Turbulence Generated through Critical Reflection of Internal Waves off the Seafloor due to Nontraditional Effects

1. Introduction

The abyssal meridional overturning circulation (MOC) of the ocean is thought to be driven primarily by diapycnal mixing processes that act to lift the abyssal waters produced through deep convection around Antarctica. Therefore, the magnitude as well as the spatial and temporal distribution of diapycnal mixing rates in the abyssal ocean are crucial for the correct modeling of the ocean circulation (Wunsch and Ferrari 2004). Inverse methods have suggested that much of the abyssal upwelling occurs in the equatorial regions (Lumpkin and Speer 2007). However, the processes that could be responsible for driving mixing predominantly in the abyssal equatorial ocean are still largely unknown and are absent in global circulation models (de Lavergne et al. 2016).

Recently, Delorme and Thomas (2019, hereafter, DT19) used idealized numerical simulations of downward-propagating equatorial waves (EWs) to show that, when so-called nontraditional (NT) effects associated with the horizontal component of the Coriolis parameter $\tilde{f}$ are taken into account, EWs can energize mixing in the abyss when they reflect off the seafloor, even over smooth topography. DT19’s work was motivated by previous studies of NT physics that focused on the midlatitudes (e.g., Phillips 1966; Saint-Guily 1970; Needler and LeBlond 1973; Beckmann and Diebels 1994; Kasahara 2003; Gerkema and Shrira 2005a,b; Winters et al. 2011). These past studies showed that $\tilde{f}$ acts to create a class of subinertial waves (i.e., waves whose frequency is less than the inertial frequency f) but also to produce a nonvanishing group velocity at the inertial latitude, where the wave’s frequency is equal to f. These effects, which are stronger when the stratification is weaker, lead to critical reflection at the inertial latitude and wave trapping poleward of the inertial latitude, both of which could potentially induce mixing. DT19 extended this theory to the equatorial regions and showed that EWs would also undergo critical reflection when reflecting off the seafloor if NT effects are taken into account. In the equatorial case, the critical reflection mechanism is triggered by a change in the meridional scale of the reflected waves: NT effects break the vertical symmetry of downward-propagating EWs, and by doing so, yield a more complex bottom reflection that requires higher meridional modes to satisfy the bottom boundary condition. These higher meridional modes are well described by the two-dimensional, NT wave equation for inertia-gravity waves (IGWs), which has been used for midlatitude applications in the studies mentioned above, and which predicts the occurrence of critical reflection at the inertial latitude as well as subinertial wave propagation.

DT19 used the KPP mixing scheme (Large et al. 1994) to parameterize the mixing in their idealized simulations. Because of the critical reflection mechanism, strong values of the vertical shear develop at the bottom of the water column in the NT simulations, lowering the Richardson number significantly, which in turn activates the interior mixing scheme of the KPP parameterization and yields elevated eddy diffusivities close to the seafloor. DT19 showed that most of the mixing was located at the inertial latitude of the wave, where critical reflection occurs and focuses the wave’s energy into a thin layer against the seafloor. Weaker mixing rates were found poleward of the inertial latitude suggesting that the wave trapping mechanism might not be as efficient as critical reflection at driving mixing in the abyss. This finding is consistent with results from Tort and Winters (2018) who demonstrated that the subinertial energy flux due to NT effects is small in numerical simulations of IGWs propagating over a β plane at midlatitudes.

The fact that EWs are prone to critical reflection when NT effects are taken into account is of particular interest since these waves are known to efficiently transport significant amounts of energy vertically in the water column (Eriksen 1980; Eriksen and Richman 1988; Smyth et al. 2015; Tuchen et al. 2018). Using values of the downward flux of energy that are typical of the deep equatorial ocean and a mean stratification profile from the eastern equatorial Pacific, DT19 found that the critical reflection process could contribute to ≈10 Sv (1 Sv ≡ 10⁶ m³ s⁻¹) of diapycnal upwelling in the abyssal ocean, consistent with the transports in the abyssal circulation inferred by the inverse model of Lumpkin and Speer (2007).

In a follow-up paper, Delorme et al. (2021, hereafter, DT21) found elevated mixing rates in realistic, quasi-hydrostatic (QH) simulations of the eastern equatorial Pacific. The QH equations do not include the vertical acceleration but do take into account the NT terms [i.e., $\tilde{f}w$ in the zonal momentum equation and $\tilde{f}u$ in the vertical momentum equation, the latter of which modifies the hydrostatic pressure field; hereafter, u = (u, υ, w) is the velocity vector]. DT21’s simulations were driven by climatological forcing and used a one-way grid nesting approach from a parent, basinwide simulation of the Pacific Ocean with realistic bathymetry. The simulations have been shown to exhibit both a realistic deep circulation and a realistic EW field at the surface that propagates well into the deep ocean. At the bottom, DT21 found enhanced mixing between 8°S and 14°N in the QH version of the model while it is confined to the ±2° equatorial band in simulations with $\tilde{f}=0$. In both sets of simulations, the KPP mixing scheme was used to parameterize the mixing. DT21 were able to associate the elevated mixing rates in the QH case to enhanced inertial vertical shear whose spatiotemporal properties matched the NT dispersion relation for IGWs, demonstrating that the critical reflection of EWs described in DT19 was responsible for the off-equatorial abyssal mixing. In this realistic setup, the enhanced mixing yields upwelling in the abyss, similar to the idealized simulations of DT19, and ultimately contributes to closing the deep cell of the MOC.

DT19 and DT21 highlighted that critical reflection of EWs due to NT effects can energize mixing in the abyssal ocean even over smooth topography in both idealized and realistic numerical simulations. The realistic simulations suggest that this could be a widespread phenomenon in the tropics that should be accurately parameterized in order to quantify its contribution to the global mixing budget of the ocean. In both DT19 and DT21, mixing is parameterized using the interior mixing scheme of KPP, which allows for an enhancement of the turbulent diffusivity in areas where critical reflection occurs through a decrease of the Richardson number below a critical level. However, parameters used in the KPP mixing scheme (such as the critical Richardson number and the maximum diffusivity coefficients) have not been optimized for the application of critical reflection of EWs modified by NT effects. Therefore, the amount of mixing as well as its spatial and temporal variability in the numerical simulations may not be accurate. In addition, the energy cascade from the large-scale wave to the smaller scales of the turbulence has not been resolved in past simulations of critical reflection of NT waves (such as the ones carried out by Winters et al. 2011; DT19) and thus, it is still not well understood.

In this paper, we address these limitations by running high-resolution, nonhydrostatic (NH) simulations of the critical reflection of IGWs modified by NT effects over a flat bottom. Our simulations resolve the larger spatial and temporal scales of the turbulence in two dimensions but not the smaller scales that are parameterized through a viscosity coefficient. The spatial resolution has been chosen such that the primary instabilities triggered by the critical reflection mechanism are well resolved, allowing us to characterize the nonlinear dynamics to smaller scales involved during the critical reflection of the wave and to estimate the mixing it generates. The overarching goal of this study is to provide the key elements needed to develop a parameterization for diapycnal mixing induced by critical reflection of internal waves due to NT effects that could be incorporated into larger-scale ocean models. To accomplish this goal, we ran a suite of numerical simulations with different parameter values to investigate the relationship between the resulting mixing and two key properties of the waves, namely, their frequency and their mean downward energy flux.

The remainder of this paper is organized as follows. The theory behind the critical reflection of IGWs modified by NT effects and the analytical formulation of the problem are described in section 2. In section 3, the numerical setup, the set of experiments and the methods used to estimate the mixing from the numerical simulations are introduced. In section 4, results from the main experiment are shown first. We describe in particular the evolution of the flow field, the energy transfer across scales and the magnitude and spatial structure of the effective mixing that results from the critical reflection mechanism. Then, in section 5, we explore the sensitivity of this mechanism to key parameters using the results from the suite of numerical experiments that were performed. We conclude in section 6 with a discussion of the strengths and limitations of the theory and potential pathways toward developing a parameterization of these effects.

2. Formulation of the problem

b. Critical reflection mechanism

Taking the derivative of (12) with respect to the wave vector gives the group velocity vector of the wave:

$(c_g^y,c_g^z)=\frac{(N^2+{\tilde{f}}^2-f^2)\frac{l}{m}+f\tilde{f}\left(1-\frac{l^2}{m^2}\right)}{\omega m{\left(1+\frac{l^2}{m^2}\right)}^2}\left(1,-\frac{l}{m}\right).$(13)

The slope of the rays along which waves propagate is −l/m. Using (12), it can be shown that

${\left(\frac{l}{m}\right)}^{\pm }=-\frac{B\pm \sqrt{B^2-AC}}{A},$(14)

where $A=N^2+{\tilde{f}}^2-{\omega }^2$, $B=\tilde{f}f$ and C = f² − ω².

Hence, at the inertial latitude of the wave where ω = f, C = 0 and

${\left(\frac{l}{m}\right)}_{\omega =f}^{-}=0,$(15)

${\left(\frac{l}{m}\right)}_{\omega =f}^{+}=-\frac{2B}{A}.$(16)

In the NT case, B ≠ 0. As a consequence, energy can be transported vertically on the steep ray at the inertial latitude with slope $-{(l/m)}_{\omega =f}^{+}$. Since the second ray where energy can propagate is flat, the flux of energy is compressed into an infinitesimally small layer when the wave reflects off a flat bottom boundary: this is the mechanism of critical reflection. The key element for this mechanism to be triggered is thus the nonzero vertical group velocity along the steep characteristic at the inertial latitude, ${(c_g^z)}_{\omega =f}^{+}$, which is allowed in the NT case only. The larger the magnitude of ${(c_g^z)}_{\omega =f}^{+}$, the stronger the critical reflection mechanism for waves with the same energy density.

As shown in (13) the magnitude of the vertical group velocity is a function of the properties of the medium (such as the stratification and the local Coriolis parameters) and of the wave. To understand how these parameters influence the critical reflection mechanism for inertial waves with ω = f, we calculated the sensitivity of the magnitude of ${(c_g^z)}_{\omega =f}^{+}$ to latitude, the background stratification, and the vertical wavenumber (Fig. 1).

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Magnitude of the vertical group velocity along the steep characteristic at the inertial latitude ${(c_g^z)}_{\omega =f}^{+}$ as a function of (a) latitude (with a stratification of N² = 10⁻⁷ s⁻² and a vertical wavelength of λ_z = 1000 m), (b) stratification (at 40° latitude and with a vertical wavelength of λ_z = 1000 m), and (c) meridional and vertical wavelength (at 40° latitude and with N² = 10⁻⁷ s⁻²). The red lines indicate the parameters used in the reference experiment.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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A maximum in the amplitude of ${(c_g^z)}_{\omega =f}^{+}$ is found at midlatitude around 39°. This might seem counterintuitive to the reader since DT21 found that much of the mixing due to the NT effects takes place between 2° and 14°. The reason is that the energy density of EWs E is much larger in the tropical regions than at higher latitudes yielding larger energy fluxes $c_g^{z}E$ in this part of the world (Pollmann et al. 2017; Eden et al. 2020).

Stratification is another parameter that influences the critical reflection mechanism. We found that the magnitude of ${(c_g^z)}_{\omega =f}^{+}$ follows exponentially the value of the background stratification (Fig. 1). As shown in previous works, low values of the stratification are needed for the critical reflection mechanism to be triggered. Such conditions are typically found in the abyssal ocean.

Finally, we find that as the vertical wavelength λ_z (and similarly the meridional wavelength λ_y), of the incoming wave increases for a fixed wave frequency, the amplitude of ${(c_g^z)}_{\omega =f}^{+}$ increases exponentially as well. For short waves, the critical reflection mechanism should be significantly weaker. However, for waves with typical scales of both EWs (Ascani et al. 2010) and midlatitude IGWs (Alford et al. 2016), critical reflection is expected to occur. EWs have particularly long wavelengths, making them even more prone to the mechanism.

Our results are in agreement and extend the calculations from Gerkema and Shrira (2006) that looked at the diurnal and semidiurnal internal-tide bands and also provide an expression for the critical slope. In this study, we focus on a flat bottom since our first objective is to explain how mixing can be generated through critical reflection off a flat bottom.

The results of this analysis were used to help design a numerical experiment aimed at investigating the mixing generated by critical reflection. In the next sections, we introduce these high-resolution simulations and study the resulting mixing and its dependence on the properties of the waves and the medium.

3. Methodology

The numerical simulations consist of one main experiment that is used to investigate the nonlinear dynamics, energy transfer across scales and resulting mixing coming from the critical reflection mechanism, as well as a set of additional experiments where the sensitivity of these results to key parameters are investigated. All simulations are performed using the MITgcm (Marshall et al. 1997) in NH mode. The model is run in a Cartesian domain, on an f plane, with a flat bottom. The experiments are performed in two dimensions and with constant stratification to simplify the interpretation and for computational constraints.

The simulations have a resolution of 10 m in both the horizontal and vertical directions resulting in a grid cell aspect ratio of 1. A sponge layer is used on the southern boundary over the first few kilometers (Fig. 2), where the velocity is smoothly damped to zero and the stratification relaxed to the initial conditions. At the bottom, a free slip boundary condition is used to avoid the interference of mixing induced by bottom drag with mixing induced by critical reflection.

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Mask $\mathcal{M}$ that modulates the forcing for the main experiment (color). In areas where the mask is equal to 1, the flow field is fully relaxed to the wave solution. In areas where the mask is equal to zero, the flow field evolves freely. The white dotted line shows the location where the sponge layer begins and is where the flow field is damped toward the initial conditions near the southern boundary. Three control volumes are defined within the numerical domain and are shown here: CV1, CV2, and CV3.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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In the model, the salinity is uniform. As a result, temperature alone governs buoyancy through a linear equation of state, such that b = gα_TT, with T the temperature anomaly field and α_T the thermal expansion coefficient. A linear, centered, second-order, temperature advection scheme is used in conjunction with a Laplacian lateral diffusion, with a coefficient ${\kappa }_T^h=5\times {10}^{-2}{\hspace{0.17em}\mathrm{m}}^2{\hspace{0.17em}\mathrm{s}}^{-1}$. We tested higher-order advection schemes with both biharmonic and Laplacian diffusion, and the results are qualitatively similar to the ones presented here.

c. Numerical experiments

Our approach consists of an in-depth investigation of a reference experiment followed by a comparison of the results with a set of other experiments where the wave’s frequency and its downward energy flux are slightly modified. Table 1 summarizes all the experiments performed in this work.

Table 1

Parameter values for the different numerical experiments. The reference experiment (in bold) is the first one on the list, and subsequent experiments consist of a change in one of the mean downward flux of energy of the wave and the frequency of the waves ω.

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The reference experiment (first on the list in bold in Table 1) consists of an f plane at 48° of latitude. We chose this relatively high latitude because IGWs at the corresponding inertial frequency have steep vertical rays (based on the analysis in section 2b) and a short meridional wavelength, in addition of having a short period, which allows us to decrease substantially the computational requirements for the simulations. The domain is 3000 m deep and 38.4 km long (i.e., 3840 × 300 grid points). The frequency of the wave is set to the inertial frequency (i.e., 0.67 days), and its vertical wavelength to 1000 m, which is representative of the deep ocean. These properties result in a meridional wavelength of 4.62 km. The simulation is run for 75 wave periods (around 50 days).

Three different control volumes are specified within the numerical domain (Fig. 2). These control volumes are used in the subsequent section for analyses focusing on different aspects of the simulation: CV1 is used to conduct energy budgets, CV2 is used to assess the effective mixing associated with the wave breaking, and CV3 is used to investigate the generation of higher harmonics. These control volumes lie outside of the sponge layer and the generation layer.

4. Results from the main experiment

In this section, we describe in detail the dynamics involved in the turbulence that is generated in the main experiment (REF in Table 1).

a. Flow field

The zonal velocity of the flow field that develops in the simulation is shown in Fig. 3. The wave is generated in the upper right corner and propagates down and to the left in the water column. When it reflects off the bottom, the wave undergoes critical reflection following the mechanism described in section 2.

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Evolution of the zonal velocity in the first 21 days of the REF simulation from left to right, top to bottom. Isopycnals are contoured in black (with an interval of 8.5 × 10⁻⁴ kg m⁻³). The two vertical dashed black lines show the area that is highlighted in Figs. 7–9. For day 14.4, rays corresponding to frequencies of the incident wave (magenta) and to the second harmonic reflected wave (green) are also shown. For day 24, rays corresponding to an inertial wave are shown in magenta to highlight the PSI mechanism.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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Rapidly, a reflected wave corresponding to the second harmonic of the incident wave reflects back up. We use ray tracing to highlight both the incident and reflected waves (Fig. 3, middle-right panel). Interestingly, the second harmonic destabilizes quickly (around day 17) and its steep phase lines are replaced by flat phase lines corresponding to a wave at the inertial frequency (magenta lines in the bottom-right panel of Fig. 3), suggesting that parametric subharmonic instability (PSI) is responsible for this phenomenon.

This process is well illustrated by a wavelet analysis of the zonal velocity conducted over CV3. The wavelet power spectrum shows that the reflected wave goes from a frequency of 2f to a frequency of f, indicating that PSI is the mechanism responsible for the destabilization of the reflected wave (Fig. 4). The question of how PSI participates in driving mixing is investigated further in this section.

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Log₁₀ of the wavelet power spectral density of the zonal velocity over the area where the second harmonic develops at the beginning of the REF simulation (CV3 in Fig. 2). Over this area, the second harmonic dominates between 10 and −17 days, but then becomes weaker than the first harmonic, suggesting that PSI destabilizes the second harmonic.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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In addition to the generation of higher harmonics, the flow field destabilizes and becomes turbulent at the reflection point around day 8 (i.e., after about 12 wave periods). Later in the simulation, the turbulent region expands both laterally and vertically up to a certain point where it stops evolving (around day 21, bottom panels in Fig. 3).

The focusing of wave energy at the reflection point yields enhanced vertical shear (Fig. 5). Initially, enhanced vertical shear over the scale of the wave is observed in the abyss. As the flow field becomes unstable, the vertical shear transitions to smaller spatial scales. However, it can be seen that maxima in the vertical shear a few hundred meters above the seafloor are found over horizontal lines that resemble the ones highlighted in Fig. 3 and that are associated with PSI. The enhanced shear that develops in the abyss ultimately lowers the Richardson number below the critical value of 0.25 (Fig. 6), suggesting that shear instability is the ultimate instability responsible for the turbulence that develops in the simulation.

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As in Fig. 3, but for the vertical shear [$\text{Sh}=\sqrt{(\partial u^2/\partial z)+(\partial {\upsilon }^2/\partial z)}$].

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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As in Fig. 3, but for the Richardson number, Ri = N²/Sh².

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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To better understand the characteristics of the instability that develops, we focus on the region between the two black dashed lines shown in Figs. 3, 5, and 6, around times close to the onset of the instability when the flow goes from laminar to turbulent (Fig. 7). As the instability develops, ripples appear in the flow and density fields. These ripples are Kelvin–Helmholtz (KH) billows that develop because of the elevated shear that leads to small values of the Richardson number in these areas (Fig. 8).

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Snapshots of the zonal velocity when the instability develops from left to right, top to bottom. This happens between day 7 and 8 into the simulation, or around 12 wave periods into the simulation. Isopycnals are contoured in black (with an interval of 2.7 × 10⁻⁴ kg m⁻³). This corresponds to the area between the two vertical dashed lines in Fig. 3.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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As in Fig. 7, but for the Richardson number Ri. The green areas correspond to location where N² < 0 (i.e., convectively unstable areas where overturning occurs).

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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The KH billows are more apparent in the vertical velocity field (Fig. 9). At the bottom of the domain, distinct cells develop with upwelling and downwelling on each side. These KH billows have an aspect ratio of 1, suggesting that they are indeed fully NH. In some areas, they generate density inversions (green spots in Fig. 8) highlighting that they are responsible for actively mixing the density field.

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As in Fig. 7, but for the vertical velocity. Density inversions lead to overturning cells that are clearly visible here and correspond to enhanced vertical velocities.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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These primary results suggest that we are nominally resolving the instabilities that develop during critical reflection. In the next section, we describe the energy transfers across scales that take place during the process.

b. Energetics

We define the kinetic energy (KE) E_k as

$E_k=\frac{{\rho }_0}{2}(u^2+{\upsilon }^2+w^2)=\frac{{\rho }_0}{2}{\left|\mathbf{u}\right|}^2.$(23)

Here, E_k encompasses the three components of the velocity.

To form the NH kinetic energy equation we take the dot product of the momentum equations with the velocity vector:

${\rho }_0\mathbf{u}\cdot \left[\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot \nabla \mathbf{u}+(\tilde{f}\mathbf{j}+f\mathbf{k})\times \mathbf{u}=-\nabla \pi +\mathbf{F}\right],$(24)

where F is a frictional force encompassing both the viscosity from the Smagorinsky scheme and from the numerical advection (a centered second-order scheme in MITgcm).

Distributing and reorganizing, we get the following KE equation:

$\frac{\partial E_k}{\partial t}=\underset{A}{\underbrace{-{\rho }_0\nabla \cdot (\mathbf{u}{E}_k)}}\underset{P}{\underbrace{-{\rho }_0\nabla \cdot (\mathbf{u}\pi )}}\underset{D}{\underbrace{+{\rho }_0\mathbf{u}\cdot \mathbf{F}}}.$(25)

The right-hand side terms are the advective energy flux convergence (A), the pressure work (P), and kinetic energy dissipation (D), respectively.

We further decompose (25) in spectral space with respect to the meridional wavenumber to get

$\mathrm{Re}\left({\widehat{\mathbf{u}}}^{*}\cdot \widehat{\frac{\partial \mathbf{u}}{\partial t}}\right)=\mathrm{Re}\left(\underset{\underset{¯}{A}}{\underbrace{-{\widehat{\mathbf{u}}}^{*}\cdot \widehat{\mathbf{u}\cdot \nabla \mathbf{u}}}}\underset{\underset{¯}{P}}{\underbrace{-{\widehat{\mathbf{u}}}^{*}\cdot \widehat{\nabla \pi }}}\underset{\underset{¯}{D}}{\underbrace{+{\widehat{\mathbf{u}}}^{*}\cdot \widehat{\mathbf{F}}}}\right),$(26)

where Re denotes the operator that selects the real part, the star denotes the complex conjugate, and the caret denotes the Fourier transform.

We integrated (25) over CV1 and looked at its evolution with time (Fig. 10). Initially, energy enters the control volume through the pressure term corresponding to the energy fluxed by the wave (blue line). At around day 8, the flow becomes turbulent and dissipation kicks in to remove energy from the system (green line). Note that the dissipation terms consist of the dissipation associated with the Smagorinsky scheme as well as the numerical dissipation coming through the time-stepping scheme and advection. The wave energy flux increases up to day 10 when it reaches a steady value. Shortly after, around day 15, dissipation fully balances the wave energy flux, both terms remain constant throughout the simulation, and a steady state is reached.

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Temporal evolution of the terms from the KE equation (25) integrated over CV1; A is the advective energy flux convergence, A = −∇ ⋅ (uE_k), P is the pressure work, P = −∇ ⋅ (uπ), and D is the energy dissipation, D = u ⋅ F.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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To understand the energy balance across spatial scales, we have calculated the spectral energy balance in the simulation as a function of the meridional wavenumber defined in (26) (Fig. 11a). The balance is spatially averaged over CV1 and temporally averaged over the entire duration of the simulation. At the meridional wavenumber of the incident wave (black dotted line), energy enters the system through the pressure term since it corresponds to the wave energy flux, and advection of KE is responsible for removing energy from this initial meridional wavenumber and transferring it across scales through nonlinear interactions. Therefore, at smaller spatial scales (Fig. 11b), the advective term becomes a source term while the pressure term becomes a sink term, showing that part of the energy leaves the control volume through higher harmonics which have smaller scales. At scales between 200 and 10 m, dissipation acts as the dominant sink term and balances the energy coming from lower wavenumbers via nonlinear advection. It is also interesting to note that the second harmonic leads to no net energy flux in or out of the control volume (i.e., P = 0 at the wavenumber corresponding to the second harmonic).

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(a) Spectral KE balance as a function the meridional wavenumber l calculated in CV1. Each term from Eq. (26) is shown. (b) As in (a), but zoomed over the shaded area corresponding to higher meridional wavenumbers. (c) KE spectral fluxes Π^X as defined in (28) of the terms from the KE equation (26) as a function of the meridional wavenumber l. The black dotted line indicates the meridional wavenumber of the first harmonic, while the gray dotted line indicates the meridional wavenumber of the second harmonic.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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Further insight into the KE transfer is gained by computing the KE spectral fluxes (i.e., the energy transfer rate in meridional wavenumber space). This is done by integrating (26) with respect to the meridional wavenumber, l, and assuming that the flux vanishes at the highest wavenumber l_max:

${\mathrm{\Pi }}^X(l)={\displaystyle {\int }_l^{l_{\max }}X}dl,$(27)

where Π^X is the spectral flux corresponding to the term X in (27).

Spectral fluxes have been calculated for each of the terms in (26) (Fig. 11c). At the meridional wavenumber of the forcing, the spectral slope is negative for the pressure term, highlighting that the flux is convergent. In contrast, the flux is divergent for the advective term, showing that energy is transferred to other wavenumbers. The point of zero crossing for Π^A is around the meridional wavenumber of the forcing. At higher meridional wavenumber, the spectral flux associated with the advective term is positive, showing that the energy cascade is directed downscale (i.e., toward higher meridional wavenumbers), and it converges for scales from around 200 to 10 m. The spectral flux Π^A is negative at lower meridional wavenumbers. Dissipation acts to counteract the effects of advection at smaller scales, fluxing energy out of the system.

c. Diapycnal mixing

At the end of the simulation, the passive tracers have been mixed vertically over the turbulent region (Fig. 12). The mixing appears to be stronger the deeper the location of the passive tracer, suggesting that the mixing is bottom intensified. Here, we explore the magnitude and vertical structure of the mixing that develops in the simulation.

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(left) Meridional average of the concentration of the six passive tracers at the beginning and end of the REF simulation. (right) Snapshots of the concentration of the six passive tracers at the end of the REF simulation.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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1) Magnitude

To quantify the mixing, we looked at the evolution of the background potential energy in the turbulent region (i.e., over CV2) following the approach described in section 3d(2) (Fig. 13a). Once the turbulence develops, the background potential energy increases steadily in the control volume. Using (23), we estimated the effective diffusivity associated with the increase in background potential energy κ_eb (Fig. 13b). We find that κ_eb increases sharply once the turbulence develops and reaches values around 1–1.5 × 10⁻³ m² s⁻¹. Note that this value is an average over CV2. In the next section, we explore the vertical structure of the mixing.

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(a) Evolution of the background potential energy E_b from the REF simulation. (b) Evolution of the effective eddy diffusivity estimated through the evolution of the background potential energy κ_eb. The red dotted line shows the moment where E_b starts to increase linearly, after 15 days.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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2) Vertical structure

Following Osborn (1980), we can relate κ_eb to the dissipation rate of turbulent kinetic energy ϵ to infer the mixing efficiency through

$\Gamma =\frac{N^2{\kappa }_{\text{eb}}}{ϵ},$(28)

where Γ is a constant coefficient.

Using this equation, we estimated the mixing efficiency coefficient over CV2 by using the dissipation rate of kinetic energy averaged over CV2. We find that the average mixing efficiency reaches values up to 0.67 in our simulation. Using this value of the mixing efficiency, it is now possible to estimate the turbulent diffusivity as a function of depth by diagnosing the dissipation rate of KE ϵ at every depth level, namely,

${\kappa }_{\text{diss}}=\Gamma ϵ/N^2.$(29)

Using this diagnostic, we find that the diffusivity is a maximum at the bottom of the domain where it reaches values up to 0.1 m² s⁻¹ and then decreases exponentially with depth (Fig. 14). However, the exponential slope is not constant, and two regimes are visible: in the lower 200–300 m, the exponential slope is larger than above in the water column.

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Effective eddy diffusivity κ_diss estimated from the dissipation rate of KE and the mixing efficiency following (30) as a function of depth (black line) from the REF simulation. Dissipation has been averaged in time from day 15 to the end of the simulation. Also shown are the eddy diffusivity estimated from the passive tracers using (20) (red stars with error bars), the theoretical exponential decay for the reflected wave (blue line), and the diffusivity coming from the KPP mixing scheme (orange line).

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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To check the consistency of the turbulent diffusivity estimated using this method with other methods, we used (19) to estimate the associated eddy diffusivity from the evolution of each passive tracer, κ_trac (red stars in Fig. 14 with confidence levels corresponding to the standard errors from time averaging). Here, κ_trac is the mean value in CV2 once the instability has developed and we do not consider the beginning of the simulation where κ_trac = 0. The time integration is thus calculated over 10 wave periods starting on day 15 of the simulation once the kinetic energy has reached a quasi–steady state (see Fig. 10). Both κ_trac and κ_diff are larger lower in the water column. However, there is a discrepancy between the two estimates close to the bottom. This is likely due to the presence of the solid boundary that limits the dispersion of the passive tracer near the bottom and therefore artificially decreases the estimation of the diffusivity. Higher in the water column, κ_trac follows κ_diss closely within the confidence bounds. The slight discrepancy between the mean value for κ_trac and κ_diss has been identified by previous studies and attributed to the approximations made to infer diapycnal diffusivities from the passive tracer (Ruan and Ferrari 2021).

To understand the bottom intensification of mixing, we looked at the 2D, linear wave equation accounting for both NT effects and vertical viscosity derived by DT19 in terms of a streamfunction ψ:

${\nu }^2{\psi }_{\mathit{zzzzzz}}+2i\omega \nu {\psi }_{\mathit{zzzz}}+(f^2-{\omega }^2){\psi }_{zz}+2\tilde{f}f{\psi }_{yz}+(N^2+{\tilde{f}}^2){\psi }_{yy}=0,$(30)

where ν is the vertical viscosity coefficient, which we assume constant for the purpose of this work. Assuming plane wave solutions of the form ψ(y, z) = Ke^ilye^rz, with r = δ + im (δ and m being real quantities), (31) becomes

${\nu }^2{r}^6+2i\omega \nu r^4+(f^2-{\omega }^2)r^2+2il\tilde{f}fr-l^2(N^2+{\tilde{f}}^2)=0.$(31)

This equation admits six solutions. DT19 showed that half of the solutions account for the incident wave while the other half represent the reflected wave that is amplified near the bottom of the domain. We solved (31) numerically using the same parameter values as the ones used in the simulation, and we kept the three roots corresponding to the reflected wave. For ν, we used a value of 1 × 10⁻³ m² s⁻¹, corresponding to the value of κ_eb of CV2, and assuming a turbulent Prandtl number of 1. Out of the three roots, we only focus on the root with the largest magnitude. This root has δ = −0.0256 m⁻¹, resulting in a decay scale for the reflected wave at the bottom. We have added a curve with an exponential decay with this scale to Fig. 14. It can be seen that while the predicted exponential profile is close to the one observed in the effective turbulent diffusivity at the bottom, the two profiles diverge quite substantially from 200 to 300 m off the bottom. In theory, the analytical prediction shows the critical layer depth over which the KE of the reflected wave is significantly enhanced because of the critical reflection mechanism. The strong KE is responsible for large values of the vertical shear that drive mixing through KH instabilities. This provides an explanation for why κ_diss is largest near the bottom. However, the theory does not explain the enhanced mixing that is observed above the critical layer. In addition, the exponential decay of the streamfunction should not necessarily translate exactly to the vertical structure of the turbulent diffusivity since the link between the two variables is not necessarily linear. Note that with ν = 1 × 10⁻³ m² s⁻¹ for ν as highlighted above, the thickness of the boundary layer is around 246 m. We further calculated the thickness of the boundary layer using ν = 1 × 10⁻⁴ m² s⁻¹ and ν = 1 × 10⁻² m² s⁻¹ and found, respectively, 114 and 523 m, highlighting its sensitivity to the viscosity coefficient.

To understand what is driving the mixing above this layer, we have diagnosed the vertical energy flux in the simulation $\overline{F_z}=\overline{pw}$ as a function of depth (Fig. 15a) to determine whether it is convergent and hence sets up the conditions for enhanced dissipation and mixing. Here, the overbar denotes an average in the meridional direction over the whole domain and in time over the entire duration of the simulation. The energy flux is downward, increasing steadily in the upper 1000 m which corresponds to the area of the wave maker where the wave is generated in our simulation. Beneath 1000 m, the energy flux is convergent, slowly decreasing toward the bottom. To quantify the contributions of different frequencies to the energy flux, we have calculated the portion associated with higher harmonics by using a high-pass filter with a cutoff of 1.5f, $\overline{F_z^f}=\overline{p^f{w}^f}$ (Fig. 15b). For higher harmonics, the flux is upward and divergent over the bottom 300 m where the higher harmonics are generated and it is then convergent above −2600 m. Subtracting the energy flux associated with higher harmonics from the total energy flux yields the energy flux associated with the incoming wave (Fig. 15c). This energy flux is nearly uniform with depth below the generation layer and decreases sharply at over the bottom 300 m, consistent with the analytical prediction for the critical layer depth associated with the critical reflection mechanism.

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(a) Total vertical energy flux, $\overline{F_z}=\overline{pw}$, averaged temporally and meridionally over the whole domain for the REF simulation, as a function of depth, and (b) its contribution from higher frequencies with a cutoff frequency of 1.5f. (c) The difference between the two quantities, which corresponds mostly to the energy flux of the first harmonic, is also shown. Note that the value of the vertical energy flux in this figure is lower than the one shown in Table 1 because it is averaged over the whole domain.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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Therefore, while critical reflection acts to increase substantially the amplitude of the wave in a bottom critical layer of about 300 m (as described in DT19), higher harmonics are generated and flux energy out of the critical layer. This energy flux is convergent above in the water column where the higher harmonics undergo PSI, which acts to elevate the vertical shear along horizontal phase lines (as shown in Fig. 5) and ultimately drives elevated mixing rates above the critical layer.

3) Comparison to the KPP mixing scheme

We ran a simulation where we activated the Ri-dependent part of the KPP mixing scheme but without convective adjustment nor a bottom boundary layer (Large et al. 1994). At this spatial resolution, KPP produces enhanced diffusivities in areas where it is expected from our previous analysis (Fig. 14). While mixing appears initially at the bottom through an enhancement of the wave field through critical reflection, it goes rapidly higher in the water column through the occurrence of PSI as predicted by our theory. Note, however, that KPP does not capture the high values of the diffusivity right at the seafloor.

While KPP seems to do a good job at capturing the spatiotemporal properties of the diffusivity in these high resolution simulations, it ultimately breaks down at lower resolution. As shown in DT19, enhanced mixing is only found right at the bottom in idealized simulations at a lower resolution. In those simulations, the generation of the second harmonics and PSI is suppressed due to the coarse resolution and thus KPP cannot capture the enhanced mixing occurring higher in the water column.

5. Sensitivity to key parameters

In the previous section, we described the processes responsible for the diapycnal mixing in the simulations. Here, we explore the dependence of the turbulent diffusivity on two key properties of the wave: its frequency and its mean downward energy flux.

We have run a set of simulations where we varied one of these two parameters at a time (Table 1). We used the approach described in section 3d(2) to estimate an effective diffusivity, κ_eb, from the increase in the background potential energy over CV2. We did not consider the first few days of each simulation where mixing had not started. Therefore, κ_eb is both a spatial and temporal average. The sensitivity of κ_eb to the two parameters is illustrated in Fig. 16.

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Values of the effective eddy diffusivity κ_eb and the local dissipation efficiency q as a function of (a) the mean downward energy flux of the wave at the inertial frequency and (b) of the wave’s frequency for a mean downward energy flux of 0.6 mW m⁻². For each panel, only one parameter is varied while the others are set to their reference value (see Table 1). In (b), the plain black line denotes the minimum frequency, and the dashed black line denotes the inertial frequency.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-22-0077.1

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As the downward wave energy flux increases, the effective diffusivity coefficients increase up to a certain value after which the diffusivity saturates (Fig. 16a). This threshold value is around 2.5 mW m⁻² in our simulations after which the effective diffusivity asymptotes to around 0.01 m² s⁻¹. In terms of the dependence on frequency, we can see that while the diffusivity is maximum at the inertial frequency (Fig. 16b), it decreases rapidly in the subinertial regime and more slowly in the superinertial regime before it dies out. The diffusivity remains high between 0.98f and 1.1f, suggesting that the critical reflection mechanism is efficient at driving mixing even for near-inertial waves.

6. Discussions and conclusions

DT19 and DT21 have shown in idealized and realistic numerical experiments that EWs experience critical reflection when reflecting off the bottom if NT effects are taken into account. The critical reflection mechanism is triggered by a shortening in the meridional scale of the reflected waves that is due to NT effects. With a shorter meridional wavelength, the reflected waves essentially become two-dimensional, and thus are governed by the NT wave equation for IGWs, which predicts the occurrence of critical reflection at the inertial latitude. Since EWs transmit a significant amount of energy in the deep ocean, this mechanism is expected to be important in the low-latitude regions as shown by DT21. Overall, DT19 showed that it could participate in around 10 Sv of diapcynal upwelling in the abyssal ocean, and thus contribute greatly to closing the deep cell of the MOC.

In this paper, we have considered the various mechanisms that are involved in the critical reflection of internal waves due to NT effects, which allows us to speculate the nature of the mixing that develops in the deep ocean. When a wave close to the inertial frequency reflects off a flat bottom, it undergoes critical reflection. As a result, the reflected wave is confined over the bottom 100–300 m of the water column. In this bottom layer, the amplitude of the reflected wave is greatly amplified. The amplification of the reflected wave results in a strong vertical shear that drives shear instability over this bottom layer. However, the interactions between the highly energetic reflected wave and the incident wave also act to generate higher harmonics that propagate upward and away from the bottom layer. These higher harmonics are then unstable to PSI and dissipate in the water column, 500–1000 m off the bottom. As a result, mixing extends a few hundred meters above the bottom layer. The mixing linked to the PSI higher in the water column is not as strong as that caused by shear instability in the bottom layer (e.g., the estimated effective diffusivity decreases from 10⁻¹ m² s⁻¹ near the bottom to 10⁻⁴ m² s⁻¹ at 1000 m off the bottom in our simulations). Meridional sections of temperature microstructure in the abyssal tropical Pacific at 110°W (e.g., Holmes et al. 2016) and at 150°W collected as part of the GO-SHIP program (J. Nash 2021, personal communication) show evidence of bottom-enhanced turbulent diffusivities in regions of smooth topography with values decreasing from 10⁻² to 10⁻⁴ m² s⁻¹ over the bottom 1000 m, which is similar to what we see in our simulations (e.g., Fig. 14).

While our simulations are two dimensional, we believe they can capture accurately the generation of the second harmonic during critical reflection and the dissipation and mixing that comes from the PSI. Fully 3D numerical simulations and laboratory experiments of critical reflection of internal waves off a slope develop second harmonics that radiate off the bottom, indicating that the phenomenon is not restricted to two-dimensional flows (Peacock and Tabaei 2005; Gayen and Sarkar 2011; Rodenborn et al. 2011). Gayen and Sarkar (2013) described simulations of PSI that can form when an internal wave beam refracts at a pycnocline and nonlinear wave–wave interactions ensue. They compared the results from 3D and 2D simulations and found that the evolution of the energetics of PSI was essentially unchanged. Gayen and Sarkar (2014) further used a fully 3D LES and found that the energy flux in the subharmonic was around 10% of the imposed flux in the beam. The net loss of kinetic energy to turbulence in the region where PSI forms is also about 10% of energy flux of the beam suggesting that the rate of energy extracted from the beam by PSI is balanced by dissipation and mixing. Therefore, if we can capture the growth of PSI in the 2D simulations, then in a steady state (where the growth is balanced by dissipation and mixing) we should be able to capture the dissipation and mixing. Having said this, if we had been able to run fully 3D simulations, we may have found that some of the quantitative details of the small-scale turbulence were off (the value of the mixing efficiency, for example) and thus our findings should be viewed as a step toward understanding the turbulence that results from critical reflection of IGWs under NT effects and not the definitive solution. However, it is unlikely that the 100–1000-fold variations of the dissipation and inferred turbulent diffusivity over the bottom 1000 m that we find is particularly sensitive to the details of the small-scale turbulence since the vertical structure of these fields is set by wave–wave interactions that can be simulated in 2D.

Data availability statement.

The numerical model simulations upon which this study is based are too large to archive. Selected subsets of the data are available by reaching out to the authors of the paper. In addition, the model setup files and processing scripts are available at https://github.com/bdelorme/DelormeThomas2022.