Directional Correlation Coefficient for Channeled Flow and Application To
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FEBRUARY 1998 KAUFMANN AND WEBER 89 Directional Correlation Coef®cient for Channeled Flow and Application to Wind Data over Complex Terrain PIRMIN KAUFMANN AND RUDOLF O. WEBER Paul Scherrer Institute, Villigen, Switzerland (Manuscript received 23 April 1996, in ®nal form 26 May 1997) ABSTRACT Analysis of vector quantities or directional data, such as the variables characterizing ¯ow, is of signi®cant interest to geophysical ¯uid dynamicists. For ¯ows with strong channeling, a new simple correlation coef®cient is de®ned. It is demonstrated by application to a model of channeled ¯ow that the new correlation captures the ¯ow features in the case of channeling better than other correlations taken from the literature. The new correlation coef®cient is applied to wind data from a mesoscale network of anemometers in complex terrain. A cluster analysis based on the correlation matrix is used to group observation sites into classes with similar behavior of the channeled ¯ow. Sites within the same class are not necessarily geographically close. A similar behavior of the wind directions indicated by these classes seems to be more closely related to the orographic features and to the altitude of the sites than to the horizontal distance between them. 1. Introduction correlations enables one to see which valleys or areas experience the same forcing of the wind. A correlation Meteorologists and oceanographers have dealt with coef®cient for wind direction of such channeled ¯ows the problem of correlating vector quantities for at least should have some special properties. When winds in 80 years (Dietzius 1916; Sverdrup 1917; Breckling two different valleys are directed down-valley, the cor- 1989; Hanson et al. 1992), and statisticians also tackled relation between the winds in the valleys should be high. the problem (Fisher 1993). A vector quantity requires When the ¯ow in one valley is in the down-valley di- both magnitude and direction for its unique character- rection and in the up-valley direction in another valley, ization. When the vector is represented by its compo- the correlation between the winds of the two valleys nents in a coordinate system like the Cartesian system should be negative. If no simultaneous channeling in or spherical coordinates, correlation coef®cients can be the two valleys is observed, the correlation should be- de®ned using these components. Many de®nitions of a come close to zero. A simple correlation coef®cient sat- single scalar value describing the correlation of vector isfying these requirements is proposed here. quantities have appeared in the literature (see the re- In section 2, several de®nitions of directional and views in Breckling 1989; Hanson et al. 1992; Crosby vector correlation coef®cients are reviewed. Section 3 et al. 1993). When the magnitude of the vector is ignored introduces a new de®nition of correlation for channeled and its direction alone is studied (which is equivalent ¯ows, which makes use of the speci®c properties of to considering vectors of unit length), problems arise channeled ¯ows. The different correlation coef®cients because the direction is a circular variable (Mardia 1972; discussed in sections 2 and 3 are compared in section Essenwanger 1986; Fisher 1993). Several de®nitions of 4 by application for an idealized situation of two wind correlation coef®cients for circular variables have been vectors showing pronounced channeling. In section 5, published (a review is given in Hanson et al. 1992). the new correlation coef®cient for channeled ¯ows is In the present paper, the highly channeled near-sur- applied to wind observations from a mesoscale ®eld face ¯ow of the atmosphere in a region with many val- experiment with the objective being to identify groups leys is the focus of attention. The determination of a of measurement sites with similar behavior of wind di- correlation coef®cient between the wind measurements rections. at different station locations allows us to compare the ¯ow in the different valleys. The examination of these 2. Review of some directional and vector correlation coef®cients Corresponding author address: Dr. Rudolf O. Weber, Paul Scherrer In the analysis of channeled wind, as discussed in Institute, CH-5232 Villigen PSI, Switzerland. section 5, we are only interested in wind direction but E-mail: [email protected] not in wind speed or the magnitude of the wind vector. q 1998 American Meteorological Society Unauthenticated | Downloaded 09/30/21 01:59 PM UTC 90 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 15 Therefore, both directional and vector correlation co- This correlation coef®cient takes values from 0 to 1. ef®cients (applied to vectors of unit length) are suitable Hanson et al. (1992) de®ne a variant of this vector cor- in our case. relation coef®cient with a sign factor, det(S12)/|det(S12)|. Among the variety of proposed directional and vector If the products p(x, y) in (4) in Sverdrup's de®nition correlation coef®cients (Hanson et al. 1992 list 17 def- (5) are replaced by the centered covariances s(x, y)in initions) we chose four different de®nitions (Sverdrup (1), the expression (7) of Breckling (1989) is recovered. 1917; Fisher and Lee 1983; Breckling 1989; Crosby et Crosby et al. (1993) discuss a vector correlation co- al. 1993). Let W1 5 (u1, y 1) and W 2 5 (u 2, y 2) be two, ef®cient de®ned by two-dimensional vectors representing the horizontal r 5 {tr([S ]21S [S ]21S )}1/2, (8) wind vector at two measurement sites. The covariance CBG 11 12 22 21 s(u1, u1) and the cross-covariance s(u1, u 2) are de®ned which is essentially the de®nition given by Hooper in the standard way as (1959) and which was further developed by Jupp and Mardia (1980). The squared correlation coef®cient s(x, y) 5 E[xy] 2 E[x]E[y], (1) 2 rCBG is the sum of squares of the canonical correlations where E[x] is the expectation value of the random vari- (Crosby et al. 1993) and ranges from 0 to 2. We use in able x. To simplify the equations later, the following the following a normalized form, covariance matrices are introduced, according to the no- /2, (9) tation of Crosby et al. (1993). rC 5 rCBG Ï which takes values from 0 to 1. Breaker et al. (1994) s(u11, u ) s(u 11, y ) studied signi®cance tests of this vector correlation. Han- S11 5 ; 12s(y 11, u ) s(y 11, y ) son et al. (1992) summarize and discuss in detail the invariance properties of these and many other correla- s(u12, u ) s(u 12, y ) tion coef®cients. S12 5 , etc. (2) 12s(y 12, u ) s(y 12, y ) 3. Directional correlation coef®cient for channeled In the same way the product matrices P11, etc, are de®ned as ¯ow All of the de®nitions in the last section apply to any p(u11, u ) p(u 11, y ) type of ¯ow and do not make use of any speci®c prop- P11 5 , etc., (3) 12p(y 11, u ) p(y 11, y ) erties of the ¯ow. In contrast, the correlation for chan- where the uncentered product moments p(x, y)oftwo neled ¯ow as de®ned in this section incorporates the random variables x and y are de®ned by properties of channeled ¯ow into the de®nition of the correlation coef®cient. This speci®c correlation coef®- p(x, y) 5 E[xy]. (4) cient becomes better suited for the channeled ¯ows but One of the oldest de®nitions of a vector correlation was is not applicable to other types of ¯ow. given by Sverdrup (1917). He de®ned a correlation by The near-surface winds over complex terrain are often channeled by valleys, even showing countercurrents to 1/2 [tr(P )]221 [p(u , y ) 2 p(y , u )] the geostrophic ¯ow (Wippermann and Gross 1981; r 5 12 1 2 1 2 , (5) S tr(P )tr(P ) Wippermann 1984; Whiteman and Doran 1993). In 5611 22 smaller valleys, often thermally induced ¯ows develop where tr(A) denotes the trace of matrix A. Sverdrup (for a review see Whiteman 1990). To compare wind- (1917) stresses that the uncentered product moments direction data from different valleys, it is desirable to p(x, y) in (4) must be taken and not the centered co- have a correlation coef®cient that indicates whether up- variances s(x, y) in (1). or down-valley ¯ow prevails in both valleys. Figure 1 Fisher and Lee (1983) developed a directional cor- shows the wind rose of a station in the Rhein Valley relation coef®cient ranging from 21 to 1. A represen- east of Basel (station E1, see section 5 for more details). tation of their correlation coef®cient in terms of the Two preferred directions (southeasterly and westerly matrices de®ned above is given in Fisher and Lee (1986) winds) are evident. This is a good example of a distri- and Breckling (1989): bution of directions in a ¯ow that is usually referred to det(P ) as channeled. We term one of them the main wind di- r 5 12 , (6) rection (southeasterly in this case) and the other the FL 1/2 [det(P11)det(P 22)] secondary wind direction (westerly in this case). It where det(A) denotes the determinant of matrix A. should be noted that these two dominant wind directions Breckling (1989) proposed the following correlation are, however, not 1808 apart. For the de®nition of a coef®cient: correlation coef®cient, each wind direction is assigned to the main or the secondary wind direction, or to the tr([SS])1/2 r 5 12 21 .