Craig-Gordon Evaporation and the Isotopic Composition of Leaf Water 1. Definitions All delta s are expressed as: (1) δ = R Multiply this delta by 10 3 to express it in per-mil units. Fractionation factors are expressed in the same way: (2) ε = α In the following equations it is assumed that the isotope ratio, R = N i, can be substituted for the N fractional abundance, F = N i N + N i. This assumption is justified when N i <<N, as is the case for 2 H- 1 H, or isotope differences are small, as is true for 18 O- 16 O (Sessions & Hayes, 2006). 2. Equilibrium Vapor-Liquid Fractionation Equilibrium fractionation is expressed as: (3) α * (T) = R L R V The vapor phase is depleted in heavy isotopes so values for α are greater than 1. Caution is necessary because this fractionation factor is sometimes defined as R V /R L and is thus less than 1. The superscript asterisk to the right of α is used to indicate an equilibrium isotopic fractionation (versus kinetic which gets a subscript K). 1
3. Craig-Gordon Evaporation 3.1. Assumptions In the Craig-Gordon evaporation model there is a diffusive sublayer above the water surface, above which is a turbulent sublayer and then the free atmosphere. At the base of the diffusive sublayer the air is saturated (relative humidity equals 1) and the water vapor is in isotopic equilibrium with the underlying liquid. This model is based upon the Langmuir linearresistance model for evaporation. The assumptions are (Gat, 1996): 1. equilibrium conditions at the air/water interface so that relative humidity h=1 and R V =α R L 2. a constant vertical flux, i.e. no divergence or convergence in the air column 3. no isotope fractionation during fully turbulent transport Figure 1 The Craig-Gordon evaporation model (drawn after Gat, 1996). 2
3.2. Evaporation Flux The flux equations for the abundant (E) and rare isotopes (E i ) of water are: (4a) (4b) ( E = C S C A ) ( E i = C SR S C A R A ) = C ( 1 h ) S ( ) i = C S R S hr A i where subscripts S and A refer to the saturated layer of water vapor immediately above the air/water surface and the free atmosphere respectively. Equations 4a and 4b describe the evaporation flux of water from the liquid to the atmosphere. The terms and i called resistance terms, and describe the rate that water isotopomers are transported from the saturated layer into the overlying free atmosphere (see section 4 on kinetic effects below). The isotopic ratio of the evaporating moisture, R E, is equal to the ratio the fluxes of the rare and abundant water isotopomers (E i /E). We can also define the isotopic ratio of water vapor in the saturated layer (R S ) in terms of the liquid water R S = R L / α. Dividing 4b by 4a and using this definition yields an equation for the isotopic composition of the evaporating moisture: (5) R E = E i E = i R L 1 h Defining the kinetic isotope effect (see section 4 on kinetic effects below) and solving for / i gives: (6a) = (1 h) i i = (1 h) +1= + (1 h) (1 h) (6b) 1 h = i + (1 h) Substituting Eq. 6b into Eq. 5 gives: 3
(7) R E = R L In δ units (δ=r-1), equation (7) becomes: ( ) α * h( δ A +1) δ E = δ L +1 δ E = δ L α * +1 α * hδ A h + h (8) δ E = δ L α * hδ A ε * α * 4. The Kinetic Isotope Factor The resistance terms, and i, are the sum of turbulent (T) and diffusive (M) resistances for the isotopomers of water: (9a) (9b) = M + T i = i,m + i,t These terms describe the rate that isotopomers are transported away from the saturated layer by diffusion and mixed into the free atmosphere by turbulent mixing (turbulent mixing can be described in a similar way to diffusion mixing, but the effective diffusivities are much higher). The term ( i /) 1 in eq. 6a can be evaluated from these equations as (Gat, 1996): i = i,m + i,t M + T = i,m M + T + i,t M + T i = i,m M M + i,t T T i = i,m M M + i,t T + M T T = i,m M M M + i,t T T T 4
(10a) i = M i,m M + T i,t T It is assumed that turbulent transport is non-fractionation, therefore i,t = T and Eq. 10a becomes: (10b) i = M i,m M and the kinetic isotope factor (Eq. 6a) becomes: (11) = (1 h) M i,m M For a fully developed diffusion layer, M is proportional to D -1 M, where D M is the molecular diffusivity of water in air. For a rough interface under strong wind conditions, the transient eddy model of Brutsaert (1965) can be applied with M proportional to D -1/2 M. At moderate wind -2/3-1/2 speeds a transition from the proportionality of D M to D M can be expected (Merlivat & Contiac, 1975). The ratio of the diffusivities in air of H 18 2 O/ H 16 2 O and 1 HDO/ 1 H 2 O have been experimentally determined by Merlivat (1978) as 0.9723 and 0.9755 respectively (Merlivat, 1978). (Note that especially for 1 HDO/ 1 H 2 O, this term differs from that calculated by the kinetic theory of gases.) Defining C D as: (12) C D = D M D i,m, re-writing i,m / M 1 from Eq. 11 in terms of diffusivities and substituting Eq. 12 gives: (13a) i,m M = ( D i,m ) q D M ( ) q = D i,m D M q This can be rewritten as: 5
(13b) i,m M = nc D with n defined as: (14) n = D i,m D M q D i,m D M Values for n depend upon the diffusion exponent and are ~0.5 for turbulent conditions (q = 1/2), ~0.66 for moderate windspeeds (q = 2/3) and 1 for a fully developed diffusion layer (q = 1). The ratio of diffusive to total transport ( M /) in Eq. 11 can be defined in terms of the relative humidity of the free atmosphere (h) and that of the boundary between the turbulent and diffusive transport layers (h ): (15) θ = M ( 1 h' ) = 1 h For small lakes whose evaporation does not affect the free atmosphere, θ is equal to 1, while for large lakes and the ocean a value 0.5<θ<1 is appropriate. Equations 13b and 15 can be substituted into Eq. 11 to define the kinetic isotope term: (16) = (1 h)θnc D 5. Enrichment of Leaf Water from Transpiration A simple model for the evaporative enrichment of leaf water uses Craig-Gordon evaporation to describe the isotopic composition of liquid water and water vapor at the site of evaporation. At steady state, the isotopic composition of water evaporating from a leaf must equal that of water flowing into the leaf from the xylem. This model calculates an enriched leafwater pool (EL), presumably at the water/vapor boundary. The bulk leaf water (BLW) consists of a mixture of this enriched pool and an unaltered pool of xylem (X) water: (17) R BLW = f l R EL + (1 f l ) 6
where f l is the fraction of enriched leaf water. Defining the kinetic fractionation factor (α K ) as: (18) α K = D i,m D M q where q is identical to Eq. 13. From Eq. 11 and 15, and noting that θ=1 for leaf evaporation (Eq. 15), the kinetic enrichment for leaf evaporation (ε K ) can be written in terms of α K as: (19) = (1 h)θ( α K ) = (1 h) ( α K ) Starting from the Craig-Gordon equation for evaporating moisture (Eq. 7), substituting α K for (Eq. 24) and noting that at steady state, the isotopic composition of evaporating water must equal that of the incoming xylem water ( =R E ): = R EL ε K R = EL (1 h) α K ( ) = R EL 1 h ( 1 h)α K = R EL (20) R EL = α * [( 1 h)α K + hr A ] Using rather than α K gives: (20a) R EL = α * [ ( ε K ) + hr A ] In delta notation, this becomes: (20b) δ EL +1= α * ( 1 h)α K δ X +1 ( )α K [ ( ) + h( δ A +1) ] Pratigya Polissar Aug., 2008 7