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Accueil > A propos du LPP > Communication > Actualités archivées > 2021 > A new 2D cascade model for plasma and atmospheric turbulence

A new 2D cascade model for plasma and atmospheric turbulence

As Ionised gases, plasmas are the seat of numerous instabilities and turbulence, which for the latter, as in gases, stems from the non-linear structure of the equations describing their dynamics. Consequently, to be correctly described, turbulence needs to take into account a wide range of different spatial scales, and therefore requires the help of powerful supercomputers. In parallel to these high-performance simulations, more frugal, reduced approaches are proving to be valuable alternatives.

The development of reduced turbulence models is a common know-how of the fusion plasma and space plasma teams, organised around the “axe transverse turbulence“ of the LPP. The differential approximation is one of these models, which makes it possible to describe energy transfers between turbulent scales by means of a flow term in Fourier space.

The original work presented here consisted, based on the analogy between the Hasegawa-Wakatani equations specific to plasmas on the one hand, and the passive scalar turbulence equations on the other, in proposing a general form of the energy flow terms, thus applicable to various contexts. The robustness of the obtained forms for these flows lies in the fact that they can be derived rigorously from shell models, but also inferred by simple symmetry and dimensional arguments.

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À gauche : spectre en vecteur d’onde de l’énergie cinétique, de forme « Nastrom Gage ». À droite : spectre en vecteur d’onde de l’énergie potentielle associée au scalaire passif, pour plusieurs localisations spectrales de l’injection associée.

As an illustration of the model’s capabilities, numerical simulations of the Navier-Stokes system of equations with a passive scalar have been performed. By injecting kinetic energy at both large and small scales, it can be seen in figure a) that a dual-slope spectrum is obtained, similar to the measurements made by Nastrom and Gage in the Earth’s atmosphere. Two cascades are at work : one, direct, transfers energy from the large scales to the smaller ones, and from the very small scales to the larger ones there is a reverse cascade of energy, which finally meet in the middle of the spectral range of the simulation. This result was already obtained by Leith in the late 1960s, but the originality of our model is that it can also deal with the spectral distribution of the potential energy of the passive scalar. For different scales of injection of the potential energy, we observe in figure b) that the energy of the passive scalar follows different slopes : Batchelor spectrum in k-1 in the case of a large-scale injection and a direct cascade of potential energy, or Corrsin-Obukhov spectrum in k-5/3 in the case of a small-scale injection, without cascade.

This first application of the model suggests many others : Hasegawa-Wakatani turbulence, where the energy injection mechanisms take place through plasma instabilities, reduced MHD turbulence useful for describing the formation of magnetic islands in tokamaks, and even gyrokinetic turbulence where the main non-linear term has the same structure as the terms modelled here.

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CNRS Ecole Polytechnique Sorbonne Université Université Paris-Saclay Observatoire de Paris
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Hébergeur : Laboratoire de Physique des Plasmas, Ecole Polytechnique route de Saclay F-91128 PALAISEAU CEDEX
Directeur de la publication : Anne Bourdon (Directrice)