1. Introduction
The development of tropical cyclones is governed by heat exchange between the air and the sea. Emanuel (1986) proposed a simple energy balance model for tropical cyclones in which the inner region of the tropical cyclone plays a role similar to that of a Carnot heat engine, extracting latent and sensible heat from the ocean and transferring that heat outward to overcome frictional dissipation in the boundary layer at large radii. Thus, extracting heat is seen as essential to maintain the intensity of a tropical cyclone.
In meteorological models, heat transfer from the sea surface is generally estimated using bulk algorithms, which have been investigated extensively using large datasets of observations. Fairall et al. (1996) proposed a method for flux estimation using bulk variables obtained by the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) for weak to moderate winds (0.5 ≤ u10 ≤ 10 m s−1). It was shown that the exchange coefficient for latent heat flux corrected for neutral stratification is an approximately constant value of 1.11 × 10−3, decreasing slightly with wind speed. DeCosmo et al. (1996) developed a bulk algorithm using the dataset of the Humidity Exchange over the Sea (HEXOS) program. The HEXOS data were collected near the Dutch coast at wind speeds of up to 18 m s−1 and are thus affected by sea spray due to shallow wave breaking. The exchange coefficients for sensible heat (CT) and latent heat (Cq) under neutral stratification determined in the study of DeCosmo et al. (1996) are CT = (1.12 ± 0.24) × 10−3 and Cq = (1.14 ± 0.35) × 10−3. In short, based on the large datasets collected under both of these programs (TOGA COARE and HEXOS), CT and Cq appear to take approximately constant values of 1.1 × 10−3 for the wind speed range of 0.5 ≤ u10 ≤ 18 m s−1, although the data exhibit considerable scatter. However, as there have been relatively few observations of latent heat flux over the ocean under stronger wind conditions, it remains unknown whether it is valid to extrapolate these bulk algorithms to stronger winds.
Sea spray has been considered a likely factor that may resolve this discrepancy (Andreas and Emanuel 2001). Under strong wind, sea spray is generated by wave breaking, and the droplets of sea spray drifting in the atmosphere continue to transfer sensible heat to the ambient atmosphere until the droplet temperature reaches the wet-bulb temperature, at which time the droplet evaporates and generates latent heat by absorbing sensible heat from the atmosphere. This heat exchange between sea spray droplets and the atmosphere mainly occurs within a droplet evaporation layer (DEL).
The possible effect of sea spray on tropical cyclone intensity has been investigated by a number of groups. For example, using a coupled ocean–atmosphere model of a tropical cyclone, Kepert et al. (1999) concluded that although sea spray had little effect on the total enthalpy flux, it could increase storm intensity by cooling and moistening the boundary layer. Wang et al. (2001) came to a similar conclusion using the parameterization of Fairall et al. (1994), but also showed that sea spray contributes directly to the total enthalpy flux if the parameterization of Andreas and DeCosmo (1999) is assumed. These early studies thus suggest that sea spray may increase the total enthalpy flux from the ocean and also cool and moisten the boundary layer.
Sensible and latent heat are transferred by turbulence in the airflow. As the airflow immediately above the ocean is disturbed by ocean waves, forming a wave boundary layer (WBL), the turbulent transfer in the WBL must be affected by ocean waves (Makin and Mastenbroek 1996; Makin 1999). A number of WBL models have been proposed to predict the drag coefficient (e.g., Janssen 1989; Makin and Kudryavtsev 1999), and most consider momentum conservation in the WBL. These studies indicate that turbulent momentum flux in the WBL is reduced by wave-induced momentum flux related to periodic airflow perturbations generated by ocean waves. Recently, Hara and Belcher (2004) proposed a WBL model based on the conservation of both momentum and kinetic energy over ocean waves, and Moon et al. (2004a) combined the model of Hara and Belcher with a wavenumber spectrum calculated by the WAVEWATCH III wave model (Tolman 2002) to investigate the momentum exchange process for young and mature seas. Application of that model under conditions of hurricane wind forcing (Moon et al. 2004b, c) revealed the same plateauing of the drag coefficient at u10 ≥ 35 m s−1 as indicated by the observations of Powell et al. (2003). This result suggests that it may soon be possible to solve many of the unanswered questions related to transfer mechanisms in the boundary layer over the ocean.
Makin and Mastenbroek (1996) and Makin (1999) evaluated heat transfer in the WBL using their WBL model.
In the present study, a boundary layer model for evaluating sensible and latent heat fluxes including sea spray–mediated heat fluxes over a mature sea is developed. The WBL model of Hara and Belcher (2004) is adopted as a basis, the wavenumber spectrum for mature sea presented by Elfouhaily et al. (1997) is used for the low-wavenumber regime, and that proposed by Hara and Belcher (2002) is used for the high-wavenumber regime. Using the information on the airflow structure obtained from the WBL model, turbulent heat transfer in the WBL is calculated using the approach of Makin and Mastenbroek (1996). The sea spray–mediated heat flux in the DEL is incorporated by introducing the parameterization of Fairall et al. (1994). The results of the developed model are used to evaluate the increase in total enthalpy under strong wind conditions.
Reduction of momentum flux in the airflow due to sea spray is not considered. The momentum of airflow is extracted by the spray droplets drifting in the atmosphere due to the acceleration of the droplets in the direction of the wind. Wu (1973) found that such momentum exchange is negligible under moderate winds, and Andreas and Emanuel (2001) showed that the contribution of this momentum exchange is three orders of magnitude less than the interfacial stress for moderate winds, although the interfacial and spray stresses become comparable for strong winds (u* ≥ 3 m s−1). The friction velocity (u* = 3 m s−1) corresponds to u10 = 60 m s−1 when CD = 0.0025. In the present study, heat transfer is investigated at wind speeds of up to u10 = 28 m s−1, at which the top of DEL reaches z = 10 m (the reference height of bulk variables). Thus, the sea spray–mediated momentum flux is considered to be negligible in the present study.
This paper is organized as follows: in section 2, the model and solving procedures are introduced. In section 3, the sensible and latent heat transfer over a mature sea without sea spray effects is evaluated, and the influence of sea spray is then discussed. The growth of the exchange coefficient of total enthalpy under tropical storm conditions is then evaluated. The main results of the present study and future work are finally presented in section 4.
2. Model description
a. Wave boundary layer model
For fast-moving waves [k > km = 0.072g/(ul*)2], the growth rate βg(k, θ) is negligible. Thus, as Eq. (9) shows, the effect of wind waves does not extend above zm = δ/km, where the height zm is defined as the top of the WBL, above which τt is constant. For detailed procedures for solving the WBL model, refer to Hara and Belcher (2004) or Moon et al. (2004a).
b. Scalar transfer in WBL
It should be noted that although the wave-induced scalar flux HwΘ in Eq. (13) is ignored, the effects of wind waves are indirectly reflected in the scalar transfer through the use of the local turbulent stress τt in Eq. (14), which is derived from the WBL model.
c. Sea spray–mediated heat transfer
The effect of sea spray on heat transfer is evaluated by applying the sea spray–mediated heat flux parameterization proposed by Fairall et al. (1994). This parameterization allows the effects of the evaporation of oceanic spray droplets on the sensible and latent heat fluxes to be determined using bulk meteorological quantities such as wind velocity, temperature, and relative humidity (RH) at a height of 10 m and at sea surface temperature.
d. Effects of stable/unstable stratification
In the present study, these stratification effects are accounted for by performing iterative calculations. First, the WBL model and sea spray model are solved under a neutral condition (i.e., ϕu = ϕT = ϕq = 1), and we obtain the turbulent momentum, sensible and latent heat fluxes, and the value of fu(z), fT(z), and fq(z). Then, L and the stratification corrections (i.e., ϕu, ϕT, and ϕq) are calculated by using the turbulent fluxes. We substitute the values of L and the stratification corrections into Eqs. (33)–(35), and we solve the WBL model and sea spray model again. The iteration continues until the fluxes reach appropriate convergence limits.
e. Solving procedure
Using bulk meteorological quantities and the wavenumber spectrum ψ(k, θ) for a mature wave state, the sensible and latent heat transfer under the effects of surface waves and sea spray is calculated. The bulk meteorological quantities include the wind velocity, temperature, and relative humidity at a height of 10 m and at the temperature and atmospheric pressure at the sea surface.
A wavenumber spectrum near the peak frequency proposed by Elfouhaily et al. (1997) is adopted here. This wavenumber spectrum is based on the wavenumber spectra proposed by Pierson and Moskowitz (1964) and the Joint North Sea Wave Project (JONSWAP) (Hasselmann et al. 1973) for a low-wavenumber regime, and the wavenumber spectra of Phillips (1985) and Kitaigorodskii (1973) for a high-wavenumber regime. The wavenumber spectrum ψ(k, θ) of Elfouhaily et al. have a relatively simple closed form and are consistent with past observational results. Further, ψ(k, θ) can be considered to represent a saturation value based on the wave age (cP/u10) or fetch, where cP is the wave velocity of the peak frequency. In the present study, the mature sea state is considered, for which cP/u10 is set at 1.19 (Elfouhaily et al. 1997).
In the high-wavenumber regime, which is considered to be an equilibrium range (Phillips 1977), the wavenumber spectrum of Hara and Belcher (2002) is employed. An equilibrium wave state is maintained by the balance of energy among the three sources of wave action: input from wind, dissipation due to wave breaking, and divergence of wave action flux by nonlinear wave interactions (Phillips 1985). The analytical model for ψ(k, θ) presented by Hara and Belcher (2002) is developed by introducing energy input sources obtained from the WBL model of Hara and Belcher (2004). Thus, ψ(k, θ) in the equilibrium range of Hara and Belcher (2002) is regarded as appropriate for solving the WBL model of Hara and Belcher (2004).
Figure 1 shows the distribution of the omnidirectional spectrum S(k) for a mature sea at several wind velocities. This omnidirectional spectrum includes the peak frequency regime of Elfouhaily et al. (1997) and the equilibrium regime of Hara and Belcher (2002). The figure shows that these two regimes are smoothly connected.
The WBL model of Hara and Belcher (2004) is then solved following Moon et al. (2004a). The upper bound of wavenumber (k1) is set at 400 rad m−1, above which Hara and Belcher (2004) showed that the calculated roughness length of momentum does not depend on the value of k1. The boundary condition at the sea surface is defined by u(z = δ/k1) = 0 at z = 0.1υa/ul*(k1). The value of δ is set at 0.008, such that the drag coefficient for the mature sea state is consistent with the parameterization of Zeng et al. (1998) for weak to moderate winds (4 ≤ u10 ≤ 18 ms−1), as shown in Fig. 2. Solving the WBL model, we can obtain the total momentum flux τtot, or friction velocity u*, and the vertical distributions of local turbulent stress τt(z) and wind velocity u(z).
3. Results and discussion
a. Heat transfer over a mature sea without sea spray
b. Heat transfer over a mature sea with sea spray
In Table 1, the present results are compared with those determined using the microphysics of Andreas (1992). The table lists the sea spray–mediated sensible and latent heat fluxes under the six conditions tabulated in Table 3 of Andreas (1992). The wind speed is fixed at u10 = 20 m s−1, while the temperatures (T10, Tw) and RH vary among cases. As can be seen in the table, the present results are generally in agreement with those of Andreas (1992), although the sensible heat flux at the lowest low in Table 1 is underestimated in the present model. This may indicate that the value of W(u10)Sυ in Eq. (19) need to be better regulated to agree with the results of Andreas (1992).
At the condition of the lowest low in Table 1, the interfacial turbulent sensible heat flux was 63.3 W m−2, which is much smaller than QL (=183.2 W m−2), and thus the total sensible heat flux is downward, even though T10 is lower than Tw. Similarly, in Andreas (1992), the corresponding interfacial turbulent sensible heat flux is 46 W m−2 using a bulk algorithm (Large and Pond 1981; Andreas and Murphy 1986), and it is also much smaller than QL (=170 W m−2). In this case, the sea spray from the warmer ocean makes the cool atmosphere cooler, and this cooling and moistening effect would influence the tropical cyclone intensity (e.g., Kepert et al. 1999; Lighthill et al. 1994).
Figure 3 shows the effects of sea spray and surface waves on the vertical profiles of temperature and specific humidity under moderate (u10 = 7 m s−1) and strong (u10 = 25 m s−1) winds for a stably stratified condition with Tw = 300 K, T10 = 302 K, P0 = 1000 hPa, and RH set at 80%. The calculated heat fluxes for the moderate wind case are QS = −0.08 W m−2, QL = 0.35 W m−2, HtotS = −12.28 W m−2, and HtotL = 27.55 W m−2, while those for the strong wind case are QS = −14.4 W m−2, QL = 855.4 W m−2, HtotS = −921.2 W m−2, and HtotL = 1006.8 W m−2. For the moderate wind conditions, the magnitudes of sea spray–mediated heat fluxes are much smaller than those of the total heat fluxes, and the sea spray would have little effect on the heat transfer mechanism. Under strong wind conditions, on the other hand, the sea spray–mediated latent heat flux is comparable in magnitude to the total heat fluxes; thus, the sea spray effect would be strong.
For moderate winds, the contributions of sea spray–mediated fluxes are much smaller than those of turbulent fluxes at the interface, and thus almost no changes are observed in the vertical gradients of T and q at the top of the DEL (i.e., at z = as). However, because of the effect of stable stratification, the vertical gradients are misaligned with a logarithmic low near z ≅ 10 m. For strong winds, on the other hand, the vertical gradient of T and q changes quickly at z = as. Strong winds generate substantial volumes of sea spray, and the evaporation of droplets cool the atmosphere in the DEL. The temperature below the DEL thus becomes lower than that without sea spray effects.
The cooling effect also increases the interfacial sensible heat fluxes (i.e., HtSz<as), because the turbulent sensible heat flux depends on the vertical gradient of temperature. This is an indirect contribution of sea spray evaporation to the total heat fluxes and would facilitate the development of tropical cyclones.
For weak to moderate winds (4 ≤ u10 ≤ 12 m s−1), the buoyancy effect on turbulence causes a difference in CG among the two cases. In the stably stratified airflow, the turbulence becomes weak, causing CG to become smaller than that in the neutral case, while under unstably stratified conditions, turbulence increases, resulting in higher CG. In the case of strong winds (u10 > 20 m s−1), the sea spray–mediated heat fluxes are responsible for the difference between the two cases. Table 2 lists the interfacial and sea spray–mediated heat fluxes for both cases under the strongest wind condition (u10 = 28 m s−1), as well as the interfacial heat fluxes without sea spray effects. Under the stably stratified condition, the total enthalpy flux becomes positive, even though the interfacial and sea spray–mediated sensible heat fluxes are negative. Although the sea spray indirectly increases the interfacial heat fluxes by 7 W m−2 because of cooling in the DEL, the sea spray–mediated heat flux directly reduces the total enthalpy flux by 36 W m−2, resulting in an overall decrease in total enthalpy flux. Under the unstably stratified condition, on the other hand, the large sea spray–mediated latent heat flux cools the DEL, and the interfacial heat fluxes are indirectly increased by 44 W m−2. In conjunction with the direct increase in heat flux of 49 W m−2, the combined effects increase the total enthalpy flux by approximately 10%. The numerical results of Wang et al. (2001) indicate that the parameterization of Fairall et al. (1994) results in a smaller direct contribution of sea spray to the total enthalpy flux than its indirect contribution in the tropical cyclone model. This is considered to be caused by the omission of the cooling effect below the reference height of bulk variables (i.e., z = 10 m), resulting in an underestimation of sea spray–mediated heat flux. Thus, the present model would give a larger total contribution by sea spray evaporation in the tropical cyclone model.
In the unstably stratified case, the increasing rate of CG with wind speed under strong wind (u10 ≥ 20 m s−1) is consistent with that for CD. In other words, the dependence of CG on wind speed is comparable to that for CD under strong wind conditions. Figure 6 shows the ratio of CG to CD under the unstably stratified condition. The parameterization of Zeng et al. (1998) for a weak to moderate wind regime (4 ≤ u10 ≤ 18 m s−1) and the extrapolation to a strong wind regime (18 ≤ u10 ≤ 28 m s−1) are also plotted. While CG/CD given by the parameterization of Zeng et al. decreases with increasing wind speed, CG/CD in the present study reaches a plateau at 0.7 for strong winds (u10 ≥ 20 m s−1). This result supports the theory of Emanuel (1995a); that is, the ratio CG/CD should be greater than 0.75 to maintain the intensity of a tropical cyclone.
The slightly lower value CG/CD = 0.7 (<0.75) in the present study may be due to an underestimation of the sea spray–mediated sensible heat flux QS. The lowest low of Table 1, which is a similar condition to Fig. 6, shows that the sea spray–mediated sensible heat flux QS in the present study is approximately half that reported by Andreas (1992). If the value of W(u10)Sυ in Eq. (19) is multiplied by a constant value of 2.2 to agree with our QS of the lowest low of Table 1 with that of Andreas (1992), the ratio increases with wind speed, reaching 0.75 at u10 = 27 m s−1 (see Fig. 6). Note that although this regulation makes the prediction of QS under the condition of Fig. 6 better, it is not appropriate for all the conditions. Thus, we have not used the regulated QS in the previous discussions.
Figure 7 shows the variation in CG/CD under various conditions. These are calculated using the value of QS without the abovementioned regulation. The temperature difference Tw − T10 in these examples varies from 0.5 to 3.0 K, and RH varies from 70% to 90%. Under the strongest wind conditions (u10 = 28 m s−1), CG/CD increases with increasing temperature difference or decreasing relative humidity. It is interesting to note that in every case, the ratio levels off at approximately 0.7 with winds exceeding 20 m s−1.
4. Conclusions
A boundary layer model over a mature sea was developed for the evaluation of sensible and latent heat fluxes at wind speeds of up to 28 m s−1. To estimate the turbulent stress, we solved the wave boundary layer model of Hara and Belcher (2004). Sea spray effects were incorporated using the parameterization of Fairall et al. (1994). It was confirmed that for weak to moderate winds (4 ≤ u10 ≤ 18 m s−1), the drag coefficient and the heat exchange coefficient without sea spray effects are consistent with a bulk algorithm developed using TOGA COARE and HEXOS observations.
The main result of the present study is that while the ratio of the exchange coefficient of total enthalpy to the drag coefficient (CG/CD) decreases with increasing wind speed below 20 m s−1, the ratio plateaus at approximately 0.7 for stronger winds (20 ≤ u10 ≤ 28 m s−1) under tropical cyclone conditions (wet air blowing over a warm mature sea). It has not been possible to obtain this result using traditional bulk algorithms, which afford a monotonic decrease in CG/CD with increasing wind speed. Our result supports the argument of Emanuel (1995a), who showed that the intensity of a tropical cyclone can only be maintained if this ratio remains greater than 0.75 in the strong wind region of intense storms.
The wave boundary layer model adopted here assumes that the perturbations due to interfacial waves reach the inner region depth, which is expressed simply as zi = δ/k. This expression is applicable for low or high wave ages, but it is not appropriate for intermediate wave ages, at which the inner region depth is thicker and dependent on wave age. The characteristics of flow perturbations differ among these cases, as shown in the previous study of Kihara et al. (2007). Although the wave-age range of intermediate wave age is not large and its effects on the momentum and heat transfer mechanism would be weak, we should confirm the effects of intermediate wave age.
Under tropical cyclones, the local wave state is determined not only by the local wind speed but also by the past wave state; that is, it is not a mature sea state, although the proposed boundary layer model in the present study is developed using wave spectrum under mature sea states. In applying our proposed model to the tropical cyclone conditions, the local wave spectrum should be used instead of the wave spectrum under a mature sea. To incorporate the local wave spectrum, it is necessary to couple our model with a wave model, such as WAVEWATCH III or WAM cycle 4, as carried out in Moon et al. (2004a, b,c).
Acknowledgments
I thank two anonymous referees for their useful comments and suggestions.
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Distributions of the omnidirectional spectrum S(k) for a mature sea for u10 = 5, 10, 15, 20, and 25 m s−1.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3828.1
Wind speed dependence of exchange coefficients CD, CT, and Cq for mature sea states obtained in the present study. Those reported by Zeng et al. (1998) are also shown. The solid line denotes CT and Cq in the present study; the dashed line is CD in the present study; the chain line is CT and Cq of Zeng et al.; and the chain double-dashed line is CD of Zeng et al.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3828.1
(a) Vertical profiles of temperature and (b) specific humidity for u10 = 7 and 25 m s−1 under a stably stratified condition, with Tw = 300 K, t10 = 30 2K, RH = 80%, and P0 = 1000 hPa. A black circle denotes the height of the top of the wave boundary layer; an open circle denotes the height of the top of the drop evaporation layer.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3828.1
(a) Vertical profiles of temperature and (b) specific humidity for u10 = 7 and 25 m s−1 under an unstably stratified condition, with Tw = 300 K, T10 = 298 K, RH = 80%, and P0 = 1000 hPa. A black dot denotes the height of the top of the wave boundary layer; an open circle denotes the height of the top of the drop evaporation layer.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3828.1
Wind speed dependence of the exchange coefficient of total enthalpy CG and the drag coefficient CD under stably and unstably stratified conditions. The solid line denotes CG under the unstably stratified condition; the dashed line is CG under the stably stratified condition; the chain line is CD under the unstably stratified condition; and the chain double-dashed line is CD under the stably stratified condition.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3828.1
Wind velocity dependence of CG/CD under an unstably stratified condition. The solid line denotes our result; the dashed line is the parameterization of Zeng et al. (1998); the dotted line is the extrapolation of the parameterization of Zeng et al.; and the chain line is our regulated result.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3828.1
Wind velocity dependence of CG/CD under unstably stratified conditions. The temperature difference Tw − T10 varies from 0.5 to 3.0 K, and the relative humidity varies from 0.7 to 0.9.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3828.1
Comparison of sea spray–mediated sensible and latent heat fluxes obtained in the present study (fourth and fifth columns) with those of Andreas (1992) (sixth and seventh columns). For every case, the wind velocity is 20 m s−1.
The interfacial and sea spray–mediated heat fluxes under both stably and unstably stratified cases for the strongest wind condition (u10 = 28 m s−1). The interfacial heat fluxes without sea spray effects are also shown. Under the stably stratified condition, we set Tw = 300 K, T10 = 302 K, RH = 80%, and P0 = 1000 hPa. On the other hand, under the unstably stratified condition, we set Tw = 300 K, T10 = 298 K, RH = 80%, and P0 = 1000 hPa.
