Turbulence at MHD and sub-ion scales
in the magnetosheath of Saturn
a comparative study between quasi-perpendicular and
quasi-parallel bow shocks using in-situ Cassini data
Thesis for the Degree of Bachelor of Science in Physics
by
Khaled Al Moulla
Supervisor: Subject reader: Examiner: Lina Hadid Emiliya Yordanova Matthias Weiszflog
Uppsala University
Department of Physics and Astronomy Bachelor Programme in Physics
in collaboration with
Swedish Institute of Space Physics
Abstract
The purpose of this project is to investigate the spectral properties of turbulence in the magnetosheath of Saturn, using in-situ magnetic field measurements from the Cassini spacecraft. According to models of incompressible, turbulent fluids, the energy spectrum in the inertial range scales as the frequency to the power of −5/3, which has been observed in the near-Earth Solar wind but not in the Terrestrial magnetosheath unless close to the magnetopause.
120 time intervals for when Cassini is inside the magnetosheath are identified — 40 in each category of behind quasi-perpendicular bow shocks, behind quasi-parallel bow shocks, and inside the middle of the magnetosheath. The power spectral density is thereafter calculated for each interval, with logarithmic regressions performed at the MHD and sub-ion scales separated by the ion gyrofrequency. The results seem to indicate similar behaviour as in the magnetosheath of Earth, without significant difference between quasi-perpendicular and quasi-parallel cases except somewhat steeper exponents at the MHD scale for the former.
These observations confirm the role of the bow shock in destroying the fully developed turbulence of the Solar wind, thus explaining the absence of the inertial range.
Keywords: Saturn, magnetosheath, bow shock, turbulence, Cassini
Sammanfattning
Syftet med detta projekt är att undersöka de spektrala egenskaperna hos turbu-lens i Saturnus magnetoskikt, med in-situ-mätningar av magnetfältet från Cassini-rymdsonden. Enligt modeller av inkompressibla, turbulenta fluider, är energispek-trumet i det intertiala omfånget proportionellt mot frekvensen upphöjd i −5/3, vilket har observerats i den jordnära Solvinden men inte i det jordiska magnetoskik-tet förutom nära magnetopausen.
120 tidsintervall för när Cassini befinner sig inuti magnetoskiktet identifieras — 40 styck i kategorierna bakom kvasi-vinkelräta bogchockar, bakom kvasi-parallella bogchockar, och inuti mellersta delen av magnetoskiktet. Effektspektraltätheten beräknas därefter för varje intervall, med logaritmiska regressioner på MHD- och subjon-skalorna som separeras av jongyrofrekvensen. Resultaten verkar tyda på lik-nande beteende som i Jordens magnetoskikt, utan märkvärdig skillnad mellan kvasi-vinkelräta och kvasi-parallella fall förutom något brantare exponenter på MHD-skalan för de förnämnda.
Dessa observationer bekräftar bogchokens roll i förstörandet av den fullt utveck-lade turbulensen i Solvinden, därmed förklarande avsaknaden av det inertiala om-fånget.
Acknowledgements
Many have been involved in enabling me to conduct this project. Therefore I would like to express my gratitude to:
Lina Hadid, my supervisor, for guiding me with her expertise in the subject and entrusting me with the assigned tasks; thank you, for without you this project would’ve remained wishful thinking. I wish you all the best with your future work. Emiliya Yordanova, my subject reader, for looking after my work and providing me with templates of code, which clarified a large portion of the methodology.
Matthias Weiszflog, my examiner and the director of my programme, for main-taining a high quality education, which respects the students’ opinions.
Rabab Elkarib, my student counsellor, for advising me throughout my bachelor studies and never limiting my options.
The staff at IRF Uppsala, for giving me a tour of the institute and offering lots of lucrative projects.
The staff at CDPP, for granting me access to their software AMDA, with which most of my data mining was conveniently executed.
Outside of the academic world, I’ve also received support from the people dear-est to me:
Jomana Makié, Sleiman Al Moulla and Mona Al Moulla, my mother, father and sister, for all the comfort and love you endow me.
My relatives, for cheering on me from afar.
Everything flows, an old Greek said. Nothing’s secure. Gold’s only lead when you stop to think. On your way up, show consideration to the ones you meet on their way down. The Latin root of condescension means we all sink.
Contents
I
Subject overview
1
1 Introduction 1 1.1 Personal involvement . . . 1 1.2 Purpose . . . 1 2 Theory 2 2.1 Turbulence in hydrodynamics . . . 2 2.1.1 Navier-Stokes equation . . . 2 2.1.2 Kolmogorov’s 5/3 law . . . 3 2.2 Turbulence in magnetohydrodynamics . . . 42.2.1 Definition and characteristics of plasma . . . 4
2.2.2 MHD scale . . . 5
2.2.3 Sub-ion scale . . . 5
2.3 The Solar wind and the planetary magnetosheath . . . 6
2.3.1 Formation of the Solar wind, bow shock and magnetosheath . 6 2.3.2 Spectral properties in the near-Earth magnetic field . . . 8
3 Background 10 3.1 Saturn . . . 10
3.2 The Cassini spacecraft . . . 11
II
Project work
12
4 Methods 12 4.1 Conventions . . . 12 4.2 Modelling . . . 13 4.3 Selecting intervals . . . 14 4.4 Analysis . . . 16 5 Results 17 5.1 Example of PSD graphs . . . 17 5.2 PSD dependence on localisation . . . 195.3 PSD dependence on bow shock geometry . . . 21
6 Discussion 23 6.1 Comparison . . . 23
6.2 Source of errors . . . 23
6.3 Applications . . . 24
7 Conclusion 24 7.1 Project achievement and future suggestions . . . 24
III
Bibliography
25
References 25 Literature . . . 25 Figures . . . 27 Tables . . . 27 Appendix 28 Table of intervals . . . 28 MATLAB code . . . 31Part I
Subject overview
1
Introduction
1.1
Personal involvement
As I’m writing this text, at the age of 21, I recall the times when my parents owned a small kiosk; it was during my final years of elementary school and on the summer breaks I would assist them in stacking the shelves with products stored in the basement. To that basement I would, when there weren’t many customers coming and going, sometimes bring with me one of the popular science magazines we sold — I never kept them, but I’d read the interesting bits. It was probably not at the same time as the milestone occurred — rather a few years later, in a compiled overview of the mission — but I remember once reading about the Cassini-Huygens mission, and in particular the story about the Huygens probe.
In late 2004, the Huygens probe departed from the Cassini orbiter, eventually landing on Saturn’s moon Titan on January 14, 2005[1]. It went on a two hour voyage through the moon’s atmosphere, then it recorded and sent data for an additional 90 minutes after touchdown. I found it remarkable that something would be sent through space for 7 seven years, only to be used for three and a half hours. It was thought of as a sacrifice, and I couldn’t help but to shed a tear thinking about this probe I had inevitably personified in my mind. In the midst of my reading — alone in the basement, yet my concious elevated — I held a minute of silence.
Today I find great joy in how this story reconnects with my present self. This opportunity of working with data from the Cassini-Huygens mission — which awed and inspired me at a young age — for my bachelor thesis, is more grand than anything my inner child could’ve dreamt of, and I’m grateful.
1.2
Purpose
The purpose of this project is to investigate the spectral properties of turbulence in the magnetosheath of Saturn, using in-situ magnetic field measurements from the Cassini spacecraft.
The power spectra are analysed at two different frequency ranges — MHD and sub-ion scales — to investigate whether the Kronian magnetosheath exhibits similar behaviour as the Terrestrial one, compared to the patterns observed in the near-Earth Solar wind. Suitable segments of Cassini data are selected and studied using online services and self-written code.
This type of investigation aims to map the magnetohydrodynamics of the inter-planetary medium, in order to understand processes of turbulence as well structural properties of the Solar system in an improved way. The goal of the project, on a personal level, is to gain insight in the methodology of space physics research and
2
Theory
2.1
Turbulence in hydrodynamics
2.1.1 Navier-Stokes equationTurbulence is often encountered in nature, and rarely fully comprehended. It is usually described as the lack of structure and order. However, even in the most chaotic systems can be found patterns and formulated relations used to describe the behaviour of turbulence.
In fluid mechanics, motion under viscous influence is described by the Navier-Stokes equation[2] ∂v ∂t + (v · ∇)v = − 1 ρ∇p + ν∇ 2v (1)
where v, p, ρ and ν are the velocity, pressure, density and kinematic viscosity of a fluid, respectively. In Eq. 1, two relevant forces are represented: the inertial force fin and the viscous force fν, both expressed in per unit volume,
fin = ρ ∂v
∂t fν = ρν∇2v
(2)
For a system of length scale l, the ratio of these two forces is given by kfink kfνk ∝ v 2/l νv/l2 = vl ν (3)
where v = kvk is the speed. This ratio is defined as the Reynolds number Re ≡ vl
ν (4)
which can be regarded as a measurement of turbulence. A flow is said to be laminar for small values of Re, i.e. when the viscous force dominates over the inertial force and causes a damping; the flow then transitions to be turbulent for very large values of Re, i.e. when the viscous force is insufficient for damping the inertial force. See Fig. 1 for an illustration on how a flow transits from laminar to turbulent.
Figure 1. Illustration by Osborne Reynolds on the development of turbulence in an initially laminar pipe flow.
2.1.2 Kolmogorov’s 5/3 law
In hydrodynamics (HD), the mechanism that converts turbulent, kinetic motion into dissipated, thermal energy is a cascade of vortices. In turbulent flows, large vortices break down into smaller and smaller ones, until they reach the minuscule scales at which the viscous forces dominate and dissipate them into heat. The different length scales of the vortices can be thought of as being transformed into a spectrum of different wave numbers, similar to a Fourier transform for time-dependent quantities (see Fig. 2). An energy spectrum would then be a representation of how energy is distributed over wave numbers.
In his famous paper from 1941 (translated to English in 1991)[3], mathematician Andrey Kolmogorov derived the third order velocity fluctuations, for a turbulent fluid in the inertial range of an energy cascade, to be
h(δv)3i = −4
5l (5)
where is the energy cascade rate per unit mass, l is the length scale in the inertial range and δv is the velocity increment defined as
δv = v(x + l) − v(x) (6)
From this, one can show that the energy spectrum should be proportional to the wave number to the power of −5/3, known as Kolmogorov’s 5/3 law. His calculations presumed the turbulent fluid to be fully developed, incompressible and having no external supplies of energy injection during the cascade. Under these presumptions, the energy spectrum per unit mass E(k) can be assumed to only depend on the energy cascade rate per unit mass and the wave number k. The result regarding the −5/3 exponent can be recreated using a simple, illustrative method. Starting off with the aforementioned variables to some unknown powers {a, b},
E(k) = Cakb (7)
where C is a dimensionless constant; through dimensional analysis of Eq. 7 E(k) =
h energy mass · wave number
i , = h energy mass · time i , k = 1 length (8) the following system of equations can be obtained and solved
m3 s2 = m2 s3 a 1 m b ⇒ ( 3 = 2a − b 2 = 3a ⇒ ( a = 2/3 b = −5/3 (9)
This results in Eq. 7 being rewritten as
Figure 2. Schematic image of the energy cascade of vortices in real-space (left) and Fourier-space (right).
2.2
Turbulence in magnetohydrodynamics
2.2.1 Definition and characteristics of plasmaPlasma — commonly known as the fourth state of matter — is highly ionized gas in which the particles can be treated as a single, conducting fluid. In Francis Chen’s textbook on the subject[4], he defines a plasma as
“... a quasi-neutral gas of charged and neutral particles which exhibits collective behaviour.”
by which quasi-neutral refers to the charge neutrality that occurs when there is an almost equal number density n of ions (protons) and electrons,
n ≈ ni ≈ ne (11)
and collective behaviour refers to how the particles are able to affect their sur-roundings. In fluids, the effect is propagated by mechanical waves through frequent collision. In plasmas, however, the movement of charged particles generate electric and magnetic fields, E and B, which exert Lorentz forces
mdv
dt = q(E + v × B) (12)
where m, q and v are the mass, charge and velocity of a particle, respectively. The study of plasmas can be performed at macroscopic or microscopic time and length scales. The temporal scale can be defined according to the gyrofrequencies of the particles; when a charged particle is placed in a magnetic field, it will gyrate perpendicular to the field direction. In the absence of an electric field, with a magnetic field in the ˆz-direction, B = Bˆz, the equations of motion for a single particle become m ˙vx = +qBvy m ˙vy = −qBvx m ˙vz = 0 ⇒ ¨ vx = +qB m ˙vy = − qB m !2 vx ¨ vy = − qB m ˙vx = − qB m !2 vy (13)
for which an angular gyrofrequency ωc and a gyrofrequency fc can be defined ωc ≡ qB m ⇒ fc= ωc 2π = qB 2πm (14) 2.2.2 MHD scale
Investigating turbulence in magnetized plasmas is more complex than in neutral fluids due to the dynamics of charged particles and long-range interactions via elec-tromagnetic fields. In the study of varying magnetic fields, known as magnetohydro-dynamics (MHD), turbulence properties are no longer solely dependant on velocity fluctuations. However, for many scenarios, MHD turbulence can be modelled as HD turbulence at sufficiently large scales. First, one can define a new quantity[5] — called the Elsässer variables — coupling the velocity field to the magnetic field,
z± ∝ v ± B (15)
Using corresponding sets of continuity and force equations, the following expression for third order increments can be obtained
h(δz±· δz±)δz∓i = −4 3
±
l (16)
which simplifies to Eq. 5 for B = 0. With this, one can therefore expect the spectral properties of magnetic turbulence to behave similarly as in incompressible fluids. 2.2.3 Sub-ion scale
The previous section discusses fluctuations at the MHD scale in the inertial range, in which the analogue vortices have not become small enough to vanish. However, for dissipative ranges, at the time scale of elementary particles’ motion — the so-called sub-ion scale — very little is known about the effective processes; they are still debated, and whether the dissipation takes place at the scale of ions or the even smaller scale of electrons is still an open question.
2.3
The Solar wind and the planetary magnetosheath
2.3.1 Formation of the Solar wind, bow shock and magnetosheath The Solar wind is a supersonic, incompressible plasma, emitted continuously from the Sun; it is expelled radially, but forms an Archimedean spiral as it travels outward due to the rotation of the Sun[6]. From what has been observed, there is a fast and a slow Solar wind — with respective, average velocities of 650 km/s and 350 km/s at 1 AU. The fast wind is thought to originate from magnetic cavities in the Solar corona, whereas the slow wind is much more complex and has been observed to originate from the Sun’s equatorial zone.Figure 3. Schematic illustration of the components of Earth’s magnetosphere with sur-roundings. Inside the magnetosheath are visible the Cluster (top) and Themis (bottom) spacecrafts.
The Solar wind propagates into interplanetary space and interacts with the plan-etary magnetospheres (see Fig. 3). Because it is supersonic, a bow shock is formed around the planet, behind which the Solar wind is slowed down, compressed and heated. A shock is a discontinuity in a physical field, through which there is a mass flux and increase of entropy[7], i.e. conversion of kinetic energy to thermal energy; in fluid dynamics, shocks occur whenever a wave propagates faster than the sound speed of its medium. The bow shock formed by the Solar wind when encounter-ing a planetary magnetosphere has different properties dependencounter-ing on its geometry, in particular the angle of incidence. One can calculate the angle θBn between the interplanetary magnetic field B and the normal n to the bow shock surface — if θBn> 45◦, the shock is referred to as a quasi-perpendicular bow shock; if θBn< 45◦, the shock is called quasi-parallel (see Fig. 4).
Figure 4. Schematic image on how the types of bow shocks — quasi-perpendicular and quasi-parallel — form depending on the angle between the interplanetary magnetic field (IMF) and the bow shock surface normal n.
A magnetosphere is the region of space surrounding a planet, where the domi-nant magnetic field is the dipole field of the planet rather than the magnetic field of interplanetary space[8]. The outermost boundary of the magnetosphere is called the magnetopause, which is where the magnetic pressures of the planet and the Solar wind balance out. The magnetosheath of a planet is defined as the turbulent region separating the magnetosphere from the Solar wind. In this intermediate region, the magnetic field, although generally weaker than the inner dipole field, experiences more drastic fluctuations. One of the reasons that make the magnetosheath impor-tant in magnetospheric physics, is its role as interface to Solar wind-magnetosphere interactions, affecting the physical processes occurring within the magnetopause. Therefore, studying properties of turbulence in the magnetosheath should lead to a better understanding of the dynamical coupling between the Solar wind and the magnetosphere.
2.3.2 Spectral properties in the near-Earth magnetic field
A correspondence of the energy spectrum, discussed for the length scales of vortices, is the power spectral density (PSD) of a time-varying signal, which represents the power distribution P (f) as a function of frequency f[9]. The PSD is defined as the Fourier transform of the autocorrelation function γ(τ) of a given signal x(t),
γ(τ ) = hX(t)X(t + τ )i (17) P (f ) =
Z +∞ −∞
γ(τ )e−i2πf τ dτ (18) where X(t) is the stochastic variable from which x(t) is measured, and τ is some time segment. As the signal is measured over long periods of time, the expected value of the squared, Fourier-transformed signal approaches the PSD,
ˆ x(f ) = Z +∞ −∞ x(t)e−i2πf t dt (19) P (f ) = lim t→+∞h|ˆx(f )| 2 i (20)
The PSD does not necessarily have the unit of power (Watt) per frequency; instead the quantity may have an arbitrary unit obtained when squaring the signal. In the case of the signal being a time-varying magnetic field, B(t) = kB(t)k, the PSD truly is proportional to the energy per frequency, since the energy density V of a magnetic field is proportional to the square of the magnetic field magnitude[10],
V = B · B 2µ ∝ B
2 (21)
where µ is the permeability of the medium. Thus, for magnetic fields, a PSD is equivalent to an energy spectrum.
In the near-Earth Solar wind, the PSD has been measured using magnetic field data from the Mariner 1, Mariner 2 and OGO 5 spacecrafts, summarized in a study from 1972[11], and more recently using Wind, Ulysses and Cluster data[12][13], from which three main power-law regions could be identified as demonstrated in Fig. 5. The PSD scaled as ∼ f−1 for f ∈ [10−6, 10−4] Hz, ∼ f−5/3 for f ∈ [10−4, 10−1] Hz, and ∼ fα where α ∈ [−4.5, −2.3] for f ∈ [1, 10] Hz.
The f−1-range is known as the energy containing scales. Its origin is still a matter of debate; there are some indications it might originate from the superposition of uncorrelated magnetic field fluctuations. The f−5/3-range is known as the inertial range, having a power-law alike the one predicted by Kolmogorov for incompressible fluids. It is thought to originate from non-linear interactions of counter-propagating, incompressible Alfvén waves. This inertial range is followed by a steeper power law at the sub-ion scales, forming the dissipation scales. The reasons for the steepening and different dissipation processes of this range are still highly debated.
Figure 5. Typical power spectrum of the near-Earth Solar wind, exhibiting three ranges: the energy containing, the inertial and the dissipative.
Similar type of measurements have been performed in the Terrestrial magne-tosheath; in a 2017 study[14], the PSD was examined using data from the Cluster spacecraft (see Fig. 6). The study showed the absence of the Kolmogorov spectrum behind the bow shock, with a direct transition from the f−1-scale to the sub-ion scale. The inertial range was shown to be formed away from the shock, and closer to the magnetopause.
Figure 6. The change in exponent (slope) as the Cluster spacecraft travels from the Solar wind into the magnetosheath and magnetopause of Earth; the −5/3 slope is present in the Solar wind, but increases to −1 near the bow shock and decreases back to −5/3 near the magnetopause.
3
Background
3.1
Saturn
Saturn, easily recognized by its giant rings (see Fig. 7), is the sixth planet from the Sun and has a volume ca. 764 times greater than that of Earth[15]. Its composition is believed to be a solid core of rock or ice, enveloped by an atmosphere of hydrogen and helium, in which layers of metallic, conducting hydrogen give rise to a magnetic dipole field[16]. Further, it has also been proposed that a process of helium depletion from the upper parts of the atmosphere may adjust the alignment of the magnetic dipole field, causing it to be axisymmetric with the rotational axis[17].
Figure 7. Saturn, photographed by Cassini.
Unlike Earth, Saturn’s environment has been much less studied. Prior to the arrival of the Cassini-Huygens mission in 2004, there was no dedicated mission to orbit Saturn continuously. In fact, Saturn’s magnetosphere was visited only three times with rapid flybys[18]: Pioneer 2 in 1979, Voyager 1 in 1980 and Voyager 2 in 1981.
The interest in exploring plasma turbulence near Saturn lies in its plasma condi-tions being different than around Earth. In Tab. 1, approximate values of different plasma parameters are listed for the Solar wind and the magnetosheaths of both Earth and Saturn. Saturn orbits the Sun at an average distance of 10 AU, and experiences a cooler and less dense Solar wind than near Earth.
Table 1. Average values of heliocentric distance r, particle density n, magnetic field strength B and ion temperature Ti in the Solar wind and magnetosheaths of Earth and
Saturn.
Region: r [AU]: n [cm−3]: B [nT]: Ti [eV]:
Near-Earth Solar wind 1 4 5 50
Terrestrial magnetosheath 1 12 20 250 Near-Saturn Solar wind 10 0.005 0.5 5 Kronian magnetosheath 10 0.5 2 300
3.2
The Cassini spacecraft
The Cassini-Huygens mission — consisting of the Cassini spacecraft and the Huygens probe — was launched on October 15, 1997, and arrived to Saturn in 2004[19]. It lasted for almost 20 years, ending on September 15, 2017, when Cassini performed its Grand Finale descent into Saturn’s atmosphere. Throughout the mission, several scientific milestones were accomplished; among them, Cassini/Huygens performed the most extensive exploration of Saturn’s atmosphere, magnetosphere, rings and moons to date[20].
The spacecraft itself was a marvel of engineering (see Fig. 8); it measured 6.7 m in length and 4 m in width, and including the probe and propellants, it weighed 5,712 kg at launch[21]. A range of in-situ and remote sensing instruments — spectrometers, magnetometers, cameras and radars — allowed Cassini to sample its surroundings and send the data back to Earth.
For the relevance of this project, only the in-situ magnetic field data will be utilized, which was sampled by the Fluxgate Magnetometer (FGM)[22]. The FGM consisted of three fluxgate sensors, orthogonally placed on a block of ceramic glass, which in turn was mounted halfway along an 11 m boom stick pointing away from the main body in order to minimize magnetic interference from other instruments. To detect an external magnetic field, coils in each sensor would induce voltages in the presence of such. Further, the FGM could measure magnetic field strength in four dynamic ranges — from 40 to 44,000 nT — with the resolution of the smallest one being 4.9 pT.
Figure 8. Schematic image of the Cassini spacecraft, with the FGM marked on the boom stick.
Part II
Project work
4
Methods
4.1
Conventions
In the models and data handling of this project, a preferred coordinate system when dealing with other planets is used. The Kronocentric Solar Magnetospheric (KSM) coordinate system (see Fig. 9) is right-hand Cartesian and has its origin at the centre of Saturn; the positive ˆx-axis always points toward the Sun, the ˆxˆz-plane is oriented such that the magnetic dipole axis of Saturn is located in it, and the ˆy-axis completes the right-hand system[23].
Figure 9. The KSM coordinate system applied in models of Saturn. The magnetic dipole axis, which coincides with the rotational axis, is labelled M.
Positions in the KSM coordinate system are expressed in units of Saturn radii, 1 RS = 60, 268 km[24]. Distances from the Sun and interplanetary distances are given in astronomical units, 1 AU = 149, 597, 870 km[25], equal to the length of the semi-major axis of Earth’s orbit around the Sun. Magnetic field strengths, or magnitudes of magnetic flux density, B = kBk, in the context of the Solar wind and Saturn’s magnetosheath, are appropriately expressed in units of nano-Teslas, 1nT = 10−9 T.
4.2
Modelling
The size of the magnetosheath for Saturn can be estimated using semi-empirical models for its boundaries. In a recent model[26] based on Cassini data, the magne-topause could be modelled by balancing the dynamic pressure pD of the Solar wind with the thermal pressure of Saturn, resulting in the following radius vector
R = R0 2 1 + cos θ K , with ( R0 = c1p−cD2 K = c3+ c4pD (22) where the radius R0 toward the Sun and exponent K were calculated by fitting the coefficients {c1, c2, c3, c4}and dynamic pressure pD to the data from Cassini,
c1 = 10.30 , c2 = 0.20 , c3 = 0.73 , c4 = 0.40 , pD = 3.60 · 10−2 nPa (23) In similar fashion, a model[27] based on several spacecraft missions has provided a lower and upper limit on the bow shock line, with its radius vector given by
R = R0 1 +
1 + cos θ , with R0 = c1p −1/c2
D (24)
where the radius R0 toward the Sun was calculated using fitted values of the coef-ficients {c1, c2} and an estimated range of the dynamic pressure pD; the same data was also used to fit the eccentricity ,
c1 = 15.0 , c2 = 5.40 , = 0.84 , pD = 2 · 10−3− 2 · 10−1 nPa (25) For the remainder of this project, a size in-between the extremes — generated by a pressure value of pD = 2 · 10−2 nPa — will be used for all visual purposes, demonstrated in Fig. 10. x KSM [RS] -50 -40 -30 -20 -10 0 10 20 30 40 50 y KSM [R S ] -50 -40 -30 -20 -10 0 10 20 30 40 50 Saturn Magnetopause Bow shock, p D = 2*10 -2 nPa Bow shock, p D = 2*10 -1 nPa Bow shock, p D = 2*10 -3 nPa
4.3
Selecting intervals
All recorded data of the magnetic field components and ephemerides for the entirety of Cassini’s mission is publicly available for download on the web archive of Planetary Data System (PDS), developed by National Aeronautics and Space Administration (NASA). An effective tool for plotting, mining and downloading selected parts of the data is the web service Automated Multi Dataset Analysis (AMDA), developed by Centre de Données de la Physique des Plasmas (CDPP).
For this project’s analysis, in-situ magnetic field data measured by the Cassini space-craft is used. Whenever the spacespace-craft enters the magnetosheath is recognized by a characteristic increase of field strength, compared to the Solar wind; the magnetic field has a magnitude of order < 1 nT in the Solar wind, and ∼ 2 nT in in the magnetosheath.
In order to distinguish between a quasi-perpendicular and quasi-parallel bow shock, correctly one has to calculate the angle of incidence. However, since Cassini is a single spacecraft, one cannot obtain this information. As a consequence, the shock geometry is distinguished only by looking at the transition profile from the Solar wind to the magnetosheath; a quasi-perpendicular shock is characterized by a very sharp transition, unlike a quasi-parallel one which has a more gradual transition. In Fig. 11, these different cases are presented for two events; in both examples, the spacecraft enters and exists the magnetosheath from and to the Solar wind.
Figure 11. Change in magnetic field strength between the Solar wind and the magne-tosheath for a quasi-perpendicular (left) and quasi-parallel (right) bow shock.
The searched time period is narrowed down to the years 2015–2016 to expand the research done by my supervisor Dr. Hadid, in which she analysed data from 2004–2014; see Fig. 12 for the orbit of Cassini during that total time period. The magnetosheath is divided into three main regions based on the aforementioned mod-els (see Fig. 13):
• Region 1: The sub-Solar nose restricted by xKSM ∈ [+20, +30] RS, yKSM ∈ [−20, +20] RS • Region 2: The dusk flank restricted by
xKSM ∈ [+00, +20] RS, yKSM ∈ [+20, +40] RS • Region 3: The dawn flank restricted by
xKSM [RS] -50 -40 -30 -20 -10 0 10 20 30 40 50 y KSM [R S ] -50 -40 -30 -20 -10 0 10 20 30 40 50 Saturn Magnetopause Bow shock Cassini orbit
Figure 12. Orbit of the Cassini spacecraft around Saturn for the investigated time period, from 2004–01–01 T00:00:00 to 2017–01–01 T00:00:00. x KSM [RS] -50 -40 -30 -20 -10 0 10 20 30 40 50 y KSM [R S ] -50 -40 -30 -20 -10 0 10 20 30 40 50
1
2
3
Saturn Magnetopause Bow shockFigure 13. Division of the magnetosheath into regions appropriate for interval searching. The searched regions are the nose (1), the dusk flank (2) and the dawn flank (3).
Data mining for when Cassini is inside any of these regions is performed with AMDA. In addition, Dr. Hadid provided me with her time table, from which more intervals are selected in order to achieve a statistically significant number of exam-ined cases. After the search is finished, the correctly cropped intervals are tabulated chronologically (see Appendix “Table of intervals”).
The identified events from the search and provided time table are segmented into three categories of intervals: behind quasi-perpendicular bow shocks, behind quasi-parallel bow shocks, and away from the shock inside the middle of the mag-netosheath. Between 2004–2016, a total of 120 intervals — 40 from each category — are selected and analysed.
4.4
Analysis
The analysis of all time intervals is performed in MathWorks’ MATLAB software (see Appendix “MATLAB code”). For each selected interval, the PSD P (f) will be retrieved numerically using MATLAB’s pwelch function, which takes a discrete time series as input and gives a discrete spectral series as output. The motivation is to investigate which energy ranges are present in the turbulence of the magnetosheath. The PSD is expected to vary as the frequency to some power, and to be characterized by two frequency ranges separated by the ion gyrofrequency fci,
P (f ) = (
C1fα1 , for f < fci C2fα2 , for f > fci
(26)
where {C1, C2} are constants of appropriate dimension; α1 and α2 will hereafter be called the MHD slope and the sub-ion slope, respectively. In each range, the slopes αi, i = 1, 2, are calculated by taking the logarithm of the data points and performing a linear regression using MATLAB’s polyfit function,
log10(P (f )) = αilog10(f ) + log10(Ci) , i = 1, 2
⇒ y = αix + βi (27)
where y ≡ log10(P (f )), x ≡ log10(f ) and βi ≡ log10(Ci). For each interval the start and end points of the linear fits have to be decided visually, due to irregularities in the PSD as well as unclear transitions between the two ranges.
5
Results
5.1
Example of PSD graphs
After performing the analysis on all 120 intervals, it is evident that the PSD of almost all cases is characterised by two different ranges — the MHD and sub-ion scales — separated by the ion gyrofrequency. In Fig. 14 is presented the distribution of ion gyrofrequencies, having a peak value of fci= 0.02Hz; and in Fig. 15 the distribution of slope values, whose peaks are discussed in greater detail in Section 5.3.
In Fig. 16 is shown three PSD graphs, chosen to be typical of their respective category. The blue and red lines represent the logarithmic fits at the MHD and sub-ion scales, and the dashed lines mark the ion gyrofrequencies.
Ion gyrofrequency f ci [Hz] 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Number of intervals 0 5 10 15 20 25 30 35 40 45 50
Figure 14. Distribution of ion gyrofrequencies for all intervals.
Absolute value of MHD slope |α
1| 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of intervals 0 1 2 3 4 5 6 7 8 9 10
Absolute value of sub-ion slope |α
2| 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Number of intervals 0 2 4 6 8 10 12 14 16 18 20
Frequency f [Hz]
10-3 10-2 10-1 100 101
Power spectral density P(f) [nT
2/Hz] 10-6 10-4 10-2 100 102 f-1.2 f-2.5
Interval 34 of quasi-perpendicular cases Signal
Regression at MHD scale Regression at sub-ion scale
Ion gyrofrequency fci
Frequency f [Hz]
10-3 10-2 10-1 100 101
Power spectral density P(f) [nT
2/Hz] 10-6 10-4 10-2 100 102 f-0.5 f-2.5 Interval 6 of quasi-parallel cases
Signal
Regression at MHD scale Regression at sub-ion scale Ion gyrofrequency f
ci
Frequency f [Hz]
10-3 10-2 10-1 100 101
Power spectral density P(f) [nT
2/Hz] 10-6 10-4 10-2 100 102 f-1.7 f-2.3
Interval 33 of inside-middle cases Signal
Regression at MHD scale Regression at sub-ion scale
Ion gyrofrequency fci
5.2
PSD dependence on localisation
In Fig. 17 is studied the dependence of the slope values on the localisation within the magnetosheath of Saturn. The intervals are indicated with circles centred at their average position and coloured according to the their absolute slope values represented on the colour bar; empty circles are intervals whose MHD or sub-ion slope could not be determined due to indistinct lines in the PSD graphs. Although it may be difficult to distinguish from the figures, the number of empty circles is significantly higher for the MHD scale than the sub-ion one.
As can be seen, the MHD slope values are lower (predominantly blue) than the sub-ion values (predominantly yellow/red), as expected. Moreover, it seems like the cases characterised by the Kolmogorov law are localised toward the flanks of the magnetosheath, and the shallower values are around the nose, i.e. more light blue circles closer to the flanks than near the nose. The sub-ion slope values seem to indicate no dependence on localisation within the magnetosheath.
Additionally, the accuracy of the applied magnetosheath model (or rather magne-topause and bow shock models) can be commented; most intervals are encountered within the bounds of the model, although there are a few outliers. The quasi-perpendicular and quasi-parallel cases (together 80 out of 120 cases) should all be located in the vicinity of the bow shock — however, most intervals are closer to the middle. This gives rise to the belief that the value of the Solar pressure may have been underestimated or that the magnetosheath is highly dynamic and constantly shape-shifting.
xKSM [RS] -50 -40 -30 -20 -10 0 10 20 30 40 50 y KSM [R S ] -50 -40 -30 -20 -10 0 10 20 30 40 50
Coloured according to absolute value of MHD slope |α
1| Saturn Magnetopause Bow shock Intervals 0 0.5 1 1.5 2 2.5 3 3.5 4 xKSM [RS] -50 -40 -30 -20 -10 0 10 20 30 40 50 y KSM [R S ] -50 -40 -30 -20 -10 0 10 20 30 40 50
Coloured according to absolute value of sub-ion slope |α
2| Saturn Magnetopause Bow shock Intervals 0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 17. Location of all 120 detected intervals, indicated by circles centred at their average positions. The circles have been coloured to present the absolute value of MHD and sub-ion slopes; empty circles are missing values.
5.3
PSD dependence on bow shock geometry
In the last stage of this work, the slope values at MHD and sub-ion scales are investigated with respect to the bow shock geometry. The compared categories are either behind a quasi-perpendicular or a quasi-parallel bow shock, as well inside the magnetosheath as a reference. In Fig. 18, the distribution of the determined slope values for each category is presented in histograms. The slopes that were unable to be determined are excluded, hence the total count adds up to all the coloured circles from Fig. 17.
Looking closer at the histograms, it is clear that the MHD slopes are more spread than the sub-ion ones. It is then difficult to present clear trends for the different types of bow shock crossings. However, some remarks can be attributed the MHD scale. The quasi-perpendicular cases seem to be distributed around absolute slope value 1, recalling the energy containing scales, and peaking at |α1|= 0.9 & 1.5. Whereas the quasi-parallel cases tend toward lower values, peaking at |α1|∈ [0.6, 0.7]. The inside-middle cases are shifted toward higher values and peak at |α1|= 1.6, which is close to the Kolmogorov law of 5/3 ≈ 1.7. At the sub-ion scales, the slope values are similar in all categories, with |α2|∈ [2.3, 2.6] roughly.
The MHD slope distributions of quasi-perpendicular and inside-middle cases are most resemblant, regarding peak values and overall histogram structure. In both categories, there exist peaks around the characteristic slope of the inertial range, contrary to the quasi-parallel cases having almost no instances where |α1|> 1.4.
Absolute value of MHD slope |α 1| 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of intervals 0 1 2 3 4 5 6 7 8 Quasi-perpendicular cases
Absolute value of sub-ion slope |α
2| 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Number of intervals 0 1 2 3 4 5 6 7 8 Quasi-perpendicular cases
Absolute value of MHD slope |α
1| 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of intervals 0 1 2 3 4 5 6 7 8 Quasi-parallel cases
Absolute value of sub-ion slope |α
2| 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Number of intervals 0 1 2 3 4 5 6 7 8 Quasi-parallel cases
Absolute value of MHD slope |α
1| 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of intervals 0 1 2 3 4 5 6 7 8 Inside-middle cases
Absolute value of sub-ion slope |α
2| 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Number of intervals 0 1 2 3 4 5 6 7 8 Inside-middle cases
Figure 18. Distribution of MHD and sub-ion slope values, for intervals detected behind quasi-perpendicular or quasi-parallel bow shocks and in the middle of the magnetosheath.
6
Discussion
6.1
Comparison
In comparison to previously conducted studies concerning the near-Earth Solar wind and Terrestrial magnetosheath, the results are quantitatively similar. In this in-vestigation, the inertial range also seems missing near the Solar wind bow shock, considering the peaks in the histograms over MHD slopes. The 5/3 exponent is also more likely encountered further away from the bow shock, as it is close to the peak value for the cases sampled deeper inside the magnetosheath. Furthermore, no clear distinction can be made between the slope values of quasi-perpendicular and quasi-parallel cases, other than somewhat steeper MHD slopes for the former. The determined values at the sub-ion scale are similar to the ones observed in the Solar wind and Terrestrial magnetosheath.
This study confirms the results observed in the magnetosheath of Earth, and shows the important role of the bow shock in destroying correlation between mag-netic field fluctuations. In doing so, the shock resets the fully developed turbulence that existed in the Solar wind, thus most likely explaining the absence of the inertial range — in both Earth’s and Saturn’s magnetosheath.
6.2
Source of errors
While searching for suitable time intervals, a consequence of the simplified magne-tosheath model was the often-times fruitless data mining. Since the modelling of a constant shape for the magnetosheath differs from reality, each search for a given time period becomes a hopeful wish that at least one crossing will be encountered. The categorization of types of bow shocks could also generate confusion, as the shocks were not always of one type or the other, but oblique as they are referred to then. In those situation, decisions had to be made in regard to which extreme case they resemble the most.
In the process of determining the slope values, the start and stop frequency for both scales in each interval was determined visually and individually. The reason for some MHD slopes missing is due to the numerical power spectrum not having enough data points in the regions of lower frequency, making it impossible to perform regression. Other times, the spectrum appeared to have no distinct power-laws. Furthermore, whenever the MHD slopes were very steep, there arose difficulty in separating the two scales from each other; either they could be told apart using the ion gyrofrequency, or — if still ambiguous — the case was assumed to be missing an MHD slope completely. Given the relative size of the histogram bins to the total number of intervals in each category, the sample size appears to be too small to avoid influence from statistical variance. The spread of, for example, the MHD slopes could be random deviation caused by uncertainties in the data analysis.
Finally, an assumption made throughout this investigation has been the so-called Taylor hypothesis, which states that the reference frame of the spacecraft is
approx-6.3
Applications
Understanding turbulence is essential in many physical fields, not least the energy transport and heating of plasmas. Moreover, studying turbulence properties around Saturn allows for the investigation of this phenomena in a different plasma régime. Some properties of the near-Saturn plasma is comparable to those of other astro-physical mediums — such as the interstellar medium and supernovas — for which in-situ observations are currently unavailable. To grasp the mechanism of energy transport in the interplanetary and interstellar medium allows us to prepare our-selves for future spacecraft missions — carrying machines or humans.
7
Conclusion
7.1
Project achievement and future suggestions
The aim of this project was to investigate the spectral properties of the Kronian magnetosheath. To that extent, the project has been successful and the results indicate that the behaviour compared to what is already known about the Terrestrial magnetosheath may be of similar nature.
The statistical analysis, however, is lacking in sample size and the importance of other parameters has to be taken into consideration in order to further justify the results. As a suggestion for future research, one could conduct investigations using larger sets of time intervals and Cassini data from other instruments.
7.2
Learning outcome
For the duration of this project’s making, I have been introduced to a variety of novelties — not to mention the field of space physics, which to me had only been approached through courses in subjects from which it is built upon, such as electro-magnetism and fluid mechanics. I have learned the usefulness of empirical data in studies of not fully understood topics, where patterns and relations are still searched for; and it is delightful that — in some sense — I have been able to contribute in this exploration.
As one might expect for astrophysics, which aims to study the mighty vast, that property also translates to the collection of data; I have gained an appreciation for the troublesome ordeal of storing and sending huge amounts of informations using small spacecrafts far away, as well for the limitations when distributing and han-dling that information back on Earth. Furthermore, I have learned the systematic approach and appropriate selection one must conduct whilst working with these enormous quantities.
Perhaps the most valuable lesson has been the method of scientific work, whose rules I have tried my best to follow. A project of this scale — regarding time and effort — has been the first of its kind for me, and it has taught me the importance of critical thinking and truthful representation.
Part III
Bibliography
References
Literature
[1] NASA. 2018. Cassini Legacy 1997–2017: Timeline.https://saturn.jpl.nasa.
gov/the-journey/timeline (accessed on 2018–05–14).
[2] P.A. Davidson. 2001. An Introduction to Magnetohydrodynamics. Cambridge University Press.
[3] A.N. Kolmogorov. 1991. The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers. Proceedings of the Royal Soci-ety of London A, 434, 9–13.
[4] F.F. Chen. 1984. Introduction to Plasma Physics and Controlled Fusion. Plenum Press. 2nd ed.
[5] R. Bruno & V. Carbone. 2013. The Solar Wind as a Turbulence Laboratory. Living Reviews in Solar Physics, 10.
[6] L. Hadid. 2016. In-situ observations of compressible turbulence in planetary mag-netosheaths and solar wind. Université Paris-Saclay.
[7] L.F. Burlaga. 1995. Interplanetary Magnetohydrodynamics. Oxford University Press.
[8] M. Blanc, R. Kallenbach & N.V. Erkaev. 2004. Solar System Magnetospheres. Space Science Reviews, 116, 227–298.
[9] D.W. Ricker. 2003. Echo Signal Processing. Kluwer Academic Publishers. [10] D.J. Griffiths. 2017. Introduction to Electrodynamics. Cambridge University
Press. 4th ed.
[11] C.T. Russell. 1972. Comments on the Measurement of Power Spectra of the Interplanetary Magnetic Field. NASA.
[12] J.J. Podesta, D.A. Roberts & M.L. Goldstein. 2007. Spectral Exponents of Ki-netic and MagKi-netic Energy Spectra in Solar Wind Turbulence. Astrophysical Journal, 664, 543–548.
[14] S.Y. Huang, L.Z. Hadid, F. Sahraoui, Z.G. Yuan & X.H. Deng. 2017. On the Existence of the Kolmogorov Inertial Range in the Terrestrial Magnetosheath Turbulence. Astrophysical Journal Letters, 836.
[15] NASA. 2018. Solar System Exploration: Saturn by the numbers. https://
solarsystem.nasa.gov/planets/saturn/by-the-numbers(accessed on 2018–
05–15).
[16] M.K. Dougherty, L.W. Esposito & S.M. Krimigis. 2009. Saturn from Cassini-Huygens. Springer.
[17] D.J. Stevenson. 1980. Saturn’s Luminosity and Magnetism. Science, 208, 746– 748.
[18] P. Bond. 2012. Exploring the Solar System. John Wiley & Sons. [19] Ibid. 1.
[20] C.T. Russell. 2003. The Cassini-Huygens mission: Overview, Objectives and Huygens Instrumentarium. Springer.
[21] A. Coustenis & F.W. Taylor. 2008. Titan: Exploring an Earthlike World. World Scientific. 2nd ed.
[22] M.K. Dougherty, S. Kellock, D.J. Southwood, A. Balogh, E.J. Smith, B.T. Tsurutani, B. Gerlach, K.H. Glassmeier, F. Gleim, C.T. Russell, G. Erdos, F.M. Neubauer & S.W.H. Cowley. 2004. The Cassini Magnetic Field Investigation. Space Science Reviews, 114, 331–383.
[23] A.H. Sulaiman. 2017. The Near-Saturn Magnetic Field Environment. Springer. [24] N.F. Comins & W.J. Kaufmann. 2008. Discovering the Universe. W. H.
Free-man and Company. 8th ed. Appendices A–7. [25] Ibid. Appendices A–15.
[26] S.J. Kanani, C.S. Arridge, G.H. Jones, A.N. Fazakerley, H.J. McAndrews, N. Sergis, S.M. Krimigis, M.K. Dougherty, A.J. Coates, D.T. Young, K.C. Hansen & N. Krupp. 2010. A new form of Saturn’s magnetopause using a dynamic pressure balance model, based on in situ, multi-instrument Cassini measurements. Journal of Geophysical Research, 115.
[27] D.R. Went, G.B. Hospodarsky, A. Masters, K.C. Hansen & M.K. Dougherty. 2011. A new semiempirical model of Saturn’s bow shock based on propagated solar wind parameters. Journal of Geophysical Research, 116.
Figures
Figure 1. O. Reynolds. 1883. An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels. Philosophical Transactions of the Royal Society of London, 174, 935–982.
Figure 2. L. Hadid. 2016. In-situ observations of compressible turbulence in planetary magnetosheaths and solar wind. Université Paris-Saclay.
Figure 3. ESA. 2018. The magnetosheath in Earth’s magnetic environment.
http://sci.esa.int/jump.cfm?oid=59948 (accessed on 2018–05–18).
Figure 4. National Research Council. 2004. Plasma Physics of the Local Cos-mos. National Academies Press.
Figure 5. R. Bruno & V. Carbone. 2013. The Solar Wind as a Turbulence Laboratory. Living Reviews in Solar Physics, 10.
Figure 6. S.Y. Huang, L.Z. Hadid, F. Sahraoui, Z.G. Yuan & X.H. Deng. 2017. On the Existence of the Kolmogorov Inertial Range in the Terrestrial Magnetosheath Turbulence. Astrophysical Journal Letters, 836.
Figure 7. NASA. 2016. Saturn, Approaching Northern Summer. https://saturn.
jpl.nasa.gov/resources/7504/saturn-approaching-northern-summer/?category=
images (accessed on 2018–05–14).
Figure 8. M.K. Dougherty, S. Kellock, D.J. Southwood, A. Balogh, E.J. Smith, B.T. Tsurutani, B. Gerlach, K.H. Glassmeier, F. Gleim, C.T. Russell, G. Erdos, F.M. Neubauer & S.W.H. Cowley. 2004. The Cassini Magnetic Field Investigation. Space Science Reviews, 114, 331–383.
Figure 9. A.H. Sulaiman. 2017. The Near-Saturn Magnetic Field Environment. Springer.
Figure 10. Self-created in MATLAB. Figure 11. Self-created in AMDA. Figure 12–18. Ibid. Fig. 10.
Tables
Table 1. L. Hadid. 2016. In-situ observations of compressible turbulence in plane-. Université Paris-Saclayplane-.
Appendix
Table of intervals
Quasi-perpendicular cases 2004-06-27T09:47:25 2004-06-27T10:07:12 2004-06-27T10:08:26 2004-06-27T10:27:54 2004-06-28T02:07:11 2004-06-28T02:57:25 2004-12-11T23:48:09 2004-12-12T00:55:55 2004-12-12T02:45:32 2004-12-12T03:59:41 2004-12-23T17:59:41 2004-12-23T20:24:09 2005-03-05T16:40:29 2005-03-05T16:51:45 2005-03-05T16:52:24 2005-03-05T17:02:27 2005-03-27T09:06:17 2005-03-27T10:14:54 2005-04-21T08:48:50 2005-04-21T09:29:49 2005-04-21T09:58:35 2005-04-21T10:44:58 2005-06-23T05:35:01 2005-06-23T05:55:26 2005-10-08T21:40:06 2005-10-08T23:26:26 2005-10-09T06:51:00 2005-10-09T08:38:21 2005-11-19T16:25:38 2005-11-19T16:55:18 2005-12-17T13:04:10 2005-12-17T14:00:19 2005-12-17T14:30:19 2005-12-17T15:24:52 2006-02-19T04:44:26 2006-02-19T05:13:49 2006-03-12T21:33:09 2006-03-12T22:25:32 2006-03-12T23:00:23 2006-03-12T23:44:24 2007-03-12T15:07:54 2007-03-12T16:03:19 2007-04-28T13:01:49 2007-04-28T15:10:05 2007-04-28T18:44:07 2007-04-28T20:55:19 2007-06-01T18:43:42 2007-06-01T19:29:57 2007-06-14T13:35:10 2007-06-14T14:53:04 2007-06-14T17:47:02 2007-06-14T19:19:36 2008-02-11T00:31:58 2008-02-11T01:36:23 2008-02-15T00:10:50 2008-02-15T01:22:15 2008-02-15T04:47:11 2008-02-15T06:12:13 2011-10-04T11:38:49 2011-10-04T12:01:40 2011-10-04T23:47:03 2011-10-05T01:00:18 2011-10-05T04:37:11 2011-10-05T05:53:18 2011-12-06T20:18:52 2011-12-06T21:16:17 2012-07-02T22:20:30 2012-07-02T23:23:58 2012-07-03T00:54:00 2012-07-03T01:47:28 2014-02-22T08:22:08 2014-02-22T09:17:42 2014-05-27T06:08:09 2014-05-27T07:26:19 2014-05-27T10:09:39 2014-05-27T11:35:20 2015-03-11T12:07:50 2015-03-11T12:57:51 2016-06-01T11:47:39 2016-06-01T12:57:29Quasi-parallel cases 2004-10-25T06:07:23 2004-10-25T06:20:05 2004-10-25T06:40:07 2004-10-25T06:52:19 2005-01-24T00:00:07 2005-01-24T01:00:20 2005-03-14T13:00:13 2005-03-14T13:55:28 2005-03-14T13:56:07 2005-03-14T14:49:25 2005-03-15T19:42:21 2005-03-15T20:31:19 2005-03-19T19:23:32 2005-03-19T20:03:48 2005-04-11T05:58:27 2005-04-11T07:24:45 2005-04-20T00:57:30 2005-04-20T01:35:55 2005-04-20T19:03:22 2005-04-20T20:49:07 2005-04-24T04:19:07 2005-04-24T04:41:04 2005-05-09T10:20:56 2005-05-09T12:11:52 2005-05-27T14:19:42 2005-05-27T15:32:50 2005-08-08T16:32:06 2005-08-08T17:43:32 2005-08-08T19:15:59 2005-08-08T20:01:37 2005-08-29T06:01:13 2005-08-29T06:58:58 2005-10-20T08:02:33 2005-10-20T09:09:42 2005-10-21T08:35:33 2005-10-21T09:18:17 2005-10-26T12:11:22 2005-10-26T12:56:50 2005-12-16T17:50:00 2005-12-16T18:50:56 2006-01-31T00:57:56 2006-01-31T01:51:32 2006-02-06T22:01:03 2006-02-06T23:10:25 2006-02-08T12:09:18 2006-02-08T13:20:30 2006-03-13T04:15:23 2006-03-13T05:01:32 2006-03-13T14:51:33 2006-03-13T15:45:39 2007-03-12T00:46:46 2007-03-12T01:47:19 2007-03-15T04:25:56 2007-03-15T05:20:47 2007-04-27T18:38:39 2007-04-27T19:10:39 2007-04-27T22:17:12 2007-04-27T23:13:42 2007-04-27T23:52:31 2007-04-28T00:20:10 2007-04-28T00:59:34 2007-04-28T01:14:46 2007-05-23T01:35:55 2007-05-23T02:25:00 2007-06-03T14:24:04 2007-06-03T16:13:43 2011-04-08T08:56:44 2011-04-08T10:04:16 2011-12-07T10:10:15 2011-12-07T11:49:12 2011-12-08T01:43:54 2011-12-08T03:35:37 2012-01-22T18:55:37 2012-01-22T20:50:28 2012-08-15T22:50:58 2012-08-16T01:29:10 2015-03-11T14:11:25 2015-03-11T15:25:34 2016-07-17T08:58:54 2016-07-17T10:32:15
Inside-middle cases 2004-06-27T09:55:35 2004-06-27T10:20:37 2004-10-25T06:19:45 2004-10-25T06:40:44 2004-12-23T15:23:27 2004-12-23T18:17:42 2005-03-05T16:45:50 2005-03-05T16:58:19 2005-03-14T15:10:39 2005-03-14T16:14:12 2005-03-16T00:00:10 2005-03-16T01:00:17 2005-03-19T00:12:07 2005-03-19T01:11:17 2005-03-20T00:18:44 2005-03-20T01:17:12 2005-03-27T12:52:21 2005-03-27T14:19:19 2005-04-20T00:16:17 2005-04-20T01:01:47 2005-04-24T04:43:06 2005-04-24T05:21:10 2005-05-09T13:17:33 2005-05-09T15:09:05 2005-06-23T12:56:59 2005-06-23T13:42:57 2005-07-29T08:23:30 2005-07-29T10:08:47 2005-08-28T23:01:26 2005-08-29T00:28:56 2005-10-09T01:45:50 2005-10-09T03:03:34 2005-10-26T12:33:34 2005-10-26T13:26:00 2005-11-20T07:00:14 2005-11-20T08:00:37 2006-01-31T02:56:42 2006-01-31T04:01:26 2006-02-08T13:21:35 2006-02-08T14:46:55 2006-03-12T22:12:47 2006-03-12T23:10:41 2007-03-12T01:48:25 2007-03-12T02:46:52 2007-03-15T00:14:29 2007-03-15T01:18:39 2007-04-27T17:47:35 2007-04-27T18:30:35 2007-04-28T15:57:49 2007-04-28T17:49:58 2007-05-30T13:36:25 2007-05-30T15:02:45 2007-06-03T17:14:00 2007-06-03T18:50:30 2007-06-14T15:07:59 2007-06-14T16:25:18 2008-02-10T23:27:28 2008-02-11T00:15:00 2008-02-15T03:27:23 2008-02-15T04:41:13 2011-10-03T23:46:47 2011-10-04T01:47:30 2011-10-04T21:48:21 2011-10-04T22:17:26 2011-12-06T22:02:04 2011-12-06T23:29:53 2012-01-22T23:40:35 2012-01-23T01:22:24 2012-02-24T07:38:27 2012-02-24T08:21:50 2012-09-15T01:37:17 2012-09-15T03:10:32 2014-02-22T06:46:19 2014-02-22T07:45:59 2014-02-23T12:10:38 2014-02-23T12:49:14 2015-03-11T12:34:11 2015-03-11T13:54:20 2016-06-22T11:43:33 2016-06-22T14:22:29
MATLAB code
Code_S.m% Script for Saturn . % Plot :
pS = plot (0,0,'ok ',' MarkerFaceColor ','k');
xlabel ('x_{KSM} [R_{S}] '); ylabel ('y_{KSM} [R_{S}] '); set(gca ,'xtick ',[ -50:10:50]) ;
set(gca ,'ytick ',[ -50:10:50]) ;
axis ([ -50 ,50 , -50 ,50]); grid on; hold on; Code_MP.m
% Script for magnetopause . % Parameters :
c = [10.30 ,0.20 ,0.73 ,0.40]; pD = 3.60*10^ -2; R0 = c(1)*pD^(-c(2)); K = c(3)+c(4)*pD;
% Radius vector :
theta = linspace ( -140 ,140 ,100) '*pi /180; R_MP = R0 *(2./(1+ cos( theta ))).^K;
R_MPx = R_MP .* cos( theta ); R_MPy = R_MP .* sin( theta ); % Plot :
pMP = plot (R_MPx ,R_MPy ,'--k'); Code_BS.m
% Script for bow shock . % Parameters :
c = [15.00 ,5.40]; pD = 2*10.^( -[1:3]) ; e = 0.84; R0 = c(1)*pD .^( -1/c(2));
% Radius vector :
theta = linspace ( -140 ,140 ,100) '*pi /180;
for i = 1: length (pD)
R_BS (:,i) = R0(i) *(1+ e) ./(1+ e*cos( theta )); R_BSx (:,i) = R_BS (:,i).* cos( theta );
R_BSy (:,i) = R_BS (:,i).* sin( theta );
Code_MS.m
% Script for magnetosheath . % Region 1: R1x = [20 ,30 ,30 ,20 ,20]; R1y = [ -20 , -20 ,20 ,20 , -20]; plot (R1x ,R1y ,'-k'); text (23 ,1 ,'1','fontsize ',20); % Region 2: R2x = [0 ,20 ,20 ,0 ,0]; R2y = [20 ,20 ,40 ,40 ,20]; plot (R2x ,R2y ,'-k'); text (8 ,31 ,'2','fontsize ',20); % Region 3: R3x = [0 ,20 ,20 ,0 ,0]; R3y = [ -40 , -40 , -20 , -20 , -40]; plot (R3x ,R3y ,'-k'); text (8 , -29 ,'3','fontsize ',20); Code_CO.m
% Script for Cassini orbit . % Orbit (2004 -2016) :
COdatax = load (' Interval_tot_x .txt '); COdatay = load (' Interval_tot_y .txt '); COx = COdatax (: ,2);
COy = COdatay (: ,2); % Plot :
Code_PSD.m
% Script for power spectral density . % Plot ? 1 for yes , 0 for no:
%( only intended for plotting interval n) plotyesno = 0; n = 1;
% Frequency ranges :
freq = load (' Intervals_Frequency .txt '); % Analysis : if plotyesno == 1 istart = n; istop = n; elseif plotyesno == 0 istart = 1; istop = 120; end
for i = istart : istop % Data :
if i < 10
ID = fopen (['Interval_00 ' num2str (i) '_B.txt ']);
elseif i >= 10 && i < 100
ID = fopen (['Interval_0 ' num2str (i) '_B.txt ']);
elseif i >= 100
ID = fopen (['Interval_ ' num2str (i) '_B.txt ']);
end
data = textscan (ID ,'%s %f','HeaderLines ',4); fclose (ID); % Time :
time = data (1); time = time {1}; time = char ( time ); ndata = length ( time );
dT = time (2: ndata ,:) - time (1: ndata -1 ,:);
T = dT (: ,12) *10*60^2 + dT (: ,13) *60^2 + dT (: ,15) *10*60 + dT (: ,16) *60 + dT (: ,18) *10 + dT (: ,19) + dT (: ,21) *10^ -1 + dT (: ,22) *10^ -2 + dT (: ,23) *10^ -3; Tm = mean (T); % Magnetic field : B = data (2); B = B {1}; Bm = mean (B); dB = B - Bm; % Parameters :
% Power spectral density :
[P,f] = pwelch (dB ,ws ,ol ,[] , fs); logP = log10 (P); logf = log10 (f); % Frequency ranges :
start1 = find (f >=10^ freq (i ,1) ,1,'first '); stop1 = find (f <=10^ freq (i ,2) ,1,'last '); start2 = find (f >=10^ freq (i ,3) ,1,'first '); stop2 = find (f <=10^ freq (i ,4) ,1,'last '); cut = find (f <=10^+2.0 ,1 ,'last ');
fci(i) = (10^ -9*1.602*10^ -19) /(2* pi *1.673*10^ -27) *Bm; % Plot :
if plotyesno == 1
pSig = loglog (f(1: cut),P(1: cut),'-k'); xlabel ('Frequency f [Hz]');
ylabel ('Power spectral density P(f) [nT ^{2}/ Hz]');
if n >= 01 && n <= 40
title (['Interval ' num2str (n) ' of quasi -perpendicular cases ']);
elseif n >= 41 && n <= 80
title (['Interval ' num2str (n -40) ' of quasi -parallel cases ']);
elseif n >= 81
title (['Interval ' num2str (n -80) ' of inside -middle cases ']);
end
axis (10.^[ freq (i ,5) ,freq (i ,6) ,freq (i ,7) ,freq (i ,8) ]); grid on; hold on;
pfci = loglog ( linspace (fci(i),fci(i) ,100) ,linspace (10^ -10 ,10^10 ,100) ,'--k');
end
% Linear regression : if freq (i ,1) == -10 alpha1 (i) = NaN;
else
c1 = polyfit ( logf ( start1 : stop1 ),logP ( start1 : stop1 ) ,1);
alpha1 (i) = c1 (1); beta1 = c1 (2);
if plotyesno == 1
x1 = f( start1 : stop1 );
y1 = x1 .^ alpha1 (i) *10^ beta1 ; pMHD = loglog (x1 ,y1 ,'-b');
text1y = 10^(( log10 (y1 (1))+ log10 (y1(end))) /2+1/2) ;
text (text1x ,text1y ,{['f^{ ',num2str ( round ( alpha1 (i ) ,1)),'}']},'fontsize ',15,'Color ','b');
end end
if freq (i ,3) == -10 alpha2 (i) = NaN;
else
c2 = polyfit ( logf ( start2 : stop2 ),logP ( start2 : stop2 ) ,1);
alpha2 (i) = c2 (1); beta2 = c2 (2);
if plotyesno == 1
x2 = f( start2 : stop2 );
y2 = x2 .^ alpha2 (i) *10^ beta2 ; pIon = loglog (x2 ,y2 ,'-r');
text2x = 10^(( log10 (x2 (1))+ log10 (x2(end)))/2); text2y = 10^(( log10 (y2 (1))+ log10 (y2(end))) /2+1) ; text (text2x ,text2y ,{['f^{ ',num2str ( round ( alpha2 (i
) ,1)),'}']},'fontsize ',15,'Color ','r');
end end
if plotyesno == 1
if freq (i ,1) ~= -10 && freq (i ,3) ~= -10 legend ([ pSig ,pMHD ,pIon , pfci ],{'Signal ','
Regression at MHD scale ','Regression at sub -ion scale ','Ion gyrofrequency f_{ci}'});
elseif freq (i ,1) ~= -10 && freq (i ,3) == -10
legend ([ pSig ,pMHD , pfci ],{'Signal ','Regression at MHD scale ','Ion gyrofrequency f_{ci}'});
elseif freq (i ,1) == -10 && freq (i ,3) ~= -10
legend ([ pSig ,pIon , pfci ],{'Signal ','Regression at sub -ion scale ','Ion gyrofrequency f_{ci}'});
elseif freq (i ,1) == -10 && freq (i ,3) == -10
legend ([ pSig , pfci ],{'Signal ','Ion gyrofrequency f_{ci}'});
end end end
% Slope values :
Code_Pos.m
% Script for positions . % Positions :
for i = 1:120
if i < 10
xID = fopen (['Interval_00 ' num2str (i) '_x.txt ']); yID = fopen (['Interval_00 ' num2str (i) '_y.txt ']);
elseif i >= 10 && i < 100
xID = fopen (['Interval_0 ' num2str (i) '_x.txt ']); yID = fopen (['Interval_0 ' num2str (i) '_y.txt ']);
elseif i >= 100
xID = fopen (['Interval_ ' num2str (i) '_x.txt ']); yID = fopen (['Interval_ ' num2str (i) '_y.txt ']);
end
xdata = textscan (xID ,'%s %f','HeaderLines ',4); ydata = textscan (yID ,'%s %f','HeaderLines ',4); fclose (xID); fclose (yID);
xdata = xdata (2); ydata = ydata (2); xdata = xdata {1}; ydata = ydata {1};
x(i) = mean ( xdata ); y(i) = mean ( ydata );
end % Sort :
ialpha1 = 1; jalpha1 = 1; ialpha2 = 1; jalpha2 = 1;
for i = 1:120
if isnan ( alpha1 (i)) == 1
empty1 ( ialpha1 ) = i; ialpha1 = ialpha1 +1;
else
filled1 ( jalpha1 ) = i; jalpha1 = jalpha1 +1;
end
if isnan ( alpha2 (i)) == 1
empty2 ( ialpha2 ) = i; ialpha2 = ialpha2 +1;
else
filled2 ( jalpha2 ) = i; jalpha2 = jalpha2 +1;
end end % Plot :
if alpha == 1
pPos = plot (x( empty1 ),y( empty1 ),'ok ');
elseif alpha == 2
pPos = plot (x( empty2 ),y( empty2 ),'ok ');
Code_Hist.m
% Script for histograms . % Slope values :
int = [1 ,120;1 ,40;41 ,80;81 ,120];
tit = {'','Quasi - perpendicular cases ','Quasi - parallel cases ','Inside - middle cases '};
for i = 1:8
if i <= 4
histogram ( alpha1 (int(i ,1):int(i ,2)),'Binedges '
,0.05:0.1:1.95 ,'FaceColor ','b');
xlabel ('Absolute value of MHD slope |\ alpha_ {1}| '); ylabel ('Number of intervals ');
title (tit(i)); if i == 1 axis ([0 2 0 10]) ; else axis ([0 2 0 8]); end elseif i >= 5
histogram ( alpha2 (int(i -4 ,1):int(i -4 ,2)),'Binedges '
,2.05:0.1:3.95 ,'FaceColor ','r');
xlabel ('Absolute value of sub -ion slope |\ alpha_ {2}| '
);
ylabel ('Number of intervals '); title (tit(i -4)); if i == 5 axis ([2 4 0 20]) ; else axis ([2 4 0 8]); end end figure ; end %Ion gyrofrequencies :
histogram (fci ,'Binedges ',0.005:0.01:0.095 ,'FaceColor ','k'
);
xlabel ('Ion gyrofrequency f_{ci} [Hz]'); ylabel ('Number of intervals ');
Code_Plot.m
% Script for plots . % Plot of bow shock :
Code_S ; Code_MP ; Code_BS ;
pBSmin = plot ( R_BSx (: ,1) ,R_BSy (: ,1) ,' -.b'); pBSmax = plot ( R_BSx (: ,3) ,R_BSy (: ,3) ,' -.r');
legend ([pS ,pMP ,pBS ,pBSmin , pBSmax ],{'Saturn ','Magnetopause ','Bow shock , p_{D} = 2*10^{ -2} nPa ','Bow shock , p_{D}
= 2*10^{ -1} nPa ','Bow shock , p_{D} = 2*10^{ -3} nPa '},
'Location ','northwest '); % Plot of magnetosheath :
figure ; Code_S ; Code_MP ; Code_BS ;
legend ([pS ,pMP ,pBS ],{'Saturn ','Magnetopause ','Bow shock '
},'Location ','northwest '); Code_MS ;
% Plot of Cassini orbit :
figure ; Code_S ; Code_MP ; Code_BS ; Code_CO ;
legend ([pS ,pMP ,pBS ,pCO ],{'Saturn ','Magnetopause ','Bow shock ','Cassini orbit '},'Location ','northwest '); % Plot of positions :
Code_PSD ;
figure ; alpha = 1; Code_S ; Code_MP ; Code_BS ; Code_Pos ; title ('Coloured according to absolute value of MHD slope
|\ alpha_ {1}| ');
legend ([pS ,pMP ,pBS , pPos ],{'Saturn ','Magnetopause ','Bow shock ','Intervals '},'Location ','northwest ');
scatter (x( filled1 ),y( filled1 ) ,[], alpha1 ( filled1 ),'filled '
,' MarkerEdgeColor ','k');
caxis ([0 4]); colormap (jet (256) ); cb = colorbar ; set(cb ,' YDir ','reverse ');
figure ; alpha = 2; Code_S ; Code_MP ; Code_BS ; Code_Pos ; title ('Coloured according to absolute value of sub -ion
slope |\ alpha_ {2}| ');
legend ([pS ,pMP ,pBS , pPos ],{'Saturn ','Magnetopause ','Bow shock ','Intervals '},'Location ','northwest ');
scatter (x( filled2 ),y( filled2 ) ,[], alpha2 ( filled2 ),'filled '
,' MarkerEdgeColor ','k');
caxis ([0 4]); colormap (jet (256) ); cb = colorbar ; set(cb ,' YDir ','reverse ');
% Plot of histograms : figure ; Code_Hist ;
