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Global Guide to Tropical Cyclone Forecasting:
CHAPTER 4: THE HABITATION LAYER

4.3 SEA STATE (WIND WAVES)

4.3.1 Wave Generation Mechanisms

The sea state at any location is normally comprised of a combination of wind waves, or "sea", and swell. Sea waves are short gravity waves generated by local wind conditions. They have a chaotic appearance with sharp, angular tops, irregular heights, short distances between crests and many smaller waves are superimposed onto the larger ones. The individual crests have a short lifetime, it is not possible to follow any one wave for a distance greater than a few times the distance between crests, and they move in many directions, at times up to 50o from the wind direction.

When the wind blows across an initially calm ocean surface, small eddies near the surface generate local pressure fluctuations that perturb the water surface and generate small ripples, of a few millimetres in height. Those ripples that have phase velocities equal to the velocities of the atmospheric eddies will continue to grow to the stage where different energy transfer mechanisms take over. The wind exerts a shear stress on the windward side of a finite amplitude wave and also induces a pressure differential across the wave. Both mechanisms contribute to continued growth of the wave. In addition, smaller waves form on the larger wave by local eddies and can contribute to its growth by non-linear interactions. Thus the growth of a sea in response to an imposed wind starts with small/high-frequency waves and develops towards larger/lower frequency waves.

As the sea develops, the longer wavelength waves will begin to move faster than the wind and lose energy back to the atmosphere, so that the maximum wavelength is limited by the wind speed. Other waves break and dissipate their energy as turbulence and heat in the ocean. Ultimately, a quasi-balanced state is reached in which the energy being supplied to waves from the wind is equal to that being lost by the waves, and the sea is said to be "fully developed". Seas that have not reached this steady state are said to be "partially developed". Whether a sea becomes fully developed depends on the duration of reasonably constant winds over a suitably large area of ocean, or "fetch". In tropical cyclone conditions, highly variable winds blow over limited fetches, so that the seas are normally in a constant state of change and remain partially developed.

To a first order approximation, the total energy (potential and kinetic) of a wave is proportional to twice the square of its wave height (Kinsman, 1965). Thus the maximum wave height is limited by the available kinetic energy in the wind, and it is roughly proportional to wind speed.

As sea waves propagate away from their generation area, they transform gradually into swell. This can occur by a combination of non-linear interaction leading to growth of certain waves at the expense of others, and gradual separation of waves of different wavelength that propagate at different speeds. Swell is thus characterised by regularly spaced, smooth wave crests which are of similar height, move in a uniform direction, and can be followed for long periods of time.

4.3.2 Coastal Effects

For major tropical cyclones of Saffir-Simpson category 3 or higher, it is not unusual to find wave heights of over 15 m in deep water. These waves break in water depths of approximately 20 m and "feel" the ocean bottom at much greater depths, well offshore on shallow continental shelves.

At the coastline the waves respond to the local bathymetry within hundreds of metres. If the near shore waters are deep and the shore itself a near vertical cliff rising from deep waters, then waves can reflect from the cliff instead of breaking. They increase in height, however, gaining potential energy at the expense of kinetic energy, and may overtop the cliff causing extensive damage immediately inland. When combined with lifted water from a storm surge, the extreme consequences may result. Typical of this behaviour are occurrences along some of the northern shores of the island of Jamaica. The shore has vertical cliffs which drop from several meters above mean sea level to tens of meters below sea level. Wind-waves generated by historical cyclones tracking north of the island have suffered reflection with their upper portions overtopping the cliffs and causing considerable destruction.

Figure 4.12. Illustration of the effects of breaking waves and wave set-up, superimposed on a storm surge (after Bureau of Meteorology, 1978).

The swell propagating away from a tropical cyclone can affect coastal regions hundreds of kilometres away. Frequently the resulting damage to coastal structures arises from the long duration of such swell waves, which causes erosion of foundations and undermining of coastal structures. This is especially a danger with large and slow moving tropical cyclones.

If the coast has an extensive and shallow continental shelf, the large wind waves will break well offshore and cause only minor residual shore effects. New wind waves can reform over the continental shelf, but these are limited in height by the shallow water. Breaking waves at the shore often force water inshore leading to a phenomenon known as wave set-up (Fig. 4.12). For shallow beach frontages, such wave set-up can significantly increase the impact of storm tides. Local obstructions, such as angled sea-walls or small inlets can focus the wave energy into a small region with dramatically increased effects.

4.3.3 Wave Forecasting Methods

Theories have been formulated for the initial generation of short gravity wind waves across a placid sea surface. However, estimating further development is fraught with difficulties. Application of simplified techniques usually requires specification of fetch length and wind duration. Unfortunately, in tropical cyclones the winds constantly change, fetches are far from ideal, and it is extremely difficult to estimate fetch length for a given situation. Further, forecasting of tropical cyclone induced sea state across coastal waters is a formidable task due to the complexity of the dynamics and of the inshore coastal topography and offshore bathymetry. Considerable understanding is required on how the sea surface responds to meteorological and bathymetric influences, together with detailed observations of the surface wind structure of the tropical cyclone.

Although sophisticated techniques exist for wave forecasting, most require computing facilities, together with an input data base for local geography, calibration and testing, that are well beyond the resources of many weather offices. The tenaciousness of scientists to formulate simplified products or models in this regard must be applauded, but engineers and forecasters need to be aware that the real world can be far different from the idealised cases that are used.

In the fetch region, the sea is chaotic and from a multitude of measurements statistics can be applied to specify a probability distribution of the wave periods and heights. The significant wave height is defined as the average of the highest one third of all waves in the region, and the significant wave period is the corresponding mean period It should be recognised that there is considerable scatter with the statistics for "significant" wave values.

Empirical nomograms can provide a first order approximation of the significant wave heights and periods according to fetch length and duration of wind (e.g. the SMB method of Bretschneider, 1957). The SMB method is semi-empirical but more sophisticated discrete spectral and parametric approaches have been developed, (Kinsman, 1965). Most such nomograms are restricted to deep water regions that are unaffected by a coast or land. They consider wave generation with a uniform and constant wind in deep, or else in shallow and constant depth waters (Shore Protection Manual, 1973).

The empirical work is biased toward extratropical storms where fetch and duration parameters can be determined, albeit coarsely. These parameters, however, are difficult to ascertain for tropical cyclones because of their small size, movement (i.e., a moving fetch), and the nearly circular and rapidly shifting wind field. The details of the core region are difficult to estimate accurately over the ocean, though new observing methods, such as microwave imagers, may help in the future. The small tropical cyclone core creates large gradients in wind-wave characteristics, the swell from one region may be propagating against the wind in another part, ensuring chaotic wave dynamics. Furthermore, near shore phenomena such as shoaling, refraction, breaking waves, wave run-up, reflection from shore obstacles, interactions with local currents, etc. are extremely difficult to treat empirically. These phenomena require specialised, local treatment of an engineering nature that is time consuming and difficult to manage in real-time.

A crude but quick estimate of the maximum significant wave height, in metres, generated by a tropical cyclone in deep water may be obtained from:

  (4.1)

where P_n-P_c is the pressure drop from the environment to the cyclone centre in hPa (Hsu, 1991). This maximum value applies only near the maximum wind region, and wave height rapidly attenuates outside this region. In shallow coastal waters, these maximum height waves normally will break well before reaching the coast.

Alternatively, Table 4.1 provides a quick reference for estimating the state of the sea under a variety of wind and fetch conditions (Bureau of Meteorology, 1978; see also Section 8.4)).

A simple, coarse, first-order estimate of wave heights near the coast can be obtained from the idealised formula:

   (4.2)

where H is the breaking wave height (crest to trough), d is an estimate of the average bathymetric depth to a couple of hundred metres offshore, T is the tidal deviation from mean sea level, and h is the storm-surge height (Shore Protection Manual, 1984). Thus, Eq. 4.2 indicates that the maximum height of waves in shallow water is, to a first approximation, about 75% of the local oceanic depth.

The sensitivity to timing of cyclone landfall, relative to the tidal cycle and the degree of storm surge are readily apparent. For example, in a region of 3 m average oceanic depth and ±1 m tidal variation, the maximum wave height varies from 1.6 to 3.2 m between high and low tide. If we add a surge of 5 m at high tide the waves could exceed 7 m. Thus, in this illustrative example, near-shore structures could experience ocean water to a height of 13 m above mean sea level (Fig. 4.12)¹. Such breaking waves can be devastating for structures that are just offshore, such as jetties and buildings on stilts. Inland from the coast, sand dunes, levees, etc., complicate the breaking wave height, but, assuming reformation of wind waves over flooded land, our surge depth of 5 m could support 4 m waves, which would be rapidly attenuated inland.

* For Beaufort scale, see section 9.4

These simple methods of forecasting sea state generated by tropical cyclones are imprecise and substantial errors must be tolerated and accounted for. One approach is to use sea-state models to estimate the worst case possibilities for a given type of tropical cyclone (in a manner similar to the storm-surge MEOW concept). Using a sophisticated wave model, wave forecasts are generated for a family of landfalling tropical cyclones with required characteristics and an understanding of the potential forecast errors. An envelope of worst-case wave conditions may then be produced for forecast guidance. Using the MEOW (Section 4.2.4) to estimate the maximum surges height, Eq. 4.2 can be used to quickly estimate the worst case effects of breaking waves.

Forecasters also may wish to record areas of particular concern for wave damage. Where coastal cliffs rise from deep waters, the expectation should be of wave activity overtopping the cliffs and causing potentially serious damage a short distance inland. Regions protected by outer reefs, or by shallow water extending several kilometres from the shore, may be protected from massive ocean waves, but may also regenerate local, shallow-water waves. Local inlets may cause wave focussing. Some structures may be particularly vulnerable to waves breaking on top of a local storm surge.

¹ Note that the height of the waves above MSL is only about half their height as given in equation 4.2

Contents Chapter 4.4