## Numerical modelling of the ocean and climate system

Increasingly important tools for climate study and forecasting are numerical models of the general circulation of the atmosphere and ocean. Numerical models, however, are far from perfect. Accurate inclusion of the many feedbacks discussed in §§1.7 and 7.2.4, even those whose physical, chemical and biological mechanisms are well understood, is problematic because of the contrasting spatial scales of models and processes. The results of models will play a prominent role in our discussion so here we will examine the fundamental ideas underlying how these models are constructed.

Circulation in the ocean and atmosphere can be described by well-established sets of mathematical equations valid at every point in each fluid. These equations describe the evolution of the density field and velocity in each direction. However, except in very special situations not encountered in the real world, exact solutions to these equations cannot be found. Techniques for numerically approximating the equations were developed as early as World War I by L. F. Richardson, who attempted the first, unsuccessful, numerical weather prediction while driving a military ambulance in the battlefields of France. The advent of electronic computers in the 1940s and 1950s allowed numerical solution of these equations to become technically feasible.

The equations of motion cannot be numerically solved at every point in space - an infinite set - so an approximation to these equations is made by solving for the essential variables of velocity, temperature, salinity (in the ocean), and surface pressure and humidity (in the atmosphere) at discrete points of a grid in space8 at discrete moments in time. Approximations to the terms in the equations are then made, linking the values at neighbouring grid points. A characteristic grid network for an ocean general circulation model (GCM) is shown in Fig. 1.30. Typical climate models currently have horizontal resolutions - the separation between grid points - of 2-5° in the atmosphere and 1-5° in the ocean and average vertical resolutions of several hundred metres in the ocean and a kilometre or two in the atmosphere, while timesteps are tens of minutes. Both ocean and atmosphere are usually more highly resolved vertically near the air-sea interface. All these parameters are dictated by the memory and disk capability of current computers, and are continually reducing. In atmospheric GCMs it is often numerically advantageous to rewrite the vertical variation of the equations in terms not of altitude but a coordinate that smoothly follows the surface. Oceanic GCMs sometimes use a vertical coordinate that depends on the density of the sea water.

Unfortunately for modellers, the climate system involves many more processes than just atmospheric and oceanic motion (see Fig. 1.1). Radiation absorption and emission, cloud formation and precipitation, chemical reactions in the atmosphere, biological processes in the ocean, sea-ice formation, deep convection, heat and nutrient transfer by oceanic eddies, snow melt and river run-off, the transfer of particles and gases between ocean and atmosphere, the storage and release of heat and moisture from terrestrial surfaces and vegetation: all of these processes occur on fundamentally smaller spatial scales than are likely to be fully resolvable within models for the foreseeable future. Often the basic mechanisms driving such processes are not well understood. These physical, chemical and biological aspects of the climate system must therefore be simplified in order to be represented on GCM grids. This is known as parameterization. Parameterization schemes differ between groups of climate modellers and pose a crucial uncertainty in current predictions. Few current climate models include marine biological processes.

Until the mid-1990s most climate model experiments used either an atmospheric GCM or an ocean GCM, but did not couple the two. The rapid increase in computer power over the last ten years has enabled all climate modelling groups to move to fully coupled models, although important sensitivity studies

8 Other techniques can be used, such as a spectral formulation, but for our purposes these are equivalent to the standard approximation.

Fig. 1.30. Allocation of model variables in space within a typical ocean GCM. Note that temperature (T), salinity (S) and streamfunction are staggered relative to the velocity vector (V ). [Fig. 1 of Han (1988). Reproduced with kind permission of Kluwer Academic Publishers.!

Fig. 1.30. Allocation of model variables in space within a typical ocean GCM. Note that temperature (T), salinity (S) and streamfunction are staggered relative to the velocity vector (V ). [Fig. 1 of Han (1988). Reproduced with kind permission of Kluwer Academic Publishers.!

can still be carried out using single fluid models. Coupled models have the added problem of correctly transferring heat, moisture and momentum, at least, between the ocean and atmosphere; more sophisticated models parameterize the exchange of carbon and sulphur. As these exchange processes are often not well understood, and certainly occur on scales smaller than typical model grids, they are difficult to correctly parameterize. Thus there is a tendency for coupled models to drift away from the stable climates that the individual components attain independently. Many models reduce the climate drift problem by imposing a flux correction - an essentially artificial removal or addition of energy - to keep the coupled model under control. Having to resort to such a technique makes interpretation of coupled climatic change experiments problematic, although the consistency of transient response simulations between different models with and without flux correction is encouraging. Increasingly, however, climate models are overcoming this drift problem and remain stable for centuries of model time without requiring a flux correction.

Nevertheless, coupled runs of advanced GCMs attempting to simulate the present climate are rarely more than a thousand years long, at most. As we have already seen (§1.3), key timescales within the ocean can be of the order of several thousand years, thus modelled climate in current coupled models is unlikely to have reached true equilibrium. This has created a niche for reduced complexity coupled models, where typically one or more major components of the climate system is approximated. This can be done by, for example, discretizing space in a much coarser manner, having an effectively one-layer atmosphere, or through using simplified equations of motion in the ocean. Such models can then test the long-term impact of changing forcing fields. Some of these models reveal that large adjustments to a mean climate can occur very rapidly, but may not begin for perhaps 1000-2000 years after the forcing has been altered. Sub-millennial experiments, therefore, run the risk of not simulating the full impact of particular changes to the Earth system. As partial mitigation of this problem one can argue that the concept of an equilibrium climate is probably flawed; nevertheless this current drawback of full coupled models needs to be remembered for Chapter 7.