impact of soil heterogeneity in a mixed-layer model of the

Hydrological Sciences—Journal—des Sciences Hydrologiques, 43(4) August 1998 Special issue: Monitoring and Modelling of Soil Moisture: Integration over Time and Space 633

Impact of soil heterogeneity in a mixed-layer model of the planetary boundary layer

CORNELIS P. KIM Boston Consulting Group, J. F. Kennedylaan 100, 3741 EH Baarn, The Netherlands

DARA ENTEKHABI 48-331 Ralph M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA e-mail: [email protected]

Abstract The dynamics of soil moisture and temperature states are forced by land surface fluxes that form the coupling between the soil and the atmosphere. This two-way interaction between the soil and the atmosphere results in feedback mechanisms that affect the temporal patterns of variability of the system. To understand these characteristics of soil moisture and temperature variability, a one-dimensional coupled mixed-layer model of the land surface and the planetary boundary layer energy and humidity budgets is constructed. To allow integrations on the order of days, the model includes both the growth phase and the collapse phase of mixed layer evolution. The model includes the radiative effects due to the presence of clouds. Given the incoming shortwave radiation, lapse rates of temperature and humidity above the mixed layer and lateral windspeed, the model predicts all longwave radiative and turbulent surface heat fluxes. The model is shown to capture accurately observations taken during the FIFE field experiment. With this model, the impact of soil heterogeneity on the evolution of the surface and mixed-layer energy and humidity budgets is examined, using hydraulic properties from sand, loam and clay soils. It is assumed that the small correlation length scale of soil hydraulic properties and surface inhomogeneities are averaged out in the mixed layer. Relative to a uniform soil, heterogeneity increases the spatial mean latent heat flux for conductive soils and decreases it for unconductive soils, due to decreased and increased percolation respectively. Locally decreasing latent heat fluxes cause a warmer and drier mixed layer and through that an increase of the latent heat flux over other areas. This mixed-layer feedback reduces the impact of heterogeneous surface fluxes.

Influence de l'hétérogénéité du sol sur un modèle de couche de mélange de la couche limite planétaire Résumé Cet article présente un modèle de couche de mélange couplant les bilans d'eau et d'énergie de la surface terrestre et de la couche limite planétaire. Pour simuler des périodes de l'ordre de plusieurs jours, le modèle prend en compte l'épaississement et l'amincissement de la couche de mélange. Le modèle prend également en compte le rayonnement dû à la présence de nuages. A partir du rayonnement incident, des taux de variation de la température et de l'humidité au-dessus de la couche de mélange, et de la vitesse latérale du vent, le modèle prédit le rayonnement thermique et le flux turbulent de chaleur dans la couche de surface. Le modèle reproduit précisément les observations réalisés sur le terrain au cours des expériences FIFE. A partir de ce modèle, il a été possible d'étudier l'influence de l'hétérogénéité du sol sur l'évolution des bilans d'eau et d'énergie de la surface terrestre et de la couche de mélange en utilisant les propriétés hydrauliques des sables, limons et argiles. Les effects dûs aux propriétés hydrauliques des sols et aux hétérogénéités de surface sont moyennes dans la couche de mélange. Par rapport à un sol uniforme, l'hétérogénéité provoque une augmentation du flux moyen de

Open for discussion until I February 1999

mailto:[email protected]

634 Cornells P. Kim & Dara Enlekhabi

chaleur latente pour les sols conducteurs, et sa diminuation pour les sols non conducteurs, étant donné que la percolation subit respectivement une croissance et une décroissance. Les flux de chaleurs localement plus faibles sont à l'origine d'une couche de mélange plus chaude et plus sèche, et de ce fait, d'une augmentation du flux de chaleur latente vers les autres régions. Cette rétroaction de la couche de mélange réduit l'influence des flux des surfaces hétérogènes.

NOTATION

a soil scale factor (-) ac cloud shortwave albedo (-) ag ground shortwave albedo (-) ax empirical constant in equation (15) (W m2) a2 empirical constant in equation (16) (-) B Bowen ratio (HUE) (-) Cm volumetric heat capacity of minerals (J m 3 K 4 ) Cs volumetric soil heat capacity (J iff3 K4) Cw volumetric heat capacity of water (J m 3 K4) cp dry air specific heat (J kg"1 K4) D, 2 dissipation of mechanical turbulent energy (m3 s"3)

did relative distance to the sun (-)

E évapotranspiration flux (kg m 2 s4) Etop dry air entrainment (kg mf2 s4)

Psl e vapour pressure (~zr7zr) (Pa)

RdIRV

fc cloud fraction (-) G soil heat flux (Rn - XE - H) (W m"2) G* production of mechanical turbulent energy (m3 s3) g acceleration of gravity (m s"2) H sensible heat flux (W mf2) Htop sensible heat entrainment (W m"2) Hv virtual heat flux (W nT2) h mixed-layer height (m) / incoming solar radiation at h (W rrf2) i soil column indicator (-) k hydraulic conductivity (m s"1) ks saturated hydraulic conductivity (m s"1) L Monin-Obukhov length (m) m exponent in longwave radiation expressions (-) N number of soil columns (-) n parameter in equations (36) and (37) (-) ph pressure at h (Pa) ps surface pressure (Pa) Qp percolation flux (m s4) q specific humidity (kg kg4)

Impact of soil heterogeneity in a mixed-layer model of the planetary boundary layer

q* saturated specific humidity (kg kg"1) Ra total downwelling longwave radiation at h (W m"2) Rad clear sky downwelling longwave radiation above h (W m"2) Rcd downwelling longwave radiation from clouds (W m 2 ) Rcs cloud shortwave radiation absorption (W m"2) Rd gas constant for dry air (J kg"1 K"1) Rgr surface reflected longwave radiation (W m"2) Rgu upwelling longwave radiation (W m"2) R„ surface net radiation (W m"2)

1 , Rmr photosynthetically active radiation (-R ) (W m )

2 -v

Rs incoming shortwave radiation (W m"2) Rsd downwelling longwave radiation within PBL (W m"2) Rsu upwelling longwave radiation within P B L (W m"2) Rv gas constant for water vapour (J kg"1 K"1) ra aerodynamic resistance (s m"1) rs stomatal resistance (s m"1) rs m mminimal stomatal resistance (s m"1) S solar constant (W m"2) s soil saturation (-) Th temperature at h (K) Ts surface temperature (K) u lateral windspeed (m s"1) u* friction velocity (m s"1) z, soil thermal depth (m) zw soil water depth (m) z0m roughness length for momentum (m) z0h roughness length for heat (m) y psychrometric constant (Pa K"1) yq lapse rate of q above h (m"1) ye lapse rate of 9 above h (K m"1) ô ? inversion strength of q (kg kg"1) ôg inversion strength of 0 (K) ea mixed-layer bulk emissivity (-) EC cloud emissivity (-) ed mixed-layer downward emissivity (-) es soil emissivity (-) zu mixed-layer upward emissivity (-) 9 mixed-layer potential temperature (K) S5 saturated moisture content (-) K von Karman constant (-) X latent heat of vapourization (J kg"1) £, parameter in equation (20) (m"1) pa air density (kg m"3)

636 Cornells P. Kim & Dara Entekhabi

plv water density (kg m~ ) T„ shortwave transmissivity above h (-) xc cloud shortwave transmissivity (-) o relative humidity (-) ucrit relative humidity at onset of cloudiness (-) ulot relative humidity at total cloudiness (-) <|> solar zenith angle (rad) v|/ soil matric head (m) \|/lim limiting matric head in equation (17) (m) i|/wiltwilting point matric head (m) \\is length scale in equation (36) (m)

INTRODUCTION

The land surface and the planetary boundary layer (PBL) form a coupled system: the exchanges of water and energy depend on, as well as change, the temperature and humidity profiles in both. Depending on the space and time scales, this mutual interaction has to be considered with varying degrees of completeness. On the characteristic space scale that soil physicists are concerned with (<102 m), the meteorological conditions can be safely regarded as exogenous. Likewise, on the typical time scale of short-term weather forecasting, the dynamics of the soil moisture state of the land surface is generally small enough to allow for only a rudimentary treatment in atmospheric models. For longer range weather forecasting and climate studies however, the complete coupled system must be taken into account.

The interaction between (uniform) soil hydrology and PBL development was studied by Pan & Mahrt (1987). They showed that the time evolution of the surface heat fluxes and, therefore, the development of the PBL, greatly depend on the soil properties. For a wide range of initial soil moisture conditions, Ek & Cuenca (1994) examined the effect of varying the exponent in the Clapp & Hornberger (1978) parameterization of soil hydraulic properties. Despite the small time scale of their computations (6 h), they found a significant dependence of the surface fluxes on the soil properties, especially for moderately dry initial conditions.

In coupling the land surface and the PBL, one invariably runs into the problem of heterogeneity along the land surface. The most common route taken around this is the specification of large-scale mean or effective parameters, as is done for example in the big leaf soil-vegetation-atmosphere-transfer schemes (SVATs). Due to the nonlinearity of the processes involved, mean parameters are by definition invalid and effective parameters are unlikely to be effective under all conditions and for all fluxes. Indeed, Lhomme et al. (1994) showed that there is no single way to define effective parameters over heterogeneous terrain; effective parameters depend on the quantity that is conserved. For the water budget of the unsaturated zone, Kim et al. (1997) reached similar conclusions. Quinn et al. (1995) linked a model of daytime PBL growth to a spatially-distributed model of macroscale hydrology that includes

Impact of soil heterogeneity in a mixed-layer model of the planetary boundary layer 637

topographical controls. They found spatial heterogeneity to have a distinctive impact on the evolution of the PBL during daytime growth.

The objective of this paper is to examine the impact of soil heterogeneity on the development of the PBL and the subsequent effect of the PBL response to the heterogeneity of the surface fluxes. Since, as mentioned before, the characteristic time scale of soil hydrology is long, we consider a time scale of the order of days, similar to Pan & Mahrt (1987). Because of the long time scales, detailed account is given to the parameterization of the longwave radiative fluxes. As shown by Kim & Entekhabi (1997a), longwave radiative feedbacks are important on longer time scales. We examine the impact of heterogeneity of the hydraulic properties present within three soil types. This type of heterogeneity is characterized by small correlation scales (10°~102 m). This scale is much smaller than the scale of the main boundary layer eddies and heterogeneity of the surface fluxes is therefore assumed to be averaged out by the PBL (e.g. Mahrt, 1996).

MODEL DEVELOPMENT

Budget equations

The model consists of N soil columns with an overlying perfectly mixed layer, as shown in Fig. 1. The principal state variables are the soil moisture and temperature states in column i, respectively s; and Tsi; the mixed layer height, h; potential

4 h\

£ 7 P ,

Q,

So.

o q

T

T<£>/f)

EJp

9n

A R. R„,

H XEt R f f f i I

f

asA

Fig. 1 Diagram illustrating the four budget equations (1M4): (a) the moisture budgets of the TV soil columns and the mixed layer; (b) the energy budgets for the N soil columns and the mixed layer, which is shown in more detail for soil column i and the mixed layer in (c).


temperature, 6; and specific humidity, q. The diagram in Fig. 1 leads to the following budget equations for non-

precipitating inter-storm conditions:

As ( E \

^>t = \t + Q») t = 1-N (1)

àt

àq d7

Pah^ = (E) + Elop (3)

de R,.+(Rg„ +Rgr Za-R^-K+W + H^ (4)

where a quantity (x) denotes the spatial mean

1 N

and the soil heat capacity is given by

Q,=.vAA,+(i + * , » X (5)

with the heat capacity of water (C,„) and mineral soil (C,„) respectively 4.19 and 1.94MJm3K4.

Equation (1) simply states that a soil column loses water through évapotranspiration and percolation. Equation (2) expresses that the soil is heated by shortwave and longwave radiation and dissipates heat by emission of longwave radiation and through the turbulent sensible heat (H) and latent heat (XE) fluxes. The downwelling longwave radiation partly originates from above the mixed layer (RJ), of which only a fraction sa (dependent on q) is absorbed in the mixed layer, and partly from radiation emitted by the mixed layer itself (Rsd). The sum of all the radiative terms is normally referred to as net radiation, Rn. We note that Ts is a "lumped" temperature state of the surface, the effective temperature at which the fluxes have a realistic diurnal behaviour. By adjusting the soil thermal depth, z„ a realistic diurnal behaviour of Ts and the fluxes, both with respect to amplitude and phase can be achieved, as will be discussed later. The mixed-layer humidity budget (equation (3)) consists of the spatially-average surface evaporation and the entrain-ment of overlying air due to mixed layer growth (normally Etop < 0). Finally, equation (4) indicates that the longwave radiative heating of the atmosphere has three components: (a) downwelling longwave radiation from above the mixed layer or overlying atmosphere (Ra, which may include the contribution of clouds, discussed later); (b) the average longwave radiation emitted by the land surface (Rgl); and (c) the longwave radiation that is reflected at the land surface (Rgr, which is zero in case the surface is a perfect black body). The sensible heating of the mixed layer results from the surface sensible heat flux (H) and the entrainment of warm air due to


convective growth of the mixed layer (Htop). Radiative cooling of the mixed-layer slab occurs through downward and upward longwave radiation (Rsd and Rm). Condensation of water vapour is assumed to occur above the mixed layer and does not appear in the energy budget of the mixed layer.

The parameterization of the radiative and turbulent fluxes, the growth and collapse of the mixed layer and entrainment of overlying air, the presence of clouds and the soil properties and heterogeneity are discussed below.

Radiation fluxes

The solar radiation received at the top of the mixed layer is defined as (Liou, 1980):

I = X"S[dd) C ° S < | ) (6)

where xa indicates the transmissivity of the overlying atmosphere (typically around 0.75, see Brutsaert, 1982), S is the solar constant, d the distance to the sun and <j) the solar zenith angle. We do not consider the absorption of shortwave radiation in the mixed layer, since absorption occurs mainly in the higher atmosphere (e.g. Liou, 1980). Thus when clouds are absent, the incoming solar radiation at the land surface, Rs, is equal to /. The model allows for the presence of clouds (discussed later) which complicates the computation of Rs, as shown in Appendix A.

The shortwave albedo of the land surface depends on the soil moisture status. We here parameterize that effect through the linear relationship:

a„ = 0.2 - 0.1s (7)

Because of the long time scales for which the model is intended, a detailed parameterization of the radiative fluxes is required. Brubaker & Entekhabi (1995) extended the solution of Brutsaert (1975) for atmospheric longwave radiation by allowing for a separate mixed layer. The various expressions are shown in Appendix B.

Turbulent fluxes

The resistance formulations for the sensible and latent heat flux read respectively:

tf^fc-e) (8)

^ = ^ ( 9 * ( « ) - ? ) (9)

where pa is the air density and q* the saturated specific humidity at the land surface:

640 Cornells P. Kim & Dara Entekhahi

Pc, = P.s

RaQ(\ + Q.6\q)

Rd eo q*(T,p) = ~~—exp

Rv p R. _L I T ~ T

0 1 .

(10)

(11)

with e0 = 611 Pa at T0 = 273K. The aerodynamic resistance is given by:

In L

\zo + In |^»L | -1^8

In;

KM*

+ 5 — + l i J J

L U Kll*

L<0

L>0

(12)

with the Monin-Obukhov length L given by:

L = -U* Pa

VQc„

(13)

The derivation of equation (12) for unstable conditions (L < 0) is based on the Dyer-Hicks profile expression and can be found in McNaughton & Spriggs (1986). The expression for stable conditions (L > 0) occurs along the same lines but now using the Dyer-Hicks profile expression for stable conditions (e.g. Brutsaert, 1982; pp. 73-74), assuming the surface layer (hs) to be equal to the thickness of the inversion layer at the surface. This shallow height is equal to the residual height of the nocturnal layer (h). During the day when H > 0, the problematic assignment of hs to residual h is not invoked since L < 0 in equation (12). In the analyses presented here only the results of Fig. 1 are affected by this approximation, and then only at night-time. Finally, the method for determining u* from the observed windspeed is based on the results of Sugita & Brutsaert (1990).

The stomatal resistance is formulated following Kim & Verma (1991):

If, (14)

with

/ . R„

v a , + « p a r

f2=(l + a2[q*(Ts,ps)-q]~

(15)

(16)


h log y - l o g y ,

logN>v 0

- l o g y ,

¥ s \|/lim

Vlim ^ ¥ ^ ¥ w i l <

¥ ^ ¥ w i l ,

(17)

where, following Kim & Verma (1991), a, and a2 are set to 50 W m"2 and 0.18 respectively. The increase of stomatal resistance related to the water availability is parameterized following Feddes et al. (1978). We here use logi|/lim = -5.0 m and

logent = " 1 6 0 m -

Growth and collapse of the mixed layer

The height of the PBL has a strong diurnal evolution. During the day the PBL grows mainly in response to the surface virtual heat flux. When the virtual heat flux vanishes the turbulence dissipates and the boundary layer collapses. During the night a stable layer is present with a height determined by the mechanically induced turbulence. Smeda (1979) presented a model that describes this diurnal evolution of the PBL height. Except for the short transition time period, the PBL height is governed by:

dh

àt

2(G*-D{+8D2)Q Hv

gh§e P«c,ôe (18)

where we use the virtual heat flux Hv = H + O.ôlOc^ (« H + 0.07XE) rather than the sensible heat flux H (as was done by Smeda, 1979) in order to also account for the buoyancy generated by the surface moisture flux. The inversion strength is ô9 and it is governed by an evolution equation that is specified below.

The turbulent energy production and dissipation terms are given as:

Gt u * u

A =w2*w(l-e~'")

A = 0.4 gh Hv

V P«£

(19)

(20)

(21)

where ô = 0 in stable conditions and 8 = 1 in unstable conditions. The value of '% is set to 0.0 nf ' throughout this paper to assure a sufficient collapse of the PBL.

In the transition from unstable to stable conditions (i.e. when the surface virtual heat flux vanishes) the PBL collapses because of the decay of turbulence. By assuming that H[op = 0 when Hv vanishes, Smeda (1979) derived that during the transition the PBL height is given by:

2Pacp(G* -D,)9 h = -

Hvg (22)


Equation (22) is to be applied just after the transition for a period of time until:

éh _ Hv

àt ~ p c 59 <0.05

H Pacp5Q

In order to determine the longwave radiative fluxes and the relative humidity at the mixed-layer top (both discussed later), the pressure at h is required. By assuming an adiabatic lapse rate in the mixed layer and a negligible difference between the geopotential and the actual height, the pressure at the mixed layer top is given by:

Ph = Ps gh

(23)

Entrainment of overlying air

The mixed layer is capped by inversions in the potential temperature and specific humidity profiles. The strengths of these step inversions, 8q and 69, determine the entrainment of the normally warm and dry (i.e. 8g < 0 and 88 > 0) overlying air:

dh £t0p = P A

" t o p = PaCp*B At

However, the inversion strengths themselves change over time according to:

dô(/ dh dq

~dT = Y« d7~d7

d§8 dh_ de ~d7~ = Ye d7~d7

The lapse rates yq and ye are forcing parameters.

(24)

(25)

(26)

(27)

Lapse rates and inversion strengths

When the lapse rate of temperature and humidity are not known, we assume exponential profiles for pressure, thermodynamic temperature, T, and vapour pressure, e (see Brutsaert, 1975) above the mixed layer:

P(z) = Ph exPl RdQ

[z-h] z> h (28)


T(z) = Th+exp\--~r[z-h] z>h (29)

e(z) = eh+ exp| 5.8-103~f + 5.5-10" [z-h] >h (30)

where Th+ = (Q + de)(p,,/ps) "Cp and yT is the lapse rate of T (positive when

temperature decreases with height). From equations (28)-(30) the profiles for 9 and q can be derived:

G(z) = 9;,+ exp h]

q(z) = q„+ e x P y 5.8-10J-™- + 5.5-10" 6 RdQ

[z-h]

z > h

>h

(31)

(30)

where Qh+ = 0 + 5e and qh+ = q + bq. From equations (31) and (32), the lapse rates ye and yq can be determined:

r \

Ye = d0 àz

àq

~àz

= 1 + -J CP

'Y:

/ = - 1 h

\ v 5 . 8 . 1 0 ^ + 5 . 5 . 1 0 ^ ^

(33)

(34)

Cloud topped mixed layer

When the relative humidity rises above 1, cloud formation will occur. However, since the mixed layer exhibits spatial inhomogeneities that are not incorporated in the model, cloud formation occurs at relative humidities smaller than 1 and complete cover may occur at relative humidities larger than 1. Ek & Mahrt (1991) examined the phenomenon of subgrid-scale cloud by assuming a normal probability distribution of relative humidity. They show that their results for cloud fraction (fc) can be expressed in a form that is often used for subgrid cloud parameterization:

fM u - u „ - t o t " c r i t

(35)

where the relative humidity at the mixed layer top is expressed as:

u = te-,/0 with q* given by equation (11), the temperature just underneath the inversion,


, «,,/<••„

Th_ =Q(p,Jps) " '', and ph is given by equation (23). Ek & Mahrt (1991) suggest u„,t = 0.9, otol = 1.05 and/ = 2.

Since the model provides no information on e.g. cloud liquid water content, cloud thickness and cloud temperature, we only include the impact of clouds on the shortwave radiation and extend the parameterization of longwave radiation fluxes by assuming clouds that behave as perfect black bodies. The parameterization of these effects can be found in Appendices A and B. The effect of clouds in the model is the reduction of incoming shortwave radiation during the day, and the reduction of radiative cooling at night.

Soil hydraulic properties, percolation and heterogeneity

The soil hydraulic characteristics are parameterized after Brooks & Corey (1966):

s(w) = \\}J (36)

1 v|/, <\ | /<0

k(s) = kss~ (37)

The parameterization of Clapp & Hornberger (1978), better known in the meteorological literature, is identical to that of Brooks & Corey (1966) with the parameter b = IIn.

By assuming downward water flow due to gravity alone, the Darcy equation that governs the percolation flux Qp in equation (1) simplifies to:

2+3»

Qp = k(s) = kss " (38)

Miller & Miller (1956) introduced the concept of geometrical scaling of soil hydraulic properties. Based on this theory, Kim et al. (1997) show that the local parameters \\>sj and ks?i are related to reference values according to:

¥.,, =a,"V,.ref (39)

ks,=<kXK( (40)

The heterogeneity of soil hydraulic properties can now be represented by means of probability distributions of the soil scale factor a and the parameter n. It is well established that the soil scale factor is a lognormally distributed random variable (e.g. Warrick et al., 1977; Hopmans & Strieker, 1989). Brakensiek et al. (1981) and Russo & Bresler (1981) found that within a textural class the exponent n can be represented by a lognormal distribution as well. The saturated moisture content 95 is taken as uniform in order to focus exclusively on the impact of heterogeneous soil water flow.


The properties of the three soil types considered here are summarized in Table 1. The lognormal probability distributions of a and n are characterized through the mean values (a) (= 1.0) and («) (summarized in Table 1) and the coefficient of variation CV. Correlation between the scale factor a and the exponent n is not considered here, although it may be easily incorporated.

Table 1 Brooks and Corey parameters for the three soils (after Bras, 1990).

Parameter

£s,ref (m day1) MVr (m) », <«>

Clay

2.94 x 10'2

-0.90 0.45 0.44

Loam

2.94 x 10 ' -0.45 0.35 1.2

Sand

2.94 x 10° -0.25 0.25 3.3

MODEL VERIFICATION

The above model is entirely forced by incoming shortwave radiation, the lapse rates ye and yq and the observed windspeed. The model has been extensively verified using observations of three so-called "golden days" during the First ISLSCP Field Experiment (FIFE). Because of the limited dynamics of soil moisture over one day, we set ds/di — 0. The impact of water availability in equation (17) is accounted for by specifying a spatially uniform value r!min.

Figure 2 shows some of the verification results of the model for the cloudless day 15 August 1987. The atmospheric turbidity ia in equation (6) was determined from using the shortwave radiation measurements directly. The lapse rates ye and yq, the windspeed and the initial atmospheric conditions were determined from the radiosonde measurements. The friction velocity u* was determined according to Sugita & Brutsaert (1990). Using z, = 0.10 m and rjmin = 70 s m"1, a realistic diurnal behaviour of the model was achieved. The model results were compared with the spatially average surface measurements compiled by Betts & Ball (1998) and the PBL observations of W. Brutsaert (see Strebel et al., 1994).

The downwelling longwave radiation at the surface (Rgd, equal to the second term in equation (2)) shown in Fig. 2(a) is somewhat high, explaining the slight overestimation of R„ in Fig. 2(b). Due to the determination of the ground heat flux as a residual in equation (2), the model slightly overpredicts IE and H (Fig. 2(b)) but the Bowen ratio is in good agreement with the measurements (Fig. 2(c)). The surface and PBL temperatures in Fig. 2(d) closely follow the measurements. The evolution of specific humidity (Fig. 2(e)) and PBL height (Fig. 2(f)) are in reasonable agreement with the observations. Overall, the correspondence between the model and the observations is satisfactory, especially since the effects of lateral advection were not incorporated. Similar results were obtained for the FIFE days 6 June and 11 July 1987, but these are not shown for brevity.


\Rs(i-ag:

u.uz

0.019

0.018

0.017

0.016

0.015 /À-A

+

_/-^t- /

+ \y

e

600

500

400

300

200

100

0

-100

315r

/ xARn

/ • ' / ' > . E ' - . \

f /H '^\ \L /> \ \

^ — T ^ T - v ^

b

200 228 227 227.2 227.4 227.6 227.8 228 227 227.2 227.4 227.6 227.i

Julian time [cj] Julian time [d] Fig. 2 Verification using FIFE data collected on 15 August 1987 (solid lines indicate the model results, dotted lines the average surface measurements and + the PBL observations; solar noon is at 227.5).

RESULTS

Computations have been performed for the three soil types in Table 1, both for the cases CVff = CV„ = 0 (uniform soil) and CVa = CV„ = 1 (heterogeneous soil). For the heterogeneous case, 900 realizations were generated, equidistant in a - n probability space. Depending on the soil type, computations were performed for 10-20 days using the forcing of 15 August repeatedly.

On this long time scale, the build up of temperature and humidity are avoided by controlling the lapse rates ye and y within a residual layer that is left behind when the


PBL collapses (e.g. Stall, 1988). The height of the residual layer is taken as 3/4/zmax, where hmiK is the maximum PBL height of the previous day. Within this residual layer y9 is set to 1 x 10 "3 Km"1 (i.e. 9 almost constant with height) and yq is set to half the value predicted by equation (34). Although these choices are arbitrary, they result in a stable model behaviour when soil saturation is kept constant. This ensures that the model results reflect the impact of soil heterogeneity alone. The soil water depth zw was set to 0.3m, a typical root depth for grass vegetation.

First, the model results are discussed with respect to the impact of soil heterogeneity on the mean surface fluxes and development of the PBL. The evolution of spatial heterogeneity of the surface fluxes is subsequently examined using a few statistical indicators. Finally, the role of interactive clouds is explored briefly.

Impact of soil heterogeneity on the mean surface fluxes and PBL development

By comparing the model results corresponding to the uniform and heterogeneous cases, the impact of soil heterogeneity can be inferred. It is important to keep in mind that the mean soil properties are identical for both cases. Hence, if the response of the model to soil heterogeneity were linear, both cases would give identical

uniform

188 189 190 191 192 Julian day

Fig. 3 Time evolution of the net radiation Rn (solid lines), latent heat flux XE (dashed lines) and sensible heat flux H (dotted lines) for (a) uniform loam soil and (b) heterogeneous loam soil. Drying of fully saturated soil columns started at day 180.


results. The results shown in this section were obtained by assuming saturated soil columns (s — 1) at Julian day 180. Clear skies (fc = 0) were imposed in order not to complicate the interpretation of the results. The effect of cloudy skies will be addressed later.

In Fig. 3, the surface energy budgets associated with the uniform and heterogeneous cases are depicted. It is clear that the latent heat flux declines rapidly for the uniform case. The corresponding higher soil temperatures increase the sensible heat flux and decrease the net radiation. For the heterogeneous case the changes are more gradual. Figure 4 shows the time evolution of PBL height,

(a) 4000

c)u.uit) ^T*1

roO.014 .̂ D)

^ >,0.012 T5

"E j2 0.01 o

1 0.008 CL

n nnK

A / y\ r~ „—

\r V

-

i(\ A

j \ r-n \ \ /,—--̂ ! Y y~~^ \

^"^— \ ^^ — j u_^-^

\ \ ' X ___ - " •>

\ v ..^ , - v

\ 185 186 187 188 189 190 191 192

Julian day

Fig. 4 Time evolution of (a) mixed-layer height h; (b) potential temperature 9; and (c) specific humidity q corresponding with the results shown in Fig. 3 for the heterogeneous loam soil (solid lines) and the uniform loam soil (dashed lines).


temperature and humidity for the two cases. The more sudden decrease of the latent heat flux over the uniform soil causes a faster deepening, warming and drying of the PBL. The deep boundary layers that develop as the soil dries out more would in the real world be constrained by large-scale phenomena and lateral advection, not incorporated in the model. The diurnal behaviour of specific humidity, an increase in the early morning followed by a decrease when the fast growth of the mixed layer causes larger dry-air entrainment, is in accordance with the FIFE observations (Betts & Ball, 1995).

Figure 5 summarizes the above behaviour for the three soils by means of the

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Fig. 5 Daily mean latent heat flux for heterogeneous soil properties (solid lines with o) and uniform soil hydraulic properties (dashed lines with +) for (a) sand soil, (b) loam soil and (c) clay soil.


daily mean latent heat flux. For the sand and the loam soil (Fig. 5(a),(b)), the latent heat flux decreases more slowly for the heterogeneous case than for the uniform case. The explanation is that for these soils, the mean properties (see Table 1) are quite favourable for percolation. Because of the lognormal distribution, it holds for the majority of soil columns that a < 1 and n<{n). These values result in smaller percolation losses and a higher spatial mean latent heat flux. For the clay soil in Fig. 5(c), the opposite is true. Here percolation rates associated with the mean soil properties in Table 1 are small. The majority of soil columns with a < 1 and n < (n) have only slightly smaller percolation losses, whereas the minority of columns for which a > 1 and n > (n) loose much more water due to percolation. The net result is an increase of the spatial mean percolation and, therefore, a smaller mean latent heat flux. For a model of the unsaturated zone with prescribed meteorological conditions and for the same mean soil hydraulic properties, Kim et al. (1997) found similar results for the average soil water budget over 15 years.

Evolution of surface flux heterogeneity

It is worthwhile to examine statistical indicators that characterize spatial heterogeneity of the surface fluxes other than the mean value discussed above. This will provide insight into the nature of spatial heterogeneity and, more importantly, identify

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feedbacks between the various soil columns due to their interaction through the PBL. The most complete description of surface flux heterogeneity is contained in the

histogram. As an example, in Fig. 6 the time evolution of the histogram of the daily mean latent heat flux is shown for the clay soil. It can be seen that over time, probability mass typically moves from higher to lower latent heat fluxes. However, from day 190 onwards, one can see probability mass moving to higher values as well. The mean, median, minimum and maximum values indicated above the histogram will be used to characterize the surface flux heterogeneity in a more condensed fashion.

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652 Cornel is P. Kim & Dara Entekhabi

For the three soil types, Fig. 7 shows the time evolution of the four statistical indicators. The daily mean latent heat flux was used previously for comparison with the uniform soil cases. As can be seen in Fig. 6, a median value that is larger than the mean value indicates a negatively skewed distribution and a median smaller than the mean a positively skewed distribution. Accordingly, one can see the skewness of the distribution of latent heat fluxes in Fig. 7 to change sign over time.

The most striking feature of all the graphs is that the maximum latent heat flux first decreases, only to increase later. The slight decrease is caused by cooling of the soil due to évapotranspiration and an increase of the surface albedo due to drying of the soil. The increase of the latent heat flux is explained in the diagram in Fig. 8 for the simple case of two soil columns. It can be seen that drying of column 1 causes a higher, drier and warmer mixed layer and causes an increase of the latent heat flux of column 2. Following the same logic for the results here, the decrease of the latent heat flux over a (large) fraction of the considered area leads to a drying and warming of the PBL. This causes the sensible heat flux to decrease and the latent heat flux to increase over the (small) fraction where évapotranspiration is limited by available energy rather than soil moisture. Thus, in the case of spatially heterogeneous surface fluxes, drying over one (relatively dry) region enhances the drying over the other (relatively wet) region, thereby initially enhancing and then reducing heterogeneity of the surface fluxes (see Fig. 7).

It is useful to draw a parallel with the heuristic complementary relationship introduced by Bouchet (1963), which states that the potential evaporation under stressed (i.e. "non-potential") conditions is higher than under unstressed (i.e. potential) conditions. Here a manifestation of a similar effect can be observed: in the columns where évapotranspiration is limited by available energy (i.e. potential évapotranspiration) the évapotranspiration increases due to the decrease of évapotranspiration in other columns (Kim & Entekhabi, 1997b).

Fig. 8 Diagram showing some feedbacks between a dry and a wet soil column. (For example, a decrease of soil saturation in column 1 decreases the latent heat flux, which decreases the mixed-layer specific humidity; this increases the latent heat flux in column 2 which enhances the depletion of soil moisture. Feedbacks related to changes in longwave radiation due to the changed conditions in the mixed layer and the land surface are not shown but are present in the model).

Column 1

Mixed Layer

Column 2


Effect of cloudiness

In order not to complicate the interpretation of the results, so far clear skies were imposed. In Fig. 9(a), however, one can see that this leads to a situation where the lifted condensation level (LCL) is lower than the mixed layer and clouds will therefore be present. Clouds form when the mixed layer starts to rise rapidly at the end of the morning and the mixed-layer top cools to the point that the air becomes saturated.

Inclusion of the effects of clouds on shortwave and longwave radiation results in a substantially different model. As mentioned, the effect of clouds is to reduce the available energy at the land surface, which reduces the surface heat fluxes. The reduced sensible heat flux causes a much smaller PBL growth relative to clear sky conditions (Fig. 9(b)) and therefore restricts the entrainment of the dry and warm air above. The daily mean latent heat flux in Fig. 10(a) is significantly smaller relative to the clear-skies case. Although, from day 190 onwards, the latent heat flux is slightly higher, due to the slower depletion of the soil water reservoir, overall more water is lost to percolation. The differences between the cases with and without the effect of clouds is reduced when the specific humidity has fallen low enough and the cloudiness disappears.

In Fig. 10(b) the daily mean latent heat flux corresponding to the clay soil case with interactive clouds is shown for comparison. For this case, the LCL and the

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mixed-layer height rise simultaneously, the LCL being a little higher during most of the drying period. Hardly any clouds are present therefore and the radiation budget of the surface is only slightly affected. This explains the only slightly lower latent heat fluxes between day 185 and day 192.

The more sudden decrease of the latent heat flux for all three uniform soil types discussed earlier leads to a faster growth of the mixed layer and, because of the lower temperatures at the mixed-layer top, to a more sudden and larger increase of cloudiness (not shown). In general, cloudiness tends to prolong the drying time (in the absence of percolation). This increases the influence of percolation, and therefore soil heterogeneity, on the soil water conditions. The results obtained emphasize the effect of both soil type and heterogeneity on the presence of clouds.

CONCLUSIONS

The impact of soil heterogeneity has been examined using a model of the coupled energy and humidity budgets of the land surface and the mixed layer. Heterogeneity of soil hydraulic properties is represented by N separate soil columns, each with their own hydraulic properties and soil moisture and soil temperature states. The characteristic time scale of soil water dynamics in the absence of precipitation (inter-storm


periods) is slow in comparison with mixing processes in the PBL. The model therefore contains a parameterization of hoth the mixed-layer growth and the collapse at the end of the day (Smeda, 1979). The long time scale also necessitates the inclusion of an interactive parameterization of longwave radiation (Brubaker & Entekhabi, 1995). The length scale at which soil heterogeneity is present is small so that the associated heterogeneity of surface fluxes is averaged out by the mixed layer. The presence of clouds and their impact on the shortwave and longwave radiative transfer is taken into account.

The model is forced only by incoming shortwave radiation, lateral windspeed and the lapse rates of temperature and humidity above the mixed layer. The model was shown to capture accurately observations taken during the FIFE experiment.

Based on the model results one can conclude that: (a) The impact of soil heterogeneity on the evolution of the surface fluxes energy

budget depends on soil type. For conductive soils, the heterogeneity decreases the spatially average percolation rate which results in an increase of the latent heat flux. For unconductive soils, the large spatially average percolation rates of the heterogeneous soil in comparison with the uniform soil lead to a decrease of the latent heat flux.

(b) Restricted évapotranspiration by soil moisture over a certain fraction leads to a warmer and drier mixed layer. This increases the évapotranspiration over the fraction where évapotranspiration can occur at an unrestricted rate. This constitutes a negative feedback: the response of the mixed layer to soil heterogeneity is extra drying over wet regions.

(c) The effect of cloudiness is to prolong the evaporative drying time and therefore increase the water loss due to percolation. The presence of cloudiness itself depends on the soil type and soil heterogeneity.

APPENDIX A

Shortwave radiation in the presence of clouds

Over the cloudy fraction, shortwave radiation is either transmitted (x(), reflected (ac) or absorbed (ec). For the shortwave optical properties of the cloud we approximate the results obtained by Liou (1976) for fair weather cumulus clouds by;

ac = 0.85-0.17cos<|) (Al)

xc = 0.12 + O.lOcoscj) (A2)

Sc = 0.03 + 0.07cos(|> (A3)

The total amount of shortwave radiation received at the land surface is given as:

*, = / ( 1 - / C [ 1 - T J ) Z ( / A W ) " (A4)

656 Cornel is P. Kim & Dara Entekhabi

where the normally rapidly converging series accounts for multiple reflections between the land surface and the clouds.

The total amount of shortwave radiation absorbed by the clouds, normalized by fc

reads:

R,.. = Ie. l + (0( l - / c [ l -T c ] )X/ £ ack (A5)

APPENDIX B

Longwave radiation

The longwave radiation emitted from the land surface is given by:

^ = ^ . x 4 (Bl)

where the surface emissivity is a parameter. The longwave radiation from the overlying atmosphere depends on the pressure,

temperature and humidity profiles, defined in equations (28)-(30). For clear skies, following Brutsaert (1975), Brubaker & Entekhabi (1995) derived the expression:

R„ 1.24a1 Ph9n+

IOYV rj-,4

(B2)

where m = 1/7, and the specific humidity and temperature just above the mixed layer. The psychrometric constant is y.

The incoming and outgoing longwave radiation are attenuated by absorption in the mixed layer. The expression for the column emissivity reads:

e„ =0.75 2 qpx

3 10g 1-

P. J (B3)

The term within the brackets represents the scaled amount of (mainly water vapour) mass in the air column in centimetres.

The expression for the longwave radiative fluxes at the top and bottom of the atmosphere due to radiation within the mixed layer may be given in terms of the potential temperature:

R* = ^ 9 "

R„, =CT£„

(B4)

(B5)

The effective emissivities of the mixed layer ed and e„ depend on the specific humidity and take into account the assumed profile of thermodynamic temperature. Following Brubaker & Entekhabi (1995) once more, the expressions for the


emissivities read:

2 qps Hi,,

^ = o.75|-^j «JyNi-j/r'dy (B6)

, 2 qp, <]/"'• (y-y h)"-Aày (B7)

where y = (p/ps)3'2- It holds that ed > s„.

The amount of total downward longwave radiation that is reflected at the land surface is:

*ff = (*J1-e„]+ * „ , ) ( ! - O (B8)

For reasons of simplicity we assume that the clouds: (a) are at thermodynamic equilibrium, (b) behave as perfect black bodies for longwave radiation, and (c) have a uniform temperature. Then, the downwelling longwave contribution for clouds, normalized by/c, reads:

*«, = ^ K + *«, + * -+ Rgu) + Rgr ( i -O (B9)

Finally, the total downwelling longwave radiation from above the mixed layer, Ra, is taken as the weighted average of the contributions from the fractions of clear sky and clouds.

Ra=d-L)Rad+LRcd (Bio)

Substitution of equations (8) and (9) in equation (10) and solving for Rd results in:

R, = •

(2-fc)Rad+fc(Rm+Rcs+V-za) « + ( l - e j ^

2 - / £ ( l - E f l )2 ( l - s , )

(BID

As expected, Ra = Rad when/c = 0. In case the soil surface is a perfect black body (ES = 1), no longwave reflection at the land surface occurs and equation (Bll) simplifies to:

Ra = Rud + v l R.s» + Res + a - 6a ){Rgll ) - R, (B12)

Equation (B12) is more easily interpreted than (Bll). It can be seen that half of the energy from the radiative fluxes absorbed by the cloud is emitted to the surface (the other half is emitted upwards). In equation (Bll), slightly more than half of the absorbed radiation is transmitted downwards, due to reflection at the soil surface, which also explains the presence of the (1 -es)i?srf term. Under most conditions, the correction term in the denominator in equation (Bll) is negligible.


REFERENCES

Betts, A. K. & Ball, J. H. (1995) The FIFE surface diurnal cycle climate. /. Geophys. Res. 100, 25,679-25,693. Belts, A. K. & Ball, J. H. (1998) FIFE surface climate and site-average dataset 1987-1989. J. Atmos. Sci. 55(7),

1091-1107. Bouchet, R. J. (1963) Evapotranspiration réelle et potentielle, signification climatique. In: General Assembly of Berkeley

(19-31 August 1963), 134-142. IAHS Publ. no. 62. Brakensiek, D. L., Engleman, R. L. & Rawls, W. J. (1981) Variation within texture classes of soil water parameters.

Trans. Am. Soc. Agric. Engrs 24(2), 335-339. Bras, R. L. (1990) Hydrology, An Introduction to Hydrologie Science, 643. Addison-Wesley, Reading, Massachusetts,

USA. Brooks, R. H. & Corey, A. T. (1966) Properties of porous media affecting fluid flow. /. Irrig. Drain. Div. ASCE

92(IR2), 61-88. Brubaker, K. L. & Entekhabi, D. (1995) An analytic approach to modeling the land-atmosphere interaction: 1. construct

and equilibrium behavior. Wat. Resour. Res. 31(3), 619-632. Brutsaert, W. (1975) On a derivable formula for long-wave radiation from clear skies. Wat. Resour. Res. 11, 742-744. Brutsaert, W. (1982) Evaporation into the Atmosphere. Reidel, Dordrecht, The Netherlands. Clapp, V. & Homberger, G. M. (1978) Empirical equations for some soil hydraulic properties. Wat. Resour. Res.

14(4), 601-604. Ek, M. & Cuenca, R. H. (1994) Variation in soil parameters: implications for modeling surface fluxes and atmospheric

boundary-layer development. Boundary-Layer Meteorol. 70, 369-383. Ek, M. & Mahrt, L. (1991) A formulation for boundary-layer cloud cover. Ann. Geophysicae 9, 716-724. Feddes, R. A., Kowalik, P. J. & Zaradny, H. (1978) Simulation model of the water balance of a cropped soil. PUDOC,

Wageningen, The Netherlands. Hopmans, J. W. & Strieker, J. N. M. (1989) Stochastic analysis of soil water regime in a watershed. / . Hydrol. 105,

57-84. Kim, C. P. & Entekhabi, D. (1997a) Analysis of feedbacks in the uncoupled and coupled land-surface and atmospheric

mixed-layer energy budgets. Boundary-Layer Meteorol. (in press). Kim, C. P. & Entekhabi, D. (1997b) Examination of two methods for estimating regional évapotranspiration using a

coupled mixed-layer and surface model. Wat. Resour. Res. 33(9), 2109-2116. Kim, C. P., Strieker, J. N. M. & Feddes, R. A. (1997) Impact of soil heterogeneity on the water budget of the

unsaturated zone. Wat. Resour. Res. 32(12), 3475-3484. Kim, I. & Verma, S. B. (1991) Modeling canopy stomatal conductance in a temperate grassland ecosystem. Agric. For.

Meteorol. 55, 149-166. Lhomme, J.-P., Chehbouni, A. & Monteny, B. (1994) Effective parameters of surface energy balance in heterogeneous

landscape. Boundary-Layer Meteorol. 71, 297-309. Liou, K.-N. (1976) On the absorptions refelection and transmission of solar radiation in cloudy atmospheres. /. Atmos.

Sci. 33, 798-805. Liou, K.-N. (1980) An Introduction to Atmospheric Radiation. Academic Press, New York. Mahrt, L. (1996) The bulk aerodynamic formulation over heterogeneous surfaces. Boundary-Layer Meteorol. 78, 87-

119. McNaughton, K. G. & Spriggs, T. W. (1986) A mixed layer model for regional evaporation. Boundary-Layer Meteorol.

34, 243-262. Miller, E. E. & Miller, R. D. (1956) Physical theory for capillary flow phenomena. / . Appl. Phys. 27, 324-332. Pan, H.-L. & Mahrt, L. (1987) Interaction between soil hydrology and boundary-layer development. Boundary-Layer

Meteorol. 38, 185-202. Quinn, P., Beven, K. & Culf, A. (1995) The introduction of macroscale hydrological complexity into land surface-

atmosphere transfer models and the effect on planetary boundary layer development. /. Hydrol. 166, 421-444. Russo, D. & Bresler, E. (1981) Soil hydraulic properties as stochastic processes: I. an analysis of field spatial

variability. Soil Sci. Soc. Am. J. 45, 682-687. Smeda, M. S. (1979) A bulk model for the atmospheric planetary boundary layer. Boundary-Layer Meteorol. 17, 411-

427. Strebel, D. E., Landis, D. R., Huemmrich, K. F. & Meeson, B. W. (1994) Collected data of the First ISLSCP Field

Experiment. In: Surface Observations and Non-image Data Sets (vol. 1). CD-ROM, NASA, GSFC, Greenbelt, Maryland, USA.

Stull, R. B. (1988) An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, Dordrecht, The Netherlands.

Sugita, M. & Brutsaert, W. (1990) Regional surface fluxes from remotely sensed skin temperature and lower boundary layer measurements. Wat. Resour. Res. 26(12), 2937-2944.

Warrick, A. W., Mullen, C. J. & Nielsen, D. R. (1977) Scaling field-measured soil hydraulic properties using a similar media concept. Wat. Resour. Res. 13(2), 355-362.

impact of soil heterogeneity in a mixed-layer model of the

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