Abstract

Low-mode internal tides, a dominant part of the internal wave spectrum, carry energy over large distances, yet the ultimate fate of this energy is unknown. Internal tides in the Tasman Sea are generated at Macquarie Ridge, south of New Zealand, and propagate northwest as a focused beam before impinging on the Tasmanian continental slope. In situ observations from the Tasman Sea capture synoptic measurements of the incident semidiurnal mode-1 internal-tide, which has an observed wavelength of 183 km and surface displacement of approximately 1 cm. Plane-wave fits to in situ and altimetric estimates of surface displacement agree to within a measurement uncertainty of 0.3 cm, which is the same order of magnitude as the nonstationary (not phase locked) mode-1 tide observed over a 40-day mooring deployment. Stationary energy flux, estimated from a plane-wave fit to the in situ observations, is directed toward Tasmania with a magnitude of 3.4 ± 1.4 kW m−1, consistent with a satellite estimate of 3.9 ± 2.2 kW m−1. Approximately 90% of the time-mean energy flux is due to the stationary tide. However, nonstationary velocity and pressure, which are typically 1/4 the amplitude of the stationary components, sometimes lead to instantaneous energy fluxes that are double or half of the stationary energy flux, overwhelming any spring–neap variability. Despite strong winds and intermittent near-inertial currents, the parameterized turbulent-kinetic-energy dissipation rate is small (i.e., 10−10 W kg−1) below the near surface and observations of mode-1 internal tide energy-flux convergence are indistinguishable from zero (i.e., the confidence intervals include zero), indicating little decay of the mode-1 internal tide within the Tasman Sea.

1. Introduction

Internal (baroclinic) tides are internal waves that are generated where surface (barotropic) tides push isopycnals up and down sloping topography (Garrett and Kunze 2007). In most locations, a large fraction of internal-tide variance is explained by the first few vertical modes (e.g., Hendry 1977; Nash et al. 2006). Because low-mode internal tides have large energy densities, fast group speeds, and weak velocity shear (i.e., long horizontal and vertical wavelengths), they account for most of the internal-tide energy flux in the ocean at any frequency (Simmons et al. 2004; Alford and Zhao 2007a) and can be tracked thousands of kilometers across ocean basins before dissipating (Ray and Mitchum 1996; Zhao and Alford 2009; Dushaw et al. 2011; Shriver et al. 2012; Zhao et al. 2016). The eventual fate of these internal waves impacts the general circulation of the ocean (Munk and Wunsch 1998; Melet et al. 2013; Waterhouse et al. 2014), effectiveness of sonar (Dushaw et al. 1995), stability of offshore structures (Osborne et al. 1978), and dynamics of coastal ecosystems (Sharples et al. 2009).

During the last 50 years, numerous studies have improved predictions of internal-tide generation (Garrett and Kunze 2007). At present, numerical predictions (Simmons et al. 2004; Arbic et al. 2010; Müller et al. 2012; Niwa and Hibiya 2014; Shriver et al. 2014) are quantitatively consistent with satellite-derived maps of surface-tide energy loss (Egbert and Ray 2000, 2001, 2003) and internal-tide radiation (Ray and Mitchum 1996; Zhao and Alford 2009; Zhao et al. 2016). Moreover, comprehensive field campaigns at the Hawaiian Ridge (e.g., Rudnick et al. 2003) and in Luzon Strait (e.g., Alford et al. 2015) have obtained in situ measurements that confirm the underlying dynamics of internal-tide generation.

However, the fate of the low-mode internal tide remains uncertain. Satellites observe a decay in internal-tide amplitude with distance from the generation sites, but because satellites only measure tidal signals that are stationary over several years, the decay may be explained by either internal-tide dissipation or loss of stationarity. Satellite and in situ observations can indicate different internal-tide amplitudes (Chiswell 2006), but the causes and distributions of these differences are largely unknown. As a result, satellite observations, which are used to map surface-tide energy losses, cannot alone map internal-tide dissipation.

Satellite observations of internal-tide decay imply very small energy-flux convergence in the open ocean (e.g., an internal tide with a depth-integrated energy flux of 1 kW m−1 that decays in 1000 km will produce an average energy-flux convergence of only 1 mW m−2). Several dissipation processes may produce this energy-flux convergence, such as scattering over rough topography (Bell 1977; Bühler and Holmes-Cerfon 2011), wave–mesoscale interactions (Dunphy and Lamb 2014; Kelly and Lermusiaux 2016), and wave–wave interactions (MacKinnon et al. 2013; Eden and Olbers 2014). However, nondissipative internal-tide interactions with temporally variable background flows can also produce small energy-flux convergences in the stationary internal tide (Rainville and Pinkel 2006; Park and Watts 2006; Zaron and Egbert 2014; Kelly et al. 2016). Because these dissipative and/or decohering processes are weak in the open ocean, any conclusions about a given process must take into account uncertainties associated with 1) imperfect in situ sampling, 2) potential biases in satellite estimates of stationary internal-tide amplitudes, and 3) the contamination of tidal signals by nontidal motions.

In essence, our understanding of the fate of the internal tide depends on whether satellite-observed internal-tide amplitudes decay because of dissipation or loss of stationarity. If the decay in satellite internal-tide amplitudes is primarily due to dissipation, then the interpretation of the data suggests that as much as 90% of the internal-tide may dissipate in the deep ocean (Buijsman et al. 2016). On the other hand, if the observed decay in altimetric fluxes is primarily due to a loss of stationarity, then internal tides may be capable of propagating across ocean basins and dissipating at distant continental slopes and midocean ridges (Johnston et al. 2003; Kelly et al. 2013; Mathur et al. 2014). Unfortunately, global observations of turbulence do not single out a dominant dissipative process but instead suggest that mechanisms vary from region to region (Waterhouse et al. 2014). Importantly, identifying the distribution of internal-tide dissipation on slopes and in the basins is crucial to determining tidal effects on the present and future global ocean climate (Melet et al. 2016).

Internal tides in the Tasman Sea

To better understand the fate of the low-mode internal tide, a large observational experiment was conducted in the Tasman Sea from January to March 2015 (Pinkel et al. 2016). In the southeastern Tasman Sea, Macquarie Ridge generates energetic M2 internal tides at several locations. These internal tides then propagate northwestward from the ridge, interfering to form a horizontally focused internal-tide “beam” that has been both modeled (Simmons et al. 2004; Pinkel et al. 2016; Klymak et al. 2016) and observed using gliders (Johnston et al. 2015) and satellite altimetry (Zhao et al. 2016, their Fig. 1). The beam is somewhat similar to that observed west of the Mariana Arc (Zhao and D’Asaro 2011); however, the internal-tide in the Tasman Sea has a larger signal-to-noise ratio because it is shielded from the remotely generated multidirectional internal-tide “swell” that fills the major ocean basins (e.g., Hendry 1977). Thus, the Tasman Sea is ideal for studying an internal tide as it propagates more than 1000 km over small-scale topographic roughness and through mesoscale eddies and the background internal-wave continuum. This paper examines in situ observations of the semidiurnal tide in the center of the Tasman Sea in order to address the following questions:

  • What fraction of the semidiurnal internal tide is mode 1 and stationary?

  • How well do in situ measurements of the stationary tide agree with satellite-derived measurements?

  • How much energy-flux convergence and turbulent kinetic-energy dissipation occur?

The remainder of the paper is organized as follows: section 2 describes the in situ observations; sections 3 and 4 document the methods used to estimate tidal quantities and turbulent dissipation, respectively; section 5 presents the results, including in situ satellite data comparisons; and section 6 states the conclusions.

2. Observations

The Tasmanian Tidal Beam Experiment (T-Beam) collected observations from the Research Vessel (R/V) Falkor (19 January to 13 February 2015) and with a full-depth mooring deployed and recovered on 10 January to 28 February 2015, respectively, by the R/V Revelle (Fig. 1). At eight stations, repeat CTD-lowered acoustic Doppler current profiler (LADCP) casts resolved the lateral extent of the internal-tide beam, by providing snapshots of the spatially integrated energy flux across two transects spanning the central basin (stations F5–F4 and F8–F9; Fig. 1). The mooring centered in the middle of the beam (A1) provided a 49-day time series of internal-tide variability and helped dealias space–time variability in energy fluxes observed from the shipboard CTD-LADCP stations. Glider measurements (Johnston et al. 2015) and satellite altimetry (Zhao et al. 2016) also add context to the ship-based and moored observations by providing regional estimates of beam variability and stationary energy flux, respectively.

Fig. 1.

Internal tide beam in the Tasman Sea. Contours show the (a) depth of the Tasman Sea (km) and (b) amplitude (mm) and (c) phase (degrees) of the semidiurnal baroclinic sea surface from satellite altimetry (Zhao et al. 2016). T-Beam observations from January to March 2015 include eight CTD-LADCP stations (F2–F9; white diamonds) and one full-depth mooring (A1; red circle). Black dots offshore of Tasmania (TAS) indicate coincident moored observations from the associated T-TIDE experiment (Pinkel et al. 2016). The internal tide generation site is along Macquarie Ridge, south of New Zealand (NZ) in (a). Mainland of Australia (AUS) is noted in (a).

Fig. 1.

Internal tide beam in the Tasman Sea. Contours show the (a) depth of the Tasman Sea (km) and (b) amplitude (mm) and (c) phase (degrees) of the semidiurnal baroclinic sea surface from satellite altimetry (Zhao et al. 2016). T-Beam observations from January to March 2015 include eight CTD-LADCP stations (F2–F9; white diamonds) and one full-depth mooring (A1; red circle). Black dots offshore of Tasmania (TAS) indicate coincident moored observations from the associated T-TIDE experiment (Pinkel et al. 2016). The internal tide generation site is along Macquarie Ridge, south of New Zealand (NZ) in (a). Mainland of Australia (AUS) is noted in (a).

At each of the eight CTD-LADCP stations, the shipboard CTD-LADCP profiled continuously for a duration ranging from 17 to 29 h. The LADCP system consisted of up-looking and down-looking Teledyne RD Instruments (TRDI) 300-kHz Workhorse ADCPs affixed to the ship’s rosette. The LADCP data were combined with data from a hull-mounted 75-kHz ADCP on the R/V Falkor and processed according to Visbeck (2002). This procedure produced a vertical profile of velocity and density every few hours, resolving the semidiurnal and diurnal tidal cycles. Owing to inclement weather, most profiles only extended from 200- to 2000-m depth. Full water-column profiles to 4700 m were obtained during the final cast at stations F6, F7, and F9.

The mooring deployed at A1 from 10 January 2015 to 28 February 2015 was equipped with upward-looking 300- and 75-kHz ADCPs at 38- and 784-m depth, respectively. These instruments captured mesoscale, tidal, and high-frequency currents by sampling once every 10 min. Point measurements of velocity were also made by four Aanderaa model 8 recording current meters (RCM8), which sampled once every 2 (at 1479- and 4739-m depth) or 5 min (at 2307- and 3166-m depth).

The mooring collected temperature, salinity, and pressure measurements at 1-min intervals using six Sea-Bird Scientific SBE37s and one RBR-Concerto placed at the near top (37 m), middle (285, 493, 975, 1680, and 2707 m), and bottom of the mooring (4726 m). An additional 31 temperature sensors (SBE56s and RBR Solo-Ts), which sampled at 1 Hz, were placed at various points along the mooring line from near surface (53 m) to near bottom (20 sensors above and 11 below 1000-m depth). The unambiguous relationship between temperature and density, which was measured from the shipboard CTD profiles and moored SBE37s, allowed the conversion of temperature to density. Mooring knockdown was accounted for by interpolating all density and velocity measurements to a constant vertical coordinate system.

Meteorological data were collected from the R/V Falkor and R/V Revelle, which were a maximum of 600 km from the T-Beam site. Additional wind estimates were obtained from the 1/2°, 3-h resolution Navy Global Environmental Model (NAVGEM; downloaded from hycom.org) from 153°E, 44°30′S. Wind stresses were calculated from each dataset using Large and Pond (1981).

During January–March 2015, numerous mooring- and ship-based measurements were also collected on the Tasmanian continental slope. These data, which document internal-tide breaking, scattering, and reflection (Pinkel et al. 2016), will be examined in subsequent manuscripts. However, glider observations (Johnston et al. 2015; Boettger et al. 2015) and numerical simulations (Klymak et al. 2016) have already highlighted the importance of reflection on this slope.

3. Tidal analyses

Tidal amplitudes and phases were extracted from the in situ observations through a combination of vertical mode and harmonic analyses. Because we make the hydrostatic approximation, the vertical modes are independent of frequency, so that vertical mode and harmonic analyses can be conducted in either order, or simultaneously in an ideal case.

a. Vertical modes

Horizontal velocity u and dynamical pressure p can be represented as a sum of vertical modes n:

 
formula

where x is the horizontal position vector and ϕn and pn are the horizontal velocity and pressure modes, respectively. The horizontal and vertical velocity modes are related via the continuity equation, which requires ϕn = dΦn/dz. The vertical velocity modes Φn are determined by an eigenvalue problem with linear-free-surface and flat-bottom boundary conditions (Wunsch 2015):

 
formula
 
formula
 
formula

where cn are the eigenspeeds, N(z) is the observed buoyancy frequency, g is gravity and H is the local depth. The theoretical group speed is , where f is the inertial frequency and ω is the tidal frequency.

Using the orthogonality of the modes (see Kelly 2016), the modal amplitudes can be defined:

 
formula

In practice, dynamic pressure must be estimated from density perturbations ρ′ using the hydrostatic balance, which leads to

 
formula

where ρ0 is the reference density and η is the surface displacement. Here, we neglect the second term in (4) because η/H ≪ 1, Φn(0) ≪ 1, and η was not measured in situ.

b. Harmonic analysis

At the mooring, the modal amplitudes can be computed every 10 min, leading to modal time series un(x, t) and pn(x, t) that contain both tidal and nontidal variability. Fourier filtering (i.e., bandpassing) around the M2 frequency [1.9 ± 0.4 cycles per day (cpd)] isolates the semidiurnal tide while retaining the nonstationary “tidal cusp” (Munk and Cartwright 1966; Kelly et al. 2015). Next, harmonic fits to the M2 and S2 tides, over the entire 49-day record, define the stationary tide. The residual time series then represents the nonstationary tide.

The precise definition of stationarity can impact the resulting fractions of stationary and nonstationary variance. For example, the variance captured by a harmonic fit generally decreases with increasing length of the fitting window (e.g., Nash et al. 2012). Here, the time scales of the nonstationary signals (defined by the tidal bandwidth) are 2.5 to 49 days. The stationary tides computed using the response method (Munk and Cartwright 1966) showed negligible differences.

c. Energy, energy flux, and apparent group velocity

Complex harmonic amplitudes, denoted with hats, are defined such that

 
formula

Therefore, tidal-averaged depth-integrated energy flux Fn, energy En, and “observed” group speed for each mode n are defined as

 
formula
 
formula
 
formula

(Alford and Zhao 2007b), where is the complex conjugate and only the real parts have physical meaning.

d. Practical considerations

The preceding framework implicitly assumes continuous time series with full-depth coverage. Since the actual data have limited temporal and depth coverage, the methods were modified in the following ways:

  • The moored pressure record has the best depth and temporal coverage of all the data collected. Pressure was calculated by integrating the density perturbations, removing the depth average at each time, bandpassing the time series at each depth, and then performing a simultaneous least squares fit to ϕ1(z) and ϕ2(z) (direct projection yielded almost identical results). The moored velocity record was least squares fit to ϕ0, ϕ1, and ϕ2 (direct projection yields poor results because of vertical gaps). Stationary M2 and S2 complex amplitudes, for both pressure and velocity, were obtained from 49-day harmonic fits to the modal-amplitude time series.

  • The CTD-LADCP records at each station are short (<30 h) and infrequently sampled (i.e., 9–14 round-trip casts) typically from 200- to 2000-m depth. Therefore, TPXO8 surface tides were subtracted from the raw velocities, and velocity was simply fit to a combined semidiurnal and mode-1 pattern. Attempts to fit more vertical modes resulted in increased uncertainties and implausible results (e.g., mode-2 amplitudes that were larger than mode-1 amplitudes). To avoid integrating density, mode amplitudes (of pressure) were determined by the least squares regression of density perturbations to a semidiurnal signal with a depth structure of N2Φ1. Owing to more irregular temporal sampling at station F2, density perturbations were defined relative to the cruise average density, rather than the station-averaged density.

In all of the datasets, a linear regression using least squares determined and and provided 95% confidence intervals on the real and imaginary parts of the complex harmonic amplitudes (Bendat and Piersol 2000). When necessary, these confidence intervals were converted to confidence intervals on amplitude, energy, energy-flux amplitude, and group speed using the standard propagation of uncertainty formula

 
formula

where xi and δxi are known and y(xi) is an arbitrary function of xi. In general, the mode-1 velocity fits have larger uncertainties than mode-1 pressure fits because (i) velocity has inherently more (unresolved) high-mode variance than pressure, (ii) near-inertial velocities cannot be separated from semidiurnal velocities at the CTD-LADCP stations (near-inertial pressure signals are typically negligible), (iii) velocity measurements on the mooring do not have extensive depth coverage, and (iv) velocities (but not densities) from the up- and downcasts at the CTD-LADCP stations were merged together in order to optimally remove the motion of the CTD rosette (Visbeck 2002). Averaging a semidiurnal signal over 3 h also biases the velocity estimate low by 0%–10%.

The CTD-LADCP stations were occupied at different phases of the spring–neap cycle but were too short to separate M2 and S2 tides. Therefore, M2 amplitudes at each CTD-LADCP station were estimated by multiplying the semidiurnal harmonic amplitudes by the complex ratio r of from the mooring record divided by the semidiurnal fit to the stationary mooring record over the same sampling period as each CTD-LADCP station. Because the mode-1 spring–neap cycle is relatively weak in the mooring record, these adjustments resulted in modest changes (<15%) to the amplitude and phase of and at each CTD-LADCP station.

4. Dissipation estimates

Turbulent kinetic energy dissipation rate ε (W kg−1) is estimated using the finescale parameterization of shear and strain from the CTD-LADCP profiles (Gregg 1989; Polzin et al. 1995a; Gregg et al. 2003). Finescale variations on the order of tens to hundreds of meters from vertical profiles of shear uz and strain ζ are linked to the rate of the downscale energy cascade of the internal wave spectrum. Here strain is defined as

 
formula

where is calculated from a quadratic fit to each 300-m profile segment (Polzin et al. 1995b; Huussen et al. 2012; Whalen et al. 2015) and is the mean of the quadratic fit. As the shipboard stations are far away from abrupt and rough topography, we expect that the finescale parameterization calculated from shear and strain will provide a suitable estimate of mixing as the assumptions required by this technique are valid (Polzin et al. 1995a).

To calculate the parameterized dissipation from shear and strain, vertical profiles of buoyancy-normalized shear and strain are divided into half-overlapping vertical segments of 300 m. Following Gregg et al. (2003), dissipation rate can be calculated as

 
formula

where ε0 is a constant of 6.73 × 10−10 W kg−1, N(z) is the stratification of the segment, N0 is a constant stratification of 5.24 × 10−3 rad s−1, and E is the spectral energy level given by

 
formula

where is the shear spectra integrated between a minimum wavenumber kmin of 1/300 m and maximum wavenumber kmax of 1/100 m. The GM subscript indicates shear from the Garrett–Munk spectra. The same wavenumber range is used when calculating the integrated strain spectra. The ratio of horizontal kinetic to potential energy or shear to strain variance Rw is used to reduce the distortion where the spectral energy levels are modified by regions of enhanced mixing (Polzin et al. 1995a) where

 
formula

The function L describes the theoretical dependence on downscale energy transfer rate on both average wave field content (through Rw) and latitude (where f and f30 are Coriolis frequencies at the local latitude and 30°N, respectively) such that

 
formula

Because of excessive ship roll and heave during the cruise, we were unable to make reliable estimates of turbulent dissipation rate from Thorpe-scale overturns from the CTD-LADCP profiles (Thorpe 1977; Dillon 1982; Ferron et al. 1998; Alford et al. 2006). However, vertical motion of the CTD package does not contaminate the finescale estimates, which depend on 100- to 300-m vertical wavelengths.

5. Results

a. In situ baroclinic pressure

Baroclinic pressure at the mooring (A1) is the best-resolved component of the internal-wave field in the dataset. The semidiurnal bandpassed 49-day record of baroclinic pressure has a clear semidiurnal mode-1 signal with a zero crossing at approximately 1500 m (Fig. 2), in agreement with the modal shapes (Fig. 3). The signal persists over the full record and contains only small perturbations in the zero-crossing depth.

Fig. 2.

Bandpassed semidiurnal baroclinic pressure (Pa) observed at mooring A1 from the full record of observations from yearday 9.72 to 58.5 (10 Jan to 28 Feb 2015).

Fig. 2.

Bandpassed semidiurnal baroclinic pressure (Pa) observed at mooring A1 from the full record of observations from yearday 9.72 to 58.5 (10 Jan to 28 Feb 2015).

Fig. 3.

(a) Time averaged buoyancy frequency N (s−1), from the mooring (A1; black) and CTD-LADCP station at F2 (gray). (b) The first two vertical modes calculated from the stratification at the mooring (A1). Notation along the centerline in (b) denotes the various instrument locations with temperature sensors (filled circles), CTDs (open circles), single point current meters (x’s), and the total ADCP coverage (dashed line).

Fig. 3.

(a) Time averaged buoyancy frequency N (s−1), from the mooring (A1; black) and CTD-LADCP station at F2 (gray). (b) The first two vertical modes calculated from the stratification at the mooring (A1). Notation along the centerline in (b) denotes the various instrument locations with temperature sensors (filled circles), CTDs (open circles), single point current meters (x’s), and the total ADCP coverage (dashed line).

Baroclinic pressure is predominantly stationary and mode 1 (Fig. 4), with mode-2 and nonstationary mode-1 signals having about 1/4 the amplitude of the stationary mode-1 tide. A notable exception is the inexplicably large nonstationary internal tide that appears after yearday 55 and shortly before the end of the mooring deployment. The mode-1 spring–neap cycle is relatively weak because S2 pressure is only 14% of M2 pressure (not shown). The mode-1 tide appears to have a remote origin because its spring–neap cycle lags the local surface-tide spring–neap cycle by 6.7 days (i.e., the “age” of the mode-1 tide is 6.7 days greater than that of the surface tide; Holloway and Merrifield 2003). Traveling for 6.7 days at the local theoretical group speed, cg,1 = 2.0 m s−1 (computed from in situ stratification), the mode-1 tide transits about 1100 km before reaching the mooring, which is roughly the distance from Macquarie Ridge.

Fig. 4.

The total, stationary, and nonstationary modal contributions to the amplitude of the baroclinic pressure (Pa) from modes 1 and 2.

Fig. 4.

The total, stationary, and nonstationary modal contributions to the amplitude of the baroclinic pressure (Pa) from modes 1 and 2.

b. Plane-wave fits to surface displacement

The moored pressure record suggests a predominantly mode-1 M2 tide that originates at Macquarie Ridge and propagates toward Tasmania, in agreement with the satellite data (Zhao et al. 2016). The internal tide is approximately a plane wave in the center of the beam (which has a half-width of about 200 km; Zhao et al. 2018), so mode-1 M2 pressure amplitudes at each station were converted to surface displacements using and fit to plane waves using the least squares method with different directions and wavelengths. Pressure at station F5, which was sampled after a large storm, was omitted from these fits because its pressure was more than two standard deviations from the mean observed during the experiment and the station length was too short to separate the semidiurnal and near-inertial signals (Table 1). For comparison, satellite data (Zhao et al. 2016) sampled at the same locations were also fit to plane waves. Approximately 90% of the spatial variance in both the in situ and satellite estimates of the mode-1 M2 internal tide is explained by a wave propagating to the northwest (i.e., a heading of θ = 145° counterclockwise from east) with a wavelength of about λ1 = 180 km (Fig. 5). The regression function returned amplitude and phase uncertainties. We estimated uncertainties in wavelength and direction from the widths of the coefficient of determination peaks. These uncertainties are robust because the coefficient curves are smooth and have visually distinct peaks between the locations where the coefficient of determination drops below its maximum by 1%.

Table 1.

Mode-1 M2 surface displacement , PE and KE, energy flux magnitude |F|, and observed group speed at each station (A1 and F2–F9). The theoretical group speed is cg = 20 m s−2. Uncertainties at A1 are less than the stated precision.

Mode-1 M2 surface displacement , PE and KE, energy flux magnitude |F|, and observed group speed  at each station (A1 and F2–F9). The theoretical group speed is cg = 20 m s−2. Uncertainties at A1 are less than the stated precision.
Mode-1 M2 surface displacement , PE and KE, energy flux magnitude |F|, and observed group speed  at each station (A1 and F2–F9). The theoretical group speed is cg = 20 m s−2. Uncertainties at A1 are less than the stated precision.
Fig. 5.

Amplitude and phase of the coefficient of determination r2 for the plane wave fits to (a) satellite and (b) in situ observations.

Fig. 5.

Amplitude and phase of the coefficient of determination r2 for the plane wave fits to (a) satellite and (b) in situ observations.

The plane-wave fits to the in situ and satellite data have statistically indistinguishable amplitudes of η1 = 1.1 ± 0.3 and 1.0 ± 0.2 cm for the satellite and in situ observations, respectively (Fig. 6). The agreement is somewhat surprising because the satellite data only contain the stationary signal, while the CTD-LADCP surface displacements contain both the stationary and nonstationary signals (although an effort was made to correct for the known spring–neap cycle). One explanation is that the CTD-LADCP stations were sufficiently spread out in time and space so that the nonstationary signals did not have any plane-wave character. If this is the case, the fit effectively isolates the stationary signal, and one could expect the nonstationary signal to add noise to the in situ plane wave fit but not the satellite fit (because the satellite data is purely stationary). However, the uncertainties from the in situ and satellite fits are similar, indicating the signals may be stationary but just not perfect plane waves.

Fig. 6.

(top) Amplitudes and (bottom) phases of plane wave fits of (left) satellite and (right) in situ observations from each T-Beam station (open circles). Background contours are the amplitude and phase from the satellite altimetry of Zhao et al. (2016). The (top) total flux and (bottom) heading and wavelength are noted in text within the panels, while the elevation and phase of the internal tide beam are noted above each panel with uncertainties. F5 is not included in this analysis as the observations were obtained immediately after a storm, making the semidiurnal and near-inertial signals inseparable.

Fig. 6.

(top) Amplitudes and (bottom) phases of plane wave fits of (left) satellite and (right) in situ observations from each T-Beam station (open circles). Background contours are the amplitude and phase from the satellite altimetry of Zhao et al. (2016). The (top) total flux and (bottom) heading and wavelength are noted in text within the panels, while the elevation and phase of the internal tide beam are noted above each panel with uncertainties. F5 is not included in this analysis as the observations were obtained immediately after a storm, making the semidiurnal and near-inertial signals inseparable.

The satellite wavelengths λ1 = 183 ± 6 km are consistent with those derived from the station eigenspeeds: = 183 ± 2 km (the uncertainty here is defined as the standard deviation between stations). The Greenwich phases ϕ and headings θ are ϕ = 140 ± 10° and 64 ± 11° and θ = 141 ± 2° and 149 ± 3° for the satellite and in situ observations, respectively. The differences between the two estimates is unclear, but small deviations in the wave propagation from Macquarie Ridge due to seasonal or interannual stratification may account for the slight difference in heading and phase. Additionally, the record length differences between the two sets of observations may account for these differences as the satellite observations comprise a 20-year mean, while in situ observations are 49 days long.

c. Energy flux, group speed, and energy-flux convergence

Mode-1 M2 energy flux can be estimated from pressure alone, assuming plane-wave propagation (see, e.g., Ray and Cartwright 2001), or from velocity–pressure correlations [(6a); see also Nash et al. 2005]. Flux estimates from pressure alone based on the plane-wave fits are |F1| = 3.4 ± 1.4 and |F1| = 3.9 ± 2.2 kW m−1 for the in situ and satellite data, respectively. The fractional uncertainty in |F1| [computed via (7)] is greater than that of because energy flux depends on the square of surface displacement.

Mode-1 energy fluxes estimated at each station via velocity–pressure correlations are noisy (i.e., they range in magnitude from 0.5 to 7.4 kW m−1) but are consistently directed toward Tasmania (Fig. 7 and Table 1). The average magnitude across all of the stations is |F1| = 4.5 ± 0.3 kW m−1. The mean AVISO sea surface height between 10 January and 28 February 2015 displays a large anticyclonic eddy (positive SSH anomaly) in the northern half of the internal-tide beam (Fig. 7). This eddy could alter internal tide energy and energy fluxes [note the variability in Table 1 and see Dunphy and Lamb (2014)], but we have not investigated this interaction.

Fig. 7.

Energy flux (kW m−1; black arrows) from CTD-LADCP stations F2–F9 and the mooring at A1 with altimetry-derived energy fluxes (gray arrows) and averaged sea surface height (cm) from AVISO altimetry from 1 Jan until 31 Feb 2015 (contours). The blue arrow is the energy flux from the plane-wave fit in Fig. 6.

Fig. 7.

Energy flux (kW m−1; black arrows) from CTD-LADCP stations F2–F9 and the mooring at A1 with altimetry-derived energy fluxes (gray arrows) and averaged sea surface height (cm) from AVISO altimetry from 1 Jan until 31 Feb 2015 (contours). The blue arrow is the energy flux from the plane-wave fit in Fig. 6.

The stationarity of the mode-1 semidiurnal tide (sum of S2 and M2 fits over 49 days relative to fits over 2.5 days) can be quantified at mooring A to provide insight into the predictability of internal tides. At this location, 67% of kinetic energy and 91% of potential energy in the semidiurnal band is associated with the stationary tide (98% and 89% of the stationary potential and kinetic energy is M2; see Table 1). These values are similar to those observed in Luzon Strait (Pickering et al. 2015), although the dynamical reason that kinetic energy is less stationary than potential energy is unclear. Satellite observations indicate the central beam has wiggles, which implies multiwave interference could produce a spatially varying kinetic energy/potential energy (KE/PE) ratio (Zhao et al. 2018). Most (93%) of the time-mean energy flux is associated with the stationary tide; however, the moored time series of energy flux illustrates how nonstationary deviations in velocity and pressure can produce large short-term variations in energy flux (i.e., 1–9 kW m−1; Fig. 8). In this case, the modest nonstationary pressure, which is 30% of the total pressure (Fig. 4), leads to factor-of-2 variability in energy flux.

Fig. 8.

(a) Total (gray; shaded with the uncertainty) and stationary (black) mode-1 semidiurnal energy flux amplitude (kW m−1) and (b) heading (°) from the moored record at A1.

Fig. 8.

(a) Total (gray; shaded with the uncertainty) and stationary (black) mode-1 semidiurnal energy flux amplitude (kW m−1) and (b) heading (°) from the moored record at A1.

The theoretical group speed cg,1 for a mode-1 M2 tide at 45°S is 2.0 m s−1, given the observed stratification. The observed group speed, calculated via (6c), is greater at the mooring and smaller at CTD-LADCP stations than the theoretical values (Table 1). These discrepancies are an indication that observed pressures and velocities do not obey the polarization relations for plane wave (see, e.g., Alford and Zhao 2007b; Wunsch 2015). One explanation is a sampling artifact; kinetic energy may be underestimated at the mooring as a result of limited depth coverage and overestimated at the CTD-LADCP stations (Table 1) because the stations were too short to separate near-inertial and M2 signals. A second explanation is that, because the beam is a superposition of several mode-1 plane waves originating from different sites along Macquarie Ridge (Zhao et al. 2018), velocity and pressure have spatially periodic nodes and antinodes (e.g., Fig. 1b; Johnston et al. 2015), which have a relatively weak impact on local plane-wave fits to pressure but a more significant impact on higher-order analyses such as flux.

Mode-1 M2 energy-flux convergence is estimated in two ways. First, the mode-1 pressures at each station are converted to energy fluxes (see, e.g., Ray and Cartwright 2001) and averaged along the northern and southern transects. Then the decrease in flux between the southern and northern lines is divided by the distance between the lines, yielding −∇⋅F1 = 6 ± 28 mW m−2. Second, net energy flux into the box connecting the in situ stations is computed from velocity–pressure correlations [(6a); see also Nash et al. 2005), including fluxes through the along-beam boundaries. The total flux through the boundaries is then divided by the area of the box, yielding −∇ ⋅ F1 = −5 ± 6 mW m−2 (i.e., a flux divergence). Both calculations indicate that energy flux convergence at the study site is statistically indistinguishable from zero (i.e., the confidence intervals include zero). However, the large uncertainties do not greatly constrain local dissipation or even preclude the possibility of (physically inexplicable) local internal-tide generation.

d. Energy dissipation

The mean vertical profile of parameterized dissipation is estimated as an average from the set of vertical profiles of dissipation from each CTD-LADCP station. Enhanced dissipation of 1 × 10−8 W kg−1 is observed above 450-m depth, decreasing to 1–2 × 10−10 W kg−1 below (Fig. 9, thick green line). The majority of the vertical profiles occurred in the top 2000 m of the water column, and dissipation below 2000 m is only computed from six profiles over the entire cruise.

Fig. 9.

Dissipation rate (W kg−1) as calculated using the finescale parameterization (Gregg et al. 2003) using buoyancy normalized shear and strain from all of the CTD-LADCP stations (gray dots), station averages (thin green lines), and mean dissipation rate from station averages (thick green line).

Fig. 9.

Dissipation rate (W kg−1) as calculated using the finescale parameterization (Gregg et al. 2003) using buoyancy normalized shear and strain from all of the CTD-LADCP stations (gray dots), station averages (thin green lines), and mean dissipation rate from station averages (thick green line).

The depth-integrated dissipation rate D ranges from

 
formula

where the spread, denoting the middle 66% of possible D values, is estimated from a Monte Carlo bootstrap method (Efron and Gong 2014). Specifically, ε(z) is integrated 100 000 times with each mean vertical profile of dissipation taken from a random set of the number of observed ε in each depth bin (drawn with replacement). Omitting the enhanced dissipation above 450 m, which may be primarily due to mesoscale and/or wind-forced motions, results in D = 0.9–1.2 mW m−2.

e. Near-inertial kinetic energy and wind work

During January to March 2015, there were several storms that produced wind stresses exceeding 1 N m−2 (Fig. 10a). These events generated near-inertial motions in the upper 200 m of the water column, which are quantified by near-inertial kinetic energy , where the NI subscript indicates a time series bandpassed between 0.9f to 1.15f (Fig. 10c). In addition, the rate of wind work in the near-inertial frequency band was estimated using from wind stress and surface velocity (i.e., the top bin of the current meter at 10-m depth; Silverthorne and Toole 2009; Plueddemann and Farrar 2006). Using the R/V Falkor winds and moored surface velocities, the wind work alternates between positive and negative values, depending on the alignment of the wind and surface currents, but has a mean of approximately 0.23 mW m−2 (Fig. 10b). Estimates of wind work using the NAVGEM wind product (Hogan et al. 2014) gave a comparable-magnitude wind work with a mean of 0.60 mW m−2, which is similar in magnitude to the estimate of the wind-energy input for the South Pacific between 50°N and 50°S of 0.57 mW m−1 using a slab model (Alford 2001).

Fig. 10.

(a) Wind stress from the R/V Falkor (red), three cruise legs from the R/V Revelle (gray, ship was within a maximum of 600 km of the T-Beam site), and the NAVGEM 3-hourly winds from 153°E, 44°30′S (black). (b) Near-inertial wind work (mW m−2) as calculated from the observed near-inertial currents and winds from the R/V Falkor (red), from the observed near-inertial currents and winds from NAVGEM (black), and from the observed currents and the R/V Revelle winds (gray). (c) The time integral of the near-inertial wind work (cumulative energy input to the mixed layer; kJ m−3). (d) Near-inertial kinetic energy as calculated from bandpassed currents at the near-inertial frequency between 0.9f and 1.15f from the mooring.

Fig. 10.

(a) Wind stress from the R/V Falkor (red), three cruise legs from the R/V Revelle (gray, ship was within a maximum of 600 km of the T-Beam site), and the NAVGEM 3-hourly winds from 153°E, 44°30′S (black). (b) Near-inertial wind work (mW m−2) as calculated from the observed near-inertial currents and winds from the R/V Falkor (red), from the observed near-inertial currents and winds from NAVGEM (black), and from the observed currents and the R/V Revelle winds (gray). (c) The time integral of the near-inertial wind work (cumulative energy input to the mixed layer; kJ m−3). (d) Near-inertial kinetic energy as calculated from bandpassed currents at the near-inertial frequency between 0.9f and 1.15f from the mooring.

6. Discussion and conclusions

In situ observations from nine locations in the center of the Tasman Sea confirm the presence of a mode-1 M2 internal tide, with a surface displacement of about η1 = 1 cm propagating toward Tasmania. The 6.7-day lag in the mode-1 spring–neap cycle is consistent with topographic internal-tide generation at Macquarie Ridge, 1100 km to the southeast. Although observed group speeds indicate that the internal-tide beam is not a simple plane wave, plane-wave fits to in situ and satellite mode-1 M2 surface displacements explain approximately 90% of the observed variance.

Plane-wave fits to in situ and satellite data indicate surface displacements that agree to within an uncertainty of about 0.3 cm, which is also the amplitude of the nonstationary mode-1 tide observed at the mooring. However, the T-Beam mooring is located between two very active mesoscale eddy regions (Chelton et al. 2011), and, thus, satellite data from these nearby areas may have more bias than data at the T-Beam site. Moreover, a nonstationary mode-1 internal tide was clearly observed at the mooring that could not have been observed by a satellite. The nonstationary tide had 30% the surface displacement and 10% the time-mean energy flux of the stationary tide. Numerical simulations are needed to determine if the nonstationary tide was generated by changes in internal-tide generation at Macquarie Ridge (D. Brahznikov et al. 2018, unpublished manuscript) or by changes in internal-tide propagation that caused the beam to wobble back and forth across the mooring.

The most notable effect of the nonstationary internal tide was that it occasionally caused total energy flux to decrease by half or double. A practical result of this variability is that dissipative processes, which can be proportional to u2, u3, or u4 (e.g., Bühler and Holmes-Cerfon 2011; Polzin et al. 1995a), may be even more intermittent.

Direct estimates of mode-1 energy-flux convergence have O(10) mW m−2 uncertainties and are indistinguishable from zero. The depth-integrated energy dissipation rate from a finescale parameterization of wave–wave interactions is approximately 2 mW m−2. However, observations of strong winds and surface-intensified turbulence suggest that much of the depth-integrated dissipation is wind driven. Omitting the near-surface peak in dissipation yields a depth integral of about 1 mW m−2, which may be characteristic of the local tidally driven turbulence.

To gain some context for these observations, a simple analytical model of a decaying mode-1 internal tide is useful:

 
formula

where x is distance relative to the mooring in the direction of wave propagation, F1 = |F1| is the magnitude of mode-1 energy flux, and D1 is the time-averaged depth-integrated mode-1 dissipation rate. The left-hand side represents energy-flux divergence along the wave path. The right-hand side must be parameterized to solve the problem analytically. Here, we find solutions for and , where r1 and r2 are unknown constants. While neither scaling is perfectly accurate for a propagating low-mode tide, the former approximates dissipation by scattering over rough topography (i.e., flux decays exponentially; Bühler and Holmes-Cerfon 2011) and the latter approximates dissipation by wave–wave interactions (i.e., dissipation is proportional to E2; Polzin et al. 1995a). The solutions to (14) for the two parameterizations are and , where observations of F1 and dF1/dx at the mooring (x = 0) can be used to determine r1, r2, F0, and x0.

Given an observed energy flux of F1 ≈ 3.5 kW m−1 at the study site, energy fluxes can be estimated at both Tasmania and Macquarie Ridge (i.e., x ≈ −1000 km and x ≈ 500 km, respectively) for different dissipation rates at the mooring (Fig. 11). For local decay rates of 1 mW m−2, the results from the different parameterizations are nearly indistinguishable; both imply that 1/3–1/2 of mode-1 energy flux is lost during the transit across the Tasman Sea. This is in rough agreement with altimetry data, where Zhao et al. (2018) predict that the semidiurnal internal tide loses little energy in its first 1000 km of propagation from Macquarie Ridge but quickly loses almost half of its energy upon reaching the Tasmanian slope. On the other hand, a modestly higher decay rate of 2 mW m−2 is unphysical, as it predicts 6–8 kW m−1 of energy flux at Macquarie Ridge (the wave–wave dissipation parameterization produces the higher estimate), about 50% higher than inferred from plane wave fits to satellite altimetry. These results suggest that the decay rate of the mode-1 tide in the Tasman Sea is more sensitive to the specified dissipation rate at x = 0 than the dissipation parameterization.

Fig. 11.

Analytical models of energy-flux decay with a scattering (black) and wave–wave (gray) dissipation parameterization, and different estimates of depth-integrated dissipation rates at the T-Beam site (1 and 2 mW m−2; solid and dashed lines). The energy flux is set to 3.5 kW m−1 at x = 0 km. Satellite flux (red; Zhao et al. 2018) is smoothed over 2°, the size of the in situ array.

Fig. 11.

Analytical models of energy-flux decay with a scattering (black) and wave–wave (gray) dissipation parameterization, and different estimates of depth-integrated dissipation rates at the T-Beam site (1 and 2 mW m−2; solid and dashed lines). The energy flux is set to 3.5 kW m−1 at x = 0 km. Satellite flux (red; Zhao et al. 2018) is smoothed over 2°, the size of the in situ array.

Together the data and analytical solution emphasize the conclusion that 4 kW m−1 mode-1 energy fluxes can only be observed far from Macquarie Ridge if open-ocean mode-1 dissipation is extremely weak (i.e., about 1 mW m−2). If mode-1 internal tides decay over shorter spatial scales in other regions, the processes responsible for their decay must be more vigorous than those that occur in the Tasman Sea.

Acknowledgments

A. F. Waterhouse and S. M. Kelly acknowledge funding from NSF-OCE1434722 and NSF-OCE1434352 as well as ship time aboard the R/V Falkor supported by the Schmidt Ocean Institute. LR and ZZ acknowledge NSF-OCE1129246. HLS and DB acknowledge support from the Tasman Sea Tidal Dissipation Experiment (TTIDE) NSF Grants OCE1130048. We are grateful to our collaborative T-TIDE scientists and, in particular, Gunnar Voet who deployed the A1 mooring from the R/V Revelle as well as the scientists and volunteers aboard the R/V Falkor: Pete Strutton, Randall Lee, Spencer Kawamoto, Gabriela Pilo, Ryan McDougall-Fisher, Hayley Dosser, Judy Lemus, Annalena Lochte, and Danielle Mitchell. Shipboard LADCP observations are available from the Interdisciplinary Earth Data Alliance (https://doi.org/10.1594/IEDA/322415; https://doi.org/10.1594/IEDA/322414). We thank two anonymous reviewers for their comments, which helped to improve the manuscript.

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Footnotes

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