Understanding plasma turbulence through DNS data analysis

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Description of the compressible Hall MHD model

In the previous section I presented for IHMHD the derivation of a new exact law with a typical Yaglom-like expression thanks to the use of a second-order structure function, dierent from the correlator used for past laws, to initiate the calculation. The same can be done for all other compressible models, however deriving an exact law while taking compressibility into account is a much more time-consuming and tedious operation than for an incompressible model, be it an HD or MHD one. Thus, I will only present the derivation of the law for the most complete model of (isothermal) CHMHD and use it as a basis to infer the laws in Compressible Magnetohydrodynamics (CMHD) and CHD (see Simon and Sahraoui (2021) for a more general derivation for isentropic ows that encompasses polytropic and isothermal closures, and for the related discussion about the thermodynamics behind exact laws). To describe this model we consider the three-dimensional isothermal CHMHD equations (Galtier, 2016) that were presented in chapter 1, and that we slightly rewrite as: @t +r (u) =0 ;

Limit cases and alternative compressible laws

From F21 one can retrieve other equations for various simpler models. We mentioned earlier strong similitudes between F21 and the incompressible laws. It is possible to retrieve F19 by taking the incompressible limit ! 0 and renormalizing (2.100) by the density, which leads to considering the normalized variables  » = « =0, b = B=p00 and the ion skin depth di = =p00. It is immediate to see that, under this limit, FMHD ! Y and SMHD ! 0 (as ! 0). The case of the two Hall terms is less straightforward as neither tends to zero. However, we do nd that the sum of the ux and source is: r` FHall + SHall = r` H􀀀 2A = 1 2r` H; (2.106) as was demonstrated in section 2.2. With this we indeed retrieve equation (2.39): 􀀀4 » = r` (Y + 1 2 H): (2.107).
While the previous limit merely serves as a validation of F21 with regard to F19, new results can also be obtained by considering other limits. Indeed, one can consider the CMHD limit of equation (2.100) by taking di=L ! 0. This is simply akin to removing the Hall components of the law, which yields.

Axisymmetry hypothesis: 3D increments

For simulations where a strong enough background magnetic eld is present the cascading modes are expected to be located predominantly in the plane orthogonal to B0. Consequently, we will usually prefer to the isotropy assumption a symmetry of revolution around B0 (usually chosen to be the z-axis). In this case we adopt cylindrical coordinates: the increment vector is dened as ` = (`?; ; `z) and the derivative operator becomes r` hFi = 1 `? @`? [`? hF`?i (`?; `z)]+@`z hF`z i (`?; `z). However, the discrete choice of directions adopted here makes it impossible to eectively compute @`z hF`z i (`?; `z) at arbitrary values of (`?; `z) without resorting to multi-dimensional interpolation on irregular grids, bringing lots of additional calculations and more imprecision to the results. Consequently, we will only study transfers in transverse directions by considering the ux averaged in the z direction, i.e. the average of the uxes calculated for increments with a xed `? and a variable `z (see gure 3.2). The derivative operator is thus reduced to: r` hFi = 1 `? @`? [`? hF`?i;`z (`?)].
We compute the projection of F on the direction of `? as: hF`?i (`?; ; `z) = hcos()Fx + sin()Fyi .

Parallelization paradigm

The rst implementations of the program required around an hour to compute the CMHD exact law components on a small 2563 dataset with only the 2D increments. Considering CHMHD would take even more time due to the presence of more terms to compute and that larger data cubes would have to be studied after, the need to optimize the code to reduce the execution time rapidly arose. The rst attempts at a « naive » parallelization, consisting on parallelizing the directions loop (and so computing the terms for several directions simultaneously) only managed to cut the execution time in half at best, and so did not yield satisfying results.
The method I nally chose to implement consists in a Message Passing Interface (MPI) parallelization that divides the data itself and distributes it evenly between all processors. The idea is to write the code as a function of the rank of the processors (i.e. its ID number) so that each processor interprets the instructions dierently and only executes a part of the total work. However, when working with MPI, each processor possesses its own memory, that is not shared with the others. This requires sending the relevant data to each processor so they can proceed with the calculations, and regrouping the nal results at the end of the execution. Of course, one cannot simply send all the data used to each processor as the memory usage will be multiplied by the total number of processors (which will most certainly trigger a memory error). Thus, a bit of memory management is required to do things right. First, a single processor opens the base data and computes all the additional arrays. For each of these arrays, the base processor sends only a slice of the data (usually cut along the mean magnetic eld axis, by default the z-axis) to every other processor (see gure 3.4). This ensures that, at any given time, the memory allocated to store n variables corresponds to at most (n+1) data cubes (n cubes for the variables and an additional empty one scattered between processors destined to receive the data sent). The initial data cubes are ultimately deleted so that the base processor too only keeps the slice it needs.

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Table of contents :

Contents
1 Introduction 
1.1 Theory of collisionless plasmas
1.1.1 What is a plasma ?
1.1.2 Kinetic and uid description of a plasma
1.1.3 The equations of MHD
1.2 From space to astrophysical plasmas
1.2.1 The Solar Wind: a turbulence laboratory
1.2.2 The Interstellar Medium and star-forming regions
1.3 Theory of turbulence
1.3.1 Basics of turbulence
1.3.2 The zeroth law of turbulence
1.3.3 Spectral approach of turbulence
1.4 Overview of the thesis
2 Exact laws in Hall MHD turbulence 
2.1 Introduction
2.2 Exact law in the incompressible Hall MHD model
2.2.1 Description of the model
2.2.2 Calculation of the dynamical equation
2.2.3 From the dynamical equation to the exact law
2.2.4 Equivalence of IHMHD exact laws
2.3 Exact law in compressible models
2.3.1 Description of the compressible Hall MHD model
2.3.2 Calculation of a compressible dynamical equation
2.3.3 From the dynamical equation to the exact law
2.3.4 Limit cases and alternative compressible laws
2.4 Conclusion
3 Methodology: laws computation and numerical implementation 
3.1 Introduction
3.2 Numerical calculation of exact laws
3.2.1 Generic framework
3.2.2 Isotropy hypothesis: 3D increments
3.2.3 Axisymmetry hypothesis: 3D increments
3.2.4 Axisymmetry hypothesis: 2D increments
3.3 Implementation of the numerical methods
3.3.1 Main steps of the calculation
3.3.2 Parallelization paradigm
3.3.3 Memory constraints
3.4 Conclusion
4 Understanding plasma turbulence through DNS data analysis 
4.1 Introduction
4.2 Numerical test of IHMHD exact laws
4.2.1 Convergence of exact laws and model choice
4.2.2 Inuence of the mean magnetic eld
4.2.3 Summary of the study on the IHMHD laws
4.3 In-depth study with generalized CHMHD exact laws
4.3.1 Context of the study
4.3.2 Presentation of the data
4.3.3 Calculation of law F21
4.3.4 Non-stationary CHMHD laws
4.3.5 Summary of the GHOST study
4.4 Application to Landau-uid simulations
4.4.1 Context of the study
4.4.2 Theory of CGL and LF models
4.4.3 Description of the numerical code
4.4.4 Evaluation of the exact law and dissipations
4.4.5 Methods of calculation of the hyperdissipation
4.4.6 Energy cascade in presence of Landau damping
4.4.7 Calculation of the heating due to Landau damping
4.4.8 Summary: relations between uid and kinetic models
4.5 Supersonic CHD turbulence
4.5.1 Context of the study
4.5.2 Framework and presentation of the data
4.5.3 Calculation of the energy cascade rate
4.5.4 Filamentary structures and locality of turbulence
4.5.5 Insights into a two-regimes turbulence
4.5.6 Summary and link to the ISM
4.6 Conclusion
5 Application to Virtual spacecraft and MMS data in Earth’s magnetosheath 
5.1 Introduction
5.2 Working with in situ data
5.2.1 Space-time ambiguity and Taylor hypothesis
5.2.2 Calculation of 3D divergences: the curlometer / gradiometer
5.3 MMS data analysis
5.3.1 Context and presentation of the data
5.3.2 Selection of the time intervals
5.3.3 Results
5.4 Gradiometer: evaluation and limitations of the method
5.4.1 Quality factor of the gradiometer
5.4.2 Causes of error of the gradiometer
5.5 Simulated MMS y-by
5.5.1 Description of the method
5.5.2 General precision of the gradiometer
5.5.3 Reaction of the gradiometer to non-linearity
5.6 Conclusion
6 Conclusions and perspectives 
6.1 Summary of the thesis
6.2 Tools for turbulence analysis
6.2.1 Derivation of exact laws
6.2.2 Numerical implementation of the exact laws
6.3 Studying turbulent ows through DNSs
6.3.1 In-depth analysis of exact laws
6.3.2 Relation between the uid cascade and kinetic dissipation
6.3.3 Supersonic CHD turbulence in the ISM
6.4 MMS in situ data analysis
6.4.1 Selection and exploitation of the data
6.4.2 Investigation of the gradiometer
6.5 Final words and perspectives
A List of Acronyms 
B Papers published and in preparation 

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