## Abstract

A new wavelet energetics technique, based on *best-shift* orthonormal wavelet analysis (OWA) of an instantaneous synoptic map, is constructed for diagnosing nonlinear kinetic energy (KE) transfers in five observed blocking cases. At least 90% of the longitudinal variance of time and latitude band mean 50-kPa geopotential is reconstructed by only two wavelets using best shift. This superior efficiency to the standard OWAs persists for time-evolving structures. The cases comprise two categories, respectively dominated by zonal-wavenumber sets {1} and {1, 2}. Further OWA of instantaneous residual nonblocking structures, combined with new “nearness” criteria, yields three more orthogonal components, representing smaller-scale eddies near the block (upstream and downstream) and distant structures. This decomposition fulfills a vision expressed to the author by Saltzman. Such a decomposition is not obtainable by simple Fourier analysis.

Eddy patterns apparent in the components’ contours suggest inferring geostrophic energetic interactions, but the component Rossby numbers may be too large to support the inference. However, a new result enabled by this method is the instantaneous attribution of blocking strain-field effects to particular energetically interactive eddies, consistent with Shutts’ hypothesis. Such attribution was only possible before in simplified models or in a time-average sense. In four of five blocks, the upstream eddies feed KE to the block, which in turn, in three of four cases, transmits KE to the downstream eddies. The small case size precludes statistically significant conclusions. The appendixes link low-order blocking structure and dynamics to some wavelet design principles and propose a new interaction diagnosis, similar to *E*-vector analysis, but instantaneous.

## 1. Introduction

Saltzman (1957) was one of the first investigators to argue that upscale energetic transfers “fed from the intermediate wavelengths . . . to the longer wavelengths which correspond in part to the so-called ‘semi-permanent’ centers” were “of importance in maintaining the stationary long waves.” His ideas were later applied to atmospheric blocking by several authors, including Hansen and Sutera (1984), Shutts (1986), Haines and Malanotte-Rizzoli (1991), and Wiin Christensen and Wiin-Nielsen (1996). See Fournier (2003, hereafter FO3) for further references. A major obstacle to investigating nonlinear-scale interactions arises because blocks are localized structures, which are not well represented by truncated Fourier series, since the Fourier coefficients of longitude-localized structures are not zonal-wavenumber localized. Thus, it was suggested by Fournier (2002, hereafter FO2) to translate the traditional Fourier analysis–based atmospheric energetics of Saltzman (1957) into a periodic orthonormal wavelet energetics.

Fournier (1999, hereafter FO99; 2000, hereafter FO0) analyzed observed Northern Hemisphere data from the National Meteorological Center (NMC). He found that the second-largest-scale (east and west hemisphere) wavelet coefficients of a multiresolution analysis of isobaric height *Z*(*λ*, *ϕ*, *t*) at 70 kPa could make up a significant component of the blocking state, as that state was defined (see Table 1 for symbol definitions). In the present paper, several refinements to the techniques of FO99 and FO3 are discussed. Section 2a describes how the *best-shift* orthonormal wavelet analysis (OWA) algorithm (reviewed by FO0) may be applied to yield a better representation of the structure of atmospheric blocking than the standard OWA. How well that representation performs instantaneously is tested in section 2b. Then maps for instantaneous geopotential height observations at 50 kPa are partitioned into four fields, which represent the block structure, nearby downstream (east) and upstream (west) eddies, and the residual structures. From this partition, particular energetic interactions between pairs of patterns may be quantified, by partially summing the *triad interactions* (described by FO2) involving that pair. Section 3 contains the results and discussion of these analyses. The goal of this paper is not to present statistically significant analyses (as FO3 does), but rather to further illustrate the adaptability of OWA methods for resolving structure components and addressing questions that are not amenable to Fourier analysis.

Appendixes A and B present an interpretation of the blocking representation with regard to signal-processing and physical points of view. Appendix C outlines a new diagnosis of synoptic maps, which in some ways is similar to E-vector analysis of Hoskins et al. (1983).

## 2. Methods

### a. Representation of blocking by wavelets with best shift

The representation of the blocking structure used by FO0, FO2, and FO3 may be refined as follows. The top curves of Figs. 1a–j show the 50-kPa height fields 〈*H**〉^{b}_{ϕ}(*λ*) averaged over blocking time intervals from the NMC analyses [as explained by Fournier (1996, 1999) and FO3; see Table 1 for notations]. The next curve below is the part of 〈*H**〉^{b}_{ϕ} reconstructed from the 10 largest-magnitude best-shift OWA components 〈*H̃*^{ε}_{j,k}〉^{b}_{ϕ}*W*^{ε}_{j,k}, where the 〈*H̃*^{ε}_{j,k}〉^{b}_{ϕ} are coefficients and the *W*^{ε}_{j,k}(*λ*) are orthonormal wavelet basis functions at “scale” 2^{−}* ^{j}* and “location” 2

^{−}

*(see Table 1 for details). This curve is labeled by the percentage of variance contributed by those 10 components. Below follow curves for each of those individual components, for both the standard [*

^{j}k*ɛ*= (0, . . . 0), odd-lettered figures] and best-shift (

*ɛ*=

*ɛ*

^{+}, even-lettered figures) OWA, as explained by FO0. The components are sorted by |〈

*H̃*

^{ε}

_{j,k}〉

^{b}

_{ϕ}|, descending, and labeled by percentage of variance contributed and by longitude resolution and location indexes (

*j*,

*k*). All functions are

*b*-averaged over one blocking event in the Pacific (Figs. 1e–f) or individually over each of four blocking events in the Atlantic (other letters). The dates for each blocking time interval may be inferred from Table 2.

Figure 2 shows the percentage of variance contributed cumulatively by the four largest |〈*H̃*^{ε}_{j,k}〉^{b}_{ϕ}| components, for the standard (lower stacks) and best-shift (upper stacks) OWA of the five blocking events. The best shift offers great improvement in every case. Over all but the second event, only two optimally shifted wavelets contribute more variance than four or more standard wavelets.

### b. Partitioning into block and eddy structures

The top panels of Fig. 3 show Hovmöller (1949) diagrams of unanalyzed 〈*H**〉* _{ϕ}*(

*λ*,

*t*). The light regions clearly depict the nonpropagating blocking structure (features parallel to the time axis) and some progressing synoptic cyclone waves (west-to-east “diagonal” structures).

Interpretation of all the multiscale details in these top panels is not easy; can the best-shift basis reveal more structure in 〈*H**〉* _{ϕ}*(

*λ*,

*t*)? In the lower two panels of Fig. 3, for each particular block the constant best-shift

*ɛ*

^{+}determined from 〈

*H**〉

^{b}

_{ϕ}(

*λ*) (Figs. 1b–j, even-lettered figures) has been used in the OWA of 〈

*H**〉

*(*

_{ϕ}*λ*,

*t*) at

*each t*. The middle panels of these figures depict the fields generated by projection onto the basis elements, which partition 〈

*H**〉

^{b}

_{ϕ}(

*λ*) into contributions from the four largest-magnitude 〈

*H̃*

^{ε+}

_{j,k}〉

^{b}

_{ϕ}. Similarly, the bottom panels depict the contributions from the residual 〈

*H̃*

^{ε+}

_{j,k}〉

^{b}

_{ϕ}. Denote this partition of the zonal-deviation field 〈

*H**〉

*in the top panel into the sum of the two fields in the lower panels by*

_{ϕ}where *Z*^{bl} and *Z*^{ed} ≡ *Z** − *Z*^{bl} are orthogonal w.r.t. the zonal mean: = 0. Here bl stands for *block* and ed stands for *eddy*. The lack of argument (*λ*, *ϕ*, *p*, *t*) means that (2.1) may be performed at any step of a linear analysis. Here *Z*^{bl} uses only best-shifted wavelets *W*^{ε+}_{j,k} with indexes from the *block set* ℬ_{4}, where ℬ_{0} ≡ 0 and

Equation (2.2) is a bulky notation for collecting the *ν*+1 largest coefficients but is useful below.

In section 3c, maps of *Z*(*λ*, *ϕ*) at 50 kPa on particular days are further analyzed. To prepare for this, section 2c contains a summary of the energetics formulated by FO2, here applied to four orthogonal projections, called *modes*, labeled by *a* ∈ {bl, do, up, re} ≡ ℳ, the *mode set*. First, (2.1) is further analyzed by

as follows. The simultaneous space-and-scale resolution of wavelets enables the nearby eddies downstream of the block to be extracted as

where the set 𝒟 contains nearby downstream wavelet indexes that are not part of the block set ℬ_{4 }(2.2) [and are not (0, 0), since wavelet *W*_{0,0} is not localizable within a hemisphere]. That is,

The nearby eddies upstream of the block are given in like manner by

The residual eddies are defined to be *Z*^{re} ≡ *Z*^{ed} − *Z*^{do} − *Z*^{up}. All modes are zero mean and orthogonal: = = 0 for distinct *a*, *b* ∈ ℳ.

“Nearness” parameters such as Δ*λ* ≡ (*π*/3) and Δ*j* ≡ 0 are somewhat arbitrary. They were chosen to reasonably well define downstream and upstream wavelets, resolve straining effects (discussed below and by FO2), and reduce blocking-sector variance leakage into *Z*^{re}. The location *λ _{j}*

_{,}

*of wavelet*

_{k}*W*

_{j,k}is defined by

Similarly, the block crest longitude *λ*_{bl} is defined by

An upper bound on the eddy scale is given by the block-scale index *j*_{bl}, defined by

Since in different cases, different reference origins *λ _{ɛ}* are used (see Table 1 and FO0), Fig. 4 is provided. It shows the location, scale, and shape of some of the most variance-explaining

*W*

^{ε+}

_{j,k}, as their patterns would appear in a polar-projection weather map. For each of the “

*j*= 1 blocks” discussed in section 3a and appendix B, and rows 1, 3, and 5 of Fig. 4, the longitude of

*W*

^{ε+}

_{1,2}extremum (minimum or maximum depending on the sign of

*Z̃*

^{ε+}

_{1,2}) in Figs. 1b,f,j is aligned with the block crest longitude. Likewise, for both of the

*j*= 0 blocks, rows 2 and 4 of Fig. 4, the longitude of maximum

*W*

^{ε+}

_{0,0}is aligned with the block crest. These extrema are labeled A and P for Atlantic and Pacific blocks, respectively.

The eight times selected for the energetics analysis are those of Table 2 and FO99. The best-shift *ɛ*^{+} corresponding to the interval containing each time was used for the blocking days. For comparison, the *nonblocking*, maximum-zonal-index times were partitioned [(2.1) and (2.3)] using the *blocking*-time best shifts corresponding to the second and fifth of the blocking time intervals. The best shift corresponding to the first block was used for the minimum zonal index day.

### c. Energetics

For a map at any given time, once the partition [(2.1) and (2.3)] has been performed and energetics derived by FO2 have been computed using the OWA with appropriate shift *λ*^{+}_{ε}, then it becomes possible to ask what structures in *Z*^{ed} [(2.3)] feed or draw kinetic energy (KE) from *Z*^{bl}, and vice versa. To address that question, energetic transfers are assigned to the maps by writing the KE transfer TK_{a}_{|}* _{b}* to mode

*a*from any other mode

*b*as follows:

[cf. Iima and Toh 1995, their Eq. (3.6)]. This follows from the formulas of FO2 (with *ω* = 0), in units of m^{3} s^{−3}, and then multiplying by

to convert to mW m^{−2} (= erg cm^{−2} s^{−1}). (This was the unit of early energetics studies, which reported values for a single level similar to the present study.) The factor *g*^{−1}Δ*p* is the mass per unit area of a column of thickness Δ*p* = 15 kPa around *p* = 50 kPa. Because many contemporary investigators, including FO3, use integrals over all *p*, the transfers reported in the present paper are comparatively smaller.

Polar maps of *Z ^{a}*(

*λ*,

*ϕ*) (3 Dm intervals; gray-filled contours) and arrows connecting

*Z*to

^{b}*Z*showing TK

^{a}

_{a}_{|}

*(2.4) are arranged in a square matrix as shown in Fig. 5a. In this schematic diagram, the arrows are drawn as if all of TK*

_{b }_{up|bl}, TK

_{do|bl}, TK

_{re|bl}, TK

_{do|up}, TK

_{re|up}, and TK

_{do|re}were positive. In the following figures, negative values of these transfers are drawn with reversed arrows.

## 3. Results: Observed energetics between block and downstream, upstream, and other eddy modes

### a. Representation of blocking by wavelets with best shift

In the Atlantic blocking cases (Figs. 1a–d, g–j) the largest-magnitude wavelet component can contribute considerably more variance to 〈*H**〉^{b}_{ϕ}, merely by choosing the best-shift *λ*^{+}_{ε}. The resolution index *j* of that component does not change because of the shift. Looking at the *j*-index of the largest-magnitude component, it appears that in the wavelet picture there are two types of blocks: a *j* = 1 type (Figs. 1a–b, i–j), corresponding to a superposition mostly of zonal wavenumbers *m* = 1, 2; and a *j* = 0 type (Figs. 1c–d, g–h), corresponding primarily to zonal wavenumber *m* = 1. The Pacific block (Figs. 1e–f) was another *j* = 1 type. This case shows the greatest improvement afforded by the best shift (Fig. 2).

### b. Persistence of best shift

For each block, the middle panels of Fig. 3 show that 〈*H*^{bl}〉_{ϕ} is dominated by steady (appearing parallel to the time axis), high (light shaded) regions at the block location. Therefore the basis elements, which have been shifted by *ɛ*^{+} to best represent the *time-average* block structure, *also* very well represent that structure *instantaneously*. Every bottom panel shows that 〈*H*^{ed}〉_{ϕ} is dominated by eastward-progressing diagonal patterns. Thus the residual of the *block* basis elements captures the progressing, smaller-scale, cyclonic wave or *eddy* part of the motion. As explained by FO0, this joint location-scale resolution could *not* be achieved by simple Fourier-based methods.

In all but one case of blocking, 〈*H*^{bl}〉_{ϕ}(*λ*, *t*) accounts for at least 84.5% of the combined (*λ*, *t*) variance of 〈*H**〉* _{ϕ}*. The lower two panels’ titles give variance-contribution percentages. The exceptional case is the fourth blocking event (Fig. 3d) in which “only” 69.4% is accounted for. As noted by FO99, this block underwent retrograde (westward) motion. It is therefore not surprising that a representation with fixed

*best reference origin*

*λ*

^{+}

_{ε}is less successful.

### c. Instantaneous partition results

Figures 5b–g show the instantaneous maps and associated energetics for the eight times in Table 2 in chronological order. These maps show that OWA enables one to construct modes of any given field that localize features in space and in scale, which is not possible using standard Fourier analysis. Letting **V*** ^{a}* ≡

*u*

^{a}**i**+

*υ*

^{a}**j**, and

*K*

_{a}≡ 2

^{−1}denote the contribution of mode

*a*to the total KE, note that although

*K*

_{re}is always comparable to

*K*

_{bl}, nevertheless the

*Z*

^{re}contours always display less variability in the sectors of most intensely variable

*Z*

^{do}+

*Z*

^{up}.

#### 1) What more can this partition show by mere inspection?

Heuristically, one might hope to infer some energetic information from inspection of *Z*(*λ*, *ϕ*) maps as follows. The geostrophic velocity

points along isohypses, with speed proportional to isohypse closeness (*Z* gradient). An approximately elliptical *Z ^{a}* isohypse implies a positive correlation between

*u*

^{a}_{g}and

*υ*

^{a}_{g}if the angle

*α*between the semimajor axis and

**i**falls in ]0, (

*π*/2)[, and a negative correlation if

*α*falls in ]−(

*π*/2), 0[. In appendix C, a novel proof is introduced to show that the covariance is maximum when

*α*= (

*π*/4), under certain assumptions. From that covariance may be inferred a northward advective transport of eastward geostrophic momentum of mode

*a*, or equivalently an eastward transport of northward momentum.

To an *arbitrarily* oriented ellipse in field *Z ^{a}*, one may associate a transport in the direction

**T**about 45° to the left of its semimajor axis direction

**A**, of a geostrophic momentum component in a direction

**D**≡

**T**×

**k**to the right of

**A**, or vice versa. Depending on the sign of the gradient

**T**·

**∇**of the flow component

**D**·

**V**

*, mode*

^{b}*b*will either receive KE from

*a*or else lose KE to

*a*. This remark is in part a generalization to arbitrary modes

*a*–

*b*of the classical eddy–zonal-mean kinetic energetics discussion as appears in the literature [e.g., Saltzman 1957, p. 516b in section 3; Holton 1992, p. 340, seventh equation; Wiin-Nielsen and Chen 1993, p. 73, their (7.22) and their Fig. 6.3].

However, the reliability of these inferences depends on the quality of the geostrophic approximation (3.1) itself. That quality is measured by the smallness of the Rossby number. Here an appropriate definition (Dutton 1986) of the Rossby number is the ratio of a characteristic magnitude of **V** − **V**_{g} to a characteristic magnitude of **V**. Typically, the Rossby number is much less than unity for large-scale atmospheric flows. To evaluate the appropriateness of invoking the geostrophic approximation for the mode-partitioned fields, and moreover its corollary energetics inferences, the Rossby number for each individual mode *a* ∈ ℳ is estimated as the ratio

of horizontal means.

Each polar map of mode *a* in Figs. 5b–i is labeled by *K _{a}* and Ro. Each figure is labeled by

*t*and the overall Rossby number Ro(

**V**,

*Z*), which is almost always significantly smaller than the Rossby number of any mode. Rossby numbers on the order of one or greater indicate that the above inferences from

*Z*structure to geostrophic wind and energetics may not be reliable. The answer to the question, “What more can this partition show by mere inspection?” is that the sign and very approximate relative magnitude of KE transfers TK

^{a}

_{b}_{|}

*might be determined by inspection of overlapping*

_{a}*Z*,

^{a}*Z*contours, but that this technique is severely limited by the quality of the geostrophic approximation, measured by the smallness of the mode Rossby numbers. It turns out not to be a consistently reliable technique for the maps analyzed here.

^{b}#### 2) Energetic diagnostics

Figure 5b shows that the northeastern Atlantic block pattern on 3 December, described by *Z*^{bl} (top left), was being energetically sustained by all three other modes. Most of the KE infusion comes from *Z*^{re} (top right). Some eddies upstream of the block, in *Z*^{up} (bottom left), were giving about 42% of the KE that *Z*^{re} did. The upstream eddies nearest to the block, south and southwest of Greenland, show the meridional elongation brought about by the strain field of the block, as discussed, for example, by Shutts (1986).

In contrast, the minimum zonal index day, 12 December (Fig. 5c), shows a *Z*^{bl} (top left) that somewhat resembled that on 3 December, but was *losing* KE to *Z*^{re} and *Z*^{do}, which had fed the 3 December block. FO99 explains that this anomalous high *did not persist long enough to be classified as a block*. Its energetic gain from upstream modes was similar to that on 3 December, so perhaps the loss to *Z*^{re} + *Z*^{do} was an important difference. [Other physical effects such as available potential energy (APE) transfer have not been computed, partly because 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 December (Fig. 5d). The residual eddies *Z*^{re} were feeding the block part *Z*^{bl} 58% more KE than on 3 December. The local eddies *Z*^{do} + *Z*^{up} were storing 92% more KE, but together give 79% less to *Z*^{bl} at this time. The pattern would change in 36 h on 23 December, when the second Atlantic block would be formed (shown in Fig. 5e). It was more longitudinally steady and compact, judging from the Hovmöller diagrams shown in Fig. 3b and by FO0; *Z*^{bl} exchanged *very little* KE with *Z*^{re} at this time. Instead there was a more vigorous transfer from downstream to upstream both directly and through the block.

The Pacific block (6 January; Fig. 5f) has the most vigorous transfer to *Z*^{bl} from *Z*^{up} of all eight cases. The relative strength of the Pacific block energetics is also evident in the energetics analyses of Hansen and Sutera (1984), Nakamura et al. (1997), and FO99. Again, the upstream eddies show the straining effects due to the block.

On 25 January (Fig. 5g), the block that underwent retrograde motion westward across the Atlantic was coming to an end. This was accompanied by a dispensing of KE from *Z*^{bl} to *Z*^{do} and *Z*^{re}. Upstream eddies fed KE to the block and residual modes. Residual eddies received KE from every other mode.

Finally, 19 (Atlantic block; Fig. 5h) and 26 (maximum zonal index; Fig. 5i) February were analyzed using the same best-shift wavelet basis. While the roles played by *Z*^{re} were very *similar* on these two days, the transfers between the other three modes were each *reversed*. The block got comparable donations of KE from the residual and upstream eddies. The strong zonal-direction flow, mostly captured by the *Z*^{bl} coefficients during nonblocking, was equally fed by the smaller, global residual eddies, as is typically the case in the *climatic* energy cycle described by classical means.

## 4. Discussion

### a. Application of wavelet energetics to blocking

Using the best-shift algorithm, an efficient low-dimensional representation has been constructed of each block structure in the National Meteorological Center data for the intense blocking winter of 1978/79. The time-*average* optimal representation proved to remain well representative as time progressed over the duration of each individual block. In fact, the very strong resemblance of the time- and latitude-averaged height field of the block to a single wavelet prompted speculation in appendixes A and B about the *physical interpretation* of wavelet design.

A criterion for nearness of any wavelet to a given block location has been presented in section 2b. Using this criterion, instantaneous weather map–like fields have been partitioned into four “modes,” representing the block, nearby downstream and upstream eddies, and distant eddies. Such a construction would not be possible by simply truncating, thresholding, or windowing a Fourier analysis (see FO0). This partition and the energetics techniques enabled the computation of the KE budget for these four modes.

In four out of five blocks, the upstream eddies feed KE to the block. In turn, in three out of four cases, the block transmits KE to the downstream eddies. Only five blocks were considered in this paper, and even the whole 90 day dataset considered by FO99 is too brief to make a statistically significant analysis, as explained by FO3 (which *does* present a statistically significant analysis, using 53 winters’ data). Such an analysis was not the goal of this paper; rather the goal was to illustrate how OWA allows structure components to be resolved and questions to be addressed, which could not be by Fourier analysis. Although statistically significant conclusions could not be drawn about blocking energetics [as for any “case study” such as Hansen and Sutera (1984)], a variety of interaction mechanisms are implied by the interactions between the elements of the new, four-mode partition. Drawing deeper conclusions also requires the consideration of other physical mechanisms, for instance involving the temperature and vertical motion fields, and vertical shear. Eventually statistically robust energetics measurements could be correlated with blocking duration time. But the present research is really another pilot study into the use of OWA and energetics, similar to the Fourier pilot study of Saltzman (1957); in this tradition it was deemed more important to develop an understanding of the correct interpretation, and perhaps limitations, of wavelet energetics, than to consider *all* the various kinds of interactions that are available to the physical system.

### b. General remarks about OWA methods and extensions

The tools presented in the sequence of papers by the present author may be applied to any global fields. One would hope that their application to datasets, much longer in time, would yield results with small enough time variance to allow for statistically significant statements, as was the case progressively over a decade of analogous studies by Saltzman and others. As the simplest kinetic energetics become clearly understood, other terms of the energy budget, such as APE, may be added to obtain a broader picture. Indeed, this extension was made by Hasegawa (2000), inspired by Fournier (1995b, 1996, 1998; B. Saltzman and H. L. Tanaka 1996, personal communication). Hasegawa (2000) found triadic eddy KE interactions to be the most important process in 8 out of 10 Pacific block onsets. It may also be useful to consider other budgets, such as momentum, enstrophy, potential vorticity, and Eliassen–Palm flux.

The wavelet methodology may also be refined, for example, by using wavelet analysis also in the latitude direction (with boundary conditions at the poles) or by using biorthogonal wavelets, which are (anti) symmetric. [Although all real *W* with finite *N* (appendix Aa) are strictly *a*symmetric (Daubechies 1992), Beylkin (1995) created such *W*s with arbitrarily small asymmetry.] Another option is to use wavelet packets, which separate indexes of wavenumber *and* scale at each location, or *local cosines*, which possess wavenumber *m* and an *arbitrary* increasing sequence of locations 2^{−1} (*λ _{k}* +

*λ*

_{k}_{+1})

*and*scales

*λ*

_{k}_{+1}−

*λ*. The researcher who develops these refinements should expect to make many technical choices and trade-offs in regard to method. The present author’s choices included the following:

_{k}to use relatively conceptually simpler wavelets than the above choices, trading symmetry and complexity for orthogonality and simplicity;

to use compactly supported rather than only quickly decaying wavelets, trading symmetry for exactness and transform speed [speed being irrelevant for small data analysis, but the author desired to use the same wavelets for numerical modeling (Fournier 1995a) as for data analysis];

to use smoother (larger

*N*) rather than more localized (smaller*N*) wavelets;to use the entropy criterion (FO0) rather than another best basis;

the number (four) of large-amplitude wavelets to retain in order to represent blocking structure; and

the nearness criteria of the nearby eddies to the block, trading energy for localization;

as well as other choices.

It is clear that wavelet analysis offers an alternative to traditional Fourier analysis that is relatively advantageous for certain types of structures, such as those of atmospheric blocking. In accordance with the Heisenberg Uncertainty Principle (see FO0), it is necessary to sacrifice some resolution in the wavenumber domain in order to gain resolution simultaneously in the location domain and also to deal with complications inherent in a mixed, yet completely orthonormal, representation. Nevertheless, the tools introduced to the dynamical meteorological community here and in Fournier (FO0) are offered in part to encourage researchers to complement and possibly transcend the limitations of pure location or wavenumber representations, and finally as an example of both the advantages and complexities in regard to physical interpretation that await the user of such methods.

## Acknowledgments

I thank A. R. Lupo for an informative and useful review and B. Saltzman, R. B. Smith, K. R. Sreenivasan, R. R. Coifman, A. R. Hansen, and P. D. M. Parker for comments on an earlier manuscript leading to this paper. I also thank D. D. Shepherd for advice on Figs. 2 and 5, G. Beylkin and L. Monzon for advising on part of appendix Aa, the Meteorology Department at the University of Maryland (funded by the Office of Biological and Environmental Research, Office of Science, U.S. Department of Energy and by the National Science Foundation, Advanced Computational Infrastructure Organization, Information Technology Research Program) and the Advanced Study Program at National Center for Atmospheric Research for support during the writing process, and the National Center for Atmospheric Research for providing the NMC analyses data.

This material is based upon work supported by the National Science Foundation under Grant 9420011 and through NCAR Project 36211014.

## REFERENCES

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**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

### APPENDIX A

#### Interpretation of Minimum Phase Filters

This appendix investigates the connection between wavelet design signal-processing considerations and the *shape* of wavelets. This connection concerns *physics* inasmuch as by efficiently *representing* a physical structure, one may *illuminate* that structure’s physical role.

##### Definitions and properties

FO0 explains that the particular wavelets *W _{j,k}*(

*λ*) used here are completly determined by a finite low-pass filter sequence

**w**

^{⊥}of length 2

*N*= 20. This filter was designed by Daubechies (1988) to yield a compactly supported

*wavelet mother*(Table 1)

*W*with the maximum number (

*M*) of vanishing moments ∫

^{∞}

_{−∞ }

*t*(

^{m}W*t*)

*dt*= 0, and to have minimum phase (MP). It turns out that MP greatly distinguishes the

*shape*of

*W*from that of other compactly supported wavelets (Fig. A1, discussed below). In this appendix we are concerned with the signal processing and physical interpretations of

*wavelet shape vis-à-vis atmospheric blocking.*Here, MP is a term from signal processing implying that the phase of the Fourier series (inverse transform),

[Daubechies 1992, her (5.1.18)], varies minimally. Daubechies achieved *M* = *N* and made

have periodic MP (Fig. A2c). She did so by requiring that the complex *z* transform *Q*(*z*) be a polynomial of degree *N* − 1 with all roots outside the unit circle 𝒮 in the complex plane. Then the curve *Q*(S) does not enclose the origin, and so as *ξ* parameterizes the curve *Q*(𝒮), the phase

never accumulates a winding of 2*π* (Robinson et al. 1986, p. 154).

Figure A1 shows the roots of four *w̆*^{⊥}(*x* + *iy*) of equal degree 2*N* − 1 = 17. Only the Beylkin and Daubechies (left) filters have no roots inside Σ and hence have MP. Superposed on each panel is a rescaled graph of *W*_{2,3}(*y*) (thick curve), showing how *W*’s *shape* is affected by the roots’ positions. Observe that by placing roots inside Σ, *W* can be made more symmetric about its peak (corresponding to nearly linear phase in Fig. A2c), but in that case **w**^{⊥} no longer has MP.

Figure A2b shows how, regardless of MP, Daubechies and “symmlet” filters have the greatest number of vanishing moments of the high-pass filter *w*_{2−ℓ} ≡ (−1)^{ℓ}*w*^{⊥}_{ℓ+1}, leading to the vanishing moments of *W*. Since

this is also seen by comparing the apparent orders of the common root at *ξ*=*π*, Fig. A2d.

Two more general properties of MP are of physical relevance. First, MP is robust to the addition of interference or noise **a**, that is, **a + w**^{⊥} is also MP if |*ă*| < |*w̆*^{⊥}| on Σ. Second, for fixed |*w̆*^{⊥}|, the phases of MP *w̆*^{⊥} are arranged so that *w*^{⊥}_{ℓ} has its variance as localized at low ℓ (in the *physical* sense: spatially, westwardly, i.e., temporally *antecedentally*) as possible. [Robinson’s (1962) “minimum delay wavelet theorem.” He and other solid-earth geophysicists define a “wavelet” merely as a function *w _{t}* satisfying

*w*

_{t<1}= 0 < Σ

^{∞}

_{t=1}|

*w*

_{t}|

^{2}< ∞, which is very different from the definition reviewed by FO0.] This localization is shown in Fig. A2a.

From the normalization ∫^{∞}_{−∞ }*W*^{⊥}(*t*) *dt* ≡ 1 of the “wavelet father” *W*^{⊥}, and the *two-scale* relations (see FO0), one derives

(Daubechies 1992, p. 193), showing that at each wavenumber *ξ*, *W* may be represented as an infinite succession of MP filters *w̆*^{⊥}, and one non-MP filter,

(Daubechies 1992, p. 163). [The filter **w** evidently cannot have MP, because if

then *w̆*(−*z*^{−1}_{0}) = 0 for |*z*^{−1}_{0}| < 1.]

##### Why might 〈H*〉^{b}_{ϕ}(*λ*) be related to M-P processes?

The physical field 〈*H**〉^{b}_{ϕ}(*λ*) resembles wavelets derived from M-P filters. This implies possible physical interpretations. One possible physical interpretation would be to posit an invertible linear relation between 〈*H**〉^{b}_{ϕ}(*λ*) and another physical quantity. Another would be that the field 〈*H**〉^{b}_{ϕ}(*λ*) effectively deconvolves two other linearly related quantities. Such inferences from signal processing to physics are precedented in geophysics. An example from solid-earth geophysics is the invertible linear relationship between stress and strain, reflected by the invertibility of the filter relating their respective time series (Claerbout 1992). Also, seismic pulse train waveforms are M-P because of a multitude of attenuating reflection events analogous to the process [(A.2); Robinson and Treitel 1980, 253–254]. Perhaps the succession of processes that construct a given atmospheric pattern described by 〈*H**〉^{b}_{ϕ}(*λ*) might be thought of as a long chain of M-P processes. The *causality* (*w*^{⊥}_{ℓ<1} = 0) might be interpreted as reflecting the spatiotemporal asymmetry due to the earth’s rotation (G. Beylkin 1998, personal communication).

What might be the relevant physical quantities for 〈*H**〉^{b}_{ϕ}(*λ*) to have this structure by this reasoning? A conclusive investigation is beyond the scope of this paper, but my literature review inspires one speculation. The linear relationships indicated here may be related somehow to the piecewise linear relationship between streamfunction *ψ* (geostrophically proportional to *Z*) and (potential) vorticity. This relationship is part of the modon theory (Tribbia 1984; Haines and Marshall 1987; Haines and Malanotte-Rizzoli 1991) and diagnosis (Butchart et al. 1989) of blocking.

##### Inverse problem: Design of wavelets

These questions may be inverted. Assume some physically motivated assumptions about the process chain that constructs a certain atmospheric pattern. How can the freedom of choice inherent in the design of the wavelet filter **w** be exploited to create especially suitable wavelets? The variety of wavelets shown in Fig. A1 are the result of that freedom being exploited for other purposes. From theorem 6.3.6 of Daubechies (1992), all that is required is the existence of *any* function *w̆* verifying

[hence *w̆*(1) = 0]. The rest of the OWA apparatus follows from equations given above. Likewise, any *physical* requirements on the wavelet *W* may be translated into additional requirements on *w̆*. That translation would use the same equations in reverse, compatible with the existence of solutions to (A.5).

##### Effects of topography on 〈H*〉^{b}_{ϕ}(*λ*) phases

It also should be pointed out that the influence of earth’s topography (Fig. A3) on 〈*H**〉^{b}_{ϕ}(*λ*) must be considered. Although the 〈*H**〉^{b}_{ϕ}(*λ*) that resemble *W*^{ε+}_{1,k}(*λ*) each involve different *λ*^{+}_{ε}, in each case there is 〈*H**〉^{b}_{ϕ} = 0 < ∂_{λ}〈*H**〉^{b}_{ϕ} near *λ* ≈ 50°W, which may be related to the Greenland ridge, just to the east. Similarly, there usually was 〈*H**〉^{b}_{ϕ} > 0 > ∂^{2}_{λ}〈*H**〉^{b}_{ϕ} near *λ* ≈ 150°W, just upstream of the Canadian Rockies, and a broad sector of 〈*H**〉^{b}_{ϕ} < 0 < ∂^{2}_{λ}〈*H**〉^{b}_{ϕ} across the Eurasian continent. It seems that topography may control certain phase aspects of 〈*H**〉^{b}_{ϕ}, but not necessarily magnitudes.

### APPENDIX B

#### Minimal Fourier Representation of the Blocking Wavelet

In this appendix, we try to use a simpler function *B*(*λ*) to describe the previous findings that used wavelets. Consider the function

This function *B*(*λ*) projects 98% of its variance onto the single wavelet *W*^{ε+}_{1,k}(*λ*). This wavelet in turn *represents most of the blocking variance*, that is, *B* ≈ *W*^{ε+}_{1,k} to within 2% for all *J*. The appropriate *ε*^{+} is empirically determined by

Figure A4 shows various closely agreeing estimates of *λ*^{+}_{ε}, and Fig. A5 (left) shows *B* and *W*^{ε+}_{1,k} for various **w**. One can see that *B* ≈ *W*^{ε+}_{1,k}.

Figure A5 (right) provides evidence of how the *m* > 2 zonal-wavenumber phasors, which distinguish different **w**, contribute to the similarity of *B* to *W*^{ε+}_{1,k}. Both Daubechies wavelets perform similarly. The symmlet does a little more poorly, perhaps because wavenumber 3 interferes with wavenumbers 1 and 6 more destructively than does either Daubechies wavelet. (Destructive interference is indicated by the symmlet phasors being mutually opposed.) The Beylkin wavelet has phasors similar to the symmlet, but with wavenumber 1 less reinforced by 5–6 and 10. The “Coiflet” simply has too much magnitude at *m* > 2.

##### Signal processing interpretation

The resemblance of *W ^{ε}*

_{1,k}to

*B*is roughly explained as follows. If

*f*=

*W*

^{ε}_{1,k}, then by construction, the wavelet coefficients are Kronecker sequences

*f̃*

_{j,k′}=

*δ*

_{j,1}

*δ*

_{k′,k}. Also, the coarse-scale coefficients

*f*

_{j≤1,k′}= 0. Putting these values into the OWA

*reconstruction*convolution algorithm (see FO0) and performing the inverse Fourier transform [(A.1) with

*z*=

*e*] yields

^{iξ}(Daubechies 1992, p. 161). We are free to let *ɛ*_{0} = 1 − *k*. The Fourier series coefficient (B.1b) of the periodic function |_{𝒮} of *ξ*, which has wavenumber index nearest to 2^{J−1}(*π*^{−1}*λ* + 1), approaches *W ^{ε}*

_{1,k}(

*λ*) as

*J*→ ∞. Conversely, if

*ξ*→ 2

^{−J }

*m*, then

(B.1c–d) as follows. Equations (A.3), (A.5), and (B.2) imply that (*e*^{−21−Jπim}) = 0 if 2^{−1}*m* is an even number; otherwise

From (A.3) and the grayscale along Σ in Fig. A1, one may infer that the product of the low-pass (*w̆*^{⊥}) and high-pass (*w̆*) filters, dilated by the 2* ^{j}*s in (B.2), retains support only at low positive wavenumbers. Thus the ratio just computed and the overall normalization serve to roughly explain the resemblance. For each root (A.4), the product in (B.2) introduces a sequence of roots

*z*

^{2−j}

_{0}of that tend to the 2

*th roots of unity on Σ. Although each*

^{J}*w̆*

^{⊥}(

*z*) factor has MP, the full product (B.2) tends not to.

##### Consequences in terms of Fourier spectral dynamics

A distinguishing characteristic of (B.1a) is that the only two wavenumbers |*m*| = 1, 2 have Fourier coefficients of equal magnitude. This may be relevant to the physical structure of the geostrophic zonal wind coefficients *û _{m }*(3.1) as follows. As a result of nonlinear advection −

*uu*, the relative phase velocity

_{λ}where all terms involving only wavenumbers *m* and 2*m* have been separated from the sums. Then equal magnitude implies that the phase difference is constant, unless other wavenumbers or effects enter. In fact, advection by itself would introduce those other wavenumbers, but in the sense that they do not affect their *own* relative phase, such equal-magnitude pairs are conducive to a kind of “blocking,” that is, quasi-stationary ridges.

##### Consequences in terms of low-order blocking models

Another consequence of the simple expression (B.1a) concerns a study [descended from Platzman’s (1962)] by Wiin Christensen and Wiin-Nielsen (1996), in which a low-order model based on the inviscid barotropic vorticity equation with Newtonian forcing was used to investigate blocking dynamics. Their blocking-type equilibrium solutions (their Fig. 7) strongly resemble (B.1a). In fact, because of the (*π*/2) phase difference, several nonlinear terms in their low-order model are simplified, and (B.1a) is an equilibrium solution for a choice of Newtonian forcing similar to theirs.

### APPENDIX C

#### Maximization of Momentum Transport with Respect to Contour Orientation

Let 𝒞* _{α}* denote a streamfunction

*ψ*= 0 contour loop, which has an “axis” (in some general sense of having an elongated asymmetry) at an angle

*α*north from east. Let

*R*(

*α*) be the covariance between rotational wind components

*u*

_{0}≡ −

*ψ*and

_{ϕ}*υ*

_{0}≡ sec

*ϕψ*around 𝒞

_{λ}*. Then*

_{α}*R*(

*α*) also equals the covariance between

*u*≡

_{α}*u*

_{0}cos

*α*−

*υ*

_{0}sin

*α*and

*υ*≡

_{α}*υ*

_{0}cos

*α*+

*u*

_{0}sin

*α*around 𝒞

_{0}, the same loop parameterized by coordinates (

*x*,

*y*) rotating the axis to point along

**∇**

*x*. Denoting the complex value

*q*≡

_{α}*u*+

_{α}*iυ*=

_{α}*e*

^{iα}q_{0}, one can easily show that

obtains a maximum value

for

Then *α*_{m} = (*π*/4) as long as in the mean around 𝒞_{0} one has 1) *u*^{2}_{0} > *υ*^{2}_{0} and 2) the net correlation between *u*_{0} and *υ*_{0} is zero. The conditions turn out to be physically intuitive and reasonable; condition 1 almost amounts to 𝒞* _{α}* being

*elongated*along its axis, while condition 2 may follow from simple symmetry assumptions on 𝒞

_{0}. For example, consider elliptical coordinates

Then a *ψ* = 0 ellipse may be written as

Then at any point (*x*, *y*),

Reinterpreting **∇***ψ* → (∂_{x}υ_{0}, −∂_{y}u_{0}), conditions 1 and 2 are satisfied under the weak assumptions

To the extent that the conditions fail, then *α*_{m} deviates from (*π*/4) according to (C.1).

This result formalizes a heuristic argument about momentum transport contributed by a streamfunction line element, consistent with discussions going back to Starr (1948). It also implies a diagnostic tool similar to the E-vector analysis of Hoskins et al. (1983), with the advantage that no temporal statistics are necessary; for any given *instantaneous* distribution of streamfunction, every contour [not encompassing |*ϕ*| = (*π*/2)] may be assigned a vector of magnitude *R*(*α*_{m}) directed *α*_{m} north of east, which signifies the transport of momentum contributed by that contour, in a direction that is significant w.r.t. both the contour’s orientation and the structure of flow along the contour. However, it should be mentioned that such an analysis may be of limited utility, as is the E-vector analysis, in case local conservation of momentum is only poorly maintained (R. Saravanan 1999, personal communication). The author is currently investigating how to extend the ideas sketched out in this appendix to the transport of a conserved or quasi-conserved quantity such as potential vorticity.

## Footnotes

* Current affiliation: Department of Meteorology, University of Maryland, College Park, College Park, Maryland

*Corresponding author address:* Aimé Fournier, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. Email: fournier@ucar.edu