1. Introduction
In this submesoscale regime, NIWs can exhibit unexpected behavior. For example, NIW velocity hodographs do not trace inertial circles but are instead elliptical and, for a given wave frequency, the direction of energy propagation is symmetric about the slope of isopycnals not the horizontal and approaches the isopycnal slope as the frequency nears ωmin (Mooers 1975; Whitt and Thomas 2013). This unusual wave physics can facilitate energy transfers between NIWs and balanced currents (Thomas 2012) and allow for the formation of critical layers along the sloping isopycnals of fronts (Whitt and Thomas 2013).
There have been several field campaigns designed to study NIW-balanced flow interactions in the mesoscale regime, that is, with 
2. Observed frontal structure
a. Data from ships and towed profilers
Observations were made while underway on two global class research vessels, R/V Knorr and R/V Atlantis, traveling at approximately 8 kt (1 kt = 0.51 m s−1). Both ships were equipped with two shipboard-mounted acoustic Doppler current profilers (ADCPs). A 300-kHz Teledyne RDI Workhorse sampled the top 100–150 m with 4-m vertical resolution, while a 75-kHz Teledyne RDI Ocean Surveyor surveyed the top 500–600 m with 8-m vertical resolution. In addition, the R/V Knorr was equipped with a towed MacArtney TRIAXUS that undulated between the surface and 200 m to provide profiles of temperature, conductivity, and pressure along a sawtooth pattern with a resolution of approximately 1 km in the direction of travel. The R/V Atlantis was equipped with a Rolls Royce moving vessel profiler (MVP) with a free-falling AML Oceanographic micro-CTD to provide vertical profiles of temperature, conductivity, and pressure, again with a resolution of approximately 1 km in the direction of travel.
Part of the analysis requires velocity and density interpolated to the same time-depth grid, covering the top 200 m. The velocity data for these grids are obtained by merging data from the 300- and 75-kHz ADCPs on each ship and interpolating that data to the density grid derived from the profiler data collected on that ship. In particular, current measurements are first averaged over approximately 1 km in the direction of travel in depth bins of 4 (300 kHz) and 8 m (75 kHz). Then, the data from the 75-kHz ADCP are linearly interpolated to the 1 km by 4 m 300-kHz grid and gaps in the 300-kHz data are filled with the interpolated 75-kHz data. The resulting merged velocity data are then smoothed with a two-dimensional Gaussian kernel with a standard deviation of 4 m in depth and approximately 1 km in the direction of ship travel. The half-power wavelengths of this two-dimensional Gaussian kernel are 30 m in the vertical and approximately 7.5 km in the direction of ship travel (i.e., 30 min at 8 kt). Finally, the smoothed velocity profiles are linearly interpolated (along the ship track) to the positions of the density profiles.
The density is gridded slightly differently, depending on the vessel/instrument. Each TRIAXUS profile is averaged into 2-m vertical bins (ups and downs are included as separate profiles). The horizontal position/time of each profile is defined by the ship position/time at the midpoint of each TRIAXUS profile. For example, gray dots in Fig. 3a (shown below) mark the ship positions at the midpoint of each TRIAXUS profile in one section across the Gulf Stream. Each downgoing vertical MVP profile is averaged into 1-m vertical bins at a given position/time (upward profiles from the MVP are excluded). For example, gray dots in Fig. 2a (shown below) mark the MVP vertical profile locations in one section across the Gulf Stream. The gridded density from each profiler is then filtered by applying the same two-dimensional Gaussian smoother that was applied to the velocity data, and the resulting filtered density is linearly interpolated (in depth) to the 4-m bins of the velocity grid.
After this filtering and gridding, density and velocity are collocated at the horizontal positions of the density profiles in uniform 4-m vertical bins, and features with wavelengths less than about 30 m in the vertical and 7.5 km in the direction of ship travel have been filtered out of both the density and velocity datasets.
b. Experimental description and observed frontal structure
Measurements focused on the North Wall of the Gulf Stream, past the point where it separates from Cape Hatteras, North Carolina, between 38° and 40°N and 60° and 70°W. Here, cold subpolar water and warm subtropical water converge to form a strong front. The campaign was broken into a series of drifts, each following an acoustically tracked Lagrangian float (D’Asaro 2003) placed in the surface mixed layer in the strongest part of the front for 2 to 4 days. In each drift, the ships made repeated frontal cross sections following the float (e.g., Fig. 1). The float was tracked using a transducer pole, which was mounted on the R/V Knorr, extended 2 m below the keel, and provided a maximum range of about 4 km. To track the float accurately, the R/V Knorr remained within about 5 km of the float position. The tracking equipment is otherwise similar to that used in D’Asaro et al. (2011), and the uncertainties associated with the float position estimates are thought to be small compared to other uncertainties in the analysis.
(a) Sea surface temperature (color) and the paths of the R/V Knorr (black) and R/V Atlantis (gray). (b) The position of the observational platforms relative to the Lagrangian float [magenta line in (a)] over the duration of the drift. The approximate locations of the ships and float at yearday 61 and 61.7 are marked with arrows.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
In this paper, attention is focused on the second drift, which occurred from 1 March through 3 March 2012 (yeardays 60 through 62), near the strongest part of the front (Fig. 1a). While the Atlantis made wider ~30-km cross sections around the float, the Knorr made narrower ~10-km cross sections, closely following the float (Fig. 1b). The survey strategy involved intensive sampling of the water around a Lagrangian float in the mixed layer in an attempt to minimize the convoluting effects of advection on the analysis. But, it is important to note the observations were obtained in a region of strong lateral and vertical gradients in velocity, and therefore the observed fluid is not exactly in a Lagrangian frame of reference at all depths and cross-stream locations observed.
A typical cross-front section from the Atlantis (Fig. 2) exhibits a strong surface-intensified jet with velocities exceeding 2 m s−1 in the streamwise direction x, order-one vorticity Rossby numbers (Ro = ζ/f ≈ −∂u/∂y/f, where f is the Coriolis frequency, u is the along-stream velocity, and y points in the cross-stream direction), isopycnal slopes sb of order 0.01, and geostrophic Richardson numbers Rig ranging from 1 to 10. To estimate Ro and Rig, the associated gradients are calculated using central differences over approximately 2 km in the horizontal and 8 m in the vertical, but variability on scales less than about 30 m in the vertical and 7.5 km in the horizontal is suppressed by the processing (see section 2a). In any case, the resolved flow is in the submesoscale part of (Ro, Ri) parameter space suggesting that if there are NIWs in the current they should be substantially modified by its vorticity and baroclinicity. As described in the next section, there were indeed features in the observed flow field during the drift with properties that are consistent with NIWs, giving us the opportunity to study the interaction of the waves with a submesoscale current.
Data from the second Atlantis section obtained at about yearday 61.3 at a mean latitude and longitude of 38.2°N, 66.1°W; see Fig. 1a. Cross-front/depth maps of (a) streamwise velocity u, (b) vorticity Rossby number Ro = −∂u/∂y/f, and (c) inverse geostrophic Richardson number 
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
3. Observations of near-inertial shear
a. Structure and evolution of the vertical shear
Banded patterns of strong vertical shear ∂u/∂z [where u = (u, υ) is the horizontal velocity vector] were observed during the drift. To analyze these features, we rotate the coordinate system so that (x, y) denotes the streamwise and cross-stream directions respectively, and x is defined to point in the direction of the local float path, whereas y points in the direction perpendicular to the local float path.
Sections of the streamwise and cross-stream shear, ∂u/∂z and ∂υ/∂z, respectively, reveal shear anomalies (i.e., relative to a linear fit in z) that are arranged into banded structures parallel to slanted isopycnals (Figs. 3a,b). The shear anomalies have a dominant vertical wavelength between 50 and 100 m and magnitudes of up to 0.005 s−1, which is comparable to the depth-averaged alongfront shear (Figs. 3a,b). However, the associated velocity anomalies, ua ~ (1/m)(∂ua/∂z) 
Data from the fifth Knorr section shown in Fig. 1a, which were obtained at yearday 61.1 at a mean latitude and longitude of 38.1°N, 66.3°W. Maps are presented of (a) the streamwise component of the vertical shear ∂u/∂z, (b) the cross-stream component of the vertical shear ∂υ/∂z, and (c) the inverse Richardson number 
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
The time series of the wind stress together with maps of the streamwise and cross-stream vertical shear observed from the ships at points near the float path illustrate the temporal evolution of the banded shear and the atmospheric forcing during the drift (Fig. 4). Individual time series of the shear averaged between −45 and −65 m and −65 and −85 m further highlight this evolution (Fig. 5). The time series show that the shear anomalies evolve significantly with time/downstream position at essentially all depths/isopycnals during the drift. Yet, the shear anomalies exhibit a consistent dominant vertical wavelength between 50 and 100 m and achieve a similar maximum amplitude of 0.005 s−1, which is similar in magnitude to the depth-averaged alongfront shear, throughout the drift. Likewise, the shear anomalies are associated with regions of low Richardson number Ri ≈ 1 below the mixed layer at various times throughout the drift (Fig. 4d).
Several time series of data collected along the whole drift. (a) The 1-min moving average of the wind stress derived from observations of wind speed at a meteorological station on the Knorr and converted to a stress using the algorithm of Large and Pond (1981). (b) The streamwise component of the vertical shear ∂u/∂z, (c) the cross-stream component of the vertical shear ∂υ/∂z, and (d) the inverse Richardson number 
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
(a),(c) The streamwise vertical shear and the (b),(d) cross-stream vertical shear observed (top) between 45- and 65-m depth and (bottom) between 65- and 85-m depth along the float path. As in Fig. 4, the observations are from the Knorr but restricted to cross-stream positions within 350 m of the float path. Data are binned into 4-m depth bins, so there are five time series in each panel. Colored lines mark the average of all five depth bins at each time.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
The structure and time/downstream evolution of the vertical shear field illustrated in Figs. 2–5 suggests qualitatively that the shear is associated with downward-propagating NIWs that are potentially modified by the vorticity and baroclinicity of the Gulf Stream, as described by Whitt and Thomas (2013). In particular, the temporal variability of the observed shear appears to be dominated by oscillations with frequencies that are slightly higher than the local Coriolis frequency f ≈ 9 × 10−5 s−1 (the local inertial period corresponds to ~0.8 days). In addition, lines of constant shear anomaly propagate upward with time, which is consistent with downward-propagating NIW physics. To test this hypothesis more quantitatively, rotary vertical wavenumber and frequency spectra will be used to decompose the shear variance into clockwise and counterclockwise components and estimate the dominant wave frequency in sections 3b and 3c, and the results will be compared to the theoretical predictions of Whitt and Thomas (2013) in sections 4–5.
b. Rotary vertical wavenumber spectra
The vertical variation of ∂u/∂z seen in Fig. 3 suggests that the shear vector rotates primarily clockwise with increasing depth at that location. A quantitative estimate of the shear vector’s average sense of rotation with depth along the float path is derived from the rotary components of the Froude number vertical wavenumber spectrum [i.e., a buoyancy frequency normalized shear spectrum (e.g., Munk 1981; Alford and Gregg 2001; Rainville and Pinkel 2004)]. We calculate Froude number spectra rather than shear spectra because the former provides more information about the local stability of the waves, and observations collected at locations with different stratifications can be more easily compared with each other (Munk 1981).
The rotary components of the Froude wavenumber spectra (plotted in Fig. 6) show that the average CW Froude variance is 2–5 times greater than the average CCW Froude variance at vertical wavelengths of 40 to 80 m. The enhanced CW Froude variance is in contrast to the canonical internal wave continuum, in which waves propagate upward and downward in equal proportion (e.g., Garrett and Munk 1972), and is consistent with the presence of energetic and coherent, downward-propagating NIWs (e.g., Leaman and Sanford 1975; Kunze and Sanford 1984; Mied et al. 1987; Alford and Gregg 2001; Shcherbina et al. 2003; Inoue et al. 2010; Thomas et al. 2010; Jaimes and Shay 2010). However, having said this, this conclusion is derived using the polarization relations of NIWs in the absence of a background flow. Whitt and Thomas (2013) show that submesoscale fronts strongly modify the polarization of NIWs, which can affect the ratio of CW to CCW Froude variance for downward-propagating NIWs. We will explore this physics in section 4.
The float path average CW and CCW rotary components of the Froude number spectrum 
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
c. Rotary frequency spectra
The temporal variation of the shear vector is most wavelike just below the base of the mixed layer. There, the shear varies nearly sinusoidally with a similar amplitude and a fixed phase relationship between the streamwise shear ∂u/∂z and cross-stream shear ∂υ/∂z (Fig. 5). In particular, ∂υ/∂z leads ∂u/∂z by a quarter of a cycle (Fig. 5), indicating that the shear vector rotates clockwise with increasing time, which is consistent with the dynamics of NIWs in the Northern Hemisphere (e.g., Gonella 1972).
(a) The depth average (from z = −41 to −189 m) of the CW and CCW rotary components of the Froude number spectrum 
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
(a) The average Froude number variance within 350 m of the float path 
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
Together, the analysis of this and the previous section support the hypothesis that a large fraction of the total shear variance observed during this roughly 2-day Lagrangian survey of the Gulf Stream Front is associated with coherent NIWs. In particular, the shear vector is polarized so that it rotates clockwise with both increasing depth and increasing time on average, and at some depths the total shear variance is dominated by a narrow band of superinertial frequencies 
4. Comparison to theory of NIWs in submesoscale fronts
Given that a large fraction of the shear variance is apparently explained by NIWs, we use the dispersion and polarization relations for NIWs in a submesoscale front, as defined in Whitt and Thomas (2013), to assess the consistency between the observed spatial variability of the shear, the observed temporal variability of the shear, and the theory for NIWs in submesoscale fronts with Ro ~ Ri ~ 1. This theory makes several assumptions about the waves and the balanced flow, namely, that the NIWs are hydrostatic and low-amplitude relative to the geostrophic flow (i.e., a linear perturbation) and uniform in the alongfront direction and that the balanced flow is steady, geostrophic, and uniform in the alongfront direction as well (Whitt and Thomas 2013). In addition, the use of a dispersion relation implies an assumption that the properties of the wave field (such as wavelength and frequency) and balanced flow are slowly varying in space and time (e.g., Lighthill 1978). Hence, the theory is expected to be accurate in a submesoscale front like the North Wall of the Gulf Stream to the extent that these assumptions about the waves and balanced flow are reasonable. In this case, the theoretical results should be interpreted with some caution because the NIW shear is approximately comparable in magnitude to the balanced shear (see section 3a). The degree of symmetry in the alongfront direction is difficult to assess from this nearly Lagrangian dataset, but the balanced flow is clearly not a perfectly axi-symmetric jet (Fig. 1a). Finally, the medium (i.e., the balanced flow) varies relatively rapidly on the scales of the waves in both space and time. Despite the caveats, we proceed to analyze the observations using this theory.
To use this theory, the ageostrophic waves and geostrophic-balanced flow must be decomposed. The cross-stream shear ∂υ/∂z is assumed to be entirely ageostrophic, while the streamwise shear ∂u/∂z can contain both balanced and unbalanced components. The method for isolating the geostrophic shear in the streamwise direction ∂ug/∂z is described in the appendix.
In an effort to obtain a precise estimate for sshear = −α from the observed shear, an algorithm was developed to identify the cross-front and vertical coordinates of large simply connected regions of ageostrophic shear bounded by contours of constant shear and compute their principal components, from which sshear = −α is derived. The procedure is outlined in the appendix and is performed in a region from z = −30 m to −170 m depth in each Knorr cross section from the drift (see Fig. 1a). A histogram of the measurements of sshear determined via this method is shown in Fig. 9a along with a histogram of the isopycnal slope sb (Fig. 9b). The ratio of sshear to sb in Fig. 9c shows that on average the most coherent shear structures are nearly parallel to or just slightly steeper than isopycnals.
Histograms of the (a) inferred slope of constant wave shear sshear = −α, (b) isopycnal slope sb, (c) logarithm of the ratio sshear/sb, (d) inferred frequencies obtained using sshear = −α and the dispersion relation [(9)], and (e) rotary coefficient [(11)]. Bootstrap 95% confidence intervals for the arithmetic mean of each histogram are indicated by black hash marks. The histograms were calculated data from 30 to 170 m below the surface along the track of the R/V Knorr (i.e., not merely those data within 350 m of the float).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
From these observations of sshear, the average wave frequency inferred from the dispersion relation (9) is ω = (1.11 ± 0.14)f, where 1.11f is the sample mean and 0.14f is the sample standard deviation of all the frequency estimates inferred via this method. The histogram of all the frequency estimates obtained from the dispersion relation is shown in Fig. 9d. This estimate of the dominant wave frequency derived from the dispersion relation (9) is consistent with the dominant frequency measured in the spectral analysis of section 3c, where the frequency bin ωk = 1.2f has the highest amplitude (see Fig. 7). This frequency bin represents the frequency range 
5. Discussion
a. Wave propagation, trapping, and amplification
Motivated by the consistency between the measured properties of the NIWs and the predictions of Whitt and Thomas (2013), we apply the theory to explore the propagation, trapping, and amplification of NIWs in the density and vorticity fields observed during the drift. In particular, we ask the following questions:
How might superinertial NIWs with the observed dominant frequencies and wavenumbers propagate in the spatially variable medium of the Gulf Stream Front (e.g., Fig. 2)?
Why might superinertial NIWs have the highest amplitude in the North Wall of the Gulf Stream?
We start with the second question. As shown in Fig. 2, the North Wall of the Gulf Stream is characterized by order-one Ro and Ri. As a result, the minimum frequency 
(a) The minimum frequency of inertial gravity waves in a front [defined in (1)] for the Atlantis section shown in Fig. 2. (b) Four example ray paths for a wave with frequency ω = 1.13f and a positive vertical wavenumber m > 0. The magnitude of the group velocity normalized by its maximum value on a ray is indicated in color in (b). The contours of constant ωmin are in some places nearly parallel to isopycnals, which is the characteristic signature of a slantwise critical layer where the group velocity goes to zero and ray paths cannot escape. Magenta contours in both panels indicate where ωmin = 1.13f and the gray contours in (a) indicate where ωmin = f.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
While trapping could lead to amplification of the superinertial waves, wave shoaling could play a role as well. The magnitude of the group velocity [defined in (13)] varies significantly along ray paths (Fig. 10b) and at the separatrix the magnitude of the group velocity equals zero since ω = ωmin and α = −sb. Consequently, as wave packets approach the separatrix their energy density should increase to conserve the energy flux, namely, the waves shoal. If the separatrix aligns with the tilted isopycnals of a front, then the reflected ray cannot escape from the separatrix and a slantwise critical layer forms, leading to especially intense wave shear (Whitt and Thomas 2013). There are several locations in the Atlantis section where this criterion is nearly met for ω = 1.13f, for example, y ≈ 2 km and z ≈ −80 m and y ≈ 10 km and z ≈ −175 m (Fig. 10b). Hence, it is plausible that superinertial NIWs can be trapped and amplified in the North Wall of the Gulf Stream, which could explain the observed superinertial peak in the Froude variance associated with downward-propagating NIWs.
b. Superinertial wave generation
(a) Time series of mixed layer near-inertial velocities calculated using the slab mixed layer model, as described in section 5b. (b) CW and CCW components of the rotary kinetic energy spectrum associated with the time series in (a).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0079.1
The calculation supports the hypothesis that the local wind stress could have excited the observed NIWs. In particular, the modeled mixed layer inertial currents exhibit speeds ranging from 0.05 and 0.1 m s−1, consistent with the observed shear, which has a velocity scale of (1/m)(∂|u|/∂z) ~ 0.1 m s−1. Moreover, like the observed shear, the modeled mixed layer velocities are dominated by clockwise rotation at slightly superinertial frequencies and present a similarly shaped spectrum (cf. Figs. 7 and 11b). In addition, it is worth noting that this slab mixed layer model also predicts that wind-generated, near-inertial oscillations in the ocean mixed layer become incoherent across the front over a time scale of order 1/f because of the order-one cross-front variations in 
6. Comparison with other observations
The observations presented here are unique in that we are observing the temporal evolution of near-inertial shear in a submesoscale front in a nearly Lagrangian frame of reference with a spatial resolution of O(1) km and a temporal resolution of O(10) min. That said, our results are qualitatively similar to a variety of earlier observations of banded, near-inertial shear interacting with mesoscale and submesoscale flow features. In this section, we will briefly describe some of the previously published observations of NIWs and discuss how those observations compare to the observations of NIWs discussed here.
For example, the shear bands (e.g., Figs. 3, 4) appear qualitatively similar to observations of near-inertial waves trapped in anticyclonic warm-core rings (e.g., Kunze 1986; Kunze et al. 1995; Jaimes and Shay 2010; Joyce et al. 2013), although the geometry is different. Whereas the shear features may have been confined in three dimensions inside the anticyclonic rings, the evidence here suggests that the amplified shear features in the North Wall are rather two-dimensional, confined in the cross-front and vertical directions but not along the front. Furthermore, Kunze et al. (1995) observed enhanced energy dissipation associated with near-inertial shear in the center of a warm-core ring but not on the edges. Our observations, on the other hand, exhibit strong near-inertial shear at the north edge of the Gulf Stream in the same location as the strongest lateral density gradients and strong cyclonic vertical vorticity (see Fig. 2). Our observations further contrast with those in rings because the observed frequency is superinertial here (Fig. 7), whereas it is generally assumed [or—more rarely—observed, as in Perkins (1976) and Jaimes and Shay (2010)] to be subinertial in anticyclonic rings.
Several studies have also observed elevated near-inertial wave energy in mesoscale and submesoscale fronts. For example, Rainville and Pinkel (2004) observed elevated shear variance and coherent structures characteristic of near-inertial waves blocked by the strong vertical vorticity and high effective Coriolis frequency feff in the Kuroshio. D’Asaro et al. (2011) observed banded shear in a strong submesoscale front in the Kuroshio and hypothesized that the shear was associated with near-inertial waves, although the wave dynamics were not explored in detail. Kunze and Sanford (1984) and Alford et al. (2013) observed strong, coherent, banded, near-inertial shear modulated by the Pacific Subtropical Front, where Ro ~ Ri−1 ~ 0.1. Pallàs-Sanz et al. (2016) observed trapped near-inertial waves at a critical layer at the base of the Loop Current after a hurricane. Mied et al. (1987) observed a near-inertial wave packet where an eddy impinged on a mesoscale frontal jet in the North Atlantic subtropical zone. The wave packet was confined to a 25-km-wide region in the front and was associated with coherent patches of elevated temperature variance (Marmorino et al. 1987; Marmorino 1987) and superinertial frequencies, similar to our observations. Whitt and Thomas (2013) observed banded shear in a ~200-km-wide section (with ~10-km resolution) across the winter Gulf Stream and noted that the properties of the shear were consistent with the spatial structure of a subinertial wave at a slantwise critical layer. Collocated shear microstructure profiles showed enhanced dissipation between 400 and 600 m in the upper thermocline in the same location as these shear structures (Inoue et al. 2010), suggesting a link between the shear and enhanced turbulence. None of the above studies considered the possibility that superinertial waves may be trapped between filaments of cyclonic vertical vorticity, as discussed here.
7. Conclusions
A high-resolution Lagrangian survey of the North Wall of the Gulf Stream made during strong and variable wintertime atmospheric forcing provided the unique, new opportunity to observe the interaction of NIWs with a current in the submesoscale dynamical regime and dominated by cyclonic vorticity. The ageostrophic vertical shear inferred from the survey has features characteristic of coherent, downward-propagating NIWs. Namely, the shear is polarized in space and time, such that it rotates primarily clockwise with increasing depth and increasing time. Frequency spectra exhibit a distinct peak in the clockwise rotary component at frequencies 
The high-resolution sections across the front allow us to check the consistency of the properties of the observed near-inertial shear with the theory of NIWs propagating perpendicular to strongly baroclinic currents (Mooers 1975; Whitt and Thomas 2013). In particular, we can estimate the aspect ratio of the observed banded shear structures and use it in the dispersion relation of Whitt and Thomas (2013), which is designed for near-inertial waves in a submesoscale front to obtain frequency estimates that are independent of the spectral analysis and contrast the results. The observed consistency between the two frequency estimates, both of which are superinertial, motivate an extension to the theoretical analysis of Whitt and Thomas (2013), which focuses on the properties and propagation of subinertial waves in submesoscale fronts, to explore how superinertial waves propagate in the North Wall of the Gulf Stream, where the vertical vorticity is strongly cyclonic and the minimum frequency for inertia–gravity waves is greater than the local Coriolis frequency. Ray tracing/characteristic paths suggest that the propagation of superinertial wave energy is strongly modified by narrow filaments of strong, cyclonic vertical vorticity in the North Wall of the Gulf Stream. Superinertial waves can bounce between the cyclonic filaments and become trapped and amplified at slantwise critical layers between these filaments, a scenario that has not been previously considered in the literature (see Fig. 10 for an illustration). In addition, a slab mixed layer model suggests that variable winds are a plausible local source of energy for the observed superinertial waves because they resonantly force superinertial motions, owing to the cyclonic vorticity of the current. The slab model also suggests that large cross-front variations in the cyclonic vertical vorticity cause mixed layer inertial oscillations to become incoherent on short time and space scales, facilitating more rapid downward propagation of wave energy between cyclonic vorticity filaments at the North Wall caused by a submesoscale inertial chimney effect. However, these theoretical results should be viewed with caution because some of the assumptions used to obtain the theoretical results are not met. In addition, we have not proved that other theories cannot explain the observations. For example, the balanced flow could also be a source of the observed downward-propagating internal wave energy (e.g., Danioux et al. 2012; Alford et al. 2013; Nagai et al. 2015; Barkan et al. 2017; Shakespeare and Hogg 2017), but quantifying the magnitude of the energy flux between the balanced flow and the waves is beyond the scope of this article. Nevertheless, the consistency between the proposed description of the physics and the observations is encouraging and motivates further efforts to better constrain the observations and to explore the theoretical physical scenario in a more realistic context using numerical simulations.
Although we have not explicitly explored the link between the NIWs observed here and microscale turbulence, a number of previously published studies have demonstrated that coherent downward-propagating NIWs may be associated with enhanced turbulence (e.g., Marmorino et al. 1987; Marmorino 1987; Hebert and Moum 1994; Kunze et al. 1995; Polzin et al. 1996; Alford and Gregg 2001; Inoue et al. 2010). These previously published observations show that coherent near-inertial waves can directly modify turbulence and that wave–mean flow interactions can facilitate the concentration, amplification, and ultimate decay of near-inertial waves via turbulent dissipation. The observations collected here show that a large fraction of the Froude variance is associated with downward-propagating NIWs in the North Wall of the Gulf Stream. The superposition of this wave shear with the thermal wind shear of the front results in banded regions of low Richardson number (e.g., Fig. 3c), which could be associated with enhanced turbulence. As turbulent mixing may be important for driving water mass transformation at the North Wall of the Gulf Stream (e.g., Klymak et al. 2016), coherent downward-propagating NIWs may also play an important role in water mass transformation in the upper pycnocline here. In addition, similar behavior could also occur at other western boundary currents, such as the Kuroshio, where filaments of strong cyclonic vorticity and near-inertial waves are prominent features of the circulation (e.g., Nagai et al. 2009, 2012). However, future work will have to explore these hypotheses, which are beyond the scope of this article.
Acknowledgments
The data were obtained during the Scalable Lateral Mixing and Coherent Turbulence (LatMix) Department Research Initiative, funded by the United States Office of Naval Research. DBW and LNT were supported by ONR Grant N00014-09-1-0202 and NSF Grant OCE-1260312. DBW was also supported by an NSF postdoctoral fellowship OCE-1421125. JMK was supported by ONR Grant N00014-11-1-0165. EAD was supported by ONR Grant N00014-09-1-0172. CML was supported by ONR Grant N00014-09-1-0266. Data and visualization scripts are available from the lead author (dwhitt@ucar.edu). The authors thank the crew and officers of the R/V Knorr and R/V Atlantis as well as the rest of the LatMix 2012 science team and the staff of the Applied Physics Laboratory at the University of Washington for making this remarkable experiment possible. DBW would particularly like to thank Dave Winkel for his help with the processing of the shipboard ADCP data.
APPENDIX
Algorithm for Computing the Shear Aspect Ratio
To estimate the aspect ratio of the waves, α = l/m, and infer their frequency from the dispersion relation [(9)], we use the following procedure:
- The coordinate system is rotated into a streamwise/cross-stream (x, y) coordinate system, as described in section 2. Then, the thermal wind shearis inferred in each section made by the R/V Knorr (e.g., Fig. 3) from the cross-stream buoyancy gradient, after additionally smoothing the density field ρ with a 40 m (vertical) by 7 km (lateral) averaging filter, that is, bg = −g〈ρ〉/ρo, where the brackets denote the filter, g is the gravitational acceleration, and ρo is a reference density. Then the streamwise ageostrophic shear ∂ua/∂z is inferred in each section by subtracting this thermal wind shear from the observed shear. The cross-stream shear is assumed to be entirely ageostrophic, that is, ∂υa/∂z = ∂υ/∂z.
In each section, the two components of the ageostrophic shear as well as the y (cross stream) and z (vertical) coordinate grids are extracted and normalized by scaling so that the mean and variance of all four fields are zero and one, respectively.
In each section the set of simply connected regions with shear magnitude greater than a threshold magnitude is identified [each region is a set of (y, z) pairs]. The chosen threshold may depend on the distribution of the shear in a given section, but we used just one value for the observations (0.5 in normalized shear units).
One thousand bootstrap samples are created for each simply connected region.
For each bootstrap sample, the slope of the first principle component of the set of (y, z) pairs was used as a value for −α (after rescaling y and z to physical units). The background flow parameters in (9), including
If the 95% confidence interval for the bootstrap frequency estimates in a given region is narrower than 0.2f, then the result is considered robust and the frequency estimate is retained in the histograms in Fig. 9.
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