George Salman | Searcharxiv

arXiv2026

Stratosphere Model Verification with Manufactured Geometry

We propose an exact solution for a stratosphere dynamical core formulated in geopotential/pressure coordinates with a time-evolving lower boundary supplied by the troposphere. Rather than constraining the stratospheric circulation via specified dynamics (``nudging'') to a reanalysis, we treat the tropopause as a moving geometric boundary. The stratospheric domain thus expands, contracts, and undulates in response to tropospheric variability while preserving familiar hybrid $σ$--$p$ structure and pressure-gradient calculations. The approach integrates naturally with arbitrary Lagrangian--Eulerian (ALE) updates and conservative remap to maintain positive layer thickness and tracer monotonicity. We outline the formulation, highlight analytical properties (well-posedness, energetics, wave propagation), and sketch a verification/validation path based on modified standard test cases and reanalysis-driven experiments.

physics.ao-ph↗

arXiv2025

Geometry of Deformed Cellular Spaces

We present an adaptive geometry in which the yardstick co-deforms with space itself, formulated on cellular spaces where length is a count: distances are shortest cell-crossing counts. No cell shape, angles, or embedding are assumed; the framework is deliberately micro-agnostic. Curvature and deformation are inferred operationally by comparing a measured radius to a radius reconstructed from boundary/area/volume counts; the linear dimension of a cell serves as the single universal unit of length, yielding unified small-ball/small-sphere estimators in 2D/3D/4D. We prove that the count metric on locally finite complexes is geodesic, show flatness on uniform lattices, and establish stability of distances and curvature estimators under small local perturbations. As a bridge to the smooth setting, a line-density field induces a conformal metric $g = e^{2u} g_0$ that reproduces the same operational quantities. We outline a Ricci-like construction from directional slices and give a spherically symmetric illustrative example consistent with Schwarzschild-type spatial behavior. Overall, the model provides an intrinsic, micro-agnostic calculus linking discrete measurements to continuum notions with guarantees, including Gromov--Hausdorff control under mild regularity assumptions.

math.GM↗