## 1. Introduction

The relative role of inertio-gravity (IG) energy in atmospheric energetics with respect to the balanced energy on various scales is still uncertain. Observational studies based on high-resolution radiosonde data (e.g., Wang and Geller 2003; Wang et al. 2005; Vincent and Alexander 2000) and GPS occultation data (e.g., Tsuda et al. 2000) describe some aspects of the gravity waves energetics. However, a precise quantitative separation of the balanced (i.e., nearly geostrophic or Rossby type) and IG (unbalanced) circulation from observational data is difficult as the two regimes are usually studied by using different sets of simplified equations. Furthermore, observational studies most often use temperature observations; this applies especially to the tropics where the direct wind measurements are relatively few.

On the other hand, global data assimilation and forecasting systems have reached resolutions at which IG waves across many scales are resolved (e.g., Shutts and Vosper 2011). Compared to the early analyses that did not consider mesoscale and that described tropical flows rather poorly (e.g., Heckley 1985), present-day analysis datasets are considerably better thanks to large amounts of satellite observations assimilated, to improved assimilation methodology, and to better models (e.g., Simmons and Hollingsworth 2002). In particular, model resolution has increased [in 2011 the European Centre for Medium-Range Weather Forecasts (ECMWF) system has horizontal resolution T1279, which corresponds to about a 16-km grid] and precipitation forecast has been improved (e.g., Andersson et al. 2005). Analysis datasets thus can provide the global distribution of IG energy as allowed by remaining uncertainties associated with difficulties in atmospheric data assimilation at the mesoscale and in the tropics and by the lack of wind observations (e.g., Žagar et al. 2004, 2005). The present study discusses the impact of the vertical resolution of analysis data on the estimates of the IG energy by performing sensitivity experiments with the operational analyses of ECMWF. In particular, we discuss the difference between the model-level and the standard-pressure-level analysis data.

A study by Daley (1983) estimated that the average percentage of ageostrophic motions in early analyses of ECMWF was about 10%; this implies that about 1% of atmospheric wave energy is associated with unbalanced flows. A better description of IG circulations in the middle atmosphere and in the tropics in the current analyses implies that the percentage of IG energy is different from 1% (i.e., the percentage should be larger). Accordingly, Žagar et al. (2009a, hereafter ZTAR09) reported based on analysis data for July 2007 that the level of IG energy in the large-scale wave circulation (without the zonally averaged state, i.e., with zonal wavenumber >0) in current analysis systems is about 10% of the total wave energy. Nearly the same number was obtained for the ECMWF and the National Centers for Environmental Prediction (NCEP) operational analyses in spite of their different resolutions, parameterizations, and model depths (80 km resolved by 91 levels and 60 km resolved by 60 levels in the case of ECMWF and NCEP, respectively). The obtained percentage is about 3 times larger than the previous estimate by Tanaka and Kimura (1996, hereafter TK96) based on about 15-yr-older analysis systems. While it was impossible to precisely identify reason(s) for the different results, ZTAR09 argued that more recent datasets with higher horizontal and vertical resolutions reflect advancements in the quality of atmospheric analyses, especially the large-scale tropical flow.

Both ZTAR09 and TK96 applied the normal-mode function (NMF) expansion to divide energy into the balanced and IG parts. However, ZTAR09 used the model-level analysis data whereas TK96 employed datasets on standard-pressure levels. The present paper compares the energy percentages of IG flow obtained from the model-level and the standard-pressure-level analyses for the same period of July 2007 to show that an increased vertical resolution of input analysis data provides more details about the vertical structure of inertio-gravity circulation resulting in larger energy percentages of the IG waves.

The paper is organized as follows. In section 2 we summarize the normal-mode expansion methodology with focus on the vertical structure equation and we describe the numerical experiments. Results are presented in section 3 while discussion and conclusions are given in section 4.

## 2. Methodology and data

### a. Derivation of NMFs

The derivation of NMFs relies on the assumption that the solutions to the linearized primitive equations can be expressed as a product of a horizontally independent part and horizontal eigensolutions. The horizontal and vertical structures are coupled through the separation constant—the equivalent depth. The expansion of three-dimensional atmospheric data into normal modes starts with the vertical projection followed by the horizontal expansion for each equivalent depth. The horizontal expansion functions are analytical solutions of the linearized shallow-water equation on the sphere, called the Hough functions (Kasahara 1976). The vertical structure functions can be obtained analytically only for some special cases such as the isothermal atmosphere (Daley 1991, chapter 6) or a constant stability profile (Terasaki and Tanaka 2007). For realistic temperature and stability profiles, solutions need to be obtained numerically.

For the purpose of investigating the atmospheric energetics (i.e., the relative role of IG and balanced flows), NMFs need to be three-dimensionally orthogonal. The first such solution was derived by Kasahara and Puri (1981) who obtained the solution of the vertical structure equation in the *σ* coordinates by applying the top and bottom boundary conditions that the vertical velocity vanishes. The same approach was followed in ZTAR09. In that case input data are presented on *σ* levels and temperature and stability profiles use globally average values on *σ* levels. The three-dimensionally orthogonal NMFs in the system with pressure as the vertical coordinate were derived by Tanaka (1985) and Tanaka and Kung (1988). In this case input data are presented on the standard-pressure levels and the bottom boundary condition is that the surface wind vanishes. This approach was followed in TK96. In both ZTAR09 and TK96 the finite-difference method was used; the obtained numerical solution results in zigzag shapes of the vertical structure functions. The accuracy of numerical solutions in *σ* coordinates is discussed in ZTAR09 and references there; for example, the accuracy improves with more levels and the top model level located closer to 0 Pa (Staniforth et al. 1985).

Another method for solving the vertical structure equation numerically is the spectral approach developed by Kasahara (1984). In this case, the solution involves the use of the Legendre polynomials, derivatives in the equation are computed analytically, and the Galerkin procedure is applied to determine the expansion coefficients. The vertical levels are located on the Gaussian grid, which corresponds to the input pressure values divided by the surface pressure. For details, see the appendix of Kasahara (1984). The top boundary condition is that *ω* vanishes at *p* = 0. The bottom boundary condition requests that the linearized version of the vertical velocity *dz*/*dt* vanishes at the constant pressure *p _{s}* near the surface.

Both the finite difference and the spectral method result in structure functions that have larger amplitudes close to the model top than lower down for the leading modes, and the peak amplitude moves downward toward the surface as the vertical mode index increases. This is illustrated in Fig. 1 for the 91-level discretization of the ECMWF model. The main difference in the appearance of solutions is that the finite-difference solution for higher vertical modes have many zero crossings, which are nearly identical to zero value (Fig. 1a), while the spectral solution for higher modes have small amplitudes also in the upper part of the model domain (Fig. 1b).

In each case, the derived three-dimensional NMFs form a complete expansion set that can be used to accurately represent the atmosphere in terms of the series of NMFs. The NMFs analyze the wind field and the mass field simultaneously and offer an attractive way to quantify the role of gravity waves on various scales by splitting the energy into parts projecting on the Rossby-type motions and on the IG circulation. This is particularly suitable for the tropics where the IG circulation dominates on all scales.

### b. Sensitivity experiments

Several experiments are performed to check the sensitivity of the results to the method used to solve the vertical structure equation and to study the difference in the results due to the vertical resolution of input analysis data and due to the top model level.

The data used are operational analyses of ECMWF for July 2007, which are results of the 12-h window four-dimensional variational data assimilation (4DVAR) scheme and the model cycle Cy32r2. The original horizontal data resolution is defined by the horizontal truncation T799 (~25 km), but the data used in this study are interpolated to a regular (Gaussian N64) grid with 256 and 128 points in the longitudinal and latitudinal directions, respectively. The study is concentrated on large scales where the IG energy is associated primarily with the tropical circulation.

The ECMWF system uses the hybrid (*σ*–*p*) vertical coordinate, which follows the orography in the lower troposphere (i.e., *σ* coordinate) and becomes flat higher up (i.e., *p* coordinate). There are 91 vertical levels and the top model level is located at 0.01 hPa (about 80 km). ZTAR09 interpolated 91 hybrid model levels onto their equivalent *σ* levels assuming that changes introduced to the distribution of levels and the data by the interpolation were small. Globally averaged temperature and stability profiles on *σ* levels were used in the vertical structure equation, which was solved by the finite-difference method followed by the horizontal projection for each equivalent depth. The results were discussed in ZTAR09 and Žagar et al. (2009b). This is the reference experiment for the present study and it is denoted M91FD.

Eight sensitivity experiments listed in Table 1 are presented to answer the addressed questions about the methodology and data resolution. Each experiment is denoted by a label that describes the input data type (M and P for the model-level and standard-pressure-level data, respectively), a number of input vertical levels, and a method for the numerical solution of the vertical structure equation [i.e., finite-difference (FD) and the spectral method (SP)]. A sensitivity experiment denoted M91SP applies the same hybrid model-level data as M91FD, but the method is the spectral method of Kasahara (1984) as outlined above; this experiment aims to check sensitivity to the solution method. In M91SP, 91 model-level data are interpolated to a 90-level Gaussian (pressure) grid with the top level located at 0.18 hPa.

List of numerical experiments discussed in the paper; M stands for model levels whereas P stands for the standard-pressure-level input data. Methods for the solution of the vertical structure equation are denoted FD and SP for the finite-difference method and the spectral approach, respectively. Number of input levels and the average pressure of the top level are listed. Further details are in the text.

The sensitivity experiments denoted P21SP and M21FD aim at presenting the importance of the standard-pressure levels as compared to the more dense model levels in the M91 experiments. Thus, P21SP uses the spectral solution method and the pressure-level data extracted directly from the ECMWF Meteorological Archive and Retrieval System (MARS). The standard-pressure data include 21 levels distributed between 1000 and 1 hPa: 11 levels up to 100 hPa (1000, 925, 850, 700, 500, 400, 300, 250, 200, and 100 hPa), 5 levels between 100 and 10 hPa (70, 50, 30, 20, and 10 hPa), and 5 levels above 10 hPa (7, 5, 3, 2, and 1 hPa; Fig. 3). In the P21SP experiment, the standard-pressure levels are interpolated to the Gaussian *p* grid of 42 levels with the top level located at 0.8 hPa. In both P21SP and M91SP experiments the extrapolation below the surface requests zero undersurface winds and a constant geopotential deviation from the global mean. The vertical interpolation is carried out by using the cubic spline to compute a value at a new level. The interpolation is carried out in the log-*σ* coordinate to avoid concentration of the levels at the top and the bottom of the atmosphere.

The M21FD experiment makes use of 21 model levels that are selected as the levels that are closest to the standard-pressure levels; the top level in this case is level 9 located at about 0.92 hPa and the bottom level is model level 88 that is on average found at 999.8 hPa. As in M91FD and all other experiments based on the model-level data and the FD method, the 91-level data are first interpolated onto the corresponding *σ* surfaces and the levels selected for the particular experiment are then picked out for the normal-mode expansion.

The structure of the four vertical modes of the P21SP and M21FD experiments is presented in Fig. 2. The four displayed modes have the equivalent depths closest to the depths of the modes presented in Fig. 1 for the same method; for the SP method, these are the equivalent depths 9.9 km, 40 m, 3.5 m, and 17 cm whereas for the FD method the depths are 10 km, 145 m, 26 m, and 2 m. Similar to Fig. 1, it can be seen in Fig. 2 that the spectral approach produces the barotropic mode (*m* = 1) with a large amplitude near the model top compared to other vertical modes. The circles along the zero line in Fig. 2b denote locations of the standard-pressure levels of the input data. Levels are added in between the pressure levels by the interpolation; for example, six levels are added by the cubic spline interpolation between the 700- and 500-hPa data levels. In the same pressure range, the 91 model-level data contain seven levels (see also Fig. 3).

Vertical eigenstructures based on (a) the ECMWF 21 model-level data selected as levels closest to the standard-pressure levels for the finite-difference method, and (b) the ECMWF 21 standard-pressure-level data interpolated to the Gaussian *p* grid of 42 points for the spectral method. The displayed four vertical modes have the same equivalent depths as the four modes presented in Fig. 1 for the same method. The round circles in (b) correspond to the locations of the standard-pressure levels.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Vertical eigenstructures based on (a) the ECMWF 21 model-level data selected as levels closest to the standard-pressure levels for the finite-difference method, and (b) the ECMWF 21 standard-pressure-level data interpolated to the Gaussian *p* grid of 42 points for the spectral method. The displayed four vertical modes have the same equivalent depths as the four modes presented in Fig. 1 for the same method. The round circles in (b) correspond to the locations of the standard-pressure levels.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Vertical eigenstructures based on (a) the ECMWF 21 model-level data selected as levels closest to the standard-pressure levels for the finite-difference method, and (b) the ECMWF 21 standard-pressure-level data interpolated to the Gaussian *p* grid of 42 points for the spectral method. The displayed four vertical modes have the same equivalent depths as the four modes presented in Fig. 1 for the same method. The round circles in (b) correspond to the locations of the standard-pressure levels.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Distribution of the vertical levels in the three experiments. P21 denotes the locations of the standard-pressure levels that are input for the P21SP experiment. Distribution of levels (left) below 200 hPa and (right) above 200 hPa.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Distribution of the vertical levels in the three experiments. P21 denotes the locations of the standard-pressure levels that are input for the P21SP experiment. Distribution of levels (left) below 200 hPa and (right) above 200 hPa.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Distribution of the vertical levels in the three experiments. P21 denotes the locations of the standard-pressure levels that are input for the P21SP experiment. Distribution of levels (left) below 200 hPa and (right) above 200 hPa.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

The remaining five experiments are carried out with the FD method and different numbers of the model levels. The goal is to quantify the changes in the energy percentage of IG waves as a function of the vertical data density and the model top. The experiment with only 12 model levels (M12FD) that are selected as levels closest to the 12 standard-pressure levels is prepared for the comparison with TK96 that used 12 standard-pressure levels up to 50 hPa. Two more levels (levels 81 and 18 at about 925 and 10 hPa, respectively) are added in the experiments M14FD. The difference between M12FD and M21FD is eight levels above 50 hPa and model level 81 (about 925 hPa) in the M21FD experiment. The impact of the level density is addressed in two experiments that contain the same levels as M21FD with the addition of other levels in between (experiment M42FD) and with all levels in between levels 9 and 88 (experiment M80FD). The experiment M87FD contains all model levels from the surface up to level 5 (0.17 hPa) and it is thus better comparable with M91SP, which used the model top level at 0.18 hPa.

There are 50 waves in the zonal direction in addition to the zonally averaged state. In the meridional direction, 18 modes are used for each motion type: the balanced (R), the westward-propagating IG (WIG), and the eastward-propagating IG (EIG) modes. For comparison, TK96 and some other previous studies used a different number of balanced and IG modes; however, there is no a priori reason for a different truncation and we always keep the same number of meridional modes for each motion type. For the comparison of the FD and SP methods, we round the energy percentages to the nearest integer as the precision of computations for 91 vertical modes depends somewhat on the number of vertical modes included in the expansion and because of differences between the two methods, especially near the surface. For both M91SP and M91FD experiments, 70 vertical modes are used and the energy percentage of the IG waves is not different if the number of vertical modes is reduced to 50 suggesting that the energetics of the large-scale IG circulation reproduced by NMFs is primarily due to large-scale waves in the upper troposphere and the middle atmosphere. For experiments that test sensitivity to the level density and the model top, changes in the energy percentage from one experiment to another are small requesting us to present results with 0.1% precision in order to show the changes among the experiments.

## 3. Results

### a. Impact of the numerical method for solving the vertical structure equation

Results of the numerical experiments are summarized in Tables 2 and 3, which show the energy distribution in wave motions split among the three motion types. We keep consistence with the presentation in ZTAR09 and TK1999 and count the mixed Rossby–gravity wave (MRG) among the balanced modes. Many studies of tropical circulations consider it to be an IG mode and it is an important component of the tropical energetics. The spatiotemporal variability of the MRG mode energy in the upper troposphere and the middle atmosphere is a subject of a separate study; here, we only mention the MRG mode as needed for the discussion of various sensitivity experiments.

Distribution of wave energy among the balanced (R), the eastward inertio-gravity (EIG), and the westward inertio-gravity (WIG) motions as percentages of the total wave energy.

The first result of the study is that the M91FD and M91SP experiments agree on the level of the IG energy in wave motions in July 2007; the level is about 10% of the total wave energy up to 0.01 hPa and up to 0.18 hPa in the M91FD and M91SP experiments, respectively. There is 1% difference in the distribution of IG energy between EIG and WIG parts. Below 1 hPa, there is 6%–7% of balanced energy based on the two methods and 21 input levels. Within the range of uncertainties in the two methodologies, Table 2 confirms that the method for the solution of the vertical structure equation does not matter significantly for the result as it has been expected. It also suggests that a lower vertical resolution of input data and the location of the top level have an important impact on the result, issues to be further discussed with the help of additional experiments.

The experiments agree that the EIG energy exceeds the WIG energy. The difference is due to the large-scale Kelvin waves as discussed in Žagar et al. (2009b) for the same dataset. The Kelvin wave is the most energetic IG mode of the atmosphere and its vertical structure is illustrated below based on model-level data of different density. We note here that when the MRG mode is counted among the WIG modes in the FD experiments, the levels of the westerly and easterly propagating IG wave energy become equal.

### b. Impact of the vertical discretization and top level

Now we discuss the remaining five experiments with the FD method. The results are summarized in Table 3 along with the M91 and M21 experiments. The comparison with TK96 is provided by the experiment denoted M12FD, which contains 12 model levels closest to the standard-pressure levels of TK96. The M12FD experiment resulted in 6% of energy in the IG modes that is twice as much as TK96. In addition to the more advanced ECMWF modeling system and a dataset with a higher horizontal resolution, a better vertical resolution is another reason for advancements in representation of the large-scale IG waves. In other words, the 12 levels used in the present study are selected among the 60 levels from the surface to 50 hPa.

The comparison of M12FD with other experiments in Table 3 shows that the IG energy percentage steadily increases as the resolution of vertical levels increases and the top model levels move higher up. Three experiments concerning the vertical density of input data—M21, M42, and M80—have bottom and top levels at model levels 88 and 9, respectively. Increasing the vertical density of model levels from 21 to 42 produces a twice-larger relative increase of the EIG energy than increasing the number of levels from 42 to 80. Interestingly, the relative level of WIG energy remains the same. A small relative increase is at least partly associated with the fact that the IG energy in the M21 experiment contains the impact of computations on all 80 levels included in M80. Another reason for small changes is most likely the fact that the vertical scales of large-scale IG waves below 1 hPa do not need a better vertical resolution in order to be resolved. This will be illustrated in section 3c on the Kelvin wave example. Lifting the top level from 50 to 10 hPa and including a level close to 925 hPa (experiment M14FD) has reduced the balanced energy percentage for 0.2% by equally increasing the EIG and WIG energy. The reduction resulted from the surface levels (925 hPa) as proved in the experiment without the 10-hPa level (not shown).

However, the largest relative increase of IG energy is associated with the mesospheric circulation as it appears from the experiments M87 and M91. The latter experiment is especially relevant for the relative increase of the WIG energy. The relative change of IG energy in the troposphere and the stratosphere that occurs in connection with the higher vertical resolution of model-level data is about 1.2% of the total wave energy (i.e., the difference between M12 and M80 experiments). However, we need to keep in mind here that the IG wave structures in the M21 and M42 experiments as well as in the experiments with even smaller number of levels originate from computations on 80 ECMWF model levels. In comparison with the level density, lifting the model top into the mesosphere increases the energy percentage of IG waves more than twice.

Lifting the top level from 10 to 1 hPa increases both EIG and WIG energy across all scales. At the same time, it reduces their relative difference on the large scales as illustrated in Fig. 4. On the other hand, adding the upper stratosphere introduces the difference between the EIG and WIG waves on scales smaller than the zonal wavenumber 7 (Figs. 4a,b). The slope of the IG spectra is close to

Global energy spectra in July 2007 for the three experiments listed in Table 1 for balanced (R), eastward- (E), and westward (W) inertio-gravity waves as a function of the zonal wavenumber. Two additional dashed lines describe −3 and

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Global energy spectra in July 2007 for the three experiments listed in Table 1 for balanced (R), eastward- (E), and westward (W) inertio-gravity waves as a function of the zonal wavenumber. Two additional dashed lines describe −3 and

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Global energy spectra in July 2007 for the three experiments listed in Table 1 for balanced (R), eastward- (E), and westward (W) inertio-gravity waves as a function of the zonal wavenumber. Two additional dashed lines describe −3 and

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Another view of the relative increase of IG energy in different experiments is provided in Fig. 5, which shows the ratio between the energy percentage in EIG and WIG waves in various experiments and in the M21 experiment as a function of the zonal scale. To discuss this figure, we note that majority of the IG energy is in largest scales; for example, *k* = 1 contains around 30% of the total IG energy, 10%–20% of IG energy pertains to *k* = 2, 5%–10% belongs to *k* = 3–6, while other zonal wavenumbers contain less than 5% of the total IG wave energy. This implies that the relative increase of EIG energy in the lowest *k* in Fig. 5a is far more important than its relative decrease beyond wavenumber 5 in the M42 and M80 experiments. Figure 5 also shows that the WIG and EIG waves behave differently. Increasing the vertical data density in the experiments M42 and M80 adds the EIG energy on the largest zonal scales but not the WIG energy. In the WIG case, including the mesosphere (M87 experiment) increases energy also at scales *k* = 1–2, while M91 adds IG energy in all scales.

Ratio between the energy percentages in (a) EIG and (b) WIG waves in various experiments and the corresponding energy percentage in the M21FD experiment as a function of the zonal wavenumber.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Ratio between the energy percentages in (a) EIG and (b) WIG waves in various experiments and the corresponding energy percentage in the M21FD experiment as a function of the zonal wavenumber.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Ratio between the energy percentages in (a) EIG and (b) WIG waves in various experiments and the corresponding energy percentage in the M21FD experiment as a function of the zonal wavenumber.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Another difference between EIG and WIG waves visible in Fig. 5 is that the EIG curve for M91 follows those of the EIG curves for other experiments: namely, a relative energy increase is smaller for larger *k*. On the contrary, in the WIG case, the M91 experiment increases energy across all scales beyond *k* = 3 for about the same factor with respect to M21. It is difficult to interpret reasons for the different behavior of the two IG wave types. It is known that the ECMWF analyses above 1 hPa of July 2007 are of poor quality and dominated by the first-guess (forecast) information that has been characterized by large biases (Orr et al. 2010). Žagar et al. (2010) applied the normal-mode function projection to analyze biases in terms of systematic analysis increments and found out that the largest systematic increments in July 2007 are at the top model levels with the absolute maximum at level 4 (about 0.10 hPa). It is possible that model biases cause a stronger projection onto the WIG modes than EIG modes, but at this stage we cannot explain exact reasons for such behavior. Furthermore, in July 2007 the model applied the Rayleigh friction at top levels starting from level 18 (~9.5 hPa) with the friction coefficient increasing upward so that the top few levels had between one and three orders of magnitude stronger friction than the levels below. The friction was applied to all wavenumbers with the relaxation time scale of 3 days. In the later cycle Cy35r3 the friction was applied only to *k* = 0 and in the coming new model discretization with 137 levels the friction is planned to be switched off all together (P. Bechtold 2011, personal communication).

### c. Vertical structure of large-scale gravity waves: An example of the Kelvin wave

The impact of vertical data resolution is compared in Table 4 for several most energetic equatorial waves. In this case we included the MRG mode as it is a major mode of the tropical atmosphere and subject of numerous studies. Three experiments that analyze the same layer of the ECMWF model atmosphere show a clear tendency for the large-scale EIG modes to become more dominant over the WIG modes as the atmosphere gets better resolved. In these experiments the Kelvin mode made over 20% of the global IG wave energy in July 2007. Most of the energy is in the zonal wavenumber 1 in the stratosphere (not shown). However, we need to keep in mind that the present study is concentrated to the large scales in the tropics (Žagar et al. 2009b). Therefore, our estimate of the IG energy percentages with respect to the balanced energy is probably at the lower limit of real IG energies and new experiments with better models with higher resolution and a better middle atmosphere will most likely result in larger numbers. Similar results for the large-scale tropical waves are obtained from the FD and SP methods (not shown). The largest difference between M91FD and M91SP is for the MRG mode and it may be associated with the treatment of the top model levels (mesosphere) in the two methods and the difference between the pressure and *σ* levels.

Energy percentages (%) of different IG waves in the IG + MRG wave energy. The meridional mode index is denoted by *n*.

The inverse projection of IG waves back to the physical space was presented in Žagar et al. (2009b) as a way to study the four-dimensional structure of IG circulation. Here we show an example of such analysis in the case of the equatorial Kelvin wave in the experiments M21, M42, and M80. The example shown in Fig. 6 is taken from 0000 UTC analysis 1 July 2007. The upper-troposphere/lower-stratospheric Kelvin wave with the easterly wind peak of about 15 m s^{−1} is displayed in the altitude range 200–30 hPa. In the M80 experiment there are 24 model levels in this pressure range making it possible for the NMF expansion to reproduce the wave structure including vertical gradients and phase change at about 100 hPa. In the M42 and M21 experiments nearly the same structure is obtained by using different vertical structure functions based on 12 and 6 levels, respectively. It shows that the vertical wave structure is reproduced from a smaller number of selected levels such as the standard-pressure levels from a multilevel state-of-the-art system; however, the wave details such as the amplitude and the vertical gradients cannot be reproduced as well as from the model-level data. A larger figure from M91 shows a large-scale structure of the wave revealing characteristics typically found in observations and numerical simulations such as an eastward tilt with height and a vertical wavelength around 30 km in the mesosphere and around 10 km in the lower stratosphere (e.g., Wallace and Kousky 1968; Salby et al. 1984, figure not shown).

Zonal wind of the Kelvin wave along the equator at 0000 UTC 1 Jul 2007, between 200 and 30 hPa and 40°E and 170°W. Isolines are drawn every ±2 m s^{−1} and the zero isoline is omitted. Thick gray lines correspond to positive and thin black lines correspond to negative values. Results are based on the (a) M21, (b) M42, and (c) M80 experiments.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Zonal wind of the Kelvin wave along the equator at 0000 UTC 1 Jul 2007, between 200 and 30 hPa and 40°E and 170°W. Isolines are drawn every ±2 m s^{−1} and the zero isoline is omitted. Thick gray lines correspond to positive and thin black lines correspond to negative values. Results are based on the (a) M21, (b) M42, and (c) M80 experiments.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

Zonal wind of the Kelvin wave along the equator at 0000 UTC 1 Jul 2007, between 200 and 30 hPa and 40°E and 170°W. Isolines are drawn every ±2 m s^{−1} and the zero isoline is omitted. Thick gray lines correspond to positive and thin black lines correspond to negative values. Results are based on the (a) M21, (b) M42, and (c) M80 experiments.

Citation: Monthly Weather Review 140, 7; 10.1175/MWR-D-11-00103.1

A similar result is obtained for other large-scale IG waves suggesting that the vertical resolution of the ECMWF model resolves well the large-scale IG circulation in the tropics. The quantitative comparison of these results with observations is difficult as most of observational and modeling studies deal with the mesoscale gravity waves. For example, Tsuda et al. (2000) calculated the global distribution of potential energy of mesoscale gravity waves from the GPS occultation data at 20–30 km and the largest monthly mean values of potential energy around 10 J kg^{−1} were found in the tropics. This magnitude corresponds to that reported in Žagar et al. (2009b) for the zonal wavenumbers of about 30 and larger. A precise comparison between the energetics based on observations and the NMF expansion is not straightforward as the NMF procedure considers both potential and kinetic energy and should be extended to smaller horizontal scales, which is an effort under way (Terasaki et al. 2011).

## 4. Conclusions

We have compared two methods for the normal-mode function (NMF) expansion of atmospheric analyses to address the difference in levels of IG energy reported in recent studies. It is shown that the different methods lead to very similar results provided the same input data are used.

For the selected dataset of operational analyses of ECMWF in July 2007 it was found that the 91 model-level data contain about 10% of the global wave energy in terms of inertio-gravity waves. For the 21 standard-pressure-level data, the energy level of IG waves was 6%–7%. In the studied dataset, the IG energy is primarily associated with the large-scale tropical motions (Žagar et al. 2009b). The energy level associated with the eastward-propagating IG waves systematically exceeds that of the westward-propagating IG modes. Sensitivity experiments addressed the importance of the vertical density of model levels and the depth of the model atmosphere. The results show the systematic increase of the IG energy percentage with the vertical level density from the standard-pressure levels toward the model-level density. A relatively larger increase of the IG energy is obtained when levels in the mesosphere are included; however, the quality of analyses at these levels in July 2007 was not good due to model biases and the applied Rayleigh friction.

The obtained energy percentages apply to a single month of data. An ongoing study with the 20-yr reanalysis data of ECMWF (ERA-Interim) suggests the existence of seasonal and interannual variability of the large-scale equatorial waves, which will be the subject of a separate study.

The application of the NMF expansion provides the four-dimensional structure of large-scale equatorial waves simultaneously in the mass field and the wind field; the latter is still poorly observed. The vertical wave structures and their climatology in the tropics may be useful for the verification of the coupled climate models, which have the largest uncertainties in the tropics.

## Acknowledgments

The authors thank Dr. Peter Bechtold of ECMWF for the discussion related to the treatment of the top levels in the ECMWF model and to anonymous reviewers for their insightful comments, which led to improvements in the paper. The visit of N. Žagar to the University of Tsukuba was funded by the Center for Computational Sciences, by the Japan Society for the Promotion of Science, and by the Slovenian Research Agency. The Centre of Excellence for Space Sciences and Technologies SPACE-SI is an operation part financed by the European Union, European Regional Development Fund, and the Republic of Slovenia, Ministry of Higher Education, Science and Technology.

## REFERENCES

Andersson, E., and Coauthors, 2005: Assimilation and modeling of the atmospheric hydrological cycle in the ECMWF forecasting system.

,*Bull. Amer. Meteor. Soc.***86**, 387–402.Daley, R., 1983: Spectral characteristics of the ECMWF objective analysis system. ECMWF Tech. Rep. 40, 117 pp.

Daley, R., 1991:

*Atmospheric Data Analysis.*Cambridge University Press, 460 pp.Heckley, W. A., 1985: Systematic errors of the ECMWF operational forecasting model in tropical regions.

,*Quart. J. Roy. Meteor. Soc.***111**, 709–738.Kasahara, A., 1976: Normal modes of ultralong waves in the atmosphere.

,*Mon. Wea. Rev.***104**, 669–690.Kasahara, A., 1984: The linear response of a stratified global atmosphere to a tropical thermal forcing.

,*J. Atmos. Sci.***41**, 2217–2239.Kasahara, A., and K. Puri, 1981: Spectral representation of three-dimensional global data by expansion in normal mode functions.

,*Mon. Wea. Rev.***109**, 37–51.Orr, A., P. Bechtold, J. Scinocca, M. Ern, and M. Janiskova, 2010: Improved middle atmosphere climate and forecasts in the ECMWF model through a non-orographic gravity wave drag parametrization. ECMWF Tech. Memo. 625, 30 pp.

Salby, M. L., D. L. Hartmann, P. L. Bailey, and J. C. Gille, 1984: Evidence for equatorial Kelvin modes in

*Nimbus-7*LIMS.,*J. Atmos. Sci.***41**, 220–235.Shutts, G. J., and S. B. Vosper, 2011: Stratospheric gravity waves revealed in NWP model forecasts.

,*Quart. J. Roy. Meteor. Soc.***137**, 303–317.Simmons, A. J., and A. Hollingsworth, 2002: Some aspects of the improvement in skill of numerical weather prediction.

,*Quart. J. Roy. Meteor. Soc.***128**, 647–677.Staniforth, A., M. Beland, and J. Coté, 1985: An analysis of the vertical structure equation in sigma coordinates.

,*Atmos.–Ocean***23**, 323–358.Tanaka, H., 1985: Global energetics analysis by expansion into three-dimensional normal-mode functions during the FGGE winter.

,*J. Meteor. Soc. Japan***63**, 180–200.Tanaka, H., and E. Kung, 1988: Normal-mode expansion of the general circulation during the FGGE year.

,*J. Atmos. Sci.***45**, 3723–3736.Tanaka, H., and K. Kimura, 1996: Normal-mode energetics analysis and the intercomparison for the recent ECMWF, NMC, and JMA global analyses.

,*J. Meteor. Soc. Japan***74**, 525–538.Terasaki, K., and H. Tanaka, 2007: An analysis of the 3-D atmospheric energy spectra and interactions using analytical vertical structure functions and two reanalyses.

,*J. Meteor. Soc. Japan***85**, 785–796.Terasaki, K., H. Tanaka, and N. Žagar, 2011: Energy spectra of Rossby and gravity waves.

,*SOLA***11**, 45–48.Tsuda, T., M. Nishida, C. Rocken, and R. Ware, 2000: A global morphology of gravity wave activity in the stratosphere revealed by the GPS occultation data (GPS/MET).

,*J. Geophys. Res.***105**(D6), 7257–7273.Vincent, R., and M. J. Alexander, 2000: Gravity waves in the tropical lower stratosphere: An observational study of seasonal and interannual variability.

,*J. Geophys. Res.***105**(D14), 17 971–17 982.Wallace, J. M., and V. E. Kousky, 1968: Observational evidence of Kelvin waves in the tropical stratosphere.

,*J. Atmos. Sci.***25**, 900–907.Wang, L., and M. A. Geller, 2003: Morphology of gravity-wave energy as observed from 4 years (1998–2001) of high vertical resolution U.S. radiosonde data.

,*J. Geophys. Res.***108**, 4489, doi:10.1029/2002JD002786.Wang, L., M. A. Geller, and M. J. Alexander, 2005: Spatial and temporal variations of gravity wave parameters. Part I: Intrinsic frequency, wavelength, and vertical propagation direction.

,*J. Atmos. Sci.***62**, 125–142.Žagar, N., N. Gustafsson, and E. Källén, 2004: Variational data assimilation in the tropics: The impact of a background error constraint.

,*Quart. J. Roy. Meteor. Soc.***130**, 103–125.Žagar, N., E. Andersson, and M. Fisher, 2005: Balanced tropical data assimilation based on a study of equatorial waves in ECMWF short-range forecast errors.

,*Quart. J. Roy. Meteor. Soc.***131**, 987–1011.Žagar, N., J. Tribbia, J. L. Anderson, and K. Raeder, 2009a: Uncertainties of estimates of inertia–gravity energy in the atmosphere. Part I: Intercomparison of four analysis systems.

,*Mon. Wea. Rev.***137**, 3837–3857; Corrigendum,**138**, 2476–2477.Žagar, N., J. Tribbia, J. L. Anderson, and K. Raeder, 2009b: Uncertainties of estimates of inertia–gravity energy in the atmosphere. Part II: Large-scale equatorial waves.

,*Mon. Wea. Rev.***137**, 3858–3873; Corrigendum,**138,**2476–2477.Žagar, N., J. Tribbia, J. L. Anderson, K. Raeder, and D. T. Kleist, 2010: Diagnosis of systematic analysis increments by using normal modes.

,*Quart. J. Roy. Meteor. Soc.***136**, 61–76.