Convectively Coupled Equatorial Waves: A New Methodology ...roger/Lectures/Tropical… · Corresponding author address: Gui-Ying Yang, Dept. of Meteo-rology, University of Reading,

VOL. 60, NO. 14 15 JULY 2003J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2003 American Meteorological Society 1637

Convectively Coupled Equatorial Waves: A New Methodology for Identifying WaveStructures in Observational Data

GUI-YING YANG, BRIAN HOSKINS, AND JULIA SLINGO

Centre for Global Atmospheric Modelling, Department of Meteorology, University of Reading, Reading, United Kingdom

(Manuscript received 30 July 2002, in final form 27 January 2003)

ABSTRACT

Convectively coupled equatorial waves are fundamental components of the interaction between the physicsand dynamics of the tropical atmosphere. A new methodology, which isolates individual equatorial wave modes,has been developed and applied to observational data. The methodology assumes that the horizontal structuresgiven by equatorial wave theory can be used to project upper- and lower-tropospheric data onto equatorial wavemodes. The dynamical fields are first separated into eastward- and westward-moving components with a specifieddomain of frequency–zonal wavenumber. Each of the components for each field is then projected onto thedifferent equatorial modes using the y structures of these modes given by the theory. The latitudinal scale yo ofthe modes is predetermined by data to fit the equatorial trapping in a suitable latitude belt y 5 6Y. The extentto which the different dynamical fields are consistent with one another in their depiction of each equatorialwave structure determines the confidence in the reality of that structure. Comparison of the analyzed modeswith the eastward- and westward-moving components in the convection field enables the identification of thedynamical structure and nature of convectively coupled equatorial waves.

In a case study, the methodology is applied to two independent data sources, ECMWF Reanalysis and satellite-observed window brightness temperature (Tb) data for the summer of 1992. Various convectively coupledequatorial Kelvin, mixed Rossby–gravity, and Rossby waves have been detected. The results indicate a robustconsistency between the two independent data sources. Different vertical structures for different wave modesand a significant Doppler shifting effect of the background zonal winds on wave structures are found anddiscussed.

It is found that in addition to low-level convergence, anomalous fluxes induced by strong equatorial zonalwinds associated with equatorial waves are important for inducing equatorial convection. There is evidence thatequatorial convection associated with Rossby waves leads to a change in structure involving a horizontal structuresimilar to that of a Kelvin wave moving westward with it. The vertical structure may also be radically changed.

The analysis method should make a very powerful diagnostic tool for investigating convectively coupledequatorial waves and the interaction of equatorial dynamics and physics in the real atmosphere. The resultsfrom application of the analysis method for a reanalysis dataset should provide a benchmark against whichmodel studies can be compared.

1. Introduction

The complete picture of the tropical atmosphere in-volves the interaction of convection and other physicalprocesses with each other and with the dynamics, bothlocally and globally. Equatorial waves have been usedto explain some fundamental phenomena of the tropicalclimate, such as the Walker circulation (e.g., Gill 1980),atmospheric teleconnection patterns (e.g., Lim andChang 1983), the Madden–Julian oscillation (MJO; e.g.,Lau and Peng 1987), and the El Nino–Southern Oscil-lation (ENSO; e.g., Lau 1981), as well as the forcingof the equatorial quasi-biennial oscillation (QBO) andsemiannual oscillation (SAO; e.g., Lindzen and Holton

Corresponding author address: Gui-Ying Yang, Dept. of Meteo-rology, University of Reading, Earley Gate, Reading RG6 6BB, Unit-ed Kingdom.E-mail: [email protected]

1968). During the Tropical Ocean Global AtmosphereCoupled Ocean–Atmosphere Response Experiment(TOGA COARE), for example, a substantial fraction ofthe large-scale variability in convection at timescalesless than 30 days has been attributed to equatoriallytrapped modes (Redelsperger et al. 1998). Hence, con-vectively coupled equatorial waves are a key part of thetropical climate system and as such, it is essential thatglobal circulation models (GCM) are able to representthem adequately.

However, at present there is limited knowledge ofthese waves, their structures and how well they are treat-ed in state-of-the-art models. A key question here is therelationship between the dynamical structure and theconvection. This is of practical importance and a faithfulrepresentation of it is needed for predictions on all time-scales. The research described in this paper is intendedto provide a basic framework that will enable the un-

1638 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. An example of horizontal winds from the ERA daily data at 1200 UTC 13 Jul 1992 for (a) 200 hPa and (b) 850 hPa.Superimposed colors show the CLAUS brightness temperature Tb for the same day. Units are m s21 for wind and K for Tb.

derstanding of convectively coupled waves and help theanswering of such questions.

Tropical convection contains variability on a varietyof space and timescales, ranging from the individualclouds, through the cloud clusters associated with syn-optic-scale disturbances, to the super clusters or ensem-bles of clusters. Synoptic activity in the Tropics is oftenassociated with disturbances, which can be related tothe preferred equatorially trapped modes of the atmo-spheric circulation based on shallow water theory (e.g.,Takayabu 1994a,b; Wheeler and Kiladis 1999; Wheeleret al. 2000). Various modes (Kelvin, mixed Rossby–gravity, equatorial Rossby, and gravity waves) can bedetected, in many cases related to the active phase ofthe MJO (Madden and Julian 1972), thus demonstratingthe important interaction between synoptic and intra-seasonal timescales.

It is clear from various studies (e.g., Takayabu1994a,b) that analysis of high-resolution satellite datacan provide information on equatorial waves and thediurnal cycle that should be essential validation forGCMs. An extended record of global cloud variabilityon a wide range of spatial and temporal scales has beenprovided by the Cloud Archive User Service (CLAUS),a European Union part-funded initiative to establish along-term, continuous archive of global brightness tem-perature (Tb) data, spanning the period 1984 to 1999.The source data for CLAUS are the intermediate-level(B3) products generated by the International SatelliteCloud Climatology Project (ISCCP) from meteorolog-ical and other satellite observations from 1984 onward.The data for CLAUS are generated by resampling thewindow brightness temperatures from the B3 data onto a global grid at a resolution of 0.58 and 3 h. Moredetailed description of the algorithm for generating theCLAUS dataset and of the dataset itself can be foundin Hodges et al. (2000) and Yang and Slingo (2001).

In addition to the CLAUS dataset, the European Cen-

tre for Medium-Range Weather Forecasts (ECMWF)Reanalysis (ERA-15) is used. ERA has been performedat relatively high resolution (spectral truncation at 106wavenumbers, equivalent to grid resolution of ;1.1258)and is contemporary with the CLAUS data. The com-bination of the ERA and CLAUS data represents aunique opportunity to diagnose the relationship betweenthe dynamical structure and convective organization ofequatorial waves.

To identify and describe the structure of convectivelycoupled equatorially trapped modes in observationaldata, an essential step is to separate the various equa-torial wave modes. Figure 1 gives an example of theconvection field, as described by the window brightnesstemperature, and ERA wind fields in the upper and lowertroposphere on 13 July 1992. The monthly mean hasbeen removed from each field to highlight the variabilityon timescales of less than 1 month. It can be seen thatthere are rich features in the convection and dynamicalfields. Some possible signatures of equatorial waves canbe seen, for example, the strong cross-equatorial me-ridional wind in the upper troposphere around the east-ern Pacific and Central America, and in the lower tro-posphere over Central America, together with convec-tion that is asymmetric about the equator, suggestive ofa mixed Rossby–gravity wave. However, it is difficultto identify, uniquely, individual wave structures andtheir associated convection.

Equatorial waves, coupled with convection, havebeen studied for several decades since satellite data be-came available (e.g., Gruber 1974; Zangvil 1975; Zang-vil and Yanai 1981; Hendon and Liebmann 1991; Tak-ayabu 1994a,b; Dunkerton and Baldwin 1995; Maganaand Yanai 1995; Wheeler and Kiladis 1999; Wheeler etal. 2000). Most of these studies used spectral analysisof the convection and dynamical fields to provide evi-dence of equatorial waves. Various methods, such asspace–time cross-spectral analysis, linear regression,

15 JULY 2003 1639Y A N G E T A L .

and lag correlation, have been used to investigate therelationship between convection and the dynamicalfields. Takayabu (1994a,b) used space–time spectralanalysis and theoretical equatorial wave dispersioncurves to isolate various equatorial wave modes. Wheel-er and Kiladis (1999), based on space–time spectralanalysis of satellite-observed outgoing longwave radi-ation (OLR), designed a selective filtering of OLR toisolate the convective variations contributing to spectralpeaks, which coincide with the equatorial wave disper-sion curves for various equivalent depths. Their studysuggested that a portion of the space–time variability ofdeep tropical cloudiness inferred from OLR could bedescribed quite well in terms of equatorially trappedwave modes from shallow water theory. Their followingcompanion paper (Wheeler et al. 2000) used linear re-gression between selectively filtered OLR data and var-ious dynamical fields to obtain dynamical structures ofthe waves. These dynamical structures showed an im-pressive agreement with those obtained from equatorialwave theory.

When the CLAUS data became available, we initiallyused space–time spectral analysis (Hayashi 1982) anda similar technique of selective filtering based on the-oretical dispersion relations (Wheeler and Kiladis 1999)to examine the convectively coupled equatorial wavesduring the summer of 1992. In agreement with Wheelerand Kiladis (1999), analysis of the Tb data (not shown)demonstrated that tropical convection is organized onpreferred space and timescales, which coincide approx-imately with those of theoretical equatorial waves. How-ever, in the real atmosphere the complicated space–timedependence of the ambient state can be expected to leadto Doppler shifting and distortion of the theoretical dis-persion curves, and to variations in the vertical heatingprofile and hence the equivalent depth (h). Consequent-ly, different equatorial modes may not, in reality, bewell separated in their frequency and zonal wavenumberdomain. Also, convectively coupled modes might beexpected to behave rather differently from dry waves,and it is unclear how much of the shallow water theoryof equatorial waves can be carried over to the real world.It is evident, therefore, that a new, less constrainingmethodology for objectively separating equatorial wavestructures would be very beneficial and this is the stim-ulus for this research.

The main aim of the whole study is to develop a newmethodology for isolating various equatorial waves, todevelop further our understanding of the physics anddynamics of the tropical atmosphere in the context ofconvectively coupled equatorial waves, and to use thatunderstanding to evaluate the performance of climateand weather forecasting models. This paper describesthe first stage of that study, that is, the development ofthe methodology and demonstration of its utility in acase study of equatorial waves in 8 days of observationaldata. In a companion paper we will apply the meth-odology to the entire season to provide statistics on

observed equatorial wave characteristics. It is hoped thatthis will serve as a test bed for weather and climateprediction models.

The outline of this paper is as follows. The theoreticalbackground for equatorial waves will be briefly intro-duced in the next section. The details of the method-ology will be described in section 3. In section 4, themethodology is applied to the ERA and CLAUS dataand examples of westward- and eastward-moving equa-torial wave activities isolated from the observationaldata are shown. Conclusions and discussion will appearin section 5.

2. Equatorial wave theory

Following Matsuno (1966), Gill (1980, 1982), andmany others, equatorially trapped waves are obtainedas solutions to the adiabatic, frictionless equations ofmotion on an equatorial b plane linearized about a stateof rest. The representation of the two horizontal-velocity(u, y) and geopotential-height (f) fields are of the form

{u9, y9, f}

5 A(z){u(y), y(y), f(y)} exp[i(kx 2 vt)], (2.1)

where k is the zonal wavenumber and v is the frequency.The common, separated vertical structure function A(z)is a solution of an equation that can also satisfy relevantsurface and upper boundary conditions; this is possibleonly for discrete values of the separation constant, c.The first internal mode has one zero in A in the tro-posphere and extrema of opposite sign near the tropo-pause and the surface. A typical value for c for thismode is in the range of 25–50 m s21. Higher verticalmodes with additional zeros (nodes) in A correspond tosuccessively smaller values of c.

The horizontal and temporal behavior of {u9, y9, f9}satisfies the linearized shallow water equations withgravity wave speed c, the separation constant from thevertical structure equation. However, the equatorialwave solutions are most easily formulated in terms ofnew variables, q, r, and y (Gill 1980), where

q 5 (g /c)f 1 u and r 5 (g /c)f 2 u. (2.2)

The dispersion relation for the waves is found to be

v 5 kc, for the Kelvin wave and (2.3a)2v bk b

22 k 2 5 (2n 1 1) ,2c v c

for n 5 0, 1, 2, . . . , (2.3b)

where n is the meridional wavenumber and b is ] f /]y.Solutions of this dispersion relation with c 5 20 m s21

are summarized in Fig. 2a.The Kelvin wave (K) satisfies Eq. (2.3b) with n 5

21 and is often referred to in this manner. For n $ 1,Eq. (2.3b) has three roots. The first, the Rossby wave(Rn), corresponds to small westward phase speed; theother two roots correspond to fast-moving westward and


FIG. 2. Dispersion curves of equatorial waves for (a) a restingatmosphere, (b) 210 m s21, and (c) 10 m s21 basic states. The boxesin each of the panels indicate the wavenumber and frequency bandsthat will be investigated in this paper, the dashed boxes are for east-ward-moving waves and the solid boxes are for westward.

TABLE 1. The structures of theoretical equatorial wave solutionsfor the Kelvin (K), the WMRG, EMRG, the R1, the WG1, EG1, andthe R2 waves. Parameter n is the meridional mode number, Dn is aparabolic cylinder function as in Eq. (2.4), y is meridional wind, andq and r are as in Eq. (2.2).

Value ofn Westward Eastward q y r

2101

n . 1

WMRGR1, WG1Rn, WGn

KEMRGEG1EGn

q0D0

q1D1

q2D2

qn11Dn11

0y 0D0

y 1D1

y nDn

00r0D0

rn21Dn21

eastward gravity waves (WGn and EGn, respectively).For n 5 0 there is one root (v 5 2kc), which is anartifact of the elimination procedure, and two furtherroots corresponding to westward- and eastward-movingcomponents of the so-called mixed Rossby–gravitywave (WMRG and EMRG). At high wavenumbers,WMRG resembles a Rossby wave and EMRG, a gravitywave. In Fig. 2, the westward-moving waves are shownfor negative k and positive v, this being a reflection ofthe situation in the positive k, negative v quarter plane.

The meridional (y) structure of the waves is given interms of parabolic cylinder functions (D), which takethe form

2D (y/y ) 5 exp[2(y/2y ) ]P (y/y ),r o o r o (2.4)

where Pr is a polynomial of degree r, and yo is thetrapping scale given by

1/2y 5 (c/2b) .o (2.5)

The structures of the waves are summarized in Table1. The coefficients of the structures for q, y, and r arerelated through

Ï2bcq 5 y and (2.6a)n11 ni(ck 2 v)

nÏ2bcr 5 y . (2.6b)n21 ni(ck 1 v)

Some aspects of the structure of the gravest horizontalmodes are illustrated in Fig. 3. Westward-moving wavesare shown in the left column, and eastward-movingwaves in the right column. The vectors indicate thelower-tropospheric wind fields. (Note that, for the firstinternal mode, the upper-tropospheric wind fields wouldbe in the opposite direction.) The colors indicate thelower-tropospheric divergence field, and for the first in-ternal mode, also show the midtropospheric vertical mo-tion. The Kelvin wave is composed of purely zonal mo-tion and has obvious equatorial convergence. The mixedRossby–gravity wave has a maximum in meridionalwind on the equator. Off the equator for WMRG, thismeridional wind is part of a horizontal, predominantlyrotational, circulation, whereas for EMRG the off-equa-torial zonal wind enhances the convergence and diver-gence. In each case, the divergence field is in the samesense as implied by the convergence of the meridionalwind, but for a given meridional wind strength it is muchstronger for the EMRG. The R1 wave has rotationalflow centered off the equator with convergence in theregion of poleward-moving air, as required by vorticity(Sverdrup) balance. For the R2 wave, the main rota-tional flow and divergence extrema are further from theequator and have symmetry across the equator which isopposite to that for R1. The westward- and eastward-moving inertio-gravity waves, WG1 and EG1, are dom-inated by their divergent meridional components withopposite senses for the zonal wind in the two waves.

If the low-level convergence and upper-level diver-gence in the first internal mode, with associated mid-tropospheric ascent, provide the organization for con-vection, then we would expect this convection to occurin the blue–green shaded regions in Fig. 3. However,for convectively coupled waves, wind-dependent sur-

15 JULY 2003 1641Y A N G E T A L .

FIG. 3. Dimensional horizontal wind (vector) and divergence (color) solutions of theoreticalequatorially trapped Kelvin wave with zonal wind uoDo(y/yo) exp[i(kx 2 vt)] and other n 5 0 ;2 waves with meridional wind y nDn(y/yo) exp[i(kx 2 vt)], where n is the meridional mode number,uo 5 y n 5 1 m s21, Dn is a parabolic cylinder function, the trapping scale yo 5 68 and zonalwavenumber k 5 6. The yo and k are chosen based on an analysis of observational data in section3 for the convenience of comparison. The waves shown are the Kelvin, the WMRG, the EMRG,the R1, the WG1, the EG1, and the R2 waves. Units are m s21 for wind and s21 for divergence.Color circles for Kelvin and Rossby waves indicate the possible convection region induced bywind-dependent surface energy fluxes.

face fluxes of moist entropy may play an important or-ganizational role for convection. Emanuel (1987) andNeelin et al. (1987) proposed a mechanism of wind-induced surface heat exchange (WISHE) or evapora-tion–wind feedback, as being crucial in driving MJO-related convection. In situ observational studies andglobal model analyses have shown that in the western

Pacific region positive anomalies of atmospheric deepconvection associated with the MJO tend to occur inregimes of enhanced surface westerly flow, rather thanin phase with the low-level convergence (e.g., Zhang1996; Zhang and McPhaden 2000; Hendon and Salby1994). These studies concentrated on the intraseasonaltimescale and very low zonal wavenumber, and the role


FIG. 4. ERA zonal mean zonal winds at 200 and 850 hPa averagedover 108N–108S for 1992 summer (May–Oct).

of the wind-dependent surface fluxes on shorter time-and smaller space scales of convection is not clear butmay be important. It is suggested, therefore, that con-vection associated with equatorial waves may, in somecases, occur where there are anomalous zonal winds, aswell as in the expected regions of low-level conver-gence. Remembering that in the real world, the warmestSSTs occur in the near-equatorial region and hence pre-dispose the atmosphere to convection, it is here thatzonal wind anomalies associated with equatorial wavesmay make the most significant contribution to the wind-induced surface energy fluxes and hence affect the con-vection.

Regions where wind-induced surface energy flux ex-trema and hence equatorial convection are likely to oc-cur in the Kelvin and R1 waves, and possibly also theR2 wave, are shown as the blue and red ovals in Fig.3. Whether the convection would coincide with the blueor red oval would depend on the sign and magnitudeof the background zonal wind. As discussed by Wang(1988), Hendon and Salby (1994), and more recentlyMoscowitz and Bretherton (2000), frictional conver-gence may also be important for organizing equatorialconvection. In this case the convection would occur inthe region of near-surface easterlies (the blue ovals) andwould be suppressed in the westerlies (the red ovals).It is possible that the heating associated with the equa-torial convection induced by the wind-induced fluxes orfrictional convergence may alter the theoretical struc-tures of the Kelvin and Rossby waves. Anticipating theresults to be shown later, for the Kelvin wave, this couldbe just a modification of the dry structure. However, forthe Rossby waves, the associated equatorial convectioncould lead to a radical change in structure and behavior.

In the real atmosphere, many of the approximationsmade in equatorial wave theory may be expected tobreak down. For example, the ambient flow in the Trop-ics is not, in general, close to a state of rest; physicalprocesses such as surface friction and convection areclearly important; and nonlinearity could also be im-portant. Considering, first, the ambient flow, Fig. 4shows as an example, the seasonal mean zonal wind at200 and 850 hPa, averaged over 108N–108S for the sum-

mer of 1992 (May–October). In general, there are upper-tropospheric westerlies in the Western Hemisphere (08–1808W) and stronger easterlies in the Eastern Hemi-sphere (08–1808E), and the lower-tropospheric winds areweaker than those at the upper troposphere and tend tohave an opposite sign to those of the upper level. Thereis also considerable temporal variability in the magni-tude and direction of the wind, with strengths exceeding20 m s21 in the upper troposphere and 10 m s21 in thelower troposphere, in some instances. Such variationsin the ambient flow are likely to have a significant im-pact on the phase speed and structure of equatorialwaves.

For any nonzero basic state, the separation of vari-ables used in the equatorial wave analysis is not strictlypossible, although, as in Hoskins and Yang (2000), theeffects of Doppler shifting can be suggested by replac-ing v by (v 2 ku) in the dispersion relations Eq. (2.3).The results for basic zonal winds of 610 m s21 areshown in Figs. 2b and 2c, respectively. The boxes shownin Fig. 2 indicate the range of frequencies (3–30 days)and spatial wavenumbers (2–10) used in this study. Notethat, for a resting atmosphere (Fig. 2a) only the Kelvinwave exists in the box for eastward-moving waves.However, an easterly basic state (Fig. 2b) reduces thefrequency of eastward-moving waves so that the east-ward box also includes low-wavenumber EMRGs, dem-onstrating the difficulty in uniquely separating individ-ual wave modes using simple space–time filtering. Inaddition, Fig. 2c shows that a westerly ambient flowadvects the westward-moving waves to such an extentthat R1 and high-wavenumber WMRG waves are foundin the lower left quadrant of negative k and v (eastward-propagating waves). Whereas the westward box in-cludes R1 and WMRG waves for a resting atmosphere(Fig. 2a) and for an easterly basic state but with in-creased frequency (Fig. 2b), only low-wavenumberWMRG are found in this box when the ambient flowis westerly (Fig. 2c).

How Doppler shifting will affect equatorial waveswill also depend on whether the wave senses the lower-or upper-tropospheric flow or even an average flow. Thevertical structure of the ambient flow around the Tropicshas already been noted in Fig. 4 and the reversal in signof the zonal flow between the upper and lower tropo-sphere may suggest that the effects of Doppler shiftingmay inhibit vertical coupling of the waves. As sum-marized in Table 2, the zero basic-state dispersion di-agram suggests that, in the wavenumber–frequency boxunder consideration, the westward-moving waves wouldbe the WMRG and Rn, and the eastward-moving wavesK and perhaps long EMRG. If there is an element ofDoppler shifting, then in the ambient westerlies at lowlevels in the Eastern Hemisphere and at upper levels inthe Western Hemisphere, Rn and short WMRG couldbe shifted from westward to eastward propagating. Inthe same region, long EMRG become less likely to befound in the eastward-propagation box. In the regions

15 JULY 2003 1643Y A N G E T A L .

TABLE 2. Westward- and eastward-moving equatorial waves in the Eastern (EH) and Western (WH) Hemispheres and the upper and lowertropospheres. The first column shows the expectation from basic theory for zonal wavenumbers 2–10 and periods 3–30 days. The secondpair of columns give possible changes suggested by notions of Doppler shifting with the winds at the level. The third pair of columns showthe convectively coupled waves actually identified. For the westward-moving waves, the order of the rows is WMRG, Rn, and WG1, andfor the eastward, K, EMRG, WMRG, Rn, and EG1. In the basic theory column, ‘‘?’’ means that the wave is on the edge of the wavenumber-frequency box. In the Doppler changes columns, ‘‘more’’ and ‘‘less’’ refer to more or less likely to occur, and ‘‘??’’ means that the wavemoves closer to the box but does not occur inside for reasonable wind speeds. In the observed columns, waves are also marked in thedateline (DL) region if they occur there and not in one of the hemispheres. The square bracket means that the sign is often the same as thatin the lower troposphere, as opposed to a first baroclinic mode structure. Basic theory suggests that the Kelvin wave may be linked to near-equatorial convective but that the others will have associated off-equatorial convection. Not included in the table is a lower-troposphericeastward-moving R1 wave with associated equatorial convection only.

Basic Possible Doppler changes

EH WH

Observed in 2 weeks

EH DL WH

WESTWARDUpper

WMRGRn

WMRG lessRn lessWG1 ??

WMRGRn

WMRG[R1], R2

LowerWMRGRn

WMRG lessRn lessWG1 ??

RnWMRG WMRG

Rn

EASTWARDUpper

KEMRG? EMRG more

EG1??

EMRG lessWMRG moreRn more

KEMRG

RnEG1

K

WMRGRn

LowerKEMRG? EMRG less

WMRG moreRn more

EMRG more

EG1 ??

K

R2

K

R2

of ambient easterlies the eastward-moving EMRG maybe more prevalent. Although they are not shifted intothe box for the winds shown, EG1 gets closer for am-bient easterly winds and WG1 for westerlies. The pos-sible Doppler shift changes are also summarized in Ta-ble 2.

The waves actually found in July 1992 using themethodology described in the next section are alsoshown in Table 2 and will be discussed in section 4. Asstated above, the observed waves can be expected todiffer from those given by basic linear theory for avariety of reasons and not just because of advection bythe ambient flow.

3. Development of the methodology

a. Overview

As noted earlier, the approximations underlying equa-torial wave theory may not be applicable when lookingfor such waves in the real atmosphere, especially thosecoupled with deep convection. However, given the re-sults of earlier studies by, for example, Wheeler andKiladis (1999), which demonstrated the qualitative suc-

cess of applying equatorial wave theory to real data, wedo not wish to reject the theory completely. In this studywe will impose a minimum from the details of the theoryand see what structure emerges from the data. The firstoperation we will perform is a spectral analysis sepa-rating brightness temperature and dynamical fields intoeastward- and westward-moving components, and re-taining a band of wavenumbers and frequencies. Thenwe will use Tb variance as a guide for specifying thebounds 6Y for the tropical strip in which the analysiswill be performed. Outside this strip the variation in allthe fields will be set to zero.

Parabolic cylinder functions form a complete set offunctions and we may represent the basic dynamicalfields at any level of the atmosphere in terms of a se-verely truncated series of them. The basic trapping scaleyo for the functions will be determined by the criterionthat the truncated series best fit the complete meridionalwind field. There will be no assumption that yo is relatedby (2.5) to a speed c that is itself determined from avertical structure.

In equatorial wave theory, in addition to meridionalwind y, the basic dynamical fields that are represented


FIG. 5. Standard deviation of the symmetric component of the brightness temperature Tb in1992 summer for (a) westward-moving (k 5 22 ; 210) and (b) eastward-moving (k 5 2 ;10) components with a period of 3 ; 30 days. Unit is K.

by parabolic cylinder functions are q 5 af 1 u and r5 af 2 u, where a 5 g/(2b ). For specified yo, the2yo

fitting of u and f by truncated series is independent ofthe choice of a. However, if we assume that the Kelvinwave is represented by qoDo and has zero r, then thecondition a 5 g/(2b ) is necessary for its zonal wind2yo

field to be in geostrophic balance. This value for a willbe used here, but again not assuming any relationshipto a speed c [although according to Eq. (2.5) a 5 g/(2b ) 5 g/c]. In section 4a, a brief discussion of al-2yo

ternative choices will be given.It will not be assumed here that the coefficients of y,

q, and r are related by (2.6) for some c and v. Instead,y, q, and r will be analyzed separately. The componentsof the truncated expansions that, according to equatorialwave theory and summarized in Table 1, would corre-spond to particular waves are then considered together.For example for the westward-moving features we con-sider the fields

1u 5 (q D 2 r D ), y 5 y D ,2 2 0 0 1 12

1f 5 (q D 1 r D )2 2 0 02a

as potentially describing a collection of R1 and WG1waves present on that day. Given that in same regionthe u and y fields combine and are consistent with thef field then we consider that we may describe a waveas being present. Coherent development and movementof the feature over consecutive analysis periods furthersupports this interpretation.

We will not assume that there is a particular verticalstructure, but instead a separate analysis will be per-

formed in the upper and lower troposphere. If analysesat different levels produce similar waves with a con-sistent phase relationship we consider that they maydescribe a 3D structure, developing and moving. If thecomponent of the Tb moving in the same direction showsa feature that has a consistent spatial relationship withthe dynamical wave then we consider that we have iden-tified a convectively coupled wave.

b. Application

The space–time spectral analysis, similar to that ofHayashi (1982), is used to separate eastward- and west-ward-moving waves. The eastward/westward data arethen spatially and temporally filtered using fast Fouriertransforms (FFT) to retain only frequencies of 3 to 30days and zonal wavenumbers 2 to 10.1 The chosen do-main contains most of the low-frequency equatorialwaves, which are relevant to synoptic and intraseasonalvariability in the Tropics. This space–time filtering isapplied to the CLAUS Tb and the ERA 200- and 850-hPa u, y, and f fields for 1992 summer (May–October).

To determine the bounds 6Y of the tropical strip, theTb coefficients are further decomposed into those sym-metric and antisymmetric about the equator.2 Standarddeviations are then calculated for both westward- andeastward-moving components. As an example, Fig. 5

1 Wavenumber 1 is not included in the analysis because of problemswith the ERA data. We found an unrealistic level of power at allwestward frequencies, particularly in lower-tropospheric geopoten-tial. The reasons for this are not understood.

2 This equatorial symmetry decomposition of Tb is not used belowwhen looking for convectively coupled aspects of equatorial waves.

15 JULY 2003 1645Y A N G E T A L .

FIG. 6. Space–time mean analysis error in the fit of (a) westward-and (b) eastward-moving 200-hPa meridional wind components as afunction of yo for a range of truncations, N 5 0–10, in the seriesdefined by Eq (3.1). Numbers indicate the values of N. Solid linesare for N even and dashed lines are for N odd.

shows the eastward and westward components sym-metric about the equator. Large variations associatedwith westward-moving waves appear over a latitude beltthat varies from about 6238 over the Indian Ocean/western Pacific to about 6178 over the eastern Pacific(Fig. 5a). In addition to large variations over the tropicalregion, the eastward-moving waves also show signifi-cant activity poleward of 258, associated with eastward-moving, extratropical Rossby waves (Fig. 5b). Betweenthese two regions, a minimum in activity is seen at about208, which separates the activity of equatorial wavesfrom that of extratropical Rossby waves. The corre-sponding plots for the components antisymmetric aboutthe equator (not shown) give similar results. This sug-gests that equatorial convectively coupled waves arefound primarily between 208N and 208S, and thereforeit is reasonable to set Y 5 208 to calculate the best fitfor yo.

The best fit yo is chosen in such a way that minimizesthe meridional wind analysis errors in the tropical strip.The time-mean analysis error E is defined as follows:

E (x , y , y )N i j o

2N yjy(x , y , t ) 2 y (x , t )DO Oi j k n i k n1 2[ ]yk n50 o

5 , (3.1)2[ y(x , y , t ) 2 y (x , y )]O i j k i j

k

where (xi, yj, tk) is a space–time grid point of the data,and represents the variable y filtered to give its east-yward-/westward-moving component in the specified fre-quency–zonal wavenumber domain {v, k}; n is theycoefficient of projected on to the nth parabolic cylinderyfunction Dn(y/yo) and, as shown in Table 1, is relatedto the nth wave mode with meridional wavenumber n;

is the seasonal mean of and hence the denominatory yof Eq. (3.1) is the variance of ; N is the truncationylimit of the series.

Figure 6 shows the error in the fit of the westward-and eastward-moving 200-hPa meridional wind com-ponents as a function of yo for a range of truncationsin the series. The 850-hPa results are very similar. Forsmall yo, the parabolic cylinder functions are stronglyequatorially trapped and a large number of terms is re-quired to represent spatial structures in the 6208 equa-torial strip. For large yo, structures away from the equa-tor are well represented, but nearer the equator the errorsare larger. In all except the N 5 0 westward case, thereis a minimum total error at intermediate values of yo.The reduction in error is much smaller with the additionof extra terms beyond N 5 4. At this truncation theminimum is at yo 5 68, and this is the basic value thatwill be used in this paper. However, any values for yo

between, say, 58 and 88 could be argued as being rel-evant.

A range of yo has been used for reruns of the analysisto be presented in the next section. For any value of yo

in the range 58 to 88, at each day and level the samephysical structures are represented as, for example, n 51 Rossby waves. The smaller the value of yo, the moreequatorially trapped the structures appear. Identifyingthe structure at consecutive days gives a phase speedfrom our technique that is, therefore, independent of theparticular values of yo used. This contrasts with the clas-sic equatorial wave theory for which all the wave speedsare derived from c 5 2b , and where c is also related2yo

to an equivalent depth h by c 5 (gh)1/2. However, it isinteresting to use these relations and also to comparewith the result given in other studies. The trapping scaleyo 5 68 would correspond to c ø 20 m s21 and h ø 43m. These are also close to the preferred values deducedby Wheeler and Kiladis (1999). Earlier, the work ofKiladias and Wheeler (1995) suggested a rather largervalue for yo, in the range 78 to 148, for the R1 wave (asdetermined from their equatorial Rossby radius of de-formation of 108–208). In contrast, the study of Tak-ayabu (1994a) suggests a value of yo 5 58, at the lowerend of the range given here.


4. Identifying structures in the real atmosphere—A case study

This section describes the application of the meth-odology to a case study of equatorial waves during the8-day period 12–19 July 1992. Throughout, the valuesY 5 208, yo 5 68, and a 5 g/(2b ) will be used, except2yo

where stated. To make the computation more tractable,the CLAUS Tb data and ERA dynamical fields have beenregridded to a coarser grid of approximately 2.58. Ex-periments show that the coarse resolution significantlyreduces computing time without affecting the analyzedwave structures.

The complex and rich array of equatorial waves iden-tified during the period is summarized in the third pairof columns in Table 2. The reader may find it useful torefer to this as the waves are identified in the followingsections.

a. Westward-moving waves associated with off-equatorial convection at 58–208N

As suggested by the summary in Table 2, variousconvectively coupled WMRG, R1, and R2 waves havebeen identified at a wide range of locations across theAtlantic, Pacific, and Indian Oceans. The majority ofthese waves are closely linked to convection, and mostof them are present in both the lower and upper tro-posphere, indicating strong vertical coupling.

As an example of the results, we show the WMRGand R1 waves identified in the upper- and lower-tro-pospheric winds over the eastern Pacific–Atlantic regionduring the 8-day period (Fig. 7). These waves have anaverage zonal wavenumber of about 6, with a longerwavelength over the eastern Pacific region than over theCentral American to Atlantic region. They are associ-ated with westward-moving areas of convection, ‘‘A’’and ‘‘B,’’ identifiable in the westward-moving bright-ness temperature anomalies, which provide the back-ground to each panel in Fig. 7. The WMRG wave isprevalent for the first 4 days (Figs. 7a,b) and the R1waves dominate for the latter 4 days (Figs. 7c,d).

On 12 July, Fig. 7a shows a maximum upper-tropo-sphere equatorial southward wind over western SouthAmerica with, from Fig. 7b, a northward wind at lowlevels. It is clear that these maxima form part of hori-zontal wind fields that are mainly rotational. The cor-responding height fields at both levels show that thecirculations are around height field extrema (not shown).The structure therefore looks like a textbook first in-ternal, baroclinic WMRG. Over the next 3 days it retainsits structure and moves westward some 88 per day (about10 m s21). The brightness temperature clearly containsa variety of components that are symmetric and anti-symmetric about the equator. However, the convectionin the region of the meridional wind extrema discussedis predominantly antisymmetric. Its spatial relationship,and in particular that of convection region A, with re-

spect to the WMRG is very close to that expected if itis associated with low-level convergence in the wave(cf. Fig. 3). As indicated by the blue line, this relation-ship is retained over the period shown. We thus considerthis to be an example of a convectively coupled WMRG.

The lower troposphere (Fig. 7d) in the region 1358–908W in the following 4-day period exhibits a westward-moving R1 structure, with the winds being predomi-nantly rotational and circulating around geopotentialmaxima (not shown). In the upper troposphere (Fig. 7c),a similar structure is indicated. The amplitude is smallerand the sign is the same. With just two levels the detailsof the vertical structure cannot be determined but in thiscase it may be barotropic. Such a barotropic verticalstructure in the R1 wave was found also over the equa-torial central Pacific by Kiladis and Wheeler (1995) andWheeler et al. (2000). The convection region A hasmoved further away from, and is more symmetric about,the equator than during the earlier period shown for theWMRG. On 16 July its spatial relationship with the R1wave is again consistent with the low-level convergencein the wave shown in Fig. 3. Over the next 3 days, thedynamical structure and the convective signature movewestward, retaining their integrity. The movement ofconvection region A is shown by a blue line, which isdashed in the upper troposphere because of the reversedcirculation there from that anticipated. Convection re-gion A continues to move further from the equator and,as discussed below, by 19 July is actually associatedalso with an R2 wave.

Seen also in Figs. 7c and 7d from 17–19 July is awestward-moving R1 structure in the Atlantic, whichappears to be coupled with convection region B. ThisR1 wave has a reversal in circulation through the tro-posphere suggestive of a first internal baroclinic modestructure.

One of the tests proposed above for the usefulness ofthe analysis technique is that the separately analyzed uand y fields for the R1 wave, for example, should bepredominantly rotational and that these wind compo-nents should be approximately in geostrophic balancewith the analyzed R1 height field. Quantitative measuresof these balances have been obtained, averaged over thetropical strip and over season. By considering thesemeasures as a function of a we can check that the choicemade for a on the basis of the geostrophy of the Kelvinwave is more generally applicable. Shown in Fig. 8 are850-hPa normalized variances of the divergence ]u/]x1 ]y/]y and the ageostrophic wind speed | fv 1 gk 3=f | . In each case the normalization is performed usingthe variances of the individual terms. At the chosenvalue of a (1 on the abscissa), there is on average,indeed, a high level of cancellation between the twoterms in each case so that the flow is predominantlyrotational and geostrophic. It is also clear that the chosenvalue of a is nearly optimum in this regard.

Other waves are found during this 8-day period. Ac-companying the WMRG and R1 waves shown in Fig.

15 JULY 2003 1647Y A N G E T A L .

FIG. 7. Horizontal winds analyzed from the ERA daily data projected onto the n 5 0 and 1 waves at 308–1508W,which are identified as the WMRG and R1 waves. The WMRG is for 12–15 Jul 1992 at (a) 200 hPa and (b) 850 hPa;the R1 wave is for 16–19 Jul 1992 at (c) 200 hPa and (d) 850 hPa. Superimposed colors show Tb for the same days.Both data are for westward-moving waves. Characters A and B show convection coupled with the WMRG and R1waves. Blue arrow line shows the Northern Hemisphere branch of the dynamical wave coherent with convection centerswith the first baroclinic structure, and the broken blue line indicates the Northern Hemisphere dynamical branchconnected with convection but with a reversed circulation from that of the first internal baroclinic mode. Units are ms21 for wind and K for Tb.


FIG. 8. The 850-hPa normalized variances of the horizontal di-vergence, ]u/]x 1 ]y/]y (solid line), and the f times ageostrophicwind speed | fv 1 gk 3 =f | (dashed line) for the westward-movingn 5 1 analysis at 850 hPa. This is expected to be dominated by R1waves but could also include small-amplitude low-frequency WG1waves. In each case the normalization is performed using the vari-ances of the individual terms. The abscissa shows values of a21

normalized by ao 5 g/(2b ), the chosen value used in this analysis.2yo

7, a convectively coupled R2 wave is also found andlinked to the convective region A (not shown), althoughthe convection associated with the R2 wave was orig-inally further north of A near 158–208N, 308–1008W, inagreement with the convergence in the dry wave shownin Fig. 3. The coexistence of the n 5 0 WMRG and n5 2 Rossby waves over South America and the easternPacific suggests that a linear regression between con-vection and dynamical fields, even after a separation ofsymmetric and antisymmetric components and space–time filtering (as done by Wheeler et al. 2000), maystill fail to identify/separate dynamical structures of par-ticular waves. This emphasizes the value of our meth-odology.

The summary given in Table 2 shows that in generalthe expected waves are present. The suggestion fromDoppler shifting, that the WMRG may not be found atlow levels in the Eastern Hemisphere is supported dur-ing this period. However, Rossby waves are found there.In the upper troposphere in the Western Hemisphere,the WMRG is found. The Rossby wave is also foundthere but it often has the same sign as at the lower level,suggesting that the ambient flow could indeed be im-portant. A modeling study by Wang and Xie (1996)showed that in a two-level model differing verticalshears can alter the vertical coupling of baroclinic andbarotropic components of equatorial waves, signifi-cantly affecting the Rossby and WMRG waves, espe-cially the former. Their results suggested that the ratioof the amplitude of the barotropic versus baroclinicmodes all increased with increasing strength of the ver-tical shear, independent of the sign of the shear. Ourresults for this period do not agree with those of Wangand Xie (1996). The vertical shear in the Western Hemi-sphere, where the barotropic structure is dominant, is

actually weaker than that in the Eastern Hemispherewhere the baroclinic structure is dominant. However,this result is based only on a limited case study andmerits further investigation with an extended record toprovide more robust statistics, as will be reported on ina later paper.

b. Eastward-moving waves associated with off-equatorial convection at 58–208N

A similar analysis has been performed for the east-ward-moving components of the Tb and ERA dynamicalfield data for the same period. The summary of off-equatorial, eastward-moving waves provided in Table 2indicate that structures related to EMRG, WMRG, R1,R2, and EG waves can all be identified as eastward-moving waves, as hinted at by the Doppler shift argu-ment.

As an example, Fig. 9 shows the eastward-moving Tb

and the upper-tropospheric eastward-moving n 5 0, 1waves for 13 and 14 July. Since for the same meridionalwavenumber n, different wave structures may occur atdifferent longitudes due to variations in the ambientflow, the whole tropical field needs to be shown forthese waves. Here we focus on the off-equatorial con-vection associated with the n 5 0 and 1 waves, althoughthe convection in Fig. 9 is dominated by the equatorialsignature of the Kelvin waves (e.g., regions ‘‘C’’ and‘‘D’’), particularly in the Eastern Hemisphere. TheseKelvin waves will be discussed in detail in section 4c.

Looking at Fig. 9a in detail, EMRG wave structures,with a zonal wavenumber of 5, appear clearly in theEastern Hemisphere, between 508 and 1208E, where theflow is easterly and signs of WMRG wave structureswith a zonal wavenumber of 6 can been seen between08 and 908W, where the ambient flow is westerly (Fig.4). The appearance of these different types of MRGwaves, dependent on the sign of the basic state, is en-tirely consistent with the suggestions given by the Dopp-ler shift argument. As indicated by the two solid bluelines, these EMRG and WMRG waves are convectivelycoupled with the two off-equatorial regions of convec-tion, ‘‘E’’ and ‘‘F,’’ respectively. Both waves last formore than 3 days. Corresponding structures are notfound at 850 hPa.

Figure 9b shows the upper-tropospheric structures ofthe eastward-moving R1 and EG1 waves (note that theTb field is the same as in Fig. 9a). The presence of anR1 wave is clearly evident in the Western Hemisphereand a weaker sign of an EG1 wave is seen between 508and 1008E. Both waves have a wavenumber of about 6.An upper-level R1 wave also appears over the MaritimeContinent even though the seasonal mean flow is east-erly there. However, during the season there are signif-icant variations in the ambient flow that can also haveimplications for the presence of different wave struc-tures. During the period 6–14 July, a westerly flow isprevalent in the region 08–108S, 1008–1508E, and on 9

15 JULY 2003 1649Y A N G E T A L .

FIG. 9. As in Fig. 7, but for the eastward-moving (a) n 5 0 and (b) n 5 1 waves and Tb on 13–14 Jul 1992. Characters E and F showconvection coupled with EMRG and WMRG waves, and G, H, and I coupled with the EG1 and R1 waves. Characters C and D are coupledwith the Kelvin waves and will be shown in Fig. 10.

July near 1208E, the westerly wind average between108N and 108S is 9 m s21. These westerlies can thereforesupport eastward moving R1 waves. Similarly, althoughFig. 2 suggests that the dispersion curve for EG1 wavesis unlikely to enter the space–time domain of interest,if the ambient flow is sufficiently easterly, say 220 ms21, then EG waves are possible. Such an ambient flowis not unrealistic. During the period 12–19 July, easterlywinds exceeding 20 m s21 do occur in the 508–808Eequatorial region. These subseasonal variations in theambient flow therefore allow a mixture of eastward-moving EG1 and R1 waves in the upper troposphere ofthe Eastern Hemisphere, which again are consistent withthe effects of Doppler shifting.

The two blue solid lines around 1208E and 1208W inFig. 9b indicate coupling of the R1 waves with the areasof convection, ‘‘H’’ and ‘‘I.’’ As in the case of theeastward-moving MRG waves, these coupled featuresalso last for more than 3 days. The blue solid line around608E indicates the EG1 wave, which is also coupled

with an equatorially centered region of convection,‘‘G,’’ which will be discussed further in section 4c.

In contrast to the upper troposphere, significant east-ward-moving R1 waves can only be seen in the lowertroposphere of the Eastern Hemisphere. However, theseR1 waves are not clearly coupled with off-equatorialconvection. Instead they appear to be closely linked withequatorial convection as discussed below in section 4c.The EG1, WMRG, and EMRG waves cannot be clearlyseen at the lower level. This difference in the wavebehavior between the upper and lower troposphere maybe explained by the weaker ambient zonal flow in thelower troposphere (Fig. 4) and the correspondinglysmaller impact of Doppler shifting.

Eastward-moving R2 waves in the upper and lowertroposphere are identified and are also associated withthe convection, E and F. These waves have a first bar-oclinic structure, but are less coherent in the upper tro-posphere of the Eastern Hemisphere and the lower tro-posphere of the Western Hemisphere where, in both


cases, the ambient flow is not favorable for the eastward-moving waves. As for the westward-moving waves de-scribed in the previous section, the n 5 0 MRG and n5 2 Rossby waves again coexist and are associated withthe same regions of convection.

The three eastward-moving waves coupled with off-equatorial convection (WMRG, R1, and R2) do not ap-pear over the central Pacific region during this period.This may be due to the fact that the westerly windsrequired for the appearance of such waves are not prev-alent in this region, in either the upper or lower tro-posphere.

The effects of Doppler shifting may inhibit verticalcoupling of eastward-moving R1, R2, and MRG waveswhen the ambient flow in the upper and lower tropo-sphere differs significantly, as is the case for much ofthe Tropics. As mentioned above and shown in Table2, compared with the eastward-moving WMRG and R1waves, the eastward-moving R2 wave is more likely tooccur at the both levels and is more vertically coupled.However where the ambient flow is easterly, the wavetends to be less coherent. The reason for the differencebetween WMRG, R1, and R2 waves is not clear butmay be partly that the R2 wave can be Doppler shiftedto move eastward in a weaker westerly flow than wouldbe required by the WMRG and R1 waves.

c. Eastward-moving waves coupled with equatorialconvection at 58N–58S

As shown in Fig. 3, basic theory suggests that equa-torial convection linked to low-level convergence ismainly associated with the Kelvin wave. It could alsobe linked with the other gravest waves, for instance then 5 1 gravity wave, but these generally occur on thehigher-frequency range.

It is found that, in this period, Kelvin waves dominatein the region 608E–1508W. Figure 10 gives examples ofthese waves during 12 to 15 July. Over the Indian Ocean,the region of deep convection C is accompanied by areasof suppressed convection to the west and, to a lesserextent, to the east. These regions of enhanced and sup-pressed convection move eastward and are mainly sym-metric about the equator, with a maximum on the equa-tor. The wind field shown in Fig. 10 confirms that thisconvection can be associated with an upper-troposphericKelvin wave, which has a wavenumber of 6 and moveseastward with a similar phase speed to the convection.Consistently, its convergence and divergence centers co-incide, respectively, with the regions of suppressed andenhanced convection, as indicated by the red and bluesolid lines.

The lower-tropospheric wind field, shown in Fig. 10b,suggests that the vertical structure of the Kelvin waveis not exactly that of the first baroclinic mode, but hasa westward phase tilt with height, consistent with thevertical structure observed in Wheeler et al. (2000). Asshown by the red and blue lines linking the region of

convection C in Fig. 10b, the lower-tropospheric regionsof divergence and convergence are located to the eastof the centers of suppressed and enhanced convection.In fact, the enhanced convection is in phase with thelower-tropospheric maximum westerly wind anomaly,as indicated in the red circle of Fig. 3. This is in agree-ment with the possible importance for convective or-ganization of wind-dependent surface energy fluxes. As-suming that the near surface and 850-hPa zonal windshave similar phases, it does not appear to support thefrictional convergence hypothesis. Also the lower-leveldivergence maximum to the west of convection C movesfaster than the convergence to its east and nearly mergesinto the convergence center at 15 July. This may beresponsible for the weakening of the convergence andthe associated convection C.

Over the central Pacific, to the east of a region ofsuppressed convection, there is an area of slightly en-hanced convection D, which is closely connected to alower-tropospheric Kelvin wave. On 12 July, when theconvection center is to the north of the equator the low-er-level convergence is weak. Subsequently, the con-vection D moves to the equator, the lower-level con-vergence intensifies, and the convection and conver-gence centers move eastward together with a similarphase speed. As the wave moves eastward, the phaseshift between the upper and lower troposphere decreasesso that by 14–15 July, a convectively coupled Kelvinwave with a first baroclinic structure is clearly seen.

Figures 10a and 10b also show the presence of aregion of convection G over the western Indian Ocean.This is not clearly related to a Kelvin wave. As shownin Fig. 9b it is related to the upper-level EG1 wave there.Further analysis indicates that this convection is alsoclosely associated with a lower-level R1 wave (Fig.10c). As noted in section 4b, Doppler shifting can giverise to lower-tropospheric eastward-moving Rossbywaves in the Eastern Hemisphere. An example of thiscan be seen on 13 and 14 July between 458 and 908Ewhere the low-level ambient flow is westerly. However,compared with the corresponding theoretical wavestructure (Fig. 3), it is clear that this R1 wave is notcoupled with off equatorial convection. Instead, it ap-pears to be closely linked with the region of equatorialconvection G, which in turn is coincident with the regionof enhanced westerly winds in the R1 wave. Noting thatthe ambient flow at this time is westerly, in excess of5 m s21, it seems likely that the wind-induced surfacefluxes are again playing an important role in determiningthe location of the convection associated with the wave.In addition, it is interesting to note that the westerlywind anomalies of the R1 wave, which coincide withthe convection, are much stronger than their easterlycounterparts suggestive perhaps of feedback betweenthe convection and the dynamics. Furthermore, it couldbe argued that the convective response to the equatorialzonal winds associated with the R1 wave modifies the

15 JULY 2003 1651Y A N G E T A L .

FIG. 10. As in Fig. 7, but for the eastward-moving Kelvin wave and Tb at 608E–1508W during 12–15 Jul at (a) 200hPa and (b) 850 hPa, and (c) for an eastward-moving R1 wave at 850 hPa at the Eastern Hemisphere during 13–14Jul. Characters C and D show convection coupled with the Kelvin wave, and G coupled with the lower level R1 wave.In (a) and (b) the blue (red) arrow lines indicate the upper-level equatorial divergence (convergence) or lower-levelconvergence (divergence) centers, which are coherent with convection (clear sky). In (c) the green arrow line showsthe equatorial maximum westerlies of the R1 wave associated with convection G.

structure of the wave so that it no longer closely resem-bles its theoretical counterpart.

d. Westward-moving waves coupled with theequatorial convection at 58N–58S

The theoretical structures shown in Fig. 3 suggest thatonly the WG1 wave would connect through its con-vergence with westward-moving equatorial centeredconvection. However, unlike EG1, it is not found withinthe frequency range considered here. Also, as indicatedin Table 1, equatorial wave theory suggests that, in con-trast to eastward-moving Kelvin waves, there should beno corresponding westward qo component. However,our analysis of the ERA data in terms of parabolic cyl-

inder functions gave an appreciable westward-movingqo. Redoing the whole analysis with different values ofyo did not lead to a reduction in this component. It couldbe reduced by approximately a factor of 1.7 at upperlevels and 3.8 at lower levels by decreasing a by a factorof 2.2–3. However, such values are not consistent witheither Kelvin wave geostrophy or as can be seen in Fig.8 comparing the values for ao/a 5 1 and ao/a 5 2.22 3, with the rotational and geostrophic nature of theR1 wave.

It has been found that the appearance of the westward-moving qo is often accompanied by a lower-troposphericwestward-moving R1 wave, with equatorial convectionthat is consistent with being induced by the wind-de-pendent fluxes of the R1 wave. Figure 11 shows an


FIG. 11. As in Fig. 7, but for the westward-moving n 5 21 and1 waves and Tb on 13–14 Jul 1992 in the Eastern Hemisphere: (a)200-hPa u associated with n 5 21 westward qo (u/wqo); (b) 850-hPaR1 waves. Characters J and K show convection connected with u/wqo divergence and u maximum of R1 waves. The blue (red) arrowlines in (a) show the divergence (convergence) of u/wqo. The greenarrow lines in (b) show equatorial maximum easterly and westerlyof the R1 wave associated with convection J and K, respectively.

example of the relationship between the upper-tropo-spheric zonal winds from the westward qo (labeled u/wqo) and the lower-tropospheric R1 wave. It is seen thatthe upper-level u/wqo is closely associated with equa-torial convection, with its divergence connected withregions of enhanced convection, ‘‘J’’ and ‘‘K,’’ and con-vergence over the region of suppressed convection overthe eastern Indian Ocean, as indicated by the blue andred solid lines.

As shown by the two green arrow lines in Fig. 11b,the lower-tropospheric R1 wave is closely connectedwith the same two regions of equatorial convection Jand K. These occur in regions of enhanced equatorialzonal flow. For J, this coincides with the region of max-imum easterly winds in an easterly ambient flow, where-as for K, it coincides with the maximum westerly windsin a westerly ambient flow during this period. This isagain consistent with the hypothesis that wind-inducedsurface heat fluxes give the association between the R1wave and equatorial convection.

The method of separating individual wave modes alsoproduces a westward qo term in the lower troposphereduring 17 to 19 July near 908W. In this case, the con-vergence of u/wqo is connected with equatorial convec-

tion and again this event is associated with an R1 wave.Returning to Fig. 7d, we can see that while the R1 waveis coupled with the off-equatorial convection A, it isalso connected with a region of equatorial convectionto the southeast of A. On 18 and 19 July, when signif-icant easterlies in the lower-tropospheric R1 wave arein phase with the equatorial convection, the convectionintensifies. As was the case for the eastern Pacific re-gion, the ambient flow is easterly, suggesting that wind-induced surface fluxes associated with the lower-levelR1 wave give rise to an intensification of the equatorialconvection and the appearance of a flow described bythe u/wqo term. It is also found, but not shown here,that some cases of activity in u/wqo may be associatedwith the convection induced by the enhanced zonalwinds of the R2 waves. Although these are off the equa-tor (see Fig. 3), they are sufficiently close to it to sig-nificantly modify near-equatorial convection.

5. Conclusions and discussion

In this study, a new methodology has been developedbased on shallow water theory that isolates individualequatorial modes in the real atmosphere. It has beenused to identify various equatorially trapped waves inobservational data. The methodology is different froma space–time spectral filter analysis, which assumes thatthe linear adiabatic theory for equatorial waves on aresting atmosphere is applicable. Here, it is assumedonly that we can use the horizontal structure functionsgiven by equatorial wave theory. The dynamical fieldsare first separated into eastward- and westward-movingcomponents using space–time spectral analysis and thenprojected on to different equatorial modes with a bestfit trapping scale yo deduced from the observational datain a suitable latitude belt.

The methodology is applied to the ERA dynamicalfields and the CLAUS satellite observed window bright-ness temperature data. The spatial and temporal con-sistency of the results suggests that the technique hasbeen successful in identifying equatorial wave structuresand in detecting some important characteristics of equa-torial modes. A best fit trapping scale yo 5 68 for tropicallatitudes 208N–208S has been deduced. By comparingthe eastward- and westward-moving components of theconvection field, inferred from the satellite Tb data, wehave been able to detect convectively coupled equatorialwaves.

The general consistency of the spatial and temporalstructure of the individual wave components identifiedin the 8-day period and of the convection with them hasgiven confidence that the methodology has been suc-cessful. As summarized in Table 2, we have found ev-idence of off-equatorial convection limited not only tothe expected westward-moving WMRG and Rn waves.Consistent with some validity of the notion of Dopplershifting by the ambient wind, at the upper level we havealso found eastward-moving EMRG, Rn, and EG1

15 JULY 2003 1653Y A N G E T A L .

waves in the Eastern Hemisphere, and WMRG and Rnwaves in the Western Hemisphere. Near-equatorial con-vection has been found to be associated not just witheastward-moving Kelvin waves but also with the zonalwind in modified Rossby waves moving in both direc-tions. Most of the waves show first internal mode be-havior or amplitude concentrated at one level. However,R1 waves in the eastern Pacific and eastern Atlanticregions have the same sign in the upper and lower tro-posphere, evidence of barotropic or mode-2 behavior.

All the waves mentioned above, apart from the R1wave, appear to have convection associated with theirlow-level convergence or, in the case of the Kelvinwave, possibly with a low-level zonal wind maximum.It appears that such convection does not radicallychange the dry wave structure, though the Kelvin wavehas a tilt in the vertical. However, it appears that equa-torial or near equatorial convection can be associatedwith the low-level zonal wind extrema in these waves,consistent with it being associated with a maximum insurface energy fluxes. The westward-moving qo, whichis found in the data but not given by dry theory, appearsto be associated with the Rossby wave. Therefore, thehypothesis is made that Rossby waves coupled withequatorial convection can exist and have a new structureinvolving a westward-moving qo, as well as the drywave components. The different vertical structure ofsome of the Rossby waves may also be evidence of achanged structure due to the convective coupling. Fur-ther investigation into those aspects is required with amore extensive set of data. This is the aim of a com-panion paper on this study.

It is hoped that the analysis method will make a verypowerful diagnostic method for investigating convec-tively coupled equatorial waves and the interaction ofequatorial dynamics and physics in the atmosphere.Comparison of the results of applying similar analysesto climate and weather forecast models should give con-siderable insights into their behavior. A parallel diag-nosis of the ECMWF 40-Year Reanalysis (ERA-40) dataand model validation will also be performed in a sub-sequent study. Key questions will be addressed con-cerning the interaction between dynamics and convec-tion, and how well this is represented in models.

Acknowledgments. Dr. G.-Y. Yang acknowledges thesupport of NERC through Grant B480700. Prof. J. Slin-go is supported by the NERC Centres for AtmosphericScience (NCAS).

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Convectively Coupled Equatorial Waves: A New Methodology ...roger/Lectures/Tropical… · Corresponding author address: Gui-Ying Yang, Dept. of Meteo-rology, University of Reading,