Damping of Inertial Motions through the Radiation of Near-Inertial Waves in a Dipole Vortex in the Iceland Basin

Leif N. Thomas aDepartment of Earth System Science, Stanford University, Stanford, California

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Eric D. Skyllingstad bCollege of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Luc Rainville cApplied Physics Laboratory, University of Washington, Seattle, Washington

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Verena Hormann dScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Luca Centurioni dScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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James N. Moum bCollege of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Olivier Asselin eOuranos, Montreal, Quebec, Canada

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Craig M. Lee cApplied Physics Laboratory, University of Washington, Seattle, Washington

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Abstract

Along with boundary layer turbulence, downward radiation of near-inertial waves (NIWs) damps inertial oscillations (IOs) in the surface ocean; however, the latter can also energize abyssal mixing. Here we present observations made from a dipole vortex in the Iceland Basin where, after the period of direct wind forcing, IOs lost over half their kinetic energy (KE) in two inertial periods to radiation of NIWs with minimal turbulent dissipation of KE. The dipole’s vorticity gradient led to a rapid reduction in the NIW’s lateral wavelength via ζ refraction that was accompanied by isopycnal undulations below the surface mixed layer. Pressure anomalies associated with the undulations were correlated with the NIW’s velocity yielding an energy flux of 310 mW m−2 pointed antiparallel to the vorticity gradient and a downward flux of 1 mW m−2 capable of driving the observed drop in KE. The minimal role of turbulence in the energetics after the IOs had been generated by the winds was confirmed using a large-eddy simulation driven by the observed winds.

Significance Statement

We report direct observational estimates of the vector wave energy flux of a near-inertial wave. The energy flux points from high to low vorticity in the horizontal, consistent with the theory of ζ refraction. The downward energy flux dominates the observed damping of inertial motions over turbulent dissipation and mixing.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Leif Thomas, leift@stanford.edu

Abstract

Along with boundary layer turbulence, downward radiation of near-inertial waves (NIWs) damps inertial oscillations (IOs) in the surface ocean; however, the latter can also energize abyssal mixing. Here we present observations made from a dipole vortex in the Iceland Basin where, after the period of direct wind forcing, IOs lost over half their kinetic energy (KE) in two inertial periods to radiation of NIWs with minimal turbulent dissipation of KE. The dipole’s vorticity gradient led to a rapid reduction in the NIW’s lateral wavelength via ζ refraction that was accompanied by isopycnal undulations below the surface mixed layer. Pressure anomalies associated with the undulations were correlated with the NIW’s velocity yielding an energy flux of 310 mW m−2 pointed antiparallel to the vorticity gradient and a downward flux of 1 mW m−2 capable of driving the observed drop in KE. The minimal role of turbulence in the energetics after the IOs had been generated by the winds was confirmed using a large-eddy simulation driven by the observed winds.

Significance Statement

We report direct observational estimates of the vector wave energy flux of a near-inertial wave. The energy flux points from high to low vorticity in the horizontal, consistent with the theory of ζ refraction. The downward energy flux dominates the observed damping of inertial motions over turbulent dissipation and mixing.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Leif Thomas, leift@stanford.edu

1. Introduction

The maintenance of the abyssal stratification and deep branch of the meridional overturning circulation by small-scale turbulent mixing hinges on an influx of kinetic energy (KE) into the ocean by the tides and the winds, with the latter estimated to contribute around half of the power needed to sustain the turbulence (Munk and Wunsch 1998). Away from equatorial waters, the energy pathway from the wind work at the surface to turbulent dissipation and mixing in the deep has been hypothesized to involve three steps: 1) KE input by the winds to inertial motions in the mixed layer, 2) conversion of inertial motions to downward-propagating near-inertial waves (NIWs), and 3) the generation of turbulence through internal wave breaking in the abyss. Boundary layer turbulence driven by near-inertial shear is thought to be a major contributor to mixed layer deepening (Pollard and Millard 1970; Pollard et al. 1973; Price et al. 1986), with implications for air–sea coupling and the climate system (Jochum et al. 2013). However, these energy losses to boundary layer turbulence can shunt the energy pathway described above and reduce the contribution of wind-driven NIWs to abyssal mixing.

Quantifying boundary and interior mixing associated with wind-driven internal waves begins with understanding the energetics of near-inertial (NI) motions near the sea surface. This energetics involves energy gain via wind-work and losses to dissipation, entrainment, and NIW radiation. Much focus has been placed on quantifying and modeling the losses of NI KE to boundary layer turbulence during the initial acceleration of inertial oscillations by the winds when the mixed layer deepens (Crawford and Large 1996; Skyllingstad et al. 2000; Alford 2020). Less attention has been placed on the later stages when the winds subside and surface NI kinetic energy can be lost to NIW radiation and turbulence in the stratified “transition” layer beneath the mixed layer and above the more weakly stratified water column below. Strong near-inertial shear has been observed in transition layers with Richardson numbers that are order one, suggestive of conditions favorable for shear-driven turbulence (Dohan and Davis 2011; Johnston et al. 2016). Hebert and Moum (1994) directly measured such turbulence in a decaying NIW and attributed a fraction of the KE loss of the wave to turbulent dissipation. Direct measurements of enhanced turbulence associated with NIWs propagating through the thermocline in the Banda Sea made by Alford and Gregg (2001) similarly imply that turbulent dissipation in the stratified interior can significantly damp NIWs. Finally, simulations run with the Price–Weller–Pinkel mixed layer model (Price et al. 1986) suggest that such turbulence in the transition layer plays an important role in damping NI oscillations on short time scales, consistent with diagnostics of NI KE budgets from observations (Plueddemann and Farrar 2006).

On longer time scales, NI energy loss is dominated by NIW radiation. The process is triggered by convergent/divergent NI oscillations in the mixed layer, which induce vertical motions and displace isopycnals in the transition layer. The corresponding pressure and vertical velocity anomalies can be correlated and thus generate a wave energy flux that drains NI KE out of the mixed layer. This is the inertial pumping mechanism envisioned by Gill (1984) and it depends strongly on the lateral scale of the waves. This follows from the expression for the vertical component of the group velocity
cg,zN2|kh|2fm3,
(where N is the buoyancy frequency, f is the Coriolis frequency, and m is the vertical component of the NIWs’ wavevector) that varies quadratically with the waves’ horizontal wavenumber kh. When kh is set by the O(100–1000) km footprint of the winds that initiate inertial motions, the resulting group velocities are small, i.e., O(1100) cm day−1, for typical values of N, f, and m. However, refraction caused by gradients in f or the vertical vorticity of a background flow, ζ¯ (which in the limit of low Rossby number modulates the effective inertial frequency as feff=f+ζ¯/2; Mooers 1975; Kunze 1985) can act to shrink the lateral scale of NIWs and enhance their vertical propagation and energy flux. These gradients in f or ζ¯ set up lateral differences in wave phase since inertial oscillations separated by a short distance from one another oscillate at slightly different frequencies, a process known as β and ζ refraction, respectively (D’Asaro 1989; Young and Ben-Jelloul 1997). Through this refraction, |kh| increases with time at a rate that depends on the gradient in f or feff (D’Asaro et al. 1995; van Meurs 1998), with a concomitant amplification in energy flux. During the process energy is fluxed down the gradient in f or feff (i.e., equatorward or into regions of anticyclonic vorticity).
Theoretical predictions for the drainage of NI KE out of the surface ocean by β and ζ refraction have been made using the formalism of Young and Ben-Jelloul (1997) [e.g., Moehlis and Llewellyn Smith (2001) and Asselin et al. (2020), respectively]. For either type of refraction, a similar scaling for the decay time of the NI KE results:
Tdecay=(fγ2HNI2No2)1/3,
where No is a measure of the buoyancy frequency below the mixed layer, HNI is the depth over which the NI energy initially spans, and γ is the gradient of either f or feff. The theory predicts that the near-surface NI KE decays as an error function and drops by ∼50% over one decay time Tdecay (Moehlis and Llewellyn Smith 2001). Not surprisingly, the time scale Tdecay is shorter the larger the gradient in f or feff. Tdecay also decreases with HNI and No following the dependence of the group velocity (1) on the stratification and m (noting that HNI sets the vertical wavenumbers of the dominant modes).

Observational evidence of the damping of mixed layer NI oscillations by NIW radiation has been inferential, not direct. Studies using surface drifter derived velocities have documented the decay of NI oscillations and described its connection to NIW propagation. The most intensive study undertaken was the Ocean Storms Experiment in the North Pacific (D’Asaro et al. 1995). The experiment documented the generation of mixed layer NI oscillations after the passage of a storm and their decay to background levels over 20 days. The decay was correlated with a downward-propagating NIW beam measured by an array of moorings. Estimates of the wave energy flux were not reported; therefore, a direct connection between NIW radiation and the drop in NI KE could not be made. However, the NI KE decay was correlated with a decrease in the meridional wavenumber of the NIWs consistent with β refraction. In addition, the predicted damping time scale (2) calculated using parameters representative of the Ocean Storms Experiment is 11.5 days, consistent with the observed decay time (Moehlis and Llewellyn Smith 2001).

The scaling of Moehlis and Llewellyn Smith (2001) has also been tested globally by comparing it to decay time scales of NI oscillations estimated from temporal correlation functions of surface drifter velocities (Park et al. 2009). The predicted increase in decay time with latitude due to a decrease in β was observed in most ocean basins except the North Atlantic, with the discrepancy attributed to the meridional variations in stratification and mixed layer depth found in that basin (Park et al. 2009). While the theoretical prediction (2) captured these large-scale trends, it generally overestimated the decay time scale by a factor of 2, suggesting that other more rapid damping mechanisms such as ζ refraction could be at play. Elipot et al. (2010) saw evidence for damping by ζ refraction in a global drifter dataset, observing a decrease in the decay time scale of NI motions with geostrophic vorticity gradient (the latter estimated using satellite altimetry). Another set of observations that provided evidence of damping by ζ refraction came from a profiling float deployed near a front within a mesoscale eddy field in the Bay of Bengal (Johnston et al. 2016). While the float did not measure velocity, it observed fluctuations in density near the inertial frequency that propagated down in the water column as their amplitude near the surface decreased in time. The inferred seven day decay time scale of the near-surface fluctuations was much shorter than Tdecay estimated for β refraction, leading Johnston et al. (2016) to posit that ζ refraction and other mesoscale–NIW interactions contributed to the faster damping that they observed, along with enhanced dissipation and mixing in the transition layer (although they did not have microstructure measurements to show this conclusively).

In this article we report on the rapid damping of near-surface NI motions after a period of direct wind forcing in a dipole vortex in the Iceland Basin that we can quantitatively attribute to NIW radiation associated with ζ refraction. Unlike the previous studies described above, we have the elements needed to estimate wave energy fluxes and hence link the damping to NIW radiation. We also assess NI KE loss terms due to turbulence in the transition layer using a combination of microstructure measurements and large-eddy simulations (LES). The observations of NI motions in the dipole vortex have been presented elsewhere (Thomas et al. 2020) but the focus there was on quantifying the evolution of the horizontal wavenumber of the motions and its relation to the strain and vorticity gradients of the vortex, not on quantifying the energetics, which is the objective here. We find that for the period after the NI motions were generated by the winds, turbulence in the transition layer plays a negligible role in the energetics of the NI motions and that radiation of NIWs into the dipole’s anticyclone can explain the observed decay in NI KE.

2. Methods

a. Survey of the axial jet of a dipole vortex

The measurements were made in the Iceland Basin as part of the Near-Inertial Shear and Kinetic Energy in the North Atlantic experiment (NISKINe), the goal of which is to study the dynamics and energetics of NIWs, with a focus on the higher modes, and their interaction with mesoscale and submesoscale flows. The observations presented here are from two surveys made from the R/V Neil Armstrong taken near the axis of a dipole vortex (Fig. 1a). The location of the surveys was selected using satellite altimetry to find a site where the gradient in vorticity was large and ζ refraction was likely to be effective. The first survey (orange line in Figs. 1a,c) straddled the axial jet of the dipole, while the second survey (purple line in Fig. 1a) involved transects between the dipole’s jet and anticyclone. A given location on the surveys was reoccupied typically every 3–4 h and hence the measurements were synoptic with respect to the local inertial period (14.2 h). The observations were timed with the passage of an atmospheric front during Survey 1, which will be referred to as the wind event, with a peak wind stress of 0.7 N m−2 at 0000 UTC 30 May 2019. Weaker yet more steady winds followed during Survey 2 (Fig. 1b).

Fig. 1.
Fig. 1.

Overview of the surveys in the Iceland Basin used to study the damping of NI KE by NIW radiation associated with ζ-refraction. (a) The two surveys (ship tracks in orange and purple) were made in a dipole vortex with surface geostrophic velocity and sea surface height anomaly indicated by the vectors and contours, respectively. A transect of the vertical shear of the meridional velocity (colors) made during Survey 2 and the path (green line) of the underway CTD transect made prior to Survey 1 are also shown. (b) Time series of the zonal (black) and meridional (gray) components of the wind stress during the two surveys whose duration is indicated by the colored shading. Time is in days in 2019 on the bottom axis and inertial periods from the start of the wind event (i.e., 0000 UTC 30 May) on the upper axis. (c) Map of the vertical vorticity of the horizontally nondivergent component of the near-surface flow normalized by f. The red star indicates the location SE where the time series in (d) were made. The direction of the horizontal component of the wave energy flux at this location and another to the northeast (NE) (indicated by the blue star) is denoted by the red arrows, and the range of energy flux orientations within the 95% confidence interval is indicated by the gray arrows. Two drifter tracks spanning from the time they were deployed on Survey 1 to ∼3 inertial periods later are indicated by the solid and dotted black lines. (d) Hovmöller diagram of ∂υi/∂z (colors) and isopycnals (contoured every 0.05 kg m−3 with the thick contour denoting the 27.15 kg m−3 isopycnal) at SE. The black triangles indicate the times when SE was occupied.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

The main observations used in the analysis consist of shipboard velocity measurements and hydrography and microstructure from a towed profiler. Near-surface velocities inferred from Lagrangian drifters drogued at 15-m depth that were deployed just prior or during Survey 1 were also used to characterize the background flow and inertial motions; note that the drifters featured here had their drogue attached throughout the study period. An additional underway CTD transect crossing the dipole (green line in in Fig. 1a) was made before Survey 1 and provides a characterization of the stratification prior to the wind event. Velocity profiles were made with a 150-kHz underway ADCP in the upper 400 m at a resolution of 8 m. A Triaxus-towed, undulating profiler collected measurements from the R/V Neil Armstrong through part of Survey 1 and for all of Survey 2. The Triaxus began profiling 0930 UTC 31 May when the winds and waves of the wind event had weakened enough for the profiler to be deployed safely. Triaxus profiled from the sea surface to 170-m depth at vertical speeds of 0.8–1.0 m s−1 and typical tow speeds of 4–8 kt (1 kt ≈ 0.51 m s−1). The profiler carried an extensive payload of physical and bio-optical sensors, including a Seabird SBE 9 plus CTD equipped with dual, pumped temperature (SBE 3plus) and conductivity (SBE 4C) sensors sampled at 24 Hz. A GusT probe (Becherer et al. 2020) attached to Triaxus was used to measure the temperature and velocity microstructure of flow undisturbed by the instrument package and from which the dissipation rate of turbulence KE (ϵ) was estimated.

b. Large-eddy simulations

To aid the interpretation of the microstructure measurements and to quantify the NI KE losses due to turbulent dissipation and entrainment, LES experiments were conducted over a 5-day period starting on 28 May 2019 using the LES model described in Skyllingstad et al. (1999). A horizontally periodic domain with dimensions 720 × 720 and 100 levels was used with rigid upper and lower boundaries and uniform grid spacing of 0.75 m. Model initial conditions were prescribed using an average profile of temperature and salinity measured using the underway CTD during the transect made prior to Survey 1. Surface forcing of wind stress and buoyancy flux was obtained from a coupled mesoscale atmosphere–ocean simulation using the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) model (Warner et al. 2010) that was performed over the time period of the NISKINe field project. For the LES case, the simulation was initiated on 28 May 2019, to allow for a boundary layer spinup period in advance of the storm that generates the inertial response examined here. We found that using the COAWST forcing data yielded a more accurate boundary layer current structure in comparison with comparable forcing data from the R/V Neil Armstrong, which did not arrive at the survey site until 29 May 2019. The LES does not simulate ζ refraction because there is no background flow in the model (neither implemented as an initial condition, nor included as additional advection terms in the equations of motion). As such, the LES provides a “control run” without NI radiative damping by ζ refraction to contrast to the observations.

3. Results

a. Overview of observations

The wind event on 30 May generated inertial motions in the upper ∼50 m as evident in the vertical shear of the meridional velocity at the southeast corner (SE) of Survey 1 (Fig. 1d). The inertial shear extends into a stratified, transition layer where the square of the buoyancy frequency N2 is order 1 × 10−4 s−2 (Fig. 2). Beneath this layer the flow was nearly barotropic. This nearly barotropic flow was subtracted from velocity profiles to isolate the inertial response with velocity (ui, υi) (Thomas et al. 2020). Inertial velocities from drifters were quantified using a different method. The background flow was estimated by fitting a fourth-order polynomial to a drifter velocity time series over the duration of Survey 1 and the inertial motions were inferred from the residual velocity.

Fig. 2.
Fig. 2.

Hovmöller diagram of the logarithm of the square of the buoyancy frequency N2 at SE (colors) and isopycnals (contoured every 0.5 kg m−3 with the thick contour denoting the 27.15 kg m−3 isopycnal).

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

To quantify the lateral gradients in the background flow, the shipboard ADCP and drifter velocities were used to construct maps of the horizontally nondivergent component of the velocity field [see the supporting information of Thomas et al. (2020) for more details on the methodology], a proxy for the background flow (u¯,υ¯). Within the dipole, the vertical component of the vorticity calculated from the background flow, ζ¯=υ¯xu¯y, was characterized by variations of order 0.1f distributed over distances of ∼10 km, yielding a gradient in vorticity > 100β (Fig. 1c; see Thomas et al. 2020).

Thomas et al. (2020) showed that the vorticity gradient of the background flow generated differences in phase in the inertial motions across the axial jet of the dipole. The growth in horizontal wavenumber inferred from the gradient in phase was consistent with ζ refraction, yielding a wavelength of ∼40 km in ∼2.5 inertial periods after the wind event. Evidence of the expected, subsequent vertical propagation of NIWs was seen on Survey 2 where a NIW beam, evident in the vertical shear (Fig. 1a), was radiating down and into the anticyclone of the dipole. Here we report on additional evidence of NIW radiation using the observations from Survey 1, which show a decay in NI KE accompanied by weak turbulence and inertially undulating isopycnals in the transition layer.

b. Observed damping of near-inertial motions

To estimate the NI KE, inertial velocities were averaged in the vertical
(ui¯z,υi¯z)1(zozt)ztzo(ui,υi)dz
from the nominal depth of the transition layer, zt = −50 m, to the depth where there were ADCP data, zo = −20 m. The resulting time series from all locations on Survey 1 is characterized by a clean, near-inertial signal (Fig. 3a). Inertial velocities from two drifters deployed on opposite sides of the axial jet (e.g., the solid and dotted black lines in Fig. 1c) have a similar phasing and amplitude to those inferred from the ADCP early in the record, but by ∼2.5 inertial periods when the drifters had moved to the northeast of the ship, the inertial motions measured by the drifters weaken while those measured by the ship retain their initial strength (Fig. 3a).
Fig. 3.
Fig. 3.

NI velocities in the surface layer from the observations and LES and their corresponding KE. (a) Zonal velocity of the inertial motions averaged over the upper 50 m from the shipboard ADCP on Survey 1, ui¯z, (gray dots) and the LES (blue line). Sinusoids oscillating at f fit to ui¯z every half an inertial period are indicated by the curves of various colors with the 95% confidence interval given by the error bars. Zonal velocity of the inertial motions from drifters 1 and 2 are indicated by the circles and “x” marks, respectively. (b) The NI KE per unit volume, KEi, calculated from the amplitude of the fits (black stars), the LES (blue stars), and the NI velocities from the tracks of drifter 1 and 2 (dashed black and gray lines, respectively, with the error bars indicating the 95% confidence interval). The shading indicates the drop in energy (within the 95% confidence interval) starting at 2.5 inertial periods that would arise from vertical radiation of NIWs using the vertical energy flux estimates paw¯t made at locations SE (light red) and NE (light blue). The theoretical prediction for the decay in NI KE by ζ refraction (6) is indicated by the blue line.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

Velocities from the LES averaged over the upper 50 m closely match the phasing and amplitude of the inertial motions measured by the ship early in the time series, but differences become apparent after three inertial periods. The amplitude of the inertial motions was quantified by fitting sinusoids to the velocity of the form (uf, υf) = [uo cos(ft + ϕu), υo cos(ft + ϕυ)] every half an inertial period (e.g., colored lines with error bars in Fig. 3a). This method was also applied to the drifter velocities but the fits are not shown in Fig. 3a. The amplitudes of the velocity components (uo, υo) were used to estimate the NI KE as KEi=ρo(uo2+υo2)/4, where ρo = 1000 kg m−3 is a reference density (Fig. 3b). While the NI KE in the LES remains fairly constant (except for a slight increase after 5 inertial periods due to a strengthening in the winds, see Fig. 1b), the NI KE estimated from the ADCP velocities shows a marked decrease, dropping by 2.5 J m−3 between 3 and 5 inertial periods after the wind event. The NI KE along the drifter trajectories experiences a similar drop, but over shorter time scales.

The discrepancy between the observed and simulated NI KE highlights how there is some damping mechanism or mechanisms that are not captured by the LES. We posit two hypotheses to explain the discrepancy. One, NI KE loss due to turbulent dissipation and mixing in the transition layer is too weak in the LES, and two, NIW radiation associated with ζ refraction (which the LES cannot simulate) is responsible for the observed damping. These hypotheses are tested in the next two sections.

c. Turbulence in the transition layer

Individual estimates of ϵ estimated from GusT probe measurements on Triaxus were further averaged into 10 m vertical bins and in time for the periods when the ship was in the proximity of location SE (Fig. 4a). In the stratified transition layer between 2.5 and 5 inertial periods, the magnitude of the dissipation is less than 1 × 10−8 W kg−1 for the majority of the time and does not exceed 1 × 10−7 W kg−1. During this same period, the value of the dissipation in the LES does not exceed 1 × 10−8 W kg−1 [a value similar to what Hebert and Moum (1994) measured in a decaying NIW] for depths greater than 20 m, in line with the observations. The weak dissipation in the transition layer coincides with enhanced inertial shear (Fig. 1d) yet stronger stratification. These two competing factors in setting the conditions for shear-driven turbulence can be quantified by the sign of the reduced shear squared Sred2=(u/z)2+(υ/z)24N2. Between 2.5 and 5 inertial periods Sred2 is negative in the transition layer in both the LES and observations (Figs. 4a,c), suggesting that the flow is stable to shear instabilities, thus providing an explanation for the weak dissipation found there. The similarities in the magnitude and vertical structure of the dissipation in the LES and observations suggests that turbulence in the transition layer cannot explain the damping of NI motions evident in the observations but not in the LES.

Fig. 4.
Fig. 4.

Diagnostics of turbulence in the boundary and transition layers. (a) The reduced shear squared Sred2 (color), dissipation of KE ϵ (magenta bars in W kg−1), and potential density (contoured every 0.5 kg m−3 with the thick contour denoting the 27.15 kg m−3 isopycnal) observed at location SE. The dissipation has been averaged in 10-m bins in the vertical. (b) The reduced shear squared [plotted on the same scale as in (a)] and (c) the logarithm of the dissipation (in W kg−1) from the LES.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

d. Inertial pumping and near-inertial wave radiation

If the decay in the NI KE observed in the surface layer is attributable to NIW radiation, anomalies in pressure should be present and correlated with the NI velocities so as to drive a wave energy flux. Undulations of isopycnals in the transition layer evidence such pressure anomalies.

The vertical position of potential density surfaces, zσ, was found to oscillate over an inertial period. The oscillations are most prominent in the pycnocline near z = −40 m (see, e.g., the 27.15 kg m−3 isopycnal surface in Fig. 1d), and are characterized by isopycnal displacements of around 10 m in the vertical (Fig. 5c). This was evident at the northeastern-most location of the survey (NE) (indicated by the blue star in Fig. 1c), as well as at SE (Fig. 6c). If these displacements are adiabatic and lateral advection of density is negligible, then they can be used to infer the vertical velocity, i.e., w = ∂zσ/∂t. To estimate the near-inertial vertical velocities under these assumptions, sinusoids of the form zf=zocos(ft+ϕσ)+z¯σ were fit to zσ (zo, ϕσ, and z¯σ are fitting parameters) and differentiated to yield wf = −zof sin(ft + ϕσ). The fit was performed using a nonlinear regression from which a 95% confidence interval for the isopycnal position and vertical velocity could be estimated. The method yields vertical velocities of order 1 mm s−1 (Figs. 5c and 6c). These values can be compared to inertial pumping, i.e., the vertical motions driven by convergences and divergences of near-inertial horizontal velocities in the surface boundary layer (Gill 1984). Given a NIO with velocity Ui, horizontal wavenumber kh, and boundary layer depth H, inertial pumping scales as wip = UikhH. For the NI oscillation measured on the survey, Ui = 0.1 m s−1, H = 40 m, and ki = 1.5 × 10−4 rad m−1 (Thomas et al. 2020), yielding inertial pumping, wip = 0.6 mm s−1, that is strong enough to explain the observed near-inertial isopycnal displacements.

Fig. 5.
Fig. 5.

Evidence of NIW radiation at location NE. (a) Temporal variability in the component of the inertial velocity measured parallel to the direction of wave propagation, u, evaluated at location NE (black asterisks) and all along the ship track (gray dots). The pressure anomaly pa evaluated at z = −40 m and at NE is indicated by the red asterisks. Sinusoidal fits to the observed pa and u are also shown (red and black lines, respectively) with the 95% confidence interval given by the error bars. (b) The temporal correlation of the pressure anomaly and the component of the inertial velocity in the direction θ, i.e., pf(ufcosθ+υfsinθ)¯t, as a function of that angle, with the 95% confidence interval given by the error bars. The direction where the correlation is maximum, i.e., the inferred direction of NIW propagation, is indicated by the red arrow in Fig. 1c. The range of the NIW propagation directions within the 95% confidence interval is indicated by the gray arrows in Fig. 1c and the red dashed lines in (b). (c) Vertical position of the σθ = 27.15 kg m−3 isopycnal, zσ, at NE (black asterisks) and its vertical velocity w = ∂zσ/∂t (blue line) calculated from a sinusoidal fit to zσ, zf (black line). The error bars denote the 95% confidence interval. (d) Histogram of the vertical component of the wave energy flux pfwf¯t at the mean depth of the σθ = 27.15 kg m−3 isopycnal estimated using the bootstrap method described in the appendix. The mean, upper, and lower endpoint values at the 95% confidence level (indicated by the solid and dashed red lines) are −0.96, −0.33, and −1.6 mW m−2, respectively.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

Fig. 6.
Fig. 6.

Evidence of NIW radiation at location SE. The variables plotted are identical to those shown in Fig. 5. The mean, upper, and lower endpoint values at the 95% confidence level of the vertical energy flux in (d) (indicated by the solid and dashed red lines) are −0.43, −0.17, and −0.68 mW m−2, respectively.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

Pressure anomalies associated with the isopycnal displacements are quantified by integrating the hydrostatic balance
pa=Dzρgdz,
where ρ is the density, g is the acceleration due to gravity, and it has been assumed that there are no pressure anomalies at a depth D. Here D is set to the base of the Triaxus profiles, 160 m, with the justification that the NIWs generated by the wind event have not had enough time to reach this depth over the short duration of Survey 1 (∼2 days) given their slow vertical group velocity (∼30 m day−1; e.g., Thomas et al. 2020).
The pressure anomaly (4) evaluated in the transition layer at NE oscillates over an inertial period and is anticorrelated with the isopycnal displacements (cf. Figs. 5a,c). The oscillations have an amplitude po = 9 N m−2 inferred by fitting a sinusoid, i.e., pf = po cos(ft + ϕp), to pa. If associated with a propagating NIW, these pressure anomalies should be correlated with near-inertial velocities, and in particular, the correlation should be maximum for the component of the velocity that runs parallel to NIW propagation, u. This fact can be used to infer the direction of NIW propagation. To this end, the correlation of pf with the sinusoidal fits to the velocity of the inertial motions (uf, υf)
pf(ufcosθ+υfsinθ)¯t
[where ()¯t denotes an average over an inertial period] was maximized with respect to the angle θ. The correlation is maximum for an angle θprop = 6.5° (e.g., Fig. 5b). The range of NIW propagation directions within the 95% confidence interval (estimated using a bootstrap method described in the appendix) is between −6° and 19° and points from high to low vertical vorticity, consistent with the theory of ζ refraction (Fig. 1c). The NIW propagation direction inferred at SE using the same method is similarly oriented nominally antiparallel to the vorticity gradient (Figs. 1c and 6b).

By definition, the component of the inertial velocity parallel to wave propagation, uui cosθprop + υi sinθprop, fluctuates in phase with the pressure anomaly and thus fluxes NIW energy horizontally (Fig. 5a). The amplitude of this energy flux, Fe,h, can be estimated by fitting a sinusoid to u, u‖,f = u‖,o cos(ft + ϕp), and calculating the correlation (5), yielding Fe,h = 0.5(pou‖,o) = 310 mW m−2. Assuming that the ratio of the vertical to horizontal components of the wave energy flux scales as khH following the continuity equation, the magnitude of the vertical component of the energy flux, Fe,υ, should be of order 1 mW m−2.

A more direct estimate of the vertical component of the wave energy flux can be made by calculating the correlation of the wave pressure anomaly and the vertical velocity inferred from isopycnal displacements: Fe,υ=pfwf¯t. A histogram of the vertical energy flux estimated at NE using the bootstrap method described in the appendix yields a vertical energy flux that is most likely downward with mean, upper, and lower endpoint values at the 95% confidence level of −0.96, −0.33, and −1.6 mW m−2, respectively (Fig. 5d). These values are entirely in line with the estimate above made using the horizontal energy flux and are large enough to explain the observed drop in NI KE on Survey 1. To illustrate this, the upper and lower endpoint values of Fe,υ were divided by the depth of the transition layer (50 m) and integrated in time from 2.5 to 5 inertial periods (i.e., the period over which the correlation pfwf¯t was calculated) to yield the range of changes in NI KE that would result from the vertical radiation of NIW energy out of the transition layer at NE (e.g., the light blue shaded area in Fig. 3b). A similar calculation was performed using the estimates of Fe,υ at SE (e.g., the light red shaded area in Fig. 3b). The reduction in NI KE observed on Survey 1 falls squarely within the range of radiative energy loss at NE and is slightly larger than the loss at SE.

4. Discussion and conclusions

While other observational studies have calculated the vertical energy flux using estimates of the wave group velocity and energy density (Kunze et al. 1995; Alford et al. 2012; Johnston et al. 2016; Sanford et al. 2021), the monochromatic nature of the NIW described here combined with the fact that we captured the wave early in its evolution before it had time to propagate too deep permitted the direct calculation of the energy flux as Fe,υ=pfwf¯t. The inferred O(1) mW m−2 vertical wave energy flux is strong enough to explain the ∼2.5 J m−3 decrease in NI KE over two inertial periods seen on Survey 1. In contrast, the turbulence observed in the transition layer, with dissipation rates of 1 × 10−8 W kg−1 or less would require more than five inertial periods to damp the NI KE by this amount (assuming that the turbulence derives its KE exclusively from the inertial shear), implying that turbulence plays a secondary role in the energetics of the NIWs in the postforcing phase that our analysis focused on. In comparison, Hebert and Moum (1994) estimated that 14%–44% of the decrease in KE of the NIW that they observed over four days (five inertial periods) could be explained by turbulent dissipation. The dominance of radiation over turbulent dissipation in the energetics of the NI motions in our study likely has more to do with the postforcing timing of the observations rather than the strength of the stratification in the transition layer (N2 ∼ 1 × 10−4 s−2), which is not atypical of the stratification in other transition layers (e.g., Dohan and Davis 2011).

Commenting further on the generality of our findings, the vorticity gradients that we observed (which are ultimately responsible for the intensified NIW radiation) are not unusually strong, but are typical of mesoscale eddies with Rossby numbers of order 0.1 or less. It should be highlighted that the mesoscale eddies that we observed are quite barotropic. This does not detract, however, from the generalizability of our findings. While baroclinity can significantly affect the propagation and properties of NIWs in the interior (e.g., Whitt and Thomas 2013), it is likely of secondary importance relative to vertical vorticity in modifying near-inertial motions in the boundary layer and enhancing the downward radiation of NI energy away from the surface. For example, a study of NIW–front interactions targeting the sharp fronts associated with the Mississippi/Atchafalaya River plume in the northern Gulf of Mexico shows that ζ refraction plays a critical role in NIW radiation in these strongly baroclinic currents as well (Qu et al. 2021). Therefore, if the focus is on NI motions in the surface boundary and transition layers, results from a study made in a barotropic background current, like ours, should be relevant to the energetics of NI motions in the boundary/transition layers of baroclinic flows as well.

The rapid decay time of NI KE in the dipole can be compared to the theoretical scaling (2) using an analytical solution for the radiative decay of a NI oscillation in a slab layer bounded below by a fluid with a uniform buoyancy frequency No, in a background flow with a uniform vorticity gradient, and initialized at time to with a kinetic energy KEo:
KE=KEo|erfc{(1+i)23[(tto)Tdecay]3/2}|2tto
(Moehlis and Llewellyn Smith 2001; Asselin et al. 2020). Using parameters representative of the observations (γ = 1 × 10−9 m−1 s−1, HNI = 50 m, No2=1×105s2, and to = 0.5 inertial periods) the analytical solution (6) qualitatively captures the measured decay in NI KE (e.g., Fig. 3b), suggesting that the scaling (2) is relevant in this setting. This combined with the fact that the inferred horizontal wave energy flux is directed from high to low vorticity is further evidence that ζ refraction explains the behavior of the observed NIWs.

One feature of the observations that is still open for interpretation is the lateral variability in the NI KE and its decay time, as revealed by the spatial heterogeneity in the drifter-based estimates of these quantities (e.g., Figs. 1c and 3b). For example, the inertial motions measured by both of the drifters appeared to have a faster decay time than those observed by the ship on Survey 1. The former experienced the largest decay between two and three inertial periods when the drifters had moved to the northeast of Survey 1. We therefore speculate that the faster decay could be due to spatial variability in the key parameters that set the decay time (2), such as the vorticity gradient.

The two drifters straddling the axial jet of the dipole also had significantly different NI KE. This in spite of being separated by less than 10 km. In particular, early in the record the drifter on the cyclonic side of the jet measured a NI KE that was less than half the KE of the other drifter. We propose two hypotheses that might explain the spatial variations in NI KE on these scales. One, the depth of the boundary layer varied across the jet allowing for a stronger acceleration of inertial motions by the winds where the layer is thinner. The stratification was weaker and the transition layer was indeed observed to be deeper on the cyclonic side of the jet on the Survey 1 (e.g., Fig. 7), which could have yielded weaker wind-driven inertial motions with less NI KE, consistent with the pattern in the drifter velocities. Alternatively, or in addition, lateral radiation of wave energy from regions of cyclonic to anticylonic vorticity could explain the differences in NI KE across the jet. Simulations run solving the phase-averaged model of Young and Ben Jelloul for NIWs in a quasigeostrophic barotropic flow representative of the observations yet with uniform stratification [see Asselin et al. (2020) for more details] show that lateral radiation can lead to variations in NI KE of 50% over scales 10 km or less in a few inertial periods (e.g., Fig. 8). The rapid evolution and heterogeneity of the NI KE in the simulations results from the combination of strong wave radiation and stirring of vorticity by the barotropic flow.

Fig. 7.
Fig. 7.

Comparison of profiles of potential density across the axial jet of the dipole vortex. (a) Profiles of potential density averaged between 2.5 and 5 inertial periods after the wind event made at the northwest (NW; black), and southeast (SE; red) corners of Survey 1 [black and red stars in (b)]. (b) Map of the vertical vorticity of the horizontally nondivergent component of the near-surface flow normalized by f. The red and black stars indicate the SE and NW locations where the profiles in (a) were made. Two drifter tracks spanning from the time they were deployed on Survey 1 to ∼3 inertial periods are indicated by the black and gray lines.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

Fig. 8.
Fig. 8.

Evolution of the surface NI KE (color) over five inertial periods (IP) in the YBJ-QG simulations. Vorticity contours [−0.2, −0.1, 0, 0.1, 0.2]f are also shown, with dashed (solid) contours indicating regions of anticyclonic (cyclonic) vorticity. The inertial wave is initialized with a uniform eastward velocity of 0.10 m s−1, such that the initial NI KE per unit mass is 0.005 m2 s−2 everywhere at the surface at t = 0. The QG flow, assumed barotropic, is obtained from satellite altimetry refined by in situ velocity from ship-mounted and drifting instruments, and thus is meant to capture the horizontally nondivergent flow during the NISKINe survey. The inertial wave is initially confined to a surface layer of about 30-m depth [Eq. (4) in Asselin et al. 2020], but refraction, and subsequently straining [Fig. 11b in Asselin et al. (2020)] introduce horizontal wave scales and thus allow radiation of NIWs below the mixed layer.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0202.1

The impacts of NIW energy fluxes extend beyond the energetics of the waves themselves by affecting the mean flows in which they propagate and in driving turbulence and mixing. NIWs generate buoyancy anomalies through ζ refraction and in so doing gain potential energy. Assuming wave kinetic energy (wave action) to be conserved in the process, Xie and Vanneste (2015) argue that this wave potential energy gain comes at the expense of the KE in the mean flow, which is drained at a rate proportional to the correlation of the vertical vorticity (normalized by f) with the divergence of the horizontal energy flux (Rocha et al. 2018). Thus, the tendency for NIWs to flux energy out of regions of cyclonic vorticity and into areas of anticyclonic vorticity, as observed in the dipole vortex here (Fig. 1c), implies a KE sink for the mean flow. We can make a rough estimate for the strength of this sink using the values of the wave energy flux and vorticity from the observations. Namely, if the O(100) mW m−2 horizontal energy fluxes were to converge over the ∼10-km lateral scales of the O(0.1f) vorticity variations, then the NIWs would damp the KE of the mean flow at a rate of O(1×109) W kg−1. Since the observational sampling strategy that was employed does not allow us to estimate the lateral scale over which the wave energy flux varies, this estimate for the mean-flow damping rate is speculative. Having said this, global, mesoscale-resolving simulations forced by high-frequency winds show a similar correlation between vorticity and the divergence of the horizontal energy flux of NIWs (e.g., Fig. 8e from Raja et al. 2022), with values that would result in comparable damping rates for the mean flow.

Raja et al. (2022) also calculated the distribution of NIW vertical energy fluxes and found downward energy fluxes out of the upper ocean of O(1) mW m−2 in regions of high mesoscale KE, similar to what we observed. Integrated over the World Ocean, the vertical energy flux out of the upper 500 m yields 0.04 TW in their simulation, suggesting that the type of downward radiation of NIWs that we observed could contribute modestly to driving turbulence in the abyss and maintaining the deep stratification and meridional overturning circulation. Having said this, the simulations of Raja et al. (2022) were run only for a 30-day period during May–June outside of the winter months when the generation of NIW by the winds is strongest. Hence their estimate of the net downward radiation of NIW is likely biased low. Another possible cause for the modest energy flux in their simulations could be dissipative mechanisms that are effective at damping NIWs above 500 m. NIW waves radiated out of the mixed layer via ζ refraction can be trapped in regions of anticyclonic vorticity and encounter critical layers at the base of anticyclones or in fronts (Kunze 1985; Whitt and Thomas 2013). Observational evidence of enhanced turbulence associated with trapped NIWs in regions of anticyclonic vorticity and fronts in the upper ocean has been reported in the literature (Kunze et al. 1995; Inoue et al. 2010; Sanford et al. 2021) and wintertime-enhanced dissipation inferred using global Argo float data has been attributed to such processes (Whalen et al. 2018). Whether the downward-radiated NIWs described here followed a similar fate and were trapped and dissipated in the anticyclone of the dipole vortex, or propagated into the abyss is an open question and one that we plan to address in future studies using the full suite of observations made as part of NISKINe.

Acknowledgments.

We are grateful to the captain and crew of the R/V Neil Armstrong who made the collection of these observations possible. This work was supported by ONR Grants N00014-18-1-2798 (L.N.T.), N00014-18-1-2780 (L.R. and C.M.L.), N00014-18-1-2083 (E.D.S.), N00014-18-1-2788 (J.N.M.), and N00014-18-1-2445 (L.C. and V. H.) under the Near-Inertial Shear and Kinetic Energy in the North Atlantic experiment (NISKINe) Departmental Research Initiative. The drifters used in this study were funded by ONR Grant N00014-17-1-2517 and NOAA Grant NA150AR4320071 “The Global Drifter Program.” Yuan-Zheng Lu performed the initial processing of GusT data. Discussions with Matthew Alford were very helpful. The comments from Tom Farrar and an anonymous reviewer helped to improve the article.

Data availability statement.

The ADCP and drifter data used in the analyses described in this article can be found at http://hdl.handle.net/1773/45636, while the GusT and Triaxus CTD data can be accessed at http://hdl.handle.net/1773/49464.

APPENDIX

Method for Estimating Confidence Intervals on the Near-Inertial Kinetic Energy, Wave Propagation Direction, and Vertical Energy Flux

As described in the main text, sinusoids oscillating at the inertial frequency were fit to the relevant quantities (i.e., the two components of the inertial velocities, the pressure anomaly, and the vertical displacement of the 27.15 kg m−3 isopycnal). The fits were performed using the MATLAB function nlinfit.m, a nonlinear regression made using least squares estimation from which a 95% confidence interval could be estimated (e.g., indicated by the error bars in Figs. 3a and 5a,c). A bootstrap method was then used to infer confidence intervals on the near-inertial kinetic energy (NI KE), wave propagation direction, and the vertical energy flux. The method entailed (i) adding normally distributed random noise with the same 95% confidence intervals of the original fits to the fits themselves to create a new time series, (ii) fitting sinusoids to the new time series with the noise, (iii) obtaining the NI KE, horizontal energy flux, and vertical energy flux from these secondary fits by calculating the relevant correlations, and (iv) repeating the process Nens = 1000 times to fill out histograms of the quantities of interest (e.g., Figs. 5d and 6d for the vertical energy flux). Normal distributions were fit to the histograms of the energy fluxes, while a gamma distribution was fit to the NI KE since it is a positive definite quantity. The confidence intervals on the quantities of interest could then be estimated from the distributions.

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  • Alford, M. H., 2020: Revisiting near-inertial wind work: Slab models, relative stress, and mixed layer deepening. J. Phys. Oceanogr., 50, 31413156, https://doi.org/10.1175/JPO-D-20-0105.1.

    • Search Google Scholar
    • Export Citation
  • Alford, M. H., and M. C. Gregg, 2001: Near-inertial mixing: Modulation of shear, strain and microstructure at low latitude. J. Geophys. Res., 106, 16 94716 968, https://doi.org/10.1029/2000JC000370.

    • Search Google Scholar
    • Export Citation
  • Alford, M. H., M. F. Cronin, and J. M. Klymak, 2012: Annual cycle and depth penetration of wind-generated near-inertial internal waves at ocean station papa in the northeast Pacific. J. Phys. Oceanogr., 42, 889909, https://doi.org/10.1175/JPO-D-11-092.1.

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  • Fig. 1.

    Overview of the surveys in the Iceland Basin used to study the damping of NI KE by NIW radiation associated with ζ-refraction. (a) The two surveys (ship tracks in orange and purple) were made in a dipole vortex with surface geostrophic velocity and sea surface height anomaly indicated by the vectors and contours, respectively. A transect of the vertical shear of the meridional velocity (colors) made during Survey 2 and the path (green line) of the underway CTD transect made prior to Survey 1 are also shown. (b) Time series of the zonal (black) and meridional (gray) components of the wind stress during the two surveys whose duration is indicated by the colored shading. Time is in days in 2019 on the bottom axis and inertial periods from the start of the wind event (i.e., 0000 UTC 30 May) on the upper axis. (c) Map of the vertical vorticity of the horizontally nondivergent component of the near-surface flow normalized by f. The red star indicates the location SE where the time series in (d) were made. The direction of the horizontal component of the wave energy flux at this location and another to the northeast (NE) (indicated by the blue star) is denoted by the red arrows, and the range of energy flux orientations within the 95% confidence interval is indicated by the gray arrows. Two drifter tracks spanning from the time they were deployed on Survey 1 to ∼3 inertial periods later are indicated by the solid and dotted black lines. (d) Hovmöller diagram of ∂υi/∂z (colors) and isopycnals (contoured every 0.05 kg m−3 with the thick contour denoting the 27.15 kg m−3 isopycnal) at SE. The black triangles indicate the times when SE was occupied.

  • Fig. 2.

    Hovmöller diagram of the logarithm of the square of the buoyancy frequency N2 at SE (colors) and isopycnals (contoured every 0.5 kg m−3 with the thick contour denoting the 27.15 kg m−3 isopycnal).

  • Fig. 3.

    NI velocities in the surface layer from the observations and LES and their corresponding KE. (a) Zonal velocity of the inertial motions averaged over the upper 50 m from the shipboard ADCP on Survey 1, ui¯z, (gray dots) and the LES (blue line). Sinusoids oscillating at f fit to ui¯z every half an inertial period are indicated by the curves of various colors with the 95% confidence interval given by the error bars. Zonal velocity of the inertial motions from drifters 1 and 2 are indicated by the circles and “x” marks, respectively. (b) The NI KE per unit volume, KEi, calculated from the amplitude of the fits (black stars), the LES (blue stars), and the NI velocities from the tracks of drifter 1 and 2 (dashed black and gray lines, respectively, with the error bars indicating the 95% confidence interval). The shading indicates the drop in energy (within the 95% confidence interval) starting at 2.5 inertial periods that would arise from vertical radiation of NIWs using the vertical energy flux estimates paw¯t made at locations SE (light red) and NE (light blue). The theoretical prediction for the decay in NI KE by ζ refraction (6) is indicated by the blue line.

  • Fig. 4.

    Diagnostics of turbulence in the boundary and transition layers. (a) The reduced shear squared Sred2 (color), dissipation of KE ϵ (magenta bars in W kg−1), and potential density (contoured every 0.5 kg m−3 with the thick contour denoting the 27.15 kg m−3 isopycnal) observed at location SE. The dissipation has been averaged in 10-m bins in the vertical. (b) The reduced shear squared [plotted on the same scale as in (a)] and (c) the logarithm of the dissipation (in W kg−1) from the LES.

  • Fig. 5.

    Evidence of NIW radiation at location NE. (a) Temporal variability in the component of the inertial velocity measured parallel to the direction of wave propagation, u, evaluated at location NE (black asterisks) and all along the ship track (gray dots). The pressure anomaly pa evaluated at z = −40 m and at NE is indicated by the red asterisks. Sinusoidal fits to the observed pa and u are also shown (red and black lines, respectively) with the 95% confidence interval given by the error bars. (b) The temporal correlation of the pressure anomaly and the component of the inertial velocity in the direction θ, i.e., pf(ufcosθ+υfsinθ)¯t, as a function of that angle, with the 95% confidence interval given by the error bars. The direction where the correlation is maximum, i.e., the inferred direction of NIW propagation, is indicated by the red arrow in Fig. 1c. The range of the NIW propagation directions within the 95% confidence interval is indicated by the gray arrows in Fig. 1c and the red dashed lines in (b). (c) Vertical position of the σθ = 27.15 kg m−3 isopycnal, zσ, at NE (black asterisks) and its vertical velocity w = ∂zσ/∂t (blue line) calculated from a sinusoidal fit to zσ, zf (black line). The error bars denote the 95% confidence interval. (d) Histogram of the vertical component of the wave energy flux pfwf¯t at the mean depth of the σθ = 27.15 kg m−3 isopycnal estimated using the bootstrap method described in the appendix. The mean, upper, and lower endpoint values at the 95% confidence level (indicated by the solid and dashed red lines) are −0.96, −0.33, and −1.6 mW m−2, respectively.

  • Fig. 6.

    Evidence of NIW radiation at location SE. The variables plotted are identical to those shown in Fig. 5. The mean, upper, and lower endpoint values at the 95% confidence level of the vertical energy flux in (d) (indicated by the solid and dashed red lines) are −0.43, −0.17, and −0.68 mW m−2, respectively.

  • Fig. 7.

    Comparison of profiles of potential density across the axial jet of the dipole vortex. (a) Profiles of potential density averaged between 2.5 and 5 inertial periods after the wind event made at the northwest (NW; black), and southeast (SE; red) corners of Survey 1 [black and red stars in (b)]. (b) Map of the vertical vorticity of the horizontally nondivergent component of the near-surface flow normalized by f. The red and black stars indicate the SE and NW locations where the profiles in (a) were made. Two drifter tracks spanning from the time they were deployed on Survey 1 to ∼3 inertial periods are indicated by the black and gray lines.

  • Fig. 8.

    Evolution of the surface NI KE (color) over five inertial periods (IP) in the YBJ-QG simulations. Vorticity contours [−0.2, −0.1, 0, 0.1, 0.2]f are also shown, with dashed (solid) contours indicating regions of anticyclonic (cyclonic) vorticity. The inertial wave is initialized with a uniform eastward velocity of 0.10 m s−1, such that the initial NI KE per unit mass is 0.005 m2 s−2 everywhere at the surface at t = 0. The QG flow, assumed barotropic, is obtained from satellite altimetry refined by in situ velocity from ship-mounted and drifting instruments, and thus is meant to capture the horizontally nondivergent flow during the NISKINe survey. The inertial wave is initially confined to a surface layer of about 30-m depth [Eq. (4) in Asselin et al. 2020], but refraction, and subsequently straining [Fig. 11b in Asselin et al. (2020)] introduce horizontal wave scales and thus allow radiation of NIWs below the mixed layer.

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