MASTER'S THESIS
Investigation of the Anisotropy of Solar Wind Turbulence with Multiple Point Spacecraft Observations
Daniel Vech 2016
Master of Science (120 credits) Space Engineering - Space Master
Luleå University of Technology Department of Computer Science, Electrical and Space Engineering Investigation of the anisotropy of solar wind turbulence with multiple point spacecraft observations
Daniel Vech
Lule˚aUniversity of Technology Department of Computer Science, Electrical and Space Engineering
Supervisor: Dr. Christopher H K Chen Department of Physics, Imperial College London ABSTRACT
Turbulence is ubiquitous in the terrestrial environment and interplanetary space as well. In space and astrophysics turbulence is of particular interest because its understanding is essential to describe a very wide range physical processes such as operation of the solar dynamo, heating of the solar corona, origin of the solar wind, dynamics of accretion disks and even creation of controlled fusion. In quasi-neutral fluids with high conductivity the turbulence is anisotropic, which is the most significant difference as compared to turbulence in neutral fluids. The anisotropy arises because of the mean magnetic field breaking the isotropic symmetry. Besides the mean magnetic field other mechanisms may affect the anisotropy as well. Recent theoretical studies focused on the possible effects of solar wind expansion on the anisotropy of the turbulence as the plasma propagates from the Sun and stretches gradually. The goal of this thesis is to test these predictions. The study is based on multiple spacecraft data, which is necessary to overcome the limitations of the single spacecraft observations and to be able to reveal three-dimensional (3-D) features of the turbulence along different sampling directions with respect to the solar wind flow and on different spatial scales as well. The results of this work are consistent with expansion playing a role in the evolution of the turbulent cascade at large scales by causing anisotropic energy distribution among the magnetic field components. It is demonstrated that the local 3-D anisotropy shows dependence on the sampling direction and the magnitude of the second-order structure functions is reduced when the anisotropy is measured in the perpendicular plane with respect the the radial, which can be a signature of expansion as well. However, on smaller scales it
ii was not possible to identify the signature of expansion. It implies that on small scales the turnover of the eddies is quicker than the effect of the expansion.
iii ACKNOWLEDGEMENTS
I would like to thank my advisor Christopher H K Chen, for all his help and guidance that he has given me during the preparation of this thesis. I am also grateful for the useful suggestions and comments of Tim Horbury and Lorenzo Matteini. I would like to thank the Department of Physics at Imperial College London for accepting me as a visiting student researcher. Finally, I would like to thank the support of the Education, Audiovisual and Culture Executive Agency of the Commission of the European Communities under the Erasmus Mundus Framework.
iv CONTENTS
Chapter 1 – Introduction 2
Chapter 2 – Basic Plasma Physics 5 2.1 Motion of charged particles in electromagnetic fields ...... 5 2.2 Waves in MHD plasmas ...... 6
Chapter 3 – Heliospheric Physics 9 3.1 Origin of the Solar Wind ...... 9 3.2 Interplanetary Magnetic Field ...... 10 3.3 Transient events in the solar wind ...... 13
Chapter 4 – Turbulence 16 4.1 Hydrodynamic Turbulence ...... 16 4.1.1 Cascade Theory and the Kolmogorov Hypotheses ...... 17 4.2 Magnetohydrodynamic Turbulence ...... 20 4.2.1 Turbulence in the Solar Wind ...... 20 4.3 Anisotropy of the Turbulence ...... 22 4.3.1 Critical Balance Theory ...... 23 4.3.2 Variance Anisotropy ...... 25 4.3.3 Wavevector and Power Anisotropy ...... 26 4.3.4 Spectral Index Anisotropy ...... 27 4.4 The Effects of Solar Wind Expansion on the Anisotropy of Turbulence . . . 28
Chapter 5 – Spacecraft description 38 5.1 Advanced Composition Explorer ...... 38 5.2 Wind ...... 39 5.3 Cluster ...... 39 5.4 ARTEMIS ...... 41
Chapter 6 – Data Analysis 43 6.1 ACE-Wind and ARTEMIS data selection ...... 45 6.2 Cluster Data Selection ...... 46 6.3 Test of the Radial Anisotropy ...... 46 6.4 Variance Anisotropy ...... 48 6.5 Local 3-D spectral anisotropy ...... 50
Chapter 7 – Discussion and Conclusion 55
vi Disclaimer
This project has been funded with support from the European Commission. This publica- tion [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.
http://ec.europa.eu/dgs/education_culture/publ/graphics/beneficiaries_all.pdf
1 Chapter 1
Introduction
Turbulence is observable in a wide range of geophysical flows in the terrestrial environment such as the atmosphere and oceans. Turbulence has been studied for over centuries, however several of its features remained poorly understood. The importance of turbulence to classical physics and to the sister sciences has been emphasized even by Richard Feynman [Feynman, 1963]. Turbulence arises when the viscosity in a fluid is weakened so that the flow becomes highly nonlinear and this transition from the laminar state introduces chaotic and seemingly random motion of the particles in the fluid. In a turbulent flow the velocity varies significantly and irregularly both as a function of location and time [Matthaus & Velli, 2011; Pope, 2001]. Turbulence can occur not only in neutral fluids but in plasmas as well, which at large scales can be considered as quasi-neutral fluids with high conductivity described by magnetohydro- dynamics (MHD). Understanding MHD turbulence is particularly important to describe the energy transfer across different scales for example in the solar dynamo, corona, solar wind, accretion disks and even fusion devices. The reversal of the solar dynamo must be related to nonlinear processes and turbulence is a reasonable candidate contributing to this phenomenon. Turbulence in the corona can be considered as a major driver of the heating and even of the solar wind itself [Mattheus & Velli, 2011]. In the case of the solar wind, turbulence can add up to 1000 J/kg/sec energy to the flow, which can have a major impact on the structure of the heliosphere [Coleman, 1968]. On even larger scales turbulence plays a major role in the transport of angular momentum during the formation of accretion disks [Biskamp, 2008]. MHD turbulence has implications
2 3
for fusion research as well. In order to achieve controlled fusion, it is essential to maintain the desired high temperature in the tokamak. Small fluctuations in the magnetic field however, cause turbulent transfer of the heat from the inner core toward the edge of the device and as a result the overall plasma performance is degraded. Knowing the turbulent transport is important for the future of fusion research [Howard et al., 2014]. A remarkable difference between the neutral and MHD fluids is that the turbulence is anisotropic in the latter case. The anisotropy is introduced by the magnetic field, which provides a preferred direction. Anisotropy is essential to understand how the turbulent cascade operates. One may assume axisymmetry with respect to the magnetic field, however this also breaks down and leads to the conclusion that the turbulence is three-dimensionally anisotropic [Chen et al., 2012] and may have different scaling along all the three spatial directions [Boldyrev et al., 2006]. Recent studies proposed new factors that should be taken into account when the anisotropy of the solar wind turbulence is studied. The papers of Dong et al. [2014] and Verdini et al. [2015] concerned the effect of the expansion of the solar wind as it propagates through the heliosphere. These authors proposed that stretching of the plasma volume along the transverse direction (e.g. perpendicular to the direction of propagation) can also give a pre- ferred direction to the turbulence affecting the small scale fluctuations and three-dimensional nature of the turbulence. The goal of this thesis is to study the effects of solar wind expansion on the anisotropy of the turbulence and to test the proposed mechanisms of the previous papers. The effect of the expansion is explored with a multiple point spacecraft technique including the data of the Advanced Composition Explorer (ACE), Wind, Cluster and ARTEMIS (Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon’s Interaction with the Sun) data. By exploiting the different separation distances and directions of these spacecraft, the turbulence can be studied when the pair of spacecraft are separated along the radial direction (e.g. along the spacecraft - Sun line) and when they are along the transverse direction (e.g. perpendicular to the propagation of the solar wind). The comparison of the observed features for the two different cases may reveal the effect of the expansion and the stretching of the plasma. The structure of the thesis is the following: Chapter 2 gives a short introduction to magne- 4
tohydrodynamic (MHD) theory. Chapter 3 concerns the solar wind and the structure of the Interplanetary Magnetic Field (IMF) including the origin and basic properties of the solar wind. The turbulence is in the scope of the 4th Chapter. Both the hydrodynamic and MHD turbulence are discussed in details with emphasizes on the different types of anisotropies arising in magnetized fluids. Chapter 5 gives a short review of the ACE, ARTEMIS, Cluster and Wind spacecraft. The mission goals, orbit parameters and the magnetometer instru- ments of each spacecraft are explained. The analysis of the magnetic field data is described in Chapter 6. Finally, the discussion of the results along with the conclusions are in Chapter 7. Chapter 2
Basic Plasma Physics
The purpose of this chapter is to provide an introduction to space plasma physics and to give the basic definitions, which are used in the later sections of this thesis. Section 2.1 concerns the motion of charged particles in electromagnetic fields. The concept of MHD theory is discussed in Section 2.2. The summary presented in this chapter is based on the works of Kivelson and Russell [1994], Gombosi [1994], Cravens [2004], Biskamp [2008] and Jackson [1974].
2.1 Motion of charged particles in electromagnetic fields
Charged particles in the presence of an electromagnetic field experience the Lorentz force defined with Equation (2.1), where m is the mass of the particle, v the velocity, q the charge, E the electric field, B the magnetic field,
dv m = q(E + v × B). (2.1) dt In the case when the electric field is zero and the magnetic field is uniform and constant, the particle will gyrate around the field line. The gyrofrequency (also known as the cyclotron frequency) of the motion is given by
|q|B Ω = 0 , (2.2) c m where q denotes the charge and B0 the magnitude of the ambient magnetic field. The radius of the guiding center is called Larmor radius and it is given by
5 2.2. Waves in MHD plasmas 6
v⊥ rc = , (2.3) Ωc where v⊥ is the velocity component of the particle perpendicular with respect to the magnetic field. In the presence of a magnetic and electric field the charged particles will have drift motion according to Equation (2.4),
E × B v = 0 0 . (2.4) E B2 It can be seen that the direction of the drift motion is independent of the particle’s charge, v⊥, q and no net current can be created by this drift. A test charge can attract particles with opposite charge while it repels the ones with same charge. As a result, in the vicinity of the test particle an oppositely charged shielding cloud appears, which cancels the electric field generated by the test charge. This is called Debye shielding and its length scale is called Debye length: r 0kBT λ = 2 , (2.5) e n0 where 0 denotes the permittivity of vacuum, kB the Boltzman constant, T the temperature of the plasma, e the elementary charge and n0 is the number density of the plasma.
2.2 Waves in MHD plasmas
MHD is used for describing electrically conducting fluids on macroscopic scales, typically scales larger than the Debye length and Larmor radius of the plasma. The MHD equations can be derived from the kinetic theory. The assumptions made in the derivation of the MHD formulas are valid when the characteristic speeds are much slower than the speed of light, the scale lengths of interest are much larger than the mean free path of particles and finally the time-scales of interest are significantly larger than the collision times, although aspects of MHD can also be applied to collisionless plasmas. The linear MHD dispersion relation has three different roots corresponding to three dif- ferent wave modes, which can propagate in the MHD plasma. These are the slow magne- tosonic, Alfv´en(also called as shear Alfv´enwave) and fast magnetosonic waves (also called as magneto-acoustic or fast Alfv´enwave). The basic properties of these waves are summarized below. The slow magnetosonic wave is the simplest MHD wave form. The plasma is considered 2.2. Waves in MHD plasmas 7
to be a compressible fluid and the velocity of the wave can be given in the form of s 5k(Te + Ti) vS = , (2.6) 3mion where Te and Ti denote the electron and ion temperatures, respectively and mion is the mass of the ions. In the equation 5/3 is the adiabatic index γ, which is the ratio of the specific heats for constant pressure and volume. The second wave mode is the Alfv´enwave, which is a transverse wave meaning that the fluctuations only have components perpendicular to the magnetic field. The Alfv´enwaves can be interpreted as waves propagating along magnetic field lines and the magnetic tension force act as a restoring force. The velocity of the Alfv´enwaves (vA) depends on the magnetic
field (B0), the mass density of the plasma (ρ) and the vacuum permeability (µ0):
B0 vA = √ . (2.7) µ0ρ The third wave mode is the fast magnetosonic wave, its velocity depends on the sonic and Alfv´envelocities: q 2 2 vMS = vS + vA. (2.8) The differences between the magnetosonic and Alfv´enwaves are presented in Figure 2.1a and b, respectively. The basic feature of the magnetosonic waves is the movement perpen- dicular to the magnetic field and it is also noted that the direction of the field is not changed by these type of waves. The Alfv´enwaves, in contrast, cause the oscillation of the field lines without generating perturbations in the plasma pressure and density. Both for the Alfv´en and magnetosonic waves the field lines are “frozen in” and are moving together with the fluid. From the aspect of this thesis the Alfv´enwaves are particularly important and their implications for the turbulence will be discussed in detail in Chapter 4.3. 2.2. Waves in MHD plasmas 8
Figure 2.1: Propagation of magnetosonic (a) and Alfv´en(b) waves in MHD plasma [Figure from Jackson, 1975]. Chapter 3
Heliospheric Physics
This thesis focuses on the solar wind turbulence therefore to understand turbulence mea- surements it is essential to introduce the structure of the heliosphere. The description starts with the origin of the solar wind (Section 3.1) and afterwards in Section 3.2 its physical pa- rameters and the structure of the interplanetary magnetic field (IMF) are explained. Finally, Section 3.3 gives a review on transient events in the solar wind. The presented summary is based on the works of Kivelson and Russell [1995], Biskamp [2008], Cravens [2004] and Gombosi [1994].
3.1 Origin of the Solar Wind
The concept of the continuous solar wind was first proposed by Biermann [1951] while the underlying physics was developed by Parker [1958]. Parker’s explanation was based on the fact that hydrostatic equilibrium is not possible between the Sun and the interplanetary medium. In the simplest case when hydrostatic equilibrium is assumed, the predicted pres- sure at infinite distance from the Sun is 10 orders of magnitude larger than the expected pressure in the interstellar medium. Parker investigated the case when non-zero radial veloc- ity and steady-state spherically symmetric isothermal corona were assumed. The equation describing this model is:
1 R2 u2 − a2 ln u = 2a2 ln r + g + C, (3.1) 2 s s r where C is an integration constant, u is the velocity of the solar wind, r is the distance from the Sun, R is the radius of the Sun, g is the gravitational acceleration at the base of the
9 3.2. Interplanetary Magnetic Field 10
corona, “a” is a constant meaning isothermal corona. It is noted that the equation describes a hydrodynamic model and does not include magnetic effects. The five possible solutions of Equation (3.1) as a function of radial distance and velocity are presented in Figure 3.1. Solutions I and II are clearly unphysical and can be discarded. Solution III implies that the solar wind is supersonic at the base of the corona, which is also unphysical. Solution V is physically possible, however it predicts finite pressure at infinite distances from the Sun and the predicted pressure is significantly higher than the expected pressure in the interstellar medium. Solution IV matches the observations and its plasma version is used to describe the solar wind. This type of solution is subsonic at the base of the corona and then it is accelerated to supersonic speeds.
Figure 3.1: The five possible solutions of the equation (3.1), rc and vc denote the distance and velocity at which the flow turns into supersonic [Figure from Gombosi, 1994].
3.2 Interplanetary Magnetic Field
The source of the interplanetary magnetic field is the dynamo of the Sun; a schematic of the magnetic field at solar minimum is shown in Figure 3.2. It can be seen that the closed field lines are prevailing in the vicinity of the Sun near the solar equator while the open field lines (connecting to the Sun with one end) extend at larger distances. The open field lines originating from the north and south magnetic poles form a boundary, which is called the 3.2. Interplanetary Magnetic Field 11
heliospheric current sheet (gray area in Figure 3.2) corresponding to the magnetic equator of the heliosphere. The very first spacecraft measurements of the IMF revealed that the radial component of the IMF occasionally changes sign and points either toward the Sun or away from it. This feature is due to the sector structure of the heliospheric current sheet, the neighboring sectors are separated with tangential discontinuities [Biskamp, 2008].
Figure 3.2: The structure of the heliospheric current sheet near solar minimum. The gray area separates the open field lines originating from the north and south magnetic poles [Figure from Smith, 2001].
The solar wind is a tenuous plasma continuously emanating from the Sun. The major components of the solar wind are hydrogen ions (≈ 96%), helium ions (≈ 4%) and electrons. One possible classification of the solar wind is based on its speed: the fast solar wind originates from the coronal holes can reach speed up to 700 km/s while slow solar wind is emitted from the closed field line regions and has typical speed around 400 km/s. The coronal holes are regions with unusually low temperature and density in the solar corona [Zirker et al., 1977]. The typical parameters of the solar wind at 1 AU are shown in Table 1. Due to the solar rotation and frozen in field, the magnetic field direction forms an angle with respect to the solar wind stream. This is known as the Parker’s spiral (Figure 3.3). The angle (φ) between the solar wind flow and the magnetic field (Equation 3.2) is a function of the rotational speed of Sun (ΩS), the distance from the Sun (R) and the solar wind velocity 3.2. Interplanetary Magnetic Field 12
Proton density 6.6 cm−3 Flow speed 450 km/s Proton temperature 1.2 · 105 K Electron temperature 1.4 · 105 K Magnitude of the magnetic field 7 nT Sound speed 60 km/s Alfv´enspeed 40 km/s Proton gyroradius 80 km/s Debye length 7 m Table 3.1: Summary of the average solar wind parameters at 1 AU [Kivelson and Russell, 1995].
◦ (VS). In the case of Earth, φ is approximately 45 . However, this angle can have large fluctuations due to waves and turbulence in the solar wind.
Ω R φ = S (3.2) VS
Figure 3.3: The outward streaming solar wind together with the solar rotation leads to the formation of the Parker spiral [Figure from Parker, 1963].
The distribution of the solar wind parameters for various solar latitudes was revealed by the Ulysses spacecraft, which was the first mission studying the solar wind at high latitudes (up to 80◦) [McComas et al., 2000]. The obtained distribution of the plasma parameters at solar minimum is shown in Figure 3.4. It can be seen that the slow solar wind (≈ 450km/s) 3.3. Transient events in the solar wind 13
is confined to the plane of ecliptic, while for higher latitudes the solar wind is faster, typically above 700 km/s. The solar wind density shows anticorrelation with the speed: the highest density regions are observed near the plane of ecliptic, while for higher latitudes the density is lower. It is noted, however, that the solar wind flux is approximately constant along any latitude.
Figure 3.4: Distribution of the solar wind speed and density for various heliospheric latitudes based on the Ulysses spacecraft data. [Figure from McComas et al., 2000].
3.3 Transient events in the solar wind
The average solar wind conditions shown in Table 3.1 can be significantly perturbed by transient events, which include Interplanetary Coronal Mass Ejections (ICME), magnetic clouds and Corotating Interaction Regions (CIR). ICMEs are large scale structures emerging from the solar corona. They drive the space weather conditions in the heliosphere and are characterized by enhanced solar wind velocity (occasionally exceeding 1000 km/s) and increased magnetic fields. The structure of a fast 3.3. Transient events in the solar wind 14
ICME consists of an interplanetary shock, which is followed by shocked and heated solar wind plasma. In some cases, the ICMEs are also accompanied with magnetic clouds, which can be considered as substructures [Gossling et al. 1990]. Burlaga et al. [1981] defined the magnetic clouds as an interplanetary structure with size of the order of 0.25 AU accompanied with higher than average magnetic field. Magnetic clouds display large scale rotation of the magnetic field and they are characterized with low proton temperature (due to the expansion of the plasma) and plasma beta below 1 (ratio of the plasma pressure and the magnetic pressure). Previous statistical studies found that nearly 30% of the ICMEs are accompanied with magnetic clouds [Gossling et al., 1990]. The interaction between the fast and slow solar wind streams is called a CIR (Figure 3.5) [Gossling et al., 1995]. The signatures of a CIR typically include a forward shock, stream interface and reverse shock. The forward shock moves gradually into the slow wind and it is characterized with sudden increase of the solar wind speed, density and magnetic field magnitude. The stream interface is the boundary between the fast and slow solar wind streams. The reverse shock moves backward into the fast stream and it is accompanied with the sudden decrease of the magnetic field strength and increase of the solar wind velocity [Heber et al., 1999]. This study focuses on the turbulence in the quiet solar wind thus ICMEs, CIRs and magnetic clouds are not desired in the data since they strongly influence the fluctuations in the magnetic field during their passages. 3.3. Transient events in the solar wind 15
Figure 3.5: Formation of CIRs; the high speed stream reaches the slow solar wind and compression regions are formed [Figure from Hundhausen, 1972]. Chapter 4
Turbulence
This chapter provides a summary of both hydrodynamic and magnetohydrodynamic turbu- lence. First in Section 4.1 hydrodynamic turbulence is described including the cascade theory and Kolmogorov hypotheses. The key concepts of turbulence theory are discussed, which are the basis of the following section 4.2, in which the properties of the MHD turbulence are explained. Finally, in 4.3 the anisotropy of the solar wind turbulence is reviewed.
4.1 Hydrodynamic Turbulence
Based on the motion of particles in a flow we can distinguish turbulence and laminar flows. Laminar flows have ordered flow features while the turbulent ones display chaotic motion. One important feature of turbulence is the ability to mix and transport particles in a much more effective way than laminar flows [Pope, 2001]. In nature the flows are generally turbu- lent and as it will be discussed in the later sections this is also the case for plasmas. One of the most important parameters to describe a flow is the Reynolds number. It is defined as UL/v where U denotes the characteristic speed, L is the characteristic length and v is the viscosity. For example, a flow with a Reynolds number of 2300 cannot be accurately described with laminar fluid theory because the viscosity is negligible and a flow with Reynolds number of 4300 is completely in the turbulent regime [Matthaeus & Velli, 2011]. For the discussion of the turbulence in the next sections always very high Reynolds numbers are considered.
16 4.1. Hydrodynamic Turbulence 17
4.1.1 Cascade Theory and the Kolmogorov Hypotheses
Toward the description and the understanding of hydrodynamic turbulence pioneering work was done by Kolmogorov [1941a, 1941b and 1941c] and Richardson [1922]. In this section the findings of these papers are summarized based on the work of Pope [2001]. Richardson [1922] introduced the concept of the energy cascade. According to this idea, the turbulence is made of eddies of different sizes. The eddies can be defined as turbulent motion and they are characterized with three parameters, which are the size l, the velocity u(l) and the timescale τ(l) = l/u(l). Eddies with the largest size have lengthscale l0 and characteristic velocity u0 = u(l0). Both of these parameters are comparable to U and L and under these circumstances the Reynolds number (Re0) of the eddies is large, which means that the effect of viscosity can be ignored. The energy is transferred to smaller and smaller scales due to the fact that the large eddies are unstable and tend to break up. This process is repeated until the Reynolds number becomes small enough that the eddies can be stable and the kinetic viscosity can dissipate the energy. Due to the transfer of energy across scales, the rate of dissipation is a function of the energy of the eddies on the largest scale. It can be shown that in the
3 case of high Reynolds number the rate of dissipation is proportional to u0/l0 and it does not depend on the turbulence viscosity. Kolmogorov [1941a, 1941b and 1941c] improved the concept of Richardson [1922] from several aspects. Kolmogorov suggested that large eddies are anisotropic, however, as they break up and transfer the energy to smaller and smaller scales the daughter eddies become isotropic. This process can be understood as the loss of directional information. Kolmogorov proposed that the statistical properties of small scale motion is similar in every turbulent flow with sufficiently high Reynolds number. This concept is also known as the first similarity hypothesis. Let lEI denote the lengthscale above which the eddies are anisotropic but below they are isotropic. In the case of l < lEI (also called as the universal equilibrium range) the two relevant processes are the energy transfer toward smaller scales and the viscous dissipation. The rate of dissipation is a function of the energy transfer from larger to smaller scales (denoted with TEI ) and the relationship between them is ≈ TEI . The statistics of the small scale motion has a universal form when the Reynolds number is high and it can be uniquely described with v and . Equation (4.1), (4.2) and (4.3) can be 4.1. Hydrodynamic Turbulence 18
used to describe the eddies on the smallest (dissipative) scales and they are known as the
Kolmogorov length (η), time (uη) and velocity (τη) scales.
v3 η = ( )1/4, (4.1)
1/4 uη = (v) , (4.2)
v τ = ( )1/2, (4.3) η In order to justify the meaning of the “similarity hypothesis” and the “universal form” phrases, let x0 be a point in a turbulent flow at time t0, then the non-dimensional coordinates are defined by
x − x y = 0 . (4.4) η Additionally, the non-dimensional velocity-difference field is defined by
U(x, t ) − U(x , t ) w(y) = 0 0 0 . (4.5) uη It can be seen that the non-dimensional field w(y) cannot depend on v and since it is not possible to construct a non-dimensional parameter with the latter two variables. As a result, when w(y) is considered on small scales the flow is statistically isotropic and identical at all points if the Reynolds number of the turbulent flow is high.
There exists a lengthscale l, which is significantly smaller than l0, however, larger than
η: l0 >> l >> η. On this scale it is a reasonable assumption that the Reynolds number is large since the size of the eddies is much larger than on the dissipation scale. An additional consequence is that the motion of the eddies is hardly affected by the viscosity. Kolmogorov’s second similarity hypothesis states that the statistics of motion on the lengthscale l has a universal form and it is uniquely determined by and does not depend on v.
Let lDI be a lengthscale, which splits the equilibrium range (l < lEI ) into two subranges.
These are the dissipation range (lDI > η) and the inertial subrange (lEI > l > lDI ). Due to the consequence of the second similarity hypothesis these two subranges have the following properties: the motions in the inertial subrange strongly depend on the inertial effects and are not affected by the viscosity. On the other hand, the viscous effects play major role in the dissipation range and they are responsible for all the dissipation. The previously discussed lengthscales are presented in Figure 4.1. The vertical dashed line 4.1. Hydrodynamic Turbulence 19
at lEI separates the inertial and so called energy-containing range, in which the bulk of the energy is stored. The vertical dashed line at lDI indicates the boundary between the inertial subrange and the dissipation range.
Figure 4.1: Various lengthscales and ranges. L: characteristic scale of the turbulence, l0: lengthscale of the largest eddies, lEI : lengthscale above which the eddies are anisotropic, lDI := 60η, η: Kolmogorov’s length scale [Adopted figure from Pope, 2001].
Equation (4.6) and Figure 4.2 summarize the cascade of the energy from the largest to the dissipation scales. In Equation (4.6) T denotes the ratio of energy transfer from larger to smaller scales starting from the energy-containing range through the inertial range and the dissipation range.
TEI = T (lEI ) = T (l) = TDI = T (lDI ) = (4.6) This equation means that the energy transfer along different scales (T) is constant and it equals to the dissipated energy . If this was not the case, the energy would accumulate on a specific scale. The kinetic energy of the turbulence is distributed along different eddy scales. The first and second Kolmogorov hypotheses have the following implications for the energy distribution: in the universal equilibrium range (k > kEI where k denotes the wavenumber) the spectrum is a universal function of and v. Furthermore, based on the second Kolmogorov hypothesis in the inertial range (kEI < k < kDI ) the spectrum is defined by Equation (4.7).
E(k) = C2/3k−5/3, (4.7) where C is a universal constant, k denotes the wavenumber (k = 2π/l). This is known as the Kolmogorov −5/3 spectrum. 4.2. Magnetohydrodynamic Turbulence 20
Figure 4.2: The energy cascade when the Reynolds number is very high [Adopted figure from Pope, 2001]. 4.2 Magnetohydrodynamic Turbulence
MHD turbulence is different from hydrodynamic turbulence in several ways [Verma, 2014, Horbury et al., 2012]. First, the velocity and the magnetic field are coupled and viscosity and resistivity are introduced as dissipative parameters. Second, in the analysis of the hy- drodynamic turbulence the mean velocity field can be removed with the Galilean transform, however, in MHD turbulence the mean magnetic field cannot be transformed away. Fi- nally, due to the presence of the magnetic field anisotropy arises, which breaks the isotropic symmetry. Turbulence is ubiquitous in space plasmas and it affects the solar dynamo, the chromo- sphere, the solar corona and ultimately the solar wind. The study of the MHD turbulence is difficult in terrestrial experiments due to the very large dissipation ranges [Verma, 2004]. This is one of the reasons that the solar wind is considered to be a natural laboratory for studying MHD turbulence and the spacecraft observations have particular importance in this field.
4.2.1 Turbulence in the Solar Wind
Among the earliest evidence for the turbulent nature of the solar wind plasma is the ob- servations of the solar wind proton temperature. The expansion of the solar wind is highly non-adiabatic meaning that the decay of the proton temperature is much slower than would be expected if there was no heating [Richardson et al., 1995]. Wolfe et al. [1966] and Hundhausen et al. [1967] found that the parallel to transverse (with respect to the direc- 4.2. Magnetohydrodynamic Turbulence 21
tion of the IMF) ratio of the proton temperature (Tp/Tt) is approximately 3 at 1 AU. The obtained value is nearly one order of magnitude less than the expected ratio based on a laminar solar-wind expansion. Another compelling piece of evidence for the presence of a non-thermal process is related to the typical value of the solar wind proton temperature at
5 1 AU, which is T = 1/2(Tp + Tt) ≈= 10 K. Sturrock and Hartle [1966] suggested that the two-fluid model of the solar wind expansion would predict proton temperatures nearly an order of magnitude less than this value and therefore there is need for a process heating the protons in the plasma and also increasing the parallel temperature. Later the measurements of the Voyager spacecraft revealed the decrease of the proton temperature as a function of the radial distance from the Sun. Instead of the expected T ∼ r−4/3 ratio corresponding to spherically symmetric adiabatic expansion, the decay of the proton temperature was closest to T ∼ r−1/2 [Richardson et al., 1995]. Coleman [1968] studied the Mariner-2 spacecraft magnetic field data with a spectral anal- ysis of the magnetic field and the radial component of the solar wind velocity. The spectra of the magnetic field fluctuations revealed that fluctuations are qualitatively different between above 10−4 Hz and below 10−5 Hz. The range below 10−5 Hz correspond to the fluctuations in the unperturbed field. The fluctuations above 10−4 Hz are typically transverse to the field direction being nearly isotropic in the transverse plane and they correspond to the pertur- bations of the magnetic field. Coleman [1968] concluded that the energy range is between 10−5 and 10−4 Hz, the inertial range is between 10−4 and 10−1 Hz and finally the dissipation range is from 10−1 Hz. The dissipation of the Alfv´enwaves was thought to be ion cyclotron damping and this process would lead to the heating of the solar wind protons. Based on qualitative calculations, Coleman [1968] suggested that the internal energy added by the turbulence to the solar wind plasma is approximately 1000 J/kg/sec at 1 AU. The wavenumber spectrum provides additional evidence for the turbulence in the solar wind [Horbury et al., 2005]. A typical magnetic field spectrum of the solar wind is shown in Figure 4.3. In Figure 4.3 the frequencies are interpreted in the spacecraft frame. The solar wind flow is significantly faster than the waves propagating in it thus the spacecraft observations can be understood as a spatial cut through the plasma. This assumption is known as the Taylor hypothesis [Taylor, 1938] and it can relate to the plasma frame wavenumber as k = 2πf/vsw where vsw is the solar wind velocity. 4.3. Anisotropy of the Turbulence 22
Figure 4.3: Typical power spectrum of the solar wind [Adopted figure from Horbury et al., 2005].
The presented spectrum can be split into two parts: for the smaller frequencies (corre- sponding to larger scales) the slope is proportional to f −1, while for lower frequencies the spectral index is approximately −5/3 being very close to the Kolmogorov spectrum. How- ever, the −5/3 spectral index is a necessary but not sufficient condition for the presence of active turbulence. Alternative explanations may include that the power spectrum is only remnant of the turbulence taking place in the corona and transported through the heliosphere as a non-interacting population. The presence of the active turbulence was given support with measurements at various solar distances. The fluctuations interact with each other as they propagate from the Sun therefore their evolution can be inferred by measuring them at various distances. Figure 4.4 shows the relationship between the spectral indexes and the frequencies for three different solar distances. It can be seen clearly that as the solar distance increases the f −5/3 part of the spectra extend to lower frequencies. As a consequence, the inertial range is gradually extending to lower frequencies and the energy is transferred across different scales.
4.3 Anisotropy of the Turbulence
In plasmas, anisotropy arises due to the presence of the magnetic field breaking the isotropic symmetry and as a result there is a preferred direction in the inertial range [see review paper 4.3. Anisotropy of the Turbulence 23
Figure 4.4: The spectral index as a function of the frequency for three different solar distances: 0.3 AU (circle), 0.4 AU (triangle) and 0.9 AU (square) [Figure from Horbury et al., 2005]. by Horbury et al., 2012]. The anisotropy of the turbulence can be considered to be the most significant difference between MHD and hydrodynamic turbulence and it has important implications for a wide range of physical processes such as the transport of particles and cosmic rays. In this section some types of anisotropies are reviewed with emphasis on the variance and spectral anisotropies since they are in the scope of the analysis described in the later sections.
4.3.1 Critical Balance Theory
Early studies of MHD turbulence estimated the spectral index (k−α, where α is the spectral index and k is the wavenumber of the power spectra, see equation 4.7) of the energy spectrum. Iroshnikov [1963] and Kraichnan [1965] proposed that the energy spectrum of the weak Alfv´enicturbulence is E(k) ∼ k−3/2. They assumed that the turbulence is isotropic and their concept later became known as the Iroshnikov-Kraichnan turbulence. The anisotropic nature of the MHD turbulence was first suggested by Montgomery and Turner [1981]. They studied the MHD turbulence in the presence of a strong dc magnetic field and concluded that it might be reasonable to assume rotational isotropy with respect to the mean magnetic field but not for the other two directions. Later it was shown by Goldreich and Sridhar [1995] that the Iroshnikov-Kraichnan turbulence model is invalid and the scaling of the energy spectrum can be best described with k−5/3. Goldreich and Sridhar [1995] proposed the concept called “critical balance” theory. This term refers to the idea 4.3. Anisotropy of the Turbulence 24
that the Alfv´enicturbulence is in critical balance, in which the timescale of the nonlinear decay is equal to the timescale of fluctuations propagating along the field lines. The critical balance theory predicts larger wavenumbers along the perpendicular direction than along 2/3 the parallel one (k⊥ > kk), and the scaling between the two wavevectors is kk ∼ k⊥ . An additional consequence of the theory is that the anisotropy of the eddies becomes more and more pronounced as the scale of the eddies is decreasing. The critical balance theory predicts spectral indexes -2 and −5/3 for the parallel and perpendicular directions, respectively, which were experimentally confirmed (see section 4.3.4). In a later work Boldyrev [2006] extended the concept of the critical balance theory for the 3-D anisotropy as well. The differences between shape of the eddies in the critical balance theory and the concept of Boldyrev [2006] are shown in Figure 4.5. In both panels the large scale magnetic field is along the vertical direction and l, λ, ζ denote the scaling of the eddies along the three spatial directions. According to the Goldreich-Sridhar theory (left panel) the scaling of the eddies along the two perpendicular directions (denoted with λ) are equal. As the size of the eddies is decreasing λ goes to 0 and the final shape of the eddy resembles a filament. In the concept of Boldyrev [2006] (right panel) the two perpendicular directions (ξ and λ) have different scaling, which are ξ ∼ λ3/4 and l ∼ λ1/2. As the size of the eddies is decreasing λ goes to 0 quicker than ξ, which means that the shape of the eddy turns into a current sheet.
Figure 4.5: Different eddy shapes and scaling along the three spatial directions as proposed by Goldreich-Sridhar (left) and the concept of Boldyrev [2006]. Figure from Boldyrev [2006]. 4.3. Anisotropy of the Turbulence 25
4.3.2 Variance Anisotropy
The variance (also known as amplitude) anisotropy was first measured by Belcher and Davis [1971]. These authors studied the Mariner-5 spacecraft data with the so called minimum variance analysis. This method was developed by Sonnerup and Cahil [1968] and it based on the following principles. In the first step, the variance matrix of the measured magnetic field vector components is calculated for a given interval. The variance matrix is defined as
Sij =< BiBj > − < Bi >< Bj >, (4.8) where i,j denote the vector components of the magnetic field B and <> denotes the averaging over the chosen time interval. The trace of the tensor S corresponds to the sum of the eigenvalues and it is invariant to rotation of axes. Based on matrix S the eigenvalues
(P1 ≥ P2 ≥ P3) and the corresponding eigenvectors (S1, S2, S3) can be derived. The interpretation of the S1 and S3 vectors is that they are in the direction of the maximum and minimum variations of the field, respectively. The eigenvalues give information about the direction of the anisotropy. In the case when P1, P2, P3 are in the same order of magnitude the direction of the eigenvectors have less relevance since the fluctuations are nearly isotropic. In contrast, when P1 and P2 are comparable and P3 is significantly smaller, the fluctuations are anisotropic in the planes, which are aligned with S3. Similarly, if P2 and P3 are significantly smaller than P1 the perturbations are most prevailing in the plane along S1. Belcher and Davis [1971] observed that the magnetic field components display fluctuations, however the magnitude of the field remains nearly constant. This implies that S3 is parallel with respect to the mean magnetic field and that S1 and S2 are perpendicular to the mean magnetic field. The authors interpreted the results in a field-velocity coordinate system, which is defined the following way: Z axis is along the mean magnetic field eB, X is defined as eB × eR, where eR is the direction of the solar wind (approximately the radial direction along the spacecraft - Sun line) and the Y axis is in the direction of eZ × eX . The observed fluctuations were the smallest along the Z direction and largest along the X axis. Based on the analysis of the correlation between the radial magnetic field and velocity, they found that the observed Alfv´enwaves are propagating outward from the Sun. To explain the origin of the observed Alfv´enwaves Belcher and Davis [1971] considered the following three mechanisms: 1) the waves are generated locally by instabilities, 2) the waves are the 4.3. Anisotropy of the Turbulence 26
result of large scale velocity differences in the solar wind and 3) the waves originate from the solar chromosphere and photosphere. For 1), the observed wavelengths are in the range of 105 − 106 km and instabilities (such as the firehose instability) are unlikely to generate waves with wavelengths much larger than the ion cyclotron radius (102 km). 2) The stream-stream collisions provide energy for large- amplitude waves, however they are unlikely to contribute to the observed Alfven waves, which have much smaller amplitudes. 3), in the chromosphere and photosphere both Alfv´enand magnetosonic waves are generated, however at 1 AU only Alfv´enwaves were observed. For the lack of the magnetosonic wave modes a possible explanation is that they are damped at a very high rate. The theoretical study of Barnes [1966] showed that in the case of collisionless plasmas with plasma beta > 0.5 the magnetosonic MHD waves are damped while the Alfv´en mode remain undamped. The damping of the Alfv´enwaves is less pronounced since they are nearly one order of magnitude slower than the solar wind itself and their wavelength is in the order of 0.01 AU, which means that only 10 oscillations occur until they reach 1 AU. The variance anisotropy was investigated with the Ulysses spacecraft data by Horbury et al. [1995]. The study focused on the measurements near the polar heliospheric magnetic field. 5-minute intervals were analyzed and it was found that the minimum variance directions are within 26◦ from the mean magnetic field in around 75% of the cases. The fluctuations perpendicular to the mean field are nearly 30 times larger than along the field.
4.3.3 Wavevector and Power Anisotropy
The wavevector anisotropy is a consequence of the anisotropic (with respect to the mean magnetic field) energy transfer through different scales. As explained in Section 4.1.1, the Kolmogorov similarity hypothesis states that the eddies in neutral fluids become isotropic during the process as they break up and become smaller and smaller. In plasmas, the field- perpendicular wavevector increases at a larger rate than the parallel one, which is known as the wavevector anisotropy. This process distorts the shape of the daughter eddies, which do not become statistically isotropic but they are longer along the field than across it [see review paper by Horbury et al., 2012]. The power anisotropy is closely related to the wavevector anisotropy. It means that the distribution of the power on a particular scale is anisotropic with respect to the mean field 4.3. Anisotropy of the Turbulence 27
[Horbury et al., 2012].
4.3.4 Spectral Index Anisotropy
The spectral index anisotropy is of great importance for studying the turbulence since in many cases this is the sole variable, which can be directly measured [Horbury et al., 2012]. When discussing this type of anisotropy, we make difference between 2-D and 3-D anisotropy and both of them can be measured with respect to the local and global mean magnetic fields. The terms “local” and “global” mean that the frame of reference in which the anisotropy is investigated is either fixed to the local magnetic field (e.g. the frame of reference follows the fluctuations in the direction of the field at a similar scale) or it is fixed to an average magnetic field direction measured over a longer period typically the correlation length. Horbury et al. [2008] studied the 2-D spectral anisotropy with respect to the local mean
field. They found evidence for varying spectral index with respect to ΘB denoting the angle between the magnetic field and the solar wind stream direction. When the ΘB was in the range of 45 − 90◦ the spectral index was approximately 5/3. On the other hand, for the case ΘB → 0, the spectral index approached 2. These results are consistent with the critical balance theory, however, it is pointed out that the small difference in the spectral index can only be detected when the anisotropy is measured with respect to the local and not to the global mean field. Chen et al. [2011] studied the 2-D spectral index and wavevector anisotropies with respect to the local and global mean fields using pairs of Cluster spacecraft data. The analysis was based on the second order structure functions (also called B-trace structure function), which can be derived as
2 X 2 δB (l) = (Bi(r + l) − Bi(r)) , (4.9) i where Bi is the ith component of the magnetic field, l is the separation vector between the two spacecraft and r is a vector pointing from the center of the coordinate system to the spacecraft. Equation (4.9) describes that the instantaneously measured magnetic field components at two different locations are subtracted from each other, the differences are squared and are summed up. The local mean magnetic field for a scale l is defined as 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 28
B(r + l) + B(r) B = . (4.10) loc 2 It simply means that the magnetic field vectors measured with the pair of spacecraft at two different locations separated with the vector l are averaged. In the later sections the “local mean magnetic field” term always refers to Equation (4.10). The calculated B-trace structure values were binned according to the spacecraft separation vectors (parallel and perpendicular components) with respect to the direction of the local mean magnetic field. The results are summarized in Figure 4.6. 9-9 equally spaced bins can be seen along the parallel and perpendicular separation distances with respect to the local mean field. The coloring corresponds to the power of the average second-order structure function values in each bin. The interpretation of the figure is that the power of the fluc- tuations (e.g. value of the second order structure functions) is largest when the spacecraft are separated perpendicular with respect to the local mean magnetic field and lowest for the parallel case. This finding is in agreement with the critical balance theory (see Section 4.3.1), which predicts that the wavenumbers are larger along the perpendicular than parallel 2/3 directions (k⊥ > kk and kk ∼ k⊥ ). Chen et al. [2011] also studied the spectral index and wavevector anisotropies with respect to the global mean field and found that turbulence is much less anisotropic.
4.4 The Effects of Solar Wind Expansion on the Anisotropy of Turbulence
The goal of this thesis is to test the predictions of recent theoretical studies about the anisotropy of the turbulence. The selected works by Verdini et al. [2015] and Dong et al. [2014] discussed the effects of expanding solar wind on the variance and spectral index anisotropies and both studies described the results of three-dimensional MHD simulations for various input parameters. Several studies reached the conclusion that the axisymmetry (rotational symmetry with respect to the mean field) of the anisotropy breaks down in the solar wind [e.g. Saur & Bieber, 1999, Narita et al. 2010, Chen et al., 2012]. Verdini et al. [2015] proposed that this can be partly due to the fact that the investigated length scales were large enough so the expansion of the solar wind could affect the results. 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 29
Figure 4.6: Binned second order structure functions for the parallel and perpendicular separation directions with respect to the local mean magnetic field [Figure from Chen et al., 2011].
The applied method to study the effect of the expansion is based on the following principles: the simulation box consists of a plasma volume, which is placed into a flow with constant speed. In this model the flow represents the solar wind, which is advected with velocity Vsw and the different t times correspond to different heliospheric distances as R(t) = R0 + Vsw · t, where R0 = 0.2 AU denotes the initial heliospheric distance of the simulation box.
In the simulation box the magnetic field is measured at two different coordinates (x1 and x2) which are separated by the increment vector l = x1 − x2. The mean magnetic field for these two points is defined as Bl = (B1 + B2)/2, while the fluctuation in the field is given by
δB = B2 − B1. The anisotropy is studied in the local frame of reference presented in Figure 4.7b: in this frame ζ is oriented along the mean field, ξ is defined as the perpendicular
fluctuation direction and it is given by Bl × [δB × Bl] while λ completes the right hand system. The angles ΘB and ΘδB⊥ in Figure 4.7b give the direction of increment with respect to the mean field. In the analysis, for a given increment vector l the corresponding second- order structure functions were calculated. The results were binned according to the angles
◦ ◦ of ΘB and ΘδB⊥ from 0 − 90 in 5 bins and the average of each bin was taken. In the first run a uniform parallel flow was used without expansion, which means that 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 30
the solar wind flow is along the R=x axis (Figure 4.7a) and the spatial dimensions of the plasma (Lx, Ly, Lz measured along the x = R, y = T , z = N axes, respectively) remain constant as the flow propagates and reaches larger heliospheric distances. For run B, the effect of the expansion was included, which means that the radial dimension of the plasma volume (Lx) remains unchanged while the lateral dimensions (Ly and Lz) increase gradually with increasing heliospheric distance. This effect corresponds to the stretching of the plasma volume.
Figure 4.7: Reference frames used in the simulation of Verdini et al. [2015]. On panel a) the global reference frame is shown. The coordinate system is based on the RTN system in which R points from the Sun to Earth, N corresponds to the axis of solar rotation and T completes the right hand ◦ rule. The mean magnetic field B0 is directed 45 from the R axis. The magnetic fluctuations are denoted with B1 and B2 and they perpendicular with respect to the B0. On panel b) the local frame can be seen, which was used for studying the 3-D anisotropy [Figure from Verdini et al., 2015].
Figure 4.8 shows the results for run A without expansion and run B with expansion. These figures were obtained the following way: the second-order structure function values were binned for a given separation distance of x1 and x2 along the radial direction (x=R axis) and the average of the following three bins were plotted against the corresponding wavenumber:
◦ ◦ ◦ ◦ the green data point corresponds to the bin 85 < ΘB < 90 , 0 < ΘδB⊥ < 5 , the blue is 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 31
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 85 < ΘB < 90 , 85 < ΘδB⊥ < 90 and the red is 0 < ΘB < 5 , 85 < ΘδB⊥ < 90 . In the next step the separation distance between x1 and x2 was decreased (corresponding to larger wavenumber), the binning was repeated and the data points corresponding to a larger wavenumber were calculated. Every time the binning is repeated for a different wavenumber
(e.g. different separation distance between x1 and x2) an additional “slice” of the blue, red, green curves can be obtained. In this way it is possible to study the evolution of the spectral anisotropy on various scales.
Figure 4.8: Second-order structure functions for the run without expansion (a) and with expansion as a function of the wavenumber [Figure from Verdini et al., 2015].
It can be seen that on panel a) for wavenumbers smaller than 10 (corresponding to large scales) the three components have nearly the same second-order structure function value. For small scales the order of the three components is λ > ξ > ζ, while the scaling of these parameters is approximately λ1/2 ∼ ξ2/3 ∼ ξ1. In the case of run B with expansion the three components display differences for small wavenumbers (k < 10) and for large wavenumbers (k ≈ 102 − 103) λ dominates over all wavenumbers. The large scale ordering of the three structure functions in Figure 4.8b is caused by the expansion. In the non-expanding case the energy distribution is isotropic between the three magnetic field components. However, when expansion is introduced the radial component of the field has by a factor of 2 less power than the two perpendicular ones. The further implication of this difference can explain the large scale (k < 10) ordering of the structure function power levels in Figure 4.8b: in order to accumulate power in ξ, the local mean field should line in the y-z plane, the fluctuations in the field are perpendicular with respect to 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 32
the mean field and lie along the x axis. This component has the smallest energy therefore the fluctuations are the smallest as well. Contribution to ζ occurs when the local mean field lies along the x axis, which means that the Alfv´enicfluctuations are in the y-z plane. The y-z components have more energy than the x one, therefore the overall power level of ζ is larger than ξ. Finally, to have power in λ both the local mean field and the fluctuations should be in the y-z plane, which is the reason λ has the highest structure function value in Figure 4.8b. Verdini et al. [2015] also studied the case when the increments are taken along the trans- verse direction corresponding to y = T axis in Figure 4.7a. The results are shown in Figure 4.9, which has the same format as Figure 4.8. The results were compensated with k1/2 in order to highlight differences in the inertial range.
Figure 4.9: Second-order structure function of the expanding run when the increments are taken along the radial (a) direction and for the transverse case (b) [Figure from Verdini et al., 2015].
The figures show that the overall characteristics of the anisotropy depends on the sampling direction. For example, the large scale anisotropy (k < 10) measured along the radial direction completely disappears along the transverse direction. For the sampling along the transverse direction the shape of the eddies is qualitatively similar on all scales, while for the radial direction it is significantly different on different scales. Verdini et al. [2015] also concluded that the expansion of the solar wind with increasing heliocentric distances will confine the magnetic fluctuations more and more into the y-z plane, moreover the local mean magnetic field will tend to lie in this plane as well. In conclusion, the predictions of Verdini et al. [2015] about the expanding solar wind can 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 33
Run Mean magnetic field in Alfv´enunits B0 = (Bx,By) Expansion A (0,0) ×
B (0,0) X C (2,0) ×
D (2,0) X Table 4.1: Summary of the initial parameters for the different runs [Dong et al., 2014]. be summarized as follows:
• Both for the x and y separations the λ component dominates, however λ is larger for the separation along the x axis than for the y.
• For large scales the anisotropy in the perpendicular plane is reduced for the y sepa- ration, which means that the second order structure function bins show less variation than for the x separation. This also implies that the ratio of λ and ξ are smaller for the y separation.
• When the anisotropy is measured for various separation distances along the y axis, the distribution of the second-order structure functions is qualitatively similar. In contrast, the distribution of the second-order structure functions varies significantly for sampling on different scales along the radial direction.
Dong et al. [2014] studied the variance and global spectral anisotropies in the expanding solar wind from 0.2 to 1 AU. The used model is similar to the one of Verdini et al. [2015]: the turbulence is studied in a stretching box with constant flow, however in this case the anisotropy is measured with respect to the global mean field. The simulation started as isotropic turbulence (with k−1 spectrum) and then the spectra in the middle range steepened toward k−3/2 and spectral anisotropy developed. The input parameters for the four different runs are summarized in Table 4.1. Runs A and C are carried out without expansion, while runs B and D include expansion. In the model
Bx corresponds to the radial magnetic field while By is the perpendicular one. Dong et al. [2014] studied the fluctuations of the velocity and magnetic field components for different heliospheric distances. The results about this process are shown in Figure 4.10. In Figure 4.10 the root-mean-square (rms) amplitudes of the 3-D modes are presented in 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 34
Alfv´enspeed units for increasing heliospheric distances. The ordering of the rms amplitudes of the magnetic field (B) and velocity (U) are the following:
Figure 4.10: decay of the root-mean-square (rms) amplitudes with increasing heliospheric distance [Figure from Dong et al., 2014]
Ux < Bx < Uy < By. (4.11)
Due to symmetry reasons Uz and Bz behave like Uy and By. Equation (4.11) indicates that the stretching of the plasma volume causes selective decay of the velocity and magnetic field components. Dong et al. [2014] suggested that the different damping rates are due the additional term that appears in the MHD equations when the expansion is included and it transfers the energy to the perpendicular components. The results of runs A-D about the global spectral anisotropy are shown in Figure 4.11.
The horizontal and vertical axes of the figure correspond to the radial (kx) and perpendicular
(k⊥) wavenumbers, respectively. The isocontours denote the 3-D magnetic energy spectrum averaged around the kx axis. This type of averaging was possible because the energy spec- trum is gyrotropic with respect to kx in the global frame. The thick black line with the squares measures the anisotropy, if the turbulence was isotropic this line would be kx = k⊥ for all wavenumbers. In panel b) and d) the dotted lines show the anisotropy, which would arise only from the expansion.
In the case of run A (no mean field, no expansion), kx and k⊥ are equal for all scales, the thick line is diagonal and no anisotropy can be seen. Run B (expansion without mean field) 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 35
Figure 4.11: Results about the global spectral anisotropy. The isocontours correspond to the magnetic energy spectrum averaged around the kx axis. For details of the runs see Table 4.1 [Figure from Dong et al., 2014]. is more elongated along the radial direction, which is due to the expansion. Run C (mean field but no expansion) is isotropic for large scales (corresponding to small wavenumbers, typically k < 5) but for high wavenumbers the perpendicular direction has most of the energy flux. Finally, for run D (mean field and expansion) the competing effects of the expansion and the mean field cause some spectral anisotropy with respect to the radial direction. The concept of Dong et al. [2014] about the effects of expansion and mean magnetic field is illustrated with Figure 4.12. In the first row (a) it can be seen that the eddies become more and more elongated along the mean magnetic field as the size of the lengthscale decreases. On the other hand, the effect of expansion (b) causes stretching of the eddies along the perpendicular direction with respect to the radial. When the effects of expansion and mean magnetic field are combined and the mean magnetic field points toward the radial direction weak anisotropy can be seen along the perpendicular direction (c). To summarize, the following parameters should be investigated to test the predictions of 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 36
Figure 4.12: schematic illustration of the prediction of Dong et al. [2014] about the competing effects of mean magnetic field and the expansion. In panel (a) the eddies are stretched along the mean magnetic field. Panel (b) shows the effect of the radial direction, which causes stretching along the perpendicular direction. Finally, panel (c) shows the combined effect of the mean magnetic field and the radial direction resulting in weak anisotropy along the perpendicular direction.
Dong et al. [2014]. For the variance anisotropy the amplitudes of the magnetic field fluctuations should be studied separately for each component to reveal the possible reduced power of the radial one. To achieve this, two spacecraft should be used, which makes possible to study the fluctuations along the radial and transverse sampling directions. It is reasonable to restrict the analysis for the periods when the mean magnetic field is out of the plane in which the anisotropy is studied. In this way the Alfv´enicfluctuations can be compared in the x-y plane for the radial and transverse sampling directions and the effect of the mean magnetic field can be ruled out. For the 3-D spectral anisotropy it is necessary to separate the effects of the expansion and the mean magnetic field, which can be achieved with the use of two spacecraft. The 4.4. The Effects of Solar Wind Expansion on the Anisotropy of Turbulence 37
second-order structure functions should be binned as a function of ΘB (angle between the spacecraft separation vector and the mean field) and ΘR (angle between the spacecraft separation vector and the radial direction). If expansion plays a role, the second-order structure functions should display variations depending on ΘR, in contrast if the second- order structure functions are independent of ΘR then it implies that on the investigated scale the expansion does not affect significantly the turbulence. Chapter 5
Spacecraft description
In this thesis the magnetic field data of the Advanced Composition Explorer (ACE), Wind, Cluster and ARTEMIS (Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon’s Interaction with the Sun) satellites are used for the analysis of the anisotropy of the solar wind turbulence. The following sections give an overview of these missions with emphasis on the orbit parameters and magnetometer instruments.
5.1 Advanced Composition Explorer
The Advanced Composition Explorer (ACE) is a NASA mission, which was launched on 25th August 1997 [Stone et al., 1997]. ACE is located at the L1 point (Figure 5.1), which is approximately 1.5 million km from the Earth (measured along the Earth-Sun line). Among the primary goals of the mission are the investigation of the solar wind structures such as magnetic clouds, ICMEs, CIRs and the study of solar wind turbulence. The instrument package consists of six sensors for studying high energetic particles including galactic cosmic rays and solar energetic particles. The spacecraft provides real-time measurements of the solar wind, which is particularly important for space weather research. The Magnetic Field Experiment (MAG) of ACE includes two sensors, which are mounted on booms attached to the solar panels. The instrument operates continuously and it samples the IMF with 15 sec resolution [Smith et al., 1998].
38 5.2. Wind 39
Figure 5.1: Orbit of the ACE spacecraft at the L1 point [nasa.gov]. 5.2 Wind
The Wind spacecraft was launched on 1st November 1994 [Lepping et al., 1995], in the first phase of the missions the spacecraft had a so called lunar swing-by orbit, which made possible to keep the apogee of the spacecraft on the dayside to survey the terrestrial magnetosphere. At later point of the mission, after a series of maneuvers Wind reached the L1 point in May 2004 and it has been operating there since then. The orbit of the spacecraft at the L1 point is similar to the one presented in Figure 5.1. The spacecraft carried a comprehensive instrument package for studying waves, magnetic fields, cosmic rays and suprathermal electrons. The design of the magnetometer is based on the heritage of past missions such as Voyager and Giotto, Mars Surveyor. The magnetic field experiment provides continuous measure- ments of the IMF. The instrument samples the solar wind with up to 22 vector s−1 resolution.
5.3 Cluster
The European Space Agency’s Cluster mission consists of 4 spacecraft flying in a tetrahedron formation [Balogh et al., 2001]. The spacecraft were launched in 2000 and the inclination of the spacecraft is approximately 135◦, the altitude of the perigee and apogee are 16000 km 5.3. Cluster 40
and 120 000 km, respectively. The relative distance of the four spacecraft with respect to each other was modified several times during the mission. The closest distance ever was 4 km, while the largest is approximately 10000 km at the apogee. The mission offered unique opportunity to survey three-dimensional processes such as turbulence, boundary layers and their dynamics. Each spacecraft carried 7 instruments including two tri-axial fluxgate and search-coil mag- netometers. Both pre-flight and in-flight calibrations were carried out the reduce the noise and errors in the measurements. The highest available sampling rate of the fluxgate magne- tometer is 67 s−1 [Balogh et al., 1997]. Due to the drift of the spacecraft orbit there is limited interval in the year when the undis- turbed solar wind can be studied. Such a configuration is shown in Figure 5.2. Despite the fact that the spacecraft are outside the terrestrial bow shock the magnetic field measure- ments can be affected by the foreshock. In the data analysis it was necessary to identify its signatures, which can significantly affect the small scale fluctuations of the magnetic field.
Figure 5.2: Orbit of the Cluster spacecraft when the undisturbed solar wind is surveyed as well [esa.net] 5.4. ARTEMIS 41
5.4 ARTEMIS
THEMIS (Time History of Events and Macroscale Interactions during Substorms) is NASA mission, which was launched on 14th February 2007. The mission consists of 5 spacecraft flying in a formation studying the trigger and large scale evolution of geomagnetic substorms [Angelopoulos, 2009]. The THEMIS spacecraft are equipped with insturments to study mag- netic, electric fields and plasma in the Earth’s magnetosphere. After the two-years nominal mission two of the THEMIS spacecraft were inserted into an Earth-Moon libration orbit after a serious of maneuvers ending in July 2011 [Angelopoulos, 2011]. The new mission called ARTEMIS studies the distant magnetotail and the lunar plasma environment. The orbit of the two spacecraft is illustrated with Figure 5.3. In this configuration the two space- craft spend significant amount of time in the undisturbed solar wind and the measurements are rarely disturbed by the terrestrial magnetotail and the lunar plasma environment. The fluxgate magnetometer of THEMIS is capable of sampling with up to 64 s−1 [Auster et al. 2009]. 5.4. ARTEMIS 42
Figure 5.3: Illustration of the ARTEMIS orbit around the Earth-Moon libration points [nasa.gov]. Chapter 6
Data Analysis
This chapter summarizes the analysis of the solar wind turbulence with the ACE, Wind, Cluster and ARTEMIS magnetic field data. The data analysis had three major goals:
• Test the prediction of Dong et al. [2014] about the radial spectral anisotropy. The solar wind plasma gradually expands while reaching larger heliospheric distances. The stretching of the plasma volume might have an imprint onto the turbulence. The fluctuations in the plasma are continuously interacting with each other and this pro- cess might counteract the effect of expansion due to the fact that the turnover of the eddies might be faster then the expansion itself. Studying this feature is important because this makes it is possible to infer the importance of expansion in the operation of the turbulent cascade and to compare the time scale of the expansion and the eddy turnover.
• Analyze the variance anisotropy and test whether the fluctuations are con- fined in the Y-Z plane. When expansion is included in the MHD equations, a new term appears, which causes transfer of energy from the radial component toward the two perpendicular ones. The anisotropic energy distribution among the three magnetic field components confines the fluctuations in the Y-Z plane and overall it can affect the turbulent cascade as well. To get a better understanding of this process, the magnitude of the fluctuations are compared and the possible differences are quantified.
• Investigate the local 3-D spectral anisotropy for the radial and transverse
43 44
sampling directions to determine the possible differences caused by the expansion. The anisotropic energy distribution among the magnetic field componenets might have an effect not only on the global structure as noted previously, but on the local structure as well. Previously Boldyrev [2006] proposed local 3-D anisotropy in the sense that the scaling of the eddies is different along all the three spatial directions. In contrast, according to Verdini et al. [2015] the observed 3-D anisotropy is primarily caused by the expansion. It might be possible to identify which of these theories can explain the observations.
To achieve the goals above a multi spacecraft approach is required. With a single spacecraft it is only possible to sample the radial component of the solar wind (kx). This can be understood as a 1-D cut of the solar wind plasma along the radial direction. To investigate the
3-D nature of the turbulence ky (perpendicular component with respect to the radial) should be considered as well. It can be studied with a pair of spacecraft which are separated along the transverse direction with respect to the solar wind. When they perform simultaneous measurements of the solar wind turbulence they can effectively sample a 1-D cut along ky. In addition to the different sampling directions, the anisotropy should be investigated on various length scales (spacecraft separation distances) as well since the properties of the anisotropy strongly depend on which length scale is studied. The past and present multi-spacecraft missions were investigated and ACE, Wind, Clus- ter and ARTEMIS were chosen for this study. The ACE and Wind spacecraft provide a large amount of data (over 260 days) when they were separated 110-135 Earth radius (RE) along the transverse direction, which made possible to study the turbulence on the largest achievable scales. The ARTEMIS mission provided substantial data for carrying out a similar analysis on the scale of 1-6 RE. For studying the turbulence on even smaller scales, the Clus- ter mission was a reasonable choice since the typical separation distance is approximately
0.7 RE. 6.1. ACE-Wind and ARTEMIS data selection 45
6.1 ACE-Wind and ARTEMIS data selection
For the selection of the ACE-Wind data the period from May 2004 to September 2015 was used. The Wind spacecraft arrived at the L1 point in May 2004 therefore this was the first time when the two spacecraft were sufficiently close to each other and the constellation was favorable. Prior May 2004 the Wind spacecraft was primarily in the vicinity of Earth and it was radially separated from ACE. The ARTEMIS spacecraft reached the equatorial orbit in the vicinity of Moon in July 2011. The data from then until July 2015 was used here. Both the ACE-Wind and ARTEMIS data was available in the Geocentric Solar Ecliptic coordinate system (GSE) in which X axis points toward the Sun, Z is perpendicular with respect to the plane of Earth’s orbit around the Sun and Y completes the orthogonal system by being perpendicular to both X and Z.
The selection criteria for the separation along the YGSE axis was that the separation vector of the two spacecraft (l = (X1 − X2,Y1 − Y2,Z1 − Z2), where X1,2,Y1,2,Z1,2 denote ◦ the coordinates of the first and second spacecraft) should be within 10 from the YGSE axis. When this requirement was met, the separation along the YGSE axis was always in the range of 110-135 and 1-6 RE for the ACE-Wind and ARTEMIS spacecraft, respectively. The data was inspected by eye and it was ensured that the measurements are not contaminated with ICMEs, CIRs and magnetic clouds, which are accompanied with interplanetary shocks, display large scale rotation of the magnetic field and as a result significantly disturb the fluctuations of the magnetic field. After this further down selection 58 intervals were identified with the ACE-Wind data, which correspond to 263 days of measurements. ACE and Wind have different sampling frequency therefore the 3-second Wind was averaged to match to the 15-second ACE data. The ARTEMIS data was inspected and besides the ICME/CIR events, the effect of the distant terrestrial magnetotail, the lunar wake region and the lunar magnetic anomalies were removed as well. This further down selection resulted in 26 days of data of interest.
For the direct comparison of the X and YGSE separations it would have been necessary to
find intervals when the ACE-Wind spacecraft are approximately 110-135 RE along the XGSE axis, however, along this direction the largest separation was approximately 40 RE. In order to solve this problem, the Taylor hypothesis was used [Taylor, 1938]. A single spacecraft 6.2. Cluster Data Selection 46
data was taken and a time lag was applied to it therefore it can correspond to any arbitrary separation distances along the radial direction and by choosing appropriate time lag it can match to the 110-135 RE separation distances. The accuracy of this assumption will be discussed in Section 6.4.
6.2 Cluster Data Selection
For the Cluster data selection, the entire data set was investigated since the launch of the spacecraft in July 2000 to 2015 September. The relative distance of all the 6 pairs of spacecraft was investigated and the orbit requirements were that the separation vector
◦ between two spacecraft should be within 10 from the YGSE axis and the distance between the two spacecraft should be larger than 0.5 RE. When both of these requirements were met, the average separation along the YGSE axis was 0.7 RE and it was never larger than
1 RE. Further down selection was needed to filter out the intervals when the terrestrial foreshock disturbed the magnetic field. Signatures that can indicate such activity are high- energy ions, Langmuir waves and increased magnetic field fluctuations. Furthermore, the events containing indications of shocks and magnetic clouds were discarded as well. Overall 19 hours of measurements met all of the requirements above.
6.3 Test of the Radial Anisotropy
To investigate the predictions discussed in Section 4.4 the second-order structure functions were calculated and binned according to ΘB (angle between the local mean magnetic field and the separation vector of the two spacecraft) and ΘR (angle between the radial direc- tion and the separation vector of the two spacecraft). For testing the radial anisotropy, it was important to study the second-order structure functions not only along the radial and
◦ transverse spacecraft separation directions but for all angles (ΘR = [0, 90 ]) from the radial.
This makes it possible to get an overall view of the power levels for various ΘR angles and to unambiguously identify any systematic changes of the power levels that might be related to expansion. The second-order structure functions were calculated across all the 6 pairs of Cluster spacecraft, which significantly improved the coverage of the ΘB-ΘR bins. To further improve 6.3. Test of the Radial Anisotropy 47
the coverage and to increase the number of data points in each bin the multi-spacecraft method of Osman and Horbury [2007] was used. This method is based on assumption that the sampling frequency of the solar wind is much larger than the frequency on which it varies, which means that Taylor hypothesis is satisfied [Taylor, 1938]. For multiple spacecraft measurements, this criterion is met when
v ∆t v sw A << 1, (6.1) |d1,2 − vsw∆t|vsw where vsw the solar wind velocity, va denotes the Alfv´enspeed (Equation (2.7)), d1,2 is the separation vector between the two spacecraft, | ... | denotes the magnitude of the vector and ∆t is the time lag. The separation vector r can be given as a function of the applied time lag:
r(∆t) = d1,2 − vsw∆t. (6.2) Equation (6.2) means that the position of one spacecraft is fixed while the other one is moved ’virtually’ to another location due to the time lag (the same time lag is applied to the magnetic field data as well). As a consequence, the separation vector between the two spacecraft will change as well. By varying the time lag additional second-order structure functions can be derived and added to the bins. It is noted that the solar wind is along the radial direction therefore the time lag only increases the separation of the two spacecraft along this direction. Increasing the length of the separation vector will reduce the angle ΘR (since longer separation vectors will be more and more aligned with the radial direction) therefore using very large time lags (e.g. > 30 sec) is not reasonable because the new data
◦ ◦ points will be added to the bins with ΘR < 10 and the overall coverage along ΘR=[0,90 ] will not be improved. For the analysis it was ensured to keep range between 5000-10000 km.
Figure 6.1a shows the average of the second-order structure functions according to ΘB and ΘR for one of the investigated periods. It can be seen that the power level displays variation with the angle from the IMF but there is no variation with angle from the radial.
The anisotropy along ΘB is well-known and it was expected based on previous studies such as Chen et al. [2011]. Figure 6.1b shows the average of all the investigated periods. There were 35 different intervals and during these short intervals (< 1 hour) the structure function power levels varied from case to case, which introduced additional noise in the averaged data. Despite 6.4. Variance Anisotropy 48
the scattering in the averages, it can be seen that the power levels vary with angle from ΘB but there is no significant systematic trend with angle from the radial. This implies that the eddy turnover is quicker than the effect of the expansion on the investigated scale. The expansion and the mean magnetic field have counteracting effects. The expansion causes stretching along the perpendicular direction with respect to the radial while the mean magnetic field causes elongation of the eddies along the mean magnetic field. The fact that no systematic trend was found with respect to ΘR implies that the expansion is slow as compared to the turnover of the eddies.
Figure 6.1: Second-order structure functions measured across all the four Cluster spacecraft for a single 45 min period (panel a) and for all the selected data (panel b).
6.4 Variance Anisotropy
For testing the predictions of Dong et al. [2014] and Verdini et al. [2015] about the variance anisotropy the following analysis was prepared: the selected 263 days of ACE-Wind data was investigated when the sampling was done along the radial and transverse directions (110-135
RE separation). Further down selection was done to identify periods when the local mean ◦ magnetic field was +/− 15 from the ZGSE axis. Overall 48 hours of data (approximately 11000 data points) was considered for the analysis. In the next step the fluctuations of
2 2 each component was calculated as δBi = (Bi,1 − Bi,2) where Bi1,2 are the simultaneously measured magnetic field components with two spacecraft. 6.4. Variance Anisotropy 49
Turner et al. [2011] studied the power spectra of the three magnetic field components in the solar wind based on single spacecraft measurements and found that the radial component has reduced power as compared to the two perpendicular ones. These authors pointed out that this ordering is simply due to a geometrical effect given by ∇ · B = 0. The equivalent of ∇ · B = 0 in Fourier space is k · δB = 0. When a single spacecraft is used, the sampling is done along kx therefore the magnitude of δBx is smaller as compared to δBy and δBz. In order to overcome the ∇ · B = 0 constraint a pair of spacecraft should be used because the
fluctuations of the different components then can be measured independently along kx and ky, respectively.
Figure 6.2: Radial and transverse sampling configurations for studying the variance anisotropy.
The applied sampling configuration in the GSE system is shown in Figure 6.2. S/C1 and
S/C2 denote the position of the two spacecraft and the direction of the local mean magnetic
field is marked with B0. The primary benefit of this configuration is that the local mean magnetic field points outward from the X − YGSE plane in both cases and the radial and 2 transverse spacecraft separations makes it possible to compare the magnitudes of δBx(kx) 2 and δBy (ky), which would not be possible with a single spacecraft.
As mentioned earlier, due to orbit restrictions the 110-130 RE separation distance was achievable only along the transverse direction and along the radial 30-40 RE was the largest distance between ACE and Wind. The 110-130 RE radial separation was reconstructed with the Taylor hypothesis [Taylor, 1938]. A single spacecraft data was taken, a time shift was 6.5. Local 3-D spectral anisotropy 50
applied to it and the second-order structure functions were derived.
2 2 2 This assumption was tested the following way: δBx, δBy and δBz were derived separately with the actual double spacecraft data along the radial direction (30-40 RE separation dis- tance) and using exactly the same data, these three variables were derived with the ACE and Wind data separately using a time lag, which was calculated based on the solar wind velocity and distance between ACE and Wind. The results are summarized in Table 6.1. It can be seen that the difference between the mean values is less than 10% thus the Taylor hypothesis is considered to be a good approximation.
2 2 Table 6.2 summarizes the results about δBy and δBx when the local mean magnetic field was approximately along the ZGSE axis. The pair of spacecraft made it possible to study the fluctuations along different sampling directions thus their correct comparison was possible.
2 When the sampling was done along XGSE, the mean value of δBx (component along the 2 sampling direction) is 5.56 +/− 0.09, while for the transverse case δBy was 7.00 +/− 0.14. Considering the error bars, the difference by the factor of 1.25 is statistically significant and can be a signature of expansion, which reduces the power of the radial component. The
2 value of δBy in the case of the radial separation (component perpendicular both to the mean 2 field and the sampling direction) is 10.60 +/− 0.16, while δBx for the transverse case is 9.43 +/- 0.17. In this case the difference is smaller, however, the results are also consistent with
2 2 expansion since δBx < δBy .
6.5 Local 3-D spectral anisotropy
The predictions of Verdini et al. [2015] about the local 3-D spectral anisotropy were in- vestigated with the ACE-Wind and ARTEMIS data. The second-order structure functions were derived and binned when the two spacecraft were along the YGSE axis (110-130 and 1-6
RE separation distances, respectively) and using the single spacecraft technique described in Section 6.1 the radial separation was reconstructed with a time lag. The binning was done in the local reference frame described with Figure 6.3. In the Figure
S/C1 and S/C2 denote the position of the pair of spacecraft, which are separated by the 6.5. Local 3-D spectral anisotropy 51
Radial separation: single ACE and Wind spacecraft data (30-40 RE) Mean Median
2 δBx 2.14 + / − 0.05 0.34 2 δBy 2.75 + / − 0.07 0.41 2 δBz 3.12 + / − 0.08 0.52
Radial separation: double spacecraft (30-40 RE) 2 δBx 2.17 + / − 0.03 0.4 2 δBy 2.51 + / − 0.05 0.37 2 δBz 3.10 + / − 0.04 0.51 2 Table 6.1: Comparison of the derived δBx,y,z components with the actual double spacecraft data along the radial direction and as a reference the single spacecraft time shifted data. The results are based on 22 days of data. The error bars correspond to the standard deviation. Radial separation Transverse separation Mean Median Mean Median
2 δBx 5.56 + / − 0.09 1.35 9.43 + / − 0.17 2.32 2 δBy 10.60 + / − 0.16 2.76 7.00 + / − 0.14 1.75 2 2 Table 6.2: Mean and median values of δBy and δBx for the radial and transverse sampling directions
(110-130 RE), respectively. The error bars correspond to the standard deviation. vector r in spatial domain. The local coordinate system including the local mean field (B0) and the fluctuations in the field (δB = B1 − B2) were re-defined for every single pair of magnetic field measurement. The second-order structure functions were binned according to
ΘB (angle between the local mean field and r) and ΘδB⊥ (angle between the projection of r onto the perpendicular plane with respect to B0 and B0 × [δB × B0]). The angles larger than 90◦ were reflected below 90◦. The reflection has been found to be a good approximation [Podesta et al., 2009]. Figure 6.4a and b show the averaged second-order structure function bins for the radial and transverse sampling directions based on the ACE-Wind (panels a and b) and ARTEMIS data (panels c and d). In Figure 6.4a and c the patterns show that the power of the structure functions increases toward larger angles of ΘB and ΘδB⊥. This tendency and the obtained pattern are in good agreement with the previous study of Chen et al. [2012]. Three bins are highlighted with colored boxes and their average and error are given: the green and blue 6.5. Local 3-D spectral anisotropy 52
Figure 6.3: Description of the used coordinate system to bin the second-order structure functions with the ACE-Wind and ARTEMIS data. bins are in the plane perpendicular with respect to the local mean magnetic field and the red one is along the mean field. For the calculation of the error bars a Monte-Carlo method was used: from each bin n number of data points were chosen randomly where n corresponds to the total number of data points in that bin, however, repetition was allowed thus the same data point could be chosen multiple times. The selection process was repeated 1000 times and each case the sample average was taken. In this way a distribution of the averages was obtained and the standard deviation of this distribution corresponds to the error bars.
The power levels of the structure functions for the YGSE separation are shown in Figure 6.4b and d. Clear differences can be seen as compared to Figure 6.4a and c. For the
YGSE separation the power levels are generally more homogenous and show significantly less variability.
◦ Figure 6.5a shows the average of the 9 bins in Figure 6.4a and 6.4b when ΘB=90 (e.g. top row of the binned data). It can be seen that the second-order structure functions are approximately in the range of 10 - 20 for the radial sampling direction. For the transverse sampling the variation is significantly reduced and the power levels are in the range of 8-13. It is noted that if the structure functions had the same value along the angles between 0 and
◦ 90 ΘδB⊥ , the turbulence would be isotropic in the plane perpendicular with respect to the local mean magnetic field. 6.5. Local 3-D spectral anisotropy 53
Figure 6.4: ACE-Wind and ARTEMIS results about the average of second-order structure function bins for the radial (a and c) and transverse sampling directions (b and d). Clear differences can be seen between the patterns: the power levels are significantly reduced for the transverse sampling direction as compared to the radial one.
The radial sampling direction was reconstructed with the Taylor hypothesis using a single spacecraft data. To investigate the accuracy of this assumption the following analysis was prepared: the second-order structure functions were derived with the actual ACE-Wind double spacecraft data along the radial separation (30-40 RE). Then using exactly the same data the structure functions were derived with a single spacecraft. To calculate the time lag (e.g. how much time it takes until the solar wind propagates from the first to the second spacecraft) was calculated based on the average separation distance and average solar wind speed in each period. Figure 6.5b shows the comparison of the ACE-Wind single and double spacecraft results along the radial sampling direction (30-40 RE separation distances). It can be seen that the 6.5. Local 3-D spectral anisotropy 54
results are converging to each other thus the figure demonstrates that the Taylor hypothesis is a reasonable assumption to reconstruct a given separation distance along the radial direction. Additional benefit of the Taylor hypothesis is that the transverse and radial sampling results in Figure 6.4 are based on exactly the same data, which would not be possible otherwise and as a consequence the effect of the different data sets can be ruled out.
Figure 6.5: Panel a shows a comparison of the top rows of the binned data in Figure 6.4a and 6.4b, while in panel b the direct comparison of the double spacecraft data and the single spacecraft can be seen. Chapter 7
Discussion and Conclusion
In this thesis a multi-spacecraft study of the solar wind turbulence was presented. The goal of the thesis was to test the predictions of recent theoretical papers of Verdini et al. [2015] and Dong et al. [2014] about the possible effects of expansion on the radial, variance and local 3-D anisotropies. The turbulence was studied with the ACE, Wind, Cluster and ARTEMIS spacecraft magnetic field measurements. Dong et al. [2014] performed an MHD simulation using an expanding box model. The goal of their study was to analyze the turbulence in a stretching plasma volume as the solar wind propagates and reaches larger heliospheric distances. They suggested that besides the well-known anisotropy with respect to the mean magnetic field the radial direction also affects the turbulent cascade and causes stretching of the eddies along perpendicular direction with respect to radial. When the mean magnetic field is along the radial direction, the competing effect of expansion and the mean magnetic field results in weak anisotropy along the perpendicular direction. To study these features of the turbulence and to separate the effect of the expansion and the mean magnetic field a multi-spacecraft method was used, which exploited all the 6 pairs of Cluster spacecraft data (Section 6.3). The second-order structure functions were binned according to ΘB and ΘR (Figure 6.1). The averaged bins showed that the power levels vary with angles from the mean field but there is no significant variation with respect to ΘR.
These findings imply that on the investigated scales (0.5-1 RE) the turnover of the eddies is faster than the effect of the expansion. As a consequence, the imprint of the expansion cannot be detected and the turbulence is primarily driven by the mean magnetic field.
55 56
Dong et al. [2014] predicted that the expansion causes selective decay of the magnetic field components and suggested that the Y, Z components have more power than the X. Verdini et al. [2015] proposed that the fluctuations are confined in the perpendicular plane with respect to the radial. To test these predictions, the fluctuations of each magnetic field component was studied separately for the radial and transverse sampling directions in the GSE coordinate system (Section 6.4). The single spacecraft measurements are affected by the ∇ · B = 0 constrain thus using a pair of spacecraft was necessary. The magnitude of the fluctuations along the radial and transverse sampling directions
2 (kx and ky, respectively) were compared (Table 6.2). It was found that δBx(kx) is smaller 2 by a factor of 1.25 than δBy (ky). Considering the error bars this difference is statistically significant. These findings are consistent with the concept of expansion and indicate that the radial component has reduced energy as compared to the two perpendicular directions and the fluctuations are confined in the Y-Z plane. This finding also implies that besides
2 2 2 the previously known δBx(kx) < δBy (kx) constrain, there is an additional one (δBx(kx) < 2 δBy (ky)), which is not given by ∇ · B = 0 and is not there in the standard non-expanding simulations. The local 3-D spectral anisotropy was studied for the radial and transverse sampling di- rections to test the prediction of whether the large scale anisotropy is reduced when mea- sured along the transverse direction. The ACE-Wind and ARTEMIS constellations made it possible to study the turbulence on 110-130 and 1-6 RE separation along the transverse direction and a single spacecraft technique was used to reconstruct radial sampling case. The second-order structure functions were binned in a local reference frame and the obtained dis- tributions were compared. The structure function bins showed significantly less variability along the transverse direction. These findings are consistent with the concept of expansion and indicate that the observed anisotropy for the radial sampling is primarily caused by the expansion of the solar wind. It is also noted that some anisotropy was found along the transverse direction, which implies that other mechanisms contribute and the observations can be consistent with the concept of Boldyrev [2006] as well. Previously several studies have discussed the effects of the local mean magnetic field on the turbulent cascade. In this thesis a new approach was used to identify additional mechanisms 57
affecting the turbulence at large scales, which is the expansion of the solar wind. The results of this thesis show that the expansion of the solar wind can affect the turbulence and is consistent with anisotropic energy distribution among the three magnetic field components. In the case of the local 3-D anisotropy, expansion has an effect on second-order structure functions but at smaller scales this effect is not significant. For future MHD simulations concerning the large scale evolution of the turbulence, the effect of the expansion should be added to the code in order to make the simulations more realistic and to be able to give better predictions based on them. References
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