Vertical Transports by Plumes within the Moderately ... - AMS Journals

15 APRIL 2002

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MASON ET AL.

Vertical Transports by Plumes within the Moderately Convective Marine Atmospheric Surface Layer RICHARD A. MASON,* HAMPTON N. SHIRER, ROBERT WELLS,

AND

GEORGE S. YOUNG

Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania (Manuscript received 5 February 2001, in final form 21 September 2001) ABSTRACT Bursts in the kinematic vertical transports of heat and horizontal momentum in a moderately convective marine atmospheric surface layer are studied by applying the variable interval time averaging (VITA) detection method to principal components analysis (PCA)–decomposed datasets obtained from the Floating Instrumentation Platform (FLIP) moored vessel during the 1995 April–May Pacific Marine Boundary Layer (PMBL) experiment. For convective plumes, a well-defined dimensionless relationship is shown to exist between the vertical transports of heat and horizontal momentum; this relationship cannot be easily deduced if PCA and VITA are not both applied. PCA decomposes a dataset using correlations within that dataset instead of bandpass filtering it to retain energy in a predetermined range of scales; PCA thus respects all scales contributing to the phenomena retained in the dataset. Subsequent use of cross-spectral techniques to group the PCA-decomposed dataset into coherent structure types leads to, among other types of coherent structures, PCA-derived plumes. The VITA method is applied to a decomposed dataset in order to identify updrafts (bursts) and downdrafts (sweeps) in the time series of correlated variables by searching the signal for events that satisfy user-specified criteria. With proper use of PCA, surface-layer plumes can be reassembled in a way that yields the same transport relationships no matter which of the two different detecting variables is used.

1. Introduction Turbulent flow in the convective surface layer often includes organized, spatially coherent motions in eddy updrafts and downdrafts (Hall et al. 1975; Kaimal et al. 1976; Schols 1984). These organized, coherent motions, or structures, in the surface layer span a large range of spatial and temporal scales and can be substructures of the larger, boundary layer–spanning eddies (Kaimal and Businger 1970). These larger eddies take the form of three-dimensional convective plumes (thermals), quasitwo-dimensional convective rolls, and shear instability– driven eddies (Brown 1980; Haack and Shirer 1992). The merging of positively buoyant surface-layer plumes, known as bursts or ejections, sustains the eddy updrafts, while downdrafts appear as horizontal sweeps (Kaimal and Businger 1970; Wilczak and Tillman 1980). Extracting plumes from a time series observed in the convective marine surface layer is complicated by the superposition of larger boundary layer–spanning * Current affiliation: DynTel, Research Triangle Park, North Carolina. Corresponding author address: Dr. Hampton N. Shirer, Department of Meteorology, The Pennsylvania State University, 503 Walker Building, University Park, PA 16802-5013. E-mail: [email protected]

q 2002 American Meteorological Society

eddies and the coexistence of dynamically distinct phenomena in the same energy-containing subrange, such as shallow structures associated with the wavy boundary layer (Edson et al. 1998). Conditional sampling, spectrally dependent filtration, and joint probability distribution estimation are techniques that have been used to obtain information related to coherent structures present in a dataset. Of these, the conditional sampling technique has been the most prevalent. Applied to observations from both the laboratory and the atmosphere, it has produced much of our increased understanding of surface- and mixed-layer events over the past 30 yr (Frisch and Businger 1973; Blackwelder and Kaplan 1976; Khalsa 1980; Lenschow and Stephens 1980; Greenhut and Khalsa 1982; Johansson and Alfredsson 1982; Schols 1984; Zecchetto et al. 1998). Successful conditional sampling requires good a priori knowledge of both the phenomena under study and the phenomena with which they can be confused (Shirer et al. 1999). Conditional sampling requires a priori empirical inputs such as the threshold level, the size of the structure of interest, and the detecting variable (or set of variables) that best isolates the coherent structure from the ambient turbulence. In this article, we focus on two variable sets used to identify surfacelayer plumes: the vertical transport of horizontal mo-

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mentum, and the vertical transport of heat. Different plume identification rules used in two separate studies (Greenhut and Khalsa 1987; Young 1988) of the same coherent structure type are shown by Schumann and Moeng (1991) to produce apparently conflicting results, when in fact the studies merely provide different views of the same structure. This spurious production of conflicting results is one major criticism of the use of conditional sampling to identify coherent structures in a dataset. We show in this article that when our data are suitably decomposed using principal components analysis (PCA), we see the dominant structures, one of these being plumes. Moreover, analyses of these plumes using a standard conditional sampling technique does in fact produce unambiguous results even when different identification rules are used. Objective preprocessing of a dataset is accomplished using the statistical technique of PCA to retrieve components of coherent structures based on variable correlations within the entire dataset (Richman 1986). When preconditioning a dataset by applying PCA, no a priori knowledge of the range of scales of the coherent structure of interest is necessary. This versatility, or insensitivity to scale, makes PCA a useful preprocessing tool for extracting coherent structures whenever identification of prominent variance patterns in complex datasets is desired. In boundary layer meteorology, PCA has been used successfully to identify coherent structures in the mixed- and surface-layers (Weijers et al. 1995). Examples include using PCA to identify convective updrafts, vortices, and horizontal rolls for different stability and shear regimes in large eddy simulations (Rinker and Young 1996; Shirer et al. 1999), and isolating surface-layer plumes and wavy boundary layer structures from datasets obtained in the marine atmospheric surface layer (Shirer et al. 1999). We focus here on analysis of the same surface-layer datasets studied by Shirer et al. (1999), which were measured using instruments on FLIP (Floating Instrumentation Platform) during the 1995 Pacific Marine Boundary Layer (PMBL) experiment. We analyze four 110-min moderately convective cases from this experiment. The initial step in our analysis involves decomposing, or preconditioning, our data subsets using obliquely rotated PCA (Richman 1986). PCA yields linear transformations of a family of correlated nonlinear measurements (variables) through variance pattern separation. Each new variable, or principal component (PC), is temporally varying and explains a certain fraction of the total variance in a dataset. Like most complex sets of physical data, measurements of the surface layer contain some variables that are well-correlated. Therefore, the first few PCs, when ordered with respect to decreasing variance, explain most of the variance and thus capture the predominant features in the dataset. The number of PCs retained for further analysis (13 here) is obtained by applying the scree test outlined by Cattell (1966). The discarded PCs

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explain little variance in the data and are assumed to be dominated by features of little interest. In our study, datasets reassembled using all 13 retained PCs are labeled PCA-decomposed. Because PCA-decomposed datasets may include effects from multiple coherent structures, we seek ways to isolate one coherent structure; to do so, we perform linear cross-spectral analysis on the PC score series (Sorbjan 1989; Shirer et al. 1999). As indicated by the coherence, phase, and power spectra of the score series, Shirer et al. (1999) find five PCs that consistently group together in all four cases. Nonlinear fractal dimension analysis described by Takens (1981) and implemented on our dataset using the algorithm of Shirer et al. (1997) reinforces the grouping of PCs for these cases. This subset of PCs represents surface-layer plumes, which, after reassembly procedures (Shirer et al. 1999), composes the PCA-plume dataset. We compare the transports of plumes estimated from PCA-plume and PCA-decomposed datasets with the complete, or raw, dataset. The purpose of this article is to demonstrate that when compared with analysis of the raw surface-layer dataset, the PCA-decomposed and PCA-plume datasets yield estimates of the turbulent transports produced by the particular coherent structure(s) present. Once groups of PCs describing surface-layer plumes are obtained, we investigate rapid and significant events (the burst–sweep cycle) in the reassembled time series using a variable interval time averaging (VITA) technique (Kaplan and Laufer 1968) similar to that used by Blackwelder and Kaplan (1976). The VITA technique uses the short-term variance for a detecting (defining) variable in the dataset. The short-term variance is a local (in time) measure of turbulence, and an event, either a burst (upward), or a sweep (downward), is said to occur when the variance exceeds a given threshold. We are interested in contrasting the correlation coefficients of the variables composing the transport of momentum and heat for plumes extracted from the events within the raw, PCAdecomposed, and PCA-plume datasets. Section 2 reviews the PMBL experiment, and section 3 discusses PCA, cross-spectral analysis, and the VITA detection technique. Results of the VITA detection technique applied to raw, PCA-decomposed, and PCAplume datasets from FLIP are discussed in section 4. Section 5 contains the conclusions and recommendations for further research. 2. The experiment The dataset analyzed in this article was obtained from the FLIP moored vessel during the April–May 1995 Pacific Marine Boundary Layer experiment (Miller et al. 1997, 1999). As discussed in Shirer et al. (1999), FLIP measured a total of 54 variables at heights up to 18 m above the sea surface. Application of PCA to eliminate data obtained from bad sensors reduces the number of retained variables to 46 (Table 1). The re-

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TABLE 1. Variable numbers, symbols, and descriptions for quantities measured on FLIP during the 1995 PMBL experiment. The application of VITA to detect plumes is based on variables 8, 11, 14, and 17 (see Fig. 3). Variable 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 43 44 45 46

31 32 33 34 35 36 37 38 39 40 41 42

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Variable description Eastward FLIP displacement Northward FLIP displacement Vertical FLIP displacement Eastward FLIP velocity Northward FLIP velocity Vertical FLIP velocity Eastward wind speed (sonic anemometer) at 8.73 m Eastward wind speed (sonic anemometer) at 13.78 m Eastward wind speed (sonic anemometer) at 18.12 m Northward wind speed (sonic anemometer) at 8.73 m Northward wind speed (sonic anemometer) at 13.78 m Northward wind speed (sonic anemometer) at 18.12 m Vertical wind speed (sonic anemometer) at 8.73 m Vertical wind speed (sonic anemometer) at 13.78 m Vertical wind speed (sonic anemometer) at 18.12 m Temperature (sonic anemometer) at 8.73 m Temperature (sonic anemometer) at 13.78 m Temperature (sonic anemometer) at 18.12 m Wind speed and direction (respectively) at 2.63 m Wind speed and direction (respectively) at 3.55 m Wind speed and direction (respectively) at 4.46 m Wind speed and direction (respectively) at 5.37 m Wind speed and direction (respectively) at 6.59 m Wind speed and direction (respectively) at 7.51 m Wind speed and direction (respectively) at 9.43 m Wind speed and direction (respectively) at 11.44 m Wind speed and direction (respectively) at 12.46 m Wind speed and direction (respectively) at 14.44 m Wind speed and direction (respectively) at 15.46 m Wind speed and direction (respectively) at 16.37 m Wave-wire sensor located below instrument mast Wave-wire sensor located along southward facing boom Wave-wire sensor located along southward facing boom Wave-wire sensor located along southward facing boom

tained variables provide 50-Hz measurements of the following: wind speed and direction at 12 heights using cup anemometers and wind vanes, all three wind components and temperature at three heights using sonic anemometers, ocean wave heights at four locations using wave wires, and FLIP displacement and velocity using accelerometers (Shirer et al. 1999). Prior analyses of FLIP data use the algorithm of Edson et al. (1998) to correct FLIP motion and use a new calibration algorithm of Hristov et al. (2000) to account for the slow response time of the cup anemometers to wind accelerations (Shirer et al. 1999). The sampled data are averaged to 1-Hz values once these corrections are applied. It is this averaged and calibrated raw dataset that we use to begin our analysis. Eastward wind speed and

northward wind speed measurements obtained from sonic anemometers (variables 7 through 9 and variables 10 through 12 respectively in Table 1) are converted to along-mean wind speed u9 and cross-mean wind speed y 9 perturbations. Four similar, 110-min periods of 6600 observations, corresponding to four of the cases that Shirer et al. (1999) studied, are chosen for analysis in this article. The 110-min periods are the largest time periods in the FLIP data for which stationarity and air–sea surface temperature differences are not compromised. Table 2 outlines the time periods and selected boundary layer scales for cases A–D. Case A begins about 36 h after a synoptic-scale cold front passed through the area. Each 110-min-period is quasi-steady and moderately convective, with Monin–Obukhov lengths L ranging from 2161 m to 2345 m (Table 2). Formulas for determining the scaling quantities in Table 2 are provided below in Table 3 in section 3e. The similar vertical transports of horizontal momentum and heat lead to the isolation of similar convective coherent structures from case to case (Shirer et al. 1999). 3. Investigative procedures The investigative procedures described here follow four steps. 1) They perform principal components analysis to decompose the raw, calibrated dataset objectively; 2) they classify the resulting PC factors composing surface-layer plumes by applying cross-spectral analysis (coherence, phase, and power spectra); 3) they apply reassembly procedures to the individual PCA-derived temporal (score) series and PCs describing surface-layer plumes; and 4) they apply the VITA detection technique to the reassembled plumes to determine the fine structure of the turbulent eddies; this structure is quantified by the calculated transports of the resulting extracted plumes. Section 4 discusses a comparison of the results obtained from the application of steps 1–4 with those obtained from application of the VITA detection technique directly to the raw (nondecomposed) dataset. a. Principal components analysis The raw data used as input for PCA are obtained by subtracting the 110-min mean value and removing the

TABLE 2. Time periods and scaling quantities for cases A–D. Case

Julian day

Begin (UTC)

End (UTC)

L (m)

u ∗ (m s21)

T ∗ (K)

A B C D

124 124 125 125

0909 1412 0054 0336

1059 1602 0244 0517

2264 2161 2308 2345

0.324 0.296 0.451 0.472

0.029 0.032 0.048 0.047

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linear trend from each variable in Table 1. For our analysis, PCA isolates coherent structures by decomposing the raw time series of multivariate data vectors into a temporally varying linear combination of temporally constant empirical basis vectors called PC factors (Shirer et al. 1999). These orthonormal basis vectors, or empirical orthogonal functions (EOFs), are the eigenvectors of a covariance matrix computed from the data matrix containing multivariate profiles from 110 min of FLIP data. While this classical PCA transformation yields the same number of new variables, the EOFs, as original variables, the new variables are statistically independent and the first few capture most of the variance in the original dataset. The remaining EOFs now describe only a small amount of the variance and so are usually discarded. We use the scree test of Cattell (1966) to determine which of the EOFs need to be retained to explain most of the variance. The scree test helps us separate the EOFs that isolate the majority of the coherent structure physics from those that represent noise (Rinker and Young 1996). This test shows that the first 13 EOFs explain at least 93% of the variance in each of the four cases investigated. This result reduces the dimensionality of our dataset from 46 (the number of original acceptable variables and EOFs) to 13, and consequently, makes the subsequent cross-spectral and VITA analyses inherently more efficient. Analytic algorithms have led to various forms of PCA (Richman 1986). The classical version we briefly discussed above yields EOFs that are orthogonal in space and time. Unfortunately, these PCs are prone to describe the shape of the underlying domain even when the shape is unrelated to the organization of the embedded structures (Buell 1975). For example, in our one-dimensional FLIP datasets, EOFs may too closely resemble the natural waves for an interval, for example, constant-amplitude sine waves (Shirer et al. 1999). To avoid this unfortunate result, we apply to the 13 retained EOFs, Varimax orthogonal rotation (Kaiser 1958) followed by Promax oblique rotation (Hendrickson and White 1964). The choice of Varimax then Promax rotation of PCs follows that of Richman (1986), who tested several rotation schemes for their ability to isolate coherent structures in meteorological data. Varimax orthogonal rotation, by definition, retains the perpendicular set of weighted eigenvectors (EOFs) but points these vectors approximately towards clusters of data. Subsequent Promax rotation relaxes this orthogonality constraint, thereby allowing the PC factors contributing to a coherent structure to point more closely towards clusters but to be correlated to some degree. It is important to note that the total amount of variance explained by the 13 EOFs and the 13 Promax PCs—hereafter simply called PCs— is the same. In summary, we find that Promax PCs best express the dataset in terms of its simple structure (Thurstone 1947; Richman 1986); that is, each PC factor contributes primarily to one coherent structure type. How well the Promax PCs separate the raw dataset

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into the underlying coherent structures is determined through cross-spectral analysis (e.g., Sorbjan 1989) of the score (time) series for each PC. Briefly, the score matrix comprises columns of time series quantifying the evolution of the features captured by the corresponding PCs. In our analysis, the Promax PCs contain information about coherent structures in the surface layer. As discussed by Rinker and Young (1996) and Shirer et al. (1999), it is uncommon for any one PC to completely describe a phenomenon because the structure is often tilted or evolving. Analyzing different PCs and grouping those that vary together accomplishes the isolation of a coherent structure. b. Grouping procedures We use cross-spectral analysis to group pairs of PCs having high coherence and relatively fixed phase at scales (frequencies) in which they possess significant power (Shirer et al. 1999). Prior to implementing crossspectral analysis, we partition the 6600-s score series for each particular case into thirty 220-s segments so that average values of coherence and phase can be found for each score-series pair. We split each case into 30 segments because it provides enough samples for crossspectral analysis and, with an average wind speed of around 10 m s 21 , a 220-s time series is long enough to capture several surface-layer plumes. Averaging reduces noise in the phase and coherence. Such partitioning, however, limits the phase and coherence that can be identified to the frequency ranges resolved in the segments. Figure 1, discussed in greater detail below, illustrates that many surface-layer plumes can be found in these 220-s windows. After partitioning the score series, we individually detrend the resulting segments to prevent longer-period structures from falling inside the 220-s sampling window. Coherence, phase, and power spectra are obtained using standard cross-spectral analysis techniques discussed in greater detail in Shirer et al. (1999). After the phase and coherence spectra of a pair of score series are found, we impose a power spectrum constraint that requires both score series to exhibit significant power in the available frequency ranges for which there is relatively high coherence and nearly fixed phase difference. If this constraint is met, then we consider PCs to be spectrally linked and hence members of the same group or coherent structure type. This power spectrum constraint is imposed because coherence is normalized by the product of the total powers of the two series, giving unreliable estimates for score series that only have significant power at different frequencies (Shirer et al. 1999). Spectral peaks from two series occurring at two different scales are unlikely to be manifestations of the same phenomenon. Thus, for a set of PCs to be considered spectrally grouped, they each must be coherent with a majority of other PCs in that grouping.

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FIG. 1. Selected case B time series of raw FLIP data of (a) detrended streamwise wind speed u9, (b) vertical wind speed w9, and (c) temperature perturbation T 9 as calculated from the 13.78-m sonic anemometer (Table 1). Vertical transports of (d) streamwise wind momentum u9w9 and (e) temperature w9T 9 are also shown. Some of the selected burst and sweep events, denoted by ‘‘b’’ and ‘‘s,’’ respectively, are discussed in the text. Horizontal bars refer to approximate periods of large-scale transport; these bars are also shown in Fig. 2a.

c. Reassembly of surface-layer plumes Once PCs composing surface-layer plumes are grouped according to cross-spectral analysis, their score series for each group are combined to reassemble the decomposed data corresponding to the group and so to reveal its spatiotemporal properties. This reassembled data series for surface-layer plumes is obtained by following procedures outlined in Richman (1986) and described in detail by Shirer et al. (1999). Reassembly of the dataset using all 13 retained PCs yields the PCA-decomposed dataset. In contrast, reassembly using cross-spectral analysis selects a subset of five of the 13 PCs to yield the PCA-plume dataset. The PCA-decomposed dataset has an advantage over the raw dataset in that many features of little interest, accounting for less than 7% of the variance in the data, are removed. The PCA-plume dataset is more advantageous, however, as it isolates coherent (through cross-spectral analysis) PCs that explain much of the variance in the variables

that compose the vertical transport of heat and streamwise momentum. We show that this dataset provides the cleanest description of the plume transports and intermittency. d. Surface-layer dataset Raw, PCA-decomposed and PCA-plume datasets are subjected to the VITA technique. Coherent structure types such as plumes do not occur at all times or in all places in the convective surface layer; therefore, detecting them by means of averages, such as transports, could be problematic. The VITA technique successfully overcomes this difficulty via examination of the intermittent structure of the evolving transports (Phong-Anant et al. 1980; Subramanian et al. 1982; Schols et al. 1985; Zecchetto et al. 1998). We examine the surface-layer dataset prior to discussing the VITA technique, beginning with the analysis

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of the turbulent signals of streamwise wind, vertical velocity, and temperature. Figure 1 illustrates the intermittency of surface-layer plumes in a detrended raw (nondecomposed) dataset. The streamwise-wind and lateral-wind perturbations are hereafter denoted by u9 and y 9, respectively. The kinematic and thermal aspects of plumes are evident in the vertical transports of streamwise-wind momentum u9w9 (Fig. 1d) and heat w9T9 (Fig. 1e), respectively. Bursts of negatively correlated streamwise wind u9 (Fig. 1a) and vertical wind w9 (Fig. 1b), indicated by horizontal bars in Fig. 1, are consistent with the downward transport of positive, streamwisewind momentum. Bursts of positively correlated vertical wind w9 (Fig. 1b) and temperature T9 (Fig. 1c), also indicated by underlying horizontal bars, are apparent in the signals; they represent the vertical transport of warmer air by updrafts and cooler air by downdrafts. Note that the episodes indicated with the horizontal bars in Fig. 1 are those during which appropriate vertical transports of both momentum and heat occur. While illustrating both the intermittency and the varying size and intensity common in surface-layer plumes, Fig. 1 also reveals their typical ramplike structure, particularly in the temperature and streamwise-wind signals. The temperature ramps exhibit the characteristic increase in temperature with time (e.g., at times 640– 665 s in Fig. 1c), followed by a sharp decrease (e.g., at time 665 s) in temperature at its upwind edge (Kaimal and Businger 1970). In the conditional sampling literature (e.g., Schols 1984), large-scale transport is said to occur when two correlated variables, one of which is a velocity component, sufficiently depart from their zero mean. As indicated in Figs. 1d and 1e by the horizontal bars, evidence of such transport appears as substantial perturbations in the time series of their products. In reference to the u9w9 signal shown in Fig. 1d, two scenarios produce transport inside the larger-scale burst of negative vertical transfer of horizontal momentum. The upward transport of slower air with u9 , 0 and w9 . 0 is known as a burst or ejection (denoted by the symbol ‘‘b’’ in Fig. 1d) and the downward transport of faster air with u9 . 0 and w9 , 0 is known as a sweep (denoted by the symbol ‘‘s’’) (Blackwelder and Kaplan 1976; Subramanian et al. 1982; Schols 1984). Similarly, in a signal of w9T9, bursts are characterized by w9 . 0 and T9 . 0, and sweeps by w9 , 0 and T9 , 0. Therefore, the terms burst and sweep distinguish the plume updrafts and downdrafts that lead to heat and momentum transports of the appropriate sign. Hereafter, the term event refers to either an individual burst or sweep within the plume structure. The large variability in both the size and streamwisewind asymmetry of the temperature and streamwisewind speed ramps partially results from the wide variation of scales inherent to coherent structures in turbulent flow and from the sampling of only the edge of some events due to measurements at a fixed location

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(Davison 1974; Schols 1984; Wilczak 1984). Estimating heat and momentum transports by surface-layer plumes requires preprocessing the data to isolate the surfacelayer plumes from dynamically distinct phenomena, which can coexist in time and space (Rinker and Young 1996). The PCA-plume datasets, obtained in an objective, scale-independent manner, meet this requirement and so are used in the subsequent VITA technique analysis. e. The VITA technique Surface-layer plumes, apparent in the time series presented in Fig. 1, are detected by applying the VITA technique to the datasets. Kaplan and Laufer (1968) first developed the VITA technique to study the simulation of the structure associated with the motion of the interface between the boundary layer and the free atmosphere. Given a series of observations of the variable d (detecting/defining variable) in the t (time) domain, the generalized VITA technique relies on the computation of the normalized short-term variance s 2s,d(t, t):

s 2s,d (t, t ) 5

5

1 1 s 2d t 2

E

t1(1/2)t

d 2 (x) dx

t2(1/2)t

[E 1 t

t1(1/2)t

t2(1/2)t

]6 2

d(x) dx

,

(1)

where t is the duration of the short time interval over which the integrations are performed, and s 2d is the longterm variance (that is, calculated over the 6600-s duration of each FLIP case) of the detecting (defining) variable d. Table 3 provides a list of symbols, particularly those relating to the VITA analysis. Johansson and Alfredsson (1982) find that (1) has, regardless of the Reynolds number of the flow, a bandpass filter character, retaining events with timescales no longer than approximately 1.3t. As discussed in Zecchetto et al. (1998), choosing the detecting variable d for the analysis is the first step in most conditional sampling techniques. The choice of the detecting variable is typically case-specific, depending on the atmospheric stability, measurement height, and whether observations are taken over land or ocean (Manton 1977; Antonia et al. 1979; Lenschow and Stephens 1980; Greenhut and Khalsa 1982; Schols 1984; Schols et al. 1985; Young 1988; Zecchetto et al. 1998). As in Zecchetto et al. (1998), we initially investigate the events detected in the raw u9w9 time series (i.e., using d 5 u9w9) because of the small values of 2z/L (;0.1) in our four moderately convective FLIP cases (Table 2). We also investigate the events detected in the PCA-decomposed and PCA-plume time series. We use data from the midlevel 13.78-m sonic anemometer in all analyses because several PC factor profiles (Fig. 3, discussed below, and Table 1) show that the 13.78-m sonic anemometer data explain more variance

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MASON ET AL. TABLE 3. List of symbols, particularly in regards to the VITA technique.

u9, w9, T 9 u9w 9 , w 9T9 pl i u9w 9 , u9w 9 pl i w9 T9 , w9 T 9 splu9, splw9, splT9 t s 2s,u9w9, s 2s,w9T9 K t ib, t ie, Ti tmin z L u*, T* T pl d G C, S ab GC ab GSdab Gmd Gbd n a

Fluctuating streamwise wind, vertical velocity, and temperature Long-term (6600 s) momentum and heat transports (covariances) Momentum transport over duration of plume pl and over event i Temperature transport over duration of plume pl and over event i Std dev of u9, w9, and T9 over duration of plume pl Integration, or short-interval, time in VITA technique (usually 6 s) Normalized short-term variances of detecting variables u9w9 and w9T9 Threshold normalized variance for event detection Beginning time, ending time, and duration of ith event Minimum time interval for event detection Height above ocean surface Monin–Obukhov length [5 2u3*/k(g/T¯ )w9T9 ] calculated from 13.78-m sonic anemometers (Table 1) Scaling velocity and scaling temperature for the surface layer, calculated from 13.78-m sonic anemometers (Table 1) Duration of plume Detecting (defining) variable in VITA technique Data type discriminator: P (PCA-plume); D (PCA-decomposed); or R (raw) Correlation coefficients for long-term (6600 s) and short-term (VITA technique plume) transports, respectively Product of variables a and b for dataset type G; ab represents either u9w9 or w9T9 in our analyses Long-term correlation coefficient (4) of ab for dataset type G Correlation coefficient (3) of ab calculated over the duration of a plume extracted by the VITA technique using detecting variable d Slope of best-fit line for plots of GSdw9T9, vs GSdu9w9 GSdw9T9 intercept of the best-fit line for plots of GSdw9T9, vs GSdu9w9 Number of plumes extracted from VITA-technique events General trimean of GSdab

than do those at either 18.12 m or 8.73 m. A plot of the short-term variance of u9w9, s 2s,u9w9 (t, t 5 6 s), corresponding to the same time series in Fig. 1, is presented in Fig. 2a. The brief negative values of s 2s,u9w9 (t, t 5 6 s) in Fig. 2a are an error in the numerical integration, as (1) is positive semidefinite. Our choice of t 5 6 s is explained below in section 4c. Figure 2a shows that the time intervals indicated by horizontal bars in Fig. 1, representative of large-scale transport, possess relatively large values of s 2s,u9w9(t, t 5 6 s). Detection of events in the datasets depends on a priori numerical criteria. We essentially follow the scheme used by Zecchetto et al. (1998) because of their success in extracting plumes from a moderately convective marine atmospheric surface layer. Events in a time series are manifested as peaks in the s 2s,d signal. Detection of either a burst or a sweep requires that s 2s,d increases with time t and exceeds a specified threshold normalized variance K for a minimum time interval tmin . As seen in Schols et al. (1985) and Zecchetto et al. (1998), the characteristic shape of the events is not affected greatly by the chosen values of t and K. Larger values of K impose higher thresholds and hence sacrifice some detail. We find that K 5 0.5, the same value as that used by Zecchetto et al. (1998), detects most well-resolved plumes (dashed line in Fig. 2a). In contrast, the horizontal bars in Fig. 2a indicate plumes chosen from visual inspection of the time series. These visually detected

surface-layer plumes are approximately those that would be detected using K 5 1; this choice yields half as many events as when K 5 0.5. Because FLIP data are averaged to 1 Hz, we use the relatively large value of tmin 5 4 s; as seen in Fig. 2b, this choice still allows us to detect numerous events. f. Extracting plumes from detected events We are not interested in the small number of weak events for which u9w9 . 0 or w9T9 , 0. We refine the Zecchetto et al. (1998) criterion above by appending the following one: u9w9 , 0,

i

u9 , 0,

w9 . 0

i

w9 . 0,

T9 . 0

w9T9 . 0,

burst–upward flux of: lower-momentum, warmer air; i

u9w9 , 0,

u9 . 0,

w9 , 0

i

w9 , 0,

T9 , 0

w9T9 . 0,

and

sweep–downward flux of: higher-momentum, cooler air,

(2)

where the ith average is applied only over the duration T i , of the ith event. The inclusion of (2) into the pre-

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FIG. 2. (a) Short-term variance of the u9w9 signal s 2s,u9w9 (t, t 5 6 s), (b) VITA technique–detected events satisfying (2) that are composed of either sweeps (denoted ‘‘s’’) or bursts (denoted ‘‘b’’), and (c) plumes extracted from the events as calculated from the 13.78-m sonic anemometer for times corresponding to Fig. 1. The value of K is 0.5 and tmin is 4 s. Horizontal bars in (a) are the same as those in Fig. 1 and refer to approximate times of large-scale transport; only these plumes would be captured if the visually apparent K 5 1 were used. The dashed line in (a) indicates the lower threshold for event detection (K 5 0.5) used in (b) and (c).

viously discussed a priori numerical criteria provides the desired refinement, as well as a sorting of events into the bursts and sweeps that are responsible for most of the heat and momentum transport in the surface layer (Zecchetto et al. 1998). Visually detected surface-layer plumes for times corresponding to those in Fig. 1 are indicated by the horizontal bars (K is about 1.0) in Fig. 2a, while the short-term variance for sweeps and bursts (events) satisfying (2) and meeting the detection criteria given by the dashed line in Fig. 2a (K 5 0.5 and tmin 5 4 s) is shown in Fig. 2b. Once events are detected using (2), extracting the duration of the associated plumes is performed (Fig. 2c). The onset of a plume is regarded as corresponding to the time t at the beginning of an event, and a plume ends at the first subsequent time t at which s 2s,u9w9(t, t 5 6 s) # K (Zecchetto et al. 1998). Figure 2c shows

that plumes often consist of two or more events. For example, two events first detected at 644 and 650 s (Fig. 2b) are found to be part of the same plume (Fig. 2c). In this case, and many like it, the integration over a time interval of duration t in the definition (1) of the short-term variance leads to the identification of plumes as extensions of the first event detected (at time 644 s). The appearance of u9 , 0, w9 . 0, in Figs. 1a and 1b (at time 644 s), respectively, indicates that this particular plume is extracted from a burst. Once a detecting variable d is used to identify sweeps and bursts, we calculate and compare the correlation coefficients of the heat and momentum transports of the corresponding plumes (see Fig. 7 in section 4d). This procedure is similar to that used by Schols et al. (1985), in which heat and momentum transports are calculated for ensemble structures obtained from VITA technique-

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detected events using perturbation temperature as the detecting variable. For the intermittent plumes captured by the VITA technique (Phong-Anant et al. 1980; Khalsa and Greenhut 1985; Schols et al. 1985; Zecchetto et al. 1998), the (short-term) correlation coefficients GS dab of the individual plumes are, on average, greater in magnitude than the long-term correlation coefficient GC ab of the analyzed time series (Davison 1974). The correlation coefficients of the individual plumes are defined as

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have preprocessed the dataset properly, and if there actually is a relationship between vertical transports of heat and streamwise momentum, then we may anticipate that the choice of detecting variable will not matter, and the same relationship, the true one, between vertical transports of heat and streamwise momentum will be seen whether d 5 u9w9 or d 5 w9T9. As we demonstrate in section 4, this robust and physically expected result is made possible by applying PCA to the dataset prior to using VITA.

pl

GS

d ab

ab 5 pl pl , sa sb

(3)

pl

where ab , s pla , and s plb are the covariance (transport) and standard deviations of the fluctuation signals a and b over the duration of the arbitrary plume identified by the detecting variable d. In contrast, GC ab is the covariance of a and b over the entire time series, normalized by s a and s b calculated over the entire time series, GC a b 5

ab . sa sb

(4)

Once we obtain distributions of GS 9 9 , we calculate correlation coefficients for GS uw9T9 9w9 occurring at the same times as those for plumes extracted from u9w9 –VITA technique–detected events. We perform this calculation because applying the VITA technique with w9T9 as the detecting variable d produces slightly different event, and hence plume, locations in the time series from those for the u9w9 plumes. Physically, the youngest surfacelayer plumes should have the greatest negative correlation between fluctuating vertical and streamwise wind. As the plumes age, the dynamic pressure perturbation tilts the plume. The fluctuating streamwise velocity responds at the speed of sound, much faster than fluctuating temperature. Consistent with this response, we will find (in Fig. 7 of section 4) that the correlation coefficients for heat transport are larger in magnitude than those for momentum transport. Once values of GS uw9T9 9w9 u w are obtained, we compare them with those of GS u9w9 9 9; plumes now have unique GS uw9T9 9w9 and GS uu9w9 9w9 values. For well-resolved plumes (those with correlation-coefficient magnitudes of at least 0.5), plots of GS uw9T9 9w9 as a function u w of GS u9w9 9 9 tend to approximate lines. The significance of this result is explained in section 4d, in which we perform simple linear regression to obtain the mean-square errors (Draper and Smith 1981; Neter et al. 1985). To determine whether PCA and grouping techniques have successfully isolated plumes, we also perform the VITA technique using d 5 w9T9 on the signals obtained from raw, PCA-decomposed, and PCA-plume time series. In a method similar to that discussed above, we calculate correlation coefficients GS wu9w9 9T 9 for the u9w9 signal during w9T9 –VITA technique–detected plumes. Also in section 4d, we perform linear regression analyses for GS ww9T9 9T 9 plotted as a function of GS wu9w9 9T 9. If we u w u9w9

4. Plume transports Applied identically to each of four, moderately convective, quasi-steady cases (Table 1) obtained from FLIP during the 1995 PMBL experiment, we compare the statistics of surface-layer plumes using three forms of input to the VITA technique: 1) PCA-plume dataset, 2) PCA-decomposed but ungrouped dataset, and 3) the raw dataset. Our analysis begins by extracting PCs describing surface-layer plumes. a. Isolating plumes The first step in our investigation consists of performing PCA on the raw dataset. After the rotation schemes (discussed in section 3a) are complete, we plot the resulting Promax PCs as factor profiles. For a particular variable, the product of PC factors and the score (time) series (discussed below) provides a measure of how much variance that PC explains. Figure 3 provides an illustration of PCs 1, 3, and 7; when weighted by the square root of the number of observations, Ï6600, we obtain what we call the deviance for each PC factor. Deviance has the desirable trait of retaining the sign of the components of the PC, thereby allowing interpretations of the correlations between variables in a PC (Rinker and Young 1996). The three PCs plotted in Fig. 3 are candidate members of the surface-layer plumes group because they each contain significant variance in the variables composing the streamwise momentum and heat transports in the surface layer, for example, u9, y 9, w9, and T9. The kinematic aspect of the surface-layer plumes appears to be well-represented by PCs 1, 3, and 7 because of the large deviance in u9, y 9, and w9 for these PC factors. Their thermal aspect, given by the correlation between the vertical velocity and temperature perturbations, appears to be captured by PC 7. Unfortunately, the shape of the PC plots does not provide sufficient information to group the PCs by coherent structure type because these plots contain no temporal information. Thus, there is no way to know if the variables they represent are in fact correlated in the same frequency range, as must occur if the PCs represent the same coherent structure type. The necessary temporal information is in the score series associated with the PCs under scrutiny. One successful grouping strategy requires that all score series

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FIG. 3. Factor profiles, weighted by a factor of the square root of the number of observations, Ï6600, of Promax PCs 1, 3, and 7 from case B. The variable numbers in the ordinate correspond to the order of the variables listed in Table 1. Variables used in our analysis, labeled on the right, include 13.78-m perturbations of temperature T 9 (17), vertical wind w9 (14), northward wind speed V9 (11), and eastward wind speed U9 (8). PCs 1 (solid curve) and 7 (dotted curve) are members of the plumes group.

for the retained PCs exhibit high coherence and nearly fixed phase over frequencies having sufficient power (Shirer et al. 1999). The coherence and phase spectra of PCs 1 and 7 and PCs 1 and 3 are provided in Figs. 4 and 5, respectively. All three PCs exhibit spectral peaks (not shown, but apparent in Fig. 1) in the lowest frequency range. Corresponding to periods of approximately 20–110 s, this occurrence is consistent with the periods observed in the signals of short-term variance s 2s,u9w9 in the detected events (Figs. 2a and 2b). By virtue of their relatively high coherence (Fig. 4a)—greater than 0.5—and nearly fixed phase difference (Fig. 4b) over this long-period range, we regard PCs 1 and 7 as part of the same group (plume). In contrast, the low coherence over all periods (Fig. 5a) between PCs 1 and 3 indicates that PC 3 describes an unrelated, dynamically distinct phenomenon. For the four cases we investigate, chaotic temporal analysis reaffirms this cross-spectral (linear) analysis grouping of PCs 1 and 7 as describing surface-layer plumes

(Shirer et al. 1999), with PC 3 describing a not-yetdetermined phenomenon. Three additional PCs (numbers 4, 6, and 12) are necessary to supplement PCs 1 and 7 for describing surface-layer plumes (Shirer et al. 1999). Factor profiles and cross-spectral plots for all five of these PCs, in our four cases, are discussed in greater detail in Rishel (1998). As seen, in part, in Fig. 3 and Figs. 17.8 and 17.10 of Shirer et al. (1999), PCs 1, 4, and 6 primarily describe the flow into and out of the plume via the fluctuations in the horizontal wind speed in various portions of the surface layer, while PCs 7 and 12 primarily describe the vertical motion and thermal structures of the plume. Rinker and Young (1996) find a similar result in their analysis of large eddy simulations (LES) data. All five PCs are thus needed to produce a complete description of the vertical transport of horizontal momentum and heat by the plumes. We reassemble the data series from the five PCs composing surface-layer plumes using procedures detailed

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FIG. 4. (a) Coherence and (b) phase spectra for the score series associated with PCs 1 and 7 in case B.

in Shirer et al. (1999); this reassembled series produces the PCA-plume time series. To produce the reassembled series, the q 5 5 Promax PCs describing surface-layer plumes are arranged into an N 3 q score matrix Fp and an M 3 q PC factor matrix Ap , where N is the number of observations (6600) and M is the number of original variables (46). This score matrix Fp is found from the original N 3 M data matrix Z using the formula (Shirer et al. 1999), Fp 5 ZAp (ATp Ap ) 21 .

(5)

This form arises because Ap is not a square matrix. The reassembled data matrix describing surface-layer plumes Zp is then given by Zp 5 FpATp .

(6)

A similar procedure yields the reassembled, PCA-decomposed data series ZD , in which q 5 13, from score matrix FD and factor matrix AD .

The PCA-plume datasets are much reduced in small variance and large-scale irrelevant features (e.g., wavy boundary structures). PCA-plume datasets are generated by the data themselves without recourse to any preconception of surface-layer plume structure. We only assume that wind and temperature variance should be mostly explained by some combination of the retained (coherent) PC factors in this moderately convective marine surface layer. Plumes of varying sizes are retained in the PCA-plume dataset because PCA does not seek a priori phenomena, but rather, identifies correlations in the multivariate dataset (Richman 1986). The variations in both size and intensity of the plumes are likely to be a result of both the properties of the plumes and the manner by which they are sampled. Plumes vary in size and intensity based on their age and height, and random portions of the plumes, ranging from edges to cores, pass across the sensors. PCA decomposes the data based on correlations between measured variables, and im-

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FIG. 5. (a) Coherence and (b) phase spectra for the score series associated with PCs 1 and 3 in case B.

poses no assumptions about the size and intensity of the plumes under investigation; the reassembled dataset fully recovers these inherent variations in plume scales. Because we are investigating a moderately unstable regime, we expect that most of the wind speed variance results from the convective eddies. The largest of these eddies spans the depth of the boundary layer and contributes the greatest amount of wind speed and temperature variance in the dataset. Demonstrated by the high coherence for longer periods in Fig. 4, and the long duration between temperature ramps in Fig. 1, the time between temperature ramps is successfully captured by the PCs that describe surface-layer plumes. b. Long-term transports Vertical transports of heat and streamwise horizontal momentum w9T9 and u9w9 , respectively, are calculated at 13.78 m over the duration (6600 s) of each case

investigated. These long-term transports are computed separately for reassembled PCA-plume time series given by (6), the reassembled PCA-decomposed time series, and the raw time series. Comparisons between w9T9 and u9w9 for all three datasets are given in Table 4. In every case, the variances of u9, w9, and T9 are smaller in the PCA-plume reassembled datasets than in the raw dataset. Despite this, Table 4 demonstrates that the longterm transports w9T9 and u9w9 are slightly larger in magnitude for the PCA-plume datasets (G 5 P) than they are for raw datasets (G 5 R). This result indicates that these plumes support most of the transport and that the other components are either small or tend to cancel. Long-term correlation coefficients GC w9T 9 and GC u9w9 in Table 4 illustrate that the PCA-plume dataset (G 5 P) consistently best isolates those variables that describe surface-layer plumes. Significantly, in every case, weak to moderate correlations deduced from variables composing heat and momentum transports in the raw da-

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TABLE 4. Long-term transports w9T9 and u9w9 and correlation coefficients GCw9T9 and GCu9w9 computed using (4) for each dataset type G [P (PCA-plume); D (PCA-decomposed); R (raw)] in cases A–D. G

Case

w9 T9 (m K s21 )

u9w 9 (m 2 s22 )

GC w9 T9

GC u9w9

P D R

A A A

0.010 0.009 0.009

20.108 20.107 20.105

0.659 0.549 0.503

20.539 20.448 20.401

P D R

B B B

0.013 0.012 0.012

20.103 20.092 20.088

0.601 0.490 0.472

20.456 20.368 20.322

P D R

C C C

0.025 0.021 0.022

20.227 20.207 20.204

0.666 0.560 0.476

20.600 20.496 20.452

P D R

D D D

0.024 0.022 0.022

20.238 20.227 20.223

0.524 0.499 0.433

20.486 20.469 20.431

tasets (G 5 R) become moderate to strong correlations after PCA, cross-spectral analysis, and reassembly procedures are performed (G 5 P). We also see that the PCA-decomposed datasets (G 5 D) yield, when compared with the raw datasets (G 5 R), consistently larger magnitudes of GC w9T 9 and GC u9w9 in all cases. As discussed above, the larger magnitudes of DC ab for the PCA-decomposed datasets over RC ab for the raw datasets are primarily the result of the removal of components in the data that contribute to the total variance of each variable, but not greatly to the covariance between them. c. VITA technique-detected transports Surface-layer plumes are extracted from raw, PCAdecomposed, and PCA-plume datasets using the VITA detection technique outlined in section 3e. As noted there, after the detecting variable d is chosen, event detection requires assigning values for the integration time t, threshold time tmin and the threshold normalized variance K. A typical property of turbulent structures like surface-layer plumes is their intermittency. As a result, it is impossible to define one unique timescale or frequency of occurrence. As discussed in Johansson and Alfredsson (1982) and Schols et al. (1985), the number n of events detected strongly varies with t and K. Most of the temperature ramps in our datasets, like those identified in Fig. 1, are detectable with K 5 0.5 (see Fig. 2b), a value also used by Schols et al. (1985) and Zecchetto et al. (1998). The seven plumes detected visually in Fig. 1 correspond to a value of K close to 1.0 (see Fig. 2a). In contrast, using K 5 0.5 captures eight additional, more weakly correlated, smaller timescale, and hence smaller amplitude, plumes (see Fig. 2c). A completely objective means for choosing the best value of K remains to be developed. Once a value for K is assigned, the value for t is identified by the maximum (not shown) of the number n of detected events as a function of t. As discussed in

TABLE 5. Characteristics of the distributions of GSdab over all four cases for each combination of dataset type G [P (PCA-plume); D (PCA-decomposed); R (raw)], detecting variable d, and transport ab. Gsdab

a

n

PSww99TT99 DSww99TT99 RSww99TT99

0.77 0.75 0.67

747 774 775

PSwu99wT99 DSwu99wT99 RSwu99wT99

20.65 20.64 20.57

747 774 775

PSuu99ww99 DSuu99ww99 RSuu99ww99

20.67 20.66 20.61

712 701 677

PSuw99wT99 DSuw99wT99 RSuw99wT99

0.78 0.74 0.66

712 701 677

section 3e, when using the VITA technique, bursts and sweeps will be restricted to approximately 1.3t. We find, after testing values of t that ranged from tmin 5 4 to 32 s, that, along with Schols (1984) and Schols et al. (1985), t 5 6 s produces the largest value of n for all three cases. For K 5 0.5, n is a maximum at t 5 6 s regardless of the type of dataset (PCA-plume, PCAdecomposed, or raw) or detecting variable (d 5 u9w9 or w9T9) selected. Physically, when the value of t becomes too large, only the largest, long-duration events and plumes are retained, and smaller plumes are not detected (Johansson and Alfredsson 1982). We quantify the correlation coefficients GS dab for all surface-layer plumes in cases A–D (Table 2) by referring to the trimean a of their distributions. The trimean a is a weighted average of the median (q 0.5 ) and first and third quartiles (q 0.25 , q 0.75 ), with the median receiving twice the weight of the other two quartiles: a 5 (q 0.25 1 2q 0.5 1 q 0.75 )/4. Wilks (1995) notes that a is both a robust and resistant measure of the central tendency of the distribution of GS dab, and so is insensitive to uncharacteristic, outlier values in the time series. It is apparent in Table 5 that the magnitudes of a are largest for the PCA-plume and smallest for the raw datasets. Table 5 also indicates that the number of plumes n extracted from VITA technique-detected events is about the same in the three dataset types and depends only on the detecting variable d. Histograms for extracted plumes in all four cases restricted to those plumes having at least moderate correlations | GS du9w9 | $ 0.5 are shown in Fig. 6. Counts are based on the bin into which the plume correlation falls. For example, in Fig. 6a, there are approximately 60 PCA-plume plumes with momentum correlation coefficients between 20.875 and 20.925; in contrast, there are less than 40 raw plumes in the same 20.90 bin. We are comfortable with our comparisons of the counts derived from both the raw and PCA-plume datasets because, as Table 5 shows, the total number n of extracted plumes is approxi-

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FIG. 6. PCA-plume (G 5 P : light bars) and raw (G 5 R : dark bars) histograms of (a) GS uu9w9 9w9, (b) GS uw9T9 9w9, (c) GS ww9T9 9T 9, and (d) GS wu9w9 9T 9 for all extracted plumes in all four cases restricted to plumes having at least moderate correlations | GS dab | $ 0.5. Magnitudes of | GS dab | increase to the right in all plots.

mately the same for each dataset type. Therefore, Fig. 6 also demonstrates that, when compared with the raw datasets, the PCA-plume datasets contain a larger proportion of well-resolved plumes, with the best-resolved having | GS dab | $ 0.8. d. Inference of plume transports Using the procedure outlined in section 3f, we plot the values of all plume correlation coefficients GS dw9T9 for variables composing the vertical heat transport against the plume correlation coefficients GS du9w9 for the variables composing the vertical transport of horizontal streamwise momentum defined in the same time interval. We restrict our investigation of individual surfacelayer plumes to those that are well-developed, exhibiting at least a moderate correlation | GS du9w9 | $ 0.5 between variables composing the detecting variable d (vertical

heat or momentum transport). Values of GS dab 5 GS uu9w9 9w9 plotted against GS dab 5 GS uw9T9 9w9 for case D are shown in Fig. 7, where the abscissas are GS uu9w9 9w9 and the u w ordinates are GS w9T9 9 9 for the G appropriate to each case (P, D, and R, as defined in Table 3). For example, an arbitrary plume from the raw dataset in case D obtained with d 5 u9w9 has a pair of unique values of RS uu9w9 9w9 and u w RS w9T9 9 9 that are plotted in Fig. 7c. A measure of the success of the burst–sweep VITA technique to separate the plumes cleanly from the other coherent structures in a dataset is given by the arrangement of the plume coordinates (GS uu9w9 9w9 ,GS uw9T9 9w9 ) in Fig. 7. From this figure, it is clear that quantifying the correlation coefficient of the vertical transport of horizontal momentum GS uu9w9 9w9 of an arbitrary surface-layer plume through VITA also produces a rough estimate of its correlation coefficient for vertical heat transport GS uw9T9 9w9. Plumes that strongly transport horizontal mo-

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FIG. 7. Plots of GS uw9T9 9w9 as a function of GS uu9w9 9w9 for the (a) PCA-plume, G 5 P, (b) PCA-decomposed, G 5 D, and (c) raw, G 5 R datasets from case D. The x axis is reversed such that the best-resolved plumes reside in the upper-right portion of the plot. The regression lines (7) that minimize the distances between the points and the line are obtained through a least squares regression technique applied to the case of moderate correlations | GS du9w9 | $0.5, as discussed in the text. Each point represents the values of GS uu9w9 9w9 and GS uw9T9 9w9 for a plume extracted using detecting variable d 5 u9w9 in the VITA technique.

mentum vertically also strongly transport thermal energy vertically. But the VITA technique alone (Fig. 7c, G 5 R) does not produce so strong a linear relationship as do either case with time series preconditioned with PCA. As the transport representation in the two PCAdecomposed cases is carried respectively by different sets of 13 (PCA decomposed) and five (PCA plume) PCs, these linear relations are not likely to be artifacts of the PCA but rather fundamental relationships governing the plumes themselves. The best relationship, as given by the small spread in the plume coordinates (GS uu9w9 9w9,GS uw9T9 9w9) about the regression line (Draper and Smith 1981; Neter et al. 1985), occurs for PCA-plumes in Fig. 7a. We regard Fig. 7 as a ‘‘plume organization plot,’’ in which young, very well-organized plumes, apparent as bursts and sweeps in the time series (Fig. 1), have the

largest magnitudes of the correlation coefficients for heat and momentum transport, and disorganized plumes are those that have small-magnitude correlation coefficients. In between, we believe that young, well-organized plumes are growing into a sheared environment in which dynamic pressure perturbations by the new plumes eventually tilt the plumes and systematically weaken the momentum correlation. The faster response of the pressure field, compared with the temperature field, to perturbations from plumes decreases the strength of the linear relationship for older plumes. We invite LES modelers to test plume correlation coefficients of heat and momentum as a function of plume age, and compare the scatter they obtain with that in Fig. 7. The quality of the linear relationship between GS dw9T9 and GS du9w9 for each dataset type is quantified by

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TABLE 6. Mean-square errors (31023 ) of GS dw9 T9 as a function of GS dw9 T9 for each dataset type G [P (PCA-plume); D (PCA-decomposed); R (raw)], detecting variable d, and case investigated for plumes exhibiting at least a moderate correlation |GSdu9w9| $ 0.5. G

Case

MSE: d 5 u9w9

MSE: d 5 w9T9

P D R

A A A

2.1 4.5 115.1

1.6 2.6 27.1

P D R

B B B

2.3 3.4 38.1

1.8 3.1 15.0

P D R

C C C

0.5 3.5 53.2

0.5 4.0 26.3

P D R

D D D

0.9 6.0 28.6

0.9 5.5 17.1

calculating the corresponding mean-square errors (MSEs) of the plume coordinates (GS du9w9 ,GS dw9T9 ) with respect to their regression lines (Wilks 1995): u9w9 u9w9 GS u9w9 3 GS u9w9 , w9T9 5 Gm u9w9 1 Gb

and

(7)

w9T9 w9T9 GS w9T9 3 GS w9T9 . w9T9 5 Gm u9w9 1 Gb

Here, Gm u9w9 and Gb u9w9 are the slope and GS uw9T9 9w9 intercept of the regression line obtained when d 5 u9w9. Similarly, the slope Gm w9T 9 and the GS ww9T9 9T 9 intercept Gb w9T 9 are defined for the regression line obtained when d 5 w9T9. Only data for which | GS du9w9 | $ 0.5 holds are used to find the lines in Fig. 7. Calculations of the MSEs for all data types in all investigated cases are presented in Table 6 for plumes exhibiting at least a moderate correlation | GS du9w9 | $ 0.5. Overall, the PCA-plume (G 5 P) and raw (G 5 R) datasets yield the best (smallest MSE) and worst (largest MSE) linear relationships respectively, between the correlation coefficients of heat and momentum transport over the time intervals in which plumes are extracted. The differences in the MSEs between PCA-plume and raw datasets are most pronounced in cases C and D, where MSEs are more than an order of magnitude smaller for the PCA-plume dataset than for the raw dataset. The smaller MSEs for the PCA-plume dataset demonstrate a stronger independence of the heat and momentum transport correlation coefficient estimates on the choice of detecting variable. The significance of decreasing the MSE by performing PCA and the subsequent cross-spectral grouping re-

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lates to how well conditional sampling techniques create composite surface-layer plumes using various detecting variables in a dataset. In contrast to raw data, using PCA-plume data allows crisper composites of plumes to be obtained because the values of the plume transports PS dab are less dependent on the detecting variable d. It is important to note that while the inclusion of the more weakly correlated extracted plumes somewhat weakens the differences in the MSEs between datasets, as seen in Fig. 7, the same ordering of the MSEs with the datasets is found. The degree to which the regression lines given by (7) model the transport correlation coefficients GS uw9T9 9w9 and GS uu9w9 9w9 may be regarded as a measure of how well the analysis captures a fundamental physical property of convective plumes. The veracity of this property is made far more convincing if both the slopes Gm u9w9 and Gm w9T 9, and the intercepts Gb u9w9 and Gb w9T 9 are equal, respectively, as discussed below. In this case, the transport relationship is independent of detecting variable. As we see below, preconditioning the dataset (G 5 P) is a critical step in removing the artificial dependence of the result on the detecting variable. For each case, values for Gm u9w9 2 Gm w9T 9 and Gb u9w9 2 Gb w9T 9 are closest and furthest from zero for the PCAplume and raw datasets, respectively. Therefore, because of the small case-to-case variability in Gm u9w9 2 Gm w9T 9 and Gb u9w9 2 Gb w9T 9, we use the average slopes Gm d and average GS dw9T9 intercepts Gb d in our remaining discussion of the transport relationships. The slopes and intercepts are averaged over all four cases and are given in Table 7. For the linear algebraic relationships in (7) to represent the same physical relationship independent of detecting variable, the values of Gb u9w9 2 Gb w9T 9 and Gm u9w9 2 Gm w9T 9 must be zero; that is, Fig. 7 would look identical for GS ww9T9 9T 9 plotted as a function of w T because the VITA technique would capture the GS u9w9 99 same plumes with d 5 w9T9. As seen in Table 7, both (Gb u9w9 2 Gb w9T 9 ) and (Gm u9w9 2 Gm w9T 9 ) are smallest in magnitude, thus indicating the strongest linear relationships between GS d5ab and GS d±ab ab ab , for PCA-plume datasets. When combined with the MSE values in Table d 6, this demonstrates that, when | PSu9w9 | $ 0.5, we can infer the value of PS uw9T9 9w9 directly from PS uu9w9 9w9 , and the value of PS ww9T9 9T 9 directly from PS wu9w9 9T 9 . The same cannot be said for raw datasets, where both (Gb u9w9 2 Gb w9T 9 ) and (Gm u9w9 2 Gm w9T 9 ) are largest in magnitude, thus indicating the weakest linear relationships. The advan-

TABLE 7. Average coefficients in linear relations (7) between GSdw9 T9 and GSdu9w9 for each dataset type G [P (PCA-plume); D (PCAdecomposed); R (raw)], detecting variable d, and case investigated for plumes exhibiting at least a moderate correlation | GSdu9w9| $ 0.5. G

Gmw9 T9

Gmu9w9

Gbw9 T9

Gbu9w9

P D R

20.687 20.731 20.595

20.687 20.743 20.619

0.316 0.264 0.314

0.315 0.252 0.252

(Gmu9w9 2 Gmw9 T9 ) 20.001 20.013 20.023

(Gbu9w9 2 Gbw9 T9 ) 20.001 20.012 20.062

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tage of cross-spectral analysis is also seen in Tables 6 and 7, as PCA-plume datasets possess smaller values of MSE, (Gb u9w9 2 Gb w9T 9 ) and (Gm u9w9 2 Gm w9T 9 ) than those in PCA-decomposed datasets. Unrelated coherent structures and small-variance features are removed in the PCA-plume data, and subsequent VITA extraction of plumes is essentially independent of the choice of heat or momentum transport for the detecting variable. The well-defined collapse of the correlation coefficients in a moderately convective marine surface layer onto a line of slope Gm u9w9 5 Gm w9T 9 ù 20.7 for | GS du9w9 | $ 0.5 suggests that a similarity relation (7), which can be rewritten as pl

w9T9 5 20.7

s plT9 pl u9w9 1 0.3s plw9 s plT9 , s plu9

(8)

exists between the momentum and heat transports and the standard deviations of streamwise momentum, vertical velocity, and perturbation temperature in surfacelayer convective plumes. At least for the moderately convective range considered, the constants in (7) or (8) for the correlation coefficients of heat and momentum transports are independent of scaling parameters L, u*, and T* given in Table 2. To see how universal this relationship is, clearly a wider range of conditions must be investigated. We have shown here that PCA preceding VITA provides an excellent means for analyzing time series to derive such transport relationships for coherent structures in the surface layer. We conclude from these results that our success in using VITA to capture plumes within a dataset is not tied to whether the chosen detecting variable d is u9w9 or w9T9 but rather to whether the dataset has been preconditioned using PCA combined with cross-spectral analysis. 5. Summary and conclusions We compare both the long-term and short-term vertical transports of heat and momentum calculated from the raw time series and two objectively preconditioned series. We perform our analyses on four similar, 110min moderately convective cases (cases A, B, C, and D) of marine atmospheric surface layer data taken during the 1995 Pacific Marine Boundary Layer (PMBL) experiment. A set of five PCs describing surface-layer plumes is grouped according to cross-spectral analysis. The reassembled PCA-plume data series shows plumes of various sizes and intensities; positive vertical heat transport and negative vertical transport of horizontal momentum are typically found in these plumes. The magnitudes for the temperature, streamwise wind, and vertical velocity perturbations, measured with the sonic anemometer at 13.78 m (Shirer et al. 1999) are similar in both the raw and PCA-plume datasets. In all four cases, the PCA-plume and raw data series are observed to contain the largest and smallest mag-

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nitudes, respectively, of the long-term correlation coefficients for both heat GC w9T 9 and momentum GC u9w9 transports. PCA-plume datasets (where G 5 P) isolate plumes, which contain most of the overall transport, from unrelated and small-variance phenomena. All four cases are moderately convective with Monin–Obukhov lengths L between 2160 and 2360 m. We find that most important surface-layer plumes are detected in the time series of s 2s,d(t, t) when values of 0.5 and 6 s are assigned to the normalized variance threshold K and integration time t parameters required in the Variable Interval Time Averaging (VITA) technique. We find that the largest and smallest weighted medians (trimeans) of | GS dab | occur when plumes are extracted from VITA technique–detected PCA-plume and raw datasets, respectively. Plotting the values of all plume correlation coefficients GS dw9T9 for variables composing the vertical transport of heat against the plume correlation coefficients GS du9w9 for variables composing the vertical transport of horizontal momentum, defined over the same time interval, produces a linear relationship between GS dw9T9 and GS du9w9 when we restrict our investigation to well-resolved plumes for which the magnitudes of GS du9w9 are greater than 0.5. When using PCA-plume datasets, we find that the average slopes Pm w9T 9 are nearly the same as the average slopes Pm u9w9 . This relationship between average slopes Gm d is less apparent in PCA-decomposed (G 5 D) datasets, and virtually nonexistent in raw (G 5 R) datasets. Therefore, based on the linear relationships between GS dw9T9 and GS du9w9 in surface-layer plumes for our three dataset types, we conclude that detecting plumes is not dependent on whether the detecting variable is u9w9 or w9T9. Rather, as seen by the improved correlations between the variables in the correlation coefficient definitions that are applied over both large and short (VITA method) averaging times, it depends primarily on whether PCA combined with cross-spectral analysis is performed on the dataset prior to the application of the VITA technique. Once this preconditioning of the time series is performed, similarity relations for coherent structures, such as (7) and (8) that we found here between the vertical heat and momentum transports for surface-layer convective plumes, become much easier to obtain. Direct analysis of the correlation coefficients for the streamwise wind-temperature transport GS du9T 9 of surface-layer plumes would provide a test of the robustness of the results we have obtained. Here we perform PCA on a multivariate dataset composed of the 46 measured variables in Table 1. Derived quantities such as transports are subsequently calculated from these reassembled datasets. In contrast to this, we suggest subjecting the raw u9w9, w9T 9, and u9T 9 signals observed at multiple levels directly to PCA and crossspectral analysis. Finally, performing our analysis on datasets in different stability regimes and over land

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would further test the repeatability of our results, including the generality of our proposed similarity relation for unstable conditions. Acknowledgments. We thank Jeremy Rishel and Dr. Tihomir S. Hristov for their advice on how to work with the FLIP dataset, which was kindly supplied to us by Drs. Jim Edson and Carl A. Friehe. Prof. John C. Wyngaard provided insightful comments at crucial times during the project. Finally, three anonymous reviewers provided a large number of helpful comments and questions, the answering of which led to significant improvements in the manuscript. The Office of Naval Research, through Grants N00014-93-1-0252 and N00014-96-1-0375 to the Pennsylvania State University, funded this work. REFERENCES Antonia, R. A., A. J. Chambers, C. A. Friehe, and C. W. Van Atta, 1979: Temperature ramps in the atmospheric surface layer. J. Atmos. Sci., 36, 99–108. Blackwelder, R. F., and R. E. Kaplan, 1976: On the wall structure of the turbulent boundary layer. J. Fluid Mech., 76, 89–112. Brown, R. A., 1980: Longitudinal instabilities and secondary flows in the planetary boundary layer. A review. Rev. Geophys. Space Phys., 18, 683–697. Buell, C. E., 1975: The topography of empirical orthogonal functions. Preprints, Fourth Conf. on Probability and Statistics in Atmospheric Sciences, Tallahassee, FL, Amer. Meteor. Soc., 188–193. Cattell, R. B., 1966: The scree test for the number of factors. Multivariate Behav. Res., 1, 35–49. Davison, D. S., 1974: The translation velocity of convective plumes. Quart. J. Roy. Meteor. Soc., 100, 572–592. Draper, N. R., and H. Smith, 1981: Applied Regression Analysis. Wiley, 709 pp. Edson, J. B., A. A. Hinton, K. E. Prada, J. E. Hare, and C. W. Fairall, 1998: Direct covariance flux estimates from mobile platforms at sea. J. Atmos. Oceanic Technol., 15, 547–562. Frisch, A. S., and J. A. Businger, 1973: A study on the convective elements in the atmospheric surface layer. Bound.-Layer Meteor., 3, 301–328. Greenhut, G. K., and S. J. S. Khalsa, 1982: Updraft and downdraft events in the atmospheric boundary layer over the equatorial Pacific Ocean. J. Atmos. Sci., 39, 1803–1818. ——, and ——, 1987: Convective elements in the marine atmospheric boundary layer. Part I: Conditional sampling techniques. J. Climate Appl. Meteor., 26, 813–822. Haack, T., and H. N. Shirer, 1992: Mixed convective–dynamic roll vortices and their effects on initial wind and temperature profiles. J. Atmos. Sci., 49, 1181–1201. Hall, F. F., Jr., J. C. Edinger, and W. D. Neff, 1975: Convective plumes in the planetary boundary layer investigated with an acoustic sounder. J. Appl. Meteor., 14, 513–523. Hendrickson, A. E., and P. O. White, 1964: Promax: A quick method to oblique simple structure. Br. Stat. Psychol., 17, 65–70. Hristov, T. S., S. D. Miller, and C. A. Friehe, 2000: Linear timeinvariant compensation of cup anemometer and vane inertia. Bound.-Layer Meteor., 97, 293–307. Johansson, A. V., and P. H. Alfredsson, 1982: On the structure of turbulent channel flow. J. Fluid Mech., 122, 295–314. Kaimal, J. C., and J. A. Businger, 1970: Case studies of a convective plume and a dust devil. J. Appl. Meteor., 9, 612–620. ——, J. C. Wyngaard, D. A. Haugen, O. R. Cote, Y. Izumi, S. J. Caughey, and C. J. Readings, 1976: Turbulence structure in the convective boundary layer. J. Atmos. Sci., 33, 2152–2169.

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