Exchanges at the Sea Surface

While the boundary between the ocean and the atmosphere has been extensively studied it is still not well understood. Heat, mass and momentum cross this boundary at a rate determined by many features of not only the sea surface motion but also the properties of the atmosphere and the ocean boundary layers on each side of the interface. There are some simplifications that can be made because the sharp variations are predominantly in the vertical, but there is a hierarchy of scales and processes at play which cannot be ignored in many applications. Central to understanding the processes at the boundary is gaining knowledge about the flux of momentum between the water and the air. The flux, which is the rate of transport of momentum across unit area, can be in either direction. In a frame of reference fixed to the earth, flux is mostly from the wind to the ocean currents, but less frequently, the flux is from the wind-waves or currents to the atmosphere.

The winds and currents have gradients of horizontal momentum. If we treat the sea surface as a sharp boundary between two fluids of different properties, we can model the flux of momentum from one of the fluids to the other as a drag force per unit area at the sea surface. This is the surface shear stress.

Introduction

Traditionally, the wind has been considered as the driving force for all ocean dynamics phenomena. Thus, we have the classical works on wind generated ocean circulation (Robinson 1963), and wind generated waves (Kinsman 1965). Almost half a century has elapsed, yet the prevailing thinking in the ocean community remains unchanged: Scientists engaged in ocean model development still fall into two categories – ocean circulation modellers who produce General Circulation Models (CGMs) and wind-wave modellers who have constructed, for example, Wave Modelling (WAM). In each respective endeavour, wind is considered as given and unchanged. This viewpoint is now changing. Research seems to be moving in the direction of treating the atmosphere and the oceans as a single system.

Wind-generated waves and currents are fundamental features of the world oceans. As the wind starts to blow over a resting ocean surface, it first generates small-scale wind-waves. These wind-waves extract momentum and energy from the wind field and modify the effective momentum flux into currents and also influence the wind field itself. The momentum flux will generate the drift currents, which in turn begin to influence the amplitude and directionality of the surface wind-wave field. The general ocean circulation pattern will also transport heat from one region to another to modify the global atmospheric dynamics. Thus, they form a closely knitted interacting trinity.

Introduction

Swell is formally defined as old wind sea that has been generated elsewhere. The term “old” is meant to signify that at some past time the swell energy, propagating through a given defined point, had been directly forced by the wind elsewhere. In view of the rather specific notion of “wave age” it might be better to think of swell as “escaped” wind sea. Having come from elsewhere, bearing the imprint of a different storm, swell may propagate at any speed relative to the wind or at any angle to the wind. Indeed, the vector difference in speed of the swell and peak wind sea may provide the only unambiguous criterion for identifying and separating swell from actively growing wind sea. Frequency dispersion separates the components of swell as they propagate away from the source area, and so swell tends to have a narrower spectrum than wind sea; but this provides only a qualitative selection criterion since the bandwidth of wind sea and swell may have considerable variation. For clarity we consider only two clearly defined cases of swell: (1) a distinct peak in the spectrum having peak phase speed greater than the wind component in the direction of propagation of the peak; (2) a distinct peak in the spectrum having peak phase velocity at an angle greater than 90 degrees to the wind.

Parametrization of the wind stress (drag) over the ocean is an essential issue in numerical analysis of the ocean–atmosphere interactions for climate modelling, satellite observations of air–sea fluxes, and other purposes. While wind stress over the ocean has been the subject of study for 50 years, the parametrization of the ocean surface momentum flux by a drag coefficient is still an uncertain process. There are many uncertainties not only in the way of proper parametrization, but in understanding the physical processes of the generation of stress by the wind system over the complicated nature of the ocean surface.

The drag coefficient has traditionally been treated as a function of the mean wind speed at a certain level, say at 10 m. Alternatively the coefficient can be represented by an aerodynamic roughness parameter. However, the spread of the observed values indicates that the question is not so simple. In 1955 Henry Charnock proposed a disarmingly simple expression for aerodynamic roughness, which was expressed in terms of the air friction velocity and the acceleration of gravity, independent of the state of ocean waves. While this expression has been widely used, there are important deviations and these were studied as a function of the wave age, a parameter representing the state of growth of wind waves relative to the local wind speed. Alarmingly, the trend of the observational values of aerodynamic roughness could be interpreted as opposite to the theoretical prediction according to the experiments considered.

Introduction

Before addressing the issues of the effects of surface tension on drag over the ocean, it is necessary to describe the physical properties and their controlling influences that make surface tension important to the air–sea boundary conditions. Surface tension is a special case of interfacial tension, that is, the pull between the molecules at the interface between two immiscible fluids. This pull exists because the molecules in the interfacial layer have fewer nearest neighbouring molecules of their own kind than do molecules in the bulk phase of either fluid. It is a physical quantity important to air–sea interaction because it affects many hydrodynamic phenomena, most notably, capillary and capillary–gravity waves. The single most important reason that these short waves (having wavelengths between about 0.1 to 30 cm) are influenced by surface tension is not really directly due to surface tension, but to surface elasticity, which is caused by the lowering of surface tension introduced by the addition of surfactants. Surfactant is a term coined by F. D. Snell that is short for surface-active agent. It describes molecular species that are more thermodynamically favoured to reside at the surface of a liquid. Typically, molecules that act as surfactants in aqueous solutions have two moieties, one hydrophilic (water-liking) and one hydrophobic (water-avoiding). In both terrestrial and aquatic environments that contain biological organisms, molecules composed of two such moieties are common.

This monograph is an attempt to address the theory of turbulence from the points of view of several disciplines. The authors are fully aware of the limited achievements here as compared with the task of understanding turbulence. Even though necessarily limited, the results in this book benefit from many years of work by the authors and from interdisciplinary exchanges among them and between them and others. We believe that it can be a useful guide on the long road toward understanding turbulence.

One of the objectives of this book is to let physicists and engineers know about the existing mathematical tools from which they might benefit. We would also like to help mathematicians learn what physical turbulence is about so that they can focus their research on problems of interest to physics and engineering as well as mathematics. We have tried to make the mathematical part accessible to the physicist and engineer, and the physical part accessible to the mathematician, without sacrificing rigor in either case. Although the rich intuition of physicists and engineers has served well to advance our still incomplete understanding of the mechanics of fluids, the rigorous mathematics introduced herein will serve to surmount the limitations of pure intuition. The work is predicated on the demonstrable fact that some of the abstract entities emerging from functional analysis of the Navier–Stokes equations represent real, physical observables: energy, enstrophy, and their decay with respect to time.

Introduction

As mentioned earlier in this text, we take for granted that the Navier–Stokes equations (NSE), together with the associated boundary and initial conditions, embody all the macroscopic physics of fluid flows. In particular, the evolution of any measured property of a turbulent flow must be relatable to the solutions of those equations. In turbulent flow regimes, the physical properties are universally recognized as randomly varying and characterized by some suitable probability distribution functions. In this and the following chapter, we discuss how those probability distribution functions (also called probability distributions or measures, or Borel measures, in the mathematical terminology; see Appendix A.1) are determined by the underlying Navier–Stokes equations. Although in many cases such distributions may not be known explicitly, their existence and many useful properties may be readily established. For many practical purposes, such partial knowledge may be all that is needed. Thus we note that the issue of an explicit form of the distribution function – in particular, whether this measure is unique or depends on the initial data – is still an incompletely solved mathematical problem. But there are enough firm results available assuring that many of the widely accepted experimental results are meaningful and in consonance with the theory of the Navier–Stokes equations.

For instance, measurements of various aspects of turbulent flows (e.g., the turbulent boundary layer) are actually measurements of time-averaged quantities.

Introduction

In principle, the idea that solutions of the Navier–Stokes equations (NSE) might be adequately represented in a finite-dimensional space arose as a result of the realization that the rapidly varying, high-wavenumber components of the turbulent flow decay so rapidly as to leave the energy-carrying (lower-wavenumber) modes unaffected. With the understanding gained from Kolmogorov's [1941a,b] phenomenological theory (see also Section 3), it appeared that, in 3-dimensional turbulent flows, only wavenumbers up to the cutoff value κd = (∈/ν3)1/4 need be considered. This is the boundary between the inertial range, which is dominated by the inertial term in the equation, and the dissipation range, which is dominated by the viscous term. As explained by Landau and Lifshitz [1971], the question is then reduced to finding the number of resolution elements needed to describe the velocity field in a volume – say, a cube of length ℓ0 on each side. Clearly, if the smallest resolved distance is to be ℓd = 1/κd, then the number of resolution elements is simply (ℓ0/ℓd)3. On adducing some phenomenological and intuitive arguments, it was argued that this ratio is Re9/4, where Re is the Reynolds number. An alternate way to count the number of active modes is as follows: since these modes are those in the inertial range, their frequency κ satisfies κ0 < κ < κd, with κ0 = 1/ℓ0; we conclude that, for κd/κ0 large, that number is of the order of (κd/κ0)3 = (ℓ0/ℓd)3.