Model of Vertical Mixing Induced by Wind Waves

Abstract—Formulas for the vertical wind-wave-induced mixing coefficient are obtained. For this purpose, in the Navier–Stokes equations the flow velocity is decomposed into four components, namely, mean flow, wave orbital motion, wave-induced turbulent flow fluctuations, and background turbulent fluctuations. Such a decomposition makes it possible to distinguish the wave stress Re_w in the Reynolds equations as an addition to the background stress Re_b. For the background turbulent fluctuations the Prandtl approximation is used for closure of Re_w. This leads to an implicit expression for the wave-induced vertical mixing function ${{B}_{v}}$. The finite expression for ${{B}_{v}}$ is determined using author’s results for the turbulent viscosity in the wave zone found earlier within the framework of the three-layer conception for the air-water interface. The explicit expression for the function ${{B}_{v}}(a,{{u}_{*}},z)$ is linear with respect to the wave amplitude a(z) at the depth z and the friction velocity u_* in air. Since the wave amplitude decreases exponentially as a function of the depth, this result for ${{B}_{v}}(a)$ means the possibility of strengthening the wave impact on vertical mixing as compared with the well-known cubic dependence of ${{B}_{v}}(a)$.

EXPLANATION OF FORMULA (1.18)

Formula (1.18) is constructed on the basis of three theoretical and experimental facts [26, 28]:

1) on the water surface the wind drift velocity is of the order of $0.5{{u}_{*}}$;

2) in the wave zone the drift velocity profile is linear with respect to z;

3) the thickness of the wave zone is of the order of 3а₀, where а₀ is the mean wave height [27, 28] (see explanation to Fig. 1).

Under the assumption that in the wave zone the flow velocity varies from U_d0 (on the boundary with the driving atmosphere layer) to the friction velocity in water ${{u}_{{*w}}} \approx {{(ro)}^{{1/2}}}{{u}_{*}} \approx 0.03{{u}_{*}} \ll {{U}_{{d0}}}$ (on the boundary with the upper water layer), in the wave zone we can write the balance equation for the flux of momentum of the form:

$${{\tau }_{w}}{\text{/}}{{\rho }_{w}} = {{K}_{{\text{t}}}}_{{\text{w}}}\frac{{\partial {{U}_{d}}(z)}}{{\partial z}} \approx {{K}_{{\text{t}}}}_{{\text{w}}}({{U}_{{d0}}}{\text{/}}3{{a}_{0}}).$$

(A1)

Here,

$${{\tau }_{w}}{\text{/}}{{\rho }_{w}} \approx ro\,u_{*}^{2} = u_{{*w}}^{2}$$

(A2)

is the vertical flux of momentum in water divided by the water density; $ro = ({{\rho }_{a}}{\text{/}}{{\rho }_{w}}) \approx {{10}^{{ - 3}}}$ is the ratio of the air and water densities; ${{u}_{{*w}}}$ is the friction velocity in water; and ${{K}_{{\text{t}}}}_{{\text{w}}}$ is the unknown turbulent mixing (or viscosity) coefficient in the wave zone. On the right-hand side of (A1) the flow velocity gradient is determined by the drift velocity gradient in the wave zone. In accordance with the above, the latter is of the order $\partial {{U}_{d}}{\text{/}}\partial z \approx {{U}_{{d0}}}{\text{/}}3a$ since on the lower boundary of the wave zone the drift velocity is of the order of the friction velocity in water ${{u}_{{*w}}}$, i.e., much less than U_d0 ≈ 0.5${{u}_{*}}$.

Substituting expression (A2) and U_d0 in Eq. (A1), we obtain the expression for the turbulent viscosity coefficient ${{K}_{{\text{t}}}}_{{\text{w}}}$ of the form (1.18).