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The Generation of Large-Amplitude Unsteady Lee Waves by Subinertial K1 Tidal
Flow: A Possible Vertical Mixing Mechanism in the Kuril Straits
TOMOHIRO NAKAMURA, TOSHIYUKI AWAJI, TAKAKI HATAYAMA, AND KAZUNORI AKITOMO
Department of Geophysics, Kyoto University, Kyoto, Japan

TAKATOSHI TAKIZAWA
Japan Marine Science and Technology Center, Yokosuka, Japan

TOKIHIRO KONO AND YASUHIRO KAWASAKI
Hokkaido National Fisheries Research Institute, Kushiro, JapanMASAO FUKASAWA
School of Marine Science and Technology, University of Tokai, Shizuoka, Japan(Manuscript received 11 August 1998, in final form 19 July 1999)
ABSTRACT
Numerical experiments with a two-dimensional nonhydrostatic model are performed to investigate tidally
generated internal waves in the Kuril Straits and their effect on vertical mixing. The results show that sill-scaleinternal waves at the K1 tidal frequency are confined to the sill slopes because the K1 tide is subinertial in the
Kuril Straits. In contrast to previous theories, the authors show that intense short internal waves generated at
the sill breaks by the subinertial K1 tidal current can propagate upstream as the tidal current slackens. Theoreticalconsiderations identify these short waves as unsteady lee waves, which tend to be trapped at the generation
region and grow into large-amplitude waves, eventually inducing vigorous mixing along their ray paths. In
particular, superposition of a propagating unsteady lee wave and a newly generated lee wave over a sill causes
significant wave breaking leading to a maximum vertical diffusivity of ;10 3 cm 2 s21 . This quite intense mixing
reaches down to the density layer of the North Pacific Intermediate Water (NPIW). In contrast, the M 2 tidal
current does not cause such strong vertical mixing, because most of generated internal waves propagate away
as first-mode internal tides and because the barotropic flow amplitude is small. The authors therefore suggest
the possibility that generation of lee waves through interactions between the K1 current and the bottom topographyof the Kuril Straits contributes to the observed modification of the Okhotsk Sea water required in the formationof the NPIW.

1. Introduction
In the Okhotsk Sea, subduction due to sea ice formation produces considerably low-salinity and recently
ventilated water at intermediate depths, and sea ice melting freshens the surface layer (Kitani 1973; Ohtani
1989; Talley and Nagata 1991). Many studies have indicated that the outflow of such remarkably low-salinity
water from the Okhotsk Sea plays an important role in
the production of the North Pacific Intermediate Water(NPIW), which is characterized by a salinity minimum(Wüst 1930; Talley 1991; Kono 1996; Yasuda 1997).

Corresponding author address: Dr. Tomohiro Nakamura, Department of Geophysics, Kyoto University, Kyoto 606 8502, Japan.
E-mail: nakamura@kugi.kyoto-u.ac.jp

q 2000 American Meteorological Society

Since the density range of the NPIW (;26.8 s u ) does
not outcrop in the open North Pacific Ocean, a driving
mechanism is required for downward motion of fresh
subarctic surface water, which in turn may induce effective subduction of other materials such as carbon
dioxide (CO 2 ). Since the NPIW is distributed in theentire subtropical gyre and even extends to the Tropics(Talley 1993), the formation mechanism of the NPIW
has important implications for the dynamics of the middepth circulation in the North Pacific and the sources
and paths through which anomalies and materials in thesurface layer are conveyed into the intermediate layer.
Therefore, the interaction process between the Okhotsk
Sea and the North Pacific needs to be clarified in order
to better understand the circulation and water propertiesof the North Pacific intermediate layer.
To assess the impact of the Okhotsk Sea on the North

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VOLUME 30

FIG. 1. Temperature sections (XBT measurements) across Bussol Strait in (a) 1991 and (b)1992 (after Kawasaki and Kono 1994). The units are degrees Celsius.

Pacific, it is necessary to study the water modification
process in the Kuril Straits in relation to the water exchange taking place between the two basins. Kitani(1973) showed that in the northern Okhotsk Sea, thefreezing of fresh surface waters can form cold saline
waters as dense as 27.05 s u , which still have lower
salinity than the ambient waters at the same density.
Associating an analysis of historical hydrographic datawith the Kitani (1973) result, Talley (1991) proposedthat the North Pacific at mid-depth is ventilated throughsea ice formation in the innermost part of the Okhotsk
Sea. From potential vorticity maps, Talley (1993) andYasuda et al. (1996) commented that the Okhotsk Sea
supplies low salinity water to the NPIW. These conclusions were also supported by recent studies. Warner etal. (1996) showed that the Okhotsk Sea is an important
location for the ventilation of the intermediate water of
the North Pacific based on chlorofluorocarbon (CFC)
observations. Yasuda (1997) and Watanabe and Wakatsuchi (1998) showed the presence of a pycnostad (i.e.,
low potential vorticity) at 26.8 s u in the Okhotsk Sea
and suggested that low potential vorticity water flowing
out into the North Pacific even determines the densityof the NPIW. In this process, turbulence and mixing is
considered to be driven by conversion of the relatively
high potential energy of the outflowing water into kinetic energy.
However, questions still remain concerning the roleof the Okhotsk Sea and its significance for the NPIW
formation. Recent detailed observations across Bussol
Strait (Kawasaki and Kono 1994), where most of theoutflow from the Okhotsk Sea is considered to occur,
have clearly revealed the presence of a deep verticallyuniform temperature distribution, leading to a horizontal
discontinuity in the temperature distribution around thesill region (Fig. 1). These facts are more likely to be
explained by vertical mixing produced by interactions
between swift tidal currents and sill topography in the
Kuril Straits and hence suggest the presence of tidal
mixing as another possible mixing mechanism. In fact,
several previous studies pointed to the importance oftidal mixing on the water modification in the Kuril
Straits (e.g., Talley 1991). However, as the actual physical mechanisms are still unknown, further investigationof the mechanism of tidal mixing at the Kurils is necessary.
According to the observations, the currents are dominated by the diurnal tidal components in the Kuril
Straits, and the semidiurnal components are rather weak(Thomson et al. 1997). Our numerical simulation ofbarotropic tides around the Kuril Straits shows that theK1 tide is predominant and induces very swift currents
of up to 1.5 m s21 (Nakamura et al. 2000). Such strong
tidal currents have been thought to cause intense verticalmixing by interactions with the large amplitude sills in
the Kuril Straits. However, this situation is out of therange of previous theories for the growth of oceanicinternal waves, which assume that an oscillating tidal
flow over an obstacle excites only internal waves at its
tidal frequency (internal tides). Since the diurnal tides
are subinertial around this high latitude (;478N), internal tides at the K1 tidal frequency are not freely propagating waves. This fact prevents us from using themodel described by Hibiya (1986), which assumes thatinternal tides propagating upstream are trapped at thegeneration region and amplified when the barotropic

JULY 2000

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NAKAMURA ET AL.

flow is critical (i.e., when the Froude number F n is unity,
where F n is the ratio of the barotropic tidal flow speed
to phase speed of the nth mode). For the same reason,interpretations of wave growth by the critical slope theory (e.g., Wunsch 1969) or the nonlinear steepening
effect (e.g., Lee and Beardsley 1974; Gerkema and Zimmerman 1995) are also inapplicable.
Thus, as a first step toward clarifying the physics
responsible for the vertical mixing in the Kuril Straits
and its influence on the North Pacific intermediate layer,
we have performed numerical simulations of tidally generated internal waves and their nonlinear evolution in
the Kuril Straits, and have estimated the vertical mixing
induced by those waves. Our work is primarily a modeling study of one possible mixing process. The relative
importance of other processes (e.g., potential energy
dissipation described above) will be assessed in later
work. For simplicity, we use a two-dimensional nonhydrostatic f -plane model with realistic topography andstratification, because inclusion of the rotation effect isnecessary to investigate the difference between the superinertial M 2 tide and the subinertial K1 tide and because nonhydrostatic models can reproduce shear instability and/or convective instability. The latter processes
are important in vertical mixing induced by waves(Scinocca and Peltier 1993), thus making the use of a
two-dimensional model meaningful, although it cannot
fully reproduce mixing because mixing is indeed athree-dimensional process. We are therefore able to estimate the tidal mixing reasonably, which is a subgridscale process and is not properly taken into account ingeneral circulation models.
The outline of this paper is as follows. In section 2,
the numerical model and parameters used are described.
The model results are presented in section 3 for the M 2
and K1 cases. In section 4, the excitation mechanism ofinternal waves is investigated by focusing on the excitation of propagating lee waves by oscillating tidal
currents over topography, and their subsequent evolution leading to wave breaking in the K1 case in section
5. The vertical mixing induced by tidally generated internal waves is estimated in section 6, and the energetics
of internal waves are discussed in section 7. Our conclusions are summarized and discussed in section 8.
2. Numerical model
The model bottom topography (Fig. 2) is representative of the sills in the northeastern part of the Kuril
Straits (in particular, the sill in Nadezhdy Strait), wherestrong tidal currents can cause considerable mixing. Although there are a variety of sill depths (ranging between
tens to several thousands m) in the Kuril Straits, thepresent sill topography is of a basic form so that simulation results can be qualitatively applied to cover othercases. Recent observations reveal that both water exchange and mixing occur between the Okhotsk Sea andthe North Pacific through the northeastern part of theKuril Straits as the western boundary current of thesubarctic gyre (the East Kamchatka Current) flows along
the Kuril Islands (Kawasaki and Kono 1994; Kono andKawasaki 1997). They also indicate that the water in
Bussol Strait, considered to be the main path of theoutflow, has already been subjected to strong verticalmixing in the northeastern part of the Kuril Straits. The
sill topography is determined according to the DBDB5
topographic data (National Geophysical Data Center,
Boulder, Colorado) with a minimum depth of 550 m at
the sill top. The bottom topography on both sides of thesill is set to be flat with a maximum depth of 2000 m.
We can adopt this flat bottom simplification since thedensity core of the NPIW (s u ; 26.8) is around a depthof 300 m.
In order to simulate vertical mixing by internal waves,
we used a nonhydrostatic model with horizontal andvertical grid sizes of 500 and 10 m, respectively. The
eddy viscosity and diffusivity coefficients are assigned
to the relatively small values of 2 3 10 5 cm 2 s21 horizontally and 0.1 cm 2 s21 vertically so that their effecton mixing is small enough to demonstrate the wave
mixing clearly. As basic forcing terms for internal wave
generation, barotropic K1 and M 2 currents are given atboth lateral boundaries. Their maximum speeds at thesill top are 0.5 m s21 for the K1 case and 0.2 m s21 forthe M 2 case, as determined from our preceding barotropic tidal simulations (details in the accompanying
paper, Nakamura et al. 2000). We took account of theeffect of rotation by adopting an f -plane approximation
to distinguish the physics of waves generated by thesubinertial K1 flow from that by the superinertial M 2
flow in the Kuril Straits. Thus the governing equationsof our two-dimensional model are
]u
]u
]u
1 ]p
]2 u
1 u 1 w 2 fy 5 2
1 nH 2
]t
]x
]z
r0 ]x
]x
1 nZ

]2 u
,
]z 2

]y
]y
]y
] 2y
] 2y
1 u 1 w 1 fu 5 n H 2 1 n Z 2 ,
]t
]x
]z
]x
]z

(1)
(2)

]w
]w
]w
1 ]p
r
1u
1w
52
2 g
]t
]x
]z
r0 ]z
r0
1 nH

]2 w
]2w
1 nZ 2 ,
2
]x
]z

]u
]w
1
5 0,
]x
]z

(3)
(4)

]u
]u
]u
] 2u
] 2u
1 u 1 w 5 kH 2 1 kZ 2 ,
]t
]x
]z
]x
]z

(5)

]S
]S
]S
]2 S
]2 S
1 u 1 w 5 kH 2 1 kZ 2 ,
]t
]x
]z
]x
]z

(6)

r 5 r (u, S, p),

(7)

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where the relative vorticity z and the streamfunction c
are defined, respectively, as

z5FIG. 2. The model topography, representing the vertical cross section of a strait in the northeastern part of the Kuril Islands, wheretidal currents are strong enough to cause considerable mixing.

where (x, y, z) are the across-sill, along-sill, and verticalcoordinates, respectively; (u, y , w) are the velocity inthe (x, y, z) directions; p is pressure; r is density variation; u is potential temperature; S is salinity; f (51.07
3 1024 s21 ) is the Coriolis parameter at 478N; r 0 is thereference density; g is the gravity acceleration; and n
and k are the coefficients of eddy viscosity and eddy
diffusivity, respectively, with the subscripts H and Z for
the horizontal and vertical direction. The equation ofstate used is based on Bryan and Cox (1972). In performing the model calculation, we solve the vorticity
equation derived from Eqs. (1) and (3) as
]z
]z
]z
]y
1u 1w 1 f
]t
]x
]z
]z
52

g ]r
] 2z
] 2z
1 nH 2 1 nZ 2
r0 ]x
]x
]z

(8)

and the Poisson equation for the stream function c as
] 2c
] 2c
1 2 5 z,
2
]x
]z

(9)

]w
]u
2 ,
]x
]z

u52

]c
,
]z

(10)
w5

]c
.
]x

(11)

At the bottom boundary, a no-slip condition is imposed in the sill region and a free-slip condition is imposed in the deep region with the flat bottom. A rigidlid approximation is used at the surface. Although thetime variation of the tidal flow in the model occurs
completely in phase throughout the entire domain as aresult of the rigid lid approximation and the incompressibility assumption, the approximation is still valid
because we are concerned only with the internal wave
generation and resulting mixing over a sill with horizontal scale of ;100 km so that the phase of the barotropic tidal flow varies only slightly over this distance.
In fact, the incoming K1 tidal waves take at most 10
minutes to propagate across the sill region, much smaller
than the K1 period. For this reason, many previous studies such as Hibiya (1988) and Lamb (1994) successfully
reproduced topographic internal wave processes using
the rigid-lid approximation. To prevent artificial wave
reflection at the two side boundaries, the horizontal eddy
viscosity is gradually increased near the lateral boundaries up to a value of 6 3 10 6 cm 2 s21 .
The basic vertical profiles of potential temperatureand salinity (Fig. 3) in the dataset of the U.S. National
Oceanographic Data Center correspond to a summertime climatology in the Kuril Basin of the Okhotsk Sea.
Crudely speaking, this stratification is thought to provide a lower estimate of the effect of tidal mixing on
intermediate layer freshening because the water out-

FIG. 3. Initial vertical profiles of (a) potential temperature and (b) salinity, which correspondto summer time climatology in the Kuril Basin of the Okhotsk Sea, determined from the datasetof the U.S. National Oceanographic Data Center.

JULY 2000

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FIG. 4. Streamfunction and current vectors obtained by setting g 5 0 in the K1 case after 1.25 periods (themaximum rightward flow). The contour interval is 5.0 3 10 5 cm 2 s21 .

flows from the Kuril Basin to the North Pacific andbecause salinity in the surface layer is lowest in summertime, producing a strong stratification (certainly, thisis more in the realm of an educated guess). It is notparticularly clear how the generation of different waves
and their different propagation due to the difference instratification would impact on the amount of mixing.
With these vertical profiles, the initial stratification is
set to be horizontally uniform. Model calculations startat the beginning of rightward flow (i.e., from the North
Pacific to the Okhotsk Sea) so that wave generation
processes can be seen clearly. According to Lamb
(1994), the along-sill velocity is initialized as y 5
f s 21
f U 0 (x) where U 0 (x) and s f are the maximum acrosssill velocities and the frequencies of the barotropic tides,
respectively.
3. Model results
We first describe the behavior of internal waves in
one period, using an internal mode streamfunction, defined as the difference between the original solutions
and the solutions obtained by setting g 5 0. Since thesolution for g 5 0 gives the barotropic flow, which acts
as the forcing (Lamb 1994), internal mode processes
can be seen in terms of internal mode streamfunctions.
This is not exactly true, since the g 5 0 flow satisfies
the no-slip condition at the sill by viscous effects,
whereas in the full solution this boundary condition issatisfied by a combination of viscous effects and internal
waves. Hence, the viscous effects are not the same in
the two cases, and hence internal mode streamfunctions
do not represent the true baroclinic response but also
include some reminiscent effects of viscosity from theg 5 0 case. However, since the thickness of the bottom
Ekman layer [;(n z / f )1/2 ] is less than 10 m in the present
case, the effect of a no-slip condition is significant in
only a few grids from the bottom. Thus, the current fieldof the g 5 0 solution in the K1 case at the time of

maximum rightward flow (Fig. 4) is vertically uniform
except very near the bottom. Therefore, the baroclinic
response can be fairly well described by internal mode
streamfunctions.
a. The M 2 case
Figure 5 shows the internal mode streamfunction
forced by the barotropic M 2 flow during the second
tidal period when the initial spinup processes are almost complete. As previous studies show (e.g., Rattray
et al. 1969; Baines 1982; Craig 1987), first-mode internal waves are generated on the slopes of the sill
after 1.25 periods (after an interval of time corresponding to 1.25 periods) at the time of the maximum rightward flow (Fig. 5a) and propagate away from the sill.
The first-order modal waves outside of these cells weregenerated at the sill in the last half period and havepropagated away. Wave generation on the sill slopesby rightward flow continues for up to 1.5 periods when
the flow stops (Fig. 5b), and the generated waves propagate away from the sill and reach x . 660 km after
1.75 periods. In the half period of leftward flow (Figs.
5c,d), almost the same sequence of events can be observed, but with their phase reversed.
From Fig. 5, the horizontal and vertical wavelengthsof the first-mode waves and the squared buoyancy frequency N 2 (averaged over whole depth) are about 100
km, 4 km, and 5 3 1026 s22 , respectively. Using thesevalues and the linear dispersion relation for frequencys and horizontal and vertical wavenumbers k and m,

s2 5

N 2k2 1 f 2m2,
k2 1 m2(12)

the frequency of first-mode waves is estimated as 1.4
3 1024 s21 , almost equal to the M 2 frequency. Theirhorizontal phase speed estimated from the dispersion
relation is 2.2 m s21 , consistent with that of 2.1 m s21
estimated from the figure. Thus the first-mode waves

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FIG. 5. The time series of the internal mode stream function in the M 2 case during the second
tidal period after (a) 1.25, (b) 1.5, (c) 1.75, and (d) 2.0 periods. The contour interval is 1.0 3
10 5 cm 2 s21 . Values in nonshaded areas and the thicker shaded areas are positive and negative,
respectively, and the absolute values in the thinner shaded areas are less than 1.0 3 10 4 cm 2 s21 .

seen in Fig. 5 can be identified as typical internal tides
at the M 2 frequency.
As described above, most of internal mode energy
generated by the M 2 flow propagates away as first-mode
internal tides, and dissipates gradually. In addition, since
the amplitude of the barotropic M 2 flow is small, largeamplitude internal waves are not formed and breaking
is absent. Consequently, vertical mixing induced bywaves generated by the M 2 flow is probably not strong
enough to cause significant freshening in the Kuril
Straits. This is confirmed later in section 6.

b. The K1 case
The time series of the internal mode streamfunctionin the K1 case (Fig. 6) shows a quite different behavior
from that of the M 2 case since the Kuril Straits arelocated north of the critical latitude for the K1 tide andbecause the amplitude of the barotropic K1 current islarger than that of the M 2 current. For example, sillscale cells (cells on the scale of the sill topography) do
not propagate away, and, in contrast to the M 2 case,intense disturbances exist on small scales.

JULY 2000

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FIG. 6. Same as in Fig. 3 but for the K1 case.

First we describe the time evolution of the sill-scale
cells. Although the K1 tide is subinertial and has no
characteristic curves of propagation, clockwise cells areproduced on both sides of the sill after 1.25 periods (themaximum rightward flow). These cells almost vanish at
the end of rightward (leftward) flow. Thus the sill-scale
cells have the same frequency as the tidal frequency anddo not propagate away because the K1 tidal period islonger than the inertial period at the Kurils. The amplitudes of these cells are larger than those in the M 2case since the barotropic flow is stronger for the K1 case(note that the cell on the right-hand slope of the sill isslightly stronger than that on the left since the steeper
slope produces a larger forcing of internal motion).
Therefore, their effect on mixing is expected to be morevigorous and confined closer to the sill region than in
the M 2 case. Note that problems associated with theartificial lack of K1 related longitudinal radiation withina context of a two-dimensional model will be discussed
later.
Next, we direct our attention to intense smaller-scaledisturbances detected in Fig. 6. These short waves may
cause considerable mixing because the process of wave
breaking takes place at smaller amplitudes for waves of1608

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shorter wavelength. After 1.25 periods, the superposition of small-scale disturbances makes sill-scale cells
wavy and distorted. In Fig. 6, such small-scale disturbances are labeled as A1 , B1 , A 2 , B 2 , a1 , and b1 fromleft to right, with the tidal period during which they are
produced indicated by subscripts and with the directionof the tidal flow at the time of their production indicatedby capital (rightward) and small (leftward) letters. At
the sill top, rightward flow produces disturbances A 2
and B 2 . The leftward flow in the last half period has
produced distortions a1 and b1 , which are now on theright side of the sill, whereas disturbances A1 and B1
were produced by rightward flow in the previous period.
These small-scale disturbances are clearly distinguished,as sill-scale cells disappear after 1.5 periods (Fig. 6b).
While B 2 stays at almost the same place, A1 , B1 , andA 2 move leftward, and a1 and b1 move rightward andare gradually dispersed. They continue to move until
they reach one of the side boundaries where they thenvanish. In the next half period, distortions a 2 and b 2 areproduced by leftward flow and remain around the sill
top until the end of the flow, in which B 2 at the sill top
is superposed on a 2 .
The propagation speeds of these small-scale disturbances estimated from Fig. 6. (0.3;1.4 m s21 ) are different from the tidal flow speed (less than 0.125 m s21
in the deep water region). This fact means that they arefreely propagating internal waves. Thus, their frequencies must be larger than the inertial frequency, despitethat the K1 frequency is subinertial. Therefore, previous
theoretical models cannot explain the generation of suchintense short waves because these models assume thatan oscillating tidal flow over topography can excite onlyinternal tides corresponding to the tidal frequency. Furthermore, these features are produced by rightward (leftward) flow and propagate leftward (rightward), indicating that they move only in the upstream direction as
determined at their generation time. This is also inconsistent with the character of internal tides that propagate
in both directions. From these considerations, it is inferred that the intense short waves in our model must
represent another class of wave. In fact, the above twoqualities are characteristics of lee waves, and thus wereinvestigate the excitation mechanism of internal waves
by a tidal flow in the next section to show that unsteadylee waves (which are capable of propagating away) canbe excited by an oscillating flow.
4. The excitation mechanism of unsteady lee waves
A tidal flow over an obstacle produces vertical currents whose time variations excite internal waves. To

investigate the wave excitation properly, it is necessary
to consider the total time-variation of the forcing towhich fluid parcels are subjected. Representing the forcing as F, the total time variation of the forcing DF/Dt
is given in Eulerian variables as
DF
]F
]F
5
1 Ui ,
Dt
]t
]x i

(13)

where U i represents the velocity of the basic flow in thex i direction. The first term on the right-hand side expresses the fact that the local time variations of forcing
are capable of exciting internal waves. When the forcing
is given by an oscillating tidal flow, this effect excites
only internal tides of the same frequency as the basic
flow. On the other hand, the second term, which represents the effect of advection, shows that the spatial
variations of forcing in the frame fixed to the groundcan also excite internal waves. This is because there is
the temporal variation of forcing in the frame moving
with the fluid at speed U i in the presence of the spatial
variations of forcing in the fixed frame.
Most of the previous theories of internal waves generated by an oscillating tidal flow have neglected theadvection effect in wave excitation, assuming that the
tidal excursion is infinitesimal compared with the horizontal scale of the spatial variations of the forcing (i.e.,a linear approximation). However, Eq. (13) shows thatin regions where the tidal flow is strong and the horizontal scale of forcing is small, such as sill tops or shelf
breaks, wave excitation cannot be fully taken into account unless the advection effect on wave excitation isincorporated. In fact, steady lee waves, which have quite
different properties from internal tides, are generated
from the advection effect as investigated in detail byatmospheric physicists (e.g., Queney 1947; Scorer 1949;
Long 1953). However, the detailed dynamics of internal
waves in the presence of both effects is still poorly
understood. For this investigation, a description in themoving frame is appropriate to reveal the physics ofinternal wave excitation. Hence in the following description, frequency, phase velocity, and group velocity
denote quantities in the moving frame.
To specify the nature and structure of tidally generated internal waves, we make our discussion correspondto the numerical model result performed in the vertical
two-dimensional case (]/]y 5 0). For simplicity, we
assume that the flow varies as a single harmonic component of frequency s f in time and that the amplitudeof the topography is small in comparison with its horizontal scale; then the vertical velocity W at the bottom
is given by←
FIG. 7. The evolution of vertical velocity (w) in the K1 case during the first period after (a) 0.125, (b) 0.25, (c) 0.375, (d) 0.5, (e) 0.625,(f ) 0.75, (g) 0.875, (h) 1 period. The contour interval is 0.5 cm s 21 . Values in nonshaded areas and the thicker shaded areas are positive
and negative, respectively, and the absolute values in the thinner shaded areas are less than 0.1 cm s 21 .

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W 5 W(x, h(x), t) 5 e isf tE`ˆ 0 (k)e 2ikx dk,
W

(14)

2`

E

t

U(x 0 , t) dt . X 1 U(x 0 , t 0 )(t 2 t 0 ).

(15)

t0

using a Fourier transformation. We can describe thewave forcing (14) in the neighborhood of x 5 x 0 and t
5 t 0 in the moving frame X using the first-order approximation in a Taylor expansion:W(x, t) 5

x5X1

VOLUME 30

EThis is equivalent to a local Doppler shift in time andspace. Substituting this into (14) and transforming thevariable t 2 t 0 to t , we obtain`ˆ 0 (k)e isf t0 exp[2i(kX 1 (kU(x 0 , t 0 ) 1 sf )t )] dk,
W

(16)

2`

which splits into the sum of travelling waves propagating with two different phase velocities (corresponding toupper and lower signs) as
W(x, t) 5E`ˆ 0 (6k)e isf t0 exp{7i[kX 1 (kU(x 0 , t 0 ) 6 sf )t ]} dk.
W

(17)

0

This expression shows that the internal wave excited in
the neighborhood of x 5 x 0 and t 5 t 0 by the basic flow
(U, W) is composed of a sum of monochromatic waves
with frequency of 2kU(x 0 , t 0 ) 6 s f and horizontal
wavenumber of k. Their vertical wavenumber m is determined so as to satisfy the governing equations andthe condition that their energy propagates upward. The
total internal wave field generated by a tidal flow overtopography is formed by the superposition of these
monochromatic waves excited at various times and positions.
The frequencies of the monochromatic waves consistof the term kU(x 0 , t 0 ), which arises from the spatial
variation of forcing and relates to the steady lee waves,and of terms 6s f , which stem from the time variationof the tidal flow and represent the internal tides. Therefore, according to the nondimensional parameterkU 0 /s f , we can classify internal waves excited by an
oscillating flow into unsteady lee waves (when kU 0 /s fk 1) and ‘‘mixed tidal lee waves’’ (when kU 0 /s f ; 1),
in addition to the internal tides (when kU 0 /s f K 1)
considered in previous theoretical models. This nondimensional parameter is a ratio of the effect of spatial
to temporal forcing variations.
These three types of ‘‘topographic internal waves’’
have the following properties: Unsteady lee waves include classical steady lee waves in the limit of kU 0 /s f
→ `. Their frequencies are 2kU(t), and their phasevelocities are 2U(t). Thus, unsteady lee waves can propagate in the upstream direction at their excitation time(in the upstream sense as it exists at the time of their
excitation), relative not only to the fluid but also to theground as the flow varies in time and/or space. The
amplitudes of unsteady lee waves depend on the mag-nitude of the forcing at the time of their excitation sothat the waves generated around the time of maximum
flow have maximum amplitudes. Such unsteady lee
waves are excited in a region where the curvature of
topography is sufficiently large, such as shelf breaks. In
contrast, internal tides have a frequency of s f and aconstant phase velocity of 6s f /k, and propagate in both
directions, as shown in previous studies. In the intermediate range, excited waves have intermediate properties between those of lee waves and internal tides, andwe name such waves ‘‘mixed tidal lee waves.’’
Bell (1975) considered a similar situation to ours.
However, since the waves in his treatment can be, in asense, regarded as an infinite series of modes in thetemporal domain, the unsteady lee waves and mixed
tidal lee waves newly proposed here are not clearly
described in his model. Consequently, his approach
might be inappropriate to identify the nature of thesewaves. On the other hand, Lott and Teitelbaum(1993a,b) studied atmospheric lee waves using a similar
approach to ours. Though they described many interesting features of unsteady lee waves, the effect of timevarying forcing on wave excitation [i.e., the first termon the right-hand side of Eq. (13)] was neglected, because tidal forcing is out of scope in most atmosphericinternal waves, and hence the existence of mixed tidal
lee waves could not be discussed.
In fact, consideration of the frequency of these newwaves allows us to explain the reason why the subinertial K1 flow can generate freely propagating internal
waves at the sill top. Figure 7c shows the distributionof vertical velocity after 0.375 period, when waves excited at the bottom around the time of maximum flow
have already propagated upward. The horizontal andJULY 2000

NAKAMURA ET AL.

vertical wavelengths of the intense short wave formedat the break on the right side of the sill can be estimatedas 8 km and 800 m, respectively. Accordingly, the nondimensional parameter kU 0 /s f becomes 5.4, indicatingthat the monochromatic components constituting thiswave can be classed as mixed tidal lee waves but arevery close to unsteady lee waves. An estimate of its
frequency based on Eq. (17) yields 3.9(60.73) 3 1024
s21 , while an estimate based on the dispersion relation(11) becomes 4.6 3 1024 s21 using N 2 5 2 3 1025 (s22 )averaged from the surface to the sill top. The comparison shows good agreement between our theoreticalmodel and the numerical model result. In addition, Fig.
7c shows that the cophase lines of the internal wave
generated at the right-hand break extend only in one
direction, confirming that the phase of these intense
short waves propagate only in the upstream direction attheir generation time, with little, if any, component
propagating downstream. These features are consistent
with the character of unsteady lee waves.
5. The evolution of large-amplitude unsteady lee
waves
Figures 7 and 8 show vertical velocity (w) and thepotential density (s u ) in the K1 case, respectively, up to
one period, when the generation process of the intense
short waves can be seen clearly.
After 0.25 period, the barotropic K1 current flows
rightward with a maximum speed of 0.5 m s21 . The
vertical velocity induced by this flow is up(down)ward
on the up(down)stream side of the sill, and its magnitudeis larger at the sill breaks where changes in the sill slope
are rapid and the horizontal flow is strong because of
the shallow water around the sill top. As expected fromour theoretical model, small-scale vertical velocity distributions produced at sudden changes in the sill slope
generate a relatively large depression of small horizontal
length at the right-hand break and adjacent weak elevation and depression over the left-hand break and sill
top. These features are consistent with the intense shortinternal waves described in section 3b.
The displacement at the right-hand break continues
to grow until the end of the rightward flow to form alarge amplitude (;100 m) unsteady lee wave (Fig. 8d).
Similar wave growth in the decelerating stage is often
observed both in field and laboratory studies (e.g., Lansing and Maxworthy 1984). On the other hand, the displacements at the left-hand break do not grow appreciably and propagate somewhat upstream as is also seenin the vertical velocity distribution. It is noteworthy that,while the lower part of the wave stays at almost thesame place, the upper part propagates upstream (leftward) and the wave leans toward the upstream side. The
induced cophase line of vertical velocity also tilts
obliquely, indicating that modal waves have not yet been
formed. Therefore, the modal wave approximation usedin the Hibiya (1986) model is not appropriate for un-

1611

derstanding the amplification mechanism of this wave.
A detailed discussion of the wave growth mechanism
is given in another paper (Nakamura and Awaji 2000,
manuscript submitted to J. Phys. Oceanogr.).
As the rightward flow vanishes after 0.5 period, thelarge-amplitude unsteady lee wave begins to propagate
in the upstream direction at the generation time (leftward) (e.g., Fig. 7). Its speed of motion relative to theground becomes faster as the barotropic flow accelerates
to the left. Such propagation processes of short internal
waves generated on the downstream side of a sill areconsistent with observations (e.g., Farmer and Smith
1980). As the upper part of this unsteady lee wave isreflected at the surface, there appears a third modelike
structure in the wave field over the sill top after 0.625
period (Fig. 7e). Similar higher-order modal waves generated at abrupt topographic changes are often observed
around shelf breaks (e.g., Kuroda and Mitsudera 1995).
The correspondence with observations indicates that theprocesses described above are not artifacts of our numerical model but are common phenomena in real seas.
When propagating across the sill, this large amplitude
wave encounters an unsteady lee wave newly generated
at the left-hand break by the leftward flow (after 0.75
period). This is clearly seen in the distribution of verticalvelocity (Fig. 7f), where the propagating lee wave (athird-modelike structure) is located on the right side ofdownwelling associated with the new unsteady lee
wave. These lee waves are superposed and interact witheach other (after 0.875 period). This event produces astrong bottom-intensified current over the sill top(shown in Fig. 9) and a large amplitude isopycnal elevation at the foot of the unsteady lee waves after 1.0
period (above the left-hand break in Fig. 8h).
Figure 10 shows the subsequent evolution of isopycnal displacements. As the flow turns to the right, thelarge amplitude elevation at the foot of the lee wave
gradually breaks (density inversion is apparent in Fig.
10a), propagating rightward, and causes vigorous vertical mixing over the sill top after 1.125 periods. As thislarge amplitude elevation passes over the right-handbreak, it produces a strong overflow accompanied by astrong bottom-intensified current, which may strengthen
the excitation of unsteady lee wave at the right-handbreak (after 1.25 periods). In this superposition of newly
generated and rightward propagating lee waves, the raypaths form a distinct V shape (Fig. 10c), as in the casewith internal tides. Note that unsteady lee waves canpropagate only to one side in contrast to internal tides
that can propagate towards both sides. The superpositionof the unsteady lee waves forms a large amplitude elevation like a hydraulic jump, inducing intense verticalmixing (density inversion is apparent after 1.375 periods). Again, an unsteady lee wave grows into a large
amplitude wave at the right-hand break as the barotropic
K1 current flows rightward (after 1.5 periods), showingthat the lee wave response is not an artifact of the initial
spinup of the model. In this way, large-amplitude un-

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JULY 2000

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NAKAMURA ET AL.

steady lee waves, as shown in Figs. 8 and 10, induce
intense vertical mixing, and relatively vertically uniform
water is produced over and around the sill top in the K1
case.
Figure 11a shows an observed potential temperature
distribution across Kruzenshterna Strait in the northernpart of the Kurils where a large amplitude sill is present(Fig. 11c). The duration time of the observation (XBT
measurement by the Hokkaido National Fisheries Research Institute in Japan) is about 2.5 hours so that it
roughly represents a snapshot. Comparison between theobserved and modeled results is hampered by the factthat the actual sill is deeper (;950 m) than the sill depth
in the model (;550 m) and the temperature field is
influenced by the temperature difference between the
Okhotsk Sea (left side) and the North Pacific (right side).
However, this time series is the best available experimental dataset in this region. A conspicuous feature in
Fig. 11a is the presence of large-amplitude short internal
waves in the surface layer above the sill, which may
cause the significant disturbance of the temperature distribution. Moreover, the elevation and depressions ofisotherms around the sill suggest their presence even at
mid depth. To see this more clearly, we attempted to
identify the ray paths of these waves. Firstly, we applieda bandpass filter to the horizontal distribution, whose
length scale roughly corresponds to a lee wave length(2;15 km). Then, assuming a linear relationship between the vertical displacement of isotherms (h9) andthe perturbation from the basic temperature field (u9)similar to Eq. (21) (shown later), we calculated the vertical displacement from the bandpassed potential temperature. In order to avoid erroneous displacements arising from dividing by ]u/]z, we derived the signal of
downward and upward displacements in terms of|]u/]z| h9 [h9 . 2u9/(]u/]z)] (Fig. 11b). In Fig. 11b, it
can be seen that a ray path extends upward from close
to the sill top to regions with significant wavy patterns
in the surface layer, consistent with our model conclusions. As a result, we consider the possibility that unsteady lee waves are contributing to the mixing process.
6. Vertical mixing induced by internal waves
The numerical model results show that large amplitude internal waves produced in the K1 case can cause
vigorous vertical mixing over the sill. Here, we estimate
the intensity of the resulting vertical mixing in terms ofdiffusivity. To relate the result to the parameterizations
used in general circulation models, we first estimate thevertical diffusivity K Ez from an Eulerian point of viewusing the following Fickian formulation,

FIG. 9. The velocity vector around the sill top in the K1 case after
0.875 period, when a bottom intensified current over the sill top is
produced.

) @ )K zE 5 S9w9

dS0
,
dz

(18)

where dS 0 /dz is the vertical gradient of the initial salinity
profile and S9w9 is salinity flux induced by perturbations, averaged over one tidal period. Figure 12 showsthe estimated diffusivity using the calculated velocity
fields in the second period. In the M 2 case, vertical
diffusivity is not so large (;10 cm 2 s21 ), as expected
in section 3a. In contrast, the value in the K1 case isvery large around the sill (;10 3 cm 2 s21 ) and relatively
small in other regions.
However, in terms of the Fickian formulation the estimated vertical diffusivity depends fairly strongly on
the vertical gradients of salinity used. Therefore, we
next estimate the intensity of vertical mixing from a
Lagrangian point of view. For this, we tracked numerous
labeled particles in the calculated velocity field in thesecond period and estimated the vertical diffusivity
K Lz fromK zL 5 0.5 3

dF
F 2 F0
. T
,
dt
2T

(19)

where F is the variance of the particle spread, F 0 and
F T are the initial (after one period) and the final (after
two periods) variance, respectively, and T is the tidal
period (Awaji 1982). This parameter represents the apparent diffusivity due to the distortion of water particle
distributions, though the value varies somewhat with
the initial phase of velocity fields used in the calculation.
The estimated apparent diffusivity is shown in Fig.
13. Vertical mixing induced by the M 2 current is relatively strong along the characteristic curves of M 2 in-←
FIG. 8. The evolution of potential density (s u ) around the sill top in the K1 case during the first period after (a) 0.125, (b) 0.25, (c) 0.375,(d) 0.5, (e) 0.625, (f ) 0.75, (g) 0.875, (h) 1 period, when the generation process of the large-amplitude unsteady lee wave can be clearly
seen. For s u , 26.5, the contour interval is 0.1 s u . For s u $ 26.5, the contour interval is 0.02 s u .

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ternal tides due to the significant velocity and is widelydistributed, reflecting the character of internal tides.
However, its magnitude may be too small to cause theobserved water modification in the Kuril Straits. In contrast, the diffusivity in the K1 case is very large but
concentrated near the sill. For example, the values are;10 2 cm 2 s21 around the sill top, the maximum beingof the order of 10 3 cm 2 s21 . This is mainly due to largeamplitude unsteady lee waves propagating over the sill
and their superposition on unsteady lee waves newly
generated by the reversed flow leading to wave breaking. Although the magnitude is smaller than that of the
Fickian diffusivity, it is still much larger than that in
the open oceans (about 10 3 times larger). Because suchintense vertical mixing takes place repeatedly, relatively
vertically uniform water down to the depth of the density
core of the NPIW (s u ; 26.8) can be produced over
and around the sill top. Indeed, after six periods, thecalculated potential temperature has a relatively homogeneous distribution over the sill (Fig. 14a). Similar
distributions can be found in the shallow region aroundthe Kuril Straits (region A1 in Fig. 14b), over a shallowbank just offshore of the Kruzenshterna Strait. In addition, in the Kruzenshterna Strait (region A 2 ), there are
relatively fewer warm/cold patches as compared withthe adjacent lateral regions. This implies the presenceof relatively strong vertical mixing there, consistent withthe ray path analysis described earlier. Therefore, it issuggested that vertical mixing from large-amplitude unsteady lee waves generated by the interaction of the K1
current with sills is one possible mechanism contributing to the observed water modification, and in particular its refreshening in the Kuril Straits.
7. Energetics
It is instructive to examine the energetics of the calculated field in order to understand the energy conversion from the input K1 flow to the internal waves andthe subsequent intense vertical mixing described in thepreceding section. In doing so, it is convenient to use
the velocity field obtained from the g 5 0 solution,(U, V, W), as the barotropic flow. Since the velocity fieldobtained from the difference between the original solution and the g 5 0 solution, (u9, y 9, w9), gives approximately the baroclinic response, as discussed in section 3. The energy equation for the uw component ofkinetic energy [KE uw 5 r 0 (u9 2 1 w9 2 )/2] is derived bymultiplying Eqs. (1) and (3) by u9 and w9, respectively,
and adding them together. Similarly, the energy equation
for the y component of kinetic energy (KE y 5 r 0 y 9 2 /2)
is derived by multiplying Eq. (2) by y 9. [Note that the←
FIG. 10. The subsequent evolution of potential density (s u ) aroundthe sill top in the K1 case, after (a) 1.125, (b) 1.25, (c) 1.375, (d) 1.5
periods. The contours are the same as in Fig. 8.

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NAKAMURA ET AL.

terms in the momentum equation for (U, V, W), such as
U t 1 UU x 1 VU y 1 WU z , drop out since (U, V, W) is
the solution for the g 5 0 case.] The energy equation
for the available potential energy (APE) is derived bymultiplying the advective–diffusive equation for potential density]r]r]r
1 u 1 w 5 dissipation
]t
]x
]z

(20)

by gh, where h is the vertical displacement of isopycnals. The dissipation in Eq. (20) represents an effect ofdiffusion terms in the equations for both potential temperature u and salinity S. In the calculations, we use thelinear approximation for h and APE (Gill 1982):

h5

g r9
,
r0 N 2

(21)

APE 5

1 g 2 r 92,
2 r0 N 2(22)

where r9 is the perturbation from the initial density field.
Integrating over the domain with periphery l (from x 5
215 km to x 5 10 km and from surface to 800-m depth
in the sill region) using the operator ^ & 5 ## dx dz, we
obtain^KE uw & t 5 CP→K 2 Cu→y 1 Buw 1 R uw
1 ^dissipation&,^KE y & t 5 Cu→y 1 By 1 R y 1 ^dissipation&,

(23)
(24)

^APE& t 5 CB→P 2 CP→K 1 BAPE 1 CN(z)1 ^dissipation&,

(25)

where
CB→P 5 ^r 9gW&,(26)

CP→K 5 ^2r 9gw9&,(27)

Cu→y 5 ^2r0 fu9y 9&,(28)

Buw 5 2[(u9n 1 Un )KE uw 1 u9n p] dl,(29)

By 5 2(u9n 1 Un )KE y dl,(30)(u9n 1 Un )APE dl,

(31)

R
R
R

BAPE 5 2

R uw 5 2r0 ^u92 Ux 1 u9w9(Uz 1 Wx ) 1 w92 Wz &,(32)←
FIG. 11. (a) An observed potential temperature distribution acrossKruzenshterna Strait (in 1997). Contour interval is 18C. (b) The signalof upward (black) and downward (white) displacements of isotherms
from the bandpassed potential temperature. (c) The topography in
Kruzenshterna Strait.

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FIG. 12. Vertical diffusivity estimated from the Eulerian point of view (Fickian) for (a) the M 2case and (b) the K1 case, using the calculated velocity fields in the second period. Contours arelogarithmic (101 , 10 2 , 10 3 cm 2 s21 ).

R y 5 2r0 ^u9y 9Vx 1 y 9w9Vz &,7

CN(z) 5 2 (w9 1 W )APE

(33)

8

1 dN 2.
N 2 dz(34)

Here, C B→P is the potential energy supply from the barotropic flow. The term C P→K is the energy conversion
from ^APE& to ^KE uw &, whereas C u→y is the conversion
from ^KE uw & to ^KE y & due to rotation effects. The quantities u9n and U n are the velocity components normal tothe boundary of the domain considered; B uw is a boundary flux term for ^KE uw & with a pressure term, and B yand BAPE are boundary flux terms for ^KE y & and ^APE&,
respectively. The terms R uw and R y are the Reynolds
stress for the (u9, w9) and y 9 components, respectively,
and represent the kinetic energy supply from the barotropic flow. The term C N(z) , arising from the verticalvariation of N 2 , can be interpreted as a correction term
for the linear approximation of h. Adding Eqs. (23),(24), and (25) gives the temporal change of the totalenergy within the domain and shows that it is determined by the potential and kinetic energy supply, theboundary flux, and the dissipation.
Figure 15 shows the temporal changes of the energybalances for ^KE uw &, ^KE y &, and ^APE& up to the second
tidal cycle. The value of C N(z) is relatively small throughout the period, showing that the linear approximation
for h works well. The energy flow responsible for thegeneration of internal waves is clearly seen during the
spinup process in the first half period. As the barotropic
flow accelerates, the potential energy supply from thebarotropic flow C B→P increases ^APE& (Fig. 15c), andthe resulting internal waves release their available potential energy ^APE& into the uw-component kinetic energy ^KE uw & through the energy conversion term C P→K(Fig. 15a). While most of the energy converted into
^KE uw & (i.e., C P→K ) is compensated by the boundary flux
B uw , the residual can increase the value of ^KE uw &, andis then converted to the y -component kinetic energy^KE y & by the Coriolis effect through the energy conversion term C u→y (Fig. 15b). Close examination showsthat B uw is determined primarily by the pressure term at
the lower boundary (depth 800 m) of the domain definedin Eq. (29) (not shown). Since vertical velocity associated with unsteady lee waves is mostly confined above
700 m (Fig. 7), this compensation of C P→K by B uw mayarise from a large outflux of the energy by internal tides.
This in turn indicates that the net effect of internal tides

JULY 2000

NAKAMURA ET AL.

1617

FIG. 13. Vertical diffusivity (cm 2 s21 ) estimated from the Lagrangian point of view for (a) the M 2 case and (b) theK1 case, using the result of particle tracking in the calculated velocity fields for the second period.

on the accumulation of ^KE uw & is small in the sill top
region. Thus, the increase of ^KE uw & around the sill isa direct consequence of the generation of unsteady lee
waves.
After 0.5 period, the curves of the energies begin tofluctuate with a period shorter than the inertial period.
This is more marked in ^KE uw & and ^KE y &, and interestingly, both temporal changes are greatly influenced
by the energy conversion due to rotational effects C u→y .
Such fluctuations are attributed to the propagation of
unsteady lee waves from the sill, which is clearly seenin Fig. 7. If unsteady lee waves were trapped in thegeneration region, their frequency in the frame fixed to
the ground is almost the same as the tidal one. In contrast, since lee waves are inertia–gravity waves, the velocity field of propagating unsteady lee waves can varywith a period shorter than the inertial period due to the
Coriolis effect (i.e., C u→y ). This result indicates that
propagating lee waves as well as newly generated unsteady lee waves have a significant impact on the energybalances around the sill top.
The increase of ^KE uw & after about 0.75 period (and1.75 period) in Fig. 15a corresponds to the formationof the strong bottom-intensified current over the sill top(Fig. 9). Figure 15a shows that this increase of ^KE uw & t

also takes place through the Coriolis effect. Figure 7f
confirms unsteady lee waves passing over the sill at thismoment, implying that the bottom-intensified current is
produced by the lee waves. The significant energy conversion from ^KE uw & to ^KE y & and ^APE& after around
one period (also around two periods) is associated withthe generation of large amplitude isopycnal elevation
by superposition of propagating and newly generated
lee waves at the left-hand break (Fig. 8h), which leads
to wave breaking. After around 1.3 period, as the lee
waves generated by the leftward flow propagate rightward and exit the domain, the boundary flux terms B uw
and B y appreciably decrease, and the enhanced overflow
at the right-hand break (Figs. 10b–d) prolongs the duration of the energy conversion from ^APE& to ^KE uw &in the second tidal cycle as compared with that in thefirst cycle (Fig. 15c), thus contributing to the enhancement of internal wave fields.
The mean energy flow in the second tidal period isshown in Fig. 16. This also indicates weak interactions
between ^APE& and ^KE uw & due to the long period of
the K1 tide. The temporal change of ^APE& is 0.4 kJ m21
s21 on average and the net increase of ^APE& is 40 MJ
m21 . Most of this large increase of ^APE& is thought tobe due to the formation of the relatively uniform density

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FIG. 14. (a) The calculated potential temperature distribution after
six periods (contour interval is 0.28C) and (b) the observed potential
temperature distribution in Kruzenshterna Strait (A 2 ) together witha shallow bank (A1 ), observed by the Hokkaido National Fisheries
Research Institute in 1994. Dashed lines indicate isopycnal surfacesof 145, 125, and 100 cl/ton (26.6, 26.8, 27.1 s u ).

structure over the sill top in the model. However, theabove energy equations do not include a considerationof energy radiated through the longitudinal boundaries,
so that further simulations tested through detailed comparison with experimental observations are required to
fully clarify the energetics in the real Kuril Straits. For
example, three dimensional effects enable the K1 internal tides to propagate along the isobaths (e.g., Brink1989). In addition, the presence of vertically well-mixed
water around the sill due to tidal mixing may affect theintensity of internal wave generation.
8. Summary and discussion
The Okhotsk Sea has been thought of as the originof low salinity in the North Pacific intermediate layer
whose manifestation is the North Pacific Intermediate
Water (NPIW). In order to better understand this process, we need to clarify the mechanism responsible forthe intense vertical mixing in the Kuril Straits and theassociated modification of water properties, particularly
the refreshening of the intermediate layer. To this end,

VOLUME 30

we investigated internal waves generated through theinteraction between the tidal currents (predominant in
the Kuril Straits) and the sill and their effect on mixing,
using a nonhydrostatic f -plane numerical model.
The model results reveal the following. Internal
waves at the tidal frequency (internal tides) generated
by the K1 flow are trapped to the sill because the K1 tide
is subinertial in the Kuril Straits. Since the strong barotropic flow is further intensified near the bottom by thetrapped internal tides, large amplitude free-propagatinginternal waves of short wavelengths are repeatedly generated and break around the sill top, thus producing
relatively vertically uniform water over and around thesill top. On the other hand, in the M 2 case, most of thegenerated internal waves propagate away as dissipating
first-mode internal tides, conveying energy away fromthe sill. Moreover, because the barotropic flow amplitude is small and its period is short, their amplitudes do
not become so large (thus wave breaking does not occur). As a result, the M 2 tidal current does not have thepotential to cause significant vertical mixing to explain
the re-freshening in the Kuril Straits.
Our theoretical consideration identifies the large-amplitude short waves in the K1 case as unsteady lee waves,whose existence has been neglected in previous oceanicinternal wave theories. Because the intrinsic frequencies
of unsteady lee waves are higher than the tidal frequency, the excitation of freely propagating waves (superinertial) by a subinertial K1 flow is naturally explained. These unsteady lee waves, and particularly
those excited at the break on the downstream side of
the sill, are effectively trapped at the excitation region
to form large amplitude internal waves. Furthermore,
the superposition of a propagating unsteady lee wave
and a newly generated lee wave over the sill generates
a large amplitude internal wave, which causes wave
breaking and induces intense vertical mixing over thesill top.
Although internal tides have a secondary effect onvertical mixing in the Kuril Straits at high latitudes, they
are expected to induce significant vertical mixing in thesubtropical and tropical regions, where the superinertial
semidiurnal tides dominate. In such regions, not only
unsteady lee waves but also internal tides are likely tobe effectively amplified. The latter is especially so when
the condition of the critical slope (e.g., Wunsch 1969)
and/or the critical Froude number (Hibiya 1986) is satisfied. The combined effects of these waves are interesting problems for future theoretical and observational
studies.
We estimated the vertical diffusivity from both Lagrangian and Eulerian points of view. Vertical diffusivity in the K1 case is very large around the sill in bothcases, reaching a maximum value of 10 3 cm 2 s21 in theintermediate layer and 101;10 2 cm 2 s21 in the surface
and deep layers. Such intense mixing occurs from thesill to the surface, reaching down to the density core of
the NPIW (s u ; 26.8). Thus it is suggested that large-

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1619

FIG. 15. The temporal change in energy balance equations for (a) ^KE uw &, (b) ^KE y &, and (c)^APE& in the K1 case. Terms in the figures are defined in Eqs. (23) through (34).

amplitude unsteady lee waves generated by the K1 tidal
current and their breaking is one possible mechanism
for the observed water modification in the Kuril Straits.
In addition, the occurrence of such intense tidal mixing
has the potential to explain the formation of lower temperature and higher productivity water in the Kuril
Straits than the surroundings. This seems to contributeto the short ventilation time of the deep water in the
Kuril Basin discussed by Riser (1996).

Since we use a vertically two-dimensional model, thesubinertial K1 internal tides are not properly reproduced.
To examine the influence of this problem, we performeda simple numerical experiment in which the horizontal
eddy viscosity and diffusivity coefficients are increased
by a factor of 4 relative to those in the original case.
This, in a sense, is a crude incorporation of a decreasein wave energy density as the waves propagate away,
which could be caused by longitudinal wave radiation

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on the FACOM M1800 and VP2600 at the Data Processing Center of Kyoto University.
REFERENCES

FIG. 16. The energy flow diagram time-averaged over the second
cycle in the K1 case. Terms in the figures are defined in Eqs. (23)through (34). The values are in units of kJ m21 s21.

in certain cases. The result reveals that although the
maximum elevation amplitude is reduced to about halfof the original case due to rapid dissipation, the lee wave
processes are basically the same and the estimated diffusivity is still of the order of 10;100 (cm 2 s21 ). Thissuggests the importance of tidal mixing on the re-freshening of the intermediate layer in the actual Kuril Straits.
In reality, however, the along-sill propagation of waves
has a much more complex impact on wave amplitudeand causes a wider range of secondary effects than canbe modeled by the viscosity/diffusivity change alone.
Further modifications are required to our numericalmodel to approach full three-dimensionality. A flow
along sills and the propagation of subinertial K1 internal
tides may also play an important role in water modification, for example, by transporting vertically mixed
water along isobaths. Thus, quantitative estimation of
the actual role of vertical mixing at the Kurils on theformation of the NPIW is beyond the power of twodimensional model and should be investigated using athree-dimensional model. Further, wave excitation may
become more complicated due to the presence of currents which flow around islands, sills, or banks. The
basic state in and around the Kuril Straits is actually
determined by wind-driven and thermohaline circulations, which may affect the excitation and evolution ofwaves. These problems are beyond the scope of thepresent paper and are left for future work as well as the
problem of clarifying the relative importance to otherprocesses postulated so far. Nevertheless the qualitative
similarities between the observational results and oursimulations point the importance of strong vertical mixing in the water modification between the Okhotsk Sea
and the North Pacific.
Acknowledgments. We wish to acknowledge Dr. T.
Satomura and Dr. J. P. Matthews for their useful comments. We also thank anonymous reviewers for theirvaluable comments. Numerical Calculations were done

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