Issues related to low resolution modeling of soil moisture ...

E L S E V I E R Global and Planetary Change 13 (1996) 161-181

GLOBAL AND PLANETARY CHANGE

Issues related to low resolution modeling of soil moisture: experience with the PLACE model

Aaron Boone a,b, Peter J. Wetzel b

a Science Systems and Applications, Inc., 5900 Princess Garden Parkway, Lanham, MD 20706, USA b Mesoscale Atmospheric Processes Branch, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

Received 5 May 1995; accepted 28 August 1995

Abstract

This study documents the new PLACE soil hydrology model, and examines the effects of various parameterization schemes on the solution of the Richards equation. Richards equation is the basis upon which many of the land surface schemes participating in the PILPS experiments model soil water transport. Generally, the integration is carried out using a coarse model grid, which makes the solution more sensitive to particulars of the parameterization scheme. Parameterization schemes for the lower boundary condition, lateral interflow, and for moisture fluxes between model layers are tested in PLACE using both high and low resolution grids. Simulations were made using PILPS-HAPEX forcing data and soil and vegetation parameters. The soil hydrology model is validated against the annual observed HAPEX soil moisture profiles. The predicted evapotranspiration is also compared to a value computed from the PILPS-HAPEX forcing data using the Penman-Monteith equation.

When testing a low-resolution soil grid typical of land surface schemes, predicted soil moisture was found to be highly sensitive to the interpolation method for computing vertical moisture fluxes between model layers. A new interpolation method for low resolution models is proposed and tested. It reproduces the high resolution model results more faithfully, over the entire range of soil moisture, than two methods commonly applied in the literature. Further tests demonstrate that by varying the parameterizations for lower boundary condition and the treatment of lateral flow (collectively called drainage), the predicted total annual evapotranspiration may range between 74% and 97% of the incident precipitation in this case. Both of these parameterizations involve one free parameter, and both are largely unconstrained by the available observations. Good overall agreement between the PLACE predicted and HAPEX observed soil moisture profiles was attained by varying these two PLACE drainage parameters over their respective ranges for a series of model simulations. Root-mean square error tests were then used to determine the set of parameters which corresponded to the best predicted soil moisture profile. However, the best predicted soil moisture profiles do not correspond with the best predicted evapotranspiration. This inconsistency occurs not only for PLACE, but, to varying degrees, for all of the land-surface schemes participating in PILPS-HAPEX.

1. Introduct ion

The effect of SVAT (Surface-Vegeta t ion-Atmo- sphere-Transfer) schemes on climate and regional scale modeling has received growing attention in

0921-8181/96/$15.00 Published by Elsevier Science B.V. SSDI 0921-8181(95)00044-5

recent years. These schemes vary both in complexity and emphasis from highly detailed canopy models (Sellers et al., 1986) and models which incorporate soil and vegetation heterogeneity and mosaic patches (Wetzel and Boone, 1995), to models which empha-

162 Aaro Boone, P.J. Wetzel/ Global and Planetary Change 13 (1996) 161-181

size increased computational efficiency while still capturing the most important physical processes (eg. Noilhan and Planton, 1989; Xue et al., 1991; Milly, 1992). PILPS (Project for Intercomparison of Land- surface Parameterization Schemes) was conceived in order to address the differences among the various SVAT parameterizations currently in use as research tools (Henderson-Sellers et al., 1993). The variability among the SVAT's in predicted surface fluxes and the water budget for PILPS has been shown to be on the same order as that which would occur due to climate change (Pitman et al., 1993), so that a detailed analysis of the various models has been undertaken in an attempt to identify the primary mechanisms which are causing the schemes to diverge.

One of the most important components of the SVAT scheme is the soil hydrology scheme, which is used to determine the partitioning of rainfall into infiltration, runoff, drainage, and storage within the soil (the soil water profile). This portion of the SVAT will be the focus of this paper. The vertical distribution of soil water within the model domain is related to the surface fluxes through bare soil evaporation and transpiration by plants. The soil water profile has a large effect on the surface energy budget through the partitioning of available surface energy between sensible and latent heat fluxes. This in turn has a large effect on the numerical simulation by the parent atmospheric model, both on climate scales (Mitchell and Warrilow, 1987; Abramopoulos et al., 1988; Meehl and Washington, 1988; En- tekhabi and Eagleson, 1989) and regional scales (Mahfouf et al., 1987; Pinty et al., 1989; Avissar and Pielke, 1989; Bougeault et al., 1991; Betts et al., 1993).

Soil hydrological models used in SVAT's for climate simulations and operational weather forecast- ing tend to be low resolution models in order to keep computational costs down. Low-resolution models usually use 1-5 layers to represent the layer from the surface down to the root zone or bedrock (generally ranging from one to tens of meters total depth). Physically realistic hydrological models incorporated into SVAT's which integrate Richards equation to determine the time evolution of the vertical soil moisture profile are plentiful (McCumber and Pielke, 1981; Mahrt and Pan, 1984; Dickinson, 1984; Sellers et al., 1986; Mahfouf et al., 1987; Abramopoulos et

al., 1988; Avissar and Pielke, 1989; Mihailovic and Rajkovic, 1991; Verseghy, 1991; Koster and Suarez, 1992).

While many of the SVAT's participating in PILPS use Darcian motion to describe soil water flow (Pit- man et al., 1993), there does not seem to be a consensus opinion among modelers on how to pa- rameterize certain processes: the drainage out of model base, treatment of flow in saturated layers, and the soil water content interpolation scheme used to compute water fluxes between soil layers. The drainage parameterizations are mathematically similar among participating schemes, but the physical interpretation of associated parameters differ significantly. Treatment of saturated flow affects total runoff which in turn affects the rate at which a soil's moisture reserve is depleted. Some models treat soil layers as closed buckets with no lateral outflow, while others allow differing degrees and forms of sub-surface " runoff" . Choice of an interpolation scheme is a significant issue, perhaps even primary because most models use a layer averaged form of Richards equation which is extremely non-linear in soil water content. Moisture values are required at soil layer interfaces in order to evaluate fluxes between layers, and these values must be estimated from the layer mean quantities which are related to volumetric water content. The layer mean hydraulic conductivity can be dependent on soil water content taken to a power as high as 30 (Cosby et al., 1984), so that the parameterization can be highly sensitive to the way the layer mean quantities are interpolated.

The PLACE (Parameterization for Land-Atmo- sphere-Cloud Exchange) model is an active participant in PILPS. The PLACE model is a detailed process model of the boundary layer and the underly- ing heterogeneous land surface. Thorough descrip- tions of the boundary layer evolution, cloud forma- tion, and the parameterization of land surface processes are presented by Wetzel and Boone (1995). PLACE can incorporate the effects of soil heterogeneity and different land surfaces within a single grid box through the use of so-called mosaic tiles (Wetzel and Chang, 1988; Wetzel et al., 1996-this issue), but this is beyond the scope of this paper. In keeping with the PILPS guidelines, these two forms of representing heterogeneity within a grid box have been "turned off" . This paper presents a new ver-

Aaro Boone, P.J. Wetzel / Global and Planetary Change 13 (1996) 161-181 163

sion of the PLACE soil hydrology scheme which uses Darcian motion to describe soil water flow by integrating Richards equation (Richards, 1931) with a different numerical implementation than the version described in Wetzel and Boone (1995).

The goal of this paper is to document the new version of the PLACE hydrology model, and to demonstrate the sensitivities of Richards equation to grid resolution and to the parameterization of drainage, of fluxes across grid cells, and of flow in saturated layers. All testing presented within this paper applies PILPS specified parameters and uses forcing data from the HAPEX-MOBILHY (Hydro- logic Atmospheric Pilot Experiment and Modelisa- tion du Bilan Hydrique), although the model and the related parameterization schemes have been tested over a broad range of soil hydraulic properties.

The number of soil water layers in the PLACE model can be varied, as can the depth of the lowest model layer. In this paper, a high-resolution model, incorporating 50 exponentially distributed layers was tested and compared with the low-resolution model containing 5 layers in the same model domain. In both versions, grid resolution is greatest near the soil surface to better model the larger moisture gradients experienced there (Dickinson, 1984). The high-resolution model solutions are used as a guideline for determination of the method for computing inter-layer fluxes in the low-resolution PLACE model. Three simple inter-layer interpolation schemes which represent those most commonly used in SVAT's are tested. It will be shown that there is a strong dependence on model geometry and that care must be exercised in choosing the proper parameterization scheme for a particular model.

2. PILPS-HAPEX hydrology

The new PLACE hydrology model was developed and tested using the HAPEX-MOBILHY dataset as part of PILPS. The model was run using PILPS- HAPEX soil-hydrological parameters and shelter- level forcing data, and the resulting soil moisture profiles were compared to observations taken approximately once per week during the year of the study. Detailed information on the HAPEX experi-

Table 1 The PILPS-HAPEX soil model parameters computed from Cosby et al. (1984). ~0 s is the matric potential at saturation, O s is the volumetric water content at saturation, k s is the hydraulic conductivity at saturation, and b is an empirical scaling parameter

Parameter Value Units

qJs - 30.0 (cm) 69 s 0.446 ( m m 3 / m m 3)

k s 4.0X 10- 4 ( c m / s ) b 5.66 -

ments and data can be found in Andr~ et al. (1988) and Goutorbe et al. (1989).

The primary goal of HAPEX was to gather hydrological and meteorological information on the spatial scale of a typical general circulation model grid box, usually on the order of 10 4 km 2 (Andr6 et al., 1988). The total annual observed precipitation was 85.6 cm. The total annual evapotranspiration Ev~ computed from the HAPEX forcing using the Penman-Monteith equation is estimated at 61.5 cm. The rest of the precipitation is assumed to run off. The PILPS- HAPEX soil parameters were computed using data from Cosby et al. (1984) for the Caumont site with a soil texture comprised of 46% silt, 37% sand and 17% clay (Goutorbe et al., 1989). The HAPEX soil hydrological parameters are shown in Table 1. The atmospheric state variables were assumed to be forced at shelter level. Incoming short-wave fluxes, long- wave fluxes and precipitation were also provided at 30 minute intervals for an entire year. The entire set of vegetation and soil parameters for HAPEX is listed in Shao et al. (1995).

The soil moisture data used as verification for the PILPS-HAPEX experiments was computed by aver- aging 4 sets of observations taken by neutron probes at the Caumont site roughly once per week (J. Noil- han, 1995, pers. comm.) with fewer observations during the winter months. The soil moisture profiles consist of water content values measured at 10 cm intervals from 5 cm below the soil surface down to a depth of 155 cm. The soil model grid requested by PILPS and the corresponding PLACE low-resolution grid are shown in Table 2. The volumetric soil water content for the three HAPEX-prescribed soil layers are the model variables which are used to describe the water profile of the top 160 cm of the soil.


Table 2 The PILPS specified model grid and the PLACE low-resolution model grid. The subscript i = 1 . . . . . N, where N = 5 for the PLACE grid. The corresponding soil depths are given by zi (cm)

and and z 0 = 0

i HAPEX z, i PLACE z~

- - ! 1 . 0

l 10.0 2 10.0 2 50.0 3 50.0 3 160.0 4 160.0 - - 5 1 0 0 0 . 0

PLACE incorporates 2 additional model layers: the uppermost thin layer is used to compute bare soil evaporation (Wetzel and Chang, 1988). The other additional layer extends from the base of the root zone down to a depth of 10 m and acts as a moisture source for the root zone during extended periods of water loss from the root zone due to plant transpiration.

3. The PLACE hydrology model

In this section, the basic equations for water flux and a discussion of the boundary conditions are presented. The numerical solution method is then presented, and two key problems that arise from the numerics of low-resolution models are discussed.

3.1. Model equations

The PLACE model uses Darcy's law to compute soil moisture flux. The vertical component of this equation is written as

0 V = - k - - ( t p + z ) (1)

OZ

where F is soil water flux, tp is the matric potential, k is the hydraulic conductivity, and z is the depth in the soil (positive) and is directed downward. The change in volumetric water content O as a function of time is written as

O0 OF S (2)

Ot Oz

where S is a soil moisture sink term. the moisture sink term for PLACE is

S = EsM/A + U + D (3)

where EsM represents evaporation from bare soil and transpiration from vegetation, and where h is the latent heat of vaporization. The evaporization of soil moisture is written as

ESM = F[ 95Eveg "Jr- (1 - qS) Es] (3a)

where F is the fractional coverage by open water, dew, and intercepted precipitation, and 95 is the fractional coverage by transpiring, photosynthetically active vegetation. Eveg represents evaporation from vegetation, so that soil moisture is extracted directly from the root zone by the plant roots. This term does not include evaporation from the plant store. E 8 represents evaporation from bare-soil, and moisture is removed from a thin surface soil layer. Evapora- tion of intercepted precipitation is not included in Eq. (3a). U is proportional to the positive difference between the rate of plant uptake of water and evaporative loss: 1( o) U = Pw A 1 ( E r r Epv)(U>_O ) (3b)

Qmax

where Pw is the density of water, Q is the plant internal water storage reservoir, Err is the threshold or supply limited evaporation rate, and Epv is the potential evaporation rate from vegetation. Eveg is proportional to the minimum of these two rates. The maximum plant storage is given by Qmax = 0-1bM, where b M is the above-ground biomass. U extracts moisture from the root zone layer, and channels the moisture into the plant internal store. Topographical horizontal discharge, D, is a function of the horizontal gradient of soil water potential:

D = k Ox 2 (3c)

where x is the direction perpendicular to z. D is parameterized as a function of the local topography and soil hydrological parameters. The topographic relief at HAPEX was very small, so that the topographical-horizontal discharge comprised a negligible portion of the PLACE computed water budgets and will not be discussed in any depth in this paper.


Detailed information on the processes represented by Eq. (3) can be found in Wetzel and Boone (1995).

Assuming the soil profile is homogeneous, Eq. (2) can be substituted into Eq. (1) to yield Richards equation

. . . . S (4) ot Oz J + Oz

where the diffusion coefficient d = kOO/O0. Soil temperature diffusion is assumed to have a negligible effect on the volumetric water content. Hydraulic conductivity and matric potential are related to volumetric soil water content as a function of soil type through the empirical relationships (Brooks and Corey, 1966)

k = ks( O/6),)zb + 3 (5a)

and

t#= ~O,(O/O,) -b (5b)

where the subscript s indicates values at saturation. The diffusion coefficient can then be written as a function of (9 using Eq. (5b);

d O, (6)

The top boundary condition in PLACE represents infiltration and is written as

F ( z = 0) = - r a i n ( I0, P ) (7a)

where I 0 is the maximum infiltration rate which depends upon the moisture content of the soil and the soil hydrological parameters (see Appendix A). P is the rate at which water reaches the surface after interaction with the surface interception store. The details related to interception and evaporation of precipitation by the soil and vegetated surfaces are beyond the scope of this paper and can be found (for PLACE) in Wetzel and Boone (1995).

Surface runoff is generated primarily by 2 mechanisms (Entekhabi and Eagleson, 1989), both of which are accounted for in the PLACE model. Runoff occurs if the water reaching the model surface exceeds the maximum infiltration rate (P > I 0) during a model time step. The excess water becomes so called Horton runoff. The maximum infiltration rate is specified as zero if the surface temperature is

below freezing, so that runoff from a frozen surface is considered to be Horton runoff. A second form of runoff occurs if the surface layer is saturated and P > 0. Only enough water to maintain saturation can enter the layer, and the excess runs off. This represents the so called Dunne runoff mechanism.

The lower boundary condition represents model drainage and is expressed as

F( Z = Zbase ) "~" - f k ( z = zb,~s~) (0 <_f<_ 1) (7b)

where Zbase is the depth of the lower boundary of the model domain, and the coefficient f is a model parameter which is related to fractures in the bedrock and defines the permeability of the lower model boundary. The lower boundary condition for most hydrological models used in SVATs is parameterized as gravitational drainage, so that the model should extend deep enough into the soil to assure that the neglect of vertical diffusion is a valid assumption.

3.2. Parameterization of drainage

Most hydrological models use a drainage parameterization as a lower boundary condition. Drainage is defined as the moisture flux across the model base, but in PILPS land-surface schemes it is often lumped with lateral drainage or interflow. In general, a hydrological model with N soil layers ( N > 1) has a moisture flux, F N, at the lower boundary which is a function of the hydraulic conductivity in the lowest model layer k N. Using Eq. (5a), the hydraulic conductivity at the model base can be written as

ON/e),) kN = ks ( . ~ \ 2 b + 3

where O N is the volumetric water content of the lowest model layer. The vertical diffusion term is oftentimes neglected at the model base [see Eq. (1)]. The base of the model is assumed to be at a level where the soil moisture is nearly constant with depth, so that the vertical gradient of soil moisture is assumed to be negligible compared to the gravitational drainage.

Examples of lower boundary conditions from PILPS-participant models which use Darcy motion to govern soil water movement are listed in Table 3. BATS (Dickinson et al., 1993), LAPS (Mihailovic and Rajkovic, 1991), SSiB (Xue et al., 1991), CLASS (Verseghy, 1991), and PLACE are land-surface

166 Aaro Boone, P.J. We t ze l / Global and Planetary Change 13 (1996) 161-181

Table 3 Drainage parameterizations of various land-surface schemes which are similar to PLACE. The lower boundary conditions are a function of volumetric water content of the lowest soil model layer

Model Drainage parameterization

S i B F u = - sin g kN

LAPS F N = - sing k N SSiB F N = - - s i n g k u - c ( [ ~ u / 0 ~ )

G I S S F N = - - ~ k N

C L A S S F u = - - k N

PLACE F N = - f k N BATS F N = - - k~,o( ~ ) N / / ~)s ) 2 b + 3

schemes which participated in the P I L P S - H A P E X experiments. GISS (Abramopoulos et al., 1988) did not participate in P I L P S - H A P E X but was involved with other PILPS experiments [see Pitman et al. (1993)]. The SiB (Sellers et al., 1986) hydrology model has been widely used and is the basis for the SSiB hydrology scheme.

The physical interpretations of the parameters in Table 3 vary among most of the models. The SiB and LAPS models use a parameter X which is related to the slope of the terrain. The SSiB model uses a slightly modified form of the SiB method in the P I L P S - H A P E X experiments (Shao et al., 1995) where c is defined as a model constant (Table 3). The GISS model uses a delta function 3 which depends on whether the lowest model level is above or below bedrock. The coefficient f (Eq. 7b) is related to the permeability of the lowest model layer in PLACE. It should be noted that GISS and PLACE treat lateral drainage separately. It is a function of topographic slope and an assumed average distance between stream channels in GISS [see Abramopou- los et al. (1988)]. Lateral drainage is a function of both topographic slope and amplitude in PLACE [see Wetzel and Boone (1995)].

The BATS model uses a value for hydraulic conductivity at saturation, k s (Eq. 5a), at the lower boundary which is constant for all soil types, k~o =

4.0 × 10 -5 cm s -1. This represents a different approach from the other models mentioned here. In fact, use of constant soil hydrological parameters throughout the soil column is not a very realistic assumption (Wetzel and Boone, 1995), and changes to this approach are planned for the future. The use

of a constant soil hydraulic conductivity value deep within the soil (Dickinson et al., 1993) would introduce another parameter into the PLACE model, so that the present method will be retained until future observations and studies can be used to determine a better choice of parameterization.

The drainage methods in Table 3 can be written in the same mathematical form as F u = - f f 2 k u , where 12 is a coefficient which controls the amount of water flowing through the model base. Drainage in SSiB can be cast into this form if the model constant c = 0. The parameter 12 ranges from 0 to 1 in SiB, LAPS, PLACE, and SSiB. The parameter 12 is either 0 or 1 in GISS and is equal to 1 in CLASS. The 12 coefficient is equivalent to k ~ o / k ~ for BATS, but this ratio is 1 for P ILPS-HAPEX. 12 can be estimated based on theoretical or observational grounds, or it can be used as a tunable model parameter (as in PLACE). It will be shown that 12 (or f in PLACE) has a profound impact on the soil moisture profile using the layer-averaged form of Richards equation in the PLACE model.

3.3. N u m e r i c a l s o l u t i o n

Eq. (4) is solved numerically by dividing the soil domain into N parallel layers in the vertical and then integrating downward over each of the model layers (Mahrt and Pan, 1984);

( oo i i,_,oo - - Z i

- z , , Ok . . . . .

+f - - 2 i z,

where i = 1 . . . . . N, and i (z) is increasing downward. The average volumetric water content within a layer is defined as

l - - Z i _ 1

O i = - - ~ z i f 6 ) d z (9) - - Z i

where A z i = zi - z i_ ~. T h e layer-averaged quantity is defined as being at the center of the layer. Eq. (8) can then be written as

A z ~ O 0 = d - - - d - - 0Z zi

Aaro Boone, pd. Wetzel / GIobal and Planetary Change 13 (1996) 161-181 167

d - - = - o , i - C

where S~ is the layer-averaged moisture sink term which has been integrated in the same manner as the volumetric water content in Eq. (9). The remaining four terms on the RHS of Eq. (10) are defined at the interface between the model layers. The index i refers to the center of the layer for the mean variables, and to the lower boundary of a layer for the flux terms.

The diffusion and gravitational drainage terms in Eq. (10) are written as

(Oi - - ~)i+ 1) ~-. - - A i

and

[ ~ '~2b+3

\ s !

( l l a )

( l i b )

where 6 Z i ~" (Zi+ 1 --Z/_l)//2. K~ and A~ represent the across-layer or inter-facial gravitational drainage and diffusion terms, respectively. ~,. is a coefficient related to the fraction of the layer over which the

N

_= 0 -.~-- - - - ~ z0

F0 1 ................. Of . . . . . . . . . . . . z

E ' " ~ ' - - ~ F1(~1) - ~ z,., -1

I K1 All ] Di ~ . . . . . . . . . . . . . Oi . . . . . . . . . . . . . AZ i

Ei "~"" ~ - - Fi (~i) -- ~ z i J Ki Ai[

DN"~-- . . . . . . . . . . . . . O~ . . . . . . . . . . .

N Z N FN= - fK s

Fig. 1. The PLACE hydrology model grid. Azi represents a model layer thickness, where z increases downward. The moisture flux across the boundary at zi is denoted by F i, where Fi = A i - K~. A~ is defined as positive upward, but it can be directed downward. K i is positive downward. The value of O at zi is given by O~. Lateral flows for each layer consist of saturated runoff (~i), and topographically- induced horizontal drainage (D~). Moisture loss from the soil due to evapotranspiration and plant storage are denoted by E~ and Ui, respectively.

fluxes are evaluated in which the temperature is below freezing. This coefficient ranges from 0 to 1:

= 1 if no portion of the layer is frozen, and ~ = 0 if the entire layer is frozen. The soil temperature profile is determined using a diffusion relation (Wetzel and Boone, 1995).

~g is defined as the inter-facial volumetric water content. The PLACE hydrology model grid is shown in Fig. 1. The inter-facial volumetric water content is defined at the boundary between two model layers, and in general ~i ~-- f( l~i ,~g+l ). The method for computing the water content at the model layer interfaces can be chosen from experimental or theoretical considerations. The approximation for ~g for non-saturated conditions needs to be carefully selected due to the highly non-linear relation between volumetric water content and hydraulic conductivity, and the irregular grid spacing used in PLACE. Eq. (10) is integrated on a stretched grid using an implicit time differencing scheme. The solution method is outlined in detail in Appendix A.

3.4. Super-saturat ion

Numerically induced super-saturation occurs when the water storage capacity of a layer is exceeded during a time step ((9 i > (gs). It is usually small but must be accounted for in the water budget. In PLACE, it is treated as lateral runoff (an addition to the topographically controlled horizontal discharge). Super-saturation occurs for 2 reasons primarily related to the numerics: the time step may be sufficiently large to cause runoff, or the drainage parameter f is set at a critical value below 1 which causes the water flux into the lowest saturated model layer to exceed the compensating flow out. This critical value of f is a function of grid cell thickness and soil hydrological parameters.

Super-saturation resulting from an excessively large time step during periods of heavy precipitation is rare and is generally negligible in the cases simu- lated to date. Super-saturation due to the flux imbal- ance of the lowest model layer is a consequence of using a zero-diffusion lower boundary condition with f < 1, and it becomes more serious as the model resolution becomes more coarse and model layers get thicker. This is because layer thickness appears only in the vertical diffusion term, and not in the

168 Aaro Boone, P.J. Wetzel / Global and Planetary Change 13 (1996) 161-181

gravitational drainage term (Eqs. l la and 1 lb), so when a layer is thick the former will be less than the latter term over a large range of O. Model layers below the root zone do not have water loss due to root uptake, and topographic horizontal discharge for HAPEX is nearly negligible so that to a good approximation the time rate of change in volumetric water content is equal to the difference in the inter- facial fluxes.

Darcian motion cannot be used to describe the moisture flow in saturated layers (Capehart, 1992), and the empirical relationships relating volumetric water content to the soil hydraulic properties [Eq. (5)] break down at soil water content values near saturation (Dickinson, 1984). Saturation can occur using Richards equation, so that a method which limits the moisture flux into a saturated layer was developed. It can be envisioned as assuming that Darcy flow ceases at the point within the time step at which a layer becomes saturated. Fluxes into saturated zones (below the water table) are reduced so that only enough water enters to maintain saturation. Details of the methodology are presented in Ap- pendix B.

It is then convenient to introduce a parameter M defined as the index corresponding to the model level at and below which the inter-facial flux is limited, so that the fluxes across z i for layers i = M . . . . . N - 1 are limited if the layers below are saturated. Physically, the parameter M defines the lowest level at which super-saturation can cause a horizontal discharge. It thus opens or closes a hypo- thetical passageway from a model layer to a stream channel. The saturation-limited flux acts to keep the layers above a saturated zone wetter than when the flux is not limited. The effects of limiting the inter- facial flux (as opposed to allowing excess water resulting from super-saturation to leave the model as lateral runoff) will be illustrated in the next section using the PLACE model for various values of the drainage parameter f.

3.5. Low-resolution treatment o f layer interfaces

Problems arise in low-resolution models because large moisture gradients can develop which are han- dled poorly by a coarse grid representation of the

highly non-linear Richards equation [Eq. (4)]. The choice of inter-facial interpolation scheme is very important to ensure proper fluxes, and the methods used by land-atmosphere schemes vary. Existing schemes generally linearly interpolate either the hydraulic conductivity or the volumetric water content itself. Some noteworthy schemes [such as BATS (Dickinson et al., 1993)] compute inter-facial fluxes using parameterized expressions derived from the solutions of high-resolution models based on Richards equation. The formulation of the inter-facial fluxes and the grid for this type of scheme vary from those used with the layer-averaged approach described here, so this type of scheme will not be addressed in this paper.

Examples of methods which interpolate hydraulic conductivity are Sellers et al. (t986), which use the thickness weighted value of conductivity to determine the value at the interface, and Abramopoulos et al. (1988). The latter tested linear and logarithmic interpolation of conductivity, and chose logarithmic interpolation for the results presented.

Methods which interpolate the volumetric water content include the scheme developed by Mahrt and Pan (1984). In this scheme, the inter-facial water content is computed using the maximum volumetric water content between two neighboring layers in their 2-layer model. Integrations were performed over relatively small time scales (on the order of a few days to a week). This approximation was chosen because it was found that fluxes computed using this method compared well to those given by a high-resolution soil model. Verseghy (1991) uses the arithmetic mean of the volumetric water content of the surrounding model layers to determine the inter-facial value. This method is used in the CLASS land- atmosphere scheme, which is integrated over much longer time scales with more model layers which extend deeper into the soil. Both of these interpolation schemes behave similarly unless there is a strong vertical moisture gradient.

A third method was developed for this study which linearly interpolates the logarithm of the absolute value of matric potential. The distance-dependent linear interpolation of the logarithm of matric potential takes into account the grid spacing so that the inter-facial moisture fluxes are determined more by the hydraulic characteristics of the wetter (upper)


soil layer in the presence of a wetting front. This is closer to the method used by Mahrt and Pan (1984) when precipitation is percolating into the soil, but is closer to the Verseghy (1991) method for intermedi- ate soil moisture and weak gradients.

However, during periods of prolonged drying of the soil due to evaporation, the inter-facial flux values are determined more by the dryer (upper) layer. This is in opposition to both approaches discussed above, but it is increasingly important in models which have very thick model layers below the root zone. The logarithmic interpolation of matric potential method handles this contingency. For very thick layers deep within the soil, the computed soil moisture diffusion is small relative to gravitational drainage. This results in downward fluxes of soil water even for very small values of inter-facial volumetric water content. Over sufficiently long time scales, excessive drying of the root zone can occur unless this new scheme is used, because the water contents of the sub-root zone layers vary much more slowly with time due to their relatively large storage capacities. Using either of the other two methods, interpolated values of inter-facial water content will cause gravitational drainage to dominate over vertical diffusion even if the overlying layer has become very dry.

Given sufficiently high-resolution, the inter-facial volumetric water content ¢9~ may be safely determined by Verseghy's method alone. As will be shown, this straightforward interpolation scheme is adequate for the high-resolution model because the water contents and grid-cell thicknesses do not vary significantly across neighboring grid cells, and the stretched grid enables adequate resolution of large near-surface moisture gradients during precipitation events.

The three interpolation schemes discussed above were tested using both the high-resolution and low resolution grids in PLACE. Note that the inter-facial water content computed using the hydraulic conduc- tivities of the surrounding layers [Sellers et al. (1986), Mihailovic and Rajkovic (1991), Xue et al. (1991)] is nearly identical to the values computed using the maximum volumetric water content because of the highly nonlinear dependence of hydraulic conductivity on water content. Inter-facial volumetric water content computed using the maximum volumetric

water content of two layers is called Method I and is written as

Oi = max(O i,Oi+ ,) (12a) Using the mean of the volumetric water contents of the two surrounding layers is called Method II and is

0i = (O i '~ Oi+ 1) /2 (12b)

Linear interpolation of the logarithm of the matric potential from the two surrounding layers is called Method III and is given as

0i = ~)s( ~i/~/s)--I/b (12c)

where Oi in Eq. (12c) is determined using Eq. (5b), and the interpolated matric potential is computed from

~i = lo8]-o'[( Azi+ ,loglol ~bil + az, log 10l q%,l)

/(az,+l +

4. Numerical experiments

A series of experiments was undertaken using the new PLACE hydrology model to show the sensitivities of Richards equation to three parameterization schemes; a gravitational drainage coefficient, the computation of inter-facial water content, and prevention of super-saturation of a soil layer. The simulations were done using both high and low-resolution grids in order to determine the impact of these parameterization schemes on the predicted soil water profile using a coarse resolution version of the model. Preliminary tests were done using the hydrology model outside of PLACE (in stand-alone mode) with simple forcing in order to determine the best inter-facial interpolation scheme for use in the low-resolution model. The next stage of testing used PILPS- HAPEX forcing and parameters to drive the PLACE model. Different values of the drainage ( f ) and flux-limit (M) parameters were used, and HAPEX observed soil moisture profiles were used to provide validation for the hydrology scheme. A comparison was also made between PLACE computed and HAPEX "observed" total annual evapotranspiration, Ev,.

The PLACE low-resolution grid corresponds to

170 Aaro Boone, PJ. Wetzel/ Global and Planetary Change 13 (1996) 161-181

0.0

200.0

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a=

D

N

400.0

600.0

800.0

\

. . . . . . Base of Root Zone \ I -- = High Resolution Grid ¶ '

Low Resolution Grid

1000.00. 0 10.0 20.0 30.0 40.0

Index for High Resolution Grid

0

Fig. 2. The PLACE high and low resolution grids. The depth of the root zone is indicated by the dashed line (z = 160 cm). The index corresponding to the root zone depth for the low resolution model is 4 and is 30 for the high resolution grid.

the HAPEX specifications, except that PLACE incorporates two additional layers (Table 2) for a total of five layers ( N = N L = 5). The top PLACE layer is used for the computation of baresoil evaporation. The lowest PLACE model layer extends from the base of the root zone (160 cm) to 1000 cm. Betts et al. (1993) showed that the inclusion of a model layer below the root zone can have a large effect on the surface fluxes on an annual basis. This layer acts as a water source when the root zone becomes dry due to large evaporative demand, because water can be diffused upward into the root zone. Deeply extending model layers can also incorporate the water table, although observational data containing such information is scarce. The high-resolution grid encompasses the same model domain as the low-resolution grid, except that it is an exponential grid with 50 layers (N = N H = 50). This stretched grid was used in order to better model the strong moisture gradients generally experienced near the surface of the soil. The two grids are shown in Fig. 2. Comparisons between the two model grid resolutions were made by examining

the volumetric water contents of the three HAPEX- prescribed soil layers (see Table 2).

The drainage parameter f and the super-saturation flux limit act to decrease model drainage and lateral runoff in the absence of any slope of the model domain. The drainage parameter was varied logarith- mically over its entire range (0 to 1) because gravitational drainage is a non-linear function of soil water content. Results using six values of f are presented in this paper. Two values of the super-saturation flux-limit parameter are presented. M was set to the index corresponding to the base of the root zone (i corresponding to z = 160 cm), which limits the vertical moisture flux downward at or below the root zone as the soil nears saturation. This parameter has a larger effect as f becomes smaller because water leaves the model through drainage at a slower rate, and the moisture content of the lower model layers increases so that more water is available for horizontal discharge, evaporation, and plant uptake. Gener- ally, super-saturation will not occur within the root zone because diffusion can balance or exceed gravitational drainage more readily, and the extraction of soil moisture by plants due to evaporation can dominate over vertical fluxes. Most of the super-saturation will occur near the lower model boundary. The other value for M was N L (N H) using the low (high) resolution model. This value "turns of f" the super- saturation prevention option so that as the model parameter f becomes smaller, any excess soil moisture will leave the model as lateral flow. When the parameter f = 1, the value of M causes very little (if any) difference in fluxes, and super-saturation will be negligible.

The interaction between evaporation and soil moisture is very crucial to the overall water budget. The PLACE results were compared to both the estimated annual evapotranspiration, Evt, and the observed soil moisture profiles from the HAPEX site, but the soil moisture values were used as the primary validation of the soil model. E~, is not as good a validation variable for the soil moisture model as the observed soil moisture profiles because it was not directly measured. Also, there are many physical processes that affect Evt which are not directly related to the soil moisture profile, such as evaporation of precipitation intercepted by the vegetation canopy. The predicted soil moisture, runoff, and

Aaro Boone, PJ. Wetzel/ Global and Planetary Change 13 (1996) 161-181 171

evapotranspiration varied greatly among the models participating in PILPS-HAPEX (Shao et al., 1995). The discrepancy between soil hydrology and evaporative flux will not be addressed here. It is the subject of ongoing PILPS work.

4.1. Interpolation scheme intercomparison

The hydrology model was initially tested outside of the PLACE model (in stand-alone mode) in order to determine the sensitivity of the time evolution of the soil water profile [described by Eq. (4)] to inter- facial interpolation schemes. The PILPS-HAPEX soil hydrology parameters were used with Eq. (5), and the moisture sinks comprising S were set to zero [Eq. (3)]. The model parameters f and M were varied using both the low and high-resolution grids with the three interpolation schemes [Eq. (12)]. A strong moisture gradient was used to test the limits of the three schemes.

A two-sided exponential function was used to impose a precipitation event. The precipitation rate was defined as

where t is the time of peak precipitation and o-, determines the duration of the precipitation event. The total integration period was 10 days, and the soil was initialized as saturated (as specified in the PILPS-HAPEX experiments). A 10 day integration period was selected because it is representative of the time scale for near-surface soil hydrological processes at the HAPEX site (Andr6 et al., 1988). The value of t = 2 (days) was used so that the precipitation event occurs at a sufficient time interval after the initialization to prevent a significant portion of the precipitation from running off. Also, the precipitation event occurs early enough during the integration period in order to examine the effects of drainage and diffusion over approximately a one-week period. The parameter o-,= 0.24 causes the precipitation event to last approximately one day. The amplitude was set as k s, so that infiltration will approach the maximum rate. Interception by a surface store and evaporation were assumed to be zero so that the

0.45

0.42 ~ ;

l i . . . . Layer 1 . . . . Layer 2

/; Layer 3 ! o4o ili ° i 8 0.38 t

N

0.33 ' ""

i , k , i _ , i ,

0'3%.0 2.0 4.0 6.0 8.0 10.0 Time (days)

Fig. 3. The soil water prof i le for a 10 day integration using PILPS-HAPEX soil hydrological parameters. HAPEX forcing is not used for this simulation, and the hydrology model is being run off-line from the rest of the PLACE model. The soil profile is initialized as saturated, and a two-sided exponential function is used to describe precipitation. Model sinks Q are set to zero. The high-resolution grid is used for this simulation. The volumetric water contents for layer I (0-10 cm), layer 2 (10-50 cm) and layer 3 (50-160 cm) are shown.

precipitation either infiltrated or left the model domain as surface runoff.

The high-resolution results were nearly identical using all three interpolation schemes for the same set of f and M values. The relative insensitivity to interpolation scheme results because volumetric water content does not vary a great deal across neighboring grid cells, and the soil moisture gradients are adequately resolved. The soil moisture profile for the high-resolution grid is shown in Fig. 3 for the case f = 1 and M = N H. The volumetric water contents for the three HAPEX-specified layers were computed using the grid thickness-weighted sums. Lay- ers 1 and 2 become nearly saturated during the precipitation event, resulting in a wetting front.

The low-resolution model, on the other hand, is very sensitive to the interpolation scheme. The low-


resolution model using Method I and the parameter values f = 1 and M -- N L is shown in Fig. 4. Method I results in too much drainage from the root zone into the lowest model layer relative to the high-resolution model. This is primarily due to the large thickness of the lowest layer. The volumetric water content of the lowest root zone layer becomes unre- alistically dry after only a few days if the profile is initialized as saturated. This scheme cannot be used with the PLACE-HAPEX low-resolution grid geometry.

Method II has the opposite effect on the profile as Method I. The downward propagation of the wetting front is much slower than in the high-resolution case, and the soil moisture profile at day 10 is much more dry (Fig. 5). The moisture in the upper layer should be weighted more heavily in determining the inter- facial value during the precipitation event to speed the propagation of the front. On the other hand, too much water has drained from the root zone by day 10 because the moisture content of the lowest layer

0,45

0,42

0,40 E g

~ 0.38

o . r _

0,35

I

0.33 ~\ li '

0,30 ~ ~ '

- - Layer1 . . . . Layer2

Layer 3

. . . . i

0.0 2.0 4.0 610 8.0 10.0 Time (days)

Fig. 4. As in Fig. 3 but for the low-resolution grid. Method I is used as the inter-facial interpolation scheme. The volumetric water contents for layer 1 (0-10 cm), layer 2 (10-50 cm) and layer 3 (50-160 cm) are shown.

E

E g

c

0.45

0.42

0.40

!

i i

Layer1 . . . . . . . Layer 2

Layer 3

0.38

0.33 i

I

0.30 - - ' i 0,0 2.0 4.0 6.0 8.0 10.0

Time (days)

Fig. 5. As in Fig. 3 but for the low-resolution grid. Method 11 is used as the inter-facial interpolation scheme. The volumetric water contents for layer 1 (0-10 cm), layer 2 (10-50 cm) and layer 3 (50-160 cm) are shown.

is weighted too much relative to the value above this layer.

Method III produces the profile which is in the best agreement with the high-resolution result (Fig. 6). The wetting front propagates downward slightly more slowly than in the high resolution case. Also, the amplitude of the moisture wave has been damped relative to the high-resolution grid wave by the time it reaches the lowest root-zone soil layer. This is due primarily to the coarse grid resolution within the root zone of the low-resolution model. The moisture profile is in good agreement with the high-resolution case by day 10.

The forcing was very simple in order to isolate the effects of the particular inter-facial moisture interpolation scheme on the evolution of the soil water profile. The differences in fluxes and the water profiles between the three interpolation schemes (for the same values of f and M) were negligible for the high-resolution grid runs. The high-resolution results were then used to determine the best interpolation

Aaro Boone, P.J. Wetzel / GIobal and Planetary Change 13 (1996) 161-181 173

0.45

0.42

E ~- 0.40 E

0.38 $

=~ 0.35

>

0.33

--~ Layer1 . . . . Layer2

Layer 3

i i i k I

0'3%.0 2.0 4.0 6.0 8.0 10.0 Time (days)

Fig. 6. As in Fig. 3 but for the low-resolution grid. Method lll is used as the inter-facial interpolation scheme. The volumetric water contents for layer 1 (0-10 cm), layer 2 (10-50 cm) and layer 3 (50-160 cm) are shown.

scheme for the low-resolution model. The cases presented here (Figs. 3-6) are just a few examples from extensive testing using all three interpolation meth-

ods and various values of the parameters f and M. Similar testing was also done using a wide range of soil hydrological parameters, with the same results. Method III consistently provided the best agreement between the low and high-resolution results.

4.2. PLACE results using HAPEX forcing

The next set of experiments utilized the entire land-atmosphere-interaction portion of the PLACE model. The high-resolution model was run using the full range of f and the two M parameter values and the three inter-facial interpolation methods [Eq. (12)]. The three schemes produced annual water budgets with differences in soil water fluxes less than 1% for the same set of PLACE parameters, so that the high-resolution grid was insensitive to the interpolation scheme even with the inclusion of the moisture sinks and more complex forcing than that from the previous section. Method III [Eq. (12c)] was then used as the interpolation scheme for the low-resolution model, which was used to determine the sensitivity of the model results to a more coarse soil grid using the same parameter values ( f and M).

The root mean square (RMS) error in soil water content for the 3 PILPS-prescribed layers, the RMS error in total soil water (TSW), and the absolute error in total annual evapotranspiration Eot were used to test model sensitivities to the PLACE param-

Table 4 The PLACE high-resolution model results as a function of f and M. The root mean square (RMS) errors are computed using the PILPS-HAPEX observed soil moisture profiles. The RMS error in f9 is the mean value for the three HAPEX model layers. The RMS errors in total soil water (TSW) and the total evapotranspiration Evt are shown in the last two columns, respectively. The "observed" E,. t is 61.5 cm

M f RMS error in ~9(mm3/mm 3) RMS error TSW(cm) Evt(cm)

30 1.0 0.05023 5.12 69.18 30 0.1 0.04997 5.05 69.29 30 0.01 0.05062 5.08 69.26 30 0.001 0.04638 4.07 72.94 30 0.0001 0.04506 3.59 83.78 30 0.0 0.04538 3.63 84.72 50 1.0 0.05023 5.12 69.18 50 0.1 0.05014 5.10 69.23 50 0.01 0.05003 5.07 69.26 50 0.001 0.05003 5.07 69.28 50 0.0001 0.05003 5.07 69.28 50 0.0 0.05003 5.07 69.28

174 Aaro Boone, P_I. Wetzel / Global and Planetary Change 13 (1996) 161-181

eters and are shown for the high-resolution grid in Table 4 and the low-resolution grid in Table 5. The magnitudes of the errors in soil water and total annual evapotranspiration vary between the two resolutions, but the overall trends are very similar. The infiltration and surface runoff were nearly the same for all of the numerical experiments.

The low-resolution runs show some sensitivity to f even when M = N L, whereas the high-resolution runs showed virtually no sensitivity to f when M = N n. This is due to the fact that the area extending from the bottom of the root zone down to the model base consists of a single grid box in the low-resolution runs. Since topographical discharge is negligible and there is no evaporation in the lowest model layer, the root zone moisture flux is strongly coupled to the magnitude of the lower boundary flux (which is modulated by f) . The coupling between the root- zone water flux and the base of the high-resolution model is much weaker because of the relatively large number of grid cells contained in this layer (20).

There is considerably more sensitivity to certain values of the parameter f when M corresponds to the base of the root zone for both grid resolutions. The parameter f has very little effect on the soil moisture errors and the evapotranspiration using either the low or high-resolution models until f_< 0.001. This represents the critical value of f for this

set of hydrological parameters (Table 1) and atmospheric forcing: the region below the root zone is nearly saturated for at least part of the year. This result is consistent for both grid resolutions. The model layers become more moist in successively higher layers as f decreases. This causes more moisture in the upper layers of the model and increased plant uptake and evaporation from the soil as evi- denced by the larger values of Evt as f decreases.

The drainage parameter f value which results in the best agreement with the soil moisture observations for the low-resolution model is f = 0.01. The RMS errors are the same for this value of f for M = 4 and M = N L, but, as stated before, the value of M does not become an important factor for either the high or low resolution grids until f < 0.001. Note that when M = 4, evaporation ranges from 74% of the precipitation for f = 1, to 97% of the precipitation for f = 0. This shows that specification of the lower boundary condition can have a significant impact on the annual water budget.

The annual soil water cycle is shown in Fig. 7. This represents the run used for the PILPS-HAPEX intercomparison (see Wetzel et al., 1996-this issue). The model tended to underpredict soil moisture in the spring and early summer, and to over-predict moisture in the late summer to fall. Because this is linked to evaporation, a detailed discussion of the

Table 5 The PLACE low-resolution model results as a function of f and M. The root mean square (RMS) errors are computed using PILPS-HAPEX observed soil moisture profiles. The RMS error in O is the mean value for the three HAPEX model layers. The RMS errors in total soil water (TSW) and the total evapotranspiration E,, t are shown in the last two columns, respectively. The "observed" Evl is 61.5 cm. The * indicates the set of parameter values for f and M used for the standard PILPS-HAPEX run

M f RMS error in ~9(mm3/mm 3) RMS errorTSW (cm) E,,t(cm)

4 1.0 0.04946 4.61 63.25 4 O. 1 0.04612 3.79 64.46 4 0.01 * 0.04440 3.74 66.52

4 0.001 0.04794 4.71 74.49 4 0.0001 0.05980 6.99 82.85

4 0.0 0.06073 7.21 82.99 5 1.0 0.04962 4.63 63.08 5 O. 1 0.04625 3.81 64.29 5 0.01 0.04440 3.74 66.52 5 0.001 0.04768 4.71 68.64 5 0.0001 0.04796 4.78 68.78 5 0.0 0.04800 4.79 68.82


reasons for the differences between the model and observations will be left for future PILPS work which will address this issue. The high-resolution run with the lowest soil moisture RMS values used the flux limit parameterization below the root zone, but the best f value was two orders of magnitude smaller. This results because vertical diffusion (upward) for the higher resolution grid occurs over a larger range of volumetric water content than for the low-resolution grid, due to the dependence of diffusion on 1 / A z [see Eq. (11)]. This enables good representation of early spring soil moisture, but the

summer soil water was over-predicted which leads to an anomalously large total annual evapotranspiration.

The PLACE water budget for the "bes t " standard PILPS-HAPEX run is shown in Table 6, selected primarily using the soil observed soil moisture data. The standard PILPS run was specified to homogeneous with respect to soil moisture and hydrological properties. E o is evaporation of water from the interception store, E s is evaporation from the plant store (Q), and E is the total PLACE evapotranspiration (see Wetzel and Boone, 1995). Topographical discharge in the top layer, D~, evaporation below the

0.40

0 .30

c Layer 1 O [] O 0.20 Layer 2

~ Layer 3 m o Observed layer 1

• Observed layer 2 "~ 0.10 o Observed layer 3 > O

D

0.00

500.0

E 4 0 0 . 0

. c

$ 300.0

200.0

- - Total Soil Water • Observations

• •O • • • • •

100.0 J h , , ,

0.0 60.0 120.0 180.0 240.0 300.0 360.0

T ime (days)

Fig. 7. The annual water cycle for HAPEX using the low-resolution grid. The volumetric water content for the PILPS-HAPEX specified layers and the total soil water are shown with the observations. The parameters f = 0.01 and M = 4. This case corresponds to the lowest

errors in soil moisture, and is submitted as the PLACE PILPS-HAPEX run.


Table 6 Water budget for the run corresponding to the PLACE model parameter values f = 0.01 and M = 4 (see Table 4). Results are for the standard PILPS-HAPEX run. Variables are defined in the text. The three totals in the table comprise the annual water budget. The difference in PLACE modeled total annual evapotranspiration and the HAPEX-observed value is shown in the last row of the table

Budget element Variable Value(cm)

Total precipitation 85.6 Surface runoff -~i (i = 0, 1) 0.7 Lateral flow ~i + Di (i = 2, N ) 0.1 Drainage F N 18.3 Total runoff ~ + D + F~, 19.1 Root uptake and baresoil loss EsM + U 62.3 Interception and plant store E D + E, 4.3 Total evaporation E 66.5 Evaporation difference I E,, t - El 5.0

root z o n e , (EsM)N, and plant storage uptake in the uppermost and lowest model layers, U 1 and U N, are all currently fixed as zero. Topographical horizontal discharge and super-saturated runoff comprise the nearly negligible lateral flow. The primary runoff mechanism is drainage, while evapotranspiration ac- counts for the majority of the precipitation. A comparison of this water budget with other SVATs which participated in PILPS-HAPEX can be found in Wet- zel et al. (1996-this issue).

5. Conclusions

The PLACE hydrology model uses Richards equation with three parameterization schemes to determine the evolution of the soil water profile. Two grid resolutions were used to determine the sensitivity of the modeled soil moisture profile and total annual evapotranspiration to a coarse vertical soil model grid. PILPS-HAPEX forcing and parameter data was used to drive the model, and the resulting annual water cycles were compared to the observed soil water profiles. Comparison was also made between the predicted and total annual "observed" evapotranspiration.

The PLACE hydrological-model drainage parameter f and moisture flux limit parameter M have a large effect on the modeled soil moisture profile and

evapotranspiration. The parameter f is a tunable model parameter which is consistent mathematically with other land-surface (SVAT) hydrology models formulations for the lower boundary condition, although the physical meaning of this parameter varies. The super-saturation prevention parameter M is tested as a way of limiting vertical or lateral drainage in the presence of the water table. Because knowl- edge of observed soil macropore networks is virtually never available, tuning of these parameters will usually be the best method to obtain results which match observations closely. The minimum number of model layers N could be tuned to find how small it could be while still giving results close to the high resolution grid, although most models are restricted to having from two to five layers for computational efficiency.

The choice of inter-facial interpolation scheme is very important, and it depends upon model grid structure. In this paper, three simple schemes were tested; the maximum volumetric water content (Method I), arithmetic mean of volumetric water content (Method II), and linear interpolation of the absolute value of matric potential (Method III) of the corresponding layer mean quantities from the two surrounding layers [Eqs. (12a), (12b), and (12c), respectively]. The high-resolution model grid was shown to be insensitive to the choice of interpolation scheme. Its results were then used as a baseline to determine the best interpolation scheme for inter-facial water content for use with the low-resolution grid. The best scheme for the low-resolution model was found to be Method III. The three interpolation schemes were also tested with both the low and high-resolution grids for a broad range of soil hydraulic properties using the PILPS-specified forcing and parameters for the Grassland, Tropical Rain Forrest, Tundra, and Cabauw (the Netherlands) cases. Method III produced soil profiles using the low-resolution model which matched the high-resolution model results the best. The low-resolution water budgets were not exactly the same as the results from the corresponding high-resolution runs, but the overall patterns and trends duplicated rather well. This indicates that low-resolution models can be used in place of more computationally expensive high-resolution models, but parameterization schemes must be determined judiciously.

Aaro Boone, P.I. Wetzel / Global and Planetary Change 13 (1996) 161-181 177

The RMS errors in total soil water were slightly better for the best high-resolution run than for the best low-resolution run, as would be expected from using a more dense model grid. On the other hand, the total annual evapotranspiration values are in slightly better agreement with the HAPEX observations using the low resolution grid, but this may be due to the fortuitous interactions between modeled evaporation and soil moisture. The fact that the PLACE modeled best values for total annual evapotranspiration and annual soil water cycle did not correspond to the same set of parameter values ( f and M) could be related to many factors, the two most obvious being: the "observed" evapotranspiration is too low, the PLACE evapotranspiration scheme removes too much water from the soil. A combination of these two factors is also likely to be causing the discrepancy.

The neglect of soil heterogeneity is also another possibility for the discrepancy between evapotranspiration and soil moisture. Additional runs were made with the PLACE model which incorporated soil moisture heterogeneity and three mosaic tiles in which the clay and sand fractions of the soil were varied. Both the soil moisture profile and the total annual evapotranspiration estimates were improved (see Wetzel et al., 1996-this issue). This will be the subject of future work, as PILPS specifications required an assumption of a homogeneous surface (PILPS standard run) for all but one of the experiments to date since not all SVATs incorporate heterogeneity. The majority of the SVAT's participating in PILPS had similar inconsistencies between the predicted total annual evapotranspiration, runoff, and soil water profile cycle. Further examination of the feedback mechanisms will be the topic of continuing PILPS work.

Acknowledgements

We gratefully acknowledge the inspired leader- ship of Ann Henderson-Sellers, and thank Yaping Shao and the entire PILPS team (both the "home team" and the "schemers") for their support, dis- cussions, and insights before, during and after the November 1994 workshop.

Appendix A. Numerical solution of the PLACE hydrology model

Substitution of Eqs. ( l i a ) and ( l l b ) into Eq. (10) yields

A Z i ~ = F i - F i_ 1 -- S i (A.1)

where the water flux across an interface is defined as

F, = A i - K i (A.2)

The time differenced form of Eq. (A.1) is then written using an implicit time scheme as

O ) i " + l = O i " + ~ ( F i "+1 -- F"+I + F i " - F i -1 )

- ~,S 7 (A.3)

where sci = A t / A z i and S 7 = (EsM)']/A + 0 7 + Ui". This scheme is stable for all time steps. This equation is solved using the Newton-Raphson method (Abramopoulos et al., 1988). The equations are solved simultaneously for the entire soil profile.

A.1. Solution method

At each time step, an iterative procedure is set up in order to solve Eq. (A.3) for the N variables at the future time step (n + 1). Eq. (A.3) represents a non-linear system of N equations, so that we write the Newton-Raphson method as

~kjk = _ ~ (A.4)

where y and y are vectors of length N, J is the N X N Jacobian coefficient matrix, and the super- script k is an index which represents the iteration. The solution is given by

~k = _ ( j k ) - I ~k (g .5 )

The solution vector ~ is used to compute the new estimate of 07 ÷ J;

(o/ .+,1 '+ ' (o:+, = ) +Yi (A.6)

where i = 1 .... N. Eq. A.5 is iterated until a sufficient tolerance is reached, defined by the norm of the vector ~k.


The forcing vector ~ is obtained by setting Eq. (A.3) equal to zero for i = 1 . . . . . N;

~i ( Fin+ l -- Fn-J-i I + Fi" - F." g , = ~)i n + l - Oi n - - 2 '-- ' - 1 )

+ ~iS.~ (A.7)

The upper and lower boundaries need special treatment. The flux at the top of the model domain F 0 at z0 = 0 represents infiltration and is computed as F 0 = -min[P ,10] . Precipitation and dew first enter the interception store, where evaporation can occur. Fractional coverage of water on the surface is also computed. Excess moisture after interaction with the surface store then reaches the hydrological model surface as P. I 0 is the maximum rate of infiltration defined as

10 = ~0k+ 1 ~ 5 ~/--2 ( 1 . 8 )

This top boundary condition is similar to those used by Abramopoulos et al. (1988) and Mahrt and Pan (1984). During precipitation or condensation events, the surface is assumed to be saturated. The gradient of soil moisture is assumed to vary linearly from the surface to the middle of the uppermost model layer. The delta function ~0 depends upon the surface temperature T s. It is equal to zero if the surface is frozen, otherwise it is equal to one.

Surface runoff occurs if the water reaching the surface exceeds the maximum infiltration rate so that

='-'0 = max(0 ,P - I0) (A.9)

where ="0 denotes Horton surface runoff. If the surface is frozen, ,-,~0 = P so that Horton runoff occurs using Eq. (A.9). If the moisture content of the surface layer exceeds saturation at the end of the time step, then runoff is computed as

~' = max(0,O~ + ' - O+) /~ 1 (A.10) ~-" 1

where ~ t denotes Dunne surface runoff. This form of surface runoff occurs, in general, if either the water table is near the surface (the entire soil water profile is at or near saturation) or there is a subsur- face layer which is partially or totally frozen. During time periods when precipitation exceeds evaporation, water can collect in the model layers above a frozen

layer because gravitational drainage and diffusion (downward) are hindered. This can result in saturated layers near the surface. Total surface runoff is obtained by simply adding Eqs. (A.10) and (A.9).

The flux at the lower boundary is F N = - f K N, where diffusion at the lower boundary is neglected ( A N = 0).

The Jacobian matrix is of the form

O + / O 0 ~ '+ I

0 n+l J = g2//O01

OgN//O0~ +1

~ , / a o ; + ' .. a , , / a o ; +1

• . g2/V,.]N

a ~ N / a O U ' .. o~N/aO~ +'

(A.11

The Jacobian matrix is tridiagonal which results in rapid convergence of Eq. (A.5). The non-zero ele- ments of the matrix are evaluated by taking derivatives of Eq. (A.7) with respect to the (9 n+l values at i - l, i, and i + l ;

Ogg ~ i ( OFin_+tl ) 06) i" +l I 2 ~ -i i -]

c~Oi,,+------ [ - 1 - "-~ 00, "+1 O0 i'+ ]

Ogi e i ( OFi n+ l )

0(~i++11 2 I 0(~i++11

For the purpose of taking derivatives, it is convenient at this point to define the diffusion term as a product of two terms, A i = 2 i A i. .~g represents the diffusion coefficient [but is opposite in sign to d in Eq. (6)], and A~ is the vertical gradient of volumetric water content. The product .-~A represents the so- called restore term because it is opposite in direction (sign) to gravitational drainage K for the same level except in the presence of a wetting front. The inter- facial diffusion coefficient, inter-facial gravitational drainage, and gradient of volumetric water content are written as

~'~i : ~"~i~i b+ 2, K i = 7~) i~ i 2b+ 3, and

(Oi- 0i+1) A i --

8Zi

Aaro Boone, PJ. Wetzel / Global a'nd Planetary Change 13 (1996) 161-181 179

where the constants ~" and ~7 are defined as

= b¢~k ,Os b-3 and n = k~072b-3

The flux derivatives can now be written as

OFin+ 1 ~ in+ 1 n + l

- - -- - - "JF ~Bri( A i ~X i , i - ~13i,i) O~)in+ 1 8Z i

OFf+, .~f+ - - + ,~,( a ; + 'cx,,,+ , - ,7~o,,,+ ~ )

OO in++l l ~ Z i

O(O:+l) x,~= 0o;+ ~ =(b+2)(O;+l) ~+ o0;--=~

O( ~in+ l ) 2b+ 3

~lPi,j = ¢9~)jn + I

~ n + I " 2 b + 2 0 ~ i n + 1 = ( 2 b + 3 ) ( 0 , ) - - ~

o6); +

where the derivative 00/~+ l /0Op+l can be evaluated either analytically, or approximated using for- ward differences (if the function is too cumbersome to evaluate analytically). The inter-facial volumetric water content derivatives q~,) and X~j are non-zero when j = i o r j = i + l .

If the updated value of volumetric water content exceeds the saturation value at the end of a time step, this excess water is removed as saturated runoff. It is computed at each time step for each layer as

~' = max(0, 0 i - Os )J~ i (A.12)

where ~ is the runoff due to super-saturation. Super-saturation can occur if the time step is large, but this form of super-saturation is usually a negligible portion of the water budget. If the lower boundary flux coefficient f is very small, super-saturation can occur unless the super-saturation prevention option is in force (see Appendix B).

The total lateral runoff for each model layer is

Ri = Di + ~Z (1 .13)

The PLACE soil water budget at each time step n can then be written as

N [ (O# - O#- P " - - ~ " ; = E -~-i

i = 1

( FV + F7 - l ) +

2

~) +_=,"+s; ']

(A.14)

where the LHS of Eq. (A.14) represents the model water source, the first term on the RHS is the change in model storage, and the remaining terms on the RHS represent the various sinks.

A p p e n d i x B . P r e v e n t i o n o f s u p e r - s a t u r a t i o n

Once Eq. (A.6) has been iterated to a sufficient tolerance and the volumetric soil water profile at the new time step has been determined, one final pass is made through the profile. This final check is for super-saturated soil layers.

The procedure to prevent super-saturation starts with the lowest model layer (i = N). The layer-averaged volumetric water content tendency equation [Eq. (1.3)] can be written for the lowest model layer a s

07+ 1 ~N, , + , _ F ~ + ~ + F 7 _ F T _ l ) = o7 + T ( F ; N-

-- ~ u S ~ ( B . 1 )

The lowest model layer is outlined in Fig. 1. The lower boundary flux terms, F N, and the moisture sink term, S N, can cause moisture loss from this layer, so the only moisture sources for this layer are the upper boundary flux terms F N_ r These are the terms which can cause super-saturation.

If O7 + 1 > O 5 at the end of the time step and N > M, this excess moisture is put back into the source layer by limiting the upper-boundary flux into this layer, and recalculating the layer-averaged volumetric water content. The upper boundary flux at the end of the current time step is recalculated as

?'+~u- = F'+IN-I + 2(O~ +1 -- 05)~SEN (8.2)

where F is the modified flux which does not follow Darcian motion. The upper-boundary flux F" is N - I not altered because it was used in the calculations during the previous time step. Substitution of K"+~ * N - l [Eq. (B.2)] for FAT + i I in Eq. (B.1) yields the modified layer-averaged volumetric water content in the lowest model layer, - - ' + l 0 N = 0 5 ,

The layer-averaged volumetric water content 0 , + 1 = ¢ ¢ F . + 1"~ N-~ "" u -~ ' , SO that Eq. (A.3) for the layer


i = N - 1 m u s t be a l te red to inc lude the m o d i f i e d

flux ,~:

N - l N - 1

+ ~:N-,2 ~'N-~/'Kn+~--F;+~ +FN, _ , _ F ~ , _ 2 )

-- ~N_~S~_, ( B . 3 )

S u b s t i t u t i o n o f Eq. (B.2) into Eq. (B .3) y ie lds the

m o d i f i e d l aye r - ave raged v o l u m e t r i c wa te r c o n t e n t in

the l ayer a b o v e the sa tura ted layer:

-~v+l = 0•+-I + (O~v +' - O , )SCN- , /~N ( B . 4 )

Th i s e x a m p l e is for M = N - I . I f M < N - 1,

the va lues o f Oi "+ ~ are m o d i f i e d in each success ive

m o d e l l ayer m o v i n g u p w a r d us ing the appropr ia te

index wi th Eqs . (B .2) and (B .4 ) for i > M. R u n o f f

due to super - sa tu ra t ion , Hi, is ca lcu la ted [see Eq.

(A.12) ] a f te r the f luxes h a v e b e e n m o d i f i e d and the

new l a y e r - a v e r a g e d v o l u m e t r i c wa te r con t en t prof i le

has b e e n de t e rmined . Th i s r u n o f f is equa l to zero for

all l ayers in w h i c h Eq. (B.2) has b e e n appl ied to the

u p p e r b o u n d a r y flux.

Phys ica l ly , this a d j u s t m e n t s c h e m e only a l lows

e n o u g h w a t e r to m a i n t a i n sa tu ra t ion to f low d o w n

into a layer . The lateral f low and g rav i t a t iona l

d ra inage are r educed in the sa tu ra ted mode l layers .

In the m o d e l layers i < M, m o r e mo i s tu re is avai l -

able for evapora t i on , p lan t s torage , and h o r i z o n t a l

d i scharge .

References

Abramopoulos, F., Rosenzweig, C. and Choudhury, B., 1988. Improved ground hydrology calculations for Global Climate Models (GCMs): Soil water movement and evapotranspiration. J. Climate, 1: 921-941.

Andrr, J.-C., Goutorbe, J.-P., Perrier, A., Becker, F., Besse- moulin, P., Bougeault, P., Brunet, Y., Brutsaert, W., Carlson. T., Cuenca, R., Gash, J., Gelpe, J., Hildebrand, P., Lagouarde, J.-P., Lloyd, C., Mahrt, L., Mascart, P., Mazaudier, C., Noil- han, J., Ottlr, C., Payen, M., Phulpin, T., Stull, R., Shuttle- worth, J., Schmugge, T., Taconet, O., Tarrieu, C., Thepenier, R.-M., Valencogne, C., Vidal-Madjar, D. and Weill, A., 1988. Evaporation over land-surfaces: First results from HAPEX- MOBILHY special observing period. Ann. Geophys., 6: 477- 492.

Avissar, R. and Pielke, R.A., 1989. A parameterization of heterogeneous land surfaces for atmospheric numerical models and

its impact on regional meteorology. Mon. Weather Rev., 117: 2113-2136.

Betts, A.K., Ball, J.H. and Beljaars, A.C.M., 1993. Comparison between the land surface response of the ECMWF model and the FIFE-1987 data. Q. J. R. Meteorol. Soc., 119: 975-1001.

Bougeault, P., Bret, B., Lacarrbre, P. and Noilhan, J., 1991. An experiment with an advanced surface parameterization in a meso-beta-model. Part II: The 16 June 1986 simulation. Mon. Weather Rev., 119: 2374-2392.

Brooks, R.H. and Corey, A.T., 1966. Properties of porous media affecting fluid flow. J. lrrig. Drain. Am. Soc. Civ. Eng., IR, 2: 61-88.

Capehart, W.J., 1992. Construction of a meteorologically driven substratum hydrology model. Thesis. Penn. State Univ., 151 pp.

Cosby, B.J., Homberger, G.M., Clapp, R.B. and Ginn, T.R., 1984. A statistical exploration of the relationships of soil moisture characteristics to the physical properties of soils. Water Re- sour. Res., 20: 682-690.

Dickinson, R.E., 1984. Climate Processes and Climate Sensitivity. (Geophys. Monogr., 29; Maurice Ewing Ser., 5). Am. Geo- phys. Union, Washington, DC.

Dickinson, R.E., Henderson-Sellers, A. and Kennedy, P.J., 1993. Biosphere-Atmosphere Transfer Scheme (BATS) Version le as coupled to the NCAR Community Climate Model. NCAR Tech. Note, TN-387 + STR, 72 pp.

Entekhabi, D. and Eagleson, P.S., 1989. Land surface hydrology parameterization for atmospheric general circulation models including subgrid scale spatial variability. J. Climate, 2: 816- 831.

Goutorbe, J.-P., Noilhan, J., Valancogne, C. and Cuenca, R.H., I989. Soil moisture variations during HAPEX-MOBILHY. Ann. Geophys., 7: 415-426.

Henderson-Sellers, A., Yang, Z.-L. and Dickinson, R.E., 1993. The project for intercomparison of land-surface parameterization schemes. Bull. Am. Meteorol. Soc., 74: 1335-1349.

Koster, R. and Suarez, M., 1992. Modeling the land surface boundary in climate models as a composite of independent vegetation stands. J. Geophys. Res., 97: 2697-2715.

Mahfouf, J.-F., Richard, E. and Mascart, P., 1987. The influence of soil and vegetation on the development of mesoscale circu- lations. J. Climate AppI. Meteorol., 26: 1483-1495.

Mahrt, L. and Pan, H., 1984. A two-layer model of soil hydrology. Bound.-Layer Meteorol., 29: 1-20.

McCumber, M.C. and Pielke, R.A., 1981. Simulation of the effects of surface fluxes of heat and moisture in a mesoscale numerical model 1. Soil Layer. J. Geophys. Res., 86: 9929- 9938.

Meehl, G.A. and Washington, W.M., 1988. A comparison of soil moisture sensitivity in two global climate models. J. Atmos. Sci., 45: 1476-1492.

Mihailovic, D.T. and Rajkovic, B., 1991. A coupled soil moisture and surface temperature prediction model. J. Appl. Meteorol., 30: 812-822.

Milly, P.C.D., 1992. Potential evaporation and soil moisture in general circulation models. J. Climate, 5: 209-226.

Mitchell, J.F.B. and Warrilow, D.A., 1987. Summer dryness in

Aaro Boone, PJ. Wetzel / Global and Planetary Change 13 (1996) 161-181 181

northern mid-latitudes due to increased CO 2. Nature, 330: 238-240.

Noilhan, J. and Planton, S., 1989. A simple parameterization of land surface processes for meteorological models. Mon. Weather Rev., 117: 536-549.

Pinty, J.P., Mascart, P., Richard, E. and Rosset, R., 1989. An investigation of mesoscale flows induced by vegetation inho- mogeneities using an evapotranspiration model calibrated against HAPEX-MOBILHY data. J. Appl. Meteorol., 28: 976-992.

Pitman, A.J., Henderson-Sellers, A., Abramopoulos, F., Avvisar, R., Bonan, G., Boone, A., Cogley, J.G., Dickinson, R.E., Ek, M., Entekhabi, D., Famiglietti, J., Garratt, J.R., Frech, M., Hahmann, A., Koster, R., Kowalczyk, E., Laval, K., Lean, L., Lee, T.J., Lettenmaier, D., Liang, X., Mahfouf, J.-F., Mahrt, L., Milly, C., Mitchell, K., De Noblet, N., Noilhan, J., Pan, H., Pielke, R., Robock, A., Rosenzwig, C., Running, S.W., Schlosser, A., Scott, R., Suarez, M., Thompson, S., Verseghy, D., Wetzel, P., Wood, E., Xue, Y., Yang, Z.-L. and Zhang, L., 1993. Results from the off-line control simulation phase of the Project for Intercomparison of Landsurface Parameterization Schemes (PILPS). GEWEX Tech. Note, IGPO Publ. Ser., 7, 47 pp.

Richards, L.A., 1931. Capillary conduction of liquids through porous mediums. Physics, 1 : 318-333.

Sellers, P.J., Mintz, Y., Sud, Y.C. and Dalcher, A., 1986. A Simple Biosphere (SiB) for use within general circulation models. J. Atmos. Sci., 43(6): 505-531.

Shao, Y., Anne, R.D., Henderson-Sellers, A., Irannejad, P., Thor- ton, P., Liang, X., Chen, T.H., Ciret, C., Desborough, C., Barachova, O., Haxeltine, A. and Ducharne, A., 1995. Soil moisture simulation, A report of the RICE and PILPS Work- shop. GEWEX Tech. Note, IGPO Publ. Ser., 14, 179 pp.

Verseghy, D.L., 1991. CLASS-A Canadian land surface scheme for GCMs I. Soil model. Int. J. Climatol., 11:111-133.

Wetzel, P.J. and Boone, A., 1995. A parameterization for Land- Atmosphere-Cloud-Exchange (PLACE): Documentation and testing of a detailed process model of the partly cloudy boundary layer over heterogeneous land. J. Climate, 8: 1810- 1837.

Wetzel, P.J. and Chang, J.T., 1988. Evapotranspiration from nonuniform surfaces: A first approach for short-term numerical weather prediction. Mon. Weather Rev., 116: 600-621.

Wetzel, P.J., Liang, X., Irannejad, P., Boone, A., Noilhan, J., Shao, Y., Skelly, C., Xue, Y. and Yang, Z.L., 1996. Modeling vadose zone liquid water fluxes: Infiltration, runoff, drainage and interflow. Global Planet. Change, 13: 57-71.

Xue, Y., Sellers, P.J., Kinter, J.L. and Shukla, J., 1991. A simplified biosphere model for global climate studies. J. Cli- mate, 6: 345-364.

Issues related to low resolution modeling of soil moisture ...

Documents

Transcript of Issues related to low resolution modeling of soil moisture ...