## 1. Introduction

Tropical convection varies coherently over a large range of spatial and temporal scales. It includes phenomena ranging from mesoscale weather systems up to planetary-scale signals such as the Madden–Julian oscillation (MJO). The importance of convective activity in the tropics and especially coherent structures such as convectively coupled equatorial (CCE) waves has led to an extensive ongoing effort to incorporate these processes into both weather and climate models (e.g., Arakawa 2004). While considerable progress has been made, there still remain significant limitations both in simulations as well as theoretical understanding.

There is general agreement that the understanding and simulation of CCE waves and the MJO forms an important part of this effort. The work of Wheeler and Kiladis (1999, hereafter WK99) and Wheeler et al. (2000), as well as more recent studies (Yang et al. 2007a,b,c), has led to considerable progress in recent years in using observational analyses to deepen understanding of convective phenomena. There remain, however, significant difficulties with their simulation in general circulation models (GCMs). Such models tend to underestimate intraseasonal variability and the simulated propagation speeds are often inaccurate (Lin et al. 2006). Many GCMs manage to produce CCE waves but have difficulties in reproducing the correct frequencies and phase speeds (Yang et al. 2009). Since a considerable amount of the variability in the tropics is associated with these low-frequency phenomena that models that cannot adequately simulate, these clearly will have deficiencies in their simulations and predictions.

Theoretical studies of the above variability have been extensive but it is fair to say as yet inconclusive. Many different proposed mechanisms exist that serve to explain various aspects of the observational evidence but a comprehensive theory remains elusive. Early efforts focused on linear theory via moist feedback processes such as conditional instability of the second kind (CISK) and wind-induced surface heat exchange (WISHE) [for a recent review with an MJO emphasis, see Zhang (2005)], whereas in recent years nonlinear mechanisms of scale transfer have been emphasized because certain features of convection tend to show self-similarity across a range of horizontal scales (Mapes 2000; Majda and Shefter 2001; Lovejoy et al. 2008). Another feature of tropical convection is its organization into convective systems on synoptic to large scales. The use of multiscale cloud models has shown some success in reproducing these coherencies (Tao and Moncrieff 2009, and references therein), but convective organization is still not very well represented in general by convective parameterizations.

Given the deficiencies in model simulations as well as the wide range of theoretical explanations, it seems natural to turn to increasingly reliable and comprehensive observational datasets and perform theoretically informed analyses. Because of the increased availability of long-time satellite observations in recent years, it has been possible to collect evidence that baroclinic equatorial waves can produce large-scale coherent features in the tropics. Takayabu (1994) and WK99 show evidence that space–time spectra of outgoing longwave radiation (OLR) have statistically significant peaks that lie along dispersion relations of linear shallow-water (LSW) waves corresponding to a baroclinic vertical mode. We will refer to these as baroclinic LSW waves in the following.

A number of observational studies have investigated the structure of tropical waves (e.g., Wheeler et al. 2000; Straub and Kiladis 2003; Yang et al. 2007a). These studies focused on finding the horizontal structures through regressions between the OLR data and reanalysis and found that the horizontal structures of the dynamical fields are generally consistent with LSW theory. While the study of WK99 focused on symmetric and antisymmetric features of the waves, further work has been done since that points out the meridional asymmetry of tropical waves not seen in linear theory (Roundy and Frank 2004; Chao et al. 2009). A recent review on CCE waves that addresses many of the issues mentioned above can be found in Kiladis et al. (2009).

Another important aspect of the work by Takayabu (1994) and WK99 is that the spectrum of deep tropical convection is red in frequency and zonal wavenumber space. On top of a red background spectrum there are peaks which follow the theoretical dispersion relations or are locally concentrated (the MJO). These modes show red behavior as well, especially in zonal wavenumber *k*. Previous spectral studies of these waves have used diverse techniques to remove the background spectrum and find the statistically significant peaks. However, it is not clear what physical processes set the background spectrum (Hendon and Wheeler 2008; Chao et al. 2009).

The observed peaks in the space–time spectra align with dispersion curves of waves that can be obtained theoretically by solving the LSW equations on an equatorial *β* plane. The wave solutions then have the parabolic cylinder functions (PCFs) as their meridional basis. This motivates using these functions as a basis for studying CCE waves (Matsuno 1966; Gill 1980; Lindzen 2003; Yang et al. 2003). The work of WK99 suggested the result that LSW dynamics are crucial but did not elaborate it thoroughly through a parabolic cylinder decomposition and through the study of the full range of LSW dynamical variables. Yang et al. (2003) did consider the PCF decomposition and coherent wave structures; however, that study was for a short time period only and excluded in its formulation the noncoupled equatorial waves and other phenomena. We view our work as a more comprehensive demonstration that relatively simple LSW dynamics are dominant in tropical convection. This study covers the longest full observational record available (23 yr) and the full range of relevant LSW dynamical variables. The projection of all relevant dynamical variables onto PCFs allows us to test the hypothesis that LSW dynamics are dominant. This follows since LSW dynamics implies characteristically different projections of the various dynamical variables and such behavior can be directly detected (or not) from the data using the PCF projection method.

The paper is organized as follows. Section 2 gives a brief review of linear shallow-water theory and sets the notation for the rest of the paper. The data and methodology are discussed in section 3. The single PCF spectra for the brightness temperature *T _{b}* and the reanalysis variables are presented in section 4 and 5, respectively. A summary is given in section 6.

## 2. Linear equatorial waves

*β*plane goes back to Matsuno (1966) and a detailed derivation can be found in Holton (2004). The LSW solutions then have the PCFs, here denoted by

*D*(

_{m}*y*), as basis functions in the meridional direction. The first six PCFs are shown in Fig. 1; the first PCF has no zeros and the shape of a Gaussian centered at the equator. Higher PCFs have an increasing number of zero crossings and larger meridional extent. The solutions for zonal wind

*U*, meridional wind

_{n}*V*, and geopotential

_{n}*Z*(

_{n}*n*= −1, 0, 1, 2, …) can be written in the form

*c*is the shallow-water speed for the

_{l}*l*th vertical structure function. In Eq. (1) indices

*n*less than 0 implicitly imply vanishing of the PCF and the respective term in the equation. For example, for

*n*= −1 only the

*D*

_{n}_{+1}terms in the

*U*and

*Z*equations are nonzero and there is no meridional flow. Here

*ξ*=

*y*/

*y*

_{0}and the choice of the trapping scale factor

From Eq. (1) it is important to observe that the variables show different behavior with respect to the PCFs. For each *n*, *U _{n}*,

*V*, and

_{n}*Z*show characteristically different projections onto the PCFs

_{n}*D*

_{n}_{−1},

*D*, and

_{n}*D*

_{n}_{+1}. Additionally for

*n*= −1 only the Kelvin wave exists with projection onto the first PCF

*D*

_{0}in

*U*and

*Z*and no meridional flow. For

*n*= 0 the mixed Rossby–gravity (MRG) wave occurs, which has a second PCF structure in

*U*and

*Z*and a first PCF structure in

*V*. For

*n*> 0 both Rossby and gravity waves occur with differing projections given by Eq. (1).

It is thus possible to infer using the set of PCF space–time spectra whether or not observed CCE waves project in the same characteristic way as the corresponding theoretical LSW modes. For example, if only linear processes were present, we would expect to see the CCE Kelvin wave signal in the first PCF spectrum of *U* and *Z* (providing the trapping scale is chosen appropriately; see below). Similarly, the CCE MRG waves should then have a signal in the second PCF spectrum for *U* and *Z* and in the first PCF spectrum of *V*. The CCE Rossby and gravity waves project onto all PCFs for *U* and *Z*, but for *V* these waves should not project onto the first PCF.

## 3. Data and methodology

### a. Data

We used the Cloud Archive User Service (CLAUS) *T _{b}* dataset (Hodges et al. 2000), since

*T*is a proxy for precipitating deep convection, and the data have a horizontal resolution of ⅓° with records available 8 times daily. The dataset has records starting on 1 July 1983 and ending 30 June 2006. The advantage of this dataset is that there are very few data missing in the equatorial region and therefore there is a very long continuous record available for spectral transformations.

_{b}As there are no direct global observations of the dynamical variables wind, divergence, and geopotential, we instead make use of reanalysis products. These are available globally at a lower resolution than the CLAUS data of 2.5° in the zonal and meridional direction and are updated every 6 h.

The records of the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) for these variables are available for download at the National Center for Atmospheric Research (NCAR) [for more information on the ERA-40 data, see Uppala et al. (2005)]. We chose the period from 1 July 1983 through 30 June 2002 because that overlaps the most with the dates where CLAUS data are available. The National Centers for Environmental Prediction (NCEP) reanalysis data (Kalnay et al. 1996) were obtained from the National Oceanic and Atmospheric Administration (NOAA) and we used the same start and end dates as ERA-40.

Since the data depend to some extent on the reanalysis methodology, we used both the ERA-40 and the NCEP reanalysis data and critically compared the results in order to address robustness issues. If similar results are seen with both datasets, this suggests that we are seeing actual features of the data rather than reanalysis technique artifacts.

### b. Decomposition into parabolic cylinder functions

The first step was to project the data onto the PCFs that form a basis in the meridional direction. This was done at each longitude grid point separately. Outside of ±20° the data vector is extended with a constant value for each zonal data point. Experiments show that the coefficients computed from the *T _{b}* data by this method are not sensitive to the value used outside of ±20°, but for the reanalysis wind and divergence data the value has to lie in the range of values the data assume between +20° and −20°, so here we set the value outside of ±20° equal to zero.

**f**, which contains all the values at grid points between +20° and −20° (

*N*

_{0}points), is padded with constant entries at additional meridional grid points outside of ±20° to bring the total length of

**f**to

*N*, and is then projected onto the first

*M*PCFs by finding the vector of coefficients

**a**(

*x*,

*t*) of length

*M*that satisfies

*N*,

*M*) matrix with the columns containing the values of the first

*M*PCFs evaluated at the same grid points as

**f**. In general

**a**(

*x*,

*t*) then contains the PCF coefficients that best approximate the data

**f**in a least squares sense.

### c. Computing the spectrum

To compute the spectrum we mainly follow WK99, but instead of using the raw data we use the PCF coefficients that we obtained in the previous subsection and perform the fast Fourier transform (FFT) on these. Note that these coefficients now depend only on time and zonal direction.

Since we are interested in synoptic time scales and would like to include the MJO, we split the record into 128-day-long segments. As in WK99 we let these segments overlap by 2 months to reduce the loss of information through windowing. We then remove the mean and linear trend in time from each segment and calculate the spectral coefficients by using a 2D FFT in time and zonal direction. The spectrum can then be obtained from these coefficients by squaring them and taking the mean over all available segments. The effective bandwidth of the result is *ω* and 1 unit zonal wavenumber.

The advantage of computing the PCF decomposition first and then computing the Fourier transforms to find the zonal wavenumber–frequency spectra is that we can easily compute spectra for any subset of PCFs. For example, using only the coefficients for symmetric PCFs we recover the symmetric spectrum and by using only coefficients of the antisymmetric PCFs we recover the antisymmetric spectrum. In the same way we can compute frequency–zonal wavenumber spectra for single PCFs.

It has become customary to remove a background spectrum to facilitate the identification of statistically significant spectral peaks in the spectra (WK99; Hendon and Wheeler 2008). Nevertheless there are several reasons for also including the full spectra. First, the red background spectrum is interesting in its own right and indeed there is no convincing explanation that we are aware of in the literature for why the spectrum is red in either the frequency or zonal wavenumber direction. Second, the precise way in which this redness manifests itself and varies between dynamical variables and pressure levels is of interest. Third, it is not clear what process defines the red background spectrum, and choosing one process over another to model the background always adds an implicit physical model. Thus, here we present the PCF spectra in two ways: without removing a background, and again for visual clarity with the background removed. In particular we compute the background spectra for each PCF spectrum separately, as done by Chao et al. (2009) for symmetric and antisymmetric spectra. They pointed out that the asymmetric nature of, for example, CCE Kelvin waves is better represented by comparing the “raw” spectra to their separate backgrounds. Using one background for all PCF spectra can lead to small peaks in one of the spectra being overpowered by the contribution of the spectrum of a different PCF with more power in that region.

### d. Discussion of the trapping scale

For the choice of the trapping scale (Gill 1982) we use the value *y*_{0} = 6° indicated in WK99 and Yang et al. (2003). The latter found the trapping scale by minimizing the meridional wind mean square error and showed that the best fit gives a *y*_{0} around 6°. This corresponds to a gravity wave speed of *c* = 20 m s^{−1}. They also experimented with a range of values for *y*_{0} between 5° and 8° and found that the same structures were represented as the same theoretical wave.

Features with a trapping scale that is close to the assumed *y*_{0} will project onto only one or two lower PCFs. Signals of features with a wider meridional scale than the assumed *y*_{0} will be visible in the higher PCFs as well. If these features are symmetric (antisymmetric) then they will project onto symmetric (antisymmetric) PCFs. This allows the identification of equatorially trapped structures from the PCF spectra and inferences about the meridional structure and scale of nonequatorially trapped phenomena.

Figure 2 shows the first three PCF spectra for *T _{b}* for

*y*

_{0}= 4° (Figs. 2a–c) and 8° (Figs. 2d–f). In both cases the Kelvin wave peak is visible in both the symmetric PCF-1 and -3 spectra. A similar duplication holds for the MRG wave dispersion curve, which is apparent in both antisymmetric PCF-2 and -4 spectra (not shown). Thus, in general choosing a trapping scale that differs significantly from

*y*

_{0}= 6° leads to leakage of the power of these waves into higher PCFs with the same symmetry. When using

*y*

_{0}= 6° this seems to be only minimally the case (see the following sections), confirming this as the optimal choice for the trapping scale.

Zonal wavenumber–frequency *T _{b}* log power spectra for single PCFs with (a)–(c)

*y*

_{0}= 4° and (d)–(f)

*y*

_{0}= 8°. The panels show PCF-1–3 spectra with the background removed. PCFs 1 and 3 are symmetric across the equator; PCF 2 is antisymmetric. Superimposed are the dispersion curves for theoretical linear shallow-water waves.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency *T _{b}* log power spectra for single PCFs with (a)–(c)

*y*

_{0}= 4° and (d)–(f)

*y*

_{0}= 8°. The panels show PCF-1–3 spectra with the background removed. PCFs 1 and 3 are symmetric across the equator; PCF 2 is antisymmetric. Superimposed are the dispersion curves for theoretical linear shallow-water waves.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency *T _{b}* log power spectra for single PCFs with (a)–(c)

*y*

_{0}= 4° and (d)–(f)

*y*

_{0}= 8°. The panels show PCF-1–3 spectra with the background removed. PCFs 1 and 3 are symmetric across the equator; PCF 2 is antisymmetric. Superimposed are the dispersion curves for theoretical linear shallow-water waves.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

In contrast to Yang et al. (2003) we do not filter the data to retain only frequencies of 3–30 days and wavenumbers 2–10, but we work with longer records and as many wavenumbers as are possible due to the longitudinal resolution to compute the spectra. Their technique further uses linear regression between *T _{b}* and the dynamical variables and focuses on finding and analyzing coherent structures of CCE waves in the summer of 1992. This excludes the noncoupled equatorial waves and other phenomena that are visible in the spectra produced here. By averaging over the segments, the spectra here present a climatology of the distribution of power in wavenumber–frequency space over the analyzed years.

As the computed spectra show signals not only of CCE waves but also other waves (not necessarily trapped at the equator), the background spectrum and extratropical disturbances, it is necessary to consider how these signals project onto the PCFs. An important factor to consider is the modification of equatorial waves by a nonresting basic state. The LSW solutions are solutions for a resting atmosphere and dispersion curves and meridional structure of the waves will change with a nonzero mean wind. Doppler effects will modify the dispersion relations while non-Doppler effects can also affect the meridional structure. In particular zonal shear flows can expand or reduce the meridional trapping scale of equatorial waves. However, the main change in the meridional structure happens outside of ±20° for most of the waves (Zhang and Webster 1989; Webster and Chang 1997) while the near-equatorial meridional structures remain close to the LSW structures. Furthermore, there is evidence of extratropical forcing projecting directly onto equatorial wave modes (Hoskins and Yang 2000) and nonlinear energy transfer from extratropical Rossby waves to equatorial waves (Majda and Biello 2003). Kasahara and Dias (1986) found that the equatorially trapped baroclinic modes are less affected by inclusion of the zonal flow and remark that the findings of Gill (1980) may be applicable to more general situations than the zero background flow. This indicates that although there are differences in the meridional structures and dispersion relations of CCE and LSW waves, these differences may be small. There are additional factors such as nonlinearities, surface friction, vertical shear, and convection that can modify the equatorial wave structures. This should be kept in mind when interpreting the results.

## 4. CLAUS data

The PCFs form an orthogonal basis on the equatorial *β* plane. If we restrict ourselves to the tropics and use the *β*-plane approximation we can represent any function by its (infinite) PCF series. This means that by decomposing data into PCFs in the meridional direction and keeping enough terms of this series we can approximate the original data arbitrarily well.

In reality not very many terms of the PCF series are necessary to approximate well the *T _{b}* data. Figure 3 shows good agreement between the spectra computed by WK99 (see their Fig. 1) and the spectra computed using the decomposition into 10 PCFs. Small differences between the spectra are to be expected since the plot shows

*T*instead of OLR, the spectra here were calculated using longer time segments and only the part of the data that projects onto the first 10 PCFs contributes.

_{b}Zonal wavenumber–frequency log power spectra of brightness temperature data computed using 10 PCFs for the entire length of the record from 1983 to 2006. (left) Symmetric spectrum with superimposed theoretical dispersion curves for the first two odd-meridional-mode linear equatorial waves for three different shallow-water speeds: 10, 15, and 25 m s^{−1}. (right) Antisymmetric spectrum with superimposed even-meridional-mode linear equatorial wave dispersion curves. The power has been summed over all available latitudes between +20° and −20°.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency log power spectra of brightness temperature data computed using 10 PCFs for the entire length of the record from 1983 to 2006. (left) Symmetric spectrum with superimposed theoretical dispersion curves for the first two odd-meridional-mode linear equatorial waves for three different shallow-water speeds: 10, 15, and 25 m s^{−1}. (right) Antisymmetric spectrum with superimposed even-meridional-mode linear equatorial wave dispersion curves. The power has been summed over all available latitudes between +20° and −20°.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency log power spectra of brightness temperature data computed using 10 PCFs for the entire length of the record from 1983 to 2006. (left) Symmetric spectrum with superimposed theoretical dispersion curves for the first two odd-meridional-mode linear equatorial waves for three different shallow-water speeds: 10, 15, and 25 m s^{−1}. (right) Antisymmetric spectrum with superimposed even-meridional-mode linear equatorial wave dispersion curves. The power has been summed over all available latitudes between +20° and −20°.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

By considering the spectra of the projection onto the first PCFs separately (Fig. 4), we obtain an idea about where in zonal wavenumber–frequency space each of these PCFs peak and how important the single PCFs are compared to each other. The maximum power decreases with increasing PCF number and the spectra of PCF 5 or higher do not show any peaks above the background. The spectrum of PCF 4 shows peaks above the background only occasionally; therefore these are not shown but will be mentioned where applicable.

Zonal wavenumber–frequency *T _{b}* log power spectra for single PCFs. (a),(c),(e) PCF 1–3 spectra, respectively; (b),(d),(f) as in (a),(c),(e), but with the background removed. PCFs 1 and 3 are symmetric across the equator; PCF 2 is antisymmetric. Superimposed are the dispersion curves for theoretical linear shallow-water waves.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency *T _{b}* log power spectra for single PCFs. (a),(c),(e) PCF 1–3 spectra, respectively; (b),(d),(f) as in (a),(c),(e), but with the background removed. PCFs 1 and 3 are symmetric across the equator; PCF 2 is antisymmetric. Superimposed are the dispersion curves for theoretical linear shallow-water waves.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency *T _{b}* log power spectra for single PCFs. (a),(c),(e) PCF 1–3 spectra, respectively; (b),(d),(f) as in (a),(c),(e), but with the background removed. PCFs 1 and 3 are symmetric across the equator; PCF 2 is antisymmetric. Superimposed are the dispersion curves for theoretical linear shallow-water waves.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

The spectrum of the first PCF is similar to the symmetric spectrum. After dividing by a background spectrum peaks aligning with the theoretical dispersion curve for the 15–25 m s^{−1} LSW Kelvin, westward inertio-gravity (WIG) and Rossby waves become visible. The main feature in the PCF-2 spectrum is the peak of the MRG wave. It has been observed that CCE Kelvin waves are not necessarily symmetric across the equator (e.g., Straub and Kiladis 2002; Roundy and Frank 2004; Chao et al. 2009). This would imply a possible CCE Kelvin wave signal in the PCF-2 spectrum and although there is a signal in the region *k* = 5−10, *ω* = 0.1–0.3 cpd, it is not possible to tell from the spectra whether these are Kelvin or extratropical Rossby wave signals.

Interestingly, the shape of the spectrum of the higher modes (not shown) has some similarity to the background spectrum calculated in WK99 and more power in the negative wavenumbers than in the positive. This is possibly due to westward moving extratropical Rossby waves adding power.

## 5. Reanalysis

The main observation about the spectra for all the variables considered here is that they are red with respect to zonal wavenumber, frequency, and PCF number. This last feature can be seen in Fig. 5, which shows the PCF power spectrum of zonal wind at 850 hPa. The PCF spectra for the other variables and heights are not shown but exhibit the same qualitative behavior.

The log_{10} PCF power spectrum of ERA-40 zonal wind at 850 hPa. PCF coefficients are averaged over all times and zonal grid points. Spectrum is calculated for the entire reanalysis record from 1983 to 2002 and latitude range is ±20°.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

The log_{10} PCF power spectrum of ERA-40 zonal wind at 850 hPa. PCF coefficients are averaged over all times and zonal grid points. Spectrum is calculated for the entire reanalysis record from 1983 to 2002 and latitude range is ±20°.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

The log_{10} PCF power spectrum of ERA-40 zonal wind at 850 hPa. PCF coefficients are averaged over all times and zonal grid points. Spectrum is calculated for the entire reanalysis record from 1983 to 2002 and latitude range is ±20°.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

To obtain a measure of how well the PCFs approximate the original data we projected the data at one instant in time onto the first 20 PCFs and then reconstructed the data from the PCF coefficients. The fit is very good; the meridional wind and geopotential reconstructions with 20 PCF are of comparable accuracy. The divergence *D*, on the other hand, which has small-scale features, is not reconstructed as well.

Another characteristic of the frequency–zonal wavenumber spectra of all variables is the change from 850 to 200 hPa. The 200-hPa spectra all show more power in the positive zonal wavenumbers than the 850-hPa spectra (Fig. 6). We do not show the frequency–zonal wavenumber spectra at 925 hPa, as these are very similar to the 850-hPa spectra.

The log_{10} power spectra of ERA-40 divergence at different heights [(top) 850 and (bottom) 200 hPa] for (left) symmetric and (right) antisymmetric parts. Record length and latitude range are as in Fig. 5.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

The log_{10} power spectra of ERA-40 divergence at different heights [(top) 850 and (bottom) 200 hPa] for (left) symmetric and (right) antisymmetric parts. Record length and latitude range are as in Fig. 5.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

The log_{10} power spectra of ERA-40 divergence at different heights [(top) 850 and (bottom) 200 hPa] for (left) symmetric and (right) antisymmetric parts. Record length and latitude range are as in Fig. 5.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Comparing Fig. 6 of the symmetric and antisymmetric divergence spectra with the *T _{b}* spectra (Fig. 3), one observes reasonable similarity between the 200-hPa spectra of divergence and

*T*particularly for the CCE wave signals. This is also apparent in the single PCF spectra (Figs. 4 and 7) and in the 850-hPa divergence (not shown). The main differences between the divergence and

_{b}*T*spectra are weaker signals for WIG waves in the divergence PCF-2 spectra, stronger extratropical signals in the divergence spectra, and slower decay in the divergence spectra with zonal wavenumber for low frequencies. The reanalysis procedure assimilates satellite data and in particular brightness temperature satellite measurements into the analyses. Brightness temperature is considered a good proxy for diabatic heating, and divergence in the upper troposphere tends to be located above that, which could explain the close resemblance between these spectra.

_{b}Zonal wavenumber–frequency divergence power spectra at 200 hPa. (a),(c),(e) “Raw” PCF-1–3 spectra, respectively, without color bars; higher intensity is indicated by darker shading. (b),(d),(f) PCF-1–3 spectra with the background removed; shading starts at a value of 1.1 and darker shading signifies higher intensity. Superimposed are the theoretical dispersion lines of the LSW waves that project onto these PCFs. Solid lines indicate 15 and 25 m s^{−1} LSW waves, dotted lines 316 m s^{−1}, and dash-dotted lines include a 15 m s^{−1} zonal background flow. For visual clarity *n* = 1 and *n* = 2 Rossby and gravity waves are plotted with separate PCFs. Record length and latitude range are as in Fig. 5.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency divergence power spectra at 200 hPa. (a),(c),(e) “Raw” PCF-1–3 spectra, respectively, without color bars; higher intensity is indicated by darker shading. (b),(d),(f) PCF-1–3 spectra with the background removed; shading starts at a value of 1.1 and darker shading signifies higher intensity. Superimposed are the theoretical dispersion lines of the LSW waves that project onto these PCFs. Solid lines indicate 15 and 25 m s^{−1} LSW waves, dotted lines 316 m s^{−1}, and dash-dotted lines include a 15 m s^{−1} zonal background flow. For visual clarity *n* = 1 and *n* = 2 Rossby and gravity waves are plotted with separate PCFs. Record length and latitude range are as in Fig. 5.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Zonal wavenumber–frequency divergence power spectra at 200 hPa. (a),(c),(e) “Raw” PCF-1–3 spectra, respectively, without color bars; higher intensity is indicated by darker shading. (b),(d),(f) PCF-1–3 spectra with the background removed; shading starts at a value of 1.1 and darker shading signifies higher intensity. Superimposed are the theoretical dispersion lines of the LSW waves that project onto these PCFs. Solid lines indicate 15 and 25 m s^{−1} LSW waves, dotted lines 316 m s^{−1}, and dash-dotted lines include a 15 m s^{−1} zonal background flow. For visual clarity *n* = 1 and *n* = 2 Rossby and gravity waves are plotted with separate PCFs. Record length and latitude range are as in Fig. 5.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

After dividing the single PCF spectra by their background spectra (Figs. 7–10), signals of three types of phenomena become visible. The first is the convectively coupled equatorial waves, where the signals roughly follow the theoretical dispersion curves for the baroclinic LSW waves with a shallow-water speed around 15–25 m s^{−1}. The second type is barotropic linear waves with a phase speed around 300 m s^{−1}. These are not associated with convection and there are no signals of these in the *T _{b}* spectra. The third phenomenon is the signal of extratropical storm track activity in the eastward wavenumbers. In the following we go through these phenomena individually.

As in Fig. 7, but for zonal wind at 850 hPa.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for zonal wind at 850 hPa.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for zonal wind at 850 hPa.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for geopotential at 850 hPa.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for geopotential at 850 hPa.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for geopotential at 850 hPa.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for meridional wind at 850 hPa. The Kelvin wave curve is included for reference, even though the theoretical Kelvin wave does not project onto the meridional wind.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for meridional wind at 850 hPa. The Kelvin wave curve is included for reference, even though the theoretical Kelvin wave does not project onto the meridional wind.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

As in Fig. 7, but for meridional wind at 850 hPa. The Kelvin wave curve is included for reference, even though the theoretical Kelvin wave does not project onto the meridional wind.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

### a. Convectively coupled equatorial waves

The CCE waves that appear in the brightness temperature spectra and align with the dispersion curves of the LSW waves with reduced wave speed around 15–25 m s^{−1} have signals in the dynamical variables as well. How these waves project onto the PCFs for each variable depends on their meridional structure and thus allows inferences about this structure and the agreement between the observed waves and LSW theory.

The 15–25 m s^{−1} CCE Kelvin wave projects onto the first PCF for zonal wind, divergence, and geopotential (Figs. 7–9), which is consistent with LSW theory. The theoretical LSW Kelvin wave also has zero meridional velocity, but previous studies have found evidence that the CCE Kelvin wave can have a meridional circulation due to some asymmetry in its structure across the equator (Straub and Kiladis 2002, 2003; Chao et al. 2009). As discussed in the previous section, it is not straightforward to identify this from the spectra (Fig. 10).

Figures 7–10 show that the CCE MRG wave signals in the dynamical variable spectra are also largely consistent with LSW theory. Zonal wind, divergence, and geopotential show signals of this wave in their PCF-2 spectra and the meridional wind in the PCF-1 spectrum. This signal is very weak in the *Z* PCF-2 spectrum (Fig. 9b), but experiments with the number of passes used to find the background spectrum reveal that there is a signal. The observed CCE MRG wave should again have an asymmetric component across the equator (Roundy and Frank 2004), but this is not visible in the spectra here. A possible reason for this may be that a slight asymmetry in meridional structure will result in the data projecting mainly onto one symmetric or antisymmetric PCF and very weakly onto one of the adjacent antisymmetric or symmetric PCFs. Small peaks in the spectra are harder to identify above a smoothed background and thus may not be visible in the residual spectra. The fact that the signals of the CCE MRG and Kelvin waves only appear in one PCF for each variable does suggest that the chosen trapping scale of 6° fits the actual trapping scale of these waves well.

The *n* = 1 WIG waves are clearly visible in the *D* PCF-1 spectrum and in the PCF-2 spectrum for *V* in agreement with LSW theory. However, there are no signals in the PCF-1 spectra for *U* and *Z* in contrast to the prediction by LSW theory. Eastward inertio-gravity (EIG) waves have significant peaks in the divergence PCF-3 spectrum at 850 hPa (not shown) and less clearly at 200 hPa (Fig. 7) and in the zonal wind PCF-3 spectrum. There are other peaks in the wavenumber–frequency region of EIG waves, but these peaks are more aligned with the dispersion curves for faster MRG waves than the CCE waves.

These observations indicate that the meridional structure of CCE waves can be very well represented by the first three PCFs with the chosen trapping scale. This is consistent with previous observational studies describing the meridional scale and structure of CCE waves (Roundy and Frank 2004; Yang et al. 2007a; Kiladis et al. 2009).

### b. Barotropic waves

In addition to the CCE waves described above, the spectra show signals of barotropic linear waves. Kasahara (1976) showed these to be solutions to the linearized primitive equations over a sphere. In Figs. 7–10 the dispersion curves for 316 m s^{−1} LSW waves are shown with the PCFs corresponding to their theoretical structure in addition to the 15 and 25 m s^{−1} curves. A feature of the geopotential signals in the right panels of Fig. 9 is the very localized nature of some of the peaks. This is due first to the difficulty of removing the background red spectrum (cf. the corresponding left panel) but is also due to the fact that spectra are only available at discrete wavenumbers and frequency. These wave signals are less equatorially trapped than the CCE waves, as indicated by the slow decay of these signals with increasing PCF number. Similar to the brightness temperature, the divergence spectra, as expected, do not show any signals of these nondivergent waves.

The spectra show signals of several different barotropic waves. To aid succinct presentation of these, a detailed description of only the more frequently seen waves is included here. Refer to Table 1 for additional information on all baroclinic waves visible in the spectra.

Barotropic wave signals in the reanalysis spectra. Wave types are Kelvin (K), Rossby (R), and mixed Rossby–gravity (MRG). Also shown are estimated frequencies and zonal wavenumbers of the waves and onto which PCFs the waves project for each dynamical variable.

The 33-h barotropic Kelvin wave (Hamilton 1984) with a frequency of about *ω* = 0.75 cpd and *k* = 1 can be seen to have the meridional structure of a Gaussian across the equator in *U* and to be symmetric in *Z*. This wave has been noted to have a trapping scale of 34° and a phase speed of 340 m s^{−1} for an equivalent depth of around 10 km (Salby 1979; Matthews and Madden 2000). The results here are consistent with that; the projection of this wave onto multiple symmetric PCFs gives an indication of the large meridional extent. Barotropic Rossby waves (Madden 2007) have the strongest signals for *n* = 1 and *n* = 2 waves and follow the Rossby–Haurwitz dispersion relations (e.g., Lindzen 1967; Hendon and Wheeler 2008). For *k* = −1 these are known as 5- and 10-day waves, respectively (Madden 1978; Kasahara 1980). The 5-day wave has a similar meridional structure to the barotropic Kelvin wave as they appear in the same PCF spectra whereas the 10-day wave has signals in the (antisymmetric) PCF-2 and -4 spectra for *U* and *Z* and in PCF 1 for *V*. The 4-day wave (Salby 1984) at *k* = −2 and *ω* = 0.25 cpd has the same meridional structure as the 5-day wave mentioned above. The 316 m s^{−1} LSW MRG wave dispersion curve has several peaks aligned with it. In the case of the 2-day wave (*ω* = 0.45 cpd, *k* = −3) there has been some discussion over the exact nature of the dynamics, namely whether it corresponds to a MRG wave (Salby 1981, 1984) or an inertio-gravity wave (Haertel and Kiladis 2004). The fact that these peaks line up with the theoretical dispersion relation for a 316 m s^{−1} LSW MRG wave, however, would suggest that the former is the case.

The meridional structure of all these waves inferred from the PCFs is closely related to that of the theoretical solutions from Kasahara (1976) with an equivalent depth of 10 km near the equator. The main differences between the signals of CCE and barotropic waves are the very localized nature of the signal in the spectra and the much wider meridional extend of the latter.

### c. Extratropical storm track activity

The signals of extratropical storm track activity are visible in the higher PCF spectra for all variables in the positive wavenumbers mainly between *k* = 5 and *k* = 10 and frequencies up to *ω* = 0.6 cpd. These signals are stronger at 200 hPa and in the meridional wind spectra. Previous studies have shown that these are stronger in the Southern Hemisphere as well and place the power spectral peaks at around *k* = 1–8 and *ω* = 0.04–0.5 cpd (Fraedrich and Boettger 1978; Fraedrich and Kietzig 1983). That these signals are visible in the equatorial belt between +20° and −20° is likely due to the propagation of extratropical Rossby waves into the tropical region to latitudes much smaller than the ±20° cutoff (Kiladis 1998). These waves retain their spectral signature when traveling into the tropics and that signal is visible in the spectra computed here.

### d. Power of higher PCFs

Another interesting issue is where the higher meridional modes add power to the spectrum. Focusing only on CCE waves, Fig. 11 corroborates that low PCF numbers explain most of the power of such waves. Projecting onto higher PCF numbers adds the most power in the eastward-gravity-wave frequency region and almost no power along the signals of low-frequency CCE waves for *T _{b}* (Fig. 11a). The analysis of the dynamical variables reveals similar behavior (Figs. 11b–d) and shows that the range of frequencies and wavenumbers where higher PCFs add power changes between variables. In contrast to

*T*the higher PCFs mainly add power at larger wavenumbers for the dynamical variables. Along the CCE wave peaks the ratio between the 10- and 3-PCF spectra is close to one, indicating that the first three PCFs contains most of the power contributing to these peaks.

_{b}Comparison of spectra obtained by projection onto the first 10 and 3 PCFs. Ratio of the two spectra for (a) brightness temperature, (b) zonal wind at 850 hPa, (c) meridional wind at 850 hPa, and (d) geopotential at 850 hPa. Superimposed are the LSW dispersion curves for shallow-water speeds of 15 and 25 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Comparison of spectra obtained by projection onto the first 10 and 3 PCFs. Ratio of the two spectra for (a) brightness temperature, (b) zonal wind at 850 hPa, (c) meridional wind at 850 hPa, and (d) geopotential at 850 hPa. Superimposed are the LSW dispersion curves for shallow-water speeds of 15 and 25 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

Comparison of spectra obtained by projection onto the first 10 and 3 PCFs. Ratio of the two spectra for (a) brightness temperature, (b) zonal wind at 850 hPa, (c) meridional wind at 850 hPa, and (d) geopotential at 850 hPa. Superimposed are the LSW dispersion curves for shallow-water speeds of 15 and 25 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 7; 10.1175/JAS-D-10-05008.1

### e. Comparison of spectra from ERA-40 and the NCEP reanalysis

The overall shape and the change with height of the spectra obtained from the ERA-40 data and the NCEP reanalysis data are very similar for all variables considered. Unfortunately there is no divergence product from the NCEP reanalysis to compare with the ERA-40 divergence. It would be interesting to see if the NCEP reanalysis divergence matches as well with the observed *T _{b}* as the ERA-40 divergence.

The main difference between the datasets is that there appears to be a gap in the NCEP spectra at *k* = 0 and high frequencies that is not in the ERA-40 spectra. Also, the 925- and 850-hPa NCEP spectra are much more symmetric with respect to zonal wavenumber at high frequencies than the ERA-40 spectra. Interesting features in the ERA-40 spectra are the artificial spectral peaks identified by WK99 at *k* = 14 and periods around 4.5 and 8.5 days. These peaks in the spectrum are due to the satellite tracks around the globe and are not part of the actual spectrum.

## 6. Summary and discussion

Based on the hypothesis that convectively modified LSW theory plays an important role in tropical convection, we decomposed brightness temperature and dynamical variables from reanalysis data into the meridional basis functions from LSW theory, the PCFs. To study the consistency of signals of CCE waves in these variables with LSW theory zonal and meridional wind, geopotential and divergence reanalysis data fields were projected onto the PCFs to calculate space–time spectra for each PCF separately. For each PCF number there is a corresponding interpretation in terms of equatorial waves (Yang et al. 2007a). That is, we can say for each PCF which LSW wave projects onto it and check whether there is a signal of the corresponding CCE wave in these spectra. In most cases the prediction by the LSW solutions proved accurate and the CCE wave projected onto the same PCFs as the corresponding LSW wave.

The main feature of all spectra we computed is the red decay with respect to frequency, zonal wavenumber, and PCF number. Conventionally it is common to assume that geophysical variables have such a spectrum as a null hypothesis; however, we would like to emphasize that this ubiquitous effect requires a clear physical explanation and the data shown here may aid in the development of such an explanation. Indeed, in the midlatitudes the precise behavior of the red spectrum may be explained by various turbulence models. To date there has not been a corresponding convincing physical explanation for the tropical red spectrum.

It has been shown in many previous studies that a red spectrum can arise in climate variables through the forcing of large-scale, low-frequency climate phenomena by small-scale weather processes. The simplest model of a red spectrum was that proposed by Hasselmann (1976), who considered a heat flux–damped ocean mixed layer stochastically forced by atmospheric boundary layer wind speed fluctuations. This model mathematically is equivalent to a one-dimensional Ornstein–Uhlenbeck linear stochastic process. Other similar but higher-dimensional models have been proposed to explain the ENSO spectrum (e.g., Kleeman 2002) or midlatitude decadal variability (e.g., Saravanan and McWilliams 1998). It remains an open question as to whether the red spectrum documented here can be explained in a similar linear fashion or whether it is intrinsically a nonlinear turbulent phenomenon that also manifests a red spectrum with somewhat different characteristics. We believe the results shown here may be useful in settling this in the future.

An important consideration when comparing data and theory are possible differences between the meridional scale and shape of CCE and LSW waves. The theoretical meridional structure can be nontrivially modified by processes in the earth’s atmosphere that are neglected in the formulation of the LSW equations. The main factors include the nonzero basic state of the equatorial atmosphere, nonlinear interactions with barotropic waves, and the interaction between the tropics and the extratropics. With this in mind, it is rather striking how much the CCE waves reflect the properties (meridional structure and dispersion curves) of theoretical LSW waves. This may be the case because the spectra show statistical properties of tropical disturbances and single episodes may be much more strongly affected by mean winds.

Experiments on the effect of the trapping scale *y*_{0} on CCE waves show that 6° is a good choice. Choosing a smaller or larger trapping scale leads to “leakage” of the spectral power into higher PCF modes. With a 6° trapping scale the signals of CCE waves appear mainly in single PCF spectra and are confined to the first three PCFs. This confinement to the first few PCFs suggests possibilities for reducing noise when filtering for these signals and has implications for future modeling and theoretical analysis.

Furthermore, the spectra show that divergence and *T _{b}* behave reasonably similarly. Both

*T*and divergence are related to convection, so it is therefore to be expected that these two variables should have similar spectral properties, but nonetheless reassuring to see in the reanalysis data. Convection is driven by the vertically integrated low-level divergence; however, because of the complex vertical structure of the boundary and cloud layers, divergence at one particular level (e.g., 850 hPa) may not represent the total low-level field very well. On the other hand, the upper-level divergence can be considered to be the consequence rather than the cause of convection, which may explain why it is better related to brightness temperature then the low-level divergence.

_{b}A limitation of this method is the inability to produce coherent wave structures and to differentiate between phenomena with a similar wavenumber–frequency range. Cross-spectral analysis can be used to identify coherent structures between the different variables and heights, and this is part of planned future investigations. An established method to identify wave structures is the filtering in wavenumber–frequency space. Here this method can be applied to only the part of the data that projects onto the PCFs with signals of the wave of interest, thus potentially reducing the amount of noise included in the filtered data. It would be valuable to quantify the possible improvement gained by this.

As part of ongoing work, we plan to use the information derived from the observed PCF spectra and the properties of the CCE waves to build a simple stochastic model of tropical convection. This model should be able to reproduce a red background spectrum and the CCE waves in a consistent manner. The spectra computed here will be used as validation for the simple model.

## Acknowledgments

The authors thank George Kiladis for his detailed insights and comments. Anonymous reviewers provided valuable comments on earlier versions of this manuscript that led to significant improvements. These results were obtained using the CLAUS archive held at the British Atmospheric Data Centre, produced using ISCCP source data distributed by the NASA Langley Data Center. The ERA-40 data for this study are from the Research Data Archive (RDA), which is maintained by the Computational and Information Systems Laboratory (CISL) at the National Center for Atmospheric Research (NCAR). NCEP reanalysis data were provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA. Both authors acknowledge the support of the National Science Foundation through Grant ATM-2574200.

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