Mathematical Modeling of the Caspian Sea Geometry Under Water Level Decline: Evidence from Iran’s Mazandaran Coast

The Caspian Sea is Earth’s largest endorheic water body and a natural laboratory for studying coupled hydroclimatic forcing and coastal geomorphic response. Over the last century, the basin has undergone multi-decadal oscillations and an accelerated decline since the mid-1990s, with strong signals projected to continue through the twenty-first century. This meta-analysis synthesizes peer-reviewed evidence on horizontal shoreline migration, vertical (level) change, and areal transformation, and evaluates how these changes reconfigure the basin’s large-scale geometry—well approximated locally by parabolic or saddle-shaped (hyperbolic-paraboloid) surfaces—while cascading into socioeconomic risk. We combine quantitative shoreline metrics (Digital Shoreline Analysis System, DSAS), spectral water delineation (NDWI/MNDWI), sediment-transport theory (Exner equation), and water-balance diagnostics to: (i) characterize recent and projected Caspian water-level trajectories; (ii) resolve planform curvature and alongshore variability along Iran’s Mazandaran coast; (iii) contrast Caspian responses to those observed in China’s coastal systems (South China Sea littoral and the Yangtze River delta); and (iv) assess risk pathways for agriculture, shipping, fisheries, and wetlands. Results from the literature indicate a long-term negative water balance dominated by increased evaporation relative to precipitation and inflows, superimposed on high interannual variability; the Volga—regulated by reservoirs including Volgograd—remains the principal control on riverine supply. Shoreline retreat and shallow-water expansion are already disrupting Mazandaran’s port operations (e.g., Amir-Abad), accelerating maintenance dredging needs, and exposing wetlands such as Gorgan Bay to desiccation and dust-storm hazards. Comparative analysis shows that while China’s open-coast margins are influenced by marine processes, the Caspian’s closed-basin geometry transmits water-level anomalies more uniformly, amplifying parabolic/saddle-like morphodynamic adjustments. We conclude with actionable adaptation options for Mazandaran (channel realignment, dynamic zoning, nature-based buffers, and flexible port layouts) aligned with realistic twenty-first-century water-level scenarios.