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  • View in gallery
    Fig. 1.

    Schematics of horizontal wavenumber spectra for (a) isopycnal slope ξx and (c) isopycnal salinity gradient Sx, and vertical wavenumber spectra for (b) vertical strain ξz. Theoretical spectral slopes in wavenumber ranges of GM internal gravity waves (IW, appendix C), anisotropic stratified turbulence (ANISO), and isotropic turbulence (ISO) are labeled, with spectral slopes indicated where relevant. Anisotropic turbulence is bound by the horizontal Coriolis wavenumber kxf=(f3/ε)1/2 and the Ozmidov wavenumber kO = (N3/ε)1/2 in horizontal wavenumber and vertical Coriolis wavenumber kzf=(fN2/ε)1/2 and kO in vertical wavenumber, where mean measured N = 9.6 × 10−3 s−1, f = 8 × 10−5 s−1, ε = 6 × 10−9 W kg−1, and kzf canonical internal wave vertical cutoff wavenumber kzc0.1 cpm (Gargett et al. 1981) were used. Isotropic turbulence lies above the Ozmidov wavenumber. Panel (c) displays the predicted +1/3 spectral slopes for surface quasigeostrophy and +1 slope for subsurface quasigeostrophy (light blue), the flat spectrum for frontogenesis (black), normalized GM IW spectrum for horizontal strain χx (dash–dot), and predicted anisotropic turbulence −1 spectral slope (blue), as well as the predicted +1/3 slope for isotropic turbulence horizontal shear (dark blue). Green shading spans the wavenumber ranges resolved by the towed CTD chain measurements in this study.

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    Fig. 2.

    (a) Map of AVHRR sea surface temperature (SST) (color contour inset) on 19 July 2018 with towed CTD chain (TCTD) survey ship track (thick black lines within the inset’s black rectangle) and ETOPOP2v2 bathymetry contours every 1000 m (thin gray). (b) Expansion of the black rectangle in (a) showing TCTD1 (blue line), TCTD2 (red dotted line), and TCTD3 (black lines) surveys. Blue shading bracketing the TCTD1 line illustrates salinity on isopycnal σ = 25 kg m−3. Green curves are trajectories of six EM-APEX floats, while dotted and solid gray curves trajectories of two surface drifters with drogues at 50- and 70-m depths, respectively.

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    Fig. 3.

    Depth profiles of TCTD and mean EM-APEX float-measured (a) temperature T, (b) salinity S, (c) potential density σ0, and (d) buoyancy stratification N2, with instantaneous values lighter and averages darker. Yellow shading in (c) spans the depth window with ∼1-m TCTD vertical resolution. (e) Depth profiles of EM-APEX float turbulent thermal diffusivity KT during TCTD3 survey.

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    Fig. 4.

    Temperature-salinity diagram for the 136 TCTD3 sections with depth in color (green through yellow corresponding to the ∼1-m vertical resolution TCTD depth window between 69- and 117-m depth). Black contours are potential density. There is a salinity maximum at ∼110-m depth (σ0 = 25.25 kg m−3) and higher water-mass variability deeper than 70 m (denser than σ0 = 24.75 kg m−3).

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    Fig. 5.

    Example TCTD3 section of (a) potential density σ0(x, z) from which isopycnal displacement ξ, and hence isopycnal slope ξx and vertical strain ξz are inferred; (b) salinity isopycnal anomaly (x, σ0) with depth contours in gray, illustrating large salinity gradients along both density and depth.

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    Fig. 6.

    Frequency spectra for (a) horizontal kinetic energy (HKE, blue) and available potential energy (APE, red), and (b) clockwise CW (dark green) and counterclockwise CCW (light green) velocity from the six EM-APEX profiling floats during TCTD3 (see Fig. 2b), averaged over 20–180-m depth. Light shading indicates GM spectra for the measured range of N. The M2 semidiurnal frequency is bracketed by the floats’ 1-h frequency resolution.

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    Fig. 7.

    Average 2D horizontal and vertical wavenumber spectrum for isopycnal slope Φ[ξx](kx,kz) from 136 TCTD3 (x, z) sections (color and solid gray contours, 20% noise level uncertainty in dashed gray contours; high kx and high kz dominated by measurement noise in white and excluded from subsequent analysis). GM spectrum using mean measured N and f (white–color–white contours). Burger number Bu = (Nkx/fkz)2 = 1 (dashed red line), and frequency isolines at 1.05f, M2 semidiurnal tide, and N/2 for linear internal waves (solid red lines).

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    Fig. 8.

    (a) Average 1D horizontal wavenumber spectra for isopycnal slope Φ[ξx](kx) from TCTD1–3, in vertical wave bands: 2 < λz < 48 m (red), λz > 48 m (green), 10 < λz < 48 m (dark blue), 2 < λz < 10 m (light blue), λz > 2 m (black), and (b) average 1D vertical wavenumber spectra for isopycnal slope: 50 < λx < 200 m (yellow), 200 < λx < 1000 m (orange), 1000 < λx < 5000 m (red), and λx > 50 m (black). Dark shading indicates standard errors, light shading ±20% noise level uncertainty. Olive shading shows the spectral range of the GM model for λz > 10 m and λx > 50 m, for 90% of the observed N. Colored dashed lines show GM spectra in the corresponding wavenumber bands. Vertical gray shading shows 90% of the ranges of Coriolis wavenumbers kxf=(f3/ε)1/2 and kzf=(fN2/ε)1/2 based on float-measured ε. Black dotted diagonal lines indicate different spectral slopes for reference. Black triangles on upper x axis in (a) show maximum λx for TCTD1–3, and black dots on lower axes show the center of bins used in spectral averaging. The 2 < λz < 10 m and λz > 2 m kx spectra [light blue and black in (a)] are underestimated at high kx because measurement noise precludes including the high-kx and high-kz parts of the 2D spectrum (Fig. 7). Likewise, the λx > 50 m and 50 < λx < 200 m kz spectra [black and yellow in (b)] are underestimated at high kz because of omission of the noise-dominated part of the 2D spectrum (Fig. 7).

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    Fig. 9.

    Average 2D horizontal and vertical wavenumber spectrum for vertical strain Φ[ξz](kx,kz) from 136 TCTD3 (x, z) sections (color and solid gray contours, 20% noise level uncertainty in dashed gray contours; high kx and high kz dominated by measurement noise in white and excluded from subsequent analysis). GM spectrum using mean measured N and f (white–color–white contours). Burger number Bu = (Nkx/fkz)2 = 1 dashed red line), and frequency isolines at 1.05f, M2 semidiurnal tide, and N/2 for linear internal waves (solid red lines).

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    Fig. 10.

    (a) Average 1D horizontal wavenumber spectra for vertical strain Φ[ξz](kz) from TCTD1–3, in vertical wave bands: 2 < λz < 48 m (red), λz > 48 m (green), 10 < λz < 48 m (dark blue), 2 < λz < 10 m (light blue), and λz > 2 m (black), and (b) 1D vertical wavenumber spectra for vertical strain in horizontal wave bands: 50 < λx < 200 m (yellow), 200 < λx < 1000 m (orange), 1000 < λx < 5000 m (red), and λx > 50 m (black). The vertical wavenumber spectrum from EM-APEX floats in (b) (solid green) is corrected for lost variance at high kz due to lower vertical resolution (dotted green), using transfer function H from the ratio between the 1-m resolution TCTD spectrum 2.6-m subsampled TCTD spectrum (equivalent to the average depth-sampling interval of the floats). Dark shading indicates standard errors, light shading ±20% noise-level uncertainty. Olive shading shows the spectral range of the GM model for λz > 10 m and λx > 50 m, for 90% of the observed N. Colored dashed lines show GM spectra in the corresponding horizontal wave bands. Vertical gray shading shows 90% of the ranges of Coriolis wavenumbers kxf=(f3/ε)1/2 and kzf=(fN2/ε)1/2 based on float-measured ε. Black dotted diagonal lines indicate different spectral slopes for reference. Black triangles on upper x axis in (a) show maximum λx for TCTD1–3, and black dots on lower axes show the center of bins used in spectral averaging. The 2 < λz < 10 m and λz > 2 m kx spectra [light blue and black in (a)] and 50 < λx < 200 m kz spectra [yellow in (b)] are underestimated at high kx and high kz, respectively, because measurement noise precludes including the high-kx and high-kz part of the 2D spectrum (Fig. 9).

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    Fig. 11.

    Average 2D horizontal and vertical wavenumber spectrum for isopycnal salinity gradient Φ[Sx](kx,kz) from 136 TCTD3 (x, z) sections (color and solid gray contours, 20% noise level uncertainty in dashed gray contours). GM spectrum using mean measured N and f (white–color–white contours). Burger number Bu = (Nkx/fkz)2 = 1 (dashed red line), and frequency isolines at 1.05f, M2 semidiurnal tide, and N/2 for linear internal waves (solid red lines).

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    Fig. 12.

    (a) Average 1D horizontal wavenumber spectra for isopycnal salinity gradient Φ[Sx](kx) from TCTD1–3, in vertical wavelength bands: 2 < λz < 48 m (red), λz > 48 m (green), 10 < λz < 48 m (dark blue), 2 < λz < 10 m (light blue), and λz > 2 m (black), and (b) 1D vertical wavenumber spectra for isopycnal salinity gradient in horizontal wave bands: 50 < λx < 200 m (yellow), 200 < λx < 1000 m (orange), 1000 < λx < 5000 m (red), and λx > 50 m (black). Dark shading indicates standard errors, light shading ±20% noise level uncertainty. Olive shading shows the spectral range of the GM model for λz > 10 m and λx > 50 m, for 90% of the observed N. Colored dashed lines show GM spectra in the corresponding horizontal wave bands. Vertical gray shading shows 90% of the ranges of Coriolis wavenumbers kxf=(f3/ε)1/2 and kzf=(fN2/ε)1/2 based on float-measured ε. Black dotted diagonal lines indicate different spectral slopes for reference. Black triangles on upper x axis in (a) show maximum λx for TCTD1–3, and black dots on lower axes show the center of bins used in spectral averaging. The 2 < λz < 10 m and λz > 2 m kx spectra [light blue and black in (a)] and 50 < λx < 200 m kz spectra [yellow in (b)] may be biased at high kx and high kz, respectively, because of the choice of noise level.

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    Fig. 13.

    Average horizontal wavenumber spectra of isopycnal slope Φ[ξx](kx) over 136 TCTD3 (x, z) sections integrated over all vertical wavelengths (black), resolved vertical wavelengths 2 < λz < 48 m (red), small vertical wavelengths 2 < λz < 10 m (light blue), and vertical wavelengths λz > 48 m (green), as well as GM spectrum for 90% of the observed N for λz > 10 m (olive shading). Dark shading indicates standard errors, light shading indicates ±20% noise level uncertainty. The purple solid line and shading represent the Batchelor (1959) and Klymak and Moum (2007) model spectra of inertial convective anisotropic turbulence ΦξxIC(kx) [Eq. (8)] for kx > 5 × 10−3 cpm using the mean (1.2 × 10−5 m2 s−1) and 90% of the float-measured KT, respectively. The purple dotted, dashed, and dash–dotted lines represent linear least squares fits of observed Φ[ξx](kx) for λz > 2 m, 2 < λz < 48 m, and 2 < λz < 10 m, respectively, to the model spectrum ΦξxIC(kx), corresponding to KT = 0.8 × 10−4 m2 s−1, KT = 4.9 × 10−5 m2 s−1, and KT = 1.7 × 10−5 m2 s−1, respectively. Vertical purple lines indicate ±20% noise-level uncertainty. The dotted black diagonal line indicates a +1/3 spectral slope for reference.

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    Fig. 14.

    Comparison of isopycnal-slope-inferred diapycnal diffusivities K[ξx2] and vertical-strain-inferred diffusivities K[ξz2] from TCTD3 measurements. K[ξx2] uses isopycnal slope variance ξx2 from 50 < λx < 200 m for the Batchelor (1959) parameterization [Eq. (9); Klymak and Moum 2007], for 2 < λz < 48 m (dark blue) and 2 < λz < 10 m (green). Circles and crosses show section-median values and vertical lines indicate 20% noise uncertainty. K[ξz2] uses vertical strain variance ξz2 from 1 < λx < 5 km and 10 < λz < 48 m following the finescale parameterization [Eq. (10); Kunze et al. 2006]. Black diagonal lines show one-to-one dependence (solid), within factor of 2 (dashed) and within factor of 5 (dotted). The solid red line shows the orthogonal linear least squares fit in log10 space for 2 < λz < 48 m, log10K[ξx2]=(1.19±0.31)log10K[ξz2]+(1.6±1.42). The orange line is the fit for 2 < λz < 10 m, log10K[ξx2]=(1.09±0.29)log10K[ξz2]+(0.84±1.34).

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    Fig. 15.

    Comparison of isopycnal-slope-inferred diapycnal diffusivities K[ξx2] and salinity gradient-inferred diffusivities K[Sx2] from TCTD3 measurements; K[Sx2] is inferred from the Batchelor (1959) parameterization [Eq. (9); Klymak and Moum 2007] using the variance of the scaled 1D isopycnal salinity gradient kx spectrum β2α2Φ[Sx](kx), for 50 < λx < 200 m and λz > 2 m. K[ξx2] is inferred from Eq. (9) using the variance of the isopycnal slope 1D kx spectrum, over the same kx bands. Dots show section-median values and bars each section’s 25th–75th percentiles. Black diagonal lines show one-to-one dependence (solid), within factor of 2 (dashed) and within factor of 5 (dotted). The solid red line shows the orthogonal linear least squares fit in log10 space, log10K[Sx2]=(1.15±0.11)log10K[ξx2]+(1.24±0.41).

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    Fig. A1.

    Internal wave phase speed C = ω/kx as a function of alongtrack wavenumber kx (solid gray) for (a) across-track wavenumber ky = 0 (anisotropic), (b) ky = kx (isotropic), and (c) kykx such that ωN and CN/kx (anisotropic). Dashed and dash–dotted gray horizontal lines denote mean sensor sampling speed U and U/3 inferred from the alongtrack water and ship speeds. Space–time aliasing will impact low alongtrack wavenumbers kx where C > U/3 (to the left of the vertical lines), which are confined to the lowest measured wavenumbers of TCTD1 for the most plausible alongtrack and isotropic scenarios in (a) and (b), respectively.

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    Fig. B1.

    Schematic illustrating individual CTD sensors instantaneous horizontal displacement δxu − Δx (blue) at t = δt, constructed from (i) movement of the ship and water relative to Earth δxu = (ushipu)δt (red) where uship is ship speed, u ADCP-measured alongtrack water speed, and δt the sampling period, and (ii) chain displacement behind the ship Δx (black). Circles on towed chain represent CTD sensors, black cross shipboard ADCP, and light blue triangle ADCP cone illustrating that the CTD chain and ADCP measurements were not collocated.

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Two-Dimensional Wavenumber Spectra on the Horizontal Submesoscale and Vertical Finescale

Anda VladoiuaApplied Physics Laboratory, University of Washington, Seattle, Washington

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Ren-Chieh LienaApplied Physics Laboratory, University of Washington, Seattle, Washington

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Eric KunzebNorthWest Research Associates, Seattle, Washington

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Abstract

Horizontal and vertical wavenumbers (kx, kz) immediately below the Ozmidov wavenumber (N3/ε)1/2 are spectrally distinct from both isotropic turbulence (kx, kz > 1 cpm) and internal waves as described by the Garrett–Munk (GM) model spectrum (kz < 0.1 cpm). A towed CTD chain, augmented with concurrent Electromagnetic Autonomous Profiling Explorer (EM-APEX) profiling float microstructure measurements and shipboard ADCP surveys, are used to characterize 2D wavenumber (kx, kz) spectra of isopycnal slope, vertical strain, and isopycnal salinity gradient on horizontal wavelengths from 50 m to 250 km and vertical wavelengths of 2–48 m. For kz < 0.1 cpm, 2D spectra of isopycnal slope and vertical strain resemble GM. Integrated over the other wavenumber, the isopycnal slope 1D kx spectrum exhibits a roughly +1/3 slope for kx > 3 × 10−3 cpm, and the vertical strain 1D kz spectrum a −1 slope for kz > 0.1 cpm, consistent with previous 1D measurements, numerical simulations, and anisotropic stratified turbulence theory. Isopycnal salinity gradient 1D kx spectra have a +1 slope for kx > 2 × 10−3 cpm, consistent with nonlocal stirring. Turbulent diapycnal diffusivities inferred in the (i) internal wave subrange using a vertical strain-based finescale parameterization are consistent with those inferred from finescale horizonal wavenumber spectra of (ii) isopycnal slope and (iii) isopycnal salinity gradients using Batchelor model spectra. This suggests that horizontal submesoscale and vertical finescale subranges participate in bridging the forward cascade between weakly nonlinear internal waves and isotropic turbulence, as hypothesized by anisotropic turbulence theory.

© 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: Anda Vladoiu, avladoiu@apl.uw.edu

Abstract

Horizontal and vertical wavenumbers (kx, kz) immediately below the Ozmidov wavenumber (N3/ε)1/2 are spectrally distinct from both isotropic turbulence (kx, kz > 1 cpm) and internal waves as described by the Garrett–Munk (GM) model spectrum (kz < 0.1 cpm). A towed CTD chain, augmented with concurrent Electromagnetic Autonomous Profiling Explorer (EM-APEX) profiling float microstructure measurements and shipboard ADCP surveys, are used to characterize 2D wavenumber (kx, kz) spectra of isopycnal slope, vertical strain, and isopycnal salinity gradient on horizontal wavelengths from 50 m to 250 km and vertical wavelengths of 2–48 m. For kz < 0.1 cpm, 2D spectra of isopycnal slope and vertical strain resemble GM. Integrated over the other wavenumber, the isopycnal slope 1D kx spectrum exhibits a roughly +1/3 slope for kx > 3 × 10−3 cpm, and the vertical strain 1D kz spectrum a −1 slope for kz > 0.1 cpm, consistent with previous 1D measurements, numerical simulations, and anisotropic stratified turbulence theory. Isopycnal salinity gradient 1D kx spectra have a +1 slope for kx > 2 × 10−3 cpm, consistent with nonlocal stirring. Turbulent diapycnal diffusivities inferred in the (i) internal wave subrange using a vertical strain-based finescale parameterization are consistent with those inferred from finescale horizonal wavenumber spectra of (ii) isopycnal slope and (iii) isopycnal salinity gradients using Batchelor model spectra. This suggests that horizontal submesoscale and vertical finescale subranges participate in bridging the forward cascade between weakly nonlinear internal waves and isotropic turbulence, as hypothesized by anisotropic turbulence theory.

© 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: Anda Vladoiu, avladoiu@apl.uw.edu

1. Introduction

In the stratified ocean interior away from boundaries, internal wave breaking is the primary source of isotropic turbulence and diapycnal mixing (e.g., Kunze 2017). However, there is an intermediate wavenumber subrange extending 1–2 decades in vertical wavenumber kz and several decades in horizontal wavenumber kx below the Ozmidov (1965) wavenumber kOO(110) cpm (wavelengths λ ∼ 0.1–1 m) that is spectrally distinct from both isotropic turbulence (kz, kx > kO) and internal gravity waves (kz < 0.1 cpm) as described by the Garrett and Munk (1979) model spectrum, suggesting different dynamics apply. The Ozmidov (1965) wavenumber kO = (N3/ε)1/2 is the lower bound wavenumber for isotropic turbulence, where ε is turbulent kinetic energy dissipation rate and N buoyancy frequency.

Vertical wavenumber kz spectra of vertical shear and strain in the finescale subrange kzc<kz<kO exhibit a −1 spectral slope (Gregg 1977; Dewan 1979; Gargett et al. 1981; Dewan and Good 1986; Smith et al. 1987; Fritts et al. 1988; Gregg et al. 1993; Dewan 1997), where the observed roll-off wavenumber kzc0.1 cpm (vertical wavelengths λz ∼ 10 m). This differs from both the flat or slightly blue (positive slope) spectra below kzc where weakly nonlinear internal waves are thought to dominate, and the +1/3 slope of gradient spectra for isotropic turbulence above the Ozmidov wavenumber kO (Fig. 1b).

Fig. 1.
Fig. 1.

Schematics of horizontal wavenumber spectra for (a) isopycnal slope ξx and (c) isopycnal salinity gradient Sx, and vertical wavenumber spectra for (b) vertical strain ξz. Theoretical spectral slopes in wavenumber ranges of GM internal gravity waves (IW, appendix C), anisotropic stratified turbulence (ANISO), and isotropic turbulence (ISO) are labeled, with spectral slopes indicated where relevant. Anisotropic turbulence is bound by the horizontal Coriolis wavenumber kxf=(f3/ε)1/2 and the Ozmidov wavenumber kO = (N3/ε)1/2 in horizontal wavenumber and vertical Coriolis wavenumber kzf=(fN2/ε)1/2 and kO in vertical wavenumber, where mean measured N = 9.6 × 10−3 s−1, f = 8 × 10−5 s−1, ε = 6 × 10−9 W kg−1, and kzf canonical internal wave vertical cutoff wavenumber kzc0.1 cpm (Gargett et al. 1981) were used. Isotropic turbulence lies above the Ozmidov wavenumber. Panel (c) displays the predicted +1/3 spectral slopes for surface quasigeostrophy and +1 slope for subsurface quasigeostrophy (light blue), the flat spectrum for frontogenesis (black), normalized GM IW spectrum for horizontal strain χx (dash–dot), and predicted anisotropic turbulence −1 spectral slope (blue), as well as the predicted +1/3 slope for isotropic turbulence horizontal shear (dark blue). Green shading spans the wavenumber ranges resolved by the towed CTD chain measurements in this study.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

Submesoscale horizontal wavenumber kx spectra for horizontal temperature gradient or isopycnal slope (Katz 1973; Marmorino et al. 1985; Samelson and Paulson 1988) are flat for 100 m < λx < 100 km. However, McKean and Ewart (1974) identified a +1/3 gradient spectral slope horizontal wavenumber subrange extending several decades below the Ozmidov wavenumber kO (Nastrom and Gage 1985; Klymak and Moum 2007; Sheen et al. 2009; Holbrook et al. 2013; Falder et al. 2016; Fortin et al. 2016), similar to that of isotropic turbulence (Fig. 1a). Moreover, the spectral level for isopycnal slope exhibits the same dependence on turbulent kinetic energy dissipation rate ε as in isotropic turbulence (Klymak and Moum 2007), implying a turbulent energy cascade to dissipation at wavenumbers well below kO.

If quasigeostrophic (QG) turbulence was the sole lateral stirring mechanism in the ocean, horizontal wavenumber spectra for passive tracer gradients, such as spice or salinity anomalies on isopycnals, would have +1 spectral slopes on length scales smaller than the Rossby deformation radius (NH/f for quasigeostrophy and Ndz/f for surface quasigeostrophy, where f is the Coriolis frequency) due to nonlocal stirring (Batchelor 1959; Scott 2006; Smith and Ferrari 2009). However, numerical simulations and measurements find submesoscale spectral slopes for isopycnal salinity gradients ranging from −1 to +1 over horizontal wavenumbers 10−5 < kx < 10−3 cpm (Capet et al. 2008a,b; Molemaker et al. 2010; Cole and Rudnick 2012; Callies and Ferrari 2013; Klymak et al. 2015; Jaeger et al. 2020), signifying that other straining processes contribute to the tracer anomaly cascade, reddening tracer gradient spectra from a +1 QG spectral form (Fig. 1c). Possible processes include reversible but substantial internal wave horizontal strain (Kunze et al. 2015), frontogenesis, and anisotropic stratified turbulence (Kunze 2019).

While both internal waves and turbulence have been well studied spectrally and dynamically with observations, theory, and numerical modeling, the intermediate finescale vertical subrange kzc<kz<kO is not well understood. Nonlinear kinematics, potential-vorticity-carrying fine structure (e.g., balanced vortical motions), double-diffusively-driven thermohaline intrusions, nonlinear internal waves, and anisotropic stratified turbulence have all been proposed to explain this intermediate subrange. It is challenging to study this subrange observationally because it requires separating weak signals from stronger internal wave fluctuations at larger scales, and theoretically because of its inherent nonlinearity. In the following, the vertical wavenumber subrange kzc<kz<kO will be referred to as the vertical finescale or anisotropic stratified turbulence subrange, although other dynamics may apply, under the assumption that nonlinear turbulence with horizontal shears exceeding f (Ro > 1) overcomes internal waves and controls time scales in this subrange (Kunze 2019). Lower vertical wavenumbers kz<kzc are thought to be dominated by weakly nonlinear internal waves and balanced motions. Higher wavenumbers kz > kO are dominated by strongly nonlinear isotropic turbulence. The corresponding horizontal wavenumber band ∼0.1(f/N) cpm < kx < kO, assuming Burger number Bu=[Nkx/(fkz)]21 at kzc, will be referred to as the horizontal submesoscale or anisotropic stratified turbulence subrange.

The processes contributing to the kx+1/3 horizontal wavenumber subrange in isopycnal slope, the broad range of horizonal wavenumber spectral shapes for isopycnal tracer gradient, and the kz1 vertical wavenumber subrange in vertical shear and strain have been interpreted as anisotropic stratified turbulence (Riley and Lindborg 2008; Kunze 2019). Recent numerical modeling (Billant and Chomaz 2000, 2001; Waite and Bartello 2004; Lindborg 2005, 2006; Brethouwer et al. 2007) and observations (Lindborg and Cho 2000; Sheen et al. 2009; Klymak and Moum 2007; Fortin et al. 2016; Falder et al. 2016; Poje et al. 2017) suggest that anisotropic turbulence participates in a forward energy cascade at horizontal wavenumbers below the Ozmidov wavenumber kO. Kunze (2019) equated this forward energy cascade rate with the turbulent dissipation rate ε and argued that anisotropic turbulence fed into isotropic turbulence.

Anisotropic turbulence is characterized by vertical shears O(N) (Billant and Chomaz 2001; Riley and DeBruynkops 2003) and horizontal shears exceeding f (Lindborg 2006; Galperin et al. 2021). Using dimensional scaling arguments, Kunze (2019) formulated an anisotropic stratified turbulence spectral model that reproduces the observed kz1 vertical wavenumber subrange of vertical strain and shear, and the kx+1/3 horizontal wavenumber subrange of isopycnal slope (Fig. 1). In this spectral model, the lower horizontal and vertical wavenumber bounds for the anisotropic stratified turbulence subrange are horizontal and vertical Coriolis wavenumbers (f3/ε)1/2 and (fN2/ε)1/2, respectively. The horizontal Coriolis wavenumber marks the transition to turbulent horizontal shears exceeding f (Ro > 1) and where nonlinear horizontal momentum terms exceed linear terms. The vertical Coriolis wavenumber appears to correspond to the internal wave upper-bound roll-off vertical wavenumber kzc (Gargett et al. 1981). The upper bound for the anisotropic turbulence subrange is the Ozmidov wavenumber kO = (N3/ε)1/2, marking the transition to isotropic turbulence and shears exceeding N. The lower horizontal wavenumber bound for anisotropic stratified turbulence has not been observationally identified and will be discussed in this study. Likewise, the role of finescale processes in the forward energy and passive tracer cascades has not been confirmed observationally and will be examined here.

Previous measurements of the submesoscale and finescale subranges have been mostly 1D either in the horizontal or vertical. In this study, 2D measurements from a towed CTD (TCTD) chain spanning wavelengths (wavenumbers) λx ∼ 50 m–250 km (4 × 10−6 < kx < 2 × 10−2 cpm) and λz = 2–48 m (2 × 10−2 < kz < 0. 5 cpm), that is, straddling the vertical roll-off wavenumber ∼0.1 cpm, are investigated. This study will 1) examine 2D horizontal and vertical wavenumber spectral properties for isopycnal slope, vertical strain and isopycnal salinity gradient, 2) discuss the relevance of the vertical strain kz1 vertical wavenumber subrange and isopycnal slope kx+1/3 horizontal wavenumber subrange for anisotropic stratified turbulence, 3) compare predicted properties of internal waves and anisotropic stratified turbulence with observed spectra, and 4) quantify the role of finescale processes in the energy and tracer variance cascades.

Section 2 outlines the data and methods. Section 3 presents the background hydrography, as well as frequency spectra. Sections 46 describe 2D and 1D spectra for isopycnal slope, vertical strain, and isopycnal salinity gradient. Section 7 compares diapycnal diffusivities inferred from the finescale strain parameterization (Polzin et al. 1995; Gregg et al. 2003) with those inferred from anisotropic stratified turbulence scaling (Batchelor 1959; Klymak and Moum 2007). Significant correlation of these independent estimates suggests that energy transfers occur from internal waves to isotropic turbulence through the intermediate anisotropic turbulence subrange. In other words, internal waves, anisotropic turbulence, and isotropic turbulence form a bucket brigade across wavenumber in the forward energy cascade to eventual dissipation. Last, section 8 provides a summary of the results, the interpretation of which is discussed further in section 9.

2. Experiment, data, and methods

a. Experiment

The measurements were collected during July 2018 in the upper 200 m of the eastern North Pacific (Fig. 2) on the edge of the California Current system off Baja California, a region of strong compensated temperature–salinity variability and moderate density contrasts, where intense mesoscale and submesoscale variability are well documented (e.g., Flament et al. 1985; Chereskin et al. 2000). Data presented here were collected with a TCTD chain and Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats.

Fig. 2.
Fig. 2.

(a) Map of AVHRR sea surface temperature (SST) (color contour inset) on 19 July 2018 with towed CTD chain (TCTD) survey ship track (thick black lines within the inset’s black rectangle) and ETOPOP2v2 bathymetry contours every 1000 m (thin gray). (b) Expansion of the black rectangle in (a) showing TCTD1 (blue line), TCTD2 (red dotted line), and TCTD3 (black lines) surveys. Blue shading bracketing the TCTD1 line illustrates salinity on isopycnal σ = 25 kg m−3. Green curves are trajectories of six EM-APEX floats, while dotted and solid gray curves trajectories of two surface drifters with drogues at 50- and 70-m depths, respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

b. Towed CTD chain

The TCTD chain is a 200-m-long armored coax sea cable terminated with a V-fin depressor. The cable was mounted with 56 Sea-Bird Scientific SBE37 MicroCATs, sampling temperature, conductivity, and pressure every 12 s. A mounting system was designed to allow MicroCATs to swing freely on the cable. Fairings were added along the entire cable to reduce drag. MicroCAT data were transmitted to the deck in near–real time through an inductive cable during the survey to allow in situ analysis and water-mass tracking.

The temperature resolution of the MicroCATs is 10−4°C and salinity resolution 10−3 psu. Forty-eight of the MicroCATs were distributed at 1.5-m intervals along the cable to span the targeted 69–117-m depth aperture within the pycnocline. Six MicroCATs at 15-m intervals were mounted above the targeted depth window, and two MicroCATs at 11- and 25-m intervals below. Because of the tow angle due to cable drag, the actual depth separation between sensors in the targeted depth window is ∼1 m.

Towed at ∼2 m s−1, TCTD chain measurements have a horizontal resolution of ∼25 m, resolving fluctuations with horizontal wavelengths > 50 m. Space–time aliasing is minimal (appendix A), so that fluctuations of TCTD measurements are interpreted as spatial variability. TCTD sampling patterns (Fig. 2b) include

  • one 260-km, 36-h-long section along a straight northeast–southwest line (TCTD1);

  • one 60-km, 8-h-long section backtracking TCTD1 along a straight southwest–northeast line (TCTD2); and

  • an ensemble of 136 7-km, 1-h-long sections with 120 sections in 30 box patterns and 16 sections in radiator patterns (TCTD3).

MicroCAT conductivity time series were despiked using an iterative method similar to that provided by the Rockland Scientific ODAS (Ocean Dissipation Acquisition System) processing software (Douglas and Lueck 2015). The signal was high-pass filtered and smoothed with a low-pass zero-phase filter, then compared to a local standard deviation computed by high-pass filtering in order to identify spikes. Identified spikes and their two adjacent data points were replaced with the mean value of the neighboring points inferred from the low-pass filter to minimize bias from unipolar pulses. Spikes account for less than 0.5% of the data.

c. EM-APEX floats

During the TCTD3 surveys, two drifters were drogued to 50- and 70-m depths to follow water movement in the pycnocline. Six EM-APEX floats (Sanford et al. 1985, 2005) equipped with dual FP07 thermistors were used to compute turbulent thermal variance dissipation rate χ by integrating the microscale temperature-gradient spectrum beyond the Batchelor roll-off wavenumber (ε/ν/κT2)1/4, and infer turbulent kinetic energy dissipation rates ε=χN2/(2ΓTz2) and thermal diffusivities KT=χ/(2Tz2) (Lien et al. 2016). Floats also measured finescale horizontal velocity (u, υ), temperature T, salinity S, and pressure. The floats profiled repeatedly in the upper 200 m with ∼1-km horizontal and 1-h temporal resolution, while moving with the depth-mean flow. Vertical resolutions are 1, 2.6, and 7 m for χ, (T, S), and (u, υ), respectively. Uncertainty in EM-APEX float horizontal currents is 0.015 m s−1. The floats moved coherently with the drifters with little dispersion. TCTD3 surveyed around float and surface-drifter trajectories with additional guidance from the TS properties of the near-real-time TCTD measurements (Fig. 2b).

d. Estimation of 1D and 2D wavenumber spectra

1) Spatial transformation of TCTD data

The TCTD measurements were converted to time series of latitude, longitude, and pressure using shipboard GPS positions, then interpolated onto a regular alongtrack horizontal grid with δx = 25 m and vertical grid with δz = 1 m for computation of 2D wavenumber spectra. The interpolation is performed using measured pressure, placement of individual sensors along the cable, distances of CTD sensors from shipboard ADCP, and ship and water current speeds. Details of this projection are described in appendix B. Correcting for water motion relative to Earth does not impact the spectra but is included for completeness.

2) Estimates of vertical isopycnal displacement

Gridded TCTD-measured potential density σ is depth-sorted to remove overturns in 6.7% of the data. For each (x, z) section, sorted σ is averaged over time at each 1-m depth grid level zσ¯ to obtain section-averaged density profiles σ¯(z). Vertical displacements of σ¯ are computed as
ξ(x,zσ¯)=z(σ¯)zσ¯.
Displacements ξ are computed for isopycnals fluctuating within the ∼69–117-m depth-average window with ∼1-m vertical resolution.

3) Estimates of 1D and 2D spectrum of isopycnal slope

The 1D horizontal wavenumber spectra of isopycnal displacement Φ[ξ](kx) are computed for each depth zσ¯. Horizontal wavenumber spectra for isopycnal slope were computed as Φ[ξx](kx)=(2πkx)2Φ[ξ](kx), where the alongtrack horizontal wavenumber kx is in cycles per meter (cpm).

For the 2D wavenumber spectra, resolved and larger-scale spatial contributions are separated. The vertical isopycnal displacements ξ(x,zσ¯) are decomposed into the depth-mean ξ¯zσ¯(x), section-mean ξ¯x(zσ¯), 2D mean ξ¯xzσ¯, and resolved perturbation ξ(x,zσ¯) as
ξ(x,zσ¯)=ξ(x,zσ¯)+ξ¯x(zσ¯)+ξ¯zσ¯(x)+ξ¯xzσ¯,
where overbars indicate averages over the superscript dimension and ξ(x,zσ¯) is used to compute the 2D (kx, kz) spectrum. Vertical means ξ¯zσ¯(x) represent horizontal variations with vertical scales longer than 48 m, primarily dominated by low- and intermediate-mode internal waves. Horizontal means ξ¯x(zσ¯) are small. They do not include unresolved horizontal scales longer than the section length because displacements are computed relative to the section-mean density profile.

The 2D horizontal and vertical wavenumber spectra Φ[ξ](kx,kz) and 1D horizontal wavenumber spectra Φ[ξ¯zσ¯](kx) for each section are computed after applying 2D and 1D single sinusoidal tapers, respectively (Riedel and Sidorenko 1995).

4) Estimates of 1D and 2D spectrum of vertical strain

Vertical strain ξz is computed as
ξz(x,zσ¯)=(zzσ¯)zσ¯,
where zσ¯ and ∂z are the thickness between two isopycnals at the reference state and at x, respectively. This is identical to the definition proposed by Pinkel et al. (1991)
ξz(x,zσ¯)=Nb2(zσ¯)N2[x,z(σ¯)]1,
with Nb2(zσ¯) the background stratification and N2[x,z(σ¯)] stratification at local depths.
Vertical strain ξz is also 2D-demeaned
ξz(x,zσ¯)=ξz(x,zσ¯)+ξz¯x(zσ¯)+ξz¯zσ¯(x)+ξz¯xzσ¯
prior to computing its 2D wavenumber spectrum Φ[ξz](kx,kz). This spectrum is identical to (2πkz)2Φ[ξ](kx,kz).

5) Estimates of 1D and 2D spectrum of isopycnal salinity gradient

TCTD-measured salinity is projected onto isopycnals to examine the spectra of a passive tracer, for example, isopycnal spice 2βS′ (where β is the saline-contraction coefficient). It is similarly decomposed into means and perturbation components
S(x,zσ¯)=S(x,zσ¯)+S¯x(zσ¯)+S¯zσ¯(x)+S¯xzσ¯.
The 2D and 1D spectra of salinity gradient spectrum are computed in the same manner as isopycnal slope.

6) 2D spectra characteristics

TCTD-inferred 2D wavenumber spectra are nearly symmetric across zero kx and kz [Φ(kx,kz)Φ(kx,kz)], suggesting horizontal isotropy and vertical symmetry, so only the kx, kz > 0 quadrant is presented for the TCTD3 2D wavenumber spectra for isopycnal slope (Fig. 7), vertical strain (Fig. 9), and isopycnal salinity gradient (Fig. 11). The 1D wavenumber spectra for isopycnal slope (Fig. 8), vertical strain (Fig. 10), and isopycnal salinity gradient (Fig. 12) were obtained by integrating over one dimension of the 2D spectra including negative wavenumbers.

2D wavenumber spectra for the TCTD3 sections span 2 × 10−4 < kx < 2 × 10−2 cpm (50 < λx < 5000 m) and 2 × 10−2 < kz < 5 × 10−1 cpm (2 < λz < 48 m). This corresponds to the submesoscale band in the horizontal and finescale band in the vertical, straddling the canonical internal wave vertical cutoff wavenumber kzc0.1 cpm (Gargett et al. 1981).

The 2D spectra allow examination of spectral properties in different wavenumber bands and different dynamic aspect ratios, e.g., with respect to Burger number Bu=(Nkx/fkz)2 (where N ∼ 9.5 × 10−3 s−1 and f ∼ 8 × 10−5 s−1), which is expected to be >1 for 2D stratified turbulence and small-scale vortical motions (Müller 1984; Lindborg 2005; Kunze 2019). Previous observations of small-scale oceanic motions have Burger numbers of O(101102) (Kunze 1993; Polzin et al. 2003; Pinkel 2014; Lien and Sanford 2019). TCTD3 2D wavenumber spectra (Figs. 7, 9, and 11) span aspect ratios kxkz corresponding to 1.05f < ω < N/2 for linear internal waves for kz < 0.1 cpm but only lower aspect ratios (internal waves frequencies) for kz > 0.1 cpm. Observed spectra are compared to the Garrett–Munk (GM) internal wave spectral model (appendix C) for kz < 0.1 cpm.

7) 2D spectra interpolation and noise correction

The 2D white-noise spectra associated with measurement uncertainty are computed from the average of 200 iterations for each section using artificial random noise. The ratio of spectra with and without interpolation correction was used as a transfer function to correct the observed 2D spectra, primarily for high-wavenumber attenuation due to interpolation onto a regular (x,zσ¯) grid. The interpolation correction also accounts for uncertainty in the projection due to ship ADCP spatial resolution. The EM-APEX float 1D kz vertical strain spectrum (Fig. 10b) was similarly corrected for loss of variance due to interpolation using a transfer function from 1D white-noise spectral ratios.

The noise level for vertical isopycnal displacement was estimated as Δξ=Δρ/ρ/z¯=0.45 m, from instrument uncertainty Δρ = 3 × 10−3 kg m−3 and median observed density gradient ρ/z¯. The noise level for salinity was estimated as ΔS = 6.7 × 10−3 psu from the RMS of the isopycnal projection uncertainty Δz × ∂S/∂z = 0.5 m × 0.012 psu m−1, and sensor uncertainty 3 × 10−3 psu. The 2D noise spectra computed from regularly gridded random noise with normal distribution (with mean = 0 and standard deviation ΔS and Δξ) were subtracted from the observed interpolation-corrected 2D spectra. Wavenumbers where (ΦTCTD − Φnoise)∕ΦTCTD < 0.2 were discarded from analysis (Figs. 7, 9, and 11). This leads to underestimation of spectral levels and slopes near the Nyquist wavenumbers in 1D spectra (Figs. 8, 10, and 12). The effect of noise-level uncertainty on the corrected spectra was estimated by subtracting 2D noise spectra using nominal noise levels Δξ and ΔS ± 20%.

3. Background conditions

The TCTD3 measurement site was chosen from the long TCTD1 and TCTD2 sections because of its strong compensated temperature and salinity variability in the 1-m resolution 69–117 m depth window (Figs. 3a–c and 4). TCTD depth profiles reveal a local subsurface salinity maximum at 100–120 m depth (Fig. 3b) associated with confluence of the surface southeastward California Current and the relatively warmer, more saline subsurface northwestward California Undercurrent. In a typical TCTD3 section, density fluctuations span a broad range of vertical and horizontal scales (Fig. 5a). There are strong salinity anomalies along both depth and density isolines (Fig. 5b).

Fig. 3.
Fig. 3.

Depth profiles of TCTD and mean EM-APEX float-measured (a) temperature T, (b) salinity S, (c) potential density σ0, and (d) buoyancy stratification N2, with instantaneous values lighter and averages darker. Yellow shading in (c) spans the depth window with ∼1-m TCTD vertical resolution. (e) Depth profiles of EM-APEX float turbulent thermal diffusivity KT during TCTD3 survey.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

Fig. 4.
Fig. 4.

Temperature-salinity diagram for the 136 TCTD3 sections with depth in color (green through yellow corresponding to the ∼1-m vertical resolution TCTD depth window between 69- and 117-m depth). Black contours are potential density. There is a salinity maximum at ∼110-m depth (σ0 = 25.25 kg m−3) and higher water-mass variability deeper than 70 m (denser than σ0 = 24.75 kg m−3).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

Fig. 5.
Fig. 5.

Example TCTD3 section of (a) potential density σ0(x, z) from which isopycnal displacement ξ, and hence isopycnal slope ξx and vertical strain ξz are inferred; (b) salinity isopycnal anomaly (x, σ0) with depth contours in gray, illustrating large salinity gradients along both density and depth.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

Frequency spectra for horizontal kinetic energy (HKE) and available potential energy (APE) from EM-APEX float time series exhibit strong semidiurnal peaks (Fig. 6). HKE spectra also show a weak near-inertial peak. Spectral slopes in the internal wave continuum above tidal frequencies are consistent with the GM model (appendix C), with spectral levels a factor of 2 below canonical GM. Clockwise and counterclockwise energy ratios are consistent with linear internal wave theory (Fofonoff 1969; Lien and Müller 1992), suggesting energy is dominated by linear internal waves.

Fig. 6.
Fig. 6.

Frequency spectra for (a) horizontal kinetic energy (HKE, blue) and available potential energy (APE, red), and (b) clockwise CW (dark green) and counterclockwise CCW (light green) velocity from the six EM-APEX profiling floats during TCTD3 (see Fig. 2b), averaged over 20–180-m depth. Light shading indicates GM spectra for the measured range of N. The M2 semidiurnal frequency is bracketed by the floats’ 1-h frequency resolution.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

4. Isopycnal slope spectra

a. 2D spectrum

The mean 2D wavenumber spectrum for isopycnal slope Φ[ξx](kx,kz) for all 136 TCTD3 sections (Fig. 7) increases with increasing kx (blue spectrum) and decreasing kz (red spectrum), i.e., increasing with increasing aspect ratios (kx/kz), such that there is negligible variance at the lowest resolved aspect ratios. Spectral values at the highest wavenumbers are influenced by noise. For kx > ∼ 0.01 cpm and kx > ∼ 0.2 cpm, where (ΦTCTD − Φnoise)/ΦTCTD < 0.2, spectra are considered dominated by noise and discarded from the analysis.

Fig. 7.
Fig. 7.

Average 2D horizontal and vertical wavenumber spectrum for isopycnal slope Φ[ξx](kx,kz) from 136 TCTD3 (x, z) sections (color and solid gray contours, 20% noise level uncertainty in dashed gray contours; high kx and high kz dominated by measurement noise in white and excluded from subsequent analysis). GM spectrum using mean measured N and f (white–color–white contours). Burger number Bu = (Nkx/fkz)2 = 1 (dashed red line), and frequency isolines at 1.05f, M2 semidiurnal tide, and N/2 for linear internal waves (solid red lines).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

For kz<kzc0.1 cpm, the 2D isopycnal slope spectrum resembles the GM internal wave model, suggesting that the observed isopycnal slope spectrum can be explained by internal waves (Fig. 7). Spectral levels are a factor of 3 lower than GM, consistent with the float-measured internal wave field (Fig. 6). For kz > 0.1 cpm, horizontal wavenumber spectra become progressively bluer (more positive slope) at higher kz and vertical wavenumber spectra become less red (less negative slope) at higher kx.

The 2D wavenumber spectra for isopycnal slope were integrated over different wavenumber bands to investigate 1D kx and kz spectral properties (Figs. 8a and 8b, respectively). The noise-dominated high wavenumbers of the spectrum are assumed equal to zero for the integrations.

Fig. 8.
Fig. 8.

(a) Average 1D horizontal wavenumber spectra for isopycnal slope Φ[ξx](kx) from TCTD1–3, in vertical wave bands: 2 < λz < 48 m (red), λz > 48 m (green), 10 < λz < 48 m (dark blue), 2 < λz < 10 m (light blue), λz > 2 m (black), and (b) average 1D vertical wavenumber spectra for isopycnal slope: 50 < λx < 200 m (yellow), 200 < λx < 1000 m (orange), 1000 < λx < 5000 m (red), and λx > 50 m (black). Dark shading indicates standard errors, light shading ±20% noise level uncertainty. Olive shading shows the spectral range of the GM model for λz > 10 m and λx > 50 m, for 90% of the observed N. Colored dashed lines show GM spectra in the corresponding wavenumber bands. Vertical gray shading shows 90% of the ranges of Coriolis wavenumbers kxf=(f3/ε)1/2 and kzf=(fN2/ε)1/2 based on float-measured ε. Black dotted diagonal lines indicate different spectral slopes for reference. Black triangles on upper x axis in (a) show maximum λx for TCTD1–3, and black dots on lower axes show the center of bins used in spectral averaging. The 2 < λz < 10 m and λz > 2 m kx spectra [light blue and black in (a)] are underestimated at high kx because measurement noise precludes including the high-kx and high-kz parts of the 2D spectrum (Fig. 7). Likewise, the λx > 50 m and 50 < λx < 200 m kz spectra [black and yellow in (b)] are underestimated at high kz because of omission of the noise-dominated part of the 2D spectrum (Fig. 7).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

b. 1D kx spectra

The 1D horizontal wavenumber spectrum from vertical wavelengths λz > 48 m resembles GM but has lower spectral level for kx < ∼ 10−3 cpm (λx > 1 km), where both internal wave and balanced contributions are expected (Fig. 8a). However, the GM spectrum has a broad spectral peak centered around kx = 2 × 10−3 cpm, while the observed spectrum exhibits a plateau between 10−4 < kx < 2 × 10−3 cpm, rolling off at 2 × 10−3 cpm, and flattening again for kx > 4 × 10−3 cpm.

The total spectrum, including all vertical scales > 2 m, has a roughly +1/3 slope for kx > ∼3 × 10−3 cpm, consistent with previous observations (Klymak and Moum 2007; Holbrook et al. 2013; Falder et al. 2016; Fortin et al. 2016), as well as anisotropic stratified turbulence numerical simulations (Riley and DeBruynkops 2003; Waite and Bartello 2004; Lindborg 2006; Brethouwer et al. 2007) and theoretical predictions (Kunze 2019). Integrated over resolved vertical wavenumbers (2 < λz < 48 m), the 1D kx spectrum for isopycnal slope is blue with + 2/3 spectral slope (Fig. 8a). Further decomposition by vertical wavenumber band reveals that spectral levels are set by the 10 < λz < 48 m internal wave subrange for kx < ∼2 × 10−3 cpm. The 2 < λz < 10 m finescale subrange only contributes significant variance for kx > 5 × 10−3 cpm (λx < 200 m). The kx spectrum in the vertical finescale (2 < λz < 10 m) is blue with spectral slopes of +2 at low horizontal wavenumber, reflecting the lack of variance at low kx and high kz in the 2D spectrum (Fig. 7), transitioning to ∼+1/2 for horizontal wavenumbers kx > ∼3 × 10−3 cpm. However, this spectral slope may be underestimated since the highest wavenumbers have been discarded because of measurement noise. Nevertheless, it is a robust feature that the spectral slopes at high kx become more positive for increasing vertical wavenumber bands.

Horizontal wavenumber spectral slopes in all vertical wavenumber bands, as well as their relative contribution, change above the horizontal Coriolis wavenumber (f3/ε)1/2 which has been argued to represent a transition (i.e., the lower-bound horizontal wavenumber) to anisotropic turbulence (Kunze 2019).

c. 1D kz spectra

The 1D vertical wavenumber spectra of isopycnal slope are partitioned into different horizontal wavenumber bands (Fig. 8b). Integrated over all resolved horizontal wavenumbers (λx > 50 m), kz spectra have slopes of ∼−1 at the lowest resolved vertical wavenumbers, rolling off with a −3 slope for kz > 0.15 cpm ∼(fN2/ε)1/2. The spectral flattening above kz > 0.3–0.4 cpm has large uncertainty and is likely due to imperfect correction for interpolation and noise. Vertical wavenumber spectra of isopycnal slope are dominated by the highest horizontal wavenumbers (50 < λx < 200 m), sensitive to interpolation and noise. In lower horizontal wavenumber bands, they are decades lower, reflecting that large horizontal fluctuations contribute little to isopycnal slope variance at small vertical scales (Fig. 7).

d. Summary

In summary, 2D wavenumber spectra for isopycnal slope are dominated by the highest measured horizontal wavenumbers, with spectral slopes ranging from 0 to +1 in kx and 0 to −2 in kz. They are consistent with the GM internal wave model for λz > 48 m at lower horizontal wavenumbers kx < 1 × 10−3 cpm (λx > 1 km) (Figs. 7 and 8). High-kx horizontal wavenumber spectra for all measured kz (λz > 2 m) have roughly +1/3 spectral slopes (Fig. 8a), consistent with previous measurements and an anisotropic stratified turbulence theory (Kunze 2019). The +1/3 kx spectral slope is contributed by 2 < λz < 48 m, encompassing the vertical finescale subrange and the high-wavenumber end of weakly nonlinear internal waves. However, red vertical wavenumber spectra (Fig. 8b) are not consistent with kz+3 theoretical predictions. Changes in spectral shapes above ∼( f3/ε)1/2 and ( fN2/ε)1/2 suggest a shift to more nonlinear dynamics.

5. Vertical strain spectra

a. 2D spectrum

The mean 2D wavenumber spectrum for vertical strain Φ[ξz](kx,kz) from the TCTD3 sections is dominated by low horizontal and vertical wavenumbers (Fig. 9). For kz < 0.1 cpm, the spectral distribution in 2D wavenumbers resembles GM, but with a weaker spectral level, consistent with the isopycnal slope spectrum (Fig. 7) and float frequency spectra (Fig. 6). For kz > 0.1 cpm, kx spectra become bluer and kz spectra redder. The vertical strain spectrum is below the estimated noise level at the highest wavenumbers. The spectral peak shifts to higher kx for kz > 0.1 cpm, roughly following Bu = 1 with increasing kx and kz.

Fig. 9.
Fig. 9.

Average 2D horizontal and vertical wavenumber spectrum for vertical strain Φ[ξz](kx,kz) from 136 TCTD3 (x, z) sections (color and solid gray contours, 20% noise level uncertainty in dashed gray contours; high kx and high kz dominated by measurement noise in white and excluded from subsequent analysis). GM spectrum using mean measured N and f (white–color–white contours). Burger number Bu = (Nkx/fkz)2 = 1 dashed red line), and frequency isolines at 1.05f, M2 semidiurnal tide, and N/2 for linear internal waves (solid red lines).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

b. 1D kz spectra

The 1D vertical wavenumber spectrum for vertical strain, integrated over all horizontal wavenumbers kx < 2 × 10−2 cpm (λx > 50 m), is flat below kz ∼ 0.15 cpm and rolls off with a −1 slope at higher kz, in agreement with previous measurements (Fig. 10b; Gregg 1977; Dewan 1979; Gargett et al. 1981; Fritts 1984; Fritts et al. 1988; Gregg et al. 1993). The flat spectrum for kz < 0.15 cpm is a factor of ∼2–3 lower than GM, consistent with Fig. 6 and the EM-APEX float vertical strain spectrum. Because of float CTD measurements’ lower vertical resolution relative to the TCTD chain, a transfer function was applied to the float vertical strain spectrum to correct for lost variance at high kz. The transfer function is derived as the ratio between the 1-m resolution TCTD spectrum and 2.6-m subsampled TCTD spectrum (equivalent to the average depth sampling interval of the floats). The corrected float spectrum reproduces the TCTD spectrum. The roll-off wavenumber of ∼0.15 cpm is consistent with the prediction for saturated internal waves kzc=(0.1 cpm)(EGM/E)=(0.1cpm)(εGM/ε)1/2 (Gargett et al. 1981; Henyey et al. 1986), where EGM is the canonical GM spectral level and E the lower observed spectral level. It is also within the range of the measured vertical Coriolis wavenumber (fN2/ε)1/2 ∼ 0.15–0.45 cpm, which is the predicted lower vertical wavenumber bound for anisotropic turbulence (Kunze 2019), with scaling consistent with the finescale parameterization ε ∼ (E/EGM)2 (McComas and Müller 1981; Henyey et al. 1986).

Fig. 10.
Fig. 10.

(a) Average 1D horizontal wavenumber spectra for vertical strain Φ[ξz](kz) from TCTD1–3, in vertical wave bands: 2 < λz < 48 m (red), λz > 48 m (green), 10 < λz < 48 m (dark blue), 2 < λz < 10 m (light blue), and λz > 2 m (black), and (b) 1D vertical wavenumber spectra for vertical strain in horizontal wave bands: 50 < λx < 200 m (yellow), 200 < λx < 1000 m (orange), 1000 < λx < 5000 m (red), and λx > 50 m (black). The vertical wavenumber spectrum from EM-APEX floats in (b) (solid green) is corrected for lost variance at high kz due to lower vertical resolution (dotted green), using transfer function H from the ratio between the 1-m resolution TCTD spectrum 2.6-m subsampled TCTD spectrum (equivalent to the average depth-sampling interval of the floats). Dark shading indicates standard errors, light shading ±20% noise-level uncertainty. Olive shading shows the spectral range of the GM model for λz > 10 m and λx > 50 m, for 90% of the observed N. Colored dashed lines show GM spectra in the corresponding horizontal wave bands. Vertical gray shading shows 90% of the ranges of Coriolis wavenumbers kxf=(f3/ε)1/2 and kzf=(fN2/ε)1/2 based on float-measured ε. Black dotted diagonal lines indicate different spectral slopes for reference. Black triangles on upper x axis in (a) show maximum λx for TCTD1–3, and black dots on lower axes show the center of bins used in spectral averaging. The 2 < λz < 10 m and λz > 2 m kx spectra [light blue and black in (a)] and 50 < λx < 200 m kz spectra [yellow in (b)] are underestimated at high kx and high kz, respectively, because measurement noise precludes including the high-kx and high-kz part of the 2D spectrum (Fig. 9).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

For kz < 0.15 cpm, most of the variance is contributed by kx < 1 cpkm (λx > 1 km; Fig. 10b). Vertical wavenumber spectra from higher horizontal wavenumber bands are weaker and progressively bluer. Above the mean vertical Coriolis wavenumber (fN2/ε)1/2 ∼ 0.3 cpm, the shortest resolved horizontal wavelengths 50 < λx < 200 are below the noise threshold so their spectrum rolls off excessively. Spectral levels and slopes are increasingly underestimated with increasing kz and kx. The −1 spectral slope characterizing the finescale (kz > 0.15 cpm) is contributed by variance from increasing horizontal wavenumbers at increasing vertical wavenumbers, that is, from a roughly constant aspect ratio (Fig. 9).

c. 1D kx spectra

Horizontal wavenumber spectra of vertical strain are red over the resolved kx range (Fig. 10a). For larger vertical wavelengths λz > 48 m, spectral slopes lie between −1 and −2, resembling the GM spectral model. Except for the lowest kx, the bulk of vertical strain variance is contributed by the vertical finescale (2 < λz < 10 m) which has −1 spectral slope, steepening at the highest wavenumbers where the spectra are underestimated because of discarded high-wavenumber variance (Fig. 9). A −1 spectral slope for 10−4 < kx < 3 × 10−3 cpm is consistent with anisotropic turbulence theory (Kunze 2019).

d. Summary

Spectra for vertical strain are largely consistent with the GM spectral model for kz < 0.15 cpm and kx < 10−3 cpm, flat in vertical wavenumber with most variance contributed by low horizontal wavenumbers. For kz > 0.15 cpm and kx > 10−3 cpm, both horizontal and vertical wavenumber spectra are red, with roughly −1 spectral slopes consistent with an anisotropic stratified turbulence model (Kunze 2019), though many other plausible explanations have been proposed.

6. Isopycnal salinity gradient spectra

a. 2D spectrum

The average 2D wavenumber spectrum for isopycnal salinity gradient Φ[Sx](kx,kz) from the 136 TCTD3 sections (Fig. 11) is dominated by low kz and high kx, resembling the 2D isopycnal slope spectrum (Fig. 7), especially for kz > 0.1 cpm. Additionally, it exhibits a peak at the lowest kx and kz.

Fig. 11.
Fig. 11.

Average 2D horizontal and vertical wavenumber spectrum for isopycnal salinity gradient Φ[Sx](kx,kz) from 136 TCTD3 (x, z) sections (color and solid gray contours, 20% noise level uncertainty in dashed gray contours). GM spectrum using mean measured N and f (white–color–white contours). Burger number Bu = (Nkx/fkz)2 = 1 (dashed red line), and frequency isolines at 1.05f, M2 semidiurnal tide, and N/2 for linear internal waves (solid red lines).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

b. 1D kx spectra

Kunze et al. (2015) suggested that internal wave horizontal deformation might explain their observed isopycnal salinity gradient spectrum at horizontal scales of O(1) km. The kx spectral shape for vertical wavelengths λz > 48 m (Fig. 12a) resembles the GM horizontal strain spectrum normalized by the observed large-scale salinity gradient for kx < ∼10−3 cpm (λx > 1 km), but has a lower level. However, while the GM spectrum rolls off at higher kx, the measured spectrum remains flat, suggesting other processes take over the cascade of tracer variance. Comparing to Fig. 1c, the measured spectrum most resembles either GM plus anisotropic stratified turbulence or GM plus frontogenesis.

Fig. 12.
Fig. 12.

(a) Average 1D horizontal wavenumber spectra for isopycnal salinity gradient Φ[Sx](kx) from TCTD1–3, in vertical wavelength bands: 2 < λz < 48 m (red), λz > 48 m (green), 10 < λz < 48 m (dark blue), 2 < λz < 10 m (light blue), and λz > 2 m (black), and (b) 1D vertical wavenumber spectra for isopycnal salinity gradient in horizontal wave bands: 50 < λx < 200 m (yellow), 200 < λx < 1000 m (orange), 1000 < λx < 5000 m (red), and λx > 50 m (black). Dark shading indicates standard errors, light shading ±20% noise level uncertainty. Olive shading shows the spectral range of the GM model for λz > 10 m and λx > 50 m, for 90% of the observed N. Colored dashed lines show GM spectra in the corresponding horizontal wave bands. Vertical gray shading shows 90% of the ranges of Coriolis wavenumbers kxf=(f3/ε)1/2 and kzf=(fN2/ε)1/2 based on float-measured ε. Black dotted diagonal lines indicate different spectral slopes for reference. Black triangles on upper x axis in (a) show maximum λx for TCTD1–3, and black dots on lower axes show the center of bins used in spectral averaging. The 2 < λz < 10 m and λz > 2 m kx spectra [light blue and black in (a)] and 50 < λx < 200 m kz spectra [yellow in (b)] may be biased at high kx and high kz, respectively, because of the choice of noise level.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

For λz > 2 m, the kx spectrum has a roughly +1 slope for kx > 2 × 10−3 cpm, consistent with nonlocal stirring and some previous observations (e.g., Klymak et al. 2015; Jaeger et al. 2020). The salinity gradient spectrum for resolved vertical wavelengths 2 < λz < 48 m also has a slope of ∼+1. At kx < 10−3 cpm, this is primarily contributed by vertical wavelengths larger than 10 m, i.e., the internal wave kz subrange. For kx > ( f3/ε)1/2 ∼2 × 10−3 cpm, the finescale (2 < λz < 10 m) contribution to the salinity gradient spectrum dominates, with a +3/2 slope.

A horizontal shear peak near kxO(104) cpm, where the vertical finescale spectral slope changes, might produce a +1 spectral slope through nonlocal stirring (Batchelor 1959; Scott 2006). Double diffusion might also influence the salinity spectra on isopycnals in this subrange (Ruddick and Richards 2003). However, the measured spectral slopes may be biased by the choice in noise level close to Nyquist wavenumbers.

c. 1D kz spectra

The vertical wavenumber spectrum for isopycnal salinity gradient (Fig. 12b) for λx > 50 m is dominated by the shortest wavelengths (50 < λx < 200 m) at all kz. The spectrum for λx > 50 m is roughly flat for kz < 0.1 cpm and rolls off slightly at kz > 0.1 cpm, close to the vertical Coriolis wavenumber (fN2/ε)1/2. Spectra for longer horizontal wavelengths roll off more steeply. The spectrum for λx > 50 m above kz ∼ 0.3 cpm is relatively flat but unreliable because of uncertainty in the noise level.

d. Summary

The horizontal wavenumber spectrum for isopycnal salinity gradient has slopes of ∼+1 above kx ∼ 10−3 cpm. Previous observations have reported spectral slopes ranging from −1 to +1 at these horizontal scales (Cole and Rudnick 2012; Callies and Ferrari 2013; Klymak et al. 2015; Kunze et al. 2015; Jaeger et al. 2020). Kunze et al. (2015) argued that the spectra for the decade below kx ∼ 10−3 cpm could be explained by GM internal wave horizontal deformation, which is consistent with this wavenumber subrange being dominated by vertical wavelengths exceeding 48 m (Fig. 12a). However, for kx ∼ 10−3 cpm, the GM strain spectrum rolls off as kx2, while the measured kx spectrum is blue. All kz contribute significantly for kx ∼10−3 cpm. The kx spectra for the resolved vertical wavenumbers (2 < λz < 48 m) have slopes between +1 and +3/2, which differ from both the shear +1/3 slope and strain −1 spectral slope predictions for anisotropic turbulence (Kunze 2019) but might be explained by nonlocal stirring (Batchelor 1959; Scott 2006). However, the vertical finescale contribution is likely biased low by the removal of noise.

7. Turbulence scaling and energy cascade

In this section, turbulent diapycnal diffusivity K scalings in the anisotropic turbulence (2 < λz < 10 m) band are compared with the well-established vertical-strain-based parameterization (Gregg and Kunze 1991; Polzin et al. 1995; Whalen et al. 2012; Kunze 2017) in the internal wave band (λz > 10 m) to test whether the finescale (anisotropic turbulence) subrange participates in the forward energy and passive tracer variance cascades to isotropic turbulence and dissipation.

Towed temperature measurements at fixed depths near the Hawaiian Ridge found +1/3 spectral slopes, with spectral levels that scaled with diapycnal diffusivity (Klymak and Moum 2007). Though their resolved wavenumbers extended decades below the Ozmidov wavenumber, they interpreted the +1/3 spectral slope as the inertial convective (IC) subrange of stratified turbulence (Batchelor 1959)
ΦξxIC(kx)=4πΓεN02[CTε1/3(2πkx)1/3] (cpm1),
where Γ = 0.2 is the mixing coefficient (Gregg et al. 2018), N0 the mean buoyancy frequency, and CT = 0.4. This model has been widely used to estimate turbulent kinetic energy dissipation rate ε using horizontal wavenumber spectra of isopycnal slope inferred from seismic measurements (e.g., Sheen et al. 2009; Holbrook et al. 2013; Falder et al. 2016; Fortin et al. 2016; Tang et al. 2020, 2021).
Assuming turbulent diapycnal diffusivity K=ΓεN02 (Osborn 1980), the inertial convective spectrum (7) can be expressed in terms of K as
ΦξxIC(kx)=4πCTΓ1/3N02/3K2/3(2πkx)1/3.

Diffusivity K[ΦξxIC], estimated from an RMS fit of model spectrum (8) to the mean measured Φ[ξx](kx) for kx > 5 × 10−3 cpm (Fig. 13), is (i) 6 × 10−5 m2 s−1 for vertical wavelengths λz > 2 m, (ii) 4.9 × 10−5 m2 s−1 for vertical wavelengths 2 < λz < 48 m, and (iii) 1.7 × 10−5 m2 s−1 for vertical wavelengths 2 < λz < 10 m [estimates from maximum-likelihood fitting, e.g., Ruddick et al. (2000), are identical]. Estimates are larger than the average 1.2 × 10−5 m2 s−1 diffusivity inferred from the EM-APEX float χ measurements (Figs. 3e and 13), reflecting uncertainties in the correct wavenumber band to include and in O(1) nondimensional coefficients. This suggests a possible overestimation bias in the scaling induced by contributions from large vertical scales. This may impact published seismic estimates which include all vertical wavenumbers.

Fig. 13.
Fig. 13.

Average horizontal wavenumber spectra of isopycnal slope Φ[ξx](kx) over 136 TCTD3 (x, z) sections integrated over all vertical wavelengths (black), resolved vertical wavelengths 2 < λz < 48 m (red), small vertical wavelengths 2 < λz < 10 m (light blue), and vertical wavelengths λz > 48 m (green), as well as GM spectrum for 90% of the observed N for λz > 10 m (olive shading). Dark shading indicates standard errors, light shading indicates ±20% noise level uncertainty. The purple solid line and shading represent the Batchelor (1959) and Klymak and Moum (2007) model spectra of inertial convective anisotropic turbulence ΦξxIC(kx) [Eq. (8)] for kx > 5 × 10−3 cpm using the mean (1.2 × 10−5 m2 s−1) and 90% of the float-measured KT, respectively. The purple dotted, dashed, and dash–dotted lines represent linear least squares fits of observed Φ[ξx](kx) for λz > 2 m, 2 < λz < 48 m, and 2 < λz < 10 m, respectively, to the model spectrum ΦξxIC(kx), corresponding to KT = 0.8 × 10−4 m2 s−1, KT = 4.9 × 10−5 m2 s−1, and KT = 1.7 × 10−5 m2 s−1, respectively. Vertical purple lines indicate ±20% noise-level uncertainty. The dotted black diagonal line indicates a +1/3 spectral slope for reference.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

To examine interdependences, turbulent diapycnal diffusivity is estimated from the isopycnal slope variance ξx2 integrated over kxmin<kx<kxmax and 0.02 < kz < 0.5 cpm (2 < λz < 48 m) for each TCTD3 section, following the spectral model in Eq. (8), as
K[ξx2]=ξx23/2N0[3π(2π)1/3CTΓ1/2(kxmax4/3kxmin4/3)]3/2.
where kxmin=5×103 cpm and kxmax=2×102 cpm (50 < λx < 200 m). EM-APEX float microstructure measurements have 75% of their diapycnal diffusivities in the range 3 × 10−7 – 2 × 10−5 m2 s−1 (Fig. 3e). Individual estimates of K[ξx2] from each section do not correlate well with simultaneous floats KT estimates (not shown), likely because the 6 ± 4 km separations between concurrent TCTD3 and float profile measurements are longer than the expected turbulence patch horizontal length scale xzN/f0.11 km for patch vertical length scales z110 m (Marmorino 1987; Marmorino et al. 1987; Itsweire et al. 1989; Rosenblum and Marmorino 1990; Marmorino and Trump 1991). This same issue prevents section-by-section comparisons between strain-based finescale parameterization estimates from the floats and TCTD chain.
Turbulent diapycnal diffusivity can also be inferred from TCTD-measured vertical strain variance ξz2 in the internal wave band (λz > 10 m) using the finescale parameterization
K[ξz2]=K0ξz22ξzGM22g(Rω)h(f/N)
(Polzin et al. 1995; Gregg et al. 2003; Kunze et al. 2006), where ξzGM2 is GM vertical strain variance, K0 = 0.05 × 10−4 m2 s−1 the diapycnal diffusivity for the canonical GM internal wave field,
g(Rω)=162Rω(Rω+1)Rω1,
Rω=Vz2N¯2ξz2,
h(f/N)=farccosh(N/f)f30arccosh(N0/f30),
with g(Rω) = 2.03 for the mean float-measured shear-to-strain variance ratio Rω = 5.6 for 0.02 < kz < 0.1 cpm and 70–120-m depth, f30 = f(30°), and N0 = 5.2 × 10−3 rad s−1. Internal wave strain variance is calculated from the TCTD3 2D vertical strain spectra (Fig. 9) integrated over 2 × 10−4 < kx < 10−3 cpm (1 < λx < 5 km) and 2 × 10−2 < kx < 0.1 cpm (10 < λz < 48 m). GM strain variance ξzGM2 is calculated over the same kx and kz wavenumber bands.

These two independently inferred turbulent diapycnal diffusivities K[ξx2] and K[ξz2] are correlated (Fig. 14), with 49% of K[ξx2] within a factor of 5 of K[ξz2]. This is larger than the factor of 2–3 scatter between microstructure measurements and the finescale parameterization in vertical profiles (Polzin et al. 1995; Whalen et al. 2012), but that is to be expected since the comparison is between two independent parameterizations. An orthogonal linear least squares fit in log10 space using K[ξx2] estimated from 2 < λz < 48 m yields log10K[ξx2]=(1.19±0.31)log10K[ξz2]+(1.6±1.42), with mean K[ξx2]/K[ξz2]=7.24±6.72. Estimating K[ξx2] over a more restrictive 2 < λz < 10 m yields log10K[ξx2]=(1.09±0.29)log10K[ξz2]+(0.84±1.34), with mean K[ξx2]/K[ξz2]=2.58±3.38 and 78% of K[ξx2] within a factor of 5 of K[ξz2]. This correlation between K[ξz2] inferred from internal wave vertical strain and K[ξx2] inferred from the finescale subrange isopycnal slope variance is consistent with the hypothesis that the internal wave subrange (λz > 10 m) is connected to the isotropic turbulence subrange (λ < 0.1 m) by a forward energy cascade through the intermediate anisotropic stratified turbulence subrange (Kunze 2019).

Fig. 14.
Fig. 14.

Comparison of isopycnal-slope-inferred diapycnal diffusivities K[ξx2] and vertical-strain-inferred diffusivities K[ξz2] from TCTD3 measurements. K[ξx2] uses isopycnal slope variance ξx2 from 50 < λx < 200 m for the Batchelor (1959) parameterization [Eq. (9); Klymak and Moum 2007], for 2 < λz < 48 m (dark blue) and 2 < λz < 10 m (green). Circles and crosses show section-median values and vertical lines indicate 20% noise uncertainty. K[ξz2] uses vertical strain variance ξz2 from 1 < λx < 5 km and 10 < λz < 48 m following the finescale parameterization [Eq. (10); Kunze et al. 2006]. Black diagonal lines show one-to-one dependence (solid), within factor of 2 (dashed) and within factor of 5 (dotted). The solid red line shows the orthogonal linear least squares fit in log10 space for 2 < λz < 48 m, log10K[ξx2]=(1.19±0.31)log10K[ξz2]+(1.6±1.42). The orange line is the fit for 2 < λz < 10 m, log10K[ξx2]=(1.09±0.29)log10K[ξz2]+(0.84±1.34).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

In the presence of large-scale isopycnal water-mass gradients, turbulent mesoscale, submesoscale, and finescale flow fields will stir the gradients to cascade tracer variance to dissipative scales (e.g., Stern 1975; Ferrari and Polzin 2005). Isopycnal salinity gradient and isopycnal slope spectra exhibit similar horizontal and vertical spectral shapes (Figs. 8, 12). At horizontal wavelengths 50 < λx < 200 m, salinity gradient and isopycnal slope variances are weakly correlated, with correlation coefficients r = 0.64 ± 0.02 for 2 < λz < 48 m and r = 0.61 ± 0.02 for 2 < λz < 10 m. The isopycnal salinity gradient spectrum is blue in the same kx range interpreted as anisotropic stratified turbulence where the isopycnal slope spectrum has a +1/3 spectral slope. Thus, the correlation between variances of isopycnal slope and salinity gradient may imply a role for anisotropic turbulence in this wavenumber range in a forward salinity (passive tracer) variance cascade toward dissipation. Diffusivity K[Sx2] can be derived from scaled isopycnal salinity gradient spectra β2α2Φ[Sx](kx)Φ[Tx](kx), where α and β are the thermal-expansion and saline-contraction coefficients, respectively, from Eq. (9) using the variance of the theoretical Batchelor (1959) spectrum
ΦTxIC=4πCTΓ1/3N2/3Tz¯2K2/3(2πkx)1/3,
where Tz¯ is the section-average vertical temperature gradient. The variance was calculated from TCTD3 1D kx spectra (including all λz) over 5 × 10−3 < kx < 2 × 10−2 cpm (50 < λx < 200 m). Section-mean K[Sx2] are correlated with K[ξx2] similarly inferred from the variance of the observed isopycnal slope 1D kx spectrum over the same wavenumber band, with log10K[Sx2]=(1.15±0.11)log10K[ξx2]+(1.24±0.41) (Fig. 15). This agreement holds for different horizontal wavenumber bands above kx > 10−3 (λx < 1 km). This suggests that the horizontal wavenumber subrange between internal waves and isotropic turbulence also participates in a turbulent scalar variance cascade.
Fig. 15.
Fig. 15.

Comparison of isopycnal-slope-inferred diapycnal diffusivities K[ξx2] and salinity gradient-inferred diffusivities K[Sx2] from TCTD3 measurements; K[Sx2] is inferred from the Batchelor (1959) parameterization [Eq. (9); Klymak and Moum 2007] using the variance of the scaled 1D isopycnal salinity gradient kx spectrum β2α2Φ[Sx](kx), for 50 < λx < 200 m and λz > 2 m. K[ξx2] is inferred from Eq. (9) using the variance of the isopycnal slope 1D kx spectrum, over the same kx bands. Dots show section-median values and bars each section’s 25th–75th percentiles. Black diagonal lines show one-to-one dependence (solid), within factor of 2 (dashed) and within factor of 5 (dotted). The solid red line shows the orthogonal linear least squares fit in log10 space, log10K[Sx2]=(1.15±0.11)log10K[ξx2]+(1.24±0.41).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

8. Summary

The 2D wavenumber (kx, kz) spectra of upper-ocean isopycnal slope ξx, vertical strain ξz, and isopycnal salinity gradient Sx over 50 m < λx < 250 km and 2 < λz < 48 m, from towed CTD chain measurements, have been presented (Figs. 7, 9, and 11), affording a unique look of spectral properties in the underexplored horizontal mesoscale and submesoscale, and vertical finescale. Resolved vertical wavenumbers straddle the vertical strain kzc0.1 cpm roll-off (Gregg 1977; Gargett et al. 1981) that marks the lower bound of the decade-wide subrange lying between weakly nonlinear internal waves (kz < 0.1 cpm) and fully nonlinear isotropic turbulence (kx, kz > 1–10 cpm) and which has been variously attributed to strongly nonlinear internal waves, kinematic distortion, vortical motions, thermohaline interleaving and anisotropic stratified turbulence.

Isopycnal slope kx spectra exhibit a ∼+1/3 spectral slope for kx > 3 × 10−3 cpm (Fig. 8), consistent with previous observations (e.g., Klymak and Moum 2007; Sheen et al. 2009; Holbrook et al. 2013; Fortin et al. 2016; Falder et al. 2016), numerical modeling (e.g., Lindborg 2006; Brethouwer et al. 2007), and an anisotropic stratified turbulence spectral model (Kunze 2019). Salinity gradient kx spectra have a +1 slope for kx > 2 × 10−3 cpm (Fig. 12a), as reported in some previous studies (e.g., Klymak et al. 2015; Jaeger et al. 2020) and consistent with nonlocal stirring. The 1D vertical wavenumber spectra for vertical strain roll off at kz > 0.15 cpm with a ∼−1 spectral slope (Fig. 10b) as found in previous studies (e.g., Gregg 1977; Gargett et al. 1981; Gregg et al. 1993). The largest horizontal scales dominate below and near the roll-off vertical wavenumber kzc. Smaller horizontal scales contribute progressively more variance with increasing kz.

For kz < 0.1 cpm, observed 2D spectra are largely consistent with the GM internal wave model spectrum (Figs. 712), suggesting that internal waves dominate.

For isopycnal slope, isopycnal salinity gradient as well as vertical strain spectra, the smallest vertical (horizontal) scales contribute progressively more variance with increasing kx (kz), relative to larger λz (λx). This suggests a cascade of variance from lower to higher horizontal and vertical wavenumbers along a more constant aspect ratio or Bu trajectory, consistent with anisotropic turbulence theory. These results have significant uncertainties at high wavenumbers (∼kx > 10−2 cpm, kz > 0.3 cpm) where signals are noisy (e.g., Figs. 13, 10, and 12). Interpolation- and noise-corrected spectra presented here are likely redder than the true spectra close to Nyquist wavenumbers since some high-wavenumber variance is discarded to avoid noise contamination. The interpolation correction at the highest wavenumbers also is uncertain.

The measured spectra share many but not all properties of theoretical anisotropic stratified turbulence spectra (Kunze 2019). Anisotropic turbulence theory suggests that the horizontal (ε/f3)1/2 and vertical (fN2/ε)1/2 Coriolis wavenumbers are lower bounds of the anisotropic turbulence subrange (Kunze 2019). These roughly coincide with observed shifts in spectral shapes for isopycnal slope, isopycnal salinity gradient, and vertical strain 1D spectra (Figs. 8, 10, and 12). The roughly +1/3 spectral slope for kx > (ε/f3)1/2 ∼ 2 × 10−3 cpm for isopycnal slope and −1 spectral slope for kz > (fN2/ε)1/2 ∼ 0.15 cpm for vertical strain are consistent with anisotropic turbulence predictions. Observed slopes for kx > (ε/f3)1/2 ∼ 2 × 10−3 cpm for vertical strain are slightly redder than the predicted −1, but may be biased by noise removal. Theory predicts a +3 spectral slope in vertical wavenumber for isopycnal slope while negative slopes are observed (Fig. 8b). The +1 spectral slope for kx > (ε/f3)1/2 cpm for isopycnal salinity gradient is consistent with nonlocal stirring but does not agree with the anisotropic turbulence model for horizontal shear (+1/3) or horizontal strain (−1). Thus, theoretical slope predictions are only partially consistent with measurements. These discrepancies may reflect inadequacies in anisotropic stratified turbulence theory, the presence of other dynamics, or measurement errors.

Independent inferences of turbulent diapycnal diffusivity K based on (i) vertical strain variance ξz2 in the internal wave band (10 < λz < 48 m, 1 < λx < 5 km) using a finescale parameterization (Gregg et al. 2003; Kunze et al. 2006), (ii) isopycnal slope variance ξx2 in the anisotropic stratified turbulence band (2 < λz < 48 m, 50 < λx < 200 m) (Batchelor 1959; Klymak and Moum 2007), and (iii) scaled isopycnal salinity gradient spectra variance in the anisotropic stratified turbulence band (50 < λx < 200 m, all λz) (Batchelor 1959; Klymak and Moum 2007) are correlated (Figs. 14, 15). This is consistent with the premise that the finescale band (λz < 10 m) participates in the forward energy and scalar variance cascades toward isotropic turbulence and dissipation.

Overall, these results support the hypotheses that internal waves can become anisotropically unstable at the finescale before vertical shear can overcome stratification, and that anisotropic turbulence at horizontal and vertical wavenumbers decades below the Ozmidov wavenumber bridges the forward energy and scalar variance cascades between internal waves and isotropic turbulence (Kunze 2019). Future and more accurate measurements spanning a wider range of turbulent intensity are needed to fully validate these conclusions.

9. Discussion

The horizontal and vertical wavenumber bands extending one decade in the vertical and several decades in the horizontal below the Ozmidov wavenumber (N3/ε)1/2 (the lowest wavenumber of isotropic turbulence) are spectrally distinct from the GM model spectrum at lower wavenumbers and isotropic turbulence at higher wavenumbers. Horizontal wavenumber spectra for isopycnal slope in this band exhibit a +1/3 spectral slope and scale with turbulent kinetic energy dissipation rate ε in the same manner as isotropic turbulence (Klymak and Moum 2007). Vertical wavenumber spectra for vertical strain in this band have a −1 spectral slope and are invariant with respect to changes in spectral levels both above the Ozmidov wavenumber and below the roll-off wavenumber ∼0.1 cpm that is the upper bound of weakly nonlinear waves (Gargett et al. 1981). While finescale isopycnal salinity gradient spectral levels scale with those of finescale isopycnal slope, observed spectral slopes of +1 differ from anisotropic turbulence predictions, suggesting additional processes, such as nonlocal horizontal deformation or double diffusion, may be active in this subrange. Since previous observations found spectral slopes between −1 and +1 for 10−4 < kx < 10−3 cpm (e.g., Klymak et al. 2015; Jaeger et al. 2020), the slopes reported in this study may be a smeared signature of superimposed intermittent processes at different stages of development.

While the 2D spectra of the horizontal submesoscale and vertical finescale wavenumber subranges presented here are consistent with past available 1D spectra and support some of the predictions of a theoretical anisotropic stratified turbulence spectrum (Kunze 2019), they have many limitations that require further investigation.

Nonnegligible uncertainty in the noise levels (e.g., Figs. 8, 12, and 10) may change the high-wavenumber spectral shapes significantly. Some noise in the signal is introduced by the projection of (x, y, z) measurements onto a regular (x, z) grid. The interpolation correction and noise subtraction have competing impacts on the highest wavenumbers. These limitations could be improved with higher-resolution measurements that would allow weaker isopycnal displacement and salinity gradient to be better resolved. Due to the finite size of the MicroCATs, cable sensor separation less than 1–1.5 m is not feasible. However, towing smaller temperature sensors in a region with a tight T–S relation could provide smaller uncertainties close to Nyquist wavenumbers.

Only water-mass measurements were made in the spectrally distinct subrange kx > (f3/ε)1/2 ∼ 10−3 cpm and kz > (fN2/ε)1/2 ∼ 0.15 cpm. Measurements on similar scales of horizontal and vertical shear are needed to validate anisotropic turbulence dynamics, in particular, that vertical shears are O(N), and horizontal shears exceed f and increase to O(N) as k approaches the Ozmidov wavenumber.

The measurements spanned the predicted lower horizontal and vertical wavenumber bounds of anisotropic turbulence, (f3/ε)1/2 and (fN2/ε)1/2, but not those transitioning to isotropic turbulence at the Ozmidov wavenumber O(110) cpm.

Microstructure measurements from floats were not sufficiently collocated to test scaling with dissipation rate ε directly so that parameterizations were compared (Figs. 14, 15). Only a decade range of dissipation rate was sampled (Figs. 14, 15). Measurements are needed over a wider range of turbulent dissipation rates to test the dissipation rate scaling, with collocated microstructure measurements. Moreover, measurements allowing quantification of the spectral transfer rate (Lindborg and Cho 2000; Poje et al. 2017) are needed to determine if the cascade rate below the Ozmidov wavenumber matches the microscale dissipation rate ε.

While it is well established that most turbulence in the stratified ocean interior arises from shear instability of finescale low-frequency internal waves (Gregg et al. 1986; Hebert and Moum 1994; Peters et al. 1995; Polzin et al. 1995), how this instability can generate anisotropic stratified turbulence patches with O(f/N) aspect ratios is unknown since most numerical simulations have included neither horizontal background shear nor sufficiently low aspect ratios. Our present understanding of the wavenumber subranges extending up to several decades below the Ozmidov wavenumber, i.e., between weakly nonlinear internal gravity waves and isotropic turbulence, remains rudimentary. Sufficient measurements to provide guidance for rigorous theoretical advances are lacking, making this an area ripe for further research.

Acknowledgments.

The authors thank Barry Ma, Avery Snyder, Ryan Newell, Jesse Dosher, and Tim McGinnes for instrument preparation and operation, Tom Sanford for the loan of EM-APEX floats, and the mates and crew of the R/V Oceanus. Undergraduate volunteers Rachel Mckenzie Scott and Ian Anderson Borchert, and graduate volunteers Zhihua Zheng, Erin Broatch, and Noah Shofer provided invaluable assistance in deploying and recovering the instruments. The authors also thank Jules Hummon of Shipboard ADCP Support Services at University of Hawaii for ADCP processing, two anonymous reviewers whose comments improved the manuscript, and Eric D’Asaro for inspiring discussions. This research was funded by NSF Grants OCE-1734160 and OCE-1734222.

Data availability statement.

The data supporting the results presented in this study is available at https://digital.lib.washington.edu/researchworks/handle/1773/48343.

APPENDIX A

TCTD Space–Time Aliasing Verification

For internal waves, the horizontal and vertical wavenumber are connected to intrinsic frequency ω through the dispersion relation for linear internal waves
ω2=N2kh2+f2kz2kh2+kz2,
where kh=(kx2+ky2)1/2 is the horizontal wavenumber magnitude. The TCTD chain measurements were taken in one horizontal wavenumber direction and horizontal isotropy is assumed.
Space–time aliasing is an issue for alongtrack internal wave phase speeds C = Cx > U, where U ∼ 2 m s−1 is the tow speed. From dispersion relation (A1), the alongtrack internal wave phase speed is
C=Cx=[f2kz2+N2(kx2+ky2)]1/2kx(kz2+kx2+ky2)1/2,
where kx and ky are the along- and across-track wavenumbers with ky unknown. Figure A1 shows the kx ranges where (i) C < U/3 with negligible space–time aliasing, (ii) U/3 < C < U where space–time aliasing may be an issue, and (iii) C > U where space–time aliasing dominates. Three cases are considered: an anisotropic internal wave field with ky = 0 (Fig. A1a), an isotropic field with ky = kx (Fig. A1b), and an anisotropic field with kykx such that ωN and CN/kx (Fig. A1c). There is little difference between the first two more plausible scenarios, which suggests that only the lowest kx < ∼10−5 cpm of TCTD1 might be impacted by space–time aliasing. Thus, the high horizontal wavenumbers of primary interest in this study should be free of space–time aliasing.
Fig. A1.
Fig. A1.

Internal wave phase speed C = ω/kx as a function of alongtrack wavenumber kx (solid gray) for (a) across-track wavenumber ky = 0 (anisotropic), (b) ky = kx (isotropic), and (c) kykx such that ωN and CN/kx (anisotropic). Dashed and dash–dotted gray horizontal lines denote mean sensor sampling speed U and U/3 inferred from the alongtrack water and ship speeds. Space–time aliasing will impact low alongtrack wavenumbers kx where C > U/3 (to the left of the vertical lines), which are confined to the lowest measured wavenumbers of TCTD1 for the most plausible alongtrack and isotropic scenarios in (a) and (b), respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

APPENDIX B

TCTD Measurements Projection onto 2D Horizontal–Vertical Grid

The TCTD chain surveys are augmented with ocean current measurements of 8-m vertical and 250-m horizontal resolution from a shipboard 75-kHz acoustic Doppler current profiler (ADCP) used to infer the alongtrack water velocity. The TCTD time series, collected as a function of time and pressure every δt = 12 s, were projected onto an alongtrack horizontal–vertical grid. For each MicroCAT, the alongtrack horizontal displacement is a combination of (i) displacement due to movement of the ship and water relative to Earth δxu(z,t)=[uship(t)u(z,t)]δt23.5 m, where ship speed uship ∼ 2 m s−1 and alongtrack water speed uǁ ∼ 0.1 m s−1, and (ii) chain displacement behind the ship Δx(z, t) ∼ −(50–250) m (Fig. B1), which are computed separately for each section. The horizontal displacement of individual sensors relative to the shipboard ADCP system point of reference Δx is trigonometrically inferred from measured pressure and placement of sensors on the cable. Because of varying ship speed and cable drag, the chain changes its shape so that Δx is not invariant with time and depth. The position of each sensor on the horizontal grid is then x(z,t)=(ushipu)dtΔx.

Fig. B1.
Fig. B1.

Schematic illustrating individual CTD sensors instantaneous horizontal displacement δxu − Δx (blue) at t = δt, constructed from (i) movement of the ship and water relative to Earth δxu = (ushipu)δt (red) where uship is ship speed, u ADCP-measured alongtrack water speed, and δt the sampling period, and (ii) chain displacement behind the ship Δx (black). Circles on towed chain represent CTD sensors, black cross shipboard ADCP, and light blue triangle ADCP cone illustrating that the CTD chain and ADCP measurements were not collocated.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1

The data, as a function of x(t) and z(t), are then interpolated onto a regular 2D grid with δx ∼ 25 m (median of the lowest 10% of δxu) and δz = 1 m for computing 2D wavenumber spectra. Alongtrack water displacements due to water flow relative to Earth uδt were included for completeness but their inclusion has no significant impact on the spectra. Neglected across-track water displacements are likewise assumed not to have significant impact on the spectra.

APPENDIX C

Garrett–Munk Internal-Wave Spectral Model

The Garrett–Munk (GM) internal wave spectral model was empirically developed to describe the distribution of ocean internal wave energy in frequency–wavenumber space (Garrett and Munk 1972, 1975). The GM frequency spectra for horizontal kinetic energy ΦGM[HKE], available potential energy ΦGM[APE] are
ΦGM[HKE](ω)=1πb2NN0E0f(<