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نویسنده

  • Wiin Christensen
چکیده

A new `Wavelet Energeti s' te hnique, based on `best-shift' orthonormal wavelet analysis (OWA) of a given instantaneous synopti map, is onstru ted for diagnosing nonlinear kineti -energeti transfers in observed blo king ases. Best-shift analysis represents longitudinal varian e of latitude-mean blo king stru tures at 50 kPa mu h more eÆ iently than does standard OWA, even for stru tures evolving in time although the analysis is optimized to the time-average. The ve blo king events omprise two ategories, respe tively dominated by zonal-wavenumber sets f1g and f1; 2g. Further OWA of instantaneous residual nonblo king stru tures, ombined with new `nearness' riteria, yield three more orthogonal omponents, representing smaller-s ale eddies near the blo k (upstream and downstream) and distant stru tures. Su h omponents ould not be obtained by simple Fourier analysis. Eddy patterns apparent in these omponents' ontours suggest inferring geostrophi energeti intera tions, but the omponent Rossby numbers seem too large for the inferen e to bear out. However a new result enabled by this method may be the instantaneous attribution of blo king strain eld e e ts to parti ular energeti ally intera tive eddies, onsistent with Shutts' hypothesis. Su h attribution has only been possible before in simpli ed models or in a time-average sense. In four out of ve blo ks, the upstream eddies feed kineti energy to the blo k, whi h in turn, in three out of four ases, transmits kineti energy to the downstream eddies. The small set of ases allows no statisti ally signi ant on lusions. The appendi es link loworder blo king stru ture and dynami s to some prin iples of wavelet design, and propose a new intera tion diagnosis, similar to E-ve tor analysis but instantaneous. -21. Introdu tion Atmospheri blo king is thought to involve the nonlinear intera tion of a range of s ales (Hansen and Sutera 1984; Shutts 1986; Haines and Malanotte-Rizzoli 1991; Wiin Christensen and Wiin-Nielsen 1996). Blo ks are lo alized stru tures, whi h are not well represented by trun ated Fourier series, sin e the Fourier oeÆ ients of longitude-lo alized stru tures are not wavenumber-lo alized. This suggested to Fournier (1999: \Atmospheri energeti s in the wavelet domain I: Governing equations and interpretation for idealized ows," submitted to J. Atmos. S i., hereafter F2) to translate the traditional Fourier analysis-based atmospheri energeti s of Saltzman (1957) into a periodi orthonormal Wavelet Energeti s. In analyzing observed Northern Hemisphere data from the National Meteorologi al Center, Fournier (1999) and Fournier (1999: \An introdu tion to orthonormal wavelet analysis with shift invarian e: Appli ation to observed atmospheri -blo king spatial stru ture," submitted to J. Atmos. S i., hereafter F1) found that the se ond-largests ale (east and west hemisphere) wavelet oeÆ ients of a multiresolution analysis of geopotential height eld Z( ; '; t) on the 70 kPa isobari surfa e ould a ount for a signi ant amount of the blo king state, as that state was de ned. In the present paper are dis ussed several re nements to the te hniques of Fournier (1999) and Fournier (1999: \Atmospheri energeti s in the wavelet domain II: Time-averaged observed atmospheri blo king," submitted to J. Atmos. S i., hereafter F3). Se tion 2a des ribes how the `best-shift' orthonormal wavelet analysis (OWA) algorithm (reviewed by F1) may be applied to yield a better representation of the stru ture of atmospheri blo king than the standard OWA. How well that representation performs instantaneously is tested in se tion 2b. In se tion 2 maps for instantaneous geopotential height observations at 50 kPa are partitioned into four elds, whi h represent the blo k stru ture, nearby east and west eddies, and the residual stru tures. From this partition, parti ular energeti intera tions between pairs of patterns may be quanti ed, by -3partially summing the triad intera tions, des ribed by F2, involving that pair. Se tion 3 ontains the results and dis ussion of these analyses. Appendi es A and B present an interpretation of the blo king representation with regard to signal-pro essing and physi al points of view. Appendi es C and D mainly provide de nitions, while appendix E outlines a new diagnosis of synopti maps, in some ways similar to E-ve tor analysis. 2. Methods a. Representation of blo king by wavelets with best shift The representation of the blo king stru ture used by F1,3 may be re ned as follows. The top urves of Figs. 1a{j show the \b-average" hH?ib' ( ) over blo king time-interval T [as explained by Fournier (1996, 1999, F3)℄ of the 55-80ÆN mean (D2) hH?i' ( ; t) of the departure H?( ; '; t) from zonal mean hHi ('; t), where H( ; '; t) se ' hse i'Z( ; '; t) 2 4) hHi' ( ; t) = 1 11 11 X m=1Z( ; 'm+22; t)35 denotes the weighted geopotential height at 50 kPa. The fa tor se ' an els the areal weight os' in h i' (D2). The next urve below is the part of hH?ib' re onstru ted from the ten largest-magnitude best-shift OWA omponents Df H e j;kEb'W e j;k [ oeÆ ients Df H e j;kEb' times orthonormal wavelet basis fun tion W e j;k( )℄. This urve is labeled by the per entage of varian e ontributed by those ten omponents. Below follow urves for ea h of those individual omponents, for both the standard [ e = (0; 0), odd-lettered gures℄ and best shift ( e = e+, even-lettered gures) OWA, as explained by F1. The omponents are sorted by Df H e j;kEb' , des ending, and labeled by per entage of varian e ontributed, and by longitude resolution and lo ation indexes (j; k 1). All fun tions are b-averaged over one blo king event in the Pa i (Figs. 1e{f), or individually over ea h of four blo king events in the Atlanti (other letters). The dates for ea h T may be inferred from Table 1. -4b. How well is this representation maintained as time evolves? The top panels of Fig. 3 show Hovm oller (1949) diagrams of unanalyzed hH?i' ( ; t). The light regions learly depi t the blo king stru ture (verti al shape), and some progressing synopti y lone waves (upper-west to lower-east diagonal stru tures). However, interpretation of these top panels is not easy. . Partitioning into blo k and eddy stru tures Can the best-shift basis reveal more stru ture in hH?i' ( ; t)? In the lower two panels of Fig. 3, for ea h parti ular blo k the best shift e+ for hH?ib' ( ) (Figs. 1b{j, evenlettered gures) has been used in the OWA of hH?i' ( ; t) at ea h t. The middle and bottom panels of these gures depi t the elds generated by proje tion onto the basis elements whi h partition hH?ib' ( ) into ontributions from the four largest-magnitude f H e+ j;k b' (middle) and the residual (bottom). Denote this partition of the top panel into the sum of the two lower panels by Z? = Z + Ze; Z X (j;k)2B4 e Z e+ j;k W e+ j;k : (2.1) Z uses only best-shifted wavelets W e+ j;k with indexes from the `blo k set' B4 , where B0 ;, B +1 B [((j; k) f H e+ j;k b' = max (j0;k0)= 2B f H e+ j0;k0 b' ) (2.2) and Z and Ze Z? Z are orthogonal w.r.t. the zonal mean: DZ ZeE = 0. The symbol denotes the typi al blo k pattern ontour shape, dis ussed by F3, and e stands for `eddy'. The la k of argument ( ; '; p; t) means that (2.1) may be performed at any step of a linear analysis. In se tion 3 , maps of Z( ; ') at 50kPa on parti ular days are further analyzed. To prepare for this, se tion 2d ontains a summary of the energeti s formulated by F2, here applied to four orthogonal proje tions, alled modes, a 2 f ;E;W; rg M , the `mode -5set'. First, (2.1) is further analyzed by Ze = ZE + ZW + Zr: (2.3) The simultaneous spa e-and-s ale resolution of wavelets enables the nearby eddies east (downstream) of the blo k to be extra ted as ZE X (j;k)2E e Z e+ j;k W e+ j;k ; where the set E of nearby downstream wavelet indexes whi h are not part of the blo k set B4 (2.2) [nor (0; 1), sin e wavelet W0;1 is not lo alizable within a hemisphere℄ is E (j; k) = 2 B4 [ f(0; 1)g j;k; 3 ; j j > 0 : Parameters su h as 3 and the nal 0 are somewhat arbitrary, but were hosen to reasonably well de ne east and west wavelets, resolve straining e e ts (dis ussed below and by F2) and redu e blo king-se tor varian e in Zr. `Nearness' is measured by the displa ement ( ; 0) from longitude 0 to , des ribed in appendix C, along with the lo ation j;k of wavelet Wj;k, and the blo k rest longitude and blo k s ale and lo ation indexes j and k . The nearby eddies west (upstream) of the blo k are given in like manner by ZW X (j;k)2W e Z e+ j;k W e+ j;k ; W (j; k) = 2 B4 [ f(0; 1)g ; j;k 3 ; j j > 0 : The residual eddies are de ned to be Zr Ze ZE ZW. All modes are zero-mean and orthogonal: hZai = DZaZbE = 0 for distin t a; b 2 M . Sin e in di erent ases, di erent referen e origins o lo are used, where l h21 J (l 1) 1i and lo 1 + J 1 X j=0 j2J j 1 (2.4) (F1), Fig. 4 is provided. It shows the lo ation, s ale and shape of some of the important W e+ j;k in the polar proje tion. Note that for ea h of the \j = 1 blo ks" dis ussed in se tion 3a and appendix B, rows 1; 3; 5 of Fig. 4, the longitude of W e+ 1;2 extremum (min or max depending on sgn e Z e+ 1;2 in Figs. 1b,f,j) is aligned with the blo k rest longitude. -6Likewise, for both of the \j = 0 blo ks," rows 2; 4 of Fig. 4, the longitude of maximum W e+ 0;1 is aligned with the blo k rest. These extrema are labelled A,P for Atlanti and Pa i blo ks. The eight times sele ted for the energeti s analysis are those of Table 1 and F3. The best shift e+ orresponding to the interval ontaining ea h time was used for the blo king days. For omparison, the nonblo king, maximum-zonal-index times were partitioned (2.1,2.3) using the blo king-time best shifts orresponding to the se ond and fth of the blo king time-intervals. The best shift orresponding to the rst blo k was used for the minimum zonal index day. d. Energeti s For a map at any given time, on e the partition (2.1,2.3) has been al ulated and energeti s derived by F2 have been omputed using the OWA with appropriate +o , then it be omes possible to ask, what stru tures in Ze (2.3) feed or draw kineti energy (KE) from Z , and vi e versa. To address that question, energeti transfers are assigned to the maps by writing the KE transfer L jb to mode from any other mode b as de ned in appendix D. Polar maps of Z ( ; ') (0.3 hm intervals, gray lled ontours) and arrows onne ting Zb to Z showing L jb (D1) are arranged in a square matrix as shown in Fig. 5a. In this s hemati diagram the arrows are drawn as if all the L jb were positive. In the following gures negative L jb are drawn with reversed arrows. 3. Results: observed energeti s between blo k and east, west and other eddy modes a. Representation of blo king by wavelets with best shift In the Atlanti blo king ases, Figs. 1a{d,g{j, the largest-magnitude wavelet omponent an ontribute onsiderably more varian e to hH?ib', merely by hoosing the best shift +o (2.4). The resolution index j of that omponent does not hange due to the shift. Looking at the j-index of the largest-magnitude omponent, it appears that in -7the wavelet pi ture there are two types of blo ks: a j = 1 type (Figs. 1a{b,i{j), orresponding to a superposition mostly of wavenumbers n = 1; 2; and a j = 0 type (Figs. 1 {d,g{h), orresponding primarily to wavenumber n = 1. The Pa i blo k, Figs. 1e{f, was another j = 1 type. This ase shows the greatest improvement a orded by the best shift. Figure 2 shows the per entage of varian e ontributed umulatively by the four largest Df H e j;kEb' omponents, for the standard (lower sta ks) and best-shift (upper sta ks) OWA of the ve blo king events. The best shift o ers great improvement in every ase. In all but the se ond ase, only two optimally shifted wavelets ontribute more varian e than four or more standard wavelets. b. Persisten e of best shift For ea h blo k, the middle panels of Fig. 3 show that DH E' is dominated by steady (appearing verti al), high (light shaded) regions at the blo k lo ation. Therefore the basis elements whi h have been shifted by e+ to best represent the time-average blo k stru ture, also very well represent that stru ture instantaneously. Every bottom panel shows that hHei' is dominated by eastward progressing (diagonal) patterns. Thus the residual of the `blo k' basis elements apture the progressing, smaller-s ale, y loni wave or `eddy' part of the motion. As explained by F1, this joint lo ation-s ale resolution ould not be a hieved by simple Fourier-based methods. In all but one ase of blo king, DH E' ( ; t) a ounts for at least 84.5% of the ombined ( ; t)-varian e of hH?i'. The lower two panels' titles give varian eontribution per entages. The ex eptional ase is the fourth blo king event, Fig. 3d, in whi h \only" 69.4% is so a ounted. As noted by F3, this blo k underwent retrograde (westward) motion. It is therefore not surprising that a representation with xed `best referen e origin' +o is less su essful. -8. Instantaneous partition results Figures 5b{g show the instantaneous maps and asso iated energeti s for the eight times in Table 1 in hronologi al order. These maps show that OWA enables one to onstru t modes of any given eld whi h lo alize features in spa e and in s ale, whi h is not possible using standard Fourier analysis. Letting Ve b ub i e + vb je , and Eb 2 1 DVe b Ve bE denote the ontribution of b to the total KE, note that although Er is always omparable to E , nevertheless the Zr ontours always display less variability in the se tors of most intensely variable ZE + ZW. 1) WHAT MORE CAN THIS PARTITION SHOW BY MERE INSPECTION? Heuristi ally, one might hope to infer some energeti information from inspe tion of Z( ; ') maps as follows. The geostrophi velo ity Ve g (2 sin') 1 gke rfZ Ve (3.1) points along isopleths of Z, with speed proportional to isopleth loseness. An approximately ellipti al Zb isopleth implies a positive orrelation between ubg and vb g if the angle between the semi-major axis and i e falls in i0; 2 h, and a negative orrelation if falls in i 2 ; 0h. In appendix E a novel proof is introdu ed to show that the ovarian e is maximum when = 4 , under ertain assumptions. From that ovarian e may be inferred a northward adve tive transport of eastward geostrophi momentum of mode b, or equivalently an eastward transport of northward momentum. To an arbitrarily oriented ellipse in eld Zb, one may asso iate a transport in the dire tion Te about 45Æ to the left of its semi-major axis dire tion Ae , of a geostrophi momentum omponent in a dire tion Df Te ke to the right of Ae , or vi e versa. Depending on the sign of the gradient Te rf of the ow omponent Df Ve , mode will either re eive KE from b or else lose KE to b. This remark is in part a generalization to arbitrary modes b{ of the lassi al eddy{zonal-mean kineti energeti s dis ussion as -9appears in the literature [e.g. Saltzman 1957, p. 516b {3; Holton 1992, p. 340, seventh Eq.; Wiin-Nielsen and Chen 1993, p. 73 (7.22) & Fig. 6.3℄. However, the reliability of these inferen es depends on the quality of the geostrophi approximation (3.1) itself. That quality is measured by the smallness of the Rossby number. Here an appropriate de nition (Dutton 1986) of the Rossby number is the ratio of a hara teristi magnitude of Ve Ve g to a hara teristi magnitude of Ve . Typi ally the Rossby number is mu h less than unity for large-s ale atmospheri ows. In order to evaluate the appropriateness of the geostrophi approximation, and moreover its orollary energeti s inferen es, the Rossby number for ea h individual mode b 2 M is estimated as the ratio Ro R Ve b; Zb D Ve b Ve bg E ' D Ve b E ' of horizontal means. Ea h polar map of mode b in Figs. 5b{g is labeled by Eb and Ro. Ea h gure is labeled by t and the overall Rossby number R(Ve ; Z), whi h is almost always signi antly smaller than the Rossby number of any mode. Rossby numbers on the order of one or greater indi ate that the above inferen es from Zb stru ture to geostrophi wind and energeti s may not be reliable. The answer to the question titling this se tion is that the sign and very approximate relative magnitude of KE transfers L jb might be determined by inspe tion of overlapping Zb, Z ontours, but that this te hnique is severely limited by the quality of the geostrophi approximation, measured by the smallness of the mode Rossby numbers. It turns out not to be a onsistently reliable te hnique for the maps analyzed here. 2) ENERGETIC DIAGNOSTICS Figure 5b shows that the Northeastern Atlanti blo k pattern on 3 De ember, des ribed by Z (top left), was being energeti ally sustained by all three other modes. Most of the KE infusion omes from Zr (top right). Some eddies upstream of the blo k, -10in ZW (bottom left), were giving about 42% of the KE whi h Zr does. The upstream eddies nearest to the blo k, south and southwest of Greenland, show the meridional elongation brought about by the strain eld of the blo k, as dis ussed e.g. by Shutts (1986). In ontrast, the minimum zonal index day, 12 De ember (Fig. 5 ), shows a Z (top left) whi h somewhat resembled that on 3 De ember, but was losing KE to Zr and ZE, whi h had fed the blo k. F3 explains that this anomalous high did not persist long enough to be lassi ed as a blo k. Its energeti gain from mode W was similar to that on 3 De ember, so perhaps the loss to Zr + ZE was an important di eren e. [Other physi al e e ts su h as available potential energy (APE) transfer have not been omputed, partly be ause Hansen and Sutera (1984) found APE transfer to be less distinguishing for these same events.℄ Compare with these the very `zonal' (maximum zonal index) day, 18 De ember, Fig. 5d. The residual eddies Zr were feeding the blo k part Z 58% more KE than on 3 De ember. The lo al eddies ZE + ZW were storing 92% more KE, but together give 79% less to Z at this time. The pattern would hange in 36 h on 23 De ember, when the se ond Atlanti blo k would be formed, shown in Fig. 5e. It was more longitudinally steady and ompa t, judging from the Hovmoller diagrams shown in Fig. 3b and by F1. Z ex hanged very little KE with Zr at this time. Instead there was a more vigorous transfer from E to W both dire tly and through . The Pa i blo k (6 January, Fig. 5f) has the most vigorous transfer to Z from ZW of all eight ases. The relative strength of the Pa i blo k energeti s is also evident in the energeti s analyses of Hansen and Sutera (1984) and F3. Again, the W eddies show the straining e e ts due to the blo k. On 25 January (Fig. 5g) the blo k that underwent retrograde motion westward a ross the Atlanti was oming to an end. This was a ompanied by a dispensing of KE from -11Z to ZE and Zr. Westward eddies fed KE to the and r modes. Residual eddies re eived KE from every other mode. Finally, 19 (Atlanti blo k, Fig. 5h) and 26 (maximum zonal index, Fig. 5i) February were analyzed using the same best-shift wavelet basis. While the roles played by Zr were very similar on these two days, the transfers between the other three modes were ea h reversed. The blo k got omparable donations of KE from the residual and upstream eddies. The strong zonal dire tion ow, mostly aptured by the Z oeÆ ients during nonblo king, was equally fed by the smaller, global residual eddies, as is typi ally the ase in the limati energy y le des ribed by lassi al means. 4. Dis ussion and Con lusions a. Appli ation of Wavelet Energeti s to blo king Using the best-shift algorithm, an eÆ ient low-dimensional representation has been onstru ted, of ea h blo k stru ture in the data, whi h proved to remain wellrepresentative for the duration of ea h individual blo k. In fa t the very strong resemblan e of the timeand latitude-averaged height eld of the blo k to a single wavelet prompted spe ulation in appendi es A and B about the physi al interpretation of wavelet design. A riterion for nearness of any wavelet to a given blo k lo ation has been designed, appendix C, by whi h instantaneous weather map-like elds have been partitioned into four terms, representing the blo k, nearby east and west eddies, and distant eddies. Su h a onstru tion would not be possible by simply pruning or windowing a Fourier analysis. This partition and the energeti s te hniques enabled the omputation of the KE budget for these four terms. In four out of ve blo ks, the upstream eddies feed KE to the blo k, whi h in turn, in three out of four ases, transmits KE to the downstream eddies. Only ve blo ks were onsidered in this paper, and even the whole 90d dataset onsidered by F3 is too brief to make statisti ally signi ant statements, as explained by F3. Su h statements were -12not the goal of this paper, whi h rather was to illustrate how OWA allows stru ture omponents to be resolved, and questions to be addressed, whi h ould not be by Fourier analysis. Although statisti ally signi ant on lusions ould not be drawn about blo king energeti s [as for any \ ase study" su h as Hansen and Sutera (1984)℄, a variety of intera tion me hanisms are implied by the intera tions between the elements of the new, four-way partition. Drawing deeper on lusions may also require the onsideration of other physi al me hanisms, for instan e involving the temperature and verti al motion elds, and verti al shear. Eventually statisti ally robust energeti s measurements ould be orrelated with blo king duration time. But the present resear h is really a pilot study into the use of OWA and energeti s, similar to the Fourier pilot study of Saltzman (1957), and in this tradition it was deemed more important to develop an understanding of the orre t interpretation, and perhaps limitations, of Wavelet Energeti s, than to onsider all the various kinds of intera tions readily available. b. General remarks about OWA methods and extensions The tools presented in the sequen e of papers by the present author may be applied to any global elds. One would hope that their appli ation to datasets, mu h longer in time, would yield results with small enough time varian e to allow for statisti ally signi ant statements, as was the ase over a de ade of analogous studies by Saltzman and others. As the simplest kineti energeti s be ome learly understood, other terms of the energy budget, su h as APE, may be added, to obtain a broader pi ture. It may also be useful to onsider other budgets, su h as momentum, enstrophy, potential vorti ity and Eliassen-Palm ux. The wavelet methodology may also be re ned, for example by using wavelet analysis also in the latitude dire tion (with boundary onditions), or by using biorthogonal wavelets, whi h are (anti)symmetri . [Although all real W with nite N (appendix Aa) are stri tly asymmetri (Daube hies 1992), Beylkin (1995) reated su h W s with arbitrarily small asymmetry.℄ Or use wavelet pa kets, whi h separate indexes of -13wavenumber and s ale at ea h lo ation, or `lo al osines', whi h possess wavenumber n and an arbitrary in reasing sequen e of lo ations 2 1 ( k + k+1), and s ales k+1 k. The resear her who develops these re nements should expe t to make many te hni al hoi es and trade-o s in regard to method. The present author's hoi es in luded: to use relatively on eptually simpler wavelets than the above hoi es, trading symmetry and omplexity for orthogonality and simpli ity to use ompa tly supported rather than only qui kly de aying wavelets, trading symmetry for exa tness and transform speed [speed being irrelevant for small data analysis, but the author desired to use the same wavelets for numeri al modeling (Fournier 1995) as for data analysis.℄ to use smoother rather than more lo alized wavelets to use the entropy riterion (F1) rather than another best basis the numbers of large-amplitude wavelets to retain in order to represent blo king stru ture the nearness riteria of the nearby eddies to the blo k, trading energy for lo alization as well as other hoi es. It is lear that wavelet analysis o ers an alternative to traditional Fourier analysis whi h is relatively advantageous for ertain types of stru tures, su h as those of atmospheri blo king. In a ordan e with the Un ertainty Prin iple (See F1), it is ne essary to sa ri e some resolution in the wavenumber domain in order to gain resolution simultaneously in the lo ation domain, and also to deal with ompli ations inherent in a mixed, yet ompletely orthonormal, representation. Nevertheless, the tools introdu ed to the dynami al meteorologi al ommunity here are o ered in part en ourage trans ending the limitations of pure lo ation or wavenumber representations, and nally as an example of both the advantages and omplexities in regard to physi al interpretation whi h await the user of su h methods. -14A knowledgements. I thank B. Saltzman, R. B. Smith, K. R. Sreenivasan, R. R. Coifman, A. R. Hansen and P. D. M. Parker for omments on an earlier version of this paper, D. D. Shepherd for assistan e with Figs. 2and 5, G. Beylkin and L. Monzon for advising on part of appendix Aa, the Yale University Departments of Physi s and Geology & Geophysi s for support while the presented resear h was arried out, and the National Center for Atmospheri Resear h for providing the data. This material is based upon work supported by the National S ien e Foundation under Grant No. 9420011. APPENDIX A Interpretation of minimum phase lters a. De nitions and properties F1 explains that the parti ular wavelets Wj;k( ) used here are ompletly determined by a nite lter sequen e we ? of length 2N = 20. This lter was designed by Daube hies (1988) to yield a ompa tly supported `wavelet mother' W with the maximum number (M) of vanishing moments Z 1 1 tmW (t)dt = 0, and to have minimum phase (MP). It turns out that MP greatly distinguishes the shape of W from that of other ompa tly supported wavelets (Fig. A1, dis ussed below). `MP' is a term from signal pro essing implying that the phase of the Fourier series (inverse transform) w? ei 1 X l= 1w? l+1eil = 2N 1 X l=0 w? l+1eil (A1) [Daube hies 1992, (5.1.18)℄ varies minimally. Daube hies a hieved M = N and made Q(ei ) (1 + ei ) N w?(ei ) have periodi MP (Fig. A2 ) by requiring that the omplex `z transform' Q(z) be a polynomial of degree N 1 with all roots outside the unit ir le S in the omplex plane. Then the urve Q(S) does not en lose the origin, and so as parametrizes Q(S) the phase argQ ei = arg w? ei 2 1N never a umulates a winding of 2 (Robinson et al. 1986, p. 154). -15Figure A1 shows the roots of four w?(x + iy) of equal degree 2N 1 = 17. Only the Beylkin and Daube hies (left) lters have no roots inside S, and hen e have MP. Superposed on ea h panel is a res aled graph of W2;3(y) (thi k urve), showing how W 's shape is a e ted by the roots' positions. Observe that by pla ing roots inside S,W an be made more symmetri ( orresponding to nearly linear phase in Fig. A2 ), but in that ase we ? no longer has MP. Figure A2b shows how, regardless of MP, Daube hies and \symmlet" lters have the greatest number of vanishing moments of w2 l ( 1)lw? l+1, leading to the vanishing moments of W . Sin e 2N 1 X l=0 ( 1)llmw? l+1 = i mdm w? d m (ei ) this is also seen by omparing the apparent orders of the ommon root at = , Fig. A2d. Two more general properties of MP are of physi al relevan e. MP is robust to the addition of interferen e ae , i.e. ae + we ? is also MP if j aj < w? on S. For xed w? , the phases of M-P w? are arranged so that w? l has its varian e as lo alized at low l (i.e. spatially, westwardly) as possible [Robinson's (1962) `minimum delay wavelet theorem'. He and other solid-earth geophysi ists de ne a \wavelet" merely as a fun tion wt satisfying wt<1 = 0 < 1 X t=1 jwtj2 <1, very di erent from the de nition reviewed by F1.℄ This lo alization is shown in Fig. A2a. From the normalization of the `wavelet father' Z 1 1W?(t)dt 1 and the `two-s ale' relations in F1 one derives W (t) = 1 Z 1 1 e 2i t2 1=2 w ei 1 Y j=1 2 1=2 w? e2 j i d ; (A2) (Daube hies 1992, p. 193) showing that at ea h wavenumber , W may be represented as an in nite su ession of M-P lters, and one non-M-P lter w(z) = z w? z 1 (A3) (Daube hies 1992, p. 163). The lter we evidently annot have MP, be ause if w?(z0) = 0 for jz0j > 1 (A4) -16then w z 1 0 = 0 for z 1 0 < 1. b. Why might hH?ib' ( ) be related to M-P pro esses? hH?ib' ( ) resembles wavelets derived from M-P lters, implying possible physi al interpretations su h as the des ription of an invertible linear relation between hH?ib' ( ) and another physi al quantity, or that hH?ib' ( ) de onvolves two other linearly related quantities. An example from solid-earth geophysi s is the invertible linear relationship between stress and strain, re e ted by the invertibility of the lter relating their respe tive time series (Claerbout 1992). Also, seismi pulse-train waveforms are M-P be ause of a multitude of attenuating re e tion events analogous to the pro ess (A2) (Robinson and Treitel 1980, pp. 253{254). Perhaps the su ession of pro esses whi h onstru t a given atmospheri pattern des ribed by hH?ib' ( ) might be thought of as a long hain of M-P pro esses. The ` ausality' (w? l<1 = 0) might be interpreted as re e ting the spatiotemporal asymmetry due to the earth's rotation (G. Beylkin 1998, personal ommuni ation). What might be the relevant physi al quantities for hH?ib' ( ) to have this stru ture by this reasoning? A on lusive investigation is beyond the s ope of this paper, but my literature review inspires one spe ulation. The linear relationships indi ated here may be related somehow to the pie ewise linear relationship between streamfun tion (geostrophi ally proportional to Z) and (potential) vorti ity, whi h is part of the modon theory (Tribbia 1984; Haines and Marshall 1987; Haines and Malanotte-Rizzoli 1991) and diagnosis (But hart et al. 1989) of blo king. . Inverse problem: design of wavelets These questions may be inverted. Given some physi ally motivated assumptions about the pro ess hain whi h onstru ts a ertain atmospheri pattern, how an the freedom of hoi e inherent in the design of the wavelet lter we be exploited to reate espe ially suitable wavelets? The variety of wavelets shown in Fig. A1 are the result of -17that freedom being exploited for other purposes. From Theorem 6.3.6 of Daube hies (1992), all that is required is the existen e of any fun tion w verifying w(z) w(z 1) + w( z) w( z 1) = 2 = 21=2 w( 1); for all z 2 S (A5) [hen e w(1) = 0℄ and the rest of the OWA apparatus follows from equations given above. Likewise, any physi al requirements on the wavelet W may be translated into additional requirements on w, using the same equations in reverse, ompatible with the existen e of solutions to (A5). d. E e ts of topography on hH?ib' ( ) phases It also should be pointed out that the in uen e of earth's topography (Fig. A3) on hH?ib' ( ) must be onsidered. Although the hH?ib' ( ) whi h resemble W e+ 1;k ( ) ea h involve di erent +o , in ea h ase there is hH?ib' = 0 < hH?ib' near 50ÆW, whi h may be related to the Greenland ridge, just to the east. Similarly there usually was hH?ib' > 0 > 2 hH?ib' near 150ÆW, just upstream of the Canadian Ro kies, and a broad se tor of hH?ib' < 0 < 2 hH?ib' a ross the Eurasian ontinent. It seems that topography may ontrol ertain phase aspe ts of hH?ib', but not ne essarily magnitudes. APPENDIX B Minimal Fourier representation of the blo king wavelet Observe that 98% of the varian e of the expression B( ) os sin 2 = Re ei + ie2i ( real) (B1a) , b Bn 1 2 Z B( )e in( + )d (B1b) = 2 1 Æjnj;1 + iÆjnj;2sgnn (B1 ) = 2 J B e 21 J in ; where Bl 8<:B ( l) ; l = 1; 2J , 0; otherwise, (B1d) is ontained the single wavelet W e+ 1;k ( ), whi h in turn represents most of the blo king varian e. That is, B W e+ 1;k to within 2% for all J . The appropriate e+ is empiri ally -18determined by (2.4) and +o 34 212 J 2 10 81:37 4227 2 J 2 10 14:2Æ (least squares t). Figure A4 shows various losely agreeing estimates of +o , and Fig. A5 l. shows B and W e+ 1;k for various we . Figure A5 r. provides eviden e of how the n > 2 spe trum, whi h distinguishes different we , ontibutes to the similarity of B to W e+ 1;k . Both Daube hies wavelets perform similarly. The \symmlet" does a little more poorly, perhaps be ause wavenumber 3 interferes with 1,6 more destru tively than either Daube hies wavelet. The Beylkin wavelet has a spe trum similar to the \symmlet," but with wavenumber 1 less reinfor ed by 5{6,10. The \Coi et" simply has too mu h magnitude at n > 2. a. Signal pro essing interpretation The resemblan e of W e 1;k to B is roughly explained as follows. If f = W e 1;k+1 then the wavelet oeÆ ients e fj;k0+1 = Æj;1Æk0;k and the oarse-s ale oeÆ ients f j 1;k0+1 = 0. Putting these values into the OWA `re onstru tion' onvolution algorithm (F1) and performing the inverse Fourier transform [(A1) with z = ei ℄ yields f2(z) = w(z) e f1 z2 z 1 = w(z)z2k 1 ; f3(z) = w?(z) f2 z2 z 2 = w?(z) w z2 z4k 2 1 2; f4(z) = w?(z) f3 z2 z 3 = w?(z) w? z2 w z4 z8k 4 1 2 2 3; ... fJ (z) = w?(z) fJ 1 z2 z J 1 = J 3 Y j=0 w? z2j w z2J 2 z2J 1 lo+1 (B2) (Daube hies 1992, p. 161), free to let 0 = 1 k. The Fourier series oeÆ ient (B1b) of fJ S of index nearest to 2J 1 1 + 1 approa hes W e 1;k+1( ) as J !1. Conversely 2 J fJ e 21 J in ! J!1 \ W e 1;k+1n b Bn -19(B1 {d) as follows. Eqs. (A3,A5,B2) imply fJ e 21 J in = 0 if 2 1n is an even number, otherwise fJ e 22 J in fJ e 21 J in = w?(i n) w[( 1)n℄ein o w?(e 21 J in) w(i n) ! n odd 21=2i nein o w?(e 21 J in) ! J!1 i nein o ! n!1 B( + o)b2 B( + o)b1 : From (A3) and the grays ale along S in Fig. A1, one may infer that the produ t of the low-pass ( w?) and high-pass ( w) lters, dilated by the 2js in (B2), retains support only at low positive wavenumbers. Thus the ratio just omputed and the overall normalization serve to roughly explain the resemblan e. Although ea h w?(z) fa tor has MP, the full produ t (B2) tends not to. b. Consequen es in terms of Fourier spe tral dynami s A distinguishing hara teristi of (B1a) is that the only two wavenumbers jnj = 1; 2 have Fourier oeÆ ients of equal magnitude. This may be relevant to the stru ture of the geostrophi zonal wind oeÆ ients b un (3.1) sin e due to nonlinear adve tion uu the relative phase velo ity d dt arg b u2n b un = nRe 24 1 jb unj2 1 jb u2nj2! b u2nb u2 n + X n0 6= n;2n b un n0 b un0 2b u2n X n00 6=n b u2n n00 b un00 b un 35 ; extra ting from the sums all terms involving only wavenumbers n,2n. Then equal magnitude implies the phase di eren e is onstant unless other wavenumbers or e e ts enter. In fa t adve tion by itself would introdu e those other wavenumbers, but in the sense that they do not a e t their own relative phase, su h equal-magnitude pairs are ondu ive to a kind of \blo king," i.e. quasi-stationary ridges. . Consequen es in terms of low-order blo king models Another onsequen e of the simple expression (B1a) on erns a study by Wiin Christensen and Wiin-Nielsen (1996), in whi h a low-order model based on the invis id barotropi vorti ity equation with Newtonian for ing was used to investigate blo king dynami s. Their blo king-type equilibrium solutions (their Fig. 7) strongly resemble (B1a). In fa t, be ause of the 2 phase di eren e, several nonlinear terms in their -20low-order model are simpli ed, and (B1a) is an equilibrium solution for a hoi e of Newtonian for ing similar to theirs. APPENDIX C De nitions for wavelet index sets The displa ement from 0 to is 0 ( ; 0) 8<: 0; 0, 2 + 0; otherwise, de ned to handle orre tly the ase 0 < 0 2 , where the bran h ut intervenes. The lo ation j;k of wavelet Wj;k is de ned by Wj;k j;k max Wj;k( ) : Similarly, the blo k rest longitude is now de ned by DH Eb'( ) max DH Eb' ( ) and an upper bound on the eddy s ale is given by the blo k s ale index j , de ned by f H e+ j ;k b' max j=0; J 1 max k=1; 2j f H e+ j;k b' ! : APPENDIX D Wavelet Transfer Fun tions between two arbitrary modes (maps) The Wavelet Transfer Fun tion to mode from any other mode b is L jb = Lbj (gr ) 1 p7 X a2M DL ja;bE' ; (D1) hfi' 0 33 X m=23 os'm1A 1 33 X m=23 f('m) os'm; 'm+1 2:5Æm (D2) [ f. Iima and Toh 1995, (3.6)℄, using the results of F2 (with ! = 0), in units of m3 s 3, and multiplying by (gr ) 1 p7 = 0:24 g m 3 -21to onvert to mW m 2 (= erg m 2 s 1, the unit of early energeti s studies, whi h reported values for a single level similar to the present ase). This fa tor times the earth radius r is the mass per unit area of a olumn of the thi kness p7 around p7 = 50 kPa (seventh mandatory level, F3). Be ause many ontemporary investigators, in luding F3, use integrals over all p, the transfers reported in the present paper are omparatively smaller. APPENDIX E Maximization of momentum transport w.r.t. ontour orientation Let R( ) be the ovarian e between rotational wind omponents u ' and v se ' around a streamfun tion ontour loop C whi h has any kind of axis at an angle north from east. Then R( ) also equals the ovarian e between u0 u os v sin and v0 v os + u sin around C 0 , the same loop parameterized by oordinates (x; y) rotating the axis to point along rfx. One an easily show that R( ) IC uv = IC0 u0v0 = 2 1Im e2i IC0 (u+ iv)2 obtains a maximum value R + = 2 1 IC0 (u+ iv)2 for = + 4 12 arg IC0 (u+ iv)2 : (E1) Then + = 4 as long as in the mean around C 0 one has u2 > v2 and the net orrelation between u and v is zero. The onditions are reasonable; the former almost amounts to C being \elongated" along its axis, while the latter may follow from simple symmetry assumptions on C 0 . For example the ellipse C 0 = f(x; y) jx = x0 + a os ; y = y0 + b sin ; < g -22implies (x0 + ar os ; y0 + br sin ) = 1 X p=0 p! 1(r 1)pUp(1; ); Up(r; ) p r (x0 + ar os ; y0 + br sin ); and the onditions are satis ed under the weak assumptions Z U1(1; )2 sin 2 d = 0 < Z U1(1; )2 b 2 sin2 a 2 os2 d : To the extent the onditions fail, + adjusts a ording to (E1). This result formalizes a heuristi argument about momentum transport ontributed by a streamfun tion line element, onsistent with dis ussions going ba k to Starr (1948). It also implies a diagnosti tool similar to the E-ve tor analysis of Hoskins et al. (1983), with the advantage that no temporal statisti s are ne essary; for any given instantaneous distribution of streamfun tion, every ontour (not en ompassing j'j = 2 ) may be assigned a ve tor of magnitude R( +) dire ted + north of east, whi h signi es the transport of momentum ontributed by that ontour, in a dire tion whi h is optimal w.r.t. both the ontour's orientation and the stru ture of ow along the ontour. However it should be mentioned that su h an analysis may be of limited utility, as is the E-ve tor analysis, in ase lo al onservation of momentum is only poorly maintained (R. Saravanan 1999, personal ommuni ation). The author is urrently investigating how to extend the ideas sket hed out in this appendix to the transport of a betteronserved quantity su h as potential vorti ity. -23REFERENCES Beylkin, G., 1995: On fa tored FIR approximation of IIR lters. Appl. Comput. Harm. Anal., 2, 293{298. But hart, N., K. Haines, and J. Marshall, 1989: A theoreti al and diagnosti study of solitary waves and atmospheri blo king. J. Atmos. S i., 46, 2063{2078. Claerbout, J. F., 1992: Earth Soundings Analysis: Pro essing versus Inversion. Bla kwell, 320 pp. Daube hies, I., 1988: Orthonormal bases of ompa tly supported wavelets. Commun. Pure Appl. Math., 41, 909{996. , 1992: Ten Le tures on Wavelets. SIAM, 357 pp. Dutton, J. A., 1986: The Ceaseless Wind: An Introdu tion to the Theory of Atmospheri Motion. Dover, 617 pp. Fournier, A., 1995: Wavelet representation of lower atmospheri long nonlinear wave dynami s, governed by the Benjamin-Davis-Ono-Burgers equation. Wavelet Appli ations II, H. H. Szu, Ed., SPIE, 672{681. , 1996: Wavelet analysis of observed geopotential and wind: blo king and lo al energy oupling a ross s ales. Pro . Wavelet Appli ations in Signal and Image Pro essing IV, Denver, CO, SPIE, 570{581. , 1999: Transfers and uxes of wind kineti energy between orthogonal wavelet omponents during atmospheri blo king. Wavelets in physi s, Cambridge, 263{298. Haines, K., and P. Malanotte-Rizzoli, 1991: Isolated anomalies in westerly jet streams: A uni ed approa h. J. Atmos. S i., 48, 510{526. , and J. Marshall, 1987: Eddy-for ed oherent stru tures as a prototype of atmospheri blo king. Quart. J. Roy. Meteor. So ., 113, 681{704. Hansen, A. R., and A. Sutera, 1984: A omparison of the spe tral energy and enstrophy budgets of blo king versus nonblo king periods. Tellus, 36A, 52{63. Holton, J. R., 1992: An Introdu tion to Dynami Meteorology. A ademi Press, 511 pp. -24Hoskins, B. J., I. N. James, and G. H. White, 1983: The shape, propagation and meanow intera tion of large-s ale weather systems. J. Atmos. S i., 40, 1595{1612. Hovmoller, E., 1949: The trough-and-ridge diagram. Tellus, 1, 62{66. Iima, M., and S. Toh, 1995: Wavelet analysis of the energy transfer aused by onve tive terms: Appli ation to the Burgers sho k. Phys. Rev. E, 52, 6189{6201. Robinson, E. A., 1962: Random Wavelets and Cyberneti Systems. GriÆn's Stat. Monogr., No. 9, Hafner, 125 pp. , and S. Treitel, 1980: Geophysi al Signal Analysis. Prenti e Hall, 466 pp. , T. S. Durrani, and L. G. Peardon, 1986: Geophysi al Signal Pro essing. Prenti e Hall, 481 pp. Saltzman, B., 1957: Equations governing the energeti s of the larger s ales of atmospheri turbulen e in the domain of wave number. J. Meteor., 14, 513{523. Shutts, G., 1986: A ase study of eddy for ing during an Atlanti blo king episode. Adv. Geophys., 29, 135{162. Starr, V. P., 1948: An essay on the general ir ulation of the earth's atmosphere. J. Meteor., 5, 39{43. Tribbia, J., 1984: Modons in spheri al geometry. Geophys. Astrophys. Fluid Dyn., 30, 131{168. Wiin Christensen, C., and A. Wiin-Nielsen, 1996: Blo king as a wave-wave intera tion. Tellus, 48A, 254{271. Wiin-Nielsen, A., and T.-C. Chen, 1993: Fundamentals of Atmospheri Energeti s. Oxford, 400 pp. -25Table Captions TABLE 1. Sele ted days exemplifying blo king and zonal global ow states. Column six indi ates Atlanti , Pa i or nonblo king, zonal index minimum or Maximum ondition. Column seven indi ates the time interval T. t (d) UTC date month yr blo ked? t interval (d) of best shift 2.5 1200 3 De ember 1978 A [0,6.5℄ 11.5 1200 12 00 00 m 00 17.5 1200 18 00 00 M [19,26.5℄ 22.0 0000 23 00 00 A 00 36.0 0000 6 January 1979 P [28.5,40℄ 55.0 0000 25 00 00 A [44,56.5℄ 80.0 0000 19 February 00 A [77,83℄ 87.5 1200 26 00 00 M 00 -26-

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تاریخ انتشار 1999