Planetary period oscillations in Saturn s magnetosphere: Evolution of magnetic oscillation properties from southern summer to post-equinox

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011ja017444, 2012 Planetary period oscillations in Saturn s magnetosphere: Evolution of magnetic oscillation properties from southern summer to post-equinox D. J. Andrews, 1,2 S. W. H. Cowley, 1 M. K. Dougherty, 3 L. Lamy, 4 G. Provan, 1 and D. J. Southwood 3 Received 8 December 2011; revised 8 February 2012; accepted 8 March 2012; published 28 April [1] We investigate the evolution of the properties of planetary period magnetic field oscillations observed by the Cassini spacecraft in Saturn s magnetosphere over the interval from late 2004 to early 2011, spanning equinox in mid Oscillations within the inner quasi-dipolar region (L 12) consist of two components of close but distinct periods, corresponding essentially to the periods of the northern and southern Saturn kilometric radiation (SKR) modulations. These give rise to modulations of the combined amplitude and phase at the beat period of the two oscillations, from which the individual oscillation amplitudes and phases (and hence periods) can be determined. Phases are also determined from northern and southern polar oscillation data when available. Results indicate that the southern-period amplitude declines modestly over this interval, while the northern-period amplitude approximately doubles to become comparable with the southern-period oscillations during the equinox interval, producing clear effects in pass-to-pass oscillation properties. It is also shown that the periods of the two oscillations strongly converge over the equinox interval, such that the beat period increases significantly from 20 to more than 100 days, but that they do not coalesce or cross during the interval investigated, contrary to recent reports of the behavior of the SKR periods. Examination of polar oscillation data for similar beat phase effects yields a null result within a 10% upper limit on the relative amplitude of northern-period oscillations in the south and vice versa. This result strongly suggests a polar origin for the two oscillation periods. Citation: Andrews, D. J., S. W. H. Cowley, M. K. Dougherty, L. Lamy, G. Provan, and D. J. Southwood (2012), Planetary period oscillations in Saturn s magnetosphere: Evolution of magnetic oscillation properties from southern summer to postequinox, J. Geophys. Res., 117,, doi: /2011ja Introduction [2] Although Saturn s internal planetary magnetic field has been found to be highly axisymmetric [Smith et al., 1980; Connerney et al., 1982], with an upper limit to the dipole tilt of 0.1 according to recent determinations [Burton et al., 2010], pronounced modulations of the magnetic field, plasma properties, plasma waves, and auroral radio and ultraviolet (UV) emissions near the 11 h planetary period have nevertheless been found to be ubiquitous throughout Saturn s magnetosphere [e.g., Warwick et al., 1981, 1982; Gurnett et al., 1981, 2005, 2007, 2010a; Desch and Kaiser, 1981; Sandel and Broadfoot, 1981; Sandel et al., 1982; Carbary and Krimigis, 1982; Espinosa 1 Department of Physics and Astronomy, University of Leicester, Leicester, UK. 2 Now at Swedish Institute for Space Physics, Uppsala, Sweden. 3 Blackett Laboratory, Imperial College, London, UK. 4 LESIA, Observatoire de Paris, CNRS, UPMC, Université Paris Diderot, Meudon, France. Copyright 2012 by the American Geophysical Union /12/2011JA and Dougherty, 2000, 2003a, 2003b; Krupp et al., 2005; Kurth et al., 2007, 2008; Southwood and Kivelson, 2007; Carbary et al., 2007a, 2007b, 2008a, 2008b, 2009, 2010, 2011; Andrews et al., 2008, 2010a, 2010b, 2011; Nichols et al., 2008, 2010a, 2010b; Burch et al., 2009; Provan et al., 2009a, 2009b; Clarke et al., 2006, 2010a, 2010b; Wang et al., 2010; Ye et al., 2010]. The perturbations in the field and plasma are found to rotate around the planet in the sense of planetary rotation, such that they are Dopplershifted by the azimuthal motion of the observing spacecraft [Cowley et al., 2006; Giampieri et al., 2006; Provan et al., 2009a; Andrews et al., 2010a], with recent evidence also being presented that the modulations in auroral hiss and Saturn kilometric radiation (SKR), believed to be associated with downward- and upward-directed auroral field-aligned currents respectively, rotate similarly around the northern and southern auroral ovals [Gurnett et al., 2009b; Andrews et al., 2011; Lamy, 2011]. [3] It has also been found that the periods of the northern and southern SKR and auroral hiss modulations are distinct, with a shorter period of 10.6 h in the north compared with 10.8 h in the south during the Saturn southern summer 1of31

2 interval investigated during the initial phase of the Cassini mission between 2004 and 2008 [Kurth et al., 2008; Gurnett et al., 2009a, 2009b]. Correspondingly, the magnetic field perturbations in the high-latitude polar magnetosphere and magnetospheric tail are also found to oscillate with distinct periods in the north and south that closely match the SKR periods [Andrews et al., 2010b; Southwood, 2011; Provan et al., 2012]. The field oscillations within the central quasidipolar magnetosphere to equatorial distances of 15 R S, referred to here as the core region [e.g., Andrews et al., 2010a], are then found to be dominated by the southern period over the same southern summer interval, as are the modulations in the plasma properties cited above. (R S is Saturn s 1 bar equatorial radius equal to 60,268 km.) However, a distinct phase jitter is also found to be present in the core region oscillations, which is due to the simultaneous presence of northern-period oscillations of lower amplitude [Provan et al., 2011]. [4] Both the northern and southern periods of the SKR and magnetic field oscillations have also been found to vary slowly with time, by fractions of a percent per annum during the southern summer conditions observed initially by Cassini [Galopeau and Lecacheux, 2000; Gurnett et al., 2005; Kurth et al., 2007, 2008; Andrews et al., 2010b]. However, starting near the beginning of 2009, ahead of vernal equinox in August 2009, the periods of the two SKR modulations have been observed to more rapidly converge, reaching nearcommon values of 10.7 h by early 2010 and either coalescing [Lamy, 2011] or crossing [Gurnett et al., 2010b, 2011] in the interval beyond. This behavior has also been followed to mid-2010 in the magnetic oscillation data, but only in respect of the southern-period oscillations [Andrews et al., 2011]. [5] The purpose of the present paper is to examine the evolution of the properties of the magnetic field oscillations over the interval from southern summer to post-equinox, from late 2004 to early 2011, both in phase and for the first time in amplitude, focusing on the combined northern- and southern-period oscillations within the quasi-dipolar core region. Using the phase jitter effect found by Provan et al. [2011], it is shown that the phase and hence period of both southern and northern field oscillations can be independently retrieved from the data, together with measures of the amplitude of the two oscillations. We show that from conditions of southern-oscillation dominance during southern summer as indicated above, the amplitude of the northern oscillations in the core grows by a factor of two to become comparable with that of the southern oscillations during the equinox interval, with significant consequences for the properties of the oscillations observed from pass to pass. It is also shown that the periods of the two magnetic oscillations strongly converge during the equinox interval, but do not coalesce or cross during the interval examined to early Planetary Period Magnetic Field Oscillations Overview 2.1. Physical Picture of Southern and Northern System Oscillations [6] We begin by discussing the overall nature of the field oscillations in Saturn s magnetosphere that has emerged from the studies cited in section 1. This is illustrated in Figure 1 taken from Andrews et al. [2010b] and Provan et al. [2011], which forms the basis of the analysis presented in this paper. Figures 1a and 1b show cuts through the principal magnetospheric meridian plane of the perturbations at some instant, where the black dashed lines indicate the near-axisymmetric unperturbed magnetospheric field with closed field lines in the equatorial region (gray) and open field lines at high latitudes (clear). The red lines in Figure 1a and the blue lines in Figure 1b then indicate the perturbation fields in these meridians associated with the southern and northern oscillatory field systems, respectively. The perturbation field lines out of this plane at these instants can be obtained to a first approximation simply by displacing these field line loops directly into or out of the plane of the figure. The two patterns of field perturbations then rotate independently around the planetary spin axis (Z) at the southern and northern periods, respectively. [7] In the high-latitude magnetosphere the perturbations have the form of a planet-centered rotating transverse dipole [Provan et al., 2009a], the instantaneous effective direction of which is shown by the large planet-centered colored arrows in each panel. The near-equatorial field perturbations within the closed-field core region then take the form of rotating quasi-uniform fields lying approximately parallel to the equatorial plane [Espinosa et al., 2003a, 2003b; Southwood and Kivelson, 2007], combined with north-south fields such that the field lines are arched with apices pointing to the north for the southern system, and pointing to the south for the northern system [Andrews et al., 2008, 2010a; Provan et al., 2011]. We note that at any instant the quasi-uniform equatorial field and the effective transverse dipole describing the high-latitude field point in the same direction as each other in each system. The specific configurations shown in Figure 1 correspond approximately to SKR maxima in the related hemisphere when the Sun is located to the right in each case. The quasi-uniform field and transverse dipole of the southern system in Figure 1a thus point down-tail (and somewhat dawnward) at southern SKR maxima, while the quasi-uniform field and transverse dipole of the northern system in Figure 1b point sunward at northern SKR maxima [Andrews et al., 2008, 2010b; Provan et al., 2011]. We note that Ampère s law applied to the field loops requires the presence of current flow out of the plane of the diagram in each case, which we take to have the form of separate rotating northern- and southern-period field-aligned current systems flowing into and out of the ionosphere on opposite sides of the planet [Andrews et al., 2010b]. [8] The form of the core region equatorial field perturbations in the two systems is considered in more detail in Figures 1c and 1d, where we view the equatorial plane from the north in both cases. The colored solid lines indicate the quasi-uniform perturbation field that rotates anti-clockwise with time in the sense of planetary rotation, as shown by the arrowed dashed circle in each panel, at the southern period in Figure 1c and the northern period in Figure 1d. The circled dots and crosses also show the simultaneous northward- and southward-directed perturbation fields respectively. As in previous studies, here we employ spherical polar magnetic field components in a planet-centered system referenced to Saturn s northern spin and magnetic axis. If we then consider the field for the southern system at point P in Figure 1c, 2of31

3 Figure 1. (a) Sketch showing the spatial structure of southern-period magnetic field oscillations. The red lines show the perturbation field lines in the principal meridian of the perturbation at some instant, where the vertical axis Z represents the spin (and magnetic) axis of the planet. The black dashed lines show the quasi-static background magnetospheric field, with closed field lines at lower latitudes (gray region) and open field lines at high latitudes (clear region) mapping into the northern and southern polar regions. To a first approximation the perturbation field lines out of the meridian plane shown can be obtained by displacing the colored loops directly into or out of the plane of the figure. This perturbation field pattern then rotates approximately rigidly about the axis at the southern SKR modulation period as indicated, giving rise to magnetic field oscillations at that period at a fixed point. In the equatorial region the perturbation field takes the form of a rotating quasi-uniform field with additional north-south components such that the field lines form arches with apices pointing to the north. In the southern high-latitude region the perturbation instead has the form of a planet-centered rotating transverse dipole, whose instantaneous direction is indicated by the large red arrow, having the same direction as the equatorial quasi-uniform field. (b) As in Figure 1a but for the northern-period field oscillations indicated by the blue lines. (c) Sketch of the perturbation fields in the equatorial plane corresponding to the southern-period oscillations shown in Figure 1a, viewed from the north. The red solid lines indicate the equatorial quasi-uniform field, which at the instant depicted has a maximum in the r component at point P, and a zero in the φ component, this defining the principal meridian in Figure 1a. The red circled dots and crosses represent the direction of the perturbation field in the north-south direction at this instant, which also has a maximum in the q component (positive southward) at point P. (d) As in Figure 1c except for the northern-period oscillations shown in Figure 1b. In this case the r component oscillation has a minimum at point P, while the q component again has a maximum. (From Andrews et al. [2010b] and Provan et al. [2011].) it can be seen that the radial component (r, positive outward) has a maximum at the instant depicted, as has the co-latitudinal component (q, positive southward), while the azimuthal component (φ, positive anticlockwise in the sense of planetary rotation) is zero. Corresponding minima in the r and q components then occur on the opposite side of the planet where the φ component is also zero, thus defining the principal meridian shown in Figure 1a. As the field pattern then rotates anti-clockwise with time, a maximum in the φ component then follows at point P one quarter cycle later, as the r and q components decline to zero. While the r and q components thus oscillate in phase in the southern system, the φ component oscillates in lagging quadrature with both [Espinosa et al., 2003a, 2003b; Southwood and Kivelson, 2007; Andrews et al., 2008, 2010a]. Similar examination of the corresponding perturbations for the northern system in Figure 1d shows that while the oscillations in the φ component remain in lagging quadrature with those in r, 3of31

4 corresponding to a quasi-uniform field in the equatorial plane, the oscillations in the q component are reversed to anti-phase with r and hence to leading quadrature with φ [Provan et al., 2011]. [9] We thus note with regard to the core region field data that the southern oscillation is taken to correspond specifically to that in which the r and q components are in phase, while the northern oscillation is taken to correspond to that in which the r and q components are in anti-phase, the φ component being in lagging quadrature with r in both cases. We also note on the basis of Figure 1 that the r and q components in the southern and northern polar regions should oscillate in phase with the corresponding components of the southern and northern oscillations in the core, while the φ components should oscillate in anti-phase, the latter corresponding to quasi-dipolar polarization. Such behavior is in agreement with the prior results of Provan et al. [2009a, 2011] and Andrews et al. [2010b], and will be further demonstrated here. We may then expect the oscillation periods of these two systems to agree with the corresponding southern and northern modulation periods determined from SKR data (within the limits identified by Andrews et al. [2011]), as will be shown here generally to be the case, though with some significant deviations post-equinox Consequences of Combined Southern and Northern Oscillations in the Core Region [10] The perturbation fields within the core thus consist of the vector superposition of the southern and northern systems, albeit dominated by the southern system during the southern summer conditions prevailing during the initial phase of the Cassini mission [e.g., Andrews et al., 2008; Provan et al., 2009a]. In this section we thus give an initial discussion of the consequences of this superposition for the core field oscillations, prior to the detailed theoretical development given in section 5. At the time depicted in Figure 1, for example, the oppositely directed quasi-uniform fields of the two systems corresponding to the r and φ components will partially cancel, while simultaneously the north-south perturbation fields corresponding to q will add. However, due to the different rotation periods of the two systems these relative orientations are not fixed, but instead undergo a cycle of in-phase and anti-phase behavior at the beat period of the two oscillations. Thus half a beat period later compared with Figure 1, the quasi-uniform fields will add, while the north-south fields will partly cancel. Thus the amplitudes of the combined oscillations will undergo cyclic variations at the beat period, with the variations in the r and φ components being in anti-phase with those in q. The beat frequency is equal to the difference between the individual oscillation frequencies, such that with periods of 10.6 h for the northern system and 10.8 h for the southern during southern summer, the beat period is 20 days encompassing 50 oscillations [e.g., Provan et al., 2011], which we note is of comparable order to the Cassini orbital period. [11] In the intervals between such extreme amplitude conditions, such as halfway between when the quasi-uniform fields of the two systems are orthogonal to each other, the presence of the weaker field causes the combined oscillation to deviate in phase relative to that of the stronger field, in opposite senses for the (r, φ) and q components [Provan et al., 2011]. The senses of these phase deviations then reverse every half beat period, changing sign across the times of extremal amplitude conditions. When the beat period is comparable with the orbital period, as during the initial southern summer Cassini interval as indicated above, these deviations are perceived as pass-to-pass phase jitter, having opposite senses for the (r, φ) and q components. Phase deviations of typically 25 as found by Provan et al. [2011] during this interval imply that the southern system is stronger than the northern by factors of 2 3 during this interval. The expected corresponding variations in amplitude, by factors of 2 over the beat cycle, have yet to be demonstrated. [12] Across the equinoctial interval considered here, however, two developments of this picture may be anticipated. First, the periods of the two oscillations may converge before possibly crossing, as reported in SKR modulation data [Gurnett et al., 2010b, 2011; Lamy, 2011], such that the beat period will increase, possibly to an extent resolvable in pass-to-pass oscillation data. Second, the relative amplitudes of the two oscillations may also change in favor of the northern oscillation, with the latter eventually becoming dominant during northern summer. This would result first in increased phase and amplitude variations as the two systems approach equal amplitudes, followed by reducing variations once more but about dominant northern system values. The most extreme beat cycle variations occur when the amplitudes of the two systems are close to each other, in which case the amplitude modulations during each beat cycle will vary between doubling of the amplitude and complete cancellation, with the maximum amplitude in (r, φ) occurring at the times of zeros in q, and vice versa. Observation of such conditions would represent a clear indicator of near-equal amplitudes of the two systems. In addition, it is shown in section 5 below that under such conditions the phase deviations in (r, φ) and q, in opposite directions, always sum to 90, flipping sign between the two halves of the beat cycle at the times of extreme amplitude conditions. The implication is that the combined oscillation in the q component shifts in phase by 90 relative to the r and φ oscillations, flipping sign every half-beat period to oscillate in lagging or leading quadrature with r, and hence either in phase or antiphase with φ. The occurrence of such altered relative phases between the oscillations in the three field components would similarly represent an indicator of near-equal amplitudes of the two systems. Examples will be presented in section 4 that demonstrate such behavior during the equinoctial interval. 3. Data Overview [13] In this section we first provide an overview of the data set employed, consisting of magnetic field oscillation data obtained over the whole 7 year Cassini orbital tour to date, the nature of which depends crucially on the orbit of the spacecraft and the consequent magnetospheric regions traversed. Figure 2 thus gives an overview of key orbital and physical parameters, spanning the interval from just prior to Saturn orbit insertion (SOI) in mid-2004, to mid Specifically, the figure covers the interval t = days (June 2004 to July 2011), where t = 0 corresponds to 00 UT on 1 January Year boundaries are shown at the top of 4of31

5 Figure 2. Overview of orbital and physical parameters during the interval considered in this study spanning the Cassini tour over t = days, where t = 0 corresponds to 00 UT on 1 January Year boundaries are shown at the top of the figure, along with alternating black and white bars indicating the duration of each Cassini Rev defined from apoapsis to apoapsis, numbered every ten Revs. (a) The latitude of spacecraft periapsis (deg) for each Rev (black dots) plotted at the time of periapsis, together with the latitude range on that Rev (vertical bars). (b) The radial distance of periapsis (R S ) for each Rev (black dots), together with the radial range to apoapsis (within a 30 R S plot limit). The horizontal dashed lines show from top to bottom the radial limit of 25 R S for the polar field data employed, the equatorial limit of 12 R S for near-equatorial core region field data (corresponding to L 12 applied more generally to such data), and the equatorial limit of 6 R S for core region field data on highly inclined orbits (corresponding to L 6 applied more generally to such data). (c) The latitude of the Sun (deg) at Saturn, passing through zero at vernal equinox at t = 2049 days marked by the vertical black dotted line. (d) The periods (hours) of the southern (red) and northern (blue) SKR modulations determined by Lamy [2011], the black dashed line showing the mean of the two. (e) The difference (hours) between these periods (black line), i.e., Dt SKR = t SKRs t SKRn. (f) The beat period (days) corresponding to the two modulation periods t SKRB = (t SKRn t SKRs )/Dt SKR (black line), while the black dots show the orbital period of Cassini, plotted for each Rev at the times of periapsis. The blue background shows the intervals A, C, and E of near-equatorial Cassini orbits that allow detailed study of the field oscillations in the core region using band-pass filtered residual data, while the yellow background shows intervals B and D of highly inclined orbits from which polar oscillation data can be obtained. Interval D is then split into two sub-intervals by the vertical orange dashed line, D1 where core region oscillation parameters can also be obtained from unfiltered φ component data only, and D2 where in general they cannot due to the raised periapsis. 5of31

6 the figure, along with alternating black and white bars indicating the duration of each Cassini orbital revolution (Rev) defined from apoapsis to apoapsis, numbered every ten Revs. [14] Figures 2a and 2b show key orbital parameters, namely the latitude and radial distance of periapsis for each Rev shown by the black dots, together with the ranges of these parameters over each Rev shown by the vertical lines. The orbital period is also shown by the black dots in Figure 2f. The tour can be divided by the orbit latitude coverage shown in Figure 2a into five basic intervals shown by the colored stripes, labeled A-E. During intervals A, C, and E shown by the blue stripes, the orbit is near-equatorial with the periapsis distance sufficiently small that few-day passes through the core region occur on each Rev, the latter being typically of days duration. The core region is specifically defined for these data as dipole L 12, as indicated for near-equatorial locations by the central horizontal dashed line in Figure 2b. Core region oscillation phase and amplitude values can then generally be obtained from these Revs for all three field components, the data first having the internal planetary field subtracted followed by band-pass filtering to extract the planetary period oscillations (see section 4.1). Exceptions occur due to data gaps, and on Revs in which the perturbation fields of the steady ring current vary due to spacecraft motion on time scales comparable to the oscillations, these data then being excluded from further analysis. The latter issue may effect both r and q components, but not φ, because the ring current within the core, carried principally by cool Enceladus plasma, is closely axisymmetric and produces essentially no azimuthal field [e.g., Bunce et al., 2007; Kellett et al., 2011]. In practice, it is mainly the r component that is affected on Revs that are modestly inclined with respect to the equatorial plane [e.g., Andrews et al., 2008]. [15] These near-equatorial intervals are then separated by two intervals of inclined orbits that reach high latitudes, shown by the yellow stripes in Figure 2 labeled B and D. These intervals then contain the Revs from which oscillation phase and amplitude data can be obtained on open polar field lines, identified by requiring the warm electron flux between energies of 50 to 500 ev to remain close to background for intervals at least comparable to the planetary period [e.g., Andrews et al., 2010b]. These data are limited to radial distances of 25 R S (upper dashed line in Figure 2b), thus excluding data from the more distant tail, and are screened to eliminate magnetosheath and solar wind intervals. In general, oscillation data can again be obtained from all three field components, except in cases affected by ring current field perturbations extending to high latitudes, when data are derived only from the φ component. In addition, it is also possible on such orbits to determine oscillation data from briefer few-hour inclined traversals through the core region, the determinations then being obtained from unfiltered φ component data only, due to the presence of variable ring current fields in both the r and q components. In such cases the region employed is restricted to dipole L 6, indicated by the lower dashed line in Figure 2b, in order to clearly avoid the high-latitude region where the principal auroral field-aligned current systems flow, leading to strong spatial structuring of the azimuthal fields [see, e.g., Andrews et al., 2010b, Figure 3]. Such core region φ component data were consistently obtained during the first part of interval D, denoted D1 in Figure 2, but not in subsequent interval D2 delimited by the vertical orange dashed line after the spacecraft periapsis was raised as shown in Figure 2b. For further discussion of data availability and limitations [see Andrews et al., 2010b, Figure 4]. [16] Figures 2c 2f then show the variation of relevant physical parameters. Figure 2c shows the latitude of the Sun at Saturn, varying over the range 26.7 during each Saturn year of 29.5 terrestrial years. As indicated in sections 1and 2, southern summer conditions prevailed at the start of the interval, solstice occurring at t = 431 days (27 October 2002) somewhat off the plot to the left. We then move through vernal equinox where the latitude of the Sun passes through zero at t = 2049 days (11 August 2009) marked by the vertical black dotted line, into northern spring at the end of the interval. Northern summer solstice will occur at t = 4893 days (25 May 2017), well off the plot to the right. Figure 2d then shows the periods of the southern (red) and northern (blue) SKR modulations determined by Lamy [2011] (the values determined by Gurnett et al. [2011] do not differ significantly over the somewhat more restricted ranges of their present availability). These show relatively constant values of 10.6 h for the northern modulations and 10.8 h for the southern during southern summer, before more rapidly converging toward a near-common value of 10.7 h over a 1 year interval centered near equinox. The black dashed line in Figure 2d shows the mean of the two SKR periods, approximately constant at 10.7 h, while Figure 2e shows their difference Dt SKR = t SKR s t SKR n ( s for south and n for north throughout). It can thus be seen that intervals A-D1 correspond to conditions of wellseparated northern and southern SKR modulation periods, with D1 occurring just prior to the interval of rapid SKR period convergence. No core data are available during the first part of the convergence interval up to the vernal equinox, corresponding to D2, but become available once more in interval E starting close to equinox and extending to the end of the interval considered here, beyond that of the presently available SKR data. [17] The beat period t SKR B of the southern and northern oscillations as determined from the SKR periods is shown by the black line in Figure 2f, given by the difference in frequencies t SKRB = t SKR n t SKR s as indicated in section 2.2, thus yielding t SKR B =(t SKR n t SKR s )/Dt SKR. This is seen to be near-constant at 20 days during the southern summer interval as also indicated in section 2.2, but then increases to markedly larger values as the periods converge across equinox. As indicated above, the black dots in this panel also show the orbital period of Cassini, plotted for each Rev at the time of periapsis as in Figures 2a and 2b. It can be seen that the beat period is comparable with the orbital period during intervals A and C, such that pass-to-pass phase and amplitude variations determined from few-day core region periapsis pass data appear as jitter under such circumstances. However, during at least the early part of interval E the SKR data suggest beat periods that become much longer than the orbital period, such that the phase and amplitude variations may then become resolved. We note that the phase data analyzed in these terms by Provan et al. [2011] correspond to interval A. Here we thus extend this analysis to the whole 6of31

7 interval shown in Figure 2, within the limits of data availability. 4. Core Region Oscillations During Southern Summer and Post-Equinox Conditions [18] We now illustrate the nature of the field oscillations observed within the core region on individual periapsis passes, and describe the techniques employed to determine phase and amplitude parameters. We also demonstrate the significant differences in oscillation properties for southern summer conditions reported, e.g., by Southwood and Kivelson [2007], Andrews et al. [2008], and Provan et al. [2009a], and those newly reported here for post-equinox conditions Southern Summer Conditions [19] In Figure 3a we first show field data from two consecutive periapsis passes during southern summer conditions, specifically for Revs 17 and 18 in the left and right plots as indicated, occurring near the middle of interval A in Figure 2. Each plot shows four days of data centered on periapsis, which occurred on days 302 and 331 of 2005, respectively, thus 30 days and 1.5 beat periods apart. Spacecraft radial distance, latitude, and LT information is given at the bottom of each plot. Pairs of panels in each plot show residual and band-pass filtered magnetic field data for each spherical polar field component as indicated. The residual data have the Cassini SOI model internal planetary field subtracted [Dougherty et al., 2005], though use of any of the usual empirical models would not significantly affect the results. Since the model field is axisymmetric, with zero azimuthal component, we note that the B φ data shown in the bottom two panels of each plot is that directly measured. Planetary-period field oscillations at few-nt amplitudes are evident throughout both passes, superposed in the case of the residual q component on a large more slowly varying negative perturbation that is due to the ring current. To isolate signals in the planetary-period band the residual data are band-pass filtered between 5 and 20 h using a standard Lanczos filter. The filtered data are shown in the panels beneath the residual data for each field component. [20] The core region intervals in Figure 3a are those between the black dashed lines in each plot, defined here in common with previously related studies as the region with dipole L 12, as indicated in section 3. The filtered residual data for each field component i within these intervals have then been least squares fitted to the function B i ðφ; t Þ ¼ B 0i cos F g ðþ φ t y i ; ð1aþ where F g (t) is some suitably chosen guide phase whose related period is close to the period of the field oscillations, and φ is azimuth measured from noon positive toward dusk (i.e., increasing in the sense of planetary rotation). A list of principal mathematical symbols employed here is provided for easy reference in the notation section at the end of the paper. The amplitude B 0i and relative phase y i (modulo 360 ) of the observed field oscillations are then determined from the least squares fit. In Figure 3a we have chosen the guide phase to correspond to a fixed oscillation period t g = 10.7 h, which according to Figure 2 is close to the mean period of the northern and southern systems throughout. This phase is thus given by F g ðþ¼ t 360 t deg; ð1bþ t g where the zero of time t is defined as in section 3 and we have made the arbitrary choice F g (0) = 0 deg. The phase differences between the components then indicate the polarization of the oscillations in the three field components as discussed in section 2, while the variations of these phases from pass to pass is governed by the difference between the field oscillation period and the guide phase period. Thus if the chosen guide phase is F g (t) and the actual oscillation phase for field component i is F i (t), then the variation of phase parameter y i obtained from pass-to-pass fits to the field data using this guide phase will follow y i ðþ t F g ðþ F t i ðþ; t such that the same F i (t) will be determined from the chosen F g (t) and measured y i (t) in each case. The results will therefore be independent of the choice of the guide phase provided that the period corresponding to the guide phase is not too far away from the actual oscillation period. An appropriate limit on the difference in periods may be obtained from the consideration that the pass-to-pass change in y i should not be susceptible to the modulo 360 uncertainty inherent in the determination of this phase angle, i.e., that it should be comparable with or smaller than 360. Thus if the periods associated with the guide phase and the field oscillation are t g and t i (over some interval of time), and the interval between determinations is the orbital period t O, the limiting difference in periods is Dt lim ¼ t i t g lim t gt i : ð1cþ t O With t i t g 10.7 h and t O 20 days (Figure 2f) we thus find Dt lim 0.24 h. This limit should be well satisfied for a guide period of 10.7 h if the SKR periods are indicative (as is indeed the case), in which case typically Dt < 0.1 h < Dt lim. This limit is also well satisfied for the other guide phases employed for particular purposes later in section 6. In this case we further note that the variation in y i occurring within each periapsis interval of 1.5 days used for data fitting is then also automatically small, typically 10 or less. [21] Thus using a guide phase of 10.7 h period, the fitted model curves are shown by the red lines in the filtered data panels, encompassing 3 oscillations of the field for both Revs. Vertical blue dot-dashed lines have been drawn through the peaks in the fitted r component, so that the phase relations with the oscillations in the other components can readily be examined. It is first seen that the oscillations in the φ component are close to lagging quadrature with those in r in both cases, such that, as discussed in section 2.1, the peaks in φ occur one quarter cycle later than those in r. Quantitatively, the phase difference (y φ y r ) of the fits is +91 for both Revs 17 and 18. Uncertainties in such phase determinations are difficult to assign robustly on an individual basis. However, empirical results presented in section 4.3 suggest overall uncertainties in such phase differences of 7of31

8 Figure 3a. Periapsis pass data from Cassini Revs 17 and 18 (left and right panels) during southern summer interval A, showing four days of data in each case centered on periapsis. From top to bottom in each plot we show residual and band-pass filtered magnetic data (nt) for the r, q, and φ field components as indicated. The residual data (DB i for field component i) have the Cassini SOI internal planetary field model subtracted, while the filtered residual data have also been band-pass filtered between 5 and 20 h to extract the planetary-period signal. The interval between the pair of vertical black dashed lines corresponds to the core region defined by dipole L 12, inside which the data have been least squares fitted to the model field given by equation (1a) using a guide phase corresponding to a fixed period of 10.7 h as given by equation (1b). The fitted model is shown by the red lines superposed on the filtered residual data. The vertical blue dot-dashed lines indicate the position of the maxima in the fitted r component model, allowing the relative phases of the two other components to be readily examined. Values given at the foot of the figure indicate the time (year/day of year (DOY)), and the spacecraft radial distance (R S ), latitude (deg), and LT (h). 10, with 7 in individual values (comparable with the phase variations over the fitting interval as indicated above), these estimates having the nature of upper limits. It can also be seen in Figure 3a that the oscillations in the q component are to a first approximation in phase with those in r, in agreement with the results of Andrews et al. [2008], while closer inspection shows a somewhat lagging phase for Rev 17 and a somewhat leading phase for Rev 18. The corresponding phase differences (y r y q ) are 45 for Rev 17 and +50 for Rev 18. These deviations correspond to the phase jitter effect analyzed by Provan et al. [2011], with peak phase deviations of 25 for each component, in opposite senses for (r, φ) and q, giving rise to peak deviations in the phase differences of 50. Revs 17 and 18 happen to lie close to the condition of peak phase deviations in opposite senses, 1.5 beat periods apart, being chosen for display on that basis (see Figure 4a introduced in section 4.3). The superposed southern and northern system r and φ components maintain their close quadrature relation under these circumstances, however, since both components are deviated by approximately the same amount and in the same sense as each other. [22] We also note that the fitted amplitudes in each case are 1 nt for r, 2 nt for q, and 1.5 nt for φ. These values are comparable to those derived by Andrews et al. [2010a] from a wider statistical study of such data Post-Equinox Conditions [23] Pass-bypass examination of the core region field data obtained during post-equinox interval E immediately demonstrates differing oscillation characteristics compared with the southern summer data in intervals A and C. This is illustrated in Figure 3b with data from four periapsis passes, having the same format as Figure 3a. The top two plots show 8of31

9 Figure 3b. Periapsis pass data from four post-equinox Revs during interval E, i.e., Revs 120, 126, 142, and 146 as indicated. The format is the same as Figure 3a. 9of31

10 Figure 4 10 of 31

11 data from Revs 120 and 126, with periapses on days 306 of 2009 and 44 of 2010, respectively. In these cases we again observe oscillations in all three field components with amplitudes comparable to those in Figure 3a. We also see that the oscillations in the φ component are again approximately in lagging quadrature with r, with (y φ y r ) values of 77 for Rev 120, and 95 for Rev 126. However, far from being approximately in phase with r, the q component oscillations are now approximately in lagging quadrature with r for Rev 120, and hence approximately in phase with φ, while being approximately in leading quadrature with r for Rev 126, and hence approximately in anti-phase with φ. Specifically, the phase differences (y r y q ) are 70 for Rev 120 and +95 for Rev 126. These deviations in effect represent an extreme form of the jitter phenomenon for nearequal amplitudes of the two oscillations, when the phase difference between r and q remains fixed near values of +90 and 90 during the two halves of the beat cycle as indicated in section 2.2, as will be proven in section 5. Such observations thus suggest that the relative amplitude of the northern-period oscillations within the core has grown to become comparable with that of the southern-period oscillations during the equinox interval. [24] If so, we may also expect to observe significant effects in the oscillation amplitudes, specifically passes in which the amplitudes of both the r and φ components, but not q, cancel to small values, and vice versa, these being the points about which the above phase differences switch sign each half beat cycle. Such examples are shown in the lower two plots in Figure 3b, from Revs 142 and 146, with periapses on days 354 of 2010 and days 79 of 2011, respectively. For Rev 142 the amplitude of the q component has a typical value of 1.8 nt, while only weak oscillations in r and φ are observed with fitted amplitudes of 0.3 and 0.2 nt, respectively. Similarly for Rev 146 the amplitudes of the r and φ components have usual values of 1.3 and 2.4 nt, respectively, while only weak oscillations are observed in q, with a fitted amplitude of 0.3 nt Overview of Phase Difference Data [25] We conclude this section with an overview of the phase difference data obtained as illustrated in Figure 3a from all the near-equatorial Revs over the interval considered here. The amplitude data will be analyzed in section 6. Results are shown in Figure 4, where for ease of visibility we have divided the overall interval into three equal intervals of 860 days each, shown from top to bottom of the figure. Year boundaries and Rev indicators are shown along the top of each plot in the same format as in Figure 2, while the colored bars underneath indicate the intervals of nearequatorial (blue) and highly inclined orbits (yellow) also as in Figure 2. It can be seen that Figure 4a corresponds essentially to interval A, Figure 4b contains intervals B, C, D1, and part of D2, while Figure 4c contains the remainder of D2 and interval E. The phase differences shown correspond only to the core region data obtained in intervals A, C, and E, since only φ component data are available for the core region in interval D1. [26] Figures 4a 4c (top) show the phase difference between the φ and r component oscillations (y φ y r ) (black dots). Almost without exception, these data are consistent with the value of +90 shown by the dotted line, such that φ is consistently in lagging quadrature with r as in the examples in Figure 3a, indicative of a rotating quasi-uniform field. The mean and standard deviation of such modulo 360 angular data, e.g., of a set K of phase values Dy k, is given by the complex sum 1 PK K e jdy k ¼ Re jdy, where Dy is the k¼1 p directional mean value, and s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lnR (in radians) is the circular standard deviation [see, e.g., Mardia and Jupp, 2000; Andrews et al., 2011, Appendix]. Application to the (y φ y r ) data in intervals A and E yields (after removal of two outliers in interval E indicative of additional unrelated effects) mean values of 85.3 and 92.0, respectively, with standard deviations of 9.1 and Since the variability in these values contains the random measurement errors as well as possible physical effects, these results indicate phase difference uncertainties of 10 or less, with consequent implied uncertainties in individual phase determinations of 7 or less. These are the values quoted in section 4.1 above. [27] Figures 4a 4c (bottom) show the phase difference (y r y q ) (red dots), combined in view of the above results with ((y φ 90 ) y q ) (blue dots), such that when both data values are available they often overlap. The black lines show a model phase difference that will be derived and discussed in section 6. It can be seen in intervals A and C in Figures 4a and 4b that these phase difference values scatter significantly about zero through typically 50, this being the jitter effect analyzed by Provan et al. [2011]. For interval A, for example, the mean and standard deviation are 4.1 and 48.8, respectively, the latter being comparable to the individual phase difference values for Revs 17 and 18 derived in section 4.1. However, during interval E in Figure 4c the magnitude of the deviations increases markedly, with many values concentrated around 90 (modulo 360 ) as illustrated in the upper panels of Figure 3b (We note that for later convenience of plotting the phase model, the 360 range shown in Figure 4c is centered on 180, rather than on 0 as in Figures 4a and 4b). The mean and standard deviation of these data are 35.1 and 100.3, respectively, though in view Figure 4. Overview of phase difference data obtained from near-equatorial core region passes in intervals A, C, and E. (a c) The overall interval shown in Figure 2 has for this purpose been divided into three equal intervals. Year boundaries and Rev indicators are shown along the top of each plot in the same format as Figure 2, while the colored bars underneath indicate the intervals of near-equatorial and highly inclined orbits labeled A-E, also as in Figure 2. The top data panel in Figures 4a 4c shows the (y φ y r ) (deg) phase difference data (black dots), over one phase cycle centered on +90. The bottom data panel in Figures 4a 4c similarly shows (y r y q ) (red dots) and ((y φ 90 ) y q ) (blue dots), again shown over one phase cycle centered on 0 in Figures 4a and 4b, and on +180 in Figure 4c. The black lines in these panels show the phase difference model derived in section 6, based on equation (6b). 11 of 31

12 of the magnitude of the standard deviation, the value of the mean is of limited significance. 5. Theoretical Analysis 5.1. Derivation of Phase and Amplitude Modulation Formulae [28] We now derive the theoretical formulae required to analyze the phase and amplitude values obtained from the magnetic field data. Following Provan et al. [2011], we consider the effect of adding two sinusoidal field oscillations that rotate around the planet with differing periods and amplitudes, taking account of the differing polarization of the field components in the two systems as discussed in relation to Figure 1 in section 2.1. As in equation (1a), we thus take the northern (n) and southern (s) system oscillations for field component i to be the real parts of the complex expressions B in ðφ; t Þ ¼ B 0in e jðfinðþ φ t Þ and B is ðφ; tþ ¼ B 0is e jðfis t ðþ φ Þ ; ð2aþ where φ is again azimuth measured from noon positive toward dusk, and the phase functions for component i are given by F in ðþ¼f t n ðþ g t in and F is ðþ¼f t s ðþ g t is : ð2bþ Here F n (t) and F s (t) are the overall magnetic phase functions for the northern and southern systems, respectively, which define the corresponding rotation periods t n; s ðþ¼ t 360 ; ð2cþ F n; s ðþ t d dt where the phase functions are expressed in degrees. The fixed component phase angles g in,s in equation (2b) define the relative phases of the oscillations in the three field components in the two systems, one of which in each system may be assigned arbitrarily. Here we define the phase angles of the r components to be identically zero g rs ¼ g rn 0; ð2dþ such that F n (t) and F s (t) then correspond by definition to the phases of the northern and southern r field components. These phases thus also give the azimuth φ = F n,s (t) (modulo 360 ) at time t of the directions of both the quasi-uniform field in the core region and the effective transverse dipole describing the polar field perturbations in the two systems (see Figure 1). From the discussion in section 2.1 we then have for the remaining components g φ s ¼ g φ n ¼ 90 ; ð2eþ such that φ is in lagging quadrature with r in both systems corresponding to rotating quasi-uniform fields in the equatorial plane, while g q s ¼ 0 and g q n ¼ 180 ; ð2fþ such that q is in phase with r for the southern system while being in anti-phase with r for the northern system. [29] The combined oscillation in field component i is then given by summing the contributions in equation (2a) B i ðφ; t Þ ¼ B 0is e jðfis t ¼ B 0i t ðþ φ ðþe j F i ðþ φ t Þ þ B 0in e jðfin t ðþ φþ ð Þ : ð3aþ Eliminating the common factor e jφ and putting B 0in ¼ kb 0is ; ð3bþ where for simplicity we assume that the northern over southern amplitude ratio k is the same for all three field components, the amplitude B 0i (t) and phase F i (t) of the combined oscillations are given by the expression B 0i ðþe t jf i ðþ t ¼ B 0is e jfisðþ t þ ke jfinðþ t : ð3cþ We first evaluate equation (3c) to determine the phase and amplitude modulations of individual field components. With regard to the phase modulation, if we take the complex conjugate of equation (3c) and multiply it by e jfis(t), then the real and imaginary parts of the resulting expression yield the following result for field component i for the combined phase F i (t) relative to the southern phase F is (t) d is ðþ¼f t is ðþ F t i ðþ¼ t tan 1 k sindf i 1 þ k cosdf i : ð4aþ Here the sign convention of the difference has been taken to agree with equation (1a), as becomes clear in section 5.2. We also note that since B 0i (t) and B 0is are both positive definite quantities in equation (3c), we take the signs of the numerator and denominator separately within the arctangent in equation (4a) to define its value over the full angular range of 360. In this expression DF i ðþ¼f t in ðþ F t is ðþ¼df t ðþ t ðg in g is Þ; ð4bþ where we have employed equation (2b) and DFðÞ¼F t n ðþ F t s ðþ t ð4cþ is the beat phase between the two oscillations. The beat period t B, given as in section 3 by the difference in frequencies t B 1 = t n 1 t s 1, is then t B ðþ¼ t t st n ðt s t n Þ ¼ 360 ; ð4dþ d ðdfðþ t Þ where the phase angles are again in degrees. Similarly, taking the complex conjugate of equation (3c) and multiplying it by e jfin(t) yields the following equivalent expression for the combined phase F i (t) relative to the northern phase F in (t) d in ðþ¼f t in ðþ F t i ðþ¼ t tan 1 ð1=kþsindf i ; ð4eþ 1 þ ð1=kþcosdf i where again we take the signs of the numerator and denominator within the arctangent separately to define its value over the full 360 range. Multiplying equation (3c) by its complex conjugate also yields the following expression for the dt 12 of 31

13 amplitude of the combined oscillation B 0i (t) relative to either the northern- or southern-period oscillations A is ðþ¼ t B 0i B 0i p ¼ k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ k 2 þ 2k cosdf i : ð4fþ B 0is B 0in From these general formulae we can then derive expressions for the combined phase and amplitude of the individual field components, noting from equations (2d) (2f) that ðg rn g r s Þ ¼ g φ n g φ s ¼ 0 while ðg q n g q s Þ ¼ 180 : ð5aþ From equations (4a), (4e), and (4f) we thus have for both r and φ components d r; φ s ðþ¼f t r;φ s ðþ F t r; φ ðþ¼ t tan 1 k sindf ; ð5bþ 1 þ k cosdf d r; φ n ðþ¼f t r;φ n ðþ F t r; φ ðþ¼ t tan 1 ð1=kþsindf ; ð5cþ 1 þ ð1=kþcosdf and A r; φ s ðþ¼ t B 0r; φ ¼ k B 0r; φ s B 0r; φ B 0r; φ n p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ k 2 þ 2k cosdf; ð5dþ while for the q component d q s ðþ¼f t q s ðþ F t q ðþ¼ t tan 1 k sindf ; ð5eþ 1 k cosdf and d q n ðþ¼f t q n ðþ F t q ðþ¼ t tan 1 ð1=kþsindf ; ð5fþ 1 ð1=kþcosdf A q s ðþ¼ t B 0q B 0q s ¼ k B 0q B 0qn p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ k 2 2k cosdf: ð5gþ [30] For comparison with the results shown in Figure 4, it is also of interest to derive compact expressions for the phase differences between the three field components for the combined oscillations. These are conveniently obtained by multiplying equation (3c) for one field component by the complex conjugate of the same equation for the other. We then extract the real and imaginary parts to determine the sine and cosine, and hence the tangent, of the phase difference. For the r and φ components this simply gives F r ðþ F t φ ðþ¼90 t ; ð6aþ since with the same value of k for the two field components and the same relative phases of the two components in the southern and northern systems as given by equations (2d) and (2e), the phase perturbations in these two components are equal to each other, such that the combined oscillations remain in quadrature, as was shown to be the case in the data in Figure 4. However, for the (r, φ) and q components we find F r ðþ F t q ðþ t ¼ F φ ðtþ F q ðtþ þ k sindf ¼ tan 1 k 2 ; ð6bþ where again the signs of the numerator and denominator in the arctangent function are taken separately to define the phase difference over the full 360 range Relation to Phases Determined From Cassini Data [31] As discussed in section 4, the phases y i obtained from Cassini data are determined by fitting the cosine function given by equation (1a) to the core region filtered residual field data, expressed in terms of some suitably chosen exact guide phase F g (t). Comparison of equations (1a) and (3a) then shows that the y i are related to the phase functions defined above by F i ðþ¼f t g ðþ y t i ðþ; t ð7aþ where y i (t) represents the phase values for field component i determined from pass to pass over time. Using equations (4a) and (4e) we then have y i ðþ¼f t g ðþ F t i ðþ¼ t F g ðþ F t is ðþ t þ dis ðþ t ð7bþ ¼ F g ðþ F t in ðþ t þ din ðþ: t ð7cþ In the latter two expressions the initial term within the brackets represents a phase difference that varies smoothly with time depending on the difference in period between the guide phase period and the southern or northern field period, while the remaining term varies with the beat period as given by equations (4a) and (4e). If we choose the guide phase to be equal to that of the southern field oscillation, i.e., F g (t)=f is (t), then given the sign convention in equation (4a) we have y i ðþ¼y t is ðþ¼d t is ðþ; t ð7dþ while if we choose the guide phase to be equal to that of the northern field oscillation, i.e., F g (t)=f in (t), we similarly have y i ðþ¼y t in ðþ¼d t in ðþ: t ð7eþ Irrespective of the choice of F g (t), the phase difference between field components p and q is y p ðþ y t q ðþ¼ t F p ðþ F t q ðþ t ; ð7fþ where the right side is given by equations (6a) and (6b), and the negative sign should be recalled in later discussion Model Implications [32] Implications of the theoretical model are illustrated in Figure 5, where we plot phase and amplitude modulation parameters versus the beat phase DF = F n F s. We note that DF increases with time if the northern period is shorter than the southern, as applies to most of the interval as indicated by the SKR data in Figure 2. However, should the two 13 of 31

14 Figure 5. Plots illustrating the theoretical results of section 5, each row corresponding to a given parameter of the combined southern and northern oscillations plotted versus the beat phase DF (deg), while each column corresponds to a given north over south amplitude ratio k, from left to right equal to 0.4, 0.8, 1.0, 1.25, and 2.5. The rows show (a) the phase deviation y is of individual field components i relative to the southern oscillation phases, red for r and φ, and green for q, (b) the corresponding phase deviations y in relative to the northern oscillation phases, (c) the phase difference between the r and q field components (y r y q ), the value of (y φ y q ) being just 90 larger, and (d) the oscillation amplitudes relative to the southern oscillation, red for r and φ, and green for q, the amplitude relative to the northern oscillation being (1/k) times these values. periods cross, reversing the inequality, it will then decrease with time. Successive columns from left to right in Figure 5 show results for northern over southern amplitude ratio k equal to 0.4, 0.8, 1.0, 1.25, and 2.5, respectively. The first of these values is typical of the southern-dominated preequinox conditions studied by Provan et al. [2011], while the last two values are the reciprocals of the first two, corresponding to equivalent northern dominance. The central column then represents the important special case of equal amplitudes. [33] The red and green lines in Figure 5a show the phase deviations of the (r, φ) and q component oscillations, respectively, relative to the southern phase, i.e., y r,φ s and y qs given by equations (5b) and (5e). For modest k as shown in the first column, the variations are near-sinusoidal over the beat cycle given approximately by k sin DF (in radians). However, as k increases toward unity the growing variations become increasingly asymmetric, until at k = 1 the phase deviations in both components decrease linearly with DF between +90 and 90, switching abruptly between these values at DF = 180 for (r, φ) and at DF =0 for q. For equal amplitudes of the two oscillations, therefore, the phases of the combined oscillations raster linearly between 90 about the southern phase during each beat period, i.e., over half the full phase range, with the variations in (r, φ) and q being out of phase by a half beat period. As k increases beyond unity, corresponding to dominant northern oscillations, the deviations from the southern phase then vary over the full 360 range during each beat period, though for k only modestly larger than unity, as for k = 1.25 in the fourth column, they are still partly organized relative to the southern phase over the beat period, falling rapidly from 90 to 14 of 31

15 +90 via 180 near DF = 180 for the (r, φ) components and near DF =0 for q, and then decreasing more slowly from +90 and 90 over the remainder of the beat period. [34] Figure 5b then similarly shows the phases of the combined oscillations relative to the northern phase, i.e., y r,φ n and y q n given by equations (5c) and (5f), which are seen to exhibit exactly equivalent behavior to Figure 5a, but with the sign of the phases and the sequence of k values reversed. In particular, when k = 1 the phase deviations in both components increase linearly with DF between 90 and +90, switching abruptly between these values at DF = 180 for (r, φ) and at DF =0 for q. Thus as k approaches unity, the phases of the combined oscillations raster linearly between 90 about both the northern and southern phases during each beat cycle, one rising and the other falling with the beat phase. This suggests that the phases of both oscillations can be recovered from such phase data when the oscillation amplitudes are comparable, as will be discussed further in section 5.4 below. [35] Figure 5c then shows the phase difference between the r and q field components, (y r y q ) given by the negative of equation (6b), equivalent to the difference between the red and green lines in Figures 5a and 5b (with 180 added to y q in the latter). Separate plots of (y φ y q ) are not shown since its value is simply 90 larger than (y r y q ), and neither are plots of (y φ y r ) equal to +90 throughout. For small k the phase difference (y r y q ) varies approximately as 2k sin DF (in radians) about 0, while for large k it similarly varies as (2/k)sin DF about 180, the 180 shift reflecting the corresponding phase difference between the (r, φ) and q field oscillations for the northern and southern systems. However, when k is close to unity, the nearsinusoidal variations transform into near-square wave variations of amplitude 90 about 0 and 180, corresponding to the near-linear variation of the individual phases in Figures 5a and 5b. Specifically, for the half beat period with 0 DF 180 we have (y r y q )= 90 (hence (y φ y q )=0 ), such that q is in lagging quadrature with r and hence in phase with φ, while for the other half beat period with 180 DF 360 we have (y r y q ) = +90 (hence (y φ y q ) = 180 ), such that q is in leading quadrature with r and hence in anti-phase with φ. These are the altered polarization states of the combined oscillations characteristic of near-equal amplitudes of the northern and southern systems, switching each half beat cycle, which were anticipated in section 2.2 and demonstrated in the data in Figure 3b. [36] The red and green lines in Figure 5d show the amplitudes of the combined (r, φ) and q oscillations relative to the southern amplitude (equations (5d) and (5g)), respectively, the amplitudes relative to the northern amplitude being simply (1/k) times these values. The modulated amplitudes vary between maximum and minimum values of (1 + k) and 1 k in each case, thus reaching a global minimum of zero corresponding to complete cancellation when k = 1. As anticipated in the discussion in section 2.2, we note that the (r, φ) amplitudes reach a maximum when the q amplitude reaches a minimum, at DF =0, and vice versa at DF = 180. We also note that the intervals of rapid phase variation for a particular field component when k is close to unity seen in Figures 5a and 5b occur close to the minima in amplitude for that component Determination of Southern and Northern Oscillation Phases [37] In this section we now demonstrate the principles, based on the above analysis, that allow the determination of both northern and southern phases, F n (t) and F s (t), independently from the empirical magnetic phase data. This is illustrated in Figure 6 through a simple specific example, where we show the phase variations versus time for two combined oscillations of fixed period, 10.8 h for the southern oscillation (r and q in phase) and 10.6 h for the northern (r and q in anti-phase), relative to a guide phase whose frequency is the mean of these two frequencies, with a consequent period close to 10.7 h. Two cycles of phase are shown in each plot to emphasize the modulo 360 continuity of the phase values, such that each value is plotted just twice, and each plot encompasses four beat cycles of days each. Results for increasing values of amplitude ratio k are shown from the top to the bottom of the figure, using the same values as employed in Figure 5. [38] Given the above choice of guide phase, the rising dashed lines in each plot then correspond to the r component phase for southern oscillations alone, equal to F g (t) F s (t) (see equation (7b)), while the falling dot-dashed lines of equal negative slope correspond to the r component phase for northern oscillations alone, equal to F g (t) F n (t) (see equation (7c)). The φ component phases are then +90 relative to r for both systems, while the q component phases are equal to r for the southern system while being displaced by 180 (modulo 360 ) for the northern system. To determine the phase versus time of southern system data alone we would thus plot y r and y q values as measured relative to the guide phase, while subtracting 90 from the y φ values to bring them to a common value, and then fit the resulting values versus time in some suitable way (addressed below) to determine F g (t) F s (t), and hence F s (t). This is the data format adopted in Figure 6 (left), where the red lines show the phase of the combined oscillations for both the r component and the φ component with 90 subtracted (equation (7b) with (5b)), while the green lines show the phase of the combined oscillations for the q component (equation (7b) with (5e)). The relation of these lines for each k relative to the rising dashed lines during each beat cycle is then exactly that of the corresponding red and green lines in Figure 5a. [39] Similarly, to determine the phase versus time of northern system data alone we would plot y r as measured relative to the guide phase while subtracting 90 from y φ as before, but now shifting y q by 180 to again bring all the data to a common value, and then fit the resulting phase values versus time to determine F g (t) F n (t), and hence F n (t). This is the data format adopted in Figure 6 (right), where we emphasize that the only change relative to the corresponding plots on the left is that the q component phase data have been shifted by 180 to reflect the opposite polarization of q relative to r in the northern system. The red and green lines in the latter plots for each k then have exactly the same relation to the falling dot-dashed lines during every beat cycle as the corresponding red and green lines in Figure 5b. [40] If we then examine the results shown in Figure 6 (left), it is evident that a suitable fit to the phase data for k = 0.4 in Figure 6a over at least one beat cycle would yield 15 of 31

16 Figure 6. (a e) The independent determination of southern and northern oscillation phases from combined oscillation phase data. For definiteness, the northern and southern oscillations are taken to have fixed periods of 10.6 and 10.8 h, respectively, while the northern over southern amplitude ratio k varies in Figures 6a 6e through the values 0.4, 0.8, 1.0, 1.25, and 2.5, as employed in Figure 5. Each plot shows the oscillation phase y i of the combined oscillations of field component i versus time t, relative to a guide phase corresponding to a fixed period whose frequency is the mean of the two oscillation frequencies, the period thus being close to 10.7 h. The rising black dashed lines correspond to the phase of southern period oscillations alone (specifically for the r component), while the falling black dot-dashed lines of equal negative slope correspond to the phase of the northern period oscillations alone (again specifically for the r component). The red lines correspond to the phases of the combined oscillations for both r and φ components, the latter with 90 subtracted to account for the lagging quadrature, while the green lines correspond to the phases of the combined oscillations for the q component. These are shown in southern format in the left column, i.e., with r and q as measured, but in northern format in the right column, i.e., with r as measured but with q shifted by 180. Two phase cycles are shown on the vertical axes of each plot, each line thus being shown just twice at a given time, with each plot encompassing four beat cycles of days in time. Dots are drawn at equal intervals of time along the colored lines to represent idealized phase data, thus emphasizing the banding of these data about the southern phase in the left column, and about the northern phase in the right column. 16 of 31

17 the southern phase versus time relative to the guide phase, shown by the rising dashed lines. This evidently remains true for k = 0.8 and 1.0 in Figures 6b and 6c, even though in the latter case the phase values vary linearly by 90 about the southern phase during each beat cycle. When the amplitude of the northern oscillations exceeds the southern for k = 1/0.8 = 1.25 in Figure 6d, the phases of the combined oscillations then vary across the full 360 range relative to the southern phase. Nevertheless the values are still clustered about the southern phase, as emphasized by the dots plotted on the lines representing idealized spot values determined at equal increments of time. A suitable fit to a sufficient interval of such data encompassing several beat cycles will thus again yield the southern phase. The same remains true in principle for even larger k = 1/0.4 = 2.5 in row (e), though it is clear that the phase banding effect will eventually become masked in the presence of measurement uncertainties in the phase data. [41] If we then examine Figure 6 (right), it is evident we can make exactly the same commentary concerning these data in relation to the northern phase shown relative to the guide phase by the falling dot-dashed lines, if taken in reverse order from Figure 6e Figure 6a. In particular, for k = 1.0 the phase data again vary linearly by 90 about the northern phase during each beat cycle, but with the opposite relative slope as in the left column. It is thus clear that the southern phase can be determined from data plotted in the format of the left column, as for southern oscillations alone, provided k is not too large compared with unity, while the northern phase can similarly be determined from data plotted in the format of the right column, as for northern oscillations alone, provided k is not too small compared with unity, and that both can be determined simultaneously for some intermediate range of k around unity. In principle, of course, the phase banding effect is present in the data for each field component separately, as can be seen from the red and green lines considered individually in Figure 6. However, the southern and northern phase values can most reliably be determined by combining together the phase data for all three field components, in southern format as in Figure 6 (left) to determine the southern phase, and in northern format as in Figure 6 (right) to determine the northern phase. [42] We now address the issue of obtaining the best linear fit y = at + b to a set of K phase data points y k at t k (in general including suitably shifted values for all three field components as discussed above), taking account of the modulo 360 nature of the data. Here we again follow the directional statistics approach of Mardia and Jupp [2000], a brief summary of relevant results being given by Andrews et al. [2011, Appendix]. A suitable measure of the variance V q0 of a set of K angles q k about some angle q 0 is given by V q0 ¼ 1 K K k¼1 ð1 cosðq k q 0 ÞÞ; ð8aþ which we note is equal to half the mean square deviation of the angles (in radians) from q 0 in the case of small deviations, and goes to a maximum value of unity when for every q k there is a corresponding value q k in the data set. We then wish to find the values of a and b that minimize the variance about zero of (y k (at k + b)), in other words the values that minimize V 0 ¼ 1 K K ð1 cosððy k at k Þ bþþ k¼1! ¼ 1 R þ 2R sin 2 ðy k at k Þ b ; ð8bþ 2 where R and ðy k at k Þ are the modulus and phase of the complex sum 1 K K e j ð y k at k Þ ¼ Re j ð y k at k Þ : ð8cþ k¼1 We note in equation (8c) that value of R lies between zero and unity, and ðy k at k Þ is the directional mean of the (y k at k ) values. It can then be seen from equation (8b) that for a given value of slope a, the minimum variance occurs when b ¼ ðy k at k Þ, and that this variance is 1 R given by equation (8c). The best fit is thus obtained by varying a in equation (8c) to determine the value that gives the maximum value of R, equivalent to the minimum value of the variance, with b then being given by the corresponding directional mean ðy k at k Þ. From equation (7) the corresponding magnetic phase function over the interval corresponding to the data is then F n;s ðþ¼f t g ðþ t ðat þ bþ; ð8dþ where F g (t) is the guide phase relative to which the phase data y k were determined. From equation (2c) the corresponding oscillation period over the interval is then given by t n;s ðþ¼ t 360 ; ð8eþ F g ðþ t a where the phases are again expressed in degrees. d dt 6. Data Analysis 6.1. Determination of Southern and Northern Phases and Periods [43] We now apply this discussion to the determination of the southern and northern oscillation phases and periods over the interval of our study. The results are shown in Figure 7, where Figures 7a and 7b relate to southern oscillations, and Figures 7c and 7d to northern oscillations. Figure 7a shows all the phase values that pertain to the southern oscillations relative to a guide phase corresponding to a fixed period of h, namely the core region data (solid circles) together with southern polar data (open circles). Two cycles of phase are again shown on the vertical axis to emphasize the modulo 360 continuity of the data, with each phase data point thus being plotted twice. Phase values for the r, q, and φ field components are plotted using red, green, and blue symbols, respectively, and are shown in southern format. Thus the r and q phases are shown as measured, while 90 has been subtracted from the core region φ phases to account for their lagging quadrature, and 90 has been added to the polar region φ phases to account 17 of 31

18 for their leading quadrature (see section 2.1 and Figure 1). The guide phase period of h was chosen as approximately central to the range of southern periods occurring over the interval, as indicated in Figure 2 and shown by the black dashed line in Figure 7b. [44] The linear fit analysis of section 5.4 has been applied to these data taken 25 points at a time, each such data set typically spanning a 200 day interval, similar to the comparable analysis of SKR modulation data by Lamy [2011]. Time is incremented by stepping the set of 25 points by one Figure 7 18 of 31