Temperature instabilities in the solar wind plasma

Transcription

1 Temperature instabilities in the solar wind plasma DISSERTATION zur Erlangung des Grades eines»doktors der Naturwissenschaften«in der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum vorgelegt von Michał Michno aus Bochum Institut für Theoretische Physik IV: Theoretische Weltraum- und Astrophysik Bochum, 2014

2 1. Gutachter: Prof. Dr. R. Schlickeiser 2. Gutachter: PD Dr. H. Fichtner Abgabe der Dissertation: Tag der Disputation:

3 Contents Acknowledgements v 1 Introduction Motivation Outline Solar wind and plasma waves basics Solar wind and its anisotropy Basic plasma description Instabilities MHD firehose Kinetic firehose: cyclotron resonance Derivation of the dispersion relation Kinetic description of the plasma Linearized equations Maxwell tensor Particle orbits Fourier-Laplace transform Dispersion relation Simplification of the dyad ˆQ Reduction for parallel modes in a thermal plasma Weak amplification limit WIND measurements WIND spacecraft WIND s orbit Instruments on WIND MFI SWE SWE components i

4 4.4 Faraday cup Calculation of moments Data processing Concluding remarks Kinetic firehose Kinetic firehose classification Dispersion relation MHD limit Approximations of the plasma dispersion function Two peak structure Proton peak ξ p Proton peak ξ p Empirical solutions PFHI 1 - ξ p PFHI 2 - ξ p EFHI - ξ p PFHI vs EFHI Summary Firehose instability numerical solutions Electron effects on PFHI Proton effects on EFHI A p > A p < Conclusion and Summary Double-polytropic MHD model Double-polytropic closure Dispersion relation Stability analysis Slow mode: 1. case A < 0, B > Slow mode: 2. case A > 0, B > Slow mode: 3. case A < 0, B < Threshold conditions ϑ dependence Slow firehose mode Interlude: oblique firehose Mirror mode Comparison of growth rates Summary ii

5 8 Effects of damping Weakly amplified solutions Alfvén wave dissipation Instability condition Summary and outlook 147 Appendices 153 Appendix A Fourier transformation 155 A.1 Fourier transformation A.2 Laplace transformation Appendix B Plasma dispersion function 157 Appendix C Empirical solutions 159 C.1 PFHI C.2 EFHI Appendix D Numerical solutions 163 D.1 Plasma dispersion function D.2 Dispersion relation D.3 Additional figures List of figures Bibliography Curriculum vitae I V XVII iii

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7 Acknowledgements In the following sentences I would like to thank all the people that have helped me throughout this thesis. I do wish to express my gratitude to a few in particular. First and foremost, my thanks go to my supervisor Prof. Reinhard Schlickeiser for the possibility to work at the Institute for Theoretical Physics IV, for the helpful discussions and all his guidance. I sincerely appreciate the trust Prof. Schlickeiser placed in me by allowing me to pursue areas I found of interest. I am very grateful for his extensive support and for giving me the chance to attend many national and international schools and conferences. I also want to thank PD Dr. Horst Fichtner, who willingly agreed to be the second reviewer of this thesis. His thorough and detailed comments helped to considerably improve this work. I am indebted to my roommate Stefan Artmann for his support and boundless expertise in data analysis, L A TEX-typesetting, and organizing lecture exercise sessions. Thank you for all the inspiring discussions and for your friendship. I would like to thank Gisela Buhr and Verena Kubiak for providing me with all the help and assistance I needed to navigate the bureaucracy. I am also thankful to our system administrator Bernd Neubacher. I thank Marian Lazar for the support and excellent collaboration in our work, the stimulus and also the honest criticism. A huge thanks goes to the Institute TP IV in general for the open doors, good times, and for improving my table soccer skills. I am more than grateful to Prof. Peter H. Yoon for his interesting talks and discussions which inspired the work presented in Chapters 5 and 6, and for providing me with his Vlasov solver. Finally, I would like to express my gratitude to my friends and my family for their love, unconditioned support and motivation. v

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9 Chapter 1 Introduction To write a general introduction, it is probably a good idea to start with the title and break it down into the concepts of instabilities, temperature, a plasma, and the solar wind. An unstable system can be intuitively explained by comparing it to a stable system. While a stable system will react weakly to a weak disturbance and returns to its equilibrium state, an unstable system will show a strong response and be in a completely different state afterwards. The notorious example for instability learned by generations of physics students is the ball on the top of a hill, which will roll down when pushed slightly, while a ball nudged in a valley returns to the valley floor and represents a stable situation. It is obvious that stability is a desired property in engineering, and instabilities have to be studied in order to avoid them, for example a bridge or building should be constructed to prevent a resonance disaster. The concept of instability is widely studied beyond engineering and technology, as every sufficiently complex system can become unstable - and not only at the level of merely using this terminology, but often applying a full-fledged mathematical formalism. At present, a prominent example is the stability of the market, where a stable economy is characterized by a constant growth and low inflation (Bordo and Jeanne (2002) [13] ). Other sciences are medicine, where not only the unbound growth of a cancer is an unstable system, but also cancerous cells are identified by the stability of their genome (Alderton (2013) [5] ), or meteorology, where various atmospheric instabilities are associated with a high variability of the weather (Zhao et al. (2006) [114] ). The fluid instabilities 1 encountered in meteorology lead us to the perhaps most prominent example of an unstable system, the magnetically confined plasmas in tokamak fusion reactors. A plasma is basically a hot gas, except 1 For a definition, see Section

10 1. Introduction that it is not only composed of neutral but also contains charged particles and, thus, subject to electric and magnetic forces. One of the many obstacles in realizing a power plant which produces quasi-unlimited, cheap, clean, and secure energy is that the confinement time is limited due to a multitude of instabilities which destroy the equilibrium where the fusion process can take place, and even damage the reactor wall (Graves et al. (2012) [37] ). Much of the plasma instability research has been and still is driven and motivated by the goal of finally building a working fusion device. The inspiration for such a device is the Sun, which also consists of plasma but works somewhat differently. In this thesis, we discuss the Sun s wind, which is a constant emission of mainly protons and electrons which fills the interplanetary space and extends far beyond the planetary system. This wind - the solar wind - mainly consist of charged elementary particles and is, thus, a plasma, although much cooler and more dilute than a fusion plasma. To give a comparison, the number of particles per cubic centimeter in the solar wind in the vicinity of Earth is of order one, while air in an "earthly wind" has about molecules in the same volume. How can such a dilute system, which easily beats the best man-made vacua on Earth by orders of magnitude 2, be unstable? Temperature Anisotropy Ap = T,p/T,p Parallel plasma beta β,p Figure 1.1: Data distribution of WIND proton measurements in β,p -A p parameter space. The colored contours indicate the number of measurements in the respective parameter region (resolution 60x60, cuto at 50, compare Chapter 4 for more information). 2 Ultra-high vacuum chambers can reach densities of 100 particles per cm 3, see Gabrielse et al. (1990) [27]. 2

11 1.1. Motivation The two key ingredients are the high temperature of roughly T 10 5 Kelvin and the interplanetary magnetic field. Temperature as defined by statistical mechanics is a measure of the mean kinetic energy per degree of freedom, k B T E kin = 1 2 mv2, where v denotes the particle velocity. Due to the magnetic field, the charged particles are subject to the Lorentz force 3, F = q v B, (1.1) c which does not act on particles moving parallel to the magnetic field. Thus, it makes sense to define a parallel temperature T, which measures the mean velocity in parallel direction, and a complementary perpendicular temperature T. It is intuitively clear that a system in equilibrium should have roughly the same temperature in both orientations. However, there are certain processes in the solar wind which increase one temperature component with respect to the other, so that T = T, which is called temperature anisotropy. In such a configuration, there is an excess of energy stored in one of the temperature components which may be released due to a triggered instability - just like the potential energy of the ball on the hill. 1.1 Motivation Figure 1.1 illustrates measurements of protons in the solar wind. The temperature anisotropy on the ordinate is defined as the ratio of the two temperatures 4 A p := T,p /T,p, where the subscript p denotes the particle species, the parallel plasma beta on the abscissa is the ratio of parallel kinetic to magnetic pressure, β,p = p kin p mag = n pk B T,p B 2 /8π, (1.2) which describes whether kinetic (β,p 1) or magnetic (β,p 1) phenomena will dominate. The contours indicate how often the plasma was measured at a given configuration (β,p, A p ) in parameter space. Note how the data is sharply confined in a rhomb-like configuration around isotropy A p = 1 and β,p = 1. Such a confinement of solar wind particles is a well-established feature from the observational point of view, and has been confirmed by different spacecraft measurements for protons, He 2+ (Vela: Hundhausen et al. (1967) [50], Hundhausen (1968) [49], Helios: Marsch et al. (1982) [72], ACE: Gary et al. (2001) [35], WIND: Kasper et al. (2006) [54], Maruca et al. (2012) [75] ) and 3 Note that we use Gaussian cgs-units throughout this thesis. Vector quantities are written in bold face, a = (a x, a y, a z ) R 3. 4 Note that some studies use a different definition A = 1 T /T or A = T /T 1. 3

12 1. Introduction electrons (Vela: Montgomery et al. (1968) [82], Pioneer: Formisano (1969) [25], Helios, Cluster, and Ulysses: Štverák et al. (2008) [107] ). This visualizes two challenging fundamental problems of plasma astrophysics: the explanation of the relaxation of cosmic collision-poor plasmas with temperature anisotropic distribution functions to isotropy (A p = 1), and the equipartition of the magnetic field energy density and the parallel kinetic energy density (β,p = 1). This equipartition is not only measured in the solar wind, it is a well-known but so far unexplained property of many dilute cosmic sources, including especially the interstellar medium (Parker (1966) [88] ). The explanation of solar wind equipartition conditions by plasma instabilities might remedy this situation. The basic idea is as follows: as an isotropization by elastic two-body Coulomb collisions can be ruled out due to the low density of the solar wind, the plasma is isotropized by temperature anisotropy driven instabilities. The remarkably sharp edges of the rhomb-like data distribution in Figure 1.1 define thresholds of these instabilities, and the plasma is only stable for parameters inside the rhomb-like configuration. If a plasma would start outside the rhombus, an instability would set in, generate fluctuations and quickly relax the plasma distribution back to the stable regime within the rhombic configuration. At present, the borders for large beta β,p > 1 are understood to be the thresholds of firehose type instabilities (A p < 1), and cyclotron or mirror instabilities (A p > 1), see Hellinger et al. (2006) [45], Bale et al. (2009) [9]. However, these instabilities do not fit particularly well to the data and the importance of the respective instability types is still under debate. Furthermore, the established thresholds are derived for idealized conditions which may not be appropriate in the solar wind. In particular, it is assumed that the electron population is isotropic when studying proton instabilities. In this study, we inspect the influence of anisotropic electrons on the proton firehose and the effect of anisotropic protons on the electron firehose, and provide new analytical descriptions of the two instability types. We also find that the proton firehose instability is markedly changed by the effect of anisotropic electrons and the fit to the data improved. We also investigate the mirror instability in an MHD model with a double-polytropic closure (after Hau and Wang (2007) [41] ), which improves the fit to the data and provides new insights on the so-called oblique firehose. The low beta regime (β,p < 1) is presently poorly understood. Only recently, it has been demonstrated that the confinement limits at small beta in principle can be provided by electromagnetic instabilities (Schlickeiser and Škoda (2010) [97] ), although the thresholds did not match the observations particularly well. We continue and improve this work to agree better with the 4

13 1.2. Outline Figure 1.2: Outline of the instability models presented in this thesis. measured data by including collisional damping. Figure 1.2 summarizes the models presented in this thesis. 1.2 Outline This work is organized as follows: in Chapter 2, we briefly introduce the basic concepts of the solar wind and plasma instabilities. The dispersion relation for parallel electromagnetic plasma modes, which is needed to calculate the kinetic firehose instability, is derived in Chapter 3. Proton measurements of the WIND mission, which are the longest 5 available in-situ observations of the solar wind plasma near 1 AU, are used to test our results in this study. The WIND mission and data analysis is introduced in Chapter 4. In Chapter 5, new analytical solutions describing kinetic firehose instabilities are derived, and in the subsequent Chapter 6 the threshold conditions are investigated numerically. In Chapter 7, a double-polytropic MHD model is used to study oblique instabilities. The improved low beta regime threshold solution is briefly discussed in Chapter 8. Finally, we give a short summary and outlook in the last chapter. 5 Data is recorded since November

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15 Chapter 2 Solar wind and plasma waves basics The goal of this section is to give a brief introduction to the solar wind and plasma instabilities to an extent which makes it possible to understand the motivation of the subsequent instability analysis, including a brief review of the current state of knowledge. Unless otherwise noted, Section 2.1 is based on Chapter 6 from Prölss (2004) [92], Section 2.2 is based on the diploma thesis Michno (2009) [79], which is based on the books by Seshadri (1973) [100] and Krall and Trivelpiece (1973) [58], where a more detailed and in-depth introduction can be found. 2.1 Solar wind and its anisotropy For a long time it was assumed that the space between the Sun and planets is an empty void. Evidence that this might not be true have been provided by the observations of solar flares closely followed by geomagnetic storm. The idea of a continuously emitted solar wind was first formulated by Biermann (1951) [12] in order to explain the orientation of the ion tail of comets, see Figure 2.1. However, it was Parker (1958) [87] who postulated and explained the existence of the solar wind 1 in his famous analysis why static models of the Sun s atmosphere cannot be correct as they are not in hydrostatic equilibrium with the interstellar medium. The reality of the solar wind was finally proven by direct measurements by the Soviet satellite Lunik 1 in Parker also coined the term solar wind, while Biermann called it Korpuskularstrahlung which roughly translates to "particle radiation". 2 See for more information. 7

16 2. Solar wind and plasma waves basics Today, it is understood that the solar wind is accelerated in the hot corona region. The source of coronal heating is the kinetic energy of the convection in the underlying photosphere which is moving around the magnetic field lines which reach up into the corona (Marsch (2006) [70] ). The energy is then transferred to heat through magnetic reconnection, which can, together with convection, drive Alfvén and ion-cyclotron waves which resonantly heat the plasma (Isenberg (2001) [52] ) 3. The solar wind is mainly composed of protons, electrons and α particles (He 2+ ). Of the positively charged particles, the protons make up 96% and the α particles 4% on average, but this ratio can change considerably in certain situations. As the solar wind is quasi-neutral, the number of electrons equals the total number of positive charges. The number density is about n = n p n e 6 cm 3 near Earth s orbit, however it can strongly fluctuate between 0.1 and 100 cm 3. The velocity of the solar wind at 1 AU is on average u = 470 km/s, but there can be also fluctuations between 170 and 2000 km/s. The reason for this is that the solar wind can originate from different coronal regions with different properties. Regions with closed magnetic loops are associated with a slow solar wind of high density, while regions with open magnetic field lines (coronal holes) are the source of the fast solar wind of low density. Table 2.1 compares typical physical quantities of the fast and slow wind near Earth. The fast wind Figure 2.1: Image of Hale Bopp form has a low density while the slow wind has a high density because the particle Comet_Hale-Bopp. The yellowish diffuse flux nu is conserved. Assuming that the solar wind is expanding approximately tail is the gas tail, whose orientation can be explained by the interplay of the Sun's gravitation and radiation pressure. radially from the Sun, the spatial dependence on the radial distance from the The blue ion tail is directed radially away from the Sun. Sun r is nu r 2. The density near the corona starts at cm 3 and decreases monotonically with r while the velocity starts at 1 km/s and radially increases up to the values near 1 AU (Fichtner and Fahr (1989) [23] ). 3 This is different for massive stars (O and B type), which most likely power their stellar wind with their radiation (line driven winds, compare Lamers and Cassinelli (1999) [61] ). 8

17 2.1. Solar wind and its anisotropy np [cm 3 ] T,p [10 5 K] T,p [10 5 K] B [10 5 G] β,p = 8πnpkBT,p/B 2 Ap = T,p/T,p Time [h] Figure 2.2: Solar wind data measured on July 5, 2011 by WIND spacecraft (see Chapter 4 for more information). 9

18 2. Solar wind and plasma waves basics The next important feature is the large-scale interplanetary magnetic field. Due to the high electric conductivity of the solar wind plasma, the solar magnetic field lines are dragged away by the flow 4. This is due to the frozen flux theorem 5, which states that two plasma elements connected by a magnetic field line will always stay connected. Thus, the magnetic field lines are carried away and wound up due to the Sun s rotation. The spiral structure of the magnetic field is called Parker spiral, as it was first described in the groundbreaking work of Parker (1958) [87]. Parker s model was extended and refined by abandoning some simplifications, see Scherer et al. (2010) [94], where several models are reviewed and compared. However, for the scope of this study it is sufficient to assume a large-scale constant magnetic field B. As stated in the introduction, it is meaningful to define a parallel and perpendicular temperature with respect to the magnetic field, 1 2 k BT = mα2 2 k B T = mα2 2 = m 2 <(v <v >) 2 >, (2.1a) = m 2 <(v <v >) 2 >, (2.1b) where α, α are thermal velocities, and k B is the Boltzmann constant and < > indicates a moment of the velocity distribution f (v), <v 2 > = d 3 v f (v)v 2. (2.2) Using this notation, the bulk velocities in parallel and perpendicular directions are u = <v > and u = <v >. Figure 2.2 shows the evolution of the proton temperatures over one day as measured by the WIND Parameter Slow wind Fast wind spacecraft at 1 AU, together Mean velocity u 300 km/s 700 km/s with the number density and Density n p 8 cm 3 3 cm 3 the magnetic field strength, and Temperature T p K K the derived parallel plasma beta β,p and anisotropy A p. We can Table 2.1: Properties of the solar wind, adapted see that the temperatures show a from Prölss (2004) [92]. strong variability, and so do the plasma beta and anisotropy. This implies that the distribution function f p (v) is not isotropic, and this is indeed the case, see Figure This is valid as long as the kinetic pressure is dominating over the magnetic pressure, that is β 1. 5 The frozen flux theorem can be derived using ideal magnetohydrodynamics (MHD). The MHD formalism will not be rigorously derived or discussed in this thesis, it will be briefly introduced and applied in Chapter 7. 10

19 2.1. Solar wind and its anisotropy Figure 2.3: Excerpt of Figure 4 in Marsch et al. (1982) [72]. Shown are reduced proton velocity distributions as measured by Helios 2 spacecraft near Earth's orbit. The full threedimensional distribution is projected on a plane dened by the bulk velocity vector u and the magnetic eld B, which is indicated with the dashed straight lines. The contour lines correspond to fractions 4/5, 3/5, 2/5, 1/5 of the maximum (solid lines) and fractions lower than 0.1 (dashed lines). 4 a) A = 1 b) A = 2 c) A = 2, κ = 4 Velocity component v Velocity component v Figure 2.4: Isocontour lines at 0.9, 0.7, 0.5, 0.3 (solid lines), 10 1, 10 2, and 10 3 (dotted lines) of the maximum for bi-maxwellian and a κ-distribution (right-hand panel). 11

20 2. Solar wind and plasma waves basics The left-hand panel shows a distribution where the contour lines are in general compressed along the magnetic field direction, which indicates an excess of parallel temperature (A p < 1), while the right-hand panel shows, at least for the highest isocontours, an excess of perpendicular temperature (A p > 1). The most simple model to reproduce such an distribution is the bi-maxwellian, ( ) f (v, v ) exp v2 α 2 v2,a α 2. (2.3),a Figure 2.4 compares the isocontours of an isotropic Maxwellian to an anisotropic one. Such an approximation can at best only account for the highest contours in Figure 2.3 (solid lines) and cannot capture the complex structure of the lowest contours (dashed lines). The outer isocontours on the right-hand panel show a different anisotropy orientation than the inner contours, and the inner contours on the left-hand panel exhibit a two peak structure. Clearly, a much more involved model is needed to reproduce all these features. However, note that the lowest isocontour lines only represent a small fraction of the total particle distribution, while the highest isocontour lines are a good description of the bulk behaviour. Gary et al. (2001) [35] argue that the bi-maxwellian is a sufficient description of the bulk of the protons (also called core population) and the deviation of the full distribution due to supra-thermal particles is negligible. Another argument which justifies the use of bi-maxwellians in this study is the fact that the solar wind datasets used here contain only thermal proton measurements which are evaluated using the assumption of bi-maxwellian distributions 6. Consequently, there is no need to compare analytical results derived from distributions other than a bi-maxwellian to this data. This is different for electrons, which show much more pronounced nonthermal features. The electrons are divided into at least two populations, the core and supra-thermal halo (Štverák et al. (2008) [107] ), often a supra-thermal, highly collimated strahl has to be considered as well. The supra-thermal features can be approximated with so-called κ-distribution functions which have a flatter slope than Maxwellians at higher velocities 7 (Lazar et al. (2013) [65] ), more realistic models fit a different distribution function to each population (Maksimovic et al. (2005) [69] ). But even in such models, the core electrons make up the majority of the total population, which also justifies the use of an bi-maxwellian, at least for a first order approximation. But where do these anisotropies originate? On the one hand, the solar wind can be already anisotropized when accelerated in the corona by 6 This is discussed in detail in Chapter 4. 7 Compare right-hand panel of Figure

21 2.2. Basic plasma description wave heating, for example protons and ions by Alfvén and cyclotron waves (Marsch and Tu (2001) [71], Laitinen et al. (2003) [60], Kasper et al. (2013) [55] ), electrons by whistler waves (Vocks and Mann (2003) [109] ). On the other hand, the anisotropization can be attributed to the so called Chew-Goldberger-Low (CGL) mechanism (after Chew et al. (1956) [18] ), where in the case of an adiabatic expansion with constant flow velocities the anisotropy and beta change like A p r 2 and β,p r 2 with the radial distance from the Sun (Matteini et al. (2007) [76] ). However, it turns out that the observations do not agree with an adiabatic expansion as the anisotropies are much lower than predicted (Camporeale and Burgess (2008) [17], Chapter 7). The idea is that the particles are pitch-scattered back to a more isotropic distribution by anisotropy driven instabilities (Yoon (2007) [112], Matteini et al. (2010) [77] ). The measured data as shown in the introductory Figure 1.1 is then a result of competing anisotropization and isotopization processes, and we want to investigate the latter in this thesis. 2.2 Basic plasma description A plasma can be described as a (often gaseous) collection of charged and neutral particles. Because the electrons or ions are not bound to an atom, but are free to a certain extent, the plasma is electrically conductive and, therefore, sensitive to electromagnetic forces. It is at once a dynamical fluid and a good conductor so that it is capable of interacting with electromagnetic fields and creating them. What are the qualitative properties defining a plasma? A free isolated particle of charge q has the electric potential Φ = q/r, where r denotes the spatial distance to the particle s location. A charged particle in a plasma attracts nearby particles of the opposite charge which shield the field of the former particle from the rest of the plasma. This shielding effect effectively reduces the field of each particle and one can show that the potential is decreased exponentially by the factor exp ( r/λ D ). The constant λ D = kb T 4πnq 2 is called the Debye length and is a measure for the influence of the charge in a plasma. If a particle is displaced a little, we have a charge separation and, thus, an electrostatic force trying to pull the particle back to its equilibrium position: the response to this small displacement is an oscillation of the particle with the so-called plasma frequency ω p = 4πnq 2 /m. With these definitions, we can postulate the requirements for our system to be a plasma: The charged particle density should be high enough, so that each particle can influence a number of particles, instead of only the closest neigh- 13

22 2. Solar wind and plasma waves basics bors, which assures that we have a collective behavior in the plasma. The Debye sphere must be significantly smaller than the spatial extent of the plasma to warrant that effects on the edge of the plasma can be neglected compared to the interactions in the inner main part of the plasma. In this case, the ideal infinite plasma is a good approximation. This also assures internal charge neutrality (quasineutrality) if we have the same amount of positive and negative charges. The electromagnetic interactions should dominate over the collisions with particles (characterizing an ordinary gas), so we demand the plasma frequency to be higher than the collision frequency. Let us quantify the last point. It can be shown that the electron-electron collision frequency can be reduced to ν ee ω p,e /λ S, where λ S = 4πn e λ 3 D /3 is the number of electrons in the Debye sphere (Schlickeiser (2002) [95] ), so that λ S must be large in order to satisfy the third requirement. We find λ S 10 8 for typical solar wind parameters, which indicates that binary collisions are not only dominated by electromagnetic turbulence, but negligible compared to it. Such a plasma, and this applies to most cosmic plasmas, is called a collisionless plasma. Now that we have a plasma, we can discuss plasma waves. A plasma is generally composed of multiple ion species and electrons, and interacts with and creates electromagnetic fields, as stated above. A local perturbation of the initial equilibrium state of a physical quantity of the plasma leads to a local oscillation. Possibly, because of the Coulomb interaction inside the plasma, the perturbation can propagate: this phenomenon is called a plasma wave. The physical quantity can be the electromagnetic field or the particle position, but because of the coupling of the plasma with the fields it is sufficient to describe plasma waves as electromagnetic waves. However, the system of charged particles and interacting electromagnetic fields gives rise to a vast amount of possible wave types. Let us consider the electric field in the plasma. A general expression for an electric field of a plasma wave would be E = E 0 e i(k r ωt), (2.4) where r, t is the position in space and time, and k, ω are the wavevector and frequency, respectively. The frequency is a function of the wavevector and the relation ω = ω(k) (2.5) is the dispersion relation, which is generally obtained from the condition for nontrivial solutions of a set of field equations describing the plasma, and 14

23 2.3. Instabilities determines the wave mode. In a spatially infinite plasma without boundaries, which is a good approximation for a space plasma, we can assume the wavevector k to be real and the frequency complex, ω = ω R + iγ. Then, the real part has a contribution of the form e iω Rt = cos(ω R t) + i sin(ω R t), which describes an oscillation of the wave. The complex part yields e γt which means an exponential damping (γ < 0) or growing (γ > 0) of the wave with increasing time, the latter case is called a plasma instability. 2.3 Instabilities An instability can only arise if there is free energy stored in the plasma which can feed the growing waves. Due to this fact, a plasma is often categorized in MHD based fluid instabilities, where the free energy can be described in terms of macroscopic quantities, or kinetic instabilities, which are driven by a plasma distribution function deviating from thermal equilibrium (Gary (1993) [28] ). An example for a fluid instability is our temperature anisotropic solar wind, where the source of instability are the different temperatures in parallel and perpendicular direction, which we will discuss in turn. A good example for a kinetic instability is a a system where a beam plasma is streaming into another plasma population 8. Such systems are also studied in astrophysics where cosmic outflows are modeled as counterstreaming systems which are unstable with respect to the filamentation instability and electrostatic instability (Michno and Schlickeiser (2010) [80] ). However, such categorization in fluid and kinetic instabilities is rather arbitrary, since the kinetic description is the more general framework and can, in principle, describe every fluid instability. We will see that instabilities obtained from the fluid description can markedly change their properties when kinetic effects are taken into account. Likewise, an MHD description may be fully sufficient, or a fluid instability analytically inaccessible in a kinetic model. Nevertheless, this is a useful classification as kinetic instabilities often operate at low wavelengths, whereas fluid instabilities are encountered in the limit of large wavelengths. Low frequency fluid instabilities are also often aperiodic instabilities, which are non-propagating modes with zero real frequency ω R = 0. As seen in the previous section, instabilities are derived by studying the dispersion relation ω(k). In practice, a solution of the dispersion relation is often a stable wave, which can be driven unstable for certain parameter sets. A condition for parameters where the mode is unstable (γ > 0) is called marginal instability condition, as the condition s graph in parameter space also 8 This can be modeled by a two peak velocity distribution function, which clearly deviates from a Maxwellian. 15

24 2. Solar wind and plasma waves basics Figure 2.5: Qualitative low frequency dispersion of parallel electromagnetic waves (adapted from Baumjohann and Treumann (1996) [11] ). represents the line where the instability ends in a state of marginal stability (γ = 0). For a more detailed classification of instabilities, it is useful to relate the stable wavemodes to the possible instabilities. In our context of the solar wind, the well established instabilities relevant for isotropization are almost all related to parallel electromagnetic waves in a magnetized plasma. Figure 2.5 shows a schematic dispersion plot for frequencies below the electron gyrofrequency 9 ω R Ω e. The dispersion shows two modes, a right-hand polarized and left-hand polarized mode which are the same in the limit of low wavenumbers and split up for higher wavenumbers and approach the electron and proton gyrofrequencies, respectively. For low wavenumbers, the Alfvén mode is destabilized by an excess of parallel temperature A < 1 and is called MHD firehose, which will be discussed in the next section. When we stay on the left-hand branch, the region marked as proton whistler (also often referred to as Alfvén/whistler or Alfvén/cyclotron branch) can be destabilized by a proton anisotropy of A p < 1 and leads to the kinetic proton firehose. The proton cyclotron mode is unstable to the opposite anisotropy A p > 1 which is then called proton cyclotron instability. On the right-hand branch, the whistler mode is unstable to the electron firehose instability driven by an excess of parallel electron temperature, A e < 0, while the electron cyclotron wave is unstable for A e > 0 and called electron cyclotron instability. The firehose modes will be qualitatively discussed in the following two sections, and in-depth in Chapter 5. 9 The gyrofrequency or cyclotron frequency Ω a = q ab cm a of particle species a describes the orbital frequency of a particle moving perpendicular to a uniform magnetic field B. 16

25 2.3. Instabilities Temperature Anisotropy Ap = T,p/T,p Parallel plasma beta β,p 97 Mirror instability Cyclotron instability 50 Kinetic firehose Oblique firehose MHD firehose Figure 2.6: Various instability thresholds compared to the WIND proton data (resolution 60x60, cuto at 50, see Chapter 4). All thresholds are adapted from Hellinger et al. (2006) [45], except for the MHD rehose which is given by equation (2.8). Figure 2.6 shows the marginal instability thresholds studied in the context of solar wind proton isotropization compared to the WIND data. The mirror instability is an aperiodic and oblique mode which can be obtained from a fluid description (see Chapter 7). The physical process is that particles become trapped in so-called magnetic mirror configurations 10 for an excess of perpendicular pressure (Treumann and Baumjohann (1997) [106] ). The oblique firehose is operating in the same regime as the parallel firehose but, like the mirror instability, has the growth rate maximum at an oblique angle and is an aperiodic mode. It was discovered by Yoon et al. (1993) [113] in an approximate qualitative analytic theory complemented with numerical results, and later independently rediscovered by Hellinger and Matsumoto (2000) [43] by means of more accurate but brute-force numerical root finding schemes and on hybrid simulations. Despite the fact that the oblique firehose is widely discussed in the literature, the analytical properties of this mode are poorly understood at present. There is solely a fluid approach which will be briefly discussed in Chapter 7. The proton cyclotron instability does not fit to the data at all, the mirror criterion fits much better, and it is altogether the best fitting threshold in this plot, although there is still room for improvement (see Chapter 7). Three fire- 10 Also called magnetic bottle, where charged particles are trapped in a region of low magnetic field density surrounded by regions of high magnetic field density which reflect the particles. 17

26 2. Solar wind and plasma waves basics hose instabilities compete in the A p < 1 regime. Again, the parallel solution does not conform to the data, while both oblique and MHD firehose fit reasonably well. The surprisingly good fit of the MHD firehose may be attributed to a result of Gary et al. (1998) [34] who calculated the threshold of the kinetic parallel firehose in 2D hybrid simulations and found that the post-saturation threshold approximately corresponds to the MHD firehose condition. However, the oblique firehose threshold is tighter and it is generally acknowledged that this is the lower bound which characterizes the solar wind plasma best. We demonstrate in Chapter 5 that there is an alternative way to interpret the data by taking anisotropic electrons into account, which considerably modifies the threshold of the parallel kinetic firehose and improves the fit. 2.4 MHD firehose The (parallel) firehose instability is the subject of the main part of this thesis. To understand the basic mechanism driving this instability, it is instructive to investigate the firehose in the MHD limit. We follow the description of Vedenov et al. (1961) [108]. The firehose instability arises from the electromagnetic Alfvén mode, which can be visualized as vibrations of the magnetic field lines, just like a plucked guitar string. The instability condition can be derived from analyzing the forces acting on such a bent field line, which is illustrated in Figure 2.7. In the MHD limit, the particles are moving along the magnetic force lines. A proton moving with velocity v along a field line curved with osculating circle of radius R, excerts a centrifugal force F c = mv 2 /Re R, where e R is the unit vector in radial direction. The centrifugal force increases the curvature of the field line. In general, such a proton is gyrating around the magnetic field Figure 2.7: Gyrating protons moving along a curved magnetic eld line to illustrate the MHD rehose mechanism. line and carries a magnetic momentum µ = mv2 2B e B. Changing the momentum with respect to the magnetic field B gives rise to a force F µ = (µ B) 18

27 2.5. Kinetic firehose: cyclotron resonance opposite to the centrifugal force. Also, the magnetic field line tension force F B = 1 4π (B )B counteracts the centrifugal force and tends to smooth out the magnetic field. Considering the full gyrotropic proton distribution f (v, µ), the total forces acting on the magnetic field line are F c,tot = dv dµ f (v, µ) mv2 R e R, F µ,tot = dv dµ f (v, µ)µ B. (2.6) If the centrifugal force is dominating over the two restoring forces, F c,tot > F µ,tot + F B, the system is firehose unstable. Integrating the forces (2.6) we can rearrange the force balance inequation to n p k B (T,p T,p ) > B 2 /4π, (2.7) which is the firehose instability criterion. Even without proof, it is clear that the destabilizing centrifugal force is proportional to the parallel temperature F c v 2 T, and the stabilizing momentum force to the perpendicular temperature F µ v 2 T. A sufficiently high parallel temperature drives a system firehose unstable, but can be balanced by the magnetic field magnitude and perpendicular temperature. The eponymous everyday analogy is the out-of-control garden hose or fire hose, where the parallel temperature corresponds to the water flowing out and the magnetic field line tension to the stiffness of the hose. Using the definition of the plasma beta (1.2) 11, the proton firehose condition can be cast into the compact form for the MHD limit. β,p β,p > 2 (2.8) 2.5 Kinetic firehose: cyclotron resonance However, the MHD threshold (2.8) is just a lower bound for the tighter threshold of the kinetic firehose, calculated with full kinetic theory of which the MHD firehose is just the low wavenumber limit. The important kinetic effect to consider is cyclotron resonance, where the wave and particles interact. Lets examine a proton at rest (v = 0) approached by an electromagnetic wave with wavevector k and frequency ω R, see Figure 2.8. The fluctuating field δe is oscillating perpendicular to the wavevector. The direction of this oscillation with respect to the wavevector is defined by the sign of ω R. 11 The perpendicular plasma beta is defined completely analogously, β,p := β,p T,p /T,p. 19

28 2. Solar wind and plasma waves basics Wave-particle interaction to illustrate the proton cyclotron resonance condi- Figure 2.8: tion. In this thesis, we use the convention ω R > 0 describes right-hand (RH) polarization, ω R < 0 describes left-hand (LH) polarization. Since the proton p carries a positive charge, it gyrates with the gyrofrequency Ω p > 0 in LH orientation with respect to the magnetic field direction. Accordingly, a resting proton will be stimulated to gyration by the incoming wave if it has the frequency ω R = Ω p (or higher multiples ω R = nω p, n N), if we assume that magnetic field and wavevector align. If the proton is moving with nonzero velocity v along the magnetic field direction, the frequency in the reference frame of the particle is Doppler shifted to ω R = ω R kv. The first order (n = 1) cyclotron resonance condition changes to ω R + Ω p = kv, and if we allow the proton to move in the opposite direction, we can formulate the resonance condition of one proton of absolute parallel velocity v as ω R + Ω p kv = 1. (2.9) The proton will not gyrate if the mode has the wrong polarization, but note that the Doppler shift can change the polarization with respect to the rest frame. In a bi-maxwell distributed plasma, f exp( v 2 /α2,p v2 /α2,p ), most protons are close to the parallel thermal velocity α,p in parallel direction. Thus, most protons of the distribution are cyclotron resonant when ω R + Ω p kα,p = 1. (2.10a) Equation (2.10a) is called resonance condition, the left-hand side resonance term. We can also formulate the resonance condition for the electron population ω R + Ω e kα,e = 1, (2.10b) 20

29 2.5. Kinetic firehose: cyclotron resonance where Ω e < 0 is the electron cyclotron frequency. In the case of a resting particle or a particle with v = 0, the wave would loose energy to the particle, which would also decrease the growth rate of the mode. However, if the population is highly energetic in perpendicular direction (large T ), the gyrating particles can just as well feed energy to the wave, increasing the growth rate of the mode. 21

30

31 Chapter 3 Derivation of the dispersion relation In this chapter, we derive the dispersion relation for parallel electromagnetic plasma waves needed to investigate the firehose instability in Chapters 5 and 6. The first sections are based on the diploma thesis Michno (2009) [79], which is following the calculations of Baldwin et al. (1969) [8]. To calculate the dispersion relation and plasma waves, we need a formal description of the plasma. A straightforward way to do so is an exact microscopic description. Consider a plasma of N charged particles. Each particle is affected by the Lorentz force, we can write down Newton s law of motion, F j = m j a j for every particle j (where the fields and, thus, the force in each equation depends on the position and velocity of every particle). Now, we only need to know the initial positions and velocities of all N particles and can solve this differential equation, obtaining N trajectories. This solution is, in principle, exact, but in practice N tends to be quite a big number so this approach is neither reasonable nor feasible. Thus, any consideration of this motion must be based on a simplified model, and there are two standard approaches: the kinetic model and the fluid or magnetohydrodynamic (MHD) model. Since one is not interested in the motion of one specific particle but the collective behavior, statistical concepts are introduced in the kinetic approach. We consider statistical ensembles of systems and obtain, by averaging out microscopic information, statistical kinetic equations. The informations r j, v j for individual particles are lost (we introduce a probability density function f (r, v, t), so that f (r, v, t)d 3 v is the probability to find a particle of a species at space-time position (r, t) within a volume d 3 v at v in velocity space), however, we still consider the now independent variables r, v and require the knowledge of the motion of a general particle, so this theory can be regarded as mesoscopic. 23

32 3. Derivation of the dispersion relation The description can be reduced even further in the MHD description. By taking moments of the kinetic equations, it is possible to obtain MHD equations which only depend on and describe macroscopic quantities (such as pressure, fluid velocity). However, MHD is only an approximation of the kinetic theory and not adequate in some situations. 3.1 Kinetic description of the plasma We consider a plasma in which collisions can be neglected 1 and assume that only electromagnetic forces act on the charged particles. We bear in mind that energies in the solar wind are sufficiently low to motivate a non-relativistic treatment. Let f (r, v, t) be the distribution function of a particle species 2 in the r-v phase space, where r and v denote the position and velocity of a particle of mass m and charge q. We calculate the time evolution of such a distribution function, supposing a solenoidal plasma. We start with the continuity equation: since there are no particle sources, the sum of the change of the particle number in a given volume plus the flux density across the surface of the volume must equal zero: d dt d 3 rd 3 v f + d 3 vd 2 r ṙ f + d 3 rd 2 v v f! = 0. (3.1) We apply Gauss theorem and assume that f and f / t are continuous functions, so that we can use Leibniz s rule, ( ) f d 3 rd 3 v t + (ṙ f ) + v ( v f ) = 0. (3.2) The above equation can hold only for arbitrary volumes if the integrand is zero. We insert the Lorentz force m v = q ( E + c 1 ṙ B ) = q ( E + c 1 v B ), and obtain f ( q t + (v f ) + v (E + v ) ) m c B f = 0. (3.3) Because v is independent of r and v (v B) = 0, we finally obtain the Vlasov equation, f t + v f + q (E + v ) m c B v f = 0. (3.4) 1 Which is a valid assumption for the solar wind plasma, compare Section In the interest of economy of notation we will refrain from a subscript now. 24

33 3.2. Linearized equations For a complete description of a plasma we also need the Maxwell equations, c E = B t, c B = 4πJ + E t, E = 4πσ, B = 0, where the charge density σ and electric current density J are σ = q d 3 v f (r, v, t) J = q d 3 vv f (r, v, t), (3.5a) (3.5b) (3.5c) (3.5d) (3.6a) (3.6b) and the summation is extended over all charged species. 3.2 Linearized equations Our set of field equations (3.4) and (3.5) is a set of coupled nonlinear integrodifferential equations for f, and not straightforward to solve. Instead of solving the exact equations, we choose a perturbative approach and examine the behaviour of a small deviation from an equilibrium solution of the system. Let f 0, E 0, B 0 denote the steady-state solution for the physical variables f, E and B. In equilibrium, the partial derivatives with respect to time of these variables are zero, f 0 t = 0, E 0 t = B 0 t = 0, (3.7) and the charge and electric current densities vanish, σ 0 = 0, J 0 = 0. The Vlasov equation for the steady state then reads v f 0 + q ( E 0 + v ) m c B 0 v f 0 = 0. (3.8) We examine the evolution of small perturbations in the variables, denoted by a subscript one, E = E 0 + E 1, B = B 0 + B 1, f = f 0 + f 1. (3.9a) (3.9b) (3.9c) Here we note the limitation of our theory: due to the assumption of small perturbations, our results will not be adequate to describe the full temporal 25

34 3. Derivation of the dispersion relation evolution of instabilities, since the perturbations are then exponentially growing. The linear approximation breaks down when the perturbed quantities will become comparable to the steady-state quantities. However, we can decide whether an instability is possible and calculate its evolution to the point where the perturbation is small enough. After inserting the perturbed variables in the Vlasov (3.4) and Maxwell equations (3.5), linearization (all perturbed terms of quadratic order or higher will be neglected) and consideration of equation (3.8) we obtain f 1 t + v f 1 + q ( m E 0 + v ) c B 0 v f 1 ( E 1 + v c B 1 = q m ) v f 0, (3.10) c E 1 = B 1 t, c B 1 = 4πJ 1 + E 1 t. (3.11a) (3.11b) Clearly, in order to linearize the Vlasov equation, it is not sufficient that the variables E 1, B 1, f 1 are small compared to the unperturbed quantities, we must also demand v f 1 v f 0. Equation (3.10) allows one to construct a formal solution for f 1 : the lefthand side can be written as a total time derivative along the trajectory of a particle, whose equations of motion are v = ṙ and m v = q [ E + 1 c ṙ B], and obtain d dt f 1 [ 1(r, v, t) = lim f 1 (r + v t, v + q ) t 0 t m (E 0 + vc B 0 ) t, t + t ] f 1 (r, v, t). (3.12) Thus, we define the vectors r (r, v, t t) and v (r, v, t t) so that they satisfy the equations of motion d dt r = v (3.13a) d dt v = q ] [E 0 (r ) + v m c B 0(r ), (3.13b) with the inital conditions r (r, v, 0) r and v (r, v, 0) v. The integration of 26

35 3.3. Maxwell tensor equation (3.10) can be formally written as f 1 (r, v, t) = f 1 [ r (r, v, t), v (r, v, t), 0] q m t 0 dt { E 1 [ r (r, v, t t), t ] + 1 c v (r, v, t t) B 1 [ r (r, v, t t), t ] } v f 0 [ r (r, v, t t), v (r, v, t t), t ]. (3.14) This equation for f 1 is a functional depending linearly on the perturbed fields E 1, B 1. Equation (3.14) is not solvable in general, since neither the particle trajectories nor the equilibrium quantities are known. However, the system we want to investigate, a homogeneous solar wind plasma, allows for some major simplifications which make a detailed analysis possible. 3.3 Maxwell tensor We construct a solution based on the so-called Maxwell tensor using additional assumptions. The high electric conductivity of space plasmas guarantees that there are no large-scale electric fields, E 0 0. Furthermore, the plasma should be spatially homogenous, f 0 = 0, and we assume a temporally and spatially constant, uniform magnetic field B 0 = B 0 e 3. Now what does it mean that the perturbed quantity E 1 is small, when E 0 is zero? We simply demand E 1 to be small enough, so that we can obtain equation (3.10) from (3.4) and (3.8), which means that the perturbed electric field E 1 is sufficiently small compared to the equilibrium magnetic field B 0 ( E 1 c 1 v B 0 ), so that we can neglect the term E 1 v f 1! The equilibrium state Vlasov equation (3.8) now reads v c B 0 v f 1 + E 1 v f 0. (3.15) qb 0 m (v v f 0 ) = 0. (3.16) Particle orbits In order to establish a further analysis of expression (3.14), we introduce the cylindrical coordinates (v, Φ, v ) in velocity space, whereas v denotes the velocity component perpendicular to the magnetic field (B 0 e 3 ) and v the 27

36 3. Derivation of the dispersion relation parallel one, v = e 1 v cos Φ + e 2 v sin Φ + e 3 v, e v = e 1 cos Φ + e 2 sin Φ, e Φ = e 1 sin Φ + e 2 cos Φ, e v = e 3. (3.17a) (3.17b) (3.17c) (3.17d) We note that v = e v v + e v v, and the nabla operator reads 1 v = e v + e Φ v v Φ + e 3. (3.18) v The Vlasov equation (3.16) then yields Ω f 0 Φ = 0, (3.19) where Ω is the gyration frequency Ω = qb 0 mc. Thus, ( this equation ) states that f 0 is independent of the gyration phase Φ, f 0 = f 0 v, v, which is called a gyrotropic system. The equations of motion (3.13) yield the trajectories ξ(v, t t) := r (r, v, t t) r = e 3 v (t t v ) + e { [ 2 cos Ω(t t ) + Φ ] cos Φ } Ω v e { [ 1 sin Ω(t t ) + Φ ] sin Φ } Ω ξ(v, t t) := v (r, v, t t) = e 3 v + e 2 v sin [ Ω(t t ) + Φ ] + e 1 v cos [ Ω(t t ) + Φ ], (3.20) which can be easily verified by insertion into the equations of motions. Observe, that r r does not depend on r, and that the gradient v can be written as v f 0 (v, v ) = (e 1 cos Φ + e 2 sin Φ) f 0 v + e 3 f 0 v. (3.21) We compare equation (3.17a) with equation (3.20) and obtain Φ = Φ + Ω(t t ). Thus, the integral on the right-hand side of equation (3.14) can be written 28

37 on comparing with equations (3.20) and (3.21) as q t [ ] { } dt E 1 + v v m c B v e 3 f 0 f e 3 v v v 0 ( = q f t 0 dt E 1 v + mv v 0 where we utilize that v, v and, thus, q mv f 0 v, f 0 v v f 0 v v f 0 v t Maxwell tensor ) e 3 [ ] dt E 1 + v c B 1, (3.22) are constants of motion Fourier-Laplace transform Now, we examine equation (3.10) and the Maxwell equations (3.11), c E 1 = B 1 t, c B 1 = 4πJ 1 + E 1 t = 4π q d 3 vv f 1 + E 1 t. (3.23a) (3.23b) This is a set of linear equations for the values f 1, E 1 and B 1, where the coefficients are independent of r and t. To transform these differential equations into algebraic equations, we apply a Fourier-Laplace transformation (compare Appendix A for the definitions used in this thesis) to the fields and the distribution function. We obtain E k,ω = 0 dte iωt E k (t) = 0 dt d 3 re i(ωt k r) E 1 (r, t), (3.24) with expressions completely analogous for the magnetic field B 1 and the distribution function f 1, where we assume without loss of generality that γ = Iω > 0 is large enough, so that the integral converges. The Maxwell equations then yield ick E k,ω = iωb k,ω + B k (0), ick B k,ω = 4πJ k,ω iωe k,ω E k (0), (3.25a) (3.25b) with the Fourier-Laplace transformed current density J k,ω = q d 3 vv f k,ω (v). To compute a specific expression for J k,ω, we need to 29

38 3. Derivation of the dispersion relation prepare the Fourier-Laplace transformation of f 1 (3.14). The Fourier transform of integral (3.22) is q t ( dt e ik[r (r,v,t t) r] 1 m 0 ( + 1 v v f 0 v v f 0 v f 0 E k (t ) v (r, v, t t) v v ) ] e 3 [E ) k (t ) + v c (r, v, t t) B k (t ). (3.26) We note that the integrand contains functions of t multiplied with functions of t t. Thus, if we Laplace transform (3.26), we can apply the convolution theorem for Laplace transforms (compare Abramowitz and Stegun (1964) [2], p. 1021, ), 0 dte iωt t 0 dt f (t )g(t t ) = 0 dte iωt f (t) dt e iωt g(t ) = f ω g ω, (3.27) 0 and obtain q f 0 E k,ω dtgv ( t) mv v + q mv ( 0 v f 0 v v f 0 v ) ] e 3 dtg [E k,ω + v c ( t) B k,ω, (3.28) 0 where we introduced the abbreviation G = exp ( iωt + ik [r (r, v, t) r ]) (3.20) = exp {i ( ) v )} ω k v t i (k Ω {sin[ωt + Φ] sin Φ}, (3.29) and k and k are defined through k k e 3 + k e 1, as we can assume without loss of generality that the wavevector lies in this specific plane for a gyrotropic system. Hence, we have an expression for f k,ω (v) and can specify the Fourier- Laplace transform of the current density J 1 with the help of the Maxwell 30

39 equations (3.25), J k,ω = q d 3 vv f k,ω (v) = q + i q2 m q2 m d 3 [ vv dtg f k v ( t), 0 ] 0 d 3 v v v d 3 vv 0 ( f 0 f 0 v v v v { dtg [ e 3 + [e 3 v ( t)] k ω 1 v f 0 ) 0 dtge 3 ( 3.3. Maxwell tensor 1 cω v ( t) B k (0) v ( t) + 1 f 0 f 0 v v v v v v ]} E k,ω. (3.30) ) Now, we can write the Maxwell equation ick B k,ω E k (0) as = 4πJ k,ω iωe k,ω ˆRE k,ω = a, (3.31) where ˆR is the dyadic ˆR = (c 2 k 2 ω 2 )1 c 2 k k + ˆQ, (3.32) with the dyadic product, the unit dyadic 1, and ˆQ = 4πiωq2 m ( + 1 v as well as the vector { 1 d 3 f 0 vv dtg v ( t) v v v f 0 v v f 0 v a = ick B k (0) iωe k (0) + 4πiω q 4πω q2 m 0 [ e 3 + [e 3 v ] ( t)] )} k, (3.33) ω d 3 [ vv dtg f k v ( t), 0 ] ( ) d 3 v v f 0 f 0 v v v v dt G v ωc e 3 v ( t) B k (0). (3.34) 0 Note that all initial conditions are in a, while ˆR contains only parameters of the steady state. That is why a is called initial condition, the dyadic ˆR is called the Maxwell operator, and the dyadic ˆQ ω 2 1 is the dielectric tensor. 0 31

40 3. Derivation of the dispersion relation Now, we can formally solve the wave equation (3.31) and have the solution for the Fourier-Laplace transformed electric field, E k,ω = ˆR 1 a. (3.35) To obtain the original field E(r, t), we must essentially perform an inverse Laplace transformation which will be discussed in the next section. 3.4 Dispersion relation Here, we will write down a formal solution for the electric field E(r, t) and additionally obtain a dispersion relation. We follow the argumentation of Schlickeiser (2002) [95]. Equation (3.35) gives us E k,ω, an inverse Laplace transformation of E k,ω delivers E k (t), and an inverse Fourier transformation of E k (t) yields E(r, t). The reverse Fourier transformation is straightforward E(r, t) = 1 (2π) 3 d 3 ke ik r E k (t), (3.36) while the inverse Laplace transformation requires some extra considerations. It is shown in Schlickeiser (2002) [95] that for our Maxwell operator ˆR, the Fourier-Laplace transform E k,ω is a meromorphic function (analytic except for a set of isolated points S) of ω in either the upper or lower complex half plane (γ > 0 or γ < 0), and it is demonstrated how to construct the analytic continuation for the lower half plane. Thus, we can use the inverse Laplace transformation (compare Appendix A.2) E k (t) = iζ+ iζ dωe k,ω e iωt, (3.37) where ζ is greater than the imaginary part of every singularity s S. We solve this integral by computing the residues. Equation (3.35) yields E k,ω = ˆR T a, (3.38) det ˆR and we conclude that the set of singularities S is determined by the zeros of the denominator, det ˆR 0. Lets name the zeros ω j, j N. Because of the Cauchy integral theorem we are free to deform the integration path in (3.37) in the complex plane as long it is homotopic to the original path. This idea was first introduced by Landau (1946) [62] into the context of kinetic plasma 32

41 3.4. Dispersion relation physics, which is why the deformed contour is sometimes called Landau contour in the literature. We change the path as denoted in Figure 3.1 and obtain the residues of E k,ω e iωt and a contour integral along a line L below all singularities ( ζ lower than the imaginary part of every s S), ) E k (t) = Res (E k,ω e iωt, ω j j N + iζ + iζ dωe k,ω e iωt, (3.39) as the paths integrals leading to each singularity cancel with the path integrals returning from the respective singularity. The common argument in the literature is that the path integral along L is exponentially damped for t with respect to the residues and can be neglected (Achatz (1989) [3] ). The idea is that the exponential in the path integral, which can be written as e iωt = e iωrt e ζ t and the integration is performed over ω R (, + ), will provide a much faster damping than for the residual terms which are damped like e Iωjt and, as constructed, Iω j > ζ for all residues. This is problematic due to our linearized equations: in the case of an instability, our solution is only valid for small timescales. We handle this problem by simply choosing ζ sufficiently large, so that the path integral can be also neglected for our small timescales. Because e iωt has no residues (Res ( e iωt, z ) = 0, z C), we obtain lim E k(t) = Res ( ) E k,ω, ω j e iω j t. (3.40) ζ j N The term e iω jt denotes a possible plasma mode, the plasma wave is damped for Iω j < 0 and growing for Iω j > 0. The frequency is a function of the wavevector, ω j = ω j (k), and can be derived from det ˆR = 0, which is also referred to as the dispersion relation. Figure 3.1: Sketch of the original and the deformed integration contour for the inverse Laplace transform, where s denotes the singularities. 33

42 3. Derivation of the dispersion relation 3.5 Simplification of the dyad ˆQ To prepare the calculation of the dyadic ˆQ for a bi-maxwellian plasma, we will cast ˆQ into a more convenient form. First, we partially derive G with respect to time (compare equation (3.29)) G t = iωg ik ṙ G = i [ ω k v ( t) ] G, (3.41) and expand the threefold cross product in equation (3.33) { [ ˆQ = 4πiq2 d 3 ω f 0 vv dtg + k ( )] f 0 f 0 v v m v v v v v ( t) v 0 ( ) 1 f 0 f 0 (ω + e 3 v v v v k v ( t) ) }, (3.42) v which allows us to integrate the last summand of ˆQ by parts with the help of equation (3.41), ˆQ = 4πiq2 m 4πq2 m 0 0 dv v dv v 2π dv 2π dv 0 0 dφ 0 + k v dφv e 3 1 v [ dtv v ω f 0 ( t)g v v ( )] f 0 f 0 v v v v ( ) v f 0 v v f 0 v We rewrite the expressions for the orbits v and v ( t), (3.20),. (3.43) v = e 3 v + v 2 (e 1 ie 2 )e iφ + v 2 (e 1 + ie 2 )e iφ, (3.44a) v ( t) = e 3 v + v 2 (e 1 ie 2 )e i(ωt+φ) + v 2 (e 1 + ie 2 )e i(ωt+φ), (3.44b) and the Jacobi-Anger identity (see Abramowitz and Stegun (1964) [2], p. 361, and set t = e iφ ), e iz sin Φ = n= e inφ J n (z), allows us to express G with Bessel functions, [ ] G = exp = e i(ω k v )t i ( ω k v ) t i k v Ω n= n = e in (Ωt+Φ) J n (sin(ωt + Φ) sin Φ) ( ) ( ) k v e inφ k v J n Ω Ω (3.45) 34

43 3.5. Simplification of the dyad ˆQ Now, we can carry out the integrations of the first sum of equation (3.43) with respect to t and Φ. In the interest of economy of notation we will refrain from writing down the argument of the Bessel functions and introduce the abbreviation e ij := e i e j. Let us consider the integration with respect to t first: because of the assumption Iω > 0 (compare equation (3.24)) we can use 0 dtei[ω+a]t = i ω+a, where a R. We obtain: I := dtv v ( t)g = { n= n = J n J n [ i e 33 v 2 ω k v Ωn eiφ(n n ) + e 13 v v /2 i ω k v Ω(n 1) i ω k v Ω(n +1) (e iφ(n n 1) + e iφ(n n +1) ) +e 23 iv v /2 (e )] iφ(n n 1) e iφ(n n +1) [ (e 31 ie 32 ) v v /2e iφ(n n +1) + (e 11 ie 12 ) v 2 /4 ( e iφ(n n ) + e iφ(n n +2) ) ( )] + (e 22 + ie 21 ) v 2 /4 e iφ(n n ) e iφ(n n +2) [ (e 31 + ie 32 ) v v /2e iφ(n n 1) + (e 11 + ie 12 ) v 2 /4 ( e iφ(n n ) + e iφ(n n 2) ) + (e 22 ie 21 ) v 2 /4 (e iφ(n n ) e iφ(n n 2) )] }. (3.46) We use the relation 2π 0 dφe iφ(n n ) = 2πδ n,n for the integration with respect to Φ and obtain 1 2π 2π 0 dφi = n i ω k v Ωn { e33 v 2 J2 n + [e 31 + e 13 + i(e 23 e 32 )]v v /2J n J n+1 + [e 31 + e 13 + i(e 32 e 23 )]v v /2J n J n 1 + 2(e 11 e 22 )v 2 /4J n 1J n+1 + [e 11 + e 22 + i(e 12 e 21 )]v 2 /4J2 n 1 + [e 11 + e 22 i(e 12 e 21 )]v 2 } /4J2 n+1 i [ = e33 v 2 n ω k v Ωn J2 n + (e 31 + e 13 )v v /2J n (J n 1 + J n+1 ) + i(e 32 e 23 )v v /2J n (J n 1 J n+1 ) + e 11 v 2 /4(J n 1 + J n+1 ) 2 + e 22 v 2 /4(J n 1 J n+1 ) 2 + i(e 12 e 21 )v 2 /4(J2 n 1 J2 n+1 )]. (3.47) 35

44 3. Derivation of the dispersion relation We utilize the relations 2n z J n(z) = J n 1 (z) + J n+1 (z), 2J n(z) = J n 1 (z) J n+1 (z), (3.48a) (3.48b) from Abramowitz and Stegun (1964) [2], first and second equation of paragraph with parameters λ = q = 1, p = 0, and can write 1 2π 2π 0 dφi = n i ω k v Ωn [ e3 e 3 v 2 J2 n + (e 3 e 1 + e 1 e 3 )v v n/zj 2 n +i(e 3 e 2 e 2 e 3 )v v J n J n + e 1 e 1 v 2 /z2 J 2 n + e 2 e 2 v 2 J 2 n Finally, we yield for the dyadic ˆQ: ˆQ + 4πq2 m 2πe 3 e 3 = 8π2 q 2 m 0 0 dv with the dyadic ˆT n given by dv dv [ +i(e 1 e 2 e 2 e 1 )v 2 n/zj n J n]. (3.49) dv v ( ω f 0 v + k f 0 f 0 v v v v ( n= ˆT n = e 3 e 3 v 2 J2 n + e 2 e 2 v 2 J 2 n + e 1 e 1 n 2 v 2 J2 n/z 2 ) v f 0 v v f 0 v )] ˆT n ω k v nω, (3.50) + i(e 1 e 2 e 2 e 1 )nv 2 J n J n/z + i(e 3 e 2 e 2 e 3 )v v J n J n + (e 1 e 3 + e 3 e 1 )v v J 2 n/z, (3.51) where the argument of each J n is z := k v Ω. Because of the relation e i(v w)e j = (e i v)(e j w), it is clear from equation (3.51), that each component of the dyadic ˆT ij n = e i ˆT n e j equals the complex conjugate of the corresponding component of the transposed dyad ˆT ij n = ˆT n, ji so we can conclude that ˆT n and, thus, ˆQ and the dielectric tensor ˆQ ω 2 1 are hermitian. Because ˆT 13 n is real and ˆT 12 n, ˆT 23 n purely imaginary ( ir), ˆQ can be cast into the form Q 1 Q 12 Q 13 ˆQ = Q 12 Q 2 Q 23. (3.52) Q 13 Q 23 Q 3 36

45 3.6. Reduction for parallel modes in a thermal plasma 3.6 Reduction for parallel modes in a thermal plasma The solar wind plasma can be approximated by a bi-maxwellian distribution for all particle species a {e, p}, ( ) ( ) n a f 0,a v, v = π 3/2 α 2,a α exp v2,a α 2 v2,a α 2, (3.53),a where n a is the number density and α,a, α,a the perpendicular and parallel thermal velocities with respect to the magnetic field direction. Let us calculate, as an example, component Q 1 of the dyad ˆQ. The first component of the dyad ˆT, (3.51), reads ˆT 11 n = n 2 v 2 J2 n/z 2 = v 2 (J2 n 1 /4 + J2 n+1 (z)/4 J n 1J n+1 /2), (3.54) where we used the relation (3.48a). Now, we can calculate the first component, Q 1 = a 8π 2 q 2 a m a n= 0 dv dv [ωv 2 f 0,a + k v ( v 3 f 0,a v v v 2 f 0,a v )] Jn 1 2 (z)/4 + J2 n+1 (z)/4 J n 1(z)J n+1 (z)/2. (3.55) ω k v nω The evaluation of the tensor and dispersion relation is complicated by the infinite sum over Bessel functions. It turns out that the dispersion relation for arbitrary wavevector directions k makes an analytical treatment unfeasible due to its high complexity. The Maxwell tensor simplifies considerably if the wavevector is parallel (k = 0 z = 0) or perpendicular (k = 0) to the magnetic field, as the transversal and longitudinal modes decouple. The latter case provides the so-called ordinary mode which is discussed in Ibscher et al. (2013) [51] and may be relevant for solar wind isotropization, although the growth rates have not been evaluated yet. Here, we exclusively discuss the parallel modes. Setting k = 0 and k := k yields J n (0) = δ n,0 (where δ n,0 is the Kronecker delta, see Abramowitz and Stegun (1964) [2], p. 360, and ), so that the infinite sum in (3.55) reduces to two non-zero summands. Inserting the distribution function provides Q 1 = a 4 πq 2 an a m a α 4,a α,a ( exp 0 dv v 3 v2 α 2 v2,a α 2,a ) ( dv [ ω + kv ( 1 ω kv Ω a + T,a T,a 1 1 )] ω kv + Ω a ), (3.56) 37

46 3. Derivation of the dispersion relation where we used T,a /T,a = α 2,a /α2,a. The integration over v is straightforward, ) 0 dv v 3 exp ( v2 α 2,a We split the integral over v in two parts, the first part yields I 1 = dv exp ( x:=v /α,a v2 α 2,a = 1 k ) ( 1 ω kv Ω a + dxe x2 = α4,a 2. (3.57) 1 x ω Ω + a kα,a 1 ) ω kv + Ω a 1 x ω+ω. (3.58) a kα,a This integral cannot be solved or reduced to elementary functions. Writing ξ a ±1 := ω±ω a kα,a, we can abbreviate 3 the integral as a function Z, π ( I 1 =: Z ( ξ a ( ) k +1) ) Z ξ a 1. (3.59) The complex function Z is well studied in plasma physics and also called the plasma dispersion function (Fried and Conte (1961) [26] ), + Z(ξ) = π 1/2 dt exp( t2 ), Iξ > 0, (3.60) t ξ valid in the upper complex half plane. The analytic continuation to the lower half plane and other properties are discussed in the Appendix B. The second integral over v reads I 2 = dv v exp ( v2 α 2,a ) ( x:=v /α,a = α,a k 1 ω kv Ω a + 1 ω kv + Ω a ( ) 1 1 dxxe x2 x ξ+1 a + x ξ 1 a. (3.61) 3 The problem of solving the integral in (3.58) still persist. In order to find analytical solutions of the dispersion relation, approximations of the integral have to be considered, see section 5.4. An alternative approach is to assume so called polynomial distribution functions (Fichtner and Sreenivasan (1993) [24] ), for which (3.58) has an algebraic solution and the dispersion relation is equivalent to a polynomial. The general formalism discussed in Fichtner and Sreenivasan (1993) [24] is also capable of approximating (bi-)maxwellians, but the high order of the polynomial form of the dispersion relation and the resulting large number of terms make this approach unfeasible for our analytical discussion. ) 38

47 Integration by parts yields so that we obtain dxxe 1 x2 x ξ = e x2 x ξ }{{} 0 I 2 = 3.6. Reduction for parallel modes in a thermal plasma = 1 2 πα,a 2k dxe x dxe x2 1 x x ξ 1 ξ x ξ = 1 πz (ξ), (3.62) 2 ( Z ( ξ a +1) Z ( ξ a 1) ). (3.63) Combining the results (3.57), (3.59), and (3.63), the tensor element Q 1 finally reads [ 2πq Q 1 = 2 an a ω ( Z ( ξ a ) ( ) ) +1 Z ξ a a m a α,a k 1 ( ) + 1 T (,a 1 Z ( ξ 2 T +1) a Z ( ξ 1) )] a. (3.64),a The remaining components can be calculated in a similar fashion. We write the nonzero entries of ˆQ in the compact form Q 1 = Q 2 = a Q 12 =: iq 4 = i a ω 2 p,a 2 (Aa +1 + Aa 1), (3.65a) ω 2 p,a 2 (Aa +1 Aa 1), (3.65b) Q 3 = ω 2 a ω 2 p,a k 2 α,a Z (ξ a ), (3.65c) where ξ a = and ω α,a k, ξa ±1 = ω±ω a, the quadratic plasma frequency ω 2 p,a = 4πn aq 2 a kα,a A±1 a = ξ az ( ξ±1 a ) 1 2 m a, ( ) T,a 1 Z ( ξ T ±1) a. (3.66),a The Maxwell tensor ˆR = (c 2 k 2 ω 2 )1 c 2 k k + ˆQ for parallel propagation reads c 2 k 2 ω 2 + Q 1 iq 4 0 ˆR = iq 4 c 2 k 2 ω 2 + Q 1 0. (3.67) 0 0 ω 2 + Q 3 39

48 3. Derivation of the dispersion relation We can immediately write down the eigenvalues λ ˆR i and eigenvectors x ˆR i of the Maxwell tensor, ˆR e 3 = ( ω 2 + Q 3 )e 3 =: λ ˆR 3 x ˆR 3, 1/ 2 ˆR ±i/ 2 = (c 2 k 2 ω 2 + Q 1 Q 4 ) 0 1/ 2 ±i/ 2 0 =: λ ˆR ± x ˆR ±. (3.68a) (3.68b) The wave equation (3.31) can be multiplied with one of the eigenvectors and we obtain λ ˆR i x ˆR i E k,ω = x ˆR i a x ˆR i E k,ω = x ˆR i a λ ˆR i, (3.69) which is an equation describing waves with an electric field amplitude parallel to x ˆR i. The dispersion relation for this wave mode is λ ˆR i 0, (3.70) the eigenvectors x ˆR i are called polarization vectors. Because x ˆR 3 is parallel to the propagation direction (k = ke 3 ), the corresponding mode is referred to as longitudinal wave mode. Since x ˆR ± e 3 = 0, these polarization vectors yield oscillations perpendicular to the wavevector, which are per definition transversal modes. 3.7 Weak amplification limit The dispersion relation is a complex equation and depends on the complex frequency ω = ω R + iγ and the real wavenumber k, λ(ω, k) = Rλ(ω, k) + iiλ(ω, k) = 0. (3.71) In general, the dispersion relation is treated by solving the joint equations Rλ = 0 and Iλ = 0 for ω as a function of k. For systems where we expect a small growth or damping rate γ ω R, we can simplify the dispersion relation by expanding around the small parameter as demonstrated by Tademaru (1969) [105]. This ansatz is also exploited in Schlickeiser (2010) [96], Schlickeiser and Škoda (2010) [97] who coined the term weak amplification limit for this approach. 40

49 3.7. Weak amplification limit This limit is obtained by expanding the real and imaginary parts of the dispersion relation (3.71) in a Taylor series around γ = 0, Rλ(ω R, γ, k) Rλ(ω R, γ=0, k) + γ Rλ(ω R, γ, k)! γ = 0, (3.72a) γ=0 Iλ(ω R, γ, k) Iλ(ω R, γ=0, k) + γ Iλ(ω R, γ, k)! γ = 0. (3.72b) γ=0 For bi-maxwellian distribution functions, the dispersion function λ is a meromorphic function of the frequency ω and we can utilize the Cauchy-Riemann relations, Rλ(ω R, γ, k) = Iλ(ω R, γ, k), ω R γ (3.73a) Rλ(ω R, γ, k) = Iλ(ω R, γ, k). γ ω R (3.73b) Inserting relation (3.73a) into (3.72b) allows us to solve for the growth rate, ( ) Rλ(ωR, γ=0, k) 1 γ = Iλ(ω R, γ=0, k), (3.74) ω R which enables us to calculate the growth rate as a function of the real frequency and wavenumber in the limit of small γ ω R. To find an expression for the real frequency, we first insert (3.73b) into (3.72a), Rλ(ω R, γ=0, k) γ Iλ(ω R, γ=0, k) ω R, (3.75) then insert (3.74) solved for Iλ and find Rλ(ω R, γ=0, k) + γ 2 2 Rλ(ω R, γ, k) = 0. (3.76) ω 2 R Assuming that 2 Rλ(ω R, γ, k)/ ωr 2 is not much larger than Rλ(ω R, γ=0, k), we can neglect the second term because of the small parameter γ 2 and find Rλ(ω R, γ=0, k) = 0. (3.77) This equation can be solved for the real frequency ω R as a function of the wavenumber and provides, together with (3.74), an accurate solution of the dispersion relation in the weak amplification limit. An analogous expansion for small real frequencies ω R γ was introduced only recently by Schlickeiser et al. (2011) [98]. This new weak propagation limit provides quasi-aperiodic solutions and is discussed in-depth in the thesis Škoda (2013) [101]. 41

50

51 Chapter 4 WIND measurements We will derive improved instability thresholds in the next sections. In order to demonstrate that our new, theoretical solutions are indeed an improvement compared to prior results and are applicable to the real world, we need a comparison to measured data. In this chapter, we discuss how data plots like Figure 1.1 are created, how they should be interpreted, and what their limitations are. This requires the understanding of data measurement and processing, which will be discussed in turn. All data used in this thesis 1 are from the NASA WIND mission, publically available through Figure 4.1: Artist's impression of the WIND the SPDF CDAWeb service, spacecraft (from cdaweb.gsfc.nasa.gov/. The main eoportal.org/web/eoportal/ emphasis lies on the SWE instrument on the WIND spacecraft, satellite-missions/v-w-x-y-z/wind). which measures proton distributions up to 8 kev and their moments, i.e. the directional temperatures T,p, T,p and the number density n p near Earth. Unless otherwise quoted, all information is from Ogilvie et al. (1995) [83], Acuña et al. (1995) [4], and Szabo and Wilson III (2013) [104]. 1 With the exception of Figures 5.13 and 6.15, where we use ready-made plots from a publication since the data therein is not available to the public. 43

52 4. WIND measurements 4.1 WIND spacecraft The WIND spacecraft was launched in November, 1994 as one component of the Global Geospace Science (GGS) Program within the International Solar- Terrestrial Physics (ISTP) Science Initiative. ISTP is an international collaboration of NASA, ESA, and ISAS with the incentive of developing a global solarterrestrial model to understand and possibly predict Earth s atmosphere s response to solar activity. WIND s objective is to make highly accurate solar wind measurements upstream of the magnetosphere, and to remotely sense interplanetary disturbances. The spacecraft is a spin-stabilized, cylindrical satellite of 1.8 m height and 2.4 m diameter, the dry mass is 895 kg (and initially 300 kg propellant) WIND s orbit Figure 4.2: WIND orbit from November, 2002 to August, 2004 (adapted from http: //pwg.gsfc.nasa.gov/wind.shtml). For the first few years, WIND measured at the L1 Lagrange point, which is one of five force-free points in the ecliptic, where the gravitational forces of Sun and Earth cancel with the centrifugal force of the spacecraft. L1 lies on the Sun-Earth line, approximately 250 R E from Earth, which is far enough from the Earth s magnetosphere to study the undisturbed solar wind, compare Figure 4.2. Then WIND performed a series of orbital maneuvers around Earth and returned to L1 in October, 1997 when the initial mission ended. The mission was extended and various maneuvers to scientifically valuable observational points followed until Among other things, WIND moved to 44

53 4.1. WIND spacecraft geomagnetic high latitudes outside the ecliptic (1999), moved away from the Sun-Earth line reaching 350 R E to the side ( ), and moved to the L2 Lagrange point 250 R E downstream of Earth (2003, see Figure 4.2). Since 2004, WIND has remaind at L1 and will stay there in the near future. Considering the large fuel reserve ( 57 kg) and the low number of four station keeping maneuvers per year to orbit around L1, the spacecraft can maintain its orbit for approximately 60 more years. However, the limiting factor seems to be the spacecraft s power systems. The three batteries are expected to maintain high enough voltages for at least 10 years, while the solar array should produce enough current well past Instruments on WIND The WIND spacecraft carries an array of eight instruments which are briefly introduced in turn. 3DP (3D Plasma and Energetic Particles) is an array of three telescopes and measures three-dimensional spectra of ions and electrons over a wide range of energies from 3 ev up to 11 MeV and 1 MeV, respectively (Lin et al. (1995) [68] ). EPACT (Energetic Particles: Acceleration, Composition and Transport) provides fluxes for highly energetic ions (helium, oxygen, iron, combined CNO). However, the highest energy telescopes broke down early, and only data up to 1 MeV/nuc is available (von Rosenvinge et al. (1995) [110] ). KONUS consists of two large, cone-shaped scintillators designed to observe solar flares and gamma-ray bursts with high time resolution (Aptekar et al. (1995) [6] ). MFI (Magnetic Field Investigation) measures the direction and magnitude of the magnetic field, and is discussed in more detail in Section 4.2 (Lepping et al. (1995) [66] ). SMS is composed of three different instruments of which one has failed since The working instruments are spectrometers of high resolution for kev/e (energy per charge) and of low resolution for supra-thermal ions for kev/e (see Gloeckler et al. (1995) [36] ). 45

54 4. WIND measurements Figure 4.3: SWE instruments on WIND spacecraft, adapted from Ogilvie et al. (1995) [83] SWE (Solar Wind Experiment) measures spectra and moments of electrons and thermal protons and ions, and is extensively discussed in Section 4.3 (Ogilvie et al. (1995) [83] ). TGRS (Transient Gamma-Ray Spectrometer) is another gamma-ray detector, but was designed for only a few years of operation and turned off due to not longer having enough coolant to operate (see Owens et al. (1995) [84] ). WAVES measures electric fields in a wide range of frequencies to investigate the properties of radio and plasma waves (Bougeret et al. (1995) [14] ). 4.2 MFI The MFI consists of two identical triaxial fluxgate magnetometers mounted at the ends of the two booms (see Figure 4.1) and associated electronics. The dual configuration does not only provide redundancy but is also needed to cancel out the spacecraft s magnetic field from the data. The triaxial magnetometer provides three-dimensional measurements of the magnetic field vector with a sample rate of 44 vectors per second for snapshots and vectors 46

55 4.3. SWE Parameter Range Precision Proton velocity 200 to 1250 km/s ±3% (3 components) Proton number density 0.1 to 200 cm 3 ±10% Proton thermal speed 0 to 200 km/s ±10% Alpha/proton number 0 to 100% ±10% density ratio Table 4.1: Key parameters from the SWE instrument, adapted from Ogilvie et al. (1995) [83]. Note that these are estimates from before the launch of the spacecraft. The error in the measured data can sometimes exceed 10%. per second for standard operation. Based on the standard measurements, averages at 3, 60, 93 seconds, and 1 hour in GSE 2 and GSM 3 coordinates are calculated by the data processing system on board. The vector components can be measured using one of eight different dynamic ranges ±4 nt, ±16 nt, ±64 nt,, ±65, 536 nt, with a respective resolution of ±0.001 nt, ±0.004 nt, ±0.016 nt,..., ±16.0 nt. The lowest range can be only activated by ground control, the remaining ranges are selected automatically by the controlling microprocessor. 4.3 SWE The SWE is a set of sensors designed to measure the distribution and its moments of electrons, proton, and ions. The interesting proton key parameters are summarized in Table 4.1. Note that the proton velocity corresponds to a kinetic energy range of 150 ev to 8 kev, so no supra-thermal protons are measured with this instrument, and that instead of temperatures T, the notation of thermal speed k B T/m is used. All these key parameters are derived from particle distributions, and we show how they are measured and processed subsequently SWE components Figure 4.3 shows how the SWE sensors are mounted on the spacecraft, the left-hand side figure shows a top and bottom view along the axis of rotation, 2 Geocentric Solar Ecliptic coordinate system, a Cartesian right-handed coordinate system with center in the Earth, x-axis pointing towards the Sun, and z-axis to the ecliptic north pole (see Hapgood (1992) [38] ). 3 Geocentric Solar Magnetospheric system, is obtain by rotating the GSE system about the x-axis so that the z-axis points to Earth s magnetic north pole (also in Hapgood (1992) [38] ). 47

56 4. WIND measurements a lateral view is presented on the right-hand side. Two Faraday cup and two Vector Electron/Ion Spectrometer (VEIS) instruments are located at each end of a spacecraft diameter, on bottom and top respectively. The group of bottom sensors is augmented by the strahl sensor. The Data Processing Unit (DPU) operates the sensors and pre-processes the measured data. The VEIS electrostatic analyzer broke down in 2001 and its functions are replaced by the strahl detector, which originally only measured the strahl component of the electron distribution. The instruments important for our analysis are the two Faraday cups. WIND rotates normal to the ecliptic plane, one Faraday cup points 15 southwards, the other 15 northwards. Both measure intensities of positively charged particles in different energy channels. Each channel is measured over the time span of one full spacecraft revolution, separately for several discrete azimuthal directions. The complete measurement of a spectrum needs, depending on the mode of operation, 7 to 31 revolutions. 4.4 Faraday cup Figure 4.4: Photography of the SWE Faraday cup instrument (from edu/space/www/wind.html) on the left-hand side. Schematic cross section on the righthand side (adapted from Ogilvie et al. (1995) [83]. The Faraday cup is a cup with around 120 aperture angle. Figure 4.4 shows a cross section, and how the intensity of positive particles in an energy channel is measured. The cup is closed by a positively charged modulator, which is basically a wire-mesh grid. With a voltage V applied to the grid, 48