Influence of the Distortion of Vertical Wavenumber Spectra on Estimates of Turbulent Dissipation Using the Finescale Parameterization: Eikonal Calculations

Anne Takahashi aDepartment of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Tokyo, Japan

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Toshiyuki Hibiya aDepartment of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Tokyo, Japan

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Alberto C. Naveira Garabato bOcean and Earth Science, National Oceanography Centre, University of Southampton, Southampton, United Kingdom

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Abstract

The finescale parameterization, formulated on the basis of a weak nonlinear wave–wave interaction theory, is widely used to estimate the turbulent dissipation rate ε. However, this parameterization has previously been found to overestimate ε in the Antarctic Circumpolar Current (ACC). One possible reason for this overestimation is that vertical wavenumber spectra of internal wave energy are distorted from the canonical Garrett–Munk spectrum by a spectral hump at low wavenumbers (~0.01 cpm). Such distorted vertical wavenumber spectra were also observed in other mesoscale eddy-rich regions. In this study, using eikonal simulations, in which internal wave energy cascades are evaluated in the frequency–wavenumber space, we examine how the distortion of vertical wavenumber spectra impacts the accuracy of the finescale parameterization. It is shown that the finescale parameterization overestimates ε for distorted spectra with a low-vertical-wavenumber hump because it incorrectly takes into account the breaking of these low-vertical-wavenumber internal waves. This issue is exacerbated by estimating internal wave energy spectral levels from the low-wavenumber band rather than from the high-wavenumber band, which is often contaminated by noise in observations. Thus, to accurately estimate the distribution of ε in eddy-rich regions like the ACC, high-vertical-wavenumber spectral information free from noise contamination is indispensable.

Current affiliation: Applied Physics Laboratory, University of Washington, Seattle, Washington.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Anne Takahashi, annetaka@uw.edu

Abstract

The finescale parameterization, formulated on the basis of a weak nonlinear wave–wave interaction theory, is widely used to estimate the turbulent dissipation rate ε. However, this parameterization has previously been found to overestimate ε in the Antarctic Circumpolar Current (ACC). One possible reason for this overestimation is that vertical wavenumber spectra of internal wave energy are distorted from the canonical Garrett–Munk spectrum by a spectral hump at low wavenumbers (~0.01 cpm). Such distorted vertical wavenumber spectra were also observed in other mesoscale eddy-rich regions. In this study, using eikonal simulations, in which internal wave energy cascades are evaluated in the frequency–wavenumber space, we examine how the distortion of vertical wavenumber spectra impacts the accuracy of the finescale parameterization. It is shown that the finescale parameterization overestimates ε for distorted spectra with a low-vertical-wavenumber hump because it incorrectly takes into account the breaking of these low-vertical-wavenumber internal waves. This issue is exacerbated by estimating internal wave energy spectral levels from the low-wavenumber band rather than from the high-wavenumber band, which is often contaminated by noise in observations. Thus, to accurately estimate the distribution of ε in eddy-rich regions like the ACC, high-vertical-wavenumber spectral information free from noise contamination is indispensable.

Current affiliation: Applied Physics Laboratory, University of Washington, Seattle, Washington.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Anne Takahashi, annetaka@uw.edu

1. Introduction

The meridional overturning circulation (MOC) of the deep ocean plays an important role in maintaining Earth’s mild climate and is considered to significantly influence long-term climate change by transporting heat and carbon (Toggweiler and Russell 2008; Marshall and Speer 2012). It is well known that the strength and structure of the MOC depend on the global distribution of diapycnal turbulent mixing (e.g., Munk 1966; Bryan 1987; Oka and Niwa 2013; Kunze 2017), which is most commonly caused by the breaking of oceanic internal waves (Waterhouse et al. 2014, and references therein).

Since direct measurements of turbulence (with vertical scales smaller than 1 m) are time-consuming and require specialized instruments, finescale parameterizations (Henyey et al. 1986; Gregg 1989; Wijesekera et al. 1993; Polzin et al. 1995; Gregg et al. 2003; Polzin et al. 2014; Ijichi and Hibiya 2015), which estimate the turbulent dissipation rate ε in terms of finescale [vertical scales O(10–100 m)] velocity and density fluctuations, have been developed. These parameterizations predict ε as the internal wave energy flux that cascades down to the turbulent scales due to nonlinear interactions within a background internal wave spectrum. Furthermore, they implicitly assume that the background internal wave field is steady state with a vertical wavenumber energy spectrum similar to the Garrett–Munk (GM) spectral model (Garrett and Munk 1972, 1975; Cairns and Williams 1976; Munk 1981). The GM spectrum is expressed, for frequency σ and vertical wavenumber m, as
EGM(σ,m)=b2N0NE0ΩGM(σ)MGM(m),
ΩGM(σ)=2π1fσ1(σ2f2)1/2,and
MGM(m)=m*(m+m*)2,
where b = 1300 m, N0 = 5.24 × 10−3 s−1, E0 = 6.3 × 10−5, m* = j*/(2b), and j* = 3 are constant coefficients, N is the buoyancy frequency, and f is the Coriolis frequency. This GM model, empirically formulated on the basis of observations in the western North Atlantic Ocean (Garrett and Munk 1972), has been used as a first-order description of the background internal wave field in the ocean interior.

The pioneering works by Gregg (1989) and Wijesekera et al. (1993) proposed the prototype of finescale parameterizations based on theoretical models (McComas and Muller 1981; Henyey et al. 1986). They assumed that background internal wave spectra are similar in shape to the GM spectrum, while changing their energy level from E0 in (1). The energy level was then estimated in terms of shear (Gregg 1989) or strain (Wijesekera et al. 1993).

Subsequently, considering the situation in which frequency spectra are distorted from the GM model, ΩGM(σ) in (1), Polzin et al. (1995) introduced a new parameter, shear/strain ratio Rω [for details, see (15) in section 2d] to improve the former parameterization. The corrected parameterization with a latitude-dependent term (Gregg et al. 2003)—called the Gregg–Henyey–Pozin (GHP) parameterization—has been widely used as the most reliable parameterization of deep-ocean mixing.

Afterward, Ijichi and Hibiya (2015) pointed out a defect in the Rω-dependent term in the GHP parameterization that led them to propose a revised parameterization. This Ijichi–Hibiya (IH) parameterization could predict ε more accurately than GHP in the Izu–Ogasawara Ridge, where the internal wave frequency spectra were significantly redder, namely, biased to lower frequencies (Ijichi and Hibiya 2015).

Finescale parameterizations have also been employed in the Antarctic Circumpolar Current (ACC) region (e.g., Naveira-Garabato et al. 2004; Wu et al. 2011; Meyer et al. 2015), where vigorous geostrophic flows coexist with energetic internal waves such as wind-induced near-inertial waves and bottom-generated lee waves (e.g., Waterman et al. 2013; Sheen et al. 2013). Finescale parameterizations are formulated based on the wave–wave interaction theory, but they do not consider other physical processes such as those associated with geostrophic currents. From in situ observations in the ACC region, the Southern Ocean Finestructure (SOFine) survey in the Kerguelen Plateau (Waterman et al. 2013, 2014) and the 19th Kaiyodai Antarctic Research Expedition (KARE19) survey south of Australia (Takahashi and Hibiya 2019), it was reported that the finescale parameterizations tended to overestimate ε by up to a factor of 10.

Takahashi and Hibiya (2020, manuscript submitted to J. Geophys. Res. Oceans) analyzed both the SOFine and KARE19 datasets. They found that this overestimation was related to the parameters characterizing an internal wave field, such as the shear/strain ratio Rω and internal wave spectral level EIW [for details, see (17) in section 2d], rather than mesoscale background current shear Uz¯. The discrepancies between the directly measured ε and that estimated with the latest finescale parameterization (Ijichi and Hibiya 2015) were more evident as EIW increased and Rω decreased.

Besides, at the locations where finescale parameterizations overestimated ε, vertical wavenumber spectra of shear and strain were distorted from the GM spectral model, MGM(m) in (1) by a spectral hump at 0.01-cpm vertical wavenumbers (Takahashi and Hibiya 2020, manuscript submitted to J. Geophys. Res. Oceans). Shear spectral humps, mostly found in the upper ocean, were thought to be caused by near-inertial wave packets superposed on GM-like internal wave spectra. These near-inertial waves might be generated by wind disturbances and trapped by mesoscale eddies (e.g., Kunze 1985) or might be spontaneously generated by instabilities of ACC frontal jets (Nagai et al. 2015). Strain spectral humps, mostly found near the seafloor, were thought to be caused by relatively high-intrinsic-frequency internal lee waves generated by geostrophic flows impinging on rough topographic features. Such distorted vertical wavenumber spectra have been observed not only in the ACC (Kilbourne and Girton 2015; Meyer et al. 2016; Takahashi and Hibiya 2019) but also in other mesoscale eddy-rich regions (Jing et al. 2011; Cuypers et al. 2012; Fernández-Castro et al. 2020).

In this study, we explore how the distortion of vertical wavenumber spectra impacts on the accuracy of the finescale parameterization, using numerical simulations called “eikonal calculations,” which quantitatively evaluate internal wave energy cascades in frequency-wavenumber space.

2. Eikonal calculations

In this study, we carry out a kind of ray-tracing simulations called “eikonal calculations” (e.g., Henyey et al. 1986; Sun and Kunze 1999a,b; Ijichi and Hibiya 2017), where the time evolution of wavenumbers, frequencies, and positions of test waves is traced. Test waves adequately sampled from a background internal wave field are Doppler-shifted by the background shear. Eikonal calculations reproduce induced diffusion interactions between a small-scale internal wave and a larger-scale background field (Henyey and Pomphrey 1983) but do not consider parametric subharmonic instabilities.

Eikonal calculations are useful in evaluating the performance of finescale parameterizations in various situations. For example, Henyey et al. (1986) used eikonal calculations to show the validity of their energy cascade model, which became the basis of finescale parameterizations. Ijichi and Hibiya (2017) also used eikonal calculations to assess their revised parameterization’s applicability to internal wave fields with frequency spectra distorted from the GM model.

We assume that the ocean has a constant depth of 2000 m and constant stratification N = N0 = 5.24 × 10−3 s−1 at latitude 30° (f = f0 = 7.29 × 10−5 s−1), where N0 and f0 are the parameters of the GM model (Munk 1981). Two kinds of eikonal calculations, the standard experiment (section 3) and hump experiments (section 4), are carried out by changing the background internal wave field. Although f = f0 and N = N0 are not typical values of the ACC region, the original parameters of the GM model are used here to make it easier to discuss the validity of the finescale parameterization. Note that, using the typical parameters of the ACC region (f = 1.0 × 10−4 s−1 and N = 1.0 × 10−3 s−1), we obtain almost the same experimental results as those presented below.

a. Initial states of the test waves

The initial state of the test waves is determined in terms of vertical wavenumber minit, intrinsic frequency σinit, and depth zinit, such that
mi,j,n,ninit=±i/400 [cpm](i=1,2,,40),
σi,j,n,ninit=1.01×20.5(j1)f(j=1,2,,13),and
zi,j,n,ninit=200 n+100 [m](n=1,2,,10).
The initial horizontal wavenumber κinit is calculated using the dispersion relation,
σ2=N2κ2+f2m2κ2+m2.
Zonal and meridional components of κinit = (kinit, linit) are determined using the angle θ, such that
ki,j,n,ninit=κinit sinθn,
li,j,n,ninit=κinit cosθn,and
θn=nπ6(n=1,2,,12).
The initial horizontal positions of the test waves (xinit, yinit) are randomly chosen.

The initial energy density of each test wave Einit is determined based on the background internal wave energy spectrum EBG(m, σ), such that Einit = EBG(minit, σinitmΔσ, with Δm and Δσ being the differences between adjacent vertical wavenumbers and frequencies. Eikonal calculations assume that the background field is in a steady state, that is, that the background internal wave spectrum is maintained in some way without specifying energy sources and sinks. In each experiment, a total of 124 800 (=40 × 2 × 13 × 10 × 12) test waves are released.

b. Ray-tracing of the test waves

The trajectory of each test wave with wavenumber k = (k, l, m) and position x = (x, y, z) is calculated using the following eikonal equations,
dkdt=kuBGand
dxdt=σk+uBG,
where uBG is the velocity of the background internal wave field. At each time step, the shorter of Δt = 1 min and Δt = 1/Cgz, with Cgz as the test wave’s vertical group velocity, is used as the time increment. The test wave’s intrinsic frequency σ is calculated from the dispersion relation (7).

To ensure the consistency of the eikonal equations and linear internal wave dispersion relation, we assume that the evolution of each test wave is only affected by background shear with spatial scales larger than the test wave scale. In other words, test waves are Wentzel–Kramers–Brillouin (WKB) scale-separated from the background internal wave field (e.g., Henyey et al. 1986; Sun and Kunze 1999a,b). Therefore, uBG is calculated at each time step by summing up the velocity of background internal waves with wavenumbers kBG = (kBG, lBG, mBG) satisfying kBG2+lBG2<0.5k2+l2 and mBG < 0.5m. Note that, although we also carry out eikonal calculations with weak scale separation (kBG2+lBG2<k2+l2 and mBG < m), the results are qualitatively unaffected.

c. Calculation of energy transfer rate

Each test wave conserves wave action A = Einit/σ along its trajectory (Bretherton and Garrett 1968). Following Ijichi and Hibiya (2017), the eikonal equations [(11) and (12)] are integrated until the test wave’s vertical wavenumber exceeds a breaking limit, mbreak = 0.2 cpm. When the test wave breaks, all the test wave energy is assumed to be transferred to turbulent energy. Values of the spectral energy transfer rate ε, which can be regarded as the turbulent dissipation rate and calculated over all the test waves that break within 40 inertial periods, are vertically averaged,
ε=1ntotal i,j,n,n|mfin|>mbreakσi,j,n,nfin Ai,j,n,nτi,j,n,n ,
where mfin and σfin are the vertical wavenumber and frequency of each test wave at the time it breaks, τ is the lifetime of each test wave, and ntotal = 10 from (6). The sensitivity of ε to the choice of breaking limit mbreak is shown in Fig. 1. The values of ε converge when mbreak > 0.2 cpm, consistent with Ijichi and Hibiya (2017).
Fig. 1.
Fig. 1.

The vertically averaged energy transfer rates ε [(13)] for different values of the breaking limit mbreak = 0.1, 0.2, 0.3, and 0.4 cpm in our eikonal simulations: the standard (black; see section 3 for details), NI-strong (red; see section 4), and lee-strong (blue; see section 4) experiments. Each error bar indicates 1 standard deviation of the vertical mean.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0196.1

d. Application of the finescale parameterization

To assess the accuracy of the finescale parameterization, we compare the spectral energy transfer rates ε obtained from the eikonal calculations [(13)] with turbulent dissipation rates εfine estimated by applying the latest finescale parameterization (Ijichi and Hibiya 2015),
εfine=ε0(N/N0)2(Uz2/N2Uz2/N2GM)2h(Rω,f,N),
to vertical wavenumber spectra of the background internal wave field. Here ε0 = 6.3 × 10−10 W kg−1, N0 = 5.24 × 10−3 s−1, and the buoyancy-normalized shear spectral variance Uz2/N2 is measured through the comparison with the GM value Uz2/N2GM.
The shear/strain ratio,
Rω=Uz2/N2ξz2,
with ξz2 as the strain spectral variance, is a parameter indicating the bulk frequency content of the internal wave field. Higher Rω implies a dominance of near-inertial internal wave energy (the frequency spectrum is redder), whereas lower Rω implies the presence of more high-frequency internal waves (the frequency spectrum is bluer). The Rω-dependent term in the parameterization (Fig. 2) is
h(Rω,f,N)={1+1/Rω4/3L1L0RωL2(Rω<9)1+1/Rω4/31L02Rω1(Rω9),
L0=2π1 arcosh(N0/f0),
μGM=2π1 arcosh(N/f),
L1=2μGM2,and
L2=log3(2μGM).
The internal wave spectral level is also calculated using shear and strain spectral variances,
EIW=Uz2/N2+ξz2Uz2/N2GM+ξz2GM,
where ξz2GM is the strain spectral variance of the GM model.
Fig. 2.
Fig. 2.

Shear/strain ratio Rω–dependent term h [(16)] in the finescale parameterization (Ijichi and Hibiya 2015) with the buoyancy frequency N = N0 = 5.24 × 10−3 s−1 and Coriolis frequency f = f0 = 7.29 × 10−5 s−1.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0196.1

Shear and strain spectral variances are calculated by integrating the spectra from the lowest wavenumber mlow (=0.0025 cpm) to the cutoff wavenumber mc. It is widely believed that mc should be the wavenumber at which the shear spectrum starts to roll off. This rolloff wavenumber mroll is thought to be a transition point from the weak nonlinear wave–wave interaction regime to the strong nonlinear wave breaking regime. Following the empirical model (Gargett et al. 1981; Munk 1981), mroll is generally inferred from the following inverse Richardson-like function:
mlowmrollΦFr(m) dm=Ric1=mlowmrollGMΦFrGM(m) dm,
where ΦFr(m) is the buoyancy-normalized shear (Froude) spectrum, Ric−1 is the inverse critical Richardson number, and mrollGM=0.1 cpm and ΦFrGM are the rolloff wavenumber and Froude spectrum of the GM model. The cutoff wavenumber determined from the inferred mroll is referred to as the Richardson number–based (Ri based) cutoff wavenumber mcRi. Previous field observations have confirmed that shear spectra roll off at lower wavenumbers as the spectral levels increase (Duda and Cox 1989; Gregg et al. 1993), consistent with the empirical model. However, previous high-resolution 2D numerical simulations showed the occurrence of a spectral rolloff not associated with shear or convective instabilities (Hibiya et al. 1996), so that the mechanism by which the rolloff arises is still unclear.

Shear spectra obtained from observations acquired by lowered acoustic Doppler current profilers (LADCPs)—most commonly used to measure the horizontal current speed—are contaminated by noise at high wavenumbers (Polzin et al. 2002; Thurnherr 2012). Therefore, when one applies the finescale parameterization to LADCP profiles, mc is typically set to a much lower value than mrollGM (e.g., Waterman et al. 2013; Sheen et al. 2013). Here, the cutoff wavenumber determined in consideration of LADCP noise is referred to as the LADCP-based cutoff wavenumber, mcLAD=0.0175 cpm. In section 4, we use either the Ri-based or LADCP-based cutoff wavenumbers for the assessment of parameterization.

3. Standard experiment

First, to confirm the reproducibility of our numerical simulations, we carry out eikonal calculations for the GM internal wave field. All the results obtained in this “standard experiment” are consistent with the previous studies (e.g., Watanabe and Hibiya 2005; Ijichi and Hibiya 2017).

a. Background internal wave field

As the background internal wave field, we assume a superposition of randomly phased, horizontally and vertically isotropic linear internal waves with their amplitudes determined from the GM model (Munk 1981). The energy density of the background internal waves, as well as that of test waves EBG, is determined from EGM [(1)]. The spatiotemporal structure of the GM velocity field is described in appendix A. Vertical wavenumber spectra of shear and strain incorporated in the standard experiment are shown in Fig. 3a.

Fig. 3.
Fig. 3.

Vertical wavenumber spectra of buoyancy-normalized shear (black) and strain (gray) incorporated in the (a) standard experiment (section 3) and four kinds of hump experiments (section 4): the (b) NI-moderate, (c) NI-strong, (d) lee-moderate, and (e) lee-strong experiments. Dashed horizontal lines indicate the GM spectra (Munk 1981); dotted vertical lines indicate the LADCP-based cutoff wavenumber mcLAD=0.0175 cpm.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0196.1

b. Results

The standard experiment results are summarized in Figs. 5a–c, showing the frequency-wavenumber distributions of wave action A, test wave’s lifetime τ, and spectral energy transfer rate ε calculated for each test wave, with respect to the initial vertical wavenumber and frequency (minit, σinit). The GM spectrum has a high energy level in the near-inertial and low-vertical-wavenumber domains (Fig. 5a). In contrast to high-frequency and high-vertical-wavenumber test waves with shorter lifetimes, near-inertial and low-vertical-wavenumber test waves do not break within 40 inertial periods (white space in the bottom left corner of Fig. 5b). Near-inertial and high-vertical-wavenumber test waves predominantly contribute to ε (Fig. 5c), as a result of the combined effect of A and τ [see (13)]. The vertically averaged energy transfer rate of ε = 6.6 × 10−10 W kg−1 is consistent with the turbulent dissipation rate obtained from field observations (Gregg et al. 2003).

4. Hump experiments

Next, we examine how internal wave energy cascades within vertical wavenumber spectra distorted from the GM spectrum. Although Ijichi and Hibiya (2017) examined energy cascades for several distorted internal wave spectra with various E0 and frequency fluctuations in (1) to assess the validity of their parameterization, they did not take into account the distortion of vertical wavenumber spectrum. In the “hump experiments,” we introduce idealized internal wave spectra with a spectral hump at low vertical wavenumbers akin to those observed in the ACC region (Takahashi and Hibiya 2020, manuscript submitted to J. Geophys. Res. Oceans). The validity of the finescale parameterization for these spectra is then assessed through several eikonal calculations.

a. Background internal wave field

To reproduce the distorted shear and strain spectra observed in the ACC, low-vertical-wavenumber (0.0025 ≤ m ≤ 0.01 cpm) internal wave packets are superposed on the standard GM internal wave field. By changing the frequencies and amplitudes of the superposed waves, we carry out four kinds of hump experiments.

In the NI-moderate and NI-strong experiments, downward-propagating (m > 0), near-inertial (σ ≤ 1.02f) internal wave packets are superposed. In the lee-moderate and lee-strong experiments, upward-propagating (m < 0), relatively high-intrinsic-frequency (2fσ ≤ 2f) internal wave packets are superposed. In the “strong” experiments, the superposed internal wave packets’ energy density is 2 times as large as in the “moderate” experiments.

Here we assume that the superposed internal wave packets are horizontally isotropic. Takahashi and Hibiya (2019) speculated that the overestimation by the finescale parameterization could be attributed to the directionality (anisotropy) of internal wave field associated with monochromatic internal wave packets. From our additional eikonal calculations, however, we have confirmed that the results are hardly affected by whether the superposed internal wave packets have a preferred direction or not (figures not shown).

The energy density distribution of the background internal wave field EBG(m, σ) and spatiotemporal structure of the background velocity uBG for the hump experiments are given in appendix B. Vertical wavenumber spectra of shear and strain incorporated in the hump experiments are shown in Figs. 3b–e. In the NI experiments, the shear spectral hump is much larger than the strain spectral hump, while both shear and strain spectral humps are significant in the lee experiments.

These idealized spectra are not unrealistic. The NI-strong spectra (Fig. 3c) resemble those obtained from the KARE19 survey (Fig. 4a) carried out in the ACC jets south of Australia (Takahashi and Hibiya 2019). The lee-moderate strain spectrum (Fig. 3d) is almost similar to the strain spectrum obtained from the SOFine survey (Fig. 4b) performed in the ACC meander around the Kerguelen Plateau (Waterman et al. 2013, 2014).

Fig. 4.
Fig. 4.

Examples of the vertical wavenumber spectra of buoyancy-normalized shear (black) and strain (gray) obtained from finestructure measurements in the ACC region. The cruise name, station, and depth of the data, details of which are given in Takahashi and Hibiya (2020, manuscript submitted to J. Geophys. Res. Oceans), are shown on the labels. Dashed horizontal lines indicate the GM spectra (Garrett and Munk 1975; Cairns and Williams 1976); dotted vertical lines indicate the LADCP-based cutoff wavenumber mcLAD=0.0175 cpm.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0196.1

b. Results

In Fig. 5, wave action A, lifetime τ, and spectral energy transfer rate ε—all calculated for each test wave with (minit, σinit)—are shown for the standard experiment (Figs. 5a–c), NI-strong experiment (Figs. 5d–f), and lee-strong experiment (Figs. 5g–i). In the NI-strong and lee-strong experiments, the values of ε are much larger than those in the standard experiment, especially due to the high-vertical-wavenumber test waves (Figs. 5c,f,i). The superposed low-vertical-wavenumber test waves constituting a spectral hump, on the other hand, hardly break throughout the calculation time (Figs. 5e,h). These results indicate that a low-vertical-wavenumber spectral hump works as a background shear, promoting the breaking of high-vertical-wavenumber internal waves, but the hump itself does not break nor directly contribute to ε.

Fig. 5.
Fig. 5.

Results of the (a)–(c) standard experiment (section 3), (d)–(f) NI-strong experiment (section 4), and (g)–(i) lee-strong experiment (section 4) are summarized for (left) wave action A, (center) reciprocal of lifetime τ−1 normalized by the inertial time scale τi = 2π/f, and (right) energy transfer rate ε, all of which are calculated for each test wave with initial vertical wavenumber and frequency (minit, σinit), as shown by the color shading.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0196.1

Since high-wavenumber test waves mostly contribute to ε in such distorted spectra, the finescale parameterization should take into account the shear spectral level in the high-wavenumber band. However, as already discussed in section 2d, when applying the parameterization to observational data—especially velocity profiles obtained using LADCPs—we cannot use shear spectra at high wavenumbers due to noise contamination (Polzin et al. 2002; Thurnherr 2012). Suppose one adopts the LADCP-based cutoff wavenumber mcLAD=0.0175 cpm. In that case, the calculated shear spectral variance normalized by the GM spectral value is much larger than that calculated in the higher-wavenumber band, resulting in an overestimation of the internal wave spectral level EIW and ε.

Besides, if near-inertial waves cause the spectral hump, the shear/strain ratio Rω in the low-vertical-wavenumber band is higher than that in the high-vertical-wavenumber band. Because of the Rω-dependent term in the finescale parameterization—higher Rω results in smaller ε (see Fig. 2)—overestimation of ε would be reduced in the NI- experiments. In contrast, if high-intrinsic-frequency internal waves cause the spectral hump, Rω calculated in the low-vertical-wavenumber band is rather lower than in the high-vertical-wavenumber band, which would not reduce the overestimation of ε.

Figures 6b and 6c show how the ratio εfine/ε, with εfine denoting the turbulent dissipation rate inferred from the finescale parameterization and ε indicating the energy transfer rate in the eikonal experiments, depends on EIW and Rω. When the LADCP-based cutoff wavenumber mcLAD is adopted, the finescale parameterization overestimates ε in all the hump experiments (Fig. 6b). The overestimation becomes more prominent as EIW increases and Rω decreases. These results are consistent with the relationships diagnosed from in situ observations in the ACC region (Fig. 6a). When the Ri-based cutoff wavenumber mcRi is adopted, overestimation is significantly reduced, except for the NI-strong experiment (Fig. 6c), where the shear spectral level is so high that mcRi shifts to a wavenumber as small as mcLAD.

Fig. 6.
Fig. 6.

(a) Relationship between the ratio of turbulent dissipation rates estimated using the finescale parameterization εfine to those obtained from microstructure measurements ε (shown by the color of dots) and internal wave parameters: shear/strain ratio Rω and internal wave spectral level EIW in the in situ observation data from the SOFine project (Waterman et al. 2013). (b),(c) Relationship between the ratio of εfine to the energy transfer rates ε (shown by the color of dots and values next to dots), Rω, and EIW obtained from five eikonal simulations. When implementing the finescale parameterization, the LADCP-based cutoff wavenumber (mcLAD=0.0175 cpm) is adopted for (a) and (b), whereas the Richardson number–based (Ri-based) cutoff wavenumber is adopted for (c).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0196.1

From these results, we can expect that the discrepancy between the directly measured ε and the parameterization inference εfine in the ACC region might be caused by the distorted shape of vertical wavenumber spectra. Such discrepancy would be reduced so long as finescale spectra free from noise contamination are available to the higher wavenumbers. However, if the spectral hump is very large, the finescale parameterization still would not estimate ε accurately as in the NI-strong case.

5. Discussion and conclusions

Although finescale parameterizations (Henyey et al. 1986; Polzin et al. 1995; Gregg et al. 2003; Ijichi and Hibiya 2015) are powerful tools to infer the spatial distribution of the turbulent dissipation rate ε in a much easier way than with microstructure measurements, they tend to overestimate ε in the ACC region (Waterman et al. 2014; Takahashi and Hibiya 2019). This overestimation was prominent at the locations where shear and strain spectra were distorted from the GM spectrum by a spectral hump at ~0.01-cpm vertical wavenumbers (Takahashi and Hibiya 2020, manuscript submitted to J. Geophys. Res. Oceans). In this paper, using eikonal simulations (e.g., Henyey et al. 1986), we have reproduced the internal wave energy cascade for several distorted vertical wavenumber spectra to quantitatively assess the validity of the latest finescale parameterization (Ijichi and Hibiya 2015) for such spectra.

In the eikonal experiments, low-vertical-wavenumber internal wave packets constituting the spectral hump promote the breaking of high-vertical-wavenumber internal waves, so that ε becomes larger than in the canonical flat-shaped GM spectrum. However, the low-vertical-wavenumber internal waves do not break within the calculation time (40 inertial periods) or directly contribute to ε because their vertical and horizontal wavenumbers are much smaller than those of breaking waves.

The finescale parameterization estimates ε in terms of the shear and strain spectral variances obtained by integrating the shear and strain spectra from the lowest wavenumber to the cutoff wavenumber mc. The upper limit of integration mc is an important parameter when the spectrum is not flat-shaped. When applying the parameterization to observational data, for example to LADCP-measured velocity profiles, we are forced to use the cutoff wavenumber mcLAD determined by the LADCP’s noise criterion (Polzin et al. 2002; Thurnherr 2012), which is usually lower than the RI-based cutoff wavenumber mcRi (Gargett et al. 1981; Munk 1981).

When one adopts mcLAD as the cutoff wavenumber, the shear and strain variances are calculated from the wavenumbers of the spectral hump, so the estimated turbulent dissipation rates εfine are larger than the spectral energy transfer rates ε in the eikonal experiments. The overestimation increases as the internal wave spectral level EIW increases and the shear/strain ratio Rω decreases, consistent with the results of previous observations in the ACC region (Takahashi and Hibiya 2020, manuscript submitted to J. Geophys. Res. Oceans). However, when mcRi is adopted instead, the above overestimation is substantially reduced in most cases.

These results are also consistent with the evaluation of the spectral energy transfer rate by Eden et al. (2019). By numerically computing the kinematic equation for various GM-like internal wave spectra, these authors found that ε depends not only on the spectral level, but also on the spectral slope of a GM-like vertical wavenumber spectrum such that ε decreases as the spectral slope decreases (i.e., as the spectrum becomes redder). This “spectral slope” is closely related to what we call a “spectral hump.”

A series of eikonal calculations supports our hypothesis that the observed discrepancies between microstructure measurements and ε inferred from the finescale parameterization in the ACC (Waterman et al. 2014; Takahashi and Hibiya 2019) were caused by the estimation of the spectral variances in the low-wavenumber band (with mcLAD), which is often distorted by a spectral hump. The most straightforward remedy to avoid this overestimation is using shear spectra free from noise contamination up to the higher wavenumbers approaching mcRi. Meyer et al. (2015) confirmed that the shear spectra obtained using the Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats in the ACC region were not affected by noise contamination and were well resolved up to mcRi for each profile.

Note, however, that the existing finescale parameterization with mcRi still overestimates ε so long as the shear spectrum has a large-amplitude spectral hump, such as in the NI-strong case in section 4. This is because, as the shear spectral level increases, mcRi shifts to almost the same value as mcLAD, which is much lower than the breaking limit mbreak. Although mcRi calculated from the empirical inverse Richardson number function is regarded as the spectral rolloff wavenumber, the mechanism by which the spectral rolloff is created is still unclear (Gregg et al. 1993; Hibiya et al. 1996) and should be investigated in the future.

Since Gregg (1989) proposed the prototype of finescale parameterizations based on a weak wave–wave interaction theory (Henyey et al. 1986; McComas and Muller 1981), several modifications have been made to take into account internal wave frequency spectra distorted from the canonical GM spectrum. Nevertheless, not much attention has been paid to the distortion of vertical wavenumber spectra. Although Polzin et al. (1995, 2014) speculated that the distortion of the vertical wavenumber spectra has little effect on the estimate of the finescale parameterization (at most a factor of 2), we have shown that (i) the influence might not be negligible in the ACC region, and (ii) shear and strain spectra free from noise up to high vertical wavenumbers are required for better estimates of ε.

Analyses in a companion paper (Takahashi and Hibiya 2020, manuscript submitted to J. Geophys. Res. Oceans) suggested that the distorted vertical wavenumber spectra observed in the ACC were associated with geostrophic shear. Shear spectral humps in the upper ocean might be caused by low-vertical-wavenumber near-inertial waves trapped by mesoscale eddies (e.g., Kunze 1985) or spontaneously generated by instabilities of ACC frontal jets (Nagai et al. 2015). Low-vertical-wavenumber internal lee waves generated by geostrophic currents impinging on rough bottom topography (e.g., Waterman et al. 2013; Cusack et al. 2017) might create humps in the near-bottom shear and strain spectra. Interestingly, such spectra biased to low-vertical wavenumbers have been reported not only in the ACC (Kilbourne and Girton 2015; Meyer et al. 2016; Takahashi and Hibiya 2019) but also under anticyclonic eddies in the subtropical western North Pacific (Jing et al. 2011), the Mediterranean Sea (Cuypers et al. 2012), and the western boundary of the subtropical North Atlantic (Fernández-Castro et al. 2020). To accurately estimate the distribution of turbulent dissipation and diapycnal mixing in the ACC and, more generally, in regions with an energetic mesoscale eddy field, special care is needed on the vertical wavenumber band used for the finescale parameterizations.

The present eikonal calculations are based on several assumptions. Although the strength of WKB scale separation between test waves and the background does not alter our results qualitatively, the evolution of internal waves in the real ocean might be affected by small-scale background structures. Furthermore, eikonal simulations calculate the spectral energy transfers based on internal wave dynamics, but they do not reproduce the dynamics of turbulent energy production. Hence, perhaps the breaking limit mbreak should be related to the transition between the linear internal wave scale and turbulent scales, which might not be constant. In the future, the appropriate setting of eikonal calculations, especially for the internal wave field distorted from the GM model, should be investigated on the basis of observational data.

Acknowledgments

The basic codes for the present eikonal calculations were kindly provided by Dr. Takashi Ijichi. The numerical simulations were carried out using the supercomputer: Fujitsu PRIMERGY CX600M1/CX1640M1(Oakforest-PACS) in the Information Technology Center of the University of Tokyo. Observational data are obtained from the 19th Kaiyodai Antarctic Research Expedition (KARE) and the Southern Ocean Finestructure (SOFine) survey. These observation data used in the preparation of this paper are publicly available (http://www-aos.eps.s.u-tokyo.ac.jp/~takahashi/SouthernOcean). We are deeply grateful to all of the participants in the KARE19 cruise of the T/V Umitaka-maru of Tokyo University of Marine Science and Technology for their support. Special thanks are given to Dr. Yujiro Kitade for his support. The SOFine data were kindly provided by Dr. Alexander Forryan at the National Oceanography Centre, University of Southampton. We also thank two anonymous reviewers for their helpful comments. This study is based on Anne Takahashi’s Ph.D. thesis and was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grants 15H02131 to T. Hibiya and 17J06060 to A. Takahashi.

APPENDIX A

Background Velocity of the GM Spectrum

In the standard experiment, the background velocity uBG is calculated from the velocity of the GM internal wave field uGM =(uGM, υGM, wGM), which is reproduced by summing over 400 frequencies σ and 400 vertical modes j. Based on the dispersion relation [(7) in this paper] and the polarization relations for linear internal waves (Gill 1982), the decomposed velocity is expressed as
uGM=σ,mc dVdzσ2f2k2+l2(k sinαl fσ cosα),
υGM=σ,mc dVdzσ2f2k2+l2(l sinα+k fσ cosα),and
wGM=σ,m cVσ2f2 cosα,
where (k, l) is the horizontal wavenumber with direction randomly determined to ensure horizontal isotropy, m is the vertical wavenumber given by
m=πj/H(j=1,2,,400),
with the depth H = 2000 m, α is the wave phase given by α = kx + lyωt + δ with δ a random value ranging from 0 to 2π, V(z) is the vertical structure function given by V = sin(mz)/(N0N)1/2, and c(σ, m) is the amplitude coefficient given by
c=2EGM(σ,m)N0N1Δσ,
with Δσ being a difference between adjacent decomposed frequencies.

APPENDIX B

Background Velocity for the Hump Experiments

In the hump experiments, the background velocity is uBG = uGM + uhump, where uGM is the velocity of the GM internal wave field (see appendix A) and uhump = (uhump, υhump, whump) is the velocity of internal wave packets superposed on the GM spectrum formulated as
uhump=σ,m dN2σ2N m1(k cosβ+l fσ sinβ),
υhump=σ,m dN2σ2N m1(l cosβk fσ sinβ),and
whump=σ,m σ2f2N d cosβ.
Here we assume that the total energy Ehump=EGMtot=b2N0NE0 (for moderate experiments) or 2 times the total energy of the GM internal wave field Ehump=2EGMtot (for strong experiments) is equally distributed over the low-vertical-wavenumber (0.0025 ≤ m ≤ 0.01 cpm), near-inertial band (σ ≤ 1.02f, for NI experiments) or a little-higher-frequency band (2fσ ≤ 2f, for lee experiments). The amplitude coefficient is
d(σ,m)=(2EhumpΔσΔmσ,mΔσΔm)1/2,
with Δσ and Δm indicating differences between adjacent decomposed frequencies and vertical wavenumbers, respectively. Unlike the GM model, the superposed waves are assumed to be plane waves, so the phase is given by β = kx + ly + mzσt + δ with δ being a random value ranging from 0 to 2π. This is because several observational works (Meyer et al. 2016; Cusack et al. 2017; Takahashi and Hibiya 2019) showed that there are many internal wave packet-like features with either upward- or downward-energy propagation creating spectral humps.

REFERENCES

  • Bretherton, F. P., and C. Garrett, 1968: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. London, 302A, 529554, https://doi.org/10.1098/rspa.1968.0034.

    • Search Google Scholar
    • Export Citation
  • Bryan, F., 1987: Parameter sensitivity of primitive equation ocean general circulation models. J. Phys. Oceanogr., 17, 970985, https://doi.org/10.1175/1520-0485(1987)017<0970:PSOPEO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cairns, J. L., and G. O. Williams, 1976: Internal wave observations from a midwater float, 2. J. Geophys. Res., 81, 19431950, https://doi.org/10.1029/JC081i012p01943.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cusack, J. M., A. C. Naveira Garabato, D. A. Smeed, and J. B. Girton, 2017: Observation of a large lee wave in the Drake Passage. J. Phys. Oceanogr., 47, 793810, https://doi.org/10.1175/JPO-D-16-0153.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cuypers, Y., P. Bouruet-Aubertot, C. Marec, and J.-L. Fuda, 2012: Characterization of turbulence from a fine-scale parameterization and microstructure measurements in the Mediterranean Sea during the BOUM experiment. Biogeosciences, 9, 31313149, https://doi.org/10.5194/bg-9-3131-2012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Duda, T. F., and C. S. Cox, 1989: Vertical wave number spectra of velocity and shear at small internal wave scales. J. Geophys. Res., 94, 939, https://doi.org/10.1029/JC094iC01p00939.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eden, C., F. Pollmann, and D. Olbers, 2019: Numerical evaluation of energy transfers in internal gravity wave spectra of the ocean. J. Phys. Oceanogr., 49, 737749, https://doi.org/10.1175/JPO-D-18-0075.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fernández-Castro, B., D. G. Evans, E. Frajka-Williams, C. Vic, and A. C. Naveira-Garabato, 2020: Breaking of internal waves and turbulent dissipation in an anticyclonic mode water eddy. J. Phys. Oceanogr., 50, 18931914, https://doi.org/10.1175/JPO-D-19-0168.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gargett, A. E., P. J. Hendricks, T. B. Sanford, T. R. Osborn, and A. J. Williams, 1981: A composite spectrum of vertical shear in the upper ocean. J. Phys. Oceanogr., 11, 12581271, https://doi.org/10.1175/1520-0485(1981)011<1258:ACSOVS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, C., and W. Munk, 1972: Space-time scales of internal waves. Geophys. Fluid Dyn., 3, 225264, https://doi.org/10.1080/03091927208236082.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, C., and W. Munk, 1975: Space-time scales of internal waves: A progress report. J. Geophys. Res., 80, 291297, https://doi.org/10.1029/JC080i003p00291.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gill, A. E., 1981: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Gregg, M. C., 1989: Scaling turbulent dissipation in the thermocline. J. Geophys. Res., 94, 96869698, https://doi.org/10.1029/JC094iC07p09686.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gregg, M. C., D. P. Winkel, and T. B. Sanford, 1993: Varieties of fully resolved spectra of vertical shear. J. Phys. Oceanogr., 23, 124141, https://doi.org/10.1175/1520-0485(1993)023<0124:VOFRSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gregg, M. C., T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from the breaking of internal waves in equatorial waters. Nature, 422, 513515, https://doi.org/10.1038/nature01507.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henyey, F. S., and N. Pomphrey, 1983: Eikonal description of internal wave interactions: A non-diffusive picture of “induced diffusion”. Dyn. Atmos. Oceans, 7, 189219, https://doi.org/10.1016/0377-0265(83)90005-2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field: An eikonal approach. J. Geophys. Res., 91, 84878495, https://doi.org/10.1029/JC091iC07p08487.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hibiya, T., Y. Niwa, K. Nakajima, and N. Suginohara, 1996: Direct numerical simulation of the roll-off range of internal wave shear spectra in the ocean. J. Geophys. Res., 101, 14 12314 129, https://doi.org/10.1029/96JC01001.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ijichi, T., and T. Hibiya, 2015: Frequency-based correction of finescale parameterization of turbulent dissipation in the deep ocean. J. Atmos. Oceanic Technol., 32, 15261535, https://doi.org/10.1175/JTECH-D-15-0031.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ijichi, T., and T. Hibiya, 2017: Eikonal calculations for energy transfer in the deep-ocean internal wave field near mixing hotspots. J. Phys. Oceanogr., 47, 199210, https://doi.org/10.1175/JPO-D-16-0093.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jing, Z., L. Wu, L. Li, C. Liu, X. Liang, Z. Chen, D. Hu, and Q. Liu, 2011: Turbulent diapycnal mixing in the subtropical northwestern Pacific: Spatial-seasonal variations and role of eddies. J. Geophys. Res., 116, C10028, https://doi.org/10.1029/2011JC007142.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kilbourne, B. F., and J. B. Girton, 2015: Quantifying high-frequency wind energy flux into near-inertial motions in the southeast Pacific. J. Phys. Oceanogr., 45, 369386, https://doi.org/10.1175/JPO-D-14-0076.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear. J. Phys. Oceanogr., 15, 544565, https://doi.org/10.1175/1520-0485(1985)015<0544:NIWPIG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., 2017: The internal-wave-driven meridional overturning circulation. J. Phys. Oceanogr., 47, 26732689, https://doi.org/10.1175/JPO-D-16-0142.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marshall, J., and K. Speer, 2012: Closure of the meridional overturning circulation through Southern Ocean upwelling. Nat. Geosci., 5, 171180, https://doi.org/10.1038/ngeo1391.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McComas, C. H., and P. Müller, 1981: The dynamic balance of internal waves. J. Phys. Oceanogr., 11, 970986, https://doi.org/10.1175/1520-0485(1981)011<0970:TDBOIW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meyer, A., B. M. Sloyan, K. L. Polzin, H. E. Phillips, and N. L. Bindoff, 2015: Mixing variability in the Southern Ocean. J. Phys. Oceanogr., 45, 966987, https://doi.org/10.1175/JPO-D-14-0110.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meyer, A., K. L. Polzin, B. M. Sloyan, and H. E. Phillips, 2016: Internal waves and mixing near the Kerguelen Plateau. J. Phys. Oceanogr., 46, 417437, https://doi.org/10.1175/JPO-D-15-0055.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Munk, W. H., 1981: Internal waves and small-scale processes. Evolution of Physical Oceanography, B. A. Warren, and C. Wunsch, Eds., MIT Press, 264–291.

  • Munk, W. H., 1966: Abyssal recipes. Deep-Sea Res. Oceanogr. Abstr., 13, 707730, https://doi.org/10.1016/0011-7471(66)90602-4.

  • Nagai, T., A. Tandon, E. Kunze, and A. Mahadevan, 2015: Spontaneous generation of near-inertial waves by the Kuroshio front. J. Phys. Oceanogr., 45, 23812406, https://doi.org/10.1175/JPO-D-14-0086.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Naveira-Garabato, A. C., K. L. Polzin, B. A. King, K. J. Heywood, and M. Visbeck, 2004: Widespread intense turbulent mixing in the Southern Ocean. Science, 303, 210213, https://doi.org/10.1126/science.1090929.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Oka, A., and Y. Niwa, 2013: Pacific deep circulation and ventilation controlled by tidal mixing away from the sea bottom. Nat. Commun., 4, 2419, https://doi.org/10.1038/ncomms3419.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterizations of turbulent dissipation. J. Phys. Oceanogr., 25, 306328, https://doi.org/10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., E. Kunze, J. Hummon, and E. Firing, 2002: The finescale response of lowered ADCP velocity profiles. J. Atmos. Oceanic Technol., 19, 205224, https://doi.org/10.1175/1520-0426(2002)019<0205:TFROLA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., A. C. Naveira Garabato, T. N. Huussen, B. M. Sloyan, and S. Waterman, 2014: Finescale parameterizations of turbulent dissipation. J. Geophys. Res. Oceans, 119, 13831419, https://doi.org/10.1002/2013JC008979.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sheen, K. L., and Coauthors, 2013: Rates and mechanisms of turbulent dissipation and mixing in the Southern Ocean: Results from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES). J. Geophys. Res. Oceans, 118, 27742792, https://doi.org/10.1002/jgrc.20217.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sun, H., and E. Kunze, 1999a: Internal wave–wave interactions. Part I: The role of internal wave vertical divergence. J. Phys. Oceanogr., 29, 28862904, https://doi.org/10.1175/1520-0485(1999)029<2886:IWWIPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sun, H., and E. Kunze, 1999b: Internal wave–wave interactions. Part II: Spectral energy transfer and turbulence production. J. Phys. Oceanogr., 29, 29052919, https://doi.org/10.1175/1520-0485(1999)029<2905:IWWIPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Takahashi, A., and T. Hibiya, 2019: Assessment of finescale parameterizations of deep ocean mixing in the presence of geostrophic current shear: Results of microstructure measurements in the Antarctic Circumpolar Current region. J. Geophys. Res. Oceans, 124, 135153, https://doi.org/10.1029/2018JC014030.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thurnherr, A. M., 2012: The finescale response of lowered ADCP velocity measurements processed with different methods. J. Atmos. Oceanic Technol., 29, 597600, https://doi.org/10.1175/JTECH-D-11-00158.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Toggweiler, J. R., and J. Russell, 2008: Ocean circulation in a warming climate. Nature, 451, 286288, https://doi.org/10.1038/nature06590.

  • Watanabe, M., and T. Hibiya, 2005: Estimates of energy dissipation rates in the three-dimensional deep ocean internal wave field. J. Oceanogr., 61, 123127, https://doi.org/10.1007/s10872-005-0025-3.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Waterhouse, A. F., and Coauthors, 2014: Global patterns of diapycnal mixing from measurements of the turbulent dissipation rate. J. Phys. Oceanogr., 44, 18541872, https://doi.org/10.1175/JPO-D-13-0104.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Waterman, S., A. Naveira Garabato, and K. Polzin, 2013: Internal waves and turbulence in the Antarctic Circumpolar Current. J. Phys. Oceanogr., 43, 259282, https://doi.org/10.1175/JPO-D-11-0194.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Waterman, S., K. L. Polzin, A. C. Naveira Garabato, K. L. Sheen, and A. Forryan, 2014: Suppression of internal wave breaking in the Antarctic Circumpolar Current near topography. J. Phys. Oceanogr., 44, 14661492, https://doi.org/10.1175/JPO-D-12-0154.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wijesekera, H., L. Padman, T. Dillon, M. Levine, C. Paulson, and R. Pinkel, 1993: The application of internal-wave dissipation models to a region of strong mixing. J. Phys. Oceanogr., 23, 269286, https://doi.org/10.1175/1520-0485(1993)023<0269:TAOIWD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wu, L. X., Z. Jing, S. Riser, and M. Visbeck, 2011: Seasonal and spatial variations of Southern Ocean diapycnal mixing from Argo profiling floats. Nat. Geosci., 4, 363366, https://doi.org/10.1038/ngeo1156.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save
  • Bretherton, F. P., and C. Garrett, 1968: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. London, 302A, 529554, https://doi.org/10.1098/rspa.1968.0034.

    • Search Google Scholar
    • Export Citation
  • Bryan, F., 1987: Parameter sensitivity of primitive equation ocean general circulation models. J. Phys. Oceanogr., 17, 970985, https://doi.org/10.1175/1520-0485(1987)017<0970:PSOPEO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cairns, J. L., and G. O. Williams, 1976: Internal wave observations from a midwater float, 2. J. Geophys. Res., 81, 19431950, https://doi.org/10.1029/JC081i012p01943.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cusack, J. M., A. C. Naveira Garabato, D. A. Smeed, and J. B. Girton, 2017: Observation of a large lee wave in the Drake Passage. J. Phys. Oceanogr., 47, 793810, https://doi.org/10.1175/JPO-D-16-0153.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cuypers, Y., P. Bouruet-Aubertot, C. Marec, and J.-L. Fuda, 2012: Characterization of turbulence from a fine-scale parameterization and microstructure measurements in the Mediterranean Sea during the BOUM experiment. Biogeosciences, 9, 31313149, https://doi.org/10.5194/bg-9-3131-2012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Duda, T. F., and C. S. Cox, 1989: Vertical wave number spectra of velocity and shear at small internal wave scales. J. Geophys. Res., 94, 939, https://doi.org/10.1029/JC094iC01p00939.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eden, C., F. Pollmann, and D. Olbers, 2019: Numerical evaluation of energy transfers in internal gravity wave spectra of the ocean. J. Phys. Oceanogr., 49, 737749, https://doi.org/10.1175/JPO-D-18-0075.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fernández-Castro, B., D. G. Evans, E. Frajka-Williams, C. Vic, and A. C. Naveira-Garabato, 2020: Breaking of internal waves and turbulent dissipation in an anticyclonic mode water eddy. J. Phys. Oceanogr., 50, 18931914, https://doi.org/10.1175/JPO-D-19-0168.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gargett, A. E., P. J. Hendricks, T. B. Sanford, T. R. Osborn, and A. J. Williams, 1981: A composite spectrum of vertical shear in the upper ocean. J. Phys. Oceanogr., 11, 12581271, https://doi.org/10.1175/1520-0485(1981)011<1258:ACSOVS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, C., and W. Munk, 1972: Space-time scales of internal waves. Geophys. Fluid Dyn., 3, 225264, https://doi.org/10.1080/03091927208236082.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, C., and W. Munk, 1975: Space-time scales of internal waves: A progress report. J. Geophys. Res., 80, 291297, https://doi.org/10.1029/JC080i003p00291.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gill, A. E., 1981: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Gregg, M. C., 1989: Scaling turbulent dissipation in the thermocline. J. Geophys. Res., 94, 96869698, https://doi.org/10.1029/JC094iC07p09686.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gregg, M. C., D. P. Winkel, and T. B. Sanford, 1993: Varieties of fully resolved spectra of vertical shear. J. Phys. Oceanogr., 23, 124141, https://doi.org/10.1175/1520-0485(1993)023<0124:VOFRSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gregg, M. C., T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from the breaking of internal waves in equatorial waters. Nature, 422, 513515, https://doi.org/10.1038/nature01507.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henyey, F. S., and N. Pomphrey, 1983: Eikonal description of internal wave interactions: A non-diffusive picture of “induced diffusion”. Dyn. Atmos. Oceans, 7, 189219, https://doi.org/10.1016/0377-0265(83)90005-2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field: An eikonal approach. J. Geophys. Res., 91, 84878495, https://doi.org/10.1029/JC091iC07p08487.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hibiya, T., Y. Niwa, K. Nakajima, and N. Suginohara, 1996: Direct numerical simulation of the roll-off range of internal wave shear spectra in the ocean. J. Geophys. Res., 101, 14 12314 129, https://doi.org/10.1029/96JC01001.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ijichi, T., and T. Hibiya, 2015: Frequency-based correction of finescale parameterization of turbulent dissipation in the deep ocean. J. Atmos. Oceanic Technol., 32, 15261535, https://doi.org/10.1175/JTECH-D-15-0031.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ijichi, T., and T. Hibiya, 2017: Eikonal calculations for energy transfer in the deep-ocean internal wave field near mixing hotspots. J. Phys. Oceanogr., 47, 199210, https://doi.org/10.1175/JPO-D-16-0093.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jing, Z., L. Wu, L. Li, C. Liu, X. Liang, Z. Chen, D. Hu, and Q. Liu, 2011: Turbulent diapycnal mixing in the subtropical northwestern Pacific: Spatial-seasonal variations and role of eddies. J. Geophys. Res., 116, C10028, https://doi.org/10.1029/2011JC007142.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kilbourne, B. F., and J. B. Girton, 2015: Quantifying high-frequency wind energy flux into near-inertial motions in the southeast Pacific. J. Phys. Oceanogr., 45, 369386, https://doi.org/10.1175/JPO-D-14-0076.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear. J. Phys. Oceanogr., 15, 544565, https://doi.org/10.1175/1520-0485(1985)015<0544:NIWPIG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., 2017: The internal-wave-driven meridional overturning circulation. J. Phys. Oceanogr., 47, 26732689, https://doi.org/10.1175/JPO-D-16-0142.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marshall, J., and K. Speer, 2012: Closure of the meridional overturning circulation through Southern Ocean upwelling. Nat. Geosci., 5, 171180, https://doi.org/10.1038/ngeo1391.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McComas, C. H., and P. Müller, 1981: The dynamic balance of internal waves. J. Phys. Oceanogr., 11, 970986, https://doi.org/10.1175/1520-0485(1981)011<0970:TDBOIW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meyer, A., B. M. Sloyan, K. L. Polzin, H. E. Phillips, and N. L. Bindoff, 2015: Mixing variability in the Southern Ocean. J. Phys. Oceanogr., 45, 966987, https://doi.org/10.1175/JPO-D-14-0110.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meyer, A., K. L. Polzin, B. M. Sloyan, and H. E. Phillips, 2016: Internal waves and mixing near the Kerguelen Plateau. J. Phys. Oceanogr., 46, 417437, https://doi.org/10.1175/JPO-D-15-0055.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Munk, W. H., 1981: Internal waves and small-scale processes. Evolution of Physical Oceanography, B. A. Warren, and C. Wunsch, Eds., MIT Press, 264–291.

  • Munk, W. H., 1966: Abyssal recipes. Deep-Sea Res. Oceanogr. Abstr., 13, 707730, https://doi.org/10.1016/0011-7471(66)90602-4.

  • Nagai, T., A. Tandon, E. Kunze, and A. Mahadevan, 2015: Spontaneous generation of near-inertial waves by the Kuroshio front. J. Phys. Oceanogr., 45, 23812406, https://doi.org/10.1175/JPO-D-14-0086.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Naveira-Garabato, A. C., K. L. Polzin, B. A. King, K. J. Heywood, and M. Visbeck, 2004: Widespread intense turbulent mixing in the Southern Ocean. Science, 303, 210213, https://doi.org/10.1126/science.1090929.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Oka, A., and Y. Niwa, 2013: Pacific deep circulation and ventilation controlled by tidal mixing away from the sea bottom. Nat. Commun., 4, 2419, https://doi.org/10.1038/ncomms3419.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterizations of turbulent dissipation. J. Phys. Oceanogr., 25, 306328, https://doi.org/10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., E. Kunze, J. Hummon, and E. Firing, 2002: The finescale response of lowered ADCP velocity profiles. J. Atmos. Oceanic Technol., 19, 205224, https://doi.org/10.1175/1520-0426(2002)019<0205:TFROLA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., A. C. Naveira Garabato, T. N. Huussen, B. M. Sloyan, and S. Waterman, 2014: Finescale parameterizations of turbulent dissipation. J. Geophys. Res. Oceans, 119, 13831419, https://doi.org/10.1002/2013JC008979.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sheen, K. L., and Coauthors, 2013: Rates and mechanisms of turbulent dissipation and mixing in the Southern Ocean: Results from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES). J. Geophys. Res. Oceans, 118, 27742792, https://doi.org/10.1002/jgrc.20217.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sun, H., and E. Kunze, 1999a: Internal wave–wave interactions. Part I: The role of internal wave vertical divergence. J. Phys. Oceanogr., 29, 28862904, https://doi.org/10.1175/1520-0485(1999)029<2886:IWWIPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sun, H., and E. Kunze, 1999b: Internal wave–wave interactions. Part II: Spectral energy transfer and turbulence production. J. Phys. Oceanogr., 29, 29052919, https://doi.org/10.1175/1520-0485(1999)029<2905:IWWIPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Takahashi, A., and T. Hibiya, 2019: Assessment of finescale parameterizations of deep ocean mixing in the presence of geostrophic current shear: Results of microstructure measurements in the Antarctic Circumpolar Current region. J. Geophys. Res. Oceans, 124, 135153, https://doi.org/10.1029/2018JC014030.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thurnherr, A. M., 2012: The finescale response of lowered ADCP velocity measurements processed with different methods. J. Atmos. Oceanic Technol., 29, 597600, https://doi.org/10.1175/JTECH-D-11-00158.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Toggweiler, J. R., and J. Russell, 2008: Ocean circulation in a warming climate. Nature, 451, 286288, https://doi.org/10.1038/nature06590.

  • Watanabe, M., and T. Hibiya, 2005: Estimates of energy dissipation rates in the three-dimensional deep ocean internal wave field. J. Oceanogr., 61, 123127, https://doi.org/10.1007/s10872-005-0025-3.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Waterhouse, A. F., and Coauthors, 2014: Global patterns of diapycnal mixing from measurements of the turbulent dissipation rate. J. Phys. Oceanogr., 44, 18541872, https://doi.org/10.1175/JPO-D-13-0104.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Waterman, S., A. Naveira Garabato, and K. Polzin, 2013: Internal waves and turbulence in the Antarctic Circumpolar Current. J. Phys. Oceanogr., 43, 259282, https://doi.org/10.1175/JPO-D-11-0194.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Waterman, S., K. L. Polzin, A. C. Naveira Garabato, K. L. Sheen, and A. Forryan, 2014: Suppression of internal wave breaking in the Antarctic Circumpolar Current near topography. J. Phys. Oceanogr., 44, 14661492, https://doi.org/10.1175/JPO-D-12-0154.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wijesekera, H., L. Padman, T. Dillon, M. Levine, C. Paulson, and R. Pinkel, 1993: The application of internal-wave dissipation models to a region of strong mixing. J. Phys. Oceanogr., 23, 269286, https://doi.org/10.1175/1520-0485(1993)023<0269:TAOIWD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wu, L. X., Z. Jing, S. Riser, and M. Visbeck, 2011: Seasonal and spatial variations of Southern Ocean diapycnal mixing from Argo profiling floats. Nat. Geosci., 4, 363366, https://doi.org/10.1038/ngeo1156.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    The vertically averaged energy transfer rates ε [(13)] for different values of the breaking limit mbreak = 0.1, 0.2, 0.3, and 0.4 cpm in our eikonal simulations: the standard (black; see section 3 for details), NI-strong (red; see section 4), and lee-strong (blue; see section 4) experiments. Each error bar indicates 1 standard deviation of the vertical mean.

  • Fig. 2.

    Shear/strain ratio Rω–dependent term h [(16)] in the finescale parameterization (Ijichi and Hibiya 2015) with the buoyancy frequency N = N0 = 5.24 × 10−3 s−1 and Coriolis frequency f = f0 = 7.29 × 10−5 s−1.

  • Fig. 3.

    Vertical wavenumber spectra of buoyancy-normalized shear (black) and strain (gray) incorporated in the (a) standard experiment (section 3) and four kinds of hump experiments (section 4): the (b) NI-moderate, (c) NI-strong, (d) lee-moderate, and (e) lee-strong experiments. Dashed horizontal lines indicate the GM spectra (Munk 1981); dotted vertical lines indicate the LADCP-based cutoff wavenumber mcLAD=0.0175 cpm.

  • Fig. 4.

    Examples of the vertical wavenumber spectra of buoyancy-normalized shear (black) and strain (gray) obtained from finestructure measurements in the ACC region. The cruise name, station, and depth of the data, details of which are given in Takahashi and Hibiya (2020, manuscript submitted to J. Geophys. Res. Oceans), are shown on the labels. Dashed horizontal lines indicate the GM spectra (Garrett and Munk 1975; Cairns and Williams 1976); dotted vertical lines indicate the LADCP-based cutoff wavenumber mcLAD=0.0175 cpm.

  • Fig. 5.

    Results of the (a)–(c) standard experiment (section 3), (d)–(f) NI-strong experiment (section 4), and (g)–(i) lee-strong experiment (section 4) are summarized for (left) wave action A, (center) reciprocal of lifetime τ−1 normalized by the inertial time scale τi = 2π/f, and (right) energy transfer rate ε, all of which are calculated for each test wave with initial vertical wavenumber and frequency (minit, σinit), as shown by the color shading.

  • Fig. 6.

    (a) Relationship between the ratio of turbulent dissipation rates estimated using the finescale parameterization εfine to those obtained from microstructure measurements ε (shown by the color of dots) and internal wave parameters: shear/strain ratio Rω and internal wave spectral level EIW in the in situ observation data from the SOFine project (Waterman et al. 2013). (b),(c) Relationship between the ratio of εfine to the energy transfer rates ε (shown by the color of dots and values next to dots), Rω, and EIW obtained from five eikonal simulations. When implementing the finescale parameterization, the LADCP-based cutoff wavenumber (mcLAD=0.0175 cpm) is adopted for (a) and (b), whereas the Richardson number–based (Ri-based) cutoff wavenumber is adopted for (c).

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