Ye and Aiki ﻿Progress in Earth and Planetary Science (2024) 11:29
https://doi.org/10.1186/s40645-024-00630-1

RESEARCH ARTICLE

Progress in Earth andPlanetary Science

Open AccessWind stress curl as a driving force of annual
waves in the upper ocean for interpreting
energetics at all latitudes
Kaiwen Ye1*   and Hidenori Aiki2,3

Abstract
The dynamics of waves and eddies in the upper ocean plays an important role in the climate variation of tropicaland subtropical regions. Previous diagnoses for annual Rossby waves in oceanic model outputs manifested zonally alternating signals (ZASs) in the time-averaged distributions of wind input as well as pressure-flux divergenceterms in the budget equation of wave energy. This is the case when the annual mean of the wind input is estimatedas the inner product of simulated velocity vector and wind stress vector in previous studies. The present study proposes a new mathematical expression for estimating the wind input that is analogous to one derived from the quasigeostrophic potential vorticity equation. Namely, the wind input is estimated as the negative of the productof pseudo-streamfunction and wind stress curl, the latter of which is associated with the horizontal divergenceof Ekman velocity. This can be interpreted as replacement of kinetic energy input with gravitational potential energy
input. Pseudo-streamfunction in the present study is inverted from Ertel’s potential vorticity anomaly and is seamlessly available at all latitudes. This contrasts with the quasi-geostrophic streamfunction which is singular at the equator. The new expression enables reducing ZASs in the horizontal distributions of both wind input and pressure-flux
divergence terms, without harming the qualitative advantage of energy flux vectors to indicate the group velocity
of waves at all latitudes.
Keywords Wave energy, Annual wave, Rossby wave, Wind input, Zonally alternating signals, Pseudo-streamfunction
1 Introduction
Variations of surface currents in the ocean are influenced
by both intrinsic dynamics and wind forcing. The inputof kinetic energy through wind forcing plays a significant role in driving ocean currents and turbulent mixing within the surface mixed layer (Ferrari and Wunsch2009). The spatial variability of winds results in Ekman*Correspondence:
Kaiwen Ye
yekaiwen1998academic@gmail.com
1
Graduate School of Environmental Studies, Nagoya University, Nagoya,Japan
2
Institute for Space–Earth Environmental Research, Nagoya University,
Nagoya, Japan
3
Application Laboratory, Japan Agency for Marine–Earth Scienceand Technology, Yokohama, Japanpumping and suction, which affect the vertical motionof water below the surface mixed layer of the ocean.
On the other hand, temporal variations in wind forcing
generate both inertia-gravity waves and planetary waves
within the ocean. The budget of mechanical energy in theupper layer of the global ocean was estimated in Lueck
and Reid (1984). They considered the sources and sinksof energy, including wind input, heat forcing, and viscous
dissipation. Their analysis revealed that the wind input is100–500 times greater than the input of available gravitational potential energy resulting from heat forcing andprecipitation/evaporation.
Previous studies put forward two mathematical expressions for estimating the energy input from wind forcing.
One is the classical expression (Wunsch 1998) which isbased on the budget of kinetic energy to calculate the© The Author(s) 2024, corrected publication 2024. Open Access This article is licensed under a Creative Commons Attribution 4.0
International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you
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were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicatedotherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To
view a copy of this licence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/.

Ye and Aiki ﻿Progress in Earth and Planetary Science (2024) 11:29

wind input by taking the dot product of surface velocity vector and wind stress vector. Using this expression,
Scott and Xu (2009) estimated that the global energy
input from wind to geostrophic flows at the sea surface
was 0.85–1.0 TW (1 TW = ­1012 watts). This may be compared with the result of Alford (2001) who estimated thatthe energy input from wind forcing to near-inertial waves
in the upper ocean amounted to about 0.29 TW. Wangand Huang (2004) estimated the wind energy input tothe Ekman layer as 3 TW. Xu et al. (2016) used the classical expression to estimate the wind input to mesoscaleeddies in the ocean, revealing a significant suppression of oceanic eddies by atmospheric winds. Anotherapproach for estimating the wind input involves quasigeostrophic dynamics. In this expression, the wind input
is calculated as the negative of the product between thequasi-geostrophic streamfunction (whose vertical variation is associated with buoyancy) and the wind stresscurl (which captures the horizontal divergence of Ekman
velocity). This second expression was used in some studies such as the paper by Berloff and McWilliams (1999a).
They investigated the spatial and temporal variability of
mid-latitude oceanic gyres generated by steady winds in
1.5-layer and 2-layer quasi-geostrophic models to study amarginal stability problem, highlighting the instability ofsteady western boundary currents at moderate Reynolds
numbers.
As a first step, the present study focuses on climatological wind forcing on ocean currents. Many studies noted
the effect of wind stress curl on annual baroclinic Rossby
waves (RWs) originating from eastern boundary regionsat mid-latitudes. Krauss and Wuebber (1982) suggested
that wind stress curl in the eastern boundary regions of
the North Atlantic Ocean effectively drives RWs. Cummins et al. (1986) investigated how RWs are influenced by
the annual cycle of wind stress curl in the North Pacific
Ocean. They noted clear wave crests and troughs in
the anomaly field of sea surface height (SSH) in regions
between 20° and 44° N, indicating that RWs originatedfrom the eastern boundary regions. While the crests andtroughs of SSH near the eastern boundary were aligned
parallel to the meridional direction, a southeast-northwest inclination was exhibited further away from theboundary. As noted by Schopf et al. (1981), the alternation in direction could be attributed to variations inwave speed with latitude depending on the deformation
radius change. The presence of high meridional wavenumbers contributes to the generation of meridionalgroup velocity. In the Southern Hemisphere, Reason et al.
(1987) demonstrated using a numerical experiment thatannual RWs are generated by the maxima of wind stresscurl in three regions at 25° S or 38° S around the easternboundary.

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We note that the estimation of wind input rates for oceanic annual waves, as demonstrated in the Indian Ocean
by Li and Aiki (2020), exhibited zonally alternating signals (referred to as ZASs in the present manuscript)that can complicate the understanding of the annual
mean energy budget. These ZASs occur near the easternboundary when the period of RWs is close to that of wind
forcing, which is often unavoidable in model experiments
using annual wind forcing. The objective of this study is
to investigate the presence of ZASs in wind input acrossother oceans, explore the factors contributing to ZASs,
and propose a method to mitigate their effects for a morecomprehensive understanding of upper ocean dynamics. This investigation will be valuable for future studies
aiming to gain insight into air-sea interactions from anenergy perspective. The structure of this manuscript is
as follows. In Sect. 2, we present the results of a modelexperiment utilizing a shallow-water equation systemassociated with the first baroclinic mode forced by climatological wind forcing in the Pacific Ocean. In Sect. 3,
we examine mathematical expressions for estimating
wind input, modifying the seamless diagnostic scheme tofacilitate the analysis of annual wave energy transfer andbudget in the upper layer of the Pacific Ocean. Finally,
Sect. 4 provides a summary of our findings and highlights
directions for future research.

2 Materials and methods
We have performed a 20-year experiment in the Pacific
Ocean using a shallow water equation system associated with the first baroclinic mode. Investigation for thesecond and greater baroclinic modes should be done ina future study following the result of the present study.
A total of 2400 snapshots have been stored in 20 years
with an interval of 3.04 (= 365/120) days. The model integration has reached a climatological equilibrium within
20 years, and all 120 snapshots in the last year have beenused in the following investigation. The present study
has used the monthly climatology of wind forcing withsubtracting a time mean component, as in Li and Aiki
(2020), to investigate the dynamics of annual waves. We
shall show for the Pacific Ocean that ZASs appear in theenergy input by wind forcing.
2.1 Model setup

The model domain of the present study is from 50°
S to 60° N and from 120° E to 70° W with realistic coastlines. The model equations are discretized inspherical coordinates using Arakawa’s C-grid with a halfdegree resolution in both the zonal and meridional directions. Basic equations in spherical coordinates are shownin Appendix A. All equations in the main body of thepresent manuscript are written in Cartesian coordinatesYe and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

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for simplicity, where the variables x and y denote thezonal and meridional directions, respectively, with values
increasing eastward and northward. We have simulated
the time evolution of zonal velocity u, meridional velocity
v , and layer thickness h using a shallow-water equation
system to read,
x
τwind
c2 ∂h
∂u
u
∂t − fv + H1 ∂x = H1 ρ0 + F ,
y

τwind
c2 ∂h
∂v
v
∂t + fu + H1 ∂y = H1 ρ0 + F ,∂h∂t + H1∂u∂v∂x + ∂y= F h,

(1a)
(1b)
(1c)

where the symbol f is the Coriolis parameter. The symbols F u and F v in Eqs. (1a) and (1b), respectively, areeddy viscosity terms as represented using the Smagorinyxsky et al. (1965) scheme. The symbols τwindand τwind in
Eqs. (1a) and (1b) represent climatological wind stressanomalies in the zonal and meridional directions, respectively, in the Pacific Ocean. These have been derived fromthe monthly climatology of ECMWF ORAS3 wind stressin 1959–2009. Here we have subtracted the time mean
component of wind stress to yield the model forcing. The
symbol F h in Eq. (1c) represents forcing for layer thickness and is set to zero in the model setup. This thickness
forcing is revisited later in the manuscript to indicate
expressions for the source/sink of gravitational potential
energy.
Gravity wave speed c in Eqs. (1a) and (1b) has been estimated to be 2.71 m/s for the first baroclinic mode using
the vertical profiles of annual mean salinity and temperature from the World Ocean Atlas (WOA). In Eqs. (1a)–(1c), the symbol H1 = 1107.5 m is the reference thickness
associatedwith the first baroclinic mode, estimated as√
H1 = HmixαHbottom based on the vertical mode decomposition. The non-dimensional parameter, α = 0.39, hasbeen used to represent the wind-induced momentum
input to the mixed-layer for the first baroclinic mode. See
Appendix A in Li and Aiki (2020) for the details of thevertical mode decomposition. The climatological annual
mean of the mixed-layer depth, Hmix = 45.5 m, has been

estimated based on a temperature difference of 0.2 °C using
Argo data in the tropical Pacific Ocean, and the symbol
Hbottom = 4100 m is the reference depth from the sea surface to the bottom of the tropical Pacific Ocean.
The zonal phase velocity of equatorial waves on an equatorial β-plane may be written as

−cβ
ω
= 2
,
2
k
k c − ω /c + (2n + 1)β(1d)where ω is wave frequency, k is zonal wavenumber, n is
meridional mode number, and β = ∂f∂y which is Rossby
parameter (Matsuno 1966). Substitution of n = −1 to
Eq. (1d) yields ω/k = c representing equatorial Kelvinwaves (KWs). Substitution of a set of n = 1 and ω = k = 0
to Eq. (1d) yields ω/k = −c/3 representing long Rossby
waves in the first meridional mode. This corresponds tothe observational result of oceanic RWs (Chelton and
Schlax 1996). Let L = 15, 000 km be the zonal distance of
the Pacific Ocean at the equator from 145° E to 80° W. It
takes Lc and 3L
c for equatorial KWs and long equatorial
RWs, respectively, to travel eastward or westward in the
Pacific Ocean basin. The sum of these periods, Lc + 3L
c , isreferred to as the basin mode period and is estimated tobe 0.70 years for the first baroclinic mode.
2.2 Basic results

We explain the seasonal sequence of simulated quantities
in the last year of the climatological experiment. Figure 1
shows snapshots in the middle of January, April, July, andOctober. The signals of zonal velocity (color shading in theleft panels of Fig. 1) deflect toward lower latitudes as RWs
propagate westward. The zonal wind stress (contours in theleft panels of Fig. 1) is much stronger in January and Julythan in April and October, indicating the significant influence of the monsoon in Southeast Asia and Northeast
Australia. In January, the wind stress is westward at 0°–20°
N and 40°–20° S, and eastward at 20°–50° N and 20° S–0°,
while the wind direction is reversed in July.
Geopotential is written as,2

p = Hc 1 (h − H1 ),(1e)(See figure on next page.)
Fig. 1 Snapshots through the last year of the basic climatological model experiment. The snapshots are taken in the middle of a, b January, c,d April, e, f July, and g, h October. Left panels a, c, e, and g color shading represents simulated zonal velocity u (with a unit of m/s) and contoursx
represent zonal wind stress τwind(with a contour interval of 0.02 N/m2, all positive values are shown by solid lines and all negative values are shownby dashed lines), and arrows represent wind stress vectors. Right panels b, d, f, and h color shading represents simulated geopotential anomalypc which has been rescaled to be compared with the magnitude of velocity (with a unit of m/s) and contours represent pseudo-streamfunction ϕinverted using Eq. (2b) (with a contour interval of 4000 ­m2/s, all positive values are shown in solid lines and all negative values are shown in dashedlines)

Ye and Aiki ﻿Progress in Earth and Planetary Science (2024) 11:29

Fig. 1 (See legend on previous page.)

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Ye and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

Page 5 of 16

whose signals are more widely spread in the meridional
direction of the Pacific Ocean (as shown by color shading in the right panels of Fig. 1) than that of zonal velocity. This indicates that gravitational potential energy is
greater than kinetic energy in mid- and high-latitude
regions, that is consistent with the quasi-geostrophic
dynamics. The ZASs of geopotential in the easternboundary regions at mid-latitude are generated by windstress curl corresponding to the simulation result of
Cummins et al. (1986). The variations in the group velocity depending on the deformation radius lead to themeridional propagation of geopotential (Schopf et al.
1981). In January, a negative and strong geopotential
anomaly area is generated in low-latitude regions, as
indicated by the red box in Fig. 1b. The negative signals
propagate to the western boundary regions in July.
Potential vorticity anomaly may be written asf (h−H1 )∂v
q = ∂x
− ∂u
∂y −
H1 .

(2a)

Contours in the right panels of Fig. 1 represent pseudostreamfunction which is derived by inverting q using thefollowing equation,
 2∇ 2 ϕ − fc ϕ = q,(2b)where ∇ 2 is the horizontal Laplacian operator. This equation has been solved using the Successive Over-Relaxation (SOR) method (Perrin 1997) with a coastal boundary
condition of ϕ = 0. Note that pseudo-streamfunction ϕ isavailable at all latitudes, in contrast to quasi-geostrophic
streamfunction which may be written as pf . The signals of
pseudo-streamfunction in Fig. 1 are positively correlatedwith that of geopotential in the Northern Hemisphereand negatively correlated in the Southern Hemisphere.
The significant anomalies of pseudo-streamfunction areconcentrated in the western equatorial regions.
Figure 2 shows a set of Hovmöller diagrams in both
mid-latitude and equatorial regions. Zonal velocity (color
shading in the left panels of Fig. 2) exhibits clear westward propagating signals at both 35° N and the equator,
indicating the influence of RWs. While zonal velocity at
35° N manifests a seasonal cycle, it takes about 2 years
to travel from 130° to 140° W. On the other hand, zonal
wind stress (contours in the left panels of Fig. 2) at 35°
N as well shows a seasonal cycle. Zonal wind stress is
positive from November to April, and negative from
May to October. These are revisited later in the manuscript. At the equator, the regions of greater magnitude
in zonal velocity appear on the western side of the basin
where zonal velocity is eastward from June to November and westward from December to May. Geopotential(color shading in the right panels of Fig. 2) as well shows

westward propagating signals. At 35° N, geopotential andpseudo-streamfunction (contours in the right panels of
Fig. 2) both show a seasonal cycle with a similar period
as zonal velocity. At the equator, the regions of greater
values in pseudo-streamfunction anomaly are foundin the western Pacific Ocean. These are consistent withthe model results of Qiu (2003). We note that correlation between geopotential and pseudo-streamfunction is
not found in equatorial regions where geostrophy is notrobust. Pseudo-streamfunction at the equator may capture the signals of mixed Rossby-gravity (Yanai) waves
associated with the semi-annual component of wind
forcing, to be explored in a future study following thespectral analysis approach of Song and Aiki (2021).

3 Results and discussion
We investigate the budget of wave energy in this section.
All results in this section have been derived by taking atime average of 120 snapshots (except for those otherwise
noted) through the last year of model integration.
3.1 Classical expression

Manipulation of shallow water Eqs. (1a)–(1c) yields anequation for the budget of wave energy to read,


 H ρ
∂ H1 ρ0  2
1 0 2
2
u +v +
p
∂t
2
2c2
∂
∂
+
(H1 ρ0 up) + (H1 ρ0 vp)
(3a)
∂x
∂y
y

x
= uτwind+ vτwind u+ H1 ρ0 uF + vF v + ρ0 pF h ,where the second line is pressure-flux divergence, thethird line is wind input rate, and the last line is dissipation rate (as well as buoyancy production rate). In thisclassical expression, the wind input is estimated as theinner product of simulated velocity vector and windstress vector, as explained in Sect. 1 of this manuscript.
The third line of Eq. (3a) can be interpreted as the inputof kinetic energy through wind forcing. This expressionof the wind input has often been used in previous studies(Wunsch 1998; Scott and Xu 2009; Ferrari and Wunsch2009; Zhai et al. 2012; Renault et al. 2016). See Aiki et al.
(2016) for the separate budget equations of kinetic energy
and gravitational potential energy to validate expressions
for the forcing for each.
We have plotted the time-averaged distribution ofsome terms in Eq. (3a) to investigate the energy budgetof annual waves in the Pacific Ocean. Despite representing the annual mean, Fig. 3a, b manifests sign-indefinite
signals in the wind input and the pressure-flux divergence, respectively, that we refer to as ZASs. The patterns of ZASs in both hemispheres converge closer to theYe and Aiki ﻿Progress in Earth and Planetary Science (2024) 11:29

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Fig. 2 Hovmöller diagrams for the results in the last year of the climatological model experiment. The research areas are at a, b 35° N, 140°–130° W
and c, d the equator, 120° E–70° W. Left panels a, c: color shading represents simulated zonal velocity u (with a unit of m/s) and contours representxzonal wind stress τwind(with a contour interval of a 0.004 N/m2, and c 0.02 N/m2, all positive values are shown in solid lines and all negative values
are shown in dashed lines). Right panels b, d: color shading represents simulated geopotential anomaly pc which has been rescaled to be comparedwith the magnitude of velocity (with a unit of m/s) and contours represent pseudo-streamfunction ϕ inverted from Eq. (2b) (with a contour intervalof b 50 ­m2/s, and d 4000 ­m2/s, all positive values are shown in solid lines and all negative values are shown in dashed lines)

equator as moving from east to west. The dissipation rate
in Fig. 3c exhibits greater values in the western equatorial
region. We note that the classical expression of the wind
input yields distinct ZASs in regions near the easternboundary around 30°–40°N and 40°–30° S in the Pacific
Ocean. The results correspond to McCreary et al. (1987)
that the annual RWs vanish around 40°N in the North
Pacific Ocean and the observations (Qiu et al. 1997). In
the Northern (Southern) Hemisphere, the isolines ofthese signals are in the northeast-southwest (southeastnorthwest) direction that is symmetric about the equator. In the Indian Ocean as well, ZASs appear in the wind
input when estimated using the classical expression (Liand Aiki 2020). Li and Aiki (2020) did not conduct an indepth investigation into the origin of ZASs, apart fromtheir general speculation that ZASs in the Indian Ocean
were believed to result from a phase relationship between
ocean flows and wind forcing. The present study focusesYe and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

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Fig. 4 Hovmöller diagrams associated with wind input
during the last year of the climatological model experiment. This
analysis is at 35° N from 140° to 130° W. Color shading represents
a wind input rate in the classical expression, which is writtenyx+ vτwind in Eq. (3a), and b the sum of additional terms
as uτwindyx− ϕx τwind , associated with the modified wind
−ϕcurltau + ϕy τwind
input in Eq. (3b). Both terms are with a unit of W/m2

Fig. 3 Annual mean energy quantities as estimated by the classicalexpression. Color shading in each panel shows a wind input rate,b pressure-flux divergence, and c energy dissipation rate. All threeterms are with a unit of W/m2 and are based on Eq. (3a)on these signals that may disturb the understanding of
the energy budget of annual waves.
To investigate the origin of these signals, we plot a
Hovmöller diagram in Fig. 4a at 35° N between 140° and
130° W, representing a typical region for ZASs. The timeseries of the classical wind input rate show westward
shifting with a seasonal cycle. The pattern of the wind
input originates from a phase relationship between ocean
flows and wind forcing, as illustrated in Fig. 2a for thezonal component. Taking the time average of signals in
the red box areas of Fig. 4a yields positive values. Likewise, taking the time average of signals in the green box
areas of Fig. 4a yields negative values. These lead to ZASs
in the annual mean field of the wind input.
3.2 The modified (phase‑resistant) expressionof the present study

As explained in Sect. 1 of this manuscript concerning thequasi-geostrophic dynamics, wind input may also be estimated as the negative of the product of streamfunction
and wind stress curl (Scott and Straub 1998; Berloff andYe and Aiki ﻿Progress in Earth and Planetary Science (2024) 11:29

McWilliams 1999a, 1999b). Streamfunction in the quasigeostrophic dynamics is associated with (the vertical integral of) temperature anomaly (through the hydrostaticbalance). Wind stress curl is associated with the horizontal
divergence of Ekman velocity (and thus the vertical convergence of velocity through the incompressibility condition). The product of streamfunction and wind stress curl
is closely linked to the budget of gravitational potential
energy, understanding vertical integration by parts. Thisshould not be confused with whether an injected energy isefficiently stored in a kinetic energy reservoir or a gravitational potential energy reservoir according to relationshipbetween wavelength and the deformation radius.
With an intent to introduce the idea of the quasi-geostrophic expression mentioned above, we propose to modify
the wind input term as in the third line of the following
equation to read,


 H ρ
∂ H1 ρ0  2
1 0 2
2
u +v +
p
∂t
2
2c2


∂ 
∂ 
y
x
+
H1 ρ0 up − ϕτwind +H1 ρ0 vp + ϕτwind
∂x
∂y

 x
y
= −ϕ curltau + u + ϕy τwind+ (v − ϕx )τwind u
+ H1 ρ0 uF + vF v + ρ0 pF h ,(3b)yx
where curltau = ∂x τwind − ∂y τwind is wind stress curl.
Pseudo-streamfunction ϕ has been defined in Eq. (2b) of
the present manuscript, and ϕx = ∂x ϕ and ϕy = ∂y ϕ.
Pressure-flux divergence term at the second line of
Eq. (3b) has also been modified from that in Eq. (3a) tomaintain mathematical consistency. The third line of
Eq. (3b) represents the modified version of the wind
input term, following the expression in the quasi-geostrophic literature. In contrast to quasi-geostrophic
streamfunction which may be written as pf , pseudostreamfunction ϕ is available at all latitudes that is a cornerstone in the present study.
Figure 5a, b shows the modified wind input and themodified pressure-flux divergence of annual wave energy
estimated by the modified expression in Eq. (3b). The modified wind input (Fig. 5a) becomes more concentrated withsmall anomaly areas at the equatorial region. The modified
pressure-flux divergence (Fig. 5b) exhibits a similar pattern to the modified wind input. It can be said that the new(modified) expression provides a phase-resistant featurefor estimating both the wind input and the pressure-flux
divergence.
Terms on the third line of Eq. (3b) may be better
explained if we decompose the horizontal component ofvelocity as,u = uekm + ugeo + uigw , v = vekm + vgeo + vigw , (4)Page 8 of 16

Fig. 5 Modified annual mean energy quantities as estimatedby the modified (phase-resistant) expression in the present study.
Color shading in each panel shows a modified wind input rate, and b
modified pressure-flux divergence. Both terms are with a unit of W/
m2 and are based on Eq. (3b)where superscripts “ekm”, “geo”, and “igw” stand for velocity components associated with Ekman flow, geostrophic
flow, and inertia-gravity waves, respectively.
One way to define the Ekman velocity is to use the reference thickness to read,
τ

y

τx

uekm = fHwind, vekm = − fHwind
,
1 ρ0
1 ρ0

(5a)

which is directed perpendicular to wind forcing vector. The result is that there is no wind input to the
Ekman component of oceanic velocity in a single-layer
framework,yxuekm τwind+ vekm τwind = 0,(5b)which is consistent with a classical framework with vertical profiles wherein wind input to Ekman spiral flow at
the sea surface (Wang and Huang 2004) is canceled inYe and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

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sharp by the vertical integral of dissipation associatedwith turbulent viscosity, as detailed in Appendix B of thepresent manuscript.
Geostrophic velocity is defined in the present study
using pseudo-streamfunction to read,ugeo = −ϕy , vgeo = ϕx ,(6a)which is available at all latitudes. Geostrophic velocity
satisfies the following, xy+ (v − ϕx )τwind− ϕcurltau + u + ϕy τwind= −ϕcurltau
 pcurltau
≈−
f


∂vekm
∂uekm
+
,
≈ −ρ0 pH1
∂x
∂y

(6b)

where the second line has been derived using Eq. (6a), thethird line has been derived using ϕ ≈ pf , the last line has
been derived using Eq. (5a). This makes it possible for thesum of additional terms in the modified expression to offset the wind input rate in the classical expression in sucha way as to reduce ZASs in the annual mean estimation.
The last line of Eq. (6b) is analogous to the last term ofEq. (3b), allowing for


∂uekm
∂vekm
,
+
−H1
(6c)
∂x
∂y
to be interpreted as a variant of thickness forcing F h whichleads to the source/sink of gravitational potential energy.
On the other hand, velocity associated with inertiagravity waves, coastal KWs, and equatorial KWs ischaracterized by no perturbation of potential vorticity
as well as pseudo-streamfunction,(7a)q = ϕ = 0,

where q has been defined in Eq. (2a). Substitution of
Eq. (7a) to the third line in Eq. (3b) yields,

 x
y
y
x
+ vigw τwind ,+ (v − ϕx )τwind = uigw τwind−ϕcurltau + u + ϕy τwind(7b)which retrieves the classical expression of wind input.
To summarize, it is only for geostrophic velocity thatthe modified expression of wind input can be considered as representing the input of gravitational potential
energy while canceling out the input of kinetic energy.
To investigate the elements of reducing ZASs, we plot
Fig. 4b, the Hovmöller diagram ofyx− ϕx τwind ,−ϕcurltau + ϕy τwind(8)

which is the sum of additional terms of the modified wind
input expression with respect to the classical expression.
The sum of additional terms in the modified expression
manifests evident consistency with the westward shifting
pattern of the classical expression, except that they have
opposite signs.
To investigate the details of the ZAS reduction in the
Pacific Ocean, we have estimated the magnitude of thewind input associated with the classical expression using

√
y2
x2
(9a)
+ τwind ,u2 + v2 τwind

as shown in Fig. 6a where the areas of greater orders inmagnitude near 30°–40°N and near the coastlines of theNorth American continent, corresponding to regions
where ZASs are significant. We have also estimated themagnitude of the modified expression using|−ϕ|curltau ,(9b)

as shown in Fig. 6b. Both terms (9a) and (9b) are in aunit of W/m2. Signals in Fig. 6b are smaller by one to twoorders of magnitude than that in Fig. 6a in the mid- andlow-latitude regions of the Pacific Ocean, especially inthe regions of ZASs. To summarize the phase-resistantfeature is given by cancelation within each of the secondand last terms on the left-hand side of Eq. (6b) as shownby its first equality. Such remedy is not required for the
product of streamfunction and wind stress curl.
3.3 Validation of the modified energy flux

We have plotted energy fluxes to help better understandthe annual mean budget of wave energy in the Pacific
Ocean. There are three schemes for illustrating the transfer routes of wave energy based on the group velocity,
as explained in Appendix C of the present manuscript.
Below we show the distributions of the energy fluxes
based on the three schemes, as their performance in the
Pacific Ocean has not been compared in the previous
studies.
Figure 7 shows energy flux vectors associated withthe pressure-flux scheme (Cummins and Oey 1997),
and the Orlanski-Sheldon scheme (Orlanski and Sheldon 1993, hereafter OS93 scheme) following expressions (16a) and (16b), respectively, in Appendix C of
the present manuscript. The energy fluxes of bothschemes are compatible with the wave energy budget
Eq. (3a). Figure 7a for the pressure-flux scheme manifests westward flux at the equator and eastward flux inregions between 8° S and 8° N. This scheme assumes
only inertia-gravity waves but not mid-latitude RWs.
On the other hand, Fig. 7b for the OS93 scheme is able
to display the westward transfer of wave energy byYe and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

Fig. 6 Comparison between the two expressions of wind input
rate in terms of magnitudes. a The classical wind input rate
as measured by term (9a), and b the modified wind input rate
as measured by term (9b). The comparison is based on the resultof the climatological model experiment during the last year. Bothmagnitudes are with a unit of W/m2

off-equatorial RWs. The distinguishing feature of theOS93 scheme is the lack of values in the vicinity of theequator and coastlines, associated with equatorial andcoastal KWs, respectively. Figure 7 illustrates the lackof validity in the pressure-flux and OS93 schemes fordetermining the group velocity of annual waves in the
Pacific Ocean. This result is in line with similar diagnoses for annual waves in the Indian Ocean (Li and Aiki2020) and the Atlantic Ocean (Song and Aiki 2020).
Figure 8a shows the annual mean distribution of theenergy flux as estimated by the Aiki-Greatbatch-Claus
scheme (Aiki et al. 2017, hereafter AGC17 scheme),
following expression (16c) in Appendix C of the present manuscript. The AGC17 energy flux is compatible

Page 10 of 16

Fig. 7 The annual mean flux of wave energy as estimatedby different diagnosis schemes. a The pressure-flux scheme assuming
the group velocity of inertia-gravity waves and b the OS93 scheme
assuming the group velocity of quasi-geostrophic waves. Colorshading shows zonal energy flux (with a unit of W/m). Arrows
represent energy flux vectors, with thin arrows for the energy
flux in the range of 200–1000 W/m, and thick arrows for greater
ranges. The result is based on the climatological model experiment
during the last yearwith the classical wind input in the wave energy budget
Eq. (3a). The AGC17 energy flux (Fig. 8a) can reveal
both the westward flux of energy associated with midlatitude RWs and the eastward flux of energy associatedwith equatorial KWs. In the Pacific Ocean, equatorial
KWs traverse toward the eastern boundary, undergoreflection, and subsequently diverge into two packets
of westward RWs. These wave packets then coalesce
into a single packet of westward propagating waves at
160° W as indicated by Toyoda et al. (2021, 2023). We
need to modify the AGC17 energy flux in such a way as
to be compatible with the modified wind input in the
wave energy budget Eq. (3b). This is referred to as themodified AGC17 scheme (hereafter, AGC17M scheme),
following expression (16d) in Appendix C of the present manuscript. Figure 8b shows the annual meanYe and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

Page 11 of 16

expression (contours in Fig. 9a, c) shows a much stronger
value than the modified wind input rate (contours in
Fig. 9b, d) generally. At 8° N, significant values in themodified wind input are distributed in the central andeastern basins. At the equator, the modified wind input
exhibits significant negative values in the western boundary regions, indicating that layer thickness anomalies are
dumped by the horizontal divergence of Ekman velocity.

Fig. 8 Same as Fig. 7 except for a the AGC17 scheme and b
the AGC17M scheme. The AGC17 scheme assumes the group velocityof all waves at all latitudes

distribution of the AGC17M energy flux. The striking
resemblance between Fig. 8a, b implies that modifying both the wind input and the energy flux vectors
has minimum impact on the qualitative superiority of
the AGC17 scheme in terms of seamless connection
between equatorial and off-equatorial regions.
Figure 9 shows the time evolution of zonal flux andwind input rate during the last year of the model experiment at both 8° N and the equator. This figure is designed
to facilitate a more comprehensive evaluation of thedisparities between the AGC17 energy flux (left panels of Fig. 9) and the AGC17M energy flux (right panels of Fig. 9). Zonal flux (color shading) in all panels of
Fig. 9 exhibits the evident signals of westward transfers
associated with RWs. Figure 9c, d exhibits the weak signals of eastward transfers in the eastern domain associated with equatorial KWs. There is almost no difference
between the zonal flux of wave energy (color shading in
Fig. 9) in the two schemes. This confirms that the new
expression in the present study continues the superiority of the AGC17 scheme in the Pacific Ocean. At both
8° N and the equator, the wind input rate of the classical4 Conclusions
The present study has focused on the annual mean energy
budget in the upper layer as a basis for understanding the
dynamics of large-scale waves. In the classical expressionof the energy budget, wind input is estimated as the innerproduct of horizontal velocity vector and wind stressvector. Application of this expression to the energy diagnosis of annual waves yields ZASs in the annual mean
wind input as shown by Li and Aiki (2020) for the Indian
Ocean and the present study for the Pacific Ocean. We
have investigated the reason why ZASs appear in theclassical expression. Velocity associated with oceanic
annual RWs contains both temporal and zonal variations
while wind stress varies mainly in time to represent a seasonal cycle. The phase relationship between velocity andwind stress causes ZASs which disturb the understandingof the annual mean wind input.
We note that, in quasi-geostrophic dynamics, wind
input is determined as the negative product of streamfunction and wind stress curl. Geostrophic streamfunction is associated with (the vertical integral of )temperature anomaly (through the hydrostatic balance),
while wind stress curl is associated with the horizontal divergence of wind-induced Ekman velocity (which
may be rewritten as the vertical convergence of velocity
using the incompressibility condition). The product of
streamfunction and wind stress curl is closely linked tothe budget of gravitational potential energy (after vertical
integration by parts). This quasi-geostrophic expressionof the wind input is singular at the equator, so the present study has modified it using pseudo-streamfunction
which is obtained from inverting the anomaly of Ertel’s
potential vorticity. The modified expression of wind input
takes the form of representing the input of gravitational
potential energy with canceling out the input of kinetic
energy. Because it uses pseudo-streamfunction, the modified (phase-resistant) expression of the wind input isapplicable to waves at all latitudes.
To maintain the mathematical consistency between
equations for the budget of wave energy, modification of
the wind input affects the expression of the wave energy
flux. The AGC17 energy flux is compatible with the classical wind input in the wave energy budget Eq. (3a), while
the AGC17M energy flux is compatible with the modified

Ye and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

Page 12 of 16

Fig. 9 Hovmöller diagrams for examining the annual mean flux of wave energy. The analyses are at a, b 8° N and c, d the equator. Color shading
represents zonal energy flux (with a unit of W/m) as estimated by a, c the AGC17 scheme and b, d the AGC17M scheme without a time average.
Contours represent the wind input rate to oceanic waves (with a contour interval of ­10–4 W/m2) as estimated by a, c the classical expression and b,d the modified (phase-resistant) expression where all positive values are shown in solid contours and all negative values are shown in dashedcontours. The diagrams represent the last year of the climatological model experimentenergy input in the wave energy budget Eq. (3b). The
striking resemblance between Fig. 8a, b implies thatmodifying both the wind input and the energy flux vectors has minimum impact on the qualitative superiorityof the AGC17 scheme in representing the group velocity
of waves at all latitudes.
The result of the present study may be applied to theinvestigation of interannual variations in such a way as
to reduce the complexity associated with a seasonal cyclein oceanic flows. We plan to utilize a model with more
realistic setup and higher vertical resolution in the future.

The realistic model would enable us to separate geostrophic flow and Ekman flow. Whether it is still effective
to reduce ZASs will be checked.
Goddard and Philander (2000) pointed out that sea
surface temperature (SST) and available potential energy
are closely correlated, implying a connection between thewind input of wave energy and El Niño events. Toyoda
et al. (2021, 2023) diagnosed an ocean reanalysis dataset
in terms of the temporal evolution of wave energy associated with ENSO. They provided an energetic view forinvestigating ENSO which had often been understoodYe and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

Page 13 of 16

in terms of raw quantities including SST, SSH, and thermocline depth in previous studies. In future studies, theenergy perspective is expected to be used more extensively to enhance our understanding of the role of waves
in tropical climate variations.

Appendix A: Equations in spherical coordinates
All equations in the main body of the present manuscript
are written in a Cartesian coordinate system for simplicity.
Both the model code of shallow water equations and the
diagnosis code of energy quantities have been written in a
spherical coordinate system, which is explained below using and θ as longitude and latitude coordinates, respectively.
The shallow-water Eqs. (1a)–(1c) in Sect. 2.1 may be
expressed in spherical coordinates to read,
x
τwind
c2
∂h
∂u
u
∂t − fv + H1 R cos θ ∂ = H1 ρ0 + F ,
y

(10a)

τwind
c2 ∂h
∂v
v
∂t + fu + H1 R ∂θ = H1 ρ0 + F ,

(10b)

H1 ∂u
H1 ∂
∂h
h
∂t + R cos θ ∂ + R cos θ ∂θ (cos θv) = F ,(10c)where R is the radius of the Earth. Manipulation of
Eqs. (10a)–(10c) yields an equation for the budget ofwave energy to read,


 H ρ
∂ H 1 ρ0  2
1 0 2
u + v2 +
p
∂t
2
2c2
∂
∂
1
1
+
(H1 ρ0 up) +(cos θ H1 ρ0 vp)
R cos θ ∂
R cos θ ∂θyx
= uτwind+ vτwind u+ H1 ρ0 uF + vF v + ρ0 pF h ,(11a)which corresponds to Eq. (3a). The pressure-flux divergence and wind-forcing terms in Eq. (11a) may be rewritten in such a way as to yield a term associated with windstress curl to read,∂1
which corresponds to Eq. (3b) and curltau = R cos
θ ∂


 y 
∂
1
x
τwind − R cosθ ∂θ cos θ τwind is understood. The inversion equation of pseudo-streamfunction may be written
in spherical coordinates to read, 2f (h−H1 )1∂v∂
1∇ 2 ϕ − fc ϕ = R cos
θ ∂ − R cos θ ∂θ (cos θu) −
H1 ,

(12)

which corresponds to Eqs. (2a) and (2b) in Sect. 2.2.

Appendix B: Case of Ekman spiral velocity
Equation (5b) shows that there is no wind input to Ekman
flows in a single-layer framework. This is revisited in thissection using a depth-dependent framework that can represent the Ekman spiral velocity. Let the symbol z represent
the vertical coordinate and increase upward. Let the symbols U ekm and V ekm be the zonal and meridional components of Ekman spiral velocity in deep water, respectively,that are defined by the balance between the Coriolis term
and the vertical viscosity term in momentum equations to
read,


−ρ0 fV ekm = ρ0 AUzekm ,
(13a)
z



ρ0 fU ekm = ρ0 AVzekm ,
z

(13b)

where A denotes to the coefficient of turbulent viscosity and subscript z denotes partial differentiation in thevertical direction. At the sea surface ( z = 0), the boundyxary condition is ρ0 AUzekm = τwindand ρ0 AVzekm = τwind .
At deeper depths ( z → −∞), the Ekman spiral velocity
decays in amplitude to zero (U ekm = 0, V ekm = 0).
Multiplying Eqs. (13a) and (13b) by U ekm and V ekm,
respectively, and summing them yields



 H ρ
∂ H1 ρ0  2
1 0 2
u + v2 +
p
∂t
2
2c2


∂ 
∂ 
1
1
y
x
+
H1 ρ0 up − ϕτwind +cos θ H1 ρ0 vp + cos θϕτwind
R cos θ ∂ Rcos θ ∂θ

1 ∂ϕ x1 ∂ϕ y= −ϕ curltau + u +τwind + v −τR ∂θR cos θ ∂ wind
+ H1 ρ0 uF u + vF v + ρ0 pF h ,(11b)

Ye and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

Page 14 of 16−U ekm ρ0 fV ekm + V ekm ρ0 fU ekm = ρ0 U ekm AUzekm + ρ0 V ekm AVzekm ,zwhere the left-hand side is clearly zero. Equation (14) may
be rewritten using vertical integration by parts to read
y
x
τwind
U ekm + τwind V ekm = ρ0

0

−∞



A Uzekm2 + Vzekm2 dz,(15)which indicates that wind input to the Ekman spiralvelocity at the sea surface is canceled in sharp by thevertical integral of dissipation associated with turbulent
viscosity.

Appendix C: Diagnostic schemes for energy fluxes
For diagnosing the horizontal flux of wave energy, basedon the direction of group velocity in linear wave theory,
three schemes were proposed in previous studies. The
pressure-flux scheme may be used in the studies of inertia-gravity waves, that is written as,
H1 ρ0 up, H1 ρ0 vp ,(16a)where the edged brackets indicate a vector in the horizontal direction. Application of this scheme to simulated
velocity and simulated geopotential in the model resultof the present study yields Fig. 7a. This scheme cannot
indicate the group velocity of mid-latitude RWs, to beconsistent with the results of Li and Aiki (2020) for the
Indian Ocean and Song and Aiki (2020) for the Atlantic
Ocean.
The OS93 scheme may be used to investigate theenergy flux of mid-latitude RWs, that is written as, 2 
 2  H1 ρ0 up + p2f, H1 ρ0 vp − p2f
,
(16b)
y

x

where subscripts x and y indicate partial differentiation in the zonal and meridional directions, respectively.
Application of this scheme to the result of the modelexperiment yields Fig. 7b. This scheme is singular at theequator, and can indicate neither the energy flux of equatorial waves nor that of coastal KWs. These problems
were resolved in the AGC17 scheme which is seamlessly
applicable to waves at all latitudes with the following
expression, pϕ   pϕ  H1 ρ0 up + 2 y , H1 ρ0 vp − 2 x ,(16c)where ϕ is defined by Eq. (2b) in the present manuscript.
Application of this scheme to the result of the model

z

(14)

experiment yields Fig. 8a. The AGC17 scheme was used
for investigating waves in the Indian Ocean (Ogata andAiki 2019; Li and Aiki 2020, 2022, 2024; Li et al. 2021),
the Atlantic Ocean (Song and Aiki 2020, 2021, 2023;
Song et al. 2023a, b), the Pacific Ocean (Toyoda et al.
2021, 2023), and the atmosphere (Aiki et al. 2021). In
Toyoda et al. (2021, 2023) and Song and Aiki (2023), thehorizontal variation of gravity wave speed c was considered in their calculation of the pseudo-streamfunction.
To be consistent with the modified (phase-resistant)
expression of the wind input in the present study, we
propose to modify the expression of the AGC17 energy
flux as follows, pϕ   pϕ  y
xH1 ρ0 up + 2 y − ϕτwind , H1 ρ0 vp − 2 x + ϕτwind ,(16d)which is referred to as the AGC17M scheme. The zonal
component of the AGC17M energy flux is shown in
Fig. 8b. Comparing Figs. 8a, b and 9, we confirm that themodification of the energy flux maintains the qualitative
superiority of the AGC17 scheme both in the horizontal
distribution and time evolution through the year.
Apart from the four schemes mentioned above, Fourier analysis and ray theory were used in previous studies for tracing the life cycle of waves. These classicalapproaches do not provide the horizontal distributionof energy fluxes in the presence of different types ofwaves.
Abbreviations
ZAS	Zonally alternating signal
RW	Rossby wave
SSH	Sea surface height
STCC​	Subtropical countercurrent
WOA	World Ocean Atlas
KW	Kelvin wave
ECMWF ORAS3	The European Centre for Medium-Range Weather Forecasts Ocean Reanalysis System 3
SOR	Successive over-relaxation
OS93 scheme	Orlanski–Sheldon scheme
AGC17 scheme	Aiki–Greatbatch–Claus scheme
AGC17M scheme	The modified Aiki–Greatbatch–Claus scheme
SST	Sea surface temperature
ENSO	El Niño Southern Oscillation
Acknowledgements
We thank Prof. Joji Ishizaka and Dr. Yoshihisa Mino for their advices in ourstudy. KY also thanks Dr. Zimeng Li for the kind suggestions in the experiment.
Author contributions
KY analyzed the data, carried out the experimental study, and constructed
the manuscript. HA proposed the topic, gave constructions of the study, andmodified the manuscript. All authors read and approved the final manuscript.

Ye and Aiki P﻿ rogress in Earth and Planetary Science (2024) 11:29

Funding
This study is funded by Japan Society for the Promotion of Science KAKENHI
Grant 22H00176.
Availability of data and materials
Readers can download the European Centre for Medium-Range Weather Forecasts Ocean Reanalysis/analysis System (ECMWF ORAS3) dataset from http://​apdrc.​soest.​hawaii.​edu:​80/​dods/​public_​data/​Reana​lysis_​Data/​ORA-​S3/​1x1_​
grid. The temperature and salinity data which are used to calculate gravity
wave speed are from http://​apdrc.​soest.​hawaii.​edu/​dods/​public_​data/​WOA/​WOA13/1_​deg/​annual. The numerical simulation code and output data usedor analyzed during the current study are available from the corresponding
author on reasonable request.

Declarations
Competing interests
The authors declare that they have no competing interests.
Received: 30 July 2023 Accepted: 25 April 2024
Published: 16 May 2024

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