Interaction of Superinertial Waves with Submesoscale CyclonicFilaments in the North Wall of the Gulf Stream

DANIEL B. WHITT

National Center for Atmospheric Research, Boulder, Colorado

LEIF N. THOMAS

Earth System Science Department, Stanford University, Stanford, California

JODY M. KLYMAK

School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada

CRAIG M. LEE AND ERIC A. D’ASARO

Applied Physics Laboratory, University of Washington, Seattle, Washington

(Manuscript received 13 April 2017, in final form 15 August 2017)

ABSTRACT

High-resolution, nearly Lagrangian observations of velocity and densitymade in theNorthWall of theGulfStream reveal banded shear structures characteristic of near-inertial waves (NIWs). Here, the current followssubmesoscale dynamics, with Rossby and Richardson numbers near one, and the vertical vorticity is positive.This allows for a unique analysis of the interaction of NIWs with a submesoscale current dominated bycyclonic as opposed to anticyclonic vorticity. Rotary spectra reveal that the vertical shear vector rotatesprimarily clockwise with depth and with time at frequencies near and above the local Coriolis frequency f. Atsome depths, more than half of the measured shear variance is explained by clockwise rotary motions withfrequencies between f and 1.7f. The dominant superinertial frequencies are consistent with those inferredfrom a dispersion relation for NIWs in submesoscale currents that depends on the observed aspect ratio of thewave shear as well as the vertical vorticity, baroclinicity, and stratification of the balanced flow. These ob-servations motivate a ray tracing calculation of superinertial wave propagation in the North Wall, wheremultiple filaments of strong cyclonic vorticity strongly modify wave propagation. The calculation shows thatthe minimum permissible frequency for inertia–gravity waves is mostly greater than the Coriolis frequency,and superinertial waves can be trapped and amplified at slantwise critical layers between cyclonic vortexfilaments, providing a new plausible explanation for why the observed shear variance is dominated bysuperinertial waves.

1. Introduction

Near-inertial waves (NIWs) and inertia–gravity waves(IGWs) with frequency close to the local Coriolisfrequency f interact strongly with mesoscale- andsubmesoscale-balanced currents for several reasons.Unlike higher-frequency IGWs, the lateral group velocityof NIWs is small compared to the speed of balancedcurrents; consequently, the ‘‘crests’’ and ‘‘troughs’’ ofNIWs can be strongly distorted by mesoscale and

submesoscale shears (Bühler and McIntyre 2005). Inaddition, the vorticity associated with these shears shiftsthe net spin of the fluid away from Earth’s rotation,leading to marked changes in the dispersion and polar-ization relations of NIWs. For example, in a geostrophiccurrent ug(y, z), the vertical vorticity zg 5 2›ug/›ymodifies the effective Coriolis frequencyfeff 5 f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(11Rog)

p(where Rog 5 zg/f is the Rossby

number), which constrains the minimum frequency ofthe NIWs. Furthermore, horizontal vorticity associatedwith thermal wind shear ›ug/›z lowers the minimumfrequency of NIWs from feff toCorresponding author: Daniel B. Whitt, dwhitt@ucar.edu

JANUARY 2018 WH I TT ET AL . 81

DOI: 10.1175/JPO-D-17-0079.1

! 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS CopyrightPolicy (www.ametsoc.org/PUBSReuseLicenses).

vmin 5 f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"feff

f

#2

21

Rig

vuut , (1)

where Rig5 N2/(›ug/›z)2 is the Richardson number of the

geostrophic flow, andN is the buoyancy frequency (Mooers1975; Kunze 1985). The modifications to the properties ofNIWs are thus particularly pronounced in submesoscale-balanced currents since they are characterized by Ri-chardson and Rossby numbers that are order one.In this submesoscale regime, NIWs can exhibit un-

expected behavior. For example, NIW velocity hodo-graphs do not trace inertial circles but are instead ellipticaland, for a given wave frequency, the direction of energypropagation is symmetric about the slope of isopycnals notthe horizontal and approaches the isopycnal slope as thefrequency nears vmin (Mooers 1975; Whitt and Thomas2013). This unusual wave physics can facilitate energytransfers between NIWs and balanced currents (Thomas2012) and allow for the formation of critical layers alongthe sloping isopycnals of fronts (Whitt and Thomas 2013).There have been several field campaigns designed to

study NIW-balanced flow interactions in the mesoscaleregime, that is, with Rog & 0:1 and Rig ! 1, that supporttheoretical predictions of trapped subinertial waves inregions of anticyclonic vorticity and the development ofvertical critical layers (e.g., Kunze and Sanford 1984;Kunze 1986; Mied et al. 1987; Kunze et al. 1995). In thisarticle we present observations of NIWs in theNorthWallof the Gulf Stream, where the balanced flow has Rossbynumbers that exceed one and Richardson numbers thatdrop below one (Klymak et al. 2016; Thomas et al. 2016).The observations were collected as part of the ScalableLateral Mixing and Coherent Turbulence (LatMix) 2012field campaign (19 February–17 March 2012), the goal ofwhich was to study submesoscale processes and their ef-fect on mixing in the Gulf Stream and northern SargassoSea. The objective of this paper is to quantitativelycharacterize the NIWs and the relevant properties of themedium in which they propagate (e.g., Ro and Ri) in ahigh-resolution Lagrangian survey of velocity and densityin the upper pycnocline of the Gulf Stream Frontobtained during the LatMix campaign. In addition, wewill use the theory ofWhitt and Thomas (2013) for NIWsin submesoscale fronts to interpret the prominent fea-tures of the wave field seen in the observations.

2. Observed frontal structure

a. Data from ships and towed profilers

Observations were made while underway on twoglobal class research vessels, R/V Knorr and R/V At-lantis, traveling at approximately 8 kt (1 kt5 0.51m s21).

Both ships were equipped with two shipboard-mountedacoustic Doppler current profilers (ADCPs). A 300-kHzTeledyne RDI Workhorse sampled the top 100–150mwith 4-m vertical resolution, while a 75-kHz TeledyneRDI Ocean Surveyor surveyed the top 500–600mwith 8-m vertical resolution. In addition, the R/VKnorr was equipped with a towed MacArtney TRI-AXUS that undulated between the surface and 200mto provide profiles of temperature, conductivity, andpressure along a sawtooth pattern with a resolutionof approximately 1 km in the direction of travel. TheR/V Atlantis was equipped with a Rolls Royce mov-ing vessel profiler (MVP) with a free-falling AMLOceanographic micro-CTD to provide vertical pro-files of temperature, conductivity, and pressure, againwith a resolution of approximately 1 km in the di-rection of travel.Part of the analysis requires velocity and density in-

terpolated to the same time-depth grid, covering thetop 200m. The velocity data for these grids are ob-tained by merging data from the 300- and 75-kHzADCPs on each ship and interpolating that data tothe density grid derived from the profiler data collectedon that ship. In particular, current measurements arefirst averaged over approximately 1 km in the directionof travel in depth bins of 4 (300 kHz) and 8m (75 kHz).Then, the data from the 75-kHz ADCP are linearlyinterpolated to the 1 km by 4m 300-kHz grid and gapsin the 300-kHz data are filled with the interpolated75-kHz data. The resultingmerged velocity data are thensmoothed with a two-dimensional Gaussian kernel with astandard deviation of 4m in depth and approximately1km in the direction of ship travel. The half-powerwavelengths of this two-dimensional Gaussian kernelare 30m in the vertical and approximately 7.5km in thedirection of ship travel (i.e., 30min at 8kt). Finally, thesmoothed velocity profiles are linearly interpolated(along the ship track) to the positions of the densityprofiles.The density is gridded slightly differently, depending

on the vessel/instrument. Each TRIAXUS profile isaveraged into 2-m vertical bins (ups and downs areincluded as separate profiles). The horizontal position/time of each profile is defined by the ship position/timeat the midpoint of each TRIAXUS profile. For exam-ple, gray dots in Fig. 3a (shown below) mark the shippositions at the midpoint of each TRIAXUS profile inone section across the Gulf Stream. Each downgoingvertical MVP profile is averaged into 1-m vertical binsat a given position/time (upward profiles from theMVPare excluded). For example, gray dots in Fig. 2a (shownbelow) mark the MVP vertical profile locations in onesection across the Gulf Stream. The gridded density

82 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

from each profiler is then filtered by applying the sametwo-dimensional Gaussian smoother that was appliedto the velocity data, and the resulting filtered density islinearly interpolated (in depth) to the 4-m bins of thevelocity grid.After this filtering and gridding, density and ve-

locity are collocated at the horizontal positions of thedensity profiles in uniform 4-m vertical bins, andfeatures with wavelengths less than about 30m in thevertical and 7.5 km in the direction of ship travel havebeen filtered out of both the density and velocitydatasets.

b. Experimental description and observed frontalstructure

Measurements focused on the North Wall of the GulfStream, past the point where it separates from CapeHatteras, North Carolina, between 388 and 408N and 608and 708W. Here, cold subpolar water and warm sub-tropical water converge to form a strong front. Thecampaign was broken into a series of drifts, each follow-ing an acoustically tracked Lagrangian float (D’Asaro2003) placed in the surface mixed layer in the strongestpart of the front for 2 to 4 days. In each drift, the shipsmade repeated frontal cross sections following the float(e.g., Fig. 1). The float was tracked using a transducerpole, which was mounted on the R/V Knorr, extended2m below the keel, and provided a maximum range ofabout 4km. To track the float accurately, the R/V Knorrremained within about 5km of the float position. Thetracking equipment is otherwise similar to that used inD’Asaro et al. (2011), and the uncertainties associatedwith the float position estimates are thought to be smallcompared to other uncertainties in the analysis.

In this paper, attention is focused on the seconddrift, which occurred from 1 March through 3 March2012 (yeardays 60 through 62), near the strongestpart of the front (Fig. 1a). While the Atlantis madewider ;30-km cross sections around the float, theKnorr made narrower ;10-km cross sections, closelyfollowing the float (Fig. 1b). The survey strategy in-volved intensive sampling of the water around aLagrangian float in the mixed layer in an attempt tominimize the convoluting effects of advection on theanalysis. But, it is important to note the observationswere obtained in a region of strong lateral and ver-tical gradients in velocity, and therefore the ob-served fluid is not exactly in a Lagrangian frame ofreference at all depths and cross-stream locationsobserved.A typical cross-front section from the Atlantis

(Fig. 2) exhibits a strong surface-intensified jet withvelocities exceeding 2m s21 in the streamwise di-rection x, order-one vorticity Rossby numbers (Ro 5z/f ’ 2›u/›y/f, where f is the Coriolis frequency, u isthe along-stream velocity, and y points in the cross-stream direction), isopycnal slopes sb of order 0.01,and geostrophic Richardson numbers Rig rangingfrom 1 to 10. To estimate Ro and Rig, the associatedgradients are calculated using central differences overapproximately 2 km in the horizontal and 8m in thevertical, but variability on scales less than about 30min the vertical and 7.5 km in the horizontal is sup-pressed 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 NIWsin the current they should be substantially modified byits vorticity and baroclinicity. As described in the next

FIG. 1. (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 theduration of the drift. The approximate locations of the ships and float at yearday 61 and 61.7 are marked witharrows.

JANUARY 2018 WH I TT ET AL . 83

section, there were indeed features in the observedflow field during the drift with properties that areconsistent with NIWs, giving us the opportunity tostudy the interaction of the waves with a submesoscalecurrent.

3. Observations of near-inertial shear

a. Structure and evolution of the vertical shear

Banded patterns of strong vertical shear ›u/›z [whereu 5 (u, y) is the horizontal velocity vector] were ob-served during the drift. To analyze these features, we

rotate the coordinate system so that (x, y) denotes thestreamwise and cross-stream directions respectively,and x is defined to point in the direction of the local floatpath, whereas y points in the direction perpendicular tothe local float path.Sections of the streamwise and cross-stream shear, ›u/›z

and ›y/›z, respectively, reveal shear anomalies (i.e.,relative to a linear fit in z) that are arranged into bandedstructures parallel to slanted isopycnals (Figs. 3a,b). Theshear anomalies have a dominant vertical wavelengthbetween 50 and 100m andmagnitudes of up to 0.005 s21,which is comparable to the depth-averaged alongfront

FIG. 2. Data from the second Atlantis section obtained at about yearday 61.3 at a meanlatitude and longitude of 38.28N, 66.18W; see Fig. 1a. Cross-front/depth maps of (a) streamwisevelocity u, (b) vorticity Rossby number Ro52›u/›y/f, and (c) inverse geostrophic Richardsonnumber Ri21

g 5 (›b/›y)2/( f 2›b/›z) show that the observations are in a submesoscale front, thatis, where Ro;Ri21

g ; 1. The mixed layer depth, defined by a 0.03 kgm23 density threshold, isindicated by a magenta line in (a). Note that the coordinate system is rotated so that thestreamwise velocity u is defined to be the velocity parallel to the float path, the coordinate yrepresents the cross-front distance in the direction perpendicular to the float path, and y5 0 kmdenotes the cross-front location of the float path. Each gray dot at z’ 0m indicates the locationof an MVP profile.

84 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

shear (Figs. 3a,b). However, the associated velocityanomalies, ua ; (1/m)(›ua/›z) & 0.1m s21, are muchsmaller than the velocity of the depth-averaged along-front flow of ;1–2ms21 (Fig. 2a). Yet, the shearanomalies are sufficiently intense to be associated withbanded patterns in the inverse Richardson numberRi21 5 j›u/›zj2/N2, which often exceeds 1 and ap-proaches 4 in some places (Fig. 3c). These high values ofRi21 occur despite the smoothing applied during dataprocessing and instrument resolution/noise effects, whichall tend to cause these estimates of Ri21 to be biased verylow at the 8-m vertical scale on which the gradients arecalculated and to be biased slightly low (by perhaps afactor of 2) at 30-m vertical scales [see Alford and Pinkel(2000) for a useful discussion of these biases]. Finally, theshear vector is polarized in z so that the direction of theshear vector rotates clockwise with increasing depth (i.e.,decreasing z; see Fig. 3d).

The time series of the wind stress together with mapsof the streamwise and cross-stream vertical shear ob-served from the ships at points near the float path il-lustrate the temporal evolution of the banded shear andthe atmospheric forcing during the drift (Fig. 4). Indi-vidual time series of the shear averaged between 245and 265m and 265 and 285m further highlight thisevolution (Fig. 5). The time series show that the shearanomalies evolve significantly with time/downstreamposition at essentially all depths/isopycnals during thedrift. Yet, the shear anomalies exhibit a consistentdominant vertical wavelength between 50 and 100m andachieve a similar maximum amplitude of 0.005 s21,which is similar in magnitude to the depth-averagedalongfront shear, throughout the drift. Likewise, theshear anomalies are associated with regions of low Ri-chardson number Ri ’ 1 below the mixed layer at var-ious times throughout the drift (Fig. 4d).

FIG. 3. Data from the fifthKnorr section shown in Fig. 1a, which were obtained at yearday 61.1 at a mean latitudeand longitude of 38.18N, 66.38W. Maps are presented of (a) the streamwise component of the vertical shear ›u/›z,(b) the cross-stream component of the vertical shear ›y/›z, and (c) the inverse Richardson numberRi21 5 [(›u/›z)2 1 (›y/›z)2]/›b/›z. Density contours are superimposed with a spacing of 0.1 kgm23. The mixedlayer depth, defined by a 0.03 kgm23 density threshold, is indicated by a magenta line in (c). (d) The hodograph ofthe shear vector ›u/›z averaged laterally between y5 1 and 4 km rotates primarily clockwise with depth. Note thatthe coordinate system is as in Fig. 2, and each gray dot at z ’ 0m indicates the location of a TRIAXUS profile.

JANUARY 2018 WH I TT ET AL . 85

The structure and time/downstream evolution of thevertical shear field illustrated in Figs. 2–5 suggestsqualitatively that the shear is associated with downward-propagating NIWs that are potentially modified by thevorticity and baroclinicity of the Gulf Stream, as de-scribed by Whitt and Thomas (2013). In particular, thetemporal variability of the observed shear appears to bedominated by oscillations with frequencies that areslightly higher than the local Coriolis frequency f’ 931025 s21 (the local inertial period corresponds to;0.8 days). In addition, lines of constant shear anomalypropagate upward with time, which is consistent withdownward-propagating NIW physics. To test this

hypothesis more quantitatively, rotary vertical wave-number and frequency spectra will be used to decomposethe shear variance into clockwise and counterclockwisecomponents and estimate the dominant wave frequencyin sections 3b and 3c, and the results will be compared tothe theoretical predictions ofWhitt andThomas (2013) insections 4–5.

b. Rotary vertical wavenumber spectra

The vertical variation of ›u/›z seen in Fig. 3 suggeststhat the shear vector rotates primarily clockwise withincreasing depth at that location. A quantitative es-timate of the shear vector’s average sense of rotationwith depth along the float path is derived from therotary components of the Froude number verticalwavenumber spectrum [i.e., a buoyancy frequencynormalized shear spectrum (e.g., Munk 1981; Alfordand Gregg 2001; Rainville and Pinkel 2004)]. Wecalculate Froude number spectra rather than shearspectra because the former provides more in-formation about the local stability of the waves, andobservations collected at locations with differentstratifications can be more easily compared with eachother (Munk 1981).The Froude wavenumber spectra are constructed as

follows: For each profile of density made within 350m ofthe float path, profiles of shear anomaly

›ua

›z(z)5

$›u

a

›z(z),

›ya

›z(z)

%

are created by subtracting a linear fit from eachcomponent of the full shear vector over the depth range(z0 2 Lz) # z # z0, where z0 5 229m and Lz 5 160m.The full shear vector is constructed from the smoothedvelocity field described in section 2a. These profiles›ua/›z are normalized by dividing by the depth-averaged buoyancy frequency N(t), which is calcu-lated over the same depth range as the shear andvaries by 40%, from 0.0056 to 0.0079 s21, during thedrift. The two components of the shear are then com-bined into a complex Froude number profile:

Fr(z)5

›ua

›z(z)1 i

›ya

›z(z)

N, (2)

where i5ffiffiffiffiffiffiffi21

p. A 20% tapered cosine (Tukey) window

is applied to Fr(z) and then the discrete Fourier trans-formcFr(mk)5Dz!2M21

n50 Fr(zn)e2imkzn is computed usingthe fast Fourier transform algorithm for each wave-number6mk56kDm (where k52M, . . .,21, 0, 1, . . . ,M21, Dm 5 2p/Lz is the smallest resolvable

FIG. 4. Several time series of data collected along the whole drift.(a) The 1-min moving average of the wind stress derived from ob-servations of wind speed at a meteorological station on the Knorrand 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-streamcomponent of the vertical shear ›y/›z, and (d) theinverse Richardson number Ri21 5 [(›u/›z)2 1 (›y/›z)2]/›b/›z.(b)–(d) are derived from Knorr ADCP and TRIAXUS data.Density contours are superimposed with a spacing of 0.1 kgm23 ineach plot. The mixed layer depth, defined by a 0.03 kgm23 densitythreshold, is indicated by a magenta line in (d). However, onlythose observations within 350m of the float path (the magenta linein Fig. 1a) are included here; gray dots indicate the location ofTRIAXUS profiles where this condition is met and data areincluded.

86 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

wavenumber, M is half the number of observations inthe depth domain, Dz 5 4m is the vertical grid spacing,and the zn are the 2M discrete depths). The resultingtransforms are rescaled to account for the variance re-duction caused by windowing and are most accurate atvertical wavelengths between 80 and 40m. This is be-cause energy in the smallest resolved wavenumberDm ismore strongly damped by the windowing, whereas thehigher wavenumbers are more severely contaminatedby instrument noise, resolution effects, and thesmoothing during processing (see section 2a). A usefuldiscussion of the relevant sampling issues related toshipboard ADCPs can be found in the appendix ofAlford and Gregg (2001). Because the data presentedhere are derived from two ADCPs with different verti-cal resolutions, no attempt is made to scale up the re-sulting Froude wavenumber spectra to account forADCP resolution or smoothing effects, both of whichmay conspire to reduce the reported Froude wave-number spectra by as much as a factor of 2 at a verticalwavelength of 40m but will impact the positive andnegative vertical wavenumbersm1 andm2 equally. Thepositive and negative vertical wavenumbersm1 andm2

represent the clockwise (CW) and counterclockwise(CCW) parts of the rotary Froude wavenumber spec-trum, respectively, which are given by

cFr2CW(m1)5hcFr(m1)

cFr(m1)ci

2pLz

, and (3)

cFr2CCW(m2)5hcFr(m2)

cFr(m2)ci

2pLz

, (4)

where the superscript c denotes the complex conjugate,and h"i denotes the ensemble average over all verticalprofiles along the float path. The spectra are normalizedby dividing by 2pLz so that

Fr2 5DzL

z

!2M21

n50

›ua(zn)›z

2

1 ›ya(zn)›z

2

N2’Dm !

M21

k52M

cFr2(m

k) (5)

by Parseval’s theorem. The contribution from differentwavenumbersmk to the depth-averaged Fr2 can then beestimated by integrating over a subset of the resolvedvertical wavenumbersmk (e.g., Munk 1981). Confidenceintervals for the average spectral estimates [(3)–(4)] arederived from a x2 distribution. The effective number ofindependent observations used to calculate the confi-dence intervals (nine in this case) is estimated by di-viding the length of the drift by the largest integral timescale calculated from the temporal correlation functionof cFr

2(mk, t) (e.g., D’Asaro and Perkins 1984; Emery

and Thomson 2001).

FIG. 5. (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 fromtheKnorr but restricted to cross-stream positions within 350m 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.

JANUARY 2018 WH I TT ET AL . 87

The rotary components of the Froude wavenumberspectra (plotted in Fig. 6) show that the average CWFroude variance is 2–5 times greater than the averageCCW Froude variance at vertical wavelengths of 40 to80m. The enhanced CW Froude variance is in contrastto the canonical internal wave continuum, in whichwaves propagate upward and downward in equal pro-portion (e.g., Garrett and Munk 1972), and is consistentwith the presence of energetic and coherent, downward-propagating NIWs (e.g., Leaman and Sanford 1975;Kunze and Sanford 1984; Mied et al. 1987; Alford andGregg 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 thepolarization relations of NIWs in the absence of abackground flow. Whitt and Thomas (2013) show thatsubmesoscale fronts strongly modify the polarization ofNIWs, which can affect the ratio of CW to CCWFroudevariance for downward-propagating NIWs. We will ex-plore this physics in section 4.

c. Rotary frequency spectra

The temporal variation of the shear vector is mostwavelike just below the base of the mixed layer. There,the shear varies nearly sinusoidally with a similar am-plitude and a fixed phase relationship between thestreamwise shear ›u/›z and cross-stream shear ›y/›z(Fig. 5). In particular, ›y/›z leads ›u/›z by a quarter of acycle (Fig. 5), indicating that the shear vector rotatesclockwise with increasing time, which is consistent withthe dynamics of NIWs in the Northern Hemisphere(e.g., Gonella 1972).To obtain quantitative measures of the frequencies

and amplitudes of the two rotary components of theobserved shear we calculate rotary frequency spectrafrom the shear vector time series collected within 350mof the float path. To do so, we define the complex vectortime series of the Froude number

Fr(t)5

›ua

›z(t)1 i

›ya

›z(t)

N(6)

at each depth bin z, where ›ua/›z(t) is the shear anomalythat is constructed by removing a linear fit (in time) fromeach component of the shear vector. After dividing bythe time-mean N(z) in each depth bin, which varies by29% from 0.0063 to 0.0081 s21 over all z betweenz 5 241 and 2189m, the resulting time series Fr(t) iswindowed with a 20% tapered cosine (Tukey) window.Since the TRIAXUS profiles within 350m of the floatpath are rather unevenly spaced in time, a least squaresapproach (e.g., Lomb 1976; Scargle 1982) is used to es-timate the Fourier transform cFr(vk) for each frequency

vk 5 kDv, where k 5 2M, . . ., 21, 0, 1, . . ., M 2 1,Dv 5 2p/T ’ 0.4f is the minimum resolvable frequency(which is constrained by the T ’ 2 days duration of thedrift), and M is now half the number of observations inthe time domain. The negative (v2 , 0) and positive(v1. 0) angular frequencies ofcFr(vk) can then be usedto calculate the clockwise and counterclockwise com-ponents of the Froude number spectrum cFr

2

CW(v2) andcFr

2

CCW(v1), respectively. The spectra are normalized by2pT so that

Fr2(t)5DtT

!2M21

n50

›ua(tn)›z

2

1›ya(tn)›z

2

N2’Dv !

M21

k52M

cFr2(v

k) , (7)

and the contribution from different frequencies vk tothe time-averaged Fr2(t) can be estimated by summingover a subset of the resolved frequencies vk. Figure 7shows the Froude number frequency spectra averagedbetween z 5 241 and 2189m. Confidence intervals forthis depth-averaged spectrum are derived from a x2

FIG. 6. The float path average CW and CCW rotary componentsof the Froude number spectrumcFr

2(m) as a function of the angular

vertical wavenumber m over the depth range z 5 229 to 2189m.Only the data shown in Fig. 4 are used in the calculation; hence, thisaverage represents data within 350m of the float path. The verticalwavelengths 2p/m that are shown range from 40 to 160m. The 90%confidence interval is shown in black.

88 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

distribution. The effective number of independent ob-servations used to calculate the confidence intervals(five in this case) is estimated by dividing the depth overwhich the spectra were calculated by the largest integraldepth scale calculated from the spatial correlationfunction of cFr

2(vk, z).

These estimates of the CW and CCW components ofcFr

2(v) in Fig. 7 reveal increased variance at frequencies

1:7f *v* f just above the Coriolis frequency. At thesefrequencies, almost all of the variance is in the CW ro-tary component, which explains about a quarter ofthe total Froude variance (CW 1 CCW) averagedfrom 241- to 2189-m depth (Fig. 7b). The maximum inthe Froude number spectrum is at vk 5 1.2f, whichrepresents frequencies 1:4f *v* f and explains about40%of the variance in the depth range240. z.280m(not shown). Depending on the depth, the two frequencybins vk with the largest variance, which represent fre-quencies 1:7f *v* f , represent between 10% and 60%of the total Froude variance Fr2(t) (which ranges be-tween about 0.1 and 0.5 here; see Fig. 8). At depthsshallower than z’2100m, the frequencies 1:7f *v* fcontain approximately 50% of the total Froude vari-ance. In this part of the water column, the two frequencybins vk with the largest variance, 1:7f *v* f , aredominated by clockwise rotary motions as quantified bythe rotary coefficient

cFr2

CW 2cFr2

CCW

cFr2

CW 1cFr2CCW(8)

(e.g., Gonella 1972), which exceeds a value of 0.8 (Fig. 8,right panel).Together, the analysis of this and the previous section

support the hypothesis that a large fraction of the totalshear variance observed during this roughly 2-day La-grangian survey of the Gulf Stream Front is associatedwith coherent NIWs. In particular, the shear vector ispolarized so that it rotates clockwise with both in-creasing depth and increasing time on average, and atsome depths the total shear variance is dominated by anarrow band of superinertial frequencies 1:7f *v* f .

4. Comparison to theory of NIWs in submesoscalefronts

Given that a large fraction of the shear variance isapparently explained by NIWs, we use the dispersionand polarization relations for NIWs in a submesoscalefront, as defined in Whitt and Thomas (2013), to assessthe consistency between the observed spatial variabilityof the shear, the observed temporal variability of theshear, and the theory for NIWs in submesoscale fronts

with Ro ; Ri ; 1. This theory makes several assump-

tions about the waves and the balanced flow, namely,

that the NIWs are hydrostatic and low-amplitude rela-

tive 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 ex-

tent that these assumptions about the waves and bal-

anced flow are reasonable. In this case, the theoretical

results should be interpreted with some caution because

the NIW shear is approximately comparable in magni-

tude 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 bal-

anced 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 inboth space and time. Despite the caveats, we proceed toanalyze the observations using this theory.In particular, looking in the cross-front plane we

can estimate the angle that bands of ageostrophic shear›ua/›zmakewith the horizontal and use it as an estimatefor the slope of lines of constant wave phase (or constantwave shear), sshear 5 2a, in the cross-front plane. Thenthe dispersion relation

v5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2eff 1 2S2a1N2a2

q, (9)

where S25 f›ug/›z, a5 l/m, and k5 (0, l,m) is the wavevector, can be used to infer the wave frequency froma and the properties of the background flow. The dis-persion relation [(9)] reaches aminimum [defined in (1)]when wave phase lines align with isopycnals, that is,sshear52a5 S2/N25 sb. This dispersion relation [(9)] isderived in the hydrostatic form used here by Whitt andThomas (2013) and in nonhydrostatic form by Mooers(1975). See also Mooers (1970) and Whitt (2015) forfurther discussion of the theory and its applications.To use this theory, the ageostrophic waves and

geostrophic-balanced flow must be decomposed. The cross-stream shear ›y/›z is assumed to be entirely ageostrophic,while the streamwise shear ›u/›z can contain both balancedand unbalanced components. The method for isolating thegeostrophic shear in the streamwise direction ›ug/›z is de-scribed in the appendix.

JANUARY 2018 WH I TT ET AL . 89

In an effort to obtain a precise estimate for sshear 5 2afrom the observed shear, an algorithm was developed toidentify the cross-front and vertical coordinates of largesimply connected regions of ageostrophic shear bounded bycontours of constant shear and compute their principalcomponents, from which sshear 5 2a is derived. The pro-cedure is outlined in the appendix and is performed in aregion from z 5 230m to 2170m depth in each Knorrcross section from the drift (see Fig. 1a). A histogram of themeasurements of sshear determined via thismethod is shownin Fig. 9a along with a histogram of the isopycnal slope sb(Fig. 9b). The ratio of sshear to sb in Fig. 9c shows that onaverage the most coherent shear structures are nearly par-allel to or just slightly steeper than isopycnals.

From these observations of sshear, the average wavefrequency inferred from the dispersion relation (9) isv 5 (1.11 6 0.14)f, where 1.11f is the sample meanand 0.14f is the sample standard deviation of all thefrequency estimates inferred via this method. The his-togram of all the frequency estimates obtained from thedispersion relation is shown in Fig. 9d. This estimate ofthe dominant wave frequency derived from the disper-sion relation (9) is consistent with the dominant fre-quency measured in the spectral analysis of section 3c,where the frequencybinvk5 1.2fhas thehighest amplitude

(see Fig. 7). This frequency bin represents the frequencyrange 1:4f *v* f ; the width of the bin is set by the dura-tion of the drift, which sets the spectral resolution 2p/T ’Dv ’ 0.4f (see section 3). Differences between the twoestimates of the frequency could also be attributable to spatialvariability since the frequency spectra were calculated within350m of the float track, while the evaluation of the dispersionrelation was made across entire cross-stream sections.The theory of Whitt and Thomas (2013) also derives the

polarization relations for NIWs and predicts how theychange with the properties of a front. For plane waves,the polarization relations set the ratio of the energy inthe clockwise to counterclockwise rotary components ofvelocity, which can be compared to the observationalestimates basedon the frequency spectra.Whitt andThomas(2013) demonstrated that a plane wave with streamwiseand cross-stream velocities uw 5Rfuexp[i(ly1mz2vt)]gand yw 5Rfyexp[i(ly1mz2vt)]g has a polarization

u5iy

vf( f 2eff 1aS2) .

(10)

Here, we use this result to derive a theoretical expres-sion for the rotary coefficient (8), which can be writtenin a variety of ways, for example,

FIG. 7. (a) The depth average (from z5241 to2189m) of the CW and CCW rotary components of the Froudenumber spectrumcFr

2(v) as a function of the angular frequency v. The 90% confidence interval is shown in black.

(b) The cumulative fraction of the total variance associated with each frequency/rotary component; each com-ponent in (b) is normalized by the total Froude variance (CW 1 CCW); hence, the sum of the two componentsapproaches one at high frequencies. Only the data shown in Fig. 4 are used in the calculation hence this averagerepresents data within 350m of the float path. The frequencies shown range from about 1/5 to 5 times the localCoriolis frequency, that is, periods of about 4 days to 4 h.

90 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

(cFr2

CW 2cFr2

CCW)

(cFr2

CW 1cFr2

CCW)5

(dKECW 2dKECCW)

(dKECW

1dKECCW

)

5ju2j

2 2 ju1j2

ju2j2 1 ju1j

2

5( f 2eff 1aS21vf )22 ( f 2eff1aS2 2vf )2

( f 2eff 1aS21vf )21 ( f 2eff1aS2 2vf )2

52( f 2eff 1aS2)vf

( f 2eff 1aS2)2 1v2f 2, (11)

where

u1 51

2f[R(u)1J(y)]1 i[R(y)2J(u)]g, and

u2 51

2f[R(u)2J(y)]1 i[R(y)1J(u)]g .

In the limit that a/ 0 and feff/ f, then (11) reduces to therotary coefficient derived by Gonella (1972), that is,2fv/( f2 1 v2). The inferred frequency from thedispersion relation, slope of the ageostrophic shearsshear 52a, and vorticity and baroclinicity of the frontwere used to calculate (11), a histogram of which isshown in Fig. 9e. The theory predicts that the kineticenergy of the NIW is primarily in clockwise rotarymotions, consistent with the findings from thefrequency spectra (cf. Fig. 8).

5. Discussion

a. Wave propagation, trapping, and amplification

Motivated by the consistency between the measuredproperties of the NIWs and the predictions of Whitt andThomas (2013), we apply the theory to explore thepropagation, trapping, and amplification of NIWs in thedensity and vorticity fields observed during the drift. Inparticular, we ask the following questions:

(i) How might superinertial NIWs with the observeddominant frequencies and wavenumbers propagatein the spatially variable medium of the Gulf StreamFront (e.g., Fig. 2)?

(ii) Why might superinertial NIWs have the highestamplitude 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 byorder-one Ro and Ri. As a result, the minimum fre-quency vmin 5 f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11Ro2Ri21

pis substantially differ-

ent from f, as shown in Fig. 10a. In particular, below themixed layer vmin is greater than the local Coriolis fre-quency in most of the region of interest (see Fig. 10a).Hence, theory suggests that propagating subinertialwaves cannot exist here, which may explain why thepeak in the observed frequency spectrum (which wascalculated near the float at y’ 0 for depths between241and 2189 m; Fig. 7) does not surround f but is

FIG. 8. (a) The average Froude number variance within 350m of the float path Fr2(t) at each depth zn; (b) the fraction of the totalFroude variance associated with the CW and CCW components of the two highest-amplitude components of cFr2CW(v), that isCW5Dv!k53,4

cFr2(2vk), CCW5Dv!k53,4

cFr2(vk), v3 5 1.2f and v4 5 1.6f; and (c) their rotary coefficient [(8)].

JANUARY 2018 WH I TT ET AL . 91

instead centered around somewhat higher superinertialfrequencies.We can better quantify the propagation of the NIWs

using ray tracing, namely, by solving for the positionrw that wave packets trace with time

drw

dt[ c

g, (12)

where

cg[=(l,m)v5

(S2 1N2a)

vm(1,2a) (13)

is the group velocity of NIWs in a front (Whitt and Thomas2013). Note, however, that the usual assumption of a slowlyvarying wave train, which is required for ray tracing to bevalid, does not hold here because themediumvaries rapidlyon the spatial and temporal scales of the waves. On theone hand, slow spatial variations are not strictly required

to characterize the propagation of the wave energyacross the front in the theory of Whitt and Thomas(2013). In particular, ray paths parallel characteristics ofthe governing partial differential equation [(14) inWhitt and Thomas (2013)], and these characteristics ac-curately characterize the two-way propagation directionof wave energy even in a medium that varies on shortspatial scales, as demonstrated by Whitt and Thomas(2013; see also Mooers 1975; Bühler and Holmes-Cerfon2011). On the other hand, the potential for temporalvariability in the medium on the time scale of the wavesposes a number of challenging problems. Therefore, theconclusions derived from the ray tracing should beviewed with caution in this context.Ray paths of wave packets with frequency v 5 1.13f

(a value close to the mean in the histogram of the dis-persion relation; e.g., Fig. 9d) initiated at a depth of70m and cross-stream locations y 5 25 and 16 kmwere calculated for theAtlantis section shown in Fig. 2.Two rays are calculated for each location,

FIG. 9. Histograms of the (a) inferred slope of constant wave shear sshear 5 2a,(b) isopycnal slope sb, (c) logarithm of the ratio sshear/sb, (d) inferred frequencies obtainedusing sshear 52a and the dispersion relation [(9)], and (e) rotary coefficient [(11)]. Bootstrap95% confidence intervals for the arithmetic mean of each histogram are indicated by blackhash marks. The histograms were calculated data from 30 to 170m below the surface alongthe track of the R/V Knorr (i.e., not merely those data within 350m of the float).

92 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

corresponding to wave propagation on the two wavecharacteristics with slopes

sc5 s

b6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 2v2

min

N2

s

. (14)

The frequency v 5 1.13f is close to feff at the initiallocations of the wave packets, and consequently theshallow characteristic in (14), that is, the negative root,is nearly horizontal, while the steep characteristic isclose to twice the isopycnal slope, as is evident in theshape of the ray paths near their origin (Fig. 10b). Thewaves can only propagate in regions where the mini-mum frequency is less than their frequency. When thewaves encounter locations where v 5 vmin (i.e., thelocus of turning points that define separatrix boundingregions where the wave equation transitions from be-ing hyperbolic to elliptic) they reflect and move alongthe opposite characteristic from which they traveled.The spatial structure of the separatrix for these su-perinertial waves is largely determined by the strongcyclonic vorticity in the North Wall of the Gulf Stream(cf. Fig. 2b and Fig. 10a), which is broken into filamentsby flow instabilities (Klymak et al. 2016). The frag-mentation of the separatrix can lead to wave trappingwith rays experiencing multiple reflections betweenvortex filaments (e.g., Fig. 10b; between 25 and 0 kmand 0 and 10 km).While trapping could lead to amplification of the

superinertial waves, wave shoaling could play a role aswell. The magnitude of the group velocity [defined in(13)] varies significantly along ray paths (Fig. 10b) and atthe separatrix the magnitude of the group velocityequals zero since v 5 vmin and a 5 2sb. Consequently,as wave packets approach the separatrix their energydensity should increase to conserve the energy flux,namely, the waves shoal. If the separatrix aligns with thetilted isopycnals of a front, then the reflected ray cannotescape from the separatrix and a slantwise critical layerforms, leading to especially intense wave shear (Whittand Thomas 2013). There are several locations in theAtlantis section where this criterion is nearly met forv5 1.13f, for example, y’ 2km and z’280m and y’10km and z ’ 2175m (Fig. 10b). Hence, it is plausiblethat superinertial NIWs can be trapped and amplified inthe North Wall of the Gulf Stream, which could explainthe observed superinertial peak in the Froude varianceassociated with downward-propagating NIWs.

b. Superinertial wave generation

How could these downward-propagating superinertialNIWs have been generated? The most obvious source ofenergy for the NIWs is the time-variable winds (e.g.,

Fig. 4a). In a geostrophic current with vertical vorticity,winds can accelerate oscillatory motions in the surfacemixed layer with frequencies that depart from the inertialfrequency because the resonant forcing frequency in such acurrent is feff 5 f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 z/f

p, not f (e.g., Weller 1982). Since

vorticity in the North Wall of the Gulf Stream is cyclonic,this resonant frequency exceeds f, suggesting that wind-driven oscillations in the surface mixed layer should besuperinertial and hence could excite NIWs of similar fre-quency. To test this hypothesis we numerically integratethe ‘‘slab mixed layer’’ momentum equations modified toaccount for the effects of vorticity (e.g., Weller 1982; Kleinand Hua 1988; Whitt and Thomas 2015):

dU

dt2 (f 1 zz)V5

txwroH

2 gU, and (15)

dV

dt1 fU5

tywroH

2 gV , (16)

where (U, V) is the velocity averaged over the mixedlayer of depthH, zz is the depth-averaged vorticity of thegeostrophic flow, g 5 1025 s21 is a damping coefficientthat parameterizes wave radiation and local dissipation,and (txw, t

yw) is a time series of the observed winds. The

initial condition isU5 V5 0, and the simulation beginsat yearday 59.5. For the calculation, we use amixed layerdepth H 5 50m and the vertical vorticity of the GulfStream averaged within 350m of the float zz 5 0:4f ,which yields an effective Coriolis frequency feff 5 1.18f.The solution for the inertial components, UI 5 U 2 UE

and VI 5 V 2 VE, is plotted in Fig. 11 along with itsrotary frequency spectrum (calculated using the methodoutlined in section 3c). Here, the steady Ekman solu-tion, that is

UE5

tywroHf

, and (17)

VE52

txwroH( f 1 zz)

, (18)

is subtracted to isolate the oscillating inertial part of the

solution (MATLAB code to solve the slab mixed layer

model is available at https://github.com/danielwhitt/

slabML).

The calculation supports the hypothesis that the localwind stress could have excited the observed NIWs. Inparticular, the modeled mixed layer inertial currentsexhibit speeds ranging from 0.05 and 0.1m s21, consis-tent with the observed shear, which has a velocity scaleof (1/m)(›juj/›z) ; 0.1m s21. Moreover, like theobserved shear, the modeled mixed layer velocities aredominated by clockwise rotation at slightly superinertial

JANUARY 2018 WH I TT ET AL . 93

frequencies and present a similarly shaped spectrum (cf.Figs. 7 and 11b). In addition, it is worth noting that thisslab mixed layer model also predicts that wind-generated, near-inertial oscillations in the ocean mixedlayer become incoherent across the front over a timescale of order 1/f because of the order-one cross-frontvariations in z/f at the Gulf Stream (see Fig. 2b). Theselarge variations in z/f can, therefore, facilitate thedownward radiation of near-inertial wave energy be-tween the vorticity filaments at the North Wall via asubmesoscale inertial chimney effect (van Meurs 1998;Lee and Niiler 1998; Whitt and Thomas 2015; Whitt2015). Although we have someADCP data from outsidethe front during the LatMix experiment, which suggests

that the CW and CCW components ofcFr2are generally

about equal there (as in Garrett and Munk 1972), thesedata are not exactly coincident in time with this survey.Hence, we do not have the data to prove that the ob-served asymmetry between CW and CCW Froude var-iance is unique to the North Wall during the survey.

6. Comparison with other observations

The observations presented here are unique in that weare observing the temporal evolution of near-inertialshear in a submesoscale front in a nearly Lagrangianframeof referencewith a spatial resolution ofO(1) kmand atemporal resolution ofO(10) min. That said, our results are

FIG. 10. (a) Theminimum frequency of inertial gravity waves in a front [defined in (1)] for theAtlantis section shown in Fig. 2. (b) Four example ray paths for a wave with frequencyv5 1.13fand a positive vertical wavenumberm. 0. The magnitude of the group velocity normalized byits maximum value on a ray is indicated in color in (b). The contours of constant vmin are insome places nearly parallel to isopycnals, which is the characteristic signature of a slantwisecritical layer where the group velocity goes to zero and ray paths cannot escape. Magentacontours in both panels indicate where vmin5 1.13f and the gray contours in (a) indicate wherevmin 5 f.

94 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

qualitatively similar to a variety of earlier observationsof banded, near-inertial shear interacting with meso-scale and submesoscale flow features. In this sec-tion, we will briefly describe some of the previouslypublished observations of NIWs and discuss how thoseobservations compare to the observations of NIWsdiscussed here.For example, the shear bands (e.g., Figs. 3, 4) appear

qualitatively similar to observations of near-inertialwaves 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 inthree dimensions inside the anticyclonic rings, the ev-idence here suggests that the amplified shear featuresin the NorthWall are rather two-dimensional, confinedin the cross-front and vertical directions but not alongthe front. Furthermore, Kunze et al. (1995) observedenhanced energy dissipation associated with near-inertial shear in the center of a warm-core ring butnot on the edges. Our observations, on the other hand,exhibit strong near-inertial shear at the north edge ofthe Gulf Stream in the same location as the strongestlateral density gradients and strong cyclonic verticalvorticity (see Fig. 2). Our observations further contrastwith those in rings because the observed frequency issuperinertial here (Fig. 7), whereas it is generally as-sumed [or—more rarely—observed, as in Perkins(1976) and Jaimes and Shay (2010)] to be subinertial inanticyclonic rings.

Several studies have also observed elevated near-inertial wave energy in mesoscale and submesoscalefronts. For example, Rainville and Pinkel (2004) ob-served elevated shear variance and coherent structurescharacteristic of near-inertial waves blocked by thestrong vertical vorticity and high effective Coriolis fre-quency feff in the Kuroshio. D’Asaro et al. (2011) ob-served banded shear in a strong submesoscale front inthe Kuroshio and hypothesized that the shear was as-sociated with near-inertial waves, although the wavedynamics were not explored in detail. Kunze andSanford (1984) and Alford et al. (2013) observed strong,coherent, banded, near-inertial shear modulated by thePacific Subtropical Front, where Ro ; Ri21 ; 0.1.Pallàs-Sanz et al. (2016) observed trapped near-inertialwaves at a critical layer at the base of the Loop Currentafter a hurricane.Mied et al. (1987) observed a near-inertialwave packet where an eddy impinged on a mesoscalefrontal jet in theNorthAtlantic subtropical zone. The wavepacketwas confined to a 25-km-wide region in the front andwas associated with coherent patches of elevated tempera-ture variance (Marmorino et al. 1987;Marmorino 1987) andsuperinertial frequencies, similar to our observations.Whittand Thomas (2013) observed banded shear in a;200-km-wide section (with ;10-km resolution) across the winterGulf Stream and noted that the properties of the shearwereconsistent with the spatial structure of a subinertial waveat a slantwise critical layer. Collocated shearmicrostructureprofiles showed enhanced dissipation between 400 and600m in theupper thermocline in the same location as these

FIG. 11. (a) Time series of mixed layer near-inertial velocities calculated using the slab mixed layer model, asdescribed in section 5b. (b) CW and CCW components of the rotary kinetic energy spectrum associated with thetime series in (a).

JANUARY 2018 WH I TT ET AL . 95

shear structures (Inoue et al. 2010), suggesting a link be-tween the shear and enhanced turbulence. None of theabove studies considered the possibility that superinertialwaves may be trapped between filaments of cyclonic ver-tical vorticity, as discussed here.

7. Conclusions

A high-resolution Lagrangian survey of the NorthWall of the Gulf Stream made during strong and vari-able wintertime atmospheric forcing provided theunique, new opportunity to observe the interaction ofNIWs with a current in the submesoscale dynami-cal regime and dominated by cyclonic vorticity. Theageostrophic vertical shear inferred from the surveyhas features characteristic of coherent, downward-propagating NIWs. Namely, the shear is polarized inspace and time, such that it rotates primarily clockwisewith increasing depth and increasing time. Frequencyspectra exhibit a distinct peak in the clockwise rotarycomponent at frequencies 1:7f *v* f that are slightlyhigher than the Coriolis frequency. At some depthsthese clockwise frequencies explain more than 50% ofthe total shear variance measured over 2 days.The high-resolution sections across the front allow us

to check the consistency of the properties of the ob-served near-inertial shear with the theory of NIWspropagating perpendicular to strongly baroclinic cur-rents (Mooers 1975; Whitt and Thomas 2013). In par-ticular, we can estimate the aspect ratio of the observedbanded shear structures and use it in the dispersionrelation of Whitt and Thomas (2013), which is designedfor near-inertial waves in a submesoscale front to obtainfrequency estimates that are independent of the spectralanalysis and contrast the results. The observed consis-tency between the two frequency estimates, both ofwhich are superinertial, motivate an extension to thetheoretical analysis of Whitt and Thomas (2013), whichfocuses on the properties and propagation of subinertialwaves in submesoscale fronts, to explore how super-inertial waves propagate in the North Wall of the GulfStream, where the vertical vorticity is strongly cyclonicand the minimum frequency for inertia–gravity waves isgreater than the local Coriolis frequency. Ray tracing/characteristic paths suggest that the propagation ofsuperinertial wave energy is strongly modified by nar-row filaments of strong, cyclonic vertical vorticity in theNorth Wall of the Gulf Stream. Superinertial waves canbounce between the cyclonic filaments and becometrapped and amplified at slantwise critical layersbetween these filaments, a scenario that has not beenpreviously considered in the literature (see Fig. 10 for anillustration). In addition, a slab mixed layer model

suggests that variable winds are a plausible local sourceof energy for the observed superinertial waves becausethey resonantly force superinertial motions, owing to thecyclonic vorticity of the current. The slab model alsosuggests that large cross-front variations in the cyclonicvertical vorticity cause mixed layer inertial oscillationsto become incoherent on short time and space scales,facilitating more rapid downward propagation of waveenergy between cyclonic vorticity filaments at the NorthWall caused by a submesoscale inertial chimney effect.However, these theoretical results should be viewedwith caution because some of the assumptions used toobtain the theoretical results are not met. In addition,we have not proved that other theories cannot explainthe observations. For example, the balanced flow couldalso be a source of the observed downward-propagatinginternal wave energy (e.g., Danioux et al. 2012; Alfordet al. 2013; Nagai et al. 2015; Barkan et al. 2017;Shakespeare and Hogg 2017), but quantifying the mag-nitude of the energy flux between the balanced flow andthe waves is beyond the scope of this article. Neverthe-less, the consistency between the proposed descriptionof the physics and the observations is encouraging andmotivates further efforts to better constrain the obser-vations and to explore the theoretical physical scenarioin a more realistic context using numerical simulations.Although we have not explicitly explored the link

between the NIWs observed here and microscale tur-bulence, a number of previously published studies havedemonstrated that coherent downward-propagatingNIWs may be associated with enhanced turbulence(e.g., Marmorino et al. 1987; Marmorino 1987; Hebertand Moum 1994; Kunze et al. 1995; Polzin et al. 1996;Alford and Gregg 2001; Inoue et al. 2010). These pre-viously published observations show that coherent near-inertial waves can directly modify turbulence andthat wave–mean flow interactions can facilitate theconcentration, amplification, and ultimate decay ofnear-inertial waves via turbulent dissipation. The ob-servations collected here show that a large fraction ofthe Froude variance is associated with downward-propagating NIWs in the North Wall of the GulfStream. The superposition of this wave shear with thethermal wind shear of the front results in banded regionsof lowRichardson number (e.g., Fig. 3c), which could beassociated with enhanced turbulence. As turbulentmixing may be important for driving watermass transformation at the North Wall of the GulfStream (e.g., Klymak et al. 2016), coherent downward-propagating NIWs may also play an important role inwater mass transformation in the upper pycnocline here.In addition, similar behavior could also occur at otherwestern boundary currents, such as the Kuroshio, where

96 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 48

filaments of strong cyclonic vorticity and near-inertialwaves are prominent features of the circulation (e.g.,Nagai et al. 2009, 2012). However, future work will haveto explore these hypotheses, which are beyond the scopeof this article.

Acknowledgments.The data were obtained during theScalable Lateral Mixing and Coherent Turbulence(LatMix) Department Research Initiative, funded bythe United States Office of Naval Research. DBW andLNT were supported by ONR Grant N00014-09-1-0202andNSFGrantOCE-1260312. DBWwas also supportedby an NSF postdoctoral fellowship OCE-1421125. JMKwas supported by ONR Grant N00014-11-1-0165. EADwas 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 leadauthor (dwhitt@ucar.edu). The authors thank the crew andofficers of the R/V Knorr and R/V Atlantis as well as therest of the LatMix 2012 science team and the staff of theApplied Physics Laboratory at the University of Wash-ington for making this remarkable experiment possible.DBWwould particularly like to thankDaveWinkel for hishelp with the processing of the shipboard ADCP data.

APPENDIX

Algorithm for Computing the Shear Aspect Ratio

To estimate the aspect ratio of the waves, a5 l/m, andinfer their frequency from the dispersion relation [(9)],we use the following procedure:

(i) The coordinate system is rotated into a streamwise/cross-stream (x, y) coordinate system, as describedin section 2. Then, the thermal wind shear

›ug

›z52

1

f

›bg

›y(A1)

is inferred in each section made by the R/V Knorr(e.g., Fig. 3) from the cross-stream buoyancy gra-dient, after additionally smoothing the density fieldr with a 40m (vertical) by 7 km (lateral) averagingfilter, that is, bg 5 2ghri/ro, where the bracketsdenote the filter, g is the gravitational acceleration,and ro is a reference density. Then the streamwiseageostrophic shear ›ua/›z is inferred in each sectionby subtracting this thermal wind shear from theobserved shear. The cross-stream shear is assumedto be entirely ageostrophic, that is, ›ya/›z 5 ›y/›z.

(ii) In each section, the two components of theageostrophic shear as well as the y (cross stream)

and z (vertical) coordinate grids are extracted andnormalized by scaling so that the mean andvariance of all four fields are zero and one,respectively.

(iii) In each section the set of simply connected regionswith shear magnitude greater than a thresholdmagnitude is identified [each region is a set of(y, z) pairs]. The chosen threshold may depend onthe distribution of the shear in a given section, butwe used just one value for the observations (0.5 innormalized shear units).

(iv) One thousand bootstrap samples are created foreach simply connected region.

(v) For each bootstrap sample, the slope of the firstprinciple component of the set of (y, z) pairs wasused as a value for 2a (after rescaling y and z tophysical units). The background flow parameters in(9), including f 2eff, S

2, and N2, were estimated usingaverages over the simply connected ageostrophicshear region for each coefficient.

(vi) If the 95% confidence interval for the bootstrapfrequency estimates in a given region is narrower than0.2f, then the result is considered robust and thefrequency estimate is retained in the histograms inFig. 9.

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