Editorial Type:
Article
Article Type:
Research Article

A Formulation of Three-Dimensional Residual Mean Flow and Wave Activity Flux Applicable to Equatorial Waves

Takenari Kinoshita Integrated Science Data System Research Laboratory, National Institute of Information and Communications Technology, Tokyo, Japan

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Kaoru Sato Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan

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Print Publication:
01 Sep 2014
DOI:
https://doi.org/10.1175/JAS-D-13-0161.1
Page(s):
3427–3438
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Abstract

The large-scale waves that are known to be trapped around the equator are called equatorial waves. The equatorial waves cause mean zonal wind acceleration related to quasi-biennial and semiannual oscillations. The interaction between equatorial waves and the mean wind has been studied by using the transformed Eulerian mean (TEM) equations in the meridional cross section. However, to examine the three-dimensional (3D) structure of the interaction, the 3D residual mean flow and wave activity flux for the equatorial waves are needed. The 3D residual mean flow is expressed as the sum of the Eulerian mean flow and Stokes drift. The present study derives a formula that is approximately equal to the 3D Stokes drift for equatorial waves on the equatorial beta plane (EQSD). The 3D wave activity flux for equatorial waves whose divergence corresponds to the wave forcing is also derived using the EQSD. It is shown that the meridionally integrated 3D wave activity flux for equatorial waves is proportional to the group velocity of equatorial waves.

Denotes Open Access content.

Corresponding author address: Takenari Kinoshita, Integrated Science Data System Research Laboratory, National Institute of Information and Communications Technology, Tokyo 184-8795, Japan. E-mail: kinoshita@nict.go.jp

Abstract

The large-scale waves that are known to be trapped around the equator are called equatorial waves. The equatorial waves cause mean zonal wind acceleration related to quasi-biennial and semiannual oscillations. The interaction between equatorial waves and the mean wind has been studied by using the transformed Eulerian mean (TEM) equations in the meridional cross section. However, to examine the three-dimensional (3D) structure of the interaction, the 3D residual mean flow and wave activity flux for the equatorial waves are needed. The 3D residual mean flow is expressed as the sum of the Eulerian mean flow and Stokes drift. The present study derives a formula that is approximately equal to the 3D Stokes drift for equatorial waves on the equatorial beta plane (EQSD). The 3D wave activity flux for equatorial waves whose divergence corresponds to the wave forcing is also derived using the EQSD. It is shown that the meridionally integrated 3D wave activity flux for equatorial waves is proportional to the group velocity of equatorial waves.

Denotes Open Access content.

Corresponding author address: Takenari Kinoshita, Integrated Science Data System Research Laboratory, National Institute of Information and Communications Technology, Tokyo 184-8795, Japan. E-mail: kinoshita@nict.go.jp
Keywords: Atmospheric circulation; Internal waves; Wave breaking; Waves, atmospheric; Forcing; Mass fluxes/transport

1. Introduction

Equatorial waves are atmospheric waves whose amplitude is maximized at the equator and decreases exponentially as the distance of the waves from the equator increases. Their dispersion relation and spatial structure were studied theoretically by Matsuno (1966). He derived eigenmodes of equatorial waves by using a plane-wave assumption in the time and zonal directions on the linear shallow-water equation. The equatorial waves that were discovered by Wallace and Kousky (1968) and Yanai and Maruyama (1966) from radiosonde observation were identified as Kelvin waves and Rossby–gravity waves, respectively.

On the other hand, quasi-biennial and semiannual oscillations (QBO and SAO) of the zonal wind exist in the equatorial stratosphere. Previous studies have shown that these oscillations are driven by atmospheric waves. The relation between the waves and the zonal-mean zonal wind can be diagnosed by the transformed Eulerian mean (TEM) equations that were derived by Andrews and McIntyre (1976, 1978). The residual mean flow is expressed as the sum of the Eulerian mean flow and Stokes drift under the small-amplitude assumption and is approximately equal to the zonal-mean Lagrangian mean flow when the wave is linear, steady, and adiabatic and when no dissipation occurs. The Eliassen–Palm (EP) flux is equal to the product of the group velocity and the wave activity density under the Wentzel–Kramers–Brillouin (WKB) approximation and is a useful physical quantity for describing the wave propagation (Edmon et al. 1980). The residual mean flow and the zonal-mean zonal wind acceleration are related to the divergence of EP flux in the zonal momentum equation. When there are no critical levels, the divergence of the EP flux is zero for linear, steady, and conservative waves. Under such conditions, the waves neither drive the residual mean flow nor accelerate the zonal-mean zonal wind. This is known as the nonacceleration theorem (Eliassen and Palm 1961; Charney and Drazin 1961). In studies using the TEM equations and equatorial wave theory, it was shown that the QBO is mainly driven by gravity waves, equatorial Kelvin waves, and Rossby–gravity waves (e.g., Sato and Dunkerton 1997; Haynes 1998; Baldwin et al. 2001), and it is recognized that the SAO is mainly driven by gravity waves, equatorial Kelvin waves, and extratropical Rossby waves (e.g., Hirota 1980; Holton and Wehrbein 1980; Hitchman and Leovy 1988; Sassi and Garcia 1997; Antonita et al. 2007).

Kawatani et al. (2010) recently investigated the zonal variation of wave forcing associated with the equatorial Kelvin waves and inertia–gravity waves using the 3D wave activity flux derived by Miyahara (2006). They showed that this variation in the stratosphere results from zonal variation of the wave sources and from the vertically sheared zonal winds associated with the Walker circulation, depending on the phase of the QBO.

However, it is not clear whether this 3D wave activity flux can describe the equatorial Kelvin waves correctly because this flux is only applicable to inertia–gravity waves, not to equatorial waves. Although Kinoshita and Sato (2013a,b) newly formulated the 3D wave activity flux on the primitive equations, their formulas are not applicable in the equator region. The present study formulates the 3D residual mean flow and wave activity flux applicable to equatorial waves and shows that these formulas are applicable to the equatorial Kelvin waves.

The paper is arranged as follows. In section 2, the 3D Stokes drift for equatorial waves (EQSD) is derived from its definition for the equatorial beta-plane equations. The 3D wave activity flux for equatorial waves (3D-EQW-flux) is formulated by using the EQSD in section 3. It is shown that the 3D-EQW-flux divergence corresponds to the equatorial wave forcing to the mean flow. It is also shown that the meridional integral of the 3D-EQW-flux accords with a product of the group velocity and the meridional integral of the wave activity density for equatorial waves. Moreover, we investigate the 3D wave energy equation for equatorial waves. A summary and concluding remarks are given in section 4.

2. The time-mean 3D Stokes drift applicable to equatorial waves

When small-amplitude perturbations in the slowly varying background horizontal flow and weak background wind shear are assumed, the perturbation equations on the equatorial beta plane are given as follows:
e2.1a
e2.1b
e2.1c
e2.1d
and
e2.1e
where z is the log-pressure height; u, υ, and w, are zonal, meridional, and vertical velocities, respectively; ρ0 is the basic density; Φ is the geopotential; N2 is the buoyancy frequency squared, which expresses static stability; β ≡ 2Ωa−1 is the beta effect; Ω is Earth’s rotation rate; a is the mean radius of Earth; the suffixes x, y, and z denote the partial derivatives; and we assume that the time-mean vertical velocity and the nonconservative and diabatic terms are negligible. The overbar and prime express the time mean and its deviation, respectively. For a perturbation, a form of plane wave is considered:
e2.2a
where A′ is the arbitrary perturbation; H is the scale height; k and m are zonal and vertical wavenumbers, respectively; and ω is the ground-based angular frequency. It should be noted that the amplitudes of perturbations are constant in the time scale of wave phase change and vary in the time scale for the background state. Basic density is expressed as
e2.2b
where ρs is a surface density. The zonal, meridional, and vertical parcel displacements (ξ′, η′, ζ′) satisfy the following relations as
e2.3
The time-mean Stokes drift is given in the following using the parcel displacements in (2.3) and perturbation wind velocities:
e2.4a
e2.4b
and
e2.4c
Here, it should be noted that the deformations on the second equal sign of each equation are made by using the relations , , , and under the assumption that the time-mean wind shear is small.

In the next section, the EQSD for equatorial Kelvin waves, Rossby–gravity waves, and other types of equatorial waves are formulated from the definition in (2.4).

a. The 3D Stokes drift for equatorial Kelvin waves

For equatorial Kelvin waves, the meridional component of perturbation wind velocity and that of parcel displacement vanish. Thus, EQSD has only the zonal and vertical components. When (2.2) is substituted into (2.1d) and (2.3), the vertical parcel displacement ζ′ is written in terms of Φ′ as
e2.5
Using (2.5) enables to be expressed as
e2.6
Thus, EQSD for the equatorial Kelvin wave is formulated in the following:
e2.7a
and
e2.7b
where the subscript (Kl) is used to distinguish other equatorial waves.
Next, the difference between and other 3D Stokes drifts is examined in terms of and (Kinoshita and Sato 2013a,b). While the term becomes equal to and is included in 3D Stokes drift for inertia–gravity waves, the term becomes equal to and is included in the one applicable to both Rossby waves and gravity waves. Note that becomes equal to zero for equatorial Kelvin waves. From zonal momentum, continuity, and thermodynamic equations, polarization and dispersion relations for equatorial Kelvin waves are expressed as follows (Andrews et al. 1987):
e2.8
e2.9
where we use and assume that is independent of the latitude. From the zonal and meridional momentum equation, the geopotential of equatorial Kelvin waves is expressed as
e2.10
where is constant. Using (2.8) and (2.10), the term is written in terms of Φ′ as
e2.11
Similarly, the term becomes
e2.12
Thus, 3D Stoke drift for equatorial Kelvin waves is equal to that applicable only to inertia–gravity waves (Kinoshita et al. 2010), not equal to that derived by Kinoshita and Sato (2013a,b).

b. The 3D Stokes drift for other types of equatorial waves

For waves having nonzero meridional components of perturbation wind velocity, slightly complex manipulation is needed to relate to the perturbation meridional velocity and other perturbation physical quantities. The dispersion relation for equatorial waves and the solution for are expressed as follows:
e2.13a
e2.13b
and
e2.13c
where , Hn(Y) are the Hermite polynomials, and is constant. Substituting (2.13b) into (2.1) and using the identities dHn(Y)/dY = 2nHn−1(Y) and Hn+1 = 2YHn(Y) − 2nHn−1(Y) make it possible to show that
e2.14a
and
e2.14b
(Andrews et al. 1987). It is noted that the latitudinal scale of the background fields is larger than an equatorial radius of deformation . First, by using (2.3), (2.14a), and (2.13b), is expressed in terms of as follows:
e2.15
Similarly, , , and the potential energy are written as follows:
e2.16a
e2.16b
and
e2.16c
The meridional derivative of the difference between (2.16a) and (2.16c) and the derivative of (2.16b) are, respectively, expressed as
e2.17a
and
e2.17b
Hereafter, the notation (Y) is omitted from Hn(Y). From the difference between (2.17b) and (2.17a), the following relation is obtained:
e2.18
Next, and are deformed in the same way as (2.6):
e2.19
Thus, EQSD is formulated as follows:
e2.20a
e2.20b
and
e2.20c
It is important that EQSD (2.20) is also applicable to the Kelvin waves (n = −1), since and , and hence . Thus, EQSD (2.20) can be used for all types of equatorial waves. It should be noted that the advantage of EQSD (2.20) is to be derived without including parcel displacements that are hardly observed and to be composed of eddy covariances. This means that the EQSD is applicable not only to monochromatic waves but also to all equatorially confined perturbations that are expressed with a superposition of sinusoidal waves.

3. A formulation of the 3D wave activity flux for equatorial waves

a. The 3D residual mean flow and wave activity flux

The time-mean zonal momentum equation on the equatorial beta plane is given by
e3.1
By substituting (2.20) into (3.1) and using the assumption that the background wind shear is negligible, we obtain
e3.2
where is the meridional component of the 3D residual mean flow associated with forcing by equatorial waves, and is the 3D wave activity flux for equatorial waves:
e3.3a
e3.3b
and
e3.3c
It should be noted that (3.3b) vanishes since u′ and υ′ are out of phase by 90°. This 3D wave activity flux [(3.3)] is related to the wave forcing for the time-mean flow. In the next section, the relation between the 3D wave activity flux [(3.3)] and the group velocity of equatorial waves is examined.

b. The relation between 3D wave activity flux and group velocity

In the 2D TEM equation system, the meridionally integrated EP flux is equal to a product of the vertical group velocity and the meridionally integrated wave activity density (Andrews et al. 1987). It can be shown that the vertical component of 3D wave activity flux [(3.3c)] satisfies this relation, as in the following.

Using (2.1d) and (2.14) enables us to write included in (3.3c) in terms of as
e3.4
The meridional integral of (3.4) is obtained by using the dispersion relation of equatorial waves (2.13a) and :
e3.5
Similarly, and its meridional integral are given in the following:
e3.6a
and
e3.6b
Thus,
e3.7
On the other hand, the wave activity density and its meridional integral are written as
e3.8a
and
e3.8b
where . The derivation of (3.5), (3.6b), and (3.8b) is given in appendix A. The zonal and vertical group velocities of equatorial waves are expressed as
e3.9a
and
e3.9b
Dividing (3.7) by (3.8b) yields
e3.10
Thus, the meridional integral of the vertical component of 3D wave activity flux for equatorial waves [(3.3c)] is proportional to the vertical group velocity.
Next, it is shown that the meridional integral of the zonal component of 3D wave activity flux for equatorial waves [(3.3a)] accords with a product of the zonal group velocity and the meridionally integrated wave activity density. Using (2.13b) and (2.14), and its meridional integral are written in terms of as
e3.11a
and
e3.11b
Dividing (3.11) by (3.8b) yields
e3.12
These results indicate that the 3D wave activity flux [(3.3)] can describe the propagation of equatorial waves. It should be noted that the terms proportional to the group velocities are not the 3D wave activity flux [(3.3)] but its meridional integral. This is similar to the case of EP flux for equatorial waves.

c. The wave energy equation for equatorial waves

This section examines how the 3D wave activity flux for equatorial waves is related to the wave activity density after the wave energy equation is derived. In this derivation, it is assumed that the time-mean wind shear is negligible.

First, taking and then using the time mean yields
e3.13
Equation (3.13) is regarded as the 3D wave energy equation for equatorial waves. Note that vanishes since υ′ and Φ′ are out of phase by 90°.
Next, by using (2.13b) and (2.14), can be written in terms of as
e3.14
From (3.11a) and (3.14)
e3.15
Similarly,
e3.16
From (3.13), (3.15), and (3.16),
e3.17
Equation (3.17) is regarded as the generalized Eliassen–Palm relation for equatorial waves under the slowly varying time-mean-flow assumption. Note that (3.2) and (3.17) express the wave–mean flow interaction as is consistent with (3.5a), (5.5a), and (5.7) in Andrews and McIntyre (1976). It should be noted that relations (3.13) and (3.17) are obtained without using the meridional integral, unlike the results of section 3b.

4. Concluding remarks

In this study, the 3D Stokes drift is formulated from its definition for equatorial beta-plane equations (EQSD) when the slowly varying background field and small-amplitude perturbations are assumed. EQSD is applicable to all equatorial waves. The 3D wave activity flux (3D-EQW-flux) is formulated by substituting the EQSD into the time-mean zonal momentum equation. These expressions are derived using the time mean and are phase independent.

Next, it is shown that the latitudinal integral of 3D-EQW-flux accords with a product of the group velocity and the latitudinally integrated wave activity density in both zonal and vertical directions. This is an extension of the relation for the Eliassen–Palm flux on the 2D TEM equations. The present study also derives the 3D wave energy equation for equatorial waves.

As it is shown that becomes equal to the 3D Stokes drift for gravity waves in section 2a, we compare and other 3D wave activity flux. The result shows that the meridional integral of is equal to that of 3D wave activity flux applicable to inertia–gravity waves [Miyahara 2006; Kinoshita et al. 2010; ]. The details are written in the appendix.

The EQSD and 3D-EQW-flux are partly different from the 3D Stokes drift and wave activity flux that are applicable to both gravity waves and Rossby waves (Kinoshita and Sato 2013a,b). The difference is due to assumptions of waves. Kinoshita and Sato (2013a,b) assume waves having meridional wavenumbers, and the term is reduced to . On the other hand, this study assumes waves whose amplitude is damped in the meridional direction, and the term is reduced to . Similar manipulations may be needed for a case of tidal waves whose meridional structures have some nodes. Thus, the 3D TEM equations applicable to tidal waves need to be derived.

Acknowledgments

We thank Hisashi Nakamura, Toshiyuki Hibiya, Masaaki Takahashi, Keita Iga, and Koutarou Takaya for their helpful comments and important suggestions. We also thank Rolando R. Garcia, Saburo Miyahara, and an anonymous reviewer for providing constructive comments. This study is supported by Grand-Aid for Research Fellow (22-7125) of the JSPS and by Grant-in-Aid for Scientific Research (A) 25247075 of the Ministry of Education, Culture, Sports and Technology, Japan.

APPENDIX A

Derivation of (3.5), (3.6b), and (3.8b)

The deformation of the first line of (3.5) is made by using (2.13c) and :
ea.1
The deformations of the first line of (3.6b) and from the first to the seconds lines of (3.8b) are also made in a similar way.
Next, by using the dispersion relation of equatorial waves (2.13a), the part included in (3.5) can be expressed as follows:
ea.2
Similarly, the parts in (3.6b) and in (3.8b) are reduced in the following:
ea.3a
and
ea.3b

APPENDIX B

Meridional Integral of 3D Wave Activity Flux Applicable to Inertia–Gravity Waves

The meridional integral of is expressed as
eb.1
The derivation of (B1) is given in the following:
eb.2
Thus, the meridional integral of becomes equal to that of 3D wave activity flux applicable to inertia–gravity waves.

APPENDIX C

Another Expression of

In this section, we introduce another expression of without using parcel displacements.

From the meridional derivative of (2.1a), the meridional derivative of (2.1b), (2.1c), and vertical derivative of (2.1d), the perturbation potential vorticity equation is expressed as follows:
ec.1
Substituting (2.13b), (2.14a), and (2.14b) into (C.1), perturbation potential vorticity q′ is expressed in terms of as follows:
ec.2a
ec.2b
ec.2c
and
ec.2d
where the dispersion relation of equatorial waves [(2.13a)] is used in the last line. Thus,
ec.3
It should be noted that this expression can be used for all equatorial waves.

REFERENCES

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  • Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and mean zonal acceleration. J. Atmos. Sci., 33, 2031–2048, doi:10.1175/1520-0469(1976)033<2031:PWIHAV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Andrews, D. G., and M. E. McIntyre, 1978: Generalized Eliassen-Palm and Charney-Drazin theorems for waves on axisymmetric mean flows in compressible atmospheres. J. Atmos. Sci., 35, 175–185.

    • Search Google Scholar
    • Export Citation

  • Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics.Academic Press, 489 pp.

  • Antonita, T. M., G. Ramkumar, K. K. Kumar, K. S. Appu, and K. V. S. Nambhoodiri, 2007: A quantitative study on the role of gravity waves in driving the tropical stratospheric semiannual oscillation. J. Geophys. Res.,112, D12115, doi:10.1029/2006JD008250.

  • Baldwin, M. P., and Coauthors, 2001: The quasi-biennial oscillation. Rev. Geophys., 39, 179–229, doi:10.1029/1999RG000073.

  • Charney, J. G., and P. G. Drazin, 1961: Propagation of planetary-scale disturbances from lower into upper atmosphere. J. Geophys. Res., 66, 83–109, doi:10.1029/JZ066i001p00083.

    • Search Google Scholar
    • Export Citation

  • Edmon, H. J., B. J. Hoskins, and M. E. McIntyre, 1980: Eliassen-Palm cross sections for the troposphere. J. Atmos. Sci., 37, 2600–2616, doi:10.1175/1520-0469(1980)037<2600:EPCSFT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Eliassen, A., and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geofys. Publ., 22 (3), 1–23.

  • Haynes, P. H., 1998: The latitudinal structure of the quasi-biennial oscillation. Quart. J. Roy. Meteor. Soc., 124, 2645–2670, doi:10.1002/qj.49712455206.

    • Search Google Scholar
    • Export Citation

  • Hirota, I., 1980: Observational evidence of the semiannual oscillation in the tropical middle atmosphere—A review. Pure Appl. Geophys.,118, 217–238, doi:10.1007/BF01586452.

  • Hitchman, M. H., and C. B. Leovy, 1988: Estimation of the Kelvin wave contribution to the semiannual oscillation. J. Atmos. Sci., 45, 1462–1475, doi:10.1175/1520-0469(1988)045<1462:EOTKWC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Holton, J. R., and W. M. Wehrbein, 1980: A numerical model of the zonal mean circulation of the middle atmosphere. Pure Appl. Geophys., 118, 284–306, doi:10.1007/BF01586455.

    • Search Google Scholar
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  • Kawatani, Y., K. Sato, T. J. Dunkerton, S. Watanabe, S. Miyahara, and M. Takahashi, 2010: The roles of equatorial trapped waves and internal inertia–gravity waves in driving the quasi-biennial oscillation. Part II: Three-dimensional distribution of wave forcing. J. Atmos. Sci.,67, 981–997, doi:10.1175/2009JAS3223.1.

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