Implementation of viscoelastic mud-induced energy attenuation in the third-generation wave model, SWAN

Abstract

The interaction of waves with fluid mud can dissipate the wave energy significantly over few wavelengths. In this study, the third-generation wave model, SWAN, was advanced to include attenuation of wave energy due to interaction with a viscoelastic fluid mud layer. The performances of implemented viscoelastic models were verified against an analytical solution and viscous formulations for simple one-dimensional propagation cases. Stationary and non-stationary test cases in the Surinam coast and the Atchafalaya Shelf showed that the inclusion of the mud-wave interaction term in the third-generation wave model enhances the model performance in real applications. A high value of mud viscosity (of the order of 0.1 m²/s) was required in both field cases to remedy model overestimation at high frequency ranges of the wave spectrum. The use of frequency-dependent mud viscosity value improved the performance of model, especially in the frequency range of 0.2–0.35 Hz in the wave spectrum. In addition, the mud-wave interaction might affect the high frequency part of the spectrum, and this part of the wave spectrum is also affected by energy transfer from wind to waves, even for the fetch lengths of the order of 10 km. It is shown that exclusion of the wind input term in such cases might result in different values for parameters of mud layer when inverse modeling procedure was employed. Unlike viscous models for wave-mud interaction, the inverse modeling results to a set of mud parameters with the same performance when the viscoelastic model is used. It provides an opportunity to select realistic mud parameters which are in more agreement with in situ measurements.

Appendices

Appendix A The procedure for solving dispersion equation of Liu and Chan (2007)

As mentioned in “Section 2.1”, the dispersion equation of Liu and Chan (2007) method is an implicit equation and needs iteration for finding out wavenumber k.

The wavenumber obtained from dispersion equation of Gade (1957), k ₀ which is an explicit equation, was used as an initial value for the iteration procedure.

$$ {k}_0={\omega}^{\ast }{\left\{\frac{\left(1+\varGamma \frac{H_{m0}}{H_{w0}}\right)-\sqrt{{\left(1+\varGamma \frac{H_{m0}}{H_{w0}}\right)}^2-4\upgamma \Gamma \frac{H_{m0}}{H_{w0}}}}{2\upgamma \mathrm{g}\Gamma {H}_{m0}}\right\}}^{\frac{1}{2}} $$

(15)

in which $\varGamma =1-\frac{\tanh \Big({m}^{\ast }{H}_{\mathrm{m}0\Big)}}{m^{\ast }{H}_{\mathrm{m}0}}$,$m=\left(1-i\right)\sqrt{\frac{\omega }{2{\nu}_{\mathrm{m}}}}$, $\gamma =\frac{\rho_{\mathrm{m}}-{\rho}_{\mathrm{w}}}{\rho_{\mathrm{m}}}$, ρ _m, ρ _w are mud and water densities, H _w0 and H _m0 are water depth and mud thickness, respectively. Moreover, ν _m is mud kinematic viscosity. The Newton-Raphson method was used as iteration process, since it can be used for complex numbers effectively. It reads as follows:

$$ {k}_{n+1}={k}_n-\frac{f\left({k}_n\right)}{f^{\prime}\left({k}_n\right)} $$

(16)

in which k _n is the wave number in the previous step, k _n + 1 is the updated wavenumber in new step, f is the implicit function of dispersion equation, and f ^′ is the derivative of function f with respect to k.

Appendix B The procedure for solving dispersion equation of Macpherson (1980)

The dispersion equation of Macpherson (1980) method is also an implicit equation and it needs iteration procedure to determine the complex wavenumber. The initial value of wavenumber can be obtained from the dispersion equation of linear wave theory:

$$ {\omega}^2=g{k}_0\ \tanh \left({k}_0\ {H}_{w0}\right) $$

(17)

in which g is the gravity acceleration, and other parameters in Eq. (17) is defined in the same as previous Appendix 1. The numerical root-finding algorithm of Muller method (Muller 1956) can be used effectively to solve dispersion equation of Macpherson (1980). However, it is a three-step method and needs three previous steps to create the new one. The second and third steps are produced using the Newton-Raphson method explained in Appendix 1.