An analysis of Pc5 pulsations observed in the SuperDARN radar data.

Where the work of others has been used, this is duly acknowledged in the text. The bottom two panels show the unwrapped phase of the analytical signal of each pulsation.

Historical outline

Hughes, in his two papers [33, 34] on the effect of the magnetosphere-ionosphere-atmosphere system, showed how disturbances in the magnetosphere would be modified as they propagate through the ionosphere and down to earth. Walker et al [72] addressed the limited spatial resolution shown by Hughes and Southwood by using the Scandinavian Dual Aurora Radar Experiment (STARE) to make in situ observations of the dynamics inside the ionosphere.

Problem statement

As noted in the previous section (6.8), the sampling period of the beams need not be uniform. In figure 12.4, the original data (in black) is plotted together with the Delaunay triangle (filled triangles) and the interpolated data (in blue).

Table 2.1 shows the limiting cases for situations where the wave is either travelling parallel to the ambient magnetic field (θ = 0 ◦ ) or travelling perpendicular (θ = 90 ◦ ) to the field.

What the thesis contains

Reduced MHD Equations

The thermodynamic nature of the plasma can be related to the rate of pressure change at. Finally, using a simplified version of Ohm's law and a low-speed version of Maxwell's equations, the time variation of the magnetic field is given as

Derivation of the wave equation

To that end, we will assume that the variations observed in our system are harmonic and of form. The form of the second equation in (2.14) is a quadratic inω2, which will give the well-known two-root solution.

Applicability

VA2+VS2 Table 2.1: Limiting cases for Alfv´en and magnetosonic phase velocity. 2.14) The form of the second equation in (2.14) is a quadratic inω2 which will give the known solution with two roots. The first follows from the first equation in (2.15) and is called the transverse Alfv'en wave.

Summary

Although too simplistic to be of any quantitative use in investigating pulsations observed with SuperDARN radars, this chapter provides a basic qualitative picture of the types of waves that can give rise to the observations. At the resonance point our model fails because some parameters describing the disturbance become singular.

The magnetosphere

In the previous chapter, the types of waves that will propagate in a homogeneous medium were defined. The boundaries in the x-direction are the magnetosphere (B-C-D-E-F in Figure 3.2) at a negative x-point and the equatorial ionosphere (A-G) in the positive x-direction.

The MHD equations for the magnetosphere

This means that the Alfv´en speed dominates over the magnetosonic speed and if we assume that the magnetic field is constant in the z-direction but dependent on x, (3.6) and (3.7) simplify to become.

Singularities in the magnetosphere

The turning point

The resonance point

Away from the singular points

Near the turning point

Near the resonance point

Equation (3.25) is of the form known as the zero-order modified Bessel equation with solutions I0(ζ) and K0(ζ). These two singularities (ie thatξ→ ∞andη→ ∞) indicate that there is a loss mechanism missing from our consideration of the system.

Applicability

At resonance, the RHS of equation (3.4) will be zero, so dψ/dx = 0, indicating that ψ is a constant.

Summary

At this point the disturbance behavior is approximated by the Airy function Ai. This is an unrealistic situation and indicates that the description of the system so far has neglected some loss mechanisms.

Introduction

Description of the standing wave perturbations

If we assume for the moment that the boundaries are infinitely conductive, the Eveld at the boundary must be zero. Spatial variation of the disturbances in the z-direction is controlled by exp{±ıkzz}, where the± is determined by the direction of propagation of the disturbance.

Ionospheric conductivity

Conductivity
Ideal plasma (collisionless)
Isotropic conductivity
Anisotropic conductivity
Conductivity variability

The motion of the charged particles will now be in the direction of the force of an applied electric field, so that. The actual current that will flow in the ionosphere is the height integration of the current density in equation (4.5).

Figure 4.2: The variation of Pedersen (S1) and Hall (S2) conductivity during 2002 for SANAE.

Ionospheric losses

Closing of the perturbations in the ionosphere

Pedersen currents have been shown to connect aligned field currents while Hall currents form closed loops around the edges of a field line. Similar to the Pedersen current, the Hall current will create a variable magnetic perturbation above and below the ionosphere asb+x and −b−x which are perpendicular to the field line perturbation.

Finite Pedersen conductivity

Ionospheric losses are then governed by the height-integrated Pedersen conductivity. Another consequence of the finite Pedersen conductivity is that if ki = 0, according to equation (4.9), Ex would be zero in the ionosphere.

Ionospheric filtering

In terms of the box model (section 3.3), the disturbance moved through the magnetosphere as a compression wave, coupled to a field line and created a transverse Alfv´en wave on that line. In Figure 12.5, the graph shows the difference in start times of the records for the overlapping cells.

Figure 5.1: Illustration of the various coordinates in a local dipole coordinate system, after Kivelson and Russell [36].

Applicability

Summary

The discussion in the previous chapters focused on a Cartesian coordinate system to find analytical solutions that could give a hint about the actual dynamics of the magnetosphere. These equations will be solved using numerical methods, the results of which will turn out to be not too far from the analytical solutions obtained in the previous chapters.

The dipole co-ordinate system

In this chapter, the problem will be approached in relation to the dipole coordinate system with the variables ν, µ and φ. This is a more accurate description of the magnetosphere, but leads to differential equations that do not have simple analytical solutions.

Dipole differential equations

Resonant perturbations

This differential equation along with the differential equation for Eν can be solved numerically using the Runge-Kutte process. Once Kr is determined, it can be increased until the resulting second order electric field matches the measured values.

Driven perturbations

Following the discussions in the box model, one can assume that Eν = 1 and Bφ = 0 at the equator. One can now vary K until Eν reaches zero at the latitude of interest, thus choosing the right K for that field line.

Figure 5.2: The variation of normalised frequency K with latitude.

Finding K i

A nonzero value of Ki was attributed to the dissipation of energy in the ionosphere, which prevented the singularity from occurring. AsKi increases so the amplitude decreases, indicating that some of the energy in the ionosphere has disappeared.

Applicability

Ki can be approximated to first order using the method outlined in Taylor and Walker [64] to get The Pedersen conductivity can be measured using incoherent scatter radars, while the variation of V withν can be measured using VLF waves or by in situ satellite measurements.

Summary

As this is the main instrument used in this analysis, a detailed description of the radar will be given. The following will be discussed: the operation of the radar, how the data is recorded, the types of data available for analysis, and finally, how the data can be used for Pc5 pulse analysis.

The SuperDARN network

If a PI has no specific use for the radar in discretionary mode, it is usually put in regular mode.

Radar targets

Ionosondes

Incoherent scatter radars

The indicated beam path of the HF signal is refracted and eventually orthogonal to the ionospheric irregularities. The discussion in this thesis focuses on the pulsation signatures observed in the scattering and not on the cause of the irregularities themselves.

Radar operation

The system components

Each delay will delay the phase of the signal by a predetermined amount depending on the direction of the beam. At the output of the phase matrix there are 16 channels with each signal having appropriate phases.

Received signal processing

Transmitted signal

The received signal

If the received signals are coherent, the signals can be summed to obtain a cleaner and more detectable signal. If the target were moving, there would be a phase difference between r1 and r2 that would be determined by the speed of the target and would manifest as a Doppler shift in the signal.

The timing

Transmitting before the last echoes have returned means that at any given time you may receive echoes from two different pulses, and from possibly two significantly different regions of the ionosphere. It is assumed that the region of the ionosphere of interest will remain coherent for the duration of the pulse pattern.

Figure 6.3: The staggered pulse pattern showing how the various ACF lags can be created from the seven pulses.

Doppler velocity

The rate of change of phase sequence is measured by observing the rate of change of phase of the successive pulses and is indicated by the Doppler shift of the signal. The Doppler velocity V depends on three terms [63], namely variations in the magnetic field and electron density and the movement of the ionosphere.

Angle of arrival

The ionospheric conditions mentioned above determine that the phase relationship between successive pulses received at separate antennas will be linear, i.e. the first term is the result of the Doppler velocity V, while the second is the instantaneous phase constant introduced due to the separation of the antennas.

The radar sounding procedure

Radar data

Applicability

The other issue regarding the effectiveness of the radar as a tool for measuring pulsations is the time lapse between successive samples of the same area. In the slow scan mode, the radar will sample the same area of the ionosphere every 120 seconds.

Summary

Southern Hemisphere

Satellites

Solar wind - ACE

To determine the state of the solar wind, use was made of the Advanced Composition Explorer (ACE) satellite data. The interplanetary magnetic field status was determined with level 2 data from the MAG project [56] and to determine the plasma electron and ion fluxes, use was made of the SWEPAM project [41] level 2 data.

Geostationary satellites

The major axis angle is shown in Figure 14.24, where 77% of the values lie in the. Figure 14.36 shows a scaled line graph of the velocity parameter over the same period.

Figure 7.2: The Lagrange libration points. [8]

On-line data sources

Storage and analysis devices

This system makes it possible to process larger amounts of data in a relatively short time, which was crucial for the analysis outlined in this thesis. In this chapter, the discussion focuses on the noise filter used to remove noise from the data, and the peak detector used to find peaks in frequency spectra.

Noise filter

The data

Throughout the analysis parts of this thesis, a significance detector has been used in various forms. The term significance detector is used to describe a routine that will process a set of data and then return those points that are either a significant part of a set of data or are not part of a set.

Figure 8.1: Data (top) and distribution histogram (bottom) for the uncleaned velocity data

The method

The result

Peak detector

Summary

The large data volume that makes up the SuperDARN dataset had to first be cleaned and then searched for pulsation events. This project used a significance detector to remove outlying data points and then detect peaks in the FFT spectra generated from the two-hour FIT velocity data.

Introduction

This gives the pure percent dispersion of the velocity values of that cell over that two-hour period for the two-hour FIT file and. Simply put, this analysis only concerns data that is suitable for the pulse-finding algorithm.

The scatter in general

Here you can see that most cells do not contain clean data, while fewer and fewer files contain more and more data. You can see that a lot of the scatter data has been ignored in the analysis.

Figure 9.2: The percentage of the scatter considered clean for all the radars in the network between 14:00 UT and 15:59 UT on 19 April 2004

Temporal variation

Daily Variation

The smoothed curve is a low-pass filtered version of the data with a cut-off period of 60 days. The best correlation was found to be with the Bz component and therefore a spectrogram similar to Figure 9.5 was produced for the IMF Bz component, which is shown in Figure 9.7.

Figure 9.4: A daily count of the number of cells that had a clean scatter percentage of greater than 80% for 2004

Locational variation

The region marked as broadband in the Bz spectrogram overlaps the peaks in the same region in the diffuse spectrogram.

Figure 9.8: The amount of FIT data considered clean shown as a function of radar station

Directional variation

The network in general

This figure shows the total number of cells in the entire network with the specified bearing. There is a similar lower peak at ±180◦, this is due to the fact that there are fewer radars in the Southern Hemisphere than in the Northern Hemisphere.

Clean scatter

Here you can see that the peak in the distribution is at 0◦ AACGM, which means that more cells in the grid point north than any other direction. These figures are similar to Figure 9.10, except that the northern and southern hemispheres are shown separately.

Figure 9.11: The bearing angle distribution for the Northern hemisphere.

Conclusion for scatter analysis

The ionosphere is not infinitely conductive with the result that there is a first-order correction for the electric field disturbances so that there are measurable pulsating electric fields in the ionosphere. The focus of this chapter is the automatic pulsation finder used to detect pulsation events in the SuperDARN data.

Figure 9.14: Number of cells with more than 80% clean scatter against bearing angle for the Southern hemisphere radars.

Data cleaning

Noise

The noise limit was determined by taking the mean of all data points±three standard deviations.

Closed field lines

Ground scatter

Pulsation finder

This was done to limit the analysis to events where the spatial variation of the event could be mapped. Single records do not allow for the determination of the spatial characteristics of the driver necessary to identify a pulsation as field line resonant.

Summary

Discussion

The number of adjacent cells per FIT file was counted and the distribution is shown in figure 11.2. The maximum number of contiguous cells in one FIT file is 67 out of a possible 1200 cells.

Conclusion

It is clear from this figure that there are only a few files with many contiguous cells. All the files that had more than 20 sets of contiguous cells (36 in total) were visually inspected and those that showed good latitudinal distribution are listed in table 11.1.

Figure 11.2: The number of FIT files with the corresponding number of contiguous cells in the file.

Frequency distribution

Discussion

It was shown in section 11.1.1 that files with more than 20 contiguous cells are rare. All files are also listed in table 11.1, except the file with the 3.8 mHz pulsation.

Figure 11.5: The average of the daily frequency distributions for 2004.

Conclusion

Since 39% of the files fall into these 5 bands and the remaining 61% fall into the remaining 50 frequency bands, there is a 6.5:1 ratio for these bands to contain files with a large number of contiguous cells. This means there is a 6.5:1 chance that a file with a large number of contiguous cells will be in one of these bands.

Temporal variations

Discussion

There was a greater correlation between pure scattering and the Bz component than either of the other two components, see Table 9.1. This correlation shows that the amount of net scattering is affected by Bz, with Bz in the south giving less net scattering.

Conclusion

It follows that in this case the changes in the pulsation characteristics are in fact a consequence of the distribution variations and are not directly driven by the IMF. An attempt to remove the temporal characteristics of the distribution from the pulsation characteristics by decoupling the distribution from the pulsations yielded a data set with no discernable characteristics and thus is not included.

Directional variations

Discussion

In this figure, the red trace is a polar plot of the northern hemisphere black line in Figure 11.11. In Figure 11.12 you can see the peak of the pulsation incidence curve (red) is slightly east of north corresponding to the peak in Figure 11.11.

Figure 11.10: The ACE on board magnetometer showing the status of the interplanetary magnetic field in GSM coordinates

Conclusion

Work has been done and [80], to use single radars to determine the direction of the pulsating irregularities by fitting amplitude and phase measurements with the assumption that the velocity sensed by the radar is given by. Ponomarenko et al [46] were able to partially model this behavior with a single radar, but these two relationships can only be correctly identified if there are two independent composite measurements of the pulsation.

Occurrence by station

The radar that has the most cells in this direction is the Stokkseyri radar and should be monitored for pulsations. It should be remembered that an FLR pulsation will also have a latitude-amplitude and phase relationship.

Figure 11.14: The number of events identified at each station.

Introduction

Preparing high resolution data

Filling the data gaps

Scattered data is data that does not fill the regular grid, such as the data in figure 12.3. The Delaunay triangle is included for illustrative purposes, here each triangle is filled with the average of the values.

Figure 12.1: The Antarctic network of SuperDARN radars with the fields of view of the Halley Bay (a) and SANAE (b) radars highlighted.

Time Differences

Using a method based on the time difference between samples means that some data will be discarded. The data was resampled so that there were 7200 samples between the first and last samples of the two-hour record.

Figure 12.5: The time difference in starting times for records from co-located cells in the Halley and SANAE data.

The merge process

To find these components, the fields of view of the Halley and SANAE radars were sectioned with a wedge that departed. The magnetic bearing for the green cells in Figure 12.8 is shown in the upper left corner of the plot and is indicated as east of north.

Figure 12.7: Re-sampled Halley data from figure 12.6 (a).

Data description

Complex demodulation

The factor 90◦ is due to the fact that the bearing is measured east of north and not north of east, as is the case with polar angles in Cartesian geometry.

Analytic Signal

Polarisation characteristics

Summary

The extent of the pulsations
The pulsations
Discussion
Conclusion

The GOES 12 data in Figure 14.10 show a lower frequency pulsation and a higher frequency pulsation. The higher frequency velocity data for 10 minutes in the two blue cells in Figure 14.3 are shown in Figure 14.12, with the more easterly cell in red.

Figure 14.1: Pulsation occurrence histogram for 2 September 2004.

Merged data on 5 August 2002

Single radar observations
Merged radar data
Characteristics of events
Conclusion

The bearing angles for each of the longitude cells are shown in Figure 14.17. The bearing angles for each of the geographic line cells are shown in figure 14.19.

Figure 14.14: Locations of pulsation events in SANAE and Halley Bay data for 5 August 2002.

Aliasing on 4 April 2004

Spectra for the radar data

Other instruments

Discussion

The positions of the Tsyganenko field line (in black), GOES 12 (in red), and the ACE satellite in the solar wind are indicated.

Figure 14.31: The GSM x − y plane for 4 April 2004. The locations of the Tsyganenko field line (in black), GOES 12 (in red) and the ACE satellite in the solar wind are indicated.

Sounding frequency changes on 5 April 2004

Ray tracing

Data

Discussion

The ray tracing shows that it is not possible to receive ground scattering from the ranges as indicated in figure 14.34. If there is no scattering, the pulsation record is broken up as is the case with the rest of the series in figure 14.36.

Figure 14.34: A summary plot of the Prince George radar on 5 April 2004 showing how the scatter changes for different transmission frequencies.

Summary of the thesis

It was found that the temporal variability of the number of pure entries correlated best with the temporal variability of the IMFBz component. The data is compared with ground and satellite data and it is concluded that the higher frequency pulsation is possibly a higher harmonic of the lower frequency pulsation data.

Summary of the results

Significance detectors
Scatter analysis
Pulsation finder
Merging
Interesting events

It is proposed that this is due to the fact that there are more clean records in the poloidal direction. The second event presented shows how the merge method can be used to determine the components of the pulsation and confirm that this was a field line resonant pulsation.

Further work

Other significance detectors

The first event shows that because the radar is measuring the pulsation in the ionosphere, it is not subjected to the spatial filtering that limits ground-based studies of pulsations, see section 4.5. The last event uses the fact that the change of the sound frequency of the radar can be used to show that ionospheric scattering is wrongly labeled as ground scattering.

Pulsation event studies

Real-time magnetospheric plasma densities

FIT file extension containing the fitted ACF parameters FITACF fitting process of an ACF. ΣP Integrated height Pedersen conductance ΣH Integrated height Hall conductance ki the imaginary part of the wave number.

Why use a FIR filter

The second was necessary for the determination of the analytical signal of a pulsation event where the pulsations are assumed to be monochromatic. To study the amplitude and phase characteristics of the signal, the data were filtered with a narrowband filter.

Filter choice in this research

Kaiser found that a window based on the modified zeroth-order Bessel function of the first kind was much easier to calculate [43, chapter 7]. The window is parameterized by two parameters, M the length of the kernel and in turn the sharpness of the transition and which controls the shape of the kernel and in turn the ripple in the pass and stop band.

Figure C.1: A windowed sync filter [57, ch 16].

Time domain aliasing

Parameters for the function include the lower and upper cutoff frequencies as a function of the sample rate and the two window parameters M and A.

IDL convolution options

Conclusion

The radar uses an oscillating pulse pattern (see figure 6.3) that transmits pulses before the more distant echoes are returned. What follows is an explanation of the process of creating an ACF with an oscillating pulse pattern, using a three-pulse pattern for simplicity.

Figure D.1: A three-pulse pattern shown together with the lag and sampling times.

The transmitted pulses

The ACF

The bearing of the cell is then the direction of the line connecting the center two points. Interpretation of the interphase spectrum of geomagnetic pulsations from field line resonance theory.

Figure E.1: The SuperDARN network field of view plot.

FITACF

Limitations

Range Aliasing

The part of the sample that is far from c4T2 will not be correlated and is considered noise. Similarly, when the bis sample is used to generate the ACF for the c4T2 range, part of the cT2 signal will be noise and therefore uncorrelated.

Bad Lags

When the ACF is generated using the sample at point b, the signal received from cT2 away will be correlated with other samples that have signals from cT2 away.

Multiple frequencies in the ACF

Summary

Thus, the center of the plots represents the AACGM pole with dotted lines for latitudes 80◦ and 70◦ respectively. The color for each cell represents the intensity of the displayed parameter, dark colors indicate low intensity while bright colors indicate higher intensity.

Method

Change of sign

The sign of Southern Hemisphere longitude is changed so that the field of view of a Northern and Southern Hemisphere station can be displayed for comparison on the same plot. A natural way to represent a station's field of view in the Northern Hemisphere would be as seen over the North Pole.

The Components

To merge data from two overlapping cells in the SuperDARN network, it is necessary to calculate the bearing angles for the cells. If one is interested in the components of the motion in the poloidal and toroidal magnetic directions, then one must calculate the magnetic bearing angles.

Figure F.2: Radar bearing angles together with an ionospheric bulk motion.

The Error in the Components

It is possible to calculate the poloidal and toroidal components of a patch of anomalies moving in the polar ionosphere from the line-of-sight velocities measured by two overlapping SuperDARN radars. A decade of the Super Dual Auroral Radar Network (SuperDARN): scientific results, new techniques and future directions.