Solar Wind 

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Sources of the wind and solar cycle

The solar atmosphere, mainly composed of protons, electrons and alpha particles, is strongly stratified in the vertical (radial) direction: density drops from n ’ 1017cm 3 at the surface down to about 5 15 cm 3 at the Earth’s orbit. The strongest stratification occurs close to the Sun. One identifies at least two layers, the thin, dense chromosphere and the hot, rarefied corona which extends up to several solar radii. The chromosphere is separated from the corona by an abrupt transition (chromospheric transition), at about 2000 km above the photosphere. While the chromosphere is relatively cold (between 6000K and 10000K) the corona has a proton temperature around 2MK. High in the corona, at several solar radii, the plasma begins to be accelerated to outer distances from the Sun and to form the solar wind.
Fig. 1.1 shows the average density and temperature profiles around the chromospheric transition (panel a) and bulk velocity measurements using radio source scintillation mea-surements ([92], panel b).
Not all the solar surface is a source of wind. In fact, sources of the wind are largely controlled by the topology of the magnetic field. The atmosphere shows in white light density contrasts which actually trace the magnetic structures, more specifically, closed and open structures. An example of density structures during an eclipse is shown in fig. 1.2 (left), while a reconstruction of magnetic field lines is shown in fig. 1.2 (right).
In the whole corona (up to about 2.5 Rs), the magnetic field energy dominates, and the magnetic field is thought to be organized in flux tubes that force the flow to follow the magnetic field direction. One can distinguish open field lines (that extend to infinity) and closed lines that are anchored at their two ends in solar surface. Along the former, the flow can escape the Sun, while the plasma is trapped in magnetic flux tubes with closed lines. In fig. 1.2 (left), open flows fill the dark regions with low densities, while flows are trapped in white regions with large densities: the main source of the solar wind are thus in dark regions with low densities and open field lines (although some closed fields can open and launch slow wind). The alternance of closed and open magnetic field regions on the Sun thus controls how the solar wind maps into the heliosphere.
This global magnetic pattern changes in a quasi-regular way during the solar cycle (see fig 1.3). The Sun’s magnetic field adopts a quasi-dipolar pattern, with two magnetic poles with opposite polarities, during the so-called Solar minimum activity, and a multipolar pat-tern during Solar maximum activity. The quasi-dipole associated with minimum activity reverses each eleven years in average. At the time of maximum activity on the contrary, no clear polarity is defined, Solar spots, cold regions of the photosphere with a high concentra-tion of magnetic field lines are frequently seen. Strong solar events such as Coronal Mass Ejections (CME) are abundant during this period and perturb properties of the pristine solar wind.
Independently of the period of solar activity, the intensity of the magnetic field decreases with distance. From about 10 solar radii onwards, the magnetic field energy becomes sub-dominant compared to the kinetic energy of the flow (which defines the Alfvén point or Alfvén surface), and the magnetic field lines cease to control the flow.
From the Alfvén point onwards, the mean magnetic field lines no longer co-rotate with the Sun, but spiral around it. This forms what is known as Parker’s spiral.
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Fast and slow winds

One can see important variations in the properties of the plasma depending on its source region, that is, the place where the solar wind originates. The winds originating from sources near the closed magnetic regions are characterized by a mean speed around 350 km s 1 when measured at 1AU. At the same distance, the particle density per unit volume is in average 15 cm 3, the proton temperature is around 5 104K and the mean magnetic field modulus is 6 nT ([12]).
On the other hand, winds from sources well within open magnetic regions reach speeds close to 600 kms 1, lower particle densities, 4 cm 3, larger proton temperatures, 2 105K, but the mean magnetic field is the same as when the source is close to closed regions. Due to these differences we shall group solar winds into slow and fast winds, or into cold and hot winds respectively. Note that this difference between cold and hot winds is only based on proton temperature: electrons have similar temperature in fast and slow winds Te 1 105K, (see table 1.1). The radial variation of electron temperature between 0.3 and 1 AU, Te / R [56] is also slower than that of protons, Tp / R [98] (see section 4.4.1 for a further discussion on the measurements of proton temperature evolution). Other ion species, such as He+2 (alpha particles) also show some differences between slow and fast solar wind streams. At 1AU, their relative abundance with respect to protons is around the 4% and the temperature ratio, T =Tp, is around 1.2 for the more collisional slow winds and around 4.5 for fast winds [63]. Bulk speed of alpha particles also shows differences between slow and fast streams, as it is smaller than that of protons for slow streams and higher for fast winds, the difference reaching 170km s 1 [58].
Until now we have just specified the amplitudes of macroscopic (i.e. fluid) properties in fast and slow winds. There are also differences in microscopic quantities, such as velocity distribution functions. While proton velocity distribution functions (VDF) for slow winds are similar to a Maxwellian distribution, fast winds VDFs are gyrotropic with respect to the mean magnetic field, as can be seen in figure 1.5. In contrast, electron VDFs do not present strong temperature anisotropies such as proton VDFs (see table 1.1). Protons, electrons and alpha particles present departures from a gaussian distribution in the form of non-thermal tails, as the one shown in figure 1.4. The generation of non-thermal tails has been linked to the wave-particle interactions of high frequency waves in plasmas such as Kinetic Alfvén Waves (KAWs) and whistler waves: KAWs are invoked in the generation of the non-thermal tail of proton distributions ([79] and references within), and whistler waves in the generation of the non-thermal tail of electron distributions [105, 104][78]. In this thesis, we restrict ourselves to the study of solar wind plasma at low frequencies (correspondingly large scales), that can be well described by MHD equations (see section 2.1). Thus, high frequency waves and their effects on the particles VDFs are excluded from our description of the plasma. Also, the deviations of the VDFs from Maxwellian distributions will be minimized: in particular, perpendicular and parallel temperatures will be considered equal, as well as electron and proton temperatures.

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From corona to Earth

Starting from the corona, the solar wind speed becomes larger than the sound speed beyond one solar radius, and larger than the Alfvén speed beyond ten solar radii (see fig. 1.6). After this distance, the wind flow is quasi-radial. The wind speed then reaches a value close to its cruise speed, e.g., it doesn’t change much between 0.2 and 1 astronomical units (AU), Figure 1.4: Scheme of an electron velocity distribution function at 1AU for high speed solar wind electrons. Bottom panel: isocontours in the plane of velocities parallel and perpendicular to the mean magnetic field (Green circle represents the limit of the Maxwellian core). Top panel: parallel (solid black line) and perpendicular (dashed blue line) cross section of the VDF in the bottom panel. Dashed red line is a Maxwellian distribution that fits the core of the VDF. From [77].
Slow winds Observations Fast winds Observations
Bulk velocity (km s 1) 320 667
Number density (cm 3) 5:4 3
Proton Temperature (K) 4:8 104 2:8 104
Electron Temperature (K) 1:1 105 1:3 105
Proton anisotropy Tpk=Tp? 3:4 1:2
Electron anisotropy Tek=Te? 1:2 1:2
Table 1.1: Average properties of fast and slow solar winds measured at 1AU made by [43] and [55] Adapted from a table from [52].
Figure 1.5: Velocity distribution functions of protons in the Solar Wind at 1AU (top row) and 0.3 AU (Bottom row). Left panels correspond to slow winds and right panels to fast winds. The dashed line corresponds to the mean magnetic field axis. From: [60].
Figure 1.6: Model profiles of the solar wind speed (U) and the Alfvén wave speed (Va) with distance from the Sun. The vertical bar separates the source, or sub-Alfvénic region, from the super-Alfvénic flow. “Previous Missions” marks the region explored by the Helios mission. From [26] i.e., between 42 and 210 solar radii: this will allow us to use the expanding box model to be described below (see section 2.2). The quasi-radial flow and the quasi-constant speed forces a plasma volume to expand in the two directions perpendicular to the radial and not in the radial direction (see fig. 2.1), so that the volume expands as R2 instead of R3.
This increase of a plasma volume embedded in the wind has two consequences (i) density should decrease as 1=R2, which is indeed observed; (ii) applying the one-fluid adiabatic law, i.e., assuming pressure forces are the only ones against which the expanding plasma volume is working, with no extra internal energy source, one obtains for the temperature T/ 1/ 2=3/R 4=3 (1.1)
However, as we shall see in the heating section 4.4.1 below, the proton temperature gradient is flatter than that, which requires a heat source to slow down the plasma cooling during expansion.

Table of contents :

I Introduction 
1 Solar Wind
1.1 Sources of the wind and solar cycle
1.1.1 Fast and slow winds
1.2 From corona to Earth
2 Plasma description
2.1 The MHD equations
2.2 MHD equations with large scale radial flow (EBM)
3 Turbulence
3.1 Homogeneous turbulence
3.1.1 Shock formation as a simplified model of turbulence
3.1.2 3D hydrodynamic turbulence
Fourier space description: K41 phenomenology
3.1.3 3D MHD turbulence
Incompressible MHD using Elsasser variables, cross-helicity
Anisotropy of the cascade with mean magnetic field B0
Local and nonlocal interactions
3.1.4 Spectra, autocorrelations and structure functions
3.2 Turbulence with large scale radial flow
3.2.1 Inhibition of the turbulent cascade by expansion
3.2.2 Fluctuations decay: strong versus weak expansion
3.2.3 Cascade rate and turbulent heating
4 Solar Wind turbulence
4.1 In situ observations
4.2 MHD inertial range
4.3 Spectral anisotropy
4.3.1 The Maltese Cross
4.4 Proton temperature gradient and turbulent heating
4.4.1 Turbulent amplitude and proton temperature variations in the inner heliosphere Temperature gradient: a long series of studies
Possible explanations of the slow proton cooling in the inner heliosphere
4.4.2 Measuring turbulent heating via second order moments
4.4.3 Measuring turbulent heating via third order moments
5 Plan of this thesis
5.1 Spectral anisotropy: understanding the Maltese Cross
5.2 Turbulent heating in slow and fast winds
II The Maltese Cross revisited 
6 Parameters and initial conditions
6.1 Physical parameters and initial spectra
6.1.1 Expansion parameter, cross helicity, Mach number
6.1.2 Initial spectra
6.1.3 Domain aspect ratio
7 Defining spectral properties in EBM simulations
7.1 Anisotropy of 3D spectra
7.2 1D spectral slopes
8 Results
8.1 Initially ISO spectrum: expansion and cross helicity effects
8.1.1 Varying expansion at zero cross-helicity
8.1.2 Varying expansion at large cross-helicity
8.2 Systematic comparison of zero vs large cross helicity
8.2.1 ISO
8.2.2 GYRO
8.2.3 Gyro-Alfvén model
9 Discussion
9.1 Summary
9.2 Explaining the discrepancy with VG16
III Can the Maltese Cross heat? 
10 1D turbulent heating
10.1 1D HD equations with expansion
10.1.1 Modified Burgers equation and semi-analytical solutions
10.2 Simulations of shock turbulence with transverse waves
10.2.1 Initial conditions
10.2.2 Results
10.3 Discussion
11 Paper ApJ 2018: « Turbulent Heating between 0.2 and 1 au: A Numerical Study »
12 Heating fast winds
12.1 Initial conditions
12.2 Results
12.2.1 Spectral anisotropy with Mach=1
12.2.2 Turbulent heating
13 Discussion
Numerical parameters vs Helios data
IV Conclusions and future work 
14 Conclusions
14.1 Obtention of Maltese Cross components at 1AU
14.2 Can the Maltese Cross heat the solar wind?
14.3 Expectations from Solar Orbiter and Parker Solar Probe
14.4 Open questions and (partial) answers
15 Future work: Anisotropy temperature description
A Liste of symbols
B Numerical method
B.1 Computation of spatial gradients
B.2 Time integration method

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