# Effects of Wave-Induced Processes in a Coupled Wave-Ocean Model on Particle Transport Simulations
## Appendix A Wave-Current Interaction Processes
### Momentum and Energy Flux from the Wave Model  
Provided that current gradients are sufficiently weak, the energy and momentum fluxes can be calculated from the energy balance equation (with an approximation from [36]:  
\[\frac{\partial}{\partial t}E+\frac{\partial}{\partial\overset{\rightarrow}{ \underset{\mathbf{z}}{\underset{\mathbf{z}}{\underset{\mathbf{z}}{\underset{ \mathbf{z}}{\underset{\mathbf{z}}{\underset{\mathbf  
where \(E=E(\omega,\theta)\) is the two-dimensional wave spectrum which gives the energy distribution of the ocean waves over angular frequency \(\omega\) and propagation direction \(\theta\), \(v_{g}\) is the group velocity. On the right-hand side of the action balance equation are the source terms that represent physical processes which generate, redistribute or dissipate wave energy in the WAM model. These terms denote, respectively, wave growth by the wind \(S_{in}\), non-linear transfer of wave energy through four-wave interactions \(S_{nl4}\) and wave dissipation caused by white capping \(S_{diss}\) and bottom friction \(S_{diss}\). In the present calculations, we also took into account depth-induced wave breaking \(S_{br}\).  
Making use of the energy balance in the Equation (A4) the wave-induced stress is given by  
\[\overrightarrow{\tau_{in}}=\rho_{w}g\int_{0}^{2\pi}\;\int_{0}^{\infty}\frac{ \overrightarrow{\omega}}{\omega}S_{in}\;d\omega\;d\theta\] (A4)  
while the dissipation stress is given by  
\[\overrightarrow{\tau_{diss}}=\rho_{w}g\int_{0}^{2\pi}\;\int_{0}^{\infty}\frac {\overrightarrow{\omega}}{\omega}S_{diss}\;d\omega\;d\theta.\] (A5)  
Similarly, the energy flux from wind to waves is given by  
\[\Phi_{in}=\rho_{w}g\int_{0}^{2\pi}\;\int_{0}^{\infty}\;S_{in}\;d\omega\;d\theta\] (A6)  
and the energy flux from waves to the ocean, \(\Phi_{diss}\), is defined as  
\[\Phi_{diss}=\rho_{w}g\int_{0}^{2\pi}\;\int_{0}^{\infty}S_{diss}\;d\omega\;d\theta.\] (A7)  
Under stationary and homogenous conditions the momentum and energy balance reduces to  
\[\int_{0}^{2\pi}\;\int_{0}^{\infty}\frac{k}{\omega}(S_{in}+S_{diss}+S_{NL})\;d \omega\;d\theta=0\] (A8)  
and  
\[\int_{0}^{2\pi}\;\int_{0}^{\infty}(S_{in}+S_{diss}+S_{NL})\;d\omega\;d\theta=0.\] (A9)  
The momentum flux to the ocean column, denoted by \(\tau_{oc}\), is the sum of the flux transferred by turbulence across the air-sea interface which was not used to generate waves \(\tau_{a}-\tau_{in}\) and the momentum flux transferred by the ocean waves due to wave breaking \(\tau_{diss}\). This leads to  
\[\overrightarrow{\tau_{oc}}=\overrightarrow{\tau_{a}}-\rho_{w}g\int_{0}^{2\pi }\;\int_{0}^{\omega_{c}}\frac{\overrightarrow{k}}{\omega}(S_{in}+S_{diss}+S_ {NL})d\omega\;d\theta.\] (A10)  
The contribution to the energy flux is  
\[\Phi_{oc}=\rho_{w}g\int_{0}^{2\pi}\;\int_{\omega_{c}}^{\infty}S_{in}\;d\omega \;d\theta-\rho_{w}g\int_{0}^{2\pi}\;\int_{0}^{\omega_{c}}(S_{diss}+S_{NL})\;d \omega\;d\theta.\] (A11)  
It is important to note that while the momentum fluxes are mainly determined by the high-frequency part of the wave spectrum, the energy flux is to some extent also determined by the low-frequency waves. The high frequency (\(\omega>\omega_{c}\)) contribution to the energy flux (first term of Equation (A11)) is  
\[\Phi_{oc_{bf}}=\rho_{w}g\int_{0}^{2\pi}\;\int_{\omega_{c}}^{\infty}S_{in}\;d \omega\;d\theta.\] (A12)In NEMO, the wave-induced turbulent kinetic energy (TKE) flux introduced at the sea surface depends on the wave energy factor \(\alpha\)[69] and is set to a constant value regardless of the sea state. Authors in [69] argued that the turbulent kinetic energy flux is relatively insensitive to the sea state and is well approximated by \(\omega u^{3}_{w^{\prime}}\) (\(u_{w^{\prime}}\) is the water-side friction velocity), and \(\alpha=100\) was thought to be representative of a mid-range of sea states between young wind seas and fully developed situations.