In the present chapter, the first of two chapters dealing with surface-piercing bodies, we impose an essential restriction that no bounded part of the free surface is separated from infinity. For the three-dimentional problem, this means that the free surface is a connected two-dimensional region (possibly multiply connected). In two dimensions, the assumption requires that there is only one surface-piercing body. However, a finite number of totally submerged bodies might be present in both cases. Supplementing this general restriction by one condition of technical nature or another, a method was developed (essentially by John) for proving the uniqueness theorem for various geometries and all values of ν > 0 (see Section 3.2). Provided the uniqueness is established, the machinery of integral equations developed in Section 3.1 leads to the unique solvability of the water-wave problem. Without the assumption about uniqueness, the integral equations method possibly does not guaranee the solvability for a certain sequence of values tending to infinity. Moreover, application of integral equations is rather tricky for semisubmerged bodies even when the uniqueness holds because of so-called irregular frequencies, which are also investigated in Section 3.1.

Integral Equations for Surface-Piercing Bodies

The essential point in application of the integral equation techniques to the case of a surface-piercing body is that the wetted boundary S is not a closed surface (contour) in three (two) dimensions, and it is bounded by a curve (a finite set of points) along the body's intersection with the free surface.

Results presented in Chapter 9 provide no details of the transient behavior of flows and do not yield direct hydrodynamic corollaries. However, there are situations in which information about developing waves in time can be extracted so that it leads to specific properties of hydrodynamic characteristics. In particular, an asymptotic analysis allows us to do this at least for two classes of disturbances. One of these classes constitutes rapidly stabilizing disturbances (this class includes brief disturbances as an important subclass), and the second class is formed by high-frequency disturbances. Both of these classes can be treated by using the same technique of two-scale asymptotic expansions for velocity potentials. The latter allows us to derive principal terms in asymptotics of some hydrodynamic characteristics.

Rapidly Stabilizing Surface Disturbances

In this section we are concerned with the effect of rapidly stabilizing disturbances on magnitudes characterizing unsteady water waves. For this purpose we consider several initial-boundary value problems describing waves caused by surface and underwater disturbances. The main example of the first kind is given by a pressure system applied to the free surface at the initial moment and rapidly stabilizing to a given distribution (a particular case is an impulsive pressure system). Underwater disturbances are presented by a source having a strength rapidly stabilizing in time to a constant value, and a rapidly stabilizing bottom movement. Complete asymptotic expansions in powers of a nondimensional small duration of disturbance are constructed for velocity potentials.

It was demonstrated in Section 3.1 that in the presence of a surface-piercing obstacle the water-wave problem is solvable for an arbitrary right-hand-side term in the Neumann condition on the obstacle's surface. However, there is an uncertainty about the set of frequencies providing the solvability. According to the proof given in Subsection 3.1.1, a sequence νn → ∞ (n = 1, 2, …) possibly exists such that for these exceptional values the solvability could be violated for some data given on the obstacle's surface. In particular, this must occur for values νn that are point eigenvalues of the water-wave problem embedded in the continuous spectrum (the latter is known to be the whole positive half-axis as is shown in the Examples section of the Introduction). If a value of the spectral parameter ν belongs to the point spectrum, then the homogeneous problem possesses a nontrivial solution with finite energy, or in other words, there is no uniqueness of solution for the nonhomogeneous problem.

In this chapter (see Section 4.1), we give examples of such non-uniqueness for the two-dimensional and axisymmetric problems, and so the exceptional values of ν do exist at least for some obstacle geometries. Moreover, for every ν > 0 a certain family of obstacles exhibiting the non-uniqueness property can be obtained. An essential point in all these examples is the presence of an isolated portion of the free surface inside the obstacle where the eigenmode waves are trapped.

The simplest “obstacle” to be placed into water is a point source. The corresponding velocity potential (up to a time-periodic factor) is usually referred to as the Green's function. This notion is crucial for the theory we are going to present in this book, since a wide class of time-harmonic velocity potentials (in particular, solutions to the water-wave problem) admit representations based on Green's function (see Section 1.3).

Potentials constructed by using Green's functions form the basis for such different topics as proving solvability theorems (see Chapters 2 and 3) and constructing examples of trapped waves (nontrivial solutions to homogeneous boundary value problems given in Chapter 4).

The plan of this chapter is as follows. Beginning with Green's functions of point sources in water of infinite (Subsection 1.1.1) and finite (Subsection 1.1.2) depths, we proceed with straight line sources and ring sources (Section 1.2) arising in two-dimensional problems and problems with axial symmetry, respectively. Green's representation of velocity potentials and related questions are given in Section 1.3. Bibliographical notes (Section 1.4) contain references to original papers treating the material of this chapter as well as other related works.

Three-Dimensional Problems of Point Sources

Point Source in Deep Water

In the present subsection, we consider in detail Green's function describing the point source in deep water. In Subsection 1.1.1.1, we define it as a solution to the water-wave problem having Dirac's measure as the right-hand-side term in the equation.

Seawater and air are viscous, conducting, compressible fluids. Yet large-scale oceanic and atmospheric circulations have such high Reynolds' numbers and such low aspect ratios that viscous stresses, heat conduction and nonhydrostatic accelerations may all be neglected. (The Mach number of ocean circulation is so low that the compressibility of seawater may also be neglected, but will be retained here in the interest of generality.) Subject to these approximations, the Navier–Stokes equations simplify to the so-called “Primitive Equations”. It is often convenient to express these equations in a coordinate system that substitutes pressure for height above or below a reference level. The Primitive Equations were for many years too complex for operational forecasting. They were further reduced by assuming low Rossby number flow, leading to a single equation for the propagation of the vertical component of vorticity – the “quasigeostrophic” equation. Now obsolete as a forecasting tool, this relatively simple equation retains great pedagogical value. To its credit, it is still competitive at predicting the tracks of tropical cyclones, if not their intensity.

The astronomical force that drives the ocean tides is essentially independent of depth, and so its effects may be modeled by unstratified Primitive Equations: the Laplace Tidal Equations. The external Froude number for the tides is so low that the “LTEs” are essentially linear. Combining the linear LTEs with the vast records of sea level elevation collected by satellite altimeters makes an ideal first test for inverse ocean modeling. […]

The dynamics are those of the standard σ-coordinate, Primitive-Equation model of a moist atmosphere on the sphere (Haltiner and Williams, 1980, p. 17). A penalty functional and the associated Euler–Lagrange equations are given in continuous form; CMFortran code for finite-difference forms is available at an anonymous ftp site. Details of the measurement functionals for reprocessed cloud-track wind observations (see §5.4), and the associated impulses in the adjoint equations, have been suppressed here. The details may be found in the code.

A data space search is the most efficient way to solve a linear, least-squares smoothing problem defined over a fixed time interval. The method exploits linearity, and so is unavailable for nonlinear dynamics, or for penalties other than least-squares. As discussed in Chapter 3, a data space search may be conducted on linear iterates of the nonlinear Euler–Lagrange equations. The existence of the nonlinear equations implies that the penalty is a smooth functional of the state, in which case a state space search may always be initiated. The nature of state space searches is intuitively clear, and their use is widespread. Conditioning degrades as the size of the state space gets very large. Collapsing the size of the state space by assuming “perfect” dynamics is the basis of “the” variational adjoint method: only initial values, boundary values and parameter values are varied. Preconditioning may in principle be effected by use of second-order variational equations, but even iterative construction of the state space preconditioner is unfeasible for highly realistic problems. Technique for numerical integration of variational equations is not a paramount consideration, but deserving of attention since it can be consuming of human time.

Operational forecasting is inherently sequential; data are constantly arriving and forecasts must be issued regularly. In such an environment, it is more natural to filter a model and data sequentially than to smooth them over a fixed interval. […]

Chapter 1 is a minimal course on assimilating data into models using the calculus of variations. The theory is introduced with a “toy” model in the form of a single linear partial differential equation of first order. The independent variables are a spatial coordinate, and time. The well-posedness of the mixed initial-boundary value problem or “forward model” is established, and the solution is expressed explicitly with the Green's function. The introduction of additional data renders the problem ill-posed. This difficulty is resolved by seeking a weighted least-squares best fit to all the information. The fitting criterion is a penalty functional that is quadratic in all the misfits to the various pieces of information, integrated over space and time as appropriate. The best-fit or “generalized inverse” is expressed explicitly with the representers for the penalty functional, and with the Green's function for the forward model. The behavior of the generalized inverse is examined for various limiting choices of weights. The smoothness of the inverse is seen to depend upon the nature of the weights, which will be subsequently identified as kernel inverses of error covariances. After reading Chapter 1, it is possible to carry out the first four computing exercises in Appendix A.

It is a long road from deriving the formulae for the generalized inverse of a model and data to seeing results. First experiments (McIntosh and Bennett, 1984) involved a linear barotropic model separated in time, simple coarsely-resolved numerical approximations, a handful of pointwise measurements of sea level and a serial computer. Contemporary models of oceanic and atmospheric circulation involve nonlinear dynamics and parameterizations, advanced high-resolution numerical approximations, vast quantities of data often of a complex nature, and parallel computers. Chapter 3 introduces some general principles for travelling this long road of implementation.

The first principle is accelerating the representer algorithm by task decomposition, that is, by simultaneous computation of representers on parallel processors. The objective may be either the full representer matrix as required by the direct algorithm, or a partial matrix for preconditioning the indirect algorithm. The calculation of an individual representer, or indeed any backward or forward integration, may itself be accelerated by domain decomposition, but this is a common challenge in modern numerical computation (Chandra et al., 2001; Pacheco, 1996) and will not be addressed here. Even without considering the coarse grain of task decomposition or the fine grain of domain decomposition, the direct and indirect representer algorithms for linear inverses are highly intricate. Schematics are provided here in the form of “time charts”.

Dynamical errors and input errors may be correlated in space or in time or in both. Error covariances must be convolved with adjoint variables. […]

An ocean data assimilation system in miniature

The pages of this book are filled with the mathematics of oceanic and atmospheric circulation models, observing systems and variational calculus. It would only be natural to ask: What is going on here, and is it really new? The answers are “regression” and hence “no”: almost every issue of any marine biology journal contains a variational ocean data assimilation system in miniature.

Linear regression in marine biology

The article “Repression of fecundity in the neritic copepod Acartia clausi exposed to the toxic dinoflagellate Alexandrium lusitanicum: relationship between feeding and egg production”, by Jörg Dutz, appeared in Marine Ecology Progress Series in 1998.

The calculus of variations uses Green's functions and representers to express the best fit to a linear model and data. Mathematical construction of the representers is devious, and the meaning of the representer solution to the “control problem” of Chapter 1 is not obvious. There is a geometrical interpretation, in terms of observable and unobservable degrees of freedom. Unobservability defines an orthogonality, and the representers span a finite-dimensional subspace of the space of all model solutions or “circulations”. The representers are in fact the observable degrees of freedom.

A statistical interpretation is also available: if the unknown errors in the model are regarded as random fields having prescribed means and covariances, then the representers are related, via the measurement processes, to the covariances of the circulations. Thus the representer solution to the variational problem is also the optimal linear interpolation, in time and space, of data from multivariate, inhomogeneous and nonstationary random fields. The minimal value of the penalty functional that defines the generalized inverse or control problem is a random number. It is the χ2 variable, if the prescribed error means or covariances are correct, and has one degree of freedom per datum. Measurements need not be pointwise values of the circulation; representers along with their geometrical and statistical interpretations may be constructed for all bounded linear measurement functionals.

Analysis of the conditioning of the determination of the representer amplitudes reveals those degrees of freedom which are the most stable with respect to the observations. […]