• Baines, P. G., 1982: On internal tide generation models. Deep-Sea Res.,29, 307–338.

  • Craig, P. D., 1987: Solutions for internal tide generation over coastal topography. J. Mar. Res.,45, 83–105.

  • ——, 1988: A numerical model study of internal tides on the Australian North West Shelf. J. Mar. Res.,46, 59–76.

  • Foreman, M. C., 1978: Manual for tidal current analysis and prediction. Pacific Marine Science Rep. 78-6, Institute of Ocean Science, Victoria, BC, Canada, 70 pp.

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Holloway, P. E., 1983: Tides on the Australian North West Shelf. Aust. J. Mar. Freshwater Res.,34, 212–230.

  • ——, 1984: On the semidiurnal internal tide at a shelf-break region on the Australian North West Shelf. J. Phys. Oceanogr.,14, 1787–1799.

  • ——, 1985: A comparison of semidiurnal internal tides from different bathymetric locations on the Australian North West Shelf. J. Phys. Oceanogr.,15, 240–251.

  • ——, 1988: Climatology of internal tides at a shelf-break region on the Australian North West Shelf. Aust. J. Mar. Freshwater Res.,39, 1–18.

  • ——, 1994: Observations of internal tide propagation on the Australian North West Shelf. J. Phys. Oceanogr.,24, 1706–1716.

  • ——, E. Pelinovsky, and T. Talipova, 1999: A generalised Korteweg-de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res.,104, 18 333–18 350.

  • Huthnance, J. M., 1989: Internal tides and waves near the continental shelf edge. Geophys. Fluid Dyn.,48, 81–106.

  • Leaman, K. D., 1980: Some observations of baroclinic diurnal tides over a near-critical bottom slope. J. Phys. Oceanogr.,10, 1540–1551.

  • LeBlond, P. H., and L. A. Mysak, 1978: Waves in the Ocean. Elsevier, 602 pp.

  • Munk, W., 1981: Internal waves and small scale mixing processes. Evolution of Physical Oceanography, B. Warren and C. Wunsch, Eds., The MIT Press, 264–291.

  • Osborn, T. R., 1980: Estimates of local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr.,10, 83–89.

  • Pingree, R. D., and A. L. New, 1989: Downward propagation of internal tide energy into the Bay of Biscay. Deep-Sea Res.,36, 735–758.

  • ——, G. T. Mardell, and A. L. New, 1986: Propagation of internal tides from the upper slopes of the Bay of Biscay. Nature,321, 154–158.

  • Pugh, D. T., 1987: Tides, Surges and Mean Sea-Level. John Wiley and Sons, 472 pp.

  • Rosenfeld, L. K., 1990: Baroclinic semidiurnal tidal currents over the continental shelf off northern California. J. Geophys. Res.,95, 22 153–22 172.

  • Sherwin, T. J., 1988: Analysis of an internal tide observed on the Marlin Shelf, North of Ireland. J. Phys. Oceanogr.,18, 1035–1050.

  • View in gallery

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

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    A cross section showing moorings and instrumentation

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

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    Stratification determined from a 27 day (18 Jan–14 Feb 1995) time average of temperatures at each current meter location and the thermistor chain. The tidal-cycle-averaged temperature from CTD station C13 on 22 Jan 1995 is also shown

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    First and second theoretical modal functions for vertical displacement (solid lines) and horizontal velocity (dotted lines) calculated for M2 tidal frequency and observed stratification from each of three locations. Buoyancy frequency profiles, calculated from observed temperature and salinity data, are also plotted

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    Profiles 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

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    Profiles 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

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    Depth integrated M2 energy fluxes 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

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    Time sequence of isopycnal depths over 13-h periods from CTD measurements at locations C3, C6, and C9. Plots are over the total water depth. Dates are listed in Table 2 and coincide with Fig. 10

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    Time sequence of cross-shelf component of baroclinic velocity over 13-h periods from ship ADCP measurements at locations C3, C6, and C9. Plots are over the total water depth, with solid contours showing onshore flow and dotted contours showing offshore flow. Dates are listed in Table 2 and coincide with Fig. 9. Contour intervals are 5 cm s−1 for C3 and C9 and 10 cm s−1 for C6

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    Vertical profiles of vertical displacement amplitude and phase at the semidiurnal period, calculated from CTD data between 14 and 22 Jan 1995

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    Vertical profiles 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

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    Cross sections of vertical displacement amplitude, cross-shelf baroclinic current amplitude, and internal tide energy flux at the semidiurnal period calculated from CTD and ship ADCP data obtained over the period 14–22 Jan 1995. Positive flux values are onshore (135° east of north), and alongshore values are to the southwest. Shading shows vertical displacement greater than 20 m (contour interval 5 m), currents greater than 8 cm s−1 (contour interval 4 cm s−1), and energy flux greater than 5 W m−2 (contour interval 5 W m−2)

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    Sections of temperature, salinity, and buoyancy frequency (100 s−1) calculated from time averages over a tidal cycle at each location from C1 to C13. Dates are listed in Table 2. Shading shows salinity greater than 35.3 psu and buoyancy frequency greater than 0.0125 s−1

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    Cross section of bathymetry showing M2 and K1 internal wave characteristic paths. Computed using Eq. (3), with f = −5 × 10−5 s−1 and buoyancy frequency calculated from observations at locations C1 to C13

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

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  • 1 School of Geography and Oceanography, University College, University of New South Wales, Australian Defence Force Academy, Canberra, Australia
  • | 2 CSIRO Marine Research, Hobart, Tasmania, Australia
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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 profile 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 first vertical mode, whereas in deeper water over the slope the vertical structure is more complicated with currents and vertical displacements intensified in the lower part of the water column. The internal tide is largely confined 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 field, 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.

*Current affiliation: Fisheries Research Services Marine Laboratory, Aberdeen, United Kingdom.

Corresponding author address: Dr. Peter Holloway, School of Geography and Oceanography, University College, University of New South Wales, Australian Defence Force Academy, Canberra ACT 2600, Australia.

Email: _Peter.Holloway@adfa.edu.au

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 profile 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 first vertical mode, whereas in deeper water over the slope the vertical structure is more complicated with currents and vertical displacements intensified in the lower part of the water column. The internal tide is largely confined 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 field, 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.

*Current affiliation: Fisheries Research Services Marine Laboratory, Aberdeen, United Kingdom.

Corresponding author address: Dr. Peter Holloway, School of Geography and Oceanography, University College, University of New South Wales, Australian Defence Force Academy, Canberra ACT 2600, Australia.

Email: _Peter.Holloway@adfa.edu.au

1. Introduction

Internal tides over continental shelf and slope regions are characterized by high spatial and temporal variability, have phases that are not locked to that of the barotropic tide and propagate both shoreward and seaward. The waves are often energetic and can significantly contribute to ocean mixing (e.g., Pingree et al. 1986). Most observations have shown internal tides at semidiurnal frequencies rather than diurnal. An exception are the observations presented by Leaman (1980) of a diurnal internal tide, at near-critical bottom slope, off the west Florida continental shelf. The propagation of internal tides, seaward from steep continental slope topography, can be described in terms of energy propagation along internal wave characteristics, that is, along the path of the group velocity vector. For example, Pingree and New (1989) present observations from the Bay of Biscay and suggest wave propagation along characteristics. In this situation, the signal propagates along a narrow beam with significant vertical phase propagation and reflects off the sea surface and seabed, consistent with simple theoretical models of internal wave generation over critical slopes (e.g., Baines 1982). However, in many instances, and perhaps in regions of less steep topography, internal tides are well described in terms of vertical modes, usually dominated by the first mode. For example, Sherwin (1988) analyzes observations from the Marlin Shelf north of Ireland and Rosenfeld (1990) analyzes observations from the shelf off northern California. In both cases the vertical structure of the internal tide is dominated by the first vertical mode. While internal wave “beams” can be described as the sum of high-order 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 continental 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 attempt to help rectify this gap in observational detail and to address some of the questions raised above, a field program was run on the Australian North West Shelf (NWS), obtaining detailed observations of the internal tide over a cross section where energetic internal waves have previously been reported (Holloway 1984, 1985, 1988, 1994). The aim was to measure the internal tide from offshore of the generation region and to track the complete cycle of evolution of the internal tide as it propagated shoreward and finally dissipated over the outer continental shelf. Measurements were obtained from moored current meters, an acoustic Doppler current profiler (ADCP), and a thermistor chain as well as from detailed shipborne CTD and ADCP profile measurements, all made during the summer months (Jan–Mar) of 1995 when the internal tide is strongest (see Holloway 1988).

In this paper, observations are analyzed in order to define the vertical structure of the internal tide in terms of both vertical displacement and horizontal currents at M2 (12.42 h), S2 (12.00 h), K1 (23.93 h), and M4 (6.21 h) tidal frequencies. Also investigated are the changes in horizontal structure across the continental slope, and the energetics of the internal tide.

2. Measurement program

Moorings containing current meters, ADCPs, thermistor chains, and water level recorders were deployed on the NWS in January 1995. A number of instrument failures reduced the array to four mooring locations and instruments at depths as indicated in Table 1. Bathymetry, CTD, and mooring locations are shown in Fig. 1, and Fig. 2 is a cross section showing instrument positions. Locations are referred to as C1 to C13, where all are CTD stations and some are mooring locations as well. Despite instrument failures, the mooring array provides results ranging from 750-m depth on the slope 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 m from 26 to 114 m in 125-m water depth. The moored instruments collected data for approximately 2 months at sampling rates of 2, 5, or 10 min. (Table 1).

A detailed CTD survey was conducted in January 1995 during the mooring deployment. At each of 13 locations in a section across the shelf and slope (Fig. 1), repeated CTD profiles were measured over a 13-h period in order to measure one complete cycle of the semidiurnal tide. At most locations profiles were measured every 30 min, but less frequently at deep stations. Simultaneously, the ship’s hull-mounted ADCP provided current profiles in 8-m bins from 16 to approximately 230 m depth, or to 14% of depth above the seabed in shallower water, for example, to 86 m in 100 m. Table 2 lists the times and dates the profiles were measured, and the repeat frequency of the profiles. Note that it took 7 days to complete the section from C1 to C13.

For the analysis of the moored instrument time series, 27 days of data are used, chosen as a common block of data to all instruments. This spans the period from 18 January to 14 February 1995.

3. The barotropic tide

The moored current meter data are analyzed in order to separate the barotropic tide from the baroclinic or internal tidal signals. The analysis of 27 days data resolves the M2 and S2 constituents in the semidiurnal band along with diurnal constituents and higher-frequency harmonics.

The barotropic currents at each mooring are defined as the depth-averaged time series. Tidal harmonic analysis (Foreman 1978) is then used to define barotropic tidal currents. Results are presented in Table 3 with M2 and K1 values plotted in Fig. 3 as tidal ellipses. Tidal currents are largely semidiurnal, and diurnal speeds are approximately 10% of semidiurnal values. The ratio of S2/M2 semimajor axis lengths average 0.65 so that the spring–neap cycle is pronounced. The M2 and S2 ellipses are aligned cross-shelf for the shallower moorings C6 and C2 but turn and become more aligned alongshelf in deeper water. Cross-shelf alignment of the ellipses is important for internal tide generation, which is most effective for a large flux of water up and down the topographic slope. There are only small changes in phase (10°) between the moorings for the barotropic constituents. At location C6, two estimates of the barotropic tidal currents are made, one as a depth average of the four current meters and the other from a depth average of the ADCP measurements. The differences in semimajor axis lengths and phases from these independent estimates are only slight for the semidiurnal constituents, the largest discrepancy in the ellipse properties being a 15° difference in ellipse orientation. For reference, the major tidal elevation constituents are listed in Table 4 for three locations. All of the constituents show phase propagation towards the coast, consistent with the cross-shelf alignment of the tidal ellipses.

4. Internal tide from moored instruments

For each current meter record a tidal analysis is carried out on the residual time series, computed as the total current minus the depth-averaged value. This analysis defines the constituents of the internal tide. In addition, temperature data are used to compute vertical displacements. Assuming continuity of temperature, and neglecting horizontal advection of horizontal temperature gradients, the vertical velocity is defined as
i1520-0485-31-5-1182-e1
where T(z, t) is the temperature as a function of depth (z) and time (t). The computed vertical velocity time series are harmonically analyzed and amplitudes converted to vertical displacements assuming sinusoidal motion and using w = −iωζ, where ω is the wave frequency and ζ the vertical displacement. Vertical displacement phase is computed as the vertical velocity phase plus 90°.

The average stratification for this time period can be seen from the 27-day averages at each current meter, producing the temperature profiles plotted in Fig. 4. These profiles are compared to the 13-h time-averaged CTD temperature profile measured at the offshore location C13 at the start of the mooring time series. The stratification shows an approximate linear gradient in the upper 300 m, corresponding to a buoyancy frequency of 0.013 s−1, with a weaker gradient in deeper water. There is only a thin, ∼20 m, surface mixed layer. The similarity in all plotted profiles suggests fairly constant stratification during the measurement period.

The first 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 profile, computed from the repeated CTD profiles 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 first mode having maximum vertical displacements and zero horizontal velocity at middepth and equal maxima in velocity at the sea surface and seabed. The upper and lower water column velocities are 180° out of phase. These symmetric modal functions are expected on the NWS for water depths less than ∼300 m with the symmetry lost in deeper water where buoyancy frequency varies substantially with depth.

Baroclinic tidal current properties and displacement amplitudes and phases for M2, S2, and K1 constituents are listed in Tables 5–8 for mooring locations C12, C10, C6, and C2 respectively. In addition, cross-section plots of the cross-shelf component of the baroclinic currents and phases are plotted in Fig. 6 and displacement amplitudes and phases are plotted in Fig. 7 for the M2 constituent. A strong semidiurnal internal tide is seen at moorings C6 and C10, in water depths of 125 and 300 m, with much weaker signals at the shallower mooring C2 and the deep mooring C12. Maximum baroclinic currents are at the M2 frequency and reach 0.13 m s−1 at C6 and 0.16 m s−1 at C10 with vertical displacement amplitudes of 27 m at mooring C6 and 13 m at C10. The S2 currents and displacements are largest at C6 reaching 0.08 m s−1 and 13 m, respectively. At mooring C12 in water depth 750-m, baroclinic M2 currents reach 0.04 m s−1 and vertical displacements reach 9 m. Mooring C6 shows a simple first-mode structure with a 180° phase change in currents at approximately middepth, corresponding to the maximum in the vertical displacements. There is little phase change through depth in the displacements, consistent with a first-mode structure. At 300-m depth, mooring C10 shows a similar picture of velocities and displacements, suggesting a dominance of first-mode waves. At the deeper mooring C12, M2 and S2 displacements are largest in the lower part of the water column, and there is an increase in phase lag from the upper to lower measuring depths.

While the ratio of S2/M2 semimajor axis lengths is approximately constant between moorings (0.65) for the barotropic tide, this is not found to be true for the internal tide where the ratios vary substantially. For example, at location C2, S2 is larger than M2. This is most likely a result of energy smearing around the semidiurnal tidal frequencies.

An M4 internal tide, generated through nonlinear interactions of the semidiurnal (M2) constituent, is observed at all mooring locations (values are not listed in the tables) with currents up to 0.03 m s−1 and displacements of 3 m. Maximum values in M4 are seen near the surface at mooring C10 in water depth 300 m. Again the vertical structures, particularly at mooring C6, appear to follow those of first-mode waves. A significant K1 internal tide is also observed, strongest at moorings C10 and C6. Displacements reach 10 m and currents 0.09 m s−1. Note that at 20° latitude, K1 is still a superinertial frequency, where |f/ω| = 0.69, f is the Coriolis parameter (−5 × 10−5 s−1), and ω is the wave frequency. Hence, diurnal internal tides can exist as freely propagating waves.

The results from the mooring data show a large-amplitude internal tide confined to an approximately 100 km wide section of the upper continental slope. The largest waves occur between 125 and 300 m depths and appear to be predominantly simple first-mode waves. At the deepest mooring (750 m) the phase of the elevations increase with depth, and the depth of the maximum in vertical displacements and zero crossing in velocity are inconsistent with a first-mode internal wave.

The phase relationship between vertical displacements and baroclinic currents determines the direction of wave propagation. If wave propagation is in the positive x direction, then there will be a 180° phase difference between the vertical displacements and the near-surface currents when defined as positive in the x direction (e.g., see Gill 1982, p. 125). For the M2 constituents presented above there is an approximately 180° phase difference between the vertical displacements and upper cross-shelf currents at moorings C6, C10, and C12, indicating shoreward wave propagation. This also tends to be true for the other constituents.

The energy flux associated with the internal tide is an important quantity, as some of the energy will be available for ocean mixing. The depth-integrated energy fluxes are calculated for each mooring location for the M2 tidal constituent. The method, discussed in the appendix, uses velocity and vertical displacement amplitudes and phases and stratification to define the buoyancy frequency. Resulting values computed using (A6) are listed in Table 9 and plotted in Fig. 8. Fluxes are approximately perpendicular to the bathymetry contours and directed onshore. At the shallow mooring C2 the energy flux is essentially zero. Values are largest between depths of 125 and 300 m with a maximum of 1089 W m−1 at 300-m depth. Holloway (1984) found an average onshore energy flux of 326 W m−1 at a mooring close to mooring C6 in water depth 125 m, derived as an average from observations over the period January–July 1982. This is consistent with values listed in Table 9.

The energy flux of the barotropic tide is calculated as (Pugh 1987)
i1520-0485-31-5-1182-e2
where ρ is the water density, η0 and u0 the barotropic tidal elevation and velocity amplitudes, and gη and gu the corresponding phases. Using values from Tables 3 and 4, M2 barotropic energy fluxes for moorings C2 and C6A are 1.7 × 104 and 2.5 × 104 W m−1, respectively, and are directed approximately onshore (in the direction of ellipse orientation as listed in Table 3). The baroclinic energy fluxes are then approximately 4% of the barotropic values.

5. Internal tides from ship CTD and ADCP profiles

The repeated CTD profiles over a tidal cycle reveal internal wave activity, to varying degrees, at all sites. Example observations from locations in water depths 65, 120, and 240-m are shown in Fig. 9 as depth–time contoured distributions of sigma-t. At the deeper location (C9), isopycnals at middepth oscillate with a height of about 40 m over the 13 h. Although there is some higher-frequency variability, there is clearly an internal tide oscillation that is approximately sinusoidal. At 120-m depth (C6) a large amplitude internal tide is seen and is strongly distorted from a sinusoidal shape. Part of the wave shows shocks on both the front and back face of the wave, giving an approximately square waveform. The 23.4 isopycnal varies from 30 to 105 m depth, giving a wave amplitude of 38 m. In water depth 65 m (C3), an internal solitary wave is seen, with positive polarity and an amplitude of 27 m. The longer-period internal tide is weak at this location. These examples show the changing nature of the internal waves from deeper to shallow water. The nonlinear steepening of the internal tide and the formation of internal solitary waves from the internal tide are important characteristics of the internal wave field on this and other continental shelf and slope regions. These nonlinear processes are also most probably important in determining the mechanisms of dissipation of the internal tide. However, these details are not addressed in this paper. The nonlinear features may also produce some aliasing of the measured internal tide, although this is suspected to be a small effect in fitting a tidal harmonic to measured signals. Similarly, assuming linear dynamics may lead to some small error in estimating the energy contained in the internal tide.

The hull-mounted ADCP on RV Franklin provided 20-min values of current to a depth of approximately 230 m in 8-m depth bins. For the periods corresponding to the CTD profile measurements, and for stations out to depth 250 m, the depth-averaged current for each time measurement is removed from the total current to provide an estimate of the baroclinic signal. Example plots of depth–time sections of cross-shelf baroclinic currents are plotted in Fig. 10 for locations C3, C6, and C9, corresponding to the sigma-t sections in Fig. 9. The velocities clearly show the reverse of phase between upper and lower layers of the water at locations C3 and C6 and correspond well with the elevations seen from the CTD data. The phase relationship between elevations and velocities (offshore surface flow at the peak of the wave) shows the waves to be propagating onshore. Baroclinic velocities reach 0.40 m s−1 at location C6, substantially larger than the M2 value calculated from the 27-day time series (Table 7a). The soliton character of the wave at C3 is seen in the velocity as is the sharp front of the internal tide at C6. A more complex velocity field is seen at C9 with multiple flow reversals with depth.

Vertical displacement amplitudes and phases are calculated for each CTD station using the time series of sigma-t over a 13-h period. Vertical velocities are computed at 2-m depth intervals using Eq. (1) with temperature replaced by sigma-t. The data are first smoothed in depth using a running-mean filter in order to reduce noise in the vertical velocity time series that can arise 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 total semidiurnal signal. Resulting vertical profiles of displacement amplitude and phase are plotted in Fig. 11 and a contoured distribution of amplitude in the upper 500 m is shown in Fig. 13.

Harmonic analysis of hull-mounted ADCP current time series provides the baroclinic cross-shelf and alongshelf M2 amplitudes and phases for the periods corresponding to the CTD profile measurements. Profiles of cross-shelf amplitude and phase are plotted in Fig. 12 and contours of the amplitude in the upper 500 m are plotted in Fig. 13. In the shallower regions, to a depth of approximately 150 m, vertical displacements appear as simple first-mode distributions with maxima occurring near middepth and uniform phase with depth. Amplitudes are large, up to 40 m in water 150 m deep. However, the profile at 400-m depth has a maximum in the lower part of the water column, and for the profile at 1400 m the maximum occurs around 800 m. There is considerable change in phase with depth at these deeper locations. Strong cross-shelf internal tide currents are seen in the shallower region coinciding with the large amplitude waves. Currents have maxima near the sea surface and seabed, a zero point at middepth, and a 180° change in phase with depth. This is consistent with the first-mode structure. The contoured section of velocities (Fig. 13) shows a distinct change in the vertical structure between internal tides inshore and offshore of approximately 200-m depth. The vertical displacements at 400-m depth (C11) and 1400-m depth (C13) cannot be described by a first-mode structure. In particular, in the deeper water the vertical displacements are largest near the seabed or in the lower part of the water column. It is noted that the observations are not synoptic but are from different times over a period of 7 days and some caution must be exercised in interpreting cross-shelf correlations in the data. However, there are strong similarities in the values of the M2 vertical displacement phases from the moorings C10 and C6 (Fig. 7) and the corresponding CTD data (Fig. 11). Also, the variations in phase with depth are similar from mooring and CTD data at location C12. This suggests a quite stable phase over time for the vertical displacements.

Energy flux is calculated from the elevations and baroclinic currents using Eq. (A6), and a section of cross-shelf values is plotted in Fig. 13. The dominant feature is a strong onshore flux in the lower half of the water column on the upper slope, between depths 80 and 150 m where peak values are 40 W m−2. The offshore flux in the upper part of the water column is much weaker, reaching 8 W m−2. Other patches of onshore flux are seen as well as weaker regions of offshore flux.

Depth-integrated energy fluxes are calculated for locations in water less than 250 m (C1 to C9), with values tabulated in Table 10 and plotted as vectors in Fig. 8. Fluxes are predominantly onshore, reaching 1821 W m−1 in 132-m water depth. The values are negligible at the shallow inshore moorings and also weaken in deeper water (beyond C8 in 162 m). The cross-shelf component is generally stronger than that alongshelf. Fluxes are consistent with those calculated from the moorings, where values are computed from 27-day data segments, although little over double the strength at location C6.

6. Energy dissipation

The baroclinic energy flux vectors plotted in Fig. 8 show a strong onshore flux between approximately 300 and 100 m water depths, and this indicates strong internal tide generation in this region. There is little energy propagating onto the shelf and little energy propagating from the deep water onto the continental slope. This indicates that much of the internal tide energy is dissipated in the upper slope region, close to the region where the internal tide is being generated. The energetics of the internal tide is examined to estimate the contribution to mixing and also to compare the energy with that of the barotropic tide.

The energy density of the internal tide averaged over a wave period is given as
i1520-0485-31-5-1182-e3
where ζ0, u0, and υ0 are the amplitudes of vertical displacement and horizontal velocity (LeBlond and Mysak 1978). The barotropic energy density averaged over a wave period is given as
i1520-0485-31-5-1182-e4
where η0 is the amplitude of the surface elevation, Ub and Vb are the amplitudes of the barotropic tidal currents, ρ0 is the depth-averaged density, and H is the water depth. For each of the four moorings, the M2 barotropic and baroclinic energy densities are calculated using (3) and (4). Velocity and elevation values are from values listed in Tables 3, 4, 5, 6, 7a, and 8. In addition, M2 tidal elevation amplitudes of 0.61 and 0.63 m for locations C12 and C10, respectively, were used in (4), obtained from a numerical tidal model of the region (P. Holloway, unpublished manuscript). Further, the baroclinic energy density is integrated over depth, and results are listed in Table 11. It can be seen that the barotropic energy density steadily increases from offshore as tidal velocities increase in shallower water. In comparison, the energy density of the baroclinic tide reaches a maximum and is approximately equal to that of the barotropic tide between locations C6 and C10 (depth 125 to 300 m), but is substantially less both over the shelf and in deeper water.
From the above results, it is reasonable to assume that most of the energy of the baroclinic tide is dissipated close to the region where it is generated. Only a small amount of the energy propagates onto the shelf and in deep water the energy flux is small. Assuming that all of the energy is dissipated close to the region of generation, the energy dissipation rate can be computed as ε = ∫ E dz/T, where T is the tidal period. These values are listed in Table 11 and show strongest dissipation (∼0.04 W m−2) at C6 and C10, corresponding to the largest energy density values. This dissipation rate can be compared to that expected from bottom frictional dissipation. For an oscillatory tidal flow of amplitude U0, Pugh (1987) shows that the dissipation rate due to a quadratic bottom stress is
i1520-0485-31-5-1182-e5
where Cd ∼ 0.0025 is the quadratic bottom friction coefficient. Applying (5) to the internal tide, where U0 is taken as the semimajor axis length of the near-bottom velocities (Tables 5, 6, 7a, 8) produces the values listed in Table 11. It can be seen that the observed dissipation is much larger, ∼20 times at C6 and C10, than would be expected from bottom friction. This suggests that energy might be dissipated and contribute to mixing throughout the water column, rather than only mixing 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
i1520-0485-31-5-1182-e6
Dividing the dissipation rates from C6 and C10 by local water depth and density, Eq. (6) gives values of Kρ of 4.1 × 10−4 and 1.9 × 10−4 m2 s−1, respectively, where N2 is taken as 0.012 s−1 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 significant mixing could take place in this region. Note that Holloway (1984) derived a value of Kρ of 1.4 × 10−4 m2 s−1 from energy flux estimates near the shelf break in the same region.

Background stratification is defined at each CTD location by time averaging the profiles measured over a tidal cycle. Resulting sections of temperature, salinity, and buoyancy frequency are plotted in Fig. 14. Although there is considerable structure in the salinity with warm high salinity water on the shelf, presumably due to evaporation over the broad shelf in summer, and several cores of high salinity water over the slope, the density distribution (not shown) is similar to the temperature. The distribution of buoyancy frequency shows considerable horizontal variability. On the shelf there is a peak in buoyancy frequency around middepth with mixed layers above and below. Over most of the slope region there is a surface mixed layer about 20 m thick and a maximum buoyancy frequency at a depth of ∼40 m. The nature of the stratification changes in the region between 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 stratification at the upper slope region is being mixed and modified 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 seafloor. Regions where these slopes are equal (critical slopes), or where the seafloor 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 defined as
i1520-0485-31-5-1182-e7
where ω 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 stratification at each CTD station C1 to C13. For much of the region the seafloor 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 ∼600 m the topography passes through a critical point to become supercritical. Internal tide generation also requires barotropic tidal flow across the topographic slope at these critical and supercritical slopes. From Fig. 3 it can be seen that the M2 barotropic tidal ellipses are large and aligned cross-shelf on the upper slope and over the shelf. In water deeper than ∼200 m, the ellipses weaken and become more aligned alongshelf. Combining this information with the characteristic plots, it can be expected that internal tide generation will be strong at the critical slope points at 125

Figure 15 also shows the characteristics for the diurnal constituent K1. These characteristics are not as steep as for the semidiurnal tide. However, there is a broad region between water depths of approximately 150 and 500 m where the continental slope topography is at near-critical slope. Although the K1 barotropic tidal flow is weak, the ellipses are aligned largely in the cross-slope direction. These favorable conditions for diurnal internal tide generation are consistent with the observations from the region, as given in Tables 5, 6, and 7c. In particular, the largest K1 internal tide is seen at location C10 in water depth 300-m where characteristics are closest to critical. The observations at this site also show bottom intensification of the baroclinic currents, consistent with generation at the site.

The observations presented in this paper show a predominantly semidiurnal internal tide over an approximately 100-km wide section of the continental slope. A weaker diurnal internal tide is also observed. The semidiurnal motion is most energetic between water depths of approximately 100 and 300 m. The slope of the topography is largely subcritical compared to the group velocity of the internal tide resulting in onshore energy propagation. The internal tide appears to be generated at a number of distinct locations that have close-to-critical slopes and large cross-slope barotropic tidal flows. On the upper parts of the continental slope the vertical structure of the internal tide is well approximated by a first baroclinic mode. However, beyond approximately 200-m depth, the vertical structure changes and is characterized by intensification of vertical displacements and velocities in the lower part of the water column. A large proportion of the energy flux of the internal tide is dissipated over the upper continental slope and shelf break, close to the generation regions, and the dissipated energy appears to enhance the mixing of the stratification throughout the water column in this region.

Acknowledgments

This work has been supported by an Australian Research Council grant. The field work was undertaken on RV Franklin and the assistance of the captain, crew, and scientific party was greatly appreciated. For mooring C6B, Woodside Energy Ltd. provided the instrumentation, while WNI Science and Engineering provided the mooring hardware and completed initial data processing. Woodside Energy Ltd. also provided logistics support, which is gratefully acknowledged.

REFERENCES

  • Baines, P. G., 1982: On internal tide generation models. Deep-Sea Res.,29, 307–338.

  • Craig, P. D., 1987: Solutions for internal tide generation over coastal topography. J. Mar. Res.,45, 83–105.

  • ——, 1988: A numerical model study of internal tides on the Australian North West Shelf. J. Mar. Res.,46, 59–76.

  • Foreman, M. C., 1978: Manual for tidal current analysis and prediction. Pacific Marine Science Rep. 78-6, Institute of Ocean Science, Victoria, BC, Canada, 70 pp.

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Holloway, P. E., 1983: Tides on the Australian North West Shelf. Aust. J. Mar. Freshwater Res.,34, 212–230.

  • ——, 1984: On the semidiurnal internal tide at a shelf-break region on the Australian North West Shelf. J. Phys. Oceanogr.,14, 1787–1799.

  • ——, 1985: A comparison of semidiurnal internal tides from different bathymetric locations on the Australian North West Shelf. J. Phys. Oceanogr.,15, 240–251.

  • ——, 1988: Climatology of internal tides at a shelf-break region on the Australian North West Shelf. Aust. J. Mar. Freshwater Res.,39, 1–18.

  • ——, 1994: Observations of internal tide propagation on the Australian North West Shelf. J. Phys. Oceanogr.,24, 1706–1716.

  • ——, E. Pelinovsky, and T. Talipova, 1999: A generalised Korteweg-de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res.,104, 18 333–18 350.

  • Huthnance, J. M., 1989: Internal tides and waves near the continental shelf edge. Geophys. Fluid Dyn.,48, 81–106.

  • Leaman, K. D., 1980: Some observations of baroclinic diurnal tides over a near-critical bottom slope. J. Phys. Oceanogr.,10, 1540–1551.

  • LeBlond, P. H., and L. A. Mysak, 1978: Waves in the Ocean. Elsevier, 602 pp.

  • Munk, W., 1981: Internal waves and small scale mixing processes. Evolution of Physical Oceanography, B. Warren and C. Wunsch, Eds., The MIT Press, 264–291.

  • Osborn, T. R., 1980: Estimates of local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr.,10, 83–89.

  • Pingree, R. D., and A. L. New, 1989: Downward propagation of internal tide energy into the Bay of Biscay. Deep-Sea Res.,36, 735–758.

  • ——, G. T. Mardell, and A. L. New, 1986: Propagation of internal tides from the upper slopes of the Bay of Biscay. Nature,321, 154–158.

  • Pugh, D. T., 1987: Tides, Surges and Mean Sea-Level. John Wiley and Sons, 472 pp.

  • Rosenfeld, L. K., 1990: Baroclinic semidiurnal tidal currents over the continental shelf off northern California. J. Geophys. Res.,95, 22 153–22 172.

  • Sherwin, T. J., 1988: Analysis of an internal tide observed on the Marlin Shelf, North of Ireland. J. Phys. Oceanogr.,18, 1035–1050.

APPENDIX

Baroclinic Energy Flux

The energy flux vector is the transport of energy past a point in space and, averaged over a tidal period (T = 2π/ω), is given as (LeBlond and Mysak 1978)
i1520-0485-31-5-1182-ea1
where u(z, t) = (u, υ) is the horizontal velocity vector, p(z, t) is the fluctuating component of pressure, z is the depth coordinate measured positive upwards, and t is time. This is the energy flux per unit meter of wave crest and has SI units of J m−2. Alternatively, the average flux per unit time (F/T) has SI units of W m−2. For sinusoidal motion in time (eiωt) the linearized equations of motion relate pressure and vertical velocity by
i1520-0485-31-5-1182-ea2
where p0 and w0 are the amplitude functions for pressure and vertical velocity. If the velocity and pressure are each considered to have a real and imaginary component (ur + iui and pr + ipi respectively) the cross-shelf energy flux component can be written from (A1) as
i1520-0485-31-5-1182-ea3
Further, from tidal analysis, the horizontal and vertical velocity amplitudes and phases [(U, gu) and (W, gw) respectively] can be found and are related to the real and imaginary components by
i1520-0485-31-5-1182-ea4
Then (A3) gives, using (A2), (A4), and (A5), the cross-shelf energy flux as
i1520-0485-31-5-1182-ea6
i1520-0485-31-5-1182-ea7

Fig. 1.
Fig. 1.

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

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 2.
Fig. 2.

A cross section showing moorings and instrumentation

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 3.
Fig. 3.

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

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 4.
Fig. 4.

Stratification determined from a 27 day (18 Jan–14 Feb 1995) time average of temperatures at each current meter location and the thermistor chain. The tidal-cycle-averaged temperature from CTD station C13 on 22 Jan 1995 is also shown

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 5.
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 stratification from each of three locations. Buoyancy frequency profiles, calculated from observed temperature and salinity data, are also plotted

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 6.
Fig. 6.

Profiles 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

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 7.
Fig. 7.

Profiles 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

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 8.
Fig. 8.

Depth integrated M2 energy fluxes 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

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 9.
Fig. 9.

Time sequence of isopycnal depths over 13-h periods from CTD measurements at locations C3, C6, and C9. Plots are over the total water depth. Dates are listed in Table 2 and coincide with Fig. 10

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 10.
Fig. 10.

Time sequence of cross-shelf component of baroclinic velocity over 13-h periods from ship ADCP measurements at locations C3, C6, and C9. Plots are over the total water depth, with solid contours showing onshore flow and dotted contours showing offshore flow. Dates are listed in Table 2 and coincide with Fig. 9. Contour intervals are 5 cm s−1 for C3 and C9 and 10 cm s−1 for C6

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 11.
Fig. 11.

Vertical profiles of vertical displacement amplitude and phase at the semidiurnal period, calculated from CTD data between 14 and 22 Jan 1995

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 12.
Fig. 12.

Vertical profiles 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

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 13.
Fig. 13.

Cross sections of vertical displacement amplitude, cross-shelf baroclinic current amplitude, and internal tide energy flux at the semidiurnal period calculated from CTD and ship ADCP data obtained over the period 14–22 Jan 1995. Positive flux values are onshore (135° east of north), and alongshore values are to the southwest. Shading shows vertical displacement greater than 20 m (contour interval 5 m), currents greater than 8 cm s−1 (contour interval 4 cm s−1), and energy flux greater than 5 W m−2 (contour interval 5 W m−2)

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 14.
Fig. 14.

Sections of temperature, salinity, and buoyancy frequency (100 s−1) calculated from time averages over a tidal cycle at each location from C1 to C13. Dates are listed in Table 2. Shading shows salinity greater than 35.3 psu and buoyancy frequency greater than 0.0125 s−1

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Fig. 15.
Fig. 15.

Cross section of bathymetry showing M2 and K1 internal wave characteristic paths. Computed using Eq. (3), with f = −5 × 10−5 s−1 and buoyancy frequency calculated from observations at locations C1 to C13

Citation: Journal of Physical Oceanography 31, 5; 10.1175/1520-0485(2001)031<1182:ITOFTA>2.0.CO;2

Table 1.

Details of moored instrumentation

Table 1.
Table 2.

Details of CTD profile measurements. Local times are used. The asterisks indicate a mooring location in addition to the CTD measurements

Table 2.
Table 3.

Barotropic tidal current properties. Ellipse parameters are semimajor (a) and semiminor (b) axis lengths in cm s−1, where +υe (b) indicates anticlockwise and −υe(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (θ) in degrees anticlockwise from true north. Mooring C6 values are calculated separately from the moored ADCP data (C6A) and current meters (C6B)

Table 3.
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

Table 4.
Table 5.

Mooring C12 baroclinic tidal currents and vertical displacements (ζ). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm s−1, where +υe(b) indicates anticlockwise and −υe(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (θ) in degrees anticlockwise from true north. Velocities are in centimeters per second and displacements in meters

Table 5.
Table 6.

Mooring C10 baroclinic tidal currents and vertical displacements (ζ). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm s−1, where +υe(b) indicates anticlockwise and −υe(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (θ) in degrees anticlockwise from true north. Velocities are in centimeters per second and displacements in meters

Table 6.
Table 7.

Mooring C6 baroclinic tidal currents and vertical displacements (ζ). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm s−1, where +υe(b) indicates anticlockwise and −υe(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (θ) 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

Table 7.
Table 8.

Mooring C2 baroclinic tidal currents and vertical displacements (ζ). Ellipse properties are semimajor (a) and semiminor (b) axis lengths in cm s−1, where +υe(b) indicates anticlockwise and −υe(b) clockwise rotation of the velocity vector, phase (g) in local time, 8 h ahead of UTC, and orientation (θ) in degrees anticlockwise from true north. Velocities are in centimeters per second and displacements in meters

Table 8.
Table 9.

Depth-integrated M2 baroclinic energy fluxes calculated from mooring data over the period 18 Jan–14 Feb 1995. Cross-shelf is positive towards the coast (135° east of north) and direction is in degrees east of north

Table 9.
Table 10.

Depth-integrated semidiurnal baroclinic energy fluxes calculated from CTD and ship ADCP data. Cross-shelf is positive towards the coast (135° east of north) and direction is in degrees east of north

Table 10.
Table 11.

Computed M2 values of barotropic energy density (Eb), depth-integrated baroclinic energy density (∫ E dz), baroclinic energy dissipation rate (ϵ), and estimated baroclinic energy dissipated rate due to bottom friction (ϵcd), as discussed in the text. Water depth (H) and mean density (ρo) are given for each mooring location

Table 11.
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