a. Atmospheric stability parameterization

The flux parameterization discussed herein was developed in parallel with a sea state model that required that the atmospheric stability terms be well behaved when the reference height (z) approaches the wave height. Consequently, we use Benoit’s (1977) parameterizations for an unstable atmosphere, which is a slightly more detailed solution than the usual Businger–Dyer parameterizations (Dyer 1974; Liu et al. 1979). For a stable boundary layer, the Beljaars and Holtslag (1991) parameterization is applied. The Monin–Obukhov scale length is calculated as described by Liu et al.

At very low wind speeds, several of the assumptions related to a shear stress break down. One problem is that a 10-m reference height is well above this layer, which implies that convection induces a significant fraction of the mixing at the reference height. Another problem is that, contrary to an assumption typically used to derive a log-wind profile, wind perturbations can apply more surface stress than the mean wind. Schumann (1988) has developed a rather complex scheme for accounting for these effects. A simple approximation of Schumann’s work produces reasonable results (Godfrey and Beljaars 1991; Fairall et al. 1996a,b) for determining an effective wind speed (U′):

UUU_s²γw²_∗^1/2(7)

where

View Expanded

γ is a constant of order one, −b∗u∗ is the buoyancy flux, Z_bl is the height of the convective boundary layer, g is gravitational acceleration, and θ_υ is the virtual potential temperature. Schumann suggested that γ = 0.7;however, a value of γ = 1.25, suggested by Fairall et al. (1996a,b) for tropical oceans is used herein. The value of Z_bl is rarely known and when not reported it is assumed to be a constant equal to 1000 m (Bradley et al. 1991), which is typical for the Tropics. The magnitude of the increase in latent heat fluxes due to wind perturbations will be compared (section 8a) to the increase due to capillary waves.

b. Roughness length parameterizations

Both field and wind tunnel observations show that, for conditions of local wind–wave equilibrium, there is a low wind speed cutoff below which surface waves do not exist. This is usually observed in the range of 1–4 m s⁻¹ (Ursell 1956). The sea surface is aerodynamically smooth for wind speeds less than this cutoff and is aerodynamically rough for speeds greater than the cutoff.

1) Aerodynamically smooth regime

Aerodynamically smooth surfaces, which are amenable to laboratory experiments, have been found to have different roughness lengths for momentum, potential temperature, and water vapor (Nikuradse 1933; Kondo 1975; Brutsaert 1982):

View Expanded

Smith (1988, 1989) suggests that waves have no direct effect on the functional form of roughness lengths for temperature and moisture; thus, the above temperature and moisture roughness lengths apply to both aerodynamically smooth and rough surfaces.

2) Aerodynamically rough regime

For U₁₀ > 7 m s⁻¹, the momentum roughness length is well described with gravity waves as roughness elements. Roughness elements are the objects causing the turbulence; they have heights much larger than the roughness length. In the case of gravity waves, the roughness elements that generate most of the turbulence appear to be the short waves (Banner and Phillips 1974;Dobson et al. 1994). The additional physics considered in our roughness length parameterization (hereafter referred to as BVW) will be discussed in comparisons to fluxes and drag coefficients based on the physics considered in several dimensionally sound roughness length parameterizations, as well as comparisons to several sets of observations. Charnock’s (1955) relation is the most widely used roughness length parameterization:

z₀au²_∗g,(13)

where a is an empirically derived constant. In near-shore environments a ≈ 0.0185 (Wu 1980); however, in open ocean observations a ≈ 0.011 (Smith 1988; Dobson et al. 1994) is usually found to be smaller than in the near-shore observations. Recently, there have been several examinations of the dependence of Charnock’s constant on sea state (Kusaba and Masuda 1988; Geernaert et al. 1990; Toba et al. 1990; Perrie and Toulany 1990; Maat et al. 1991; Smith et al. 1992; Yelland et al. 1998), for which Smith et al. found

View Expanded

where w_a = c_p/u∗ is the wave age, and c_p is the phase speed of the dominant waves. These analyses are restricted to the subset of observations without significant swell (long gently sloping waves generated by distant events). The drag coefficients and stresses determined from this subset are approximately 20% larger than the open ocean results. A simple inverse wave age relationship (14) does not adequately describe the variation in C_D(U₁₀) observed when swell is present (Smith et al. 1992; Dobson et al. 1994). Dobson et al. suggest that a model with two horizontal dimensions is required to resolve the complexities introduced by swell.

Both of the above roughness length parameterizations (Smith 1988; Smith et al. 1992) are based upon field observations at U₁₀ > 6 m s⁻¹ and do not adequately model the processes that govern low wind speeds. Smith (1988) attempted to rectify this shortcoming by assuming that the roughness length for an aerodynamically smooth surface (Nikuradse 1933; Kondo 1975; Brutsaert 1982) could be added linearly to the gravity wave roughness length:

View Expanded

A summary of the physics considered in these parameterizations is provided in Table 1.

a. Capillary wave roughness length

For high winds, the momentum roughness length is well described with gravity waves as roughness elements: however, at low wind speeds or short fetches, capillary waves can dominate the generation of turbulence (Wu 1968). These waves are much shorter than the gravity waves; however, they are much steeper and more densely packed. Wu (1968, 1994) has suggested that capillary waves contribute to turbulent transfer and hence to the momentum roughness length. This reasoning is supported by Byshev and Kuznetsov’s (1969) observations that, at similar wind velocities, turbulent momentum fluxes were much greater when capillary waves were observed. Wu’s (1968) observations showed that roughness length associated with capillary waves decreases monotonically as wind speed increases. For similar wind speeds, the roughness length associated with capillary waves was larger than the roughness length for aerodynamically smooth surfaces. On the basis of dimensional analysis, Wu suggested that the momentum roughness length for capillary waves was related to friction velocity and the kinematic surface tension (σ/ρ_w, where σ is the surface tension, ρ_w is the density of water):

View Expanded

where b is an empirically derived constant that he determined to be approximately 0.18.

We propose that roughness length for combined gravity and capillary waves can be approximated as the root-mean-square sum of the roughness lengths. There have been many investigations into the relationship between the characteristics of roughness elements and the corresponding roughness length (e.g., Lettau 1969; Kondo and Yamazawa 1985). Kondo and Yamazawa’s findings support an additive rule; however, the exact nature of the weighting remains uncertain. The roughness length is likely to be either the sum of the components (z_0g and z_0c) or the rms sum of the components. There is little difference between the two sums except when z_0g ≈ z_0c (6 < U₁₀ < 8 m s⁻¹):

View Expanded

In either case, the total roughness length is usually approximately equal to the sum of the component roughness lengths. The shape of C_D(U₁₀) for the rms sum is qualitatively similar to an empirically derived parameterization (Large et al. 1994), which is a synthesis of Large and Pond’s (1981) 25 years of ocean weather ship observations, and a GCM-based estimate (Trenberth et al. 1989) of C_D(U₁₀) for low wind speeds. The BVW C_D(U₁₀) for wind–wave equilibrium is 10%–20% greater than that of Large et al. (1994) and other open ocean observations (Smith 1980; Anderson 1993; Dobson et al. 1994; Dupuis et al. 1997); it will be shown that these differences are consistent with the influence of sea state.

There is a further complication to the above solution. Roughness length [Eq. (2)] is highly dependent on the Newtonian frame of reference. For a solid surface the frame of reference is obviously earth relative (U_s = 0). For aerodynamically rough fluids, it is not clear that the mean surface current is the appropriate reference velocity. In practice, a frame of reference is chosen, and roughness length is determined for that frame of reference. The complication is that the frame of reference for capillary waves observed in a wave tank could differ from the frame of reference of capillary waves superimposed on a field of gravity waves in the open ocean.

We propose that the momentum roughness length for an aerodynamically rough surface is the weighted rms sum of the roughness lengths for capillary waves and for gravity waves:

View Expanded

where the β terms are weights based on several physical considerations. This parameterization does not contradict previous results for high winds: at high wind speeds, the capillary wave roughness is insignificant in comparison to gravity wave roughness. The weights will account for the above-mentioned frame of reference (section 3c) as well as modifications due to directions of wave propagation and surface currents that are nonparallel to the wind direction (section 3b).

b. Anisotropic roughness length

It is difficult to describe the parameterization for the weights without first describing the log-wind relation in vector form. A scalar form has been commonly used because the problem is greatly simplified when all the mean horizontal motion vectors were considered parallel. In such a case, a single roughness length (for the vertical plane that was parallel to the wind) could be used to model the mixing and the wind profile. When the mean horizontal flows are not all parallel, another roughness length is required to describe mixing due to the cross-flow.

We propose that the general form of the log-wind profile for a rough surface becomes

View Expanded

where ê_i are orthogonal horizontal basis vectors (e.g., i = 1 for flow parallel to the direction of wave propagation, and i = 2 for cross-flow), and z_0i are roughness length components for the corresponding basis. This approach requires reconsideration of the meaning of u∗ in wave age and the roughness lengths. There are few plausible interpretations, the number of which can be reduced by two limitations: the roughness length must be positive, and the magnitude of stress should be invariant to the choice of orthogonal basis vectors. The second constraint is difficult to apply, so the limiting cases of the stress associated with individual roughness components (z_0c and z_0g) are used to reinterpret the roughness length parameterizations. For z_0g there are two cases that meet this requirement:

View Expanded

In the first case z_0g is invariant to direction. Equation (19) shows that z₀ can be independent of bases only when (U − U_s)/u∗ is independent of wind speed, which is inconsistent with the vast majority of observations. Consequently, z₀ must be anisotropic.

The roughness length in (21) also has the advantage of being consistent with a zero-magnitude stress component when there is no wind shear in that component. That is, the roughness length approaches zero as the wind shear and stress both approach zero. Additional support for this form comes from its similarity to the stress vector (1): |u∗|u∗ · ê_i are the components of stress parallel to ê₁ and ê₂. The form of Eq. (21) is physically more reasonable; consequently, it is used in the BVW roughness length.

Similar reasoning is used to interpret the roughness length components for capillary waves. Again the same two forms meet the requirement of invariance to the choice of basis vectors; the form similar to (21) is chosen for similar reasons:

View Expanded

In the case of capillary waves, an infinite roughness length corresponds to nonexistent wind shear and a zero stress component.

c. Weights for roughness lengths

Traditionally, z_0g is considered determined for the frame of reference moving with the surface current. Consequently, there is no frame of reference adjustment to β_g. However, if swell is present, and if it is considered in the roughness for the direction it is propagating, then gravity wave roughness is either that of the swell (z₀₁) or that of gravity wave moving perpendicular to the swell (z₀₂). That is, airflow parallel to the “crests” of the swell will not “feel” the swell as gravity wave roughness elements. Neither the Smith et al. (1992) nor BVW parameterization of roughness length considers complications such as multiple sources of swell. Such consideration can be added to the BVW model; several prerequisite considerations are discussed in section 8. The wave field assumptions used herein are an improvement over the common assumptions in that they consider roughness elements for airflows that are parallel and perpendicular to swell.

Any stress, due to an airflow component perpendicular to the direction of swell propagation, is related to relatively short wavelength wind-induced waves. The waves in the cross-flow direction are assumed to be in local equilibrium with the wind. Winds tend to be approximately parallel to the direction of wave motion except at low wind speeds; therefore, the phase speed for the perpendicular component of the flow is generally small in comparison to the phase speed of the dominant waves. This assumption is most easily conceptualized when ê₁ is parallel to the direction of wave propagation for the dominant wave, and when ê₂ is perpendicular to ê₁ (the positive direction is consistent with a right-handed coordinate system, where the vertical axis is positive in the upward direction). This assumption results in one wave age for the dominant waves, and a smaller wave age for the cross-flow.

d. Capillary cutoff

The capillary cutoff is used to determine if wind-induced capillary waves are present and contributing to the surface roughness. We have assumed that surface waves do not exist if the modeled phase speed of the dominant waves is less than the minimum phase speed (c_pmin) for water waves:

View Expanded

The minimum phase speed for surface waves is

c_pmingσρ_w^1/4(27)

which is approximately equal to 23 cm s⁻¹. The wind speed corresponding to the capillary cutoff is dependent on the relation between z₀ and u∗. For the BVW model, with b = 0.18 (Wu 1968), the cutoff is U₁₀ = 2.0 ± 0.05 m s⁻¹; however, with b = 0.06 (the value determined in section 4) the cutoff becomes U₁₀ = 1.8 ± 0.05 m s⁻¹.

The parameterization of roughness length for capillary waves generated through wave–wave interaction may be different from that of capillary waves generated through wind–wave interaction. This modeled capillary cutoff applies only to wind-generated capillary waves. Radar observations of the sea surface indicate that, in the presence of swell, parasitic capillary waves can exist at wind speeds less than that of the cutoff for conditions of local equilibrium (D. E. Weissman 1996, personal communication).

e. Summary of roughness length parameterization

In summary, the BVW momentum roughness length parameterization is

View Expanded

where ê₁ is parallel to the direction of wave propagation, and ê₂ is perpendicular to ê₁. For point fluxes, the β′ values are equal to unity if the corresponding type of surface is contributing to the roughness length, and equal to zero if that type of surface is not contributing to the roughness length. The wave age for flow parallel to the direction of wave propagation can be associated with wave evolution (e.g., growing waves or swell), whereas the wave age for ê₂ is always assumed to be that for local equilibrium.

The momentum roughness length components that have been discussed herein are shown in Fig. 1 as functions of U₁₀ for conditions of neutral atmospheric stability. A wave age equal to 28 (corresponding to a fully developed sea), has been assumed for all curves except that labeled “BVW with local equilibrium wave age,” for which the wave age for local equilibrium was determined with a Phillips sea state parameterization, where the ratio of U(H_s) to the mean wind speed over the crest is set at 0.82 (based on the imposed requirement of c_p/u∗ ≈ 28 for U₁₀ > 7 m s⁻¹; Bourassa et al. 1998, submitted to J. Phys. Oceanogr.). Both BVW roughness lengths are calculated for parallel wind, surface currents, and wave propagation. The differences in roughness length between the BVW parameterization with w_a = 28 and w_a determined by the equilibrium sea state parameterization are shown to be small for high wind speeds, but the differences can be large for light winds, where the capillary cutoff changes from 1.8 m s⁻¹ for the equilibrium sea state parameterization to an unrealistically low value of 0.8 m s⁻¹ when w_a = 28 is assumed. The discontinuities in the roughness length for the BVW curves correspond to this capillary cutoff, and are due to the assumption of local wind-wave equilibrium: the discontinuity is analogous to the jump in stored energy as a function of temperature when H₂O changes phase from water to vapor. For U₁₀ between the capillary cutoff and approximately 5 m s⁻¹, the roughness length determined with the BVW model is much greater than that of the Smith (1988) and Smith et al. (1992) parameterizations. The differences arise because neither the Smith (1988) nor Smith et al. (1992) parameterizations are based on observations within this range of wind speeds (they are based on higher wind speed observations), and neither parameterization considers capillary waves as roughness elements. Furthermore, the Smith (1988) parameterization is based on open ocean observations: they are likely to be confounded by swell and complex sea states, which reduces the roughness length approximately a factor of three.

a. Reevaluation of wave tank observations

Our technique for determining u∗ and z₀ differs from traditional wave tank applications of the profile technique in that we consider the importance of surface current and displacement height (d) in deriving u∗ and z₀. The displacement height is a vertical offset of the logarithmic profile (Covey 1983; Stull 1988): z in Eq. (2) is replaced with z − d. Both U_s and d are negligible when a known relation between u∗ and z₀ is used to determine U(z ≫ d), u∗, or z₀ from theory; however, they are not negligible when the profile method is used to determine u∗ and z₀ (Covey 1983; Stull 1988). Covey’s (1983) technique is used to determine u∗, z₀, and d, where U is replaced by U − U_s. Wu’s (1968) observed values of U and z are obtained from his Fig. 6, and the surface currents from his Fig. 3. Several of Wu’s near-surface observations were in a region that is not characterized by a log-profile (Large et al. 1995); these observations are not used herein to determine log-profile parameters.

The importance of displacement height in determining log-profile values of u∗ and z₀ is demonstrated with observations from the wind–wave-current facility at the University of Delaware’s Air–Sea Interaction Laboratory. Profiles of mean wind speed were measured and for the same conditions that friction velocity was determined through the eddy correlation method (Tseng 1987). Measurements from one instrument were used in the eddy correlation and profile methods: fluctuating wind velocities (and mean winds) were determined with a hot-film anemometer. The friction velocities determined from the eddy correlation method and both profile methods are listed in Table 2. The errors in profile methods’ friction velocities (assuming the eddy correlation values are correct) are given by ε_d=0 (average of +43% when d is assumed to equal zero) and ε_d (average of 3% when displacement height is considered). The rms percentage differences in friction velocity are 45% and 7%, respectively. The differences in z₀ between the two profile methods are large: in two of the three examples there are an order of magnitude difference in roughness length. Proper consideration of displacement height greatly reduces errors in log-profile values of friction velocity and roughness length.

The frame of reference modification may be responsible for a portion of the large variances from mean observed fluxes, as well as an additional explanation for how sea state contributes to differences in z₀ between field and wave tank observations. The frame of reference modification reduced the values of z₀ calculated from the wave tank velocity profiles. It is presumed that the relationship between z₀ and u∗ in (16) applies to roughness lengths that have not been modified to the frame of reference of the surface current. These roughness lengths can be calculated by inverting (24). The values of T_p and H_s are taken from Wu’s Fig. 4. The resulting roughness lengths are shown as squares in Fig. 2; uncertainties in z₀ are not shown because uncertainties in T_p and H_s are unknown. The y intercept is found for the best-fit line with a slope of −2 (the solid line in Fig. 2), and b is found to equal 0.06 (which is used throughout this study). This value is approximately one-third of Wu’s original value. In the next section, it will be shown that this value of b is similar to the value determined from field observations.

b. Capillary wave roughness in SCOPE observations

Observations from the San Clemente Ocean Probing Experiment (SCOPE; Fairall et al. 1996a) were used to test the model for extremely old age seas. These observations were taken from the Scripps Institute Floating Instrument Platform: R/P FLIP. The advantage of the SCOPE dataset over the Coupled Ocean–Atmosphere Response Experiment (COARE) datasets is that SCOPE observations include the dominant wave phase speed. Consequently, there is no need for any assumptions about wave age or local wind–wave equilibrium. The SCOPE phase speeds and wave ages (Fig. 3) are extremely large compared to local equilibrium values; therefore, the influence of capillary waves is emphasized. The greatest difficulty with SCOPE data is that the angle between the direction of wave propagation and the wind was not recorded in the available dataset, other than the qualitative statement that the angle was often large. The SCOPE observations were kindly provided by Chris Fairall.

Observations from SCOPE were used to determine the value of b required for the model to match the observations. Observed wind speeds, phase speeds, temperatures, and humidities were used as input; the values of b were chosen to match the measured friction velocities. Observations of the direction of wave propagation were unavailable; therefore, it was assumed that the waves propagated parallel to the winds. For 5 of the 131 cases it was found that b = 0, indicating that capillary waves were not present. For most of the cases, the values of b were between 0.02 and 0.3. The geometric average of the values is 0.05 with confidence limits of 0.04 and 0.06 (a factor of 1.2). This value of b is one standard deviation (of the field value) lower than the value determined from the laboratory observations. More detailed field observations, including vector stresses and the direction of wave propagation, will be required before field observations can be used to make a more accurate evaluation of b. Nevertheless, the near-consistency between laboratory and field observations strongly supports the concepts of capillary wave roughness and frames of reference used to develop the BVW model.

a. Stress

The BVW stress is similar to that of the gravity wave-based parameterizations (Charnock 1955; Smith 1988; Smith et al. 1992) for U₁₀ > 7 m s⁻¹ (Fig. 5); however, for U₁₀ < 5 m s⁻¹, capillary waves can make important contributions to the stress. The mean winds in the Tropics are typically between 1 and 5 m s⁻¹. Therefore, tropical stresses and wind-induced currents are often underestimated. The local influence of this increase in stress is probably small; however, when the increase is integrated over that area of the Tropics the cumulative effect is likely to have an important impact on the tropical general circulation.

Wave age has little effect on the stress for U₁₀ < 7 m s⁻¹ except to alter the value of the capillary cutoff for wind-induced capillary waves. Wave ages as large as 120 have been observed in the Tropics (Deleonibus 1972) and off of the California coast (Fairall et al. 1996a); the corresponding cutoff is 0.4 m s⁻¹, which is much less than U₁₀ = 1.8 m s⁻¹ for the BVW model with equilibrium wave age. Some caution must be used when applying Eq. (14) to wave ages greater than 30 because the gravity wave roughness length parameterization has not been verified for such conditions. Nevertheless, due to the influence of sea state on the capillary cutoff, it is clear that the model indicates that sea state (directional information as well as wave age) is an important parameter at low wind speeds as well as at high wind speeds.

b. Drag coefficients

The sensitivity of the drag coefficient to wave age is relatively low for lower wind speeds and older seas. In section 5, and in the observations of Donelan et al. (1997), it was shown that for these conditions there is considerable variation in the drag coefficient due to the angle between the mean wind and the mean direction of wave propagation. These results are consistent with the observations that open ocean drag coefficients (old seas with swell from multiple sources) have little apparent dependence on wave age (e.g., Yelland et al. 1998). The consideration of directional aspects of sea state is required to model most open ocean conditions.

c. Sensible and latent heat fluxes

The neutral terms of the modeled sensible and latent heat fluxes (Figs. 6a,b) behave similarly to the corresponding neutral stress. The figures correspond to values of T₁₀ − T_s = 1.5 K and 90% relative humidity, which are typical of tropical differences of temperature and moisture with height. Changes of less than 25 W m⁻² in the sum of the latent and sensible heat fluxes have been shown to have a qualitative impact of the shape and strength of the European Centre for Medium-Range Weather Forecasts (ECMWF) GCM’s tropical general circulation (Miller et al. 1992; Carrington and Anderson 1993). For low wind speed regions, such as the Tropics, the results shown in Fig. 6 suggest that increases in fluxes due to capillary waves can be of similar magnitude. The version of the ECMWF model discussed in the above studies had a Charnock roughness length parameterization, which is similar to the curve (Fig. 6) for “gravity waves with c_p/u∗ = 28.” The curves in Fig. 6, which include fluxes for an aerodynamically smooth surface and for gravity waves, provide an indication of the impact of capillary waves. The impact of capillary waves will be larger for nonneutral fluxes (section 7b), and may be sufficiently large to have a substantial effect on the tropical general circulation.

a. Accuracy of various models

All of the roughness length parameterizations discussed herein accurately model the general trend of increasing stress with increasing wind speed. In the case of the SCOPE data the linear correlation coefficients between modeled and observed stresses (for 0 < U₁₀ < 7 m s⁻¹) are 0.86 ± 0.01; indicating similar accuracy in the models’ predictions of the trend in the change of stress with wind speed. A comparison of modeled stresses and observed stresses (Fig. 7) indicates that the modeled stresses are usually within 0.01 N m⁻² of the observed stresses. The vast majority of larger differences show the modeled stress underestimating the observations. Neither the SCOPE nor the R/V Moana Wave datasets include the direction of the wind relative to the direction of propagation of the dominant waves. This absence, coupled with the observations of Donelan et al. (1997) and the modeling in section 5, suggests that the larger underestimations of the model could be due to winds, waves, and currents that do not have parallel mean motion vectors.

The drag coefficient has relatively little dependence on wind speed (Fig. 8) for 2 < U₁₀ < 6 m s⁻¹: the wind speed dependent trend is tiny in comparison to that of stress. However, the use of C_D rather than stress does not remove the influence of sea state (Donelan et al. 1997): wave age and differences in direction of wind and wave propagation result in additional variability in C_D. The correlation between drag coefficients is a much better indication of a model’s capability in predicting departures from the trend. As indicated earlier, the uncertainty in the direction of wave propagation is likely to be responsible for a large fraction of the variation in the drag coefficients. Nevertheless, it will be shown that the accuracy of the modeled drag coefficients improves for the models with more detailed physical considerations, despite the assumption of parallel wind and wave propagation vectors. The BVW model without capillary waves (β^′_c = 0, and U//U_s//e₁; similar to Smith et al. 1992) has a poor correlation (r = −0.20) to SCOPE observations; the negative value indicates that increases (decreases) in modeled C_D are associated with decreases (increases) in the observed C_D. The Smith (1988) parameterization, which considers the roughness length to be the sum of the roughness lengths for gravity waves and an aerodynamically smooth surface, has a correlation coefficient of 0.15. Despite this low correlation, the Smith (1988) parameterization is a great improvement (Δr = 0.35) over the base correlation of −0.20. The correlation with the BVW model is 0.31, which is a great improvement (Δr = 0.51) over the base value and doubles the correlation of the Smith (1988) parameterization. Clearly, more detailed observations will be required to further test and improve the modeled physics, particularly the anisotropic roughness length.

b. Changes in magnitudes of modeled fluxes for tropical nonneutral conditions

The changes in fluxes due to capillary waves are examined using three datasets from tropical field observations. One set is the SCOPE data, the second is Chris Fairall’s Tropical Ocean Global Atmosphere (TOGA) COARE observations from the R/V Moana Wave (Fairall et al. 1996b), and the third set combines Frank Bradley’s observations from the R/V Franklin (Bradley et al. 1991; Bradley et al. 1993). These datasets were chosen because they emphasize low wind speeds. Neither wave age nor phase speed observations were available with the observations from the Moana Wave and the Franklin; local equilibrium values of wave age, from the equilibrium sea state parameterization, were used to determine the fluxes. It is likely that swell was present for most of the observations. Therefore, the local-equilibrium wave ages underestimate the true wave ages, which results in an underestimation of the importance of capillary waves. The observed winds, temperatures, humidities, and phase speeds (for SCOPE data) are used to model stresses, drag coefficients, sensible heat fluxes, and latent heat fluxes. Mean fluxes modeled with capillary waves are compared to observations and mean fluxes without capillary waves (Table 3). In all cases, consideration of capillary waves results in increases in the modeled drag coefficient and mean fluxes of momentum, heat, and moisture. Furthermore, the mean modeled result considering capillary waves are always a better match to the observations. The observations from the R/V Franklin do not include stresses because of difficulties removing ship motion from the winds used in the eddy correlation method (F. Bradley 1994, personal communication). The large differences between modeled heat fluxes and those observed from the R/V Franklin tend to be associated with low wind speeds. Improvement of the model through the inclusion of additional physics, such as differences between the skin temperature and observed temperature due to evaporative cooling and solar heating, increases the accuracy of modeled fluxes (Clayson et al. 1996). The physics discussed herein leads to improvements in the modeled stress, and the physics discussed by Clayson et al. improves the modeled fluxes in heat and moisture. Combining these improvements would lead to improved modeling of atmospheric stability, and consequently improvements in the modeled fluxes.

The capillary wave-related increases in modeled heat fluxes are sufficiently large that they would cause a qualitative change in the ECMWF’s tropical general circulation. It is expected that these increases in fluxes are sufficiently large that there will also be qualitative changes in the results of GCMs with better representations of the Tropics.

a. Importance of capillary waves vs convection

Gustiness has been suggested by Fairall (Fairall et al. 1996b) to cause a large increase (∼5 W m⁻²) in heat fluxes for low wind speed fluxes (U₁₀ < 3 m s⁻¹). A similar result is found when Fairall’s gustiness parameterization [Eq. (7)] is applied in the calculation of latent heat fluxes for the Franklin observations (Fig. 9). Only a small fraction of observed wind speeds are sufficiently small that there are large increases due to gustiness. The probability distributions of wind speeds for the three datasets are shown in Fig. 10. The average changes in latent heat fluxes (Table 4) related to gustiness, for all wind speed observations less than 7 m s⁻¹, are between 0.9 and 2 W m⁻².

The increase in latent heat flux due to capillary waves is also a function of wind speed (Fig. 9). For wind speeds less than the capillary cutoff there are no changes in the fluxes. For wind speed near 2 m s⁻¹ the increases are near 10 W m⁻², and the magnitudes of the changes are reduced as the wind speeds increase. The average change in latent heat flux (Table 4), for all wind speed observations less than 7 m s⁻¹, is between 4 and 9 W m⁻². The mean change in latent heat flux due to capillary waves is approximately three to six times larger than the increase due to convective overturning. Furthermore, the interaction between the two processes lowers the capillary cutoff and causes the mean increase in latent heat fluxes to be greater than the sum to the independent changes. Clearly, the increased heat flux due to capillary waves is more important than that due to convective processes.

b. Bulk flux applications of the BVW model

Bull fluxes represent transfers over large areas that may be nonhomogeneous, whereas point fluxes represent transfers over small homogeneous surfaces. For most low wind speeds (U₁₀ < 5 m s⁻¹), large water surfaces cannot be described as being uniformly aerodynamically smooth or rough (β′ = 0 or β′ = 1). Such surfaces are patchy due in part to small-scale convection, and in part due to insufficient stress to maintain a rough surface. It is these features that smooth the discontinuity near the capillary cutoff in the point flux model. In such cases, the binary logic applied to the point flux model is an inadequate description of the surface. It is likely that fuzzy logic (McNeill and Frieburger 1993) can be applied to improve the description. With a fuzzy logic, the β′ values correspond to the fraction of the surface covered with the corresponding type of roughness elements. There are two constraints:

View Expanded

The first constraint indicates that the total surface area is equal to the area that is aerodynamically smooth plus the area with gravity waves. The second constraint results from the condition that in the open ocean capillary waves rarely exist in the absence of gravity waves (albeit very small gravity waves for low wind speeds). At this time, there are insufficient low wind speed flux observations to determine how these fuzzy weighting parameters vary as functions of wind speed and atmospheric stability.

Inhomogeneities on the ocean surface could serve to enhance convective processes. Smaller roughness lengths in regions without capillary waves will cause the atmospheric stability to be further from neutral: in unstable boundary layers convective processes will be enhanced. Furthermore, the greater stress over the rougher surface will reduce the influence of warm surface layers. Consequently, updrafts are expected over smooth patches, and downdrafts over the areas with capillary waves. The parameterization of the influence of surface inhomogeneities on convection will likely require knowledge of the fraction of the surface covered by each type of roughness element. These considerations are likely to be important in modeling the interactions between surface fluxes, boundary layer convection, and boundary layer stability.

c. Gravity wave and frame of reference

It seems likely that a frame of reference correction should also be applied to gravity waves; however, there is little observational evidence that can be used to determine such a relationship. There have been numerous suggestions (e.g., Munk 1955; Phillips 1977; Al-Zamadi and Hui 1984; Donelan 1990) that the “natural” frame of reference for gravity waves is that which moves with the phase speed of the gravity waves that interact with the airflow. Many of these suggestions are related to the concept of form drag due to the gravity waves, and the effectiveness of wave evolution models (e.g., Miles 1957; Komen et al. 1994) based on form drag.