**1. Introduction**

The Earth is often referred to as the water planet, although water accounts for only 0.023% of the mass of the planet. Nevertheless, water is found mainly at or near the surface and in the atmosphere and therefore is a very prominent planetary feature when viewed from space. Water as a substance appears in all three physical phases – solid, liquid, and gas. Under the present day climatic conditions, ice is found mainly in the polar regions, at latitudes north of 60°N and south of 60°S. Liquid water is found in the hydrosphere which includes the oceans, marginal seas, lakes, and rivers. The oceans cover nearly 70% of the surface of the Earth, with an average depth of ~ 4000 m. Water vapor, the gaseous phase, appears in the atmosphere and accounts for up to 4% of the mass. The hydrologic cycle describes the continuing transfer of water among these three components. All three forms of water also play important roles in the climate system. Water vapor is the main absorber of infrared radiation and therefore is a major contributor to the greenhouse effect. Clouds and ice are the major factors that determine the albedo of the Earth and therefore are mostly responsible for the reflection of approximately 30% of the incoming solar radiation. The specific heat capacity of water is nearly four times that of air and therefore the oceans serve as a major heat reservoir and regulator of the climate system. Furthermore ocean currents are responsible for more than one third of the heat transport from the equator to the poles and therefore affect the horizontal temperature gradients in the atmosphere which are closely linked to the development of major weather systems on various temporal and spatial scales.

The oceans also serve as a major source of food and natural resources and are important for commerce and transportation. For hundreds and perhaps even thousands of years, mariners intuitively understood some of the salient features of the surface circulation in the most highly traversed parts of the ocean. In 1770 Benjamin Franklin and Timothy Folger published the first map of the Gulf Stream, the major ocean current that flows northward along the east coast of North America and then turns northeastward and flows across the North Atlantic Ocean. The purpose of this map was to help mail ships sailing from Europe to North America to avoid this current and thereby shorten the duration of their trip. Yet despite the interest in and the importance of the oceans, oceanography as a formal science is relatively young, being only slightly more than a century old. In the early years, it was

Numerical Modeling of the Ocean Circulation:

coordinates (x, y, z), includes seven equations as follows:

�� + � ��

�� + � ��

�� ���

horizontal and vertical diffusion coefficients, respectively;

�� �� + � ��

�� + � ��

�� + � ��

�� ��� + �

�� �� <sup>+</sup> ��

�� + � ��

��

�� � �� = � �

�� + �� = � �

Where u, v, w are the velocity components in the x, y, z directions, t is time, ρ is the density (the subscript 0 indicates the mean value), � = ������ is the Coriolis parameter (Ω is the rotation rate of the Earth and φ is the latitude), p is the pressure, and *DIFF*(ψ) is the

�� ���

�� <sup>+</sup> ��

**Conservation of internal energy** (can be written in terms of density or temperature and

�� + � ��

�� ��� + �

�� ��

�� ��

�� ���

��

�� = ��� (3)

�� = 0 (4)

�� = ����(�) (5)

�� + ����(�) (1)

�� + ����(�) (2)

�� �, where *Ah* and *Az* are the

�� �� + � ��

�� �� + � ��

diffusion given by ����(�) <sup>=</sup> �

*Vertical momentum* (hydrostatic equation)

*Horizontal momentum* 

Where g is gravity; *Mass continuity* 

salinity)

From Process Studies to Operational Forecasting – The Mediterranean Example 113

the three dimensional equation for the conservation of momentum which is essentially an expression of Newton's second law of motion. The two fundamental forces that must be considered are the pressure gradient force and gravity. For geophysical fluids, rotation of the Earth is also important and therefore Coriolis force must also be added to the equations. To complete the description of the motion equations for mass conservation (continuity) and for the conservation of internal energy must also be added. The latter can be expressed in terms of density or in terms of temperature and salinity. To make these equations more tractable and directly applicable to the ocean circulation, various simplifications and approximations are applied. These simplifications are usually based on a scale analysis of the various terms in the equations. The two most common approximations are: (1) the vertical extent or depth of the fluid layer is much smaller than the horizontal scale of motion, and (2) the Boussinesq approximation in which the density variations are assumed to be small compared to the mean value and are therefore neglected except in the buoyancy term of the equation. As a result of the first approximation, the vertical component of the conservation of momentum can be reduced to a diagnostic equation for hydrostatic balance (i.e., the vertical component of the pressure gradient force exactly balances gravity or the weight of the fluid). The second approximation, which is roughly equivalent to assuming that seawater is incompressible, means that mass continuity can be reduced to a diagnostic equation for the conservation of volume (i.e., three dimensional nondivergence). The final set of the governing equations (usually referred to as the primitive equations) in Cartesian

primarily a descriptive science based on sparse and scattered observations. The quantitative aspects of physical and dynamical oceanography saw a major breakthrough with the publication of Henry Stommel's seminal work on the North Atlantic circulation (Stommel, 1948). With a simple mathematical model of the wind driven circulation he was able to elegantly explain the phenomenon of westward intensification (i.e., the formation of strong western boundary currents such as the Gulf Stream) as a result of the meridional variation of the Coriolis force.

The idea of using numerical models to further expand the understanding of the intricacies and complexities of the ocean circulation was introduced nearly twenty years later in the pioneering work of Bryan & Cox (1967). As with Stommel's research, they too investigated the circulation of the North Atlantic Ocean which at the time was the most highly observed ocean basin. The purpose of the model was to solve an initial value problem based on a simplified version of the Navier-Stokes equations. Through their model they were able to study the interaction between the wind driven and the thermohaline components of the circulation. Their work drew heavily from the experience of numerical weather prediction which took nearly thirty years to develop the capability of producing skillful forecasts beginning with Richardson's (1922) original concept but unsuccessful attempt and continuing to Charney et al. (1950) producing the first successful 24 hr forecast. As computational capabilities have increased exponentially over the past thirty years, so too has ocean modeling developed from a tool for simplified and focused process studies to fully operational forecasting systems. In this sense, the distinction between process studies (or simulations) and a forecasting system can be explained as follows. In the former, the goal is to understand the physical basis of the process without regard to reproducing specific details at any particular instant in time. In the latter, attention is focused on being able to produce the most accurate simulation of a particular realization of the flow at a specific time. The development of models for process studies and simulations was a necessary step in the development of forecasting systems. Furthermore, the useful range of a forecast, which is closely related to the limit of predictability, is limited by the chaotic behavior of the fluid flow. One the other hand, longer term simulations for the projection of future climate change is perhaps the most common example today of a process study. In both modern process studies and forecasting systems, the initial focus of model development has been the circulation, but today major progress has been made in developing components for simulating and predicting the fundamental biogeochemical processes of the oceanic ecosystem as well.

The goal of this chapter is to present an overview of modern ocean modeling as a tool for basic research as well as for operational forecasting. Considering the rapid developments and extensive experience of the Mediterranean oceanographic research community from recent years, we will use the Mediterranean as the prototype to explain and demonstrate these capabilities and successes in ocean modeling.
