1. Introduction
Oceanic internal waves are ubiquitous. These waves are the result of the restoring action of buoyancy and Coriolis forces on water parcels displaced from their equilibrium position. Internal waves induce isopycnal oscillations in the ocean interior as opposed to surface gravity waves that induce oscillations of the free surface of the ocean. Internal waves are characterized by vertical length scales from a few meters to 1 km, horizontal length scales from a few meters to tens of kilometers, group velocities of 5 cm s−1, amplitudes from meters to tens of meters, and periods from several minutes to several hours (Thorpe 2005; Kantha and Clayson 2000, chapter 6). Along their pathway, nonlinear interactions among the waves (wave–wave interactions), interactions with the mean flow (wave–mean interactions), and other processes cascade the energy of internal waves to smaller scales until the energy from internal waves is dissipated. Ultimately, most of the energy of internal waves dissipates as turbulent mixing. The remaining fraction of energy is exchanged with the mean flow. While some internal waves are sufficiently nonlinear to break immediately, enhancing diapycnal mixing, many internal waves radiate away from their generation point, and their breaking contributes to the background mixing in the ocean interior. Importantly, internal waves provide a link between large-scale forcing and small-scale dissipation. The transport of momentum and energy by internal waves within the ocean is a key process in the meridional overturning circulation (Osborn 1980). Here we analyze upper ocean horizontal velocity observations from the Kerguelen Plateau region in the Southern Ocean to describe the local internal wave field and its relation to mixing. Coherent features are identified in velocity profiles from EM-APEX Argo floats and interpreted in terms of internal wave kinematics.
Perturbations of the density field that generate internal waves originate at the surface of the ocean because of the wind or at the bottom of the ocean because of tides and mean flow interaction with topography. One of the most important sources of kinetic energy in the ocean is the wind stress force τw exerted by the wind on the ocean surface (Wunsch and Ferrari 2004). The global net transfer of kinetic energy from the wind to the ocean is estimated to be between 7 and 36 TW (Lueck and Reid 1984; Wunsch and Ferrari 2004), most of which is directly dissipated in the surface mixed layer. A residual of some 0.3–1.2 TW is transferred to near-inertial motions (Alford 2001; Watanabe and Hibiya 2002; Scott and Xu 2009) that can propagate away from the surface mixed-layer before breaking and generating mixing at depth (Pollard and Millard 1970). Observations of near-inertial wave energy propagation from the mixed-layer into the ocean interior suggest that wind-generated inertial motions are an important part of the ocean mixing budget, particularly in the Southern Ocean (Alford et al. 2012). A second source of internal waves in the ocean is the interaction of currents with the sea floor: internal waves are generated when tidal currents and geostrophic currents flow over a topographic obstacle. Globally, the flow of the barotropic tide over topographic features such as continental slopes, shelf breaks, sills, abyssal hills, and seamounts generates more internal waves than geostrophic currents do. The estimated global tidal energy input into internal tidal waves is 1 TW, while the global energy conversion from geostrophic currents into internal waves is 0.2 TW (Nikurashin and Ferrari 2013). In the Southern Ocean geostrophic motions such as the Antarctic Circumpolar Current are very efficient at generating internal waves (Nikurashin and Ferrari 2011, 2013). Thus, the internal wave field is composed of many waves from various sources, locally generated or from the global internal wave continuum.
Despite the multiplicity of sources and boundary conditions for oceanic internal waves, there is a notion of universality of the internal wave spectrum (Munk 1981). Garrett and Munk (1975) were the first to describe and model the background oceanic internal wave field in terms of a vertical wavenumber and frequency spectrum using observations from site D in the western North Atlantic (Garrett and Munk 1972; Fofonoff 1969). The energy spectrum produced by Garrett and Munk evolved (e.g., Garrett and Munk 1975; Cairns and Williams 1976) as observational tools became more sophisticated. Ever since it was proposed, it has been used as the reference background wave field and has influenced global analyses of the internal wave field. In the past 25 years, however, regional internal wave fields have been found to vary subtly from the Garrett and Munk reference spectrum (D’Asaro and Morehead 1991; Polzin and Lvov 2011). Such variations are attributed to regional differences in the balance between nonlinearity, generation, and propagation. The tendency of internal waves to propagate large distances is one of the most ill-constrained issues that has delayed the progress of regional models of the internal wave spectrum.
The documentation of regional internal wave fields is limited. Indeed, direct observations and sampling of the spatially inhomogeneous and temporally intermittent internal wave features in four dimensions is very complex. Much of the available observations come from remote sensing of continental shelves and other shallow regions, where the vertical displacement of the thermocline due to internal waves modulates the roughness of the ocean surface (e.g., Apel et al. 1975; Fu and Holt 1982; Alpers 1985; Liu et al. 1998; Zheng et al. 2002; Susanto et al. 2005; Zhao et al. 2012; Da Silva et al. 2012). Individual internal waves are observed in echograms, acoustic backscattering, and other sonar images, as well as time series of density, velocity, and temperature variance. Globally, there are many previous observations of individual internal wave packets (e.g., Ziegenbein 1970; Liu et al. 1998; Stanton and Ostrovsky 1998). There are a few reports presenting observations of several internal wave packets (e.g., Cuypers et al. 2013), but no study has yet explored the composition of a regional internal wave field by analyzing the characteristics of dozens of individual internal waves. Instead, previous observations of internal waves have often focused on the spectral characteristics of the internal wave field (e.g., Levine et al. 1997; Kunze et al. 2002; Fer et al. 2010; Dosser et al. 2014; Hennon et al. 2014; Martini et al. 2014).
Here we characterize 46 internal waves observed with Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats. The EM-APEX floats, which sample two dimensions in a quasi-Lagrangian framework, give a unique perspective on the persistence of internal waves propagating through the mean flow. The large number of internal waves analyzed provides a quantitative framework for resolving generation versus propagation dynamics. We show that the internal wave properties are a direct result of regional dynamics. Our key finding is that in the Subantarctic/Subtropical Front, some of the internal waves and their energy are advected away from the Kerguelen Plateau, likely affecting the mixing and stratification budget in the far field. The data are described in section 2, along with aspects of the Kerguelen Plateau regional oceanography. Linear wave theory and the method to identify and analyze internal waves are described in section 3. In section 4, we present the general properties of internal waves, their regional characteristics, and their consistency with the mixing field. The implications in terms of Doppler shift and wave–mean flow interactions are discussed in section 5. We include some results from a companion paper that explores the impact of the internal wave field on local mixing characteristics (Meyer et al. 2015).
2. Data and regional oceanography
a. Observations
1) Survey
The observations for this study were collected north of the Kerguelen Plateau in late 2008 during the Southern Ocean Finestructure (SOFine) project. The objective of SOFine was to investigate the relation between finescale processes and the momentum balance in the Antarctic Circumpolar Current (Naveira Garabato 2009; Waterman et al. 2013; Damerell et al. 2013; Waterman et al. 2014; Phillips and Bindoff 2014; Meyer et al. 2015; Forryan et al. 2015). In this study, the analysis of the internal wave field is based primarily on data from eight EM-APEX Argo-equivalent floats. The EM-APEX floats were deployed along the ship track, on the northern edge of the Kerguelen Plateau during the SOFine oceanographic survey. Programmed to surface twice a day, they were advected eastward by the Antarctic Circumpolar Current. We use a subset of 914 profiles of temperature, salinity, pressure, and horizontal velocity from the ocean surface to 1600 m depth sampled over a period of 10 weeks in the vicinity of the Kerguelen Plateau. The profiles obtained downstream of the Kerguelen Plateau are not examined.
2) EM-APEX floats
The EM-APEX profiling float is an innovative instrument that provides inexpensive, autonomous, and high-resolution observations of horizontal velocity, conductivity, temperature, and pressure. It combines a standard Teledyne Webb Autonomous Profiling Explorer float with a velocity-sensing electromagnetic subsystem. A Sea-Bird Electronics SBE-41 CTD instrument measures the temperature, salinity, and pressure. The EM-APEX electromagnetic subsystem has a compass, accelerometer, and five electrodes to estimate the magnitude of the horizontal velocity (Sanford et al. 1978) at 3-m vertical intervals.
3) Quality control
The temperature, salinity, and pressure data from the EM-APEX floats were quality controlled by applying the CSIRO Marine and Atmospheric Research Argo delayed-mode quality-control procedure. This was followed by extensive processing to convert the relative velocity measured by the floats to absolute horizontal velocity (Phillips and Bindoff 2014).
The Argo delayed-mode quality control is a rigorous and internationally standardized procedure to ensure that the temperature and salinity profiles are free of detectable errors and biases (Wong et al. 2013). The CSIRO software was adapted to fit the sampling strategy and data format of the EM-APEX floats, which are different than that of the Argo profiling floats. The procedure includes identifying bad pressure points, checking for density inversions, testing for surface pressure drifts, checking for salinity drifts, and applying a thermal lag correction. The thermal lag correction is tailored for the EM-APEX floats following the method of Johnson et al. (2007), and values for the correction parameters are α = 0.023 and τ = 25.0 s. The quality control of the EM-APEX float velocity data is described in more detail in Phillips and Bindoff (2014).
b. Regional oceanography
The profiles from the EM-APEX profiling floats stretch over 6500 km of float trajectories in the vicinity of the Kerguelen Plateau (41°–50°S, 61°–79°E), sampling the upper water column (0–1600 m) between November 2008 and January 2009 (Fig. 1). The EM-APEX float profiles of potential temperature, salinity, horizontal velocity, and buoyancy frequency help us identify oceanographic features near the Kerguelen Plateau (Fig. 2). Profiles are shown in sampling order and are grouped by floats, and floats are organized by decreasing mean sampling latitude. This gives the composite trajectories the appearance of a meridional transect, with the southernmost float on the left (Fig. 2).
Trajectories of EM-APEX floats overlying topographic contours (gray) ranging from 500 to 5000 m at 800-m increments. Each black dot denotes the surface location of a float profile. Float numbers are indicated next to their deployment locations (red dots). To identify frontal jets, mean surface geostrophic speed over 0.35 m s−1 is shown for the deployment period (color scale). The “front” label indicates the presence of the Subantarctic/Subtropical Front (SAF/STF region).
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
Vertical distribution of (a) potential temperature, (b) salinity, (c) current speed, and (d) buoyancy frequency squared (N2) along the trajectories of the eight EM-APEX floats. The position of the SAF/STF region is indicated by the vertical red line in (a) and (b). Selected labeled potential density (σθ) contours delimit water masses north and south of the SAF/STF region: Subantarctic Mode Water (SAMW) is defined as σθ ≤ 27.1 kg m−3, Antarctic Intermediate Water (AAIW) is defined as 27.1 ≤ σθ ≤ 27.5 kg m−3, and Upper Circumpolar Deep Water (UCDW) is defined as σθ ≥ 27.5 kg m−3. The 0.55 m s−1 current speed contour (red) identifies the location of jets in (c). The vertical dashed lines separate floats.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
Fronts can be identified using hydrographic criteria such as velocity maxima location, water-mass distribution, and locations where certain hydrographic properties cross particular depths (Belkin and Gordon 1996). Using surface velocity maxima determined from satGEM (Meijers et al. 2011), a weekly geostrophic velocity product, Meyer et al. (2015) found a coherent and intense jet north of the Kerguelen Plateau over the sampling period of the EM-APEX floats (18 November 2008 to 30 January 2009). The jet was identified as the middle and northern branches of the Subantarctic Front, likely merged with the southern branch of the Subtropical Front. We refer to this Subantarctic/Subtropical Front as the SAF/STF region (Figs. 1, 2). Another feature identified by Meyer et al. (2015) from the EM-APEX data and the satGEM product is a large meander-like structure pinching off the SAF/STF region, hereinafter referred to as the eddy. The eddy is located between 44° and 42°S and 69° and 73°E (Fig. 1). The EM-APEX floats do not sample the Polar Front (Meyer et al. 2015).
We sort the EM-APEX float profiles according to their position relative to the SAF/STF region. They are either in the region north of the SAF/STF region (Subantarctic Zone), in the SAF/STF region, in the region south of the SAF/STF region (Polar Front Zone), or in the eddy. Profiles that are located in a region of geostrophic flow velocity larger than 0.35 m s−1 are considered to be in the SAF/STF region, or in the eddy if not along the main axis of the SAF/STF region. The rest of the profiles are assigned to either the Subantarctic Zone or to the Polar Front zone. Half the profiles are found to be in the Polar Front Zone, 23% in the SAF/STF region, 10% in the Subantarctic Zone, and 17% in the eddy.
The characteristics of the oceanographic fronts in the area and eddy observed in this dataset are described in more detail in Meyer et al. (2015) (section 4a).
3. Linear wave theory
a. Governing equations
b. Dispersion relation
The direction of propagation of a plane wave phase is given by the three-dimensional wave vector p = (k, l, m), where m is the vertical plane component, also called the vertical wavenumber, and where k and l combine as the horizontal wave vector k = (k, l) (Fig. 3). The azimuth φ of the horizontal wave vector is φ = tan−1(l/k), and the magnitude of the horizontal wave vector is the horizontal wavenumber kh = |k| = (k2 + l2)1/2 (Fig. 3).
Coordinate system used to describe internal waves. The wave vector is p = (k, l, m) while the horizontal wave vector is k = (k, l) in the direction of increasing (x, y, z). The horizontal azimuth φ of the horizontal wave vector determines the orientation of the wave vector p in the horizontal plane.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
c. Polarization relations
Assuming a constant stratification and plane wave solutions where the variables are proportional to 
- The first diagnostic provides an estimate of the ratio of horizontal kinetic energy Ek to potential energy Ep for a single internal wave:
- The second diagnostic provides an estimate of the ratio of velocity variance in the clockwise (ECW) and counterclockwise (ECCW) rotating components (rotary ratio). The velocity vector u draws an elliptical helix with depth (Fig. 3) due to the rotation for the earth. In this near-inertial rotation of the velocity vector, the ratio of velocity variance in the ECW and in the ECCW rotating component iswhere the convention is that a positive intrinsic frequency implies downward phase (and therefore upward energy) propagation of the internal wave and corresponds to the dominant CW rotation of the velocity vector in the Southern Hemisphere (Polzin and Lvov 2011).
- The third diagnostic provides an estimate of the azimuth of the horizontal wave vector k from the rotary-buoyancy coherence. Using the linear polarization relations for a single plane wave and near-inertial interpretations of clockwise (CW = u − iυ) and counter clockwise (CCW = u + iυ) phase rotation with depth, the observed clockwise-buoyancy phase isand the counterclockwise-buoyancy phase iswhere
for waves with a velocity vector rotating CCW and asfor waves with a velocity vector rotating CW. See appendix A for derivations.
d. Properties of internal waves
Internal wave parameters and relations based on single plane wave solutions of the form 
4. Method of analyzing internal waves
Identifying internal waves and extracting their properties from observations is challenging. In this section, we explain how we obtain the properties of 46 internal waves in the region of the Kerguelen Plateau, applying linear wave theory to coherent features identified in vertical profiles of horizontal velocity. We first identify coherent features in real space; next, we interpolate variables onto a Wentzel–Kramers–Brillouin (WKB) coordinate and then execute a spectral analysis in the WKB stretched coordinate (see appendix C for details). Doing so, we make the assumption that observed coherent features are internal waves, as previously done (Polzin 2008; Müller et al. 1978). This analysis is restricted to depths at which the theory can be applied and where velocity observations are available (200–1600 m). We demonstrate the method by presenting the analysis of one of the 46 internal waves identified as internal wave 5. We define internal wave 5 as the coherent feature that ranges from approximately profile 64 to profile 76 between 600 and 1000 m in float 3761 velocity data (Fig. 4c). This wave has a very clear signal in the velocity anomaly profiles and a corresponding large peak in the energy spectrum. As a result, internal wave 5 makes a great case example. The method described below was also applied to the other 45 internal waves identified.
(a) Float 3761 eastward horizontal velocity (dark blue) and northward horizontal velocity (cyan) from profiles 64–76. (b) Corresponding smoothed velocity profiles where the smoothed velocity is the result of a sliding second-order polynomial regression with an increasing vertical fit window length. (c) Time series of corresponding horizontal velocity anomaly. The velocity anomaly profile is defined as the horizontal velocity minus the smoothed velocity. Internal wave 5 is identified by the coherence in the velocity anomaly between approximately 400 and 1200 m depth (gray shading). Missing profile numbers correspond to profiles interrupted by a drift at constant depth that were excluded from this analysis.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
a. Estimating the buoyancy frequency and buoyancy perturbation
We decompose the observed buoyancy b into a mean buoyancy 
b. Identifying internal waves from coherent features
The time–depth series of the velocity profiles show the presence of many internal waves. Altogether, internal waves are found in 400 profiles out of 914 velocity profiles available. To locate internal waves, we first visually identify regions of coherence in the horizontal velocity profiles. Such a region is highlighted in gray on Fig. 4c. Next, we compute the energy density and rotary spectra of the selected velocity profiles as well as the rotary-buoyancy coherence and rotary-buoyancy phase plots (Fig. 5). If the energy spectra and the rotary-buoyancy coherence have peaks at the same wavenumber and the coherence is statistically significant, we proceed to analyze that group of velocity profiles to extract internal wave properties.
(a) Horizontal potential (Ep) and kinetic (Ek) energy as well as CW (ECW) and CCW (ECCW) rotary spectra in terms of stretched vertical wavenumber of horizontal velocity profiles 64–76 (float 3761) computed using data between 402 and 1126 m. Internal wave 5 is identified by the peak in both kinetic and downward (CCW) energy at vertical wavenumber ~1 × 10−2 cpm (gray shading). (b) Rotary-buoyancy coherence 
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
Demodulation methods to identify wave packets in velocity datasets have been applied in previous studies (Cuypers et al. 2013). The time resolution of the velocity dataset in this study puts fundamental limits on our ability to apply such methods. To estimate the error associated with the initial choice of velocity profiles on the derived internal wave properties and the sensitivity of internal wave properties to the spectral coherence, we apply a bootstrap-like method to a subset of six internal waves. For a given wave, we compute its vertical wavenumber and energy spectra several times, withholding a different single velocity profile from the analysis each time. If a wave is made up of 10 horizontal velocity profiles, we compute the spectra 10 times, each time withholding a different velocity profile. We then compare the estimated internal wave properties from each run. From the range of obtained values, we compute the standard deviation σ (Table 2, values in bold). This method shows that the vertical wavenumber estimates are very robust and that the Ek/Ep energy ratio estimates are also robust.
Properties of analyzed internal waves from the EM-APEX dataset: number of the internal wave, float number, profiles in which the coherent feature was identified, spectral peak and energy propagation, Ek/Ep energy ratio, depth of wave, vertical wavelength λ, horizontal wavelength λh, vertical wavenumber m, horizontal azimuth ϕ, Doppler shift (Dshift), horizontal group velocity Cgh, period T, dissipation time scale τε, and propagation time scale τprop. For internal wavenumbers 5, 10, 15, 30, 36, and 47, the standard deviation of the derived property is indicated in bold.
The eastward and northward horizontal velocity anomaly profiles are defined as the horizontal velocity profiles (eastward and northward) subtracted from the smoothed velocity profiles (eastward and northward), where the smoothed velocity profiles are the result of a sliding second-order polynomial regression with an increasing vertical fit window length (Fig. 4). These velocity anomaly profiles provide information about the location of internal waves and about their phase propagation.
The coherent features used to define internal waves in the EM-APEX data might be partially biased toward internal waves with lower frequencies and large horizontal scales. This is the result of using coherence in the amplitude of the observations to identify internal waves. Higher frequency internal waves have smaller amplitude and are therefore harder to identify with the method applied. Analyzing data resolving the full water column would increase the confidence of statistics, in particular for the upward propagating internal waves generated in deeper waters.
c. Wavelength, frequency, and period
Next we consider the vertical wavenumber and energy density spectra for the profiles in which internal wave 5 is identified (Fig. 5a). The equation E = Ek + Ep is the energy density composed of a kinetic (Ek) and a potential (Ep) component. We find a peak in kinetic energy (Ek > Ep) with a vertical wavenumber m = 0.0049 cpm, which is matched by a peak in CCW rotary motions (ECCW > ECW; Fig. 5a, gray shading). These peaks are identified as the spectral signature of internal wave 5. The CCW rotary peak in the spectra confirms the earlier finding of upward phase propagation. Note that in stretched coordinates m = 0.01 cpm (Fig. 5a).
From the vertical wavenumber m of internal wave 5, we estimate its vertical wavelength λ = 206 m. We also estimate the intrinsic frequency ω0 of internal wave 5 by rearranging the energy ratio equation: ω0 = f[(Ep + Ek)/(Ek − Ep)]1/2 = 1.1 × 10−4 rad s−1 (Table 1), where we use observed energy estimates from the spectra. Subsequently, we estimate the period T = 2π/ω0 = 5.7 × 104 s = 15.8 h of internal wave 5.
d. Horizontal wavenumber
The horizontal wavenumber kh of internal wave 5 can be estimated using the dispersion relation (Table 1) where the hydrostatic approximation 
Given the rotary-buoyancy coherence, we estimate the horizontal wavenumber bias associated to background noise in potential energy for a subset of six internal waves (Polzin 2008). We find that this bias translates in the horizontal wavenumber being overestimated by a mean factor of 0.62 for this subset.
e. Horizontal wave vector azimuth and horizontal group velocity
From the spectral coherence, we can examine the relation between the rotary, either CCW or CW, and buoyancy perturbation b′. The analysis of the coherence between the rotary (CCW in this case) and the buoyancy perturbations shows a peak in the rotary-buoyancy coherence that matches the peak in the energy spectrum at approximately 0.01 cpm (Fig. 5b). The peak in the rotary-buoyancy coherence corresponds to a mean phase ϕ = 1.7 rad (Fig. 5c). From this peak, we estimate the horizontal wave vector k components (k, l), where k = kh sin(ϕ) = 4 × 10−4 rad and l = −kh cos(ϕ) = 5 × 10−5 rad (for CCW rotation).1
For a subset of six internal waves, we derive the uncertainty in the mean phase estimate ϕ given the rotary-buoyancy coherence (Table 2, bold values in ϕ column). The standard deviation of the phase for internal wave 5 is σ5 = ±0.1, and the mean standard deviation over the subset is σmean = ±0.25.
The horizontal azimuth φ of the horizontal wave vector k of internal wave 5 is estimated with the observed phase ϕ: φ = tan−1[−cos(ϕ)/sin(ϕ)] = 7.2°, where ϕ = 1.7 ± 0.1 rad [see third diagnostic in polarization relations: Eqs. (13)–(16)]. The horizontal azimuth indicates that internal wave 5 is propagating westward. The horizontal intrinsic group velocity Cgh of internal wave 5 is estimated, using the k and l vectors derived above: Cgh = 0.024 m s−1 (Table 1). We set N = N0 = 0.005 24 s−1 to account for WKB stretched coordinates.
f. Wave–mean interactions
We can estimate properties of the internal waves with regards to the flow field, such as the Doppler shift associated with advection by the mean flow and wave–mean interaction time scales.
The Doppler shift is defined as 
5. Results
All the identified internal waves and their properties are individually listed in Table 2. In theory, this analysis only applies to single linear internal waves.
a. General properties of internal waves
We observe 50 internal waves in the data from the EM-APEX floats, of which numbers 1, 2, 4, and 14 have ambiguous spectral signals and are discarded. On average, eight vertical profiles of horizontal velocity define the internal waves. The internal waves are scattered along the trajectories of the floats, with most internal waves located near the northern edge of the Kerguelen Plateau, in the SAF/STF region, and in the eddy (Fig. 6). The location of the identified internal waves in the water column is relatively evenly distributed between 200 and 1000 m, with the maximum number of internal waves observed at 600 m depth.
Properties of the propagation and location of internal waves over topographic contours (gray). The arrow size is proportional to the horizontal intrinsic group velocity of the wave (larger arrows correspond to larger group velocities) and its color indicates the energy propagation direction (red is upward; blue is downward). Also indicated is the mean location of the SAF/STF region and the eddy during the sampling period (blue shading), corresponding to mean surface geostrophic speeds larger than 0.38 m s−1 between 18 Nov 2008 and the 30 Jan 2009. Topographic contours range from 600 to 3200 m at 200-m intervals. The black dashed lines indicate the trajectories of the eight EM-APEX floats.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
The identified internal waves in the vicinity of the Kerguelen Plateau have a mean vertical wavelength λ of 200 m (σ = ±20), a mean horizontal wavelength λh of 15 km (σ = ±0.9), and a mean horizontal group velocity Cgh of 3 cm s−1 (σ = ±0.3), where the mean standard deviation σ of derived parameters is given in parentheses. Internal waves that are inertially forced at the ocean surface are expected to have intrinsic frequencies close to the local inertial frequency f. We find that the average period T is 16 h (σ = ±0.2), close to the local inertial period of 17 h at 45°S. The aspect ratio of ocean surface waves is typically one. Here, we find that the average aspect ratio (λh/λ) of the internal waves is 0.015 (Fig. 7a). Internal waves with a smaller period (higher frequency) have higher horizontal group velocity (Fig. 7b).
(a) Ratio of λh/λ presents the relationship between the vertical wavelength λ and the horizontal wavelength λh. (b) Relationship between the horizontal group velocity Cgh and the period T of each of the 46 internal waves.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
Of the 46 internal waves analyzed, 20 have energy propagating upward and 26 have energy propagating downward (Fig. 6). Downward-propagating internal waves travel in all directions, but upward-propagating internal waves have a tendency to propagate southeast on the horizontal plane (not shown). Internal waves with upward energy propagation are on average 100 m deeper than internal waves with downward energy propagation and appear clustered in the southern range of the sampling area (higher latitudes), closer to the Kerguelen Plateau, while internal waves with downward energy propagation are mostly located away from the plateau (lower latitudes). Deeper internal waves, where deep is defined as between 750 and 1200 m, propagate 25% faster than shallow waves that are shallower than 550 m depth.
The observed deep upward-propagating internal waves closer to the Kerguelen Plateau could be generated at the seafloor through the interaction of the flow (either the Antarctic Circumpolar Current or tidal) and rough topography. Downward-propagating internal waves can be wind generated but can also be the result of background wave field internal waves that have been refracted, for example, in a critical layer situation or in the case of internal wave capture (Polzin 2008).
For each internal wave, we estimate the topographic roughness of the seafloor below the wave as the variance of bottom height. The topographic roughness is computed over a 0.1° longitude × 0.1° latitude box, which is approximately 8 km × 11 km, using the topography dataset from Smith and Sandwell (1997, version 13.1).2 For each internal wave, we also estimate the associated mean wind speed within a 24-h time scale and a radius of 0.5° around the location of the wave. The wind data are a blended, satellite-derived 6-hourly wind product provided by European Remote Sensing Satellite Processing and Archiving Facility at the French Research Institute for Exploitation of the Sea.3 Finally, we estimate the mean shear-to-strain variance ratio Rω. These estimates are based on the results from a shear-strain finescale parameterization analysis applied to this EM-APEX dataset (Meyer et al. 2015). Variance ratio Rω is an estimate of the mean aspect ratio of the internal wave field from which a measure of the bulk frequency of the wave field content can be derived (Polzin et al. 2014). Higher Rω values imply a dominant presence of near-inertial waves, while lower Rω values can be attributed to the presence of more high-frequency internal waves at high vertical wavenumber or the presence of shear instabilities when m > mc (Polzin et al. 2003).
b. Regional characteristics of internal waves
To identify the potential dynamics behind the generation of the observed internal waves, we separate the internal waves by their location: either in the Polar Front Zone, SAF/STF region, eddy, or Subantarctic Zone. Median values of parameters of observed internal waves and other associated parameters for each dynamical region are summarized in Table 3.
Median values of parameters of observed internal waves and other associated parameters for each dynamical region. The topographic roughness estimates are based on topography data from Smith and Sandwell (1997, version 13.1). The wind speed estimates are based on a blended satellite-derived wind product provided by the European Remote Sensing Satellite Processing and Archiving Facility at the French Research Institute for Exploitation of the Sea. The rotary-with-depth shear variance (ϕCCW/ϕCW), the shear-to-strain variance ratio Rω, and the dissipation rate estimates result from a shear-strain finescale parameterization applied to the EM-APEX dataset in Meyer et al. (2015). The mean standard deviation σ of derived parameters is indicated in parentheses.
Most of the internal waves in the Polar Front Zone are propagating upward, while all of the internal waves in the Subantarctic Zone are propagating downward. Internal waves are on average deepest in the Polar Front Zone and shallowest in the Subantarctic Zone. The shortest horizontal wavelengths are on average observed in the SAF/STF region. Internal waves found in both the eddy and in the Subantarctic Zone have a period close to the local inertial period of 17 h.
Mean horizontal group velocities are higher in the Polar Front Zone [Cgh = 3 cm s−1 (σ = ±0.3)] and in the eddy [Cgh = 3 cm s−1 (σ = ±0.3)], but they are particularly low in the Subantarctic Zone [Cgh = 1 cm s−1 (σ = ±0.3)]. We find that waves in the Polar Front Zone, where the topographic roughness is the highest, are associated with the highest percentage of upward propagating waves and the largest dissipation time scale. The internal waves in the SAF/STF region also have the highest local dissipation rate and the lowest local shear-to-strain ratio (Rω, Table 3). In the eddy, internal waves are observed at the shallowest depth, mostly downward propagating and associated with relatively high local dissipation values. In the Subantarctic Zone, where the mean wind speed is the highest, the observed internal waves all propagate downward and are relatively shallow, slow, associated with high Rω values, and associated with the lowest local dissipation values (Table 3). We find that internal waves in the eddy and in the Subantarctic Zone areas are shallower and have intrinsic frequencies closer to the local inertial frequency f. Waves in the SAF/STF region and in the Polar Front Zone are deeper and have higher vertical wavelength.
This suggests that in the SAF/STF region, the interaction of the topography and the strong flow generates upward-propagating internal waves associated with very large values of energy dissipation. Away from the Kerguelen Plateau in the Subantarctic Zone, wind forcing might be the source of downward-propagating, near-inertial internal waves associated with smaller values of energy dissipation.
c. Doppler shift
The Doppler shift of the flow, 
A ratio of Doppler shift to f equal to or larger than one suggests the Doppler shift is significant for that internal wave. We find that the Doppler shift is significant in the SAF/STF region, as well as in the eddy, while it is low in the Subantarctic Zone (Table 3). We estimate the orientation of the internal waves with regard to the flow gradient as the angular displacement between the horizontal wave vector and horizontal velocity vector of the local flow (Fig. 8). Angular displacements close to 90° indicate the internal wave is perpendicular to the flow while angular displacements close to 0° or 180° indicate the internal wave is parallel to the flow. We find that particularly large Doppler shift to f ratios are associated with internal waves parallel to the flow (median angle is 30°) while small Doppler shift to f ratios are associated with internal waves near perpendicular to the flow (median angle is 70°; Fig. 8). This confirms that the filamentation process associated with a large Doppler shift reduces the angle between the horizontal wave vector and the flow gradients.
Angular displacement (°) between the azimuth of the waves and the flow azimuth vs Doppler shift to f ratio for each internal wave. Internal waves with angular displacements close to 90° (waves are perpendicular to the flow) and with Doppler shift to f ratio smaller than 1 are likely affected by relative vorticity (blue shading); internal waves with angular displacements close to 0° (waves are parallel to flow) and with Doppler shift to f ratio larger than 1 are likely dominated by the Doppler shift (orange shading). The colors denote the dynamical zone to which the internal waves belong.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
d. Internal wave and mixing fields
We compare the location of the identified internal waves with the local mixing distribution as estimated from a shear-strain finescale parameterization (Meyer et al. 2015). Large-amplitude internal waves are likely associated with large vertical shears leading to large dissipation rates under the assumptions of the finescale parameterization (Meyer et al. 2015). Since most of the internal waves observed in the EM-APEX dataset have large wave amplitudes necessary for their identification, we expect the position of observed internal waves (blue and red arrows) to match areas of enhanced dissipation rate (color contour; Fig. 9). Nearly all observed waves are found in areas where finescale parameterization predicts higher dissipation values. This confirms that the identification of internal waves is consistent with the shear-strain parameterization analysis.
Distribution of the dissipation rate ε along float trajectory estimated from the EM-APEX floats with a shear-strain finescale parameterization (Meyer et al. 2015). The overlaying arrows indicate both the location of the identified internal waves and the direction of wave propagation where red is upward and blue is downward. The vertical dashed lines separate floats.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
6. Discussion
a. Doppler shift and relative vorticity
The EM-APEX data show that Doppler shift dynamics dominate in the SAF/STF region. Internal waves associated with large Doppler shift values tend to be aligned with the flow, while some internal waves perpendicular to the flow are associated with small Doppler shift to f ratios (Fig. 8). The latter is a characteristic of near-inertial wave trapping (Kunze 1985) where the relative vorticity gradient dominates the wave dynamics. Some internal waves in the Polar Front Zone and in the eddy are likely candidates for waves influenced by relative vorticity.
b. Wave–mean flow interactions
To investigate how much the internal waves interact with the mean flow based on their location, we analyze the spatial distribution of the dissipation time scale τε and propagation time scale τprop of each wave. We find that, on average, internal waves propagate away faster than they dissipate in the SAF/STF region and in the Polar Front Zone, while in the Subantarctic Zone internal waves are likely to dissipate faster than they propagate (Table 3). Studying individual waves, we find that some internal waves in the SAF/STF region are unstable enough to dissipate locally, while other internal waves are advected away by the mean flow (Fig. 10).
Dissipation (τε) and propagation (τprop) time scales of the observed internal waves. The colors denote the dynamical zone to which the internal waves belong. Waves under the black dashed line (1:1 ratio) are more likely to propagate away from the site, while waves above are more likely to dissipate locally.
Citation: Journal of Physical Oceanography 46, 2; 10.1175/JPO-D-15-0055.1
This local versus far-field dissipation of internal waves has implications for the Southern Ocean stratification. That some internal waves propagate in the SAF/STF region implies they dissipate downstream, driving far-field mixing. The EM-APEX dataset strongly indicates some internal wave energy is advected away from the Kerguelen Plateau, thereby enhancing mixing and affecting the ocean stratification downstream.
7. Conclusions
The EM-APEX dataset provides a unique view of the spatial distribution and variability of the internal wave field north of the Kerguelen Plateau. Analyzing the internal waves in multiple dimensions (wavenumber and frequency domain) but also resolving the variability in the temporal and vertical wavenumber domain allows for an increasingly sophisticated characterization of the internal wave field. The results from the analysis of 46 internal waves in the area are consistent with the mixing analysis of the same dataset (Meyer et al. 2015). The high sampling density of the EM-APEX floats in space and time over a range of dynamical regions considerably enhances the information in terms of the generation and propagation of observed internal waves.
Internal waves observed near the Kerguelen Plateau have on average a vertical wavelength of 200 m (σ = ±20), a horizontal wavelength of 15 km (σ = ±0.9), a period of 16 h (σ = ±0.2) close to the local inertial period, and a horizontal group velocity of 3 cm s−1 (σ = ±0.3). We plan to apply ray tracing theory to estimate the temporal evolution of the wavenumber and frequency of the observed internal waves. This would provide key evidence about the sources of the internal waves identified in this dataset.
The internal wave properties, their location, and their direction of propagation are dependent on regional dynamics, suggesting that different generation mechanisms of internal waves dominate in different dynamical zones. The wave field in the SAF/STF region and the Polar Front Zone is influenced by the local small-scale topography and flow strength; the eddy-wave field is influenced by the large-scale flow structure, while the internal wave field in the Subantarctic Zone is influenced by atmospheric forcing. The SAF/STF region has an energetic wave field with the local dissipation rate associated with the waves at least twice as high as anywhere else in the area.
The analysis of the internal wave properties has far-reaching implications. We find evidence that in the SAF/STF region, only some of the internal waves generated at the Kerguelen Plateau dissipate their energy locally. Other internal waves are advected away from the region and their energy is dissipated far from the generation site. This far-field dissipation is important in setting the Southern Ocean stratification and driving the overturning circulation of the Southern Ocean.
The observed interaction of internal waves with an eddy leads to questions about energy transfers between internal waves and eddies, ultimately leading to the dissipation of internal wave energy in the form of mixing. Regions of enhanced, large-amplitude eddy activity are found in the Southern Ocean (Chelton et al. 2011). Because of the transfer of energy from internal waves to eddies, such regions have the potential to be hot spots of water mass transformation and to impact the regional overturning circulation. With the predicted intensification of wind stress and potential increase of eddy energy in the Southern Ocean over the coming decades (Meredith and Naveira Garabato 2012), it remains an open question if the eddy contribution to the mixing will increase as well.
We have shown that the local generation of internal waves is likely impacting the large-scale stratification of the Southern Ocean. Whether the observed internal waves are controlled by Doppler shift dynamics or relative vorticity dynamics has consequences for the regional divergence of momentum and therefore the Southern Ocean momentum balance. The role of the frontal region is key in advecting internal wave energy and momentum away from the generation site.
Acknowledgments
We thank the scientists and crew from the RRS James Cook JC029. We acknowledge the valuable contributions of the SOFine team, Stephanie Waterman and Alberto Naveira Garabato in particular, for providing advice and data on numerous occasions. Comments and suggestions from Beatriz Pena-Molino are gratefully acknowledged. A.M. was supported by the joint CSIRO-University of Tasmania Quantitative Marine Science (QMS) program and the 2009 CSIRO Wealth from Ocean Flagship Collaborative Fund. K.L.P.’s salary support was provided by Woods Hole Oceanographic Institution bridge support funds. B.M.S. was supported by the Australian Climate Change Science Program. We thank John Dunlap, Steve Rintoul, Susan Wijffels, and Beck Owens for their support in mounting the EM-APEX component of the SOFine project. Funding for the EM-APEX experiment was provided by the Australian Research Council Discovery Project (DP0877098) and Australian Government DEWHA Projects 3002 and 3228.
APPENDIX A
Polarization Relations
APPENDIX B
Group Velocity Approximations
APPENDIX C
The WKB Approximation
The WKB approximation is used to obtain an approximate solution to a time-independent, one-dimensional differential equation. It involves recasting the wave function as an exponential power series to recover approximate solutions for the wave phase with slowly varying amplitude. The problem was first solved by Lord Rayleigh (Rayleigh 1912). Subsequently, the solution was rediscovered almost simultaneously by Wentzel (1926), Kramers (1926) and Brillouin (1926).
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