The vorticity equation

Irrotational flow can be established from a state of rest in an ideal incompressible fluid by the instantaneous transmission throughout the fluid of impulsive pressures from a moving boundary. If the boundary motion is subsequently arrested the motion everywhere ceases immediately. Kelvin's theorem (§2.10), that the kinetic energy of an irrotational flow is always smaller than that of any other flow consistent with the same boundary conditions, is a consequence of the fact that the number of degrees of freedom of irrotational motion is exactly the same as the number of degrees of freedom of the boundary itself. In a real fluid, however, there are typically an unlimited number of degrees of freedom, the flow is rotational, and the motion continues after the boundary stops moving. Kelvin (1867) therefore proposed the following definition of a vortex in a homogeneous incompressible fluid: ‘… a portion of fluid having any motion that it could not acquire by fluid pressure transmitted from its boundary’. Vorticity is actually a derived kinematic quantity, but its introduction greatly increases understanding of a complex flow and a knowledge of its distribution frequently permits the description of the fluid motion to be simplified.

When a small fluid particle is imagined to be suddenly solidified without change in its angular momentum, it continues to translate and rotate as a solid body. Its initial angular velocity of rotation is determined by its moment of inertia tensor, which depends on the particle shape.

Fluid mechanics impinges on practically all areas of human endeavour. But it is not easy to grasp its principles and ramifications in all of its diverse manifestations. Industrial applications usually require the numerical solution of the equations of motion of a fluid on a very large scale, perhaps coupled in a complicated manner to equations describing the response of solid structures in contact with the fluid. There has developed a tendency to regard the subject as defined solely by its governing equations whose treatment by numerical methods can furnish the solution of any problem.

There are actually many practical problems that are not yet amenable to full numerical evaluation in a reasonable time, even on the fastest of present-day computers. It is therefore important to have a proper theoretical understanding that will permit sensible simplifications to be made when formulating a problem. As in most technical subjects such understanding is acquired by detailed study of highly simplified ‘model problems’. Many of these problems fall within the realm of classical fluid mechanics, which is often criticised for its emphasis on ideal fluids and potential flow theory. The criticism is misplaced, however: For example, potential flow methods provide a good first approximation to airfoil theory, and ‘free-streamline’ theory (pioneered in its modern form by Chaplygin) permits the two-dimensional modelling of complex flows involving separation and jet formation.

Introduction

Steady free-streamline flows of water when gravitational forces can be neglected have been discussed in §3.7. Most unsteady free-streamline problems are intractable except by numerical means and generally become more so when gravitational forces are important. However, flows involving gravity where the unsteady motion is a ‘small’ perturbation of a relatively simple mean state occur frequently in the form of surface waves. In the absence of motion the free surface of a liquid in equilibrium under gravity is often ‘horizontal’. A disturbance applied locally that distorts the surface brings into play gravitational restoring forces that cause the disturbance to spread out over the surface in the form of ‘waves’. The waves carry energy away from the source region, propagating parallel to the mean free surface. The agitation produced by a passing wave and the energy flux is generally in the form of a transient disturbance of the fluid particles (around approximately closed particle paths), which are not in themselves transported to any great extent by the wave, and the influence of the wave on fluid at depths exceeding a characteristic wavelength tends to be negligible. In this section these general properties of surface gravity waves are discussed and illustrated by simple examples.

Conditions at the free surface

Consider the simplest case of water whose free surface in equilibrium can be regarded as horizontal and in the plane z = 0 of the coordinate axes (x, y, z), where z increases vertically upwards (Figure 5.1.1).

The fluid state

Consider a fluid that can be regarded as continuous and locally homogeneous at all levels of subdivision. At any time t and position x = (x1, x2, x3) the state of the fluid is defined when the velocity v and any two thermodynamic variables are specified. A fluid in unsteady motion, in which temperature and pressure vary with position and time, cannot strictly be in thermodynamic equilibrium, and it will be necessary to discuss how to define the thermodynamic properties of the small individual fluid particles of which the fluid may be supposed to consist.

The distinctive fluid property possessed by both liquids and gases is that these fluid particles can move freely relative to one another under the influence of applied forces or other externally imposed changes at the boundaries of the fluid. Five scalar partial differential equations are required for determining these motions. They are statements of conservation of mass, momentum, and energy, and they are to be solved subject to appropriate boundary and initial conditions, dependent on the problem at hand. This book is concerned with the use of these equations to formulate and analyse a wide range of model problems whose solutions will help the reader to understand the intricacies of fluid motion.

The balance equations describe changes of extensive quantities, the amounts of water, salt, momentum, and internal energy within a (infinitesimal) volume. They represent basic physical principles. Since the internal energy is not a directly measured quantity and since sea water can for many purposes be regarded as a nearly incompressible fluid it is convenient to introduce pressure and temperature as prognostic variables, instead of the density and specific internal energy. The basic equations for oceanic motions then consist of prognostic equations for the:

• pressure;

• velocity vector;

• temperature; and

• salinity.

There are six equations. They contain:

• “external” fields and parameters like the gravitational potential and earth's rotation rate, which need to be specified (or calculated);

• molecular diffusion coefficients, which need to be specified;

• thermodynamic coefficients, which need to be specified or are given by the equilibrium thermodynamic relations of Chapter 2. Most importantly, the density or equation of state must be specified.

These equations have to be augmented by appropriate boundary conditions. The equations, together with the boundary conditions, then determine the time evolution of any initial state. Nothing else is needed. It can, of course, be elucidating to study the evolution of other quantities, such as the circulation and vorticity. The evolution of such quantities is governed by theorems that are consequences of the basic equations. One important theorem is Ertel's potential vorticity theorem.

In this chapter we discuss the gravitational potential whose gradient enters the momentum equation. The potential is the sum of two terms, the potential of the Earth and the tidal potential caused by the Moon and Sun. For many problems, the gradient of the Earth's potential can be assumed to be a constant gravitational acceleration g0. However, the actual equipotential surfaces (the geoid) have a fairly complicated shape. The gravitational potential is determined by the mass distribution, as the solution of a Poisson equation. For a prescribed mass distribution this solution can be expressed in terms of the Green's function of the Poisson equation. The determination of the gravitational potential then becomes a mere matter of integration, with well-known solutions for a sphere and other simple distributions. For a self-attracting rotating body, like the Earth, the mass distribution is not known a priori but needs to be determined simultaneously with the gravitational potential. For a fluid body of constant density, the solution to this implicit problem is MacLaurin's ellipsoid. For the Earth, the geoid needs to be measured.

The basic geometry of the geoid is an oblate ellipsoid. This suggests oblate spheroidal coordinates as the most convenient coordinate system. Since the eccentricity of the geoid is small one can approximate the metric coefficients of this coordinate system such that they look like the metric coefficients of spherical coordinates.

In this chapter we discuss free linear waves on a sphere, again for an ideal fluid. Waves are a basic mechanism by which a fluid adjusts to changes and propagates momentum and energy. Many of the approximations that are applied to the basic equations of oceanic motions and that form the remainder of this book can be characterized by the wave types that they eliminate. Free linear waves, as opposed to forced waves, are solutions of homogeneous linear equations. They can be superimposed. Mathematically, these homogeneous linear equations form an eigenvalue problem. The eigenfunctions determine the wave form, the eigenvalues the dispersion relation. Specifically, we consider linear waves on a stably stratified motionless background state. The linearized equations have separable eigensolutions. The vertical eigenvalue is characterized by the speed of sound, gravitation, and stratification. The horizontal eigenvalue problem is characterized by rotation and the spherical geometry of the earth. We then classify the various wave solutions into:

• sound (or acoustic) waves;

• surface and internal gravity waves; and

• barotropic and baroclinic Rossby waves;

and consider the limit of short-wave solutions. This chapter owes much to Kamenkovich (1977) where more details and proofs can be found.

Basic kinematic concepts of waves, such as propagating and standing waves, and the geometric optics approximation are covered in Appendix E.