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Global-observations-of-nonlinear-mesoscale-eddies_2011_Progress-in-Oceanography

Global observations of nonlinear mesoscale eddies

Dudley B.Chelton ?,Michael G.Schlax,Roger M.Samelson

College of Oceanic and Atmospheric Sciences,104COAS Administration Building,Oregon State University,Corvallis,OR 97331-5503,United States

a r t i c l e i n f o Article history:

Received 22February 2010

Received in revised form 6January 2011Accepted 10January 2011

Available online 28January 2011

a b s t r a c t

Sixteen years of sea-surface height (SSH)?elds constructed by merging the measurements from two simultaneously operating altimeters are analyzed to investigate mesoscale variability in the global ocean.The prevalence of coherent mesoscale features (referred to here as ‘‘eddies’’)with radius scales of O(100km)is readily apparent in these high-resolution SSH ?elds.An automated procedure for identify-ing and tracking mesoscale features based on their SSH signatures yields 35,891eddies with lifetimes P 16weeks.These long-lived eddies,comprising approximately 1.15million individual eddy observa-tions,have an average lifetime of 32weeks and an average propagation distance of 550km.Their mean amplitude and a speed-based radius scale as de?ned by the automated procedure are 8cm and 90km,respectively.

The tracked eddies are found to originate nearly everywhere in the World Ocean,consistent with pre-vious conclusions that virtually all of the World Ocean is baroclinically unstable.Overall,there is a slight preference for cyclonic eddies.However,there is a preference for the eddies with long lifetimes and large propagation distances to be anticyclonic.In the southern hemisphere,the distributions of the amplitudes and rotational speeds of eddies are more skewed toward large values for cyclonic eddies than for anticy-clonic eddies.As a result,eddies with amplitudes >10cm and rotational speeds >20cm s à1are preferen-tially cyclonic in the southern hemisphere.By contrast,there is a slight preference for anticyclonic eddies for nearly all amplitudes and rotational speeds in the northern hemisphere.

On average,there is no evidence of anisotropy of these eddies.Their average shape is well represented as Gaussian within the central 2/3of the eddy,but the implied radius of maximum rotational speed is 64%smaller than the observed radius of maximum speed.In part because of this mismatch between the radii of maximum axial speed in the observations and the Gaussian approximation,a case is made that a quadratic function that is a very close approximation of the mode pro?le of the eddy (i.e.,the most frequently occurring value at each radius)is a better representation of the composite shape of the eddies.This would imply that the relative vorticity is nearly constant within the interiors of most eddies,i.e.,the ?uid motion consists approximately of solid-body rotation.

Perhaps the most signi?cant conclusion of this study is that essentially all of the observed mesoscale features outside of the tropical band 20°S–20°N are nonlinear by the metric U /c ,where U is the maximum circum-average geostrophic speed within the eddy interior and c is the translation speed of the eddy.A value of U /c >1implies that there is trapped ?uid within the eddy interior.Many of the extratropical eddies are highly nonlinear,with 48%having U /c >5and 21%having U /c >10.Even in the tropics,approx-imately 90%of the observed mesoscale features are nonlinear by this measure.

Two other nondimensional parameters also indicate strong degrees of nonlinearity in the tracked eddies.The distributions of all three measures of nonlinearity are more skewed toward large values for cyclonic eddies than for anticyclonic eddies in the southern hemisphere extratropics but the opposite is found in the northern hemisphere extratropics.There is thus a preference for highly nonlinear extra-tropical eddies to be cyclonic in the southern hemisphere but anticyclonic in the northern hemisphere.Further evidence in support of the interpretation of the observed features as nonlinear eddies is the fact that they propagate nearly due west with small opposing meridional de?ections of cyclones and anticy-clones (poleward and equatorward,respectively)and with propagation speeds that are nearly equal to the long baroclinic Rossby wave phase speed.These characteristics are consistent with theoretical expec-tations for large,nonlinear eddies.While there is no apparent dependence of propagation speed on eddy polarity,the eddy speeds relative to the local long Rossby wave phase speeds are found to be about 20%faster in the southern hemisphere than in the northern hemisphere.The distributions of the propagation directions of cyclones and anticyclones are essentially the same,except mirrored about a central azimuth

Corresponding author.Tel.:+15417374017;fax:+15417372064.

E-mail address:chelton@https://www.wendangku.net/doc/d716635899.html, (D.B.Chelton).

angle of about1.5°equatorward.This small,but we believe statistically signi?cant,equatorward rotation of the central azimuth may be evidence of the effects of ambient currents(meridional advection or the effects of vertical shear on the potential vorticity gradient vector)on the propagation directions of the eddies.

While the results presented here are persuasive evidence that most of the observed westward-propa-gating SSH variability consists of isolated nonlinear mesoscale eddies,it is shown that the eddy propaga-tion speeds are about25%slower than the westward propagation speeds of features in the SSH?eld that have scales larger than those of the tracked eddies.This scale dependence of the propagation speed may be evidence for the existence of dispersion and the presence of features that obey linear Rossby wave dynamics and have larger scales and faster propagation speeds than the nonlinear eddies.The amplitudes of these larger-scale signals are evidently smaller than those of the mesoscale eddy?eld since they are not easily isolated from the energetic nonlinear eddies.

1.Introduction

High-resolution sea-surface height(SSH)?elds constructed by merging measurements from two simultaneously operating altim-eters(Ducet et al.,2000;Le Traon et al.,2003)have revealed that SSH variability is dominated by westward-propagating nonlinear mesoscale eddies throughout most of the World Ocean(Chelton et al.,2007).Prior to the availability of this merged dataset,inter-pretations of westward-propagating SSH variability were based on SSH?elds constructed from TOPEX/Poseidon(T/P)data alone.The ground track spacing of the T/P orbit was too coarse to resolve the mesoscale variability that is evident in the merged altimeter data-set.The merged dataset is thus enabling observational studies of mesoscale ocean variability that were not previously possible using altimetry data.This investigation extends the global analysis of10years of these high-resolution SSH?elds by Chelton et al. (2007)to include an additional6years of data,and presents a re-?ned and more comprehensive summary of the characteristics of the observed mesoscale eddies detected using an improved eddy identi?cation and tracking procedure.

While it is arguably a matter of semantics,the terminology adopted here refers to features that obey linear dynamics,perhaps modi?ed by ambient conditions of mean?ow or bottom topogra-phy,as Rossby waves.The term eddy is reserved for the coherent mesoscale features that are the focus of this study,which are shown in Section6.1to have maximum rotational?uid speeds U that exceed their translation speed c,and are therefore character-ized by an advective nonlinearity ratio U/c>1.The possible alter-native terminology‘‘nonlinear wave’’for these features is purposely avoided in order to emphasize the distinction from lin-ear waves.The mesoscale features for which U/c>1can advect a parcel of trapped?uid as they translate.

From an historical perspective,it is important to note that west-ward-propagating SSH variability could not be unambiguously identi?ed by satellite altimetry prior to the launch of T/P.The T/P orbit was carefully designed to minimize the effects of tidal alias-ing,thus allowing the detection of westward propagating features without the aliased tidal errors that contaminated SSH?elds con-structed from measurements from the Geosat altimeter that pre-ceded T/P(Schlax and Chelton,1994a,b,1996;Parke et al.,1998). It was evident from the?rst few years of the SSH?elds from T/P that westward propagation is nearly ubiquitous in the World Ocean(Chelton and Schlax,1996),con?rming the conclusions from previous analyses of upper-ocean thermal observations in the North Paci?c during the1970s and1980s(see Fig.9of Fu and Chelton,2001).Subsequent global analyses from a10-year T/P data record(Fig.14of Fu and Chelton,2001)and from the16-year merged dataset analyzed in this study(see Section7.2and Fig.22below)have validated the strong tendency for westward propagation of SSH variability and?ne-tuned the estimates of the propagation speeds.

The qualitative similarity between the latitudinal variation of the observed westward propagation speeds and the phase speeds expected for long baroclinic Rossby waves has led to widespread interpretation of the westward propagation as linear Rossby waves. Close scrutiny reveals that the propagation speeds outside of the tropics are somewhat faster than predicted by the classical theory for Rossby waves(Chelton and Schlax,1996;Fu and Chelton, 2001,and references therein;Osychny and Cornillon,2004,and numerous subsequent studies;see also Section7.2and Fig.22be-low).This has inspired numerous theoretical studies to understand the dynamics responsible for the speedup.While the relevance of these theories to the nonlinear mesoscale features that are shown here and by Chelton et al.(2007)to dominate the SSH variability is unclear,these theoretical studies have led to important improve-ments in the understanding of the dynamical effects of ambient conditions on Rossby waves.In particular,it has been shown that much of the discrepancy between the observed westward propaga-tion speeds and those predicted by the classical theory may be ac-counted for by the vertical shear of background mean currents(e.g., Killworth et al.,1997;Dewar,1998;de Szoeke and Chelton,1999; Liu,1999;Yang,2000;Colin de Verdière and Tailleux,2005), small-scale bottom roughness(Tailleux and McWilliams,2001)or the combined effects of vertical shear and variable large-scale bot-tom topography(Killworth and Blundell,2004,2005,2007).

To date,little attention has been paid to other inconsistencies between the observations and classical Rossby wave theory.In particular,

(1)It was apparent even from the T/P data that the observed

westward-propagating SSH over most of the ocean(espe-cially poleward of about20°of latitude)was dominated by ‘‘blobby’’structures rather than the latitudinally b-refracted continuous crests and troughs that are expected for long Rossby waves and are sometimes evident in the altimeter data.

(2)The observed variability comprises a broad continuum of

time and space scales,with little or no evidence in most regions of a spectral peak at the annual period that might have been expected for long Rossby waves forced by the strong annual cycles of wind and thermal forcing and has been sought in numerous past studies.

(3)There is little evidence in the time-longitude structure of

SSH variability for the dispersion expected for linear Rossby waves;the blobby features propagate westward for long dis-tances as coherent structures.

(4)There is little evidence for the meridional propagation

expected for Rossby waves with the?nite meridional scales of the blobby structures apparent in the SSH?elds.

These characteristics are all consistent with the conclusion of Chelton et al.(2007)that the westward propagating variability

168 D.B.Chelton et al./Progress in Oceanography91(2011)167–216

consists mostly of nonlinear eddies rather than linear Rossby waves.The focus on Rossby wave interpretations in earlier studies was a consequence of the coarse resolution of SSH?elds constructed from T/P data alone(see the top panel of Fig.1and Appendix A.1).The distinction between linear Rossby waves and nonlinear eddies is important since the latter can transport water parcels and their associated physical,chemical and biological prop-erties,while linear Rossby waves cannot.Eddies can thus have important in?uences on heat and momentum?uxes and on marine ecosystem dynamics.

The SSH?elds analyzed here span the16-year period14Octo-ber1992through31December2008and were constructed by SSALTO/DUACS at7-day intervals on a Mercator grid with a nom-inal spacing of1/3°using measurements from two simultaneously operating altimeters,one in a10-day exact repeat orbit(T/P,fol-lowed by Jason-1and presently by Jason-2)and the other in a 35-day exact repeat orbit(ERS-1followed by ERS-2and presently by Envisat).The SSALTO/DUACS processing(see Appendix A.2)in-cludes removal of the7-year mean SSH(1993–1999)to eliminate the unknown geoid.These SSH?elds are distributed and referred to by AVISO as the‘‘Reference Series.’’(See the Acknowledgments for de?nitions of the above acronyms associated with the dataset analyzed here.)The analysis presented here is based on the version of this Reference Series that was available in early2010and in-cluded SSH?elds for the14October1992–31December2008time period(see the footnote in Appendix A.2).Except in the left panels of Fig.A3in Appendix A.3for reasons explained in the caption,the analysis presented here is based on the anomaly SSH?elds that were interpolated by SSALTO/DUACS from their1/3°Mercator grid onto a globally uniform1/4°latitude by1/4°longitude grid.

It is shown in Appendix A.3that the objective analysis proce-dure used to construct the SSH?elds of the AVISO Reference Series has half-power?lter cutoff wavelengths of about2°in latitude by 2°in longitude.For eddies with Gaussian shape,this corresponds approximately to an e-folding radius of about0.4°,or roughly 40km(see Appendix A.3).Even a cursory comparison of the SSH ?elds of the AVISO Reference Series with the low-resolution SSH ?elds from T/P data alone reveals a fundamentally different per-spective on the nature of SSH variability(Fig.1).The T/P data re-solve only very large scales while the SSH?elds of the AVISO Reference Series are rich in mesoscale cyclonic and anticyclonic features(negative and positive SSH,respectively)with O(100km) radius scales that are too small to be detected by the T/P sampling pattern,except when these features are near the crossovers of ascending and descending ground tracks.The eddy detection algo-rithm developed and applied to the AVISO dataset for this study (Appendix B.2)identi?es3291mesoscale eddies in Fig.1alone, 2398of which were trackable for4weeks or longer.This is typical of the number of eddies that are detectable and trackable at any gi-ven time in this dataset.

We note that AVISO also provides SSH?elds with higher accu-racy and potentially higher resolution than the Reference Series analyzed here.These?elds,referred to by AVISO as the‘‘Updated Series,’’were constructed from measurements by all of the altime-ters available at any given time.The most well-sampled time per-iod is the3-year period October2002through September2005 during which four altimeters were operating simultaneously (Jason-1,T/P in an orbit interleaved with the Jason-1ground tracks, Envisat and Geosat Follow-On).While three altimeters were oper-ating simultaneously at various other times during the16-year data record analyzed here,only two altimeters were operating most of the time,in which case the SSH?elds of the Updated Series are identical to those of the Reference Series.The superiority of the SSH?elds of the Updated Series when more than two altimeters were in operation has been demonstrated by Pascual et al. (2006).For the purpose of this investigation,however,the homogeneous resolution of the SSH?elds of the AVISO Reference Series over the16-year data record is preferable to the temporally varying resolution of the Updated Series.The potentially larger amplitudes and smaller scales(because of improved resolution) of eddies in the Updated Series during periods when more than two altimeters were operating could complicate the statistical analysis of mesoscale eddies presented here.A comparison of the results of this study with the eddy characteristics deduced from the Updated Series during periods of higher resolution SSH?elds is deferred to a future investigation.

While it is visually apparent from the middle panel of Fig.1that much of the SSH variability is composed of energetic mesoscale features,there is also evidence at latitudes lower than about20°in the Paci?c for the long crests and troughs that are expected for Rossby waves,distorted into westward-pointing chevron pat-terns by b refraction.Although they are relatively small in ampli-tude and are‘‘speckled’’by much more energetic mesoscale features,these telltale chevron patterns are identi?able across much of the South Paci?c,arguably to latitudes as high as50°S in the eastern,and possibly the central,part of the basin.In the North Paci?c,they are less evident in the middle panel of Fig.1be-cause of an overall higher SSH in the northern hemisphere in this map,as expected from the steric effects of summertime heating of the upper ocean during the August time period of the map. Depending on the details of the?ltering,the chevron patterns can become more evident in the eastern North Paci?c as far north as about50°N when the SSH?elds are spatially high-pass?ltered to remove the steric effects of large-scale heating and cooling (e.g.,the one-dimensional zonal high-pass?ltering used for Fig.1 of Chelton et al.,2007),but they generally do not penetrate more than about2000km westward from the eastern boundary in the North Paci?c(Fu and Qiu,2002).These chevron patterns are mostly eliminated with the two-dimensional high-pass?ltering applied to isolate the mesoscale eddies for this investigation(bot-tom panel of Fig.1;see Section2for a description of this?ltering).

The abundance of mesoscale features in the SSH?elds of the AVISO Reference Series con?rms globally the view of the ocean posited from regional?eld programs during the1970s,referred to by Wunsch(1981)as the‘‘decade of the mesoscale’’.Observa-tions in the western North Atlantic from the Mid-Ocean Dynamics Experiment(MODE Group,1978)and POLYMODE(McWilliams et al.,1983)were interpreted as evidence that mid-ocean variabil-ity is dominated by mesoscale eddies(see also Robinson,1983). Satellite altimetry has thus advanced to the point where observational studies of mesoscale dynamics that have been feasi-ble only from regional in situ datasets can now be addressed glob-ally from multiple satellite altimeters operating simultaneously, with the caveat that only the surface characteristics can be ob-served by altimetry.

This paper is organized as follows.The resolution of the SSH ?elds of the AVISO Reference Series and the automated eddy iden-ti?cation procedure developed for this study are summarized in Section2;the details of the assessment of the resolution are given in Appendix A and the details of the eddy identi?cation and track-ing procedure and an assessment of the biases of the eddy ampli-tude estimates are presented in Appendices B and C,respectively. Census statistics for the$36,000eddies with lifetimes of16weeks and longer identi?ed and tracked in the16-year data record by this automated procedure are presented in Section3:their lifetimes, propagation distances,trajectories,geographical distributions, and polarities(cyclonic versus anticyclonic).The kinematic proper-ties(amplitudes,scales,rotational speeds and estimated Rossby numbers)of these robust mesoscale eddies are summarized in Sec-tion4and the composite average eddy shape is investigated in Sec-tion5.The nonlinearity of the mesoscale eddies is assessed in Section6from three different metrics,and their propagation

D.B.Chelton et al./Progress in Oceanography91(2011)167–216169

characteristics(direction and speed)are summarized in Section7. This wealth of information about mesoscale eddies deduced from the16-year dataset is summarized in Section8.

2.Feature resolution and automated eddy detection

It is shown in Appendix A.3that features with wavelength scales shorter than3°are attenuated in the SSH?elds of the AVISO Reference Series analyzed in this study.The variance attenuation is about a factor of2at a wavelength of about2°,which we interpret as the approximate half-power?lter cutoff of the objective analysis procedure used to construct the AVISO?elds.This?lter cutoff wavelength can be expressed in terms of the approximate scales of the mesoscale features that can be resolved by considering an idealized eddy that has the form of a two-dimensional axisymmet-ric Gaussian,which is shown in Section5to be a reasonable approximation on average,at least over the central2/3of the ed-dies,and is adequate for present purposes.A wavelength resolu-tion of2°corresponds to a Gaussian eddy with an e-folding radius of about0.4°(see Appendix A.3).We thus conclude that the SSH?elds of the AVISO Reference Series have been?ltered to attenuate Gaussian-like features with e-folding radii shorter than roughly40km.It should be kept in mind,however,that only those features with e-folding radii larger than about60km are unatten-uated by the?ltering.(This is the e-folding scale of a Gaussian that corresponds to the3°wavelength at which there is no attenuation of the SSH?elds of the AVISO Reference Series.)The amplitudes of features with smaller e-folding radii are increasingly attenuated with decreasing scale;the amplitude attenuation(as opposed to variance attenuation)is about a factor of2à1/2at the e-folding ra-dius of$40km.

The oceanic mesoscale can be characterized as consisting of var-iability with radius scales of10–500km.The lower end of the range of spatial scales of mesoscale variability is thus not address-able from the$40km feature resolution of the SSH?elds analyzed here.The conclusions of this study are therefore restricted to mesoscale eddies with relatively large radii.

Evidence is presented in Appendix A.3that the smoothing in the objective analysis procedure used to produce the SSH?elds of the AVISO Reference Series(see Appendix A.2)may not be quite suf?-cient.The SSH variance in these?elds is locally higher near the T/P crossovers where SSH variability is best resolved in the merged dataset,and lower in the centers of the diamonds formed by the T/P ground track pattern where SSH observations are limited to

on28August1996constructed from TOPEX/Poseidon(T/P)data only(top)and from the merged

panel is the SSH?eld from the merged T/P and ERS-1data after spatially high-pass?ltering

automated procedure described in Appendix B.2identi?es3291eddies in the bottom panel,of which 170 D.B.Chelton et al./Progress in Oceanography91(2011)167–216

just the altimeter in the35-day repeat orbit(see the bottom panels of Fig.A1).These inhomogeneities in SSH variance from spatially varying resolution capability of the sampling pattern of the two simultaneously operating altimeters from which the SSH?elds of the AVISO Reference Series are constructed can spuriously modu-late estimates of the amplitude,radius and the location of the cen-troid of a propagating eddy(see,for example,Pascual et al.,2006), which can interrupt the tracking of the eddy.It is not possible to say how frequently this occurs in the16-year global dataset ana-lyzed here,but we believe that any such problems do not strongly in?uence the general conclusions of this study.

A new eddy identi?cation procedure was developed for this study.As described in Appendix B.2,this procedure is based on de?ning the eddies in terms of SSH,thus obviating the need to dif-ferentiate the SSH?elds and avoiding the associated deleterious ef-fects of noise that are problematic in the procedure based on products of second derivatives of SSH in the Okubo-Weiss param-eter(see Appendix B.1)that has been used in numerous previous studies,including Chelton et al.(2007).The assessment in Appendix A.3of the?ltering properties of the objective analysis procedure used to construct the SSH?elds of the AVISO Reference Series that are analyzed in this study is incorporated into the SSH-based eddy identi?cation and tracking procedure to set a minimum area for the mesoscale features that are tracked from one time step to the next.The new procedure yields signi?cantly more eddies than found by Chelton et al.(2007)and these eddies have longer lifetimes(see Fig.B3).For the eddies with lifetimes of16weeks and longer that are the focus of this study(see Section3.1),the new procedure yields about twice as many eddies.For lifetimes longer than a year,there are about?ve times more eddies in the new eddy dataset.

Automated detection and tracking of mesoscale features in the SSH?elds is complicated by several factors.The most apparent of these is the presence of large-scale SSH variability from the steric effects of heating and cooling of the upper ocean that result in large-scale SSH variations that often mask the more subtle signa-tures of eddies,especially in the open ocean away from the regions of energetic mesoscale variability associated with unstable cur-rents.While such signals occur on a wide range of time scales,they are dominated by the large-scale annual cycle of summertime heating and wintertime cooling.The effects of this annual steric heating and cooling are readily apparent from the hemispheric dif-ferences in large-scale sea level in the top and middle panels of Fig.1(high in the northern hemisphere and low in the southern hemisphere in these summertime maps).A cyclonic eddy(concave upward SSH)that is clearly identi?able during the wintertime can be more dif?cult to detect in summer when SSH is high over the entire ocean basin.Likewise,anticyclonic eddies can be more dif?-cult to detect during the wintertime when SSH is low over the en-tire ocean basin.These dif?culties are easily seen from time-longitude plots of SSH variability,in which the angled patterns associated with westward propagating eddies are interrupted annually by horizontal bands of zonally coherent and large-ampli-tude SSH?uctuations from seasonal heating and cooling(see,for example,Fig.11of Fu and Chelton,2001).To facilitate eddy iden-ti?cation,the AVISO SSH?elds were therefore spatially high-pass ?ltered in two dimensions to remove variability with wavelength scales larger than20°of longitude by10°of latitude.This?ltering is very effective at removing steric heating and cooling effects,as well as other large-scale variability(compare the middle and bot-tom panels of Fig.1).Analogous high-pass?ltering has been ap-plied in all previous investigations of westward-propagating SSH variability,although usually in the form of one-dimensional (zonal)?ltering.

Other factors that limit the accuracy of eddy identi?cation are more dif?cult to contend with.The most signi?cant is the practical dif?culty of de?ning an eddy boundary.Since eddies are continu-ally evolving,time-dependent?uid structures that do not have persistent,clearly demarcated boundaries,there is inevitably some arbitrariness both in the fundamental de?nition of an eddy struc-ture and in the speci?c eddy boundaries de?ned by any automated procedure.Because of the mesoscale complexity of the SSH?eld that generally surrounds an identi?ed eddy feature,SSH along the outer boundary of a compact feature as de?ned by the auto-mated procedure usually has a non-zero value of SSH.As a conse-quence,the residual SSH?elds after subtracting the eddy contributions to SSH within all of the de?ned eddy boundaries con-sists of‘‘plateaus’’of constant SSH across the interior of the eddy (locally higher than the ambient SSH for anticyclones and locally lower for cyclones).These plateaus are typically interconnected to the residual plateaus of other eddies by ridges and valleys that are part of the overall mesoscale variability,presumably arising from the spectral continuum of the up-scale energy cascade of geostrophic turbulence(e.g.,Kraichnan,1967;Batchelor,1969; Charney,1971;Rhines,1975,1979;Stammer,1997;Scott and Wang,2005).From a dynamical or kinematical standpoint,some or all of this residual variability should perhaps be more properly considered to be part of the eddy structures,although this would result in eddy boundaries with non-compact form.Exclusion of this residual SSH from the interiors of de?ned eddy perimeters thus introduces what arguably might be interpreted as bias in the estimated amplitudes of the tracked eddies.

The adequacy of the estimated amplitudes of the eddies ob-tained from the automated procedure described in Appendices B.2–B.4is assessed in Appendix C.It is concluded that the above-noted mesoscale variability with non-compact form accounts for much of the SSH variance.Any eddy identi?cation algorithm that de?nes eddies based on a conceptual notion of compact structures in SSH will therefore unavoidably leave much of the total SSH var-iance unaccounted for,even if each de?ned eddy-like feature encapsulates all of the SSH topography within the portion of the de?ned eddy that has compact form.The bias of the estimated amplitudes of the mesoscale features with compact form is shown in Appendix C to be usually less than1cm outside of the regions of most energetic mesoscale variability(see Figs.C2and C3).The bias in regions of energetic mesoscale variability may sometimes be1 or2cm,but only occasionally more than that,in the energetic regions.

3.Census statistics of mesoscale coherent structures

3.1.Eddy lifetimes and propagation distances

Global histograms and upper-tail cumulative histograms(i.e., the number of eddies with lifetimes greater than or equal to each particular value along the abscissa)of the eddy lifetimes are shown separately for cyclones and anticyclones in Fig.2.In total,the auto-mated procedure summarized in Appendices B.2–B.4detected $177,000eddies with lifetimes of4weeks or longer over the16-year data record;eddies with lifetimes shorter than4weeks were discarded.The eddy counts drop off rapidly with increasing life-time.The numbers of eddies with lifetimes that exceeded16,26, 52,78and104weeks were35,891,17,252,4396,1494and620, respectively.There is a slight preference for cyclonic over anticy-clonic eddies with lifetimes of60weeks or less(bottom left panel of Fig.2).Eddies with lifetimes of78weeks and longer were pref-erentially anticyclonic.Overall,there were6%more cyclones than anticyclones for lifetimes of16weeks and longer,but21%more anticyclones than cyclones for lifetimes of78weeks and longer (bottom right panel of Fig.2).

D.B.Chelton et al./Progress in Oceanography91(2011)167–216171

To alleviate concerns that imperfections of the detection and tracking procedure may affect the conclusions of this study,we fo-cus attention on the robust eddies,which we de?ne to be eddies with lifetimes of16weeks and longer.These$36,000eddies have an average lifetime of32weeks and their average propagation dis-tance is550km.Histograms and upper-tail histograms of the eddy propagation distances for these eddies are shown separately for cy-clones and anticyclones in Fig.3.The eddy counts drop off rapidly with increasing propagation distance.The numbers of eddies with propagation distances that exceeded500,1000,2000,3000and 4000km were13,211,4886,1066,326and129,respectively.Con-sistent with the notion that eddy lifetime and propagation distance are correlated,there is a small preference for cyclones over anticy-clones for propagation distances of2500km or less(bottom left panel of Fig.3).Eddies that propagated longer distances were pref-erentially anticyclonic.Overall,there were59%more anticyclones than cyclones with propagation distances of3500km and longer.

3.2.Eddy trajectories

More than75%of the$36,000eddies analyzed here(Fig.4a) propagated westward.The8606eddies that had a net eastward displacement(Fig.4b)occurred primarily in the strong eastward Antarctic Circumpolar Current,the Gulf Stream and its extension northeast of Grand Banks,and in the region of con?uence of the Kuroshio and Oyashio Currents and their eastward extensions. These regions of eastward propagation have previously been noted by Fu(2009),and were identi?ed for the Antarctic Circumpolar Current region by Hughes et al.(1998).This is to be expected from advection of the eddies by the strong eastward currents in these regions.Histograms and upper-tail histograms of the lifetimes and propagation distances of the eastward propagating cyclonic and anticyclonic eddies are shown by the lower thick and thin lines,respectively,in the top panels of Figs.2and3.They account for about25%of the tracked eddies with short lifetimes,decreasing to about15%for the longest lifetimes(Fig.2).The most notable characteristic of the eastward-propagating eddies is that they have much shorter propagation distances than the westward propagat-ing eddies(Fig.3).Fewer than20eddies of each polarity propa-gated eastward more than1000km and none propagated eastward more than1800km.

Except in some of the low-latitude regions,the high density of the eddy tracks in Fig.4a obscures individual eddy trajectories. To see the propagation of individual eddies better,the eddy trajec-tories with successively longer minimum lifetimes of26,52,78 and104weeks are shown in Figs.4c–f.The numbers of eddies of each polarity are labeled at the top of each panel.The anticyclonic preferences for eddies with long lifetimes and large propagation distances are visually evident from Fig.4e and f.Close inspection reveals that cyclones and anticyclones tend to have opposing small meridional de?ections,poleward for cyclones and equatorward for anticyclones.As shown by Morrow et al.(2004),this phenomenon is most clearly evident off the west coasts of Australia,North Amer-ica,and South Africa.It can also be seen off the west coast of South America.While more dif?cult to see in the mid-ocean regions be-cause of the high density of eddy tracks,opposing meridional drifts of cyclones and anticyclones occur throughout the World Ocean. These meridional de?ections are investigated in detail in Section7.1.

3.3.Geographical distribution of the eddies

A census of the$36,000eddies with lifetimes P16weeks that are analyzed here is shown in Fig.5.The upper panel shows the number of eddy centroids that propagated across each1°?1°re-gion over the16-year data record.Within the eddy-rich

regions, 172 D.B.Chelton et al./Progress in Oceanography91(2011)167–216

about15–30eddy centroids were typically observed,i.e.,about one to two per year,on average.Almost no eddy centroids were ob-served in the regions centered at about50°N,160°W in the north-east Paci?c and at about50°S,95°W in the southeast Paci?c.These ‘‘eddy deserts’’have been noted previously by Chelton et al.(2007). If eddies exist in these regions,their amplitudes or scales are too small to be detected and tracked or their lifetimes were shorter than the minimum16-week lifetime considered here.

A notable feature of the upper panel of Fig.5is the small num-ber of eddies observed throughout the equatorial region.The spa-tial scales of eddies and eddy-like features at these low latitudes are large because of the large Rossby radius of deformation(see the right panel of Fig.12below).As represented in the SSH?elds of the AVISO Reference Series,the SSH topography within the inte-riors of these large-scale features often consists of considerable small-scale irregularity with small amplitude,likely due at least in part to noise in these SSH?elds constructed from two simulta-neously operating altimeters(Pascual et al.,2006;see also the bot-tom panels of Fig.A1and the related discussion in Appendix A.3). As a consequence,the automated procedure described in Appendix B.2often identi?es multiple small eddies within the interiors of the large low-latitude features.In combination with the observed rapid evolution of the irregular SSH topography(again suggestive of noise)and the fast westward propagation of the large-scale features at these low latitudes,this poses dif?cult challenges for automated eddy identi?cation and tracking procedures.The procedure described in Appendix B.2could be modi?ed to track only the large-scale aspects of the features that are of interest at these low latitudes(e.g.,tropical instability waves)but this is deferred to a fu-ture study.The emphasis here is therefore on mesoscale variability at latitudes higher than about10°at which propagation speeds and irregularities in the SSH topography within eddy interiors are less problematic for automated eddy identi?cation and tracking.

The eddy centroid census in the upper panel of Fig.5is a con-servative estimate of the number of eddies that in?uence SSH at any given location.The lower panel shows the census of eddy inte-riors that propagated across every1°?1°region during the16-year data record.The interiors are de?ned here to be the portion of each identi?ed eddy within the closed contour of SSH around which the average geostrophic speed is maximum(see Appendix B.3),which corresponds approximately to a contour of zero relative vorticity.On average,this speed-based eddy scale is about70%of the effective radius of the area enclosed by the eddy boundary as de?ned by the automated eddy identi?cation procedure(see Fig.B2b).The geographical pattern of the census of eddy interiors is very similar to that of the census of eddy centroids.The number of tracked eddies during the16-year data record that in?uence SSH within any particular1°?1°region is four to six eddies per year, on average,in the eddy-rich regions,i.e.,about three times larger than the number of eddy centroids that propagate through the re-gion as shown in the top panel of Fig.5.

The reduced number of eddies along the axes of the Gulf Stream,the Agulhas Return Current,and the Kuroshio Extension (especially the latter two)that is evident in Fig.5is noteworthy. Comparatively large numbers of eddies occurred in bands that straddle both sides of the cores of these jet-like currents,presum-ably from the shedding of eddies in the form of detached meanders on both sides of the meandering currents.The mesoscale variabil-ity within the cores of the currents consists mostly of meanders of the jet-like currents rather than isolated vortices.Because the anal-ysis is performed on20°?10°spatially high-pass?ltered anomaly SSH,meanders resemble vortices in any particular snapshot.Unlike coherent vortices,however,the shapes of meanders in the anomaly SSH?elds can deviate from the compact forms assumed by our eddy identi?cation procedure.Moreover,they can evolve more rapidly in time and may therefore not be trackable for the mini-mum lifetime of16weeks for the mesoscale features analyzed in this study.For both of these reasons,the numbers of tracked eddies are somewhat lower in the cores of the jet-like currents than on their

?anks.

D.B.Chelton et al./Progress in Oceanography91(2011)167–216173

3.4.Eddy origins and terminations

A census of eddy origins is shown in the upper panel of Fig.6.The most clearly de?ned regions of frequent eddy formation are along the eastern boundaries of the ocean basins.These eddies most likely form as meanders that pinch off of the eastern bound-ary currents and undercurrents or from other manifestations of baroclinic instability in these regions of vertically sheared currents.Outside of these eastern boundary regions,eddies apparently form throughout most of the open-ocean regions where propagating ed-dies are observed (Figs.4a and 5).This is consistent with the con-clusions of Gill et al.(1974),Robinson and McWilliams (1974),Stammer (1998)and Smith (2007b)and others that nearly all of the World Ocean is baroclinically unstable,particularly in regions where the ?ow is non-zonal (Spall,2000;Arbic and Flierl,2004;Smith,2007a ).The large number of eddies formed along the various seamount chains northwest of Hawaii is notable.This may be an indication of interaction between bottom topography and the ?ow ?eld,which could include Rossby waves incident from the eastern basin.Or it may be attributable to abrupt ampli?cation of westward propagat-ing eddies that are too small to detect in the eastern basin and only become trackable when their amplitudes increase as they encoun-ter these bathymetric features.

It should be kept in mind that some of the apparent eddy formations in the upper panel of Fig.6may actually be the reappear-ance of eddies that are temporarily lost to the tracking procedure be-cause of a variety of factors (e.g.,noise in the SSH ?elds or because the shapes of the eddies become temporarily too distorted from interac-tions with other nearby mesoscale features).Based on animations of the tracked eddies,we do not feel that this is a major problem,but we are not able to quantify how frequently this

occurs.

The trajectories of cyclonic (blue lines)and anticyclonic (red lines)eddies over the 16-year period October 1992–December 2008for (a)lifetimes lifetimes P 16weeks for only those eddies for which the net displacement was eastward.The numbers of eddies of each polarity are labeled at the top

A census of eddy terminations is shown in the lower panel of Fig.6.Like the eddy formation census in the upper panel,eddy terminations occur throughout the regions where eddies are ob-served.Not surprisingly,terminations are more frequent near most of the western boundaries,especially in the Gulf Stream and Brazil Current and along the Hawaiian Island Chain.The ter-minations in the open ocean could occur for a variety of reasons, including frictional decay and coalescence with other eddies as a consequence of the up-scale energy cascade of geostrophic turbu-lence.Some of these terminations may also occur from temporary or permanent loss of an eddy by the tracking procedure because of noise in the SSH?eld or imperfections of the tracking algorithm.3.5.Eddy contributions to SSH variance and eddy kinetic energy

Because of the spatial high-pass?ltering described in Section2 that was applied to the raw SSH?elds obtained from AVISO,fea-tures with wavelength scales larger than about20°of longitude by10°of latitude have been attenuated.As previously discussed in Section2and shown in detail in Appendix A.3,features with wavelength scales smaller than about3°?3°have been attenu-ated by the?ltering of the objective analysis procedure used to construct the SSH?elds of the AVISO Reference Series.The half-power?lter cutoff of this?ltering is estimated in Appendix A.3 to be about2°?2°.The?ltered SSH?elds analyzed here thus re-tain variability with zonal wavelength scales between about2

°Fig.4c and d.The same as Fig.4a,except:(c)lifetimes P26weeks and(d)lifetimes P52weeks.

and20°of longitude and meridional wavelength scales between about2°and10°of latitude.Numerous physical processes contrib-ute to the variance and eddy kinetic energy of the mesoscale vari-ability that is resolvable by these SSH?elds.In addition to the eddies with compact form that are the focus of this study,there are meanders of hydrodynamically unstable currents and the inter-connecting ridges and valleys between eddies discussed previously that are part of the up-scale energy cascade.There is also large-scale variability of SSH that is unrelated to the eddy?eld.The objective of this section is to assess the contributions to the?ltered SSH variability from the resolvable vortices with compact form.

The fraction of SSH variance that is accounted for by compact eddies is more dif?cult to determine than it may at?rst seem.This depends critically on how well the eddy boundaries can be de?ned. To appreciate the dif?culty,consider the20°?10°high-pass?l-tered SSH at a location(x,y)and time t i,which we denote as h(x,y,t i).The sample mean of this SSH over the N=847gridded ?elds at7-day intervals in the16-year AVISO Reference Series ana-lyzed here is

hex;yT?1

X N

i?1

hex;y;t iT:e1T

In view of the removal of the7-year(1993–1999)average SSH as part of the processing by SSALTO/DUACS(see Appendix A.2)and the20°?10°high-pass?ltering applied here to the raw AVISO SSH?elds,it is not surprising that this mean is found to be less than a few centimeters everywhere.The total variance of the20°?10°high-pass?ltered SSH?elds is

tot

ex;yT?

X N

i?1

?hex;y;t iTà hex;yT 2:e2

TFig.4e and f.The same as Fig.4a,except:(e)lifetimes P78weeks and(f)lifetimes P104weeks.

D.B.Chelton et al./Progress in Oceanography91(2011)167–216177

eddy centroids(top)and eddy interiors(bottom)for eddies with lifetimes P16weeks that passed

1992–December2008.The eddy interiors are de?ned by the contour of SSH around which the average

contour of zero relative vorticity).

(a)

(b)

lifetimes P16weeks showing the numbers of(a)eddy originations and(b)eddy terminations for

A very liberal estimate of the contributions of compact eddies to SSH variance can be obtained by assuming that all of the SSH within the boundary of each eddy can be attributed to that eddy(e.g., Chelton et al.,2007).In this case,the eddy variance is

r2 hi ex;yT?

1X N

i?1

dex;y;t iT?hex;y;t iTà hex;yT 2;e3T

where d(x,y,t i)=1if the location(x,y)is within an eddy at time t i and0otherwise.This effectively assumes that the basal value of each eddy is zero.If an eddy is embedded in a region of non-zero larger-scale ambient SSH from other causes or is interconnected to neighboring eddies by non-compact mesoscale variability in the form of the ridges and valleys of the spectral continuum of the up-scale energy cascade,as is often the case(see Figs.C2and C3),this ascribes too much SSH variance to the eddies.The eddy contribution to SSH variance given by Eq.(3)that was reported by Chelton et al.(2007)is therefore biased high.

An estimate of the variance of eddies with compact form that takes into account any local background SSH outside the de?ned eddy boundaries includes only the portion of SSH within each eddy relative to the basal value of the eddy.For a particular eddy,de?ne h0(x,y,t i)to be the basal value associated with every location(x,y) that is interior to the eddy at time t i.In practice,h0is de?ned to be the average SSH over the piecewise continuous perimeter pixels of each eddy(see Appendix B.3).The estimate of eddy variance ad-justed for the basal value of each of the compact eddies in the data-set is

r2 lo ex;yT?

1X N

i?1

dex;y;t iT?hex;y;t iTàh0ex;y;t iT 2:e4T

If this concept of localized eddies superimposed on a regional back-ground SSH unrelated to the eddy?eld were correct and it were possible to de?ne and determine each eddy boundary unambigu-ously and accurately,this approach would provide an accurate assessment of the eddy contributions to SSH variance.

In practice,eddies do not have well-de?ned boundaries.As dis-cussed in Appendix C,the outer perimeter of an eddy becomes especially dif?cult to de?ne when the eddy is interacting with other eddies nearby.Estimated eddy boundaries may therefore not encapsulate all of the SSH attributable to the eddies,i.e.,the h0(x,y,t i)are conservative estimates of the true basal heights; the estimates are likely to be biased high for anticyclonic eddies and low for cyclonic eddies,resulting in underestimation of eddy amplitudes and residual plateaus after subtracting the SSH relative to the estimated basal heights.The eddy variance in Eq.(4)is thus necessarily biased low because of this incomplete eddy removal.It is shown in Appendix C that these biases in the estimated ampli-tudes of eddies with compact form are seldom more than1cm out-side of regions of energetic mesoscale variability and may be1or 2cm,but only occasionally more than that,in the energetic regions (Figs.C2and C3).Much of the background SSH outside of these compact features consists of larger-scale variability that is unre-lated to the eddy?eld,as well as the previously noted intercon-necting ridges and valleys of SSH between eddies(Figs.C2and C3).The eddy identi?cation procedure summarized in Appendix B.2has been speci?cally designed to exclude these non-compact structures since they do not have a form that resembles the usual notion of an eddy.

The two approaches in Eqs.(3)and(4)to estimating the eddy contributions to SSH variance could be cautiously interpreted as upper and lower bounds on the actual SSH variance that is attrib-utable to eddies.However,a better assessment of the performance of the automated eddy identi?cation and tracking procedure can be obtained from the eddy kinetic energy determined from the velocity components u and v computed from SSH by the geostropic Eqs.(B.3a)and(B.3b)in Appendix B.1.The large-scale SSH and much of the non-compact mesoscale variability are effectively?l-tered out by the spatial high-pass?ltering of the derivative opera-tions applied to the SSH?elds to compute geostrophic velocity to determine the eddy kinetic energy.The total eddy kinetic energy from the20°?10°high-pass?ltered SSH at location(x,y)is

EKE totex;yT?

q1X N

i?1

?uex;y;t iTà uex;yT 2t?vex;y;t iT

à vex;yT 2;e5Twhere q is the water density and uex;yTand vex;yTare de?ned fol-lowing Eq.(1)for the mean SSH, hex;yT,to be the sample mean geo-strophic velocity components at location(x,y)over the N=847 gridded SSH?elds analyzed here.The eddy kinetic energy attribut-able to the mesoscale features with compact form(eddies)can be estimated by including only the geostrophic velocities within eddy interiors,

EKE loex;yT?

X N

i?1

dex;y;t iT?uex;y;t iTà uex;yT 2

tvex;y;t iTà vex;yT 2

;e6T

where d(x,y,t i)is de?ned as in Eq.(3).Since the estimated eddy boundaries may not encapsulate all of the SSH attributable to the eddies as discussed above(see also Appendix C),this is a lower-bound estimate of the eddy kinetic energy attributable to the eddies.

The value of EKE lo computed by Eq.(6)and expressed as a per-centage of the total eddy kinetic energy EKE tot of the20°?10°high-pass?ltered SSH?elds computed by Eq.(5)is shown in the top panel of Fig.7for the case of the eddies with lifetimes of 16weeks and longer that are the focus of this study.In the eddy-rich regions,the eddies typically account for more than40%of the eddy kinetic energy,with more than70%explained in some re-gions(e.g.,the regions southwest of Australia,to the west of South Africa,in the Alaska Stream along the Aleutian Island Chain,in por-tions of the Antarctic Circumpolar Current in the South Paci?c and South Indian oceans,and in the Kuroshio Current and Gulf Stream just east of where they separate from the western boundaries). About30%of the eddy kinetic energy is accounted for in the more quiescent regions.

Some of the unexplained eddy kinetic energy in the top panel of Fig.7is attributable to the$140,000compact eddies that have life-times between4and16weeks(see Fig.2)that,except where noted,are not considered in the analyses presented in this study. The value of EKE lo expressed as a percentage of EKE tot for eddies with lifetimes of4weeks and longer(the minimum retained by the tracking algorithm)is shown in the bottom panel of Fig.7. The percentage of eddy kinetic energy explained increases to more than60%in most of the eddy-rich regions and is typically about 40%or more in the quiescent regions.Since the bias of the esti-mated amplitudes of the compact forms that we de?ne to be ed-dies is sometimes1or2cm and only occasionally more than that(Appendix C),much of the remaining unexplained eddy ki-netic energy in the bottom panel of Fig.7consists of the elongated interconnecting ridges and valleys between eddies.Some is attrib-utable transient eddies with lifetimes shorter than the4-week minimum retained by the tracking algorithm.

3.6.Geographical distribution of eddy polarity

The?nal census statistic presented here reveals an interesting inhomogeneity in the geographical distribution of eddy polarity. While the histograms in Figs.2and3indicate only small

178 D.B.Chelton et al./Progress in Oceanography91(2011)167–216

differences between the numbers of cyclones and anticyclones in a global census for the lifetimes P16weeks considered here,the partitioning of eddy polarity can be very inhomogeneous region-ally.Overall,the ratio of cyclonic to anticyclonic eddies is rather noisy(Fig.8),but some patterns do emerge.For example, mesoscale variability is predominantly cyclonic on the equator-ward sides of strong,meandering eastward currents such as the Gulf Stream,the Kuroshio Extension and the Agulhas Return Cur-rent.Likewise,there is a predominance of anticyclonic eddies on the poleward sides of these currents,although these regions of preference for anticyclones are narrower and somewhat less well de?ned than their counterpart regions of preference for cyclonic eddies.

The parallel bands of preference for opposing eddy polarity on the equatorward and poleward sides of the meandering?ows are to be expected because the meanders that pinch off to form closed vortices have cyclonic vorticity on the equatorward side and anti-cyclonic vorticity on the poleward side of the unstable?ows.The less well-de?ned bands of anticyclonic eddies may be an indication that the anticyclones on the poleward sides of the currents have shorter lifetimes than the cyclones on the equatorward sides,per-haps because they tend to be reabsorbed into the currents,as is known to be the case for the Gulf Stream region.In part,this is likely due to the tendency for westward propagating anticyclonic eddies to de?ect equatorward,and hence toward the cores of the currents.

percentage of eddy kinetic energy that is accounted for by eddies with lifetimes P16weeks

to anticyclonic eddy centroids for eddies with lifetimes P16weeks that propagated through

logarithmic scale is used for the color bar in order to give equal emphasis to the ratios r and

D.B.Chelton et al./Progress in Oceanography91(2011)167–216179

Analogous patterns of preference for eddy polarity are found in association with other major meandering currents in the World Ocean.For example,mesoscale variability off the west coast of the US is predominantly cyclonic on the offshore side of the equa-torward California Current,presumably from offshore meanders of the?ow pinching off to form cyclones.Patches of preferred anticy-clonic eddies occur on the inshore side of the California Current near the major capes.

An intriguing feature of Fig.8is the pattern of quasi-zonal bands of alternating preference for cyclones and anticyclones in many mid-ocean regions.There are reasons to expect a preferred eddy polarity in some regions.Examples include a preference for anticyclones to the west of the Central American wind jets(e.g., Palacios and Bograd,2005;Zamudio et al.,2006;Willett et al., 2006)and the anticyclones that form at the Agulhas Retro?ection and propagate across the entire South Atlantic(e.g.,Byrne et al., 1995;Schouten et al.,2000).The two bands of preferentially anti-cyclonic eddies separated by a band of preferentially cyclonic ed-dies to the west-southwest of the Hawaiian Islands are generated by wind forcing from the wind stress curl patterns that are asso-ciated with the westward wind jet at the southern tip of the is-land of Hawaii and the gap winds between the islands of Hawaii and Maui(e.g.,Holland and Mitchum,2001;Lumpkin and Flament,2001;Calil et al.,2008;Yoshida et al.,2010).The band of preferential cyclonic eddies along34°N in the North Atlantic coincides with the Azores Front(e.g.,Pingree and Sinha, 2001;Mouri?o et al.,2003).In most other open-ocean regions, no simple explanations exist for the banded structures of alter-nating polarity preference.

Whether the bands of alternating preference for eddy polarity in the open ocean in Fig.8are persistent features or are attribut-able to inadequate sampling of the energetic mesoscale eddy?eld with the16-year data record analyzed here is an open question. These quasi-zonal structures are reminiscent of the alternating quasi-zonal jets or striations in time averages of velocity that have received a great deal of attention in recent years(e.g.,Maximenko et al.,2008and references therein).Such alternating jets are predicted as the end result of the up-scale cascade of energy from geostrophic turbulence theory(Rhines,1975).Dynamical interpre-tation of these features is complicated by the presence of the ener-getic mesoscale eddy?eld that is the subject of this study. Striations with characteristics very similar to those reported in the literature as quasi-zonal jets can arise purely as artifacts of the limited sampling of a completely random eddy?eld(Schlax and Chelton,2008).The amplitudes of these artifacts diminish as the record length of the time averages increase.The present dura-tion of the SSH?elds of the AVISO Reference Series is not suf?cient to resolve the issue of whether the apparent alternating jets are real or artifacts of an inadequately sampled eddy?eld.

The bands of alternating preference for eddy polarity evident in Fig.8further complicate the interpretation of quasi-zonal jets.An isolated propagating eddy generates a pair of opposing zonal veloc-ity structures in time averages of the velocity?eld owing to the opposing zonal velocities in the northern and southern portions of the eddy(see Fig.10of Scott et al.,2008).The opposing merid-ional velocities in the western and eastern portions of the eddy cancel from the combination of westward propagation and time averaging,thus resulting only in the pair of opposing zonal velocity structures in the time average.While these are true features of the time-averaged velocity?eld,they clearly cannot be interpreted as quasi-zonal jets since they do not exist in jet-like form in any instantaneous snapshot.A zonal band of preferred eddy polarity can likewise be expected to result in a pair of opposing zonal veloc-ity structures in long time averages.To at least some extent,the quasi-zonal jets deduced from time averages of the SSH?elds of the AVISO Reference Series could thus be simply the inevitable consequence of the bands of alternating polarity preference in Fig.8.

4.Kinematic properties of the observed eddies

The automated eddy identi?cation and tracking procedure de-scribed in Appendices B.2–B.4provides estimates of eddy ampli-tude,scale and rotational speed as de?ned in Appendix B.3at each7-day time step along an eddy trajectory.These kinematic properties and estimates of the Rossby numbers of the tracked ed-dies over the16-year data record analyzed here are summarized in this section.

4.1.Eddy amplitudes

The amplitude A of an eddy is de?ned here to be the magnitude of the difference between the estimated basal height of the eddy boundary and the extremum value of SSH within the eddy interior (Appendix B.3).Since the eddy identi?cation procedure strives to identify the eddies as compact mesoscale features,the basal height represents the larger-scale ambient SSH around the perimeter of the compact features,thus resulting in amplitudes that are much smaller than those relative to a reference of zero SSH,as discussed previously in Sections2and3.5(see also Appendix C).The ampli-tudes de?ned in this manner are mostly quite small.Histograms and upper-tail cumulative histograms of the eddy amplitudes are shown separately for cyclones and anticyclones in the top two pan-els of the left column of Fig.9a for the northern hemisphere and Fig.9b for the southern hemisphere.The modes of the distributions (i.e.,the most frequently occurring amplitude)are about4cm for both polarities in both hemispheres.The distributions are strongly skewed toward large values;the mean amplitudes are about dou-ble the modes of the distributions.Globally,about40%of the tracked eddies have amplitudes of A<5cm and25%have A>10cm.

From the map of average eddy amplitude in the top left panel of Fig.10,the large-amplitude eddies occur only in the relatively con-?ned regions of highly unstable currents such as the Gulf Stream and its extension around Grand Banks,the Kuroshio Extension, the Agulhas Current and the Agulhas Return Current,the Antarctic Circumpolar Current,the Brazil–Malvinas Con?uence region,the East Australia Current,and the Loop Current in the Gulf of Mexico.

A band of large mean eddy amplitude extends west of the Central American wind jets.Over the rest of the ocean,the mean eddy amplitudes are generally less than10cm.

The predominance of small eddy amplitudes may raise concerns that the distribution of observed eddy amplitudes is in?uenced by the unavoidable bias toward underestimation of eddy amplitude discussed in Section3.5.From the binned scatter plot in the bottom right panel of Fig.10,however,there is close geographical agree-ment between the average eddy amplitudes in the top left panel and the standard deviation of20°?10°high-pass?ltered SSH in the bottom left panel.The standard deviation can be considered a measure of the magnitude of the typical value of the SSH anom-aly contributing to variability in the spatially high-pass?ltered SSH ?elds.If the standard deviation were larger than the average eddy amplitudes,this would be a clear indication that the observed ed-dies do not account for much of the mesoscale SSH variability.That the standard deviation and the average eddy amplitude are nearly the same suggests that the standard deviation of spatially high-pass?ltered SSH is largely attributable to the eddies.It also lends con?dence that the bias of the estimated eddy amplitudes is not large in the open-ocean regions,at least not for the eddies with lifetimes P16weeks that are the focus of this investigation.The ?attening of the relation between average eddy amplitude and

180 D.B.Chelton et al./Progress in Oceanography91(2011)167–216

SSH standard deviation in the bottom right panel of Fig.10for average amplitudes larger than about20cm is likely attributable to the tendency of our eddy identi?cation procedure to underesti-mate the amplitudes of eddies in the regions of most

energetic D.B.Chelton et al./Progress in Oceanography91(2011)167–216181

mesoscale variability because the size constraint imposed by our algorithm for global identi?cation of eddies is too restrictive in these regions.From the discussion in Appendix C,this can lead to biases of1or2cm in these regions but seldom more than that, at least for the compact forms of mesoscale eddies assumed by our eddy identi?cation procedure.

The lifetime distributions of eddies for three different classes of eddy amplitude(averaged over the lifetime of each eddy)are shown in Fig.11.The observed eddies with the25%smallest ampli-tudes(A<3.6cm)have short lifetimes;74%have lifetimes 68weeks and only7%have lifetimes longer than the16-week threshold of the eddies analyzed in this study.Eddies within the middle25%of the distribution of amplitudes(4.6cm6A67.6cm) are much longer lived;43%have lifetimes68weeks and29%have lifetimes P16weeks.For eddies with the25%largest amplitudes (A>10.1cm),34%have lifetimes68weeks and47%have lifetimes P16weeks.

The latitudinal variation of the zonally averaged eddy ampli-tude is shown along with the zonally averaged SSH standard devi-ation in the upper right panel of Fig.10.Like the overall distributions of the eddy amplitudes in Fig.9a and b,a positive skewness of the distribution of eddy amplitudes within each lati-tude band is evident from the relationship between the latitudinal pro?les of the mean and the interquartile ranges of the distribu-tions that are shown by gray shading(i.e.,the25%and75%points of the distribution at each latitude);the mean at each latitude is skewed toward large values within the interquartile range of variability.

The peak in the latitudinal pro?le of mean eddy amplitude in the upper right panel of Fig.10that occurs between about36°N and42°N corresponds to the latitude range of the Gulf Stream Extension and the Kuroshio Extension.The broad peak of smaller amplitude centered near15°N is associated with the eddies to the west of Central America(e.g.,Palacios and Bograd,2005;Wil-lett et al.,2006)that are generated by the Central American wind jets over the Gulfs of Tehuantepec and Papagayo and possibly by instabilities of the nearshore currents triggered by downwelling coastally trapped waves of equatorial origin(Zamudio et al.,

average amplitude of eddies with lifetimes P16weeks(top left)and the standard deviation identi?ed(bottom left)for each1°?1°region.The upper right panel shows meridional pro?les amplitudes within each1°latitude bin(i.e.,the25and75%points of each distribution,

(dashed line).The lower right panel shows binned averages of the eddy amplitudes from

?ltered SSH from the bottom left panel,with the interquartile range of the distribution in

182 D.B.Chelton et al./Progress in Oceanography91(2011)167–216

2006).These eddies originate near the coast but often reach their maximum amplitude well away from the coast,suggesting the importance of baroclinic instability of the North Equatorial Current (Farrar and Weller,2006).The narrow peak centered at about 12°S in the upper right panel of Fig.10is associated with eddies that are generated between Australia and Indonesia and propagate west-ward across much of the south tropical Indian Ocean (Birol and Morrow,2001;Feng and Wijffels,2002;Nof et al.,2002).The nar-row peak at about 38°S and the broader peak centered near 50°S correspond to the Agulhas Return Current and the Antarctic Cir-cumpolar Current,respectively.

A signi?cant characteristic of the amplitude histograms in Fig.9b is that the distribution of the amplitudes of cyclonic eddies in the southern hemisphere is more skewed toward large values than for anticyclonic eddies.As a result,eddies with amplitudes larger than 10cm are preferentially cyclonic (see the bottom left panel of Fig.9b).This result is perhaps surprising in view of the previously discussed fact that the eddies with the longest lifetimes and the largest propagation distances are preferentially anticy-clonic (Figs.2and 3).Amplitude is evidently not the sole factor that determines the longevity of an eddy.A preference for large-amplitude eddies to be cyclonic is expected from the gradient wind effect of centrifugal force that pushes ?uid outward in rotating eddies (e.g.,Gill,1982),thus intensifying the low pressure at the centers of cyclones and weakening the high pressure at the centers of anticyclones.However,this cannot account for the differences between cyclonic and anticyclonic eddies in the northern hemi-sphere where there is a slight preference for anticyclonic eddies for nearly all amplitudes (see the bottom left panel of Fig.9a).We are not able to explain this hemispheric difference in the amplitude dependence on polarity.4.2.Eddy scales

The characterization of eddy scale used here is the speed-based radius estimate L s ,de?ned in Appendix B.3to be the radius of a cir-cle with area equal to that within the closed contour of SSH in each eddy that has the maximum average geostrophic speed.This eddy scale corresponds approximately to the radius at which the relative vorticity within the eddy is zero.If the eddies had Gaussian shapes,this would occur at a radius of 2à1/2L e ,where L e is the e-folding scale of the radial dependence r of a Gaussian pro?le of height h ex-pressed in the form h er T?A exp àr 2

=L 2e

We also investigated an alternative de?nition of eddy scale based on a Gaussian approximation of each eddy.It is shown in Appendix B.3,however,that estimates of the Gaussian e-folding scale L e resulted in radii of corresponding maximum rotational speed that were 64%smaller on average than the empirically deter-mined radius L s of actual maximum speed within each eddy.This indicates that,while a Gaussian shape is a good approximation for the average eddy pro?le over the inner 2/3of the eddies (see Section 5),the rotational speeds within eddy interiors are not well represented by Gaussian approximations,at least not over their en-tire interiors.The shapes of the observed eddies are examined in detail in Section 5.

Histograms and upper-tail cumulative histograms of the radius scales of the eddies are shown separately for cyclones and anticy-clones in the top two panels of the middle column of Fig.9a for the northern hemisphere and Fig.9b for the southern hemisphere.Un-like the amplitude distributions in Fig.9,there are no signi?cant differences between the distributions of the speed-based eddy scale L s for cyclones and anticyclones in either the northern or the southern hemisphere (bottom panels of the middle columns of Fig.9a and b,respectively).The modes of both distributions oc-cur at about 75km (somewhat larger in the northern hemisphere and smaller in the southern hemisphere).Globally,more than 90%of the tracked eddies had scales between 50and 150km.For both eddy polarities,the mean values of L s are 96km in the north-ern hemisphere and 87km in the southern hemisphere.

The geographical distribution of the mean eddy scale (left panel of Fig.12)is characterized as an essentially monotonic decrease from about 200km in the near-equatorial regions to about 75km at 60°latitude in both hemispheres.This simple latitudinal depen-dence is further evident from the relatively narrow interquartile range of variability within each latitude band shown by gray shad-ing in the right panel of Fig.12.As previously found by Stammer (1997)and Chelton et al.(2007)and others,this approximate fac-tor-of-2.5decrease in eddy scale is much smaller than the approx-imate factor-of-25decrease of the Rossby radius of deformation over the same latitude range (Chelton et al.,1998),which is shown by the dotted line in the right panel of Fig.12.The large scales of the observed mesoscale eddies compared with the Rossby radii at middle and high latitudes are consistent with the up-scale trans-fer of kinetic energy from a source with scales near the Rossby ra-dius of deformation that is expected from geostrophic turbulence theory (e.g.,Kraichnan,1967;Batchelor,1969;Charney,1971;Rhines,1975,1979;Stammer,1997;Scott and Wang,2005).

There is a signi?cant hemispheric distinction between the distributions of eddy scale.For both cyclonic and anticyclonic eddies,the distributions are less skewed toward large values in

average speed-based radius scale L s for eddies with lifetimes P 16weeks (left)for each 1and the interquartile range of the distribution of L s (gray shading)in 1°latitude bins.The long Gaussian approximation of each eddy (see Appendix B.3).The short dashed line represents Series for the zonal direction (see Appendix A.3)and the dotted line is the meridional pro?le D.B.Chelton et al./Progress in Oceanography 91(2011)167–216183

the southern hemisphere than in the northern hemisphere.This explains the smaller mean value of L s in the southern hemisphere noted above.The hemispheric difference in the distributions of eddy scale is likely attributable to the general decrease in L s with increasing latitude and the much greater expanse of ocean at high latitudes in the southern hemisphere.

An important point to be noted from the right panel of Fig.12is that the estimated eddy scales are much larger everywhere than the minimum scale of features that can be resolved by the SSH ?elds of the AVISO Reference Series,which is shown in Appendix A.3to be equivalent to an e-folding radius of about0.4°(corre-sponding to approximately40km)for an eddy with Gaussian shape(shown by the bottom dashed line in the right panel of Fig.12).This is true regardless of whether the eddy scales are char-acterized by the speed-based scale L s that is our preference(the so-lid line in the right panel of Fig.12)or the e-folding scale L e of a Gaussian approximation of each eddy estimated as described in Appendix B.3(the long dashed line in Fig.12).The large scales of the eddies obtained from the automated eddy identi?cation and tracking procedure are thus not artifacts of resolution limitations of the SSH?elds of the AVISO Reference Series.On the other hand, the histograms of eddy scales in Fig.9would likely change sub-stantially at the lower range if the SSH?elds analyzed here were capable of resolving features with scales smaller than40km.

The lifetime distributions of eddies for three different classes of eddy scale L s(averaged over the lifetime of each eddy)are shown in Fig.13.To avoid misinterpretation from latitudinal variations in the eddy scales and lifetimes,only midlatitude eddies between latitudes of20°and40°of both hemispheres are considered in Fig.13.Eddies at lower latitudes were excluded because they are predominantly large scale with fast propagation speeds and rela-tively short lifetimes because of the dif?culties tracking eddies at low latitudes discussed in Section3.3.Likewise,eddies at higher latitudes have been excluded because they are also dif?cult to track owing to their smaller scales,very slow propagation speeds, and their interactions with other eddies that often result in sub-stantial distortions of the eddy boundary from one time step to the next.

It is evident from Fig.13that the observed midlatitude eddies with the25%smallest scales(L s<67km)have very short lifetimes; 81%have lifetimes68weeks and only4%have lifetimes longer than the16-week threshold of the eddies analyzed in this study.This may be attributable at least in part to the attenuation of these small-scale eddies by the?ltering inherent in the objective analy-sis procedure used to produce the SSH?elds of the AVISO Refer-ence Series(see Appendices A.2and A.3).Eddies within the middle25%of the distribution of scales(75km6L s695km)are much longer lived;39%have lifetimes68weeks and33%have life-times P16weeks.For eddies with the25%largest scales (L s>107km),only23%have lifetimes68weeks and53%have life-times P16weeks.

It is seen from Figs.11and13that the lifetimes of the meso-scale eddies depend on the horizontal scales of the eddies in a manner that might have been anticipated:eddies with small amplitude or horizontal scale have short lifetimes while eddies with large amplitude or horizontal scale generally have longer lifetimes.

The global joint distribution of eddy amplitudes A and scales L s in Fig.9c shows a surprisingly weak correlation between the two. The eddy amplitudes A are broadly distributed for any particular eddy scale L s.The overall correlation between A and L s over the $36,000eddies with lifetimes P16weeks is only0.13.There is very little latitudinal variation of this low correlation;when com-puted within10°bands of latitude,the correlation is less than0.16 at all latitudes.This indicates that there is not a universal,self-sim-ilar structure for these eddies;instead,amplitude and horizontal scale apparently vary independently.

4.3.Eddy rotational speeds

The rotational speed U of an eddy is characterized here by the maximum of the average geostrophic speeds around all of the closed contours of SSH inside the eddy,i.e.,the average geostrophic speed around the same SSH contour that de?nes the eddy scale L s discussed in Section4.2and described in detail in Appendix B.3. Histograms and upper-tail cumulative histograms of U are shown separately for cyclones and anticyclones in the top two panels of the right column of Fig.9a for the northern hemisphere and Fig.9b for the southern hemisphere.The modes of the skewed dis-tributions are about10cm sà1for both polarities in both hemi-spheres.Globally,about50%of the observed eddies had U values between10and20cm sà1and5%had U P40cm sà1.

Eddy rotational speed can be roughly characterized as propor-tional to the ratio of the eddy amplitude A to the eddy scale L s.In view of the cyclonic preference of eddies with large amplitudes in the southern hemisphere(Section4.1)and the lack of polarity preference for eddy scale(Section4.2),it is not surprising that the distribution of rotational speeds in the southern hemisphere is more skewed toward large values for cyclonic eddies than for anticyclonic eddies.As a result,eddies with fast rotational speeds of U>20cm sà1were preferentially cyclonic in the southern hemi-sphere(see the bottom right panel of Fig.9b).Consistent with the distributions of eddy amplitude discussed in Section4.1,there is a slight preference for anticyclonic eddies for all rotational speeds in the northern hemisphere(see the bottom right panel of Fig.9a). We are not able to explain this hemispheric difference in the dependence of rotational speeds on polarity.

4.4.Rossby number

The Rossby number is de?ned to be the ratio of material advec-tion to the Coriolis term in the momentum https://www.wendangku.net/doc/d716635899.html,ing the speed-based eddy scale L s and maximum rotational speed U sum-marized in Sections4.2and4.3,the Rossby number can be charac-terized as Ro=U/(fL s),where f is the Coriolis parameter.The histograms and upper-tail cumulative histograms of Ro de?ned in this way are shown in Fig.14for three different latitude bands: the northern hemisphere extratropics(20°N–60°N),the

tropics 184 D.B.Chelton et al./Progress in Oceanography91(2011)167–216

(20°S–20°N),and the southern hemisphere extratropics(60°S–20°S).The percentages of combined cyclonic and anticyclonic mesoscale features in these three latitude bands with Ro>0.05 are9%,35%and6%,respectively.Only about1%of the extratropical eddies and about10%of the tropical eddies had Ro>0.1.While there is no objective criterion for de?ning when the Rossby num-ber is‘‘large,’’the distributions in Fig.14indicate that the observed eddies are not highly nonlinear or ageostrophic by this measure, except at low latitudes where f approaches zero.

The smallness of Ro implies that the momentum equation is dominated by the geostrophic balance between the pressure gradi-ent force and the Coriolis acceleration.This quasi-geostrophic approximation of the dynamics in turn implies that altimetry is adequate for investigating the dynamics of the observed mesoscale features that are resolved by the SSH?elds of the AVISO Reference Series since surface velocity can be computed by geostrophy from SSH.

The Rossby number can alternatively be interpreted as the ratio of relative vorticity(which can be characterized by U/L s)to the lo-cal planetary vorticity,which is equal to the Coriolis parameter f. The smallness of Ro thus implies that the relative vorticities of the rotating eddies are small compared with the planetary vorticity.

5.Eddy shapes

The shapes of the mesoscale eddies were investigated from the combined cyclones and anticyclones by normalizing the SSH within each eddy by its(positive)amplitude A,and then nor-malizing its spatial coordinates by the speed-based scale L s of the eddy.Speci?cally,if the SSH within an eddy at time step t i is h(x,y,t i),we formed the doubly normalized SSH de?ned by

h0ex0;y0;t iT?Aà1hex=L s;y=L s;t iT:e5:1TEach observation of an eddy is thus transformed to have unit ampli-tude and scale,allowing them to be composited to investigate the shapes of the eddies.The$36,000tracked eddies with lifetimes P16weeks comprise$1.15million individual eddy observations. For each of these observations,h(x0,y0,t i)was binned at each nor-malized coordinate(x0,y0),yielding a distribution of the doubly nor-malized SSH,h0(x0,y0),from which inferences about eddy shape may be drawn.

The distributions of doubly normalized SSH along east–west and north–south pro?les,that is,along the x0and y0axes,respec-tively,are shown in Fig.15.The east–west and north–south pro-?les of the average of doubly normalized SSH are shown by the blue lines,and the modes of the distributions in each bin are shown by the red lines.The interquartile range for the east–west pro?le is shown as the gray shaded area(the interquartile range for the north–south pro?le is essentially the same).While the interquartile range increases away from the center of the eddies, indicating less coherent structure with increasing radius,the gen-eral shape of the composited eddies is unambiguously‘‘bell-shaped,’’as expected.Consideration of cyclones and anticyclones separately yields composites that do not differ signi?cantly from those shown in Fig.15for the combined cyclonic and anticyclonic eddies.

The two blue lines in Fig.15are nearly the same,indicating that, on average,there is no evidence for anisotropy of the eddy shape in the orthogonal east–west and north–south cross sections of the average of the doubly normalized SSH.The pro?les of the averages are well approximated across the central2/3of the eddy interiors by the axisymmetric Gaussian pro?le with an e-folding normalized radius of L e=0.64that is overlaid as the thick short dashed line. The associated radius of maximum speed L=2à1/2L e is shown

D.B.Chelton et al./Progress in Oceanography91(2011)167–216185

the vertical dotted lines.The?anges of the east–west and north–south pro?les of the averages are seen to be?atter than the Gauss-ian approximation for normalized radii greater than approximately the e-folding scale of the Gaussian,i.e.,for unnormalized radii greater than about0.64L s.

The distribution of doubly normalized SSH at radii within most of the interior of the composite eddy is consistently skewed toward small values,resulting in a mode that is larger than the average across approximately85%of the normalized eddy interior.This skewness is nearly identical in both the east–west and the north–south cross sections,as evidenced by the fact that the modes as a function of normalized radius that are shown by the red lines in Fig.15are essentially the same for both cross sections.This is further evidence that,on average,there is no apparent anisotropy of the eddy shape.The mode pro?les are well approximated by the quadratic function that is overlaid as the thick long dashed line, which has zero crossings at a normalized radius of0.95,i.e.,very near the radius corresponding to the speed-based eddy scale L s.

The average value is a good characterization of any variable that has an approximately symmetric distribution.For a variable with a skewed distribution,however,the mode is arguably a better char-acterization since it corresponds to the most frequently occurring value of the variable.The choice of the mode as the preferred rep-resentation of the binned pro?les of eddy shape in Fig.15is further motivated by the large discrepancy between the speed-based eddy scale L s and the radius of maximum rotational speed,which occurs at L=2à1/2L e for a Gaussian eddy(shown in normalized coordi-nates by the vertical dotted lines in Fig.15).From the comparison between estimates of L s and L in Appendix B.3,L s%1.64L(see Fig.B2c).The observed radius L s of maximum rotational speed in the observed eddies is thus typically64%larger than it would be if the eddies had Gaussian shape.This is consistent with the differ-ences between the quadratic and Gaussian approximations of the eddy shapes in the composite eddy cross sections in Fig.15.

It is noteworthy that the average and mode pro?les of doubly nor-malized eddies in the Parallel Ocean Program global ocean circula-tion model run with a nominal grid spacing of1/10°at Los Alamos National Laboratory(Maltrud and McClean,2005)are very similar to those shown in Fig.15.The details of this analysis will be reported elsewhere.For present purposes,this should alleviate concerns that the shapes of the eddies in the SSH?elds of the AVISO Reference Ser-ies are imposed by the covariance function in the objective analysis procedure used to construct these SSH?elds(see Appendix A.2).

While the gray shaded region in Fig.15shows that there is con-siderable variability in the shapes of the large eddies analyzed here,it is clear that Gaussian approximations are not valid over the full interiors of the eddies.It is likewise clear that a substantial number of the eddies have the quadratic structure of the mode pro?les.For axially symmetric rotation,a radial pro?le that is qua-dratic implies that the relative vorticity is constant within the eddy interior,i.e.,out to radii close to the speed-based scale L s.beyond which the?anges of the eddy become?at.To the extent that the typical eddy can be characterized as having a quadratic pro?le,this has important dynamical implications;the associated?uid motion consists of solid-body rotation.

6.Nonlinearity

The nonlinearity of the eddies identi?ed in the SSH?elds of the AVISO Reference Series is assessed in this section from the statis-tics of three different nondimensional parameters.

6.1.Advective nonlinearity parameter

A common measure of nonlinearity for the rotating vortices that are of interest here is the nondimensional ratio U/c,where U is the maximum rotational speed summarized in Section4.3and c is the translation speed of the eddy estimated at each point along the eddy trajectory from centered differences of the(x,y)coordinates of successive centroid locations.For Gaussian eddies,a value of U/c that exceeds1implies that there is trapped?uid within the eddy interior that is advected with the eddy as the eddy translates. This can be most easily seen from a transformation to a coordinate frame moving with the eddy(e.g.,Samelson,1992).More gener-ally,values of U/c P1that occur when typical rotational?uid speeds are as large as or larger than the eddy translation speed im-ply that the eddy cannot be regarded as a linear wave disturbance propagating through a nearly stationary medium,but instead is capable of modifying the medium by advecting a trapped?uid par-cel as it translates.Eddy advection of trapped?uid implies that the eddies can transport water properties such as heat salt and poten-tial vorticity,as well as biogeochemical characteristics such as nutrients and phytoplankton.,While we feel that this advective measure of nonlinearity is the most germane of all nonlinearity parameters for the present study since the trapping of?uid is a fundamental distinction between linear waves and nonlinear ed-dies,we also investigate two other commonly used measures of nonlinearity in Sections6.2and6.3.

The distributions of U/c de?ned in the above manner are shown in the?rst column of Fig.16for the same three latitude bands as the Rossby number considered in Fig.14:the northern hemisphere extratropics(20°N–60°N),the tropics(20°S–20°N),and the south-ern hemisphere extratropics(60°S–20°S).The estimates of U/c ob-tained here are nearly a factor-of-2larger than our previous estimates(Chelton et al.,2007).It is shown in Appendix B.5that this is because the estimate of U used in our earlier study was overly conservative.

It is apparent from Fig.16that virtually all of the observed mesoscale eddies outside of the tropics had U/c>1and hence were nonlinear by this measure.Some of the mesoscale eddies were highly nonlinear.For example,48%of the U/c values for the extra-tropical eddies in both hemispheres exceeded5and21%exceeded 10.Even within the tropics where the translation speeds c are very fast and hence U/c tends to be smaller,about90%of the combined cyclonic and anticyclonic mesoscale features had U/c>1.Fewer of the tropical eddies are highly nonlinear;only14%of the U/c values exceeded5and4%exceeded10.

Close inspection of the top and bottom left panels of Fig.16re-veals that the distributions of U/c for extratropical eddies are more skewed toward high values for cyclones than for anticyclones in the southern hemisphere but the opposite is found in the northern hemisphere.This puzzling result is consistent with the hemi-spheric asymmetries of the dependencies of amplitudes and rota-tional speeds on eddy polarity discussed in Sections4.1and4.3.

A map of the geographical distribution of the average advective nonlinearity parameter U/c in each1°square region is shown in the top panel of Fig.17.Not surprisingly,the largest average U/c values are found in all of the major unstable,meandering currents:the Gulf Stream and its extension across most of the high-latitude North Atlantic,the Kuroshio and Oyashio Currents and their east-ward extension half way across the North Paci?c,the Agulhas Re-turn Current,the Brazil–Malvinas Con?uence and the Antarctic Circumpolar https://www.wendangku.net/doc/d716635899.html,rge average values of U/c are also evident in the East Australia Current and in some of the eastern boundary current systems(e.g.,the California Current,the Alaska Current, and the Leeuwin Current off the west coast of Australia).A narrow band of high nonlinearity is found along the Azores Front centered near34°N in the central North Atlantic(Pingree and Sinha,2001; Mouri?o et al.,2003)and in the region south and east of Madagas-car(Schouten et al.,2003;Quartly et al.,2006).In contrast,the average U/c values are less than2everywhere equatorward of about15°latitude except very near the Central American wind jets,

186 D.B.Chelton et al./Progress in Oceanography91(2011)167–216