Wave Properties

3.6.3.1 Wave Number, Propagation Direction and Wavelength

The azimuthal and radial wave numbersky and kx determined by the model fitting can be used to determine the direction that the wave is propagating. Figure 3.17 shows the probability distribution built from 1-D kernel smoothing of the angle of propagation from the Sun-Saturn line and its uncertainty. An overwhelming majority of sectors display a positive radial skew with a small additional positive azimuthal component. This means that the majority of the waves in the bin are travelling radially away from the planet.

This probability distribution is tested against an isotropic distribution equal to the mean of each bin as null hypothesis. All sectors, except the noon sector, give a probability of less than 5% that the null hypothesis is correct. An additional check used is that the χ2 value of the isotropic distribution compared with the probability distribution is much greater than the mean of the distribution plus two standard deviations, this process is a standard check for the reliability and evaluation of aχ2 value. Below, table 3.2 shows these values to show that all sectors except noon reject the null hypothesis of isotropic propagation.

Waves in the noon sector are propagating in all directions, however the other sec- tors are all skewed towards a particular direction, that direction being radially outwards from Saturn. All radial bins of the evening sector give a peak of the distribution at

around 10◦ - 130◦ showing a radially outwards propagations direction, with a small ad- dition in the azimuthal direction with corotation. This is also true for the night sector where the global maximum of the distribution occurs at 210◦ (travelling down tail away from Saturn).

The morning sector has only one large global maximum in each radial bin, and outwards of 20 RS that direction is again radially outwards with an addition in the azimuthal direction. However, inwards of 20 RS the probability distribution peaks at 340◦, which is again in the direction of corotation.

The final sector, noon, shows lower event numbers coupled with multiple possible wave sources. Hence, this sector shows a much larger spread in the probability dis- tribution. Important to consider in this sector is that the magnetopause is generally seen between 21 and 26RS (Pilkington et al., 2015a) thus the majority of events in the >20RS bin are assumed to occur when the magnetopause is in an extremely expanded phase. This bin shows a large spread mainly across the 210−0◦ area. This bin is formed from events on the dusk side of the noon magnetosphere and hence this may show these events propagating perpendicular to the magnetopause. A small number of events are situated close to the morning sector, and these account for the two thinner peaks at 60◦ and 30◦ which also show propagation perpendicular to the magnetopause.

Table 3.2: Table ofχ2values, probability of null hypothesis being correct and whether

or not the null hypothesis was rejected for each section

Sector Radial Distance [RS] χ2 µ+ 2σ Probability Reject

Morning R≤20 409 13 <1% Yes - 20< R≤40 113 46 <1% Yes - R >40 549 10 <1% Yes Noon R≤20 472 12 39% No - 20< R≤40 347 14 9% No Evening R≤20 329 11 <1% Yes - 20< R≤40 440 96 <1% Yes - R >40 195 25 <1% Yes Night R≤20 144 31 <1% Yes - 20< R≤40 147 32 <1% Yes - R >40 160 29 <1% Yes

Inwards of 20RSa primarily bimodal distribution of inwards and outwards radially travelling waves is observed, however this bin is not statistically significantly different to an isotropic propagation distribution. These signatures suggest a possible magnetopause source for a percentage the waves found. However, the majority of all waves are prop- agating radially, concluding that a source of waves is found in the inner magnetosphere and propagate outwards, further discussed in section 3.6.5.

3.6.3.2 Amplitude

Figure 3.16 shows a radial dependence on the amplitude of waves in the morning and night sectors. The largest negative values of amplitude are found at large radial distances in the morning and night sectors, and the largest positive amplitudes are found in the evening sector. Initially, this may be due to the large scale bowl shape of the current sheet at the times the events are found. In the morning sector, the bowl is above the equator and so only negative values of amplitude may be seen, and conversely, the evening sector events are seen during 2010-2012 where the bowl is expected to be below the equator and hence a mixture of positive and negative amplitudes are seen as Cassini has a varied trajectory in ˆz.

This leads to the discussion of amplitude having a radial dependence due to the bowl shape, meaning that as the bowl shape causes the current sheet to become further from the equator, only increasingly larger amplitudes may be seen as the current sheet will need to be deformed more to be able to be seen at the equator, i.e. an observer bias is present. This seems likely due to the positive and negative dependence on the bowl position. However, during this time, Cassini is not constrained to the equator, but travels between +10RS and −10RS therefore smaller waves should still be sampled at larger radial distances. The only change will be that the bowl is more likely to be above or below Cassini, thus seeing more positive amplitudes in the evening sector (post-equinox, northern summer, downwards bowl), and more negative amplitudes in the morning and night sectors (pre-equinox, southern summer, upwards bowl). The conclusions here being that the radial dependence of the amplitude of the waves is not affected by the bowl shape, however the sign of the amplitude seen is. Additionally, this could mean that

almost double the number of waves seen could actually have been propagating at the time as only one sign of amplitude can be viewed at a specific time.

Additionally, now having ruled out the observer bias of an increasing amplitude, a physical interpretation is examined. To understand conceptually why the amplitude is increasing with radial distance, the physical analogy of a wave on a string or a wave upon water is used. A comparison is drawn from the physics of water shoaling, where in a dispersive medium the energy of a wave must remain constant. This energy is related to the linear mass density, amplitude and frequency of the aperiodic wave. As the amplitude is increasing, then the linear mass density or the frequency must decrease. Frequency has been shown to have no radial distribution in this study, and Arridge et al. (2011) shows a decrease in linear mass density with radial distance, hence it is suggested that amplitude increase is due to a medium that is decreasing in density.