# Numerical modeling

## numerical modeling

In oceanography, the prediction of flow evolution via numerical construction of approximate solutions to the governing equations.

Solutions are obtained by assigning discrete values to temporal and spatial derivatives in order to convert the governing differential equations into algebraic equations that can be solved by using computational methods. Because computational resources are finite, no one technique is ideal for all applications. Some models define the equations on very fine spatial intervals (
see direct numerical simulation). This approach furnishes solutions that are very accurate, but that span only small spatial regions (spatial scales of a few meters, at present). At the other extreme, some models span entire ocean basins by using large spatial intervals (hundreds of kilometers). Here, approximation of unresolved motions is a crucial and difficult issue (
see very large eddy simulation). Similar trade-offs must be made with respect to temporal solutions. Numerical models also differ in the equations and boundary conditions that are employed. The most general model commonly used in oceanography includes momentum conservation via the incompressible Navier–Stokes equations with the Boussinesq approximation, mass conservation via the incompressibility condition, and equations expressing conservation of heat energy and salt (e.g., Gill 1982). For large-scale applications, the hydrostatic approximation is usually made. The vertical coordinate may be the geometric height, or a convenient substitute such as density, pressure, logarithm of pressure, or potential temperature. Surface boundary conditions generally express fluxes of momentum, heat, and freshwater from the atmosphere. Basin-scale models use boundary conditions that approximate the effects of bottom topography. Smaller-scale models typically specify periodic conditions at the side boundaries and an energy radiation condition at the bottom. 