The hurricane-induced decay of mesoscale eddies: an energy source for near-inertial waves

Abstract

Oceanic mesoscale eddies represent the most energetic component of the ocean circulation; however, their decay routes remain elusive, leaving a key gap in our understanding of the oceanic energy cascade. Here we combine satellite data with mooring measurements from the South China Sea in 2014 to reveal a rapid decay of oceanic eddies following the passage of typhoons. This decay is accompanied by a substantial generation of internal waves with near-inertial frequency in the ocean interior. The temporal correspondence and quantitatively commensurate energy changes indicate a direct energy transfer from eddy to internal wave field. We propose that typhoon-induced perturbations trigger an adjustment process within the eddy, resulting in energy loss through the radiation of near-inertial waves. Quantitatively supported by numerical model simulations, this mechanism plays a crucial role in the evolution of mesoscale eddies and points to a previously overlooked interior source of oceanic internal waves.

Introduction

Geostrophic eddies dominate the oceanic kinetic energy, comprising over 90% of the total near-surface kinetic energy¹. These coherent structures are fundamental to oceanic heat, carbon, and nutrient transport, with profound implications for global climate². Advances in observational technologies over recent decades have elucidated key kinematic properties of eddies such as translation velocities, spatial structures, and interactions with the mean currents^3,4,5,6. Nevertheless, the pathways through which this massive energy reservoir is ultimately dissipated remain poorly constrained.

Quasi-geostrophic turbulence theory predicts an inverse energy cascade, transferring mesoscale eddy energy toward larger scales⁷. To maintain a statistically steady state, this energy must be effectively dissipated to prevent unlimited energy accumulation on large scales. However, the direct dissipation through boundary friction is estimated to be insufficient to close the energy budget of the eddy field⁸. Thus, other processes must provide an effective pathway for a forward energy cascade, transferring energy toward smaller scales where it can eventually be dissipated through irreversible mixing¹. Among proposed mechanisms^{9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}, the energy transfer to internal waves via loss of balance has attracted increasing attention^{19,20,21,22,23,24,25,26,27,28,29,30}.

Statistical analyses in a tropical-cyclone-centered coordinate frame have established that the passage of strong storms leads to a noticeable decay of both cyclonic eddies (CEs) and anticyclonic eddies (AEs) outside the storm’s vorticity-dominated core, where wind-induced perturbations remain substantial^31,32. These studies attribute the eddy decay to geostrophic adjustment: mesoscale eddies are perturbed out of geostrophic balance by strong storms and subsequently adjust back toward equilibrium, losing energy in the process. This interpretation is grounded in a well-established theoretical foundation that geostrophic adjustment releases non-geostrophic energy in the form of inertia-gravity waves^33,34,35. In the oceanic observation, the kinetic energy of the high-frequency internal wave field—within which these theoretical waves reside—is dominated by near-inertial frequencies¹, i.e., the near-inertial waves (NIWs). A canonical example bridging the statistical phenomena to the NIW field is the observed upward propagation of near-inertial energy (NIE) within a weakening CE in the Gulf of Mexico following its interaction with Hurricane Katrina³⁶. While these correlative observations are highly suggestive, direct and quantitative evidence for a causal energy transfer pathway from decaying eddies to NIWs remains elusive, owing to the scarcity of concurrent interior observations.

Here, we provide direct observational and numerical evidence for this energy transfer mechanism. Analysis of satellite altimetry revealed rapid AE decay in the South China Sea after the passage of Typhoon Rammasun (July 2014) and Kalmaegi (September 2014). To investigate the underlying mechanism, we combined the data from a moored array and found a concurrent, substantial enhancement of NIE in the ocean interior during both events. Through targeted numerical experiments, we isolated the geostrophic adjustment process and demonstrated that the energy lost from the mesoscale eddy was quantitatively transferred to NIWs. Our findings establish geostrophic adjustment as a direct and efficient pathway for channeling mesoscale energy into the internal wave field, with broad implications for the energetics of oceanic circulation.

Results

Responses of oceanic eddies and NIWs to typhoons

To investigate the upper ocean response to typhoons, a cross-shaped moored array comprising five moored buoys and four subsurface moorings was deployed in the northern South China Sea from June 2014 to April 2015. The synchronous atmospheric and oceanic observations are compiled into the MASCS 1.0 dataset³⁷. Key instruments included Acoustic Doppler Current Profilers (ADCPs) that measured current velocities covering the upper 850 m water column at 15-min resolution, and SeaBird 37 recorders that sampled temperature and salinity from 0 to 400 m depth and near the bottom at 2-min intervals. During the observation period, two major typhoons—Typhoon Rammasun (July) and Typhoon Kalmaegi (September)—passed over the observation area. The 3-h typhoon track and intensity data were obtained from the International Best Track Archive for Climate Stewardship (IBTrACS)^38,39, indicating maximum sustained winds of approximately 44 m s⁻¹ and 33 m s⁻¹, respectively, at their closest approach to the array. With five stations adjacent to both storms’ tracks (Fig. 1), the deployment of the array provided an exceptional opportunity to observe the upper ocean’s response to both typhoons.

Fig. 1: Sea level anomaly (SLA) snapshots during two typhoon events.

Relative times are given in days with respect to each typhoon’s passage. a–c SLA field on 12 Jul (−5 days), 17 Jul (0 days), and 22 Jul (+5 days) 2014 during Typhoon Rammasun (blue track). d–f SLA field on 10 Sep (−5 days), 15 Sep (0 days), and 20 Sep (+5 days) 2014 during Typhoon Kalmaegi (green track). Black pentagrams mark the five stations.

Based on daily gridded sea level anomaly (SLA) data distributed by Archiving, Validation and Interpretation of Satellite Oceanographic Data in Oceanography (AVISO), we identified a pre-existing AE covering most of the array prior to each typhoon’s arrival (Fig. 1a, d). This dataset is produced by integrating multiple altimeter missions using optimal interpolation to reconstruct mesoscale signals⁴⁰, a methodology that makes it widely applied in oceanic eddy research^19,31,32,41 and fully appropriate for this study. Following the passage of each storm, the data revealed a marked SLA decrease at the core of the respective eddy (Fig. 1b, c for Rammasun; Fig. 1e, f for Kalmaegi), signaling a noticeable weakening of these AEs. The reduction was more pronounced during Kalmaegi, likely due to its closer proximity to the eddy core. This clear signal of typhoon-induced eddy attenuation raises a fundamental question: What is the fate of the energy lost from these weakening eddies?

By a rather preliminary analysis, we were led to a crucial dynamical element in the typhoon-ocean interaction process, the NIWs. The near-inertial velocities at each station were extracted by bandpass filtering the ADCP-measured currents, with station-specific filtering bandwidths based on the e-folding scale of Gaussian-fitted near-inertial peaks in the frequency spectra (Supplementary Fig. 1; Methods). The NIWs are commonly observed phenomena in the upper ocean following the passage of typhoons. Surface drifter trajectories trapped within the AE during Rammasun’s passage revealed near-inertial oscillations superimposed on the eddy’s clockwise advection (Supplementary Fig. 2). According to the widely accepted paradigm^42,43,44,45, by inciting oscillations at near-inertial frequencies, typhoons inject NIE into the oceanic mixed layer. Then, by convergence or divergence in association with these oscillations, the isopycnals below the mixed layer are perturbed, transmitting NIE from the surface to the interior. This paradigm of surface-generated NIWs was reflected in the time-depth distribution of near-inertial kinetic energy (NIKE) at station 4 following Rammasun, characterized by a distinct, narrow band that propagated downward over time and took about 30 days to reach 300-m depth (Fig. 2a). In stark contrast, the response at station 2 was immediate and dramatic. After the passage of Rammasun, NIKE increased substantially and nearly simultaneously across the entire upper 400 m of the water column within just 2–3 days (Fig. 2b). This rapid, bulk energization within the ocean interior is incompatible with the slow downward-propagation process. Notably, this energy burst was reproducible. During Kalmaegi, station 2 exhibited a qualitatively similar response despite differing large-scale conditions, underscoring the robustness of this phenomenon (Fig. 2c).

Fig. 2: Spatio-temporal evolution of near-inertial kinetic energy (NIKE) during typhoon passages.

Time-depth distribution of NIKE (log10 scale) at station 4 after Typhoon Rammasun (a) and at station 2 after Typhoon Rammasun (b) and Kalmaegi (c). Dashed black lines mark the typhoon arrival on 17 July 2014 (a, b) and on 15 Sep 2014 (c). Local inertial periods (IPs) are 1.5 days and 1.6 days for stations 4 and 2, respectively. The NIKE values were averaged over the respective IP at each station.

The observed eddy weakening and concomitant NIKE enhancement following typhoons suggested a potential energy transfer between the eddy field and the internal wave field. To assess this conjecture, we focused on data from stations 2 and 4 during the first typhoon event (Rammasun). This selection was made to ensure robustness, as the second event (Kalmaegi) and other stations were compromised by extensive data gaps or physical processes that confounded the signal of interest (see Methods for full selection criteria). The first event provided a complete dataset and the clearest contrast between the anomalous response at station 2 and the classical wind-driven response at station 4, allowing a definitive quantification of our conjecture.

Interior NIW generation coincident with surface-to-depth eddy decay

Based on the representative first event, we evaluated whether surface-generated NIWs could exclusively account for the post-typhoon NIKE enhancement observed at station 2 through a proof-by-contradiction approach. Assuming the observed NIKE originated entirely from surface wind and propagated downward, we used a ray-tracing model⁴⁶ that traces rays backward in time from the observed energy peak at 300 m depth of station 2 to examine whether they can return to the sea surface within the observed energization period of 2–3 days (Methods). The surface-only generation assumption, combined with the observed vertical phase structure of the NIWs, set the allowable vertical-wavenumber band and thus the maximum downward group velocity. Placing the assumption under the most favorable conditions, we initialized the model with the vertical wavenumber that gives the largest possible wavelength—and hence the fastest downward propagation. Even with this optimal configuration, the simulated rays took over 10 days to connect surface origins to the 300 m depth at station 2. Specifically, after a 10-day backward integration, all rays remained deeper than 70 m, with fewer than 1% reaching depths shallower than 100 m (Fig. 3a). Moreover, the 10 fastest rays required 12–16 days to ascend above 50 m (Fig. 3b). This timescale vastly exceeded the observed energization window. Given this fundamental discrepancy, we conclusively reject the surface-only generation assumption, confirming that an interior source must be involved.

Fig. 3: Ray-tracing analysis and rotary spectral analysis.

a Pathways of all simulated rays within 10-day backward integration. The location of station 2 is denoted by the red pentagram. b Three dimensional trajectories of the 10 fastest rays. Gray plane indicates the 50 m isobath. c Rotary spectrum of the near-inertial motions isolated from ADCP data at station 4 averaged within the first two inertial periods after Rammasun. The unit nm represents normalized meters, sm denotes stretched meters, and c.p.sm denotes cycles per stretched meter³⁶. Blue line is for clockwise (CW) rotating component and red line is for counterclockwise (CCW) rotating component. Colored shadings represent 95% confidence intervals computed using bootstrap method. d Same as c but at station 2.

Rotary spectral analysis provided further evidence. For NIWs, the vertical propagation is expected to be predominantly downward for wind-generated waves and upward for lee waves, with bidirectional propagation possible during loss of balance or wave-wave interactions⁴⁷. In the Northern Hemisphere, downward-propagating NIE exhibits clockwise (CW) polarization, whereas upward propagation corresponds to counter-clockwise (CCW) rotation⁴⁸. We computed rotary spectra for near-inertial motions at stations 2 and 4 in the first event, with WKBJ approximation and stretched vertical coordinates applied^36,48. At station 4, the dominance of CW over CCW component suggested the wave energy was predominantly surface-generated by wind (Fig. 3c). In comparison, the difference between two components is greatly narrowed at station 2 (Fig. 3d). The diminished dominance of the CW component indicated the prominence of the upward energy propagation at station 2, corroborating an interior-generation process of NIWs in the first event.

Having confirmed the existence of an interior energy source, we next identified the weakening mesoscale eddy as its most probable source. To this end, we quantified the energy loss of the AE in the first event using SLA data, in which the sea surface height signature of NIWs is negligible⁴⁹. Specifically, we computed the surface geostrophic kinetic energy by first deriving geostrophic velocities from the SLA data based on geostrophic balance, and then averaging the resulting energy over the eddy area identified with the Okubo-Weiss parameter^50,51 (Methods). Prior to typhoon’s arrival (16 July 2014), the surface geostrophic kinetic energy was 42.9 ± 2.5 J m⁻³. Following the passage of the typhoon, the surface eddy kinetic energy (EKE) dropped rapidly for several days and then declined more slowly over the subsequent 10 days, eventually decreasing to less than half of its initial value (Fig. 4a). Correspondingly, the kinetic-energy changing rate of the eddy plummeted to about −3.5 ± 1.7 J m⁻³ day⁻¹ before gradually returning to a minimal background level (Fig. 4b). The surface eddy available potential energy (EPE) also exhibited marked decay, with a pre-typhoon value of 29.2 ± 1.2 J m⁻² and a peak decay rate at around −1.7 ± 0.8 J m⁻² day⁻¹ (Fig. 4c, d).

Fig. 4: Evolution of surface eddy energy and subsurface eddy velocity during the first typhoon event.

a Surface eddy kinetic energy (EKE) averaged over the anticyclonic eddy area. b Temporal rate of change of surface EKE. c, d Same as a, b but for eddy available potential energy (EPE). All energy metrics are calculated from daily sea level anomaly data, with the dashed lines denoting the arrival of Typhoon Rammasun on 17 July 2014. Blue shadings represent the uncertainty range derived from propagating the formal mapping error of the altimetry data. e Cross-track component of eddy swirl velocity at station 2, low-pass filtered from ADCP data recorded at 15-min intervals. The top horizontal axis represents the distance between station 2 and the eddy center, normalized by the radius of maximum swirl velocity, R_max. The cross-track direction was defined perpendicular to the typhoon’s path with positive values indicating direction away from the typhoon track.

To determine whether this energy loss extended into the ocean interior, we analyzed the cross-typhoon-track velocity component at station 2. This component, defined as perpendicular to the typhoon’s path, was derived from ADCP measurements processed with a 5-day low-pass filter (Methods). The resulting eddy velocity revealed that the energy decline indeed penetrated into the ocean interior. As the anticyclone translated northwestward toward station 2, the evolution of the cross-track velocity component at station 2 exhibited three distinct phases (Fig. 4e). Prior to the typhoon’s closest approach (12–17 July), velocity intensified as the station converged upon the eddy’s maximum velocity radius (R_max), consistent with expected dynamics. Critically, during the immediate post-typhoon period (17–22 July), the distance between station and eddy continued to decrease to 1.1 R_max, yet velocity at all depths declined substantially—a sharp reversal of the pre-typhoon trend of stronger flow observed at closer proximity. This anomalous reduction provided unambiguous evidence of typhoon-induced eddy energy loss in the ocean interior. Thereafter (22 July–1 August), the velocity evolution resumed its expected relationship with distance as the station moved through and inside the R_max.

The clearly comprehensive decay of the AE during the first event warranted a full three-dimensional quantification of its total energy loss and an assessment of the energy transfer from the decaying AE to the NIW field. We selected a control volume encompassing the AE (Supplementary Fig. 3), with its boundary positioned to include station 2. Although located outside the anticyclonic vorticity core, station 2 lay within a critical transition zone where strong eddy flow could potentially supply energy for NIWs, allowing the control volume to capture the complete energy pathway, including local enhancement and lateral radiation of NIE. The total eddy energy, comprising both kinetic and available potential energy components, was derived from a three-dimensional reconstruction that combined the horizontal structure from SLA data with an empirically determined vertical structure from normalized, low-pass filtered ADCP velocity profiles (Methods).

Our energy balance analysis showed that the net eddy energy decline of (5.7 ± 0.9) × 10¹⁴ J was largely balanced by the total generation of NIE. Two weeks after the typhoon’s passage, the local NIKE within the control volume reached (3.0 ± 0.2) × 10¹⁴ J, while the cumulative lateral energy flux amounted to (3.3 ± 0.2) × 10¹³ J. Together, these NIE contributions accounted for more than one-half of the energy decrease in the central AE, providing quantitative evidence for the energy transfer mechanism. Temporal analysis validated that the AE’s energy loss was well compensated by the growth of NIE over time (Fig. 5). This conclusion is robust to the choice of control volume, as comparisons between fixed and eddy-following volumes showed negligible differences due to eddy propagation during this period (Supplementary Fig. 4).

Fig. 5: Comparison between eddy decay and near-inertial wave generation.

Blue line shows volume-integrated eddy energy (EE) based on the combination of SLA data and ADCP data. Red line shows total interior-generation of the near-inertial energy (NIE) derived from ADCP data. Blue shading represents the uncertainty range derived from propagating the formal mapping error of the altimetry data. Red shading represents the uncertainty determined from sensitivity tests of the inertial-period averaging window.

To bound the contribution of direct wind spin-down, a canonical eddy decay pathway^9,10, we performed an upper-bound estimate using the conventional drag law under extreme winds (Methods). This yielded an energy dissipation rate (~10¹²J day⁻¹), an order of magnitude smaller than the total eddy energy loss rate, ruling it out as the primary driver. In contrast, the temporal correspondence and commensurate energy changes between eddy decay and NIW growth provided compelling observational evidence for a direct energy transfer. This led us to propose geostrophic adjustment as a plausible mechanism. As theorized, middle-to-large scale fields adjust toward geostrophic equilibrium via geostrophic adjustment, shedding energy as inertia-gravity waves^33,34,35. Accordingly, we conjecture that the loss of geostrophic balance, triggered by typhoon-induced perturbation, initiated an adjustment process that released energy in the form of inertia-gravity waves, resulting in the observed eddy decay. Given the horizontal scale of mesoscale eddies, the radiated wave field could be dominated by frequencies near the inertial band.

Energy transfer from perturbed eddies to NIWs in numerical experiments

To test this mechanistic interpretation, we conducted a series of idealized numerical experiments using the Massachusetts Institute of Technology general circulation model (MITgcm), which has been widely applied in studying geostrophic-internal wave interactions^30,41,52. The simulations are performed on an f-plane with f = 1 × 10⁻⁴ s⁻¹, and the configuration is designed to isolate the energy transfer pathway by suppressing viscous dissipation, topography scattering, barotropic gravity waves, Rossby-wave dispersion, and boundary reflections (see Methods for details).

We designed six cases with varying background eddies and perturbation types to assess the robustness of the energy pathway (Table 1). All experiments employ mesoscale eddies with a Gaussian-shaped sea surface height anomaly described by $\eta \left(r\right)={\eta }_{0}\left(1-\frac{{r}^{2}}{2{R}_{0}^{2}}\right)\cdot \exp \left(-\frac{{r}^{2}}{2{R}_{0}^{2}}\right)$⁵, where ${\eta }_{0}$ is the amplitude at the eddy center, $r$ is the radial distance, and ${R}_{0}=$50 km is the characteristic radius. The eddies share an identical initial intensity, characterized by a maximum swirl velocity of 0.5 m s⁻¹, corresponding to a Rossby number of ~0.1. Given a first baroclinic Rossby deformation radius of 27 km in our experimental setup, the horizontal scale of the eddies satisfies the condition for geostrophic balance.

Table 1 List of model setup and resulting energy transfer for the experiments

To systematically dissect the energy pathway, our experimental strategy follows a progression from simplified to realistic perturbations. Cases 1–4 employ full-depth flow perturbations, while Case 5 adopts a surface-intensified perturbation confined to the upper 500 m. Furthermore, Case 6 introduces surface wind forcing to represent a more realistic scenario, thereby testing the mechanism under the hurricane-induced perturbation. In each case, an initial impulsive perturbation disrupts the geostrophic balance of the eddy, triggering an unbalanced eddy field that subsequently undergoes geostrophic adjustment and returns toward equilibrium.

We illustrate the adjustment process using Case 1, in which a geostrophic CE is perturbed by a flow field with identical spatial structure but reduced amplitude (Supplementary Fig. 5). Following the initial perturbation, geostrophic adjustment proceeds through two dynamical regimes³³: an initial rapid phase (T ~ f⁻¹) marked by the establishment of quasi-geostrophic balance and the excitation of inertial-gravity waves that disperse non-geostrophic energy, followed by a slower phase wherein the waves radiate outward while the eddy evolves gradually. The short duration of the initial phase hinders separation via conventional temporal filters, motivating our recourse to a spatial diagnostic method.

The total energy field during the slower phase reveals the propagation of waves and their spatial decoupling from the eddy. The radiated waves exhibit cylindrical wavefronts with a ripple-like modulation, their amplitudes diminishing radially due to geometric spreading and dispersion (Fig. 6a–c). Their vertical structure reflects a modal separation: faster mode-1 waves lead the propagation, followed by slower higher-mode components (Fig. 6d–f). Leveraging this spatial decoupling, we quantify the energy flux across a cylindrical boundary at a distance of 250 km, optimized to avoid contamination by proximal eddy signals and distal numerical dissipation. The retained wave energy and the net eddy energy change within the eddy region are evaluated after three inertial periods, by which time temporal filtering becomes viable for separating the signals (Methods). The total wave energy generated (1.21 × 10¹³J) effectively balances the eddy energy loss (1.31 × 10¹³J) in Case 1. Spectral analysis confirms that wave energy is predominantly concentrated in the near-inertial band (Fig. 7a). Extending this analysis to all cases reveals consistent behavior: wave energy generation accounts for more than 90% of the perturbed eddy’s energy loss across cases, with spectral peaks near the inertial frequency (Table 1 and Fig. 7).

Fig. 6: Evolution of total energy distribution across perspectives and cases in numerical experiments.

Sea-surface horizontal distribution for Case 1 at day 4 (a), day 6 (b), and day 8 (c). Vertical distribution along the Y = 0 transect for Case 1 at day 4 (d), day 6 (e), and day 8 (f). Vertical distribution along the Y = 0 transect for Case 5 at day 4 (g), day 6 (h), and day 8 (i). Vertical distribution along the Y = 0 transect for Case 6 at day 4 (j), day 6 (k), and day 8 (l). The total energy is calculated as the sum of kinetic energy and available potential energy. Red color shading remains at the center represents the non-propagating, energetic eddy field. Ripple-like structures denote the propagating waves.

Fig. 7: Frequency spectra of velocity field in numerical experiments.

a Spatially averaged power spectral density (PSD) for Case 1. b–f Same as a but for Cases 2–6. Dashed lines denote the Coriolis frequency f = 1 × 10⁻⁴ s⁻¹.

To resolve the spatial structure of wave generation during the early adjustment when conventional filtering fails, we employ an envelope-mean method. It isolates wave signals via their near-zero local mean, incorporates signal preprocessing to suppress boundary oscillations⁵³, and operates without predefined spectral bands. Although quantitative accuracy is limited initially, the method reliably captures the dominant spatial pattern of the wave source term, derived diagnostically from the local energy tendency and flux divergence (Methods).

In Case 1, the intensive wave generation is confined primarily within the eddy core during the first inertial period, with negligible values beyond a 100 km radius (Fig. 8a), confirming that the waves are generated solely by the adjusted eddy itself. The faint source activity in the second period is an order of magnitude weaker (Fig. 8b). Apart from the diminishing adjustment, the residual weak signal may stem from either a numerical artifact of the fixed integration window or tenuous eddy-wave nonlinear interactions. This rapid, marked decline of source strength indicates that wave generation is essentially complete within a few inertial periods. Thereafter, the wave field evolves primarily through the propagation of the energy released during this early adjustment.

Fig. 8: Vertical pattern of the wave energy source/sink during the early adjustment phase in numerical experiments.

a Distribution of source/sink term integrated over the first inertial period (IP; 1IP = 17.5 h) for Case 1. Same as a but over the second inertial period (b), for Case 5 (c), and for Case 6 (d).

Notably, even shallow perturbations can excite a full-depth wave response. In Case 5, a perturbation confined to the upper 500 m triggers wave generation extending below 500 m (Fig. 8c). As the perturbation structure modulates wave properties, the waves are radiated with energy concentrated in higher vertical modes, contrasting with the mode-1 dominance in Cases 1–4 (Fig. 6d–i).

In the more realistic wind-forced scenario (Case 6), a geostrophic AE subjected to 6 h of cyclonic wind forcing attenuated by 7.41 × 10¹³ J upon wind cessation, about 95% of which is converted into wave energy. Wave generation during early adjustment penetrates below the surface, with its spatial structure organized by and anchored to the pre-existing eddy (Fig. 8d). This demonstrates that surface wind forcing activates a full-depth adjustment process mediated by the eddy. Subsequent wave propagation patterns further corroborate the radiation mechanism (Fig. 6j–l). These findings thus confirm that the intrinsic energy pathway operates effectively under the hurricane-driven conditions that mirror our observations.

Collectively, these results demonstrate a consistently high fraction (>90%) of energy transfer from eddy decay to wave generation through geostrophic adjustment, though the intensity of the adjustment process varies with cases. Crucially, even shallow perturbations excite a surface-to-depth wave generation. This simulated deep response is consistent with the observed rapid energization and provides the interior wave source inferred from ray-tracing analysis, underscoring the efficiency of this pathway in transferring geostrophic imbalance into wave-mediated dissipation.

Discussion

Wunsch and Ferrari⁸ have given a rough summary of the energy budget for the global ocean circulation, suggesting that the potential importance of the loss of balance in the geostrophic mesoscale merits further investigation. Our study documented a reproducible case of this phenomenon: the rapid decay of strong AEs following the passage of Typhoon Rammasun (July 2014) and Typhoon Kalmaegi (September 2014), which was accompanied by a substantially subsurface-generated NIE. This repeatability motivated a further investigation into the link between these events.

We quantitatively examined this underlying energy pathway. Ray-tracing and rotary spectral analyses confirmed an interior NIW source, coinciding with the observed attenuation of the eddy. While our analyses provide compelling evidence for the energy transfer, several inherent limitations in the observational dataset must be acknowledged. First, the estimate of the eddy’s energy loss relied on satellite altimetry, which captured primarily the surface expression of the process. Second, the calculation of the NIE generation was constrained by the spatial coverage of the two stations, introducing uncertainty into the absolute energy quantification. Despite these limitations, the consistent temporal covariation and the commensurability of energy changes between the two reservoirs provided a robust foundation for a direct energy transfer.

To explain this, we turned to the classical theory of geostrophic adjustment, which had established that unbalanced flows radiate energy as waves to regain balance^54,55. We propose that mesoscale eddies, when perturbed by external perturbations as they probably constantly are, may undergo this adjustment process. Given the horizontal scale of the observed eddies, the waves radiated outward from them probably had a frequency close to the inertial frequency, consistent with our observations of NIWs. Although the typhoon acts as the essential external trigger, our analyses focus on the subsequent intrinsic energy pathway.

Numerical simulations provide strong quantitative support for our proposal. The model results indicate that over 90% of the energy lost from the perturbed eddies is released as NIWs. This confirms that the self-adjustment of the perturbed eddies, a process that could be initiated by any sufficiently disruptive agent, represents the primary mechanism channeling their decaying energy into the wave field. Notably, the efficiency of this adjustment process is modulated by the perturbations’ characteristics. Although the adjustment is rapid in our idealized simulation with impulsive perturbations, its duration is expected to be prolonged in the presence of a complex oceanic background. This expectation, combined with the limitations of observational data, reconciles the discrepancy between the model and observations. Our framework, supported jointly by observations and modeling, thus resolves a fundamental question of the fate of energy during eddy decay, potentially explaining the universal decay of eddies following typhoons^31,32.

The energy transfer pathway we identify not only explains the observed eddy decay but also has broader implications for large-scale ocean circulation. This is particularly relevant given that mesoscale eddies dominate oceanic kinetic energy at sub-inertial frequencies and play a crucial role in the energy cascade^1,8,20. If a substantial portion of eddy energy is indeed converted to NIWs through the geostrophic adjustment process described here, it would represent a previously underappreciated sink for the eddy field and a source for the internal wave field. Furthermore, the NIWs generated via this process may represent a major contributor to ocean mixing through strong shear and breaking, thereby influencing climate and biogeochemistry^8,47. Additionally, AEs can trap NIWs^{46,47,56,57,58}, providing an effective route for them to propagate into the deep ocean and consequently contribute to the abyssal mixing that sustains the thermohaline circulation.

Methods

Temperature and salinity data

The cross-shaped array consisted of five moored buoys and four subsurface moorings. Temperature and salinity were measured every 2 min by SeaBird 37 recorders from 0 to 400 m depth (on the buoys) and near the bottom (on the moorings). However, the strings of SeaBird recorders from stations 1 and 3 were lost during the deployment, and records from stations 2 and 5 lacked coverage of the first typhoon event. Given that only station 4 provided data during Typhoon Rammasun, the background density field for energy calculations and ray-tracing analysis was reconstructed by integrating the station 4 data with contemporaneous Argo float profiles⁵⁹ and the World Ocean Atlas 2018 (WOA18, ref. ⁶⁰).

Surface eddy energy

The Okubo-Weiss parameter^50,51, defined as

$$W={\left(\frac{\partial {v}_{{{{\rm{g}}}}}}{\partial x}+\frac{\partial {u}_{{{{\rm{g}}}}}}{\partial y}\right)}^{2}+{\left(\frac{\partial {u}_{{{{\rm{g}}}}}}{\partial x}-\frac{\partial {v}_{{{{\rm{g}}}}}}{\partial y}\right)}^{2}-{\left(\frac{\partial {v}_{{{{\rm{g}}}}}}{\partial x}-\frac{\partial {u}_{{{{\rm{g}}}}}}{\partial y}\right)}^{2},$$

(1)

where ${u}_{{{{\rm{g}}}}}=-\frac{g}{f}\frac{\partial \left({{{\rm{SLA}}}}\right)}{\partial y}$ and ${v}_{{{{\rm{g}}}}}=\frac{g}{f}\frac{\partial \left({{{\rm{SLA}}}}\right)}{\partial x}$ are the zonal and meridional geostrophic velocities, respectively, was used to identify the eddy from the SLA map⁶¹. The area $A$, enclosed by the contour $W=-0.2\,{\sigma }_{w}$ (where ${\sigma }_{w}$ is the spatial standard deviation of $W$ in the study region), defined the horizontal scale of the eddy ${R}_{0}=\sqrt{A/\pi }$.

The geostrophic kinetic energy per unit volume at the surface was calculated by

$${{{\rm{GKE}}}}=\frac{1}{2}{\rho }_{0}\left({u}_{{{{\rm{g}}}}}^{2}+{v}_{{{{\rm{g}}}}}^{2}\right),$$

(2)

where ${\rho }_{0}$ is the surface seawater density. The surface available potential energy per unit area associated with the eddy was given by

$${{{\rm{SAPE}}}}=\frac{1}{2}{\rho }_{0}g{\left({{{\rm{SLA}}}}-{{{{\rm{SLA}}}}}_{0}\right)}^{2},$$

(3)

where ${{{{\rm{SLA}}}}}_{0}$ is the SLA value around the eddy edge (the outermost closed SLA contour enclosing the eddy⁴). By tracking the local maximum of SLA each day as the center of the anticyclone, the surface EKE and EPE were computed as the area-average over the eddy:

$${{{{\rm{EKE}}}}}_{{{{\rm{s}}}}}=\frac{1}{A}{\int }_{A}{{{\rm{GKE}}}}{ds},$$

(4)

$${{{{\rm{EPE}}}}}_{{{{\rm{s}}}}}=\frac{1}{A}{\int }_{A}{{{\rm{SAPE}}}}{ds}.$$

(5)

Geostrophic relative vorticity is a key attribute of eddies, and can also be used to identify eddy and validate the result obtained from the Okubo-Weiss method. Here, the e-folding scale of geostrophic relative vorticity at the eddy center is used to isolate the eddy. The resulting eddy energy changes are consistent with those obtained using the Okubo-Weiss method (Supplementary Fig. 6).

The uncertainty ranges for the surface eddy energy are quantified by propagating the formal mapping error of the daily gridded SLA data through the standard error propagation formula.

Eddy cross-track component

To further validate the energy evolution and dynamical response of the eddy to the typhoon, we analyzed the time evolution of the cross-track velocity component at station 2. The eddy velocity was obtained by applying a 5-day low-pass filter to the raw current measurements. The cross-track direction was defined perpendicular to the typhoon’s path with positive values indicating direction away from the typhoon track. The distance from each station to the eddy center was normalized by the radius of maximum velocity R_max of the anticyclone, where R_max was determined as the radial distance at which the depth-averaged magnitude of eddy velocity reaches its maximum.

Three-dimensional reconstruction of eddy

To estimate the full three-dimensional energy distribution, the vertical structure of the eddy was considered, while the horizontal structure was determined by the SLA data. According to the vertical structure of mesoscale eddies proposed by ref. ⁵, the horizontal eddy velocities ${u}_{{{{\rm{e}}}}}$ and ${v}_{{{{\rm{e}}}}}$ share a vertical function decrease with the depth. Here, this vertical structure function, $H\left(z\right)$, was obtained empirically: velocity data from stations 2, 4, and 5 were 5-day low-pass filtered, normalized by the surface geostrophic velocity, and then temporally averaged over the typhoon passage period. The resulting profiles from the three stations were averaged to define a unified vertical structure $H\left(z\right)$ for the AE. The horizontal eddy velocity components at depth were then derived as ${u}_{{{{\rm{e}}}}}\left(z\right)={u}_{{{{\rm{g}}}}}\cdot H\left(z\right)$ and ${v}_{{{{\rm{e}}}}}={v}_{{{{\rm{g}}}}}\cdot H\left(z\right)$. The density anomaly ${\rho }_{{{{\rm{e}}}}}$ associated with the eddy was derived hydrostatically from the ocean pressure anomaly. The pressure anomaly ${p}_{{{{\rm{e}}}}}$, which varies with depth as ${p}_{{{{\rm{e}}}}}={p}_{0}\cdot H\left(z\right)$, originates from the sea surface pressure anomaly ${p}_{0}={\rho }_{0}g\left({{{\rm{SLA}}}}-{{{{\rm{SLA}}}}}_{0}\right)$. The density anomaly is then given by ${\rho }_{{{{\rm{e}}}}}=\frac{1}{g}\frac{\partial {p}_{{{{\rm{e}}}}}}{\partial z}$. Finally, the EKE and EPE are given by the volume integrals:

$$\left\langle {{{\rm{EKE}}}}\right\rangle ={\int }_{V}\frac{1}{2}{\rho }_{0}\left({u}_{{{{\rm{e}}}}}^{2}+{v}_{{{{\rm{e}}}}}^{2}\right)d\tau ,$$

(6)

$$\left\langle {{{\rm{EPE}}}}\right\rangle ={\int }_{V}\frac{1}{2}{\rho }_{0}{N}^{2}{\zeta }^{2}d\tau .$$

(7)

where angle brackets denote the volume-integrated quantity, $N=\sqrt{-\frac{g}{{\rho }_{0}}\frac{\partial \rho }{\partial z}}$ is the buoyancy frequency obtained from the background potential density field $\rho$ and $\zeta =-\frac{{\rho }_{{{{\rm{e}}}}}}{\partial \rho /\partial z}$ is the isopycnal displacement of the eddy.

Near-inertial energy at stations

The horizontal velocity observed from the moored array was first examined to obtain the frequency spectra (Supplementary Fig. 1). Gaussian fitting results were employed to determine the e-folding scale of the near-inertial frequency band. Subsequently, bandpass filtering was applied to extract the near-inertial velocity components, denoted as ${u}_{{{{\rm{i}}}}}$ and ${v}_{{{{\rm{i}}}}}$. The frequency passbands of 0.92–1.11 f₂ at station 2 and 0.87–1.19 f₄ at station 4 were used, respectively and the baroclinic structure of NIWs at two stations is shown in Supplementary Fig. 7. Hence, the NIKE per unit volume was computed as:

$${{{\rm{NIKE}}}}=\frac{1}{2}\rho \left({u}_{{{{\rm{i}}}}}^{2}+{v}_{{{{\rm{i}}}}}^{2}\right).$$

(8)

It is noteworthy that the observational temperature and salinity data were insufficient to reliably compute the potential energy of NIWs. However, given that the kinetic energy of near-inertial motions is typically much stronger than their potential energy component⁴⁷, the use of NIKE alone is deemed sufficient to characterize the energy variability of NIWs in the observational study.

Data and events selection criteria

To substantiate the existence of an internal NIW source linked to eddy decay, we focused our quantitative analysis on specific stations and events that provided the clearest signal. We primarily utilized data from stations 2 and 4 solely on the first typhoon event (Rammasun) because of three practical and physical constraints that compromised other stations or the second event (Kalmaegi).

• 1. Data availability. The upper 200 m of data were unavailable at stations 1 and 4 following Kalmaegi. Furthermore, the buoy at station 3 was lost after its wire rope was snapped by Kalmaegi, resulting in a data absence for both events.

• 2. Signal strength and purity. During the first event, station 1 was located far from both the eddy core and the typhoon track, resulting in an overly weak near-inertial signal that was unsuitable for quantitative analysis. More critically, station 5—situated within the AE core—was strongly affected by negative vorticity, which trapped NIWs and made it difficult to separate interior-generated NIWs from surface-generated, accelerated downward-propagating signals. Prior to the second typhoon, a strong pre-typhoon NIE at station 5 further complicated the isolation of the post-typhoon generated signal (Supplementary Fig. 8).

• 3. Eddy energy estimation. Although an AE was present during the second event, its core was too distant from the moored array to allow a robust estimate of the eddy’s energy loss.

Thus, the first event provided the most complete and interpretable dataset, with stations 2 and 4 offering clearly contrasting and representative responses, allowing a definitive quantitative analysis.

Ray tracing analysis

To determine whether surface-generated NIWs were the exclusive source of the simultaneous NIKE increase within the upper 300 m depth post typhoon, a backward ray-tracing analysis was performed based on the theoretical framework of wave-mean flow interaction⁴⁶ that has been successfully applied to identify the propagation trajectories of NIWs in mesoscale eddies^36,57,58. In this ray-tracing model, the wave position ${{{\boldsymbol{r}}}}$ and the wavenumver ${{{\boldsymbol{k}}}}=({k}_{x},{k}_{y},{k}_{z})$ are governed by

$$\frac{d{{{\boldsymbol{r}}}}}{{dt}}={{{{\boldsymbol{C}}}}}_{{{{\rm{g}}}}}+{{{{\boldsymbol{V}}}}}_{{{{\rm{g}}}}},$$

(9)

$$\frac{d{{{\boldsymbol{k}}}}}{{dt}}=-\nabla \omega ,$$

(10)

where ${{{{\boldsymbol{C}}}}}_{{{{\rm{g}}}}}=({C}_{{{{\rm{g}}}}x},{C}_{{{{\rm{g}}}}y},{C}_{{{{\rm{g}}}}z})$ is the group velocity of the NIWs, ${{{{\boldsymbol{V}}}}}_{{{{\rm{g}}}}}=({U}_{{{{\rm{g}}}}},{V}_{{{{\rm{g}}}}})$ is the backgound geostrophic velocity and $\omega$ is the Eulerian frequency. The dispersion relation is given by

$$\omega ={\omega }_{0}+{{{\boldsymbol{k}}}}{{{\boldsymbol{\cdot }}}}{{{{\boldsymbol{V}}}}}_{{{{\rm{g}}}}},$$

(11)

with the intrinsic frequency ${\omega }_{0}$ approximated as⁴⁶

$${\omega }_{0}={f}_{{{{\rm{eff}}}}}+\frac{{N}^{2}{k}_{{{{\rm{H}}}}}^{2}}{2f{k}_{z}^{2}}+\frac{1}{{k}_{z}}\left(\frac{\partial {U}_{{{{\rm{g}}}}}}{\partial z}{k}_{y}-\frac{\partial {V}_{{{{\rm{g}}}}}}{\partial z}{k}_{x}\right),$$

(12)

where ${k}_{{{{\rm{H}}}}}^{2}={k}_{x}^{2}+{k}_{y}^{2}$, and ${f}_{{{{\rm{eff}}}}}$ is the effective Coriolis frequency. Thus, the group velocities can be given by

$${C}_{{{{\rm{g}}}}x}=\frac{\partial {\omega }_{0}}{\partial {k}_{x}}=\frac{{N}^{2}{k}_{x}}{f{k}_{z}^{2}}-\frac{1}{{k}_{z}}\frac{\partial {V}_{{{{\rm{g}}}}}}{\partial z},$$

(13)

$${C}_{{{{\rm{g}}}}y}=\frac{\partial {\omega }_{0}}{\partial {k}_{y}}=\frac{{N}^{2}{k}_{y}}{f{k}_{z}^{2}}+\frac{1}{{k}_{z}}\frac{\partial {U}_{{{{\rm{g}}}}}}{\partial z},$$

(14)

$${C}_{{{{\rm{g}}}}z}=\frac{\partial {\omega }_{0}}{\partial {k}_{z}}=\frac{{-N}^{2}{k}_{{{{\rm{H}}}}}^{2}}{f{k}_{z}^{3}}-\frac{1}{{k}_{z}^{2}}\left(\frac{\partial U}{\partial z}{k}_{y}-\frac{\partial V}{\partial z}{k}_{x}\right).$$

(15)

The set of Eqs. (9, 10) was numerically integrated backward in time with a fourth-order Runge–Kutta method³⁶ with a fixed time step of 1 h. The integration was initiated at 10:30 UTC on 19 July 2014—when NIKE at 300 m depth at station 2 exceeded pre-typhoon levels by an order of magnitude—and carried out over a 20-day period. The three-dimensional geostrophic velocity field ${{{{\boldsymbol{V}}}}}_{{{{\rm{g}}}}}$ was derived from the reconstructed eddy structure as described in Three-dimensional reconstruction of eddy Section, representative of the conditions on 19 July 2014. The initial vertical wavenumber was constrained by the observed vertical phase structure θ = arctan(u_i/v_i) under surface-generation assumptions, yielding k_z = ∂θ⁄∂z = 0.11 rad m⁻¹ (ref. ⁵⁸; Supplementary Fig. 9), corresponding to the maximal vertical wavelength and the fastest possible downward propagation. To ensure comprehensive coverage of possible ray paths, we sampled intrinsic frequencies from 0.8 f₂ to 1.2 f₂ in increments of 0.01 f₂, and all 360° azimuthal angles at 1° resolution.

Energy balance analysis

Step 1: Fixed control volume for simplicity.

The control volume was selected as a cylindrical column with a radius of one degree of longitude and latitude extending vertically from the surface to a depth of 800 m. The lateral boundary of this column is shown in Supplementary Fig. 3. The eddy propagation within our study period of about 2 weeks is relatively small and has little effect on our analysis (Supplementary Fig. 4), justifying the use of a fixed control volume.

Step 2: Local enhancement of NIE

To eliminate the interference from surface-input, we first quantified wind-driven NIE injection. We computed the typhoon-induced enhancement of NIKE by subtracting pre-typhoon background levels from the average NIKE within the mixed layer (0–60 m) over a 10-day period following the typhoon’s passage. The resulting enhancement was 2.9 J m⁻³ at station 2 and 2.6 J m⁻³ at station 4, indicating that both stations received comparable amounts of wind-generated NIKE input during the typhoon. Similar results were obtained for different averaging periods (Supplementary Fig. 10), collectively confirming nearly consistent typhoon-generated energy inputs at both stations. Thus, the additional depth-integrated NIKE at station 2 beyond that at station 4 provided a rough estimate of the interior NIKE enhancement at station 2. Multiplying this enhancement by the volume of the control volume yielded a locally enhanced NIE of about (3.0 ± 0.2) × 10¹⁴J.

Step 3: Cumulative lateral energy flux

Station 2, located within a frontal zone, provided an optimal location for observing NIWs radiating from the anticyclone. The peak frequency was determined from ADCP measurements, while horizontal wavenumbers were estimated based on the baroclinic deformation radius. The horizontal group velocity was subsequently estimated for each vertical mode. To quantify the energy flux, the measured velocity data were decomposed into vertical modes to determine the modal distribution of NIKE. The energy flux for each mode was then calculated as the product of its kinetic energy and the corresponding horizontal group velocity. These modal contributions were summed to obtain the total energy flux. Under the assumption that the NIE distribution along the lateral boundaries of the control volume resembles that observed at station 2, the total outward energy flux through these boundaries was estimated to be approximately (3.3 ± 0.2) × 10¹³J over the 2-week period following the typhoon’s arrival.

Step 4: Energy budget balance

Under the assumption of no other substantial sources or sinks, the energy propagation equation takes the following form:

$$\frac{\partial {{{\rm{EE}}}}}{\partial t}+\frac{\partial {{{\rm{NIE}}}}}{\partial t}+\nabla \cdot \left({{{{\boldsymbol{C}}}}}_{{{{\rm{g}}}}}\cdot {{{\rm{NIE}}}}\right)=0.$$

(16)

Integrating Eq. (16) over a cylindrical column with the anticyclone in the interior and applying the divergence theorem, we obtain:

$$\frac{\partial \left\langle {{{\rm{EE}}}}\right\rangle }{\partial t}=-\frac{\partial \left\langle {{{\rm{NIE}}}}\right\rangle }{\partial t}-{\int }_{V}\nabla \cdot \left({{{{\boldsymbol{C}}}}}_{{{{\rm{g}}}}}\cdot {{{\rm{NIE}}}}\right)d\tau =-\frac{\partial \left\langle {{{\rm{NIE}}}}\right\rangle }{\partial t}-{\oint }_{A}{{{{\boldsymbol{C}}}}}_{{{{\rm{g}}}}}\cdot {{{\rm{NIE}}}}d{{{\boldsymbol{s}}}},$$

(17)

where angle brackets denote the volume-integrated quantity. Equation (17) basically says the energy loss from the central anticyclone should be balanced by the sum of the enhanced NIE within the control volume and the outward flux of wave energy through the side surface.

The total energy loss from the central anticyclone over the 2-week period after the typhoon’s passage was (5.7 ± 0.9) × 10¹⁴J. The local NIE enhancement plus fluxed NIE is about (3.3 ± 0.2) × 10¹⁴J, collectively accounting for more than one-half of the energy decrease of the central anticyclone. The uncertainty ranges for the volume-integrated eddy energy are quantified by propagating the formal mapping error of the daily gridded SLA data through the standard error propagation formula. The uncertainty ranges for the interior-generated NIE are determined from sensitivity tests of the inertial-period averaging window (1–4 periods).

A canonical pathway for eddy decay is direct spin-down by large-scale winds. To bound its possible contribution here, we computed the wind stress τ_r = ρ_aC_d|u_a − u_g | (u_a − u_g) with ρ_a = 1.225 kg m⁻³ and C_d = 1.5 × 10⁻³, using a uniform background wind |u_a | = 40 m s⁻¹ (the maximum sustained speed of Rammasun) and the eddy’s surface geostrophic velocity u_g = (u_g, v_g) on 17 July. The work done by the wind, P = τ_r $\cdot$ u_g, thus yields an energy dissipation rate on the order of 10¹²J day⁻¹. Even under such an extreme-wind upper bound, the wind-induced dissipation is no more than ~10% of the observed three-dimensional eddy-energy drop (on the order of 10¹³J day⁻¹; Fig. 5). Direct wind damping is therefore secondary to the observed energy transfer into the NIW field.

Model setup and initial conditions

The nonlinear inviscid hydrostatic governing equations are simulated in a flat-bottomed square basin on f-plane (f = 1 × 10⁻⁴ s⁻¹) with a temporal resolution of 1 h. The model domain has a length of 3400 km with a uniform horizontal resolution of 10 km and a depth of 2 km with a uniform vertical resolution of 40 m. The rigid lid boundary is applied to exclude surface gravity waves, and the sponge boundary layers are set to weaken the wave reflection. For simplicity’s sake, the background buoyancy frequency is set to be vertically constant N = 4.2 × 10⁻³ s⁻¹, given the uniform salinity profile and linear temperature profile.

A series of idealized experiments, consist of Cases C1–C6, are executed (Table 1). The initial condition of cases is an exactly geostrophic eddy subjeted to diversing perturbations, entering a perturbed, unbalanced state. The background eddy was simulated with the structure proposed by ref. ⁵ in all cases. The sea surface height follows a Gaussian distribution given by

$$\eta \left(r\right)={\eta }_{0}\left(1-\frac{{r}^{2}}{2{R}_{0}^{2}}\right)\cdot \exp \left(-\frac{{r}^{2}}{2{R}_{0}^{2}}\right),$$

(18)

where ${\eta }_{0}$ is the sea surface height at the eddy center, $r$ is the radial coordinate, and ${R}_{0}=$50 km (Supplementary Fig. 5a). Thus, the pressure anomaly is also horizontally Gaussian distributed, whereas it is set vertically sinusoidal with the depth^5,30. Based on the hydrostatic relation, the temperature field can be obtained as follows:

$$T\left(z\right)={T}_{0}\cos \left(\frac{z}{2H}\right)+{T}_{{{{\rm{ref}}}}},$$

(19)

where ${T}_{0}$ denotes the temperature anomaly, ${T}_{{{{\rm{ref}}}}}$ denotes the reference temperature profile, and $H=2000{m}$ is the depth of the domain. Consequently, the horizontal velocity field can be derived by geostrophic balance. For example, the vertical structure of the CE in Case 1 is shown in Supplementary Fig. 5b.

In Cases 1–4, the perturbations take the form of the full-depth flow fields identical in spatial structure to the eddies, but scaled to 20% of the eddy’s maximum velocity. In other words, the azimuthal velocity of perturbation radially satisfies:

$$R\left(r\right)=\left|r\left(2-\frac{{r}^{2}}{2{R}_{0}^{2}}\right)\exp \left(-\frac{{r}^{2}}{2{R}_{0}^{2}}\right)\right|.$$

(20)

Vertically, the perturbations are set to decline sinusoidally from the surface ($z=0$) to the bottom ($z=-H$) in Cases 1–4:

$$Z\left(z\right)=1+\sin \left(\frac{\pi z}{2H}\right).$$

(21)

In summary, the perturbations can be described as follows:

$${U}_{{{{\rm{P}}}}}\left(r,z\right)={U}_{\max }\cdot R\left(r\right)\cdot Z\left(z\right),$$

(22)

where ${U}_{\max }=0.1{{{\mathrm{ms}}}}^{-1}$ denotes maximum velocity. The vertical structure of the perturbation in Case 1 is shown in Supplementary Fig. 5c. The perturbation in Case 5 (Supplementary Fig. 5d) is also described as Eq. (22), but ${U}_{\max }=0.5{{{\mathrm{ms}}}}^{-1}$ and vertical structure turns into

$$Z\left(z\right)=\left\{\begin{array}{cc}1+\sin \left(\frac{2{{{\rm{\pi }}}}z}{H}\right), & 0\ge z\ge -\frac{H}{4}\\ 0, & -\frac{H}{4}\ge z\ge \,-H\end{array}\right..$$

(23)

In Case 6, the wind field is set cyclonic, applied at the ocean surface for 6 h, with tangential stress $\tau$ given by

$$\tau \left(r\right)={\tau }_{\max }\left(\frac{r}{L}\right)\exp \left(\frac{1}{2}-\frac{{r}^{2}}{2{L}^{2}}\right),$$

(24)

following the structure in ref. ⁶². Here, ${\tau }_{\max }=\,1{{{\rm{N}}}}{{{{\rm{m}}}}}^{-2}$ is the maximum stress and $L$ $=$50 km.

Eddy energy quantification in model

The eddy energy (EE) is comprised of the EKE and EPE. In order to analyze the decay of EE, we apply a fourth-order Butterworth filter with a three-inertial-period cutoff period to obtain the eddy velocity field $({U}_{{{{\rm{e}}}}},\,{V}_{{{{\rm{e}}}}})$ and density field ${\rho }_{{{{\rm{e}}}}}$. Accordingly, the EE can be expressed as

$${{{\rm{EE}}}}={{{\rm{EKE}}}}+{{{\rm{EPE}}}}=\int \frac{1}{2}{\rho }_{{{{\rm{ref}}}}}\left({U}_{{{{\rm{e}}}}}^{2}+{V}_{{{{\rm{e}}}}}^{2}+{N}^{2}{\zeta }_{{{{\rm{e}}}}}^{2}\right)d\tau ,$$

(25)

where ${\rho }_{{{{\rm{ref}}}}}$ is the reference density and ${\zeta }_{{{{\rm{e}}}}}=-\frac{{\rho }_{{{{\rm{e}}}}}}{\partial {\rho }_{{{{\rm{ref}}}}}/\partial z}$ denotes the isopycnal displacement due to the eddy.

Wave energy quantification in model

With regard to the total generation of wave energy (WE), we consider two primary contributions: the net outward flux of wave energy from the eddy field (WE₁), and the residual wave energy remaining within the eddy field (WE₂).

At a sufficient distance from the eddy core, as low-frequency eddy signals become negligible, the distal oscillations are attributed to propagating waves. Here, the net outward energy flux can be estimated with ${{{\boldsymbol{F}}}}={{{\boldsymbol{U}}}}{P}^{{\prime} }$, where ${{{\boldsymbol{U}}}}=(U,V)$ is the velocity vector and ${P}^{{\prime} }$ is the pressure anomaly⁶³ obtained directly from model output. By integrating this flux vector over the entire closed control surface and time, the total internal wave energy radiated outward through the control surface is obtained:

$${{{{\rm{WE}}}}}_{1}=\iint {{{\boldsymbol{F}}}}d{{{\boldsymbol{S}}}}{dt}=\iint {{{\boldsymbol{U}}}}{P}^{{\prime} }d{{{\boldsymbol{S}}}}{dt}.$$

(26)

Crucially, control surfaces defined by closer contours remain strongly contaminated by eddies, while those defined by farther contours introduce unnecessary dissipation biases. To determine the optimal control surface, we analyzed the radial distribution of total energy across multiple time slices:

$${{{\rm{TE}}}}\left(r\right)={\int }_{-H}^{0}{\int }_{0}^{2\pi }\frac{1}{2}{\rho }_{{{{\rm{ref}}}}}\left({U}^{2}+{V}^{2}+{N}^{2}{\zeta }^{2}\right){rd}\phi {dz}.$$

(27)

Here, $\zeta =-\frac{{\rho }^{{\prime} }}{\partial {\rho }_{{{{\rm{ref}}}}}/\partial z}$ denotes the total isopycnal displacement and ${\rho }^{{\prime} }=\rho -{\rho }_{{{{\rm{ref}}}}}$ represents the density anomaly, where the potential density $\rho$ can be obtained from the potential temperature directly output by the model. Compared with the wave energy subsequently radiated outwards, the initial eddy energy within 250 km is unneglectable, which evidently revealed that low-frequency eddy signals contaminate the wave field within radii less than 250 km (Supplementary Fig. 11). Consequently, 250 km represents the minimal radius where uncontaminated wave energy flux can be reliably measured. Our calculations thus employ the 250 km contour to minimize signal loss.

In addition to the radiated flux, the residual wave energy present within the control volume is quantified. By subtracting low-frequency eddy components from the total field, the velocity field $\left({U}_{{{{\rm{w}}}}},{V}_{{{{\rm{w}}}}},{W}_{{{{\rm{w}}}}}\right)$ and density field ${\rho }_{{{{\rm{w}}}}}$ of internal waves are extracted. The residual wave energy is calculated as the sum of wave kinetic energy (WKE) and available potential energy (WPE), given by

$${{{{\rm{WE}}}}}_{2}={{{\rm{WKE}}}}+{{{\rm{WPE}}}}=\int \frac{1}{2}{\rho }_{{{{\rm{ref}}}}}\left({U}_{{{{\rm{w}}}}}^{2}+{V}_{{{{\rm{w}}}}}^{2}+{W}_{{{{\rm{w}}}}}^{2}+{N}^{2}{\zeta }_{{{{\rm{w}}}}}^{2}\right)d\tau ,$$

(28)

where ${\zeta }_{{{{\rm{w}}}}}=-\frac{{\rho }_{{{{\rm{w}}}}}}{\partial {\rho }_{{{{\rm{ref}}}}}/\partial z}$ denotes the isopycnal displacement due to the internal waves. Accordingly, the total generation of wave energy is obtained by ${{{\rm{WE}}}}={{{{\rm{WE}}}}}_{1}+{{{{\rm{WE}}}}}_{2}$.

Envelope-mean method

The core algorithm involves identifying local extrema (maxima and minima) of the signal $x\left(t\right)$, constructing upper and lower envelopes (${e}_{\max }(t),{e}_{\min }(t)$) via cubic spline interpolation, and deriving the envelope mean $m\left(t\right)=({e}_{\max }\left(t\right)+{e}_{\min }(t))/2.$ The high-frequency component is then obtained by $h\left(t\right)=x\left(t\right)-m\left(t\right).$ However, this envelope averaging method lacks sufficient extremum constraints at the endpoints, which can lead to abnormal oscillations⁵³. To suppress oscillations at initial points, we preprocess the signal as follows:

Step 1: Temporal domain extension.

Extend the signal backward from $[0,{T}]$ to $\left[-{T}_{{{{\rm{e}}}}},{T}\right]$ and initialize the extended signal with centered values:

$${x}_{1}\left(t\right)=\left\{\begin{array}{cc}0, & t\in \left(-{T}_{{{{\rm{e}}}}},0\right)\\ {x}_{0,} & t\in \left[0,T\right]\end{array}\right.,$$

(29)

where ${x}_{0}\left(t\right)=x\left(t\right)-\frac{1}{T}{\int }_{0}^{T}x\left(t\right){dt}.$

Step 2: High-frequency superposition.

Inject an auxiliary wave across the entire extended domain:

$${x}_{2}\left(t\right)={x}_{1}\left(t\right)+{A}^{* }\cos \left(\omega t\right),\,t\in \left[-{T}_{{{{\rm{e}}}}},\,T\right],$$

(30)

where amplitude ${A}^{* }$ is chosen much larger than the amplitude of x(t) to meet the constraint on the original signal. Notably, each envelope-mean operation excludes the highest-frequency component. By setting angular frequency ω exceeds the dominant frequency of x(t), the auxiliary wave’s high frequency ensures it functions as a sacrificial filter: it absorbs boundary artifacts during initial processing and is subsequently eliminated, leaving the intrinsic signal dynamics uncontaminated.

Step 3: Periodicity constraints.

To ensure the zero-mean property of the auxiliary wave over the entire extended domain, the combined duration of the extended domain must be an integer multiple of the auxiliary wave period:

$${T}_{{{{\rm{tot}}}}}={T}_{{{{\rm{e}}}}}+T=\frac{2{{{\rm{\pi }}}}k}{\omega },$$

(31)

where k belongs to the set of positive integers. Furthermore, samples per wave cycle must be an even integer to prevent bias in the discrete implementation.

Following the pre-processing, two sequential envelope-mean operations are applied:

$$m\left(t\right)=\varepsilon \left(\varepsilon \left({x}_{2}\left(t\right)\right)\right),$$

(32)

where ${{{\mathcal{E}}}}$ denotes the envelope-mean operator. High-frequency wave field can be obtained by:

$$h\left(t\right)={x}_{0}\left(t\right)-m\left(t\right),\,t\in \left[0,T\right].$$

(33)

Wave energy source spatial pattern

Using the wave field $({{{{\boldsymbol{U}}}}}_{{{{\bf{w}}}}},{P}_{{{{\rm{w}}}}},{\zeta }_{{{{\rm{w}}}}})$ separated by the envelope-mean method, the wave energy source can be assessed from to the internal wave conservation of energy equation:

$$\frac{\partial E}{\partial t}+\left({{{\boldsymbol{U}}}}\cdot \nabla \right)E+\nabla \cdot \left({{{{\boldsymbol{U}}}}}_{{{{\rm{w}}}}}{P}_{{{{\rm{w}}}}}\right)=Q,$$

(34)

where $E=\frac{1}{2}{\rho }_{{{{\rm{ref}}}}}\left({U}_{{{{\rm{w}}}}}^{2}+{V}_{{{{\rm{w}}}}}^{2}+{W}_{{{{\rm{w}}}}}^{2}+{N}^{2}{\zeta }_{{{{\rm{w}}}}}^{2}\right)$ is the wave energy density and Q represents source/sink⁶³.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.