On the Impact of Long Wind-Waves on Near-Surface Turbulence and Momentum Fluxes

Abstract

We propose a new phenomenological model to represent the impact of wind-waves on the dissipation of turbulence kinetic energy near the sea surface. In this model, the momentum flux at a given height results from the averaged contribution of eddies attached to the sea surface whose sizes are related to the surface geometry. This yields a coupling between long wind-waves and turbulence at heights of about 10 m. This new wind-and-waves coupling is thus not exclusively confined to the short wave range and heights below 5 m, where most of the momentum transfer to the waves is known to occur. The proposed framework clarifies the impact of wind-waves on Monin–Obukhov similarity theory, and the role of long wind-waves on the observed wind-wave variability of momentum fluxes. This work reveals which state variables related to the wind-wave coupling require more accurate measurements to further improve wind-over-waves models and parametrizations.

Appendices

Appendix 1: Coupling Between Short Wind-Waves and Atmospheric Turbulence

Details are provided on the coupling between short wind-waves and atmospheric turbulence, following the wind-over-waves model presented in Kudryavtsev et al. (2014) and references therein. The short wind-wave model is first described, and expressions for wave-induced stress are then presented.

Waves are described by their wavenumber k, frequency $\omega$, phase speed c, and direction of propagation $\psi$, and follow the dispersion relation $\omega ^2 = gk + T_{sw} k^3$ where $T_{sw}$ is the dynamical surface water tension. The wave field is specified by means of the directional spectrum $S^d(k, \psi )$. We also introduce the saturation spectrum $B(k, \psi ) = k^4S^d(k, \psi )$, which will be used in the following.

As proposed by Kudryavtsev et al. (2014), the full wave spectrum can be defined as a composition of a short-wave spectrum $B_{sw}$ and a long-wave spectrum $B_{lw}$ (in this study, the fetch-dependent spectrum of Donelan et al. 1985, is used). The weighted sum between $B_{lw}$ and $B_{sw}$ represents a wind-driven sea spectrum, without the presence of non-local waves (swell). It is in a one-to-one relation with the local atmospheric state. The short-wave spectrum is coupled to atmospheric turbulence through form drag, and further affects the momentum WBL through airflow separation stresses. The long-wave part is prescribed given some parameters (here spatial fetch).

The short-wave component $B_{sw}$ describes both gravity waves and parasitic capillary waves. The latter are generated on the forward face of shorter gravity waves (in the wavelength range 0.03–0.3 m), as they approach their maximum steepness, which, for longer gravity waves, would lead to breaking (Longuet-Higgins 1963).

The gravity short wind-wave spectrum results from a balance between wind forcing ($\beta$), non-linear energy losses due to wave breaking (or generation of parasitic capillary waves for shorter waves), and generation of short waves by large breakers (or of parasitic capillary waves by steep and shorter waves, $Q_b$). The balance equation reads

$$\begin{aligned} \beta _v(k, \psi ) B(k, \psi ) - B(k, \psi )\left( \frac{B(k, \psi )}{a}\right) ^{n_g} + Q_b(k, \psi ) = 0, \end{aligned}$$

(34)

with $\beta _v(k, \psi ) = \beta (k, \psi ) - 4\nu k^2/\omega$ the effective growth rate (with $\nu$ air viscosity), and $a = 2.2 \times 10^{-3}$ and $n_g = 10$ two tuning constants fitted to observations (from Yurovskaya et al. 2013). Expression for the source term $Q_b$ can be found in Appendix A of Kudryavtsev et al. (2014).

The short parasitic capillary waves, corresponding to waves of wavelengths of $3 \times 10^{-4}$ m or less, follow the balance Eq. 34 without the wind input term, and with modified constants a and $n_g$. For this range of waves for which wave breaking does not occur, the non-linear term is associated to a non-linear saturation of the wave spectrum.

Both equations are solved by iterations, given a wind forcing resulting from the WBL model (Eq. 28), and expressed as

$$\begin{aligned} \beta (k, \psi ) =\left\{ \begin{aligned}&c_{\beta } \left\{ \frac{u_*^l[h(k)]}{c}\right\} ^2 \cos \psi |\cos \psi |&\text {for } U[h(k)] > c \\&0&\text {for } U[h(k)] < c\\ \end{aligned}\right. \end{aligned}$$

(35)

where $c_\beta = 3 \times 10^{-2}$ is Plant’s constant and $h(k) = 0.1\,{k}^{-1}$ is the inner region height. Note that since wind input depends on the ratio between friction velocity and wave phase speed, it is supported mostly by slow (and short) waves (Plant 1982).

To solve Eq. 28, the wave-induced stress must be specified. Let ${{\tilde{T}}}$ and ${{\tilde{T}}}_a$ be the intensity of form drag and airflow separation induced by a wave component of wavenumber k. Both these effects act over a shallow atmospheric layer, up to heights $h(k) \sim 0.1 k^{-1}$ and $h_{a} (k) \sim 0.3 k^{-1}$, respectively (Kudryavtsev et al. 2014). We further assume, for simplicity, that form drag (respectively airflow separation) is constant up to h (resp. $h_a$) and cancels for $z>h$ (resp. $z>h_a$). This yields the following expression for the total wave-induced stress

$$\begin{aligned} (u_*^w)^2(z)= & {} \int {{\tilde{T}}}(k)\text{ He }[h(k) - z] \mathrm{d}k \nonumber \\&+ \int {{\tilde{T}}}_a(k)\text{ He }[h_a(k) - z] \mathrm{d}k \end{aligned}$$

(36)

where He(x) is the Heaviside step function ($\hbox {He}(x)=1$ for $x>0$ and 0 otherwise). This expression couples the short wind-wave model (Eq. 34) to the SBL model (Eq. 28).

Form drag describes the impact of the wind-to-waves energy transfer on atmospheric turbulence, and is expressed as

$$\begin{aligned} {{\tilde{T}}}(k) = \left\{ \begin{aligned}&\frac{c_\beta }{k} \frac{\rho _w}{\rho _a} \{u_*^l[h(k)]\}^2 \int B(k,\psi )~ (\cos \psi )^3~ \mathrm{d}\psi&\text {for } U[h(k)] > c \\&0&\text {for } U[h(k)] < c\\ \end{aligned}\right. , \end{aligned}$$

(37)

where $\rho _w$ and $\rho _a$ are the density of water and air respectively.

Waves of wavelength greater than 0.3 m generate an additional stress due to airflow separation on top of breaking waves (Reul et al. 1999). The expression for airflow separation stress for a given wavenumber depends on wave-breaking statistics. However, following Phillips (1985), wave-breaking statistics can be related to wave energy dissipation (the second term from the left in Eq. 34). For waves in the equilibrium range, on top of which most of airflow separation events occur, the spectral balance (Eq. 34) is further assumed to be reduced only to a balance between wind input and dissipation. This results in the following expression for airflow separation for $U[h_a(k)] > c$

$$\begin{aligned} T_a(k) = \frac{2 c_{db} c_{\beta }}{a}h_a(k)k f_g(k) \left( \frac{\displaystyle U[h_a(k)]}{\displaystyle c} - 1 \right) ^2 \int B(k, \psi ) (\cos \psi )^5~ \mathrm{d}\psi \end{aligned}$$

(38)

where $f_g(k)$ is a cut-off function restricting airflow separation in the equilibrium range, and $c_{db}$ is the local roughness on top of breaking crests, which has a mean value of 0.35 (see Kudryavtsev and Makin 2001). For $U[h_a(k)] < c$, airflow separation is assumed to vanish (i.e. $T_a = 0$) which limits airflow separation to slow (short) waves (similar to form drag).

Appendix 2: Expressions for the Eddy Anisotropy and the Businger–Dyer Momentum Function

The Businger–Dyer universal momentum function (Businger 1988), derived from the Kansas measurements, reads

$$\begin{aligned} \phi _m^B(\zeta ) = \left\{ \begin{aligned}&1 + 4.7 \zeta&\text {for } \zeta >0 \\&\left( 1 - 15 \zeta \right) ^{-1/4}&\text {for } \zeta < 0\\ \end{aligned}\right. . \end{aligned}$$

(39)

This empirical function was recovered by Katul et al. (2011), by considering an eddy anisotropy of the form

$$\begin{aligned} f_a(\zeta ) = \left\{ \begin{aligned}&\left( 1 - \frac{0.38}{0.55}[1 - \text{ exp }(15\zeta )]\right) ^{-1}&\text {for } \zeta \le 0 \\&\left( 1 + \frac{1}{0.55}\zeta \right) ^{-6}&\text {for } \zeta > 0 \\ \end{aligned}\right. . \end{aligned}$$

(40)

This expression was obtained from measurements of turbulent vertical velocity spectra (from Kaimal et al. 1972).