## 1. Introduction

**F**

^{(EP)}satisfies the following relation under the Wentzel–Kramers–Brillouin (WKB) approximation (Edmon et al. 1980):where

**C**

_{g}is the group velocity and

*A*is the zonal-mean wave activity density. The conservation relation of wave activity density is expressed using the divergence of EP flux,where

*D*denotes nonconservative terms such as frictional and diabatic effects. This equation is called the generalized EP theorem (Andrews and McIntyre 1978). The time derivative of the wave activity density expresses the wave transience. For linear, steady, and conservative waves in a purely zonal basic flow, the divergence of EP flux vanishes. Under such conditions, the waves neither drive the meridional circulation nor accelerate the zonal-mean zonal wind. This is called the nonacceleration theorem (Eliassen and Palm 1961; Charney and Drazin 1961). Thus, the EP flux is a powerful tool for examining the wave propagation and wave-mean flow interaction in the meridional cross section.

On the other hand, there are many studies that generalize the TEM equations to three dimensions in order to examine the 3D wave propagation and local wave activity. Hoskins et al. (1983), Trenberth (1986), and Plumb (1986) extended the TEM equations to 3D using the time mean instead of the zonal mean under the quasigeostrophic (QG) approximation. Hoskins et al. (1983) derived the 3D wave activity flux based on the horizontal velocity correlation tensor. While successfully representing the interaction between the time-mean flow and waves, their wave activity flux is not parallel to the group velocity of Rossby waves. Trenberth (1986) derived the 3D wave activity flux by adding the zonal and meridional derivatives of the perturbation kinetic energy to the zonal and meridional momentum equations, respectively. His wave activity flux is parallel to the horizontal group velocity of Rossby waves only for the barotropic case. Plumb (1986) derived the 3D wave activity flux using the potential vorticity conservation theorem. This wave activity flux is equal to the product of the group velocity and the wave activity density of Rossby waves like the TEM (2D) EP flux under the QG approximation. Takaya and Nakamura (1997, 2001) derived a phase-independent 3D wave activity flux applicable to quasi-stationary Rossby waves in zonally asymmetric flow using the wave activity density and wave energy conservation laws. Their flux can be calculated without first computing the time mean to eliminate the wave phase structure, and their flux is equal to the product of the group velocity and the generalized wave activity density for QG perturbations.

From the primitive equations, Miyahara (2006) and Kinoshita et al. (2010) derived a 3D wave activity flux applicable to inertia–gravity waves using the time mean when the Coriolis parameter is constant. Since their wave activity flux is equal to the product of the group velocity and wave activity density of inertia–gravity waves under the WKB approximation, it is similar to the TEM (2D) EP flux under the constant Coriolis parameter assumption. Note that their wave activity flux can be reduced to the 3D wave activity flux describing the propagation of Rossby waves, which was derived by Plumb (1986) under the small Rossby number assumption.

Noda (2010) formulated a generalized 3D-TEM equation for a plane wave under the WKB approximation. He also showed that his wave activity flux is equal to the product of the group velocity and the wave activity density. However, since the covariance of perturbations is included in the denominator of the formulas for the 3D wave activity flux, his formulas can only be used for a purely monochromatic wave.

In ocean dynamics, Lee and Leach (1996) derived the 3D wave activity flux for a nonquasigeostrophic time-mean flow in isopycnic coordinates from the time-mean horizontal momentum and continuity equations. While the divergence of their 3D wave activity flux is related to the wave forcing of the mean flow, it is not parallel to the group velocity. Gent and McWilliams (1996) derived the time-mean 3D wave activity flux from the primitive equations by replacing the mean flow with the 3D residual mean flow of the time-mean horizontal momentum equation in all terms except for the acceleration term. However, they did not examine the relation between the direction of their 3D wave activity flux and the group velocity.

These previous studies [except for Noda (2010), Lee and Leach (1996), and Gent and McWilliams (1996)] assumed either the QG approximation or a constant Coriolis parameter. Thus, their formulas are applicable either to Rossby waves or inertia–gravity waves. Our companion paper Kinoshita and Sato (2013) gives a formulation of the 3D residual mean flow and wave activity flux that is applicable to both Rossby waves and inertia–gravity waves. The divergence of this 3D wave activity flux corresponds to the wave forcing of the mean flow in the horizontal momentum equation.

On the other hand, the present study derives a form of the 3D wave activity flux that is proportional to the group velocity. This wave activity flux has different forms from that formulated by Kinoshita and Sato (2013). However, this does not mean that our formulations are incomplete. It rather indicates that different 3D wave activity fluxes are needed for the study of the wave forcing to the mean flow and for the study of the wave propagation.

This paper is organized as follows. In section 2, after a unified dispersion relation for inertia–gravity waves and Rossby waves is derived, the relation between the 3D wave activity flux and the group velocity is examined. To relate the 3D wave activity flux to the group velocity, the wave activity density and 3D wave activity flux are modified. It is shown that the modified wave activity density reduces to the wave activity density for inertia–gravity waves under the constant Coriolis parameter assumption and reduces to the wave activity density for Rossby waves under the small Rossby number assumption. Section 3 shows that the modified 3D wave activity flux is equal to the product of the group velocity and the wave activity density for inertia–gravity waves and Rossby waves under the appropriate assumptions. In section 4, the relation between the 3D residual mean flow and the divergence of 3D wave activity flux is discussed. Moreover, the relation between the 3D wave activity flux and that derived by Plumb (1986) is investigated. A summary and some concluding remarks are given in section 5.

## 2. The 3D wave activity flux describing the wave propagation for the primitive equation system

### a. The time-mean 3D Stokes drift and 3D wave activity flux

*z*is the log-pressure height;

*u*,

*υ*, and

*w*are the zonal, meridional, and vertical wind components, respectively;

*ρ*

_{0}is the basic density; Φ is the geopotential;

*f*is the Coriolis parameter;

*N*

^{2}is the buoyancy frequency squared (expressing static stability); the suffixes

*x*,

*y*, and

*z*denote the partial derivatives; and we define

**F**

_{1}(

*F*

_{11},

*F*

_{12},

*F*

_{13}) and

**F**

_{2}(

*F*

_{21},

*F*

_{22},

*F*

_{23}) are included in the time-mean zonal and meridional momentum equations, respectively. Note that (2.2) is named 3D-flux-M because it is the flux whose divergence corresponds to the wave forcing to the time-mean flow (Kinoshita and Sato 2013). However, the 3D wave activity flux describing the wave propagation is different from 3D-flux-M and is called 3D-flux-W in this article. We examine how 3D-flux-W should be written so as to take the form of the product of the group velocity and the wave activity density using the dispersion relation, which describes both inertia–gravity waves and Rossby waves.

### b. The dispersion relation for the primitive equation system

*f*depends on the latitude (and hence the horizontal momentum equation for perturbations is nonlinear for the meridional direction), a perturbation

*a*′ having the following form is considered:where

*H*is the constant-scale height;

*k*and

*m*are the zonal and vertical wavenumbers, respectively; and

*ω*is the ground-based angular frequency. The basic density is expressed aswhere

*ρ*is the surface density. Substituting (2.4) into (2.3a) and (2.3b) and eliminating

_{s}*u*′ yieldswhere

*u*′ yieldswhere

*f*=

*βy*and

*β*is constant. Assuming that the square bracket in (2.7) is independent of

*y*, the sinusoidal wave form for

*υ*

_{0}is an arbitrary constant, and

*l*is the meridional wavenumber. Then, a dispersion relation for the primitive equations is obtained:This relation agrees with the dispersion relation of Rossby waves under a small Rossby number assumption

*f*=

*f*

_{0}+

*βy*and

*f*

_{0}is the Coriolis parameter at the reference latitude. On the other hand, this relation agrees with the dispersion relation of inertia–gravity waves when the Coriolis parameter is constant (

*f*= 0):Thus, (2.9) can be considered as a unified dispersion relation for inertia–gravity waves and Rossby waves.

_{y}### c. The meridional and vertical components of 3D-flux-W and the modified wave activity density

In this section, it is shown that the meridional and vertical components of 3D-flux-M in (2.2) describe the wave propagation by examining the relation between 3D-flux-M and the group velocity. Moreover, in order to relate the flux to the group velocity, the modified wave activity density is derived. The relation for the zonal component will be examined in the next section.

*υ*in (2.8) also have a sinusoidal wave form in the meridional direction, (2.4) can be written asSubstituting (2.12) into (2.3), the wind perturbations are written in terms of Φ′ asBy using (2.13), 3D-flux-M in (2.2) is written in terms of Φ′ asDefining the wave activity density as the ratio of the perturbation energy to the intrinsic phase velocity, the wave activity density is written aswhere

*F*

_{12}like 2D-EP flux in the TEM equations but is not identical to

*F*

_{12}. To relate

*F*

_{12}to (2.17), an additional

### d. The zonal component of 3D-flux-W

*F*

_{11}. To relate

*F*

_{11}to (2.20), the term

*F*is defined as follows:Using (2.21), we obtain the zonal component of 3D-flux-WIt is important to note that

_{u}*F*is rewritten using

_{u}*F*vanishes under the constant Coriolis parameter assumption (i.e., for inertia–gravity waves) and agrees with the perturbation energy under the QG assumption (i.e., for Rossby waves) as shown in the next section.

_{u}In summary, by defining the modified wave activity density as (2.19b) and (2.19d) and 3D-flux-W as (2.22a), (2.22b), (2.19a), and (2.19c), we derived a unified relation for the 3D propagations of inertia–gravity waves and Rossby waves; that is, 3D-flux-W is equal to the product of the group velocity and the modified wave activity density in all directions.

## 3. The relation between 3D-flux-W and the other 3D wave activity flux

### a. The relation between 3D-flux-W and the 3D wave activity flux for inertia–gravity waves (IG-flux)

### b. The relation between 3D-flux-W and the 3D wave activity flux for Rossby waves (QG-flux)

In this section, first, the 3D QG-flux is obtained by converting 3D-flux-W in (2.22a), (2.22b), (2.19a), and (2.19c) under the small Rossby number assumption. Then, it is confirmed that the 3D QG-flux is proportional to the group velocity of the Rossby waves.

*g*and

*a*denote the geostrophic and ageostrophic components, respectively. Moreover,

**F**

_{P}= (

*F*

_{P}_{1},

*F*

_{P}_{2},

*F*

_{P}_{3},) derived by Plumb [1986, his (4.4)]:where

**F**

_{P}is expressed with

**M**

_{R}defined by Plumb (1986) and is equal to the product of the intrinsic group velocity (rather than the ground-based group velocity) and the wave activity density of the Rossby waves (Plumb 1986). Note also that neither the 3D QG-flux nor Plumb’s wave activity flux is equal to the flux whose divergence corresponds to the wave forcing to the mean flow. This point is discussed in more detail in the next section.

## 4. Discussion

In the TEM equations, the 2D-EP flux included in the zonal momentum equation can be written as the product of the wave activity density and the group velocity under the WKB approximation. However, the 3D formulation of the wave activity flux given by (2.2) cannot be written as the product of the group velocity and the wave activity density. Most previous studies first formulated the 3D wave activity flux, which describes the wave propagation from the standpoint of waves, and then formulated the 3D residual mean flow by using the 3D wave activity flux in the time-mean horizontal momentum equation. However, the 3D residual mean flow derived in such a way is not equal to the sum of the time-mean flow and the Stokes drift. This is due to the fact that the 3D wave activity flux related to the wave forcing to the time-mean flow is different from that describing the wave propagation. Kinoshita and Sato (2013) and the present study formulate both types of 3D wave activity flux (3D-flux-M and 3D-flux-W, respectively).

In this section, in order to show that the divergence of 3D-flux-M corresponds to the wave forcing to the mean flow and that 3D-flux-W describes the wave propagation, we show an analysis of European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) data to compare the 3D residual mean flow derived by Kinoshita and Sato (2013) to the divergences of three kinds of 3D wave activity flux (3D-flux-M, 3D-flux-W, and Plumb’s wave activity flux). The direction and magnitude of the three kinds of 3D wave activity flux are compared.

### a. Data description

As in Kinoshita and Sato (2013), the ERA-Interim data for 19 yr from 1990 to 2008 are used for the analysis. The time-mean field is obtained by applying a low-pass filter with a cutoff period of 60 days to the data. The disturbance field is defined as the deviation from the time-mean field. We calculate the climatological time-mean quantities over those 19 yr. The results are shown for 15 April, when the transient disturbances are strong in the Northern Hemisphere (Nakamura 1992; Sato et al. 2000). Note that the low-pass-filtered data for 15 April roughly corresponds to the data averaged over 30 days with a center at 15 April.

### b. Comparison between the residual mean flow and the three kinds of 3D wave activity flux

The difference between 3D-flux-M and other kinds of 3D wave activity flux mainly appears in the zonal component. We focus on the storm-track region in the upper troposphere, where the longitudinal variation is large. It should be noted that this section uses all equations in spherical coordinates, not Cartesian coordinates, and they are introduced in the appendix. Figure 1 shows the longitude–pressure cross section of the meridional component of the 3D residual mean flow associated with disturbances **F**_{1} is 3D-flux-M (*F*_{11}, *F*_{12}, *F*_{13}) (Fig. 1c); **F**_{W1} is 3D-flux-W [*F _{W}*

_{11}(=

*F*

_{11}+

*F*),

_{u}*F*

_{12},

*F*

_{13}] (Fig. 1d);

*F*is large (Fig. 1f). Note that Plumb’s wave activity flux corresponds to 3D-flux-M and does not describe the wave forcing to the time-mean flow under the QG assumption (Plumb 1986; Kinoshita and Sato 2013).

_{u}Figure 2 shows the longitude–pressure cross section of **F**_{1} (Fig. 2b), **F**_{W1} (Fig. 2c), and **F**_{P} (Fig. 2d), which are averaged over the latitudes from 30° to 60°N. The color shows *F _{u}*. The contours show the variance of the geopotential height disturbances. Note that all quantities except for the variance of the geopotential height are divided by the square root of

*ρ*

_{0}to emphasize their differences. It is found that

**F**

_{W1}is in good agreement with

**F**

_{P}. Note that

**F**

_{P}is equal to the product of the group velocity and the wave activity density of the Rossby waves (Plumb 1986). On the other hand,

**F**

_{1}and

**F**

_{P}, especially around the storm tracks. This is because

**F**

_{1}is not proportional to the group velocity in the zonal component.

These results indicate that the divergences of 3D-flux-M correspond to the wave forcing causing the 3D residual mean flow associated with the disturbances and that 3D-flux-W describes the wave propagation. The difference between 3D-flux-M and 3D-flux-W is large around the storm tracks. The storm-track region corresponds to the region where the energy of the Rossby waves is large since *F _{u}* agrees with

## 5. Concluding remarks

In this study, we have formulated a 3D wave activity flux (3D-flux-W) that is proportional to the group velocity. 3D-flux-W and 3D-flux-M, whose divergence describes the wave forcing to the mean flow by Kinoshita and Sato (2013), differ by an additional term. A unified dispersion relation for inertia–gravity waves and Rossby waves and the modified wave activity density have been formulated in order to relate 3D-flux-W to the group velocity. The modified wave activity density agrees with the wave activity density for inertia–gravity waves under the constant Coriolis parameter assumption and agrees with that for Rossby waves under the small Rossby number assumption.

Next, we examined the relation between 3D-flux-W and the other 3D wave activity flux. It was shown that 3D-flux-W is equal to 3D IG-flux when the Coriolis parameter is constant and is equal to 3D QG-flux when the Rossby number is small. Thus, 3D-flux-W describes the propagation of both inertia–gravity waves and Rossby waves.

To examine the difference between 3D-flux-M and 3D-flux-W, an analysis was made of the disturbances in the storm-track region of the upper troposphere in April using the ERA-Interim data. The distribution of the 3D-flux-M divergence is in good agreement with the meridional component of the 3D residual mean flow associated with the disturbances that were derived by Kinoshita and Sato (2013), while the divergence of 3D-flux-W is slightly different from that in the upstream and downstream regions of the storm tracks. On the other hand, the direction and magnitude of 3D-flux-W are in good agreement with those of Plumb’s wave activity flux describing the wave propagation, while 3D-flux-M is different from that around the storm tracks. The formulas of Hoskins et al. (1983), Trenberth (1986), Plumb (1986), Miyahara (2006), Kinoshita et al. (2010), Noda (2010), and this study are summarized and compared in Kinoshita and Sato (2013, their Table 1).

It is important to note that the difference between the formulas of Kinoshita et al. (2010) and those of this study is *F _{u}* in the zonal component) is also due to the Rossby waves and is equivalent to the Rossby wave energy for a small Rossby number limit.

The analysis performed in this article focused the 3D wave activity flux associated with synoptic-scale disturbances in the storm-track region. For future work, other types of waves should be analyzed. It should be noted that 3D-flux-W is derived by using small-amplitude theory in a slowly varying mean flow and is not applicable to atmospheric waves whose amplitudes increase quickly, such as unstable waves. It is also important to note that 3D-flux-W should not be applied near the critical level since the modified wave activity density diverges.

Nevertheless, our formulation and Kinoshita and Sato (2013) are applicable to both Rossby waves and inertia–gravity waves. It is emphasized again that the 3D wave activity flux corresponding to the wave forcing to the mean flow is different from that describing the wave propagation in the zonal component, though the difference in their divergence is small. Thus, we need to use 3D-flux-M for the analysis of 3D mass transport and 3D-flux-W for the analysis of wave propagation.

## Acknowledgments

We thank Matthew H. Hitchman and R. Alan Plumb for their helpful comments and fruitful discussions. ERA-Interim data were used for the analysis. Yoshihiro Tomikawa and Kazue Suzuki helped in the treatment. Thanks are due to Rolando R. Garcia, M. Joan Alexander, and an anonymous reviewer for constructive comments. The GFD-DENNOU library was used for drawing figures. This study is supported by Grand-in-Aid for Research Fellow (22-7125) of the JSPS and by Grant-in-Aid for Scientific Research (B) 22340134 of the Ministry of Education, Culture, Sports and Technology, Japan.

## APPENDIX

### Formulas in the Spherical Coordinates

*λ*, latitude

*φ*, and Earth’s radius

*a*, the transformed time-mean zonal momentum equation (4.1) and

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