Analysis of rate-limiting processes in soil evaporation with
implications for soil resistance models
Thambirajah Saravanapavan
a, Guido D. Salvucci
b,*a
Department of Geography, Boston University, 675 Commonwealth Ave., Boston, MA 02215, USA
b
Departments of Earth Sciences and Geography, Boston University, Boston, MA 02215, USA
Received 15 March 1999; accepted 7 September 1999
Abstract
Numerical integrations of coupled equations of moisture, vapor and heat diusion in soil are analyzed to explore the relative roles of vapor and liquid ¯uxes in rate-limiting the transfer of water to the soil±atmosphere interface. Approximate analytical in-tegrations of a simpler isothermal system are then introduced to explore the interactions of vapor and liquid transport. Although vapor diusion dominates total moisture ¯ux near the soil surface, both models indicate that liquid transport deeper in the soil limits evaporation at daily time scales for all but very dry soils. The rate-limiting role of the liquid ¯ow is demonstrated by the insensitivity of the integrated-coupled equations to the molecular diusivity of water vapor (Da). The mechanism underlying the insensitivity is that the depth of the drying front (L) shrinks in order to compensate for reductions inD
ain such a way that the capillary rise to the drying front can still be transported to the surface. The feedback mechanism that causesLto shrink is the mass imbalance that
would occur for reduced vapor transport out of the drying front and essentially unchanged liquid transport into it. Implications for the utility and interpretation of soil resistance terms (de®ned as proportional to the ratio ofLtoDa) employed in SVAT models are
discussed. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction
Evaporation occurs when liquid water is converted into water vapor and transported in this form into the atmosphere. In the case of soil evaporation, this con-version may take place at or below the soil surface [22]. Ecient modeling of this and other processes in the unsaturated zone remains a signi®cant challenge in hy-drology and meteorology.
One approach to this problem, based on analogies to electrical resistance networks, simpli®es the complexity of soil evaporation with a term representing soil resis-tance to vapor diusion (e.g. [4,6,10,11,15,16,19,28]). Chanzy and Bruckler [5], Wallace [29], and Salvucci [25] provide recent reviews of the relative merits and draw-backs of resistance-based and alternative approaches to modeling soil evaporation. Most of the above citations document diculties associated with the de®nition, im-plementation, uniqueness, and formulation (e.g. mois-ture dependence) of resistance terms. With the goal of
understanding why the resistance approach can be problematic, the present study explores the role of vapor diusion in rate-limiting soil evaporation.
1.1. Numerical and analytical analysis of moisture ¯ow
Numerical models have been widely used to explore ¯ow processes in the unsaturated zone. In such an ap-proach the underlying physics of the transport process-es, for example gradient ¯ow laws within a thin layer of soil, are assumed to be well known. The model is used only to explore the system behavior that results from the interaction of many such layers. Of particular impor-tance are the resulting net surface ¯uxes.
In the analyses below, a numerical model [17,18] is used to simulate water and heat ¯ow in soil through integration of a matric potential based ®nite-element formulation of the Philip and de Vries [24] and de Vries [7] (henceforth PdV) framework. This framework in-cludes both thermal and isothermal liquid and vapor ¯uxes. Next, a simpli®ed analytical model of isothermal liquid and vapor ¯ow is derived. This model divides the unsaturated zone into three dynamic layers: a dry sur-face layer, a moist layer, and a layer at the initial *
Corresponding author. Tel.: 8344; fax: +1-617-353-8399.
E-mail address:[email protected] (G.D. Salvucci).
moisture content (i.e. not yet in¯uenced by the presence of surface evaporation). The analytic form of the simple model helps isolate the relative roles of liquid and vapor transport in limiting soil evaporation. Furthermore, it provides insight into why soil resistance terms may be ill de®ned to constrain soil water evaporation.
1.2. Previous investigations and limits of present study
One clear shortcoming of any analysis based on the numerical or analytical integration of presumed process physics equations (in this case the PdV framework) is that unconsidered physical processes may be missed al-together. This is especially relevant herein because ®eld and laboratory evidence for the adequacy of the PdV framework is somewhat mixed. A few selected studies that report on the adequacy of PdV, review prior liter-ature critically, and discuss the relative role of vapor and liquid transport in evaporation, are reported below.
An early review of the evolution of the PdV frame-work (including developments and modi®cations) is provided by Milly [17], who recast the PdV transport equations in terms of matric potential and developed a ®nite-element model for integration. In a further review, Milly [18] found that the literature on applications of PdV to ®eld and laboratory studies mostly supported the framework, with the relative importance of thermal and isothermal liquid and vapor ¯ows dependent on the particular experimental conditions and soil depths con-sidered. To explore the issue of relative importance further, he applied his numerical model to a set of wetting and drydown conditions. From these experi-ments he concluded that the impact of including vapor processes in soil evaporation is minor with respect to predicted surface evaporation at daily or longer time scales, even though vapor transport may dominate at given times and depths within the soil.
In general terms Milly's [18] ®ndings are consistent with both: (1) Gardner's [12] analysis, which showed good agreement between predictions based on ¯ow equations that neglected the vapor transport mechanism and laboratory measurements of drying columns; and (2) the column drying experiments by Hanks et al. [13], in which the contributions of vapor transport were in-directly controlled (for a silt loam, a sand loam and a sand) by adjusting either the wind speed, which cools the surface and induces upward thermal vapor transport, or the radiation intensity, which heats the surface and in-duces downward thermal vapor transport. The 40 day cumulative evaporation totals for each treatment were within about 10% of each other, with the more ecient evaporating condition dependent on soil texture.
Later work with Milly's [17] model includes the study by Scanlon and Milly [26], who evaluated and explained liquid and vapor ¯uxes in a shallow layer of unsaturated soil in the Chihuahuan Desert of Texas. In this
vapor-dominated arid environment, the PdV framework was found to be adequate to capture the seasonal dynamics of measured temperature and matric potential gradients. Numerical simulations based on the PdV framework also agree reasonably well with the results of the ®eld experiment by Yamanaka et al. [31].
More recently, however, Cahill and Parlange [2] demonstrated through a ®eld experiment that changes in moisture content due to vapor ¯ux convergence could not be predicted by the PdV framework. Speci®cally the predicted moisture changes in a 7±10 cm layer were too small in magnitude and opposite in sign compared to those estimated as a residual of the energy and mass balance equations. Furthermore, Parlange et al. [3] showed that this disagreement could be explained by convectively enhanced water vapor transport.
With the caveat that the PdV framework may be in-complete, we present below an analysis of rate-limiting processes of soil evaporation with a focus on the relative roles of vapor and liquid transport. The analysis is similar in its goal to that of Milly [18], but builds upon his results by identifying, through a simpli®ed model, some of the feedback mechanisms responsible for the relative insensitivity of evaporation to vapor transport in moist soils.
2. Methodology
2.1. Numerical simulations under meteorological forcing
A numerical experiment with two dierent water va-por molecular diusivities (Da 0.27 cm2/s andDa/10) was conducted to explore the sensitivity of soil evapo-ration to water vapor transport in soils. In this experi-ment, one year (1983) of ¯uxes was simulated using Milly's [17,18] model, including both thermal and iso-thermal vapor and liquid ¯uxes. The hourly climate forcing data [21] (wind speed, radiation, precipitation, air temperature, and air humidity) are from Jackson-ville, Florida. Milly's [17,18] model uses these data to couple the boundary layer and the soil via energy and water balance of the ¯uxes of sensible heat, latent heat, radiation, ground heat and precipitation. In this simple representation, the boundary layer is prescribed and does not change in response to the soil moisture and thermal state. The hydraulic parameters used for the simulation represent the soil water retention and con-duction of a silty-clay and a loamy-sand soil measured by Scanlon and Milly [26]. Details of the retention and conduction models, as well as the parameters used, are summarized in Table 1.
water vapor molecular diusivities Da and Da/10. The RMSE is de®ned as follows:
RMSE 1
To determine the sensitivity at dierent time-scales, the simulated ¯uxes were aggregated as E1
T
PtT=2 tÿT=2Edt, whereTde®nes the time scale of interest (e.g. 24 h).
2.2. Simpli®ed analytical model
A second analysis was designed to explore the inter-action of liquid and vapor transport mechanisms in soil evaporation. A simple analytical model was derived which approximately captures both the movement of the drying front as soil evaporation progresses and the contribution of vapor and liquid transports. The sensi-tivity to water vapor diusivity of both the drying front depth and of the contributions of vapor and liquid ¯uxes was examined for the same two soils as above. In order to test the precision of the simple model, the analytic integrations were compared with those using the ®nite element model of Milly [17,18].
To simplify the problem, gravity and non-isothermal transports were neglected. The soil column was initially set at 83% saturation and let dry for 500 h with the surface maintained at a constant relative humidity of 0.2. Holding the surface at a known humidity to draw moisture out of the soil de®nes the soil water ex®ltration capacity [8,12]. By studying only this condition, the analysis ignores the so-called ®rst stage of evaporation, which is limited by neither soil liquid nor vapor trans-port, but rather by available atmospheric transport ca-pacity [22]. Furthermore, the simulations for dierent initial soil saturations with the same surface forcing were
carried out in order to examine the sensitivity of vapor transport to dierent drying conditions.
2.3. Model derivation
The total ¯ux of moisture in a porous medium can be expressed as the sum of liquid ¯ux and vapor ¯ux. The liquid ¯ux (ql, LTÿ1) is modeled by the Buckingham± Darcy equation, which can be written (neglecting grav-ity) as
ql ÿK ow
oz: 2
In (2), K (LTÿ1) is the unsaturated hydraulic conduc-tivity andw(L) is matric potential.
The vapor ¯ux relation (qv, LTÿ1) is based on the PdV framework as presented in Milly [17]
qv water vapor molecular diusivity; h (dimensionless) is the volumetric water content; X (dimensionless) is the tortuosity of the air-®lled pore domain, which in Milly's [18] analysis is set to nÿh2=3;n(dimensionless) is the soil porosity; ha (dimensionless) is the volumetric air content (equal to (nÿh)); andql(MLÿ3) andq
v(MLÿ3) are densities of water in the liquid and vapor phases respectively.
The vertical soil column is divided into two zones: a vapor-¯ow dominant zone (zone-1) and liquid-¯ow dominant zone (zone-2) (Fig. 1). The depth of zone-2 (L0(t)) extends from the bottom of zone-1 to the depth at which the matric potential is still equal to its initial value (w0), i.e. the penetration depth of the changed surface boundary condition. The depth of zone-1 (L(t)) extends from the surface, where the matric potential (wsurf) and Table 1
Hydraulic parameters of soils used in this study [26]
Parametersa Silty clay Loamy sand
Saturated hydraulic conductivity (Ks, cm sÿ1) 3.20´10ÿ6 3.70´10ÿ3
aThe main drying curve was determined by Scanlon and Milly [26] by ®tting experimental data on soil water desorption to:
hdminfhu;hu w=a b
ÿ ÿ105=ab
c5ÿlog ÿwg, wherewis in meters,huis the water content obtained upon rewetting, anda,b, andcare
®tting parameters. The unsaturated hydraulic conductivity was calculated by integration of Mualem's [20] expression as
vapor density (qsurf) are held constant, to the depth at which isothermal vapor and liquid transport are equally eective. As such L(t) locates the (arbitrarily de®ned but nonetheless useful) drying front.
With this de®nition, L(t) is located where the matric potential takes on the critical value (w) for which liquid and vapor transports due to a unit gradient of matric potential are equal. Following Milly [17], the latter can be expressed by multiplying (3) by the derivative of z with respect to w. The critical potentialw can thus be found by solving:
The dependence ofqvon matric potential (w), saturation vapor density (qs), and temperature (T) is modeled from the thermodynamic relation of Edlefson and Anderson [9]
qv w;T qs Texp wg=RvT 5
In (5), g (LTÿ2) is the acceleration of gravity and R v (L2Tÿ2 Hÿ1
) is the gas constant for water vapor. Approximating the derivatives in (2) and (3) as ®nite dierences, the total ¯uxes in zone-1 (q1) and zone-2 (q2) can now be expressed as
q1
In review, wsurf and qsurf are, respectively, the matric potential and water vapor density at the surface,wand
q are matric potential and water vapor density at the drying front, and w0 and q0 are the initial matric po-tential and water vapor density.L(t) andL
0(t) are the depths of zone-1 and zone-2 at a time t (Fig. 1),
re-spectively, and K1 and K2 are the eective hydraulic conductivities over each zone. In the results presented below, the conductivities are approximated from the following expressions, which are exact for steady-state horizontal ¯ow [30]:
Conservation of mass for the model system (Fig. 1) can be expressed with zas the dependent variable (fol-lowing [23]) as: (8b) represent the dynamic water stored in zones 1 and 2 combined, and zone 2, respectively. Zeroth order ap-proximations of these integrals would be given by trap-ezoidal integration as L h0ÿh 12L hÿhsurf
approximations were found to generate considerable error as compared with high-resolution numerical solu-tions. To address this error in a manner consistent with the method for approximating eective conductivities, the moisture pro®le dynamics in each zone are approxi-mated as a series of successive steady states. The pre-sumed steady moisture pro®les (z(h;q)) are found by integrating (2) and (3) using the chain rule, the soil water retention curve, and Eq. (5) to relatewandqvtoh. Be-cause the limits of integration are ®xed according to Eqs. (8a) and (8b) (see also Fig. 1), this integration needs to be done only once. If the resulting integrals for zones 1 and 2 are divided by the zeroth-order trapezoidal approxi-mations above and denotedv1andv2, Eqs. (8a) and (8b)
C ÿ1 v2 h0ÿh
w
ÿw0K2
qÿq
0
ql
Da nÿh25=3
: 12
The simultaneous solutions of the coupled dierential Eqs. (9a) and (9b) are:
Lbt1=2; 13a
L0ct1=2; 13b
where
b 8Cÿ4Ac
2
4B
1=2
14
and
c 8C
ÿ8ABÿ4C ÿh 4Aÿ8Cÿ8AB2
ÿ64 ÿB 1BA2i
1=2
2 Bÿ1 1=2
: 15
With Eqs. (7), and (13a)±(15), the ¯uxes in (6) can be quanti®ed.
This conceptual framework for the movement of the drying front and its in¯uence on evaporation is based in part on Monteith [19] and Choudhury and Monteith [6], who modeled the depth of the dry layer to be propor-tional to cumulative evaporation. A key dierence here is that the growth of the drying front due to evaporation is oset by the (mostly liquid) ¯ux entering from below. Brisson and Perrier [1] also proposed a model along these lines, although they later neglected the in¯uence of the entering moisture. The critical role played by the incoming water is discussed in detail later in this paper.
3. Results and discussion
3.1. Sensitivity of simulated evaporation to vapor diu-sivity at various time-averaging scales
The numerical simulation experiment explores the sensitivity of soil evaporation to vapor transport at various time scales (hourly, daily, and monthly). The experiment is designed after that by Milly [18], where the role of vapor transport was analyzed by comparing simulations with and without thermal and isothermal
vapor diusion. The present study analyzes vapor transport by comparing the simulations with two dif-ferent water vapor molecular diusivities (Da and Da/ 10).
The climate conditions [21] at Jacksonville, FL during 1983 yielded an average precipitation rate of 3.85 mm/ day and a maximum, minimum, and average tempera-ture of 36.1°C, 11.1°C, and 18.7°C respectively. With the
above-mentioned climate forcing applied to a silty-clay soil and a loamy-sand soil (Table 1), the numerical simulations predicted that soil-controlled (or stage two) evaporation [14] occurred approximately 52% and 64% of the time, respectively. This condition is appropriate to study the limiting processes of soil evaporation because stage-two evaporation is limited by neither available energy nor atmospheric transport capacity, but rather by moisture diusion within the soil [22].
The root mean square error between evaporation predictions using Da and Da/10 and the fraction of RMSE relative to the average evaporation are given in Table 2. The RMSE values for hourly averaged evapo-ration for silty-clay and loamy-sand are about 11% and 21% of mean evaporation, respectively. The error is mostly attributable to the dierences in thermally induced downward vapor ¯ux during mid-day and up-ward vapor ¯ux at night. The RMSE values for daily mean evaporation are approximately 3% and 10% of the mean evaporation for the silty-clay and loamy-sand soils, respectively. For monthly mean conditions, these values are reduced to approximately 2% and 4%, respectively. The small RMSEÕs for daily and monthly mean evaporation show that the eect of water va-por molecular diusivity in soil evava-poration becomes small (oq=oDa0) once the diurnal variations have been averaged out. These results are generally similar to those found by Milly [18], although he found larger RMSE when neglecting all vapor transport at the hourly time scale.
At least for these soil-climate combinations, the re-sults imply that vapor transport within the soil is not the rate-limiting process for soil evaporation at daily or larger time scales. Even though thermal- and matric potential-induced vapor diusion can dominate at var-ious times and depths in the soil, insensitivity of the total eux to Da indicates that it is the deeper, dominantly liquid processes (diusion and drainage) that ultimately limit soil evaporation. For this to be true, diurnally varying downward and upward thermal and isothermal
Table 2
Relative errors calculated in evaporation for simulations with two dierent water vapor molecular diusivities (DaandDa/10)
Soil Type Silty Clay (q2:97 mm/day) Loamy sand (q1:91 mm/day)
Time scale 1 h 1 day 1 month 1 h 1 day 1 month
RMSE (mm/day) 0.324 0.094 0.049 0.400 0.185 0.082
vapor ¯ows must adjust themselves such that they transmit, on average, the deeper and dominantly liquid ¯ow from below. This adjustment process is explored below for a simpler isothermal system undergoing a drydown from uniform initial moisture content.
3.2. Single 500 h dry-down and results of the simple analytical model
For the isothermal drydown from uniform initial moisture, the total ¯ux predicted by the simple model is in close agreement with that predicted by Milly's [17] model for both molecular diusivity values (Da andDa/ 10) (Fig. 2). Most importantly, estimated ¯uxes from both models show negligible variations when Da is re-duced by an order of magnitude. The growth of the drying front (L(t)), however, shows great sensitivity to
Da (Fig. 3). Taken together, Figs. 2 and 3 illustrate that vapor transport has a strong in¯uence on the movement of the drying front, but not on the total eux to the atmosphere.
This interpretation is further illustrated by the ¯ux pro®les at the 100 h of simulation (Fig. 4). In the drying front, the isothermal vapor ¯ux (denoted by pluses (simple model) and solid dots (Milly model)) contributes more signi®cantly than that of the liquid ¯ux (denoted by circles (simple model) and dashed line (Milly model)). Below the drying front, the liquid ¯ux completely con-tributes to the total ¯ux. Though the contributions of vapor and liquid ¯uxes dominate in and below the drying front respectively, the total ¯ux (denoted by crosses (simple model) and solid line (Milly model)) is approximately equal to the liquid ¯ux below the drying front. The simple model provides a good approximation of the extent of the vapor zone, withL approximately
Fig. 2. Simulated total ¯uxes. ``Milly'' indicates the simulation using Milly's [16] model and ``Simple'' indicates the simulation by simple model. Da and Da/10 indicate the cases of dierent water vapor diusivity.
Fig. 3. Movement of drying in both Milly's [16] model and simple models for dierent water vapor diusivities. Asterisks and dots represent the case ofDa, and the continuous and dashed lines represent the case ofDa/10.
at the depth where the vapor ¯ux curve exceeds the liquid ¯ux curve predicted by Milly's model [17]. The overall sharpness of the drying fronts in each model are in qualitative agreement with experimental results such as those reported in Hanks et al [13]. Furthermore, the pro®les (Fig. 4) appear similar to the results of experi-mental and numerical simulations of Yamanaka et al. [31].
Comparison of Fig. 4(a) and (b) demonstrates the shift of the extent of the drying front with respect to water vapor molecular diusivity. Note that the extent of the drying front is brought from approximately 4±0.4 mm when the water vapor molecular diusivity is re-duced by one order of magnitude. Thus the drying front shrinks by one order of magnitude when the water vapor molecular diusivity is reduced by one order of magni-tude. Based on this observation, the rate limiting role of the deeper liquid ¯ows appears to result from the fol-lowing constraint: if the vapor diusion in the near surface dry zone were limiting the transport of moisture to the surface (i.e., qv;1<ql;2), then the extent (L) and the in¯uence of the drying front would be diminished by the net in¯ux of liquid water from below over the eux of vapor to the atmosphere (since dL=dt/qv;1)ql;2).
Given the standard de®nition of bare soil resistance (rs) as proportional toL/Da(e.g. [6]), this compensation by L to changes in D
a implies that rs is insensitive to
water vapor diusivity, even under the soil-limited evap-oration conditions modeled above. At least for the case analyzed here, this strange result occurs because vapor transport through the drying front is not the rate-lim-iting mechanism and thus is ill-de®ned as a means of predicting soil limitations on evaporation.
As will be discussed below, this feedback mechanism depends on capillary rise from below recharging the drying front. In the absence of this recharge, either be-cause it is simply left out of the model (as in most re-sistance models) or because the underlying soil is very dry, the feedback disappears and evaporation will strongly depend on bothL andD
a.
3.3. Interaction of transport processes
If the limiting process of soil evaporation is liquid ¯ow from below and not vapor transport within the drying front, then one would expectoq1=oDa0. Forq1 given by (6a), this may be written
o
oDa
wsurfÿw
L K1
qsurfÿq
L
Da
ql nÿh1
5=3
0:
16
To explore this relation, the simple model is evaluated for another three intermediate values ofDa. Becausew,
L,K
1,q, andh1are all directly or indirectly dependent on the value ofDa, it is useful to separately illustrate the
variations of each of the components in Eq. (16) with respect to water vapor diusivity.
The sensitivity of each component is plotted in Fig. 5. Note that as the water vapor molecular diusivity in-creases, the vapor ¯ux (denoted by stars) increases and the liquid ¯ux (denoted by circles) decreases. These opposing eects keep the total ¯ux (denoted by crosses) essentially constant, i.e., the condition, oq1=oDa0, is met. In addition, the terms of (nÿh1)5=3 (denoted by diamonds) and (qÿq
surf) (denoted by squares) vary inversely, while the term Da/L (denoted by pluses) is constant. Constant Da/L further con®rms the observa-tions of Fig. 4(a) and (b) that the extent of the drying front shrinks in order to compensate for the reduction in
Da.
In summary, the sensitivity analysis indicates that soil evaporation (under the initial moisture conditions and hydraulic characteristics applied here) is not sensitive to the water vapor molecular diusivity, and that the rea-son underlying this insensitivity is that the drying front will shrink or grow to whatever size is required such that it can transmit the liquid ¯ow from below. The feedback mechanism that causes L to shrink is the mass imbal-ance that would occur for reduced vapor transport out of the drying front (qv;1) and essentially unchanged liq-uid transport into it (ql;2).
This explanation requires that ql;2 be insensitive to changes in L, which can be explained heuristically as follows. For moist soils, both experience and Eqs. (13a)±(15) indicate thatLis much smaller thanL
0. Furthermore, when L grows, re¯ecting the excess of evaporation over ql;2, L0 is largely unaected. To see this, imagine a limiting case where all of the growth in
L came at the expense of L
0 (i.e. the depth wherew0 occurs, L+L
0, is held ®xed). So long asL L0, the
relativechange inL0will be small, and thusql;2will stay more or less ®xed (see Eq. (6b)). In essence, the moisture gradients ((wÿw0=L0) driving liquid ¯ow up to the drying front are largely unaected by the growth of the drying front. As is discussed further below, this mech-anism is dependent on the moisture conditions in zone 2.
3.4. Sensitivity of ®ndings to initial moisture content and relation to previous studies
It was observed in the previous sections that the movement of the drying front (L), but not the total evaporation, is sensitive to vapor transport in moist soils. As will be shown below, the total mass error in ignoring vapor transport (e.g. whenDatends toDa/10) is roughly proportional to the product of the extent of the drying front (L) and the moisture stored at potentials below which vapor transport is more ecient (h). Essentially this represents moisture that cannot readily escape by liquid transport. To understand the magnitude of this error, note that when the soil is reasonably moist: (1) the growth of drying front is reduced by replenishment of liquid water from below ( dL=dt/qv;1ÿql;2); and (2) the relative amount of available water that is stored be-lowhis small. For both of these reasons, the mass error stays small for initially moist soils.
Conversely, if the soil is very dry initially (e.g. when
h0 is close to h),ql;2 becomes small and the growth of
drying front will depend only on vapor ¯ux
(dL=dt/q
v;1). Essentially this is the condition applied by Monteith [19] and Choudhury and Monteith [6], al-though they do not explicitly state this restriction. In this case,L grows rapidly, re¯ecting the loss of tightly held water to the atmosphere through vapor diusion. Because this tightly held water (i.e., h6h) cannot readily escape by liquid transport mechanisms (see Eq. (4)), the vapor ¯uxisthe limiting transport mechanism. Under these dry conditions, one would expect that the sensitivity of soil evaporation to water vapor molecular diusivity will be signi®cant (i.e., oq1
oDa60).
In order to quantify how dry a soil needs to be in order for the vapor transport to limit overall evapora-tion, the calculations using the simple model are re-peated in a similar manner to that reported above, but for a range of initial soil moisture contents (h00.35, 0.30, 0.25, and 0.20). Table 3 lists the cumulative mass error caused by reducing Da to Da/10 (at t100 h), along with the product of (hÿhsurf) andL(calculated usingDa), for both soil types. Eqs. (9a) and (9b) predict that the matric potential in equilibrium with a relative humidity of 0.2 is about-22 000 m. For the silty-clay soil, that matric potential corresponds to a surface moisture content (hsurf) of 0.05. The potential vapor transport exceeds liquid transport at a matric potential (w) of
ÿ1700 m, for whichh0.14. The dierence (hÿhsurf) is thus approximately 0.09 and there is, therefore, sig-ni®cant water stored that can only escape eciently through vapor transport. As expected, the product of this dierence and L (listed in the last column of Table 3) is roughly proportional to the cumulative mass error and provides an upper bound to it.
Note that in the driest cases where the error is largest in a relative sense, the total evaporation at 100 h is only a few millimeters. As discussed above, the relative error induced by neglecting vapor transport increases as (hÿhsurf) becomes a larger fraction of (h0ÿhsurf), i.e., for drier soils. Also note that hsurf;h and their dierence (approximately 0.03) are each smaller for the sand-loam soil than for the silty-clay, but that: (1) the product of (hÿhsurf) and L is cor-respondingly smaller in the dry cases, but not in the wet cases where the sand-loam has a largerL; and (2) the actual magnitude of the errors (column 6) are about the same for each soil.
Although vapor transport can be signi®cant in very dry conditions, it appeared to have little in¯uence in the year long simulation forced with meteorological data (Table 2). Most likely this is because the initial moisture conditions of the drydowns were always determined by the previous rain storm, keeping the surface soil rela-tively moist and the total errors relarela-tively small.
Table 3
Cumulative evaporation dierence after 100 h of drying for simple model with two dierent water vapor molecular diusivities (DaandDa/10) compared with estimated volume of water not readily available for liquid transport
Soil h0 q1(mm/day) Pq1(mm) DPq1(mm) L(mm) L(hÿhsurf) (mm)
Da Da/10 Da Da/10
Silty-clay 0.20 0.21 0.12 1.59 0.96 0.63 27.79 2.46
0.25 0.35 0.29 2.73 2.25 0.48 16.38 1.45
0.30 0.68 0.65 5.26 5.01 0.25 8.44 0.75
0.35 1.50 1.50 11.52 11.33 0.19 3.86 0.34
Loamy-sand 0.20 0.23 0.20 1.94 1.49 0.49 24.2 0.78
0.25 0.50 0.47 3.86 3.46 0.40 15.2 0.49
0.30 0.71 0.68 5.53 5.28 0.25 10.7 0.35
3.5. Implications for resistance models
For conditions in which vapor transport does not limit bare soil evaporation, one would expect diculties in de®ning, validating, and implementing models using soil resistance to vapor diusion (rs) to calculate soil-limited evaporation. The resistance term rs was de®ned by Monteith [19] as part of a conceptual framework to explain the square root of time behavior of evaporation and its relation to the movement of the drying front. Shuttleworth and Wallace [28] applied the concept in a multi-layer energy balance model, but stated that the dependence of rs on empirical models might lead to problems in physical interpretation. Later Choudhury and Monteith [6] de®ned the resistance term explicitly as a function of Da, L, porosity (n), and tortuosity (X) throughrs (XL)/(n Da). When applied in a four-layer land surface energy balance model, they encountered diculties in implementingrsinsofar as the model pre-dicted transition from potential to soil-limited evapo-ration to occur after a few hours instead of a few days (as observed). They ascribed the problem to variations in atmospheric surface pressure. It may be, however, that the problem was that their implementation neglects the replenishment of water to the drying front through liquid transport (ql;2). Ignoring the upward ¯ow causes
L to grow much too fast and consequently causes
evaporation to be reduced too soon.
A common alternative approach (e.g. [4,15,27]) is to represent rs as a function of volumetric soil moisture content in a constant depth of soil (for example the 0± 0.5-cm layer). Although this approach may indirectly include liquid ¯ow from below, the drying front is ar-ti®cially kept at a constant depth. Moreover, the value ofrsin these applications has been shown to depend on moisture through site-speci®c empirical relations, which again leads to the problems of physical interpretation.
In either approach the actual source of diculty (i.e. non-uniqueness and/or poor prediction) may simply be that vapor transport in the dry surface layer is simply not the rate-limiting factor, and thus is not the best suited part of the process to model when trying to esti-mate soil-limited evaporation. One can anticipate sub-stantial improvements to the soil resistance approach if the liquid ¯ux from below the drying front is accounted for when determining movement of the drying front.
4. Summary and conclusion
Results of numerical and approximate analytic inte-grations of coupled moisture and vapor diusion in soil were used to explore the question of whether or not there is a single, dominant rate-limiting process in the transfer of water to the soil surface. The simulations and analysis were designed in similar form to the study by
Milly [18]. The sensitivity of soil evaporation to water vapor transport was examined by analyzing the root mean square error between simulated ¯uxes (under measured meteorological forcing) using two dierent water vapor molecular diusivities (Da0.27 cm2/s and
Da/10). A simple model was derived which approxi-mately captures both the movement of the drying front as soil evaporation progresses and the contribution of vapor and liquid transport in the unsaturated zone. The model is introduced to explore the hypothesis that liquid water ¯ow is generally the rate limiting process in bare soil evaporation.
For the soil and boundary conditions studied here, an insensitivity of total eux to water vapor diusivity is found whenever soil is not very dry. The lack of sensitivity appears to result from the drying front shrinking such that it transports the net in¯ux of liquid water from below to the atmosphere above. The feed-back mechanism that causes the drying front to shrink is the mass imbalance that would occur for reduced vapor transport out of the drying front and essentially unchanged liquid transport into it. Most strikingly, the compensation between the depth of the drying front and the water vapor diusivity implies that the often used bare soil resistance terms are independent of va-por diusivity. This curious result highlights the di-culty associated with trying to estimate soil-limited evaporation using resistance factors under conditions when liquid capillary rise, and not vapor transport, is the rate-limiting factor.
Cumulative errors from neglecting isothermal vapor transport are approximately equal to the product of the depth of the drying front and the moisture held at matric potentials below which liquid transport is inecient. This product is only on the order of a few millimeters. The results of this study indicate that cumulative evap-oration is mainly limited by the liquid water ¯ux from the deeper, wetter, soil in all cases except those in which the soil is so dry that total evaporation may be negligible anyway.
Acknowledgements
This work was supported by NASA grant NAG5-6716.
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