SEA Files

A SEA file represents the sea state as a Fourier series of regular waves. The surface elevation is therefore:

\[ \begin{equation} \eta(x,y,t) = \sum_{n = 1}^{N}{a_{n}\cos\left (k_{n}\left( x\cos\theta_n + \ y\sin\theta_n \right) + \omega_{n}t + \phi_{n} \right)} \end{equation} \]

where
\(a_n\) is the amplitude of the \(n^{th}\) component,
\(k_{n} = 2\pi/\lambda_{n}\) is the wave number of the \(n^{th}\) component and \(\lambda_{n}\) is the wavelength,
\(\theta_n\) is the direction of the \(n^{th}\) component in cartesian system (see the definition in wave particle kinematics),
\(\omega_{n} = 2\pi f_{n}\) is the angular frequency of the \(n^{th}\) component,
and \(\phi_{n}\) is the phase of the \(n^{th}\) component.

The amplitude (\(a_{n}\)) of the each Fourier component is a function of the spectral density (if deterministic). The JONSWAP spectrum is described in JONSWAP/Pierson Moskowitz Spectrum. In the SEA file generator, the period parameter can be defined in several ways other than the peak period:

• Energy period, the period of a regular wave with the same power per metre crest, \(T_{e} = \frac{m_{- 1}}{m_{0}}\)
• Mean period, the reciprocal of the mean spectral frequency, \(T_{m} = \frac{m_{0}}{m_{1}}\)
• Zero up-crossing period, the average period at which the sea surface elevation cross the mean water level in an upward direction, \(T_{z} = \sqrt{\frac{m_{0}}{m_{2}}}\)

where \(m_{n}\) is the \(n^{th}\) moment of the spectrum:

\[ \begin{equation} m_{n} = \int_{0}^{\infty}{f^{n}E(f)df} \end{equation} \]

Directional distributions

The direction of each component (\(\theta_n\)) is a random number distributed according to the directional distribution. The various options of directional distributions \(D(\theta)\) available in Bladed are defined below. In the following, the mean direction (\(\theta_{m}\)) is assumed to be constant with frequency.

Cosine 2s (spreading is assumed constant with frequency):

\[ \begin{equation} D(\theta) = \frac{1}{2\sqrt{\pi}}\frac{\Gamma(s + 1)}{\Gamma\left( s + \frac{1}{2} \right)}\cos^{2s}\left( \frac{\theta - \theta_{m}}{2} \right) \end{equation} \]

where \(s\) is spreading parameter, \(\Gamma\) is the gamma function. The spreading parameter \(s\) is related to the RMS spread defined in terms of circular moments \(\sigma_{c}\) by:

\[ \begin{equation} \sigma_{c}^{2} = \frac{2}{1 + s} \end{equation} \]

Wrapped Normal (spreading is assumed constant with frequency):

\[ \begin{equation} D(f,\theta) = \frac{1}{\sigma_{l}\sqrt{2\pi}}\sum_{j = - \infty}^{\infty}{\exp\left\lbrack - \frac{1}{2}\left( \frac{\theta - \theta_{m} - 2\pi j}{\sigma_{l}} \right)^{2} \right\rbrack} \end{equation} \]

where \(\sigma_{l}\) is RMS spread defined in terms of line moments. The line and circular RMS spread for the wrapped normal distribution are related by:

\[ \begin{equation} \sigma_{c}^{2} = 2\left\lbrack 1 - \exp\left( - \frac{\sigma_{l}^{2}}{2} \right) \right\rbrack \end{equation} \]

Ewans Wind Sea (spreading is variable with frequency) (Ewans K.C. 2002):

\[ \begin{equation} D(f,\theta) = \frac{1}{\sigma(f)\sqrt{8\pi}}\sum_{j = - \infty}^{\infty}\left\{ \exp\left\lbrack - \frac{1}{2}\left( \frac{\theta - \theta_{1}(f) - 2\pi j}{\sigma(f)} \right)^{2} \right\rbrack + \exp\left\lbrack - \frac{1}{2}\left( \frac{\theta - \theta_{2}(f) - 2\pi j}{\sigma(f)} \right)^{2} \right\rbrack \right\} \end{equation} \]

with

\[ \begin{align} \theta_1(f) &= \frac{\theta_{m} + \Delta\theta(f)}{2} \\ \theta_2(f) &= \frac{\theta_{m} - \Delta\theta(f)}{2} \\[1.5ex] \Delta\theta(f) &= \begin{cases} 14.93 & \text{ for } f < f_p \\ \exp\left\lbrack 5.453 - 2.750\left( \frac{f}{f_{p}} \right)^{- 1} \right\rbrack & \text{ for } f\geq f_p \end{cases}\\[1.5ex] \sigma(f) &= \begin{cases} 11.38 + 5.5357\left( \frac{f}{f_{p}} \right)^{- 7.929} & \text{ for } f < f_p\\ {32.13 - 15.39\left( \frac{f}{f_{p}} \right)^{- 2}} & \text{ for } f \geq f_p \end{cases} \end{align} \]

where \(f_p\) is the spectral peak frequency.

Last updated 06-09-2024