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Vertically Propagating Kelvin Waves and Tropical Tropopause Variability

JUNG-HEE RYU AND SUKYOUNG LEE

Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

SEOK-WOO SON

Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York

(Manuscript received 15 March 2007, in final form 1 November 2007)

ABSTRACT

The relationship between local convection, vertically propagating Kelvin waves, and tropical tropopauseheight variability is examined. This study utilizes both simulations of a global primitive-equation model andglobal observational datasets. Regression analysis with the data shows that convection over the westerntropical Pacific is followed by warming in the upper troposphere (UT) and cooling in lower stratosphere(LS) over most longitudes, which results in a lifting of the tropical tropopause. The model results reveal thatthese UT–LS temperature anomalies are closely associated with vertically propagating Kelvin waves, indi-cating that these Kelvin waves drive tropical tropopause undulations at intraseasonal time scales.

The model simulations further show that regardless of the longitudinal position of the imposed heating,the UT–LS Kelvin wave reaches its maximum amplitude over the western Pacific. This result, together withan analysis based on wave action conservation, is used to contend that the Kelvin wave amplification overthe western Pacific should be attributed to the zonal variation of background zonal wind field, rather thanto the proximity of the heating. The wave action conservation law is also used to offer an explanation as towhy the vertically propagating Kelvin waves play the central role in driving tropical tropopause heightundulations.

The zonal and vertical modulation of the Kelvin waves by the background flow may help explain theorigin of the very cold air over the western tropical Pacific, which is known to cause freeze-drying oftropospheric air en route to the stratosphere.

1. Introduction

The established view is that the height of the time-mean tropical tropopause is controlled by deep convec-tion up to a height of !12–13 km (Riehl and Malkus1958), and that radiative balance explains the continueddecrease in temperature up to the cold-point tropo-pause at !17 km. In contrast, the temporal variabilityof the tropical tropopause height, and related variablessuch as tropopause temperature, are known to be as-sociated with fluctuations in upwelling induced bystratospheric extratropical wave breaking (Holton et al.1995) and perhaps in the strength of the Hadley circu-lation (Plumb and Eluszkiewicz 1999).

At intraseasonal time scales, there is increasing evi-dence that tropical tropopause temperature fluctua-tions are associated with Kelvin waves. Zhou and Hol-ton (2002) found that the cooling of the cold-pointtropopause in the tropics is associated with the Mad-den–Julian oscillation (MJO; Madden and Julian 1971),and they attributed this cooling to the presence ofKelvin waves. Based on radiosonde data, collected overa period of 1 month over eastern Java, Indonesia,Tsuda et al. (1994) reported that the tropopause de-scends following the phase lines of 20-day Kelvinwaves, which appear to be excited by tropical convec-tion. With a linearized primitive-equation model Salbyand Garcia (1987) and Garcia and Salby (1987) showedthat localized tropical heating can generate similar ver-tically propagating Kelvin waves in the stratosphere.Salby and Garcia (1987) found that short-term heatingfluctuations generate vertically propagating waves,which resemble observed Wallace and Kousky (1968)

Corresponding author address: Sukyoung Lee, Department ofMeteorology, The Pennsylvania State University, UniversityPark, PA 16802.E-mail: [email protected]

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DOI: 10.1175/2007JAS2466.1

© 2008 American Meteorological Society

JAS2466

Kelvin waves, while seasonal time-scale heating pro-duces a Walker circulation–like Kelvin wave, which isconfined to the troposphere. The relationship betweenthe convection and Kelvin waves was observationallyestablished by Takayabu (1994), Wheeler et al. (2000),Kiladis et al. (2001), and Randel and Wu (2005).

The above studies collectively suggest that convec-tive heating, equatorial Kelvin waves, and tropicaltropopause height undulations are closely related. Infact, with Challenging Minisatellite Payload (CHAMP)/global positioning system (GPS) radio occultationmeasurements collected between May 2001 and Octo-ber 2005, Ratnam et al. (2006) showed that the seasonaland interannual variation in the zonal wavenumber 1and 2 Kelvin wave amplitudes coincide with those ofthe tropical tropopause height. They also show that sta-tistically significant correlations exist between theseplanetary-scale Kelvin wave amplitudes and outgoinglongwave radiation (OLR) in the tropics. However, be-cause each of these studies focused on either one or twoof these processes (Zhou and Holton 2002; Wheeler etal. 2000; Kiladis et al. 2001), and because the time pe-riod of some of these analyses was limited (Tsuda et al.1994; Randel and Wu 2005), it is still difficult to draw ageneral coherent picture of how each of these processesare related. While the results of Ratnam et al. (2006)establish the relationship between convection, Kelvinwaves, and variability of tropical tropopause height,their focus was on seasonal and interannual time scales,not on the global-scale Kelvin wave and its relationshipwith intraseasonal tropopause height variations.

In a more recent study, Son and Lee (2007) showedthat the lifting of the zonal mean tropical tropopause ispreceded first by MJO heating over the Pacific warmpool, followed by the excitation of a large-scale circu-lation, which rapidly warms (cools) most of the tropicaltroposphere (the lowest few kilometers of the tropicallower stratosphere). However, they could not discernwhether the lower stratosphere (LS) cooling is associ-ated with vertically propagating Kelvin waves (Tsuda etal. 1994; Randel and Wu 2005), or if it reflects thermalwind adjustment in response to Gill-type Kelvin andRossby waves (Highwood and Hoskins 1998). In addi-tion, because the analysis of Son and Lee (2007) wasbased on the height of the zonal mean tropical tropo-pause, their analysis may have selectively highlightedconvective activity, which excites a global-scale re-sponse.

In this study, we investigate the relationship betweenconvective heating, vertically propagating equatorialKelvin waves, and the tropical tropopause height. Aswill be shown in this study, the LS Kelvin wave is a

robust response to localized tropical heating, and theKelvin wave response is greatest in the western tropicalPacific. By presenting a plausible mechanism that canaccount for the Kelvin wave’s amplitude modulation inboth the zonal and vertical directions, this study alsoattempts to provide an explanation for why the Kelvinwave response is greatest in the western and centraltropical Pacific, and why vertically propagating Kelvinwaves play a pivotal role for tropical tropopause undu-lations.

The primary tool in this study is a multilevel primi-tive-equation model on the sphere. Observationalanalyses are also presented. Section 2 briefly describesthe data and the model. Results from a regressionanalysis with a global dataset will be presented in sec-tion 3, and the model simulations will be described insection 4. Section 5 presents a wave action analysis onthe effect of the background flow on vertically propa-gating Kelvin waves. This will be followed by the sum-mary and conclusions in section 6.

2. Data and model

a. Data and analysis method

The data and regression methods are identical tothose used in Son and Lee (2007). The National Cen-ters for Environmental Prediction–National Center forAtmospheric Research (NCEP–NCAR) reanalysisdataset is utilized. Daily data on sigma levels, whichspan the time period from 1 January 1979 to 31 Decem-ber 2004, are interpolated with a cubic spline ontoheight coordinates with a 0.5-km interval. The tropo-pause height is then identified with the standard lapse-rate criterion—the lowest level at which the tempera-ture lapse rate remains below 2 K km!1 for a verticaldistance of 2 km. The tropical tropopause identified bythis procedure compares very well with the tropicaltropopause pressure provided by NCEP–NCAR (notshown). The intensity of the convection is estimatedfrom the daily OLR dataset, archived by the NationalOceanic and Atmospheric Administration (NOAA).

In all datasets, the seasonal cycle, which is defined bysmoothed, calendar-day mean values, is removed. In-terannual variability is also removed by subtracting theseasonal mean for each year. To more precisely de-scribe these operations, we write the variable of inter-est, say T, in the following manner:

T"i, j# $ Tc"i# % Ts" j# % T !"i, j#.

The first term on the right-hand side, Tc(i), is thesmoothed calendar mean, defined as L [(1/N)&N

j$1 T(i,j)], where L is a low-pass filter with a 20-day cutoff, N

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the total number of years, i the date within each year,and j the year index. The second term, Ts( j), is theseasonal mean for each year after the seasonal cycle hasbeen removed, Ts(j) ! (1/M)"M

i!1 [T(i, j) # Tc(i)],where i ! 1 corresponds to 1 December, and i ! M to28 February. The filtered temperature anomalies aredenoted by T $ and the filtered tropical tropopauseheight anomalies by Z$, where the prime denotes a de-viation from the seasonal cycle and seasonal mean foreach year. These anomalies are averaged from 10°S to10°N, which will be indicated with an angled bracket.

The above tropical mean anomalies are regressedagainst the OLR anomalies of the western Pacific,which are defined to span the region between 120°Eand the date line, and from 10°S to 10°N (hereafterdenoted as %OLRwp&$). Smoothing is not applied, andthe resulting values are set to correspond to a one stan-dard deviation value of %OLRwp&$. We consider only theNorthern Hemisphere winter, when the MJO convec-tion is strongest [see Son and Lee (2007) for a discus-sion on the seasonal differences].

b. Model and experiment design

The numerical model used in this study is the drydynamical core of the Geophysical Fluid DynamicLaboratory (GFDL) global spectral model (Feldstein1994; Kim and Lee 2001; Son and Lee 2005). Rhom-boidal 30 resolution is used and there is no topography.The vertical levels are defined in ' coordinates. Thereare 60 levels, starting from 0.975 near the surface up to0.0012 (about 45 km at the equator). To accurately re-solve upper-troposphere (UT)–LS features, fine verti-cal resolution, corresponding to about 90 m, is usednear the tropopause (see Fig. 1b, which shows the ver-tical heating profile). Throughout the troposphere, thevertical resolution is approximately 1 km.

Two different basic states are used. One state is azonally asymmetric (3D) and the other is zonally sym-metric (2D). The former state is the December–February (DJF) time mean flow for the years 1948through 1996, and the latter state is the correspondingzonal mean. Because the climatological 3D state is notbalanced, a forcing term is added to ensure that the 3Dstate is a solution of the model equations (e.g., James etal. 1994; Franzke et al. 2004). Franzke et al. (2004)discuss the limitations of this approach and provide arough estimate for the length of the integration periodfor which the basic state can be regarded as a steadysolution. The length of this integration is inversely pro-portional to the strength of the climatological eddyfluxes, where eddy is defined as the deviation from theclimatological flow. According to their estimation, thisperiod lasts for about 20 days in the North Atlantic.

While this time scale is expected to be greater in thetropics, we take the 20-day limit as rough guidance forour model integration.

Newtonian cooling is applied to the perturbationtemperature, where a perturbation is defined as thedeviation from the basic state. The time scale of theNewtonian cooling is 5 days in the stratosphere and 20days in the middle and upper troposphere (cf. Jin andHoskins 1995; Taguchi 2003). Throughout the lowertroposphere (' ! 0.725), the Newtonian cooling timescale gradually decreases to 8 days at the lowest modellevel. There is also friction at the lowest four modellevels (' ! 0.825), with the time scale decreasing from4.5 to 1.5 days at the lowest level. The surface value ofthe vertical diffusion coefficient ( is set to 0.5 m2 s#1

and the value of ( is chosen to keep the dynamic dif-

FIG. 1. (a) The horizontal structure of the idealized heating at400 hPa. The contour interval is 1 K day#1. (b) The vertical profileof the idealized heating at 150°E and the equator. Open circlesindicate the vertical grid points used for the model.

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fusion coefficient !" constant with height, where " isdensity. To prevent contamination by vertical wave re-flection, we apply very large values of the diffusioncoefficient to the highest eight levels (# ! 0.01). For thehorizontal diffusion, an eighth-order diffusion schemeis used with a horizontal diffusion coefficient of 8 $1037 m8 s%1.

For the diabatic heating, an elliptic horizontal distri-bution is utilized, which is similar to the intertropicalconvergence zone (ITCZ) proxy used by Nieto Ferreiraand Schubert (1997). The heating field at 400 hPa,which is centered at 150°E and the equator, is shown inFig. 1a. The zonal and meridional extents of the heatingare 100° and 10°, respectively. The vertical profile ofthe heating takes the form of two parabolas joined at400 hPa, as shown in Fig. 1b. It has a maximum value of4.8 K day%1 at 400 hPa and decreases to 0 at 975 and175 hPa. The heating is gradually turned on (off) during

the 1st (10th) day. The response of the model atmo-sphere to the idealized heating is defined as the devia-tion from the basic state prescribed at the initial time.

3. Observations

a. Tropical tropopause fluctuations

Figure 2 shows the temporal evolution of &ZT'( (themain panel) regressed against the convective activityover the Pacific warm pool, measured by a negative onestandard deviation in &OLRwp'( at lag 0. As indicated insection 2, the angled bracket denotes a latitudinal av-erage from 10°S to 10°N. The positive (negative) timelag denotes the days after (before) the minimum daily&OLRwp'( value. Superimposed upon the contours inFig. 2 is the &OLR'( field regressed against &OLRwp'(.To indicate the location of the convection, values lessthan %2 W m%2 are shaded. It is worth noting that while

FIG. 2. Linear regression of &ZT'( (contours) and &OLR'( (shading) against &OLRwp'( for the periodfrom December to February. The positive (negative) time lags denote the days after (before) theminimum &OLRwp'( value. The contour and shading intervals are 10 m and 2 W m%2, respectively. Zerolines are omitted and &OLR'( values less than %2 W m%2 are shaded. The thick dashed line indicates anapproximate linear fit of the longitude and time at which the &ZT'( value first exceeds 50 m. (right) Thezonal mean &ZT'( (solid line) and &OLR'( (shading), averaged between 120°E and the date line, arepresented. As stated in the text, the angle bracket denotes a latitudinal average from 10°S to 10°N.


no filtering is performed other than the removal of theseasonal cycle and the interannual averages, the re-gressed !OLR"# field shows eastward propagation at theMJO time scale. In fact, Son and Lee (2007) performedspectral analysis on the daily !OLRwp"#, and found aspectral peak at 45 days, which is statistically significantabove the 95% confidence level against the red-noisenull hypothesis.

The autoregression of !OLRwp"# is shown with shad-ing in the right panel of Fig. 2. The maximum value ofthe zonal mean !ZT"# occurs at a lag of 7–8 days (see theright panel in Fig. 2). Because the value of zonal mean!ZT"# is close to zero at lag $2 days, it takes about 10days for the zonal mean !ZT"# to reach its maximumvalue. At the time of the maximum zonal mean !ZT"#, apositive !ZT"# contribution comes from most longitudes.This tropics-wide contribution results from an eastwardpropagation of !ZT"#, as indicated by the thick dashedline, which is an approximate linear fit of the longitudeand the time at which the !ZT"# value first exceeds 50 m.The local maxima in !ZT"# are statistically significantabove the 99% confidence level. In addition, similarresults are obtained when the regressions are per-formed against the NCEP–NCAR tropopause pressure(not shown).

b. Temperature perturbations

The results described above indicate that (i) the west-ern Pacific convective heating typically gives rise to alifting of the zonal mean !ZT"#, which takes place overa 10-day time period; (ii) fluctuations in !ZT"#over thewestern Pacific play a key role for modulating the zonalmean !ZT"# response; and (iii) the contribution to thepositive zonal mean !ZT"# comes from most longitudes.

The dynamical link between the western Pacific heat-ing and the tropopause height can be grasped in a morecoherent manner by examining the associated tempera-ture anomalies, because the tropopause is typicallyraised by UT warming–LS cooling. Figure 3 shows theanomalous temperature !T "# and wind vector field (!u"#,!w"#), regressed against !OLRwp"#. The vertical compo-nent of the wind !w"# is scaled by multiplying by a factorof 1440, so that the slope of the wind vector is consistentwith the aspect ratio of the zonal and vertical axes. Asexpected, based on the location of the convective heat-

FIG. 3. Longitude–height cross section of the regression of !T "#(contours) and (!u"#, !w"#) (arrows) against !OLRwp"# at lag (a)$5, (b) 0, and (c) 5 days. The contour interval is 0.1 K and thevalues that are statistically significant above the 99% confidencelevel are shaded. Wind vectors (m s$1) are plotted only wheneither !u"# or !w"# is statistically significant at the above the 99%confidence level. The vertical wind is multiplied by 1.44 % 103 sothat the slope of the wind vector is consistent with the aspect ratioof the figure. The thick solid line in each panel indicates the DJF

←

climatological tropical tropopause height, and the dotted line isfor the perturbation tropopause height. The scale for the pertur-bation tropopause height is shown in the vertical axis on the rightside. The horizontal bar shown under each panel indicates thelocation of the minimum !OLR"# presented in Fig. 2.

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ing, the maximum !T "# occurs in the midtroposphereover the Pacific warm pool. In addition to this tropo-spheric warming, there is also a wavy pattern in !T "# inthe vicinity of the tropopause, which bears resemblanceto Fig. 6 of Kiladis et al. (2005), who display tempera-ture fields regressed against an MJO index. The MJOindex in Kiladis et al. (2005) is defined by the eastward-propagating component of the MJO time-scale OLRanomalies. Given that our index !OLRwp"# has a spec-tral peak at the MJO time scale (Son and Lee 2007), itis not surprising that the two temperature fields—oneregressed against !OLRwp"# and the other against anMJO index—resemble each other.

The lack of vertical resolution in the reanalysis datain the LS region makes it difficult to identify the plan-etary-scale wavy features seen in the UT–LS region.However, given the observational and theoretical ex-pectations that the LS region is dominated by verticallypropagating planetary-scale Kelvin waves (Wallace andKousky 1968) and mixed Rossby–gravity waves (Yanaiand Maruyama 1966), the UT–LS features in Fig. 3appear to be accounted for, in part, by the planetary-scale vertically propagating Kelvin waves. A visual in-spection of Fig. 3 indicates that the zonal scale is domi-nated by zonal wavenumbers 1–2, the vertical wave-length in the LS region is about 6 km, and the wavyfeatures are tilted eastward with height. These charac-teristics are consistent with vertically propagatingKelvin waves, not with the mixed Rossby–gravity waves[see Table 4.1 in Andrews et al. (1987) for a summary ofthe wave characteristics].

For the sake of clarity, it needs to be mentioned thatalthough excited by convective heating, verticallypropagating LS Kelvin waves in the tropics are notcoupled to convection. Chang (1976) demonstrated thatrandomly distributed convective heating can generatevertically propagating Kelvin waves, and above theforcing region Kelvin waves of the largest zonal scaleare most effectively excited. Because of the absence ofcoupling, Randel and Wu (2005) described the LS ver-tically propagating Kelvin waves as “free” waves, asopposed to convectively coupled waves (Takayabu1994; Wheeler and Kiladis 1999; Wheeler et al. 2000).

4. Numerical model experiments

The key features highlighted in the previous sectionwill be examined with an idealized model. Because thevertical resolution in the NCEP–NCAR reanalysis isvery low above the troposphere, regardless of the reli-ability of the analysis, the representation of the LS fea-tures is not adequate. However, the analysis in the pre-vious section indicates that the LS features are critical

for tropical tropopause height fluctuations. While acomparison with direct observations, such as radio-sonde data (Kousky and Wallace 1971; Tsuda et al.1994, etc.), is the most persuasive means to verify thefeatures revealed by the regression analysis, availabilityof such direct observations is limited both in time andspace. Although simulations from a numerical modelcannot be used to verify the reanalysis data, with asufficiently high resolution in the vertical direction, themodel simulation can help reveal the LS features,which are not well represented in the reanalysis. Moreimportantly, as will be described in section 4b, themodel can be used to investigate the mechanism behindUT–LS Kelvin wave modulation.

a. Western Pacific heating

In the simulation to be presented in this section, theidealized heat source (see section 2b) is placed in thewestern equatorial Pacific in order to model the con-vective heating indicated by the regressed OLR field.To mimic the eastward propagation of the OLR field,the center of the heating is prescribed to stay at 150°Efor the first 4 days, and then to move eastward at aspeed of 30° (5 days)$1.

1) MODEL-SIMULATED KELVIN WAVE STRUCTURE

Prior to presenting the model simulation of the tropi-cal tropopause undulation, we examine the three-dimensional structure of the model response both forcomparison with those results obtained from previousstudies, and to help understand the model’s tropopauseresponse to the heating. For ease of comparison withthe previous studies (Gill 1980; Jin and Hoskins 1995),we first show the results with the 2D basic state. Theleft column in Fig. 4 displays the UT perturbation pres-sure and horizontal wind fields. A Gill-type response isevident, with the anticyclonic Rossby wave dipole tothe west of the heating, and the Kelvin wave to the eastof the heating. The Rossby wave signal is stronger inthe Northern Hemisphere because of the asymmetry ofthe background flow about the equator. The Kelvinwave can be identified by the equatorial westerlies. Theleading edge of the westerlies advances to 60°W at day5, to the Greenwich, United Kingdom, meridian by day7, and to 30°E by day 9. This eastward expansion of theUT equatorial westerlies (seen in Fig. 4) can be inferredin the streamfunction field shown in Fig. 4a of Jin andHoskins (1995). Likewise, the UT wind fields, simu-lated with the 3D basic state (the left column in Fig. 5),are also consistent with the streamfunction field shownby Jin and Hoskins (cf. day 9 in Fig. 5 with their Fig.12a). Although there are differences in both the 2D andthe 3D model runs, the UT Kelvin waves are anchored


to the local heating, as for the time mean Walker cir-culation (e.g., Gill 1980; Salby and Garcia 1987; Jin andHoskins 1995).

In contrast to the above UT response, the LS Kelvinwave response is not anchored to the heat source. Atthe 20-km level, a patch of equatorial westerlies (the

right column in Figs. 4 and 5) travels eastward andencircles the globe in 10 days. Figure 6 shows the UT–LS vertical cross section of the perturbation pressure(shaded), temperature (contours), and wind vectorsfrom the 3D simulation. (Having demonstrated that the2D simulation conforms to that of previous studies, and

FIG. 4. The perturbation pressure (contours) and wind (arrows) at (left) 14 and (right) 20 km simulated with the2D basic state for (top to bottom) days 3, 5, 7, 9. The contour interval is (left) 10 and (right) 2 Pa. The location ofthe heating is indicated with shading. Wind vectors are in m s!1.

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because the essential feature of the equatoriallytrapped Kelvin wave structure is also faithfully simu-lated in the 3D run, for the sake of brevity, the verticalcross section from the 2D simulation is not shown.) Thephase relationships between the pressure and windfields, and between the wind and temperature fields,are consistent with the theoretical phase relationships

of a vertically propagating Kelvin wave (Wallace andKousky 1968). As can be seen, the westerly perturba-tions are in phase with high pressure perturbations,consistent with the fact that the equatorial Kelvin waveis in geostrophic balance. As expected by the fact thatthe Kelvin wave is driven by buoyancy, the upwardmotion leads the negative temperature perturbation by

FIG. 5. The same as Fig. 4, but for the 3D basic state.


1⁄4 of a cycle. This phase relationship can be seen bycomparing the day-5 wind with the day-7 temperaturefield, and the day-7 wind field with day-9 temperaturefield. As expected for a Kelvin wave, which is excitedfrom below, downward phase propagation can be seen.

2) TROPICAL TROPOPAUSE UNDULATION

To investigate how this Kelvin wave response is as-sociated with the tropopause undulation, Fig. 7 recastsFig. 6, ranging from 5 to 25 km, showing simulations outto 17 days. Simulations from days 9 to 17 show that the

vertical scale of the LS Kelvin wave is about 6–10 kmand the period is about 10–15 days, further conformingto the observed Wallace–Kousky Kelvin wave. Moreimportantly for the tropopause undulation, it can beseen that when the heating is turned on (recall that theheating is turned off on day 10), a shallow layer ofcooling occurs at the tropopause level directly abovethe heating. In particular, the day-5 temperature re-sponse resembles the buoyancy solution of Pandya etal. (1993) for an idealized heating in a Boussinesq fluid.For all the tropical locations and time scales that they

FIG. 6. Longitude–height cross sections of pressure (Pa, shading), temperature (contours), and wind vector(m s!1) perturbations (arrows), generated with the 3D basic state. All fields shown are averaged from 10°S to 10°N.The contour interval is 0.3 K. The vertical wind is multiplied by 3.6 " 103 so that the slope of the wind vectors isconsistent with the aspect ratio of the figure.

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FIG. 7. Longitude–height cross section of perturbation temperature (K, contours), wind vectors (m s!1,arrows), and tropopause height (scale is indicated on the right side), generated with the 3D basic statefor (top to bottom) days 3–17. The thick solid line indicates the DJF climatological tropopause. Allquantities shown are averaged between 10°S and 10°N. The vertical wind is multiplied by 1.44 " 103 sothat the slope of the wind vectors is consistent with the aspect ratio of the figure. The location of heatingis indicated with a horizontal bar under each frame.


examined, Holloway and Neelin (2007) found with sat-ellite and reanalysis data that a shallow layer of tropo-pause-level cooling occurs in conjunction with tropo-spheric heating. Using a Boussinesq equation set, theyprovided an explanation as to why a shallow layer ofcooling must occur above a tropospheric heat source.Because their explanation describes how a heat sourcegenerates internal gravity waves in a hydrostatic atmo-sphere, the same explanation can be given for the gen-eration of tropopause-level cooling in our simulations.

In the simulation presented in this study, the tropo-pause-level cooling explained above plays the key rolefor the tropopause height variability. Figure 7 showsthat the tropopause rises (dotted lines indicate the de-viation from the time mean tropopause) in regionswhere the cooling occurs just above the time meantropopause (thick horizontal lines). Because the Kelvinwave response occurs to the east of the heat source, thecenter of the cooling does not occur directly above theheating region, but instead is shifted to the east. Inaddition, because the phase lines for vertically propa-gating Kelvin waves tilt eastward with height, cooling

occurs just above (below) the tropopause in the eastern(western) part of the heating. This causes the tropo-pause to ascend in the eastern side of the heating, andto descend in the western side. This descent is furtheraided by warming, which begins to appear on day 5 justabove and to the west of the cooling. In the same re-gion, the wind field on day 3 (see Fig. 6) shows that thiswarming is preceded by sinking motion, consistent withthe earlier analysis that the LS response represents avertically propagating Kelvin wave.

To the east of the heating, the LS cooling and theattendant elevated tropopause rapidly propagate to theeast. Referring back to Fig. 5 and the foregoing discus-sion, we recognize that this eastward propagation re-flects the LS Kelvin wave. In Fig. 8, this eastwardpropagation of the elevated tropopause is indicatedwith a thick dashed line. It can be seen that it takesabout 10 days for the elevated tropopause to encirclethe entire tropics. During the eastward expansion of theelevated tropopause, the height of the zonal meantropopause gradually increases (right panel of Fig. 8).However, because the LS cooling is no longer forced

FIG. 8. The temporal evolution of the tropical tropopause height perturbation (contours), averagedbetween 10°S and 10°N, simulated by the idealized heating. The contour interval is 30 m. The 400-hPaheating is indicated by the shading. The right panel displays the zonal mean tropical tropopause heightperturbation. As in Fig. 2, the thick dashed line indicates an approximate linear fit of the longitude andtime at which the !ZT"# value first exceeds 50 m.

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after the heat source is turned off at day 10 (notice thatthe LS cold anomaly no longer exist above the heatingon day 11), the zonal mean tropopause height declinesafter day 10.

With the exception of the negative perturbation near60°W on days 12 and 13, the regressed and simulatedtropopause height perturbations resemble each other(cf. Fig. 2 with Fig. 8, while noting that lag-N day in Fig.2 should be compared with a model day, which isgreater than N, say N ! 5, in Fig. 8). This is because thelag-0 day corresponds to the time of the maximumOLR, meaning that the OLR anomaly amplitude is stillsubstantial at small " lag days.) In addition, in the im-mediate vicinity of the tropopause, below 20 km, theregressed LS temperature field (Fig. 3) compares wellwith the simulated LS temperature field (Fig. 7).

b. Sensitivity to the location of heat source

The foregoing analysis indicates that the simulatedtropopause undergoes undulation as the LS Kelvinwave propagates both vertically and zonally. In addi-tion, the LS Kelvin wave amplitude is greatest over thewestern Pacific. This large Kelvin wave amplitude overthe western Pacific may be attributed to the proximityto the heating. However, the subsequent evolutionfrom day 13 to 15 shows that as the Kelvin wave packetreenters the western Pacific, the LS cold anomaly inthat region strengthens once more, despite the fact thatthe heating has been turned off 5 days earlier. Thesefindings raise the interesting possibility that the zonaland vertical variation of the LS Kelvin wave structuremay hinge on the ambient background flow, more sothan on the proximity to the heat source. This possibil-ity will be explored further in sections 4b and 5.

To test the idea that the LS Kelvin wave structuremay be influenced more directly by the backgroundflow than by the proximity of the heat source itself,additional simulations are performed. In one simula-tion, the heat source is prescribed in the eastern Pacificcentered at 120°W, but the remaining model setup isexactly the same as before. Figure 9 shows the result,which is analogous to Fig. 7. Unlike the run with west-ern Pacific heating, the LS Kelvin wave response, ifany, is very weak directly above the heating. The LSKelvin wave signal does not become apparent until theKelvin wave packet nears the Greenwich meridian. Asthe wave packet enters the Indian Ocean and westernPacific, the wave amplifies and the wave signal extendsfurther into the stratosphere (days 13–17). The overallKelvin wave structure in the stratosphere during days15–17 is remarkably similar to that on day 9 of thewestern Pacific heating simulation, with largest ampli-tude over the western Pacific.

Although not shown, five additional experiments areperformed, with the prescribed heat source centered atthe Greenwich meridian, 60°E, 120°E, 180°, and 60°W.In all of these simulations, the strongest LS Kelvin waveresponse once again occurs over the western Pacific.This result demonstrates that the background flow overthe western Pacific is responsible for the large Kelvinwave signal in that region. This, however, is not to saythat the western Pacific heating is irrelevant for theKelvin wave modulation. Because the characteristics ofthe background flow must be determined in part bytropical convective heating, the western Pacific heatingmust influence the Kelvin wave structure, but appar-ently its impact is made indirectly through the back-ground flow.

c. Zonally symmetric basic state

The impact of the zonal modulation of the Kelvinwave on the zonal mean tropopause undulation istested with a third simulation in which the basic state isthe zonal mean of the DJF climatological state. Asshown in Fig. 10, there is very little zonal variation inthe Kelvin wave amplitude. Once again, the tropopauserises as cold Kelvin wave perturbations position them-selves just above the tropopause (not shown). How-ever, the maximum zonal mean tropopause height per-turbation in this simulation is 38 m (not shown), about15% smaller than those from the two previous simula-tions with a zonally varying basic state. This indicatesthat, all else being equal, the zonal variation of thebackground flow enables the tropopause height to un-dulate by a greater margin.

5. Influence of background state on Kelvin waves

The foregoing model simulations indicate that, re-gardless of the geographical location of the imposedtropical heating, the temperature perturbation associ-ated with the LS Kelvin wave is greatest over the west-ern Pacific. This western Pacific amplification of theKelvin wave signal may also be highly relevant for theissue of stratosphere–troposphere exchange (STE),namely, the origin of the cold region, known as the“cold trap.” According to the “stratospheric fountain”hypothesis of Newell and Gould-Stewart (1981), tropo-spheric air enters the stratosphere through the coldtrap, which is a relatively small, cold, confined regionover the western tropical Pacific. The temperature inthe cold trap is so low that the moisture within thetropospheric air “freeze dries.” This leaves behind verydry air that then enters the stratosphere. Variousmechanisms have been suggested to explain this cold


FIG. 9. As in Fig. 7, but the heating is centered at 120°W.

JUNE 2008 R Y U E T A L . 1829

FIG. 10. As in Fig. 7, but the zonally averaged DJF flow is used as the basic state.


trap, including convective detrainment (Sherwood andDessler 2001; Sherwood et al. 2003; Kuang andBretherton 2004) and a hydrostatic gravity wave re-sponse (Holloway and Neelin 2007) to a heat source.

Tsuda et al. (1994) and Randel and Wu (2005) rec-ognized that the LS cold perturbations associated withthe Kelvin wave may be an answer to this question.Because the Kelvin wave is one particular type of grav-ity wave, the Kelvin wave mechanism and the mecha-nism of Holloway and Neelin (2007) are not mutuallyexclusive. An important difference, though, is thatwhile the Holloway–Neelin mechanism explains a layerof cooling directly above the heat source, a near-fieldresponse, the Kelvin wave mechanism identified byTsuda et al. (1994) and Randel and Wu (2005) is, inprinciple, applicable both for the near- and far-fieldresponses. The results from the current simulation takethe ideas of these two studies one step further to sug-gest that the low temperature in the cold trap may relyon the Kelvin wave intensification over the western Pa-cific. Therefore, it is important to ask what basic-stateproperty is responsible for this zonal modulation.

In addition to the zonal modulation, we also recog-nize that the vertical modulation of wave amplitude bythe background state is an important feature that callsfor a better understanding. This stems from the ques-tion of why the LS Kelvin wave is so important for thetropopause undulation.

With these questions in mind, in this section, we ap-peal to linear wave theory to gain insight into how theUT–LS Kelvin wave amplitude varies with the back-ground state. To do so, we consider the dispersion re-lation for a hydrostatic Kelvin wave, which takes theform of

! ! Uk " Nk"m, #1$

where U is the background zonal wind, N the back-ground buoyancy frequency, % the ground-based fre-quency, and k and m the zonal and vertical wavenum-bers, respectively. Taking the sign of % as positive defi-nite, a positive value of k represents eastward phasepropagation, and vice versa. Similarly, a positive (nega-tive) value of m indicates upward (downward) phasepropagation. With this sign convention, k & 0 for equa-torially trapped Kelvin waves. Because the Kelvin wavein question is excited from below, the vertical compo-nent of the group velocity must be positive. This con-straint requires that m ' 0, which results in the negativeroot in (1) being the only valid dispersion relation (Hol-ton 1992; Andrews et al. 1987). It follows then that

c ( U # 0, #2$

for Kelvin waves, where c is phase speed.

To examine how the Kelvin wave amplitude modu-lation depends upon the zonal and vertical variation ofthe background state, we consider the wave action con-servation law (Lighthill 1978). For equatorially trappedwaves (Andrews and McIntyre 1976), wave action con-servation takes the form of

$A$T

)$

$X#ACgx$ )

$

$Z#ACgz$ ! 0, #3$

following a ray, which is the path parallel to the localgroup velocity, Cgx(X, Z, T) and Cgz(X, Z, T), where Tand (X, Z) are the slowly varying time and spatial scales,as required by the Wentzel–Kramers–Brillouin (WKB)approximation T ! *(1(2+/%), X ! *(1(2+/k), and Z !*(1(2+/m), where * is a small parameter (Andrews et al.1987). For the wave field in question,

Cgx#X, Z, T $ ! U#X, Z$ ( N#X, Z$"m#X, Z, T $, #4$

Cgz#X, Z, T $ ! N#X, Z$k#X, Z, T $"m2#X, Z, T $, #5$

and the wave action A ! ,E -%(1r , where %r is the rela-

tive frequency and ,E- is the meridionally integratedwave energy density (in units of wave energy per area),that is,

,E- ! !(%

% &o

2 #u2 ) N(2'z2$ dy, #6$

in log-pressure coordinates (Andrews et al. 1987).Therefore, the use of (3), the wave action conservationlaw, allows us to determine how ,E-, which we use as ameasure of the Kelvin wave amplitude, depends uponthe background state. In (6), u is the zonal wind and .is the geopotential associated with the wave, /o is thebasic-state density, and .z ! RH(1T, where R is the gasconstant and H the density-scale height. In practice, themeridional integral in (6) should be taken over finitelimits. Given that the Kelvin wave decays exponentiallyin the meridional direction, a natural choice for theselimits is the e-folding distance (separately determinedfor the Northern and Southern Hemispheres) from theequator. In calculating (6), we used the u e-folding dis-tance at each longitude for both hemispheres and foreach model day. The resulting values were averagedfrom day 7 to 17, and from 17 to 25 km. It turned outthat zonal variation in the e-folding distance was small.As a result, the temperature and wind fields, integratedover this zonally varying e-folding distance (notshown), were found to be very similar to those obtainedby integrating from 10°S to 10°N, as in Figs. 7 and 9.

In principle, given U and N, and the initial values forA, k, and m, A(X, Z, T ) can be solved by integrating (3)and the following two equations: dgk(X, Z, T )/dt !(0%(X, Z, T )/0X, dgm(X, Z, T )/dt ! (0%(X, Z, T )/0Z,

JUNE 2008 R Y U E T A L . 1831

where dg /dt denotes the rate of change following thelocal group velocity (Andrews et al. 1987). However, togain physical insight into how A (and therefore !E") isinfluenced by U(X, Z) and N(X, Z), we take an alterna-tive approach and consider an approximate wave con-servation equation for which there is an analytical ex-pression for A. To do so, we assume that the fractionalvariation of k and m with respect to U and N is muchless than 1, and therefore that k and m can be treated asconstants. The fully developed stratospheric Kelvinwave structure, shown in the right panel of Fig. 7, indi-cates that this is a reasonable approximation, because kand m do not vary by more than a factor of 2, whilevariations in both U and N (U is shown in Fig. 11, andN can be inferred from Fig. 12a) are much greater.

To determine which term in (3) dominates, we cal-culate the slope of the ray [see (4) and (5)],

S #Cgz

Cgx$

N%x, z&ko !mo2

U%x, z& ' N%x, z&!mo, %7&

where x and z are used in place of X and Z, respectively,to indicate that observed values of U and N are used,and the subscript o emphasizes that k and m are treatedas constants.

Figure 12 shows an estimation of Cgz and Cgx for theKelvin wave and the slope of the ray S at 3.35°N. Forregions where |Cgx | " 10'3 m s'1, contours of S are notdrawn. This latitude is chosen because the largest valueof S, excluding the regions where |Cgx | " 10'3 m s'1, is

found at 3.35°N. The values of ko and mo used for theestimation in Fig. 12 are 1.57 ( 10'7 m'1 and '6.28 (10'4 m'1, respectively, corresponding to a zonal wave-number 1 Kelvin wave with a vertical scale of 10 km.The structure of Cgz largely reflects the static stability,which has both large values just above the tropopauseand little zonal variation (Fig. 12a). In contrast, thezonal variation of Cgx is much greater, with large valuesover the eastern Pacific where U ) 0 and small valuesover the western Pacific where U * 0 (cf. Fig. 12b withFig. 11). As a result, the zonal variation of the slope Sis determined mostly by Cgx, with the steepest slopeoccurring over the western Pacific (Fig. 12c). Except inthe region where the value of U approaches '10 m s'1,the slope S ranges from 10'4 to 10'3 (Fig. 12c).

a. Horizontal structure

For a conservative steady wave, the first term in (3)is zero and, to determine whether the primary balanceis achieved by the horizontal flux divergence term,Cgx(+A/+X) , A(+Cgx /+X), or by the vertical flux diver-gence term, Cgz(+A/+Z) , A(+Cgz /+Z), we perform thefollowing scaling analysis. By assuming that the hori-zontal scale of A and Cgx are equal, we write the hori-zontal scale L such that L $ | (1/A)(+A/+X) |'1 $ | (1/Cgx)(+Cgx /+X) |'1. Making a parallel assumption for thevertical scale, we write the vertical scale D $ | (1/A)(+A/+Z) | '1 $ | (1/Cgz)(+Cgz /+Z) | '1. It follows thatO[Cgx(+A/+X)] $ O[A(+Cgx /+X)] $ ACgx /L, and

FIG. 11. The DJF climatological zonal wind, averaged over the latitudes between 10°S and 10°N. Thecontour interval is 5 m s'1. (right) The zonal mean of the DJF climatological zonal wind, averagedbetween 10°S and 10°N.


O[Cgz(!A/!Z)] " O[A(!Cgz /!Z)] " ACgz /D. There-fore, if

Cgz

Cgx

LD

K 1, #8$

the primary balance in the wave action equation is be-tween Cgx(!A/!X) and A(!Cgx /!X), and (3) can be re-duced to

CgxA % constant along x. #9$

Figure 12 indicates that the inequality (8) is valid forthe model simulations, except for the region between

110° and 165°E, and between &10 and &17 km; else-where in the domain, O(Cgz /Cgx) % 3 ' 10(4 (see Fig.12c), O(L) % 1 ' 107 m (Fig. 12b), and O(D) % 1 ' 104

m, yielding O[(Cgz /Cgx)(L/D)] % 3 ' 10(1. Thus, ex-cept for the limited region over the western Pacific (theregion not contoured in Fig. 12c),

CgxA " #U ( N!mo$)(mo!#Nko$*E

" cko( 1#c ( U$(1E % constant along x. #10$

This relation reveals that if U + 0, (c ( U)(1 must besmall and positive [see (2)], and therefore the value ofE must be large. Likewise, E must take on a smallervalue in regions where 0 + U + c.

If one relaxes the assumption of k being constant andallows k to evolve following the ray-tracing equationdgk(X, Z, T )/dt " (!,(X, Z, T )/!X under condition (8),it can be shown that

k2#X$ " k2#Xr$!U#Xr$ (Nm

#Xr$

U#X$ (Nm

#X$ " ,

where Xr is a reference longitude. Provided that theratio N/m does not vary with X, this relation implies thatk(X) is small (large) in the eastern (western) Pacificwhere U(X) is positive (negative). Because (10) indi-cates that small (large) k makes E even smaller (larger),allowing k to vary in X further strengthens the conclu-sion that E amplifies over the western Pacific whereU + 0.

Because the basic-state density -o and the buoyancyfrequency N vary by a small amount in the zonal direc-tion, variation in E mostly reflects variation in |u | andin |.z | (or in |T | from the relation .z " RH(1T).Therefore, to the extent that (10) can be applicable,assuming that the partition between wave kinetic en-ergy and wave available potential energy is constant,both the temperature and the zonal wind fields of thewave are expected to be proportional to (c ( U)1/2 inthe zonal direction. This prediction is indeed consistentwith the simulations. In the western1 (eastern) Pacificwhere U + 0 (U / 0), both temperature and windperturbations are relatively large (small). This pre-dicted amplification of the Kelvin wave in the wester-lies is also consistent with the observational findings ofNishi and Sumi (1995) and Nishi et al. (2007). Theyfound that because of the Wallace–Kousky Kelvinwave, the zonal wind near the tropical tropopause un-

1 Recall, however, that this analysis is invalid for the regionbetween 110° and 165°E.

FIG. 12. (a) The vertical Kelvin wave group velocity (Cgz,m s(1), (b) the zonal Kelvin wave group velocity (Cgx), and (c) theslope of the ray (S) at 3.35°N; Cgz and Cgx are evaluated assumingthat k and m take on constant values. The values for S are notindicated in regions where |Cgx | " 10(3 m s(1.

JUNE 2008 R Y U E T A L . 1833

dergoes rapid fluctuations, and the magnitude of thiszonal wind fluctuation is greatest in the Eastern Hemi-sphere where the climatological zonal wind is easterly.

b. Vertical structure

In a manner parallel to the previous subsection, inregions where the rays are sufficiently steep so that

Cgz

Cgx

LD

k 1, !11"

we may approximate the wave action conservation lawas

CgzA # constant along z.

Between 110° and 165°E and between 10 and 17 km,Fig. 12c indicates that O(Cgz /Cgx) ! 10$1. Therefore,for O(L) # 1 % 107 m and O(D) # 1 % 104 m, we findthat O[(Cgz /Cgx)(L/D)] ! 102, satisfying the inequalitycondition for the above equation. Neglecting the wavekinetic energy, the conservation law can be further ap-proximated as

Cgz"r$1#oN$2$z

2 # constant along z, !12"

where &oN$2'2z is the wave potential energy per unit

volume. As mentioned in the previous subsection, be-cause 'z is proportional to T, we treat 'z as tempera-ture. Substituting Nko /m2

o for Cgz, $mo/(Nko) for ($1r ,

and using the dispersion relation c $ U ) $N/mo, theleft hand side of (12) can be rewritten as (c $U)&oN$3'2

z. Thus, (12) implies that

|$z!z" |

) #o$ 1%2!z"#o

1%2!zr"!c $ U!zr"

c $ U!z""1%2!N!z"

N!zr""3%2

|$z!zr" |

) e!z$zr"%2H! c $ U!z"

c $ U!zr""$1%2!N!z"

N!zr""3%2

|$z!zr" | , !13"

with the understanding that c $ u(z) * 0 [cf. (2)], withzr being a reference height. The relation (l3) predictsthat away from the region of the forcing, in the absenceof dissipation, and within a depth of less than twice thedensity-scale height, the Kelvin wave temperature ismodulated by the background buoyancy frequencyN(z) to the power of 3/2 and by the Doppler-shiftedphase speed c $ U(z) to the power of $1⁄2. Because Nrapidly increases above the tropopause and reaches alocal maximum near 20 km, and because the tempera-ture depends on N to a higher power than on c $ U(z),Kelvin wave temperature fluctuations are expected tobe greatest in the vicinity of #20 km, and to dependweakly on the background wind speed U(z). Figures13a–c display the height dependency of the three fac-

tors on the right-hand side of (13): e(z$zr)/2H, {[c $U(z)]/[c $ U(zr)]}$1/2, and [N(z)/N(zr)]3/2. To be con-sistent with Fig. 12, climatological data at 3.35°N areused for this evaluation. The value for H is set equal to8 km, and the reference height zr is taken to be 12 km.The perturbation fields shown in Figs. 7 and 9 suggestthat the direct impact of the imposed heating on thetemperature field is very weak above 12 km. As ex-pected, Fig. 13 shows that the impact of the back-ground-state static stability is the most pronounced ofthe three factors. Therefore, the combination of thesethree factors, shown in Fig. 13d, closely follows the pro-file in Fig. 13c. Above 20 km, the rate of increase in|'z(z) | gradually declines. This profile shows qualita-tive agreement with the simulated temperature profileover the western Pacific (e.g., see day 9 in Fig. 7) where(12) is valid. This finding suggests that the large Kelvinwave temperature perturbation in the LS can be attrib-uted to the strong static stability in the LS region.

If m is allowed to vary as governed by dgm(X, Z, T )/dt ) $+((X, Z, T )/+Z, with the following three condi-tions: (i) the inequality represented by (11), where Land D are now the horizontal and vertical scales overwhich m varies; (ii) O[(1/k)(+k/+z)] K O{[1/(U $N/m)][+(U $ N/m)/+z]}; and (iii) O(+U/+z) K O{[+(N/m)/+z]}, one arrives at the relation

m2!Z" ) m2!Zr"N!Z"%N!Zr".

The factor {[c $ U(z)]/[c $ U(zr)]}1/2[N(z)/N(zr)]3/2 in(13) becomes [N(z)/N(zr)]5/4, using (1) and the aboverelation for m with Z being switched to z. Thus, thewave energy now depends on N to the power of 5/4 ratherthan 3/2, but still becomes larger with increasing N.

Above 25 km, the density factor becomes increas-ingly important. However, the time scale associatedwith radiative damping becomes comparable to thetime that the Kelvin wave packet takes to reach thislevel. Because the mean vertical group velocity be-tween 15 and 25 km is about 7.0 % 10$3 m s$1 (Fig.12a), it takes approximately 17 days for a Kelvin wavepacket to propagate from 15 to 25 km. Given that theradiative damping time scale is about 5 days in the LS(Jin and Hoskins 1995; Taguchi 2003), one expects thatwave attenuation resulting from radiative damping willbe significant as the wave packet propagates vertically.Because regions higher than 25 km are beyond thescope of this study, these processes are not further con-sidered.

6. Conclusions and discussion

Both regression analyses with global observationaldatasets and numerical model simulations show that


tropical tropopause undulation is a robust response toconvection over the western tropical Pacific. In re-sponse to the convective heating, there is warming inthe upper troposphere (UT) and cooling in lowerstratosphere (LS). With the exception of one region,immediately to the west of the heating, this UT warm-ing and LS cooling occur over most longitudes, andresults in a lifting of the tropical tropopause. The modelresults show that these UT–LS temperature anomaliesare closely associated with vertically propagating inter-nal Kelvin waves. As the phase of this internal Kelvinwaves progresses downward, the tropopause undulates.The tropopause rises (descends) when cold (warm)Kelvin wave perturbations position themselves imme-diately above the tropopause.

Based on an analysis of wave action conservation, wesuggest that Kelvin waves play a prominent role for thetropical tropopause undulation because temperatureperturbations associated with the Kelvin wave attainlarge values in the vicinity of the tropopause where thestatic stability is strong. The background zonal windmust also modulate the amplitude of the temperatureperturbation, but the amplitude dependency on staticstability is much greater.

The model simulations also show that the LS Kelvinwave response is greatest over the western Pacific, re-gardless of the location of the heating. This result indi-cates that the LS Kelvin wave attains its largest ampli-tude over the western Pacific because of wave modula-tion by the zonal wind of the background state, and notbecause of the proximity to the heating. Thus, the effectof the heating on the LS Kelvin wave amplitude is re-alized indirectly through the impact of the heating onthe background zonal wind field. This wave modulationin the zonal direction is again consistent with the waveaction conservation law, which predicts that, followinga zonal ray, the Kelvin wave energy must be propor-tional to c ! U. Because the strongest easterlies arefound over the western Pacific, the highest wave energymust occur in this region.

There are two nontrivial consequences that arisefrom the above vertical and zonal modulation of theKelvin wave: First, a comparison between simulationswith the DJF climatological flow and a zonal average ofthis state indicates that the Kelvin wave modulation inthe zonal direction causes a greater amount of zonalmean tropopause displacement. Second, the physicallink between the tropical convection, vertically propa-

FIG. 13. The vertical distribution of the three factors on the right-hand side of Eq. (13) taken separatelyand together: (a) e(z!zr)/2H, (b) {[c ! U(z)]/[c ! U(zr)]}!1/2, (c) {[N(z)/N(zr)]}3/2, and (d) e(z!zr)/2H{[c !U(z)/]/[c ! U(zr)]}!1/2[N(z)/N(zr)]3/2, averaged from 110° and 165°E.

JUNE 2008 R Y U E T A L . 1835

gating Kelvin waves, and tropopause height has impli-cations for the long-term cooling trend of the tropicalcold-point tropopause identified by Zhou et al. (2001).These authors provided evidence that the cooling trendis associated with rising sea surface temperature andincreasingly frequent and/or stronger convection. Theysuggested that the tropical tropopause cooling may oc-cur through the mechanism of Highwood and Hoskins(1998) in which the cold tropopause over the westernPacific is caused by thermal adjustment to the Gill-typecirculation. In this picture, stronger and/or more nu-merous convective activity causes a stronger Gill-typeresponse, and thus increased cooling. The findings inthis study indicate that the cooling may not only arisethrough the mechanism of Highwood and Hoskins(1998), but also through the excitation of verticallypropagating Kelvin waves.

More importantly, these results may explain the for-mation of the cold UT–LS region over the western Pa-cific, known as the cold trap. As Randel and Wu (2005)pointed out, while the low temperature is also consis-tent with cooling caused by convective detrainment(Sherwood and Dessler 2001; Kuang and Bretherton2004), Kelvin waves can also account for the cooling.Tsuda et al. (1994) estimate that the cooling associatedwith Kelvin waves over the western Pacific may be suf-ficient to bring the LS air temperature very close to thevalue necessary for freeze-drying in the cold trap. Thewave action analysis presented in this paper indicatesthat the large Kelvin wave response over the westernPacific can be attributed to the ambient easterlies inthat region and to the large static stability in the UT–LS. These results collectively suggest that the existenceof the cold trap may be dependent upon Kelvin wavemodulation, caused by zonal variation in the back-ground zonal wind and vertical variation in the back-ground static stability.

Acknowledgments. SL was supported by the NationalScience Foundation under Grants ATM-0324908 andATM-0647776. We acknowledge Drs. George Kiladisand Steven Feldstein for their comments on the manu-script, which vastly improved the content and presen-tation. Dr. Takeshi Horinouchi was instrumental in im-proving the analysis presented in section 5. We alsoacknowledge Dr. William Randel and one anonymousreviewer for helpful comments. We thank the ClimateAnalysis Branch of the NOAA/Earth System ResearchLaboratory/Physical Sciences Division (formerly theNOAA Climate Diagnostics Center) for providing uswith the NCEP–NCAR reanalysis dataset. The multi-level primitive-equation model used in this study was

originally provided by Dr. Isaac Held of the NOAA/Geophysical Fluid Dynamics Laboratory.

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