1

Observed Structure of Convectively Coupled Waves as a Function of Equivalent Depth:

Kelvin Waves and the Madden Julian Oscillation

Paul E. Roundy1

University at AlbanyState University of New York

1 Corresponding author address: Paul Roundy, Department of Atmospheric and Environmental Sciences, 1400 Washington Ave., Albany, NY, 12222.E-mail: roundy@atmos.albany.edu

1

2

3

4

5

6

7

8

9

10

1112

1234

2

Abstract

The view that convectively coupled Kelvin waves and the Madden Julian oscillation are

distinct modes is tested by regressing data from the Climate Forecast System Reanalysis

against satellite outgoing longwave radiation data filtered for particular zonal wave

numbers and frequencies by wavelet analysis. Results confirm that nearly dry Kelvin

waves have horizontal structures consistent with their equatorial beta plane shallow water

theory counterparts, with westerly winds collocated with the lower tropospheric ridge,

while the MJO and signals along Kelvin wave dispersion curves at low shallow water

model equivalent depths are characterized by geopotential troughs extending westward

from the region of lower tropospheric easterly wind anomalies through the region of

lower tropospheric westerly winds collocated with deep convection. Results show that as

equivalent depth decreases from that of the dry waves (concomitant with intensification

of the associated convection), the ridge in the westerlies and the trough in the easterlies

shift westward. The analysis therefore demonstrates a continuous field of intermediate

structures between the two extremes, suggesting that Kelvin waves and the MJO are not

dynamically distinct modes. Instead, signals consistent with Kelvin waves become more

consistent with the MJO as the associated convection intensifies. This result depends

little on zonal scale. Further analysis also shows how activity in synoptic scale Kelvin

waves characterized by particular phase speeds evolves with the planetary scale MJO.

131415

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

3

1. Introduction

The tropical atmosphere organizes moist deep convection over a broad range of spatial

and temporal scales. The Maddan-Julian oscillation (MJO) dominates variability in

convection on intraseasonal timescales of roughly 30-100 days (Madden and Julian 1994;

Zhang 2005). Rainfall associated with the local active convective phase of the MJO

(hereafter, active MJO) is in turn organized into smaller scale wave modes and mesoscale

convective systems. Convectively coupled Kelvin waves are widely recognized as a

leading signal among the population of modes that comprise the sub scale anatomy of the

MJO. These waves produce the highest amplitude signals in outgoing longwave radiation

(OLR) data near the equator (Wheeler and Kiladis 1999 (hereafter WK99); Straub and

Kiladis 2002; Roundy 2008). MacRitchie and Roundy (2012) showed that roughly 62%

of rainfall that occurs in the negative OLR anomalies of the MJO between 10N and 10S

over the Indo-Pacific warm pool regions occurs within the negative OLR anomalies of

the Kelvin wave band (after excluding those negative anomalies that do not enclose

signals less than -0.75 standard deviation). That result represents nearly twice the average

rainfall rate per unit area outside of the Kelvin waves but still within the active MJO.

MacRitchie and Roundy also showed that potential vorticity (PV) accumulates in the

lower to middle troposphere in wakes along and behind the Kelvin wave convection on

its poleward sides, and that this PV remains in the environment for longer than the period

of the Kelvin waves. The enhanced PV spreads pole ward behind the waves, and it

becomes part of the rotational structure of the MJO itself. Another portion of the

rotational response to convection coupled to Kelvin waves propagates eastward with the

waves, yielding low-level cyclones poleward of the equatorial convection (Roundy

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

4

2008). The response to deep convection moving eastward with convectively coupled

Kelvin waves makes them similar in many respects to the geographically larger MJO. On

the other hand, these patterns distinguish observed convectively coupled Kelvin waves

from theoretical Kelvin waves of Matsuno (1966) and Lindzen (1967), which do not

include meridional circulation. Nevertheless, many authors acknowledge that Kelvin

wave dynamics dominate their evolution because of their dispersion characteristics and

because of the relationship between wind and pressure observed in the lower stratosphere

away from the deep convection, which consistently shows westerly wind in the ridge and

easterly wind in the trough, with little meridional circulation. Although the MJO clearly

modulates Kelvin wave activity, amplitudes, and propagation speeds (Straub and Kiladis

2003; Roundy 2008), these waves occur independent of the MJO.

Although several authors during the 1980s and 1990s suggested that the MJO

itself might be a modified moist Kelvin mode (e.g., Lau and Peng 1987; Wang 1988; Cho

et al. 1994), the idea has since fallen out of favor for several reasons. First, the

relationship between zonal wind and pressure anomalies in the MJO appears to be

reversed or dramatically offset from that of Kelvin waves, with westerly wind anomalies

frequently appearing in the pressure trough collocated with the deep convection (e.g.,

Madden and Julian 1994; Zhang 2005). Second, a spectral peak associated with

convectively coupled Kelvin waves appears to be distinct from that of the MJO (Kiladis

et al. 2009), suggesting that the two have phase speed distributions that might not

overlap. Third, zonal wave number frequency spectra of OLR data suggest that the

spectral peak of the MJO extends across a broader range of wave numbers at a given

frequency than does the spectral peak associated with the Kelvin waves, giving the

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

5

impression of a flat dispersion relationship, even though most of that signature can be

explained by geographical variation in MJO propagation rather than true dispersion. This

perspective is supported by composite MJO events plotted in the longitude-time lag

domain (such as by Hendon and Salby 1994), which show structures favoring wave

number 2 over the warm pool (consistent with opposite signed anomalies of convection

over the Indian and western Pacific basins) and a half wave number 1 across the western

hemisphere. Such half wave number 1 signals project more onto wave number 1 than any

other wave number, as shown by a simple application of the Fourier transform in space

and time to a perfect eastward-propagating wave number 1 sine wave that is set to zero in

one hemisphere and left alone in the other (a synthetic half wave number 1 pattern). Such

geographical variations in MJO propagation must project onto different portions of the

spectrum. Seasonal variations in MJO propagation must also project onto different

portions of the spectrum. A global wave number-frequency spectrum analysis

conglomerates all of these varying signals together, such that a spectral peak aligned in a

particular pattern does not necessarily imply wave dispersion.

A more careful look at each of these characteristics casts some doubt on the

assertion that the MJO and Kelvin waves are distinct. First, the algorithm of WK99

would artificially enhance the extent of the spectral gap between the MJO and Kelvin

peaks. WK99 normalized their OLR spectra by dividing by a smoothed background

spectrum. This background spectrum was obtained by smoothing the original spectrum

by an arbitrary number of repeated applications of a 1-2-1 filter in frequency and in wave

number. This approach conserves the total power in the spectrum but redistributes power

in the MJO peak into its surrounding neighborhood, including the region of the spectral

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

6

gap. This artificial increase in background power would reduce the normalized power

there, making the MJO and Kelvin peaks appear better separated. For reference, Fig. 1

shows a wave number frequency spectrum of OLR calculated in a similar manner. The

more objective spectrum analysis of Hendon and Wheeler (2007) confirms the presence

of a local minimum in power in the spectrum, but not as pronounced as suggested by

WK99.

Recently, Roundy (2012) integrated wavelet power in the zonal wave number

frequency domain over geographical regions where the 100-day low pass filtered 850 hPa

zonal winds are easterly or westerly. He found that the gap in the global OLR spectrum

derives entirely from regions of easterly low-level background flow. The spectrum

integrated over regions low-level westerly winds has power decline smoothly from the

maximum in the MJO band, with no evident spectral gap. Thus the source of the spectral

gap is not over warm pool zones where MJO convective signals attain their greatest

amplitude. The lowest rate of decline of power occurs along Kelvin wave dispersion

solutions between equivalent depths of 5 and 8m. This result suggests that Kelvin waves

also propagate eastward more slowly over the warm pool than over trade wind zones.

Signals in trade wind zones dominate the global spectrum because these zones occur over

more of the global tropics for more of the time than do signals in warm pool westerly

wind zones, even though the individual events over the warm pool zones average higher

in amplitude.

Equatorial beta plane theories suggest that Kelvin waves are non dispersive

except at the shortest wavelengths (e.g., Roundy and Janiga 2012), but variation in

coupling between the waves and deep convection apparently results in a large range of

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

7

phase speeds across the full population of events (Roundy 2008). Although many of these

Kelvin waves propagate eastward at 15-17 ms-1 around much of the globe, or roughly

twice the phase speed of the MJO, Roundy (2008 and 2012) showed that they tend to

propagate more slowly over the warm pool zones. He also showed that synoptic scale

Kelvin waves propagate eastward even more slowly as they move through the local

active convective phase of the MJO. The same Kelvin wave disturbance can

circumnavigate the entire globe, with its phase speed changing continuously with the

amplitude of the associated convective signal. This observed variation in phase speeds

leaves open the possibility that long Kelvin waves and the MJO may have overlapping

dynamics because their phase speed distributions might overlap. These results

demonstrate that the spectral characteristics of the MJO and the Kelvin waves are not as

distinguishable as previously thought.

The relationship between zonal wind and pressure remains a factor whereby

Kelvin waves and the MJO might be distinguishable. Since the pressure wind relationship

differs between dry Kelvin waves and the observed MJO, and since the observed

distribution of frequencies associated with Kelvin waves are higher than the comparable

distribution for the MJO, the pressure wind relationship associated with eastward-moving

signals in OLR data must vary with frequency. If the prevailing view that Kelvin waves

and the MJO are distinct modes is correct, the presence of both modes would yield a

particular pattern of transition in frequency between the spatial patterns associated with

Kelvin waves and those associated with the MJO. At low wave number, the spectral

peaks of Kelvin waves and the MJO are proximate to each other. Since the spectral

characteristics of both the MJO and Kelvin waves vary substantially from event to event,

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

8

proximity of the two peaks suggests that there must be overlap between the spectra of the

two phenomena. If the MJO and Kelvin waves represent distinct modes in which the

pressure-wind relationship is not a function of frequency (consistent with the prevailing

view), then at some point in spectrum between the peaks of the two modes, both signals

would be present and explain roughly the same amount of variance in geopotential height

anomalies. Both modes would have low-level westerly wind anomalies collocated with

negative OLR anomalies, but the associated geopotential height anomalies are strongly

offset or opposite. Thus, active convective anomalies at that frequency would be

associated with negative geopotential height anomalies with one mode and positive with

the other mode. Statistical analysis to extract the average coherent pattern associated with

the active convection at that frequency without distinguishing between the modes would

thus yield significant lower tropospheric westerly wind anomalies associated with active

convection but no significant geopotential anomalies because the two opposite signals

would wash each other out. If, however, only one dominant coherent mode exists, with

structure modulated by the intensity of the associated rainfall, then the phase relationship

between wind and geopotential might shift as a continuous function of frequency, with no

geopotential amplitude minimum associated with signals at frequencies between the two

extremes.

Statistical analysis of observations and reanalysis data might shed light on the

nature of the transition of spatial structures as a function of frequency between the

spectral peaks that we associate with Kelvin waves and the MJO. Recently, Roundy and

Janiga (2012) applied zonal wave number-frequency wavelet analysis and simple linear

regression to assess the structure of convectively coupled mixed Rossby gravity (MRG)

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

9

waves characterized by specific zonal wave numbers and frequencies. They applied

wavelet analysis at a particular zonal wave number and a specified frequency to generate

a time index of the corresponding signals. Regression of fields of data against that index

reveals the space-time structures of the patterns corresponding to those signals. By

choosing frequencies consistent with a selected equivalent depth at several different

individual wave numbers, they showed how MRG wave structures vary with wave

number along particular shallow water model dispersion curves. A similar analysis of

signals proximate to the Kelvin wave peak in the OLR spectrum might suggest how

Kelvin wave structures change with equivalent depth (h), or how structures of coherent

disturbances change between the Kelvin and MJO spectral peaks. The purpose of this

work is to apply this technique to better understand what observations suggest about the

extent to which the MJO and long convectively coupled Kelvin waves can be considered

independent phenomena and to enhance our understanding of interactions between short

Kelvin waves and the MJO.

2. Data

Daily-interpolated outgoing longwave radiation (OLR) data on a 2.5-degree grid are

applied as proxy for moist deep convection (Liebmann and Smith 1995). These OLR data

have been updated every few months since 1995 following the original algorithm. Daily

mean zonal and meridional wind, temperature, and geopotential height data are obtained

from the Climate Forecast System Reanalysis (Saha et al. 2010). The mean and first four

harmonics of the seasonal cycle are subtracted from the OLR, wind, and geopotential

height data to generate anomalies. All data are analyzed for the period January 1, 1979

through December 2009.

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

10

3. Methods

a. Space-time Wavelet Decomposition

Signals in atmospheric convection obtained from the regions of the zonal wave

number frequency domain near equatorial beta plane shallow water model solution

dispersion curves at h=25m are associated with spatial patterns similar in many respects

to those obtained from shallow water theory (Matsuno 1966; Lindzen 1967; WK99). For

reference, Fig. 1 shows dispersion lines of shallow water model Kelvin wave solutions at

h=8 and 90m superimposed on a normalized OLR spectrum. All subsequent

observational analyses of Kelvin wave signals in this study are reported for points in the

OLR spectrum along the Kelvin dispersion curves with h ranging from 5 to 90m. It is

important to point out that other signals and noise occur along the dispersion curves of

the shallow water model Kelvin wave solutions. Extratropical waves advected eastward

by westerly winds project substantially onto similar regions of the spectrum as Kelvin

waves. Such signals tend to be small over the warm pool zones and large in regions of

upper tropospheric westerly winds such as the eastern Pacific and Atlantic basins. In spite

of other signals, the OLR spectral peaks between the dispersion curves of Kelvin wave

solutions of h=5 and 90m include those of convectively coupled Kelvin waves (centered

on roughly 25m, but ranging from roughly 8 to 90m), and the MJO, which extends

roughly from wave numbers 0-9 eastward and periods of roughly 30-100 days. Keep in

mind, however, that these spectral peaks are not distinct when the spectrum is integrated

only over the low-level westerly wind zones of the warm pool (Roundy 2012). The

Kelvin wave dispersion curves intersect with the traditionally defined MJO spectral peak

at low wave number (Wheeler and Kiladis 1999).

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

11

Most previous observational analyses of Kelvin wave signals conglomerate structures

of a broad range of wavenumbers and frequencies through filtering in the wave number

frequency domain across broad bands (e.g., Wheeler et al. 2000; Roundy and Frank

2004). Interpretation of the results is complicated because the vertical and meridional

structures of the observed waves might vary with wavenumber and frequency. This

project applies zonal wave number-frequency wavelet analysis to extract signals from

OLR data at specified wave numbers and frequencies, following Roundy and Janiga

(2012). When combined with regression or composite analysis, this more specific

approach diagnoses how spatial structures change with frequency at a given wave

number. A detailed description of space-time wavelet analysis is beyond the scope of this

paper, but Kikuchi and Wang (2010) and Wong (2009) offer overviews of the technique.

The space-time wavelet transform is the wavelet transform in longitude of the wavelet

transform in time of the OLR anomalies. This analysis applies the Morlet wavelet

Ψ (s )= 1√ (πB )

exp (iσs )exp (−(s2 )B ), (1)

where s represents x or t for the spatial or temporal transforms, respectively, and

represents angular frequency or wavenumber k. B, the bandwidth parameter, was

assigned a value of 4( ν2π )

−3/2

for the temporal transform and 1.5( k2π )

−3/2

for the temporal

transform. Conclusions are not sensitive to these arbitrarily assigned values of B, but

much larger values reduce the amplitude contrast in time of signals and enhance a ringing

effect, and substantially smaller values do not sufficiently resolve large-scale or low

frequency waves. The transform is obtained by taking the time-centered dot product of

the wavelet and all daily consecutive overlapping time series segments at each grid point

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

12

around the globe, then applying a similar transform in longitude to the result by the same

approach. The transform for a selected wave number and frequency is calculated for

every day and at every longitude grid point in the 7.5S to 7.5N averaged OLR anomaly

data from 1975 to 2009. Averaging OLR over 7.5S to 7.5N increases the likelihood that

the dominant coherent signals are Kelvin waves because this average acts as a filter for

cross equatorial symmetry, and proximity to the equator reduces the net contribution of

extratropical waves.

b. Linear Regression Models

Simple linear regression is frequently applied to diagnose structures that are coherent

with filtered signals (e.g., Hendon and Salby 1994; Wheeler et al. 2000). In this analysis,

the space-time wavelet transform at a selected longitude, wave number, and frequency,

becomes a base index time series for regression models at each grid point over a range of

longitudes and pressure levels. Either the real or imaginary parts of the transform work

for this index. The imaginary part produces convenient zonal phasing in the regression

maps, but the conclusions are the same regardless of this choice. Base longitudes are at

each 2.5 grid point from 60E to 90E. This focus on the Indian basin reduces the

contribution of extratropical features that are much more pronounced over the western

hemisphere. Calculating regression models at each base point, then averaging over all of

them reduces local disconformities, yielding conclusions less sensitive to geography. The

time series from each of those points serve as predictors in regression models at each grid

point across a broad geographical domain to diagnose the associated structures. One grid

of regression models is calculated for each base point time series. To illustrate, the

algorithm models the variable Y at the grid point S as

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

13

Ys Px As , (2)

where Px is a matrix whose first column is a list of ones and second column is the base

index at the longitude grid point x. As is a vector of regression coefficients at the grid

point S. After solving for As at each grid point by matrix inversion, (2) is then applied as a

scalar equation to diagnose wave structure by substituting a single value for the second

column of

Px that is representative of a crest of a wave located at the base longitude (its

value is set here at +1 standard deviation). These regression models are applied to create

‘composite’ anomalies of OLR, u and v winds, and geopotential height. Results are

calculated for the region 180 to the east and west of each base longitude, and then the set

of results from all base points are averaged, following Roundy and MacRitchie (2012).

The statistical significance of the difference from zero of the result at each point on the

map is assessed based on the correlation coefficient (e.g., Wilks 2011), and I analyze and

discuss only those regressed signals that are deemed to be significantly different from

zero at the 90% level. This significance test is completed for each individual regression

map before averaging over results from each base point, so that the number of degrees of

freedom is not inflated by inclusion of the same wave events at multiple grid points.

Since the regression is accomplished in the time domain, some signal from wave numbers

other than the target wave number can appear in results if they tend to occur together in a

particular pattern (Wheeler et al. 2000). Such regression results will be most reliable

close to the centers of the composites because spreading will occur due to the episodic

nature of convection and variations in the background state. Amplitudes of regressed

anomalies following this approach are much smaller than those of other authors who have

regressed signals against OLR data filtered for the broader Kelvin band of Wheeler and

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

14

Kiladis (1999) because the full band includes much more variance than is retained at just

one wave number and frequency.

The above approach diagnoses the spatial structures associated with signals along

the dispersion curves of shallow water model Kelvin wave solutions at particular

equivalent depths. For reference, we also calculate a composite MJO following a similar

approach, by replacing the base index time series with MJO band pass filtered OLR

anomalies averaged from 10N to 10S. The MJO band signal is averaged over a broader

latitude band than are Kelvin signals in this work in order to capture the signals of some

MJO events that have OLR anomalies shifted farther into one hemisphere. The MJO band

is defined as wave numbers 0-9 eastward and periods of 30-100 days. The value of −1

standard deviation in the base index is then substituted to generate a map. Base points for

the MJO composite are selected from 70E to 110E in order to reduce contamination of

the western portion of the regression maps by Africa. Kelvin band signals at higher wave

numbers did not require such an eastward shift because the wavelengths assessed for

them yielded less contamination from Africa.

A variation on the above regression approach is also applied here to diagnose how

signals along the Kelvin wave dispersion curves at particular equivalent depths vary with

the phase of the MJO. An index of OLR anomaly data filtered for the wave number

frequency band of the MJO at 80E is assigned as P in equation (2) and applied to predict

Y, which is assigned to be the absolute value of the sum of the real and imaginary parts of

the wavelet transform at a selected wave number and frequency at a time lag. Averaging

regression maps over multiple base points is not applied for this analysis since the

associated geographical signals is the target outcome. The value of −1 standard deviation

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

15

is then substituted for the second column of P at each grid point and time lag. After

discarding the time local mean of the result, it shows how the timing of changes in

activity in Kelvin waves characterized by particular phase speeds varies with the MJO.

4. Results

a. Regression at Various Equivalent Depths Along Kelvin Dispersion Curves

Figure 2 shows regressed geopotential height anomalies (contours) and zonal

wind anomalies (shading, with westerly anomalies in red) for panels a-e, equivalent

depths of 90, 25, 12, 8, and 5m (respectively) for zonal wave number 4 Kelvin waves.

These data are plotted against regressed total geopotential height instead of pressure to

facilitate measurement of the vertical tilts of the regressed anomalies. Thus, the plotted

geopotential height anomalies represent the displacement of isobars at a given height

from their climatological positions. The results show patterns that tilt toward the west

with height between the surface of the earth and roughly 10,000m, with tilt reversing

toward the east above (consistent with Kelvin wave composites by Kiladis et al. 2009 and

references therein). Each panel shows eastward flow in the ridges and westward flow in

the troughs above 104m, but structures vary with equivalent depth below that level. At the

equivalent depth of 90m (panel a), westerly wind anomalies are collocated with positive

geopotential height anomalies near the center of the composite. Comparison of all panels

shows that the westerly wind anomalies near the centers of the composites are nearly an

order of magnitude stronger at h=5m (panel e) than at h=90m, but the as equivalent depth

decreases, the geopotential trough in the easterlies on the east side of the domain extends

westward until at h=5m it encompasses nearly all of the westerly anomalies near the

center of the composite below 10,000m. The differences between the composites for

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

16

large and small equivalent depths occur smoothly across the equivalent depths plotted

here. The regressed geopotential height anomalies do not vanish at some equivalent

depth, as would occur if the MJO and Kelvin band signals include two distinct modes

characterized by opposite pressure wind relationships. The vertical cross sections for the

other wave numbers are similar to those for k=4.

Figure 3 shows the horizontal maps of the regressed geopotential height and

winds for wave number 4 at 900 hPa along the Kelvin wave dispersion curves for the

same equivalent depths as in Fig. 2. Regressed OLR anomalies are shaded, with active

convection suggested in blue, and regressed geopotential height anomalies are contoured

with positive anomalies in red. At h=90m, westerly wind anomalies are collocated with

positive geopotential height anomalies and slightly negative OLR anomalies. Easterly

wind anomalies occur in the trough, consistent with the shallow water model Kelvin

wave. With increasing equivalent depth, the negative OLR anomalies strengthen in the

vicinity of the equatorial westerly wind anomalies. At the same time, locally positive

geopotential height anomalies shift westward from the active convection toward the

suppressed convection. Trough anomalies shift westward from east of the negative OLR

anomalies at h=90m (panel e) through the convective region by h=5m. The increased

amplitude of the OLR anomalies with decreasing equivalent depth suggests that lower

equivalent depths are associated with higher rainfall rates. TRMM 3B42 rain rate data

available since 1999 confirm this observation (not shown). The structure of the OLR and

geopotential height anomalies also changes with equivalent depth. The negative OLR

anomaly at h=90m nearly forms an ellipse centered on the equator, but at smaller

equivalent depths, the negative OLR and geopotential anomalies distort increasingly

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

17

westward with distance form the equator, forming boomerang patterns across the equator.

Regression maps for other wave numbers show similar structures (not shown). Thus, both

long and short Kelvin waves become more like the MJO with increasing precipitation

rates. This statement also holds true for Kelvin waves of wave number 6 and 8 (not

shown). Although the h=5m result at wave number 6 has a period of about 11 days, well

outside of the traditional MJO band of the wave number frequency domain, the

associated regression maps still show pronounced westward shifting of the geopotential

height anomalies relative to the OLR anomalies, along with pronounced westward

distortion with latitude. In other words, signals along the Kelvin wave dispersion curve at

h=5m and wave number 6 are associated with structures similar to those of the MJO, but

with smaller zonal scale. Within the traditional MJO band, structures observed along the

dispersion curve for the h=5m Kelvin wave at zonal wave number 2 also shows similar

traits. That signal propagates at about 7ms-1 and has a period of about 30 days.

b. Regressed MJO Structure

For comparison with Fig. 3, Fig. 4a shows a horizontal map of geopotential height

anomalies and zonal wind regressed against MJO-filtered OLR anomalies at 900 hPa.

These results show a geopotential trough collocated with easterly wind anomalies on the

eastern side of the domain. That trough also extends westward across much of the region

of low-level westerly winds collocated with the negative OLR anomaly. That

geopotential trough and the negative OLR anomalies form a triangle pattern with one side

perpendicular to and bisected by equator on the west and the two other legs meeting to

the east on the equator. This pattern is consistent with distortion of the OLR and

geopotential height anomalies westward with distance form the equator at low equivalent

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

18

depths in Fig. 3d and e. Figure 4b shows the corresponding vertical cross section of

regressed geopotential height anomaly and zonal wind anomaly on the equator, for

comparison with Fig. 2. The result compares well with Fig. 2d and 2e.

c. The Association Between Synoptic Kelvin Wave Activity and the MJO

Straub and Kiladis (2003) evaluated the evolution of signals in the broader Kelvin

wave band with the northern hemisphere summer MJO. The present work expands on

their analysis by demonstrating how that evolution depends on the phase speeds of the

Kelvin waves in a generalized MJO without explicit assessment of seasonality. Figure 5

shows regressed activity in Kelvin waves at zonal wave numbers 3-8 (shading) along

with regressed MJO-filtered OLR. Panels (a)-(e) represent results for equivalent depths of

90, 25, 12, 8, and 5m, respectively. Enhanced convection in the MJO band is indicated by

blue contours. Fast Kelvin waves (~30ms-1) at 90m equivalent depths (Fig. 5a) are

characterized by lower amplitude signals in OLR anomalies than all other equivalent

depths (consistent with the expectation that such Kelvin waves should be nearly dry).

Figure 5a suggests that prior to onset of convection in the MJO band over the Indian

basin (hereafter called “MJO initiation” for simplicity), fast Kelvin waves are prevalent

over the Atlantic basin and Africa, but quiet over the Pacific basin. This activity extends

eastward early in the lifetime of the negative OLR anomaly in the MJO band over the

Indian basin. This activity then declines to below average over the Indian basin after lag

= +5 days. Activity in these fast Kelvin waves then grows over the Pacific Ocean to the

east of the active MJO. Kelvin waves characterized by h=25m also show enhanced

activity in OLR anomalies over the Atlantic basin and Africa prior to MJO initiation, but

substantially more than for h=90m. After lag = 0, enhanced activity occurs at the eastern

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

19

edge of the negative OLR anomalies in the MJO band, a little farther west than for

h=90m. At h=12m, similar to at h=90m and h=25m, activity begins over the Atlantic

basin and Africa prior to MJO initiation, but the level of activity becomes much stronger

over the Indian basin within the active MJO and then extends only slightly eastward from

the negative OLR anomaly of the MJO after lag = +5 days. Although signal at h=8m and

h=5m is also suggested over the Atlantic basin and Africa leading up to the active MJO,

most of the signal in these bands concentrates within the negative OLR anomaly of the

MJO over both the Indian and western Pacific basins. These slow Kelvin wave signals

are more consistent with the slow eastward-moving supercloud clusters of the active

convective phase of the MJO noted by Nakazawa (1988) than are the faster Kelvin

waves. Figure 5a confirms the previous result of Kikuchi and Takayabu (2003) that dry

Kelvin waves radiate eastward from the active convective phase of the MJO over the

western Pacific basin, but Fig. 5 b-d also shows that a substantial convectively coupled

Kelvin wave signal at h=12m and h=25m (about 11 and 16 ms-1 respectively) also occurs

over the Pacific basin east of the active MJO. The slowest Kelvin waves at wave numbers

3-8 are largely confined to the active convective phase of the MJO over the Indo Pacific

warm pool. Although the local amplitudes of OLR anomalies at h=8m and 5m are

substantially higher than for OLR anomalies at 25m, the isolation of these low h signals

largely within active convective phases of the MJO over the warm pool reduces their net

contribution to the OLR spectrum, leading to the more global signals near 25m standing

out in the OLR spectrum. These results are especially interesting in the context of Figs. 2

and 3, which suggest that these synoptic scale Kelvin waves themselves have spatial

structures similar to those of the planetary scale MJO.

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

20

5. Conclusions

A wave number frequency wavelet analysis of OLR anomaly data and simple linear

regression reveal how the structures associated with signals along the dispersion curves

of Kelvin waves change with equivalent depth. Results suggest that the phase relationship

between geopoential height and wind anomalies for signals along Kelvin wave dispersion

curves adjusts continuously westward with decreasing equivalent depth from patterns

consistent with Kelvin waves of equatorial beta plane shallow water theory (which have

westerly wind anomalies in the geopotential ridge) to patterns that look more like the

MJO (with westerly wind anomalies extending westward through the geopotential

trough). If there were two distinct modes present with opposite pressure wind

relationships overlapping in the spectrum, with one mode dominant at low frequencies

and the other dominant at higher frequencies, then at some frequency in between the two,

the geopotential signals would wash out of the regression while regressed wind and OLR

signals would remain. Instead, the regression analysis reveals a continuous shift of the

phase between zonal wind and pressure signals. High wave number Kelvin waves whose

signals are far in the spectrum from the MJO band follow similar patterns at low

equivalent depths. These results thus do not support the perspective that the MJO and

Kelvin waves are distinct modes like the present consensus suggests. This continuous

evolution instead supports the perspective that more intense convection modifies the

convectively coupled Kelvin wave to take on characteristics more consistent with the

MJO. In that sense, the low wave number portion of the disturbance traditionally labeled

as the MJO might be a planetary scale Kelvin wave modified by the influence of intense

convection. Analysis of the power spectrum by Roundy (2012) further confirms that no

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

21

separation between the Kelvin and MJO spectral peaks occurs over the low-level westerly

wind zones over the warm pool. Thus in those regions, MJO signals cannot be

distinguished from a continuum of disturbances that begin at high frequencies in

association with dry Kelvin waves.

This work also demonstrates how synoptic scale Kelvin waves characterized by

particular phase speeds (or equivalent depths) vary with the MJO. Kelvin wave activity at

all phase speeds tends to be enhanced over the Atlantic basin and Africa prior to

development of deep convection in the MJO band over the Indian basin. Fast Kelvin

waves are also prevalent well to the east of MJO convection when that convection is

located over the western Pacific basin. The slowest Kelvin waves characterized by

equivalent depths of less than 12m are strongest within the active convective phase of the

MJO over the Indian basin, consistent with the assessment of the associated supercloud

clusters by Nakazawa (1988) and slow Kelvin waves by Roundy (2008). These slow

synoptic scale Kelvin waves themselves have vertical and horizontal structures similar to

those of the planetary scale MJO.

Acknowledgments.

Funding was provided by the National Science Foundation Grant# 1128779 to Paul

Roundy. The NOAA PSD provided OLR data, and the NOAA CPC provided CFS

reanalysis data.

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465466

467

468

22

References

Cho, H.-R., K. Fraedrich, and J. T. Wang, 1994: Cloud clusters, Kelvin wave CISK, and

the Madden-Julian Oscillations in the equatorial troposphere. J. Atmos. Sci., 51,

68-76.

Hendon, H. H., and M. L. Salby, 1994: The life cycle of the Madden-Julian Oscillation.

J. Atmos. Sci., 51, 2225-2237.

Hendon, H. H., and M. C. Wheeler, 2008: Some space-time spectral analyses of tropical

convection and planetary scale waves. J. Atmos. Sci. 65, 2936-2948.

Kikuchi, K., and Y. N. Takayabu, 2003: Equatorial circumnavigation of moisture signal

associated with the Madden-Julian oscillation (MJO) during Boreal winter. J.

Meteorol. Soc. Japan, 81, 851-869.

Kikuchi, K., and B. Wang, 2010: Spatiotemporal wavelet transform and the multiscale

behavior of the Madden Julian oscillation. J. Climate, 23, 3814-3834.

Kiladis G. N., M. C. Wheeler, P. T. Haertel, K. H. Straub, and P. E. Roundy, 2009:

Convectively coupled equatorial waves. Reviews of Geophysics. 47, RG2003,

doi:10.1029/2008RG000266.

Lau, K.-M., and L. Peng, 1987: Origin of low-frequency (intraseasonal) oscillations in

the tropical atmosphere: Part I, Basic theory. J. Atmos. Sci., 44, 950-972.

Liebmann, B., and C. Smith, 1996: Description of a complete (interpolated) outgoing

longwave radiation dataset. Bull. Amer. Meteor. Soc. 77 : 1275-1277.

Lindzen R. S., 1967: Planetary waves on beta-planes. Mon. Wea. Rev., 95, 441-451.

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

23

MacRitchie, K., and P. E. Roundy, 2012: Potential vorticity accumulation following

atmospheric Kelvin waves in the active convective region of the MJO. J. Atmos.

Sci., In Press.

Madden, R. and P. R. Julian, 1994: Observations of the 40-50-day tropical oscillation—A

review. Mon. Wea. Rev., 122, 814-837.

Matsuno T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc.

Japan, 44, 25-43.

Nakazawa, T., 1988: Tropical super clusters within intraseasonal variations over the

western Pacific. J. Meteor. Soc. Japan, 66, 823-839.

Roundy, P. E., 2008: Analysis of convectively coupled Kelvin waves in the Indian Ocean

MJO. J. Atmos. Sci., 65, 1342-1359.

Roundy, P. E., 2012: The spectrum of convectively coupled Kelvin waves and the

Madden Julian oscillation. Manuscript submitted to J. Atmos. Sci.

Roundy, P. E., and W. M. Frank 2004: A climatology of waves in the equatorial region.

J. Atmos. Sci. 61, 2105-2132.

Roundy, P. E., and M. Janiga, 2012: Analysis of vertically propagating convectively

coupled equatorial waves using observations and a non-hydrostatic Boussinesq

model on the equatorial beta plane. Q. J. Roy. Meteorol. Soc. In Press.

Saha, S. Nadiga, C. Thiaw, and J. Wang, W. Wang, Q. Zhang, and H. M. Van den Dool,

H.-L. Pan, S. Moorthi, D. Behringer, D. Stokes, M. Peña, S. Lord, and G. White,

W. Ebisuzaki, P. Peng, and P. Xie, 2006: The NCEP climate forecast system. J.

Climate, 19, 2483-3517.

Straub, Katherine H., George N. Kiladis, 2002: Observations of a Convectively Coupled

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

24

Kelvin Wave in the Eastern Pacific ITCZ. J. Atmos. Sci., 59, 30–53.

Straub, K. H., and G. N. Kiladis, 2003: Interactions between the Boreal summer

intraseasonal oscillation and higher-frequency tropical wave activity. Mon. Wea.

Rev., 131, 945-960.

Wang, B., 1988: Dynamics of tropical low frequency waves: An analysis of the moist

Kelvin wave. J. Atmos. Sci., 45, 2051-2065.

Wheeler M., and G. N. Kiladis, 1999: Convectively-coupled equatorial waves: Analysis

of clouds in the wavenumber-frequency domain. J. Atmos. Sci., 56, 374-399.

Wheeler M., G. N. Kiladis, and P. J. Webster, 2000: Large-scale dynamical fields

associated with convectively-coupled equatorial waves. J. Atmos. Sci., 57, 613-

640.

Wong, M. L. M., 2009: Wavelet analysis of the convectively coupled equatorial waves in

the wavenumber-frequency domain. J. Atmos. Sci., 66, 209-212.

Wilks, D. S., 2011: Statistical Methods in the Atmospheric Sciences. Third Edition.

International Geophysics Series. Elsevier Academic Press. Oxford, UK.

Zhang, C., 2005: The Madden-Julian Oscillation. Rev. Geophys. 43, RG2003,

doi:10.1029/2004RG000158.

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

25

List of Figures

Figure 1. Shallow water model dispersion curves for various equatorial wave modes

plotted on a spectrum of OLR anomalies. The spectrum was normalized by

dividing by a smoothed background spectrum.

Figure 2. Longitude-height cross sections of regressed zonal wind anomalies (shading,

ms-1) and geopotential height anomalies (contours, negative in blue, with an

interval of 0.25m) for signals along the Kelvin wave dispersion curves at zonal

wave number 4. Panels a-e represent results for equivalent depths of 90, 25, 12, 8,

and 5m, respectively. The vertical axis is labeled in terms of regressed total

geopotential height to facilitate measurement of vertical tilts. Positive longitude is

represented as degrees east of the base points.

Figure 3. Horizontal maps of regressed OLR anomalies (shading, Wm-2), geopotential

height anomalies (positive in red, contour interval 0.15m), and wind anomalies at

900 hPa for signals along the Kelvin wave dispersion solutions for zonal wave

number 4. Panels correspond to equivalent depths of 90, 25, 12, 8, and 5m,

corresponding to the same panels of Fig. 2.

Figure 4. a. Anomalies of 900 hPa wind (vectors), OLR (shading, with negative in blue),

and geopotential height (with negative anomalies in blue) regressed against OLR

anomalies filtered in the wave number frequency domain for the MJO. b. Vertical

cross section of zonal wind (shading) and geopotential height anomalies

(contours, with negative in blue) on the equator, plotted against regressed total

geopotential height.

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

26

Figure 5. Shading shows the result of regressing absolute value of OLR anomalies along

the Kelvin wave dispersion curves for zonal wave numbers 3-8 against MJO-

filtered OLR anomalies at 80°E (Wm-2). The local mean is subtracted at each grid

point. The shading thus provides a measure of how Kelvin wave activity at

particular equivalent depths varies with the local phase of the MJO. Red (blue)

shading thus represents anomalously active (suppressed) mean OLR anomaly

amplitude at the equivalent depth noted in the panel title. Contours represent

regressed MJO-filtered OLR anomalies, with negative in blue (the interval is

5Wm-2 with the zero contour omitted). Panels a through e show results for signals

along Kelvin wave dispersion solutions at equivalent depths of 90, 25, 12, 8, and

5m (respectively).

553

554

555

556

557

558

559

560

561

562

563

564

27

Figure 1. Shallow water model dispersion curves for various equatorial wave modes

plotted on a spectrum of OLR anomalies. The spectrum was normalized by dividing by a

smoothed background spectrum. The MJO band is outlined in a rectangle, and wave

number 4 is marked with a vertical dashed line. Equivalent depths of 5, 12, and 25m are

marked along that line in addition to the plotted dispersion curves.

565

566

567

568

569

570

571

28

Figure 2. Longitude-height cross sections of regressed zonal wind anomalies (shading, ms-1) and geopotential height anomalies (contours, negative in blue, with an interval of 0.25m) for signals along the Kelvin wave dispersion curves at zonal wave number 4. Panels a-e represent results for equivalent depths of 90, 25, 12, 8, and 5m, respectively. The vertical axis is labeled in terms of regressed total geopotential height to facilitate measurement of vertical tilts. Positive longitude is represented as degrees east of the base points.

572573574575576577578579

29

Figure 2. Longitude-height cross sections of regressed zonal wind anomalies (shading, ms-1) and geopotential height anomalies (contours, negative in blue, with an interval of 0.25m) for signals along the Kelvin wave dispersion curves at zonal wave number 4. Panels a-e represent results for equivalent depths of 90, 25, 12, 8, and 5m, respectively. The vertical axis is labeled in terms of regressed total geopotential height to facilitate measurement of vertical tilts. Positive longitude is represented as degrees east of the base points.

580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609

30

Figure 3. Horizontal maps of regressed OLR anomalies (shading, Wm-2), geopotential height anomalies (positive in red, contour interval 0.15m), and wind anomalies at 900 hPa for signals along the Kelvin wave dispersion solutions for zonal wave number 4. Panels correspond to equivalent depths of 90, 25, 12, 8, and 5m, corresponding to the same panels of Fig. 2.

610

611

612613614615616

31

Figure 3. Horizontal maps of regressed OLR anomalies (shading, Wm-2), geopotential height anomalies (positive in red, contour interval 0.15m), and wind anomalies at 900 hPa for signals along the Kelvin wave dispersion solutions for zonal wave number 4. Panels correspond to equivalent depths of 90, 25, 12, 8, and 5m, corresponding to the same panels of Fig. 2.

617618619620621622623624625626627628629630631632633634635636637638639640641642643644645646647648649650651

32

Figure 4. a. Anomalies of 900 hPa wind (vectors), OLR (shading, with negative in blue), and geopotential height (with negative anomalies in blue) regressed against OLR anomalies filtered in the wave number frequency domain for the MJO. b. Vertical cross section of zonal wind (shading) and geopotential height anomalies (contours, with negative in blue) on the equator, plotted against regressed total geopotential height.

652

653654655656657658

659

660661662663664

33

Figure 5. Shading shows the result of regressing absolute value of OLR anomalies along the Kelvin wave dispersion curves for zonal wave numbers 3-8 against MJO-filtered OLR anomalies at 80E (Wm-2). The local mean is subtracted at each grid point. The shading thus provides a measure of how Kelvin wave activity at particular equivalent depths varies with the local phase of the MJO. Red (blue) shading thus represents anomalously active (suppressed) mean OLR anomaly amplitude at the equivalent depth noted in the panel title. Contours represent regressed MJO-filtered OLR anomalies, with negative in blue (the interval is 5Wm-2 with the zero contour omitted). Panels a through e show results for signals along Kelvin wave dispersion solutions at equivalent depths of 90, 25, 12, 8, and 5m (respectively).

665666

667668669670671672673674675676677