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Über dieses Buch

Large-scale winds and currents tend to balance Coriolis and pressure gradient forces. The time evolution of these winds and currents is the subject of the quasi-geostrophic theory.

Chapter 1 presents concepts and equations of classical inertial fluid mechanics.

Chapter 2 deals with the equations of thermodynamics that close the governing equations of the fluids. Then, the motion is reformulated in a uniformly rotating reference frame.

Chapter 3 deals with the shallow-water model and the homogeneous model of wind-driven circulation. The chapter also describes a classical application of the Ekman layer to the atmosphere.

Chapter 4 considers the two-layer model, as an introduction to baroclinic flows, together with the concept of available potential energy.

Chapter 5 takes into account continuously stratified flows in the ocean and in the atmosphere.





Chapter 1. Basic Continuum Mechanics

At all scales, from galaxies to elementary particles, matter is discrete. Yet, the “continuum” paradigm is still of paramount importance, both in theory and in practice. This chapter briefly elucidates this paradox, with emphasis on fluid mechanics, and presents the basic conceptual tools needed to approach heuristically the physics of fluids. Both the Lagrangian and the Eulerian descriptions of motion are introduced. The link between these two descriptions stems from the request that a property of an Eulerian field, at a given point and at a given time, coincides with the corresponding property of the Lagrangian particle that occupies that point at that time. In the Lagrangian framework, the governing equations of fluid mechanics, the vorticity dynamics and the parameterization of turbulence are discussed. Illustrative applications of these results to simple models (e.g. flow in a channel and 2-D breezes) are also given.
Fabio Cavallini, Fulvio Crisciani

Chapter 2. Basic Geophysical Fluid Dynamics

Geophysical Fluid Dynamics (GFD) governs the behaviour of large bodies of air in the atmosphere and of seawater in the ocean. Thus, the equations of mass and momentum balance must be supplemented with suitable information about the thermodynamics of these fluids. In this way, a complete set of governing equations, indeed hardly mathematically tractable, is obtained.
Moreover, the atmosphere and the ocean rotate as they were almost fixed with the Earth; so, large-scale winds and ocean currents, as detected by a terrestrial observer, are just the result of small departures of these fluids from the rest state. On the other hand, Earth’s rotation heavily influences the motion of the winds and the ocean currents because of the Coriolis acceleration, and, therefore, the latter enters into the momentum equation in a way that deeply characterizes the dynamics of geophysical flows.
One of the most fascinating aspects of this dynamics lies in the common obedience of the atmospheric and the oceanic currents to two main constraints:
  • The geostrophic balance, which is typical of rotating systems and involves the pressure gradient acting in the fluid interior
  • The hydrostatic equilibrium, which holds in the absence of significant vertical accelerations of the fluid
All these ingredients contribute to the formulation of the so-called quasi-geostrophic dynamics, which, luckily, is (partially) amenable to an analytical treatment.
Fabio Cavallini, Fulvio Crisciani



Chapter 3. Quasi-Geostrophic Single-Layer Models

The crudest representation of currents and winds is that given by a single, constant-density fluid layer in relative motion with respect to the rotating Earth. In the case of the ocean, the layer is bounded from below by the sea floor and from above by the free surface of the sea. In the case of the atmosphere, the layer is bounded from below by the ground and from above by a hypothetical surface, above which the density of the atmosphere goes abruptly to zero. In both cases, the horizontal pressure gradient arises from the modulation of such surfaces with respect to the geoid, while the Coriolis force tends to arrange the flow in geostrophic balance with the pressure gradient.
The resulting motion is inertial, and potential vorticity is conserved. If Earth’s curvature is taken into account, for instance, by considering the beta-plane approximation, a fundamental consequence of potential vorticity conservation for motions crossing circles of latitude is the formation of Rossby waves. The above-described model is also the first step towards the formulation of the homogeneous model of wind-driven ocean circulation in which both the wind-stress forcing and vorticity dissipation are taken into account. In this case, the modulation of the surfaces which sandwich the geostrophic layer is ascribed, according to the classical Ekman theory, to the convergence/divergence of the marine current just below the free surface, caused by the wind stress, and just above the sea floor in the benthic layer, caused by friction. Also, the lateral diffusion of relative vorticity in the interior may be considered as a dissipative mechanism.
The solutions of the homogeneous model explain many fundamental phenomena of large-scale ocean circulation, in spite of the very simple picture which is usually adopted in describing the structure of the planetary wind field over the oceans. It is worth noting that the Ekman model of the benthic layer applies also, as it stands, to the lower atmosphere from the ground up to about one kilometre. The related convergence/divergence of the wind is responsible of the vertical motion of air masses, whose vicissitudes influence the weather.
Fabio Cavallini, Fulvio Crisciani

Chapter 4. Quasi-Geostrophic Two-Layer Model

The two-layer model is the first step towards the picture of the dynamics of a continuously stratified ocean and atmosphere, which are closer to common intuition. Models with a finite, and greater than two, number of layers are just generalizations of the simpler two-layer one – and their formulation is left to the interested reader.
In each layer, the dynamics is basically that of the homogeneous model illustrated in Chap. 3, with the exception of the interface between them, whose motion makes the layers mutually interacting. Thus, the quasi-geostrophic governing equations are coupled and may also include a reciprocal frictional retardation produced at the interface.
In deriving the energetics of this kind of system, a new quantity appears, the available potential energy. This is given by the difference between the potential energy of the flow, which presupposes a modulated interface, and the potential energy of the two-layered fluid in the absence of motion, which implies a flat interface. Thus, available potential energy can be transformed into kinetic energy and vice versa according to the thermal wind relation, while the potential energy in the absence of motion cannot be transformed into kinetic energy. For instance, the production of eddies and vortices in the ocean and the atmosphere is based on the transformation of the available potential energy of a certain “basic state” into the kinetic energy of time-dependent “disturbances”, which are shaped just like eddies and vortices.
Fabio Cavallini, Fulvio Crisciani

Chapter 5. Quasi-Geostrophic Models of Continuously Stratified Flows

The transition from layered flows to continuously stratified flows requires the use of thermodynamic equations basically because, in the latter models, neither horizontal current is depth or height independent nor density fluid is a constant. Thus, vertical integration of the incompressibility equation is no longer fit for explicitly yielding the vertical velocity field, and the latter must be inferred on the basis of further assumptions about large-scale geophysical flows. These assumptions concern the resort to the thermodynamic equations already expounded in Sec. 2.4.2. In particular, in the core of the oceanic water body, sea water density is conserved up to a high degree of approximation, so that (2.588) applies there; on the other hand, near the ground, the atmosphere undergoes thermal forcing because of the long-wave radiation emitted by the Earth and the greenhouse effect, so that (2.589) applies in the troposphere. In Chap. 5, the adiabatic quasi-geostrophic circulation on two typical ocean scales will be developed together with the analysis of steady, quasi-geostrophic atmospheric waves under the effect of a spatially modulated thermal forcing. Moreover, the effect of bathymetry on ocean currents and of topography on adiabatic winds are also taken into account in some typical cases.
Fabio Cavallini, Fulvio Crisciani


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