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
Vertical wavenumber kz spectra of vertical shear and strain in the finescale subrange
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
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
While both internal waves and turbulence have been well studied spectrally and dynamically with observations, theory, and numerical modeling, the intermediate finescale vertical subrange
The processes contributing to the
Anisotropic turbulence is characterized by vertical shears
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
Section 2 outlines the data and methods. Section 3 presents the background hydrography, as well as frequency spectra. Sections 4–6 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.
(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
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one 260-km, 36-h-long section along a straight northeast–southwest line (TCTD1);
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one 60-km, 8-h-long section backtracking TCTD1 along a straight southwest–northeast line (TCTD2); and
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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
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
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
The 2D horizontal and vertical wavenumber spectra
4) Estimates of 1D and 2D spectrum of vertical strain
5) Estimates of 1D and 2D spectrum of isopycnal salinity gradient
6) 2D spectra characteristics
TCTD-inferred 2D wavenumber spectra are nearly symmetric across zero kx and kz
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
The 2D spectra allow examination of spectral properties in different wavenumber bands and different dynamic aspect ratios, e.g., with respect to Burger number
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 (
The noise level for vertical isopycnal displacement was estimated as
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).
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
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
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.
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
Average 2D horizontal and vertical wavenumber spectrum for isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1
For
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.
(a) Average 1D horizontal wavenumber spectra for isopycnal slope
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
5. Vertical strain spectra
a. 2D spectrum
The mean 2D wavenumber spectrum for vertical strain
Average 2D horizontal and vertical wavenumber spectrum for vertical strain
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
(a) Average 1D horizontal wavenumber spectra for vertical strain
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
Average 2D horizontal and vertical wavenumber spectrum for isopycnal salinity gradient
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
(a) Average 1D horizontal wavenumber spectra for isopycnal salinity gradient
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
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
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.
Diffusivity
Average horizontal wavenumber spectra of isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1
These two independently inferred turbulent diapycnal diffusivities
Comparison of isopycnal-slope-inferred diapycnal diffusivities
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0111.1
Comparison of isopycnal-slope-inferred diapycnal diffusivities
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
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
For kz < 0.1 cpm, observed 2D spectra are largely consistent with the GM internal wave model spectrum (Figs. 7–12), 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
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
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
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
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
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) ky ≫ kx such that ω ∼ N and C ∼ N/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
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 = (uship − u∥)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 u∥δt 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
A vertical wavenumber cutoff at 0.1 cpm was applied to ΦGM[ξ](ω, kz) with the assumption that higher kz are not weakly nonlinear internal waves. The ΦGM[ξ](kz, kx) was obtained by numerically integrating the nonseparable ΦGM[ξ](kx, ky, kz) over ky at each kx and kz. The 2D horizontal–vertical wavenumber spectrum of isopycnal slope is
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