Turbulent momentum flux characterization using extended multiresolution analysis

Transcription

1 Quarterly Journalof the RoyalMeteorologicalSociety Q. J. R. Meteorol. Soc. 4: , July 24 A DOI:.2/qj.2252 Turbulent momentum flux characterization using extended multiresolution analysis Erik Olof Nilsson,* Erik Sahlée and Anna Rutgersson Department of Earth Sciences, Uppsala University, Sweden *Correspondence to: E. O. Nilsson, Department of Earth Sciences, Uppsala University, Villavägen 6, Uppsala, Sweden. erik.nilsson@met.uu.se The variability of turbulent momentum flux in neutral and unstable atmospheric boundary layers is characterized by analyzing surface-layer measurements and data from large-eddy simulations (LES). The method involves multiresolution (MR) decomposition of vertical wind and advected variables into eddy fluctuations on different scales. It provides a measure of the amount of flux variability that stems from same-scale correlations and from combinations of different-scale eddy fluctuations. Combining two analysis methods enabled MR component cospectra to be introduced, in order to study the contribution of and upward flux on different scales. These component cospectra were used to investigate at which scales most of the upward and momentum flux occurs. By using MR spectra, cospectra and flow visualization, this investigation provides insights into turbulence structure and fluxes in neutral and unstable stratification. It is shown that most of the flux variability in the lower part of the boundary layer can be characterized as a combination of larger scale streamwise elongated horizontal wind streaks and smaller scale vertical wind fluctuations. These streaks are found to account for a large part of momentum flux at relatively large, energy-containing scales. Most of the upward momentum flux is found to occur at smaller scales. This can be interpreted as showing that upward momentum flux in these conditions is caused by the generation of smaller scale secondary motions when larger scale turbulence elements break down and dissipate. Differences in the height dependence of turbulence structure and momentum flux for neutral and unstably stratified conditions are also investigated and related to the existence of wind streaks and horizontal rolls in these different conditions. Key Words: neutral and unstable atmospheric boundary-layer turbulence; large-eddy simulation; multiresolution flux decomposition; scale of turbulent transfer Received 3 January 23; Revised 3 August 23; Accepted 8 September 23; Published online in Wiley Online Library 9 November 23. Introduction.. Turbulence structure in neutral and unstably stratified flows over flat terrain Numerical simulations (Khanna and Brasseur, 998; Foster et al., 26) and observations (Baas, 28; Iwai et al., 28) have shown that in near-neutral conditions the dominant shearinduced motions are streamwise low-speed motions close to the ground that align with the mean wind. These motions are often separated by streaky high-speed motions. In the moderately unstable boundary layer, large-scale turbulent motions in the form of horizontal convective roll vortices, also called longitudinal rolls, may be created through the interaction between buoyancy and shear. The major axes of these rolls are aligned with the mean boundary-layer wind-shear vector and are typically oriented in the mean wind direction. In this work, longitudinal denotes orientation in the mean wind direction, whereas lateral means perpendicular to the mean wind direction. The rolls originate through the organization of thermal plumes that form in the unstable boundary layer (i.e. regions of buoyancy-dominated motions having a positive vertical velocity). The horizontal scale of the thermal plumes increases with distance from the surface (see for example Khanna and Brasseur, 998). As the plumes merge with neighbouring plumes, they form a larger scale region of upward-moving warmer fluid in the mixed layer, surrounded by broader regions of gentler -moving cooler fluid that separate the individual updraughts. In the absence of strong wind shear, these thermal plumes have no preferred horizontal orientation. However, when there is strong wind shear the thermal plumes get stretched out, creating a pattern of streaks of upward and motions along the mean wind direction. With increasing height, these streaks merge, creating larger scale longitudinal rolls. The details of this process are described in, for example, Moeng and Sullivan (994). Buoyancy-induced updraughts and downdraughts are also responsible for much of c 23 Royal Meteorological Society

2 76 E. O. Nilsson et al. the vertical flux of momentum, heat and passive scalars in the convective boundary layer (CBL: Etling and Brown, 993; Khanna and Brasseur, 998)..2. Effects of buoyancy and wind shear in atmospheric boundary layers Monin Obukhov similarity theory is typically used to represent atmospheric boundary layers in models relating fluxes to vertical profiles of wind and temperature in the surface layer. Wind shear and convection are the two most important forcing mechanisms for atmospheric boundary-layer flows in general. The Obukhov length L is a length-scale with absolute value approximately equal to the height where mechanical and thermal production of turbulent energy are equal. The Obukhov length is defined as L= u3 T gkw θv, where u is the friction velocity, w θv is the near-surface kinematic virtual heat flux, θv is the virtual potential temperature, κ =.4 is the von Ka rma n constant, g /T is the buoyancy parameter, T is the surface-layer temperature constant under the Boussinesq approximation and g is the acceleration of gravity. The nondimensional heights z/l and zi /L have frequently been used as similarity variables in the analysis of measurements and models to diagnose atmospheric stability. Here z is the height above the surface and zi is the height of the top of the planetary boundary layer (PBL). For large negative values of z/l, the surface-layer behaviour is like that in free convection. Tennekes (97) calls this local free convection. Above the surface layer, the size of turbulence elements tends to scale with zi ; in this region zi replaces z as the appropriate length-scale. In many situations, Monin Obukhov similarity theory correctly describes the relationship between averaged fluxes and vertical profiles of wind and temperature in the surface layer. The theory does not, however, provide information about the local character of fluxes and turbulence structure. These will be examined in this work using a multiresolution analysis technique. The relative roles of buoyancy and shear were previously studied using large-eddy simulations (LES). Moeng and Sullivan (994), Khanna and Brasseur (998) and Conzemius and Fedorovich (26) showed how the neutral atmospheric boundary layer changes its character in conditions with unstable stratification over flat terrain. An extensive review of the conceptual understanding of wind shear in CBLs is found in Fedorovich and Conzemius (28), who considered wind-shear effects using a variety of different approaches including numerical simulations, field measurements and laboratory experiments. However, as discussed in Young et al. (22), the mechanisms by which horizontal rolls interact with gravity waves in the overlying atmosphere and with streaks in the underlying surface layer are by no means fully explained. Baas (28) also reviewed some of the challenges of direct surface-layer field observations of wind streaks or aeolian streamers with pronounced spatial and temporal variability in wind-blown sand, for example, over beaches. One of the issues emphasized was the lack of quantitative methods to analyze local shapes and patterns. Another issue that needs to be explored further is the influence of organized turbulent structures such as horizontal convective rolls on the horizontal distribution of shear in the entrainment zone (Conzemius and Fedorovich, 26)..3. Vertical fluxes in the atmospheric surface and boundary layer Our purpose is to investigate the variability of a turbulent flux time series w φ. If the time series wi represents vertical wind and the time series φi the transported variable, the turbulent flux time series can be defined as w φ (i) = (wi wi )(φi φi ), () where i denotes the ith observation. When this time series is timeaveraged, the resulting averaged flux w φ describes the vertical eddy flux of φ. Figure shows an example of such a time series corresponding to the vertical kinematic eddy flux of u momentum w u (with φ = u) in a situation with neutral stratification. In general, momentum flux is a second-order tensor with nine momentum-flux components. The vertical flux of horizontal momentum near the surface is often dominated by the stress component in the mean wind direction τxz = ρ u w. The lateral stress component τyz = ρ v w is often smaller, as the mean wind direction and mean wind-shear vector are typically reasonably well-aligned. The vertical kinematic eddy flux of u momentum will be used to demonstrate the use of a multiresolution analysis technique to study the variability of turbulent momentum flux in natural and numerically simulated atmospheric boundary-layer flows. This technique can also be applied to the study of other turbulent fluxes such as heat flux (φ = θ ) or scalars such as carbon dioxide (φ = CCO2 ) or methane (φ = CCH4 ). In these cases, φ corresponds to the concentration C of the scalar. We thus choose to use the general notation variable φ in the presentation of the analysis methods but restrict examples of the application of the methods to u momentum (φ = u). Applying multiresolution analysis to time series decomposes the record into averages on different time-scales. The method is described in detail, including a simple example, in Vickers and w u [m2s 2] time [s] Figure. Example of a turbulent flux time series for longitudinal stress w u in neutral stratification at a measurement height of 4 m. c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24) 6

3 Turbulent Flux Characterization 77 Table. Details of the analyzed field measurements and LES data. Case Date and time U 6 (ms ) U 3.9 (ms ) L (m) u (ms ) H (Wm 2 ) Neutral 2 June 986, 5. SNT Convective 2 June 986,.6 SNT ZN ,.3. ZC Here U is the mean wind speed at 6 and 3.9 m, L is the Obukhov length-scale, u is the surface friction velocity and H is the surface heat flux. The starting time for each analysis is given as Swedish Normal Time (SNT). Mahrt (23). We extend their analysis of turbulent fluxes by introducing a new representation that maps all the variability of the turbulent flux time series w u to fluctuations of vertical and longitudinal wind on different time-scales. New filters based on this representation are defined to illustrate which combination of time-scales for longitudinal and vertical wind fluctuations accounts for most of the flux variability. It is clear from Figure that the example time series shown there includes both negative and positive fluctuation values. The negative fluctuation values are often referred to as ejection and sweep events and the positive values are referred to as outward and inward interactions (Katul et al., 26). Turner (998) introduced a partitioning of flux events into counter-gradient and downgradient contributions by averaging flux contributions at each scale separately over and upward flux contributions. Combining the work by Turner (998) with the multiresolution decompositioning of Vickers and Mahrt (23) enabled the creation of multiresolution (MR) component cospectra, which can be used to investigate at which scales most of the upward and momentum fluxes occur in neutral and unstably stratified conditions. By using local averaging of turbulent quantities over different time or spatial scales, the MR technique allows for an investigation of variances and fluxes, with emphasis on the local nature of turbulent events or structures and the scale of turbulent transfer. In section 2 we present the field measurements and LES data that were used in this work. In section 3, the MR technique of Vickers and Mahrt (23) is explained. Thereafter the new representation, new filters and MR component cospectra are described. This is followed by the results in section 4. The results and advantages of the introduced analysis method are discussed in section 5 before we summarize and conclude in section Description of data 2.. Measurements The field data come from the agricultural flat terrain site Lövsta, situated about km southeast of Uppsala in Sweden (59 5 N). We selected two periods with data representing neutral and unstable stratification from a data set previously described in Högström (99) and used in Högström et al. (22). Table provides parameters for the two selected cases. For each case, the time series of 2 Hz turbulence data (for the three wind components and temperature) consisting of 2 5 data points (27.3 min) was rotated into a coordinate system such that u fluctuations are in the direction of the mean wind. These data were then used in the multiresolution analysis discussed in section LES data Large-eddy simulations were used to study the combined effect of wind shear and convection in an idealized way and to compare these numerical results qualitatively with field measurements taken in the surface layer. The LES results are used to extend the discussion of the height dependence of momentum flux in neutral and unstably stratified flows. Table summarizes some parameters for the two largeeddy simulations ZN and ZC3 (for neutral and unstable stratification) from Nilsson et al. (22) used in this work. We used a computational domain (2, 2, 8) m discretized using (25, 25, 96) grid points. The horizontal resolution is thus reasonably high (4.8, 4.8) m. The vertical grid spacing is variable to give a high vertical resolution of about m near the surface. A high Reynolds number surface drag law based on a z boundary condition was applied at the lower boundary with a small surface roughness value z =.2 m. This boundary condition was used in Nilsson et al. (22) for simulations of open ocean conditions, which typically have less roughness than conditions over land. This may at least partly explain the lower friction velocities and wind variances in the LES data compared with the measured data used in this study. We used a flux boundary condition for the surface heat flux Q =. (.) km s for neutral (unstable) stratification. The following parameter settings were used for both cases: geostrophic wind forcing (U g = 5.ms, V g = ms ), Coriolis parameter f = 4 s, initial depth of the atmospheric boundary layer (ABL) z i = 4 m and an initial temperature profile with dθ/dz = uptoz i with a strong stable inversion dθ/dz =. km above z i. Approximately 75 grid levels were located between the surface and the PBL inversion. We employed a subgrid-scale (SGS) model using a transport equation for subgridscale energy e. The SGS model and LES code are described in Sullivan et al. (994, 28). Additional information about these simulations can be found in Nilsson et al. (22). All statistics shown were determined by spatial and temporal averaging using the previously archived 3D data (Nilsson et al., 22). The resolved field variables were used to analyze fluxes and variances using the methods presented in the next section. The resolved flow variables were linearly interpolated from 25 to 256 data points along the x-direction of the model domain (which is approximately in the mean wind direction) before multiresolution analysis was performed. The ratio of SGS and resolved momentum flux is a rapidly decreasing function of height in these simulations, with 85% of the flux being within the resolved scales at a height of about m for neutral conditions and 8 m for unstable conditions. 3. Analysis method 3.. A multiresolution approach Multiresolution decomposition partitions the record of w (and similarly φ) into simple block averages on different scales (segments) of width, 2, 4,...,2 M consecutive data points. For a given scale m, the averaging segments of width 2 m points are sequenced as n =, 2,...,2 M m,wheren identifies the position of the segment within the series. In the following description, we have adopted the notation of Vickers and Mahrt (23). The average for the nth segment at scale m (Vickers and Mahrt, 23) is given by w n (m) = 2 m J wr i (m), (2) where wr i (m) is the residual series after removing the segment averages for windows of width > 2 m points. In Eq. (2), I and J i=i c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

4 78 E. O. Nilsson et al. Figure 2. (a) Example time series of the longitudinal (blue dots) and vertical (black dots) wind component for neutral stratification at a measurement height of 4 m. (b) Mapped time series decomposition φ(i, m) for the longitudinal wind component and (c) w(i, m) for the vertical wind component. denote the position of the first and last points of segment n (see Vickers and Mahrt (23) or Voronovich and Kiely (27) for an illustration of the calculation procedure). Another way of expressing this is to say that, for m =,,..., M, w n (m) is the average vertical wind deviation for segment n at scale m relative to an average of the vertical wind taken over the segment at one larger scale (m + ), which consists of 5% of segment n at scale m. This means that we can interpret w n (m) as an average eddy fluctuation speed at scale m relative to an average taken over a time record twice as long as scale m. Eddies are in general three-dimensional entities and so our use of the term eddy fluctuations needs to be explained. We use this term for deviations or fluctuations from mean values taken over longer time-scales (or larger spatial scales in the case of LES data). Such deviations reflect the different turbulence structure (or eddies) found in turbulent flows during various conditions. Therefore we choose to call these deviations eddy fluctuations. In Eq. (2), if wr i (M) becomes the original time series w i then w n (M) becomes the record mean. Thus we cannot define eddy fluctuation speeds at scale M given a record of 2 M points (we would need 2 M+ points). We can define fluctuation speeds at scale m = with the above formula, in which case w n () is simply equal to the residual series wr i (). The eddy fluctuation speed w n (m) atscalem is calculated using information from the 2 m points that belong to a segment n. Therefore each data point (corresponding to a time i) that is part of the segment n is associated with this eddy fluctuation speed at scale m. Thus a mapped time series w(i, m) of the eddy fluctuation speed can be constructed at scale m with m =,,..., M. For scale M, the two half-records of the mapped time series will have two different eddy fluctuation speeds; for scale M 2, the four record quarters will have four different eddy fluctuation speeds; and so on up to scale m =, which has one separate value of the eddy fluctuation speed associated with each of the 2 M times in the record. Note that we have dropped the segment index n for the mapped time series w(i, m) because it has a defined value at each scale m for each time i; thus the segment index is not needed to locate its position relative to the original time series w i and φ i. In Figure 2(a) we show time series of longitudinal and vertical wind components from measurements taken at 4 m height in neutral stratification, as described in section 2.. The corresponding flux time series for u momentum flux is shown in Figure. Figure 2(b) and (c) show the corresponding mapped time series u(i, m) and w(i, m). Similar time frequency representation can also be achieved by using various forms of wavelet transforms or the windowed Fourier transform. The eddy time-scale m on the y-axis in (b) and (c) is given in units of seconds by using the relation 2 m. The values of the eddy fluctuation speed or amplitude (in m s ) are shown by the colours in the scale in the colour bar. We note from Figure 2(c) that the most intense vertical wind fluctuations occur on small scales of about 2 s or less. The longitudinal wind fluctuation values are shown in (b) to be most intense on time-scales of about 5 s and longer. c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

5 Turbulent Flux Characterization 79 The MR spectra are the second moment about the mean of the segment averages (Vickers and Mahrt, 23), given by D w (m + ) = 2 M m 2 M m w 2 n (m) = 2 M n= 2 M i= w 2 (i, m), (3) where the expression on the far right is the new expression for MR spectra in terms of the introduced mapped time series w(i, m) instead of the more compact form summed over the segments n (Vickers and Mahrt, 23; Howell and Mahrt, 997). The MR cospectra are given by (Vickers and Mahrt, 23) D wφ (m + ) = 2 M m 2 M m w n (m)φ n (m) = 2 M 2 M i= n= w(i, m)φ(i, m), (4) where again the expression on the far right is a new expression given in terms of the introduced mapped time series w(i, m) and φ(i, m). This calculation procedure using mapped time series rather than segments is, of course, more inefficient if only MR spectra and cospectra are to be evaluated. However, we will use the mapped time series representations to discuss the variability of the w φ (i) flux time series defined in Eq. (). This time series, based on the two time series w i and φ i, can be expressed in terms of the mapped eddy fluctuation speed time series w(i, m) and φ(i, m) as M M w φ (i) = w(i, j)φ(i, k). (5) j= k= Figure 3 shows a schematic illustration of how w and u fluctuations (φ = u) of different scale combine to contribute to the turbulent flux time series w u (i) at each instant in time, in accordance with Eq. (5). Figure 3 and Eq. (5) are based on consideration of any instant fluctuation w u (i) as the result of superpositioned averaged fluctuation values w and u on different scales, similarly to the way in which the surface elevation at a point on the sea surface can be considered to consist of a large number of waves with different amplitudes. The longitudinal wind fluctuations on different scales (from the second quadrant) are multiplied by the vertical wind fluctuations (from the fourth quadrant) to produce the local contribution to w u (i) from each combination of different scales k and j (in the first quadrant of Figure 3). From Eq. (5) it follows that the time-averaged flux, typically calculated from one-point measurements as a substitute for spatial or ensemble averaged Reynolds flux using the ergodicity assumption (pp in Wyngaard, 2), is given by w φ = M D wφ (m) m= = 2 M = 2 M 2 M M M i= j= 2 M M i= m= k= w(i, j)φ(i, k) w(i, m)φ(i, m), (6) where the last expression on the right is true since, as discussed in Howell and Mahrt (997), only same-scale fluctuations j = k (= m) contribute to the MR cospectra D wφ (m). That is, for j k the w(i, j) andφ(i, k) eddy fluctuations contribute to w φ (i) variability. However, the time average over the full record is equal to (i.e. for j k): w φ (j, k) = 2 M i= w(i, j)φ(i, k) =. (7) In terms of Figure 3, this implies that off-diagonal contributions to w φ (i) (in the first quadrant) sum to when averaging over the full record. The linear correlation between φ(i, k) and w(i, j) is equal to zero for off-diagonal contributions (j k). We will show that combinations of different scale fluctuations (j k) contribute significantly to the flux variability w φ of a turbulent time series by defining some filters in section 3.2 based on the mapped time series decomposition w φ (i, j, k), which is defined as w φ (i, j, k) = w(i, j)φ(i, k). (8) This is a function of three parameters that map the contribution to w φ at times i =, 2,...,2 M from combinations of w and φ eddy fluctuations of scale j and k =,,..., M, respectively. It is important to study the variability of fluxes, as the variability influences sampling strategies. Larger variability requires longer records to reduce the random flux error to acceptable levels. The representation in Eq. (8) allows investigation of which scales much of the flux variability is located at Quantification of flux variability on different scales by filtering In order to summarize statistically the scales at which (with respect to vertical wind fluctuations and fluctuation scales of an advected variable) the variability in the flux time series w φ (i) occurs, we designed four different filters based on the mapped time series decomposition w φ (i, j, k) defined in Eq. (8) Filter A This filter is defined to remove correlations between w eddy fluctuations of scale m or larger with φ eddy fluctuations of all scales to form a new filtered time series. For m =,,..., M the filter is written M M M M w φ A (i, m) = w φ (i, j, k) w φ (i, j, k). (9) j= k= j=m k= Figure 3. A schematic representation of how flux time series variability can be seen as a combination of vertical and horizontal wind fluctuations on different scales. For the shortest scale m =, all energy is removed from the original time series and so the filtered time series is equal to at all times. This can be considered as high-pass filtering of c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

6 72 E. O. Nilsson et al. the original time series with respect to the scale of vertical wind fluctuations. In addition to providing physical insights, filter A has practical applications because low attack angles mean that w is often not trusted on larger scales near the surface. This issue is studied in Kochendorfer et al. (22). Including mesoscale transport in calculated fluxes also potentially degrades similarity relationships (Smedman, 988; Vickers and Mahrt, 23). Later, we will compare this type of high-pass filtering with respect to w to high-pass filtering with respect to the scale of the transported variable u (φ = u, in the case of vertical flux of u momentum) for different atmospheric conditions Filter B Filter B is designed to filter out interscale correlations between w eddy fluctuations of scale m or larger and φ eddy scales less than m.form =, 2,..., M this filter is written M M M w φ B (i, m) = w φ (i, j, k) w φ (i, j, k). j= k= j j=m k= () For the shortest scale m =, no additional information from the original flux time series is removed in comparison to scale m =. We will apply this filter both with w and φ = u as the conditioned variable in section to analyze w u (i) flux time series variability from interscale contributions Filter C The same-scale contributions to w φ (i.e. the part with j = k (= m), from which the MR cospectra can be formulated) is removed by filter C, defined in Eq. () below. For m =,,..., M : M M M w φ C (i, m) = w φ (i, j, k) w φ (i, m, m). () j= k= l=m This filter allows for analysis of same-scale contribution effects on the w φ flux time series Filter D We also define a high-pass filter that removes all contributions from scale m and larger with respect to both w and φ fluctuations. For m =, 2,..., M : M M w φ D (i, m) = w φ (i, j, k) j= k= w φ (i, j, k) M M j=m k= m M w φ (i, j, k). (2) j= k=m For the shortest scale m =, all energy is removed from the original time series and the filtered time series is equal to at all times Component cospectra for and upward fluxes Fourier cospectra have traditionally been used in many atmospheric surface and boundary-layer studies to interpret how much net (upward or ) flux occurs at different scales. Turner (998), however, studied turbulence and mixing in and over a forest using wavelets and introduced a partitioning of flux events into counter-gradient and down-gradient contributions by averaging flux contributions at each scale separately over and upward flux contributions. Fluxes are typically counter-gradient; momentum flux for the situations studied in this article is dominated by flux. Turner (998) discusses this partitioning further as the most physically meaningful separation available, in the absence of a clear way to define a set of coherent structures. In terms of the orthogonal multiresolution decomposition used here, we can introduce the upward MR cospectra as D up wφ (m + ) = 2 M 2 M i= w φ upward (i, m, m), (3) where w φ upward (i, m, m) isequaltow φ (i, m, m) when positivevalued and otherwise. The MR cospectra are similarly defined as D down wφ (m + ) = 2 M 2 M i= w φ (i, m, m), (4) where w φ (i, m, m) is equal to w φ (i, m, m) when negative-valued and otherwise. The relationship between the upward and cospectra and the original MR cospectra (Vickers and Mahrt, 23) is given by D wφ (m) = D down (m) + Dup (m). (5) wφ These upward and cospectra can be used to assess and investigate how much upward and flux occurs on average on scale m. For example, a net flux value close to zero can occur in several ways. If the turbulence level is very low, net flux naturally becomes small; alternatively significant upward and flux events may cancel each other out. Turner (998) characterized the spectral slopes of the and upward cospectra at small scales as having values close to 2/3. We will examine this result using data for neutral and unstable conditions. We will also qualitatively compare spectra and cospectra from measurements and LES in the surface layer. Spectral and cospectral length-scales determined from the scales with the most variance or flux from LES are then used to illustrate the dominant scales of upward and momentum flux in neutral and unstable conditions. 4. Results 4.. Flux variability with respect to the scale of vertical and horizontal wind fluctuations We can illustrate which combinations of longitudinal and vertical wind eddy scales contribute to the variability in the flux time series w u (i) by calculating the standard deviation of w u (i, j, k) for each combination of scales j and k and normalizing the results by the standard deviation of the flux time series w u (i). Figure 4(a) shows the results for neutral stratification with measurements taken at 4 m. In Figure 4(b), the corresponding LES results are shown. It can be seen from Figure 4 that most of the flux variability is due to large-scale u fluctuations combined with smaller scale w fluctuations. The larger standard deviation values are mostly located above the diagonal (same-scale fluctuation) elements. This is clear from the measurements shown in (a) and can also, but less clearly, be seen in the LES results in (b). The simulated flux variability is distributed among the modelled scales given by the domain size and resolution of the model. This does not cover as wide a range of scales as are measured in a 3 min time series. Numerical simulations with a larger domain size covering additional scales may wφ c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

7 Turbulent Flux Characterization 72 (a) σ(w u jk ) / σ(w u ).25 u scale [m] (b) w scale [m] σ(w u jk ) / σ(w u ).25 Figure 5. The resolved scale u component of the horizontal wind field in the surface layer is shown for simulation results with neutral stratification. About half the computational domain is shown (the y-axis only spans 6 m). u scale [m] w scale [m] Figure 4. Standard deviation of the w u (i, j, k) decompositioned time series for each combination of the scales j and k, shown normalized by the standard deviation of the total flux time series w u (i). In (a) a case with measurements taken at 4 m and neutral stratification is shown and in (b) LES results at 4 m above the surface for case ZN are shown. be more successful in showing that little flux variability in neutral conditions is explained by large-scale w fluctuations in combination with small-scale u fluctuations, as shown by the measurements in Figure 4(a). We note from comparing Figure 4(a) and (b) that the simulation results are much smoother than the measurement results. This result is not unexpected because (b) represents the averaging of results from numerical data sampled over the last 3 h of each simulation, whereas (a) represents a measurement result for one analyzed 3 min period. The natural variability of fluxes at scales much larger than typical boundary-layer depth scales is known to be large and difficult to characterize. The difficulty may be that these fluxes depend more on parameters external to the boundary layer than on internal parameters. Our intention with this work is to characterize some of the large-scale flow features that are responsible for fluxes on boundary-layer scales that are simulated in our previous LES runs and compare them qualitatively with measurements. Fluctuations related to mesoscale or submesoscale motions are of course very important for atmospheric flows in general but are not the focus of this study. elongations in roughly the mean wind direction. We have here included an isosurface with u = 4.3ms for this part of the domain. The vertical x z plane slice and the isosurface reveal that there are regions of high wind speed reaching down from above, causing strong local wind shear. The y z vertical plane slice shows that these high wind speed eddies are separated by regions of lower wind speed contributing to the low wind speed streaky structures that occur on horizontal plane slices. Shear is crucial for the formation of streaks (Khanna and Brasseur, 998; Lee et al., 99) and has been shown to carry most of the vertical turbulent transport of momentum in neutrally stratified flows. It is important to note that the u eddy fluctuations are mostly on a larger scale (especially in the x-direction, due to their elongation) than the vertical wind eddies, which are shown in Figure 6. The structure of some of the most intense updraughts and downdraughts is shown using two isosurfaces (red and blue) with w =.4 and.4ms in Figure 6. The vertical wind eddies are of smaller scale than the u eddy structures shown in Figure 5, with no clear elongation in the mean wind direction. The vertical wind fluctuations shown on the horizontal planes contrast with the larger scale elongated u fluctuations shown on the horizontal planes in Figure 5. At model levels below m, much of the turbulent stress is subgrid-scale and most of the resolved vertical wind fluctuations are less intense (between. and.ms ) but with a similar small-scale structure, as shown on the horizontal slice at 4 m Comparison of filters between measurements and LES for neutral stratification Figure 7 shows the results of applying the filters introduced in section 3.2 to data with neutral atmospheric stratification for 4.2. Turbulence structure of the neutral surface layer from LES Longitudinal wind streaks are a common flow feature over homogeneous land conditions and in turbulent flows with wind shear. These flow features have been observed as aeolian streamers on beaches by Baas (28), in LES of the neutral boundary layer (Foster et al., 26) and in high Reynolds number laboratory flows by Kim and Adrian (999) and others. Figure 5 shows the longitudinal wind component of the horizontal velocity field on two vertical and three horizontal slices in the surface-layer part of the computational domain (.z i is about 54 m), for neutral stratification. As discussed in Nilsson et al. (22), a partial horizontal slice (covering half the illustrated domain) shows low and high wind speed u eddies at about 4 m that form band-like Figure 6. The resolved scale vertical wind in the surface layer is illustrated for a neutrally stratified simulation, corresponding to the horizontal wind field in Figure 5. About half the computational domain is shown. c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

8 722 E. O. Nilsson et al. (a) (b) (c) σ u w filtered /σ u w.6.4 Filter A on w Filter B on w.2 Filter A on u Filter B on u Filter C Filter D 4 2 k x [m ] σ u w filtered /σ u w.6.4 Filter A on w Filter B on w.2 Filter A on u Filter B on u Filter C Filter D 4 2 k x [m ] σ u w filtered /σ u w k x [m ] Figure 7. Measurement results for filters A D are shown in (a) for a case with neutral stratification and measurement height 4 m. In (b) the results from filters A D are shown for LES case ZN and in (c) the measurements and LES results are shown after applying a simple band-pass filtering of the measurements by removing the three largest and four smallest scales m ={, 2, 3, 4, 3, 4, 5} in the w(i, m) andu(i, m) decomposition. (a) measurements and (b) LES. The standard deviation of each filtered flux data series is normalized by the standard deviation of the unfiltered flux and is shown on the y-axis. The x-axis shows the longitudinal wave number k x. A value of is displayed at the largest scales (smallest wave number), when no flux contributions have been filtered out. As the various filters successively filter out more and more of the smaller scale fluctuations, the ratio of the standard deviation of filtered and unfiltered flux data typically decreases. This is even more evident at scales where a large amount of flux variability exists. We see from Figure 7(a) and (b) that filter D, which is a highpass filter that successively removes both u and w fluctuations at a given scale m, is the filter that causes the fastest decrease towards. Application of filter A to the longitudinal wind component u (purple line) causes a decrease almost as quickly at small wave numbers because most of the flux variability at these scales is duetolarge-scaleu fluctuations and smaller scale w fluctuations. The light blue curve shows filter A applied to the vertical wind component w. In this case, the decrease of the filtered line towards occurs mainly at scales corresponding to about m (or a time-scale of about 2 s), where we have most of the intense w fluctuations (see Figure 2). Application of filter C, which removes only the same-scale fluctuations, shows that such fluctuations account for only a small amount of the total w u (i) flux variability. It is the off-diagonal elements in quadrant of Figure 3 that are responsible for most of the variability in the flux time series. These contributions can be investigated further by using filter B. The red lines in Figure 7(a) and (b) correspond to filter B applied to u and the dark blue line corresponds to filter B applied to w. The results show that the red lines decrease more rapidly than the blue lines for these conditions, indicating that it is the above-diagonal contributions of large-scale u structures combined with smaller scale w structures that contribute to most of the w u variability (see also Figure 4). Scale disparities make it difficult to compare the LES results and measurements directly and only qualitative analogies can be drawn. To obtain a more detailed comparison, we apply a simple band-pass filter to the measurements by removing the three largest and four smallest scales in the w(i, m)andu(i, m) decomposition and then perform a filter analysis on the remaining signal. The remaining measured signal has reduced influence from mesoand submesoscale motions, which are not simulated in LES with its limited domain size. The measurement results (full lines) and LES results (dashed lines) are shown in Figure 7(c). From this figure we see that the filtered results of the measurements and LES follow each other qualitatively for all filters. If it had been possible to include additional measured data representing similar neutrally stratified atmospheric conditions, the curves would probably have been smoother. It can be seen that the LES results typically decrease more at larger scale than the measurements do. This could be related to subgrid closure effects, numerical diffusion or other numerical factors involved in the deterioration of small-scale turbulence features. Using Fourier spectral analysis, Khanna and Brasseur (998) and others have shown that energy levels often decrease more quickly in LES at small scales than would be expected from theory and measurements. In addition, the application of Taylor s hypothesis to translate time-scales into spatial scales in the case of one-point measurements could be considered questionable. Alternately, measurement results could also have been shown as functions of frequency (without the need to use Taylor s hypothesis). Information about variances and fluxes corresponding to long time-scales could then have been discussed in the context of the large spatial scales provided by the LES. In either case it is only possible to make qualitative comparisons between LES and measurements Height and stratification dependence on momentum flux variability Figure 8 shows results from application of filter B for (a) neutrally stratified conditions and (c) unstable conditions for two measurement heights, 6 and 4 m, in the atmospheric surface layer. The measurement results shown in (a) and (c) are bandpass filtered as previously described to include a similar range of scales to those simulated using LES. The corresponding LES results are shown for neutral stratification in Figure 8(b) and unstable stratification in Figure 8(d). Two additional heights at and 2 m above the surface are included to illustrate further the height dependencies discussed below. We first note that the dashed lines exhibit a faster decrease than full lines for both measurements and LES at 6 and 4 m height, since most of the w u variability is due to large-scale horizontal wind fluctuations and smaller scale vertical wind fluctuations at these heights. In unstable stratification, as shown in Figure 8(c), a slightly larger amount of momentum flux variability at 4 m is accounted for by large-scale w structures combined with smaller scale u structures than at the 6 m measurement height. This is indicated by the full line with circles (4 m) decreasing more quickly than the full line with crosses (6 m). As can be seen from the dashed lines in Figure 8(c) for the same measurement period, less flux variability can be explained by large-scale u structures combining with smaller scale w structures at 4 m than at 6 m. Thus the measurements show that with increasing height more of the c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

9 Turbulent Flux Characterization 723 (a) (b) σ u w filtered /σ u w σ u w filtered /σ u w (c) σ u w filtered /σ u w Filter B on w, z = 6 m k [m, z = 4 m ] x Filter B on u, z =6 m, z = 4 m (d) σ u w filtered /σ u w Filter B on w, z = 6 m k [m, z = 4 m ] x, z = m, z = 2 m Filter B on u, z = 6 m, z = 4 m, z = m, z = 2 m k x [m ] 3 2 k x [m ] Figure 8. Measurement and LES results for filter B are shown for neutral stratification in (a) and (b) respectively and for unstable stratification in (c) and (d). Solid (dashed) lines denote that filter B has been applied with respect to the vertical (longitudinal) wind component. Crosses, circles, triangles and squares denote 6, 4, and 2 m height above the surface, respectively. w u variability is due to large-scale w structures combining with smaller u eddies. In relation to unstable conditions with wind shear, Moeng and Sullivan (994) and Khanna and Brasseur (998) discussed the formation of large-scale buoyancy-driven circulations with pronounced updraughts and downdraughts. Thus more flux variability related to large-scale vertical wind motions can be expected in such conditions. The LES results for unstable conditions in Figure 8(d) are very similar to the measurement results at 6 and 4 m. The additional levels at and 2 m show that the dependence on height for the surface layer continues up to almost the middle of the boundary layer. With increasing height, more flux variability is explained by large-scale w structures. At m, about the same decrease is seen for filter B applied to w (solid line with triangles) or to u (dashed line with triangles) in Figure 8(d). This indicates that at this height about the same amount of flux variability is explained by large-scale vertical and large-scale longitudinal wind fluctuations respectively. At 2 m, in contrast to conditions near the ground, the greater flux variability is mainly due to large-scale w fluctuations combined with smaller scale u fluctuations. In neutral conditions, when no formation of large-scale buoyancy-induced circulations occurs, we expect a different result with less pronounced height dependence, due to the lack of large-scale vertical wind fluctuations in terms of updraughts and downdraughts. As shown in Figure 8(a), the measurements indicate this difference in turbulence structure by displaying the opposite pattern from that found in unstable conditions. Here, slightly less flux variability is due to large-scale w structures combined with smaller scale u structures at 4 m than at 6 m measurement height. This is indicated by the solid line with circles (4 m) decreasing less than the solid line with crosses (6 m) in Figure 8(a). The dashed lines indicate about the same total decrease at both 6 and 4 m measurement height, but the decrease occurs at slightly different scales. The LES results for neutral stratification in Figure 8(b) show a small degree of height dependence. The same combinations of scales of vertical and horizontal wind fluctuations account for most of the flux variability. The turbulence structure is thus more z/z i Neutral: Filter B applied on w Filter B applied on u Filter C Unstable: Filter B applied on w Filter B applied on u Filter C σ /σ u w u w filtered Figure 9. Vertical profiles of the standard deviation of filtered fluxes are shown, normalized by the standard deviation of the corresponding total flux w u.also shown are measured values at 4 m height, with filled circles denoting a neutral case and empty circles an unstable case. height-independent than for unstable stratification, with most of the w u variability being explained by large-scale longitudinal wind fluctuations and smaller scale vertical wind fluctuations both in the surface layer and in the middle of the boundary layer. In Figure 9, the height dependence of the filter B results is illustrated further by showing the normalized total decrease at each vertical level in the LES as a vertical profile of height z normalized by boundary-layer height z i. These lines correspond to filtering out either all above- or all below-diagonal contributions to the flux variability in Figure 3. Also shown are results for filtering out all same-scale u and w fluctuations by using filter C. The corresponding measured values for all filters and 4 m height are also shown for neutral (filled circles) and unstable conditions (empty circles). c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)

10 724 E. O. Nilsson et al. The full red and blue lines in Figure 9 show that for neutral conditions (in accordance with the discussion of Figure 8) a larger decrease occurs when filter B is applied to the horizontal wind component u than when it is applied to the vertical wind component w. This means that in neutrally stratified conditions, at each height in the boundary layer, the flux variability is mostly due to large-scale u structures and smaller scale w structures. The full green line in Figure 9 is the total normalized decrease after having filtered out all same-scale contributions using filter C. It can be seen that only a small amount of the flux variability is accounted for by u and w fluctuations of the same scale. The green dashed line shows that the same applies in unstable conditions. As seen in Figure 8, most of the flux variability near the surface is due to combinations of larger scale u fluctuations and smaller scale vertical wind fluctuations. This is also indicated by the dashed red line in Figure 9. The large decrease for the dashed red line near the surface is indicative of large-scale u fluctuations and smaller scale vertical wind fluctuations. This becomes less pronounced in the middle of the boundary layer for unstable conditions. Instead, a larger decrease can be seen for the dashed blue line. This is interpreted as meaning that the greater flux variability is due to organization of large-scale updraught and downdraught regions in moderately unstable conditions Downward and upward momentum fluxes in neutral and unstably stratified ABLs Measured MR cospectra (black full line for neutral and dashed black line for unstable conditions) are shown in Figure (a). Also shown are upward and cospectra as red and blue lines for neutral (solid line) and unstable (dashed line) conditions. These are compared with the cospectra calculated from LES in Figure (c). There is qualitative agreement between the LESgenerated and measured cospectra in several aspects for the scales simulated using LES. In general, there is more net flux in unstable conditions (dashed black lines) than in neutral conditions. From the upward and cospectra we see that the increased net flux for the unstable case, in comparison to neutral conditions (solid black line), is due to an increase in negative or flux (dashed blue line) in the energy-containing scales, which is only slightly compensated for by an increase in average upward flux (dashed red line). Both the measurements and LES indicate that the flux occurs mainly at larger (energy-containing) scales, whereas the upward flux has its largest values at smaller scales. In unstable conditions the peak value for the upward cospectra at 4 m remains at the same scale as in neutral conditions. However, an increasing amount of upward flux also occurs at larger scales in both LES and measurements. Figure (b) shows measured and Figure (d) LES-generated MR u spectra (in blue) and w spectra (in red) for neutral (solid lines) and unstable (dashed lines) conditions. The u spectra have most of their energy located on a larger scale than the w spectra. The peak in the w spectra is at nearly the same scale as the upward flux peak. In addition, for the LES results in neutral conditions, the peak scale of the u spectra is equal to the peak scale of the cospectra. For the measurements shown in Figure (a) and (b), most of the flux occurs at scales with higher u variance. However, the results are less smooth than the LES results due to some fluctuations (in longitudinal wind in particular) that may be specific to the analyzed time period. In Figure (a), the absolute values of the MR cospectra and upward and cospectra are shown for neutral (solid lines) and unstable conditions (dashed lines). Both axes are logarithmic. At small scales the MR cospectral energy has a slope of nearly 4/3, as found in many previous measurement results (Kaimal et al., 972; Su et al., 24). For the upward and cospectra, the slopes are instead close to 2/3, in agreement with previous measurement results (Turner, 998). At small scales, the u spectra and w spectra shown in Figure (b) also have an approximately 2/3 slope.the w spectra also have more energy than the u spectra at small scales. The MR spectra are proportional to wave-number-weighted Fourier spectra (Vickers and Mahrt, 23). (a) D uw [m 2 s 2 ] Neutral, MR cospectra upward.2 Unstable, MR cospectra.25 upward (b) k x [m ] k x [m ] D [m2 s 2 ].3.2. Neutral, MR w spectra MR u spectra Unstable, MR w spectra MR u spectra (c) D uw [m 2 s 2 ] 2 2 x 3 3 Neutral, MR cospectra 4 upward 5 Unstable, MR cospectra 6 upward (d) D [m2 s 2 ] k x [m ] k x [m ] Neutral, MR w spectra MR u spectra Unstable, MR w spectra MR u spectra Figure. In (a) and (c), multiresolution cospectra (black) and upward and cospectra (red and blue) at 4 m are shown for measurements and LES, respectively. In (b) and (d), the multiresolution w and u spectra are shown in red and blue for measurements and LES. Results for neutral stratification are shown by full lines and for convective conditions by dashed lines. c 23 Royal Meteorological Society Q. J. R. Meteorol. Soc. 4: (24)