## 1. Introduction

Four-dimensional analysis datasets are the main information source for understanding the atmospheric general circulation and its variability on time scales from days to decades. The reanalysis datasets, such as those of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR; Kalnay et al. 1996) and the European Centre for Medium-Range Weather Forecasts (ECMWF; Uppala et al. 2005), have been applied to a broad range of research topics requiring information about atmospheric motions. Furthermore, these reanalyses are used to tune and validate climate models for their ability to reproduce the present-day climate, and thus to provide some confidence about their simulated circulations into the future.

A region where differences between various analyses, as well as differences between the climate models, are largest is the tropics. Analysis differences arise from the model differences, differences in the observational datasets used, and the data assimilation methodology. Numerous studies dealing with analysis intercomparisons (e.g., Trenberth et al. 2001; Hodges et al. 2003; Song and Zhang 2007) point out difficulties in drawing conclusions about tropical weather systems based on analyses because of the large uncertainty in the representation of tropical systems and difficulties in identifying these systems in the first place.

Major tropical circulation systems, like the Hadley cell and Walker circulation, are largely divergent. In particular, zonally averaged cross-equatorial (meridional) winds cannot be represented without involving motions of inertia–gravity type. At the same time, quantitative estimates of the relative contribution of inertia–gravity waves to the energetics of the large-scale flow and its four-dimensional structure are not well understood. The present study aims to fill this gap by analyzing the large-scale circulation in terms of normal-mode functions and by providing estimates of the uncertainty of large-scale divergent flows in present-day (re)analysis datasets.

Normal modes are obtained from eigensolutions of the linearized primitive equations. They have been used for almost 40 yr (e.g., Dickinson and Williamson 1972) but their application has mainly been concentrated on the initialization problem for numerical weather prediction (NWP; e.g., Baer and Tribbia 1977; Daley 1979; Temperton and Williamson 1981; Errico 1997). In NWP applications of nonlinear normal-mode initialization (NNMI) only the first few vertical modes were subjected to initialization in which the time tendencies of inertia–gravity modes were set to zero. In the tropics, the procedure led to the weakening of the Hadley and Walker circulations and significant efforts have been devoted to improve the diabatic circulation (e.g., Heckley 1985; Wergen 1988).

As the horizontal and vertical resolutions increased, the model top level moved higher up and convective parameterizations improved, a successful application of the NNMI became more complex. In the current NCEP analysis system, the tangent-linear normal-mode constraint is applied to analysis increments during each inner loop integration of the minimization process in their three-dimensional variational data assimilation (3DVAR; Kleist et al. 2009). However, questions related to the derivation of normal modes for models with many levels and the choice of what vertical and horizontal modes to initialize remain open. An alternative initialization method for NWP is the digital filter initialization (Lynch and Huang 1992). It can easily be implemented as another control term in a four-dimensional variational data assimilation (4DVAR) cost function and it was shown to have a small impact on 4DVAR analysis increments (Gauthier and Thepaut 2001).

The application of normal-mode functions (NMFs) in the present study has a different purpose from that described above. We apply an orthogonal set of normal modes with the goal of representing most of the input dynamical information in the wind and geopotential fields on all model levels. This is measured by evaluating the correlation coefficient and variance ratio between the input and projected fields. A reliable projection allows us to represent the circulation in terms of balanced (quasi-rotational or Rossby type) and unbalanced (eastward- and westward-propagating inertia–gravity) motions of different vertical and horizontal scales. Differences between derived spectra for various analysis datasets represent a measure of uncertainty of the present descriptions of large-scale tropical circulations in terms of various scales and wave types.

The applied normal-mode set was derived by Kasahara and Puri (1981). They performed normal-mode analysis of Northern Hemisphere daily data for a single month in the late 1970s by projecting data onto a few vertical modes and a small number of corresponding symmetric horizontal modes. In agreement with the quasigeostrophic scaling and the poor quality of the tropical circulation in early analyses (e.g., Heckley 1985), the contribution of the inertia-gravity (IG) modes to the global energy was found to be small. More recent studies of global energetics (Terasaki and Tanaka 2007; Tanaka and Kimura 1996) also indicate low percentages of IG energy; for example, Tanaka and Kimura (1996) report about 0.7% IG energy in the total and around 3% in the wave part of the global motions. These energy levels are significantly smaller than those reported in this study. Terasaki and Tanaka (2007) also showed that, in order to represent the source of the available potential energy, higher-order modes of the vertical structure functions are required. This last result corroborates an earlier numerical model study by Ko et al. (1989), which revealed that the gravity waves associated with the shallow vertical modes and long zonal waves play an important role in the balanced gravitational energy.

An important part of the tropical variability is associated with large-scale IG waves, which are characterized by small phase speeds and are equatorially trapped. Understanding these modes, their role in the transient tropical circulations, and their impact on the extratropics has been a subject of numerous observational (e.g., Takayabu 1994; Wheeler and Kiladis 1999) and modeling (e.g., Majda et al. 2004) studies in the last decade; it is also a topic addressed by research programs such as The Observing System Research and Predictability Experiment (THORPEX; Shapiro and Thorpe 2004). Observational evidence of equatorial waves most often rely on time series of satellite observations of convection (i.e., convection proxy data such as the outgoing longwave radiation and brightness temperature). This is mass-field information; the dynamical structure of the waves has been identified by applying linear regressions between the proxy data and reanalysis fields (e.g., Yang et al. 2003). Space–time spectral analysis points out several equatorially trapped waves coupled to the convection; besides the Madden–Julian oscillation (MJO), which does not correspond to a particular normal mode, identified modes include Kelvin, equatorially trapped Rossby, mixed Rossby–gravity (MRG), eastward inertia–gravity (EIG), and westward inertia–gravity (WIG) waves.

Contrary to the space–time spectral analysis, diagnostics by means of normal modes analyze the mass and wind field simultaneously. On output, the energy and variability related to each mode are quantified. Furthermore, a comparison between various datasets provides an uncertainty range of equatorial wave analysis, information important for improving climate models that have significant problems in simulating tropical intraseasonal variability (Lin et al. 2006).

Since atmospheric motions are nonlinear, we cannot clearly separate them into the high-frequency (ageostrophic or unbalanced) IG and low-frequency, quasigeostrophic or balanced (ROT) motions except in case of linearized equations around some specific background states. Nevertheless, the normal modes, which are orthogonal and complete in functional representation and handle both the velocity and mass consistently, are useful for analyzing the observed states of the real atmosphere, as shown by numerous studies.

Our estimate of the uncertainty of inertia–gravity wave energy is based on four analysis datasets that are presented in section 2. Section 3 deals with the methodology for normal-mode expansion, the derivation of the vertical structure functions, and the selection of the truncation limits, which define the modal basis. Results of the normal-mode diagnostics are organized in two parts. The present paper (Part I) presents the monthly averaged energetics of the balanced and IG circulation and compares four datasets. In Žagar et al. (2009, hereafter Part II), we look in detail at the spectra of various equatorial waves and at their spatial and temporal structure, especially the Kelvin wave. Discussion and conclusions of the present paper are presented in section 5.

## 2. Analysis datasets

Results are presented based on the analysis of a single month of data, July 2007. The four analysis datasets are the NCEP–NCAR reanalysis, the operational NCEP and ECMWF analyses, and the analysis produced by the ensemble assimilation system at NCAR.

NCAR’s assimilation system, the Data Assimilation Research Testbed (DART), is based on the ensemble adjustment Kalman filter (EAKF) described in Anderson (2001) and Anderson (2003). In ensemble data assimilation, a short-range forecast ensemble is used to derive the forecast-error covariances and it is the flow dependency of these covariances that is expected to provide improved analyses with respect to present variational methods (e.g., Lorenc 2003; Kalnay et al. 2007). We applied the ensemble assimilation with the Community Atmospheric Model (CAM), which is the atmospheric component of the Community Climate System Model (CCSM; Collins et al. 2006a,b). Version 3.1 of the CAM is used, which is an improved version of the model documented in a number of papers in a special issue of the *J. Climate* (2006, Vol. 19, No. 11). Dynamical simulations of the model are documented in Hurrell et al. (2006), whereas Tribbia and Baumhefner (2004) showed that the model’s skill in short-range forecasts is comparable to the conventional statistical measures that are used for NWP models (i.e., anomaly correlation at 500 hPa). In its implementation for data experiments reported in this study, the spectral version of the model was used with T85 truncation. There are 26 hybrid vertical levels extending up to 3.7 hPa. The atmospheric model is coupled to a land surface model and forced by monthly mean sea surface temperature fields produced at NCEP.

The initial 80-member ensemble is taken from various CCSM simulations for July done within the Atmospheric Model Intercomparison Project (AMIP); the initial ensemble is thus characterized by a large spread that reduces after the first couple of days of assimilation. For this reason, we are not considering the first 2 days of the assimilation and all DART–CAM-related analysis is based on the period 3–31 July. We only use the ensemble mean fields; the study of uncertainties as measured by the ensemble spread and their flow dependency are the subject of a separate paper. Here we aim primarily at comparing this relatively new system to three well-established datasets.

Observations used are a subset of the observations used in the NCEP–NCAR reanalysis project; DART uses radiosondes, aircraft measurements, and satellite cloud motion wind vectors. As the most valuable information source is the radiosondes, there is a large spatiotemporal inhomogeneity in the observation coverage. Analysis increments are much more significant at 0000 and 1200 UTC everywhere but especially in the Northern Hemisphere extratropics. Observations are allowed to influence the model variables up to 100 hPa. Above this height, increments are gradually decreased to become zero within several levels. Significant damping applied to the upper levels in this model version makes EAKF assimilation efforts in the stratosphere difficult.

The NCEP–NCAR reanalysis project is described in Kalnay et al. (1996); various features of this dataset have been presented in numerous studies (e.g., Kistler et al. 2001). Here we utilize the analysis fields on the model levels and regular Gaussian grid. The reanalyses are based on the 1995 version of the NCEP model and the spectral statistical interpolation method (Parrish and Derber 1992). The balance of analysis increment fields is imposed through the background-error term of the three-dimensional variational assimilation. Besides all conventional data, the satellite cloud drift winds and satellite soundings (retrievals) are assimilated. The assimilation outputs used here were postprocessed horizontally from a T62 grid to a Gaussian N47 grid and they are provided on 28 vertical sigma (*σ*) levels in the range between 0.995 and 0.0027. The Gaussian grid thus includes 192 and 94 points in the longitudinal and latitudinal directions, respectively (corresponding to the grid spacing of 1.875° × 1.915°).

There are numerous differences between the operational NCEP assimilation system and that employed for the reanalysis project. The differences include a change from the spectral to the gridpoint statistical interpolation (Wu et al. 2002) and numerous changes of the forecast model including a change of the vertical coordinate and increased horizontal and vertical resolutions. The model used in July 2007 has 64 vertical hybrid levels with the model top level located at 0.32 hPa and T382 truncation. While the reanalysis fields are not initialized, operational products include the tangent linear normal-mode constraint described in Kleist et al. (2009).

Operational analyses of ECMWF for July 2007 are the result of the 12-h window 4DVAR assimilation scheme and the model cycle Cy32r2. The large number of satellite observations assimilated and 4DVAR assimilation method are important factors contributing to the model’s overall forecast success in the medium range over the last decade (e.g., Simmons and Hollingsworth 2002). Compared to the other systems, the ECMWF model has a relatively high resolution; the horizontal truncation is T799 (∼25 km) and there are 91 vertical levels with a model top level located at 0.01 hPa.

For a better comparison with the DART–CAM system, and because we are interested mainly in the large-scale flows, the operational NCEP and ECMWF fields are interpolated to the grid used by the CAM model—a regular (Gaussian N64) grid with 256 and 128 points in the longitudinal and latitudinal directions, respectively. The ECMWF fields were extracted on this grid from the meteorological archival and retrieval system of ECMWF while the NCEP T382 data were processed using the spherical harmonic interpolation within the NCAR command language (NCL).

A comparison of the zonal wind field in the tropical upper troposphere in the four datasets is shown in Fig. 1. It can be seen that, besides the overall similarity in the alternating regimes of westerlies and easterlies, there are some significant differences between the analysis solutions, especially over the Pacific. The NCEP–NCAR reanalyses are produced at significantly lower resolution than the other three datasets and they almost completely lack episodes of westerlies over the Pacific. Figure 1 also shows that the DART–CAM analyses produce solutions for the large-scale tropospheric flow comparable to the operational systems, in spite of poor observation coverage.

## 3. Application of the normal-mode expansion

*σ*coordinates. The equations are linearized around a motionless mean state with a vertical stratification as a function of

*σ*. Solutions to the adiabatic and inviscid linearized equations are then sought by assuming separability of the vertical and horizontal dependences of the dependent variables which leads to the two standard systems of horizontal and vertical structure equations. The latter takes the following form:

*H*

_{eq}, which has the dimension of length, and it couples the horizontal and vertical equations. A subscript “eq” stands for the “

*equivalent depth*” as this parameter is best known; this comes from the equivalency of the horizontal equations to linearized shallow-water equations with the fluid depth of

*H*

_{eq}. The solutions of Eq. (1) require two boundary conditions, at the model top and bottom half levels (

*σ*= 0 and

*σ*= 1, respectively):

*σ*levels, Eq. (1) requires as input the vertical stability profile Γ

*, defined as*

_{o}*κ*is defined as

*κ*=

*R*/

*C*where

_{p}*C*is the specific heat at constant pressure and

_{p}*R*is the gas constant. The globally averaged temperature on model levels is denoted by

*T*and

_{o}*g*is gravity. The spectrum of solutions of (1) is discrete (Cohn and Dee 1989) and it is given in terms of vertical eigenfunctions Π

*(*

_{m}*σ*), where

*m*ranges between 1 and

*N*, the number of vertical levels.

_{σ}Solutions of the horizontal system are given in terms of Hough functions and the reader is referred to Kasahara and Puri (1981) for details. Here we discuss the application of the normal-mode expansion to our four datasets. A discussion of the importance of the model top boundary conditions is provided in the appendix.

### a. Expansion of discrete data into normal modes

*u*,

*υ*,

*g*

^{−1}

*P*)

^{T}is represented by the following finite series:

*P*is a modified geopotential height variable, defined as

*gh*is the geopotential, and

*p*is the surface pressure field. At the surface,

_{s}*P*=

*gh*+

_{s}*R*(

*T*)

_{o}*ln(*

_{s}*p*). The subscript “

_{s}*s*” refers to the surface values, and

*h*stands for the height of orography. Parameters

_{s}*k*,

*n*, and

*m*denote the zonal wavenumber, meridional mode index, and vertical mode index, respectively. The index

*p*is associated with various wave types; value 1 indicates EIG, 2 is WIG, and

*p*= 3 corresponds to the balanced modes, denoted by ROT. A single mode is denoted by a four-component index,

*v*= (

*k*,

*n*,

*m*,

*p*). The scaling matrix 𝗦

*removes dimensions from the input data vector after the vertical projection is performed; it is a diagonal matrix with elements (*

_{m}*gH*

_{eq})

^{1/2}, (

*gH*

_{eq})

^{1/2}, and

*H*

_{eq}. The Hough function

*k*≠ 0 and the case

*k*= 0 is treated in Kasahara (1978). Independent variables

*λ*,

*θ*, and

*σ*define location in terms of the latitude, longitude, and vertical level, respectively.

The truncation indices *N _{k}* and

*N*correspond to the number of waves along a latitude circle and a maximal number of meridional structure functions for a given (

_{n}*k*,

*m*,

*p*) combination. The number of vertical modes

*N*is equal to the number of vertical model levels. Each vertical mode

_{m}*m*is associated with equivalent depth

*H*

_{eq}(

*m*), which couples the horizontal motion and vertical structures Π

*(*

_{m}*σ*). For each

*H*

_{eq}(

*m*) and each zonal wavenumber

*k*, there exist a set of 3 ×

*N*frequencies, one per wave type. The dispersion curves for two values of

_{n}*H*

_{eq}, 10 km and 11 m, are shown in Fig. 2; it illustrates the frequency gap that exists between the IG motions and balanced waves, especially for large equivalent depths. Figure 2 also illustrates that as the equivalent depth decreases, the frequency gap between the ROT and IG modes reduces; on the other hand, the gap at the lowest

*k*becomes clearer. This gap has been extensively used as the basic criterion in the initialization procedures. In agreement with observations of large-scale equatorial waves, it can be seen that for the longest zonal scales and

*H*

_{eq}of the order of 10 m the Kelvin and MRG modes appear more clearly distinguishable from all other modes. The meridional trapping of the modes to the equator associated with the reduction of the equivalent depth also occurs with increasing zonal wavenumber.

Following Kasahara (1976), the MRG mode is included in balanced motion as the *n* = 0 ROT mode. The Kelvin mode is the eastward-propagating *n* = 0 IG wave. The *n* = 1 EIG mode corresponds to the wave often denoted as the eastward-propagating MRG; in such cases the Kelvin mode is denoted as the *n* = −1 EIG mode, the eastward MRG becomes the *n* = 0 EIG mode, and the meridional indices for WIG modes are increased by one so that the *n* = 0 WIG is denoted *n* = 1 WIG and so forth. The *n* = 0 WIG mode is thus associated with the westward-propagating MRG (our *n* = 0 rotational mode). This can be seen in Fig. 2 where the dispersion curves for the MRG mode connect to the *n* = 1 EIG mode. The *n* = 0 WIG mode connects to the *n* = 2 EIG mode in the limit *k* → 0, and similarly for larger *n*.

*χ*are nondimensional. To make values dimensional again (i.e., to calculate the energy in a particular mode),

_{ν}*χ*coefficients are multiplied by

_{ν}*gH*

_{e}### b. Derivation of vertical eigenmodes

Vertical structure functions are derived on *σ* levels. The DART–CAM, ECMWF, and NCEP analyses all have hybrid *σ*-pressure levels that follow *σ* levels close to the surface and become pressure levels in the upper troposphere and higher up. For these models the winds and geopotential data are thus first interpolated to their corresponding *σ* levels. The vertical temperature and stability profiles needed to solve the vertical structure in Eq. (1) are derived by averaging over the whole dataset for each analysis system (116 samples for DART–CAM and 124 samples for ECMWF, NCEP, and NCEP–NCAR data). Then Eq. (1) is solved using the finite-difference approach as in Kasahara and Puri (1981).

NMFs are derived for the bounded atmosphere with lid located at the model top half-level (*σ* = 0 or *p* = 0). This means that the analysis systems are characterized by different values of the depth (Δ*σ*) and stability (Γ* _{o}*) for the top model layer. With the surface temperature value taken as the same global constant, the

*H*

_{eq}for

*m*= 1 is about 10 km for all systems. This is expected since the barotropic equivalent depth depends only on the surface temperature and the model depth (Cohn and Dee 1989). Furthermore, in agreement with the findings by Staniforth et al. (1985), equivalent depths for the subsequent internal modes appear relatively insensitive to the value of the surface boundary condition, but they are sensitive to the top boundary conditions.

The resulting 26 vertical eigenmodes for DART–CAM are presented in Fig. 3. Each panel shows the structure for three subsequent modes and the modal index *m* is associated with *m* − 1 zero crossings in the vertical profile. The mode with lowest vertical index, *m* = 1, does not change sign and is often referred to as the barotropic mode. The lowest 10 modes have two zero crossings in the troposphere and more structure in the stratosphere. As the value of *m* increases, the relevant structure of Π* _{m}* functions moves downward toward the surface. The increasing value of

*m*is associated with the reduction of the equivalent depth,

*H*

_{eq}. In the case of DART–CAM, the equivalent depths have values from 10 km (

*m*= 1) down to 0.3 m (

*m*= 26). Solutions for the NCEP–NCAR reanalysis system, with its 28

*σ*levels distributed similarly to the discretization adopted in CAM, look similar both in the shapes of Π

*functions and the values of corresponding*

_{m}*H*

_{eq}. For the ECMWF and NCEP systems, the zigzag shape of the vertical eigenstructures moves downward more slowly because of the large number of levels and many levels in the stratosphere.

As illustrated in Fig. 3, modes with small equivalent depths are associated with the model levels in the lower troposphere and close to the surface. This implies that, in order to represent the vertical structure of the motions in these layers in terms of normal modes, it is crucial to keep higher vertical modes in the expansion. This especially applies to the ECMWF and NCEP systems that have many modes with small *H*_{eq}; in each of these two systems one-forth of the modes are characterized by equivalent depths smaller than 1 m. For example, in the ECMWF model the planetary boundary layer is represented by at least 12 levels (those below 900 hPa) and there are 30 levels above the tropopause. Discretization and stability profiles for these layers and the numerical solution of Eq. (1) result in many vertical modes with small equivalent depths and zigzag shapes.

Contrary to the horizontal eigenmodes, which have a well-defined physical significance and can be derived analytically, the structure of the vertical modes is related to the finite-difference approximation made to solve (1) with the boundary conditions in (2). The accuracy of the numerical solution of Eq. (1) has been discussed in a number of studies (e.g., Sasaki and Chang 1985; Staniforth et al. 1985; Castanheira et al. 1999; Terasaki and Tanaka 2007). In comparison to these earlier studies, our solution involves much deeper model atmospheres; for example, realistic stability profiles we used in solving (1) have amplitude variations of five orders of magnitude in the case of ECMWF and NCEP profiles (two orders of magnitude for DART–CAM and NCEP–NCAR data). Large changes in the stability profile across the tropopause and in the middle atmosphere result in more condensed structures of Π* _{m}*(

*σ*) across these layers, as noticed in Sasaki and Chang (1985). On the other hand, better discretization compared to previous studies improves the accuracy of the finite-difference solution. The solution is also more accurate when the top model level is closer to 0 Pa (Staniforth et al. 1985). Higher vertical modes, which are not computed accurately, are not of particular interest here; this study concentrates on the leading vertical modes and associated horizontal large-scale motions that are computed sufficiently accurately; as shown in the results section, these modes contain the majority of atmospheric energy.

### c. Selection of truncation parameters for the expansion

*N*,

_{k}*N*, and

_{n}*N*, which are chosen to maximize the percentage of the input flow variance, which is represented by normal modes. Namely, when the forward projection in (6)–(7) is followed by the inverse in (4), the output vector of values for wind and geopotential is somewhat different from the input. If we denote the resulting fields after the inverse projection by the superscript

_{m}*i*, (

*u*,

^{i}*υ*,

^{i}*g*

^{−1}

*P*)

^{i}^{T}, the mismatch between (

*u*,

*υ*,

*g*

^{−1}

*P*)

^{T}and (

*u*,

^{i}*υ*,

^{i}*g*

^{−1}

*P*)

^{i}^{T}can be used to define the accuracy of the projection. Our goal is to represent as much of the input flow for all levels, regions and all three variables as possible; therefore, we tune the choice of the three truncation parameters to meet this criterion. The accuracy is measured by two parameters: the correlation coefficient and the variance ratio between the input fields and their inverses. The correlation coefficient is defined as it is usually done while the variance ratio at level

*σ*for a single variable, for example

*u*wind, is defined as

*u*

*u*variable.

Figure 4 presents the results of testing the projection quality for several variables and various datasets. For each dataset, the projection accuracy is computed separately for three regions of the globe in order to emphasize different issues related to the selection of truncation parameters *N _{m}*,

*N*, and

_{n}*N*, which provide the optimal results in various areas and levels. The three regions correspond to the tropics (20°S–20°N), the Southern (south of 20°S), and the Northern (north of 20°N) Hemisphere extratropics. With truncations fixed, the value of

_{k}*ϵ*varies little from one analysis time to another. The correlation coefficient produces significantly better scores than the variance ratio [e.g., Eq. (9)]; above the planetary boundary layer the correlation coefficient is over 0.9 for the wind components and around 0.99 for the mass-field variable. Both coefficients show better results for the meridional wind component in the tropics than for the zonal component and other regions since the meridional wind has smaller variability in the tropics than in the midlatitudes.

As seen in Fig. 4, the representation of the lowest troposphere, particularly in the tropics, is most difficult since we do not include all vertical modes in the expansion because of the accuracy problems with solving (1) and because of the equatorial trapping of horizontal solutions for the highest vertical modes. There is relatively little energy at scales above zonal wavenumber 20, but nevertheless it is important to keep *N _{k}* sufficiently large to represent the variance properly. The disadvantage of involving high zonal truncation, together with many vertical modes, is that the zonal wind variance in the tropical lower troposphere becomes overestimated because of the equatorial trapping at higher vertical modes. On the other hand, for the same reason the boundary layer variance in the midlatitudes can be underestimated, especially when an insufficient number of the meridional modes is used. On average, the ECMWF and NCEP projections contain too much zonal wind variance at the lowest model levels in the tropics and not enough variance in the midlatitudes close to the jet stream regions because of significant spatial gradients in the input wind field. The worst results close to the surface are also due to the local nature of the orography field entering the

*P*variable, which has the largest effect on the ECMWF projection.

Selection of *N _{n}* requires special care. For larger

*n*, the meridional structure of Hough functions is characterized by large amplitudes near the poles (Kasahara 1976, 1978) that can easily give rise to false signals in high latitudes in the projection. This is particularly true for the zonal wind. The final choice of the truncation parameters is thus a trade-off between the levels, variables, and regions of most interest.

An example of the input and inverse field is shown in Fig. 5 for the zonal wind in the DART–CAM datasets at a randomly chosen date. At model level 20 (∼602 hPa) the zonal wind variance in the tropics is about 10% too large, which can be seen from the more zonally aligned contours in the inverse field than in the input; this is a consequence of the equatorial trapping of high vertical modes, which becomes more serious in the ECMWF and NCEP datasets and lower troposphere. Overall, it can be seen that most flow features are represented very well.

Choices of truncations in Fig. 4 are those used to obtain results discussed in the following sections. For DART–CAM the values used for truncations are as follows: 25 modes were used in the vertical direction, the zonal truncation was set to *N _{k}* = 80 while the meridional truncation was set to

*N*= 25 for all three wave types. For the NCEP–NCAR reanalysis, the choice of parameters was

_{n}*N*= 26,

_{m}*N*= 18, and

_{n}*N*= 46. Selection of the expansion basis for the ECMWF and NCEP models was more difficult because of difficulties with representing the lower troposphere. At the same time, these systems include many vertical levels in the middle atmosphere where a small number of horizontal and vertical modes suffices to represent the input data nearly perfectly. The results discussed further for the ECMWF datasets are obtained by using

_{k}*N*= 79 vertical modes,

_{m}*N*= 54 zonal waves, and

_{k}*N*= 18 meridional modes for each motion type. For the operational NCEP analysis, the chosen truncation set is

_{n}*N*= 54,

_{m}*N*= 18, and

_{n}*N*= 54. Horizontal truncations are the same for ECMWF and NCEP; for DART–CAM, a better result was obtained with a larger meridional truncation and the zonal truncation was kept close to the maximal value (T85) because of the follow-on study that deals with the ensemble spread and requires better representation of small-scale features. The selected orthogonal bases for the four datasets account for nearly all variance above the boundary layer. This allows us to continue with the discussion of the average July 2007 energetics and the energy partition among various motions.

_{k}## 4. Energetics of atmospheric circulation in July 2007

First we discuss the one-dimensional energy spectra separated into ROT, EIG, and WIG components. Total energy as a function of the zonal wavenumber is displayed in Fig. 6. Since the input model resolution and model top significantly differ, energy amplitudes are not to be compared quantitatively. Instead, we concentrate on the slope of the spectra and energy distribution in terms of scales and motion types. For each dataset, the ROT spectrum has −3 slope over many scales smaller than the synoptic injection scale (*k* = 6–7), well known from theoretical prediction (e.g., Leith 1968) and diagnosis of atmospheric datasets (e.g., Boer and Shepherd 1983; Tanaka and Kung 1988). For the scales larger than those of the synoptic disturbances, the slope is close to

There are several differences between the slopes for ROT spectra and those for the IG modes. The slope of the IG spectra at the longest scales (*k* = 1–5) is close to −1 and at synoptic and shorter scales the IG energy spectra shift toward ^{1} This surface variability enters the potential energy integration via the surface pressure term of the *P* variable [Eq. (5)] and it is responsible for the noisy appearance of the ECMWF spectra beyond *k* = 20. The interpolation of input fields from the hybrid to the *σ* surfaces acts to spread this effect vertically. In Part II we show the ECMWF energy spectrum resulting only from the wind variables; in this case the spectrum has a slope between −3 and *k* > 7 and a smooth appearance.

In general, it is not easy to discuss the IG spectra in Fig. 6 as they represent the whole atmosphere and consist of both potential and kinetic energy. Theoretically it is not completely clear what the IG slope should be, especially at the mesoscale, although various studies suggest the *k*^{−5/3} slope (e.g., Dewan 1997; Waite and Snyder 2009). The slope can be different for various vertical modes, in agreement with scales that are projected onto that mode. Thus, steeper slopes better fit lower vertical modes at synoptic forcing scales (not shown). Normal-mode expansion of earlier datasets by Tanaka and Kimura (1996) shows slopes for the IG spectra between −1.8 and −1.0 for the IG modes with *m* = 0–4. In the present case differences between IG spectra in Fig. 6 are related to differences in the quality of tropical analyses that are influenced by the assimilation procedure and the model much more than the corresponding midlatitude solutions (e.g., Žagar et al. 2005). Furthermore, various datasets represent significantly different layers of the atmosphere; for example, the ECMWF model extends up to 0.01 hPa while the CAM model has a lid at 2.3 hPa. With data provided at a 6-h frequency, the IG spectra contain both the westward-propagating diurnal oscillation, including the diurnal sun-synchronous tide and its harmonics, and the eastward-propagating diurnal oscillation (non-sun-synchronous or non-migrating tide) and its harmonics. To separate these signals from the IG spectra in Fig. 6 is beyond the scope of the present study. Further discussion of the IG spectra is provided, however, in Part II.

Figure 6 shows that the energy contained in ROT modes far exceeds the energy in IG motion, in agreement with basic understanding of the large-scale balanced dynamics. However, the percentage of total IG energy is not negligible; for *k* > 0 the IG energy percentage ranges from ∼9% in NCEP to 12% in DART–CAM, and 15% and 45% in ECMWF and NCEP–NCAR data, respectively (Table 1). Corresponding percentages in the total flow (all *k*) range between 1.3% in NCEP and 1.4% in DART–CAM to 1.8% and 4.1% in ECMWF and NCEP–NCAR data, respectively. These percentages are larger than those reported in Tanaka and Kimura (1996); these authors found about 0.7% and about 3% of energy in the total and wave flow, respectively, associated with IG waves in the winter of 1998/99. We speculate that the difference is due to differences in the datasets and different methodology of the normal-mode expansion. The earlier include higher-resolution models and improved model parameterizations as well as deeper model atmospheres. In particular, in later years the description of the tropics in the operational analysis systems has improved significantly (e.g., Bechtold et al. 2008). The methodology differences include use of model levels instead of standard pressure levels and selection of the modal basis, which provides the optimal fit to the input data. For example, mentioned study of Tanaka and Kimura (1996) used data on the 2.5° × 2.5° horizontal grid and 12 standard pressure levels up to 50 hPa. In the next section we discuss that the IG circulation on model levels between two standard pressure levels can vary significantly, even when time averaged. Furthermore, in contrast to earlier implementations of NMFs for the global energetics (i.e., Tanaka and Kung 1988; Tanaka and Kimura 1996), we keep the same truncation for the Rossby and for the EIG and WIG modes. There is no a priori reason for truncating these modes at values smaller than those for balanced motions.

Associated with the higher resolutions of the NCEP and ECMWF models compared to earlier studies, increased levels of IG energy may also be related to the resolved large-scale drag due to internal Rossby–gravity waves existing in a stably stratified westerly flow in the midlatitudes (Teixeira and Grisogono 2008). A possible reason for increased IG energy in DART–CAM at large scales may be covariance localization performed in the ensemble Kalman filter assimilation system that has been shown to impact synoptic-scale Rossby wave balance (Kepert 2009). For a particular dataset, and especially for the ECMWF system with its many vertical levels in the lower troposphere, there is an uncertainty of percentages shown in Table 1 related to the numerical solution of Eq. (1) and the number of vertical modes used for the projection. For example, if only 30 vertical modes is kept in the expansion (e.g., 6–7) for ECMWF data, the percentage of IG wave motion drops from ∼14% to ∼10%. But the projection fit to the input data on model levels in the troposphere deteriorates.

It is worth remarking that our results for the contribution of IG motions are in agreement with recent laboratory experiments of Williams et al. (2008), which suggest that IG waves emitted from balanced fluid flow might make a significant contribution to the energy budgets of the atmosphere and oceans (i.e., 9% of the total energy; Table 1 in Williams et al. 2008).

At the largest scales (*k* = 1–3) eastward IG dominates westward, especially at *k* = 1 and in the DART–CAM analyses. This is due to signals that project onto the Kelvin wave, as will be illustrated below. The dominance of EIG over WIG energy associated with the Kelvin wave has been noted also in the earlier study by Tanaka and Kung (1988). At larger wavenumbers, the energy content is very small in all modes but there are significant differences between the datasets concerning the scales at which ROT and IG energy became equally important.

The two-dimensional spectra *E*(*n*, *m*) presented in Fig. 7 illuminate further details about the energy distributions. The spectra are shown as percentages of the total wave energy (i.e., excluding the zonally averaged state for a particular model). The largest difference exists between the NCEP–NCAR reanalyses and other three systems in the distribution of energy for the IG modes and the contribution of IG energy to the total wave energy. In the two operational systems (ECMWF in Figs. 7d–f and NCEP in Figs. 7i,j,m) the main properties of the *E*(*n*, *m*) distribution appear similar. The maximum of ROT energy is in the barotropic (*m* = 1) mode and *n* = 3 meridional mode, which is symmetric. A secondary maximum is seen in the asymmetric (*m*, *n*) = (2, 2) mode. While DART–CAM ROT energy peaks at the same *n* as the operational systems, the NCEP–NCAR solution is characterized by two maxima in the barotropic mode, *n* = 4 and *n* = 7. Results are similar for the IG spectra; the spectra as a function of the meridional mode have maxima at *n* = 0 for both EIG and WIG for ECMWF and NCEP. The EIG spectra in DART–CAM peak at *n* = 0 and the WIG part is maximized across modes *n* = 0–2. The spectra as a function of vertical mode have maximal amplitudes in *m* = 3 and *m* = 4 for the EIG and WIG components, respectively, in DART–CAM. The ECMWF spectra maximize at *m* = 6–10 and *m* = 2 for EIG and WIG motions, respectively; similar shapes characterize the operational NCEP spectra, which have energy peaks at *m* = 2–5 for WIG and *m* = 6–8 for EIG modes. In the NCEP–NCAR reanalyses both EIG and WIG spectra have maximum IG energy at *m* = 1. Consistent with a lack of a strong Kelvin wave signal the EIG distribution in this case is maximal at *n* = 4 with a secondary maximum at *n* = 0 whereas the WIG *E*(*n*) distribution is maximal at *n* = 1. While we do not have clear evidence why the NCEP–NCAR reanalyses are significantly different from the other three datasets, it can be noted that meridional modes have a different structure for NCEP–NCAR data because of their lower resolution.

It should also be noted that the ROT spectra in Figs. 6 –7 include the MRG energy since in our expansion this mode is the *n* = 0 ROT mode. This does not affect the slope of the spectra in Fig. 6. In Fig. 7 the energy in the MRG mode falls below the lowest contour interval plotted, except for the ECMWF system as this mode contains much less energy than ROT modes. In Part II the MRG mode is discussed as an IG mode and studied in more detail.

ROT energy in the zonally averaged state (*k* = 0), not included in Figs. 6 –7, appears similar in different datasets; the bulk of energy is concentrated in the barotropic vertical mode maximizing at *n* = 2 and the secondary maximum is located at intermediate vertical modes (e.g., 7–10 for DART–CAM and 9–11 for ECMWF) and the global Rossby mode *n* = 1 (figure not shown). The IG *k* = 0 energy has identical (*n*, *m*) distributions in eastward and westward components and the IG energy is a very small percentage of the *k* = 0 ROT energy (e.g., for DART–CAM it is less than 0.5%).

### a. Tropical inertia-gravity circulations

Figure 8 presents circulations associated with IG modes averaged over the whole sample for each dataset for a single level in the lower troposphere close to 750 hPa, at slightly different levels for each system. This figure illustrates features of the large-scale inertia–gravity flow related to the easterlies in the Pacific, westerlies in the Indian Ocean, and large-scale orographically induced IG winds due to the Andes, Himalayas, Rocky Mountains, and eastern African plateau. The four datasets differ significantly in spite of averaging; on the other hand, differences are expected given the lack of direct wind observations and the complexity of the data assimilation problem in the tropics (e.g., Žagar et al. 2005). However, a prominent difference exists between NCEP–NCAR and other three datasets in the midlatitude regions where the NCEP–NCAR reanalyses contain a significant amount of IG energy, especially in the Southern Hemisphere. This excessive IG flow in NCEP–NCAR data is in agreement with the total contribution of IG modes to the energy spectra for these data shown in Fig. 6. Figure 8 can be further separated into westward- and eastward-propagating modes to reveal that the meridional winds at the lowest model levels originating from the Southern Hemisphere and crossing the equator in all three ocean basins project onto the WIG modes (not shown). In the DART–CAM analyses, the westerlies over equatorial Africa and the western Indian Ocean, as well as the easterlies over the equatorial Pacific are part of the EIG signal. These low-level easterlies in the western Pacific are replaced by westerlies at about the 750-hPa level, which become stronger higher up (figures not shown).

The IG circulation in the Indian Ocean in NCEP–NCAR appears in a form of a gyre while in other datasets the flow is predominantly westerly. It should be mentioned that we selected to show a particular level at which this feature is most intense; it appears only at several models levels and represents the largest deviations of NCEP–NCAR from three newer analysis systems. For comparison with earlier studies of energetics, it is worth mentioning that these model levels are located between two standard pressure levels (i.e., 850 and 700 hPa) used in other studies. By submitting the total flow obtained from Eq. (4) to a decomposition into rotational and divergent wind components based on the Helmholtz theorem it is verified that the Indian Ocean gyre belongs to the nondivergent circulation (not shown). When Fig. 8c is split into various IG components, it is found that the circulation over the Indian Ocean is mainly in EIG modes and most of it is associated with the meridional modes *n* = 0–4. The flow north of the equator is represented by the Kelvin and *n* = 1 EIG (so-called eastward MRG) modes (figures not shown).

The IG flow in Fig. 8 is constituted of both rotational and divergent components. In particular, large-scale equatorially trapped IG waves can be vorticity dominated; for example, the ratio between the vorticity and the divergence in the *k* = 1 Kelvin wave with phase speed of 15 m s^{−1} is as large as 4 (Hendon 1986). Another example is the MRG mode that has maximum vorticity at the equator, and rotational and divergent wind components of similar magnitude. Thus, the depiction of these waves in Fig. 8 depends on the correct determination of both the rotational and divergent flow in the analysis procedure. We can only speculate that the assimilation methodology of NCEP–NCAR reanalyses is the reason for the Indian Ocean gyre partly projecting onto the IG modes in this and not in other datasets.

## 5. Conclusions

We have applied the normal-mode functions (NMFs) as a diagnostic tool for separating the atmospheric energy spectra into balanced and inertia-gravity contributions. A set of three-dimensional orthogonal normal modes derived by Kasahara and Puri (1981) was applied to four analysis datasets for a single month (July 2007). Each analysis dataset is taken at its respective model levels that extend from the surface to the upper stratosphere (DART–CAM and NCEP–NCAR) and the mesosphere (ECMWF and NCEP). While this prevents us from comparing the energy in different datasets quantitatively, it allows us to discuss results in a physically consistent manner and to draw conclusions with some relevance for each analysis system.

The most relevant results are summarized in the following four points:

In the present application of NMFs, the truncation parameters (zonal, meridional, and vertical truncation) that define the modal basis are selected in order to represent most of the input dynamical information in the wind and geopotential fields. This approach is different from previous applications of the normal modes and it allows us to apply the inverse projection to obtain a part of the circulation associated with selected modes and, in particular, to study details of the flow in the vertical domain, a subject of Part II. For the successful reproduction of three-dimensional IG circulations by the normal modes it is important that the expansion includes many vertical modes based on realistic temperature and stability profiles.

In the operational analysis systems of NCEP and ECMWF in July 2007, the percentage of IG motion in the total energy field is between 1% and 2%. In the wave part of the flow (k ≠ 0), the IG contribution is between 9% and 15%. This range is defined by the forecast model including its vertical atmosphere depth, resolution, and physical parameterizations, as well as by the assimilation methodology. The DART–CAM ensemble system, which has the top model level significantly lower than the NCEP and ECMWF systems (3.7 versus 0.32 and 0.01 hPa), is characterized by an average energy distribution similar to that of the operational variational systems. Obtained levels of the IG energy are 2 and 3 times larger than previously reported (Tanaka and Kimura 1996) for the total and wave part of the global flow; the reasons for differences are most likely the different methodology of the normal-mode representation, especially the use of model levels in the present study compared to standard pressure levels used earlier, and properties of the datasets, which in the present case have higher resolution and significant convective dynamics in the tropics.

The NCEP–NCAR reanalyses contain more IG motion in July 2007 compared to the other three datasets. In particular, there are significant IG motions in the Southern Hemisphere extratropics. There are several possible reasons for this result including the fact that the reanalysis system assimilates retrievals and that analyses are not initialized; over the Southern Hemisphere extratropical oceans where other observations are not available this might have caused increased imbalances. However, neither this nor other possible reasons can easily be investigated in detail by the authors and it is unknown to what extent this result is specific for the period studied.

The time-averaged large-scale IG circulation is confined to the tropics. There is a difference between the levels of energy associated with eastward and westward IG waves of about 2%–3% of the total wave energy in all datasets. The difference is associated with the motions projecting onto the Kelvin wave signal, the most energetic IG wave, which is further studied in Part II.

## Acknowledgments

We are most indebted to Akira Kasahara, who developed the Hough function expansion used in this study, for his interest, discussions, and his constructive comments that helped to improve this paper. We also thank Peter Gent and Hiroshi Tanaka for reading the manuscript and providing comments. The operational NCEP analyses were kindly provided by Russ Treadon and John Derber of the Environmental Modeling Center of NCEP. We are grateful to Dennis Shea for providing a NCL script for horizontal data interpolation and to Tim Hoar and Nancy Collins for their support in running DART–CAM. We also thank George Kiladis and two anonymous reviewers for providing valuable comments on an earlier draft of this paper.

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## APPENDIX

### The Impact of the Model Depth

One difference between the present study and other NMF applications is our use of the model levels and the whole model depth, which in the operational models of ECMWF and NCEP extends to the mesosphere. To illustrate the importance of using realistic stratification and the full model depth, the vertical structure equation is solved with the model lid placed at three different levels in the ECMWF system. The reference solution is one including the whole model atmosphere where the model top full level is located at 0.01 hPa (∼80 km). The second solution is obtained by lowering the top to model level 12, which roughly corresponds to the top level in the DART–CAM and NCEP–NCAR systems, at ∼2.4 hPa. In the third experiment the top level is taken to be at model level 39, which is located at about 102.6 hPa. This last experiment addresses the classical picture of atmospheric normal modes in which the model top level is placed at the tropopause. Both additional solutions are based on the same realistic temperature and stability profile up to the assumed model top and they have the same bottom boundary condition; that is, the only difference between experiments is the location of the top full model level.

Solutions are compared in Fig. A1. Three vertical eigenmodes presented in a single panel are characterized by approximately the same value of the equivalent depth. Corresponding modal indices for each *H*_{eq} and model top levels are displayed in Table A1. Figure A1 and Table A1 together illustrate the impact of the model depth on the solution of Eq. (1). It can be seen that placing the model top level at the tropopause and applying realistic stratification leads to the vertical mode *m* = 2 having equivalent depth around 400 m. The only zero crossing of this mode is located at about 350 hPa (Fig. A1a). This location coincides with the first zero crossing (from the surface) for the other two cases in which the model top level is located higher. In other words, vertical modes characterized by the same equivalent depth share the shape and location of the zero crossings up to the (*m* − 1)^{th} zero crossing in the case of the lowest top level. Figures A1b,c illustrate this property for the other two equivalent depths for which the difference in *m* indices becomes larger. Solutions for the top level located at 2.4 and at 0.01 hPa are very similar for the lowest *m* (Fig. A1a,b) and they become identical once the nonzero structure of solutions for the case with top at 0.01 hPa moves below 2.4 hPa (Fig. A1c). It can also be noted that many vertical modes have so-called “first baroclinic mode” structure characterized by a single zero crossing in the midtroposphere and opposite sign in the lower and upper troposphere.

The downward movement of the first zero crossing with simultaneous increase in the equatorial trapping of the Hough functions is illustrated in Fig. A2 for the MRG mode. The meridional velocity component in the vertical plane is displayed as a function of latitude for 3 vertical modes with equivalent depths 3.8 km, 406 m, and 21 m. While *H*_{eq} = 3.8 km is associated with a circulation spanning nearly the whole atmosphere from pole to pole, the mode with *H*_{eq} = 406 m is trapped within 20°S–20°N and further trapping occurs for smaller equivalent depths.

The sensitivity of the *H*_{eq} estimate to the model depth affects the accuracy of the projection through the solution of the horizontal equations. For example, it is difficult to use the operational ECMWF data to analyze the midlatitude tropospheric circulation by using only model levels below 100 hPa because representation of high latitudes becomes difficult in this case since the second vertical mode (*m* = 2) is already confined in the tropics (i.e., *H*_{eq} ∼ 400 m).

In conclusion, we consider the equivalent depth as an eigenvalue of (1) and we do not make an explicit connection between the magnitude of *H*_{eq} and the depth of the shallow fluid. The numerical solutions of Eq. (1) for realistic stratification and atmosphere including both the stratosphere and the mesosphere have a single zero crossing in the troposphere for a number of vertical modes and a zigzag shape that moves toward the surface as *m* increases.

Dispersion curves for lowest ROT and IG modes for vertical modes *m* = 1 and *m* = 17 and the DART–CAM dataset. Corresponding equivalent depths are 10 km and 11 m. For EIG modes, *n* = 0 (KW) to *n* = 3 modes and the lowest three meridional modes (*n* = 0, 1, 2) for WIG are shown. Other westward-propagating modes are the MRG mode (*n* = 0 ROT mode) and *n* = 1, 2, 3, and 4 ROT modes.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Dispersion curves for lowest ROT and IG modes for vertical modes *m* = 1 and *m* = 17 and the DART–CAM dataset. Corresponding equivalent depths are 10 km and 11 m. For EIG modes, *n* = 0 (KW) to *n* = 3 modes and the lowest three meridional modes (*n* = 0, 1, 2) for WIG are shown. Other westward-propagating modes are the MRG mode (*n* = 0 ROT mode) and *n* = 1, 2, 3, and 4 ROT modes.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Dispersion curves for lowest ROT and IG modes for vertical modes *m* = 1 and *m* = 17 and the DART–CAM dataset. Corresponding equivalent depths are 10 km and 11 m. For EIG modes, *n* = 0 (KW) to *n* = 3 modes and the lowest three meridional modes (*n* = 0, 1, 2) for WIG are shown. Other westward-propagating modes are the MRG mode (*n* = 0 ROT mode) and *n* = 1, 2, 3, and 4 ROT modes.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Vertical eigenstructures for DART–CAM July 2007 dataset. Three subsequent vertical modes are (a) *m* = 1–3, (b) *m* = 4–6, (c) *m* = 7–9, (d) *m* = 10–12, (e) *m* = 13–15, (f) *m* = 16–18, (g) *m* = 19–21, (h) *m* = 22–24, and (i) *m* = 25–26. The lowest *m* in each plot is the thick gray line, the middle mode is the thin black line with dot symbols, and the third mode is shown by the dashed line with diamonds.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Vertical eigenstructures for DART–CAM July 2007 dataset. Three subsequent vertical modes are (a) *m* = 1–3, (b) *m* = 4–6, (c) *m* = 7–9, (d) *m* = 10–12, (e) *m* = 13–15, (f) *m* = 16–18, (g) *m* = 19–21, (h) *m* = 22–24, and (i) *m* = 25–26. The lowest *m* in each plot is the thick gray line, the middle mode is the thin black line with dot symbols, and the third mode is shown by the dashed line with diamonds.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Vertical eigenstructures for DART–CAM July 2007 dataset. Three subsequent vertical modes are (a) *m* = 1–3, (b) *m* = 4–6, (c) *m* = 7–9, (d) *m* = 10–12, (e) *m* = 13–15, (f) *m* = 16–18, (g) *m* = 19–21, (h) *m* = 22–24, and (i) *m* = 25–26. The lowest *m* in each plot is the thick gray line, the middle mode is the thin black line with dot symbols, and the third mode is shown by the dashed line with diamonds.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Projection quality for various datasets and coefficients. (a) Variance ratio between the inverse and input variables in the Northern Hemisphere (20°–80°N) in the NCEP–NCAR reanalyses. (b) As in (a), but the correlation coefficient. (c) As in (a), but DART–CAM analyses for the Southern Hemisphere (80°–20°S). (d) Zonal wind scores for the ECMWF analyses. Variance ratio globally (80°S–80°N, thick gray) and in the tropics (20°S–20°N, thin black), and correlation coefficient in the tropics (dashed).

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Projection quality for various datasets and coefficients. (a) Variance ratio between the inverse and input variables in the Northern Hemisphere (20°–80°N) in the NCEP–NCAR reanalyses. (b) As in (a), but the correlation coefficient. (c) As in (a), but DART–CAM analyses for the Southern Hemisphere (80°–20°S). (d) Zonal wind scores for the ECMWF analyses. Variance ratio globally (80°S–80°N, thick gray) and in the tropics (20°S–20°N, thin black), and correlation coefficient in the tropics (dashed).

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Projection quality for various datasets and coefficients. (a) Variance ratio between the inverse and input variables in the Northern Hemisphere (20°–80°N) in the NCEP–NCAR reanalyses. (b) As in (a), but the correlation coefficient. (c) As in (a), but DART–CAM analyses for the Southern Hemisphere (80°–20°S). (d) Zonal wind scores for the ECMWF analyses. Variance ratio globally (80°S–80°N, thick gray) and in the tropics (20°S–20°N, thin black), and correlation coefficient in the tropics (dashed).

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

An example of the projection input and inverse for the zonal wind (m s^{−1}) at model level 20 (∼602 hPa) in DART–CAM dataset for a randomly chosen time: (a) Input *u* wind and (b) *u* wind after the projection and its inverse. Isolines are drawn every ±4 m s^{−1} and the zero isoline is omitted.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

An example of the projection input and inverse for the zonal wind (m s^{−1}) at model level 20 (∼602 hPa) in DART–CAM dataset for a randomly chosen time: (a) Input *u* wind and (b) *u* wind after the projection and its inverse. Isolines are drawn every ±4 m s^{−1} and the zero isoline is omitted.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

An example of the projection input and inverse for the zonal wind (m s^{−1}) at model level 20 (∼602 hPa) in DART–CAM dataset for a randomly chosen time: (a) Input *u* wind and (b) *u* wind after the projection and its inverse. Isolines are drawn every ±4 m s^{−1} and the zero isoline is omitted.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Total energy in rotational (ROT), eastward- (EIG), and westward (WIG) inertia–gravity motions as a function of the zonal wavenumber. Spectra are obtained by averaging over (a) 29 days of July for DART–CAM, and the whole month for (b) ECMWF, (c) NCEP–NCAR, and (d) NCEP datasets. Summation is performed over all (*m*, *n*) and the mean state (*k* = 0) is not included.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Total energy in rotational (ROT), eastward- (EIG), and westward (WIG) inertia–gravity motions as a function of the zonal wavenumber. Spectra are obtained by averaging over (a) 29 days of July for DART–CAM, and the whole month for (b) ECMWF, (c) NCEP–NCAR, and (d) NCEP datasets. Summation is performed over all (*m*, *n*) and the mean state (*k* = 0) is not included.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Total energy in rotational (ROT), eastward- (EIG), and westward (WIG) inertia–gravity motions as a function of the zonal wavenumber. Spectra are obtained by averaging over (a) 29 days of July for DART–CAM, and the whole month for (b) ECMWF, (c) NCEP–NCAR, and (d) NCEP datasets. Summation is performed over all (*m*, *n*) and the mean state (*k* = 0) is not included.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

As in Fig. 8, but energy is represented as a function of the meridional and vertical modes. Drawn are percentages of the total energy summed across all modes except the zonally averaged state (*k* = 0). (a),(d),(g),(j) ROT modes values are multiplied by 10 and for the (b),(e),(h),(k) EIG and (c),(f),(i),(l) WIG modes the multiplication factor is 100. (a)–(c) DART–CAM, (d)–(f) ECMWF, (g)–(i) NCEP–NCAR, and (j)–(l) NCEP.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

As in Fig. 8, but energy is represented as a function of the meridional and vertical modes. Drawn are percentages of the total energy summed across all modes except the zonally averaged state (*k* = 0). (a),(d),(g),(j) ROT modes values are multiplied by 10 and for the (b),(e),(h),(k) EIG and (c),(f),(i),(l) WIG modes the multiplication factor is 100. (a)–(c) DART–CAM, (d)–(f) ECMWF, (g)–(i) NCEP–NCAR, and (j)–(l) NCEP.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

As in Fig. 8, but energy is represented as a function of the meridional and vertical modes. Drawn are percentages of the total energy summed across all modes except the zonally averaged state (*k* = 0). (a),(d),(g),(j) ROT modes values are multiplied by 10 and for the (b),(e),(h),(k) EIG and (c),(f),(i),(l) WIG modes the multiplication factor is 100. (a)–(c) DART–CAM, (d)–(f) ECMWF, (g)–(i) NCEP–NCAR, and (j)–(l) NCEP.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Average July 2007 IG circulation (without MRG motion) at a sigma level in the lower troposphere according to (a) DART–CAM, level 22 (∼788 hPa), (b) ECMWF, level 73 (∼753 hPa), (c) NCEP–NCAR, level 20 (∼751 hPa), and (d) NCEP, level 48 (∼763 hPa). Shaded contours are the models’ orography with shades every 1000 m, starting from 500 m to 5.5 km.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Average July 2007 IG circulation (without MRG motion) at a sigma level in the lower troposphere according to (a) DART–CAM, level 22 (∼788 hPa), (b) ECMWF, level 73 (∼753 hPa), (c) NCEP–NCAR, level 20 (∼751 hPa), and (d) NCEP, level 48 (∼763 hPa). Shaded contours are the models’ orography with shades every 1000 m, starting from 500 m to 5.5 km.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Average July 2007 IG circulation (without MRG motion) at a sigma level in the lower troposphere according to (a) DART–CAM, level 22 (∼788 hPa), (b) ECMWF, level 73 (∼753 hPa), (c) NCEP–NCAR, level 20 (∼751 hPa), and (d) NCEP, level 48 (∼763 hPa). Shaded contours are the models’ orography with shades every 1000 m, starting from 500 m to 5.5 km.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Fig. A1. Vertical eigenstructures for the ECMWF model levels, mean stability, and temperature profiles in July 2007. Different panels correspond to structure functions for a single equivalent depth and various model top levels as described in the legend (full gray line corresponds to the top-level pressure of 1 Pa, black dotted line to 2.4 hPa, and dashed line with diamonds denotes lid at 102.6 hPa): *H*_{eq} = (a) 402, (b) 120, and (c) 21 m.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Fig. A1. Vertical eigenstructures for the ECMWF model levels, mean stability, and temperature profiles in July 2007. Different panels correspond to structure functions for a single equivalent depth and various model top levels as described in the legend (full gray line corresponds to the top-level pressure of 1 Pa, black dotted line to 2.4 hPa, and dashed line with diamonds denotes lid at 102.6 hPa): *H*_{eq} = (a) 402, (b) 120, and (c) 21 m.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Fig. A1. Vertical eigenstructures for the ECMWF model levels, mean stability, and temperature profiles in July 2007. Different panels correspond to structure functions for a single equivalent depth and various model top levels as described in the legend (full gray line corresponds to the top-level pressure of 1 Pa, black dotted line to 2.4 hPa, and dashed line with diamonds denotes lid at 102.6 hPa): *H*_{eq} = (a) 402, (b) 120, and (c) 21 m.

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Fig. A2. Structure of the meridional velocity (m s^{−1}) of the mixed Rossby–gravity mode in the vertical–meridional plane for three different vertical modes in the case with model lid at 2.4 hPa. Isolines every 1 m s^{−1}, with positive values drawn thick, negative values thin, and the zero isoline dashed–dotted: (a) *m* = 2 (*H*_{eq} = 3.8 km), (b) *m* = 5 (*H*_{eq} = 406 m), and (c) *m* = 9 (*H*_{eq} = 21 m).

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Fig. A2. Structure of the meridional velocity (m s^{−1}) of the mixed Rossby–gravity mode in the vertical–meridional plane for three different vertical modes in the case with model lid at 2.4 hPa. Isolines every 1 m s^{−1}, with positive values drawn thick, negative values thin, and the zero isoline dashed–dotted: (a) *m* = 2 (*H*_{eq} = 3.8 km), (b) *m* = 5 (*H*_{eq} = 406 m), and (c) *m* = 9 (*H*_{eq} = 21 m).

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Fig. A2. Structure of the meridional velocity (m s^{−1}) of the mixed Rossby–gravity mode in the vertical–meridional plane for three different vertical modes in the case with model lid at 2.4 hPa. Isolines every 1 m s^{−1}, with positive values drawn thick, negative values thin, and the zero isoline dashed–dotted: (a) *m* = 2 (*H*_{eq} = 3.8 km), (b) *m* = 5 (*H*_{eq} = 406 m), and (c) *m* = 9 (*H*_{eq} = 21 m).

Citation: Monthly Weather Review 137, 11; 10.1175/2009MWR2815.1

Percentages of energy in various motions with respect to the energy in (top row) all motions except *k* = 0 and (bottom row) with respect to the total energy.

Table A1. Indices of vertical modes corresponding to selected equivalent depths from various solutions of the vertical structure equation.

^{1}

After the paper had been accepted, we discovered that the retrieval procedure used to obtain the orography field of the ECMWF model was responsible for the noisy spectra in Fig. 6b. The correct orography field resulted in smooth spectra over all scales. This has no significance for the main results and conclusions of the study.

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* The National Center for Atmospheric Research is sponsored by the National Science Foundation.