Decoding the Geodynamo: Quantitative Dependence of Magnetic Field Diagnostics on Dimensionless Control Parameters

Understanding how planetary magnetic fields are generated and sustained requires careful analysis of how key physical quantities respond to changes in the underlying control parameters. In this work, we focus on the scaling behaviour of magnetic energy, dipolarity, field symmetry, and temporal variability with respect to the Rayleigh number (Ra), a key driver of convective vigour. Using the XSHELLS simulation framework, we carry out a series of dynamo simulations in rotating spherical shells, systematically varying Ra along with the Ekman number (E) and magnetic Prandtl number (Pm). Our results highlight clear transitions in magnetic field structure—from strong, stable dipoles to weaker, multipolar states—as Ra increases. We identify scaling laws that relate the intensity and geometry of the magnetic field to Ra and Pm, and track how spectral energy distributions shift with these parameters. In particular, we find that decreasing E leads to the development of rotationally constrained flows and quasi-geostrophic dynamics, which strongly influence magnetic field morphology. Overall, this study emphasizes the role of Ra as a central control parameter in dynamo models and provides a clearer picture of how convective forcing shapes planetary magnetic fields.