## 1. Introduction

*H*is defined as [see, e.g., Smith 1980, Eq. (4)]

_{s}*H*

_{s}

*ρC*

_{p}

*C*

_{T}

*T*

_{sea}

*T*

_{air}

*U*

_{10}

*ρ*is the air density and

*C*

_{p}the specific heat capacity at constant pressure,

*T*

_{sea}is the “bucket” seawater temperature in the wave-mixed layer,

*T*

_{air}is the mean air temperature at the 10-m reference height,

*C*

_{T}is the sensible heat flux coefficient, and

*U*

_{10}is the wind speed at the 10-m reference height.

*H*

_{l}is defined as (Roll 1965)

*H*

_{l}

*L*

_{T}

*E*

*L*

_{T}

*C*

_{E}

*ρ*

*q*

_{sea}

*q*

_{air}

*U*

_{10}

*L*

_{T}is the latent heat of vaporization,

*E*is the evaporation,

*C*

_{E}is the latent heat flux coefficient, and

*q*

_{sea}and

*q*

_{air}are the specific humidity for the sea and air, respectively.

*H*

_{s}and

*H*

_{l}and since values of

*q*need the input of vapor pressure, which is not always measured, the so-called Bowen ratio

*B*has been suggested (see, e.g., Roll 1965) and from Eqs. (1) and (2), we have

*B*may vary from 0.1 to 0.3 in the Tropics (e.g., Pond et al. 1971) and from 0.61 to 0.78 during cold air outbreaks over a midlatitude coastal water (Chou et al. 1986).

The Bowen ratio in the marine environment has been studied by many investigators (see, e.g., Roll 1965; Pond et al. 1971; Kondo 1976; Hicks and Hess 1977; Rao et al. 1986; Liu and Niiler 1990; Konda and Imasato 1996). The spatial variation of *B* is large, particularly for shallow seas during winter when cold air outbreaks occur frequently. An example is shown in Fig. 1. During the Air Mass Transformation Experiments (AMTEX) from 14–28 February 1974, in the sea areas of the southwest islands of Japan, the value of *B* was about 0.7–0.8 over the Yellow Sea, whereas it was 0.3–0.4 over the AMTEX region, and 0.1–0.2 over the subtropical Pacific Ocean. Kondo (1976) speculated that the saturation vapor pressure (*e*_{s}) is very low at low temperatures, but as the temperature increases, *e*_{s} increases exponentially, so that the ratio of the sea–air vapor pressure difference to the sea–air temperature difference takes a large value for high temperatures. The main purpose of this research is to further substantiate this Clausius-Clapeyron effect physically and mathematically.

*H*

_{total}can be estimated from

*B*and

*H*

_{s}and from Eq. (3),

*B*is needed in order to compute the Monin–Obukhov stability length, which takes the form (see Panofsky and Dutton 1984, p. 132)

*u*∗ is the friction velocity,

*κ*

_{a}is the von Kármán constant, and

*g*is the gravitational acceleration.

The objectives of this paper are 1) to offer a possible explanation regarding the variation of B based on thermodynamic considerations, 2) to formulate a relationship between (*T*_{sea} − *T*_{air}) and (*q*_{sea} − *q*_{air}), and 3) to develop a formula that relates *B* to (*T*_{sea} − *T*_{air}) only so that one may “bypass” the requirement of (*q*_{sea} − *q*_{air}) measurements.

## 2. Thermodynamic considerations

*q*

_{sea}is related to the saturation vapor pressure

*e*

_{sea}through (see, e.g., Hsu 1988, pp. 20–21)

*e*

_{sea}= 6.1078 × 10

^{[7.5Tsea/(237.3+Tsea)]}

*p*is the atmospheric pressure. Thus,

*q*

_{sea}is related nonlinearly to

*T*

_{sea}.

*e*

_{air}= 6.1078 × 10

^{[7.5Tdew/(237.3+Tdew)]}

*T*

_{dew}is the dewpoint temperature in degrees Celsius. So,

*q*

_{air}is related to

*T*

_{dew}nonlinearly.

The problem now is to find a relationship between *T*_{dew} and *T*_{air} so that *q*_{air} can be related to *T*_{air}. This is accomplished as follows.

*z*is the altitude. In a well-mixed atmospheric boundary layer from surface to the lifting condensation level (LCL: Hsu 1988, pp. 26–27),

*H*

_{LCL}is the height of the LCL. For dewpoint lapse rate (see, e.g., McIlveen 1986, p. 151),

*R*

_{υ}and R is the gas constant for water vapor and dry air, respectively. For typical low tropospheric

*T*

_{dew}between 283 K (or 10°C) and 293 K (or 20°C), Eq. (12) becomes approximately

*T*

_{dew LCL}=

*T*

_{LCL}, and Eqs. (14)–(11) become

*H*

_{LCL}is in meters and the dewpoint depression at the surface is in degrees Celsius.

Therefore, *T*_{dew} is linearly related to *T*_{air} near the sea surface through the parameterization of *H*_{LCL}. From Eqs. (8), (9), and (16), it is inferred that *q*_{air} is related nonlinearly to *T*_{air} so that the composite quantity of (*q*_{sea} − *q*_{air}) may also be related nonlinearly to that of (*T*_{sea} − *T*_{air}). A proof of this idea is done in the next section.

## 3. Field results

*q*

_{sea}−

*q*

_{air}) as a function of the quantity (

*T*

_{sea}−

*T*

_{air}) must be verified for the idea as discussed above. Between 18 and 22 December 1996, a severe cold air outbreak occurred over the northern Gulf of Mexico, producing a large range of both (

*q*

_{sea}−

*q*

_{air}) and (

*T*

_{sea}−

*T*

_{air}) as shown in Fig. 3. Hourly data for both

*T*

_{air}and

*T*

_{dew}were obtained for Buoy 42040. The results are plotted in Fig. 3. Two curves are obtained, one is linear and the other nonlinear. It can be seen that the linear correlation between (

*q*

_{sea}−

*q*

_{air}) and (

*T*

_{sea}−

*T*

_{air}), with a very high correlation coefficient (

*r*= 0.98), is superior to the nonlinear one (

*r*= 0.96). That is, the linear equation provided in Fig. 3 can directly account for

*r*

^{2}= (0.98)

^{2}= 96.0% of the variability in (

*q*

_{sea}−

*q*

_{air}) using (

*T*

_{sea}−

*T*

_{air}) alone. This correlation is very compelling. If we accept that the linear correlation between (

*q*

_{sea}−

*q*

_{air}) and (

*T*

_{sea}−

*T*

_{air}) does exist, we may then infer from Eq. (3) that

*B*is also related to (

*T*

_{sea}−

*T*

_{air}) only. In fact, this idea is verified in Fig. 4 using the same dataset and Eq. (3) for the same period shown in Fig. 3. In other words, we postulate that

*B*

*a*

*T*

_{sea}

*T*

_{air}

^{b}

*B*

*A*

*b*

*T*

_{sea}

*T*

_{air}

*a*and

*b*need to be determined from field data. The results from all available pairs of

*B*as obtained from Eq. (3) with

*C*

_{T}= 1.1 × 10

^{−3}(see Large and Pond 1982) and

*C*

_{E}= 1.12 × 10

^{−3}(Smith et al. 1994) versus (

*T*

_{sea}−

*T*

_{air}) are shown in Fig. 2. Values of

*a*and

*b*along with the correlation coefficient

*r*from the linear regression of Eq. (17b) between

*B*and (

*T*

_{sea}−

*T*

_{air}) are obtained as follows according to the curve numbers for buoy/station

GDIL1, 1995–96:

*B*= 0.087(*T*_{sea}−*T*_{air})^{0.76}with*r*= 0.79GDIL1, 1996–97:

*B*= 0.087(*T*_{sea}−*T*_{air})^{0.72}with*r*= 0.7742002, 1996–97:

*B*= 0.077(*T*_{sea}−*T*_{air})^{0.69}with*r*= 0.8542019, 1993–94:

*B*= 0.077(*T*_{sea}−*T*_{air})^{0.71}with*r*= 0.8942040, 1996–97:

*B*= 0.077(*T*_{sea}−*T*_{air})^{0.70}with*r*= 0.8942002, 1995–96:

*B*= 0.078(*T*_{sea}−*T*_{air})^{0.67}with*r*= 0.87.

If one accepts these high correlation coefficient values between *B* and (*T*_{sea} − *T*_{air}), one may say that Eq. (17) is verified. Note that hourly data for these four locations for the period given in Fig. 2 were based on measurements provided by the National Data Buoy Center, and for brevity and comparison purposes each individual point from each hour is not plotted in Fig. 2. According to Breaker et al. (1998), uncertainties in the calculated values of specific humidity from these buoys were estimated and ranged between 0.27% and 2.1% of the mean values as compared to nearby ship reports. Note that these hourly buoy data are also available on the World Wide Web. Note also that although the transfer coefficient of *C*_{T} and *C*_{E} may vary with stability (e.g., Konda 1996, Table 2), according to Garratt (1992, pp. 101–104), at present *C*_{E} = *C*_{H} ≃ 1.1 × 10^{−3} (±15%) under near neutral conditions. This is supported most recently by DeCosmo et al. (1996) that *C*_{EN} = *C*_{HN} = 1.1 × 10^{−3} and by Fairall et al. (1996) that *C*_{E} = 1.11 × 10^{−3}. Futhermore, according to Smith (1988, pp. 15470), Friehe and Schmitt (1976) summarized eddy correlation data then available to find *C*_{E} = 1.32 × 10^{−3}, corresponding to *C*_{EN} = 1.18 × 10^{−3}; the difference between *C*_{E} and *C*_{EN} was within the 15% range as suggested by Garratt (1992). Therefore, we use *C*_{T} = 1.1 × 10^{−3} based on Large and Pond (1982) and *C*_{E} = 1.12 × 10^{−3} from Smith et al. (1994) for our computations.

The results as assembled in Fig. 2 are interesting in that for different years at different locations from shelf break to the deeper Gulf of Mexico shown by curves 3 through 6, variations of *a* are only between 0.077 and 0.078 and *b* between 0.67 and 0.69 along with *r* between 0.85 and 0.89. In the nearshore area as represented by GDIL1 (for Grand Isle, Louisiana), however, both *a* and *b* values are higher as shown by curves 1 and 2 due possibly to land effects.

In order to apply Eq. (17) to the deep ocean, Fig. 5 is provided. All data points from Pond et al. (1971) were incorporated except number 12 from the Oregon State University run, which was questionable, as noted by the authors. It is surprising that with only 19 data points the correlation coefficient reaches 0.84. If one accepts this high correlation between *B* and (*T*_{sea} − *T*_{air}), Eq. (17) is also applicable to the deep ocean; although coefficients *a* and *b* may vary, the generic relationship does exist. Certainly, more data are needed to further verify Eq. (17) to deep ocean conditions.

## 4. Conclusions

On the basis of thermodynamic considerations, it is postulated that the quantities of (*q*_{sea} − *q*_{air}) and (*T*_{sea} − *T*_{air}) are related. This postulation was verified during a severe cold air outbreak over the northern Gulf of Mexico under which large ranges of (*q*_{sea} − *q*_{air}) versus (*T*_{sea} − *T*_{air}) along with a very high-correlation coefficient of 0.98 were found. With this compelling correlation, we further postulate that the Bowen ratio *B* is also related to the quantity of (*T*_{sea} − *T*_{air}) such that *B* = *a*(*T*_{sea} − *T*_{air})^{b}. Based on simultaneous hourly measurements of all three temperatures, that is, of *T*_{air}, *T*_{sea}, and *T*_{dew} in the Gulf of Mexico at four stations from 1993 through 1997, we found that in the Gulf of Mexico, values of *a* varied from 0.077 to 0.078, *b* from 0.67 to 0.71, and correlation coefficient *r* from 0.85 to 0.89. Because curve 5 covers the range of (*T*_{sea} − *T*_{air}) from 0° to 18°C, it is recommended for operational use. Equations for the nearshore environment are also obtained. Limited data from the open ocean also support this generic relationship between *B* and (*T*_{sea} − *T*_{air}). Certainly, similar measurements are needed from other ocean basins to further substantiate our proposed formulas.

## Acknowledgments

This study was supported in part by the Louisiana/Texas Shelf Physical Oceanography Program funded by the U.S. Minerals Management Service (MMS) under Contract 14-35-0001-30509 for work to be performed by the Texas A&M University System and subcontractors. The contents of this paper do not necessarily reflect the views or policies of the MMS.

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