Internal Tide Observations from the Australian North West Shelf in Summer 1995

1182 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

PETER E. HOLLOWAY AND PAUL G. CHATWIN* School of Geography and Oceanography, University College, University of New South Wales, Australian Defence Force Academy, Canberra, Australia

PETER CRAIG CSIRO Marine Research, Hobart, Tasmania, Australia

(Manuscript received 7 January 2000, in ®nal form 6 July 2000)

ABSTRACT Observations are presented of the internal tide over the continental shelf and slope from a cross section on the Australian North West Shelf. Data collected from moored instruments and repeated pro®le measurements during the summer months of 1995 show an energetic, large amplitude, shoreward propagating semidiurnal internal tide. Multiple generation sites are suggested, coinciding with near-critical bottom slopes. In water less than approximately 200 m deep, the vertical structure of the internal tide is predominantly a ®rst vertical mode, whereas in deeper water over the slope the vertical structure is more complicated with currents and vertical displacements intensi®ed in the lower part of the water column. The internal tide is largely con®ned to a region approximately 100 km wide, from water depths between 70 and 1000 m. Strong generation and dissipation of the internal tide energy is observed over this region and there is evidence that the dissipated energy impacts on the vertical mixing of the density ®eld, particularly near the shelf break and upper continental slope. Even though the diurnal barotropic tidal currents are weak, a diurnal internal tide is observed and appears to be generated

over a section of the continental slope that is at the critical slope for the K1 tidal frequency. The M4 harmonic

is also observed and this results from nonlinear interactions of the M2 baroclinic tide.

1. Introduction from the Bay of Biscay and suggest wave propagation Internal tides over continental shelf and slope regions along characteristics. In this situation, the signal propa- are characterized by high spatial and temporal variability, gates along a narrow beam with signi®cant vertical phase have phases that are not locked to that of the barotropic propagation and re¯ects off the sea surface and seabed, tide and propagate both shoreward and seaward. The consistent with simple theoretical models of internal wave waves are often energetic and can signi®cantly contribute generation over critical slopes (e.g., Baines 1982). How- to ocean mixing (e.g., Pingree et al. 1986). Most obser- ever, in many instances, and perhaps in regions of less vations have shown internal tides at semidiurnal frequen- steep topography, internal tides are well described in terms cies rather than diurnal. An exception are the observations of vertical modes, usually dominated by the ®rst mode. presented by Leaman (1980) of a diurnal internal tide, at For example, Sherwin (1988) analyzes observations from near-critical bottom slope, off the west Florida continental the Marlin Shelf north of Ireland and Rosenfeld (1990) shelf. The propagation of internal tides, seaward from steep analyzes observations from the shelf off northern Cali- continental slope topography, can be described in terms fornia. In both cases the vertical structure of the internal of energy propagation along internal wave characteristics, tide is dominated by the ®rst vertical mode. While internal that is, along the path of the group velocity vector. For wave ``beams'' can be described as the sum of high-order example, Pingree and New (1989) present observations vertical modes, there seems to be little observational evidence of transition from regions dominated by high modes to those dominated by low modes. Although numerous surveys of internal tides near con- * Current af®liation: Fisheries Research Services Marine Labora- tory, Aberdeen, United Kingdom. tinental margins have been reported in the literature [see Huthnance (1989) for a review], most studies have been limited in their spatial and/or temporal coverage. In an Corresponding author address: Dr. Peter Holloway, School of Ge- attempt to help rectify this gap in observational detail and ography and Oceanography, University College, University of New South Wales, Australian Defence Force Academy, Canberra ACT to address some of the questions raised above, a ®eld 2600, Australia. program was run on the Australian North West Shelf E-mail: ࿞[email protected] (NWS), obtaining detailed observations of the internal tide

᭧ 2001 American Meteorological Society

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TABLE 1. Details of moored instrumentation. Water Sampling Moor- depth Current meter depths Temperature sensor depths interval ing (m) Instrument type (m) (m) (min) C12 750 Aanderaa RCM5/5S current meters 150, 300, 450, 600 150, 300, 450, 600 5 C10 300 Interocean S4 current meters 40, 90, 140, 190, 240, 290 40, 90, 140, 190, 240, 290 5 C6A 125 RDI 150 KHz ADCP every 4 from 22±114 5 C6B 125 Steedman Acoustic current meters 12, 39, 66, 120 12, 39, 66, 120 2 C6C 125 Aanderaa thermistor chain every 10 from 20±120 10 C2 65 Steedman Acoustic current meters 25, 55 25, 55 5 C2 65 Applied Micro Systems Tide gauge 5 over a cross section where energetic internal waves have In this paper, observations are analyzed in order to previously been reported (Holloway 1984, 1985, 1988, de®ne the vertical structure of the internal tide in terms 1994). The aim was to measure the internal tide from of both vertical displacement and horizontal currents at offshore of the generation region and to track the complete M2 (12.42 h), S 2 (12.00 h), K1 (23.93 h), and M 4 (6.21 cycle of evolution of the internal tide as it propagated h) tidal frequencies. Also investigated are the changes shoreward and ®nally dissipated over the outer continental in horizontal structure across the continental slope, and shelf. Measurements were obtained from moored current the energetics of the internal tide. meters, an acoustic Doppler current pro®ler (ADCP), and a thermistor chain as well as from detailed shipborne CTD 2. Measurement program and ADCP pro®le measurements, all made during the summer months (Jan±Mar) of 1995 when the internal tide is Moorings containing current meters, ADCPs, therm- strongest (see Holloway 1988). istor chains, and water level recorders were deployed

FIG. 1. Map showing bathymetry, mooring, and CTD/ADCP stations on the Australian North West Shelf.

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FIG. 2. A cross section showing moorings and instrumentation. on the NWS in January 1995. A number of instrument tions. Locations are referred to as C1 to C13, where all failures reduced the array to four mooring locations and are CTD stations and some are mooring locations as instruments at depths as indicated in Table 1. Bathym- well. Despite instrument failures, the mooring array pro- etry, CTD, and mooring locations are shown in Fig. 1, vides results ranging from 750-m depth on the slope and Fig. 2 is a cross section showing instrument posi- into 65-m depth on the outer shelf. Moorings provide good vertical coverage, particularly at location C6 where the ADCP gives current measurements every 4 TABLE 2. Details of CTD pro®le measurements. Local times are m from 26 to 114 m in 125-m water depth. The moored used. The asterisks indicate a mooring location in addition to the CTD measurements. instruments collected data for approximately 2 months at sampling rates of 2, 5, or 10 min. (Table 1). Water Distance Repeat A detailed CTD survey was conducted in January Loca- depth from C1 cycle 1995 during the mooring deployment. At each of 13 tion Start time Finish time (m) (km) (min) locations in a section across the shelf and slope (Fig. C1 1300 15 Jan 0100 16 Jan 65 0 30 1), repeated CTD pro®les were measured over a 13-h C2* 0200 16 Jan 1500 16 Jan 65 10.5 30 C3 1600 16 Jan 0500 17 Jan 70 18.8 30 period in order to measure one complete cycle of the C4 2200 17 Jan 1100 18 Jan 86 27.0 30 semidiurnal tide. At most locations pro®les were mea- C5 1230 18 Jan 0130 19 Jan 116 35.0 30 sured every 30 min, but less frequently at deep stations. C6* 0730 17 Jan 1830 17 Jan 124 43.3 30 Simultaneously, the ship's hull-mounted ADCP provid- C7 1830 14 Jan 0730 15 Jan 132 53.3 30 ed current pro®les in 8-m bins from 16 to approximately C8 0330 19 Jan 1230 19 Jan 162 63.5 30 C9 1800 19 Jan 0700 20 Jan 242 76.3 30 230 m depth, or to 14% of depth above the seabed in C10* 0830 20 Jan 2130 20 Jan 302 89.3 60 shallower water, for example, to 86 m in 100 m. Table C11 1930 13 Jan 0830 14 Jan 416 103.8 60 2 lists the times and dates the pro®les were measured, C12* 0000 21 Jan 1300 21 Jan 764 122.8 60 and the repeat frequency of the pro®les. Note that it C13 1500 21 Jan 0430 22 Jan 1382 140.8 90 took 7 days to complete the section from C1 to C13.

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TABLE 3. Barotropic tidal current properties. Ellipse parameters are semimajor (a) and semiminor (b) axis lengths in cm sϪ1, where ϩ␷e (b) indicates anticlockwise and Ϫ␷e(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (␪) in degrees anticlockwise from true north. Mooring C6 values are calculated separately from the moored ADCP data (C6A) and current meters (C6B). M S K O Moor- 2 2 1 1 ing abg␪ abg␪ abg ␪ abg␪ C12 5.6 0.6 209 109 3.7 Ϫ0.5 290 117 1.4 0.1 234 66 1.1 0.3 168 75 C10 9.6 Ϫ1.4 225 82 6.7 Ϫ0.6 305 86 2.7 0.3 217 82 0.9 Ϫ0.1 208 110 C6A 16.0 4.6 223 43 9.6 1.3 301 45 3.3 0.9 245 49 0.8 0.3 135 68 C6B 17.6 4.0 220 58 10.7 1.2 301 61 3.7 0.6 224 56 0.9 Ϫ0.2 99 151 C2 26.0 4.5 219 51 16.3 2.2 297 53 4.0 1.1 231 58 1.6 Ϫ0.9 185 48

For the analysis of the moored instrument time series, and K1 values plotted in Fig. 3 as tidal ellipses. Tidal 27 days of data are used, chosen as a common block of currents are largely semidiurnal, and diurnal speeds are data to all instruments. This spans the period from 18 approximately 10% of semidiurnal values. The ratio of

January to 14 February 1995. S 2/M 2 semimajor axis lengths average 0.65 so that the spring±neap cycle is pronounced. The M2 and S 2 ellipses are aligned cross-shelf for the shallower moorings C6 3. The barotropic tide and C2 but turn and become more aligned alongshelf The moored current meter data are analyzed in order in deeper water. Cross-shelf alignment of the ellipses is to separate the barotropic tide from the baroclinic or important for internal tide generation, which is most internal tidal signals. The analysis of 27 days data re- effective for a large ¯ux of water up and down the solves the M 2 and S 2 constituents in the semidiurnal topographic slope. There are only small changes in band along with diurnal constituents and higher-fre- phase (10Њ) between the moorings for the barotropic quency harmonics. constituents. At location C6, two estimates of the bar- The barotropic currents at each mooring are de®ned otropic tidal currents are made, one as a depth average as the depth-averaged time series. Tidal harmonic anal- of the four current meters and the other from a depth ysis (Foreman 1978) is then used to de®ne barotropic average of the ADCP measurements. The differences in tidal currents. Results are presented in Table 3 with M2 semimajor axis lengths and phases from these indepen-

FIG. 3. Barotropic M2 (solid) and K1 (dotted) tidal ellipses at moorings C12, C10, C6B, and C2.

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TABLE 4. Tide elevation constituents for Dampier (from Holloway 1983), mooring C2, and North Rankin, a natural gas production platform approximately 4 km south of mooring C6 (constituents supplied by S. Buchan, 1999 personal communication). Phase is in local time, 8 h ahead of UTC.

M2 S2 K1 O1 A A A A Location (cm) g (cm) g (cm) g (cm) g Dampier 111.9 306 64.6 14 23.2 292 14.5 274 C2 83.9 295 51.8 15 24.7 308 13.1 269 North Rankin 73.3 293 41.3 359 21.1 290 13.7 270 dent estimates are only slight for the semidiurnal con- total current minus the depth-averaged value. This anal- stituents, the largest discrepancy in the ellipse properties ysis de®nes the constituents of the internal tide. In ad- being a 15Њ difference in ellipse orientation. For ref- dition, temperature data are used to compute vertical erence, the major tidal elevation constituents are listed displacements. Assuming continuity of temperature, and in Table 4 for three locations. All of the constituents neglecting horizontal advection of horizontal tempera- show phase propagation towards the coast, consistent ture gradients, the vertical velocity is de®ned as with the cross-shelf alignment of the tidal ellipses. tץ/Tץ w(z, t) ϭ , (1) zץ/Tץ Internal tide from moored instruments .4 For each current meter record a tidal analysis is car- where T(z, t) is the temperature as a function of depth ried out on the residual time series, computed as the (z) and time (t). The computed vertical velocity time series are harmonically analyzed and amplitudes con- verted to vertical displacements assuming sinusoidal motion and using w ϭϪi␻␨, where ␻ is the wave frequency and ␨ the vertical displacement. Vertical displacement phase is computed as the vertical velocity phase plus 90Њ. The average strati®cation for this time period can be seen from the 27-day averages at each current meter, producing the temperature pro®les plotted in Fig. 4. These pro®les are compared to the 13-h time-averaged CTD temperature pro®le measured at the offshore location C13 at the start of the mooring time series. The strati®cation shows an approximate linear gradient in the upper 300 m, corresponding to a buoyancy frequency of 0.013 sϪ1, with a weaker gradient in deeper water. There is only a thin, ϳ20 m, surface mixed layer. The similarity in all plotted pro®les suggests fairly constant strati®cation during the measurement period. The ®rst and second vertical modal functions for vertical displacement and horizontal velocity at locations C6, C10, and C12 are computed (e.g., Munk 1981) and plotted in Fig. 5. For each location a time-averaged density pro®le, computed from the repeated CTD pro- ®les over 13 hours, is used to calculate the modal functions. At locations C6 and C10 in depths 125- and 300 m respectively, the modal functions are symmetrical with maximum displacement and zero velocity at middepth. At C12 in 750-m depth, the maximum displacement is at 250 m with stronger velocities in the upper part of the water column than in the lower part. An ocean with constant buoyancy frequency leads to symmetric modal functions for internal waves with the ®rst mode having maximum vertical displacements and zero hor- FIG. 4. Strati®cation determined from a 27 day (18 Jan±14 Feb 1995) time average of temperatures at each current meter location izontal velocity at middepth and equal maxima in ve- and the thermistor chain. The tidal-cycle-averaged temperature from locity at the sea surface and seabed. The upper and CTD station C13 on 22 Jan 1995 is also shown. lower water column velocities are 180Њ out of phase.

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FIG. 5. First and second theoretical modal functions for vertical displacement (solid lines) and horizontal velocity

(dotted lines) calculated for M2 tidal frequency and observed strati®cation from each of three locations. Buoyancy frequency pro®les, calculated from observed temperature and salinity data, are also plotted.

These symmetric modal functions are expected on the are listed in Tables 5±8 for mooring locations C12, C10, NWS for water depths less than ϳ300 m with the sym- C6, and C2 respectively. In addition, cross-section plots metry lost in deeper water where buoyancy frequency of the cross-shelf component of the baroclinic currents varies substantially with depth. and phases are plotted in Fig. 6 and displacement am-

Baroclinic tidal current properties and displacement plitudes and phases are plotted in Fig. 7 for the M2 amplitudes and phases for M2, S 2, and K1 constituents constituent. A strong semidiurnal internal tide is seen

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TABLE 5. Mooring C12 baroclinic tidal currents and vertical displacements (␨). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm sϪ1, where ϩ␷e(b) indicates anticlockwise and Ϫ␷e(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (␪) in degrees anticlockwise from true north. Velocities are in centimeters per second and displacements in meters. M S K Depth 2 2 1 (m) abg␪ abg ␪ abg␪ 150 4.5 Ϫ0.4 209 26 2.3 0.9 101 173 2.8 1.5 274 77 300 2.2 Ϫ0.1 334 153 2.3 0.7 218 32 1.5 0.8 241 20 450 2.5 Ϫ0.3 50 17 2.1 1.0 307 15 1.4 0.6 119 131 600 2.1 Ϫ1.1 178 87 1.8 1.0 235 53 0.9 Ϫ0.4 93 70 ␨ g ␨ g ␨ g 150 4.4 142 1.9 348 0.3 237 300 3.9 246 1.8 121 1.7 351 450 9.1 280 4.3 204 4.7 295 600 6.4 299 2.0 8 6.6 290

at moorings C6 and C10, in water depths of 125 and While the ratio of S 2/M2 semimajor axis lengths is 300 m, with much weaker signals at the shallower moor- approximately constant between moorings (0.65) for the ing C2 and the deep mooring C12. Maximum baroclinic barotropic tide, this is not found to be true for the in- Ϫ1 currents are at the M2 frequency and reach 0.13 m s ternal tide where the ratios vary substantially. For ex- at C6 and 0.16 m sϪ1 at C10 with vertical displacement ample, at location C2, S 2 is larger than M2. This is most amplitudes of 27 m at mooring C6 and 13 m at C10. likely a result of energy smearing around the semidi- The S 2 currents and displacements are largest at C6 urnal tidal frequencies. reaching 0.08 m sϪ1 and 13 m, respectively. At mooring An M 4 internal tide, generated through nonlinear in- C12 in water depth 750-m, baroclinic M 2 currents reach teractions of the semidiurnal (M Ϫ1 2) constituent, is ob- 0.04 m s and vertical displacements reach 9 m. Moor- served at all mooring locations (values are not listed in ing C6 shows a simple ®rst-mode structure with a 180Њ the tables) with currents up to 0.03 m sϪ1 and displace- phase change in currents at approximately middepth, ments of 3 m. Maximum values in M are seen near the corresponding to the maximum in the vertical displace- 4 ments. There is little phase change through depth in the surface at mooring C10 in water depth 300 m. Again displacements, consistent with a ®rst-mode structure. At the vertical structures, particularly at mooring C6, ap- 300-m depth, mooring C10 shows a similar picture of pear to follow those of ®rst-mode waves. A signi®cant velocities and displacements, suggesting a dominance K1 internal tide is also observed, strongest at moorings C10 and C6. Displacements reach 10 m and currents of ®rst-mode waves. At the deeper mooring C12, M2 Ϫ1 and S 2 displacements are largest in the lower part of the 0.09 m s . Note that at 20Њ latitude, K1 is still a su- water column, and there is an increase in phase lag from perinertial frequency, where | f/␻| ϭ 0.69, f is the Cor- the upper to lower measuring depths. iolis parameter (Ϫ5 ϫ 10Ϫ5 sϪ1), and ␻ is the wave

TABLE 6. Mooring C10 baroclinic tidal currents and vertical displacements (␨). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm sϪ1, where ϩ␷e(b) indicates anticlockwise and Ϫ␷e(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (␪) in degrees anticlockwise from true north. Velocities are in centimeters per second and displacements in meters. M S K Depth 2 2 1 (m) abg␪ abg␪ abg␪ 40 15.9 6.4 284 38 3.3 2.0 233 129 2.4 1.0 102 52 90 10.5 1.8 129 49 3.4 2.3 44 25 5.0 2.4 209 43 140 2.3 1.8 328 21 1.5 1.1 85 26 5.3 2.4 193 16 190 5.3 1.4 284 59 2.1 0.8 208 29 1.8 0.2 73 96 240 13.5 5.0 131 22 3.2 0.3 226 41 6.0 4.4 55 11 290 11.2 5.6 291 58 2.7 Ϫ0.6 97 137 8.6 5.3 209 63 ␨ g ␨ g ␨ g 40 7.6 358 2.2 90 4.5 281 90 7.8 330 1.8 90 3.7 286 140 13.1 331 2.8 118 1.9 299 190 9.5 335 5.3 142 4.7 297 240 6.8 348 2.8 162 2.8 265 290 2.2 343 0.5 66 1.9 273

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TABLE 7. Mooring C6 baroclinic tidal currents and vertical displacements (␨). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm sϪ1, where ϩ␷e(b) indicates anticlockwise and Ϫ␷e(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (␪) in degrees anticlockwise from true north. Velocities are in centimeters per second and displacements in meters. Currents are from mooring C6A and vertical displacements from the thermistor chain at mooring C6C. M S K Depth 2 2 1 (m) abg␪ abg ␪ abg␪ 26 12.5 2.3 352 30 8.0 3.3 120 19 4.3 0.6 319 29 30 11.6 2.1 351 30 7.5 3.1 116 18 3.5 1.1 320 24 34 10.6 1.9 349 30 6.9 2.8 116 18 2.7 1.5 324 23 38 9.6 1.8 346 29 6.2 2.4 115 18 2.2 1.5 350 40 42 8.4 1.8 346 28 5.6 1.9 114 18 2.3 1.1 191 48 46 7.2 1.8 348 29 5.1 1.6 113 17 2.4 0.8 22 39 50 5.5 1.4 359 29 4.4 1.3 112 11 2.3 0.5 37 26 54 4.6 0.9 6 29 3.8 1.1 113 4 2.2 0.4 47 27 58 3.7 0.2 13 26 2.8 0.9 115 179 2.3 0.2 57 34 62 2.7 Ϫ0.4 26 16 1.9 0.7 123 173 2.4 Ϫ0.3 65 37 66 2.3 Ϫ0.6 56 169 0.9 0.5 124 156 2.5 Ϫ0.9 77 28 70 2.9 0.4 78 154 0.6 0.3 103 65 2.7 Ϫ1.2 91 14 74 3.2 Ϫ0.8 191 144 3.7 0.4 293 161 2.1 Ϫ1.4 253 69 78 3.5 Ϫ1.0 169 175 4.1 0.6 289 167 1.5 Ϫ1.3 111 41 82 4.5 Ϫ0.6 161 15 4.4 0.9 288 174 1.5 Ϫ0.7 151 23 86 6.0 Ϫ0.1 160 26 4.8 1.1 286 179 1.9 Ϫ0.4 165 32 90 7.7 0.4 161 33 5.4 1.4 288 3 2.5 Ϫ0.1 172 45 94 9.3 0.7 164 37 5.7 1.5 288 9 3.1 0.5 4 55 98 10.9 0.9 166 39 5.8 1.7 290 16 3.5 0.6 25 67 102 12.3 1.1 166 40 5.9 2.1 295 21 3.7 0.3 36 76 106 13.0 1.2 170 39 6.7 2.5 303 25 3.2 Ϫ0.0 42 73 110 13.3 0.9 165 36 6.7 3.1 301 17 3.0 0.4 45 51 ␨ g ␨ g ␨ g 20 4.1 351 2.2 35 3.0 35 30 9.2 19 3.6 114 1.7 19 40 13.4 23 4.3 122 4.5 352 50 17.6 22 6.3 136 6.3 345 60 21.8 22 5.8 136 9.7 330 70 23.0 8 10.2 141 8.0 342 80 26.7 12 13.3 146 9.9 332 90 18.2 8 11.1 137 8.2 331 100 16.2 17 11.3 129 8.5 325 110 14.8 346 12.0 140 1.3 318 120 8.3 359 6.1 122 1.9 247 frequency. Hence, diurnal internal tides can exist as free- mode waves. At the deepest mooring (750 m) the ly propagating waves. phase of the elevations increase with depth, and the The results from the mooring data show a large- depth of the maximum in vertical displacements and amplitude internal tide con®ned to an approximately zero crossing in velocity are inconsistent with a ®rst- 100 km wide section of the upper continental slope. mode internal wave. The largest waves occur between 125 and 300 m The phase relationship between vertical displace- depths and appear to be predominantly simple ®rst- ments and baroclinic currents determines the direction

TABLE 8. Mooring C2 baroclinic tidal currents and vertical displacements (␨). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm sϪ1, where ϩ␷e(b) indicates anticlockwise and Ϫ␷e(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (␪) in degrees anticlockwise from true north. Velocities are in centimeters per second and displacements in meters. M S K Depth 2 2 1 (m) abg␪ abg␪ abg␪ 25 3.2 Ϫ1.1 111 39 4.2 Ϫ0.1 355 65 1.1 Ϫ0.9 251 72 55 3.2 Ϫ1.1 291 39 4.2 Ϫ0.1 175 65 1.1 Ϫ0.9 71 72 ␨ g ␨ g ␨ g 25 2.5 262 1.8 6 1.3 284 55 1.5 275 0.6 170 1.1 310

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FIG. 6. Pro®les of M2 internal tide cross-shelf current amplitude and phase from moored current meters from a 27-day data analysis over the period 18 Jan±14 Feb 1995. of wave propagation. If wave propagation is in the pos- velocity and vertical displacement amplitudes and phases itive x direction, then there will be a 180Њ phase dif- and strati®cation to de®ne the buoyancy frequency. Re- ference between the vertical displacements and the near- sulting values computed using (A6) are listed in Table 9 surface currents when de®ned as positive in the x di- and plotted in Fig. 8. Fluxes are approximately perpen- rection (e.g., see Gill 1982, p. 125). For the M2 con- dicular to the bathymetry contours and directed onshore. stituents presented above there is an approximately 180Њ At the shallow mooring C2 the energy ¯ux is essentially phase difference between the vertical displacements and zero. Values are largest between depths of 125 and 300 upper cross-shelf currents at moorings C6, C10, and m with a maximum of 1089 W mϪ1 at 300-m depth. C12, indicating shoreward wave propagation. This also Holloway (1984) found an average onshore energy ¯ux tends to be true for the other constituents. of 326 W mϪ1 at a mooring close to mooring C6 in water The energy ¯ux associated with the internal tide is an depth 125 m, derived as an average from observations important quantity, as some of the energy will be avail- over the period January±July 1982. This is consistent able for ocean mixing. The depth-integrated energy ¯uxes with values listed in Table 9. are calculated for each mooring location for the M2 tidal The energy ¯ux of the barotropic tide is calculated constituent. The method, discussed in the appendix, uses as (Pugh 1987)

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FIG. 7. Pro®les of M2 internal tide vertical displacement amplitude and phase from moored current meters from a 27-day data analysis over the period 18 Jan±14 Feb 1995.

1 the corresponding phases. Using values from Tables 3 Fb ϭ ␳gH␩00u cos(g␩ Ϫ gu), (2) 2 and 4, M 2 barotropic energy ¯uxes for moorings C2 and C6A are 1.7 ϫ 104 and 2.5 ϫ 104 WmϪ1, respectively, where ␳ is the water density, ␩ 0 and u 0 the barotropic and are directed approximately onshore (in the direction tidal elevation and velocity amplitudes, and g␩ and gu of ellipse orientation as listed in Table 3). The baroclinic energy ¯uxes are then approximately 4% of the baro-

TABLE 9. Depth-integrated M2 baroclinic energy ¯uxes calculated tropic values. from mooring data over the period 18 Jan±14 Feb 1995. Cross-shelf is positive towards the coast (135Њ east of north) and direction is in degrees east of north. 5. Internal tides from ship CTD and ADCP pro®les Cross-shelf Alongshelf Magnitude Direction Location (W mϪ1) (W mϪ1) (W mϪ1) (deg) The repeated CTD pro®les over a tidal cycle reveal C2 Ϫ1 0 1 316 internal wave activity, to varying degrees, at all sites. C6A 555 Ϫ54 558 141 Example observations from locations in water depths C10 1076 168 1089 126 65, 120, and 240-m are shown in Fig. 9 as depth±time C12 195 Ϫ94 217 161 contoured distributions of sigma-t. At the deeper loca-

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FIG. 8. Depth integrated M2 energy ¯uxes of the internal tide calculated from mooring data over the 27 day period 18 Jan±14 Feb 1995, and from CTD/ADCP data collected between 14 and 22 Feb 1995. tion (C9), isopycnals at middepth oscillate with a height 20-min values of current to a depth of approximately of about 40 m over the 13 h. Although there is some 230 m in 8-m depth bins. For the periods corresponding higher-frequency variability, there is clearly an internal to the CTD pro®le measurements, and for stations out tide oscillation that is approximately sinusoidal. At to depth 250 m, the depth-averaged current for each 120-m depth (C6) a large amplitude internal tide is seen time measurement is removed from the total current to and is strongly distorted from a sinusoidal shape. Part provide an estimate of the baroclinic signal. Example of the wave shows shocks on both the front and back plots of depth±time sections of cross-shelf baroclinic face of the wave, giving an approximately square wave- currents are plotted in Fig. 10 for locations C3, C6, and form. The 23.4 isopycnal varies from 30 to 105 m depth, C9, corresponding to the sigma-t sections in Fig. 9. The giving a wave amplitude of 38 m. In water depth 65 m velocities clearly show the reverse of phase between (C3), an internal solitary wave is seen, with positive upper and lower layers of the water at locations C3 and polarity and an amplitude of 27 m. The longer-period C6 and correspond well with the elevations seen from internal tide is weak at this location. These examples the CTD data. The phase relationship between eleva- show the changing nature of the internal waves from tions and velocities (offshore surface ¯ow at the peak deeper to shallow water. The nonlinear steepening of of the wave) shows the waves to be propagating on- the internal tide and the formation of internal solitary shore. Baroclinic velocities reach 0.40 m sϪ1 at location waves from the internal tide are important character- C6, substantially larger than the M 2 value calculated istics of the internal wave ®eld on this and other con- from the 27-day time series (Table 7a). The soliton char- tinental shelf and slope regions. These nonlinear pro- acter of the wave at C3 is seen in the velocity as is the cesses are also most probably important in determining sharp front of the internal tide at C6. A more complex the mechanisms of dissipation of the internal tide. How- velocity ®eld is seen at C9 with multiple ¯ow reversals ever, these details are not addressed in this paper. The with depth. nonlinear features may also produce some aliasing of Vertical displacement amplitudes and phases are cal- the measured internal tide, although this is suspected to culated for each CTD station using the time series of be a small effect in ®tting a tidal harmonic to measured sigma-t over a 13-h period. Vertical velocities are com- signals. Similarly, assuming linear dynamics may lead puted at 2-m depth intervals using Eq. (1) with tem- to some small error in estimating the energy contained perature replaced by sigma-t. The data are ®rst smoothed in the internal tide. in depth using a running-mean ®lter in order to reduce The hull-mounted ADCP on RV Franklin provided noise in the vertical velocity time series that can arise

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FIG. 9. Time sequence of isopycnal depths over 13-h periods from FIG. 10. Time sequence of cross-shelf component of baroclinic CTD measurements at locations C3, C6, and C9. Plots are over the velocity over 13-h periods from ship ADCP measurements at loca- total water depth. Dates are listed in Table 2 and coincide with Fig. tions C3, C6, and C9. Plots are over the total water depth, with solid 10. contours showing onshore ¯ow and dotted contours showing offshore ¯ow. Dates are listed in Table 2 and coincide with Fig. 9. Contour intervals are 5 cm sϪ1 for C3 and C9 and 10 cm sϪ1 for C6. when the vertical gradient is small. Only one semidiurnal constituent is resolved from the harmonic analysis and this is at the M2 period. However, this is really the at 1400 m the maximum occurs around 800 m. There total semidiurnal signal. Resulting vertical pro®les of is considerable change in phase with depth at these deep- displacement amplitude and phase are plotted in Fig. 11 er locations. Strong cross-shelf internal tide currents are and a contoured distribution of amplitude in the upper seen in the shallower region coinciding with the large 500 m is shown in Fig. 13. amplitude waves. Currents have maxima near the sea Harmonic analysis of hull-mounted ADCP current surface and seabed, a zero point at middepth, and a 180Њ time series provides the baroclinic cross-shelf and change in phase with depth. This is consistent with the alongshelf M2 amplitudes and phases for the periods ®rst-mode structure. The contoured section of velocities corresponding to the CTD pro®le measurements. Pro- (Fig. 13) shows a distinct change in the vertical structure ®les of cross-shelf amplitude and phase are plotted in between internal tides inshore and offshore of approx- Fig. 12 and contours of the amplitude in the upper 500 imately 200-m depth. The vertical displacements at m are plotted in Fig. 13. In the shallower regions, to a 400-m depth (C11) and 1400-m depth (C13) cannot be depth of approximately 150 m, vertical displacements described by a ®rst-mode structure. In particular, in the appear as simple ®rst-mode distributions with maxima deeper water the vertical displacements are largest near occurring near middepth and uniform phase with depth. the seabed or in the lower part of the water column. It Amplitudes are large, up to 40 m in water 150 m deep. is noted that the observations are not synoptic but are However, the pro®le at 400-m depth has a maximum in from different times over a period of 7 days and some the lower part of the water column, and for the pro®le caution must be exercised in interpreting cross-shelf cor-

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FIG. 11. Vertical pro®les of vertical displacement amplitude and phase at the semidiurnal period, calculated from CTD data between 14 and 22 Jan 1995. relations in the data. However, there are strong simi- cations in water less than 250 m (C1 to C9), with values larities in the values of the M 2 vertical displacement tabulated in Table 10 and plotted as vectors in Fig. 8. phases from the moorings C10 and C6 (Fig. 7) and the Fluxes are predominantly onshore, reaching 1821 W corresponding CTD data (Fig. 11). Also, the variations mϪ1 in 132-m water depth. The values are negligible at in phase with depth are similar from mooring and CTD the shallow inshore moorings and also weaken in deeper data at location C12. This suggests a quite stable phase water (beyond C8 in 162 m). The cross-shelf component over time for the vertical displacements. is generally stronger than that alongshelf. Fluxes are Energy ¯ux is calculated from the elevations and bar- consistent with those calculated from the moorings, oclinic currents using Eq. (A6), and a section of cross- where values are computed from 27-day data segments, shelf values is plotted in Fig. 13. The dominant feature is although little over double the strength at location C6. a strong onshore ¯ux in the lower half of the water column on the upper slope, between depths 80 and 150 m where 6. Energy dissipation peak values are 40 W mϪ2. The offshore ¯ux in the upper part of the water column is much weaker, reaching The baroclinic energy ¯ux vectors plotted in Fig. 8 8WmϪ2. Other patches of onshore ¯ux are seen as well show a strong onshore ¯ux between approximately 300 as weaker regions of offshore ¯ux. and 100 m water depths, and this indicates strong in- Depth-integrated energy ¯uxes are calculated for lo- ternal tide generation in this region. There is little energy

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FIG. 12. Vertical pro®les of cross-shelf baroclinic current amplitude and phase at the semidiurnal period, calculated from the ship-mounted ADCP for the period between 14 and 22 Jan 1995. propagating onto the shelf and little energy propagating Holloway, unpublished manuscript). Further, the baro- from the deep water onto the continental slope. This clinic energy density is integrated over depth, and results indicates that much of the internal tide energy is dis- are listed in Table 11. It can be seen that the barotropic sipated in the upper slope region, close to the region energy density steadily increases from offshore as tidal where the internal tide is being generated. The ener- velocities increase in shallower water. In comparison, getics of the internal tide is examined to estimate the the energy density of the baroclinic tide reaches a max- contribution to mixing and also to compare the energy imum and is approximately equal to that of the baro- with that of the barotropic tide. tropic tide between locations C6 and C10 (depth 125 to The energy density of the internal tide averaged over 300 m), but is substantially less both over the shelf and a wave period is given as in deeper water. From the above results, it is reasonable to assume that 1 22 2 2 E(z) ϭ ␳(z)[N (z)␨000(z) ϩ u (z) ϩ ␷ (z)], (3) most of the energy of the baroclinic tide is dissipated 4 close to the region where it is generated. Only a small amount of the energy propagates onto the shelf and in where ␨ 0, u 0, and ␷ 0 are the amplitudes of vertical displacement and horizontal velocity (LeBlond and Mysak deep water the energy ¯ux is small. Assuming that all 1978). The barotropic energy density averaged over a of the energy is dissipated close to the region of gen- wave period is given as eration, the energy dissipation rate can be computed as ␧ϭ# Edz/T, where T is the tidal period. These values 1 E ϭ ␳ [g␩222ϩ H(U ϩ V )], (4) are listed in Table 11 and show strongest dissipation b 4 00 bb (ϳ0.04 W mϪ2) at C6 and C10, corresponding to the largest energy density values. This dissipation rate can where ␩ is the amplitude of the surface elevation, U 0 b be compared to that expected from bottom frictional and V are the amplitudes of the barotropic tidal cur- b dissipation. For an oscillatory tidal ¯ow of amplitude rents, ␳ is the depth-averaged density, and H is the 0 U , Pugh (1987) shows that the dissipation rate due to water depth. For each of the four moorings, the M 0 2 a quadratic bottom stress is barotropic and baroclinic energy densities are calculated using (3) and (4). Velocity and elevation values are from 4 ␧ϭ ␳CU3, (5) values listed in Tables 3, 4, 5, 6, 7a, and 8. In addition, cd3␲ d 0 M2 tidal elevation amplitudes of 0.61 and 0.63 m for locations C12 and C10, respectively, were used in (4), where Cd ϳ 0.0025 is the quadratic bottom friction co- obtained from a numerical tidal model of the region (P. ef®cient. Applying (5) to the internal tide, where U 0 is

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TABLE 10. Depth-integrated semidiurnal baroclinic energy ¯uxes calculated from CTD and ship ADCP data. Cross-shelf is positive towards the coast (135Њ east of north) and direction is in degrees east of north. Loca- Cross-shelf Alongshelf Magnitude Direction tion (W mϪ1) (W mϪ1) (W mϪ1) (deg) C1 Ϫ22 Ϫ11 25 289 C2 52 Ϫ28 59 163 C3 20 36 41 74 C4 457 Ϫ174 489 156 C5 459 Ϫ95 469 147 C6 1247 72 1249 132 C7 1794 309 1821 125 C8 1234 Ϫ239 1257 146 C9 5 413 413 46

in the bottom boundary layer. The mixing could result from strong shear throughout the water column and could possibly be associated with nonlinear aspects of the internal tide or internal solitary waves that form in this region (e.g., Holloway et. al 1999). Of the energy that is being dissipated, some will contribute to the vertical mixing of density. For a given energy dissipation rate, Osborne (1980) suggests an upper bound on the vertical eddy diffusion as ␧ K ϭ 0.2 . (6) ␳ N2 Dividing the dissipation rates from C6 and C10 by local

water depth and density, Eq. (6) gives values of K␳ of 4.1 ϫ 10Ϫ4 and 1.9 ϫ 10Ϫ4 m 2 sϪ1, respectively, where N 2 is taken as 0.012 sϪ1 at each location (Fig. 5). Values are much lower at both the shallower and deeper moorings. The maximum vertical eddy viscosity at the 125 m deep location suggests signi®cant mixing could take place in this region. Note that Holloway (1984) derived Ϫ4 2 Ϫ1 a value of K␳ of 1.4 ϫ 10 m s from energy ¯ux estimates near the shelf break in the same region. Background strati®cation is de®ned at each CTD location by time averaging the pro®les measured over a tidal cycle. Resulting sections of temperature, salinity, and buoyancy frequency are plotted in Fig. 14. Although FIG. 13. Cross sections of vertical displacement amplitude, cross- there is considerable structure in the salinity with warm shelf baroclinic current amplitude, and internal tide energy ¯ux at high salinity water on the shelf, presumably due to evap- the semidiurnal period calculated from CTD and ship ADCP data oration over the broad shelf in summer, and several cores obtained over the period 14±22 Jan 1995. Positive ¯ux values are onshore (135Њ east of north), and alongshore values are to the south- west. Shading shows vertical displacement greater than 20 m (contour TABLE 11. Computed M values of barotropic energy density (E ), interval 5 m), currents greater than 8 cm sϪ1 (contour interval 4 cm 2 b depth-integrated baroclinic energy density ( Edz), baroclinic energy sϪ1), and energy ¯ux greater than5WmϪ2 (contour interval 5 W # dissipation rate (⑀), and estimated baroclinic energy dissipated rate mϪ2). due to bottom friction (⑀cd), as discussed in the text. Water depth (H)

and mean density (␳o) are given for each mooring location. taken as the semimajor axis length of the near-bottom ␳o

velocities (Tables 5, 6, 7a, 8) produces the values listed Moor- H (kg Eb # Edz ⑀ ⑀cd in Table 11. It can be seen that the observed dissipation ing (m) mϪ3) (J mϪ2) (J mϪ2) (W mϪ2) (W mϪ2) is much larger, ϳ20 times at C6 and C10, than would C2 65 1023 2890 31 0.7 ϫ 10Ϫ3 0.04 ϫ 10Ϫ3 be expected from bottom friction. This suggests that C6A 125 1023 2166 1689 37.8 ϫ 10Ϫ3 2.55 ϫ 10Ϫ3 energy might be dissipated and contribute to mixing C10 300 1026 1708 1911 42.7 ϫ 10Ϫ3 1.53 ϫ 10Ϫ3 Ϫ3 Ϫ3 throughout the water column, rather than only mixing C12 750 1028 1543 481 10.8 ϫ 10 0.01 ϫ 10

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water depths of approximately 100 and 150 m, where buoyancy frequency is strong throughout the water column and has a maximum near the seabed. This region of near-uniform buoyancy frequency corresponds to the region of strong internal mixing inferred from baroclinic energy estimates. It is reasonable to suggest that the strati®cation at the upper slope region is being mixed and modi®ed by the internal tide.

7. Discussion An indication of the potential generation sites of the internal tide can come from a comparison of the slope of the internal wave characteristics, the path of the group velocity vector, to the slope of the sea¯oor. Regions where these slopes are equal (critical slopes), or where the sea¯oor is steeper than the characteristics (supercritical slopes), are likely internal tide generation sites (Baines 1982; Craig 1987, 1988). The slope of the characteristics can be de®ned as

␻22Ϫ f 1/2 c ϭ , (7) ΂΃N22Ϫ ␻ where ␻ is the wave frequency, f the Coriolis parameter, and N(z) is the buoyancy frequency as a function of depth (z). Figure 15 shows a bathymetry cross-section

with the M2 characteristics plotted and where N(x, z)is calculated from a 13-h time-averaged strati®cation at each CTD station C1 to C13. For much of the region the sea¯oor slope is subcritical. However, there are several sites over the upper continental slope that become critical. Dips in the topography at approximately 120 and 200 m produce critical conditions, and in water deeper than ϳ600 m the topography passes through a critical point to become supercritical. Internal tide generation also requires barotropic tidal ¯ow across the topographic slope at these critical and supercritical

slopes. From Fig. 3 it can be seen that the M 2 barotropic tidal ellipses are large and aligned cross-shelf on the upper slope and over the shelf. In water deeper than ϳ200 m, the ellipses weaken and become more aligned alongshelf. Combining this information with the char- FIG. 14. Sections of temperature, salinity, and buoyancy frequency (100 sϪ1) calculated from time averages over a tidal cycle at each acteristic plots, it can be expected that internal tide gen- location from C1 to C13. Dates are listed in Table 2. Shading shows eration will be strong at the critical slope points at 125 salinity greater than 35.3 psu and buoyancy frequency greater than and 200 m and that, due to the weak cross-slope com- 0.0125 sϪ1. ponent of the barotropic tide, there may only be weak generation in water deeper than ϳ600 m. This is consistent with the semidiurnal internal tide observed, for of high salinity water over the slope, the density dis- example, Figs. 6 and 7, and also with the energy ¯ux tribution (not shown) is similar to the temperature. The calculations shown in Fig. 8. This is also consistent with distribution of buoyancy frequency shows considerable the ®ndings of Craig (1988). Using similar topography horizontal variability. On the shelf there is a peak in and constant buoyancy frequency, Craig's modeling buoyancy frequency around middepth with mixed layers showed strongest generation at a change in slope at depth above and below. Over most of the slope region there ϳ400-m with weaker generation at the change in slope is a surface mixed layer about 20 m thick and a max- at 200 m and over the steep topography at 100 m. imum buoyancy frequency at a depth of ϳ40 m. The Figure 15 also shows the characteristics for the di- nature of the strati®cation changes in the region between urnal constituent K1. These characteristics are not as

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FIG. 15. Cross section of bathymetry showing M2 and K1 internal wave characteristic paths. Computed using Eq. (3), with f ϭϪ5 ϫ 10Ϫ5 sϪ1 and buoyancy frequency calculated from observations at locations C1 to C13. steep as for the semidiurnal tide. However, there is a column. A large proportion of the energy ¯ux of the broad region between water depths of approximately internal tide is dissipated over the upper continental 150 and 500 m where the continental slope topography slope and shelf break, close to the generation regions, is at near-critical slope. Although the K1 barotropic tidal and the dissipated energy appears to enhance the mixing ¯ow is weak, the ellipses are aligned largely in the cross- of the strati®cation throughout the water column in this slope direction. These favorable conditions for diurnal region. internal tide generation are consistent with the observations from the region, as given in Tables 5, 6, and Acknowledgments. This work has been supported by

7c. In particular, the largest K1 internal tide is seen at an Australian Research Council grant. The ®eld work location C10 in water depth 300-m where characteristics was undertaken on RV Franklin and the assistance of are closest to critical. The observations at this site also the captain, crew, and scienti®c party was greatly ap- show bottom intensi®cation of the baroclinic currents, preciated. For mooring C6B, Woodside Energy Ltd. pro- consistent with generation at the site. vided the instrumentation, while WNI Science and En- The observations presented in this paper show a pre- gineering provided the mooring hardware and com- dominantly semidiurnal internal tide over an approxi- pleted initial data processing. Woodside Energy Ltd. mately 100-km wide section of the continental slope. A also provided logistics support, which is gratefully ac- weaker diurnal internal tide is also observed. The semi- knowledged. diurnal motion is most energetic between water depths of approximately 100 and 300 m. The slope of the to- APPENDIX pography is largely subcritical compared to the group velocity of the internal tide resulting in onshore energy Baroclinic Energy Flux propagation. The internal tide appears to be generated The energy ¯ux vector is the transport of energy past at a number of distinct locations that have close-to- a point in space and, averaged over a tidal period (T ϭ critical slopes and large cross-slope barotropic tidal 2␲/␻), is given as (LeBlond and Mysak 1978) ¯ows. On the upper parts of the continental slope the vertical structure of the internal tide is well approxi- T mated by a ®rst baroclinic mode. However, beyond ap- F ϭ ͵ u(z, t)p(z, t) dt, (A1) proximately 200-m depth, the vertical structure changes 0 and is characterized by intensi®cation of vertical dis- where u(z, t) ϭ (u, ␷) is the horizontal velocity vector, placements and velocities in the lower part of the water p(z, t) is the ¯uctuating component of pressure, z is the

Unauthenticated | Downloaded 09/26/21 12:12 AM UTC MAY 2001 HOLLOWAY ET AL. 1199 depth coordinate measured positive upwards, and t is REFERENCES time. This is the energy ¯ux per unit meter of wave Baines, P. G., 1982: On internal tide generation models. Deep-Sea crest and has SI units of J mϪ2. Alternatively, the av- Res., 29, 307±338. erage ¯ux per unit time (F/T) has SI units of W mϪ2. Craig, P. D., 1987: Solutions for internal tide generation over coastal For sinusoidal motion in time (eϪi␻t) the linearized equa- topography. J. Mar. Res., 45, 83±105. , 1988: A numerical model study of internal tides on the Aus- tions of motion relate pressure and vertical velocity by tralian North West Shelf. J. Mar. Res., 46, 59±76. 0 Foreman, M. C., 1978: Manual for tidal current analysis and predic- ␳0 tion. Paci®c Marine Science Rep. 78-6, Institute of Ocean Sci- p (z) ϭ iw(z)(N22(z) Ϫ ␻ ) dt, (A2) 0 ␻ ͵ 0 ence, Victoria, BC, Canada, 70 pp. z Gill, A. E., 1982: Atmosphere±Ocean Dynamics. Academic Press, 662 pp. where p0 and w0 are the amplitude functions for pressure Holloway, P. E., 1983: Tides on the Australian North West Shelf. and vertical velocity. If the velocity and pressure are Aust. J. Mar. Freshwater Res., 34, 212±230. each considered to have a real and imaginary component , 1984: On the semidiurnal internal tide at a shelf-break region on the Australian North West Shelf. J. Phys. Oceanogr., 14, (ur ϩ iui and pr ϩ ipi respectively) the cross-shelf energy ¯ux component can be written from (A1) as 1787±1799. , 1985: A comparison of semidiurnal internal tides from different bathymetric locations on the Australian North West Shelf. J. T F ϭ (up ϩ up). (A3) Phys. Oceanogr., 15, 240±251. xrrii2 , 1988: Climatology of internal tides at a shelf-break region on the Australian North West Shelf. Aust. J. Mar. Freshwater Res., Further, from tidal analysis, the horizontal and vertical 39, 1±18. , 1994: Observations of internal tide propagation on the Aus- velocity amplitudes and phases [(U, gu) and (W, gw) re- tralian North West Shelf. J. Phys. Oceanogr., 24, 1706±1716. spectively] can be found and are related to the real and , E. Pelinovsky, and T. Talipova, 1999: A generalised Korteweg- imaginary components by de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res., 104, 18 333±18 350. 222 Ϫ1 Huthnance, J. M., 1989: Internal tides and waves near the continental U ϭ uriϩ u and g uϭ tan (u ir/u ), (A4) shelf edge. Geophys. Fluid Dyn., 48, 81±106. W 222ϭ w ϩ w and g ϭ tanϪ1 (w /w ). (A5) Leaman, K. D., 1980: Some observations of baroclinic diurnal tides ri w ir over a near-critical bottom slope. J. Phys. Oceanogr., 10, 1540± Then (A3) gives, using (A2), (A4), and (A5), the cross- 1551. LeBlond, P. H., and L. A. Mysak, 1978: Waves in the Ocean. Elsevier, shelf energy ¯ux as 602 pp. Munk, W., 1981: Internal waves and small scale mixing processes. ␳ TU 0 Evolution of Physical Oceanography, B. Warren and C. Wunsch, 0 22 Fxuϭ sin(g ) W cos(gw)(N Ϫ ␻ ) dz Eds., The MIT Press, 264±291. 2␻ ͵ Osborn, T. R., 1980: Estimates of local rate of vertical diffusion from ΂ z dissipation measurements. J. Phys. Oceanogr., 10, 83±89. 0 Pingree, R. D., and A. L. New, 1989: Downward propagation of 22 internal tide energy into the Bay of Biscay. Deep-Sea Res., 36, Ϫ cos(gu) W sin(gw)(N Ϫ ␻ ) dz , ͵ 735±758. z ΃ , G. T. Mardell, and A. L. New, 1986: Propagation of internal (A6) tides from the upper slopes of the Bay of Biscay. Nature, 321, 154±158. and this allows the energy ¯ux to be computed from the Pugh, D. T., 1987: Tides, Surges and Mean Sea-Level. John Wiley results of the tidal analysis of horizontal and vertical and Sons, 472 pp. Rosenfeld, L. K., 1990: Baroclinic semidiurnal tidal currents over the velocities. The depth-integrated energy ¯ux is de®ned as continental shelf off northern California. J. Geophys. Res., 95, 22 153±22 172. 0 Sherwin, T. J., 1988: Analysis of an internal tide observed on the F ϭ Fdz. (A7) x ͵ x Marlin Shelf, North of Ireland. J. Phys. Oceanogr., 18, 1035± Ϫh 1050.

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