Foundational Mathematics and Discretization

Climate models are systems of Partial Differential Equations (PDEs) based on fundamental laws of physics, specifically the conservation of mass, momentum, and energy. The core fluid dynamics are governed by the Navier-Stokes equations applied to a rotating sphere, which describe the evolution of velocity, pressure, density, and temperature. Radiative transfer, the driver of the climate system, is often calculated using Schwarzschild’s equation, quantifying how electromagnetic radiation interacts with the atmosphere via absorption and emission.

Because these equations cannot be solved analytically for the Earth's complex geometry, they are solved numerically using discretization. The continuous atmosphere and ocean are partitioned into 3D grids or spectral modes. Spectral methods, which use s ...