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[1] Recent laboratory measurements have shown that the stationary-state vapor pressures of aniline and n-heptanol are enhanced by the application of a positive temperature gradient in the vapor, that the vapor pressure of water over 50 percent sulfuric acid is enhanced by a positive temperature gradient and diminished by a negative temperature gradient, and that values of the Onsager heat of transport derived from these measurements are of similar magnitude to the latent heats of vaporization. When the vapor gap over which the temperature difference is applied is less than about 0.8 mean free paths, the liquid behaves as though its surface were at the temperature close to that of the gas on the other side of the Knudsen layer. These results are discussed in relation to field measurements of the rate of air-sea exchange of carbon dioxide.

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[2] The problem of making accurate measurements of the rate of exchange of carbon dioxide between atmosphere and the ocean is a notoriously difficult one, because the fluxes of interest are very small and correspondingly hard to measure and because the magnitude and direction of the flux are variable in space and time, on both large and small scales. Hence it is not surprising that this problem has been the source of a certain amount of controversy. The use of inventories of ^{14}C or radon in surface waters is problematic because the two tracers average the fluxes over different time scales and they do not agree well. The approach via parameterisation in terms of a wind-speed dependent transfer velocity, as presented by Liss and Merlivat [1986], is attractively simple to use but is vulnerable to the effects of uncontrolled variables. The roles of a number of such variables have been characterised [e.g., Wanninkhof, 1992; Ho et al., 2000] but others probably remain. The dual tracer method of Watson et al. [1991] is not an independent method because it assumes the same kind of dependence on wind speed in order to allow different gases to be compared on the basis of their Schmidt numbers. The approach via eddy-correlation measurements [Jones and Smith, 1977; Smith and Jones, 1985; Tsukamoto et al., 2004], is experimentally difficult and involves the application of corrections that are of similar magnitude to the quantity being measured, but its directness is clearly an advantage. Unfortunately, the results of eddy-correlation and wind-speed parameterisation usually disagree, for reasons that are not entirely clear. The present author has suggested, on theoretical grounds, that a significant problem with the parameterisation approach is its neglect of temperature effects associated with the irreversible thermodynamic coupling of heat and matter fluxes through the air-sea interface [Phillips, 1997]. This idea was not generally accepted, probably because there had been no experimental measurements of the heat of transport at a gas-liquid interface. That situation has now changed; hence the present paper.

2. Irreversible Thermodynamics of the Gas-Liquid Interface

[3] The simultaneous steady-state fluxes J_{1} and J_{2} of heat and matter through a narrow region of thickness δ are given by the heat-flux equation

and the gas-flux equation

where the coefficient L_{11} is the average value of a thermal conductivity multiplied by temperature and coefficient L_{22} is the average of a diffusion coefficient multiplied by the mean concentration or pressure in the region δ. The temperature and pressure changes across δ are ΔT and ΔP, respectively, R is the ideal gas constant, the Onsager heat of transport Q* is defined by equation (1), and the quantities ΔT/T and ΔP/P are the limiting forms of Ln(T_{2}/T_{1}) and Ln(P_{2}/P_{1}) for small temperature and pressure changes [Denbigh, 1951; Phillips, 1991a, 1991b].

[4] Setting J_{2} = 0 in the gas-flux equation gives the stationary-state equation

which has been used to obtain the heat of transport for gases passing through natural-rubber membranes [Denbigh and Raumann, 1952] and through the liquid-vapor interface [Mills and Phillips, 2002; Mills et al., 2004; R. A. James and L. F. Phillips, manuscript in preparation, 2004]. In our work, the stationary-state equation and, by implication, the gas-flux equation, were found to be obeyed at the liquid-vapor interface for aniline and n-heptanol, with positive values of ΔT, and for water vapor at the surface of 50% (by weight) sulfuric acid with both positive and negative values of ΔT. Negative values of ΔT could not be used with the pure liquids because of condensation. Values of Q* obtained by Denbigh and Raumann were positive or negative, depending on the nature of the gas. Values found so far for the liquid-vapour interface are all negative. If we identify Q* as the heat released at the interface by unit gas flux, this requires that a positive flux must be directed out of the liquid.

[5] With n-heptanol, for which the low triple-point vapour pressure allowed measurements to be made at pressures as low as 0.003 Torr, the dependence of ΔP on ΔT was found to be linear up to ΔT values of at least 3 K when the thickness of the vapor gap across which the temperature difference was applied was 0.8 mean free paths or less. At higher pressures (the physical width of the vapor gap was fixed by the apparatus) the dependence became non-linear but Q* could still be determined from the initial slope of a plot of ΔP against ΔT. In the transition region, both linear and curved plots were obtained, which implies that the curvature is almost certainly an artifact. The non-linearity is tentatively attributed to a form of small-scale turbulence arising from persistence-of-velocity effects [Alder and Wainwright, 1970], but more work is needed to settle this point.

[6]Figure 1 shows the dependence of the magnitude of the measured heat of transport for n-heptanol on the number of mean free paths in the vapor gap, n_{λ}. The empirical curve has been drawn so as to extrapolate to the latent heat of vaporization ΔH_{vap}, as derived from the temperature dependence of vapor pressure, at n_{λ} = 0. The largest measured ∣Q*∣ value is 88 percent of ΔH_{vap}. These results are consistent with the view that the important part of the temperature gradient is located in the Knudsen layer immediately adjacent to the liquid surface. However, the effect of the temperature gradient is observable over a much greater distance. With aniline, at the highest pressure used, the vapor gap amounted to 38 mean free paths and the measured value of ∣Q*∣ was still more than 20 percent of ΔH_{vap}.

[7] A referee has suggested that our Q* measurements could be subject to a systematic error, due to neglect of heating of the liquid surface by radiative transfer across the vapour gap. Such an effect would make our measured Q* values too large. There are three strong arguments against this being a significant source of error. (i) A radiative effect should vary strongly with temperature, in proportion to T_{2}^{4}-T_{1}^{4} ∼ 4T^{4} ΔT/T and, because the gas pressure was controlled by the working temperature, would therefore produce the opposite dependence of Q* on gas pressure from that which is evident in Figure 1. (ii) The temperature of the liquid surface, which was in a steady-state with respect to the heat flux, was regulated to within a few hundredths of a degree Celsius by reference to a very small thermistor whose active element was located right at the surface. (iii) The stationary-state equation (2), from which Q* is obtained, is independent of the mode of heat transfer, requiring only that it vary as ΔT/T, so a small radiative effect would appear simply as an enhanced thermal conductivity in the L_{11} factor of equation (1). Nevertheless, further experimental work is now planned to establish the magnitude of the radiative effect, if any.

[8] The application of the gas-flux equation to the air-sea exchange of carbon dioxide [Phillips, 1991a, 1991b, 1992] was criticised by Doney [1994, 1995a] on two main grounds. Doney's first criticism, that the heat of transport was incorrectly derived as Q-C_{p}T, where Q is the latent heat of vaporization or solution and C_{p} is the heat capacity of the gas being transferred, has been shown by our experiments to be correct. The error arose from regarding the gas-liquid interface as the boundary between two semi-infinite regions. Correct cancellation of the C_{p}T term is obtained by considering a portion of the interface locally as a closed adiabatic system. Elimination of this error makes the heat of transport numerically larger and brings it almost into agreement with the theoretical work of Spanner [1954], who predicted that Q* for a two-surface system should be equal to the latent heat of vaporization. Spanner's work appears to have been overlooked by physical chemists, presumably because it is part of the plant physiology literature.

[9] Doney's second point, that the heat of transport should be an order of magnitude smaller than the latent heat, is clearly contradicted by our experiments. A much smaller value of Q* would indeed be expected if the effect were due to simple heating of the liquid surface, but that evidently is not the case. The existence of a large heat of transport appears to be understandable in terms of the effect of the applied temperature gradient on a free energy barrier to evaporation which is located in the narrow (∼16 Å for water at 300 K) zone of thermally-excited capillary-waves at the surface of the liquid [Phillips, 2003]. It is interesting to note that recent molecular dynamics calculations for n-octane predict a Q* value of the order of 30 percent of the latent heat [Simon et al., 2004].

[10] A qualitative plot of the gas-flux equation is shown in Figure 2 for two different flux directions and for the stationary state, with zero flux, that prevailed during our measurements of Q*. A similar plot has been used to account for three kinds of paradoxical behavior at the surface of a liquid [Mills et al., 2004]. The slopes of the lines are equal to Q*P/RT^{2}, so ΔT and ΔP are required to be small in comparison with T and P. In the laboratory, linear plots like the zero-flux line in Figure 2 are obtained either when ΔT is small (<0.5 K), or when the thickness of the vapor gap over which ΔT is applied is 0.8 mean free paths or less. In the field, the fact that the important region is very thin (the mean free path is approximately 1000 Å at atmospheric pressure) implies that the steady-state gas flux is established on a very short time-scale, of the order of a nanosecond, as estimated by dividing the mean free path by the mean speed of a carbon dioxide molecule and multiplying by a safety factor of 4. Therefore the gas-flux equation (2) should apply even when conditions are changing very rapidly, as during the evolution of a wave with entrained air. The main point of Figure 2 is, of course, that not just the magnitude but even the direction of the gas flux can be altered by varying the temperature difference across the Knudsen zone. Because the response to the temperature gradient is so rapid, this is likely to remain true even under windy conditions or in a surf zone.

3. Field Measurements of Air-Sea Gas Exchange

[11] A number of measurements of air-sea fluxes of carbon dioxide have been based on the plot of transfer velocity versus wind speed presented by Liss and Merlivat [1986], with refinements to take account of the effects of such variables as wind fetch [Wanninkhof, 1992], or the effect of the ‘cool-skin of the ocean’ on surface solubility [Robertson and Watson, 1992].

[12] Direct measurements of CO_{2} fluxes by eddy correlation have frequently disagreed with the measurements based either on ^{14}C or radon inventories or on wind-speed-related transfer velocities combined with bulk concentration differences, the eddy-correlation fluxes typically being one or two orders of magnitude larger. The pioneering work of Jones and Smith [1977] and Smith and Jones [1985] was strongly criticised on this account [Broecker et al., 1986; Smith and Jones, 1986; Wesely, 1986]. However, as eddy-correlation equipment and techniques have steadily been refined, the difference between eddy-correlation and wind-speed based CO_{2} fluxes has not gone away. The discrepancy was not as large in the ASGAMAGE [Jacobs et al., 1999] and GasEx results (GasEx 98, http://www.cmdl.noaa.gov/hats/ocean/gasex98/gasex98.html, and GasEx 2001, http://www.pmel.noaa.gov/co2/gasex2/), but it seems to have reappeared with undiminished vigor in the recent work of Tsukamoto et al. [2004]. The larger fluxes obtained by Smith and Jones could in part be attributed to their shore-based measurements being adjacent to a surf zone; however, the more recent measurements were made from an ocean platform or aboard a ship at sea.

[13] Model calculations, using the field data of Smith and Jones [1986] and the wind-tunnel data of Liss et al. [1981], showed fairly convincingly that the Onsager gas-flux equation was required in order to understand the results [Phillips, 1994], although this too was challenged [Doney, 1995b; Phillips, 1995]. At present there appears to be a movement away from wind-speed based transfer velocities towards eddy-correlation measurements, which the present author regards as a positive trend. The considerable scatter of the wind-speed based flux measurements and the disagreement with recent direct measurements suggests that the indirect measurements are affected by one or more uncontrolled variables, in addition to the variables that have been considered so far. The present paper reiterates that one such variable is likely to be the air-sea temperature difference.

4. The Effect of −Q* Being Approximately Equal to ΔH_{vap}

[14] For a small temperature change ΔT, the equilibrium vapor pressure change ΔP is given by the approximate form of the Clapeyron equation as

where the approximations consist of neglecting the molar volume of the liquid in comparison with that of the gas and assuming that the vapor obeys the ideal gas equation, these same approximations being involved in the derivation of equation (2). Equation (4) has the same form as the stationary-state equation (3) with ΔH_{vap} in place of −Q*. Hence, recalling that Q* is negative and ΔH_{vap} is positive, if the thickness of the vapor gap over which the temperature difference is applied is 0.8 mean free paths or less, so that −Q* ∼ ΔH_{vap}, then the effective vapour pressure of the liquid is the same as it would be if the surface temperature of the liquid were equal to the temperature of the gas on the other side of the vapor gap. This result, which is a consequence of Onsager's irreversible thermodynamics, can also be justified on the basis of the requirement of detailed balance for the fluxes of molecules arriving at and leaving the liquid surface [Phillips, 2004], which is not surprising in view of the fact that Onsager originally derived his reciprocal relation by arguments about detailed balance. However, the conclusion that, in the presence of a temperature gradient, the actual temperature of the ocean surface, cool skin or otherwise, is almost irrelevant to the effective vapor pressures of water and dissolved gases, is as unexpected as it is interesting. It suggests that the differences in gas fluxes between areas of the ocean in contact with very warm air and areas in contact with very cold air are likely to be considerable, and perhaps even comparable with the difference between a flat sea and a surf zone.

Acknowledgments

[15] The author gratefully acknowledges the support of the US National Science Foundation (Grant No. 0209719).