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  • View in gallery
    Fig. 1.

    (a) Theoretical curves (thick solid lines) of the ratio to counterclockwise-with-time (Pccw) and clockwise-with-time (Pcw) average rotary spectra at 10° (blue), 30° (red), and 60° (yellow) latitude as a function of frequency [Eq. (1)]. Inertial frequencies are indicated by the blue, red, and orange vertical lines. Black dashed vertical lines denote the semidiurnal frequency (ωM2) as well as typical abyssal (N4000m) and upper-ocean (N50m) buoyancy frequencies. Variability in the abyssal and upper-ocean stratification are noted by the gray bars (the standard deviation of climatological stratification at those two depths). (b) Theoretical curves for the vertical asymmetry ratio [Eq. (6)] for a downgoing monochromatic wave at three different wave frequencies (ωM2, ωK1, and ωNMax where Nmax is 0.0145 rad s−1).

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    Fig. 2.

    Profiles of the magnitude of vertical asymmetry, |Ω(z)| for a surface-forced near-inertial wave on a β plane as it propagates downward and equatorward (blue), then reflects to propagate upward and equatorward (red) without dissipation. The black curve corresponds to their sum assuming that wave energy depends inversely on vertical group velocity |cgz| and the rotary ratio as (ω2f2)/(ω2 + f2). Depth z has been WKB-normalized to constant stratification N. Numbers on the upper axis are corresponding frequencies for the vertically propagating near-inertial waves based on Eq. (6).

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    Fig. 3.

    (a) Map of GO-SHIP LADCP [light orange from Kunze et al. (2006); dark orange 2004–19], HDSS (green), 75-kHz JASADCP (blue), and 38-kHz JASADCP (purple) measurements. Total number of profiles from LADCP, HDSS, 75-kHz JASADCP, and 38-kHz JASADCP in 4° bins by (b) latitude and (c) longitude. (d)–(g) Distribution of observations per month from each platform (dark and light bars denote data from Northern and Southern Hemispheres, respectively).

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    Fig. 4.

    Global distribution of 4°-binned depth-averaged (left) total GM76- and buoyancy-frequency-normalized shear variance (ϕccw + ϕcw), and (right) vertical asymmetry ratio Ω [Eq. (6)]. Depth averages span (a),(b) 200–600-m depth (HDSS, JASADCP, and LADCP), (c),(d) 600 m–1000 mab (LADCP), and (e),(f) the bottom 1000 m (LADCP). Integration wavelengths vary depending on instrument (Figs. A2 and A4). Positive Ω (blue) indicates excess downgoing shear variance while negative (red) excess upgoing. Critical latitude for semidiurnal parametric subharmonic instability (M2 PSI) at 28.8° is noted with black dashed line in (a)–(f).

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    Fig. 5.

    Variance of the downgoing (dark) and upgoing (light) GM76- and buoyancy-frequency-normalized normalized rotary shear in 4° latitude bins from the depth-average over (a) 200–600-m depth (HDSS, JASADCP, LADCP), (b) 600 m–1000 mab (LADCP), and (c) below 1000 mab (LADCP). Standard error (σ/n ) is plotted for LADCP variances. Binned shear variances with standard errors greater than twice the observed mean are not plotted [as in (c)]. Vertical lines at 28.8°N and °S denote the latitude at which semidiurnal parametric subharmonic instabilities and western boundary intensification may occur.

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    Fig. 6.

    Histograms of the 4°-binned vertical asymmetry ratio, Ω [Eq. (6)] for 200–600 m, 600 m–1000 mab, and below 1000 mab. When Ω = 0, there is equipartition of down- and upgoing shear variance (vertical dashed line). The mean of the independent observations Ω¯ is statistically greater than zero for 200–600-m depth.

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    Fig. 7.

    Latitude and wavenumber dependence of the vertical asymmetry ratio Ω [Eq. (6)] from 200 to 600 m. Mean Ω vs wavenumber from each of the three platforms: (a) HDSS (green), (b) LADCP (orange), and (c) JASADCP (blue). (d) Arithmetic mean Ω over the wavenumber bands used for integration vs latitude. Distribution of Ω (colors) vs latitude and wavenumber for (e) HDSS, (f) LADCP, and (g) JASADCP. Blue (red) indicates downgoing (upgoing) shear variance. (h) The distribution of measurements for the HDSS (green), LADCP (orange), and JASADCP (blue) with latitude. For JASADCP and HDSS, Ω above 100 and 56 m, respectively, are excluded due to noise (see associated spectra in Fig. A3b).

  • View in gallery
    Fig. 8.

    Global average forcing distribution vs latitude of the global power input from (a) wind, (b) tides plotted as log10 (W m−2) (Waterhouse et al. 2014), and (c) roughness variance (m2) from Whalen et al. (2012) (gray lines). The average power input and roughness variance only from locations sampled by the JASADCP, HDSS and LADCP observations are indicated in blue, green and orange lines, respectively. Associated PDFs of sampling obtained over the global range of (d) wind, (e) tide power inputs, and (f) roughness variance.

  • View in gallery
    Fig. A1.

    Comparison of 1D and 2D spectra of buoyancy-frequency-normalized shear over 4.2 days during 2006 IWAP (Zhao et al. 2010; MacKinnon et al. 2013a; Alford et al. 2017). 2D spectra of buoyancy-normalized shear from (a) MP3 and (b) coincident 10-min averaged HDSS observations. (c) 1D buoyancy-normalized wavenumber spectra of shear computed from the 2D spectra [in (a) and (b)] by integrating in frequency (fm) space from MP3 (black) and HDSS (green). In (c), average 1D HDSS spectrum [as calculated in section 2c(1)] during the 4.2-day occupation by the R/V Revelle are also shown (blue).

  • View in gallery
    Fig. A2.

    Coincident sections of LADCP, HDSS, and JASADCP (a)–(c) zonal and (d)–(f) meridional 10-min averaged shear over 200–600 m from a subset of observations from the 2016 I08S/I09N GO-SHIP Repeat Hydrography Cruise (see Fig. A3).

  • View in gallery
    Fig. A3.

    (a) Map of shear sampling from Fig. A2. (b) Associated average normalized rotary-with-depth shear spectra from LADCP (orange), HDSS (green), and JASADCP (blue) data where light and dark lines of each spectrum indicate counterclockwise-with-depth (ccw) and clockwise-with-depth (cw) rotary components, respectively. Each spectrum has been normalized by the buoyancy frequency, appropriate transfer functions and by the Garrett–Munk (GM76) spectrum (Garrett and Munk 1975) as implemented by Cairns and Williams (1976) at the same stratification (gray line). Integration limits for LADCP, HDSS, and JASADCP are noted with the horizontal bars in (b). Bathymetry contours in (a) are plotted every 2000 m.

  • View in gallery
    Fig. A4.

    Rotary spectra and transfer functions for LADCP, JASADCP, and HDSS profiles extracted from the sample data shown in Figs. A2 and A3. A subset of individual spectra from each shear profile are plotted as thin lines, and means are thick lines. Clockwise (counterclockwise) spectra are plotted in darker (lighter) lines. From top to bottom are the rotary spectra of (a)–(c) measured shear (uz); (d)–(f) buoyancy-frequency-normalized shear (uz/N); (g)–(i) sinc2 transfer functions associated with range averaging (RA), depth binning (BIN), velocity inversion (VI), bin size (DZ), and pulse length (PL; section 2); (j)–(l) buoyancy-frequency-normalized shear normalized by transfer functions (uz/N/T) with integration limits for LADCP, JASADCP, and HDSS noted as the horizontal bars; and (m)–(o) shear normalized by transfer functions with noise spectra (black line). The associated noise (cm s−1) is noted in each subpanel.

  • View in gallery
    Fig. B1.

    Analytical solutions to near-inertial ray paths generated at 45° and z = 0 through depth H = 1200 m on a β plane for vertical modes n = 1–6 (colored lines).

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Global Observations of Rotary-with-Depth Shear Spectra

Amy F. WaterhouseaScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Tyler HennonaScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Eric KunzebNorthWest Research Associates, Redmond, Washington

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Jennifer A. MacKinnonaScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Matthew H. AlfordaScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Robert PinkelaScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Harper SimmonscApplied Physics Laboratory, University of Washington, Seattle, Washington

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Caitlin B. WhalencApplied Physics Laboratory, University of Washington, Seattle, Washington

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Elizabeth C. FineaScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Jody KlymakdUniversity of Victoria, Victoria, British Columbia, Canada

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Julia M. HummoneUniversity of Hawai‘i at Mānoa, Honolulu, Hawaii

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Open access

Abstract

Internal waves are predominantly generated by winds, tide–topography interactions, and balanced flow–topography interactions. Observations of vertical shear of horizontal velocity (uz, υz) from lowered acoustic Doppler current profilers (LADCP) profiles conducted during GO-SHIP hydrographic surveys, as well as vessel-mounted sonars, are used to interpret these signals. Vertical directionality of intermediate-wavenumber [λzO(100) m] internal waves is inferred in this study from rotary-with-depth shears. Total shear variance and vertical asymmetry ratio (Ω), i.e., the normalized difference between downward- and upward-propagating intermediate wavenumber shear variance, where Ω > 0 (<0) indicates excess downgoing (upgoing) shear variance, are calculated for three depth ranges: 200–600 m, 600 m–1000 mab (meters above bottom), and below 1000 mab. Globally, downgoing (clockwise-with-depth in the Northern Hemisphere) exceeds upgoing (counterclockwise-with-depth in the Northern Hemisphere) shear variance by 30% in the upper 600 m of the water column (corresponding to the globally averaged asymmetry ratio of Ω¯=0.13), with a near-equal distribution below 600-m depth (Ω¯0). Downgoing shear variance in the upper water column dominates at all latitudes. There is no statistically significant correlation between the global distribution of Ω and internal wave generation, pointing to an important role for processes that redistribute energy within the internal wave continuum on wavelengths of O(100) m.

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

Hennon’s current affiliation: College of Fisheries and Ocean Sciences, University of Alaska Fairbanks, Fairbanks, Alaska

Corresponding author: Amy F. Waterhouse, awaterhouse@ucsd.edu

Abstract

Internal waves are predominantly generated by winds, tide–topography interactions, and balanced flow–topography interactions. Observations of vertical shear of horizontal velocity (uz, υz) from lowered acoustic Doppler current profilers (LADCP) profiles conducted during GO-SHIP hydrographic surveys, as well as vessel-mounted sonars, are used to interpret these signals. Vertical directionality of intermediate-wavenumber [λzO(100) m] internal waves is inferred in this study from rotary-with-depth shears. Total shear variance and vertical asymmetry ratio (Ω), i.e., the normalized difference between downward- and upward-propagating intermediate wavenumber shear variance, where Ω > 0 (<0) indicates excess downgoing (upgoing) shear variance, are calculated for three depth ranges: 200–600 m, 600 m–1000 mab (meters above bottom), and below 1000 mab. Globally, downgoing (clockwise-with-depth in the Northern Hemisphere) exceeds upgoing (counterclockwise-with-depth in the Northern Hemisphere) shear variance by 30% in the upper 600 m of the water column (corresponding to the globally averaged asymmetry ratio of Ω¯=0.13), with a near-equal distribution below 600-m depth (Ω¯0). Downgoing shear variance in the upper water column dominates at all latitudes. There is no statistically significant correlation between the global distribution of Ω and internal wave generation, pointing to an important role for processes that redistribute energy within the internal wave continuum on wavelengths of O(100) m.

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

Hennon’s current affiliation: College of Fisheries and Ocean Sciences, University of Alaska Fairbanks, Fairbanks, Alaska

Corresponding author: Amy F. Waterhouse, awaterhouse@ucsd.edu

1. Introduction

Internal waves in the ocean are predominantly generated at surface and bottom boundaries, then propagate into the stratified interior. How far these waves propagate before dissipating determines the spatial distribution of the turbulent mixing to which ocean climate models are sensitive (Jochum 2009; De Lavergne et al. 2014; Jochum et al. 2012; Melet et al. 2013a,b, 2016; MacKinnon et al. 2017; Vic et al. 2019; Whalen et al. 2020). Internal waves are thought to be generated by three main mechanisms. Wind forcing generates near-inertial waves which propagate downward from the surface (D’Asaro 1984, 1985; Alford 2001). Tidal currents (Bell 1975; Garrett and Kunze 2007) and geostrophic flows over topography (Bell 1975; Nikurashin and Ferrari 2011; Melet et al. 2016; Scott et al. 2011; Wright et al. 2014) generate internal waves with energy propagating upward (downward if their characteristic slopes are less than topographic slopes; Thorpe 1992). Lower modes that are not dissipated near their generation sites can reflect off topography (Eriksen 1982) or the surface (Althaus et al. 2003; Nash et al. 2006), dissipate at critical layers (Lueck and Osborn 1985; Kunze et al. 1995; Whalen et al. 2018), scatter to higher modes (Müller et al. 1992; Johnston and Merrifield 2003), or cascade to turbulence (McComas and Müller 1981a; Henyey et al. 1986; Gregg 1989).

For the low-frequency internal waves that dominate internal wave energy and shear in the deep ocean (e.g., Müller et al. 1978), upward and downward energy propagation with depth can be diagnosed by separating shear profiles into counterclockwise- and clockwise-with-depth components in the Northern Hemisphere (Leaman and Sanford 1975; Pinkel 1985; Shcherbina et al. 2003; Pinkel 2008a, 2014; Waterman et al. 2014; Takahashi and Hibiya 2019). From a compendium of 1983–2002 shear and strain observations in the upper 680 m of the North Pacific, Pinkel (2014) found that the vertical wavenumber spectrum of clockwise-with-depth shear exceeded the counterclockwise-with-depth shear (and strain) spectra at all wavelengths greater than 10 m. This is at odds with the Garrett–Munk (GM) spectrum (Munk 1981), which assumes vertical symmetry. However, the global distribution of vertical internal wave directionality in the full water column has not been documented. Determining the vertical asymmetry ratio in different ocean basins may help steer future work in closing the budget of wave generation, propagation, and dissipation.

Here, we use vertical shear profiles from full-depth lowered acoustic Doppler current profilers (LADCP) profiles from GO-SHIP repeat hydrography sections [a subset was previously used to compute 150–320-m rotary shear variances by Kunze et al. (2006)], as well as two different shipboard sonar datasets to generate global maps of shear variance and diagnose upward and downward energy propagation from each component of the rotary shear. The first shipboard sonar dataset is from the Hydrographic Doppler Sonar System (HDSS) mounted on the R/V Revelle during 2006–19 with near-continuous data to 800–1000-m depth. The second is from RDI ADCPs during 1994–2016 mounted on U.S. oceanographic research fleet vessels with data to 600–1000-m depth. Details of each sonar dataset are discussed in section 2.

Excess downgoing shear variance is found in the upper water column (200–600 m) while there is equipartition of upward-to-downgoing shear variances over the full water column and bottom kilometer. In the upper water column, downward rotary shear dominates at intermediate wavenumbers (wavelengths of hundreds of meters). The intermediate wavenumber range used here is similar to that used in various finescale parameterizations (Gregg 1989; Gregg and Kunze 1991; Polzin et al. 1995; Gregg et al. 2003; Kunze et al. 2006). Although low modes of wind-forced near-inertial waves may contribute up to half of the near-inertial energy and energy flux (Nash et al. 2004; Alford 2020; Raja et al. 2022), the intermediate wavenumber range contains shear that is more strongly correlated with dissipation rates and mediates the transfer of energy from large to dissipative scales (e.g., Gregg 1989).

The manuscript describes various shipboard measurements used to obtain these results in section 2, followed by the global distribution of rotary-with-depth shear spectra (section 3) with a brief discussion in section 4. Detailed comparisons between example datasets confirm the analysis methods used here (appendix A).

2. Observations

a. Theoretical background of vertically propagating internal waves

Internal waves can propagate vertically in the ocean provided their frequency ω is between N and |f|, where N is the stratification and |f| the absolute value of the Coriolis frequency. In the limit as ω approaches |f|, oscillating motion describes a clockwise circle in time in the Northern Hemisphere (counterclockwise in the Southern Hemisphere), with no vertical propagation. Low-frequency internal waves (ω slightly above |f|) have horizontal velocities that describe a nearly circular ellipse that is clockwise-rotary-with-time in the Northern Hemisphere and counterclockwise-rotary-with-time in the Southern Hemisphere (Mooers 1970; Gonella 1972; Müller et al. 1978; Fu 1981; Alford 2001), where clockwise is defined with the z axis pointing upward. With increasing frequency, the dominant sense of rotation remains the same but ellipticity increases. This ellipticity can be expressed using the ratio of the rotary spectra components that quantify purely circular counterclockwise-with-time and clockwise-with-time motion as
PccwPcw=(ω|f|ω+|f|)2
(Fig. 1a; Müller et al. 1978; Lien and Müller 1992), where Pccw and Pcw are the counterclockwise- and clockwise-with-time variances, respectively. Note that any elliptical motion can be described as the sum of two purely circular motions, one clockwise and one counterclockwise. For near-inertial waves (ω ∼ |f|) in the Northern (Southern) Hemisphere, almost all the variance is clockwise (counterclockwise) in time and the ratio [Eq. (1)] is near zero (Pinkel 1984). For continuum and near-N waves (ω ≫ |f|), energy is nearly equally distributed between Pccw and Pcw so velocities are rectilinear in the direction of horizontal propagation, where rectilinear is defined as velocities that do not turn with depth, such that the ratio [Eq. (1)] is 1. For internal tides generated at a latitude where ω = 2|f| or at low latitudes where ω = 4|f|, the ratio [Eq. (1)] is 1/9 and (3/5)2, respectively.
Fig. 1.
Fig. 1.

(a) Theoretical curves (thick solid lines) of the ratio to counterclockwise-with-time (Pccw) and clockwise-with-time (Pcw) average rotary spectra at 10° (blue), 30° (red), and 60° (yellow) latitude as a function of frequency [Eq. (1)]. Inertial frequencies are indicated by the blue, red, and orange vertical lines. Black dashed vertical lines denote the semidiurnal frequency (ωM2) as well as typical abyssal (N4000m) and upper-ocean (N50m) buoyancy frequencies. Variability in the abyssal and upper-ocean stratification are noted by the gray bars (the standard deviation of climatological stratification at those two depths). (b) Theoretical curves for the vertical asymmetry ratio [Eq. (6)] for a downgoing monochromatic wave at three different wave frequencies (ωM2, ωK1, and ωNMax where Nmax is 0.0145 rad s−1).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

For vertically propagating waves that can be described with exp[i(kzzωt)], rotary-with-time behavior implies rotary-with-depth behavior that depends on the sign of the vertical wavenumber kz (Leaman and Sanford 1975; D’Asaro and Perkins 1984), e.g., in the Northern Hemisphere, velocities turn clockwise in the direction of energy propagation. The rotary-with-depth behavior of internal waves gives a good estimate of propagation direction for near-inertial waves (with nearly circular hodographs). For higher-frequency waves, the difference in upgoing and downgoing shear variances becomes increasingly muted, as hodographs become more elliptical.

There is no simple relationship between the ratio of upgoing or downgoing shear measured here [Eq. (1)] with energy since the ratio of kinetic to potential energy changes with wave frequency
EKEEPE=(ω2+f2)(ω2f2)N2ω2N2,
assuming constant stratification (cf. Polzin and Lvov 2011). Additionally, intermediate-wavenumber rotary-with-depth shear cannot be equated with energy flux, nor dominance of a rotary component in shear or energy with energy flux (Pinkel 1984). This is because vertical group velocity cgz is a function of frequency (or horizontal wavenumber) as well as vertical wavenumber
cgz=N2f2ω|k|4kh2kz,
where |k|4=(kx2+ky2+kz2)2 and kh=kx2+ky2 is the horizontal magnitude (Vallis 2017). In most oceanic situations fN and in the hydrostatic limit where kx, kykz and N2ω2:
cgzN2f2kz2+N2kh2kz2kh2kz3=ω2f2ωkz,
where the dispersion relation is
ω2f2N2kh2kz2.

For example, through estimates of horizontal wavelengths and vertical group velocities, D’Asaro and Perkins (1984) demonstrated equal up- and downgoing near-inertial energy flux in the summer Sargasso Sea pycnocline, although downgoing (clockwise-with-depth) energy exceeded upgoing because faster upward group velocities with larger horizontal wavelengths (and less energy) compensated for slower downward group velocities with smaller horizontal wavelengths (and more energy).

An additional caveat is included for understanding the separate influences of tides and lee waves, both generated at the bottom at superinertial frequencies. For internal tides ω/f is a function of latitude so that internal tides and equatorward-propagating near-inertial waves become increasingly rectilinear toward the equator (Fig. 1a). Lee wave energy flux, ω2f2N2ω2S[h](k) where S[h](k) is the horizontal wavenumber spectrum for topographic height, tends to peak at intermediate frequency in the continuum band [see Fig. 6 of Kunze and Lien (2019)], where rectilinear flow with depth is expected. Although lee wave generation of 0.2 TW (Nikurashin and Ferrari 2011) is much weaker than internal tide generation of 1 TW (Ray and Egbert 2004), disentangling the two generation mechanisms remains difficult.

Vertical profiles of vertical shear described here are decomposed into their clockwise-with-depth and counterclockwise-with-depth components. Following Gonella (1972), the vertical asymmetry ratio,
Ω=ϕcwϕccwϕcw+ϕccw=±2ω|f|ω2+f2,
is the normalized difference between downward- and upward-propagating intermediate-wavenumber shear variance in the Northern Hemisphere. ϕcw and ϕccw are the integrated in wavenumber clockwise- and counterclockwise-with-depth rotary shear components, respectively, where
ϕccw=kz,minkz,cSccw[Uz/N¯](kz)dkzUzGM2/N¯2, and
ϕcw=kz,minkz,cScw[Uz/N¯](kz)dkzUzGM2/N¯2,
where Sccw and Scw are the values of the dimensionless stratification-normalized rotary shear spectrum, kz,min and kz,c are the integration limits, and both rotary components are normalized by the canonical GM76 shear spectrum (Garrett and Munk 1975) as implemented by Cairns and Williams (1976) integrated over the same wavenumber range, UzGM2/N¯2. The positive (negative) right-hand-side of Eq. (6) is expected for a downgoing (upgoing) monochromatic wave. This assumes that the signals are dominated by near-inertial waves as is typical for shear (e.g., Kunze et al. 1990; Alford et al. 2017), dependent on frequency and latitude. The rotary coefficient C in Gonella (1972) is the Ω of Eq. (6). In the Southern Hemisphere, the vertical asymmetry ratio Ω is calculated with the clockwise- and counterclockwise-with-depth shear components reversed.

For vertically standing waves where ωf (which are rectilinear with depth and have near-vertical energy propagation in both up- and downgoing waves), Pcw = Pccw, giving Ω ≃ 0 regardless of wavenumber. For purely wind-forced near-inertial waves on a β plane (see appendix B), shear variance associated with downgoing (upgoing) near-inertial waves (Leaman and Sanford 1975) gives Ω = 1 (−1) in the upper ocean decreasing with depth (Fig. 2). However, since the ocean is comprised of a continuum of higher-frequency waves, the solution of propagating near-inertial waves on a β plane does not explain the full frequency content of the ocean nor the resulting observed Ω, as we will explore below. As calculated here, Ω is an average ratio over all the waves in the vertical wavenumber band used for the integration. Because the calculation uses shear instead of velocity or energy spectra, the relative contributions of each wave to that average are weighted based on their contribution to shear variance, that is, toward higher wavenumbers.

Fig. 2.
Fig. 2.

Profiles of the magnitude of vertical asymmetry, |Ω(z)| for a surface-forced near-inertial wave on a β plane as it propagates downward and equatorward (blue), then reflects to propagate upward and equatorward (red) without dissipation. The black curve corresponds to their sum assuming that wave energy depends inversely on vertical group velocity |cgz| and the rotary ratio as (ω2f2)/(ω2 + f2). Depth z has been WKB-normalized to constant stratification N. Numbers on the upper axis are corresponding frequencies for the vertically propagating near-inertial waves based on Eq. (6).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

Rotary shear spectra are calculated for observations obtained from three shipboard platforms: 1) LADCP, 2) HDSS, and 3) RDI ADCPs. Shipboard measurements are more frequent during summer so that periods of intense winter storms are less well represented in these data (Figs. 3d–g). Details of instrumentation-specific analysis are presented below to confirm that the analysis presented herein gives consistent cross-instrument results since each instrument, while measuring horizontal velocity acoustically, has different measurement ranges and resolutions. See appendix A for 1) an example dataset comparing rotary spectra from HDSS and moored observations (Fig. A1), 2) comparison of rotary spectra between the three shipboard platforms from simultaneous observations (Figs. A2, A3), and 3) discussion of associated spectral normalization and corrections for each instrument discussed below (Fig. A4).

Fig. 3.
Fig. 3.

(a) Map of GO-SHIP LADCP [light orange from Kunze et al. (2006); dark orange 2004–19], HDSS (green), 75-kHz JASADCP (blue), and 38-kHz JASADCP (purple) measurements. Total number of profiles from LADCP, HDSS, 75-kHz JASADCP, and 38-kHz JASADCP in 4° bins by (b) latitude and (c) longitude. (d)–(g) Distribution of observations per month from each platform (dark and light bars denote data from Northern and Southern Hemispheres, respectively).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

b. Full water column profiles of shear: GO-SHIP LADCP

Rotary shear spectra are calculated from half-overlapping 320-m intervals of full water column profiles of vertical shear from LADCP profiles collected during the WOCE/CLIVAR and GO-SHIP hydrographic cruises in the Indian, Pacific, Atlantic, and Southern Oceans from 1991 to 2019. Rotary shear spectra include those presented in Kunze et al. (2006), as well as more recent data from 2004 to 2019, covering latitudes between 80°N and 80°S, and longitude (Figs. 3a,b). Three datasets from 2003 (sections A16N, A20, and A22; Kunze et al. 2006) are reanalyzed using a broader wavenumber range by decreasing the lower vertical wavelength integration limit. Data were downloaded from the LADCP GO-SHIP archives (see Data availability statement) where the velocity-inversion method has been used to calculate horizontal velocities following Visbeck (2002) with a set of linear equations including navigational data, shipboard ADCP measurements, and bottom tracking (Thurnherr 2010). Due to noisy shear variance signals from several more recent GO-SHIP LADCP sections involving newer 150- or 300-kHz ADCPs, which offer poor performance when there are low scatterers (E. Firing 2015, personal communication), data were not included below the upper 1000 m if shear variance exceeded 10−6 s−1.

LADCP profiles are broken into half-overlapping 320-m-long segments starting from the bottom (e.g., Kunze et al. 2006; Alberty et al. 2017). Shear components are normalized by the segment-averaged stratification N (rad s−1) from the associated CTD cast with spectral corrections following Kunze et al. (2006). This scaling accounts for depth variations in the buoyancy frequency enabling examination of variability not due to changes in stratification (Bell 1974; Leaman and Sanford 1975; Duda and Cox 1989; Polzin and Ferrari 2004; Alford and Whitmont 2007). For analysis of more recent GO-SHIP lines (2004–19), as well as the three reanalyzed pre-2004 lines, shear spectra are corrected for range-averaging and depth-binning, and velocity inversion following Thurnherr (2010). Tilt correction is ignored since bin sizes are ≤10 m (Thurnherr 2010). Shear spectra of clockwise- and counterclockwise-with-depth components are then normalized by the GM76 shear spectra (Garrett and Munk 1975) as implemented by Cairns and Williams (1976).

Normalized shear spectra are integrated from the minimum vertical wavenumber (kz,min = 1/λz,max) to a maximum cutoff wavenumber (kz,c = 1/λz,c) to obtain the shear variance. Analysis from Kunze et al. (2006) used minimum and maximum vertical wavelengths λz,max = 320 m and λz,c = 150 m. For more recent analyses (2003–19), the integration range is extended to λz,c = 100 m to coincide with the wavelength limit of shipboard sonar data (described below) since Thurnherr (2010) indicates that a broader wavenumber band can be used. The narrow vertical wavelength band (100 or 150–320 m) excludes the energy- and energy-flux-containing low modes, as well as the shear-variance-containing finescale.

c. Upper-ocean profiles of shear: Shipboard sonars

Shear in the upper water column was obtained from two sources: 1) the HDSS mounted on the R/V Revelle (Pinkel et al. 2020) and 2) shipboard ADCP current meter, a compilation of all observations from research vessels participating in the Joint Archive for Shipboard ADCP (JASADCP). These two systems were analyzed similarly to provide the most consistent comparison between the three shear products.

1) Hydrographic Doppler Sonar System

The HDSS consists of two sonars operating at 140 and 50 kHz. The 50-kHz sonar used for this study resolves profiles of velocity in 8–14-m bins (depending on the year; Table 1), sampling to 1000-m depth when the ship is stationary (less while in motion), with a 2-s ping rate. The true depth resolution of both the HDSS and JASADCP sonar systems is set by the vertical extent of their finite-width beam, which becomes coarser with increasing range, not by the pulse length or the bin size. For the HDSS, ship-coordinate shears (∂us/∂z, ∂υs/∂z), averaged in 10-min intervals to reduce noise, are used to calculate rotary spectra. Along-ship track velocities from the HDSS can be contaminated by daytime vertical inhomogeneous planktonic scattering layers (Pinkel 2012), an issue of finite angular width of Doppler sonar beams on moving ships. To mitigate the errors in the scattering intensity field associated with these planktonic layers, HDSS shear sampled between sunrise and sunset when the ship was moving at a speed greater than 2.5 kt (1 kt ≈ 0.51 m s−1) was excluded in this analysis.

Table 1

HDSS date and bin size (m).

Table 1

Vertical profiles of shear are normalized by climatological buoyancy frequency, N, from the WOCE Global Hydrographic Climatology (WOCE GHC; Gouretski and Koltermann 2004) which consists of 45 depth levels with a 0.5° spatial resolution. The bin size of the climatology may result in an overly smooth pycnocline at low latitudes, depending on the distribution of these depth bins. Shear variances are normalized using the equivalent GM76 variance (Garrett and Munk 1975; Cairns and Williams 1976). Climatological buoyancy frequency does not include seasonality.

The rotary spectrum is calculated from a single 10-min profile of buoyancy-normalized shear over 200–600-m depth. With a ship speed of 5 m s−1 (or 10 kt), horizontal averaging with 10-min-averaged shear occurs over approximately 3 km. While the HDSS measures to 1000-m depth, a more restricted range is chosen to correspond with the JASADCP measurements. The minimum resolvable vertical wavenumber is determined by the length of the segment (kz,min = 1/λz,max ∼ 1/400 cpm). While HDSS has a relatively high kz,c compared to JASADCP, variances are calculated over the same wavenumber band (described below; from kz,min = 0.0025 cpm to kz,cJASADCP = 0.01 cpm).

A Hanning window was applied to the data to reduce sidelobe effects and resulting spectra normalized accordingly. Each component of the rotary spectra was divided by three sinc2 functions to correct for (i) finite differencing with sinc(kzΔzsamp)2, where Δzsamp is the bin size (see Table 1 for bin size variation with year); (ii) finite spatial extent of the transmitted pulse with sinc(kzΔzres)2 where Δzres = cT/2 cos(α), where c is the speed of sound in water (∼1490 m s−1), T is the transmitted pulse duration where T = St with S = 264 samples in a sampling period t of 80 ms per sample, and α is the beam angle (60°) (Pinkel 2008b), which results in Δzres = 13.63 m; and (iii) range-averaging with sinc(kzΔzavg)2, where Δzavg = Δzres/2.

2) Shipboard ADCP: JASADCP

The shipboard ADCP dataset consists of Teledyne RDI ADCP measurements during 1994 to 2016. The primary frequencies in the JASADCP database are 38 and 75 kHz, which have nominal depth ranges of 1000 and 600 m and vertical bin sizes of 24 and 16 m, respectively, in broadband mode. The narrowband and broadband 75 kHz (16- and 8-m bin sizes, respectively) and narrowband 38 kHz (12-m bin size) offer the best blend between vertical coverage and resolution for comparison with 50-kHz HDSS. Hereafter, references to JASACP will refer to all 75-kHz data and narrowband 38-kHz data.

JASADCP velocity (in Earth-relative coordinates, where u is positive east and υ positive north) is averaged over 10-min temporal blocks to reduce noise, similar to the HDSS. Visual inspection is used to discard some transects with unusually high noise levels. After this step, there remain 265 transects with a total of 483 099 10-min blocks of usable data. Most often, Δzsamp = 16 m for the JASADCP data (broadband 75 kHz), but is sometimes 8 m (narrowband 75 kHz) or 12 m (narrowband 38 kHz). As in section 2c(1), JASADCP velocities from 200- to 600-m depth are transformed into buoyancy-frequency-normalized shear profiles using the corresponding stratification from WOCE GHC, which are then used to calculate rotary shear spectra. Spectra are divided by sinc(kzΔzsamp)4 to correct for finite-differencing and range-averaging. Rotary variances are normalized by the equivalent GM76 spectrum.

d. Synthesis of integrated shear estimates

After spectral corrections, counterclockwise- and clockwise-with-depth variances, ϕcw and ϕccw, are obtained for approximately 106 independent estimates. Two products are obtained from the synthesis of these observations. The first is the total depth-average GM76- and buoyancy-frequency-normalized shear variance (ϕcw + ϕccw) in 4° bins (i) from 200 to 600 m (LADCP, HDSS, and JASADCP), (ii) from 600 m to 1000 mab (meters above the bottom) (LADCP), and (iii) below 1000 mab (LADCP; section 3a). To avoid any overlapping depth ranges, profiles from water depths less than 1600 m are excluded, removing roughly 10% of the observations. Wavenumber bands used in the spectral integrations vary depending on instrumentation. The vertical asymmetry ratio Ω is calculated from average downgoing to average upgoing shear variances in 4° bins (section 3b). The global distribution and wavenumber dependence of vertical asymmetry ratio are examined for each depth range and instrument platform.

3. Results

a. Global patterns of normalized shear variance

Globally, the total GM76- and buoyancy-frequency-normalized shear variance (ϕccw + ϕcw) varies between the upper, middle, and bottom of the water column (Fig. 4). Between 200 and 600 m, shear variance is enhanced along the equator, in the Bay of Bengal, North Atlantic, and east and west of Drake Passage in the Southern Ocean. Between 600 m and 1000 mab, shear variance is enhanced in the Southern Ocean south of Australia, offshore of western South America, offshore of the southwest of Africa, and offshore of the east coast of the United States and Canada. In the bottom 1000 m, shear variance is enhanced in isolated regions such as at the equator in the Atlantic, in the Southern Ocean south of the Indian Ocean, in the Indian Ocean in the vicinity of Madagascar, and offshore of western South America.

Fig. 4.
Fig. 4.

Global distribution of 4°-binned depth-averaged (left) total GM76- and buoyancy-frequency-normalized shear variance (ϕccw + ϕcw), and (right) vertical asymmetry ratio Ω [Eq. (6)]. Depth averages span (a),(b) 200–600-m depth (HDSS, JASADCP, and LADCP), (c),(d) 600 m–1000 mab (LADCP), and (e),(f) the bottom 1000 m (LADCP). Integration wavelengths vary depending on instrument (Figs. A2 and A4). Positive Ω (blue) indicates excess downgoing shear variance while negative (red) excess upgoing. Critical latitude for semidiurnal parametric subharmonic instability (M2 PSI) at 28.8° is noted with black dashed line in (a)–(f).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

Across all latitudes, the relative dominance of down- to upgoing shear variance varies (Fig. 5). In the upper 200–600 m, shear variance exhibits peaks at the equator, high southern latitudes, and less prominently at midlatitudes (Fig. 5a). High northern latitudes exhibit more upgoing than downgoing shear variance, possibly a signature of near-inertial semidiurnal tides. Between 600 m and 1000 mab, both down- and upgoing shear variances peak at the equator and between 20° and 35°, near the 28.8° critical latitude for semidiurnal parametric subharmonic instability (M2 PSI) (e.g., Hibiya et al. 1998; MacKinnon and Winters 2005; van Haren and Millot 2005; Tian et al. 2006; Simmons 2008; MacKinnon et al. 2013b; Ansong et al. 2015; Kunze 2017; Alford et al. 2019). This also corresponds to the latitude where western boundary current intensification dominates shear dynamics along the basin margins (e.g., Lazaneo et al. 2020). In the bottom 1000 m, there is a peak in the Northern Hemisphere between 20° and 35° and no noticeable peak at the equator. Note that shear variance observations (Figs. 5b,c) are sparser at the depths with the prominent signal between 20° and 35°.

Fig. 5.
Fig. 5.

Variance of the downgoing (dark) and upgoing (light) GM76- and buoyancy-frequency-normalized normalized rotary shear in 4° latitude bins from the depth-average over (a) 200–600-m depth (HDSS, JASADCP, LADCP), (b) 600 m–1000 mab (LADCP), and (c) below 1000 mab (LADCP). Standard error (σ/n ) is plotted for LADCP variances. Binned shear variances with standard errors greater than twice the observed mean are not plotted [as in (c)]. Vertical lines at 28.8°N and °S denote the latitude at which semidiurnal parametric subharmonic instabilities and western boundary intensification may occur.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

b. Global patterns of the vertical asymmetry ratio

The globally averaged asymmetry ratio Ω¯ [Eq. (6)] is greater than zero between 200 and 600 m, Ω¯=0.131± 0.006 (mean ± standard error; Fig. 6), indicating a preponderance of downgoing shear variance in the upper water column, consistent with previous work (e.g., Leaman and Sanford 1975; Alford and Gregg 2001). While downgoing shear variance dominates, there are isolated regions where upward components dominate such as south of the Aleutian Islands, adjacent to Antarctica, south of Japan, west of central South America, and in the eastern North Atlantic (Fig. 4b).

Fig. 6.
Fig. 6.

Histograms of the 4°-binned vertical asymmetry ratio, Ω [Eq. (6)] for 200–600 m, 600 m–1000 mab, and below 1000 mab. When Ω = 0, there is equipartition of down- and upgoing shear variance (vertical dashed line). The mean of the independent observations Ω¯ is statistically greater than zero for 200–600-m depth.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

In the mid water column (from 600 m to 1000 mab), while the distribution of down- to upgoing shear is geographically variable (Fig. 4d), the global mean asymmetry ratio is closer to zero and only slightly downward (Ω¯=0.020± 0.008). Signals dominated by upgoing shear cover a greater geographic area than in the upper water column (Fig. 4d), including sections across the North Pacific (30°), between 40° and 60°N in the North Atlantic, and south of the equator in the Pacific along 160°W.

Below 1000 mab, the average vertical asymmetry ratio is not statistically different from zero (Ω¯=0.010± 0.009; Figs. 4e and 6). As in the midwater column, there are preferential regions of positive (downgoing dominant; western North Pacific and central Indian Ocean) and negative (upgoing dominant; North Atlantic south of Greenland and Iceland, and the Southern Ocean south of Australia) vertical asymmetry ratio.

c. Wavenumber and latitude dependence of the vertical asymmetry ratio

The vertical asymmetry ratio in the upper ocean varies with both latitude and wavenumber with a preponderance of positive vertical asymmetry ratio (associated with more downgoing shear variance) at lower wavenumbers equatorward of 40°, consistent across platforms (Figs. 7a–d), possibly linked to semidiurnal internal tides or storm forced near-inertial waves. At high northern latitudes (>40°), Ω is positive. At higher wavenumber (above the integration limits used in the global estimates calculated here), there is a distributed pattern of both positive and negative Ω (LADCP and HDSS; Figs. 7e,f).

Fig. 7.
Fig. 7.

Latitude and wavenumber dependence of the vertical asymmetry ratio Ω [Eq. (6)] from 200 to 600 m. Mean Ω vs wavenumber from each of the three platforms: (a) HDSS (green), (b) LADCP (orange), and (c) JASADCP (blue). (d) Arithmetic mean Ω over the wavenumber bands used for integration vs latitude. Distribution of Ω (colors) vs latitude and wavenumber for (e) HDSS, (f) LADCP, and (g) JASADCP. Blue (red) indicates downgoing (upgoing) shear variance. (h) The distribution of measurements for the HDSS (green), LADCP (orange), and JASADCP (blue) with latitude. For JASADCP and HDSS, Ω above 100 and 56 m, respectively, are excluded due to noise (see associated spectra in Fig. A3b).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

Roughness variance and power input from winds (Simmons and Alford 2012) and tides (Simmons et al. 2004) are used to determine whether these three platforms have consistently sampled low-frequency internal-wave forcing and topographic roughness. The barotropic-to-baroclinic semidiurnal tidal conversion is 0.7–1.3 TW (Munk and Wunsch 1998; Egbert and Ray 2001; Nycander 2005). The total global power input from winds to near-inertial waves is 0.3–1.5 TW (Alford 2001; Watanabe and Hibiya 2002; Alford 2003; Jiang et al. 2005; Furuichi et al. 2008; Simmons and Alford 2012; Rimac et al. 2013; Alford 2020). The magnitude of both wind and tide power inputs likely needs to be reduced by a factor of 2–3 due to near-field dissipation (Alford 2020). Roughness variance, which has been linked to enhanced turbulent mixing independent of the details of internal wave generation or destruction (Decloedt and Luther 2010; Kunze 2017), is the variance calculated in a 30 km × 30 km square using Smith and Sandwell ship-sounding bathymetry, version 14.1 (Smith and Sandwell 1997) as calculated by Whalen et al. (2012). From global maps of the total energy input into winds and tides and roughness presented in Waterhouse et al. (2014), the latitudinal distribution and pdf sampled by each of the three instruments is presented (Fig. 8).

Fig. 8.
Fig. 8.

Global average forcing distribution vs latitude of the global power input from (a) wind, (b) tides plotted as log10 (W m−2) (Waterhouse et al. 2014), and (c) roughness variance (m2) from Whalen et al. (2012) (gray lines). The average power input and roughness variance only from locations sampled by the JASADCP, HDSS and LADCP observations are indicated in blue, green and orange lines, respectively. Associated PDFs of sampling obtained over the global range of (d) wind, (e) tide power inputs, and (f) roughness variance.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

Mean wind-power input is enhanced at midlatitudes, associated with seasonal storms, and reduced at the equator (gray line; Fig. 8a). Tidal power input has no average latitudinal dependence (gray line; Fig. 8b). Roughness variance decreases close to the equator and at high latitudes (Fig. 8c). Sampling of each of these quantities is distributed differently for each instrument platform but covers the full range. While the vertical asymmetry ratio Ω increases poleward (Fig. 7d), direct links with wind power or topographic roughness variance are not discernible. Likewise, no correlation was found with tides. This lack of correlation with forcing may arise from lateral and vertical redistribution of low-mode wave energy through propagation or nonlinear wave–wave interactions.

LADCP sampling closely resembles the PDF of the wind and tide power inputs, and topographic roughness, indicating a well-sampled dataset (Figs. 8d–f; orange line). JASADCP observations undersample regions of high wind power while the HDSS observations cover the full range of wind power. Both JASADCP and HDSS oversample regions of enhanced tide power input. As noted in Figs. 3d–g, data were biased toward undersampling periods of storm generation (winter), particularly for the LADCP. The reduced forcing captured by the observations may underestimate downgoing asymmetry in the upper ocean both above and below 600-m depth (Dohan and Davis 2011).

4. Summary

Intermediate-wavenumber (λz = 100–350 m) rotary-with-depth shears from global acoustic profiling of velocity are interpreted as a ratio of down- versus upgoing shear variance of the low-frequency internal-wave field, with clockwise-with-depth (counterclockwise-with-depth) shears interpreted as downgoing internal waves in the Northern (Southern) Hemisphere. Excess downgoing shear variance (30% or Ω = 0.13) is found in the upper 600 m while equipartition characterizes greater depths (Figs. 4, 6, and 7). Excess downgoing near-inertial energy in the upper ocean is well established (Leaman and Sanford 1975; Müller et al. 1978) and attributed to wind generation (Pollard and Millard 1970; Fu 1981; D’Asaro and Perkins 1984; D’Asaro et al. 1995; Alford et al. 2015b; Alford 2020). But this is, to our knowledge, the first global census of this ratio.

Comparison between the rotary-with-depth results and global patterns of near-inertial and tidal internal wave generation reveals no simple correlation. For example, while winds are the likely generating mechanism for near-inertial waves (Alford et al. 2015a; Alford 2020), the distribution of the asymmetry ratio in the upper water column is not strongly related to the distribution of inertial wind-work with latitude (Fig. 7d compared to Fig. 8a). There is also no clear relationship to internal tide generation with near-symmetry between up- and downgoing shear in the abyssal ocean.

5. Discussion

The resolved 100–350-m band represents neither the energy- and energy-flux-bearing low modes nor the finescale responsible for intermittent breaking and turbulence production. Shear at these wavenumber scales will be governed by competition between propagation, wave–wave interactions (McComas and Bretherton 1977; McComas and Müller 1981a; Henyey et al. 1986; Le Boyer and Alford 2021) and dissipation (Gregg 1989; Polzin et al. 1995). The question remains whether the observed shear Ω and forcing allows us to put bounds on the combinations of these processes. For example, the frequency dependence of Eq. (6) suggests that, for example, at 22° latitude, a vertically propagating semi diurnal tide would have only two-thirds the contribution to asymmetry Ω as a near-inertial wave. Some insight into the relative role of those different processes can come from comparison of associated time scales. Propagation time scales will be ∼H/cgzfkzH/(ω2f2) ∼ nπf/(ω2f2), where n is the mode number (which will vary with stratification (depth) for the resolved band).

Of the wave–wave interaction mechanisms identified by McComas and Bretherton (1977), elastic scattering tends to symmetrize the internal-wave field, as assumed in recent parameterizations of propagation and dissipation in ocean global climate models (Eden et al. 2019, 2020). Elastic scattering time scales vary depending on wavelength and frequency with time scales of O(10) days for waves of intermediate-wavelength and near-inertial frequency. Time scales are O(3) days for midlatitude semidiurnal internal waves, decreasing for internal tides at lower latitudes (McComas and Müller 1981b). Dissipation time scales are ∼E/ϵ where E is the wave energy, ϵ = KN2/Γ, diapycnal diffusivities K are O(105) m2 s1, and mixing efficiencies Γ ∼ 0.2 (Gregg et al. 2018). However, dissipation is not expected to impact wave symmetry.

For surface-generated intermediate-wavenumber near-inertial waves, all three time scales are O(10) days. The base of the upper-ocean layer (600-m depth) can be interpreted as the depth at which elastic scattering has had time to symmetrize the downgoing near-inertial wave field.

At mid and low latitudes, intermediate-wavenumber semidiurnal internal waves propagate at least 10 times faster than near-inertial waves; they thus have the potential to reflect from the surface and return to the bottom before elastic scattering can act. As a result, symmetrization of intermediate-scale internal tides throughout the water column is likely due primarily to surface and bottom reflections rather than wave–wave interactions. At lower latitudes, the noncircular nature of internal tide velocity hodographs also leads to a greater up/down symmetry. This is not the case at higher latitudes where ωM2f and net upward propagation is observed (Figs. 7d–g). Higher-frequency waves (ωf) with rectilinear rather than rotary behavior with depth will tend to further enhance vertical symmetry as measured throughout the ocean in Ω.

Thus, the only observed asymmetry appears to be due to slow-moving wind-generated near-inertial waves. This asymmetry appears to be eradicated by elastic scattering by roughly 600-m depth. At mid and low latitudes, internal tides have less circular velocity ellipses and propagate rapidly enough to reflect from the surface and bottom so that, even near generation sites, there is little vertical asymmetry Ω in the abyssal ocean. Determining the processes leading to the distribution of the asymmetry ratio will help close the global budget of wave generation, propagation, and dissipation. Recent theoretical work by Shakespeare et al. (2021) provides a mathematical framework to account for both the specific geography of wave generation and estimates of wave decay rates.

Acknowledgments.

AFW and JAM were supported by the Climate Process Team (CPT) on internal wave-driven mixing through NSF OCE 0968721. Additional funding support for AFW came from ONR N00014-18-1-2423. TDH and MHA were supported by ONR N00014-18-1-2404 and N00014-19-1-2635. EK’s participation was made possible by NSF OCE 1459173 and OCE 1756093. CBW was supported by NASA 80NSSC19K1116. ECF was supported by NSF OCE 1850762. We are grateful to the MOD engineering group at Scripps Institution of Oceanography for the design, construction, and maintenance the HDSS, in particular Mike Goldin, Tyler Hughen, Mai Bui, Jonathan Ladner, San Nguyen, Spencer Kawamoto, Sara Goheen, and Tony Aja. Operation of the HDSS is overseen by the Shipboard Technical Services Group at Scripps Institution of Oceanography. Development of the HDSS was funded by the National Science Foundation and the Office of Naval Research. The authors thank Sam Kelly, Callum Shakespeare, and Andy Hogg for various helpful discussions over the years. We are grateful to Nicolas Grisouard and an anonymous reviewer for providing helpful and extremely thoughtful feedback. The paper is much improved with their input, and we are grateful for their contribution.

Data availability statement.

All shipboard sonar data are publicly available through the Joint Archive for Shipboard ADCP Online Inventory (https://uhslc.soest.hawaii.edu/sadcp/) and the HDSS Online data repository (https://library.ucsd.edu/dc/collection/bb3383985h; doi:10.6075/J07W69MN). GO-SHIP LADCP data are available from the University of Hawaii GO-SHIP LADCP website with observations post-2014 (https://currents.soest.hawaii.edu/go-ship/ladcp/) and pre-2014 (https://currents.soest.hawaii.edu/clivar/ladcp/index.html). A collection of Post-WOCE Hydrography from JAMSTEC is available from http://www.jamstec.go.jp/iorgc/ocorp/data/post-woce.html.

APPENDIX A

Comparisons of Observed Shear and Spectra

a. Moored and HDSS shear comparison

Examples of oceanic shear from locations coincident in space and time from various platforms are presented to assess the (i) analysis method and (ii) resolution of each instrument. The first example is comparison of data from the Internal Waves Across the Pacific (IWAP) Experiment (Alford et al. 2007; Zhao et al. 2010; MacKinnon et al. 2013a,b; Alford et al. 2017) using vertical shear from a McLane profiler moored north of the Hawaiian Ridge (Fig. A1). The McLane profiler (MP3 in Zhao et al. 2010; Alford et al. 2017) recorded vertical profiles of horizontal currents for 50 days at 28.9°N, 196.5°E. Shipboard shear was obtained by the HDSS on the R/V Revelle during mooring deployment. The R/V Revelle was in the vicinity of MP3 for 4.2 days between yearday 120.5 and 124.7 in 2006. Horizontal velocity shear from MP3 and HDSS observations are both normalized by WOCE buoyancy frequency, to be consistent with the analysis methods described above.

Fig. A1.
Fig. A1.

Comparison of 1D and 2D spectra of buoyancy-frequency-normalized shear over 4.2 days during 2006 IWAP (Zhao et al. 2010; MacKinnon et al. 2013a; Alford et al. 2017). 2D spectra of buoyancy-normalized shear from (a) MP3 and (b) coincident 10-min averaged HDSS observations. (c) 1D buoyancy-normalized wavenumber spectra of shear computed from the 2D spectra [in (a) and (b)] by integrating in frequency (fm) space from MP3 (black) and HDSS (green). In (c), average 1D HDSS spectrum [as calculated in section 2c(1)] during the 4.2-day occupation by the R/V Revelle are also shown (blue).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

Two-dimensional (2D) spectra of the observed shear are calculated from (i) 10-min-averaged shipboard HDSS shear and (ii) a subset of the McLane profiler (MP3) shear from the 4.2-day overlap of the two instruments. Results from the individual one-dimensional (1D) rotary spectra (as described in section 2c(1)] from the same subset of HDSS data are compared with the integrated 2D spectra to ensure consistency between all three methods.

In both MP3 and HDSS measurements, 2D spectra indicate that clockwise-with-depth (negative wavenumber) signals dominate near f (Figs. A1a,b, bottom-left quadrants). Integrating the 2D spectra over all frequencies, the dominant wavelength is 120–200 m (kz = 0.0083–0.005 m−1) from MP3 in both (1D and 2D) methods of estimating spectral levels from HDSS shear (Fig. A1c). At this location, there is near-equal clockwise-with-depth (negative kz) and counterclockwise-with-depth (positive kz) in this wavenumber band from all three methods, in agreement with Alford et al. (2017). The similarity of the 1D and integrated 2D spectra suggests that the method of calculating 1D rotary spectra in 10-min averaged sonar observations used hereafter is consistent with standard methods.

b. LADCP, HDSS, and JASADCP shear comparison

The second set of shear and associated spectra from coincident LADCP profiles, HDSS sonar, and JASADCP observations is from the 2016 occupation of the I08S/I09N GO-SHIP Repeat Hydrography Cruise in the Indian Ocean (Fig. A2). Shear from 200 to 600 m from the HDSS (Figs. A2b,e), JASADCP (Figs. A2c,f), and from individual LADCP profiles along a subset of the 2016 occupation are plotted (Figs. A2a,d). Both the observed vertical shear and the resulting rotary spectra highlight a dominant 100-m wavelength signal (Figs. A2 and A3b). The clockwise and counterclockwise rotary components between 200 and 600 m from each platform has approximately the same power level at the lowest wavenumber with higher clockwise-with-depth (downward) levels than counterclockwise-with-depth (upward) (Fig. A3b). The spectral roll-off for each instrument depends on instrument noise.

Fig. A2.
Fig. A2.

Coincident sections of LADCP, HDSS, and JASADCP (a)–(c) zonal and (d)–(f) meridional 10-min averaged shear over 200–600 m from a subset of observations from the 2016 I08S/I09N GO-SHIP Repeat Hydrography Cruise (see Fig. A3).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

Fig. A3.
Fig. A3.

(a) Map of shear sampling from Fig. A2. (b) Associated average normalized rotary-with-depth shear spectra from LADCP (orange), HDSS (green), and JASADCP (blue) data where light and dark lines of each spectrum indicate counterclockwise-with-depth (ccw) and clockwise-with-depth (cw) rotary components, respectively. Each spectrum has been normalized by the buoyancy frequency, appropriate transfer functions and by the Garrett–Munk (GM76) spectrum (Garrett and Munk 1975) as implemented by Cairns and Williams (1976) at the same stratification (gray line). Integration limits for LADCP, HDSS, and JASADCP are noted with the horizontal bars in (b). Bathymetry contours in (a) are plotted every 2000 m.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

c. Spectral corrections and transfer functions

To illustrate the variation in spectra from raw shear to buoyancy-frequency-normalized shear corrected for the various transfer functions, we demonstrate the procedure of converting the profiles of observed shear extracted from the data shown in Fig. A2 to a normalized rotary spectra corrected with appropriate transfer functions (Fig. A3, respectively). Raw shear (top row of Fig. A4) and the buoyancy-frequency-normalized shear (second row) spectra are the normalized by the transfer functions (row 3) to give the resulting buoyancy-normalized transfer function corrected spectra (row 4). LADCP data has 3 associated transfer functions (three sinc2 associated with range averaging, binning, and velocity inversion), the JASDACP are divided by a sinc4 to normalize the spectra and the HDSS has three sinc2 transfer functions associated with range averaging, binning, and pulse length. Spectra of different noise levels are presented for each instrument (row 5) along with the spectra of the observed shear normalized by the transfer functions.

Fig. A4.
Fig. A4.

Rotary spectra and transfer functions for LADCP, JASADCP, and HDSS profiles extracted from the sample data shown in Figs. A2 and A3. A subset of individual spectra from each shear profile are plotted as thin lines, and means are thick lines. Clockwise (counterclockwise) spectra are plotted in darker (lighter) lines. From top to bottom are the rotary spectra of (a)–(c) measured shear (uz); (d)–(f) buoyancy-frequency-normalized shear (uz/N); (g)–(i) sinc2 transfer functions associated with range averaging (RA), depth binning (BIN), velocity inversion (VI), bin size (DZ), and pulse length (PL; section 2); (j)–(l) buoyancy-frequency-normalized shear normalized by transfer functions (uz/N/T) with integration limits for LADCP, JASADCP, and HDSS noted as the horizontal bars; and (m)–(o) shear normalized by transfer functions with noise spectra (black line). The associated noise (cm s−1) is noted in each subpanel.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

APPENDIX B

Ω(z) for an Equatorward-Propagating Near-Inertial Wave

Considering the idealized case of purely wind-generated near-inertial waves in the ocean, we calculate the asymmetry ratio Ω(z) for an equatorward-propagating near-inertial wave on a β plane (Garrett 2001), where f(y) = βy and the meridional wavenumber is ky = −βt. To solve this problem analytically, we assume constant stratification of N=5×103 rad s1 and bottom depth of H = 1200 m, where the vertical wavenumber kz = /H can be posed as modes n. The hydrostatic Lagrangian frequency (Nf) is
ωL=kz2f2+ky2N2kz=kz2f2+β2t2N2kz
is invariant. The meridional group velocity is
cgy=N2kyfkz2=N2H2βtπ2n2f,
and the vertical group velocity
cgz=N2ky2fkz3=N2H3β2t2π3n3f,
such that the wave packet’s meridional position y evolves as
y=y0N2H2βtπ2n2f,
and the vertical position is
z=N2H3β2t33π3n3f
assuming that f is approximately constant (mid- and high latitudes).

The analytical solution is initialized with modes n = 1–6 at the surface (z = 0) at a latitude near 45°N. These modes propagate under the influence of β, reflecting from the bottom (Fig. B1). Time integrations of the analytical solutions are performed numerically and do not assume constant f or β. Modal waves take approximately n × 10 days to reach the bottom. These results are used to calculate the vertical structure of Ω(z) (Fig. 2).

Fig. B1.
Fig. B1.

Analytical solutions to near-inertial ray paths generated at 45° and z = 0 through depth H = 1200 m on a β plane for vertical modes n = 1–6 (colored lines).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0015.1

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