Basic Numerical Methods in Meteorology and Oceanography

The purpose of this book is to provide an introduction to numerical modelling of the ocean and the atmosphere. It originates from courses given at Stockholm University and is intended to serve as a textbook for students in meteorology and oceanography with a background in mathematics and physics.

Focus is on numerical schemes for the most commonly used equations in oceanography and meteorology as well as on the stability, precision and other properties of these schemes. Simple equations capturing the properties of the primitive equations employed in models of the ocean and atmosphere will be used.

These model equations are solved numerically on a grid by discretisation, the derivatives of the differential equations being replaced by finite-difference approximations. The focus will be on the basic numerical methods used for oceanographic and atmospheric modelling. These models are based on the Navier-Stokes equations (including the Coriolis effect) and a tracer equation for heat in both the atmosphere and ocean and tracer equations for humidity and salt in the atmosphere and ocean, respectively. A coupled atmospheric and oceanic general circulation model represents the core part of an Earth System climate model.

The book starts by presenting the most common types of partial differential equations and finite difference schemes used in meteorology and oceanography. Subsequently the limitations of these numerical schemes as regards stability, accuracy, presence of computational modes and accuracy the computationally determined phase speed are discussed.

The shallow-water equations are discretised for different spatial grids and friction and diffusion terms are introduced. Hereafter implicit and semi-implicit schemes are discussed as well as the semi-Lagrangian technique. Coordinates for atmospheric as well as oceanic models are presented as well as a highly simplified 3D model. A brief description is given of how some atmospheric general circulation models use spectral methods as "horizontal coordinates". Finally, some "pen-and-paper" theoretical exercises and a number of GFD computer exercises are given.

Kristofer Döös is a professor of climate modelling. He has a PhD in oceanography from Université de Pierre et Marie Curie in Paris.He has previously worked at Southampton Oceanography Centre and at the Institute of Oceanographic Sciences in the U.K. His main research has been on ocean and climate numerical modelling with particular emphasis on the overturning circulation and the Lagrangian tracking of heat and water masses in both the ocean and the atmosphere.

Peter Lundberg is a professor emeritus of oceanography at Stockholm University. He defended his doctoral thesis at Göteborg University. His main research focus is on physical processes in the Baltic and the North Atlantic.

Aitor Aldama Campino is an ocean and atmosphere modeler with a PhD from Stockholm University. His main interest has been on overturning circulation and development of the Lagrangian trajectory model TRACMASS.