Progress in Oceanography 206 (2022) 102850

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Progress in Oceanography
journal homepage: www.elsevier.com/locate/pocean

Incoherence of the M2 and K1 internal tides radiated from the Luzon Strait
under the influence of looping and leaping Kuroshio
Anzhou Cao a, *, Zheng Guo b, Shuya Wang c, Xinyu Guo d, Jinbao Song a, *a

Ocean College, Zhejiang University, Zhoushan, China
Marine Science and Technology College, Zhejiang Ocean University, Zhoushan, Chinac
Key Laboratory of Physical Oceanography, Ocean University of China, Qingdao National Laboratory for Marine Science and Technology, Qingdao, ChinadCenter for Marine Environmental Study, Ehime University, Matsuyama, Japanb

A R T I C L E I N F O

A B S T R A C T

Keywords:
Incoherence
Internal tide
Looping Kuroshio
Leaping Kuroshio
Luzon Strait

The Kuroshio in different paths is considered to have different impacts on the generation, propagation and en­ergetics of internal tides (ITs) originated from the Luzon Strait (LS). However, its influence on the incoherence ofITs remains unknown. Based on a regional model simultaneously simulating ITs and background circulations,this study examines the incoherence of the M2 and K1 ITs radiated from the LS under the influence of looping andleaping Kuroshio. The looping Kuroshio causes stronger incoherence of the M2 and K1 ITs than the leaping
Kuroshio. The reason is that the looping Kuroshio causes a larger variation of eigenspeed, which results instronger incoherence of ITs than the leaping Kuroshio does. For both the looping and leaping Kuroshio, the lowfrequency background currents play a more important role than the stratification in changing the eigenspeed. We
believe that the results of this study can deepen our understanding of the incoherent nature of ITs near the LS.

1. Introduction
Incoherence is a common characteristic of internal tides (ITs) aroundthe world (Nash et al., 2012a; Shriver et al., 2014; Ansong et al., 2017).
Compared with the coherent ITs with temporal invariant harmonic
constants, the incoherent ITs (also named as nonstationary ITs) arecharacterized by high temporal variability (Ray and Zaron, 2011).
Under the influence of varying background currents and stratification,
part of the internal tidal energy is transferred to frequencies outside thedeterministic tidal frequencies and incoherence is generated (Eich et al.,
2004; van Haren, 2004; van Aken et al., 2007; Zhao et al., 2010; Pick­ering et al., 2015; Kelly and Lermusiaux, 2016; Kelly et al., 2016; Kerryet al., 2016; Buijsman et al., 2017). The existence of incoherence makes
the ITs in coastal areas difficult to be predicted (Nash et al., 2012a;
2012b). Moreover, in situ observations have indicated that comparedwith the coherent ITs, the incoherent ITs contain a larger proportion ofhigher modes (Liu et al., 2016) and contribute more to the local shear(Cao et al., 2019). Hence, the incoherent ITs may have a considerable
contribution to the turbulent mixing in the ocean.
The Luzon Strait (LS), composed by two meridional submarine ridges
and connecting the South China Sea (SCS) and Philippine Sea, is afamous generation area of ITs (e.g. Alford et al., 2011, 2015; Kerry et al.,2013; 2014a; 2014b; Zhao, 2014; Xu et al., 2016; Wang et al., 2018,2020; Guo et al., 2020a; 2020b). The ITs generated at the LS are very
intense (Alford et al., 2011; 2015) and can propagate into the SCS andPhilippine Sea for more than 1000 km (Zhao, 2014; Xu et al., 2016;
Wang et al., 2018, 2020). Due to the variable background currents(especially the Kuroshio) and active mesoscale activities (e.g. Nan et al.,2011; Chen et al., 2011; Zhang et al., 2017), the incoherence of ITs in the
SCS and Philippine Sea is significant. In situ observations have shownthat at the LS, the proportion of incoherent ITs is generally smaller than
20% away from the large topographic gradients (Pickering et al., 2015).
In the SCS Basin, the proportion of incoherent ITs is generally on thelevel of 20–40% and show apparent changes in space (Xu et al., 2014;
Cao et al., 2017). On the continental slope of the northern SCS, theproportion of incoherent ITs could increase to 60–90% (Lee et al., 2012;
Xu et al., 2013). In the Philippine Sea, the proportion of incoherent ITs isa little smaller than that in the SCS Basin (Cao et al., 2017). Based on a
year-long simulation, Kerry et al. (2016) quantified the incoherence ofthe M2 energy fluxes in the SCS and Philippine Sea. They found that the
time-averaged incoherent energy fluxes account for 30% of the west­
ward coherent energy fluxes in the SCS and 25% of the eastward
coherent energy fluxes in the Philippine Sea. They concluded that thevariability in the generation and propagation of ITs under the influence* Corresponding authors.
E-mail addresses: caoanzhou@zju.edu.cn (A. Cao), songjb@zju.edu.cn (J. Song).
https://doi.org/10.1016/j.pocean.2022.102850
Received 9 September 2021; Received in revised form 6 June 2022; Accepted 5 July 2022
Available online 8 July 2022
0079-6611/© 2022 Elsevier Ltd. All rights reserved.

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Fig. 1. Bathymetry (shading, unit: m) of the simulated domain. Six black pluses denote the locations of six moorings (MP1-MP6) in Cao et al. (2017).

of background currents accounts for the generation of incoherent ITs.
The variability of ITs radiated from the LS is strongly related to the
Kuroshio whose presence affects the generation and propagation of ITs.
Because of the lack of long-term in-situ observations, this issue has been
investigated by some numerical studies. By using an idealized model,
Jan et al. (2012) pointed out that the presence of Kuroshio enhances(weakens) the generation and energy radiation into the SCS for semi­
diurnal (diurnal) ITs. However, Ma et al. (2013) demonstrated that thediurnal energy flux into SCS is intensified due to the strengthened
density front of Kuroshio, while the semidiurnal energy flux is notaffected. Li et al. (2016) suggested that the occurrence of Kuroshio
intrusion is associated with westward zonal currents, which can cause
the suppression of semidiurnal IT generation at the LS. The above results
are somewhat controversial, but they suggest that the impact of Kur­oshio on the ITs radiated from the LS is complicated due to the variationof Kuroshio in path, strength and vertical structure. Recent work
revealed that the generation, propagation and energetics of ITs responddifferently to the Kuroshio in the leaping, looping and leaking paths,
implying a complex spatial and temporal variability of ITs in the SCS andPhilippine Sea (Kerry et al., 2014a; Xu et al., 2021). These studies pro­vide insights on the IT dynamics under the influence of Kuroshio.
Nevertheless, how the Kuroshio in different paths affects the incoher­
ence of ITs remains unknown, which provides the motivation of thisstudy.
In this study, based on the simultaneous simulations of ITs andbackground circulations, we investigated the incoherence of the M2 andK1 ITs radiated from the LS under the influence of looping and leaping
Kuroshio. The paper is organized as follows. The model configurations
and data analysis methods are introduced in Section 2. The simulated
results including barotropic tides, background currents and internal
tidal currents are validated in Section 3. The baroclinic energetics of theM2 and K1 ITs are analyzed in Section 4. In Section 5, the incoherence ofenergy fluxes and horizontal kinetic energy (HKE) of the M2 and K1 ITs
under the influence of looping and leaping Kuroshio is shown. The
reason for the difference of incoherent ITs induced by the looping andleaping Kuroshio is discussed in Section 6. Finally, the paper is sum­
marized in Section 7.

2. Methodology
2.1. Model configurations
The regional ocean modeling system (ROMS) under the frameworkof coastal and regional ocean community model (CROCO, version 1.0,
https://www.croco-ocean.org) was adopted in this study to individuallysimulate the M2 and K1 ITs generated at the LS under the influence of
background currents. The CROCO solves the free-surface, primitive
equations on the curvilinear grid and terrain-following vertical coordi­
nate under the hydrostatic and Boussinesq approximations. The model
has been known to perform well in different scenarios including thesimulation of ITs in the SCS (e.g. Renault et al., 2017; Marchesiello et al.,2019; Guo et al., 2020a; 2020b). As we mainly focus on the incoherenceof ITs near the LS where the water is deep and the horizontal scale of ITs
is much greater than the vertical scale, the hydrostatic approximation isvalid.
The simulated domain was 115.5–126.5◦ E and 15.5–24.5◦ N, with auniformly horizontal resolution of 1/20◦ (approximate 5 km). A total of
25 uneven sigma layers were considered in the vertical direction with afiner resolution near both the surface and bottom. The bathymetry usedin the model (Fig. 1) was obtained from the ETOPO1 data (Amante andEakins, 2009) and smoothed to minimize the horizontal pressure
gradient error. At the open boundaries of the simulated domain, thesurface elevations and barotropic currents of the M2 or K1 extracted fromthe Oregon State University global models of ocean tides (TPXO7.2;
Egbert and Erofeeva, 2002) were introduced into the model via the
Flather radiative conditions. To avoid artificial reflection of ITs at the
open boundaries, a 0.5◦ sponge layer was used to absorb the baroclinic
energy.
The Large/McWilliams/Doney K-profile parameterization mixing
scheme (Large et al. 1994) was used for vertical turbulent mixing of
momentum and tracers. The Laplacian horizontal mixing of momentum
was applied to handle the subgrid-scale turbulence. The explicit lateralviscosity was set to 10 m2/s and the Bilaplacian background diffusivity
was 20 m4/s, accompanying the implicit diffusion in the advection
schemes(CROCOuser
guide,
https://www.croco-ocean.
org/documentation/roms_agrif-user-guide/).
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The Kuroshio and ITs were simultaneously simulated from 1 January to
31 March 2010 (a total of 90 days). The simulated Kuroshio exhibitedlooping and leaping characteristics in January and February, respectively(see Section 3.2). The simulation was forced by either the M2 or K1 baro­
tropic tide at the open boundaries, i.e. the M2 run and the K1 run. In eachrun, the initial conditions including the water temperature, salinity, sea
surface height and background currents on 1 January 2010 were inter­
polated from the hybrid coordinate ocean model (HYCOM) reanalysis re­sults downloaded from the Asia-Pacific Data-Research Center (apdrc.soest.
hawaii.edu/data/data.php), which have a spatial resolution of 0.08◦ . At
the open boundaries, in addition to the tidal forcing, the daily temperature,salinity and currents from the HYCOM reanalysis results were introduced
into the model via a nudging scheme (CROCO user guide, https://www.
croco-ocean.org/documentation/roms_agrif-user-guide/). At the sea sur­face, the atmospheric forcing data including the air temperature and spe­cific humidity at 2 m height, zonal and meridional wind speeds at 10 mheight, skin temperature, precipitation rate, longwave radiation flux andsolar radiation flux downloaded from the National Centers for Environ­mental Prediction Reanalysis II (NCEP2) data (psl.noaa.gov/data/gridded/data.ncep.reanalysis2.gaussian.html) were interpolated onto
the model grids. The NCEP2 data have a spatial resolution of 1.875◦ and atemporal interval of 6 h.
In each run, the simulated results were output every one hour. To
ensure that the ITs were sufficiently developed, the simulated results inthe first 10 days were discarded for analysis. Therefore, the simulated
results from 11 January to 31 March 2010 (a total of 80 days) were used
for harmonic analysis and calculation of conversion rate, energy flux andHKE of ITs. According to previous studies (e.g. Zhao et al., 2010; Pick­ering et al., 2015), the 80-day record is sufficient to determine thecoherent ITs. Because the simulated Kuroshio took a looping and aleaping path in January and February, respectively, the simulated re­sults from 11 to 31 January (1 to 21 February) were used to extract theincoherent ITs under the influence of looping (leaping) Kuroshio.

1
H+η

1 ′
C = P ( − H)Wbt ( − H)cos[θP ( − H) − θW ( − H) ]
2

′

∫η
F=

′

′

p (z, t)u (z, t)dz
− H

(5)

(6)

where P′ , Wbt, θP and θW are the amplitudes and phases of pressure
perturbation and vertical barotropic velocity with respect to the M2 (K1)
ITs in the M2 (K1) run. Similarly, the zonal and meridional componentsof tidal-period-averaged energy flux were calculated as:∫η
1 ′′P (z)U (z)cos[θP (z) − θU (z) ]dz
Fx =
(7)
− H 2
∫η
Fy =

1 ′
′
P (z)V (z)cos[θP (z) − θV (z) ]dz

− H2

(8)

where U′ , V′ , θU and θV are the amplitudes and phases of zonal andmeridional baroclinic velocities. In this study, P′ , Wbt, U′ , V′ , θP, θW, θU
and θV were calculated through harmonic analysis of p’, wbt, u’ and v’
with the Matlab package T_tide (Pawlowicz et al. 2002). The HKE wascalculated as.
∫
1 η ( ′2
′ )
HKE =
(9)
ρ u + v 2 dz
2 −H
According to previous studies, there are two methods to evaluate theincoherence of ITs. Method one is widely used in the analysis of mooredobservations (e.g. Zhao et al., 2010; Xu et al., 2013, 2014; Cao et al.,2017). In this method, because observations contain signals of all fre­quency bands, a bandpass filter is first adopted to isolate the signals inthe semidiurnal (usually [1.73, 2.13] cpd) and diurnal frequency bands(usually [0.80, 1.20] cpd). The hindcasted signal of the M2, S2, N2 and K2(K1, O1, P1 and Q1) through harmonic analysis is regarded as thecoherent semidiurnal (diurnal) component. Subtracting the coherent
semidiurnal (diurnal) component from the bandpass filtered semi­
diurnal (diurnal) signal, the incoherent semidiurnal (diurnal) compo­
nent is obtained. Method two was used by Kerry et al. (2016) whenanalyzing the year-long data of simulated M2 ITs in the Philippine Sea. In
this method, the harmonic analysis result of the entire year-long data isregarded as the coherent component. The harmonic analysis result of
every 3-day data reflects the influence of background currents on the M2
ITs. The difference between them is regarded as the incoherent
component. Note that because of the Rayleigh criterion, the 3-day
window is too short to separate the predominant constituents (e.g. theM2, S2, K1 and O1). Therefore, method two is suitable to extract theincoherent ITs from numerical simulations with single tidal forcing
rather than extract the incoherent ITs from oceanic observations,whereas method one is feasible in both cases. Because only the M2 or K1
tidal forcing was considered in the numerical simulations of this study,
both methods were suitable. The Appendix compares the two methods
and explores the sensitivity of method two to harmonic analysis win­
dow. According to the Appendix, method two with 1-day harmonic
analysis window has a comparable performance to method one. There­
fore, method two with 1-day harmonic analysis window was used toevaluate the incoherence of ITs. The coherent energy fluxes and HKE
were calculated from the harmonic analysis results of the 80-day (from11 January to 31 March 2010) records. The calculation of incoherent
energy fluxes was similar to Kerry et al. (2016) except the harmonic
analysis window. The incoherent HKE was calculated from the inco­
herent baroclinic currents according to Eq. (9).

The incoherence of the M2 and K1 ITs under different Kuroshio paths
is the main focus of this study. The conversion rate, baroclinic energy
flux and HKE were calculated to quantify the incoherence of ITs. Ac­
cording to Kerry et al. (2014a), the barotropic-to-baroclinic tidal con­version rate (C) and depth-integrated baroclinic energy flux (F) were
calculated as:′′

p (z, t)dz = 0− H

Following Zilberman et al. (2011), the tidal-period-averaged con­version rate was calculated as.

2.2. Data analysis methods

C = p ( − H, t)wbt ( − H, t) = p ( − H, t)[ubt ⋅∇( − H) ]

∫η

(1)
(2)

where t is the time, z is the depth, η is the surface elevation, H is the total
water depth, p’ is the pressure perturbation, ubt=(ubt, vbt) and wbt are thehorizontal and vertical barotropic velocities and u’=(u’, v’) is the hor­izontal baroclinic velocity. The barotropic and baroclinic velocities ubt
and u’ were calculated as:
∫η
1
′
ubt =
udz and u = u − ubt
H+η − H
where u=(u, v) is the simulated horizontal velocity. The pressure
perturbation p’ was calculated as:∫η′′p (z, t) = psurf (t) +ρ (̂z , t)gd̂z(4)z

where ρ’ is the density perturbation which is determined by the
instantaneous density (ρ) minus the time-averaged density, g is the ac­celeration due to gravity, and psurf is the surface pressure which isdetermined by.
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Fig. 2. Cotidal charts (amplitude: shading, unit: m; phase: dashed lines, unit: ◦ ) of the (a and b) M2 and (c and d) K1 surface tides for the (a and c) TPXO results and (band d) CROCO simulations. The domain-averaged RMSEs between CROCO simulations and TPXO results are listed in panels (b) and (d).

Fig. 3. Vertical averaged PSDs of zonal currents at 120◦ E, 21◦ N in the two runs. The vertical dashed lines indicate the local Coriolis (f), K1 and M2 frequencies. The
shading indicates the frequency range ([0, 0.5] cpd) to extract background currents.

3. Validation of simulated results

the cotidal charts extracted from TPXO are also shown. On the whole,
the M2 and K1 runs reproduce the characteristics of the M2 and K1 sur­face tides around the LS, respectively. Both the M2 and K1 surface tides
in the northern SCS are propagated from the Philippine Sea. The M2
surface tide has larger amplitudes in the Philippine Sea and Taiwan
Strait and smaller amplitudes in the SCS Basin, whereas the amplitudesof the K1 surface tide are larger in the SCS Basin than in the Philippine
Sea. To further assess the accuracy of the simulated M2 and K1 surface

3.1. Barotropic tides
Before exploring the influence of looping and leaping Kuroshio on
the incoherence of the M2 and K1 ITs generated at the LS, it is necessary
to validate the simulated results of CROCO. Fig. 2 shows the cotidal
charts of the M2 and K1 surface tides in the two runs. As a comparison,4

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Fig. 4. Time-averaged surface currents in (a) January, (b) February and (c) March. The looping and leaping Kuroshio paths in January and February are indicated byblue curves in panels (a) and (b), respectively. Zonal sections of (d) meridional background currents (unit: m/s), (e) temperature (unit: ◦ C) and (f) salinity along 22◦ N
averaged in the three months.

Fig. 5. Comparison of amplitudes of zonal internal tidal currents between in situ observations (black) and simulated results for the M2 (blue) and K1 ITs (red).
Observations at the six moorings (MP1-MP6) are obtained from Cao et al. (2017).

tides, the root-mean-square error (RMSE, Cummins and Oey, 1997) is
calculated as.
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
)1( 2RMSE =A + A2s − Ao As cos(Go − Gs )
(10)
2 o

RMSEs for the M2 and K1 surface tides are 4.2 and 2.8 cm, respec­tively, which demonstrate the reliability of the simulated M2 and K1
surface tides. From Fig. 2b and 2d, we can detect small fluctuations
superposed on the co-amplitude and co-phase lines, which are the causeof the RMSEs. Actually, these fluctuations are the surface modulations of
the M2 and K1 ITs radiated from the LS (e.g. Niwa and Hibiya, 2004; Janet al., 2008; Xu et al., 2016), which are absent from TPXO.

where Ao and Go are the M2 (K1) surface amplitude and phase from TPXO
and As and Gs are those from the M2 (K1) run. The domain-averaged

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Fig. 6. (a and b) Conversion rates (shading, unit: W/m2) and (c and d) baroclinic energy fluxes (magnitude: shading, unit: kW/m; direction: quivers) of the (a and c)
M2 and (b and d) K1 ITs. The gray contours indicate the 1000, 2000 and 3000 m isobaths. In panels (a) and (b), the LS is indicated by the dashed box and theconversion integrated at the LS is shown.

3.2. Background currents

3.3. Internal tidal currents

Before validating the background currents simulated by CROCO, weneed to extract them from the raw model outputs at first. Fig. 3 displays
the vertical averaged power spectral densities (PSDs) of zonal currents at
120oE, 21oN in the two runs. In the spectra, the M2 (K1) and its higher
harmonics exhibit apparent peaks in the M2 (K1) run. In addition to thetidal peaks, significant peaks can be found at a very low frequency,
which correspond to the background currents. We can also find from
Fig. 3 that all the PSDs in the two runs reach a local minimum near 0.5
cpd. Therefore, a fourth-order lowpass Butterworth filter with the cutoff
frequency of 0.5 cpd is adopted to extract the low-frequency backgroundcurrents. This filtering band ([0, 0.5] cpd) is wide enough to maximally
extract the background currents and narrow enough to remove thesignals from tidal and near-inertial frequencies. The filtered lowfrequency background currents in the two runs are nearly the same.
Fig. 4 shows the simulated monthly surface currents and zonal sec­
tions of meridional background currents, temperature and salinity along22◦ N. The simulated Kuroshio took a looping path (Fig. 4a) and aleaping path (Fig. 4b) in January and February, respectively (Nan et al.,2011). The simulated meridional background currents, temperature andsalinity along 22◦ N are consistent with the climatological-mean obser­
vations in winter (Rudnick et al., 2011; Fig. 4 therein). Therefore, we
believe that the simulated results can be used to explore the incoherenceof the M2 and K1 ITs under the influence of looping and leaping
Kuroshio.

The simulated ITs are validated by comparing with in situ observa­
tions. Fig. 5 displays the simulated M2 and K1 amplitudes of zonal in­
ternal tidal currents at six moorings in the SCS (Fig. 1) as well as theobserved results (Cao et al., 2017). On the whole, the simulated M2 andK1 internal tidal currents are consistent with observations at the six
moorings, especially for the vertical structure. However, the magnitudeof simulated M2 and K1 internal tidal currents deviates a little fromobservations at some moorings. This deviation is attributed to thefollowing causes. First, the simulated amplitudes are obtained throughharmonic analysis of 80-day records from January to March, whereas
the observed amplitudes are obtained from longer time series (nearly
5–8 months). Second, the nonlinear interaction of ITs at different fre­quencies, which has a crucial impact on the intensity of simulated ITs(Guo et al., 2020a), is not considered in the simulations of this study.
Based on the above reasons, we believe that the simulated M2 and K1 ITs
are reasonable.
4. Baroclinic energetics
Fig. 6 shows the conversion rates and baroclinic energy fluxes of theM2 and K1 ITs, which are calculated from the harmonic analysis resultsof the last 80 days. It is easy to find that the LS is the main source of theM2 and K1 ITs. However, the M2 and K1 tidal conversion rates show
different spatial patterns at the LS. For the M2 ITs, large conversion rates

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Fig. 7. Time-averaged incoherent baroclinic fluxes (magnitude: shading, unit: kW/m; direction: quivers) of the (a and b) M2 and (c and d) K1 ITs under the influenceof (a and c) looping and (b and d) leaping Kuroshio near the LS. Note that the colorbar range is different for the M2 and K1 incoherent energy fluxes.

Fig. 8. Meridionally (18.5-22◦ N) integrated zonal energy fluxes for the (a) M2 and (b) K1 ITs under the influence of the looping and leaping Kuroshio near the LS. The
shading denotes the location of the LS.

mainly appear upon the east ridge and the eastern flank of the west ridgeof the LS (e.g. Kerry et al., 2013; 2014a; Xu et al., 2016; Guo et al.,2020a). Whereas for the K1 ITs, large conversion rates mainly appear
upon the east ridge, especially the southwestern part, as well as thewestern flank of the west ridge, and the conversion rates upon theeastern flank of the west ridge are negative (Xu et al., 2016). The M2
conversion integrated at the LS is 19.5 GW, which is consistent with the13.01–19.95 GW estimated by Kerry et al. (2016). The K1 conversion
integrated at the LS is 12.5 GW, which is comparable to the 12.3 GW

estimated by Xu et al. (2016).
The energy fluxes show the propagation paths of the M2 and K1 ITs.
The M2 ITs generated at LS mainly propagate in three directions: west­
ward to the Dongsha Island, southwestward to the SCS Basin and east­
ward to the Philippine Sea, whereas the K1 ITs generated at LS mainly
propagate eastward and westward. These features agree with thosederived from altimetry data (Zhao, 2014) and previous numerical sim­
ulations (e.g. Xu et al., 2016; Guo et al., 2020a).

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Fig. 9. Proportions (shading, unit: %) of the (a and b) M2 and (c and d) K1 incoherent HKE under the influence of (a and c) looping and (b and d) leaping Kuroshio
near the LS. The gray contours indicate the 1000, 2000 and 3000 m isobaths.

Fig. 10. Meridionally (18.5-22◦ N) averaged proportions of incoherent HKE for the M2 and K1 ITs under the influence of the looping and leaping Kuroshio near the LS.
The shading denotes the location of the LS.

5. Incoherence(2016) and observations of Pickering et al. (2015).
The looping and leaping Kuroshio indeed have different impacts on
the M2 and K1 incoherent energy fluxes. Both the M2 and K1 incoherent
energy fluxes are stronger under the looping Kuroshio than under theleaping Kuroshio. The maximum values of the M2 and K1 incoherent
energy fluxes are 11.3 and 7.3 kW/m under the looping Kuroshio, while
they are 8.1 and 4.6 kW/m under the leaping Kuroshio. The spatial
patterns of the M2 and K1 incoherent energy fluxes under the looping
and leaping Kuroshio also show significant differences. The M2 inco­
herent energy fluxes near the LS mainly have two branches (westward
and northeastward) under the looping Kuroshio, while there are threebranches (westward, northeastward and southeastward) under the

5.1. Incoherence of energy fluxes
Fig. 7 illustrates the time-averaged M2 and K1 incoherent energy
fluxes near the LS under the influence of looping and leaping Kuroshio.
Compared with the M2 and K1 coherent energy fluxes shown in Fig. 6cand 6d, the M2 and K1 incoherent energy fluxes are weaker. The
coherent energy fluxes of the eastward and westward M2 beams andwestward K1 beams radiated from the LS could reach 30 kW/m (Fig. 6cand 6d), while the incoherent energy fluxes are smaller than 12 kW/m
(Fig. 7). This result is consistent with the simulations of Kerry et al.
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Fig. 11. (a and b) Time-averaged mode-1 eigenspeed (shading, unit: m/s) and (c and d) its normalized standard deviation under the influence of (a and c) loopingand (b and d) leaping Kuroshio near the LS. The gray contours indicate the 1000, 2000 and 3000 m isobaths.

Fig. 12. The mode-1 eigenspeed in the SCS (120oE, 21oN) under the influence of the (a) looping and (b) leaping Kuroshio. The combined and individual effects ofstratification and background currents on eigenspeed are shown by solid curves with different color.

leaping Kuroshio. In contrast to the M2 incoherent energy fluxes which
are mainly east-westward, the K1 incoherent energy fluxes are mainly
southward near the LS. The K1 incoherent energy fluxes exhibited amore complicated pattern in the SCS under the looping Kuroshio thanthe leaping Kuroshio. These results suggest that the Kuroshio in different
paths have different impacts on the generation and propagation of ITs atthe LS (Kerry et al., 2016; Xu et al. 2021). In addition, whether under the
influence of looping or leaping Kuroshio, the magnitude of the M2 andK1 incoherent energy fluxes are stronger to the west of the LS (in theSCS) than to the east of the LS (in the Philippine Sea).

Fig. 8 shows the zonal energy fluxes meridionally integrated between
18.5oN and 22oN, from which the decay patterns of the M2 and K1 en­ergy fluxes away from the LS are detected. The changes of the M2
coherent and incoherent energy fluxes (Fig. 8a) generally agree withthose in Kerry et al. (2016). For both the M2 and K1 ITs, the eastward andwestward coherent energy fluxes reach their maximum on the east andwest ridges of the LS, respectively. With the increase of distance awayfrom the LS, both the M2 and K1 coherent energy fluxes gradually
decrease. The 1-day mean energy fluxes of the M2 and K1 ITs exhibit asimilar pattern as their coherent ones, respectively. However, the9

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Progress in Oceanography 206 (2022) 102850

incoherent energy fluxes of the M2 and K1 ITs show different tendencies
from their coherent energy fluxes. Because the K1 incoherent energy
fluxes are mainly southward near the LS (Fig. 7c and 7d), the meridio­
nally integrated K1 incoherent energy fluxes are relatively small. The M2
incoherent energy fluxes generally show an increasing tendency withthe distance away from the LS. As a result, the ratio between the M2
incoherent and coherent energy fluxes Fin/Fco (Fin and Fco represent themagnitude of meridionally integrated incoherent and coherent energy
fluxes) generally increases with the distance away from the LS. Under
the looping (leaping) Kuroshio, the Fin/Fco of the westward propagating
M2 ITs increases from 13.7% (4.2%) at 121oE to 47.0% (45.7%) at
118.5oE and the Fin/Fco of the eastward propagating M2 ITs increases
from 14.7% (7.7%) at 122oE to 52.1% (29.2%) at 124oE. These results
are generally consistent with the estimation of Kerry et al. (2016) andindicate that the looping Kuroshio indeed causes larger incoherent en­ergy fluxes than the leaping Kuroshio does. Actually, the leaping Kur­oshio causes a domain averaged Fin/Fco of 22.4%, which is elevated by
the looping Kuroshio to 28.8%. Because the Fco is the same, this result
suggests that compared with the leaping Kuroshio, the looping Kuroshio
elevates the proportion of the M2 incoherent energy fluxes by 29% near
the LS.

the mode-1 eigenspeed (or phase speed) can be used to diagnose theincoherence of ITs. Hence, the Taylor-Goldstein (TG) equation (e.g.
Buijsman et al., 2017; Xu et al., 2021; Wang et al., 2021) is introduced:
]
[
d2 W
N2
1 d2 U2
W=0(11)+− |k| −dz2
U − c dz2
(U − c)2where W is the vertical velocity eigenfunction, c is the eigenspeed, N isthe buoyancy frequency, U is the background currents and k is thehorizontal wave number. The phase speed (cp) of ITs is a function of theeigenspeed (e.g. Zhao et al., 2010):

ω

cp = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ cω2 − f 2(12)Eqs. (11) and (12) reflect the influence of stratification and lowfrequency background currents on the propagation of ITs. Fig. 11
shows the time-averaged mode-1 eigenspeed and its normalized stan­
dard deviation under the looping and leaping Kuroshio near the LS. The
spatial patterns of time-averaged eigenspeed under the looping andleaping Kuroshio are generally the same: the eigenspeed is a little larger
in the Philippine Sea than in the SCS. However, the looping Kuroshio
causes a larger variation of eigenspeed than the leaping Kuroshio, which
is clearly shown in Fig. 11c and 11d. This result explains why the looping
Kuroshio causes larger proportions of incoherent ITs than the leaping
Kuroshio. In addition, because the variation of eigenspeed is greater in
the SCS than in the Philippine Sea especially under the looping Kuroshio(Fig. 11c and 11d), the incoherence of the M2 and K1 ITs in the SCS is
more significant.
According to Eq. (11), the eigenspeed is modulated by both lowfrequency background currents (U) and stratification (N). To investi­
gate the relative importance of background currents and stratification in
determining the eigenspeed, we also solved the TG equation under theindividual influence of stratification or background currents (Fig. 12shows an example). Compared with the influence of stratification, theeigenspeed under the influence of background currents is more close tothat under the combined effects of stratification and background cur­rents, for both the looping and leaping Kuroshio. This result indicatesthat whether under the looping or leaping Kuroshio, the low-frequency
background currents play a more important role than the stratificationin modulating the eigenspeed. This is consistent with the conclusions of
Buijsman et al. (2017) and Xu et al. (2021).
The simulated results in this study are qualitatively consistent with insitu observations. For example, the north-south asymmetry of incoher­
ence of ITs in the SCS Basin reported by Xu et al. (2014) is visible inFig. 9. The proportions of the M2 and K1 incoherent HKE is stronger tothe west of the LS (in the SCS) than to the east of the LS (in the PhilippineSea), which is consistent with the observations of Cao et al. (2017).
Nevertheless, because the remote generation sites of ITs (e.g. Mariana
Island Arc; Kerry et al., 2013; Wang et al., 2018) and nonlinear in­teractions of ITs at different frequencies (e.g. Guo et al., 2020a) are notconsidered in this study, the quantitative estimations of incoherence of
ITs in this study still have a bias from those obtained from in situ
observations.

5.2. Incoherence of HKE
Fig. 9 displays the proportions of the M2 and K1 incoherent HKE
under the looping and leaping Kuroshio near the LS, and Fig. 10 illus­
trates their meridionally averaged values. From these results, we caneasily obtain the same conclusion as that obtained from the M2 and K1
energy fluxes, i.e. the looping Kuroshio causes stronger incoherent ITs
than the leaping Kuroshio. Under the looping Kuroshio, the domain
averaged proportions of the M2 and K1 incoherent HKE are 24.0% and17.7%, which are greater than those (15.2% and 12.8%) under theleaping Kuroshio. Compared with the leaping Kuroshio, the looping
Kuroshio enhances the proportions of the M2 and K1 incoherent HKE by
58% and 38%, respectively.
The proportions of the M2 and K1 incoherent HKE show spatial dif­ferences. According to Fig. 9, the proportions of incoherent HKE along
the main propagation paths of the M2 and K1 ITs (Fig. 6c and 6d) areapparently smaller than those away from the main propagation paths.
This is because that the coherent HKE are stronger along the main
propagation paths of the M2 and K1 ITs, which results in smaller pro­portions of incoherent HKE. This may to a certain extent explain thenorth–south asymmetry of incoherence of ITs in the SCS Basin reported
by Xu et al. (2014). Moreover, because the main propagation paths of
the K1 ITs (Fig. 6d) are wider than those of the M2 (Fig. 6c), the pro­portions of the K1 incoherent HKE are smaller than those of the M2. In
addition, the proportions of the M2 and K1 incoherent HKE are greater tothe west of the LS (in the SCS) than to the east of the LS (in the Philippine
Sea). To the west of the LS, the averaged proportions the M2 and K1
incoherent HKE are 29.5% and 22.6% under the leaping Kuroshio(20.2% and 17.2% under the looping Kuroshio), whereas to the east of
the LS, the values become 25.3% and 12.0% (15.7% and 8.5%). Com­
parison of the two ITs reveals that their incoherence are comparable atthe LS, whereas in the SCS and Philippine Sea, the proportions of the K1
incoherent HKE become a little weaker than those of the M2.

7. Summary
In this study, we investigated the impacts of looping and leaping
Kuroshio on the incoherence of the M2 and K1 ITs radiated from the LS,
based on the simultaneous numerical simulations of ITs and backgroundcurrents. Results indicate that the looping Kuroshio causes stronger
incoherence of the M2 and K1 ITs near the LS than the leaping Kuroshio
does. Compared with the leaping Kuroshio, the looping Kuroshio ele­
vates the proportions of the M2 incoherent energy fluxes and the M2 andK1 incoherent HKE by 29%, 58% and 38% near the LS. By solving the TG

6. Discussion
Results of this study indicate that compared with the leaping Kur­oshio, the looping Kuroshio elevates the proportions of the M2 and K1
incoherent ITs. To understand why the looping Kuroshio causes larger
proportions of incoherent ITs than the leaping Kuroshio, attention ispaid to the propagation of ITs modulated by the Kuroshio. According to
Buijsman et al. (2017) and Savage et al. (2020), the relative change of10

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Progress in Oceanography 206 (2022) 102850

equation, we find that the looping Kuroshio causes a larger variation of
eigenspeed, which accounts for the stronger incoherence of ITs than theleaping Kuroshio does. In this process, the low-frequency backgroundcurrents play a more important role than the stratification in changing
the eigenspeed, for both the looping and leaping Kuroshio. Moreover,
the observed north-south asymmetry of incoherence of ITs in the SCS
Basin and the asymmetry of incoherence of ITs to the east and west of the
LS are reproduced in the numerical simulations of this study. We believethat the results of this study can deepen our understanding of theincoherent nature of ITs near the LS.

analysis window and compared them with the result of method one.
The simulated results of the M2 and K1 runs at 120◦ E, 21◦ N weretaken as examples. First, the semidiurnal ([1.73, 2.13] cpd) and diurnal([0.80, 1.20] cpd) baroclinic currents were extracted through a fourthorder Butterworth filter from the raw simulated results of the M2 andK1 runs, respectively. Because the filtering has a boundary effect on theextracted signals, the filtered currents in the last three days were notused. Then, harmonic analysis was performed to the baroclinic currents
with different windows (1, 3 and 80 days). For the results of the M2 (K1)run, only the M2 (K1) constituent was considered in the harmonic
analysis. Note that the harmonic analysis result with the 80-day window
is the coherent component in both methods. Finally, all these currents
were used to calculate the HKE according to Eq. (9).
Fig. A1 shows the filtered HKE and those obtained through harmonic
analysis with different windows. Both the filtered semidiurnal anddiurnal HKE significantly varies under the influence of backgroundcurrents and these variations have a scale of one to several days. The 3day window used by Kerry et al. (2016) can generally capture thesevariations as the hindcasted M2 and K1 HKE with the 3-day window
accounts for 97% of the semidiurnal and diurnal HKE, respectively.
Compared with the 3-day window, the 1-day window has a better per­formance: The hindcasted M2 and K1 HKE accounts for nearly 100% of
the semidiurnal and diurnal HKE, respectively. In this case, the two
methods can be regarded as the same. This conclusion is further
confirmed by the proportions of the 1-day HKE in the filtered HKE under
the influence of looping and leaping Kuroshio (Fig. A2), which exceed
98% at most areas. Based on these results, method two with 1-day
harmonic analysis window was used to evaluate the incoherence ofthe energy flux and HKE of ITs in this study.

Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
This study is supported by the National Natural Science Foundationof China through grant 41806012, the State Key Laboratory of Tropical
Oceanography, South China Sea Institute of Oceanology, Chinese
Academy of Sciences through grant LTO2108, and the China ScholarshipCouncil through grant 202006325025.
Appendix. . Comparison of the two methods to evaluate theincoherence of ITs
Kerry et al. (2016) used method two with a 3-day window whenevaluating the incoherence of the M2 ITs in the Philippine Sea. However,when analyzing the simulated data in this study, we found some varia­
tions within 3 days in both semidiurnal and diurnal bands. Therefore, weinvestigated the sensitivity of results of method two to harmonic

Fig. A1. Comparison between the bandpass filtered HKE and hindcated HKE through harmonic analysis with different windows (1, 3 and 80 days) at 120◦ E, 21◦ N forthe (a) M2 and (b) K1 ITs. Note that the blue curve is almost overlapped on the black curve. The proportions of hindcasted HKE in the filtered HKE are shown in
the labels.

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Progress in Oceanography 206 (2022) 102850

Fig. A2. Proportions (shading, unit: %) of the 1-day HKE in the filtered HKE for the (a and b) M2 and (c and d) K1 ITs under the influence of (a and c) looping andleaping Kuroshio. The black dashed box indicates the region near the LS. The gray contours indicate the 1000, 2000 and 3000 m isobaths.

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