Vi

I 28 -- 30 ¡26 -- 28 ¡24 -- 26 ¡22 -- 24 ¡20 -- 22 18 -- 20 16 -- 18 14 -- 16 12 -- 14 10 -- 12 8.0 - 10 6 0 - 8.0 4 0 - 6.0 | 2.0 - 4.0 0 - 2.0 -2.0 -- 0

Temperature in deg. C at 150 deg.W

Temperature in deg. C at 150 deg.W

Figure 2.65a,b. Annual average temperature T (above) and salinity S (below) along a south-north section (150°W) through the Pacific Ocean (based on Levitus and Boyer, 1994; data source: University of Columbia, 1998).

To provide a rough picture of the classes of wave motion, Fig. 2.66 indicates the frequency spectrum of wave motion in the oceans (Roll, 1957). The harmonic waves in the spectral decomposition may be characterised by their wavelength, Xw, the phase velocity, Uw, and the amplitude at the water surface, a. Figure 2.66 shows a time-averaged estimate of the distribution of amplitudes, a, on cycle times, Tw = AwUw'1. For waters of depth h much larger than the wavelength Xw, harmonic waves governed by the gravity and surface tension forces are characterised by the relation gK + ^nL

2n KvP

Figure 2.66. Sketch of the main features of the spectral decomposition of ocean wave amplitudes, averaged over time and position (based on Roll, 1957).

between wave velocity and wavelength, provided that the viscous forces in the water are neglected. The gravitational acceleration at the Earth's surface is g = 9.81 m s-1, and the surface tension of water against air, yw, has a value of about 0.073 N m-1. It follows from (2.19), which is shown graphically in Fig. 2.67, that Uw must exceed a minimum value of about 0.23 s-1 and that two values of Xw are associated with each Uw above the minimum value. The branch of wavelengths smaller than about 0.017 m (corresponding to the minimum Uw ) is called capillary waves, since they are mainly governed by the surface tension. The branch of wavelengths above Xw = 0.017 m are the gravity waves, for which the term involving g in (2.19) dominates.

At cycle times around 2 min, pulsations in the amplitude of gravity waves formed earlier by interaction with the wind field may give rise to the surf beat phenomenon observed at shores (Sverdrup and Munk, 1947; Munk, 1980). The broad peak in the wave spectrum in Fig. 2.66, centred around Tw = 20 min, includes the flood-producing tsunamis occurring quite frequently,

e.g. in the Pacific Ocean, as a result of sudden movements of the ocean floor caused by earthquakes (Neumann and Pierson, 1966).

Figure 2.67. Theoretical relationship between wave (phase) velocity, Uw, and wavelength, Xw, under the influence of gravity and surface tension, but neglecting viscosity (based on Roll, 1957).

The peaks at 12 and 24 h represent the tidal waves created by the time-dependence of the gravitational fields of various celestial bodies. The most important tides are caused by the Moon and the Sun in connection with the variations in distance from a given spot on the Earth's surface, resulting from the rotation of the Earth. Much longer tidal periods may be associated with orbital motion (lunar month, year, etc.). The theoretical description of waves and tides is discussed in section 2.D.

Joint models of general circulation in oceans and atmosphere

Early models coupling atmospheric and ocean circulation models did not include any detailed topography, had fixed amounts of cloud cover, and prescribed a radiation field corresponding to constant carbon dioxide levels, constant water vapour and constant ozone. Oceanic equations included temperature, salinity and ice, and the hydrological model was fairly detailed on precipitation, evaporation, soil moisture, snow cover, sea ice, melting processes and river run-off (e.g. assuming that when soil moisture exceeded a certain value, the excess would run off to the nearest ocean). The spatial grids were typically around 500 km by 500 km at the Equator, with around 10 vertical layers, and the time steps were of the order of a month for oceans, but of the order of 6 h for the atmosphere (Bryan, 1969; Manabe, 1971; Wetherald and Manabe, 1972).

As illustrated by e.g. the ocean salinity calculation (Fig. 2.68), the general results were quite realistic (compare with an average of the salinity data shown in Figs. 2.63-2.65). The models also allowed a first orientation into the structure of major current systems in the oceans (Fig. 2.69).

Figure 2.68. Average salinity (g kg-1) calculated with an early joint ocean-atmosphere model without seasonal variations in radiation (based on Bryan, 1969).

Figure 2.69. Streamfunction of meridional transport of water, averaged over year and longitude, calculated by a joint ocean-atmosphere model (unit: 109 kg of water per second). The sign convention is such that a meridional cell with positive streamfunction has northward transport at the top and southward below (based on Wetherald and Manabe, 1972).

Figure 2.70. Coupled ocean—atmosphere model results for ocean salinity, longitudinally averaged (England et al, 1993).

Figure 2.69. Streamfunction of meridional transport of water, averaged over year and longitude, calculated by a joint ocean-atmosphere model (unit: 109 kg of water per second). The sign convention is such that a meridional cell with positive streamfunction has northward transport at the top and southward below (based on Wetherald and Manabe, 1972).

Over the years, models have been refined in terms of effects included, as well as in terms of the mesh sizes used. Some models replace the latitude-longitude compartments by a spectral model for the Fourier components of the variables [i.e. expanding all functions in terms of sine and cosine functions of a base frequency times n (n = 1,...nmax), with nmax typically around 30] (Manabe and Stouffer, 1988). This speeds up computing times, but once realistic topography is included, the advantage of this representation is diminished.

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