NOVEMBER 2019

MASUNAGA ET AL.

2917

Strong Internal Waves Generated by the Interaction of theKuroshio and Tides over a Shallow Ridge
EIJI MASUNAGA
Center for Water Environment Studies, Ibaraki University, Mito, JapanYUSUKE UCHIYAMA
Department of Civil Engineering, Kobe University, Kobe, JapanHIDEKATSU YAMAZAKI
Department of Ocean Sciences, Tokyo University of Marine Science andTechnology, Tokyo, Japan(Manuscript received 20 November 2018, in final form 22 August 2019)
ABSTRACTThe Kuroshio and tides significantly influence the oceanic environment off the Japanese mainland andpromote mass/heat transport. However, the interaction between the Kuroshio and tides/internal waves has
not been examined in previous works. To investigate this phenomenon, the two-dimensional high-resolution
nonhydrostatic oceanic Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes
Simulator (SUNTANS) model was employed. The results show that strong internal tides propagating upstream in the Kuroshio are generated at a near-critical internal Froude number (Fri 5 0.91). The upstream
internal wave energy flux reaches a magnitude of 12 kW m21, which is approximately 3 times higher than thatof internal waves without the Kuroshio. On the other hand, under supercritical conditions, the Kuroshio
suppresses the internal wave energy flux. The interaction of internal tides and the Kuroshio also generates
upstream propagating high-frequency internal waves and solitary wave packets. The high-frequency internal
waves contribute to the increase in the total internal wave energy flux up to 40% at the near-critical Fri value.
The results of this study suggest that the interaction of internal tides and the Kuroshio enhances the upstream
propagating internal tides under the specified conditions (Fri ; 1), which may lead to deep ocean mixing andtransport at significant distances from the internal wave generation sites.

1. Introduction
Internal waves play major roles in the global ocean
circulation and Earth’s climate system (Munk andWunsch 1998). Internal waves in the ocean are generated mainly by barotropic tides and winds. Tidal forcing
is a major contributor to internal waves, and internal
waves with tidal frequencies are called internal tides.
The interaction of the tidal flow and shallow ridges/sills
is one of the major factors contributing to internal
tide generation (Alford et al. 2015; Da Silva et al.
2015; Masunaga et al. 2018). The energy of internal
waves increases when the topographic slope angle
matches the beam angle of internal waves known as theCorresponding author: Eiji Masunaga, eiji.masunaga.office@vc.ibaraki.ac.jp

critical slope (Cacchione et al. 2002). Internal waves
transport energy from generation sites and finally break,inducing turbulence and mixing. Shoreward propagatinginternal waves increase their nonlinearity and generatestrong turbulent mixing on shallow slopes (Richards
et al. 2013; Masunaga et al. 2015), occasionally causing
sediment resuspension and mass transport (Masunaga
et al. 2017a).
A shallow ridge located in Luzon Strait in the South
China Sea is well known as one of the world’s strongest
internal tide generation sites (Alford et al. 2015). The
intensified tidal flow in the shallow strait increases wave
steepness, resulting in the release of internal solitary
packets from the internal wave generation site (Zhang
et al. 2011; Guo and Chen 2014). Such internal solitary
waves are formed when the internal Froude number(Fri) exceeds a critical value of one (Zhang et al. 2011),

DOI: 10.1175/JPO-D-18-0238.1
Ó 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright
Policy (www.ametsoc.org/PUBSReuseLicenses).

2918

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

FIG. 1. (a),(b) Maps of the study area with bathymetry, (c) time-averaged surface velocity, and (d) time-averaged
internal wave energy flux in August 2013 provided by Masunaga et al. (2018). The solid (gray) black contoursin (b) and (c) denote the sea surface heights (isobaths) plotted at intervals of 0.1 m (500 m). The red dash–dot linein (b) indicates the transect line used for the bathymetry studies.

where the internal Froude number is defined as the ratio
between the current speed u and the internal wave speed
ci, Fri 5 u/ci.
In addition to Luzon Strait, recent studies havefound that energetic internal waves are generated at the
Izu–Ogasawara Ridge stretching over 600 km off theJapanese mainland (Fig. 1a). Barotropic tides generatestrong tidal flows across the ridge. The interaction between barotropic tidal currents and the ridge generates
internal waves (Masunaga et al. 2017b, 2018). Masunaga
et al. (2018) conducted numerical simulations considering all possible processes (three-dimensional flows,tides, winds, and heat, radiation, and freshwater water
fluxes). They found that 92% of the total internal wave
flux was caused by internal tides, which contributed to
10% of the total kinetic energy in the study area.
Therefore, internal tides play a significant role in physical processes that occur in the Izu–Ogasawara Ridge
region. The Kuroshio strongly influences various physical processes over the Izu–Ogasawara Ridge as well astides and internal tides. It crosses the ridge from west toeast, and its flow speed reaches a magnitude of 2.0 m s21
along the main axis. The interaction of the Kuroshio andshallow ridges promotes turbulent mixing (Hasegawa
et al. 2008; Nikurashin and Ferrari 2010; Nagai et al. 2017)
Physical processes associated with the Kuroshio andinternal tides have been discussed in much detail but
separately. The interaction between these two processes
has not been systematically investigated yet. The recent
high-resolution numerical simulations with a horizontal
resolution of 1 km conducted by Masunaga et al. (2018)
showed for the first time that the interaction of theKuroshio with internal tides likely induced an upstream
strong internal tide energy flux (westward) from the Izu–
Ogasawara Ridge. Figure 1d shows the intensified upstream internal tide energy flux along the Kuroshio axis
off the Izu–Ogasawara Ridge. The maximum value ofthe internal tide energy flux reaches 8.4 kW m21 at the
point of intersection between the Kuroshio and theridge. The strong internal tide energy flux reaches anorder of 10 kW m21 in other ridges, such as Hawaiian
Ridge (Rudnick et al. 2003) and ridges in Luzon Strait(Alford et al. 2015). Thus, the Izu–Ogasawara Ridge is
one of the most energetic internal wave generation sites
in the ocean. Masunaga et al. (2018) took into accountall possible forcings (tides, the Kuroshio, winds, and heat,

NOVEMBER 2019

MASUNAGA ET AL.

2919

FIG. 2. (a) A schematic image of the model configuration with bathymetry and (b) time seriesof boundary forcing for the background flow (thick gray line) and barotropic tides (thin blackline). Notations P1–5 in (a) show the data locations for Figs. 9 and 10. The light gray shaded
area in (a) represents the region used for energy budget analysis.

radiation, and freshwater fluxes); however, it was difficult to extract the details of the interaction between the
Kuroshio and tides due to the complexity of modelingnumerous physical processes in realistic oceanic simulations. In contrast, Lamb and Dunphy (2018) conducted idealized two-dimensional (2D) (x–z) numericalsimulations of the internal waves generated over a sillwith background shear flows and found that their interactions could enhance an asymmetrical internal wave
structure on the upstream and downstream sides of thesill with enhanced upstream internal wave energy flux
toward the upstream side. Although the authors foundthe existence of such interactions, their simplified model
consisted of a small Gaussian-shape sill with a subcriticalinternal Froude number.
In this study, we conducted numerical simulations
over realistic sill topography using two-dimensional
nonlinear Stanford Unstructured Nonhydrostatic
Terrain-Following Adaptive Navier–Stokes Simulator(SUNTANS) model (Fringer et al. 2006). We present
the model configuration in section 2. Section 3 analyzes
the physical processes and energetics related to the interaction of the Kuroshio with internal tides. Finally,
section 4 states the results of our study and summarizes
the major conclusions.

2. Model configuration
a. Model setup
Two-dimensional modeling has been frequently used
to investigate internal wave dynamics (Bourgault et al.
2014; Arthur and Fringer 2016; Masunaga et al. 2017a)

because this approach significantly reduces computational efforts and time. On the other hand, 2D models
cannot reproduce many physical processes related tolarge-scale geostrophic motions. However, it allows us
to significantly increase the grid resolution and thus
more accurately characterize the broadband nature ofinternal waves as compared to the coarser resolutionof three-dimensional (3D) models. To investigate relatively small-scale physical processes related to the interaction of the Kuroshio and internal tides, a 2D (x–z)
version of the SUNTANS model (Fringer et al. 2006)
was used to solve the Reynolds-averaged nonhydrostaticNavier–Stokes equation with the Boussinesq approximation, depth-averaged continuity equation, free surface,and salinity and temperature transport equations. The
water density is calculated from temperature, salinity,and pressure via an equation of the state (Millero andPoisson 1981). The Mellor–Yamada level 2.5 turbulent
closure model is employed to compute the vertical eddy
viscosity and diffusivity (Mellor and Yamada 1982). The
horizontal eddy viscosity and eddy diffusivity are set to1 3 1022 m2 s21 and 0, respectively. The quadratic bottom
drag coefficient CD 5 0.0025 is kept constant throughout
the model domain. The time step is set to 10 s in order tomaintain numerical stability. The horizontal grid spacing
is set to 100 m in the middle of the domain covering theridge region (Fig. 2a, 2250 , x , 250 km); it is also
stretched to 10 km at the western and eastern boundaries to avoid the reflection of high-frequency internal
tides from the model boundaries. As a result, the model
consists of 5098 horizontal cells containing 150 stretchedvertical z layers with grid height refinement near the sea
surface. The minimum (surface) and maximum (bottom

2920

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

FIG. 3. Initial (a) temperature, (b) salinity, and (c) density, (d) N2 profiles, and (e) normalized background flow atboundaries. The dots represent points on the z-level grid. Gray lines in (e) represent time-averaged current profiles
taken at six locations in the upstream of the Kuroshio shown as white X marks in Fig. 1c.

at a depth of 3000 m) vertical gridcell sizes are equal to
3.2 and 62.0 m, respectively. Earth’s rotation is ignored
to simplify the model setup. Although the model isidealized by using the 2D domain without taking into
account Earth’s rotation, it is expected to reproduce theprimary physical processes related to the Kuroshio andinternal tides. Surface heat, freshwater fluxes, and wind
forcing are not considered in this study since we focus on
the interaction between the Kuroshio and internal
waves with relatively short time scales.

b. Bathymetry and initial conditions
We selected the bathymetry along latitude 32.78N(Figs. 1b and 2a), where the main Kuroshio axis flowed
approximately across the Izu–Ogasawara Ridge inAugust 2013, (Masunaga et al. 2018). The bathymetry
data were taken from the J-EGG500 dataset distributed
by Japan Oceanographic Data Center. The maximum
far-field depth from the ridge is set to 3000 m to simplify
the model and avoid computational efforts of resolving
the deep trench in the far field. According to literature,internal tide generation and dissipation occur in depths
shallower than 3000 m in the study area (Masunaga et al.
2017b, 2018). Thus, the assumption of the maximum
depth (3000 m) is a reasonable approximation to investigate internal tide dynamics. As shown in Fig. 2, the originof the horizontal coordinate axis (x) is located at thesummit of the ridge, and the positive (negative) direction
corresponds to the eastward (westward) direction.
To simulate internal tides in the summer season presented in Masunaga et al. (2018), the initial temperatureand salinity vertical profiles are taken from the summer seasonal climatological database World Ocean Atlas
2009 (WOA09; National Oceanographic Data Center;longitude: 141.58E, latitude: 34.58N). The buoyancy
frequency is defined assﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g dr
,
(1)
N5 2
r0 dz
where g is the gravitational acceleration, r is thewater density, and r0 is the reference water density(1024 kg m23). The squared buoyancy frequency N2,
shows the maximum vertical density gradient located at
the depth of approximately 50 m (Fig. 3d). The horizontal distributions of the initial salinity and temperature are assumed to be uniform. It should be noted thatthe salinity minimum is observed in the subsurface layer
at z ; 2700 m due to the intrusion of the cold lowsalinity Oyashio water from the northern area of Japan.
The initial velocity and free surface height are set to zerothroughout the entire model domain.

c. Boundary forcing
The Kuroshio is generated by the effects of Earth’s
rotation and winds on the large/global oceanic scale. The
proposed 2D model is not able to reproduce it spontaneously. Instead, a mean background flow is imposed torepresent the Kuroshio in the 2D domain, whose eastern
and western boundaries are forced by the prescribed
idealized flow mimicking the Kuroshio as an exponentialfunction with respect to depth as

NOVEMBER 2019 2zuBK 5 UBK exp 2 2 ,s(2)where UBK is the maximum background velocity at thesea surface, z is the vertical coordinate, and s (5300 m)
is the thickness of the Kuroshio. The western andeastern boundaries correspond to x 5 2500 and 500 km,
respectively. The background flow is imposed on theboundaries as a clamped boundary velocity forcing. The
exponential velocity profile used in the model is compared with velocity profiles at six locations taken fromthe upstream of the Kuroshio presented in Masunaga
et al. (2018) (Fig. 3e, the six locations are shown in
Fig. 1c as white X marks). Although the velocities obtained by Masunaga et al. (2018) vary with space anddepth, the simplified exponential velocity appears to
be a reasonable approximation to reproduce the Kuroshio
in the model. To reduce the degree of numerical instability and unrealistic oscillations at the beginning ofsimulations, the boundary background velocities aregradually increased with time as
uBK

2921

MASUNAGA ET AL.

actual5 uBK [1 2 exp(2t/tBK )] ,(3)

where t is the time and tBK is the spinup time scale set to
2 days. To generate barotropic tidal waves propagatingfrom east to west, the eastern and western model
boundaries are also forced by the barotropic tidal flows
defined as
uBtide 5 A sin(kx 2 vt) ,(4)

where A is the barotropic velocity magnitude, k is thehorizontal wavenumber, x is the horizontal distancecoordinate, and v is the tidal frequency. The amplitude
and phase from a global tidal model were not employedin this study because of the idealized model domain in
2D. In this study, the parameters for Eq. (4) are chosen
to reproduce the sea surface tidal oscillations with theM2 tidal frequency and amplitude of 0.5 m over the ridge.
To meet this condition, the barotropic velocity amplitude
A and tidal frequency v are set to 2.8 3 1022 m s21 and1.4 3 1024 rad s21, respectively. The horizontal wavenumber k is set to 8.2 3 1027 rad m21 determined fromthe tidal frequency and linear longwave speed at a
total depth of 3000 m (171.5 m s21), which can induce
realistic barotropic tidal waves in the study area
(Masunaga et al. 2017b). Tidal forcing is also gradually
increased with time similarly to the background flow
forcing in Eq. (3); however, in this case, the spinup
time scale is set to 1 day.
The model was spun up for 10 days with the background flow; then, tidal forcing was added. The tidal

TABLE 1. Boundary conditions, internal Froude numbers, andminimum Richardson numbers used for various model runs inthis study.

Run1
Run2 (Run2w/oTide)
Run3 (Run3w/oTide)
Run4 (Run4w/oTide)
Run5 (Run5w/oTide)
Run6 (Run6w/oTide)
Run7 (Run7w/oTide)

UBK (m s21)a (m)ci (m s21)

0
0.25
0.5
0.75
1
1.25
1.5

0.5
0.5 (0)0.5 (0)0.5 (0)0.5 (0)0.5 (0)0.5 (0)

1.09
1.09
1.09
1.09
1.09
1.09
1.09

Fri

Ri

0
—
0.23 85.7
0.46 21.4
0.68 9.5
0.91 5.4
1.14 3.4
1.36 2.4

spinup phase includes six M2 cycles (74.52 h), and thetotal spinup period is equal to approximately 13 days
(Fig. 2b). We used the computational results of two M2
cycles (24.48 h) after the spinup for the data analysis in
this study. The condition t 5 0 represents the moment
when tidal forcing was imposed after the 10-day background flow spinup. Seven conditions for UBK ranging
from 0 to 1.5 m s21 were conducted (Table 1). In addition
to these runs involving tidal forcing, runs without tidal
forcing were also conducted (Run2w/oTide–Run7w/oTide).
The internal Froude number (Fri) 5 UBK/ci, UBK, tidal
amplitude (a), and minimum gradient Richardson
number (Ri) determined at the model boundaries for allcases are presented in Table 1. Fri is computed from theratio between UBK and the linear mode-one internal
wave speed ci at the top of the ridge (ci 5 1.09 m s21). Its
magnitude ranges from 0 to 1.36 m s21. The backgroundcurrent was not taken into the computation of c i;thus, c i is determined from the background stratification and depths of the ridge. Note that Fr i definedin this study is the maximum internal Fri at theshallowest summit on the ridge. The internal wave
speed increases as the depth increases, so that Fr i
decreases away from the summit. The phase speed ofthe mode-one internal waves at the depth of 3000 m is
2.57 m s 21 , namely, Fri is less than one in deep regions
for super critical cases (Run6 and Run7). Therefore,
internal waves cannot cross the ridge from the east to
the west for super critical cases, but they can propagate westward in deep regions far away from theshallow summit. The Ri value is calculated from thevertical density gradient and buoyancy frequency as
follows
Ri 5 N 2 /(du/dz)2 .

(5)The condition of Ri , 1/4 is necessary for instability, inwhich background flow is not linearly stable. The minimum Ri value is higher than 1/4 for all cases; hence, thebackground flow does not cause instability at the model
boundaries.

2922

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

3. Results and discussion
In the beginning of this section, we present the model
results of the Kuroshio–internal tide interaction for four
different cases: Run1 (a no-current case, Fri 5 0), Run5
(a near-critical case, Fri 5 0.91), Run7 (a supercriticalcase, Fri 5 1.36), and Run5w/oTide (a case without tidal
forcing). Afterward, we discuss the energy budgets ofinternal tides, high-frequency internal waves, and mixing due to internal tides.

a. No-current case (Fri 5 0)
The case without the background Kuroshio flow
(Run1) shows internal tide generation over the ridge
and off-ridge radiation of internal tides. The dash–dotlines in Figs. 4b and 4c indicate the internal wave raypaths (characteristic slopes) calculated via the following
dispersion relation
k
s5 5
m

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
v2,
N 2 2 v2

(6)

where s is the angle of the internal wave ray path, k is thehorizontal wavenumber, m is the vertical wavenumber,and v is the M2 tidal frequency. The horizontal velocity
patterns shown in Fig. 4c follow the internal tide raypaths. The maximum internal Froude number computedfrom the tidal flow and internal wave speed is equal to0.22 (below the critical value of one), which indicatesthat internal hydraulic jumps do not occur.
To estimate the intensity of internal tides, the depthintegrated internal wave energy flux is estimated via thefollowing formula (Kang and Fringer 2012; Masunaga
et al. 2017b):
FIW 5

ðh
2d

p0 uBC dz ,(7)

where d is the depth, h is the sea surface elevation, p0 is
the high-frequency perturbation pressure excluding thepressure due to the free surface (p0 5 p 2 p0, where p0 is
the low-frequency pressure component with a cutoff
frequency of 3.9 3 1026 s21 corresponding to a 3-day
cycle), and uBC is the lateral baroclinic (BC) velocity(uBC 5 u 2 uBT, where uBT is the barotropic velocity).
The pressure p contains both the hydrostatic andnonhydrostatic components. The barotropic velocity isdefined as
uBT 5

ð
1 h
u dz.
h 1 d 2d(8)

The internal wave energy flux is computed in a fixed
Eulerian frame. The maximum tidal excursion length

FIG. 4. Simulation results for Fr 5 0 (Run1). (a) Density, (b),(c)horizontal velocity at t 5 84 h (t/T0 5 6.76), (d) internal wave energy flux, (e) BT–BC conversion rate, and (f) BC kinetic energy
flux. Panel (b) is an expanded version of (c). The dash–dot blacklines in (b) and (c) denote the beams of internal waves withfrequency M2 .

over the shallow ridge is approximately 1700 m and ismuch smaller than the scale of the ridge. Figure 4d shows
the time-averaged depth-integrated internal wave energy flux averaged over two M2 cycles after the spinup.
The internal wave energy flux was compared over several different time scales for time-averaging operations(two, three, and four M2 cycles) to evaluate significanceof low-frequency biases. Because low-frequency biases
were found to be much smaller than that of internal wave
energy flux, we assume that the time average of the two
M2 cycles is adequate in this context (see the appendix).

NOVEMBER 2019

MASUNAGA ET AL.

2923

The intensities of its eastward and westward components are distributed almost symmetrically and decrease
with distance from the ridge; however, the magnitude ofthe eastward flux is slightly higher than that of the
westward flux (their maximum intensities are equal to
5.4 and 4.1 kW m21, respectively). This difference canonly be due to the asymmetric shape of the topography.
The intensity of internal wave generation can be estimated from the barotropic–baroclinic (BT–BC) conversion rate. The vertically integrated BT–BC conversion
rate C is defined as (Kang and Fringer 2012)
ðh
r0 gW dz ,
(9)
C5
2d

where W is the barotropic vertical velocity represented as
W 5 2=  [(d 1 z)uBT ] .

(10)Figure 4e shows the time-averaged depth-integrated
BT–BC conversion rate averaged over two M2 cycles.
The conversion rate is significant mostly in the shallow
area at depths of less than 1000 m with a maximum ofaround 0.8 W m22 at x ; 18 km (Fig. 4e). The magnitudes of the internal wave flux and BT–BC conversion
rate are consistent with the results of previous 3D numerical simulations that did not consider the direct influences of the Kuroshio (Masunaga et al. 2017b, 2018).

b. Near-critical case (Fri 5 0.91)
Figure 5 displays the results of Run5 which includes
the background Kuroshio flow at Fri 5 0.91. It showsa significantly different internal wave distribution ascompared to that for the case without the backgroundflow (Run1). In particular, it contains sharp small-scaleinternal solitary wave-like vertical isopycnal displacements on the west side of the ridge (Fig. 5a), which are
not observed on the east side. The velocity distribution
consists of a horizontally compressed structure in theupper layer (z . 21000 m) on the west side (Figs. 5b,c).
At the same time, the velocity distribution on the eastside is horizontally stretched by the Kuroshio to
produce a ‘‘fan-like’’ shape (Lamb and Dunphy 2018).
These data for both the west and east sides are consistent with the results of numerical simulations conducted at subcritical Froude numbers by Lamb andDunphy (2018).
The internal wave energy flux and BT–BC conversion
rate for the cases that include a background current canbe also estimated using Eqs. (7) and (8). However, thelow-frequency background currents or oscillations haveto be removed to compute these variables associatedwith the tidal oscillation, because the background flow

FIG. 5. Simulation results for Fr 5 0.91 (Run5). The horizontal
velocity in (b) and (c) represents the perturbation component u0 .

also includes the vertically sheared BC component. The
high-frequency (perturbation) components u0 and W0
are used in computations of the internal wave energy
flux and BT–BC conversion. High-frequency velocities
are extracted at a cutoff frequency of 3.9 3 1026 s21(period of 3 days). The internal wave energy flux for
Run5 exhibits an asymmetric structure on the west andeast sides of the ridge (Fig. 5d). Further, the westward
flux is strongly enhanced, and its maximum amplitude
reaches a value of 12.0 kW m21, which is approximately
3 times higher than the magnitude for the case without
the Kuroshio (Run1). The maximum westward flux
matches a steep peak of the BT–BC conversion rate withan intensity of up to 1.8 kW m22 at x ; 232 km (Fig. 5e).
The maximum intensity of the internal wave energy flux

2924

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

and BT–BC conversion is comparable to that by the
Hawaiian Ridge where strong internal tides were observed (Merrifield and Holloway 2002). On the other
hand, the eastward flux is smaller than the value for
Run1 (its maximum amounts to 3.7 kW m21, which isonly 69% of the magnitude obtained from Run1).
Consequently, the background Kuroshio flow enhances the BT–BC conversion and the upstream propagation of internal tides and reduces the downstream
internal tides, which is consistent with the findings of therecent study conducted by Lamb and Dunphy (2018).
Such wave energy enhancement caused by the interaction of the background currents with upstream waves
has been considered in studying the dynamics of surface
gravity waves (Mei 1992). The numerical results obtained in this work suggest that the interaction of theKuroshio and internal tides has a significant effect on the
upstream internal wave flux, which is similar to that
produced by the current–surface wave interaction.

c. Supercritical case (Fri 5 1.36)
The numerical simulations conducted for the supercritical case (Run7, Fig. 6) produce water density andcurrent structures similar to those for the near-criticalcase on the east side (Run5, Fig. 5). Meanwhile, thecorresponding magnitude of horizontal velocity andwestward internal wave energy flux are noticeably lower
on the west side (see Figs. 5b,d and 6b,d). The maximum
amplitude of the westward internal wave energy flux is
5.9 kW m21, which is roughly half of the magnitude forthe near-critical case. This low upstream internal wave
flux results from the fact that the surface velocity exceeds the internal wave phase speed over the top of theridge, suggesting that the Kuroshio flushes out internal
wave energy generated around the top of the ridge
downstream. The maximum eastward flux is 3.9 kW m21,
which is close to the value for the near-critical case. Although the internal wave radiation from the ridge is relatively low, the BT–BC conversion rate is apparently
close to that of the near-critical case (Figs. 5e and 6e),indicating that a large amount of converted internal wave
energy is dissipated over the ridge due to the Kuroshio(the details of this process are discussed in section 3e).

d. Case without tidal forcing
The cases without tidal forcing are also considered
to investigate the effects exclusively caused by the
Kuroshio. Figure 7 shows the results of numerical simulations for Run5w/oTide (UBK 5 1, a 5 0). Here, themaximum lateral velocity at the side boundaries is
1.0 m s21. The surface current speed is enhanced by theshallow ridge reaching a magnitude of 1.25 m s21, andthe maximum Froude number over the ridge exceeds

FIG. 6. Simulation results for Fr 5 1.36 (Run7). The horizontal
velocity in (b) and (c) represents the perturbation component u0 .

unity. A hydraulic jump is observed over the ridge, resulting in abrupt isopycnal displacements and velocity
perturbations near the two shallow peaks of the sill
(Fig. 7, x 5 ;220 and 10 km). The internal wave intensity is much weaker than the values for the cases thattake into account tidal forcing. The internal wave energy
flux and BT–BC conversion are less than 0.1 kW m21
and 2 3 1023 W m22, respectively (Figs. 7f,g), which arethree orders of magnitude smaller than the values computed for runs which include tidal forcing (Run5).

e. Energy budget
Several previous studies have reported that most of
the total BC energy flux can be explained by the internal

NOVEMBER 2019

2925

MASUNAGA ET AL.

FBC 5 FIW 1 FKEBC .

(11)

The BC KE flux is given by (Kang and Fringer 2012)
FKEBC 5ðh2duBC EKBC dz 1

1
EKBC 5 r(u2BC 1 w2 ) ,
2

ðh
2d

uBC EKBTBC dz ,andEKBTBC 5 ruBT uBC ,FIG. 7. Simulation results for the case without tidal forcing(Run5w/oTide). The contents of (f) and (g) are identical to those of(d) and (e), respectively, but their vertical ranges are different. The
horizontal velocity in (b) and (c) represents the perturbation
component u0 .

wave energy flux (pressure work) FIW (Venayagamoorthy
and Fringer 2006; Kang and Fringer 2012), but strictly
speaking the total BC energy flux is the sum of advection (kinetic energy and available potential energy),
pressure (internal wave) and diffusion terms. By contrast to these studies, a recent work done by Lamb andDunphy (2018) showed that the kinetic energy (KE)advection term also contribute to the total energy flux
when internal waves interact with background flows.
Thus, we assume that the total BC energy flux FBC is
the sum of the internal wave energy flux FIW and the
BC KE advection term FKEBC as follows

(12)
(13)
(14)

where w is the vertical velocity, and EKBC and EKBTBC
are known as the BC kinetic energy and cross-term kinetic energy, respectively. To extract high (tidal) frequency motions from modeled data, the perturbation
component of the velocity u0 and density r0 is used tocompute FKEBC with a cutoff frequency of 3.9 3 1026 s21(the same approach as the computation of FIW). The
time-averaged FKEBC for Run1, Run5, and Run7 areplotted in Figs. 4f, 5f, and 6f, respectively. For Run1, themagnitude FKEBC is less than 0.2 kW m21 and the direction of FKEBC is not consistent with FIW. The interaction of the background flows and tides intensifies
FKEBC over the shallow summits, reaching 20.8 and20.7 kW m21 for Run5 and Run7, respectively (Figs. 5f
and 6f). However, most of the generated BC kinetic
energy decays over the shallow ridge. Overall, themagnitude of FKEBC is an order of 0.1–1 kW m21, which
is roughly 10 times smaller than that of FIW (Figs. 5d,f).
To investigate the internal tidal energetics in moredetail, the time-averaged and area-depth-integrated
energy terms, BT–BC conversion rate, BC radiation,
and BC dissipation are calculated over the ridge for
Run1–Run7. The analyzed area over the ridge is definedas the shallow part with depths of less than 700 m(depicted as the light gray shaded area in Fig. 2a). The
BT–BC conversion within the analyzed area explains
89% of the total BT–BC conversion in the whole model
domain for Run5, thus, we assume that the analyzed
area is suitable for the energy budget analysis. The BC
radiation corresponds to the outgoing total BC energy
flux from the analyzed area. The schematic of the energy
budget analysis is shown in Fig. 8a. The BC dissipation isdefined as«BC 5 =H  FBC 2 C ,(15)where =H is the horizontal gradient operator, and theoverbar denotes the time averaging operation. The
width of the 2D domain (in the y direction) is assumed to be 1 m to compute area-integrated values.
The time averaging period includes two M 2 cycles
after the spinup.

2926

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

at which the upstream BC energy flux is approximately 6times larger than that on the east side determined for thenear-critical case. The BC dissipation does not exhibit
an apparent relation to Fri under subcritical conditions;however, its value clearly increases with increasing Fri
under supercritical conditions (Fig. 8d). Roughly 0.3%(34%) of the total BC energy generated in Run5
is dissipated over the ridge at Fri 5 0.91 (Run7 with
Fri 5 1.36) (Fig. 8e). This suggests that a large amount of
the generated BC (or internal tidal) energy is radiated(dissipated) from the ridge (over the ridge) by the
Kuroshio under the near- (super) critical conditions.
Figure 8d compares the BC dissipation estimated by the
total BC energy flux («BC) and that estimated only by
the internal wave energy flux («IW 5 =H  FIW 2 C). The
difference between «BC and «IW («BC 2 «IW) ranges
between 20.41 and 0.16 kW. The BC kinetic energy
term reduces «BC by a factor of approximately 10. Thus,
although the contribution of KE to the BC flux is muchsmaller than internal tides, the BC kinetic energy has tobe considered when computing the tidal energy budget
associated with the interaction between the backgroundflows and tides.

f. High-frequency internal waves
FIG. 8. (a) Schematic illustrating the energy budget analysis
procedure. (b) BT–BC conversion rate over the ridge, (c) internal
wave energy radiation from the ridge, (d) total BC dissipation («BC;black dash–dot line) and BC dissipation explained by internal
waves («BC; red dash dot line) over the ridge, and (e) ratio between
the BC dissipation and BT–BC conversion.

The BT–BC conversion increases as Fri increases
under subcritical conditions and drops once Fri reaches
the critical value (Fri 5 1, Fig. 8b). The Kuroshio has apotential to enhance the BC energy (internal wave)
generation by a factor of 1.75 near the critical point(Run5). On the other hand, the fast Kuroshio under thesupercritical conditions sharply reduces BC energy
generation. The BC radiation on the west side of theridge increases with Fri for subcritical cases and decreases for the supercritical cases demonstrating a trend
similar to that for the BT–BC conversion. The westward
BC radiation for the near-critical case (Run5) is 4.2
times greater than that for the case without the background current (Run1) (Fig. 8c). Meanwhile, on the eastside of the ridge, the BC radiation slightly decreases(increases) with increasing Fri under sub- (super) criticalconditions (Fig. 8c). The minimum eastward BC radiation for the near-critical case (Run5) is roughly half ofthe magnitude for the case without the background flow
(Run1). As described above, the upstream (westward)
BC energy flux (BC radiation) is remarkably enhancedby the background current under subcritical conditions,

High-frequency internal waves are generated in our
numerical simulations. Figures 9b–f shows the temporal
evolution of the bandpass filtered vertical displacements of the main pycnocline (sT 5 24.2) for Run5 in
the frequencies of M2 –4M 2 (f 1 –f 4) and higher than
5M2 (f 5). The filter band range f i (i denotes thenumber of the harmonics) is 1.5 3 10 25 , f 1 , 3.0 31025 , f 2 , 5.4 3 1025 , f3 , 7.7 3 1025 , f4 , 9.9 31025 , f5 , ‘ s21.The internal wave phase speed along
the pycnocline appears to be slower on the west sidethan that on the east side, which cannot be simply explained by the sum of the first mode internal wave speed
ci and the background flow speed UBK, because thebackground flow is not uniform vertically. The blackdash–dot lines (C1–C4) in Fig. 9b represent the phasespeeds of the first four vertical internal wave modes with
the M2 tidal frequency estimated using normal mode
analysis, whose magnitudes are equal to 2.6, 1.3, 0.9, and
0.7 m s21, respectively. The calculated phase speed fits
the slope of the fourth vertical internal wave (C4) on thewest side of the ridge (Fig. 9b). On the same side, thevelocity distribution represents a multivertical mode
structure (Fig. 9g), which implies that high vertical modeinternal waves explain the displacements of the isotherm
depth. The multivertical modal structure and associatedslow internal wave phase speed also appear in the casewithout background flows (Run1, Figs. 10a–c). The internal wave phase speed is almost the same for Run1 andNOVEMBER 2019

MASUNAGA ET AL.

2927

FIG. 9. (a)–(f) Vertical isopycnal displacements (sT 5 24.2) and (g),(h) BC velocities in the frequency band of
M2 on the (g) west (P1) and (h) east (P5) sides of the ridge for Run5 (UBK 5 1.0 m, a 5 0.5). The bandpass filteredisopycnal displacements with frequencies (b) M2, (c) 2M2, (d) 3M2, (e) 4M2, and (f) higher than 5M2. The blackdash–dot lines in (b) indicate the internal wave speed at a depth of 3000 m, and subscripts 1–4 denote the verticalmode numbers. The wave phase speed slope in (f), the black dashed lines, shows the first mode wave speed considering the bathymetry (C1), and the bathymetry and barotropic background flow (C1 2 UBKBT).

Run5 far away from the ridge (x , 2100 km and x .
100 km). BC horizontal velocities for the first four vertical modes estimated from the initial stratification areplotted in Figs. 10d–g. The BC velocity structure on thewest side of the ridge can be roughly explained by thethird or fourth mode. The second mode explains the BC

velocity on the east side. Thus, the slow wave phasespeed observed for the vertical isopycnal displacements
can be explained by the presence of multivertical internal tides and the wave phase speed is not significantly
influenced by the background flows far away from the
top of the ridge. Previous observations and models

2928

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

FIG. 10. (a) Vertical isopycnal displacements (sT 5 24.2), (b),(c) BC velocities in the frequency band of M2 onthe (b) west (P1) and (c) east (P5) sides of the ridge for Run1 (UBK 5 0 m, a 5 0.5), and (d)–(g) normalized
BC horizontal velocities for first four vertical modes.

showed that the rough topography results in high-mode
internal tides radiated from ridges (St. Laurent andNash 2004; Zilberman et al. 2009; Vic et al. 2019). St.
Laurent and Nash (2004) and Vic et al. (2019) showeddominant internal wave energy in high internal wave
mode over the Mid-Atlantic ridge. In our model case,the shallow and rough topography causes the dominanthigh mode internal tides radiating off-ridge, which results in the slow propagation speed of internal tides. The
difference of the modal structure on the west and eastsides is due to the difference in the topography.
The isopycnal displacements produced at higher frequency than 5M2 (f . 9.9 3 1025) show that internal
solitary waves are released upstream (westward) fromthe eastern shallow peak of the ridge during every tidal
cycle (Fig. 9f, x ; 10 km). The amplitude of these waves
increases as they propagate westward over the shallow
area of 260 , x , 10 km. Such waves generate wave
packets containing several smaller internal waves
through a process known as ‘‘wave fission,’’ which is acommon feature of the internal solitary waves with relatively large amplitudes (Nash and Moum 2005). Two
first mode internal wave speeds are plotted in Fig. 9f
as the black dash dotted lines taking into account of 1)
the bathymetry (C1) and 2) both of the bathymetry andbarotropic (depth averaged) background flow (C1 2UBKBT). The BT background flow speeds are 0.09 and0.72 m s21 at the depth of 3000 m and at the shallowest
summit, respectively. The BT background flow isroughly 30 times slower than the internal wave speed
far from the ridge (C1 5 2.57 m s21), meanwhile, itreaches 66% of the wave speed over the ridge (C1 5
1.09 m s21). Although the two wave speeds do notcompletely match to the solitary wave speed appearing the isotherm displacements, the slope of C1 2UBKBT seems to be a better approximation than C1.
Therefore, the phase speed of the internal solitary
waves is modulated by the background flow and canbe roughly explained by the first mode internal wave
speed taking into account of the BT background flow.
The time series data obtained at three different locations P2–4 with depths of 1500, 1000, and 500 m (seeFig. 2a) illustrates the growth of the internal solitary
waves, which consists of the following steps (Fig. 11). 1)
Single wave packets are formed during each tidal cycle
at locations with depths of 500 and 1000 m and increasing
wave amplitude as they propagate away from the ridge. 2)
The internal solitary waves with large amplitudes are
separated into several wave packets (wave fission) in thearea deeper than 1500 m. The internal solitary waves

NOVEMBER 2019

MASUNAGA ET AL.

2929

FIG. 11. (a)–(c) Densities, (d)–(f) horizontal perturbation velocities, and (g)–(i) vertical velocities obtained at
locations (left) P2, (center) P3, and (right) P4 on the west side of the ridge during Run5 (UBK 5 1.0 m, a 5 0.5).

produce strong vertical currents reaching a speed of
0.1 m s21 at depths greater than 1000 m.
The amplitude of high-frequency internal waves isrelatively small on the east side of the ridge (Fig. 9f).
The wave phase speed along the main isopycnal line isclose to the speed of the second mode on the east side
(Figs. 9b,f). The horizontal current distribution consistsof a thin BC modal structure confined in the surface
layer, which is significantly different from that on thewest side. This structure may result from the fan-likestretched current distribution due to the multiple internal wave mode structure and their phases modulatedby the vertical shear caused by the Kuroshio (see Fig. 5).
The high-frequency internal wave energy flux (FIWhigh)at frequencies higher than 2M2 (f . 3.0 3 1025) is computed via Eq. (7). Figures 12a and 12b compare the valuesof FIW and FIWhigh for Run5. The internal wave energy
flux at the frequency of M2 corresponds to the difference
between FIW and FIWhigh (FIW 2 FIWhigh). The maximum
magnitude of FIWhigh reaches 3.1 kW m21 on the west sideof the ridge (Fig. 12a, x ; 250 km). The contribution of
the high-frequency internal wave energy flux to the total
flux ranges from 10% to 40% on the west side, which isconsistent with the value shown by Masunaga et al.
(2018), indicating that 14% of the total internal wave
energy flux can be explained by the existence of higher
harmonic internal tides over the Izu–Ogasawara Ridge.
Meanwhile, similar to the isopycnal displacements(Fig. 9), the high-frequency internal wave energy flux ismuch lower on the east side than on the west side. The
maximum eastward high-frequency internal wave flux is
below 6.2 3 1022 kW m, which is 50 times smaller thanthat on the west side.
Figure 12e compares the radiation of the high-frequency
internal waves at the side boundaries of the analysis area
(near the ridge summit with a depth of 700 m as schematically illustrated in Fig. 8a) detected in Run1–Run7.
The westward radiation peaks near the critical Fri value,
which is consistent with the total BC radiation (Fig. 8c).
The contribution of high-frequency waves to the total

2930

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

FIG. 12. (a) Total (blue) and high-frequency (yellow) internal wave energy fluxes, (b) contribution of the high-frequency internal wave
flux to the total internal wave energy flux, (c) divergence of the high-frequency internal wave energy flux (d) high-frequency BT–BC
conversion rate for Run5 (UBK 5 1.0 m, a 5 0.5), (e) high-frequency internal wave radiation on the west (red) and east (black) sides of theridge, and (f) contribution of the high-frequency internal wave radiation to the total radiation, area-integrated and time averaged (g) highfrequency BT–BC conversion and (h) the divergence of high-frequency internal waves, and (i) ratio between the high-frequency conversion and internal wave divergence as a function of the internal Froude number.

internal wave energy flux exceeds 20% for Run5 andRun6 (at Fri 5 0.91 and 1.14, respectively, Fig. 12f). It is
well known that large internal Froude numbers result in
the formation of nonlinear high-frequency internal
waves from the internal tides with lower frequencies
through a hydraulic jump (Farmer and Armi 1999;
Masunaga et al. 2017b). The large contributions of highfrequency waves observed near the critical conditions
likely result from the enhancement of wave’s nonlinearity due to the background current. On the other
hand, the eastward high-frequency wave flux decreases
as Fri increases (Fig. 12f). Nevertheless, the contributionof high-frequency internal waves observed for Run7
is less than 0.1% (Fri 5 1.36), which indicates thatthese waves barely propagate on the downstream sideat Fri . 1.
The generation of high-frequency internal wave canresult from the nonlinear steepening of M2 internal
tides or directly from BT tides (high-frequency BT–BC
conversion Chigh), and the generation can be roughly
measured by the positive part of the divergence of highfrequency internal wave flux (=H  FIWhigh ). The spatial
distribution of the divergence of high-frequency internal

wave energy flux shows enhanced generation ofhigh-frequency internal tides over the shallow ridge(Fig. 12c, 250 , x , 20). Noisy positive and negativevalues of the high-frequency conversion next to eachother indicates that the high-frequency BT–BC or
BC–BT energy conversion occurs at a small scale on anorder of 1–10 km (Fig. 12d). Negative values of theconversion indicate dissipation of high-frequency internal tides, or energy transfer to lower frequency internal tides or BT tides. The area integrated Chighand =H  FIWhigh over the analysis area for Run1–7 areplotted in Figs. 12g and 12h. The range of the areaintegrated Chigh is 0.02–0.07 kW, which is less than 1% of
the total BT–BC conversion. The area-integrated =H 
FIWhigh is more than 10 times higher than the highfrequency BT–BC conversion for Run1–6, which implies
that the generation of high-frequency internal waves isdominated by the conversion from M2 internal tides.
The conversion from M2 internal tides to high-frequency
internal waves is probably due to nonlinear steepeningof M2 internal tides caused by the Kuroshio–tides interaction. For the case with the strongest Kuroshio(Run7, Fri 5 1.36), the contribution of the high-frequency

NOVEMBER 2019

MASUNAGA ET AL.

2931

BT–BC conversion becomes larger than other cases andit reaches 28% of the divergence of the high-frequency
internal wave flux (Fig. 12i).

g. Deep ocean mixing
It is well known that internal tides significantly contribute to mixing in the ocean (e.g., Munk and Wunsch1998). A recent study of Vic et al. (2019) reported thathigh-mode internal tides significantly contribute to local
mixing. In addition to local mixing near internal wave
generation sites, the internal tides/waves radiate offshore and presumably contribute to deep ocean mixing
far from their generation sites, known as far-field internal tide dissipation (MacKinnon et al. 2017). Internal
tides radiate away from generation sites through lowmode internal waves, which contributes to roughly
20%–80% of the total internal tide dissipation in theocean (MacKinnon et al. 2017). The vertical eddy diffusivity estimated using the Mellor–Yamada level 2.5
turbulent closure model (Mellor and Yamada 1982) isused to evaluate the mixing intensity in the present
study. The time-averaged vertical eddy diffusivities for
Run1, Run5w/oTide, and Run5 are plotted in Fig. 13. The
no-current case (Run1) exhibits strong diffusivity
around the internal wave ray paths over the ridge(Fig. 13a) with a nearly symmetric pattern and equalmagnitudes on both the west and east sides of the ridge.
For Run5w/otide, the vertical shear due to the Kuroshio
produces a remarkable increase in diffusivity in thesubsurface layer (Fig. 13b), and the eddy diffusivity
reaches a magnitude of 1 3 1024 m2 s21. In a previous
study, an eddy diffusivity of O(1024–1023) m2 s21 wasobserved along the Kuroshio axis (Nagai et al. 2012),
which was consistent with the results of our simulations.
The Kuroshio produces an asymmetric mixing structure on the west and east sides of the ridge, in which
the vertical mixing is enhanced on its east side atz , 21000 m (Fig. 13b). Such a structure is commonly
formed over a sill and has been frequently observed in theatmosphere and ocean (Farmer and Denton 1985;
Edwards et al. 2004; Liu et al. 2016). The enhanced vertical mixing on the east (downstream) side of the ridge
observed in our simulations may reflect effects of the
background flow and an internal hydraulic jump. In the
run that takes into account both the background flow andtides (Run5, Fig. 13c), the eddy diffusivity on the east side
is almost identical to that in Run5w/oTide; however, its
magnitude calculated for the deep area (z , 21000 m) on
the west side exceeds the values in Run1 and Run5w/oTide.
To further investigate the deep ocean mixing due tothe interaction of the Kuroshio with internal tides, thetime and spatially averaged vertical eddy diffusivities inbottom cells computed at depths ranging between 1500

FIG. 13. Time-averaged vertical eddy diffusivity estimatedusing the Mellor–Yamada turbulent closure model for (a) Run1,(b) Run5w/oTide , and (c) Run5.

and 2500 m are compared for all model cases. The rangeof 1500–2500 m is selected, because the near surface
layer (depth , 1500 m) is directly influenced by the
background flows and the deep layer (depth . 2500 m)
does not show significant differences of the eddy diffusivity between the three cases. Figures 14a and 14b show
the averaged eddy diffusivities for the west and eastsides of the ridge. The eddy diffusivity on the west side
exhibits a peak of 7.5 3 1026 m2 s21 at Fri ; 1 and tidal
forcing, which is much larger than the sum of the eddy
diffusivities obtained in Run1 (with tidal forcing andFri 5 0) and Run2w/oTide –Run7w/oTide. In addition, thebottom eddy diffusivity is almost proportional to the
upstream internal wave energy flux under subcriticalconditions (Fri , 1; see Fig. 14c). In the turbulent closure scheme of the present model, the shear production
term (} du/dz)2 typically transfers energy from the mean
flow into the turbulent kinetic energy (TKE), while thebuoyancy flux term (} N2) reduces the TKE. The differences in the vertical eddy diffusivity between themodel cases may largely be caused by the differences inthe vertical shear because the background stratification
for all cases is the same. The interaction of the Kuroshio
and tides enhances the BC velocity in deep regions(cf. Fig. 9g with Fig. 10b). Thus, near-bottom currents enhanced by internal tides due to the interaction
result in elevated mixing in the deep bottom boundary
layer. In turn, the eddy diffusivity on the east side simply

2932

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 49

FIG. 14. Time and spatially averaged eddy diffusivities at the bottom boundary layer anddepths ranging from 1500 to 2500 m on the (a) west and (b) east sides of the ridge estimated for
Run1–Run7 (red) and Run2w/oTide–Run7w/oTide (black). (c) Bottom eddy diffusivity on the westside plotted as a function of the westward internal wave flux radiation at Fri , 1.

increases as Fri increases (Fig. 14b). The contribution of
tidal forcing to the total bottom eddy diffusivity is roughly
1 3 1026 m2 s21 (see the differences in the eddy diffusivity
between Run2–Run7 and Run2w/oTide–Run7w/oTide).
The vertical eddy diffusivity estimated by our simulations suggests that the enhancement by the interactionof the Kuroshio and tides is quite small compared withthat in energetic oceans directly influenced by geostrophic flows, tides and internal waves, reaching anorder of 1024–1022 m2 s21(e.g., Masunaga et al. 2015;
Nagai et al. 2017). By contrast to such energetic oceans,the vertical eddy diffusivity is low in calm deep oceans,an order of 1026 m2 s21 (Ferron et al. 2014; Shang et al.
2017). The intensified vertical shear in the strong internal tides generated by the Kuroshio–tides interaction
is able to contribute to the deep ocean mixing in the
upstream region of the Kuroshio.

4. Conclusions
In this study, the generation of strong internal tides
influenced by the interaction of the Kuroshio and tides

over a shallow ridge is described using the fully nonlinear nonhydrostatic hydrodynamic SUNTANS model.
The results of several runs conducted at different internal Froude numbers show that the upstream propagating internal tides enhanced by the Kuroshio with
the internal wave energy flux reaching a magnitude of
12.0 kW m21 near the critical condition (Fri ; 1) due tothe interaction of the Kuroshio and tides. The strong
upstream propagating internal tides are accompanied
by high-frequency and solitary internal waves, which
likely amplify deep ocean mixing in the upstream region
relative to the ridge. In contrast, the downstream internal tides are suppressed by the Kuroshio and tides
interaction.
Results shown in this study support the strong upstream internal tide energy flux from the Izu–Ogasawara
Ridge found by Masunaga et al. (2018) (Fig. 1d). The
first mode internal wave phase speed around the Izu–
Ogasawara Ridge is plotted in Fig. 15a, in which theresults from Masunaga et al. (2018) were used for thephase speed estimation. The phase speed of internal
waves ranges between 1 and 2 m s21(Fig. 15a), except for

FIG. 15. (a) First vertical mode of the internal wave phase speed and (b) internal Froude number plotted forAugust 2014 using the numerical simulation data provided by Masunaga et al. (2018). The gray contours denote theisobath intervals of 500 m.

NOVEMBER 2019

2933

MASUNAGA ET AL.

shallow coastal regions, and the current speed of theKuroshio also ranges roughly between 1 and 2 m s21(Fig. 1a). This matching of the internal wave speed andthe Kuroshio flow speed results in Fri ; 1 along theKuroshio path (Fig. 15b). The combination of the critical Fri condition and strong tidal flows over the Izu–
Ogasawara Ridge causes strong internal tide energy flux
upstream of the Kuroshio shown in Fig. 1d. Furthermore, the strong internal tides formed by the interactionof the Kuroshio and tides may promote mixing andtransport along the Kuroshio flow.
The majority of previous studies have separately
focused on geostrophic flows (such as the Kuroshio
and Gulf Stream) and internal tides. However, bothphenomena coexist in world’s oceans, for example,
Tokara Strait (Nagai et al. 2017), Luzon Strait (DaSilva et al. 2015), and Straits of Florida (Davis et al.
2008). Strong internal tides can be generated by theinteraction of the background flow and tides in any
areas other than the Izu–Ogasawara Ridge at critical
Fri values. The internal tides transport heat and massacross the oceans and are known as an important
contributor to maintaining the global ocean circulations and ecosystems (e.g., Walsh 1991; Sharples et al.
2009; Masunaga et al. 2017a). Hence, the strong internal tides generated by the background flow andtides are anticipated to contribute to circulations andecosystems in the ocean.
Acknowledgments. This study was supported by
JST-CREST (grant No. JPMJCR12A6). E. Masunaga
and Y. Uchiyama acknowledge the JSPS Grant-in-Aid
for Scientific Research (No. 18H03798).

APPENDIX
Validation Test for the Averaging Time Scale
To validate the averaging time scale used for computing the internal wave energy flux and energy budget,the internal wave energy flux is compared with time
averages over two, three, and four M2 cycles (t/T0 5 6–8,
6–9 and 6–10) for Run1. Time series of the internal wave
energy flux on the west and east sides of the ridge (depthof 700 m) and the time-averaged internal wave flux forthe three periods are shown in Fig. A1. The time series
data represent similar patterns of the internal wave
energy flux repeated in each tidal phase without any
substantial low-frequency biases. The estimated timeaveraged internal wave energy flux also shows a consistency for the three averaging periods. The time-averaged
internal wave flux on the west (east) side of the ridge
is 22.41 (3.79), 22.49 (3.82), and 22.53 (3.70) kW m22

FIG. A1. (a) Internal wave flux as a function of t/T0 on the west
(solid line) and east (dash–dot line) sides, and (b) time-averaged
internal wave flux computed for periods of 6–8 (thick gray line), 6–9(thin black line), and 6–10 (black dash–dot line) M2 cycles.

for the period of t/T0 5 6–8, 6–9, and 6–10, respectively.
Therefore, we assume that two tidal cycles are sufficient
to obtain reasonable averaged quantities.
REFERENCES
Alford, M. H., and Coauthors, 2015: The formation and fate ofinternal waves in the South China Sea. Nature, 521, 65–69,
https://doi.org/10.1038/nature14399.
Arthur, R. S., and O. B. Fringer, 2016: Transport by breaking internal waves on slopes. J. Fluid Mech., 789, 93–126, https://doi.org/10.1017/jfm.2015.723.
Bourgault, D., M. Morsilli, C. Richards, U. Neumeier, and D. E.
Kelley, 2014: Sediment resuspension and nepheloid layers
induced by long internal solitary waves shoaling orthogonallyon uniform slopes. Cont. Shelf Res., 72, 21–33, https://doi.org/
10.1016/j.csr.2013.10.019.
Cacchione, D. A., L. F. Pratson, and A. S. Ogston, 2002: The
shaping of continental slopes by internal tides. Science, 296,
724–727, https://doi.org/10.1126/science.1069803.
Da Silva, J. C. B., M. C. Buijsman, and J. M. Magalhaes, 2015:
Internal waves on the upstream side of a large sill of the
Mascarene Ridge: A comprehensive view of their generation
mechanisms and evolution. Deep-Sea Res. I, 99, 87–104,
https://doi.org/10.1016/j.dsr.2015.01.002.
Davis, K.A., J. J. Leichter, J. L. Hench, and S. G. Monismith, 2008:
Effects of western boundary current dynamics on the internal
wave field of the Southeast Florida shelf. J. Geophys. Res., 113,
C09010, https://doi.org/10.1029/2007JC004699.
Edwards, K. A., P. MacCready, J. N. Moum, G. Pawlak, J. M. Klymak,
and A. Perlin, 2004: Form drag and mixing due to tidal flow
past a sharp point. J. Phys. Oceanogr., 34, 1297–1312, https://doi.org/10.1175/1520-0485(2004)034,1297:FDAMDT.2.0.CO;2.
Farmer, D. M., and R. A. Denton, 1985: Hydraulic control offlow over the sill in Observatory Inlet. J. Geophys. Res., 90,9051–9068, https://doi.org/10.1029/JC090iC05p09051.
——, and L. Armi, 1999: The generation and trapping of solitary
waves over topography. Science, 283, 188–190, https://doi.org/
10.1126/science.283.5399.188.

2934

JOURNAL OF PHYSICAL OCEANOGRAPHY

Ferron, B., F. Kokoszka, H. Mercier, and P. Lherminier, 2014: Dissipation rate estimates from microstructure and finescale internal wave observations along the A25 Greenland–Portugal
OVIDE line. J. Atmos. Oceanic Technol., 31, 2530–2543, https://doi.org/10.1175/JTECH-D-14-00036.1.
Fringer, O. B., M. Gerritsen, and R. L. Street, 2006: An
unstructured-grid, finite-volume, nonhydrostatic, parallel coastal
ocean simulator. Ocean Modell., 14, 139–173, https://doi.org/
10.1016/j.ocemod.2006.03.006.
Guo, C., and X. Chen, 2014: A review of internal solitary wave
dynamics in the northern South China Sea. Prog. Oceanogr.,
121, 7–23, https://doi.org/10.1016/j.pocean.2013.04.002.
Hasegawa, D., H. Yamazaki, T. Ishimaru, H. Nagashima, andY. Koike, 2008: Apparent phytoplankton bloom due to island
mass effect. J. Mar. Syst., 69, 238–246, https://doi.org/10.1016/
j.jmarsys.2006.04.019.
Kang, D., and O. Fringer, 2012: Energetics of barotropic and baroclinic tides in the Monterey Bay area. J. Phys. Oceanogr., 42,
272–290, https://doi.org/10.1175/JPO-D-11-039.1.
Lamb, K. G., and M. Dunphy, 2018: Internal wave generation by
tidal flow over a two-dimensional ridge: Energy flux asymmetries induced by a steady surface trapped currents. J. Fluid
Mech., 836, 192–221, https://doi.org/10.1017/jfm.2017.800.
Liu, Z., T. Ishihara, T. Tanaka, and X. He, 2016: LES study ofturbulent flow fields over a smooth 3-D hill and a smooth 2-Dridge. J. Wind Eng. Ind. Aerodyn., 153, 1–12, https://doi.org/
10.1016/j.jweia.2016.03.001.
MacKinnon, J. A., and Coauthors, 2017: Climate process team oninternal wave–driven ocean mixing. Bull. Amer. Meteor. Soc.,
98, 2429–2454, https://doi.org/10.1175/BAMS-D-16-0030.1.
Masunaga, E., H. Homma, H. Yamazaki, O. B. Fringer, T. Nagai,
Y. Kitade, and A. Okayasu, 2015: Mixing and sediment resuspension associated with internal bores in a shallow bay. Cont.
Shelf Res., 110, 85–99, https://doi.org/10.1016/j.csr.2015.09.022.
——, R. S. Arthur, O. B. Fringer, and H. Yamazaki, 2017a: Sediment resuspension and generation of intermediate nephew
layers by shoaling internal bores. J. Mar. Syst., 170, 31–41,
https://doi.org/10.1016/j.jmarsys.2017.01.017.
——, O. B. Fringer, Y. Kitade, H. Yamzaki, and S. M. Gallager,
2017b: Dynamics and energetics of trapped diurnal internal
Kelvin waves around a mid-latitude island. J. Phys. Oceanogr.,47, 2479–2498, https://doi.org/10.1175/JPO-D-16-0167.1.
——, Y. Uchiyama, Y. Suzue, and H. Yamazaki, 2018: Dynamics ofinternal tides over a shallow ridge investigated with a highresolution downscaling regional ocean model. Geophys. Res.
Lett., 45, 3550–3558, https://doi.org/10.1002/2017GL076916.
Mei, C. C., 1992: The Applied Dynamics of Ocean Surface Waves.
Advanced Series on Ocean Engineering, Vol. 1, World Scientific, 768 pp.
Mellor, G. L., and T. Yamada, 1982: Development of a turbulence
closure model for geophysical fluid problems. Rev. Geophys.,
20, 851–875, https://doi.org/10.1029/RG020i004p00851.
Merrifield, M. A., and P. E. Holloway, 2002: Model estimates of M2
internal tide energetics at the Hawaiian Ridge. J. Geophys.
Res., 107, 3179, https://doi.org/10.1029/2001JC000996.
Millero, F. J., and A. Poisson, 1981: International one-atmosphereequation of state of seawater. Deep-Sea Res., 28A, 625–629,
https://doi.org/10.1016/0198-0149(81)90122-9.

VOLUME 49

Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics oftidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, https://doi.org/10.1016/S0967-0637(98)00070-3.
Nagai, T., A. Tandon, H. Yamazaki, M. J. Doubell, and S. Gallager,
2012: Direct observations of microscale turbulence and thermohaline structure in the Kuroshio Front. J. Geophys. Res.
Oceans, 117, C08013, https://doi.org/10.1029/2011JC007228.
——, D. Hasegawa, T. Tanaka, H. Nakamura, E. Tsutsumi,
R. Inoue, and T. Yamashiro, 2017: First evidence of coherent
bands of strong turbulent layers associated with highwavenumber internal-wave shear in the upstream Kuroshio.
Sci. Rep., 7, 14555, https://doi.org/10.1038/s41598-017-15167-1.
Nash, J. D., and J. N. Moum, 2005: River plumes as a source oflarge-amplitude internal waves in the coastal ocean. Nature,
437, 400–403, https://doi.org/10.1038/nature03936.
Nikurashin, M., and R. Ferrari, 2010: Radiation and dissipation ofinternal waves generated by geostrophic motions impinging
on small-scale topography: Theory. J. Phys. Oceanogr., 40,
1055–1074, https://doi.org/10.1175/2009JPO4199.1.
Richards, C., D. Bourgault, P. S. Galbraith, A. Hay, and D. E.
Kelley, 2013: Measurements of shoaling internal waves
and turbulence in an estuary. J. Geophys. Res. Oceans,
118, 273–286, https://doi.org/10.1029/2012JC008154.
Rudnick, D. L., and Coauthors, 2003: From tides to mixing along
the Hawaiian Ridge. Science, 301, 355–357, https://doi.org/
10.1126/science.1085837.
Shang, X., Y. Qi, G. Chen, C. Liang, R. G. Lueck, B. Prairie,and H. Li, 2017: An expendable microstructure profiler for
deep ocean measurements. J. Atmos. Oceanic Technol., 34,
153–165, https://doi.org/10.1175/JTECH-D-16-0083.1.
Sharples, J., C. M. Moore, A. E. Hickman, P. M. Holligan, J. F.
Tweddle, M. R. Palmer, and J. H. Simpson, 2009: Internal
tidal mixing as a control on continental margin ecosystems.
Geophys. Res. Lett., 36, L23603, https://doi.org/10.1029/
2009GL040683.
St. Laurent, L. C., and J. D. Nash, 2004: An examination of theradiative and dissipative properties of deep ocean internal
tides. Deep-Sea Res. II, 51, 3029–3042, https://doi.org/10.1016/
j.dsr2.2004.09.008.
Venayagamoorthy, S. K., and O. B. Fringer, 2006: Numericalsimulations of the interaction of internal waves with a shelf
break. Phys. Fluids, 18, 076603, https://doi.org/10.1063/
1.2221863.
Vic, C., and Coauthors, 2019: Deep-ocean mixing driven bysmall-scale internal tides. Nat. Commun., 10, 2099, https://doi.org/10.1038/s41467-019-10149-5.
Walsh, J. J., 1991: Importance of continental margins in the marine
biogeochemical cycling of carbon and nitrogen. Nature, 350,
53–55, https://doi.org/10.1038/350053a0.
Zhang, Z., O. B. Fringer, and S. R. Ramp, 2011: Threedimensional, nonhydrostatic numerical simulation of nonlinear
internal wave generation and propagation in the South China
Sea. J. Geophys. Res., 116, C05022, https://doi.org/10.1029/
2010JA016287.
Zilberman, N. V., J. M. Becker, M. A. Merrifield, and G. S. Carter,
2009: Model estimates of M2 internal tide generation over
Mid-Atlantic Ridge topography. J. Phys. Oceanogr., 39,
2635–2651, https://doi.org/10.1175/2008JPO4136.1.