Underlying structures in orographic flow: An analysis of measurements made on the Isle of Arran

Transcription

1 Q. J. R. Meteorol. Soc. (26), 132, pp doi: /qj Underlying structures in orographic flow: An analysis of measurements made on the Isle of Arran By R. R. BURTON 1,S.B.VOSPER 2 and S. D. MOBBS 1 1 Institute for Atmospheric Science, University of Leeds, UK 2 Met Office, Exeter, UK (Received 6 September 25; revised 7 December 25) SUMMARY In a recent field experiment on the Isle of Arran, Scotland, measurements of surface pressure and near-surface wind fields were made along a roughly west east transect on the hill Tighvein. A principal components analysis of these measurements has been made to determine any significant atmospheric modes of variability, and this (objective) method successfully identifies several physically meaningful flows. KEYWORDS: Flow separation Orography Principal components analysis 1. INTRODUCTION Principal components analysis (PCA) is a widely used technique in the atmospheric sciences and is used to determine significant modes of variability and important underlying structure (Preisendorfer 1988; Wilks 1995). For a review of the method in meteorology and an extensive list of references, see Richman (1986). Hitherto, most applications of PCA have been to large spatial (e.g. northern hemisphere) and temporal scale (e.g. tens of years) datasets. Notable exceptions are scarce. Chen and Li (1995) applied PCA to near-surface pressure and wind measurements taken during the Taiwan Area Mesoscale Experiment (TAMEX), based on and near the island of Taiwan, a country with significant orography. By comparison with previous synoptic, nonstatistical studies of the TAMEX data, Chen and Li showed that the results of their PCA could be interpreted in a physically meaningful manner. Also, DíazdeArgandoña et al. (23) applied PCA to wind and pressure structures across the Pyrenees; this was a synoptic-scale treatment, using daily means of the field variables. More recently, Ludwig et al. (24) used PCA to analyse wind patterns in valleys. This latter study is closest to what is presented here, although their study sought to identify diurnal variations in empirical orthogonal function intensity and thermally induced flows. Thus, there is a precedent for applying PCA to much smaller spatial and temporal scales than is commonly the case, although the sample rate and spatial resolution in the studies mentioned above were too low to resolve features such as the detailed nature of the pressure field across a hill. The data analysed here were originally sampled at 3-second intervals and then subsequently post-processed into 1-minute averages; thus they have the advantage of being sampled at a relatively high rate (Vosper et al. 22, hereafter V2) and thus short-lived features can be captured by the PCA. In particular, events such as adverse pressure gradient episodes (i.e. with a pressure gradient which opposes the flow, and found to be quite transient in this experiment) may be observed. The structure of this paper is as follows. In section 2 the field experiment is described; the results from the PCA are given in section 3, and section 4 contains a discussion and summary of this work. Corresponding author: Institute for Atmospheric Science, School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK. ralph@env.leeds.ac.uk c Royal Meteorological Society,

2 1458 R. R. BURTON et al. Figure 1. (a) The Isle of Arran and the location of the hill Tighvein. (b) Orography of Tighvein, showing the locations of the eight measurement sites. Terrain heights in both (a) and (b) are shown in metres. (Note that the numbering is slightly different to that in V2.) In both figures the direction of true north is indicated. 2. THE ARRAN FIELD EXPERIMENT (a) General background In late 1997 and early 1998, two separate three-week field campaigns took place on the Isle of Arran, located off the west coast of Scotland. These experiments were designed to improve understanding of orographic flows, by making measurements of surface pressure and 2 m wind speed across Tighvein ( N, 5 1 W), an 8 km wide hill with a summit height of 458 m (see Fig. 1). This height is typical of the hills to the south of the island, whereas in the north the topography reaches a maximum of 874 m. The upper slopes of Tighvein are covered in heather; below about 25 m the terrain is partially forested. The approaches to the summit are fairly gentle with typical slope values of.5; close to the summit the orography steepens and slope values are approximately.25. Eight stations were positioned along a roughly west east transect across the hill (the experiment was mainly concerned with westerly flow). They were in continuous operation for the periods 27 October 1997 to 17 November 1997, and 5 March 1998 to 28 March At each station was a 2 m mast holding a cup anemometer and wind vane, and a microbarograph to make highly sensitive surface pressure measurements. From an analysis of the measurements from the campaign, V2 concluded that the flow across the hill could be classified according to the upstream Froude number (see below as to how this was calculated). The type of flow actually occurring was found to depend upon the Froude number F = U/NL, whereu is a representative wind speed, N is a characteristic Brunt-Väisälä frequencyandl the horizontal length scaleof the hill. The pressure field across the hill for F>.25 was found to contain a pressure minimum at the summit of the hill, consistent with weakly stratified or neutral flow over orography. For F.25 the pressure was found to be negative for greater distances downwind of the summit than for the F>.25 case and this represents a high-drag, or gravity-wave, surface pressure field. However, the calculation of the Froude number itself is somewhat subjective and the values obtained, noisy. The use of objective PCA techniques is found to be a much

3 STRUCTURES IN OROGRAPHIC FLOW 1459 easier way of classifying the flows over Tighvein, as shown below. Moreover the PCA analysis can be related to the findings of V2. (b) Technical background The technical aspects of instrument calibration and accuracy were fully covered in V2 and so will not be repeated here. We will, however, briefly mention how the synoptic and hydrostatic components of the pressure signal were removed. For each tenminute segment of data, the synoptic pressure p S was simply estimated to be the average pressure derived from all eight microbarographs. The calculation of the hydrostatic pressure contribution δp is rather more complicated. During periods when the wind speed was small at all sites the difference between the average pressure at a particular site and the synoptic pressure is taken to be caused by the height above sea level for that site (since the flow-induced component of pressure is small due to the low winds). Taking an average over as many low-wind situations as is feasible was found to give a good estimate for δp. Then, the flow-induced pressure contribution p to the actual measured pressure p is given by p = p p S δp. Throughout the following discussion, it is this flow-induced contribution we are concerned with. For brevity we shall simply refer to the pressure, but it should be remembered we are always dealing with a pressure perturbation. Itisalsoworth mentioning that the Froude numbers used in V2 were calculated from data from twicedaily radiosonde releases made at Machrie (on the west coast of Arran, approximately 5 km north of Tighvein.) The Froude numbers were calculated from the value of U at the middle-layer height (Hunt et al. 1988) and mean values of N below that height; the length scaling L was calculated by a best geometric fit to the shape of Tighvein. 3. RESULTS OF THE ANALYSIS (a) PCA of pressure (i) Data treatment. A common time series was compiled containing synchronized pressures and winds at the eight sites. This time series was then subjected to a PCA. We shall concentrate here on the second phase of the experiment, 5 March 1998 to 23 March 1998, during which time the technical problems encountered in the first phase had been corrected, and during which a large percentage of the observed flows were from the westerly direction. Cases where the wind direction at site 8 was between 26 and 3 were selected (the same directional spread as studied in V2), giving 725 ten-minute samples, or approximately 12 hours of data. (ii) The underlying structures. The mean pressure field is shown in Fig. 2. The distribution is near-symmetric with a minimum at the summit. V2 showed that this mean pressure is composed of two parts: a gravity-wave surface pressure field and a weakly stratified (or neutral) flow (see above, section 2(a)). The proportion of variance explained by each empirical orthogonal function (EOF) is shown in Table 1. From this, and according to conventional PCA wisdom, only the first two EOFs can be said to represent a signal; all the rest are noise. Thus, objectively, these are the governing regimes, as determined by the analysis. Both EOF1 and 2 are The language of PCA is not consistent in the literature. In this paper, EOFs refer to spatial patterns in the data; principal component coefficients refer to temporal variations in the data.

4 146 R. R. BURTON et al. TABLE 1. P ERCENTAGE OF THE VARIANCE IN THE DATA EXPLAINED BY EACH EOF EOF (pressure) no. POVE (%) <1 8 <1 POVE = proportion of variance explained. mean pressure perturbation (Pa) height /m (a) x /km EOF value (Pa) height /m (b) EOF 1 EOF x /km 2 15 (c) PC 1 PC 2 1 PC coefficient time (hr) Figure 2. (a) The mean flow-induced pressure perturbation (Pa), together with the orography along the transect; (b) the first two EOFs for pressure, with the orography; (c) the time series of the first two PC coefficients for pressure. significant, in the sense that they satisfy various objective selection criteria and together explain nearly 9% of the variance in the data. Technically, (1) they both satisfy Rule N (Preisendorfer 1988), based on a set of 1 Monte-Carlo simulations, and (2) they both satisfy Kaiser s rule with T = 1 (Kaiser 1959). Criteria (1) and (2), despite the unilluminating terminology, are standard in the literature of PCA. Rule N seeks to compare the spread of eigenvalues with that from a matrix composed of purely random data. Many such selection rules exist, including the rather more subjective scree test (Cattell 1966), which also points to EOF1 and 2 forming the signal. The ultimate test, and proof that these two EOFs are important, however, is that they can be assigned physically meaningful interpretations.

5 STRUCTURES IN OROGRAPHIC FLOW 1461 From these selection rules, the first two EOFs represent, to a large extent, the dominant structures embedded in the data and are worthy of inspection. The first two EOFs are shown in Fig. 2(b) where, following convention, we have scaled the EOFs such that the magnitude of the EOF vector is equal to unity; the time series of principal component (PC) coefficients is shown in Fig. 2(c). It can be seen that EOF1 (with a positive PC coefficient) corresponds to a highdrag pressure pattern, the direction of the force determined by the sign of PC 1.EOF1 is qualitatively similar to the mean pressure perturbation value (Fig. 2(a)) and can be seen as reinforcing/reducing the mean pressure, depending upon the sign of the first PC coefficient. EOF2 (with a positive coefficient) represents a low-drag state, with a pressure minimum at the summit, accompanied by a maximum in wind speed (see section 3(b) for a discussion of the PCA of winds). The type of flows associated with these two EOFs would correspond, in a qualitative sense, to the two types of pressure field found in V2. The lower and higher F regimes correspond to EOF1 and 2, respectively. In general, it is usually the high-drag component (with higher pressure upstream), in combination with a neutral-type component, which prevail. The adverse pressure gradient component, where the pressure is higher downstream than upstream (corresponding to a negative PC coefficient for EOF1), is often present in the signal though, albeit buried ; when it dominates the other signals, we might expect to have interesting flow patterns, and this will be explored further below. In the sections which follow, use is made of the relative PC coefficients. Consider forming a normalized PC coefficient for the ith EOF (i = 1or2)attimet, S(PC P,i (t)) where the subscript P refers to pressure: S(PC P,i (t)) = S(PC P,i (t)) S(PC P,1 (t)) + S(PC P,2 (t)) where S(PC P,i ) is the PC coefficient for the ith EOF. Then S(PC P,i (t)) will determine the contribution of the ith EOF to the overall flow at time t; thus, if S(PC P,i (t)) is large (small), then the ith EOF makes a significant (insignificant) contribution to the flow at that time. Note that in the definition of S(PC P,i (t)), the summation in the denominator is carried over only the first two principal component coefficients, since the first two EOFs explain nearly 9% of the variance in the data. In what follows, the time-dependence of S is implicitly assumed, for ease of notation. (iii) Correlations with wind speed. It is instructive to consider the evolution of the pressure field during a period of extended slack lee-side flow. Such a period is shown in Fig. 3, which shows the measured 1-minute-averaged wind vectors for times hours. At the beginning of this episode, the flow pattern follows broadly the neutral-type flow characterized by EOF2 (see the inset figures); with increasing time, this pattern is seen to be accompanied by relatively high pressures downwind of the summit, until at the end of the episode there is a significant adverse pressure gradient. As in V2, an upstream reference wind speed U REF is taken, the zonal wind speed at the most upstream measurement location (site 8). The relationship between the normalized westerly surface wind component U 1 /U REF at site 1 (the most downstream site) and the normalized first PC coefficient for pressure during this episode is shown in Fig. 4. There is clearly a robust correlation between these two normalized parameters (the correlation coefficient r =.98). During periods when EOF1 is negative and makes

6 1462 R. R. BURTON et al. Figure 3. Observed wind vectors for the times (a) 33, (b) 38, (c) 311 and (d) 313 hours. The shaded contours have the same values as in Fig. 1. Inset into each plot is the corresponding pressure perturbation, P, across the hill. The summit of Tighvein is located at x = km. a significant contribution to the overall flow (i.e. large, negative S(PCP,1 )) the normalized zonal wind speed is correspondingly low: the slackened lee-side flow prevails here and appears to be associated, in a linear manner, with the adverse pressure gradient. This slackened lee-side flow is potentially accompanied by lee-side separation (Mason and King 1984; Wood 1995; Doyle and Durran 22), though whether this is truly the case is impossible to tell given the absence of measurements away from the surface. As will be seen in the next section, this relationship is confirmed when considering the PCA of zonal winds. It would be a natural and logical step to examine the relationship between the upstream Froude number and the PC coefficients during this event (and times shortly before and after it.) However, the sparsity of radiosonde releases during this period (three sondes were released during hours ) means that the comparison is inconclusive. Even after interpolating (the typically noisy) sonde data onto the 1-minute

7 STRUCTURES IN OROGRAPHIC FLOW U 1 / U REF S(PC P,1 ) normalised parameter S(PC P,1) U 1 / UREF time (hr) Figure 4. Normalized downstream wind speed, U 1 /U REF and normalized PC 1 for the surface pressure for an extended period of slack flow. The inset figure is a scatter plot of the two parameters; the solid line represents the best linear regression, with a correlation coefficient of r =.98. PC data, the amount of scatter was found to be very large, and the relationship indeterminable. (b) PCA of winds We now briefly present the PCA of zonal wind component U and show that the structures in the U wind can be related to structures in pressure. The mean zonal winds are shown in Fig. 5(a). There is a maximum at the summit, and downwind of the summit the winds are relatively lower than those at the most upwind sites. This is consistent with the mean pressure measurements (Fig. 2). Table 2 shows the proportion of variance explained by each eigenvalue; the significant EOFs are shown in Fig. 5(b) for positive PC coefficients and the time series of PCs in Fig. 5(c). The first two EOFs explain 95% of the variance in the data and can be considered as representing the underlying signal. They satisfy the same objective selection criteria as discussed in section 3(a)(ii). As for pressure, the first EOF can be seen as increasing/reducing the mean profile, depending upon the sign of the first PC coefficient. The second EOF is negative downstream of the summit; thus we would expect this component to be relatively large (with a positive PC coefficient) during a slackened flow episode. As shown by Fig. 6, the second normalized PC for U is well correlated (r =.97) with the first PC coefficient for pressure for the slack flow episode (hours ). As can be seen, when S(PC P,1 ) is large and negative, S(PC U,2 ) is large and positive; illustrating again the link between slackened flow and the adverse pressure gradient. The proportion of variance explained by the first EOFs for wind and pressure are different: this is due to the fact that EOF1 (pressure) frequently has a coefficient of either sign and represents both the high-drag state and the reversed flow state. However EOF2 (wind) is generally positive and corresponds to the slack lee-side flow case only, which occurs relatively infrequently.

8 1464 R. R. BURTON et al (a).4 (b) mean zonal wind (ms -1 ) EOF value (ms -1 ) EOF 1 EOF 2 height /m x /km height /m x /km 6 5 (c) PC 1 PC 2 4 PC coefficient time (hr) Figure 5. (a) The mean zonal wind speed (m s 1 ), together with the orography along the transect; (b) the first two EOFs for U, with the orography; (c) the time series of the first two PC coefficients for U. TABLE 2. P ERCENTAGE OF THE VARIANCE IN THE DATA EXPLAINED BY EACH EOF EOF (U) no. POVE (%) <1 6 <1 7 <1 8 <1 POVE = proportion of variance explained. Note that we might expect flow separation episodes to occur when S(PC U,2 ) is large and positive. This may be accompanied by a change in turbulence characteristics at the downwind sites: for example, Mobbs et al. (25) showed that the downwind standard deviation of the wind field, normalized by the magnitude of the mean downwind wind vector, was high during periods of flow separation (for data collected on hills in the Falkland Islands). However this is beyond the scope of this paper; and determining if the flow had truly separated would be impossible anyway. All of the measurement sites considered here were situated at the surface in an approximately straight line, and would be unlikely to capture such complex, three-dimensional structures as occur during periods of flow separation.

9 STRUCTURES IN OROGRAPHIC FLOW r =.97.2 S(PC P,1 ) S(PC U,2 ) Figure 6. The second (normalized) PC coefficient for U against the first (normalized) PC coefficient for P. The straight line shows the best fit to the data. 4. CONCLUSIONS To categorize a large quantity of measured data into fundamental structures, without recourse to PCA, can be a daunting task indeed. The data studied here give an example of how the PCA method can be used to classify flow regimes derived from measured orographic flows. The principal components analysis revealed three basic flow patterns, derived from two EOFs: (a) neutral-type flow; (b) high-drag flow and (c) a flow similar in structure to (a), but with higher pressure upstream than downstream (adverse pressure gradient). Flow patterns (a) and (c) are distinguished by the sign of the first PC coefficient. Note that it is beyond the scope of this paper to identify what determines which flow regime will occur. This would require further study, perhaps involving relating the PCA analysis to the upstream conditions. It seems possible, however, that the occurrence of regimes (a) and (c) might depend on the phase of gravity waves generated by Tighvein or hills located upwind. During the major and sustained lee-side slackened flow event which occurred during the experiment, the adverse pressure gradient was seen to be associated with a slackening of the downstream zonal wind, when considering both normalized wind speeds and PC coefficients. The extremely good linear correlations obtained suggest a robust relationship; the occurrence of separation downwind of the summit could well be occurring during these adverse pressure gradient events, but is difficult to ascertain. Furthermore, this study has identified two significant flow regimes which can be related qualitatively to the two regimes found in V2 without knowledge of, or reference to, the upstream wind and stability parameters: the method does not rely upon arbitrarily filtering the data based upon carefully chosen values of F. This work also provides a powerful model validation tool. If an extended series of numerical model runs, using a realistic orography dataset and appropriate initial conditions, cannot reproduce the EOFs found here (and in the correct relative proportions), then the model may not be accurately describing the underlying dynamical processes.

10 1466 R. R. BURTON et al. ACKNOWLEDGEMENTS During the course of this work Ralph Burton was sponsored by the Natural Environment Research Council, under grant number NER/A/S/21/514. The authors would like to thank Francis L. Ludwig and an anonymous referee for many helpful comments and suggestions which have improved this paper. REFERENCES Cattell, R. B The scree test for the number of factors. Multivariate Behav. Res., 1, Chen, Y.-L. and Li, J Characteristics of surface airflow and pressure patterns over the island of Taiwan during TAMEX. Mon. Weather Rev., 123, Díaz de Argandoña, J., Ezcurra, A. and Bénech, B. 23 Surface pressure disturbance in the Ebro valley (Spain) produced by the Pyrenees mountains during PYREX. Q. J. R. Meteorol. Soc., 129, Doyle, J. D. and Durran, D. R. 22 The dynamics of mountain-wave-induced rotors. J. Atmos. Sci., 59, Hunt, J. C. R., Leibovich, S. and Richards, K. J Turbulent shear flows over low hills. Q. J. R. Meteorol. Soc., 114, Kaiser, H. F Computer program for varimax rotation in factor analysis. Educational and Psychological Measurement, 19, Ludwig,F.L.,Horel,J.and Whiteman, C. D. 24 Using EOF analysis to identify important surface wind patterns in mountain valleys. J. Appl. Meteorol., 43, Mason, P. J. and King, J. C Atmospheric flow over a succession of nearly two-dimensional ridges and valleys. Q. J. R. Meteorol. Soc., 11, Mobbs, S. D., Vosper, S. B., Sheridan, P. F., Cardoso, R., 25 Observations of downslope winds and rotors in the Falkland Islands. Q. J. R. Meteorol. Soc., 131, Burton, R. R., Arnold, S. J., Hill, M. K., Horlacher, V. and Gadian, A. M. Preisendorfer, R. W Principal component analysis in meteorology and oceanography. Elsevier, Amsterdam Richman, M. B Review article: Rotation of principal components. J. Climatol., 6, Measurements of the near-surface flow over a hill. Q. J. R. Meteorol. Soc., 128, Vosper, S. B., Mobbs, S. D. and Gardiner, B. A. Wilks, D. S Statistical methods in the atmospheric sciences. Academic Press, San Diego, CA Wood, N The onset of separation in neutral, turbulent flow over hills. Boundary-Layer Meteorol., 76,