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 k_{z} and several decades in horizontal wavenumber k_{x} below the Ozmidov (1965) wavenumber
Vertical wavenumber k_{z} spectra of vertical shear and strain in the finescale subrange
Submesoscale horizontal wavenumber k_{x} spectra for horizontal temperature gradient or isopycnal slope (Katz 1973; Marmorino et al. 1985; Samelson and Paulson 1988) are flat for 100 m < λ_{x} < 100 km. However, McKean and Ewart (1974) identified a +1/3 gradient spectral slope horizontal wavenumber subrange extending several decades below the Ozmidov wavenumber k_{O} (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 k_{O}.
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) λ_{x} ∼ 50 m–250 km (4 × 10^{−6} < k_{x} < 2 × 10^{−2} cpm) and λ_{z} = 2–48 m (2 × 10^{−2} < k_{z} < 0. 5 cpm), that is, straddling the vertical rolloff 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 (EMAPEX) 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 EMAPEX floats, while dotted and solid gray curves trajectories of two surface drifters with drogues at 50 and 70m depths, respectively.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
(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 EMAPEX floats, while dotted and solid gray curves trajectories of two surface drifters with drogues at 50 and 70m depths, respectively.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
(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 EMAPEX floats, while dotted and solid gray curves trajectories of two surface drifters with drogues at 50 and 70m depths, respectively.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
b. Towed CTD chain
The TCTD chain is a 200mlong armored coax sea cable terminated with a Vfin depressor. The cable was mounted with 56 SeaBird 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 watermass tracking.
The temperature resolution of the MicroCATs is 10^{−4}°C and salinity resolution 10^{−3} psu. Fortyeight of the MicroCATs were distributed at 1.5m intervals along the cable to span the targeted 69–117m depth aperture within the pycnocline. Six MicroCATs at 15m intervals were mounted above the targeted depth window, and two MicroCATs at 11 and 25m 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 260km, 36hlong section along a straight northeast–southwest line (TCTD1);

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

an ensemble of 136 7km, 1hlong 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 highpass filtered and smoothed with a lowpass zerophase filter, then compared to a local standard deviation computed by highpass 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 lowpass filter to minimize bias from unipolar pulses. Spikes account for less than 0.5% of the data.
c. EMAPEX floats
During the TCTD3 surveys, two drifters were drogued to 50 and 70m depths to follow water movement in the pycnocline. Six EMAPEX 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 temperaturegradient spectrum beyond the Batchelor rolloff 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](k_{x}) 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
TCTDinferred 2D wavenumber spectra are nearly symmetric across zero k_{x} and k_{z}
2D wavenumber spectra for the TCTD3 sections span 2 × 10^{−4} < k_{x} < 2 × 10^{−2} cpm (50 < λ_{x} < 5000 m) and 2 × 10^{−2} < k_{z} < 5 × 10^{−1} cpm (2 < λ_{z} < 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 whitenoise 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 highwavenumber 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 1m 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 EMAPEX floatmeasured (a) temperature T, (b) salinity S, (c) potential density σ_{0}, and (d) buoyancy stratification N^{2}, with instantaneous values lighter and averages darker. Yellow shading in (c) spans the depth window with ∼1m TCTD vertical resolution. (e) Depth profiles of EMAPEX float turbulent thermal diffusivity K_{T} during TCTD3 survey.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Depth profiles of TCTD and mean EMAPEX floatmeasured (a) temperature T, (b) salinity S, (c) potential density σ_{0}, and (d) buoyancy stratification N^{2}, with instantaneous values lighter and averages darker. Yellow shading in (c) spans the depth window with ∼1m TCTD vertical resolution. (e) Depth profiles of EMAPEX float turbulent thermal diffusivity K_{T} during TCTD3 survey.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Depth profiles of TCTD and mean EMAPEX floatmeasured (a) temperature T, (b) salinity S, (c) potential density σ_{0}, and (d) buoyancy stratification N^{2}, with instantaneous values lighter and averages darker. Yellow shading in (c) spans the depth window with ∼1m TCTD vertical resolution. (e) Depth profiles of EMAPEX float turbulent thermal diffusivity K_{T} during TCTD3 survey.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Temperaturesalinity diagram for the 136 TCTD3 sections with depth in color (green through yellow corresponding to the ∼1m vertical resolution TCTD depth window between 69 and 117m depth). Black contours are potential density. There is a salinity maximum at ∼110m depth (σ_{0} = 25.25 kg m^{−3}) and higher watermass variability deeper than 70 m (denser than σ_{0} = 24.75 kg m^{−3}).
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Temperaturesalinity diagram for the 136 TCTD3 sections with depth in color (green through yellow corresponding to the ∼1m vertical resolution TCTD depth window between 69 and 117m depth). Black contours are potential density. There is a salinity maximum at ∼110m depth (σ_{0} = 25.25 kg m^{−3}) and higher watermass variability deeper than 70 m (denser than σ_{0} = 24.75 kg m^{−3}).
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Temperaturesalinity diagram for the 136 TCTD3 sections with depth in color (green through yellow corresponding to the ∼1m vertical resolution TCTD depth window between 69 and 117m depth). Black contours are potential density. There is a salinity maximum at ∼110m depth (σ_{0} = 25.25 kg m^{−3}) and higher watermass variability deeper than 70 m (denser than σ_{0} = 24.75 kg m^{−3}).
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Example TCTD3 section of (a) potential density σ_{0}(x, z) from which isopycnal displacement ξ, and hence isopycnal slope ξ_{x} and vertical strain ξ_{z} are inferred; (b) salinity isopycnal anomaly (x, σ_{0}) with depth contours in gray, illustrating large salinity gradients along both density and depth.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Example TCTD3 section of (a) potential density σ_{0}(x, z) from which isopycnal displacement ξ, and hence isopycnal slope ξ_{x} and vertical strain ξ_{z} are inferred; (b) salinity isopycnal anomaly (x, σ_{0}) with depth contours in gray, illustrating large salinity gradients along both density and depth.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Example TCTD3 section of (a) potential density σ_{0}(x, z) from which isopycnal displacement ξ, and hence isopycnal slope ξ_{x} and vertical strain ξ_{z} are inferred; (b) salinity isopycnal anomaly (x, σ_{0}) with depth contours in gray, illustrating large salinity gradients along both density and depth.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Frequency spectra for horizontal kinetic energy (HKE) and available potential energy (APE) from EMAPEX float time series exhibit strong semidiurnal peaks (Fig. 6). HKE spectra also show a weak nearinertial 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 EMAPEX profiling floats during TCTD3 (see Fig. 2b), averaged over 20–180m depth. Light shading indicates GM spectra for the measured range of N. The M_{2} semidiurnal frequency is bracketed by the floats’ 1h frequency resolution.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
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 EMAPEX profiling floats during TCTD3 (see Fig. 2b), averaged over 20–180m depth. Light shading indicates GM spectra for the measured range of N. The M_{2} semidiurnal frequency is bracketed by the floats’ 1h frequency resolution.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
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 EMAPEX profiling floats during TCTD3 (see Fig. 2b), averaged over 20–180m depth. Light shading indicates GM spectra for the measured range of N. The M_{2} semidiurnal frequency is bracketed by the floats’ 1h frequency resolution.
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
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/JPOD210111.1
Average 2D horizontal and vertical wavenumber spectrum for isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Average 2D horizontal and vertical wavenumber spectrum for isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
For
The 2D wavenumber spectra for isopycnal slope were integrated over different wavenumber bands to investigate 1D k_{x} and k_{z} spectral properties (Figs. 8a and 8b, respectively). The noisedominated 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/JPOD210111.1
(a) Average 1D horizontal wavenumber spectra for isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
(a) Average 1D horizontal wavenumber spectra for isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
b. 1D k_{x} spectra
The 1D horizontal wavenumber spectrum from vertical wavelengths λ_{z} > 48 m resembles GM but has lower spectral level for k_{x} < ∼ 10^{−3} cpm (λ_{x} > 1 km), where both internal wave and balanced contributions are expected (Fig. 8a). However, the GM spectrum has a broad spectral peak centered around k_{x} = 2 × 10^{−3} cpm, while the observed spectrum exhibits a plateau between 10^{−4} < k_{x} < 2 × 10^{−3} cpm, rolling off at 2 × 10^{−3} cpm, and flattening again for k_{x} > 4 × 10^{−3} cpm.
The total spectrum, including all vertical scales > 2 m, has a roughly +1/3 slope for k_{x} > ∼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 < λ_{z} < 48 m), the 1D k_{x} 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 < λ_{z} < 48 m internal wave subrange for k_{x} < ∼2 × 10^{−3} cpm. The 2 < λ_{z} < 10 m finescale subrange only contributes significant variance for k_{x} > 5 × 10^{−3} cpm (λ_{x} < 200 m). The k_{x} spectrum in the vertical finescale (2 < λ_{z} < 10 m) is blue with spectral slopes of +2 at low horizontal wavenumber, reflecting the lack of variance at low k_{x} and high k_{z} in the 2D spectrum (Fig. 7), transitioning to ∼+1/2 for horizontal wavenumbers k_{x} > ∼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 k_{x} 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 (f^{3}/ε)^{1/2} which has been argued to represent a transition (i.e., the lowerbound horizontal wavenumber) to anisotropic turbulence (Kunze 2019).
c. 1D k_{z} spectra
The 1D vertical wavenumber spectra of isopycnal slope are partitioned into different horizontal wavenumber bands (Fig. 8b). Integrated over all resolved horizontal wavenumbers (λ_{x} > 50 m), k_{z} spectra have slopes of ∼−1 at the lowest resolved vertical wavenumbers, rolling off with a −3 slope for k_{z} > 0.15 cpm ∼(fN^{2}/ε)^{1/2}. The spectral flattening above k_{z} > 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 < λ_{x} < 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 k_{x} and 0 to −2 in k_{z}. They are consistent with the GM internal wave model for λ_{z} > 48 m at lower horizontal wavenumbers k_{x} < 1 × 10^{−3} cpm (λ_{x} > 1 km) (Figs. 7 and 8). Highk_{x} horizontal wavenumber spectra for all measured k_{z} (λ_{z} > 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 k_{x} spectral slope is contributed by 2 < λ_{z} < 48 m, encompassing the vertical finescale subrange and the highwavenumber 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/JPOD210111.1
Average 2D horizontal and vertical wavenumber spectrum for vertical strain
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Average 2D horizontal and vertical wavenumber spectrum for vertical strain
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
b. 1D k_{z} spectra
The 1D vertical wavenumber spectrum for vertical strain, integrated over all horizontal wavenumbers k_{x} < 2 × 10^{−2} cpm (λ_{x} > 50 m), is flat below k_{z} ∼ 0.15 cpm and rolls off with a −1 slope at higher k_{z}, 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 k_{z} < 0.15 cpm is a factor of ∼2–3 lower than GM, consistent with Fig. 6 and the EMAPEX 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 k_{z}. The transfer function is derived as the ratio between the 1m resolution TCTD spectrum and 2.6m subsampled TCTD spectrum (equivalent to the average depth sampling interval of the floats). The corrected float spectrum reproduces the TCTD spectrum. The rolloff 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/JPOD210111.1
(a) Average 1D horizontal wavenumber spectra for vertical strain
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
(a) Average 1D horizontal wavenumber spectra for vertical strain
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
For k_{z} < 0.15 cpm, most of the variance is contributed by k_{x} < 1 cpkm (λ_{x} > 1 km; Fig. 10b). Vertical wavenumber spectra from higher horizontal wavenumber bands are weaker and progressively bluer. Above the mean vertical Coriolis wavenumber (fN^{2}/ε)^{1/2} ∼ 0.3 cpm, the shortest resolved horizontal wavelengths 50 < λ_{x} < 200 are below the noise threshold so their spectrum rolls off excessively. Spectral levels and slopes are increasingly underestimated with increasing k_{z} and k_{x}. The −1 spectral slope characterizing the finescale (k_{z} > 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 k_{x} spectra
Horizontal wavenumber spectra of vertical strain are red over the resolved k_{x} range (Fig. 10a). For larger vertical wavelengths λ_{z} > 48 m, spectral slopes lie between −1 and −2, resembling the GM spectral model. Except for the lowest k_{x}, the bulk of vertical strain variance is contributed by the vertical finescale (2 < λ_{z} < 10 m) which has −1 spectral slope, steepening at the highest wavenumbers where the spectra are underestimated because of discarded highwavenumber variance (Fig. 9). A −1 spectral slope for 10^{−4} < k_{x} < 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 k_{z} < 0.15 cpm and k_{x} < 10^{−3} cpm, flat in vertical wavenumber with most variance contributed by low horizontal wavenumbers. For k_{z} > 0.15 cpm and k_{x} > 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/JPOD210111.1
Average 2D horizontal and vertical wavenumber spectrum for isopycnal salinity gradient
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Average 2D horizontal and vertical wavenumber spectrum for isopycnal salinity gradient
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
b. 1D k_{x} 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/JPOD210111.1
(a) Average 1D horizontal wavenumber spectra for isopycnal salinity gradient
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
(a) Average 1D horizontal wavenumber spectra for isopycnal salinity gradient
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
For λ_{z} > 2 m, the k_{x} spectrum has a roughly +1 slope for k_{x} > 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 < λ_{z} < 48 m also has a slope of ∼+1. At k_{x} < 10^{−3} cpm, this is primarily contributed by vertical wavelengths larger than 10 m, i.e., the internal wave k_{z} subrange. For k_{x} > ( f^{3}/ε)^{1/2} ∼2 × 10^{−3} cpm, the finescale (2 < λ_{z} < 10 m) contribution to the salinity gradient spectrum dominates, with a +3/2 slope.
A horizontal shear peak near
c. 1D k_{z} spectra
The vertical wavenumber spectrum for isopycnal salinity gradient (Fig. 12b) for λ_{x} > 50 m is dominated by the shortest wavelengths (50 < λ_{x} < 200 m) at all k_{z}. The spectrum for λ_{x} > 50 m is roughly flat for k_{z} < 0.1 cpm and rolls off slightly at k_{z} > 0.1 cpm, close to the vertical Coriolis wavenumber (fN^{2}/ε)^{1/2}. Spectra for longer horizontal wavelengths roll off more steeply. The spectrum for λ_{x} > 50 m above k_{z} ∼ 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 k_{x} ∼ 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 k_{x} ∼ 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 k_{x} ∼ 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 < λ_{z} < 10 m) band are compared with the wellestablished verticalstrainbased parameterization (Gregg and Kunze 1991; Polzin et al. 1995; Whalen et al. 2012; Kunze 2017) in the internal wave band (λ_{z} > 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/JPOD210111.1
Average horizontal wavenumber spectra of isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Average horizontal wavenumber spectra of isopycnal slope
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
These two independently inferred turbulent diapycnal diffusivities
Comparison of isopycnalslopeinferred diapycnal diffusivities
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Comparison of isopycnalslopeinferred diapycnal diffusivities
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Comparison of isopycnalslopeinferred diapycnal diffusivities
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Comparison of isopycnalslopeinferred diapycnal diffusivities
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Comparison of isopycnalslopeinferred diapycnal diffusivities
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
Comparison of isopycnalslopeinferred diapycnal diffusivities
Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPOD210111.1
8. Summary
The 2D wavenumber (k_{x}, k_{z}) spectra of upperocean isopycnal slope ξ_{x}, vertical strain ξ_{z}, and isopycnal salinity gradient S_{x} over 50 m < λ_{x} < 250 km and 2 < λ_{z} < 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 k_{x} spectra exhibit a ∼+1/3 spectral slope for k_{x} > 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 k_{x} spectra have a +1 slope for k_{x} > 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 k_{z} > 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 rolloff vertical wavenumber
For k_{z} < 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 k_{x} (k_{z}), relative to larger λ_{z} (λ_{x}). 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 (∼k_{x} > 10^{−2} cpm, k_{z} > 0.3 cpm) where signals are noisy (e.g., Figs. 13, 10, and 12). Interpolation and noisecorrected spectra presented here are likely redder than the true spectra close to Nyquist wavenumbers since some highwavenumber 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/f^{3})^{1/2} and vertical (fN^{2}/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 k_{x} > (ε/f^{3})^{1/2} ∼ 2 × 10^{−3} cpm for isopycnal slope and −1 spectral slope for k_{z} > (fN^{2}/ε)^{1/2} ∼ 0.15 cpm for vertical strain are consistent with anisotropic turbulence predictions. Observed slopes for k_{x} > (ε/f^{3})^{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 k_{x} > (ε/f^{3})^{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 (N^{3}/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 rolloff 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} < k_{x} < 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 highwavenumber 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 higherresolution 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 watermass measurements were made in the spectrally distinct subrange k_{x} > (f^{3}/ε)^{1/2} ∼ 10^{−3} cpm and k_{z} > (fN^{2}/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, (f^{3}/ε)^{1/2} and (fN^{2}/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 lowfrequency 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 EMAPEX 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 OCE1734160 and OCE1734222.
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/k_{x} as a function of alongtrack wavenumber k_{x} (solid gray) for (a) acrosstrack wavenumber k_{y} = 0 (anisotropic), (b) k_{y} = k_{x} (isotropic), and (c) k_{y} ≫ k_{x} such that ω ∼ N and C ∼ N/k_{x} (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 k_{x} 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/JPOD210111.1
Internal wave phase speed C = ω/k_{x} as a function of alongtrack wavenumber k_{x} (solid gray) for (a) acrosstrack wavenumber k_{y} = 0 (anisotropic), (b) k_{y} = k_{x} (isotropic), and (c) k_{y} ≫ k_{x} such that ω ∼ N and C ∼ N/k_{x} (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 k_{x} 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/JPOD210111.1
Internal wave phase speed C = ω/k_{x} as a function of alongtrack wavenumber k_{x} (solid gray) for (a) acrosstrack wavenumber k_{y} = 0 (anisotropic), (b) k_{y} = k_{x} (isotropic), and (c) k_{y} ≫ k_{x} such that ω ∼ N and C ∼ N/k_{x} (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 k_{x} 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/JPOD210111.1
APPENDIX B
TCTD Measurements Projection onto 2D Horizontal–Vertical Grid
The TCTD chain surveys are augmented with ocean current measurements of 8m vertical and 250m horizontal resolution from a shipboard 75kHz 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 δx_{u} = (u_{ship} − u_{∥})δt (red) where u_{ship} is ship speed, u_{∥} ADCPmeasured 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/JPOD210111.1
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 δx_{u} = (u_{ship} − u_{∥})δt (red) where u_{ship} is ship speed, u_{∥} ADCPmeasured 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/JPOD210111.1
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 δx_{u} = (u_{ship} − u_{∥})δt (red) where u_{ship} is ship speed, u_{∥} ADCPmeasured 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/JPOD210111.1
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 δx_{u}) 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 acrosstrack water displacements are likewise assumed not to have significant impact on the spectra.