Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue5.2000:

Advances in Water Resources 23 (2000) 493±502

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 di€usion 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 integrations of a simpler isothermal system are then introduced to explore the interactions of vapor and liquid transport. Although
vapor di€usion 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 di€usivity 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 in Da in such a way that the capillary rise to the
drying front can still be transported to the surface. The feedback mechanism that causes L 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. Implications for

the utility and interpretation of soil resistance terms (de®ned as proportional to the ratio of L to Da ) 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 conversion may take place at or below the soil surface [22].
Ecient modeling of this and other processes in the
unsaturated zone remains a signi®cant challenge in hydrology 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 resistance to vapor di€usion (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 drawbacks of resistance-based and alternative approaches to
modeling soil evaporation. Most of the above citations
document diculties associated with the de®nition, implementation, uniqueness, and formulation (e.g. moisture dependence) of resistance terms. With the goal of
*

Corresponding author. Tel.: +1-617-353-8344; fax: +1-617-3538399.
E-mail address: gdsalvuc@bu.edu (G.D. Salvucci).

understanding why the resistance approach can be
problematic, the present study explores the role of vapor
di€usion 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 approach the underlying physics of the transport processes, 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 importance 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 includes 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 surface layer, a moist layer, and a layer at the initial

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 4 5 - 7

494

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502

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 altogether. 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 literature 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 framework (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 considered. To explore the issue of relative importance
further, he applied his numerical model to a set of
wetting and drydown conditions. From these experiments 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 indirectly 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 induces downward thermal vapor transport. The 40 day
cumulative evaporation totals for each treatment were
within about 10% of each other, with the more ecient
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 incomplete, 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 di€erent water vapor molecular di€usivities (Da ˆ 0.27 cm2 /s and Da /10)
was conducted to explore the sensitivity of soil evaporation to water vapor transport in soils. In this experiment, one year (1983) of ¯uxes was simulated using
Milly's [17,18] model, including both thermal and isothermal vapor and liquid ¯uxes. The hourly climate
forcing data [21] (wind speed, radiation, precipitation,
air temperature, and air humidity) are from Jacksonville, 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 conduction 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.
The sensitivity of soil evaporation to water vapor
transport is examined by analyzing the root mean
square error (RMSE) between simulated ¯uxes using

495

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502
Table 1
Hydraulic parameters of soils used in this study [26]
Parametersa

Silty clay
ÿ1

Saturated hydraulic conductivity (Ks , cm s )
n
hu
a (cm)
b
c
hk

3.20 ´ 10
0.47

0.42
)50
)2.067
0.07875
0.11

ÿ6

Loamy sand
3.70 ´ 10ÿ3
0.45
0.41
)2
)0.43
0.00898
0.034

a
The main drying curve was determined by Scanlon and Milly [26] by ®tting experimental data on soil water desorption to:
hd ˆ minfhu ; hu ‰…w=a†b ÿ …ÿ105 =a†b Š ‡ c‰5 ÿ log …ÿw†Šg, where w is in meters, hu is the water content obtained upon rewetting, and a, b, and c are

®tting parameters. The unsaturated hydraulic conductivity was calculated by integration of Mualem's [20] expression as
82




32 2
32 9
bÿ1
bÿ1
>
 ,

 >

=
w…hu †bÿ1 ÿ w…hk †bÿ1
p< hu b w ÿ w…hk †
1
1

h
b
1
1
u
5
5 Ks ;
4
K…w† ˆ Se 4 b
‡ 0:4343c
ÿ
ÿ
‡ 0:4343c
b
>
>
…b ÿ 1†
…b ÿ 1†
w w…hk †
a
w…hu † w…hk †
;
: a

where Se is the e€ective saturation: Se ˆ …h ÿ hk †=…hu ÿ hk † and hk is the value of water content at which liquid ¯ow becomes negligible.

water vapor molecular di€usivities Da and Da /10. The
RMSE is de®ned as follows:
RMSE ˆ

"

N

2
ÿ
1X
EDa =10 ÿ EDa
N iˆ1

#1=2

:

…1†

To determine the sensitivity at di€erent time-scales,
Pt‡T =2 the
simulated ¯uxes were aggregated as E ˆ T1 tÿT =2 E dt,
where T de®nes the time scale of interest (e.g. 24 h).
2.2. Simpli®ed analytical model
A second analysis was designed to explore the interaction 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 sensitivity to water vapor di€usivity 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 transport, but rather by available atmospheric transport capacity [22]. Furthermore, the simulations for di€erent
initial soil saturations with the same surface forcing were

carried out in order to examine the sensitivity of vapor
transport to di€erent 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 gravity) as
ow
:
…2†
oz
In (2), K (LTÿ1 ) is the unsaturated hydraulic conductivity and w (L) is matric potential.
The vapor ¯ux relation (qv , LTÿ1 ) is based on the PdV
framework as presented in Milly [17]
ql ˆ ÿK

qv ˆ

Da Xha oqv Da …n ÿ h†
ˆ
ql
ql
oz

5=3

oqv
:
oz

…3†

The variables in (3) are as follows: Da (L2 Tÿ1 ) is the
water vapor molecular di€usivity; h (dimensionless) is
the volumetric water content; X (dimensionless) is the
tortuosity of the air-®lled pore domain, which in Milly's
2=3
[18] analysis is set to …n ÿ h† ; n (dimensionless) is the
soil porosity; ha (dimensionless) is the volumetric air
content (equal to (n ÿ h)); and ql (MLÿ3 ) and qv (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

496

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502

spectively, and K1 and K2 are the e€ective 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]:
R w
K…w† dw
w
;
…7a†
K1 ˆ surf
…w ÿ wsurf †
K2 ˆ

Fig. 1. Schematic illustration of the vapor and liquid dominant zones
in the simple model at a time t (when t ˆ 0; L0 ˆ L ˆ 0).

vapor density (qsurf ) are held constant, to the depth at
which isothermal vapor and liquid transport are equally
e€ective. 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 potential w can thus be
found by solving:

5=3
Da …n ÿ h† oqv 
ˆ K …w †:
…4†
ql
ow wˆw

The dependence of qv on matric potential (w), saturation
vapor density (qs ), and temperature (T ) is modeled from
the thermodynamic relation of Edlefson and Anderson
[9]
qv …w; T † ˆ qs …T † exp…wg=Rv T †

…5†

In (5), g (LTÿ2 ) is the acceleration of gravity and Rv
(L2 Tÿ2 Hÿ1 ) is the gas constant for water vapor.
Approximating the derivatives in (2) and (3) as ®nite
di€erences, the total ¯uxes in zone-1 (q1 ) and zone-2 (q2 )
can now be expressed as




q1 ˆ

…wsurf ÿ w †
…q ÿ q † Da …n ÿ h1 †
K1 ‡ surf

ql
L …t†
L …t†

q2 ˆ

…w ÿ w0 †
…q ÿ q0 † Da …n ÿ h2 †
K2 ‡
L0 …t†
L0 …t†
ql

R w0
w

K…w† dw

…w 0 ÿ w  †

:

…7b†

Conservation of mass for the model system (Fig. 1)
can be expressed with z as the dependent variable (following [23]) as:
Z h0
o
…8a†
zdh^ ˆ qjhsurf ÿ qjh0 ˆ q1 ;
ÿ
ot hsurf
ÿ

o
ot

Z

h0
h

z dh^ ˆ qjh ÿ qjh0 ˆ q2 :

…8b†

R
The integrals z dh^ on the left-hand side of Eqs. (8a) and
(8b) represent the dynamic water stored in zones 1 and 2
combined, and zone 2, respectively. Zeroth order approximations of these integrals
would be given by trap
ezoidal integration as L …h0 ÿ h † ‡ 12 L …h ÿ hsurf †‡
1
L …h ÿ h †g and fL …h0 ÿ h †‡ 12 L0 …h0 ÿ h †g. Such
2 0 0
approximations were found to generate considerable
error as compared with high-resolution numerical solutions. To address this error in a manner consistent with
the method for approximating e€ective conductivities,
the moisture pro®le dynamics in each zone are approximated as a series of successive steady states. The presumed 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 relate w and qv to h. Because 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 approximations above and denoted v1 and v2 , Eqs. (8a) and (8b)
can be reduced to:
A

oL 1 oL0
B
ˆ ;
‡
2 ot
L
ot

…9a†

5=3

;

…6a†

oL 1 oL0 C
ˆ ;
‡
ot
2 ot
L0

…9b†

where

5=3

:

…6b†

In review, wsurf and qsurf are, respectively, the matric
potential and water vapor density at the surface, w and
q are matric potential and water vapor density at the
drying front, and w0 and q0 are the initial matric potential and water vapor density. L (t) and L0 (t) are the
depths of zone-1 and zone-2 at a time t (Fig. 1), re-

1 …h ÿ hsurf †
;
2 … h0 ÿ h †

ÿ1
…wsurf ÿ w †K1
Bˆ
v1 …h0 ÿ h †

…qsurf ÿ q †
5=3
‡
Da …n ÿ h1 †
;
ql
Aˆ1‡

…10†

…11†

497

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502


ÿ1
…q ÿ q0 †
…w ÿ w0 †K2 ‡

v2 …h0 ÿ h †
ql

5=3
Da …n ÿ h2 †
:

Cˆ

…12†

The simultaneous solutions of the coupled di€erential
Eqs. (9a) and (9b) are:
L ˆ b t1=2 ;

…13a†

L0 ˆ c t1=2 ;

…13b†

where

1=2
8C ÿ 4A ‡ c2
bˆ
4B

…14†

and
cˆ



h
2
…8C ÿ 8AB ÿ 4C† ÿ …4A ÿ 8C ÿ 8AB†

ÿ 64… B ÿ 1†BA2

i1=2 

2…B ÿ 1†

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 proportional to cumulative evaporation. A key di€erence here
is that the growth of the drying front due to evaporation
is o€set 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 di€usivity 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 di€usion. The present study analyzes vapor
transport by comparing the simulations with two different water vapor molecular di€usivities (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 temperature 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 di€usion 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 evaporation for silty-clay and loamy-sand are about 11% and
21% of mean evaporation, respectively. The error is
mostly attributable to the di€erences in thermally
induced downward vapor ¯ux during mid-day and upward 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 e€ect of water vapor molecular di€usivity in soil evaporation becomes
small (oq=oDa  0) 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 results 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 di€usion can dominate at various times and depths in the soil, insensitivity of the total
e‚ux to Da indicates that it is the deeper, dominantly
liquid processes (di€usion 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 di€erent water vapor molecular di€usivities (Da and Da /10)
Soil Type

Silty Clay (q ˆ 2:97 mm/day)

Time scale

1h

1 day

1 month

Loamy sand (q ˆ 1:91 mm/day)
1h

1 day

1 month

RMSE (mm/day)
RMSE/q

0.324
0.109

0.094
0.032

0.049
0.017

0.400
0.209

0.185
0.097

0.082
0.043

498

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502

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 di€usivity values (Da and Da /
10) (Fig. 2). Most importantly, estimated ¯uxes from
both models show negligible variations when Da is reduced 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 e‚ux 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 contributes 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, with L 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 di€erent water vapor
di€usivity.

Fig. 3. Movement of drying in both Milly's [16] model and simple
models for di€erent water vapor di€usivities. Asterisks and dots
represent the case of Da , and the continuous and dashed lines represent
the case of Da /10.

Fig. 4. Flux pro®les at t ˆ 100 h, (a) for Da and (b) for Da /10. Vapor
¯ux dominates in the drying front and liquid ¯ux dominates below.

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502

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 experimental 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 di€usivity. Note that the extent
of the drying front is brought from approximately 4±0.4
mm when the water vapor molecular di€usivity is reduced by one order of magnitude. Thus the drying front
shrinks by one order of magnitude when the water vapor
molecular di€usivity is reduced by one order of magnitude. Based on this observation, the rate limiting role of
the deeper liquid ¯ows appears to result from the following constraint: if the vapor di€usion 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 e‚ux
of vapor to the atmosphere (since dL =dt / qv;1 ) ql;2 ).
Given the standard de®nition of bare soil resistance
(rs ) as proportional to L /Da (e.g. [6]), this compensation
by L to changes in Da implies that rs is insensitive to
water vapor di€usivity, even under the soil-limited evaporation 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-limiting 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 because it is simply left out of the model (as in most resistance models) or because the underlying soil is very
dry, the feedback disappears and evaporation will
strongly depend on both L and Da .

499

variations of each of the components in Eq. (16) with
respect to water vapor di€usivity.
The sensitivity of each component is plotted in Fig. 5.
Note that as the water vapor molecular di€usivity increases, the vapor ¯ux (denoted by stars) increases and
the liquid ¯ux (denoted by circles) decreases. These
opposing e€ects keep the total ¯ux (denoted by crosses)
essentially constant, i.e., the condition, oq1 =oDa ˆ 0, is
met. In addition, the terms of (n ÿ h1 )5=3 (denoted by
diamonds) and (q ÿ qsurf ) (denoted by squares) vary
inversely, while the term Da /L (denoted by pluses) is
constant. Constant Da /L further con®rms the observations 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 di€usivity, and that the reason 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 imbalance that would occur for reduced vapor transport out
of the drying front (qv;1 ) and essentially unchanged liquid 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 that L is much smaller than L0 .
Furthermore, when L grows, re¯ecting the excess of
evaporation over ql;2 , L0 is largely una€ected. To see
this, imagine a limiting case where all of the growth in
L came at the expense of L0 (i.e. the depth where w0
occurs, L + L0 , is held ®xed). So long as L  L0 , the

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 expect oq1 =oDa  0. For q1
given by (6a), this may be written


o …wsurf ÿ w †
…qsurf ÿ q † Da
5=3
K1 ‡
…n ÿ h1 †
 0:
ql
oDa
L
L
…16†

To explore this relation, the simple model is evaluated
for another three intermediate values of Da . Because w ,
L , K1 , q , and h1 are all directly or indirectly dependent
on the value of Da , it is useful to separately illustrate the

Fig. 5. Components of the total ¯ux in the drying front (zone-1). Note
that the total ¯ux and Da /L are insensitive to the changes in the water
vapor molecular di€usivity (Da ).

500

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502

relative change in L0 will be small, and thus ql;2 will stay
more or less ®xed (see Eq. (6b)). In essence, the moisture
gradients ((w ÿ w0 †=L0 ) driving liquid ¯ow up to the
drying front are largely una€ected by the growth of the
drying front. As is discussed further below, this mechanism 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. when Da tends to Da /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 ecient (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;1 ÿ ql;2 ); and (2)
the relative amount of available water that is stored below h is 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 / qv;1 ). Essentially this is the condition applied
by Monteith [19] and Choudhury and Monteith [6], although 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 di€usion.
Because this tightly held water (i.e., h 6 h ) cannot
readily escape by liquid transport mechanisms (see Eq.
(4)), the vapor ¯ux is the limiting transport mechanism.
Under these dry conditions, one would expect that the
sensitivity of soil evaporation to water vapor molecular
oq1
6ˆ 0).
di€usivity will be signi®cant (i.e., oD
a

In order to quantify how dry a soil needs to be in
order for the vapor transport to limit overall evaporation, the calculations using the simple model are repeated in a similar manner to that reported above, but
for a range of initial soil moisture contents (h0 ˆ 0.35,
0.30, 0.25, and 0.20). Table 3 lists the cumulative mass
error caused by reducing Da to Da /10 (at t ˆ 100 h),
along with the product of (h ÿ hsurf ) and L (calculated
using Da ), 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 which h ˆ 0.14. The di€erence (h ÿ hsurf )
is thus approximately 0.09 and there is, therefore, signi®cant water stored that can only escape eciently
through vapor transport. As expected, the product of
this di€erence 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 (h ÿ hsurf ) becomes a larger fraction of
(h0 ÿ hsurf ), i.e., for drier soils. Also note that hsurf ; h
and their di€erence (approximately 0.03) are each
smaller for the sand-loam soil than for the silty-clay,
but that: (1) the product of (h ÿ hsurf ) and L is correspondingly smaller in the dry cases, but not in the
wet cases where the sand-loam has a larger L ; 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 relatively moist and the total errors relatively small.

Table 3
Cumulative evaporation di€erence after 100 h of drying for simple model with two di€erent water vapor molecular di€usivities (Da and Da /10)
compared with estimated volume of water not readily available for liquid transport
P
P
q1 (mm/day)
q1 (mm)
Soil
h0
D q1 (mm)
L (mm)
L (h ÿ hsurf ) (mm)
Da

Da /10

Da

Da /10

Silty-clay

0.20
0.25
0.30
0.35

0.21
0.35
0.68
1.50

0.12
0.29
0.65
1.50

1.59
2.73
5.26
11.52

0.96
2.25
5.01
11.33

0.63
0.48
0.25
0.19

27.79
16.38
8.44
3.86

2.46
1.45
0.75
0.34

Loamy-sand

0.20
0.25
0.30
0.35

0.23
0.50
0.71
0.81

0.20
0.47
0.68
0.75

1.94
3.86
5.53
6.03

1.49
3.46
5.28
5.77

0.49
0.40
0.25
0.26

24.2
15.2
10.7
9.4

0.78
0.49
0.35
0.30

T. Saravanapavan, G.D. Salvucci / Advances in Water Resources 23 (2000) 493±502

3.5. Implications for resistance models
For conditions in which vapor transport does not
limit bare soil evaporation, one would expect diculties
in de®ning, validating, and implementing models using
soil resistance to vapor di€usion (rs ) to calculate soillimited 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)
through rs ˆ (X L )/(n Da ). When applied in a four-layer
land surface energy balance model, they encountered
diculties in implementing rs insofar as the model predicted transition from potential to soil-limited evaporation 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 arti®cially kept at a constant depth. Moreover, the value
of rs in 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 diculty (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 estimate soil-limited evaporation. One can anticipate substantial 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 integrations of coupled moisture and vapor di€usion 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

501

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 di€erent
water vapor molecular di€usivities (Da ˆ 0.27 cm2 /s and
Da /10). A simple model was derived which approximately 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 e‚ux to water vapor di€usivity
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 feedback 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 di€usivity implies that the often
used bare soil resistance terms are independent of vapor di€usivity. 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 inecient.
This product is only on the order of a few millimeters.
The results of this study indicate that cumulative evaporation 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 NAG56716.
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