# Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509

LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France

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Article dans Atmosphere (2020)

## Nonlinear Effects on the Precessional Instability in Magnetized Turbulence

Abdelaziz Salhi, Amor Khlifi & Claude Cambon

By means of direct numerical simulations (DNS), we study the impact of an imposed uniform magnetic field on precessing magnetohydrodynamic homogeneous turbulence with a unit magnetic Prandtl number. The base flow which can trigger the precessional instability consists of the superposition of a solid-body rotation around the vertical ($x_3$) axis (with rate $\Omega$) and a plane shear (with rate $S=2\varepsilon\Omega$) viewed in a frame rotating (with rate $\Omega_p=\varepsilon\Omega$) about an axis normal to the plane of shear and to the solid-body rotation axis and under an imposed magnetic field that aligns with the solid-body rotation axis $(\mathbf{B}\parallel \mathbf{\Omega})$. While rotation rate and Poincaré number are fixed, $\Omega=20$ and $\varepsilon=0.17$, the $\mathbf{B}$ intensity was varied, $B=0.1,0.5$ , and $2.5$, so that the Elsasser number is about $\Lambda=0.1,2.5$ and $62.5$, respectively. At the final computational dimensionless time, $St=2\varepsilon\Omega t=67$, the Rossby number $Ro$ is about $0.1$ characterizing rapidly rotating flow. It is shown that the total (kinetic + magnetic) energy ($E$) , production rate (${\mathcal P}$) due the basic flow and dissipation rate (${\mathcal D}$) occur in two main phases associated with different flow topologies : (i) an exponential growth and (ii) nonlinear saturation during which these global quantities remain almost time independent with ${\mathcal P}\sim{\mathcal D}$. The impact of a "strong" imposed magnetic field ($B=2.5$) on large scale structures at the saturation stage is reflected by the formation of structures that look like filaments and there is no dominance of horizontal motion over the vertical (along the solid-rotation axis) one.
The comparison between the spectra of kinetic energy $E^{(κ)}(k_\perp)$, $E^{(κ)}(k_\perp,k_\parallel=1,2)$ and $E^{(κ)}(k_\perp,k_\parallel=0)$ at the saturation stage reveals that, at large horizontal scales, the major contribution to $E^{(κ)}(k_\perp)$ does not come only from the mode $k_\parallel=0$ but also from the $k_\parallel=1$ mode which is the most energetic. Only at very large horizontal scales at which $E^{(κ)}(k_\perp)\sim E^{(κ)}_{2D}(k_\perp)$, the flow is almost two-dimensional. In the wavenumbers range $10\le k_\perp\le40$, the spectra $E^{(κ)}(k_\perp)$ and $E^{(κ)}(k_\perp,k_\parallel=0)$ respectively follow the scaling $k^{−2}_\perp$ and $k^{−3}_\perp$. Unlike the velocity field the magnetic field remains strongly three-dimensional for all scales since $E^{(m)}_{2D}(k_\perp)\ll E^{(m)}(k_\perp)$. At the saturation stage, the Alfvén ratio between kinetic and magnetic energies behaves like $k^{−2}_\perp$ for $Bk_\parallel/(2\varepsilon\Omega)<1$.

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