Annales Geophysicae Dipole tilt effects in plasma sheet B y : statistical model and extreme values

With 11 years of Geotail measurements we construct a model of plasma sheet By , depending on IMFBy , coordinatesX, Y , and geodipole tilt angle. At midnight and pre-midnight local timesBy is positively correlated with tilt (positive in summer). Thus in summer By is shifted towards positive values and in winter towards negative values, so that up to several nT could be added to the IMF influence. The dawn side plasma sheet By generally does not exhibit any tilt dependence, but within 15 RE the weaker negative correlation with tilt was revealed. The tilt dependence is just a useful parametrization and several mechanisms actually affecting plasma sheet By were previously suggested. In particular, similar coupling between tilt and IMFBy was earlier found in the ionospheric convection patterns. Besides this average response, extreme By (|By |>5 nT, By>IMF By) were often observed (up to 20–25% of cases during solar maximum and in the pre-midnight sector within 20 RE). They can not be explained by our statistical model and are preliminary interpreted as an “over-reaction” of the magnetosphere in some individual events. LargeBy field radically changes dynamics of the current sheet and has to be taken into account during substorm-related studies.


Introduction
Magnetic field component B y appears in the magnetotail due to a number of reasons (GSM frame of reference is used hereafter).The field line flaring creates B y proportional to local B x , while the neutral sheet twist creates B y proportional to Correspondence to: A. A. Petrukovich (apetruko@iki.rssi.ru)local B z (e.g., Fairfield, 1979;Tsurutani et al., 1984;Kaymaz et al., 1994a;Kullen and Janhunen, 2004).These two effects could be corrected with some further adjustment of the proper reference frame and do not alter principally the geometry, since the cross-tail current remains perpendicular to the magnetic field.Regular magnetotail B y other than that is usually described in terms of penetration of IMF B y (hereafter B i y ) (e.g., Cowley and Hughes, 1983).It should be noted however, that physical mechanisms actually responsible for appearance of the IMF-related B y may be rather complicated.Therefore the term "penetration" is not rigorously justified, though it is frequently used in a rather broad sense as a measure of proportionality between IMF and plasma sheet field.Statistically determined penetration efficiency varies from 0.1 to 0.8 (see a review in Kaymaz et al., 1994b).In the lobes it is small ∼0.1-0.15, while in the plasma sheet it increases towards Earth from ∼0.2 in the distant tail to ∼0.6 in the ISEE-1,2 data set (Sergeev, 1987) and to ∼0.8 at the GOES orbit (Wing et al., 1995).IMF-related B y is parallel to the (horizontal) cross-tail current and thus the tail magnetic structure becomes essentially three-dimensional.
Physics of B y formation in the plasma sheet is generally related with the magnetospheric dynamics.Cowley (1981) suggested that IMF B y induces asymmetries in the tail adding reconnected flux preferentially to different sides of northern and southern lobes (see also Birn, 1990).Asymmetric convection cells in the opposite hemispheres may move the ionospheric ends of the field lines differently, thus tilt the lines and drive additional increase of B y in the plasma sheet (Moses et al., 1985;Voigt and Hilmer, 1987).It was also suggested that Earthward plasma sheet convection may be responsible for enhancement of B y (Hau and Erickson, 1995).Distribution of B y across the tail is therefore interpreted in terms of two components, a smaller one filling the whole tail, and a larger one induced only in the plasma sheet.Magnetospheric field-aligned currents (region 1) may create more localized B y in the plasma sheet (Tsyganenko et al.,  1993).Though the effect was not explicitly described, it is actually included in T96 and T01 magnetospheric models (see Sect. 5.2) (Tsyganenko and Stern, 1996).
With such penetration efficiency and average (IMF) |B i y |∼3 nT, plasma sheet B y is believed to be a minor component, generally smaller than B z .However, recently in a Cluster project study of the growth phase-related thin current sheets at −15<X<−20 R E it was shown that extreme B y (larger than 5 nT and larger than corresponding IMF B y ) appears in almost 30% of cases (Petrukovich et al., 2007).Large plasma sheet B y was reported also previously in some event studies (Sergeev et al., 1993;McComas et al., 1986;Nakamura et al., 2008), but this aspect was not addressed in detail so far.Presence of a large current-aligned magnetic component substantially changes particle dynamics, mapping, etc and therefore affects substorm studies in a variety of ways.Several aspects of this problem are discussed in Sect.5.4.Geotail has by far superior coverage of the magnetotail than any other mission.Therefore we analyze these observations with a specific aim to revisit formation of plasma sheet B y and to determine occurrence of large B y .

The data and the approach
Geotail magnetic field data (Kokubun et al., 1994) with the 12-s sampling, measured during 1995-2005 in the range were used in this investigation.Geotail orbit offers relatively even coverage of this domain (Fig. 1).The flaring-related component of B y is proportional to B x and may be quite large at the flanks and for larger B x .Flaring does not contribute to B y in the neutral sheet, but selection of only |B x |<5 nT with relatively small flaring effect substantially reduces amount of usable data and hence limits statistical significance of results.There is no ultimate solution to this problem.Making averages between the spatial bins, in a hope that effects of B x of the opposite signs will cancel each other, may smear out fine details, related with anomalous B y .
Therefore the following algorithm was designed.All data were sequentially divided in 15-min intervals, containing at least 75% of 12-s samples with |B x |<15 nT.This criterion also selects only the inner (high-β) plasma sheet.In each such interval the linear fit coefficient R defining the flaring effect, B y ∼R•B x was determined.R is supposed to depend on Y , but the actual scatter is rather large (Fig. 2) and it was not possible to use these values directly.Instead the model coefficient R m i (X i , Y )=A i tanh(Y /D i ) depending on the Y coordinate was set up.The free parameters A i and D i were determined during nonlinear fits of R m i to R in the set of 2radii wide bins along the X coordinate (denoted by the subscript i).Only the intervals with the maximum B x variation larger than 5 nT were used.A sample R m i (Y ) is shown in Fig. 2 by a red curve.The maximum R m i (at the flanks) varies from ∼0.2 in the tail-most bin to ∼0.7 in the Earth-most bin.Thus we define the model flaring-related B f y =R m i (X i , Y )•B x and subtract it from the measured B y .All following analysis was performed with such flaring-corrected values.
Finally the data were averaged in each 15-min interval only over the samples with |B x |<15 nT.The final set contains 29 574 of 15-min points.Each such point was augmented with the current and previous (up to 10 h) solar wind and IMF data taken from the 5-min OMNI data set.A number of factors of lesser importance (such as the tail twist or the solar wind aberration) might also affect our analysis.They were not taken into account in the primary model and will be discussed in Sect.5.1.1: 15-20 R E downtail, the thin plasma sheet with the small B z <3 nT and the years after 2000), the percentage became higher than 20%.For |B y |>3 nT then it was almost 40%.Therefore large B y is indeed quite common in some sectors of the magnetotail and in some conditions.
The last subset (the last line of Table 1) has <|B i y |>∼3.7 nT, higher than that for the whole set (3 nT).Therefore higher IMF, in particular during solar maximum, may contribute to more frequent appearance of large |B y |.In  Selection Table 2).The average IMF B z for the large B y was a bit more negative than for the whole data set, but both B i z distributions were similar (not shown here).
Figure 3a shows a histogram of IMF |B i y | for |B y |>5 nT and for the whole data set.The maximum of IMF occurrence for the large plasma sheet field is at the moderately large |B i y | of the order of 5 nT.Extremely large |B i y |>12 nT, required by the nominal penetration efficiency, are only ∼5% of observations.The IMF distribution for the whole data set is distinctly different and has no maximum at |B i y |∼5 nT.|B y |>5 nT were not associated with a preceding substantially larger B i y .On average IMF B i y remained within 5 nT during preceding 10 h (Fig. 3b).Even for the cases with the rather small current IMF |B i y |<2 nT, the preceding IMF was within 3 nT on average (the green curve in Fig. 3b).Therefore in the most of cases large plasma sheet |B y | were not related with proportionally larger IMF B i y , required by nominal penetration and another explanation is necessary.
One possible factor, contributing to such extreme plasma sheet B y , is revealed in Table 3, where two spatial subsets from Table 1 are split to positive and negative B i y and positive and negative geodipole tilt angle τ .Configurations with the same signs of tilt angle and B i y significantly increase probability of appearance of large B y with the corresponding sign.If tilt and B i y signs are opposite, the effect is much weaker, but sometimes the large B y with the "wrong" sign with respect to IMF was detected.About 50% of such peculiar cases were with |B i y |>3 nT, so this effect occurs not only when |B i y | is close to zero and it's sign is not so important (not shown here).In order to further investigate origins of extreme B y , one needs a basis, a comprehensive statistical model of B y , defining an average reaction of the magnetosphere to IMF B y and other parameters.This is done in the next section.

The statistical model of plasma sheet B y
The main driver of B y in the plasma sheet is IMF B y , but their correlation is rather moderate (Fig. 4).The grey solid line in Fig. 4 shows the linear regression model for the whole data set B y =a•B i y with the coefficient (penetration rate) a=0.3544.The next step is to determine the spatial dependence a=a(x, y).Therefore we tested it in several spatial bins with sizes X=5 R E and Y =5 R E .a(x, y) increases towards Earth and towards midnight (Fig. 5, solid lines and diamonds) and is almost symmetrical with respect to the sign of Y GSM (not shown here).Basing on this test, the spatially resolved relation between B y and B i y was defined as: Magnetic field values are in nT, X, Y coordinates -in R E .The numerical coefficients were determined with the nonlinear fit.Their values with the 95% confidence ranges are: a 1 =0.3229±0.013, a 2 =0.5615±0.039, a 3 =0.7779±0.027, a 4 =−0.0547±0.026.Confidence ranges were confirmed also with the bootstrap method (e.g., Tsyganenko and Fairfield, 2004).The error for the last coefficient a 4 is quite large, but its absolute value is rather small in consistency with the standard assumption that B y is zero, when IMF B y is zero.
The model (dotted lines in Fig. 5) fits quite well to our initial estimate.
Besides IMF, B y was found to depend on the geodipole tilt angle τ .The tilt effect was introduced in the model as Here τ is in degrees and magnetic field is in nT.Coefficient of proportionality a t was determined via the linear fit.After a manual inspection of data, four major zones with the substantially differing tilt effect were identified in the considered plasma sheet domain (Figs.6, 7, Table 4).
Positive B y correlation with the tilt (a t >0) appearing in Table 3 was actually limited to the pre-midnight and midnight zones.At −31<X<−20 R E and −5<Y <15 R E (zone "+", Fig. 7) the average addition to B y at the maximum tilt was about 1 nT (Fig. 6a).At X>−20 R E and in the narrower local time sector 0<Y <10 R E (zone "++", Fig. 7) the effect was four times stronger (Fig. 6a).
In the post-midnight sector the tilt dependence was almost absent (zone "0", Fig. 6).Only in its near-Earth part X<−15 R E and −15<Y <−5 R E (zone "−", Fig. 7) the small reverse effect of tilt appeared with average addition up to −1.5 nT (Fig. 6b).The mixed colors in Fig. 7 indicate areas, where the tilt effect was intermediate.
The tilt effect is uneven and it is difficult to describe it in a simple way.We construct a unified model only for 0<Y <10 R E , where the tilt dependence X is more uniform.
This model has larger error ranges due to the smaller data set.The only statistically significant difference of numerical coefficients in Eqs. ( 1) and ( 3) is between a 2 and a t2 .
It results in a small decrease of IMF penetration, e.g., at X=−10 R E from 0.69 to 0.6.The statistical quality of these models is following.The correlation coefficient of B y and IMF B y is 0.51, for B y and B m y it is 0.54.In the zone with the maximum tilt dependence ("++", Fig. 8) the correlation of B y and B mt y is substantially higher -0.71.The correlation of B y and B mt1 y is 0.68.

Applicability of the statistical model
The large volume of Geotail observations as well as the algorithm of the flaring removal helped to create an improved model of plasma sheet B y with four driving parameters (IMF B y , X, Y , τ ).Inclusion of (rather flat) spatial dependence of the IMF influence results in a very modest increase of correlation coefficient, in comparison with a simplest expression B y ∼a•B i y .The major improvement is due to the geodipole tilt dependence.The model is essentially limited to the inner (high β) plasma sheet, to −31<X<−8 R E , −15<Y <15 R E and describes the quasi-stationary configuration, since the 15-min averaging removes most of transient effects, such as flux ropes.
Our model is not in a contradiction with the previous estimates of penetration efficiency.However, Kaymaz et al. (1994b) suggested that according to IMP-8 data the penetration was somewhat higher at the flanks (|Y |>12 R E ) than in the center of the magnetotail.This region is outside our domain and it is indeed reasonable to assume that IMF influence may increase at more distant flanks adjacent to LLBL.
All main features of the model were confirmed with the smaller data set |B x |<5 nT (not shown here) and therefore our flaring removal procedure and criterion of the inner plasma sheet (|B x |<15 nT) are reliable.Extending the model with the same technique to the outer plasma sheet and the lobe is not straightforward, since the flaring component is then larger, while B y in question is smaller.
Our first model (Eq. 1) is the most general one and describes only the IMF influence (penetration).Though the tilt effect was not accounted for explicitly, it was effectively averaged out.Indeed the model does not depend on the sign of Y and contains data from all seasons, thus tilt effects of opposite signs are canceling each other.There exists a number of other minor factors affecting B y .The first one to discuss is deviation of the tail axis from GSM Y due to solar wind aberration (∼4 • ) and transient changes of solar wind direction (so-called GSM-corrected and GSW frames of reference) (e.g., Tsyganenko and Fairfield, 2004).The aberration creates the 2-R E shift in Y coordinate at the outermost part of our tail domain.Since our model is only rather weakly sensitive to spatial coordinates, this factor is of the lesser importance.
In a tilted current sheet magnetic component along the electric current, which is important for particle dynamics, is different from B y GSM.The tilt at the tail flanks due to the warp can reach 40 • (Tsyganenko and Fairfield, 2004;Petrukovich et al., 2005).It is not quite clear however, whether the warp tilts field lines in the neutral sheet (thus affecting B y ) or it only shifts the neutral sheet surface.
The tail twist may result in a rotation of the whole tail.The twist is driven by IMF B y and does tilt field lines.Thus plasma sheet B y contains also a small contribution, equal to tan φ•B z , where φ is the twist angle.In the Tsyganenko and Fairfield (2004) model the twist angle is proportional to X, B i y , and in a lesser extent to B i z .In our data set (taking real values of B z , X, B i z ) the twist was generally within 10 • and this contribution to B y was equal on average to ∼−0.1•B i y .Thus the actual IMF penetration might be somewhat underestimated in our model.
Besides these factors, a more refined model should also take into account possible FAC contribution, including tilting of the FAC sheets, as well as the north-south asymmetry of flaring.Implementation of all such smaller factors in simple model similar to our's is unpractical.On the other hand they are more or less automatically accounted for in more general models like T96 or global MHD.However, the simple model might be more convenient to study specific effects.

Tilt-related asymmetry of B y
The geodipole tilt dependence of B y could be revealed only with the sufficiently extensive data set, covering all locations during variety of seasons.The major tilt effect is concentrated in the premidnight and midnight plasma sheet.Positive B y -s have preference during (northern) summer season (positive tilt) while negative -during winter.Closer to Earth the effect is stronger and is larger than that of the IMF driving.The tilt effect generally disappears at the post-midnight sector but has the reverse sign in the relatively narrow near-Earth post-midnight zone.Here we present a preliminary qualitative discussion of a problem.Some tilt effects on B y are basically included in the T96 and T01 magnetospheric models, though it was not described explicitly.The direct point-by-point comparison of our statistics with T96 is impossible since the difference in magnetic field could be not only due to the difference in the sources (currents), but also due to the difference in location relative to these sources.Thus we just compute the neutral sheet B y (Y ) in T96 at B x =0, X=−15 R E , for IMF B i y =−5 nT, and for tilts −30 • , 0 • , 30 • (Fig. 8).Beyond an IMF-related component (most clearly seen at the zero tilt), T96 B y has a significant tilt-related component, positive for positive tilts and negative for negative tilts at Y >0.The amplitudes and the tailward extent of the tilt-induced B y are consistent with the observations.At Y <0 in T96 the effect has the opposite sign, but it's magnitude is practically the same as at the dusk-side.
The source of such B y in T96 are field-aligned currents (region 1), flowing northward and southward from the plasma sheet.At equinox a perfectly symmetric pair of FAC does not produce any B y between them, in the plasma sheet.At non-zero tilts unequal currents generate the non-uniform B y across the magnetotail, including some (slowly changing) net B y in the plasma sheet.The only tilt-related modification of FAC R1 in T96 is distortion of coordinates, smoothly converting the frame in which FAC is defined (Z aligned with the dipole axis) to GSM (Tsyganenko and Stern, 1996).Then at the same X GSM the northward and southward FAC sheets have different shapes and amplitudes depending on the season.Note that initially amplitude of FAC in T96 does not depend on tilt, therefore any conductance related effects are excluded.In the later T01 model such FAC-related effects in B y vanish well within 15 R E (not shown here).The similar B y profile across the magnetotail (as in Fig. 8) was also obtained from the run of BATS-R-US global MHD model, accessed via CCMC at NASA GSFC.The effect remains even if the sunlit-related conductance is switched off (not shown here).
The strong dawn-dusk asymmetry in the real data in comparison with the models could be related with the general asymmetry of the real magnetotail (e.g., ion and electron gradient drifts, thicker plasma sheet at the dawn, etc.), which at the modern level is not accounted for.Specifically it might be, e.g., due to an earlier FAC closure or to a less pronounced FAC north-south asymmetry at the dawn side.An alternative interpretation seems also reasonable: antisymmetric in Y B y pattern within 15 R E could be generated in accordance with the models, while the excess of the B y -tilt effect at positive Y and further downtail might be due to some another effect.
B y in the plasma sheet is a self-consistent reaction of the whole magnetosphere-ionosphere system and the tilt effect is just a convenient parametrization.IMF and solar wind flow impose an outer boundary condition on the magnetosphere, but local B y is formed by a number of mechanisms, such as magnetotail convection and large-scale FAC, which in turn should match global convection and current patterns.In more detail comparison between B y in experiment and models as well as interpretation of the effect will be accomplished in a next publication.In particular the IMF B y /season asymmetry was revealed earlier also in the polar convection.The mirror symmetry of two hemispheres with respect to IMF B y is a well known phenomenon (the Svalgaard-Mansurov effect) (e.g., Nishida, 1978).However, Heppner and Maynard (1987) showed that convection patterns were clearer for the pair of the north polar cap and the positive IMF B y as well as for the pair of the south polar cap and negative IMF B y .Papitashvili et al. (2002) reported that southern summer IMF B y <0 and northern summer IMF B y >0 FACs were slightly larger.Finally Ruohoniemi andGreenwald (1995, 2005) explicitly formulated a "reinforcement of IMF influence" in forming convection cells at the combinations of B y +/summer and B y -/winter.This seasonal effect was interpreted as a result of interaction of the initially mirror-symmetric IMF B y pattern with the day-night conductivity gradient depending on the degree of solar illumination (e.g., Tanaka, 2001) or with the single lobe cell appearing only in summer (Crooker and Rich, 1993).For an extended discussion and review of relevant studies an interested reader is referred to the papers of Tanaka (2001) and Ruohoniemi and Greenwald (2005).In the frame of our findings, such a preference might be produced by the combination of the tilt and IMF B y effects on the FAC system.

Formation of extreme B y
Geotail observations confirmed occurrence of extremely large B y , reported previously by the Cluster project.The share of |B y |>5 nT can reach 20-25%, while the share of |B y |>3 nT -40%.The most of such cases are not related with anomalously large IMF B y .Our statistical model was not capable to reproduce the majority of them as well as the majority of Cluster cases (Petrukovich et al., 2007, their  Table 1).Indeed, the zone −31<X<−8 R E , 0<Y <10 R E contained 671 points of |B y |>5 nT, but only 125 points were reproduced by the model with the tolerance ±2 nT.In the most of cases large B y was substantially underestimated.Indeed in the most favorable case the IMF B i y ∼5 nT plus the effect of maximum tilt 30 • create B y ∼7 nT at X=−15 R E .With a smaller IMF and a smaller tilt it is impossible to obtain comparable values.
Since extreme B y happen with the proper combination of tilt and IMF B y signs, their formation may be related to the same mechanism as the statistical tilt effect.It can be due to a stronger FAC imbalance or due to amplification of existing B y by plasma sheet convection (Hau and Erickson, 1995).Therefore the magnetospheric reaction is in each case individual and may be different in different parts of the magnetotail.In some circumstances an apparent over-reaction to IMF may occur, in other (more rare) cases IMF direction may be overridden, if the opposite sign is favored.Further advance of this problem might be achieved with better understanding of the regular tilt effect.

Role of B y in the magnetotail
B y creates an important asymmetry in the magnetotail and when B y is larger than B z , plasma dynamics may change substantially.We list and discuss just a few aspects of this problem: 1. Large B y tilts field lines and the difference between the south and north foot-points may reach 2 h MLT (e.g., Sergeev, 1987).Indeed observations of nonconjugate auroras were often related with large IMF B y (Stenbaek-Nielsen and Otto, 1997;Østgaard et al., 2004).
2. Non-zero B y decreases the κ parameter, a ratio of larmor radius to minimum curvature radius, describing the degree of non-adiabaticity of particles (Büchner and Zelenyi, 1989).In particular it is often used in precipitation estimates (Newell et al., 1996).For |B y | B z the magnetic curvature radius is multiplied by (B y /B z ) 2 (e.g., Shen et al., 2003).Thus for B y =8 nT and B z =2 nT κ may change by a factor of 16.
3. Charged particle dynamics becomes asymmetric in the north-south direction, when B y is larger than B z .In particular electrons and protons tend to precipitate in opposite hemispheres depending on the sign of B y .The effect in the adiabatic approximation was described by, e.g., Sivukhin (1965), while Delcourt et al. (2000); Zhu and Parks (1993);Holland et al. (1996) performed the numerical modeling of non-adiabatic particle trajectories in a sheared current sheet.
4. New current sheet instabilities may arise in case of large B y (Hau and Voigt, 1992).On the other hand, instabilities usually suggested for substorm onset do not include B y , or assume that it is small.
In a particular example, strongly tilted current sheets, discovered by Cluster (Sergeev et al., 2004;Petrukovich et al., 2006;Sharma et al., 2008) were actually observed only when B y was small.All three cases, identified by Petrukovich et al. (2006) as a slippage wave, happened with B y less than 2 nT.The twelve intervals with the higher frequency (10 mHz) waves, identified by Zelenyi et al. (2009) as a kink drift mode, occurred with B y less than 3 nT.A more general check was done with the full Cluster data set of fast current sheet crossings (∼360 cases during 2001 and 2004) (Runov et al., 2006).Figure 9 contains B y and B z values, taken in the middle of each crossing.A substantial amount of events had large B y (larger than B z or larger than 5 nT).However most of them belong to more horizontal sheets (with normals tilted by less than 45 • from the Z GSM, shown by the green crosses).More vertical sheets (more than 45 • , shown by the blue crosses) usually have B y either within 2 nT, or smaller than B z .

Conclusions
The unique Geotail data set helped to built a statistical model of plasma sheet B y .Beyond it, observations of extremely large B y , which is not related with the appropriately large IMF, were not rare during solar maximum and in the premidnight near tail, where substorms are believed to originate.The B y relation with the geodipole tilt is generally consistent with FAC Region 1 phenomenological structure and some previous ionospheric convection studies.However it's specific driving mechanism remains unclear and further studies are necessary.Comparison with the global MHD simulations will be helpful.The successful model should explain also the asymmetry in Y and origin of extreme B y .On the other hand the picture of plasma sheet B y may give a new insight in understanding the global magnetospheric structure.Ground-based studies rarely encompass two hemispheres, while plasma sheet B y includes integrated effect of both polar caps.

Fig. 1 .
Fig. 1.Coordinates of Geotail observations during 1995-2005 used in the analysis.Each point corresponds to a 15 min interval.

Fig. 2 .
Fig. 2. Grey dots -the linear fit coefficients R of B y w.r.t. to B x in 15-min long intervals at −20 R E <X<−18 R E .Blue curve -the box average.Red curve -the approximation with the hyperbolic tangent.See text for details.

Fig. 3 .
Fig. 3. (a) Histogram of IMF |B y | for the whole data set and for |B y |>5 nT.(b) IMF B y for 10 h preceding to observation of |B y |>5 nT for all values of IMF B y (blue) and for only small IMF B y (green).

Fig. 4 .
Fig. 4. IMF B y -plasma sheet B y scatter-plot.The grey line corresponds to a penetration efficiency of 0.3544.

Fig. 5 .
Fig. 5. Linear fit coefficients of B y w.r.t.IMF B i y in several spatial bins (diamonds and solid lines).Dashed lines -the nonlinear spatial model.

Table 4 .
Regression coefficients a t between B y −B m y and tilt and their 95% confidence ranges.The Zone names correspond to Fig.7.Zone Selection a t , nT/deg error "

Fig. 6 .
Fig. 6.Tilt angle dependence of plasma sheet B y with subtracted IMF B y influence.Magnetic field values are averaged in 5-deg bins of τ .(a) Pre-midnight and midnight sectors.(b) Post-midnight sector.

Fig. 7 .
Fig. 7.The sketch of the tilt-angle effect zones.See text for details.

Fig. 7 .
Fig. 7.The sketch of the tilt-angle effect zones.See text for details.

Table 1 .
Occurrence of large By for several data subsets.# -number of points with |B y |>5 nT; percentage of points |B y |>5 nT and |B y |>3 nT; <|B i y |> -average IMF in a subset.

3 Occurrence of extreme B y
). Outside this Y sector the occurrence drops by a factor of two.With the additional selections, bringing the Geotail data closer to the Cluster cases (the last line in Table

Table 2 .
Relation between large B y and IMF B i y , B i z .

Table 2 .
Table 2 compares <|B y |>, <|B i y |>, and <|B i z |> for several data subsets.Nominally plasma sheet B y is expected to be of the order of 40-50% of IMF.Indeed for the full data set IMF B y is larger than plasma sheet B y (the first line of Table 2).However, for the selection |B y |>5 nT average IMF <|B i y |> was 5.3 nT, that is smaller than the plasma sheet component (the second line of Table2).The result for |B y |>3 nT was similar.On the other hand the selection of large IMF |B i y |>5 nT had relatively small <|B y |> again in agreement with the nominal penetration (the last line of

Table 3 .
Occurrence of large B y (in %) with respect to signs of tilt τ and IMF B i y .