Eric B. Kraus’s scientific contributions

Both Cartesian tensor and vector notation will be used in this text. The notation xi means the i-component of the vector x = (x1 x2, x3). When used in the argument of a function [e.g., f(xi)], xi, represents the whole vector, so that f(xi) stands for f(xl,x2,x3). Repeated indices indicate summations over all coordinate directions, (uiui = u2i = u21 + u22 + u23). Two special and frequently used tensors are the unit tensor δij and the alternating tensor ∊ijk. The unit tensor has components equal to unity for i = j and zero for i ≠ j. The alternating tensor has components equal to +1 when the indices are in cyclical sequence 1, 2, 3 or 2, 3, 1 or 3, 1,2; equal to -1 when the indices are not cyclical; and equal to zero when two indices are the same. The vorticity vector is defined by the relation The symbol = is used throughout to represent a definition or identity. Conditions near the sea surface are usually very anisotropic. It is often desirable to distinguish between the horizontal and vertical directions. We shall do so by using an x, y, z coordinate system with the origin at mean sea level and the z -axis pointing upward. Unless otherwise specified, the x and y directions will be toward east and north. The vertical velocity will be denoted by W; the horizontal velocity by the vector U with components U and V. Unity vectors in the x, y, z directions are denoted by i, j, k. The usual vector operation symbols will be used only to represent operations within the horizontal plane. For example, In a fluid one has to distinguish between local changes and changes that are experienced by an individual fluid element as it moves about. The former can be recorded by a fixed sensor and is represented by the partial time differential. The individual change could only be recorded by a sensor that would float with the element. It is denoted by the total time differential In a treatise that covers such a variety of topics, some use of the same symbols for different properties is inevitable.

This chapter deals with convective fluxes of sensible heat, moisture, and salinity that originate at the sea surface. In Section 8.1 we consider the relative influence of oceanic and atmospheric variability upon these fluxes. The general character of deep convection and its occurrence in the polar oceans is discussed in Section 8.2. The case of deep convection over the ocean in the tropical atmosphere, which is somewhat more complicated because of compressibility and cloud formation, is discussed in Section 8.3. Finally, in Section 8.4, we consider some of the long-term ocean-atmosphere feedback processes. Kinetic energy in the atmosphere-ocean system is derived mainly from an upward flux of buoyancy. The resulting redistribution of mass reduces available potential energy APE and lowers the centre of gravity. In turn, APE is generated, primarily by non-adiabatic processes: unequal absorption and emission of radiation; local release of latent heat in the atmosphere; local salinity changes in the ocean; and unequal heat conduction from the boundaries. The total mass of the oceans is about 280 times that of the atmosphere; their heat capacity is nearly 1200 times larger. Oceanic response times to external forcing are correspondingly slower. Although the annual irradiation cycle affects only a small part of the water mass, the thermal inertia is strong enough to prevent large or fast temperature variations. It is well known that this has a dominant influence on the whole terrestrial climate. This influence is particularly strong in the marine temperate regions. Figure 5.10 showed that even the daily temperature changes of the surface waters are smaller than those in the air. By virtue of their mechanical and thermal inertia, the oceans tend to play the role of a flywheel in the air-sea system. The atmosphere is the more volatile and more variable partner. It supplies mechanical energy to the oceans at a rate that has a very skewed distribution in space and time because the work of the wind stress is proportional to the third power of the windspeed. This creates a strong bias in favour of restricted stormy areas.

The earth receives virtually all of its energy from the sun in the form of electromagnetic radiation. This radiation is absorbed, reflected, and scattered by the earth’s surface, the ocean, and the atmosphere. The absorbed radiation is transformed into heat and other forms of energy, and eventually it is returned to space as low-temperature terrestrial radiation. It is clear that radiation is of fundamental importance to atmosphere-ocean interaction. There exists an adequate body of literature on the subject from an introductory treatment by Fleagle and Businger (1980) to specialized monographs by Kondratjev (1969), Liou (1980), and Goody and Yung (1989). Here it will suffice to introduce the basic concepts and focus on the applications to the air-sea interface. Radiation in the atmosphere and ocean comes from all directions simultaneously. The radiation energy per unit time coming from a specific direction and passing through a unit area perpendicular to that direction is called the radiance, I. The irradiance, Fi, or radiant flux density, is the radiant energy that passes through a unit horizontal area per unit time coming from all directions above it. Therefore where θ is the zenith angle and dω is an infinitesimal solid angle. The cos θ reflects the projection of the horizontal unit area into the direction from where I comes. The limits 0 and 2π of the integral reflect the hemisphere of directions above the unit area. When the radiance is independent of direction it is called isotropic. Equation may then be integrated to yield The irradiance from below the unit area is also called exitance and is denoted by Fe. The net irradiance, Fn, is defined by For isotropic radiance, the net irradiance Fn = 0. The fluxes are positive when upward and negative when downward. The interactions between radiation and matter may take various forms. They include refraction, reflection, scattering, diffraction, absorption, and emission. All these interactions are described by the theory of electromagnetic waves (e.g., Panofsky and Phillips, 1962). The full theory will not be developed here, but a number of basic and useful relations will be introduced to describe the characteristics of the interactions mentioned previously.

Kinetic energy flows almost exclusively downward, from the atmosphere into the ocean. The upward flux of energy is thermal, and that will be the topic of our concluding chapter. In the present chapter, we shall deal with the effects of kinetic energy inputs into the ocean. Although this requires some discussion of the different types of oceanic perturbations, our treatment of these topics is necessarily brief and incomplete. We are not concerned with details of the motion pattern in the deeper ocean or with processes involving friction and non-adiabatic mixing in the interior. These processes are essential for an understanding of ocean circulations. They are treated in general oceanographic textbooks and in many monographs that deal specifically with these subjects. Surface stress and air pressure variations produce surface wind waves along with a variety of other wave forms. Most of these waves are relatively slow, with periods that can be measured in hours, days, or even years. The amplitude of internal gravity waves in the oceans is often much larger than that of surface waves and their wavelengths tend to be in the kilometer range. The square of the amplitude-wavenumber product is usually a very small quantity. This makes first order approximations appropriate for many purposes. It justifies use of the hydrostatic approximation and of the linear equations as a basis for the following discussion. To do so, it is necessary to represent the unspecified forcing terms on the right-hand side of those equations in a linearized form. An algorithm for the inclusion of the various atmospheric inputs as a linearized boundary condition in the equations of motion for the ocean is discussed in Section 7.1. Section 7.2 describes a two-layer ocean model. Systems of this type are convenient for the conceptual consideration of atmosphere-ocean interactions, because the wind affects the ocean primarily through action upon the surface mixed-layer. Internal waves, the topic of Section 7.3, are ubiquitous both in the ocean and in the atmosphere. Essentially, sea surface gravity waves can be viewed as internal waves at the interface between two fluids of very unequal density.

The term planetary boundary layer (PBL) is often used as a synonym for the Ekman layer (i.e., for the region in which the convergence of the vertical flux of momentum is of the same order as the Coriolis force and the pressure gradient). The definition favored in this chapter is somewhat broader and includes the regions on both sides of the interface in which the vertical fluxes, not only of momentum but also of heat, moisture, and salinity, determine the vertical distribution of these properties. Such a definition may suggest as many different boundary layers as there are transported properties. This may be the case, but the various fluxes are coupled with each other to such an extent that it is usually possible to define a single layer in which interface effects remain significant. In Section 6.1 we shall deal first with the classic Ekman treatment of the steady-state motion field above and below the boundary of two incompressible, rotating laminar fluids. This will be followed by a discussion of transients and of integral horizontal transports. Section 6.2 deals with coherent structures—longitudinal rolls, thermal plumes, convection cells, and so on—that are common phenomena, particularly in the gravitationally unstable PBL. This is followed in Section 6.3 by a discussion of various parameterization schemes and models that have been used either to represent vertical fluxes or vertical profiles of conservative properties. Mixed-layer models, which are the topic of Section 6.4, are distinguished from these parameterization schemes by use of vertical integrals of the conservative properties. The resulting gain in simplicity is offset to some extent by a loss of detail. In Section 6.5 we shall discuss the merits and drawbacks of the different approaches in the two preceding sections. During his arctic expeditions on the ship Fram, late in the nineteenth century, Nansen noticed that the pack ice drifted at an angle of about 20-40° to the right of the surface wind. He interpreted this correctly as being due to the deflecting force of the earth’s rotation, and inferred further that the water below, which is dragged along by the ice, must be driven even further to the right.

A thermodynamic phase specifies a substance or mixture of substances that occupies a limited volume with a characteristic temperature, T, a definite pressure, p, at its boundary, and a composition that can be described at any moment by the masses of its constituents. A well-mixed sample of air or of sea water in contact with our instruments, therefore, is defined as a phase. Different phases, brought together in a thermodynamic system, tend to change until an equilibrium has been established. This involves an exchange of energy and matter between the originally different phases. If the process is isolated from external influences, then it results in an increase of entropy. In fact, entropy is always generated by mixing (i.e., by the transport of a conservative property down its own gradient). This can be said to include the transport of momentum by viscosity. Entropy can be diminished locally when gradients are sharpened by a flux in the opposite direction. In nature this can be brought about by a coupling of the transports of different properties. The potential for change that exists in a physical system is the subject of the thermodynamics of irreversible processes. The ocean and the atmosphere are open systems which can exchange matter through their boundaries. Though the system that they form together is closed—at least on the time scales with which we are concerned—it is not isolated, being subject to heat exchange with the surrounding universe. The inhomogeneity and variable temperature of both oceans and atmospheres cause further complications. Although equilibrium thermodynamics alone cannot provide information about the rate or the mechanism of energy transformations, it does provide, in the First Law, a constraint that must always be obeyed. It also allows us to predict the general direction of development in limited regions that only interact slowly with their surroundings. The properties or ‘coordinates’ which specify a thermodynamic phase are not independent. Anyone of them can be expressed as a function of all the others by an equation of state. In a liquid, the state is strongly affected by molecular interaction.

The atmosphere and the ocean are in intimate contact at their interface, where momentum, water substance, heat, and trace constituents are exchanged. This exchange is often modest when a light breeze strokes the surface; sometimes the processes are violent, when gale force winds sweep up ocean spray into the atmosphere and when braking waves engulf air into the ocean. It may even appear that the transition between ocean and atmosphere becomes gradual and indistinct. The transition from ocean to atmosphere is usually an abrupt transition of one fluid to another. The interface may then be considered a continuous material surface. On both sides of the interface the fluids are usually in turbulent motion and properties are transported readily, but upon approaching the interface turbulence is largely suppressed so that on both sides of the interface a very thin layer exists where the molecular diffusion coefficients play a major role in the transport. The interface is consequently a significant barrier to the transport from ocean to atmosphere and vice versa, with little or no turbulent transport of scalar quantities across it. The quantitative determination of the thickness of the molecular sublayers and the strength of the gradients and shear layers within them are discussed in Section 5.1. We also examine the transition from the molecular sublayers to the well-mixed turbulent layers that exist beyond them, and the structure of these turbulent layers on either side of the interface. In Section 5.2 we discuss the effect of stratification on the structure of these surface layers. Some of the nonstationary interactions between the wind and the sea surface are described in Section 5.3. Sections 5.4 and 5.5 deal with practical applications: a formulation of gas transfer across the interface and of the sea surface temperature. Several observational techniques are discussed in Section 5.6.

Rhythmic and monotonously repetitive, but quite unpredictable in its details, the structure of the sea surface is an epitome of the natural world. Surface waves have been studied actively by mathematicians and physicists since the dawn of modern science. Though the phenomenon seems deceptively simple, it cannot be explained or predicted rigorously by existing theories. Nonlinear interactions between wind, waves, and currents cause theoretical problems as well as make it difficult to obtain comprehensive, interactive data sets. In response to wind and pressure changes at the air-sea interface, the ocean reacts with waves that occupy some nine spectral decades: from capillary waves, which undulate within a fraction of a second over distances smaller than one centimeter, to planetary waves with periods measured in years and wavelengths of thousands of kilometers. The dynamics of all these waves can be related to the set of equations discussed in Section 4.1. For that reason, a consideration of all wave forms could have been combined in the same chapter, but we found it more convenient to divide the subject into two parts. The present chapter deals exclusively with wind-generated waves at the sea surface. They determine the small-scale configuration of the air-sea interface and that affects the turbulent transfers, which are the topic of the following chapter. On the other hand, information and energy transports from the sea surface into the ocean interior by internal and inertial waves, depend upon the state of the upper layers of the ocean. This made it desirable to discuss these wave forms in Chapter 7, after the consideration of planetary boundary layers in Chapter 6. Small-amplitude or linear, harmonic surface waves are considered in Section 4.2. Analysis of these waves has been the classic approach to the topic. Linear waves represent an idealized abstraction, but their analysis does provide basic insights into actual wave dynamics. Linear approximations have to be abandoned when one considers the energy and the momentum of wave fields. This is the topic of Section 4.3. In Section 4.4 we discuss the various sources and sinks of wave energy and momentum.

With both the growing importance of integrating studies of air-sea interaction and the interest in the general problem of global warming, the appearance of the second edition of this popular text is especially welcome. Thoroughly updated and revised, the authors have retained the accessible, comprehensive expository style that distinguished the earlier edition. Topics include the state of matter near the interface, radiation, surface wind waves, turbulent transfer near the interface, the planetary boundary layer, atmospherically-forced perturbations in the oceans, and large-scale forcing by sea surface buoyancy fluxes. This book will be welcomed by students and professionals in meteorology, physical oceanography, physics and ocean engineering.