1. Introduction
One type of “balanced dynamics” in the tropical atmosphere occurs on planetary zonal scales; accordingly it is called the equatorial long-wave dynamics (Heckley and Gill 1984; Majda 2003; Majda and Klein 2003).1 These dynamics are marked by several related characteristics, including small meridional winds, geostrophic balance in the meridional direction, and inertio-gravity waves of small amplitude. Theories that exploit an assumed small ratio of meridional to zonal length scales suggest that these dynamics are valid on “long” zonal and temporal scales.
The main goal of this paper is to assess how long the zonal and temporal scales must be in order for equatorial long-wave dynamics to exist. Such an assessment is important as many low-dimensional models of planetary-scale phenomena in the tropical atmosphere are derived in part using the long-wave approximation. Examples include models of equatorial waves and their interactions (Majda and Biello 2003) and the Madden–Julian oscillation (MJO; Majda and Biello 2004; Biello and Majda 2005; Majda and Stechmann 2009). Other models incorporate a steady-state version of the long-wave theory, such as models for the Walker circulation (Gill 1980; Stechmann and Ogrosky 2014) and El Niño–Southern Oscillation (ENSO; Cane and Zebiak 1985).
In addition to justifications based on scale analysis or formal asymptotics, there are rigorous mathematical proofs of the convergence to long-wave dynamics under the appropriate limit (Dutrifoy and Majda 2006; Dutrifoy et al. 2009). What is more, fast-wave averaging has been used to show the existence of balanced dynamics even with imbalanced initial conditions in the tropics (Dutrifoy and Majda 2007). These results have been proven for the equatorial shallow-water equations; a remaining mathematical challenge is to prove similar results for the three-dimensional primitive equations.
Using observational data, assessing the scales on which long-wave dynamics exist is fraught with challenges. A substantial disconnect exists between tropical dynamics in nature and in idealized fluid dynamics models. For example, in idealized fluid dynamics models, the effects of water vapor and convection must be included in some way, perhaps through an imposed forcing or a parameterization. In nature, in contrast, water vapor and convection are part of a highly complex turbulent dynamics. Also, idealized models typically either neglect forcing or prescribe forcing functions of a particular order of magnitude; the magnitude of forcing in nature need not satisfy such constraints. For these reasons and others, one might not expect to be able to accurately assess the validity of the long-wave approximation using observational or reanalysis data. Nevertheless, it will be shown below that a sensible assessment can in fact be made.
This assessment is achieved by applying a spectral data analysis and wave projection technique introduced recently in Stechmann and Majda (2015). Spatial projections onto theoretical basis functions in x, y, z, and t are used. Previous studies have also used some combination of spatial projections (e.g., Wheeler and Kiladis 1999) and many others have utilized Fourier modes in x and t. Yang et al. (2003, 2007) and Gehne and Kleeman (2012) utilized parabolic cylinder functions in y, and projecting onto vertical basis functions has a long history (e.g., Kasahara 1976; Kasahara and Puri 1981; Fulton and Schubert 1985). The spectral approach employed here uses a unique projection in the vertical direction that isolates the first baroclinic mode and does not include the stratosphere. Combined with projections in the other directions, this approach systematically isolates (i) the first baroclinic mode, (ii) individual meridional modes, and (iii) individual zonal and temporal Fourier modes in the data. This spectral isolation of spatiotemporal scales in each dimension allows for a clear assessment of the scales on which the dynamics are long-wave in character.
Wave amplitudes are also identified here for individual wave types, providing a systematic way to combine data from multiple fields (such as wind and geopotential height) into a single meaningful multivariate field. This method was presented by Stechmann and Majda (2015) for the long-wave equations; here it is introduced for the traditional (i.e., no long-wave assumption) shallow-water equations as well.
Three perspectives will be used here, each corresponding to a different choice of model variables: (i) primitive 
The rest of the paper is thus organized as follows. Section 2 contains a review of the equatorial long-wave approximation. A simple prediction is given of the scales on which this approximation may be expected to be valid based on physical and mathematical arguments. Section 3 describes the data and methods used for the observational data analysis. The data are then analyzed from three perspectives: (i) primitive variables, (ii) characteristic variables, and (iii) wave variables, in sections 4, 5, and 6, respectively. Some discussion of the results is given in section 7, and a summary of the main findings of the paper is given in section 8. Additional results from the viewpoint of wave variables can be found in the supplementary materials.
2. Background on the equatorial long-wave approximation
a. First baroclinic mode dynamics
Values of scaling parameters and constants.
b. Long-wave scaling in primitive variables
c. Estimating long-wave scales from theory
A natural question to ask is “Over what range of spatial and temporal scales can the long-wave equations [(2.13)] be expected to accurately describe the linear dynamics of the first baroclinic mode of the tropical atmosphere?” Before turning to reanalysis data to address this question, this range of spatiotemporal scales is briefly estimated in two ways from theoretical considerations.
Figure 1 shows the regions of wavenumber–frequency space where conditions (2.17) and (2.21) are satisfied. The dispersion curves of some of the linear solutions to (2.7) are also shown. The first estimate [i.e., (2.17)] encompasses a broad range of wavenumbers and frequencies, including pieces of both the mixed Rossby–gravity (MRG) and inertio-gravity (IG) wave dispersion relations. The second estimate [i.e., (2.21)] is a much more restrictive assumption; the difference in dimension size of the two regions is a factor of 
Regions in wavenumber–frequency space where conditions (2.17) (light shading) and (2.21) (dark shading) are satisfied. Dispersion relations for linear solutions to (2.7) are depicted by solid lines.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
We pause to briefly discuss the underlying difference between these two perspectives. In the first perspective, the zonal-to-meridional aspect ratio has been defined by comparing a whole zonal wavelength with the meridional trapping scale 
Note that the width and height of the region in Fig. 1 associated with (2.21) is proportional to the chosen value of 
3. Data and data analysis methods
We now turn to reanalysis data to provide an observation-based estimate of the spatial and temporal scales on which long-wave dynamics occur. NCEP–NCAR reanalysis daily zonal winds, meridional winds, and geopotential height are used to estimate u, υ, and θ, respectively (Kalnay et al. 1996). These datasets have a horizontal spatial resolution of 2.5° × 2.5°. The time period used in this study is the 34-yr period from 1 January 1980 to 31 December 2013. The data are made dimensionless by the scales in (2.5) using the parameter values shown in Table 1.
To test the robustness of the results, a second reanalysis dataset with higher resolution was used. All the results presented here were also calculated using the ERA-Interim winds and geopotential during the same time period with 1° × 1° horizontal resolution and four-times-daily temporal resolution (Dee et al. 2011).
Meridional basis functions 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
Next, for each spectral coefficient 
Finally, the power contained in the anomalies of 
4. Primitive variables
We now turn to reanalysis data to assess the range of spatial and temporal scales on which long-wave dynamics occur in nature. These scales are quantified first from the perspective of the primitive variables 
Observational data analysis
Figure 3 shows a Hovmöller plot of the zeroth meridional mode of horizontal wind and potential temperature anomalies from a seasonal cycle for the 1-yr period from 1 July 2009 to 30 June 2010. The zonal winds contain evidence of Kelvin waves, seen in the rapid eastward propagation of anomalies throughout the figure. The much slower eastward propagation of anomalies centered between 90°E and 150°W during the months of October 2009–February 2010 are evidence of the well-documented MJO activity during the Year of Tropical Convection (YOTC) (e.g., Moncrieff et al. 2012; Waliser et al. 2012). The meridional wind and potential temperature anomalies exhibit both eastward and westward propagation with the largest anomalies occurring over the Pacific Ocean from approximately 180° to 90°W. For both the zonal winds and potential temperature, large anomalies appear to occur primarily on large spatial and temporal scales. In contrast, the meridional wind anomalies appear to occur primarily on small spatial and temporal scales.
Hovmöller plots of dimensionless (a) 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
A power spectrum confirms that meridional winds are indeed weaker than zonal winds for small wavenumber and frequency. Figure 4 shows the power spectrum for the zeroth and first meridional mode of each variable [i.e., the quantities 
(a)–(c) Power spectrum of the zeroth meridional mode of the first baroclinic dimensionless (a) zonal wind, (b) meridional wind, and (c) potential temperature anomalies during 1 Jan 1980–31 Dec 2013. Anomalies are from a seasonal cycle. (d)–(f) As in (a)–(c), but for the first meridional mode.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
The smallness of υ can be seen more clearly in the ratio of power (i.e., 
(a),(c) Power spectrum ratios 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
The spatiotemporal scales that satisfy condition (2.21) are also indicated on each plot by the thick dashed rectangular box. The dark line depicting the 0.3 contour lies entirely within the box in Figs. 5b–d, while a portion lies outside the box in Fig. 5a along a Kelvin wave–type dispersion curve. This box region corresponding to condition (2.21) thus appears in general to slightly overestimate the region where the long-wave approximation holds, but the agreement is reasonable. This slight overestimation persists over a range of values for 
5. Characteristic variables
The results in section 4 suggest that one aspect of the long-wave approximation, small meridional winds, holds only over a very narrow range of scales. We next examine the data from the perspective of characteristic variables. The equations expressed using characteristic variables are first summarized in sections 5a and 5b; observational analysis results are given in section 5c.
a. Definition of characteristic variables
b. Long-wave theory with characteristic variables
c. Observational data analysis
Turning again to reanalysis data, we first briefly examine the spatiotemporal scales on which υ is small from this viewpoint of characteristic variables. Figure 6 shows the ratios of each quantity 
(a)–(c) Power spectrum ratios 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
Another feature readily apparent in Fig. 6 is the expanding region of small ratio with increasing meridional mode number m. This can be qualitatively anticipated from the following physical considerations. As m increases, the average distance between one local maximum and an adjacent local minimum in the basis function 
Before continuing on, we note that one might expect the opposite trend of that seen in Fig. 6 (i.e., one might expect that the region of long-wave scales should shrink with increasing m). This expectation could arise by considering that meridional modes of higher number have a wider base of support (i.e., these modes extend farther away from the equator). Thus, it might be expected that higher meridional modes have a larger length scale, which would suggest that the long-wave approximation should only hold over a narrower region of zonal length scales. However, as discussed in the previous paragraph, it is also the case that the distance between peaks in these meridional basis functions decreases with increasing meridional mode number. If this distance between peaks is taken as a meridional length scale 
Hovmöller plots of (a) 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
This approximate geostrophic balance is confirmed quantitatively, scale by scale, by examining the power spectrum of the quantity MGIm, shown in Figs. 8a–c for m = 1–3, respectively. In all three panels, the power present in wavenumbers 
(a)–(c) Power spectrum of MGIm anomalies during 1 Jan 1980–31 Dec 2013 for m = 1–3, respectively. (d)–(f) Power spectrum ratios 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
Another way to assess the degree of the imbalance term MGIm is to compare its magnitude with that of 
(a) The log of the average low-frequency power of meridional geostrophic imbalance (MGI) as defined in (5.13) with 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
6. Wave variables
The results in section 5 suggest that the second aspect of long-wave dynamics considered here (meridional geostrophic balance) is seen in the data over a slightly larger range of spatiotemporal scales than the first aspect (weak meridional winds). The long-wave approximation is next examined from a third viewpoint: wave variables. These wave variables are defined in section 6a, their behavior in the long-wave limit is discussed in section 6b, and their structure is isolated in reanalysis data in section 6c.
a. Definition of wave variables
Equations (5.4) and (5.5) describe the evolution of the Kelvin wave, and mixed Rossby–gravity (MRG) and inertio-gravity (EIG0) waves, respectively. Equations (5.6) for 
Components of normalized (a) R1, (b) WIG1, and (c) EIG1 eigenvectors as a function of zonal wavenumber k.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
In contrast, many previous studies have identified equatorial waves using the eigenvalues from the linear theory. In these studies, space–time filtering of a single variable is used to identify anomalous peaks in its power spectrum. Anomalies that are in close proximity to the eigenvalues 
Other techniques for wave identification have also been used, of which a small sampling is discussed here. A spatial projection technique that made use of spherical harmonics at a single pressure level was used by Madden (2007) to identify free large-scale Rossby waves in the upper troposphere. Matthews and Madden (2000) used Fourier analysis of sea level pressure at nine locations in the tropics to study the 33-h barotropic Kelvin wave; see Salby (1984) for a discussion of earlier observational studies of barotropic equatorial waves. Tindall et al. (2006) used both space–time filtering and a projection technique that does not require selection of an equivalent depth a priori to study the long-term climatology of equatorial waves in the lower stratosphere. Hendon and Wheeler (2008) studied the spatial structure of convectively coupled waves by studying the space–time coherence spectrum of OLR and zonal winds; see Kiladis et al. (2009) for further discussion of the features of convectively coupled waves.
b. Long-wave theory with wave variables
The normalized long-wave versions of eigenvector components shown in Fig. 10.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
We note that the leading order of the long-wave Rossby eigenvalue in (6.10a) is only expected to be a good approximation of the eigenvalue for small zonal wavenumber k. This long-wave approximate Rossby eigenvalue is also a feature of other balance models that are based on the equatorial long-wave scaling (Stevens et al. 1990; Chan and Shepherd 2013, 2014). For any wavenumber k, not necessarily small, the Rossby eigenvalue is approximately 
c. Observational data analysis
Figure 12 shows a Hovmöller plot of the standard Rossby and inertio-gravity wave structures from (6.9) in reanalysis data using the projection technique described in section 6a. The most obvious feature is that the inertio-gravity waves clearly have much smaller amplitudes than the Rossby wave. The variability that does exist in the inertio-gravity waves appears to occur primarily on short length and time scales. Also apparent is that the Rossby and westward inertia-gravity (WIG) wave structures each exhibit both periods of eastward and westward propagation, while the eastward inertia-gravity (EIG) wave structure exhibits mostly eastward propagation.
Hovmöller plot of (a) R1, (b) WIG1, and (c) EIG1 anomalies from a seasonal cycle. The time period shown is 1 Jul 2009–30 Jun 2010.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
These features are confirmed by the power spectrum of each wave structure, shown in Fig. 13, and deserve additional comment. We note that studies that use space–time filtering to identify free equatorial waves have shown that EIG and WIG waves have less power than Rossby waves, a feature consistent with the redness of the background spectrum. Here, however, the waves are defined solely by their spatial structure; no temporal filtering has been used. While it is reasonable to conjecture that the spatial structures of inertio-gravity waves would contain less power than Rossby waves at low frequencies, it is not clear a priori that this must necessarily be the case with the method employed here.
Power spectrum of (a) R1, (b) WIG1, and (c) EIG1 anomalies during 1 Jan 1980–31 Dec 2013.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
Also, while free Rossby and WIG waves propagate strictly westward and free EIG waves propagate strictly eastward, the picture is more complicated for forced waves. Nonlinear advection, heating and cooling, dissipation, etc. all can contribute to the forcing terms 
Figure 14 shows the ratio of power in the inertio-gravity waves WIG1 and EIG1 to the Rossby wave 
Power spectrum ratios (a) 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
The features of Fig. 14 discussed above are all present in the ERA-Interim data as well (not shown). Use of this higher-resolution dataset does result in a slightly larger ratio of WIG1 to 
Given the strength of the Rossby wave relative to the IG waves, it is natural to wonder if a further simplification can be made by using the long-wave form of the Rossby wave 
Hovmöller plot of (a) 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
Power spectrum of (a) 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
This analysis was also conducted for the 
Power spectrum of the (a) Kelvin, (b) MRG, and (c) EIG0 wave anomalies during 1 Jan 1980–31 Dec 2013.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0065.1
In summary, all of these figures demonstrate that reanalysis data projects weakly onto the spatial structures of inertio-gravity waves over a broad range of wavenumbers and frequencies.
7. Discussion
We note that there are many other facets of the tropical atmosphere that have been neglected here. For one, the effects of both dissipative mechanisms (e.g., Rayleigh friction and Newtonian cooling) and forcing (e.g., convective heating and radiative cooling) have not been directly quantified here. The results presented, however, do contain contributions from both free and forced waves in the tropical troposphere, and the impact of forcing on the results was discussed in section 6. While some aspects of the tropical circulation have been modeled well without the traditional damping terms [e.g., the Walker circulation (Stechmann and Ogrosky 2014) and the MJO (Majda and Stechmann 2009)], both forcing and dissipative mechanisms have been shown to play a significant role in the tropical atmosphere; it would be interesting to directly study the role these mechanisms play in setting the long-wave scales presented here.
In addition, the first baroclinic mode has been studied exclusively here, in part due to the primary role it plays in many models of the tropical atmosphere; it would be interesting to adapt the data analysis methods used here for other vertical modes. The nonlinear interactions between different vertical modes have also been neglected here. Neglecting these nonlinearities allows for the clear spectral methods presented here, but any role these nonlinearities play in setting the long-wave scales has not been considered here [see, e.g., Lin et al. (2005) and Lin et al. (2008) for estimates of the role these nonlinearities play in the tropical atmosphere, and Stechmann et al. (2008) for a model of nonlinear interactions between two baroclinic modes]. The results presented here, however, suggest that when a snapshot of the tropical atmosphere is described in terms of the solutions to Matsuno’s linear shallow-water theory, the degree to which each solution is present in the data is in good agreement with the long-wave approximation statistically.
Also, this study has focused solely on the dry variables used in Matsuno (1966); since convectively coupled waves tend to exhibit different scaling than their dry counterparts, it would be interesting to extend this quantitative assessment to include the role of moisture in setting these long-wave scales. Last, the focus here has been on the tropical atmosphere; it would be interesting to adapt this assessment technique to oceanic long-wave dynamics considered in Harvey and Patzert (1976), Legeckis (1977), and Legeckis et al. (1983) or to midlatitude or global atmospheric dynamics.
8. Conclusions
This paper has provided a quantitative assessment of the spatiotemporal scales on which long-wave dynamics are seen in reanalysis data. Specifically, three interconnected aspects of the equatorial long-wave approximation were considered: (i) the smallness of meridional wind anomalies relative to those of zonal winds and potential temperature, (ii) the leading-order dynamics being in meridional geostrophic balance, and (iii) the filtering out of inertio-gravity waves. This assessment was achieved by using a spectral method that allows for analysis from three different perspectives: primitive variables 
This assessment illustrated that different aspects of long-wave dynamics may exist over different ranges of spatiotemporal scales. Specifically, it was shown that while meridional winds are small for a very narrow range of length scales (
The results here do suggest that the largest-scale features of the tropical atmosphere (e.g., the Walker circulation and the MJO), whose main features are well described with small zonal wavenumbers, say, k = 1–4, may be effectively modeled using the long-wave approximation, provided the model and data are compared using a wave perspective [see, e.g., Stechmann and Majda (2015) (MJO) and Stechmann and Ogrosky (2014) (Walker circulation)]. This wave perspective was identified as one of the factors contributing to the good agreement found between a simple model and observations of the Walker circulation in Stechmann and Ogrosky (2014). Of course, even for scales where the tropical atmosphere exhibits long-wave dynamics statistically, the dynamics of the atmosphere at a given moment may not be well described by the long-wave approximation.
Several further issues were also raised and described in more detail in section 7. For example, the present study did not explicitly account for many effects such as nonlinearity, water vapor, convection, and other forcing and dissipative mechanisms. Such effects are major challenges for idealized models and major challenges for comparing theory with observational data. Despite these simplifications, the main characteristics of long-wave dynamics could still be identified here.
An important ongoing task is assessing the accuracy of the many low-dimensional models of the tropical atmosphere that make use of equatorial long-wave theory (e.g., Majda and Biello 2003; Majda and Stechmann 2009; Stechmann and Ogrosky 2014). Such assessments are typically made by comparing model results with observational or reanalysis data (Stechmann and Ogrosky 2014; Stechmann and Majda 2015; Ogrosky and Stechmann 2015). If significant discrepancies between model and observations exist, it is important to understand whether these discrepancies are due to the long-wave approximation or to some other simplifying assumption (e.g., treatment of convective heating). It is our hope that the quantitative assessment presented here provides an additional resource for such assessments.
The data for this paper are available from NOAA/OAR/ESRL PSD, Boulder, Colorado, from their website at http://www.esrl.noaa.gov/psd/. The research of S.N.S. is partially supported by ONR Young Investigator Award N00014-12-1-0744, ONR MURI Grant N00014-12-1-0912, and by the Sloan Research Fellowship. H.R.O. is supported as a postdoctoral researcher by ONR MURI Grant N00014-12-1-0912. The authors thank A. Majda, M. Gehne, and two anonymous reviewers for their helpful comments.
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Another class of balanced dynamics in the tropics includes a variety of weak temperature gradient (WTG) approximations (Charney 1963; Sobel et al. 2001; Majda and Klein 2003; Stechmann and Stevens 2010).
This method for measuring the width of a Gaussian is one standard approach, although others could certainly be used as well. For example, if the half-width of the Gaussian is estimated by two standard deviations, the result is 
The meridional trapping scale is one standard deviation of the Gaussian on which the parabolic cylinder functions are based, and not a full wavelength in the meridional direction. Thus, an argument could be made that a more appropriate estimate could be constructed by requiring that the ratio of one zonal wavelength to one meridional wavelength be small. (Note that this notation “meridional wavelength” is not well defined without further clarification since the distance from one crest to an adjacent crest is not uniform within a given meridional basis function.)
The definition of the long-wave Rossby structure 
