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GEOPHYSICAL RESEARCH LETTERS, VOL. 37, L04602, doi:10.1029/2009GL041456, 2010

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ArticleBreaking of unsteady lee waves generated by diurnal tides
T. Nakamura,1 Y. Isoda,2 H. Mitsudera,1 S. Takagi,3 and M. Nagasawa4
Received 21 October 2009; revised 11 January 2010; accepted 15 January 2010; published 20 February 2010.

[1] Diapycnal mixing caused through breaking of large‐
amplitude internal lee waves generated by sub‐inertial
diurnal tides, which are modulated with a 18.6‐year period,
is hypothesized to be fundamental to both the intermediate‐
layer ventilation and the bi‐decadal oscillation around theNorth Pacific Ocean. The first observational evidence ofsuch wave breaking is presented here. The breaking wave
observed had ∼200 m height and ∼1 km width, and its
associated diapycnal mixing was estimated to be ∼1.5 m2
s−1, with a temporal average ∼104 times larger than typicalvalues in the open oceans. Our estimate suggests that asimilar mixing process occurs globally, particularly aroundthe Pacific and Antarctic Oceans. Citation: Nakamura, T.,
Y. Isoda, H. Mitsudera, S. Takagi, and M. Nagasawa (2010),
Breaking of unsteady lee waves generated by diurnal tides, Geophys. Res. Lett., 37, L04602, doi:10.1029/2009GL041456.

1. Introduction[2] Diapycnal mixing induced by oceanic internal waves
is a driving force of the global thermohaline and material
circulations and affects the climate. A major energy sourceof internal waves is tides. Diurnal tides, however, have been
paid much less attention than semidiurnal tides in terms of
diapycnal mixing. This may be because semidiurnal tides
are dominant in most regions, and because diurnal tides are
subinertial at latitudes higher than ≈30°, and hence diurnalinternal tides cannot be internal waves. Nevertheless, subinertial diurnal tides are also important around the North
Pacific rim and the Antarctic Ocean. In these regions, nearly
half of the kinetic energy of barotropic diurnal tides isconcentrated [Egbert and Ray, 2003], and diurnal tidal
currents often dominate the local current field through resonant generation of topographically trapped waves [e.g.,
Chapman, 1989]. Also, with high current speed, subinertial
diurnal tides can generate unsteady lee waves, which canhave an intrinsic frequency higher than the local inertial
frequency and hence can be internal waves [Nakamura etal., 2000].
[3] In particular, numerical studies suggested that diurnal
tides cause strong diapycnal mixing through the generation
and breaking of large‐amplitude unsteady lee waves in the
Kuril Straits, located between the North Pacific and the
Okhotsk Sea (Figure 1a), and that this mixing enhances
the ventilation of the North Pacific intermediate layer through1
Pan-Okhotsk Research Center, Institute of Low Temperature
Science, Hokkaido University, Sapporo, Japan.
2
Faculty of Fisheries, Hokkaido University, Hakodate, Japan.
3
School of Fisheries, Hokkaido University, Hakodate, Japan.
4
Department of Earth and Planetary Science, Graduate School of
Science, University of Tokyo, Tokyo, Japan.

Copyright 2010 by the American Geophysical Union.
0094‐8276/10/2009GL041456$05.00

the following process [Nakamura et al., 2004]. First, thediapycnal mixing in the Kuril Straits causes both direct
water‐mass modification in the Straits and strengthened
convection in the Okhotsk Sea—i.e., saline subsurface water
brought up to the surface by mixing is subsequently transported by the wind‐driven gyre to the northern part of theSea, where the resulting salt flux increases the density of theDense Shelf Water, which is produced as a result of coolingand sea‐ice formation in winter and is the major source ofNorth Pacific Intermediate Water. Second, the effects of
tidal mixing are spread by a dynamical adjustment throughcoastally trapped waves and Rossby waves and by advective
transport of heat, salt, and materials, leading to modificationof the circulation and water mass properties in the North
Pacific.
[4] Another reason why diapycnal mixing caused by
diurnal tides is of interest is that the 18.6‐year nodal cycle
significantly modulates diurnal lunar tidal forces (K1, ±14%;O1, ±19%) [e.g., Ray, 2007]. The associated modulation of
the barotropic tidal currents is expected to modulate the amplitude of lee waves generated, thus leading to the 18.6‐year
variation in the resulting mixing. Because the modulation ofsemidiurnal lunar tidal forces is smaller (M2, ±4%), the effects of the 18.6‐year cycle occur more significantly in regions where diurnal tides are dominant. In particular, acombination of recent studies suggests the hypothesis that the
mixing due to diurnal tides in the Kuril Straits could induce
18.6‐year oscillations, which are found both in the ocean andthe atmosphere around the North Pacific [e.g., Wilson et al.,2007], through the following process: (1) the modulation in
the mixing strength causes the 18.6‐year variation in the
tidally enhanced ventilation process described above[Yasuda et al., 2006], (2) the coastally trapped waves generated through this process reach the equator and vary thesubsurface temperature there, thus regulating El Niño
Southern Oscillation [Hasumi et al., 2008], and (3) atmospheric teleconnections, such as the Pacific North American
pattern, spread its influence [McKinnell and Crawford,
2007].
[5] Previous observations have found breaking events ofinternal waves caused by semidiurnal tides in fjords [e.g.,
Farmer and Freeland, 1983], off Hawaii [Klymak et al.,2008], and off Oregon [Nash et al., 2007], as well as density inversion associated with hydraulic jumps in the Straitof Gibraltar [e.g., Wesson and Gregg, 1994]. However,large‐amplitude unsteady lee waves generated by diurnal
tides and their breaking—i.e., a major mixing mechanism
for subinertial diurnal tides—have not been observed in thereal ocean.
[6] To look for evidence of such wave generation andbreaking, observations were conducted in the Amchitka
Pass, the Aleutian arc (Figure 1), where the dynamical
conditions for tidally generated internal waves are similar to
those in the Kuril Straits: Diurnal tides are subinertial andL04602

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Figure 1. Locations and times of observations. Topographic maps showing (a) the location of the Amchitka Pass and(b) the observation sites in the Pass. Tidal height (TPXO7.1) in the Amchitka Pass during (c) 2008, (d) June 2008, and(e) around 16 June 2008 (UTC). Figure 1e also shows the durations of XCTD/XBT observations (shaded areas), vertically‐
averaged cross‐ridge flow observed over the ridge top (crosses), and northward tidal velocity (dashed line) predicted based on
TPXO7.1, which was lagged by −1.35 hours, amplified 1.94 times, and had a mean value of 0.055 m s−1 added.
dominant (Figures 1c and 1d); a ridge of ∼10 km width and∼35 km length is present (Figure 1b); and the density
stratification is similar in the sense that both areas are in thesubarctic Pacific. In addition, mixing in the Amchitka Passmight affect the ventilation and variability around the Bering Sea and the North Pacific, as is similar to mixing in the
Kuril Straits. The Pass allows a significant water exchange
between the two basins [Stabeno et al., 1999], which would
spread the effects of mixing, and the energy flux of barotropic tides through the Pass has been estimated to increaseby 36% due to the 18.6‐year cycle [Foreman et al., 2006].
Motivated by the observational results obtained, we thenmade a global estimate of the occurrence of a similar process
due to the K1 tide.

2. Observation[7] According to numerical experiments [Nakamura et al.,2000], breaking of unsteady lee waves tends to occur aroundthe time when the barotropic tidal flow in the cross‐ridge
direction ceases. To estimate the time of zero cross‐ridge
flow, we first measured currents in the cross‐ridge (i.e.,
meridional) section shown in Figure 1b six times, and in
three of these measurements (indicated by i–iii in Figure 1e),the vertical sections of temperature and salinity were alsomeasured. The results suggested that cross‐ridge barotropic
flow over the top of the ridge stops near the time of high/low
water. Extensive observations were thus conducted aroundthe time of high water (indicated by iv in Figure 1e). As it
turned out, the cross‐ridge barotropic flow was well correlated with a tidal model prediction (TPXO7.1 [Egbert andErofeeva, 2002]) and agreed well after fitting (Figure 1e),indicating the dominance of tidal currents.
[8] The measurements were conducted using a ship‐board
acoustic Doppler current profiler (ADCP, Furuno Electoric
CI‐3500AD), 44 expendable conductivity temperature
depths (XCTDs, Tsurumi‐Seiki XCTD‐1), and 41 expendable bathythermographs (XBTs, Tsurumi‐Seiki XBT‐7).
The ADCP has a nominal depth range of 10–706 m with avertical resolution of 6 m. XCTD and XBT data was visually checked for spikes, low‐pass filtered, corrected for thefinite‐time responses of the sensors and their mismatches,

and then averaged into 1‐m bins. The XBT data was further
calibrated and utilized to calculate potential temperatureusing the adjacent XCTD data. Errors associated with thelatter calculation were within 0.001°C. To exclude spurious
overturns resulting from observation errors, tests of verticaland density resolutions, run length, and water mass
[Galbraith and Kelley, 1996] were carried out with revised
criteria for run length tests [Johnson and Garrett, 2004] andwater mass tests [Martin and Rudnick, 2004]. The main
overturn discussed below cleared all the tests.

3. Breaking Wave
[9] The extensive observations captured a breaking wave
having a height of ∼200 m (Figure 2a) and causing inversionof the potential density (s, hereafter referred to as density;
Figure 2b). The horizontal scale of the major density inversion reached ∼1.5 km with a core width of ∼500 m in the
cross‐ridge direction. Its density ranged from 26.48 to 26.71
s (kg m−3) in the meridional range of 51°34.75′–35.25′N.
The breaking wave had various‐scale structures andcontained smaller‐scale density inversions (Figures 2a and2b), indicating the occurrence of instability and associatedenergy transfer from the wave to turbulent motion.
[10] The time evolution of isopycnal surfaces suggests thatthe breaking wave was tidal in origin (Figure 3). After thebarotropic current flowed southward (leftward in Figure 3),
isopycnal surfaces were lifted over the northern slope of theridge and depressed over the southern slope (Figure 3a). As
the barotropic current changed direction and flowed northward, isopycnal surfaces began to rise on the southern sideand to descend on the northern side (Figure 3b). A large‐
amplitude elevation appeared over the ridge top as water (orthe crest of the wave generated by the southward flow)crossed the ridge (Figure 3c). When the northward barotropic
flow nearly ceased, internal waves generated during thenorthward flow grew (Figure 3d), resulting in the wave
breaking shown in Figure 2. Similar wave generation was
also seen over a small peak at 51°37.5′N and over a lower
peak at 51°40.5′N (Figure 3e).
[11] The breaking wave had a characteristic feature of lee
waves, that is, co‐phase lines slanted to the upstream side

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L04602indicate that the breaking wave was an unsteady lee wave
generated by tides.
[12] Diapycnal mixing associated with the main overturnat 51°35.0′N (Figure 2) was estimated by two approaches.
One is the Thorpe scale method [e.g., Thorpe, 2005]. The
turbulence dissipation rate was estimated as " ≈ cN3s L2T,
where c ∼ 1, Ns is the mean sorted buoyancy frequency, andLT is the Thorpe scale. The diapycnal diffusivity coefficient,r, was then estimated as r = gɛ/N2s , where the conventional value of g is 0.2. This approach yields r of 1.5 m2
s−1 with calculated values of LT (60 m), Ns (0.0021 rad s−1),
and " (3.1  10−5 W kg−1). The other approach assumed
that the statically unstable region will be vertically mixed,
following past numerical experiments [e.g., Scinocca andPeltier, 1993]. Equating the resulting density flux to a vertical diffusive flux, z∂rs/∂z where rs is sorted density, we
obtain a vertical diffusivity coefficient, z, that almost corresponds to r. Here, the density flux was estimated as thedensity transport divided by the time required for mixing,

Figure 2. Breaking internal wave seen in the extensive
observations (iv in Figure 1e). (a) Cross‐ridge section ofpotential temperature and the XCTD (}) and XBT (+) sites.
White areas represent the bottom or missing data. The
observations took 38 min for this whole section and 14 minfor the main part of the wave (34.75′–35.5′N), during which
the wave may have moved 0.2 km or less. The potential
temperature behaves almost as a passive tracer in this area
below the seasonal thermocline, where the density stratification is determined mostly by salinity. It was neverthelesscorrelated with density, except for the alternating layers
located 100–300 m deep on the left hand side of the ridge
top, which did not show associated density inversions of
this vertical scale. (b) Vertical profile of potential density(s) at 51°35′N. (c) Cross‐ridge (northward) baroclinic
currents (the deviation from the vertically averaged flow
over the ridge top) and the horizontal resolution of thetemporally averaged ADCP data (r). Selected isotherms
are superimposed.
(i.e., south), as seen in both current and density fields(Figures 2c and 3d). In fact, the ratio of the Doppler shift (orlee wave frequency) to tidal frequency (kU/w) was estimatedto be 11, suggesting that the wave was an unsteady lee wave
[Nakamura et al., 2000], where U, w, and k are barotropic
tidal flow speed, tidal frequency, and horizontal wavenumber
estimated from Figure 2. Also, the intrinsic frequency estimated from the dispersion relation (8  10−4 rad s−1) roughly
agreed with the lee wave frequency (kU ≈ 7.9  10−4 rad s−1).
Actually, the observed time evolution had similarities to thoseof unsteady lee waves seen in numerical experiments[Nakamura et al., 2000]. In addition, the inverse Froudenumber (N h/U) and the wave Froude number (U/N D) were
estimated to be 3 and 0.2, respectively, where N is the depth
average of buoyancy frequency, and D is the water depth at
the ridge top, and wave height, h, was used instead of topographic height because the tidal excursion is finite. The two
estimated numbers suggest that the wave generating force
was strong enough for the wave to break but that the flow was
not hydraulically controlled [e.g., Baines, 1995]. All of these

Figure 3. (a–d) Temporal evolution of potential density
surfaces. The observation times of Figures 3a, 3b, 3c, and3d are shown by shaded areas numbered i, ii, iii, and iv inFigure 1e, respectively. Arrows show the direction of barotropic cross‐ridge flow at the ridge top. (e) The measured
bottom topography.

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Figure 4. (a) The condition for lee‐wave generation (∣k · U0∣/w), (b) maximum possible wave height (hmax), and (c) the condition for hydraulic jump (∣U0∣/N D) for the K1 tide. Here, ∣k ·U0∣/w and hmax are set to zero, where Nl ≤ ∣k · U0∣/w. The calculation used ETOPO1 bathymetric data, TPXO7.1 tidal data,
and density data (s0, s2, s4) of WOCE Global Hydrographic
Climatology [Gouretski and Koltermann, 2004]. Abnormal
values found in the last in some marginal seas were corrected.
The calculation was conducted on 1′  1′ grids and maximum
values in 1°  1° boxes are shown.

where the density transport was calculated as the vertical
integration of the magnitude of the difference between thedensity observed and the density that would be achieved byvertical mixing to a statically stable state. This approachgave z of 1.2 m2 s−1 when mixing occurred in the mean
buoyancy period (2p/Ns) of 3000 s.
[13] Both approaches gave diffusivity ∼104 times larger
than typical values found in the open oceans. (The estimatesyield a temporal average of 0.08–0.1 m2 s−1 if the mixing
lasts for the buoyancy period and occurs twice a day, whileobservational estimates are ∼0.1  10−4 m2 s−1 away from
topographic features or at latitudes higher than ≈30° or
lower than ∼15° [e.g., Gregg et al., 2003; Hibiya et al.,2006].) The rough agreement of the two estimates together
with the wide applicability of the Thorpe scale method[Wesson and Gregg, 1994] supports these estimates, though
the estimates should have a margin of error.

4. Global Distribution[14] Figure 4a shows values of ∣k · U0∣/w estimated for thelargest diurnal constituent, the K1 tide, where lee wave
generation is expected when ∣k · U0∣/w ≥ 1 and Nl > ∣k ·U0∣. Here, U0 is the amplitude vector of horizontal, barotropic tidal velocity, w is the corresponding tidal frequency,
and Nl is buoyancy frequency averaged from the bottom in
one vertical wavelength of hydrostatic lee waves (2p(∣k ·U0∣/∣k∣)/Nl [e.g., Baines, 1995]). Horizontal wavenumber

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vector, k, was estimated from the horizontal scale of thevertical velocity amplitude (W0 = −U0 · rD, where D is
depth). Note that lee wave generation is likely to be underestimated particularly in areas of weak tidal currents,
because of the horizontal resolution (1′  1′) of the topographic data used (ETOPO1 [Amante and Eakins, 2008]).
For ∣U0∣ ∼ 0.01 m s−1, the condition ∣k · U0∣/w ≥ 1 requires a
topographic length scale of less than 1 km, which is hardly
resolved because 1′ in latitude is 1.852 km.
[15] The global map suggests that lee waves are expected
in other regions along the Aleutian and Kuril Islands, andalso around the Antarctic and Pacific Oceans. The generation
condition is also met in the Indian Ocean, the IndonesianSeas, the northwest of Canada, and around the Caribbean and
Svalbard Islands.
[16] As a rough measure of the significance of the lee
wave process, an upper limit of the wave height of tidally
generated lee waves (hmax) was estimated together with the
Froude number (∣U0∣/N D; Figures 4b and 4c). The upper
limit was estimated as the minimum of the vertical excursionof barotropic tidal flow estimated at a fixed point (2∣W0∣/w),
the depth (D), the height of topographic features (DH), anda half vertical wavelength (lz/2). This is based on two facts:(1) the maximum vertical displacement of isopycnal surfaces (i.e., wave height) is nearly equal to the maximum
vertical excursion of the fluid moving with the flow, whichis less than or equal to min(2∣W0∣/w, D, DH) in a Boussinesq fluid; and (2) the wave height of internal waves shouldbe less than or comparable to lz/2, as long as the wave ray
slope is small and the horizontal scale of the density inversion is smaller than the horizontal wavelength. Here, lz
was approximated as the vertical wavelength of hydrostaticlee waves (2p(∣k · U0∣/∣k∣)/Nl). DH was estimated as DH =max(H′) − min(H′), where H′ is the deviation of topographic
height (H) from the topographic mean slope (H′ = H − l ·rH, where the overbar denotes the horizontal average andl is horizontal displacement). The calculations for DH andH′ were performed within the distance of the local horizontal tidal excursion from each point.
[17] Estimated values of hmax exceed 100 m along the
Aleutian and Kuril Islands, and around the Pacific andAntarctic Oceans, particularly around the Ross and Weddell
Seas. The distribution is similar to that of ∣k · U0∣/w, but ismore concentrated in areas of a sharp change in depth, suchas shelf breaks, ridges, and seamounts. The distribution of
the Froude number is quite similar to that of hmax. In regionsof strong tidal currents (e.g., the Kuril and Aleutian Islands),
areas having large hmax are interwoven with areas where thejump condition is met (∣U0∣/N D > 1/p). Vigorous mixing
might be caused in such areas by the K1 tide, though estimation of the mixing strength needs future work.
[18] In contrast to the K1 case, hmax for the M2 tide, thelargest semidiurnal constituent, is insignificant around theAntarctic, except for the Weddell Sea, but is significant in
the Atlantic Ocean, according to the map of W0 calculated
by Legg and Klymak [2008].

5. Summary and Discussion[19] This study showed evidence that subinertial tides are
causing vigorous diapycnal mixing through the breaking of
unsteady lee waves in the real ocean. This process is fundamental to the tidally enhanced ventilation and the tidally

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induced bi‐decadal oscillation around the North Pacific. We
also made global maps of the areas where lee waves or
hydraulic jumps are expected. These revealed further implications: (1) Mixing due to lee waves might be significant
for a global estimate of diapycnal mixing because such areas
are distributed globally. (2) Diurnal tides could affect theformation of the Antarctic Bottom Water (AABW) throughthe generation of lee waves or hydraulic jumps, which is
expected around the Ross and Weddell Seas, two of the
major sources of the AABW. (3) The 18.6‐year variation
could be significant in areas having a large Froude number
or hmax. (4) The combination of the last two suggests thepossibility that the 18.6‐year cycle could affect the formation of the AABW. A quantitative understanding of these
possible effects would be obtained from a global estimate ofdiapycnal mixing caused by lee wave breaking (and jumps),
although problems concerning nonlinearity of waves andtidal currents still remain, and higher resolution data of the
topography, tides, and stratification are desired.
[20] Acknowledgments. We thank scientists who joined the cruise
and the captain and crew of the R.V. Oshoro‐Maru for their support in
the observations. This work was supported by Grants‐in‐Aid for Scientific
Research and for Young Scientists, and by NEDO. Numerical analysis wasdone on the Pan‐Okhotsk Information System of ILTS.

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M. Nagasawa, Department of Earth and Planetary Science, Graduate
School of Science, University of Tokyo, Tokyo 113‐0033, Japan.
T. Nakamura and H. Mitsudera, Institute of Low Temperature Science,
Hokkaido University, Sapporo 060‐0819, Japan. (nakamura@lowtem.
hokudai.ac.jp)
Y. Isoda, Faculty of Fisheries, Hokkaido University, Hakodate 041‐8611,
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S. Takagi, School of Fisheries, Hokkaido University, Hakodate 041‐8611,
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