Deep–Sea Research I 186 (2022) 103828

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

Modulation of submesoscale motions due to tides and a shallow ridge along
the Kuroshio
Eiji Masunaga a, b, *, Yusuke Uchiyama c, Xu Zhang d, Waku Kimura e, Taichi Kosako faGlobal and Local Environment Co-creation Institute, Ibaraki University, Japan
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA, USAc
Department of Civil Engineering, Kobe University, Japand
Department of Atmospheric and Oceanic Sciences, School of Physics, Peking University, ChinaeKajima Corporation, Japanf
Port and Airport Research Institute, Japanb

A R T I C L E I N F O

A B S T R A C T

Keywords:
Submesoscale motions
Kuroshio
Tides
Kinetic energy budget
The Izu-Ogasawara ridge

This study investigated submesoscale motions (SMs) and transport over the Izu-Ogasawara Ridge, located off
mainland Japan, using a double-nested Regional Oceanic Modeling System (ROMS). Both the Kuroshio and tides
largely influence physical processes in this area. The mean and submesoscale kinetic energies (MKE and SKE,
respectively) were calculated using a horizontal Gaussian filter. Tidal forcing enhances SKE through Reynolds
stress due to mean velocity shear (barotropic conversion); meanwhile, tides suppress the MKE and total kinetic
energy. This implies that tidal forcing contributes to energy transfer from large-scale motions to smaller scales,
which results in accelerated kinetic energy dissipation. Energy conversion from submesoscale potential energy to
SKE (baroclinic conversion) is reduced by tidal forcing. In the summer, SKE is generated by barotropic con­version and suppressed by negative baroclinic conversion. On the other hand, in winter, SKE is enhanced by bothbarotropic and baroclinic conversions.

1. Introduction
The Kuroshio Current (‘the Kuroshio’) is a western boundary current
and a major source of kinetic energy that significantly contributes toocean circulation and ecosystems in the northwestern Pacific Ocean (e.
g., Guo et al., 2013; Kodama et al., 2014). The Kuroshio flows from thesouthwest toward the east of mainland Japan, and its flow magnitude
and transport volume reach 2.0 m s− 1 and 65 Sv (1 Sv = 106 m3 s− 1),
respectively (Andres et al., 2015) . Several shallow straits and ridges arelocated along the Kuroshio path, including the Tokara Strait andIzu-Ogasawara Ridge (Fig. 1). Instability and strong mixing occur when
the Kuroshio passes over shallow regions (e.g., Hasegawa et al., 2004;
Nagai et al., 2017; Tsutsumi et al., 2017). Observations conducted by
Hasegawa et al. (2004) showed strongly enhanced mixing due to the
Kuroshio passing around an island, leading to upwelling of deepplankton-rich water. Tsutsumi et al. (2017) reported that an interaction
between the Kuroshio and a shallow ridge resulted in near-inertial in­
ternal waves accompanied by strong diapycnal mixing. The Kuroshio
also transports fish larvae from tropical oceans toward mid-latitude

coastal regions (e.g., Chang et al., 2018; Miyake et al., 2020), andMiyake et al. (2020) showed that the catch of glass eels increases as theKuroshio path shifts toward the coast.
Oceanic eddying motions are typically categorized into two scales:(1) mesoscale motions with horizontal scales of O (100) km or larger and(2) submesoscale motions (SMs) with horizontal scales of less than tensof kilometers. SMs with scales of O (0.1–10) km include eddies, fila­
ments, and fronts, and are referred to as submesoscale coherent struc­tures (e.g., McWilliams, 2016; Dauhajre and McWilliams, 2018).
Recently developed high-resolution observational techniques and nu­
merical simulations can resolve SMs, and these approaches haverevealed the importance of SMs in energy transports/cascades and masstransport (e.g., Thomas et al., 2008; McWilliams, 2016). SMs are
dynamically defined by the Rossby number, Ro > O(1) (Ro = U/fL, where
U is the velocity, f is the Coriolis parameter, and L is the length scale)(Thomas et al., 2008); that is, SMs are not bounded by the Earth’s
rotational effect. The vertical length and time scales of SMs range fromapproximately 10 to 1000 m and less than several days, respectively(McWilliams, 2016). SMs are accompanied by intensified vertical* Corresponding author. Global and Local Environment Co-creation Institute, Ibaraki University, 4-12-1, Nakanarusawa, Hitachi, Ibaraki, Japan.
E-mail address: eiji.masunaga.office@vc.ibaraki.ac.jp (E. Masunaga).
https://doi.org/10.1016/j.dsr.2022.103828
Received 19 July 2021; Received in revised form 7 June 2022; Accepted 17 June 2022
Available online 20 June 2022
0967-0637/© 2022 Elsevier Ltd. All rights reserved.

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motion in the surface mixed layer, which significantly contributes tovertical mass and heat fluxes (Su et al., 2018). The vertical flux due to
SMs is occasionally larger than that caused by mesoscale eddies andturbulent mixing (e.g., Mahadevan, 2016). Importantly, the upward
nutrient flux associated with SMs contributes to maintaining primary
production in the surface layers (e.g., Lévy et al., 2012; Uchiyama et al.,2017), and submesoscale eddy-driven subduction transports organiccarbon from surface layers toward the deep oceans (Omand et al., 2015).
Mesoscale motions and SMs are present in the vicinity of the Kur­oshio path (e.g., Waseda et al., 2003; Sasaki et al., 2014; Uchiyama et al.,2017). The Kuroshio, accompanied by large mesoscale cyclonic eddies,
induces upwelling, which intensifies primary production in the upper
ocean (e.g., Kimura et al., 1997). High-resolution numerical simulations
by Sasaki et al. (2014) showed enhanced SMs in the surface mixed layer
due to baroclinic instability, referred to as mixed-layer instability(Boccaletti et al., 2007), in the winter season. Uchiyama et al. (2017)
found that the SMs induce vertical nutrient supply and promote primary
production in the Kuroshio extension. Yang et al. (2021a) reported two
major processes for generating SM-induced vertical heat transport:
geostrophic deformation and vertical mixing of momentum in the Kur­oshio extension. In addition to vertical transport, Kamidaira et al. (2017)
suggested that SMs significantly contribute to lateral heat transport fromthe Kuroshio toward the coast in the vicinity of the Ryukyu Islands.
In addition to mesoscale currents and SMs, tides significantly
contribute to ocean dynamics and transport. Interaction between
topography and barotropic tides generates internal waves (e.g., Alford
et al., 2015; Masunaga et al., 2019) and SMs (e.g., Nakamura et al.,2012; Cheng et al., 2020). The typical lateral wave phase speed of in­
ternal tides is of the order of 1 m s− 1. Assuming a wave speed of 1 m s− 1
and a wave period of M2 (12.42 h), the horizontal wavelength isapproximately 45 km, which is within the range of the submesoscale.
Nonlinear internal tides exhibit small-scale physical processes withhorizontal scales of less than O (10) km (Masunaga et al., 2019b). Rocha
et al. (2016) also suggested that internal tides influence mesoscale andsubmesoscale dynamics in the upper ocean in the Kuroshio extension.
Nakamura et al. (2012) reported SMs generated by barotropic tides
around islands near the Kuril Strait, based on satellite images and nu­
merical simulations. Furthermore, numerical simulations conducted byCheng et al. (2020) showed that SMs generated by the Kuroshio at a
sharp cape, located south of Taiwan, are modulated by tidal flows. While
tides may contribute to the generation of SMs, tidally generated SMs are
not well understood.
This study examines the physical processes and transport associatedwith SMs around a shallow ridge, the Izu Ogasawara Ridge (Fig. 1),
where both the Kuroshio and tides have a significant influence. Nu­
merical simulations conducted by Masunaga et al. (2018) showed that
67%, 20%, and 13% of the total kinetic energy over the northern area of
this ridge is generated by the Kuroshio, tides, and wind stress, respec­tively. SMs are enhanced by the Kuroshio flow over the ridge owing tolateral mean shear stress (Uchiyama et al., 2017). In addition, numerous
islands are located along the Kuroshio path and von Karman vortex-likeSMs are generated in island wakes (Liu and Chang, 2018). An interaction
between bathymetry and barotropic tides generates strong internal tides
that propagate from the ridge (Varlamov et al., 2015; Masunaga et al.,2018). Diurnal internal tides are trapped around islands located onridges, leading to vertical mixing and lateral mass transport (Masunaga
et al., 2020). Masunaga et al. (2019) found that an interaction between
the background Kuroshio and tides strongly enhance internal wave
propagation toward the Kuroshio downstream.
We employed a double-nested Regional Oceanic Modeling System(ROMS; Shchepetkin and McWilliams, 2005) with a horizontal grid
resolution of 1 km to investigate SMs in the study area; this is anappropriate resolution for simulating submesoscale dynamics andassociated transport processes in the upper ocean (e.g., Capet et al.,2008; Kamidaira et al., 2018). It is well known that submesoscale dy­
namics vary seasonally (Sasaki et al., 2014; Uchiyama et al., 2017).
Tides and internal tides significantly influence the kinetic energy in deep
layers (depth >1000 m; Kang and Fringer, 2012; Masunaga et al., 2017,2020). In this study, we present a full-depth kinetic energy budget
analysis associated with SMs for two seasons and two forcing cases,
summer and winter, with and without tidal forcing. The remainder ofthis paper is organized as follows. Section 2 describes the configurationand validation of the proposed model. Section 3 presents the back­ground physical conditions, submesoscale kinetic energy (SKE) budget,and generation mechanism of SKE Finally, section 4 summarizes thestudy and presents the main conclusions.
2. Model configuration
2.1. ROMS model
We employed a double-nested ROMS covering the northern area of
the Izu-Ogasawara Ridge (Fig. 1a and b). The outer nested domain (L1)of the ROMS was embedded in the 7-year-averaged daily climatologicaloceanic data (2008–2014) obtained from the JCOPE2 reanalysis product(Miyazawa et al., 2009). The horizontal resolution was 3 km, and 32
vertical s-coordinate layers were used for the L1 domain (see Uchiyama
et al., 2017 for more details on L1 model configuration). The innerdomain (L2), with a horizontal grid spacing of 1 km, was nested in theparent L1 model using the one-way offline nesting method proposed by
Mason et al. (2010). Vertical S-coordinates with 40 layers were used forvertical gridding. The model topography was derived from the
J-EGG500 (Japan Oceanographic Data Center) and SRTM30_PLUS(Becker et al., 2009) products. The open boundaries were forced byhorizontal currents, tracers (temperature and salinity), and sea surface
height from the parent L1 model output after a 1-year spin-up. Baro­
tropic (BT) tidal forcing was also imposed at the L2 open boundaries
using the TPXO 7.0 tidal constituents (Egbert and Erofeeva, 2002). The

Fig. 1. Maps of the model domains of the (a) parent L1 and (b) downscaled L2. The red dots show the locations of tidal gauge stations used for model validation. The
black chain-dotted box in panel (b) is the analysis area for this study.
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amplitude and phase of the TPXO tide model were slightly modified to
improve the reproducibility of tidal signals (see Masunaga et al., 2018).
The ocean surface boundary condition was imposed using the daily
climatological grid point value of the global spectral model (GPV-GSM)
product (Japan Meteorological Agency, JMA; e.g., Roads, 2004) for
wind, and monthly climatological surface heat, freshwater fluxes, andradiation fluxes from the NOAA-COADS dataset (Woodruff et al., 1987).
Freshwater discharge from 31 rivers was also imposed using monthly
averaged climatological data provided by the Japan River Association(http://www.japanriver.or.jp/publish/book/nenpyou_dvd.htm). Verti­
cal eddy viscosity and diffusivity were estimated using the K-Profile
Parameterization (KPP) mixed-layer turbulence model (Large et al.,1994). The horizontal eddy viscosity and diffusivity were set to 0 m2 s− 1,
and the baroclinic (BC) time step was set to 10 s.
To investigate the effects of background geostrophic flows (i.e., theKuroshio), tides, and seasonal variations, four model runs were con­ducted: summer without tides (Run #1), summer with tides (Run #2),
winter without tides (Run #3), and winter with tides (Run #4). The
model cases with and without tides allowed us to discriminate the effectsof tides/internal tides and evaluate the effects of tidal forcing. The L2model was spun-up approximately 1 and 1.5 years without tidal forcing
for the winter and summer runs, respectively. The analysis period was
set to 28 days (i.e., approximately two spring–neap tidal cycles) in allcases, and the days with the warmest and coldest water temperatures
were set in the middle of the summer and winter periods, respectively.
Tidal forcing was imposed 14 days prior to the analysis period, and thehourly averaged model outputs were used in subsequent analyses.

cases in Fig. 2. The warm Kuroshio water flowing into the westernboundary and out of the eastern boundary is well downscaled to the L2
domain from the L1 domain. Large temperature and velocity gaps are
not observed between the L2 and L1 models. Therefore, we assume thatthe double-nested L2 model is successfully nested in the parent L1
model. Validations of the outer L1 domain have been reported by
Uchiyama et al. (2017).
The amplitude and phase of the four tidal constituents (S2, M2, K1,
and O1) between our model and those provided by the JMA for 43 tidal
gauge stations are compared in Fig. 3 (the locations of the tidal gauge

2.2. Validation of the ROMS model
It has been reported that double-nested ROMS-L1 and L2 model re­sults show good agreement with field observations around the Kuroshio
stream (Kamidaira et al., 2017; Uchiyama et al., 2018). Snapshots of thesurface temperature and velocity magnitude from the L2 model areplotted on those from the parent L1 model for both summer and winterFig. 3. Model validation for the four principle tidal constituents. S2 (blue), M2(red), K1 (green), and O1 (gray); (a, b) amplitude and (c, d) phase for (a, c)summer and (b, d) winter.

Fig. 2. Snapshots of L2 model results on those from the L1 parent model at the midpoint of the analysis periods. (a, b) Surface temperature; and (c, d) surface velocity
magnitude for (a, c) summer and (b, d) winter.
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stations are shown as red dots in Fig. 1b). The root-mean-square error(E) between the modeled and observed tidal constituents is given by thefollowing formula (Carter et al., 2008):√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1 2
E=(a + a2M ) − aO aM cos(GO − GM )(1)
2 O

SMs are enhanced and suppressed at 32.3◦ E in summer and winter,
respectively. This seasonal variation in effects of tidal forcing is
consistent with previously reported SMs in the Kuroshio Extension(Rocha et al., 2016). The kinetic energy spectra follow a slope of − 2 in
the fully developed submesoscale turbulent regime (Capet et al., 2008;
Kamidaira et al., 2018). The kinetic energy spectra are close to a slope of− 2 in winter and close to a slope of − 3 in summer (Fig. 5c) Therefore,
fully developed SMs are established for winter cases.

where a is the amplitude of the tides, G is the tidal phase, and the sub­scripts O and M denote the values of the observational constant from the
JMA and our model, respectively. The resulting error is less than 5.4 ×
10− 2 m, and the averaged error over the four tidal constituents iscomputed as 2.9 × 10− 2 and 3.1 × 10− 2 m for summer and winter,
respectively (Table 1). The average relative errors (E/aO) for the summer
and winter cases are 11% and 12%, respectively.

3.2. Extraction of SMs
We used a horizontal two-dimensional Gaussian smoothing filter todecompose the submesoscale and mean (or mesoscale) velocities. The
filter length scale (standard deviation) and the size of the filter window
were set to 9 km and 30 × 30 km, respectively. We refer to the sub­mesoscale and mean velocity components as uS and uM , respectively
̃, where ̃(uS = u − uM , uM = u
u indicates the filtered spatial mean ve­
locity with the spatial Gaussian filter). The horizontal kinetic energy
spectra computed from uE and uM for Run #3 are compared in Fig. 5d.
The two spectra from uS and uM cross at 2.2 × 10− 5 cpm (with a hori­zontal scale of approximately 45 km). Thus, uS and uM dominate thekinetic energy at scales smaller and larger than approximately 45 km,
respectively. Although the cutoff wave number/scale cannot be clearly
defined when we use the Gaussian filter, the cutoff length scale seems tobe approximately 45 km for our case. Snapshots of the relative surface
vorticity from the mean (ζM /f) and submesoscale (ζS /f) velocities areplotted in Fig. 6. The vorticity from submesoscale velocities clearlyshows enhanced SMs along the coast of mainland Japan and in the
Kuroshio downstream during both seasons, and in the southern area ofthe model domain in winter (Fig. 6c and d). During the summer season,
the submesoscale vorticity shows thin filament-like structures nearcoastal regions, which are supposed to be partly caused by topographic
effects. Although the mean vorticity shows numerous vortices/filamentswith scales larger than ~50 km in winter, the intensity of the meanvorticity components is similar for both seasons (Fig. 6a and b).
Table 2 shows the mean square (MS) of the relative vorticity,

3. Results and discussion
3.1. Background physical conditions
The time-averaged surface temperatures and velocities from themodel cases without tidal forcing (Runs #1 and #3) are plotted in Fig. 4.
The surface temperature along the Kuroshio axis is approximately 10 ◦ Chigher in summer than in winter; however, the horizontal temperature
gradient is consistent in both cases. The Kuroshio passes over the IzuOgasawara Ridge with a maximum velocity magnitude exceeding 2 m
s− 1 and is separated into several branches due to the presence of islands(Fig. 4c and d, branches are clearer in the summer case). The inflow ratesof the Kuroshio at the western boundary are approximately 54 and 55 Sv(1 Sv = 106 m3 s− 1) for the summer and winter cases, respectively. The
Kuroshio path is broadly consistent between the summer and winterseasons in the upstream region (136◦ E− 139◦ E). However, the Kuroshio
tends to pass over the northern area of the ridge during summer. The
surface velocity along the Kuroshio is slightly lower in winter than insummer because the surface mixed layer deepens in winter, resulting indeceleration of the flow magnitude in the surface layer. A cyclonic eddy
exists along the coast upstream of the Kuroshio (34◦ N, 137◦ E− 139◦ E) inboth summer and winter, which has been frequently observed in pre­vious studies (e.g., Nakata et al., 2000; Uchiyama et al., 2017). Two
additional model cases with tidal forcing (Runs #2 and #4) also exhibit
background conditions that are similar to the results shown in Fig. 4(Runs #1 and #3), indicating that tidal forcing does not largely influ­
ence the background conditions.
The surface relative vorticity, ζ/f, is plotted in Fig. 5a and b, where ζ= ∂v/∂x − ∂u/∂y and f is the Coriolis frequency. In particular, in winter,small-scale eddies with ζ/f > 1 occur across the entire model domain.
Topography–current-generated vortices and filaments also appear near
the coast and in island wakes on the Kuroshio downstream during bothseasons. Horizontal kinetic energy spectra were computed for the fourmodel cases (Runs #1–4) from the horizontal velocity components (uand v) obtained along the 32.3◦ E latitude line (Fig. 5c), shown as black
chain-dotted lines in Fig. 5a and b. The power of the spectra where thewavenumber is higher than 10− 5 cpm and is approximately one order ofmagnitude higher in winter than in summer. Tidal forcing slightly en­
hances and suppresses the power at higher wavenumbers in summer andwinter in the presence of tidal forcing, respectively. This indicates that

< (ζM /f)2 > and < (ζS /f)2 >, computed from uM and uS, respectively,
for all four model cases (Runs #1–4), where brackets and overbars
denote horizontal and time averaging operators, respectively. MS wascomputed for the entire 28-day analysis period. According to the t-test,
differences in the MS of vorticity are significantly different for all modelcases (p < 0.01), where the t-test compare the time series of the hori­zontally averaged MS of relative vorticity. The MS of the mean vorticity
does not change largely between the model cases, with differences of lessthan 6%. In comparison, submesoscale vorticity varies owing to seasonal
differences and tidal forcing; submesoscale vorticity is found to beapproximately twice as high in winter as it is in summer, which is also
clearly seen in the vorticity maps (Fig. 6c and d). Furthermore, tidal
forcing enhances submesoscale vorticity by 17% during the summer.
Thus, the Gaussian filter successfully extracts SMs that are influenced byseasonal variation and tides.
3.3. Submesoscale kinetic energy
The total, mean, and submesoscale kinetic energy (KE, MKE, andSKE) terms are given by

Table 1Root-mean-square error (E) between modeled tidal elevations and tidal con­stants provided by the Japan Meteorological Agency.
Summer

Winter

E [m]
M2
S2
K1
O1
AverageE [m]− 2

5.2 × 10
2.6 × 10− 2
1.8 × 10− 2
2.1 × 10− 2
2.9 × 10− 2− 2

5.4 × 10
4.2 × 10− 2
1.4 × 10− 2
1.6 × 10− 2
3.1 × 10− 2

KE : KTotal =)1( 2u + v 2 + w2
2

(2)

MKE : KM =)1( 2u + v2M + w2M
2 M

(3)

)
1( 2
u + v2S + w2S
2 S

(4)

SKE : KS =Note that the sum of KM and KS is not equal to KTotal because of the4

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Fig. 4. Time-averaged (a, b) surface temperature and (c, d) surface velocity over the 4-week (28-day) analysis period for (a, c) summer and (b, d) winter. Timeaverages were obtained for the entire 28-day analysis period. The gray and black contour lines are isobaths at intervals of 500 m and sea-surface height at intervals of
0.1 m, respectively.

Fig. 5. (a, b) Snapshots of surface relative vorticity for (a) summer and (b) winter, and (c, d) kinetic energy spectra along the black chain-dotted lines in panels (a, b).
Panel (d) compares the kinetic energy spectra of mean and submesoscale components extracted using the Gaussian filter for the winter case (Run #3). The kinetic
energy spectra are averaged over the 4-week (28-day) analysis period.

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Fig. 6. Snapshots of surface relative vorticity computed from (a,b) mean and (c,d) submesoscale components for the (a,c) summer (Run #1) and (b,d) winter cases(Run #3). The times of these snapshots are the same as those in Fig. 5a and b.

tidal forcing by 15% and 13% in summer and winter, respectively(Figs. 7f and 8f, Table 3). These kinetic energies are found to have sta­tistically significant differences based on the t-test (p < 0.01), which
compares the time series of the area-averaged and depth-integrated ki­
netic energies. These results demonstrate that tidal forcing deaccelerates
the Kuroshio and promotes the generation of SKE and dissipation of KE.
A significant enhancement of SKE appears in the southwest area of themodel domain during the summer season (Fig. 7f; 136◦ E− 13◦ E and<33◦ N). It has been reported that internal wave energy increases insummer in the study area owing to highly stratified conditions (Rocha
et al., 2016; Masunaga et al., 2020). Enhanced SKE due to tides far fromthe coast and ridge in our model results might be influenced by internal
tides (see Section 3.4).
To further investigate regional-scale variations in kinetic energy, thearea- and time-averaged and vertically integrated KE, MKE, and SKE
values were computed for the following six regions (Fig. 9, Table 3): (1)coastal upstream (CU), (2) coastal downstream (CD), (3) Kuroshio up­
stream (KU), (4) Kuroshio downstream (KD), (5) offshore upstream(OU), and (6) offshore downstream (OD). The “coastal”, “Kuroshio”, and“offshore” regions are defined by time-averaged sea-surface heights ofless than − 0.1 m, − 0.1–0.6 m, and higher than 0.6 m, respectively, forthe summer cases. For the winter cases, these three regions are separated
by time-averaged sea-surface heights of − 0.3 and 0.3 m. The “upstream”
and “downstream” regions are separated by a straight line passing theislands positioned on the Izu-Ogasawara Ridge (the line is given by Lat= − 4.4 × Long + 648.12, where Lat and Long are latitude and longitude,
respectively, see Fig. 9).
The KU region is the most energetic MKE region, and MKE is reducedby 29% and 40% between the KU and KD regions in summer and winter,
respectively (Table 3). Meanwhile, SKE is 2.2 (1.6) times higher in the
KD region than in the KU region for summer (winter). Assuming that
MKE is mainly transported in the upstream-downstream direction, aninteraction between the Kuroshio and ridge results in the dissipation of
MKE and energy transfer from MKE to SKE. The differences in kinetic

Table 2
Mean square (MS) of relative vorticity in the surface layer for the summer andwinter model cases.
Summer< (ζM /f)2 >2< (ζS /f) >Winterw/o tides
Run #1

with tides
Run #2w/o tides
Run #3

with tides
Run #4

8.6 × 10− 2

8.1 × 10− 2

8.4 × 10− 2

8.2 × 10− 2

2.3 × 10− 1

2.7 × 10− 1

4.7 × 10− 1

5.0 × 10− 1cross-term KC (KTotal = KM + KS + KC, where KC = uM uS + vM vS + wM wS ),which is not considered in this study. The time-averaged and depthintegrated KE, MKE, and SKE are shown in Fig. 7a–c and Fig. 8a–c forthe summer and winter cases, respectively. In addition, Fig. 7d–f andFig. 8d–f shows the ratio of kinetic energy between the model cases withand without tidal forcing (i.e., kinetic energy from a model with tidal
forcing divided by that without tidal forcing). High KE regions appear
along the Kuroshio stream, with the KE reaching O(103) m3 s− 2 (Figs. 7aand 8a). The MKE shows a distribution similar to that of KE (Fig. 7a andb and Fig. 8a and b), which implies that MKE dominates the total kinetic
energy. The SKE is much lower than the MKE (Fig. 7b and c and Fig. 8b
and c); the domain-time-averaged and depth-integrated SKE is approx­
imately 8% of the MKE (Table 3). On the other hand, SKE is enhanced
over the ridge and is comparable to MKE downstream of the Kuroshio(Figs. 7c and 8c; ~140◦ E− 141◦ E and 34◦ N). In the southern area of themodel domain, SKE is higher in winter than in summer (Fig. 7b and c
and Fig. 8b and c), which probably reflects the enhanced SMs during thewinter season, as shown in Fig. 6.
Tides reduce the MKE in the Kuroshio stream and enhance the MKEalong the coast and in the vicinity of the islands over the ridge (Figs. 7eand 8e). The area- and time-averaged vertically integrated MKE is sup­pressed by tidal forcing by 8% and 6% in the summer and winter,
respectively (Table 3). The total kinetic energy (KE) is also decreased bytides, similar to that of the MKE. In contrast, the SKE is intensified by6

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Fig. 7. Time-averaged and depth-integrated (a) total kinetic energy, (b) mean kinetic energy, and (c) submesoscale kinetic energy for the summer model case (Run#1); (d–f) lower panels show the ratio of kinetic energy between model cases with and without tidal forcing (kinetic energy from Run #2 is divided by that from Run#1). Gray and black contour lines are isobaths spaced at intervals of 500 m and iso-sea-surface height at intervals of 0.1 m, respectively.

Fig. 8. As for Fig. 7 but for the winter season.

energy between the model cases with and without tidal forcing werevalidated using the t-test. Tidal forcing significantly modulates kinetic
energy, except for KE and MKE in the CD region in winter and SKE in the
KD and OD regions in winter (Table 3). KE and MKE in the CU (CD)region are enhanced (reduced) by tides during the summer (Table 3).
Although both the CU and CD regions are coastal, the effect of tides on
kinetic energy varies. The higher KE (or MKE) in the CU region than in
the CD region reflects a cyclonic eddy generated along the north side of

the Kuroshio (Figs. 4, Figure 7, and Fig. 8) and tidally enhanced KE in
the CU region (Table 3), indicating that the CU region is an energetic
coastal region largely influenced by both Kuroshio-induced cycloniceddies and tides. Tidal forcing enhances the SKE, except in the OD regionin summer, and the KD and OD regions in winter (Table 3). In particular,
SKE is considerably enhanced in the KU by 25% and 37% in summer andwinter, respectively.
To further investigate the modulations of kinetic energy due to tidal
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Table 3Area- and time-averaged and depth-integrated kinetic energy for the model runs with tidal forcing (unit of m3 s− 2; Runs #2 and #4). Values in brackets are differences
between model cases with and without tidal forcing. Positive and negative values in brackets indicate kinetic energy is increased or decreased by tidal forcing,respectively. Asterisks (*) denote no significant difference between results from models with and without tidal forcing (t-test, p > 0.01).
SummerAll areas
CU
CD
KU
KD
OU
OD

Winter

KE

MKE

SKE

KE

MKE

SKE

70.3 (− 4.5)
47.6 (+7.8)
25.3 (− 0.6)
230.9 (− 36.6)
181.7 (− 17.9)
30.3 (− 3.2)
46.5 (− 5.5)

60.1 (− 4.9)
36.8 (+5.9)
18.0 (− 1.5)
212.3 (− 36.7)
150.3 (− 17.0)
26.1 (− 3.6)
39.1 (− 4.7)

4.7 (+0.5)
5.2 (+0.8)
3.9 (+0.6)
7.0 (+1.4)
15.2 (+0.6)
2.3 (+0.4)
3.2 (− 0.2)

75.2 (− 3.4)
38.9 (+3.0)
32.0 (− 0.4) *239.7 (− 52.6)
156.7 (− 26.4)
43.7 (− 2.6)
52.4 (− 2.9)64.0 (− 3.9)
30.4 (+2.1)
24.0 (− 0.3) *219.2 (− 52.1)
131.6 (− 25.3)
37.2 (− 3.0)
42.3 (− 2.7)

4.4 (+0.5)
4.6 (+0.5)
4.2 (+0.5)
7.4 (+2.0)
11.8 (− 0.2) *3.7 (+0.3)
4.5 (0.0) *Fig. 9. Six separate regions used for kinetic energy budget analysis in (a) summer (Run #1) and (b) winter (Run #3). Abbreviations: CU, coastal upstream; CD,coastal downstream; KU, Kuroshio upstream; KD, Kuroshio downstream; OU, offshore upstream; OD, offshore downstream.

Fig. 10. Probability density functions (PDFs) of (a, b, e, f) mean kinetic energy (MKE) and (c, d, g, h) submesoscale kinetic energy (SKE) in (a–d) for all analyzed
areas and along (e–h) the Kuroshio path (upstream, KU, and downstream, KD, regions) for the (left panels) summer and (right panels) winter model cases. Vertical
black solid and chain-dotted lines show means for the model cases with and without (w/o) tidal forcing, respectively.

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forcing, we compared the probability density functions (PDFs) of thetime-averaged and vertically integrated MKE and SKE (Fig. 10). The
PDFs of SKE shift to the higher energy side owing to tidal forcing; inparticular, the densities at the lower energy tails of SKE are reduced in
the cases with tidal forcing (Fig. 10cd). Although modulations of MKEs
are not clear from the data in the entire analysis domain, the PDFs of
MKEs are clearly shifted to the lower energy side by tidal forcing in the
KU and KD regions (Fig. 10ef). According to the Kolmogorov–Smirnov
test, the PDFs of the SKE and MKE for the six separate regions are
significantly different between model cases with and without tidal
forcing (p < 0.01). These results imply that tidal forcing significantly
modulates the distribution of the SKE and MKE.

viscous effects) including the wind stress (surface drag), interior dissi­
pation and bottom drag. We applied this general TKE equation to our
spatially (Gaussian) filtered model output and estimated the SKE
budget. The shear production (KMKS) and buoyancy production (PSKS)
terms are expressed as follows:

3.4. SKE budget

FSKE = u⋅( ′ 2)∂u1gρ′ w′′∇(u′ p′ ) − uh ⋅(u′ ⋅∇)uh −+ε= − ∇ u⋅ h −
2
ρ0
ρ0
∂t 2

(6)

gρS wS(7)

PS K S = −by:

The turbulent kinetic energy (TKE) budget equation in a time domain
is generally represented as follows (e.g., Gula et al., 2016b; Wang et al.,2018),u′ 2h

KM KS = − uS h ⋅(uS ⋅∇)uMh

ρ0

The SKE flux (FSKE) and submesoscale pressure flux (FSP) are given

FSP =

u2Sh
2

(8)

uSh pS(9)ρ0Under a steady-state condition (i.e., ∂uS 2h /2∂t ∼ 0), the dissipation
term can be written as follows:(5)

ε = ∇⋅FSKE + ∇⋅FSP − KM KS − PS KS

where p is pressure, ρ is density, ρ0 is the reference density of 1027.5 kg
m− 3, g is the gravitational acceleration, and h denotes the horizontal
operator. The overbar and prime indicate time-averaged and fluctuation
components in a time domain. The first and second terms on the righthand side are the advection of SKE by total flows and submesoscalepressure flux terms, respectively. The third and fourth terms on theright-hand side represent shear production and buoyancy production,
respectively; that is, energy conversion from MKE to SKE (KMKS) andfrom submesoscale potential energy to SKE (PSKS), respectively. These
two terms, KMKS and PSKS, are often referred to as barotropic (BT)conversion and baroclinic (BC) conversion, respectively (e.g., Donget al., 2007). The last term is the energy sink/dissipation term (viz.,(10)

where a negative ε indicates net dissipation in the model. Strictly

speaking, ∂uS 2h /2∂t is not zero during the analysis periods (∂uS 2h /2∂t =− 2.2 × 10− 7, − 4.8 × 10− 7, 2.1 × 10− 7, and 1.5 × 10− 7 m3s− 3 for Runs
#1–4, respectively. However, these changes in SKE are approximately
one to two orders of magnitude smaller than the estimated dissipation.

Therefore, we assume that ∂uS 2h /2∂t = 0 to estimate the dissipation. The
cross term associated with KC is ignored in the SKE budget analysis.
High KMKS appears in the vicinity of the islands along the Kuroshio
path, and the distribution of the KMKS is similar for the summer andwinter seasons (Fig. 11a and b). At the same time, PSKS shows negative
conversion near the islands, although PSKS is enhanced in offshore

Fig. 11. Time-averaged and depth-integrated (a, b) shear production (KMKS) and (c, d) buoyancy production (PSKS) for the (a, c) summer and (b, d) winter modelcases with tidal forcing. For visualization purposes, the plotted data were smoothed using a horizontal Gaussian filter.
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Deep-Sea Research Part I 186 (2022) 103828

regions in winter (Fig. 10c and d). This enhanced PSKS in the winterseason is thought to be due to instability resulting from surface wintercooling (Sasaki et al., 2014; Uchiyama et al., 2017). Recent studies of
Yang et al. (2021a; 2021b) investigated details of vertical eddy heattransport in the Kuroshio extension in winter. They suggested that thebuoyancy flux (~PSKS) is promoted by strong turbulent mixing owing tostrong surface cooling with intensified winds through a strong ageo­
strophic secondary circulation. The strongly enhanced KMKS and nega­tive PSKS around the islands are consistent with the numericalsimulation conducted by Dong et al. (2007). The distributions of the
KMKS and PSKS are not largely altered by tidal forcing (data not shown).
The time-averaged and depth-integrated FSKE and FSP values for thesummer are plotted in Fig. 12. The SKE is mainly advected by theeastward Kuroshio and is enhanced near islands over the ridge from O(10) to O (102) m4 s− 3 (Fig. 12a and b). The divergence of the FSKE near
the islands is consistent with that of the KMKS, as shown in Fig. 11a. The
direction of the submesoscale pressure flux is mainly westward andopposite to that of the Kuroshio and FSP (Fig. 12c and d). The distribu­
tions of FSKE and FSP are not largely different between the summer andwinter (data not shown). The submesoscale pressure flux should not becaused by barotropic motions, but is thought to be caused by internal
gravity waves. Masunaga et al. (2018) reported that internal waves
contribute 10% of the total kinetic energy in the study area. The nu­
merical simulations conducted by Rocha et al. (2016) showed that in­
ternal gravity waves influence SMs. High FSP areas appear near the
Kuroshio path for both model cases, with and without tides (Fig. 12c andd). One possible explanation for the high FSP without tides isnear-inertial internal waves (NIIWs) generated by the interaction be­
tween background flows and topography (e.g., Waterman et al., 2013;
Nagai et al., 2017). Furthermore, frontal instabilities in geostrophic
flows can transform energy into NIIW energy via spontaneous genera­tion (e.g., Nagai et al., 2015; Alford et al., 2016). On the other hand,tidal forcing significantly enhances the FSP on the southwest side of theridge, which is thought to be due to the propagation of internal tides(compare Fig. 12c and d; between 138◦ E and 140◦ E). Therefore, theenhanced SKE on the southwest side of the ridge, shown in Fig. 7f, is
probably caused by internal tides. The intensity of the internal tide en­ergy flux ranges from approximately 1 to 5 m4 s− 3 (Masunaga et al.,2018, 2020), which is similar to the submesoscale pressure flux esti­
mated in our model (Fig. 12c, d, O (1–10) m4 s− 3).
The area- and time-averaged and depth-integrated SKE budgets areshown in Table 4 (with area averaging over the entire analysis domainshown by the black chain-dotted box in Fig. 1b). KMKS and PSKS are
positive and negative in summer, respectively. Approximately half of the
SKE generated by KMKS is converted back to submesoscale potential
energy via negative PSKS during summer. The negative buoyancy pro­duction can be explained by a highly stratified condition in summer,
leading to the suppression of the SKE. In the winter season, KMKS isapproximately 20% smaller than that in summer, and PSKS is positiveand comparable with that of KMKS.
According to these results, the generation mechanisms of SKE are
different in summer and winter, as follows: (1) for the summer season,shear production dominates SKE generation, and buoyancy production
suppresses SKE through negative conversion; (2) for the winter season,
both shear and buoyancy production equally dominate in SKE
Table 4
Submesoscale kinetic energy budgets. Area- and time-averaged and verticallyintegrated ∇⋅FSKE , ∇⋅FSP , KMKS, PSKS, and ε for all model cases (unit = 10− 7
m3 s− 3). Asterisks denote differences between models with and without tidal
forcing that are not significant (t-test, p > 0.01).
Summer

Winterw/o tides (Run#1)
with tides (Run#2)
w/o tides (Run#3)
with tides (Run#4)∇⋅FSKE

∇⋅FSP

KMKS

P SK S

ε

− 1.6

− 0.6

105.4

− 47.8− 59.9

0.4

6.3

112.3

− 58.3− 47.4− 3.6− 0.04 *

78.4

97.1

− 179.2

− 0.9

0.8 *

89.0

79.4

− 168.5

Fig. 12. Time-averaged and depth-integrated (a, b) submesoscale kinetic energy (SKE) flux (FSKE) and (c, d) submesoscale pressure flux (FSP) for the summer modelcases (a, c) with and (b, d) without tidal forcing. For visualization purposes, the plotted data were smoothed using a horizontal Gaussian filter.
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Deep-Sea Research Part I 186 (2022) 103828

generation. KMKS and PSKS increases and decreases with tidal forcing by
7% and 14%, respectively, in summer, and by 22% and 18%, respec­tively, in winter (these differences are statistically significant (t-test, p <0.01). The enhancement of KMKS owing to tides can be explained bylateral shear flows through the interaction between tidal flows andtopography. One possible reason for the reduction in PSKS owing to tides
is tidal mixing. In addition, the buoyancy production might be reduced
by energy toward submesoscale internal tides. Indeed, tide-induced andinternal tide-induced mixing significantly influence physical processes
and transport in the study area (Masunaga et al., 2020). In our modelcases, the vertical eddy viscosity computed from the KPP scheme in­creases by 19% and 8% owing to tidal forcing in summer and winter,
respectively. Therefore, tides contribute to the SKE budget in two ways:
first by SKE generation through shear production, KMKS, and second by
the suppression of buoyancy production, PSKS, caused by tide-induced
mixing.
The divergence of FSKE and FSP is much smaller than that of the otherterms, which implies that most of the generated SKE is dissipated in the
study area; that is, the SKE budget is mostly closed or balanced. This also
implies that intensified mixing due to enhanced SKE mainly influences
local transport in the study area. The divergence of FSP in the summercase with tides (Run #2) is significantly higher than that in other cases(Table 4). For the winter season, there is no significant difference in the
FSP between model cases with and without tidal forcing (Table 4). The
enhanced submesoscale pressure flux of the FSP due to tides in summer
should be the baroclinic pressure flux (internal tide energy flux) because
the wavelength of barotropic tidal waves is O (103) km and cannot be
exhibited as a small-scale submesoscale component. Therefore, theintensified divergence of FSP caused by tides represents internal tide
generation and propagation, which enhances the radiation of SKE farfrom internal tide-generation sites.
The dissipation and sink of the SKE varies seasonally. During thesummer season, SKE sinks via two pathways: approximately half of the
SKE from shear production sinks to negative potential energy production(PSKS) and the other half sinks as dissipation. During the winter season,
most of the SKE generated owing to shear and buoyancy production
sinks into dissipation. Based on our modeling, winter dissipation isapproximately three times higher than that in the summer and is
weakened by tidal forcing owing to negative buoyancy production(PSKS) by tides.

further considered to fully understand physical and ecosystem processes
and dynamics in the North Pacific. This study reveals the effects of tides
on the kinetic energy budget and SMs; however, internal tides were notseparated by our filtering approach using a horizontal Gaussian filter. To
further investigate the effects of tides and internal tides on SMs, moreadvanced filtering techniques for both the temporal and spatial domains
are needed (e.g., Torres et al., 2018).
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.
Data availability
Data will be made available on request.
Acknowledgements
This study was supported by a JSPS Grant-in-Aid for Scientific
Research (18H03798, 19K20432 and 22K18018). E. Masunaga also
acknowledges a grant from the Fundamental Research Developing As­
sociation for Shipbuilding and Offshore (REDAS).
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