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

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, $V_{A}k,N$, 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 $V_{A}k/f\ll 1,$ 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 $V_{A}k/f\gg 1,$ 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 ($V_{A}k/N\ll 1),$ there is an equipartition of energy between magnetic and potential components, while at small scales ($V_{A}k/N\gg 1),$ 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 ($V_{A}k/\sqrt{N^2+f^2}\gg 1),$ the ratio $S_p(k,t)/S_{\kappa }\left(t\right)$ behaves like $k^{-1}$ and $S_{\kappa }(k,t)/S_m(k,t)=1.$

Figure 1

Variation of the Alfvén ratio, $S_{\kappa }(k,t)/S_m(k,t),$ in function of the scale-dependent Lehnert number ${\mathcal{L}}_k=V_{A}k/f$ in a rotating homogeneous turbulence submitted to a uniform magnetic field aligning with the rotation axis (axisymmetric case). The figure compares the LST results given by Eq. (38) and the DNS data collected from Fig. 9 in Ref. [20] displaying the radial spectrum of kinetic and magnetic energies obtained at $t^{+}=(u_0/{\ell }_0)t=\text{Ro}\left(ft\right)=5$ and three values of the Elsasser number $\mathrm{\Lambda }=0.5,2.0,8.0$ (see Table 1). The figure shows that, at $V_{A}k/f<1,$ LST is in agreement with DNS. At $V_{A}k/f>1,$ LST predicts an equipartition of energy between magnetic and kinetic components while DNS data indicate that the Alfvén ratio is smaller than one characterizing a non-Alfvénic regime.

Figure 2

Variation of the Alfvén ratio, $S_{\kappa }(k_3,t)/S_m(k_3,t),$ in function of the vertical scale-dependent Lehnert number $V_A{k}_3/f$ at $ft=100$ in a rotating homogeneous turbulence submitted to a uniform magnetic field aligning with the rotation axis (axisymmetric case). The green curve represents the numerical results for the ratio $S_{\kappa }(k_3,t)/S_m(k_3,t).$ These one-dimensional spectra in the vertical direction have been determined numerically by integrating the spectral density of energy for a diffusive fluid with $P_{\text{rm}}=\nu /\eta =1$ and $P_{\text{rt}}=\nu /\kappa =1$ using the initial spectrum $S_{\kappa }(k,0)\propto k^2{\mathrm{e}}^{-k_0^{-2}{k}^2}$ [see Eq. (61)]. As expected, the numerical results fluctuate around the black curve $\left[1+{\left(2V_A^2{k}_3^2{f}^{-2}\right)}^{-1}\right]$ (details on the derivation of the above expression are reported in Appendix pp3). It is clear that, at large vertical scales $(V_A{k}_3/f\ll 1),$ the Alfvén ratio behaves like $k_3^{-2}$ and, at small scales $(V_A{k}_3/f\gg 1),$ there is an equipartition of energy.

Figure 3

Variation of the Alfvén ratio, $S_{\kappa }(k,t)/S_m(k,t),$ in function of the scale-dependent Lehnert number ${\mathcal{L}}_k=V_{A}k/f$ in a rotating homogeneous turbulence submitted to a uniform magnetic field which is either parallel (axisymmetric case) or perpendicular (asymmetric case) to the rotation axis. The figure illustrates the fact that, in the axisymmetric case, the inertial waves affect more the dynamics of large scales (induced by the Alfvén waves in the absence of rotation) than in the asymmetric case, but in both cases the equipartition of energy is broken.

Figure 4

Case of magnetic Archimedes wave weak turbulence. Panel (a) shows, scale by scale, the long-time behavior of the energy ratios potential/kinetic and potential/magnetic. It indicates that, at large scales ($V_{A}k/N\ll 1),$ these ratio exhibit a $k^0$ form and there is an equipartition of energy between magnetic and potential components. At small scales ($V_{A}k/N\gg 1),$ these energy ratios potential/kinetic and potential/magnetic behave like $k^{-1}.$ At small scales, the gravity waves do not alter the equipartition of energy between kinetic and magnetic components induced by the linear dynamic of the Alfvén waves. Panel (b) displays the long-time behavior of the radial spectrum of kinetic, magnetic, and potential energies normalized by the initial spectrum $S_{\kappa }(k,0)$ for a diffusionless fluid. It indicates that, at small scales, the potential energy approaches zero, while both the kinetic and magnetic components approach $\frac{1}{2}{S}_{\kappa }(k,0).$ The kinetic energy is the same $(=\frac{1}{2}{S}_{\kappa }(k,0))$ at all scales.

Figure 5

Case of magnetic Archimedes Coriolis wave turbulence. Panel (a) shows, scale by scale, the long-time behavior of the energy ratios potential/kinetic and potential/magnetic. It indicates that, at large scales ($V_{A}k/\sqrt{f^2+N^2}\ll 1),$ these ratio exhibit a $k^0$ form, while at small scales ($V_{A}k/\sqrt{f^2+N^2}\gg 1),$ they behave like $k^{-1}$ as in the magnetic Archimedes weak wave turbulence. At small scales, neither the gravity waves nor the inertial waves alter the equipartition of energy between kinetic and magnetic components induced by the linear dynamic of the Alfvén waves. Panel (b) displays the the long-time behavior of the radial spectrum normalized by the initial spectrum $S_{\kappa }(k,0)$ at large times for a diffusionless fluid. It indicates that, at small scales, the potential energy approaches zero, while both the kinetic and magnetic components approach $\frac{1}{2}{S}_{\kappa }(k,0).$

Figure 6

Time history of kinetic and magnetic energies normalized by the initial kinetic energy in a rotating homogeneous turbulence submitted to a uniform magnetic field aligning with the rotation axis (axisymmetric case) for several values of the Elsasser number, $\mathrm{\Lambda }=0.5,2.0,8.0,\infty$ (nonrotating case). Top panel: LST results. Bottom panel: DNS results by Favier et al. [20].

Figure 7

Time history of the Alfvén ratio $E_{\kappa }\left(t^{+}\right)/E_m\left(t^{+}\right)$ in a rotating homogeneous turbulence submitted to a uniform magnetic field aligning with the rotation axis (axisymmetric case) for several values of the Elsasser number, $\mathrm{\Lambda }=0.5,2.0,8.0,\infty$ (nonrotating case). Top panel: LST results. Bottom panel: DNS results by Favier et al. [20].