed to have a paraboloidal According to our investigations, the typical period ofmir- shape with the focus placed halfway between the Earth and ror type !uctuations observed by the ISEE-1=2 spacecraft the nose ofthe magnetopause. The boundaries ofthe magne- was around 30–40 s. No characteristic change ofthe period tosheath are fully determined by two parameters: the stand- was seen with increasing distance from the bow shock and o. distance ofthe magnetopause and that ofthe bow shock approaching the magnetopause. As discussed in the Intro- which can be calculated by ÿtting a paraboloid to the actu- duction, the period ofmirror mode !uctuations observed in ally observed bow shock crossing and another paraboloid to Saturn’s magnetosheath was increasing towards the magne- the observed magnetopause crossing. topause (Bavassano Cattaneo et al., 1998), while no similar The steady-state model ofKobel and Fl uckigerQ (1994) e.ect was observed in Jupiter’s magnetosheath (Erd˝os and provides a unique relation between the upstream interplan- Balogh, 1996). etary magnetic ÿeld components and the downstream ÿeld values at any place in the magnetosheath. The plasma !owlines can be determined considering the special case when 3. Discussion the upstream ÿeld vector is parallel to the Sun–Earth line. In that case, the ÿeld lines are identical with the plasma Mirror mode waves can originate in a number ofplaces !owlines also in the magnetosheath. With the approxima- where the plasma is subject to the mirror mode instability tion that the !ow properties are independent ofthe inter- as discussed in the Introduction. Based on numerical sim- planetary magnetic ÿeld direction, we can use the !owline ulations, Omidi et al. (1994) and McKean et al. (1995) model for oblique ÿeld direction as well. As for the veloc- suggested that mirror mode and ion cyclotron waves are ity ofthe plasma !ow, the situation is more complex and generated at the quasi-perpendicular bow shock and then needs further modelling which involves hydrodynamical pa- convected downstream without su.ering much damping. rameters. Lacking those parameters, some approximations Some investigations showed that even ifmirror mode waves were applied. It was assumed that the normal component of were observed across quasi-perpendicular shocks, they dis- the upstream plasma velocity is reduced by a factor of four appeared downstream ofthe undershoot and ion cyclotron downstream ofthe shock while the tangential component and Alfvenic waves were found in the outer region of the ter- remains unchanged. Further downstream the plasma veloc- restrial magnetosheath (Lacombe et al., 1992). From ISEE-1 ity changes according to the divergence=convergence ofthe and ISEE-2 data, Hubert et al. (1998) also identiÿed ion !owlines. Although this assumption is questionable, the re- cyclotron waves in the outer sheath and pure mirror mode sulting plasma velocities are in a reasonably good agreement waves in the inner magnetosheath, while there was a region with the more sophisticated, hydrodynamical calculation of in the middle where mixed !uctuations were observed. As Spreiter and Stahara (1980). discussed in the Introduction, however, mirror mode !uctu- Fig. 3 shows a representative example ofhow the plasma ations were seen in di.erent regions ofthe magnetosheath !owlines and velocity isolines can be determined between ofJupiter and Saturn (Erd˝os and Balogh, 1996; Bavassano the bow shock and the magnetopause applying the above Cattaneo et al., 1998). discussed model. The bow shock and magnetopause location Assuming that the ÿeld strength perturbation B varies measured by ISEE-1 on orbit 7 outbound (shown by circles) in time as exp(−i!t), the amplitude ofthe corresponding were used for this calculation. When comparing these results waves will grow with time if ! has a positive imaginary with the values ofthe same !owÿeld parameters ofSpreiter part, i.e. ! = !0 +i where ¿0 is the growth rate. The and Stahara (1980), it can be seen that the only important mirror instability is a purely growing wave with !0 =0. Mir- di.erence is that the dashed velocity isolines are more par- ror mode structures are frozen in the plasma, they are con- allel than perpendicular to the streamlines close to the mag- vected from the bow shock to the inner regions of the magnetopause at higher SZA. In this region the approximation netosheath along the streamlines ifthe source ofinstability applied in our model that the gradient ofthe plasma den- is somewhere around the bow shock. Usually, the observ- sity is perpendicular to the streamlines is not fulÿlled. This ing satellite’s orbit is not parallel to a particular streamline, simpliÿcation results in an overestimation ofthe propaga- therefore it will detect mirror mode !uctuations propagating tion time (maximum 10–15%) when following the stream- along di.erent plasma streamlines and the propagation time line up to the magnetopause since the calculated velocity is will depend on the length ofthis !owline and on the local smaller than provided by the Spreiter model (Spreiter and plasma velocity. Stahara, 1980) close to the magnetopause at higher SZA. In order to estimate the propagation time ofplasma pack- For each magnetosheath pass, the !owÿeld parameters ages from the bow shock to the orbit of the satellite, model can be estimated in the same way ifthe bow shock and the calculations were performed. Plasma !owlines were deter- magnetopause crossings were observed and the upstream M. Tatrallyay,Ã G. Erdos˝ / Planetary and Space Science 50 (2002) 593–599 597

Fig. 4. Relationship between the measured amplitudes ofmirror type dips NB and the calculated propagation time ofsolar wind plasma fromthe bow shock to the observation point for the outbound leg of orbit 7 of ISEE-1. The growth rate is provided by the tangent ofthe regression line.

Fig. 3. Simulated plasma !owlines (continuous lines) and velocity isolines normalized to the upstream value (dashed lines) between the bow shock and the magnetopause using the measured bow shock and magnetopause location forthe outbound leg oforbit 7 ofISEE-1. The orbit is shown in cylindrical coordinates.

solar wind velocity was measured. Ifthere were multiple bow shock crossings, then the calculations can be performed for more than one bow shock location providing di.erent streamlines. The propagation time from the bow shock to any place in the magnetosheath can be calculated by integrating dS=V along the !owline where dS is the inÿnitesimal length ofthe streamline and V is the local velocity. When compar- Fig. 5. The same parameters as in Fig. 4 for the inbound leg of orbit 24 ing the measured average wave amplitudes at two di.erent ofISEE-1. locations along the orbit, can be determined from the dif- ference between the propagation times calculated from the number ofthe selected mirror type magnetic ÿeld dips was bow shock to the two locations ofobservation based on the 20–25% ofall detected dips. above discussed assumption that B ∼ exp(t). Figs. 4 and 5 illustrate how the measured amplitude of As discussed in the previous section, our investigation the !uctuations depend on the calculated propagation time found mirror type !uctuations (mixed with other type of ofthe plasma fromthe bow shock to the observation point !uctuations) in some cases also in the outer regions ofthe during the two magnetosheath passes presented in Figs. 1 magnetosheath. In order to estimate the growth rate, four and 2, respectively. The regression line determined by the magnetosheath passes were selected when a relatively large method ofleast squares provides the value of growth rate. number ofmirror type !uctuations were foundin the outer, In orbit 24 inbound, a minimum and a maximum propagation middle, and inner regions ofthe sheath and the growth ofthe time was calculated corresponding to the location ofthe NB=B values was obvious with increasing distance from the innermost and outermost bow shock crossing. Horizontal bow shock. All measured amplitudes were used neglecting lines connecting the minimum and maximum propagation only a few !uctuations not ÿtting the general trend, i.e. large time values suggest that the mirror instability might have amplitudes at the shock (overshoot, downstream waves) originated anywhere between the innermost and outermost and small amplitudes at the magnetopause (a.ected by the bow shock crossing. The intermediate values (shown by depletion layer). In these four magnetosheath passes, the circles) were used for determining . Only the amplitudes 598 M. Tatrallyay,Ã G. Erdos˝ / Planetary and Space Science 50 (2002) 593–599

of!uctuations observed downstream ofthe innermost bow !uctuations observed close to the magnetopause may be shock are shown here. the magnetopause itself. The growth rate values estimated for the four selected (2) The source is around the bow shock, but it is lo- magnetosheath passes (orbit 1 inbound, orbit 25 inbound, calized in time and space (e.g. nose ofthe shock, and the above presented two passes) were in the same quasi-perpendicular regions), therefore the mirror struc- range: 0:002 s−1 ¡¡0:0035 s−1 values were obtained. tures convected from the bow shock to the point of The above discussed overestimation ofthe propagation observation along di.erent streamlines originate from times in the magnetopause region produced a reduction of di.erent sources active at di.erent times. It has to be less than 15% in the calculated growth rate values. It has to mentioned here, however, that our investigations do be mentioned here that out ofthe 12 magnetosheath passes not support the plausible simple suggestion that large when mirror type magnetic ÿeld !uctuations were found in amplitude mirror mode !uctuations originate from the di.erent regions ofthe magnetosheath, the growth ofthe quasi-perpendicular region ofthe bow shock and waves amplitudes was the most striking in these four cases. In ofsmaller amplitudes originate fromother regions. The other cases the growth rate must be smaller. observed amplitudes did not show any correlation with The plasma parameters were measured by the Fast Plasma the actual angle between the model bow shock normal Experiment (Bame et al., 1978) aboard ISEE-2 (located a and the ‘mapped-back’ upstream magnetic ÿeld vector few 100 km away from ISEE-1) for the complete magne- calculated from the downstream measured ÿeld using tosheath pass oforbits 24 and 25 inbound and with a data the model ofKobel and Fl uckigerQ (1994). gap of45 mins fororbit 7 outbound. No plasma data were (3) When the wave amplitude grows to large values com- available for orbit 1, the upstream solar wind velocity was pared to the background ÿeld, the linear approximation determined from IMP-8 measurements. The plasma ÿ was applied in the calculations ofGary et al. (1993) may !uctuating between 1 and 10 in the inbound leg oforbits 24 not be appropriate to describe the growth rate ofthe ob- and 25 reaching the largest values in ÿeld minimum, while served mirror type !uctuations. ÿ was somewhat larger (in average around 10) in the out- (4) Finally, the possible e.ect ofthe particles ofsmall paral- bound leg oforbit 7. The proton temperature anisotropy was lel velocity should be mentioned here. In a linear kinetic 1:5 ¡T⊥=T ¡ 2 in all three passes. In all four cases the treatment, Southwood and Kivelson (1993) showed that upstream solar wind velocity was very low (¡ 350 km=s). these particles behave di.erently from the rest of the When taking into account that p proton cyclotron distribution as the instability is growing. They move lit- frequency is usually larger than 1 s−1 throughout the ter- tle along the ÿeld and cannot ÿll in the weak ÿeld re- restrial magnetosheath, it is evident that the growth rate gions between two mirroring points in a magnetic bottle values obtained in this study are almost an order ofmagni- (expected to exist when mirror waves develop) produc- tude smaller than the maximum growth rates calculated by ing distribution functions di.erent from bi-Maxwellians. Gary (1992) and Gary et al. (1993) based on a fully kinetic Since the growth rate ofthe instability is inversely pro- linear Vlasov dispersion theory. The obtained small growth portional to the number ofparticles ofsmall parallel rate ofthe mirror instability raises the question why other velocity, they may serve as a break to prevent the insta- instabilities, especially the ion cyclotron mode is not the bility to grow faster as their number is growing. dominant one in an anisotropic plasma. According to the calculations ofGary et al. (1993), the maximum growth rate ofthe mirror instability is = p ≈ 4. Summary 0:02 in a plasma containing 5–10% particles when T⊥=T =1:5, T =4Tp, and ÿp = 4. Since this value is In this study the evolution ofmirror type !uctuations was signiÿcantly larger than the maximum growth rate ofthe analyzed in the terrestrial magnetosheath based on magnetic proton cyclotron instability, mirror waves can grow faster. ÿeld and plasma observations aboard the ISEE-1=2 space- Hubert et al. (1998) obtained about the same max value craft. The occurrence of the mirror mode !uctuations was for a plasma of slightly di.erent parameters (n=np = 6%, determined from the characteristics of the magnetic ÿeld T⊥=T =1:5, T =5Tp, ÿp = 5) and = p was increasing variations applying minimum variance analysis. Mirror !uc- with increasing ÿ. tuations were found in di.erent regions of the sheath from We can ÿnd di.erent explanations for the obtained small the bow shock to the magnetopause in about 20% ofthe growth rates ofthe observed mirror type !uctuations. magnetosheath passes ofthe ÿrst 30 orbits. The amplitude and period ofthe !uctuations were de- (1) The mirror instability is not originating from the bow termined individually. Mirror type dips (ÿeld depressions) shock, there may be di.erent sources inside the mag- were selected by taking into account the change in the to- netosheath. In this case the growth rate cannot be de- tal ÿeld direction and the direction ofthe variance vector. termined by comparing the amplitudes ofmirror type Our investigations provided similar amplitudes and periods !uctuations in the outer or middle regions ofthe sheath when selecting mirror type peaks (ÿeld increases) showing to those in the inner region. The source ofmirror type that the !uctuations were close to quasi-sinusoidal waves. M. Tatrallyay,Ã G. Erdos˝ / Planetary and Space Science 50 (2002) 593–599 599

Assuming that the ÿeld strength perturbation B varies in References time as exp(t) for mirror waves, growth rate was determined from four selected magnetosheath passes when mir- Balogh, A., Daugherty, M.K., Forsyth, R.J., Southwood, D.J., Smith, ror type !uctuations were found in the outer, middle, and E.J, Tsurutani, B.T., Murphy, N., Burton, M.E., 1992. Science 257, inner region ofthe sheath and the growth ofthe amplitudes 1515. Bame, S.J., Asbridge, J.R., Felthauser, H.E., Glore, J.P., Paschmann, G., was the most striking with increasing distance from the bow Hemmerich, P., Lehmann, K., Rosenbauer, H., 1978. IEEE Trans. shock. The period ofthe mirror type !uctuations did not Geosci. Electron. GE-16, 216. show any characteristic change during the magnetosheath Bavassano Cattaneo, M.B., Basile, C., Moreno, G., Richardson, J.D., passes. The obtained growth rate values were in the range 1998. J. Geophys. Res. 103, 11,961. 0:002 s−1 −1 Erd˝os, G., Balogh, A., 1996. J. Geophys. Res. 101, 1. ¡¡0:0035 s being almost an order ofmag- Gary, S.P., 1992. J. Geophys. Res. 97, 8519. nitude smaller than the maximum growth rate values pre- Gary, S.P., Fuselier, S.A., Anderson, B.J., 1993. J. Geophys. Res. 98, dicted by the calculations ofGary et al. (1993) based on a 1481. fully kinetic linear Vlasov dispersion theory. Hasegawa, A., 1969. Phys. Fluids 12, 2642. This signiÿcant di.erence between the observed and cal- Hubert, D., Lacombe, C., Harvey, C.C., Moncuquet, M., 1998. J. Geophys. Res. 103, 26,783. culated growth rates can be explained in di.erent ways. A Kobel, E., Fluckiger,Q E.O., 1994. J. Geophys. Res. 99, 23,617. plausible explanation is that the mirror type !uctuations ob- Lacombe, C., Pantellini, F.G.E., Hubert, D., Harvey, C.C, Mangeney, A., served along the trajectory of the satellite originate from dif- Belmont, G., Russell, C.T., 1992. Ann. Geophys. 10, 772. ferent sources active at di.erent times, therefore the growth Lucek, E.A., Dunlop, M.W., Balogh, A., Cargill, P., Baumjohann, W., rate ofthe instability cannot be determined by comparing Georgescu, E., Haerendel, G., Fornacon, K.-H., 1999. Geophys. Res. Lett. 26, 2159. all amplitudes. Possible sources may develop in local re- McKean, M.E., Omidi, N., Krauss-Varban, D., 1995. J. Geophys. Res. gions ofthe bow shock, in the magnetosheath, or around 100, 3427. the magnetopause. Another possibility is that the linear ap- Ogilvie, K.W, von Rosenvinge, T., Durney, A.C., 1977. Science 198=4313, proximation applied in the calculations ofGary et al. (1993) 131. is inappropriate to describe the growth rate ofthe observed Omidi, N., O’Farrell, A., Krauss-Varban, D., 1994. Adv. Space Res. 14=7, mirror type !uctuations. 45. Russell, C.T., et al., 1978. IEEE Trans. Geosci. Electron. GE-16, 239. Ifthe growth rate ofthe mirror type !uctuations is really Russell, C.T., Song, P., Lepping, R.P., 1989. Geophys. Res. Lett. 16, so small, the question remains why other instabilities cannot 1485. grow faster and dominate over the mirror mode !uctuations. Russell, C.T., Huddleston, D.E., Strangeway, R.J., Blanco-Cano, X., Kivelson, M.G., Khurana, K.K., Franck, L.A., Paterson, W., Gurnett, D.A., Kurth, W.S., 1999. J. Geophys. Res. 104, 17,471. Acknowledgements Southwood, D.J., Kivelson, M.G., 1993. J. Geophys. Res. 98, 9181. Spreiter, J.R., Stahara, S.S., 1980. J. Geophys. Res. 85, 7715. The authors wish to thank the Fluxgate Magnetometer team TÃatrallyay, M., Erd˝os, G., 2000. ESA SP-449, 311. and the Fast Plasma Experiment team for providing us data. Tsurutani, B.T., Smith, E.J., Anderson, R.R., Ogilvie, K.W., Scudder, J.D., Baker, D.N., Bame, S.J., 1982. J. Geophys. Res. 87, 6060. This work was supported by OTKA grant T020371 ofthe Violante, L., Bavassano Cattaneo, M.B., Moreno, G., Richardson, J.D., Hungarian Science Fund. 1995. J. Geophys. Res. 100, 12,047.