Key concepts

• In this book, oceanic waters are deep waters (such that the waves are unaffected by the seabed) with straight or gently curving coastlines, without currents or obstacles such as islands, headlands and breakwaters.

• Under certain idealised conditions (constant wind blowing perpendicularly off a long and straight coastline over deep water), the significant wave height is determined by the wind, the distance to the upwind coastline (fetch) and the time since the wind started to blow (duration). So are the significant wave period and the energy density spectrum.

• Under these idealised conditions, the one-dimensional frequency spectrum has a universal shape: the JONSWAP spectrum for young seastates or the Pierson-Moskowitz spectrumfor fully developedsea states. The (one-sided) directional widthof the corresponding two-dimensional spectrumis typically 30°.

• To model waves under more realistic, arbitrary oceanic water conditions, the concepts of fetch and duration cannot be used. Instead, the spectral energy balance of the waves is used. It represents the time evolution of the wave spectrum, based on the propagation, generation, wave-wave interactions and dissipation of all individual wave components at the ocean surface.

• Conceptually, a Lagrangianapproach (based on wave rays) or an Eulerianapproach (based on a grid that is projected onto the ocean) can be used to formulate this energy balance. Owing to the interaction amongst the various wave components, the Eulerian approach is better suited for computations than the Lagrangian approach.

• Waves are generated by air-pressurefluctuations at the sea surface (not by wind friction), which are almost entirely due to wave-induced variations in the airflow(wind) just above the waves.

• […]

Key concepts

• In this book, coastal waters are waters that are shallow enough to affect the waves, adjacent to a coast, possibly with (small) islands, headlands, tidal flats, reefs, estuaries, harbours or other features, with time-varying water levels and ambient currents (induced by tides, or river discharge).

• Horizontal variations in water depth cause shoaling and refraction. Horizontal variations in amplitude cause diffraction.

• Shoaling is the variation of waves in their direction of propagation due to depth-induced changes of the group velocity in that direction. These changes in group velocity generally increase the wave amplitude as the waves propagate into shallower water (the propagation of wave energy slows down, resulting in ‘energy bunching’).

• Refraction is the turning of waves towards shallower water due to depth- or current-induced changes of the phase speed in the lateral direction (i.e., along the wave crest). For harmonic, long-crested waves in situations with parallel depth contours, Snel's law can be used to compute the wave direction. If the depth contours are not parallel, the wave direction should be computed with wave rays.

• Diffraction is the turning of waves towards areas with lower amplitudes due to amplitude changes along the wave crest. Diffraction is particularly strong along the geometric shadow line of obstacles such as islands, headlands and breakwaters. For long-crested, harmonic waves, propagating over a horizontal bottom, Huygens' principle, or a generalisation thereof, can be used to compute the diffraction pattern.

• A long-crested, harmonic wave that reflects off an obstacle, with or without energy dissipation, creates a (partially) standing wave.

• The simultaneous occurrence of shoaling, refraction, diffraction and reflection of long-crested, harmonic waves can be computed with the mild-slope equation.

• […]

Key concepts

• The conventional short-term description of ocean waves requires statistical stationarity. A time record of actual ocean waves (the fluctuating sea-surface elevation as a function of time at one location) needs therefore to be as short as possible. However, characterising the waves with any reliability requires averaging over a duration that is as long as possible. The compromise at sea is a record length of 15--30 min. If the record is longer, it should be divided into such segments (possibly overlapping; each assumed to be stationary).

• The wave condition in a stationary record can be characterised with average wave parameters, such as the significant wave height and the significant wave period.

• The significant wave height is fairly well correlated with ‘the’ wave height as estimated visually by experienced observers. This is not true for the significant wave period.

• A more complete description of the wave condition is obtained by approximating the time record of the surface elevation as the sum of a large number of statistically independent, harmonic waves (wave components). This concept is called the random-phase/amplitude model.

• The random-phase/amplitude model leads to the concept of the one-dimensional variance density spectrum, which shows how the variance of the sea-surface elevation is distributed over the frequencies of the wave components that create the surface fluctuations.

• If the situation is stationary and the surface elevations are Gaussian distributed, the variance density spectrum provides a complete statistical description of the waves.

• […]

Key concepts

• In this book, coastal waters are waters that are shallow enough to affect the waves, adjacent to a coast, possibly with (small) islands, headlands, tidal flats, reefs, estuaries, harbours or other features, with time-varying water levels and ambient currents (induced by tides, storm surges or river discharge).

• Under certain idealised conditions (constant wind blowing perpendicularly off a long and straight coastline, over shallow water with a constant depth), the significant wave height is determined by the wind speed, the distance to the upwind coastline (fetch), the time elapsed since the wind started to blow (duration) and the depth. So are the significant wave period and the energy density spectrum.

• Under these idealised conditions, the spectrum has a universal shape: the TMA spectrum, which is a generalised version of the JONSWAP spectrum (see Chapter 6). The directional width of this spectrum seems to be the same as in deep water (30°, one-sided width).

• Under more realistic, arbitrary coastal-water conditions, the spectral energy balance of the waves is used to compute the wave conditions. This shallow-water version of the energy balance is conceptually a straightforward extension of the energy balance in oceanic waters (see Chapter 6). It represents the time evolution of the wave spectrum, based on the propagation, generation, wave-wave interactions and dissipation of all spectral wave components individually.

• As in oceanic waters, an Eulerian representation (based on a computational grid projected onto the coastal region) should be used for computations with the spectral energy balance.

• Ambient currents can be accounted for by replacing the energy density with the action density (i.e., the energy density divided by the relative frequency) in the energy-balance equation and taking some other relatively simple (conceptually) measures.

• […]

Introduction

Measurements of the sea-surface elevation are almost always obtained with an electrical current in some instrument. This analogue signal can be transformed into an estimate of the variance density spectrum of the waves, using analogue systems, such as electronic circuits or optical equipment. However, with today's small and fast computers the analogue signal can also be transformed into a digital signal for a subsequent numerical analysis. The latter option has been accepted widely and it will be treated here.

The numerical analysis depends on the type of measurement. The most common and simplest measurement in this respect is a record of the sea-surface elevation at one location as a function of time (i.e., a one-dimensional record). Records like these are produced by instruments such as a heave buoy, a wave pole or a low-altitude altimeter. These can be analysed with a one-dimensional Fourier transform. Other types of measurements generate multivariate signals (i.e., several, simultaneously obtained, time records), e.g., the two slope signals of a pitch-and-roll buoy. Such signals require a cross-spectral analysis (e.g., Tucker and Pitt, 2001), or some other, advanced method (e.g., Hashimoto, 1997; Young, 1994; Pawka, 1983; Lygre and Krogstad, 1986 and many others). Two-dimensional images, e.g., from a surface-contouring radar, require a two-dimensional Fourier transform (e.g., Singleton, 1969) and moving images (e.g., those produced by a ship's radar) require a three-dimensional Fourier transform. Here, we consider only the simplest possible measurement: the sea-surface elevation at one location as a function of time.

Specification of the primitive equation model

The primitive equations, commonly understood as the Navier–Stokes equations in a rotating reference frame, with the vertical momentum equation replaced by the hydrostatic relation, form the basis of the most commonly used models of the ocean on regional to global scales. The flow is generally presumed to be incompressible but stratified; density variations are neglected save in the buoyancy terms. This is known as a shallow Boussinesq approximation. It is much more suitable for the oceans than for the atmosphere.

As in a truly incompressible fluid, there are no exchanges of energy between the flow and the internal energy of any fluid parcel, and a cubic meter of bottom water traveling at 1ms–1 is assumed to have the same momentum as a cubic meter of surface water traveling at the same speed. This has the effect of filtering sound waves, which would otherwise render practical large-scale models impractical.

It is sometimes useful to discard the hydrostatic approximation. This is necessary for the study of the details of deep convection in high latitudes and in some coastal ocean modeling applications. Marshall and co-workers (Marshall et al., 1997a, b) have suggested that nonhydrostatic models may be practical for simulation of the ocean on large scales, given the speed of modern computers. This would have the advantage of a single model for the global circulation, including formation of deep water.

The primitive equations by their nature present a number of obstacles to practical computation. For the most part, practical models differ in the fashion in which they cope with these problems.

Introduction

In recent years interest in the coastal ocean has increased throughout the world scientific community. Knowledge of the coastal oceans is directly relevant to issues of resource management and security, among others, and is therefore of broad societal interest, since a large proportion of the world's population lives near coastlines. In a purely scientific context, new instruments such as surface velocity mapping radars have been developed, and advances in computers and computing techniques have made detailed models of the coastal ocean practical.

The essential physical mechanisms that determine the most interesting aspects of coastal flow differ from season to season and from location to location. Coastal flows are affected by tides. Nonlinear effects can include rectification, so periodic tidal forcing of the coastal ocean can lead to significant residual steady flows. Interaction of the barotropic tide with topography can result in significant baroclinic motion. Interaction with motions on longer timescales can be nontrivial, and simply averaging over tidal periods or implementing other strategies for filtering out the relatively high frequency tidal motions may not be sufficient to deal with tidal interactions. River outflow, with its associated buoyancy fluxes, can be important.

Coastal upwelling, with its ecological implications, is an important feature of coastal circulation in many areas, as are the coastal jet and the ubiquitous coastally trapped waves, which propagate with the coast on their right as you face in the direction of propagation in the northern hemisphere (see Exercises 6.2 and 6.3).

Surface and bottom boundary layers are of the order of tens of meters thick, and often contain many of the phenomena of interest, beyond simple vertical transfer of momentum and heat.

This text grew out of notes for a course taught to graduate students in physical oceanography at the College of Oceanic and Atmospheric Sciences at Oregon State University. The students are typically in their second year of graduate school, having passed introductory courses in theoretical physical oceanography. This is the background assumed for the course. Most of the students at this point have seen some numerical analysis.

The course, and hence this text, is intended for all students of physical oceanography. Major emphasis is on those features that distinguish models of the ocean from other models in computational fluid mechanics. The intent is to examine ocean models critically, and determine what they do well and what they do poorly. We will ask when we can be confident that the model reflects nature, and when we can say that it is likely that we are looking at a feature of the model itself.

This is not a mathematics text as such, but it has a high mathematical content. Numerical analysis is introduced as needed. The reader may wish to consult supplementary references for basic numerical analysis of partial differential equations such as Sod (1985) (many typos, but reasonably current on fundamentals) or Richtmyer and Morton (1967), which is useful and commonly cited, though outdated. The reader might also find Isaacson and Keller (1966) or Allen et al. (1988) useful as general references. We will take examples from the two-volume work by Fletcher (1991) and the monograph by Leveque (1992). Durran (1999) contains useful and closely related material, from a slightly different viewpoint.