# 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 Phys. Rev. E (2017)

## Energy partition, scale by scale, in magnetic Archimedes Coriolis weak wave turbulence

A. Salhi, F. S. Baklouti, F. Godeferd, T. Lehner, and C. Cambon

Magnetic Archimedes Coriolis (MAC) waves are omnipresent in several geophysical and astrophysical flows such as the solar tachocline. In the present study, we use linear spectral theory (LST) and investigate the energy partition, scale by scale, in MAC weak wave turbulence for a Boussinesq fluid. At the scale $k^{−1}$, the maximal frequencies of magnetic (Alfvén) waves, gravity (Archimedes) waves, and inertial (Coriolis) waves are, respectively, $V_Ak$, $N$, and $f$. By using the induction potential scalar, which is a Lagrangian invariant for a diffusionless Boussinesq fluid [Salhi et al., Phys. Rev. E 85, 026301 (2012)], we derive a dispersion relation for the three-dimensional MAC waves, generalizing previous ones including that of $f$-plane MHD “shallow water” waves [Schecter et al., Astrophys. J. 551, L185 (2001)]. A solution for the Fourier amplitude of perturbation fields (velocity, magnetic field, and density) is derived analytically considering a diffusive fluid for which both the magnetic and thermal Prandtl numbers are one. The radial spectrum of kinetic, $S_\kappa(k,t)$, magnetic, $S_m(k,t)$, and potential, $S_p(k,t)$, energies is determined considering initial isotropic conditions. For magnetic Coriolis (MC) weak wave turbulence, it is shown that, at large scales such that $V_Ak/f\ll1$, the Alfvén ratio $S_\kappa(k,t)/S_m(k,t)$ behaves like $k^{−2}$ if the rotation axis is aligned with the magnetic field, in agreement with previous direct numerical simulations [Favier et al., Geophys. Astrophys. Fluid Dyn. (2012)] and like $k^{−1}$ if the rotation axis is perpendicular to the magnetic field. At small scales, such that $V_Ak/f\gg1$, there is an equipartition of energy between magnetic and kinetic components. For magnetic Archimedes weak wave turbulence, it is demonstrated that, at large scales, such that ($V_Ak/N\ll1$), there is an equipartition of energy between magnetic and potential components, while at small scales ($V_Ak/N\gg1$), the ratio $S_p(k,t)/S_\kappa(k,t)$ behaves like $k^{−1}$ and $S_\kappa(k,t)/S_m(k,t)=1$. Also, for MAC weak wave turbulence, it is shown that, at small scales ($V_Ak/\sqrt{N^2+f^2}\gg1$), the ratio $S_p(k,t)/S_\kappa(t)$ behaves like $k^{−1}$ and $S_\kappa(k,t)/S_m(k,t)=1$.