1
1
2
3
Observed Structure of Convectively Coupled
Waves as a Function of Equivalent Depth:
Kelvin Waves and the Madden Julian Oscillation
4
5
6
7
8
9
10
11
12
Paul E. Roundy1
University at Albany
State 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
2
13
14
15
Abstract
The view that convectively coupled Kelvin waves and the Madden Julian oscillation are
16
distinct modes is tested by regressing data from the Climate Forecast System Reanalysis
17
against satellite outgoing longwave radiation data filtered for particular zonal wave
18
numbers and frequencies by wavelet analysis. Results confirm that nearly dry Kelvin
19
waves have horizontal structures consistent with their equatorial beta plane shallow water
20
theory counterparts, with westerly winds collocated with the lower tropospheric ridge,
21
while the MJO and signals along Kelvin wave dispersion curves at low shallow water
22
model equivalent depths are characterized by geopotential troughs extending westward
23
from the region of lower tropospheric easterly wind anomalies through the region of
24
lower tropospheric westerly winds collocated with deep convection. Results show that as
25
equivalent depth decreases from that of the dry waves (concomitant with intensification
26
of the associated convection), the ridge in the westerlies and the trough in the easterlies
27
shift westward. The analysis therefore demonstrates a continuous field of intermediate
28
structures between the two extremes, suggesting that Kelvin waves and the MJO are not
29
dynamically distinct modes. Instead, signals consistent with Kelvin waves become more
30
consistent with the MJO as the associated convection intensifies. This result depends
31
little on zonal scale. Further analysis also shows how activity in synoptic scale Kelvin
32
waves characterized by particular phase speeds evolves with the planetary scale MJO.
3
Introduction
33
1.
34
The tropical atmosphere organizes moist deep convection over a broad range of spatial
35
and temporal scales. The Maddan-Julian oscillation (MJO) dominates variability in
36
convection on intraseasonal timescales of roughly 30-100 days (Madden and Julian 1994;
37
Zhang 2005). Rainfall associated with the local active convective phase of the MJO
38
(hereafter, active MJO) is in turn organized into smaller scale wave modes and mesoscale
39
convective systems. Convectively coupled Kelvin waves are widely recognized as a
40
leading signal among the population of modes that comprise the sub scale anatomy of the
41
MJO. These waves produce the highest amplitude signals in outgoing longwave radiation
42
(OLR) data near the equator (Wheeler and Kiladis 1999 (hereafter WK99); Straub and
43
Kiladis 2002; Roundy 2008). MacRitchie and Roundy (2012) showed that roughly 62%
44
of rainfall that occurs in the negative OLR anomalies of the MJO between 10N and 10S
45
over the Indo-Pacific warm pool regions occurs within the negative OLR anomalies of
46
the Kelvin wave band (after excluding those negative anomalies that do not enclose
47
signals less than -0.75 standard deviation). That result represents nearly twice the average
48
rainfall rate per unit area outside of the Kelvin waves but still within the active MJO.
49
MacRitchie and Roundy also showed that potential vorticity (PV) accumulates in the
50
lower to middle troposphere in wakes along and behind the Kelvin wave convection on
51
its poleward sides, and that this PV remains in the environment for longer than the period
52
of the Kelvin waves. The enhanced PV spreads pole ward behind the waves, and it
53
becomes part of the rotational structure of the MJO itself. Another portion of the
54
rotational response to convection coupled to Kelvin waves propagates eastward with the
55
waves, yielding low-level cyclones poleward of the equatorial convection (Roundy
4
56
2008). The response to deep convection moving eastward with convectively coupled
57
Kelvin waves makes them similar in many respects to the geographically larger MJO. On
58
the other hand, these patterns distinguish observed convectively coupled Kelvin waves
59
from theoretical Kelvin waves of Matsuno (1966) and Lindzen (1967), which do not
60
include meridional circulation. Nevertheless, many authors acknowledge that Kelvin
61
wave dynamics dominate their evolution because of their dispersion characteristics and
62
because of the relationship between wind and pressure observed in the lower stratosphere
63
away from the deep convection, which consistently shows westerly wind in the ridge and
64
easterly wind in the trough, with little meridional circulation. Although the MJO clearly
65
modulates Kelvin wave activity, amplitudes, and propagation speeds (Straub and Kiladis
66
2003; Roundy 2008), these waves occur independent of the MJO.
67
Although several authors during the 1980s and 1990s suggested that the MJO
68
itself might be a modified moist Kelvin mode (e.g., Lau and Peng 1987; Wang 1988; Cho
69
et al. 1994), the idea has since fallen out of favor for several reasons. First, the
70
relationship between zonal wind and pressure anomalies in the MJO appears to be
71
reversed or dramatically offset from that of Kelvin waves, with westerly wind anomalies
72
frequently appearing in the pressure trough collocated with the deep convection (e.g.,
73
Madden and Julian 1994; Zhang 2005). Second, a spectral peak associated with
74
convectively coupled Kelvin waves appears to be distinct from that of the MJO (Kiladis
75
et al. 2009), suggesting that the two have phase speed distributions that might not
76
overlap. Third, zonal wave number frequency spectra of OLR data suggest that the
77
spectral peak of the MJO extends across a broader range of wave numbers at a given
78
frequency than does the spectral peak associated with the Kelvin waves, giving the
5
79
impression of a flat dispersion relationship, even though most of that signature can be
80
explained by geographical variation in MJO propagation rather than true dispersion. This
81
perspective is supported by composite MJO events plotted in the longitude-time lag
82
domain (such as by Hendon and Salby 1994), which show structures favoring wave
83
number 2 over the warm pool (consistent with opposite signed anomalies of convection
84
over the Indian and western Pacific basins) and a half wave number 1 across the western
85
hemisphere. Such half wave number 1 signals project more onto wave number 1 than any
86
other wave number, as shown by a simple application of the Fourier transform in space
87
and time to a perfect eastward-propagating wave number 1 sine wave that is set to zero in
88
one hemisphere and left alone in the other (a synthetic half wave number 1 pattern). Such
89
geographical variations in MJO propagation must project onto different portions of the
90
spectrum. Seasonal variations in MJO propagation must also project onto different
91
portions of the spectrum. A global wave number-frequency spectrum analysis
92
conglomerates all of these varying signals together, such that a spectral peak aligned in a
93
particular pattern does not necessarily imply wave dispersion.
94
A more careful look at each of these characteristics casts some doubt on the
95
assertion that the MJO and Kelvin waves are distinct. First, the algorithm of WK99
96
would artificially enhance the extent of the spectral gap between the MJO and Kelvin
97
peaks. WK99 normalized their OLR spectra by dividing by a smoothed background
98
spectrum. This background spectrum was obtained by smoothing the original spectrum
99
by an arbitrary number of repeated applications of a 1-2-1 filter in frequency and in wave
100
number. This approach conserves the total power in the spectrum but redistributes power
101
in the MJO peak into its surrounding neighborhood, including the region of the spectral
6
102
gap. This artificial increase in background power would reduce the normalized power
103
there, making the MJO and Kelvin peaks appear better separated. For reference, Fig. 1
104
shows a wave number frequency spectrum of OLR calculated in a similar manner. The
105
more objective spectrum analysis of Hendon and Wheeler (2007) confirms the presence
106
of a local minimum in power in the spectrum, but not as pronounced as suggested by
107
WK99.
108
Recently, Roundy (2012) integrated wavelet power in the zonal wave number
109
frequency domain over geographical regions where the 100-day low pass filtered 850 hPa
110
zonal winds are easterly or westerly. He found that the gap in the global OLR spectrum
111
derives entirely from regions of easterly low-level background flow. The spectrum
112
integrated over regions low-level westerly winds has power decline smoothly from the
113
maximum in the MJO band, with no evident spectral gap. Thus the source of the spectral
114
gap is not over warm pool zones where MJO convective signals attain their greatest
115
amplitude. The lowest rate of decline of power occurs along Kelvin wave dispersion
116
solutions between equivalent depths of 5 and 8m. This result suggests that Kelvin waves
117
also propagate eastward more slowly over the warm pool than over trade wind zones.
118
Signals in trade wind zones dominate the global spectrum because these zones occur over
119
more of the global tropics for more of the time than do signals in warm pool westerly
120
wind zones, even though the individual events over the warm pool zones average higher
121
in amplitude.
122
Equatorial beta plane theories suggest that Kelvin waves are non dispersive
123
except at the shortest wavelengths (e.g., Roundy and Janiga 2012), but variation in
124
coupling between the waves and deep convection apparently results in a large range of
7
125
phase speeds across the full population of events (Roundy 2008). Although many of these
126
Kelvin waves propagate eastward at 15-17 ms-1 around much of the globe, or roughly
127
twice the phase speed of the MJO, Roundy (2008 and 2012) showed that they tend to
128
propagate more slowly over the warm pool zones. He also showed that synoptic scale
129
Kelvin waves propagate eastward even more slowly as they move through the local
130
active convective phase of the MJO. The same Kelvin wave disturbance can
131
circumnavigate the entire globe, with its phase speed changing continuously with the
132
amplitude of the associated convective signal. This observed variation in phase speeds
133
leaves open the possibility that long Kelvin waves and the MJO may have overlapping
134
dynamics because their phase speed distributions might overlap. These results
135
demonstrate that the spectral characteristics of the MJO and the Kelvin waves are not as
136
distinguishable as previously thought.
137
The relationship between zonal wind and pressure remains a factor whereby
138
Kelvin waves and the MJO might be distinguishable. Since the pressure wind relationship
139
differs between dry Kelvin waves and the observed MJO, and since the observed
140
distribution of frequencies associated with Kelvin waves are higher than the comparable
141
distribution for the MJO, the pressure wind relationship associated with eastward-moving
142
signals in OLR data must vary with frequency. If the prevailing view that Kelvin waves
143
and the MJO are distinct modes is correct, the presence of both modes would yield a
144
particular pattern of transition in frequency between the spatial patterns associated with
145
Kelvin waves and those associated with the MJO. At low wave number, the spectral
146
peaks of Kelvin waves and the MJO are proximate to each other. Since the spectral
147
characteristics of both the MJO and Kelvin waves vary substantially from event to event,
8
148
proximity of the two peaks suggests that there must be overlap between the spectra of the
149
two phenomena. If the MJO and Kelvin waves represent distinct modes in which the
150
pressure-wind relationship is not a function of frequency (consistent with the prevailing
151
view), then at some point in spectrum between the peaks of the two modes, both signals
152
would be present and explain roughly the same amount of variance in geopotential height
153
anomalies. Both modes would have low-level westerly wind anomalies collocated with
154
negative OLR anomalies, but the associated geopotential height anomalies are strongly
155
offset or opposite. Thus, active convective anomalies at that frequency would be
156
associated with negative geopotential height anomalies with one mode and positive with
157
the other mode. Statistical analysis to extract the average coherent pattern associated with
158
the active convection at that frequency without distinguishing between the modes would
159
thus yield significant lower tropospheric westerly wind anomalies associated with active
160
convection but no significant geopotential anomalies because the two opposite signals
161
would wash each other out. If, however, only one dominant coherent mode exists, with
162
structure modulated by the intensity of the associated rainfall, then the phase relationship
163
between wind and geopotential might shift as a continuous function of frequency, with no
164
geopotential amplitude minimum associated with signals at frequencies between the two
165
extremes.
166
Statistical analysis of observations and reanalysis data might shed light on the
167
nature of the transition of spatial structures as a function of frequency between the
168
spectral peaks that we associate with Kelvin waves and the MJO. Recently, Roundy and
169
Janiga (2012) applied zonal wave number-frequency wavelet analysis and simple linear
170
regression to assess the structure of convectively coupled mixed Rossby gravity (MRG)
9
171
waves characterized by specific zonal wave numbers and frequencies. They applied
172
wavelet analysis at a particular zonal wave number and a specified frequency to generate
173
a time index of the corresponding signals. Regression of fields of data against that index
174
reveals the space-time structures of the patterns corresponding to those signals. By
175
choosing frequencies consistent with a selected equivalent depth at several different
176
individual wave numbers, they showed how MRG wave structures vary with wave
177
number along particular shallow water model dispersion curves. A similar analysis of
178
signals proximate to the Kelvin wave peak in the OLR spectrum might suggest how
179
Kelvin wave structures change with equivalent depth (h), or how structures of coherent
180
disturbances change between the Kelvin and MJO spectral peaks. The purpose of this
181
work is to apply this technique to better understand what observations suggest about the
182
extent to which the MJO and long convectively coupled Kelvin waves can be considered
183
independent phenomena and to enhance our understanding of interactions between short
184
Kelvin waves and the MJO.
185
2. Data
186
Daily-interpolated outgoing longwave radiation (OLR) data on a 2.5-degree grid are
187
applied as proxy for moist deep convection (Liebmann and Smith 1995). These OLR data
188
have been updated every few months since 1995 following the original algorithm. Daily
189
mean zonal and meridional wind, temperature, and geopotential height data are obtained
190
from the Climate Forecast System Reanalysis (Saha et al. 2010). The mean and first four
191
harmonics of the seasonal cycle are subtracted from the OLR, wind, and geopotential
192
height data to generate anomalies. All data are analyzed for the period January 1, 1979
193
through December 2009.
10
194
3. Methods
195
a. Space-time Wavelet Decomposition
196
Signals in atmospheric convection obtained from the regions of the zonal wave
197
number frequency domain near equatorial beta plane shallow water model solution
198
dispersion curves at h=25m are associated with spatial patterns similar in many respects
199
to those obtained from shallow water theory (Matsuno 1966; Lindzen 1967; WK99). For
200
reference, Fig. 1 shows dispersion lines of shallow water model Kelvin wave solutions at
201
h=8 and 90m superimposed on a normalized OLR spectrum. All subsequent
202
observational analyses of Kelvin wave signals in this study are reported for points in the
203
OLR spectrum along the Kelvin dispersion curves with h ranging from 5 to 90m. It is
204
important to point out that other signals and noise occur along the dispersion curves of
205
the shallow water model Kelvin wave solutions. Extratropical waves advected eastward
206
by westerly winds project substantially onto similar regions of the spectrum as Kelvin
207
waves. Such signals tend to be small over the warm pool zones and large in regions of
208
upper tropospheric westerly winds such as the eastern Pacific and Atlantic basins. In spite
209
of other signals, the OLR spectral peaks between the dispersion curves of Kelvin wave
210
solutions of h=5 and 90m include those of convectively coupled Kelvin waves (centered
211
on roughly 25m, but ranging from roughly 8 to 90m), and the MJO, which extends
212
roughly from wave numbers 0-9 eastward and periods of roughly 30-100 days. Keep in
213
mind, however, that these spectral peaks are not distinct when the spectrum is integrated
214
only over the low-level westerly wind zones of the warm pool (Roundy 2012). The
215
Kelvin wave dispersion curves intersect with the traditionally defined MJO spectral peak
216
at low wave number (Wheeler and Kiladis 1999).
11
217
Most previous observational analyses of Kelvin wave signals conglomerate structures
218
of a broad range of wavenumbers and frequencies through filtering in the wave number
219
frequency domain across broad bands (e.g., Wheeler et al. 2000; Roundy and Frank
220
2004). Interpretation of the results is complicated because the vertical and meridional
221
structures of the observed waves might vary with wavenumber and frequency. This
222
project applies zonal wave number-frequency wavelet analysis to extract signals from
223
OLR data at specified wave numbers and frequencies, following Roundy and Janiga
224
(2012). When combined with regression or composite analysis, this more specific
225
approach diagnoses how spatial structures change with frequency at a given wave
226
number. A detailed description of space-time wavelet analysis is beyond the scope of this
227
paper, but Kikuchi and Wang (2010) and Wong (2009) offer overviews of the technique.
228
229
230
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)
231
where s represents x or t for the spatial or temporal transforms, respectively, and 
232
represents angular frequency  or wavenumber k. B, the bandwidth parameter, was
233
assigned a value of 4(
234
temporal transform. Conclusions are not sensitive to these arbitrarily assigned values of
235
B, but much larger values reduce the amplitude contrast in time of signals and enhance a
236
ringing effect, and substantially smaller values do not sufficiently resolve large-scale or
237
low frequency waves. The transform is obtained by taking the time-centered dot product
238
of the wavelet and all daily consecutive overlapping time series segments at each grid
ν
-3/2
)
2π
for the temporal transform and 1.5(
k
-3/2
)
2π
for the
12
239
point around the globe, then applying a similar transform in longitude to the result by the
240
same approach. The transform for a selected wave number and frequency is calculated for
241
every day and at every longitude grid point in the 7.5S to 7.5N averaged OLR anomaly
242
data from 1975 to 2009. Averaging OLR over 7.5S to 7.5N increases the likelihood that
243
the dominant coherent signals are Kelvin waves because this average acts as a filter for
244
cross equatorial symmetry, and proximity to the equator reduces the net contribution of
245
extratropical waves.
246
b. Linear Regression Models
247
Simple linear regression is frequently applied to diagnose structures that are coherent
248
with filtered signals (e.g., Hendon and Salby 1994; Wheeler et al. 2000). In this analysis,
249
the space-time wavelet transform at a selected longitude, wave number, and frequency,
250
becomes a base index time series for regression models at each grid point over a range of
251
longitudes and pressure levels. Either the real or imaginary parts of the transform work
252
for this index. The imaginary part produces convenient zonal phasing in the regression
253
maps, but the conclusions are the same regardless of this choice. Base longitudes are at
254
each 2.5 grid point from 60E to 90E. This focus on the Indian basin reduces the
255
contribution of extratropical features that are much more pronounced over the western
256
hemisphere. Calculating regression models at each base point, then averaging over all of
257
them reduces local disconformities, yielding conclusions less sensitive to geography. The
258
time series from each of those points serve as predictors in regression models at each grid
259
point across a broad geographical domain to diagnose the associated structures. One grid
260
of regression models is calculated for each base point time series. To illustrate, the
261
algorithm models the variable Y at the grid point S as
13
262
Ys  Px As ,
(2)
263
where Px is a matrix whose first column is a list of ones and second column is the base

264
index at the longitude grid point x. As is a vector of regression coefficients at the grid
265
point S. After solving for As at each grid point by matrix inversion, (2) is then applied as
266
a scalar equation to diagnose wave structure by substituting a single value for the second
267
column of Px that is representative of a crest of a wave located at the base longitude (its
268
value is set here at +1 standard deviation). These regression models are applied to create
269

‘composite’
anomalies of OLR, u and v winds, and geopotential height. Results are
270
calculated for the region 180 to the east and west of each base longitude, and then the set
271
of results from all base points are averaged, following Roundy and MacRitchie (2012).
272
The statistical significance of the difference from zero of the result at each point on the
273
map is assessed based on the correlation coefficient (e.g., Wilks 2011), and I analyze and
274
discuss only those regressed signals that are deemed to be significantly different from
275
zero at the 90% level. This significance test is completed for each individual regression
276
map before averaging over results from each base point, so that the number of degrees of
277
freedom is not inflated by inclusion of the same wave events at multiple grid points.
278
Since the regression is accomplished in the time domain, some signal from wave numbers
279
other than the target wave number can appear in results if they tend to occur together in a
280
particular pattern (Wheeler et al. 2000). Such regression results will be most reliable
281
close to the centers of the composites because spreading will occur due to the episodic
282
nature of convection and variations in the background state. Amplitudes of regressed
283
anomalies following this approach are much smaller than those of other authors who have
284
regressed signals against OLR data filtered for the broader Kelvin band of Wheeler and
14
285
Kiladis (1999) because the full band includes much more variance than is retained at just
286
one wave number and frequency.
287
The above approach diagnoses the spatial structures associated with signals along
288
the dispersion curves of shallow water model Kelvin wave solutions at particular
289
equivalent depths. For reference, we also calculate a composite MJO following a similar
290
approach, by replacing the base index time series with MJO band pass filtered OLR
291
anomalies averaged from 10N to 10S. The MJO band signal is averaged over a broader
292
latitude band than are Kelvin signals in this work in order to capture the signals of some
293
MJO events that have OLR anomalies shifted farther into one hemisphere. The MJO band
294
is defined as wave numbers 0-9 eastward and periods of 30-100 days. The value of −1
295
standard deviation in the base index is then substituted to generate a map. Base points for
296
the MJO composite are selected from 70E to 110E in order to reduce contamination of
297
the western portion of the regression maps by Africa. Kelvin band signals at higher wave
298
numbers did not require such an eastward shift because the wavelengths assessed for
299
them yielded less contamination from Africa.
300
A variation on the above regression approach is also applied here to diagnose how
301
signals along the Kelvin wave dispersion curves at particular equivalent depths vary with
302
the phase of the MJO. An index of OLR anomaly data filtered for the wave number
303
frequency band of the MJO at 80E is assigned as P in equation (2) and applied to predict
304
Y, which is assigned to be the absolute value of the sum of the real and imaginary parts of
305
the wavelet transform at a selected wave number and frequency at a time lag. Averaging
306
regression maps over multiple base points is not applied for this analysis since the
307
associated geographical signals is the target outcome. The value of −1 standard deviation
15
308
is then substituted for the second column of P at each grid point and time lag. After
309
discarding the time local mean of the result, it shows how the timing of changes in
310
activity in Kelvin waves characterized by particular phase speeds varies with the MJO.
311
4.
Results
312
a. Regression at Various Equivalent Depths Along Kelvin Dispersion Curves
313
Figure 2 shows regressed geopotential height anomalies (contours) and zonal
314
wind anomalies (shading, with westerly anomalies in red) for panels a-e, equivalent
315
depths of 90, 25, 12, 8, and 5m (respectively) for zonal wave number 4 Kelvin waves.
316
These data are plotted against regressed total geopotential height instead of pressure to
317
facilitate measurement of the vertical tilts of the regressed anomalies. Thus, the plotted
318
geopotential height anomalies represent the displacement of isobars at a given height
319
from their climatological positions. The results show patterns that tilt toward the west
320
with height between the surface of the earth and roughly 10,000m, with tilt reversing
321
toward the east above (consistent with Kelvin wave composites by Kiladis et al. 2009 and
322
references therein). Each panel shows eastward flow in the ridges and westward flow in
323
the troughs above 104m, but structures vary with equivalent depth below that level. At the
324
equivalent depth of 90m (panel a), westerly wind anomalies are collocated with positive
325
geopotential height anomalies near the center of the composite. Comparison of all panels
326
shows that the westerly wind anomalies near the centers of the composites are nearly an
327
order of magnitude stronger at h=5m (panel e) than at h=90m, but the as equivalent depth
328
decreases, the geopotential trough in the easterlies on the east side of the domain extends
329
westward until at h=5m it encompasses nearly all of the westerly anomalies near the
330
center of the composite below 10,000m. The differences between the composites for
16
331
large and small equivalent depths occur smoothly across the equivalent depths plotted
332
here. The regressed geopotential height anomalies do not vanish at some equivalent
333
depth, as would occur if the MJO and Kelvin band signals include two distinct modes
334
characterized by opposite pressure wind relationships. The vertical cross sections for the
335
other wave numbers are similar to those for k=4.
336
Figure 3 shows the horizontal maps of the regressed geopotential height and
337
winds for wave number 4 at 900 hPa along the Kelvin wave dispersion curves for the
338
same equivalent depths as in Fig. 2. Regressed OLR anomalies are shaded, with active
339
convection suggested in blue, and regressed geopotential height anomalies are contoured
340
with positive anomalies in red. At h=90m, westerly wind anomalies are collocated with
341
positive geopotential height anomalies and slightly negative OLR anomalies. Easterly
342
wind anomalies occur in the trough, consistent with the shallow water model Kelvin
343
wave. With increasing equivalent depth, the negative OLR anomalies strengthen in the
344
vicinity of the equatorial westerly wind anomalies. At the same time, locally positive
345
geopotential height anomalies shift westward from the active convection toward the
346
suppressed convection. Trough anomalies shift westward from east of the negative OLR
347
anomalies at h=90m (panel e) through the convective region by h=5m. The increased
348
amplitude of the OLR anomalies with decreasing equivalent depth suggests that lower
349
equivalent depths are associated with higher rainfall rates. TRMM 3B42 rain rate data
350
available since 1999 confirm this observation (not shown). The structure of the OLR and
351
geopotential height anomalies also changes with equivalent depth. The negative OLR
352
anomaly at h=90m nearly forms an ellipse centered on the equator, but at smaller
353
equivalent depths, the negative OLR and geopotential anomalies distort increasingly
17
354
westward with distance form the equator, forming boomerang patterns across the equator.
355
Regression maps for other wave numbers show similar structures (not shown). Thus, both
356
long and short Kelvin waves become more like the MJO with increasing precipitation
357
rates. This statement also holds true for Kelvin waves of wave number 6 and 8 (not
358
shown). Although the h=5m result at wave number 6 has a period of about 11 days, well
359
outside of the traditional MJO band of the wave number frequency domain, the
360
associated regression maps still show pronounced westward shifting of the geopotential
361
height anomalies relative to the OLR anomalies, along with pronounced westward
362
distortion with latitude. In other words, signals along the Kelvin wave dispersion curve at
363
h=5m and wave number 6 are associated with structures similar to those of the MJO, but
364
with smaller zonal scale. Within the traditional MJO band, structures observed along the
365
dispersion curve for the h=5m Kelvin wave at zonal wave number 2 also shows similar
366
traits. That signal propagates at about 7ms-1 and has a period of about 30 days.
367
b. Regressed MJO Structure
368
For comparison with Fig. 3, Fig. 4a shows a horizontal map of geopotential height
369
anomalies and zonal wind regressed against MJO-filtered OLR anomalies at 900 hPa.
370
These results show a geopotential trough collocated with easterly wind anomalies on the
371
eastern side of the domain. That trough also extends westward across much of the region
372
of low-level westerly winds collocated with the negative OLR anomaly. That
373
geopotential trough and the negative OLR anomalies form a triangle pattern with one side
374
perpendicular to and bisected by equator on the west and the two other legs meeting to
375
the east on the equator. This pattern is consistent with distortion of the OLR and
376
geopotential height anomalies westward with distance form the equator at low equivalent
18
377
depths in Fig. 3d and e. Figure 4b shows the corresponding vertical cross section of
378
regressed geopotential height anomaly and zonal wind anomaly on the equator, for
379
comparison with Fig. 2. The result compares well with Fig. 2d and 2e.
380
c. The Association Between Synoptic Kelvin Wave Activity and the MJO
381
Straub and Kiladis (2003) evaluated the evolution of signals in the broader Kelvin
382
wave band with the northern hemisphere summer MJO. The present work expands on
383
their analysis by demonstrating how that evolution depends on the phase speeds of the
384
Kelvin waves in a generalized MJO without explicit assessment of seasonality. Figure 5
385
shows regressed activity in Kelvin waves at zonal wave numbers 3-8 (shading) along
386
with regressed MJO-filtered OLR. Panels (a)-(e) represent results for equivalent depths of
387
90, 25, 12, 8, and 5m, respectively. Enhanced convection in the MJO band is indicated by
388
blue contours. Fast Kelvin waves (~30ms-1) at 90m equivalent depths (Fig. 5a) are
389
characterized by lower amplitude signals in OLR anomalies than all other equivalent
390
depths (consistent with the expectation that such Kelvin waves should be nearly dry).
391
Figure 5a suggests that prior to onset of convection in the MJO band over the Indian
392
basin (hereafter called “MJO initiation” for simplicity), fast Kelvin waves are prevalent
393
over the Atlantic basin and Africa, but quiet over the Pacific basin. This activity extends
394
eastward early in the lifetime of the negative OLR anomaly in the MJO band over the
395
Indian basin. This activity then declines to below average over the Indian basin after lag
396
= +5 days. Activity in these fast Kelvin waves then grows over the Pacific Ocean to the
397
east of the active MJO. Kelvin waves characterized by h=25m also show enhanced
398
activity in OLR anomalies over the Atlantic basin and Africa prior to MJO initiation, but
399
substantially more than for h=90m. After lag = 0, enhanced activity occurs at the eastern
19
400
edge of the negative OLR anomalies in the MJO band, a little farther west than for
401
h=90m. At h=12m, similar to at h=90m and h=25m, activity begins over the Atlantic
402
basin and Africa prior to MJO initiation, but the level of activity becomes much stronger
403
over the Indian basin within the active MJO and then extends only slightly eastward from
404
the negative OLR anomaly of the MJO after lag = +5 days. Although signal at h=8m and
405
h=5m is also suggested over the Atlantic basin and Africa leading up to the active MJO,
406
most of the signal in these bands concentrates within the negative OLR anomaly of the
407
MJO over both the Indian and western Pacific basins. These slow Kelvin wave signals
408
are more consistent with the slow eastward-moving supercloud clusters of the active
409
convective phase of the MJO noted by Nakazawa (1988) than are the faster Kelvin
410
waves. Figure 5a confirms the previous result of Kikuchi and Takayabu (2003) that dry
411
Kelvin waves radiate eastward from the active convective phase of the MJO over the
412
western Pacific basin, but Fig. 5 b-d also shows that a substantial convectively coupled
413
Kelvin wave signal at h=12m and h=25m (about 11 and 16 ms-1 respectively) also occurs
414
over the Pacific basin east of the active MJO. The slowest Kelvin waves at wave numbers
415
3-8 are largely confined to the active convective phase of the MJO over the Indo Pacific
416
warm pool. Although the local amplitudes of OLR anomalies at h=8m and 5m are
417
substantially higher than for OLR anomalies at 25m, the isolation of these low h signals
418
largely within active convective phases of the MJO over the warm pool reduces their net
419
contribution to the OLR spectrum, leading to the more global signals near 25m standing
420
out in the OLR spectrum. These results are especially interesting in the context of Figs. 2
421
and 3, which suggest that these synoptic scale Kelvin waves themselves have spatial
422
structures similar to those of the planetary scale MJO.
20
Conclusions
423
5.
424
A wave number frequency wavelet analysis of OLR anomaly data and simple linear
425
regression reveal how the structures associated with signals along the dispersion curves
426
of Kelvin waves change with equivalent depth. Results suggest that the phase relationship
427
between geopoential height and wind anomalies for signals along Kelvin wave dispersion
428
curves adjusts continuously westward with decreasing equivalent depth from patterns
429
consistent with Kelvin waves of equatorial beta plane shallow water theory (which have
430
westerly wind anomalies in the geopotential ridge) to patterns that look more like the
431
MJO (with westerly wind anomalies extending westward through the geopotential
432
trough). If there were two distinct modes present with opposite pressure wind
433
relationships overlapping in the spectrum, with one mode dominant at low frequencies
434
and the other dominant at higher frequencies, then at some frequency in between the two,
435
the geopotential signals would wash out of the regression while regressed wind and OLR
436
signals would remain. Instead, the regression analysis reveals a continuous shift of the
437
phase between zonal wind and pressure signals. High wave number Kelvin waves whose
438
signals are far in the spectrum from the MJO band follow similar patterns at low
439
equivalent depths. These results thus do not support the perspective that the MJO and
440
Kelvin waves are distinct modes like the present consensus suggests. This continuous
441
evolution instead supports the perspective that more intense convection modifies the
442
convectively coupled Kelvin wave to take on characteristics more consistent with the
443
MJO. In that sense, the low wave number portion of the disturbance traditionally labeled
444
as the MJO might be a planetary scale Kelvin wave modified by the influence of intense
445
convection. Analysis of the power spectrum by Roundy (2012) further confirms that no
21
446
separation between the Kelvin and MJO spectral peaks occurs over the low-level westerly
447
wind zones over the warm pool. Thus in those regions, MJO signals cannot be
448
distinguished from a continuum of disturbances that begin at high frequencies in
449
association with dry Kelvin waves.
450
This work also demonstrates how synoptic scale Kelvin waves characterized by
451
particular phase speeds (or equivalent depths) vary with the MJO. Kelvin wave activity at
452
all phase speeds tends to be enhanced over the Atlantic basin and Africa prior to
453
development of deep convection in the MJO band over the Indian basin. Fast Kelvin
454
waves are also prevalent well to the east of MJO convection when that convection is
455
located over the western Pacific basin. The slowest Kelvin waves characterized by
456
equivalent depths of less than 12m are strongest within the active convective phase of the
457
MJO over the Indian basin, consistent with the assessment of the associated supercloud
458
clusters by Nakazawa (1988) and slow Kelvin waves by Roundy (2008). These slow
459
synoptic scale Kelvin waves themselves have vertical and horizontal structures similar to
460
those of the planetary scale MJO.
461
Acknowledgments.
462
Funding was provided by the National Science Foundation Grant# 1128779 to Paul
463
Roundy. The NOAA PSD provided OLR data, and the NOAA CPC provided CFS
464
reanalysis data.
465
466
467
468
22
469
References
470
Cho, H.-R., K. Fraedrich, and J. T. Wang, 1994: Cloud clusters, Kelvin wave CISK, and
471
the Madden-Julian Oscillations in the equatorial troposphere. J. Atmos. Sci., 51,
472
68-76.
473
474
475
476
477
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
478
associated with the Madden-Julian oscillation (MJO) during Boreal winter. J.
479
Meteorol. Soc. Japan, 81, 851-869.
480
481
Kikuchi, K., and B. Wang, 2010: Spatiotemporal wavelet transform and the multiscale
behavior of the Madden Julian oscillation. J. Climate, 23, 3814-3834.
482
Kiladis G. N., M. C. Wheeler, P. T. Haertel, K. H. Straub, and P. E. Roundy, 2009:
483
Convectively coupled equatorial waves. Reviews of Geophysics. 47, RG2003,
484
doi:10.1029/2008RG000266.
485
486
487
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
488
longwave radiation dataset. Bull. Amer. Meteor. Soc. 77 : 1275-1277.
489
Lindzen R. S., 1967: Planetary waves on beta-planes. Mon. Wea. Rev., 95, 441-451.
23
490
MacRitchie, K., and P. E. Roundy, 2012: Potential vorticity accumulation following
491
atmospheric Kelvin waves in the active convective region of the MJO. J. Atmos.
492
Sci., In Press.
493
494
495
496
497
498
499
500
501
502
503
504
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.
505
Roundy, P. E., and M. Janiga, 2012: Analysis of vertically propagating convectively
506
coupled equatorial waves using observations and a non-hydrostatic Boussinesq
507
model on the equatorial beta plane. Q. J. Roy. Meteorol. Soc. In Press.
508
Saha, S. Nadiga, C. Thiaw, and J. Wang, W. Wang, Q. Zhang, and H. M. Van den Dool,
509
H.-L. Pan, S. Moorthi, D. Behringer, D. Stokes, M. Peña, S. Lord, and G. White,
510
W. Ebisuzaki, P. Peng, and P. Xie, 2006: The NCEP climate forecast system. J.
511
Climate, 19, 2483-3517.
512
Straub, Katherine H., George N. Kiladis, 2002: Observations of a Convectively Coupled
24
513
Kelvin Wave in the Eastern Pacific ITCZ. J. Atmos. Sci., 59, 30–53.
514
Straub, K. H., and G. N. Kiladis, 2003: Interactions between the Boreal summer
515
intraseasonal oscillation and higher-frequency tropical wave activity. Mon. Wea.
516
Rev., 131, 945-960.
517
518
519
520
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.
521
Wheeler M., G. N. Kiladis, and P. J. Webster, 2000: Large-scale dynamical fields
522
associated with convectively-coupled equatorial waves. J. Atmos. Sci., 57, 613-
523
640.
524
525
526
527
528
529
530
531
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.
25
532
List of Figures
533
Figure 1. Shallow water model dispersion curves for various equatorial wave modes
534
plotted on a spectrum of OLR anomalies. The spectrum was normalized by
535
dividing by a smoothed background spectrum.
536
Figure 2. Longitude-height cross sections of regressed zonal wind anomalies (shading,
537
ms-1) and geopotential height anomalies (contours, negative in blue, with an
538
interval of 0.25m) for signals along the Kelvin wave dispersion curves at zonal
539
wave number 4. Panels a-e represent results for equivalent depths of 90, 25, 12, 8,
540
and 5m, respectively. The vertical axis is labeled in terms of regressed total
541
geopotential height to facilitate measurement of vertical tilts. Positive longitude is
542
represented as degrees east of the base points.
543
Figure 3. Horizontal maps of regressed OLR anomalies (shading, Wm-2), geopotential
544
height anomalies (positive in red, contour interval 0.15m), and wind anomalies at
545
900 hPa for signals along the Kelvin wave dispersion solutions for zonal wave
546
number 4. Panels correspond to equivalent depths of 90, 25, 12, 8, and 5m,
547
corresponding to the same panels of Fig. 2.
548
Figure 4. a. Anomalies of 900 hPa wind (vectors), OLR (shading, with negative in blue),
549
and geopotential height (with negative anomalies in blue) regressed against OLR
550
anomalies filtered in the wave number frequency domain for the MJO. b. Vertical
551
cross section of zonal wind (shading) and geopotential height anomalies
552
(contours, with negative in blue) on the equator, plotted against regressed total
553
geopotential height.
26
554
Figure 5. Shading shows the result of regressing absolute value of OLR anomalies along
555
the Kelvin wave dispersion curves for zonal wave numbers 3-8 against MJO-
556
filtered OLR anomalies at 80°E (Wm-2). The local mean is subtracted at each grid
557
point. The shading thus provides a measure of how Kelvin wave activity at
558
particular equivalent depths varies with the local phase of the MJO. Red (blue)
559
shading thus represents anomalously active (suppressed) mean OLR anomaly
560
amplitude at the equivalent depth noted in the panel title. Contours represent
561
regressed MJO-filtered OLR anomalies, with negative in blue (the interval is
562
5Wm-2 with the zero contour omitted). Panels a through e show results for signals
563
along Kelvin wave dispersion solutions at equivalent depths of 90, 25, 12, 8, and
564
5m (respectively).
565
27
566
567
Figure 1. Shallow water model dispersion curves for various equatorial wave modes
568
plotted on a spectrum of OLR anomalies. The spectrum was normalized by dividing by a
569
smoothed background spectrum. The MJO band is outlined in a rectangle, and wave
570
number 4 is marked with a vertical dashed line. Equivalent depths of 5, 12, and 25m are
571
marked along that line in addition to the plotted dispersion curves.
572
28
573
574
575
576
577
578
579
580
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.
29
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
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.
30
611
612
613
614
615
616
617
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.
31
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
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.
32
653
654
655
656
657
658
659
660
661
662
663
664
665
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.
33
666
667
668
669
670
671
672
673
674
675
676
677
678
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).