(t) crest Period = T A H t trough wave length = Note the following definitions of the properties of waves. The wavelength is the distance between two

Transcription

1 Lecture 15 Waves in the Ocean In previous lectures we have learned about the effects on the ocean caused by Coriolis, density (pressure) gradients and wind forcing. We have seen how these agents alter oceanic currents. That, however, is not the complete story because density gradients tend to produce internal waves when influenced by sheared currents. Wind forcing tends to produce wind waves, ripples and swell, and when combined with Coriolis acceleration, bathymetry effects, and other atmospheric forcing, it tends to produce planetary and topographic waves. Gravitational forces (mostly due to the moon and sun) produce tides and tectonic forces such as earthquakes that cause vertical displacements of the ocean floor cause tsunamis. The typical periods, wavelengths, and forcing mechanisms of waves in the ocean are presented in the following table. Name Typical Periods Wave lengths Forcing Mechanism -2 Ripples < 0.2 s 10 m wind on sea surface Sea s 130 m " Swell 9-30 s 100s of m " Internal minutes to several m current shear on hours stratification Planetary and hours to days s km bathymetry/atmospheric Topographic pressure/coriolis Tsunamis 15 minutes to 1 hour few 100s km seismic Tides several hours several 100s - few gravitational (moon and 1000s km sun mainly) Types of Waves in the Ocean The first class of waves that we will study are those produced by wind. Wind can generate waves locally --sea--, which travel in different directions, or can generate waves that travel from a remote location in the open ocean --swell--, which travel in one direction and are more regular than sea waves. To study wind waves, we should take ideal waves with sinusoidal shape. We define the properties of waves from these ideal waves, as presented in the following diagram. 1

2 (t) crest Period = T A H t trough wave length = Note the following definitions of the properties of waves. The wavelength is the distance between two crests (or two troughs or two inflexion points with the same curvature above and below the points). The period is the time it takes two consecutive troughs (or crests or any other wave reference point) to pass by a fixed position. The speed or celerity C of the wave is the quotient of the wavelength over period. The wave amplitude A equals one half of the wave height H, which is the distance between the crest and the trough. The wave crest is the point of maximum elevation, and the wave trough is the point of minimum elevation. The wave height does not depend on C,, or T. It depends on the energy transferred to the surface by the wind. As we will see, the stronger the wind, he higher the waves. The period and wave length can be expressed in terms of their reciprocals, the wave fequency 3 = 2!/T, and the wave number = 2!/. The wave number vector has magnitude and points in the direction of wave propagation, i.e., perpendicular to the surface wave crests. With the definitions of 3 and, the wave speed C equals It should be noted that waves do not change their period T. The wave length is a function of T, and C is a function of. Thus, long waves travel faster than short waves. The study of wind waves is simplified by treating them as "small amplitude" waves. Small amplitude means that the wave steepness (ratio of of wave height H to wave length ) is smaller than one twentieth in idealized situations. In real cases, the steepness of small amplitude waves is 1/50. For a progressive wave (cosine wave), the vertical displacement of the free-surface is given by 2

3 (15.1) where x is position and t is time. The argument [x - 3t] is the phase of the wave and acquires values between 0 and 2! going from one crest to the next at a fixed point or from one crest to the next at a given time. The speed at which the argument propagates or phase speed C is obtained from the equations of motion, assuming that the wave is progressive, incompressible, and irrotational and equals (15.2) Before explaining what happens to the phase speed in shallow and deep waters, let us digress for a moment to review the behavior of hyperbolic functions as presented in the following diagram. Note first the behavior of the hyperbolic tangent. It is zero when the argument is zero. It equals the argument when the argument is small. It equals 1 when the argument is large. Next note the behavior of the hyperbolic sine. It is zero when the argument is zero, it equals the argument when the argument is small, and it is much greater than the argument when the argument is large. Finally note the behavior of the hyperbolic cosine. It is 1 when the argument is zero or small, and it is much greater than the argument when the argument is large. Let us now return to explain the behavior of the phase speed under different conditions. A deep 3

4 water wave or short wave is found where the wave length is smaller than twice the water column depth, i.e., < 2h. This means that is small and h is large, and therefore tanh h 1. Then, it is seen from (15.2) that for a deep water wave, (15.3) and depends on the wave length. This concept is illustrated in the following diagram that shows that speed changes as a function of depth, for waves of diverse, according to (15.2). Shallow Water Waves Intermediate Waves Deep Water Waves Deep water waves occur underneath the dash-dot line and their phase speed C is given by (15.3). s Note that C does not change with depth variations. s A shallow water wave or long wave is found where > 20h. This means that the wave length is long and therefore tanh h h. Then, for shallow water waves, the phase speed (15.4) 4

5 does not depend on the wave length but on the local water depth. This is an important relationship because it says that long waves such as tidal waves propagating over coastal areas will have phase speeds that are proportional to the square root of the water depth. The figure shows only one value of C l for each value of h. It also shows that C l is independent of. Intermediate waves are those that fall between long and short waves (shaded area in the figure). Its phase speed is given by (15.2). The range of depths at which these waves may exist increases with wavelength. There is an alternative way of representing (15.2), which is: So that the wave frequency 3 is related to the wave number by: (15.5) This is called the dispersion relation because it relates the wave period to the wave length. This relation describes how different waves of different periods travel at different speeds and get polarized. Deep water waves are dispersive as their speed depends on and on their period T. This means that short waves will travel at different speeds over a given depth. Hence, the waves that travel fast will separate from the slow ones. In contrast, shallow water waves are nondispersive because their speed is independent of and T. The dispersion relation is useful to classify waves as we will see when we talk about planetary waves. The dispersion relation leads to the concept of group speed of waves. Group Speed The group speed is the speed at which a group of waves travel. The quantitative notion of group speed is derived form the dispersion relation (15.5), by estimating d3/d, or the change of the wave length with respect to the period: (15.6) For long waves, h is very small and sinh 2h = 2h, so that the group speed is identical to the phase speed, i.e., C g = C. In other words, long (shallow water) waves are non-dispersive and carry the energy of the wave. For short waves, h is large and sinh 2h N 2h, so that the group speed equals one half of the phase speed, i.e., C g = ½C. Short (deep water) waves are then dispersive and tend to form trains of waves of the same period that travel in groups. Group Velocity 5

6 The group velocity is analogous to the group speed but also describes the direction of wave travel. For surface waves (air-water interface) the group velocity and the phase velocity are in the same direction. For other types of waves (as in internal and planetary waves) may be different from. The concept of group velocity and group speed is useful to determine the distance to the region where surface waves were produced. This distance d is determined with (15.7) where t is the time of arrival at the point of measurement and t is the time of wave generation. 0 The waves measured at the beach must have been deep water waves at the time of generation so that their phase speed was C = g/3 because = 3/C, and their group speed was C = C/2 = g/23. (15.8) g From (15.7) and (15.8), it is seen that the wave frequency is inversely related to the distance travelled, and directly related to the time of arrival (high frequency waves arrive later), i.e., (15.9) At the beach, we can measure 3 as a funcion of time and then plot 3 vs. t. If a straight line is fitted to the distribution of 3 vs. t, then the slope of that line is g/2d according to (15.9) and the time of wave generation will be the value at 3 = 0 (abscissa intercept). The distance of wave generation is obtained from the slope, as illustrated below. 6

7 3 t t 0 The main characteristics of small amplitude waves can be summarized in the table on the following page. We have already discussed C, and C g. The behavior of is straightforward. The horizontal and vertical velocities of wave particles, and their displacements describe circular trajectories for short waves (same amplitude for u,w and,) and elliptical trajectories for long waves (u and have different amplitudes than w and ). This is illustrated in the example in class. Note that the shape of the wave is the feature that moves continuously forward (at speed C s). The wave particle velocities contribute little to the currents, except when the waves have large amplitudes in shallow waters. Wave-current interactions is a topic that is receiving increased attention. It has been proposed that waves may actually act to retard mean flows in coastal regions because thay tend to augment bottom stresses. In reality, wave shapes are not perfectly sinusoidal nor their orbits are perfectly closed. Other types of waves include Stokes, cnoidal, solitary, trochoidal. This is illustrated with another figure in class. Stokes drift becomes important as wave grows and breaks. There is a net onshore transport produced by Stokes drift. This net transport must be balanced by offshore transport in the way of rip currents, which are narrow seaward jets. Long waves also cause important Stokes drift in coastal areas, as seen later. The energy (per unit of crest length) E generated by waves is proportional to the square of the height H, or, 7

8 Small amplitude waves: Steepness < 1/20 (1/50 in real ocean) = A cos(x - 3t); A=H/2, = 2!/, 3=2!/T deep waves (short) intermediate shallow waves (long) < 2h 2h < < 20h > 20h phase speed, or wave celerity C C = = C = C = (gh) 1/2 wave length = = tanh h = CT group velocity C g = 0.5 C C g= C g = C C g (h N 1) (h M 1) wave particle horizontal velocity u vertical or orbital velocity w Horizontal Displacements Vertical Displacements ' Subsurface Pressure p p = #g - #gz p = #g(-z) 8