Abstract

Shipboard ADCP and towed CTD measurements are presented of a near-inertial internal gravity wave radiating away from a zonal jet associated with the Subtropical Front in the North Pacific. Three-dimensional spatial surveys indicate persistent alternating shear layers sloping downward and equatorward from the front. As a result, depth-integrated ageostrophic shear increases sharply equatorward of the front. The layers have a vertical wavelength of about 250 m and a slope consistent with a wave of frequency 1.01f. They extend at least 100 km south of the front. Time series confirm that the shear is associated with a downward-propagating near-inertial wave with frequency within 20% of f. A slab mixed layer model forced with shipboard and NCEP reanalysis winds suggests that wind forcing was too weak to generate the wave. Likewise, trapping of the near-inertial motions at the low-vorticity edge of the front can be ruled out because of the extension of the features well south of it. Instead, the authors suggest that the wave arises from an adjustment process of the frontal flow, which has a Rossby number about 0.2–0.3.

1. Introduction

Internal gravity waves are a ubiquitous and important phenomenon in the ocean and the atmosphere. In both media, breaking internal waves transport significant amounts of heat and momentum. In the ocean, internal waves are most energetic near the inertial frequency (f), where a pronounced peak in the spectrum often contains a substantial portion of the total kinetic energy. These waves with frequency near f are commonly referred to as near-inertial waves (NIW). Because of the large amount of energy they contain and their general tendency to have significant vertical shear, they are known to be important in mixing the shallow ocean (Hebert and Moum 1994; Alford and Gregg 2001) and are thought to be important to mixing the deep ocean as well (Ferrari and Wunsch 2009). Hence, an understanding of their sources and subsequent propagation is important for parameterizing their effects in large-scale models.

Various mechanisms can generate internal gravity waves, with differing relative importance in the ocean and atmosphere. It is known that the wind blowing on the surface mixed layer is an important source of internal gravity waves (specifically, near-inertial waves) in the ocean. However, other sources are known to be important in the atmosphere, such as direct generation via Kelvin–Helmholtz instability (Chandrasekhar 1961) and topographic interactions, leading to mountain waves (Eliassen and Palm 1961). In the ocean, topographic interactions lead to internal tides (e.g., St. Laurent and Garrett 2002) and may also lead to near-inertial waves in the deep sea, as suggested by some evidence (Fu 1981; Alford and Whitmont 2007). Surface waves can theoretically force internal gravity waves via nonlinear interactions (Watson 1990), but the mechanism is not thought to be very important energetically. Finally, parametric subharmonic instability (PSI) can transfer energy from other frequencies (usually the internal tides) to near-inertial waves (MacKinnon and Winters 2005; Carter and Gregg 2006; Alford et al. 2007; Chinn et al. 2012; MacKinnon et al. 2013).

Theoretically, low-frequency flows such as the general circulation, mesoscale eddies, and submesoscale features such as fronts can also generate internal gravity waves (usually near-inertial waves). Since the waves derive their energy from geostrophically balanced flows, these mechanisms are broadly termed “loss of balance” or “spontaneous adjustment” (Ford 1994; Ford et al. 2000; Williams et al. 2008), which are summarized in a recent review by Vanneste (2012). Internal gravity waves can be generated through a rich variety of wave–mean flow interactions, and the parameter space is not fully explored (Vanneste 2012). However, many of them may be loosely grouped into two categories: those involving frontogenesis (Williams 1967; Hoskins and Bretherton 1972), wherein waves and ageostrophic circulations arise as a large-scale strain field sharpens a near-surface front, and wave radiation by time-dependent instabilities of the balanced flow (Ford 1994). Wave generation in the presence of frontogenesis has been modeled in the atmosphere (Griffiths and Reeder 1996; Reeder and Griffiths 1996) and the ocean (Thomas 2005; Buhler and McIntyre 2005; Capet et al. 2008; Thomas 2012; Danioux et al. 2012; Nagai et al. 2012, manuscript submitted to Nature). Laboratory studies demonstrating internal gravity wave radiation from two-layer geostrophic flows in an annulus (Williams et al. 2003, 2005, 2008) appear consistent with the Ford mechanism.

These mechanisms are distinct from the classical adjustment problem first described by Rossby (1938), wherein an initially unbalanced flow radiates internal gravity waves during its adjustment to a geostrophically balanced flow, conserving potential vorticity in the process. Once balance is reestablished, radiation stops. [See Ou (1984) and references therein for more recent treatments of the Rossby problem including layered flows and continuous stratification.] In spontaneous adjustment, energy is continually drained from the “balanced” flow (Molemaker et al. 2005) into the internal gravity waves. In spite of these theoretical, laboratory, and numerical studies, radiation of internal gravity waves by loss of balance has not been clearly observed either in the ocean or in the atmosphere (McIntyre 2009) [a possible exception may be a reinterpretation of data from the MidOcean Dynamics Experiment (MODE) by Polzin (2008) as near-inertial waves interacting with the mesoscale eddy field via the “wave capture” mechanism of Buhler and McIntyre (2005)].

Unambiguous observation of internal gravity waves generated by spontaneous adjustment in the ocean, with quantification of energy transfer rates relative to other mechanisms, would be a major advance in the understanding of energy flow in the ocean for two reasons. First, Wunsch and Ferrari (2004) argue that 2 TW of energy are required to maintain the abyssal stratification of the oceans, of which breaking internal tides may provide about half. Estimates of the work done by the wind on near-inertial waves range from 0.3–1.4 TW (Alford 2003; Watanabe and Hibiya 2002; Jiang et al. 2005), indicating that near-inertial waves can also generate a significant portion of this power by propagating into the deep sea and breaking. If, on the other hand, spontaneous adjustment provides a significant additional source of internal waves, then they could be correspondingly elevated in importance.

Second, spontaneous adjustment is a potentially important means by which mesoscale flows lose energy to turbulence. It is thought that mesoscale flows arise from baroclinic instability of the general circulation (Ferrari and Wunsch 2009), whose energy source is the ~1 TW of work done on it by the wind (Wunsch 1998). Global calculations using the Gent and McWilliams (1990) parameterization indicate that a similar amount is transferred to the mesoscale eddy field (Ferrari and Wunsch 2009). However, while bottom friction (Sen et al. 2008; Arbic et al. 2009) and internal wave drag (Nikurashin et al. 2013) likely dissipates some of their energy, the means by which the mesoscale flows are dissipated is not well understood (Ferrari and Wunsch 2009; Ferrari 2011; Nikurashin et al. 2013). A back-of-the-envelope calculation by Williams et al. (2008) based on an observed decay rate of 100 days in their laboratory study suggests that 1.5 TW could be dissipated by spontaneous adjustment globally. Because the process transfers energy to internal gravity waves, which can transfer the energy far from the region of initial instability prior to breaking, knowing how much of the energy lost to eddies enters the internal gravity wave field is important.

One of the difficulties in observing these processes in the ocean is that the natural place to look for the phenomena is near fronts, where the Rossby number (Ro ≡ ζ/f, where ζ is the relative vorticity) is large. While a large number of studies have found high near-inertial energy near fronts (Kunze and Sanford 1984; Weller et al. 1991; Salat et al. 1992; Granata et al. 1995; Rainville and Pinkel 2004; D’Asaro et al. 2011; Nagai et al. 2013), interpretation is complicated because of the strong, vertically- and laterally-sheared flows typical of fronts. Specifically, time series long enough to establish the frequency content of the waves are challenging because the waves are quickly blown past fixed moorings. Shear complicates the collection and analysis of spatial data. The strong and often meandering flows make shipboard measurements difficult to reference correctly to the front in question and can Doppler shift observed frequencies, giving confusing results even in models (Nagai et al. 2012, manuscript submitted to Nature). Finally, the strongest fronts are often associated with western boundary currents where winds forcing near-inertial motions tend to be the strongest (Alford 2001), making it difficult to discern whether generation is by wind or spontaneous adjustment, as in Kunze and Sanford (1984), Salat et al. (1992), and D’Asaro et al. (2011).

A second complication is that the vorticity of the frontal jet can modulate the effective inertial frequency, feff = f + ζ/2 (Kunze 1986), allowing near-inertial waves to be trapped in the region where feff < f. Therefore, observation of high near-inertial energy near a front does not imply generation (Kunze and Sanford 1984; Weller et al. 1991; Granata et al. 1995; Rainville and Pinkel 2004; Bouruet-Aubertot et al. 2005; Nagai et al. 2013, 2012, submitted to Nature). In none of these cases could trapping could be ruled out, as near-inertial energy was confined to the region of negative relative vorticity on one side of the front.

In this paper we present three-dimensional observations of velocity, shear, and density in the vicinity of the Subtropical Front in the North Pacific. Unlike the situation at stronger fronts such as the Kuroshio, the flows (0.35 m s−1) and Rossby number (0.2–0.3) here are modest, making observations easier and more easily interpreted. While our observational techniques of shipboard ADCP plus towed undulating CTD are similar to some past efforts, simultaneous resolution of spatial and temporal aspects of near-inertial waves in a front is nonetheless rare. Additionally, the data are more consistent with generation by a frontal adjustment process than past studies. Specifically, the observations indicate a near-inertial wave propagating downward and toward the equator from the vicinity of the front. The time series is of long enough duration to establish the wave’s frequency as near f. The wave has many of the same properties as the near-inertial wave observed in the same region by Kunze and Sanford (1984), but here the wave extends at least 100 km south of the front, eliminating the possibility that the elevated shear is due to trapping at the front. Wind work computed from a slab model driven with shipboard and reanalysis winds is substantially less than the estimated vertical energy flux of the wave. Since trapping is ruled out and wind generation appears much too weak, the front remains as the most likely energy source of the wave. Although the data are not sufficient to determine the specific instability mechanism, some sort of frontal instability appears to be the most likely explanation.

We first present the data and techniques, followed by the observations of the front and the wave. We then summarize and conclude with a discussion of possible generation mechanisms.

2. Observations

a. Data

As part of an effort to map thermohaline intrusions (Shcherbina et al. 2009; Shcherbina et al. 2010), R/V Wecoma conducted repeated spatial surveys near the North Pacific Subtropical Front (Fig. 1). During the portion of the cruise discussed in this paper (5–11 July), the strategy was to run a long north–south line along 158°W (5–7 July), followed by various radiator patterns mapping different intrusive features observed in the front until 11 July. After that, the ship began doing smaller-scale surveys around a Lagrangian float, which are not discussed here.

Fig. 1.

(a) Ship track during our observations (black) overplotted on a map of sea surface temperature on 7 July based on Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) imagery [courtesy of National Aeronautics and Space Administration/Jet Profulsion Laboratory/Physical Oceanography Distributed Active Archive Center (PODAAC)]. (b) Shear magnitude, averaged from the surface to 250 m, plotted as a function of location. The x, y coordinate system (km) relative to 31°N, 158°W used in the paper is shown at top and right. In both panels, the subsurface locations, determined from the towed surveys, of the two fronts, F1 and F2, discussed in the paper, are indicated with dashed arcs; the surface locations are also shown, labeled with the superscript “s.”

Fig. 1.

(a) Ship track during our observations (black) overplotted on a map of sea surface temperature on 7 July based on Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) imagery [courtesy of National Aeronautics and Space Administration/Jet Profulsion Laboratory/Physical Oceanography Distributed Active Archive Center (PODAAC)]. (b) Shear magnitude, averaged from the surface to 250 m, plotted as a function of location. The x, y coordinate system (km) relative to 31°N, 158°W used in the paper is shown at top and right. In both panels, the subsurface locations, determined from the towed surveys, of the two fronts, F1 and F2, discussed in the paper, are indicated with dashed arcs; the surface locations are also shown, labeled with the superscript “s.”

Velocity data are from a hull-mounted 75-KHz ADCP, which transmitted alternating narrowband and broadband pings each 1.2 s in 8-m and 16-m bins, respectively. The two systems are complementary in offering higher resolution and longer range, respectively. Shallow data (z < 150 m) are from a 150-KHz ADCP, collected in 4-m bins. Velocity estimates between about 20-m and 800-m depth are obtained by merging all three datasets. Shear is then computed by first differencing and smoothing vertically over a 30-m interval. This choice smooths out some of the sharper shear features in the upper 100 m or so where the stratification is the strongest, but minimizes noise and emphasizes the near-inertial wave of interest.

Temperature and salinity together with other quantities such as dissolved oxygen and chlorophyll, which are not discussed in this paper, were measured from redundant, pumped Sea-bird Electronics sensors mounted on the nose of a towed system known as the Shallow Water Integrated Mapping System (SWIMS III). Towed at 4–5 kt, the system was winched up and down between 50 and 150 m each few minutes, resulting in profiles every few hundred meters in the horizontal.

b. The front

The North Pacific Subtropical Front is part of a large-scale system of fronts spanning 30°N that separate the warmer, saltier water of the North Pacific Central Water from cooler, fresher waters of subpolar origin (Roden 1980). During our measurements, the front is evident in sea surface temperature (SST) from a Moderate Resolution Imaging Spectroradiometer (MODIS) image on 7 July (Fig. 1a). SST decreases sharply in two steps of about half a degree each. The two fronts appear as arcs and appear to be part of a larger meandering instability of the front of O(100 km) scale (Shcherbina et al. 2009), consistent with past observations (Van Woert 1982; Kunze and Sanford 1984). Following Shcherbina et al. (2010), the southern and northern fronts are referred to as F1 and F2, respectively, with the surface signatures indicated in Fig. 1 with the superscript “s.”

The SWIMS and shipboard ADCP data provide a very detailed view of the subsurface frontal structure between 0 and 150 m (Fig. 2). Near-surface temperature and salinity (Figs. 2a,b) along the long meridional line at 158°W both show sharp features at approximately the same locations as indicated in the SST map. The subsurface locations of F1 and F2, determined following Shcherbina et al. (2010) from the lateral salinity gradient shallower than 100 m, are indicated in Fig. 2 with dashed lines. We only crossed F2 on the long meridional section, but crossed F1 at many zonal locations (Fig. 1, black ship tracks). The location of each front determined from all transects is plotted in Fig. 1.

Fig. 2.

Cross-frontal structure from the long meridional section (location in Fig. 1). (a)–(d) Temperature, salinity, potential density, and zonal velocity measured on the first occupation. (e) Mean zonal velocity averaged over the three occupations of the line, partially eliminating internal waves. (f) Geostrophic velocity computed from the thermal wind relations, referenced to measured velocity at 150 m. (g) Rossby number, ζ/f, computed from the meridional gradient of mean zonal velocity (e). The locations of the two subsurface fronts F1 and F2 (dashed lines) are indicated in all panels.

Fig. 2.

Cross-frontal structure from the long meridional section (location in Fig. 1). (a)–(d) Temperature, salinity, potential density, and zonal velocity measured on the first occupation. (e) Mean zonal velocity averaged over the three occupations of the line, partially eliminating internal waves. (f) Geostrophic velocity computed from the thermal wind relations, referenced to measured velocity at 150 m. (g) Rossby number, ζ/f, computed from the meridional gradient of mean zonal velocity (e). The locations of the two subsurface fronts F1 and F2 (dashed lines) are indicated in all panels.

The frontal locations F2 and F2s are close to each other, but the surface feature at F1s is about 20 km to the north of F1. In the following, we will use the Shcherbina et al. (2010) definition of the two subsurface fronts F1 and F2 (Fig. 2, vertical dashed lines), recognizing that the surface feature at F1 is somewhat to the north. The misalignment at F1 could be due to vertical structure, temporal evolution between the time of the SWIMS surveys and the SST image, or both.

In potential density (Fig. 2c) the fronts are less sharp owing to partial compensation of temperature and salinity (Ferrari and Rudnick 2000; Shcherbina et al. 2009). However, significant baroclinicity still exists, with isopycnals sloping upward toward the north general region between the two fronts. The greatest isopycnal slopes occur near F1 and F2.

Observed stratification is weaker south of the front, as seen by the widened isopycnals in Fig. 2c. Buoyancy frequency (Fig. 3, thick lines) is reduced and shows a deeper peak on the south side of the front. The peak is just below the mixed layer, which is about 40 m deep on the south side, shoaling to about 25 m on the north side. At all locations, observed stratification near the base of the mixed layer is stronger than climatological values (thin lines, Levitus and Boyer 1994), but similar below 70 m.

Fig. 3.

Stratification N(z) from observations (thick lines) and Levitus and Boyer (1994) (thin), south (gray) and north (black) of the front.

Fig. 3.

Stratification N(z) from observations (thick lines) and Levitus and Boyer (1994) (thin), south (gray) and north (black) of the front.

Measured velocity from the shipboard ADCP (Fig. 2d) is toward the east in this same broad frontal region, as expected from geostrophy. Pronounced maxima of about 0.4 m s−1 are seen near both F1 and F2. Because velocity in Fig. 2d is from only a single transect, it includes internal gravity waves of all frequencies as well as the mean flow. The prominent feature extending downward toward the south is the primary focus of the paper; namely, a downward-propagating near-inertial wave extending downward and toward the equator from the frontal region.

Since we desire a picture of the mean flow for our basic description of the front, in (Fig. 2e) we average together data from all transects occurring from 7 to 11 July (locations in Fig. 1). For locations ± 20 km, the average is over about 20 occupations of the grid patterns conducted over the four days, which is sufficient to fairly reliably average out near-inertial waves and other higher-frequency signatures. For locations farther north and south, only our two or three occupations of the long line from 30° to 32°N are averaged together, depending on latitude, so that some wave motions may remain depending on the phasing of the observations at each latitude relative to the NIW motions. Nonetheless, the downward-sloping feature is greatly reduced in (Fig. 2e), confirming it is not an aspect of the low-frequency frontal flow. By contrast, the zonal velocity within the two fronts is only slightly reduced, indicating that these flows are dominated by the mean rather than the wave signatures.

Geostrophic velocity relative to 150 m is computed from the observed density field and added to the observed mean flow (Fig. 2e) at 150 m. The result (Fig. 2f) is close to that observed (Fig. 2e). Depth profiles of mean geostrophic and measured velocity within ±5 km of F1 are quite similar (Fig. 4a), indicating that the frontal jet is in approximate geostrophic balance. The associated kinetic energy and shear profiles (Figs. 4b,c) also demonstrate the similarity in the average observed velocity and that expected from geostrophy.

Fig. 4.

Frontal zonal velocity, kinetic energy and shear. (a) Mean measured (black) and geostrophic (gray) zonal velocity averaged over periods when the ship was within 5 km of F1. The mean measured velocity at 150 m was added to the geostrophic velocity. (b) Measured and geostrophic kinetic energy of the front (thin black and gray, respectively). The wave energy is plotted for comparison (thick black). (c) Measured and geostrophic shear in the front (black and gray, respectively).

Fig. 4.

Frontal zonal velocity, kinetic energy and shear. (a) Mean measured (black) and geostrophic (gray) zonal velocity averaged over periods when the ship was within 5 km of F1. The mean measured velocity at 150 m was added to the geostrophic velocity. (b) Measured and geostrophic kinetic energy of the front (thin black and gray, respectively). The wave energy is plotted for comparison (thick black). (c) Measured and geostrophic shear in the front (black and gray, respectively).

The vorticity of the total mean flow, approximated as , is computed by differencing Fig. 2e in the y direction and smoothing over 5 km. It is plotted in Fig. 2g normalized by the inertial frequency at 30°. The vorticity ζ reaches about −0.2f (+0.3f) on the negative (positive) vorticity side of the front, respectively (Ro of 0.2 and 0.3).

Shear increases sharply on the south side of the front, as shown in plan view by plotting depth-averaged shear magnitude above 250 m as a function of location (Fig. 1b). Shear magnitude is < 10−3 s−1 north of F2 at all locations except for the corner of one survey. A sharp increase is then seen transiting the subsurface location of each front (dashed lines), particularly F1.

This tendency is quantified and examined relative to the depth-average structure of the front by plotting versus north–south distance (Fig. 5a). The depth-mean zonal velocity (Fig. 2b) and Ro (Fig. 2c) from the snapshot, mean and geostrophic fields are plotted for comparison. Data are now plotted in a coordinate system where x and y are east and north, respectively, relative to 31°N, 158°W (see Fig. 1b). In this reference frame, F1 and F2 at x = 0 occur at −9 and 52 km, respectively. Shear magnitude increases by about a factor of 2 crossing the frontal system, particularly near F1. Relative to that on the northern side, shear remains high all the way to the southern end of our sections.

Fig. 5.

Sections across the front of (a) shear magnitude averaged over depths > 100 m; all three occupations of the section are plotted. (b) Velocity and (c) Rossby number averaged over z < 100 m, computed from the measured mean velocity (gray shading), the measured instantaneous velocity from section 1 (thick black), and the geostrophic velocity (thin gray). The frontal locations F1 and F2 are indicated in all panels.

Fig. 5.

Sections across the front of (a) shear magnitude averaged over depths > 100 m; all three occupations of the section are plotted. (b) Velocity and (c) Rossby number averaged over z < 100 m, computed from the measured mean velocity (gray shading), the measured instantaneous velocity from section 1 (thick black), and the geostrophic velocity (thin gray). The frontal locations F1 and F2 are indicated in all panels.

Internal waves undergo refraction as they encounter different regions of stratification, resulting in shear magnitude generally scaling as N or N3/2, depending on the assumptions made (Leaman and Sanford 1975; Pinkel 1985). To ensure that the observed lateral distribution of shear is not affected by refraction, we note that, because shallow stratification is greater on the north side of the front, this effect would tend to reduce the shallow shear south of the front relative to that north of it, opposite to observed. In the depth range of the main propagating feature (200–250 m), which is well below the front, climatological stratification does not change greatly (Fig. 3), so the increased shear south of the front is not due to refraction. Instead, it is associated with the NIW evident in Fig. 2d, which will be be discussed in detail next.

c. Time series

The time series of shear (Fig. 6b) is dominated by motions near the inertial frequency. The motions are best described in terms of shear rather than velocity owing to the contamination of velocity by tides and lower-frequency flows, which have longer vertical scales. Since the ship was conducting spatial surveys over 20–40 km scales during this time (Fig. 6a), spatial information is aliased onto the time series, such as the zigzag features near 600 m on 7 July that are visually correlated with y. Nonetheless, the record is dominated by depth-alternating shear layers with 250-m vertical wavelength and frequency very close to the local inertial frequency f = 1.03 cycles per day.

Fig. 6.

Time series during the radiator patterns: (a) ship location relative to 30°N, 158°W, (b) observed meridional shear, and (c) inertially back-rotated shear.

Fig. 6.

Time series during the radiator patterns: (a) ship location relative to 30°N, 158°W, (b) observed meridional shear, and (c) inertially back-rotated shear.

Phase propagation is upward. This, together with the observed clockwise rotation of the shear vector with depth (not shown), indicates downward propagation for a linear internal wave in the Northern Hemisphere (Leaman and Sanford 1975). The overall coherence of the oscillations, in spite of the lateral movements of the ship, imply that the wave has broad lateral extent relative to our ~40 km survey scale. The sawtooths superimposed on the upward phase propagation are correlated with latitude (Fig. 6a, green), indicating that they are associated with meridionally sloping features, which will be examined below. Their association with lateral slopes is supported by their relative absence during the later part of the record when the survey scale was much smaller.

Given the short record length, the Rayleigh criterion limits the precision with which we can determine the wave frequency from the time series alone, Δω ~ T−1, where T is the time series length (Bendat and Piersol 1986). Our time series of 5–7 days gives Δω of 15%–20%, or ω = (1 ± 0.2)f. The wave is near inertial, but its frequency is uncertain to about 5%–20%.

The space–time aliasing of our data precludes isolation of the near-inertial signals by bandpass filtering. Taking advantage of the observed near-inertial dominance of the shear, we use a form of complex demodulation known as inertial back-rotation (Leaman and Sanford 1975) to separate the space and time dependence of the signals. Defining the complex shear vector as wzuz + z we suppose that it has time/space structure given by

 
formula

which would describe a wave with spatial structure rotating clockwise at radian frequency ω. The structure is assumed to evolve slowly relative to an inertial period, with a “slow time” τ. If the wave dominates the spatial structure of shear, then “back-rotation” or multiplication by eiωt gives , providing a means of describing the structure of the near-inertial wave by means of spatial surveys such as ours that take a wave period or longer to conduct. Shear at frequencies other than ω introduces errors. The reference time tref is arbitrary, and is here set to 0000 on 1 January 2007.

Back-rotated shear is computed assuming ω = f in the bottom panel of Fig. 6. The operator removes the sense of temporal phase propagation. The remaining structure should be the depth, lateral, and slowly-varying part of the signal, which we will endeavor to map below.

d. Spatial structure

Time series of upward near-inertial phase propagation similar to those shown are often interpreted as downgoing wind-generated waves. Here, the variation of shear and back-rotated shear during the spatial surveys gives additional information that, instead, connect it to the front. Spatial and temporal information are unraveled by plotting shear along the three occupations of the long north–south leg (Fig. 7, location in Fig. 1), which occurred immediately prior to the period plotted in Fig. 6. Meridional shear (Figs. 7d–f) and back-rotated shear (Figs. 7g,h) are plotted versus latitude for each occupation of the section. At the top (Figs. 7a–c), total shear magnitude, which is unaffected by the back-rotation operator, is plotted.

Fig. 7.

Shear plotted vs north–south distance y on each occupation of the line from 30° to 32°N along 158°W (see Fig. 1 for location): (a)–(c) shear magnitude averaged from 0 to 250 m, (d)–(f) meridional shear, (g)–(i) and back-rotated shear. In (d)–(i) isopycnals with mean spacing of 25 m are overplotted in black. Gray regions indicate zonal velocity greater than 0.3 m s−1. In (g)–(i) rays for frequency ω = (1.01, 1.04, 1.08)f are overplotted in gray. The front locations from Fig. 2 are indicated with vertical lines.

Fig. 7.

Shear plotted vs north–south distance y on each occupation of the line from 30° to 32°N along 158°W (see Fig. 1 for location): (a)–(c) shear magnitude averaged from 0 to 250 m, (d)–(f) meridional shear, (g)–(i) and back-rotated shear. In (d)–(i) isopycnals with mean spacing of 25 m are overplotted in black. Gray regions indicate zonal velocity greater than 0.3 m s−1. In (g)–(i) rays for frequency ω = (1.01, 1.04, 1.08)f are overplotted in gray. The front locations from Fig. 2 are indicated with vertical lines.

Velocity measurements from an ADCP on a moving ship are subject to biasing in the along-ship direction in the presence of strong gradients in scattering intensity (King et al. 2001; Pinkel 2012). The bias worsens at high speeds, leading to the possibility of spurious velocity features at the depth of the gradient. Though our speed was always 6 kt or less, we did see evidence of this biasing near the top edge of a deep scattering layer near 400 m, common in this part of the ocean. These regions are masked out and appear in all three sections as white gaps near 400 m. Luckily, the problem regions were on the north side of the front and also well below the primary features of interest. Therefore, they do not significantly impact our analysis.

The three occupations of the section are all dominated by shear features sloping downward toward the equator. Since the sections take about two days to complete (about two inertial periods), the phase of shear (Figs. 7d–f) varies as the ship steams along. (Note, in particular, the ~180° phase shift between the first and third legs, explaining the success of averaging in removing the wave evident in Fig. 2d from the mean fields shown in Fig. 2e.) The inertially back-rotated fields (Figs. 7g–i) show a much more constant phase following the sloping features, again indicative that shear is dominantly near inertial. The features persist and maintain nearly constant phase over the two days taken to conduct the three sections. Comparing their overall depth structure and phase of back-rotated shear with the time series, it is apparent that they are the same feature as shown in Fig. 6. Therefore, the near-simultaneous time series allow identification of the shear sections as near-inertial waves, unlike most previous studies that did not have time series information.

The slope of the shear features indicates a wave near the inertial frequency (consistent with the time series), as demonstrated by overplotting rays in the lower panels (gray lines). Following Garrett (2001), rays are computed by solving

 
formula

where kh and m are the horizontal and vertical wavenumber; N(z) is from observations above 120 m and climatology for the southern side of the front below that. The transition depth of 120 m is used since the observations and climatology match there (Fig. 3). The buoyancy frequency N(z) is assumed spatially constant south of the front. Rays are initiated at F2 at 60 m, the approximate depth of the jet at that location. The deeper sloping features appear to have slopes consistent with the more superinertial rays, ω = 1.04f and 1.08f.

The strongest feature, between 200 and 300 m on the south side of the front, is consistent for a wave with frequency ~1.01f, emanating from the vicinity of the strongest isopycnal slopes (black lines), which are near 100 m at F1. The zonal velocity maximum occurs somewhat shallower, at about 60 m (thick gray contours and Fig. 2d). The near-inertial shear feature begins with a slope similar to the slope of the isopycnals in the front but continues propagating downward and equatorward south of the front.

The wave propagates at least 100 km south of the front, sloping from about 100-m to 300-m depth over that range. Since the front is essentially nonexistent in this region, the high shear can therefore be distinguished from near-inertial motions trapped at the front. This contrasts with the situation described by Kunze and Sanford (1984), where shear was high in a 10-km band on the immediate warm side of the front and therefore could not be distinguished from trapping. Here, the wave extends well past and deeper than the region of high vorticity and is therefore not trapped but, rather, radiating downward and away from the front.

Single lines can only detect the along-track component of features. We conducted two identical radiator patterns in the immediate vicinity of the front. These three-dimensional surveys allow estimation of the zonal wavenumber as well as the meridional wavenumber. Since they were separated by two days, we also gain some sense of the time evolution of the wave, though it is not fully resolved. Successive north–south lines from the 7 and 9 July radiator patterns occupied are shown at top and bottom, respectively, in Fig. 8. In each survey, successive panels indicate north–south transects at the x location indicated, moving from west to east. In spite of the shorter meridional extent of these sections relative to Fig. 7, all of the same general features are seen, including the main shallow feature and the deeper and steeper features. Though there is a tendency for the wave to weaken moving west, no discernible slope is seen in the east–west direction. Hence, the wave is oriented in the north–south direction, perpendicular to the front. In the second occupation two days later, the wave is noticeably weaker in shear magnitude and shows a more complicated structure, possibly due to loss of energy as the wave radiates away.

Fig. 8.

As in Fig. 7 but for two occupations (7 and 9 July) of the radiator pattern shown in Fig. 1. The x location of each transect is indicated at the bottom of each panel.

Fig. 8.

As in Fig. 7 but for two occupations (7 and 9 July) of the radiator pattern shown in Fig. 1. The x location of each transect is indicated at the bottom of each panel.

e. Energy and energy flux

We next compute the energy and group velocity in the wave to obtain an estimate of its downward energy flux. Horizontal kinetic energy is given by

 
formula

where the angle brackets indicate some type of average and ρ is the density of seawater. As stated above, the near-inertial portion of the velocity signals cannot be isolated via bandpass filtering owing to the record’s aliased spatial and temporal information. Instead, near-inertial velocity is estimated by removing the mean in depth and time over the period shown in Fig. 6, and then evaluating (3). Removal of the time mean largely eliminates the signature of the front, while removal of the depth mean minimizes contributions from longer-wavelength features associated with the internal tide and other motions. The HKE of the wave averaged over the period from 7 to 11 July (Fig. 4b, thick black) is about 10 J m−3 near the mixed layer base, decreasing to about one tenth that value at 250 m.

To ensure that motions at additional frequencies do not contaminate our estimate, we also use an alternate expression from Alford and Gregg (2001) involving the shear:

 
formula

where the observed wavenumber m = 2πλ−1 is used to obtain velocity from the observed shear. The two methods yield very similar estimates because of the dominance of the energy by the single near-inertial wave.

HKE averaged over the upper 250 m (Fig. 9b) begins near 7 J m−3 (velocity amplitude of 0.1 m s−1) and decreases to about half that value over the 4-day period, for a time-mean value of 5.2 J m−3. For comparison, the wave is weaker than those observed by Kunze and Sanford (1984) and Alford and Gregg (2001), which had HKE of about 20 and 25 J m−3, respectively.

Fig. 9.

Wind speed, wave energy, and wave energy flux and wind work prior to and during the observations: (a) time series of wind speed from the shipboard anemometer (black) and NCEP winds from 30°N, 157.5°W; (b) wave horizontal kinetic energy (HKE); and (c) vertical energy flux (black with gray shading) and wind work estimated from the Pollard and Millard (1970) slab model. Wind work is estimated at 30°N (thick gray) and 32.5°N (black).

Fig. 9.

Wind speed, wave energy, and wave energy flux and wind work prior to and during the observations: (a) time series of wind speed from the shipboard anemometer (black) and NCEP winds from 30°N, 157.5°W; (b) wave horizontal kinetic energy (HKE); and (c) vertical energy flux (black with gray shading) and wind work estimated from the Pollard and Millard (1970) slab model. Wind work is estimated at 30°N (thick gray) and 32.5°N (black).

To estimate the vertical energy flux, we use the expression, Fz = cgzHKE, where the vertical group velocity is computed from the dispersion relationship:,

 
formula

Since cgz varies strongly near the inertial frequency, Fz is sensitive to our estimates of wave frequency, which are uncertain. For our estimate of ω = 1.01f from the wave slope, we obtain 3.5 m s−1 (3 m day−1). In Fig. 9c, we plot flux computed using this value, with shading indicating values for vertical group velocity half and twice that value. The mean vertical energy flux is about 1.2 × 10−3 W m−2.

3. Summary and discussion

We have presented ADCP and towed-CTD observations of a downward- and equatorward-propagating near-inertial wave (NIW) on the south side of the Subtropical Front. Similarly elevated shear has been observed near many fronts, but is not always identifiable as near inertial owing to the lack of time series information (Rainville and Pinkel 2004). Other studies (e.g., Kunze and Sanford 1984) have been able to demonstrate that the shear is near inertial, but unable to determine the wave origin because of the possibility of other processes including 1) trapping rather than generation at the front and 2) wind generation. Here, time series identify the waves as near inertial (within about 20% of f). The shear extends over 100 km south of the front, allowing trapping to be ruled out since the wave persists beyond the region of negative vorticity.

Depth-averaged shear increases sharply on the south side of the front, strongly suggesting that the waves emanate from the frontal region. Because trapping has been ruled out and our observations are well north of the “critical latitude” for PSI and far from rough topography, the only plausible possibilities are generation by the wind—and some sort of frontal instability process. In this section we first consider wind generation, discuss the energetics of the wave and the front, and then conclude with speculations on possible generation mechanisms.

a. Wind work

To investigate the possibility that the wind generated the wave, the work done by the wind on inertial motions in the mixed layer is estimated, using the slab model of Pollard and Millard (1970), and compared to the downward energy flux of the wave. Many past studies have demonstrated the remarkable skill of the slab model in predicting the amplitude and phase of the observed inertial mixed layer response. The work done by the wind on the inertial motions can then be simply computed either from an energy equation derived from the model momentum equations (D’Asaro 1985), or more simply and intuitively from the dot product of the inertial parts of the wind stress and the inertial currents (Alford 2001; Silverthorne and Toole 2009; Alford et al. 2012). The difference between the two methods is O(10%) (Alford 2003), substantially smaller than other uncertainties involving the choice of wind product (Jiang et al. 2005) or the possible high bias of the slab model because of its incorrect treatment of damping (Plueddemann and Farrar 2006).

The model is forced with shipboard winds while the ship was on site, and with the National Centers for Environmental Prediction (NCEP) reanalysis winds (Kalnay et al. 1996) prior to that. Observed and NCEP winds are in reasonable agreement during the cruise with both showing light and variable winds less than 8 m s−1 (Fig. 9a, gray and black). Though the 6-h sampling of the NCEP winds can bias wind work estimates low by as much as a factor of 2 at higher latitudes since the inertial frequency is poorly resolved (Alford 2003; Alford et al. 2012), no evidence for a systematic bias has been found at midlatitudes such as this site (Alford 2001, 2003). Since Plueddemann and Farrar (2006) demonstrated that the slab model can be biased high by as much as a factor of 2–3 relative to the observed wind work, here we take the slab-model estimate as an upper bound on the actual wind work.

The model is run with the observed mean mixed layer depth of 40 m and a Rayleigh damping parameter r = (6 days)−1. Previous work (e.g., D’Asaro 1985; Alford 2001) has demonstrated that the calculation is not sensitive to the exact choice of r. The model is solved spectrally using the method described in Alford (2003), which gives very similar results to the traditional time-stepping method.

Slab-model wind work is plotted for shipboard and NCEP winds in Fig. 9c (thin gray, black, respectively) and compared to the downward energy flux estimated from the wave (black with gray shading). To account for generation at an earlier time and subsequent downward propagation at 3 m day−1 as computed earlier, wind work is plotted back to 3 June (30 days prior). Wind work is also computed from NCEP winds from 250 km to the north are plotted (thick gray) to examine generation at higher latitude. As typical for midsummer conditions (Alford 2003), wind work is generally only (0.1 − 0.3) × 10−3 W m−2, substantially less than the wave energy flux of (0.6 − 2.4) × 10−3 W m−2 estimated above.

Though suggestive, these calculations do not categorically rule out wind forcing for two reasons. First, the slab model is strictly invalid in lateral gradients owing to its neglect of lateral pressure gradients in the mixed layer. While a numerical study by Klein et al. (2004) found that the wind work computed from the slab model did not differ in the presence of eddies, it is conceivable that sharper fronts could lead to greater errors. Second, since our observations are near 30°, the possibility exists that diurnal mixed layer modulations could force near-inertial waves. No diurnal modulations in mixed layer depth were observed during our cruise, but we cannot rule out the possibility that they existed prior to our observations.

b. Energetics

If wind generation can be ruled out, the front itself, from which the wave appears to emanate based on the distribution of shear (Figs. 1, 5a, and 7), is the wave’s most likely energy source. Frontal kinetic energy near F1 (Fig. 4b, thin) is substantially greater than the wave (thick line). Averaging over the upper 150 m gives a value of KEfront ={27, 30} J m−3 for the measured and geostrophic flows, respectively, or around five times the energy in the wave. Since potential energy exceeds kinetic energy for fronts with Ro < 1, the total energy in the front is still greater. Using APEfront ≈ Ro−1 KEfront (Gill 1982), we estimate a total energy for the front of about Efront = (1 + Ro−1)KEfront ≈ {180, 120} J m−3 assuming Ro = {0.2, 0.3}.

A rough time scale can be estimated for the wave to drain energy from the jet. We assume decay by vertical energy radiation through a ray tube extending from the bottom of the front. Then

 
formula

where H = 100 m is the depth of the front. Using the values of Efront computed above and Fz = 1.2 × 10−3 W m−2 we obtain τrad = 186 days and 124 days, respectively.

These values are of the same order as a 100-day time scale for wave growth and decay of the geostrophic flow observed in the laboratory by Williams et al. (2008). Based on this measured time scale, the authors speculated that the process could lead to a total global energy loss of 1.5 TW from the eddy field, comparable to the energy input to the mesoscale circulation and the internal wave fields. While their calculation was only intended to demonstrate the potential importance of the process and little importance should be attached to its value, the similarity of our decay time scale to that observed in the laboratory may be significant. Though our wave and front are comparatively weak, stronger fronts would presumably radiate more energetic waves. These would be more difficult to observe, for the reasons discussed in the introduction. If these too have similar implied growth rates, then the process could well be energetically important in the ocean as a sink for the mesoscale circulation and a source for internal waves.

c. Generation mechanisms

Any mechanism transferring energy from the balanced flow requires the front to be evolving, either via frontogenesis [Hoskins and Bretherton (1972)-type mechanisms] or time-dependent flow instabilities [Ford (1994)-type mechanisms]. The front’s much greater energy than the wave’s suggests that only small fluctuations in its magnitude or direction would be required to give deviations of the same order as the observed wave energy, making either type plausible. Because our observations were not taken for this purpose, we collected little information on the large-scale structure and evolution of the front that would allow an unambiguous identification of the mechanism. Here we examine the clues that we can identify from the data regarding the possible mechanism of the waves.

1) Frontogenesis

To examine the possibility that large-scale strain is sharpening the front, the observed velocity was averaged over the upper 100 m and plotted as vectors (Fig. 10, left, gray). The mean overall occupations of the line (gray) and the first occupation only (black) are plotted. Flow is, indeed, confluent near F1, with meridional convergence ∂υ/∂y (right) of about 0.05f. Therefore, the possibility exists that frontogenesis is involved.

Fig. 10.

(left) Velocity vectors and (right) y-strain f −1υ/∂y, averaged over the upper 100 m during the long survey. Gray is the mean over all occupations of the line from 6 to 8 July; black is the first occupation of the line.

Fig. 10.

(left) Velocity vectors and (right) y-strain f −1υ/∂y, averaged over the upper 100 m during the long survey. Gray is the mean over all occupations of the line from 6 to 8 July; black is the first occupation of the line.

2) Time dependence

Comparison between the mean (gray) and the snapshot from the first occupation of the line (black) indicates some time variability of the flow, which might be consistent with the Ford (1994) class of mechanisms. However, the snapshot includes wave motions with wavelengths longer than 100 m and therefore might over-represent time variability of the frontal flow.

3) Rossby number

Because the Rossby number is the ratio of the wave and balanced time scales, many of the generation studies predict weak radiation at low Rossby number. The laboratory studies by Williams et al. (2008) generate internal gravity waves from a balanced flow, even at Rossby numbers lower than ours (0.05–0.1).

4) Radiation

A significant feature of our observations is the radiation of the wave well away from the front. Extant model studies show a variety of spatial structures in the generated waves, including distant radiation (Griffiths and Reeder 1996; Reeder and Griffiths 1996) and local dissipation (Thomas 2012; Nagai et al. 2012, manuscript submitted to Nature). Our observations have similar time and space scales as the waves modeled by Nagai et al., but the waves radiate far away, in contrast to their study.

These wave–mean flow interactions are of potential importance for the energy budget of not only the mesoscale circulation but also the internal wave field and its mixing (Ferrari and Wunsch 2009). Before their role can be assessed, more direct comparisons are needed between realistic simulations and observations. Distinguishing between the distant-radiating and local-generation scenarios, as well as predictions of the expected space and time scales of the generated waves for a range of Rossby numbers, should be a high modeling priority. We hope that our observations will catalyze these efforts.

Acknowledgments

The authors are grateful to the Captain, crew, and marine technician of R/V Wecoma for their expert ship handling, hard work, and flexibility that made the operations possible, and to Paul Aguilar, Steve Bayer, Eric Boget, Andrew Cookson, Jack Miller, Avery Snyder, and Dave Winkel for their skill in developing and operating the SWIMS system. We thank Jules Hummon (University of Hawaii) for her help in setting up the UHDAS ADCP data acquisition system on Wecoma. We sincerely thank the two anonymous reviewers for their comments, which greatly improved the manuscript. This work was supported by the National Science Foundation under Grant OCE054994.

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