To describe the flow of geophysical fluids, equations are needed for conservation of momentum (or Newton's second law) and conservation of energy. The former leads to the Navier–Stokes equations which cannot be solved analytically (yet). Therefore, further approximations are generally made in order to be able to solve these equations. First, the fluid is assumed to be incompressible. Remarkably, this works well even for a highly compressible fluid like air as long as sound and shock waves can be ignored. Second, the fluid is assumed to be a Newtonian fluid, meaning that there is a linear relation between the shear stress  and the strain , for example
where  is the viscosity. Under these assumptions the Navier-Stokes equations are
The left hand side represents the acceleration that a small parcel of fluid would experience in a reference frame that moved with the parcel (a Lagrangian frame of reference). In a stationary (Eulerian) frame of reference, this acceleration is divided into the local rate of change of velocity and advection, a measure of the rate of flow in or out of a small region.
The equation for energy conservation is essentially an equation for heat flow. If heat is transported by conduction, the heat flow is governed by a diffusion equation. If there are also buoyancy effects, for example hot air rising, then natural convection, also known as free convection, can occur. Convection in the Earths outer core drives the geodynamo that is the source of the Earths magnetic field. In the ocean, convection can be thermal (driven by heat), haline (where the buoyancy is due to differences in salinity), or thermohaline, a combination of the two.