Energy Partition between Submesoscale Internal Waves and Quasigeostrophic Vortical Motion in the Pycnocline

Anda Vladoiu aApplied Physics Laboratory, University of Washington, Seattle, Washington

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

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Eric Kunze bNorthWest Research Associates, Redmond, Washington

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Abstract

Shipboard ADCP velocity and towed CTD chain density measurements from the eastern North Pacific pycnocline are used to segregate energy between linear internal waves (IW) and linear vortical motion [quasigeostrophy (QG)] in 2D wavenumber space spanning submesoscale horizontal wavelengths λx ∼ 1–50 km and finescale vertical wavelengths λz ∼ 7–100 m. Helmholtz decomposition and a new Burger number (Bu) decomposition yield similar results despite different methodologies. While these wavelengths are conventionally attributed to internal waves, both QG and IW contribute significantly at all measured scales. Partition between IW and QG total energies depends on Bu. For Bu < 0.01, available potential energy EP exceeds horizontal kinetic energy EK and is contributed mostly by QG. In contrast, energy is nearly equipartitioned between QG and IW for Bu ≫ 1. For Bu < 2, EK is contributed mainly by IW, and EP by QG, while, for Bu > 2, contributions are reversed. Finescale near-inertial IW dominate vertical shear variance, implying negligible QG contribution to vertical shear instability. In contrast, both QG and IW at the smallest λx ∼ 1 km contribute large horizontal shear variance, so that both may lead to horizontal shear instability, while QG, with its longer time scales, likely dominates isopycnal stirring. Both QG and IW contribute to vortex stretching at small vertical scales. For QG, the relative vorticity contribution to linear potential vorticity anomaly increases with decreasing horizontal and increasing vertical scales.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. 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

Shipboard ADCP velocity and towed CTD chain density measurements from the eastern North Pacific pycnocline are used to segregate energy between linear internal waves (IW) and linear vortical motion [quasigeostrophy (QG)] in 2D wavenumber space spanning submesoscale horizontal wavelengths λx ∼ 1–50 km and finescale vertical wavelengths λz ∼ 7–100 m. Helmholtz decomposition and a new Burger number (Bu) decomposition yield similar results despite different methodologies. While these wavelengths are conventionally attributed to internal waves, both QG and IW contribute significantly at all measured scales. Partition between IW and QG total energies depends on Bu. For Bu < 0.01, available potential energy EP exceeds horizontal kinetic energy EK and is contributed mostly by QG. In contrast, energy is nearly equipartitioned between QG and IW for Bu ≫ 1. For Bu < 2, EK is contributed mainly by IW, and EP by QG, while, for Bu > 2, contributions are reversed. Finescale near-inertial IW dominate vertical shear variance, implying negligible QG contribution to vertical shear instability. In contrast, both QG and IW at the smallest λx ∼ 1 km contribute large horizontal shear variance, so that both may lead to horizontal shear instability, while QG, with its longer time scales, likely dominates isopycnal stirring. Both QG and IW contribute to vortex stretching at small vertical scales. For QG, the relative vorticity contribution to linear potential vorticity anomaly increases with decreasing horizontal and increasing vertical scales.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. 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, velocity and density fields with Rossby numbers Ro = Uh/f < 1 and gradient Froude numbers Fr = Uz/N < 1 can be linearly decomposed into three eigenmodes, where Uh=(ux2+uy2+υx2+υy2)1/2,Uz=(uz2+υz2)1/2 are horizontal and vertical shears with u zonal and υ meridional velocity components, respectively, and subscripts denote partial derivatives, N is buoyancy frequency and f is Coriolis frequency. Two modes correspond to linear internal gravity waves described by a dispersion relation, and one to vortical mode including geostrophy and quasigeostrophy (e.g., Müller 1984; Müller et al. 1988). Vortical mode has zero frequency and, contrary to internal waves, carries nonzero perturbation potential vorticity
P V=(υ/xu/y)relativevorticityfξ/zvortexstretching,
where ξ is vertical isopycnal displacement. Commonly, interior meso- and submesoscale oceanic motions have been categorized into geostrophic or quasigeostrophic (QG, sometimes referred to as balanced) and ageostrophic, or unbalanced, such as tides and internal gravity waves (e.g., Torres et al. 2018).

Internal waves (IW) and vortical motions (VM) can coexist on the submeso- and finescale (Müller et al. 1986). Separating and quantifying their energy is important because they have distinct kinematic properties and dynamics, and therefore different contributions to vertical mixing and lateral dispersion. Internal-wave energy is generated mostly at boundaries (e.g., Althaus et al. 2003; Nash et al. 2007) and propagates into the interior from forcing sites (e.g., Garrett and Munk 1979; Lozovatsky et al. 2003; Alford et al. 2016) where weakly nonlinear wave/wave interactions transfer energy to fill in the frequency-wavenumber spectrum (McComas and Müller 1981). Internal waves are thought to be responsible for most turbulence production, thus dissipation and diapycnal mixing, in the stratified interior (e.g., Gregg and Sanford 1988; Polzin et al. 1995; Waterhouse et al. 2014; Kunze 2017), while the role that VM plays in turbulence production is poorly understood. The geostrophic eddy field stirs properties along isopycnals (e.g., Polzin and Ferrari 2004; Sundermeyer et al. 2005; Sundermeyer and Lelong 2005), while the IW contribution to lateral dispersion, which is sensitive to nonlinear perturbation of purely oscillatory flow, is uncertain (Ledwell et al. 1998). In the following, VM, as resolved by the measurements presented herein with Ro < 1 and Fr < 1, will be referred to as QG (justified in Fig. 14), and VM and QG will be used interchangeably.

IW have intrinsic frequencies ωf. The conventional theoretical framework is the Garrett and Munk (1979) spectral model (GM). Conversely, QG has ω < f. Vertically Doppler-shifted IW and QG have been separated in fine-resolution time series through transformation to isopycnal coordinates on the finescale to identify Lagrangian or intrinsic frequencies (Sherman and Pinkel 1991; Pinkel 2014). Previous measurements have identified QG and IW occupying the same length scales in spatial measurements in the Sargasso Sea with typical midocean background internal-wave fields (Müller et al. 1988; Lien and Müller 1992b; Lien and Sanford 2019), the Beaufort Sea in a weak internal-wave field (D’Asaro and Morehead 1991), in the wake of a seamount (Kunze 1993; Kunze and Sanford 1993), in the Labrador Sea during deep convection (Lilly and Rhines 2002), as well as in an ensemble of midlatitude North Pacific profile time series (Pinkel 2014).

When along–ship track velocity and buoyancy measurements are available, total energy can be decomposed into IW and QG contributions using Helmholtz decomposition along with the kinematic properties of linear internal waves and vortical mode (Bühler et al. 2014). Helmholtz decomposition of divergent and rotational components of the flow has been used primarily with 1D along-track ADCP data to identify the transition length scale between dominantly balanced and unbalanced motions in the most energetic low modes (e.g., Rocha et al. 2016; Soares et al. 2022; Chai and Zhao 2024), motivated by interpretation concerns for the SWOT sea surface altimetry satellite mission (Fu and Ubelmann 2014), though there is no one-to-one correspondence between horizontal kinetic energy and sea surface height (available potential energy) as a function of horizontal wavenumber alone as shown here. Transition length scales between dominant QG and IW have been studied observationally as well as in simulations. Balanced motions have been found to dominate on scales exceeding 8–20 km in western boundary currents (Callies and Ferrari 2013; Qiu et al. 2017, 2022) and the Subtropical Countercurrent (Qiu et al. 2022), 40 km in Drake Passage (Rocha et al. 2016), 70 km in the California Current (Chereskin et al. 2019), to in excess of 150 km within ±10° of the equator and parts of the Southern Ocean (Qiu et al. 2018; Soares et al. 2022; Qiu et al. 2022) and subtropical gyres (Callies and Ferrari 2013). The transition length scale between QG and IW has been found to depend on eddy energy (Qiu et al. 2017, 2022), internal tide energy (Callies and Ferrari 2013; Qiu et al. 2018, 2022) and season (Torres et al. 2018; Chai and Zhao 2024).

Different QG dynamics have been characterized with distinct energy spectral slopes in horizontal wavenumber. Low-wavenumber spectral slopes of −3 for QG horizontal kinetic energy were found in strong eddy fields associated with western boundary currents and the ACC, consistent with interior QG (Charney 1971). But in the California Current, where the mesoscale is weak and submesoscale strong (Chereskin et al. 2019), slopes of −2 may be a signature of frontogenesis, streamers and filaments (Boyd 1992; Molemaker et al. 2010; Molemaker and McWilliams 2010). No clear evidence for surface QG spectral slopes of −5/3 (Blumen 1978) has been reported.

Previous studies applied Helmholtz decomposition in 1D along-track wavenumber, which resolved the most energetic vertical scales. Here, Helmholtz decomposition will be extended to 2D in along-track and vertical wavenumbers (kx, kz) to examine the vertical finescale and horizontal submesoscale, not just the most energetic vertical scales.

A new Burger number decomposition method is also used to separate QG and IW energies based on their distinct functional dependence of the ratios of available potential to horizontal kinetic energy EP/EK on Burger number Bu=[Nkh/(fkz)]2, where kh and kz are horizontal and vertical wavenumbers, respectively (Müller et al. 1978; D’Asaro and Morehead 1991; Lien and Müller 1992a; Kunze 1993).

This study investigates energy contributions from IW and QG as functions of horizontal submesoscale and vertical finescale wavelengths. Shipboard ADCP and towed CTD chain data from the summer eastern North Pacific pycnocline resolving horizontal wavelengths 1 < λx < 50 km and vertical wavelengths 7 < λz < 100 m are used to compute 2D (kx, kz) wavenumber spectra for horizontal kinetic and available potential energy (sections 2a and 2b). Unresolved λz > 100 m from 1D kx spectra are also investigated assuming they correspond to a vertical wavenumber band 100 < λz < 1000 m. Energy partition into IW and QG components is performed in (kx, kz) space using (i) Helmholtz decomposition (Bühler et al. 2014, 2017, section 2c) and (ii) Burger number decomposition (section 2d). IW and QG signatures are identified in terms of total available potential to horizontal kinetic energy ratio EP/EK and Bu (section 3). The two decomposition methods yield nearly identical relative contributions of IW and QG total energies (section 4). IW and QG total energies are further partitioned into kinetic and potential components using the Bu decomposition (section 5). Sections 6 and 7 discuss 1D horizontal and vertical wavenumber spectra of total, horizontal kinetic, and available potential energies, as well as implications of the decomposition. IW and QG contributions to horizontal and vertical shears are examined in different horizontal and vertical wave bands (section 8). Last, section 9 investigates the linear components of potential vorticity, relative vorticity, and vortex-stretching, as well as the relative contributions of QG vortex stretching and relative vorticity to potential vorticity, as functions of Bu. Conclusions are presented in section 10. Section 11 reviews results in light of previous studies and discusses the outlook for future research.

2. Data and methods

Data were collected below the <10-m-thick summertime mixed layer at pycnocline depths of 20–120 m on the western edge of the California Current System off Baja California, 32.8°–34.8°N, 125.5°–127°W, during July 2018 (Fig. 1; Vladoiu et al. 2022). This region is characterized by a rich submesoscale (e.g., Chereskin et al. 2000). Strong isopycnal water-mass variability on a wide range of scales suggests significant isopycnal stirring. Along-track and vertical wavenumber (kx, kz) spectra for vertical displacement, isopycnal slope, and salinity gradient in this region can mostly be explained by internal waves at horizontal wavelengths exceeding 1 km (Vladoiu et al. 2022). However, the possibility of comparable contributions from QG was not excluded. Segregation of the energy into IW and QG is the primary purpose of this paper. Because the summertime measurements in the pycnocline only resolve Ro < 1, Fr < 1 and hydrostatic aspect ratios, as well as the mixed-layer being less than 10 m thick, nonlinear motions or non-internal-wave divergence such as mixed-layer instabilities, fronts, and eddies (Fox-Kemper et al. 2008) are not expected to have a significant impact (Torres et al. 2022; Cao et al. 2023).

Fig. 1.
Fig. 1.

(a) Map of five 50-km-long sections (chronologically blue, green, yellow, red, and gray; start and end times indicated) with ship speed and direction (arrows; red and black arrows are shifted from center by 0.01°). (b) Comparison of internal-wave phase speed C assuming a horizontally isotropic internal-wave field (blue curves for different kz), mean ship speed Uship (dashed black) ± one standard deviation (shading), and (1/3)Uship (dash–dotted black). Space–time aliasing is negligible for kx > 2 × 10−5 cpm (λx < 50 km) where C<(1/3)Uship.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

a. Shipboard ADCP velocity (kx, kz) spectra

Velocity measurements from five 50-km-long sections (Fig. 1) were obtained from two shipboard ADCPs. Composite velocities between 20- and 120-m depth using 2-m resolution 20–75-m depth data from 300-kHz and 8-m resolution 75–120-m depth data from 75 kHz were linearly interpolated onto a 2-m vertical resolution grid. ADCP data were ensemble-averaged at 1-min intervals, corresponding to ∼120-m horizontal resolution for the ∼2 m s−1 ship speed. The noise level of 1-min averaged ADCP velocity is ∼0.01 m s−1. Velocities were projected onto along- and across-track components (u, υ).

Along-track and vertical wavenumber (kx, kz) spectra for along- (Cu) and across-track (Cυ) velocities were computed using a single 2D sinusoidal taper (Riedel and Sidorenko 1995). Wavenumbers will be used to describe spectral properties throughout, while wavelengths where possible to be more intuitive. A transfer function, inferred from the ratio of ensemble-averaged white-noise spectra with and without interpolation (Vladoiu et al. 2022), was applied to correct for high-wavenumber attenuation due to along-track interpolation onto a regular grid. Analysis is restricted to vertical wavelengths λz > 7 m to avoid noise contamination. While 7 m is resolved by the 300-kHz ADCP (Nyquist 4 m), it is not by the 75-kHz ADCP (Nyquist 16 m). Inclusion of 7 < λz < 16 m in the analysis does not impact results, but skepticism is advised in interpreting spectra in this wave band. Horizontal kinetic energy spectra EK(kx, kz) were computed as EK=(1/2)(Cu+Cυ) (Fig. 2a). In the following, spectra are presented in positive 2D wavenumbers (kx, kz) from the sum of the four quadrants. Space–time aliasing is not significant because the ∼2 m s−1 ship speed exceeds typical internal-wave phase speeds (Fig. 1b; Vladoiu et al. 2022).

Fig. 2.
Fig. 2.

Measured 2D wavenumber (kx, kz) spectra for (a) horizontal kinetic energy EK and (b) available potential energy EP, averaged over all five sections. White diagonals denote constant Burger numbers Bu(Nkx/fkz)2 (assuming ky = 0), dashed-black diagonals internal-wave frequencies ω = f(1 + Bu)1/2 = 1.1f and =M2 (semidiurnal tidal frequency).

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

b. Towed CTD chain isopycnal displacement (kx, kz) spectra

A towed CTD chain consisting of 56 SeaBird MicroCats provides simultaneous temperature, salinity and pressure measurements at 8-s temporal resolution in the pycnocline. The dataset and projection onto a regular depth-along-track grid are described in Vladoiu et al. (2022). Over the 20–73-m depth range where vertical resolution is ∼10 m, vertical isopycnal displacement ξ is calculated from the potential density σ anomaly and background vertical density gradient as ξ(x,zi)=[σ(x,zi)σ¯x(zi)]/[(σ¯/z)(zi)] for each depth zi, where the overbar denotes along-track averaging. Over the 73–120-m depth where vertical resolution is ∼1 m, ξ is computed directly following the vertical displacement of isopycnals ξ(x,zσ¯)=z(σ¯)zσ¯ (Vladoiu et al. 2022).

Composite isopycnal displacements between 20- and 120-m depth are interpolated onto a 1-m vertical grid. Isopycnal displacement spectra Cξ(kx, kz) were computed and corrected for interpolation similarly to the velocity spectra. Available potential energy spectra were inferred from isopycnal displacement spectra as EP(kx,kz)=(1/2)N2Cξ(kx,kz) (Fig. 2b). The buoyancy frequency varies little over this depth range, with mean N2 = 7.8 × 10−5 s−2 for the five sections.

Measurements are decomposed into depth-means and perturbations u=u+u¯z,υ=υ+υ¯z,ξ=ξ+ξ¯z, where overbars denote depth means and primes are perturbations. The 1D horizontal wavenumber spectral analysis was performed on the depth-mean time series that captures but does not resolve vertical wavelengths larger than 100 m, while 2D spectral analysis was conducted on the perturbation time series that resolves vertical wavelengths of 7–100 m. The 1D along-track horizontal wavenumber kx spectra for the depth means ( u¯z,υ¯z,ξ¯z) were computed and corrected for along-track horizontal interpolation and noise in the same manner as the 2D spectra. The 1D horizontal (vertical) spectra for the resolved wavenumbers are obtained by integrating 2D (kx, kz) spectra over different kz (kx) wave bands.

c. Helmholtz decomposition of internal wave (IW) and quasigeostrophic (QG) energies

Helmholtz decomposition is used to segregate IW and QG energy from along-track measurements of horizontal velocity and buoyancy (e.g., Bühler et al. 2014; Lindborg 2015; Bühler et al. 2017; Li and Lindborg 2018). To our knowledge, Helmholtz decomposition has previously only been applied to 1D ship track (u, υ) data either at a single depth or depth-averaged so only isolated the most energetic signal in vertical wavenumber, presumably low mode. Here, the depth variability of ADCP profiles is used to apply Helmholtz to resolved vertical scales as well as the depth-average. The 2D wavenumber (kx, kz) spectra for divergent Kdiv and rotational Krot horizontal kinetic energies can be computed from Helmholtz decomposition of along-track velocity spectra Cu(kx, kz) and across-track velocity spectra Cυ(kx, kz) from the shipboard ADCP data, allowing for horizontal anisotropy
Kdiv(kx,kz)=12{Cu(kx,kz)1kxkx[Cυ(kx,kz)Cu(kx,kz)+σu2συ2σuυ2Cuυ(kx,kz)]dkx},
Krot(kx,kz)=12{Cυ(kx,kz)+1kxkx[Cυ(kx,kz)Cu(kx,kz)+σu2συ2σuυ2Cuυ(kx,kz)]dkx}
(Bühler et al. 2017). For isotropic flow with cross-spectrum C = 0, the last terms in Eqs. (2) and (3) vanish (Bühler et al. 2014). In this dataset, isotropic (Bühler et al. 2014) and anisotropic (Bühler et al. 2017) methods yield nearly identical results (appendix A) so only isotropic results are presented.

Divergent (Cdiv) and rotational (Crot) velocity spectra are twice their corresponding kinetic energy spectra by definition, i.e., Cdiv = 2Kdiv, Crot = 2Krot. Helmholtz decomposition formally requires integration to infinite wavenumbers [Eqs. (2) and (3)] so results may be biased by finite Nyquist wavenumber or measurement noise at high kx (appendix A). Because QG motion is horizontally nondivergent and the sum of IW rotational kinetic energy and available potential energy is equal to IW divergent kinetic energy (Bühler et al. 2014; Lindborg 2015), total internal-wave energy EIW (horizontal kinetic plus available potential energy) can be equated to twice the divergent kinetic energy spectrum, i.e., EIW = Cdiv = 2Kdiv, under hydrostatic conditions. The expectation is that total energy E = EK + EP (where EK and EP are measured horizontal kinetic and available potential energy spectra, respectively), contributed by both IW and QG, will exceed EIW = Cdiv, with the residual representing QG contributions. However, this is not always the case for the Helmholtz decomposition of ocean data because of inherent measurement errors and statistical uncertainty of the spectra. Instances of EIW > E were discarded (accounting for 10.3% of the data). The QG total energy spectrum is estimated as the residual of measured total energy from the estimated IW total energy spectrum, i.e., EQG = EEIW, of the remaining data.

d. Burger number decomposition of internal wave (IW) and quasigeostrophic (QG) energies

An alternative method for segregating IW and QG energy uses the different IW and QG functional dependences (consistency relations) of energy ratios EP/EK on Burger number Bu=[Nkh/(fkz)]2. For Rossby numbers Ro = |Uh|/f < 1 and gradient Froude numbers Fr = |Uz|/N < 1, the energy ratio EP/EK for IW is
RIW=EIWP(kh,kz)EIWK(kh,kz)=Bu(1+Buf2N2)Bu+2BuBu+2,
and for QG is
RQG=EQGP(kh,kz)EQGK(kh,kz)=1Bu
(D’Asaro and Morehead 1991; Lien and Müller 1992a; Kunze 1993), where the approximation for IW potential-to-kinetic energy ratio RIW on the RHS of Eq. (4) applies to hydrostatic waves [ωN, kh=(kx2+ky2)1/2kz]. Although the IW hydrostatic approximation is shown here and in appendix B for simplicity, the full nonhydrostatic expressions were used in the analysis. With increasing Bu, the ratio of potential-to-kinetic energy decreases for QG and increases for IW (Fig. 3). At Bu = 2, the two consistency curves intersect so that separation into IW and QG energies is not possible.
Fig. 3.
Fig. 3.

2D histogram of measured available potential to horizontal kinetic energy ratio EP/EK as a function of Bu for resolved 7 < λz < 100 m (yellow–brown shading) and unresolved λz > 100 m (assuming kz wave band 10−3–0.01 cpm, empty gray circles), with means in different vertical and horizontal wave bands (large green symbols). Theoretical EP/EK dependences on Bu for internal waves [Eq. (4); red, with reference frequencies marked], geostrophy [Eq. (5), thermal wind; blue], and log10(EIW/EQG) [Eq. (B7); blue–red contours]. Black dotted lines denote EP/EKf/N Bu−1 for Bu < 10−2, EP/EK ∼ 0.44 for 10−2 < Bu < 1, and EP/EK ∼ 0.41 for Bu > 1. The smaller dark green triangle and circle (without black edges) indicate the largest horizontal scales for 7 < λz < 30 m and λz > 100 m, respectively.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

IW total energy EIW, QG total energy EQG, their available potential ( EIWP,EQGP) and horizontal kinetic ( EIWK,EQGK) components, as well as their energy ratios [Eqs. (B1)(B7)], were derived from the consistency relations [Eqs. (4) and (5)] assuming linearity (appendix B).

Unlike Helmholtz decomposition, the Burger number decomposition does not rely on spectral integration to infinite kh but is applied to each wavenumber locally. However, the Burger number decomposition requires resolving horizontal and vertical length scales, and is degenerate for Bu ∼ 2. To apply the Burger number decomposition, measured (kx, kz) spectra are converted to (kh, kz) assuming horizontal isotropy [Eq. (D2); Henyey 1991; Williams et al. 2002]. Bu is computed with bulk N = 8.8 × 10−3 s−1 and f = 8.1 × 10−5 s−1 averaged over all sections and depths.

Total energy decomposition into IW and QG is performed independently with the Helmholtz and Burger number methods. In the following, the tilde symbol indicates IW and QG energies inferred from the Burger number decomposition [Eqs. (B1)(B4)]. Consistency relations for EP/EK as a function of Bu [Eqs. (4) and (5)] are used to separate EIW and EQG inferred from the Helmholtz decomposition into their kinetic and potential components.

High-wavenumber EK and EP noise, affecting Cdiv, Crot and the energy ratio, get convolved by both decompositions. As confirmed using theoretical GM spectra with added white noise (not shown), the measurement noise uncertainty reflected by the Bu decomposition is largest at high wavenumbers where it is estimated to be at most 5%. Noise in the Helmholtz decomposition will lead to equipartition of Cdiv and Crot (appendix A) and potentially more equipartition between EIW and EQG. Therefore, energy equipartition at high wavenumbers found by both Helmoholtz and Bu decompositions (Figs. 6 and 7) may be due to measurement noise.

3. Observed 2D available potential and horizontal kinetic energy and their ratio

Before segregating energy between quasigeostrophy (QG) and internal waves (IW), it is instructive to examine available potential to horizontal kinetic energy ratios EP/EK as a function of Burger number Bu=N2kx2/(f2kz2) (Fig. 3). This reveals that EP dominates at low Bu and EK plays an increasing role with increasing Bu, and that energy ratios tend to lie between the consistency curves [Eqs. (4) and (5)], indicating a superposition of QG and IW.

Observed 2D wavenumber (kx, kz) spectra for horizontal kinetic energy EK and available potential energy EP span BuO(104100), corresponding to internal-wave frequencies ωf(1+Bu)1/2O(1.0005f17f) (Fig. 2). Most of the energy lies at low kx and kz, while signal to noise is low at high kx and kz.

Observed energy ratios EP/EK exhibit distinct dependences on Bu (Fig. 3):

  • For low BuO(1040.01), corresponding to layers of resolved small vertical and large horizontal wavelengths, 7 < λz < 30 m and 25 < λx < 50 km, EP exceeds EK (median EP/EK = 1.66 ± 0.02, standard error), consistent with the density layering reported by Pinkel (2014), with EP/EKf/(N Bu).

  • For Bu > 0.01 and resolved vertical wavelengths, median EP/EK = 0.43.

For measured Rossby numbers Ro < 1 and gradient Froude numbers Fr < 1 (to be described later), pycnocline processes are mostly linear so will consist of QG eddies and internal gravity waves (Müller et al. 1978; Lien and Müller 1992a,b)

  • For large unresolved vertical wavelengths λz > 100 m and horizontal wavelengths 10 < λx < 50 km, energy ratios lie roughly along the IW curve.

  • For resolved vertical wavelengths 7 < λz < 100 m, most energy ratios lie between the internal-wave (IW) and geostrophic (QG) consistency curves, implying that observed energy is a superposition of IW and QG.

4. QG and IW energy partition and Burger-number dependence

Partitioning of total energy EP + EK reveals that QG dominates for Bu < 0.01 while IW for Bu > 0.01. Total energy spectra for IW and QG computed from Helmholtz and Burger number decompositions (sections 2c and 2d) exhibit similar shapes and levels (Fig. 4). Most of the variance is at low kh and low kz for both IW and QG. The most notable differences are for Bu < 0.01 where Bu decomposition exhibits less IW energy, and for Bu ∼ 2 where Bu decomposition is degenerate so it cannot distinguish IW and QG. IW and QG energies reveal strong distinct dependence on Bu in two different Bu regimes, Bu < 0.01 and Bu > 0.01.

Fig. 4.
Fig. 4.

2D (kh, kz) wavenumber spectra for (a),(c) IW and (b),(d) QG total energies inferred from (top) Helmholtz and (bottom) Burger number decompositions, averaged over all five sections. White diagonals denote constant Bu and gray diagonal internal-wave frequency ω = f(1 + Bu)1/2. Gray shading masks degenerate Burger number decomposition at Bu ∼ 2.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

For Bu < 0.01:

  • QG dominates with 68% ± 17% of the total energy;

  • measured EP/EKf/(N Bu) (Fig. 3) and the IW-to-QG total energy ratio EQG/EIWf/(N Bu) (Figs. 7a,b) as expected because EQGEQGP [Eq. (5)] and EIWEIWK for Bu ≪ 1 [Eq. (4)].

For Bu > 0.01:

  • IW dominates with 61% ± 18% on average (Figs. 5 and 7a,b), by as much as a factor of ∼15 more total energy than QG (Fig. 6a);

    Fig. 5.
    Fig. 5.

    Percentage of total energy spectra contributed by (a) IW and (b) QG, as inferred from Helmholtz decomposition. Black diagonals denote constant Bu and internal-wave frequency ω = f(1 + Bu)1/2.

    Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

    Fig. 6.
    Fig. 6.

    Ratios of internal-wave (IW) to quasigeostrophic (QG) (a) total energy E, (b) horizontal kinetic energy EK, and (c) available potential energy EP from the Helmholtz decomposition. Black diagonals denote constant Burger numbers Bu and internal-wave frequencies ω = f(1 + Bu)1/2.

    Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

  • for 0.01 < Bu < 2, EP/EK ∼ 0.44 (Fig. 3) and EQG/EIW ∼ 0.44 (Figs. 3 and 7a,b), as expected from Eqs. (4), (5), and (B7);

    Fig. 7.
    Fig. 7.

    (a) Ratios of quasigeostrophic (QG) to internal-wave (IW) total energy from Helmholtz (blue) and Burger number (red) decompositions. (b) Ratios of internal wave (red) and quasigeostrophic (blue) to total energy. (c) Ratios of internal-wave to quasigeostrophic horizontal kinetic (red) and available potential (blue) energy. In all panels, curves are bin averages (Δlog10 Bu = 0.08), and shading indicates standard deviations. In (b) and (c), darker colors are inferred from Helmholtz, and lighter colors are from Burger number decompositions. Black curves in (a) are analytic expressions [Eq. (B7)] derived using median observed ratios from Fig. 3: EP/EKf/N Bu−1 for Bu < 10−2 (black dash–dotted line), EP/EK ∼ 0.44 for 10−2 < Bu < 1 (black solid line), and EP/EK ∼ 0.41 for Bu > 1 (black dotted line), with thin black lines marking ratios 25th and 75th percentiles; red and blue thin curves in (b) and (c) are equivalent expressions. Horizontal gray lines indicate ratios of 0.5 in (b) and 1 in (a) and (c); vertical gray lines are Bu = 0.01 (ω = 1.005f) and Bu = 2 (ω = 1.73f).

    Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

  • for large Bu > 2, EP/EK ∼ 0.41 (Fig. 3) and EQG/EIW ∼ 0.74 (Figs. 7a,b), but may be contaminated by proximity to the Nyquist and low ADCP signal-to-noise ratio.

5. QG and IW components of horizontal kinetic and available potential energy

Further partitioning of EP and EK reveals that QG is dominated by EP, i.e., density layers, and IW by EK, i.e., near-inertial waves, at low Bu < 2, while the roles are reversed for Bu > 2. Horizontal kinetic and available potential energies are decomposed into IW and QG components using consistency relations [Eqs. (4) and (5)] and reveal clear dependence on Bu:

  • for Bu < 1, IW have up to three orders of magnitude more horizontal kinetic energy than QG (Figs. 6b and 7c), while QG contributes most of the available potential energy (Figs. 6c and 7c);

  • for Bu > 1, IW contribute increasingly more EP than QG, while QG contributes increasingly more EK than IW (Fig. 7c).

6. QG and IW total energy 1D spectral properties

Spectral levels and slopes for both QG and IW total energy EP + EK resemble the Garrett and Munk (1979) model spectrum with spectral slopes between −2.5 and −1, kx spectra becoming bluer at higher kz, and kz spectra becoming bluer at higher kx. QG spectra tend to have less energy at low kx and kz, reflected by slightly bluer slopes.

a. 1D total energy horizontal wavenumber spectra

The 1D horizontal wavenumber kx spectra for observed total energy EIW + EQG, total EIW and total EQG exhibit spectral slopes between −2.5 and −1, tending to be bluer for smaller vertical wavelengths (Figs. 8a,c,e). For unresolved vertical wavelengths λz > 100 m (Fig. 8a), IW spectra have similar shapes to the GM model (appendix C), but lower levels EIWEIWGM/3, consistent with the weaker internal-wave field in this dataset (Vladoiu et al. 2022). Mean spectral slopes estimated from first-order polynomial fits are −2.4 ± 0.2. For resolved vertical wavelengths 7 < λz < 100 m (Figs. 8c,e), IW spectra exhibit slopes of −1.5 ± 0.1. Higher spectral level than GM at low kx may be the result of a storm passing one to two days prior, energizing intermediate vertical wavelengths.

Fig. 8.
Fig. 8.

Average 1D along-track wavenumber kx spectra for total energy E (black), internal-wave energy EIW (solid red), and quasigeostrophic energy EQG (solid blue) from Helmholtz decomposition, IW energy EIW˜ (dash–dotted orange) and QG energy EQG˜ (dash–dotted light blue) from Burger number decomposition, and IW energy EIŴ (dotted dark red) and QG energy EQĜ (dotted dark blue) from VS and RV [Eq. (E8)]. The kx spectra were inferred from (a) 1D kx spectra for depth-means λz > 100 m, and from 2D (kx, kz) spectra by integrating over vertical wavelengths (c) 30 < λz < 100 m and (e) 7 < λz < 30 m. Shading shows standard errors. Thin black curves are GM spectra. Black dotted diagonals show spectral slopes for reference. (b),(d),(f) Corresponding IW (shades of red) and QG (shades of blue) percentages relative to total energy.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

QG kx spectral slopes are similar to IW irrespective of vertical wave band, in contrast to results from western boundary currents (Callies and Ferrari 2013; Rocha et al. 2016) but consistent with previous analysis in the California Current (Chereskin et al. 2019). QG total energy spectra for large λz > 100 m have slopes of −2.1 ± 0.2, similar to theoretical slopes for front formation (Boyd 1992), as well as streaks and filaments (Molemaker et al. 2010; Molemaker and McWilliams 2010). QG spectra for intermediate 30 < λz < 100 m have −1.8 ± 0.6 spectral slopes, consistent with either fronts or −5/3 surface QG (Blumen 1978), which has not been previously reported. QG spectra for small 7 < λz < 30 m have −1 ± 0.3 spectral slopes, consistent with theoretical −1 slopes expected from observed Cυ/CuO(1) for nondivergent flow (not shown; Batchelor 1959), but not with interior QG. This could be due to stirring of passive variance at small vertical scales by horizontal shears on larger (Rossby) scales (Scott 2006), since EQG is dominated by EQGP on these scales (Fig. 6c). Interior QG slopes of −3 driven by deep baroclinic instabilities (Charney 1971) are not observed on the 1–50-km horizontal scales resolved here.

IW contributes ∼70% and QG ∼ 30% to the total energy for (i) all kx and 30 < λz < 100 m, (ii) kx < 2 × 10−4 cpm and λz > 100 m, as well as (iii) kx > 6 × 10−5 cpm and 7 < λz < 30 m (Figs. 8d,b,f), consistent with previous Helmholtz decomposition results at similar kx in the Sargasso Sea (Lien and Sanford 2019). Near-equipartition of QG and IW for kx > 3 × 10−4 cpm and λz > 100 m (Fig. 8b) is consistent with Bu > 0.01 in Fig. 3. There is some ambiguity associated with unresolved large vertical scales due to the choice of vertical wave band assumed 100 < λz < 1000 m. Near-equipartition for 7 < λz < 30 m arises from integrating in kz over different Bu (Fig. 5). This raises a concern about interpreting 1D wavenumber spectra which smear across different Bu regimes for IW and QG.

b. 1D total energy vertical wavenumber spectra

Observed IW total energy kz spectra exhibit spectral shapes similar to GM (Fig. 9a). Internal waves contribute ∼70% of the energy for kx < 5 × 10−2 cpm (Fig. 9b). At higher kz, the QG contribution increases, with near-equipartition at kz ∼ 10−1 cpm. There is little agreement between the different methods for kz > 10−1 cpm where velocity is not well-resolved, so this wave band is not reliable.

Fig. 9.
Fig. 9.

(a) Average 1D vertical wavenumber kz spectra for total energy E (black), internal-wave energy EIW (solid red), and quasigeostrophic energy EQG (solid blue) from Helmholtz decomposition, IW energy EIW˜ (dash–dotted orange) and QG energy EQG˜ (dash–dotted light blue) from Burger number decomposition, and IW energy EIŴ (dotted dark red) and QG energy EQĜ (dotted dark blue) from VS and RV [Eq. (E8)]. The kz spectra were estimated from (kx, kz) spectra integrated over 1 < λx < 50 km. Variances in the depth-mean (λz > 100 m) from 1D kx spectra normalized by the vertical wavenumber band 10−3–10−2 cpm (100 < λz < 1000 m), provide spectral levels at the lowest kz ∼ 5.5 × 10−3 cpm (circles and crosses). Shading shows standard errors and thin black curve the GM spectrum. The gray band masks unreliable 7 < λz < 16 m. (b) Corresponding IW (shades of red) and QG (shades of blue) percentages relative to total energy.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

7. QG and IW horizontal kinetic and available potential energy 1D spectral properties

The different Bu dependences for the QG and IW energy ratios leads to more varied EK and EP spectra. QG EK spectra are bluer and lower than IW for decreasing vertical wavelengths while QG EP are redder with lower levels at large λz and higher at small λz.

a. 1D horizontal wavenumber spectra

Previous analyses were often performed in 1D horizontal wavenumber space at a fixed depth or averaged over a vertical layer (e.g., Rocha et al. 2016; Soares et al. 2022). Here, we investigate horizontal wavenumber kh spectra in different vertical bands. For vertical wavelengths λz > 100 m, which contain the most energy and may be best compared with previous 1D along-track results, both QG and IW EK spectral slopes are ∼−2 (−1.7 ± 0.4 for QG, −2.1 ± 0.3 for IW from first-order polynomial fits to Helmholtz spectra; Fig. 10a). In contrast, IW EP spectral slopes of −1.6 ± 0.2 are very different from QG EP spectral slopes of −3.7 ± 0.4 (Fig. 10d). All EK and EP spectral slopes become bluer at smaller vertical scales.

Fig. 10.
Fig. 10.

Horizontal wavenumber kh spectra for internal-wave (IW) and quasigeostrophic (QG) (top) horizontal kinetic and (bottom) available potential energy, from Helmholtz decomposition (red and dark blue) and Bu decomposition (orange and light blue), for vertical wave bands (a),(d) λz > 100 m; (b),(e) 30 < λz < 100 m; and (c),(f) 7 < λz < 30 m. Shading shows standard errors, and thin dark red curves are the equivalent GM spectra. Bu values are indicated along the upper axes.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

IW EK and EP kh spectra resemble GM for all λz wave bands (Fig. 10). In contrast, QG spectra are harder to interpret using existing QG theories. For λz > 100 m, QG EP spectral slopes of −3.7 are redder and EK spectral slopes of −1.7 bluer than the −3 predicted by interior QG (Charney 1971). That the spectral slopes for EP and EK differ is also inconsistent with interior QG theory. Chereskin et al. (2019) previously reported −2 spectral slopes for IW and QG EK in the California Current, noting that the region has a weak mesoscale and strong submesoscale. Less red slopes than the expected −3 for interior QG may arise because the measurements are on the eastern boundary where mesoscale eddies have to be generated locally and cannot propagate from the east as in the basin interior.

b. 1D vertical wavenumber spectra

At large horizontal scales 2.5 < λx < 50 km, 1D vertical wavenumber kz spectra for EK are mostly contributed by IW while EP mostly contributed by QG (Figs. 11a,c). At small horizontal scales 1 < λx < 2.5 km, EIWKEQGK while EP is mostly contributed by IW for kz < 0.05 cpm. IW EK and EP resemble GM within a factor of 2–3 for all λx.

Fig. 11.
Fig. 11.

Vertical wavenumber kz spectra for internal-wave (IW) and quasigeostrophic (QG) (top) horizontal kinetic and (bottom) available potential energy, from Helmholtz decomposition (red and dark blue) and Bu decomposition (orange and light blue), for horizontal wave bands (a),(c) 2.5 < λx < 50 km and (b),(d) 1 < λx < 2.5 km. Shading shows standard errors, and thin dark red curves are GM spectra.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

For the smallest resolved vertical wavelengths 7 < λz < 30 m, Bu < 1 and EQGP>EQGK at all kh (Figs. 10c,f and 11), resembling the finescale density layering reported by Pinkel (2014). Therefore, interpreting total measured EP spectra for small vertical scales, especially smaller than 30 m, as internal waves, as customary in the finescale shear and/or strain parameterization for turbulent dissipation rates and diffusivities (Gregg and Kunze 1991; Wijesekera et al. 1993; Polzin et al. 1995; Whalen et al. 2012; Kunze 2019), may be inappropriate.

8. QG and IW horizontal and vertical shear

The partition of kinetic energy into IW and QG can be used to infer their relative contributions to horizontal shear CUh(kh,kz)=2(2πkh)2EK(kh,kz) and vertical shear CUz(kh,kz)=2(2πkz)2EK(kh,kz), in 2D as well as 1D wavenumber space. IW horizontal and vertical shears are consistent with GM (not shown), as expected from Figs. 10 and 11.

The total normalized rms horizontal shear Uh/f is ∼0.66 for QG and 0.59 for IW, for resolved 16 < λz < 100 m and 1 < λx < 50 km. The largest QG horizontal shear variance is contributed by kh ∼ 10−3 cpm at all kz (Figs. 12 and 6b). Comparable high kh IW CUh is limited to high kz. At lower kh, IW CUh is progressively larger than QG with increasing kz in the near-inertial band. Even with comparable IW and QG horizontal shear variance, QG is expected to dominate stirring because of its longer time scales compared to transient internal waves. Smaller vertical wavelengths 7 < λz < 16 m are displayed in Fig. 12 but, though they show the same trends, low signal to noise and proximity to the Nyquist make them quantitatively suspect so magnitudes are not quoted from this band.

Fig. 12.
Fig. 12.

2D (kh, kz) wavenumber spectra for (a) IW and (b) QG horizontal shear from the Helmholtz decomposition, averaged over all five sections. Black diagonals denote constant values of Bu or internal-wave frequency ω = f(1 + Bu)1/2. 1D kh spectra for unresolved λz > 100 m are shown as bars below the 2D panels. Gray shading masks unreliable 7 < λz < 16 m.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

Total normalized rms (for resolved 16 < λz < 100 m, 1 < λx < 50 km) vertical shear Uz/N is 0.35 for QG and 0.67 for IW. The largest vertical shear variance is contributed by IW at high kz in the near-inertial band (Fig. 13). Large IW and insignificant QG CUz near kz ∼ 0.06 cpm for low kh are consistent with Polzin et al. (1995). IW CUz is larger than QG except at the smallest resolved vertical and horizontal wavelengths, where the spectra are unreliable.

Fig. 13.
Fig. 13.

2D (kh, kz) wavenumber spectra for (a) IW and (b) QG vertical shear from the Helmholtz decomposition, averaged over all five sections. Black diagonals denote constant values of Bu or internal-wave frequency ω = f(1 + Bu)1/2. The gray shading masks unreliable 7 < λz < 16 m.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

IW and QG shear dynamics can be summarized in Rossby number Ro = |Uh|/f and gradient Froude number Fr = |Uz|/N parameter space, intrinsically related through Bu = (Ro/Fr)2 (Fig. 14). For Ro<O(1), vortical motions can be characterized as geostrophic for Bu < Ro, and quasigeostrophic for Bu > Ro (Müller et al. 1978). With Ro, Fr < 1 (Fig. 14), measured shears are only weakly nonlinear. Our observations reside mostly in the quasigeostrophic regime (Müller 1984) with measured QG roughly following Fr20.1Ro, implying U0.1N2kh/(fkz2) over nearly 2 decades in Fr and 4 decades in Ro. Any physical implication of this relation needs further investigation.

Fig. 14.
Fig. 14.

Gradient Froude number Fr = |Uz|/N and Rossby number Ro = |Uh|/f from 2D Helmholtz internal-wave (pink clouds) and quasigeostrophic (light blue clouds) EK. Averages in different wave bands are indicated by symbols corresponding to λz and colors denoting λx. Diagonals correspond to constant Bu and internal-wave frequencies ω = f(1 + Bu)1/2. Unstable (gray) and turbulent (yellow, purple) regimes are shaded. Balanced regimes are indicated. With Fr < 0.5 and Ro < 1, processes at the measured length scales are stable. Balanced motions are in the quasigeostrophic regime with Bu ∼ 10Ro. Internal waves occupy f < ω < ∼17f.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

IW have higher Fr than QG for the same horizontal wave band at the smallest λz (Figs. 14 and 13). IW and QG have the highest Ro at small horizontal scales (Figs. 14 and 12). Inclusion of 7 < λz < 16 m, which are not resolved by the 75-kHz ADCP, does not qualitatively impact these patterns. While the measurements did not resolve unstable shear conditions, Figs. 13 and 14 suggest that small vertical scale IW are most prone to vertical instability. If the trends in Fig. 6b were extended to higher unresolved kh, then QG would be most prone to horizontal instability, but this is conjecture.

9. QG and IW vortex-stretching and relative vorticity, and QG potential vorticity

Spectra for QG energy, vortex stretching (CVS) and relative vorticity (CRV) are related through consistency relations (Müller et al. 1988; Lien and Müller 1992a,b). In the following, relative vorticity estimates from along-track velocity measurements and vertical strain from concurrent density measurements are used to infer QG potential vorticity anomaly spectra (CPV; Müller et al. 1988, appendix E).

The 2D (kh, kz) wavenumber spectra for VS, RV, and PV reveal clear dependence on Bu. Total vortex stretching CVS variance decreases, while total relative vorticity CRV variance increases, with increasing Bu (Figs. 15a,d) as is to be expected from Eq. (5). Horizontal divergence CHD resembles CRV (Fig. 15g). This might indicate that RV is mostly contributed by near-inertial waves which have identical HD and RV, reflecting the similarity between Cdiv and Crot (Fig. A2) since near-inertial wave EK dominates.

Fig. 15.
Fig. 15.

Average 2D (kh, kz) wavenumber spectra for (a) total vortex stretching (CVS) and its (b) QG ( CVSQG) and (c) IW ( CVSIW) components; (d) total relative vorticity (CRV) and its (e) QG ( CRVQG) and (f) IW ( CRVIW) components; (g) horizontal divergence (CHD); (h) potential vorticity (CPV); and (i) QG energy ( EQĜ) inferred from CVS and CRV [Eq. (E8)]. Black diagonals denote constant Bu (solid) and internal-wave frequencies ω = f(1 + Bu)1/2 (dashed). The gray shading masks degeneracy at Bu ∼ 2.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

Total observed CVS/CRV ratios lie between the theoretical consistency curves for IW CVSIW/CRVIW=1 and QG CVSQG/CRVQG=1/Bu2 (Fig. 16a), as expected from Fig. 3. CVS larger than CRV and CHD for Bu < 1 supports Chereskin et al. (2019) who found higher rms strain than divergence and vorticity in an 8-km resolution LLC4320 MITgcm simulation of the California Current.

Fig. 16.
Fig. 16.

(a) Ratios of vortex-stretching CVS to relative vorticity CRV as a function of Burger number (total in green, IW component in red, QG component in blue). Black curves indicate consistency relations for IW CVSIW/CRVIW=1 and QG CVSQG/CRVQG=1/Bu2. (b) Ratios CVSQG/CVSIW (green) and CRVQG/CRVIW (blue) as functions of Bu. (c) Theoretical contributions from CVSQG (green) and CRVQG (blue) to CPV as functions of Bu. In all panels, curves are bin averages (bin width in log space Δlog10 Bu = 0.08), and shadings standard deviations.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

For Bu < 1, spectral levels and patterns are similar between CVSQG and CVS (Figs. 15b, 16b, 17, and 18), and between CRVIW and CRV (Figs. 15f, 16b, 17, and 18), i.e., most of the vortex stretching is due to QG (like EP, Fig. 6c) and most of the relative vorticity due to IW (like EK, Fig. 6b). Conversely, for Bu > 1, most of CVS is IW and most of CRV is QG. Since IW have zero PV anomalies (assuming a Lagrangian frame), CPV has similar patterns and spectral levels as CVSQG for Bu < 1 (Figs. 15h and 16c), and CRVQG for Bu > 1. In other words, CPV and CVS are largest at small vertical and large horizontal scales, while the contribution of CRV to CPV increases with increasing Bu and kh (Figs. 15, 17, and 18). Vortex stretching at small vertical scales is contributed by both IW and QG (Fig. 18).

Fig. 17.
Fig. 17.

Normalized 1D kh spectra for (a),(b) IW: CHD (red), CRVIW (light blue), and CVSIW (green crosses); and (c),(d) QG: CPV (black), CRVQG (light blue), and CVSQG (light green) for (left) 30 < λz < 100 m and (right) 7 < λz < 30 m. In all panels, dash–dotted blue curves are total CRV, dash–dotted green curves are total CVS, and dotted black lines reference −1 and +2/3 slopes. Bu values are indicated along the upper axes.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

Fig. 18.
Fig. 18.

Normalized 1D kz spectra for (top) IW: CHD (red), CRVIW (light blue), and CVSIW (green crosses); and (bottom) QG: CPV (black), CRVQG (light blue), and CVSQG (light green) for (a),(d) 25 < λx < 50 km; (b),(e) 10 < λx < 25 km; and (c),(f) 1 < λx < 10 km. In all panels, dash–dotted blue curves are total CRV, dash–dotted green curves are total CVS, and dotted black lines reference −1 and +2/3 slopes.

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

1D spectral properties

The 1D wavenumber spectra of CPV, CRV, CPV, and CHD are presented to provide reference for future studies. For resolved 1 < λx < 50 km, CHD and CRV kh spectral slopes are ∼+2/3 and agree with GM (not shown), but CVS spectral slopes are ∼−1 (Fig. 17). The CPV spectral slopes are close to −1 for kh < 10−4 cpm, flattening for kh > 2 × 10−4 cpm.

Müller et al. (1988) estimated 1D kh spectral slopes of ∼+2/3 for CPV, CHD, CRV and CVS, for 50 < λx < 104 m, integrated over all λz as they did not resolve vertical scales; Lien and Müller (1992a) showed that Müller et al.’s CHD and CRV likely contaminate each other. Shcherbina et al. (2013) report relatively flat CRV kx spectra in the North Atlantic Mode Water winter pycnocline at the same λx resolved here, with shapes similar to GM, consistent with the IW component at larger λz (Fig. 17a). Wang et al. (2023) used MITgcm HD and RV from the South China Sea surface to separate internal waves and vortical motion in (kx, ky, ω); they found CHD kh spectral slope of +2/3 but with higher level than in Fig. 17, and red CRV spectral slope as opposed to ∼ flat (IW) or blue (QG) spectra in Fig. 17, although of similar level.

The 1D kz spectra for CPV are white for all resolved 1 < λx < 50 km, (Figs. 18d–f), consistent with Lien and Sanford (2019).

10. Summary

Concurrent shipboard ADCP and towed CTD chain profile sections along a ∼250-km long ship track from the western edge of the California Current (33°–35°N, 125.5°–127°W) provide velocity and isopycnal displacement measurements from 20–120-m pycnocline depths (Fig. 1). 2D (kx, kz) wavenumber spectra for horizontal kinetic energy EK and available potential energy EP were computed for 1 < λx < 50 km and 7 < λz < 100 m, corresponding to Burger numbers O(104100) (Fig. 2). In addition, 1D along-track wavenumber spectra for unresolved vertical wavelengths > 100 m were presented and compared to previous 1D along-track spectra.

Measured energy ratios EP/EK are functions of Bu and lie between the theoretical consistency relations for QG and IW (D’Asaro and Morehead 1991; Kunze 1993, Fig. 3), indicating the presence of both dynamics. Using Helmholtz (Bühler et al. 2014, section 2c), and Burger number (section 2d) decompositions under assumptions of hydrostatics, linearity and horizontal isotropy, total energy E, horizontal kinetic energy EK and available potential energy EP were segregated between QG and IW. The two methods yield similar results, demonstrating that 2D horizontal velocity and density measurements can separate QG and IW at submesoscales when Ro and Fr are small; we caution that both approach 1 at high wavenumbers in our measurements (Fig. 14) so the linear decompositions may become unreliable (e.g., Kunze 1993). The use of 2D measurements better delineates the partition and reveals Bu dependence between (as well as within) the two dynamic modes. Previous 1D measurements only captured the energetic signal, probably associated with low modes, while here the vertical finescale contributions have also been quantified. Our analysis raises a warning about interpreting 1D spectra that represent integrals over vertical or horizontal wavenumber, so may smear over different dynamical regimes delineated by Bu.

The energy ratio EQG/EIW exhibits strong dependence on Burger number (Figs. 6 and 7), which may prove to be a useful diagnostic. Overall, IW contribute ∼2/3 of the total energy, with QG contributing more for Bu < 0.01 (i.e., low kh and high kz), and IW for Bu > 0.01. The energy ratio EQG/EIW was nearly constant at ∼0.5 over a two-decade range in Burger number, 0.01 < Bu < 5. Three Bu regimes are identified for EQG/EIW. For Bu < 0.01, EQG/EIWEP/EKf/(N Bu) because EQGEP and EIWEK at low Bu. For 0.01 < Bu < 1, EQG/EIWEP/EK ∼ 0.4. For Bu > 1, EQG/EIW ∼ 0.7 and EP/EK ∼ 0.4 (Figs. 7 and 3). For Bu < 2, IW contribute most of the horizontal kinetic energy while QG most of the available potential energy (Figs. 6 and 7c), so the QG contribution at high kz and low kx is mostly passive density finestructure with little EK or shear signal. For Bu > 2, EP is dominantly IW, while EK equipartitioned between QG and IW. Burger numbers Bu ≫ 10 are not well-resolved.

For λz > 16 m, QG Froude number Fr = |Uz|/N < 1 and Rossby number Ro = |Uh|/f < 1 (Fig. 14) are consistent with quasigeostrophy (Müller 1984). For the measured Bu range, internal waves have frequencies ω < 17f. Unstable horizontal shears (|Uh| > f) and vertical shears (|Uz| > 2N) were not resolved. IW vertical shear is largest at the highest vertical wavenumbers in the near-inertial band (Fig. 13) and agrees with GM (Fig. 10). While QG vertical shear is comparable to IW at the highest vertical and horizontal wavenumbers, these signals are unreliable because they are near the vertical and horizontal Nyquists. Independent of vertical wavelength, the largest measured horizontal shear is from QG at the highest horizontal wavenumbers (Fig. 12); IW horizontal shear is comparably elevated at high horizontal wavenumbers. There is ambiguity for λz > 100 m at the highest resolved kx where Ro and aspect ratio approach 1 (Fig. 14) so that linear decompositions may not correctly partition QG and IW energies in this wave band.

IW relative vorticity CRV and horizontal divergence CHD are smaller than QG vortex stretching CVS for Bu < 1, while their relative magnitudes are reversed for Bu > 1 (Fig. 15). IW and QG vortex stretching (available potential energy) are similar at small vertical scales (Fig. 18) as also found by Pinkel (2014). The QG density layering represents a nearly passive finescale analog to the gyre-scale circulation with little horizontal kinetic energy. Spectra for vertical strain and relative vorticity were used to compute CPV and QG energy (Müller et al. 1988, Fig. 15), agreeing between the Helmholtz and Bu decompositions. CPV is dominated by CVS for Bu < 1 and CRV for Bu > 1 (Fig. 16), as expected given QG Bu dependences (B3), (B4), and (B6).

11. Discussion

QG not only dominates for Bu < 0.01 (large horizontal and small vertical wavelengths, Fig. 7a) but significant quasigeostrophic contributions were found on small horizontal and vertical wavelengths conventionally attributed to internal gravity waves (e.g., Garrett and Munk 1979; Kunze et al. 1990). Evidence for finescale QG perturbations has been previously reported (D’Asaro and Morehead 1991; Kunze 1993; Pinkel 2014). Lien and Sanford (2019) applied Helmholtz decomposition and found vortical motions contributing ∼30% of the total energy at similar horizontal wavenumbers in the Sargasso Sea. Kunze et al. (1990) inferred that 1–5-m shear was dominated by finescale near-inertial waves based on shear/strain variance ratio exceeding 1 but did not measure Bu so could not partition IW and QG to identify QG dominance of finescale EP. In our measurements, energy partition exhibits no depth dependence (not shown). Pinkel (2014) also reported comparable IW and QG vertical strain for Bu ∼ 0.1 and λz = 20 m in a similar depth range, but found 3–10 times more IW than QG below 500-m depth where stratification was weaker.

a. Impact of QG on energy and enstrophy cascades

The path to dissipation appears to be through both vertical and horizontal shear instability for IW while horizontal shear instability appears more likely for QG (Figs. 1214). Traditionally, QG and IW dynamics have been treated separately. Low-mode internal waves can propagate from boundary forcing to fill the stratified ocean interior (Garrett and Munk 1979) where wave/wave interactions fill in wavenumber–frequency space (e.g., McComas and Müller 1981; Henyey et al. 1986). QG has zero energy-flux divergence in physical space (Nagai et al. 2015) and its energy cascade is traditionally thought to be upscale (inverse cascade) rather than downscale (forward cascade; e.g., Charney 1971; Ferrari and Wunsch 2009). Recent numerical modeling suggests that the coexistence of QG and IW catalyzes energy exchange and a mutual forward cascade (e.g., Lelong and Riley 1991; Barkan et al. 2017; Sinha et al. 2019; Thomas and Daniel 2021; Hernández-Dueñas et al. 2021), particularly if EIW exceeds EQG (Thomas and Daniel 2020) as here. Numerical simulations suggest that internal waves modulate vortex spectral shapes (Hernández-Dueñas et al. 2021).

QG carries potential vorticity, a conserved quantity like potential temperature or salinity, which can only be altered by boundary forcing or molecular dissipation. Fine- and submesoscale PV in the upper ocean could be (i) a subducted relict from boundary generation (though how long it can persist is unknown), (ii) due to a forward potential enstrophy cascade from the mesoscale toward PV dissipation perhaps catalyzed by the more energetic internal-wave field, or (iii) an inverse cascade from microscale PV anomalies generated at dissipative (Kolmogorov and Batchelor) length scales (Sundermeyer and Lelong 2005; Watanabe et al. 2016). Our observations show stronger PV at higher wavenumbers (Figs. 15h, 17, and 18) so support a sorting of higher PV anomalies to smaller length scales.

b. Impact of QG on IW finescale parameterizations

Finescale shear-and/or-strain parameterizations based on the convention that IW dominate small scales (Garrett and Munk 1979) and the nonlinear cascade through wave–wave interaction theory (McComas and Müller 1981; Henyey et al. 1986) have demonstrated robust skill at predicting microstructure turbulent dissipation rates (e.g., Gregg 1989; Polzin et al. 1995; Gregg et al. 2003; Whalen et al. 2015). While finescale vertical shear is dominated by IW (Fig. 13), vortex-stretching (vertical strain) has comparable contributions from both IW and QG (Figs. 17 and 18), implying that strain-based parameterizations may be biased. It may be that the finescale parameterizations are insensitive to whether the finescale strain is internal wave or vortical mode, or that the empirical coefficients are tuned to canonical ocean states. But this still points to the need for more realistically forced and damped regional numerical simulations to better address QG/IW exchange, the spectral pathways and cascade rates of QG and IW in both vertical and horizontal wavenumber, and the dominant instabilities and relative contributions to turbulence production, particularly because the sinks for the balanced circulation and eddy field are comparatively uncertain (Ferrari and Wunsch 2009). Significant intermediate and finescale QG shear (Fig. 11b) would also impact the ratio of CW- to CCW-with-depth rotary shear, which has been interpreted as a signature of vertical near-inertial energy propagation (e.g., Waterhouse et al. 2022).

c. Impact of QG on lateral dispersion

Submesoscale vortical motions, with their longer time scales and larger lateral strains, are expected to dominate lateral dispersion (e.g., Ledwell et al. 1998; Sundermeyer and Lelong 2005) and the cascade of water-mass variance to dissipation scales (Polzin and Ferrari 2004). Sundermeyer et al. (2005) reported that lateral dispersion is largest for vortices of Rossby deformation radius RD=N/(fkz)=Bu1/2/kx. In our observations, comparable QG and IW horizontal shears were found at RD (RD ∼ 2 km for 7 < λz < 30 m, RD ∼ 7 km for 30 < λz < 100 m, RD ∼ 60 km for λz > 100 m; these roughly correspond to 2D bands 1 < λx < 2 km and 7 < λz < 30 m, 1 < λx < 5 km and 30 < λz < 100 m, 1 < λx < 30 km and λz > 100 m; Fig. 12).

d. Measurement and methods limitations

Each 50-km section of our survey took 7–8 h to complete. Lower-wavenumber internal waves with higher phase speeds might be affected by time–space aliasing (Fig. 1b), although this was argued to be negligible (section 2a).

The Helmholtz decomposition assumes that IW are the sole contributors to divergent kinetic energy. This was argued to be valid for observed Ro < 1 and Fr < 1. Since the mixed layer is thinner than 10 m and the measurements are in the pycnocline, submesoscale balanced mixed layer frontal instability dynamics should have little influence (e.g., Torres et al. 2022; Cao et al. 2023). Resolved λz < 100 m do not include nonhydrostatic aspect ratios but unresolved λz > 100 m 1D kh spectra include λz ∼ 1000 m so nonhydrostatic aspect ratios may contribute at high kh. The nonhydrostatic effect on the energy partition needs further investigation for balanced motions.

The Helmholtz decomposition was developed for noise-free signals with infinitesimal resolution, which is not the case for ocean measurements. Small vertical scales λz < 16 m not resolved by the 75-kHz shipboard ADCP are unreliable. Thus, there is high-wavenumber uncertainty associated with instrument noise and the Nyquists, as well as low-wavenumber uncertainty associated with few degrees of freedom (e.g., Li and Lindborg 2018). Effects of measurement limitation and uncertainty of the Helmholtz decomposition are investigated in appendix A. Discarding high horizontal wavenumbers kx > 1 × 10−3 cpm subsequent to testing and validating the Helmholtz decomposition on GM spectra with added white noise (appendix A) suggests that potential high-kx bias is small. Equipartition of rotational and divergent velocity spectra is one spectral property of white noise (Fig. A2). Agreement between Helmholtz and Bu results suggests these issues are not a significant concern though both methods are susceptible to low signal to noise at high wavenumbers.

At small λx where Ro approaches 1 (Fig. 14), linear decomposition of QG and IW may no longer be reliable (Kunze 1993), potentially explaining the increased tendency for equipartition between QG and IW at high wavenumbers. The Burger number decomposition is degenerate and unable to separate IW and QG for Bu ∼ 2 (Fig. 3).

e. Outlook

Our results represent preliminary quantification of fine- and submesoscale QG signals over broad previously rarely characterized horizontal and vertical wavenumber bands in the stratified upper ocean interior, identifying a strong Bu dependence for QG-to-IW energy ratios (Figs. 57). They can provide ground truthing for future numerical modeling of this parameter space and guide future measurements. More measurements in different dynamical settings (e.g., central ocean subtropical and subpolar gyres, western boundary currents, Antarctic Circumpolar Current, etc.) are needed to establish how widespread the partition dependencies are. Measurements and modeling over a broader range of horizontal and vertical wavenumbers (Burger numbers) are needed, particularly higher horizontal wavenumbers and lower vertical wavenumbers, to determine the higher-Bu dependence of EQG/EIW ratios. More realistically forced and damped numerical simulations would better identify and quantify the pathways of IW and QG energy from forcing to dissipation, including QG and IW cascade rates, IW/QG exchange, as well as dominant instability mechanisms leading to turbulence production in this relatively underexplored vertical and horizontal wavenumber space.

Acknowledgments.

We thank Barry Ma, Ryan Newell, Jesse Dosher, and Tim McGinnes for instrument preparation and operation, 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. We also thank Jules Hummon of Shipboard ADCP Support Services at University of Hawaii for ADCP processing. We are grateful to Eric D’Asaro, Andrey Shcherbina, and two reviewers whose comments helped improve the manuscript. This research was funded by NSF Grants OCE-1734160 and OCE-1734222.

Data availability statement.

The data supporting the results presented in this study are available at https://hdl.handle.net/1773/51263.

APPENDIX A

Helmholtz Decomposition

Bühler et al. (2017) proposed a method to account for horizontal anisotropy in the Helmholtz decomposition method for horizontal isotropic flow fields previously presented in Bühler et al. (2014). The anisotropic Helmholtz decomposition model [Eqs. (2) and (3)] is suitable if all following conditions are met: (i) velocity variances σu2συ2; (ii) covariance σ ≠ 0; (iii) the imaginary part of cross-spectrum C(kx) is negligible; and (iv) the real part of C(kx) is single-signed. Our measurements satisfy (i)–(ii) based on F and t tests, with σ on average a factor of ∼4 smaller than σu2 and συ2, but not (iii)–(iv), with RuυO(Iuυ) and Ruυ>0O(|Ruυ<0|). Bühler et al. (2017) point out that oceanic datasets are subject to measurement noise and sampling errors, resulting in model assumptions not being formally met so that the decomposition produces nonphysical negative values for divergent or rotary power spectra. In our analysis, both the isotropic and anisotropic models were computed for comparison: 73% of all KdivISO are within a factor of 2 of KdivANISO (87% within a factor of 3), and 73% of KrotISO are within a factor of 2 of KrotANISO (86% within a factor of 3). The discrepancies between the ISO and ANISO methods are small for section- and depth-averaged spectra: 77% of mean KdivISO are within a factor of 1.5 of mean KdivANISO and 78% of mean Krot within a factor of 1.5 of mean KrotANISO. Any spurious negative spectral values at a particular wavenumber in the decomposition are deemed nonphysical and discarded.

To investigate the effect of finite Nyquist wavenumbers on the decomposition, the Helmholtz decomposition isotropic model was applied to GM along- and across-track velocity spectra ( CuGM and CυGM, Fig. A1). The GM canonical energy was normalized by 1.41, the ratio of the mean observed kinetic energy spectrum to GM at kx < 4 × 10−4 cpm. The decomposition reproduces theoretical CdivGM, but biases CrotGM high at kx > 1 × 10−3 cpm (where Nyquist wavenumber is 2 × 10−3 cpm), due to finite Nyquist wavenumber effect.

Fig. A1.
Fig. A1.

(a) GM kx spectra for along-track velocity Cu [Eq. (C7), solid blue] and across-track velocity Cυ [Eq. (C8), solid red] using measured spectral level 1.4 times lower than canonical GM, with and without a superimposed white spectrum for velocity noise 0.01% ± 50% m s−1 (dashed ± shading). (b) GM kx spectra for divergent energy Cdiv [Eq. (C10), solid purple] and rotational energy Crot [Eq. (C11), solid green], from Helmholtz decomposition on Cu and Cυ integrated to kxc=2×103cpm, with and without noise (dashed and circles, respectively).

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

The impact of measurement noise Δυ on the decomposition was estimated by adding a white-noise spectrum equivalent to Δυ = 0.01% ± 50% m s−1 to CuGM and CυGM. At high wavenumbers, CuGM>CυGM so that CυGM is more affected by noise due to smaller signal-to-noise ratio. The added noise effect on the decomposition is that CdivGM diverges from the theoretical spectrum for kx > 5 × 10−4 cpm, while CrotGM diverges for kx > 4 × 10−4 cpm and becomes white for kx > 1 × 10−3 cpm. Close to the Nyquist kx = 2 × 10−3 cpm, CdivGMCrotGM. For the observed spectra, Cu > Cυ and Cdiv > Crot does not necessarily hold as for GM. Therefore, estimates of both Cdiv and Crot are likely unreliable for kx > 4 × 10−4 cpm due to ADCP noise. However, this conclusion only applies for GM-like internal-wave spectra. Oceanic spectra deviating from GM would likely have different impacts of measurement noise and the Nyquist on the Helmholtz decomposition of rotational and divergent velocity spectra.

Observed velocity spectra do not exhibit a clear white-noise floor. However, to avoid any high-wavenumber bias, only wavenumbers kx < 1 × 10−3 cpm are shown for the 2D (kx, kz) spectra, with the caveat that kx > 4 × 10−4 cpm may be biased by proximity to Nyquist or measurement noise. Agreement with the Burger number decomposition (section 2d) at high kx suggests that this bias may be insignificant. Spectra for the depth-means are less affected by noise so are displayed at high 10−3 < kx < 2 × 10−3 cpm.

Section-averaged 1D kx spectra for Cdiv and Crot agree between the isotropic and anisotropic models (Figs. A2a,b). For 7 < λz < 100 m, Cdiv is slightly redder than CdivGM, while CrotCrotGM. The depth-mean (i.e., vertical wavelengths λz > 100 m) 5-section-mean Cdiv and Crot kx spectra contribute most of the variance and have slopes of ∼−2. CdivCdivGM for low kx < 4 × 10−5 cpm and CdivCdivGM/2 for kx > 4 × 10−5 cpm. The Crot is a factor of ∼2 above GM for kx < 10−4 cpm and diverges even more with increasing kx. Bluer Crot spectra than GM at high kx may be due to bias in the decomposition to velocity measurement noise (Fig. A1), QG, imperfect GM velocity kx spectrum, or wavenumber bandwidth (we assume j*=3).

Fig. A2.
Fig. A2.

Average along-track wavenumber kx spectra for (a) Cdiv and (b) Crot, from the Bühler et al. (2014) isotropic (black) and Bühler et al. (2017) anisotropic (red) Helmholtz decompositions of (kx, kz) spectra for 7 < λz < 100 m and average 1D kx spectra for λz > 100 m (thick blue). GM spectra for 7 < λz < 100 m (thin black) and λz > 100 m (thin blue). (c) Ratios Crot/Cdiv from (a) and (b). Shading shows standard deviation. Horizontal gray lines mark ratios for IW frequencies ω = f (solid) and ω = M2 (dashed).

Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0090.1

Measured Crot/Cdiv are O(1) on average (Fig. A2c), consistent with Chereskin et al. (2019). The Crot/Cdiv are larger than the GM model. This can be explained by QG contributions since Crot/Cdiv=(CrotIW+CrotQG)/(CdivIW+CdivQG)=(CrotIW+CrotQG)/CdivIW because CdivQG=0, where superscripts IW and QG denote internal wave and quasigeostrophic components, respectively.

For linear internal waves, Crot/Cdivf2/ω2 [Eqs. (C10) and (C11)] so that Crot/Cdiv ∼ 1 suggests near-inertial dominance, or the presence of QG motion. Mean phase differences ∼90° between along- and across-track vertical shears, in both horizontal and vertical wavenumber (not shown), also suggest near-inertial waves. For λz > 15 m, the mean polarization ratio for counterclockwise-to-clockwise rotating vertical shear vertical wavenumber spectra CCW/CW(kz) = 0.76 ± 0.09, indicating net downward near-inertial energy propagation (Leaman and Sanford 1975).

APPENDIX B

Burger Number Decomposition of IW and QG Energies

Under the hydrostatic approximation, total energy E is the sum of horizontal kinetic EK and available potential energy EP. Horizontal kinetic and available potential energy can be linearly decomposed into IW and QG components, i.e., EK=EIWK+EQGK and EP=EIWP+EQGP so the kinetic and potential energy of IW and QG can be separated using the consistency relations and measured ratio of available potential to horizontal kinetic energy r = EP/EK:
EIWK(kh,kz)=EKEQGK=1RIWRQGrRQG1+rE=Bu+21+BurBu1(Bu2)(1+r)E,
EIWP(kh,kz)=EPEQGP=RIWRIWRQGrRQG1+rE=Bu1+BurBu1(Bu2)(1+r)E,
EQGK(kh,kz)=EPRIWEKRQGRIW=1RQGRIWrRIW1+rE=Bu1+BuBur(Bu+2)(Bu2)(1+r)E,
EQGP(kh,kz)=RQGEQGK=RQGRQGRIWrRIW1+rE=11+BuBur(Bu+2)(Bu2)(1+r)E,
where Burger number Bu=[Nkh/(fkz)]2,RIW=EIWP/EIWKBu/(Bu+2), and RQG=EQGP/EQGK=1/Bu. Total IW and QG energies are
EIW=1+RIWRIWRQGrRQG1+rE=2(rBu1)(Bu2)(1+r)E,
EQG=1+RQGRQGRIWrRIW1+rE= Bur(Bu+2)(Bu2)(1+r) E,
and the ratio of IW to QG energy
EIWEQG=(1+RIW)(rRQG)(1+RQG)(RIWr)=2(rBu1)Bur(Bu+2).

APPENDIX C

GM Spectra

The polarization relation for internal waves links velocity and displacement to a common parameter a (the following expressions are in rad s−1 for simplicity):
(uυξ)=(ωkx+ifkyωkyifkxikx2+ky2kz)a,
kz2=(N2ω2)ω2f2kh2.
The divergent and rotational components of velocity are velocity components projected parallel and perpendicular to the horizontal wavevector direction. The polarization relation between divergent and rotational components of velocity is
(udivurot)=1kh(kxkykykx)(uυ)=(ωkhifkh)a.
This relates spectra of velocity, potential energy, divergent velocity, rotational velocity, along-track velocity, across-track velocity, and total energy. The divergent and rotational components of velocity spectra are thus related to the total velocity spectrum Cvel = Cu + Cυ:
Cdiv(ω,kx,ky)=ω2ω2+f2Cvel(ω,kx,ky),
Crot(ω,kx,ky)=f2ω2+f2Cvel(ω,kx,ky).
Similarly, the horizontal divergence CHD and relative vorticity CRV spectra are
CHD=kh2Cdiv,CRV=kh2Crot.
Garrett and Munk (GM) internal-wave spectra were computed using canonical b = 1300 m, N0 = 5.2 × 10−3 s−1, nondimensional spectral level E0 = 6 × 10−5, reference mode number j*=3 and vertical wavenumber cutoff at 0.1 cpm (Garrett and Munk 1972, 1975, 1979).
GM kx spectra for along-track velocity ( CuGM), across-track velocity ( CυGM), potential energy ( EPGM), divergent energy ( CdivGM) and rotational energy ( CrotGM) were obtained by integrating Eqs. (C7)(C11), respectively, over ky and ω:
CuGM(ω,kx,ky)=12b2NN0E0(ω2kx2+f2ky2)2πω2(kx2+ky2)3/2[2πfω(ω2f2)1/2][2π(1+kh2kh*2)1/kh*],
CυGM(ω,kx,ky)=12b2NN0E0(ω2ky2+f2kx2)2πω2(kx2+ky2)3/2[2πfω(ω2f2)1/2][2π(1+kh2kh*2)1/kh*],
EPGM(ω,kx,ky)=12(ω2f2)ω2b2NN0E02π(kx2+ky2)1/2[2πfω(ω2f2)1/2][2π(1+kh2kh*2)1/kh*],
CdivGM(ω,kx,ky)=12b2NN0E0π(kx2+ky2)1/2[2πfω(ω2f2)1/2][2π(1+kh2kh*2)1/kh*]=EGM(ω,kx,ky),
CrotGM(ω,kx,ky)=f2ω212b2NN0E0π(kx2+ky2)1/2[2πfω(ω2f2)1/2][2π(1+kh2kh*2)1/kh*],
kh*=(ω2f2)1/2N1kz*=(ω2f2)1/2N1πbNN0j*=(ω2f2)1/2N0πbj*.

APPENDIX D

Horizontal Wavenumber Magnitude Spectra from Single-Track Measurements

The 1D along-track wavenumber spectra W1(kx) from single-track measurements were converted to 1D horizontal wavenumber magnitude spectra W3(kh), where kh=(kx2+ky2)1/2 with kx = kh cosθ and kυ = kh sinθ along- and across-track horizontal wavenumbers, to be used for the Burger number decomposition and consistency relations (section 2d, appendix B). First, the two-sided 2D wavenumber spectrum W2(kx, ky) was derived from W1(kx) assuming horizontal isotropy:
W2(kx,ky)=1π0{W1(kx2+ky2)W1[(kx2+ky2+q2)1/2]}dq/q2
(Henyey 1991; Williams et al. 2002). The measured W1 spectrum was log-linearly interpolated onto its arguments in Eq. (D1), so (kx2+ky2)1/2 and (kx2+ky2+q2)1/2 lie in the measured range of discrete along-track wavenumber kx, and integration over q is limited by the Nyquist. The 1D wavenumber magnitude spectrum W3(kh) was then inferred from W2(kx, ky) as
W3(kh)=02πW2(khcosθ,khsinθ)khdθ.
The integrations are sensitive to wavenumber resolution.

APPENDIX E

Vortex-Stretching, Relative Vorticity, and Potential Vorticity Anomaly Spectra

Vortex-stretching spectra CVS were computed directly from vertical displacement spectra Cξ(kh, kz) inferred from TCTD data as
CVS(kh,kz)=f2(2πkz)2Cξ(kh,kz),
where the along-track wavenumber spectrum has been converted to wavenumber magnitude spectrum following appendix D. Vertical relative vorticity CRV and horizontal divergence CHD spectra can be expressed as functions of Cdiv and Crot derived from the Helmholtz decomposition [Eqs. (C4)(C6)]:
CHD(kh,kz)=(2πkh)2Cdiv(kh,kz),
CRV(kh,kz)=(2πkh)2Crot(kh,kz).
The linear decompositions assume internal waves account for all the horizontal divergence ∇huh and vortical mode all potential vorticity [Eq. (1)]. Following consistency relations between spectra of VS and RV, total CRV and CVS can be separated into IW and QG components as a function of Bu (Lien and Müller 1992b):
CRV=CRVIW+CRVQG,
CVS=CVSIW+CVSQG,
CVSIW/CRVIW=1,
CVSQG/CRVQG=1/Bu2.
The system of equations above can be rearranged to express CVSQG,CRVQG,CVSIW,CRVIW as functions of total CRV, CVS and Bu.
The linear PV anomaly includes relative vorticity and vortex-stretching. The potential vorticity anomaly spectrum CPV and total QG energy spectrum are related through Bu (Müller et al. 1988):
CPV=(1+Bu)2CVSQG=2f2N2(1+Bu)(2πkz)2EQĜ.
Thus, spectra of linear PV can be inferred from QG energy spectra or vice versa (Figs. 15i, 8 and 9; Müller et al. 1988).

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