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Qingyang Song
and
Hidenori Aiki

Abstract

Intraseasonal waves in the tropical Atlantic Ocean have been found to carry prominent energy that affects interannual variability of zonal currents. This study investigates energy transfer and interaction of wind-driven intraseasonal waves using single-layer model experiments. Three sets of wind stress forcing at intraseasonal periods of around 30, 50, and 80 days with a realistic horizontal distribution are employed separately to excite the second baroclinic mode in the tropical Atlantic. A unified scheme for calculating the energy flux, previously approximated and used for the diagnosis of annual Kelvin and Rossby waves, is utilized in the present study in its original form for intraseasonal waves. Zonal velocity anomalies by Kelvin waves dominate the 80-day scenario. Meridional velocity anomalies by Yanai waves dominate the 30-day scenario. In the 50-day scenario, the two waves have comparable magnitudes. The horizontal distribution of wave energy flux is revealed. In the 30- and 50-day scenarios, a zonally alternating distribution of cross-equatorial wave energy flux is found. By checking an analytical solution excluding Kelvin waves, we confirm that the cross-equatorial flux is caused by the meridional transport of geopotential at the equator. This is attributed to the combination of Kelvin and Yanai waves and leads to the asymmetric distribution of wave energy in the central basin. Coastally trapped Kelvin waves along the African coast are identified by alongshore energy flux. In the north, the bend of the Guinea coast leads the flux back to the equatorial basin. In the south, the Kelvin waves strengthened by local wind transfer the energy from the equatorial to Angolan regions.

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Zimeng Li
and
Hidenori Aiki

Abstract

The present study investigates the interannual variability of the tropical Indian Ocean (IO) based on the transfer routes of wave energy in a set of 61-yr hindcast experiments using a linear ocean model. To understand the basic feature of the IO dipole mode, this paper focuses on the 1994 pure positive event. Two sets of westward transfer episodes in the energy flux associated with Rossby waves (RWs) are identified along the equator during 1994. One set represents the same phase speed as the linear theory of equatorial RWs, while the other set is slightly slower than the theoretical phase speed. The first set originates from the reflection of equatorial Kelvin waves at the eastern boundary of the IO. On the other hand, the second set is found to be associated with off-equatorial RWs generated by southeasterly winds in the southeastern IO, which may account for the appearance of the slower group velocity. A combined empirical orthogonal function (EOF) analysis of energy-flux streamfunction and potential reveals the intense westward signals of energy flux are attributed to off-equatorial RWs associated with predominant wind input in the southeastern IO corresponding to the positive IO dipole event.

Significance Statement

The present study gains a new insight into the mechanism of the Indian Ocean dipole events using a new diagnostic scheme for wave energy based on 61-yr hindcast experiments. The results have shown the existence of two sets of westward transfer of wave energy at the equator during 1994. One set of westward signals shows the same group velocity with theoretical equatorial Rossby waves that appear reasonably along the equator. The other set of westward signals at the equator represents a slightly slower group velocity than the theoretical equatorial Rossby waves, which is associated with abnormally extended southeasterly winds during the Indian Ocean dipole event.

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Zimeng Li
and
Hidenori Aiki

Abstract

The present study adopts an energy-based approach to interpret the negative phase of Indian Ocean dipole (IOD) events. This is accomplished by diagnosing the output of hindcast experiments from 1958 to 2018 based on a linear ocean model. The authors have performed a composite analysis for a set of negative IOD (nIOD) events, distinguishing between independent nIOD events and concurrent nIOD events with El Niño–Southern Oscillation (ENSO). The focus is on investigating the mechanism of nIOD events in terms of wave energy transfer, employing a linear wave theory that considers the group velocity. The proposed diagnostic scheme offers a unified framework for studying the interaction between equatorial and off-equatorial waves. Both the first and third baroclinic modes exhibit interannual variations characterized by a distinct packet of eastward energy flux associated with equatorial Kelvin waves. During October–December, westerly wind anomalies induce the propagation of eastward-moving equatorial waves, leading to thermocline deepening in the central-eastern equatorial Indian Ocean, a feature absent during neutral IOD years. The development of wave energy demonstrates different patterns during nIOD events of various types. In concurrent nIOD–ENSO years, characterized by strong westerly winds, the intense eastward transfer of wave energy becomes prominent as early as October. This differs significantly from the situation manifested in independent nIOD years. The intensity of the energy-flux streamfunction/potential reaches its peak around November and then rapidly diminishes in December during both types of nIOD years.

Significance Statement

The present study provides an interpretation of wave energy transfer episodes in the upper ocean during the negative phase of the Indian Ocean dipole (IOD) based on the diagnosis of hindcast experiments. The results suggest that the reflection of Kelvin and Rossby waves at the eastern and western boundaries of the Indian Ocean (IO), respectively, accompanied by variations in thermocline depth, plays a crucial role in the development process of IOD events. Specifically, during the negative phase of the IOD, the tropical IO exhibits positive signals of energy-flux streamfunction in the Northern Hemisphere, along with positive signals of energy-flux potential associated with westerly wind anomalies occurring in October–December. These findings highlight the significance of these factors in shaping the characteristics of negative IOD events.

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Hidenori Aiki
and
Richard J. Greatbatch

Abstract

There is an ongoing discussion in the community concerning the wave-averaged momentum equations in the hybrid vertically Lagrangian and horizontally Eulerian (VL) framework and, in particular, the form stress term (representing the residual effect of pressure perturbations) that is thought to restrict the handling of higher-order waves in terms of a perturbation expansion. The present study shows that the traditional pressure-based form stress term can be transformed into a set of terms that do not contain any pressure quantities but do contain the time derivative of a wave-induced velocity. This wave-induced velocity is referred to as the pseudomomentum in the VL framework, as it is analogous to the generalized pseudomomentum in Andrews and McIntyre. This enables the second expression for the wave-averaged momentum equations in the VL framework (this time for the development of the total transport velocity minus the VL pseudomomentum) to be derived together with the vortex force. The velocity-based expression of the form stress term also contains the residual effect of the turbulent viscosity, which is useful for understanding the dissipation of wave energy leading to a transfer of momentum from waves to circulation. It is found that the concept of the virtual wave stress of Longuet-Higgins is applicable to quite general situations: it does not matter whether there is wind forcing or not, the waves can have slow variations, and the viscosity coefficient can vary in the vertical. These results provide a basis for revisiting the surface boundary condition used in numerical circulation models.

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Hidenori Aiki
and
Richard J. Greatbatch

Abstract

The residual effect of surface gravity waves on mean flows in the upper ocean is investigated using thickness-weighted mean (TWM) theory applied in a vertically Lagrangian and horizontally Eulerian coordinate system. Depth-dependent equations for the conservation of volume, momentum, and energy are derived. These equations allow for (i) finite amplitude fluid motions, (ii) the horizontal divergence of currents, and (iii) a concise treatment of both kinematic and viscous boundary conditions at the sea surface. Under the assumptions of steady and monochromatic waves and a uniform turbulent viscosity, the TWM momentum equations are used to illustrate the pressure- and viscosity-induced momentum fluxes through the surface, which are implicit in previous studies of the wave-induced modification of the classical Ekman spiral problem. The TWM approach clarifies, in particular, the surface momentum flux associated with the so-called virtual wave stress of Longuet-Higgins. Overall, the TWM framework can be regarded as an alternative to the three-dimensional Lagrangian mean framework of Pierson. Moreover, the TWM framework can be used to include the residual effect of surface waves in large-scale circulation models. In specific models that carry the TWM velocity appropriate for advecting tracers as their velocity variable, the turbulent viscosity term should be modified so that the viscosity acts only on the Eulerian mean velocity.

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Hidenori Aiki
and
Richard J. Greatbatch

Abstract

Previous attempts to derive the depth-dependent expression of the radiation stress have led to a debate concerning (i) the applicability of the Mellor approach to a sloping bottom, (ii) the introduction of the delta function at the mean sea surface in the later papers by Mellor, and (iii) a wave-induced pressure term derived in several recent studies. The authors use an equation system in vertically Lagrangian and horizontally Eulerian (VL) coordinates suitable for a concise treatment of the surface boundary and obtain an expression for the depth-dependent radiation stress that is consistent with the vertically integrated expression given by Longuet–Higgins and Stewart. Concerning (i)–(iii) above, the difficulty of handling a sloping bottom disappears when wave-averaged momentum equations in the VL coordinates are written for the development of (not the Lagrangian mean velocity but) the Eulerian mean velocity. There is also no delta function at the sea surface in the expression for the depth-dependent radiation stress. The connection between the wave-induced pressure term in the recent studies and the depth-dependent radiation stress term is easily shown by rewriting the pressure-based form stress term in the thickness-weighted-mean momentum equations as a velocity-based term that contains the time derivative of the pseudomomentum in the VL framework.

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Hidenori Aiki
and
Kelvin J. Richards

Abstract

Understanding the role of mesoscale eddies in the global ocean is fundamental to gaining insight into the factors that control the strength of the circulation. This paper presents results of an analysis of a high-resolution numerical simulation. In particular, the authors perform an analysis of energetics in density space. Such an approach clearly demonstrates the role of layer-thickness form drag (residual effects of hydrostatic pressure perturbations), which is hidden in the classical analysis of the energetics of flows. For the first time in oceanic studies, the global distribution of layer-thickness form drag is determined. This study provides direct evidence to verify some basic characteristics of layer-thickness form drag that have often been assumed or speculated about in previous theoretical studies. The results justify most of the previous assumptions and speculations, including those associated with (i) the presence of an oceanic energy cycle explaining the relationship between layer-thickness form drag and wind forcing, (ii) the manner in which layer-thickness form drag removes the energy of vertically sheared geostrophic currents, and (iii) the reason why the work of layer-thickness form drag nearly balances the work of eddy-induced overturning circulation in each vertical column. However, the result of the analysis disagrees with speculation in previous studies that the layer-thickness form drag in the Antarctic Circumpolar Current is the agent that transfers the wind-induced momentum near the sea surface downward to the bottom layers. The authors present a new interpretation: the layer-thickness form drag reduces (and thereby cancels) the vertical shear resulting from the eddy-induced overturning circulation (rather than the vertical shear resulting from the surface wind stress). This interpretation is consistent with the results of the energy analysis conducted in this study.

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Hidenori Aiki
,
Koutarou Takaya
, and
Richard J. Greatbatch

Abstract

Classical theory concerning the Eliassen–Palm relation is extended in this study to allow for a unified treatment of midlatitude inertia–gravity waves (MIGWs), midlatitude Rossby waves (MRWs), and equatorial waves (EQWs). A conservation equation for what the authors call the impulse-bolus (IB) pseudomomentum is useful, because it is applicable to ageostrophic waves, and the associated three-dimensional flux is parallel to the direction of the group velocity of MRWs. The equation has previously been derived in an isentropic coordinate system or a shallow-water model. The authors make an explicit comparison of prognostic equations for the IB pseudomomentum vector and the classical energy-based (CE) pseudomomentum vector, assuming inviscid linear waves in a sufficiently weak mean flow, to provide a basis for the former quantity to be used in an Eulerian time-mean (EM) framework. The authors investigate what makes the three-dimensional fluxes in the IB and CE pseudomomentum equations look in different directions. It is found that the two fluxes are linked by a gauge transformation, previously unmentioned, associated with a divergence-form wave-induced pressure . The quantity vanishes for MIGWs and becomes nonzero for MRWs and EQWs, and it may be estimated using the virial theorem. Concerning the effect of waves on the mean flow, represents an additional effect in the pressure gradient term of both (the three-dimensional versions of) the transformed EM momentum equations and the merged form of the EM momentum equations, the latter of which is associated with the nonacceleration theorem.

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Hidenori Aiki
,
Yoshiki Fukutomi
,
Yuki Kanno
,
Tomomichi Ogata
,
Takahiro Toyoda
, and
Hideyuki Nakano

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.

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Takahiro Toyoda
,
Hideyuki Nakano
,
Hidenori Aiki
,
Tomomichi Ogata
,
Yoshiki Fukutomi
,
Yuki Kanno
,
L. Shogo Urakawa
,
Kei Sakamoto
,
Goro Yamanaka
, and
Motoki Nagura

Abstract

A method is introduced for diagnosing the time evolution of wave energy associated with ENSO from an ocean reanalysis. In the diagnosis, time changes of kinetic and available potential energy are mainly represented by energy inputs caused by surface wind stress and horizontal energy fluxes for each vertically decomposed normal mode. The resulting time evolutions of the wave energy and vertical thermocline displacements in the 1997/98 and 2014–16 El Niño events are consistent with our previous knowledge of these events. Further, our result indicated that representation of several vertical modes is necessary to reproduce the broadly distributed downward thermocline displacements in the central to eastern equatorial Pacific, generated by a westerly wind event in the western equatorial Pacific (e.g., in March 1997), that are preconditioning for El Niño development. In addition, we investigated the wave energy budget, including the influence of data assimilation, on the complicated time evolution of equatorial thermocline displacements caused by repeated westerly and easterly wind events during the 2014–16 El Niño event. Our result suggests that noise from a momentum imbalance near the equator associated with data assimilation, which possibly affected the El Niño prediction failure in 2014, was much reduced by our developed ocean data assimilation system and reanalysis. This study, which provides a new connection between the theoretical works and reanalysis products that use sophisticated systems for synthesizing OGCMs and observations, should be useful for climate research and operational communities interested in ENSO.

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