Deep–Sea Research I 192 (2023) 103946

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Deep-Sea Research Part I
journal homepage: www.elsevier.com/locate/dsri

Refraction of the M2 internal tides by mesoscale eddies in the South
China Sea
Zheng Guo a, Shuya Wang b, Anzhou Cao c, b, *, Jieshuo Xie d, e, Jinbao Song c, Xinyu Guo baMarine Science and Technology College, Zhejiang Ocean University, Zhoushan, China
Center for Marine Environmental Studies, Ehime University, Matsuyama, Japanc
Ocean College, Zhejiang University, Zhoushan, ChinadState Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, ChinaeSouthern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou, Chinab

A B S T R A C T

Mesoscale eddies (MEs) are active in the northern South China Sea, yet it is hard to separate their impacts on the internal tides (ITs) from those of other subtidal
circulation. This study examines the role of an anticyclonic eddy and a cyclonic eddy in modulating the M2 ITs based on numerical simulations. In idealized ex­
periments, MEs that differ only in polarity are located on the main beam of the M2 ITs in the northern South China Sea. Results show that the anticyclonic eddy andcyclonic eddy cause northward and southward refractions of the M2 ITs, respectively, which echoes the realistic simulation. The most dramatic changes in the M2 IT
energy fluxes occur on the continental slope to the west of the MEs. Theoretical analysis based on the ray-tracing model and the empirical model for estimating wave
front locations reveals that the MEs cause refraction of the M2 ITs by changing the phase speeds. Further investigation shows that the currents related to the MEs
make a greater contribution to the IT refraction than the ME-associated stratification. These results have important implications for investigation of spatial andtemporal variations in the magnitude and direction of the M2 IT energy fluxes.

1. Introduction
Internal tides (ITs) are ubiquitous in the stratified ocean. As an in­termediate step of the tides-to-turbulence cascade, ITs contribute greatly
to dissipating surface tidal energy and enhancing mixing (Rudnick et al.,2003). ITs are generated in stratified waters by the interaction of bar­
otropic tidal currents with variable bottom topographies (Garrett andKunze, 2007) and could propagate thousands of kilometers away fromtheir generation site (Zhao, 2014). In this process, ITs inevitably interact
with other oceanic flows and processes, such as the wind-driven circu­
lations and mesoscale eddies (MEs). The modification of ITs by MEs has
been an ongoing subject of study.
MEs can impact the radiation of ITs as they are associated withvarying stratification and sheared currents. Previous oceanic observa­
tions showed direct evidence for the interaction between ITs and MEs.
Chavanne et al. (2010) found that in the Kauai Channel, Hawaii, thecyclonic eddy caused the energy of IT rays propagating through its core
to increase near the surface (up to a factor of 15), with surfacing timedelayed by up to 5 h (~150◦ phase lag). Liang (2014) suggested that thevariation of baroclinic semidiurnal tides over the East Pacific Rise wasrelated to the eddy-induced low-frequency flows in the deep ocean,particularly their cross-ridge components. In the Atlantic Ocean, Löb

et al. (2020) noticed a pronounced damping of the semidiurnal IT energy
fluxes coinciding with the passing of two eddies. In the presence of asurface-intensified eddy, the coherent part of the IT in the first twomodes was lowered. However, a subsurface eddy caused decease of IT
energy flux mainly in the second mode. Whereas, due to the temporaland spatial limitation, it is hard to figure out the dominant mechanism
on how MEs affect the ITs directly from observations.
For the interpretation of observational results, theoretical models aredeveloped by researchers. The ray tracing model of Rainville and Pinkel(2006) could explain the refraction of ITs through a barotropic meso­
scale current field. Duda et al. (2018) improved that ray tracing modelby using extended Jones equation and considering anisotropism of wave
speed. Their results indicated that the IT rays could be strongly redir­ected and even trapped by baroclinic currents. In addition to the ray
tracing model, other theoretical models such as “hydrostatic wave
equations” (Wagner et al., 2017) were also established for correspond­
ing investigations. However, it should be noted that approximations are
introduced in the above theoretical models. As a result, they may lead todivergent results and still need further modification.
Apart from theoretical models, numerical models are another effec­
tive approach for exploring the IT-ME interaction. Compared withtheoretical ones, numerical models provide three-dimensional wave

* Corresponding author. Ocean College, Zhejiang University, Zhoushan, China.
E-mail address: caoanzhou@zju.edu.cn (A. Cao).
https://doi.org/10.1016/j.dsr.2022.103946
Received 10 August 2022; Received in revised form 7 November 2022; Accepted 4 December 2022
Available online 7 December 2022
0967-0637/© 2022 Elsevier Ltd. All rights reserved.

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Deep-Sea Research Part I 192 (2023) 103946

fields of ITs without neglecting several terms in the governing equations.
Based on idealized experiments, Dunphy and Lamb (2014) showed that
hot and cold spots of energy fluxes were produced in beam-like patterns
after mode-1 ITs passed through a barotropic eddy. The pass of mode-1
ITs through a mode-1 baroclinic eddy resulted in IT energy transfer tomodes two and higher. Ponte and Klein (2015) demonstrated that as ITs
propagate across a front, they lose coherence due to the interaction witheddies generated by the unstable jet. By using realistic topography andforcing, numerical models could also reproduce the radiation of ITs in an
eddying field for specific regions. Corresponding results revealed thatmesoscales eddies result in time-variable refraction of ITs and hence thevariation of IT energy (Zaron and Egbert, 2014; Varlamov et al., 2015;
Savage et al., 2020).
In the South China Sea (SCS), both the ITs and MEs are active. The
SCS has been identified as a site of intense ITs due to the double
meridional submarine ridges, strong stratification, and astronomicaltides at the Luzon Strait (LS). The evolution of ITs in the SCS has been
explored widely (e.g., Xu et al., 2016; Wang et al., 2021). They are
generated at the LS, propagate westward in the deep central basin, andthen shoal onto the continental slope (Alford et al., 2015). At each stageof their evolution, ITs are susceptible to MEs that are frequent and en­ergetic in the SCS. Altimetric observation reveals that MEs are present in
the northern SCS during 35%–60% of the time (Chen et al., 2011). The
low-frequency current related to the MEs could reach 1 m/s at the sur­face and the temperature anomaly resulting from anticyclonic eddies(AE) and cyclonic eddies (CE) near the main thermocline are up to 7.5 ◦ Cand − 3.0 ◦ C, respectively (Zhang et al., 2013). Consequently, inevitable
modulation of ITs by MEs in the SCS deserves attention. Hydrographic
observations reveal the changes in propagation speed and the energy
refraction of the mode-1 semidiurnal ITs, as well as strengthened
mode-2 ITs when a pair of eddies passed the northern SCS (Huang et al.,2018). However, because the Kuroshio is strong and takes variable
intrusive pathways within the LS (Nagai et al., 2019), its influence on theITs in the SCS can’t be ignored (Kerry et al., 2014; Xu et al., 2021; Caoet al., 2022) but it is hard to be separated from the influence of MEs.
Separately investigating the influence of MEs is the motivation for ourstudy.
In this study, the influence of MEs on the M2 IT refraction in thenorthern SCS is numerically investigated with specially designed real­
ized and idealized experiments. The configuration for both realized andidealized experiments as well as the analysis methods are presented in
Section 2. In Section 3, the responses of the M2 ITs to a pair of eddies (anAE and a CE) are first simulated in the real context where the Kuroshio
and other subtidal circulation remains. To exclude the influence of theKuroshio and other subtidal circulation and quantify the influence of theMEs, a series of idealized experiments are conducted. In Section 4,
theoretical analysis is performed based on the ray-tracing model and theempirical model for estimating wave front. The conclusions summarize
the paper in Section 5.

ITs and other non-hydrostatic processes are not our focus, the hydro­
static approximation is valid for this study.
The domain of simulation covers the northern SCS and part of theWestern Pacific with the LS located in the center (Fig. 1). The model hasa uniform horizontal resolution of 1/20◦ (approximately 5 km) and 25
uneven vertical sigma layers with a higher resolution near the surface
and bottom. The topography used in the simulation is extracted from the
1-min gridded elevations/bathymetry for the world (ETOPO1) data
(NOAA National Geophysical Data Center, 2009) and has beensmoothed to minimize the horizontal pressure gradient errors. The M2
tidal forcing is introduced into the model at the open boundaries withsurface elevations and barotropic currents extracted from Oregon State
University global models of ocean tides (TPXO7.2; Egbert and&Erofeeva, 2002). A ten-cell-wide (0.5◦ ) sponge layer is used to absorb
the baroclinic energy to avoid artificial reflection of ITs at the open
boundaries. The Large-McWilliams-Doney k-profile parameterization
mixing scheme (Large et al., 1994) is used for vertical turbulent mixingof momentum and tracers. The Laplacian horizontal mixing of mo­
mentum is used for the subgrid-scale turbulence.
2.1.2. Realized experiments
In the realized experiment, we simultaneously simulate the M2 ITs
and subtidal circulation including MEs and the Kuroshio. The boundary
and initial conditions including temperature, salinity, sea surface heightand background currents from the HYbrid Coordinate Ocean Model(HYCOM) are downloaded from the Asia-Pacific Data-Research Center(apdrc.soest.hawaii.edu/data/data.php). The atmospheric forcing isprovided by the National Centers for Environmental Prediction Climate
Forecast System Version 2 data (https://cfs.ncep.noaa.gov/).
The simulation is integrated for 57 days, from 6 December 2013 to 31
January 2014, when an AE and a CE passed the simulation domain
successively. Detailed description of the eddy evolution can be found in
Zhang et al. (2016). The simulated results are output every 1 h. The first
10 days are excluded from analysis to ensure that the ITs are sufficiently
developed.
To validate the simulated results, the subtidal motions are extractedwith a fourth-order lowpass Butterworth filter. The cutoff frequency is
set as 0.5 cpd, which is wide enough to extract the background currents
and narrow enough to remove higher-frequency signals such as tides(Zhang et al., 2016; Cao et al., 2022). The surface velocity averaged forthe entire simulation period is similar to the altimetry observation, andthe characteristics of the mean meridional background currents, tem­
perature, and salinity along 22◦ N are consistent with those of the

2. Methodology
2.1. CROCO model configurations
2.1.1. Common settings
This study employs the Coastal and Regional Ocean COmmunity(CROCO) model (version 1.0; www.croco-ocean.org) to explore theimpacts of MEs on the M2 ITs in the northern SCS. The CROCO model isbuilt on the regional ocean modelling system from Institut de Recherche
pour le Developpement (ROMS_AGRIF) and, by default, solves theprimitive equations under the hydrostatic equilibrium and Boussinesq
approximation. The model performs well in different scenarios including
the ITs in the northern SCS (Marchesiello et al., 2019; Renault et al.,2017; Guo et al., 2020a; 2020b, 2021; Cao et al., 2022). Since the hor­izontal scale of ITs is much larger than the vertical scale in the deep
water of the northern SCS (Zhao, 2014; Xu et al., 2016), and breaking ofFig. 1. Bathymetry (shading, unit: m) in the northern SCS, with gray contours
indicating − 3000, − 2000, − 1000, and − 500 m isobaths.
2

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Deep-Sea Research Part I 192 (2023) 103946

climatological mean observations in winter (Rudnick et al., 2011, Fig. 4
therein). The M2 barotropic tide extracted from the model output is also
validated by comparing with the TPXO7.2. Details can be found in the
supporting information.

1
H+η

− H

where u′ is the horizontal baroclinic velocity obtained by′u (z, t) = u(z, t) − u(z) − u0 (t)2.3. Calculation of phase speed
We calculate the eigenspeed cn using the Taylor-Goldstein equation
as in Xu et al. (2021):[/ ]d2 U dz2
̂
d2 w
N2
2
w
̂ =0
(10)
+
− |k| −dz2
U − cn(U − cn )2
̂ is the eigenfunction of vertical velocity, and U is the back­where w
ground flow component parallel to the IT direction. Thereafter, thephase speed cp can be derived from the eigenspeed following Zhao et al.
(2010).

(3)

where k=(k, l) is the horizontal wave number, N is the buoyancy fre­quency that represents the stratification, ω is the frequency of the M2tide, f is the Coriolis frequency, and H is the total water depth. A nu­
merical method is used for the general N(z) prescribed in this study (Caoet al., 2015). The initial horizontal velocities are (u, v)=(-Ψ y, Ψ x), andthe density field is prescribed by cyclo-geostrophic and hydrostaticbalances. Based on the characteristics of the pair of eddies in the realexperiments, the eddy center (xc, yc) is set at (119◦ E, 21◦ N), |UE| is 0.9
m/s, LE is 1 × 105 m, and the eddy is composed of the first baroclinic
mode and a barotropic mode such that the velocity is zero at the bottom
level. Each idealized experiment is run for 15 days and the results are
stored every 1 h. Only the results of the last 5 days are used for analysis.

cp = (

ωω2 − f 2)1/2 cn(11)

In the calculation, if U is set to be zero, cp would be determined by N.
In other words, we exclude the effects of background current and obtaina solution under the sole influence of stratification. If N is set to be
horizontally homogenous, the spatial mean stratification of the northern
SCS in our case, then we obtain the solution only determined by the
background current. In the following, we use the phase speed of the firstmode to calculate the wave path, considering that most of the IT energy
is transported away by the first mode (Zhao, 2014).
2.4. Energy ray-tracing model

2.2. Analysis methods for baroclinic tidal energetics

The ray trajectories of the M2 ITs are calculated following Duda et al.
(2018):
⃒ ]
[
⎧
dx
∂S⃒⃒
⎪
⎪
=
Q
S(
α
α
+
sin
α
)cos
⎪
⎪
ds
∂θ⃒θ=α
⎪
⎪
⎪
⎪
⃒ ]
[
⎪
⎪
dy
∂S⃒
⎪
⎪
⎪
⎨ ds = Q S(α)sin α − cos α∂θ⃒⃒
θ=α
(12)
⎪
⎪
dp
∂
S
x
⎪
⎪
= QS(α)
⎪
⎪
⎪
ds
∂x
⎪
⎪
⎪
⎪
⎪
dpy
∂S
⎪
⎩
= QS(α)ds∂yThe barotropic-to-baroclinic energy conversion rate Ebt2bc is calcu­
lated with the pressure perturbation and the vertical barotropic velocity
at the seafloor:′(9)where the three terms on the right-hand side are instantaneous, timeaveraged, and depth-averaged horizontal velocities, respectively.

subject to the boundary conditions′(6)

where P′ (Wbt) and θp (θw) are the amplitude and phase of the pressure
perturbation (vertical barotropic velocity) at the seafloor of the M2 ITs,
respectively.
The depth-integrated baroclinic energy flux Fbc is calculated
following Kerry et al. (2014, 2016):
∫η
′
′
Fbc =
p (z, t)u (z, t)dz
(8)

where r = [(x-xc)2+(y-yc)2]1/2 is the distance away from the eddy center,ψ (r) is the horizontal structure of the eddy, and Φ(z) prescribes thevertical structure. The scaling factor of − 55/2/64 ensures that ψ (r) yields
a peak azimuthal velocity of UE at r = LEtanh− 1(51/2/5)≈0.48LE. The
parameter LE is the length scale of the eddy, and the peak velocity has
dropped to below 2% at a radial distance of 2LE. Φ(z) is composed ofsolutions to the Sturm-Liouville problem (Michael Dunphy, 2014)( 2)
dΦn (z)N (z)Φn (z) = 0+ |k|2 2
(2)
2
2
dz
ω − f

Ebt2bc (t) = p (− H, t)wbt (− H, t) = p (− H, t)[ubt (− H, t) ⋅ ∇(− H)]′

p (z, t)dz = 0− H

The tidal-period-averaged conversion rate is obtained following
Zilberman et al. (2011):)(′〈Ebt2bc 〉 = 0.5P (− H)Wbt (− H)cos θp − θw
(7)

2.1.3. Idealized experiments
Since MEs are hard to be separated from the Kuroshio and othersubtidal circulation in the real experiments, we conduct three idealized
experiments: no-eddy experiment, AE experiment, and CE experiment.
In each experiment, the M2 constituent is introduced at the open
boundaries. For the no-eddy experiment, a horizontally homogenous
stratification is specified. The stratification is calculated with annual
temperature and salinity from the World Ocean Atlas 2009 (WOA09)
and averaged over the domain. In the experiments with eddies, thedomain is initialized with a single isolated eddy (Dunphy and Lamb,2014; Xie et al., 2015). The eddy centered at (xc, yc) is prescribed by thestream function
( )
5
52
r
Ψ (r, z) = ψ (r)Φ(z) = −Φ(z)(1)
UE LE sech4
LE64Φn (0) = Φn (− H) = 0

∫η

(4)

where t is the time, p’ is the pressure perturbation, wbt and ubt are thevertical and horizontal barotropic velocities. According to Nash et al.
(2005), p′ is calculated as∫η′′p (z, t) = psurf (t) +ρ (̂z , t)gd̂z(5)z

where the coordinates (x, y) define the ray trajectory which has length
increment ds; the angle α = arc tan (px/py) is the direction of a vector
normal to the phase front; θ is the angle of the wave vector k; Q is thenormalization factor; and S = k/ω is the slowness vector.
To examine IT propagation behavior in this study, we first calculate

where z is the depth, η is water elevation at the surface, ρ′ is the density
perturbation, g is the acceleration due to gravity, and psurf is the surfacepressure, which is determined by3

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Deep-Sea Research Part I 192 (2023) 103946

the phase-speed field using the background current and stratification
over the domain. These are obtained by averaging the 50-h CROCO
output records starting from Day 11 in each idealized experiment.
Thereafter, Equation set (12) are integrated to compute IT ray trajec­
tories with initial α (set to be 180◦ pointing westward in the SCS).

the two periods is approximately 17.52 and 18.07 GW, respectively,
which lies within the range (13.01–19.95 GW) estimated by Kerry et al.
(2016).
Our focus is then put on the energy fluxes of ITs in the northern SCS
where the AE and CE appeared. The propagation of ITs changes in
respondence to the MEs (Fig. 3). The M2 ITs propagated from LS to the
SCS in two beams and the north beam bifurcated to some extent uponleaving LS and merged again after reaching 119◦ E. On Day 16 when an
AE was located to the west of the LS, the north beam propagatednorthwestward after passing the eddy center and impinged on the con­tinental slope at latitudes north of 21◦ N. However, on Day 38 when the
AE was replaced by a CE, the north beam refracted south, directing due
west along 21◦ N. The different patterns indicate refraction of M2 IT
energy by the eddies, which is consistent with observations frommoorings (Huang et al., 2018).
The changes of ITs are certainly related to the MEs. However, vari­ation occurring at the IT generation site and before ITs encounter theeddies can also be attributed to the Kuroshio and other backgroundcirculation (Kerry et al., 2014; Song and Chen, 2020; Xu et al., 2021; Caoet al., 2022). It is hard to separate and quantify their contribution in thereal context. In addition, the AE and CE also differed in positions andstrengths. To determine to what extent the polarity of MEs account for
the above changes in ITs, we conduct idealized experiments and analyze
the results in Section 3.2.

2.5. Empirical model for estimating wave front locations
The empirical model proposed by Jackson (2009) is used to estimate
the IT wave front positions in this study. First, the background current
and stratification are extracted by averaging the 50-h CROCO output
records starting from Day 11 in idealized experiment and the phasespeed of the M2 IT is calculated following Equations (10) and (11). The
next step is to determine the propagation time and propagation path fora wave whose speed varies depending on its location along the propa­
gation path. The problem is solved using the two-dimensional eikonal
differential equation( )2 ( )2∂T∂T+= S2 (x, y)
(13)
∂x
∂y
where T(x, y) is the travel time between an origin (x0, y0) and location(x, y). Only the positions of the mode-1 IT are calculated for the same
reason as that in Section 2.3. The geographic origin for the waves andother settings are the same as those in Jackson (2009).

3.2. Idealized experiments

3. Results

A time window of 50 h starting from Day 11 is selected to calculate
the M2 energy fluxes in each experiment. The start date of the time
window ensures that the IT field has well developed and its length en­sures that the background field changes very slightly. The backgroundfields in the AE and CE experiments on Day 11 are shown in Fig. 4.
Compared to their initial state, both the AE and CE moved a bit west­
ward and their peak velocity showed some decrease. But the two eddies
were still quite similar except having different polarity.
The results of no-eddy experiment serve as the benchmark. The M2
tidal conversion integrated around the LS is 18.52 GW, and the energy
propagating into the SCS (5.07 GW) is a bit larger than that into theWestern Pacific (4.60 GW). The energy budgets are largely consistent
with previous studies (Kerry et al., 2013; Xu et al., 2016). Fig. 5a shows astrong westward beam of the M2 ITs pointing to the Dongsha Island anda much weaker beam propagating southwestward to the SCS basin. The
patterns are generally consistent with previous research (e.g., Kerryet al., 2013; Xu et al., 2016; Zhao, 2020).
The M2 energy flux beam pointing to the Dongsha Island is stronger
than that propagating southwestward and the eddy is located just on its
path. Consequently, focus is put on its changes. Compared to the noeddy experiment, the changes of the M2 energy fluxes in the northernand southern part of the AE are opposite: increasing in the north anddecreasing in the south (Fig. 5d). The most dramatic changes occur tothe west of the eddy on the continental slope, decreasing to 15% of its
original value or nearly doubling at some areas. However, the CE causes
the M2 energy fluxes to decrease in the eddy’s northern half and near thesouthern edge but to increase in the remaining area (Fig. 5e). Changes
on the continental slope are also more significant. Besides, the directionof the energy fluxes also changes, indicating that the M2 ITs are refracted
to the north by the AE and to the south by the CE.

3.1. Realized experiment
The MEs and other subtidal motions are first extracted with a fourthorder lowpass Butterworth filter. The eddy evolution is shown in Fig. 2.
On Day 16, an AE was centered at approximately 118.5◦ E, 21◦ N with apeak velocity near 1 m/s. Thereafter, it moved westward. A CE appearedat the end of December and gradually developed but the associatedcurrents were slightly weaker than the AE. The successful simulation of
the eddy pair enables us to explore the ITs’ energetics in respondence tosubtidal circulation.
We selected two 50-h time windows starting from Days 16 and 38,
respectively, to conduct harmonic analysis on the simulated results tocalculate the M2 tidal conversion rates and energy fluxes. The M2 con­version integrated within the LS (120◦ − 122.5◦ E, 18.5◦ − 22◦ N) during

4. Analysis of radiation rays and wave fronts
In this section, we will further explore the mechanism for refractionof the M2 ITs by MEs based on propagation of energy radiating rays andwave front. The background current and stratification used in analysis
are extracted from the corresponding idealized simulation (shown in
Fig. 4).
Fig. 6 shows the theoretical energy radiating rays for the mode-1 M2

Fig. 2. Evolvement of the subtidal circulation during the simulation period.
The surface elevation is indicated by shadings (unit: m) and surface currents byarrows. The magenta and cyan circles indicate locations of the AE and CE
nearest to the LS, respectively.
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Deep-Sea Research Part I 192 (2023) 103946

Fig. 3. Depth-integrated baroclinic energy fluxes (quiver) and their magnitude (shading, unit: kW/m) of the M2 ITs when encountering (a) an AE and (b) a CE. The
magenta and cyan circles indicate locations of the AE and CE, respectively.

Fig. 4. Subtidal circulation on Day 11 in the (a) AE and (b) CE experiments with the surface elevation indicated by shadings (unit: m) and surface currents by arrows.
The magenta and cyan circles indicate locations of the AE and CE, respectively.

Fig. 5. Depth-integrated baroclinic energy fluxes (quiver) and their magnitude (shading, unit: kW/m) of the M2 ITs in the (a) no-eddy, (b) AE, and (c) CE exper­
iments. The relative changes of the energy flux magnitude in the (d) AE and (e) CE cases compared with the no-eddy case are also displayed, where only areas withmagnitude greater than 5 kW/m are shown for clarity. The magenta and cyan circles indicate locations of the AE and CE, respectively.

ITs, which are calculated based on the ray-tracing model (Equation set(12); Duda et al., 2018). The CROCO simulated energy fluxes in ideal­
ized experiments are superposed for comparison. The ray trajectories
follow the CROCO simulated energy flux vectors, although there is not
one-to-one agreement because only the mode-1 westward propagating
M2 IT is considered in the theoretical model. When an AE exists to thewest of the LS, the rays bifurcate near the southern edge of the eddy andthe northern subbranch shows stronger refraction. However, thechanges of ray trajectories within the CE are opposite to those within the
AE: they become more convergent, focused within 20.5◦ N-21◦ N after
passing the eddy center, which corresponds to the areas where energy
fluxes increase in the CROCO simulation. It should be noted that, as theM2 energy fluxes to the south of 20◦ N are fairly weak in the CROCO
simulation, they show northward (southward) refraction by the AE (CE)rather than bifurcation (convergence) revealed by the ray tracing model(Fig. 6a and d). Consider that both the background currents and strati­
fication associated with MEs contribute to the refraction of ITs bychanging the phase speed, we further compare their contribution.
Fig. 6b (6e) show that AE- (CE-) induced changes in stratification result
in refraction of rays but it is not obvious. On the contrary, in either the
AE or the CE experiment, the changes in phase speed caused by currents
have a significant influence on the ray trajectories (Fig. 6c and f). The AE
causes most ray trajectories passing through the eddy to turn north andthose passing by the southern edge to turn slightly south. The changes
are opposite when a CE exists. In summary, the rays calculated with thecurrents-determined phase speed show a pattern more similar to those in
Fig. 6a and d. In other words, the currents related to the eddies
contribute more to the refraction of the M2 IT than the stratification.
This is consistent with the results of previous studies that backgroundflow instead of stratification have a more significant influence on IT
5

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Deep-Sea Research Part I 192 (2023) 103946

Fig. 6. M2 IT ray trajectories (green lines) calculated with the background fields of the (a–c) AE and (d–f) CE experiments. During the calculation, the influence ofstratification and currents is considered simultaneously in (a,d), but separately in (b,e) and (c,f), respectively. The depth-integrated energy fluxes of the M2 IT in eachexperiment are shown by quivers in (a,d) for comparison. The magenta and cyan circles indicate locations of the AE and CE, respectively.

Fig. 7. Snapshots of the zonal velocity of the M2 IT at 100 m depth (shadings, unit: m/s) and the wave front locations estimated from the empirical model (blacklines) in the (a) no-eddy, (b) AE, and (c) CE experiments. The magenta and cyan circles indicate locations of the AE and CE, respectively.

refraction (Duda et al., 2018; Savage et al., 2020; Xu et al., 2021; Caoet al., 2022).
The mechanism of the M2 IT refraction can be also confirmed by theempirical model for estimating the wave front (Jackson, 2009). In
snapshots of the zonal velocity of the M2 IT at 100 m depth, the 0 con­tours along the north branch of the M2 westward energy fluxes show
good agreement with the model wave fronts in terms of shape andorientation (Fig. 7). The AE causes convex of the wave front and the CE
causes a concave pattern by changing the phase speed of the M2 IT.

slope besides within the eddy. The CROCO simulation shows good
agreement with model ray trajectories and wave fronts locations, indi­
cating that the MEs cause refraction of the M2 IT by changing the phasespeed field. Based on the raying-tracing model, we find that the back­ground currents related to the MEs contribute more to the refraction
than the stratification.
The role of subtidal circulation on internal waves has been discussed
in previous studies. However, focus has been mainly put on the Kuroshiowith different paths (Jan et al., 2012; Xie et al., 2021; Xu et al., 2021;
Cao et al., 2022). This study makes an attempt to investigate the impacts
of MEs on IT propagation separately from that of other subtidal circu­
lation. The results have important implications for investigation of
spatial and temporal variations in the magnitude and direction of the M2
IT energy fluxes. In addition, since the formation of internal solitary
waves is related to the semidiurnal ITs via nonlinear steeping process(Helfrich and Grimshaw, 2008; Cai et al., 2012), refraction of the M2 IT
by the eddies has great implication for internal solitary waves prediction(Park and Farmer, 2013; Li et al., 2016). Impacts of eddies with different
location and intensity may be further quantified in future studies.

5. Summary and discussion
Background currents could change the generation and propagationof ITs. In the SCS, the Kuroshio, MEs and other subtidal currents are allintense and their impacts on ITs are hard to separate from each other. In
this study, a pair of eddies successively passed the northern SCS were
simulated in the realized experiment and results preliminarily indicatedthe modulation of IT by MEs. To further investigate the mechanism of IT
refraction, we conducted idealized experiments, setting MEs on the main
beam of the M2 IT propagating from the LS into the SCS. The results
show that the M2 IT encountering an AE in its path would refract
northward and a CE would cause refraction to the south. The refraction
leads to significant changes of the M2 energy flux on the continental

Declaration of competing interest
The authors declare that they have no known competing financial
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Deep-Sea Research Part I 192 (2023) 103946

interests or personal relationships that could have appeared to influence
the work reported in this paper.

Kerry, C.G., Powell, B.S., Carter, G.S., 2016. Quantifying the incoherent M2 internal tidein the Philippine Sea. J. Phys. Oceanogr. 46 (8), 2483–2491.
Large, W.G., McWilliams, J.C., Doney, S.C., 1994. Oceanic vertical mixing: a review anda model with a nonlocal boundary layer parameterization. Rev. Geophys. 32 (4),363–403.
Li, Q., Wang, B., Chen, X., Chen, X., Park, J.-H., 2016. Variability of nonlinear internal
waves in the South China Sea affected by the Kuroshio and mesoscale eddies.
J. Geophys. Res.: Oceans 121 (4), 2098–2118. https://doi.org/10.1002/
2015JC011134.
Liang, X., 2014. Semidiurnal tidal currents in the deep ocean near the East Pacific Rise
between 9◦ and 10◦ N. J. Geophys. Res.: Oceans 119 (7), 4264–4277. https://doi.
org/10.1002/2013JC009522.
Löb, J., Köhler, J., Mertens, C., Walter, M., Li, Z., von Storch, J.-S., Rhein, M., 2020.
Observations of the low-mode internal tide and its interaction with mesoscale flow
south of the azores. J. Geophys. Res.: Oceans 125 (11), e2019JC015879. https://doi.
org/10.1029/2019JC015879.
Marchesiello, P., Nguyen, N.M., Gratiot, N., Loisel, H., Anthony, E.J., Dinh, C.S.,
Kestenare, E., 2019. Erosion of the coastal Mekong delta: assessing natural against
man induced processes. Continent. Shelf Res. 181, 72–89. https://doi.org/10.1016/
j.csr.2019.05.004.
Michael Dunphy, 2014. Focusing and Vertical Mode Scattering of the First Mode Internal
Tide via Mesoscale Eddy Interactions. Doctoral dissertation, University of Waterloo,
Waterloo, Canada). Retrieved from. http://hdl.handle.net/10012/8865.
Nagai, T., Saito, H., Suzuki, K., Takahashi, M., 2019. Kuroshio Current: Physical,
Biogeochemical, and Ecosystem Dynamics, vol. 243. John Wiley & Sons.
Nash, J.D., Alford, M.H., Kunze, E., 2005. Estimating internal wave energy fluxes in theocean. J. Atmos. Ocean. Technol. 22 (10), 1551–1570.
NOAA National Geophysical Data Center. 2009: ETOPO1 1 Arc-Minute Global Relief
Model. NOAA National Centers for Environmental Information.
Park, J.-H., Farmer, D., 2013. Effects of Kuroshio intrusions on nonlinear internal waves
in the South China Sea during winter. J. Geophys. Res.: Oceans 118 (12),7081–7094. https://doi.org/10.1002/2013JC008983.
Ponte, A.L., Klein, P., 2015. Incoherent signature of internal tides on sea level inidealized numerical simulations. Geophys. Res. Lett. 42 (5), 1520–1526. https://doi.
org/10.1002/2014GL062583.
Rainville, L., Pinkel, R., 2006. Propagation of low-mode internal waves through theocean. J. Phys. Oceanogr. 36 (6), 1220–1236.
Renault, L., McWilliams, J.C., Penven, P., 2017. Modulation of the agulhas current
retroflection and leakage by oceanic current interaction with the atmosphere incoupled simulations. J. Phys. Oceanogr. 47 (8), 2077–2100. https://doi.org/
10.1175/jpo-d-16-0168.1.
Rudnick, D.L., Boyd, T.J., Brainard, R.E., Sanford, T.B., 2003. From tides to mixing along
the Hawaiian ridge. Science 301, 355–357. https://doi.org/10.1126/
science.1085837.
Rudnick, D.L., Jan, S., Centurioni, L., Lee, C.M., Lien, R.-C., Wang, J., Chern, C.-S., 2011.
Seasonal and mesoscale variability of the Kuroshio near its origin. Oceanography 24(4), 52–63.
Savage, A.C., Waterhouse, A.F., Kelly, S.M., 2020. Internal tide nonstationarity andwave–mesoscale interactions in the tasman sea. J. Phys. Oceanogr. 50 (10),2931–2951. https://doi.org/10.1175/jpo-d-19-0283.1.
Song, P., Chen, X., 2020. Investigation of the internal tides in the northwest pacific ocean
considering the background circulation and stratification. J. Phys. Oceanogr. 50(11), 3165–3188. https://doi.org/10.1175/jpo-d-19-0177.1.
Varlamov, S.M., Guo, X., Miyama, T., Ichikawa, K., Waseda, T., Miyazawa, Y., 2015. M2
baroclinic tide variability modulated by the ocean circulation south of Japan.
J. Geophys. Res.: Oceans 120 (5), 3681–3710. https://doi.org/10.1002/
2015JC010739.
Wagner, G., Ferrando, G., Young, W., 2017. An asymptotic model for the propagation ofoceanic internal tides through quasi-geostrophic flow. J. Fluid Mech. 828, 779–811.
https://doi.org/10.1017/jfm.2017.509.
Wang, S., Cao, A., Li, Q., Chen, X., 2021. Reflection of K1 internal tides at the continental
slope in the northern South China sea. J. Geophys. Res.: Oceans 126 (7),e2021JC017260. https://doi.org/10.1029/2021JC017260.
Xie, J., He, Y., Chen, Z., Xu, J., Cai, S., 2015. Simulations of internal solitary wave
interactions with mesoscale eddies in the northeastern south China sea. J. Phys.
Oceanogr. 45 (12), 2959–2978. https://doi.org/10.1175/jpo-d-15-0029.1.
Xie, J., Fang, W., He, Y., Chen, Z., Liu, G., Gong, Y., Cai, S., 2021. Variation of internal
solitary wave propagation induced by the typical oceanic circulation patterns in thenorthern South China sea deep basin. Geophys. Res. Lett. 48 (15), e2021GL093969
https://doi.org/10.1029/2021GL093969.
Xu, Z., Liu, K., Yin, B., Zhao, Z., Wang, Y., Li, Q., 2016. Long-range propagation andassociated variability of internal tides in the South China Sea. J. Geophys. Res.:
Oceans 121 (11), 8268–8286.
Xu, Z., Wang, Y., Liu, Z., McWilliams, J.C., Gan, J., 2021. Insight into the dynamics of theradiating internal tide associated with the Kuroshio current. J. Geophys. Res.:
Oceans 126 (6), e2020JC017018. https://doi.org/10.1029/2020JC017018.
Zaron, E.D., Egbert, G.D., 2014. Time-variable refraction of the internal tide at the
Hawaiian Ridge. J. Phys. Oceanogr. 44 (2), 538–557.
Zhang, Z., Zhao, W., Tian, J., Liang, X., 2013. A mesoscale eddy pair southwest of Taiwan
and its influence on deep circulation. J. Geophys. Res.: Oceans 118 (12), 6479–6494.
https://doi.org/10.1002/2013JC008994.
Zhang, Z., Tian, J., Qiu, B., Zhao, W., Chang, P., Wu, D., Wan, X., 2016. Observed 3D
structure, generation, and dissipation of oceanic MEs in the South China Sea. Sci.
Rep. 6, 24–349. https://doi.org/10.1038/srep24349.
Zhao, Z., 2014. Internal tide radiation from the Luzon Strait. J. Geophys. Res.: Oceans
119 (8), 5434–5448.

Data availability
The data used in this study are available in the Zenodo at https://doi.
org/10.5281/zenodo.6912171
Acknowledgements
This study is supported by the National Natural Science Foundationof China (Grant numbers: 42176002; 42130404; 41776008), the StateKey Laboratory of Tropical Oceanography, South China Sea Institute of
Oceanology, Chinese Academy of Sciences (Grant number: LTO2108),
the China Scholarship Council (Grant number: 202006325025), and theYouth Science and Technology Innovation Talent of Guangdong TeZhi
Plan (Grant number: 2019TQ05H519).
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.dsr.2022.103946.
References
Alford, M.H., Peacock, T., MacKinnon, J.A., Nash, J.D., Buijsman, M.C., Centurioni, L.R.,
Fringer, O.B., 2015. The formation and fate of internal waves in the South China Sea.
Nature 521 (7550), 65.
Cai, S., Xie, J., He, J., 2012. An overview of internal solitary waves in the South China
sea. Surv. Geophys. 33 (5), 927–943. https://doi.org/10.1007/s10712-012-9176-0.
Cao, A.-Z., Li, B.-T., Lv, X.-Q., 2015. Extraction of internal tidal currents andreconstruction of full-depth tidal currents from mooring observations. J. Atmos.
Ocean. Technol. 32 (7), 1414–1424.
Cao, A., Guo, Z., Wang, S., Guo, X., Song, J., 2022. Incoherence of the M2 and K1 internal
tides radiated from the Luzon Strait under the influence of looping and leaping
Kuroshio. Prog. Oceanogr. 206, 102850 https://doi.org/10.1016/j.
pocean.2022.102850.
Chavanne, C., Flament, P., Carter, G., Merrifield, M., Luther, D., Zaron, E., Gurgel, K.W.,
2010. The surface expression of semidiurnal internal tides near a strong source at
Hawaii. Part I: observations and numerical predictions. J. Phys. Oceanogr. 40 (6),1155–1179.
Chen, G., Hou, Y., Chu, X., 2011. MEs in the South China Sea: mean properties,
spatiotemporal variability, and impact on thermohaline structure. J. Geophys. Res.:
Oceans 116 (C6).
Duda, T.F., Lin, Y.-T., Buijsman, M., Newhall, A.E., 2018. Internal tidal modal ray
refraction and energy ducting in baroclinic gulf stream currents. J. Phys. Oceanogr.
48 (9), 1969–1993.
Dunphy, M., Lamb, K.G., 2014. Focusing and vertical mode scattering of the first modeinternal tide by mesoscale eddy interaction. J. Geophys. Res.: Oceans 119 (1),523–536.
Egbert, G.D., Erofeeva, S.Y., 2002. Efficient inverse modeling of barotropic ocean tides.
J. Atmos. Ocean. Technol. 19 (2), 183–204.
Garrett, C., Kunze, E., 2007. Internal tide generation in the deep ocean. Annu. Rev. Fluid
Mech. 39, 57–87.
Guo, Z., Cao, A., Lv, X., Song, J., 2020a. Impact of multiple tidal forcing on thesimulation of the M2 internal tides in the northern South China Sea. Ocean Dynam.
70 (2), 187–198.
Guo, Z., Cao, A., Lv, X., Song, J., 2020b. Impacts of stratification variation on the M2
internal tide generation in Luzon Strait. Atmos.-Ocean 58 (3), 206–218.
Guo, Z., Cao, A., Wang, S., 2021. Influence of remote internal tides on the locallygenerated internal tides upon the continental slope in the South China sea. J. Mar.
Sci. Eng. 9 (11), 1268.
Helfrich, K.R., Grimshaw, R.H.J., 2008. Nonlinear disintegration of the internal tide.
J. Phys. Oceanogr. 38 (3), 686–701. https://doi.org/10.1175/2007jpo3826.1.
Huang, X., Wang, Z., Zhang, Z., Yang, Y., Zhou, C., Yang, Q., Tian, J., 2018. Role of MEs
in modulating the semidiurnal internal tide: observation results in the northern
South China sea. J. Phys. Oceanogr. 48 (8), 1749–1770.
Jackson, C.R., 2009. An empirical model for estimating the geographic location of
nonlinear internal solitary waves. J. Atmos. Ocean. Technol. 26 (10), 2243–2255.
Jan, S., Chern, C.S., Wang, J., Chiou, M.D., 2012. Generation and propagation of
baroclinic tides modified by the Kuroshio in the Luzon Strait. J. Geophys. Res.:
Oceans 117 (C2).
Kerry, C.G., Powell, B.S., Carter, G.S., 2013. Effects of remote generation sites on model
estimates ofM2Internal tides in the philippine sea. J. Phys. Oceanogr. 43 (1),120927134059000.
Kerry, C.G., Powell, B.S., Carter, G.S., 2014. The impact of subtidal circulation oninternal tide generation and propagation in the Philippine Sea. J. Phys. Oceanogr. 44(5), 1386–1405.

7

Z. Guo et al.

Deep-Sea Research Part I 192 (2023) 103946

Zhao, Z., 2020. Southward internal tides in the northeastern south China sea. J. Geophys.
Res.: Oceans 125 (11), e2020JC016554. https://doi.org/10.1029/2020JC016554.
Zhao, Z., Alford, M.H., MacKinnon, J.A., Pinkel, R., 2010. Long-range propagation of thesemidiurnal internal tide from the Hawaiian ridge. J. Phys. Oceanogr. 40 (4),713–736.

Zilberman, N., Merrifield, M., Carter, G., Luther, D., Levine, M., Boyd, T., 2011.
Incoherent nature of M 2 internal tides at the Hawaiian Ridge. J. Phys. Oceanogr. 41(11), 2021–2036.

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