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Engineering › Fluid Mechanics › Waves › Spectra ›

JONSWAP

The JONSWAP spectra in the wave-frequency domain

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Contents

Interface
JONSWAP Gnnw
JONSWAP Wp
JONSWAP Alpha
JONSWAP Gnnk
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Name	HTML Code
JONSWAP_Gnnw Defines the JONSWAP spectra in the wave-frequency domain	<script type="text/javascript" src="http://www.codecogs.com/ic_json-463&u=5"></script>
JONSWAP_wp Compute the peak frequency for a JONSWAP spectrum based on wind speed and fetch length.	<script type="text/javascript" src="http://www.codecogs.com/ic_json-463&u=6"></script>
JONSWAP_alpha Compute the constant \alpha that defines the intensity of the JONSWAP spectra from wind speed and fetch length.	<script type="text/javascript" src="http://www.codecogs.com/ic_json-463&u=7"></script>
JONSWAP_Gnnk Defines the Jonswap spectra in the wave-number domain	<script type="text/javascript" src="http://www.codecogs.com/ic_json-463&u=8"></script>

Overview

The JONSWAP (Joint North Sea Wave Project) spectra is an empirical relationship that defines the distribution of energy with frequency within the ocean.

The JONSWAP spectrum is effectively a fetch-limited version of the Pierson-Moskowitz spectrum, except that the wave spectrum is never fully developed and may continue to develop due to non-linear wave-wave interactions for a very long time. Therefore in the JONSWAP spectrum, waves continues to grow with distance (or time) as specified by the α (alpha) term, and the peak in the spectrum is more pronounced, as specified by the γ (gamma) term. Hasselmann (1966) found the latter to be particularly important as it lead to enhanced non-linear interactions.

References

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_04.htm

JONSWAP Gnnw

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doubleJONSWAP_Gnnw(	double	`w`
	double	`wp`
	double	`alpha` `= 0.0081`
	double	`gamma` `= 3.3`
	double	`beta` `= 1.25`	)

The JONSWAP (Joint North Sea Wave Project) spectra is an empirical relationship that defines the distribution of energy with frequency within the ocean.

The underlying equation is:

$S(\omega) = \frac{\alpha g^2}{\omega^5} \exp \left[-\beta \frac{\omega_p^4}{\omega^4}\right ] \gamma^a$

(1)

where

$\displaystyle a=\exp \left [-\frac{(\omega - \omega_p)^2}{2 w_p^2 \sigma^2} \right ]$
$\displaystyle \sigma = \begin{cases} 0.07 & \text{ if } \omega\leq \omega_p \\ 0.09 & \text{ if } \omega>\omega_p \end{cases}$
$\displaystyle \beta = \frac{5}{4}$
α is a constant that relates to the wind speed and fetch length, see below. Typical values in the northern north sea are in the range of 0.0081 to 0.01
ω is the wave frequency
$\omega_p$ is the peak wave-frequency

Most problems is the literature are expressed in the above form. However if a particular wind speed and fetch length are known, then α and $\omega_p$ can be estimated using the subsequent two functions.

For a range of typical north sea conditions (where α =0.0081 and $\omega_p=2 \pi/12.4$ =0.5), but with varying peak enhancements the JONSWAP spectra has the form

$\graph w=0:1.5, wp=0.5, alpha=0.0081, gamma=1:3.3:4, beta=1.25$

Standards

This function conforms to British Standards (BS 6349-1:2000), 24 July 2003.

Parameters

w	wave-frequency (2 π/s)
wp	the peak wave frequency (2 π/s)
alpha	The intensity of the Spectra. Default value = 0.0081
gamma	Peak enhancement factor. Default value = 3.3
beta	A shape factor (Rarely changed). Default value = 1.25

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JONSWAP Wp

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doubleJONSWAP_wp(	double	`wind`
	double	`length`	)

The peak of the JONSWAP spectrum is empirically define by

$\omega_p = 2.84 \; g^{0.7}\; L_F^{-0.3}\; U_W^{-0.4}$

(2)

where

$U_W$ is the wind speed at 10m above the sea surface
$L_F$ is the fetch length

Standards

This function conforms to British Standards (BS 6349-1:2000), 24 July 2003.

Parameters

wind	The wind speed 10m above the sea surface. [m/s]
length	The fetch length. [m]

Source Code

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JONSWAP Alpha

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doubleJONSWAP_alpha(	double	`wind`
	double	`length`	)

The overall energy within the JONSWAP spectrum is controlled by the α constant and is related to wind speed and the peak frequency by:

$\alpha = 0.033 \left ( \frac{\omega_p U_w}{g} \right )^{2/3}$

(3)

where

$U_W$ is the wind speed at 10m above the sea surface
$\omega_p$ is the peak frequency calculated using equation (2)

This function uses JONSWAP_wp (above) to obtain w_p for a given fetch length and wind speed.

Parameters

wind	The wind speed 10m above the sea surface. [m/s]
length	The fetch length. [m]

Source Code

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JONSWAP Gnnk

JONSWAP Gnnk Calculator

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doubleJONSWAP_Gnnk(	double	`k`
	double	`wp`
	double	`depth` `= 0`
	double	`alpha` `= 0.0081`
	double	`gamma` `= 3.3`
	double	`beta` `= 1.25`	)

Uses the description of the JONSWAP spectra described in frequency to obtain the distribution in wave-number using the 1st order dispersion relationship give in dispersion.

This conversion is thus

$G_{\eta\eta}(k) = G_{\eta\eta}(\omega) \frac{\partial \omega}{\partial k}$

(4)

where in deep water

$\frac{\partial \omega}{\partial k} = \frac{g}{2\omega}$

(5)

and in shallow water

$\frac{\partial \omega}{\partial k} = \frac{g}{2\omega} \left [ k sech^2(k d) + tanh(k d) \right ]$

(6)

For a range of north sea conditions (where α =0.0081 and $\omega_p=2 \pi/12.4$ =0.5), but with varying peak enhancements the JONSWAP spectra has the following form in wave-number:

$\graph k=0:0.1, wp=0.5, depth=0, alpha=0.0081, gamma=1:3.3:4, beta=1.25$

Parameters

k	Wave-number (2 π/m)
wp	The peak wave frequency
depth	The water depth. Default value=0 (infinite depth)
alpha	The intensity of the spectra. Default value = 0.01
gamma	The peak enhancement factor. Default value = 3.3
beta	A shape factor (Rarely changed). Default value = 1.25

Source Code

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double	`JONSWAP_Gnnw` (double w, double wp, double alpha=0.0081, double gamma=3.3, double beta=1.25) Defines the JONSWAP spectra in the wave-frequency domain
double	`JONSWAP_wp` (double wind, double length) Compute the peak frequency for a JONSWAP spectrum based on wind speed and fetch length.
double	`JONSWAP_alpha` (double wind, double length) Compute the constant α that defines the intensity of the JONSWAP spectra from wind speed and fetch length.
double	`JONSWAP_Gnnk` (double k, double wp, double depth=0, double alpha=0.0081,double gamma=3.3, double beta=1.25) Defines the Jonswap spectra in the wave-number domain