Plasma characteristic frequencies and scales in near Earth’s regions

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Interplanetary Magnetic Field interaction with Geomagnetic field lines

In this section we give an example of magnetic reconnection occurring at the magnetosphere, as it has been briefly discussed in section 1.1. In figure 6 two different configurations of the geomagnetic field are shown according to the interplanetary solar wind conditions. Magnetosphere is classified into “open” and “closed” configuration, top and bottom panel, respectively, according to how it interacts with the interplanetary magnetic field. During southward configurations, reconnection takes place both at the nose and at the tail between the solar wind field and northward Earth’s magnetic field. In particular, reconnected field lines drag solar wind particles inside the magnetosphere, causing phenomena like auroras. During northward interplanetary magnetic field, instead, there is no dayside reconnection but reconnection takes place at the south and north geomagnetic poles. We already anticipated in section 1.1 that under this condition still cold dense solar wind plasma is observed in the Earth’s magnetosphere together with a broad boundary layer at low latitude (LLBL) a region where a mixture of solar wind and magnetospheric populations, Ref. Fujimoto et al. [44], so other phenomena like Kelvin-Helmholtz instability must take place.

plasma characteristic frequencies and scales in near earth’s regions

A plasma has a large number of characteristic scales. Of particular importance is the Debye length, which is the typical distance over which any charge imbalance is shielded by the electrostatic field. An isolated charge q generates an electrostatic potential φ0(r) = q/4πr but, if we consider this particle in a plasma, then many particles with opposite sign will be attracted thus shielding its potential from the rest of the plasma (and vice versa). Let us consider a positive charge q in a neutral plasma; the number density of the electrons cloud is given by the Boltzmann distribution ne = n∞eeφ/kBTe with φ(r → ∞) = 0. One dimensional Poisson equation for the electrostatic potential is ∇²φ = d2φ/dx2 = −4πe(ni − ne). Substituting the values of ne and ni it becomes: d2φ/dx2 = 4πen∞(eeφ/kBTe − 1). Far from the influence of the electrostatic potential, i.e. when |eφ/kBTe| ≪ 1 the exponential h eφ/kBTe + 1 2 (eφ/kBTe)2 + … i .
Keeping first order terms we can write d2φ/dx2 = 4πn∞ 􀀀 e2/kBTe φ. By defining the Debye length as λD = p kBTe/4πne2, the electrostatic potential turns out to be φ = φ0e−x/λD . As result, the plasma beyond the Debye sphere is neutral. The number of particles in a Debye sphere is N = 4/3πnλ3 D ∝ nλ3 D, the shielding of individual charge is efficient if N is much larger than unity. It is usually convenient to deal  with dimensionless quantities, for this reason we define the plasma parameter g as: g = 1 nλ3.

kelvin-helmholtz and rayleigh-taylor instabilities

Let us now consider themain driver of our system, the Kelvin-Helmholtz instability. It originates any time two layers of a fluid (or two different fluids) are in relative motion. In a plasma, it can be stabilized by a magnetic field directed along the flow due to magnetic tension. In the case of the shear flow between the solar wind and the magnetosphere, when the IMF is northward, the equatorial component of the magnetic field can be negligeable at low latitude and so the K-H instability can develop eventually producing fully rolled-up vortices. Such vortices have been observed by Cluster satellites, see Ref. Hasegawa [54], Hasegawa et al. [50, 52] at the magnetopause.


earlier observations of kelvin-helmholtz

The Kelvin-Helmholtz is a fluid instability that can be found everywhere in nature from fluids to space plasma environments. In figure 9, we report some examples of K-H observed in different astrophysical regions. It is indeed it is thought to act in disk accreting stars Ref. Lovelace et al. [70], Vietri and Stella [125], cometary tails, Ref. Ershkovich [29], sheared flows in the solar corona Kopp [63], Ofman and Thompson [85], see top left panel of figure 9, during coronal mass ejections, see top right panel in figure 9 from Ref. Foullon et al. [40], in planet’s environments like Mercury, Ref. Boardsen et al. [8], Sundberg et al. [112], on Jupiter and on Saturn, bottomright and left panels respectively, Ref. Masters et al. [71], on Venus, Ref.Walker et al. [129] and at the Earth’s magnetopause, Ref. Miura [75], Hasegawa et al. [50].

spacecraft data analysis methods

In this part we will briefly introduce the methods of data analysis that are used in this thesis, to determine for example the normal to the magnetopause, see section 3.3.1 and 3.3.2, the magnetopause velocity, see section 3.3.3 and the local magnetopause reference frame, see section 3.3.4.

Table of contents :

i introduction 
1 interaction between solar wind and earth’s magnetosphere
1.1 Introduction
1.2 Solar wind and magnetospheric plasma regions
1.3 Magnetic Reconnection
1.3.1 Interplanetary Magnetic Field interaction with Geomagnetic field lines
1.4 Space Plasma Discontinuities
1.5 Plasma characteristic frequencies and scales in near Earth’s regions
1.6 Kelvin-Helmholtz and Rayleigh-Taylor Instabilities
1.6.1 Hydrodynamic case
1.6.2 Magnetohydrodynamic case
2 modelling kelvin-helmholtz instability using a two fluid code 
2.1 Numerical Simulations of Kelvin-Helmholtz instability
2.2 Two Fluid code
3 spacecraft observations in near earth’s space 
3.1 Earlier Observations of Kelvin-Helmholtz
3.2 Recent spacecraft missions
3.2.1 Cluster Mission
3.2.2 Geotail mission
3.2.3 ACE 2
3.3 Spacecraft data analysis methods
3.3.1 Minimum Variance Analysis
3.3.2 Timing method
3.3.3 DeHoffman Teller
3.3.4 The boundary normal coordinate system(LMN)
ii turbulence in kelvin-helmholtz vortices
4 two fluid numerical simulation of turbulence in k-h vortices 
4.1 Initial plasma configuration
4.2 Turbulence Analysis
4.2.1 Hann window
4.2.2 Anisotropy
4.2.3 Current sheets and magnetic reconnection regions.
4.3 Discussion and Conclusions
iii simulations of kelvin-helmholtz instability at the magnetopause initialized with experimental profiles
5 simulations based on satellites crossings of the magnetopause 
5.1 Selection of Cluster Data events
5.2 Event 2001-11-20
5.3 Velocity and density profile estimation
5.3.1 Cluster Crossing
5.3.2 Geotail data: Multi-crossings event
5.4 Simulation using experimental initial conditions
5.5 Simulations with higher velocity jump
5.6 Comparison with observational results
5.7 Discussion and Conclusions
iv conclusions and outlook 
v appendix 
a generalized ohm’s law
b the numerical code


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