Definition of the wave-age parameter

Hanely et al. (2010) aims at presenting a valid climatology of “wave age” for the entire World Ocean, emphasizing in particular “the wave-driven regime” (i.e., the regime dominated by strong swell). Then the question of proper definition of wave age becomes crucial. In the literature, the following four variants appear: c_p/U₁₀, c_p/(U₁₀ cosθ), c_p/u_*, and c_p/(u_* cosθ), where c_p is the phase speed of the waves at the spectral peak, U₁₀ is the wind speed at 10 m, u_* is the friction velocity, and θ is the angle between the wind and the direction of propagation of the dominant waves. Inclusion of the cosθ term is loosely motivated by “the need to take into account the component of the wind” (or, in the case of u_*, the component of the stress vector) in the direction of wave propagation (see below). It is often argued that θ is small, so that cosθ becomes close to unity, thus warranting the use of c_p/U₁₀ or c_p/u_*, as the case may be. This argument is most often valid in the case of “wind-driven waves.” However, in the case of the wave-driven regime, this is often far from true, as demonstrated clearly in Hanley et al. (2010). This makes the issue of the cosine term highly relevant, which can be illustrated by a simple example, where we turn to “inverse wave age” as used by Hanley et al. (2010): Assume c_p = 12 m s⁻¹, U₁₀ = 10 m s⁻¹, and θ = 80°; then U₁₀/c_p = 0.83 and (U₁₀ cosθ)/c_p = 0.14. However, U₁₀/c_p = 0.83 represents “mature sea,” whereas (U₁₀ cosθ)/c_p = 0.14 is in the regime of expectedly upward momentum transfer. If instead c_p/(U₁₀ cosθ) is used the amplification of the wave age is even more notorious, then c_p/(U₁₀ cosθ) = 6.9 and c_p/U₁₀ = 1.2.

We noticed this dilemma in our studies of the swell regime in the Baltic Sea, based on data from our Östergarnsholm station. Smedman et al. (2003) introduced an alternative parameter to characterize the wave state, based on subdivision of the one-dimensional wave spectrum S(n) into two parts, E₁ and E₂, respectively, where E₁ is the integral of the wave spectrum from n = 0 to n = n₁ = g/(2πU₁₀) and E₂ is the corresponding integral for frequencies from n₁ to infinity. The parameter E₂, which is expected to represent the contribution from wind-forced waves, was shown to be proportional to , whereas E₁, which was expected to reflect swell, showed zero correlation with the local wind, implying that the ratio E₁/E₂ can be used as a wave-state parameter. In Smedman et al. (2003), the cosine term was retained. We later plotted E₁/E₂ against c_p/U₁₀, as illustrated in Fig. 1a, and against c_p/(U₁₀ cosθ), as shown in Fig. 1b (swell data from a 35-day period in January–February 1998). These graphs clearly show that there is much less scatter in Fig. 1a than in Fig. 1b, which led us to expect that a wave-age formulation without the cosine term would be more appropriate. Note the factor of 10 difference of ordinate scale between Figs. 1a,b and, in particular, that there are many negative values of c_p/(U₁₀ cosθ) in Fig. 1b. This reflects the singularity at θ = 90°, where c_p/(U₁₀ cosθ) becomes infinite, with large positive values for θ slightly less than 90° and large negative values for θ slightly larger than 90°.

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(a) c_p/U₁₀ plotted against E₁/E₂; (b) c_p/U₁₀ cosθ plotted against E₁/E₂ for the same dataset. Measurements from the Östergarnsholm station in the Baltic sea from the time period 6 Jan–15 Feb 1998. Data were selected with the criteria c_p/U₁₀ ≥ 1.2 and U₁₀ ≥ 2 m s⁻¹.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05015.1

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In a later swell study, Smedman et al. (2009) and Högström et al. (2009), we made the observation that cases with strong swell, with similar c_p/U₁₀ values (around 4.7) but very different wind–wave angle [around 90° for C1 (“cross swell” case) and close to zero for F1 (“following swell” case)], show very similar characteristics [cf. the corresponding wind profiles in Fig. 3a of Smedman et al. (2009) and the corresponding turbulent kinetic energy (TKE) budgets of Figs. 2a,b in Högström et al. (2009)]. Similar conclusions were drawn for other cases with cross swell (cases C2 and C3) and following swell (F2), respectively, which have c_p/U₁₀ values around 1.7. These cases are similar to each other in terms of wind profile and TKE budget but are significantly different compared to the cases with c_p/U₁₀ values around 4.7. This led us to reconsider the theoretical background for inclusion of the cosine term. As demonstrated in an appendix of Smedman et al. (2009), the crucial point is the choice of coordinate system, with dramatically different results being obtained when evaluating the pressure drag term in a system following the wave compared to in a fix system. The arguments from the appendix are repeated here in a modified and generalized form.

The form stress or pressure drag over a wave with wavelength λ is obtained from

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where η is the height of the wave surface at position x and p is the pressure on the surface at x. Considering now a wind at an angle θ to the normal of the wave front, this changes the effective slope to (dη/dx)(cosθ). However, this is only true for a wave with phase speed c = 0 (i.e., for a stationary wave). For a wave with finite phase speed c, Eq. (1) must be evaluated in a coordinate system following the wave, because it is the air motion relative to the wave that matters here. It means that the angle θ is transformed into another, “relative” angle α. This is illustrated schematically in Fig. 2, where we consider a train of monochromatic two-dimensional waves traveling toward the west (i.e., to the left in the figure) with phase speed c. The relative motion of air and wave is illustrated schematically in the two graphs in Fig. 2. The wind speed U is the same for the two cases, but the angle θ is different: in case A (Fig. 2a), θ < 90°; in case B (Fig. 2b), θ > 90°. Here, c is assumed to be larger than U. The following vectors are shown in the two subplots: (i) the relative motion of the wave surface and the air that would ensue for the case of zero wind (i.e., a vector c opposite to the wave motion and of the same magnitude c as that of the wave phase speed); (ii) the wind vector U; and (iii) the resultant sum of vectors (i) and (ii), R.

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Schematic picture of wind/swell geometry for a case with monocromatic swell waves propagating toward the west with speed c for (a) a case with wind–wave angle θ < 90° and (b) a case with θ > 90°. Indicated vectors refer to a coordinate system moving with the wave: c indicates the wave phase vector, U indicates the wind vector; and R indicates the resultant vector. The angle β in (b) equals θ − 90°.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05015.1

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Case A is the following swell case: θ < 90°. The graph shows the angle between the wind and the wave direction θ and the corresponding angle α₁, between the resultant trajectory vector R and the vector c, which is obtained from the following relation:

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Case B is the counterswell case (θ > 90°), from which the angle α₂ between the resultant trajectory vector and the vector c is obtained,

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where β = 90° − θ. For θ = 90°, Eqs. (2) and (3) give α₁ = α₂ = arctan(U/c).

From Eqs. (2) and (3), it can be evaluated how the angle α, and hence the factor cosα, varies with wave age c/U and the wind–wave angle θ. It is found that, for c/U > 2.0, 0.86 < cosα < 1 for all values of θ: that is, for practical purposes, c/U cosα ≈ c/U. For c/U ≈ 1.6, the angle α varies somewhat with θ, with cosα having a minimum of 0.78 at θ = 45°.

The above result implies that, for large enough c/U, the resulting pressure drag (D_p)₀ [cf. Eq.(1)] is largely independent of the wind–wave angle θ. This result is valid for a gentle, symmetrical wave form (e.g., a sine wave) that is not modified by the wind. However, such a modification may occur if a crosswind is strong enough. Thus, Drennan et al. (1999) report about significantly enhanced 10-m neutral drag coefficient C_DN values for cases with “counter swell.” However, their cases have rather high wind speed, 9 m s⁻¹ for one case and “22 cases, over the wind speed range 4.5–13.5 m s⁻¹.” On the other hand, Smedman et al. (2009) report in their Table 1 that the wind–wave angle is 105° for their “crosswind case” C1 and 120° and 125° for their cases C2 and C3, respectively. As noted earlier, wind profile and TKE budget were closely similar for C1 and F1 (where F stands for following swell); both cases have c_p/U close to 4.7 on one hand, and c_p/U around 1.7 for C2, C3, and F2 on the other hand: that is, these data indicate a significant effect from c_p/U, but no effect from the wind–wave angle.

Additionally, Hanley and Belcher (2010) have used the mean wave direction parameter from the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) (Bidlot 2001). In the tropics, where swell waves are strongly dominant and most of the energy is concentrated in a narrow peak, the propagating direction of the peak wave can be considered close to the mean wave direction. However, along the middle to subpolar latitudes, where several wave systems from different directions can coexist at the same time, the mean wave direction can be very different from the propagating direction of the peak wave, inducing an additional error in θ.

The conclusion from this study is that, for climatological mapping of the effect of wave age on the air–sea momentum exchange, the parameter c_p/U₁₀ should be used rather than c_p/(U₁₀ cosθ), as in Hanley et al. (2010).