**Theoretical expression of a breaking wave:**

The energy dissipation rate due to wave breaking in a unit area of water column can be expressed as 

$$\mathbf{R\_{ds}} = \rho\_{\mathbf{w}} \mathbf{g} \int \mathbf{S\_{ds}}(\mathbf{k}) d\mathbf{k},\tag{A3}$$

where ρw is the density of sea water, g is the gravity constant (9.8 m/s2), and Rds is the downward input of turbulent kinetic energy flux due to wave breaking on the sea surface. In practice, a unified analytical form of wave breaking energy dissipation is derived as follows [59]:

$$\mathcal{R}\_{\rm dis} = 2.97 \gamma \rho\_{\rm w} \lg \beta\_{\ast}^{-2} \omega\_{\rm P} \mathcal{E}\_{\prime} \tag{A4}$$

where

$$
\beta\_\* = \frac{\text{g}}{\text{u}\_\* \omega\_{\text{p}}},
\tag{A5}
$$

$$\mathbf{E} = \int \mathbf{F}(\mathbf{k})d\mathbf{k},\tag{A6}$$

$$
\omega\_{\rm p} = \frac{2\pi}{\mathcal{T}\_{\rm P}} \,\prime\tag{A7}
$$

where *γ* is energy dissipated per unit of white crown accounting for the percentage of total wave energy dissipation, generally *γ* = 0.1; β∗ is the wave age; E is spectral density; **k** is wave number vector; F(**k**) is the two-dimensional wave spectrum; <sup>ω</sup>p is the spectral peak angular frequency; *u*∗ is the friction velocity (=√cdU10); cd is the drag coefficient; and Tp is the dominant wave period.

## **Theoretical expression of a nonbreaking wave:**

The parameter schemes of vertical diffusion coefficient Kh and vertical eddy viscosity coefficient Km are

$$\mathsf{K}\_{\mathsf{m}} = \frac{2\mathrm{ak}^2 \lambda}{\pi \Gamma} \mathsf{e}^{\frac{2\pi \mathsf{a}}{\lambda}},\tag{A8}$$

$$\mathbb{K}\_{\mathrm{h}} = \frac{2\mathrm{Pk}^2}{\mathrm{g}} \delta\beta^3 \mathcal{W}^3 \mathrm{e}^{\frac{\mathrm{gr}}{\beta^2 \mathrm{W}^2}},\tag{A9}$$

where k is the Kaman constant; a is the amplitude, which is twice the effective significant wave height; T is the fluctuation period; λ is the wavelength; z is the distance from the sea surface to a certain depth; β∗ is the wave age; P is a dimensionless variable related to the Richardson number coefficient, and δ is wave steepness. Generally, at the sea surface k = 0.4, β = 1.0, P = 0.1, δ = 0.1, π = 3.14, and g = 9.8 m/s2.
