A qualitative picture of turbulence in every day life is given by random patterns (usually eddies) forming in the wake of a solid obstacle facing a flow, or air turbulence forming for instance above a tea cup, as in fig. 3.1(Left). Turbulence can be found in a variety of contexts, such as geophysical flows (), gravitational waves (), interstellar medium (), or the Solar Wind (). Here, we shall concentrate on fluid turbulence, with fluids that can be described either by Navier-Stokes or MHD equations.
The simplest case is that of homogeneous and isotropic fluid turbulence, that is, with nothing destroying a priori homogeneity and isotropy of space. It shows two main properties (i) a quasi-constant flux of energy (or related quantity) flowing in general from large to small scales (in that case one speaks of direct cascade); (ii) a systematic randomness which prevents one to exhibit any analytic solution of a turbulent system and asks for a statistical description of the phenomenon, a situation analogous to that of thermodynamical equilibrium for instance.
Shock formation as a simplified model of turbulence
The flux of energy from large to small scales is well illustrated by a related phenomenon: shock formation. It is illustrated in fig.3.1(right) by a numerical solution of the 1D compressible Navier-stokes equations,
@t + @x(u) = 0 (3.1).
@t(u) + @x(u2) + @xP = @xxu (3.2).
@tP + ux@xP + P@xux = @xxT (3.3).
P = T (3.4).
Panel (a) shows how the initial smooth (cosine) profile evolves finally into a shock with a large gradient in the center of the figure. Panel (b) shows the growth of harmonics of the successive Fourier modes with time. Panel (c) shows the corresponding evolution of the energy spectrum with time. Panel (d) shows finally the kinetic energy evolution: the energy in the first Fourier mode first decreases (solid line), followed by the decrease of the overall energy content (dotted line).
3D hydrodynamic turbulence
In the previous 1D shock formation, the eddies visible in fig. 3.1(Left) are lacking completely, since only 1D movements are possible. In this case, the random character of turbulent evolution is absent, the evolution is deterministic. As a consequence, another property of turbulence which we explain now is lacking: the locality of the kinetic energy flux which allows to predict how energy is distributed among scales.
Assume indeed that we deal now with incompressible 3D hydrodynamic equations. The equations are similar to the incompressible 3D MHD equations, but with no magnetic field and with the pressure equation replaced by a divergenceless condition on the velocity field: @tu + u ru + rP = u (3.5) r u = 0 (3.6).
The divergenceless (incompressibility) condition is obtained from the compressible equations (see MHD eqs.2.1 with B = 0) by taking the limit of infinite sound speed (or very subsonic flow). The situation is not very different from the previous 1D compressible Navier- Stokes equations (3.4), except for two things: (i) the incompressibility conditions implies that the total kinetic energy of the system is conserved by nonlinear couplings, while in the 1D compressible case this conservation was only approximately satisfied; (ii) the 3D configuration in practice transforms completely the evolution by making it chaotic: the precise spatial and temporal distribution of large gradients, contrary to the shock case seen previously, is unpredictable. A consequence is that no analytic (or even semi-analytic) solution of the problem exists presently. This implies in particular that in 3D turbulence, no simple typical profile can be produced such as the one shown for a 1D shock at time t = 2 in fig. 3.1a.
3D MHD turbulence
We again consider here the incompressible limit of the equations, as it is only in this limit that a turbulent phenomenology is easily built. In this limit, one can prove that there are three inviscid invariants (by this we mean a quantity conserved by nonlinear couplings, in the limit of zero viscosity): (i) “total” energy, that is, the sum of kinetic and magnetic energy (in Alfvén speed units) Ev = u2=2 and Eb = B2=(2); (ii) cross helicity v:B= p ; (iii) magnetic helicity a:B where a is the magnetic potential (B = r a). Let us define the fluctuating magnetic field as B = B B0, where B0 is the mean magnetic field. The standard view is that the turbulent spectrum, in the MHD case, is formed as in the hydrodynamic case by the cascade of the total energy. Thus, applying the Kolmogorov phenomenology to the total energy, i.e., with E(k) replacing the kinetic energy spectrum, we obtain again E(k) / k5=3. More precisely, the total energy flux between scales reads.
Turbulence with large scale radial flow
Imposed large scale flows or magnetic fields can lead to new or modified turbulent regimes. An example in hydrodynamics is the case of rigid rotation which modifies both the scaling (the spectrum steepens) and anisotropy (the cascade is mainly active in the Fourier plane perpendicular to the rotation axis), and in MHD with mean field as seen in subsection 3.1.3. In general, large scale gradients are supposed to play the role of an energy reservoir and/or help to trigger the onset of the energy cascade. In the solar wind context, for instance, the shear between fast and slow streams has been considered in this way, and, also, to trigger the decrease of cross-helicity with heliocentric distance ().
Table of contents :
1 Solar Wind
1.1 Sources of the wind and solar cycle
1.1.1 Fast and slow winds
1.2 From corona to Earth
2 Plasma description
2.1 The MHD equations
2.2 MHD equations with large scale radial flow (EBM)
3.1 Homogeneous turbulence
3.1.1 Shock formation as a simplified model of turbulence
3.1.2 3D hydrodynamic turbulence
Fourier space description: K41 phenomenology
3.1.3 3D MHD turbulence
Incompressible MHD using Elsasser variables, cross-helicity
Anisotropy of the cascade with mean magnetic field B0
Local and nonlocal interactions
3.1.4 Spectra, autocorrelations and structure functions
3.2 Turbulence with large scale radial flow
3.2.1 Inhibition of the turbulent cascade by expansion
3.2.2 Fluctuations decay: strong versus weak expansion
3.2.3 Cascade rate and turbulent heating
4 Solar Wind turbulence
4.1 In situ observations
4.2 MHD inertial range
4.3 Spectral anisotropy
4.3.1 The Maltese Cross
4.4 Proton temperature gradient and turbulent heating
4.4.1 Turbulent amplitude and proton temperature variations in the inner heliosphere
Temperature gradient: a long series of studies
Possible explanations of the slow proton cooling in the inner heliosphere
4.4.2 Measuring turbulent heating via second order moments
4.4.3 Measuring turbulent heating via third order moments
5 Plan of this thesis
5.1 Spectral anisotropy: understanding the Maltese Cross
5.2 Turbulent heating in slow and fast winds
II The Maltese Cross revisited
6 Parameters and initial conditions
6.1 Physical parameters and initial spectra
6.1.1 Expansion parameter, cross helicity, Mach number
6.1.2 Initial spectra
6.1.3 Domain aspect ratio
7 Defining spectral properties in EBM simulations
7.1 Anisotropy of 3D spectra
7.2 1D spectral slopes
8.1 Initially ISO spectrum: expansion and cross helicity effects
8.1.1 Varying expansion at zero cross-helicity
8.1.2 Varying expansion at large cross-helicity
8.2 Systematic comparison of zero vs large cross helicity
8.2.3 Gyro-Alfvén model
9.2 Explaining the discrepancy with VG16
III Can the Maltese Cross heat?
10 1D turbulent heating
10.1 1D HD equations with expansion
10.1.1 Modified Burgers equation and semi-analytical solutions
10.2 Simulations of shock turbulence with transverse waves
10.2.1 Initial conditions
11 Paper ApJ 2018: « Turbulent Heating between 0.2 and 1 au: A Numerical Study »
12 Heating fast winds
12.1 Initial conditions
12.2.1 Spectral anisotropy with Mach=1
12.2.2 Turbulent heating
Numerical parameters vs Helios data
IV Conclusions and future work
14.1 Obtention of Maltese Cross components at 1AU
14.2 Can the Maltese Cross heat the solar wind?
14.3 Expectations from Solar Orbiter and Parker Solar Probe
14.4 Open questions and (partial) answers
15 Future work: Anisotropy temperature description
A Liste of symbols
B Numerical method
B.1 Computation of spatial gradients
B.2 Time integration method
C Résumé en français