15.2: Numerical Models in Oceanography

Numerical models are very widely used for many purposes in oceanography. For our purpose we can divide models into two classes.

Mechanistic models are simplified models used for studying processes. Because the models are simplified, the output is easier to interpret than output from more complex models. Many different types of simplified models have been developed, including models for describing planetary waves, the interaction of the flow with sea-floor features, or the response of the upper ocean to the wind. These are perhaps the most useful of all models because they provide insight into the physical mechanisms influencing the ocean. The development and use of mechanistic models is, unfortunately, beyond the scope of this book.

Simulation models are used for calculating realistic circulation of oceanic regions. The models are often very complex because all important processes are included, and the output is difficult to interpret.

The first simulation model was developed by Kirk Bryan and Michael Cox (Bryan, 1969) at the Geophysical Fluid Dynamics laboratory in Princeton. They calculated the 3-dimensional flow in the ocean using the continuity and momentum equation with the hydrostatic and Boussinesq approximations and a simple equation of state. Such models are called primitive equation models because they use the basic, or primitive form of the equations of motion. The equation of state allows the model to calculate changes in density due to fluxes of heat and water through the surface, so the model includes thermodynamic processes.

The Bryan-Cox model used large horizontal and vertical viscosity and diffusion to eliminate turbulent eddies having diameters smaller about 500 km, which is a few grid points in the model. It had complex coastlines, smoothed sea-floor features, and a rigid lid. The rigid lid was needed to eliminate ocean-surface waves, such as tides and tsunamis, that move far too fast for the coarse time steps used by all simulation models. However, the rigid lid had disadvantages. Islands substantially slowed the computation, and the sea-floor features were smoothed to eliminate steep gradients.

The first simulation model was regional. It was quickly followed by a global model (Cox, 1975) with a horizontal resolution of 2\(^{\circ}\) and with 12 levels in the vertical. The model ran far too slowly even on the fastest computers of the day, but it laid the foundation for more recent models. The coarse spatial resolution required that the model have large values for viscosity, and even regional models were too viscous to have realistic western boundary currents or mesoscale eddies.

Since those times, the goal has been to produce models with ever-finer resolution, more realistic modeling of physical processes, and better numerical schemes. Computer technology is changing rapidly, and models are evolving rapidly. The output from the most recent models of the north Atlantic, which have resolution of 0.03\(^{\circ}\), look very much like the real ocean. Models of other areas show previously unknown currents near Australia and in the south Atlantic.

Ocean and Atmosphere Models

These two models use very different spacing of grid points. As a result, ocean modeling lags about a decade behind atmosphere modeling. Dominant ocean eddies are \(1/30\) the size of dominant atmosphere eddies (storms), but ocean features evolve at a rate that is \(1/30\) the rate in the atmosphere. Thus ocean models running for, say, one year have \((30 \times 30)\) more horizontal grid points than the atmosphere, but they have \(1/30\) the number of time steps. Both have about the same number of grid points in the vertical. As a result, ocean models run 30 times slower than atmosphere models of the same complexity.