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Atmospheric Energetics in the Wavelet Domain. Part II: Time-Averaged Observed Atmospheric Blocking AIME´ FOURNIER* Department of Physics, and Department of Geology and Geophysics, Yale University, New Haven, Connecticut (Manuscript received 17 July 2001, in final form 24 July 2002) ABSTRACT Wavelet energetics (WE) is a useful generalization of traditional wavenumber energetics, for analyzing atmospheric dynamics. WE is doubly indexed by wavenumber band j and location k. The interpretation is that 2 to the jth power is proportional to zonal wavenumber bandwidth and bandcenter, and k is proportional to longitude. Here, all 44 Atlantic and 16 Pacific atmospheric blocking events that are observed in 53 winters of the NCEP– NCAR reanalysis, and last no less than 6 days, are analyzed. Temporal average and variance maps suggest that persistent blocking structures are associated with smaller-scale eddy activity concentrated on either side of the block. Wavelet energetics, partitioned by blocking state and sector, are averaged over 308–808N and 10–100 kPa. For j above two, WE indicates that kinetic energy (KE) and enstrophy increase upstream and decrease downstream of both Atlantic and Pacific blocking. At smaller j the increases are at the block; this includes zero j for Pacific, but not Atlantic. As measured by new wavelet flux functions, at j above one, on average there are localized upscale KE and enstrophy cascades upstream, and localized downscale cascades downstream of the blocks. This is not significantly determined for KE in the Atlantic. Correlating WE, blocking relative to nonblocking, suggests a similarity of Pacific to Atlantic energetic patterns if the former are shifted over the latter; this holds for all enstrophy WE, and for KE stocks, but not for other KE wavelet energetics. The theoretical conservation of wavelet flux is numerically supported. Statistical significance is strongly suggested, if not rigorously established.

1. Introduction The term atmospheric blocking refers to a large-scale, persistent, anomalous, high-isobaric geopotential height region, referred to as a block or blocking structure. In the Northern Hemisphere, such structures climatologically tend to be over the eastern Atlantic or Pacific ocean. Blocking impedes the normal, mostly zonal progress of synoptic weather patterns, and there is reason to believe that weather is more predictable during blocking situations than during ‘‘normal’’ meteorological states (Bengtsson 1981). The earliest studies were in the late 1940s (Berggren et al. 1949; Elliott and Smith 1949; Rex 1950a,b) and mainly descriptive. They included proposing block creation mechanisms using simple momentum or hydraulic arguments. Saltzman (1959) proposed that quasi-permanent disturbances in general may be maintained by an upscale kinetic energy (KE) cascade. This idea was * Current affiliation: Department of Meteorology, College of Computer, Mathematical, and Physical Sciences, University of Maryland, College Park, Maryland. Corresponding author address: Aime´ Fournier, National Center for Atmospheric Research, P. O. Box 3000, Boulder, CO 80307-3000. E-mail: [email protected]

later applied to blocking by Green (1970, 1977), Hansen and Chen (1982, p. 1162a and Figs. 5a and 15a), Riyu and Ronghui (1996, Figs. 8–10), Nakamura et al. (1997), and others cited below. From the collection of blocking studies by Benzi et al. (1986), the most relevant to the present study was by Shutts (1986), further developing his ‘‘eddy straining mechanism’’ (Shutts 1983). He sought to explain how eastward propagating smaller ‘‘eddies’’ (i.e., smaller-scale synoptic structures) are meridionally elongated by the strain field of the larger block as they approach it, and transfer their KE to it. Shutts’ conceptual model has motivated many subsequent studies (e.g., the works cited above and Butchart et al. 1989). Haines and Marshall (1987) used a simplified dynamical model to show that such eddy forcing could create and maintain block-like (also wavelet-like) structures known as modons. ‘‘Remarkable resemblance’’ between time sequences of the model output and real-world observations suggested that much of the structural development could be understood in terms of barotropic advective deformation, without considering conversions from available potential energy (APE). This approach is adopted in the present research, whose aim is to distinguish between blocking and nonblocking in the 53-yr wintertime observational record, using orthonormal wavelet analysis (OWA; Fournier 2000, here-

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after Fo0) and wavelet energetics (WE) introduced by Fournier (1995, 1996, 1998, 1999; Fournier 2002, hereafter Fo1). Hansen and Sutera (1984, hereafter HS84) studied five blocks in the 1978/79 winter, employing Hansen’s extension of Saltzman’s Fourier energetics (Saltzman 1957; Kanamitsu et al. 1972; Hansen 1981; Fo1). They found a ‘‘pronounced upscale cascade of KE and enstrophy from intermediate to planetary-scale wavenumbers during blocking.’’ Because Fourier spectra lack location information, they could not possibly associate the cascades with the blocks, spatially. Kung and Baker (1986) noted that ‘‘it is reasonable to have’’ complex energetics variations between cases, which compels an investigator ‘‘to perform an energetics analysis with the proper design . . . to identify the similarity and dissimilarity of energy processes among blocking cases.’’ Furthermore, ‘‘standard spectral energetics should be supplemented with a diagnosis of the local energy balance.’’ This motivates the present application of WE to blocking, in order to objectively combine location and wavenumber information. HS84 also found that Fourier wavenumber transfers from APE to KE were of greater magnitude than the transfers between different wavenumbers of KE, but varied less between blocking and nonblocking averages. Similarly, Mullen (1987) determined that ‘‘barotropic processes are of primary importance in the eddy forcing of the upper-level circulation.’’ These findings additionally justify the neglect of the APE–KE transfer in the present study. Fournier (1995, 1996, 1998, 1999) used WE to study the same blocking events in the 1978/79 winter as HS84 studied with Fourier energetics. He found upscale cascades of KE downstream, and downscale cascades upstream, of the mean Atlantic and Pacific blocks. This work inspired a comprehensive study (B. Saltzman and H. L. Tanaka 1996, personal communication) by Hasegawa (2000), who found triadic eddy KE interactions to be the most important process in 8 out of 10 Pacific block onsets. Both investigators found considerable temporal variability in blocking energetics, which indicates a need to examine more events, and so build up statistics. The definition of blocking is still a subject of debate (Liu 1994). Some believe that blocking is not a statistically significant anomaly from the climatological state (Lindzen 1986). The present study is aimed at characterizing blocking using the wavelet multiresolution analysis techniques reviewed by Fo0, and statistical significance is estimated here, given the limited computation feasability for such a large dataset [section 3b(3)]. An issue of interpretation of time-averaged nonlinear interactions was raised by Feldstein (1998) and summarized by Cash and Lee (2000): time averages tend to suggest an approximate balance between linear and nonlinear terms, but can obscure the relative magnitude of transient linear-term fluctuations. This is addressed in the present study by measuring the significance of each

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time-averaged nonlinear interaction mode relative to its own temporal variability. A blocking definition is adopted in section 2. Some statistics are presented in section 3, that give a general picture of blocking. Fournier (1999) and Fo0 showed that multiresolution analysis has some advantages over simple bandpass filtering (e.g., Wiin-Nielsen and Chen 1993, hereafter WNC) for representing blocking structure and dynamics; the main result of the present study is the WE presented in section 3b, that demonstrates how such global spectral transfers as computed by HS84 and others may be instantaneously distributed among multiscale interactions at different longitudes. Beyond the implicit energetics cycle, further physical interpretations are made, where that is straightforward; however, the basic goal accomplished here is to document the physical climatology of 60 blocking events in 53 winters, in a new and useful way. 2. Methodology a. Data and blocking definition The data are the wind components u, y , and geopotential height Z from National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research (NCAR) gridded Northern Hemisphere analyses (CDC Data Management Group 2001). The spatial domain consists of full latitude circles between 308 and 82.58N at 2.58 spacing, on the 12 standard p levels in [10, 100] kPa. Fifty-three years of December, January, and February (DJF), 6-hourly data are included, a total of I 5 19 220 records. The following definition of blocking is similar to that of HS84, and is inevitably somewhat arbitrary. A blocking event is defined as a time interval no shorter than 6 days, throughout which the crest height at longitude l 5 lm (t) exceeds the zonal band mean value ^Z (t)&w by .250 m and is quasi stationary (see Table 1 for notation definitions, and the appendix for a definition of quasi stationary). Figure 1 shows the rescaled crest height deviation as a function of t (day, abscissa and year, ordinate) for the 53 DJF of these data. All negative anomalies are shown in light gray, as are positive anomalies that fail the blocking criterion (are too weak or too brief ). Here, black, gray, and white indicate Atlantic, Pacific and (two) continental blocks, respectively, identified as described in the appendix. Given these time labels and an arbitrary field f , define its time averages f b and f n restricted to blocking- and nonblocking-state observations, respectively. Using these averaging operators, the corresponding biased variance estimates (s b f ) 2 and (s n f ) 2 can also be computed. Specific superscripts b 5 A or P correspond to time averages over blocks located in the Atlantic and Pacific oceans, respectively.

15 JANUARY 2003 TABLE 1. Definitions. APE b DJF

FE, FK Ix j

k K , m ME, MK n OWA p, p r, ps t TE, TK wp

WE, WFF, WTF Xm

dbX z l, lm(t) l j,k w, w l rbX sx f v ¯f f. fj X j,k, ¯f j,k ^ f &w ^f&m ^ f &p Dp r x corr(X,Y ) shiftlJ,,X j,k

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Available potential energy Superscript indicating time average over blocking events; b takes A or P values for Atlantic or Pacific block locations (see also n) Dec–Feb: days 1–60 of years 1948–2000 and days 335–365 (366 if leap year) of 1948–1999 ⇒ 60 3 53 1 31 3 39 1 32 3 13 5 4805 days [ 222j Sjj950 2j9 TEj9,[2j92jk], flux of enstrophy, similarly flux of KE Number of observations of state x 5 b, n, A or P in 1 or 53 DJF Resolution, zonal wavenumber octave band ø]2j 2 1, 2j ], scale ø 212jp r! cos w, index, in range 0, 1, 2, · · · J 2 1, · · · (r! 5 mean earth radius, conveniently [ 1) Locations (longitudes l j,k ) index, in range 0, 1, 2, · · · 2j 2 1 (mod 2j ) [ 221 (u.2 1 y .2 ), eddy KE j-independent longitude index, in range 0, 1, 2 · · · 2J 2 1 (mod 2J ) Zonal wavenumber index, in range . . . 22J21, · · · 21, 0, 1, · · · 2J21, · · · Mean-flow interaction for enstrophy, for KE Superscript indicating time average over nonblocking events (see also b) Orthonormal wavelet analysis Pressure, rth of 12 standard pressure levels, surface pressure Time Triadic interaction for enstrophy, for KE Scale factor: for KE transfer functions, [ g so that ^ & p converts (kg21 ) to (m22 ); for enstrophy, [ S12 r51 Dp r 5 96.25 kPa ø 1 atm, to retain units Wavelength energetics, wavelet flux function, wavelet transfer function Object with Fourier zonal wavenumber index m [ X b 2 X n, change in X from time-averaged nonblocking to time-averaged blocking [ secw[y l 2 (coswu )w ], relative vorticity Longitude, longitude that maximizes ^Z(l, 50 kPa, t)&w [ (212jk 2 1)p , k 5 0, · · · 2j, longitude of peak W j,k Latitude, latitude p /6 1 p l/72 b 2 21 [ ^dbX &mp /(I21 ^s nX &2mp)1/2, sigb ^s X &mp 1 In nificance criterion for dbX, the denominator being an estimated std dev of ^dbX &mp [ [( f 2 ) x 2 ( f x ) 2 ]1/2, biased estimate of std dev of state f x for x 5 b or n Isobaric velocity J21 [ (2p )21 #2p p f (l)dl 5 22J S2,50 f (l J,, ) 1 O(42J ), the zonal-mean f [ f 2 ¯f, the zonal deviation f For j 5 l, w, p, t, partial derivative of f w.r.t. j Object with wavelet indexes j and k, wavelet coefficient of f [ (1/11) S20 l510 f (w l ), latitude mean 21 [ (S20 S20 l50 cos w l ) l50 f (w l ) cos w l , sine-latitude mean 12 [ w21 Sr51 f ( p r ) Dp r , pressure mean p [ p 2 2 p1 (r 5 1), p r 2 p r21 (r 5 12), 221( p r11 2 p r21 ) (otherwise) Greatest integer # x j21 j21 j921 J21 J21 [ (Sj50 S2k50 X j,kY j,k )/(Sj,j950 S2k50 S2k950 X2j,kY2j9,k9)1/2 [ Xj,k12j2J(,22J21)

b. Wavelet energetics The relevant tools of OWA were reviewed by Fo0, and the WE methodology is fully discussed by Fo1. Briefly, K j,k is the instantaneous stock of KE, in a mode described by two indexes. The j index indicates zonal scale ø212j pr! cosw and the k index indicates longitude ølj,k as defined in Table 1. That is, Kj,k is ‘‘contained’’ in wind features of this scale at this location. This localization is rigorously equivalent to zonal wavenumber localization in the band ø]2 j21 , 2j ]. The index ranges are j 5 0, 1, 2, · · · J 2 1 for length-2 J longitude data, and k 5 0, 1, 2, · · · 2 j 2 1 (mod 2 j ) to cover a whole latitude circle from 1808W to 1808E. This localization in dual location and wavenumber domains is a powerful generalization of the traditional Fourier energetics. From the primitive equations, Fo1 derives the evolution equation (in part) ]t K j,k 5 MK j,k 1 TK j,k 1 BK ha j,k 1 · · · . Here, MK j,k and TK j,k are the from-mean and from-eddy (via so-called triadic interactions) wavelet transfer functions (WTFs) for KE. BKhaj,k comprises boundary transfers to KE from local horizontal geopotential-flux divergence, whose potential importance to this study was pointed out by an anonymous reviewer. [The subscript ‘‘a’’ recalls the derivation including such a term in coupling to APE (e.g., WNC).] When a WTF is positive or negative, the terms ‘‘gain’’ and ‘‘loss’’ are used, respectively. The characteristics of multiscale nonlinear eddy interactions can be summarized in terms of the cumulative wavelet flux function (WFF), denoted FK j,k . It is positive for k local downscale KE cascades across j (net transfers from large to small scales), and negative for upscale cascades. Similar to these expressions, one writes enstrophy stock in terms of vorticity wavelet coefficients: E j,k [ 2 21z˜ j,kz˜ j,k . Necessarily, all the j, k contributions tend in the infinite sum as

OO `

2j21

j50 k50

E j,k → 221 z . z . (see Table 1). Enstrophy dynamics is given by

]t E j,k 5 ME j,k 1 TE j,k 1 · · · (in part). This introduces new from-mean and triadic WTFs, with appropriate infinite sums: ME j,k [ 2(secw u zfl j,k 1 y˜ j,k z w )z˜ j,k → 2secw u zl z . 2 z w y .z . 5 2z w y .z . , TE j,k [ 2221 {secw [u .zl 1 (u .z . )l ] 1 y . (zw. 2 z . ]w ) 1 v . (z p. 2 z . ]p )}˜j,k z˜ j,k → 0 (and a WFF). Following B. Saltzman (1957; 1995, personal communication), in the present study terms involving v.] p were neglected relative to horizontal advection.

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FIG. 1. Fifty-three-year dataset represented as time series of 50-kPa blocking index max l^Z . &w /250 m 2 1 (light gray for nonblocking states, and black, gray, and white for Atlantic, Pacific, and continental blocks, respectively). On the ordinate there is an offset for each year, with three winter months (labeled by D-, J-, F-day filling a line. Each time series is labeled at right by number of blocking days, with numbers of A, P events in parentheses.)

The ‘‘coiflet12’’ wavelet basis functions W j,k (l) were used to analyze l dependence. The wavelet coefficients were phase corrected for j . 2 as described by Fo1. Experiments with test functions of longitude showed that the mode-of-energy shift (Fo1) was found to correct exactly the coefficients of l mod 2p, and undercorrect slightly the coefficients of sinl. 3. Results a. Basic time-averaged statistics Figure 2 shows the geopotential height time averages Z n and Z b (see section 2a and Table 1). Blocking is evidenced by the unusually strong ridges in the Atlantic

(A; Fig. 2b) and Pacific (P; Fig. 2c) cases compared to the nonblocking (Fig. 2a) cases. Haines and Marshall (1987) noted the resemblance of this phenomenon to nonlinear wave breaking on a fluid surface. This is visually striking for P, but one should not infer a mechanism similar to wave breaking. At lower pressure, the high Z contours are closed off from the Tropics. Figure 2 also shows s n,b Z. The SW–NE distortion of s n Z in places suggests the typical eastward–poleward progression of synoptic weather patterns such as Atlantic winter storm tracks. This apparent diagonal arrangement of contours is rearranged in s bZ; instead of (north-) eastward-translating structures, what makes up the variance during blocking are small regions of transient eddy activity in the vicinity of the block. Here transient means con-

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FIG. 2. Time-averaged fields (a) Z n for nonblocking, (b) Z A for Atlantic, and (c) Z P for Pacific blocking; the averaging times for (a), (b), and (c) correspond to time series segments in Fig. 1 shaded light gray, black, and gray, respectively. Maps at 70 kPa vs longitude l and latitude w use a 6-Dm contour interval (black curves) and repeat the line style sequence every 24 Dm. Note the pronounced ridges in (b) and (c), where the black bars indicate the ls that maximize the latitude means ^Z&bm. Shading indicates time std dev field s x Z (section 2a), with a 1-Dm contour interval. Titles show total numbers Ix of observations of the three states.

tributing to the time variance, but not to the time average. There are two very strong maxima of s A Z, near the southern tip of Greenland and western Russia, that is, very symmetrically on both sides of the ridge in Z A over Norway. Similarly, the dramatic ridge of Z P over Alaska is (less symmetrically) flanked by local s P Z maxima, just to the west (upstream) and (more weakly) in northeastern Canada. Both s P Z maxima are similar to, but much more concentrated than corresponding s n Z maxima, and are closer to the block sector. This preliminary analysis suggests that large blocking structures are strongly energetically involved with smaller eddies around them. This will be investigated systematically in section 3b. Having looked at the blocking record as a time series (Fig. 1) and time-averaged picture (Fig. 2), we briefly turn to a statistical summary. The time-averaged crest

longitudes lbm over all events are 3 6 48E (A) and 150 6 78W (P). Although the average lAm seems numerically sharper, this is because there are 2.5 times more A events, and 11% of the P events are outliers. One must look at the distributions of all lbm, shown in Figs. 3a,b. As a consistency check, note that the averages here approximate the crest longitudes 22.88E (A) and 1388W (P), calculated from the event-average ^Z&bm fields in Fig. 2 (see Table 1). From Figs. 3a,b one sees that lm (t) varies more for A than P. This can mask possibly significant WE in the t average (J. Tribbia 2001, personal communication), because OWA proceeds from high to low j. There is also more blocking-frequency and -duration (Fig. 3c) variability for A (D’Andrea et al. 1998), with the same effect. Discarding the two P outliers (8.75 days in 1957 and 7.5 days in 1978) reduces the l standard deviation from 288 to 68. For consistency, one A

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event .508 west of the average was also discarded (7.25 days in 1958). b. Wavelet energetics of blocking In this section we present the WE of blocking and nonblocking states. The 53-winter total numbers of observations of the various states are In 5 16 943, IA 5 1657 (44 events) and IP 5 526 (16 events). Thus A and P blocking events account for about 10% and 3% of the total DJF observations, respectively. Similar A–P disparity was found by Lejena¨s (1995) in analyzing 94 yr of sea level pressure data. The wavelet spectra, WTF and WFF statistics ^K n,b j,k & mp etc. are presented in sections 3b(4) and 3b(5). For convenience the ^ & mp-mean notation will henceforth occasionally be omitted. Resolutions j . 5 are not shown since their KE transfer magnitudes are ,2 mW m 22 . Since only means are discussed here, we note that w dependence of WE for 1978/79 DJF (the most blocked winter in Fig. 1) was discussed by Fournier (1999). 1) GLOBAL

SUMS

The global sums are in Table 2. These can be favorably compared with analogous quantities in the literature. The j, k sums of KE stock in the OWA domain, K j,k , are less than the stationary zonal eddy KE stock reported in WNC (their Fig. 7.3) by ,29% of the latter. That result came from a much briefer record and on a wider domain than the present one. The sum of interactions with the mean flow MK j,k has the opposite of the usual sign because the zone of u. y . . 0, (secwu )w . 0 south of 308N was omitted. This effect was also displayed by Oort and Peixo´to (1974, Figs. 4c, 5a) and Lee (1983, Figs. 11, 25–26). (The present latitude cutoff choices are explained in the appendix.) Comparable MK may be inferred from figures in HS84. The conservation law, that nonlinear interactions only transfer KE and enstrophy among the j, k, without creating or destroying any, is reflected in the low sums for the triadic interactions TK j,k and TE j,k . By symmetry, these sums would vanish identically if the longitude gradient ] l were applied exactly. Here, they are ,0.02% and ,0.20% of the rms values for TK j,k and TE j,k , respectively. The latter term has poorer cancelations because it includes more ] l operations than the former. The sums of BKhaj,k are close to analogous values discussed by Kung (1966). Also the E j,k sums are ø10% less than the sum of HS84’s Fig. 1b, and the ME j,k sums are comparable to the sum of HS84’s Fig. 4b. Both the present and HS84 results

← FIG. 3. Distribution of blocking events of Fig. 1: event-averaged longitudes in the (a) Atlantic, 45 events with total average 3 6 48E

(light gray line), standard deviation 248; (b) Pacific, 18 events with total average 1508 6 78W, standard deviation 288; and (c) durations of 44 A and 16 P events, stacked.

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TABLE 2. Global wavelet energetics (see Table 1 for definitions) 2j21 21 J21 2j21 x 2 1/2 SJ21 j50 Sk50 ^Xj,k&mp. Subscripts indicate uncertainty (Ix Sj50 Sk50 ^s Xj,k&mp) . Blocking state x X

(Units)

K MK TK BKha E ME TE

22

(kJ m ) (mW m22 ) (mW m22 ) (mW m22 ) (Ms22 ) (Ms23 ) (Ms23 )

n

A

P

1240.0 2 230.0 49 0.08 55 21160.0134 428.01 251.0 27 0.21 25

1260.0 6 208.0149 0.08168 21160.0 419 431.0 2 258.0 82 0.21 77

1400.012 254.0 261 0.11 328 2787.0 787 440.0 4 231.0133 0.23140

for from-mean enstrophy transfer sums are opposite the usual sign, for the reason just cited with respect to MK j,k . 2) EXAMPLE

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To provide a conceptual picture, WE are given in Table 3 for the strong A-blocking instantaneous state illustrated in Fig. 4a. First of all, note in Figs. 4c–h how each localized high and low Z feature (light or dark closed-contour packet) is associated with a wavelet with appropriate coefficient sign (up or down triangle markers). The 15 wavelet components shown there sum to the fields in Fig. 4b, comprising, for example, 92% of the Z variance. In particular, scales indexed by j 5 2 and 3 isolate wave trains upstream and downstream (east) of the block, in Figs. 4e–f. Overall, the decomposition is independently consistent with the blocking description, in that all the largest magnitude coefficients happen to be near the block (see Fo0). Several WE values in Table 3 permit ready physical interpretation. For example, the maximal gains TK 2,k.0 and TE 2,k.0 from eddies are probably compensated by losses TK 1,k , TE 1,k (larger-scale neighbors) and TK 3,4#k#5 , TE 3,4#k#6 , and TE 4,7 (smaller-scale neighbors). These losses are due to eddies that neighbor the gaining components in space and scale. The maximal value E 2,2 is associated with the apparently maximally rotational (closest packed) streamlines, that appear at the block in Fig. 4e. These and other anticyclonic geostrophic streamlines to their west exhibit a u–y anticorrelated deformation, that is, flattening of streamlines where uy , 0. This accounts for ME 2,1#k#2 , 0 physically. The qualitative arguments associated with this picture should be kept in mind in order to interpret the ensuing results. Unfortunately, it is not straightforward to infer any similar picture from the WTF values alone. 3) STATISTICAL

SIGNIFICANCE

It should be pointed out that for the available number of events, the time standard deviations of KE and especially enstrophy transfers can be of comparable magnitude to the time averages. This was also the case in the early Fourier-energetics studies by Saltzman (1958),

TABLE 3. Wavelet energetics (see Table 1 for definitions) for 0000 UTC 25 Jan 1979, 50 kPa (Fig. 4). For each scale index j, at most 2 the three longitude indexes k of largest coefficient magnitude |^Z˜j,k &m| are listed. Boldface values are discussed in the main text; for example, note the transfer from j 5 1, 3 to j 5 2 (a major block component) for both KE and enstrophy. K (dJ kg21 )

j

k

0 1 1 2 2 2 3 3 3 4 4 4 5 5 5

0 83 0 70 1 107 1 87 2 94 3 37 4 56 5 27 6 24 7 75 8 28 9 11 15 6 16 8 17 5

MK

TK

BHha

(mW kg21 ) 280 14 82 259 254 12 33 230 18 23 72 32 215 30 43

50 28 288 104 68 85 217 29 6 270 64 17 22 21 7

150 211 2249 254 215 240 2130 216 240 218 218 22 4 37 26

E (s22 ) 16 17 30 25 43 16 25 15 15 33 30 27 3 7 8

ME

TE (s23 )

2151 14 94 2109 2228 57 579 275 234 2100 192 2334 40 2762 2523

190 2232 2338 156 113 189 2237 273 290 2328 290 143 33 2305 4

Saltzman and Fleisher (1960, 1962), and Saltzman and Teweles (1964, henceforth ST64) with daily data, and by inference from the Fourier-energetics time series of Hansen and Chen (1982) and Kung and Baker (1986). Therefore even if the time statistics were Gaussian, which may not be the case, it could be far from the whole dynamical story that is described by the time averages. This is a shortcoming of studies such as HS84 and others, that present only time averages. Likewise, single- or few-case studies can be illuminating, but in general they are not conclusive. Several transfers (e.g., P & mp ) for the present 53-DJF study ^MKP2,0& mp and ^TK1,0 are opposite to the single-DJF case study of Fournier (1999), although the latter is compatible with traditional energetics studies of the same period and region. The statistical significance of the present results regarding blocking WE is limited by the shortness in time of the data, that nevertheless comprise the longest complete set available. As a conservative measure, significance shading is used in Figs. 7–17b–c to show the WE significance criterion rb X j,k defined in Table 1. This criterion is just the ratio of the change in a quantity, to the ‘‘standard error’’ of that change [ST64; Devore 1987, Eq. (5.13), Example 6.8]. Henceforth, we shall understand the term significant as follows. A quantity X bj,k is significantly greater or less than X nj,k if and only if rb X j,k is greater than two or no greater than minus two, respectively. Such X bj,k values are shaded white or dark gray, respectively (regardless of their sign, which is indicated by a bar standing up or down from the plane). Bars are shaded light gray or gray for the cases X bj,k greater or less than X nj,k , but with rb X j,k lying in ]0, 2] or ]22, 0], respectively. In later Figs. 7–17a in this article, the same shading scheme applies with significance understood as certainty, that is, ratio to the stan-

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FIG. 4. For 0000 UTC 25 Jan 1979, 50 kPa: (a) Z . (shading and 60-m contours) and u.i 1 y .j (vectors, speed # 42 m s 21 ) vs l, w; (b) reconstruction of (a) using the 15 components (c)–(h) for each zonal scale location index pair (j, k) in Table 3. Triangle markers show center ls and signs of mean wavelet coefficients ^Z˜ j,k & m , up for 1, down for 2. Here, (c)–(h) are labeled by the corresponding k-summed variance contributions to Z, u, and y , and (c)–(h) sum to (b).

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TABLE 4. Significant Atlantic-blocking (compared to nonblocking) KE (in kJ m22 ; Fig. 9) and enstrophy (Ms21 ; Fig. 10) stocks for j , 5. Here, rA is the significance operator defined in Table 1, and only values with |rA| .2 are shown. Boldface values arise from the data shown in Fig. 6.

FIG. 5. Shading code for Figs. 7–17, based on mapping the significance criterion rb X j,k in Table 1 in the four intervals ]2`, 22], ]22, 0], ]0, 2], and ]2, `[ to the shades dark gray to white; the threshold of significance is the magnitude two.

dard error estimate I 21/2 ^s n & mp . This shading scheme is n summarized in Fig. 5. An underlying assumption of 6hourly sample independence may be problematic, but this was the only feasible estimate for such a complicated 4D calculation, and the threshold value of two appears to be conservative enough. To illustrate: as observed by an anonymous reviewer, the change dA K 3,4 5 23 kJ m 22 in Table 4 is small but significant; one may get a feeling for the meaning of this significance criterion by regarding the yearly K A,n 3,4 values in Fig. 6— the error bars overlap (implying insignificant differences) in some years, but the cumulative difference is significant. This case is also illustrative because the autocorrelation e-folding time of K 3,4 for all states, all 53 winters, was computed to be about 1 day (i.e., four observations), so that the threshold value Ï4 brings this study in line with ST64 and other daily data studies. Note the following incidental observations. For nonblocking cases the magnitude of transfer kI 21/2 ^s n & m p n is apparently enough to establish significance of almost all nonblocking energetics. It was also found that the ^s P (MK, TK) j,k & mp values, and hence the significance variation too, are relatively k localized (not shown). Finally, there are fewer significant enstrophy WTFs, because z is a more heterogeneous field than are u and y . As noted above, OWA proceeds from high to low j, and thus is sensitive to heterogeneities such as the y l term in z. 4) ATLANTIC

j

k

KAj,k

Knj,k

rAKAj,k

EAj,k

Enj,k

rAE j,k

0 1 1 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4

0 0 1 1 2 2 3 4 5 6 7 0 5 6 7 8 9 10 11 12 13 15

125 103 109 121 87 64 43 32 24 31 48 13 16 17 13 10 8 7 5 5 7 11

149 83 92 107 75 58 47 35 27 37 53 14 15 13 12 11 9 8 5 6 7 12

28.6 8.4 7.2 5.4 6.5 4.0 24.0 23.3 25.1 27.3 23.8 23.9 2.9 6.7 4.0 23.0 25.3 23.8 24.8 24.6 22.7 23.9

22 21 22 23 22

26 18 20 22 20

26.0 5.0 2.9 2.1 3.9

14

13

2.1

17 8 11 11 9 7 6

18 9 10 9 8 7 6

22.4 22.9 2.6 5.7 2.6 22.0 23.6

4 4

4 5

22.5 22.8

7

8

23.8

ever, this representation has no information regarding the spatial location of these transfers. Alternately, consider the average stationary eddy KE distribution in Fig. 8. One can see here that, on average, eddy KE increases near the block in both sectors. However, there is no information about the zonal scales involved, or their interactions. The wavelet representation provides both kinds of information, that is, spatial location (longitude) and scale (wavenumber band). Thus we introduce the dual-spectrum bar graph in Fig. 9. The rest of this paper relies on

CASE

(i) KE and enstrophy stocks The traditional zonal wavenumber KE spectra are shown in Fig. 7. The power spectrum K m scales a little more steeply than } m 23 (see appendix). The exponent estimate varies around a value of 23, similarly to a finding by Baer (1972, Fig. 10). For both blocking sectors there are significant reductions (dark gray) at intermediate scales and increases (white) at large scales. This is consistent with the findings of HS84, and those presented below, that blocking is accompanied by enhanced eddy transfer of KE to the largest scales. How-

FIG. 6. DJF-averaged K x3,4 6 I x21/2s x K 3,4 (in kJ m 22 ) vs year for x 5 A (black diamonds) and n (white squares). Here, the label x corresponds to the number of observations of x for each year in Fig. 1.

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FIG. 7. Latitude–pressure–time average KE spectrum ^K xm& mp for (a) x 5 n (nonblocking), (b) A (Atlantic), and (c) P (Pacific blocking) (kJ m 22 ). The bars are shaded by their ordinates’ significance as in Fig. 5. One abscissa is log 2 wavenumber and the other represents longitude information; as expected, a Fourier spectrum contains no longitude information.

similar diagnostic pictures. The ( j 1 1)th row (scale j) has 2j bars of width } 2 2j along the k (} l) axis. (The precise k–l relationship depends on j, so that axis label 5 2 52j (k 1 2 21 ) 2 2 21 5 2 21 1 p 21 lj24,k11/2 ; however, the most germane fact is that 0 # k , 2 j periodically spans the l domain.) In the ensuing discussion it is sim-

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FIG. 8. (a)–(c) Latitude–pressure–time average KE field ^K x & mp , as in Fig. 7. One abscissa is longitude l and the other represents wavenumber information; as expected, field values contain no wavenumber information. (b), (c) In the block longitude (the black bar in Figs. 2b–c) is indicated by an outlined frame in the appropriate l plane.

plest to identify the kth bar by counting from k 5 0 as indicated in Fig. 9a. One way to check for WE consistency is by direct comparison with published Fourier spectral results. In the present study, there is agreement between the global 2 J21 J21 2 j21 statistics S m51 K m , S j51 S k50 K j,k and K to 11 decimal places. Thus both Fourier and wavelet decompositions

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FIG. 9. Average wavelet-indexed KE stock ^K xj,k& mp , as in Figs. 7 and 8. Significant changes from (a) to (b) and (c) are tabulated in Tables 4 and 7, respectively. One abscissa is scale index j (Table 1); the other is proportional to k and to l (see text). Orthonormal wavelet analysis provides combined zonal scale (wavenumber band) and longitude information. In (b) and (c), the block longitude (outlined frame in Fig. 8) is in the appropriate k (longitude–index) plane. In (a), the bars are labeled by k 5 0, 1, 2, · · · 2 j 2 1 for j 5 0, 1, 2, 3.

are consistent in the global sense. In regard to wave2 j21 number band, the KE spectrum S k50 K j,k roughly folj lows 2 K mø332j22, as would be expected by the heuristic argument of Yamada and Ohkitani (1991). The enstro2 j21 phy spectrum S k50 E j,k is roughly } 2 2j or else is almost

329

FIG. 10. Average enstrophy stock ^E xj,k& mp (Ms 22 ), as in Fig. 9. Significant changes from (a) to (b) and (c) are tabulated in Tables 4 and 7, respectively.

flat, over different bands j. These findings are compatible with expectations, and with the findings of Steinberg et al. (1971) and Boer and Shepherd (1983) in the Fourier domain. Wavelet KE energetics are compared for Atlantic versus nonblocking in Fig. 9b versus 9a, and Table 4. Generally there is a shift of KE stock from large scale (K n0,0 . K A0,0 in dark gray) to intermediate and small

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TABLE 5. Significant Atlantic-blocking KE WTFs and WFFs (in W m22 ), for j , 6. See Table 1 and Figs. 11, 13, 14, and 17.

MK 0,0 MK 2,0 MK 2,2 MK 4,7 MK 4,8 TK 0,0 TK1,1 TK 2,1 TK 2,2 TK 2,3 TK 4,7 TK 5,17 FK 0,0 BKha0,0 BKha2,2 BKha2,3

ValueA

Valuen

rA

0.23 20.33 0.20 0.05 0.05 0.11 0.27 20.00 0.24 20.11 20.04 20.00 20.11 0.04 20.66 10.00

0.33 20.25 0.14 10.00 0.02 0.32 0.35 20.13 0.06 20.26 0.02 10.00 20.32 20.53 20.25 0.34

24.8 22.0 2.1 2.3 2.1 25.1 22.1 2.2 4.3 4.6 22.6 22.3 2.8 3.9 23.2 23.2

lose (have negative r A K 3,k$3 and r A K 4,8#k#16 ). Analogously to the finding by HS84 in the Fourier domain, E j,k (Figs. 10a–b and Table 4) behaves qualitatively very similarly to K j,k (except for E 3,4 ). The suggested general picture is of Atlantic blocking and its near-upstream regions increasing KE and enstrophy stocks at the expense of other components. (ii) Mean-flow interactions Comparing Fig. 11b to 11a, and Table 5, the mean flow contributes less MKA0,0 , MKn0,0, and instead greater MKA2,2 . MKn2,2 (just downstream of the block) and MKA4,k57,8 (at the block). The MK 2,2 increase is roughly balanced by a decrease in MK 2,0 in the zonally opposite hemisphere. Referring to Figs. 12a–b and Table 6, the gain MEn0,0 has been dramatically reduced, replaced to a large extent by (less significant) gains at j . 0, downstream of the block. For both KE and enstrophy, the changes at j 5 0 constitute a major qualitative and quantitative departure for blocking from nonblocking, consistent with HS84 but with more likely significance. (iii) Eddy interactions

FIG. 11. Average KE transfers ^MKxj,k& mp from mean flow (W m 22 ), as in Fig. 9. Significant changes from (a) to (b) and (c) are tabulated in Tables 5 and 8, respectively.

scales near or upstream from the block. The KE ren ceiving components are both global (K A1,k . K 1,k for both k) and especially near the block (positive r A K 2,k51,2 and r A K 3,2 ). The region near the block at the j 5 4 scale also gains (has r A K 4,5#k#7 . 0), while many eddies of scale j 5 3 and smaller, and downstream of the block,

Like MKA0,0, also TKA0,0 5 2FKA0,0 . 0 is very reduced (Figs. 13a–b and Table 5). The normal transfer TKn1,1 . 0 from the smaller ‘‘neighbor’’ eddies TKn2,k51,3 , 0 is partly replaced by transfer TKA2,2 . 0 at the same scale j 5 2 at the block, between them. These two observations suggest eddies feeding KE to the block. The overall zonal (k) oscillation pattern of FKAj.0,k is similar to that of FKnj,k (Figs. 14a–b). The smaller scales j . 4 contain relatively negligible transfers (e.g., ,20 mW m 22 ) of any type, although dATK j,k , 0 is significant for ( j, k) 5 (4, 7), (5, 17) at the block. As seen in Fig. 15a–b and Table 6, TE j,k changes significantly, for j 5 2, 5, just upstream of the block. Therefore during A blocking, there are significantly increased upscale enstrophy cascades upstream, and downscale cascades

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TABLE 6. Significant Atlantic-blocking enstrophy WTFs and WFFs (in Ms23 ), as in Table 5. See Figs. 12, 15, and 16.

ME 0,0 TE 2,1 TE 5,13 FE 1,1 FE 2,1 FE 2,2 FE 3,5 FE 4,7 FE 4,8 FE 4,11 FE 5,11 FE 5,16 FE 5,17 FE 5,22 FE 5,23 FE 5,31

ValueA

Valuen

rA

12 14 34 1 223 16 13 22 16 5 215 4 3 6 3 7

30 221 17 215 17 212 2 21 1 24 21 26 26 0 22 21

23.1 2.1 2.5 2.0 24.3 3.9 2.1 2.1 2.2 2.8 22.1 2.8 2.5 3.1 3.6 3.5

to a moderate source BKAha0,0 . 0, and become more negative BKAha2,k$2 downstream of the block (Figs. 17a– b and Table 5). The physical interpretation of these fluxes amounts to the local effects of ageostrophic flow, as explained in the space domain by Chang and Orlanski (1993). Since ageostrophic flow may be strongly affected by the block flow structure, some significant changes of this flux near the block are not surprising. As an aside, note in Tables 5 and 6 that there is a scale gap j 5 3 in which, for KE, no significant WTF or WFF changes of any type occur between A and nonblocking, and for enstrophy, only one does. 5) PACIFIC

CASE AND COMPARISON WITH

ATLANTIC

(i) KE and enstrophy stocks

FIG. 12. Average enstrophy transfers ^MExj,k& mp from mean flow (Ms 23 ), as in Fig. 9. Significant changes from (a) to (b) and (c) are tabulated in Tables 6 and 9, respectively.

downstream, as indicated by FE j.0,k (Figs. 16a–b and Table 6).

Generally K Pj,3,k . K nj,3,k, as there is more global large-scale KE during P blocking than nonblocking (Figs. 9a,c and Table 7). The zonal deviations of K1,k change roles for k 5 0, 1 between A and P, according to hemispheric sector. The increase in K P2,0 right at the block is analogous (in the sense of rb sign, and location w.r.t. the block) to that of K A2,2, and K P2,3 (just upstream) and K P4,0 (at the block) are analogous to K A2,1 and K A4,4,k,8, respectively. The stock changes dP K j,k for ( j, k) 5 (2, 2), (3, 3), (4, 11), and (4, 12) are not quite as significant as for most other significant index pairs, and their locations suggest no relation with the block. However the reductions to K P4,1,k,6 are similar to K A4,8#k#16, both sets being downstream of the block. As in the A case, E Pj,k is qualitatively similar to K Pj,k (Fig. 10c and Table 7). This includes the striking increase E P2,0 due to the block vortex, flanked by downscale-neighbor decreases E P3,k51,7. (ii) Mean-flow interactions

(iv) Geopotential fluxes Geopotential fluxes are a global sink BK , 0 normally, but during A blocking they increase significantly n ha0,0

For the P block, Fig. 11a,c and Table 8, there is little change at the global scale, MKP0,0 ø MKn0,0 . MKA0,0. At

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significant change in ME j,k is MEP1,0, the block transferring enstrophy to the mean flow (Fig. 12a,c and Table 9). (iii) Eddy interactions There is a dramatic increase in TKP0,0 . 0, the opposite of the A case change (Fig. 13a,c and Table 8). At the second largest scale j 5 1, in the block’s hemisphere there is a loss (TKP1,0) to the eddies, opposite TKA1,1 but with the same sign of change (dark gray) relative to nonblocking. There are losses TKP2,0 near the block, and TKP5,k55,6 downstream of the block. The overall TKAj,k and TKPj,k distributions are quantitatively different. For example, consistent with the energetics measurement for Fourier zonal wavenumbers m (HS84, Figs. 2d and 3) that TKAm53 . 2TKPm53 . 0, here we have significant wavenumber m 5 3 losses. These appear in TKP2,0 (see Fig. 5b of Fo0), located at the block. From Figs. 15a,c and Table 9, the most significant triadic interaction change for enstrophy is seen to be TEP1,0 K 2TEn1,0 , 0. The high j distribution of TEbj,k around the block is similar for b 5 A or P. Whereas for higher j, A blocking, and nonblocking, there are a sector with positive FK j,k upstream and another sector with negative FK j,k at or just downstream; P blocking is different in that the FKPj,k signs are reversed relative to being up- or downstream of the block (Figs. 14a,c and Table 8). FKPj,k is positive downstream of the block and negative upstream of the block, consistently the opposite of FKnj,k. Thus, in the P case there are localized downscale KE cascades just downstream of the block, and localized upscale cascades upstream of the block, the reverse of A. However, the signs of the changes from FKnj,k to FKbj,k are mostly the same w.r.t. block proximity for b 5 A or P, suggesting similar block dynamics in both sectors, with the A sector tendency being too weak to overcome the normal pattern. The overall pattern of FEPj,k is also similar to the A case w.r.t. block proximity (Figs. 16b–c and Tables 6 versus 9). For enstrophy, for both block sectors, significant upscale and downscale cascades are in evidence, respectively, upstream and downstream of the block. (iv) Geopotential fluxes

FIG. 13. Average KE transfers ^TKxj,k& mp from triads, as in Fig. 11. Significant changes from (a) to (b) and (c) are tabulated in Tables 5 and 8, respectively.

scale j 5 2 there is a loss MKP2,0 to the mean flow from the block, and a gain MKP2,1 downstream, similar to the relative position of MKA2,2 . 0. Most of the smallerscale mean transfers are relatively negligible, but there is a significantly stronger small-scale sink (more negative MKP5,1 , MKn5,1 , 0) near the block. The only

As seen in Figs. 17a,c and Table 8, geopotential fluxes near the block are more intense in P than in nonblocking cases, especially the big gains BKPhaj51,2,k50 just downstream, and loss BKPha2,1 further downstream of the block, playing a similar role to that of BKAha2,2. (v) Shifted correlations Another way to compare A and P events is by the WE correlation defined in Table 1, including various shifts of one WE pattern over the other. This correlation maximization may be thought of as a form of pattern matching. Atlantic- and Pacific-blocking WE are similar

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FIG. 14. Average KE fluxes ^FK & , as in Fig. 11. Significant changes from (a) to (b) and (c) are tabulated in Tables 5 and 8, respectively. Note that in the Pacific case (c) there are clear reversals to upscale cascade upstream of the block (dark gray), together with downscale cascade downstream (white). x j,k mp

in the following sense. Correlation between some WE distribution dA X and one for dP X increases, often nearly maximally, if the latter is k shifted east for each j, provided that the k shift corresponds to dl 5 1358, that is,

333

FIG. 15. Average enstrophy transfers ^TExj,k& mp , from triads, as in Fig. 12. Significant changes from (a) to (b) and (c) are tabulated in Tables 6 and 9, respectively.

to the difference in block longitudes in Fig. 2. Conversely, the shift dl1 defined to maximize correlation relative to nonblocking is often close to dl. Thus it turns out that the naturally occuring phase shift from A to P is also nearly that which maximizes the A–P energetics correlation. This similarity holds for X 5 K, E, ME,

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TE, or FE. Atlantic- and Pacific-blocking WE are different in the sense that this similarity fails for other X. 4. Summary and discussion The orthonormal-wavelet-indexed wavelet transfer and flux functions for kinetic energy (KE) and enstrophy represent activity simultaneously dually localized in space (longitude) and zonal scale (wavenumber band). For certain phenomena, this offers a distinct advantage over Fourier-based analogs, because the latter can only represent global activity between scales, without spatial information. Applying this technique to the blocking and nonblocking data quantifies the energetic scale interaction characteristics of blocking, as reported by HS84 and the other works cited above with several new contributions to generally understanding these phenomena: • complementary energetics information in the joint location-scale domain, • rigorous preservation of KE and enstrophy interaction conservation rules, • conservative (‘‘two sigma’’) estimate of energetics statistical significance, • apparently especially appropriate for blocking (judging from relation of energetics to block locations), and • characteristics now documented for the full 53-winter record. Furthermore, specifically, the stocks (wavelet spectra) ^K j,k & mp of KE and ^E j,k & mp of enstrophy behave strikingly similarly in response to blocking. The eddy from-meanflow KE transfer ^MK j,k &bmp increases downstream at j 5 2 from ^MK j,k &nmp and also at j 5 4 for A. The mean flow significantly changes its contribution mainly at j 5 0 for A and j 5 1, 2 for P blocking. Except for decreasing triadic transfers ^TKb1,k & mp downstream, unshifted ^TKbj,k& mp is not very consistent across b 5 A, P, supporting the belief that A and P blocking are somewhat dynamically different. Correlating WE, blocking relative to nonblocking, suggests a similarity of Pacific to Atlantic energetic patterns if the former are shifted over the latter; this holds for all enstrophy WE, and for KE stocks, but not for other KE wavelet energetics. Depending on longitude, the triadic transfers and fluxes to smaller scales can have either sign. Specifically, within the uncertainty due to the small number of events, the analysis indicates significant downscale KE and enstrophy cascades downstream of the average P block and significant upscale cascades upstream. (The reverse holds for the average A-block KE cascades, but it is not evidently significantly different from the normal case.) The tendency from nonblocking to blocking is the same for A and P, if perhaps not significantly so. In regard P to enstrophy triads, only TEA2,1, TEA5,13, TE1,0 , and TEP5,k55,6 change significantly from nonblocking, yet this is enough to imply significant upscale enstrophy cascades upstream and downscale cascades downstream of

FIG. 16. Average enstrophy fluxes ^FExj,k & mp , as in Fig. 12. Significant changes from (a) to (b) and (c) are tabulated in Tables 6 and 9, respectively. Note that by this measure, enstrophy behaves like KE for Pacific blocking (cf. Fig. 14). Atlantic blocking behaves similarly to Pacific.

both A and P block sectors. The Atlantic block also receives intermediate to-large-scale KE from smaller neighboring eddies, the more significantly changed gains being at the intermediate scales. [Some of the findings supercede Fournier (1995) and later brief case studies by this author; these previous analyses were sub-

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TABLE 7. Significant Pacific-blocking (compared to nonblocking) KE and enstrophy stocks, as in Table 4. See Figs. 9 and 10. j

k

KPj,k

Knj,k

rPKPj,k

EPj,k

Enj,k

rPE j,k

0 1 1 2 2 2 2 3 3 3 4 4 4 4 4 4 4

0 0 1 0 1 2 3 1 3 7 0 2 3 4 5 11 12

177 134 103 151 148 85 72 47 43

149 83 92 103 107 75 63 52 47

5.0 12 3.1 11 9.0 3.2 3.4 22.3 22.2

31

26

3.7

33 25

21 22

8.5 2.6

17

19

22.2

17 10 8 9 11 6 7

14 12 13 14 15 5 6

3.0 24.5 28.9 28.2 25.3 2.4 2.1

15 10 7 6 6 8

18 9 7 8 9 10

23.8 2.4 22.2 26.3 25.5 23.7

enstrophy cascade is consistent with Shutts’ (1983, 1986) eddy-straining hypothesis, in that eddies are reduced in the longitudinal direction as the block’s strain field elongates them meridionally. The upstream–downstream asymmetry may be physically explained by a hypothetical (but reasonable) statistical predominance of typical dipolar block patterns that have S-shaped streamlines enclosing a low south of a high. (Actually the head of the S must be seen as continuing southeastward, so the overall pattern also resembles a westward-tilting V.) In such a configuration, eddies apTABLE 8. Significant Pacific-blocking KE WTFs and WFFs, as in Table 5. See Figs. 11, 13, 14, and 17.

FIG. 17. Average boundary fluxes ^BKxhaj,k& mp , as in Fig. 11. Significant changes from (a) to (b) and (c) are tabulated in Tables 5 and 8, respectively.

sequently found to include temporal variances large enough to prevent the significance attributed to the present study.] These findings support the idea that blocks are maintained by an upscale KE cascade, as was included in the ideas of Saltzman (1959) and others. The downscale

MK 2,0 MK 2,1 MK 5,1 TK 0,0 TK1,0 TK 2,0 TK 5,1 TK 5,5 TK 5,6 FK 2,0 FK 3,7 FK 4,0 FK 4,1 FK 4,14 FK 4,15 FK 5,0 FK 5,2 FK 5,3 FK 5,5 FK 5,28 FK 5,29 FK 5,30 FK 5,31 BKha1,0 BKha2,0 BKha2,1

ValueP

Valuen

rP

20.58 0.51 20.02 1.0 20.46 20.08 0.03 20.00 20.01 0.06 20.13 0.11 0.13 20.05 20.07 0.04 0.05 0.07 0.06 20.03 20.03 20.04 20.05 1.2 1.1 21.2

20.25 0.20 20.00 0.32 20.04 0.18 0.01 0.01 10.00 20.24 0.07 20.05 20.04 0.07 0.03 20.04 20.03 20.03 20.01 0.03 0.03 0.01 10.00 0.54 0.16 20.41

23.8 3.8 22.4 8.1 25.2 22.6 2.1 22.1 22.2 2.1 22.8 2.7 2.8 22.9 22.3 2.8 2.5 3.2 2.1 23.1 22.9 22.3 22.3 2.4 2.8 22.6

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TABLE 9. Significant Pacific-blocking enstrophy WTFs and WFFs, as in Table 6. See Figs. 12, 15, and 16.

ME1,0 TE1,0 TE 5,5 TE 5,6 FE1,0 FE 2,3 FE 4,13 FE 5,4 FE 5,5 FE 5,26 FE 5,27 FE 5,30

ValueP

Valuen

rP

294 261 212 210 39 1 3 13 21 21 22 28

235 2 7 0 28 30 22 21 1 9 8 3

23.5 22.8 22.7 22.1 2.1 22.4 22.3 2.5 2.8 22.0 22.1 22.1

proaching from upstream may be elongated longitudinally before they swing around south of the low, implying an upscale enstrophy cascade. This scenario is supported by the finding of Lejena¨s and Madden (1992) that the split of the upstream jet stream is not north– south symmetric as in the Shutts’ (1983, 1986) schematic, but rather is enhanced to the south. The upstream southward flow enhancement could change zonally downscale cascades to upscale, essentially by rotating the strained eddies so that their elongation is more zonal than meridional. As for the downstream downscale cascades, note that both Atlantic and Pacific blocks tend to have strong orographic upstream–downstream asymmetry: ocean upstream, land downstream. This could explain our observed upstream–downstream upscale–downscale asymmetry, as follows. The boundary layer over land could equalize the downstream jet stream confluence, so that the downstream eddy straining mechanism acts just as described by Shutts (1983, 1986), producing downscale cascades. The S-shaped streamlines would also explain the predominance of dbME j,k , 0 (Fig. 12). On average near the jet-stream, z w . 0, whereas the blocking high contributes to z. , 0. Thus each contribution to ME ; 2z wy . z. , 0 depends on local zonal assymetries of y . . Specifically, the upstream y . . 0 values will more tend to cancel because of the S-shaped flow, allowing the downstream y . , 0 to dominate and make a net contribution to ME , 0. A limitation of this study may be the exclusion of other energy forms, which was justified similarly as in the approaches reviewed in section 1. Also it should be remembered that no extra information can be generated without cost in other information; acquiring some spatial information costs a reduction in scale resolution, as explained in Fo0. Combining spatial and scale information can complicate interpretation, but it is the only way to address some questions. To better establish statistical significance it would be necessary to, for example, estimate the time autocorrelation and effective degrees of freedom of the blocking WE for all modes, as was done

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for K 3,4 . That would be an order of magnitude larger, 4D calculation than the present one—itself taking .3 teraflops. The author is preparing generalizations of this partition of the KE budget across location and scale simultaneously, to other energy forms such as APE and other budgets such as potential vorticity. Other planned enhancements include: • 2D orientable anisotropic wavelets on the sphere, • lagrangian energetics (following each event in its own reference frame) as in WNC, • energetics of observational composite, synchronized, centered events, and • ‘‘inverse energetics;’’ that is, given some energetics, visualize the corresponding synoptic weather map. These may prove to be more useful, for example by providing a more complete and better conserved budget. Promising future applications include other quasi-persistent geophysical fluid dynamic structures in the atmosphere, ocean, or other systems, such as storm tracks, oceanic vortex rings, or Jupiter’s Great Red Spot. Acknowledgments. I am very grateful to J. D. Berner, A. G. Pouquet, J. J. Tribbia, and the editor and three anonymous reviewers for many useful suggestions. I also thank B. Saltzman, R. B. Smith, K. R. Sreenivasan, R. R. Coifman, A. R. Hansen, P. D. M. Parker, J. C. van den Berg, and two anonymous reviewers for comments on an earlier version; the Meteorology Department, University of Maryland and Advanced Study Program, National Center for Atmospheric Research, for support while the presented research was carried out; the Climate Diagnostics Center for providing the observational data; and J. Caron, D. Hooper, and A. Mai for technical assistence. This material is based upon work supported by the National Science Foundation under Grant 9420011 and through NCAR Project 36211014. This paper is dedicated to Barry Saltzman: mentor and pioneer in harmonic analysis of atmospheric observations. APPENDIX Analysis Methodology Details Since the analysis is more straightforward on a grid size of a natural power 2 J , l dependence was cubically interpolated (MATLAB INTERP1) to the periodic grid l J,, , J [ log 2 144 5 7 (see Table 1). This J choice minimizes interpolation artifacts. Following HS84, a lower limit 308N is used in ^ f & m to reduce the meridional boundary transports (BKhz and BKhe defined by Fo1). In ^ f &w the lower limit 558N is used in order to focus on the block region. In the blocking event selection, transients were smoothed by removing frequencies n . 1 day 21 . To count as a block, a ridge must be quasi stationary; this is defined by requiring within each time interval that

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each associated crest longitude shift D lm (t i ) [defined as was Dp r (Table 1), with (r, p r ) → (i, lm (t i ))] had to be less than the full width at half max of the 1978/79 lm distribution [reference case of HS84 and Fournier (1999)]. The northeastern Atlantic and Pacific oceans are the climatological block locations (e.g., D’Andrea et al. 1998). In a first pass, the total averages lbm in Fig. 3 were established. From these and the distributions, a range of 6508 was selected around the averages, which labeled 60 out of 63 blocks as either A or P. It was observed that the time–Fourier spectrum of blocking events (as a binary sequence) has maximum amplitude at n 5 0, and for A at least 40% of that maximum, also for timescales n 21 ∈ {6, 12} months. Similarly for P, the time–Fourier spectrum is significant for timescales n 21 ∈ {6, 10, 12, 63} months. At no other n was there as much amplitude. Thus some interseasonal to interannual variability is not excluded by the operations f b . For this reason, a Cauer elliptic high-pass filter (MATLAB ELLIP) was applied to each DJF only for the s b Z results in Fig. 2, with zeroed phase distortion, doubled order, and minimized transients (FILTFILT). The gains in the pass band and stop band were ∈ [95, 100]% and ∈ [0, 10]%, with cutoff at 7 days. The filtered Z much better exhibits the meridional elongation of s P Z just upstream of the block, than did the unfiltered Z; but s n Z was weaker due to filtering Z. In order to best focus on new methodology and interpretation in the space domain, the present study does not further pursue these time domain analysis issues. For completeness it would be necessary to resolve interactions between different n components of the flow, that would significantly complicate the analysis. Extra steepness in Fig. 7 results from the linear interpolation error spectrum increasing as O(m) for m $ 16, and smoothing away higher m. Fourier interpolation would yield a better K m . However, it has the worst effect on K J21,k (and hence for j , J 2 1) out of the interpolation methods Fourier, spline, cubic, nearest neighbor, and linear, in that order. Linear interpolation causes 45% less K J21,k error than Fourier. In any case, spectra steeper than m 23 are commonly observed (e.g., Charney 1971 and references therein). REFERENCES Baer, F., 1972: An alternate scale representation of atmospheric energy spectra. J. Atmos. Sci., 29, 649–664. Bengtsson, L., 1981: Numerical prediction of atmospheric blocking— A case study. Tellus, 33, 19–42. Benzi, R., B. Saltzman, and A. Wiin-Nielsen, Eds.,1986: Anomalous Atmospheric Flows and Blocking. Advances in Geophysics, Vol. 29, Academic Press, 459 pp. Berggren, R., B. Bolin, and C.-G. Rossby, 1949: An aerological study of zonal motion, its perturbation and breakdown. Tellus, 1, 14– 37. Boer, G. J., and T. G. Shepherd, 1983: Large-scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci., 40, 164–184. Butchart, N., K. Haines, and J. Marshall, 1989: A theoretical and

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and transient eddies between two general circulation models: The GLAS Climate Model and the NCAR Community Climate Model. Ph.D. dissertation, Iowa State University, 185 pp. Lejena¨s, H., 1995: Long term variations of atmospheric blocking in the Northern Hemisphere. J. Meteor. Soc. Japan, 73, 79–89. ——, and R. A. Madden, 1992: Travelling planetary-scale waves and blocking. Mon. Wea. Rev., 120, 2821–2830. Lindzen, R. S., 1986: Stationary planetary waves, blocking, and interannual variability. Advances in Geophysics, Vol. 29, Academic Press, 251–273. Liu, Q., 1994: On the definition and persistence of blocking. Tellus, 46A, 286–298. Mullen, S. L., 1987: Transient eddy forcing of blocking flows. J. Atmos. Sci., 44, 3–22. Nakamura, H., M. Nakamura, and J. L. Anderson, 1997: The role of high- and low-frequency dynamics in blocking formation. Mon. Wea. Rev., 125, 2074–2093. Oort, A. H., and J. P. Peixo´to, 1974: The annual cycle of the energetics of the atmosphere on a planetary scale. J. Geophys. Res., 79, 2705–2719. Rex, D., 1950a: Blocking action in the middle troposphere and its effect upon regional climate. I. An aerological study of blocking action. Tellus, 2, 196–211. ——, 1950b: Blocking action in the middle troposphere and its effect upon regional climate. II. The climatology of blocking action. Tellus, 2, 275–301. Riyu, L., and H. Ronghui, 1996: Energetics examination of the blocking episodes in the Northern Hemisphere. Chin. J. Atmos. Sci., 20, 118–130.

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