A simple extension of “An alternative approach to sea surface aerodynamic roughness” by Zhiqiu Gao, Qing Wang, and Shouping Wang

Authors

  • Zhiqiu Gao,

    Corresponding author
    1. State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, CAS, Beijing, China
    2. 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)
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  • Linlin Wang,

    1. State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, CAS, Beijing, China
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  • Xueyan Bi,

    1. Institute of Tropical and Marine Meteorology, CMA, Guangzhou, China
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  • Qingtao Song,

    1. National Satellite Ocean Application Service, Beijing, China
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  • Yuchao Gao

    1. Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, Maryland, USA
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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)

Abstract

[1] Based on the data previously collected during the Humidity Exchange over the Sea Main Experiment (HEXMAX), the methods used to parameterize aerodynamic roughness (z0), friction velocity (u*), and the neutral drag coefficient (CDN) under moderate wind speed conditions originally proposed by Gao et al. (2006) were extended by using the nondimensional significant wave height (gHs/u*2 or gHs/U10N2) instead of wave age (cp/u* or cp/U10N), where g is the acceleration of gravity, Hs is the significant wave height, U10Nis the horizontal wind speed at 10-m height under the neutral atmospheric condition, andcp is the phase velocity of the peak wave spectrum. The results show (1) u* = 0.024U10N(gHs/U10N2)−1/4, (2) z0 = 10 × exp[−4.797(gHs/u*2)1/6] or z0 = 10 × exp[−16.613(gHs/U10N2)1/4], and (3) CDN = 0.007(gHs/u*2)−1/3 or CDN = 5.76 × 10−4(gHs/U10N2)−1/2. The present parameterization schemes were experimentally tested.

1. Introduction

[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 (cp/u* or cp/U10N), where cp is the phase velocity of the peak wave spectrum, u* the friction velocity, and U10Nthe 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 z0 on cp/u* or cp/U10N 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 z0 depends mainly on significant wave height (Hs). In order to reconcile the differences, this paper attempts (1) to extend GWW's equation derivation by using the nondimensional significant wave height (gHs/u*2 or gHs/U10N2) instead of wave age (cp/u* or cp/U10N) 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]:

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where H* is the nondimensional significant wave height; T* the nondimensional period, and B an empirical coefficient, H* = gHs/u*2, and T* = 2πcp/u*. Equation (1) thus becomes

display math

[4] Inspired by Oost et al. [2002], GWW reexamined the relationship between the dimensionless variables u*(gTp)−1 and U10N2(gHs)−1 by dimensional analyses:

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where a1 = 1.19 × 10−3for the HEXMAX experiment. GWW incorporated equation (2) into equation (3) for the HEXMAX experiment to eliminate Hs, resulting in

display math

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

display math

where k is the von Karman constant, and z0 is the sea surface aerodynamic roughness length. Combining equations (4) and (5) to eliminate U10N gives equation (6), or to eliminate u* gives equation (7). Then z0 can be expressed as follows:

display math
display math

[6] The relationship between the neutral drag coefficient (CDN), defined as CDN ≡ u*2/U10N2, and wave age was accordingly derived as follows:

display math
display math

where inline image.

2.2. Proposed Revision of the GWW Method

[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 (CDN) and the wave age parameter (cpU10N−1) against the 10 m neutral wind speed(U10N) were removed because both CDN(≡ u*2/U10N2) and cpU10N−1 are mathematically correlated with U10N.

Figure 1.

Data from the HEXMAX experiment [Janssen, 1997] plotted against the neutral 10-m wind speed,U10N: (a) friction velocity, u* (ms−1); (b) phase velocity at peak of the wave spectrum, cp (ms−1); (c) significant wave height Hs (m); and (d) wave age (cp/u*).

[8] Because the physical significance of the dimensionless variables u*(gTp)−1 and U10N2(gHs)−1 are not clearly demonstrated in Figure 4 of GWW, we reexamined the nondimensional wave period (gTp)/u* and the nondimensional significant wave height gHs/U10N2 by dimensional analyses here, resulting in

display math

where the regression coefficient b = a1−1 = 836.99. Figure 1 shows that the nondimensional significant wave height gHs/U10N2 depends mainly on wind speed and not wave height. Equation (3′) is similar to GWW's equation (3), and the correlation coefficient between (gTp)/u* and gHs/U10N2 is 0.93 (Figure 2). Equation (3′) can be rewritten as

display math

which means a quadratic dependence for u* in terms of U10N when the coefficient of U10N2 is function of wave parameters (Hs and Tp). 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 U10N rather than by the wave state, although it is influenced by wave state. For identical U10N, u* would be greatest for low frequency (long waves) with low amplitude.

Figure 2.

Nondimensional period (gTp)/u* as a function of nondimensional significant wave height gHs/U10N2 for HEXMAX data.

[9] Combining equation (2) and equation (3′) to eliminate cp gives

display math

[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*, Hs and cp, (2) measurements of Hs and cp, 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.

Figure 3.

Scatterplot of u* estimated from equations (4) and (4′) against direct measurements for HEXMAX data.

[11] Inserting equations (4′) into (5) to eliminate U10N gives equation (6′), or to eliminate u* gives equation (7′). Then z0 can be expressed as follows,

display math
display math

[12] We roughly consider the values of z0 determined by z0 = 10 exp(−kU10N/u*) as direct measurements since U10N and u*were directly measured, although Monin-Obukhov similarity theory is used.Figure 4 shows the variation of z0 relative to gHs/u*2 and gHs/U10N2 for both equations (6′7′) and the direct measurements respectively. It is obvious that z0 decreases with increasing nondimensional significant wave height (gHs/u*2 or gHs/U10N2). Accordingly, the neutral drag coefficient, CDN, can be written as:

display math
display math
Figure 4.

(a) Variation of aerodynamic roughness length z0 against nondimensional significant wave height gHs/u*2 for HEXMAX data; (b) variation of aerodynamic roughness length z0 against nondimensional significant wave height gHs/U10N2 for HEXMAX data.

[13] Figure 5 shows the variations of CDN with gHs/u*2 or gHs/U10N2 for both equations (8′9′) and the direct measurements respectively. CDN decreases with increasing gHs/u*2 or gHs/U10N2.

Figure 5.

(a) Variation of neutral drag coefficient CDN against nondimensional significant wave height gHs/u*2 for HEXMAX data; (b) variation of neutral drag coefficient CDN against nondimensional significant wave height gHs/U10N2 for HEXMAX data.

[14] Guan and Xie [2004] showed that within the normal range of wind speed, CDN remains a linear function of U10N and the slope of the function is dependent on wave steepness. Equation (9′) shows CDN remains a linear function of U10N 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 becausecp and Hs can be obtained from ocean wave model and U10N can be obtained from atmospheric models or scatterometer measurements, respectively.

Figure 6.

Scatterplot of u* estimated by equations (4) and (4′) against direct measurements for the FPN experiment.

[16] Using in the COARE algorithm [Fairall et al., 1996a, 1996b, 2003], GWW compared equations (6) and (7) with three widely used sea surface roughness schemes described in previous studies [i.e., Taylor and Yelland, 2001; Oost et al., 2002]. Because equations (6′) and (7′) and (6) and (7) are theoretically identical, it can be inferred from the results in GWW (their Figure 8) that equations (7′)do not suffer from self-correlation problems.

4. Summary Remarks

[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 (gTp)/u* and the nondimensional significant wave height gHs/U10N2 is explored by dimensional analyses at this observation site. Combining this relationship with the previous significant wave height formula results in u* = 0.0241U10N(gHs/U10N2)−1/4. When the neutral wind profile equation is applied, the results show that (1) the dependence relationship between z0 and gHs/u*2 (or gHs/U10N2) is z0 = 10 exp[−4.797(gHs/u*2)1/6 or z0 = 10 exp[−16.613(gHs/U10N2)1/4]; and (2) the dependence of CDN on gHs/u*2 (or gHs/U10N2) is CDN = 0.007(gHs/u*2)−1/3 or CDN = 5.76 × 10−4(gHs/U10N2)−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 cp/u* or cp/U10N scaling while this paper derives a gHs/u*2 or gHs/U10N2 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.

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