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    Analytical solution for an ETW in a domain that is unbounded in the vertical and uniformly stratified (left) with and (right) without the TA and for different values of N [(top) higher N, corresponding to typical value in the deep ocean; (bottom) lower N, corresponding to typical value in the abyssal ocean]. Parameter is taken at the equator where it is maximal. The scalar field [e.g., Eq. (18)] is plotted for and , corresponding to an eastward-propagating Yanai wave. Where , lines of constant phase (i.e., contours of ) are tilted upward with latitudes and tend to align with surfaces of constant absolute momentum (dotted blue lines) rather than isopycnals (dotted gray lines). Note that because of the change in stratification, the vertical wavelength in the bottom panels is in theory 10 times what it is in the top panels. This schematic is used to emphasize the meridional phase curvature in the abyss.

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    (a) Profile of the stratification used in the MITgcm simulations on a logarithmic scale (blue line). Stratification is uniform at the surface, but asymptotes toward low values in the deep as observed in the mean stratification profile across all casts measured by H16 over ±2° near 110°W in the eastern equatorial Pacific (dark brown line), and in the mean profile derived with the Gibbs Seawater Oceanographic Toolbox (Feistel 2008) from pressure, salinity, and temperature profiles obtained with 346 independent casts made near 110°W over ±5° and taken from the World Ocean Database 2013 (light brown line). Note the different y axis for the blue and brown profiles. (b) Vertical structure of the mask that modulates the forcing of the ETW. The mask is maximal at the surface where horizontal velocities and density are fully relaxed to the analytical solution for a downward-propagating ETW, and zero in the deep where the wave evolves freely.

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    Snapshots of the meridional velocity from the simulations of downward-propagating ETWs (left) with and (right) without the TA 2 years into the simulation. The inertial latitude of the wave is indicated by the dashed lines. The wave is generated in the upper layer, indicated by the mask shown in Fig. 2, and evolves freely beneath it. The downward energy flux is the same for each wave. Notice how in the nontraditional case, beams of IGWs appear to emanate from the inertial latitude.

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    The meridional velocity from the simulation of a nontraditional, downward-propagating Kelvin wave at four different times expressed in units of the wave period T. Small-scale features with enhanced shear develop in the abyss around the inertial latitude and propagate upward in the water column following two coherent beamlike structures, but also poleward of the inertial latitude in the weakly stratified abyss.

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    Snapshot of the difference in the meridional velocity (color) between the traditional and nontraditional simulations of an eastward-propagating Yanai wave taken 2 years into the simulations. Taking the difference between both runs allows us to highlight the beams that are generated when nontraditional effects are active. Ray paths corresponding to the characteristic curves of the 2D wave [Eq. (12)] with and without the TA are indicated by the gray and magenta lines, respectively. The separatrix (black line) where the nontraditional 2D wave equation changes from being hyperbolic to elliptic is also plotted. Nontraditional effects extend the separatrix poleward when the stratification is weak, allowing for 2D wave propagation past the inertial latitude of the wave, which in this case is .

  • View in gallery

    The log10 of the power spectral density of the meridional velocity as a function of frequency ω and meridional wavenumber l for the (left) traditional and (right) nontraditional runs of the westward-propagating Yanai wave, averaged over the last 1500 m of the domain. The frequency of the simulated wave is s−1 (white dashed line). In the nontraditional case, the small-scale features that develop broaden the spectral peak in wavenumber space and also result in higher harmonics in frequency (white dotted lines).

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    The meridional velocity from the simulation of a westward-propagating Yanai wave averaged over the last wave period. In the nontraditional case, mean flows are generated through wave–wave interactions and develop along the beams in the deep ocean.

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    Turbulent diffusivity in the simulations shown in Fig. 3 averaged zonally and over the last wave period (i.e., after 2 years of simulation). The sharp small-scale features in the nontraditional case ultimately increase the shear of the flow, leading to enhanced diffusivities in all the simulations that we considered. The largest values of the diffusivity are found near the inertial latitude (black dashed lines) and extend upward off the bottom and poleward of the inertial latitude.

  • View in gallery

    (a) Plan view of the diffusivity averaged over one wave period and the bottom 300 m 2 years into the simulation of a Rossby wave. (b) Corresponding Hovmöller diagram of the diffusivity averaged zonally. It takes approximately 6 months for the Rossby wave to reach the seafloor, after which enhanced mixing persists throughout the simulation. The inertial latitude of the wave is indicated by the black dashed lines.

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    Hovmöller diagram of the diapycnal volume transport per wavelength of ETW (Sv) in the last 500 m of the domain for the simulations shown in Fig. 3 in the nontraditional case. Significant diapycnal upwelling develops once the wave reaches the seafloor.

  • View in gallery

    The meridional velocity from the solution to (12) forced with with , corresponding to the frequency of the eastward Yanai wave in Fig. 3. The stratification profile used in the calculation is the same as that employed in the MITgcm simulations. Energy propagates from the bottom in beams that emanate from the inertial latitudes (indicated by the dashed vertical lines) and follow ray paths (magenta) corresponding to characteristic of Eq. (12).

  • View in gallery

    Schematic of the characteristics of Eq. (30) for the three types of waves: (left) superinertial, (center) inertial, and (right) subinertial. The schematic is drawn for the Northern Hemisphere, with north to the right. The corresponding group velocity vector is aligned with the characteristic curves, whereas the wave vector is perpendicular to them.

  • View in gallery

    (a) Traditional simulation with the MITgcm of an equatorial eastward-propagating IGW in a domain with a topographic feature. In this case, critical reflection occurs and generates sharp beams. (b) The slope of the characteristics of the 2D wave equation with the TA [i.e., Eq. (31) with ] and the slope of the topography as a function of latitude. Critical reflection occurs where the characteristic and the topography are aligned.

  • View in gallery

    (left) Vertical wavelength and (right) decay rate of 2D waves on an f plane with (colored lines) and without (black crosses and circles) viscosity as a function of frequency. Viscosity prevents the vertical wavelength from going to zero at . Parameter is the imaginary part of the roots of Eq. (38) and the real part. Solid (dashed) lines correspond to height-decreasing (increasing) solutions. In the (top) nontraditional case, the symmetry of the solutions that we observe in the (bottom) traditional case is broken. To derive these solutions, we chose a vertical wavelength of 2 km for the incident wave and a period of 30 days. The stratification is set to s−1 and the viscosity coefficient to m2 s−1.

  • View in gallery

    Streamlines (contours) and KE (colors) of the (right) reflected, (center) incident, and (left) total wave field solutions to Eq. (37). (top) A free-slip boundary condition is used; (bottom) a no-slip boundary condition is used. The steep characteristic along which the incident wave propagates is shown in red in the center panels.

  • View in gallery

    Sensitivity of the ratio in Eq. (41) of the KE of the reflected wave to the KE of the incident wave to different parameters: the period of the incident wave T, the stratification N, the viscosity coefficient ν, and the vertical wavenumber of the incident wave . (left) The variation of the KE ratio as a function of (a proxy for latitude) for three different parameter values; (right) the ratio evaluated at the inertial latitude for a range of parameter values (a dashed line with a slope of −1 is plotted for reference).

  • View in gallery

    The KE of the reflected wave as a function of wave period calculated from Eq. (41) for the ETWs from our simulations. (a) Variations in the local energy flux are taken into consideration; (b) the same local energy flux for all wave types is assumed.

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Abyssal Mixing through Critical Reflection of Equatorially Trapped Waves off Smooth Topography

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  • 1 Earth System Science Department, Stanford University, Stanford, California
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Abstract

The inferred diapycnal upwelling in the abyssal meridional overturning circulation (AMOC) is intensified near the equator, but little is known as to why this is so. In this study, it is shown that the reflection of equatorially trapped waves (ETWs) off the bottom leads to seafloor-intensified mixing and substantial diapycnal upwelling near the equator when the full Coriolis force and the so-called nontraditional effects are taken into account. Using idealized simulations run with the MITgcm of downward-propagating ETWs of various types (i.e., inertia–gravity, Yanai, Kelvin, and Rossby waves) accounting for nontraditional effects, it is demonstrated that the reflection of ETWs off a flat seafloor generates beams of short inertia–gravity waves with strong vertical shear and low Richardson numbers that result in bottom-intensified, persistent, zonally invariant mixing at the inertial latitude of the ETW through the mechanism of critical reflection. The beams are more intense with weaker stratification and, for a given wave type, are stronger for waves with shorter periods and longer vertical wavelengths. The intensity of the beams also differs between wave types because their distinct meridional structures modulate the amount of energy fluxed to the bottom at the inertial latitude. As a result, equatorial inertia–gravity, Rossby, and eastward-propagating Yanai waves yield stronger mixing than Kelvin and westward-propagating Yanai waves in the simulations. It is estimated that this process can result in order 10 Sv (1 Sv ≡ 106 m3 s−1) of diapycnal upwelling per wavelength of ETW in the abyss and thus could play an important role in closing the AMOC.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bertrand Delorme, bdelorme@stanford.edu

Abstract

The inferred diapycnal upwelling in the abyssal meridional overturning circulation (AMOC) is intensified near the equator, but little is known as to why this is so. In this study, it is shown that the reflection of equatorially trapped waves (ETWs) off the bottom leads to seafloor-intensified mixing and substantial diapycnal upwelling near the equator when the full Coriolis force and the so-called nontraditional effects are taken into account. Using idealized simulations run with the MITgcm of downward-propagating ETWs of various types (i.e., inertia–gravity, Yanai, Kelvin, and Rossby waves) accounting for nontraditional effects, it is demonstrated that the reflection of ETWs off a flat seafloor generates beams of short inertia–gravity waves with strong vertical shear and low Richardson numbers that result in bottom-intensified, persistent, zonally invariant mixing at the inertial latitude of the ETW through the mechanism of critical reflection. The beams are more intense with weaker stratification and, for a given wave type, are stronger for waves with shorter periods and longer vertical wavelengths. The intensity of the beams also differs between wave types because their distinct meridional structures modulate the amount of energy fluxed to the bottom at the inertial latitude. As a result, equatorial inertia–gravity, Rossby, and eastward-propagating Yanai waves yield stronger mixing than Kelvin and westward-propagating Yanai waves in the simulations. It is estimated that this process can result in order 10 Sv (1 Sv ≡ 106 m3 s−1) of diapycnal upwelling per wavelength of ETW in the abyss and thus could play an important role in closing the AMOC.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bertrand Delorme, bdelorme@stanford.edu

1. Introduction

Inverse models of the ocean’s circulation suggest that much of the zonally integrated diapycnal upwelling that closes the abyssal meridional overturning circulation (AMOC) occurs in the tropical oceans (Lumpkin and Speer 2007). This result implies that intense mixing takes place in these regions. However, the abyssal near-equatorial oceans are poorly sampled; therefore, the inferences from the inverse models have not been fully tested in the field.

To address this problem and study mixing in the deep equatorial regions, Holmes et al. (2016, hereafter H16) recently carried out observations in the eastern equatorial Pacific. During this field campaign, a lowered acoustic Doppler current profiler (LADCP), a conductivity–temperature–depth (CTD) cage, and microstructure sensors were used to gather information on flow velocity, stratification of the water column, and turbulence, respectively. The instruments were deployed from the surface all the way to the seafloor. Following this procedure, H16 reported evidence of strong turbulence around 0.5°S, ranging from the seafloor up to 700 m above. The rate of kinetic energy dissipation associated with this turbulence had values typical of flows over rough topography, such as midocean ridges (based on values reported by Waterhouse et al. 2014). In H16, however, the observations were carried out over smooth topography, in a region where the generation of internal tides through tidal–topographic interactions is relatively weak (St. Laurent and Garrett 2002). Hence, the common interpretation that bottom-intensified mixing is driven by internal waves breaking locally through interactions with rough topography is not applicable in the region studied by H16.

H16 also reported evidence of a downward-propagating Yanai wave in the LADCP data with a downward energy flux of about 2 mW m−2. This rate is comparable to the 1 mW m−2 average dissipation rate estimated in the abyss in the mixing region. Therefore, H16 suggested that the mixing may have been energized by an equatorially trapped wave (ETW) whose physics had been modified by the horizontal component of Earth’s rotation when the so-called nontraditional effects are active. This hypothesis was motivated by previous studies that investigated the modification of inertia–gravity waves (IGWs) at midlatitudes by nontraditional effects. In particular, Gerkema and Shrira (2005) and Winters et al. (2011) identified that acts to 1) create a class of subinertial waves (i.e., waves whose frequency is smaller than the inertial frequency f ); and 2) produce a nonvanishing group velocity at the inertial latitude, where the wave frequency is equal to f. These studies demonstrated that nontraditional 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 likely induce mixing. Following this theory, H16 solved the zonally uniform, time-periodic, forced linear wave equation on an equatorial β plane under the full Coriolis force. The solutions that H16 found display enhanced shear around the inertial latitude in the weakly stratified abyss, which ultimately acts to decrease the Richardson number and thus could support shear instabilities and subsequent mixing.

However, it remains to be seen whether this mechanism can explain the enhanced mixing observed by H16 because their calculation used several assumptions that may not be valid. Namely, the solutions to the wave equation described by H16 assume that the waves are linear and two-dimensional. Nonlinear dynamics are known to be dominant at critical reflections (Dauxois and Young 1999). As a result, they might play a significant role around the inertial latitude when nontraditional effects are permitted. In particular, these processes could act to enhance or suppress the mixing that develops or to modify its spatial structure. In addition, the two-dimensional wave equation that H16 considered fails to describe the equatorial dynamics accurately. This equation, by assuming zonally invariant motions, captures the physics of traveling waves confined to the meridional plane. It is thus suitable to study IGWs at midlatitudes, which have a rather high frequency (close to the inertial frequency in their source region) and a long zonal wavelength relative to their meridional wavelength. These waves propagate mainly toward the equator, eventually breaking or dissipating along the way (Alford et al. 2016). In contrast, at the equator, most of the variability in the deep is supported by waves that have a rather large meridional scale and that primarily propagate zonally over a much longer time period. In this study, we refer to such waves as ETWs in contrast to the midlatitude IGWs because they have fundamentally distinct physical properties (Philander 1978).

The properties of ETWs were first described in detail by Matsuno (1966) and Blandford (1966), and they have subsequently been observed and amply documented (see Wunsch and Gill 1976; Harvey and Patzert 1976; Weisberg and Horigan 1981; Brandt and Eden 2005; Farrar 2008; Smyth et al. 2015). In particular, Eriksen (1980) and Eriksen and Richman (1988) used time series from moored arrays in the central Pacific at depths of 1500 and 3000 m to provide one of the most comprehensive descriptions of the deep equatorial ocean. They found that the energy at these depths is distributed over a broad band in frequency, with wave periods ranging from days to months, but a narrow band in zonal wavenumber of order O(1000) km. In a recent review of the characteristics of the deep equatorial variability, Ascani et al. (2010) showed that these observations were consistent with results from several ocean general circulation models. Nevertheless, the propensity of these ETWs to drive mixing in the abyss when remains to be investigated.

The scant attention in the literature to the nontraditional effects on ETWs in comparison to midlatitude IGWs is somewhat surprising. At the equator, is maximal, and , raising more questions about the traditional approximation (TA) that sets to 0. To the best of our knowledge, nontraditional effects on ETWs have been addressed only analytically in a linear framework for waves propagating over an infinite domain (Raymond 2000, 2001; Fruman 2009). These past studies showed that nontraditional effects greatly affect the ETW’s spatial structure by 1) widening the equatorial trapping and 2) bending upward with latitude their lines of constant phase. However, contrary to the midlatitude IGW case, does not act to modify the dispersion of ETWs.

In the study presented here, we investigate the influence of on ETWs in a fully nonlinear and three-dimensional numerical model, and we interpret the results with analytical theory. We show that nontraditional effects induce fundamental changes in the wave bottom reflection off a flat seafloor by generating small-scale features that act to increase the shear in the abyss, inducing mixing. We argue that these features are generated predominantly at the inertial latitude because of an amplification mechanism due to critical reflection of the ETW off the flat bottom. We further characterize the spatiotemporal variability of the mixing that develops and investigate the dependence of this mechanism on the key properties of the waves and basic state to understand under which conditions nontraditional effects would be more effective at driving mixing in the real equatorial ocean. Our study suggests that because ETWs are ubiquitous in the equatorial regions, their energy input to sustain mixing and, subsequently, to control diapycnal upwelling in the abyssal equatorial oceans could be substantial and generate diapycnal volume transports of order 10 Sv (1 Sv ≡ 106 m3 s−1).

The remainder of this paper is organized as follows. In section 2, we summarize earlier theoretical contributions on nontraditional effects on ETWs and discuss the conceptual differences with the midlatitude IGW case. We then comment on the implications that nontraditional effects have for wave reflection off a flat bottom. In section 3, we introduce the numerical model and describe the design of the numerical experiments. In section 4, we report the results from the numerical experiments and compare runs with and without nontraditional effects, highlighting the spatiotemporal characteristics of the mixing that develop when nontraditional effects are active and its implication for diapycnal transport in the abyss. In section 5, we interpret the results using two-dimensional, viscous, analytical models to describe how critical reflection is responsible for the enhanced shear when nontraditional effects are active. Then, we discuss the dependence of this mechanism on the key parameters of the waves and of the medium in which they propagate, and we highlight some crucial differences across ETW types that might explain the differences in mixing that we observe in the numerical simulations. Section 6 concludes by discussing the strengths and limitations of the theory and potential topics for future research.

2. Analytical solution of ETWs accounting for nontraditional effects

We begin with a brief overview of linear equatorial wave theory when is taken under consideration. To the best of our knowledge, D. Moore (1993, unpublished manuscript), Raymond (2000, 2001), and Fruman (2009) are the only studies that have addressed this problem in the past. We should also note the contribution from Roundy and Janiga (2012), who extended the theory of Fruman (2009) by considering the fully nonhydrostatic equations of motion. The derivation described in this section largely follows these previous studies, but we highlight the features of the solution that have implications for wave reflection off the bottom and that differ fundamentally from the two-dimensional (2D) solutions.

We start with the inviscid Boussinesq equations linearized around a state of rest over an equatorial β plane under the full Coriolis force:
e1
e2
e3
e4
e5
where is the velocity vector. Term , where p is the deviation of the pressure from that of a resting, hydrostatically balanced ocean, and is a reference value for the density. The buoyancy perturbation is then , where ρ is the density deviation from a background vertical profile , and g is the acceleration due to gravity. is the Brunt–Väisälä frequency determined from the buoyancy of the background . Term is the nontraditional horizontal component of the Coriolis parameter at the equator, and is the meridional gradient of the vertical component of the Coriolis parameter at the equator, where is Earth’s angular velocity and is Earth’s radius. Subscripts x, y, z, and t indicate partial derivatives throughout the paper.
We start by considering the general case of a nonvanishing meridional velocity, but note that the Kelvin wave case can be treated in a similar way. To derive a closed-form solution to Eqs. (1)(5), N has to be constant. Assuming that this is the case, we introduce a scalar field , such that
e6
e7
e8
e9
e10
Because we are looking for time-periodic solutions, and ignoring zonal boundaries for simplicity, we set , with k and ω being the zonal wavenumber and the frequency of the wave, respectively. With this scalar representation, Eqs. (1) and (3)(5) simplify, and Eq. (2) becomes
e11
This equation captures the physics of the Rossby, Kelvin, and Yanai waves in addition to IGWs. H16 only considered IGWs by setting in Eq. (11), yielding the 2D wave equation:
e12
Note that the TA renders Eq. (11) separable by removing the mixed derivative and thus makes solving it much easier. If , we must ultimately assume an unbounded domain in the vertical direction to obtain a closed-form analytical solution. Doing so allows us to set , where m is the vertical wavenumber of the wave. Conceptually, this approximation consists in considering fully developed meridional modes propagating along vertical rays. As stated by Knox and Anderson (1985), this approach is particularly reasonable in the deep equatorial ocean because ETWs have rather shallow rays (Cox 1980; Eriksen 1981; McCreary 1984). This view is supported by the few measurements that have been made in the deep in the equatorial ocean, which suggest that ETWs propagate as beams rather than modes (Luyten and Swallow 1976; Eriksen 1980; Eriksen and Richman 1988; Ascani et al. 2010). With this ansatz, Eq. (11) becomes
e13
It is possible to further decompose the solution by setting , where
e14
represents the meridional phase dependence of the ETW, which does not exist under the TA. The equatorial Rossby radius of deformation in the nontraditional (NT) case is . It differs from the traditional (T) equatorial Rossby radius of deformation following
e15
The decomposition above allows us to remove the first-order derivative term in Eq. (13), such that
e16
Introducing the variable , Eq. (16) becomes
e17
This equation is similar to Schrödinger’s equation for a harmonic oscillator. Where the coefficient in front of ϕ in Eq. (17) is positive, the solution has a wavelike behavior, but it will eventually become negative at a turning latitude , where the equation becomes elliptical in character and solutions rapidly decay. Those solutions exist only when
e18
with . This is the dispersion relation for ETWs propagating vertically in an unbounded domain. Remarkably, nontraditional effects have no influence on the dispersion of such waves, a result first outlined by Needler and Leblond (1973) for long-period waves at midlatitudes, and that remains true at the equator. However, affects the turning latitude of the wave:
e19

Note that these turning latitudes are different from the ones derived in the 2D case. In the 2D traditional case, the turning latitude is the inertial latitude (i.e., ); in the 2D nontraditional case, the turning latitude differs from the inertial latitude only in the weakly stratified abyss because of the influence of that permits wave propagation poleward of the inertial latitude. However, in the 3D case, k acts to shift the turning latitude away from the inertial latitude everywhere in the water column, even when the stratification is high.

The full solution is then
e20
where is the Hermite polynomial of nth order and a constant.

In these calculations, we have shown that even if the dispersion relation of the wave is not affected by , is amplified and broadens the Gaussian envelope of the wave when nontraditional effects are taken into account. In the traditional case, and thus remains constant for a given wave. Without the TA, experiences changes that are governed by the ratio ; thus, as N decreases, the deformation radius of the wave increases. More importantly, acts to curve surfaces of constant phase with latitude because of the presence of the quadratic function in the phase of . These surfaces are no longer parallel to the horizontal, but they approach lines of constant absolute momentum . Here again, these nontraditional effects are more pronounced in regions where is stronger, which is typically the case in the abyss near the equator. In Fig. 1, we illustrate these differences for two values of the stratification that correspond to typical values in the deep and abyssal oceans.

Fig. 1.
Fig. 1.

Analytical solution for an ETW in a domain that is unbounded in the vertical and uniformly stratified (left) with and (right) without the TA and for different values of N [(top) higher N, corresponding to typical value in the deep ocean; (bottom) lower N, corresponding to typical value in the abyssal ocean]. Parameter is taken at the equator where it is maximal. The scalar field [e.g., Eq. (18)] is plotted for and , corresponding to an eastward-propagating Yanai wave. Where , lines of constant phase (i.e., contours of ) are tilted upward with latitudes and tend to align with surfaces of constant absolute momentum (dotted blue lines) rather than isopycnals (dotted gray lines). Note that because of the change in stratification, the vertical wavelength in the bottom panels is in theory 10 times what it is in the top panels. This schematic is used to emphasize the meridional phase curvature in the abyss.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

In the ocean, we expect the tilting of the phase lines due to to occur in the abyss, where the stratification is weak enough so that . Because depends on the sign of the vertical wavenumber m, it creates an asymmetry that makes it no longer possible to superimpose two waves with equal and opposite vertical wavenumbers to satisfy the bottom boundary condition of no normal flow at a horizontal boundary. This crucial consequence of nontraditional effects is a manifestation of the nonseparability of the problem that prevents the use of a single monochromatic wave to close the problem. We will show in the next sections that it has profound implications for the dynamics of the flow in the abyss, when the wave reflects off the bottom.

3. Numerical experiments

a. Model equations

The MITgcm is employed in this study (Marshall et al. 1997). The model is nonlinear and is used in both its hydrostatic and quasi-hydrostatic versions. In the quasi-hydrostatic version, terms associated with are incorporated in the governing equations, whereas they are not included in the hydrostatic version, allowing us to investigate nontraditional effects on ETWs by comparing runs using both versions of the model. The differences between the two versions can be easily understood by looking at the equations solved by the model:
e21
e22
e23
e24
e25
where we have used the same notation as in Eqs. (1)(5). Here, and represent the vertical turbulent viscosity and diffusivity coefficients, respectively. Note that the lateral viscosity and diffusivity coefficients have been explicitly set to zero in our study. The last term on the right-hand side in Eqs. (19), (20), and (23) is a nudging term used to force the wave by relaxing the horizontal velocities and buoyancy to prescribed fields (, , and ) over a time scale τ and in the spatial region defined by the 3D mask . Incorporating these relaxation terms in the governing equations is straightforward in the MITgcm thanks to the RBCS package [section 6.3.2 in Adcroft et al. (2016)]. In the hydrostatic version of the model, the underlined terms in Eqs. (19) and (21) are neglected, but they are retained in the quasi-hydrostatic version where the TA is relaxed. For the interested reader, more details on the energetics and numerical subtleties of each version of the MITgcm are provided by Marshall et al. (1997).

b. Model setup

In our simulations, we use a 3D Cartesian coordinate system centered at the equator, extending up to about ±8° in latitude and 4500 m deep with a flat bottom. Two sponge layers are used on the southern and northern boundaries, ranging from ±5.5° to ±8°, where the velocity is smoothly damped to zero and the stratification relaxed to the initial conditions. These regions act to absorb weak nonlinear perturbations only because the linear dynamics considered here are confined to the equatorial waveguide. Similar to the analytic derivation in section 2, periodic boundary conditions are used in the zonal direction. At the bottom, a free-slip boundary condition is used to avoid the interference of mixing induced by drag with mixing induced by nontraditional effects. However, as discussed in section 5, considering a nonzero drag coefficient at the bottom yields similar conclusions to those that we will present in the next section.

The resolution of the model is (, , ) = (20 km, 3 km, 15 m). Note that we ran sensitivity tests spanning (, , ) that show that our results are robust relative to grid resolution.

In the model, salinity is constant in the whole domain. As a result, temperature alone governs buoyancy through a linear equation of state, such that , with T the temperature anomaly field and the thermal expansion coefficient. A linear, second-order, temperature advection scheme is used that results in small numerical dissipation and diffusion; however, no explicit lateral or vertical background diffusivities/viscosities are prescribed. The KPP Ri-dependent mixing scheme (Large et al. 1994) is used to increase the vertical viscosity and diffusivity in regions where the Richardson number is small. This parameterization hence provides a proxy for the presence of shear instabilities that could drive mixing in the water column. In this study, KPP is used with the default values of the parameters: the limit Richardson number is 0.7, and the maximal viscosity and diffusivity is 50 × 10−4 m2 s−1.

c. Wave generation

To mimic the generation of a downward-propagating ETW beam by upper-ocean processes, we relax the velocity and density fields in the upper part of the domain to the analytical solution of nontraditional, downward-propagating ETWs; that is, the , , and terms in Eqs. (19)(23) are set to their corresponding expressions in Eqs. (6)(10). However, because we have shown in section 2 that such solutions can be derived only if the stratification is assumed constant, the background stratification profile remains constant in the upper layer, where the wave is generated, and it asymptotically approaches a stratification profile in the deep that is based on representative values from the equatorial ocean. Specifically, it is derived from averaging stratification profiles measured by H16 and those taken from the World Ocean Database 2013 (Boyer et al. 2013) in the eastern equatorial Pacific (Fig. 2a).

Fig. 2.
Fig. 2.

(a) Profile of the stratification used in the MITgcm simulations on a logarithmic scale (blue line). Stratification is uniform at the surface, but asymptotes toward low values in the deep as observed in the mean stratification profile across all casts measured by H16 over ±2° near 110°W in the eastern equatorial Pacific (dark brown line), and in the mean profile derived with the Gibbs Seawater Oceanographic Toolbox (Feistel 2008) from pressure, salinity, and temperature profiles obtained with 346 independent casts made near 110°W over ±5° and taken from the World Ocean Database 2013 (light brown line). Note the different y axis for the blue and brown profiles. (b) Vertical structure of the mask that modulates the forcing of the ETW. The mask is maximal at the surface where horizontal velocities and density are fully relaxed to the analytical solution for a downward-propagating ETW, and zero in the deep where the wave evolves freely.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

In our study, the mask that modulates the forcing is horizontally uniform, approaching 1 near the surface and decreasing to zero with depth where the stratification starts to vary, following a tanh profile (Fig. 2b). Hence, the wave is generated in the upper layer and propagates freely beneath it without any further forced adjustments. Once the incident wave beam reaches the bottom, it reflects back up in the water column and enters the forcing region, where it gets smoothly damped by the nudging terms in Eqs. (19) and (20). To allow for this damping effect and to prevent the reflection of wave energy off the generation layer, the relaxation time scale τ is set equal to the period of the wave.

d. Experimental design

We ran a suite of numerical simulations with five ETWs of a variety of frequencies: a mode-1 Rossby wave, a Kelvin wave, eastward- and westward-propagating Yanai waves, and a mode-1 westward-propagating equatorial inertia–gravity wave. Here, “mode” refers to the meridional mode number of the wave, which is represented by n in Eq. (18). Since these waves arise from different physical mechanisms (e.g., from buoyancy restoring forces to potential vorticity conservation), we can explore how the modifications induced by vary with the type of wave. Note that we have also run simulations of higher-mode Rossby and equatorial inertia–gravity waves that resulted in similar conclusions as the gravest mode but that are not shown in the paper.

For each type of wave, we chose the same zonal and vertical wavelengths, and we allowed the frequency to vary according to the dispersion relation of the wave [Eq. (17)]. We set the zonal wavelength to 1200 km and the vertical wavelength to 1700 m, both of which are representative of the deep equatorial ocean [see Ascani et al. (2010) for a thorough description of ETWs’ length scales]. For each simulation, the wave is forced from the beginning of the simulation for up to 2 years.

The amplitude of the waves is set by constraining their downward energy flux in the generation layer
e26
where p and w are the analytical expressions for the pressure perturbation and the vertical velocity derived in Eqs. (6)(10); is the complex conjugate of w; Re denotes the real part; and m is the depth where the energy flux is evaluated and corresponds to the bottom of the generation layer. Following Smyth et al. (2015) and Eriksen and Richman (1988), we define the net downward energy flux per unit horizontal area in the equatorial waveguide as
e27
where is the turning latitude of the wave defined in Eq. (15). To eliminate variations in dissipation and mixing due to differences in this average energy flux across type of ETWs, we set to 0.15 mW m−2 for all experiments, which constrains the amplitude of each wave. This value is equal to the annually averaged downward energy flux of Yanai waves reported by Eriksen and Richman (1988) based on observations from depths between 1000 and 3000 m in the equatorial Pacific Ocean. Note that this value is around an order of magnitude smaller than the energy flux associated with individual Yanai waves estimated by Smyth et al. (2015) and H16. Therefore, our results could be interpreted as corresponding to the mean state of the equatorial oceans and would be significantly amplified if we were to consider the peak value of the energy flux in a wave event.
However, because setting the same downward energy flux across all types of waves results in different energy density in the wave field, some waves will be more prone to nonlinearities. We thus introduce two nondimensional quantities, and , to quantify nonlinear advection to acceleration in the horizontal momentum equations for both the traditional and nontraditional cases:
e28
e29
where ω is the frequency of the wave; k is its zonal wavenumber; U is the amplitude of the horizontal velocity; and and are the inertial and turning latitudes of the wave, respectively. The difference between and lies in the fact that a new meridional length scale appears in the nontraditional case because of the development of features between the inertial latitudes of the wave that we highlight later in the paper.

Table 1 summarizes the key properties of the simulated waves in the generation layer, where the stratification is constant. Note the gap between the inertial and turning latitudes of the wave, even in the generation layer where stratification is rather high. This discrepancy is due to the zonal wavenumber of the wave, highlighting the difference between 2D and 3D wave physics. Below the generation layer, the turning latitude increases due to lower stratification values, while the inertial latitude remains constant. Note also that the Rossby and westward-propagating Yanai waves have the largest kinetic energy density, which is approximately one order of magnitude larger than the other waves. These waves are thus more prone to nonlinearities, as shown by their large values. More importantly, for the ETWs with the longest periods (i.e., the Rossby, Kelvin, and westward-propagating Yanai waves), nontraditional effects significantly enhance because the inertial latitude is close to the equator, making the dynamics of those waves more nonlinear.

Table 1.

Properties of the ETWs generated in the numerical experiments. These properties are valid only in the upper layer, where the wave is generated and where the stratification is constant. Below this layer, the variations in stratification modify the properties of the wave. The period and zonal wavelength will, however, remain constant. For the Yanai wave, both westward- (W) and eastward- (E) propagating waves are simulated. The nondimensional parameters and defined in section 3 are shown for each wave. The zonal and vertical components of the group velocity , the averaged downward flux of energy , and the averaged kinetic energy density in the equatorial waveguide are also shown.

Table 1.

4. Numerical solutions

a. Beams at the inertial latitude

Figure 3 shows snapshots of the resulting meridional velocity from each simulation, with and without , 2 years into the simulation. In the traditional case (left column), the velocity field represents a perfect vertical mode: the structures oscillate in time but do not propagate vertically. Interestingly, this is also true for the larger-scale vertical structures in the nontraditional case (right column), as low-pass-filtered (in vertical wavenumber) modes do not propagate in the vertical (not shown). Therefore, even when , the downward-propagating wave can reflect into an upward-propagating wave with similar characteristics and little loss of amplitude. However, as explained in section 2, the two waves can only satisfy the bottom boundary condition at one latitude in each hemisphere, and this results in the generation of other flow features. In particular, in addition to the large-scale vertical structures, in all of the ETW types that we consider here, the velocity field develops sharp features below the generation layer when nontraditional effects are active. However, these features are more visible for some waves than for others. In the westward-propagating Yanai wave simulation, for example, the sharp features are weaker than for the eastward-propagating Yanai wave. In all cases, these features are completely absent in simulations run with .

Fig. 3.
Fig. 3.

Snapshots of the meridional velocity from the simulations of downward-propagating ETWs (left) with and (right) without the TA 2 years into the simulation. The inertial latitude of the wave is indicated by the dashed lines. The wave is generated in the upper layer, indicated by the mask shown in Fig. 2, and evolves freely beneath it. The downward energy flux is the same for each wave. Notice how in the nontraditional case, beams of IGWs appear to emanate from the inertial latitude.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

The Kelvin wave case is particularly compelling because the meridional velocity is everywhere zero under the TA (except right at the surface, since our solution for a downward-propagating wave does not satisfy the free-surface boundary condition). As a result, the sharp features are particularly visible in the Kelvin wave simulation where they form two wave beams. Figure 4 shows the time development of the beams at four different stages in the simulation. Once the wave reaches the bottom, enhanced velocity values appear close to the seafloor around the inertial latitude of the wave. As the simulation moves forward, these features intensify and start to form two well-defined beams that emanate from the inertial latitude. The beams then propagate upward in the water column, reflecting back and forth at the inertial latitude of the wave. Eventually, they reach the surface generation layer, where they are damped by the nudging terms.

Fig. 4.
Fig. 4.

The meridional velocity from the simulation of a nontraditional, downward-propagating Kelvin wave at four different times expressed in units of the wave period T. Small-scale features with enhanced shear develop in the abyss around the inertial latitude and propagate upward in the water column following two coherent beamlike structures, but also poleward of the inertial latitude in the weakly stratified abyss.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

The beams that appear in the nontraditional runs propagate in the meridional plane, contrary to the ETWs that form a standing wave in the meridional plane. In addition, the alternating jets that develop in the abyss poleward of the inertial latitude (visible in Figs. 3, 4) indicate that the waves propagate past the inertial latitude, as predicted by the 2D nontraditional theory for IGWs (Gerkema and Shrira 2005). This suggests that the sharp beams are governed by 2D physics contrary to the ETWs that force the simulations. For 2D waves (i.e., ), the characteristic curves of Eq. (12) run parallel to wave rays. These characteristics are plotted on top of the MITgcm simulations for the eastward Yanai wave in Fig. 5. For comparison, we are also showing the shape of the characteristics in the traditional case. Here, we show the difference between the meridional velocity from the traditional and nontraditional runs to emphasize the beams that develop in the nontraditional case. The beams from the numerical model follow the characteristics of the 2D wave equation almost perfectly, thus confirming that 2D physics govern their dynamics. The dominance of 2D physics is explained by the fact that the meridional wavelength of the beams is much smaller than their zonal wavelength (which is set by the zonal wavelength of the ETW), and thus the terms associated with k in Eq. (11) are negligible relative to the y derivatives, inevitably yielding Eq. (12).

Fig. 5.
Fig. 5.

Snapshot of the difference in the meridional velocity (color) between the traditional and nontraditional simulations of an eastward-propagating Yanai wave taken 2 years into the simulations. Taking the difference between both runs allows us to highlight the beams that are generated when nontraditional effects are active. Ray paths corresponding to the characteristic curves of the 2D wave [Eq. (12)] with and without the TA are indicated by the gray and magenta lines, respectively. The separatrix (black line) where the nontraditional 2D wave equation changes from being hyperbolic to elliptic is also plotted. Nontraditional effects extend the separatrix poleward when the stratification is weak, allowing for 2D wave propagation past the inertial latitude of the wave, which in this case is .

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

Figure 5 shows how the characteristics in the nontraditional case deviate from the characteristics in the traditional case where and [see Eq. (31) for the expression of the slope of the characteristics]. In the traditional case, the characteristics are the classic trajectories of internal wave rays, with slopes , and near the inertial latitudes, the rays flatten out to the horizontal before reflection. Therefore, in the traditional case, the latitudes between the inertial latitudes can fill with rays with symmetrical vertical energy propagation, thus producing vertical modes (Philander 1978). In the nontraditional case, the symmetry is broken because of the poleward energy propagation past the inertial latitude in the abyss and the focusing reflection at the inertial latitude, thus producing the beams.

In the Rossby wave simulation in Fig. 3, beams are not clearly visible, but nonetheless features with small vertical and meridional scales develop in the deep. These features are confined to latitudes below the inertial latitude, which is quite close to the equator, given the low frequency of the wave.

b. Mean flows and higher harmonics

In our simulations, nontraditional effects significantly affect the spectral signature of the ETWs. This is illustrated in Fig. 6, which shows the spectrum for the westward-propagating Yanai wave in frequency–meridional wavenumber space, averaged over the last 1500 m of the water column, below the generation layer. In the traditional run, the spectral energy is confined to the frequency and meridional scale of the simulated wave. In the nontraditional run, however, the spectral signature is significantly broadened to higher meridional wavenumbers, reflecting the presence of the beams. In addition, higher harmonics in frequency are evident and could play a significant role in the propagation of wave energy to higher latitudes since these waves of higher frequency are not necessarily confined to the inertial latitude of the forced wave. Note, however, that despite the energy in the higher harmonics, internal wave beams at those frequencies are not clearly visible in the solutions shown in Fig. 3. It is probable that the spatial structure of the higher harmonics does not satisfy the dispersion relation for inertia–gravity waves and therefore does not generate freely propagating waves.

Fig. 6.
Fig. 6.

The log10 of the power spectral density of the meridional velocity as a function of frequency ω and meridional wavenumber l for the (left) traditional and (right) nontraditional runs of the westward-propagating Yanai wave, averaged over the last 1500 m of the domain. The frequency of the simulated wave is s−1 (white dashed line). In the nontraditional case, the small-scale features that develop broaden the spectral peak in wavenumber space and also result in higher harmonics in frequency (white dotted lines).

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

It is also interesting to notice the enhanced power at when nontraditional effects are active. This illustrates the generation of mean flows, which are clearly visible in a plot of the meridional velocity averaged over one wave period at the end of the simulation (see Fig. 7). Similar features are seen in plots of the time-averaged vertical and zonal velocities (not shown). Such mean flows must be generated through nonlinear wave–wave interactions that are enhanced near the wave beams that develop in the nontraditional case.

Fig. 7.
Fig. 7.

The meridional velocity from the simulation of a westward-propagating Yanai wave averaged over the last wave period. In the nontraditional case, mean flows are generated through wave–wave interactions and develop along the beams in the deep ocean.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

c. Abyssal mixing

The small-scale features that develop when act to enhance the shear locally, which then ultimately reduces the Richardson number and elevates the turbulent diffusivity in our simulations through the KPP mixing scheme. In particular, enhanced values of the diffusivity coefficient are observed in the proximity of the inertial latitude, where the beams are generated (see Fig. 8). The mixing region extends a bit poleward, where the beams get trapped due to low abyssal stratification and nontraditional effects, but also upward, within a few hundred meters from the bottom. As the frequency of the wave becomes lower, the mixing gets closer to the equator, tracking the inertial latitude.

Fig. 8.
Fig. 8.

Turbulent diffusivity in the simulations shown in Fig. 3 averaged zonally and over the last wave period (i.e., after 2 years of simulation). The sharp small-scale features in the nontraditional case ultimately increase the shear of the flow, leading to enhanced diffusivities in all the simulations that we considered. The largest values of the diffusivity are found near the inertial latitude (black dashed lines) and extend upward off the bottom and poleward of the inertial latitude.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

In all the simulations, the near-bottom enhancement of the diffusivity is zonally invariant (Fig. 9a) and sets up once the ETW reaches the bottom, having propagated down from the upper ocean. It persists as long as the forcing of the ETW is maintained, suggesting that no negative feedbacks act to weaken the mixing despite the enhancement of nonlinear effects due to (Fig. 9b). Conversely, over the duration of the simulation, the mixing region does not rapidly expand upward despite the induced weakening of stratification.

Fig. 9.
Fig. 9.

(a) Plan view of the diffusivity averaged over one wave period and the bottom 300 m 2 years into the simulation of a Rossby wave. (b) Corresponding Hovmöller diagram of the diffusivity averaged zonally. It takes approximately 6 months for the Rossby wave to reach the seafloor, after which enhanced mixing persists throughout the simulation. The inertial latitude of the wave is indicated by the black dashed lines.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

Figure 8 shows that the magnitude of the diffusivity and its spatial variability differ for each wave type. In particular, the westward inertia–gravity and eastward Yanai waves yield mixing regions with a large meridional extent. The Rossby wave, however, yields a mixing region with a higher vertical extent. In contrast, the westward Yanai wave induces lower diffusivities, suggesting that this wave type is less efficient at driving mixing.

d. Implications for diapycnal transport in the abyss

To assess the impact that enhanced diffusivities that develop in the abyss might have on the AMOC, we calculate the diabatic vertical velocity, defined as
e30
where , and is the turbulent diffusivity provided by the KPP mixing scheme in the model. We can then define a vertical volume transport quantity , corresponding to the meridionally integrated, diabatic vertical velocity multiplied by the zonal length scale of the mixing region Lmix. Because the mixing develops where the ETW reflects off the bottom, the zonal length scale of the mixing should correspond to the zonal length scale of the wave, which is at least one zonal wavelength . Therefore, we set such that
e31
which is expressed in Sverdrups.

Figure 10 shows Hovmöller diagrams of for each wave. For all waves, the turbulent diffusivities that develop in the model yield large-scale upwelling in the abyss rather than downwelling (except for some short localized events and on the upper edge of the mixing region). The upwelling results from the increase in magnitude of the turbulent buoyancy flux with height off the bottom due to the strengthening stratification (Fig. 2). While the diffusivity weakens with height, the variation in has a greater influence on the gradient in the buoyancy flux and hence determines the sign of wdia near the bottom. The vertical extent over which the upwelling occurs depends on the type of the wave and is strongly related to diffusivity patterns shown in Fig. 8: the Rossby wave has a larger vertical extent than the other waves because the enhanced diffusivity values develop higher in the water column, whereas the westward Yanai wave yields upwelling focused to the bottom of the domain. Note that the magnitude of the diapycnal transport that develops in the simulation is O(10) Sv per wavelength of ETW and is associated with ETWs of moderate strength, corresponding to the average state of the Pacific Ocean (as discussed in section 3) and suggesting that the process could contribute substantially to the AMOC.

Fig. 10.
Fig. 10.

Hovmöller diagram of the diapycnal volume transport per wavelength of ETW (Sv) in the last 500 m of the domain for the simulations shown in Fig. 3 in the nontraditional case. Significant diapycnal upwelling develops once the wave reaches the seafloor.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

5. Beam formation mechanism

In the numerical solutions described in the previous section, the mixing in the abyss develops around the inertial latitude, where the wave beams originate, highlighting how the beams are essential for enhancing the mixing through strengthening the shear and lowering the Richardson number of the flow. In this section, we explore the formation mechanism of the beams to understand the key parameters that set their strength. We start by showing that the beams are excited because of the nonzero vertical velocity of the downward-propagating ETWs at the seafloor when nontraditional effects are active. Then, we focus on the region around the inertial latitude of the wave to show that a vertical velocity there can drive critical reflection of the ETW off the flat seafloor, which is permitted when only, and that can lead to the intensification of the reflected wave and beam formation. We finish by investigating an analytical solution for the reflected wave accounting for viscosity to calculate an amplification factor that scales the kinetic energy (KE) of the beams to the KE of the downward-propagating ETW and determine its sensitivity to the properties of the ETW and the medium in which it propagates.

a. Beams generation through bottom reflection of ETWs

In section 2, we demonstrated that prevents the setup of a mode in the vertical direction because it is no longer possible to superimpose two waves with the same vertical wavenumber but opposite sign to respect the bottom boundary condition of no normal flow, which for a flat bottom corresponds to at the bottom. This symmetry breaking makes the reflection of ETWs off the bottom more complex and can lead to beam formation.

To illustrate that this process can generate the beams that we observe in the numerical simulations, we solved the 2D wave [Eq. (12)] subject to a nonzero vertical velocity at the bottom. More specifically, we set the vertical velocity at the bottom equal to , where H = 4500 m, to represent minus the vertical velocity of the downward-propagating ETW (thus, by adding the ETW to this solution, the total vertical velocity would be zero at the bottom). Thus, represents the meridional structure of the vertical velocity associated with a given wave. Because all the ETWs that we studied give rise to the beams, we deduce that any meridional structure could be used for . We solved the equation using second-order finite differences on a 400 × 400 point grid with m and km, given a frequency ω and the same stratification profile as the one used in the MITgcm simulations. Because of the singularities at the inertial latitude, we added Laplacian friction to the right-hand side of Eq. (12), as in H16.

Figure 11 shows the result for the case where we set constant for simplicity (other more complicated meridional structures were tried but essentially yielded the same results). With this configuration, we are able to reproduce the beams that we observe in the MITgcm simulations and that follow the characteristic curves of the 2D wave equation. Note that despite being apparently absent from Fig. 11, a larger-vertical-scale, upward-propagating wave is also generated in this 2D configuration, but this wave has different physics than the 3D solutions shown in Fig. 3. In particular, its amplitude is much lower than the beams’ amplitude, and therefore it is not visible in the figure.

Fig. 11.
Fig. 11.

The meridional velocity from the solution to (12) forced with with , corresponding to the frequency of the eastward Yanai wave in Fig. 3. The stratification profile used in the calculation is the same as that employed in the MITgcm simulations. Energy propagates from the bottom in beams that emanate from the inertial latitudes (indicated by the dashed vertical lines) and follow ray paths (magenta) corresponding to characteristic of Eq. (12).

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

This calculation confirms that it is the change in the bottom reflection process induced by nontraditional effects that gives rise to the beams and that it does not depend on the specific details of the ETW’s spatial structure. Our interpretation is that the reflection of an ETW off a flat-bottom boundary in the nontraditional case requires the generation of higher meridional modes that are ultimately governed by the 2D physics and thus amplified at the inertial latitude.

b. Critical reflection around the inertial latitude

To understand the physical mechanism that amplifies the beams, we “zoom” around the inertial latitude of the wave by considering nontraditional effects on an f plane and mimic variations in latitude by varying f. In this framework, Eq. (12) becomes
e32
The characteristic curves of Eq. (30) correspond to rays along which the wave energy propagates. They are defined by constant, where
e33
Because f and N are constant in this analysis, the slope of the characteristics is also constant.

Figure 12 shows the characteristic curves in the three limits where (i.e., the subinertial limit, corresponding to latitudes poleward of the inertial latitude), (i.e., the inertial limit, corresponding to the inertial latitude), and (i.e., the superinertial limit, corresponding to latitudes equatorward of the inertial latitude). At the inertial latitude, only the steep characteristic , which does not exist under the TA, supports a vertical velocity and downward energy propagation. Hence, the ETW that we force in our simulation can transmit energy along this characteristic when it is propagating down in the water column. Because the shallow characteristic is aligned with the seafloor, this leads to critical reflection, a singular limit where the wave energy accumulates and goes to infinity at the inertial latitude.

Fig. 12.
Fig. 12.

Schematic of the characteristics of Eq. (30) for the three types of waves: (left) superinertial, (center) inertial, and (right) subinertial. The schematic is drawn for the Northern Hemisphere, with north to the right. The corresponding group velocity vector is aligned with the characteristic curves, whereas the wave vector is perpendicular to them.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

We conclude that the beams form through this process since they are strongest near the inertial latitude and since they run parallel to the shallow characteristic , which is flat at the inertial latitude. The link between beam formation and classical critical reflection off topography can also be made in the traditional case in a configuration where the topography is steep enough that its slope and the slope of the characteristics can align. This is illustrated in Fig. 13, which shows the simulation of an eastward-propagating equatorial IGW with the MITgcm in the traditional case. The same domain, wave maker, and stratification profile as the ones used in the simulations shown in Fig. 3 have been used. In the figure, we see that beams of IGWs are generated locally, at the latitude where the slope of the topography is equal to the slope of characteristics. This analogy describes how nontraditional effects can induce similar turbulence over a flat seafloor as the one found over abrupt topography in the traditional case.

Fig. 13.
Fig. 13.

(a) Traditional simulation with the MITgcm of an equatorial eastward-propagating IGW in a domain with a topographic feature. In this case, critical reflection occurs and generates sharp beams. (b) The slope of the characteristics of the 2D wave equation with the TA [i.e., Eq. (31) with ] and the slope of the topography as a function of latitude. Critical reflection occurs where the characteristic and the topography are aligned.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

c. Analytical solution for the reflected wave accounting for viscosity

With viscosity, the amplification factor of the reflected wave under critical reflection is finite and can be calculated analytically. The amplitude of this reflected wave presumably sets the strength of the beams, therefore motivating its calculation. To do this, we solve the 2D, linear, quasi-hydrostatic, Boussinesq equations on an f plane, with constant stratification and vertical Laplacian friction and diffusion (where we assume a Prandtl number of 1):
e34
e35
e36
e37
e38
These can be combined into a single equation by introducingca streamfunction ψ, where and , yielding
e39
Assuming plane wave solutions of the form , with (δ and m being real quantities), Eq. (37) becomes
e40

This equation admits six solutions with roots () that we solved numerically. To derive these solutions, we chose an incident wave with a period of 30 days, and we vary f and set l using the formula for the steep characteristic: , with km, a vertical wavelength typical of the ETW in the deep. The solutions are shown in Fig. 14 for the traditional and nontraditional cases. The two roots associated with the inviscid theory are also shown. With viscosity, no vertical wavelength is equal to zero for any frequency displayed, illustrating that the singularity predicted by the inviscid theory for critical reflection in the nontraditional case is absent.

Fig. 14.
Fig. 14.

(left) Vertical wavelength and (right) decay rate of 2D waves on an f plane with (colored lines) and without (black crosses and circles) viscosity as a function of frequency. Viscosity prevents the vertical wavelength from going to zero at . Parameter is the imaginary part of the roots of Eq. (38) and the real part. Solid (dashed) lines correspond to height-decreasing (increasing) solutions. In the (top) nontraditional case, the symmetry of the solutions that we observe in the (bottom) traditional case is broken. To derive these solutions, we chose a vertical wavelength of 2 km for the incident wave and a period of 30 days. The stratification is set to s−1 and the viscosity coefficient to m2 s−1.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

As illustrated in Fig. 14, tracks the vertical wavenumber of the inviscid solution for the steep characteristic, while follows the vertical wavenumber of the inviscid solution for the shallow characteristic only when ω is not close to f and viscous effects are weak. Around the inertial latitude, when , the singular behavior on the shallow characteristic where is prevented by viscous effects, the inviscid dispersion relationship is not satisfied, and other roots come into play. In the traditional case, however, viscosity is not required to regularize the problem and satisfy the bottom boundary condition because and are equal and opposite since Eq. (37) is separable and does not yield a singularity at the inertial latitude.

With viscosity, waves can increase or decrease along the vertical direction of their propagation, depending on the signs of m and δ. However, the only physically relevant solutions are those where the waves decay in the direction of propagation. Therefore, is the only height-increasing solution (i.e., ) that we retain because it corresponds to the downward-propagating incident wave for our configuration. The solutions for decrease with height and are thus retained because they correspond to the potential reflected wave solutions that are excited during critical reflection:
e41
e42
where is the amplitude of the solution corresponding to ; the “inc” and “ref” subscripts correspond to the incident and reflected solutions. The amplitude of the incident wave is found by setting the downward flux of energy to a constant, like we did in section 3. In addition, two more conditions need to be taken into account at the bottom boundary: 1) the incident and reflected waves must cancel each other to respect the no normal flow boundary condition (i.e., ) and 2) a free-slip condition (i.e., ) or a no-slip condition (i.e., ). These conditions, used in conjunction with the polarization relations for the waves that can be derived from Eqs. (32)(36), yield a system of three equations with three unknowns that we solved numerically to find the coefficients , , and of the reflected wave based on the parameters of the incident wave and of the medium.

The KE and streamlines of the incident, reflected, and total solutions are shown in Fig. 15 for no-slip and free-slip bottom boundary conditions. The wave field consists of the incident wave with vertical wavenumber and group velocity along the steep characteristic, which radiates from the surface (indicated by the red line in the figure), and the corresponding reflected wave, which is confined to the shallow characteristic at the inertial latitude and subject to viscous decay as it is propagating up. This results in a focusing of the reflected wave, and that dominates the total solution in the abyss as the KE of the reflected wave is much larger than that of the incident wave. Interestingly, the solution does not change qualitatively whether a no-slip or a free-slip boundary condition is used.

Fig. 15.
Fig. 15.

Streamlines (contours) and KE (colors) of the (right) reflected, (center) incident, and (left) total wave field solutions to Eq. (37). (top) A free-slip boundary condition is used; (bottom) a no-slip boundary condition is used. The steep characteristic along which the incident wave propagates is shown in red in the center panels.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

The ratio of the phase-averaged KE between the incident wave and the reflected wave at the bottom is defined as
e43

In Fig. 16, we show the sensitivity of this ratio to different parameters: the latitude (through varying f ), the stratification, the viscosity, the period of the incident wave, and its vertical wavelength . For each panel, we fix four out of the five parameters (f, N, T, minc, ν) and vary the fifth one. Here, we use the notation for the vertical wavenumber of the incident wave. Note that except for ν, this requires adjusting the meridional wavenumber l to satisfy the dispersion relation of the incident wave. The sensitivity of the amplification factor to latitude is shown in the left panels of Fig. 16 and illustrates that the factor is maximal near the inertial latitude where . The variations of Eq. (41) with (f, N, T, minc, ν) at the inertial latitude are plotted in the right panels of Fig. 15. Only the solutions for a free-slip boundary condition are shown because setting a no-slip boundary condition at the bottom yields qualitatively the same results. Figure 16 shows that the amplification factor is larger when the stratification, the viscosity coefficient, the period of the incident wave, and its vertical wavenumber are lower. Note that the sensitivity, however, is not the same for all parameters; the vertical wavenumber and the stratification have a greater influence than the period of the wave and the viscosity on the amplification factor.

Fig. 16.
Fig. 16.

Sensitivity of the ratio in Eq. (41) of the KE of the reflected wave to the KE of the incident wave to different parameters: the period of the incident wave T, the stratification N, the viscosity coefficient ν, and the vertical wavenumber of the incident wave . (left) The variation of the KE ratio as a function of (a proxy for latitude) for three different parameter values; (right) the ratio evaluated at the inertial latitude for a range of parameter values (a dashed line with a slope of −1 is plotted for reference).

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

d. Impact of the local energy flux

The KE of the reflected wave and hence the beams at the inertial latitude is a function of both the amplification factor in Eq. (41) and the KE of the incident wave KEinc. The latter is set by the local energy flux at the inertial latitude coming from the downward-propagating ETW. The various ETW types have different meridional structures and, consequently, different local energy fluxes even if they have the same meridionally averaged downward flux of energy. Consequently, the variations in the strength of the beams and mixing with wave type seen in Figs. 3 and 8 could arise from the sensitivity of the amplification factor to both wave properties, as discussed above, and differences in the local energy flux for each wave type.

To illustrate this idea, we use the local energy flux evaluated at the inertial latitude in the abyss from the simulations of ETWs shown in Fig. 3, and we derive the corresponding KE of the incident wave using the vertical group velocity of the ETWs [i.e., , where is calculated using Eqs. (6)(10), and N is taken from the bottom value of the stratification profile used in the simulations]. For the f plane calculation described in the previous section, the parameter that varies for the different waves is the period. Therefore, using the amplification factor in Eq. (41) as a function of wave period, we can estimate the resulting KE in the reflected wave at the inertial latitude associated with the different wave type in the numerical simulations. The results are shown in the left panel of Fig. 17, while the right panel shows the corresponding results if we assume that the local energy flux is the same across wave types. By contrasting these two curves, it is possible to isolate the effects of the variable energy flux and the variations of the amplification factor associated with the wave period. It can be seen that the westward Yanai wave yields lower KE in the reflected wave than the other waves when the variations in the local energy flux are taken into account. This can explain why this wave type generates the weakest beams and the least amount of mixing in the abyss (e.g., Figs. 3, 8).

Fig. 17.
Fig. 17.

The KE of the reflected wave as a function of wave period calculated from Eq. (41) for the ETWs from our simulations. (a) Variations in the local energy flux are taken into consideration; (b) the same local energy flux for all wave types is assumed.

Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0197.1

6. Discussion and conclusions

The current paradigm for the diapycnal upwelling of the AMOC attributes the mixing necessary to drive diapycnal motions to internal waves breaking over rough topography (Armi 1978; Polzin et al. 1997; Waterhouse et al. 2014; Ferrari 2014). Recent studies have emphasized the role of mixing on sloping topography, such as the boundaries of ocean basins, in facilitating diapycnal upwelling in the deep (McDougall and Ferrari 2017; Callies and Ferrari 2018). Here, we illustrate the potential importance of ETWs in driving diapycnal upwelling in the abyssal equatorial oceans, a feature that is seen in inverse calculations of the global meridional overturning circulation (Lumpkin and Speer 2007), and we explain the mechanism by which ETWs generate near-bottom mixing.

Using numerical simulations and analytical theory, we have shown that nontraditional effects induce a fundamental change in the bottom reflection of the ETWs. In the weakly stratified abyssal equatorial ocean, the meridional and vertical structures of ETWs are shaped by absolute momentum surfaces, which are curved when . As a consequence, a simple vertical modal structure cannot form when an ETW reflects off a flat seafloor. Instead, other modes of oscillations are generated at the bottom boundary and give rise to sharp beams of IGWs formed through critical reflection at the inertial latitude of an ETW. The beams are characterized by strong shear and yield enhanced, bottom-intensified turbulent diffusivities localized around the inertial latitude of the wave. The diffusivities are persistent and zonally invariant for the spatially and temporally uniform wave maker used in our simulations. However, the variations in diffusivity do vary across ETW type. These variations are related to the sensitivity of critical reflection to the properties of the waves and of the medium in which they propagate, but also to the differences in the local energy flux of the ETW at the inertial latitude, which are determined by the particular meridional structure of each ETW type in the abyss.

While the diffusivities are bottom intensified, they induce diapycnal upwelling, not downwelling, because the stratification increases with height off the bottom, resulting in a convergent buoyancy flux. Integrated over the tropics and zonally over a typical wavelength of an ETW, we estimate that the mixing could yield of order 10 Sv of diapycnal transport in the few hundred meters above the seafloor and hence could contribute significantly to the diapycnal upwelling of the abyssal equatorial circulation inferred by inverse models. Our study is, however, potentially limited by the Richardson number–dependent mixing scheme that we used, which parameterizes the strength of the turbulent diffusivity. Having said that, the maximum value of the diffusivity employed in the mixing scheme is similar to estimates made from microstructure observations in the abyssal equatorial ocean (e.g., H16), suggesting that diffusivities in our simulations are not unrealistically large. Because the shape of the stratification profile in the abyss is crucial for the resulting upwelling, it is also necessary to investigate whether the feature shown in Fig. 2 is ubiquitous in the equatorial oceans.

The diffusivities that we see in our simulations are consistent with the localized mixing observed by H16 at 0.5°S, which corresponds to the inertial latitude of the Rossby wave in our simulations. However, H16 observed a mixing region ranging up to 700 m above the seafloor, while mixing patterns in our simulations extend up to 300 m above the seafloor. This discrepancy could be because H16 observed the mixing associated with a particular wave event, whereas we considered ETWs with energy fluxes consistent with the mean state of the equatorial ocean in our calculations.

In addition, the beams are also responsible for the generation of mean flows. This mean flow generation mechanism could influence the complex equatorial deep jets’ structure, whose origin is still debated (Luyten and Swallow 1976; Ménesguen et al. 2009; Ascani et al. 2015; Cravatte et al. 2017). Further work will investigate the link between nontraditional effects and the steady equatorial circulation.

In our simulations, we use an ETW wave maker that is zonally and temporally invariant and symmetric relative to the equator. In the real ocean, however, there is a large meridional asymmetry in the ETW field that can result in “shadow zones” where the wave energy does not converge (Schopf et al. 1981; Cane and Moore 1981). Similarly, zonal variations in the processes that generate ETWs, such as tropical instability waves (Eriksen and Richman 1988) or winds (Durland and Farrar 2012), can restrict the longitudinal extent of the ETW energy reaching the bottom (e.g., Ascani et al. 2010). Furthermore, these processes can vary on seasonal time scales and longer (e.g., associated with ENSO cycles). How these meridional, zonal, and temporal variations in the ETW wave field modulate the near-bottom mixing and the net diapycnal transport across the entire extent of the equatorial Pacific remains to be quantified. Future work will focus on running realistic simulations of the eastern equatorial Pacific with that capture ETW generation in the upper ocean by TIWs, winds, and so on to characterize and understand the spatial and temporal variability associated with critical reflection under a realistic ETW field.

Acknowledgments

This work was funded by the National Science Foundation Grant OCE-1260312. Thanks to T. Durland for his insightful comments and encouragements that have significantly improved the quality of this article. This work benefitted also from stimulating discussions with D. Moore, M. Fruman, J. Wenegrat, and L. Maas.

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