Evaluation of Three Shortwave Spectrum Representations on the Air-Sea Momentum Flux

Abstract

Studies on the shortwave spectrum, namely short-gravity, gravity-capillary, and parasitic-capillary waves, reveal that spectrum representation may modify the estimate of momentum transport at the air-sea interface. However, in numerical simulations, the shortwave spectra are usually approximated by simplified formulations. The effect of three shortwave spectrum formulations on the momentum balance at the air-sea interface was quantitatively evaluated for light to high wind speeds and fully developed seas. In the simulations, the spectra considered were: (i) obtained by an extrapolated function, (ii) dependent on the wave age derived from the observations, and (iii) from the solution of the energy balance equation. Considering computational time, the second was the fastest. while the first and third the computational time increased, respectively, by approximately 2–7% and 15–30%, depending on the wind speed. Concerning the observations, the mean square slope, the coupling parameter, and the drag coefficient, the second and third formulations showed better agreement, while the first one showed a large discrepancy. The results highlighted the importance of shortwave formulations in the analysis of the interaction between wind and wave.

Appendices

Appendix 1: Energy Balance Equation

Following Kudryavtsev et al. (1999), Kudryavtsev and Johannessen (2004), and Yurovskaya et al. (2013) the shortwaves part of the wave spectrum ($S_{s}=S_{s}^{sg}+S_{s}^{gc}+ S_{s}^{pc}$) is computed as a solution of the energy balance equation (Phillips 1977, 1985) written in the form:

$$\begin{aligned} \begin{array} {cccccc} Q(S_s)= &{}\underbrace{F_w\, S_s(k,\theta )} &{} -\, \underbrace{N_l\,S_s(k,\theta )} &{} + \, \underbrace{Q_b(k,\theta )}&{}=0\\ &{}(i)&{} \hspace{0.4cm}(ii)&{} (iii) \end{array}, \end{aligned}$$

(49)

where $Q(S_s)$ is the total energy dimensionless source represented as a sum of (i) the effective wind forcing, where:

$$\begin{aligned} F_w=k^3\,\beta _v(k,\theta ), \end{aligned}$$

(50)

(ii) the nonlinear energy losses associated with wave breaking in the gravity range, and nonlinear limitation of the spectral level in the capillary-gravity range, where

$$\begin{aligned} N_l= k^3\,\left[ k^3\,S_s(k,\theta )/\alpha (k)\right] ^{n(k)}, \end{aligned}$$

(51)

and (iii) generation of short-gravity, gravity-capillary, and parasitic-capillary waves, by breaking of longer waves. Here:

$$\begin{aligned} \beta _v(k,\theta )=\beta (k,\theta ) - 4\,\nu k^2/\sigma , \end{aligned}$$

(52)

is the effective wave growth rate dimensionless parameter, which is the difference between the wind wave growth rate $\beta$ [Eq. 33] and the rate of viscous dissipation, and $\nu =1.0 \times 10^{-6}$ m$^{2}$ s$^{-1}$ is the water viscosity coefficient.

In Eq. 51, $\alpha$ and n are the tuning dimensionless functions, defined as (Yurovskaya et al. 2013; Kudryavtsev et al. 2014):

$$\begin{aligned} \alpha (k) = 2 \times 10^{-3}\, e^{[-\ln (\bar{cb})/ n(k)]}, \end{aligned}$$

(53)

and:

$$\begin{aligned} n(k)= 10/[9\,h(k/k_{b}) +1], \end{aligned}$$

(54)

where $\bar{cb} = 0.03$ is a mean value of the growth rate parameter and h is a function as in Eq. 22, which describes the transition between the regime where the energy loss is dominated by wave breaking, by generation of capillaries, and by resonant three wave-wave interactions, respectively.

Finally, the source term $Q_b$ describes the generation of the shortwaves by breaking of longer waves, which is given by Eq. 19 for short-gravity and gravity-capillary waves and Eq. 20 for parasitic-capillary waves.

Appendix 1a: Solution of the Energy Balance Equation for $S_{s}^{sg}$ and $S_{s}^{gc}$

The spectrum $S_{s}$, which is $S_{s}^{sg}$ for short-gravity waves in the range $[k_{cut},k_{b}]$ and $S_{s}^{gc}$ for gravity-capillary waves in the range $[k_{b},k_{\gamma }]$, is computed using Eq. 49, with $Q_b$ given by Eq. 19. Then, Eq. 49 becomes:

$$\begin{aligned} k^3\,\beta _v(k,\theta )\, S_{s}(k,\theta ) - k^3\,S_{s}(k,\theta )\left( \frac{k^3\,S_{s}(k,\theta )}{\alpha (k)}\right) ^{n(k)} \,+ \,Q_{b}(k,\theta )=0. \end{aligned}$$

(55)

As the relation (55) is a nonlinear algebraic equation, the Newton iterative method is evoked. Then, $S_{s}(k,\theta )$ is calculated as:

$$\begin{aligned} \begin{array}{ll} S_{s}^{(0)}=&{} S_{s}^{(i)} \\ S_{s}^{(1)}=&{} S_{s}^{(0)} - \frac{Q(S_{s}^{(0)})}{\left[ \frac{\partial Q}{\partial S_{s}}\right] _{(0)}}\\ &{}\vdots \\ S_{s}^{(n)}=&{} S_{s}^{(n-1)} - \frac{Q\left( S_{s}^{(n-1)}\right) }{\left[ \frac{\partial Q}{\partial S_{s}}\right] _{(n-1)}} \\ \end{array}, \end{aligned}$$

(56)

where:

$$\begin{aligned} \frac{\partial Q(S_{s})}{\partial S_{s}}=\beta _v(k,\theta )\, k^3 - \frac{[n(k)+1]\,k^{3[n(k)+1]}}{\alpha (k)^{n(k)}}S_{s}(k,\theta )^{n(k)}. \end{aligned}$$

(57)

The first guess $S_{s}^{(i)}$ is calculated using three asymptotic solutions at up- ($S_{su}$), down- ($S_{sd}$), and cross-wind ($S_{sc}$) directions. At down-wind directions, the source $Q_{b}$ should be small in comparison with wind forcing. Then, the approximate solution of Eq. 49 is:

$$\begin{aligned} S_{sd}(k,\theta )=\, k^{-3}\,\alpha (k)\,\beta _v(k,\theta )^{1/n(k)}. \end{aligned}$$

(58)

At cross-wind directions, the effective wind input is small or vanishing [$\beta _v (k,\theta )\approx 0$], and we have

$$\begin{aligned} S_{sc}(k,\theta )=\, k^{-3}\,\alpha (k)^{\frac{n(k)}{n(k)+1}}\,\left[ Q_{b}(k,\theta )\right] ^{\frac{1}{n(k)+1}} . \end{aligned}$$

(59)

Finally, at up-wind directions, where $\beta _v(k,\theta )<0$, the spectral level is small enough to ignore the nonlinear term. Then, the approximate solution results from the balance between the wave breaking source and energy losses (due to viscosity and the interaction with the opposing wind):

$$\begin{aligned} S_{su}(k,\theta )=-\frac{Q_{b}(k,\theta )}{k^3\,\beta _v(k,\theta )}. \end{aligned}$$

(60)

Thus, using the Eqs. (58), (59), and (60), the first guess is defined by:

$$\begin{aligned} S_{s}^i(k,\theta )= \max [S_{sd}, \min (S_{sc},S_{su})]. \end{aligned}$$

(61)

Appendix 1b: Solution of the Energy Balance Equation for $S_{s}^{pc}$

The spectrum $S_{s}$, which is $S_{s}^{pc}$ for parasitic-capillary waves in the range $[k> k_{\gamma }]$, results from the same energy balance (Eq. 49), where $Q_b$ (Eq. 20) is balanced by viscous and nonlinear dissipation. Here, the wind forcing $\beta$ is omitted in the term $F_w$ (Eq. 50). Considering that $n(k)= 1$ at capillaries range, follows:

$$\begin{aligned} Q(S_{s})=-\left( \frac{k^6}{\alpha (k)}\right) \,S_{s}^2(k,\theta )- \left( \frac{4\,\gamma \,k^5}{\sigma }\right) \,S_{s}(k,\theta ) + Q_{b}(k,\theta )=0. \end{aligned}$$

(62)

As the relation (62) is a quadratic equation, its solution is straightforward. Hence, $S_{s}$ is calculated by:

$$\begin{aligned} S_{s}(k,\theta ) = \frac{ \alpha (k)}{2\,k^{6}}\left[ -\frac{4\,\gamma \,k^5}{\sigma }+ \sqrt{\left( \frac{4\,\gamma \,k^5}{\sigma }\right) ^2 + \frac{4\,k^{6}\,Q_{b}(k,\theta )}{\alpha (k)}} \, \right] . \end{aligned}$$

(63)