Near-Inertial Wave Interactions and Turbulence Production in a Kuroshio Anticyclonic Eddy

Sebastian Essink aApplied Physics Laboratory, University of Washington, Seattle, Washington

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Eric Kunze bNorthWest Research Associates, Redmond, Washington

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Ren-Chieh Lien aApplied Physics Laboratory, University of Washington, Seattle, Washington

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Ryuichiro Inoue cResearch Institute for Global Change, Japan Agency for Marine Earth Science and Technology, Yokosuka, Japan

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Shin-ichi Ito dAtmosphere and Ocean Research Institute, The University of Tokyo, Chiba, Japan

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Abstract

Interactions between near-inertial waves and the balanced eddy field modulate the intensity and location of turbulent dissipation and mixing. Two EM-APEX profiling floats measured near-inertial waves generated by Typhoons Mindulle, 22 August 2016, and Lionrock, 30 August 2016, near the radius of maximum velocity of a mesoscale anticyclonic eddy in the Kuroshio–Oyashio confluence east of Japan. High-vertical-wavenumber near-inertial waves exhibit energy fluxes inward toward eddy center, consistent with wave refraction/reflection at the eddy perimeter. Near-inertial kinetic energy tendencies are nearly two orders of magnitude greater than observed turbulent dissipation rates ε, indicating propagation/advection of wave packets in and out of the measurement windows. Between 50 and 150 m, εO(10−10) W kg−1, more than an order of magnitude weaker than outside the eddy, pointing to near-inertial wave breaking at different depths or eddy radii. Between 150 and 300 m, small-scale inertial-period patches of intense turbulence with near-critical Ri occur where comparable near-inertial and eddy shears are superposed. Three-dimensional ray-tracing simulations show that wave dynamics at the eddy perimeter are controlled by radial gradients in vorticity and Doppler shifting with much weaker contributions from vertical gradients, stratification, and sloping isopycnals. Surface-forced waves are initially refracted downward and inward, consistent with the observed energy flux. A turning-point shadow zone is found in the upper pycnocline, consistent with weak observed dissipation rates. In summary, the geometry of wave–mean flow interaction creates a shadow zone of weaker near-inertial waves and turbulence in the upper part while turning-point reflections amplify wave shear leading to enhanced dissipation rates in the lower part of the eddy.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sebastian Essink, sessink@uw.edu

Abstract

Interactions between near-inertial waves and the balanced eddy field modulate the intensity and location of turbulent dissipation and mixing. Two EM-APEX profiling floats measured near-inertial waves generated by Typhoons Mindulle, 22 August 2016, and Lionrock, 30 August 2016, near the radius of maximum velocity of a mesoscale anticyclonic eddy in the Kuroshio–Oyashio confluence east of Japan. High-vertical-wavenumber near-inertial waves exhibit energy fluxes inward toward eddy center, consistent with wave refraction/reflection at the eddy perimeter. Near-inertial kinetic energy tendencies are nearly two orders of magnitude greater than observed turbulent dissipation rates ε, indicating propagation/advection of wave packets in and out of the measurement windows. Between 50 and 150 m, εO(10−10) W kg−1, more than an order of magnitude weaker than outside the eddy, pointing to near-inertial wave breaking at different depths or eddy radii. Between 150 and 300 m, small-scale inertial-period patches of intense turbulence with near-critical Ri occur where comparable near-inertial and eddy shears are superposed. Three-dimensional ray-tracing simulations show that wave dynamics at the eddy perimeter are controlled by radial gradients in vorticity and Doppler shifting with much weaker contributions from vertical gradients, stratification, and sloping isopycnals. Surface-forced waves are initially refracted downward and inward, consistent with the observed energy flux. A turning-point shadow zone is found in the upper pycnocline, consistent with weak observed dissipation rates. In summary, the geometry of wave–mean flow interaction creates a shadow zone of weaker near-inertial waves and turbulence in the upper part while turning-point reflections amplify wave shear leading to enhanced dissipation rates in the lower part of the eddy.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sebastian Essink, sessink@uw.edu

1. Introduction

The global hotspot for near-inertial wind power is located at 40°N (Alford 2003) just east of Japan and the Kuroshio–Oyashio confluence. During fall and winter, a dozen tropical cyclones, typhoons, and cold-weather outbreaks pass over the region, generating the bulk of global near-inertial power (D’Asaro 1985). A 15%–25% fraction of this wind power radiates equatorward as low-vertical-mode near-inertial waves, implying that the larger fraction dissipates locally (e.g., Furuichi et al. 2008). How and where this fraction is dissipated remains unknown, but the two most likely locales are the surface layer and upper pycnocline.

Storm-forcing generates surface-layer inertial oscillations of large horizontal scale that will not propagate into the pycnocline since the vertical group velocity Cgz=N2kh2/(fkz3) is small, with stratification N2, Coriolis frequency f, horizontal wavenumber kh=(kx2+ky2)1/2, and vertical wavenumber kz. In eddy-free regions, the planetary β effect shortens the motions’ meridional wavelengths 1/ky as Dky/Dt = −β = −∂f/∂y, which allows downward and equatorward propagation (Anderson and Gill 1979; D’Asaro et al. 1995a; D’Asaro 1995; Garrett 2001). Mesoscale vorticity gradients increase horizontal wavenumber magnitudes through gradients of the effective Coriolis frequency fe = [f(f + ζ)]1/2, with eddy vertical vorticity ζ = ∂xV − ∂yU, at least an order of magnitude more quickly to rapidly duct energy from the surface into the pycnocline (Kunze 1985; D’Asaro et al. 1995a; Lee and Niiler 1998; Klein et al. 2004; Young and Ben Jelloul 1998; Sheen et al. 2015; Asselin et al. 2020; Thomas et al. 2020). This mechanism will be particularly active in the strong radial vorticity gradients on the perimeters of cyclones and anticyclones, acting to focus near-inertial waves toward more negative vorticity and away from positive (Kunze 1985). Near-inertial waves in anticyclones with Lagrangian frequencies fe < ωL < f will be trapped, reflecting from lateral turning points on the rim associated with radial vorticity gradients, and stalling at vertical critical layers at the core base due to weakening of anticyclonic vorticity with depth (Kunze 1985; Kunze et al. 1995). Waves with finite horizontal wavelengths will also be affected by mesoscale horizontal advection and Doppler shifting (Olbers 1981). Doppler shifting can enhance vorticity trapping for waves propagating against the flow and reduce or even eliminate it for waves propagating with the flow (Kunze 1985). Near-inertial wave properties can also be modified by sloping isopycnals (vertical shear through thermal wind) (Mooers 1975; Kunze 1985; Whitt and Thomas 2013) but this tends to be a weaker effect unless the eddy gradient Froude number is ∼O(1).

Since eddies focus or eject near-inertial waves, they will impact the distribution of wave breaking and turbulence. Previous microstructure measurements in anticyclones have found turbulence to be reduced in the upper core and enhanced near the core base and edges (Lueck and Osborn 1986; Kunze et al. 1995; Sheen et al. 2015, Qi et al. 2021). Whalen et al. (2018) inferred higher dissipation rates in anticyclones than in cyclones based on a finescale strain parameterization for turbulence to the Argo float dataset. Sanford et al. (2021) examined the evolution of near-inertial waves near the velocity maximum of a Sargasso Sea anticyclone where eddy confluence is strong and may influence wave trapping (Bühler and McIntyre 2005; Polzin 2008). However, microstructure measurements in eddies have tended to focus on strong signals at the base of the core while the eddy perimeter near the radius of maximum velocity is underobserved.

The paper layout is as follows: section 2 describes the float dataset, the Kuroshio–Oyashio confluence, and analysis methods. Section 3 describes the mesoscale anticyclonic eddy and typhoon forcing. Storm-forced generation of surface-layer near-inertial oscillations is described in section 4 and near-inertial wave packet propagation into the pycnocline and interaction with the eddy in section 5. Ray-tracing simulations in a model anticyclonic eddy are performed in section 6. Section 7 describes four distinct layers of turbulence variability and relates these to the near-inertial shear measurements and ray-tracing results. Results are summarized in section 8.

2. Experiment and analysis methods

a. Kuroshio–Oyashio confluence experiment

Three pairs of Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats were deployed in the Kuroshio–Oyashio confluence off the east coast of Japan during August 2016 to measure the upper-ocean response to storm forcing through subsequent fall and winter (Fig. 1a). The region lies just west of the global inertial wind power maximum at 40°N, 160°E–180° (Alford 2003) and is bound by the warm Kuroshio to the south and cold Oyashio to the north. It contains coherent anticyclonic eddies spun off from the Kuroshio and is subject to multiple storms, especially during fall and winter.

Fig. 1.
Fig. 1.

(a) Trajectories of six EM-APEX floats deployed east of Japan during late 2016. Floats 7788 (red) and 7787 (yellow) were trapped on the perimeter of an anticyclone near the radius of maximum and are subject of analysis here. (b) Photograph of an EM-APEX profiling float with Seabird Electronics SBE-41 CTD (T, S, P), two pairs of electromagnetic velocity sensors (u, υ), and a pair of FP07 microstructure temperature probes (χ) mounted on top.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

The float sampling strategy was a compromise between saving battery power to prolong the mission and profiling fast enough to resolve near-inertial wave evolution during and after storms. By monitoring near-real-time weather forecasts from the Japan Meteorological Agency, profiling rates were adjusted before the arrival of storms to profile continuously between surface and target depth, collecting a ∼500-m roundtrip every ∼1.5 h at ∼0.14 m s−1 ascent and descent speeds, fully resolving near-inertial waves. During weak wind forcing, floats were programmed to only profile twice per inertial period to capture inertial wave signals, parking between 600 and 1200 m to save power when not profiling. This approach is effective if the velocity is dominated by balanced and near-inertial motions as here where internal tides are weak.

This analysis will focus on measurements taken by two floats deployed within an anticyclonic eddy (floats 7787 and 7788 in Fig. 1). Float 7788 was trapped in the eddy during its entire 3-month mission while float 7787 escaped after 1 month. Additional shipboard acoustic Doppler current profiler (ADCP), conductivity–temperature–density (CTD), and microstructure measurements aboard R/V Shinsei-maru provide further context of ocean conditions in the vicinity of float trajectories during the deployment cruise.

b. EM-APEX floats

EM-APEX floats (Fig. 1b) measure finescale horizontal velocity (u, υ) with ∼5-m resolution, temperature T, and salinity S with 2–3-m resolution, and microscale temperature-variance dissipation rate χ with 1-m resolution as a function of depth in a semi-Lagrangian frame following the average water movement over the 0–500-m profiling depth range in this experiment. Velocity is inferred from voltage difference across an insulating body due to the electric field induced by ocean currents moving through Earth’s magnetic field (Sanford et al. 2005, 2011). FP07 (https://rocklandscientific.com/) thermistors on each float, as well as deployment of float pairs, allow cross comparison of microstructure measurements to ensure quality of turbulence estimates. While at the sea surface, floats obtain their GPS positions, transmit data, and receive new mission instructions via Iridium satellites. All EM-APEX measurements were quality controlled and compared with neighboring floats and shipboard measurements when possible.

c. EM-APEX float velocity measurements

1) Correction for magnetic declination and unknown v¯*

Magnetic declination is corrected to transform measured float velocity from geomagnetic to geographic coordinates. Float velocity measurements u are relative to an unknown depth-independent velocity offset v¯* that is computed following Lien et al. (2013) as v¯*=uGPS¯u¯, where uGPS¯ is the time-average GPS-inferred velocity from fixes before descending and after surfacing, and u¯=udt/Δt the time-average EM velocity over the pair of descent and ascent profiles between the GPS fixes of duration Δt. The computed v¯* correction of magnitude ∼O(0.1) m s−1 makes velocity u absolute.

2) Near-inertial filtering

The ∼1.5-h profiling period allows measured total horizontal velocity utot to be decomposed into subinertial usub and near-inertial uNI components. Irregular sampling in time is resampled to a 2-h target sampling interval using a piecewise cubic interpolating polynomial. To retrieve the near-inertial component, velocities are bandpass-filtered using a fifth-order Butterworth filter with cutoff frequencies 0.8f and 1.2f centered around the Coriolis frequency f=4πsinφ/Tf, where φ is latitude and inertial period Tf=2π/f about 19 h at 39°N. Diurnal and semidiurnal tidal frequencies are 0.63f and 1.26f, respectively, so are not contaminated in uNI. Near-inertial horizontal kinetic energy is then HKENI=(uNI2+υNI2)/2 and near-inertial vertical shear squared SNI2=(zuNI)2+(zυNI)2.

d. Thermal diffusivity and turbulent dissipation rate

Microscale temperature is recorded by FP07 sensors at 250 Hz and used to compute temperature-variance dissipation rate χ on 10-s (∼1-m) intervals following Lien et al. (2016) from which turbulent kinetic energy (TKE) dissipation rate ε is estimated following Osborn (1980). Frequency spectra are converted to temperature-gradient vertical wavenumber spectra ΦzT(kz) using the floats, ∼0.14 m s−1 ascent speed. The dissipation rate χ is calculated by integrating ΦzT(kz) over vertical wavenumber
χ=6κkminkmax(ΦzTΦnoise)dkz,
where the lower wavenumber bound kmin = 20 rad m−1 is chosen such that the background nonturbulent temperature-gradient variance is not included in the integration, and the upper wavenumber bound kmax = 400 rad m−1 as a compromise between including most of the turbulent temperature-gradient variance and avoiding high-wavenumber noise. κ is the molecular thermal diffusivity and Φnoise the empirical noise spectrum. Lien et al. (2016) demonstrated that float-measured microstructure temperature-gradient spectra resolve the Batchelor roll-off wavenumber so capture most of the turbulent temperature-gradient variance for typical ocean turbulence intensities. Turbulent thermal diffusivity kT (Osborn and Cox 1972) and turbulent dissipation rate ε (Osborn 1980) are then
kT=0.5χzT¯2,ϵ=kTN2Γ,
where zT¯ is the mean vertical temperature gradient, Γ = 0.2 the mixing coefficient (Gregg et al. 2018), and N2=(g/ρ0)(ρ/z) the buoyancy frequency squared. Thermal and diapycnal turbulent diffusivities are assumed identical.

FP07s were deployed in pairs, yielding two estimates per float. Data were omitted when probes failed. Several tests were used to only retain well-behaved vertical wavenumber spectra following in Lien et al. (2016). Bad values were removed using χ, background zT¯, and kT thresholds. Profiles of inferred turbulent dissipation rate ε were validated against three nearby shipboard direct microscale shear profiles during the deployment cruise.

e. Available potential energy

Available potential energy (APE) is the fraction of potential energy available for conversion into kinetic energy. We compute APE associated with near-inertial waves using the linear approximation
APE=12N2ξ˜2,
where ξ˜ is a moving-window near-inertial fit to observed vertical isopycnal displacement. Isopycnal displacements are computed as ρ[x,y,zi+ξ˜(x,y,zi,t),t]=ρb(zi), where ρb(zi) is the 1-week average prestorm density on 2-m depth intervals zi. Isopycnal displacements are demeaned with depth to isolate resolved vertical scales.
Applied to near-inertial waves in the pycnocline, (2.3) compares well with
APE2=zξzg[ρ(z)ρr(z)]dz,
where ρr is the reference density in a state of rest (Holliday and McIntyre 1981; Lamb 2008; Kang and Fringer 2010).

f. Near-inertial wave energy flux

Horizontal and vertical energy fluxes on resolved vertical wavelengths were computed using floats’ synchronous profiles of density and velocity following Kunze et al. (2002). Near-inertial energy fluxes are covariances of near-inertial velocity uNI and near-inertial baroclinic reduced pressure anomaly p˜=P/ρ0, without the need to isolate individual wave packets, along with their wavenumbers and frequencies to quantify group velocities.

Reduced pressure perturbation p˜ is computed by vertically integrating the hydrostatic relation
0=zp˜N2ξ˜,
such that
p˜(z)=z0N¯2(z)ξ˜(z)dz+p˜(0)=z0N¯2(z)ξ˜(z)dz1HH0z0N¯2(z)ξ˜(z)dzdz.
Removing the depth-average reduced pressure perturbation in the last term accounts for the surface pressure perturbation due to near-inertial waves (Kunze et al. 2002; Althaus et al. 2003; Nash et al. 2005). In the present analysis, the depth-average is limited to the floats’ 500-m profiling range so the observed p(z) only captures fluctuations with <500-m vertical scales.

Near-inertial vertical velocities are estimated as wNI=ξ˜/t=ifξ˜, where ξ˜ is estimated from moving near-inertial fits to the isopycnal vertical displacement time series over two inertial periods (Dosser and Rainville 2016). Horizontal velocity components (uNI, υNI) are bandpass-filtered velocity (section 2c). To remove unresolved modes in the 10–500-m measurement range, linear vertical fits are subtracted from all three velocity components and reduced pressure perturbation before computing energy fluxes.

Based on normal vertical modes calculated from historical deep CTD casts (Gill 1984; Eriksen 1988), modes n ≥ 5 are expected to be resolved with the 500-m float profiles while lower-mode variability is not captured. Removal of the linear fits will remove more than 80% of variance of the unresolved modes, n < 5. HYCOM simulations suggest that more than half of the wind-forced near-inertial energy and energy flux will be in modes higher than 5 (Raja et al. 2022). The depth-invariant v¯* correction, although improving energy estimates, does not factor into energy fluxes.

g. Reanalysis wind stress

Wind speed, direction, and wind stress from the reanalysis forecasting model NCEP CFSv2 (Saha et al. 2010; Saha et al. 2014) are used to quantify surface forcing during storms. Reanalysis wind stress and wind speed have 0.25° horizontal and 1-h temporal resolution. Wind-work τu is computed using reanalysis wind stress τ interpolated onto the float trajectories and float-measured near-surface currents u.

3. Background conditions

a. Eddy characteristics

Two EM-APEX floats were deployed in a strong anticyclonic mesoscale eddy on 26 August 2016 (Fig. 2a). Satellite altimetry from AVISO (www.aviso.altimetry.fr) (Le Traon et al. 1998) reveals a ∼120-km-diameter eddy with a minimum sea surface vorticity of −0.3f at eddy center and −0.2f to −0.1f near the float trajectories at the eddy perimeter. Floats circle the eddy center at about 90% of the radius of maximum azimuthal velocity. Vertical profiles of area-averaged vorticity estimated from float velocity measurements using the circulation theorem along float trajectories’ (Fig. 2b) find a nearly homogeneous vorticity layer of −0.25f at 50–200-m depth, below which it weakens rapidly over a depth scale of O(100) m. Near-surface area-averaged vorticity is about −0.2f, in agreement with AVISO. At ∼280-m depth, the float’s area-averaged vorticity equals that near the near surface. The eddy remains anticyclonic over the 500-m-deep sampling. Averaging float measurements along their trajectories, vertical profiles of subinertial shear and stratification along a fixed radius reveal maximum N2 ∼ 2 × 10−4 s−2 and S2 ∼ 10−4 s−2 at about 50 m depth. Local minima occur at 80 and 120 m. The N2 and S2 decrease nearly exponentially below 200-m depth.

Fig. 2.
Fig. 2.

Eddy structure from (a) AVISO sea level anomaly (SLA) derived vertical vorticity with superimposed ship track of the R/V Shinsei-maru (red) and the two float trajectories, 28 Aug–3 Sep 2016 (black and gray dotted). (b) Vertical profiles of time-averaged shear squared S2 and stratification N2 measured by the two floats (left), and area-averaged vorticity (right) computed from floats’ velocity using the circulation on multiple float orbits and from ADCP by averaging vorticity over the longitude range of the eddy. (c) Zonal and (d) meridional velocity sections from shipboard ADCP. (e) One-component vorticity approximation ζ/fυx/f from ADCP meridional velocity measurements. The black vertical lines in (c)–(e) indicate the intersections of the float and shipboard measurements. The yellow cross in (a) marks the float deployment location.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Shipboard ADCP measurements along a zonal section through the eddy are consistent with AVISO and float measurements (Figs. 2c–e). Section-averaged shipboard ADCP-derived vorticity agrees with area-averaged float vorticity (Fig. 2b). There are subsurface velocity maximum and one-component vorticity extremum at 100 m depth with decay below. However, due to the ∼1-day time lag between shipboard ADCP measurements and the nearest float measurements, the exact float position within the eddy relative to the ADCP section is uncertain.

b. Storm forcing

Typhoons Mindulle (category 1) and Lionrock (category 4) with wind speeds exceeding 30 m s−1 and lateral extents of several hundred kilometers, passed over the eddy with the eye of the storms skirting its western edge (Fig. 3). Mindulle traveled on a southwest–northeast trajectory and came within 50 km of the eddy on 22 August, four days prior to the floats’ deployment. Lionrock traveled on a south-southeast–north-northwest trajectory and came closest to the floats on 30 August.

Fig. 3.
Fig. 3.

Weather maps of Typhoons (a) Mindulle on 22 Aug and (b) Lionrock on 30 Aug 2016 east of Japan from 6-hourly CFSv2 reanalysis. Wind speed and direction shown with colored shading and wind barbs (one barb denotes 10 m s−1 winds), respectively. Red dots mark the positions of the typhoons’ eyes at 6-h intervals and red lines show their approximate tracks. Overlaid in blue are the trajectories of the two floats, 28 Aug–3 Sep 2016.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

As propagating low pressure systems, Mindulle and Lionrock generate a near-inertial upper-ocean response due to resonant wind-work on ocean surface currents on the right side of their propagating tracks (e.g., Price 1981, D’Asaro 1985), that is, directly over the eddy. At the time of eddy encounter, Lionrock had higher wind speed (∼30 m s−1), and a more clearly defined maximum velocity core than Mindulle (∼22 m s−1), which had weakened along the Japanese coast (Fig. 3).

4. Near-inertial wave generation

a. Local and remote waves

Profile time series of near-inertial velocity uNI suggest the presence of two near-inertial wave packets (Figs. 4d,e), (i) one at 150–200-m depth, remotely generated by northward-moving Mindulle, and (ii) one in the mixed layer and upper pycnocline, locally generated by Lionrock. Figures 4b and 4c show that the floats circled the eddy in 5 days per cycle. Near-inertial velocity is 5 times smaller than the background eddy velocity.

Fig. 4.
Fig. 4.

(a) Time series of wind stress τ and wind-work τu from CFSv2 reanalysis and the shallowest float velocity u. (b),(c) Profile time series of total zonal utot and meridional υtot velocity. (d),(e) Profile time series of near-inertial zonal uNI and meridional υNI velocity. (f) Profile time series of near-inertial horizontal kinetic energy HKENI. Red arrows in (f) delineate the approximate downward-propagating wave packets generated by Mindulle and Lionrock.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Generation of the Mindulle wave packet was not directly measured. Its generation time is estimated to be 4–5 days prior to the float deployments by assuming a constant downward-propagating speed based on the wave packet’s change in depth with time (Fig. 4f) in agreement with Mindulle’s passage by the eddy’s western perimeter. Generation by the later northward Lionrock was captured by the floats. On 30 August (Fig. 4a), reanalysis wind stress exceeded 1 N m−2 and wind-work 1 W m−2. Near-inertial energy increased rapidly in the ∼80-m-thick surface mixed layer during the first few days after Lionrock’s passage due to resonant generation of inertial oscillations. In contrast to the Mindulle wave packet, this packet is mostly confined to the surface layer and only weakly energizes the upper pycnocline.

While both storm eyes skirt the western side of the eddy, their inertially resonant right-hand sides lie directly over the eddy (Fig. 3). Because Typhoons Mindulle and Lionrock propagate in different directions, near-inertial wave trains generated by these two typhoons should initially also have different propagation directions, i.e., the Mindulle-generated wave train toward the north-northeast and the Lionrock-generated wave train toward the north-northwest. Advection by the eddy and interactions with its vorticity and Doppler shifting are expected to change their propagation directions and properties. Wave propagation in the direction of the flow will tend to reduce the trapping tendency by the vorticity, i.e., expanding the trapping radius or allowing escape (Kunze 1985). This will be further investigated with ray-tracing simulation in section 6.

5. Propagation and interaction

a. Vertical propagation

The descending subsurface Mindulle-generated wave packet is the most prominent signal in near-inertial velocity, with its maximum HKENI propagating from shallower than 200-m depth on 29 August to deeper than 210 m on 1 September (Figs. 4d–f). It is dominated by upward phase propagation, consistent with downward energy propagation. Isotachs of the downward-propagating packet are almost vertical at the beginning but become slanted with time (Figs. 4d,e), consistent with an increasing vertical wavenumber.

Near-inertial phase θu=arctan(υNI/uNI) is back-rotated to reference time tref using the local inertial frequency, such that the back-rotated phase is
θbrθu+f(ttref)
to better allow interpretation of phase variability in time and depth (Fig. 5). Blue–white–red progressing in time between 100- and 300-m depth indicates counterclockwise rotation with time, which signifies a subinertial signal while red–white–blue progressing with depth signifies clockwise rotation with depth and downward energy propagation. Increasing phase by 180° in 4 days (Fig. 5a) implies that the wave train has frequency lower than local f by 0.1f, i.e., fe = 0.9f, consistent with our estimate of fe=f+(ζ/2) from the eddy’s near-surface vorticity ζ = −0.2f.
Fig. 5.
Fig. 5.

Inertially back-rotated (a) velocity phase θu and (b) vertical wavenumber m based on velocity phase. The line plots in (a) are example profiles taken at marked times and depths (black horizontal and vertical lines) in the contour plots. Trending positive in time (blue–white–red) implies subinertial frequencies while trending negative with depth (red–white–blue) implies downward energy propagation. The line plot in (b) is the near-inertial kinetic energy-weighted vertical wavenumber mHKE.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

We estimate vertical wavenumber as mθu/z (Fig. 5b) and compute a near-inertial energy-weighted mean profile mhke(z)=m(z,t)HKE(z,t)dt/HKE(z,t)dt to increase the signal-to-noise ratio (D’Asaro and Perkins 1984). Vertical wavenumber is mostly positive with mean m ∼ 0.005 cpm above 300-m depth, increasing to a maximum of 0.01 cpm (λz = 100 m) at about 300-m depth. Minimum vertical wavenumber occurs just below 300-m depth, suggesting a transition to a different wave regime. Wenzel–Kramer–Brillouin (WKB) scaling predicts increasing near-inertial wave amplitude u and vertical wavenumber m with N (Leaman and Sanford 1975). But WKB effects are much smaller than measured variability (not shown), pointing to influences other than stratification.

The observed enhancement of near-inertial velocity, shear, and vertical wavenumber at 200–300 m depth implies wave amplification which might arise from radial caustics or vertical critical layers due to slowing group velocities or superposition of incident and reflected waves. Group velocities pass through zero at caustics while they approach zero at critical layers. Critical layers have been observed and simulated at the base of eddy cores (Kunze 1986; Kunze et al. 1995; Lelong et al. 2020; Sanford et al. 2021), whereas the turning-point caustics expected on the lateral peripheries of anticyclonic cores have rarely been studied. Ray-tracing simulations in section 6 aim to unravel wave physics near the eddy’s velocity maximum, differentiate critical layers from caustics and compare with float-measured properties.

b. Kinetic and potential energy conversion

To understand the relative roles of radiation and dissipation as the two wave packets transit the floats, we examine the time rate change of horizontal kinetic energy HKE and available potential energy APE.

In the surface mixed layer, HKE increases at rates of O(106) Wkg1, due to energization and decay of near-inertial oscillations as Lionrock passes (Fig. 6). Near-inertial energy dissipates and/or propagates away after 1 September. Surface-layer dissipation rates are ∼O(10−7) W kg−1 (section 7; Fig. 11).

Fig. 6.
Fig. 6.

Profile time series of inertially smoothed (a) near-inertial horizontal kinetic energy HKE, (b) available potential energy APE, and (c) time rates of change HKE/t and (d) APE/t. The black vertical line indicates the approximate arrival of Typhoon Lionrock, which coincides with HKE/t>0 in both surface layer and 200–300 m, though the latter is likely a coincidence.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

At 200–300-m depth in the pycnocline, the descending HKE packet appears on 29 August and disappears on 1–2 September (Fig. 6c) at rates of O(106) Wkg1, only slightly weaker than in the surface mixed layer. The observed time rate of change of HKE in the pycnocline can be due either to a laterally confined Mindulle-generated wave packet propagating in and out of the measurement window (dup/dx)+(dυp/dy) or local turbulent dissipation ε. As discussed in section 7, maximum measured ε in the same depth range is O(108) Wkg1, two orders of magnitude below HKE/t, indicating that turbulent dissipation cannot explain the observed time rate of change. Therefore, propagation or advection of near-inertial wave packets must explain the HKE change.

The time rate of change of APE is one order of magnitude weaker than that of horizontal kinetic energy, as expected for near-inertial waves. It is largest in the mixed layer, has larger vertical scales, and exhibits a similar descent slope and downward-propagating phase as the envelope of near-inertial HKE (Fig. 4f) but is too weak a signal to interpret reliably. Between 200- and 300-m depth, APE/t is out of phase with and trailing HKE/t. The APE/t oscillates with the inertial period, due to HKE–APE exchange arising from horizontal divergence of near-inertial waves. This pattern is not evident in HKE/t, because (i) the HKE exchange is overshadowed by the larger amplitude of HKE/t by radiation or advection and (ii) computation of HKE removes oscillatory phase information between the two velocity components, that is, most of the energy exchange in near-inertial waves is between the two out-of-phase velocities.

c. Inward and downward NI energy fluxes

High-mode horizontal and vertical energy fluxes are computed following the method discussed in section 2. In the upper 200 m, radial energy fluxes are inward toward eddy center (Fig. 8). Azimuthal fluxes are slightly biased cyclonic against the flow and vertical fluxes primarily downward (Figs. 7, 8). Below 200 m, radial fluxes are still predominantly inward, though again are outward in the southwest while vertical fluxes are mostly upward, and azimuthal energy fluxes mostly cyclonic, though both azimuthal directions are seen.

Fig. 7.
Fig. 7.

Inertially smoothed (a) azimuthal, (b) radial, and (c) vertical energy fluxes. Center panels show profiles of the associated time-averaged energy fluxes. Right panels show histograms of all data (blue) and above 200-m depth (yellow). Positive azimuthal fluxes are cyclonic, positive radial fluxes are outward away from the eddy center.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Fig. 8.
Fig. 8.

Vectors of inertially smoothed horizontal energy fluxes (〈up′〉, 〈υ ′p′〉) averaged over (a) 0–50 m (mixed layer), (b) 50–250 m, and (c) 250–500 m. The scale in each panel is shown by a black quiver and color denotes time. Approximate storm track directions of Typhoons Lionrock and Mindulle (Fig. 3) are shown in green.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Azimuthal and radial energy fluxes of O(102) m3 s3 are three orders of magnitude larger than vertical fluxes of O(105) m3 s3 (Fig. 7), as expected for near-inertial waves. Vertical and horizontal group speeds are estimated as Cgh=|up|/HKE with inertial averages 〈⋅〉, and Cgz=wp/HKE, since HKE′ dominates total near-inertial wave energy, HKE′ is computed with the resolved near-inertial velocity yield horizontal group speeds of 1 km day−1 and vertical group speeds of 10 m day−1, in the ballpark of the estimates above. These estimates capture the resolved high-mode components but miss low modes.

Between 200 and 250 m, azimuthal and inward radial energy fluxes increase with depth, while vertical energy fluxes change sign. The strongest vertical gradient in vertical and radial energy fluxes occurs at about 200 m, coinciding with the depth of increasing vertical wavenumber (Fig. 5), shear squared, and turbulent dissipation rate (Fig. 12). Interpretation of this variability will be presented in section 6. Horizontal gradients of energy fluxes are unknown so energy flux divergence cannot be evaluated.

Horizontal energy fluxes are averaged in the surface mixed layer, upper pycnocline, and lower pycnocline (Fig. 8). In all three layers, energy fluxes are southward at the northwestern edge, opposite to the storms’ propagation directions and inward to the eddy center, outward in the southwest. These waves could be trapped by anticyclonic vorticity. Turning-point reflection or refraction inward is expected by the large radial vorticity and velocity gradients (Doppler-shifting) on the eddy perimeter (Kunze 1985; Asselin et al. 2020) and will be explored with ray-tracing simulations in the next section.

6. Ray-tracing simulations

To gain insight into how storm-forced near-inertial wave packets interact with an anticyclone near the eddy’s radius of maximum velocity, ray-tracing simulations of surface-generated near-inertial waves propagating in a model anticyclone were performed. The eddy is modeled as steady Gaussian reduced pressure anomaly P=P0exp[(x2+y2)/r02+z2/z02] (Lelong et al. 2020) in geostrophic balance. Model parameters are chosen to approximate the observed anticyclone (though specific values have little bearing on the dynamical interpretation). An invariant background stratification Nbg = 5 × 10−3 rad s−1 is assumed; the eddy strengthens total stratification N2 at its base and weakens it in the upper part of the eddy core but N2 remains stable. The model anticyclone has surface-intensified vorticity reaching −0.3f in its core and an annulus of +0.04f outside the radius of maximum velocity rVmax (Fig. 9). It has maximal Rossby number |Vh|/f = 0.3 and gradient Froude number |Vz|/N=(ξx2+ξy2)1/2=0.13, where ξ is balanced isopycnal displacement.

Fig. 9.
Fig. 9.

Map of surface vorticity (color) and ray paths (black) with the radius of maximum velocity rVmx, eddy radial length scale r0, and maximum vorticity rZmx marked by the three green dotted circles. Rays escaping the eddy are not displayed. Rays are advected azimuthally around the anticyclone by the flow but with little dynamical impact other than filling the anticyclonic core.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Waves propagate and change their properties following the ray-tracing equations
DkDt=ωE=(kV+ωL)=[kV(i)+fe(ii)+N2kh22fm2(iii)(Bxkfm+Byfm)(iv)],
DxDt=Cg+V,
under the geometrical-optics approximation (Lighthill 1978) where wavevector k = (k, , m), ∇ is a spatial partial derivative operator, invariant Eulerian frequency ωE = ωL + kV, eddy velocity V = (U, V), and the Lagrangian or intrinsic frequency ωL varies as a wave propagates following the approximate dispersion relation for small Rossby number and gradient Froude number (Kunze 1985)
ωL=fe+N2kh22fm2BxkfmByfm,
but is invariant if there are no changes in the Doppler shift kV. The effective Coriolis frequency is fe = f + ζ/2, where ζ is the eddy vorticity. Zonal and meridional eddy buoyancy gradients (Bx, By) can be expressed as isopycnal slopes −N2(ξx, ξy) and are related to vertical shears, and hence eddy gradient Froude numbers, through thermal wind. Lagrangian frequency ωL (term i) tends to fe as kh/m → 0, i.e., group velocity becomes horizontal, and term ii has minimum frequency ωmin=feBh2/(2fN2) for kh/m = ∇hB/N2, that is, when the wavevector is parallel to the total buoyancy gradient ∇B, i.e., group velocity becomes parallel to isopycnals and vanishes. For ωminωLfe, waves’ vertical phase and energy propagation directions are the same, in contrast to typical internal-wave properties for ωLfe. However, diapycnal phase and group velocities are still in opposite directions.

In baroclinic geostrophic shear flow, the wavevector k evolves following (6.1) due to gradients in the

  1. flow (i.e., Doppler shifting kV) (Olbers 1981),

  2. vorticity ζ through fe (Mooers 1975; Kunze 1985; van Meurs 1998; Asselin et al. 2020),

  3. buoyancy frequency N2 (Munk 1981), and

  4. horizontal buoyancy gradients Bx and By (Mooers 1975; Kunze 1985; Whitt and Thomas 2013).

Changes in the wavevector k lead to changes in Eulerian group velocity CgE = ∇kωE = ∇kωL + V and hence wave amplitude and ray path r orientation.

Test waves are initiated on a horizontal grid at the sea surface spanning the model eddy. Initial meridional and vertical wavelengths are 300 km and 50 m, corresponding to passing storms and mixed layer thickness, respectively. Initial Lagrangian frequencies are ∼1.05fe and maximum Doppler shifts ∼±0.2f. Initial propagation is northward, set by the storms’ propagation direction, but strong interactions with the axisymmetric anticyclone eradicate this information as the waves radiate into the pycnocline and are advected around the eddy (Fig. 9). Ray-tracing equations (6.1) and (6.2) are integrated for one year to explore wave evolution in the eddy with a focus on behavior near the velocity maximum and time scales similar to those of the float measurements. Since measured near-inertial wave packets appear to have length scales smaller than the eddy (Figs. 4, 6), modeling them as radial/azimuthal modes (e.g., Kunze and Boss 1998; Llewellyn Smith 1999) is not appropriate, so wavenumbers and ray paths are integrated in Cartesian coordinates. Eulerian frequencies ωE are invariant as expected in a steady background field. Lagrangian frequencies ωL are largely above the effective Coriolis frequency and asymptote to fe with time, while horizontal wavenumbers kh tend to decrease, vertical wavenumbers m increase and group velocities Cg decrease (not shown). Lagrangian frequencies ωmin < ωL < fe are only found near radial turning-point cusps (<1%).

Some waves escape the anticyclone, usually those propagating with the mean flow, so are not considered further. Waves trapped in the negative vorticity core are advected azimuthally by eddy velocities (Fig. 9). Azimuthal advection has little dynamical impact because the eddy gradients affecting wave evolution are only nonzero with radius r and depth z. Thus, results will be presented with all azimuths collapsed onto the radial-depth plane, i.e., cylindrical coordinates.

Near-surface interactions with eddy vorticity and flow gradients preferentially turn radial wavenumbers inward so that trapped waves propagate downward and toward the eddy core’s center (Fig. 10), consistent with float-measured inward energy fluxes (Fig. 8). In contrast to Asselin et al. (2020) and Thomas et al. (2020), Doppler-shifting’s influence is comparable to vorticity in this evolution because the storms have imprinted finite initial horizontal wavenumber.

Fig. 10.
Fig. 10.

Depth–radius sections of vorticity (colored shading as in Fig. 9), isopycnals (green dotted), isotachs (red dotted), and ray paths (black solid). Waves escaping the anticyclone are not displayed. (a) Starting at the surface, rays propagate inward and downward, pass through the eddy center (r = 0, note that they are not reflected at the eddy center but pass through it) to propagate outward until reflected inward at turning-point caustics where ωL = ωmin inside r0. Near the radius of maximum velocity rVmx, interaction geometry results in a turning-point shadow zone in the upper half of the eddy (blue box) and turning points in the lower half [red box inset blown up in (b)]. (b) Turning-point cusps are tilted upward by sloping isopycnals so that radial reflection is parallel to isopycnals [green box inset blown up in (c)]. (c) Outward-radiating waves encounter two caustics. The first (a) lies just inside the upturned turning-point cusp (b), corresponding to radial wavenumber and vertical group velocity passing through zero (changing sign) and ωL passing through fe. The second (b) is the more prominent turning-point cusp at slightly larger radii associated with ωL grazing the minimum frequency ωmin, as well as radial and vertical group velocities changing sign as the wave is reflected inward toward the eddy center.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

The rays pass through the center of the eddy (apparent “reflection” at r = 0 in Fig. 10a) to radiate outward until they are reflected inward by turning points (cusps) near the radius of maximum velocity (Fig. 10a). In the upper half of the eddy near its perimeter, propagation and trapping geometry produce a turning-point shadow zone because of initial inward radiation due to vorticity refraction (Fig. 10a). This is consistent with weak near-inertial shear and turbulence at shallower depths found in the float measurements (section 7, Fig. 11). In the lower half of the eddy near the perimeter, the interaction geometry produces turning points (Fig. 10c) where measured turbulence is elevated (section 7, Fig. 11). Turning points at greater depths occur at larger radii because these waves originate at the surface at larger radii, and hence larger fe and ωL (Fig. 10a).

Fig. 11.
Fig. 11.

Schematic of observed turbulence distribution in the anticyclonic eddy. (a) Storm-generated turbulence in the mixed layer. (b) Layer of reduced turbulence. (c) Patches of intense inertial-period turbulence. (d) Turbulence due to near-inertial wave amplification at turning-point caustics. (e) Enhanced turbulence layer at the base of the eddy (not sampled).

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Trapped outward-propagating rays encounter two caustics where group velocities transiently pass through zero (Figs. 10b,c). The first (A, Fig. 10c) is associated with the radial wavenumber and vertical group velocity passing through zero (changing sign) and ωL passing through fe. Radial propagation continues outward, and the vertical wavenumber remains positive. The second is at slightly larger radii associated with the more visible upturned turning-point cusps (B, Fig. 10c) where the ray path reflects radially inward parallel to isopycnals. Here, the Lagrangian frequency grazes the minimum frequency, and both radial and vertical group velocities change sign (Fig. 10). The vertical wavenumber m remains positive and diapycnal group velocity negative (toward decreasing buoyancy) throughout. The two caustics are in such close proximity compared to horizontal wavelengths as to be indistinguishable.

At both caustics, vorticity gradients are the primary agents of radial wavenumber evolution with Doppler shifting, stratification and isopycnal slopes playing secondary roles (not shown). Based on 10 000 test waves seeded at different positions on the surface and with different initial horizontal and vertical wavenumbers, trapped-wave caustic behavior near the radius of maximum velocity is relatively insensitive to initial wave conditions, including setting the initial zonal and meridional wavenumber to zero to eliminate Doppler shifting. This is because, by the time waves encounter turning points at depth, strong interactions with the axisymmetric eddy have eradicated their initial conditions.

Wave amplification is expected at turning-point caustics both because of (i) transiently diminished group velocities and (ii) superposition of outward incident and inward reflected waves (Bender and Orszag 1978). Therefore, if incident waves are marginally stable, superposition will tend to create unstable shears and turbulent dissipation as measured by the floats in the lower half of the eddy (section 7).

In summary, the geometry of trapping in ray-tracing simulations is consistent with (i) measured inward energy flux (Fig. 8), (ii) a turning-point shadow zone in the upper half of the eddy perimeter (Fig. 10) compatible with weak measured near-inertial shear and turbulence in the upper pycnocline (Figs. 12, 13), and (iii) turning points in the lower half of the eddy perimeter where wave amplification due to superposition of radially incident and reflected waves could lead to elevated wave breaking and turbulence production as observed (section 7). Gradients in vorticity are the primary agents for wavenumber evolution near the radius of velocity maximum, with gradients in Doppler shifting, stratification and isopycnal slopes playing secondary roles. Near-inertial critical-layer stalling [defined as where m → ∞, Cg → 0 with time, as occurs at the base of the ring’s core (Kunze 1986; Kunze et al. 1995)] does not arise near the radius of maximum velocity or over the O(1) month time scales of the float measurements.

Fig. 12.
Fig. 12.

Profile time series of (a) zonal near-inertial vertical shear uz, (b) total near-inertial shear squared S2, (c) buoyancy frequency squared N2, (d) gradient Froude number δN=S/N (δN > 2 unstable), (e) turbulent kinetic energy dissipation rates ε, and (f) turbulent thermal diffusivity KT. The three pairs of vertical red lines demark periods of strong turbulence patches around 200-m depth.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Fig. 13.
Fig. 13.

Time-averaged vertical profiles of total (blue) and high turbulence patches bracketed by red vertical lines in Fig. 12 (orange) of (a) turbulent kinetic energy dissipation rate ε, (b) shear squared S2, (c) buoyancy frequency squared N2, (d) gradient Froude number δN = S/N, (e) vertical wavenumber m, (f) normalized vertical vorticity ζ/f, and (g) normalized vertical vorticity gradient zζ/f. A pair of horizontal dotted lines brackets the 150–300-m depth range of elevated dissipation rate ε in (a), shear squared S2 in (b), and gradient Froude number δN in (d).

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

7. Turbulence production

In the pycnocline, the approximate horizontal near-inertial wave energy budget is
HKEt=εuHKExυHKEyvp
since near-inertial APE is small, where 〈·〉 is a depth and time average, ∇〈vp′〉 the energy flux divergence. Omitted from this budget are spectral transfers in and out of the near-inertial band. Advective fluxes are primarily azimuthal in the direction of eddy velocity. In section 5 we showed that advection and propagation in and out of the measurement window are important to this HKE budget. Energy fluxes were discussed in section 5, but their divergences cannot be quantified using this dataset.

Turbulent kinetic energy dissipation rates ε inferred from float microstructure temperature measurements (section 2d) reveal four distinct turbulent layers near the anticyclone’s perimeter (Figs. 1113) modulated by near-inertial wave propagation, amplification, and breaking of the two recently generated near-inertial wave packets and eddy vertical shear.

a. Mixed layer strong dissipation

Turbulent kinetic energy dissipation rates are highest in the surface mixed layer during storm passage with εO(107) W kg−1 (Figs. 12e, 13a). The float-inferred ε in the surface layer is uncertain because floats do not measure the uppermost 10 m where turbulence is largest. Furthermore, the inference of ε from χ (section 2) has large uncertainties where local parameterizations (Osborn and Cox 1972; Osborn 1980) are not applicable because of surface-forced nonlocal turbulence production (Large et al. 1994), and weak temperature and buoyancy stratification may not allow reliable estimation of turbulent dissipation rates from Eq. (2.2).

b. Layer of suppressed turbulence

Turbulent dissipation rates ε ∼ 10−10 W kg−1 and thermal diffusivities kTO(106) m2 s−1 in 50–150-m depth (Figs. 12e, 13a) are not only the lowest inside the eddy, but more than an order of magnitude lower than turbulence levels at the same depths outside Kuroshio anticyclones (Fig. 14). The low-turbulence layer is consistent with previous findings in Gulf Stream warm-core rings (Lueck and Osborn 1986; Kunze et al. 1995; Joyce et al. 2013) and a deep Southern Ocean anticyclonic eddy (Sheen et al. 2015).

Fig. 14.
Fig. 14.

Vertical averages between the base of mixed layer and 300-m depth of (a) turbulent dissipation rates ε and (b) thermal diffusivities kT along float trajectories in the Kuroshio–Oyashio confluence during 2016 and 2017. Both ε and kT are reduced by more than one order of magnitude inside anticyclonic eddies and increase by more than an order of magnitude upon exiting anticyclones.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Near-inertial shear squared S2 and gradient Froude numbers S/N are also weaker in this depth range (Figs. 13b,d). The ray-tracing simulations of the previous section suggest that the upper eddy perimeter is a turning-point shadow zone for trapped near-inertial waves (Fig. 10a). suggesting that mechanisms for amplifying wave shear are weak in this depth range.

c. Patches of intense turbulence

Small-scale inertial-period turbulence patches with ε ∼ 10−8 W kg−1 and kT ∼ 10−4 m2 s−1 occur in 150–300 m (Figs. 12e, 13, 15). The turbulence patches appear to be associated with the remote wave packet generated by Typhoon Mindulle. They correspond to when near-inertial and subinertial vertical shear are aligned to elevate gradient Froude number above critical (δN > 2), leading to shear instability (Figs. 15a,e,f). Near-inertial waves provide 2–10 times as much shear as the eddy.

Fig. 15.
Fig. 15.

Profile time series between 150- and 250-m depth of (a) turbulent dissipation rate ε, (b) shear squared S2, (c) direction of subinertial shear αzuLOW, (d) direction of inertial shear αzuNI, (e) positive dot product of inertial and subinertial shear ∂zuLOW ⋅ ∂zuNI (gray shading) with turbulent dissipation rate ε superimposed where ε > 10−9 W kg−1 (orange shading). Five turbulent patches are highlighted by boxes in (a) and (e). (f) The two-dimensional histogram of dissipation rate (a) as a function of positive dot product (e). The pink dots show that bin-averaged dissipation rate increases with the shear dot product.

Citation: Journal of Physical Oceanography 52, 11; 10.1175/JPO-D-21-0278.1

Diminished inertial-period stratification N2 also coincides with the inertial-period turbulence patches (Figs. 12, 13, 15). This stratification variability δN2=N2N2¯105s2, where N2¯ is the mean over the entire period, might arise from either near-inertial vertical strain or turbulent mixing. Turbulent mixing δN20T2(γε)/z2dt would take one week to produce the observed stratification anomaly assuming a mixing coefficient γ = 0.2. Furthermore, it would produce anomalies out of, rather than in, phase with the turbulence. Therefore, a mixing explanation can be ruled out in favor of internal-wave vertical strain. Reduced stratification due to near-inertial wave straining in phase with the superposition of near-inertial and subinertial shear enhances gradient Froude numbers so the tendency for shear instability and turbulence production, consistent with the observed strong inertial-period patches.

Since the turbulent patches are generated as a superposition of subinertial and near-inertial shear (Fig. 15), both near-inertial and subinertial energy should be dissipated. Approximate decay time scales are Eeddy/ε450 days for the eddy, where Eeddy is estimated from subinertial HKE and PE in the upper 500 m, and ENIW/ε20 days for near-inertial waves in the 200–250 m. The eddy time scale is likely longer since the eddy contributes only a small fraction of the unstable shear. Compared to typical mesoscale e-folding time scales of ∼224 days, this is likely not a significant sink of eddy energy with patches only occurring intermittently during storm events.

Time-average profiles (i) over the entire record and (ii) over intervals of intense turbulence show that the inertial-period turbulent patches between 210- and 280-m depth (Fig. 15a) coincide with elevated shear squared S2, gradient Froude number δN, vertical vorticity gradient magnitudes |∂ζ/∂z| and vertical wavenumber kz, as well as reduced stratification N2 (Fig. 15). Elevated near-inertial shear at these depths may be due to the superposition of incident and reflected waves at a radial turning point near the eddy perimeter (Fig. 10).

d. Turning-point caustics

Turbulence ε ∼ 10−8 W kg−1 is also enhanced below the inertial-period patches, below 300-m depths (Figs. 12, 13, 15) despite decaying stratification N2 and shear squared S2. This again may be due to near-inertial wave amplification through superposition of incident and reflected waves at a radial turning point near the eddy perimeter (Fig. 10). But at these depths, superposition of near-inertial and eddy vertical shear does not modulate turbulence.

e. Microstructure summary

To summarize the microstructure measurements, while past measurements have reported the strongest concentration of turbulent dissipation in critical layers at the base of the eddy center (Lueck and Osborn 1986; Kunze et al. 1995; Joyce et al. 2013), here, the floats found both reduced and elevated turbulence on the previously undersampled perimeter of a mesoscale anticyclone. Interactions with radial gradients of vorticity are expected to trap near-inertial waves in the anticyclone’s core (section 6). The geometry of trapping produces a turning-point shadow zone in the upper half of the eddy perimeter (Fig. 10a) which appears to coincide with the layer of weak measured turbulence (Fig. 12e). Likewise, radial turning-point reflection in the lower half of the eddy perimeter (Fig. 10a) can explain the elevated turbulence (Figs. 12e, 13, 15) through amplification of near-inertial shear by transiently slow group velocities as well as the superposition of incident and reflected waves. Also found were inertial-period turbulence patches (Fig. 12e) due to periodic alignment of superposed inertial and eddy vertical shears (Fig. 15).

8. Summary

Two EM-APEX floats deployed in the Kuroshio–Oyashio confluence during fall 2016 were trapped near the radius of the velocity maximum of an anticyclonic eddy with minimum vorticity −0.3f and maximum velocity exceeding 1 m s−1 (Figs. 1, 2). Float measurements captured two storm-forced near-inertial wave packets (Figs. 3, 4), one generated remotely by Typhoon Mindulle on 22 August 2016 before the float deployments and the other generated locally by Typhoon Lionrock on 30 August 2016 during float measurements.

Inertially back-rotated phase increases with time at a rate of 180° per 4 days, implying a subinertial near-inertial frequency fe = 0.9f (Fig. 5), which is consistent with the blue shift by surface negative vorticity ζ ∼ −0.2f of the anticyclonic eddy so the waves will be trapped in the anticyclone (Perkins 1976; Kunze 1985) and expected to encounter radial turning points within the eddy perimeter (Kunze 1985). The vertical wavenumber, inferred from the vertical variation of phase, is positive (downward energy propagation) in the upper 300 m and elevated in 150–300-m depth (Fig. 5).

In the pycnocline, the time rate of change of HKE is nearly two orders of magnitude greater than local dissipation rates ε (Fig. 6), suggesting that horizontal passage of laterally confined high-vertical-wavenumber wave packets through the float measurements by lateral propagation or relative advection.

Horizontal energy fluxes are inward toward the eddy center (Figs. 7, 8), suggesting reflection or refraction of northward-propagating near-inertial waves from northern eddy perimeter, consistent with the tendency for turning-point reflection (Fig. 10) and near-inertial refraction toward more negative effective Coriolis frequency fef + ζ/2 (Kunze 1985; van Meurs 1998; Asselin et al. 2020; Thomas et al. 2020).

Ray-tracing simulations in an idealized model eddy find that near-inertial wave behavior near the eddy rim is largely controlled by radial gradients in eddy vorticity and velocity (Doppler shifting), causing trapping in the negative vorticity core through inward refraction and reflection at radial turning points (Figs. 9, 10). Near the radius of velocity maximum, rays are directed inward, consistent with observations (Fig. 8). A turning-point shadow zone in the upper half of the eddy perimeter where trapped waves are absent (Fig. 10a) is consistent with weak measured near-inertial and turbulent signals in the upper pycnocline (Figs. 12, 13). Turning points in the lower half of the eddy perimeter (Fig. 10a) are expected to lead to elevated wave shear and turbulence, as observed (Figs. 12, 13), due to transient slowing of group velocities and the superposition of incident and reflected waves (Bender and Orszag 1978).

Turbulence measured on the anticyclonic eddy perimeter exhibits four distinct layers (Figs. 1215):

  • Surface-layer turbulence is the strongest as a result from the direct surface forcing. It is likely underestimated because of nonlocal surface-forced turbulence (Large et al. 1994) and unreliability of local parameterizations when stratification is weak (Osborn and Cox 1972; Osborn 1980).

  • Between 50 and 150 m, turbulence dissipation rates ε are reduced by more than an order of magnitude, and finescale shear is also weak compared to outside the eddy. Ray tracing predicts that the upper part of the eddy perimeter is a turning-point shadow zone (Fig. 10) so there may be no mechanism to amplify near-inertial shear and turbulence production.

  • Between 150 and 300 m, inertial-period patches of enhanced turbulence have ε ∼ 10−8 W kg−1 and kT ∼ 10−4 m2 s−1 due to superposition of near-inertial and eddy shears, as well as near-inertially reduced stratification, periodically leading to shear instability and wave breaking.

  • Below 300 m, persistent turbulence ε ∼ 10−8 W kg−1 can be explained by radial turning points in the lower half of the eddy perimeter (Fig. 10). Shear amplification is expected near turning points because of transiently vanishing group velocities and the superposition of incident and reflected waves.

This paper described the first detailed measurements of both fine- and microstructure on the rim of an anticyclonic eddy that allow examination of how turning points of recently generated vortex-trapped near-inertial waves amplify shear and turbulence. The geometry of wave trapping (Fig. 10) produces a turning-point shadow zone in the upper part of the eddy perimeter where weak shear and turbulence are found. In the lower part of the eddy perimeter, near-inertial shear and turbulence are amplified by turning-point reflections. Superposition of near-inertial and eddy shear also leads to inertial-period modulation of turbulence. Dissipative time scales for the eddy are predicted to be longer than one year, and the wave packet ∼O(10) days so are significant.

The measurements presented here were in depth–time near the eddy’s radius of maximum velocity. The ray-tracing simulations suggest that more could be learned about turning-point structure, behavior, and turbulence with radial-depth surveys as a potential future area of research.

Acknowledgments.

The authors would like to express their special thanks to the captain and crew of the Japanese R/V Shinsei-maru from which the EM-APEX floats were deployed and to Avery Snyder and John Dunlap for preparation, operation, and mission management of floats. This work is funded by NSF Grant OCE-1459173 and Grant in Aid for Scientific Research on Innovative Areas (Ministry of Education, Culture, Sports, Science and Technology [MEXT] KAKENHI JP15H05818, JP15H05823, and JP15K21710).

Data availability statement.

The EM-APEX float data are available at the University of Washington ResearchWorks Archive (http://hdl.handle.net/1773/48128). The real-time weather forecasts are distributed by the Japan Meteorological Agency (http://www/jma.gp.jp/jma/). Data products from the Climate Forecast System version 2 (CFSv2) can be found on the UCAR/NCAR Research Data Archive (https://rda.ucar.edu/datasets/ds094.0/). Daily AVISO sea level anomaly is distributed by CMEMS (https://resources.marine.copernicus.eu/).

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    • Export Citation
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  • D’Asaro, E. A., and H. Perkins, 1984: A near-inertial internal wave spectrum for the Sargasso Sea in late summer. J. Phys. Oceanogr., 14, 489505, https://doi.org/10.1175/1520-0485(1984)014<0489:ANIIWS>2.0.CO;2.

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  • Dosser, H. V., and L. Rainville, 2016: Dynamics of the changing near-inertial internal, wave field in the Arctic Ocean. J. Phys. Oceanogr., 46, 395415, https://doi.org/10.1175/JPO-D-15-0056.1.

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    • Export Citation
  • Eriksen, C. C., 1988: On wind forcing and observed oceanic wave number spectra. J. Geophys. Res., 93, 49854992, https://doi.org/10.1029/JC093iC05p04985.

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    • Export Citation
  • Furuichi, N., T. Hibiya, and Y. Niwa, 2008: Model‐predicted distribution of wind‐induced internal wave energy in the world’s oceans. J. Geophys. Res., 113, C09034, https://doi.org/10.1029/2008JC004768.

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  • Holliday, D., and M. E. Mcintyre, 1981: On potential energy density in an incompressible, stratified fluid. J. Fluid Mech., 107, 221225, https://doi.org/10.1017/S0022112081001742.

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    • Search Google Scholar
    • Export Citation
  • Kang, D., and O. Fringer, 2010: On the calculation of available potential energy in internal wave fields. J. Phys. Oceanogr., 40, 25392545, https://doi.org/10.1175/2010JPO4497.1.

    • Search Google Scholar
    • Export Citation
  • Klein, P., S. Llewellyn Smith, and G. Lapeyre, 2004 Organization of near-inertial energy by an eddy field. Quart. J. Roy. Meteor. Soc., 130, 11531166, https://doi.org/10.1256/qj.02.231.

  • Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear. J. Phys. Oceanogr., 15, 544565, https://doi.org/10.1175/1520-0485(1985)015<0544:NIWPIG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kunze, E., 1986: The mean and near-inertial velocity fields in a warm-core ring. J. Phys. Oceanogr., 16, 14441461, https://doi.org/10.1175/1520-0485(1986)016<1444:TMANIV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kunze, E., and E. Boss, 1998: A model for vortex-trapped internal waves. J. Phys. Oceanogr., 28, 21042115, https://doi.org/10.1175/1520-0485(1998)028<2104:AMFVTI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kunze, E., R. W. Schmitt, and J. M. Toole, 1995: The energy balance in a warm-core ring’s near-inertial critical layer. J. Phys. Oceanogr., 25, 942957, https://doi.org/10.1175/1520-0485(1995)025<0942:TEBIAW>2.0.CO;2.

  • Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg, 2002: Internal waves in Monterey Submarine Canyon. J. Phys. Oceanogr., 32, 18901913, https://doi.org/10.1175/1520-0485(2002)032<1890:IWIMSC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lamb, K. G., 2008: On the calculation of the available potential energy of an isolated perturbation in a density-stratified fluid. J. Fluid Mech., 597, 415427, https://doi.org/10.1017/S0022112007009743.

    • Search Google Scholar
    • Export Citation
  • Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32, 363–403, https://doi.org/10.1029/94RG01872.

    • Search Google Scholar
    • Export Citation
  • Leaman, K. D., and T. B. Sanford, 1975: Vertical energy propagation of inertial waves: A vector spectral analysis of velocity profiles. J. Geophys. Res., 80, 19751978, https://doi.org/10.1029/JC080i015p01975.

    • Search Google Scholar
    • Export Citation
  • Lee, D.-K., and P. P. Niiler, 1998: The inertial chimney: The near-inertial energy drainage from the ocean surface to the deep layer. J. Geophys. Res., 103, 75797591, https://doi.org/10.1029/97JC03200.

    • Search Google Scholar
    • Export Citation
  • Lelong, M.-P., Y. Cuypers, and P. Bouruet-Aubertot, 2020: Near-inertial energy propagation inside a Mediterranean anticyclonic eddy. J. Phys. Oceanogr., 50, 22712288, https://doi.org/10.1175/JPO-D-19-0211.1.

    • Search Google Scholar
    • Export Citation
  • Le Traon, P. Y., F. Nadal, and N. Ducet, 1998: An improved mapping method of multisatellite altimeter data. J. Atmos. Oceanic Technol., 15, 522534, https://doi.org/10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2.

  • Lien, R.-C., T. B. Sanford, S. Jan, M.-H. Chang, and B. B. Ma, 2013: Internal tides on the East China Sea continental slope. J. Mar. Res., 71, 151185, https://doi.org/10.1357/002224013807343461.

    • Search Google Scholar
    • Export Citation
  • Lien, R.-C., T. B. Sanford, J. A. Carlson, and J. H. Dunlap, 2016: Autonomous microstructure EM-APEX floats. Methods Oceanogr., 17, 282295, https://doi.org/10.1016/j.mio.2016.09.003.

    • Search Google Scholar
    • Export Citation
  • Lighthill, M. J., 1978: Waves in Fluids. Cambridge University Press, 504 pp.

  • Llewellyn Smith, S. G., 1999: Near-inertial oscillations of a barotropic vortex: Trapped modes and time evolution. J. Phys. Oceanogr., 29, 747761, https://doi.org/10.1175/1520-0485(1999)029<0747:NIOOAB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lueck, R., and T. Osborn, 1986: The dissipation of kinetic energy in a warm‐core ring. J. Geophys. Res., 91, 803818, https://doi.org/10.1029/JC091iC01p00803.

    • Search Google Scholar
    • Export Citation
  • Mooers, C. N. K., 1975: Several effects of a baroclinic current on the cross‐stream propagation of inertial‐internal waves. Geophys. Fluid Dyn., 6, 245275, https://doi.org/10.1080/03091927509365797.

    • Search Google Scholar
    • Export Citation
  • Munk, W. H., 1981: Internal waves and small scale processes. The Evolution of Physical Oceanography: Scientific Papers in Honor of Henry Stommel, B. A. Warren and C. Wunsch, Eds., MIT Press, 264291.

  • Nash, J. D., M. H. Alford, and E. Kunze, 2005: Estimating internal wave energy fluxes in the ocean. J. Atmos. Oceanic Technol., 22, 15511570, https://doi.org/10.1175/JTECH1784.1.

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