Magnetic quenching of the inverse cascade in rapidly rotating convective turbulence

We present results from an asymptotic magnetohydrodynamic model that is suited for studying the rapidly rotating, low-viscosity regime typical of the electrically conducting fluid interiors of planets and stars. We show that the presence of sufficiently strong magnetic fields prevents the formation of large-scale vortices and saturates the inverse cascade at a finite length scale. This saturation corresponds to an equilibrated state in which the energetics of the depth-averaged flows are characterized by a balance of convective power input and ohmic dissipation. Quantitative criteria delineating the transition between finite-size flows and domain-filling (large-scale) vortices in electrically conducting fluids are found. By making use of the inferred and observed properties of planetary interiors, our results suggest that convection-driven large-scale vortices do not form in the electrically conducting regions of many bodies.

Figure 1

Simulation snapshots. Volumetric renderings of the geostrophic stream function $(\psi$, top row) and the depth-averaged axial vorticity $(\langle \zeta \rangle$, bottom row). All plots correspond to a reduced Rayleigh number of $\widetilde{\text{Ra}}=160$. Three values of the reduced Chandrasekhar number are shown: $\tilde{\mathrm{Q}}=0$ (first column), $\tilde{\mathrm{Q}}=1$ (second column), and $\tilde{\mathrm{Q}}=2$ (third column).

Figure 2

Spectral energy transfer functions. All functions are averaged over a period of time in which the convective (baroclinic) dynamics are statistically stationary, as indicated by the time evolution of the vertical Reynolds number $\widetilde{\text{Re}}$ (see Supplemental Material [29] for time series of kinetic energy and $\widetilde{\text{Re}}$ for representative cases). Magnetic field strength (as characterized by the Chandrasekhar number $\tilde{\mathrm{Q}})$ increases from left to right, with a fixed Rayleigh number of $\widetilde{\text{Ra}}=160$. Each plot illustrates the energetic contributions to wave number $k$ of the depth-averaged (barotropic) flow, from the other baroclinic and barotropic modes $(T_k$ and $F_k)$, from the Lorentz force $\left(L_k\right)$, and from viscous dissipation $\left(D_k\right)$. Positive (negative) values indicate energy is being transferred to (from) the $k\mathrm{th}$ barotropic mode.

Figure 3

Depth- and time-averaged barotropic kinetic energy $\left(K_{bt}\right)$ spectra for a fixed Rayleigh number of $\widetilde{\text{Ra}}=160$ and different values of the Chandrasekhar number $\tilde{\mathrm{Q}}$. Lines of slope $k^{-5/3}$ and $k^{-3}$ are shown for reference. In all cases, energy is injected through convection around $k=10$.

Figure 4

(a) Reynolds number $\widetilde{\text{Re}}$ and (b) interaction parameter $\tilde{\mathrm{N}}=\tilde{\mathrm{Q}}/\widetilde{\text{Re}}$ as a function of $\widetilde{\text{Ra}}$ for different values of $\tilde{\mathrm{Q}}$. Values of $\widetilde{\text{Re}}$ are time averaged as in Figs. 2 and 3 and the error bars in (a) represent the fluctuations around the mean value, measured by the standard deviation. Error bars on $\tilde{\mathrm{N}}$ would not be clearly visible on this plot and have been omitted. The solid red circles in (b) indicate large-scale vortex (LSV) formation. The black horizontal line marks the threshold $\tilde{\mathrm{N}}=0.013$ above which no LSV is observed.

Figure 5

Interaction parameter $\tilde{\mathrm{N}}={\widetilde{\text{Re}}}^{-1}\tilde{\mathrm{Q}}$ for Jupiter as a function of the nondimensional radius $r/R_J$, where $R_J$ is the equatorial radius. The blue and red dots are calculated by estimating the large-scale magnetic field intensity to be given by the JRM09 magnetic field model [33] truncated at spherical harmonic degrees $l=1$ and $l=8$, respectively. The case $l=1$ corresponds to a dipolar field and for $l>8$ the power in the field components drops below $1\%$ of the power contained in the dipolar field. Viscosity, electrical conductivity, and density are taken from Ref. [34], and $\widetilde{\text{Re}}$ is calculated from zonally averaged meridional flow speeds that are estimated from Cassini spacecraft observations [35]. The horizontal black line demarcates the $\tilde{\mathrm{N}}=0.013$ threshold.