The Mathematics of Models for Climatology and Environment

In the last years increasing attention has been paid to reduce the environmental impact of domestic and industrial wastewater discharges. As the objective of non-polluting effluent disposal alternatives is limited by technical or financial factors, an outfall is sometimes the only feasible solution (see Quetin & De Rouville [1986]).

“Science is now a tripartite endeavour with simulation added to the two classical components, experiment and theory (Robinson, 1987). The routine use of numerical “simulation in scientific research (numerical experimentation, sensitivity and process studies, etc.) is thought by many to represent the first major step forward in the basic scientific method since the seventeenth century”(Robinson, 1987).

In the last decades it has become increasingly evident that, in order to understand the behavior of the climate system and its response to human activities, a multi-disciplinary effort is needed to model the interactions between the atmosphere, the hydrosphere, the biosphere, the cryosphere, and the chemosphere (Trenberth 1992). Interactions between these components of the climate system have been studied for many years. Only recently, however, mathematical models (hereafter referred to as Climate System Models, or CSMs) are being sought to describe the system as a whole, so that possible feedback mechanisms can be identified and quantified.

The essential of this work is the determination of the velocity fields of the marine streams with a view of solving biologic models, pollution transport... The studied models concern phenomena specific to the seas where the main forcing is the wind. We present the study of two models: a three dimensionnal model and a shallow water model in depth-mean velocity formulation. This last, that is obtained by integration on the depth of the three dimensionnal model, allows a preliminary study that can be useful to obtain for instance initial conditions (or boundary) and even to solve the three dimensional model when the hypothesis of the rigid-lid is not done.

First of all, we present an existence result of the shallow water problem, result that we obtained thank to the help of Pr. P.L. Lions. We will not give all the details of the demonstration but the main ideas.

The equations of this model differ from classical equations by the boundary conditions that are of the kind u.n and curl u fixed on the boundary. This difference allowed us to solve the equations by the Galerkin’s method using the eigen basis of the operator △ with u.n = 0 and curl u = 0 on the boundary. These boundary conditions give to this base particular properties that allow to smoothly solve the equations of the Shallow water model herafter presented.

This basis is also used to solve the three dimensionnal model.

The important sizes of the domains and the difficulties linked to the adjustment of the parameters make the Galerkin’s method particularly well adapted. As a matter of fact, the heavy calculi linked to the dimensions are carried out once and for all.

The plant of the study is the following: We present first the models and notations. In the paragraph I we give an existence result of the shallow water problem and in paragraph II we give the numerical analysis of this problem. In the paragraph III we give the analysis of the three dimensional problem.

Our aim in this article is to present some receent results on the mathematical theory of Inertial Manifolds and Slow Manifolds.

Climate models have different characteristics than weather prediction models: the time scale is completely different (centuries versus days or weaks) and their main goal is also complementary (prognostic in the weather prediction and diagnostic in the case of climate models). Climate models were introduced in order to understand past and future climates and their sensitivity on a few of relevant features (which a quantitative analysis reduces to some parameters).

Climate models are distinguished by the relative importance they attach to the different components and processes of the climate system. One finds the so-called energy balance climate models at the bottom on a scale of models of increasing complexity, and coupled general circulation models of atmosphere and oceans at the top on that scale.

The presence of ice or snow at the surface of the Earth changes the reflectivity of the system to sunlight. When the planet is made colder by changing some control parameter such as the solar constant, ice caps grow; moreover, the increased reflectivity (albedo) of the ice-covered surface causes an enhancement of the cooling. The effect of the ice-cap growth is to amplify the effects of an externally induced climate change. The ice-clbedo mechanism is a positive feedback in the climate system. The climate system has many feedbacks, positive and negative. They play various roles in forced climate change. This lecture will concentrate on the effects of that one isolated feedback, since in this simplified form the problem becomes tractable and quasi analytical solutions can be found.

Glaciers are huge and slow moving rivers of ice which exist in various parts of the world: Alaska, the Rockies, the Alps, Spitsbergen, China, for example. They drain areas in which snow accumulates, much as rivers drain catchment areas where rain falls. Glaciers also flow in the same basic way that rivers do. Although glacier ice is solid, it can deform by the slow creep of dislocations within the lattice of ice crystals which form the fabric of the ice. Thus, glacier ice effectively behaves like a viscous material, with, however, a very large viscosity: a typical value of ice viscosity is 1 bar year (in the metre-bar-year system of units!). Since 1 bar = 10⁵ Pa, 1 year ≈ 3 × 10⁷ s, this is a viscosity of some 10¹² Pa s, about 10¹⁵ times that of water. As a consequence of their enormous viscosity, glaciers move slowly - a typical velocity would be in the range 10–100 m y^-1 (metres per year), certainly measurable but hardly dramatic. More awesome are the dimensions of glaciers. Depths of hundreds of metres are typical, widths of kilometres, lengths of tens of kilometres. Thus glaciers can have an important effect on the human environment in their vicinity. They are also indirect monitors of climate; for example, many lithographs of Swiss glaciers show that they have been receding since the nineteenth century, a phenomena thought to be due to the termination of the ‘little ice age’ in the middle ages.

The interaction between large masses of snow and ice with the earth’s climate is of fundamental importance. The strong coupling between the cryosphere, which includes the ice and snow fields, sea ice and permafrost, with the other components of the climatic system cannot be underestimated. Its influence on the global climate acts at long time scales and is a dominant factor in the variation of the planetary albedo (see, for instance, [PO]).

The Strait of Gibraltar, the link between the Atlantic Ocean and the Mediterranean Sea, plays a major role in the renovation of the Mediterranean water and in the dynamics of water masses in the Alboran Sea. Therefore, for the numerical simulation of the hydrodynamic processes that take place in the Alboran Sea a thorough knowledge and modelling of fluxes through the Strait is needed. This paper is devoted to the numerical simulation of hydrodynamical processes taking place in both the Strait of Gibraltar and the Alboran Sea. The main goal is to present the derivation of a multi-layer shallow water model and its application to this zone of the Mediterranean. This model takes into account Coriolis force, wind and bottom drag effects, interaction between two adjacent layers, etc… This has been developed from a shallow-water model developed at the University of Santiago de Compostela (see[2]).

A number of different models of varying complexity have been developed and used in the last two decades to simulate El Niño/Southern Oscillation (ENSO) interannual climate variability in the tropical Pacific (Neelin et al., 1992). These models range from simple conceptual systems, involving only one equation and one variable, to Coupled General Circulation Models (CGCMs) with a large number of degrees of freedom. All of these models have played a role in under standing ENSO. Conceptual low order models drastically reduce the spatial structure of the atmosphere and ocean. Simple models also exist which represent the ocean and/or atmosphere with simplified sets of partial differential equations. Both of these types of model are useful for illustrating fundamental processes of air-sea interaction and have been developed as tools to understand more complicated systems. Intermediate coupled models using reduce-gravity ocean and atmosphere components and realistic thermodynamics are sophisticated enough to produce realistic solutions, and are also simple enough to diagnose (Zebiak and Cane, 1987). High resolution CGCMs can be closely compared to observations, but the important processes at work in them are not always easy to determine.

It is well recognized that successful modeling of seasonal and long term climate variability requires a good representation of atmosphere-ocean heat exchange, as well as oceanic heat storage and transport. This has motivated the analysis of the main elements of heat transfer in the ocean during the last decades. In this review we look at heat transfer and formation of Warm Water Mass (WWM) in the eastern North Atlantic Subtropical Gyre (NASG), approximately between Cape Vert (15°N) and the Strait of Gibraltar (36°N) and between the African coast and 30°W.

The characterization of different air masses arriving at a baseline station is a great aid in determining atmospheric background conditions. The features of the air masses depend on their source areas as well as their trajectories. When the air masses move over the ocean, their physical and chemical properties are different from those of air masses that pass over land. For example, air masses passing over western Europe on the way to Izaña Observatory (Tenerife, Canary Islands, Spain) often show evidence of anthropogenic pollution, whereas maritime air masses are usually very clean.

Mesoscale is a dominant scale of motion in the world’s oceans. During the last decade, the wide use of infrared imagery and altimetry, has shown that mesoscale structures are ubiquitous and dominate the energy spectrum. These structures present a wide range of space and time scales, from 10 to 100 km and 10 to 100 days, respectively. These features present a large variety of patterns such as meanders, eddies, filaments or plumes and are responsible for the existence of enhanced density and velocity shears. The observation of mesoscale structures has therefore become a priority in recent years and significant efforts are now devoted to resolve or parametrize mesoscale fluctuations in ocean circulation models, both in large scale open ocean or more coastal studies. The structures observed and their dynamics are however different in these two environments. In the study of large scale open ocean circulation, adequate treatment or parametrization of turbulent mesoscale eddies is needed since they enhance mixing and horizontal transport of heat, salt, particulate matter, etc., which play an important role in the global ocean circulation. In the coastal ocean, mesoscale features are important since they represent a major forcing for cross-shore transport of properties. Another important difference between open ocean and coastal studies is that, while large scale eddies are generally isotropic, coastal features are highly anisotropic, appearing usually as elongated filaments.