Dual periodicities in planetary-period magnetic field oscillations in Saturn ’ s tail

[1] We examine magnetic field data from 10 apoapsis passes of the Cassini spacecraft during 2006 when the spacecraft explored the midnight and dawn sectors of Saturn’s magnetotail to down-tail distances of 65 RS (Saturn radius, RS, is 60,268 km). Oscillations in the radial component of the field near the 11 hour planetary period associated with north-south motions of the current sheet are ubiquitous in these data. Here, we examine and model the phase of these oscillations throughout the interval, taking account of both local time and radial propagation effects, and show that the oscillations exhibit dual periodicities. Those observed at distances exceeding 3 RS north of the modeled average center of the current sheet are found to oscillate near the modulation period of the northern Saturn kilometric radiation (SKR) emissions, while those observed south of this location oscillate near the modulation period of the southern SKR emissions. The phasing in both cases is consistent with the sense of the associated rotating quasi-uniform perturbation fields within the quasi-dipolar “core” region. We determine the structure of the current sheet as a function of the modeled phases, the results implying that the form of the modulation varies significantly over the beat cycle of the two oscillations. When the two field oscillations are in phase, the current sheet oscillates north-south with a peak-to-peak amplitude of 3 RS. When they are in antiphase, however, the thickness of the current sheet is also strongly modulated during the oscillation by factors of 2.


Introduction
[2] Despite the close axial symmetry of Saturn's planetary magnetic field, oscillations in field and plasma parameters near the planetary rotation period have been found to be ubiquitous throughout Saturn's magnetosphere.These modulations were first discovered in Voyager observations of Saturn kilometric radiation (SKR) emissions from Saturn's auroral regions, having a period of $10.7 hours [Kaiser et al., 1980[Kaiser et al., , 1981;;Desch and Kaiser, 1981;Warwick et al., 1981;Gurnett et al., 1981;Kaiser and Desch, 1982;Lecacheux and Genova, 1983;Galopeau et al., 1995].Subsequent study using Cassini data has shown that the emissions from the northern and southern hemispheres exhibit differing periods, $10.6 hours in the north and $10.8 hours in the south during the preequinox southern summer conditions encountered during the initial 2004-2008 phase of the mission [Gurnett et al., 2009a[Gurnett et al., , 2011;;Lamy, 2011].It was also first shown using Ulysses radio data and subsequently with Cassini data that these periods vary slowly with time [Galopeau and Lecacheux, 2000;Gurnett et al., 2005;Kurth et al., 2007Kurth et al., , 2008]], notably converging over the Saturn equinox interval starting in early 2009 [Gurnett et al., 2010b[Gurnett et al., , 2011;;Lamy, 2011].
[4] Given the dual periodicities observed in the magnetic field oscillations, particularly in the two high-latitude polar regions, it is of interest to examine their consequences for oscillations in Saturn's magnetic tail, the lobes of which map into the two polar regions at the planet.In this article, we focus on periodicities in the tail field observed on 10 apoapsis passes of the Cassini spacecraft, when it explored the midnight and dawn sectors of Saturn's magnetic tail to down-tail distances of $65 R S during the southern summer interval in 2006 (Saturn's radius, R S , is 60,268 km).Following an initial discussion by Jackman et al. [2009] of plasma sheet oscillations during this interval, Arridge et al. [2011] have recently studied these data on a pass-by-pass basis, showing that the oscillations observed in the southern and central tail are indeed closely governed by the southern SKR modulation phase, in a consistent relationship with the field oscillations observed in the inner quasi-dipolar magnetosphere.However, this relationship was found to break down on orbits that passed significantly into the northern tail.Here, we extend their study to consider the field oscillations in both the northern and southern tails, and the role of dual periodicities.

Physical Picture
[5] The physical background to the study is introduced with the aid of Figure 1, where we summarize the results of the magnetic field investigations cited previously by Andrews et al. [2008Andrews et al. [ , 2010aAndrews et al. [ , 2010b] ] and Provan et al. [2009aProvan et al. [ , 2011]].Figures 1a and 1b illustrate the overall form of the rotating perturbation field for the southern-and northern-period systems, respectively.Here, the black dashed lines indicate the near-axisymmetric "background" magnetospheric field with closed field lines at lower latitudes (shaded) and open field lines in the polar regions (clear), while the red lines in Figure 1a and the blue lines in Figure 1b indicate the perturbation fields in the meridian planes shown.The perturbation field lines out of these planes can then be obtained to a first approximation by displacing these field loops directly into and out of the plane of Figure 1.The two patterns of perturbation fields then rotate independently around the planet's spin and magnetic axis with a period close to the southern SKR period in Figure 1a, and close to the northern SKR period in Figure 1b, as indicated.
[6] In the polar regions, the perturbation fields are similar in form to a planet-centered rotating transverse dipole in each hemisphere [Provan et al., 2009a], the instantaneous effective directions of which are shown by the planet-centered colored arrows in each of Figures 1a-1d.The nearequatorial field perturbations then take the form of rotating quasi-uniform fields lying approximately parallel to the equatorial plane in the directions of these effective dipoles, combined with north-south fields such that the perturbation field lines are arched with apices pointing to the north for the southern-period perturbations, and pointing to the south for the northern-period perturbations [Andrews et al., 2008[Andrews et al., , 2010b]].The southern-and northern-period perturbations are superposed in the equatorial magnetosphere, giving rise to the phase "jitter" phenomenon mentioned in section 1, with the southern oscillations being dominant by factors of $3 during the preequinox interval [Provan et al., 2011].We note that the directions of the perturbation fields shown in these diagrams in Figure 1 correspond approximately to SKR maxima when the Sun is located on the right-hand side in both figures.Thus, at southern SKR maxima, the quasiuniform equatorial field and effective transverse dipole shown in Figure 1a point away from the Sun, that is, downtail, while at northern SKR maxima, the quasi-uniform field and effective transverse dipole shown in Figure 1b point toward the Sun.
[7] The similarly colored lines in Figures 1c and 1d then illustrate how these perturbation fields affect the structure of the southern-and northern-magnetospheric fields, respectively.Two principal effects are shown.The first is the displacement of the polar region of open field lines shown on the right in Figures 1c and 1d, as previously discussed by Provan et al. [2009b].The implication of these directions is that the auroral oval is displaced sunward at SKR maxima in corresponding hemispheres and tailward at SKR minima, consistent with the results in the study by Nichols et al. [2008Nichols et al. [ , 2010b]].Second, the effective magnetic equator (dotted lines) where the radial field changes sign also becomes tilted with respect to the planet's spin equator, as previously discussed in the study by Andrews et al. [2010b], in opposite senses in Figures 1c and 1d.Thus, for example, as shown in Figure 1c, at southern SKR maxima (with the Sun to the right), the magnetic equator is displaced southward on the nightside and northward on the dayside, consistent with the results in the study by Carbary et al. [2008b].We note that both these field effects are consistent with those expected from combining the effective transverse dipoles shown in Figure 1 with the planet's northward-pointing internal dipole.
[8] With regard to their physical origin, the form of these perturbation fields clearly suggests the presence of two distinct magnetosphere-ionosphere coupling current systems whose transverse magnetic dipole moments are indicated by the arrows in Figure 1, which rotate with different periods in the two hemispheres [Andrews et al., 2010a].Ampère's law applied to the perturbation field loops shown in Figure 1 implies current flow out of the plane of the diagram in both cases at the instants depicted, that is, downward into the ionosphere into the plane of the diagram and upward out of the ionosphere out of the plane of the diagram.The SKR emission in each hemisphere maximizes when the corresponding upward current is centered near dawn [Andrews et al., 2011].These field-aligned currents map to the equatorial plane and close there at a radial distance of $15 R S [Andrews et al., 2010b], thus defining the quasi-dipolar "core" region of the magnetosphere enclosing the region of quasi-uniform perturbation fields.The perturbation fields then become quasi-dipolar in form at larger distances, in the sense indicated by the transverse magnetic dipoles in the corresponding hemispheres.
[9] The implications of this picture for Saturn's tail field are evident.If we first consider Figure 1c for the southern system, likely to be dominant during the preequinox southern summer conditions considered here, the current sheet and associated plasma sheet will be displaced southward of their mean position when the quasi-uniform field and effective southern transverse dipole point tailward, corresponding to southern SKR maxima, and northward of their mean position when the quasi-uniform field and effective southern transverse dipole point sunward, corresponding to southern SKR minima.These expectations are consistent with the findings of Arridge et al. [2011].We note in passing that the "mean position" of the current sheet center referred to here is displaced consistently northward of the planet's equatorial plane during these seasonal conditions because of the action of the solar wind flow [Arridge et al., 2008a], an aspect not represented in our diagrams.Similar considerations then apply to the weaker northern system shown in Figure 1d, except that in this case, southward displacements of the tail current sheet correspond to northern SKR minima, and northward displacements to northern SKR maxima.
[10] Because the northern and southern systems rotate with differing periods, however, the corresponding perturbation fields will have relative orientations that vary over the beat cycle of the two oscillations, giving rise to the phase "jitter" in the dominant southern-period oscillation within the preequinox "core" region.For oscillations of $10.58 hour periods for the northern system and $10.79 hours for the southern, as applies to the 2006 data considered here, the beat period is $22.6 days [e.g., Provan et al., 2011].The previous discussion of Figure 1 then suggests that when the magnetic perturbations of the two systems are in phase such that the rotating quasi-uniform fields and transverse dipoles point (temporarily) in the same directions as each other as they rotate, the two systems will act to produce north-south oscillations of the current and plasma sheet that are in phase with each other.The southward displacements should occur at southern SKR maxima and northern SKR minima when the perturbation fields (and transverse dipoles) both point radially outward in the tail, acting to enhance the field in the northern tail while reducing it in the southern tail.Similarly, northward displacements should occur at southern SKR minima and northern SKR maxima when the quasi-uniform fields (and transverse dipoles) both point radially inward, enhancing the field in the southern tail while reducing it in the northern tail.Half a beat period later, however, the magnetic perturbations of the two systems will be in antiphase, producing oppositely directed field perturbations in the northern and southern tails that successively enhance and reduce the tail field simultaneously in both hemispheres, leading to modulation of the current and plasma sheet.According to this picture, field enhancement in both hemispheres will correspond to SKR minima in both hemispheres, while field reduction in both hemispheres will correspond to SKR maxima.At the same time, if the southern system is dominant, north-south oscillations of the current sheet with the phasing associated with the southern system will continue.The following analyses then constitute a search in the Cassini data for evidence of such behavior.

Spacecraft Trajectory
[11] As indicated in section 1, here we study the tail field data obtained on a sequence of 10 Cassini apoapsis passes during 2006.Specifically, the interval considered spans from the periapsis of revolution (rev) 20 on day 17 of 2006 to the periapsis of rev 30 on day 284, where we note that Cassini orbital revs are defined from apoapsis to apoapsis of the spacecraft orbit.Figure 2 presents the color-coded trajectories of these revs in the Kronian solar magnetic (KSMAG) coordinate system in which Z is aligned with Saturn's rotation axis, the X-Z plane contains the Sun, and Y completes the right-handed system.The trajectories are projected onto (1) the equatorial X-Y plane, (2) the noon-midnight X-Z plane, and (3) the dawn-dusk Y-Z plane.The black solid and dashed lines also show cuts in these planes through the magnetopause and bow shock according to the empirical models of Kanani et al. [2010] and Masters et al. [2008], respectively, both shown for a typical solar wind dynamic pressure of 0.03 nPa (see Figure 2 caption for further details).It can be seen that the earlier revs are closely equatorial and traverse the dawn magnetotail to radial distances of $65 R S .These data have been screened to flag and eliminate magnetosheath intervals from the analysis.Later revs are centered nearer to midnight reaching distances of $50 R S , and starting with the outbound pass of rev 26 are also inclined to the equatorial plane with apoapses in Saturn's northern hemisphere.

Magnetic Field Oscillations on an Equatorial Orbit
[12] We first study the oscillations observed on a nearequatorial apoapsis pass, specifically from the periapsis of rev 25 to the periapsis of rev 26.In Figure 3, we show plotted versus time (day of year (DOY) of 2006) from top to bottom: Figure 3a is an electron spectrogram color-coded according to the scale on the right covering the energy range 0.6 eV to 28 keV obtained by the ELS sensor of the Cassini Plasma Spectrometer [Young et al., 2004], shown to aid region identification, Figure 3b shows the radial component B r of the residual magnetic field obtained by the fluxgate magnetometer [Dougherty et al., 2004], from which the "Cassini SOI" internal field model has been subtracted [Dougherty et al., 2005], Figure 3c shows the residual B r field band pass filtered between periods of 5 and 20 hours using a standard Lanczos filter to isolate oscillations centered in the planetary-period band [see Andrews et al., 2008 andProvan et al., 2009a for further details], Figures 3d-3g show the field oscillation phase data to be described in detail in the following, Figure 3h shows the radial distance of the spacecraft, Figure 3i shows the distance z of the spacecraft northward of the planet's equatorial plane (black line) together with the northward displacement of the current sheet center (red line) according to the model in the study by Arridge et al. [2008a] using a hinging distance of 29 R S (the overall optimal value found in that study), and Figures 3j-3l show the spacecraft trajectory projected onto the equatorial, noon-midnight, and dawn-dusk KSMAG planes (red lines) with the model magnetopause and bow shock superposed (black lines) as in Figure 2. We note, in particular, from Figure 3i, that because of the significant displacement of the modeled current sheet center northward of the equatorial plane during the southern summer conditions prevailing, the spacecraft is expected to be positioned significantly southward of the current sheet center for most of this interval.
[13] Figures 3a and 3b immediately show that oscillations in the plasma electron flux and magnetic field are present at approximately the planetary period throughout the interval in accordance with the findings of Arridge et al. [2011], with approximately two cycles every (Earth) day.Overall, the value of the residual field is predominantly negative, indicating a location principally within the southern tail as anticipated, but with periodic excursions to smaller and sometimes positive values indicative of passages into and across the center of the current sheet that separates the two tail lobes.The electron fluxes at energies between $10 eV and $1 keV are consistently elevated at such times, indicative of the spacecraft passing periodically into the central tail plasma sheet (intense electron fluxes below $10 eV are predominantly spacecraft photoelectrons).
[14] In common with previous related studies [Andrews et al., 2008[Andrews et al., , 2010a[Andrews et al., , 2010b[Andrews et al., , 2011;;Provan et al., 2009aProvan et al., , 2011]], to quantify these oscillations we determine their phase relative to a perturbation that rotates around the planet at the SKR period, such that the oscillation in the radial component is expressed as where j is the azimuth measured from noon positive toward dusk (equivalent to LT), and F SKRn,s (t) is the phase of the northern (n)-or southern (s)-hemisphere SKR modulations as determined in the study by Lamy [2011].This phase is such that the period of the modulations is given by where the phase is expressed in degrees, and the absolute value (to modulo 360°) is such that maxima in the SKR emission from either hemisphere occur at times given by Figure 2. Color-coded plots of the Cassini trajectory from the periapsis of rev 20 to the periapsis of rev 30 shown in KSMAG coordinates in which Z is aligned with Saturn's rotation axis, the X-Z plane contains the Sun, and Y completes the right-handed system.The black solid and black dashed lines show cuts through the magnetopause and bow shock according to the models of Kanani et al. [2010] and Masters et al. [2008], respectively, for a typical solar wind dynamic pressure of 0.03 nPa.These boundaries, defined in Kronian solar magnetospheric coordinates, correspond specifically to the time of apoapsis at the beginning of rev 25 (the central rev employed in the study), although the variation over the interval of these data is small.
for successive integer values of N. At relevant (north or south) SKR maxima, therefore, the oscillatory radial field component has its maximum value at azimuth [15] In this study, we have determined the phases y rn,s relative to both northern and southern SKR modulations in successive 22 hour data intervals, marked by the vertical black dashed lines in Figure 3, such that each contains approximately two oscillation cycles.The phases have been determined by cross-correlating equation (1a) with the filtered residual data to determine which value gives the Figure 3 highest cross-correlation coefficient, to 1°resolution in phase.The phases from intervals with peak cross-correlation coefficients of less than 0.75, or containing data gaps whose length is comparable with the oscillation period, or with significant field variations that are clearly unrelated to the regular oscillations, are excluded from further analysis.The previous results in the studies by Andrews et al. [2010aAndrews et al. [ , 2011] ] and Provan et al. [2011] show that, at small radial distances within the quasi-dipolar "core," the radial field component associated with the southern-period system has a value of y rs ≈ 180°during this interval, such that from equation (1d) at times of southern SKR maxima the rotating quasi-uniform perturbation field points antisunward as discussed in section 2. Similarly, the radial field component within the "core" associated with the northern-period system has a value of y rn ≈ 0°, such that at times of northern SKR maxima, the related rotating quasi-uniform perturbation field points sunward, as also discussed in section 2. From these values at small radial distances, we may then expect the phases to increase with radial distance because of the outward propagation of the perturbations, with a phase gradient of $2°-3°R S

À1
, corresponding to an outward propagation speed of $200-300 km s À1 according to the studies conducted by Andrews et al. [2010b], Clarke et al. [2010b], and Arridge et al. [2011].The phase of the radial field oscillations associated with the southern system is thus expected to have a value of y rs ≃ 180°in the inner system, increasing systematically with radial distance by $100°-150°at typical apoapsis distances of $50 R S , while the oscillations associated with the northern system should have a value of y rn ≃ 0°in the inner system, similarly increasing systematically with radial distances to $100°-150°near apoapsis.However, the y rn values for an oscillation that is actually at the southern period will show no consistent value in the inner region, and in addition to the radial dependency, will also increase systematically with time because of the difference in period, at a rate of $15°-20°d À1 for northern and southern periods of $10.6 and $10.8 hours, respectively.Similarly, the y rs values for an oscillation that is actually at the northern period will again show no consistent value in the inner region, and will also decrease systematically with time at similar rates because of the differences in period.
[16] These expected behaviors form the basis for understanding the phase results shown in Figure 3, where in Figure 3d we show the values of y rs for each 22 hour interval (red circles), while in Figure 3f we similarly show the values of y rn (blue circles) for the same intervals.To aid visualization of the continuity of the data, the y rs values are shown over the ranges of 90°-450°, while the y rn values are shown over the ranges of 0°-360°, any 360°range being equivalent.To aid interpretation, we have also added the red and blue dashed lines, which show overall empirical phase models for the southern and northern oscillations, respectively, that are derived in section 4.These have values of y rs ≈ 180°and y rn ≈ 0°in the inner region as expected and take account of LT and radial propagation effects as discussed in the following.Figures 3e and 3g then show the differences between the phase values in Figures 3d and 3f and the overall phase models, respectively.In Figures 3d  and 3e, using the southern SKR modulation phase, it can be seen that the phase values approach a value of y rs ≈ 180°i n the inner region at both the beginning and end of the interval and vary systematically with radial distance over the orbit to values near $360°at apoapsis, with little deviation from the overall model.However, the chief feature of the phase data in Figures 3f and 3g using the northern SKR modulation phase is the slow systematic increase in phase with time superposed on the radial variation, both inbound and outbound, at a rate comparable with that estimated previously.Thus, it is evident from these results that the field oscillations observed on this pass are consistent with oscillations at the southern period throughout, in conformity with the results of the study by Arridge et al. [2011].

Magnetic Field Oscillations on an Inclined Orbit
[17] In Figure 4, we show data for the next consecutive apoapsis pass, from the periapsis of rev 26 to the periapsis of rev 27, in the same format as shown in Figure 3.The trajectory is similar to that of the previous pass as can be seen in Figures 4j-4l, but following an encounter with Titan, just prior to the periapsis of rev 26 (seen in Figure 3), the orbit is now significantly inclined to the equatorial plane with apoapsis in the northern hemisphere.If we then examine Figure 4i, it can be seen that the z position of the spacecraft (black line) now lies well northward of the modeled position of the current sheet center (red line) on the outbound pass, while approaching and crossing southward of the modeled current sheet center on the inbound pass.Correspondingly, the residual B r field in Figure 4b is predominantly positive during the outbound pass, indicating a location principally in the northern tail, but with periodic excursions to smaller and sometimes negative values indicative of passages into and across the center of the current sheet.Electron fluxes in Figure 4a are again consistently elevated at such times, indicative of the spacecraft passing periodically into the plasma sheet.On the inbound pass, however, positive and negative fields initially occur approximately equally, while negative fields predominate toward the end of the interval showing that the spacecraft has then returned to the southern tail.
[18] If we now examine the phase data shown in Figure 4d, it can be seen that, on the outbound pass, when the spacecraft was well northward of the modeled current sheet center, the phase values y rs using the southern SKR modulation phase deviate significantly from the overall model, consistent with the findings of Arridge et al. [2011] for this pass, falling continuously with time in Figure 4e.In Figure 4f, however, the phase values y rn using the northern SKR modulation phase follow the overall model very well, starting near a value of y rn ≈ 0°in the inner region and increasing systematically with radial distance, with little deviation as shown in Figure 4g.Thus, these results indicate that the field oscillations observed during this interval are at the northern period.However, this picture changes rather abruptly around day 213 as the spacecraft moves closer to the modeled current sheet center from the north, when y rs begins again to agree with the overall southern phase model as shown in Figure 3 with little deviation in Figure 4e, while y rn is not in agreement with the overall northern-phase model, instead rising consistently in value with time.These later data are thus consistent with oscillations at the southern period, the change occurring when the spacecraft was located a few R S northward of the modeled current sheet center and still observing predominantly positive radial fields.

Overall Phase Models
[19] We now derive overall phase models for the northernand southern-period oscillations, whose parameters are determined from appropriate phase data from all the revs included in the study.The division of the data into northernand southern-period values is taken to be related simply to the north-south displacement of spacecraft relative to the center of the current sheet according to the model given in the study by Arridge et al. [2008a], with a hinging distance of 29 R S , that is, where the northern-period data is taken to correspond to Dz ≥ 3 R S and the southern-period data to Dz ≤ 3 R S .This choice is in accordance with the data shown in Figure 4 and will be justified in detail in section 4.2.Because the majority of the phase data obtained through analyses exemplified in Figures 3 and 4 relate to the southern-period oscillations, here we determine the spatial phase dependency associated, for example, with radial propagation using this data alone, and then apply this also to the northern-period oscillations.
We note that the study by Andrews et al. [2010b] has previously developed a spatial phase model for the nearequatorial (southern-period) oscillations, but that this is applicable only to radial distances of $30 R S , while here we consider phase data to distances of $65 R S .
[20] In the Dz ≤ 3 R S phase data, y rs relative to the southern SKR modulation phase are shown by the red circles in Figure 5.In Figure 5a, they are plotted versus radial distance, exhibiting increasing values with radius indicative of outward propagation, while in Figures 5b-5g, they are plotted versus LT in six contiguous 10 R S radial ranges as marked, also showing some LT dependency as found previously in the studies by Andrews et al. [2010b] and Arridge et al. [2011].The functional form used to fit these data has then been chosen to be where j is again the azimuth from noon measured positive toward dusk and r is the radial distance from the planet in R S .
The term linear in r gives rise to outward radial propagation at a fixed speed, while inclusion of the quadratic term allows this speed to vary with distance.The inclusion of the azimuthal functions then allows both these effects to vary independently with LT.As indicated previously, the two rdependent terms are taken to be the same for both northernand southern-period oscillations, as seems reasonable (e.g., similar radial propagation speeds north and south), while the leading constants a 0n,s determining the phase of the oscillations in the inner region (r→0) relative to the respective SKR modulations are different, in accordance with the discussion in section 3.
[21] Equation (3a) has then been fitted to the southernperiod data using a two-step Levenberg-Marquardt least squares fitting procedure.The first fit was used to identify "outlier" data points that lie more than 90°from the fit, which might reasonably have been affected by some other varying process.These points, four in number identified by the open circles as shown in Figure 5, were then eliminated from the final fit, whose parameters are We note that the value of the leading constant term a 0s is in excellent accord with the above expectation that y rs * ≈ 180°f or small r.Equations (3a)-(3d) then define the southernperiod phase model shown in Figures 3d and 4d.In Figures 5a-5g, the solid black circles show model values determined at the same spatial positions as the actual data, exhibiting good overall agreement.In Figure 5a, the solid line also shows the model phase versus radial distance along the midnight meridian, while the dashed and dot-dashed lines correspond to 21:00 and 03:00 LTs, respectively, the latter lines being plotted only over the radial ranges where they are constrained by the data.The dependency along the midnight meridian is so constrained over essentially the whole range shown and can be seen to be almost a straight line with a slope of $2.5°R S À1 in conformity with the results cited previously, with the quadratic term being of significance only beyond $40 R S .The associated radial phase speed is given by where t s is the oscillation period in seconds (taken to be equal to the mean southern SKR modulation period of 10.81 hours during the interval) and the phase gradient is in deg R S À1 .This is shown for the midnight meridian in Figure 5h, falling from $220 km s À1 in the inner "core" region to $150 km s À1 at $70 R S .We note that these results are consistent with those derived from the pass-by-pass analysis in the study by Arridge et al. [2011], who found radial propagation speeds between $80 and $270 km s À1 , with a mean value of $170 km s À1 .The solid lines in Figures 5b-5g then show the model phase versus LT at the radial distance corresponding to the center of each range, while the dashed and dot-dashed lines show the corresponding curves at the upper-and lower-radial limits.At distances beyond $45 R S , the phases decrease somewhat with increasing LT, as also seen from the lines in Figure 5a.
[22] The Dz ≥ 3 R S northern-period data are shown by the blue circles in Figure 6, in a similar format as shown in Figure 5.The black circles and lines similarly show model values where the coefficients of the r-dependent terms are taken to be the same as in equations (3c) and (3d), but where the leading constant term a 0n has been determined by a least squares fit.The value so determined is again in excellent accord with the expectation that y rn * ≈ 0°f or small r.Equations (3a), (3c), (3d), and (5) then define the northern-phase model shown in Figures 3f and 4f.It can be seen in Figure 6 that it gives a reasonable account of the northern-period data.

Division Between Northern-and Southern-Period Oscillation Dominance
[23] We now provide detailed a posteriori justification for the north-south division of the phase data analyzed in section 4.1.In Figure 7a, the red circles show the deviation dy rs = y rs À y rs * between the phase data y rs determined using the southern SKR modulation phase and the model southern phase y rs * given by equation ( 3), plotted versus Dz.These are the quantities shown previously in Figures 3e and  4e for the particular apoapsis passes displayed in those figures.In Figure 7b, the blue circles similarly show the deviation dy rn = y rn À y rn * between the phase data y rn determined using the northern SKR modulation phase and the model northern-phase y rn * , also plotted versus Dz.These are the quantities shown previously in Figures 3g and 4g.In Figure 7a it can be seen that the dy rs values are generally well organized about 0 for all Dz less than $3 R S but become much more scattered for larger Dz values.Conversely, the dy rn values are scattered for all Dz less than $0 R S but become increasingly organized about 0 for larger Dz values.
[24] The scatter is quantified in Figure 7c, where we show a measure of the degree to which each of these phase differences are organized about 0, plotted versus Dz, determined using overlapping 2 R S intervals of Dz.An appropriate measure is given simply by where the sum is over the K values of phase difference in each Dz interval [e.g., Mardia and Jupp, 2000].We note that for small values of dy rn,s (i.e., for data that are well grouped about 0) we have M n,s ≈ 1 À 〈dy rn,s 2 〉/2, that is, the value of M n,s differs from +1 by half the mean square value (in radians) of the phase differences.Values of dy rn,s that are well scattered about 0 in any Dz interval then give M n,s ≈ 0, while phase deviations that are predominantly opposite in sense to 0 produce negative values of M n,s , with a lower limit of À1.In Figure 7c, the M values for the northern-and southern-system phase difference values are plotted versus Dz as the blue and red lines, respectively.It can be seen that M s (red line) has positive values for all negative values of Dz, peaking close to +1 for Dz ≃ À2 R S , and then falls sharply near Dz ≃ 2 R S to near 0 and negative values beyond Dz ≃ 4 R S .By contrast, M n is small or negative for Dz less than $À1 R S , but then rises to approach M s in the range 1 ≤ Dz ≤ 3 R S , before exceeding the latter just beyond Dz ≃ 3 R S .The latter Dz value, shown by the vertical dashed line in Figure 7, thus represents a suitable position to divide the data between southern and northern systems, as employed in section 4.1.However, it can be seen that the phases are also partly organized by the northern system in the range 1 ≤ Dz ≤ 3 R S , indicating that oscillations of the northern system are also present in this range, though with the southern system being the more significant.

Current Sheet Structure as a Function of Oscillation Phase
[25] We now employ the phase models y rn,s * (r,j) derived in section 4 to determine the structure of the tail field and current sheet as functions of the model oscillation phase where Y rs * is taken to apply to Dz ≤ 3 R S and Y rn * to Dz ≥ 3 R S .To exclude field variations not associated with the planetary-period oscillations in this analysis, we do not include data from 22 hour intervals (such as those shown in Figures 3 and 4) where the cross-correlation coefficient with the relevant model is less than 0.75, but otherwise we combine together all the field data beyond radial distances of 20 R S , corresponding to the main "tail" region.To do this we employ the total B r field measured (not the residual field, although the "planetary" field contribution is generally small in this regime), and normalize this to the strength of the lobe field at the corresponding radial distance determined in the study by Jackman and Arridge [2011], given by Normalized values B′ r of $+1 then correspond to the north lobe, $À1 to the south lobe, and smaller values between +1 and À1 to the central current sheet, that is, the plasma sheet.
[26] We have then averaged all the eligible normalized B′ r values, determined at 1 min temporal resolution, into 18°i ntervals of oscillation phase (thus 20 in total), and 0.5 R S intervals of Dz, with results shown in Figure 8. Figure 8a shows color-coded values of 〈B′ r 〉 plotted versus northernphase Y rn * horizontally and Dz vertically over the range between 3 and 13 R S .Figure 8b similarly shows 〈B′ r 〉 plotted versus southern phase Y rs * and Dz over the range À10 to 3 R S .The common color-scale to the right of Figures 8a and  8b shows values above +1 and below À1 as pale red and pale blue, respectively.These colors then become more intense with decreasing field magnitude to highlight the current sheet region within the plots, and to clearly delineate its center from the color transition from red to blue.Boxes with no data points are shown as white.The overall coverage of the data is shown in corresponding Figures 8c and 8d, in which the contributions from each Cassini apoapsis pass are displayed, color-coded as shown in Figure 2. The most extensive coverage is contained in the region À7 ≤ Dz ≤ 5 R S , while the coverage is notably sparse for 7 ≤ Dz ≤ 11 R S .
[27] Both Figures 8a and 8b show significant variations of the radial field with the corresponding northern and southern phases, present at essentially all distances Dz from the Arridge et al. [2008a] model center of the current sheet.Within the central region around Dz $ 0 R S , these correspond to north-south displacements of the current sheet, the displacements being northward for phase values of $180°and southward for $0°/360°in both cases, in conformity with the corresponding field oscillations varying as cos Y rn,s * .In Figure 8b, it is seen that the field reversal is usually located in the region Dz ≤ 3 R S governed principally by the southern oscillations, varying in position from Dz ≈ À0.5 R S when Y rs * ≈ 0°/360°to Dz ≈ 2.5 R S close to the northward boundary of this region when Y rs * ≈ 180°, corresponding to a peak-to-peak amplitude of $3 R S .The center of the oscillation is thus located near Dz ≈ 1.0 R S , close to but northward of the mean position of the current sheet center Figure 7. Plots showing the deviation of the phase data obtained from all the Cassini revs from the southern-and northern-phase models derived in section 4, (a) dy rs (red circles) and (b) dy rn (blue circles), plotted versus distance from the current sheet center Dz (R S ).(c) The measures M n,s of the degree to which these phase differences are organized about 0, given by equation ( 6) using overlapping 2 R S intervals of Dz, where the blue and red lines correspond to the northern and southern phase values, respectively.The vertical dashed line marks the approximate transition between phases that are best organized by the southern system for Dz ≤ 3 R S and by the northern system for Dz ≥ 3 R S .
indicated by the Arridge et al. [2008a] model employed here.The data in Figure 8a indicate that the current sheet center can also intrude into the region Dz ≥ 3 R S governed principally by the northern oscillations, but only in the region close to the southern boundary at Dz ≈ 3.5 R S for northern phases close to Y rn * ≈ 180°.
[28] The field variations with oscillation phase are explored in greater detail as shown in Figure 9.In Figure 9a, we show 〈B′ r 〉 versus northern oscillation phase Y rn * for contiguous color-coded ranges of Dz ≥ 3 R S as shown in Figure 9a, representing horizontal "cuts" through Figure 8a but now using 1 R S intervals of Dz.In Figure 9b, we similarly plot 〈B′ r 〉 versus southern oscillation phase Y rs * for color-coded ranges of Dz ≤ 3 R S as also shown in Figure 9b, representing horizontal "cuts" through Figure 8b.It can be seen in Figure 9b that major oscillations in 〈B′ r 〉 between +0.5 and À0.5, that is, between plus and minus half the lobe field strength, occur in the Dz ranges between 0 and 2 R S , in conformity with the previous discussion, while oscillations of similar phase but diminishing amplitude occur in all other Dz ranges away from this band.
[29] This is further displayed in Figure 10, where we plot 〈B′ r 〉 horizontally versus Dz vertically for fixed ranges of the oscillation phase, the northern phase in Figure 10a for 3 ≤ Dz ≤ 13 R S and the southern phase in Figure 10b for À10 ≤ Dz ≤ 3 R S , thus representing vertical cuts through Figures 8a   and 8b.As shown in Figure 8, we use 0.5 R S intervals of Dz, but now employ eight contiguous 45°intervals of phase centered on the values indicated in Figures 10a and 10b.These plots thus exhibit the north-south structure of the field through the current sheet for fixed values of the oscillation phases.The phases that correspond in the models to maximum and minimum values of 〈B′ r 〉, that is, Y rn,s * equal to 0°a nd 180°, are shown by the blue and red curves, respectively, while equally spaced phases on either side of these values are shown by dashed (for 0°< Y rn,s * < 180°) and dotted (for 180°< Y rn,s * < 360°) curves of the same color, as indicated.These curves again show field oscillations of maximum amplitude near Dz ≃ 1R S , diminishing significantly in amplitude for Dz ≥ 6 R S in the northern region and for Dz ≤ À1 R S in the southern region, though being essentially consistently present over the whole Dz range shown, from À10 to +13 R S .The curves at fixed values of the oscillation phase also indicate that the central current sheet is a relatively narrow structure with a width of $2-3 R S , although generally embedded within a region of reduced field values of overall width $6 R S , to which the temporal and spatial averaging over the natural variability of the system, no doubt, contributes.We note that Arridge et al. [2011] found pass-by-pass current sheet widths between 3 and 6 R S , more usually the former than the latter.[30] In considering these results, however, it should be recalled from section 2 that the northern and southern oscillation phases Y rn * and Y rs * are independent quantities that pass through a cycle from in phase to antiphase and back every beat period of $23 days (corresponding to $50 oscillations).When the two oscillations are in phase, the results shown in Figures 8-10 are consistent with an essentially fixed spatial structure of overall width $6 R S that oscillates north-south about Dz ≃ 1 R S with a peak-to-peak amplitude of $3 R S .This can be seen from Figure 10, for example, by continuing across Figures 10a and 10b along lines of the same color-code, as is then appropriate.In relation to the discussion of Figure 1 in section 2, in this case, the B r field perturbations of both northern and southern systems, positive for 270°< Y rn,s * < 90°and negative for 90°< Y rn,s * < 270°, act to produce north-south displacements of the current sheet that are in concert with one another.This configuration is illustrated explicitly in Figure 11a, where we replot the data in Figures 8a and 8b on a continuous Dz scale using values of the phase on the horizontal axis corresponding to in-phase behavior, Y rs * shown at the bottom and Y rn * shown at the top, equal to each other in this case.The overall oscillatory behavior is clear.We note that because of the near antiphase relationship between the field perturbations and the SKR modulations in the two systems, as found in section 4 and in previous studies, this condition corresponds to intervals when the SKR modulations in the two hemispheres are in antiphase.The current sheet near midnight is displaced southward at southern SKR maxima, and northward at northern SKR maxima in conformity with the discussion in section 2.
[31] This is no longer the case when the two oscillations are in antiphase, however, when, for example, we continue from red to blue lines, and vice versa, as shown in Figures 10a and 10b.In this case, the field perturbations associated with the two systems are of opposite sign.When Y rn * ≈ 0°/360°and Y rs * ≈ 180°, for example, the B r field perturbations are positive for the northern system and negative for the southern system, both acting to enhance the tail field on opposite sides of the current sheet.In this case, our results indicate that the current sheet narrows to an overall width of $4 R S centered near Dz ≃ 2.5 R S .However, half an oscillation cycle later when Y rn * ≈ 180°and Y rs * ≈ 0°/ 360°, the B r field perturbations are negative for the northern system and positive for the southern, both acting to reduce the tail field on either side of the current sheet.In this case, our results indicate that the current sheet expands to a somewhat asymmetric structure of overall width $10 R S located in the range À 4 ≤ Dz ≤ 6R S , with the field reversal occurring at Dz ≃ À0.5 R S .In addition to the overall oscillation of the field reversal, therefore, under this condition, the current sheet width also undergoes compressions and expansions by a factor of $2.5 during each oscillation cycle.This configuration is illustrated in Figure 11b, where the data shown in Figures 8a and 8b are again plotted on a continuous Dz scale, but now using values of the phase on the horizontal axis corresponding to antiphase behavior, Y rs * shown at the bottom and Y rn * shown at the top.While oscillations of the field reversal still continue, governed principally by the southern period at Dz ≤ 3R S , the sheet thickness is also strongly modulated during the oscillations.We note that this condition corresponds to intervals when the SKR modulations in the two hemispheres are in phase.The weakened tail field and expanded current sheet condition corresponds to SKR maxima in both hemispheres, while the strengthened tail field and contracted current sheet corresponds to SKR minima in both hemispheres, also in conformity with expectations discussed in section 2.

Summary
[32] Previous studies by Gurnett et al. [2009aGurnett et al. [ , 2011] ] and Lamy [2011] have shown that dual periodicities near the "planetary period" are present in the SKR modulations, with shorter periods of $10.6 hours in the northern modulations and longer periods of $10.8 hours in the southern modulations during the preequinox southern summer conditions considered here.Subsequently, the studies by Andrews et al.
[2010a] and Southwood [2011] have also shown that the quasi-dipolar magnetic field oscillations at high polar latitudes rotate with the periods of the SKR modulations in the corresponding hemispheres, while the study by Provan et al. [2011] has shown that both periods are present in the inner "core" region data (with dipole L ≤ 12), with the southernperiod oscillations being stronger than the northern-period oscillations by factors of $3 during the preequinox interval.Both the effective transverse dipole associated with the polar oscillations and the quasi-uniform equatorial field associated with the "core" oscillations point approximately tailward for the southern-period oscillations at southern SKR maxima, while they point approximately sunward for the northernperiod oscillations at northern SKR maxima.
[33] In this article, we have extended these studies to consider the field oscillations observed during a sequence of 10 Cassini apoapsis passes in 2006, which explored Saturn's magnetic tail to distances of $65 R S mainly in the midnight to dawn sector.Arridge et al. [2011] have modeled these data on a pass-by-pass basis, in particular, examining the oscillations with respect to the phase of the southern-period system.They showed that the predominant data in the central and southern tail are consistent with north-south oscillations of the current sheet and plasma sheet that are well organized by the southern phase.Specifically, the current sheet is displaced southward when the effective transverse dipole and quasi-uniform field of the southern system point down-tail, corresponding to southern SKR maxima, and is displaced northward when the effective transverse dipole and quasi-uniform field of the southern system point sunward, corresponding to southern SKR minima, these being consistent with the observations of Carbary et al. [2008b].However, the phase relationship so established was found to break down for oscillations observed in the northern tail.
Here we have developed an overall empirical phase model for these data, and have distinguished between oscillations that are associated with the southern-and northern-period systems, thus resolving the latter issue.
[34] The phase of the oscillations in the northern and southern systems has been modeled by a function with the same radial and LT dependence in each case, fit to the more plentiful southern-period data, but with differing constant terms defining the phase of the rotating disturbances at small distances relative to the corresponding SKR modulations.As in the previous studies of Andrews et al. [2010a] and Provan et al. [2011], the best fit constant terms are such that in the inner region the southern-period quasi-uniform field points approximately tailward at southern SKR maximum while the northern-period quasi-uniform field points approximately sunward at northern SKR maximum, as indicated previously.The principal spatial dependency is an increase in phase with increasing radial distance of $2.5°R S À1 , consistent with outward radial propagation at speeds of $200 km s À1 .Such values are consistent with the previous related results reported by Carbary et al. [2007b], Andrews et al. [2010b], Clarke et al. [2010b], and Arridge et al. [2011].
[35] Ordering the phase data by the north-south displacement of the spacecraft Dz from the mean current center modeled by Arridge et al. [2008a], our principal finding is that the oscillations observed for Dz ≥ 3 R S are governed principally by the northern period, while those for Dz ≤ 3 R S are governed principally by the southern period.Evidence has also been presented for dual control in the range 1 ≤ Dz ≤ 3 R S , indicative of comparable oscillation amplitudes in this range, but with the southern-period oscillation being the stronger.As the field reversal region is found to be centered near to Dz ≈ 1 R S , that is, 1 R S northward of the modeled value, this implies that oscillations at the southern period dominate throughout the southern portion of the current sheet and southern lobe where the radial field is negative, while oscillations at the northern period dominate in the outer portion of the northern current sheet and northern lobe where the radial field is positive, with a region of dual periodicity (though still more strongly controlled by the southern period) in the inner portion of the northern current sheet between these two regimes.We note that these results are consistent with the findings of Carbary et al. [2009], who demonstrated the existence of such dual periodicities in energetic electron fluxes in the outer (>20 R S ) part of Saturn's magnetosphere.
[36] These results imply that the modulation of the current sheet and plasma sheet during these oscillations depends significantly on the $23 day beat cycle between the northern and southern oscillations.When the two magnetic oscillations are in phase, meaning that the SKR modulations are in antiphase, the current sheet of overall width $6 R S (probably somewhat widened from the true value by data averaging) simply oscillates north-south about a central position of Dz ≈ 1 R S with a peak-to-peak amplitude of $3 R S .The field reversal region is displaced southward of its central position at southern SKR maxima and northern SKR minima when the radial perturbation fields in the tail are both positive, thus strengthening the positive field in the northern lobe and weakening the negative field in the southern lobe.Similarly, the field reversal region is displaced northward of its central position at northern SKR maxima and southern SKR minima when the radial perturbation fields in the tail are both negative, thus strengthening the negative field in the southern lobe and weakening the positive field in the northern lobe.
[37] When the two oscillations are in antiphase, however, implying that the SKR modulations are in phase, the current sheet center not only oscillates north-south principally with the southern phase as before, but also the sheet is modulated strongly in width.At joint SKR minima when the northern radial perturbation field in the tail is positive and the southern is negative, thus enhancing the lobe field in both hemispheres, the overall width is reduced to $4 R S , centered on a field reversal region near Dz ≈ 2.5 R S .Correspondingly, at joint SKR maxima when the northern perturbation field in the tail is negative and the southern is positive, thus reducing the lobe field in both hemispheres, the overall width increases to $10 R S , located in the range À4 ≤ Dz ≤ 6 R S , with the field reversal occurring at Dz ≃ À0.5 R S .Such modulations of the current and plasma sheet width might possibly account for the in-phase modulations of the field strength and energetic (tens of kiloelectron volt) electron flux reported in the dawn tail in the study by Krupp et al. [2005], opposite to the antiphase electron flux modulations that are most evident in Figures 3 and 4, and for the periodic thickening of the plasma sheet with antiphase northward displacements of its center modeled in the study by Morooka et al. [2009].We also note that modulations in current sheet thickness, with evident consequences for the sheet current density, may also have implications for plasma sheet stability, a topic that may be worthy of future exploration.However, we emphasize that in the picture discussed here, such modulation is restricted to only one part of the beat cycle, when the northern and southern oscillations are near antiphase, and the northern and southern SKR modulations are in phase.In the other part of the cycle, simple oscillations occur when the northern and southern oscillations are in phase, and the northern and southern SKR modulations are in antiphase.
[38] Acknowledgments.Work at Leicester was supported by STFC grant ST/H002480/1.A.J.C. was supported by STFC funding to MSSL/ UCL, M.K.D. by STFC funding to Imperial College, C.M.J. by a Leverhulme Trust Early Career Fellowship, C.S.A. by a STFC Post-Doctoral Fellowship, and D.J.A. by a STFC Quota Studentship.We thank S. Kellock and the Cassini team at Imperial College for access to processed magnetic field data, L. Lamy for access to SKR phase data, and G.R. Lewis and L.K. Gilbert at MSSL for CAPS/ELS data processing.
[39] Masaki Fujimoto thanks Donald Gurnett and another reviewer for their assistance in evaluating this paper.

Figure 1 .
Figure 1.Sketches illustrating the magnetic field perturbations associated with the planetary-period field oscillations at Saturn, adapted from Andrews et al. [2010a] and Provan et al. [2011].(a and b) Colored lines indicate the form of the perturbation fields of the southern-and northern-period field systems, respectively, while (c and d) similarly colored lines illustrate their effect on the overall structure of the southernand northern-magnetospheric fields.

Figure 3 .
Figure 3. Cassini data for the near-equatorial apoapsis pass from the periapsis of rev 25 to the periapsis of rev 26.From top to bottom: (a) an ELS electron spectrogram covering the energy range 0.6 eV to 28 keV; (b) the residual radial magnetic field B r (nT) from which the "Cassini SOI" internal field has been subtracted; (c) the residual radial field band pass filtered between periods of 5 and 20 hours; (d) the oscillation phase values = rs (in degrees) shown by red circles determined from 22 hour data intervals (vertical dotted lines) by cross-correlation of the filtered residual field with equation (1a) using the southern SKR modulation phase F SKRs , together with the overall southern phase model shown by the red dashed line given by equation (3); (e) the difference between the = rs phase data and the model in Figure 3d; (f) the oscillation phase values = rn (deg) shown by blue circles determined from the same 22 hour data intervals by cross-correlation of the filtered residual field with equation (1a) using the northern SKR modulation phase F SKRn , together with the overall northern-phase model shown by the blue dashed line given by equations (3) and (5); (g) the difference between the = rn phase data and the model in Figure3f;(h) the radial distance of the spacecraft (R S ); (i) the distance z (R S ) of the spacecraft northward of the planet's equatorial plane (black line) together with the northward displacement of the current sheet center (red line) according to the model ofArridge et al. [2008b] with a hinging distance of 29 R S ; and (j-l) the spacecraft trajectory (red lines) projected onto the KSMAG equatorial, noon-midnight, and dawn-dusk planes, respectively, with time (start of day of year (DOY)) plotted every 4 days and the model magnetopause and bow shock superposed (black lines).

Figure 4 .
Figure 4. Cassini data for an inclined apoapsis pass, specifically for the interval from the periapsis of rev 26 to the periapsis of rev 27, in the same format as shown in Figure 3.

Figure 5 .
Figure 5. Plots showing the radial and LT dependency of the southern-period (Dz ≤ 3 R S ) phase data y rs from all the Cassini revs employed (red circles), plotted (a) versus radial distance and (b-g) versus LT in 10 R S radial ranges.The black circles show the fitted model values given by equation (3) evaluated at the same positions as the data.The solid black line in Figure5ashows the model phase versus radial distance for the midnight meridian, while the dashed and dot-dashed lines correspond to 21:00 and 03:00 LT.The solid black lines in Figures5b-5gshow the variation of the model phase versus LT at the center radius of each range, while the dashed and dot-dashed lines correspond to the upper-and lower-radial limits.(h) Radial phase speed plotted versus radial distance for the midnight meridian, calculated from equation (4).

Figure 6 .
Figure 6.Plots showing the radial and LT dependency of the northern-period (Dz ≥ 3 R S ) phase data y rn from all the Cassini revs employed (blue circles), together with fitted model values (black circles and lines), in the same format as shown in Figure 5.

Figure 8 .
Figure 8. Plots showing color-coded values of the mean radial field normalized to the lobe field 〈B′ r 〉, plotted versus the oscillation phase Y rn,s * (deg) given by equation (7) horizontally and north-south distance from the model current sheet center Dz (R S ) vertically.Field data with 1 min cadence for all radial distances exceeding 20 R S have been combined.(a) 〈B′ r 〉 versus the northern-phase Y rn * for Dz between 3 and 13 R S ; (b) 〈B′ r 〉 plotted similarly versus the southern phase Y rs * for Dz between À10 and 3 R S .Data boxes are 18°in phase and 0.5 R S in Dz.Boxes with no data points are shown white.(c and d) The corresponding data coverage, color-coded as shown in Figure 2.

Figure 9 .
Figure 9. Plots showing the mean normalized total radial field 〈B′ r 〉 plotted versus oscillation phase Y rn,s * (deg) for various color-coded contiguous 1 R S intervals of Dz (R S ) as indicated.(a) 〈B′ r 〉 is shown versus the northern oscillation phase Y rn * , representing horizontal "cuts" through Figure 8a; (b) 〈B′ r 〉 is shown versus southern oscillation phase Y rs * , representing horizontal "cuts" through Figure 8b.

Figure 10 .
Figure 10.Plots showing the mean normalized total radial field 〈B′ r 〉 horizontally versus Dz (R S ) vertically for fixed intervals of the oscillation phase, using (a) the northern phase for 3 ≤ Dz ≤ 13 R S and (b) the southern phase for À10 ≤ Dz ≤ 3 R S , representing vertical cuts through Figures 8a and 8b.We employ 0.5 R S intervals of Dz and eight contiguous 45°intervals of phase centered on the values indicated.Phases that correspond in the models to maximum and minimum values of 〈B′ r 〉, that is, Y rn,s * equal to 0°and 180°are shown by blue and red lines, respectively, while equally spaced phases on either side of these values are shown by dashed (for 0°< Y rn,s * < 180°) and dotted (for 180°< Y rn,s * < 360°) curves of the same color.

Figure 11 .
Figure 11.The data in Figures 8a and 8b are shown replotted on a continuous Dz scale, using simultaneous values of the northern and southern phases on the horizontal axes that correspond to (a) in-phase behavior of the two systems and (b) antiphase behavior.The values of Y rs * (deg) applying to À10 ≤ Dz ≤ 3 R S are shown in the bottom scale, while the simultaneous values of Y rn * (deg) applying to 3 ≤ Dz ≤ 13 R S are shown in the top scale.