Properties of Rossby Waves in the North Atlantic Estimated...

JANUARY 2004 61O S Y C H N Y A N D C O R N I L L O N

q 2004 American Meteorological Society

Properties of Rossby Waves in the North Atlantic Estimated from Satellite Data

VLADIMIR OSYCHNY AND PETER CORNILLON

Graduate School of Oceanography, University of Rhode Island, Narragansett, Rhode Island

(Manuscript received 24 July 2002, in final form 2 July 2003)

ABSTRACT

This study uses satellite observations of sea surface height (SSH) to detect westward-propagating anomalies,presumably baroclinic Rossby waves, in the North Atlantic and to estimate their period, wavelength, amplitude,and phase speed. Detection involved a nonlinear fit of the theoretical dispersion relation for Rossby waves tothe time–longitude spectrum at a given latitude. Estimates of period, wavelength, and phase speed resulteddirectly from the detection process. Based on these, a filter was designed and applied to extract the Rossbywave signal from the data. This allowed a mapping of the spatial variability of the Rossby wave amplitude forthe North Atlantic. Results showed the familiar larger speed of observed Rossby waves relative to that expectedfrom theory, with the largest differences occurring at shorter periods. The data also show that the dominantRossby waves, those with periods that are less than annual, propagated with almost uniform speed in the westernpart of the North Atlantic between 308 and 408N. In agreement with previous studies, the amplitude of theRossby wave field was higher in the western part of the North Atlantic than in the eastern part. This is oftenattributed to the influence of the Mid-Atlantic Ridge. By contrast, this study, through an analysis of the wavespatial structure, suggests that the source of the baroclinic Rossby waves at midlatitudes in the western NorthAtlantic is located southeast of the Grand Banks where the Gulf Stream and the deep western boundary currentinteract with the Newfoundland Ridge. The spatial structure of the waves in the eastern North Atlantic is consistentwith the formation of these waves along the basin’s eastern boundary.

1. Introduction

Rossby waves play an important role in the responseof the ocean to forcing. In this paper we discuss prop-erties of the sea surface height (SSH) anomaly field inthe Rossby wave portion of the frequency–wavenumberspectrum of the Ocean Topography Experiment (TO-PEX)/Poseidon data in the midlatitude North AtlanticOcean. Our interest was initiated by, although not re-stricted to, an investigation of the impact of long bar-oclinic Rossby waves on the variability of the GulfStream path.

A number of studies have shown that westward-prop-agating anomalies (we refer to these as Rossby wavesas has been done in previous papers on this subject aswell as for brevity) can be detected in SSH data. Cheltonand Schlax (1996, hereinafter CS) provided a globaldescription of the Rossby wave field based on the 3-yrTOPEX/Poseidon time series available to them at thetime. They concentrated on major features that appearto be common to such waves in all of the major oceanbasins although most of their results were presented withexamples from the North Pacific. Chelton and Schlax

Corresponding author address: Vladimir Osychny, GraduateSchool of Oceanography, University of Rhode Island, Narragansett,RI 02882-1197.E-mail: [email protected]

(1996) observed Rossby waves with periods rangingfrom 6 to 24 months and wavelengths ranging from 500km at 508N to 10 000 km or longer in the Tropics. Theyfound these waves to propagate faster westward in mid-latitudes than expected from simple linear theory. Theyalso found the wave amplitude to be larger west of majortopographic features than east of these features, whichled them to suggest that topography may support thegeneration or amplification of the observed waves, al-though recently Fu and Chelton (2001) noted that thisincrease in the wave amplitude is observed often, butnot always. Last, CS demonstrated that the spatial struc-ture of the waves in the Tropics was consistent with brefraction: wave crests/troughs extending over a rangeof latitudes tend to curve toward midlatitudes as theycross the basin because of an increase in the phase speedtoward the equator. In contrast, outside the Tropics theyfound that the SSH anomalies lose their coherent struc-ture in the north–south direction soon after generation.

Chelton and Schlax (1996) drew their conclusions byconsidering westward-propagating features, presumablyRossby waves, seen in time–longitude diagrams of SSHanomalies (see our examples in Fig. 1). In particular,they estimated the wave phase speed by fitting straightlines to propagating features in these diagrams. Zangand Wunsch (1999, hereinafter ZW) pointed out thatCS’s approach characterizes the propagation of anom-

62 VOLUME 34J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

FIG. 1. Time–longitude diagrams of the SSH anomalies. Color scale is the same for all plots in the figure.

alies as a whole; it does not discriminate between con-stituents of wave packets—Rossby waves of differentwavenumbers and frequencies.

In a spectral analysis of the wave field, one that diddiscriminate between wave constituents, Zang andWunsch (1999) found that outside the Tropics 1) a sub-stantial fraction of energy was consistent with the clas-sical theory (mostly, at large spatial and temporal scales)and 2) short Rossby waves were faster than predictedby the theory and, thus, responsible for the speed upobserved by CS. Zang and Wunsch’s conclusions con-tradicted those of CS by stressing that the fraction ofthe total energy consistent with the classical theory wassubstantial.

Like ZW we use a spectral approach to detect Rossbywaves and estimate their parameters although details ofour data processing differ from that of ZW in severalcrucial ways. They employed a beamforming technique,that is, a high-resolution method of spectral estimation.These methods rely on a priori assumptions about thesignal structure. Most important, the methods detect thewaves on a rather widely spaced grid of frequencies. Atthese frequencies the signal is assumed to consist of asingle sinusoid buried in noise. As a result, all of theestimates (including those of uncertainties) are contin-gent on the validity of these assumptions. Instead, we

resort to classical (Fourier) spectral estimation. Its prop-erties are well known, and the basic assumptions arenot that strict; hence we sacrifice somewhat higherwavenumber resolution for a denser frequency grid, amore complicated signal composition in wavenumber,and the better understood Fourier analysis. In additionto serving our goals, classical spectra can be used toaddress the validity of the assumptions used by ZW.

We begin by considering the spectral composition ofthe Rossby wave field, characterizing the period, wave-length, and phase speed of the observed waves. Then,we describe the spatial pattern of the wave field, ad-dressing the following questions: Where are the wavesgenerated? Do they have any coherent structure? If theydo, how does that structure change as the waves prop-agate westward and especially as they cross the Mid-Atlantic Ridge? Do they show a significant increase inamplitude because of interaction with the Mid-AtlanticRidge? Last, where and why is the wave field mostenergetic?

2. Data and methods

This study uses a subset of the TOPEX/Poseidon SSHanomaly dataset provided by Van Snyder and Zlotnicki(2000). Briefly, they applied all standard corrections to


the raw TOPEX/Poseidon measurements, subtracted amodeled tidal signal and an estimated mean sea surface,interpolated data onto a uniform 18 3 18 3 5 day grid,and subtracted the temporal mean field at each gridpoint. The subset of data that we use spans the periodfrom December 1992 to December 1998 and covers theNorth Atlantic from the equator to 608N. The conclu-sions presented here relate to the region between 158and 478N where our procedure provides confident de-tection of Rossby waves.

We begin by low-pass filtering the data in time inaccordance with the large temporal and spatial scalesof interest and subsampling the resulting smoothedfields with a time step of Dt 5 29.7 days, three repeatTOPEX/Poseidon cycles. The filtering was performedwith an eighth-order Chebyshev infinite impulse re-sponse filter (Jackson 1996) with the transition bandbetween six (the Nyquist period for the new time step)and seven repeat TOPEX/Poseidon cycles. The residualtemporal mean values and linear trends were subtractedfrom the filtered, subsampled fields to obtain the as-sociated anomaly field for further analysis.

Detection of Rossby waves and estimation of theirparameters was performed using time–longitude dia-grams of data at a given latitude. (Examples are shownin Fig. 1.) Two-dimensional spectra were estimated forthe diagrams. Similar to the previous studies, Rossbywaves were identified by energy peaks in the part ofthe spectrum that represents westward propagation. Ev-ident in Fig. 1 is the large, predominantly annual, bas-inwide signal that is mainly steric in nature. Since thefocus of this work is on propagating anomalies, we re-move this basin-scale signal to facilitate subsequentanalysis. This was done by subtracting the best fitstraight line in longitude at each degree of latitude foreach 29.7-day field. Certainly a linear trend only ap-proximates a more complicated spatial structure of theannual signal. However, the goal here is to unmask thepropagating anomalies and to use the subsequent anal-ysis based on two-dimensional spectra to obtain furtherrefinements.

Estimation of the spectra began with a time–longitudeperiodogram computed at each latitude. This was ob-tained via the fast Fourier transform of the datasmoothed by a two-dimensional Chebyshev window(Emery and Thomson 1998) to suppress possible side-lobes. The size of the window was determined from thenumber of longitudinal data values at a given latitudeand from the length of the time series. Periodogramscorresponding to three adjacent latitudes were averagedand the resulting estimate was assigned to the centrallatitude. Averaging improved statistical reliability of thespectral estimates. Additional smoothing was performedin the frequency–wavenumber domain with a three fre-quency by three wavenumber moving average. Exam-ples of the spectra are shown in Figs. 2 and 3.

To determine the possible effect of nonstationarity,spectra were computed for the entire time series and for

3-yr-long overlapping subsections. The spectra of thesubsections were not significantly different from oneanother or when compared with the spectrum of theentire time series. Therefore we concluded that the timeseries was, to the accuracy of our estimates, temporallystationary over the analyzed interval.

On the other hand, inspection of the time–longitudediagrams (Fig. 1) reveals a noticeable variation in theamplitude of anomalies and, even more important, intheir speed of propagation between the eastern and thewestern part of the North Atlantic. This is seen mostclearly in the region between 258 and 408N where alarge change takes place in the anomaly field over arelatively narrow zone approximately in the middle ofthe North Atlantic (see the time–longitude diagram for358N, Fig. 1). Fortunately, the ocean at these latitudesis sufficiently wide to allow us to estimate spectra usingdata covering two nonoverlapping subsections repre-senting the western and the eastern basins as well asthe entire width of the basin.

The two-dimensional spectra, whether they be for thewidth of the basin (Figs. 2 and 3) or for the eastern orthe western basins (not shown), provide a basis for eval-uating the phase speed of the observed Rossby waves.The speed can be estimated by locating frequencies andwavenumbers of individual peaks in the spectra (as inZW). The uncertainty of such a speed estimate dependson the uncertainty of a given spectral peak and on theresolution of the spectrum. Averaging over appropriateregions in frequency–wavenumber space can improvethis uncertainty in speed estimation. However, the re-gions over which the averages are performed must beproperly chosen. Here our choice is motivated by threeobservations. First, Figs. 2 and 3 show that at manylatitudes, especially in the 208–358N range, spectralpeaks form a more or less continuous ridge of elevatedenergy, presumably revealing a form of an empiricaldispersion relationship. Second, although we do notknow the mathematical form of the dispersion relation-ship, CS, ZW, and Fu and Chelton (2001) conclude thatthe standard theory does describe a significant portion(at least 50%) of the dynamics of observed westwardpropagating anomalies. The standard theory is definedby the following dispersion relation:

2bkv 5 , (1)

2 2 2k 1 l 1 (1/R )d

where v is the frequency of the wave, k is the east–west (zonal) wavenumber, l is the north–south (merid-ional) wavenumber, b is the north–south gradient of theplanetary vorticity, and Rd is the radius of deformationof the first baroclinic Rossby wave. Third, one of thegoals here (similar to CS and ZW) is to compare ob-served Rossby wave speed with that obtained from thestandard theory and provide details about such a com-parison: for example, where is (geographically) the cor-respondence better, at what wavenumbers/frequencies,


FIG. 2. Frequency–wavenumber spectra of anomalies at 308–448N. Magnitude is normalized by the maximum value at each latitude andis shown on a linear scale. The red line depicts the estimated dispersion diagram; the blue line corresponds to the theoretical dispersiondiagram.


FIG. 3. Same as Fig. 2 but for 158–298N.


FIG. 4. The frequency–wavenumber spectrum of anomalies at 318N.The red line depicts the estimated dispersion diagram; the blue linecorresponds to a theoretical dispersion diagram; the white line is the‘‘eyeball’’-drawn line passing through the ridge of high energy in thespectrum. See text for details.

and so on. Based on this, our approach was to fit thestandard dispersion relation (1) to estimated SSH spec-tra, calculate associated phase speeds, and comparethem with those expected from theory. (Details of thefitting procedure as well as estimation of errors asso-ciated with it can be found in the appendix.)

The best-fit (observed) dispersion curves are shownin red for each degree of latitude in Figs. 2 and 3 alongwith the underlying spectrum. Also plotted in these fig-ures, in blue, are the ‘‘theoretical’’ dispersion curves.The latter were obtained from (1) with the Rossby radiusof deformation estimated by Chelton et al. (1998) usinga climatological average of vertical density profiles fromin situ observations. (Here the quotation marks, whichwe drop hereinafter, are used to emphasize that empiricaldata were used in the calculation.) This is similar to theapproach taken by CS and ZW.

The phase speed calculated from (1) depends onwavenumber/frequency. However, Figs. 2 and 3 showthat most of the spectral peaks fall within the range oflong nondispersive Rossby waves (for which the dis-persion diagram is well approximated by a straight line).Therefore, to simplify a comparison, we used the long-wave (nondispersive) approximation in calculation ofthe phase speed:

2C 5 2bR .p d (2)

For the scales considered here (wavelength larger than68 longitude), the ratio of the difference between thephase speeds calculated for the dispersive (red curvesin Figs. 2 and 3) and nondispersive (straight lines, notshown) case to the phase speed in the dispersive casedid not exceed 5% and did not affect any of the con-clusions.

Inspection of Figs. 2 and 3 shows that at some lati-tudes between 258 and 418N the observed dispersionrelation differs from that described by (1). At these lat-itudes spectral peaks were spread out over the broadestfrequency–wavenumber range. In many cases the best-fit dispersion relation (red line) passes through a chainof the most energetic peaks. Yet, in some cases therewas another zone of elevated energy that was close tobut, nevertheless, separated from the best fit. For ex-ample, at 318N the peak at the annual period falls belowthe estimated dispersion curve (Figs. 2 and 4). If onewere to draw a line passing through the ridge of elevatedenergy in this spectrum (white line in Fig. 4), this linein the vicinity of the annual period would be locatedrelatively closer to the theoretical dispersion curve (blueline). At shorter and longer periods it would turn andfollow the estimated dispersion curve (red line). This isa manifestation of the fact emphasized by ZW that ob-served Rossby waves of different frequencies propagatewith different speeds although their period and lengthscale remain within the theoretical nondispersive waverange. In light of the poor fit in some cases between (1)and the observed spectra, the procedure used to estimateCp was repeated for different frequency bands. In these

cases, the spectral energy from a particular frequencyband only, rather than from the entire spectrum, wasused to estimate Cp. This may be thought of as a piece-wise approximation to the unknown true dispersion re-lation by segments of curves (1). As a result our esti-mates may be, for instance, consistent with the theoryleading to (1) for one frequency/wavenumber bandwhile inconsistent for the other.

Last, it is important to recognize that the observedwestward propagating anomalies are only compared to,rather than strictly identified with, the first mode baro-clinic Rossby waves that satisfy the classical dispersionrelationship (1). Similar to other papers on this subject[see Fu and Chelton (2001) for a comprehensive sum-mary], this work only starts with the dynamics impliedby (1). Although the general shape of the spectra is notinconsistent with Rossby wave dynamics, completeagreement between the observed signal and the standardtheory leading to (1) is hardly expected because of anumber of limitations of the latter. Accordingly, the term‘‘Rossby waves’’ is applied to these anomalies in abroader sense than that implied by (1).

3. Results

Figures 2 and 3 show spectra for 158–448N. It isclearly seen that westward propagating anomalies dom-inate the signal at most latitudes. Energetic anomaliesthat move eastward are found between 388 and 428Nand are, most probably, related to the Gulf Stream Ex-tension.

Within the part of the spectra that characterizes west-ward propagation, most of the energy is found in theneighborhood of the estimated dispersion curve as ex-pected from the fitting procedure. In addition, however,some peaks occur between the estimated and theoreticaldispersion curves, but never at frequencies below thelatter and rarely at frequencies higher than the former.


Therefore, we identify this range as that correspondingto Rossby waves. In the following we first examine themeridional distribution of frequency and wavenumber,then the meridional distribution of phase speed, andfinally the spatial structure of Rossby waves.

a. Variation of frequency and wavenumber withlatitude

Figures 2 and 3 suggest that the North Atlantic canbe separated into three regions based on the structureof the frequency–wavenumber spectra of the westwardpropagating anomalies.

1) To the north of 418N, the spectra show a single highenergy region with periods larger than annual. Thespectral energy reaches a maximum at periods be-tween 1.5 and 2 yr and wavelengths between 108 and158 of longitude (order 800–1200 km). There is neg-ligible energy at annual or shorter periods.

2) In the region between 258 and 418N, the spectra ex-hibit peaks with periods both longer and shorter thanannual. In most cases the period of the dominantpeak is less than the annual decreasing from northto south. To the north, between 408 and 368N, thedominant peak ranges from 9 to 11 months while tothe south, between 358 and 258N, it ranges from 6to 8 months. The relative amplitude of the dominantpeak also decreases from north to south as does thewavelength, the latter ranging from 108–138 of lon-gitude (800–1100 km) in the northern part of theregion to 78–108 of longitude (700–1000 km) in thesouthern part. The dominant peak largely defines theestimated dispersion curve. A few less energeticpeaks appear in the frequency–wavenumber regionbetween the estimated and theoretical dispersioncurves. The main secondary peak has a period longerthan annual and tends to be closer than the dominantpeak to the theoretical dispersion curve indicatingthat the longer observed waves propagate more slow-ly than the shorter ones. This finding is consistentwith ZW.

3) To the south of 258N, annual waves tend to dominate.There is a trail of energy at shorter periods that grad-ually becomes negligible as one goes southward. Inmost cases this trail follows the estimated dispersioncurve largely determined by the annual peak. Thewavelength of the annual waves ranges from 158–208 of longitude at 258N (1500–2000 km) to about408 of longitude south of 158N (.4000 km).

These three regions are well delineated in Figs. 5aand 5b, which show meridional changes in the integratedspectral energy of westward-propagating anomalies ver-sus frequency, and, separately, versus wavenumber. Fig-ure 5a shows that the spectral energy is limited to pe-riods longer than annual to the north of 418N (region1). Between 258 and 418N (region 2), the spectrum isricher; that is, it is spread out in frequency, with peaks

often occurring at periods shorter and longer than annualat the same latitude. To the south of 258N (region 3)the spectrum narrows, being dominated by an annualpeak. Figure 5b shows the corresponding changes in thewavelength of the observed Rossby waves.

The North Atlantic is sufficiently wide in the 258–408N range for an analysis of zonal differences in Ross-by wave properties. To this end, we estimated spectrabased on nonoverlapping sections of data covering re-gions to the west and to the east of the Mid-AtlanticRidge. Figures 6a and 6c show the meridional depen-dence of integrated spectral energy versus frequency forthe western and for the eastern parts of the North At-lantic. These spectra demonstrate that the western sideof the basin is more energetic at all frequencies thanthe eastern side. The difference is especially large northof 338N. A zone of increased energy in the western partoccurs in the 7–11 month range between 348 and 408N.In the eastern part of the basin, a maximum in the Ross-by wave energy occurs between 328 and 348N, that is,slightly to the south of the maximum in the westernpart, and at longer periods, 12–18 months. The ratio ofthe spectral energy at a particular frequency over thetotal energy at a given latitude in the side of the basinbeing considered (Figs. 6b,d) shows that waves of short-er than annual period dominate the western part of theNorth Atlantic over the meridional range consideredhere, 258–408N. In the eastern part, this is true onlybetween 368 and 378N and south of 298N.

b. Variation of phase speed with latitude

The meridional distribution of Cp evaluated from thespectra as well as the associated estimated errors areshown in Fig. 7a. The uncertainties of the estimates(error bars in the figure) depend on the resolution of theestimated spectrum and on the shape of the spectrumitself, that is, on the underlying ocean dynamics. At agiven latitude, smaller errors result from a narrowerspectral resolution and from the dominance of Rossbywaves in dynamics. (See the appendix for a discussionof the calculation of errors.) However, this is true onlyif the spectral peaks associated with the waves are ad-equately resolved; that is, both spatial and temporalscales of the waves are small compared with the cor-responding length of the data record. Estimates of un-certainty (Fig. 7) show that the data and their processingallow confident identification of Rossby waves between158 and 478N. North of 478N the period of the wavesbecomes too long to be resolved with the data we have,while south of 158N the large zonal length scale of thewaves becomes a limiting factor. Although the errorsbetween 408 and 438N are relatively large, these errorsresulted from a broad spectrum (Fig. 2) associated withthe complicated dynamics in this area rather than frompoor spectral resolution. The smooth change of Cp inthe meridional direction lends additional credibility tothe estimates in this range of latitude. Further analysis


FIG. 5. Integrated spectral energy of westward propagating anomalies: (a) distribution of energy vs frequency as a function of latitude and(b) distribution of energy vs wavenumber as a function of latitude.

relates only to the region between 158 and 478N or tosubregions within this range.

Figure 7 demonstrates that the observed speed of whatwe believe to be Rossby waves exceeds that expectedfrom classical theory at all latitudes considered, between158 and 478N. Furthermore, this difference increases tothe north from 10%–20% between 158 and 258N to amaximum of about 300% at 428N (Fig. 7b). This is alsoseen in the difference between the slopes of the esti-mated dispersion curves (red lines in Figs. 2 and 3) and

the theoretical curves (blue lines). Between 448 and478N, observed waves are about twice as fast as thetheoretical ones. Qualitatively, this is consistent withthe estimate of CS.

The discrepancies between observations and theorymay result from inadequacies in the theory as well asfrom errors (shown in Fig. 7) in the estimates. In ad-dition, recall that the theoretical values of Cp shown inFig. 7 are estimates (Chelton et al. 1998) based on cli-matological in situ data. The errors associated with these


FIG. 6. Distribution of spectral energy in the western and in the eastern part of the North Atlantic: (a) distribution of energy vs frequencyas a function of latitude in the western part, (b) fraction of energy at a particular frequency to the total energy at a given latitude in thewestern part, and (c) and (d) similar to (a) and (b) but for the eastern part. The same color scale is used for all plots. Energy in (a) and (c)is normalized by its maximum in the western part; fraction of energy in (b) and (d) is normalized by its maximum value in the western andeastern parts, respectively.

estimates are difficult to evaluate. Chelton et al. (1998)argue that these errors do not exceed a few percent,except near intense currents. Subsequently, they con-clude that the bias in the observed values (based onSSH) is too high to be accounted for by the uncertainties.Our results support their conclusion.

The meridional distribution of observed Cp reveals aregion of almost uniform magnitude between 328 and

408N. Within this region the observed phase speed is alittle less than 4 cm s21, while the theoretical Cp de-creases monotonically from 3 to 1 cm s21. Between 428and 458N, the best-fit Cp decreases from 4 to 2 cm s21.Between 158 and 258N, the observed phase speed isstatistically consistent with theory although systemati-cally biased towards higher values.

Given the increase in phase speed from east to west


FIG. 7. Comparison of (a) Rossby wave phase speed and (b) ratio of the best-fit Rossby wavephase speed with that expected from theory. See text for more details. Stars denote independentestimates from SSH spectra. Uncertainties estimated from 95% confidence level for a peak in anintegrated spectrum (see the appendix). Accurate results are obtained for 158–478N. Regions ofunreliable estimates are shaded.

found by CS as well as the ZW observation that thephase speed varies with wavenumber, we also analyzethe phase speed by subbasin and by frequency/wave-number (Fig. 8). In strict terms, the uncertainty doesnot allow a more precise statement than that of consis-tency of the estimates with one another. (In Fig. 8, errorbars for different estimates, i.e., for those made for dif-ferent subbasins and/or for different wavenumber/fre-quency bands, overlap.) The difference between the es-timates exposes itself only via a bias, albeit a persistentone judging from the change of the estimates with lat-itude. Thus, based on a bias, shorter waves (7–11-monthperiod) were found to propagate faster than longer ones(annual and longer period) in both the eastern and west-ern parts of the North Atlantic. Also, both shorter andlonger waves were found to propagate up to 20% fasterin the western side of the basin than in the eastern side.Chelton and Schlax found a higher, up to 50%, differ-

ence in the speed of Rossby waves between the westernand the eastern basins in the Pacific, although they con-sidered the most remote east–west parts of the oceanrather than an average over the two halves. Interestingly,the difference between the speeds of waves of differentwavenumber/frequency bands in the same subbasin islarger than that of the waves of the same wavenumber/frequency band in different subbasins.

On both sides of the North Atlantic, the change inthe observed phase speed of longer waves (Fig. 8b) asa function of latitude is quite similar to that predictedby theory. By contrast the speed of the observed shorterwaves (Fig. 8a) does not decrease to the north as quicklyas the theoretical speed does: on the eastern side of thebasin between 258 and 408N the observed speed of thesewaves decreases from 5 to 3 cm s21 while the theoreticalone decreases from 4 to 1 cm s21; on the western side,the difference in the rate of change is especially large


FIG. 8. Comparison of phase speed of shorter (7–11-month period) and longer (annual and longer period) Rossby waves in the westernand in the eastern part of the North Atlantic: (a) shorter waves in two basins and (b) longer waves in two basins. Theoretical curves are thesame in (a) and (b) and are labeled only in (a). Uncertainties are shown in shades of gray: the darkest shade relates to the western basin;the lightest, to the eastern basin; the intermediate, to the region where the two overlap.

FIG. 9. An example of the frequency–wavenumber spectrum of anomalies at 398N and the magnitude of the frequency response of thefilter used to extract the Rossby wave signal at this latitude. The red line depicts the best-fit dispersion curve; the blue line corresponds tothe theoretical dispersion curve and defines the low-frequency limit of the filter pass band; the white line shows the high-frequency limit ofthe filter pass band and corresponds to a dispersion curve associated with the upper confidence limit of the best-fit dispersion curve.

between 308 and 408N where the observed speed is al-most uniform (it varies between 4.5 and a little less than4 cm s21) while the theoretical speed decreases mono-tonically from a little over 3 to 1 cm s21.

c. Spatial structure of Rossby waves

To better understand the spatial structure of the Ross-by wave field, the Rossby wave signal was extracted byapplication of a two-dimensional bandpass finite im-pulse response filter. This filter, designed in the Fourierdomain, passed all frequency–wavenumbers located inthe Rossby wave range (primarily between the observedand theoretical dispersion curves; Fig. 9). Outside thisregion the magnitude of the filter frequency response

decreased from 1 to 0 as the cosine squared to suppressthe Gibbs effect upon inversion from the Fourier to thetime–longitude domain. This step resulted in time–lon-gitude arrays at each latitude, filtered to include onlycontributions from the Rossby wave portion of the spec-trum. It is important to note that the time–longitudesection at each latitude was filtered independently fromthe others so that no additional north–south coherencywas imposed on the data in this process.

For each spatial grid point the root-mean-square ofthe filtered sea surface elevation in time was calculatedfrom the filtered arrays, yielding an estimate of the totalamplitude of the Rossby wave field. Similarly, the root-mean-square was calculated for each spatial grid pointfor the shorter and longer waves after the arrays were


FIG. 10. Estimated amplitude of Rossby waves (normalized by its maximum value). Contours show bottomdepth of 2000, 3000, 4000, and 5000 m.

additionally filtered in time to include only the relevantperiods. The results for the total Rossby wave field, forits shorter and for its longer constituents, are qualita-tively similar, and so only those associated with the totalRossby wave signal are presented here (see Fig. 10).

The largest amplitudes are found in the vicinity ofthe Gulf Stream and the North Atlantic Current (northof the Newfoundland Ridge), in particular where thestream crosses the continental slope, the New EnglandSeamounts, and the Newfoundland Ridge as well as inthe region of the Mann Eddy (Mann 1967). The area ofthe Gulf Stream and its extension was identified as ageneration region for the first baroclinic Rossby wavesin the modeling study of Hermann and Krauss (1989).They found that in this region the waves resulted fromnonlinear energy transfers. However, if indeed the west-ward-propagating anomalies are governed primarily byRossby-wave-like dynamics, then, based on classicallinear theory, their energy flux should be directed pre-dominantly westward. This means that little to no energyfrom this region penetrates the ocean interior except,possibly, for the section of the Gulf Stream between theNew England Seamounts and the Newfoundland Ridge(more on this below).

At this point, we emphasize that, consistent with thespectral plots of Fig. 6, Fig. 10 shows higher amplitudesof the Rossby waves west of the Mid-Atlantic Ridgethan east of it. However, direct influence of the Mid-Atlantic Ridge seems to be limited to 308–458N, wherethe ridge is taller and wider than in other areas. [Asimilar region of increased amplitude was reported bySchlax and Chelton (1994) and Polito and Cornillon(1997) based on TOPEX/Poseidon data and by Tok-makian and Challenor (1993) based on Geosat data.]However, this is also the area where one expects sig-nificant activity associated with the Gulf Stream and theNorth Atlantic Current. For example, the regions of thehighest wave amplitude closest to the Mid-AtlanticRidge are found in the vicinity of the Mann Eddy, anintegral part of the North Atlantic Current system, andnear the Corner Seamounts, possibly a result of their(rather than Mid-Atlantic Ridge) influence on a branchof the Gulf Stream circulation. Thus, with the data avail-able here one cannot justify the increase in the waveamplitude only by the direct influence of the Mid-At-lantic Ridge.

Analysis of the complex empirical orthogonal func-tions (CEOFs; Preisendorfer 1988) of the Rossby wave


signal provides further details about the distribution ofthe wave amplitude and propagation of the waves. Figure11 shows a sequence of the sum of the first two CEOFsin the 7–11-month period range (waves of these scalesare most energetic between 258 and 408N) extractedfrom the data. These CEOFs account for 65% of thetotal Rossby wave variance in this frequency band.

Clearly evident in Fig. 11 is the large coherence ofthe Rossby waves over a range of latitudes. In the west-ern half of the basin wave crests/troughs are well definedover at least 108 of latitude south of 358N, while in theeastern half of the basin crests/troughs are well definedover at least 158 of latitude. This differs from the CSobservation that in the midlatitudes anomalies do notexhibit coherent structure in the north–south directionoutside their generation region. Apparently, in our casea more persistent coherency was revealed due to sep-aration of the wave field into different components.

The Mid-Atlantic Ridge appears as a clear boundaryin the orientation of Rossby waves. East of the Mid-Atlantic Ridge the waves are oriented generally fromsouthwest to northeast. This orientation is consistentwith generation of the waves along the eastern boundarywith subsequent b refraction during westward propa-gation. [For a discussion on generation of Rossby wavesoff the eastern boundary see, e.g., Hermann and Krauss(1989) and Gerdes and Wubber (1991).] West of theMid-Atlantic Ridge, wave fronts are oriented in thesoutheast–northwest direction, which is unexpected ifthe waves were to continue from the eastern half of thebasin or if they were to originate along a substantialnorth–south section of the Mid-Atlantic Ridge.

Figure 11 shows that individual fronts do not smooth-ly pass from the eastern side of the North Atlantic tothe western side. In fact, after generation along a con-siderable portion of the eastern boundary, the wavespreserve a coherent structure while propagating west-ward, but then lose it upon transmission over the ridge,especially over its wider and taller part north of 308Nwhere the topographic scale in the east–west directionis larger than the wavelength. Wave fronts west of theMid-Atlantic Ridge appear to originate from a relativelysmall area southeast of the Grand Banks in the GulfStream Extension region (approximately 358–408N,408–508W). In this region the waves have the largestamplitude overall and are oriented parallel to the axisof the Newfoundland Ridge. Such a spatial configura-tion of the westward propagating anomaly field at pe-riods in the semiannual–annual range was insensitive toa variation in the details of the fashion in which thewaves were extracted such as the exact frequency band,type of filters applied before CEOF decomposition, andinclusion of one or two leading CEOFs; that is, waveorientation is a robust feature of our processing. Phys-ically, a maximum in the wave energy may result fromwave interference or from a change of the waveguidecharacteristics. However, the location and orientation ofthe maximum, in the vicinity of and parallel to the New-

foundland Ridge, as well as the existence of strong cur-rents in this region leads us to suggest that this area isa source of the waves in the western part of the NorthAtlantic. Presumably, this source originates from theinteraction of the Gulf Stream and/or the deep westernboundary current (DWBC) with the bottom topography.Apparently, this interaction leads to a deviation of theGulf Stream path from the mean west–east direction andto separation of the DWBC, or at least of some portionof it, from the continental slope. This causes significantanomalies in the water structure in the vicinity of theGrand Banks. We anticipate that at least part of theenergy of these anomalies spread in the form of Rossbywaves as a result of an adjustment process. Thereforethis spreading should have a large westward component.Details of the propagation, however, can be affected bythe complex dynamical background in the region. Figure11 shows that these disturbances spread to the southwestand their crests gradually extend to the southeast. Thesedisturbances can be traced all the way to the Antillesand Bahama Islands (see progression of the dark bluelines in Fig. 11).

Note that our suggestion that the most energeticwaves in the wavenumber–frequency portion of thespectrum corresponding to Rossby waves in the NorthAtlantic appear to result from interaction of the GulfStream/DWBC with the Newfoundland Ridge is differ-ent from the theory of Rossby wave radiation from theGulf Stream due to meandering of its path, presumably,between Cape Hatteras and the Grand Banks [Kamen-kovich and Pedlosky (1998) review this subject and pro-vide its most recent development]. Essentially, the ra-diation theory assumes that the time-dependent far fieldis represented by Rossby waves corresponding to theclassical Rossby wave theory and that disturbances thatradiate from the meandering Gulf Stream have to matchthe classical Rossby wave field. This implies that southof the stream, energy of the radiating waves propagatessouth (from the Gulf Stream) and west (to accommodatethe far field). Then, the classical theory demands north-east–southwest phase (front) orientation for radiatingwaves. Indeed, our results show that sometimes in aregion to the east of the New England Seamounts (be-tween 508 and 608W) observed wave fronts exhibit suchan orientation (e.g., see images for 1995 in Fig. 11).However, this pattern is not persistent in time and doesnot spread southward beyond 358N. By contrast, thenorthwest–southeast oriented anomalies dominate mostof the time and over a larger region of the western NorthAtlantic. Interestingly, following the classical theory,the observed (northwest–southeast) orientation impliesthat the flux of Rossby wave energy would be to thenorthwest. Hence, if these anomalies correspond to clas-sical Rossby waves, their amplitude should increase tothe northwest as the waves propagate to the southwest.However, this is not what is observed; the observedwaves appear to propagate to the southwest with littledecrease in wave amplitude to the southeast. If anything,


FIG. 11. Sum of the two first complex empirical orthogonal functions of the Rossby wave field in the 7–11-month periodrange. Color scale is the same for all plots. Blue lines demonstrate propagation of individual wave fronts in the western andin the eastern parts of the basin.


wave energy appears to spread in the direction of thephase propagation, to the southwest. This suggests, onceagain, that these anomalies do not correspond to clas-sical Rossby waves or that processes not accounted forin the simple linear theory such as sheared, time de-pendent currents, wind forcing, complex topography,and so on, play a significant role in the characteristicsof Rossby waves in this part of the North Atlantic.

4. Conclusions

Analysis of the SSH anomaly field in the Rossby waveportion of the frequency–wavenumber spectrum in theNorth Atlantic between 158 and 478N suggests that thisregion can be divided into three subregions based onthe spectral composition of the observed waves. Southof 258N annual period waves dominate. North of 408Nonly waves with periods larger than annual exist. Be-tween 258 and 408N, the spectral content of the anomalyfield is more complex. Here, constituents of annual andlonger periods play a secondary role to those of shorterthan annual period, especially in the western part of thebasin. The most energetic waves in the North Atlanticwere found in the western subbasin between 348 and408N with periods in the 7–11-month range.

Qualitatively, our result that westward propagation ofanomalies observed in altimeter-derived sea surfaceheight fields is faster than expected from classical Ross-by wave theory (1) agrees with that of CS. Also, ourresult that the difference between best-fit phase speedsand those obtained theoretically depends strongly onwavelength/period, with shorter waves (subannual pe-riods in our case) propagating substantially faster andlonger waves propagating at very nearly the theoreticalspeeds is consistent with that of ZW.

Not surprising, as the proportion of shorter waves inthe total Rossby wave field at a given latitude increasesthe difference between the theoretical and best-fit speedsof propagation averaged over all wavelengths increases,too. The observed waves propagate at speeds similar tothose expected from theory between 158 and 258N wherethe annual waves dominate, while farther to the north,shorter than annual waves come into play and, accord-ingly, the discrepancy in the phase speeds grows untilit reaches a maximum at about 408N, north of whichthe energy of the short waves becomes negligible again.

The physical mechanisms responsible for the differ-ence in the best-fit and theoretical speeds remain to bedetermined. [See review of related theories in Fu andChelton (2001).] Certainly, an environment that doesnot comply with the assumptions of the classical theory(forcing, background currents, nonuniform bottom to-pography, etc.) distorts propagation of the waves. Ourresults show that the northward decrease in the phasespeed of the observed subannual Rossby waves is sub-stantially slower than that expected from theory. More-over, in the western part of the basin the phase speedof these waves is almost uniform between 308 and 408N

(Fig. 8a). In addition, since on average the subannualwaves are dominant across the entire basin in this zone,the phase speeds estimated from the total spectra arealso almost uniform between 328 and 408N (Fig. 7b).

Last, analysis of the spatial structure reveals impor-tant differences in the nature of the observed waves tothe east and west of the Mid-Atlantic Ridge. In theeastern part of the North Atlantic, Rossby waves radiatefrom the eastern boundary where an initial anomaly canbe forced by large-scale processes, probably by a changein the large-scale wind or by large-scale topographicwaves propagating along the eastern boundary. It ap-pears that interaction with the Mid-Atlantic Ridgebreaks the coherent structure of the waves but does notlead to their significant amplification, as suggested byearlier studies. Instead, a completely different structurefor waves in the western part of the North Atlantic ac-companied by larger energy results from what appearsto be a source in the region to the southeast of the GrandBanks. Although beyond the scope of this study, wehypothesize that this source is related to temporal var-iability and/or structural changes of the Gulf Stream andthe DWBC as they interact with the NewfoundlandRidge. For instance, substantial parts of these currentsloop around the Grand Banks while some portion ofeach is thought to branch to the southeast towards theMid-Atlantic Ridge. These processes create a largeanomaly in oceanic properties oriented parallel to theaxis of the Newfoundland Ridge, from the northwest tothe southeast. [The hydrography of this region is poorlyknown; see Clark et al. (1980).] Rossby waves can ra-diate from this anomaly in a process of adjustment:excited Rossby waves try to smooth out the anomalywhile the currents replenish the lost energy to sustainit. In reality, Gulf Stream path and strength vary in thevicinity of the Grand Banks, providing an additionaltime-dependent forcing of rich frequency content avail-able for the resonant excitation of Rossby waves. Al-though the anomalies studied here have the spectral sig-nature of Rossby waves, energy propagation in the wavefield south of the Grand Banks is not consistent withthat of simple linear Rossby waves.

Acknowledgments. This research was supported byNASA Grant NS033A06. We thank Dr. Nelson Hogg,Dr. Thomas Rossby, Dr. Lewis Rothstein, and Dr. Ran-dolph Watts for their insightful comments on earlierversions of this paper.

APPENDIX

Fitting of Dispersion Curve and ItsError Estimates

Determination of the best-fit dispersion diagram wasbased on a nonlinear fit of the dispersion relation (1) tothe estimated spectrum, with the Rossby radius of de-formation Rd being the unknown parameter. The merid-


ional wavenumber l was assumed to be zero, a reason-able assumption given that the observed wavelength wasfound to be an order of magnitude larger than the typicalRossby radius of deformation in the midlatitude range.For a given spectrum, a grid of dispersion curves (1)with Rd ranging from 1 to 150 km in steps of 1 km wasconsidered. For each curve within this grid, we averagedthe spectral energy along the curve. The curve with themaximum average spectral energy was selected as thebest-fit dispersion curve.

To evaluate the uncertainty of such an estimate, it isimportant to realize that each value of the optimal func-tion, that is, the spectral energy averaged along a givendispersion curve, is simply an integrated spectrum.Thus, its uncertainty can be estimated similarly to thatof a classical spectrum, that is, from a x2 distributionwith a certain number of degrees of freedom. The errorbars for the position of the peak of the optimal functioncan be associated with those dispersion curves from thevicinity of the peak for which the optimal function isno less than the lower limit of the 95% confidence in-terval of the peak value.

To proceed, we need to determine the number of de-grees of freedom n involved in obtaining of the optimalfunction. This number results from multiplication of thenumber of spectral estimates N along a particular dis-persion curve by the number of degrees of freedom inevaluating the spectrum itself, nPSD. Thus, n 5 NnPSD.Since three time–longitude sections and averaging ofthree frequency/wavenumber bands were used to esti-mate each spectrum, nPSD 5 2 3 3 3 3 5 18, the samefor all spectra. This means that the total number of de-grees of freedom for a given value of the optimal func-tion depends mostly on N, which in turn depends onthe resolution of a given spectrum (here, the number ofspectral points per unit length is important) and on howclose a given dispersion curve is relative to the fre-quency or wavenumber axes (a number of dispersioncurves that are too close to one of the axes pass throughonly a few spectral points or cannot be resolved in agiven spectrum at all). Typically, n was larger than 100in the region between 158 and 478N, and this resultedin relatively small uncertainties. North/south of this re-

gion, most of the spectral energy was located too closeto the frequency/wavenumber axis so that the lower/upper confidence limit was zero/infinity.

REFERENCES

Chelton, D. B., and M. G. Schlax, 1996: Global observations ofoceanic Rossby waves. Science, 272, 234–238.

——, R. A. de Szoeke, M. G. Schlax, K. El Naggar, and N. Siwertz,1998: Geographical variability of the first baroclinic Rossby ra-dius of deformation. J. Phys. Oceanogr., 28, 433–460.

Clarke, R. A., H. W. Hill, R. F. Reiniger, and B. A. Warren, 1980:Current system south and east of the Grand Banks of Newfound-land. J. Phys. Oceanogr., 10, 25–65.

Emery, W. J., and R. E. Thomson, 1998: Data Analysis Methods inPhysical Oceanography. Pergamon, 634 pp.

Fu, L.-L., and D. B. Chelton, 2001: Large-scale ocean circulation.Satellite Altimetry and Earth Sciences, L.-L Fu and A. Cazenave,Eds., Academic Press, 133–169.

Gerdes, R., and C. Wubber, 1991: Seasonal variability of the NorthAtlantic Ocean—A model intercomparison. J. Phys. Oceanogr.,21, 1300–1322.

Hermann, P., and W. Krauss, 1989: Generation and propagation ofannual Rossby waves in the North Atlantic. J. Phys. Oceanogr.,19, 727–744.

Jackson, L. B., 1996: Digital Filters and Signal Processing. KluwerAcademic, 502 pp.

Kamenkovich, I. V., and J. Pedlosky, 1998: Radiation of energy fromnonzonal ocean currents, nonlinear regime. Part II: Interactionsbetween waves. J. Phys. Oceanogr., 28, 1683–1701.

Lee, T., and P. Cornillon, 1996: Propagation and growth of GulfStream meanders between 758 and 458W. J. Phys. Oceanogr.,26, 225–241.

Mann, C. R., 1967: The termination of the Gulf Stream and thebeginning of the North Atlantic Current. Deep-Sea Res., 14, 337–359.

Polito, P. S., and P. Cornillon, 1997: Long baroclinic Rossby wavesdetected by TOPEX/POSEIDON. J. Geophys. Res., 102, 3215–3235.

Preisendorfer, R. W., 1988: Principal Component Analysis in Mete-orology and Oceanography. Elsevier, 425 pp.

Schlax, M. G., and D. B. Chelton, 1994: Aliased tidal errors in TO-PEX/POSEIDON sea surface height data. J. Geophys. Res., 99,24 761–24 775.

Tokmakian, R. T., and P. G. Challenor, 1993: Observations in theCanary Basin and the Azores frontal region using Geosat data.J. Geophys. Res., 98, 4761–4773.

Van Snyder, W., and V. Zlotnicki, cited 2000: Sea surface heightuniform latitude–longitude–time grids. [Available online athttp://oceanesip.jpl.nasa.gov/sealevel/.]

Zang, X., and C. Wunsch, 1999: The observed dispersion relationshipfor North Pacific Rossby wave motions. J. Phys. Oceanogr., 29,2183–2190.

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