CHAPTER
COORDINATORS
COORDINATORS
Lotfi Aouf and Gabriel Diaz-Hernandez
8.8
Outputs and post processing
8.8.1
Post-processing of the wave model results for the final delivery
Wave models provide at each grid point two-dimensional wave spectrum F(f,θ), which describes how the wave energy is distributed as a function of frequency f and propagation direction θ. In general, the wave spectrum F is discretized in 30 frequencies and 24 directions. To simplify the study of wave conditions, integrated parameters are derived from weighted integrals of F(f,θ). The moment of order n, mn is defined as the following integral:
The integrations are performed over all frequencies and directions or over a spectral subdomain when the spectrum is split between wind sea and swell or partitioned into main components. The wind sea wave component is subject to the wind forcing, and then wave phase speed is smaller than the wind speed at the ocean surface. The remaining part is considered swell. It is established in the WAM model for instance, the spectral energy is subject to wind forcing when the following approximation is satisfied:
where u∗ is the friction velocity, c is the phase speed as derived from the linear theory of waves and φ is the wind direction. The integrated parameters are therefore also computed for wind waves and swell by only integrating over the respective components of F(f,θ) that satisfies 8.43 or not.
Significant wave height
The wave energy is the 0th order of the moment m0 and significant wave height (Hs) is defined as follows (Hs snapshot shown in Figure 8.38):
Figure 8.38. Snapshot of Hs (in meters) from Copernicus Marine Service global wave system (3 February 2022 at 21 UTC).
Mean period
The mean period (snapshot in Figure 8.39) is expressed in several ways. The most used is Tm-1 which is based on the moment of order -1, that is
Tm-1 is also commonly known as the energy mean wave period. By considering Hs, it can be used to determine the wave energy flux per unit of wave-crest length in deep water, also indicated as the wave power per unit of wave-crest length P.
Figure 8.39. Snapshot of mean period Tm-1 (in seconds) from Copernicus Marine Service global wave system (3 February 2022 at 21 UTC).
To analyse different aspects of the wave field, other moments can be used to define a mean period. Periods can be based on the first moment Tm1 given by:
Tm1 is essentially the reciprocal of the mean frequency. It can be used to estimate the magnitude of Stokes drift transport in deep water and periods based on the second moment Tm2 given by:
Tm2 is also known as the zero-crossing mean wave period, as it corresponds to the mean period that is determined from observations of the sea surface elevation using the zero-crossing method.
Peak period
The peak period is defined for total sea and can be expressed as the reciprocal peak frequency of the 1D wave spectrum F(f) integrated over directions. There is a second way to compute the peak frequency and it is obtained from a parabolic fit around the discretized maximum of the two-dimensional wave spectrum F(f,θ).
Mean wave direction
The mean wave direction is defined by weighting the wave spectrum F(f,θ). It is expressed as follows:
where S1 is the integral of sin(θ)*F(f,θ) over frequencies and directions, while C1 is the integral of cos(θ)*F(f,θ) over f and θ.
Directional spread
The wave directional spread gives the information on the directional distribution of the total sea, or it can be applied for different wave components. It is expressed as follows:
where M is I/m0 and I is the integral of cos(θ-<θ>)*F(f,θ) over f and θ. <θ> is the mean direction. The directional spread can be computed for wind, sea, and swell components.
Surface Stokes Drift
The Stokes drift impacts the turbulence in the upper ocean layers and contributes to the source of energy of the ocean circulation, particularly the Langmuir circulation. The surface Stokes drift Us is computed from the wave spectrum in deep water by the following relation:
where the integration is over all frequencies and directions. k is the unit vector in the direction of the wave component. In the high frequency range, the Phillips spectral shape is used with accounting of spectral level of the last frequency bin. Figure 8.40 shows the ratio of Stokes drift magnitude to 10 m wind speed.
Figure 8.40. Ratio (in percentage) between surface Stokes drift and wind speed from Copernicus Marine Service global wave system on 20 June 2021 at 21 UTC. Arrows show the Stokes drift direction.
Partitioning wave spectrum
In general, wave forecasters firstly analyse the integrated parameters over the full wave spectrum describing the total sea. Then, they refine their analysis by examining the different dominant wave trains representing wind, sea, and swell. Most wave models include a partitioning procedure, which aims to separate the different wave systems represented by energy peaks in the wave spectrum. The most used partitioning procedure is adapted from Hanson and Phillips (2002) and is based on the watershed method inspired from image processing. After splitting the wind sea and swell wave spectrum, the method consists in identifying the energy peaks in the wave spectrum and isolating a partition with decreasing energy from the peak to a limit corresponding to an increase in energy. Several partitions or wave systems can be detected in a wave spectrum, and they are classified by decreasing order of their wave height. An example of partitioning is shown in Figure 8.41, where three partitions are detected with two swells and one wind sea. The average height, period and direction can be calculated on each partition.
Figure 8.41. Top left: full wave spectrum for the location of the Prestige ship accident (Trulsen et al., 2015). Top right: swell partition-1, the most energetic propagating to the South East direction. Bottom left: long wind sea partition-2 propagating to the North-East direction. Bottom right: swell partition-3 propagating to the EastNorth-East direction (source: Copernicus Marine Service).
Wave energy flux
The wave energy flux per unit of wave-crest length in deep water can be computed by using the wave period Tm-1 and significant wave height Hs:
where rw is the water density and g is the acceleration due to gravity.
The wave energy flux can be expressed by integrating the flux of each spectral component.
where Cg is the group velocity in deep water.
8.8.2
Common output variables
Numerical models for wave generation and propagation can provide different variables to be used in multi-year and predictive systems. Table 8.1 lists variables that are commonly provided by numerical and that may be of special relevance for users, as well as for developers who wish to set up future wave OOFS and multi-year systems.
Common variable names (usually provided by third-generation spectral wave model and/or a mild slope approximations) | Symbol | Units |
|---|---|---|
| Significant wave height | Hs | m |
| Peak period | Tp | s |
| Mean wave period | Tm | s |
| Mean and Peak wave direction | Ө | °N |
| Complete wave spectra matrix | S | m²/Hz/ºN |
| Mean parameters of wave partitions (Hs,Tm,Tp and Dir) for: 2 Swells and 1 wind sea | Hsi, Tpi, Diri | m, s, ºn |
| Maximum wave height | Hmax | m |
| Maximum wave period | Tmax | m |
| Meridional component of Stokes drift | Us | m/s |
| Zonal component of Stokes drift | Vs | m/s |
| Drag coefficient with waves | Cd | |
| Normalised stress to ocean | Tauoc | |
| Mean square slope | mss |
Advanced variable names (usually provided by phase resolving models and CFD approaches) | Symbol | Units |
|---|---|---|
| Free-surface time series at points and maps | η | m |
| Wave breaking-induced currents | U-V | m/s |
| Instantaneous wave run-up | Ru | m |
| Instantaneous wave overtopping volume | q | m³/s per m |
| Infragravity wave oscillations | ηIG | m |
| Multi-directional wave spectra matrix (agitation) | S | m ²/Hz/ºN |
| Instantaneous wave pressures over structures | P | N/m² |
| Instantaneous forces over structures | F | N |
| Instantaneous wave currents | u-v | m/s |
Table 8.1. Common names of wave variables.
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