Fewer tropical cyclones yield more near-inertial wind work to the global ocean over the past four decades

In general, tropical cyclones (TCs) will inject energy into oceanic inertial motion‒a prevalent phenomenon in the ocean. Under global warming, the intensity of TCs is on the rise, while their frequency has exhibited a decline since 2000. However, the long-term trend of this energy infusion is an underexplored problem in this context. Using a damped-slab model, we computed the wind work exerted by TCs on the ocean’s mixed-layer inertial motions. Our results show that the global wind work has increased by approximately 50% from 1979 to 2023. The wind work increase of strong TCs (Saffir–Simpson levels 4–5) is the major contributor to the increasing trend of global wind work, primarily due to their increasing frequency and substantial wind stress. At basin scale, the wind work input of the North Atlantic TCs has increased by 2 times, owing to an increase in both their intensity and frequency. Specifically, in the South Indian and the eastern North Pacific basins, the rise in wind work is primarily attributed to the enhanced wind energy of TCs within the inertial bands.


Introduction
Wind serves as a vital mechanical energy source that maintains the general oceanic circulation through interior mixing.According to estimations by Munk and Wunsch (1998), the energy flux required for deep-ocean mixing amounts to approximately 2.1 TW (1 TW = 10 12 W), with wind forces accounting for approximately 1.2 TW of this total.Although a significant portion of the wind energy dissipates in the ocean's upper layers via surface gravity waves and sub-inertial motions (Wunsch 1998, Wang andHuang 2004), some observations have revealed that wind-generated near-inertial waves (NIWs) can propagate into the deep ocean (Simmons andAlford 2012, Ma et al 2022).Previous studies have been undertaken to quantify the energy flux from wind to mixed-layer inertial motions (Alford 2001, Watanabe and Hibiya 2002, Liu et al 2019, von Storch and Lüschow 2023).The estimates remain varied, ranging from 0.2 to 1.4 TW, largely due to the sensitivity of these calculations to the spatial and temporal resolution of wind datasets (Jiang et al 2005, Rimac et al 2013).In the distribution of global energy flux, large amplitudes are typically observed in the midlatitude storm-track regions during the winter season (Thomas and Zhai 2022).
Tropical cyclones (TCs), characterized by rapidly changing wind stress, act as excellent generators of near-inertial motions and have been documented extensively (Price 1983, Sun et al 2011, Chen et al 2013, Shu et al 2016, Wang et al 2019, Zhang et al 2022).Liu et al (2008), through their study of 1500 TCs spanning from 1984 to 2003, concluded that the energy flux introduced by TCs to ocean inertial motions is roughly 0.03 TW.Recent findings argue for a higher contribution, with values ranging from 0.028 to 0.065 TW, which represent 8% to 17% of the global energy flux (Lin et al 2023, von Storch andLüschow 2023).Compared to midlatitude storms, the vertical propagation of NIWs generated by TCs is indeed faster, primarily due to the smaller horizontal scales of tropical storms (Alford et al 2016).These rapidly propagated NIWs significantly affect the upper ocean, contributing to processes like the deepening of the mixed layer and the formation of cold wakes along TC's path, largely due to their vertical mixing capabilities (D'Asaro et al 2007, Guan et al 2014, Yang et al 2015, Wang et al 2016, Gutiérrez Brizuela et al 2023).
In global warming, the number of hurricanes has significantly decreased from 1990 to 2021 (Klotzbach et al 2022).However, the TCs intensity shows an increasing trend based on homogenized data from satellites and surface drifters (Kossin et al 2020, Wang et al 2022).Considering the potential impact of global warming on the intensity and frequency of TCs, understanding how much energy is transferred by TCs to oceanic inertial motions is critical for grasping their potential effects on meridional overturning circulation and climate change.Using a damped-slab model (Pollard and Millard Jr 1970), here we systematically evaluate the wind work from TCs to mixed-layer inertial motions during 1979-2023.We show that the wind work from TCs is increasing, even as the frequency of TCs is declining.

Data
In this study, mixed layer depth and wind speed data are used to drive the damped-slab model.Although the mixed layer depth by high latitudes shows different trends in summer and winter, it has not shown significant changes in the TCs region (Sallée et al 2021, Sugimoto 2022).Therefore, we obtained the 1 • × 1 • mixed layer depth from the global monthly climatology dataset by Holte et al (2017), as shown in supplement figure S1.The 10 m wind speed data is obtained from the fifth generation of the European Center for Medium-Range Weather Forecasts reanalysis (ERA5), with a 1 hour temporal and 0.25 • × 0.25 • spatial resolution (Hersbach et al 2020).To pinpoint the position of the TC center, we also obtain data like mean sea level pressure; 925 hPa, 850 hPa, and 700 hPa relative vorticity; and 850 hPa and 700 hPa geopotential height, all from the ERA5 dataset.For the robustness of this study, we also conducted comparative analyses using the National Centers for Environmental Prediction (NCEP)-Department of Energy reanalysis 2 dataset data.The NCEP 10 m wind data have a 6 h temporal resolution and 2.5 • × 2.5 • spatial resolutions (Kanamitsu et al 2002).
The best-track dataset of TC is obtained from the International Best Track Archive for Climate Stewardship (IBTrACS) Version v04r00 (Knapp et al 2010).IBTrACS is a collection of best-track datasets from different agencies.The positions of the TC center from this dataset are used in our study.Following D'Asaro (1985), we used the damped-slab model to estimate the wind-induced energy flux and the momentum equation is

The damped-slab model
Where t is time, Z = u + iv is the mixed layer velocity, r the empirical damping coefficient parameterizing energy decay, f the local Coriolis frequency, H the mixed layer depth, and T = (τ x + iτ y )/ρ the wind stress scaled by the density of seawater ρ.The behavior of equation ( 1) is well known, following Plueddemann and Farrar (2006), the energy flux from the wind to the mixed-layer inertial motions is given by Where Z * I is the complex conjugate of the inertial current Z I .Z I can be computed using the hourly wind stress and monthly climatology mixed layer depth, see details in the appendix of D'Asaro (1985).The integrated flux is net wind work on inertial motions.A detailed stepby-step derivation of the model is in the supporting information.

Definition of TC size
The TC center position in the ERA5 dataset is initially estimated using IBTrACS data.The final position of the TC center is derived from the average of the centers of mass for six different variables: mean sea level pressure; 925 hPa, 850 hPa, and 700 hPa relative vorticity; and 850 hPa and 700 hPa geopotential height (Marchok 2002).The TC size metric is defined as the radius R tc at which the relative vorticity decreases/increases to 1 × 10 −5 s −1 from the TC center in the northern/southern hemisphere (Liu and Chan 1999).

Extracting wind work of TCs
To calculate the wind work generated by TCs, we initially extracted the inertial energy flux along the track of each TC.The series of inertial energy flux denoted as {Π t } = {Π 1 , Π 2 , Π 3 , . . ., Π t }, where the interval t is 6 h (matching the time resolution of the TC track).The value of Π t is obtained by integrating the energy flux within the radius R tc of TC and averaging it over 6 h.Subsequently, the wind work of each TC is computed using equation ( 3) (P = ∫ {Π t } dt, with dt being 6 h).For an entire year, the collective wind work of all TCs is represented as {P n } = {P 1 , P 2 , P 3 , . . ., P n }, where n = 1, 2, 3 . . .denotes the sequence of TCs.The annual accumulated wind work (AWK) can then be determined by summing the individual contributions: sum ({P n }).

Definition of accumulated variance of inertial (AVI) bands
The power spectrum of TC is derived from ERA5 10 m wind vectors at the TC's center (see example in supplement figure S2).This implies that the number of power spectra for a single TC is equal to the number of best track points along the TC trajectory.To compute each individual spectrum, we examine a time period that encompasses two inertial periods before and after the TC reaches its center.This approach effectively captures the power spectrum associated with the TC's inertial fluctuation along its trajectory.Inertial variances are defined by summing the variance within inertial bands (0.8-1.2 f ) of the power spectrum.Finally, the annual AVI bands is determined by summing each inertial variance across all TCs in a given year.

Statistical information
Linear regression is employed for trend estimation in this study.The statistical significance of the regression is determined using a two-tailed t-test, taking into account the complete degrees of freedom within the 45 yr time series.It is crucial to emphasize that the assessment of statistical significance relies on the twosided 95% confidence intervals rather than P values, as indicated by Kossin (2018).The quantification of trend change is calculated by dividing the discrepancy between the final and initial points of the optimal trend line by the initial point.(1985).This mathematical relationship suggests that strong TCs have a significantly higher energy flux throughout their lifetime compared to medium (Saffir-Simpson categories 1-3) and weak (Saffir-Simpson categories TD-TS) TCs, as shown in supplement figure S3.Therefore, the rising number of strong TCs, as shown in figure 1(b) and discussed in Klotzbach et al (2022), drives the global increase in the AWK input of TCs.

Changes in wind work of TCs over the past four decades
Although the number of Weak TCs showed a smaller decrease and was not significant (figure 1(b), table 2), a highly significant increase in the AWK of weak TCs is evident, of 195% over the 45 yr period from 1979 to 2023 (table 1).It could be associated with a notable intensification of weaker TCs (Wang et al 2022).On the other hand, the number of weak TCs accounts for more than half of the total TC numbers (figure 1(b)).However, the proportion of wind work input by weak TCs to the total is minimal.The primary factor contributing to this phenomenon is the significantly lower energy flux within the lifecycle of weak TCs (as shown in figure S3).Furthermore, the ERA5 dataset exhibits limitations in accurately resolving the intensity and internal structures of weaker TCs, as pointed out by Bian et al (2021).This limitation potentially leads to an underestimation of weak TCs to the AWK of global TCs.
In contrast, the AWK from medium TCs does not exhibit positive contributions to the increasing trend of the AWK of global TCs.An annual decrease of −0.2 PJ yr −1 is shown in figure 1(a), but its trend was not significant (table 1).Although the intensity of medium TCs increases are identified by Kossin et al (2020), the declining trend in the frequency of medium TCs is evident, of −31% over the 45 yr period (as shown in figure 1(b), table 2).Consequently, the wind work input from medium TCs is essentially no change.

Changes in wind work of TCs in the basin scale
The distribution of AWK by TCs from 1979 to 2023 is given in figure 2. The high-value region of wind work is concentrated around 18 • N in the western North Pacific, matching the distribution of TC hazard frequency highlighted by Peduzzi et al (2012).Due to the high vertical shear of NIWs, mixing may be strong in this region and some evidence is shown in the study of Whalen et al (2018).Other ocean basins exhibit a more dispersed distribution of wind work, devoid of concentrated high-value zones.According to initial estimates, the TCs' contribution to the global overall wind work amounts to approximately 13% (The distribution of AWK by global wind is shown in supplement figure S4).In other words, the average energy flux from TCs to inertial motions is about 0.027 TW, which is almost the same as the value of 0.028 TW estimated by the recent research of Lin et al (2023).The trends and percentage changes over the period 1979-2023 are presented for various regions and different TC categories.Significance is indicated by P-values of regressions and confidence intervals of the trends.Significant trends, identified based on the confidence interval, are highlighted in bold.At the basin scale, the AWK input from the North Atlantic TCs is the most significant contributor to the increasing trend in the AWK of global TCs.The annual increment of 2.7 PJ yr −1 indicates that the TC's AWK input in the North Atlantic increased by 2 times from 1979 to 2023 (figure 2).The increasing trend of AWK in this basin is primarily correlated with the dramatic increase in TC numbers (shown in supplement figure S5).Moreover, the number of TC in the North Indian increased by 77% from 1979 to 2023 (table 2).Consequently, TC's AWK input in the North Indian shows a growth trend of 1.0 PJ yr −1 , significant at greater than the 99% confidence level (table 1).
In the eastern North Pacific, TC's AWK input exhibits an increasing trend of 1.7 PJ yr −1 (at the 90% confidence level), as depicted in figure 2 and table 1.However, from 1979 to 2023, the quantity of TCs in this basin has remained relatively stable, with little discernible variation or discernable trend during this temporal span.Similarly, in the South Indian basin, TC's AWK input is showing a growth trend of 1.1 PJ yr −1 (at the 95% confidence level), while its number of TCs is declining (figure S5).Notably, in both basins, the number of strong TCs demonstrates no significant change.The increases in AWK in these two basins are primarily influenced by other factors, which will be elaborated upon in the following section.
In the western North Pacific, despite hosting the majority of TCs (He et al 2015), the impact of TC's AWK in this basin on the overall increasing trend in global TCs' AWK is modest, with an annual increase of 0.2 PJ yr −1 (highly insignificant), as depicted in figure 2 and detailed in table 1.This is likely related to the abrupt decrease in TC frequency in this basin since 1998 (Zhao et al 2018).Furthermore, the TC's AWK from the South Pacific shows a small decreasing trend of −0.4 PJ yr −1 and is also highly insignificant (refer to table 1).Except for the South Pacific, all other basins contribute to the increasing global trend in the AWK of global TCs shown in figure 2.

Changes in inertial variance of TCs
Inertially rotating winds can resonantly force powerful inertial motions in oceanic mixed layers (Pollard and Millard Jr 1970).In this context, we introduce the AVI (see Method) to assess TC's energy changes within the inertial frequency band.The global AVI exhibits a significant increasing trend (figure 3).This suggests that the TC's energy in the inertial frequency band is increasing, which means that the fluctuations of TC wind tend to be more inertial.The increasing TC's inertial energy will drive the stronger wind work input, and the global AVI (figure 3) is visibly correlated with the AWK of global TCs shown in figure 1(a).The primary contributor to the global AVI increase is the AVI of the strong TCs, with an annual increase of 198.8 m 2 s −2 yr −1 (figure 3, table 3).The AVI of weak TCs also has an important contribution to the global AVI increase, with an annual increment of up to 141.3 m 2 s −2 yr −1 (figure 3, table 3).The AVI of medium TCs shows a declining trend, consistent with its downward trends of AWK.
Figure 4 illustrates the time series of the AVI for the individual basins.The AVI trends are consistent with the AWK trends across all ocean basins.The overall congruence highlights a strong relationship between wind work and inertial variance.This further indicates that the TC's energy in the inertial frequency band dominates the TC's wind work input to a  The trends and percentage changes over the period 1979-2023 are presented for various regions and different TC categories.
Significance is indicated by P-values of regressions and confidence intervals of the trends.Significant trends, identified based on the confidence interval, are highlighted in bold.
certain extent.Notably, the AVI in the North Atlantic shows a clear increasing trend with an annual increase of 178.3 m 2 s −2 yr −1 (figure 4(c), table 3), which may be related to the significant increase in the number of TCs (shown in supplementary figure S5).
In both the South Indian and eastern North Pacific basins, the number of total and strong TCs did not show increases, as shown in supplement figure S5.The accumulated cyclone energy (Bell et al 2000) of total and strong TCs, which incorporates both the intensity and duration information of TCs, also did not show an increasing trend, as shown in supplement figure S6.Klotzbach et al (2022) provide a similar discussion of the number of TCs and accumulated cyclone energy of TCs in these two basins.
However, the AVI (figures 4(d) and (f)) and AWK (figures 2(d) and (f)) in these two basins are experiencing increasing trends.The rise in AVI indicates an increase in the wind stress of TCs within the inertial frequency band, despite the absence of a notable increase in the overall wind stress.This rise in wind stress might be attributed to two factors: the slowing translation speed of TCs (Kossin 2018) and the decrease in the radius of maximum wind speed as TCs intensify (Carrasco et al 2014).These changes increase the likelihood that the rotation rate of the maximum wind stress of TCs matches the local inertial rotation period.This results in stronger resonance mechanisms occurring (Price 1981), which amplify inertial motions to levels that contribute to the increasing trend of AWK.Notably, the variations in the translation speed and radius of maximum wind speed of TCs are global; but the resonance mechanisms are latitude-dependent, with the period of the inertial rotation varying with latitude (Zhai 2015).Therefore, the resonance mechanism effect may exhibit differences in different basins due to latitudinal differences in the concentration of AWK.

Discussion
The robustness increase in global AWK is consistently evident in the NCEP data (figure S7) and the increasing trend almost equal with those from ERA5 are 5.9 PJ yr −1 .Due to the low spatial resolution of NCEP data, the estimated size of TCs based on their relative vorticity tends to be larger.As a result, the wind work calculated from NCEP data is greater than that computed from ERA5 data.The spatial distribution of AWK from NCEP and ERA5 data has obvious differences, especially in the western North Pacific and North Atlantic basins (figure S8).In the NCEP data, high values of wind work are more dispersed and tend to be biased towards higher latitude regions.
It is worth noting that existing all reanalysis data tends to underestimate TC intensity (Murakami 2014, Dulac et al 2022).Specifically, the reanalysis data may exhibit relatively weaker resolving intensity for weaker TCs.Specifically, the reanalysis data may show relatively weaker resolution in capturing the structures of weaker TCs.For example, in the North Indian, TC's AWK has increased by 10 times from 1979 to 2023 (table 1).The reason behind this phenomenon is that, around 1980, there were relatively few TCs in this basin, and all of them were weak TCs.As a result, the computed early-stage wind work may be underestimated.This implies that the actual contribution of TCs to global overall wind work could be greater than our current estimation.

Conclusion
In this study, we investigated the near-inertial wind work through a simple slab model driven by ERA5 wind, specifically focusing on the TC's contribution.Our analysis revealed a significant increasing trend in global AWK, showing a substantial 50% increase from 1979 to 2023.Although strong TCs make up just 15% of the total count of TCs, they contribute about 57% to the global AWK due to their huge wind stress and substantial wind stress.The increase in global AWK is attributed to the rising number of strong TCs.Zooming into basin scale, the AWK input from the North Atlantic basin has emerged as the major contributor to the increasing trend in global AWK.The AWK input in the North Atlantic has shown a 2 fold surge from 1979 to 2023, which could be attributed to a notable rise in both the intensity and frequency of TCs.The increasing AWK is inversely correlated with the Atlantic Meridional Overturning Circulation index (Rahmstorf et al 2015).Increased wind work in this basin can potentially weaken the Atlantic Meridional Overturning Circulation by intensifying the mixing and deepening of the ocean's mixed layer through NIWs (Peng et al 2022).The growth of AWK in the South Indian and eastern North Pacific basins Y Ma et al is linked to the increased wind energy of TCs within the inertial bands.The near-inertial energy within the mixed layer can penetrate the deep ocean, in the form of NIWs (Alford et al 2016).Therefore, the AWK increase at the global and basin scale may have a significant impact on global meridional overturning circulation.Its importance may grow over time, due to the significant increase in strong TCs numbers with global warming.

Figure 1
Figure1(a) illustrates the time series of the AWK (see Methods for detail), revealing a clear increasing trend in the AWK of global TCs, while its number markedly decreased after 2000 (figure1(b)).The AWK of global TCs exhibits a growth of 6.2 PJ yr −1 , as illustrated in figure1(a).This indicates an approximate 50% increase from 1979 to 2023, with statistical significance at the 98% confidence level (P value = 0.02), as highlighted in table 1.The primary contributor to the increase in the AWK of global TCs is strong TCs (Saffir-Simpson categories 4-5), with an annual increment of 5.2 PJ yr −1 (figure1(a)).These strong TCs, though comprising only 15% of the total number of TCs, are responsible for 57% of the wind work input.This is due to that the energy flux (Π ) is

Figure 1 .
Figure 1.Time series of annual-accumulated wind work (a) and annual TC numbers (b) along with their linear trend.Time series are shown for the different TC categories: Global, the category TD-TS, the category 1-3, and the category 4-5.The period of the time series is 1979-2023.The shading indicates the two-sided 95% confidence bounds of the trends.The symbols ∇ represent trend change over the period 1979-2023.(units: 1 EJ = 10 18 J; 1 PJ = 10 15 J).

Figure 2 .
Figure 2. The distribution of accumulated wind work by TCs on inertial motions from 1979 to 2023.Time series shows the annual accumulated wind work for the individual ocean basins: North Indian (N.I), western North Pacific (WN.P), North Atlantic (N.A), South Indian (S.I), South Pacific (S.P), and eastern North Pacific (EN.P).On the left of the panel, the black line is the normalized meridional distribution of the zonally integrated wind work.(units: 1 MJ m −2 = 10 6 J m −2 ).

Figure 3 .
Figure 3.Time series of annual accumulated variance of inertial bands along with their linear trend.Time series are shown for the different TC categories: Global, the category TD-TS, the category 1-3, and the category 4-5.The shading indicates the two-sided 95% confidence bounds of the trends.

YFigure 4 .
Figure 4. Time series of annual-accumulated variance of inertial bands along with their linear trend for North Indian (a), western North Pacific (b), North Atlantic (c), South Indian (d), South Pacific (e), and eastern North Pacific (f).The shading indicates the two-sided 95% confidence bounds of the trends.
Impacts ed M Meredith and A Garabato (Elsevier) pp 95-115 von Storch J-S and Lüschow V 2023 Wind power input to ocean near-inertial waves diagnosed from a 5-km global coupled atmosphere-ocean general circulation model J. Geophys.Res.128 e2022JC019111 Wang G, Li D, Wei Z, Li S, Wang Y and Xu T 2019 Observed near inertial waves in the wake of Typhoon Linfa (2015) in the Northern South China Sea J. Ocean Univ.China 18 1013-21 Wang G, Wu L, Johnson N C and Ling Z 2016 Observed three-dimensional structure of ocean cooling induced by Pacific tropical cyclones Geophys.Res.Lett.43 7632-8 Wang G, Wu L, Mei W and Xie S-P 2022 Ocean currents show global intensification of weak tropical cyclones Nature 611 496-500 Wang W and Huang R X 2004 Wind energy input to the Ekman layer J. Phys.Oceanogr.34 1267-75 Watanabe M and Hibiya T 2002 Global estimates of the wind-induced energy flux to inertial motions in the surface mixed layer Geophys.Res.Lett.29 64-61-64-63 Whalen C B, MacKinnon J A and Talley L D 2018 Large-scale impacts of the mesoscale environment on mixing from wind-driven internal waves Nat.Geosci.11 842-7 Wunsch C 1998 The work done by the wind on the oceanic general circulation J. Phys.Oceanogr.28 2332-40 Yang B, Hou Y, Hu P, Liu Z and Liu Y 2015 Shallow ocean response to tropical cyclones observed on the continental shelf of the northwestern South China Sea J. Geophys.Res. 120 3817-36 Zhai X 2015 Latitudinal dependence of wind-induced near-inertial energy J. Phys.Oceanogr.45 3025-32 Zhang H et al 2022 Observed impact of Typhoon Mangkhut (2018) on a continental slope in the South China Sea J. Geophys.Res.127 e2022JC018432 Zhao J, Zhan R, Wang Y and Xu H 2018 Contribution of the interdecadal Pacific oscillation to the recent abrupt decrease in tropical cyclone genesis frequency over the western North Pacific since 1998 J. Clim.31 8211-24

Table 1 .
Trend in accumulated inertial wind work and their statistics.
Trend (PJ yr −1 ) Change (%) P-valueThe trends and percentage changes over the period 1979-2023 are presented for various regions and different TC categories.Significance is indicated by P-values of regressions and confidence intervals of the trends.Significant trends, identified based on the confidence interval, are highlighted in bold.The symbol * represents significant uncertainty due to a small sample size.

Table 2 .
Trend in tropical cyclones numbers and their statistics.

Table 3 .
Trend in accumulated variance of inertial bands and their statistics.