Time-fractional advection-dispersion-reaction modeling framework for marine pollution: Application to cyanobacterial...

Pollutant transport-reaction dynamics in aquatic systems often exhibit environmental memory, in which the present state depends on antecedent environmental conditions. This study develops a time-fractional advection-dispersion-reaction (TF-ADR) framework in which this antecedent influence is represented through a Caputo time-fractional derivative. The fractional term is interpreted as a bounded-memory parameterization of unresolved persistence and delayed response, rather than as evidence of a specific biological memory mechanism. The numerical framework combines P1 finite elements, streamline-upwind/Petrov-Galerkin stabilization, Caputo-L1 time discretization, and a finite-memory approximation. The framework is demonstrated through an application to 2021 cyanobacterial harmful algal bloom dynamics in Lake Pontchartrain Estuary, Louisiana, USA. Advective transport is driven by depth-averaged velocity fields from the ADvanced CIRCulation (ADCIRC) model, and environmental drivers are constructed from satellite, in situ, and reanalysis products. Memory-kernel analysis showed that, for α = 0.90, a 30 d window retained nearly all of the cumulative resolved model-period Caputo-L1 history coefficient weight. Non-assimilated sensitivity analyses indicated that α = 0.70-0.90 provided acceptable tradeoffs between CyAN spatial concordance and biomass responsiveness, with α = 0.90 and W = 30 d retained for the assimilative configuration. Compared with the integer-order ADR model and 7 d and 14 d antecedent-driver alternatives, the selected TF-ADR configuration produced higher CyAN spatial concordance while producing more moderate domain-scale bloom peaks. Assimilative diagnostics showed that sparse field observations could be incorporated while maintaining continuous station-scale trajectories. The results support TF-ADR as a parsimonious bounded-memory extension of ADR modeling for aquatic transport-reaction applications.