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Ocean dynamics. (English) Zbl 1296.86001

Berlin: Springer (ISBN 978-3-642-23449-1/hbk; 978-3-642-23450-7/ebook). xxiii, 704 p. (2012).
More than two thirds of the surface of the Earth are covered by oceans. Therefore oceans “have a profound influence on our climate, our weather, and the Earth’s ecosystems. They store and transport heat, CO\(_2\), nutrients, and, of course, water, as well as other important components determining the conditions of life on our planet” (foreword by Klaus Hasselmann). Because of their large storage capacity for heat, oceans damp short-term weather variations and convert them into longer-term climate variations. The important role of oceans for the climate recently led to a strong increase in ocean research. In particularly, more detailed numerical models of the ocean circulation were developed. However, oceans are highly complex systems which are characterized by many different spatial and temporal scales coupling to the Earth’s climate. All these scales cannot be taken into account in oceans models, even if most advanced super-computers are available. Thus approximations have to be introduced.
The aim of the present book is a thorough and careful representation of the ocean dynamics. The authors explain with high accuracy, both in physical argumentation and mathematics, what is important to understand the physics of the oceans and why the recently generally used approximations are made. The book starts with the fundamental fluid dynamic equations, and then, “gradually develops a series of pictures of the ocean system obtained by focusing on different aspects of the dynamics. Each approximation is carefully and rigorously introduced. The treatise leads, finally, ..., to a number of state-of-the-art ocean models designed for particular applications. But the emphasis is always on the fundamentals, rather than the numerical models.” (foreword by Klaus Hasselmann)
The mathematical equations describing the evolution of the states of oceans are the macroscopic conservation laws of partial masses, momentum, and internal energy, as used in classical hydrodynamics and thermodynamics. Thus, in the first part of the work, a phenomenological derivation of the hydrodynamics of seawater is presented based on the conservation of mass, momentum and energy of small volume elements moving in a fluid. Appropriate boundary conditions are discussed. There, thermodynamic and molecular properties of the oceans have to be taken from experiments or empirical considerations. The macroscopic conservation laws are found within the Lagrangian picture, and then transformed into an Eulerian form, which seems to be more useful for applications. The conservation equations are presented for inertial and rotating coordinate systems. Assuming local thermodynamic equilibrium, also the thermodynamics of the volume elements is reviewed. Especially the entropy budget, temperature equations, and vorticity problems are considered. Some differences between ocean and atmospheric thermodynamics are briefly discussed.
The system of hydrodynamic conservation laws derived in part I describes all ocean processes, from sound waves with time scales of milliseconds to the thermohaline circulation with periods of up to thousands of years. But usually, one is only interested in one kind of waves. This gives the possibility to introduce approximations which considerably simplify the conservation laws and the analyses. Thus, in part II the mode filtering of a system is described, which eliminates certain wave branches from all possible wave solutions (projection on the eigenvector of the interesting mode, consideration of diagnostic limits). However, the most practical way to simplify the conservation laws is to neglect terms which are small under certain conditions. Such approximations are also presented in part II, where equations are derived describing the large-scale oceanographic circulation (the so-called “primitive equations”, Boussinesq and hydrostatic models) or geostrophically balanced systems (at small Rossby number). For all approximations, the consequences for the energy cycle and the potential vorticity are explained.
Ocean waves, having time scales between fractions of seconds and many years and spatial scales from millimeters up to thousands of kilometers, can be considered as the most basic form of ocean motion. These waves can be grouped into three types, sound (speed about 1500 m/s), inertial-gravity (speed about 1 – 200 m/s), and Rossby waves (speed a few cm/s), which are caused by different physical reasons and have different physical dynamics. Part III starts with the study of sound waves, which are important for the transmission of information in the ocean and used in various measurement methods such as echo-sounding, acoustic floats, ADCP’s, and acoustic tomography. As their mathematical description is rather simple in comparison with the inertial-gravity and Rossby waves, their discussion is used to explain general concepts of linear waves, such as e.g.plane-wave propagation, dispersion and group velocity, propagation in waveguides and in inhomogeneous media. Next, the propagation characteristics of gravity waves in oceans in variable stratification and under the influence of boundary conditions are considered. Gravity waves are important for the adjustment of ocean currents towards geostrophic equilibrium. They cause mixing processes as well as the exchange of momentum and heat. Next ocean motions with horizontal scales much larger than the water depth are studied. They may be described using the hydrostatic model and are generally called “long waves” (e.g.long gravity, planetary, and topographic waves). Much attention is here paid to the wave propagation at the equator of the Earth. Some aspects of planetary wave propagation on a mean shear flow are discussed,e.g.the Doppler shift through advection and the change of the mean potential vorticity gradient. It is shown that a mean flow can cause an instability of a long wave. Further, special attention is payd to the description of slowly varying wave trains and wave-wave interaction, both based on the Lagrangian formalism. The equations of oceanic motions are nonlinear and contain forcing terms (atmospheric pressure, wind stress), either in the boundary conditions or in the dynamical expressions. These forcing terms were neglected in the book up to this point. Thus, at the end of part III, a conceptual framework to treat forced wave problems is presented. As example the forcing of internal gravity waves is considerd, where nonlinear effects play an important role - here waves interact and one wave may be excited by another one.
Part IV of the work deals with oceanic turbulence and eddies. In oceans, in the ocean-atmosphere transition layer, but also in the surrounding atmosphere, occur large fluctuations in space and time, that means turbulence. The cause of turbulence considered “lies in the instability of flows, which tends to occur when inertial forces become large compared to other forces in the momentum balance. ... Associated is a high sensitivity to small changes in initial and boundary conditions.” (p.331) Part IV starts with the discussion of Kolmogorov’s theory of homogeneous turbulence. Up to now, no comprehensive theory of turbulence exists, but there are many concepts to describe certain aspects of turbulence. Some concepts, which are mainly directed to practical applications – and not to a basic understanding of the turbulent flows – as well as some parametrizations based on the concepts are discussed in part IV. Basic ideas for small-scale three-dimensional turbulence (with or without gravity; inertial forces are larger than the viscous or gravity forces) as well as for large-scale (spatial scale larger than the Rossby radius) and (quasi) two-dimensional turbulence (meso-scales: spatial scale around the Rossby radius, temporal scale between the inertial period and a month) are considered. Kolmogorov’s theory is applied to two-dimensional isotropic systems. The mixing effect of meso-scale turbulence in the ocean is discussed. It is dealt with differences between advective and diffusive effects of turbulent mixing.
Classical concepts and models of ocean circulation are combined with newer material in part V. There first surface forcing functions of the large-scale ocean circulation are presented. The exact boundary conditions at the sea surfacs derived in part I are approximated or replaced by the “bulk formulae” which are based on experimental observations (surface fluxes of energy, heat, momentum, and freshwater). Effects of a prescribed windstress on the ocean in the presence of stratification and topography are discussed. Further, the theory of thermokline ventilation is reviewed. The physics of the oceanic meridional overturning (e.g.by thermohaline processes) is explained. The authors analyse the shortcomings of recent overturning models (box models of various configurations and zonally averaged models) and try to suggest better alternatives if possible. The end of Part V is devoted to the circulation in the Southern Ocean, especially its two interleaving branches, the predominantly wind-driven Antarctic Circumpolar Current and the wind- and thermohaline-driven overturning circulation. Several simplified models of the Southern Ocean circulation, driven by wind and surface density flux and influenced by stratification, topography and meso-scale eddies, are constructed.
Finally, an appendix reviews some of the needed mathematical tools, such as vector and tensor analysis, differential equations, statistical concepts, and also considerations of oblate spheroidal and spherical coordinate systems. A second appendix deals with several models of ocean circulation, which are used in the book.
The excellent comprehensive work is intended for graduate students of physical oceanography, meteorology, geophysics and general physics, as well as to any reader interested in a thorough discussion of ocean dynamics. The volume is also useful for researchers working in the field of climate physics.

MSC:

86-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geophysics
86A05 Hydrology, hydrography, oceanography
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76Fxx Turbulence
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
37N15 Dynamical systems in solid mechanics
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