## 1. Introduction

In mesoscale and general circulation models the proper representation of the surface energy balance is essential for successful long-term simulations and climate studies. From among the many obstacles toward this goal only two are mentioned here: (i) the energy partitioning in general depends on a great variety of soil, surface, and vegetation parameters, of which only a few are routinely available; and (ii) those parameters that are available are usually representative of spatially homogeneous conditions, whereas the actual surface within a grid-cell area of the model is often composed of many different surface types. The parameter values applied in numerical models therefore need to be *effective* values in a double sense: they represent the net effect of a range of physical processes on the energy partitioning at the surface (like the moisture availability or the surface resistance, for example) and they need to account for subgrid-scale surface effects that are unresolved by the model. In the present study the focus lies entirely on the second aspect; however, in order to specify the surface properties most suited for spatial averaging the first aspect-should be kept in mind.

A detailed examination of textural aspects of subgrid-scale variability on the spatial aggregation of fluxes is achieved by a numerical experiment in a one-way nesting approach, where a hypothetical grid cell of a large-scale model forms the model domain of a high-resolution model that contains the complete description of the heterogeneous surface explicitly. To allow a structured analysis of the aggregation problem, this work concentrates on a simple description of the surface and its variability: at the surface, fluxes are controlled by a net surface resistance that implicitly includes any canopy, stomatal, or subsoil effects. In addition, to examine surface texture effects in isolation, all other model parameters are held constant. Thus, time-dependent and transient effects are explicitly and deliberately excluded.

A comprehensive discussion of grid-averaged surface fluxes is given by Mahrt (1987). He decomposed any given flow variable *ψ* at the lowest modeling level into three portions, the first being the value resolved by the grid; the second a local time-averaged part, including all motions on spatial scales smaller than grid scale but larger than turbulent scale (corresponding to the averaging time); and the third representing all remaining turbulent fluctuations with timescales smaller than the averaging time. Based on this decomposition he showed that the vertical grid-averaged surface flux of *ψ* is composed of three contributions, of which only one is expressed in terms of explicitly resolved values. The remaining two portions include correlations of either spatially or temporally unresolved motions and therefore need to be parameterized.

In the case of horizontal momentum it has become common practice to parameterize the vertical grid-averaged surface flux, due to subgrid-scale motions over inhomogeneous terrain, by a grid-cell-averaged effective roughness length, *z*^{eff}_{0m}*z*^{eff}_{0T}*z*^{eff}_{0m}*z*^{eff}_{0T}*blending height* and heuristic arguments. He compared his findings with the results of a selection of model scenarios, showing that the agreement between parameterized and modeled coefficients is very poor if advective effects are neglected. In a more systematic study, Blyth et al. (1993) focused on a parameterization of grid-averaged heat fluxes in terms of effective surface and aerodynamic resistances. They proposed a heuristic model based also on the concept of a blending height and compared its results with those of various averaging schemes and the detailed computations of a two-dimensional flow model. In accordance with Claussen (1991b) they conclude that consistently accurate estimates of effective surface fluxes can only be obtained from an averaging method that takes advection into account.

These approaches to aggregated fluxes express the effective surface parameters or transfer coefficients in terms of their values at each individual subgrid-scale surface type. Advective effects are taken into account implicitly by relating the individual surface fluxes to the grid-averaged flow at the blending height. Since the blending height itself is determined by the condition of dynamical balance between horizontal advection and vertical flux divergence, the individual contributions are expected to be averaged according to their actual weight. Indeed, for simple surface configurations in neutral conditions with no predominant surface type, Blyth (1995) showed very close agreement between this approach and the study of Schmid and Bünzli (1995a), who used a high-resolution flow model to evaluate the dependence of the effective roughness length on basic surface descriptors. However, Schmid and Bünzli (1995b) pointed out that this correspondence is not obvious a priori, since in the heuristic model the dependence of the blending height on basic surface descriptors does not accord with simple physical arguments, whereas the corresponding dependence of a blending height derived from the flow model does. Moreover, the results of their simulations show that the spatial relationship between different surface patches, that is, the surface texture, has a significantinfluence on the grid-averaged surface momentum flux, which is not fully captured by the implicit blending height approach.

The present study is aimed at a quantification of subgrid-scale surface inhomogeneity in terms of basic surface descriptors that are most significant for advective contributions to heat fluxes. The model approach described in Schmid and Bünzli (1995a) is applied to evaluate the grid-averaged fluxes of sensible and latent heat in unstable conditions, using a two-dimensional *E*–*ε* model with periodic boundary conditions and a description of the local surface energy balance as proposed by Penman (1948) and Monteith (1981). The influence of basic surface parameters on the grid-averaged heat fluxes is investigated for a large range of parameter values in order to clarify the potential effect of surface texture on the magnitude of the averaged fluxes. The results are compared with those obtained with the heuristic averaging scheme described in Blyth et al. (1993), and the sensitivity of their method on the blending height is discussed in detail. It is shown that an accurate evaluation of the blending height is needed in the heuristic model in order to include the advective contributions to the aggregated fluxes due to the surface texture completely. Provided that the blending height is known precisely, the results of the two methods compare very well. However, since the blending height is usually considered a height scale instead of a precise height, a reassessment of its significance and evaluation is suggested in the context of the heuristic averaging scheme.

A brief review of the basic physics involved in advective enhancement is given in the following section. In section 3 the applicability and physical reliability of the *E*–*ε* model in nonneutral conditions is analyzed, based on comparisons with surface-layer profiles that are well established experimentally. In particular, the limitations implied by its closure scheme and their effect on the accuracy of the model predictions are discussed in detail. The results are presented in section 4 and compared with the heuristic model of Blyth et al. (1993) in section 5. Section 6 contains the main conclusions, followed by a complete description of the *E*–*ε* model in the appendix.

## 2. Advective enhancement of latent heat flux

In the present study we assume the surface to be composed of numerous different surface types with definite land use categories and horizontal length scales between 10 m and 1 km. In other words, we address surface variability of the “unorganized” type (following Shuttleworth 1988), which effectively excludes nanoscale variability on the one hand and mesoscale and larger variations on the other hand. The area averaged latent heat flux of such a surface is mainly determined by the sum of the fluxes above each individual surface type, *Q*^{mean}_{e}_{i} *f*_{i}*Q*^{(i)}_{e}*f*_{i} denotes the area fraction covered by the *i*th surface type and *Q*^{(i)}_{e}*Q*_{e} is described in terms of the moisture gradient and transfer resistances (see Monteith 1981), *Q*_{e} ∼ (*q*_{sat} − *q*)/(*r*_{a} + *r*_{s}), where *q* denotes the moisture value of the air above the surface, *q*_{sat} the saturated moisture value at surface temperature, *r*_{a} the aerodynamic resistance, and *r*_{s} the surface resistance to vapor transfer, the advective increase of the flux is Δ*Q*^{+}_{e}*q*/(*r*_{a} + *r*^{moist}_{s}*Q*^{−}_{e}*q*/(*r*_{a} + *r*^{dry}_{s}*q* denotes the amount of advected moisture. Since *r*^{moist}_{s}*r*^{dry}_{s}*r*_{a} is approximately constant, Δ*Q*^{+}_{e}*Q*^{−}_{e}

An analogous asymmetry was already reported by Wood and Mason (1991), Claussen (1991a), and Schmid and Bünzli (1995a) for the flux of momentum. However, in that case the physical process is different, since advection of momentum is essentially nonlinear, whereas advection of moisture is linear with respect to *q.* Nevertheless, the phenomenon appears to be almost identical, not only qualitatively but also quantitatively. It is therefore suggested that the effect of surface texture, as described in detail in Schmid and Bünzli (1995a) for the grid-averaged flux of momentum, is also relevant for the averaged fluxes of heat (note that the sensible heat flux is symmetric to the latent heat flux, since their sum equals the net available energy flux that is prescribed in the Penman–Monteith approach and held constant here). In particular, the relative arrangement of different surface types and the number of transitions per unit area are surface characteristics on which the area-averaged heat fluxes strongly depend, in addition to the individual surface parameters and area fractions.

## 3. Model approach

In order to quantify the effect of surface texture on the area-averaged fluxes of heat, a two-dimensional modeling study was performed in which the surface was composed of only two types, denoted “wet” and “dry,” respectively, and characterized by four basic surface parameters: *f*_{wet}, the area fraction covered by wet surface;*r*^{dry}_{s}*Q*^{wet,dry}_{in}*z*_{0m}/*λ,* the ratio between roughness length and horizontal period, measuring the patchiness (i.e., the number of transition per unit area) of the composed surface. The two surface types are arranged periodically in order to ensure integral homogeneity, and an *E*–*ε* model was used to compute the flow with high spatial resolution. A comprehensive description of the model can be found in Bünzli (1995), and a summary of the basic model equations and boundary conditions is given in the appendix. Here, its physical reliability is analyzed more closely, in particular with respect to applications in nonneutral conditions and flux computations.

*E*–

*ε*model has become quite popular in recent years for numerical studies of atmospheric boundary layer flows, especially in neutral conditions (Detering and Etling 1985; Beljaars et al. 1987; Duynkerke 1988; Levi-Alvares 1991; Ayotte et al. 1994). However, an essential shortcoming of the model is the fact that the well-known similarity profiles for horizontal momentum, potential temperature, and specific humidity,(see Businger et al. 1971; Dyer 1974; Högström 1988)are not a stationary solution of the model equations. In Fig. 2 the modeled profiles are illustrated for a set of typical surface parameters, scaled with the above profiles (1) as reference. The deviation increases rapidly with increasing height, stability, or instability and becomes larger than 20% for

*z*/

*L*≥ 2 or

*z*/

*L*⩽ −5, respectively. Whereas a deviation of 20% in the profiles may still be acceptable, the resulting uncertainty in the nondimensional gradients

*ϕ*

_{m}=

*kz*∂

_{z}

*u*/

*u*∗ and

*ϕ*

_{h}=

*kz*∂

_{z}

*θ*/

*θ*∗ =

*kz*∂

_{z}

*q*/

*q*∗ (which determine the fluxes) become too large. Writing

*u*(

*z*) =

*u*

^{ref}(

*z*)[1 +

*δ*(

*z*)], the nondimensional gradient for momentum is

*ϕ*

_{m}(

*z*) =

*ϕ*

^{ref}

_{m}

*δ*+

*u*

^{ref}∂

_{z}

*δ*/∂

_{z}

*u*). The difference to

*ϕ*

^{ref}

_{m}

*z,*where ∂

_{z}

*u*becomes very small. Although this uncertainty may be reduced by adjusting the model constants appropriately (see the appendix and references therein), the discrepancy between model results and similarity profiles remains substantial and needs further clarification, since the fluxes are the primary quantities considered in the present study.

*Q*

^{ref}

_{e}

*ρL*

_{υ}

*u*∗

*q*∗ and

*Q*

^{ref}

_{h}

*ρc*

_{p}

*u*∗

*θ*∗ are determined by the following set of equations:together with Eq. (1). Here, the subscript “

_{0}” refers tovalues at the height of the roughness length,

*z*

_{0T}, and standard notation is used throughout (see, e.g., Garratt 1992). Given the flow values at the upper boundary, the above set of equations is solved for

*u*∗,

*θ*∗, and

*q*∗, from which the reference fluxes

*Q*

^{ref}

_{e}

*Q*

^{ref}

_{h}

The reference fluxes thus obtained are used to test the physical reliability of the model results. In Fig. 3 the latent heat flux computed with the *E*–*ε* model is illustrated for various surface resistances, scaled by the corresponding reference flux, *Q*^{ref}_{e}*r*_{s} ≃ 0) but increases up to 20% when the surface becomes extremly dry (upper curve in Fig. 3). However, when the surface is composed of a wet and a dry strip with equal area fractions, the arithmetically averaged flux *Q*^{mean}_{e}*Q*^{wet}_{e}*Q*^{dry}_{e}

- The model domain should be confined to the surface layer (i.e.,
*z*⩽ 100 m), since the deviation of the modeled profiles from the experimentally derived similarity profiles increases rapidly with increasing height. - In order to provide a quantitatively reliable measure of advective enhancement, the area-averaged latent heat flux should be scaled by the arithmetically averaged equilibrium flux, using equilibrium fluxes that are also computed with the numerical model.
- Quantitative results of latent heat fluxes over predominantly dry surfaces may be overestimated. These fluxes are generally small, so the absolute error is acceptable. However, if the sensible heat flux is computed as difference between the available and the latent heat flux (as in this study), the resulting values may be significantly underestimated.

*E*–

*ε*model is considered to be a suitable tool for applications in nonneutral conditions, which is able to provide results that are quantitatively reliable within a few percent.

## 4. Model results

In this section the main results of the modeling study are presented, addressing the question how the advective enhancement of grid-averaged heat fluxes depends on the four basic surface descriptors introduced at the beginning of section 3. The model was initialized by a stationary solution of the model equations above the wet strip (i.e., *r*^{wet}_{s}*Q*_{in}, the friction velocity, *u*∗, and the Bowen ratio, *β* = *Q*_{h}/*Q*_{e} (or, alternatively, by the Obukhov length, *L*). The values *u*∗ = 0.3 ms^{−1} and *β* = 0.25 are selected for all results presented here, however, the advective enhancement is not sensitive to these values for quite a large range. Given the boundary conditions described in the appendix, the model was run close to a stationary solution, from which the area-averaged heat fluxes were computed as arithmetic averages of the local fluxes.

The influence of the area fraction covered by the wet surface type, *f*_{wet} is illustrated in Fig. 4 for a model scenario in which the dry surface type is characterized by *r*^{dry}_{s}^{−1} (a moderate value, corresponding to dry grassland), and *Q*_{in} = 500 W m^{−2} is assumed for both surface types. The maximum advective enhancement occurs at an area fraction *f*_{wet} ≃ 0.25, showingthat small fractions of wet patches within dry areas increase the enhancing asymmetry more than vice versa. The magnitude of the enhancement is above 10% for 0.1 ⩽ *f*_{wet} ⩽ 0.6; that is, the effective latent heat flux is substantially underestimated in this range by the arithmetically averaged surface flux, *Q*^{mean}_{e}

In Fig. 5 the dependence of the advective enhancement on the surface resistance of the dry strip, *r*^{dry}_{s}*r*^{wet}_{s}*f*_{wet} = 0.5 constant. For the results illustrated by the upper curve, the available energy flux was again *Q*_{in} = 500 W m^{−2} for both surface types, leading to a monotonic increase of the enhancement with increasing *r*^{dry}_{s}*r*^{dry}_{s}^{−1} (corresponding to dry soil). For the results illustrated by the lower curve in Fig. 5, the available energy flux of the dry strip was reduced to *Q*^{dry}_{in}^{−2}, corresponding to a larger albedo and therefore reflectivity. In this case the advective enhancement is also reduced substantially, an effect that is not due to the overall reduction of the flux, since this modification is already accounted for by the reduced value of the arithmetically averaged flux, *Q*^{mean}_{e}*Q*^{dry}_{in}^{−2} and *Q*^{wet}_{in}^{−2}, in which case the advective enhancement increases up to 50% for *r*^{dry}_{s}^{−1}. On the other hand, if the surfaceresistance is not varied, only the available energy flux, the local fluxes are in equilibrium with the underlying surface everywhere and no advective enhancement is observed at all. The sensitivity to the available energy flux was further tested by keeping the surface resistance of the dry strip constant at *r*^{dry}_{s}^{−1} and varying the available energy flux for the entire surface between 100 W m^{−2} and 600 W m^{−2}, resulting only in a marginal decrease of the advective enhancement (with increasing *Q*_{in}) of less than 2% within the specified range. It is therefore concluded that the influence of the available energy flux on the advective enhancement is only significant for transitions for which it changes simultaneously with the surface resistance (or other thermal surface parameters).

In analogy with the advective enhancement of momentum flux (see Schmid and Bünzli 1995a), the dependence of *Q*^{eff}_{e}*Q*^{mean}_{e}*z*_{0m}/*λ,* which measures the number of transitions per unit area, is also investigated. The results are illustrated in Fig. 6. In these configurations the available energy flux is constant, *Q*_{in} = 500 W m^{−2}, as well as the surface resistance of the dry strip, *r*^{dry}_{s}^{−1}, and the area fraction of the wet surface type, *f*_{wet} = 0.5. The patchiness of the surface is successively increased by either increasing the roughness length or decreasing the period of the transitions, resulting in a substantial increase of the advective enhancement by several percent. From the local flux distribution shown in Fig. 1 this increase is expected, since the contributions from the strongly perturbed flow regions, in which the asymmetry is especially marked, become more important. Again the similarity to the momentum flux is close, at least for constant values of *z*_{0m}. However, whereas *τ*^{eff}/*τ*^{mean} in neutral conditions depends only on the ratio *p* = *z*_{0m}/*λ,* the advective enhancement of the latent heat flux, *Q*^{eff}_{e}*Q*^{mean}_{e}*z*_{0m} and *λ* separately. Reducing*z*_{0m} and *λ* simultaneously while keeping *p* = *z*_{0m}/*λ* constant reduces the advective enhancement as well, an effect that is illustrated in Fig. 6 by the split of the curves. Although several attempts were made to adapt the definition of the patchiness parameter to heat transfer in order to remove this additional dependence, the appropriate expression could not be found. The major problem lies in the fact that the surface resistance cannot be taken into account without relating the extended parameter to a velocity (or time) scale, leading to an expression that combines surface and flow properties. Whether or not the advective enhancement of the heat fluxes scales with such a dimensionless combination remains an open question. Here we conclude that the patchiness parameter *p* = *z*_{0m}/*λ* describes a surface property that is significant for the advective enhancement of momentum as well as heat fluxes. However, whereas its applicability is well suited for momentum fluxes in neutral conditions, it does not represent a scaling variable for heat fluxes.

## 5. Comparison with an analytical averaging scheme

The modeling approach applied in this study is aimed at an averaging method for effective surface fluxes that is designed for improved physical completeness and quantitative reliability. However, due to its computational expense it is more suited for preliminary studies of specific surface configurations than for direct applications in operational models. For the latter purpose alternative schemes are needed that are much more efficient but still sufficiently accurate in order to capture the major influence of surface texture on the area-averaged fluxes. For these schemes the present modeling approach provides a useful reference method to which selected test simulations can be compared.

*l*

_{b}:Here,

*L*denotes the horizontal length scale of variation, related to the period of the surface transitions,

*λ,*by

*L*=

*λ*/2

*π*(Mason 1988). An additional equation for the effective roughness length is needed in general (see Wood and Mason 1991; Blyth et al. 1993); however, since the surface roughness is not varied in the present study it can be omitted. Equations (1) and (2) are solved for the local fluxes

*Q*

_{e}= −

*ρL*

_{υ}

*u*∗

*q*∗ and

*Q*

_{h}= −

*ρc*

_{p}

*u*∗

*θ*∗above each surface type, using

*u*(

*l*

_{b}),

*θ*(

*l*

_{b}), and

*q*(

*l*

_{b}) as reference values. Finally, the effective surface fluxes are evaluated by averaging the local fluxes according to the individual area fractions,For the comparison presented here the mean flow values at the blending height are taken from the

*E*–

*ε*model, in accordance with Blyth et al., who also used a numerical model to provide these values. From a methodical point of view this procedure may be questioned, since these flow values include more information about the surface than the computation in a large-scale model can provide. In other words, the similarity profiles above the individual surface types in the heuristic model are adjusted to a fully developed flow, resulting in an averaging process that includes advection not only because of the proper height scale at which the flow is averaged, but also because the inhomogeneous flow is simulated properly. The uncertainty is related to the fact that the height

*l*

_{b}is located still within the transition zone of the flow and represents only a height scale and not a precise height. However, as will become more obvious in the following, there is no reference state that can be considered as absolute “truth.” The implicit uncertainty is of the same order as those discussed below, and the procedure is considered to be the best available at present.

*Q*

^{eff}

_{e}

*Q*

^{mean}

_{e}

*f*

_{wet}, keeping

*r*

^{dry}

_{s}

^{−1}and

*Q*

_{in}= 500 W m

^{−2}constant. The results of the heuristic model turn out to be quite sensitive to the estimate of

*l*

_{b}, the uncertainty being comparable to the contribution due to advection. If

*l*

_{b}is determined according to Eq. (3) by using

*L*=

*λ*(as in Blyth et al. 1993), the advective enhancement of the latent heat flux is less than 5% for all area fractions (dashed-dotted line in Fig. 7). On the other hand, using

*L*=

*λ*/2

*π*in Eq. (3) as proposed by Mason (1988), the agreement between the two methods is close, provided that the results of the heuristic model are scaled with (analytical) similarity profiles and those of the

*E*–

*ε*model with equilibrium profiles that are solutions of the stationary model equations (dashed and solid lines, respectively). Conversely, the advective enhancement computed by the heuristic model becomes substantially larger than predicted by the

*E*–

*ε*model if the blending height is evaluated according to the relationwhich was proposed by Wood and Mason (1991). Here,

*z*

_{0T}= 0.1

*z*

_{0m}is assumed, in accordance with Blyth et al. (1993) and references therein (not shown). The various estimates of

*l*

_{b}thus imply a considerable uncertainty for the advective enhancement predicted by the heuristic model. Given the “correct” estimate of

*l*

_{b}the method yields almost identical results to those of the

*E*–

*ε*model. On the other hand, since

*l*

_{b}is only an order of magnitude estimate, the resulting uncertainty of

*Q*

^{eff}

_{e}

*E*–

*ε*model we conclude that the use of Eq. (3), together with

*L*=

*λ*/2

*π,*represents the best choice for the periodic surface configurations considered here, in accordance with the scheme originally proposed by Mason (1988). However, from a physical point of view the significance of

*l*

_{b}in this context is closer to an appropriate averaging height than to a blending height, an interpretation that is confirmed by the fact that

*l*

_{b}increases monotonically with increasing effective roughness length, whereas a physical blending height is expected to vanish if the surface is uniform [see Eq. (3) and the discussion given in Schmid and Bünzli 1995b).

The discussion above suggests that the results of the *E*–*ε* model represent the “true” reference state, an implication that needs to be put into perspective. As mentioned in the context of Fig. 3, there is a discrepancy between the model results and the similarity profiles derived empirically. If the latter are considered as truth this discrepancy implies a quantitative uncertainty of the model results, which is also estimated here. In Fig. 7 the effective fluxes computed by the *E*–*ε* model are scaled with equilibrium values corresponding to solutions of the stationary model equations, whereas the fluxes computed by the heuristic model are scaled with values corresponding to similarity profiles (solid and dashed line, respectively). In both cases the Obukhov length is taken as reference height for the equilibrium profiles, *z*_{ref} = |*L*|. This procedure is considered to bethe most appropriate, since it compensates systematic errors most effectively and yields the correct value *Q*^{eff}_{e}*Q*^{mean}_{e}*f*_{wet} → 0 and *f*_{wet} → 1. On the other hand, the difference between the two scales provides a (conservative) estimate of the absolute error of *Q*^{eff}_{e}*E*–*ε* model based on the “wrong” scale. The difference between the dotted and the solid line indicates that *Q*^{eff}_{e}*Q*^{eff}_{e}*Q*^{mean}_{e}

## 6. Summary

In the present study the effective energy partitioning over inhomogeneous terrain is investigated with a view to quantify the influence of surface texture on the area-averaged fluxes of sensible and latent heat. The study is based on a two-dimensional model scenario in which the surface consists of two individual strips that are arranged periodically and an *E*–*ε* model is applied to evaluate the surface-layer flow with high spatial resolution. The simplicity of the surface configuration allows a description of its texture in terms of two basic parameters, the area fraction covered by the wet surface type and the number of transitions per unit area. Two additional parameters are used to characterize the individual surface properties, namely, the available energy fluxes and the surface resistance of the dry strip. For these configurations the advective contributions to the effective surface fluxes are shown to be up to 20% of the mean values corresponding to the arithmetic average of the equilibrium fluxes above each surface type. The area fraction covered by wet surface patches and the variation of the surface resistance are confirmed to be the strongest determinants, followed by the number of transitions per unit area measured by the patchiness parameter. The influence of the available energy flux on the advective contributions turns out to be significant only for surface inhomogeneities for which it changes simultaneously with the surface resistance. If the available energy flux above the dry surface type is reduced (due to a larger albedo), the advective contributions are reduced as well in these configurations.

Prior to the numerical simulations the physical reliability of the *E*–*ε* model was tested by comparing its results with similarity profiles derived empirically. The model is shown to be a trustworthy tool for surface-layer flow simulations in nonneutral conditions, provided its results are scaled properly. Due to its moderate numerical expense it is applicable in principle to fairly complex surface configurations or to flow simulationsin conjunction with surface parameterizations that are physically more complete. It is therefore considered to be suited for a variety of applications that are relevant to evaporation modeling.

The model results were finally compared with those of the heuristic model proposed by Blyth et al. (1993), which is based on a blending height approach. The agreement between the two methods is close for all surface configurations considered in this study, provided that the blending height is determined by the scheme originally proposed by Mason (1988). Using larger values for *l*_{b} results in a substantial underestimation of advective effects due to the surface texture, which indicates that an order of magnitude estimate for *l*_{b} is not sufficiently accurate for the heuristic model to represent these effects completely.

It is concluded that the description of the surface energy partitioning in mesoscale models can be significantly improved by including some textural information about the subgrid-scale surface variability in the parameterization scheme of the effective surface fluxes. It is still an open question how the texture of a more general surface can be quantified in terms of a few basic descriptors. However, a classification of land use categories that includes subgrid-scale surface properties, along with the values of the corresponding effective parameters, will certainly make a difference.

This study was supported by the Swiss National Science Foundation, Grant No. 20-29541.90. It was finalized during a visit of one of the authors (D.B.) to NCAR. The hospitality of the Mesoscale Prediction Group is greatly appreciated.

## REFERENCES

André, J.-C., and C. Blondin, 1986: On the effective roughness length for use in numerical three-dimensional models.

*Bound.-Layer Meteor.,***35,**231–245.Ayotte, K. W., D. Xu, and P. A. Taylor, 1994: The impact of turbulence closure schemes on predictions of the mixed spectral finite-difference model for flow over topography.

*Bound.-Layer Meteor.,***68,**1–33.Beljaars, A. C. M., J. L. Walmsley, and P. A. Taylor, 1987: A mixed spectral finite-difference model for neutrally stratified boundary-layer flow over roughness changes and topography.

*Bound.-Layer Meteor.,***38,**273–303.Blyth, E. M., 1995: Comments on “The influence of surface texture on the effective roughness length.”

*Quart. J. Roy. Meteor. Soc.,***121,**1169–1171.——, A. J. Dolman, and N. Wood, 1993: Effective resistance to sensible and latent heat flux in heterogeneous terrain.

*Quart. J. Roy. Meteor. Soc.,***119,**423–442.Bünzli, D., 1995: The influence of subgridscale surface variations on the atmospheric boundary layer flow. A modelling study. Verlag Geographisches Institut ETH Zürich, Vol. 64, 82 pp. [Available from Verlag Geographisches Institut ETH, Winterthurerstrasse 190, CH 8057 Zurich, Switzerland.].

Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux profile relationships in the atmospheric surface layer.

*J. Atmos. Sci.,***28,**181–189.Claussen, M., 1989: Subgridscale fluxes and flux divergencies in neutrally stratified, horizontally inhomogeneous surface layer.

*Beitr. Phys. Atmos.,***62,**236–245.——, 1991a: Local advection process in the surface layer of the marginal ice zone.

*Bound.-Layer Meteor.,***54,**1–27.——, 1991b: Estimation of areally-averaged surface fluxes.

*Bound.-Layer Meteor.,***54,**387–410.Detering, H. W., and D. Etling, 1985: Application of the

*E*–*ε*turbulence model on the atmospheric boundary layer.*Bound.-Layer Meteor.,***33,**113–133.Duynkerke, P. G., 1988: Application of the

*E*–*ε*turbulence closure model to the neutral and stable atmospheric boundary layer.*J. Atmos. Sci.,***45,**865–880.Dyer, A. J., 1974: A review of flux-profile relations.

*Bound.-Layer Meteor.,***1,**363–372.Garratt, J. R., 1992:

*The Atmospheric Boundary Layer.*Cambridge University Press, 316 pp.Högström, U., 1988: Non-dimensional wind and temperature profiles in the atmospheric surface layer: A re-evaluation.

*Bound.-Layer Meteor.,***42,**55–78.Karpik, S. R., J. L. Walmsley, and W. Weng, 1995: The mixed spectral finite-difference (MSFD) model: Improved upper boundary conditions.

*Bound.-Layer Meteor.,***75,**353–380.Launder, B. E., and D. B. Spalding, 1974: The numerical computation of turbulent flows.

*Comp. Meth. Appl. Mech. Eng.,***3,**269–289.Levi-Alvares, S., 1991: Simulation numérique des écoulements urbains à l’échelle d’une rue à l’aide d’un modèle

*k*–*ε.*Ph.D. thesis, Université de Nantes, 118 pp.Mahrt, L., 1987: Grid-averaged surface fluxes.

*Mon. Wea. Rev.,***115,**1550–1560.Mason, P. J., 1988: The formation of areally-averaged roughness lengths.

*Quart. J. Roy. Meteor. Soc.,***114,**399–420.——, and D. J. Thomson, 1987: Large-eddy simulations of the neutral-static-stability planetary boundary layer.

*Quart. J. Roy. Meteor. Soc.,***113,**413–443.Monteith, J. L., 1981: Evaporation and surface temperature.

*Quart. J. Roy. Meteor. Soc.,***107,**1–27.Pasquill, F., 1972: Some aspects of boundary-layer description, presidential address.

*Quart. J. Roy. Meteor. Soc.,***98,**469–494.Penman, H. L., 1948: Natural evaporation from open water, bare soil, and grass.

*Proc. Roy. Soc. London, Ser. A,***193,**120–195.Schmid, H. P., and D. Bünzli, 1995a: The influence of surface texture on the effective roughness length.

*Quart. J. Roy. Meteor. Soc.,***121,**1–21.——, and ——, 1995b: Reply to comments by E. M. Blyth on “The influence of surface texture on the effective roughness length.”

*Quart. J. Roy. Meteor. Soc.,***121,**1173–1176.Shuttleworth, W. J., 1988: Macrohydrology—The new challenge for process hydrology.

*J. Hydrol.,***100,**31–56.Stull, R. B., 1988:

*An Introduction to Boundary-Layer Meteorology.*Kluwer, 665 pp.Taylor, P. A., 1987: Comments and further analysis on effective roughness lengths for use in three-dimensional models.

*Bound.-Layer Meteor.,***39,**403–418.Wieringa, J., 1986: Roughness-dependent geographical interpolation of surface wind speed averages.

*Quart. J. Roy. Meteor. Soc.,***112,**867–889.Wood, N., and P. J. Mason, 1991: The influence of static stability on the effective roughness lengths for momentum and heat transfer.

*Quart. J. Roy. Meteor. Soc.,***117,**1025–1056.

# APPENDIX

## Model Description

The basic equations of the *E*–*ε* model are conservation equations for mass, momentum, heat, and moisture, together with an equation of state (ideal gas). Turbulent fluxes are approximately described by *K* theory, wherein the eddy viscosity *K* is modeled as a function of the turbulent kinetic energy and the dissipation. For these two quantities prognostic equations are also included, thus making the local turbulent length scale *l* ∼ *E*^{3/2}/*ε* a prognostic variable. The momentum equations are simplified by applying the Boussinesq approximations, and molecular diffusion as well as molecular viscosity are neglected. [A complete derivation of the basic equations can be found in Stull (1988). See also Launder and Spalding (1974) for a more detailed discussion of the closure scheme and Bünzli (1995) for a comprehensive description of the model implementation used in this study.]

### Governing equations

In the following equations *t* denotes the time; *x* and *z* the horizontal and vertical dimensions, respectively;*u, w* the corresponding velocities; *p* the pressure; *ρ* the density of air; *θ* the potential temperature; *θ*_{υ} the virtual potential temperature; *q* the specific humidity; *E* the turbulent kinetic energy; and *ε* the dissipation. Overbars denote ensemble averages (mean quantities) and primes deviations from the mean quantities (turbulent fluctuations). The variables *K*_{m}, *S, B,* and *P* denote the eddy viscosity, shear production, buoyancy, and turbulent kinetic energy production, respectively; Pr_{t} the turbulent Prandtl number; and *c*_{0}, *c*_{1}, and *c*_{2} model constants specified below. Reference values for virtual potential temperature and density are denoted *θ*_{υ,ref} and *ρ*_{ref} respectively; *p*_{eq} is the hydrostatic pressure; and *g* = 9.8 m s^{−2} the constant of gravity.

Here, *ψ* denotes any of the variables *θ, q, E,* or *ε* and Pr_{t,ψ} is the corresponding turbulent Prandtl number.

*ρ*

*ρ*

_{ref}

*ρ*

_{ref}

*θ*

_{υ}

*θ*

_{υ,ref}

*θ*

_{υ,ref}

*K*

_{m}

*c*

^{4}

_{0}

*E*

^{2}

*ε*

*E*

*p*

_{eq}

*p*

_{ref}

*ρ*

_{ref}

*g*

*z*

*z*

_{ref}

The generation and destruction terms in the *ε*_{x}(*u*′*E*′_{z}(*w*′*E*′*c*_{1}, dependent on a characteristic length scale of the flow. Based on a comparison between different parameterization schemes they showed that neglect of this contribution leads to unrealistic results in the upper part of the neutral boundary layer.

The model constants *c*_{0}, *c*_{1}, *c*_{2}, and Pr_{t,ψ} are also adopted from Duynkerke (1988), where a detailed investigation and evaluation of their proposed values is given. They are quoted here for reference.

### Discretization

*x*and

*z*direction for

*u*

*w*

*z*direction, starting at the height of the roughness length and growing with a constant stretching factor

*r*

_{z}≃ 1.2:The accuracy of the discretization scheme is increased by including correction factors in the finite difference approximations of ∂

_{z}

*u*

_{z}

*θ*

_{z}

*q*

_{z}

*ε*

*ε*

_{ref}∼ 1/

*z.*In flow direction the spacing is uniform, allowing the Poisson equation for the pressure (i.e., the divergence of the momentum equations, combined with continuity) to be Fourier transformed horizontally in order to apply periodic boundary conditions. All terms of the model equations are discretized in flux-conserving form by centered finite difference formulas, except for the products

*uu*

*uw*

*uε*

### Lower boundary conditions

*z*

_{j=0}=

*z*

_{0m}implies

*u*

_{j=0}

*z*

_{0m}, is not varied in the present study). Here

*w*

*z*

_{j=0}; however, the lowest value is located at

*z*

_{j=1/2}due to the staggered grid. Moreover, since the upper boundary is “open” (see below), the lower boundary condition for

*w*

*m*+ 1 denotes the number of grid points in flow direction.

*E*

*ε*

*u*

^{2}

_{*}

*K*

_{m}∂

_{z}

*u*

*u*∗

_{i}is located at

*x*

_{i+1/2}due to the staggered grid):The energy balance at the surface is described according to Penman (1948) and Monteith (1981),which determines the potential temperature at the boundary,

*θ*

_{j=0}, in terms of

*θ*

*q*

*j*= 1. Since the derivative of the function

*q*

_{sat}(

*θ*

*θ*

_{j=0}, the specific humidity is determined byIt is worth mentioning that the roughness length for temperature

*z*

_{0T}does not appear in the above equations, implying that the model results are independent of the (still controversial) value of this parameter. Moreover, the aerodynamic resistances for heat and moisture transfer,

*r*

_{ah}and

*r*

_{aυ}, need not to be specified but are determined by the closure scheme of the model equations:Together with the analogous relation for −

*w*′

*q*′

_{t}

*w*

*p*

_{j=0}. However, in the implementation presented here the Poisson equation is solved by a horizontal Fourier transform, resulting in a tridiagonal set of equations in vertical direction for the Fourier coefficients of

*p*

*stationary*momentum equation in discretized form:

### Upper boundary conditions

*l*

_{b}; on the other hand, the magnitude of the stability parameter |

*ζ*| = |

*z*/

*L*| should be small. As a rule of thumb,

*z*

_{top}≃ 20

*l*

_{b}turns out to be a good compromise. At this height the values for

*p*

*u*

*θ*

*q*

_{z}

*w*

*w*

_{z}(

*w*′

*u*′

*K*

_{m}∂

_{z}

*u*

*u*

^{2}

_{*}

_{z}(

*w*′

*θ*′

_{z}(

*w*′

*q*′

*ϕ*

_{h}=

*kz*∂

_{z}

*θ*

*θ*∗ =

*kz*∂

_{z}

*q*

*q*∗, does not equal

*ϕ*

_{m}. Therefore the empirically determined functions

*ϕ*

_{m}and

*ϕ*

_{h}can not be reproduced simultaneously by the

*E*–

*ε*model, making a compromise unavoidable (see the following section for a discussion). In (A20),

*ϕ*

_{m}is preferred to

*ϕ*

_{h}since it keeps the eddy viscosity smaller and therefore the velocity gradient larger (in unstable conditions), resulting in a more realistic profile for the Richardson number Ri Whereas (A20) is applied as boundary condition to stable as well to unstable flows, the second condition to be specified (i.e., for

*ε*

*ε*

*S*+

*B.*Using the relations Ri = −

*B*/

*S*≃

*ζ*

*ϕ*

^{−1}

_{m}

*w*′

*E*′

*u*

^{3}

_{*}

*αζ,*where

*α*is approximately 2.3 (see Stull 1988). The turbulent kinetic energy balance therefore becomes

*S*+

*B*−

*ε*

_{z}(

*w*′

*E*′

*α*

*u*

^{3}

_{*}

*L,*which is again solved for

*ε*

### Equilibrium profiles

*x*independence and vanishing vertical momentum:In Eq. (A24a) the individual fluxes are closed according to Eq. (A2), that is,

*w*′

*u*′

*K*

_{m}∂

_{z}

*u*

*w*′

*θ*′

*K*

_{m}∂

_{z}

*θ*

_{t}and

*w*′

*q*′

*K*

_{m}∂

_{z}

*q*

_{t}. Using Pr

_{t}= 1, the above relations are not consistent with the experimentally derived surface-layer profiles

*ϕ*

_{m}=

*kz*∂

_{z}

*u*

*u*∗ and

*ϕ*

_{h}=

*kz*∂

_{z}

*θ*

*θ*∗ =

*kz*∂

_{z}

*u*

*u*∗, as already mentioned in the previous section. The inconsistency can be removed by using Pr

_{t}(

*ζ*) =

*ϕ*

_{h}/

*ϕ*

_{m}instead of Pr

_{t}≡ 1, a modification that improves the agreement between the similarity profiles (1) and the modeled temperature and humidity profiles considerably. However, this modification implies

*K*

_{m}=

*u*∗

*kz*/

*ϕ*

_{m}(

*ζ*), which (along with

*K*

_{m}=

*c*

^{4}

_{0}

*E*

^{2}/

*ε*

*E*

*K*

_{m}=

*c*

^{4}

_{0}

*E*

^{2}/

*ε*

*ε*

*c*

_{1}(

*ζ*) [or

*c*

_{2}(

*ζ*), alternatively]. Although this procedure makes the experimentally derived similarity profiles a stationary solution of the model equations, it was not applied in the present study. The modification of the turbulent Prandtl number may be justified but increases the numerical expense without removing the inherent inconsistency. The adjustment of the model constant

*c*

_{1}(or

*c*

_{2}) as described above is considered to be too specific, since it depends on the physical parameters that specify the initial condition. Applying this procedure thus violates the requirement that the model should be universal at least for the same type of flow. Therefore, the equilibrium profiles used in this study and the empirically derived similarity profiles(1) are not identical. Figure 2 shows that their difference is small (i.e., ⩽7%) for |

*z*/

*L*| ⩽ 0.1 but increases rapidly with increasingheight, stability or instability. The limitations discussed at the end of section 3 should therefore be observed in order to ensure the physical reliability of the results. [A discussion of the model constants for applications to the entire atmospheric boundary layer can be found in Detering and Etling (1985), Duynkerke (1988), and references therein. See also Karpik et al. (1995) for a more recent discussion of the parameter values appropriate for neutral surface-layer flow over complex terrain, including topography.]