JONSWAP — Spectra

The JONSWAP spectra in the wave-frequency domain

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Interface

#include <codecogs/engineering/fluid_mechanics/waves/spectra/jonswap.h>

using namespace Engineering::Fluid_Mechanics::Waves::Spectra;

double JONSWAP_Gnnw(double w, double wp, double alpha=0.0081, double gamma=3.3, double beta=1.25)
double JONSWAP_wp(double wind, double length)
double JONSWAP_alpha(double wind, double length)
double JONSWAP_Gnnk(double k, double wp, double depth=0, double alpha=0.0081,double gamma=3.3, double beta=1.25)

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

FUNCTION

JONSWAP_Gnnw

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 in 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

Standards

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

Parameters

wave-frequency (2 π/s)

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

Interactive Calculator

alpha

gamma

beta

Result –

FUNCTION

JONSWAP_wp

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]

Interactive Calculator

wind

length

Result –

FUNCTION

JONSWAP_alpha

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]

Interactive Calculator

wind

length

Result –

FUNCTION

JONSWAP_Gnnk

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

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

(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 alpha=0.0081 gamma=1:3.3:4 depth=0

Parameters

Wave-number (2 π/m)

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

Returns

The JONSWAP spectral density in the wave-number domain

Interactive Calculator

depth

alpha

gamma

beta

Result –