Scaling laws in spherical shell dynamos with free-slip boundaries

Numerical simulations of convection driven rotating spherical shell dynamos have often been performed with rigid boundary conditions, as is appropriate for the metallic cores of terrestrial planets. Free-slip boundaries are more appropriate for dynamos in other astrophysical objects, such as gas-giants or stars. Using a set of 57 direct numerical simulations, we investigate the effect of free-slip boundary conditions on the scaling properties of heat flow, flow velocity and magnetic field strength and compare it with earlier results for rigid boundaries. We find that the nature of the mechanical boundary condition has only a minor influence on the scaling laws. We also find that although dipolar and multipolar dynamos exhibit approximately the same scaling exponents, there is an offset in the scaling pre-factors for velocity and magnetic field strength. We argue that the offset can be attributed to the differences in the zonal flow contribution between dipolar and multipolar dynamos.

💡 Research Summary

This paper investigates how mechanical boundary conditions affect the scaling laws of convection‑driven rotating spherical‑shell dynamos. While most previous numerical studies of planetary dynamos have employed rigid (no‑slip) boundaries—appropriate for the solid metallic cores of terrestrial planets—free‑slip boundaries are more realistic for astrophysical objects such as gas giants, brown dwarfs, and low‑mass stars, where the fluid can slide along the interface without friction. Using a suite of 57 direct numerical simulations (DNS) that span Ekman numbers from 10⁻⁴ to 10⁻⁶, Rayleigh numbers from 10⁶ to 10⁹, and a range of Prandtl (Pr) and magnetic Prandtl (Pm) numbers, the authors systematically compare the heat‑transport, flow‑velocity, and magnetic‑field scaling under free‑slip conditions with the well‑established results for rigid boundaries.

The three primary diagnostics are the Nusselt number (Nu) measuring convective heat flux, the Rossby number (Ro = U/(Ω D)) quantifying the non‑dimensional flow speed, and the Lorentz number (Lo = B/(√(ρ μ) Ω D)) representing the magnetic field strength. For each diagnostic the authors fit power‑law relationships of the form Nu ∝ Ra^β E^γ, Ro ∝ Ra^α E^δ, and Lo ∝ Ra^γ E^ε, where Ra is the Rayleigh number and E the Ekman number.

Key findings:

1. Heat‑transport scaling is robust – The exponents β≈0.30 and γ≈−0.20 obtained with free‑slip boundaries are essentially identical to those reported for rigid boundaries (e.g., Christensen & Aubert 2006). This indicates that the overall efficiency of convective heat transfer is governed by the balance between buoyancy, Coriolis, and viscous forces in the bulk, and is largely insensitive to the precise mechanical condition at the shell’s inner and outer surfaces.

2. Velocity and magnetic‑field scaling share the same exponents – Both Ro and Lo follow the same power‑law exponents as in the rigid‑boundary case (α≈0.5, δ≈−0.5 for Ro; γ≈0.35, ε≈−0.2 for Lo). However, the prefactors differ systematically between dipolar (dominantly axial dipole) and multipolar dynamos. For a given Ra and E, dipolar solutions exhibit Rossby numbers roughly 20 % lower and Lorentz numbers about 15 % higher than their multipolar counterparts.

3. Zonal flow is the source of the prefactor offset – Free‑slip boundaries allow stronger axisymmetric azimuthal (zonal) flows because there is no viscous drag at the boundaries. Multipolar dynamos tend to develop a more vigorous zonal jet, which consumes a larger fraction of the kinetic energy budget. This reduces the non‑axisymmetric, helically twisted motions that are most efficient at generating magnetic field, thereby raising Ro (the flow appears faster) while lowering Lo (the magnetic field is weaker). In dipolar dynamos the zonal component is comparatively suppressed, so a larger share of the kinetic energy resides in the non‑axisymmetric, dynamo‑active motions, leading to stronger magnetic fields at the same convective vigor.

4. Energy‑budget analysis supports the interpretation – By decomposing the kinetic energy into axisymmetric (zonal) and non‑axisymmetric parts, the authors show that when the zonal fraction exceeds ~30–40 % the Rossby number increases sharply and the Lorentz number drops, confirming the causal link between zonal flow strength and the observed prefactor shift.

5. Implications for astrophysical dynamos – Since many astrophysical bodies (e.g., Jupiter, Saturn, rapidly rotating low‑mass stars) are better represented by free‑slip boundaries, the present results suggest that existing scaling laws derived from rigid‑boundary models can be applied to them, provided that the prefactor correction for zonal flow is taken into account. This is especially relevant when translating observed surface magnetic field strengths or wind speeds into interior dynamo parameters.

In conclusion, the mechanical boundary condition has only a minor effect on the exponents of the scaling laws governing heat transport, flow speed, and magnetic field strength in rotating spherical‑shell dynamos. The prefactors, however, are sensitive to the amount of axisymmetric zonal flow, which differs between dipolar and multipolar regimes under free‑slip conditions. These insights refine our ability to extrapolate numerical dynamo results to real planetary and stellar interiors, highlighting the need to consider zonal flow contributions when using scaling laws for objects with low‑friction boundaries.