S-FMM: A nonlinear spatial-frequency modulation framework for structured oscillatory fields on the torus

Understanding and modeling two-dimensional fields is a fundamental challenge in many areas of science, where spatial organization, directionality, and periodicity play a central role. To address this problem, we introduce the Spatial-Frequency Modulated Möbius model (S-FMM), a structured parametric framework for modeling fields defined on the torus. The framework is motivated by the analysis of spatial patterns arising in magnetically confined plasmas, which naturally exhibit periodic geometry and coherent directional modes. S-FMM represents the field as a compact collection of nonlinear modes defined along two angular coordinates. Möbius-transformed phase functions are used to model non-sinusoidal and anisotropic spatial patterns, allowing the framework to adapt to localized and direction-dependent structures that are difficult to capture with classical spectral methods. Model parameters are estimated through a specifically designed algorithm for signal-plus-noise settings, with identifiability ensured via simple ordering constraints. The effectiveness of the proposed approach is demonstrated through numerical experiments covering a range of noise conditions and spatial configurations. In particular, an application to high-resolution plasma turbulence simulations obtained from established physics-based models shows that S-FMM successfully recovers coherent directional structures that are poorly represented by classical decomposition methods. Across all tested scenarios, S-FMM achieves higher reconstruction accuracy with fewer parameters, providing a compact and parsimonious representation of the underlying fields that facilitates analysis and dimension reduction of fields defined on periodic domains. The scope of application is not limited to magnetically confined plasmas and naturally includes other settings where toroidal or angular organization is present. For example, S-FMM is well suited for the analysis of grid cell firing patterns in neuroscience, where latent toroidal structure plays a central role.