2 The core of a climate model
The Navier-Stokes equations for fluid flow are at the heart of climate models. The first three equations represent Newton's second law and give the acceleration of the winds in the east-west (u), north-south (v) and vertical directions (w). The mass-continuity equation ensures that although the density, speed and direction of the air change as it flows around the Earth, its mass is conserved, while the thermodynamic equation allows heat-transfer processes such as heating by the Sun to be included as a parametrized source term (S). We use the same equations to model the dynamics of the ocean, but usually make further simplifying approximations. In the equations, r is the distance from the Earth's centre, Ω is the angular velocity of the Earth's rotation, φ is latitude, λ is longitude and t is time. cp is the specific heat capacity of air at constant pressure, θ is potential virtual temperature, Π is the "Exner function" of pressure and ρ is air density. The subscript "d" refers to dry air.