Advanced Ocean Modelling: Using Open-Source Software

Chapter 1IntroductionAbstract This chapter reviews the Navier-Stokes equations for an incompressiblefluid, summarises the basics of finite-difference modelling, and gives an overviewof softwares required for the conduction of exercises.1.1 Fundamental Physical Laws1.1.1 Cartesian CoordinatesFor convenience, locations are defined by means of the Cartesian coordinate system(Fig. 1.1) in which the vertical axis points upward at right angle to the undisturbedsurface of a fluid at rest. Horizontal coordinates are denoted by x and y. Thex-axispoints to the east. The y-axis points to the north. The undisturbed surface of thefluid is defined by z = 0. Use of a Cartesian coordinate system implies that thetrue curvature of the sea surface is ignored, which is a reasonable approximation foroceanic processes on spatial scales < 500 km.1.1.2 The Navier-Stokes EquationsThe Navier-Stokes equations comprise several physical conservation principles; thatis, conservation of momentum (Newton’s laws of motion), conservation of mass(which turns into conservation of volume for an incompressible fluid), and conservationof field variables such as temperature and salinity that via the equation ofstate give density (which appears in the buoyancy force). In Cartesian coordinates,the momentum equations can be written as:∂w∂t∂u∂t + Adv(u) − f v =−1 ∂ Pρ o ∂x + Diff(u)∂v∂t + Adv(v) + fu=−1 ∂ P+ Diff(v) (1.1)ρ o ∂y+ Adv(w) =− 1 ∂ Pρ o ∂z − (ρ − ρ o)g + Diff(w)ρ oJ. Kämpf, Advanced Ocean Modelling, DOI 10.1007/978-3-642-10610-1 1,C○ Springer-Verlag Berlin Heidelberg 20101