Large Eddy Simulation of an Inhomogeneous Atmospheric Boundary Layer under Neutral Conditions

Transcription

1 2479 Large Eddy Simulation of an Inhomogeneous Atmospheric Boundary Layer under Neutral Conditions CHING-LONG LIN Department of Mechanical and Industrial Engineering and IIHR Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa JOHN W. GLENDENING Naval Research Laboratory, Monterey, California (Manuscript received 28 October 2001, in final form 25 February 2002) ABSTRACT Flow structures in an inhomogeneous neutrally stratified atmospheric boundary layer flow, obtained from large eddy simulation, are analyzed and compared with homogeneous case counterparts. The inhomogeneity is imposed in the streamwise direction by using two different surface roughness heights z o, each covering a streamwise distance of 4.8 km, to produce internal boundary layers. Adjustments of the mean velocity profiles are primarily confined to the lowest 140 m of the overlying boundary layer, with parcels being accelerated and decelerated over the respective smooth and rough surfaces. For large fetches downwind of the surface roughness discontinuities, the mean velocity profiles close to the surface are approximately logarithmic, but fitting to Monin Obukhov similarity profiles using the z o and stress of the underlying surface requires that the von Kármán constant be k 0.4 over the smooth surface and k 0.37 over the rough surface. Much of this difference is attributed to velocity accelerations created by locally induced pressure gradient forces within the boundary layer, requiring k to be adjusted when horizontally homogeneous similarity expressions are utilized. Quadrant analysis indicates that ejection and sweep intensities differ from those of homogeneous surface cases but the occurrence frequencies are similar. Budget analysis of momentum flux indicates that the shear production and pressure destruction terms roughly balance, which is consistent with previous homogeneous surface findings. Flow visualization and conditional sampling demonstrate that these two terms are physically associated with ejections, sweeps, and vortical structures. 1. Introduction Flow structures under neutral conditions in the surface layer of the atmospheric boundary layer (ABL) have been studied experimentally and numerically due to their importance in turbulent transport of momentum, heat, and moisture. Observationally, gust microfronts resulting from the collision of high- and low-speed momentum fluxes were identified by Mahrt (1991), ejection and sweep events in the near-neutral atmospheric surface layer were investigated by Bergström and Högström (1989) and Högström and Bergström (1996), and streaky structures were revealed by Weckwerth et al. (1997) using high-resolution Doppler lidar in the Lidars in Flat Terrain (LIFT) project. Numerically, the large eddy simulation (LES) technique has been used to study the characteristics of coherent structures over a horizontally homogeneous surface, since it allows fully three-dimensional analysis of Corresponding author address: Dr. Ching-Long Lin, Dept. of Mechanical and Industrial Engineering and IIHR Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa ching-long-lin@uiowa.edu fields within the atmospheric boundary layer. For instance, Moeng and Sullivan (1994) and Khanna and Brasseur (1998) noted streaky structures in the surface layer of neutral or near-neutral ABL, and Glendening (1996) analyzed the coupling between lineal roll structures and surface layer fluxes in a marine boundary layer over an extensive horizontal domain for many eddy cycles. In a simulated neutral ABL, Lin et al. (1996) found two types of eddy motions that lead to the formation of vortical structures and estimated the lifetime of ejection and sweep eddies; in a simulated convective ABL, Lin (2000) identified coherent motions responsible for pressure transport, which acts as a source of turbulent kinetic energy budget in the lower part of the ABL. The real atmospheric boundary layer, however, often contains abrupt changes of surface roughness that produce internal boundary layers (IBL). The mean IBL growth has been studied extensively for relatively short fetches downwind of a surface discontinuity, as reviewed by Garratt (1990, 1997). Mahrt et al. (1998) studied the effect of IBL in the coastal zone on thermal and momentum roughness lengths. Hobson et al. (1999) studied LES-generated neutrally stratified flow over het-

2 2480 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 erogeneous surfaces where the roughness length varied sinusoidally in the direction of the geostrophic wind. Lin et al. (1997) used LES to study flow over a spatially homogeneous but temporally varying surface roughness, as a spatially inhomogeneous IBL simulacrum, and found good agreement with IBL heights predicted by the models of Elliott (1958) and Panofsky and Townsend (1964) after adjusting a constant advective velocity to relate time with fetch. An interesting but not fully understood issue is the dependence of flow structures on the underlying roughness height. Oncley et al. (1996) have reported a dependence of the von Kármán constant k on the roughness Reynolds number, finding k over a nearly uniform plowed field but k 0.42 over snow covered ground. Katul et al. (1997) studied ejection and sweep motions at the canopy atmosphere interface of two forest systems and found that their occurrence frequencies are less sensitive to the underlying surface roughness while the flux contributions of the two motions are affected by the surface roughness. It is unclear how the IBL-related spatial inhomogeneity affects these structures. The objective of this paper is to study the dependence of flow structures on an abrupt change of surface roughness in a neutral ABL using the LES technique. We consider the variation of mean speed profiles over smooth and rough surfaces, the dependence of the von Kármán constant on the surface roughness, and the characteristics of ejection and sweep eddies in the presence of spatial inhomogeneity. The results from the spatially inhomogeneous surface case will be compared with those of the spatially homogeneous counterparts. The characteristics and physical realizations of shear production and of pressure destruction of Reynolds shear stress in the IBL are also investigated using the approaches of flow visualization and conditional sampling. Glendening and Lin (2002) have used the same LES dataset to investigate the mean turbulence structure at IBL transitions. The paper is organized as follows. In section 2, the numerical method and simulation parameters are briefly described. In section 3, the budget of mean streamwise velocity is considered and the relationship between the von Kármán constant and surface roughness is discussed. In section 4, the IBL heights obtained from the inhomogeneous surface case are compared and discussed with those of homogeneous surface cases. In section 5, the structures of ejection and sweep eddies are studied via quadrant analysis, conditional sampling, momentum flux budget, and flow visualization. Concluding remarks are given in section Numerical formulation Our LES numerical code solves the following timedependent, filtered incompressible Navier Stokes equations, including large-scale pressure gradient and Coriolis forces: U i 0, (1) x i Ui P* P g ijkujk ij3 fuj t x x i g ij i3, (2) x o where U i and i denote the respective resolved-scale velocity and vorticity, with i 1, 2, 3 corresponding to the east, north, and vertical directions (x, y, z); (U, V, W) are used alternately with U i. A Coriolis parameter f appropriate to midlatitudes (10 4 s 1 ) is applied, g is the gravitational acceleration, denotes the resolvedscale fluctuating virtual potential temperature, and o represents a reference virtual potential temperature; P* p/ R kk /3 (U k U k )/2, where R kk /2 is obtained by calculating a prognostic subgrid-scale (SGS) turbulence energy equation (Moeng 1984); P g /x i is the horizontal mean pressure gradient force generated by a geostrophic wind. A geostrophic wind of (U g, V g, 0) yields a forcing of (P g /x, P g /y, 0) ( fv g, fu g, 0). The fifth term on the right-hand side of Eq. (2) represents a buoyancy force. Here, ij R ij R kk ij /3 is the deviatoric part of the SGS stress tensor and is parameterized using the two-part eddy viscosity model of Sullivan et al. (1994): ij 2tSij 2TS ij, (3) where t and T are fluctuating and mean-field eddy viscosities, respectively, and is an isotropic factor; S ij is the resolved strain-rate tensor, and the spatial averaging (overline) operation over the homogeneous direction is performed at each time step. This model uses Monin Obukhov similarity theory to provide the surface friction velocity for calculation of the SGS vertical fluxes at the first vertical grid level above the surface. Sullivan et al. (1994) and Khanna and Brasseur (1997, 1998) studied the effect of SGS parameterization on ABL structure by using the SGS model noted above and that of Moeng (1984), and demonstrated that the SGS model greatly improves the physical realization of ABL surface layer structures. Moeng (1984) originally developed the code for the simulation of homogeneous atmospheric boundary layer flows; Sullivan et al. (1996) later implemented a nested grid technique that matches total momentum and temperature fluxes in the overlap region through the use of Germano s identity (Germano et al. 1991). The SGS fluxes on the fine grid ij are computed using the twopart SGS model Eq. (3), while the SGS fluxes on the coarse grid T ij are obtained through filtering the finegrid SGS fluxes and variables: f f f f T (U U U U ), (4) ij ij i j i j where the overbar with a superscript f denotes a filtering j i

3 2481 FIG. 2. Wind hodographs. FIG. 1. Vertical profiles of the ratios TKE SGS /(TKE RES TKE SGS ) and uw SGS /(uw RES uw SGS ), where TKE is turbulent kinetic en- ergy, SGS is subgrid-scale, and RES is resolved scale. operation. Glendening (1994) further adapted the code for the simulation of inhomogeneous flows by applying homogeneous surface flux parameterization only in the spanwise direction. Periodic boundary conditions are applied in both streamwise and spanwise directions. Spectral and centered finite difference methods are used for horizontal and vertical spatial derivatives, respectively. The turbulent flow is driven by a geostrophic wind (U g, V g ) (15, 0) m s 1. The stratification is essentially neutral below a capping inversion. The domain-averaged ABL height z i, defined by the temperature flux minimum, is 525 m. The aerodynamic roughness heights z o are, respectively, 0.16 m and 0.83 m between x 0 and 4800 m, and between x 4800 and 9600 m. The abrupt change in z o first affects the surface stress and the horizontal velocity component at the first vertical grid point above the surface by satisfying the similarity theory. In this study the gradients of mean horizontal speed in the x direction at z 2.5 m near the z o discontinuities do not exhibit abrupt change, so Fourier representation of velocity components near the discontinuities should not create difficulties. Relatively large roughness heights were chosen to increase the resolved turbulent kinetic energy near the surface; the roughness height z o 0.16 m corresponds to several trees, many hedges, few buildings, while z o 0.83 m represents large towns and cities (from Fig. 9.6 of Stull 1994). Fine-scale flow structures in the surface layer are resolved by a two-grid architecture which allows two-way interaction as in Sullivan et al. (1996). An outer coarse grid of points in the (x, y, z) direction covers a physical domain of m 3, and an inner fine grid of points is placed in the lower 300 m of the former to triple-grid resolutions in all directions. The simulation evolved over a period of about 11 h to a quasi-equilibrium, with data from the last 0.86 h being analyzed. The simulation took a total of about 204 CPU days (even longer clock/ calendar time) to run on a Cray J932 computer, due to the very large x-direction grid required to allow sufficient downwind development over each surface. Resolved variables (denoted by upper case variables) are composed of horizontal means averaged in the y direction only (upper case variables with an overbar) and fluctuating parts (lower-case variables); for example, U(x, y, z, t) U(x, z, t) u(x, y, z, t). Angle brackets denote a temporal averaging, such as U (x, z) U(x, z, t). Figure 1 shows vertical profiles of the SGS contribution to energy and momentum flux on the fine grid. At the fourth grid level above the surface (z 20 m), the SGS contribution is 20% of the total, which is a typical value for a large eddy simulation. Data below this height should be treated with caution because large eddies are underresolved. A grid-refinement study on the current model for simulating a neutrally stratified planetary boundary layer was presented in Lin et al. (1997). The solution with a grid resolution of m 3 was compared with that of m 3. Two-point correlation analysis showed that average streak distances in the surface layer on both grids are about the same. For the present case, the use of a twogrid structure allows one to compare the solutions on the coarse and fine grids (a grid resolution of m 3 versus m 3 ) to estimate the effect of grid sensitivity. It was found that the average streak distances on the coarse grid are comparable to those of the fine-grid solution. The instantaneous eddy structures, however, are better resolved on the fine grid. Wind hodographs obtained by averaging U and V above the respective smooth and rough surface domains are exhibited in Fig. 2 (note that the scale for Vis enlarged for clarity). The mean wind vector below z 100 m is angled 5 counterclockwise from the x axis, while at higher levels it veers due to the Coriolis effect. Here we define the surface layer as the region below 100 m, where wind is approximately constant in direction. In the following presentation, the fine-grid results of this inhomogeneous surface case will be discussed and compared with counterparts having the same z o but for

4 2482 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 TABLE 1. Surface friction velocities for homogeneous and inhomogeneous surface cases. Note that u * for the inhomogeneous surface case is obtained by averaging those after overshooting and undershooting behaviors of surface shear stress. Cases z o (m) u * (m s 1 ) Homogeneous Homogeneous Inhomogeneous Inhomogeneous a horizontally homogeneous surface, as previously discussed in Lin et al. (1997). Both homogeneous and inhomogeneous surface cases are driven by a geostrophic wind of 15 m s 1. The surface friction velocities for these cases are listed in Table Characteristics of mean velocity profiles This section presents mean velocity profiles to demonstrate differences between the inhomogeneous and homogeneous surface cases and to illustrate the mean speed (M) response to an abrupt change of surface roughness. Figures 3a and 3b show mean speed profiles at several downwind fetches for the inhomogeneous case, for fetches over the smooth and rough surfaces respectively, while Fig. 3c presents profiles for the smooth and rough homogeneous surface cases. The smooth and rough z o employed for the inhomogeneous and homogeneous cases are identical, but for the homogeneous case, internal boundary layers were generated by instantaneously changing the surface roughness over the entire domain, thereby allowing variation of mean velocity in time but retaining spatial homogeneity (Lin et al. 1997). The solid line in Fig. 3c depicts the mean speed profile of the spatially homogeneous, but temporally inhomogeneous IBL having the rougher surface. This profile is obtained by averaging data over a period of five largeeddy turnover times which begins at the sixth largeeddy turnover time from the time of surface roughness alteration from 0.16 to 0.83 m. The mean speed over the smoother surface homogeneous case (dashed line) is about 1.5 m s 1 faster than that over the rougher surface case throughout the boundary layer. For the inhomogeneous surface case, however, the mean speed is faster over the smoother surface only within the lowest 140 m of the 525-m-deep ABL, and the velocity variation between the rough and smooth surfaces is much smaller than between the rough and smooth homogeneous cases. The mean velocity differences between the homogeneous and inhomogeneous cases reflect a fundamental difference between them. The homogeneous surface cases have no locally induced mean pressure gradient, but for the inhomogeneous case the locally induced pressure and its gradient, as shown in Fig. 4, significantly affect the mean flow. To understand the formation of the local pressure gradient, one shall consider the U budget equation: U U U uu U W t x z x I II uw 1 p f (V V g). (5) z x III IV V For a homogeneous case, terms I, II, and IV are dropped. The remaining terms contribute to the Ekman balance mean velocities and the inertial oscillation velocities (Fig. 2 in Lin et al. 1996). The unsteady term on the left-hand side approaches zero if the time averaging is performed over at least one inertial period (1/ f 2.78 h). For the present inhomogeneous case, due to the use of periodic boundary condition in the x direction and by averaging Eq. (5) in the x direction, the same U budget equation as in the homogeneous case is obtained: U uw f (V V g), (6) t z III V where denotes averaging in the x and y directions and in time. To remove the Coriolis effect from the U budget equation, one can subtract Eq. (6) from Eq. (5) to obtain U U uu 0 U W x z x I II uw 1 p f V, ˆ (7) z x III IV V where ˆ. All of the terms in the above equation arise from spatial inhomogeneity. Term V is on the order of 10 6 and is much smaller than the other terms. Thus the spatial distributions of the first four terms are considered below. Figures 5a and 5b display the contours of the two mean advection components: UU/x and WU/ z. The mean velocity advection term is caused by the mean vertical velocity induced by spatial inhomogeneity and the mean horizontal velocity perturbed by the former as shown in Fig. 6. The contours of the stress gradient terms II and III are plotted in Figs. 7a and 7b; uu/x is only significant along the IBL height since the zero contour level fills most of the domain. As for uŵ/z, it is also strongest near the IBL top, but its

5 2483 FIG. 3. (a) Mean speed M (U 2 V 2 ) 1/2 profiles over the smooth surface, (b) mean speed profiles over the rough surface, and (c) mean speed profiles for homogeneous surface cases. The figures on the left side use a linear scale for the z axis and those on the right side use a logarithmic scale. Profiles for homogeneous surface cases are superimposed on (a) and (b) on the left side for comparison. effect extends throughout the surface layer at large downwind fetches, for instance, the regions below 140 m marked by letters A and B in Fig. 7b. Above 140 m, however, its effect diminishes. With the foregoing observations, the U budget Eq. (7) in the region above 140 m and away from the IBL top at large fetches, for example, x f 3200 m near the upper contour levels 5 and 6 in Fig. 5a and marked by the letters A and B, is simplified to U 1 p 0 U. (8) x x It indicates that the induced pressure gradient above the surface layer acts to decelerate (accelerate) the flow over the smooth (rough) surface region as shown in Fig. 3a (Fig. 3b) and Fig. 6a. In the surface layer away from the IBL top at large fetches, for example, the regions marked by letters A

6 2484 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 4. Contours of (a) p/ and (b) 1/ p/x. and B in Fig. 7b, the uŵ stress gradient counters the pressure gradient, yielding the approximate balance equation U uw 1 p 0 U. (9) x z x I III IV Note that terms I and IV are of the same sign opposite to term III leading to the acceleration (deceleration) of the inhomogeneous velocity over the smooth (rough) surface region as shown in Fig. 3a (Fig. 3b) and Fig. 6a. In contrast, for the spatially homogeneous but temporally inhomogeneous IBL, the lack of an adverse pressure gradient leads to the balance between terms I and III and produces a greater velocity difference after transitions. The variation of mean velocity in the outer layer is also governed by the two terms, differing from Eq. (8). Thus the mean velocity is affected throughout the entire boundary layer. In Fig. 8, a comparison of the mean speed profiles against Monin Obukhov similarity profiles finds differing von Kármán constants over the two surfaces for the inhomogeneous case. To allow direct comparison of velocities to a single similarity profile over each of the two roughnesses, the surface stress u * is that averaged over the domain covered by each surface roughness, but neglecting regions of undershooting and overshooting near the z o discontinuities (see Table 1). Figure 8a shows that, over the smooth surface, the mean speed profiles around x x f 400 m agree with a logarithmic similarity profile when the assumed von Kármán constant k is greater than 0.4. At large fetches, for example, x x f 1200 m and x x f 3600 m, the profiles converge toward a logarithmic similarity profile with a von Kármán constant k of Over the rough surface, however, at large fetches, such as, x 6000 m (x f 1200 m) and x 8400 m (x f 3600 m), the profiles are in better agreement with a logarithmic profile when k 0.37 is used.

7 2485 FIG. 5. Contours of (a) UU/x and (b) WU/z. FIG. 6. Contours of (a) U and (b) W. We also compare the nondimensionalized mean speed gradients at several downwind fetches over the smooth and rough surfaces in the surface layer. Figure 9 shows the vertical distributions of the reciprocal of the gradient z/u * (M/z). The mean values of k over the two regions are consistently different. It is noteworthy that the average of these k values is about 0.4. It suggests that by filtering out the effect of local pressure gradients through averaging in the x direction, one can obtain the von Kármán constant for the similarity profiles that assume horizontally homogeneous conditions, with no mean pressure gradient or advection. The mean speed profiles of the homogeneous surface cases during the transition are also plotted against the similarity profiles in Fig. 8c to demonstrate that for z o 0.16 m the mean speed profile agrees with a similarity profile when k After z o is changed to 0.83 m everywhere, a fit of the mean speed profile at a large downwind fetch to the similarity profile requires k However, the variation of k with z o is significantly weaker than for the inhomogeneous surface case. The

8 2486 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 7. Contours of (a) uu/x and (b) uw /z. additional variation in k found for the inhomogeneous surface case is therefore attributable to the locally induced pressure gradients created over each surface, a feature lacking in the homogeneous surface case. Oncley et al. (1996) obtained average von Kármán constants of over a nearly uniform plowed field and 0.42 over snow covered ground. They also compiled data from their measurements and others to illustrate a dependence of the von Kármán constant upon roughness Reynolds number (see Fig. 6 in Oncley et al. 1996). Our result qualitatively agrees with theirs in that the similarity profile over a smoother surface is associated with a larger von Kármán constant. Our result also suggests that the pressure gradients associated with large-scale spatial inhomogeneity can further alter the perceived von Kármán constant over a significantly large downwind distance. 4. Effect of local pressure gradients on IBL height Several studies have derived formulas for predicting IBL height at a given fetch, including Elliott (1958), Panofsky and Townsend (1964), Townsend (1966), Panofsky and Dutton (1984) and Wood (1982), though the derivations generally employed assumptions more applicable to significantly shorter fetches than those simulated here. In Fig. 10, predictions from these formulas are superimposed on the contours of turbulent momentum flux, the sum of the resolved and SGS constituents. For comparison, squares in the figure denote the IBL height defined by flux differences (Garratt 1990). Specifically, the IBL is defined to be the height at which the difference in mean flux from its upwind value at fetch x f 1600 m is 10%, that is to say, uw 0.9uw x 1600 m. Here, the profiles at fetch x f 1600 f m (i.e., x 8000 and 3200 m) serve as the reference profiles since they are considered relatively undisturbed by the z o changes at fetch x f 0 m (i.e., x 0 and 4800 m). The IBL height can also be defined using the difference in mean speed from the upwind value; this definition is more sensitive to the selection of the difference (Garratt 1990) and is therefore not adopted in this study. A comparison of IBL heights over both sur-

9 2487 FIG. 9. Vertical profiles of 1/[z/u * (M/z)] at various downwind fetches. The mean values of each profile are shown in the figure. FIG. 8. Normalized mean speed (M/u * ) profiles at various x locations over (a) the smooth surface and (b) the rough surface. (c) Profiles obtained from homogeneous surface cases. faces indicates that IBL height dependence upon fetch is roughly comparable in both regions. Figure 10 shows that at fetches less than 1600 m, all the models except Townsend (1966) and Wood (1982) overpredict IBL heights defined by a 10% perturbation from upwind profiles, with the IBL heights predicted by Elliott (1958) and Panofsky and Townsend (1964) being deeper than those from Panofsky and Dutton (1984). The primary difference among these models lies in their approximation of the velocity distribution. Elliott (1958) adopted a logarithmic profile, Panofsky and Townsend (1964) used a logarithmic-linear profile, and Townsend (1966) introduced self-preservation concepts to improve the blending function, each resulting in different IBL-depth-fetch formulas. Comparison with experimental data tends to support the self-preservation assumptions on the downwind profiles (Garratt 1990). For large fetches, say x f 2400 m and z 100 m, all models overpredict the IBL heights defined above, suggesting that a better approximation for the outer-layer velocity profile should be employed in developing the model. Lin et al. (1997) demonstrated that the growth of the IBLs generated by a spatially homogeneous but temporally varying surface roughness are well-predicted by the IBL height models of Elliott (1958) and Panofsky and Townsend (1964). With the presence of adverse pressure gradients in the inhomogeneous surface case, the alteration of the mean velocity in response to the underlying change of surface roughness is greatly reduced, leading to the overprediction of these two models in the present case. Whether or not these models perform similarly under different geostrophic wind conditions requires further investigation. For short fetches the model of Townsend (1966) and the empirical IBL-height-fetch relationship of Wood (1982) both predict IBL depths close to those defined by uw 0.9, which roughly coincides uw x f 1600 m with the contours of maximum and minimum uw/x. Townsend s model is based on the self-preservation assumptions [Eq. (4.2) in Townsend 1966], giving lnh/z o2(lnh/zo2 N) 2 h kx, f (10) 2lnh/z N o2 where h is the IBL height, N lnz o1 /z o2, and z o1 and z o2 are the roughness lengths for x f 0 and x f 0, respectively. Substitution of N by lnz o1 /z o2 into Eq. (10) gives lnh/z lnh/z o1 o2 2 h kx, f (11) 2lnh/zo1zo2 suggesting that IBL growth at a junction of smooth and rough surfaces does not depend upon the sign of the

10 2488 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 10. Contours of mean momentum flux uw. IBL heights predicted by the formulas of Elliot (Elliot 1958), PT (Panofsky and Townsend 1964), T66 (Townsend 1966), PD (Panofsky and Dutton 1984), and Wood (Wood 1982). Open squares are uw 0.9. uw xf1600 m change in surface roughness. The IBL formula of Wood (1982) is 0.8 h x C f m i m, (12) z z o m where C i 0.28 and z o is the maximum of the two surface roughness heights before and after the abrupt change of roughness. It thus implies that the IBL height is controlled by the rougher surface, which is not necessarily the local surface. This suggests that the IBL height over the smooth surface is strongly affected by turbulent eddies that are produced over the rough surface and advected into the smooth surface region. The agreement of Eqs. (11) and (12) in the present case is apparently fortuitous since Eq. (11) is dependent upon both roughnesses but Eq. (12) is not. Both Eqs. (11) and (12) imply that IBL growth does not depend upon the ordering of the surface roughnesses. Glendening and Lin (2002) argue for Townsend s interpretation of this result by demonstrating that self similarity is achieved at fetches greater than 0.5 km and noting that the asymptotic limit of Eq. (11) also depends solely on the rougher surface as the difference in roughness becomes large. 5. Characteristics of momentum flux a. Quadrant analysis Momentum fluxes in turbulent boundary layer and channel flows can be partitioned into four components based upon the signs of the velocity fluctuations. Ejection and sweep events are the most active of these components and play a significant role in turbulent boundary layer dynamics (Robinson 1991). Statistically, ejections are more intense than sweeps except near the surface (Willmarth and Lu 1972; Kim et al. 1987; Lin et al. 1996). Since wind direction veers with height in the o ABL, we replace the x-component velocity fluctuation u by q to simplify the following analysis, where q is the horizontal fluctuating velocity in the mean momentum flux direction. Figure 11 presents vertical profiles of averaged s n qw, where (s, n) (, ), averaged within four different regions. q and q signifies q 0 and q 0, respectively; likewise for w 0 and w 0. Profiles obtained from the homogeneous surface case having the same z o are also displayed for comparison. In general, momentum fluxes in the second (q w, ejection) and fourth (q w, sweep) quadrants are stronger than those in the first (q w ) and third (q w ) quadrants. There are several interesting observations. 1) For x f 10 to 2400 m, ejections (sweeps) over the rough surface shown in Fig. 11c are stronger than those over the smooth surface in Fig. 11a only for z 100 m (50 m). At large fetches these heights increase to 190 m (ejections) and 120 m (sweeps) as shown in Figs. 11b and 11d. Thus, the expected intensity differences are found only near the surface, since ejections and sweeps located above the IBL originate at the upwind surface before the z o change, and thus do not reflect the underlying surface conditions. This is consistent with the contours of momentum flux over the smooth surface between x 0 and 2000 m, shown in Fig. 10, that exhibit an unambiguous continuity with the rough surface region contours. 2) Ejections and sweeps over the smooth surface decrease in intensity with fetch, whereas, over the rough surface, the opposite occurs. This implies that an appreciable number of ejections and sweeps over the smooth surface originate over the rough surface and subsequently either propagate outward or decrease in number and intensity after entering the smooth-surface region.

11 2489 FIG. 11. Vertical distributions of s n qw obtained by averaging data in the range of x: (a) m, (b) m, (c) m, and (d) m. Here, (s, n) (, ), (, ), (, ), (, ). Lines with solid circles are data from homogeneous surface cases with surface roughness height z o 0.16 m, for parts (a) and (b) and 0.83 m for parts (c) and (d). 3) Ejections and sweeps over the smooth surface are stronger than their homogeneous surface counterparts, except near the surface, while those over the rough surface are weaker than their homogeneous counterparts. The difference is attributable to the large-scale pressure gradient and the blending of those eddies advected from the upwind surface with those generated by the local surface. Willmarth and Lu (1972) found that ejections contribute 80% of the Reynolds stress in a flat-plate turbulent boundary layer with Re 4230 ( is momentum thickness) at z/ ( is boundary layer thickness). At the same dimensionless height, the fractional contributions of ejections in the four different regions in Figs. 11a d are 68%, 69%, 72%, and 71%, respectively. For the homogeneous-surface case, the fractional contribution is 70% (Lin et al. 1996). Thus, in the smooth and rough surface regions, the fractional contributions are only slightly larger and slightly smaller, respectively, than for the homogeneous surface case. This suggests that the fractional contribution of ejections to the total momentum flux is not overly sensitive to spatial inhomogeneity. In spite of these differences in intensity (Fig. 11), the frequencies at which the four quadrant events occur are similar in the four different regions (see Fig. 12) and are also similar to those of the homogeneous surface cases. Overall, the occurrence frequencies are 18% for the first and third quadrant events, and 34% and 31% for the ejection and sweep events. These findings are consistent with those observed at a forest atmosphere interface. For instance, Katul et al. (1997) found that the time fraction (occurrence frequencies) of the ejection and sweep events at the canopy atmosphere interface of two forest systems are less sensitive to the variation of surface roughness; they are 0.36 and 0.31, respectively, for momentum transport. b. Budget of momentum flux To understand how the spatial inhomogeneity affects the transport of momentum flux, we analyze the budget

12 2490 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 12. Frequencies of q s w n events as a function of height at x (a) m, (b) m, (c) m, and (d) m. Here, (s, n) (, ), (, ), (, ), (, ). Lines with solid circles are data from homogeneous surface cases with surface roughness height z o 0.16 m, for parts (a) and (b) and 0.83 m for parts (c) and (d). of momentum flux uw (i.e., Reynolds shear stress), which reads uw uw uw U 2 U W w t x z z I II 2 uw 1 p p w u, (13) z x z III IV where terms I, II, III, IV are referred to as advection, shear production, turbulent transport and pressure velocity interaction (pressure destruction), respectively. Here, viscosity has been assumed negligible (Lumley 1970) and buoyancy also neglected due to the neutral conditions. The shear production terms of (uww/ z uuw/x uwu /x) and the turbulent 2 transport term of uw/xare found to be negligible as compared with other terms, consistent with those of Mulhearn (1978), and are not shown. Compared with the momentum flux budget for the homogeneous surface case [for example, Eq. (17) in Wyngaard 1992], term I is a new term resulting from the spatial inhomogeneity. From surface layer measurements, Wyngaard (1992) determined that the turbulent transport term III is negligible and that the pressure velocity interaction term IV acts as a sink term, locally balancing to a first approximation, the shear production term II. In wind-tunnel modeling of a rough-to-smooth transition, Mulhearn (1978) found that shear production is roughly balanced by pressure velocity interaction and that both terms are much larger than the advection and turbulent transport terms. We first investigate the budget Eq. (13) for the IBL and then identify coherent structures taking part in the local balance of shear production and pressure destruction. Since the advection term appears only in the inhomogeneous ABL, we first examine contours of this term together with uw contours. Figure 13a indicates that the advection term maximum is closely associated with

13 2491 FIG. 13. Contours of (a) advection term and (b) shear production term in the uw budget (m 2 s 3 ). the IBL height defined by uw 0.9uw x f 1600 m (squares in Fig. 10). The advection term is negative (contour levels 1, 2) over the smooth surface but positive (contour levels 3, 4) over the rough surface. This sign alternation results from the mean momentum flux decreasing over the smooth surface but increasing over the rough surface. Contours of the shear production term are displayed in Fig. 13b. Near the surface discontinuities, x 0 and 4800 m, and below z 50 m, the shear production is at least 10 times stronger than the advection term. Along a selected height, say z 50 m, the shear production term decreases with increasing fetch to an asymptotic minimum over the smooth surface and increases to an asymptotic maximum over the rough surface. Figures 14a and 14b show the vertical profiles of four terms of Eq. (13) over the smooth surface, averaging over 200 m streamwise to obtain sufficient statistics. Two features are observed. First, the advection and turbulent transport terms are of comparable magnitude and are much smaller than the shear production (negative value) and pressure destruction (positive value) terms. Second, the magnitudes of shear production and pressure destruction over the smooth surface at a given height decrease with fetch to an asymptote (see also Fig. 13b). This suggests local correlation of these two terms. Both features are also found in the rough surface region (Figs. 14c and 14d). Although the notion of local balance between shear production and pressure destruction is well known and has led to simple models useful for surface layer calculations (e.g., Wyngaard 1992), the instantaneous physical realizations of the two terms and their association with coherent structures have not been clearly established. In the next two sections, we first use flow visualization to establish the deterministic nature of these two terms and then use conditional sam-

14 2492 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 14. Vertical distributions of momentum flux uw budget averaged in the range of x (a) m, (b) m, (c) m, and (d) m. Solid line is the shear production term; dashed line is the advection term; dashed dotted line is the turbulent transport; dotted line is the pressure velocity interaction term. pling to provide statistical evidence for their association with specific coherent structures. c. Flow visualization Because of the dominance of the pressure velocity interaction term IV in the momentum budget Eq. (13), we first identify negative low-pressure regions that tend to associate with vortical structures (Robinson 1991). Figure 15a shows four vortical structures associated with low-pressure regions, marked by letters A, B, C, D. Vortices A and B are quasi-spanwise rotational motions as revealed by the velocity vector field (Fig. 15b) on the x z plane passing through structures A, B, C (see the horizontal solid line at the center of Fig. 15a). Strong ejection motions are found at the lower left side of structures A and B. Strong sweep motions are found at their upper right side. If the x z plane containing velocity vectors is moved toward structure D, the rotational nature of structure D becomes evident. Structure C has a horseshoe shape with two legs whose directions of rotation are normal to those of structures A and B. Since the x z plane containing velocity vectors is tangent to the edge of one leg of structure C (Fig. 15a), the rampshaped ejection motions associated with this leg are observed in Fig. 15b. Another vortex that remains beneath structure C has features similar to those of the others vortices and so is not discussed here. In Fig. 15c isosurfaces of instantaneous shear production w 2 U/z (denoted by solid gray regions) are superimposed on the vortical structures identified above (represented by transparent gray regions). Structures A, C, D in Fig. 15b are clearly associated with shear production also marked by letters A, C, D in Fig. 15c, while shear production also appears in the vicinity of structure B when isosurface values are reduced. Shear production is associated with vortical structures because vortices produce strong ejections and sweeps when stretched by mean shear (Robinson 1991). It should be noted that vortical structures near the surface are underresolved as

15 2493 FIG. 16. Schematics of pressure velocity interaction. (a) A spanwise vortex, (b) quasi-streamwise counterrotating vortices. quadrant events are characterized by their spatial correlation with vortical motions. Strong pressure velocity interaction tends to occur around vortical motions due to the proximity of positive w and negative p/x on one side of the vortex and of negative w with positive p/x on the other side. These schematics, however, do not explain why the occurrence frequencies and intensity of these events are different. The surface inhomogeneity does not affect the spatial correlation between vortical motions, shear production, and pressure destruction, but does affect their intensity. FIG. 15. Isosurfaces of low-pressure fluctuation regions p/ 4 m 2 s 2 : (a) top view; (b) side view with velocity vectors. (c) Isosurfaces of shear production w 2 U/z 0.08 m 2 s 3 (solid gray). (d) Isosurfaces of pressure destruction 1/(up/z wp/x) 0.2 m 2 s 3 (solid gray). indicated by Fig. 1. Therefore the above spatial correlation is not evident as the surface is approached. Figure 15d displays isosurfaces of the instantaneous pressure velocity interaction term 1/(up/z wp/ x), which acts to destroy shear stress. The regions marked by A, B1 and B2, C1 and C2, and D are locations of pressure destruction associated with the respective vortical structures A, B, C, and D. The above observations can be illustrated by Fig. 16, which shows the schematics of pressure velocity interaction for a spanwise vortex and a pair of quasi-streamwise counterrotating vortices. The schematic in Fig. 16a represents vortical structures A and B in Fig. 15b, and the one in Fig. 16b represents structure C in Fig. 15b. The four d. Conditional sampling of shear production To provide further evidence for the physical association of shear production and pressure destruction with ejection and sweep events, we use the following formula for conditional sampling, which is based on the concept of a conditional eddy by Adrian (1990): ˆ(x, y, z) (xx, y y, z z, t), (14) where the overhat ˆ (x, y, z) denotes the conditional average of a physical quantity and is the conditioning event at all (y, t) at a selected detection location (x, z). Here, as previously, the overbar designates averaging in the y direction, for which the forcing is homogeneous, and the angle bracket denotes averaging in time. For a selected (x, z), the methodology searches along the y coordinate and in time, for events that satisfy the specified condition, and places these events at the origin of coordinates (x, y, z) that parallel the original Cartesian coordinates (x, y, z). Averaging on any physical quantity associated with these events is then performed using coordinates (x, y, z). The conditional sampling technique has been used by Kim and Moin

16 2494 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 TABLE 2. Partition of shear production w 2 U/z, m 2 s 3 at (x, z) (800, 50) m into four quadrants. Quadrant 1, q w 2, q w 3, q w 4, q w Percent of grid points Percent of contribution of shear production (1986) to study ejections, sweeps and vortical structures in a turbulent channel flow, by Schmidt and Schumann (1989) to study large-scale updrafts in a strongly convective boundary layer, and by Lin (1999) to extract coherent structures associated with energy backward and forward scatter transfer at near-grid scale in a convective boundary layer. The conditional events associated with shear production is defined as w 2 U /z m 2 s 3 at the location (x, z) (800, 50) m. The sampling height has to be inside the surface layer (100 m) and away from the surface to capture well-resolved strong events. Only about 30% of grid points meet the condition but they constitute about 80% of the local shear production. Based upon the previous flow visualization results, we partition the shear production into four momentum flux quadrants. The results, summarized in Table 2, demonstrate that strong shear production events are generally associated with ejections and sweeps. In the following analysis, therefore, the conditional event is defined as a combination of w 2 U/z m 2 s 3 and q w (or q w )at(x, z) (800, 50) m. First, we shall consider the strong shear-production event associated with ejection motion q w. Figure 17 presents the conditionally sampled velocity vectors and also contours of the shear production and pressure destruction for such a conditional event. The side view, Fig. 17a, shows that strong shear production is located near the center of a strong ejection and that strong pressure destruction is found at its upper rear side. The front view, Fig. 17b, reveals counterrotating motions associated with the ejection. This conditional field bears a FIG. 17. Conditionally sampled shear production w 2 U /z m 2 s 3 associated with ejection at (x, z) (800, 50) m. Contours of conditionally sampled shear production term (sp), pressure destruction term (pd, solid line), and velocity vectors on (a) x z plane, side view at y 0 m, and (b) y z plane, front view at x 0m.

17 2495 FIG. 18. Conditionally sampled shear production w 2 U /z m 2 s 3 associated with sweep at (x, z) (800, 50) m. Contours of conditionally sampled shear production term (sp), pressure destruction term (pd, solid line) and velocity vectors on (a) x z plane, side view at y 0 m and (b) y z plane, front view at x 0m. striking resemblance to the Fig. 16b schematic. This suggests that quasi-streamwise vortices predominate over spanwise vortices. Next we consider the strong shear-production event associated with sweep motion q w. Figure 18a indicates that strong pressure destruction is located at the lower front side of strong shear production. The front view, Fig. 18b, also reveals counterrotating motions. As illustrated in Fig. 16, strong pressure velocity interaction tends to occur around vortical motions due to the proximity of positive w and negative p/x on one side of the vortex and of negative w with positive p/x on the other side. The above analysis provides a clear physical picture of the local balance between shear production and pressure destruction of Reynolds shear stress. The production of the stress is essentially associated with ejections and sweeps. Strong pressure destruction tends to immediately follow strong ejection and sweep events. Due to the small physical extent of these coherent structures, local balance is maintained regardless of the presence of large-scale spatial inhomogeneity. 6. Conclusions We have investigated the inhomogeneous atmospheric boundary layer under neutral conditions simulated by a grid-nested LES code. Spatial inhomogeneity has been introduced through abrupt changes of surface roughness, producing local internal boundary layers that grow with downwind fetch. Since a dependence of the von Kármán constant k upon surface roughness has been previously reported in Oncley et al. (1996), we compare mean speed profiles over both the rough and smooth surfaces to logarithmic similarity profiles. Over the smooth surface, the mean speed profiles at large fetches agree with the Monin Obukhov similarity profile when k 0.40 is assumed, but over the rough surface, correspondence to the similarity profile requires k A similar qualitative

18 2496 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 dependence upon surface roughness is also found in LES results over horizontally homogeneous surfaces of corresponding roughnesses in the temporal transition phase, but the qualitative difference in the k value is much larger for the inhomogeneous surface case. Both the inhomogeneous and homogeneous surface results are qualitatively consistent with Oncley et al. (1996) in that a larger k is obtained over the smoother surface. We attribute the quantitative difference between the inhomogeneous and homogeneous surface cases to the modification of mean velocity by the local pressure gradients created by spatial inhomogeneity. Over the smooth surface the induced pressure gradient decelerates parcels and thereby reduces the velocity shear from that which would occur over a surface of identical roughness without a pressure gradient. When horizontally homogeneous similarity expressions are utilized to describe the velocity shear, this shear reduction requires that k must be increased if the expression is to match the actual velocity profile. Here it should be recognized that the k value obtained for the inhomogeneous surface is altered by the fact that the flow is not actually horizontally homogeneous, as the similarity expressions assume. While Oncley et al. (1996) felt that obstacles near their measurement site which might introduce spatial inhomogeneity had a negligible influence on their data, our results indicate that pressure gradients can alter the perceived von Kármán constant over a given surface, exaggerating differences that would occur over a horizontally homogeneous surface, and are thus a possible contaminant of field measurement data. For our IBL case, the pressure gradient influence created at a surface discontinuity extends further downwind than does the advective influence of that discontinuity. For fetches less than about three times the height of the overlying boundary layer, the IBL height based on our definition is best predicted by the models of Townsend (1966) and Wood (1982). Both models predict that the IBL growth will be similar for the rough-to-smooth and smooth-to-rough transitions as obtained in our results. However, this conclusion is based on a small value of N lnz o1 /z o2 and further investigation is needed for cases with large N value. In comparison with the homogeneous surface case, the overprediction of the IBL height in the inhomogeneous-surface case by the models of Elliott (1958), and Panofsky and Townsend (1964) may be attributable to the presence of adverse pressure gradients in the inhomogeneous surface case, which alter the mean velocity profiles. Quadrant analysis of the inhomogeneous-surface IBL finds features quite different from the homogeneoussurface counterparts. First, ejections and sweeps over the rough surface are stronger than those over the smooth surface only in the lower part of the boundary layer, not through the entire IBL as for the homogeneous surface case. Second, ejections and sweeps over the smooth surface decrease in strength with increasing fetch and increase with fetch over the rough surface. Third, ejections and sweeps over the smooth surface are stronger than their homogeneous surface counterparts except near the surface, attributed to the effect of spatial inhomogeneity. Over the rough surface, the opposite occurs. The occurrence frequencies of these ejection and sweep eddies quantitatively agree with those observed at the canopy atmosphere interface. For a homogeneous-surface boundary layer, it has been previously established that shear production and pressure velocity interaction terms of the Reynolds shear stress budget are approximately in local balance. Our analysis also shows such a balance at mesoscale fetches despite the presence of spatial inhomogeneity. Flow visualization and conditional sampling reveal that strong shear production is physically associated with strong ejection and sweep events surrounding vortical structures. The relatively small scale of these coherent structures allows local balance to be achieved over a small averaging volume, regardless of spatial inhomogeneity. Acknowledgments. J. W. Glendening was supported by the Office of Naval Research Program Element N. Computer support has been provided by the Department of Defense High Performance Computing Center at the Arctic Region Supercomputing Center and by the Naval Research Laboratory. The comments of the three reviewers are greatly appreciated. REFERENCES Adrian, R. J., 1990: Stochastic estimation of sub-grid scale motions. Appl. Mech. Rev., 43, Bergström, H., and U. Högström, 1989: Turbulent exchange above a pine forest. II. Organized structures. Bound.-Layer Meteor., 49, Elliott, W., 1958: The growth of the atmospheric internal boundary layer. Trans. Amer. Geophys. Union, 39, Garratt, J. R., 1990: The internal boundary layer A review. Bound.- Layer Meteor., 50, , 1997: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp. Germano, M., U. Piomelli, P. Moin, and W. Cabot, 1991: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids, 3, Glendening, J. W., 1994: Dependence of a plume heat budget upon lateral advection. J. Atmos. Sci., 51, , 1996: Lineal eddy features under strong shear conditions. J. Atmos. Sci., 53, , and C.-L. Lin, 2002 Large-eddy simulation of internal boundary layers created by a change in surface roughness. J. Atmos. Sci., 59, Hobson, J. M., N. Wood, and A. R. Brown, 1999: Large-eddy simulations of neutrally stratified flow over surfaces with spatially varying roughness length. Quart. J. Roy. Meteor. Soc., 125, Högström, U., and H. Bergström, 1996: Organized turbulence structures in the near-neutral atmospheric surface layer. J. Atmos. Sci., 53, Katul, G., C.-I. Hsieh, G. Kuhn, D. Ellsworth, and D. Nie, 1997: Turbulent eddy motion at the forest atmosphere interface. J. Geophys. Res., 102, Khanna, S., and J. G. Brasseur, 1997: Analysis of Monin Obukhov