88 JOURNAL OF CLIMATE VOLUMEI NOTES AND CORRESPONDENCE A Spectral Climatology EDWARD S. EPSTEIN Climate Analysis Center, NMC/NWS/NOAA. Washington, DC 26 March 1987 and 12 August 1987 ABSTRACT Using 5 years of daily initialized height fields from the National Meteorological Center, expressed as coefficientsof spherical harmonics, a climatology of the annual cycle has been formulated for the 1000, 700, 500 and 250mb surfaces. The global analyses were first separated into separate Northern and Southern Hemisphere analyses,with a rhomboidal truncation at wavenumber 12. The daily values of each of the spectral coefficients were fitwith the first four annual harmonics. Only those harmonics and spectral coefficients were retained which explaina statistically significant amount of variance in time and space. The resulting mean height fields for both theNorthern and the Southern hemispheres compare very favorably with established climatologies in spite of thelimited length ofthe record on which they are based and the use of operational analyses. The statistical selectionof spatial and temporal harmonics which contribute significantly to the annual mean and the annual cycle offersa unique insight into the structure of the climate in the two hemispheres.1. Introduction Climatoiogies of large-scale atmospheric circulationare usually constructed by averaging, over many years,monthly means evaluated at a large number of gridpoints. Most often there is no explicit smoothing inspace, but the extensive temporal filtering implicit inthe monthly mean fields results in smooth geographicrepresentations. Fields representing the climatological"norm" at times other than mid-months are most oftendetermined by interpolation. The availability of analyses expressed in a spectralformat has made another approach possible. Spatialfiltering is accomplished by the analysis itself. By increasing or decreasing the number of terms used torepresent the raw fields, any desired level of geographicsmoothing may be achieved. We have chosen to useharmonic analysis to smooth and filter the data in time.This can also be done with grid-point data, but manymore coefficients would have to be calculated; it takesfewer spectral coefficients than grid values to representthe features of large-scale circulations. Also, interdependencies are less likely among the coefficients of theharmonics of adjacent spectral coefficients than amongadjacent grid points, so judgments based on considerations of statistical significance are easier to justify.We have averaged the annual harmonics over only asmall span of years, but we are able to construct a veryreasonable annual climatology including only thoseharmonics of spectral coefficients for which the mean Corresponding author address: Dr. Edward S. Epstein, ClimateAnalysis CenterfNMC/NWS/NOAA, WWB, W/NMC5 !, Washington, DC 20233.amplitude is statistically significant in comparison toits year-to-year variations. The purpose of this note is to describe the specificsof the calculations and present some results based on5 years of operational National Meteorological Center(NMC) analyses (1982-86). We will present several setsof charts to demonstrate that the 5-yr mean does represent a reasonable climatology of the annual meanand the annual cycle. It is, in fact, an estimate of aclimatology based on a sample mean, and subject tothe idiosyncrasies of the sample period as well as thelimitations of operational data cutoffs and processing.Indeed, there are reasons to question the sample; itincludes the unusual 1982/83 El Nifio, and a numberof changes in the NMC initialization procedures wereimplemented during the 5-yr sample period. Thus thislimited analysis should not be treated as a definitiveclimatology, although the large-scale features outsidethe tropics, where our attention will be focused, shouldbe reasonably well represented. We shall describe in somewhat greater detail thespectral makeup of the annual mean and the annualcycle as determined by our statistical tests. In particular,we will contrast the significant spectral and harmoniccomponents of the annual cycles of the Northern andSouthern hemispheres.2. Calculations The basic data are daily 0000 UTC global initializedanalyses prepared at NMC. Tapes containing analysesof I000, 700, 500 and 250 mb heights, as coefficientsof spherical harmonics with zonal and meridionalwavenumbers from 0 to 12, were used (i.e., R12 truncation). In the conventional notation for spherical harJANUARY 1988 NOTES AND CORRESPONDENCE 89monics, where n is the total wavenumber, m the zonalwavenumber, and l = n - m is the meridional wavenumber, the height field on the dth day ofthejth yearZ(~b, X, j, d) is given by J Iml+J Z(rk, X,j, d) = ~ ~ Aim(j, d)ptm(sinck)ei'"x, m=-J I~lrnlwhere the A/m(j, d) are the complex spherical harmoniccoefficients, Ptm(sinqb) is the associated Legendre polynomial, and ~b is latitude. For RI2 truncation, J = 12.The condition that the heights are everywhere real implies that Atm and AFm are complex conjugates, so itis not necessary to calculate the coefficients for negativevalues of m. We will introduce the notation A~m= Re(At~) for m >~ 0, and Aim = Re(iAt~) = Re(-iAi-m)'for m < 0. The first step was to convert the data from globalanalyses to separate Northern Hemisphere-only orSouthern Hemisphere-only analyses. This was doneby mapping each field to a grid in physical space, replacing the Southern Hemisphere data with a mirrorimage of the Northern Hemisphere, or vice versa, andreconverting to spectral coefficients. When this is doneall coefficients with odd values of the meridional wavenumber vanish. This allows us to work with half thenumber of coefficients to achieve a given level of resolution. The two hemispheres were presumed to undergo distinctive seasonal cycles, with more seasonalityin the Northern Hemisphere. Treating them separatelymakes intercomparison easier. One would expect thatthe Southern Hemisphere analyses, because of morescanty data, are noisier. Eliminating this noise shouldmake it easier to discern valid Northern Hemispheresignals. Removing the burden of accounting for thestronger Northern Hemisphere cycle when dealing withthe Southern Hemisphere should make it easier to discern the weaker signals of the Southern Hemispherecycle. The next step was to fit the daily values of each spectral coefficient for each of the five available calendaryears to the first four annual harmonics (i.e., sine waveswith'periods of 12, 6, 4, and 3 months): ~'~ Dhd t, i2~rhd/L Atm(j, d) =where L = 365 days/year. Again we note that B~n,~ andB?,~~/are complex conjugates, and B~'m~ = Re(B~'m~) forh >~ 0, Btn,~ = Re(iB~,~) for h < 0. Truncating at four harmonics was arbitrary, but issomewhat justified (after the fact) by the low rate ofoccurrence of significance of the third and fourth harmonics. In this step it was necessary to accommodatea small amount of missing data (up to 16 days in ayear). This was done by resorting to least squares inlieu of harmonic analysis. Harmonic analysis requiresthat bogus data be introduced where real data are missing; when no data are missing, conventional harmonicanalysis and least squares are entirely equivalent. In each year, j = 1, - - - , 5, each combination ofthe indices m, l, and h represents a particular spectral/temporal wave defined by the four coefficientsBI, m,hdBI, m-nd, Blh,,_jm, and B f.h'~., The variance in time and space(the power) of each wave in each year is proportionalto the sum of the squares of the four coefficients. Tosimplify the notation slightly, we use a single (vector)index k = (l, m, h) and refer to the four coefficients asbk, i0, i = 1, 2, 3, 4. The variance in time and space (thepower) explained by the wave is (I/n) Z 2 bk, i0. (When l= 0 or h = 0 the factor is 1/2; when l = 0 and h = 0 thefactor is 1.) Similarly the variance attributable to theaverage wave over the five years is V~ = (I/4) Z ~,,~,where ~,i = (1/5) Z bk, i,j. The year-to-year variation of the coefficients providesa measure of the uncertainty that should be ascribedto the 5-yr mean. The appropriate measure of this uncertainty is the mean square deviation M~ = (1/4) Z [(1/4) ix ~ (b~,~,i - ~,i)2]. Given the null hypothesis that theJmeans are zero, and other appropriate assumptions asto normality and independence, the ratios Vk/M~ willeach have an F-distribution with 4 and 16 degrees offreedom (2 and 8 when either the longitudinal wavenumber or the frequency is zero; I and 4 when bothare zero). This allows us to distinguish between thoseharmonic terms that show real persistence from yearto year and those with mean values which are unreliableestimates of the multiyear average. Not all the assumptions (e.g. independence) required for the F-distribution to be valid are strictly met; e.g., the missingdata result in some loss of independence among theestimates of the coefficients. However, the critical values of the F-ratio for various levels of significance stillprovide useful criteria to determine which harmonicsto include in the climatology and which to exclude dueto lack of confidence that they contribute to the climat. ological norm or its annual cycle.3. Choice of significance level for inclusion of waves in annual cycle Figures l a and b show the Northern Hemispheremean annual cycle of the 500 mb surface as of 15 Febmary and 15 August, respectively, including all termswithout regard to statistical significance. They both resemble the expected climatological patterns, but thereare details, particularly apparent against the weakerbackground circulation of the summer, that seem tobe of greater amplitude than expected of a climatological average, or are displaced from their expected location. For example, consider the ridge extending fromthe Bering Sea over the northern coast of Siberia inFebruary, or, in August, the strong ridges over NorthAmerica, the Atlantic, and western Europe. But this isindeed the observed mean for the period. Figures 2a and b present the same fields as Figs. 1 aand b, but include only those harmonics (terms) for90 JOURNAL OF CLIMATE VOLUME IJANUARY 1988 NOTES AND CORRESPONDENCE 9192 JOURNAL OF CLIMATE VOLUME 1which the annual mean is significant at the 5% level.The differences are small but clearly the amount ofdetail is less. What constitutes an optimum criterionis arbitrary; we have made the judgment that, for themost part, ~a 1% significance level leads to too high anerror of type II (i.e., of excluding components that arein fact part of the annual cycle), while a 10% test admitstoo many spurious components. The maps presentedbelow will be based on a 5% significance criterion.4. The observed mean annual cycle Given the coefficients of the mean annual cycle,maps can be reconstructed for any day of the year.Maps one day or one week apart will resemble oneanother very closely; each will be the best availablerepresentation for that day of the state of the slowlyevolving annual cycle. For purposes of depicting theannual cycle, maps about one month apart appear adequate. Most of the maps we have constructed aredrawn for the 15th of the month to facilitate comparison with the more common monthly averages. Shown in Figs. 3a, b, c, and d are the mean annualcycle of the Northern Hemisphere 1000 mb surface asof, respectively, 15 January, 15 April, 15 July and 15October. In winter (Fig. 3a), the 1000 mb map is dominated by strong Aleutian and Icelandic lows and alarge Siberian high. The high pressure center over NorthAmerica is much weaker and merges with weak subtropical highs over the eastern Pacific and Atlanticoceans. By' spring the Aleutian and Icelandic lows areconsiderably weaker; central heights have increased byabout 60 m. Subtropical highs, still weak, are presentin the central Atlantic and Pacific oceans near 35-N.The Siberian high remains, but its central height hasdecreased by about 100 m. The pattern over NorthAmerica is quite flat. By summer the Aleutian and Icelandic lows havedisappeared. Both the Atlantic and Pacific oceans aredominated by strong subtropical highs. The central,height is greater in the Atlantic and a ridge extendswell into the southeastern United States. A very weakhigh remains just north of the Tibetan plateau, and aweak trough remains over China and eastern Siberia.The lowest heights in the hemisphere are associatedwith a low center over the Persian Gulf. The corresponding 500 mb fields for the middle ofJanuary, April, July, and October are shown in Fig. 4.Among the familiar features of these maps are thestrong influence of low wavenumbers in winter andthe very tight gradients in middle latitudes off the eastcoasts of Asia and North America, the much weakergradients in summer, and the appearance of multipleridges and troughs in middle latitudes. The polar lowis quite symmetrical in summer but in winter is splitinto two strong centers over northern Siberia and theCanadian arctic archipelago. These features are common to all standard climatologies. We have made the most detailed comparisonswith a set of 1948-70 monthly mean 700 mb chartsavailable at the National Weather Service ClimateAnalysis Center plus charts which show the departuresof the most recent 10-yr mean from these "normals.".The differences between the "normals" and the limitedspectral climatology based on only the years 1982-85are small (mostly less than 30 m), and generally in thedirection of the "anomaly" of the most recent decade. The annual cycle in the Southern Hemisphere differsmarkedly from that of the Northern Hemisphere. Ascan be seen in Fig. 5, the 1000 mb map is, in all seasons,characterized by a circumpolar trough surroundingAntarctica with lowest heights generally near 150-Wand somewhat lesser low centers near 0- and 90-E.The maps in Fig. 5 agree closely with those shown byLeMarshall et al. (1985). The differences appear to beno larger than expected from the two different sampleperiods (1982 is in both samples). Clearly, the seasonalcycle of the Southern Hemisphere differs both in character and magnitude from that of the Northern Hemisphere. At 500 mb the Southern Hemisphere circumpolarvortex is quite symmetrical throughout the year (seeFig. 6).. The essential features of the midsummer andmidwinter 500 mb charts match closely those shownin similar charts by Trenberth (1979) as well as thoseby LeMarshall et al. The only discernible difference ofnote is the presence on the NMC .map for 15 July ofthe weak closed high over the South Pole. The NMCwinter analyses near the South Pole have undergoneseveral modifications and corrections during the periodwhen these data were obtained (Bonner et al., 1986)and are not considered as reliable as other aspects ofthe analyses. - Departures from zonal symmetry provide a moresubstantial comparison of the various datasets. Figure7, showing mid-January and mid-July maps of thezonally asymmetric portion of the mean SouthernHemisphere 500 mb surface, were prepared to be directly comparable with Fig. I of Trenberth (1980), ananalysis based on an earlier 6-yr period. Ignoring theimmediate polar region in winter, the same large-scalefeatures appear in both datasets, with very much thesame amplitudes but generally somewhat displaced. To provide a more graphic description of the contrasting mean seasonal cycles of the Northern andSouthern hemispheres, Fig. 8 shows the NorthernHemisphere 500 mb asymmetric components of theheight fields for mid-January and mid-July. Comparingwinter hemispheres we see that the amplitude of theasymmetric part of the field is about three times aslarge in the Northern Hemisphere. Also, the space scaleof the features appears larger in the Northern comparedto the Southern Hemisphere. Comparison of the summer asymmetries tells a different story. The amplitudesare comparable in magnitude but the scale of theasymmetries seems, if anything, to be smaller in theNorthern Hemisphere.JANUARY 1988 NOTES AND CORRESPONDENCE 93 If, in addition to setting the zonal mean terms tozero (to get the zonally asymmetric components of thefields) we also ignore the time-mean terms (the zerothharmonic), then we are left with the asymmetric cyclicpart of the fields. The zonally asymmetric, annuallycyclic 1000 mb fields for the Northern Hemisphere areshown in Fig. 9. If only first harmonics had been included in the "climatology" these two charts would bemirror images of each other. Clearly the first harmonicsare the dominant though not the only terms. What ismostly evident in Fig. 9 is a vast hemisphere-widemonsoon centered on Siberia and North America, onthe one hand, and the North Atlantic and North Pacificoceans on the other. This is in contrast to the SouthernHemisphere (Fig. 10) where the asymmetric cyclic fieldis weak. Nevertheless, there is some hint of a geographically fixed Southern Hemisphere monsoon with threeoceanic centers and three centers more nearly associated with Australia, South Africa, and South America.5. Spectral composition of the annual cycle In constructing this representation of the annual cycle, 455 tests of statistical significance were conductedfor each level in each hemisphere. The outcomes ofthese tests provide a description of which terms are orare not essential in describing the mean annual circulation and its annual cycle. This should provide useful information to those who would construct modelsof the general circulation or of the annual cycle. Propefiy designed GCMs should be capable of replicating,in its essential features, the spectral composition of thereal atmosphere. Those who would design, simplifiedmodels of the annual cycle should bear in mind whatthe persistent components of the observed cycle are. Table I contains the results of the 3640 tests of significance. Each row of the table corresponds to a particular spherical harmonic wavenumber; L representsthe longitudinal wavenumber and N the total wavenumber. Because of the symmetry condition imposedon the data, N - L is always even, and (N - L)/2 isthe number of zero-crossings between the equator andthe pole of the particular spherical harmonic function. Each column refers to a particular harmonic (in time,0 to 4) for a given level and hemisphere. The symbolstell whether the 5-yr mean coefficient of each term wassignificant (at the 5% and 1% levels) for the givenhemisphere, level, and wavenumber. The absence of asymbol means that the 5-yr mean was not statisticallysignificant. (As the length of record increases, thenumber of terms that are deemed significant will generally increase if the level of significance is held constant.) The symbols do not indicate the magnitude of theamplitudes of the waves, only their statistical significance using the year-to-year variation as a yardstick.Criteria for significance can vary over several orders ofmagnitude. The charts shown in Figs. 2-10 are basedonly on the components indicated in Table 1 as significant at the 5% level. The zero harmonic refers to the annual mean. Thus,we can see that there are very few waves shorter thanzonal wavenumber 3 (except for wavenumber 8) thatcontribute to the annual mean Northern Hemisphere250 mb surface. A "spectral void" is particularly noticeable at wavenumber 5. On the other hand, thehigher wavenumbers are very frequently significant forthe annual mean at 1000 mb in both hemispheres.There is a tendency, in both hemispheres, for fewercomponents to be significant at higher levels, beyondzonal wavenumber 4. This may be due to poorer datacoverage at the higher levels or it may be a reflectionof stronger persistent small-scale surface influences atthe lowest levels. The "spectral void" at wavenumber5 in the annual mean is more present at the.500 mblevel of the Southern Hemisphere than at the 250 mblevel. The annual cycle at 1000 mb in the Northern Hemisphere is most complex. The first harmonic of almostevery spectral component examined passes the tests ofsignificance, suggesting that truncation at wavenumber12 neglects small but reliable details of the annual cycleof the lower troposphere. This is in sharp contrast tothe Southern Hemisphere, where beyond zonal wavenumber 1 fewer than half the possible components ofthe first harmonic at 1000 mb are significant. (But notethat at zonal wavenumber 8 the four waves with thelowest total wavenumber are highly significant.) Theweak wavenumber 3 pattern apparent in Fig. 10 showsup largely because, while weak, it is not disguised bymany competing terms. In the Northern Hemisphere the seasonal cycles at700, 500 and 250 mb each contain strong contributions from zonal wavenumbers 0 through 3, somewhatless of a contribution from wavenumbers 4 and 5, andmore from wavenumber 6. Starting at zonal wavenumber 10 there is a suggestion that the highest levelsrequire fewer components than the lower. In theSouthern Hemisphere the upper levels, similarly to1000 mb, show only scattered significant componentsbeyond zonal wavenumber 1. The contrast with theNorthern Hemisphere is particularly apparent atwavenumber 3 and at wavenumber 6 and beyond. Foreach of the 500 and 250 mb surfaces of the SouthernHemisphere, from wavenumber 6 through 12, only 2of 49 terms involving the first harmonic proved significant at the 1% level and only two or three more atthe 5% level. Second harmonics play an important role in theNorthern Hemisphere at zonal wavenumbers 0, 1 and3, and in the Southern Hemisphere at wavenumber 0and perhaps also 1. Third and fourth harmonics provesignificant much less frequently, and also much lesssystematically. There is little to suggest that extendingthe calculations to higher harmonics would be worthwhile.94 JOURNAL OF CLIMATE VOLUME 1OOOOZJANUARY 1988 NOTES AND CORRESPONDENCE 950 ,y.96 JOURNAL OF CLIMATE VOLUME IJANUARY 1988 NOTES AND CORRESPONDENCE 9798 JOURNAL OF CLIMATE ~, VOLUMEIJANUARY 1988 NOTES AND CORRESPONDENCE 99100 JOURNAL OF CLIMATE VOLUME 1OOJANUARY 1988NOTES AND CORRESPONDENCE \101102 JOURNAL OF CLIMATE VOLUME I-~% %.I^NU^RY 1988 NOTES AND CORRESPONDENCE 103104 JOURNAL OF CLIMATE VOLUME I Jj/JANUARY 1988./?'xNOTES AND \CORRESPONDENCE105106 JOURNAL OF CLIMATE VOLUME ITABLE 1. Significant harmonics of the annual cycle (heights). Open circle represents significance at 5% level; G represents significance at 1% level.WavenumberNorthern HemisphereSouthern Hemisphere1000 mb 700 mb 500 mb 250 mb 1000 mb 700 mb 500 mb 250 mb Harmonics Harmonics0 1234 0 1234 0 1234 0 1234 0 1234 0 1234L N 0 12340 12341234567 0 2 4 6 81012 1 3 5 7 91113 2 4 6 8I01214 3 5 7 9111315 4 6 810121416 5 7 911131517 61012141618 7 91113151719GGGGGG GOGG GGOGGGGGGGGGG OGGGGGGGGGGG 0GGGOG 0GG0 GOG 000GOGOG 0G 0G O0OGGGO0GGGGOGGGO G OGGGOGG OGO G OGGGG G OOGGO0 OO OG OGO G OJ^NU^RY 1988 107NOTES AND 'CORRESPONDENCE TABLE 1. (Continued)WavenumberNorthern HemisphereSouthern Hemisphere1000 mb 700 mb 500 mb 250 mb 1000 mb 700 mb 500 mb 250 mb Harmonics Harmonics0 1234 0 1234 0 1234 0 1234L N 0 12340 1234 0 1234 0 1234 8 910ll12 8101214161820 9111315171921101214161820221113151719212312141618202224rrrrrrr0r 0r0rrrr0r 0rrrr00 rrC)Orrr 00 rOOrr rrrrrrrr 0 rr rrrrOrrrr00Orr r0r rrrrr O0rr0rrrrr0rr0 0 0 Orr00 r r0 r 0 r r0 0 r0 r0 rr 0 00 rr r 0 rr r 0 rr 0 0 r0 r r0 0 r r r rr rr r r 000O00 r0 r r 00r r 0O0 r O00 rrrr0 0r 0 r0r 0 0 r 0 00 0r000r0 0r0O00rr 0 006. Conclusions Representing the annual climate cycle by a truncatedand statistically selected set of spherical harmonic andtemporal harmonic coefficients is convenient and useful. The statistical selection process itself provides usefulinformation on the physical characteristics of the annual mean and the annual cycle. It would be of interestto compare the spectral makeup of the climatology derived from the NMC analyses and similar climatologiesderived from models. To all appearances the mean annual cycles of boththe Northern and Southern hemispheres derived fromoperational NMC analyses compare favorably withother mean maps and can be treated as estimates ofthe true climate. It is our intention to use the NMCspectral climatology as a candidate null forecast forpurposes of verification and for the statistical correctionof middle- and long-range predictions. REFERENCESBonner, W. D., G. H. White, M. S. Tracton, V. E. Kousky and G. P. Cressman, 1986: Global analysis and prediction at NMC Washington. Second Int. Conf on Southern Hemisphere Me teorology (Extended Abstracts), Wellington, New Zealand, Amer. Meteor. Soc., 1-9.LeMarshall, J. F., G. A. M. Kelly and D. J. Karoly, 1985: An at mospheric climatology of the Southern Hemisphere based on ten years of dally numerical analyses (1972-82). I: Overview. Aust. Meteor. Mag., 33, 65-85.Trenberth, K. E., 1979: Interannual variability of the 500 mb zonal mean flow in the Southern Hemisphere. Mon. Wea. Rev., 107, 1515-1524. ,1980: Planetary waves at 500 mb in the Southern Hemisphere. Mon. Wea. Rev., 108, 1378-1389.