a. Spectral definitions

Space–time power and cross-power spectra are computed using two-dimensional (time and space) Fourier transforms. Following the notation of Båth (1974), the discrete space–time Fourier transforms of two fields f₁(t_n, x_m) and f₂(t_n, x_m), given at M equally spaced points around a latitude circle (x_m) and N equally spaced points in time (t_n), are

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Here i = −1 and the frequencies and zonal wavenumbers are discrete.

The powers (E₁₁ and E₂₂) and cross-power (E₁₂) spectra are

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and

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Here the asterisk indicates complex conjugate: P₁₂ and Q₁₂ are the real and imaginary parts of the cross-power spectrum and are referred to as the cospectrum and quadrature spectrum, respectively.

Unlike the power spectrum, which is always real and positive, the cross-power spectrum is in general complex, containing information about magnitude and phase of the relationship between the two fields. One standard way of presenting the cross-power spectrum is by normalizing the square of its magnitude by the individual powers, which yields the coherence squared spectrum:

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The coherence squared spectrum can be thought of as a squared correlation coefficient that is a function of frequency and wavenumber and is bounded 0 < |γ₁₂|² < 1. The squared coherence is identically equal to 1 for a single estimate at an individual frequency and wavenumber. A true estimate is obtained by sector averaging and smoothing in frequency of the individual powers and cross power prior to forming the coherence (e.g., Båth 1974).

The phase lag φ(ω, k) of F₁ with respect to F₂ is obtained from

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b. Data and calculations

Discrete Fourier transforms are computed with a fast Fourier transform. All input fields are anomalies from the climatological annual cycle, which is defined by the mean and first three harmonics of the annual cycle. We use daily mean OLR (Liebmann and Smith 1996) and zonal winds from National Centers for Environmental Prediction (NCEP)– National Center for Atmospheric Research (NCAR) reanalysis (Kalnay et al. 1996) for the period 1979–2005. These data are available on a 2.5° grid. The use of daily data means that the Nyquist period (shortest resolvable period) is 2 days. Thus, this analysis will not resolve the prominent equatorial gravity waves with periods less than 2 days that were detected by WK99. To facilitate identification of theoretical equatorial wave modes, we partition the data into equatorially symmetric and antisymmetric components, as in WK99. To adequately resolve the MJO (whose period ranges up to 90 days), the records are then broken into 256-day segments that overlap 50 days, which are each detrended and tapered to zero with a cosine-tapered rectangular window over the first and last 12 days. The segments are Fourier transformed at each equatorially symmetric and antisymmetric latitude. Power and cross power are computed and then averaged over all segments and latitudes. To improve the presentation, power and cross power are further smoothed by three passes of a 1–2–1 running mean filter in frequency only. For the calculations shown here, we use the range 2.5°–10° latitude for both symmetric and antisymmetric components. The nominal frequency bandwidth is (256 days)⁻¹, and the effective bandwidth, as a result of the frequency smoothing, is increased to ∼(90 days)⁻¹. Assuming that the frequency smoothing results in each smoothed estimate containing information from approximately three raw spectral estimates and conservatively assuming no latitudinal independence, each spectral estimate, after averaging in latitude, is assumed to have ∼2(amplitude and phase) × 3(raw spectral estimates per smoothed estimate) × 27(yr) × 365(days)/256(segment length) degrees of freedom (≈230 dof).

Significance of peaks in the space–time power spectrum is judged relative to an equivalent red (in frequency) background spectrum (e.g., Masunaga et al. 2006), which we estimate at each eastward and westward zonal wavenumber. In contrast to the technique used in WK99, estimating the background spectrum as a red noise process is arguably more objective and does not require ad hoc smoothing in wavenumber or frequency: the red background spectrum is completely determined by the decorrelation time and total power of the original spectrum. We derive the background spectrum from the average of the raw (unsmoothed) spectra for symmetric and antisymmetric latitudes from 2.5°–10° as in WK99. This is equivalent to averaging the spectra computed from the data at each original latitude 10°S–10°N (excluding the equator) and assumes (without further justification) that the background spectrum derives from fluctuations that are uncorrelated about the equator. Following Gilman et al. (1963), a red frequency spectrum is then approximated as a lag − 1 autoregressive process by using the lag − 1 autocorrelation at each eastward and westward zonal wavenumber, which is obtained by inverse Fourier transform of the averaged symmetric–antisymmetric power spectrum (but which is otherwise unsmoothed in frequency or wavenumber). Here we make use of the fact that the correlation function and power spectrum form a Fourier pair (Båth 1974). To better estimate the hypothetical red background spectrum that would exist in the absence of the MJO, the influence of the MJO in the background autocorrelation is reduced by halving the power at eastward wavenumbers 1–3 for periods 30–90 days prior to inverse Fourier transform of the averaged symmetric–antisymmetric spectrum. The resultant correlation at lag − 1 is then used in the Gilman et al. (1963) discrete formula to obtain an equivalent red frequency spectrum at each zonal wavenumber, which is normalized to have the same power as the averaged symmetric–antisymmetric spectrum for that wavenumber. As in WK99, we use this constructed red background spectrum as the null hypothesis for both symmetric and antisymmetric spectra. Estimates of significance of spectral peaks using this background spectrum (discussed below) are not sensitive to the preremoval of the MJO spectral peak prior to computing the correlation function, but the ad hoc removal of the spectral peak associated with the MJO does result in a more eastward–westward symmetric background spectrum.

Confidence thresholds for the observed spectrum exceeding the background red spectrum are estimated using a Chi-squared test, assuming that smoothed spectral estimates at each frequency possess ∼230 dof (justified above). A spectral estimate E(ω, k) is judged to significantly stand above the background red spectrum E_red(ω, k) at the 99% level if E(ω, k)/E_red(ω, k) > 1.25. Alternatively, the “signal strength,” defined as the fraction of total power that stands above the background, is deemed significant where (E(ω, k) − E_red(ω, k))/E(ω, k) > 0.2. Significance of the coherence squared statistic is judged against a null hypothesis of zero coherence using an F test, assuming that the coherence squared estimates at each frequency also possess ∼230 dof. Coherence squared greater than 0.05 (equivalently a correlation coefficient of 0.224) is deemed to be significantly different from zero at better than 99% confidence.

c. Display

We display relative spectral density, which is the power spectrum that is normalized at each wavenumber and frequency by the total power summed over all wavenumbers and frequencies. We contour the relative spectral density and shade the “signal strength” with the first significant shading level at 0.2.

The power spectrum E(ω, k) is typically displayed as a linear function of frequency and zonal wavenumber. In this fashion, the power integrated between two frequencies,

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is simply proportional to the area under the curve. In many meteorological applications, where phenomena of interest span a wide range of frequencies, a logarithmic frequency scale is more practical. However, to preserve the variance as being equal to the area under the curve, it is necessary to then plot ωE(ω, k). This is clear because

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Plotting power as a function of the logarithm of frequency (in this case, base 10 logarithm) has the advantage of providing better detail of the spectral peak associated with the MJO, which occurs near the lowest frequency typically resolved in the analysis. Plotting ωE(ω, k) also helps highlight the power associated with the Kelvin waves (or Rossby–gravity waves for the antisymmetric case), which occur over a range of periods that require a relatively large number of frequencies to resolve (e.g., WK99). It is important to observe, however, that the alternate display does not preserve the position of maxima and minima of the spectra from the linear display. Practically, maxima and minima do tend to roughly coincide in the two systems in cases of pronounced peaks (or troughs). The alternate display also distorts well-known dispersion relationships. For example, the linear Kelvin wave dispersion (i.e., frequency is a linear function of wavenumber) appears to be a nonlinear function of wavenumber when plotted against the logarithm of frequency.

The squared coherence spectrum is contoured as a function of frequency and zonal wavenumber. The coherence spectrum is only contoured where it exceeds 0.05, which we assess to be significantly greater than 0 at better than the 99% level. The phase lag φ of the second field with respect to the first field is displayed as a vector of unit length with components (sin φ, cos φ) = [Q₁₂/(P²₁₂ + Q²₁₂), P₁₂/(P²₁₂ + Q²₁₂)]. Our convention is an upward-pointing vector indicates zero phase lag, downward pointing indicates out-of-phase behavior, rightward pointing indicates that the first field leads the second field by 1/4 cycle, and leftward pointing indicates that the second field leads the first field by 1/4 cycle.

a. Properties of the background red spectrum

The red background spectrum for OLR, displayed as a linear function of frequency (Fig. 1a), exhibits a similar decay in frequency and wavenumber as that created by WK99, though the spectrum here is markedly more symmetric about wavenumber 0 due to the preconditioning to remove the MJO (section 2). A direct quantification of the “redness” of the background spectrum is obtained from the decorrelation time of the autocorrelation function. For a first-order autoregressive process the discrete autocorrelation function is cor(κ) = ρ^κ, where κ is the integer lag and ρ is the lag-1 autocorrelation [ρ = cor(1)]. The decorrelation time τ_D, defined here as the time for the correlation to drop to a value of e⁻¹, is τ_D = −( lnρ)⁻¹. In the appendix, we demonstrate that τ_D ≈ (1/2)τ_DI, where τ_DI = (1 + ρ)/(1 − ρ) is the standard estimate of the time between independent samples in an autocorrelated time series (e.g., von Storch and Zwiers 1999). We note that the implied reduction in total independent samples in a given record for τ_DI > 1 does not affect our estimation of the degrees of freedom used to assess significance of peaks in the power and coherence spectra (section 2), because significance testing of the spectrum is applied independently at each frequency.

The decorrelation time of the background OLR and zonal wind at 850 hPa (hereafter U₈₅₀), as a function of zonal wavenumber, is shown in Fig. 2. Decorrelation times are typically twice as long for U₈₅₀ as for OLR, ranging from 2 days (1 day) at high wavenumber to 8 days (3 days) at eastward wavenumbers 0 and 1. Thus, the background spectrum for U₈₅₀ is markedly more “red” than for OLR, which reflects that convective variability is more closely associated with the divergent component of the wind, which is spatially and temporally noisier than the total wind.

The alternate display of the OLR background red spectrum (power multiplied by frequency versus logarithm of frequency) is shown in Fig. 1b. As discussed above, this alternate display, while preserving variance, does not preserve the location of spectral peaks. For the spectrum of a first-order autoregressive (red) process, maximum power occurs at zero frequency. But, in the alternate display, the red spectrum peaks at frequency f_max > 0 (e.g., Weickmann et al. 2000). Here f_max is estimated by considering the correlation function for a continuous (rather than discrete) first-order autoregressive process (e.g., Jones 1975):

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where κ is the time interval or lag and the lag-1 autocorrelation is expressed as ρ = exp(−ν). The spectral density of this continuous time process can be computed analytically (e.g., Jones 1975):

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where the frequency f is in cycles per time unit. The maximum of

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is then found by differentiating fE( f ) with respect to f and then setting equal to zero. Thus, fE( f ) is maximum at f_max = (2πτ_D)⁻¹.

Estimates of f_max based on decorrelation times from Fig. 2 agree with the location of the spectral peaks in Fig. 1b. For instance, the decorrelation time for OLR at eastward wavenumber 1 is approximately 3 days from Fig. 2. This yields a f_max = ∼0.053 cycles day⁻¹ (equivalently a period of about 18.8 days), which agrees closely with the location of the spectral peak at eastward wavenumber 1 in Fig. 1b. Although the alternate display of power distorts the location of the spectral peaks, the alternate displays allow for an estimate of the decorrelation time to be inferred directly from the power spectrum without having to compute the autocorrelation function (e.g., Weickmann et al. 2000).

b. Space–time spectra of equatorial symmetric OLR and U₈₅₀

The space–time power spectra of the symmetric components of OLR and U₈₅₀ are displayed in Figs. 3 and 4, respectively. Power, as a linear function of frequency, is contoured in the top panels, while the alternate display of power multiplied by frequency as a function of the logarithm of frequency is contoured in the bottom panels.

Considering first the traditional linear display for OLR (Fig. 3a), we see the dominant spectral peak associated with the MJO at ∼50 day period for eastward wavenumbers 1 and 2 (and extending to higher wavenumbers). The shading indicates the MJO “signal” peaks at about 70%. Additional spectral peaks are apparent for the other key phenomena detected by WK99: the higher-frequency Kelvin waves (eastward wavenumbers greater than 2 and periods less than ∼15 days) and westward equatorial Rossby waves (periods greater than ∼15 days). The signal strength of the Kelvin waves is similar to that for the MJO (peaks at ∼50% of the power for wavenumbers 2–7), while that for the equatorial Rossby waves is more modest (peaks at about 30% of the power for wavenumbers 3 and 4.

Representative dispersion curves for the theoretical equatorial wave modes in a motionless basic state (e.g., Matsuno 1966) are drawn to highlight the features described above. Three Kelvin wave dispersion curves are displayed using equivalent depths 10, 40, and 200 m, which correspond to phase speeds of ∼10, ∼20, and ∼45 m s⁻¹, respectively. The 10-m (10 m s⁻¹) curve intersects the MJO peak at eastward wavenumber 1 with 50-day period. This does not imply that the MJO can be interpreted as a Kelvin wave, rather it is just indicative of the eastward phase speed of the gravest zonal component of the MJO. Note, however, that the MJO peak at ∼50 day period extends to higher wavenumbers, which is an indication of the actual phase speed (i.e., the implied phase speed for the peak at wavenumber 2 is 5 m s⁻¹) and confinement of the convective component of the MJO across the Indian Ocean and western Pacific warm pools (e.g., Hendon and Salby 1994). The 40-m (20 m s⁻¹) curve is representative of the Kelvin waves for wavenumbers 2–6, whose periods range from ∼15 to ∼4 days. At higher wavenumbers and frequencies, the Kelvin wave peak tends to shift to lower phase speed (see also WK99). The 200-m (45 m s⁻¹) curve is representative of waves with a vertical wavelength of approximately twice the depth of the troposphere, which is the anticipated peak projection response to deep convective heating (i.e., the gravest baroclinic structure in a dry troposphere; Fulton and Schubert 1985; Salby and Garcia 1987). The main point to be emphasized here is that the Kelvin waves are propagating distinctly faster than the MJO and that both the Kelvin waves and the MJO, which are associated with deep convective heating, exhibit much slower phase speed than anticipated from dry dynamics (e.g., WK99).

For low westward wavenumbers less than about 6, the spectral peak tends to fall upon the dispersion curves for the first symmetric equatorial Rossby wave with an equivalent depth between 20 and 40 m. Thus, westward power tends to peak on equatorial Rossby wave dispersion curves with lower equivalent depth than for the eastward Kelvin waves. Peaks in the power spectrum are apparent at higher westward frequencies and wavenumbers (periods less than 10 days for wavenumbers greater than 6), which WK99 associated with “asymmetric” Rossby waves (e.g., easterly waves) that propagate along the convergence zone in one hemisphere. The additional dispersion curves for Rossby–Haurwitz waves (dashed curves in Figs. 3, 4) will be discussed below.

In the alternate display of the power spectrum of OLR (Fig. 3b) the Kelvin waves appear as a local spectral maximum, which results partly from “compression” of the spectrum by use of the logarithmic frequency axis. Furthermore, in this alternate display, it is now apparent that the integrated power associated with the Kelvin waves exceeds that for the equatorial Rossby waves and is, thus, second only to the MJO for prominence. An additional benefit of the alternate display of the power spectrum is that the MJO peak is moved well away from the frequency origin, while still allowing a clear separation from the higher-frequency Kelvin waves. A downside to the alternate display is that the spectral peak for the MJO now appears to be at higher frequency. Also, the peaks in the power spectrum associated with the higher wavenumber components of the MJO are not clear, and the Kelvin waves appear to be dispersive (i.e., dispersion curves are not straight lines).

Turning to the spectrum of U₈₅₀ (Fig. 4), we see most of the features identified in the spectrum of OLR, though with some important variations. Starting with the MJO, we see that the significant spectral peak at ∼50 day period is confined to eastward waves 1–3, whereas for OLR it extends out to at least wave 10. This difference reflects an important characteristic of the dynamic response to localized convective heating (see discussion in Salby and Hendon 1994). Another interesting feature is that the Kelvin wave signal in U₈₅₀, with strength similar to that for OLR, shows more of a tendency to be concentrated along a single dispersion curve (in this case with an equivalent depth slightly greater than 40 m), whereas for OLR the Kelvin wave peak tends to shift to slower phase speeds at higher wavenumbers.

At westward wavenumbers, the equatorial Rossby waves are not prominent in the spectrum of U₈₅₀. Whether this stems from inadequacies of the atmospheric analyses is not known. In contrast to OLR, however, spectral peaks for low westward wavenumbers, but at higher frequencies, are apparent. These peaks are especially prominent in the alternate display of power, where peaks are seen at westward wavenumber 1 with ∼5 day period, westward wavenumber 4 with ∼6 day period, and westward wavenumber 1 with ∼15 day period. Previous studies have detected similar spectral peaks in fields, such as surface pressure and stratospheric winds and temperatures, and have associated them with planetary, external (barotropic) Rossby–Haurwitz waves (e.g., Madden 1978, 2007; Kasahara 1980; Salby 1984). For instance, the 5-day peak at wavenumber 1 has been associated with the gravest symmetric (n = 1) external Rossby–Haurwitz wave, while the peak at 15-day period for wavenumber 1 has been associated with the second symmetric (n = 3) external Rossby–Haurwitz wave.

We provide further evidence that these peaks are associated with external Rossby–Haurwitz waves by considering the theoretical dispersion curves (dashed curves in Figs. 3 and 4). Although the external Rossby–Haurwitz waves are not equatorially trapped and thus should be investigated using spherical geometry, Lindzen (1967) has shown that an equatorial β-plane approximation is adequate for the gravest meridional modes. The dispersion curves for the low-order external Rossby–Haurwitz waves are thus approximated by the dispersion curves for equatorial Rossby waves (Matsuno 1966), but with an equivalent depth set to 10 km (Kashara 1980). However, the global nature of the Rossby–Haurwitz waves and their relatively slow phase speeds means that advection by the background zonal wind cannot be ignored, especially for the higher wavenumber modes that have slower phase speeds (Kashara 1980). As a simple approximation, the dispersion curves for the external Rossby–Haurwitz waves are computed here using a constant westerly advecting velocity of 15 m s⁻¹, which is meant to be representative of the midlatitude “equivalent barotropic level” (Kasahara 1980). The correspondence of the aforementioned peaks in the spectrum of U₈₅₀ with the approximated dispersion curves (Fig. 4) is outstanding, especially for wavenumbers 1, 3, and 4 for n = 1 and wavenumber 1 for n = 3. While it is not the purpose of this study to fully describe the horizontal and vertical structure of these barotropic planetary waves, we will show below that these spectral peaks are also associated with pronounced peaks in vertical coherence of the zonal wind through the depth of the troposphere and into the stratosphere, which provides further evidence that these peaks are associated with the external Rossby–Haurwitz waves. The lack of a pronounced signature of these waves in convection (Fig. 3) is consistent with their barotropic structure.

c. Space–time coherence spectrum of symmetric OLR and U₈₅₀

Additional insight into the organization of convection into equatorial wave modes and into the interaction of convection with the large-scale circulation is provided by the coherence spectrum between symmetric OLR and U₈₅₀ (Fig. 5a). The most prominent feature is the peak coherence associated with the MJO, with coherence squared exceeding 0.6 at wavenumber 1 with ∼50 day period. This implies that ∼60% of the variance of U₈₅₀ can be accounted for by OLR fluctuations at this wavenumber and frequency, consistent with the MJO signal strength in the individual power spectra. Peak coherence squared of up to ∼0.3 occurs for the Kelvin waves (wavenumbers 2–6 for periods 5–10 days, concentrated along the 40-m dispersion curve). The peak signal strength for the Kelvin waves in U₈₅₀ (Fig. 4) is about 60%, which implies that about half of the signal of the Kelvin waves in U₈₅₀ can be accounted for by coherent variations with OLR. Peak coherence also falls upon the dispersion curves for equatorial Rossby waves. It is interesting that the equatorial Rossby waves show up distinctly in the coherence of OLR with U₈₅₀, but their signature in the spectrum of U₈₅₀ (Fig. 4) is relatively modest.

The vectors in Fig. 5 provide information about the phase lag between OLR and U₈₅₀. For the MJO peak, the southwest pointing vectors imply that positive U₈₅₀ lags negative OLR by about 1/8 cycle or, equivalently, by about 5 days. This agrees with previous studies of the phasing of U₈₅₀ with OLR for the MJO: maximum westerlies are shifted some 5 days to the west of maximum convection (e.g., Hendon and Salby 1994). A similar 1/8 cycle phase lag is seen for the Kelvin waves with wavenumbers 2–4. But, interestingly, at higher wavenumber the lag reduces. For instance, at wavenumber 6 and ∼4 day period, negative OLR is indicated to be in phase with positive U₈₅₀. Whether this reflects a fundamental change in the phasing of local convection and winds with increasing frequency or is an artifact of the Fourier analysis, stemming, for instance, from localization of the convective variability, is unclear. For the westward equatorial Rossby waves, the southeast pointing vectors indicate that positive U₈₅₀ leads negative OLR by ∼1/8 cycle. Considering that these waves are propagating westward, this phase structure is similar to that for the MJO: maximum westerlies are shifted ∼1/8 cycle to the west of maximum convection.

A peak in the coherence spectrum between OLR and U₈₅₀ also occurs at westward wavenumber 1 with 5-day period, which we have associated with the global 5-day wave (i.e., the barotropic Rossby–Haurwitz wave for n = 1, k = 1 with 5-day period). While the magnitude of coherence squared is relatively weak (<0.1), the signal is deemed to be significant. The phase angle indicates that negative OLR (convection) leads U₈₅₀ by 1/4 cycle. Considering the structure of the westward propagating Rossby–Haurwitz wave [equatorial westerlies in phase with low surface pressure, Madden (1978)], this implies that enhanced convection tends to occur in phase with falling pressure (i.e., enhanced convection leads low surface pressure and westerlies by 1/4 cycle). Patel (2001), building on the earlier work of Burpee (1976), detected a similar phasing of lightning modulation across equatorial Africa with respect to the 5-day wave. The present analysis cannot reveal whether the 5-day wave is modulating convection or whether random convection, with deep vertical structure such that it projects onto the barotropic structure of the 5-day wave, is exciting the 5-day wave (e.g., Miyoshi and Hirooka 1999). However, the lack of a clear spectral peak in the OLR spectrum (Fig. 3) suggests the latter.

d. Power and coherence of symmetric zonal wind at 150 and 50 hPa

Space–time power and cross-power spectra are also examined for the symmetric zonal wind at 150 hPa (U₁₅₀), which we take to be at the level of maximum divergence associated with deep convection, and at 50 hPa (U₅₀), which is representative of the lower stratosphere. The space–time power spectra for these fields are shown in Fig. 6. The spectrum of U₁₅₀ is similar to that for U₈₅₀ except there is now less of a break in the spectrum between the MJO and the higher-frequency Kelvin waves. For U₅₀, there is barely any evidence of a spectral peak associated with the MJO, but now there is a distinct peak associated with eastward wavenumber 1 at ∼15-day period, which we interpret to be the signal of the stratospheric Kelvin waves detected by Wallace and Kousky (1968). The lack of a signature of the MJO in U₅₀ is consistent with the notion that the MJO, due to its low frequency, possesses a small vertical wavelength above the tropopause (e.g., Salby and Garcia 1987; Wheeler et al. 2000): hence, it propagates slowly in the vertical and is subject to dissipation. It is also apparent in Fig. 6 that the spectral peak associated with the Kelvin waves tends to occur at higher frequencies at 50 hPa than at 150 hPa, consistent with the notion that the higher-frequency Kelvin waves are able to propagate vertically more readily into the stratosphere because they have longer vertical wavelengths and greater upward group velocity (e.g., Salby et al. 1984; Andrews et al. 1987). The spectral peaks associated with the external Rossby–Haurwitz waves are also prominent at 150 and 50 hPa, which are expected owing to their barotropic structure.

The association of the equatorial wave modes in the upper troposphere and lower stratosphere with convection is indicated by the coherence squared of U₁₅₀ and U₅₀ with OLR (Figs. 5b and 5c, respectively). The coherence of OLR with U₁₅₀ is similar to that of OLR with U₈₅₀, but with a generally opposite phase lag relationship, which reflects the deep baroclinic structure of the circulation associated with convection. The peak coherence for the MJO and Kelvin waves is similar to that for U₈₅₀, but it is weaker for the equatorial Rossby waves. This is consistent with previous work that has shown that the equatorial Rossby waves are most prominent in the lower troposphere (e.g., Kiladis and Wheeler 1995).

The coherence of OLR with U₅₀ is dominated by the Kelvin signal, but now weaker and shifted to higher frequency than for OLR with U₁₅₀ and U₈₅₀. The shift to higher frequency from that in the troposphere is again consistent with the idea that the higher frequency waves will penetrate more readily into the stratosphere because they have longer vertical wavelengths and greater vertical group velocity, while the lower frequency waves that have shorter vertical wavelengths and lower vertical group velocities will be dissipated soon after entering the lower stratosphere (e.g., Andrews et al. 1987). The vertical wavelength of the Kelvin waves that are associated with tropospheric convection could, in principal, be computed by comparing the phase angle at successive levels in the vertical. While we cannot unambiguously compute the vertical wavelength using just one level in the stratosphere, we do note that the phase angle between OLR and U₁₅₀ is about 180° shifted from that between OLR and U₅₀ along the axis of peak coherence associated with the Kelvin waves (e.g., at wavenumber 2 with ∼7 day period). Hence, an implied vertical wavelength is about twice the distance between the tropopause and the 50-hPa level (assuming the wavelength is not shorter than the distance between these two levels). This suggests a vertical wavelength of about 10 km, which is consistent with that for the Kelvin waves detected by Wallace and Kousky (1968). We also note that, at a given zonal wavenumber, the phase lag between OLR and U₅₀ decreases with increasing frequency, consistent with the notion that the higher frequency waves have longer vertical wavelengths (e.g., Andrews et al. 1987).

Comparison of the peak coherence squared along the Kelvin wave dispersion curve (∼0.3 in Fig. 5c) with the Kelvin wave signal strength in U₅₀ (∼60% from shading in Fig. 6b) suggests that about half of the Kelvin wave signal in U₅₀ can be accounted for by fluctuations of tropical convection. However, this figure is probably a lower limit because convectively forced stratospheric Kelvin waves of the same frequency and zonal wavelength will have different vertical wavelengths depending on the sign and strength of the zonal mean zonal wind, which varies significantly with the quasi-biennial oscillation (QBO) (e.g., Andrews et al. 1987). The coherence statistic will be reduced when westerly and easterly QBO years are lumped together because the phase lag of U₅₀ with respect to OLR will be systematically different for west years and east years. It is also interesting to note that there is no signal at 50 hPa of the tropospheric equatorial Rossby waves associated with convection (Figs. 5a and 5b), consistent with the notion that these westward Rossby waves, with low westward phase velocity, cannot propagate vertically into easterlies that predominate in the equatorial lower stratosphere.

e. Vertical coherence of symmetric zonal wind

We also examine the vertical coherence of the zonal wind to provide more insight into the external Rossby–Haurwitz waves and to further explore the shift to higher frequency as the Kelvin waves propagate from the troposphere into the stratosphere. Figure 7 displays the coherence squared of U₈₅₀ and U₅₀ with U₁₅₀. The coherence of U₈₅₀ with U₁₅₀ is dominated by the MJO at eastward wavenumber 1 with 50-day period. At higher eastward frequencies, the coherence falls upon the Kelvin wave dispersion curves, but with a wider range of equivalent depths than for the coherence of the zonal wind with OLR (Figs. 5a and 5b). This indicates that interaction of convection and dynamics mainly occurs within a fairly narrow range of vertical structure (i.e., peaked along the dispersion curve for 40-m equivalent depth), but Kelvin waves, with a wide range of vertical structure, can be excited by stochastic convective forcing (Salby and Garcia 1987), for instance, or by dynamical interaction with the extratropics (Hoskins and Yang 2000). The coherence of U₅₀ with U₁₅₀ further demonstrates this, as peak coherence is now shifted to even higher frequency (larger equivalent depth) than for the coherence of U₅₀ with OLR. We expect the peak coherence between the upper troposphere and lower stratosphere to be associated with waves that have the longest vertical wavelength, as they will propagate most rapidly in the vertical. Although Kelvin waves with a wide range of equivalent depth are evident in U₁₅₀, it is those with the largest equivalent depth that propagate most readily into the stratosphere. Again, the roughly out-of-phase relationship between 150 and 50 hPa along the axis of maximum coherence (e.g., wavenumber 1 at ∼15-day period and wavenumber 2 at ∼7-day period) implies a vertical wavelength of about 10 km, which is consistent with that inferred from the coherence with OLR (Fig. 5) and with Wallace and Kousky (1968). The clockwise rotation of the phase lag with increasing frequency at a given zonal wavenumber indicates decreasing phase lag, which again is consistent with an increasing vertical wavelength with frequency, as anticipated from theory (Andrews et al. 1987).

At westward frequencies, the most dominant signal for the coherence of U₁₅₀ with both U₈₅₀ and U₅₀ is the Rossby–Haurwitz waves. The near-zero phase lag indicates the barotropic nature of these waves. Although there is some indication of the equatorial Rossby waves at low wavenumbers in the coherence of U₁₅₀ with U₈₅₀ (note the out-of-phase relationship between U₈₅₀ and U₁₅₀), this analysis emphasizes the prominence of the Rossby–Haurwitz waves for coherent variations in the troposphere.

f. Antisymmetric variations

These space–time spectral analyses can also provide insight into behavior that is antisymmetric about the equator. Figure 8 displays the power spectrum of antisymmetric OLR. Similar to the analysis of WK99, the prominent spectral peak falls upon the Rossby–gravity dispersion curve for equivalent depth 20–40 m. The enhanced power seen at low eastward wavenumber and frequency is presumably associated with asymmetric behavior of the MJO (e.g., WK99). The coherence of antisymmetric U₈₅₀ with OLR also exhibits a peak that falls upon the Rossby–gravity wave dispersion curve and coincides with the spectral peak in antisymmetric OLR. At westward wavenumbers the phase lag indicates that convection leads westerlies by ∼1/4 cycle, which is consistent with previous studies of westward Rossby–gravity waves that are coupled with convection (e.g., Hendon and Liebmann 1991). Coherence of antisymmetric OLR with antisymmetric U₅₀ is shown in Fig. 8c. Again, peak coherence falls upon the Rossby–gravity dispersion curves for equivalent depths 20–40 m, which is indicative that the convectively coupled Rossby–gravity waves detected in the troposphere (Figs. 8a and 8b) propagate vertically into the stratosphere. However, the coherence is weak, indicating that the signal of the convectively coupled Rossby–gravity waves is relatively weak in the stratosphere. This is expected because the vertical wavelength (and vertical group propagation) above the tropopause of these relatively low frequency Rossby–gravity waves is expected to be small (e.g., Wheeler et al. 2000).