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LECTURE 4 Offshore waves: Capillary waves, Wind Sea and Swell. I- Capillary waves 1) Observations 2) Physical description II- Wind Sea 1 ) Principle 2) Wind-wave growth 3) Wind-wave generation spectrum models III- Offshore swell propagation and decay 1) Angular Spreading
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LECTURE 4Offshore waves: Capillary waves, Wind Sea and Swell I- Capillary waves 1) Observations 2) Physical description II- Wind Sea 1) Principle 2) Wind-wave growth 3) Wind-wave generation spectrum models III- Offshore swell propagation and decay 1) Angular Spreading 2) Dispersion 3) Dissipation IV- Non-linear wave-wave interactions V- Equations of Motion of the offshore Waves 1) Source terms 2) Equation of motion VI- Important Notions
MAR 367 - Waves and Tides I- Capillary Waves or Ripples I- 1. Observations Small ripples or capillary waves are observed first when a fresh wind blows over smooth water. They have rounded crests and v-shaped troughs. Von Arx(1974) describes these waves as the way the wind gets a "grip" on the water since they are numerous and move slowly before the wind. Stories are told of seafarers who first observed these capillary waves when the wind was rising and later experienced the swells of the gravity waves, so it was thought that the capillary waves were the first mechanism by which the wind gave energy to the water. As the gravity waves build up, their wavelength tends to lengthen and speed increase until it matches the speed of the wind, at which point they can no longer extract energy from the wind.
MAR 367 - Waves and Tides I- Capillary Waves or Ripples I- 2. Physical description Capillary waves • stretch the surface of the fluid and are generally superimposed to chop and swell • have a wavelength (L) shorter than 2cm => wave number k > 350rad/m • have the surface tension of the water as an additional controlling (or restoring) force to the gravity. Capillary waves are thus primarily influenced by the surface tension of the ocean. In a previous lecture, for the analysis of the waves on the string, we find that the angular frequency due to the tension is: where T is the tension of the string is the mass density of the string (kg.m-1) is the wave number (rad.m-1) By analogy, for capillary waves of density (kg.m-3) with = 1027kg.m-3 the density of the water, = 1.22kg.m-3 the density of the air and we define the angular frequency due to the tension T:
MAR 367 - Waves and Tides I- Capillary Waves or Ripples I- 2. Physical description Capillary waves Capillary waves generated at the interface between the atmosphere and the ocean are influenced by both the effects of surface tension and gravitywhich leads to the definition of the angular frequency and the celerity as: The capillary wave speed (or celerity) increases as the wave number gets bigger: this behavior is called “anomalous dispersion” The minimum wave speed is reached for the wave number k=363rad/mand is a 0.23m/s => start of the capillary waves
MAR 367 - Waves and Tides II- Wind Sea II- 1. Principle
MAR 367 - Waves and Tides II- Wind Sea II- 2. Wind-Wave growth Wind-Wave growth: If we consider that the only input of energy to the sea surface over the timescales comes from the wind: transfer of energy to the wave field is achieved through the surface stress applied by the wind and this varies roughly as the square of the wind speed. After the onset of a wind over a calm ocean there are thought to be two main stages in the growth of wind waves: (1) the small pressure fluctuations associated with turbulence in the airflow above the water are sufficient to induce small perturbations on the sea surface and to support a subsequent linear growth as the wavelets move in resonance with the pressure fluctuations (Phillips’ resonance (1957); only significant early in the growth of waves on a calm sea) – linear growth of the waves. (2) the wavelets have grown to a sufficient size to start affecting the flow of air above them. The wind now pushes and drags the waves with a vigor which depends on the size of the waves themselves– exponential growth of the waves.
MAR 367 - Waves and Tides II- Wind Sea II- 2. Wind-Wave growth Wind-Wave growth: Hs and T only depend on the Fetch and the wind speed (assumed constant): Fetch limited conditions Where is the fetch (m) is the gravitational acceleration (m.s-2) is the significant wave height (m) is the wave period (s) is the wind speed (m.s-1): is the wind speed at 10m height t is the duration of wind (s) The value of the Fetch only depends on the duration of the wind: time limited conditions There are many empirical formulae for wave growth which have been derived from large data sets. These formulae make no attempt to separate the physical processes involved. They represent net wave growth from known properties of the wind field (wind speed - , fetch - and duration of wind - t ). Observations made during the Joint North Sea Wave Project (JONSWAP) (Hasselmann et al., 1973) gave the following relationships for fetch-limited conditions (where wave growth under a steady offshore wind was limited by the distance from the shore):
MAR 367 - Waves and Tides II- Wind Sea II- 3. Wind-Wave generation spectrum models Wave spectrum models: The previous formulations only give the significant wave height and period, the wave spectrum is most commonly used for modeling the sea state under a blowing wind. Models of the spectrum enable the spectrum to be expressed as some functional form, usually in terms of frequency, or frequency and direction, where is the spectral wave density (m2.s). Models of the spectrum are used to obtain an estimate of the entire wave spectrum from known values of a limited number of parameters such as the significant wave height (Hs) and wave period (T). These may be obtained by hindcast calculations, by direct measurement or visual observation.
MAR 367 - Waves and Tides II- Wind Sea II- 3. Wind-Wave generation spectrum models Pierson-Moskowitz model: in m2.s in s-1 The Pierson-Moskowitz spectrum (Pierson and Moskowitz, 1964) is often used as a model spectrum for a fully developed sea (idealized equilibrium state reached when duration and fetch are unlimited). This spectrum is based on a subset of 420 selected wave measurements recorded with the ship borne wave recorder — developed by Tucker (1956) — on board British ocean weather ships during the five-year period 1955–1960. The model spectrum can be written: where the wave frequency (); is the peak frequency (); is a constant; ;the wind speed () at 19.5 m above the sea surface
MAR 367 - Waves and Tides II- Wind Sea II- 3. Wind-Wave generation spectrum models in m2.s Independent of the wind speed! non dimensional Dependent of the wind speed! Generally the wind used to force the wave is at 10m height: in m2.s Peak values at the peak frequency
MAR 367 - Waves and Tides II- Wind Sea II- 3. Wind-Wave generation spectrum models JONSWAP spectrum: in m2.s in s-1 The JONSWAP spectrum is often used to describe waves in a growing phase. Observations made during the Joint North Sea Wave Project (JONSWAP) (Hasselmannet al., 1973) gave a description of wave spectra growing in fetch-limited conditions (where wave growth under a steady offshore wind was limited by the distance from the shore). The basic form of the spectrum is: where - non dimensional, is the peak enhancement factor, which modifies the interval around the spectral peak making it much sharper than in the Pierson-Moskowitz spectrum. Otherwise, the shape is similar.
MAR 367 - Waves and Tides II- Wind Sea II- 3. Wind-Wave generation spectrum models (1) The JONSWAP spectrum results in the generation of more wave energy than the Pierson-Moskowitz spectrum! (2) All the wave generated by these spectra are in the same direction than the local wind blowing!
MAR 367 - Waves and Tides III- Offshore Swell Propagation and Decay III- 1. 2D wave energy spectrum 2D wave energy spectrum: Generally we are interested in how a local average of the energy (or a wave group) moves. This is not as simple as moving the energy at one point in a straight line (or more correctly a great circle path) across the ocean: (1) for any location the energy is actually spread over a range of directions (2) waves at different frequencies will propagate at different speeds (dispersive waves in deep water) Thus, from a point source (= Fetch area), where we assume the waves are generated in the same direction than the wind, each component of the 2D spectrum - , can be propagated in the direction with a speed .
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 1. 2D wave energy spectrum
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 2. Angular Spreading Angular Spreading: Wind direction Highest waves Along the angles to either side of the mean wind direction, smaller waves follow their own great circle track . Awave group doesn’t propagate away from the fetch area in a unique direction but spreads laterally. The lateral spread is increasing with the distance from the source region (=Fetch area). The largest wave heights of the group (in reality swell height) are located closest to the great circle path extending from the mean wind direction in the fetch area. In other words the highest waves are find in the same direction than the wind. The swell height decreases laterally each side of the mean wind direction.
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 2. Angular Spreading Mathematical formulation: Redistribution of Energy due to spreading Mathematically, it is often assumed that the directional part of the wind wave distribution can be given by: The total energy is then given by: predominant direction of the waves =direction of the spectral component concerned for E(f) In this figure, the angular spreading is considered for two points A and B at the extremity of the fetch area. The point P will receive wave energy from points A and B (in reality from points all along the front of the fetch) and thus the maximum energy will be found closest to the great circle path extending from the mean wind direction in the fetch area.
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 2. Angular Spreading Spatial variations of the angular spreading:
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 3. Dispersion Dispersion: Lower frequencies travel faster than higher frequencies and the longest the wavelength, the shortest the frequency! the longest the wavelength, the fastest the waves! We have seen that, in deep water, long waves (long wavelength) and their energy travel faster than short waves (short wavelength)and their energy. The long waves are thus also the wave with the lowest frequency and the short waves have the highest frequency! In conclusion: lower frequencies travel faster than higher frequencies. The wave field leaving a generating area (fetch area) has a mixture of frequencies . At a large distance from the generating fetch, the waves with low frequencies (long waves) will arrive first, followed by waves of increasing frequency.
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 3. Dispersion Effect of the dispersion on the spectrum: spectrum at the front edge of the generating fetch The swell spectrum is therefore limited to a narrow band of frequencies The shaded portion of the spectrum is the maximum that can be expected at the point of observation. The ratio of this area to the total area under the spectrum is called the wave-energy dispersion factor. Higher frequencies arrive later, by which time some of the fastest waves (lowest frequencies) have passed. First components to arrive at a point far downstream.
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 4. Dissipation Dissipation: Dispersion and spreading can be considered the main causes of a gradual decrease of offshore swell waves. But some energy is also lost through Dissipation. Dissipation by internal friction and air resistance This acts on all components of the wave spectrum, but is strongest on the shorter waves (higher frequencies), contributing to the lengthening of swell at increasing distance from the source. This dissipation is often small enough that swell can survive over large distances.
MAR 367 - Waves and Tides III- Offshore SwellPropagation and Decay III- 4. Dissipation Dissipation by white-capping The white-capping is the breaking of the tip of the wave crest due to the fact that the crests are being driven forward by the wind faster than the wave itself is traveling. Much of the energy dissipated by white-capping is converted into forward momentum of the water reinforcing the wind-driven surface current. This is the primary source of dissipation for deep water waves.
MAR 367 - Waves and Tides IV- Non-Linear Wave-Wave Interactions Principle: The linear wave theory from which the wave energy is derived is an approximation. The theory is restricted by a requirement that the waves do not become too steep but the weakly non-linear interactions have been shown to be very important in the evolution of the wave spectrum. These weakly non-linear, resonant, wave-wave interactions transfer energy between waves of different frequencies, redistributing the energy within the spectrum in such a way that preserves some characteristics of the spectral shape, i.e. a self-similar shape. The process is conservative, being internal to the wave spectrum and not resulting in any change to the overall energy content in the wave field. The resonance which allows this transfer between waves can be expressed by imposing the conditions that the frequencies of the interacting waves must sum to zero and likewise the wavenumbers.
MAR 367 - Waves and Tides IV- Non-Linear Wave-Wave Interactions Wave growth and Non-Linear Interactions: The growth from non-linear interactions is given by the function Snl(f) for the mean JONSWAP spectrum (E). The maximum of non-linear interactions is found before the peak frequency. This positive growth just below the peak frequency leads to the downshift in peak frequency as a wind sea develops.
MAR 367 - Waves and Tides IV- Non-Linear Wave-Wave Interactions Wave growth and Non-Linear Interactions: Another feature of the non-linear wave-wave interactions is the “overshoot” phenomenon. Near the peak, growth at a given frequency is dominated by the nonlinear wave-wave interactions. As a wind sea develops (or as we move out along a fetch) the peak frequency decreases. A given frequency, fe, will first be well below a peak frequency, resulting in a small amount of growth from the wind forcing, some non-linear interactions, and a little dissipation. As the peak becomes lower and approaches fe, the energy at fe comes under the influence of a large input from non-linear interactions. As the peak falls below fe this input reverses, and an equilibrium is reached (known as the saturation state).
MAR 367 - Waves and Tides V- Equations of Motion of the offshore Waves V- 1. Source terms: wave growth and wave dissipation by white-capping Wind-Wave growth: with A the linear growth (depending on ) B the exponential growth(depending on ) in m2 Wave dissipation by white-capping: Based on the Komen et al. (1984) formulation, the processes of white-capping is represented by the pulse-based model of Hasselmann (1974) and can be expressed as follow: in m2 with and the mean frequency and the mean wave number a coefficient depending on the wave steepness: the total wave energy non dimensional Based on the two wave growth mechanisms (resonance due to the pressure difference generated by the wind and turbulent interaction of the wind with the waves), wave growth due to wind commonly described as the sum of linear and exponential growth term of a wave component:
MAR 367 - Waves and Tides V- Equations of Motion of the offshore Waves V- 2. Equation of Motion Equation of motion: The evolution of the wave energy is mathematically represented by an equation of transport of the wave Energy and can be expressed as follow (Mei, 1983; Komen et al., 1994): The final expression of the transport of the wave energy is then given by: JONSWAP spectrum E(f) multiplied by the angular spreading: E(f) Non-linear wave-wave interactions Dissipation by white-capping Wind-wave growth
MAR 367 - Waves and Tides VI- Important Notions JONSWAP wave spectrum: where the wave frequency () is the peak frequency () is a constant the wind speed () at 19.5 m above the sea surface - non dimensional, is the peak enhancement factor L is the wavelength (m) in m2.s Angular spreading and directional spectrum: E(f) and Lower frequencies travel faster than higher frequencies the longest the wavelength, the shortest the frequency! the longest the wavelength, the fastest the waves! Dispersion:
MAR 367 - Waves and Tides VI- Important Notions Wind-Wave growth: in m2 Non-Linear Interactions: Wave dissipation by white-capping: Primary source of dissipation for deep water waves Phase 2: Exponential growth of the waves – dependent of the waves Phase 1: Linear growth of the waves – independent of the waves
MAR 367 - Waves and Tides VI- Important Notions Equation of motion: The evolution of the offshore wave energy is equal to the sum of the wind-wave growth, the dissipation by white-capping and the non-linear wave-wave interactions Non-linear wave-wave interactions Dissipation by white-capping Wind-wave growth