A review of applications of fractional calculus in Earth system dynamics

Fractional calculus has been used to model various hydrologic processes for 15 years. Yet, there are still major gaps between real-world hydrologic dynamics and fractional-order partial differential equations (fPDEs). In addition, the applicability of fPDEs in the broad field of Earth dynamics remains obscure. This study first reviews previous applications and then identifies new research directions for fPDEs simulating non-Fickian transport in both surface and subsurface hydrology. We then explore the applicability of fractional calculus in various anomalous dynamics with a wide range of spatiotemporal scales observed in the solid Earth, including internal dynamics (such as inner core rotation, outer core flow, mantle convection, and crustal deformation), large-scale surface dynamics (in fluvial, Aeolian, and glacial systems), and small vertical-scale surface kinetics (in crystal growth, rock/mineral weathering, and pedogenesis), where driven forces, previous modeling approaches, and the details of anomalous dynamics are analyzed. Results show that the solid Earth can provide an ideal and diverse base for the application of fractional calculus and fPDEs. Complex dynamics within and across spatiotemporal scales, multi-scale intrinsic heterogeneity, and intertwined controlling factors for dynamic processes in the solid Earth can motivate the application of fPDEs. Challenges for the future application of fPDEs in Earth systems are also discussed, including poor parameter predictability, the lack of mathematical specification of bounded fractional diffusion, lack of intermediate-scale geologic information in parsimonious and upscaling models, and a lack of models for multi-phase and coupled processes. Substantial extension of fPDE models is needed for the development of next-generation, solid Earth dynamic models, where potential solutions are discussed based on our experience gained in the development and application of fractional calculus and fPDEs over the last decade. Therefore, the current bottleneck in the application of fractional calculus in hydrologic sciences should not be the end of a promising stochastic approach, but could be the early stage of a decade-long effort filled with multiple new research and application directions in geology. This conclusion may shed light on the bottleneck challenging stochastic hydrogeology, where the advanced stochastic models (with more than 3500 journal publications in the last three decades) have not significantly impacted the practice of groundwater flow and transport modeling.