State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, CAS, Beijing, China

Jiangsu Key Laboratory of Agricultural Meteorology, College of Applied Meteorology, Nanjing University of Information Science and Technology, Nanjing, China

Corresponding author: Z. Gao, State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, CAS, Box 9804, Beijing 100029, China. (zgao@mail.iap.ac.cn)

[1] Based on the data previously collected during the Humidity Exchange over the Sea Main Experiment (HEXMAX), the methods used to parameterize aerodynamic roughness (z_{0}), friction velocity (u_{*}), and the neutral drag coefficient (C_{DN}) under moderate wind speed conditions originally proposed by Gao et al. (2006) were extended by using the nondimensional significant wave height (gH_{s}/u_{*}^{2} or gH_{s}/U_{10N}^{2}) instead of wave age (c_{p}/u_{*} or c_{p}/U_{10N}), where g is the acceleration of gravity, H_{s} is the significant wave height, U_{10N}is the horizontal wind speed at 10-m height under the neutral atmospheric condition, andc_{p} is the phase velocity of the peak wave spectrum. The results show (1) u_{*} = 0.024U_{10N}(gH_{s}/U_{10N}^{2})^{−1/4}, (2) z_{0} = 10 × exp[−4.797(gH_{s}/u_{*}^{2})^{1/6}] or z_{0} = 10 × exp[−16.613(gH_{s}/U_{10N}^{2})^{1/4}], and (3) C_{DN} = 0.007(gH_{s}/u_{*}^{2})^{−1/3} or C_{DN} = 5.76 × 10^{−4}(gH_{s}/U_{10N}^{2})^{−1/2}. The present parameterization schemes were experimentally tested.

[2]Gao et al. [2006] (hereinafter referred to as GWW) presented a set of equations aimed at parameterization of sea surface aerodynamic roughness length by using the wave age (c_{p}/u_{*} or c_{p}/U_{10N}), where c_{p} is the phase velocity of the peak wave spectrum, u_{*} the friction velocity, and U_{10N}the horizontal wind speed at 10-m height under neutral atmospheric conditions, for moderate wind conditions by using the dimensional analysis method for the data previously collected from the Humidity Exchange over the Sea Main Experiment (HEXMAX) [Janssen, 1997]. The dependence of sea surface aerodynamic roughness length z_{0} on c_{p}/u_{*} or c_{p}/U_{10N} was also demonstrated by previous studies [e.g., Smith et al., 1992; Donelan et al., 1993; Oost and Oost, 2004]. Meanwhile, Taylor and Yelland [2001] pointed out that z_{0} depends mainly on significant wave height (H_{s}). In order to reconcile the differences, this paper attempts (1) to extend GWW's equation derivation by using the nondimensional significant wave height (gH_{s}/u_{*}^{2} or gH_{s}/U_{10N}^{2}) instead of wave age (c_{p}/u_{*} or c_{p}/U_{10N}) from the same database, and (2) to experimentally validate the equations obtained in the present work.

2. Extension of the GWW Method

2.1. Original GWW Method

[3] After briefly introducing the HEXMAX experiment performed at the Dutch research platform 9 km offshore in the North Sea, GWW started with a similarity law proposed by Jones and Toba [2001]:

H*=BT*3/2,

where H* is the nondimensional significant wave height; T* the nondimensional period, and B an empirical coefficient, H* = gH_{s}/u_{*}^{2}, and T* = 2πc_{p}/u_{*}. Equation (1) thus becomes

Hs=Bu*2(2πcp/u*)3/2/g.

[4] Inspired by Oost et al. [2002], GWW reexamined the relationship between the dimensionless variables u_{*}(gT_{p})^{−1} and U_{10N}^{2}(gH_{s})^{−1} by dimensional analyses:

u*(gTp)−1=a1U10N2(gHs)−1,

where a_{1} = 1.19 × 10^{−3}for the HEXMAX experiment. GWW incorporated equation (2) into equation (3) for the HEXMAX experiment to eliminate H_{s}, resulting in

u*=(a12πB)2/3U10N4/3cp1/3=0.0353U10N4/3cp1/3.

[5] GWW then combined equation (4) with the neutral wind profile equation in the surface layer

u*=kU10Nln(10/z0),

where k is the von Karman constant, and z_{0} is the sea surface aerodynamic roughness length. Combining equations (4) and (5) to eliminate U_{10N} gives equation (6), or to eliminate u_{*} gives equation (7). Then z_{0} can be expressed as follows:

z0=10exp[−k(a1/2πB)−1/2(cp/u*)1/4],

orz0=10exp[−k(a1/2πB)−2/3(cp/U10N)1/3].

[6] The relationship between the neutral drag coefficient (C_{DN}), defined as C_{DN} ≡ u_{*}^{2}/U_{10N}^{2}, and wave age was accordingly derived as follows:

[7] The HEXMAX database, published by Janssen [1997], is also used in dimensional analyses in the present study in order to be consistent with GWW. Figure 1 is the same as Figure 1 of Gao et al. [2006] except that the variations of the neutral drag coefficient (C_{DN}) and the wave age parameter (c_{p}U_{10N}^{−1}) against the 10 m neutral wind speed(U_{10N}) were removed because both C_{DN}(≡ u_{*}^{2}/U_{10N}^{2}) and c_{p}U_{10N}^{−1} are mathematically correlated with U_{10N}.

[8] Because the physical significance of the dimensionless variables u_{*}(gT_{p})^{−1} and U_{10N}^{2}(gH_{s})^{−1} are not clearly demonstrated in Figure 4 of GWW, we reexamined the nondimensional wave period (gT_{p})/u_{*} and the nondimensional significant wave height gH_{s}/U_{10N}^{2} by dimensional analyses here, resulting in

(gTp)/u*=b(gHs)/U10N2,

where the regression coefficient b = a_{1}^{−1} = 836.99. Figure 1 shows that the nondimensional significant wave height gH_{s}/U_{10N}^{2} depends mainly on wind speed and not wave height. Equation (3′) is similar to GWW's equation (3), and the correlation coefficient between (gT_{p})/u_{*} and gH_{s}/U_{10N}^{2} is 0.93 (Figure 2). Equation (3′) can be rewritten as

u*=(Tp/bHs)U10N2,

which means a quadratic dependence for u_{*} in terms of U_{10N} when the coefficient of U_{10N}^{2} is function of wave parameters (H_{s} and T_{p}). Using the ASGAMAGE data, Oost et al. [2002], and Guan and Xie [2004] also found this quadratic behavior. Equations (3′) and (3″) show that u_{*} is dominated by the U_{10N} rather than by the wave state, although it is influenced by wave state. For identical U_{10N}, u_{*} would be greatest for low frequency (long waves) with low amplitude.

[10] Scatterplots of u_{*} calculated from equations (4) and (4′) and field measurements of u_{*} are shown in Figure 3. Theoretically, equations (4) and (4′) should generate the same results, but Figure 3 shows that these results are somewhat different. This may be caused by the complexities in (1) relationships among u_{*}, H_{s} and c_{p}, (2) measurements of H_{s} and c_{p}, and (3) our assumptions in the derivations above. The correlation coefficient between the values of u_{*} generated from equation (4) and those measured is 0.97.

[12] We roughly consider the values of z_{0} determined by z_{0} = 10 exp(−kU_{10N}/u_{*}) as direct measurements since U_{10N} and u_{*}were directly measured, although Monin-Obukhov similarity theory is used.Figure 4 shows the variation of z_{0} relative to gH_{s}/u_{*}^{2} and gH_{s}/U_{10N}^{2} for both equations (6′–7′) and the direct measurements respectively. It is obvious that z_{0} decreases with increasing nondimensional significant wave height (gH_{s}/u_{*}^{2} or gH_{s}/U_{10N}^{2}). Accordingly, the neutral drag coefficient, C_{DN}, can be written as:

[13]Figure 5 shows the variations of C_{DN} with gH_{s}/u_{*}^{2} or gH_{s}/U_{10N}^{2} for both equations (8′–9′) and the direct measurements respectively. C_{DN} decreases with increasing gH_{s}/u_{*}^{2} or gH_{s}/U_{10N}^{2}.

[14]Guan and Xie [2004] showed that within the normal range of wind speed, C_{DN} remains a linear function of U_{10N} and the slope of the function is dependent on wave steepness. Equation (9′) shows C_{DN} remains a linear function of U_{10N} but the slope depends on significant wave height.

3. Evaluation of Results Obtained Above

[15]Equations (4) and (4′)are inter-compared by using the data collected in the North Sea Platform (FPN) experiment in December 1985 [Geernaert et al., 1987]. Figure 6 shows the values of u_{*} obtained from equations (4) and (4′). The root-mean square (rms) difference ofu_{*} derived from equations (4) and (4′) are 0.067 and 0.094 ms^{−1} respectively against the direct measurements. The advantages of equations (4) and (4′)are that they do not suffer from self-correlation problems becausec_{p} and H_{s} can be obtained from ocean wave model and U_{10N} can be obtained from atmospheric models or scatterometer measurements, respectively.

[17] Using the HEXMAX observations, this study extended our previous analyses on the parameterizations of the sea surface roughness length [Gao et al., 2006] to investigate the dependence of sea surface roughness on nondimensional significant wave height. The relationship between the nondimensional period (gT_{p})/u_{*} and the nondimensional significant wave height gH_{s}/U_{10N}^{2} is explored by dimensional analyses at this observation site. Combining this relationship with the previous significant wave height formula results in u_{*} = 0.0241U_{10N}(gH_{s}/U_{10N}^{2})^{−1/4}. When the neutral wind profile equation is applied, the results show that (1) the dependence relationship between z_{0} and gH_{s}/u_{*}^{2} (or gH_{s}/U_{10N}^{2}) is z_{0} = 10 exp[−4.797(gH_{s}/u_{*}^{2})^{1/6} or z_{0} = 10 exp[−16.613(gH_{s}/U_{10N}^{2})^{1/4}]; and (2) the dependence of C_{DN} on gH_{s}/u_{*}^{2} (or gH_{s}/U_{10N}^{2}) is C_{DN} = 0.007(gH_{s}/u_{*}^{2})^{−1/3} or C_{DN} = 5.76 × 10^{−4}(gH_{s}/U_{10N}^{2})^{−1/2}. Finally, our new parameterizations of sea surface roughness length were validated by using an independent database.

[18] The present paper describes extension of a previous analysis [Gao et al., 2006] where friction velocity, roughness length, or drag coefficient are scaled in terms of wave variables. The earlier analysis developed a c_{p}/u_{*} or c_{p}/U_{10N} scaling while this paper derives a gH_{s}/u_{*}^{2} or gH_{s}/U_{10N}^{2} scaling. i.e., significant wave height is used to characterize the wave effect rather than wave phase speed.

[19] The HEXMAX and FPN data are both collected from offshore towers, so they are in the shallow water regime (water depth less than wavelength). Applicability to open ocean data is speculative and needs further testing.

[20] With the continuous satellite altimetry missions (e.g., TOPEX/Poseidon, Jason-1 and Jason-2) that are capable of making direct measurements of significant wave heights, the new parameterizations presented in this study are expected to have practical usage in momentum and heat fluxes calculations and numerical models in conjunction with satellite observations. Due to their non-self-correlation characteristics, the new parameterizations are computationally effective and easy to be implemented.

Acknowledgments

[21] This study was supported by the National Program on Key Basic Research Project of China (973) under grants 2011CB403501 and 2012CB417203, by China Meteorological Administration under grant GYHY201006024, by National Natural Science Foundation of China under grants 40906023 and 41076012, by the CAS Strategic Priority Research Program grant XDA05110101, and by the Priority Academic Development of Jiangsu Higher Education Institutions (PAPD). We are very grateful to three anonymous reviewers for their careful review and valuable comments, which led to substantial improvement of this manuscript.