Magnetopause properties - methods

Studying the dynamics of structures in space requires precise knowledge about their normal direction as well as their velocity. Discussion on the normal direction, veloc-ity and thickness of the magnetopause cannot be separated from the techniques used to determine these properties. When the magnetopause is a tangential discontinuity i.e. when there is no reconnection going on and therefore no normal magnetic field component, a cross product of the magnetic field vectors on either side of magne-topause would yield the normal direction. In reality this may not be the case all the time and a one dimensional magnetopause discontinuity assumption together with

∇ ·B = 0 will allow a strictly constant normal magnetic field. In this approach the normal direction would be the magnetic field direction that varies the least [Sonnerup and Cahill, 1967, Sonnerup and Scheible, 1998]. This method is called minimum variance of magnetic field (MVAB). Early studies of the magnetopause used this approach.

Earlier spacecraft such as Explorer 12 encountered frequent magnetopause cross-ings during a particular inbound or outbound orbit indicating that the magnetopause is always in motion. Based on the time difference between successive crossings, early magnetopause studies deduced the magnetopause motion to be of a few tens of kilo-metres. However, with the advent of ISEE and AMPTE missions, which provided lower order plasma moments with better time resolution, it became possible to use single spacecraft magnetic field and plasma data to estimate both the normal direc-tion and the speed of magnetopause. Berchem and Russell [1982] used successive crossings of the magnetopause by ISEE-1 and ISEE-2 spacecraft to estimate the magnetopause speed. They used the time difference between the two spacecraft serving the same magnetopause structure together with the spacecraft speed to ob-tain magnetopause speed. These results from ISEE indicated magnetopause speeds up to 200 km/s with an average around 40 km/s.

Earliest of the single spacecraft methods to estimate magnetopause speed is based on the de Hoffmann and Teller [1950] frame determination. This method in-volves in finding a frame that moves with a velocityV_HT and the plasma velocity transformed into this frame aligns with the magnetic field leading to zero electric field in this frame of reference. The component of VHT along the normal

direc-1.3. THE MAGNETOPAUSE 11 tion is the speed of the magnetopause. Aggson et al. [1983] were the first to apply this technique to estimate the magnetopause motion. Since then, MVAB technique has been used with deHoffmann–Teller frame technique [Khrabrov and Sonnerup, 1998a] to determine both magnetopause normal and velocity from single spacecraft measurements. Another single spacecraft method based on Faraday’s law called minimum Faraday residue (MFR) [Khrabrov and Sonnerup, 1998b, Terasawa et al., 1996, Terasawa et al., 1997] uses both magnetic field and plasma moments to pre-dict the magnetopause normal direction and velocity. This method used the fact that Faraday’s law allows a constant electric field tangential to a one-dimensional discontinuity of fixed structure moving with constant velocity. A comparison of the results from MFR with dual spacecraft timing estimates of magnetopause velocity using AMPTE data showed considerable disagreements [Bauer et al., 2000]. These disagreements are due to uncertainties in single spacecraft MFR method as well as to difficulties in obtaining an accurate timing of similar structures.

A more recent development in single spacecraft methods is to use the conserva-tion laws [Kawano and Higuchi, 1996]. An improved precision of plasma moments from the Cluster mission [Escoubet et al., 2001] allowed Sonnerup et al. [2004a] to present a technique called the minimum mass residue flux (MMR) method which uses only plasma moments i.e. velocity and density. This method uses residue min-imization process and predicts the normal and velocity of the magnetopause. Later, Sonnerup et al. [2006] extended the same methodology to be used with any conser-vation law and presented a generic method which they named the generic residue analysis (GRA) method. The main aim of this method is to use all available data from a single spacecraft or even combining data from multiple spacecraft and to provide magnetopause normal and velocity estimation from different conservation laws. Sonnerup et al. [2006] used the conservation of total energy (MTER), the conservation of linear momentum (MLMR), the conservation of entropy (MER) as well as MMR, MFR and MVAB together with deHoffmann–Teller (deHT) analysis.

They also extended the method used in Sonnerup et al. [2004a] to combine results from different methods to yield a single prediction. This is a composite (COM) of all other methods which weights each conservation law based on a predefined crite-ria. However, both the above cited studies using MMR and GRA are single event studies, where a comparison of the results among different conservation laws and against multispacecraft methods showed a good agreement.

InPaper IIIandPaper IVwe used the GRA method with only MVAB together with deHT, MFR, MMR and MER to arrive at a composite (COM) prediction of the magnetopause normal and speed based on eigenvalue ratios. Paper III presented a comparison of magnetopause normal and velocity derived from both multispacecraft constant velocity approach (CVA) and GRA method using data set of 28 magnetopause crossings. To our knowledge this is the first study that compared results from GRA and multispacecraft methods. Magnetopause normals and velocities were estimated for about 4000 magnetopause crossings by Cluster spacecraft inPaper IVutilizing GRA method. To our knowledge this is the largest study that has tested and used GRA methodology.

The first multispacecraft analysis technique to deduce the orientation and the speed of a discontinuity in space was presented by Russell et al. [1983]. They studied

12 CHAPTER 1. INTRODUCTION an interplanetary shock using ISEE-1, ISEE-2, ISEE-3 and Interplanetary Monitor-ing Platform (IMP) magnetic field data. This method assumes that the discontinuity is planar and does not change much over the time scales it passes by all the spacecraft and all the spacecraft observe the same structure. Then the time difference between different spacecraft and the spacecraft separation vectors are used to deduce the normal direction and the speed of the discontinuity. Russell et al. [1983] used the crossing duration of the discontinuity over each spacecraft to estimate the thickness of the discontinuity. Since this method assumes a constant velocity of the disconti-nuity, it became known as constant velocity approach (CVA) and the method was later reviewed by Harvey [1998] and Schwartz [1998]. The magnetopause crossings analyzed inPaper I,Paper IIandPaper IIIused CVA method to obtain magne-topause normal and velocity. Paper IIIpresented a small magnetopause crossing data set where CVA results were compared against GRA results and found a good agreement for majority of the crossings. This is the first study that examined and compared CVA results against the GRA method.

Haaland et al. [2004a] developed a method called a constant thickness approach (CTA) on the assumption that magnetopause thickness is constant. This method used both the crossing times and crossing durations by each spacecraft together with the spacecraft separation vectors. Haaland et al. [2004a] also compared the results on magnetopause velocity and normal against various single spacecraft methods and CVA and another multispacecraft method called discontinuity analyze (DA) [Dunlop and Woodward, 2000]. The DA method initially calculates the normal directions from all four spacecraft from MVAB method and obtains the timing information to estimate the speed and thickness of the discontinuity. Paschmann et al. [2005]

combined CVA and CTA methods by weighting them to derive a new method called minimum thickness variation (MTV) method which allows for small acceleration of the discontinuity and minimizes thickness variations. The normal direction is a weighted average of CVA and CTA methods. Haaland et al. [2004b] presented a new method based on the conservation of electric charge called minimum variance of cur-rent. This method uses magnetopause current density and then finds the minimum variance direction to the current layer, presenting the magnetopause normal.

Comparison of magnetopause normals obtained from single and multispacecraft methods using AMPTE data showed considerable disagreements [Bauer et al., 2000]

and were interpreted to be because of the errors in single spacecraft techniques and inaccurate timing. A statistical study using Cluster data [Paschmann et al., 2005]

estimated the magnetopause to be of 100 to 3000 km thick with a peak at 400–800 km. Magnetopause speeds range from smaller than 10 km/s up to 300 km/s with a peak around 20–40 km/s. These recent Cluster results are in agreement with previous results from ISEE [Berchem and Russell, 1982] and AMPTE-IRM [Phan and Paschmann, 1996] missions.

For magnetopause crossings used inPaper IandPaper II, the magnetopause velocities are about 20-40 km/s. The statistical study in Paper III resulted in similar magnetopause velocities i.e. few km/s to 100 km/s from both CVA and GRA methods. Apart from a couple of event studies [Haaland et al., 2004a, Son-nerup et al., 2006],Paper IIIis the first study to compare multispacecraft methods against single spacecraft methods. Case studies comparing the normal direction from

1.3. THE MAGNETOPAUSE 13 different methods [Haaland et al., 2004a, Sonnerup et al., 2006] indicated that dif-ferent single spacecraft methods yield normals that are within a few degrees of an average normal direction from all 4 spacecraft and are also in agreement with various multispacecraft methods. Normal direction comparison in Paper IIIshowed very good agreement between GRA and CVA normals as well as clear differences. These differences could be due to the breakdown of the assumptions such as planarity, startionarity of the magnetopause and uncertainties associated with both methods.