Electron heating in collisionless shocks observed by the ...ltu.diva-portal.org/smash/get/diva2:1188860/FULLTEXT01.pdf · Electron heating in collisionless shocks observed by the

Electron heating in collisionless shocks

observed by the MMS spacecraft

Martin Svensson

Space Engineering, master's level

2018

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

Master Thesis

Electron heating in collisionless shocks observedby the MMS spacecraft

Author:Martin Svensson

Supervisor:Andreas Johlander,

Swedish Institute of Space Physics

Examiner:Lars-Göran Westerberg,

Luleå University of Technology

Submitted for the degreeMaster’s degree in Space Engineering with specialization in Space and Atmospheric Physics

at Luleå Unversity of Technology

Luleå University of TechnologySwedish Institute of Space Physics

March 8, 2018

Martin Svensson Luleå Tekniska Universitet March 8, 2018

Acknowledgements

I want to thank my supervisor Andreas Johlander who has provided me guidance throughout thewhole project. Without Andreas expertise in shock physics, data analysis and structuring of researchwork, this thesis would not have been possible. I would also like to thank the MMS group at IRFUppsala for the inspiring and interesting meetings. Then also the rest of the personnel at IRFUppsala for a great welcoming and interesting discussion subjects at the lunch table. Lastly Iwant to thank my girlfriend Hanna Larsson for her unending support. The data used in this thesiswere obtained from the Science Data Center at the Laboratory for Atmospheric and Space Physics(LASP) at the University of Colorado, Boulder.

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AbstractShock waves are ubiquitous in space and astrophysics. Shocks transform directed particle flow

energy into thermal energy. As the major part of space is a collisionless medium, shocks in spacephysics arises through wave-particle interactions with the magnetic field as the main contributor.The heating processes are scale dependent. The large scale processes governs the ion heatingand is well described by magnetohydrodynamics. The small scale processes governs the electronheating, it lies within the field of kinetic plasma theory and is still today remained disputed. Astep towards the answer for the small scale heating would be to measure the scale, in order torelate it to a known instability or other small scale processes.

The multi-spacecraft NASA MMS spacecraft carries several high resolute particle and fieldinstruments enabling almost instantaneous 3D particle measurements and accurate measurementsof the magnetic field. Also the separation between the four MMS spacecraft is as small as ≤ 8kmfor a certain mission phase. This allows for new approaches when determining the scale which forshocks has not been possible before when using data from previous multi-spacecraft missions withspacecraft separation much larger. The velocity of the shock is large compared to the spacecraft,thus the shock width cannot be directly measured by each spacecraft. Either a constant velocityhas to be estimated or we could use gradients of a certain parameter between the spacecraft asthe shock flows over them. The usage of gradients is only possible with MMS as all the spacecraftcould for MMS be within the shock simultaneously. The change for a parameter within the shockis assumed to be linear between the spacecraft and measurements. It is also assumed that thegradient of the parameter maximizes in the shock normal direction. Using these assumptions twomethods have been developed. They have the same working principles but are using two or fourspacecraft for linear estimation at each measurement point. From the gradient and parametricdata the shock ramp width could then be found. The parameter used in this thesis is the electrontemperature.

The methods using one, two and four spacecraft were tested using electron temperature datafrom different shock crossings. Two problems with the gradient methods were found from theresults, giving false data for certain time spans. To avoid these problems, the scale of the electrontemperature gradient was determined for roughly half the shock ramp.

It was found using the two and four spacecraft methods that an assumption of constant velocityfor the shock speed is an uncertain assumption. The shock speed varies over short time scales andin the shock crossings analysed the constant velocity estimations were generally overestimated.From the two and four spacecraft methods roughly half of the temperature rise in the shockramp occurred over 10.8 km or 12.4 lpe. This is almost a factor of two greater than previous scaleestimates using Cluster data and a multi-spacecraft timing method for shock speed estimation.

From the results it is concluded that the methods when using gradients between spacecraft hassome restrictions. They can only be used for MMS data, requires quasi-perpendicular high Machnumber and will give false results if the temperature is disturbed by interacting hot plasma clouds.However, even though we have these limitations for the tested gradient methods, they were foundbetter and more reliable compared to previous methods for shock scaling.

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Contents

1 Introduction 51.1 Shocks, fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Collisionless Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Structure of supercritical quasi-perpendicular shocks . . . . . . . . . . . . . . 81.2.2 The Earth’s bow shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Reversible and irreversible heating within shocks . . . . . . . . . . . . . . . . . . . . 111.4 Magnetospheric Multiscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Fast Plasma Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.2 Fluxgate Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Scope and motivation 142.1 Scales hints the physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Previous work and the problem with scaling . . . . . . . . . . . . . . . . . . . . . . . 152.3 Improvements with MMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Methods 163.1 Shock crossing selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Shock normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Shock normal angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.3 Perpendicular and parallel temperature . . . . . . . . . . . . . . . . . . . . . 203.2.4 Mach number, beta and shock scale values . . . . . . . . . . . . . . . . . . . . 20

3.3 Temperature gradient scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 Single Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Two Spacecrafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.3 Four Spacecrafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Results 254.1 Crossing 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.1 Single Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 Two Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.3 Four Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1.4 Comparison and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Crossing 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Crossing 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Crossing 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Summary and Conclusion 44

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6 Future work 44

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1 Introduction

1.1 Shocks, fluid dynamics

Shock waves are dramatic accelerations caused by an impact of a supersonic flow. The first interestsin shocks arose in the late 19th century when high speed flows in the field of fluid mechanics weregiven a lot of attention. In 1885 (Blackmore 1972) Ernst Mach defined his famous criteria for shockdevelopment when he realized that for a shock to evolve the relative speed between a flow and anobstacle has to be greater than the local speed of sound, cs. Mach was able to visualize his idea bydrawing a sound generating projectile with speed greater than the speed of sound, creating a coneknown as the Mach Cone, see Figure 1. From Mach’s criteria were also the shock characterizationparameter, the Mach number M defined as

M =u

cs, (1)

where u is the speed of the flow.

Figure 1: Visualization of a Mach Cone. A projectile travelling to the right in a flow with a relativespeed greater than the speed of sound. Image credit: Gibson, University of Connecticut.

1.2 Collisionless Shocks

By far the most abundant state of matter in the Universe is plasma (Chen 1984). Plasma is foundin stellar interiors, the solar wind, the interplanetary space, lightnings and at certain altitudesin planetary atmospheres. A plasma consists of free charge carriers typically electrons and ions.Shock waves are ubiquitous in astrophysical plasmas. As like the shocks described in the section

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above in fluid mechanics, a plasma shock is formed when an obstacle is placed in a supersonic flow.Astrophysical plasmas are often carrying a magnetic field which, first according to De Hoffmann& Teller (1950) is the primary factor for energy exchange between different particles and waves incollisionless shocks. Collisionless means that the collision rate in the medium is small enough to benegligible. For the plasma in the interplanetary medium the typical scale for the mean free path ismuch greater than other associated scales for a shock wave (Balogh & Treumann 2013). These scaleshighly depend on the specific type of plasma, for example the scale in the interplanetary mediumis much larger than the scale in the solar atmosphere. Thus comparing shocks in fluid mechanicswhere bi-collisions between molecules are the dominant interaction with collisionless shocks in lowdense plasmas where the scales are to large for collisional interaction to be dominant. The dominantinteraction in collisionless shocks has to be wave-particle interaction mainly caused by the magneticfield. An important property for a shock is its ability to reflect incoming upstream ions at highenergy flows for high Mach numbers. The shock will then not be able to heat all the plasma fastenough through wave-particle interactions. The shock then has to reflect partitions of the inflowingions in the upstream direction, see Edmiston & Kennel (1984).

An important parameter characterizing a shock is the Mach number. The Mach number isdescribed in section 1.1 as the speed of the flow divided by the speed of sound. For a collisionlessplasma the physical circumstances is different and hence the Mach number has to be described inanother way. A common way in space physics is to use the Alfvén Mach Number MA, which is themagnitude of the upstream flow speed, Vu divided by the Alfvén speed, vA. The Alfvén speed isthe phase speed of an Alfvén wave and is given by

vA =B0√µ0ρ

, (2)

where B0 is the solar wind magnetic field, µ0 is the vacuum permeability and ρ is the mass densityof the plasma. Thus MA reads

MA =VuvA

=VuB0√µ0ρ

. (3)

An Alfvén wave is propagating with its wave group velocity parallel to the magnetic field. Inmagnetohydrodynamics, there also exist a wave type which has a group velocity that could beperpendicular to the magnetic field: the magnetosonic wave, see Balogh & Treumann (2013). For amagnetosonic wave the background magnetic field and the oscillating electric field E1, generated bythe oscillating ions, will compress the plasma by the E1 ×B0 drift which lies in the same directionas the propagating wave. Thus as not the case for Alfvén waves the compression or the pressuregradient ∇p needs to be considered for magnetosonic waves. The group speed of a magnetosonicwave is

v2ms(θBV ) =c2ms2±√c4ms4− v2Ac2scos2θBV , (4)

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where θBV is the angle between the magnetic field and the plasma bulk velocity. cms is theperpendicular magnetosonic speed given by

c2ms = v2A + c2s (5)

and cs is the speed of sound as

c2s =γekBTe + γikBTi

me +mi. (6)

The specific heat ratios for electrons and ions, γe,i, is typically three (3) for electrons and one (1)for ions (Chen 1984), kB is the Boltzmann constant, me,i are the electron and ion masses and Te,iare the electron and ion temperature. The two solutions in equation 4 refers to the fast, v+ms andslow magnetosonic wave, v−ms, where v−ms < vA < v+ms. The particle bulk speed relative to thespeed of the obstacle is in general larger than v+ms. In these common cases the shock is considereda fast-mode shock i.e a steepened fast-mode wave. Thus v+ms is used for the magnetosonic Machnumber as

Mms =Vu

vms(θBV ). (7)

Also, as Mms indicates the shock wave steepening for perpendicular flows it is the ‘real’ Machnumber when considering shocks. The magnetosonic Mach number can be calculated by combiningequations 4-6.

Another important parameter, which determines the particle motions, is the shock normal angle.The shock normal angle is the angle between the shock normal vector and the magnetic field. Asthe movement of reflected particles is highly dependent on the direction of the magnetic field in theupstream region the structure of a shock is highly dependent on this angle. Figure 2 illustrate thereflection of particles and the described shock normal angle

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Figure 2: Illustration of the Shock normal angle θBn and particle reflection for different θBn at theshock surface. Image credit: Balogh & Treumann (2013)

The third characteristic shock parameter important to the large scale structure of the shock isthe plasma beta β. The plasma β is the ratio of plasma thermal pressure to magnetic field pressure,given by

β =nkT

B2/2µ0. (8)

β simply describes the ratio of pressure issued by thermic movements to pressure issued by themagnetic field. Thus it is a measure of the relative importance between kinetics and magnetic fieldsfor a flow, see Chen (1984).

1.2.1 Structure of supercritical quasi-perpendicular shocks

Shocks of all kinds are steepened waves between an obstacle and a flow. Depending on thecharacteristics of the flow and obstacle they might, however, have a different structure in detail. Forsupercritical shocks parts of the ion flow is reflected. For quasi-perpendicular shocks the reflectedions are convected back to the shock as the magnetic field lines are tangential to the shock surface.The shock foot is the upstream region that is dominated by the incoming plasma and the shockreflected ions. The size of the shock foot is proportional to the ion gyroradius, rci of the reflected ions.On the downstream side of the foot is the ramp, which is the steepening part of the shock wave. Theramp is characterized by a sudden, sharp increase of magnetic field and density, see Figure 3. Before

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the shocked downstream region in the magnetosheath there is a large scale rise in the magnetic fieldcalled the overshoot which forms above some critical Mach number for quasi-perpendicular shocks.The overshoot region is directly followed by an undershoot region. The changing from overshootto undershoot will follow the structure of a damped oscillator and is caused by electron currentsin the shock ramp (Balogh & Treumann 2013). The spikes are then thought to represent energeticions trapped in the shock front (Saxena et al. 2005). See Figure 3 for an overview picture of thestructure using MMS ion density data from a shock crossing. The overall shock structure varieswith the characteristic parameters for shocks, a quasi-perpendicular shock is for instance typicallysteeper and has less fluctuations than a quasi-parallel shock.

Figure 3: Illustration of the different physical regions for a shock. Arrows mark these regions. Datafrom a shock crossing measured by MMS.

1.2.2 The Earth’s bow shock

The Earth’s bow shock is found at an approximate distance of 15RE (RE = the radius of the Earth)from the center of the Earth in the day side direction, see Farris et al. (1991). This distance variesand is highly dependent on the solar wind ram pressure. The distance between the magnetosphereand bow shock is a few RE and is called the stand-off distance. The region between the bow shock

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and the magnetopause is called the magnetosheath. The Earth’s bow shock has a curved geometryand is often considered symmetric around the Earth-Sun line with a close to parabolic shape. Anillustration of the Earth’s bow shock can be found in Figure 4. Here one also can see the dependenceof shock normal angle as discussed in section 1.2, with the escaping ion and electrons into theirrespective foreshock region. The angle between the Earth-Sun line and the interplanetary magneticfield also represent the realistic case in the figure as it generally is approximately 45 degrees, seeParker (1958).

Figure 4: Illustration of Earths bow shock with the solar wind in the upstream direction andmagnetosheath downstream. The dependence of the shock normal angle θBn is clearly visible asthe development of a ion and electron foreshock region is marked at the quasi-parallel side.

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Common coordinates used when working with Sun-Earth related interactions like the bow shockis the Geocentric Solar Ecliptic, GSE coordinates. For GSE coordinates the origin is placed at theEarth’s center and the x-axis points towards the Sun. The positive z-axis is the Ecliptic north poleof earth. The y-axis completes the right-handed system and points in the dusk-ward direction. TheGSE coordinate system is used throughout this thesis.

1.3 Reversible and irreversible heating within shocks

The energy distribution for an ideal gas where collisions exists is, typically, a Maxwellian to minimizethe entropy. For a collisionless medium the distribution does typically not follow a Maxwellian. Fora particle flow in a collisionless medium, the distribution is obtained by measuring the energy forevery particle in all directions. From the distribution the temperature can be considered of as thewidth. A wider distribution means that there are more particles with higher speed in the gas.Heating can then be thought as the broadening of the energy distribution. Thus if one observethe distribution at different times, with the same number of particle counts but with a broaderdistribution for the later, the gas has been heated. In plasma physics it is often interesting todiffer between the distributions for different particle velocity directions like parallel, antiparalleland perpendicular directions relative to the magnetic field lines. In Figure 5 one can see theelectron energy distributions for different times from a shock crossing of the Cluster spacecraft.Here the distribution broadening and thus the heating of the electrons can be observed for differentdirections.

Figure 5: Broadening of particle distributions for different direction and time. Data from a Clustershock crossing. Image credit: Schwartz et al. (2011).

One can differ between different heating mechanisms as if they are reversible or irreversible. Areversible heating is for shocks related to the slow change in electromagnetic field over large scales.While the irreversible heating means the small scale changes, which are thought to be generated bypitch-angle scattering within the shock that is irreversible. Heating because of changes in magnetic-and electric field strength can be performed in both directions. The heating or acceleration of

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particles by the magnetic field is related to the magnetic moment invariance

µ =mv2⊥2B

(9)

where m is the particle mass, v⊥ is the particle velocity perpendicular to the magnetic field and Bis the magnitude of the magnetic field. Heating of particles caused by the electric field is relatedto the induced electric field in the shock ramp. The electric field is induced as the ion penetratesdeeper into the ramp before being reflected compared to the electrons. The electrons will then alsobe accelerated by the Lorentz force

F = q(E + v ×B) (10)

where q is the particle charge andE is the electric field. This acceleration will for quasi-perpendicularshocks be directed perpendicular to the magnetic field. Here the magnetic field lines arequasi-tangential to the shock surface and E is directed in the shock normal direction. Thus fromequation 9 and 10 this represents the reversible heating because the equations allows for bothincreasing and decreasing B and E over large scales. One can then for shocks use the distributionfor perpendicular particle velocities to observe the the reversible heating.

In Figure 5 one can observe heating in both the perpendicular and parallel direction. Asdescribed above the reversible processes are related to the perpendicular temperature. There alsohas to be irreversible heating as the distribution for parallel velocity follows the perpendicularclosely for different time cuts. The origin of this irreversible heating has not been determined. Onetheory is that it may be caused by scattering of electrons from a plasma instability within the shockregion as Debye-scale electric fields has been observed, see Bale et al. (1998). Clues to the answerof the origin of the irreversible heating should be found by knowing how the spatial width of theshock ramp scales.

1.4 Magnetospheric Multiscale

The Magnetospheric Multiscale (MMS) mission is a NASA unmanned space mission that consists offour satellites flying in a close tetrahedron formation (Burch et al. 2016). The main science goal forMMS is to provide in-situ measurements in the ion and electron diffusion regions caused by magneticreconnection. Magnetic reconnection is a physical process that occurs when different magnetic fieldlines realign themselves into a new pattern, this transfers magnetic energy into heat and particlekinetic energy. Therefore, magnetic stored energy is released into the surrounding plasma (Freyet al. 2003). From the measurements the goal is to further understand the physics of magneticreconnection. The instruments on MMS are providing data at an unprecedented time resolution.Together with the small spacecraft separation which for phase 1b (see Figure 6) is ≤ 8km, MMS isable to quantify the physics at scales smaller than the boundary layers in the magnetosphere andfurther out in space.

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The MMS mission is divided into several phases in which the orbits and the spacecraft separationare changed. Figure 6 shows how the mean spacecraft separation at apogee has varied with time,the times for the different orbit phases 1a, 1b and 2a are marked as the shaded green areas.

Figure 6: MMS spacecraft separation as a function of time. The black line shows the meanseparation distance, the shaded yellow area shows the variance in separation and the greenshadowing marks the specific orbital phases. In the lower part the local time at apogee is plottedas a function of time. Image credit: Johlander, IRF Uppsala.

All the four spacecraft of MMS named MMS1, MMS2, MMS3 and MMS4 carries an identicalset of plasma and fields instruments where the ones used in this thesis will be described in furtherdetail below. The data products produced by the instruments of MMS are classified accordingto their degree of time resolution and calibration. The instruments generates data sets in either’burst’ mode or ’fast’ mode depending on the time resolution, by which burst mode has the higherresolution.

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1.4.1 Fast Plasma Investigation

The Fast Plasma Investigation (FPI) instruments of MMS is described by Pollock et al. (2016). TheFPI instruments are measuring the differential particle flux for MMS with an unprecedented hightime resolution. At each spacecraft the FPI consists of four dual 180 × 90◦ ion spectrometers andfour dual 180× 90◦ electron spectrometers separated around the periphery of the MMS spacecraft.This provides MMS with 3D distributions within the energy range of 10 eV/q to 30 keV/q at atime resolution of 150ms for ions and 30ms for electrons. The FPI uses the electrostatic type ofspectrometers with a hemispheric geometry for the electron instrument and a toroidal geometryfor the ion instrument. The general working principle for an electrostatic analyzer is that over twoparallel curved conducting plates with adjustable voltages an electric field is created. The incomingparticle then needs to have a specific velocity (Energy) in order to pass between the curved platesas the centripetal and electrostatic forces need to match. If the particle passes through the curvedplates it will be detected and counted by a particle detector. This process can be described bybalancing the equation

W

q=

1

2ER, (11)

where W is the particle energy, q is the charge, E is the applied electric field and R is the radius ofthe curved plates. Thus by adjusting the electric field the particle distribution for different energiescan be measured.

1.4.2 Fluxgate Magnetometers

The FluxGate Magnetometers (FGM) of MMS is described by Russell et al. (2016). Precisemeasurement of the magnetic field is of utmost importance for MMS to obtain its scientificobjectives. MMS uses two different kinds of triaxial fluxgate magnetometers provided by twodifferent manufacturers. These are mounted on separated booms and are called the Digital FluxGatemagnetometer (DFG) and the Analogue FluxGate magnetometer (AFG). Their maximum samplingfrequency is 128Hz. Data obtained from the two sensors are then continually inter-compared. Afluxgate magnetometer consists of drive windings and sense windings around a magnetic core. Thedrive windings are fed with an alternating current which is strong enough to saturate the coil foreach half-period. This periodic variation will become asymmetrical if an external magnetic field isadded to the system by which the degree of asymmetry is proportional to the added magnetic field.This is measured by means of the sense winding (Hoymork 2002).

2 Scope and motivation

2.1 Scales hints the physics

The electron temperature rise in Collisionless shocks are as described in section 1.3 an interplaybetween reversible and irreversible changes. To further understand this interplay we need to examine

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and increase our understanding of the shock ramp and what processes within the ramp that arecausing this irreversibility. An important parameter which should give clues about the physicswithin the ramp is the scale or thickness. This can then be attributed to candidate processeslike kinetic instabilities, dispersion or unstable shock reformation. One should also relate themeasured thickness to the corresponding Mach number, shock normal angle and plasma beta valueas these are believed to affect the ramp width. In this thesis the main focus is to test and comparenovel techniques for determination of the temperature gradient scale. The obtained results will becompared to previously reported results.

2.2 Previous work and the problem with scaling

A similar study as this has been made using data from the ESA Cluster mission which also was amulti-spacecraft system, exploring the interaction between the solar wind and the magnetosphereof the Earth. By statistical studies Hobara et al. (2010) argues that partitions of the electrontemperature gradient in a quasi-perpendicular shock scale as the electron inertial length, λpe.Schwartz et al. (2011) show data from one Cluster measurement of a shock, which supports thetheory of electron inertial length scaling. Also the multi-spacecraft ISEE performed measurementsof shock crossings.The main issues for both Cluster and ISEE with scaling of small scale boundary parameters (likethe temperature in a shock) is the large spacecraft separation and instrumental time resolution.For Cluster the spacecraft separation varied but was never small enough for the possibility tohave all the spacecraft situated within the ramp at once. Thus they could not directly measurespatial differences in the ramp. Schwartz et al. (2011) assumed a constant shock velocity which wasestimated using a timing method. In this timing method they used the time when the spacecraft ofCluster measured a specific magnitude of the magnetic field in the shock ramp. The time difference,together with the spacecraft positions gave a constant velocity estimate (Schwartz 1998), see Figure7.

Figure 7: Magnetic field data from all the Cluster spacecraft. The lines intersecting the curvesmarks the times for when the steep ramp crossed the spacecraft. The magnetic field data has beenshifted 20nT for clarity. Image credit: Schwartz et al. (2011)

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As the particle instrument of Cluster requires a complete rotation about the spacecraft spinaxis for a complete 3D particle distribution the time resolution for a 3D particle distribution is≈4 s, (Johnstone et al. 1997). This limits the particle measurements at thin boundary layers likethe steep ramp of a shock. Complete 3D distribution for shocks with high propagation speeds couldtherefore not be obtained. However approximations of gyrotropic particle distributions were madefor some special cases when the spacecraft spin axis were aligned with the magnetic field lines, seeSchwartz et al. (2011).

2.3 Improvements with MMS

As described in section 1.4 MMS is primary a mission for further understanding of the magneticreconnection process. Thus, as also described in section 1.4, MMS is highly suitable for in-situmeasurements at small scales giving a new tool for deeper understanding of the physics inside theplasma boundary layers. This means that MMS will provide measurements from within the steepramp of the bow shock which gives the opportunity to directly measure gradients within the shockramp. The gradients could then be used to estimate the spatial span for the steepening parameters.The FPI instrument allows MMS to make instantaneous 3D measurements of particle distributionswith a time resolution of 30ms for electrons and 150ms for ions see Pollock et al. (2016). This isa major improvement compared to previous similar missions where the full 3D distributions onlycould be obtained after the spacecraft had rotated 360◦.

3 Methods

MATLAB has been the program used for analysis of the data. It was used since it typically has shortdevelopment time and there is a large collection of routines for spacecraft data analysis available athttps://github.com/irfu/irfu-matlab from which several routines has been highly applicable for thescope of this thesis. All the used functions and scripts needed to generate the results in section 4can be found at https://www.dropbox.com/sh/9cdwtr0zvv8tbr1/AACp1jwi7Uodz_YXPwWaVkBna?dl=0.

3.1 Shock crossing selection

To obtain the desired data from each shock crossing the first thing one needs to know is when ashock crossing has occurred. The simplest way to find out is to read the reports at the MMS SITLReport website, at https://lasp.colorado.edu/mms/sdc/public/. Here a Scientist In The Loop(SITL), continuously analyses live data from MMS within the mission Region Of Interest, ROI. TheROI is the region where the main physical processes of interest for MMS occurs and the instrumentsare running in burst mode. If the SITL observes something of interest in the data, the data at thespecific time interval will be transmitted down to Earth with a higher time resolution (burst mode)

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https://www.dropbox.com/sh/9cdwtr0zvv8tbr1/AACp1jwi7Uodz_YXPwWaVkBna?dl=0

https://www.dropbox.com/sh/9cdwtr0zvv8tbr1/AACp1jwi7Uodz_YXPwWaVkBna?dl=0

https://lasp.colorado.edu/mms/sdc/public/


than otherwise. Thus more data is available at these events of interest for MMS. The SITL thenreports the time spans chosen with descriptions of these events through a text document.

When a crossing is found at the SITL website, a good methodology is to check the reportedtime for the crossing using ’Quicklook’ at the MMS science data center website for confirmation.’Quicklook’ allows for easily accessible investigation of the data showing only the fast mode(lower time resolution) data for the desired spacecraft, instrument and time span, see Figure 8for an example. From ’Quicklook’ one can then also get certain hints if the shock crossing isquasi-perpendicular or quasi-parallel by observing the magnetic field data where the magneticfield is usually more turbulent for quasi-parallel shocks. The particle distribution should also givecertain hints because the upstream distribution for a quasi-perpendicular shock should be a clearcold ion beam. For a quasi-parallel shock the distribution is more smeared out because of thereflected particles. These particles travels upstream following the interplanetary magnetic fieldlines (Schwartz 1991). If the ’Quicklook’ data showed signs for a quasi-perpendicular shock, thedesired data such as magnetic field, temperature, distributions etc. for all the MMS spacecraft wasdownloaded from the local servers at IRF Uppsala using a function in MATLAB. At this stage allthe obtained data were level 2 burst mode. The analysis methods described below are sensitive tocertain parameter values and shock wave structure. Throughout the data analysis we discoveredsome requirements for the upstream data by observing differences mainly in the density data forelectrons and ions. It is required that the upstream region has a high density to avoid randomerrors due to a low number of particle counts in the detector. A high upstream temperature forreducing the eventual over estimations due to the instrumental energy limits, see section 1.4.1. Fora high upstream density we then have more particle samples for better estimation of temperature.See the FPI instrument properties and limits in Pollock et al. (2016).

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Figure 8: Example of an image obtained by using ’Quicklook’ for fast data investigation. The blackarrow marks the time for an example of a quasi-perpendicular shock. The blue arrow marks aquasi-parallel shock example. Image credit: MMS Science data center.

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3.2 Parameters

The large scale structure of a shock is mainly characterized by the three shock parameters mentionedin section 1.2. The applicability for the shock scaling methods described below are partly determinedby the large scale structure as a clear distinction between the shock and the upstream region isrequired. Also the shock ramp should be steep, with as few plasma fluctuations in the data aspossible.

3.2.1 Shock normal vector

The shock normal vector n̂ is a unit vector with direction normal to the shock surface. In spacecraftobservations n̂ can be determined in various ways using the field and plasma measurements. Oftenused is the assumption of 1D-planar shock geometry. The method used in this report uses amix of the co-planarity assumption for both the magnetic field and the bulk velocity fpr ions(Abraham-Shrauner 1972). In Paschmann & Daly (1998) at page 255 the mixed method numberthree

n̂ = ± (∆B×∆V)×∆B|(∆B×∆V)×∆B|

(12)

where ∆B is the difference of the magnetic field vectors upstream and downstream, Bu and Bd.The same goes for the velocity difference ∆V. The ± symbol refers to if the normal should pointupstream or downstream. Throughout this report n̂ is pointing upstream. Equation 12 was thenimplemented in MATLAB and by taking averages upstream and downstream for the magnetic fieldand ion drift velocity, n̂ was found.

3.2.2 Shock normal angle

The shock normal angle, θBn, is the angle between the upstream magnetic field and the shocknormal vector. After determining n̂, θBn can easily be calculated. Simply use the scalar product

Bu · n̂ = cos θBn||Bu|| (13)

to get

θBn = cos−1Bu · n̂||Bu||

. (14)

Equation 14 was then implemented in MATLAB and with the shock normal and upstream meanof the magnetic field, θBn could be found.

19


3.2.3 Perpendicular and parallel temperature

The temperature data was for each measurement obtained as a second order tensor Tij , which canbe represented as a 3x3 matrix. The scalar mean temperature is given by

T =Tr(Tij)

3=

1

3

3∑i=1

Tii (15)

where the trace, Tr means a summation of the diagonal terms. See Paschmann et al. (1998) forfurther description of extracting parameters from velocity distributions. In collisionless plasmas,particle distribution functions are typically non-isotropic and it is for this thesis interesting toshow the rise of parallel T‖ and perpendicular temperature T⊥ separately. As described in section1.3, T‖ refers to the small scale fast irreversible heating while T⊥ refers to the large scale slowreversible heating. Parallel and perpendicular refers to the direction relative to the magneticfield. To extract these from the the temperature tensor Tij , one has to perform a tensor rotationusing transformational matrices to align Tij to the magnetic field. The magnitudes of the differentdirectional temperatures are then found as the pivot elements in the matrix representation of therotated Tij .

3.2.4 Mach number, beta and shock scale values

Besides the above mentioned shock normal angle, from section 1.2 it is mentioned that the Machnumber and beta are also characterizing parameters for a shock. These values were calculated basedon equation 3 for the Mach number and equation 8 for the beta-value.

The ion and electron plasma inertial length, lpi and lpe, used for scaling the obtained results aregiven by

lpi,e =c

ωpi,e(16)

where c is the speed of light and ωpi,e is the electron and ion plasma frequency given by

ωpi,e =

√ni,ee2

ε0mi,e. (17)

Here e is the elementary charge, ni,e is the number density, ε0 is the vacuum permittivity and mi,e

is the particle mass, see Chen (1984).

3.3 Temperature gradient scale

The orbital speed for each MMS spacecraft (≈1km/s) is typically much smaller than the speed ofthe shock, which highly varies between ≈10-150 km/s and is according to Meziane et al. (2014)dependent on the changes in upstream solar wind ram pressure. Therefore, shock crossings should

20


be thought of as the shock moving forward and backward over the spacecraft. It is thus not possibleto directly from data products measure the shock ramp width. To determine the width of the shockramp we then have to use other approaches. For this thesis three different methods are used fordetermination of the spatial scaling. They have been categorized below by how many spacecraftrequired for the specific method.

3.3.1 Single Spacecraft

For determination of the spatial scale when using only one spacecraft, one way is to first assumethat the shock is moving relative to the spacecraft with a constant velocity, Vsh. The spatial scalecould then easily be determined by multiplying with the time elapsed while the spacecraft wasinside the shock ramp. Thus when using only one spacecraft, the method used for estimating Vsh iscrucial. Schwartz et al. (2011) used a multi-spacecraft timing method which is not as efficient whenusing data from MMS. Because of the small separation between each spacecraft of MMS accuratetiming is difficult as their measurements are very similar. See Figure 9 for the measurements of thedifferent spacecraft of MMS and compare with Figure 7 for Cluster.

21


Figure 9: Magnetic field data measured by all the spacecraft of MMS, plotted on the same timeaxis. The Figure visualize the close separation between the spacecraft of MMS. Making timing aninefficient method for velocity estimation

Thus timing the exact same data value will give inaccurate results because of possible smallscale shock fluctuations. In this report another method is used developed by Gosling & Thomsen(1985), and is hereby called the Gosling-Thomsen Method (GTM). With the GTM the constantshock velocity is given by

Vsh = (Vu · n̂)x0

1± x0(18)

where ∆t is temporal width of the shock foot, Vu is the particle bulk velocity upstream and ωci thereflected ion cyclotron frequency. This reflection is assumed to be specular. x0 is the turnarounddistance for a reflected ion, given by

x0 =ωcit1(2cos2θBn − 1) + 2sin2θBn sin(ωcit1)

ωci∆t, (19)

22


where the turnaround time for the reflected ion t1 is given by

t1 = cos−1(

1− 2 cos2 θBn

2 sin2 θBn

). (20)

Thus by using equation 19-20, equation 18 could be solved and the shock velocity Vsh be found.Then for the spatial distance, the obtained Vsh was multiplied with the time span for the shockcrossing.

3.3.2 Two Spacecrafts

As mentioned in section 2.3 one of the major benefits with MMS compared to Cluster is the smallseparation distance between the spacecraft. This allows for all the MMS spacecraft to be inside theshock ramp at the same time. Hence, it is possible to measure the gradient of a specific parameterinside the shock ramp. From the gradient data we then know how the parameter spatially changeswhich we can use to determine distance by combining with means of measured changes for theparameter. An important concept when determine the change of a parameter χ at an arbitrarypoint in a flow is to consider the material derivative

Dχ

Dt=∂χ

∂t+ u · ∇χ, (21)

where u is the flow velocity and t is time. Comparing the two right hand terms in equation 21for χ in a space within a shock one can conclude that the spatial term u · ∇χ dominates. Thisas χ mainly is changed because the shock with the velocity u is flowing with a steep gradient∇χ much larger than the temporal change in χ itself. Thus we assume that the shock profile istime-independent between two measurement points. We also assumes that the gradient for χ islargest in the shock normal direction. Thus we use the simplification that shock only moves in theshock normal direction. One can then simplify equation 21 by cancelling the temporal term andalso rewrite by using the shock velocity Vsh as flow velocity with respect to an arbitrary point

Dχ

Dt= Vsh(∇χ · n̂). (22)

Shi et al. (2006) used a technique for instantaneous determination of velocity between data points,using Cluster magnetic field data. A similar method will here be applied for shocks using an arbitraryparameter χ instead of the magnetic field. We could then use equation 22 and measurements of χfrom two spacecraft for determination of the spatial width. This technique has the main assumptionthat the χ has to linearly increase or decrease between two spacecraft and two measurement points.The material derivative in equation 22 is the total derivative for the measured parameter χ betweentwo measurement points. It can be rewritten as

Dχ

Dt=〈χ〉i+1 − 〈χ〉i

dt(23)

23


where dt is the time lapsed between the measurements and 〈χ〉i is the mean value between the twospacecraft at measurement i given as

〈χ〉i =χi,m + χi,n

2. (24)

The index m and n denotes the two spacecraft used. Consider ∇χi as the gradient of χ betweentwo spacecraft as

∇χi =χi,m − χi,nxi,m − xi,n

(25)

where xi,m,n denotes the position of the spacecraft m or n for each measurement i along theshock normal. From the assumption that χ increases linearly between two measurements, the∇χ mentioned in equation 22 refers to the mean of ∇χi between two measurements which can berewritten as

〈∇χ〉i =∇χi+1 +∇χi

2. (26)

To not confuse 〈∇χ〉 with means between spacecraft, the variable name is changed as

〈∇χ〉i = βi. (27)

Vsh can be rewritten for a specific measurement i as

Vsh = Vi =〈x〉i+1 − 〈x〉i

dt(28)

where 〈x〉i is the mean position of spacecraft m and n at measurement i given as

〈x〉i =xm + xn

2. (29)

By substituting equation 23-28 in equation 22 one obtains

〈χ〉i+1 − 〈χ〉idt

=〈xi+1〉 − 〈xi〉

dtβi. (30)

Then cancel the time derivatives on each side and solve for 〈xi+1〉 which gives

〈xi+1〉 =〈χ〉i+1 − 〈χ〉i

βi+ 〈xi〉. (31)

Equation 31 can then be solved by iteration for the complete shock width along the shocknormal. The two spacecraft used should be the pair with separation lines that lies the most alongthe shock normal vector for optimizing the results. This is done by comparing the separation vectorobtained from the spacecraft positions to the shock normal vector for every possible spacecraftcombination.

24


3.3.3 Four Spacecrafts

Once again as a major benefit with MMS is allowing all spacecraft to be inside the shock ramp,a possible method to determine the gradient of a specific parameter inside the ramp is to use theconcept of reciprocal vectors, see Vogt et al. (2011) and Chanteur (1998). By the same principles asfor the two spacecraft method described above the increase or decrease of a parameter χ betweenspacecraft and between measurement is assumed linear. The difference is how to obtain the ∇χiterm. We can by using the positions for the four spacecraft construct four reciprocal vectors aslinear estimators between the spacecraft (Chanteur 1998). The reciprocal vectors are given by

kα =rβγ × rβλ

rβα · rβγ × rβλ(32)

where kα is the reciprocal vector for each spacecraft α and r is the distance vector between spacecraftα, β, γ and λ, see Vogt et al. (2011). The formula for the gradient between the four spacecraft canthen be constructed as

∇χi =

4∑α=1

kαχi,α. (33)

We can then obtain the average position of the spacecraft using

〈x〉i+1 =〈χ〉i+1 − 〈χ〉i∇χi

+ 〈x〉i (34)

where 〈χ〉 is the four-spacecraft average of χ, and ∇χi is the gradient linearly interpolated betweeni and i+ 1.

Before applying this method for spatial scale determination a good methodology is to plot thefrom the reciprocal vectors obtained gradient data. The gradient data over the shock region shouldthen be well distinguishable from the solar wind gradient data. One should also plot the gradientin the shock normal direction for evaluation of the assumption that the gradient is largest in thisdirection. For this one could also perform a maximum variance analysis (Paschmann & Daly 1998)which if the assumption is correct should give approximately the normal vector direction.

4 Results

When fulfilling the requirements described in section 3.1 the picked shock crossings were analysedas described in the method section above. The date for each picked crossing and other associatedparameters and scales are found in Table 1.

25


Table 1: List of shock crossings with corresponding upstream parameters used in this thesis.

Crossing 1 2 3 4Date 2016-11-10 2016-12-06 2016-12-18 2017-01-15MA 12.9 9.14 11.75 11.2θBn [◦] 83.21 82.7 58.58 83βe 1.68 1.68 2.45 2.18lpe [km] 0.87 1.27 1.73 1.50

The obtained results for each shock using the three methods will be presented and analysed,ordered by the given crossing names in Table 1. Since the analysis methods are the same for allthe crossings only the obtained final results for crossing 2, 3, and 4 will be shown for time/spaceefficiency.

4.1 Crossing 1

The shock crossing which occured 2016-11-10 between 16:58:48 and 16:58:51 is shown in Figure 10for different time spans. The shock is quasi-perpendicular and the upstream solar wind has a densityof 37.0 cm−3 and an electron temperature of 18.4 eV , which is unusually high, see Tribble (2003) forcomparison. The different panels in the figure represents parameters used in the methods describedin the section above. It is a common manner in space physics to plot the magnetic field data inthe top panel as a map giving hints of the location and space weather for the specific measurementtime span.

26


(a) Zoomed out view of a shock crossing. (b) Zoomed in view of a shock crossing.

Figure 10: Shock crossing which occured 2016-11-10. Grey section in Figure a) marks time spanfor Figure b)

4.1.1 Single Spacecraft

Using the method for a single spacecraft described in section 3.3.1. Using ∆t = 2.3 s and fcp =ωci/(2π) = 0.17Hz, the constant shock velocity, Vsh was estimated as -70 km/s. A negative valuebecause as seen in Figure 10 the spacecraft is first situated downstream. Then after the shockcrossing they are situated upstream. The shock is therefore moving in the negative direction as seen

27


from the spacecraft. Figure 11 was then obtained. The left figure shows the electron temperaturedata and the distance travelled by the shock. The right figure displays the electron temperature asa function of distance travelled by the shock. The shock ramp is marked as a yellow area.

(a) Upper panel shows the electron temperature dataover time. Bottom panel shows the distance travelledby the shock over time.

(b) Electron temperature vs distance.

Figure 11: Obtained distance by using the single spacecraft method. The yellow shading marks theshock ramp

The single spacecraft method is using a constant velocity thus the bottom panel is expected toshow a linear result, which it does. Using this method the obtained spatial width for the completeshock ramp is 68 km which scales to 78lpe. The temperature rose from 23.4 eV to 40.8 eV, see theborders of the yellow sections in Figure 11. Thus using this method over the complete temperaturerise one can conclude that the temperature rose 17.4 eV over 78lpe. By looking at the temperaturedata in Figure 11 one can see that the temperature rise over the complete ramp is far from linear.Notice the section in the central region of the shock ramp where the temperature is close to constantfor a short period of time. This could be fluctuations in the temperature data but it could also meanthat the shock speed is changing at small spatial and temporal scales. It is therefore reasonableto divide it into smaller parts and measure the scale over those. Schwartz et al. (2011) who usedCluster data only scaled the steepest half of the temperature rise in the shock ramp, see Figure 12.For the best possible comparison using this method we also had to scale the steepest half of thetemperature rising in Figure 13a. This section of the ramp was marked using the yellow shadingand is found in Figure 13b.

28


Figure 12: Electron temperature and magnetic field data from a shock crossing by Cluster. Imagecredit: Schwartz et al. (2011)

2016-11-10 UTC

0

5

10

15

20

25

30

35

40MMS1

16:58:47.0 .5

2016-11-10 UTC

5

10

15

20

(a) Upper panel shows the electron temperature data overtime. Bottom panel shows the distance travelled by theshock over time.

-250 -200 -150 -100 -50 0 50 100 150 200 250

x[km]

0

10

20

30

40

50

60

T[e

V]


Figure 13: Obtained distance by using the single spacecraft method. The yellow shading marks thesteep part of the shock ramp

29


Schwartz et al. (2011) as seen in Figure 12 estimated that half of the temperature rise in theshock ramp occured over a 6.4lpe. Here the temperature rose from 23.4 eV to 34.1 eV over 23.2 kmwhich corresponds to a 11.3 eV rise over 26.7lpe.

4.1.2 Two Spacecraft

When using the two spacecraft method, described in section 3.3.2 first the spacecraft pair whichwere closest separated tangential to the shock normal had to be found. Using the position data forall the spacecraft and the obtained shock normal, Figure 14 was generated.

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

2

4

6

8

10

12

13 14

23

24

34

12

13 14

23

24

34

12

13 14

23

24

34

12

13 14

23

24

34

Figure 14: Separation distance between two spacecraft along the tangential (y-axis) and normal(x-axis) direction relative the shock. Every mark consists of two numbers representing the specificspacecraft pair.

From Figure 14 it is clear that MMS2 and MMS3 are the spacecraft pair closest separatedacross the shock normal. Using the two spacecraft method with MMS2 and MMS3, Figure 15 wasobtained.

30




Figure 15: Obtained distance by using the two spacecraft method. The yellow shading marks theshock ramp

In the bottom panel of Figure 15a two lines representing the distance travelled by the shock overtime are found. The two lines are identical with an offset representing the mean spatial distancebetween the spacecraft. The total temperature rose 16.33 eV. As seen in the panel the slope turnsslightly negative at some points which demonstrates an inevitable problem when using gradients forspatial scaling. Comparing with the electron temperature data in the upper panel on can observethat these slopes represents sudden sign changes for the temperature gradient. This is most likelydue to an encounter with a small plasma cloud containing hot electrons because as seen in thepanels the sign change is highly temporal. However if the source is a plasma cloud the temporalerror will not affect the total width measured as the gradient restore itself after the encounter.Then as seen in the bottom panel of Figure 15a there is a sudden steep rise of distance travelledwhich does not restore itself. This error arises from when the temperature measurement plots forthe different spacecraft intersects. Comparing with the upper panel for the same temporal regionthis intersection is found. When the lines intersect the denominator in equation 25 approaches zero.This error is also visible in the temperature profile plot in Figure 15b. The occurrence of this error

31


is a restriction when using this method. Figure 15a was then adjusted with a cut-off for times laterthan 16:48:57.28. Figure 16 was then obtained.



Figure 16: Obtained distance by using the two spacecraft method. The yellow shading marks theshock ramp.

The unavoidable error is thereby removed. Figure 16 then shows a temperature rose of 11.24 eV(≈ 67% of the total rise) over 18.51 km or 21.23lpe

4.1.3 Four Spacecraft

The third method using four spacecraft was used and is described in section 3.3.3. The gradientdata obtained by the usage of reciprocal vectors as a linear estimator of the spatial change betweenall the spacecraft is found in Figure 17.

32


Figure 17: Temperature gradient obtained from the four spacecraft method. The upper panel isshowing the gradient in x,y, z direction. The middle shows the total gradient compared to thegradient in the normal direction. The third panel shows the electron temperature for all MMSspacecraft.

From Figure 17 the gradient within the time span for the shock crossing is well distinguishablefrom the later times within the region of the solar wind. This is an indication of the validity ofthe obtained data. Also in the middle panel one can see that the gradient in the normal directionfollows closely the magnitude of the total gradient which confirms the assumptions made for the twoand four spacecraft methods that the gradient is largest in the normal direction. From the gradientand temperature data the spatial width of the shock ramp was then obtained and is plotted as afunction of time in Figure 18 .

33


Figure 18: Distance obtained by the four spacecraft method over the whole shock ramp. Upperpanel shows temperature, middle panel shows the gradient data in the shock normal direction andthe bottom panel shows the distance profile.

The errors observed when using the two spacecraft method are present also when using the fourspacecraft method. However as the linear assumption is assumed between four spacecraft ratherthan two the errors from small plasma perturbations described above will be smoothed out. Theerror which occurs when the gradient crosses zero and change time are still a problem also for thefour spacecraft method. Thus as for the case with two spacecraft, Figure 18, the temperature dataand spatial width are plotted again with a cut-off for times later than 16:48:57.28.

34


Figure 19: Distance obtained with the four spacecraft method with a cutoff at 16:48:57.28 were themethod breaks down.

From Figure 19 one can then observe that for the 11.24 eV rise(≈ 67% of the total rise) intemperature the spatial width was measured as 16.28 km or 18.67lpe.

4.1.4 Comparison and discussion

To compare the different methods the different distance plots obtained from the three differentmethods are plotted together with the cutoff used for the two and four spacecraft methods, thisplot is found in Figure 20a. Also the broadening of the distribution for perpendicular, parallel andanti-parallel temperature at different time cuts is found in Figure 20b, symbolizing the irreversibleheating described in section 1.3.

35


(a) Comparison of the different methods for the completeshock ramp. Upper panel shows the electron temperature,bottom panel shows the profiles obtained from the threemethods.

(b) Distribution at two different time cuts. The dashed linesrepresenting the initial shock ramp close to the solar wind.The solid lines representing a time the ending of the shockramp close to the downstream region.

Figure 20: Comparison between the scaling methods.

In Figure 20a one can see that the curves for spatial widths using two and four spacecraft followeach other closely. This was expected because the methods are using the same principles but withdifferent number of spacecraft when linearizing between measurements. By looking at the plots fortwo and four spacecraft in the bottom panel it is clear that the shock velocity varies at these timescales. Thus as the spatial width obtained when using the single spacecraft method was larger, theestimated constant shock velocity was overestimated. Another property observed in Figure 20a isthat the shock appears to decelerate to a velocity similar to the spacecraft, as the distance travelledby the shock relative the spacecraft is zero for time spans in the bottom panel. This is a propertyobserved in several shock crossings but for obvious reason could not been observed when using asingle spacecraft method with a constant shock velocity estimate.

As been done for the single spacecraft method it is interesting to compare the obtained widthwhen using two and four spacecraft with recent data. Schwartz et al. (2011) as mentioned only

36


estimated the width of roughly half of the temperature rise, where the temperature gradient waslargest. For the single spacecraft method the steep temperature rise marked in Figure 13b wasused. This part of the shock ramp contains the error described in the results for both two and fourspacecraft where the temperature gradient crossed zero. Also when estimating the gradient usingthe four spacecraft method in Figure 17, it can be seen that the other steep part of the shock ramphas a larger gradient in the normal direction. Thus the other steep part of the shock ramp is usedfor comparison, see Figure 21.

Figure 21: Comparison of the different methods for roughly half the shock ramp. Upper panelshows the electron temperature, bottom panel shows the profiles obtained from the three methods.

This part of the shock ramp represents a rose of 10.1 eV (≈46% of the total rise). Then asseen in Figure 21 the distance obtained from the gradient methods using two and four spacecraftfollows each other closely while the distance obtained from the single spacecraft method appearsoverestimated. The resulted distance over this part of the shock was found as 12.4 km for the twospacecraft method and 10.8 km for the four spacecraft method or 14.2lpe and 12.4lpe. Compared

37


to previous scaling estimates by Schwartz et al. (2011) using Cluster data, the results obtained byusing the gradient methods shows a factor of two greater shock width.

4.2 Crossing 2

The third crossing which occured in 2016-12-06 had a solar wind density of 17.4 cm−3 and atemperature of 14.3 eV. The overview map is found in Figure 22 and the resulted distance profilesover the steepest half of the shock ramp in Figure 23.

(a) Zoomed out view (b) Zoomed in view.

Figure 22: 2016-12-06 shock crossing. Grey section in Figure a) marks time span for Figure b)

38



This part of the shock ramp represents a rose of 11.7 eV (≈61% of the total rise). Then as seenin Figure 23 the distance obtained from the two methods using two and four spacecraft follows eachother closely while the distance obtained from the single spacecraft method appears overestimated.The resulted distance over this part of the shock was found as 15.9 km for the two spacecraft methodand 13.8 km for the four spacecraft method or 12.5lpe and 10.8lpe.

4.3 Crossing 3

The third crossing which occured in 2016-12-18 had a solar wind density of 9.4 cm−3 and atemperature of 21.2 eV. The overview map is found in Figure 24 and the resulted distance profilesover the steepest half of the shock ramp in Figure 25.

39


(a) Zoomed out view (b) Zoomed in view.


40




4.4 Crossing 4

The fourth crossing which occured in 2017-01-15 had a solar wind density of 12.5 cm−3 and atemperature of 11.3 eV. The overview map is found in Figure 26 and the resulted distance profilesover the steepest half of the shock ramp in Figure 27.

41


(a) Zoomed out view of a shock crossing which occured20170115

(b) Zoomed in view of a shock crossing which occured20170115.


42




43


5 Summary and Conclusion

Shocks are ubiquitous in space. Through wave-particle interactions between a magnetic field anda particle flow, the flow bulk speed is decelerated and heated. The origin of the electron heatingobserved in collisionless shocks is a poorly understood subject consisting of a complex interplaybetween the large scale reversible heating and the small scale irreversible heating. To relate thesmall scale irreversible heating to a known plasma property, the scaling of the steep shock ramp hasto be determined. The electron heating in the nearby bow shock of the Earth has been measuredby the MMS spacecraft which by their small spacecraft separation and high instrumental timeresolution allowed for new approaches for the scale estimation. The new possibilities by the useof gradients within the shock obtained from MMS data has been tested with, for shocks, noveltechniques using one, two and four spacecraft. From the data presented in the result section aboveone can conclude that an assumption of a constant shock velocity will lead to errors in spatial scalingas the shock velocity changes throughout the ramp. The methods for spatial scaling using two andfour spacecraft gave similar results but the four spacecraft method showed smaller errors. In theorythe gradient methods could be generalized enabling us to use even more spacecraft than four. Theresults from this work show that for one shock crossing roughly half of the temperature increasehappens on a length scale of 10.8 km or 12.4 lpe. Three other shock crossings by MMS show similarresults. These results differ from previous results using Cluster by Schwartz et al. (2011), wherethey found a length scale 2-3 times smaller than what is found here. The results from MMS alsohighlights that the four spacecraft methods shows a clear overview of the shocks movement for eachcrossing. Thus with the high resolute data of MMS, the usage of gradients between measurementswithin a shock has been shown as a new possible method for a better scaling estimation.

6 Future work

To further confirm the reliability of the gradient methods tested in this thesis, statistical studiesare necessary with more shock crossings and varying shock parameters. Also as seen in most ofthe shock crossings analysed in this thesis, the shock appears to slow down in the center of theshock ramp. By the lack of other possible solutions we believe that it actually represents a dropin shock velocity for a short time. Then at last the distributions shown in this report, could beplotted together with the restrictions from magnetic mirroring and the Sonnerup-Wu mechanism,see Balogh & Treumann (2013). This will show a more clear result of which particles that areallowed in the distribution. From this the particles accelerated from pitch-angle scattering willmore easily be found as they should be found within this not permitted region of the distribution.

44


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