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## Numerical model for the Vlasov-Poisson system

The Vlasov-Poisson model, previously described in section 1.2, is used for the simulations described in sections 3.2 and 3.3.

The numerical scheme that solves these equations uses a partially eulerian approach. The distribution function f at a given time t is known on a space velocity grid (xi, vj). Then a Lagrangian step is used to follow f along the characteristics which at t + t ends up at (xi, vj). This characteristics started at time t at a phase space point (x⋆i , v⋆ j ) which usually is not a grid point, so that some interpolation is needed to calculate f(x⋆i , v⋆ j , t) = f(xi, vj , t + t). The numerical model uses a III order Van-Leer interpolation scheme. The code integrates the Vlasov equation by means of the so-called « splitting scheme » (i.e. it splits the Vlasov equation in several advection equations). The main points of the numerical scheme can be described as follows. It uses a fundamental property of the Vlasov equation is the Liouville theorem, which states that the phase-space distribution function is conserved along trajectories of matter elements (the characteristics curves) in phase space: f x(t), v(t), t = constant The numerical treatment for solving the Vlasov equation uses this property to advance with time the distribution function in the (x,v) phase space. The numerical scheme of the electrostatic version of the Vlasov code is based on the splitting scheme [Cheng and Knorr, 1976]. It consists in exploiting the Liouville theorem in two steps, while evolving the system during a time step t, where the space and velocity advection terms are

advanced separately: fsp(x, v) = f(x − vt/2, v, t).

### Evidence for Langmuir electrostatic decay

The electrostatic decay of Langmuir waves, described in section 1.3, is a resonant versionof Langmuir ponderomotive effects. As explained in section 2.1, three-wave coupling in general and Langmuir electrostatic decay in particular may explain the physical mechanism at the origin of radio emissions associated with Type III bursts.

I use (i) in-situ observations from the STEREO mission to show that three-wave coupling between Langmuir and ion acoustic waves indeed occur during Type III events, (ii) Vlasov simulations to compute the threshold for Langmuir electrostatic decay in typical solar wind conditions. By coupling in-situ observation and kinetic simulations, we show that the observed level of Langmuir energy matches the computed threshold for the decay process, confirming that Langmuir electrostatic decay is indeed observed during Type III bursts.

#### Low energy Langmuir cavitons, the breakdown of weak turbulence

I now concentrate on the long-time evolution3 of the Vlasov-Poisson system for moderately high initial amplitude, still corresponding to the weak turbulence domain ǫ0E2 L/nkBTe ∼ 10−3.

Though 1D-1V Vlasov-Poisson simulations, the long time evolution of Langmuir weak turbulence self-consistently generates ion cavitons. Langmuir cavitons are coherent structures in equilibrium between the total kinetic pressure force and the ponderomotive force created by the high frequency Langmuir oscillations. They are characteristic of strong Langmuir turbulence and are widely thought to be generated at high energy and to saturate when the Langmuir energy is of the order of the background plasma thermal energy. Langmuir cavitons observed for long-time evolution of Langmuir waves saturate at low energy (with an electric to kinetic energy ratio as low as three orders of magnitude), unlike the widespread belief that such structures saturate at higher energy ratios.

Electrostatic coherent structures of typical dimension much greater that a few Debye length are produced by the long time evolution of an initial relatively moderate amplitude turbulence. In particular, it gives evidence that « large » and « shallow » stable cavitons also exist, which could give new insight in the interpretation of space plasma observations of localized Langmuir waves. This result can have an important impact on the interpretation of space plasma spacecraft data and encourage the space physics community to revisit the admitted conclusion that strong turbulent Langmuir structures are formed at too high energy to be relevant in space plasma environments. This analysis and the results have been submitted [Henri et al., 2010d], the submitted version in reproduced at the end of this thesis.

Henri et al. [2010d] shows how the Langmuir electrostatic decay can also evolve toward the formation of cavitons. It illustrates the breakdown of long time evolution Langmuir weak turbulence, previously described for general wave turbulence in Biven et al. [2001].

The point there is that the system can fully evolve through weak turbulence effects, until the time scale for strong turbulence is reached. At this point the system dynamics is completely modified and govern by the evolution of spatially coherent structures, the same as the one described by strong turbulence. The main difference is that the observed cavitons saturate at much smaller energy and remain stable until the end of simulation.

**Studying wave-wave interactions with future space experiments**

Future space plasma experiments are already in preparation. Space missions such as Solar Orbiter, Solar Probe, etc, have been proposed to further investigate space plasma. What would be the ideal spacecraft for the investigations of coherent nonlinear wave-wave interactions?

First of all, three-wave couplings are resonant coherent processes. I stress that the phase coherence is the signature that enables to identify coherent wave-wave interactions. Testing the phase coherence requires that the data conserves the information on the phase of the signal. Waveform measurements are the only way to identify the phase coherence during the coupling process. It is thus mandatory to keep such measurements capabilities in future missions.

Second, to study nonlinear physics by means of observations, it is necessary to make sure that the observed nonlinearities are parts of the physical mechanisms, and have not been introduced by the instrument itself. Electronic should be free of spurious nonlinear behavior that would contaminate the data, in particular through the A/D converter. This is one of the improvements made from WIND/WAVE/TDS to STEREO/WAVE/TDS, that should be kept in mind in future similar space plasma experiments.

Wave-wave interactions couple oscillations at different frequencies. The frequencies of the different coupled waves can span over several order of frequencies. It is thus necessary to have long time series, associated with wide filters, to be able to compare the dynamics at high and low frequencies. Ideally the frequency range [1 Hz – 50 kHz] would cover signals from below the electron cyclotron frequency (∼ 100 Hz) to above the plasma frequency (∼ 10 kHz). This could enable to address the question of the possible coupling between electrostatic waves and electromagnetic waves (such as the Langmuir – whistler coupling).

**Table of contents :**

Acknowledgments

Abstract

Résumé de la thèse

Riassunto della tesi

**1 Introduction **

1.1 Studying nonlinear space plasma dynamics… What for?

1.2 Modeling collisionless space plasma: the Vlasov equation

1.3 Nonlinear evolution of Langmuir waves

**2 The complementary tools: from in-situ observations to kinetic simulations.**

2.1 The solar wind: a natural laboratory for nonlinear plasma dynamics

2.2 Observations of the electric field: STEREO/WAVES experiment

2.3 Observations of density fluctuations, a new approach

2.4 Numerical model for the Vlasov-Poisson system

2.5 Example of interpretation of in-situ observations through Vlasov simulations

**3 Results **

3.1 Observational evidence for Langmuir ponderomotive effects

3.2 Evidence for Langmuir electrostatic decay

3.3 Low energy Langmuir cavitons, the breakdown of weak turbulence

4 **Conclusions and Perspective **

**5 List of Papers **

**Appendices **

**A STEREO Mission **

**B Using the STEREO spacecraft as a density probe. **

**C Identification of waveforms of interest: application to S/WAVES **

**D Bicoherence: a powerful diagnostic for three-wave interactions **

**E Vlasov Code **

**Bibliography **