Turbulent regime from Navier-Stokes : Reynolds number 

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Triadic interaction and localness of interactions

It is clear from the equation (4.37) that the non-linear term, which efficiently transfers energy, essentially associates three modes k, p and q in Fourier space. For an effective non-zero energy transfer, those three modes should obey the condition of resonance i.e. k+p+q = 0. This introduces the notion of triadic or three-wave interaction. As the non-linear term is believed to bring about the energy cascade, the corresponding mechanism is thus believed to be driven by this type of triad interactions. Let the wave numbers be ordered by a sequence say kn. As energy cascading should be a process of inter-scale step-by step energy transfer (can be shown theoretically for hydrodynamic turbulence), it is necessarily local in wave space which means then the efficient energy transferring interactions corresponding to scale k−1 n will involve either kn or kn±1. Now practically in order to implement the locality in the triadic interactions, all the triads for which the smallest wave number is less than the half of the largest one are eliminated selectively. So if we consider an interacting triad of wave vectors kn, pn and qn with an ordering (bychoice) that |kn| > |pn| > |qn|, the selected triads will be those which would have 2|qn| > |kn|. It is noteworthy that energy is conserved in each triad interaction. Interestingly, in three dimensions, triad interactions can be reduced to pair interactions (two-mode interaction) which is not feasible in two dimensional turbulence due to enstrophy which is also conserved in each triad in 2d turbulence (Kraichnan, 1967a). This result of triad-wise conservation is also valid for incompressible MHD turbulence (Biskamp, 2008). In MHD turbulence both the energy and cross-helicity can be shown to be conserved for each interacting triad thereby validating the localness of interactions in MHD case as well.

IK phenomenology

This phenomenology depends on the image of non-linear interaction of two oppositely propagating weakly fluctuating linear modes. According to this phenomenology energy is transferred from one length scale to another length scale by the deformation of one fluctuating linear wave by the other one.
This phenomenology was independently proposed by Iroshnikov (1964) and Kraichnan (1965). They applied this concept in understanding the turbulence in a plasma fluid or more precisely in an MHD fluid. In incompressible MHD (as seen in the previous chapter), we have only one linear mode which is Alfvén mode. Now, in presence of a strong magnetic field (sometimes called the guiding magnetic field) B0, which is supposed to be constant in space and time, the dynamical equation for the Elsaässer variables (z± = v ± b) is written as (Elsässer, 1950) @z± @t ⌥(VA · r) z± + , z⌥ · r -.

Degrees of freedom of a turbulent flow

A turbulent system is said to possess infinitely large number of degrees of freedom (references). Theoretically it can serve our purpose but in practical cases, for numerical simulations for example, we need to have an idea of that infinitely large value for a real system. More precisely, an estimation of the minimum number of grid points necessary to simulate a completely developed three dimensional turbulence is required. In the following we shall give an approximate order of that quantity using the above K41 phenomenology. From the expression of the mean energy injection rate (which is equal to the mean energy dissipation rate in a stationary state) the integral scale (l0) can be written as l0 ⇠ v3 0  » .

Energy decay in non-forced turbulence

If a turbulent flow is not forced, it will dissipate energy in time due to viscous effects. An approximate law for this decay can be derived by two following assumptions:
(1) An « infrared asymptotic self similarity » (IRSS) for velocity with a negative2 scaling exponent ↵ which means for l!1, vl ⇡ Cl↵, C being a constant.
(2) Principle of permanence of large eddies which asserts that if a freely decaying turbulent flow initially possesses IRSS, this symmetry will be preserved at later instants with same scaling exponent ↵ and constant C. Using the definition of IRSS and that of the spectral density of energy, we write for l!1, v2 l = C2l2↵ ) E(k) = C2k−2↵−1, (4.66) for k ! 0. As ↵ is negative, the above law indicates a growing energy spectrum at small scales which is not physical. It is therefore necessary to have a lower cut-off scale l0 below which the turbulence is of completely developed type and the scaling law vl ⇠ l 1 3 is obeyed instead. This l0 is called the integral scale which translates also with time. Corresponding v0 scales as v0 ⇠ Cl↵ 0. For integral scale Reynolds number R0 ⇠ l0v0 ⌫ >> 1, the mean energy dissipation rate can be estimated by d dt v2 0 ⇠ − » ⇠ − v3 0 l0 .

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Table of contents :

1 Introduction 
1.1 General interest
1.2 Turbulence in space 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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