Towards a seamlessly diagnosable expression for the energy flux associated with both equatorial and midlatitude waves
 Hidenori Aiki^{1, 2}Email author,
 Richard J. Greatbatch^{3, 4} and
 Martin Claus^{3}
DOI: 10.1186/s4064501701211
© The Author(s) 2017
Received: 28 June 2016
Accepted: 6 March 2017
Published: 31 March 2017
Abstract
For midlatitude Rossby waves (RWs) in the atmosphere, the expression for the energy flux for use in a model diagnosis, and without relying on a Fourier analysis or a ray theory, has previously been derived using quasigeostrophic equations and is singular at the equator. By investigating the analytical solution of both equatorial and midlatitude waves, the authors derive an exact universal expression for the energy flux which is able to indicate the direction of the group velocity at all latitudes for linear shallow water waves. This is achieved by introducing a streamfunction as given by the inversion equation of Ertel’s potential vorticity, a novel aspect for considering the energy flux. For ease of diagnosis from a model, an approximate version of the universal expression is explored and illustrated for a forced/dissipative equatorial basin mode simulated by a singlelayer oceanic model that includes both midlatitude RWs and equatorial waves. Equatorial Kelvin Waves (KWs) propagate eastward along the equator, are partially redirected poleward as coastal KWs at the eastern boundary of the basin, and then shed midlatitude RWs that propagate westward into the basin interior. The connection of the equatorial and coastal waveguides has been successfully illustrated by the approximate expression of the groupvelocitybased energy flux of the present study. This will allow for tropicalextratropical interactions in oceanic and atmospheric model outputs to be diagnosed in terms of an energy cycle in a future study.
Keywords
Group velocity Model diagnosis Equatorial Rossby waves Equatorial mixed Rossbygravity waves Equatorial inertiagravity waves Equatorial Kelvin waves Coastal Kelvin waves Midlatitude Rossby waves Midlatitude inertiagravity waves Tropicalextratropical interactionsIntroduction
A feature of many phenomena in the equatorial oceans is the role played by equatorial Kelvin waves (KWs), examples being El Niño Southern Oscillation (ENSO; Philander 1989) and the socalled Atlantic Niño (Merle 1980). KWs propagate along the equator and are partially redirected into coastal KWs at the eastern boundary, where they can influence offequatorial latitudes (e.g., Lübbecke et al. 2010) as well as excite extratropical Rossby waves (RWs) that subsequently propagate into the ocean interior (McPhaden and Ripa 1990; Isachsen et al. 2007). A striking example of this behavior is the equatorial basin mode (Cane and Moore 1981). For the gravest basin mode, the time scale is set by the time taken for an equatorial KW to propagate across the basin and for the reflected gravest long Rossby wave to return to the western boundary (that is 4L/c where L is the basin width and c is the phase propagation speed for KWs). In addition to waves that are trapped on the equator, equatorial basin modes also feature coastal KWs that propagate along the eastern boundary and extratropical RWs that are excited by these KWs and refocus on the equator, as described by Schopf et al. (1981). There is growing evidence that equatorial basin modes play an important role in equatorial ocean dynamics. For example, basin modes have been associated with the equatorial deep jets (Johnson and Zhang 2003; Brandt et al. 2011; Claus et al. 2016) and with the semiannual (Thierry et al. 2004) and annual cycles (Brandt et al. 2016) in the equatorial Atlantic. However, the energy cycle associated with equatorial basin modes has received little attention and is an important factor when considering the forced/dissipative basin modes that one can relate to observations. A particularly interesting example is the upward energy propagation associated with the Atlantic equatorial deep jets (Johnson and Zhang 2003; Brandt et al. 2011; Mathiessen et al. 2015). Yet, the detailed energy cycle associated with the jets remains largely unknown.
One way to approach the energy flux is to use ray theory. However, ray theory is linked to the dispersion relation of a single type of wave and is not suitable for investigating the sequential connection of different types of waves that are associated with a basin mode. Likewise, a Fourier analysis is not suitable for the investigation of waves near the coastal boundaries of the ocean. In fact, it is only for midlatitude inertiagravity waves (IGWs) that the flux of wave energy has been diagnosed from oceanic model output (Cummins and Oey 1997; Niwa and Hibiya 2004; Furuichi et al. 2008). On the other hand, in the atmospheric literature, the model diagnosis of pseudomomentum (or wave activity) flux has been more popular than the model diagnosis of the energy flux (Hoskins et al. 1983; Plumb 1986; Takaya and Nakamura 1997; Nakamura and Solomon 2011).
Here, we seek a general expression that can be used to diagnose the energy flux associated with linear shallow water waves at all latitudes from model output. This manuscript is organized as follows. First we provide the theoretical background. Then, we present an analytical investigation that leads to a general expression for the energy flux that can indicate the exact profile of the group velocity times wave energy for both equatorial and midlatitude waves. The utility of the universal expression of energy flux as a model diagnostic is illustrated for a forced/dissipative equatorial basin mode simulated by a singlelayer model. The model diagnosis is achieved by introducing an inversion for the linearized version of Ertel’s potential vorticity. This is a novel aspect for considering the energy flux in the presence of a coastal waveguide that connects the equatorial and midlatitude regions.
Theoretical background
List of symbols, where A ^{∗} and A are arbitrary quantities written dimensionally or nondimensionally, respectively
\(f^{*}=f^{*}_{0} + \beta ^{*} y^{*}\)  Coriolis parameter 

c ^{∗}  Speed of long gravity wave 
x,y,t  Cartesian coordinates wherein x and y increase eastward and northward 
〈 〈a,b〉 〉  Horizontal vector with eastward and northward components a and b 
V=〈 〈u,v〉 〉  Horizontal velocity vector 
∇≡〈 〈∂ _{ x },∂ _{ y }〉 〉  Horizontal gradient operator 
p  Pressure 
q≡v _{ x }−u _{ y }−y p  Linearized Ertel’s potential vorticity: \(q^{*} \equiv v^{*}_{x^{*}}u^{*}_{y^{*}}  (f^{*}/{c^{*}}^{2}) p^{*}\) 
φ  
φ ^{ a p p }  Solution of ∇^{2} φ ^{ a p p }−y ^{2} φ ^{ a p p }=q, see (26a) & (18a) 
(u ^{2}+v ^{2}+p ^{2})/2  Wave energy: (u ^{∗} ^{2}+v ^{∗} ^{2}+p ^{∗} ^{2}/c ^{∗} ^{2})/2 
θ=k x−ω t  Wave phase 
k  Zonal wavenumber 
ω  Wave frequency 
H ^{(n)}  Hermite polynomial, see endnote 1 
n  Meridional mode number of free equatorial waves 
\(\overline {A}\)  Phase average of A 
Characteristics of different waves at various latitudes
\(\overline {\mathbf {V}^{*}p^{*}}\) parallel to group velocity  q ^{∗}=0 (φ ^{app∗}=0)  

Equatorial Rossby wave  No  No 
Equatorial mixed Rossbygravity wave  Depends on frequency  No 
Equatorial inertiagravity wave  Roughly yes  No 
Equatorial Kelvin wave  Yes  Yes 
Coastal Kelvin wave  Yes  Yes 
Midlatitude Rossby wave  No  No (φ ^{app∗}≃p ^{∗}/f ^{∗}) 
Midlatitude inertiagravity wave  Yes  Yes 
where \(\nabla ^{*} \equiv \langle \!\langle \frac {\partial ~}{\partial {x^{*}}}, \frac {\partial ~}{\partial {y^{*}}}\rangle \!\rangle \) and the overbar symbol represents a phaseaverage operator (i.e., for a sinusoidal wave, \(\overline {A^{*}}=0\) for A ^{∗}=u ^{∗}, v ^{∗}, and p ^{∗}) or a lowpass time filter (for this reason, we retain the local time derivative in (3) to allow for slow time variations in the general case).
where \(\overline {\mathbf {V}^{*} p^{*}}\) is as in the gravity wave literature (i.e., V ^{∗} is the sum of the geostrophic and ageostrophic components of velocity). The second term in (6) is the additional rotational component required to reproduce the direction of the group velocity of midlatitude RWs (z is the upward vertical unit vector). In LonguetHiggins (1964), the second term of (6) has been expressed as \(\nabla ^{*} \times [f^{*} \overline {\psi ^{*2}}/2]\mathbf {z}\) where ψ ^{∗} is a streamfunction based on the assumption of horizontally nondivergent velocity. This assumption is hardly used in modern oceanography owing to the smallness of the deformation radius. In quasigeostrophic theory, ψ ^{∗}=p ^{∗}/f ^{∗} from which the connection with (6) is clear.
The question naturally arises as to whether or not it is possible to find a general expression for the additional rotational flux, R ^{∗}, that holds for waves at all latitudes and is such that the corresponding energy flux \(\overline {{\mathbf {V}}^{*} p^{*}} + \mathbf {R}^{*}\) always points in the direction of the group velocity and thus constitutes a general expression for the energy flux associated with waves at all latitudes. This is the main subject of the present study. In this study, we focus on wave types for which the group velocity has been well formulated in the literature/textbook, as listed in Table 2. Of particular interest is the energy flux associated with equatorial RWs given that the expression in (6) is singular at the equator. The assumption of horizontally nondivergent velocity in LonguetHiggins (1964) is also inappropriate for equatorial regions. In the next section, by investigating the analytical solution of equatorial waves, we derive an exact universal expression for the rotational flux which, after being added to \(\overline {{\mathbf {V}}^{*} p^{*}}\), is able to indicate the direction of the group velocity for linear waves at all latitudes.
Analytical investigation
We begin by revisiting analytical expressions for the profile of the energy flux associated with equatorial waves. This investigation allows us to derive an expression for the energy flux that points in the direction of the group velocity for waves at all latitudes.
Energy flux associated with equatorial waves
where \(\partial _{t} \equiv \frac {\partial ~}{\partial t}\), \(\nabla \equiv \langle \!\langle \frac {\partial ~}{\partial x}, \frac {\partial ~}{\partial y}\rangle \!\rangle \), and for A=u, v, or p, \(\overline {A}=0\) for sinusoidally varying waves.
where is wave amplitude and the symbol H ^{(n)} is the Hermite polynomial with n being the meridional mode number^{3}. The subscript θ represents partial differentiation in terms of the wave phase .
where 2ω ^{3}/k in the denominator has often been ignored in previous studies when focusing on lowfrequency equatorial waves (e.g., equatorial RWs; Gill 1982).
Identification of the additional rotational flux associated with equatorial waves
where the first equality has been derived using (11b)–(11c) and \(\overline {\sin \theta \sin \theta }=\overline {\cos \theta \cos \theta }\) and the second equality has been derived using (10). Note that it is the second of the two terms whose meridional integral is zero (noting that v and yv go to zero at large distances from the equator).
where the first and second equalities have been derived using (13), the third equality has been derived using \(\overline {\cos \theta \cos \theta }=\overline {\sin \theta \sin \theta }\), and the last equality has been derived using (11b)–(11c). The last line of (14e) has been written as the meridional gradient of scalar quantities. Thus, the meridional integral of (14e) vanishes for equatorial waves (with a meridionally decaying structure) and is consistent with (14a).
where the scalar quantity φ has been introduced. We have confirmed that, as long as φ is set by (15b), the meridional profile of the zonal energy flux, \(\overline {up}+ (\overline {p\varphi }/2 + \overline {u_{tt}\varphi })_{y}\), in (15a) is precisely identical to \((\partial \omega /\partial k) (\overline {u^{2}+v^{2}+p^{2}})/2\) for all types of equatorial waves in Figs. 2 and 3. Namely, all solid black lines in Figs. 2 and 3 may be drawn using either expression. As far as we know, (15a) and (15b) have not been mentioned in previous studies and therefore constitute a new result.
Inversion equations for Ertel’s potential vorticity
where ∇^{2}≡∂ _{ xx }+∂ _{ yy } is understood, the first line has been derived using (10), and the second line has been derived using (8) [i.e., q _{ t }=−ω q _{ θ }=−v and thus −ω q _{ θ θ }=ω q=−v _{ θ }]. The new Eq. (16) of EPV is the cornerstone of the present study, because it suggests a possibility for the scalar quantity φ to be estimated without using a Fourier analysis. This feature is important for identifying the direction of the energy flux of waves in the presence of coastal boundaries.
The additional rotational flux in (17b) corrects the profile of the energy flux, without affecting the divergence of the energy flux. The quantity φ ^{∗} in (17b) is the solution of the accurate streamfunction Eq. (17a) associated with EPV in a dimensional form. We note in passing that for zonally propagating equatorial waves, as given by (11a)–(11c), \(\overline {v^{*}p^{*}}\) vanishes owing to the phase relationship between v ^{∗} and p ^{∗} [see (11a) and (11c)] and the meridional component of the additional rotational flux, \((\overline {p^{*} \varphi ^{*}}/2 + \overline {u^{*}_{t^{*}t^{*}}\varphi ^{*}}/\beta ^{*})_{x^{*}}\), also vanishes.
Equatorial KWs
So far, we have not investigated the energy flux of equatorial KWs. Since KWs are gravity waves, \(\overline {\mathbf {V}^{*} p^{*}}\) becomes equal to the group velocity times wave energy. Namely, the additional rotational flux is absent. KWs are also characterized by q ^{∗}=0; hence, the EPV equation (17a) yields φ ^{∗}=0. The result is that, in the case of KWs, the expression for the energy flux, as given by (17b) reduces to \(\overline {\mathbf {V}^{*} p^{*}}\), which is consistent with the nature of gravity waves.
Boundary conditions and the connection to midlatitude regions
In a general situation in the ocean, waves propagating eastward along the equatorial waveguide are partially redirected poleward as KWs along the eastern boundary where they can shed RWs that then propagate westward into the ocean interior (Cane and Moore 1981; Philander 1989; Chelton and Schlax 1996; Isachsen et al. 2007).
We now investigate whether or not the set of (17a) and (17b) is applicable to offequatorial regions where smallamplitude perturbations are characterized by either midlatitude RWs or IGWs. For perturbations associated with midlatitude RWs, the solution φ ^{∗} of (17a) corresponds to the geostrophic streamfunction for which φ ^{∗}≃p ^{∗}/f ^{∗} is a reasonable approximation in an interior region (i.e., far from coastal boundaries), noting that ∇^{∗} ^{2} φ ^{∗} corresponds to \(v^{*}_{x^{*}}u^{*}_{y^{*}}\). The result is that the energy flux in (17b) automatically reduces to the expression of OS93 for midlatitude RWs^{5}. On the other hand, if perturbations associated with midlatitude IGWs are given, the inversion Eq. (17a) of EPV, which equals zero, yields, with φ ^{∗}=0 on the boundaries, φ ^{∗}=0 everywhere. Thus, the energy flux in (17b) automatically reduces to \(\overline {\mathbf {V}^{*} p^{*}}\) which represents the group velocity of midlatitude IGWs times wave energy. We conclude that the set of (17a) and (17b) can represent the exact profile of the group velocity times wave energy associated with both midlatitude IGWs and RWs, which may be reconfirmed using almost the same procedure as in the “Identification of the additional rotational flux associated with equatorial waves” section. See Appendix 1 for details.
Methods/Experimental
The rest of this manuscript presents an example illustrating the diagnosis of the energy flux from a model. To be useful for our discussion, the exact universal expression for both equatorial and midlatitude waves, as given by the set of (17a) and (17b), is hereafter referred to as the level0 energy flux. In practice, the level0 expression of the energy flux is not straightforward to compute from model output, since the secondorder time derivative term in (17a) makes it difficult to solve for φ ^{∗}.
which we refer to as the level2 expression for the energy flux. As shown by the solid blue lines in Figs. 2 and 3, the level2 expression provides an approximation for the groupvelocitybased energy flux of both low and highfrequency equatorial waves, although there can be some error. Further discussion of the level2 approximation is given in Appendices 2 and 3 where it is noted that the level2 approximation is comparable in accuracy to the pseudomomentum (or waveactivity) flux used in previous studies (Randel and Williamson 1990; Brunet and Haynes 1996; Fukutomi and Yasunari 2002; Wakata and Kitaya 2002; Kawatani et al. 2010).
List of energy flux vectors and EPVbased streamfunctions in dimensional form and their location in the text and figures
Approx.  Energy flux vector  Equation  

Level0  \(\overline {\mathbf {V}^{*} p^{*}}  \nabla ^{*} \times [(\overline {p^{*} \varphi ^{*}})/2+ (\overline {u^{*}_{t^{*}t^{*}}\varphi ^{*}})/\beta ^{*}]\mathbf {z}\)  Solid black  –  
Level1  \(\overline {\mathbf {V}^{*} p^{*}}  \nabla ^{*} \times [(\overline {p^{*} \varphi ^{\mathrm {app*}}})/2+ (\overline {u^{*}_{t^{*}t^{*}}\varphi ^{\mathrm {app*}}})/\beta ^{*}]\mathbf {z}\)  Dashed orange  –  
Level2  \(\overline {\mathbf {V}^{*} p^{*}}  \nabla ^{*} \times (\overline {p^{*} \varphi ^{\mathrm {app*}}})/2\mathbf {z}\)  Solid blue  (c)  
QG  \(\overline {\mathbf {V}^{*} p^{*}} \nabla ^{*} \times [\overline {{p^{*}}^{2}}/(2f^{*})]\mathbf {z}\)  –  (b)  
fPlane  \(\overline {\mathbf {V}^{*} p^{*}}\)  Dashed green  (a)  
Definition of EPVbased streamfunctions  
\(\nabla ^{*2} \varphi ^{*}  (f^{*}/c^{*})^{2} \varphi ^{*}  (3/c^{*2})\varphi ^{*}_{t^{*} t^{*}} = q^{*}\),  
∇^{∗2} φ ^{app∗}−(f ^{∗}/c ^{∗})^{2} φ ^{app∗}=q ^{∗}, 
Model setup
To illustrate the importance of dissipation for explaining the observed crossequatorial width of the equatorial deep jets, Greatbatch et al. (2012, hereafter G12) have simulated a forced/dissipative basin mode solution using a singlelayer reducedgravity linear model. The model is set up in spherical coordinates, with a rectangular domain in latitude/longitude space of roughly the same width as the Atlantic Ocean at the equator (that is 55° in longitude) and reaching to 10°N/S on either side of the equator^{7}. All lateral boundaries are closed. In both G12 and Claus et al. (2014, hereafter C14), the model has been forced by an idealized oscillatory forcing with a period of 4.5 years in the zonal momentum equation to mimic the forcing of the jets, together with a lateral mixing of momentum that provides dissipation. [See Ascani et al. (2015) for a discussion on the forcing of the equatorial deep jets, the details of which are not important here]. It should be noted that 4.5 years is roughly the time taken for an equatorial KW and the reflected long gravest equatorial RW, to travel across the basin for the vertical mode that is closest to resonance. As noted in G12 and C14, the (westward) propagation speed of equatorial long RWs is three times less than the (eastward) propagation speed of equatorial KWs [see the dispersion relation (12)].
Parameters in the model experiment of the present study
Long gravity wave speed  c ^{∗}=0.17 m/s 
Equatorial deformation radius  \(\sqrt {c^*/\beta ^*} = 87~\text {km}\) 
Equatorial inertial period  \(2\pi /\sqrt {c^* \beta ^*}=\) 37 days 
Forcing period  T ^{∗}= 4.5 years 
Forcing amplitude  10^{−10} m/s ^{2} 
Forcing area  Full domain 
Domain size  55 ° (zonal) × 20 ° (meridional) 
Forcing Froude number  (0.0023 m/s)/c ^{∗}=0.014 
Horizontal resolution  0.1 ° (zonal) × 0.1 ° (meridional) 
Lateral eddy viscosity  10 m ^{2}/s 
Results and discussion
From Figs. 6 c and 7 c, it is clear that when the set of Eqs. (18a), (18c) and (17c) is used to estimate the energy flux, the westward flux associated with the offequatorial RWs is part of a recirculation of energy in the eastern part of the basin (Fig. 7 c) with eastward energy flux along the equator and westward energy flux off the equator. The eastward flux along the equator in Figs. 6 c and 7 c is in the opposite direction to the westward \(\overline {\mathbf {V}^* p^*}\) flux in Figs. 6 a and 7 a along the equator in the same region. This indicates the role of the rotational flux contribution in (18c) which counters the westward \(\overline {\mathbf {V}^*p^*}\) flux along the equator. This westward flux is associated with the equatorial RWs but represents an overestimation of the energy flux associated with these waves (see Fig. 2). When the rotational flux is added, what emerges is the eastward flux associated with the KW which, in turn, leads to a poleward flux arising from KWs propagating along the eastern boundary and, in turn, leads to the westward flux associated with the offequatorial RWs that are excited at the eastern boundary. Here, in terms of the transfer of wave energy, the equatorial waveguide has been connected to the eastern coastal waveguide and, in turn, to the basin interior at offequatorial latitudes, which is at the heart of the present study.
Finally, we note that the forcing period of T ^{∗}=4.5 years is much longer than the equatorial inertial period of \(2\pi /\sqrt {c^*\beta ^*}=37\) days. It can be said that the simulated equatorial basin mode consists of lowfrequency equatorial waves, as in Fig. 2, and midlatitude RWs. We recall the small difference between the solid blue and dashed orange lines in Fig. 2, the former and the latter of which may be written as \(\overline {u^* p^*} + (\overline {p^* \varphi ^{\mathrm {app*}}}/2)_{y^*}\) and \(\overline {u^* p^*} + (\overline {p^* \varphi ^{\mathrm {app*}}}/2+\overline {u^*_{t^* t^*}\varphi ^{\mathrm {app*}}}/\beta ^*)_{y^*}\), respectively, in a dimensional form (see level2 and level1, respectively, in Table 3). Since arrows in Figs. 6 c and 7 c have been plotted using the expression which corresponds to the solid blue line in Fig. 2, we have checked for any improvement by using the expression which corresponds to the dashed orange lines in the same figure. The checking has been done by comparing the distribution of \(\overline {p^* \varphi ^{\mathrm {app*}}}/2\) and \(\overline {u^*_{t^*t^*} \varphi ^{\mathrm {app*}}}/\beta ^*\), from which we have learned that the latter quantity (not shown) is three orders of magnitude smaller than the former. Thus, we conclude that, in the diagnosis of the simulated basin mode, the expression of the energy flux, as given by (18c), has provided a nice approximation for the group velocity times wave energy.
Conclusions
In previous studies of the ocean, the energy flux of waves in model output has been diagnosed using \(\overline {\mathbf {V}^* p^*}\), where V ^{∗} is the horizontal component of velocity perturbation and p ^{∗} corresponds to the pressure perturbation. This is appropriate for understanding the energy flux associated with midlatitude inertiagravity waves (IGWs). For midlatitude Rossby waves (RWs), however, the direction of \(\overline {\mathbf {V}^* p^*}\) differs from the group velocity and hence the energy flux, by a rotational vector flux with zero divergence. The rotational flux to be added to \(\overline {\mathbf {V}^* p^*}\) for estimating the group velocity of midlatitude RWs has previously been derived using quasigeostrophic equations and is singular at the equator.
By investigating the analytical solution of both equatorial waves (“Analytical investigation” section) and midlatitude waves (Appendix 1), we have derived an exact universal^{9} expression for the rotational flux which, after being added to \(\overline {\mathbf {V^*}p^*}\), is able to indicate the profile of the group velocity times wave energy for linear waves at all latitudes. This is what we call the level0 expression of the energy flux. The level0 energy flux is written using the solution φ ^{∗} of (17a), previously unmentioned in the literature, which we refer to as the accurate streamfunction associated with Ertel’s potential vorticity (EPV). Equation (17a) is the cornerstone of the present study, because it suggests a possibility for the energy flux to be estimated (i) without using a Fourier analysis nor ray theory and (ii) in the presence of coastal boundaries, which will allow for tropicalextratropical interactions in model output to be diagnosed in terms of an energy cycle in a future study. Presently, the level0 energy flux is not practical for use as a model diagnostic, since the secondorder time derivative term in (17a) makes it difficult to solve for φ ^{∗}. Thus, we hope that a future study is able to develop a numerical algorithm to solve (17a) for φ ^{∗}. We also note the need to extend the theory to a continuously stratified ocean and also to test out the theory in the presence of a sheared mean flow, both of which topics await a future study. This is a new step from the recent understanding of energetics in the atmosphere and ocean that had been focused on, for example, the global mapping of energy conversion rates associated with various physical processes (e.g., baroclinic and barotropic instabilities) and external forcing (Iwasaki 2001; Aiki and Richards 2008; Zhai et al. 2012).
The potential of our analysis as a model diagnostic is illustrated in the present study for a forced/dissipative equatorial basin mode simulated by a singlelayer model. The model result includes both midlatitude RWs (maintained by coastal KWs propagating poleward along the eastern boundary) and equatorial RWs (maintained by the reflection of equatorial KWs at the eastern boundary). We have used approximate expressions for the energy flux (what we call the level1 and level2 energy fluxes) that is based on the inversion equation (18a) of EPV and which is shown to be good approximations to the level0 expression in the case of the model run being considered. Since (18a) is seamlessly solvable at all latitudes with φ ^{app∗}=0 at coastlines, the source of the westward energy flux of midlatitude RWs in the model output has been successfully illustrated in the present study. To our knowledge, this is the first attempt to diagnose the energy cycle of a tropicalextratropical interaction associated with the connection of the equatorial and coastal waveguides.
Endnotes
^{1} While the energy flux of waves at all latitudes is considered in the present study, the pseudomomentum (or waveactivity) flux of waves at all latitudes is considered in Aiki et al. (2015, hereafter ATG15). Both the formulations of the present study and ATG15 may be reproduced even if a spherical coordinate system is used. The use of a Cartesian horizontal coordinate system in both the present study and ATG15 is for the purpose of simplicity, which will allow for the results of the two studies to be linked in a future study. A related discussion appears in Appendix 3.
^{2} What we call pressure, energy, and momentum in the present study are actually dynamic pressure, energy density, and momentum density, respectively, following ATG15.
^{3} d H ^{(n)}/d y=2n H ^{(n−1)}, H ^{(n+1)}=2y H ^{(n)}−2n H ^{(n−1)}, H ^{(0)}=1, H ^{(1)}=2y, H ^{(2)}=4y ^{2}−2, H ^{(3)}=8y ^{3}−12y, H ^{(4)}=16y ^{4}−48y ^{2}+12.
^{4} The factor ∂ ω/∂ k to calculate the energy flux is added in (14e).
^{5} The second term in the square brackets of (17b) vanishes as \(\overline {u^{*}_{t^{*}t^{*}}\varphi ^{*}}\simeq \overline {(p^{*}_{y^{*} t^{*} t^{*}}/f^{*})(p^{*}/f^{*})}=0\) where the phase relationship of plane waves is understood.
^{6} We use the term “diagnosable” to indicate that the quantity is readily estimated from quantities in model output without relying on a Fourier analysis.
^{7} In a related paper, Claus et al. (2014) also used this solution to investigate the influence of the barotropic mean flow on the Atlantic equatorial deep jets. The Atlantic equatorial deep jets are resonant with the gravest basin mode for a highorder baroclinic mode (typically the 15th vertical normal mode) and consist of vertically stacked zonal jets that oscillate at a given depth with a period of around 4.5 years.
^{8} This is lower than the value recommended by G12 for capturing the observed width of the deep jets but is chosen here since it is not so large as to prevent focusing of RWs on the equator. In the inviscid solution of Cane and Moore (1981), there is a singularity on the equator at the center of the basin due to RW focusing as described by Schopf et al. (1981).
^{9} In the present manuscript, we have used the term “exact” to refer to the level0 expression, in contrast to approximate expressions (i.e., level1 and 2). Likewise, we have used the term “universal” to indicate the ability to handle all wave types in Table 2, for which the group velocity has been well formulated in the literature/textbook.
^{10}Although it is not in the list of wave types in Table 2, IGWs on a midlatitude βplane may be characterized as α≪1,δ ^{2}≤1,γ ^{2}<1 where α≪1 corresponds to (19b). Thus, the net content in the square brackets on the last line of (24c) becomes O(1). Given α in front of \(c^* \overline {v^* v^*}\) on the last line of (24c), we may justify (23d) for IGWs on a midlatitude βplane. It can be said that the right hand side of (24c) becomes significantly nonzero when the assumption of plane waves in the meridional direction becomes inconsistent (Anderson and Gill 1979).
^{11}While the pseudomomentum flux itself \((\overline {E^*v^*v^*})\) is diagnosable from model output, the pseudomomentumfluxbased expression of the energy flux \((\overline {E^*v^*v^*})\omega ^*/k^*\) is not easily diagnosable from model output because of multiplication by the phase speed (see Appendix 3 for details).
Appendix 1
Is the streamfunction Eq. (17a) associated with EPV applicable to midlatitude waves?
which is a universal expression for the dispersion relation of the various types of waves in midlatitude regions. For example, substitution of β ^{∗}=0 to (21) yields a classical dispersion relation for midlatitude IGWs (i.e., waves on an fplane), and substitution of ω ^{∗2}≪c ^{∗2} k ^{∗2} to (21) yields a classical dispersion relation for midlatitude RWs.
where the first equality has been derived using (20b) and the second equality has been derived using (2) [i.e., q t ^{∗}∗=−ω ^{∗} q θ∗=−β ^{∗} v ^{∗} and thus −ω ^{∗} q θ θ∗=ω ^{∗} q ^{∗}=−β ^{∗} v θ∗]. As far as we know, the set of (23a) and (23c) has not been mentioned in previous studies for midlatitude waves and has turned out to be almost the same as the set of (17b) and (17a) that has been derived for equatorial waves.
It can be said that the last line of (24c) represents the contribution of higher order terms in an asymptotic expansion based on α, δ, and γ. This contribution should not be confused with the universal expression of the additional rotational flux which has already been clarified at (23a) and (23d). It should be also noted that the net content within the square brackets on the last line of (24c) is nondimensional, for which we shall make scale analysis in the next paragraph.
Thus, the last line of (24c) vanishes, which justifies (23d) for IGWs on an fplane^{10}.
To summarize, the streamfunction Eq. (17a) associated with EPV and the universal expression of the additional rotational flux in (17b) applies to both midlatitude and equatorial waves, in particular for wave types considered in the present study, as listed in Table 2.
Appendix 2
Approximate expressions for the energy flux
The exact profile of the group velocity times wave energy is given by the set of (15a) and (16), which is what we call the level0 energy flux. Owing to the last term on the left hand side of (16) that contains the secondorder partial differentiation with respect to time, the procedure of inverting EPV, without using a Fourier analysis, is still complicated.
as shown by the dashed orange lines in Fig. 2 for lowfrequency equatorial waves (e.g., equatorial RWs) and in Fig. 3 for highfrequency equatorial waves (e.g., equatorial IGWs). Since this is an analytical investigation, we have used φ ^{app}=−v _{ θ }/(k−ω ^{3}) which has been derived from the EPV inversion Eq. (26a) with the use of the characteristic Eq. (10). All panels in Fig. 2 show a nice agreement between the dashed orange line given by (26b) and the solid black line, \((\partial \omega /\partial k)(\overline {u^2+v^2+p^{2}})\). By contrast, all panels in Fig. 3 show a finite disagreement between the dashed orange line given by (26b), \(\overline {up}+(\overline {p\varphi ^{\text {app}}}/2+\overline {u_{tt} \varphi ^{\text {app}}})_{y}\), and the solidblack line, \((\partial \omega /\partial k)(\overline {u^2+v^2+p^{2}})\).
where φ ^{app}=−v _{ θ }/(k−ω ^{3}) is the solution of (26a). The profile of (26c) is shown by the solid blue lines in Figs. 2 and 3 for low and highfrequency equatorial waves, respectively. This expression provides what we think is a potentially useful approximation for the group velocity times wave energy (the solid black lines) for all types of equatorial waves, as we show in the “Methods/Experimental” section.
In the present study, (26b) and its vector and dimensional form (18b) are referred to as the level1 energy flux. Likewise, (26c) and its vector and dimensional form (18c) are referred to as the level2 energy flux.
Why do we appreciate the level2 energy flux regardless of the error? An expression for pseudomomentum (or waveactivity) flux has long been used for the model diagnosis of the direction of the group velocity of waves in the atmosphere (and also the ocean), including in lowlatitude regions (Ripa 1982; Hoskins et al. 1983; Plumb 1986; Haynes 1988; Randel and Williamson 1990; Brunet and Haynes 1996; Fukutomi and Yasunari 2002; Wakata and Kitaya 2002; Kawatani et al. 2010). Using the analytical solution of equatorial waves, we have calculated the profile of the traditional pseudomomentum flux^{11} times the phase velocity of waves (see Appendix 3), as shown by the purple dots in Figs. 2 and 3. Interestingly, for lowfrequency waves, the profile of the pseudomomentumfluxbased expression (the purple dots) is almost the same as that of the level2 energy flux (the blue solid line). On the other hand, for highfrequency waves, the profile of the pseudomomentumfluxbased expression (the purple dots) is similar to that of the level1 energy flux (the orange dashed line) and quite different from the exact, level0 energy flux to which the level2 energy flux is a better approximation. Thus, the level2 energy flux is, in general, an improvement on the traditional model diagnosis of group velocity based on the pseudomomentum flux.
Concerning extension to midlatitude waves, both the level1 and level2 energy fluxes satisfy all conditions noted in the last paragraph of the “Boundary conditions and the connection to midlatitude regions” section. Note that the inversion Eq. (18a) of EPV is seamlessly solvable at all latitudes with the boundary condition of φ ^{app∗}=0. To summarize, the set of (18a) and (18c) [together with the boundary condition (17c)]—what we call the level2 expression—originates from a tradeoff between mathematical exactness and practical accessibility. The mathematical exactness for retrieving the group velocity of equatorial waves times wave energy has been achieved by the set of (17a) and (17b)—what we call the level0 expression. However, its accessibility is harmed by the secondorder time derivative term in the streamfunction equation (16) associated with EPV. On the other hand, concerning the practical accessibility, the set of (18a) and (18c)—the level2 expression—has the advantages that (i) it is seamlessly solvable at all latitudes and (ii) it provides a unified expression for all types of waves with which to estimate the direction of the group velocity. We have noted, for equatorial waves, that the profile of the level2 energy flux is somewhat better than that of the traditional pseudomomentum flux. It should be also noted that the energy flux given by (18c) satisfies the boundary condition of no flux through coastlines [using (17c)], an issue not considered in previous studies for the pseudomomentum flux. With these requirements in mind, we hope that future studies can lead to either an improved approximation or a numerical algorithm for the level0 energy flux.
Appendix 3
Similarity between the level2 energy flux of this study and the pseudomomentum flux in previous studies
which is a prognostic equation for the wave energy wherein the zonal component of the flux is proportional to that in the IB pseudomomentum equation (27a).
It is easy to expect that the expression of the flux in (29) can indicate the direction of the group velocity of midlatitude RWs and IGWs (Hoskins et al. 1983; Plumb 1986; Haynes 1988). For equatorial waves, here, we investigate the meridional profile of \((\overline {E^*v^*v^*})\omega ^*/k^*\) as shown by the purple dots in Figs. 2 and 3 for low and highfrequency waves, respectively. For lowfrequency waves (Fig. 2), the meridional profile of \((\overline {E^*v^*v^*})\omega ^*/k^*\) (the purple dots) is almost the same as that of the level2 energy flux (the blue solid line), showing that the level2 energy flux and the IB flux are closely related. For highfrequency waves (Fig. 3), the meridional profile of \((\overline {E^*v^*v^*})\omega ^*/k^*\) (the purple dots) is nearly the same as that of the level1 energy flux (the orange dashed line), indicating that the level2 energy flux is somewhat better than the IB flux.
In fact, without relying on the level0 expression, we have arrived at the level2 expression of the energy flux by extending the investigation of ATG15 concerning the algebraic structure of the IB flux (to be explained in a future study). ATG15 have addressed the importance of a waveinduced scalar quantity and symbolized it as Λ: it vanishes for midlatitude IGWs (i.e., waves with no perturbation of EPV) and becomes nonzero for midlatitude RWs (i.e., wave with a perturbation of EPV). Here, we suggest that \(\overline {\Lambda }=(\overline {p^* \eta ^* })_{y^*}/2\) is closely linked to \((\overline {p^* \varphi ^{\mathrm {app*}}})_{y^*}/2\) in the present study (η ^{∗} is meridional displacement). This is why the level2 expression for the energy flux in the present study can indicate the direction of the group velocity of different types of waves, an issue we shall discuss in a future study.
Note that the IB flux in (27a) has already been used for the model diagnosis of waves in lowlatitude regions (Randel and Williamson 1990; Brunet and Haynes 1996; Fukutomi and Yasunari 2002; Wakata and Kitaya 2002; Kawatani et al. 2010). We suggest that, despite the certain inaccuracy associated with equatorial waves as compared with the level0 expression, the level2 expression of the energy flux in the present study will be at least as useful as the IB flux which has long been used in the atmospheric (and oceanic) literature. For oceanic applications, the level2 energy flux brings two new advantages over the IB flux: (i) the level2 energy flux satisfies a nonormalflux boundary condition at coastlines, and (ii) the wave energy is a signdefinite quantity while the IB pseudomomentum is not.
Overall, we address the balance of (i) model accessibility, (ii) unified treatment for different types of waves, (iii) mathematical accuracy, and (iv) boundary conditions at coastlines. With these requirements in mind, we hope future studies can lead to either an improved approximation or a numerical algorithm for the level0 energy flux, wherein the profile of the IB flux will provide a reference for accuracy because the IB flux has long been used in previous studies.
Abbreviations
 EPV:

Ertel’s potential vorticity
 IGW:

Inertia gravity wave
 KW:

Kelvin wave
 RGW:

Mixed Rossbygravity wave
 RW:

Rossby wave
Declarations
Acknowledgements
This manuscript has been improved by comments from two anonymous reviewers. HA thanks Paal Erik Isachsen for the helpful discussions and RJG is grateful to the GEOMAR for ongoing support.
Funding
This study was supported by JSPS KAKENHI Grant Numbers 26400474 and 15H02129 and also by the Deutsche Forschungsgemeinschaft as part of the Sonderforschungsbereich 754 “Climate  Biogeochemistry Interactions in the Tropical Ocean,” by the German Federal Ministry of Education and Research as part of the cooperative project SACUS (03G0837A), and by the European Union 7th Framework Programme (FP7 20072013) under grant agreement 603521 PREFACE project.
Authors’ contributions
HA proposed the topic and performed the analytical investigation. RJG helped write the manuscript. MC helped with the numerical investigation. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interest.
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