## Abstract

The study of the atmospheric boundary layer flow over two-dimensional low-sloped hills under a neutral atmosphere finds numerous applications in meteorology and engineering, such as the development of large-scale atmospheric models, the siting of wind turbines, and the estimation of wind loads on transmission towers and antennas. In this paper, the intermediate variable technique is applied to the momentum equations in streamline coordinates to divide the flow into regions, with each characterized by the dominance of different terms. Using a simple mixing-length turbulence closure, a simplified form of the *x* momentum equation is solved for the fully turbulent region, resulting in a modified logarithmic law. The solution is expressed as a power series correction to the classical logarithmic law that is valid for flat terrain. A new parameter appears: the *effective radius of curvature* of the hill. The modified logarithmic law is used to obtain new equations for the speedup, the relative speedup, the maximum speedup, and the height at which it occurs. A new speedup ratio is proposed to calculate the relative speedup at specific heights. The results are in very good agreement with the Askervein and Black Mountain field data.

## Introduction

The ability to predict the atmospheric boundary layer (ABL) flow over hills has long been the subject of intense studies by meteorologists, environmentalists, and engineers. Of particular importance in practical applications is the knowledge of the so-called flux–gradient relationships, which connect the fluxes of momentum, heat, and moisture at the surface to the velocity, temperature, and specific humidity gradients, respectively. Integration of these expressions produces the corresponding velocity, temperature, and specific humidity profiles. The most well known flux–profile relation is the logarithmic law for flows over flat, vegetated surfaces, that is,

where *z* is the displaced height above the ground, *u*∗ is the friction velocity, *κ* is the von Kármán parameter, and *z*_{0} is the roughness length.

In the ABL flow over hills, meteorologists apply the flux–profile relationships in the development of large-scale atmospheric models. These models depend critically on the parameterizations adopted for the ABL. Equation (1) has been widely used as a lower boundary condition, despite the fact that it is valid for flat terrain only. The reason is to avoid performing the integration all of the way down to the surface, where strong velocity gradients require a fine computational mesh. Equation (1) is also used in meso- and microscale models, for the same reasons. Other meteorological applications for flux–profile relations over hills are the direct parameterization of the momentum flux and the development of high-order turbulence closure schemes. Finnigan (1992) and Taylor and Lee (1984) list a number of other relevant meteorological and engineering applications. The most important in the engineering field are the siting of wind turbines in regions of enhanced wind speed and the estimation of wind loads on towers, antennas, and buildings located on the top of hills.

Several descriptions of the flow over hills and complex terrain are available in the literature. An important part of this study started with the paper by Jackson and Hunt (1975), who divided the flow field into two sublayers using asymptotic expansion techniques, with each layer having different flow dynamics. The validity of their analysis was then extended to three dimensions by Mason and Sykes (1979). Sykes (1980) also showed that it is necessary to include a third layer to impose the surface boundary condition consistently. Newley (1985) also contributed to the understanding of the problem in the form of a detailed discussion about the various height scales. Newley also modeled the problem numerically and compared his results very favorably with field data (see also Belcher et al. 1993). Two years later, Zeman and Jensen (1987) solved the turbulence equations in streamline coordinates, establishing the role of each term and pointing out the importance of curvature effects.

The linear analysis initiated by Jackson and Hunt (1975) was further revised by Hunt et al. (1988a), who divided the flow field into four regions. The study of Hunt et al. (1988b) extended the results to stably stratified flow. The division proposed in these two papers has deeply influenced the current understanding of the ABL flow structure. Belcher et al. (1993) refined the analysis even further, proposing a smooth match between the inner and outer regions. An excellent review of the study of the airflow over complex terrain is presented by Wood (2000), who also reviewed numerical and observational studies.

Although Eq. (1) provides a simple description of the wind field near the ground, it represents very well the atmospheric flow over a flat terrain under neutral atmosphere, even in the case of very rough surfaces. This fact suggests that the logarithmic law may be extended to cover flows over hills, but few attempts are reported in the literature. Taylor and Lee (1984) proposed an empirical exponential damping of the maximum speedup over the hilltop, which can be used to obtain a modified log law. Their result was later revised by Walmsley et al. (1989) and Weng et al. (2000). Later, Finnigan (1992), using asymptotic matching techniques, assumed the well-known buoyancy–curvature analogy and proposed the acceleration–curvature analogy to obtain another solution in the form of a modified log law. Finnigan’s solution is an implicit function because it depends on the curvature Richardson number and on the curvature of the *z* axis in streamline coordinates. Comparison with observational data showed encouraging results.

In this work, a new modified logarithmic law is proposed for the mean wind velocity distribution in flows over low-sloped 2D hills, under neutral stratification conditions. The result appears in the form of a flux–profile relationship, obtained through an order-of-magnitude analysis of the *x* momentum equation, which is carried out by employing the intermediate variable technique (IVT). The analysis also suggests a new dynamic subdivision of the flow field in regions characterized by the predominance of groups of forces. From the modified log law, new results for the flux–gradient relationship, the velocity speedup, the relative speedup, and the heights at which their maximum values occur are also derived. A new parameter, called the speed-up ratio, is proposed that allows the calculation of the relative speedup at given heights to be performed. All results are tested against field data for vegetated hills. Comparison with the data of Askervein Hill (Taylor and Teunissen 1987), for the case of short vegetation, and with the data of Black Mountain (Bradley 1980), for the case of tall vegetation, shows very good agreement. Because this work aims at obtaining a new modified log law, the structure of turbulence is not considered. As a consequence, the law proposed here is not an exact detailed solution for the velocity field in the entire ABL. It is an extension to the well-known Eq. (1) and serves essentially the same purposes.

## Definition of the problem and governing equations

Following Kaimal and Finnigan (1994), we define hill as a topographical feature with a characteristic length of less than 10 km, whereas larger features are considered mountains. A hill slope is considered low when it never exceeds 10°, leading to heights on the order of 1 km or less. This value is the maximum value for which observations and wind tunnel experiments indicate that no separation occurs, even under strong wind conditions (Kaimal and Finnigan 1994)

We consider an isolated two-dimensional low-sloped hill in the middle of an otherwise flat terrain of constant or slowly varying surface properties. Suppose the existence of a neutrally stratified atmosphere and a period of the day during which the flow can be considered statistically stationary. Figure 1 illustrates the idealized flow. The vertical coordinate *z* is defined as the height above the local terrain, and the horizontal coordinate is *x*. The main geometrical parameters of the hill are its height *h*, its surface curvature radius *R _{h}*, and its horizontal length scale

*L*. The hill’s surface radius of curvature is defined as [1 + (

_{h}*dh*/

*dx*)

^{2}]

^{3/2}/|

*d*

^{2}

*h*/

*dx*

^{2}|. It can physically be interpreted as the radius of a circle tangent to a given curve with the same local rate of variation of its direction along the curve. Figure 1 shows the local radius of curvature of the hilltop (HT). This definition implies that

*R*is less than 0 at the HT. The length scale is defined as the horizontal distance from the HT to the half-height point, and the location upwind of the hilltop at which the velocity profile is not perturbed by the presence of the hill is the reference site (RS).

_{h}The vertical profile of the mean horizontal wind at RS *u*_{0}(*z*) is considered to be essentially logarithmic and is given by Eq. (1). Over the hill, the mean velocity is given by

where the speedup Δ*u* is caused by topographic and surface property variations. The speedup is positive at HT because the flow is accelerated to satisfy the continuity equation and is negative somewhere at the upwind slope because of hill curvature effects. By dividing the speedup by the RS velocity, the relative speedup Δ*S* can be defined as

Over the last decades many works have been focused on determining the vertical profiles of Δ*u* and Δ*S*. Except for Lemelin et al. (1988), most of the results are valid only for the HT, although it has been recognized that results applicable for the upwind and downwind slopes would be exceedingly valuable.

### Governing equations for the ABL

Finnigan (1983) recommends the use of streamline coordinates to treat the ABL equations and describes its advantages in the study of flow over hills. Following Finnigan (1983), the time-averaged governing equations for 2D stationary flow in streamline coordinates, under the Boussinesq approximation, can be written as

with the characteristic lengths *L _{a}* and

*R*given by

and the viscous terms defined as

In the preceding set of equations, *x* represents the direction parallel to the streamlines and *u* and *u*′ are the mean and turbulent velocities in the *x* direction, respectively. The direction normal to the streamlines is represented by *z*, and the corresponding turbulent velocity is *w*′. The thermodynamic mean pressure is denoted by *p* and the mean temperature is *T*. Respectively, *T*_{0} and *ρ*_{0} are the reference temperature and density of the environment, which is considered to be hydrostatic with the air assumed to be an ideal gas. The difference between the mean temperature and the environmental temperature is denoted by Δ*T*. The components of gravity in the *x* and *z* directions are denoted by *g _{x}* and

*g*, respectively, and the dynamic viscosity is

_{z}*ν*. In Eq. (7), Ω represents the mean component of vorticity in the direction normal to the plane of the flow. Finnigan (1983) points out that Ω is an invariant of the transformation, and so it can be written in Cartesian coordinates as

where *w* is the *z* component of the flow velocity, which is zero in streamline coordinates. In Eqs. (4) and (5), *L _{a}* can be interpreted as a length scale for the acceleration term in the

*x*direction, whereas

*R*can be interpreted as the local radius of curvature of the streamlines in 2D flows. Finnigan et al. (1990) state, “both

*L*and

_{a}*R*are signed quantities, being positive if their local centers of curvature lie in the positive

*z*and

*x*directions, respectively.” For real hills, this means that

*R*is less than 0 in the vicinity of the HT and

*R*is greater than 0 in the remaining parts of both slopes. In the set of Eqs. (4)–(10), the mass conservation equation is not included because it is automatically satisfied by the transformation. Finnigan (1992) shows that it must be substituted by the geometrical identity

### The intermediate variable technique

In this section, an overview of the method used here to simplify Eqs. (4)–(10) is presented. The method employed here has its roots on the technique used by Prandtl (1904) to simplify the equations of the flow over a flat plate. In a modern version (e.g., Schetz 1993) the method consists in, first, making the governing equations of the problem nondimensional, which must be done in such a way that every term of the equations is rewritten as a combination of order-1 variables multiplied by a nondimensional parameter. Depending on the nature of the problem, important phenomena may occur very close to the boundaries of the flow, and so regions in which the nondimensional normal coordinate is nearly zero must be investigated. The next step is to “stretch” the nondimensional normal coordinate through a variable transformation. This is achieved by dividing the normal (nondimensional) coordinate by a (nondimensional) small parameter that is obtained during the preceding step. In Prandtl’s (1904) work, the procedure was used to magnify mathematically the thin region of the flow field in which friction forces are important. At this point, all terms of the equations are a combination of order-1 variables and nondimensional terms, and it is easy, therefore, to compare them and eventually to neglect the higher-order ones. After this simplification, the resulting equations can be transformed back to dimensional form and solved, if they are simple enough. It is evident that the choice of the parameters used to render the equations nondimensional and to stretch the vertical coordinate is crucial.

A number of extensions to the method depicted above appeared in the study of different engineering problems. One was the IVT (Kaplun 1967; Lagerstron and Casten 1972; Mellor 1972; Roberts 1984). In IVT, the governing equations are made nondimensional and are stretched, as in Prandtl’s method. The stretching parameter is then allowed to vary continuously between certain limits to magnify different regions of the problem’s domain. As a consequence, different regions of the flow field are defined, each governed by certain terms of the original equations, to first order of approximation. Again, the choice of the parameters is a crucial step.

### Order-of-magnitude analysis

We now apply the IVT to Eqs. (4) and (5). Readers not interested in the mathematical details may skip the next pages and go directly to section 2c(3).

The notation adopted here follows Tennekes and Lumley (1972). If the error involved is less than 30%, the symbol “≅” is used. Coarser approximations are represented by “∼.” This symbol is also used whenever two or more terms of an equation are compared. After the dominant terms have been established, the simpler notation “=” is used in the resulting equation, bearing in mind that the error can be made as small as necessary in some appropriate asymptotic limit. Whenever *M* is much greater (at least one order of magnitude larger) than *N*, the notation *M* ≫ *N* is used.

Equations (4) and (5) are made nondimensional by the following variable transformations: *X* ≡ *x*/*L*, *Z* ≡ *z*/*L*, *U* ≡ *u*/*U _{g}*,

*P*≡

*p*/(

*ρ*

_{0}

*U*

^{2}

_{g}), and Ω

^{ad}≡ Ω

*L*/

*U*. Here,

_{g}*L*is the horizontal length scale of the problem under study and

*U*is the geostrophic wind speed. Although ABL problems usually do not consider Coriolis force effects, the geostrophic velocity is used because it represents an upper limit of the ABL velocity in most situations. The turbulence terms in Eqs. (3), (4), and (5) can be made nondimensional with the friction velocity

_{g}*u*∗. After dropping the overbars representing the time average for simplicity, the result is

where

are the small parameters. They are indeed small, because *u*∗ ≪ *U _{g}* and Re ≫ 1 in the ABL. In Eqs. (14a,b), Re ≡

*U*/

_{g}L*ν*is the Reynolds number and Gr

*≡ (*

_{x}*g*

_{x}L^{3}/

*ν*

^{2})(Δ

*T*/

*T*

_{0}) and Gr

*≡ (*

_{z}*g*

_{z}L^{3}/

*ν*

^{2})(Δ

*T*/

*T*

_{0}) are the Grashof numbers in the

*x*and

*z*directions, respectively. The variables

*L*

^{ad}

_{a}and

*R*

^{ad}are defined as

The nondimensional viscous terms are

To clarify how the IVT works, we give a short example at this point. As Jackson and Hunt (1975), Hunt et al. (1988a), and Kaimal and Finnigan (1994) do for other purposes, suppose the existence of a region in which the mean flow advection and the cross-stream divergence of the shearing stresses balance each other in the ABL. Substitution of Eq. (15) into Eq. (12) yields *U*∂*U*/∂*Z* ∼ *ɛ*^{2}∗(∂*U*′*W*′/∂*Z*). Recalling that all variables, except possibly *Z*, must be of order 1, the previous relation suggests that Z ∼ *ɛ*^{2}∗ Because *Z* ≡ *z*/*L*, this implies that the balance between inertia and turbulence occurs in a region in which *z* ∼ *Lɛ*^{2}∗ = *L*(*u*^{2}∗/*U*^{2}_{g}).

The preceding analysis shows the existence of a region in which advection and shearing stresses balance could be assessed by substituting a stretched variable *Z** = *Z*/*ɛ*^{2}∗ of order 1 in Eq. (12). This is actually what Prandtl’s method does, but what distinguishes it from the IVT is that, in the latter, the small parameter in the denominator is allowed to vary. Therefore, we define

with variable *ɛ*. Setting *ɛ* ∼ *ɛ*^{2}∗ in Eq. (19) and assuming that after the stretching *Z** ∼1 implies that *Z* ∼ *ɛ*^{2}∗. Substituting this conclusion into Eq. (12) readily gives *U*∂*U*/∂*Z* ∼ *ɛ*^{2}∗(∂*U*′*W*′/∂*Z*), as before. From Eq. (19) and *Z** ∼ 1, it follows that *z* ∼ *ɛL*, meaning that advection and shearing stresses balance in a region defined by *z* ∼ *Lɛ*^{2}∗ = *L*(*u*^{2}∗/*U*^{2}_{g}).

The preceding example illustrates essentially how the IVT can be used to associate flow regions with the dominant terms in an equation. Varying the value of *ɛ* causes other terms in Eq. (12) to become of the same order in other regions. For example, requiring that *ɛ* ∼ *ɛ _{R}*/

*ɛ*

^{2}∗ implies that

*ɛ*

^{2}∗(∂

*U*′

*W*′/∂

*Z*) ∼

*ɛ*∂

_{R}^{2}

*U*/∂

*Z*

^{2}, which means that shearing stresses and viscous effects are of the same order in a region of the ABL defined by

*z*∼

*L*(

*u*

^{2}∗/

*U*

^{2}

_{g})/Re. The process of varying

*ɛ*can be used systematically to search for values that make the terms containing

*Z*in Eq. (12) change their order of magnitude—the range for this variation being 0 <

*ɛ*< 1. Equation (19) shows that as

*ɛ*→ 0, the surface is approached, and as

*ɛ*→ 1, the stretching effect disappears. The stretching effect is, therefore, inversely proportional to the value of

*ɛ*. One interesting characteristic of the IVT is the fact that the simplified equations that are obtained often give relatively good results for regions much larger than that established in the analysis. This general behavior is shared by other asymptotic theories—for example, the linear analyses carried out by Jackson and Hunt (1975). According to Wood (2000), “All . . . observational campaigns broadly supported the predictions of the theory . . . , even in cases when the theory is not strictly applicable.” The IVT can be applied to a broader class of phenomena, known as boundary layer problems, characterized by a distinctive mathematical behavior near the physical boundaries.

We now return to the problem. Equation (19) is substituted into Eqs. (12) and (13), and the viscous terms and *R*^{ad} are written out. The result is

The analysis of the preceding equations may be simplified with a change in notation. The advective term on the left-hand side of Eq. (20) may be denoted *A _{x}*. On the right-hand side, we denote the pressure term by

*P*, the turbulence terms (second and third) by

_{x}*T*

_{x}_{1}and

*T*

_{x}_{2}, the curvature terms (fourth and fifth) by

*C*

_{x}_{1}and

*C*

_{x}_{2}, the buoyancy term (sixth) by

*B*, and the viscous terms (the remaining) by

_{x}*V*

_{x}_{1}, . . . ,

*V*

_{x}_{5}. Analogous definitions can be used for Eq. (21). After multiplying these equations through by

*ɛ*

^{2}and

*ɛ*, respectively, the result is

Equations (22) and (23) can now be stretched and the leading-order terms can be sorted out to determine the approximate governing equations for the various flow regions. This is done in sections 2c(1) and 2c(2), which follow. There, the equations are presented in the simplified notation introduced above and their regions of validity are presented in small parameter form. In section 2c(3), equations and regions are returned to their dimensional form, and typical values for the heights bounding those regions are analyzed.

#### Analysis of the *x* momentum equation

To carry out the analysis, it is necessary to establish a relation between *ɛ*∗ and *ɛ _{R}*. In the literature, it is usual to assume that

*ɛ*≪

_{R}*ɛ*

^{2}∗. In mathematical terms, this relation is considered a working hypothesis in the context of asymptotic theories, and it is based on the fact that as Re increases,

*ɛ*decreases and

_{R}*ɛ*∗ increases. In physical terms, this assumption may be supported by comparison with field data. Typical values of the parameters involved in the definitions of

*ɛ*∗ and

*ɛ*are

_{R}*U*∼10 m s

_{g}^{−1},

*L*∼1000 m, and

*ν*∼1.5 × 10

^{−5}m

^{2}s

^{−1}. For

*u*∗, a typical value of 1.0 m s

^{−1}is assumed, although Holton (1992) suggests a lower value. The reason is that the suggested value (

*u*∗ ∼ 0.3 m s

^{−1}) holds for flat surfaces and

*u*∗ is expected to increase near the HT. Substituting these values into Eqs. (14a,b) results in

*ɛ*∼ 4 × 10

_{R}^{−5}and

*ɛ*

^{2}∗ ∼1 × 10

^{−2}, which shows that

*ɛ*≪

_{R}*ɛ*

^{2}∗ indeed. Allowing

*ɛ*to vary in Eq. (22) produces the results below.

##### Advective region (region V)

If one assumes that *ɛ*^{2}∗ ≪ *ɛ* ≤ 1 in Eq. (22), the largest turbulence and curvature terms, *ɛ*^{2}∗*ɛT _{x}*

_{2}and

*ɛ*

^{2}∗

*ɛC*

_{x}_{2}, can be neglected in comparison with the advective and pressure terms,

*ɛ*

^{2}

*A*

_{x}_{2}and

*ɛ*

^{2}

*P*. Because

_{x}*ɛ*≪

_{R}*ɛ*

^{2}∗,

*ɛ*≪

_{R}*ɛ*≤ 1 and the largest viscous terms,

*ɛ*

_{R}V_{x}_{2},

*ɛ*

_{R}V_{x}_{4}, and

*ɛ*

_{R}V_{x}_{5}, can also be neglected in comparison with the advective and pressure terms. Therefore, Eq. (22), correct to order

*ɛ*, simplifies to

To be rigorous, the buoyancy term *B _{x}* = Gr

*≡ (*

_{x}*g*

_{x}L^{3}/

*ν*

^{2})(Δ

*T*/

*T*

_{0}) in Eq. (24) can only be determined if Δ

*T*is known from the coupled solution of the energy equation, which depends on the static stability of the atmosphere. In this analysis,

*B*is seen as an external forcing term, known beforehand. Keeping

_{x}*B*in Eq. (22) means that it is of the same order of magnitude as

_{x}*A*and

_{x}*P*. This assumption implies that

_{x}*B*∼ 1/

_{x}*ɛ*

^{2}

_{R}. If

*B*≪ 1/

_{x}*ɛ*

^{2}

_{R}, the buoyancy term can be neglected and Eq. (22) simplifies to

*A*= −

_{x}*P*. If 1/

_{x}*ɛ*

^{2}

_{R}≪

*B*, the buoyancy term becomes the largest, and, therefore, it dominates Eq. (24). In these highly nonneutral cases, however, the Boussinesq approximation breaks down and Eq. (4) is no longer valid. Thus, no conclusions can be obtained from the analysis of this case, unless the energy equation is considered. Figure 2 illustrates the behavior of the terms of Eq. (22) in region V and in the other regions that are defined below.

_{x}##### Turbulent advective region (region IV)

Now suppose that *ɛ* ∼ *ɛ*^{2}∗ in Eq. (22). In this case, the largest turbulent and curvature terms are of the same order as the advective and pressure terms. Substituting *ɛ* ∼ *ɛ*^{2}∗ into *ɛ _{R}* ≪

*ɛ*

^{2}∗ yields

*ɛ*≪

_{R}*ɛ*

^{2}. Thus, the largest viscous terms can be neglected in comparison with the advective and pressure terms. Equation (22), correct to order

*ɛ*, simplifies to

As in region V, keeping *B _{x}* in Eq. (25) means that

*A*∼

_{x}*P*∼

_{x}*ɛ*

^{2}

_{R}

*B*, implying that

_{x}*B*∼ 1/

_{x}*ɛ*

^{2}

_{R}. If

*B*≪ 1/

_{x}*ɛ*

^{2}

_{R}, the buoyancy term can be neglected in Eq. (25), which simplifies to

*A*= −

_{x}*P*−

_{x}*T*

_{x}_{2}+

*C*

_{x}_{2}. If 1/

*ɛ*

^{2}

_{R}≪

*B*, the buoyancy term dominates Eq. (25) and the energy equation has to be considered.

_{x}##### Fully turbulent region (region III)

If one assumes that *ɛ _{R}*/

*ɛ*

^{2}∗ ≪

*ɛ*≪

*ɛ*

^{2}∗, the largest turbulent and curvature terms dominate Eq. (22), and the condition

*ɛ*≪

_{R}*ɛ*

^{2}∗ is implicitly satisfied. Thus, Eq. (22), correct to order

*ɛ*, reduces to

This time, keeping *B _{x}* in Eq. (26) implies that

*ɛ*

^{2}∗

*T*

_{x}_{2}∼

*ɛɛ*

^{2}

_{R}B

_{x}, which, in turn, implies that

*B*∼

_{x}*ɛ*

^{2}∗/

*ɛɛ*

^{2}

_{R}, because

*T*

_{x}_{2}∼1. Substituting

*ɛ*/

_{R}*ɛ*

^{2}∗ ≪

*ɛ*≪

*ɛ*

^{2}∗ into this expression yields 1/

*ɛ*

^{2}

_{R}≪

*B*≪

_{x}*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}. If

*B*≤ 1/

_{x}*ɛ*

^{2}

_{R}, then the buoyancy term can be neglected, and Eqs. (24)–(26) simplify to 0 = −

*T*

_{x}_{2}+

*C*

_{x}_{2}. If

*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}≤

*B*, the buoyancy term dominates Eq. (26) and the energy equation has to be taken into account.

_{x}##### Turbulent viscous region (region II)

Suppose now that *ɛ* ∼ *ɛ _{R}*/

*ɛ*

^{2}∗. In this case, the largest turbulent and curvature terms are on the same order of the largest viscous terms and Eq. (22), correct to order

*ɛ*, simplifies to

Keeping *B _{x}* implies that

*B*∼

_{x}*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}. If

*B*≪

_{x}*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}, then the buoyancy term can be neglected and Eq. (27) simplifies to 0 = −

*T*

_{x}_{2}+

*C*

_{x}_{2}+

*V*

_{x}_{2}−

*V*

_{x}_{4}+

*V*

_{x}_{5}. If

*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}≪

*B*, the buoyancy term dominates Eq. (27) and the energy equation has to be considered.

_{x}##### Viscous region (region I)

Suppose that *ɛ* ≪ *ɛ _{R}*/

*ɛ*

^{2}∗ in Eq. (22). Now, the largest viscous terms dominate, and Eq. (22), correct to order

*ɛ*, can be written as

Keeping *B _{x}* implies that

*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}≪

*B*. If

_{x}*B*≤

_{x}*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}, the buoyancy term can be neglected and Eq. (28) simplifies to 0 =

*V*

_{x}_{2}−

*V*

_{x}_{4}+

*V*

_{x}_{5}. If

*ɛ*

^{4}∗/

*ɛ*

^{3}

_{R}≪

*M*≪

*B*, with

_{x}*M*being a bounded arbitrary value, the buoyancy term dominates and the energy equation has to be considered.

#### Analysis of the *z* momentum equation

For all possible values of *ɛ* in Eq. (23), with 0 ≤ *ɛ* ≤ 1, the largest turbulent, curvature, and viscous terms can be neglected in comparison with the advective and pressure terms. In this case, Eq. (23), correct to order *ɛ*^{2}∗ with respect to the turbulent and curvature terms and to order *ɛ _{R}* with respect to the viscous terms, simplifies to

Keeping *B _{x}* in Eq. (29) means that

*A*∼

_{z}*P*∼

_{z}*ɛɛ*

^{2}

_{R}B

_{z}, which implies that

*B*∼ 1/

_{z}*ɛɛ*

^{2}

_{R}. Because the ABL has not been divided into regions in this case, the use of the condition 0 ≤

*ɛ*≤ 1 into

*B*∼ 1/

_{z}*ɛɛ*

^{2}

_{R}only imposes a restriction on

*B*valid for the ABL as a whole. A more useful result can be obtained by rewriting

_{z}*ɛ*in terms of

*B*according to

_{z}*ɛ*∼ 1/

*B*

_{z}ɛ^{2}

_{R}. In this case, the magnitude of

*B*determines the region for which the buoyancy term has to be included in Eq. (29). If

_{z}*B*≪ 1/

_{z}*ɛɛ*

^{2}

_{R}, the buoyancy term can be neglected within the region defined by

*ɛ*≪ 1/

*B*

_{z}ɛ^{2}

_{R}and Eq. (29) reduces to

*A*= −

_{z}*P*. In the region

_{z}*ɛ*≫ 1/

*B*, the buoyancy term dominates Eq. (29) and the energy equation has to be considered.

_{z}#### Simplified equations in dimensional form

It is useful to rewrite the nondimensional equations above in dimensional form so that they are ready for general meteorological use in numerical schemes or in analytical solutions. Their regions of validity must also be rewritten in physical terms. Substituting the definitions of the nondimensional variables and small parameters where appropriate and recalling that *Z** ∼ 1, the equations in dimensional form, written in streamline coordinates for the regions identified above, become as described below.

##### Advective region (region V)

In this region,

valid for (*u*∗/*U _{g}*)

^{2}≪

*z*/

*L*≤ 1 and Gr

*/Re*

_{x}^{2}∼1. Using the typical values of section 2c(1), the first condition leads to 10 m ≪

*z*≤ 1000 m.

##### Turbulent advective region (region IV)

##### Fully turbulent region (region III)

In this region,

valid for (*U _{g}*/

*u*∗)

^{2}/Re ≪

*z*/

*L*≪ (

*u*∗/

*U*)

_{g}^{2}and 1 ≪ Gr

*/Re*

_{x}^{2}≪ Re(

*u*∗/

*U*)

_{g}^{4}. Typical values yield 0.2 mm ≪

*z*≪ 10 m.

##### Turbulent viscous region (region II)

In this region,

valid for *z*/*L* ∼ (*U _{g}*/

*u*∗)

^{2}/Re and Gr

*/Re*

_{x}^{2}∼ Re(

*u*∗/

*U*)

_{g}^{4}. Typical values yield

*z*∼ 0.2 mm.

##### Viscous region (region I)

In this region,

valid for *z*/*L* ≪ (*U _{g}*/

*u*∗)

^{2}/Re and Re(

*u*∗/

*U*)

_{g}^{4}≪ Gr

*/Re*

_{x}^{2}. Typical values yield

*z*≪ 0.2 mm.

##### Entire ABL

For the entire ABL,

valid for 0 ≤ *z*/*L* ≤ 1 and *z*/*L* ∼ Re^{2}/Gr* _{z}*. Equation (35) without the buoyancy term reduces to Euler’s equation in the direction normal to the streamlines.

### Comparison with the regions found in previous studies

Here, we compare the regions found in previous studies with the results obtained in our study. Hunt et al. (1988a) proposed splitting the ABL over hills into two regions, each subsequently divided into two layers. Based on the predominant forces that define them, we establish the following correspondence: Hunt et al.’s upper and middle layers (their outer region) correspond to our region V; Hunt et al.’s shear stress layer corresponds to our region IV, and Hunt et al.’s inner surface layer corresponds to our regions I, II, and III.

## Solution for the fully turbulent region under neutral atmosphere

We now focus our attention on a solution for the equation for the fully turbulent region. We consider only the neutral atmosphere case, for which Eq. (32) reads

To solve Eq. (36) we suppose that region III is close enough to the surface that the streamlines can be assumed to have the same radius of curvature as the hill’s surface, that is, *R*(*x*, *z*) ≈ *R*(*x*, 0) ≡ *R _{h}*(

*x*). This supposition is equivalent to assuming that the flow is parallel to the surface in this region, which is supported by the experimental results of Salmon et al. (1988). Equation (36) can then be rewritten as ∂

*ψ*/∂

*z*−

*p*(

*x*)

*ψ*= 0, where

*ψ*(

*x*,

*z*) ≡ − and

*p*(

*x*) ≡ 2/

*R*. The solution to this equation is

_{h}where *C*_{1}(*x*) is the integration constant. To determine the value of *C*_{1}(*x*), one boundary condition for the fully turbulent region is necessary. In IVT (as in other types of asymptotic analysis) this condition may come from matching Eq. (37) to the solution of the layers immediately below and above, that is, regions II or IV. None of these conditions are available here because no simple analytical solution for regions II or IV could be found. Therefore, an empirical condition must be sought. If we define *ξ* ≡ *z*/*R _{h}*, we rewrite Eq. (37) as

*ψ*(

*x*,

*z*) =

*C*

_{1}(

*x*) exp(2

*ξ*).

Field results show that over flat terrain, the momentum flux () does not vary appreciably with *z* next to the surface, and so *ψ* = (*x*, *z*) = *u*^{2}∗ for small *z*. Assuming this behavior to be also valid for low hills and extending the region of validity of Eq. (37) to include the region close to the ground yields *u*^{2}∗ = *C*_{1} exp(2*ξ*_{0}), where *ξ*_{0} is the value of *ξ* on the lower boundary. Therefore, *C*_{1} = *u*^{2}∗ exp(−2*ξ*_{0}), and consequently *ψ*(*x*, *z*) = *u*^{2}∗ exp(−2*ξ*_{0}) exp(2*ξ*). Because *ξ* = *z*/*R _{h}*, we set

*ξ*

_{0}=

*z*

_{0}/

*R*

_{h}_{0}in this equation, with

*R*

_{h}_{0}being a parameter associated to the radius of curvature defined below. Equation (37) then becomes

The choice of *ξ*_{0} = *z*_{0}/*R _{h}*

_{0}is important and deserves special consideration. To clarify the reasoning involved, the reader is briefly reminded of one possible derivation of the log law, Eq. (1) (e.g., Stull 1997). If one assumes that the mixing length theory holds, the momentum flux in the surface layer can be written as −(

*x*,

*z*) = (

*κz*∂

*u*/∂

*z*)

^{2}. Supposing the flux to be approximately constant in the region considered, we can write

*u*

^{2}∗ = (

*κz*∂

*u*/∂

*z*)

^{2}. Taking the square root, separating variables, and integrating from

*z*

_{0}to

*z*with

*u*(

*x*,

*z*

_{0}) = 0 finally yields Eq. (1). Because in this integration the initial approximation −(

*x*,

*z*) = (

*κz*∂

*u*/∂

*z*)

^{2}does not hold at

*z*= 0, then

*z*

_{0}≠ 0. The roughness length

*z*

_{0}is interpreted as the height at which

*u*vanishes.

By analogy to the fact that *z*_{0} ≠ 0, our analysis suggests that *ξ*_{0} ≠ *z*_{0}/*R _{h}* because Eq. (37) also does not hold all the way down to

*z*=

*z*

_{0}, and

*ξ*

_{0}=

*z*

_{0}/

*R*

_{h}_{0}is then the logical choice. This expression defines

*R*

_{h}_{0}. Therefore, in the same way as

*z*

_{0}≠

*h*

_{0}, where

*h*

_{0}is the real height of the roughness elements,

*R*

_{h}_{0}is expected to be different from

*R*. The parameter

_{h}*R*

_{h}_{0}, hereinafter referred to as the

*effective radius of curvature*, can be physically interpreted as the radius of curvature that the hill should have for the momentum flux, extrapolated downward close to ground, to be algebraically equal to its flat terrain value

*u*

^{2}∗.

Adopting a turbulence parameterization allows Eq. (38) to be integrated to yield the velocity profile. Assuming that turbulence is appropriately modeled by the mixing length theory (in streamline coordinates), with the mixing length given by *l _{m}* =

*κz*(

*κ*being the von Kármán constant),

*ψ*may be written as

*ψ*≡ − = (

*κz*∂

*u*/∂

*z*)

^{2}. Substituting this expression into Eq. (38) and separating variables yields

The use of a mixing length parameterization implies that turbulence is in local equilibrium in the region. This can be justified by observing that this hypothesis is known to be valid close to the surface over flat terrain and that we may assume no qualitative change to occur over low-sloped hills. The assumption can ultimately be verified a posteriori by comparison with field data.

Equation (39) indicates that the velocity profile depends on both *R _{h}*

_{0}and

*R*. Because

_{h}*R*is not easily modeled and is generally not measured in field campaigns, we avoid the explicit dependence of Eq. (39) on it, replacing

_{h}*R*by

_{h}*R*

_{h}_{0}and assuming that the difference can be absorbed into

*u*∗. Hence, we rewrite Eq. (39) as

This procedure provides an approximate model for Eq. (39) and yields a practical solution to the problem, because *R _{h}*

_{0}can be estimated from field measurements, as shown in the next section. The use of such an approximation prevents an estimate of the numerical value of

*R*

_{h}_{0}from being obtained a priori, despite the physical interpretation of

*R*

_{h}_{0}.

Integration of Eq. (39) can now be performed between *z*_{0} and *z*, assuming that *u*(*x*, *z*_{0}) = 0 as in the flat-terrain case. The result is

for *z* ≥ *z*_{0}. The value of *z*_{0} in Eq. (41) is assumed to be the same as that over flat terrain. The function Ei(*x, z*) is the exponential integral function, which has well-known properties (Abramowitz and Stegun 1970). If one expresses Ei(*x, z*) as an infinite power series (Abramowitz and Stegun 1970), Eq. (41) can be rewritten in terms of the logarithmic function as

Equation (42) [or Eq. (41)] provides a new law for the ABL flow over low hills under neutral atmosphere, and it is hereinafter referred to as the *modified logarithmic law*. These equations have no restrictions regarding their application to points in *x* along the slopes of the hill, provided there is no flow separation. Considering that *R _{h}*

_{0}∝

*R*(as shown below) and that the streamline coordinates reduce to ordinary Cartesian coordinates when

_{h}*R*→ ∞ (the flat-terrain case), Eq. (42) reduces to the logarithmic law, Eq. (1), when

_{h}*R*→ ∞. Equations (41) and (42) can also be extended to include hills covered with tall vegetation simply by displacing the origin, as in the flat-terrain case. Hence, it is assumed now that

_{h}*z*denotes the displaced height instead of the height above the ground.

Equation (40) can be rearranged in flux-gradient form. Defining the nondimensional velocity gradient as *ϕ _{m}* = (

*κz*/

*u*∗)(∂

*u*/∂

*z*) results in

### Speedup and relative speedup

Equation (41) can be used to write speedup and relative speedup equations. With the recollection that the streamline coordinates reduce to Cartesian coordinates far from the hill, substitution of Eqs. (1) and (41) into Eqs. (2) and (3) yields

where the friction velocity and the roughness length at RS are denoted by *u*∗_{0} and *z*_{00}, respectively. In power series form, Eqs. (44) and (45) become

Equations (44) and (46) only hold where both the logarithmic law and the modified logarithmic law hold. In mathematical terms, they are defined for *z* ≥ *z*_{0′}, with *z*_{0′} = max(*z*_{0}, *z*_{00}). The same is true for Eqs. (45) and (47), which are defined for *z* ≥ *z*_{0′} and *z*_{0′} ≠ *z*_{00}. Equations (46) and (47) show that both Δ*u* and Δ*S* → 0 when *R _{h}* → ∞.

### Heights of maximum speedup and relative speedup

Readers who are not interested in the mathematical analysis that follows may skip most of this section and go directly to the relevant results, Eqs. (53)–(55). The heights at which the speedup and the relative speedup are maximum, *l* and *l _{s}*, respectively, can be calculated by setting ∂Δ

*u*/∂

*z*= 0 and ∂Δ

*S*/∂

*z*= 0. By assuming that these points of maximum fall into the fully turbulent region, we can apply this procedure to Eqs. (44) and (45). By noting that ∂Δ

*u*/∂

*z*= ∂

*u*/∂

*z*− ∂

*u*

_{0}/∂

*z*and using the results of Eqs. (38) and (1), ∂Δ

*u*/∂

*z*becomes

Setting Eq. (48) equal to zero and recalling that *u*∗ > 0 (no reverse flow), *u*∗_{0} > 0, and *z* ≥ *z*_{0′} >0 results in *u*∗ exp[(*z*_{crit} − *z*_{0})/*R _{h}*

_{0}] =

*u*∗

_{0}, which has the unique solution

for *z*_{crit} > 0. This solution implies that *u*∗_{0} > *u*∗ if *R _{h}*

_{0}> 0 and that

*u*∗

_{0}<

*u*∗ if

*R*

_{h}_{0}< 0. To decide whether

*z*

_{crit}calculated through Eq. (49) is a maximum or a minimum, the expression

*u*∗ exp[(

*z*

_{crit}−

*z*

_{0})/

*R*

_{h}_{0}] =

*u*

_{*0}is substituted in Eq. (48). It follows that

For *R _{h}*

_{0}> 0, the solutions for the above expressions are

which implies that *u*∗_{0} > *u*∗ and that *z*_{crit} is the absolute minimum. For *R _{h}*

_{0}< 0

which implies that *u*∗_{0} < *u*∗ and that *z*_{crit} is the absolute maximum. In most cases, we are interested in *l* at the HT, which corresponds to Eq. (52). Substituting for *z*_{crit} with *l* in Eq. (49) yields

An expression for the maximum speedup, Δ*u*_{max}, can be obtained by the substitution of Eq. (53) into Eqs. (44) or (46).

The same analysis is repeated to obtain *l _{S}* from the Δ

*S*profile, and the complete calculations can be found in Pellegrini (2001). The result is

if *z*_{0′} belongs to the domain of Δ*S* or lim_{z→z0′}(*z*) from the right if it does not. This expression represents an absolute maximum of Δ*S* when *R _{h}*

_{0}< 0 and an absolute minimum when

*R*

_{h}_{0}> 0, similar to Eq. (53). The analytical result expressed by Eq. (54) has only been obtained in the literature from extrapolation of field measurements down to the ground (Taylor and Lee 1984; Mickle et al. 1988). Substitution of Eq. (54) into Eq. (47) to obtain Δ

*S*

_{max}yields

Again, Eq. (55) represents a maximum of Δ*S* when *R _{h}* < 0 and a minimum when

*R*> 0.

_{h}### Heights of maximum speedup derived from velocity profiles available in the literature

The procedure to determine *l* used in the preceding section can also be applied to other velocity profiles available in the literature, such as the profiles of Taylor and Lee (1984), Lemelin et al. (1988), and Finnigan (1992). Taylor and Lee (1984) proposed a simple theory to calculate Δ*S*_{max} and Δ*S*(*z*) in flows over hills, which can be used to derive an expression for Δ*u*(*z*) from Eq. (3). Analysis of the expression for Δ*S*(*z*) yields

where *l*^{+} ≡ *l*/*z*_{0}, *L*^{+}_{h} ≡ *L _{h}*/

*z*

_{0},

*C*

_{2}≡ 1/

*Aκ*

^{2}, and

*A*is a constant that takes different values according to the type of hill considered. Equation (56) is identical to Jackson and Hunt’s (1975) equation,

*l*

^{+}ln(

*l*

^{+}) = 2

*κ*

^{2}

*L*

^{+}

_{h}, except that the constant 2 is substituted here by

*C*

_{2}, which varies from 1.6 to 2.2 (with

*κ*= 0.39), depending on the value adopted for

*A*. As a consequence, assuming the validity of Taylor and Lee’s (1984) velocity profile is the same as assuming the validity of Jackson and Hunt’s (1975) expression for

*l*.

Lemelin et al. (1988) also proposed an expression for the profile of Δ*S*(*z*). Analysis of their expression produces

where *C*_{3} ≡ 1/(2*Dκ*^{2}) with the constant *D* taking on different values according to the type of hill considered. Equation (57) is also similar to Jackson and Hunt’s (1975) equation.

The analysis of Finnigan’s (1992) equation for *l* furnishes

where *C*_{4} is a constant to be determined by comparison with observational data.

### Speedup ratios

In the present analysis, the relative speedup at any height can be calculated from Eq. (47). However, in some practical applications, it is not necessary to have the entire vertical velocity profile available. Knowledge of the relative speedup at certain standard heights above the ground is often sufficient and can be done by combining Eqs. (47) and (55).

Equation (47) shows that, for a generic height *a*, Δ*S*(*x, a*) + 1 = (*u*∗/*u*∗_{0})*f*(*R _{h}*

_{0},

*z*

_{0},

*z*

_{00}), where

*f*is a known function. With recollection that Δ

*S*and (

*u*∗/

*u*∗

_{0}) are nondimensional parameters, dimensional analysis of the remaining parameters shows that Δ

*S*(

*x, a*) + 1 = (

*u*∗/

*u*∗

_{0})

*F*(

*R*

_{h}_{0}/

*z*

_{0},

*R*

_{h}_{0}/

*z*

_{00}), with

*F*being an unknown function. In addition, we can derive from Eq. (55) that Δ

*S*

_{max}+ 1 = (

*u*∗/

*u*∗

_{0}). Dividing these two results yields a ratio

at height *a*, where *G* is an unknown function. From Eq. (59), Δ*S*(*x, a*) can be calculated if Δ*S*_{max} and the dependence of *R _{a}* on the hill’s geometry are known. The value of Δ

*S*

_{max}can be calculated from Eq. (55) and the dependence of the function

*R*on the hill’s geometry needs to be inferred through comparison with field data, which is presented in the next section.

_{a}## Comparison with observational data

According to Wood (2000), the experimental studies carried out on Askervein Hill (Taylor and Teunissen 1987), Black Mountain (Bradley 1980), and Cooper’s Ridge (Coppin et al. 1994) were the only ones capable of providing the near-neutral data for wind speed over isolated low-sloped hills needed for comparison. Other recent studies, such as Walmsley and Taylor (1996) and Founda et al. (1997), provide support to this conclusion. Only one experiment has been conducted under neutral atmosphere since 1996 (Reid 2003), but the density of measurements near the surface is not high enough for the purposes of this paper.

At Askervein Hill and Black Mountain, vertical profiles of mean wind velocity were measured simultaneously at RS and HT up to considerable heights. At Askervein, vertical velocity has been measured up to heights of 50 m. Askervein is an essentially elliptical hill, uniformly covered with low vegetation, where the minor axis is 1 km long, the major axis is 2 km long, *h* = 116 m, and *z*_{0} is approximately equal to 0.03 m. At Black Mountain, mean velocities were measured up to heights of 89 m above the displaced origin at the HT and 18 m at the RS. Black Mountain is a slightly asymmetric hill, with *h* = 170 m and *L _{h}* approximately equal to 275 m (in the prevailing wind direction). Its surface is covered with a close tree canopy, with average height of 10 m and

*z*

_{0}of about 1.0 m. As mentioned in connection with Eq. (39), measurements of the radius of curvature were not made over any of the hills.

At Cooper’s Ridge, measurements covered a range of stability classes, including the neutral class, and the hill’s curvature was estimated (Coppin et al. 1994). Because the original data are not available, they are not used here for comparison.

### The modified logarithmic law

Equation (41) depends on four parameters that are not known a priori: *z*_{0}, *κ*, *u*∗, and *R _{h}*

_{0}. They must be estimated before the modified log law can be used. In atmospheric flow over a flat terrain, the von Kármán parameter

*κ*is first assigned a value, generally 0.4. Then, if Eq. (1) is to be valid, field data plotted in semilogarithmic form must follow a straight line. Measurement of the linear and angular coefficients yields the values of

*z*

_{0}and

*u*∗, respectively, for each velocity profile. Application of this procedure to the data at RS determines

*z*

_{00}and

*u*∗

_{0}. In the ABL flow over hills, this same process can be used if the value of

*R*

_{h}_{0}is available beforehand. Because it is not, a different method is then employed. Following the procedure for the flat-terrain case, we define

In Eqs. (60) and (61), the parameter 1/*R _{h}*

_{0}in the argument of the logarithmic functions and the Euler constant of the series expansions are omitted because they cancel out.

As in the flat-terrain case, plotting the field data in the variables *ζ* versus *u* produces a straight line with linear coefficient *B* = *ζ*_{0} and angular coefficient *m* = [*κ* exp(*z*_{0}/*R _{h}*

_{0})]/

*u*∗. Once

*B*has been obtained,

*ζ*

_{0}becomes known and

*z*

_{0}can be calculated from Eq. (61), whereas the determination of

*m*allows

*u*∗ to be calculated from the previous relation. Both calculations, however, require

*R*

_{h}_{0}to be known. Because this condition is not the case, an initial value for

*R*

_{h}_{0}is guessed and an iterative procedure is implemented. First, the initial guess of

*R*

_{h}_{0}and

*z*

_{00}are substituted into Eqs. (60) and (61) so that

*ζ*(for each height) and

*ζ*

_{0}can be calculated. The results are then plotted in the form of

*ζ*versus

*u*and a straight line is best-fitted to them, which determines

*B*and

*m*and allows

*z*

_{0}and

*u*∗ to be calculated. Given the assumption that the hill is uniformly covered with vegetation, calculated values of

*z*

_{0}and

*z*

_{00}should be equal and a straight line should fit the observed data well. If both of these conditions are satisfied, the value assumed for

*R*

_{h}_{0}is considered to be correct. If one of the two conditions is not satisfied, a new value of

*R*

_{h}_{0}is assumed and the whole procedure is repeated. When both conditions are satisfied, the iteration stops and the last value of

*R*

_{h}_{0}is considered to be correct. In the analysis that follows, the von Kármán parameter is assumed to depend on Re* =

*u*∗

*z*

_{0}/

*ν*, (

*ν*being the kinematic viscosity), according to Frenzen and Voguel (1995), who suggest average values of

*κ*= 0.39 for 0.007 <

*z*

_{0}< 0.087 m and

*κ*= 0.37 for

*z*

_{0}≥ 1.0 m for most atmospheric flows. These values are used in our analysis.

It can be assumed that *R _{h}*

_{0}depends on the geometry of the hill (through

*R*), on viscous and pressure forces, and on

_{h}*z*

_{0}. Based on that assumption, dimensional analysis shows that

*R*

_{h}_{0}/

*h*=

*g*

_{1}(

*R*/

_{h}*h*, Re*, Ro*), where Ro* =

*U*/

_{g}*fz*

_{0}is the roughness Rossby number and

*f*is the the Coriolis factor. Because detailed hill profile shapes are not available to determine

*R*, an approximate relation between

_{h}*R*and the global geometry of the hill is used, based on the calculation of the curvature radius of two analytical functions (the bell and the Gaussian shapes) that approximate real hill shapes. Both calculations depend on

_{h}*L*

^{2}

_{h}/2

*h*as a scaling parameter, suggesting that

*R*∝

_{h}*L*

^{2}

_{h}/2

*h*and, thus, that

*R*/

_{h}*h*∝

*L*/

_{h}*h*and

*R*

_{h}_{0}/

*h*=

*g*

_{2}(

*L*/

_{h}*h*, Re*, Ro*).

The dependence of *R _{h}*

_{0}/

*h*on

*L*/

_{h}*h*is shown in Fig. 3 for the Askervein data, where different values of

*L*/

_{h}*h*correspond to different wind directions. The dependence of

*R*/

_{h}*h*on

*L*/

_{h}*h*for the HT of Black Mountain is not investigated because all of the measurements were obtained for the same incident wind direction, which gives

*R*/

_{h}*h*= −0.35 for

*L*/

_{h}*h*= 1.62. The straight line fitted to the data has a slope that is consistently high, confirming the suspected dependence between

*R*

_{h}_{0}/

*h*and

*L*/

_{h}*h*, despite the relatively large data scatter (shown by the correlation coefficient,

*R*

^{2}). No strong dependence of

*R*

_{h}_{0}/

*h*with Re* or Ro* was detected in our study. Therefore,

In the analysis above, the Askervein experimental data acquired on days 25 September 1982, 26 and 30 September 1983, and 5 October 1983 were not considered to be adequate for the purpose of this work and were eventually disregarded. On 25 September 1982 and 30 September 1983 (hereinafter referred to as the 3D days), the wind approached the hill from a direction almost parallel to the major axis, suggesting the occurrence of nonnegligible 3D effects. On 5 October 1983 (hereinafter referred to as the wake day), the wind blew from the direction of some nearby farm buildings and showed nonlogarithmic profile at RS. The data for 26 September 1982 cannot be well fitted by the modified log law, probably because the first two measurement levels (1 and 5 m) are missing. A thorough investigation of Fig. 3 to assess the effects of the 3D and wake days leads to the conclusion that 25 September 1982 and 30 September 1983 are indeed strongly affected by 3D effects, whereas 5 October 1983 is weakly (but observably) influenced by wake effects. Therefore, the 3D and wake days are disregarded in the discussion that follows. The data for 26 September 1982 are permanently neglected.

Additional analysis of Fig. 3 shows that *R _{h}*

_{0}is different from the estimated average value of

*R*, as suspected in section 3. With the values of

_{h}*R*

_{h}_{0}calculated above, the modified log law can be evaluated for all available observed profiles. Figure 4 shows the experimental data of Askervein and Black Mountain plotted in

*ζ*versus

*u*form. The experimental points are average values, weighted by their respective measuring periods. The straight line corresponds to the modified logarithmic law and shows good agreement with the observations. The influence of the wake day, excluded from the 1983 Askervein data, is shown below.

At Askervein, the results show the expected behavior up to heights around 15 m. At Black Mountain, the observed profiles agree well with the theory from 30 to 89 m. The result for Black Mountain is predictable in some sense. Kaimal and Finnigan (1994) observed that the log law is not expected to be valid at the roughness sublayer (the layer going from the ground to 3*h*_{0}, where *h*_{0} is the average height of the roughness elements). At Black Mountain, 3*h*_{0} = 30 m. At Askervein, however, 3*h*_{0} is approximately equal to 1 m and the effects of the roughness sublayer are not detected.

### The relative speedup

Figures 5 –8 show the comparison of Eq. (47) with the observational data. The experimental points are average values, weighted by their respective measuring periods. The wake day is not included in Fig. 6 and its effects are shown in Fig. 7, which supports the indication that this point should be disregarded for the relative speedup. In general, the theory agrees very well with observation above the roughness sublayer. Equation (47) is observed to be sensitive to the values of *R _{h}*

_{0},

*u*∗, and

*u*∗

_{0}, but its sensitivity to

*z*

_{0}and

*z*

_{00}is negligible. Figures 5 and 6 also show the comparison between Eq. (47) and the equations proposed by Taylor and Lee (1984), Weng et al. (2000), and Lemelin et al. (1988). Overall, Eq. (47) shows the best agreement with observation, whereas Figs. 5 and 6 indicate that Δ

*S*

_{max}is different from Δ

*S*(

*z*= 10 m), as assumed by Taylor and Lee (1984) and Weng et al. (2000).

For the Askervein data, Figs. 5 and 6 confirm that *l _{s}* =

*z*

_{0}, as predicted by Eq. (54). At Black Mountain, on the other hand, Fig. 8 shows the maximum clearly located in the jet, far from the ground. This result is expected because, as mentioned before, Eq. (44) is obtained from Eq. (54), which is valid for region III, assuming it extends down to

*z*=

*z*

_{0}. At Askervein, where the roughness sublayer is thin, Δ

*S*does not vary much in this region and, therefore, the approximation is good. At Black Mountain, however, the roughness sublayer is large, Δ

*S*varies considerably, and Eqs. (54) and (55) are no longer valid in that region. A solution for region I is likely to correct for this effect. As a consequence, no predictions for Δ

*S*

_{max}are compared with the observations at Black Mountain.

Predicted Δ*S*_{max} values are compared indirectly with the Askervein data because no velocity measurements were made at *z*_{0}. From Figs. 5 and 6 we see that the observed values of the velocity measurements close to the ground agree well with the theoretical curve. This fact allows us to infer that the predicted value for Δ*S*_{max} also agrees well with the observed value. The same reasoning indicates that Δ*S*_{max} is roughly 90% larger than Δ*S*(*z* = 10 m) for the data in Fig. 5 and is 70% larger for the data in Fig. 6.

Figure 9 shows the dependence of the calculated values of Δ*S*_{max} on the incident wind direction *ϕ*. The error bars are set as 10% of Δ*S*_{max}. The predictions of Taylor and Lee (1984) are also plotted for comparison and are seen to be substantially smaller than the values obtained with Eq. (55). The dependence of *L _{h}* on

*ϕ*is also represented in Fig. 9, and the expected negative correlation with Δ

*S*

_{max}is observed. Polynomial best-fit curves are fitted to the data, and some

*L*values are interpolated from Taylor and Lee (1984). Both the 3D and wake days are included (extreme points of the graphs) because they aid in the polynomial best-fit procedure.

_{h}### The height of maximum speedup

The most well-known expressions found in the literature during the last decades to calculate the maximum of the function Δ*u* come from the works of Jackson and Hunt (1975), Jensen et al. (1984), Claussen (1988), and Beljaars and Taylor (1989). Walmsley and Taylor (1996) present a good review of these expressions. In a recent paper, Pellegrini and Bodstein (2000) propose a new constant for Jensen’s equation based on a best fit to available field data. The result is

Pellegrini and Bodstein (2000) conclude that Eq. (64) describes the experimental data slightly better than the equations obtained in the references above. The work in this paper proposes a new expression to calculate *l* [Eq. (53)], and only this equation is compared with Eq. (64) in Fig. 10, which shows predictions for *l* for the Askervein Hill data. All of the days are included so that the influence of 3D and wake days can be accessed. It is clear that Eq. (53) provides the best description of the observed values. Equation (64) tends to overestimate field results and shows large dispersion and poor prediction for the 3D and wake days (appearing far to the left and above the line). The average difference between theoretical and observed values is 44% and the standard deviation of these differences is 117%, considering all of the days. Equation (53), on the other hand, is considerably more accurate and shows less dispersion. The average difference is close to −13% and the standard deviation is close to 20%, no matter which days are considered. A more thorough analysis of this equation is presented in Pellegrini and Bodstein (2004).

For Black Mountain, *l* is observed to be located always at a height of about 30 m. An average value for *l* is obtained through Eq. (53), using average values of *u*∗ and *u*∗_{0} calculated in section 4a. The result is *l* = 41.8 m, overestimating the observed value by 39%. The result obtained from Eq. (64) produces *l* = 14.0 m, underestimating the observed value by 53%.

### The speedup ratios

The experimental data of Askervein are now used to obtain the dependence of *R _{a}* on

*R*

_{h}_{0}/

*z*

_{0}in Eq. (59). The dependence of

*R*on

_{a}*z*

_{00}cannot be explicitly detected because

*z*

_{0}=

*z*

_{00}. Because the measurement levels were different during the 1982 and 1983 campaigns, these years are considered separately. For the 1982 data, only seven velocity profiles are available, and these results are disregarded. Because the measurements for the 1983 campaign closest to the standard levels of 2 and 10 m are 3 and 8 m, respectively, the ratios

*R*

_{3}and

*R*

_{8}are calculated instead. The results of Black Mountain are disregarded because Δ

*S*

_{max}cannot be calculated (section 4c).

Figure 11 shows the values obtained for *R*_{3} and *R*_{8} from the Askervein data. The wake day does not follow the data trend and is neglected. The curves represent a logarithmic best fit, valid for 50 < |*R _{h}*

_{0}/

*z*

_{0}| < 500. In this range,

*R*=

_{a}*A*+ ln|

_{a}*R*

_{h}_{0}/

*z*

_{0}| +

*B*, with

_{a}*A*

_{3}= 0.077,

*A*

_{8}= 0.061, and

*B*

_{3}=

*B*

_{8}= 0.47, for both heights. This equation is used to calculate Δ

*S*(

*a*= 3 m) and Δ

*S*(

*a*= 8 m), and the results are compared with the field data in Fig. 12. The theory is seen to describe well the observed data at both heights, although there is a slight tendency to underestimate them at 8 m. The data dispersion is low in both cases.

## Conclusions

In this paper, a modified logarithmic law is proposed to describe the ABL flow over a two-dimensional low-sloped hill, under neutral atmosphere. The intermediate variable technique is applied to the *x* momentum equation in streamline coordinates to divide the flow field into five regions, each characterized by the dominance of different terms. Solution of the equation for the fully turbulent region provides a modified logarithmic law, which is expressed as a power series correction to the classical logarithmic law, valid for flat terrain, and, therefore, depends on *u*∗ and *z*_{0} as the logarithmic law does. In the modified logarithmic law, a new parameter appears, the *effective radius of curvature* of the hill. Although the influence of the buoyancy term in the equations is established, the nonneutral case is not considered. Additional expressions obtained from this law include equations for the speedup and relative speedup, expressions for their maximum values and the heights at which they occur, and an equation for the speedup ratio, which allows a simple calculation of the relative speedup to be performed.

The parameter *R _{h}*

_{0}is essential to the applicability of our modified log law. Our results show that

*R*

_{h}_{0}is a physically sound parameter and that it represents the influence of the detailed shape of the hill on the flow. In general,

*R*

_{h}_{0}/

*h*=

*g*

_{1}(

*R*/

_{h}*h*, Re*, Ro*), but the analysis of Askervein and Black Mountain data indicates that the dependence with Re* and Ro* is negligible. If this conclusion proves to be correct in the future,

*R*

_{h}_{0}can be tabulated definitively as a function of the hill’s geometry given by

*R*, in analogy to the well-known existing tabulations of

_{h}*z*

_{0}as a function of the vegetation height

*h*

_{0}(e.g., Stull 1997). This task requires that experimental velocity profiles measured over hills with known

*R*be available so that values for

_{h}*R*

_{h}_{0}can be determined. Before this is done, our results show that a simple linear relation between

*R*

_{h}_{0}/

*h*and

*L*/

_{h}*h*is a good approximation.

Comparison of the modified logarithmic law with Askervein and the Black Mountain field data through the velocity and relative speedup profiles shows very good agreement. We believe that this agreement is not just a result of fitting *R _{h}*

_{0}to observational data because this parameter appears in the modified logarithmic law as the argument of the function Ei(

*z*/

*R*

_{h}_{0}) and not simply as a scaling factor. In fact, the wind velocity over the HT appears to be well described by Ei(

*z*/

*R*

_{h}_{0}) in the same sense that it is well described by ln(

*z*/

*z*

_{0}) over flat terrain.

Relative speedup profiles also agree very well with observation. The expression for the height of maximum speedup is compared with Askervein data and is seen to perform better than the others available in the literature. The same is true for the maximum relative speedup. The equation for the speedup ratio is also compared with observations and shows encouraging results.

With respect to the Askervein data, the analysis indicates that the data containing wake and 3D effects must be considered with caution because the effective radius of curvature *R _{h}*

_{0}is sensitive to these effects, particularly the three-dimensional one. The comparison also shows that

*u*, Δ

*S*, and

*R*are sensitive to the wake effect and that Δ

_{a}*S*

_{max}and

*l*are not very sensitive to any of these effects.

Some recommendations about the field measurements are also suggested by the theory. It is desirable that a higher density of measurements be obtained close to the height *l* so that its value can be more firmly determined. The lowest measurement level should be close to the ground so that the position of *l _{S}* can be inferred. It is also desirable that complete vertical velocity profiles be measured at one or two stations over the hill slopes so that the dependence of the results on

*x*can be assessed. In all cases, measurements should include the real radius of curvature of the hill

*R*so that the approximation used in Eq. (39) do not need to be made. Measurements should also be obtained at the same levels at RS and HT so that no logarithmic approximations need to be made and the speedup ratios can be calculated.

_{h}## Acknowledgments

The authors thank Dr. Peter A. Taylor for sending us the original wind velocity data of the Askervein Hill experiment, Dr. John L. Walmsley for the digital data, and Dr. Wensong Weng for the copies of the “Guidelines” papers. We are also very grateful to all three reviewers for their many insightful comments and numerous corrections of the original manuscript. The authors acknowledge the financial support obtained from CAPES (PICDT) and CNPq, under Grants AI 143041/97-5, APQ 474904/01-6, and AI 551364/2002-5, during the execution of this project.

## REFERENCES

## Footnotes

*Corresponding author address:* Prof. Gustavo C. R. Bodstein, Dept. of Mechanical Engineering-EE/COPPE, Federal University of Rio de Janeiro-UFRJ, Caixa Postal 68503, 21945-970, Rio de Janeiro, RJ-Brazil. gustavo@mecanica.coppe.ufrj.br