JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. D15, PAGES 18,619-18,630, AUGUST 20, 1999

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. D15, PAGES 18,619-18,630, AUGUST 20, 1999 An advective-diffusive model isotopic evaporation-condensation Hui He I and Ronald B. Smith Department of Geology and Geophysics, Yale University, New Haven, Connecticut Abstract. A one-dimensionm advective-diffusive model is developed for the evaporation process across an evaporating surface, providing a general theory of fractionation during evaporation and condensation. The model configuration is based on the Craig-Gordon linear evaporation model. We improve the model by allowing advection and diffusion to exist in the skin layer beneath the evaporating surface. The model is based on two fundamental physical principles, Henry's law and Fick's law. The model solutions show that diffusion of isotopes in both the atmosphere and the condensed phase plays an essential role in controlling the isotope flux ratio 5F, which is a measurable quantity. There are several dimensionless parameters that control the isotopic fractionation during evaporation. The most important parameter is N, which takes into account the evaporation rate and the depth of the advective-diffusive layer and represents the relative magnitude of the rate of advection versus the rate of diffusion. Assuming rapid surface exchange, the water vapor is always in isotopic equilibrium with the liquid water at the interface. Under the slow evaporation limit, i.e., N - 0, the back flux from the atmosphere to the liquid water reservoir is not negligible compared with the upward flux; hence the net flux ratio must take into account information from the atmosphere such as the mixing ratio and the isotope ratios of water vapor. When N is large, the back flux becomes negligible, so the upward flux dominates the net flux and brings only the information of the liquid water reservoir into the net flux; therefore the flux ratio approaches the isotope ratio of the liquid water reservoir. The model also justifies itself by matching with independent theoretical results from earlier studies. Vapor condensation onto ice is also included as a special case in the model. 1. Introduction Merlivat and Coantic [1975] conducted studies on mass transfer at the air-water interface using an isotopic The correct use of stable water isotopes, HD160 and method which makes it possible to calculate the rela- H2180, in studying various aspects of the hydrological tive contribution of molecular and turbulent transfer to cycle relies on our ability to understand and model the the total mass transfer in the boundary layer above the fractionation process affecting water molecules during water surface. With this method the observations of their atmospheric cycle. The applications of these stathe fractionation of isotopic water species in water vable isotopes as tracers for lake mass balance calculations [Dincer, 1968; Gat, 1970] have demonstrated that the por under laboratory conditions [Merlivat, 1978] show isotopic composition of evaporating water vapor is the good agreement with the evaporation theory by Brutmost difficult to estimate among all parameters which saert [1975]. Their parameterization of fractionation at the air-water interface serves as a basis for the studmust be determined. The uncertainty involved in methies of deuterium - oxygen 18 relationship in meteoric ods of estimating the isotopic enrichment of evaporation can result in an uncertainty of more than 50% of the waters [Merlivat and Jouzel, 1979] and isotopic general calculated evaporation rate [Zimmermann and Ehhalt, 1970]. x Now at Joint Center for Earth Systems Technology, University of Maryland, Baltimore County, Baltimore. Copyright 1999 by the American Geophysical Union. Paper number 1999JD /99/1999JD ,619 circulation models (GCMs) [Jouzel et al., 1987, 1994]. The isotopic fractionation accompanying the evaporation of surface waters is the result of the difference between two opposing water fluxes from the air-water interaction, i.e., one upward from the surface and one downward to the surface. If the atmosphere is saturated at the temperature of the liquid surface, the interaction will bring the atmospheric water vapor and liquid water into isotopic equilibrium. However, when the air is undersaturated, the imbalance results in a net evaporation

2 18,620 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL flux. The evaporation rate is determined by the rate of diffusion of water vapor across the air boundary layer in response to the humidity gradient between the surface and the well-mixed ambient air [Gat, 1996]. The overall isotopic fractionation is determined by three factors: (1) equilibrium fractionation at the air-liquid interface, (2) additional fractionation resulting from the different diffusion rates betwee normal water (H2160) and isotopic water (HD160 or H2 80) across the air and water boundary layers, and (3) back flux of atmospheric water vapor. The best known model for isotope fractionation during evaporation is a laminar layer diffusion model developed by Craig and Gordon [1965], which is based on the linear-resistance evaporation model. The model divides the regions above and below a liquid water surface into several zones: the turbulent liquid reservoir, the diffusive liquid boundary layer, the air-liquid interface, the diffusive air boundary layer, and the well-mixed turbulent air layer. The Craig-Gordon model considers the discrete layers above and below the air-liquid interface, and within each layer one or the other transport mechanism is totally dominant. By taking into account the diffusive laminar layers on both sides of the interface, the model allows the concentration gradients of isotopes to build up between the evaporating surface and the bulk water and between the surface and the mixed layer in the atmosphere. Such surface gradients may have an important effect in situations such as a closed basin un- dergoing excessive evaporation. The Craig-Gordon model was derived in analogy to Ohm's law. While it includes all the relevant physics, it does not represent a clear solution of the differential equations governing the processes of advection, diffusion, and evaporation. Generally, this model is not based on the fundamental physical principles of transport and evaporation. The objective of this paper is to develop a clear and universal isotopic evaporation model that takes into account the processes of evaporation, isotope fractionation, diffusion, and turbulent mixing from the fundamental physical laws. We would like to use the model to gain deeper insights into other isotopic evaporationsexchange theories such as those given by Merlivat and Jouzel [1979] and Jouzel and Merlivat [1984] and to provide a theoretical framework for the interpretation of other experimental data. 2. Physical Considerations Because of the difference in saturation vapor pressure between normal water and isotopic water, isotope fractionation takes place in all processes which involve phase change of water between the vapor phase and a condensed phase, for example, liquid water or ice. There are many natural processes that involve the transport of water between the surface and the atmosphere and the phase change of water, such as evapo- ration from an open water surface, transpiration from plants, and condensation of vapor onto ice. Although these processeseem quite different, they share a common hydrological path; that is, water moves from one reservoir (e.g., ocean) to another reservoir (e.g., atmosphere), and phase change takes place when water molecules move across the vapor-liquid or vapor-ice interface. Considering the evaporation at the Earth's surface, the first reservoir usually exists at certain depth below the surface and extends downward. In this region the composition of source water is nearly constant. Other transport processes may take water from this region but cannot alter its composition. In the ocean this reservoir is the subsurface water body except for the diffusive sur- face layer just below the surface, while the subsurface lake water or soil moisture consists of the reservoir in continental areas. This source water is transported to the air-liquid interface where evaporation takes place. Isotope fractionation occurs during the phase change of water. After that, water molecules diffuse through the first several millimeters to centimeters of the air above to move away from the interface. They are then carried farther upward into the upper reservoir, i.e., the free atmosphere, by turbulent transport through the atmospheric boundary layer (ABL). To study the nature of isotope fractionation across different types of surfaces, we develop a one-dimensional advective-diffusive isotopic evaporation model following some well-known physical principles. The basic mechanisms resulting in different degrees of isotope fractionation are advection and diffusion. This one-dimensional advective-diffusive evaporation model focuses on finding a general solution for the isotope fractionation under vertical flux of water vapor from and to the surface. Vapor condensation can also be included as a special case. 3. Henry's Law and Fick's Law The model is based on two fundamental physical laws in chemistry and physics, namely, Henry's law and Fick's law. Henry's law is for the solubility of gases in liquids. It can be expressed as a proportionality between the mole fraction of the solute and the pressure of gas above the solution [Alberty, 1987]: p-kxx (1) where X is the mole fraction of the solute, p is the pressure of gas above the solution, and kx is a constant and can be determined experimentally. The law is obeyed well for gases for which (1) the solubility is low, (2) the gas is reasonably ideal, and (3) the gas does not react with or ionize in the solvent. The composition of stable isotopes in natural water approximately satisfies these conditions. Therefore the partial pressure of the minor isotope in vapor phase shohld be proportional to the

3 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL 18,621 isotope ratio in liquid water. As we will see later in the model development, the equilibrium fractionation relationship is a natural derivation from the Henry's law. Fick's first law states that in a mixture of two sub- stances A and B, the net flux of one substance, say A, in the presence of the flux of substance B can be expressed as [Bird et al., 1960] where A and s are the net fluxes of A and B, respec- tively, and have the units of LT -1. The value As is the diffusivity of A in B, and x A is the fraction of A in the total flow. The fraction of B in the total flow will be represented by x s thereafter. Equation (2) shows that the diffusion flux FA relative to the stationary coordinates is the resultant of two vector quantities- the vector XA (I A + s), which is the molar flux of A result- ing from the bulk motion of the fluid, and the vector -- ABVXA, which is the molar flux of A resulting from the diffusion superimposed on the bulk flow. Similarly, the net flux of B can be written as where sa is the diffusivity of B in A. Under the limit where one of the two components, say B, becomes the major species and the other one, here A, becomes the minor species, Fick's law can be interpreted as a pure advective flow of the major species plus an advective-diffusive flow of the minor species. Because A is the minor species, we have xa << xs and XB = 1--XA 1. Therefore, if we denote F, - (4) 4. Theoretical Model 4.1. Model Setup The general physical domain, which involves the advective-diffusive fluxes of normal water (H21 0) and isotopic water (HD160 or H2180) across the water surface and surface layer into the open atmosphere, can be subdivided into five discrete layers. One or the other transport mechanism is dominant within each layer. The model domain is shown in Figure 1. At the bottom of the domain the liquid water reser- voir, denoted by region 1, represents the source of liquid water with constant isotopic composition. In large water bodies, such as the ocean or lakes, this region starts from just a few millimeters below the air-liquid interface downward to the rest of the water body. Over the land, region I normally represents the groundwater table. In both cases, region I represents a large and well-mixed reservoir with constant isotopic composition. Therefore the isotopic composition of liquid water at the lower boundary of region 2 right above is fixed. The advective-diffusive surface layer, denoted by region 2, starts from the top of region I and ends at the place where evaporation and/or condensation in the form of phase change between liquid water and water vapor actually takes place. In the case of the open water surface the top of region 2 is the air-liquid interface. Over the forests, region 2 has a less definite top boundary; however, the boundary can be considered as the stomatal openings at the leaf surface. Region 2 represents the advective-diffusive layer of liquid water just before evaporation takes place. In this region, heavy isotopes are allowed to diffuse down the gradient which is caused by the fractionation process during phase change at its upper boundary. The evaporation as the advective velocity of the major species, (2) be- comes FA A VXA + XA (5) where we have used the approximation that 1- x A 0 1. Equation (5) is the governing equation for the advective-diffusive flow of the minor species A in the presence of the advective flow of the major species B. For incompressible fluid flow the continuity equation states that in the absence of net sources and sinks, OX 0- +V.F-0 where F is the net flux of the tracer element with concentration x. Under the steady state situation, O/Or - O, the continuity equation becomes v.-0 (6) and in a one-dimensional case, (6) states that the net flux is constant everywhere in the domain. z z Free Atmosphere (Region 5) Turbulent (Region Mixed 4) Layer Diffusive Surface Layer (Region 3) I I z=li+l 2 I z x Advective-Diffusive (Region 2) Surface Layer I Liquid Water Reservoir (Region 1) I I - z=0 Figure 1. Sketch of the one-dimensional advectivediffusive model. Area enclosed by dashed lines is the model domain for one-dimensional advective-diffusive flow for both normal.water and isotopic water. Arrows indicate that the net fluxes for all passive tracers are conserved in regions 2, 3, and 4. The specification of each layer parallels that of Graig and Gordon [1965].

4 18,622 HE AND SMITH' ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL rate of normal water can be represented by the velocity After molecules of both normal water and isotopic water have escaped from the liquid phase, they must move upward through the first few millimeters or centimeters of air by molecular diffusion due to the lack of other transport mechanisms near the surface such as advection or turbulence. The transport is driven by the vertical gradient of the concentrations of water molecules. In this diffusive surface layer, denoted by region 3, forced advection no longer exists, and the flow becomes pure diffusive. In addition, the major carrier species becomes air, while both normal water and isotopic water molecules become minor species which diffuse through the air. For these pure diffusive fluxes the governing equations for region 3 can be obtained from (5) and (6)as - 0 (7) ness of the advective-diffusive surface layer (region 2) in the liquid side and the thickness of the diffusive surface - 0 (8) air layer (region 3) just above the air-liquid interface, respectively. where Q and Qi are the mixing ratios of normal water In region 2, H2160 molecules are transported from (H2160) and isotopic water (HD160 or H2180), respec- z - 0 to z - L1 with a velocity u in the vertical directively. As molecules diffuse upward through region 3, they enter region 4, which represents the mixed layer where turbulent mixing starts dominating the transport of passive tracers; the governing equations for the transport of normal and isotopic water have the same form as (7) and (8), but the coefficients of diffusion in the solutions now become eddy diffusivities of each species instead of molecular diffusivities. Although the individual molecular diffusivities of normal water and its isotopes are different, the eddy diffusivities of both species are tion due to external forcing such as evaporation. Since H2160 is the major species, the slight gradient of H2160 can always be neglected. In addition to the advective transport of H2160, molecules of HDi O and H2180 are also transported upward. However, since HDlCO and H2180 are the minor species, the advective transport of these molecules is also accompanied by a diffusive flux that tends to move them from the region of higher concentration to the region of lower concentration. Using the flux in the form of (5), the continuity equation (6) for the concentration of isotopic water can now be normally assumed to be equal to each other, because written as both species are passive tracers and are analogous in V. (s - nov's) - 0 (9) the way that they are brought into the atmosphere and transported. Therefore the turbulent transport is nonwhere nd is the molecular diffusivity of HD160 (or fractionating. This model does not deal with the mixed H 180) in H 160 and s is the concentration of HD160 molecular/eddy diffusivity that was studied by Brut- (or H 180) in liquid water and is given by saert [1975] and Merlivat and Coantic [1975]. At the top of the turbulent mixed layer the free atmosphere is represented by region 5. This part of the atmosphere is assumed to have constant mixing ratios of normal water and isotopic water. It functions as the other reservoir for the overall flow. Therefore the mix- ing ratios at the top of region 4 are fixed. The properties of this free atmosphere are assumed to be independent of the local evaporation processes. where R represents the isotopic concentrations of D/H and O18/O 16 by molar volume and RSMOW represents the isotopic concentrations in SMOW. The values of RSMOW are x 10-6 for D/H and x 10-6 for OlS/O 16 [Jouzel, 1986]. For reading convenience we will use 5 to represent 5/1000 hereafter in all formulae. Since fractionati0n of hydrogen and oxygen isotopes takes place during evaporation, the advective-diffusive problem must take into account the isotopic effects and can be considered in the vertically oriented onedimensional domain extending from z - 0 (bottom of region 2) to z - L1 + L2 (top of region 3), indicated by the region enclosed by dashed lines in Figure 1, by assuming that water exists in liquid form from z - 0 to z - L1 (air-water interface) and in vapor form from z - L1 to z - L1 + L2, where L1 and L2 are the thick- pi s jr P where p and pi are densities of H2160 and HDlOO (or H2180), respectively. The value j is the ratio of molecular weight of HDlOO (or H2180) to that of H2160, and R is the isotope ratio of liquid water. Equation (9) states that the divergence of the net flux of HD160 (or H2180), 4.2. Model Derivation The HD160/H2160 and H2180/H2160 isotope ratios are commonly expressed in terms of delta values, 5D and 5lSO, which are the relative deviation of the isotope concentrations from the known standard, standard mean ocean water (SMOW), as 5-(R-RsMøw) / SMOW xloo0 F- s - novs is zero everywhere in region 2, where F is the sum of the advective flux sff, which is driven by the advection of H2160,,, and the diffusive flux -nov's, which is driven by the gradient of the isotope concentration s. The contribution of the advective mass flow ff is required to compensate the loss of water at the top of the advective-diffusive layer and to keep the interface position unchanged. In the one-dimensional case, (9)

5 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL 18,623 becomes We can now define Os 02s u zz- n Oz--- (10) to represent a length scale of the depth of a subsurface skin layer within region 2 just beneath the air-water interface. There are much stronger vertical gradients of heavy isotopes within this depth than there are in the rest of region 2. Although the overall vertical profiles of delta values in region 2 are determined by the evapora- tion/condensation processes at the air-water interface, only the liquid water within the depth of œ can communicate well with the atmosphere above the interface. It is then useful to define a dimensionless constant N as / D Ll ul1 N - : (11) œ D Therefore with constant L, N represents the relative magnitude of advection versus diffusion; that is, N >> 1 when the flux is mostly advective and diffusion is negligible, and N << I when the flux is mostly diffusive and the advective flow is negligible. From (11), N also represents the relative magnitude of L1 versus œ and thus the ability to communicate between region I and region 3. Strong advection tends to push the bottom of the subsurface skin layer closer to the surface and reduce the communication between the atmosphere and the lower part of the advective-diffusive layer. For 0 < z < L1, (10) has the general solution with the boundary condition of s(z) - C + D e (z/r) (12) C + D - jrsmow (50 + 1) (13) and isotope ratio at z - L1 + L2, respectively. Since Qi = jr - jrsmow(5 q- 1) Q the boundary conditions become A (L 1 q- L2) q- B = Qb (16) A t (L 1 q- L2) q- B' = jrsmowqb(5b + 1) (17) According to (6), the flux values of each species are constant and equal in both regions 2 and 3 under steady state situations. The boundary conditions at z = L are determined by matching the net fluxes of H2160 and HD160 (or H21SO) on both sides of the air-liquid interface and by the equilibrium fractionation relationship of the isotope ratios between the liquid and the saturated vapor. The matching of net fluxes means that the net flux from the interior of region 2 to the interface must be equal to the net flux which leaves the interface into region 3. From (4) and (5) the net fluxes of H2160 and HD160 (or H21SO) at the lower (condensed phase) side of the interface (z = L -) are by assuming that liquid water has a constant HD160 (or H2180) isotope ratio of 0 at z -- 0, i.e., slz=0 = Oz where n and ;i are the molecular diffusivities of H2160 So -- jrsmow (50 + 1). and HD160 (or H21SO) in the air, respectively. In region 3 where L < z < L1 q-l2 the major carrier At the interface the net vertical flux of H2 O or species becomes air. Both normal water (H2160) and HD160 (or H21sO) is the sum of upward flux and downisotopic water (HD160 or H2180) are now minor species ward flux, and it can be parameterized as that can only diffuse away from the air-liquid interface into the atmosphere above. There is no advective flux associated with the movement of H2160 and HD160 F0 - - [Q(L1 +) - Q*] (or H2180). In the one-dimensional case the governing equations (7) and (8) for the concentrations of H2160 rj - -0 ½1 +) - (19) and HD160 (or H2 80), represented by their vapor mix- where Q(L1 +) - AL 1 q- B and Qi(LI+ ) - AtL1 q- B' are ing ratios ( and Qi, have the solutions of from the general solutions (14) and (15). Q(z) = A z + B (14) In (18) and (19), 99 and 0 are the exchange coefficients for H2160 and HD160 (or H2180) between water vapor and the condensed phase at the interface z = L1. Qi(z) = A'z + B' (15) Q* is the saturation vapor mixing ratio at the surface which are subject to the boundary conditions that the values of Q and 5 are fixed at the bottom of the turbulent mixed layer (region 4), such that Q = Qb and 5 = 5b, where Q and 5 are the constant mixing ratio temperature, and Qi, is the HD160 (or H2180) mixing ratio in the water vapor that is saturated at the surface temperature and in isotopic equilibrium with respect to the liquid water at surface. F_ -- U F / - slz:z, lu - nd zz z'-l1 - Cu where we have used (12). Note that the net transport rates upward are the result of the two-way exchange process. At the upper (atmosphere) side of the air- liquid interface, the net fluxes are diffusive fluxes and can be written as OQ -An F+ - -n Oz - F - -n i OQi At n i --

6 18,624 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL 4.3. Rapid Surface Exchange: qo-4 cx and -4 Limits at z = L, where ae is the equilibrium fractionation factor. Equation (20) states that water vapor just above the interface is in isotopic equilibrium with the liquid water at the surface; that is, the isotope ratio of water vapor is 1/a times the isotope ratio in liquid water: - 1 as follows. The mixing ratio of isotopic water is proportional to its vapor pressure In general, there are three places where resistances to the movement of any species between phases exist: the gas phase, the liquid phase, and the interface itself. The interface resistance is negligible in most applications at low mass transfer rates [Bird el' at., 1960]. Keeping in mind that and are the exchange rates of normal wa- Qi, ½x: e i* where e i* is the vapor pressure of HD O (or Ha SO) when H2 60 is saturated. On the other hand, Slz: - jrlz: ter and isotopic water at the interface, this statement indicates that the values of and are large enough to where R[z= is the isotopic composition of surface waenable negligible resistance at the interface. Laboratory ter; hence experiments [Friedman el at., 1962] verified the theoret- e i* x R]z=L - ical predictions of rapid exchange rate between liquid which states that the isotope vapor pressure above the water and water vapor at the interface. Physically, this surface is proportional to the isotope ratio of the surface means that the evaporation and condensation rates of water, i.e., Henry's law. water at the interface are so large that the vapor in contact with the liquid water is essentially saturated. Ex Solutions for Unknown Constants perimental studies by Stewart [1975] also revealed that By matching the fluxes above and below the interface, equilibrium fractionation exists between the condensed phase and the vapor at its surface even during rapid i.e., setting F_ - F+ and F! - F _, we obtain evaporation in dry atmosphere. Kinetic fractionation takes place only as water molecules diffuse away from the interface. Mathematically, this states that u - Cu - -A'n i (26) and at z = L. The model will assume and hereafter. Because of the rapid exchange at the air-liquid interface, the equilibrium fractionation relationship is tenable, i.e., Equations (13), (16),(17),(23),(24),(25), and (26) can be solved for seven unknowns: u, ALa, A'La, B, B', C, and D. Note that rather than solving for A and A', ALa and A'La are solved because they have the a (20) same dimensions as B and B'. In addition to N, it is convenient to define three other dimensionless variables: h- Qb Q, (27) P - (28) - L2 K - (29) which is equivalent to Q * 1 = e Q* Using (12), (21) now gives rise to where we have defined - [c + w:q* e With and and for finite values of F0 and F, (18) and (19) now become a + B = Q* (23) a' + [c + (a4) The equilibrium fractionation relationship (21) states Henry's law implicitly. This can be briefly illustrated Of these dimensionless parameters, h is the relative humidity defined as the ratio of the mixing ratio at z = L + La to the mixing ratio of saturated air at the temperature of the interface at z = L. P represents the ratio of the depth of the advective-diffusive liquid layer to the depth of the diffusive air layer. K is the ratio of molecular diffusivities of isotopic water to normal water in the air and has values of for H2 SO/H2 60 and for HD 60/H2160 [Gat, 1996]. Solutions of the seven unknowns in dimensionless forms in terms of the known quantities are ul2 = Q* (1 - h) (30) AL2, = -(1- h) (31) (1 + P) - hp (32)

7 B' AtL2 - --jrsmow(1 -- h) Q, HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL 18,625 [ h(sb + 1)e -N-(1/c e)(50 + 1) (33) A'L2 Q, = jrsmowh(6b + 1) - Q, (1 + P) (34) C= A'L IV Q* 1-h (35) 4.5. Isotope Flux Ratio D - so - C (36) Note that the quantities in (25) and (26) are the net 4.6. Isotopic Composition at the Interface vertical fluxes of H2-X60 and HDX60 (or H2XSO), respectively. Dividing (26) by (25) gives the ratio of the flux of HDX O (or H2XSO) to the flux of H2X O. From the With the analytical solutions of six constants, A, B, A', B', C, and D, we can use (12), (14), and (15) to left-hand sides of (25) and (26), the two fluxes are equal calculate the isotope concentration of liquid water and to u and Cu, respectively. Therefore the flux ratio is the mixing ratios of H2160 and HDX O (or H2180) at the same as the constant C, i.e., z = Lx, as well as any other derived quantities such as the delta values and the fractionation factor at the F i h(5 + 1)e -N -(I/de)(50 + 1) interface. These are - = jrsmowk -(1 - h)e -N + (K/a )(e -N - 1) (37) Figure 2 shows the variation of the deuterium flux ratio, 5DF, with the dimensionless parameter N, for different values of relative humidity h. At lower N values the flux ratio 5DF is very sensitive to the change of h. The value 5DF increases rapidly as h becomes large, indicating the disappearance of the net flux of normal water. In isotopically lighter moist air, for example, h = 75%, the back diffusion of light water makes the isotope flux ratio appear heavy; that is, exchange dominates over evaporation. For large N, advection dominates over diffusion in the liquid/solid phase, resulting in a "batch process" with no fractionation. The flux ratio becomes independent of h and approaches 5D0 when N -+ q-c o Q(L1 +) - Q* (39) In (37), 5 and 50 are the delta values of the water vapor at z = Lx + L2 and the liquid water at z = Lx. An alternative but convenient way to view the flux ratio is to convert F i/f in terms of a delta value, 5F, as 5O 1 F i 5F + I - (38) jrsmow F Qi(Li+) - jrsmowq* [-(1- h)(50 + 1)+ Kh(5, + 1)(e -N - 1)] -c (1 - h)e -N + K(e -N - s(l -) - jrsmow (40) [-(1- h)(5o + 1)+ Kh(5b + 1)(e -N- 1)] - n)e -N + (K/ae)(e-N - (41) -5O -loo h=25% \ h=50% \ h=75% \ -150 ' ' log, on \, The delta values of water vapor and liquid water on both sides of the air-liquid interface are therefore 5v(Lx+) + 1-5c(L -) + I - -(1 - h)(5o + 1)+ Kh(5 + 1)(e -N - 1) -a (1 - h)e -N + K(e -N - 1) (42) -(1 - n)(50 + 1) ) -( ) (44) Note that 5c(L ) + I normally is not equal to (50 + 1), because of the existence of the differential molecular diffusion through the surface layers in both the liquid Figure 2. Flux ratio of HDX O/H2X O as a function phase and the gas phase. These diffusive processes are of dimensionless parameter N = ul1/nd, for relative humidity h = 25%, 50%, and 75%, respectively. Values taken into account by the dimensionless parameters N used for other parameters are L1 = 0.1 m, L and K. Equations (42) and (43) result in the fractionm, Q* = 15 g kg -x, 5D0 = -60 per mil, 5D = -200 ation factor per mil, c , nd= 1.0 x 10-0 m 2 s -1, n = a- a (44) 2.57 x 10-5 m 2 s -x, and K The thin solid line represents the isotope ratio of the liquid reservoir, between the isotopic composition of water vapor and 5D0, and the thin dashed line represents its equilibrium liquid water at z - L1. Now an effective fractionation isotope ratio, 5D per mil. factor can be defined as the fractionation factor between

8 18,626 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL _6o h=25% -8o,=5o% /"- t h= :L log,on Figure 3. Variation of 5Dv(L +) (delta value of water vapor at surface) with dimensionless parameter N = ul /nd, for relative humidity h = 25%, 50%, and 75%, respectively. Other parameters are the same as those in Figure 2. The thin dashed line represents the equilibrium isotope ratio of the liquid reservoir, 5De per mil log,on Figure 5. Variation of effective fractionation factor with dimensionless parameter N - ul /nd, for relative humidity h - 25%, 50%, and 75%, respectively. The equilibrium fractionation factor is ( e Other parameters are the same as those in Figure 2. ' the condensed phase at the interface z - L and the vapor phase at the top of the diffusive layer z - L + L2' 5c(L ) + 1 -(1- h)(5o + 1)+ Kh(5 + 1)(e -Iv - 1) [-(1 - h)e -Iv + (K/ e)(e-iv - 1)](5b + 1) (45) Figures 3 and 4 show the variations of 5DvLt +) and 5Dc(L ') with N for the same parameters as those in Figure 2. Because of the two-way transfer process, the isotope ratios of water vapor and liquid water at the air-water interface are strongly affected by the relative humidity above the diffusive air layer when the evaporating flux is large. Evaporation accumulates the heavy isotopes at the surface and causes the isotopic concentration of the surface water to increase with increasing evaporation rate. In the large N limit, i.e., the so-called "batch process," the isotope ratio of the surface water will approach a value required to evaporate vapor at the isotopic flux ratio 5F - 0, and the liquid water reservoir and most of the surface advection-diffusive layer become homogeneous. The response of the effective fractionation factor & to the dimensionless parameter N is plotted in Figure Application to Various Limiting Situations -2O -4O E -6O -80 ' log,on Figure 4. Variation of 5Dc(L -) (delta value of liquid water at surface) with dimensionless parameter N - ul /nd, for relative humidity h = 25%, 50%, and 75%, respectively. Other parameters are the same as those in Figure 2. The thin solid line represents the isotope ratio of the liquid reservoir, 5Do per mil Slow Water Evaporation: N- 0 Limit Let us now consider the isotopic fractionation by evaporation from an open water surface such as the ocean. Mathematically, this is equivalent to the limit of N - 0. The length scale of the subsurface diffusive skin layer is 12 = I D/u -- oo, owing to the small evaporation rate u. Providing sufficient energy to the liquid water, the rate of evaporation is mainly dependent on the humidity above the interface [Stewart, 1975]. To approach the /2 - c limit, the air above the surface must have high humidity or be slightly undersaturated with respect to the surface temperature. This allows evaporation to take place at the surface at a slow rate. Since N is inversely proportional to u and L, there are two situations to be considered for the N - 0 limit. The first limit assumes the extreme situation that the net evaporation is shut down at the surface, which corresponds to the situation where the air above the

9 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL 18,627 surface is saturated. This gives h - 1, u - 0, and N - 0. With the approximation of e -N N, the effective fractionation factor defined in (45) now becomes which simply states the equilibrium fractionation relationship at the air-liquid interface. It should be noted here and in sections 5.2 and 5.3, however, that the definition of the effective fractionation factor & is different from the definition of the equilibrium fractionation factor ae. As the air at z - Lx +L2 is saturated, the thickness of the diffusive layer L2 essentially becomes zero since a gradient zone above the interface disap- pears. Therefore vapor at z - Lx + L2 is in direct contact with the surface and must have the equilibrium value; that is, 5b must satisfy 50+1 which is normally not equal to a and is understandable because the isotopic composition of surface liquid is the same as that of the liquid reservoir due to small L1 or large œ. From (37) the flux ratio Fi/F now becomes 1 F i K jrsmow F (1/a )(5o + 1) - h(jb + 1) (47) I - h Equation (47) is identical to equation (9) of Merlivat and Jouzel [1979] for the isotope concentrations in the evaporating water vapor as a function of the relative humidity and isotope concentrations in the liquid water and the atmospheric water vapor Strong Evaporation rom Trees: N - +c Limit For the strong evapotranspiration from the plant leaves, the diffusive layer depth is probably only of the order of the thickness of the leaves, whereas the liquid reservoir in the soil is far away from the leaf part. Such a condition indicates a small value of œ and a very large value of Lx. Although (30) indicates that u has an upper limit Umax - nq*/l., N can still approach infinity if Lx -- oe or nd Since e -N - 0 when N - +c, from (45) we have &_ a [(1-h K )(( ( :+1) + hi so & is dependent only on h, 50, and 5. The flux ratio now becomes F i -- - so- jrsmow(5o + 1) which states that the flux ratio is identical to the isotope ratio of the liquid water reservoir at the lower boundary when N is sufficiently large. This helps to address the difference between the nonfractionated isotopic flux ratio and the nonfractionated isotopic concentration during evaporation and transpiration as is pointed out by He [1998] and He and Smith [1999], which has been If one must maintain a nonequilibrium value for the flux ratio F /F under the limit of N - 0 will not have a finite value, since the flux F does not have to vanish as the flux of H2 60 approaches zero. This renrather ambiguously illustrated in previous studies. Under the N - +oe limit, it is not the isotope ratio of ders that F /F may go to +c or -c, depending on the vapor but the flux ratio that should be equal to the the sign of the HD160 (or H.loO) flux. If lighter air sits isotope ratio of the water reservoir. Trees are able to above the diffusive layer, then from (37) the flux ratio will approach +c, and vice versa. A more general consideration is to assume that air above the diffusive layer is not saturated, whereas the thickness of the actual advective-diffusive layer œ1 is very small. This allows h 0 while N - 0. This situation is more applicable to the atmosphere, since actual atmospheric humidity rarely reaches 100% above the Earth's surface. It also allows the existence of the difproduce such "heavy" flux ratios because they are characterized by large values of N; that is, the groundwater is so far away from the leaf part as the height of each tree is much greater than the depth of the diffusive liquid layer beneath the leaf surface. Figures 2, 3, and 4 illustrate an example of how the flux ratio and isotope ratios at the interface vary with the value of N Vapor Condensation on Ice: N - -c Limit fusive layer above the interface. From (45) the effective fractionation factor is The fractionation process occurring at the liquid-solid a 50+1 interface during freezing has been studied by Souchez et (46) al. [1987], in which the influence of advection and dif- 5v+l fusion in the liquid skin layer is taken into account. We now consider the condensation of water from gas phase to ice phase. Because of their lower molecular diffusivities in the air, isotope molecules tend to condense more slowly than H2160 does. Jouzel and Merlivat [1984] formulated the kinetic isotopic effect for vapor deposition on ice. They expressed the kinetic effect using a ki- netic fractionation coefficient which takes into account the supersaturation of air over ice and the differential diffusivity between normal water molecules and isotopic water molecules. In this advective-diffusive model, the case of vapor deposition on ice can be included by allowing a downward advection u ( 0. If the vertical coordinate is fixed at the depositing interface, the deposition of water vapor will cause an effective downward motion of the ice condensed. Since vapor condensation occurs when air above the ice is supersaturated, the mixing ratio of vapor at the top of the diffusive layer (region 3) must

10 18,628 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL therefore be larger than the saturation mixing ratio at the vapor-ice interface, which means that Table 1. Summary of the Solutions to the Advective- Diffusive Model h > I Quantity Solution at z = L14. L2 and (h- 1) now becomes the so-called supersaturation as defined by Jouzel and Merlivat [1984]. Because of the lack of mobility of isotope molecules in ice phase, the molecular diffusivity of HD160 (or H2180) in ice phase is essentially negligible such that v(Lx+) 4.1 General Solution nd 0 therefore the dimensionless parameter N when u < 0 is N - ul c I D and subsequently e -N The isotope fractionation factor between the deposited ice and the source vapor above the diffusive layer can be obtained from (45) as h &- c e (c e/k)(h- 1)+ 1 (48) which indicates that & is only the function of the humidity h above the ice phase. Equation (48) is identical to equation (11) of Jouzel and Merlivat [1984]. Therefore the situation of vapor condensation on ice is only a special case in this advective-diffusive model. Under the limit of N - -c the flux ratio of isotopic water to normal water now becomes, from (37), F i h(sb 4-1) -- - jrsmowk (49) F -(1 - h)+ K/c Taking the same limit for (43), which is the isotope ratio of the condensation at the interface, we get h(5b 4-1) - (50) 5c(L 1 ) > K_( 1 _ h) + K/c Comparing (49) with (50), it is clear that the flux ratio is equal to the isotope ratio of the surface condensation, as I F i 6c(L -) + 1 = jrsmow F which is also expressed in equation (8) of Jouzel and Merlivat [1984]. 5c(L ) 4. 1 (L ) 5 +1 ( +) + 1, (L ) + 1 (L ) 5 +1 ( +) + 1, (Lr) + 1 a(œ ) v( +) + 5c(L ) 4.1 (L ) c e [Sv(Lx+) 4-1] - (1- h)(5o 4-1) 4- Kh(5v 4-1)(e -N - 1) [-(1 - h)e - v 4- (K/o )(e-n - 1)](Jr 4-1) Slow Evaporation (N ) K(1/c )(5o 4-1)- h(5v 4-1) 1-h I (504-1) O/e o+1 5v+1 Strong Evaporation (N -- +cx ) 5o+1 (1 K - h )( o 4-1) 4- h(5v 4-1) cr [Sv(Lx +) + 1] O/e K )(50 Jr+l) 1 + hi Vapor Condensation (N -- -cx ) K h(5v + 1) -(1 - h) 4- K/ere Kh(5b 4-1) - c (1- h) 4. K 5F4-1 ( /K)( - ) Summary Table I summarizes the general solutions and the solutions under the limits of N - 0, N - +c, and N - -c in this advective-diffusive model. All of th solutions in Table I are functions of only three dimensionless parameters: N, h, and (Jr + 1)/(50 + 1) as f N,h, (50+1) (Jr + (51) Equation (51) indicates that the isotopic behaviors of the evaporation-condensation system are dependent on not only the evaporation mechanisms of the system (represented by N) but also on the conditions of the atmosphere above the diffusive layer in terms of its humidity h and isotope composition (Jr + 1)/(5o+ 1). However, in the three extreme cases we have discussed, the isotopic behaviors are no longer dependent on N.

11 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL 18,629 Table A1. List of Symbols Used in the Advective-Diffusive Model Symbol A A' B B' C D i. F F i h J K L L2 N P Q Qi Qo Q. Qi. R 8 80 U w z i i I D o 0 P pi Variable constant in the general solution of Q(z) constant in the general solution of Q(z) constant in the general solution of Qi(z) constant in the general solution of Qi(z) constant in the general solution of s(z) constant in the general solution of s(z) vapor pressure of HD 60 (or H21SO) in equilibrium water vapor net flux of H2160 net flux of HD160 (or H SO) relative humidity defined as Qb/Q* ratio of the molecular weight of HDi O (or H lso) to H21 O I i / l I'i; D / U thickness of the advective-diffusive surface liquid layer thickness of the diffusive surface air layer ul1/nm L1/L mixing ratio of H 1 O mixing ratio of HDi O (or H lso) Q at z = L1 + L H i o saturation mixing ratio at surface temperature HDi O (or H lso) mixing ratio in equilibrium water vapor isotope ratio in terms of molar ratio jr of the condensed phase satz=o advection velocity (evaporation rate) vertical coordinate fractionation factor between liquid and vapor equilibrium fractionation factor effective fractionation factor (i.e., [Sc(L ) + 1]/[5 + 1]) molecular diffusivity of H21 O in air molecular diffusivity of HDlOO (or H2180) in air molecular diffusivity of HDlOO (or H2180) in liquid or solid H21 O delta value of water vapor or liquid water 5atz=0 5 at z = Ll + L2 (1/j RsMow )(F i/f) exchange rate of HDlOO (or H2180) at the surface density of liquid H21 O density of liquid HDlOO (or H2180) exchange rate of H21 O at the surface 6. Conclusions ation during evaporation or condensation are analyzed under various limits of the model by adjusting several In this paper, we have developed a one-dimensional dimensionless parameters. The most important paramadvective-diffusive model for the evaporation process across an evaporating surface. This model is based on the Craig-Gordon linear evaporation model. We imeter is N = ul1/rid, which takes into accounthe evaporation rate u and the depth of the advective-diffusive layer. These parameters may cause the difference of proved the model by allowing advection and diffusion isotope ratio in evaporation between such evaporating to exist in the skin layer beneath an evaporating surface. The model is based on Henry's law and Fick's law and the assumption of equilibrium fractionation at the liquid-gas interface. We find that diffusion of isotopes in both the atsurfaces as forest and open water surfaces. Under the limit of rapid surface exchange, i.e., large values of qo and 0, the water vapor is always in isotopic equilibrium with the liquid water at the interface. Under the slow evaporation limit, i.e., N - 0, the back flux from mosphere and the condensed phase plays an essential the atmosphere to the liquid water reservoir is not negrole in controlling the isotope flux ratio 5F, which is ligible compared with the upward flux; hence the net a measurable quantity. Behaviors of isotopic Ëaction- flux ratio must take into account such information from

12 18,630 HE AND SMITH: ADVECTIVE-DIFFUSIVE ISOTOPIC MODEL the atmosphere as the mixing ratio and isotope ratio. When N is large, the back flux becomes negligible, so the upward flux dominates the net flux and brings only the information of the liquid reservoir into the net flux; therefore the flux ratio approaches the isotope ratio of the source liquid water reservoir. The model also helps to unify the independent theoretical results from earlier studies. Vapor condensation onto ice is also included as a special case in the model. Appendix: List of Symbols For reference, the mathematical symbols used in this advective-diffusive model are listed in Table A1. Acknowledgments. We are grateful to Danny M. Rye of the Department of Geology and Geophysics of Yale University, Xuhui Lee of the Yale School of Forestry and Environmental Studies, and Donald E. Aylor of the Connecticut Agricultural Experimental Station for their thoughtful discussions. We also thank the two anonymous reviewers for their useful comments that helped to improve the quality of this paper. This research is supported in part by NSF research grant ATM References Alberty, R. A., Physical Chemistry, 943 pp., John Wiley, New York, Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 780 pp., John Wiley, New York, Brutsaert, W., A theory for local evaporation (or heat transfer) from rough and smooth surfaces at ground level, Water Resour. Res., 11, , Craig, H., and L. I. Gordon, Deuterium and oxygen 18 variations in the ocean and the marine atmosphere, in Stable Isotopes in Oceanographic Studies and Paleotemperatures, edited by E. Tongiorgi, pp , Lab. Geol. Nucl., Pisa, Italy, Dincer, T., The use of oxygen-18 and deuterium concentrations in the water balance of lakes, Water Resour. Res., (, , Friedman, I., L. Machta, and R. Soller, Water-vapor exchange between a water droplet and its environment, J. Geophys. Res., 67, , Gat, J. R., Environmental isotope balance of Lake Tiberias, in Proceedings of the IAEA Symposium on Isotope Hydrology, pp , Int. At. Energy Agency, Vienna, Austria, He, H., Stable isotopes in the evaporating atmospheric water vapor, Ph.D. dissertation, 234 pp., Yale Univ., New Haven, Conn., He, H., and R. B. Smith, Stable isotope composition of water vapor in the atmospheric boundary layer above the forests of New England, J. Geophys. Res., 10, 11,657-11,673, Jouzet, J., Isotopes in cloud physics: Multiphase and multistage condensation processes, in Handbook of Environ- mental Isotope Geochemistry, vol. 2, edited by B. P. Fritz, and J. C. Foutes, pp , Elsevier, New York, Jouzel, J., and L. Merlivat, Deuterium and oxygen 18 in precipitation: Modeling of the isotopic effects during snow formation, J. Geophys. Res., $9, 11,749-11,757, Jouzel, J., G. L. Russell, R. J. Suozzo, R. D. Koster, J. W. C. White, and W. S. Broecker, Simulations of the HDO and H2 so atmospheric cycles using the NASA GISS general circulation model: The seasonal cycle for present-day conditions, J. Geophys. Res., 92, 14,739-14,760, Jouzel, J., R. D. Koster, R. J. Suozzo, and G. L. Russell, Stable water isotope behavior during the last glacial maximum: A general circulation model analysis, J. Geophys. Res., 99, 25,791-25,801, Merlivat, L., The dependence of bulk evaporation coefficients on air-water interfacial conditions as determined by the isotopic method, J. Geophys. Res., $3, , Merlivat, L., and M. Coantic, Study of mass transfer at the air-water interface by an isotopic method, J. Geophys. Res., $0, , Merlivat, L., and J. Jouzel, Global climatic interpretation of the deuterium-oxygen 18 relationship for precipitation, J. Geophys. Res., 8, , Souchez, R., J.-L. Tison, and J. Jouzel, Freezing rate determination by the isotopic composition of the ice, Geophys. Res. Left., 1, , Stewart, M. K., Stable isotope fractionation due to evaporation and isotopic exchange of falling waterdrops: Applica- tions to atmospheric processes and evaporation of lakes, J. Geophys. Res., $0, , Zimmermann, U., and D. H. Ehhalt, Stable isotopes in the study of the water balance of Lake Neusiedl, Australia, in Proceedings of the IAEA Symposium on Isotope Hydrology, pp , Int. At. Energy Agency, Vienna, Austria, H. He, Joint Center for Earth Systems Technology, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD (he@umbc.edu) R. B. Smith, Department of Geology and Geophysics, Yale University, P.O. Box , New Haven, CT (ronald.smith@yale. edu) Gat, J. R., Oxygen and hydrogen isotopes in the hydrologic (Received February 18, 1999; revised May 11, 1999; cycle, Annu. Rev. Earth Planet. Sci., 2, , accepted May 14, 1999.)