One dimensional model for discontinuous compressible flow: Burgers’ equation

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Two approaches to plasma

A plasma can be treated microscopically or macroscopically according to the interest. The microscopical approach is known as the kinetic approach whereas the macroscopic treatment is called the fluid approach.

Kinetic approach

Aplasma is a systemofmutually interacting charged particles (ions, electrons) which sometimes can contain considerable number of neutral atoms too. A kinetic theory for plasma consists in analyzing different properties of the plasma by the help of the individual probability distribution functions (PDF) of each particle species in phase space. This approach takes into account the collective behaviour of the particles as well as their individual behaviour. We thus need to know the nature of the microscopic interaction forces along with external macroscopic forces in order to construct a kinetic description of a plasma phenomenon. In the current context, a brief and schematic presentation of kinetic approach to plasma will be presented without entering into formal and detailed derivations.

Magnetohydro dynamics (MHD)

As discussed above, the fluid approach gives a much simpler method for describing a plasma compared with rigorous kinetic approach. Multi-fluid models are appropriate where (i) the departure from thermodynamic equilibrium is very weak but, (ii) the departure from quasi-neutrality is not necessarily weak (in case of sheath for example). In case of a bi-species plasma (ions and electrons) where the quasi-neutrality is perturbed very weakly, the ionic and the electronic fluids are strongly coupled by the ambipolar electric field and the electronic charge density evolves according to the ionic charge density. The plasma dynamics can, in that case, be described by a single fluid whose inertia is governed by the massive ions and mobility by the lighter electrons.
This fluid, being quasi-neutral, is almost free of any net electrostatic force and is therefore called a hydromagnetic (old nomenclature) or a magnetohydrodynamic (MHD) fluid.

Mono-fluid model: Basic equations of MHD

Here we shall carry out a general derivation of mono-fluid equations by summing the equations of a multi-fluid plasma over all the species of the plasma (denoted by index ↵). For that we have to define some new quantities. The mass density of the resulting single fluid is given by ⇢ = X ↵ n↵m↵, (3.16) and the respective charge density is defined as ⇢c = X ↵ n↵q↵.

Linear waves in ideal MHD

It is crucial to examine the existence of waves in an ideal MHD fluid for understanding the corresponding dynamics and the nature of response created by the fluid to any external attempt of perturbation with respect to a steady state flow. Of the waves, the linear modes are the simplest to obtain. These are the eigen modes corresponding to a very small or first order perturbation in an ideal MHD fluid. In order to obtain the linear waves, we have to linearize the ideal MHD equations. Under the assumption of a polytropic closure and zero steady state velocity (which can be obtained without losing generality just by a suitable Galilean transformation), these equations are written as (following the same formalism as that of 2.2).

Invariants of ideal MHD

In this section, we shall examine the invariance of some dynamical variables in an ideal MHD flow. As mentioned earlier, the main objective of finding inviscid invariants lies on the fact that the knowledge of these invariants will help us determine the possibility of cascades (explained in the next chapter) in the inertial zone (defined in the previous chapter) which is supposed to be free from any large scale forcing effect and small scale viscous effect. For simplicity, here we derive those conservation principles right from the inviscid ideal MHD equations (⌫ = 0, ⌘ = 0). The generalized expressions with the viscous terms can be found in standard text books (Galtier, 2013).
The choice of boundary conditions plays a key role in obtaining the invariants. In the following demonstrations, we use the most common and realistic boundary conditions i.e. at the surface of the chosen Eulerian volume (in which the flow is confined), the velocity and the magnetic fields are purely tangential i.e. v.n = 0 and b.n = 0 at each point of the boundary surface, n being the unit normal vector at an arbitrary point of the surface. We shall also see that some quantities, which are not invariant under the said boundary condition, can be invariant if we assume v = b = 0 at every point of the boundary surface.

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Table of contents :

1 Introduction 
1.1 General interest
1.2 Turbulence in sace and astrophysical plasmas
1.3 An outline of my thesis
2 Compressibilityinfluids 
2.1 What is compressibility ?
2.2 Measure of compressibility for a fluid in motion
2.3 Closure for compressible fluids
2.4 Invariants in compressible barotropic fluid
2.4.1 Total energy
2.4.2 Kinetic helicity
2.4.3 Mass and linear momentum
2.5 Potential flow
2.6 Two dimensional compressible flow
2.7 One dimensional model for discontinuous compressible flow: Burgers’ equation
2.8 Compressibility ratio for a polytropic gas across a normal shock
2.9 Baroclinic vector
3 Plasmaphysics andmagnetohydrodynamics 
3.1 What is a plasma ?
3.2 Two approaches to plasma
3.2.1 Kinetic approach
3.2.2 Fluid approach
3.3 Magnetohydrodynamics (MHD)
3.3.1 Mono-fluid model: Basic equations of MHD
3.3.2 Ideal MHD approximation from generalized Ohm’s law
3.3.3 Linear waves in ideal MHD
3.3.4 Invariants of ideal MHD
3.3.5 Elsässer variables in magnetohydrodynamics
4 Turbulentflow: importantnotions 
4.1 Turbulence – A phenomenon or a theory ?
4.2 Turbulent regime from Navier-Stokes : Reynolds number
4.3 Chaos and/or turbulence ?
4.4 Basic assumptions
4.4.1 Statistical homogeneity
4.4.2 Statistical isotropy
4.4.3 Stationary state
4.5 Two approaches to turbulence
4.5.1 Statistical approach
4.5.2 Spectral approach
4.6 Phenomenology
4.6.1 K41 phenomenology
4.6.2 IK phenomenology
4.6.3 Utilities of phenomenology
4.7 Dynamics and energetics of turbulence
4.7.1 Turbulent forcing
4.7.2 Turbulent cascade
4.7.3 Turbulent dissipation
4.8 Intermittency
4.8.1 ! fractal model
4.8.2 Refined similarity hypothesis : Log-Normal model .
4.8.3 Log-Poisson model
4.8.4 Extended self-similarity
5 Turbulence incompressiblefluids 
5.1 Why is it important ?
5.2 Primitive theoretical approaches
5.3 Numerical approaches using one dimensional model
5.4 Numerical Simulations in higher dimensions
5.4.1 Numerical methods for compressible turbulence
5.4.2 Piecewise Parabolic Method (PPM):
5.4.3 Compressible intermittency
5.4.4 Compressible and solenoidal forcing
5.4.5 Choice of inertial zone and sonic scale
5.4.6 Two-point closure in EDQNM model for compressible turbulence
5.5 Observational studies
6 Exact relations inturbulence 
6.1 Exact relations in incompressible turbulence
6.1.1 Incompressible hydrodynamic turbulence
6.1.2 Incompressible MHD turbulence
6.2 Previous attempts for exact relations in compressible turbulence
6.2.1 Heuristic approach by Carbone et al. (2009)
6.2.2 FFO approach for a generalized exact equation
6.3 New exact relations and phenomenologies in compressible turbulence : My research work
6.3.1 Isothermal hydrodynamic turbulence
6.3.2 A new phenomenology for compressible turbulence
6.3.3 Isothermal MHD turbulence
6.3.4 Polytropic hydrodynamic turbulence
7 Solarwinddata analysis 
7.1 Introduction
7.2 The solar wind
7.2.1 The heliosphere
7.2.2 Prediction for the solar wind
7.2.3 The fast and the slow solar wind
7.2.4 Exploration of the solar wind
7.2.5 MHD fluctuations in the solar wind
7.2.6 Nature of the solar wind turbulence
7.3 Data source
7.3.1 The THEMIS mission
7.3.2 A brief description of instruments
7.4 Judicial selection of data
7.4.1 Selection of intervals
7.4.2 Relevant spatial and temporal scales
7.5 Analysis of the selected data
8 Resuming andlooking ahead
8.1 Answered and unanswered issues of compressible turbulence .
8.2 Some future projects
References 

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