Density Enhancements in the Solar Wind Plasma Cluster Data Analysis
M.Sc. Thesis in Plasma Physics Georgios Spanopoulos
June 2010
Supervisor: Dr. Tomas Karlsson
Royal Institute of Technology Report ………
Abstract
In this study density variations in the solar wind are examined based on data from the Cluster Mission. The data are originating from the stream outside the bowshock and thus they are spanning in an interval of three to four months for each mission year up to 2006.
As the data are examined, variations above the relative electron density threshold of 1.3 are archived. The variations are analyzed in terms of position, orientation, magnetic field perturbation and scale sizes. The magnetic field perturbations are exhibiting diamagnetic and paramagnetic behavior and a possible link to similar observations inside the magnetosphere is attempted through the impulsive penetration mechanism. The final conclusion of the report is that plasma density enhancements, similar to those identified from previous studies inside the magnetosphere, are also evident in the free solar wind stream close to earth.
Acknowledgements: I would like to thank my supervisor Dr. Tomas Karlsson for his valuable guidance during the project and Mr. Ola Carlström for his assistance with the IT system while I was working in the lab.
Table of contents
1 Introduction
1.1 Sun and Earth as a system 1
1.2 Interactions between the solar wind and the geomagnetic field 2
1.3 Purpose of this report and data used 3
References 3
2 The cluster mission
2.1 General characteristics of the mission 4
2.2 History, orbital elements and data collection 5 2.3 EFW instrument measurements and calibration 7
References 8
3 Data collection
3.1 Criteria and requirements for the data 9
3.2 Cluster orbital position for solar wind measurements 10
3.3 Orbital position and foreshock effect 11
3.4 Date Catalog (dates and hours) 13
3.5 Density enhancements search with IDL programs 14
3.6 Minimum variance analysis 17
3.7 Utility of the minimum variance analysis 19
3.8 Final dates and data catalog 21
References 22
4 Analysis and results
4.1 Position and Orientation 23
4.2 Density and dates of occurrences 30
4.3 Scale sizes and magnetic field behavior 31 4.4 Pressure balance and magnetic behavior 36
References 38
5 Conclusions
5.1 Data collection 39
5.2 Position and orientation 40
5.3 Density and dates of occurrences 40
5.4 Scale sizes and magnetic field behavior 40 5.5 Pressure balance and magnetic behavior 41
5.6 Further investigation and suggestions 41
5.7 Further investigation and suggestions 42
6 Appendix A 43 7 Appendix B 47
8 Appendix C 109
CHAPTER 1
Introduction
1.1 Sun and Earth as a system
All celestial bodies in the solar system are affected not only by the suns huge gravitational attraction but also from the emitted radiation in the form of thin plasma expelled every moment known as solar wind. The latter plays a very important role as the link between the sun and the other bodies in the solar system. The solar wind is responsible for some phenomena observed on earth such as the auroras and the magnetic storms. The interplanetary magnetic field is carried to the outer solar system by the solar wind due to the “frozen in” condition which requires the magnetic field lines to be tied together with the expelled stream of plasma. This coupling is responsible for the phenomena observed close to earth and more accurately because of the interaction between the solar wind plasma, carrying the interplanetary magnetic field (IMF) and the geomagnetic field. As the solar wind interacts with the geomagnetic field it distorts the perfect dipole like shape of the geomagnetic field and a “bubble” is formed around the earth known as the magnetosphere. Fig 1.1 is giving a general idea of the magnetosphere and its various regions.
The IMF (Interplanetary Magnetic Field) is directed southward and because of the northward GMF (Geomagnetic Field) at the forefront of the bow shock there is a region where a phenomenon called “magnetic reconnection” occurs. Just before the bow shock someone can assume that the solar wind has a constitution which agrees with the average solar wind parameters given in Table1.1.
Solar wind properties at 1 AU (Close to earth vicinity)
Proton Density 6.6 cm-3
Electron density 7.1 cm-3
He2+ density 0.25 cm-3
Mean Flow Speed 450 km/s
Proton Temperature 1.2 105 K Electron Temperature 1.4 105 K Magnetic Field 7 10-9 Tesla
Table 1.1: Solar wind parameters
1.2 Interactions between the solar wind and the geomagnetic field
Inside the magnetosheath the parameters of Table 1.1 do not provide anymore adequate representation for the plasma existing there. For the plasma populating the areas inside the magnetosphere (magnetosheath, magnetopause) several explanations has been given with regard to their origin. One of the theories refers to the plasma injection into the magnetopause from the magnetosheath by proposing an explanation which assumes impulsive penetration mechanisms. According to recent observations by R. Lundin et al.
[2] these plasma transfer events (PTEs) are more easily distinguished in the magnetopause than in the magnetosheath. The impulsive penetration process can involve two different mechanisms with which it takes place, the expulsion and the polarization mechanism. The expulsion mechanism takes place through localized reconnection phenomena while the self polarization is produced in a plasmoid exhibiting diamagnetic behavior which leads to an E x B drift which facilitates the propagation across the geomagnetic field lines. However PTEs located in the magnetosheath in the magnetosheath are likely to enter by the expulsion mechanism when and if they reach the magnetopause [3].
1.3 Purpose of this report and data used
In this report the main goal is to investigate if localized plasma density enhancements exist in the solar wind before any interaction with the geomagnetic field give rise to them.
The results presented in this document are focusing on all density enhancements (above a certain threshold) in the solar wind plasma which can be identified in the close earth vicinity outside the bow shock. Such events have the potential to evolve into the PTEs observed in the magnetosheath and magnetopause. Thus their registration and classification will help further analysis regarding a possible correlation to the plasma injection mechanisms.
The data used for the analysis in the following chapters are based on observations from the Cluster mission which nowadays comes to its closure phase after nine years of service. Information for the Cluster mission is provided in chapter 2. In this report, data from the years 2001 to 2006 have been used. All data correspond to a period in each year where the Cluster satellites were moving outside the bow shock and had the opportunity to collect pure solar wind measurements. Analytical presentation of the data selection and their validation follows in chapter 3. Chapter 4 is dedicated to the analysis of all selected events with graphical tools. All previous steps lead to chapter 5 in which conclusions and recommendations from the overall analysis are drawn and further steps of investigation are reviled. The ambition of this report is to contribute with its analysis to the ongoing research in the field of space and plasmas physics.
References
[1] Margaret Galland Kivelson, Christopher T. Russell, Introduction to space physics Cambridge University Press 1995
[2] R. Lundin et al. Evidence for impulsive solar wind plasma penetration, Annales Geophysicae (2003) 21: 457-472
[3] T. Karlson et al. Localized density enhancements in the magnetosheath; 3D morphology and possible importance for impulsive penetration. Kth 2009
[4] http://space.rice.edu/IMAGE/livefrom/sunearth.html
[5] Bengt Hultqvist, Marit Øieroset, Rudolf Treumann. Magnetospheric Plasma Sources and Losses. Final Report of the ISSI Study Project on Source and Loss Processes of Magnetospheric Plasma. Space Science Reviews journal, Vol. 88/1-2, 1999, 496 p.
CHAPTER 2
The Cluster mission
2.1 General characteristics of the mission
The main goal of the Cluster mission is to provide in-situ measurements from different key locations around the magnetosphere and to investigate plasma variations close to earth’s vicinity.
These measurements have already contributed to a more precise and detailed image of the complex interaction between the solar wind and the geomagnetic field. The latter is essential under the consideration that operational credibility for spacecraft improves as our conception for the space environment becomes better. Cluster is the first scientific space mission which employs a constellation of four spacecraft in formation to perform in-situ measurements giving the opportunity for three dimensional analysis of the data. Figure 2.1 shows the Cluster satellites during the preparatory stage of the mission.
Fig.2.1 Preparatory stages of the Cluster mission at ESA.
2.2 History, orbital elements and data collection
Cluster as an idea originates back to 1982 and the first satellites were ready to place into orbit in June 1996. Unfortunately a failure of the Ariane 5 launcher destroyed this first mission. The European Space Agency’s Science Program Committee (ESA’s SPC) decided to restart the mission including the one reserve satellite (made from spare parts) and three new. In August 2000 the Cluster satellites were placed into orbit by two Soyuz rockets and the end date for the mission set to be December 2005. The mission was declared officially operational on February 2001. Four years later on February 2005 ESA’s SPC approved the extension of the mission until December 2009 allowing this way plasma measurements to be taken over a nine year period. This extension enables the constellation to cross unexplored regions of the magnetosphere and partially observe plasma changes for nine out of the eleven years of the solar cycle.
Cluster is formed by four satellites in a tetrahedral constellation which facilitates the research and study of plasma structures in three dimensions. The regions crossed by the mission are the solar wind region in front of the bow shock, the bow shock itself, the magnetosheath, the magnetopause, the polar cusps, the magnetotail and the aurora zones. The spacecraft are placed on a polar orbit with an apogee of 19.6 earth radii. As the earth is orbiting the sun the Cluster constellation has a fixed orbital plane sweeping this way all regions of the magnetosphere in a year’s period. Due to the orbital mechanics that governs the movement of the constellation the distances among the spacecraft is changing continuously during the orbit. Thus the tetrahedron shape has different dimensions in different points of the orbit around the earth. Figure 2.2 gives some details concerning the regions of the magnetosphere Cluster are visiting in a year and the change of the constellations shape in each orbit.
Fig.2.2 Cluster motion in a year period and separation change in an orbit [1].
The accumulation of data is done by scientific instruments carried onboard each satellite. Every satellite accommodates 11 instruments for in-situ measurements and this set of instruments is the same on each satellite. For this reason the Cluster constellation is capable to measure the same quantities, at different satellite positions, at the same time and provide a spatial resolution of the plasma variations at a certain position of the magnetosphere. The same regions are visited again providing data for different separation of the constellation. The instruments are capable for electric and magnetic field measurements and the detection of electron and ion distribution functions at spin resolution [6]. No other mission before Cluster was able to give such a huge amount of data from different regions of the magnetosphere. The first six years of the mission account for 300 000 compact disks of data distributed to the scientific community worldwide.
Each instrument onboard is associated with a Principal Investigator (PI). Table 2.1 is providing information about the instruments onboard [6] and some facts about the spacecraft.
Table2.1. Instruments onboard each spacecraft.
2.3 EFW instrument measurements and calibration [7]
The Cluster spacecraft are equipped with instruments in order to study small scale plasma structures in the solar wind. The current study is mainly based on solar wind electron density data and thus it is very important to present some facts regarding the instrument which performs the measurements and some of its operational modes.
Cluster EFW (Electric Field and Wave) instrument can measure electric fields with unprecedented accuracy. The instrument is using two pairs of spherical probes deployed on wired booms in the spin plane of the spacecraft with probe to probe separation 88 m.
The instrument can operate in two modes:
1. Voltage mode where it measures ambient plasma fields
2. Current mode where it measures ambient plasma electron density
Electron density and gradient of electron density are fundamental parameters and the EFW instrument can routinely produce spacecraft potential data with time resolution of 0.2 sec. The spherical sensors at the end of the booms can serve as Langmuir probes to provide information for the plasma density and the electron temperature. The collected electron current is proportional to the plasma density under the condition that the plasma temperature remains relatively constant. Then the variation in the plasma density corresponds to variations in the electron density and eventually is correlated to the electron current measured at the probes. The current mode of operation is better suited for denser plasma measurements as those encountered by the spacecraft in the solar wind regions the magnetosheath and the plasmasphere. For this mode of operation probe bias sweeps are performed every two hours.
Photoelectrons are also controlled in order to minimize errors. The spacecraft potential is a function of the emitted photoelectrons and ambient electron and a weak function of the electron energy. The potential difference Vs – Vp has previously been calibrated with a preliminary calibration against plasma density [Vaivads, private communication].
Properly calibrated spacecraft potential is essential for accurate density measurements.
Figure 2.3 is illustrating electron density data from all four spacecraft where it is apparent that Samba measurements (green line) are higher compare to the input from the other spacecraft and that this calibration is not perfect for this time interval. Further work to
Figure 2.3 Electron density measurements reproduced in IDL.
EFW can provide 4 output data rates and two main modes of operation with regard the available telemetry:
1. Normal mode telemetry with 1440 bits/sec, 10 Hz filtering and sampled at 25 samples /sec.
2. Burst mode telemetry with 15040 bits/sec, 180 Hz filtering and sample at 450 samples /sec.
References
[1]C.P Escoubet, M. Fehringer, and M. Goldstein, ‘The Cluster mission’, Annales Geophysicae (2001) 19:
1197-1200 © European Geophysical Society 2001
[2]http://www.spacedaily.com/reports/1000th_Orbit_For_The_Cluster_Mission_999.html
[3]http://www.esa.int/esaSC/120383_index_0_m.html
[4]http://en.wikipedia.org/wiki/Cluster_mission (Used as pool of information for institutions )
[5]http://www.fys.uio.no/plasma/old/norsk/forskning/Cluster/4Cluster.gif
[6]http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=44015
[7] G. Gustavsson et al., ‘First results of electric field and density observations’, Annales Geophysicae (2001) 19: 1219-1240 © European Geophysical Society 2001
CHAPTER 3
Data collection
3.1 Criteria and requirements for the data
As it has been stated in the introduction the investigation presented in this report is based on solar wind measurements. This means that the data collection from the Cluster archive must correspond to the area outside the magnetosphere. Thus the first and more important criterion in the investigation is to find and distinguish all the days which fulfill this requirement from the initiation of the mission to the most recent available date. As the Cluster constellation is changing orbit position day after day and the boundaries of the bow shock and magnetopause are not fixed in space and time, a validation process must follow after the preliminary selection. This validation procedure will be done in detail and it will involve computer aided tools for graphical representation as well as subjective estimations. More analytically the steps which will be followed in the data collection process are:
1. Cluster approximate orbit plots 2. Ion energy and foreshock region
3. Dates and hours for solar wind data outside the bow shock region
4. Date verification and event search (Solar Wind Plasma Density Enhancements) 5. Catalog of events
The final result of this procedure will be the catalog of events which will be used for the analysis presented in chapter 4. The methodology from steps 1 to 5 as it has been presented above corresponds to the general processes involved. A more detailed presentation of steps 1 to 5 is following in the next paragraphs.
3.2 Cluster orbital position for solar wind measurements
The orbiting position for the Cluster satellites can be easily acquired from the mission database CSDS webpage (Cluster Science Database System) which provides quick look data from the mission. Among the other Cluster data available, the page provides a 3D representation for the position of the satellites in regard to the bow shock and magnetopause boundaries approximation. Figure 3.1 illustrates the Cluster orbit as it can be seen in the CSDS webpage plot in the date interval 15th - 17th of March 2003.
Figure 3.1 Orbit plot for the Cluster satellites available from the CSDS [1].
The position of the constellation is not totally accurate but it is based on the orbit estimation for spacecraft 3 (Samba). Then the separation among the other spacecraft is taken into account with regard to spacecraft 3. In Fig.3.1 the constellation is shown to cross the outer boundary of the bow shock and remain for some time outside the magnetosphere. During this time interval the data collected from the spacecraft will correspond to pure solar wind measurements which are required for the purposes of this report. Consequently the orbit presented in Fig.3.1 can be considered as a reference paradigm for further date selection. It is important here to clarify that the orbit plane can deviate by an angle from the GSE z-x plane as long as there is a time window where all four spacecraft can be found outside the magnetospheric boundaries. Also the depicted boundaries of the bow shock and the magnetopause are assumed to represent a mean value but in reality there is motion and thus the corresponding distances are changing. So the plots like Fig.3.1 are not the absolute criterion but they provide a starting point for date selection, in which the research for plasma density enhancements (PDE) will focus.
3.3 Orbital position and foreshock effect
Once the orbit and the relevant position of the satellites have been checked another criterion will be used to eliminate the possibility for the apogee to be too close to the magnetospheric boundaries. The bow shock can be characterized as the boundary at which the speed of the solar wind abruptly drops as a result of its approach to the magnetopause [3]. But upstream the bow shock is the so called foreshock region which can affect the pure solar wind data. The foreshock plasma includes convected solar wind plasma as well as electrons and ions reflected by or leaking through the bow shock [3].
Thus the fluctuations of the boundaries especially during the contraction are favoring the likelihood for the measurements to fall inside the foreshock region. In this case the data from the quick look overview page correspond to Fig.3.2. The foreshock effect becomes evident in the ion energy graph where the high energy ions are concentrated in the middle and the “cold” ions occupy the edges but in between there are some ions with energies (light green spectrum) indicating the plasma reflection from the bow shock.
Figure 3.2 Quick look overview graphs available from the CSDS [1].
When the constellation encounters the solar wind stream without any effect from the bow shock the measurements are not affected from the foreshock so much. Someone can actually see that the foreshock ion energy pattern appears in the middle of the graph for some time and then it disappears. The quick look graphs are a rough method to check for clear solar wind but they are adequate for the creation of the initial date catalog.
3.4 Date Catalog (dates and hours)
From the quick look search many dates were identified as possible sources for pure solar wind data. It is important to state here that the dates and hours from the quick look investigation correspond to a good initial set which will be further analyzed in detail in order to conclude to the final set of all density enhancements. All dates and corresponding hours are presented in Table 3.1.
In the table above the eight first digits correspond to the date. The year occupies the first four digits, the next two the month and the last two the date. The remaining digits represent the time interval in which the measurements had no foreshock effects from T0 to T1. Year 2003 was the period of the mission in which Cluster gathered most of the solar wind measurements outside the bow shock. The next step is to analyze the data from each date and check if they correspond to pure solar wind measurements and find possible density enhancements.
3.5 Density enhancements search with IDL programs
From Table 3.1 the basic information for the date and time are available but a more thorough check must be done by using IDL programs which have been developed in the Space and Plasma Physics section of the Alfven Lab [5]. All the data from the Cluster satellites have been saved as ephemeris data in the server of the department. The programming language IDL is installed and IDL programs can be used to visualize the measurements from the Cluster mission. The server is accessed remotely and all the work can be performed by a remote access application from another computer inside the department. As an example Fig.3.4 is illustrating data from a particular date as they can be reproduced in the IDL environment.
Figure 3.4 Electron density measurements (absolute and relative)
In the figure above and in the plots to follow, the raw spacecraft data have been smoothed with a boxcar average of 900 points in order to define a background density level. This helps to search for density enchantments without the risk to confuse noise with actual sudden density changes. The top graph represents the electron density ne and the bottom graph represents the relative density ratio between the actual value ne on a specific time point t0 and an average value ne,av around this point (t0-a , t0+a). It is useful to remember that the time is in seconds and that 24 h (a day) corresponds to 86400 seconds.
The top graph of Fig.3.4 shows a lrge scale variation in the plasma density which changes the electron density measurement from 15 e- / cm3 at t0 ≈ 5,7E4 sec to 45 e- / cm3 at t1 ≈ 5,85E4 sec (encircled in the big black circle). The change in density detected from the spacecraft is considered to be quite big in magnitude and duration. Such solar wind plasma changes have been identified and researched [4] but their scale is far too big for the purpose of the present study. A rough estimate with regard to the plasma variation of Fig.3.4 gives a scale size of approximately 126 Re (2000 [sec] X 400 [km/sec] = 8· 105 [km] ≈ 126 [Re]). Variations of smaller size are more likely to enter the magnetosheath according to the observations and thus an investigation ratio ne / ne,av has been used instead, to identify possible density enhancements which are likely to cross the boundaries of the bowshock and move inside the magnetosheath. As initial criterion the ratio was set to 1.2 but as the amount of events satisfying it was growing bigger the threshold limit increased and eventually only events with density ratio above 1.3 were taken under consideration. The density enhancement inside the black circle illustrated in the bottom graph of Fig.3.4 has a ratio at the limit of 1.2 and it has been ruled out from the final data set but it provides a good example for the type of events to be identified in this investigation. It is also evident that the visualization of the electron density as a ratio provides a different perspective to the way the selection of events is done.
In Fig.3.4 it is encircled with red color the effect of the foreshock in the measurements. The reason why these fast and abrupt changes are attributed to the foreshock is explained with the abruptly increased ion energy while the velocity remains
encounters the solar wind region but the ion energy increase signatures the effect of the foreshock. Thus all density enchantments exhibiting very fast and abrupt changes with ion energy increase and without significant changes in the solar wind speed are regarded to be the result of the foreshock and such events aren’t perceived as pure solar wind density enhancements. Figure 3.6 is illustrating an event with very clear signature in the values of electron density, electron density ratio, and solar wind velocity and ion energy.
Figure 3.6 Electron density measurements on 4th April 2004
As it is shown in this case the electron density ratio has a peak slightly above 1.5 while the average density measurement has a value of 15 e- / cm3. The velocity and the ion energy values from the same day correspond to the nominal expected ones for this region of the orbit which is outside and close to the magnetosphere.
All IDL programs used for the Cluster data representation and analysis are available after request from the Alfvén Laboratory, Stockholm, Sweden [5].
3.6 Minimum variance analysis [6].
The minimum variance analysis (MVA) is used to estimate from spacecraft data the direction normal to one dimensional or approximately one dimensional current layer, wave front or other transition layer in plasma [6]. The minimum variance analysis applied to the magnetic field vector data measured by a spacecraft traversing a transition layer is based in the following analysis:
Initially we assume x, y, z local coordinate based in the transition layer with its z axis along the normal vector ˆn . By assuming also that the transitional layer is 1-D (one dimensional) and from time follows that:
∇ · B = Bz/ z = 0∂ ∂ (3.1)
Bz/ t = 0
∂ ∂ (3.2)
As a result the traversing spacecraft is observing a constant value of Bz and the maximum amount of measurements required in order to define the coordinates of ˆn is three. So in this ideal situation if B1, B2 and B3 are the available measurements from the two opposite sides and the middle of the layer then:
1 2
1 2 3
2 3
ˆ ˆ ˆ ˆ
ˆ
⎧ − ⎫
⎪ ⎪
⇒ ⎨ ⎬
⎪ − ⎪
⎩ ⎭
(B B )· n = 0 B · n = B · n = B · n
(B B )· n = 0 (3.3)
Thus the vectors (B1-B2) and (B2-B3) are tangential to the layer and thus their cross product satisfies the relation below:
1 2 2 3 1 2 3
1
1 2 2 3 1 2 2 3
( )
ˆ ± − × − ⋅ˆ ± ⋅ ×
− × − − × −
(B B ) (B B ) B B B
n= or B n=
(B B ) (B B ) (B B ) (B B ) (3.4)
When B ˆn = 0 is assumed to be a special condition which serves as an additional equation if the vectors (B1-B2) and (B2-B3) are aligned. This is the case where the current
vectors inside the layer are unidirectional. Another case where the additional equation is needed is when two not aligned measurements are available instead of three. If the measured vectors are nearly the same then no reliable calculation of the normal can be obtained. For this reason the measurement are taken approximately on each side and at the center of the layer. At this point it is important to remind that the model above is the basis for the variance analysis which is applied in real space measurements.
The ideal 1-D transition layer is not representative of the actual transition layers found in space. Most cases are deviating from the ideal hypothesis in many ways. The main reasons why these discrepancies exist are listed below:
1. The layer is likely to have 2- D or 3-D structures which evolve on time creating fluctuations in the temporal orientation of the normal vector.
2. Systematic measurement errors are also one possible cause of uncertainty in the normal estimation.
3. The onboard magnetometers with high resolution capabilities are able for many vector measurements B(m) where m= 1, 2, 3, …M as the spacecraft is passing through the layer.
As a result from the above affects, the most adequate way to determine the direction normal to the transition layer is to incorporate a minimum variance technique. The method identifies the direction in space for which the field component set {B (m) · ˆn } has minimum variance.
$2
2
1
1 M (Bm B ) n
σ = M
∑
− ⋅ (3.5)$ 2
1
. 1
M m s t
n
with B B
=
=
∑
Here it is important to remind that ˆn is expressed with three components along the three axis of the GSE (or GSM) coordinate system (nx , ny, nz) depending on the components of the B(m) vectors. Equation (3.5) is solved by using Lagrange multipliers and the result is the Magnetic Variance Matrix as it is expressed in the eigenvector basis shown by equation (3.6).
B
ii i i i i i
M = B B − B B =λ (3.6)
The minimum variance analysis consists of the three Langrage equations differentiated and expressed in matrix form and lead to the eigenvalues λi and the corresponding eigenvectors of the matrix. “The eigenvector x3 corresponding to the smallest eigenvalue λ3 is used as the estimator for the vector normal to the current sheet and λ3 itself represents the variance of the magnetic field component along the estimated normal” [6].
3.7 Utility of the minimum variance analysis.
For every event identified from the Cluster measurements it is important to verify its scale size. The size (width of the event) it is more convenient to be verified along the normal vector from the minimum variance analysis. Figure 3.8 is presenting density measurements (upper graph) from the Cluster satellites as they encountered an event inside the magnetosheath. This example illustrates a case where the satellites are passing through a density change and the density peaks are found to be in different time point for each satellite. This case is indicating the fact that the measurements are depending a) on the scale size of the event, b) its orientation in space relative to the spacecraft constellation, c) the relative separation among the spacecraft and their relative velocity and d) the instrument accuracy and resolution. So in this case the size and orientation of the event compare to the separation of the spacecraft is such that first interacts with s/c 3 then with s/c 4 after with s/c 2 and finally with s/c 1.
Figure 3.8: Minimum variance analysis for density enhancement.
But in order to get a good estimation of the scale size measured in earth radii a plot which overlays the density data from all spacecraft along the estimated normal direction of the event is required. This type of plot is provided in the lower graph of figure 3.8 in which the x axis is scaled in Re’s and the y axis in e- / cm3 for the density.
For the solar wind data the estimation of the scale sizes for each event is performed in the same way. Here it also useful to mention that the value in Re’s for each case is considered to be approximately the distance in the middle of the curve (Fig. 3.8 bottom graph) and not the maximum distance at the bottom. This is a rough estimation of the average distance since the peaks are associated with the minimum and the basis of the curve with the maximum but for distances measured in Re’s the effect on the overall accuracy is considered to be negligible. The data output from the minimum variance analysis which are included in the final catalog are:
1. The scale sizes of the events in Re’s, and
2. The MVA eigenvectors from spacecraft 3 as a way to confirm the effective application of the method on the data measurements.
At this point it is useful to comment on the application of the method for the events found in solar wind measurements and outside the magnetosheath. In most cases if not all for the solar wind dataset there was no time lag among the density peaks measured by the four spacecraft in contrast to Fig. 3.8 which is presented here. Or better if there was some in most cases it was much less compare to the case presented in Fig. 3.8. Although the MVA was used for the whole dataset it was impossible to detect events which had such big time lag. Some contributing factors to this fact can be summarized in the following points.
1. The spacecraft constellation when encounters the solar wind is close to its apogee.
As a result the separation closes to the minimum possible for the orbit and the velocity also reduces.
2. The density enhancements identified at this point of the orbit have on average more traverse orientation relative to the constellation and thus the effect of time lag among spacecraft measurements is reduced compare to the measurements inside the magnetosheath. The orbital representation in Fig. 3.1 is helpful for qualitative understanding.
3.8 Final dates and data catalog.
The full date and data catalog with all the events and some samples of their graphical representation in IDL can be found in appendix A and B. The events listed in the catalog are fulfilling all the criteria described in the previous sections and the selection is based solely on the solar wind density profile.
The total amount of data gathered is referring to 70 identified events in the period between the years 2001 and 2006. Although the mission continues until now the last entry in the department’s database (Alfvén Lab data server) was for the year 2006. The database is expanding to contain all data from the following years of the mission but the procedure takes some time and thus it was not feasible to perform the investigation described in this chapter for the whole Cluster dataset. It is also important to focus on the
fact that the catalog is an event list and thus several density enhancements were found on the same date. For each event in the catalog the information recorded are as follow:
1. Date
2. Time in seconds for the peak of the event 3. Density ratio ne at the peak of the event 4. Actual density Ne at the peak of the event 5. GSE coordinates X, Y, Z of the event peak 6. The velocity components Vx, Vy, Vz in GSE 7. The angle d_alfa from MVA
8. Size estimation of the event (Width) in Re (Earth Radii) along the normal vector ˆn 9. The GSE coordinates of the unit vector ˆn (e1_3x, e1_3y, e1_3z)
10. The GSE coordinates of the 3rd eigenvector (eigen3_x, eigen3_y, eigen3_z) 11. The dB/B ratio which has been calculated from the graphs in Appendix B
In the following chapter the analysis will be presented and the main attempt will be to extract useful information and results in order to convey meaning from the data.
References
[1] http://www.Cluster.rl.ac.uk/csdsweb-cgi/csdsweb_peak?P_TYPE=P4 [2] http://www.Cluster.rl.ac.uk/csdsweb/index.html
[3] Iver H. Cairns , P. A. Robinson , and G. P. Zank Progress on Coronal, Interplanetary, Foreshock, and Outer Heliospheric Radio Emissions, PASA, 17(1), 22
http://www.atnf.csiro.au/pasa/17_1/cairns/paper/node4.html
[4] P.A. Dalin et al. Large-scale solar wind density enhancement and its boundaries: Helios 1, 2 and IMP 8 observations. 2007
[5] Tomas Karlson. IDL programs. Alfven Lab - Royal Institute of Technology Sweden
[6] Bengt U. Ö. Sonnerup and Maureen Scheible. Minimum and Maximum Variance Analysis. Dartmouth College Hanover, NH, U.S.A.
CHAPTER 4
Analysis and results
The data set presented in Appendix A has been assembled according to the criteria stated in chapter 3 and the final catalog is containing all the information necessary for the analysis presented here. The focus will be given to graphs and plots which facilitate the visual inference and can lead to valuable conclusions regarding the location of the events their sizes and the estimated frequency of occurrence. The main goal of the analysis presented here is to help previous investigations related to the interactions taking place between the solar wind and the earth’s magnetosphere. The mechanisms taking place in magnetospheric interactions will be also analyzed since they are tied to the research regarding the plasma injection and the plasma structures observed inside the magnetosphere. Finally an evaluation of the present data set is to be performed in order to raise suggestions for further research and potential thesis work.
4.1 Position and orientation
In figure 4.1 the events’ positions have been plotted on the XY plane of the GSE coordinate system. The continuous lines as they appear from the right to the left of the graph are forming two half elliptic sections and they are representing the boundaries of the bow shock [1] and the magnetopause [2] respectively. These borderlines are not fixed in space and their depiction here is based on a statistical model which best describes them by accounting for the parameters which cause this continual fluxuations of the boundaries.
In the catalog are listed seventy events with their respective measurements associated to the density peaks for each case. The GSE coordinates of the peaks are plotted in the following graphs together with the boundaries of the magnetosphere. From the velocities it is also possible to create a flow profile of the events on the same plane visualizing this way the direction of the enhancements as they transverse the Cluster constellation
XZ and ZY (density peak coordinates) and for every plane the flow profile is provided.
The graphs presented here were created with the help of IDL 6.3 and with MS office [3].
Figure 4.1 Density peaks on the XY plane of the GSE coordinate system.
All distances in Fig. 4.1 are given in earth radii and as it is shown most of the events are found outside the boundaries. The two events which are placed behind the bow shock border line they are illustrating the fact that the XY plane is placed at Z = 0. So actually all of the events in Fig, 4.1 are outside the boundaries as the borders are folding around the earth’s magnetic field and thus this orbicularity is not apparent on the graph. This is also evident in the rest of the graphs with plane projections for the density enhancements.
The orientation of the events is clearly visible in Fig.4.2 where the arrows are representing the normal direction as it has been calculated from the minimum variance analysis for each event.
Figure 4.2 Normal directions in association with Fig. 4.1.
The reader should keep in mind that the arrow visualization corresponds to the coordinates of the event peaks as they are listed in the catalogue of Appendix A and they are accurate instantly and locally.
Figure 4.3 provides a view for the density variations as they appear on the XZ plane.
Again some of the events appear to cross the boundaries of the bow shock but as in the case of Fig. 4.1 this is just due to the 2D representation when actually all the events are outside the border lines. Apart from the seemingly same pattern of event distribution revealed in both images of Fig. 4.1 & 4.3 the spread1 of the data as they appear on the XY and the XZ plane is different.
1 In this paragraph the word spread is not used with its pure statistical meaning referring to statistical
Figure 4.3 Density peaks on the XZ plane of the GSE coordinate system.
On the XZ plane the data are distributed almost evenly on both sides of the X axis while they appear quite close to it from both sides (up and down) with a maximum spread of -11 Re ≤ Z ≤ 8 Re. On the XY plane although the data points appear also to be evenly distributed on both sides of the X axis the spread is -12 Re ≤ Y ≤ 18 Re much larger compare to the one observed in Fig. 4.1. This observation holds for the majority of the data and it’s not based only on the extreme points at the edges of the axes but it reveals the general trend of the data set including the points laying into the middle. But since the trend is evident on the whole data set it is then convenient to present the values from the most remote points without loss of generality. One possible explanation for this spread difference can be found again in the orbital characteristics of the Cluster mission with association to the GSE coordinate system. As it has been pointed out before in chapter 2 the spacecraft are placed on a polar orbit with an apogee of 19.6 earth radii. As the earth is orbiting the sun the Cluster constellation has a fixed orbital plane sweeping this way all
regions of the magnetosphere in a year’s period. This fixed polar orbital plane creates the difference on how the data are spread on the XY and the XZ plane of the GSE coordinate system. The data on the XY plane are gathered as the spacecraft sweeping the plane during the months for which the constellation is reaching the solar wind in an approximately three month period. Thus the data in Fig. 4.1 appear unbounded in this time period. Whereas the pattern of the data points as it appears on the XZ plane are presumed to be a manifestation of the fixed orbital plane.
From these two plots someone can possibly conclude to the point that density enhancements fulfilling the criteria, as those posed for the data set of this study can be found in the solar wind around the magnetosphere and close to the boundaries, regardless the location of the experimental equipment. The patterns of the overlapping density peak coordinates and the spread on the XY and XZ planes are mainly expressions of the orbital characteristics from the Cluster mission and consequently they should not be correlated to the solar wind interactions with the magnetosphere or additional plasma properties without further investigation.
In Fig. 4.4 a view of the orientation for the events is given on the XZ plane with the arrows representing the normal vectors to the forefront of the enhancements. The plot is similar with that in Fig.4.2 and the orientation appears parallel to the X axis. Something which is expected and is also supported by the measurements since the velocity component along the X axis Vx has the largest magnitude. But this is not the case for the XY plane where most of the forefronts of the plasma blobs are having diverging directions and they don’t appear to be aligned perpendicularly to the Y axis. This result is something to be expected since the XY plane motion is affected by the spiral shaped magnetic field lines and the radial motion of the solar wind.
Another projection for the data points with great interest is that on the YZ plane which allows someone to observe their distribution from the viewing angle against the sun.
Figure 4.5 is providing the position plot of the events while Fig. 4.6 corresponds to the orientation. In Fig. 4.5 it is important to bring into attention that the border lines for the bow shock and the magnetopause are plotted at X = 0. Therefore most of the data points are lying inside the magnetopause circle which is just a matter of representation.
Figure 4.4 Orientation of the density peaks in association with Fig. 4.3.
Figure 4.5 Density peaks on the YZ plane of the GSE coordinate system.
Though it is easy in Fig. 4.6 to check the normal orientation of the events relatively to the bow shock and the magnetopause boundaries still is not quite clear on the graph the associated movement of the peaks on the YZ plane. Most of the fronts are almost parallel to the XY plane and very few appear to have a Z component. It is obvious that the dominant trend is dictated by the IMF and its spiral motion outwards the sun.
All graphs in the present section were created in IDL 6.3 and the corresponding scripts are available in Appendix C. The measurements listed in Appendix A have been used as an input for the IDL scripts by creating a text file with ASCII format of the data.
Figure 4.6 Orientation of the density peaks in association with Fig. 4.5.
.
4.2 Density and dates of occurrences
In this section it is investigated the connection among the absolute density and the density ratio as also their correlation to the corresponding dates. For the statistical analysis of the data in Appendix A the software used here for many of the figures in this and in the following sections is @RISK [5]. In Fig. 4.7 the distribution for the relative electron density2 is given as it has been plotted from data.
Figure 4.7 Histogram for Ne / Neb and associated frequency of occurrence.
As it is shown in the graph, although the corresponding dates of the data are representing different periods in the solar cycle the histogram is best fitted by an exponential function with the parameters presented on the top of the graph. The red bar is
2 The distribution for the absolute density is almost the same with the one in Fig.4.8 and for this reason is not presented here.
indicating that 90 % of the identified density ratios lie in the interval from 1.3 to 1.5781.
The associated absolute density values, fulfilling all the criteria (Chapter 3), are in the range of 5.6 cm-3 to 71.7 cm-3. For the fitted function the 90 % corresponds to an 84.9 % as it is shown.
4.3 Scale sizes and magnetic field behavior
This section is dedicated to the analysis of the scale sizes and the magnetic field signature for each event of the data set. In Chapter 3 the MVA (Minimum Variance Analysis) was introduced and the main purpose of its utility was to compute the approximate distance along to the vector, normal to the current layer of the event. Figure 4.9 is presenting a frequency diagram for the scale sizes in the data set.
Figure 4.9 Scale sizes in (Re).
Most of the events’ sizes are falling in the range between 1 and 30 Re which corresponds to the 90 % of the data set. The outlied value corresponds to an event which alleged to have a width of 90 Re, twice the size of the second largest entry in the catalog.
It is important here to emphasize that the mean scale size is about 11.5 Re.
Some of the plasmoids which have been identified in the plasmasheath and are potentially part of transient processes into the magnetopause are exhibiting magnetic field perturbations. These potential PTEs were found from Cluster measurement inside the magnetopause region while the IMF was directed southward (-2 nT) and they represent possible magnetosheath plasma intrusion without eliminating the possibility of a magnetopause encounter [7]. It is possible that the bipolar perturbations in the magnetic field vector B (for all three components) to be a manifestation of current flux tubes similar to the FTEs (flux transfer events) but still a more thorough analysis of the magnetic field data is required [7]. Though the analysis presented here accommodates data and measurements from the solar wind region outside the forefront of the bow shock it would still be useful to analyze the magnetic field data as they appear in the plots provided in Appendix B in order to reveal similar perturbations. The correlation implied here is not proven neither the phenomena which can possibly lead to a direct link are fully understood up to this point. The PTEs inside the magnetopause originated from the magnetosheath plasma may have characteristics which support the impulsive penetration model [8] but their connection to observations from the solar wind regions is under investigation and at the moment no final conclusions are to be drawn based on the analysis provided in this document. So the purpose here is to add more information which can support and help further investigation towards the better understanding of the phenomena as it is related to the plasma injection mechanisms through the different layers of the magnetosphere. In Fig. 4.10 the measurements are illustrating an enhancement with a density peak clearly distinguished from the base value of the background (40 cm-3). The duration of the event is approximately 400 sec and the velocity Vx ≈ 363 km /sec. The magnetic field value B represents the total field and not one particular component (Bx, By or Bz). The component Bz along the Z axis of the GSE system is shown in the last box of Fig. 4.10. The sudden change of the density is coupled with a magnetic field distortion and the value of B decreases from an average background
of 6.5 nT down to 4.5 nT. The difference is -2.5 nT and its percentage ratio relative to the background measurement is -2.5 /6.5 = - 0.384. Magnetic field distortions of the same magnitude (0.30 or 30%) have been observed also for PTEs inside the magnetopause [7].
So the suggestion here is that plasma pockets or blobs with magnetic field signatures similar to those of the PTEs in the magnetopause can be found also in the solar wind.
That is something that potentially reinforces the impulsive penetration mechanisms and scenarios but this topic requires more research as factors regarding the pressure balance and the internal electromagnetic configuration of the events must become more tangible.
The approximate calculations (percent changes) of the magnetic field value for all the events identified in this study are also listed in Appendix A.
Figure 4.10 Density enhancements with magnetic field perturbations.
From the magnetic field data Fig. 4.11 has been plotted which shows all the
corresponding event. What is clear from the graph below is that all density changes are linked together with a perturbation of the magnetic field which occurs simultaneously regardless of the scale width. In most of the cases the magnetic field strength is decreasing locally while the majority has a scale length of less than 25 Re.
Figure 4.11 Magnetic field % change Vs. Scale sizes
It is also important to observe that the highest concentration of events encircled inside the red ellipse on the graph has a scale size below 25 Re and a relative decrease of the magnetic field strength (diamagnetic behavior) between -0.02 and -0.5 approximately.
Figure 4.12 depicts the associated frequency diagram for the relative changes of the magnetic field strength. The highest difference in the magnetic field value corresponds to relative increment of 1.4 (paramagnetic behavior) for an event with scale width of 20 Re.
The diamagnetic and paramagnetic behaviors are referring to the tendency of the plasma blob to oppose or reinforce the ambient magnetic field and it also shows some
characteristics with regard to the internal currents. With the help of Fig. 4.12 someone can see that the high concentration area on Fig.4.11 accounts for the 73% of the input values. A useful figure here is also the relative magnetic field change for the whole data set which has a mean value of -0.2133.
Figure 4.12 dB/B values frequency diagram.
One more plot of particular interest is the one in Fig.4.13 which is providing information as to if there is any obvious correlation between the relative change of the magnetic field dB / B and the value of the density ratio ne. From the graph apparently no such conclusion can be drawn but what becomes evident is that diamagnetic events are prominent compare to the paramagnetic ones and their density ratios are lower than 1.7 while the bulk values for the magnetic field ratio are between -0.02 and close to -0.9. The maximum relative difference though in the magnetic field strength appears for one paramagnetic event with a value of 1.4 relative to the background measurement. Since
the absolute density Ne is used for the calculation of the ne no additional plot is required in order to visualize any further relationship to the relative magnetic field change.
Figure 4.13 dB/B values Vs. Ne /Neb.
4.4 Pressure balance and magnetic behavior
The diamagnetic and paramagnetic behavior can be also explained in terms of pressure balance as a condition of conservation for the energy content on a local scale. By assuming the frozen in condition and that the IMF is “riding” the solar wing someone can assume that current flux tubes are formed as a process where the fast solar wind is catching up with the earlier emitted plasma which has been slowed down in the interplanetary space. Because of the local density enhance a pressure gradient builds up and thus if B doesn’t change spatially along the tube axis or the variation is too small
then a diamagnetic current should exist in order to balance the pressure gradient [9]. As it is stated in Chen [9] “the magnetic field must be low where the density is high”. “The decrease of the magnetic field inside the plasma is caused, of course, by the diamagnetic current”. Equation (4.1) is providing the mathematical expression for such a case of pressure balance.
2
2 o
p B const
+ μ = (4.1)
The size of the diamagnetic effect is given by equation (4.2) which represents the ratio between the terms of eq. (4.1) [9].
2 / 2 o 2/ 2 o
p nkT Particle Pressure B B Magnetic Field Pressure
β = μ =
∑
μ = (4.2)High β plasmas are common in space. If β is high the local value of B can be greatly reduced by the plasma. In that case it is customary to use the vacuum value of B in (4.2) [9].
One possible explanation why the paramagnetic behavior is much less observable in the data set should be peered to the current flux tubes and their relative resonance compare to the associated magnetic field. When there is resonance between the current and that of the IMF any frequency less than this may create a paramagnetic event. This is something which requires more thorough investigation since in this study the number of events exhibiting paramagnetic behavior is limited to 8 though the associated ne ratios are above 1.4 for all of them. What is suggested here might not be accurate but the thought behind the proposed explanation is to induce ideas and to broaden the available perspectives using established principals.
References
[1] M Paredo, J.A Slavin, E. Mazur & S.A Curtis, Three dimensional position and shape of the bow shock and their variation with Alfvénic, sonic and magneto sonic Mach numbers and interplanetary magnetic field orientation. Journal of Geophysical Research .Vol 100 No A5, PAGES 7907 -7916, MAY 1, 1995 [2]D.G. Sibeck, R.E Lopez & E.C. Roelof. Solar wind control of the magnetopause shape, location and motion. Journal of Geophysical Reasearch, Vol.96, No A4, Pages 5489-5495, April 1, 1991
[3] RSI, What’s new in IDL 6.3, IDL 6.3 Edition, April 2006 [4]http://www.esa.int/esaSC/SEM3050P0WF_index_0.html
[5]Palisade Corporation, @ RISK, Risk Analysis and Simulation Add in for MS Excel, User’s Guide V5.5 - May 2009
[6]Tomas Karlson. IDL programs. Alfven Lab - Royal Institute of Technology Sweden
[7]R. Lundin et al. Evidence for impulsive plasma penetration through the dayside magnetopause. Annales Geophysicae (2003) 21: 457-472 © European Geosciences Union 2003
[8]Lamaire, J & Roth, M, Penetration of solar wind elements into the magnetosphere, J. Atmos. Terr.
Phys., 40, 331-335, 1978
[9]Francis F. Chen. Introduction to plasma physics and controlled fusion. Second Edition, Springer 1974.
CHAPTER 5
Conclusions
This Chapter summarizes the most important points derived from the analysis presented previously in Chapter 4. The points are presented in a sequence which follows the analysis while the discussion is supported with more ideas and proposals for further analysis of the subject.
5.1 Data collection
The data gathering process from the Cluster mission archive was itself a difficult process which required a lot of attention to detail. The main target was to identify abrupt density changes as long as the Cluster constellation was taking measurements in the solar wind region. Possible issues in the final catalogue in Appendix A are related to following factors:
1. Missing data from the archive on specific dates or corrupted data measurements decreased the total amount of events in the data set.
2. The human factor is always a potential source for errors in the data set. Some of the impacts are related to misinterpretation of data, exclusion of data which were fulfilling the criteria but were very close to the limits and misjudgments.
The validation of the final catalogue was performed under the supervision of Dr.
Tomas Karlsson and thus the validated set is considered to be suitable for the analysis performed in Chapter 4.
5.2 Position and orientation
The events and their corresponding locations on the GSE coordinate system are sketching out the section of the Cluster orbit which encounters the solar wind region.
Consequently it is logical to infer that these density enhancements can be found through out the whole solar wind stream regardless the position where the measurements are performed. This result can also be verified from other missions like ACE and Wind.
What is important to emphasize here is that since the Cluster mission is intersecting the solar wind stream very close to the bow shock someone has to rule out the possibility that the data are affected or biased by the foreshock or other interactions with the geomagnetic field. This is also the point where the relevancy of the previous comment becomes evident.
One part of the analysis which was also very interesting and possibly requires further investigation is the eastward flow trend for the motion of the identified density enhancements.
5.3 Density and dates of occurrences
The density events are best fitted by an exponential function with 90 % of the identified density ratios to lie in the interval from 1.3 to 1.5781. The fitted function is quite representative of the set in order to create input data for possible Monte Carlo simulations regarding scenarios for future missions and their measurement equipment or validation of mathematical models requiring a full range of solar wing measurements.
5.4 Scale sizes and magnetic field behavior
Since the scale sizes calculation is based on the minimum variance analysis it useful to keep in mind that all widths are close approximations to the real figures. Their possible
deviation from the real numbers is acceptable due to the large scale size of the events measured in Earth Radii.
The plasma pockets or blobs identified in this study do have magnetic field signatures similar to those of the PTEs in the magnetopause but it is not quite clear that they are direct descendants of the solar wind events.
The PTEs inside the magnetopause originated from the magnetosheath plasma may have characteristics which support the impulsive penetration model [8] but their connection to observations from the solar wind regions is under investigation and at the moment no final conclusions are to be drawn based on the analysis provided in this document.
73% of the diamagnetic events identified is this study have a relative magnetic field change in the range between -0.02 to -0.5. Another useful figure here is that for the whole data set the mean value of the relative magnetic field change is -0.2133. Very close to the magnetic field perturbations observed in the magnetopause region.
5.5 Pressure balance and magnetic behavior
The diamagnetic behavior of the events is signifying the existence of current flux tube like entities which are the result of a pressure balance process. Although in the data set the majority of the density enhancements were associated with decrease in the magnetic field (diamagnetic behavior) still there were events (8 cases in total) which were exhibiting paramagnetic behavior. One possible explanation why the paramagnetic behavior is much less observable in the data set should be peered to the current flux tubes and their relative resonance compare to the associated magnetic field.
5.6 Further investigation and suggestions
One very interesting possibility would be if data could be gathered in the region of the magnetosphere and the solar wind simultaneously. Then it would be possible to identify
events in the solar wind region and track down their evolution as they are approaching the geomagnetic field. Of course such an expedition would require a lot of resources and might not be possible for the foreseeable future but still it would provide very useful insights for the magnetospheric interactions.
Another possible scenario would be to create microsatellite constellations which are cost efficient and can actually launched into orbit massively in order to perform plasma measurements. Of course the measurement equipment should also go through miniaturization modifications which might require some years or decades of technological advances.
In all cases it is important to stress that the potential to acquire multipoint measurements can be extended from a single orbital plane to multiple ones with surveillance capabilities for plasma variations. Such a mission would provide dynamical data and not stationary point measurements. The enormous possibilities and potentials would stem out from the fact that actual plasma motion could be tracked down in an analogous way the weather satellites are providing images for atmospheric phenomena on a regular basis.
5.7 Final result and contribution of this study
The overall contribution of the present study can be summarized to the fact that plasma blobs similar to those previously identified inside the magnetosphere have been discovered also in the solar wind stream close to the Earth’s vicinity. Although the precise mechanism of transfer inside the magnetosheath is not fully investigated here a connection to the impulsive penetration model is attempted as a supportive explanation.
The data presented in this study can be used for further analysis in order to correlate phenomena observed outside and inside the geomagnetic field.
