## 1. Introduction

Continental shelf waves of diurnal tidal period were first identified in current meter records from the Hebrides Islands shelf off Scotland (Cartwright 1969; Cartwright et al. 1980) and subsequently observed in current measurements from a variety of shelf regions, including the southern Kuril Islands (Kovalev and Rabinovich 1980; Yefimov and Rabinovich 1980), the west coast of Vancouver Island (Crawford and Thomson 1982, 1984), the east coast of Sakhalin Island (Rabinovich and Zhukov 1984), and the north coast of Hokkaido Island (Odamaki 1994). [For other areas, see the review by Clarke (1991).] Because diurnal shelf waves are formed through the joint effect of earth's rotation and shelf topography rather than gravity (LeBlond and Mysak 1978), their energy is predominantly kinetic (Hsieh 1982; Brink 1991). Consequently, the waves produce negligible sea level variation but significant tidal currents. Diurnal currents associated with shelf wave motions can have speeds in excess of 1 m s^{−1}, roughly 10 to 100 times greater than current speeds generated by gravitational tidal waves. Currents of this magnitude may be responsible for dissipation of tidal energy over continental shelf regions of the World Ocean (Rabinovich and Shevchenko 1984; Kowalik and Proshutinsky 1993) and for the small-scale spatial variability of diurnal tides in high-latitude oceanic margins (Cartwright et al. 1980; Yefimov and Rabinovich 1980; Rabinovich and Zhukov 1984).

Shelf waves and related forms of subinertial topographically trapped waves in the vicinity of the Kuril Islands contribute to coastal upwelling, shelf circulation, nonbarometric response of sea level to large-scale moving atmospheric disturbances, the formation of topographic eddies, and exchange processes through the Kuril Straits (Lappo et al. 1978; Yefimov and Kulikov 1978; Mysak et al. 1979; Likhacheva and Skripnik 1981; Yefimov et al. 1985; Kulikov 1987). Numerical simulations suggested that these waves also may explain the strong diurnal tidal currents observed in this region (cf. Yefimov et al. 1985; Kowalik and Polyakov 1998; Awaji et al. 1999; Nakamura et al. 2000). The possibility of strong *O*_{1} tidal currents associated with diurnal barotropic shelf waves of *O*_{1} period on the southern Kuril Islands shelf was predicted theoretically by Yefimov and Rabinovich (1978) and subsequently supported by measurements of significant (36–40 cm s^{−1}) near-bottom *O*_{1} currents off Shikotan Island (Fig. 1) (Kovalev and Rabinovich 1980). A numerical model of shelf waves with realistic topography (Yefimov and Rabinovich 1980) yielded *O*_{1} current ellipse parameters that were in good agreement with observations and indicated that *O*_{1} shelf waves should be dominated by first mode oscillations.

Contrary to Yefimov and Rabinovich (1978, 1980), Odamaki (1994) finds that both *K*_{1} and *O*_{1} diurnal shelf waves are responsible for the strong diurnal tidal currents observed along the coast north of Hokkaido Island (Fig. 1). Similarly, numerical computations by Kowalik and Polyakov (1998) and Awaji et al. (1999) reveal pronounced *K*_{1} and *O*_{1} diurnal currents off the Central and northern Kuril Islands and in certain adjoining passages, which the authors claim are associated with continental shelf waves. The latter study also suggests significant spatial variability in diurnal shelf wave currents, which we assume is caused by alongshore changes in the width and slope of the Kuril Island shelf and the presence of numerous straits connecting the Pacific Ocean and the Sea of Okhotsk. Dispersion curves for computed barotropic shelf waves for the Pacific shelf of the Kuril Islands (Fine 1980) indicate that the maximum frequency of the first mode, *ω*^{(1)}_{max},*f* to 0.78*f* alongshelf (*f* is the local Coriolis frequency). This implies a complicated wave structure for diurnal tides near the Kuril Islands, including possible effects of stratification and mean currents, similar to that found by Crawford and Thomson (1984) and Cummins et al. (2000) for the coast of Vancouver Island in the northeast Pacific. Generation of diurnal shelf waves off the Kuril Islands could arise through scattering of barotropic Kelvin waves at the numerous coastal straits along the islands (Kulikov 1979) or, alternatively, through strong diurnal currents at the entrances to adjoining straits, as demonstrated by Crawford (1984) and Flather (1988) for the west coast of Vancouver Island. The Kuril Islands region, with its convoluted coastline and many narrow straits, creates favorable conditions for both generation mechanisms.

In this paper, we show that the strong diurnal tidal currents observed off the Kuril Islands in 1993 can be interpreted in terms of freely propagating baroclinic shelf waves of *K*_{1} and *O*_{1} frequencies. Both first and second mode waves contribute to the diurnal currents, with a suggestion—based on admittedly short time series records—that second mode oscillations are more prominent than first mode oscillations. Results provide insight into tidal currents on the shelf and adjoining straits of the Kuril Islands, a region of considerable scientific and practical interest for which there are relatively few oceanographic observations (Awaji et al. 1996; Kashiwai 1996; Luchin 1996).

## 2. Drifter records

On 4 September 1993, three standard World Ocean Circulation Experiment satellite-tracked surface drifters were deployed 200–300 km east of the Kuril Islands (Fig. 1) from the Research Vessel *Alexander Nesmeyanov* as part of the WOCE Line P1W Repeat Hydrography (Freeland et al. 1998). (See Table 1 for deployment times and positions.) All three drifters were manufactured by Seimac Inc., had a 7-m holey-sock drogue centered at 15-m depth, and a nominal drogue-to-nondrogue ratio of 40:1 (Thomson et al. 1990, 1998). To reduce tracking costs, the drifters had a programmed “duty cycle” that alternated between continuous transmission for 90 days followed by transmission every third day for the next 90 days. A 90-day period of reduced transmission occurred after drifter 15371 entered the Sea of Okhotsk. Because high-resolution positional information was lost during this period, focus is confined to the first 90 days of the trajectory.

We examine the flow dynamics on the Pacific and Okhotsk shelves of the Kuril Islands using the zonal, *u* (positive eastward) and meridional *υ* (positive northward), components of drifter velocity (Fig. 2). For commonly achieved drifter position accuracy Δ*x,* Δ*y* ≈ 150–300 m and times Δ*t* ≈ 2–6 h between successive satellite passes (Table 1), velocity errors (*u*_{e}, *υ*_{e}) ≈ (Δ*x*/Δ*t,* Δ*y*/Δ*t*) associated with the tracking are less than 5 cm s^{−1}. Velocity errors arising from slippage between the water and the drogue are of comparable magnitude to those for positional accuracy, and both are small compared to the typical 10–200 cm s^{−1} current speeds we measured. Failure of the satellite-tracking system on 30 September 1993 caused a simultaneous 77-h data gap for drifters 15371 and 15374. Omitting this gap, the times between satellite positional fixes for all three drifters were irregularly spaced within a range of minutes to 18.7 h. Mean time delays Δ*t* ≈ 3 h correspond to eight fixes per day and are sufficiently rapid that the Nyquist sampling frequency (∼4 cpd) exceeds the diurnal, semidiurnal, and inertial frequencies. A rigid cubic-spline algorithm was used to interpolate from the irregularly spaced location series to a uniform time step Δ*t* ≈ 3 h [see Thomson et al. (1997, 1998) for further processing details].

## 3. Analysis of current velocity

Following deployment, drifter 15372 quickly crossed the Kuril–Kamchatka Trench (KKT) on its way toward the Kuril Islands and subsequently ran aground after 17 days on the Iturup Island shelf (Fig. 1). Drifter 15374 moved toward the seaward slope of the KKT and failed at sea after 72 days. In contrast, drifter 15371 began a four-year Pacific odyssey, detecting various oceanic features along its basinwide trajectory, including tidal currents, anticyclonic and cyclonic eddies, wind-driven inertial oscillations, and throughstrait flows (Thomson et al. 1997). Prior to its journey across the Pacific, the drifter made a few circuits in a quasi-permanent anticyclonic eddy centered over the axis of the KKT, traveled over the Urup Island shelf, moved through Friz Strait into the Sea of Okhotsk, made a few cyclonic loops, and then returned through Bussol' Strait back to the Pacific Ocean (Fig. 1).

### a. Spectral analysis

Rotary spectral analysis (Gonella 1972; Mooers 1973; Emery and Thomson 1997) reveals that the motions of drifters 15371 and 15372 were dominated by diurnal (*K*_{1}, *O*_{1}) oscillations (Figs. 3a,b) whereas corresponding inertial motions were weak and *M*_{2} semidiurnal motions insignificant. In contrast, the motion of drifter 15374 was dominated by inertial oscillations with negligible energy at diurnal and semidiurnal frequencies (Fig. 3c). These differences are directly related to the distinct paths taken by the drifters. Specifically, drifters 15371 and 15372 crossed the Kuril Island shelf while drifter 15374 remained entirely in the deep ocean (Fig. 1).

*u*

^{−}) and counterclockwise (

*u*

^{+}) components of velocity variance

*ω*< 0.9 cycles per day (cpd); the diurnal band,

*ω*≈ 0.9–1.1 cpd; the inertial band,

*ω*≈

*f*(1.10–1.85 cpd); the semidiurnal band,

*ω*≈ 1.85–2.10 cpd; and the high-frequency band,

*ω*> 2.1 cpd. Here, Δ

*ω*denotes a specified frequency band and var

_{tot}= var(

*u*

^{−}) + var(

*u*

^{+}) is the total velocity variance for a given drifter record. According to the variance partitioning, diurnal currents encountered by drifters 15371 and 15372 were seven to eight times more energetic than corresponding semidiurnal currents (Table 2) despite comparable semidiurnal and diurnal sea level oscillations in the region (see Fig. 4 and tide tables for the coasts of Iturup and Urup islands, Table 3). Results are in good agreement with Awaji et al. (1999) and Nakamura et al. (2000), who find that

*K*

_{1}currents in the straits between the Kuril Islands are significantly greater than

*M*

_{2}currents despite similar sea level amplitudes for these constituents. Clearly, direct astronomical forcing does not explain the enhanced diurnal currents in the drifter records.

### b. Complex demodulation

The rotary spectra in Figs. 3a,b represent integral characteristics for the drifter velocities. To examine temporal variations in the frequency content of the velocity records, we have turned to a rotary version of the “multiple filter technique” (MFT) first introduced by Dziewonski et al. (1969) to study nonstationary seismic signals. Modification of the MFT for vector processes enables us to examine changes in the clockwise velocity *u*^{−}(*t*; Δ*ω*) and counterclockwise velocity *u*^{+}(*t*; Δ*ω*) as functions of time *t* (and hence space) for different frequency bands (cf. Thomson et al. 1997; Bograd et al. 1997; the appendix). This rotary MFT (RMFT) is similar to wavelet analysis generalized to vector series (cf. Liu and Miller 1996).

According to the RMFT analyses (Fig. 5), drifter motions are dominated by clockwise currents in all frequency bands with good agreement between amplification of the rotary diurnal currents and decreasing water depth. For example, the motion of drifter 15372, which crossed the KKT on its way toward shore (Fig. 1), was influenced by diurnal currents only after it entered the shelf region. Diurnal current amplitudes were negligible in the deep (>4000 m) open ocean segment of the track but increased dramatically as the drifter approached the coast (Fig. 5b). Drifter 15371 was markedly affected by anticyclonic and cyclonic eddies, which caused the drifter to approach and recede from the shelf on multiple occasions (Fig. 1). Significant clockwise rotary diurnal currents were encountered on each visit to a shallow water area (Fig. 5a) and diurnal current speeds in excess of 1 m s^{−1} were observed as the drifter passed through Friz Strait into the Sea of Okhotsk. Marked diurnal currents also were observed on the Okhotsk shelf of Urup Island, a region coincident with maximum diurnal tidal dissipation estimates for the Kuril Islands based on TOPEX/Poseidon satellite observations (Kantha et al. 1995). The observational results also are in agreement with the numerical computations of diurnal tides for this region (Kowalik and Polyakov 1998; Awaji et al. 1999; Nakamura et al. 2000).

Unlike drifters 15371 and 15372, drifter 15374 remained over the deep sea portion of the KKT where its trajectory was almost entirely dominated by near-inertial currents (Fig. 5c). This finding is consistent with Thomson et al. (1998), who used 47 drifter records from 1990 to 1993 to examine current variability in the upper mixed layer of the deep northeast Pacific. According to the latter study, inertial motions accounted for 58% of the total observed current variance while semidiurnal currents accounted for roughly 10% of the observed variance. None of the open-ocean drifter records had measurable diurnal currents.

## 4. Wave model selection

*υ*for gravity-driven tidal currents off the Kuril Islands is

*g*is the acceleration of gravity,

*h*is the water depth, and

*ζ*is the sea level displacement. For

*h*= 500 m (shelf region) and

*ζ*∼ 20–35 cm (see tidal heights in Table 3),

*υ*∼ 3–5 cm s

^{−1}. Observed semidiurnal currents are in good agreement with (2) whereas diurnal currents are 5–10 times greater. The implication is that semidiurnal currents in this region arise from direct astronomical forcing whereas diurnal currents are generated through interaction of the astronomically forced diurnal tides with the regional topography and/or density structure. More specifically, because semidiurnal motions are superinertial while diurnal motions are subinertial at the latitude of the Kuril Islands, only the diurnal tidal motions can propagate as free quasigeostrophic waves in this region [see LeBlond and Mysak (1978, 219–245) and Mysak (1980) for discussions of gravity and quasigeostrophic waves]. As noted by Awaji et al. (1999) “Since the

*K*

_{1}tide is subinertial in the Okhotsk Sea, topographically trapped waves are effectively generated, contributing to strong tidal currents with a maximum amplitude of over 1.5 m s

^{−1}in the Kuril Straits.”

Based on the present analysis and the results of previous studies, it is reasonable to assume that the amplified diurnal tidal currents observed in the vicinity of the Kuril Islands are evidence for topographically trapped waves. Accordingly, we have used the Brink and Chapman (1987) algorithms to construct three possible analytical models for the cross-shore structure of these waves: 1) a barotropic model of continental shelf waves with zero mean alongshore flow, *V* = 0; 2) a barotropic model of shelf waves with cross-shore dependent mean flow, *V* = *V*(*x*); and 3) a baroclinic model with realistic density structure but zero mean flow. The coordinate system for each side of the islands has been chosen with the *x* axes positive in the offshore direction and the *y* axes directed alongshore (Fig. 6a).

### a. Barotropic shelf waves: Zero mean flow

*ω*and wavenumber

*k*in a rotating coordinate system with zero mean alongshore horizontal current

*V*(

*x*) = 0 yield the following equation for the cross-shore structure of the sea level displacement

*ζ*(LeBlond and Mysak 1978; Yefimov et al. 1985):

*u,*

*υ*) are the vertically averaged velocity components in the (

*x,*

*y*) directions,

*h*is the depth (uniform alongshore), and ′ ≡

*d*/

*dx.*For subinertial frequencies (

*ω*<

*f*), (3)–(4) describe two types of trapped coastal waves: gravity Kelvin waves and quasigeostrophic continental shelf waves (LeBlond and Mysak 1978). Kelvin waves generate large sea level oscillations with relatively weak shelf currents whereas shelf waves generate strong shelf currents with relatively small sea level oscillations (Mysak 1980; Yefimov and Rabinovich 1980).

*L*and depth

*h*

_{o}(≈0.1 km), is proportional to the phase speed of the shelf waves

*ĉ*

_{s}=

*fL*divided by the phase speed of long gravity waves over the shelf

*c*

_{l}=

*gh*

_{o}

*D*

^{2}is small, the nondivergent approximation is applicable for shelf motions (Buchwald and Adams 1968; Buchwald 1973). (The effect of divergence is to reduce the phase speed and maximum frequencies of shelf waves.) On the Pacific side of the Kuril Islands,

*f*≈ 10

^{−4}s

^{−1}and

*L*∼ 100 km, while for the Okhotsk side

*L*∼ 15 km. Thus,

*ĉ*

_{s}∼ 10 m s

^{−1}for the Pacific shelf and

*ĉ*

_{s}∼ 1.5 m s

^{−1}for the Okhotsk shelf. Since the corresponding values of

*c*

_{l}are roughly 30 m s

^{−1}for both shelves,

*D*

^{2}is small (∼0.01–0.1) and the nondivergent approximation applies to both sides of the Kuril Islands. Use of the nondivergent approximation removes the barotropic Kelvin wave.

Our nondivergent wave solutions are based on four distinct depth profiles for the Pacific shelf and one profile for the Okhotsk shelf (Fig. 6). As indicated by the computed dispersion diagrams (Figs. 7a,b), shelf waves exist only on the negative sides of these diagrams, indicating that the waves propagate toward the southwest on the Pacific shelf and toward the northeast on the Okhotsk shelf. The dispersion curves for barotropic shelf waves in the absence of a mean flow (top panel Fig. 7a) show slight differences among the different depth profiles (lines 1–4), but for all computations the maximum frequency of the first mode, *ω*^{(1)}_{max},

### b. Barotropic shelf waves: Nonzero mean flow

^{−1}. On this basis, we expect the mean flow to have a greater influence on waves for the Okhotsk shelf than the Pacific shelf.

Following Brooks and Mooers (1977), we estimated shelf waves in the presence of a mean barotropic flow. For quantitative results, we assumed *V*(*x*) = *const* and computed the dispersion curves for *V* = −50, −30, −20, −5, 0, 5, 20, 30 cm s^{−1} for the Pacific shelf and *V* = −20, −10, 0, 10, 20 cm s^{−1} for the Okhotsk shelf (top and bottom panels Fig. 7b). Negative *V* corresponds to mean flow in the same direction as shelf wave propagation and positive *V* to mean flow in the opposite direction to shelf wave propagation. The mean flow clearly modifies the shelf wave dispersion curves through Doppler shifting. Negative mean flow (i.e., in the same direction as the shelf waves) increases the phase speed of the waves and their frequency, while positive flow has the opposite effect.

The Oyashio Current, the prevailing flow on the Pacific shelf of the Kuril Islands, is directed toward the southwest in the same direction as the shelf wave propagation. Strong southwestward flowing mean currents of about 50 cm s^{−1} appear to be required for diurnal shelf waves of *K*_{1} and *O*_{1} frequency (top panel Fig. 7b). According to Ponomarev et al. (1996), the speed of the Oyashio Current decreases along the Kuril Islands from 70–80 cm s^{−1} in the northeast sector to 30–40 cm s^{−1} in the southwest sector due to scattering effects and water exchange through the straits. As a consequence, mean currents in the vicinity of the Urup and Iturup Islands are expected to be about 30–50 cm s^{−1}. Thus, the Oyashio Current might enhance the likelihood of diurnal shelf waves near the islands, but this effect is unlikely to persist along the entire shelf.

For the Okhotsk shelf of Urup Island, a northeastward flowing mean current as weak as 10 cm s^{−1} can effect the formation of diurnal shelf waves (bottom panel Fig. 7b). As shown by Ponomarev et al. (1996), the prevailing flow for this shelf is mainly to the northeast, albeit still highly variable and complex (see also the track of drifter 15371 in Fig. 1, which made several loops in this area). As a result, we do not expect persistent diurnal shelf waves on the Okhotsk shelf of the islands.

### c. Baroclinic shelf waves

*p*:

*N*(

*z*) = (−

*gρ*

_{0,z}/

*ρ*

_{0})

^{1/2}is the Brunt–Väisälä frequency,

*ρ*(

*x,*

*y,*

*z*;

*t*) is the density perturbation from the rest state

*ρ*

_{0}(

*z*),

*w*is the vertical component of velocity, and

*z*is measured vertically upward. Smoothed

*N*

^{2}profiles for the Pacific and Okhotsk shelves (Fig. 8a) have been constructed from observational data for the region (Freeland et al. 1998). The appropriate boundary conditions are

*S*can be considered the ratio of the internal Rossby radius of deformation to the width

*L*of the shelf.) Small

*S*(e.g., shallow depth or wide shelf) implies a weak influence of stratification on subinertial motions while large

*S*(e.g., strong vertical stratification or narrow shelf) indicates that internal Kelvin waves prevail (Brink 1982; Chapman 1983). When

*S*is near unity, waves are of a mixed type: specifically, shelf waves modified by stratification. For the Kuril Islands in September 1993, we estimate the mean Brunt–Väisälä frequency to be

*N*= 3.6 × 10

^{−3}s

^{−1}(Freeland et al. 1998). A typical internal Kelvin wave speed for the Pacific (Okhotsk) side of the Kuril Islands is then

*ĉ*∼ 18.7 cm s

^{−1}(11.9 cm s

^{−1}), with a corresponding value for

*S*of 1.9 (8.0). These values of

*S*suggest that stratification has a considerable influence on coastal-trapped waves in the vicinity of the Kuril Islands, especially on the Okhotsk side where shelf is comparatively narrow.

As indicated by the computed dispersion curves (Figs. 8b,c), stratification significantly increases the wavelengths, phase speeds, and frequencies of baroclinic wave modes relative to the barotropic case (cf. Mysak 1967; Huthnance 1978; Middleton and Wright 1990). For the Kuril Islands region, changes in stratification appear to affect shelf waves more than changes in mean flow. As a result, the dispersion curves for the first and the second modes intersect the *K*_{1} and *O*_{1} “thresholds” on the Pacific and Okhotsk sides of Urup Island. The baroclinic model is therefore a more plausible model for the drifter observations than the barotropic models.

### d. Sensitivity analyses

To determine whether a different choice of observed density structure would influence the shelf wave modes, we undertook several numerical experiments using different *N*^{2} profiles taken from WOCE Line P1W for September 1993 (Freeland et al. 1998). We find differences to be small, with changes in the phase speeds (wavelengths) of the first and second wave modes less than a few percent at diurnal frequencies. Similar results were obtained by Crawford and Thomson (1984), who report only 3%–4% differences in the summer and winter phase speeds for first-mode *K*_{1} shelf waves for the coast of Vancouver Island (see also Cummins et al. 2000). The near constancy of baroclinic shelf wave solutions for changes in density structure arises from the fact that the wave properties are determined primarily by the slowly varying main pycnocline rather than by the rapidly changing upper mixed layer (P. Cummins 1999, personal communication).

We used four different depth profiles for the Kuril Islands shelf (Fig. 6) to examine the effect of alongshore topographic variations on the wave modes. Unlike the negligible influence of density, alongshore changes in water depth have a profound effect on the eigenvalue solutions. For example, wavenumbers for the first-mode *K*_{1} shelf wave for cross-shelf lines 1–4 were: 1.73 × 10^{−3}, 3.53 × 10^{−3}, 4.11 × 10^{−3}, and 2.28 × 10^{−3} km^{−1}, respectively. This considerable effect of depth on the wavelength and phase speeds of coastal-trapped waves has been noted by other researchers, including Crawford and Thomson (1984), Yao et al. (1984), and Cummins et al. (2000). Despite the differences, all solutions are qualitatively similar and for all depth profiles the dispersion curves intersect the diurnal (*K*_{1} and *O*_{1}) frequency bands for both the first and the second modes. Thus, both modes can contribute to baroclinic shelf waves in the study region. As shown by Huthnance (1978), higher baroclinic shelf modes also can reach the inertial frequency band (i.e., attain diurnal tidal frequencies). However, we find that, for relatively high frequencies (>0.5*f*), these modes are unstable because of their sensitivity to small changes in stratification and bottom topography.

### e. Baroclinic wave solutions

The spatial structure of the wave variables *ζ,* *p,* *u,* and *υ* are derived once the wavenumbers for the diurnal frequencies are found from the dispersion curves. The mode number *n* determines the number of zero-crossings of the variables as functions of offshore distance. Because of the greater proximity of the drifter tracks to Line 1, this line was used to define the wave parameters for the Pacific side of the islands. For the Okhotsk side, Line 5 was used. Calculated dispersion curves for these depth profiles are shown in Figs. 8b,c and the wave properties in Table 4. As indicated by the wave structures for the first and second modes for the Pacific side (Figs. 9 and 10, respectively), the most intense diurnal currents occur in a shelf zone 80 km wide at depths less than 2 to 2.5 km. Within this zone, velocities are roughly independent of depth, similar to barotropic shelf waves. Vertical shear in the *u* and *υ* velocity components is weak and occurs only over continental slope and trench. On the Okhotsk side of the islands, significant diurnal currents occupy a narrow (15 km) near-coastal zone spanning depths less than 800 m (Figs. 11 and 12). Vertical shear in this zone is much more pronounced than on the Pacific side, apparently because of the significantly larger stratification parameter *S.*

## 5. Comparison of observations and baroclinic shelf wave model

Previous studies of amplified diurnal currents over continental shelves have generally relied on comparisons between theoretical models and alongshore variation in flow from moored current meter arrays to determine the properties of coastally trapped waves (cf. Crawford and Thomson 1984; Rabinovich and Zhukov 1984; Freeland 1988; Odamaki 1994). Unlike moored arrays, drifters typically do not provide sufficient coverage of current phase to permit determination of alongshore wave propagation. On the other hand, cross-shore current measurements are highly effective for detecting the modal structure of continental shelf waves (Hsieh 1982). Thus, provided they make one or more circuits across the shelf–slope region for periods of interest, drifters can serve as a useful tool for cross-shore current measurement.

*t*and cross-shore position

*x.*Drifter 15371 had a more complicated trajectory. Instead of approaching the coast directly, the drifter made a number of “forays” onto the shelf, each time retreating into deeper water (Fig. 1). To estimate the cross-shore structure of the flow, we therefore divided the record from 15371 into several alongshore bands with cross-shore locations

*X*

_{j}. We then found the averaged amplitudes for the clockwise

*A*

^{−}

_{j}

*ω*;

*t*) and counterclockwise

*A*

^{+}

_{j}

*ω*;

*t*) components of diurnal current for each band such that

*A*

^{±}

*X*

_{j}

*A*

^{±}

_{j}

*x*

_{k}

*y*

*x*

_{k}(

*t*) −

*X*

_{j}| < Δ

*x*/2, and Δ

*x*= 5 km is the width of each alongshore band (

*j*= 1, 2, · · · , 50). Error bars

^{±}

_{j}

*X*

_{j}as

^{±}

_{j}

^{±}(

*X*

_{j}) =

*σ*

^{±}

_{j}

*N*

*σ*

^{±}

_{j}

*A*

^{±}

_{j}

*t*) in the

*j*th band and

*N*is the number of estimates in the band. Drifter 15371 eventually passed through Friz Strait into the Sea of Okhotsk where procedure (10) was repeated for

*j*= 1, 2, · · · , 20.

*F*

^{±}

_{n}

*x*) for the clockwise (−) and counterclockwise (+) components of the diurnal currents as functions of cross-shore distance

*x*(cf. Hsieh 1982; Pugh 1987), where

*F*

_{n}functions for the amplitudes of first two baroclinic modes (

*n*= 1, 2) for the Pacific and Okhotsk shelves are presented in Figs. 13a and 14a, respectively. Corresponding plots for the observed current amplitudes for drifters 15371 and 15372 are shown in Figs. 13b and 14b.

*A*

^{−}(

*x*

_{j}) and counterclockwise

*A*

^{+}(

*x*

_{j}) components of the diurnal currents, assuming a combination of first and second baroclinic shelf modes for both sides of the Kuril Islands. Specifically, we found the best fit of the curves in Figs. 13b and 14b to the theoretical curves in Figs. 13a and 14a. Amplitudes of the first and second modes (

*a*

_{1}and

*a*

_{2}, respectively) were obtained by minimizing the expression

*σ*

^{2}with the total variance

*σ*

^{2}

_{res}

*σ*

^{2}/

*σ*

^{2}

_{0}

## 6. Discussion

According to our results, strong diurnal currents over the continental shelves on the Pacific and Okhotsk sides of Urup and Iturup Islands arise from a superposition of first and second baroclinic shelf wave modes. The prominence of shelf wave motions corroborates earlier findings (based mainly on sea level observations), which suggested the existence of low-frequency, southwestward propagating atmospherically generated shelf waves on the Pacific side of the islands (cf. Lappo et al. 1978; Yefimov and Kulikov 1978; Likhacheva and Skripnik 1981; Kulikov 1987; Kulikov and Shevchenko 1992). Similarly, Likhacheva and Rabinovich (1986) report shelf waves on the Okhotsk side of the islands, with waves moving in the same direction (to the northeast) as atmospheric cyclones and typhoons.

The wavelengths of the diurnal shelf wave modes are in the range 300–600 km for the Pacific side and 60–130 km for the Okhotsk side (Table 4), or roughly the distances between the main straits separating the islands. This would support an assumption by Kulikov (1979) and Kulikov and Shevchenko (1992) that generation of shelf modes in this region arises from large-scale resonant features of the regional topography and to the existence of deep straits connecting the Pacific Ocean with the Sea of Okhotsk (Nakamura et al. 2000). Similar results have been presented for the coast of Vancouver Island where diurnal shelf waves appear to be generated by strong tidal flow through Juan de Fuca Strait separating the island from the mainland (cf. Crawford 1984; Foreman and Thomson 1997; Cummins et al. 2000).

Bussol' Strait is the deepest (2700 m) and the widest (67 km) of the channels through the Kuril Islands. Based on the theoretical work by Kulikov (1979) and the numerical work of Nakamura et al. (2000), the strait is responsible for generation of the strong diurnal currents observed on the Pacific shelf of Urup and Iturup Islands. Indirect confirmation of this analysis is provided by the sea level tidal harmonics in Table 3. In particular, sea level phase is determined mainly by the large-scale coastal-trapped Kelvin wave, which normally dominates the diurnal sea level oscillations (cf. Cartwright et al. 1980; Crawford and Thomson 1982, 1984). Small-scale variations in phase should then be determined by diurnal shelf waves (Yefimov and Rabinovich 1980; Rabinovich and Zhukov 1984). Shelf wave dispersion gives rise to differences in the phases of the *K*_{1} and *O*_{1} tidal constituents so that near to the source area this difference should be close to zero (Freeland 1988). As indicated by Fig. 15, *K*_{1} − *O*_{1} phase differences for the Pacific coast stations increase from north to south in the direction of propagation for continental shelf waves. Zero phase difference—which we assume coincides with the source region for the shelf waves—is found near the seaward axis of Bussol' Strait.

The key role of Bussol' Strait in the formation of diurnal shelf waves in this region probably accounts for the greater importance of the second baroclinic mode compared to the first baroclinic mode (see the results of the previous section). An oscillating current through a strait would generate shelf waves with wavelengths roughly twice the width of the strait (H. Freeland 2000, personal communication). For example, diurnal tidal flow through Bass Strait between Tasmania and Australia generates several predominantly second-mode coastal-trapped waves (Freeland et al. 1986) whose wavelength is approximately twice the strait width (about 500 km). The width of Bussol' Strait, together with adjoining Urup Strait, is about 100 km. This width more closely matches the half-wavelength of the second baroclinic diurnal shelf mode (∼150 km; Table 4) than the half-wavelength of the first mode (∼300 km).

Results for the Okhotsk side of the Kuril Islands are more complicated. Here, *K*_{1} phases exceed respective *O*_{1} phases (Table 3) but there is no clear spatial tendency in the *K*_{1} − *O*_{1} phase difference. We assume this is because diurnal shelf waves off Urup Island are generated to the southwest of the observational area in the vicinity of the southern Kuril Islands. Because there are no large and deep straits in this region, the diurnal currents encountered by the drifters are probably associated with shelf waves generated through the net effect of shoreline irregularities and several small straits.

From a technological point of view, one of the encouraging aspects of the present study is that, with only a few satellite-tracked drifters, it is possible to delineate the properties of diurnal shelf wave currents in a relatively inaccessible region of the World Ocean. Aside from a degree of serendipity, two important factors contributed to our successful drifter measurements of diurnal currents on the Pacific and Okhotsk shelves of the Kuril Islands: 1) the recent improvement in drifter quality and data transmission that allows for a reduction in the time steps between positional fixes, thereby increasing Nyquist frequency and the resolution of tidal motions, and 2) the development of the rotary multiple filter technique for determining frequency variations of vector series as functions of time. Although satellite drifters were originally designed to study global ocean circulation (cf. Thomson et al. 1990), our findings clearly demonstrate their wider application.

## 7. Summary and conclusions

The trajectories of three satellite-tracked drifters deployed in September 1993 in the vicinity of the Kuril–Kamchatka Trench off the central Kuril Islands enabled us to determine the cross-shore structure in surface current velocity as a function of time. These data, together with a newly devised “rotary multiple filter technique,” further enabled use to examine variations in the frequency content of these motions as functions of drifter location. One of the drifters (15374) remained in the deep ocean where it recorded mainly energetic inertial oscillations. The other two drifters (15371 and 15372) crossed the continental shelf on the Pacific side of Urup and Iturup Islands where they encountered strong (40–50 cm s^{−1}) diurnal tidal currents. Drifter 15371 subsequently traveled through Friz Strait into the Sea of Okhotsk where it encountered intense (>100 cm s^{−1}) diurnal currents in the strait and significant (35–40 cm s^{−1}) diurnal currents on the Okhotsk shelf of Urup Island. These observed diurnal currents were roughly eight times stronger than semidiurnal currents, despite the fact that diurnal and semidiurnal sea level oscillations have about the same amplitudes.

The strong diurnal currents observed off the Kuril Islands, combined with the theoretical and observational evidence provided by Kovalev and Rabinovich (1980), Kantha et al. (1995), and Nakamura (2000), suggests that currents are the result of subinertial coastally trapped waves. Of the three different wave models we used to account for the observed tidal currents, only the baroclinic model accurately explains the main features of the diurnal currents for the Pacific and Okhotsk shelves of the Kuril Islands. Although both first and second baroclinic shelf wave modes contribute to the diurnal motions, the second mode appears to be more prevalent than the first mode. Additional current velocity data are needed to confirm this possibility. The *K*_{1} − *O*_{1} phase difference in sea level fluctuations on the Pacific side of Iturup and Urup Islands supports the existence of diurnal shelf waves in this region and suggests that Bussol' Strait is the primary source for these waves.

## Acknowledgments

We are grateful to Alexander Bychkov and the officers and crew of the Research Vessel *Alexander Nesmeyanov* for deploying the drifters. Howard Freeland kindly supplied the hydrological data for the area and made several valuable suggestions regarding the results. Special thanks to Jane Eert for helping with the numerical computations and to Patricia Kimber for drafting the figures. Helpful comments and criticism from Evgueni Kulikov, Isaac Fine, and Patrick Cummins are much appreciated. An earlier version of this work benefited from the comments of two anonymous reviewers.

## REFERENCES

Awaji, T., T. Nakamura, T. Hatayama, and K. Akitomo, 1996: Tidal exchange through the Kuril Straits.

*Proc. Int. Workshop on the Okhotsk Sea and Arctic,*Tokyo, Japan, North Pacific Marine Science Organization, 142–146.Awaji, T., T. Nakamura, T. Hatayama, K. Akitomo, and T. Takizawa, 1999: Tidal exchange through the Kuril Straits.

*Proc. Second Int. Workshop on the Okhotsk Sea and Arctic,*Sidney, BC, Canada, North Pacific Marine Science Organization, 77–82.Bograd, S. J., A. B. Rabinovich, P. H. LeBlond, and J. A. Shore, 1997: Observations of seamount-attached eddies in the North Pacific.

,*J. Geophys. Res***102****,**(C6),. 12441–12456.Brink, K. H., 1982: A comparison of long coastal trapped wave theory with observations off Peru.

,*J. Phys. Oceanogr***12****,**897–913.Brink, K. H., 1991: Coastal-trapped waves and wind-driven currents over the continental shelf.

,*Annu. Rev. Fluid Mech***23****,**389–412.Brink, K. H., and D. C. Chapman, 1987: Programs for computing properties of coastal-trapped waves and wind-driven motions over the continental shelf and slope. Woods Hole Oceanographic Institution Tech. Rep. WHOI-87-24, 119 pp.

Brooks, D. A., and C. N. K. Mooers, 1977: Free, stable continental shelf waves in a sheared, barotropic boundary current.

,*J. Phys. Oceanogr***7****,**380–388.Buchwald, V. T., 1973: On divergent shelf waves.

,*J. Mar. Res***64****,**188–193.Buchwald, V. T., and J. K. Adams, 1968: The propagation of continental shelf waves.

,*Proc. Roy. Soc. London***A305****,**235–250.Cartwright, D. E., 1969: Extraordinary tidal currents near St. Kilda.

,*Nature***223****,**928–932.Cartwright, D. E., J. M. Huthnance, R. Spenser, and J. M. Vassie, 1980: On the St. Kilda shelf tidal regime.

,*Deep-Sea Res***27A****,**61–70.Chapman, D. C., 1983: On the influence of stratification and continental shelf and slope topography on the dispersion of subinertial coastally trapped waves.

,*J. Phys. Oceanogr***13****,**1641–1652.Clarke, A. J., 1991: The dynamics of barotropic tides over the continental shelf and slope.

*Tidal Hydrodynamics,*B. B. Parker, Ed., Wiley and Sons, 79–108.Crawford, W. R., 1984: Energy flux and generation of diurnal shelf waves along Vancouver Island.

,*J. Phys. Oceanogr***14****,**1600–1607.Crawford, W. R., and R. E. Thomson, 1982: Continental shelf waves of diurnal period along Vancouver.

,*J. Geophys. Res***87****,**9516–9522.Crawford, W. R., and R. E. Thomson, 1984: Diurnal-period continental shelf waves along Vancouver Island: A comparison of observations with theoretical models.

,*J. Phys. Oceanogr***14****,**1629–1646.Cummins, P. F., D. Masson, and M. G. G. Foreman, 2000: Stratification and mean flow effects on diurnal currents off Vancouver Island.

,*J. Phys. Oceanogr***30****,**15–30.Dziewonski, A., S. Bloch, and M. Landisman, 1969: Technique for the analysis of transient seismic signals.

,*Bull. Seism. Soc. Amer***59****,**427–444.Emery, W. J., and R. E. Thomson, 1997:

*Data Analysis Methods in Physical Oceanography*. Pergamon, 634 pp.Fine, I. V., 1980: Computation of trapped waves for the region of the Kuril Islands.

*Wave Processes in the Northwestern Part of the Pacific Ocean*(in Russian), S. S. Lappo, Ed., Far Eastern Branch of Russian Academy of Sciences, Vladivostock, 87–92.Flather, R. A., 1988: A numerical model investigation of tides and diurnal-period continental shelf waves along Vancouver Island.

,*J. Phys. Oceanogr***18****,**115–139.Foreman, M. G. G., and R. E. Thomson, 1997: Three-dimensional model simulations of tides and buoyancy currents along the west coast of Vancouver Island.

,*J. Phys. Oceanogr***27****,**1300–1325.Freeland, H. J., 1988: Diurnal coastal-trapped waves on the east Australian continental shelf.

,*J. Phys. Oceanogr***18****,**690–694.Freeland, H. J., and Coauthors. 1986: The Australian Coastal Experiment: A search for coastal-trapped waves.

,*J. Phys. Oceanogr***16****,**1230–1249.Freeland, H. J., A. S. Bychkov, F. Whitney, C. Taylor, C. S. Wong, and G. I. Yurasov, 1998: WOCE section P1W in the Sea of Okhotsk. 1: Oceanographic data description.

,*J. Geophys. Res***103****,**(C8),. 15613–15623.Gidrometeoizdat, 1960: Waters of the Asiatic Part of the USSR and Neighboring Foreign Foreign Regions, 1960:. Gidrometeoizdat, Leningrad (in Russian).

Gonella, J., 1972: A rotary-component method for analysing meteorological and oceanographic vector time series.

,*Deep-Sea Res***19****,**833–846.Harris, F. J., 1978: On the use of windows for harmonic analysis with the Fourier transform.

,*Proc. IEEE***66****,**51–83.Hsieh, W., 1982: On the detection of continental shelf waves.

,*J. Phys. Oceanogr***12****,**414–427.Huthnance, J., 1978: On coastal trapped waves: Analysis and numerical calculation by inverse iteration.

,*J. Phys. Oceanogr***8****,**74–92.Kantha, L. H., C. Tierney, J. W. Lopez, S. D. Desai, M. E. Parke, and L. Drexler, 1995: Barotropic tides in the global oceans from nonlinear tidal model assimilating tides. 2: Altimetric and geophysical implications.

,*J. Geophys. Res***100****,**25309–25317.Kashiwai, M., 1996: Importance of tidal exchange through Kuril Islands.

*Proc. Int. Workshop on the Okhotsk Sea and Arctic,*Tokyo, Japan, North Pacific Marine Science Organization, 59–63.Kovalev, P. D., and A. B. Rabinovich, 1980: Bottom measurements of tidal currents in the southern part of the Kuril–Kamchatka Trench.

,*Oceanology***20****,**294–299.Kowalik, Z., and A. Y. Proshutinsky, 1993: Diurnal tides in the Arctic Ocean.

,*J. Geophys. Res***98****,**16449–16468.Kowalik, Z., and I. Polyakov, 1998: Tides in the Sea of Okhotsk.

,*J. Phys. Oceanogr***28****,**1389–1409.Kulikov, E. A., 1979: Diffraction of a Kelvin wave on coastline irregularities.

*Wave Processes in Ocean-Border Regions*(in Russian), S. S. Lappo, Ed., Sakhalin Complex Scientific Research Institute, Yuzhno-Salkalinsk, 3–11.Kulikov, E. A., 1987: Generation of continental shelf waves by atmospheric disturbances.

,*Izv., Atmos. Oceanic Phys***23****,**575–580.Kulikov, E. A., and G. V. Shevchenko, 1992: The resonant generation of shelf waves by a moving cyclone.

,*Sov. J. Phys. Oceanogr***3****,**331–341.Lappo, S. S., A. V. Skripnik, and A. B. Rabinovich, 1978: Relation between atmospheric pressure and sea level in the Northwestern Pacific Ocean.

,*Sov. Meteor. Hydrol***12****,**39–44.LeBlond, P. H., and L. A. Mysak, 1978:

*Waves in the Ocean*. Elsevier, 602 pp.Likhacheva, O. N., and A. V. Skripnik, 1981: Formation of weather fluctuations of sea level in the vicinity of the Kuril Chain.

,*Izv., Atmos. Oceanic Phys***17****,**372–377.Likhacheva, O. N., and A. B. Rabinovich, 1986: Reaction of sea level to variations in atmospheric pressure in the Kuril Ridge area.

,*Oceanology***26****,**704.Liu, P. C., and G. S. Miller, 1996: Wavelet transforms and ocean current data analysis.

,*J. Atmos. Oceanic Technol***13****,**1090–1099.Luchin, V. A., 1996: Characteristics of the tidal motions in the Kuril straits. PICES Sci. Rep. 6, Institute of Ocean Sciences, Sidney, BC, Canada, 188–193.

Middleton, J. F., and D. G. Wright, 1990: Coastally trapped waves in a stratified ocean.

,*J. Phys. Oceanogr***20****,**1521–1527.Mooers, C. N. K., 1973: A technique for the cross spectrum analysis of pairs of complex-valued time series, with emphasis on properties of polarized components and rotational invariants.

,*Deep-Sea Res***20****,**1129–1141.Mysak, L. A., 1967: On the theory of continental shelf waves.

,*J. Mar. Res***25****,**205–227.Mysak, L. A., 1980: Recent advances in shelf wave dynamics.

,*Rev. Geophys. Space Phys***18****,**211–241.Mysak, L. A., P. H. LeBlond, and W. J. Emery, 1979: Trench waves.

,*J. Phys. Oceanogr***9****,**1001–1013.Nakamura, T., T. Awaji, T. Hatayama, and K. Akitomo, 2000: Tidal exchange through the Kuril Straits.

,*J. Phys. Oceanogr***30****,**1622–1644.Odamaki, M., 1994: Tides and tidal currents along the Okhotsk coast of Hokkaido.

,*J. Oceanogr***50****,**265–279.Ponomarev, V. I., E. P. Varlaty, and M. Y. Cheranyev, 1996: An experimental study of currents in the near-Kuril region of the Pacific Ocean and in the Okhotsk Sea. PICES Sci. Rep. 6, Institute of Ocean Sciences, Sidney, BC, Canada, 131–137.

Pugh, D. T., 1987:

*Tides, Surges, and Mean Sea-Level*. J. Wiley and Sons, 472 pp.Rabinovich, A. B., and G. V. Shevchenko, 1984: Two-step mechanism for dissipation of tidal energy in the ocean.

,*Trans. (Dokl.) USSR Acad. Sci., Earth Sci. Sec***276****,**228–231.Rabinovich, A. B., and A. E. Zhukov, 1984: Tidal oscillations on the shelf of Sakhalin Island.

,*Oceanology***24****,**184–189.Thomson, R. E., P. H. LeBlond, and W. J. Emery, 1990: Analysis of deep-drogued satellite-tracked drifter measurements in the northeast Pacific.

,*Atmos.–Ocean***28****,**409–443.Thomson, R. E., P. H. LeBlond, and A. B. Rabinovich, 1997: Oceanic odyssey of a satellite-tracked drifter: North Pacific variability delineated by a single drifter trajectory.

,*J. Oceanogr***53****,**81–87.Thomson, R. E., P. H. LeBlond, and A. B. Rabinovich, 1998: Satellite-tracked drifter measurement of inertial and semidiurnal currents in the northeast Pacific.

,*J. Geophys. Res***103****,**(C1),. 1039–1052.Wang, D-P., and C. N. K. Mooers, 1976: Coastal-trapped waves in a continuously stratified ocean.

,*J. Phys. Oceanogr***6****,**853–863.Yao, T., H. J. Freeland, and L. A. Mysak, 1984: A comparison of low-frequency current observations off British Columbia with coastal-trapped wave theory.

,*J. Phys. Oceanogr***14****,**22–34.Yefimov, V. V., and E. A. Kulikov, 1978: Application of the method of adaptive estimation of space-time spectra to the analysis of trapped waves.

,*Izv., Atmos. Oceanic Phys***14****,**532–537.Yefimov, V. V., and A. B. Rabinovich, 1978: Influence of boundary waves on formation of tides in the northwestern Pacific Ocean.

*Surface and Internal Waves*(in Russian), B. A. Nelepo, Ed., Marine Hydrophysical Institute, Sevatopol, 11–22.Yefimov, V. V., and A. B. Rabinovich, 1980: Resonant tidal currents and their relation to continental shelf waves of the northwestern Pacific Ocean.

,*Izv., Atmos. Oceanic Phys***16****,**805–812.Yefimov, V. V., E. A. Kulikov, A. B. Rabinovich, and I. V. Fine, 1985:

*Ocean Waves in Boundary Regions*(in Russian). Gidrometeoizdat, Leningrad, 280 pp.

## APPENDIX

### Rotary Multiple Filter Method

*H*(

*ω*) with a Gaussian window that isolates a specific center frequency,

*ω*

_{n}= 2

*πf*

_{n}:

*H*

_{n}

*ω*

^{−α[(ω − ωn)/ωn]2}

*α*. The higher the value of

*α*, the better the resolution in the frequency domain and the worse the resolution in the time domain. In this paper, we used

*α*= 100. A system of Gaussian filters leads to a constant resolution on a log

*ω*scale. The Fourier transform of

*H*

_{n}(

*ω*) is

*A*,

*C*are, respectively, the length of the anticlockwise and clockwise rotating vectors, and

*Φ*are their phases. Results consist of a matrix of amplitudes and phases of anticlockwise and clockwise motions with columns representing time and rows representing frequency (the

*f*–

*t*diagrams).

Deployment information for the 1993 Kuril Islands drifters

Relative contribution of energy in five frequency bands derived from the drifter trajectories in the vicinity of the Kuril Islands

Tidal constituents for the coast of Iturup (I) and Urup (U) Islands, Kuril Islands (from Gidrometeoizdat 1960). Numbers in column 1 refer to the numbers in Fig. 4

Computed wavelengths and phase speeds for coastal-trapped waves of diurnal frequency for the Pacific and Okhotsk shelves of the Kuril Islands