The Vertical Structure of Internal Lee-Wave-Driven Benthic Mixing Hotspots

Ying He aKey Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Center for Ocean Mega-Science, Chinese Academy of Sciences, Qingdao, China
bDepartment of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Tokyo, Japan
cUniversity of Chinese Academy of Sciences, Beijing, China

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

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Abstract

In global ocean circulation and climate models, bottom-enhanced turbulent mixing is often parameterized such that the vertical decay scale of the energy dissipation rate ζ is universally constant at 500 m. In this study, using a nonhydrostatic two-dimensional numerical model in the horizontal–vertical plane that incorporates a monochromatic sinusoidal seafloor topography and the Garrett–Munk (GM) background internal wave field, we find that ζ of the internal lee-wave-driven bottom-enhanced mixing is actually variable depending on the magnitude of the steady flow U0, the horizontal wavenumber kH, and the height hT of the seafloor topography. When the steepness parameter (Sp = NhT/U0 where N is the buoyancy frequency near the seafloor) is less than 0.3, internal lee waves propagate upward from the seafloor while interacting with the GM internal wave field to create a turbulent mixing region with ζ that extends farther upward from the seafloor as U0 increases, but is nearly independent of kH. In contrast, when Sp exceeds 0.3, inertial oscillations (IOs) not far above the seafloor are enhanced by the intermittent supply of internal lee-wave energy Doppler-shifted to the near-inertial frequency, which occurs depending on the sign and magnitude of the background IO shear. The composite flow, consisting of the superposition of U0 and the IOs, interacts with the seafloor topography to efficiently generate internal lee waves during the period centered on the time of the composite flow maximum, but their upward propagation is inhibited by the increased IO shear, creating a turbulent mixing region of small ζ.

© 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).

He’s current affiliation: Center for Earth System Modeling and Prediction of China Meteorological Administration, Beijing, China.

Hibiya’s current affiliation: Tokyo University of Marine Science and Technology, Tokyo, Japan.

Corresponding authors: Ying He, heying@cma.gov.cn; Toshiyuki Hibiya, thib001@kaiyodai.ac.jp

Abstract

In global ocean circulation and climate models, bottom-enhanced turbulent mixing is often parameterized such that the vertical decay scale of the energy dissipation rate ζ is universally constant at 500 m. In this study, using a nonhydrostatic two-dimensional numerical model in the horizontal–vertical plane that incorporates a monochromatic sinusoidal seafloor topography and the Garrett–Munk (GM) background internal wave field, we find that ζ of the internal lee-wave-driven bottom-enhanced mixing is actually variable depending on the magnitude of the steady flow U0, the horizontal wavenumber kH, and the height hT of the seafloor topography. When the steepness parameter (Sp = NhT/U0 where N is the buoyancy frequency near the seafloor) is less than 0.3, internal lee waves propagate upward from the seafloor while interacting with the GM internal wave field to create a turbulent mixing region with ζ that extends farther upward from the seafloor as U0 increases, but is nearly independent of kH. In contrast, when Sp exceeds 0.3, inertial oscillations (IOs) not far above the seafloor are enhanced by the intermittent supply of internal lee-wave energy Doppler-shifted to the near-inertial frequency, which occurs depending on the sign and magnitude of the background IO shear. The composite flow, consisting of the superposition of U0 and the IOs, interacts with the seafloor topography to efficiently generate internal lee waves during the period centered on the time of the composite flow maximum, but their upward propagation is inhibited by the increased IO shear, creating a turbulent mixing region of small ζ.

© 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).

He’s current affiliation: Center for Earth System Modeling and Prediction of China Meteorological Administration, Beijing, China.

Hibiya’s current affiliation: Tokyo University of Marine Science and Technology, Tokyo, Japan.

Corresponding authors: Ying He, heying@cma.gov.cn; Toshiyuki Hibiya, thib001@kaiyodai.ac.jp

1. Introduction

The breaking of bottom-generated upward-propagating internal waves induces turbulent mixing that is enhanced over the seafloor topography and contributes to the diapycnal mixing required to maintain the global overturning circulation (e.g., Ledwell et al. 1993, 2000; Munk 1966; Polzin et al. 1997). The effects of near-field mixing have often been incorporated into global ocean circulation models as well as global climate models in the form of the semiempirical parameterization formulated by Jayne and St. Laurent (2001) and St. Laurent et al. (2002), which predicts the intensity of bottom-enhanced mixing by multiplying the upward energy flux of the internal tide at the seafloor topography ET by a local dissipation efficiency q and a vertical structure function F(z).

However, this parameterization has several shortcomings that should be addressed; for example, it ignores the existence of internal lee waves, which are typically generated by geostrophic flow interacting with topographic features, such as the Antarctic Circumpolar Circulation (ACC) impinging on the rough seafloor topography in the Southern Ocean (e.g., Naveira Garabato et al. 2004; Nikurashin and Ferrari 2011; Sheen et al. 2013; Sloyan 2005; St. Laurent et al. 2012). Furthermore, the vertical decay scale of the energy dissipation rate ζ in F(z) is assumed to be a global constant of 500 m based on field observations in limited areas of the ocean (St. Laurent et al. 2001), although microstructure observations have shown that ζ is highly variable in the global ocean (e.g., Waterhouse et al. 2014).

Hibiya et al. (2017) performed ray tracing calculations of upward-propagating internal lee-wave packets in the Garrett–Munk (GM) background internal wave field (Garrett and Munk 1972, 1975; Munk 1981) and found that ζ of the internal lee-wave-driven bottom-enhanced mixing is variable depending on the magnitude of the bottom currents U0 and the horizontal wavenumber of the seafloor topography kH. Motivated by this, Hibiya (2022) proposed a new parameterization for the internal lee-wave-driven bottom-enhanced mixing by replacing ζ in St. Laurent’s parameterization with a “vertical mean free path” Hfree = Cgz0τ, where Cgz0 is the vertical group velocity of the internal lee waves at the seafloor topography and τ is the resonant interaction time of the internal lee wave in the GM background internal wave field (McComas and Müller 1981). In both studies, however, the height of the seafloor topography hT was assumed to be infinitesimal and its nonlinear effects negligible.

As a measure of the nonlinearity associated with the finite-amplitude seafloor topography, Nikurashin and Ferrari (2010) used the steepness parameter, defined as the ratio of the topographic slope to the characteristic slope of the internal lee wave:
Sp=NhT/U0,
where N is the background buoyancy frequency near the seafloor topography. Using a two-dimensional numerical model in the horizontal–vertical (xz) plane, they found that, for Sp > 0.3, baroclinic inertial oscillations (IOs), characterized by a horizontally elongated structure and small group velocities, developed near the seafloor topography while promoting the breaking of bottom-generated internal lee waves; in contrast, for Sp < 0.3, such evolution of IOs was found to be very slow and the upward propagation of bottom-generated internal lee waves continued. However, the GM background internal wave field was not considered in their simulation, even though subsequent field observations in the Southern Ocean showed that the internal lee-wave activity was superimposed on the GM-like background internal wave field (e.g., Sheen et al. 2013; Takahashi and Hibiya 2021; Waterman et al. 2013).

Some previous studies assessed the impact of internal lee-wave-driven bottom-enhanced mixing on the global water mass transformation and thus on the global overturning circulation using the St. Laurent parameterization with ET replaced by EL, namely, the upward energy flux caused by the geostrophic flow interacting with the seafloor topography (e.g., Melet et al. 2014; Nikurashin and Ferrari 2013; Nikurashin et al. 2014). It was found that the internal lee-wave-driven bottom-enhanced mixing could contribute to about 1/3 or more of the global abyssal water mass transformation rate and thus affect the global overturning circulation, although this assessment was subject to the choice of several parameter values, including ζ in the St. Laurent parameterization. To improve the performance of global ocean circulation and climate models, it is therefore necessary to find the key parameters that control the vertical structure of the internal lee-wave-driven bottom-enhanced mixing.

This motivates the present study, in which a nonhydrostatic numerical model incorporating the GM background internal wave field is run for various combinations of hT, U0, and kH (section 3a). In addition, the numerically obtained vertical profiles of turbulent dissipation rates are compared with microstructure measurements from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES), as this is the only available observation that shows the close relationship between the vertical profile of turbulent dissipation rates and the amplitude of the ACC impinging on the high-wavenumber bathymetry (e.g., Sheen et al. 2013). Sheen et al. (2013) measured vertical profiles of turbulent dissipation rates over several meridional cross-ACC transects in or near Drake Passage. In particular, their Fig. 14 shows the vertical profiles of the turbulent dissipation rates averaged over the regions with steepness parameters Sp < 0.3 and Sp > 0.3 in these transects. Based on the obtained numerical results, we also revisit the evolution process of the IOs for Sp > 0.3 (section 3b).

2. Numerical model

We use the Massachusetts Institute of Technology General Circulation Model (MITgcm; Marshall et al. 1997) with its nonhydrostatic capability, which includes the GM background internal wave field and a monochromatic sinusoidal seafloor. To allow for wave interactions at different scales as much as possible, we adopt a vertical two-dimensional numerical model (i.e., ∂/∂y = 0) with a horizontal resolution of Δx = 50 m and a vertical resolution of Δz = 5 m, keeping in mind that this grid resolution is far from sufficient to directly reproduce the internal wave breaking process. The domain size is L × H = 90 km × 4 km, which is large enough to introduce the inertial component of the GM background internal wave field. Cyclic boundary conditions are applied to the lateral sides of the model, which prevents us from assessing the value of q in the near-field mixing parameterization, and the bottom of the model is assumed to be slippery.

The local inertial frequency f = 1.263 × 10−4 s−1 (corresponding to an inertial period Ti = 13.82 h) and a constant background buoyancy frequency N = 1.14 × 10−3 s−1 are assumed to be typical values for the deep ocean in the Southern Ocean (e.g., Sheen et al. 2013). To maintain the stability of the computations and suppress grid noise near the seafloor topography, we use Laplacian-type viscosity and diffusivity in the vertical (AV = 10−5 m2 s−1; KV = 10−5 m2 s−1) and biharmonic-type viscosity and diffusivity provided by MITgcm in the horizontal (A4H = 102 m4 s−1; K4H = 102 m4 s−1). Note that, although increasing the horizontal resolution is another way to suppress grid noise, we find that the model results with Δx = 10 m and Laplacian-type viscosity and diffusivity (AH = 10−2 m2 s−1; KH = 10−2 m2 s−1) are very close to those with Δx = 50 m and biharmonic-type viscosity and diffusivity, so instead of a finer horizontal resolution, a biharmonic-type viscosity and diffusivity is used to save computational resources.

In section 3a, we perform nine different numerical experiments (Exp01–09) for different values of hT, U0, and kH (Table 1). Note that, by varying hT from 10 to 80 m and adjusting U0 accordingly, Sp ranges from 0.11 to 0.46 for a fixed value of N = 1.14 × 10−3 s−1. At the same time, even for the same value of Sp, the dependence of the vertical structure of the internal lee-wave-driven bottom-enhanced mixing on hT and/or U0 (Exp01and Exp06; Exp07, Exp08 and Exp09) and on kH (Exp03, Exp04 and Exp05) is investigated. The empirical model GM79 (Munk 1981) is used as initial condition for Exp01–09. In GM79, the dimensionless energy density E^ of the internal wave with frequency ω and vertical mode number j is given by
E^(ω,j)=B(ω)H(j)EGM,
where
B(ω)=2π1fω1(ω2f2)1/2,NfB(ω)dω=1,
H(j)=(j2+j*2)11(j2+j*2)1,j=1H(j)=1,
with EGM = 6.3 × 10−5 and j*=3. Based on the dispersion relation, E^(ω,j) can be transformed to E^(n,j), where n is the horizontal mode number. To reproduce the GM background internal wave field for each numerical experiment, following Hibiya et al. (2002), we first simulate nonlinear interactions over 10 inertial periods (t = 0Ti–10Ti) between randomly phased linear internal waves of horizontal 0–900 modes and vertical 1–500 modes, each amplitude determined from the GM79 model (Munk 1981). More details on the reconstruction of the GM background internal wave field in the numerical model can be found in Iwamae et al. (2009). After achieving the quasi-steady GM internal wave field, we use the next inertial period (t = 10Ti–11Ti) to gradually establish the depth-independent steady flow throughout the model domain before running the model for the remaining 9 inertial periods (t = 11Ti–20Ti). We also add a body force U0 × f0 to the meridional momentum equation, which is necessary to keep the barotropic pressure gradient in equilibrium with the steady flow U0 at all depths.
Table 1.

The identification (ID), the height hT, wavenumber kH, and wavelength λH of the seafloor topography, the magnitude of the steady flow U0, and the steepness parameter Sp for each numerical experiment in section 3a.

Table 1.

In section 3b, another numerical experiment (Exp10) with the same hT, U0, and kH as in Exp07 is performed to revisit the evolution process of IOs. To compare the computed results of the present study with those of Nikurashin and Ferrari (2010), the GM background internal wave field is removed and the width of the model domain is reduced to 10 km to save computational resources, and a sponge layer is applied in the upper 1 km of the vertical domain. For Exp10, we first use one inertial period (t = 0Ti–1Ti) to gradually establish the depth-independent steady flow U0 throughout the model domain, and then run the numerical model with only the established steady flow for another 19 inertial periods (t = 1Ti–20Ti).

3. Results and discussion

Following Nikurashin and Ferrari (2010), we decompose the velocity field into a zonally averaged flow (including the prescribed steady flow U0 and the IOs) and deviations from the zonally averaged flow (remaining internal waves), namely,
u=U0+uIO+uIW,
υ=υIO+υIW,
w=wIW,
where uIO, υIO are the zonal and meridional components of the IOs, respectively; and uIW, υIW, wIW are the zonal, meridional, and vertical components of the residual internal waves, respectively. In practice, uIO and υIO are calculated by zonally averaging the perturbation velocity field, while uIW, υIW and wIW are calculated by subtracting the zonally averaged perturbation velocity from the perturbation velocity field.

a. Parameter dependence study

The evolution of IOs depends on the value of Sp. Figure 1 shows that, in addition to the IOs due to the prescribed GM background internal wave field, there are IOs that evolve near the seafloor topography after t = 10Ti for Sp = 0.46 (Exp07), but the evolution of IOs is not obvious for Sp = 0.11 (Exp05), which is consistent with the previous study showing that IOs evolve rapidly only for Sp > 0.3 (Nikurashin and Ferrari 2010). Therefore, we divide all the results of our numerical experiments into two categories: quasi-stationary regime (Sp < 0.3) and time-dependent regime (Sp > 0.3).

Fig. 1.
Fig. 1.

Time evolution of the zonal component of the inertial oscillations uIO during t = 0Ti–20Ti in the numerical experiments with the steepness parameter (Sp) equal to (a) 0.11 (Exp05) and (b) 0.46 (Exp07). Note that the GM79 empirical model is used as the initial condition, and the steady flow U0 is gradually established in the model after t = 10Ti.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-22-0268.1

For each numerical experiment, we evaluate the energy dissipation rate:
ε=εvisc+εdiff(Wkg1),
using the dissipation term of the kinetic energy equation given by
εvisc=A4H[(2ux2)2+(2υx2)2+(2wx2)2]+AV[(uz)2+(υz)2+(wz)2],
and the dissipation term of the potential energy equation given by
εdiff=K4Hg2ρ02N2(2ρx2)2+KVg2ρ02N2(ρz)2,
where g is the acceleration due to gravity and ρ′ is the density perturbation from the background density profile. It should be noted that Eq. (8) only gives an estimate of the energy dissipation rates at the grid scale of the numerical model, and is not strictly equivalent to an estimate of the turbulent dissipation rates associated with internal wave breaking. Nevertheless, previous studies have shown that the vertical profiles of turbulent dissipation rates estimated based on this equation generally agree well with those obtained from actual turbulence observations (e.g., Nagai et al. 2017). Furthermore, it has been found that changing the values of A4H and AV by several orders of magnitude has little effect on the value of ε, presumably because the magnitude of horizontal and vertical shear also changes to counteract this effect.
We also calculate ζ by fitting the function
ε(z)=ε0(1+z/ζ)2(Wkg1)
(Hibiya et al. 2017; Polzin 2004, 2009) to the zonal-mean vertical profile of ε averaged over t = 15Ti–20Ti, where z is the height above the seafloor topography, and ε0 is the energy dissipation rate near the seafloor topography (i.e., the bottom value of the fitted line). Note that the ε data in the vicinity of the seafloor topography is very large due to grid noise, so it is excluded when fitting Eq. (11).

We first examine the vertical distributions of ε for various U0, kH, and a fixed hT in the quasi-stationary regime (Sp < 0.3) and their fits to Eq. (11). Note that, for U0 = 0.05 m s−1, there exists a turbulent dissipation rate ε that saturates at 3 × 10−11 W kg−1 even at heights greater than 1500 m above the seafloor, presumably caused by the nonlinear interaction between the background Garrett–Munk internal waves. This implies that there is a minimum value of ε that gives the applicability limit of the fit using Eq. (11). In the numerical experiments with a fixed kH, the energy dissipation region extends farther upward from the seafloor as U0 increases, such that ζ for U0 = 0.05 m s−1 (Exp01), 0.075 m s−1 (Exp02), and 0.1 m s−1 (Exp03) becomes about 1800, 4500, and 20 000 m, respectively (Figs. 2a,b). In addition, we can see that there is a clear trade-off between ζ and ε0, namely, the increase (decrease) of ζ is accompanied by the decrease (increase) of ε0 (Fig. 2a). On the other hand, in the numerical experiments with a fixed U0, the dependence of the vertical structure of ε on kH is relatively weak (Fig. 2c). It is worth noting that, although the values of ζ for the three numerical experiments appear to be very different (Fig. 2d), they are all large enough to suggest that bottom-enhanced mixing can extend into the upper ocean without obvious decay (Fig. 2c). The parameter dependencies thus obtained from our numerical experiments in the quasi-stationary regime (Sp < 0.3) are consistent with those obtained from the ray tracing calculations of Hibiya et al. (2017). This allows us to speculate that, even if the seafloor topography is multichromatic, ζ depends approximately only on U0 as long as Sp < 0.3 everywhere.

Fig. 2.
Fig. 2.

Dependence of the zonal-mean vertical distribution of the energy dissipation rate ε averaged over t = 15Ti–20Ti in (a) Exp01–03 on the steady flow magnitude U0 when the horizontal wavenumber of the seafloor topography kH is held constant at 1 × 10−3 cpm and (c) Exp03–05 on kH when U0 is held constant at 0.1 m s−1. The dashed lines are obtained by fitting the ε profiles to Eq. (11). (b),(d) The values of the vertical decay scale ζ and the vertical mean free path Hfree corresponding to the numerical experiments in (a) and (c), respectively. Note that all five numerical experiments satisfy Sp < 0.3.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-22-0268.1

We then examine the vertical distributions of ε for various U0, hT, and a fixed kH in the time-dependent regime (Sp > 0.3) and their fits to Eq. (11). When Sp increases to 0.46, most of the internal lee-wave energy is dissipated within 1000 m above the seafloor topography (Fig. 3a), and the dependence of ζ on U0 is very small compared to that in the quasi-stationary regime (Sp < 0.3) (Fig. 3b). It is interesting to note that, although both Exp06 (Sp = 0.23) and Exp09 (Sp = 0.46) have a large U0 of 0.2 m s−1, the value of ζ in Exp09 (808 m) is much more reduced than that in Exp06 (40 000 m), reflecting a very strong dependence of ζ on Sp.

Fig. 3.
Fig. 3.

(a) Zonal-mean vertical distributions of the energy dissipation rate ε averaged over t = 15Ti–20Ti in Exp07 (U0 = 0.1 m s−1), Exp08 (U0 = 0.15 m s−1), and Exp09 (U0 = 0.2 m s−1), respectively. The dashed lines are obtained by fitting the ε profiles to Eq. (11). (b) The values of the vertical decay scale ζ and the vertical mean free path Hfree corresponding to the numerical experiments in (a). Note that all three numerical experiments satisfy Sp = 0.46.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-22-0268.1

To see more clearly the processes occurring in the time-dependent regime (Sp > 0.3), we examine the time series of the snapshots of the zonal perturbation velocity field in Exp08 (Fig. 4). At t = 10Ti, the model domain is occupied by the quasi-stationary GM internal wave field (Fig. 4a). After the addition of the horizontal steady flow (t = 12Ti), internal lee waves are found to propagate upward through the GM background internal wave field (Fig. 4b). As time progresses (t = 17Ti), IOs with strong horizontal currents develop in the 0–800 m depth range above the seafloor topography, overlapping the background internal lee waves (Fig. 4c). Figure 4d shows the distribution of the Richardson number [Ri = N2/S2, where S2 = (∂u/∂z)2 + (∂υ/∂z)2 is the squared vertical velocity shear] at t = 20Ti in Exp08. It can be seen that the values of Ri below the critical value of 0.25 occur mainly in the depth range of 0–800 m above the seafloor topography where strong IOs are found. This indicates that the internal lee waves interact with the IOs and eventually break due to vertical shear instability, producing a small ζ as shown in Fig. 3.

Fig. 4.
Fig. 4.

Snapshots of the zonal perturbation velocity field at (a) t = 10Ti, (b) t = 12Ti, and (c) t = 17Ti in Exp08. (d) The distribution of the Richardson number (Ri) at t = 20Ti in Exp08.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-22-0268.1

For each numerical experiment, we compare ζ with the “vertical mean free path” Hfree of internal lee-wave packets in the GM background internal wave field formulated by Hibiya (2022):
Hfree=Cgz0τ=1.3{1[f/(kHU0)]2}3/2{1[(kHU0)/N]2}1/2U02N2(m).
For Sp < 0.3, ζ and Hfree are generally comparable, indicating that the new parameterization proposed by Hibiya (2022) performs well in estimating ζ in the quasi-stationary regime (Figs. 2b,d). However, for Sp > 0.3, ζ becomes much smaller than Hfree (Fig. 3b).

Compared to previous observations in the Southern Ocean, the calculated results for Sp < 0.3 and U0 ≥ 0.1 m s−1 (Exp03–06) are consistent with the observed feature that bottom-enhanced mixing extends throughout the water column with no apparent attenuation near polar fronts where the ACC locally intensifies (Fig. 3 of Sloyan 2005; Fig. 3 of Sheen et al. 2013). However, the calculated results for Sp > 0.3 (Exp07–09) are consistent with the observed feature that the vertical upward extent of bottom-enhanced mixing becomes more limited as the slope gradient increases with the height of the seafloor topography (Fig. 14 of Sheen et al. 2013).

b. Revisiting the evolution process of inertial oscillations

Nikurashin and Ferrari (2010) and Labreuche et al. (2022) described the evolution process of IOs for Sp > 0.3 as follows: 1) small IOs are generated near the seafloor topography during the initial transient phase in the numerical model; 2) a steady flow and IOs interact individually with the seafloor topography to generate a pair of internal waves: The first is a steady internal lee wave with intrinsic frequency ω1 = kHU0 in the moving reference frame, and the second is a wave with intrinsic frequency ω2 = ±f + kHU0, namely, the inertial frequency Doppler shifted by kHU0; 3) the resonant triad interaction between these two waves transfers energy to the third wave with intrinsic frequency ω3 = ±f (i.e., ω3 = ω2ω1, k3 = k2k1 = 0, and m3 = m2m1, where both k1 and k2 are equal to kH); 4) the IOs are thus amplified and the feedback continues.

The time series of the IO (uIO) at 100 m above the seafloor topography obtained from Exp10 is shown in Fig. 5b, where we find that the amplitude of the IO increases by a factor of about 5 within two inertial cycles during the initial phase, which is obviously too fast in time to be explained by the resonant triad interaction between the internal waves (McComas and Müller 1981). Furthermore, what we find most unconvincing about the above explanation is that it assumes that the steady flow and the growing IO (before the amplitude of the IO becomes of the same order as that of the steady flow) interact individually with the seafloor topography to generate separate internal waves, even though they overlap to form a zonal composite flow with peak velocities of U0 ± uIOmax, where uIOmax is the zonal amplitude of the IO. Since the composite flow has an excursion parameter well above unity, we must consider the nonlinear advection effect in its interaction with the seafloor topography; the linear idea of IO and steady flow interacting independently with the seafloor topography to generate separate internal waves is invalid. Figure 5a shows the time evolution of the zonal-mean vertical energy flux Ef¯=pw¯ (hereafter overbar denotes zonal mean) in Exp10, where p′ is the baroclinic pressure and w is the vertical velocity. After t = 3Ti, we find two phase lines of Ef in the t–z plane within each Ti, with Ef¯ corresponding to the first phase line much larger than Ef¯ corresponding to the second phase line, especially within the bottom 500 m (Fig. 5a). We can also see that the large (small) Ef corresponding to the first (second) phase line occurs at the maximum (minimum) of the composite flow (Fig. 5b). This recalls the finding of the previous study (Mohri et al. 2010) that a strong tidal flow with an excursion parameter kHUtide/ωtide (Utide and ωtide are the amplitude and frequency of the tidal flow, respectively) greater than unity interacts with the seafloor topography to generate internal lee waves at a time when the acceleration and deceleration of the tidal flow almost disappear. During t = 1Ti–11Ti, the excursion parameter at the composite flow minimum, β = (U0uIOmax)kH/f remains between 1 and 2, but that at the composite flow maximum, β = kH(U0 + uIOmax)/f can reach up to 4 (Fig. 5b). This strongly suggests that the internal lee waves are most efficiently generated during the period centered on the time of the composite flow maximum (U0 + uIOmax), when it becomes transiently stationary (Mohri et al. 2010).

Fig. 5.
Fig. 5.

(a) Spatial and temporal evolution of the zonal-mean vertical energy flux Ef¯ during t = 1Ti–11Ti in Exp10. (b) Time series of the composite horizontal flow U0 + uIO (red solid) at 100 m above the seafloor topography, the zonal-mean vertical energy flux Ef¯ (blue dashed) at 200 m above the seafloor topography, and the steady horizontal flow U0 (red dashed) in Exp10. Note that the excursion parameter β of the composite horizontal flow is marked in black at its maximum and minimum.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-22-0268.1

Figure 6 shows the snapshot of the two-dimensional kinetic energy spectrum at t = 11Ti in Exp10 superimposed on the dispersion curves of the internal lee waves generated by the steady flow U0 (0.1 m s−1) and the maximum composite flow U0 + uIOmax (0.17 m s−1), respectively. We can confirm the spectral peak corresponding to the internal lee wave generated by the interaction between the maximum composite flow U0 + uIOmax and the seafloor topography with the wavenumber kH.

Fig. 6.
Fig. 6.

Snapshot of the kinetic energy spectrum in the vertical (m) and horizontal (k) wavenumber domain at t = 11Ti, calculated from the perturbation velocity within 0–3000 m above the seafloor topography in Exp10. Solid and dashed black lines represent the dispersion curves of the internal lee waves generated by the steady flow U0 (0.1 m s−1; solid) and the maximum velocity of the composite flow U0 + uIOmax (0.17 m s−1; dashed), respectively. Gray dashed lines show the contours of the internal wave frequency (ω = 1.01f, 2f, 3f, 4f, 5f, 6f) calculated from the dispersion relation. Note that kH is highlighted in red to emphasize its location.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-22-0268.1

To see the physical process responsible for the evolution of the IOs, we next analyze the zonal-mean energy budget approximately given by
E¯t=Ef¯zρ0ε¯,
where angle brackets denote the time average; ρ0ε is the energy dissipation with ρ0 the reference water density 1000 kg m−3; E is the total energy density:
E=E0+EIO+EIW,
with
E0=12ρ0U02,
EIO=12ρ0(uIO2+υIO2),
EIW=12ρ0(uIW2+υIW2+wIW2)+g2ρ22ρ0N2,
where E0 is the constant energy density of the steady horizontal flow; EIO is the energy density of the IOs; EIW is the energy density of the residual internal waves.

First, to evaluate the left side of Eq. (13), E¯ averaged over two consecutive time intervals is calculated and compared. Since Ef¯ varies during each inertial period (Fig. 5), these time intervals are chosen to be sufficiently longer than the inertial period. Figure 7a shows EIW¯ and EIO¯ averaged over t = 1Ti–6Ti and t = 6Ti–11Ti, respectively. We can see that there is no obvious difference of EIW¯ between the two consecutive time intervals, while there is a prominent difference of EIO¯ between the two consecutive time intervals, especially in the depth range of 80–250 m above the seafloor topography. This indicates that the time variation of EIO¯ is the main cause of E/t¯ during t = 6Ti–11Ti. Next, to evaluate the right side of Eq. (13), we compute Ef/z¯ and ρ0ε¯, both averaged over t = 6Ti–11Ti. We can see that Ef/z¯ρ0ε¯ at most depths (Fig. 7b) and, roughly speaking, large values of Ef/z¯(0.51.5)×105Wm3 occur in the depth range of 90–200 m above the seafloor topography, where the large differences of EIO¯1Jm3 are found between the two consecutive time intervals, which are about 5Ti (∼2.5 × 105 s) apart (Fig. 7a). This indicates that a significant fraction of the energy of the bottom-generated internal lee waves is supplied to enhance the IOs during t = 6Ti–11Ti, such that EIW¯ is conserved but EIO¯ increases.

Fig. 7.
Fig. 7.

(a) Zonal-mean energy density of inertial oscillations EIO¯ (magenta) and residual internal waves EIW¯ (black) in Exp10 averaged over t = 1Ti–6Ti (solid) and t = 6Ti–11Ti (dashed). (b) Zonal-mean energy dissipation ρ0ε¯ (black solid) and vertical energy flux divergence Ef/z¯ (magenta solid) averaged over t = 6Ti–11Ti in Exp10.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-22-0268.1

In reality, the enhancement of the IO is thought to be achieved by the intermittent supply of energy that occurs depending on the sign and magnitude of ∂uIO/∂z, as expected from the Doppler shift equation for the evolution of the vertical wavenumber of the internal lee wave propagating through the high vertical wavenumber IO:
mt=kuIOz,
where m and k are the vertical and horizontal wavenumbers of the upward-propagating internal lee wave, respectively (Hibiya et al. 2002). In our numerical simulations, both the signs of k and m are negative because the horizontal and vertical phase velocities of the generated internal lee waves are negative. In this case, during the time interval when ∂uIO/∂z is positive, ∂m/∂t becomes positive, so the absolute value of m becomes smaller and the internal lee waves continue to propagate upward through the IO. In contrast, during the time interval when ∂uIO/∂z is negative, ∂m/∂t becomes negative, so the absolute value of m becomes larger and the propagation direction of the internal lee waves becomes more horizontal, with the result that ∂Ef/∂z < 0. Thus, the internal lee-wave energy, Doppler-shifted to the near-inertial frequency, is intermittently supplied to intensify the IO not far above the seafloor. This explains the spreading of the kinetic energy of the spectral peak toward the higher vertical wavenumber along the vertical line located at kH (see Fig. 6).

Based on the analyses in this section, the following possible scenario can be derived for the IO evolution process for Sp > 0.3:

  1. Small IOs are generated in the initial transient response to the establishment of the steady flow U0.

  2. The IO overlaps the steady flow U0 to form a composite flow that interacts with the seafloor topography and efficiently generates internal lee waves during the period centered around the time of the composite flow maximum, when it becomes transiently stationary.

  3. The IO not far above the seafloor is enhanced by the intermittent supply of internal lee-wave energy Doppler-shifted to the near-inertial frequency, which occurs depending on the sign and magnitude of the background IO shear.

  4. The increased IO shear is fed back to enhance the above process (iii), which inhibits the upward propagation of the internal lee waves, creating a turbulent mixing region of small ζ.

4. Summary and conclusions

Using a numerical model incorporating the Garrett–Munk (GM) background internal wave field, we have investigated the key parameters that control the vertical structure of internal lee-wave-driven bottom-enhanced mixing.

For the steepness parameter Sp < 0.3, internal lee waves propagate upward from the seafloor topography while interacting with the GM background internal wave field to create a turbulent mixing region whose vertical decay scale ζ extends upward with increasing bottom flow U0, but is nearly independent of the horizontal wavenumber of the seafloor topography kH. When the bottom flow is sufficiently strong, e.g., U0 ≥ 0.1 m s−1, we find that the enhanced mixing extends throughout the water column without apparent decay, consistent with observed features near polar fronts in the Southern Ocean where the ACC is locally enhanced (e.g., Sloyan 2005; Sheen et al. 2013). We also confirm the good performance of Hibiya’s (2022) new parameterization of internal lee-wave-driven near-field mixing by comparing ζ obtained in the present study with the parameterized “vertical mean free path” Hfree.

In contrast, for Sp > 0.3, the inertial oscillation (IO) generated in the initial transient response is fed by the intermittent supply of internal lee-wave energy Doppler-shifted to the near-inertial frequency, which occurs depending on the sign and magnitude of the background IO shear. The enhanced IO overlaps the steady flow U0 to form a composite flow that interacts with the seafloor topography and efficiently generates internal lee waves during the period centered around the time of the composite flow maximum when it becomes transiently stationary, but their upward propagation is inhibited by the enhanced IO shear, creating a short turbulent mixing region.

As mentioned in the Introduction, previous studies have shown that bottom-enhanced mixing driven by internal lee waves may account for ∼1/3 or more of the global abyssal water mass transformation rate, but this assessment depended on the choice of several parameter values, including ζ in the St. Laurent parameterization (Nikurashin and Ferrari 2013). Since the main objective of the present study is to validate Hibiya’s (2022) new parameterization of turbulent mixing associated with the breaking of upward-propagating internal lee waves generated over infinitesimal amplitude seafloor topography, Sp = NhT/U0 was only increased from 0.11 to 0.46 and kept in the subcritical regime while satisfying f < kHU0 < N. When Sp exceeds unity and becomes supercritical, the flow is blocked on the upstream side of the seafloor topography and becomes more nonlinear, which is difficult to handle mathematically. In this case, the generation of internal lee waves is not related to the actual hT and kH of the seafloor topography, but to the effective ones experienced by the flow over the crest of the seafloor topography (Winters and Armi 2012). Interestingly, especially for kHU0 < f, both upstream blocking and downstream breaking of the “nonpropagating” stratified water response over the seafloor topography with Sp > 1 have been shown to generate large form drag forces, causing a globally extrapolated 2.7 TW of work against the low-frequency circulation, a substantial source of energy that is redistributed to open-ocean mixing (Klymak 2018; Klymak et al. 2021).

There are several issues that need to be addressed in the near future. First, due to limited computational resources, our calculations are restricted to a vertical two-dimensional (xz) plane. In the vertical two-dimensional calculation, all bottom currents are assumed to flow over, rather than around, the seafloor topography (Nikurashin et al. 2014), and the GM background internal wave shear is stronger than in the three-dimensional calculation, so both the upward-propagating lee-wave energy from the seafloor and its dissipation due to nonlinear interaction with the GM background internal wave shear are overestimated. The vertical two-dimensional calculation also misses the interaction of internal lee waves with other types of motions such as eddies. Second, this vertical two-dimensional model assumes a monochromatic sinusoidal seafloor topography and cyclic lateral boundary conditions, which precludes a quantitative discussion of the local dissipative efficiency q in near-field mixing. Third, assuming that the large-scale vertical shear associated with the ACC is weak compared to the fine-scale GM vertical shear, a background steady flow with no vertical shear is used in the numerical experiment. However, even if the magnitude of the background steady flow shear is small, its superposition on the GM background shear may cause changes in the efficiency with which internal lee waves are Doppler shifted. Fourth, the transient interaction between the growing sheared IO superimposed on the steady flow and the upward-propagating internal lee waves is only qualitatively discussed in this study, and further theoretical considerations, e.g., based on ray tracing calculations, are essential for its full understanding (Bölöni et al. 2016). The results of our research on these remaining issues will be reported elsewhere.

Acknowledgments.

The authors thank Taira Nagai of the Japan Fisheries Research and Education Agency for kindly providing the basic codes for the numerical experiments performed in this study. This study was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology through a Grant-in-Aid for Scientific Research on Innovative Areas to Toshiyuki Hibiya (JP15H05824), and was conducted while Ying He was a joint Ph.D. student at the University of Tokyo. Ying He would like to thank Fan Wang and Jianing Wang of the Institute of Oceanology at CAS, as well as the China Scholarship Council and the University of Tokyo for supporting her study abroad.

Data availability statement.

All the numerical information used in this study is publicly available at https://doi.org/10.6084/m9.figshare.20522544.v2.

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Save
  • Bölöni, G., B. Ribstein, J. Muraschko, C. Sgoff, J. Wei, and U. Achatz, 2016: The interaction between atmospheric gravity waves and large-scale flows: An efficient description beyond the nonacceleration paradigm. J. Atmos. Sci., 73, 48334852, https://doi.org/10.1175/JAS-D-16-0069.1.

    • 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.

    • 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.

    • Search Google Scholar
    • Export Citation
  • Hibiya, T., 2022: A new parameterization of turbulent mixing enhanced over rough seafloor topography. Geophys. Res. Lett., 49, e2021GL096067, https://doi.org/10.1029/2021GL096067.

    • Search Google Scholar
    • Export Citation
  • Hibiya, T., M. Nagasawa, and Y. Niwa, 2002: Nonlinear energy transfer within the oceanic internal wave spectrum at mid and high latitudes. J. Geophys. Res., 107, 3207, https://doi.org/10.1029/2001JC001210.

    • Search Google Scholar
    • Export Citation
  • Hibiya, T., T. Ijichi, and R. Robertson, 2017: The impacts of ocean bottom roughness and tidal flow amplitude on abyssal mixing. J. Geophys. Res. Oceans, 122, 56455651, https://doi.org/10.1002/2016JC012564.

    • Search Google Scholar
    • Export Citation
  • Iwamae, N., T. Hibiya, and M. Watanabe, 2009: Numerical study of the bottom-intensified tidal mixing using an “Eikonal approach.” J. Geophys. Res., 114, C05022, https://doi.org/10.1029/2008JC005130.

    • Search Google Scholar
    • Export Citation
  • Jayne, S. R., and L. C. St. Laurent, 2001: Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett., 28, 811814, https://doi.org/10.1029/2000GL012044.

    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., 2018: Nonpropagating form drag and turbulence due to stratified flow over large-scale abyssal hill topography. J. Phys. Oceanogr., 48, 23832395, https://doi.org/10.1175/JPO-D-17-0225.1.

    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., D. Balwada, A. N. Garabato, and R. Abernathey, 2021: Parameterizing nonpropagating form drag over rough bathymetry. J. Phys. Oceanogr., 51, 14891501, https://doi.org/10.1175/JPO-D-20-0112.1.

    • Search Google Scholar
    • Export Citation
  • Labreuche, P., C. Staquet, and J. Le Sommer, 2022: Resonant growth of inertial oscillations from lee waves in the deep ocean. Geophys. Astrophys. Fluid Dyn., 116, 351373, https://doi.org/10.1080/03091929.2022.2138865.

    • Search Google Scholar
    • Export Citation
  • Ledwell, J. R., A. J. Watson, and C. S. Law, 1993: Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment. Nature, 364, 701703, https://doi.org/10.1038/364701a0.

    • Search Google Scholar
    • Export Citation
  • Ledwell, J. R., E. T. Montgomery, K. L. Polzin, L. C. St. Laurent, R. W. Schmitt, and J. M. Toole, 2000: Evidence for enhanced mixing over rough topography in the abyssal ocean. Nature, 403, 179182, https://doi.org/10.1038/35003164.

    • Search Google Scholar
    • Export Citation
  • Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997: A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102, 57535766, https://doi.org/10.1029/96JC02775.

    • Search Google Scholar
    • Export Citation
  • McComas, C. H., and P. Müller, 1981: Time scales of resonant interactions among oceanic internal waves. J. Phys. Oceanogr., 11, 139147, https://doi.org/10.1175/1520-0485(1981)011<0139:TSORIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Melet, A., R. Hallberg, S. Legg, and M. Nikurashin, 2014: Sensitivity of the ocean state to lee wave–driven mixing. J. Phys. Oceanogr., 44, 900921, https://doi.org/10.1175/JPO-D-13-072.1.

    • Search Google Scholar
    • Export Citation
  • Mohri, K., T. Hibiya, and N. Iwamae, 2010: Revisiting internal wave generation by tide-topography interaction. J. Geophys. Res., 115, C11001, https://doi.org/10.1029/2009JC005908.

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

  • 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.

  • Nagai, T., T. Hibiya, and P. Bouruet-Aubertot, 2017: Nonhydrostatic simulations of tide-induced mixing in the Halmahera Sea: A possible role in the transformation of the Indonesian throughflow waters. J. Geophys. Res. Oceans, 122, 89338943, https://doi.org/10.1002/2017JC013381.

    • 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.

    • Search Google Scholar
    • Export Citation
  • Nikurashin, M., and R. Ferrari, 2010: Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory. J. Phys. Oceanogr., 40, 10551074, https://doi.org/10.1175/2009JPO4199.1.

    • Search Google Scholar
    • Export Citation
  • Nikurashin, M., and R. Ferrari, 2011: Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean. Geophys. Res. Lett., 38, L08610, https://doi.org/10.1029/2011GL046576.

    • Search Google Scholar
    • Export Citation
  • Nikurashin, M., and R. Ferrari, 2013: Overturning circulation driven by breaking internal waves in the deep ocean. Geophys. Res. Lett., 40, 31333137, https://doi.org/10.1002/grl.50542.

    • Search Google Scholar
    • Export Citation
  • Nikurashin, M., R. Ferrari, N. Grisouard, and K. Polzin, 2014: The impact of finite-amplitude bottom topography on internal wave generation in the Southern Ocean. J. Phys. Oceanogr., 44, 29382950, https://doi.org/10.1175/JPO-D-13-0201.1.

    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., 2004: Idealized solutions for the energy balance of the finescale internal wave field. J. Phys. Oceanogr., 34, 231246, https://doi.org/10.1175/1520-0485(2004)034<0231:ISFTEB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Polzin, K. L., 2009: An abyssal recipe. Ocean Modell., 30, 298309, https://doi.org/10.1016/j.ocemod.2009.07.006.

  • Polzin, K. L., J. M. Toole, J. R. Ledwell, and R. W. Schmitt, 1997: Spatial variability of turbulent mixing in the abyssal ocean. Science, 276, 9396, https://doi.org/10.1126/science.276.5309.93.

    • 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.

    • Search Google Scholar
    • Export Citation
  • Sloyan, B. M., 2005: Spatial variability of mixing in the Southern Ocean. Geophys. Res. Lett., 32, L18603, https://doi.org/10.1029/2005GL023568.

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  • Fig. 1.

    Time evolution of the zonal component of the inertial oscillations uIO during t = 0Ti–20Ti in the numerical experiments with the steepness parameter (Sp) equal to (a) 0.11 (Exp05) and (b) 0.46 (Exp07). Note that the GM79 empirical model is used as the initial condition, and the steady flow U0 is gradually established in the model after t = 10Ti.

  • Fig. 2.

    Dependence of the zonal-mean vertical distribution of the energy dissipation rate ε averaged over t = 15Ti–20Ti in (a) Exp01–03 on the steady flow magnitude U0 when the horizontal wavenumber of the seafloor topography kH is held constant at 1 × 10−3 cpm and (c) Exp03–05 on kH when U0 is held constant at 0.1 m s−1. The dashed lines are obtained by fitting the ε profiles to Eq. (11). (b),(d) The values of the vertical decay scale ζ and the vertical mean free path Hfree corresponding to the numerical experiments in (a) and (c), respectively. Note that all five numerical experiments satisfy Sp < 0.3.

  • Fig. 3.

    (a) Zonal-mean vertical distributions of the energy dissipation rate ε averaged over t = 15Ti–20Ti in Exp07 (U0 = 0.1 m s−1), Exp08 (U0 = 0.15 m s−1), and Exp09 (U0 = 0.2 m s−1), respectively. The dashed lines are obtained by fitting the ε profiles to Eq. (11). (b) The values of the vertical decay scale ζ and the vertical mean free path Hfree corresponding to the numerical experiments in (a). Note that all three numerical experiments satisfy Sp = 0.46.

  • Fig. 4.

    Snapshots of the zonal perturbation velocity field at (a) t = 10Ti, (b) t = 12Ti, and (c) t = 17Ti in Exp08. (d) The distribution of the Richardson number (Ri) at t = 20Ti in Exp08.

  • Fig. 5.

    (a) Spatial and temporal evolution of the zonal-mean vertical energy flux Ef¯ during t = 1Ti–11Ti in Exp10. (b) Time series of the composite horizontal flow U0 + uIO (red solid) at 100 m above the seafloor topography, the zonal-mean vertical energy flux Ef¯ (blue dashed) at 200 m above the seafloor topography, and the steady horizontal flow U0 (red dashed) in Exp10. Note that the excursion parameter β of the composite horizontal flow is marked in black at its maximum and minimum.

  • Fig. 6.

    Snapshot of the kinetic energy spectrum in the vertical (m) and horizontal (k) wavenumber domain at t = 11Ti, calculated from the perturbation velocity within 0–3000 m above the seafloor topography in Exp10. Solid and dashed black lines represent the dispersion curves of the internal lee waves generated by the steady flow U0 (0.1 m s−1; solid) and the maximum velocity of the composite flow U0 + uIOmax (0.17 m s−1; dashed), respectively. Gray dashed lines show the contours of the internal wave frequency (ω = 1.01f, 2f, 3f, 4f, 5f, 6f) calculated from the dispersion relation. Note that kH is highlighted in red to emphasize its location.

  • Fig. 7.

    (a) Zonal-mean energy density of inertial oscillations EIO¯ (magenta) and residual internal waves EIW¯ (black) in Exp10 averaged over t = 1Ti–6Ti (solid) and t = 6Ti–11Ti (dashed). (b) Zonal-mean energy dissipation ρ0ε¯ (black solid) and vertical energy flux divergence Ef/z¯ (magenta solid) averaged over t = 6Ti–11Ti in Exp10.

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