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Floquet Linear Analysis of Spectral MHD (review)
Our objective is to study large scale effects that can happen for two linear MHD phenomena : (i) the induction equation presented in eq.(4.19) and (ii) the linearized Navier-Stokes equation presented in equation eq.(4.20). The induction equation describes the evolution of the magnetic field b in a conductive fluid of velocity u with a magnetic diffusivity @tb = r×(u×b) + b = Lmb and r· b = 0 . (4.19).
The linearized Navier-Stokes equation describes the evolution of the velocity perturbation v in a global flow of velocity U with a kinematic viscosity @tv = U×r×v + v×r×U −rPv + v = LNSv and r· v = 0 . (4.20).
The linearity of both equations is highlighted with the linear operators Lm and LNS. These operators are spatial differential operators but are time-independent. The analogy between large scale effects of the MHD phenomenon and the Bloch wave is quite straightforward. In the case of [0; 2]3-periodic geometries, studying the large scale effect of a flow requires to connect several periodic boxes and enforces a global periodicity to group of boxes. In that sense, large scale MHD systems in periodic box have two levels of periodicity just like Bloch wave-system. The first level of periodicity is due to the presence of the same flow in every cell and the second level of periodicity is related to the periodic boundary condition applied to global system. For the induction equation, the magnetic field b and the velocity field u are the Bloch analogues to the wave-function and to the atomic potential V respectively. For the linearized Navier-Stokes equation, the perturbation v and the flow U are the Bloch analogues to the wave-function and to the atomic potential V respectively. Applying a Floquet analysis to the linear MHD equation is however different from the Bloch method used to study crystalline lattices. Unlike Bloch waves, MHD problems cannot be solved in the stationary case with the eigen-states of the Hamiltonian. In addition, the normalization condition on the modulus square of the wave-function does not apply to MHD problem, thus the fields are not bounded. These two elements imply that MHD problems have growth properties that can be related to the standard Floquet method. Because the MHD problems considered are linear, solutions can be decomposed in a basis of eigen-states of the linear operator Lm and LNS.
Turbulent equilateral ABC flows
As discussed in the introduction, the driving flow does not need to be laminar to use Floquet theory. It is only required to obey the 2`-periodicity. It is worth thus considering large scale instabilities in a turbulent ABC flow that satisfies the forcing periodicity. This amounts to the turbulent flow forced by an ABC forcing in a periodic cube of the size of the forcing period 2`. Due to the stationarity of the laminar ABC flow, it can be excluded as possible candidate for an AKA-instability. However, this is not true of a turbulent ABC flow since it evolves in time. We cannot thus a priori infer that a turbulent ABC flow results in an AKA-instability or not. To test this possibility, we consider the linear evolution of the large scale perturbations v driven by an equilateral ABC flow above Re = 50, that is beyond the onset of the small scale instability Rec S ‘ 13. The turbulent equilateral ABC flow U is obtained solving the Navier-Stokes eq. (5.3) in the domain (2`)3 driven by the forcing function FABC = UABC. The code is executed until the flow reaches saturation. The evolution of the large scale perturbations is then examined solving eq. (5.8)-(5.9) with the FLASHy code coupled to the Navier-Stokes eq. (5.3). The kinetic energy EU of the turbulent equilateral ABC flow U is shown in fig. 5.15. The energy EU strongly fluctuates around a mean value. The evolution of the energy.
Non-linear calculations and bifurcation diagram
We further pursue our investigation of large scale instabilities by examining the nonlinear behavior of the flow close to the instability onset. We restrict ourselves to the case of the equilateral ABC flow whose non-linear behavior has been extensively studied in the absence however of scale separation [15]. The linear stability of the ABC flow in the minimum domain size has been studied in [48] and more recently in [51]. These studies have shown that the ABC flow destabilizes at Rec S ‘ 13.
To investigate the non-linear behavior of the flow in the presence of scale separation, we perform a series of DNS of the forced Navier-Stokes equation (eq. (5.1)) in triple periodic cubic boxes of size 2L using the GHOST code [52, 53]. The forcing maintaining the flow is FABC = p2 p3|K|2UABC so that the laminar solution of the flow is the ABC flow [15] normalized to have unit energy. Four different box-sizes are considered: KL = 1, 5, 10 and 20. For each box size and for each value of Re, the flow is initialized with random initial conditions and evolves until a steady state is reached. Fig. 5.18 shows the saturation level of the total energy EV at steady state as a function of Re for the four different values of KL. At low Reynolds number, the laminar solution V = UABC is the only attractor and so the energy is EV = 1. At the onset of the instability the total energy decreases. A striking difference appears between the KL = 1 case and the other three cases. For the KL = 1 case the first instability appears at Rec S ‘ 13 in agreement with the previous work [48, 51]. By definition, only small scale instabilities are present in the KL = 1 case (i.e. instabilities that do not break the forcing periodicity). For the other three cases, which allow the presence of modes of larger scale than the forcing scale, the flow becomes unstable at a much smaller value: Rec ‘ 3. This value of Rec is in agreement with the results obtained in section 5.3.3 for large scale instability by a negative eddy-viscosity mechanism. The energy curves for the forcing modes KL 5 all collapse on the same curve. This indicates that not only the growth rate but also the saturation mechanism for these three simulations are similar.
Further insight on the saturation mechanism can be obtained by looking at the energy spectra. Fig. 5.19 shows the energy spectrum of the velocity field at the steady state of the simulations. Two types of spectra are plotted. In fig. 5.19, spectra plotted using lines and denoted as k-bin display energy spectrum collected in bins where modes k satisfy n1 −1/2 < |k|L n1 +1/2, with n1 a positive integer. E(k) then represents the energy in the bin n1 = k. In fig. 5.19, spectra plotted using red dots and denoted by k2-bin display the energy spectrum collected in bins where modes k satisfy |k|2L2 = n2, with n2 a positive integer. Since kL is a vector with integer components mx, my and mz, its norm k2L2 = m2 x+m2y +m2z is also a positive integer. E(k) then represents the energy in the bin n2 = k2L2. This type of spectrum provides more precise information about the energy distribution among modes. In our case, they help separate K modes from K ± 1/L modes and highlight the three-mode interaction. The k = K ± 1/L modes as well as the largest scale mode kL = 1 that were used in the three-mode model are shown by blue circles in the spectra. The drawback of k2-bin spectra is their memory consumption. They have a number of bins equal to the square of the numbeof bins of standard k-bin spectra. However, since spectra are not outputted at every time-step, this inconvenience is limited.
Absolute equilibrium theory (review)
Even though the total energy of a spectrally-truncated ideal fluid is conserved and has an expression close to the energy of an ideal gas, ideal fluids do not have a Hamiltonian formulation close to that of ideal gas [91, 92]. Instead of describing the system in the position-impulsion phase space, the system will be expressed as a function of its positive and negative Fourier helical components given in eq. (2.19). These components do not following the equations of Hamiltonian given in eq. (7.8). However using these variables simplifies the expression of the helicity which is the other conserved quantity in an ideal fluid (see eq. (2.22)). In order to have consistency between the expression of energy and helicity, they will be written as 2E = E = E+ + E− where E± = X k |u±|2 , (7.11) H = H+ + H− where H± = ± X k k|u±|2 .
Table of contents :
A Preamble
0 Présentation
0.1 Cadre d’étude
0.2 Résultats sur les instabilités grande échelle
0.3 Résultats sur les temps de corrélation
1 Introduction
2 Elements of context: Hydrodynamics (review)
2.1 Ideal fluids (review)
2.2 Magnetic equivalences (review)
2.3 Conserved quantities (review)
B Large scale instabilities
3 Elements of context: Large scale effects (new model in sec. 3.3)
3.1 Alpha-effect (review)
3.2 AKA-effect (review)
3.3 Distribution of energy (description of a new model)
4 Elements of context: Floquet analysis (review)
4.1 Floquet theory (review)
4.2 Bloch theory (review)
4.3 Floquet Linear Analysis of Spectral MHD (review)
5 Large scale instabilities of helical flows (published in PRF)
5.1 Introduction
5.2 Methods (description of new procedures)
5.3 Results (new results)
5.4 Conclusion
5.5 Appendix: FLASHy (description of a new procedure)
6 Fate of alpha-dynamos at large Rm (published in PRL)
6.1 Introduction
6.2 Results (new results)
6.3 Discussion
C Thermalized state
7 Elements of context: Thermodynamics (review)
7.1 Ideal gas distribution (review)
7.2 The Liouville theorem (review)
7.3 Absolute equilibrium theory (review)
8 Elements of context: Time correlation (new model in sec. 8.2 and 8.3)
8.1 Definition and examples (review)
8.2 Hydrodynamic application (description of a new model)
8.3 Spatio-temporal measurements (description of a new procedure)
9 Large scale correlation time (submitted)
9.1 Introduction
9.2 Results (new results)
9.3 Conclusion
9.4 Appendix (description of new procedures)
D Conclusion
E Numerical methods
10 Elements of context: Numeric methods (review)
10.1 Pseudo-spectral methods (review)
10.2 Semi-Lagrangian methods (review)
11 High-order semi-Lagrangian schemes (published in IJNMF)
11.1 Introduction
11.2 Method (description of new procedures)
11.3 Conclusion
11.4 Appendix (description of new procedures)
12 Conservative semi-Lagrangian schemes (submitted)
12.1 Introduction
12.2 Method (description of new procedure)
12.3 Perspectives
12.4 Appendix (convergence study)
F References
Acknowledgments
List of Figures
Bibliography
