The Energy Flux of Three-Dimensional Waves in the Atmosphere: Exact Expression for a Basic Model Diagnosis with No Equatorial Gap

Hidenori Aiki aInstitute for Space-Earth Environmental Research, Nagoya University, Aichi, Japan

Search for other papers by Hidenori Aiki in
Current site
Google Scholar
PubMed
Close
https://orcid.org/0000-0002-0604-9900
,
Yoshiki Fukutomi aInstitute for Space-Earth Environmental Research, Nagoya University, Aichi, Japan

Search for other papers by Yoshiki Fukutomi in
Current site
Google Scholar
PubMed
Close
,
Yuki Kanno bCentral Research Institute of Electric Power Industry, Chiba, Japan

Search for other papers by Yuki Kanno in
Current site
Google Scholar
PubMed
Close
,
Tomomichi Ogata cJapan Agency for Marine-Earth Science and Technology, Kanagawa, Japan

Search for other papers by Tomomichi Ogata in
Current site
Google Scholar
PubMed
Close
,
Takahiro Toyoda dMeteorological Research Institute, Ibaraki, Japan

Search for other papers by Takahiro Toyoda in
Current site
Google Scholar
PubMed
Close
, and
Hideyuki Nakano dMeteorological Research Institute, Ibaraki, Japan

Search for other papers by Hideyuki Nakano in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

A model diagnosis for the energy flux of off-equatorial Rossby waves in the atmosphere has previously been done using quasigeostrophic equations and is singular at the equator. The energy flux of equatorial waves has been separately investigated in previous studies using a space–time spectral analysis or a ray theory. A recent analytical study has derived an exact universal expression for the energy flux, which can indicate the direction of the group velocity for linear shallow water waves at all latitudes. This analytical result is extended in the present study to a height-dependent framework for three-dimensional waves in the atmosphere. This is achieved by investigating the classical analytical solution of both equatorial and off-equatorial waves in a Boussinesq fluid. For the horizontal component of the energy flux, the same expression has been obtained between equatorial waves and off-equatorial waves in the height-dependent framework, which is linked to a scalar quantity inverted from the isentropic perturbation of Ertel’s potential vorticity. The expression of the vertical component of the energy flux requires computation of another scalar quantity that may be obtained from the meridional integral of geopotential anomaly in a wavenumber–frequency space. The exact version of the universal expression is explored and illustrated for three-dimensional waves induced by an idealized Madden–Julian oscillation forcing in a basic model experiment. The zonal and vertical fluxes manifest the energy transfer of both equatorial Kelvin waves and off-equatorial Rossby waves with a smooth transition at around 10°S and around 10°N. The meridional flux of wave energy represents connection between off-equatorial divergence regions and equatorial convergence regions.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Hidenori Aiki, aiki@nagoya-u.jp

Abstract

A model diagnosis for the energy flux of off-equatorial Rossby waves in the atmosphere has previously been done using quasigeostrophic equations and is singular at the equator. The energy flux of equatorial waves has been separately investigated in previous studies using a space–time spectral analysis or a ray theory. A recent analytical study has derived an exact universal expression for the energy flux, which can indicate the direction of the group velocity for linear shallow water waves at all latitudes. This analytical result is extended in the present study to a height-dependent framework for three-dimensional waves in the atmosphere. This is achieved by investigating the classical analytical solution of both equatorial and off-equatorial waves in a Boussinesq fluid. For the horizontal component of the energy flux, the same expression has been obtained between equatorial waves and off-equatorial waves in the height-dependent framework, which is linked to a scalar quantity inverted from the isentropic perturbation of Ertel’s potential vorticity. The expression of the vertical component of the energy flux requires computation of another scalar quantity that may be obtained from the meridional integral of geopotential anomaly in a wavenumber–frequency space. The exact version of the universal expression is explored and illustrated for three-dimensional waves induced by an idealized Madden–Julian oscillation forcing in a basic model experiment. The zonal and vertical fluxes manifest the energy transfer of both equatorial Kelvin waves and off-equatorial Rossby waves with a smooth transition at around 10°S and around 10°N. The meridional flux of wave energy represents connection between off-equatorial divergence regions and equatorial convergence regions.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Hidenori Aiki, aiki@nagoya-u.jp

1. Introduction

The role of wave propagation in the atmosphere may be investigated by several approaches, where the group velocity vector is an important factor when tracing waves from source to sink regions (Eliassen and Palm 1960). A conservation equation for the quasigeostrophic wave activity has been used to diagnose the direction of the group velocity of quasigeostrophic Rossby waves (RWs) in off-equatorial regions (Plumb 1986; Takaya and Nakamura 2001; Wirth et al. 2018; Fukutomi 2019). Attempts to relax the limitation of quasigeostrophic dynamics have been made in recent studies for the formulation of wave activity (Aiki et al. 2015; Harada et al. 2019). A space–time spectral and filtering analysis has been used to investigate the role of equatorial waves in quasi-biennial oscillation (QBO) and Madden–Julian oscillation (MJO) (Yanai et al. 1968; Wheeler and Kiladis 1999; Lin et al. 2006; Yasunaga et al. 2019). A ray tracing theory has been used to analyze the interaction of equatorial and off-equatorial waves (Hoskins and Karoly 1981; Sobel and Bretherton 1999; Shaman and Tziperman 2005; Sato et al. 2009; Ribstein et al. 2015; Scaife et al. 2016).

The sign of the wave activity is indefinite owing to both the effective beta parameter and the direction of wave propagation. In contrast, wave energy is positive definite and is suitable for investigating its budget with source and sink terms. For off-equatorial RWs in the atmosphere, the expression for the energy flux for use in a model diagnosis, and without relying on a time–space spectral analysis or a ray tracing theory, has previously been derived using quasigeostrophic equations and is singular at the equator (Orlanski and Sheldon 1993, hereafter OS93; Cai and Huang 2013; Durland and Farrar 2020). It is important to investigate how equatorial waves and off-equatorial RWs are linked to each other and where they originate from, without subdividing the description in order to reflect the differences in dynamics between the equatorial and off-equatorial regions at around 10°S and around 10°N. A recent analytical study has derived an exact universal expression for the energy flux which can indicate the direction of the group velocity for both planetary and gravity waves at all latitudes (Aiki et al. 2017, hereafter AGC17). The expression of AGC17 is universal in that it has allowed for investigations of the direction of the group velocity when, for example, equatorial RWs and Kelvin waves (KWs) interact with each other. This is in contrast to the ray tracing theory which needs to preselect a dispersion relation for the given type of waves. The expression of AGC17 has been used in Ogata and Aiki (2019) and Li and Aiki (2020) to analyze the group velocity vector of intraseasonal and annual waves, respectively, in the tropical Indian Ocean, and also in Song and Aiki (2020) to analyze the group velocity vector of annual waves in the tropical Atlantic Ocean. Toyoda et al. (2021) have used the AGC17 expression to identify energy transfer episodes associated with El Niño–Southern Oscillation (ENSO) from an ocean reanalysis. These studies have made first attempts in oceanic literature to investigate the transfer routes of wave energy without being restricted by the quasigeostrophic approximation.

AGC17 have adopted a single-layer shallow water formulation and have not given expressions for the vertical profile nor vertical component of the energy flux. The present study extends the analytical investigation of AGC17 to yield a Boussinesq framework for the energy flux of three-dimensionally propagating waves at all latitudes in the atmosphere, for use in a model diagnosis with a smooth transition between the equatorial and off-equatorial regions at around 10°S and around 10°N. Section 2 provides the theoretical background. As explained in section 3, the mathematical development of the present study has been given in the online supplemental material.1 The exact version of the universal expression is explored and illustrated in section 4 for three-dimensional waves induced by an idealized MJO forcing. Section 5 provides a summary. The novelty of the present study is explained in section 6.

2. Theoretical background

Let an arbitrary variable with an associated physical dimension be expressed by an asterisk superscript symbol throughout this manuscript, which is aimed at minimizing complexity in the set of nondimensionalized equations (without an asterisk superscript symbol) for equatorial waves as well as the Hermite polynomial in the supplemental material. The present study uses primitive equations for a hydrostatic Boussinesq fluid appropriate to linear waves in a rotating frame of reference and in the absence of a mean flow nor a moist (convectively coupled) process. Let Cartesian coordinates be labeled by the set of independent variables x*, y*, z*, and t*, where each of x*, y*, and z* increases eastward, northward, vertically upward, respectively, and u*, υ*, and w* are the corresponding three-dimensional components of velocity (a list of variables is given in Table S1 in the online supplemental material). If necessary, the vertical coordinate may be rewritten as z*=H*log(p*/1000hPa), where H* is a height scale and p* is pressure. The equations may then be written as
u*t*f*υ*+Φ*x*=0,
υ*t*+f*u*+Φ*y*=0,
1ρ0*z*[ρ0*(1N*22Φ*z*t*)w*]+(u*x*+υ*y*)=0,
Φ*z*=g*θ*θ0*,
N*2=g*θ0*dθ0*dz*,
where f*=f0*+β*y* is the Coriolis parameter, N*=N*(z*) and g* are the buoyancy frequency and the acceleration due to gravity. The symbols ρ0*=ρ0*(z*) and θ0*=θ0*(z*) represent the reference density and the reference potential temperature of the atmosphere, respectively. The symbols Φ*=Φ*(x*,y*,z*,t*) and θ*=θ*(x*,y*,z*,t*) represent the perturbation component of geopotential and potential temperature, respectively. Readers who are interested in convectively coupled moist waves may replace the buoyancy frequency N* in the present study with, for example, one which takes into account the effective static stability associated with diabatic forcing (Kiladis et al. 2009). The incompressibility Eq. (1c) includes the definition of the vertical component of velocity, w*=w*(x*,y*,z*,t*), which may be rewritten using the time derivative of a scalar quantity ζ*=ζ*(x*,y*,z*,t*) to read
w*t*ζ*=1N*22Φ*t*z*,
ζ*1N*2Φ*z*=θ*(dθ0*/dz*),
where (1d) and (1e) have been used to derive the second equality of (1g). The right-hand side of (1g) allows for ζ* to be interpreted as an approximation based on the Taylor expansion in the vertical direction for the vertical displacement of an isentropic surface at a given horizontal point and a given reference height. Manipulation of (1a)(1c) yields a prognostic equation for potential vorticity (symbolized as q*) to read
t*[υ*x*u*y*+1ρ0*z*(ρ0*f*N*2Φ*z*)]q*+υ*β*=0,
which is applicable to waves at all latitudes, such as off-equatorial RWs, off-equatorial inertia–gravity waves (IGWs), and equatorial waves [i.e. equatorial RWs and IGWs, equatorial Rossby–gravity waves (RGWs; i.e., Yanai waves), and equatorial KWs; Matsuno 1966; Yanai and Maruyama 1966], understanding f0*=0 for an equatorial β plane and β*=0 for an off-equatorial f plane. Both off-equatorial IGWs (i.e., β*=0) and equatorial KWs (i.e., υ*=0) are characterized by q*=0. We investigate relationship between the quantity q* in (2) and Ertel’s potential vorticity (EPV). The third term in the square brackets in (2) may be rewritten as
1ρ0*z*(ρ0*f*N*2Φ*z*)=1ρ0*z*[f*θ*(1/ρ0*)(dθ0*/dz*)]=1ρ0*[f*(1/ρ0*)(dθ0*/dz*)θ*z*θ*(1/ρ0*)2(dθ0*/dz*)2z*(f*ρ0*dθ0*dz*)]=1(1/ρ0*)(dθ0*/dz*)[f*ρ0*θ*z*1N*2Φ*z*z*(f*ρ0*dθ0*dz*)].
Substitution of (3) to (2) yields a prognostic equation for the linearized version of EPV to read
t*Q*+υ*y*Q0*+w*z*Q0*=0,
Q*(υ*x*u*y*)1ρ0*dθ0*dz*+f*ρ0*θ*z*,
Q0*f*ρ0*dθ0*dz*,
where the quantities Q*=Q*(x*,y*,z*,t*) and Q0*=Q0*(y*,z*) are the perturbation and background components, respectively, of EPV in linear wave theory. We now use (3) to explain the quantity q* in (2) as
q*=1(1/ρ0*1)(dθ0*/dz*)[Q*+(1N*2Φ*z*)ζ*z*Q0*],
where ζ* has been defined at (1f). Note that Q* is the Eulerian perturbation of EPV in height coordinates. We suggest interpreting the content of the square brackets in (5) as the isentropic perturbation of EPV as approximated using a Taylor expansion in the vertical direction. Indeed vertical advection is explicit in the prognostic equation, Eq. (4a), for the Eulerian perturbation Q* of EPV, while it is implicit in the prognostic equation, Eq. (2), for the isentropic perturbation q* of EPV. Equation (5) will be useful for extending the framework of the present study to finite-amplitude waves in the presence of a mean flow or a moist process in a future study. On the other hand, a prognostic equation for wave energy may be derived from (1a) to (1c) as
t*ρ0*2[u*2+υ*2+1N*2(Φ*z*)2]¯+*ρ0*u*Φ*¯,ρ0*υ*Φ*¯,ρ0*w*Φ*¯=0,
where */x*,/y*,/z* and the overbar symbol represents a phase-average2 operator (i.e., for a sinusoidal wave, A*¯=0 for A*=u*, υ*, w*, and Φ*), or a low-pass time filter [for this reason we retain the local time derivative in (6) to allow for slow time variations in the general case]. The present study focuses on wave types for which the group velocity has been well formulated in the literature/textbooks. In what follows, the group velocity times wave energy is referred to as wave energy flux (WEF).

a. Equatorial waves

Using the classical analytical solution of zonally and vertically propagating equatorial waves (Matsuno 1966), we compare the meridional profiles of the pressure flux and the WEF. For low-frequency equatorial waves (with ω < 1, i.e., all equatorial RWs and westward-propagating RGWs), the meridional profiles of uΦ¯ and (ω/k)[u2+υ2+(Φ/z)2¯]/2 are shown by the dashed green and solid black lines, respectively, in Fig. 1. The symbols ω, k, m, and n represent wave frequency, the zonal and vertical components of wavenumber, and meridional mode number, respectively. See section A1 of the supplemental material for the details of nondimensionalization (thus represented without an asterisk superscript symbol). It is clear that, when compared at a given latitude, uΦ¯ is not equal to the group velocity times wave energy. In particular, the meridional profile of uΦ¯ is sign indefinite for low-frequency equatorial waves (see Fig. 1). For eastward-propagating RGW (Fig. 1a), WEF has a single eastward peak while pressure flux has double eastward peaks. The peak magnitudes of WEF and pressure flux are roughly comparable to each other. For short RW with n = 1 (Fig. 1b), WEF has double eastward peaks while pressure flux has both double eastward peaks near the equator and double eastward peaks in the flanks of the equator. The peak magnitude of pressure flux is several times greater than that of WEF. For long RW with n = 1 (Fig. 1c), WEF has a single westward peak while pressure flux has both a single eastward peak at the equator and double eastward peaks in the flanks of the equator. The peak magnitude of pressure flux is several times greater than that of WEF. For short RW with n = 2 (Fig. 1d), WEF has triple eastward peaks while pressure flux has the set of weak double eastward peaks near the equator, strong double westward peaks in the flanks of the equator, and strong double eastward peaks in regions further away from the equator. For long RW with n = 2 (Fig. 1d), WEF has double westward peaks while pressure flux has both double westward peaks in the flanks of the equator and double eastward peaks in regions further away from the equator.

Fig. 1.
Fig. 1.

Meridional profiles of the zonal component of phase-averaged energy flux calculated at z = 0.5. The solid black line is (ω/k)[u2+υ2+(Φ/z)2¯]/2, the dashed green line is uΦ¯, and the light-blue dots are the right-hand side of (S10a). All panels are for low-frequency equatorial waves with ω < 1: (a) eastward-propagating RGWs, (b) short and (c) long RWs in the first meridional mode, (d) short and (e) long RWs in the second meridional mode. The associated values of meridional-mode number n, zonal wavenumber k, vertical wavenumber m, wave frequency ω, and group velocity ∂ω/∂k are noted in each panel. For each of (a)–(e), the wave amplitude A in (S6a) has been set to normalize the meridional integral of wave energy: [u2+υ2+(Φ/z)2¯]/2dy=1.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

On the other hand, as shown by the solid-black and dashed-green lines in Fig. 2 for high-frequency equatorial waves (with ω > 1, i.e., all equatorial IGWs and eastward-propagating RGWs), the meridional profile of uΦ¯ provides a much better approximation for the group velocity times wave energy. The profiles of WEF and pressure flux are comparable to each other in terms of both the number and sign of peaks. The peak magnitudes of pressure flux are a few tens of percent greater than that of WEF. Namely, they are on the same order in terms of magnitude.

Fig. 2.
Fig. 2.

As in Fig. 1, but for high-frequency equatorial waves with ω > 1: (a) westward- and (b) eastward-propagating IGWs in the second meridional mode, (c) westward- and (d) eastward-propagating IGWs in the first meridional mode, and (e) eastward-propagating RGWs.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

For low-frequency equatorial waves (with ω < 1, i.e., all equatorial RWs and westward-propagating RGWs), the meridional profiles of wΦ¯ and (ω/m)[u2+υ2+(Φ/z)2¯]/2 are shown by the dashed green and solid black lines, respectively, in Fig. 3. It is clear that, when compared at a given latitude, wΦ¯ is not equal to the group velocity times wave energy. The meridional profile of wΦ¯ is single-signed for low-frequency equatorial waves (see Fig. 3). For eastward-propagating RGW (Fig. 3a), WEF has a single upward peak. For short RW with n = 1 (Fig. 3b), WEF has double upward peaks. For long RW with n = 1 (Fig. 3c), WEF has a single upward peak. For short RW with n = 2 (Fig. 3d), WEF has triple upward peaks. For long RW with n = 2 (Fig. 3d), WEF has double upward peaks. Pressure flux has double upward peaks in all panels of Fig. 3. The peak magnitudes of WEF and pressure flux are comparable to each other in all panels of Fig. 3.

Fig. 3.
Fig. 3.

As in Fig. 1 about low-frequency equatorial waves, but for the vertical component of phase-averaged energy flux. The solid black line is (ω/m)[u2+υ2+(Φ/z)2¯]/2, the dashed green line is wΦ¯, the light-blue dots are the right-hand side of (S13a), and the red squares are given by the set of (S15b), (12), and (S26).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

On the other hand, as shown by the solid-black and dashed-green lines in Fig. 4 for high-frequency equatorial waves (with ω > 1, i.e., all equatorial IGWs and eastward-propagating RGWs), the meridional profile of wΦ¯ provides a better approximation for the group velocity times wave energy. The profiles of WEF and pressure flux are comparable to each other in terms of both the number and sign of peaks. The peak magnitudes of pressure flux are a few ten of percent greater than that of WEF, and are on the same order. To our knowledge, previous observational and model studies have estimated the vertical flux of equatorial wave energy using the pressure flux without illustrating difference in their meridional profiles (Nitta 1970; Kawatani et al. 2009).

Fig. 4.
Fig. 4.

As in Fig. 2 about high-frequency equatorial waves, but for the vertical component of phase-averaged energy flux. The solid-black line is (ω/m)[u2+υ2+(Φ/z)2¯]/2, the dashed green line is wΦ¯, the light-blue dots are the right-hand side of (S13a), and the red squares are given by the set of (S15b), (12), and (S26).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

b. Off-equatorial waves

We rewrite the three-dimensional energy flux in (6) in a vector form as
ρ0*U*Φ*¯=ρ0*u*Φ*¯,ρ0*υ*Φ*¯,ρ0*w*Φ*¯,
which points in the same direction as the group velocity for off-equatorial IGWs in the atmosphere and ocean (Holton 2004). The vector in (7) does not point in the direction of the group velocity of off-equatorial RWs (Longuet-Higgins 1964). To retrieve the correct direction for the energy flux associated with off-equatorial RWs, OS93 have suggested modifying (6), without affecting the three-dimensional divergence of the energy flux, as
t*ρ0*2[u*2+υ*2+1N*2(Φ*z*)2]¯+*ρ0*[u*Φ*¯+y*(Φ*2¯2f*)],ρ0*[υ*Φ*¯x*(Φ*2¯2f*)],ρ0*w*Φ*¯=0,
where f*=f0*+β*y* is understood. The three-dimensional energy flux in (8) consists of two terms,
ρ0*U*Φ*¯+ρ0*y*(Φ*2¯2f*),x*(Φ*2¯2f*),0,
where U*Φ*¯ is as in the gravity wave literature (i.e., U* consists of both the geostrophic and ageostrophic components of velocity). The second term in (9) is the additional rotational component required to reproduce the direction of the group velocity of off-equatorial RWs.

AGC17 have shown, using a height-independent oceanic framework, how to derive a general expression for the additional rotational flux that points in the direction of the group velocity associated with waves at all latitudes. The main subject of the present study is to extend the theoretical result of AGC17 to a height-dependent atmospheric framework, with application to the investigation of interaction between equatorial and quasigeostrophic waves in mind.

3. Analytical investigation for the height-dependent energy flux of three-dimensional waves at all latitudes

Using the analytical solutions of both equatorial and off-equatorial waves, we investigate whether or not it is possible to derive an exact universal expression for the rotational flux which, after being added to U*Φ*¯, is able to indicate the direction of the group velocity for linear waves at all latitudes. This mathematical development has been given in sections A1–A3 of the supplemental material of the present manuscript, which is outlined as follows.

The analytical solution of linear waves has been separately derived for equatorial regions and for off-equatorial regions in the literature, which is why our formulation in the supplemental material separates sections A1 and A2. The outcome of section A1 is identification of a unified expression for all types of equatorial waves (i.e., RWs, RGWs, IGWs, and KWs) that represents group-velocity-based energy flux in the three-dimensional space [(S15a), (S15b), and (S17) as listed in Table 1]. Likewise the outcome of section A2 is identification of a unified expression for all types of off-equatorial waves (i.e., RWs and IGWs) that represents group-velocity-based energy flux in the three-dimensional space [(S22c), (S23b), and (S25c) as listed in Table 1]. A next step is to consider how to obtain the distribution of WEF from a model diagnosis without restriction on latitudes. For the horizontal component of WEF, the same expression has been obtained between equatorial waves [see (S15a) and (S17)] and off-equatorial waves [see (S22c) and (S25c)] in a height-dependent framework. Namely, equatorial and off-equatorial regions have already been connected in sections A1 and A2 as far as the horizontal WEF is concerned, which is a robust result and is the cornerstone of the present study. For the vertical WEF, we identify a difference3 between the results of equatorial and off-equatorial expressions [see (S15b) and (S23b), respectively]. In section A3, we reconcile expressions for the vertical component of equatorial and off-equatorial WEFs derived in sections A1 and A2, respectively.

Table 1.

Comparisons of three approaches for computing the height-dependent energy flux, as explained in section 4c. The term “low-frequency equatorial waves” refers to Rossby waves and westward-propagating mixed Rossby–gravity waves. The term “high-frequency equatorial waves” refers to inertia–gravity waves and eastward-propagating mixed Rossby–gravity waves. The term “off-equatorial waves” refers to both off-equatorial inertia–gravity and Rossby waves. Level-1 and level-2 approximations are explained in section A5 of the supplemental material.

Table 1.

Based on these results of analytical investigation, we rewrite (6) using the exact expression of energy flux for waves at all latitudes to read
t*ρ0*2[u*2+υ*2+1N*2(Φ*z*)2]¯+*ρ0*u*Φ*¯+ρ0*y*(Φ*φ*¯2+1β*2u*t*2φ*¯),ρ0*υ*Φ*¯ρ0*x*(Φ*φ*¯2+1β*2u*t*2φ*¯)(ρ0*R*)z*,ρ0*w*Φ*¯+(ρ0*R*)y*=0,
where the three-dimensional components are adapted from (S15a), (S17), and (S15b). The WEF expression in (10) includes a set of scalar quantities defined as
φ*=(υ*/μ)2m˜*2ω*3/(β*N*2)+k*,
1ρ0*z*(ρ0*ϖ*z*)=υ*z*,
χ*=υ*m˜*2ω*2/N*2k*2,
and
R*G20y*(2ϖ*y*t*χ*y*¯2υ*z*t*χ*¯f*2N*2)dy*+1β*(2w*t*2φ*¯)G2β*(3ϖ*y*t*2φ*¯),
where μ is wave phase, k* and m* are the zonal and vertical component of wavenumber [m˜*2=m*2+1/(4H*2)], and ω* is wave frequency. The definition of R* in (12) includes a nondimensional function G=G(y*) which is set to become unity in equatorial regions and vanish in off-equatorial regions. For details, see section A3 where G is referred to as meridional tapering function. Each of the second and third terms on the right hand of (12) automatically vanishes for off-equatorial waves, as shown by a scale analysis in sections A2 and A3.

The scalar quantity φ* in (10) presents a rotational flux which affects both zonal and meridional components of WEF. The definition of φ* in (11a) and the way by which it appears in the WEF expression of (10) is as in AGC17. The quantity φ* may be referred to as pseudostreamfunction. This quantity is the cornerstone of a unified treatment of gravity and planetary waves. This is clarified using inversion equations for the isentropic perturbation of EPV, as explained in the next section. The expression of the vertical component of the equatorial WEF requires computation of another scalar quantity χ*, previously unmentioned in the literature. We come back to this issue in the next section.

Light blue dots in Figs. 14 represent the meridional profiles of zonal and vertical fluxes in (10), as calculated using the analytical solution of equatorial waves where G = 1. Readers may confirm, by comparing the solid black lines and light-blue dots in Figs. 1 and 2, that the meridional profile of the zonal energy flux of (10) is precisely identical to the zonal component of group velocity times wave energy for all types of equatorial waves. Likewise readers may confirm, by comparing the solid black lines and light-blue dots in Figs. 3 and 4, that the meridional profile of the vertical energy flux of (10) is precisely identical to the vertical component of group velocity times wave energy for all types of equatorial waves. To summarize the result of sections A1–A3, we have cleared all basic steps for extending the single-layer shallow water framework of AGC17 to a Boussinesq framework in height coordinates. This is the main result of the present study.

4. Computational approach and result of model diagnoses

The scalar quantities φ* in (11a) and χ* in (11c) have been defined in a spectral form. A concern is the handling of the vertical wavenumber m* in the stretched vertical grid system of general circulation model outputs. Below we propose alternative procedures for obtaining φ* and χ*.

a. The new scalar quantity

The quantity χ* is associated with the vertical component of the energy flux that is absent in AGC17. The mathematical structure of χ* may be better understood by substituting (11c) to (S6c) associated with equatorial waves to yield
2χ*t*y*+f*χ*x*=Φ*,
which can also be derived by substituting (11c) to (S19c) associated with off-equatorial waves. Namely, (13a) is applicable to both equatorial and off-equatorial waves. Comparisons of (13a) and the zonal integral of (1a) allow for χ* to be interpreted as the zonal integral of a variant of streamfunction. Equation (13a) may be rewritten as
χ*y*=f*k*χ*+Φμ*ω*,
which may be integrated in the meridional direction using an implicit computational method to obtain χ* from a model diagnosis.

b. Inversion equation for the isentropic perturbation of EPV

AGC17 have shown in a shallow water framework that (11a) may be rewritten into an expression which does not contain μ*, k*, m˜*, or ω* to yield an inversion equation for (what they have called) EPV. This result is reproduced by a Boussinesq framework in height coordinates as
(2x*2+2y*2)φ*+1ρ0*z*[(ρ0*N*2)z*(f*2φ*+32φ*t*2)]=q*,
where q* has been defined at (2) that is the isentropic perturbation of EPV as shown by (5). The derivation of (14) has been given in (S11). While (14) has been derived from the characteristic equation, (S5), associated with equatorial waves, (14) can also be derived from the universal characteristic equation, (S18a), associated with off-equatorial waves. Namely, (14) is applicable to both equatorial and off-equatorial waves. The height-dependent inversion equations, (S11) and (14), of isentropic EPV correspond to the height-independent inversion equations, (16) and (17a), respectively, of shallow water EPV in AGC17 and are referred to as the level-0 (i.e., exact) expression. We find that what has been referred to as EPV in AGC17 is actually the isentropic perturbation of EPV as shown by (5). This terminological correction improves our understanding of the EPV-inversion equations as part of the universal formulation of the horizontal WEF.

c. Results: Rossby and Kelvin waves in an idealized MJO simulation

We have used a linear atmospheric model based on (1a)(1g), rewritten in spherical coordinates, to simulate three-dimensional waves induced by an idealized MJO forcing following Hendon and Salby (1996, hereafter HS96). There is no mean flow in our model, to be consistent with the governing equations of the present study. A heat forcing term has been added to (1f)(1g) to read
θ*t*+w*dθ0*dz*=(1.75Kday1)FtFxFyFz,
Ft=exp[(t*/40days)2]sin[2π(λ*/180°t*/40days)],
Fx=2cos[2π(λ*100°)/360°]+13,
Fy=exp[(ϕ*/20°)4],
Fz=max{cos[π(p*550hPa)/800hPa],0},
where λ* and ϕ* are longitude and latitude, respectively, in degrees. The forcing magnitude of 1.75 K day−1 has been adapted from HS96. The temporal and zonal structures of the heat forcing in (15b) and (15c), respectively, have been set in such a way as to idealize Fig. 1a of HS96. The meridional and vertical structures of the heat forcing in (15d) and (15e), respectively, follow that in HS96. The model has been integrated from day −100 to day +100 the former of which represents the initial condition of no motion.

The model adopts both Rayleigh friction and Newtonian cooling whose vertical profiles are set to be the same as that in HS96. The model domain covers all longitudes with a zonally cyclic condition. The model code has been discretized with a 2° × 2° resolution in the horizontal direction from 70°S to 70°N and uses the Smagorinsky (1963) scheme with a nondimensional coefficient of unity. The model has 28 vertical levels from 1000 to 5 hPa where the vertical profiles of ρ0* and θ0* are adapted from the Japanese 55-year Reanalysis dataset (Kobayashi et al. 2015; Harada et al. 2016) to yield globally uniform background stratification. We have performed a single experiment. Figure 5 shows Hovmöller diagrams for the set of heat forcing (color shading), simulated geopotential anomaly Φ*, and inverted pseudostreamfunction φ* (contours) from day −60 to day +60. The eastward phase translation of geopotential anomaly at z* = 2.2 km is faster than that of heat forcing (Fig. 5a). The peak magnitude of geopotential anomaly is about 100 m2 s−2 near the surface (Fig. 5b). The pseudostreamfunction is meridionally antisymmetric about the equator, which is attributed to RWs (Fig. 5c).

Fig. 5.
Fig. 5.

Hovmöller diagrams of heat forcing (color shading, unit: K day−1) on the right-hand side of (15a) from day −60 to day +60. (a) Zonal distribution at z* = 2.2 km of the equator, (b) vertical distribution at 100°E of the equator, and (c) meridional distribution at 100°E of z* = 2.2 km. Contours in (a) and (b) represent simulated geopotential anomaly Φ* with an interval of 20 m2 s−2. Contours in (c) represent inverted pseudostreamfunction φ* with an interval of 5 × 105 m2 s−1. Solid and dashed contours are positive and negative values, respectively.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

In what follows, all quantities have been calculated from the output of the same experiment with a time average between day −60 and day +60. Figure 6 presents comparisons for the horizontal component of the pressure flux in (6), the OS93 flux in (8), and the energy flux of the present study in (10), the last of which is hereafter referred to as level 0 (AGC17). The pressure flux manifests a meridionally alternating pattern (Fig. 6a), while it can represent the eastward transfer of equatorial KW energy whose magnitude is on the order of 100 W m−2. The OS93 flux is not available near the equator, while it can represent the westward transfer of off-equatorial RW energy whose magnitude is on the order of 10 W m−2 (Fig. 6b). The level-0 flux can represent the energy transfer of both equatorial KWs and off-equatorial RWs with a smooth transition at around 10°S and around 10°N (Fig. 6c). Figure 7 shows energy fluxes in the meridional-vertical section at 100°E, all of which manifest peaks in both the upper troposphere and the lower troposphere. All panels of Fig. 7 indicate that the vertical component of the energy fluxes is upward in the upper troposphere and downward in the lower troposphere. In the upper troposphere of off-equatorial regions, the magnitude of the OS93 zonal flux (color shading in Fig. 7b) is nearly the same as that of the level-0 zonal flux (color shading in Fig. 7c). In the vicinity of the surface, the OS93 zonal flux is westward whereas the level-0 zonal flux is eastward in off-equatorial regions, which indicates that the pseudostreamfunction does not approximates to the geostrophic streamfunction owing to friction.

Fig. 6.
Fig. 6.

Comparisons of three expressions for the horizontal flux (arrows) of wave energy at z* = 2.2 km and its zonal component (color shading, unit: W m−2). (a) The pressure flux [see Eq. (6)], (b) the pressure flux plus the additional rotational flux of OS93 [see Eq. (8)], and (c) the pressure flux plus the additional rotational flux of the present study [see Eq. (10)], referred to as level 0. In both (b) and (c), the additional fluxes have been calculated using the rotation operator in the spherical coordinate system to be consistent with the model formulation. All quantities in Figs. 69 have been calculated from the output of the same experiment with a time average between day −60 and day +60.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for energy flux vectors (arrows) in meridional–vertical plane at 100°E and the zonal fluxes (color shading).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

Color shading in Fig. 8 shows the three-dimensional divergence of the energy flux that is independent of difference among the pressure flux, the OS93 flux, and the level-0 flux. In the horizontal section at z* = 2.2 km, the source of wave energy (i.e., flux-divergence) is located in off-equatorial regions, while the sink of wave energy (i.e., flux-convergence) is located in the equatorial regions. These regions are connected by the equatorward flux of wave energy as manifested by the level-0 expression (contours in Figs. 8a,b). In the meridional-vertical and zonal-vertical sections, the source of wave energy (i.e., flux-divergence) is located in the middle troposphere, while the sink of wave energy (i.e., flux-convergence) is located in both the upper troposphere and near the surface where Newtonian cooling and Rayleigh friction operate. This vertical profile of divergence is consistent with the distribution of the vertical flux of wave energy, shown by arrows in Figs. 7c and 8c.

Fig. 8.
Fig. 8.

Three-dimensional divergence of the energy flux (color shading, unit: 10−5 W m−3). (a) Horizontal distribution at z* = 2.2 km, (b) meridional–vertical distribution at 100°E, and (c) zonal–vertical distribution at the equator. Contours in (a) and (b) represent the meridional component of the energy flux of the present study, referred to as level 0, with an interval of 2 W m−2. Solid and dashed contours are positive and negative values, respectively. Arrows in (c) represent the energy flux of the present study with a unit of W m−2.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

Figure 9 shows the vertical component of the level-0 flux (color shading) whose magnitude is on the order of 10−2 W m−2. Contours in Fig. 9 show the difference of the level-0 vertical flux and the pressure vertical flux. The magnitude of signals represented by these contours is on the order of 10−4 W m−2, which is two orders smaller than the vertical flux of wave energy mentioned above. Namely, the level-0 vertical flux is nearly identical to the pressure vertical flux. This may be attributed to the dominance of equatorial KWs in equatorial regions (between 10°S and 10°N: red-color areas in Fig. 6c) and the dominance of off-equatorial RWs in far regions (blue-color areas in Fig. 6c). For both equatorial KWs and off-equatorial RWs, the vertical flux of wave energy is theoretically represented by the pressure flux. See the supplemental material for details.

Fig. 9.
Fig. 9.

As in Fig. 8, but for the vertical component of the energy flux of the present study (color shading, unit: 10−3 W m−2) in Eq. (10) and its difference from the vertical component of the pressure flux with solid and dashed contours representing positive and negative values, respectively, with an interval of 4 × 10−5 W m−2. Namely, contours are given by the meridional gradient of ρ0*R*.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0177.1

The model diagnosis of the present study has illustrated the utility of the level-0 WEF for investigating the internal dynamics of MJO with no equatorial gap. The role of WEFs in MJO–extratropics interaction has not been addressed in the present study owing to the absence of a mean flow (Lau et al. 2011; Stan et al. 2017). We are planning to adopt the way by which the WEF formulation of OS93 includes the effect of a mean flow in a quasigeostrophic framework. This update will not be as mathematically heavy as the present study on connection between equatorial and off-equatorial dynamics at around 10°N and 10°S for three-dimensional waves. It will be interesting to compare the magnitudes of WEFs associated with the MJO internal dynamics and the MJO–extratropics interaction, the former of which has been shown to be on the order of 10–100 W m−2 in the present study.

5. Summary

For off-equatorial RWs in the atmosphere, the expression for the energy flux for use in a model diagnosis has previously been derived using quasigeostrophic equations and is singular at the equator. The energy flux of equatorial waves has been separately investigated in previous studies using a space–time spectral analysis or a ray theory. A recent analytical study has derived an exact universal expression for the energy flux, which can indicate the direction of the group velocity for linear shallow water waves at all latitudes (AGC17). The present study has extended the analytical investigation of AGC17 to yield a height-dependent framework for the energy flux of three-dimensionally propagating waves in a stratified dry Boussinesq fluid without a mean flow. The expression of WEF in the present study may be used for the diagnosis of general circulation model outputs with a smooth transition between equatorial and off-equatorial regions. This is a novel aspect in the progress of atmospheric science for investigating the interaction of equatorial and quasigeostrophic dynamics from a viewpoint of wave energy transfers.

As outlined in section 3, the mathematical development of the present study has been given in sections A1–A3 of the supplemental material. The present study has focused on wave types for which the group velocity has been well formulated in the literature/textbooks. The analytical solution of linear waves has been separately derived for equatorial regions and for off-equatorial regions in the literature, which is why sections A1 and A2 have been separated in the present study. The outcome of section A1 is identification of a unified expression for all types of equatorial waves (i.e., RWs, RGWs, IGWs, and KWs) that reproduces group-velocity-based energy flux in the three-dimensional space. Likewise the outcome of section A2 is identification of a unified expression for all types of off-equatorial waves (i.e., RWs and IGWs) that reproduces group-velocity-based energy flux in the three-dimensional space. For the horizontal component of WEF, the same expression has been obtained between equatorial waves [see (S15a) and (S17)] and off-equatorial waves [see (S22c) and (S25c)] in height coordinates. Expressions for the horizontal component of the energy flux are similar to each other between the height-dependent Boussinesq framework given by the present study and the height-independent shallow water framework given by AGC17, both of which are linked to a scalar quantity φ* inverted from isentropic EPV. This finding is given by interpreting the content of the square brackets in (5) as the isentropic perturbation of EPV as approximated using a Taylor expansion in the vertical direction. Equation (5) will be useful for extending the framework of the present study to finite-amplitude waves in the presence of a mean flow as well as a moist (convectively coupled) process in a future study.

The expression of the vertical component of the equatorial WEF requires computation of another scalar quantity χ*, previously unmentioned in the literature, which may be obtained by the meridional integral of (13a) in a spectral space. The quantity χ* may be interpreted as the zonal integral of a variant of streamfunction, to be investigated further in a future study for waves in the presence of a mean flow as well as a moist (convectively coupled) process. For the vertical WEF, analytical expressions (S15b) and (S23b) associated with equatorial and off-equatorial waves, respectively, are somewhat different from each other. In section A3, we have explained how to obtain the distribution of vertical WEF from a model diagnosis with a smooth transition between the equatorial and off-equatorial expressions.

In section 4, the exact version of the universal expression is explored and illustrated for three-dimensional waves induced by an idealized MJO-like forcing in a basic model experiment without a mean flow, to be consistent with the formulation of the present study. The equatorward flux of wave energy maintains connection between off-equatorial source regions and equatorial sink regions, which forms a cornerstone for a future study on the internal dynamics of MJO. In this experiment, where equatorial KWs and off-equatorial RWs dominate, the exact vertical WEF is nearly identical to the pressure vertical flux. The difference of the two quantities is two orders in terms of magnitude smaller than the vertical WEF. Overall the seamless nature of AGC17 has now been reproduced by the atmospheric formulation in a height-dependent framework. This is attributed to the pseudostreamfunction φ* which reduces to the geostrophic streamfunction in off-equatorial regions for RWs and vanishes for gravity waves, a novel aspect for considering the universal expression of the energy flux. Inclusion of a mean flow is yet to be investigated in a future study.

6. Discussion

The novelty of the present study may be discussed in terms of packaging (i.e., assortment to constitute a new diagnosis framework). AGC17 have made a major advance since the 1960s, when papers by Longuet-Higgins (1964) and Matsuno (1966) appeared, in terms of connection between equatorial and off-equatorial energetics. One may point out that the packaging of AGC17 (expressions without vertical profiles nor vertical component) is not enough to convince all readers with its applicability to the diagnoses of general circulation model outputs. We suggest interpreting the whole content of the present study as new packaging which consists of WEF expressions with both horizontal and vertical components that are ready for global model diagnoses by a range of readers who may be benefited from our classification based on the tradeoff between accuracy and complexity (section A5 in the supplementary material; see also Table 1). The present manuscript includes mathematical development, which is heavier than that in AGC17, to identify expressions for the vertical component of energy flux associated with both equatorial and off-equatorial waves. This is the last piece of state-of-the-art packaging for WEF expressions set for a broader audience in the community.

The progress of atmospheric science has long been given by accumulation of inductive knowledge from a series of model and observational diagnoses as well as accumulation of deductive mathematical development. An example of the former is an abundance of diagnosis studies using the Eliassen–Palm flux. The diagnosis framework of AGC17 has recently been used in Ogata and Aiki (2019), Li and Aiki (2020), Song and Aiki (2020, 2021), Li et al. (2021), and Toyoda et al. (2021) to illustrate well-defined energy transfer episodes in the ocean for interpreting the role of RWs and equatorial KWs in the interaction between equatorial and off-equatorial regions. On the other hand, the atmospheric diagnosis framework of the present study will be used for (i) filling an equatorial gap in the quasigeostrophic framework (as the above studies show for oceanic waves) and (ii) revisiting the connection of equatorial and off-equatorial regions where atmospheric planetary waves play an important role (Kawahira 1983; Frederiksen and Webster 1988; Horinouchi et al. 2000; Serra et al. 2014).

Major sources of tropical variability are the QBO, the MJO, and ENSO. The QBO can influence vertically propagating RWs from the troposphere to produce effects on the winter Northern Hemisphere stratospheric polar vortex (Holton and Tan 1980; Baldwin et al. 2001; Gray et al. 2018). The MJO can generate an RW train that produces significant effects on extratropical circulation and intraseasonal climate, including effects on the North Atlantic Oscillation during northern winter (Cassou 2008). Sea surface temperature anomalies in the western Pacific Ocean associated with ENSO modulate convective activity near the Philippines that induces an RW train through Japan to the west coast of North America (Nitta 1987; Huang and Sun 1992; Fukutomi and Yasunari 2002; Kosaka and Nakamura 2006). The impact of ENSO on European climate has been investigated with taking into account interaction between the troposphere and the stratosphere (Brönnimann 2007; Ineson and Scaife 2008; King et al. 2018; Warner et al. 2020). Future studies should investigate which of the exact and approximate WEF expressions and what kind of phase averaging procedures (corresponding to the overbar symbol in the present formulation) are suitable for mapping wave energy transfer routes associated with a given phenomenon in the global atmosphere.

Acknowledgments

The authors thank constructive comments from two anonymous reviewers and the editor. HA thanks Yuya Baba for helpful discussion. This study was supported by JSPS KAKENHI Grants 18H03738 and 15H02129.

Data availability statement

Readers can download the analysis code from http://cidas.isee.nagoya-u.ac.jp/databases/index.shtml.en.

REFERENCES

  • Aiki, H., K. Takaya, and R. J. Greatbatch, 2015: A divergence-form wave-induced pressure inherent in the extension of the Eliassen–Palm theory to a three-dimensional framework for waves at all latitudes. J. Atmos. Sci., 72, 28222849, https://doi.org/10.1175/JAS-D-14-0172.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Aiki, H., R. J. Greatbatch, and M. Claus, 2017: Towards a seamlessly diagnosable expression for the energy flux associated with both equatorial and mid-latitude waves. Prog. Earth Planet. Sci., 4, 11, https://doi.org/10.1186/s40645-017-0121-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Baldwin, M. P., and Coauthors, 2001: The quasi-biennial oscillation. Rev. Geophys., 39, 179229, https://doi.org/10.1029/1999RG000073.

  • Brönnimann, S., 2007: Impact of El Niño–Southern Oscillation on European climate. Rev. Geophys., 45, RG3003, https://doi.org/10.1029/2006RG000199.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cai, M., and B. Huang, 2013: A new look at the physics of Rossby waves: A mechanical-Coriolis oscillation. J. Atmos. Sci., 70, 303316, https://doi.org/10.1175/JAS-D-12-094.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cassou, C., 2008: Intraseasonal interaction between the Madden–Julian oscillation and the North Atlantic Oscillation. Nature, 455, 523527, https://doi.org/10.1038/nature07286.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Durland, T. S., and J. T. Farrar, 2020: Another note on Rossby wave energy flux. J. Phys. Oceanogr., 50, 531534, https://doi.org/10.1175/JPO-D-19-0237.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eliassen, A., and E. Palm, 1960: On the transfer of energy in stationary mountain waves. Geophys. Publ., 22, 123.

  • Frederiksen, J. S., and P. J. Webster, 1988: Alternative theories of atmospheric teleconnections and low-frequency fluctuations. Rev. Geophys., 26, 459–494, https://doi.org/10.1029/RG026i003p00459.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fukutomi, Y., 2019: Tropical synoptic-scale waves propagating across the Maritime Continent and northern Australia. J. Geophys. Res. Atmos., 124, 76657682, https://doi.org/10.1029/2018JD029795.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fukutomi, Y., and T. Yasunari, 2002: Tropical–extratropical interaction associated with the 10–25-day oscillation over the western Pacific during the northern summer. J. Meteor. Soc. Japan, 80, 311331, https://doi.org/10.2151/jmsj.80.311.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gray, L. J., J. A. Anstey, Y. Kawatani, H. Lu, S. Osprey, and V. Schenzinger, 2018: Surface impacts of the quasi biennial oscillation. Atmos. Chem. Phys., 18, 82278247, https://doi.org/10.5194/acp-18-8227-2018.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harada, Y., and Coauthors, 2016: The JRA-55 Reanalysis: Representation of atmospheric circulation and climate variability. J. Meteor. Soc. Japan, 94, 269302, https://doi.org/10.2151/jmsj.2016-015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harada, Y., K. Sato, T. Kinoshita, R. Yasui, T. Hirooka, and H. Naoe, 2019: Diagnostics of a WN2-type major sudden stratospheric warming event in February 2018 using a new three-dimensional wave activity flux. J. Geophys. Res. Atmos., 124, 61206142, https://doi.org/10.1029/2018JD030162.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hendon, H. H., and M. L. Salby, 1996: Planetary-scale circulations forced by intraseasonal variations of observed convection. J. Atmos. Sci., 53, 17511758, https://doi.org/10.1175/1520-0469(1996)053<1751:pscfbi>2.0.co;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 2004: An Introduction to Dynamic Meteorology. 3rd ed. Academic Press, 531 pp.

  • Holton, J. R., and H.-C. Tan, 1980: The influence of the equatorial quasi-biennial oscillation on the global circulation at 50 mb. J. Atmos. Sci., 37, 22002208, https://doi.org/10.1175/1520-0469(1980)037<2200:TIOTEQ>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Horinouchi, T., F. Sassi, and B. A. Boville, 2000: Synoptic-scale Rossby waves and the geographic distribution of lateral transport routes between the tropics and the extratropics in the lower stratosphere. J. Geophys. Res., 105, 26 57926 592, https://doi.org/10.1029/2000JD900281.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 38, 11791196, https://doi.org/10.1175/1520-0469(1981)038<1179:TSLROA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huang, R., and F. Sun, 1992: Impacts of the tropical western Pacific on the East Asian summer monsoon. J. Meteor. Soc. Japan, 70, 243256, https://doi.org/10.2151/jmsj1965.70.1B_243.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ineson, S., and A. A. Scaife, 2008: The role of the stratosphere in the European climate response to El Niño. Nat. Geosci., 2, 3236, https://doi.org/10.1038/ngeo381.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kawahira, K., 1983: Global structures of stationary planetary waves in the middle atmosphere. J. Meteor. Soc. Japan, 61, 695716, https://doi.org/10.2151/jmsj1965.61.5_695.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kawatani, Y., M. Takahashi, K. Sato, S. P. Alexander, and T. Tsuda, 2009: Global distribution of atmospheric waves in the equatorial upper troposphere and lower stratosphere: AGCM simulation of sources and propagation. J. Geophys. Res., 114, D01102, https://doi.org/10.1029/2008JD010374.

    • Search Google Scholar
    • Export Citation
  • Kiladis, G. N., M. C. Wheeler, P. T. Haertel, K. H. Straub, and P. E. Roundy, 2009: Convectively coupled equatorial waves. Rev. Geophys., 47, RG2003, https://doi.org/10.1029/2008rg000266.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • King, M. P., I. Herceg-Bulić, I. Bladé, J. García-Serrano, N. Keenlyside, F. Kucharski, C. Li, and S. Sobolowski, 2018: Importance of late fall ENSO teleconnection in the Euro-Atlantic sector. Bull. Amer. Meteor. Soc., 99, 13371343, https://doi.org/10.1175/BAMS-D-17-0020.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kobayashi, S., and Coauthors, 2015: The JRA-55 Reanalysis: General specifications and basic characteristics. J. Meteor. Soc. Japan, 93, 548, https://doi.org/10.2151/jmsj.2015-001.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kosaka, Y., and H. Nakamura, 2006: Structure and dynamics of the summertime Pacific–Japan teleconnection pattern. Quart. J. Roy. Meteor. Soc., 132, 20092030, https://doi.org/10.1256/qj.05.204.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lau, W. K. M., D. E. Waliser, and P. E. Roundy, 2011: Tropical–extratropical interactions. Intraseasonal Variability in the Atmosphere-Ocean Climate System, W. K. M. Lau and D. E. Waliser, Eds., Springer, 497–512.

    • Crossref
    • Export Citation
  • Li, Z., and H. Aiki, 2020: The life cycle of annual waves in the Indian Ocean as identified by seamless diagnosis of the energy flux. Geophys. Res. Lett., 47, e2019GL085670, https://doi.org/10.1029/2019GL085670.

    • Search Google Scholar
    • Export Citation
  • Li, Z., H. Aiki, M. Nagura, and T. Ogata, 2021: The vertical structure of annual wave energy flux in the tropical Indian Ocean. Prog. Earth Planet. Sci., 8, 43, https://doi.org/10.1186/s40645-021-00432-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lin, J. L., and Coauthors, 2006: Tropical intraseasonal variability in 14 IPCC AR4 climate models. Part I: Convective signals. J. Climate, 19, 26652690, https://doi.org/10.1175/JCLI3735.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, M. S., 1964: On group velocity and energy flux in planetary wave motion. Deep-Sea Res., 11, 3542, https://doi.org/10.1016/0011-7471(64)91080-0.

    • Search Google Scholar
    • Export Citation
  • Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan, 44, 2543, https://doi.org/10.2151/jmsj1965.44.1_25.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nitta, T., 1970: A study of generation and conversion of eddy available potential energy in the tropics. J. Meteor. Soc. Japan, 48, 524528, https://doi.org/10.2151/jmsj1965.48.6_524.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nitta, T., 1987: Convective activities in the tropical western Pacific and their impact on the Northern Hemisphere summer circulation. J. Meteor. Soc. Japan, 65, 373390, https://doi.org/10.2151/jmsj1965.65.3_373.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ogata, T., and H. Aiki, 2019: The pathway of intraseasonal wave energy in the tropical Indian Ocean as identified by a seamless diagnostic scheme. SOLA, 15, 262267, https://doi.org/10.2151/sola.2019-047.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Orlanski, I., and J. Sheldon, 1993: A case of downstream baroclinic development over western North America. Mon. Wea. Rev., 121, 29292950, https://doi.org/10.1175/1520-0493(1993)121<2929:ACODBD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 1986: Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci., 43, 16571678, https://doi.org/10.1175/1520-0469(1986)043<1657:TDPOTQ>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ribstein, B., U. Achatz, and F. Senf, 2015: The interaction between gravity waves and solar tides: Results from 4-D ray tracing coupled to a linear tidal model. J. Geophys. Res. Space Phys., 120, 67956817, https://doi.org/10.1002/2015JA021349.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sato, K., S. Watanabe, Y. Kawatani, Y. Tomikawa, K. Miyazaki, and M. Takahashi, 2009: On the origins of mesospheric gravity waves. Geophys. Res. Lett., 36, L19801, https://doi.org/10.1029/2009GL039908.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Scaife, A. A., and Coauthors, 2016: Tropical rainfall, Rossby waves and regional winter climate predictions. Quart. J. Roy. Meteor. Soc., 143, 111, https://doi.org/10.1002/qj.2910.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Serra, Y. L., X. Jiang, B. Tian, J. Amador-Astua, E. D. Maloney, and G. N. Kiladis, 2014: Tropical intraseasonal modes of the atmosphere. Annu. Rev. Environ. Resour., 39, 189215, https://doi.org/10.1146/annurev-environ-020413-134219.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shaman, J., and E. Tziperman, 2005: The effect of ENSO on Tibetan Plateau snow depth: A stationary wave teleconnection mechanism and implications for the South Asian monsoons. J. Climate, 18, 20672079, https://doi.org/10.1175/JCLI3391.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smagorinsky, J., 1963: General circulation experiments with the primitive equation. I. The basic experiment. Mon. Wea. Rev., 91, 99165, https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sobel, A. H., and C. S. Bretherton, 1999: Development of synoptic-scale disturbances over the summertime tropical northwest Pacific. J. Atmos. Sci., 56, 31063127, https://doi.org/10.1175/1520-0469(1999)056<3106:dossdo>2.0.co;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Song, Q., and H. Aiki, 2020: The climatological horizontal pattern of energy flux in the tropical Atlantic as identified by a unified diagnosis for Rossby and Kelvin waves. J. Geophys. Res. Oceans, 125, e2019JC015407, https://doi.org/10.1029/2019JC015407.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Song, Q., and H. Aiki, 2021: Horizontal energy flux of wind-driven intraseasonal waves in the tropical Atlantic by a unified diagnosis. J. Phys. Oceanogr., 51, 30373050, https://doi.org/10.1175/JPO-D-20-0262.1.

    • Search Google Scholar
    • Export Citation
  • Stan, C., D. M. Straus, J. S. Frederiksen, H. Lin, E. D. Maloney, and C. Schumacher, 2017: Review of tropical-extratropical teleconnections on intraseasonal time scales. Rev. Geophys., 55, 902937, https://doi.org/10.1002/2016RG000538.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Takaya, K., and H. Nakamura, 2001: A formulation of a phase-independent wave-activity flux for stationary and migratory quasigeostrophic eddies on a zonally varying basic flow. J. Atmos. Sci., 58, 608627, https://doi.org/10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Toyoda, T., and Coauthors, 2021: Energy flow diagnosis of ENSO from an ocean reanalysis. J. Climate, 34, 40234042, https://doi.org/10.1175/JCLI-D-20-0704.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Warner, J. L., J. A. Screen, and A. A. Scaife, 2020: Links between Barents-Kara sea ice and the extratropical atmospheric circulation explained by internal variability and tropical forcing. Geophys. Res. Lett., 47, e2019GL085679, https://doi.org/10.1029/2019GL085679.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wheeler, M., and G. N. Kiladis, 1999: Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber–frequency domain. J. Atmos. Sci., 56, 374399, https://doi.org/10.1175/1520-0469(1999)056<0374:CCEWAO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wirth, V., M. Riemer, E. K. M. Chang, and O. Martius, 2018: Rossby wave packets on the midlatitude waveguide—A review. Mon. Wea. Rev., 146, 19652001, https://doi.org/10.1175/MWR-D-16-0483.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yanai, M., and T. Maruyama, 1966: Stratospheric wave disturbances propagating over the equatorial Pacific. J. Meteor. Soc. Japan, 44, 291294, https://doi.org/10.2151/jmsj1965.44.5_291.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yanai, M., T. Maruyama, T. Nitta, and Y. Hayashi, 1968: Power spectra of large-scale disturbances over the tropical Pacific. J. Meteor. Soc. Japan, 46, 308323, https://doi.org/10.2151/jmsj1965.46.4_308.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yasunaga, K., S. Yokoi, K. Inoue, and B. E. Mapes, 2019: Space–time spectral analysis of the moist static energy budget equation. J. Climate, 32, 501529, https://doi.org/10.1175/jcli-d-18-0334.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
1

Section A1 presents an analytical investigation associated with the classical solution of equatorial IGWs, RWs, and mixed RGWs, and KWs in height coordinates. Section A2 is the repetition of section A1 except for the set of off-equatorial IGWs and RWs. Section A3 reconciles expressions for the vertical component of equatorial and off-equatorial wave energy fluxes. Section A4 explains the detailed derivation of a set of laborious equations. Section A5 presents the approximate versions of the wave energy flux expression.

2

The present formulation does not impose any specific requirements for the choice of a phase averaging operator that will be explored by a series of model diagnoses in future studies. For example, the time evolution of oceanic upper-layer energy flux at the equator of the Indian Ocean has been shown without using a phase averaging operator in the Hovmöller diagram of Li and Aiki (2020). The atmospheric community may begin with adopting empirical knowledge derived from a series of previous model diagnosis studies using, for example, the wave activity flux.

3

The analytical solution of equatorial waves assumes meridionally modal structure. The analytical solution of off-equatorial waves assumes meridionally propagating structure. This yields the expressional difference of vertical WEFs in the present study. On the other hand, for the horizontal WEFs of equatorial waves and of off-equatorial waves, the two expressions have turned out to be identical to each other, which is not obvious in terms of mechanical interpretation given the difference of meridionally modal and propagating structures. The seamless nature of the horizontal WEFs is closely linked to the pseudostreamfunction φ inverted from the isentropic perturbation of EPV as shown in section 4. These issues may open up a new research theme for mechanical interpretation studies (Cai and Huang 2013; Durland and Farrar 2020) that have been dedicated to, so far, off-equatorial RWs in a height-independent shallow water framework.

Supplementary Materials

Save
  • Aiki, H., K. Takaya, and R. J. Greatbatch, 2015: A divergence-form wave-induced pressure inherent in the extension of the Eliassen–Palm theory to a three-dimensional framework for waves at all latitudes. J. Atmos. Sci., 72, 28222849, https://doi.org/10.1175/JAS-D-14-0172.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Aiki, H., R. J. Greatbatch, and M. Claus, 2017: Towards a seamlessly diagnosable expression for the energy flux associated with both equatorial and mid-latitude waves. Prog. Earth Planet. Sci., 4, 11, https://doi.org/10.1186/s40645-017-0121-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Baldwin, M. P., and Coauthors, 2001: The quasi-biennial oscillation. Rev. Geophys., 39, 179229, https://doi.org/10.1029/1999RG000073.

  • Brönnimann, S., 2007: Impact of El Niño–Southern Oscillation on European climate. Rev. Geophys., 45, RG3003, https://doi.org/10.1029/2006RG000199.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cai, M., and B. Huang, 2013: A new look at the physics of Rossby waves: A mechanical-Coriolis oscillation. J. Atmos. Sci., 70, 303316, https://doi.org/10.1175/JAS-D-12-094.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cassou, C., 2008: Intraseasonal interaction between the Madden–Julian oscillation and the North Atlantic Oscillation. Nature, 455, 523527, https://doi.org/10.1038/nature07286.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Durland, T. S., and J. T. Farrar, 2020: Another note on Rossby wave energy flux. J. Phys. Oceanogr., 50, 531534, https://doi.org/10.1175/JPO-D-19-0237.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eliassen, A., and E. Palm, 1960: On the transfer of energy in stationary mountain waves. Geophys. Publ., 22, 123.

  • Frederiksen, J. S., and P. J. Webster, 1988: Alternative theories of atmospheric teleconnections and low-frequency fluctuations. Rev. Geophys., 26, 459–494, https://doi.org/10.1029/RG026i003p00459.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fukutomi, Y., 2019: Tropical synoptic-scale waves propagating across the Maritime Continent and northern Australia. J. Geophys. Res. Atmos., 124, 76657682, https://doi.org/10.1029/2018JD029795.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fukutomi, Y., and T. Yasunari, 2002: Tropical–extratropical interaction associated with the 10–25-day oscillation over the western Pacific during the northern summer. J. Meteor. Soc. Japan, 80, 311331, https://doi.org/10.2151/jmsj.80.311.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gray, L. J., J. A. Anstey, Y. Kawatani, H. Lu, S. Osprey, and V. Schenzinger, 2018: Surface impacts of the quasi biennial oscillation. Atmos. Chem. Phys., 18, 82278247, https://doi.org/10.5194/acp-18-8227-2018.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harada, Y., and Coauthors, 2016: The JRA-55 Reanalysis: Representation of atmospheric circulation and climate variability. J. Meteor. Soc. Japan, 94, 269302, https://doi.org/10.2151/jmsj.2016-015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harada, Y., K. Sato, T. Kinoshita, R. Yasui, T. Hirooka, and H. Naoe, 2019: Diagnostics of a WN2-type major sudden stratospheric warming event in February 2018 using a new three-dimensional wave activity flux. J. Geophys. Res. Atmos., 124, 61206142, https://doi.org/10.1029/2018JD030162.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hendon, H. H., and M. L. Salby, 1996: Planetary-scale circulations forced by intraseasonal variations of observed convection. J. Atmos. Sci., 53, 17511758, https://doi.org/10.1175/1520-0469(1996)053<1751:pscfbi>2.0.co;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 2004: An Introduction to Dynamic Meteorology. 3rd ed. Academic Press, 531 pp.

  • Holton, J. R., and H.-C. Tan, 1980: The influence of the equatorial quasi-biennial oscillation on the global circulation at 50 mb. J. Atmos. Sci., 37, 22002208, https://doi.org/10.1175/1520-0469(1980)037<2200:TIOTEQ>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Horinouchi, T., F. Sassi, and B. A. Boville, 2000: Synoptic-scale Rossby waves and the geographic distribution of lateral transport routes between the tropics and the extratropics in the lower stratosphere. J. Geophys. Res., 105, 26 57926 592, https://doi.org/10.1029/2000JD900281.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 38, 11791196, https://doi.org/10.1175/1520-0469(1981)038<1179:TSLROA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huang, R., and F. Sun, 1992: Impacts of the tropical western Pacific on the East Asian summer monsoon. J. Meteor. Soc. Japan, 70, 243256, https://doi.org/10.2151/jmsj1965.70.1B_243.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ineson, S., and A. A. Scaife, 2008: The role of the stratosphere in the European climate response to El Niño. Nat. Geosci., 2, 3236, https://doi.org/10.1038/ngeo381.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kawahira, K., 1983: Global structures of stationary planetary waves in the middle atmosphere. J. Meteor. Soc. Japan, 61, 695716, https://doi.org/10.2151/jmsj1965.61.5_695.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kawatani, Y., M. Takahashi, K. Sato, S. P. Alexander, and T. Tsuda, 2009: Global distribution of atmospheric waves in the equatorial upper troposphere and lower stratosphere: AGCM simulation of sources and propagation. J. Geophys. Res., 114, D01102, https://doi.org/10.1029/2008JD010374.

    • Search Google Scholar
    • Export Citation
  • Kiladis, G. N., M. C. Wheeler, P. T. Haertel, K. H. Straub, and P. E. Roundy, 2009: Convectively coupled equatorial waves. Rev. Geophys., 47, RG2003, https://doi.org/10.1029/2008rg000266.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • King, M. P., I. Herceg-Bulić, I. Bladé, J. García-Serrano, N. Keenlyside, F. Kucharski, C. Li, and S. Sobolowski, 2018: Importance of late fall ENSO teleconnection in the Euro-Atlantic sector. Bull. Amer. Meteor. Soc., 99, 13371343, https://doi.org/10.1175/BAMS-D-17-0020.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kobayashi, S., and Coauthors, 2015: The JRA-55 Reanalysis: General specifications and basic characteristics. J. Meteor. Soc. Japan, 93, 548, https://doi.org/10.2151/jmsj.2015-001.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kosaka, Y., and H. Nakamura, 2006: Structure and dynamics of the summertime Pacific–Japan teleconnection pattern. Quart. J. Roy. Meteor. Soc., 132, 20092030, https://doi.org/10.1256/qj.05.204.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lau, W. K. M., D. E. Waliser, and P. E. Roundy, 2011: Tropical–extratropical interactions. Intraseasonal Variability in the Atmosphere-Ocean Climate System, W. K. M. Lau and D. E. Waliser, Eds., Springer, 497–512.

    • Crossref
    • Export Citation
  • Li, Z., and H. Aiki, 2020: The life cycle of annual waves in the Indian Ocean as identified by seamless diagnosis of the energy flux. Geophys. Res. Lett., 47, e2019GL085670, https://doi.org/10.1029/2019GL085670.

    • Search Google Scholar
    • Export Citation
  • Li, Z., H. Aiki, M. Nagura, and T. Ogata, 2021: The vertical structure of annual wave energy flux in the tropical Indian Ocean. Prog. Earth Planet. Sci., 8, 43, https://doi.org/10.1186/s40645-021-00432-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lin, J. L., and Coauthors, 2006: Tropical intraseasonal variability in 14 IPCC AR4 climate models. Part I: Convective signals. J. Climate, 19, 26652690, https://doi.org/10.1175/JCLI3735.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, M. S., 1964: On group velocity and energy flux in planetary wave motion. Deep-Sea Res., 11, 3542, https://doi.org/10.1016/0011-7471(64)91080-0.

    • Search Google Scholar
    • Export Citation
  • Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan, 44, 2543, https://doi.org/10.2151/jmsj1965.44.1_25.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nitta, T., 1970: A study of generation and conversion of eddy available potential energy in the tropics. J. Meteor. Soc. Japan, 48, 524528, https://doi.org/10.2151/jmsj1965.48.6_524.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nitta, T., 1987: Convective activities in the tropical western Pacific and their impact on the Northern Hemisphere summer circulation. J. Meteor. Soc. Japan, 65, 373390, https://doi.org/10.2151/jmsj1965.65.3_373.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ogata, T., and H. Aiki, 2019: The pathway of intraseasonal wave energy in the tropical Indian Ocean as identified by a seamless diagnostic scheme. SOLA, 15, 262267, https://doi.org/10.2151/sola.2019-047.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Orlanski, I., and J. Sheldon, 1993: A case of downstream baroclinic development over western North America. Mon. Wea. Rev., 121, 29292950, https://doi.org/10.1175/1520-0493(1993)121<2929:ACODBD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 1986: Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci., 43, 16571678, https://doi.org/10.1175/1520-0469(1986)043<1657:TDPOTQ>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ribstein, B., U. Achatz, and F. Senf, 2015: The interaction between gravity waves and solar tides: Results from 4-D ray tracing coupled to a linear tidal model. J. Geophys. Res. Space Phys., 120, 67956817, https://doi.org/10.1002/2015JA021349.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sato, K., S. Watanabe, Y. Kawatani, Y. Tomikawa, K. Miyazaki, and M. Takahashi, 2009: On the origins of mesospheric gravity waves. Geophys. Res. Lett., 36, L19801, https://doi.org/10.1029/2009GL039908.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Scaife, A. A., and Coauthors, 2016: Tropical rainfall, Rossby waves and regional winter climate predictions. Quart. J. Roy. Meteor. Soc., 143, 111, https://doi.org/10.1002/qj.2910.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Serra, Y. L., X. Jiang, B. Tian, J. Amador-Astua, E. D. Maloney, and G. N. Kiladis, 2014: Tropical intraseasonal modes of the atmosphere. Annu. Rev. Environ. Resour., 39, 189215, https://doi.org/10.1146/annurev-environ-020413-134219.

    • Crossref