Inverse cascade of hybrid helicity in $B\mathrm{\Omega }$-MHD turbulence

We investigate the impact of a solid-body rotation ${\mathbf{\Omega }}_0$ on the large-scale dynamics of an incompressible magnetohydrodynamic turbulent flow in presence of a background magnetic field ${\mathbf{B}}_{\mathbf{0}}$ and at low Rossby number. Three-dimensional direct numerical simulations are performed in a periodic box, at unit magnetic Prandtl number and with a forcing at intermediate wave number $k_f=20$. When ${\mathbf{\Omega }}_0$ is aligned with ${\mathbf{B}}_{\mathbf{0}}$ (i.e., $\theta \equiv \widehat{\left({\mathbf{\Omega }}_0,{\mathbf{B}}_{\mathbf{0}}\right)}=0$), inverse transfer is found for the magnetic spectrum at $k<k_f$. This transfer is stronger when the forcing excites preferentially right-handed (rather than left-handed) fluctuations; it is smaller when $\theta >0$ and becomes weak when $\theta \ge {35}^{\circ }$. These properties are understood as the consequence of an inverse cascade of hybrid helicity which is an inviscid/ideal invariant of this system when $\theta =0$. Hybrid helicity emerges, therefore, as a key element for understanding rotating dynamos. Implication of these findings on the origin of the alignment of the magnetic dipole with the rotation axis in planets and stars is discussed.

Figure 1

Temporal evolution of the magnetic spectrum for simulations 20R (solid line) and 20L (dashed line). The parameters of these simulations can be found in Table 1. Colors indicate the approximate times at which the spectra have been computed. Note that simulation 20R is stopped before the formation of a condensate. Insert: spectrum of case 20R at $t\sim 62$ compared with two reference cases: without rotation (case 0R) and without magnetic field (case 20R_B0).

Figure 2

Magnetic spectra at $t\sim 69$ for simulations 20R (solid line) and 20L (dashed line) decomposed into L ($E_b^L$) and R ($E_b^R$) fluctuations. This decomposition highlights the relative importance of each type of fluctuations in a single simulation.

Figure 3

Spectral flux (in logarithmic-linear coordinates) associated with the four different nonlinear terms of $B\mathrm{\Omega }$-MHD (simulation 20R) for three typical times. The main contribution exhibiting a significant negative flux at large scales ($k<k_f$) comes from ${\mathrm{\Pi }}_{bu}^b$.

Figure 4

Magnetic spectra for situations where the angles $\theta =0^{\circ },{25}^{\circ },{35}^{\circ },{45}^{\circ }$, and ${90}^{\circ }$ (simulations $20{\mathrm{R}}_0$ to $20{\mathrm{R}}_{90}$). The spectra are plotted approximately at the same time $t\sim 105$. Hypoviscosity (${\nu }^{-}$) has been added to avoid a condensation at small $k$. At large scales, a similar shape is observed for $\theta \le {35}^{\circ }$ whereas the behavior seems different when $\theta \ge {45}^{\circ }$.

Figure 5

Time evolution (in linear-logarithmic scales) of the normalized large-scale dissipation rate of kinetic energy ${\epsilon }_u^{-}$ (dashed lines) and magnetic energy ${\epsilon }_b^{-}$ (solid lines) for angles $\theta =0^{\circ },{25}^{\circ },{35}^{\circ },{45}^{\circ }$, and ${90}^{\circ }$. ${\epsilon }^{\text{tot}}\equiv {\epsilon }^{+}+{\epsilon }^{-}$ is the total dissipation rate. For $\theta =0^{\circ }$ the large-scale dissipation rate is dominated by the magnetic contribution, while for $\theta ={90}^{\circ }$ the contribution of each field is similar.

Figure 6

Amplitude of the current density for case $20R_0$ at $t\sim 105$. The magnetic field is along $\mathbf{z}$, corresponding to the vertical axis on this representation. Despite strong rotation, no strong anisotropy is observed.

Figure 7

Time evolution (in linear-logarithmic scales) of the hybrid helicity $H_h$ for the five simulations with different values of $\theta$. Note that the particular behaviors of case $20R_{35}$ and $20R_{45}$ which cross around $t=90$ seem to have different origins (see Fig. 4). As expected, $H_h$ is much smaller for the orthogonal configuration $20R_{90}$.

Figure 8

Spectral fluxes (in logarithmic-linear scales) of the normalized magnetic helicity (top left) divided by $d$, cross-correlation (top right) and hybrid helicity (bottom) for simulation $20{\mathrm{R}}_0$ at three different times. The hybrid helicity and the magnetic helicity exhibit a plateau at large scale which widens with time.

Figure 9

Spectral fluxes (in logarithmic-linear scales) associated with the four different nonlinear terms of $B\mathrm{\Omega }$-MHD for several angles (simulation $20R_0$ to $20R_{90}$) and at time $t\sim 105$. For the flux ${\mathrm{\Pi }}_{bu}^b$, the difference between $\theta =0^{\circ }$ and the other angles is undeniable.

Figure 10

Time evolution of the magnetic helicity spectrum for simulation $20R_0$. The spectra are compared with a power law in $k^{-5}$ (segment). Note that the spectrum peaks at wave number $k=2$ from $t\sim 88$.