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    Snapshots of the vertical velocity as predicted by the Bell (1975b) theory for a combined steady and oscillatory flow, u = U + Ut cosΩtt, over a two-dimensional Gaussian ridge, h(x) = 50 exp[−(x/1 km)2]. The tidal flow speed is fixed at Ut = 2 cm s−1, with a varying steady flow speed: (a) U = 0, (b) U = 1 mm s−1, (c) U = 2 cm s−1 and (d) U = 4 cm s−1. (left) The full solution. (right) The solution split into the four wave bands identified herein: upstream beams, downstream beams, classical lee waves, and tidal lee waves. Only harmonics |n| ≤ 1 are shown.

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    The wavenumber spectrum of internal wave generation in the presence of a mean flow and barotropic tide for various parameter regimes: (a) zero frequency (nΩt = 0), (b) subinertial (nΩt < |f| < N), (c) normal internal tide wave band (|f| < nΩt < N), and (d) superbuoyancy (|f| < N < nΩt), where nΩt > 0. For the purposes of the plot, we set f = 1, N = 5, and nΩt = 0.5, 2, and 6 for (b), (c), and (d), respectively. The slope of lines of constant phase is shown in black, and the horizontal group velocity cgx is in blue (left axis), with the (approximate) nondimensionalized wave stress τ in gray (right axis). Generation is only permitted in (up to) four distinct wavenumber bands. These bands are, in order of increasing wavenumber, (i) downstream beams, (ii) upstream beams, (iii) lee waves, and (iv) tidal lee waves. (e) The vertical velocity field associated with each internal tide band generated at an isolated hill (for small excursion length and |f| < nΩt < N). The wavevector (k, −m) (slope of phase lines) is indicated by the black triangle. The direction of the energy flux (group velocity) is indicated by the blue arrow. Only upstream beams can flux energy upstream, against the mean flow, and only for small Doppler shifts [shaded region in (c)].

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    The total energy flux radiated by a Gaussian ridge h = h0 exp[−(k0x)2] of varying half-width 1/k0, where U = Ut (as pictured in Fig. 1c), Nt = 5, and ft = 0.7. (a) Energy flux for the combined mean and tidal generation calculation (solid lines) and the calculations made independently (dashed). The distribution of energy flux between harmonics as a function of excursion parameter in the (b) no-cutoff case and (c) combined/total problem. Energy fluxes are expressed relative to the maximum value given by the green line at the very top of (a). (d) The change in the distribution of energy flux (units: ρ0Ut2h02/2) between harmonics for the combined calculation as compared with summing the independent mean and tidal calculations.

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    Input model data from the ACCESS-OM2–01 0.1°-resolution global ocean–sea ice model (Kiss et al. 2020): (a) 5-day-averaged near-bottom velocities and (b) year-averaged near-bottom stratification (near bottom is the second model level from the bottom).

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    Isotropic tidal excursion parameter ε for the eight major tidal constituents; ε=k(Ut2+Vt2+2UtVtcosϕ)1/2/Ωt. The horizontal wavenumber is taken as k=(kN2+kS2)1/2, where kN are kS are the cutoff wavenumbers for the abyssal-hill topography. The actual excursion length may be larger or smaller depending on relative orientation of the tides in comparison with the abyssal hills. Tidal amplitudes are sourced from the TPX08 atlas (Egbert and Erofeeva 2002), and abyssal-hill topography is from Goff (2010).

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    Lee wave energy flux (mW m−2) radiated from abyssal hills: (a) energy flux at zero frequency (lee waves), ignoring the presence tides, as per previous calculations, (b) energy flux at zero frequency (lee waves), including the impact of the eight major tidal constituents, (c) the difference between (a) and (b), and (d) the lee wave suppression factor—i.e., the ratio of (b) to (a). The globally integrated energy flux in gigawatts is shown above (a) and (b).

  • View in gallery

    Globally integrated energy flux radiated from abyssal hills for each individual combination of tidal constituents. Frequencies are ordered by radiated flux, with the largest contributions at the bottom. (a) Individual fluxes on a logarithmic scale. (b) Cumulative sum from largest to smallest contribution. The solid gray line corresponds to the sum over all frequencies, and the dashed line is 95% of this value. For both (a) and (b) the fraction of the bar in a given color shows the proportion of the energy flux radiated as upstream (blue), lee (red), or downstream (yellow) waves.

  • View in gallery

    (a) Wavenumber spectra of the energy flux radiated from abyssal hills. Peak generation occurs at a wavelength of 10 km. (b) Cumulative sum of the energy flux from small to large wavenumber; 80% of the generation occurs between wavelengths of 2 and 40 km.

  • View in gallery

    Wave energy flux (mW m−2) radiated from abyssal hills for the eight largest frequency contributors, accounting for 95% of the total flux, ranked from largest [(a) M2] to eighth largest [(h) M2 + K1]. The K1 critical latitudes are shown by dashed red lines on (d).

  • View in gallery

    The K1 primary frequency wave energy flux (mW m−2) radiated from abyssal hills in the vicinity of Drake Passage, beyond the critical latitude.

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    The modulus of the wave stress (N m−2) generated at abyssal hills decomposed into the (a) free and (b) trapped wave components. Trapped waves are expected to dissipate in the flow that generated them, whereas free waves may propagate to the upper ocean. (c)–(f) The zonal and meridional components of the free and trapped wave stress in Drake Passage [indicated by the black box in (a) and (b)]. (g) The M2 barotropic tide magnitude in Drake Passage. (h) The geostrophic flow in Drake Passage. (i),(j) The trapped (lee) wave stress in the traditional calculation (ignoring tides) minus the trapped wave stress in the full calculation (including tides) for the zonal and meridional components. The trapped wave stress is suppressed by the addition of tides.

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    The nth-order Bessel functions of the first kind, Jn(ε), for n = 0, 1, 2, and 3.

  • View in gallery

    Sensitivity of results to the chosen 5-day analysis period: the globally integrated energy flux for twelve 5-day analysis periods (approximately one per calendar month over a year).

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Interdependence of Internal Tide and Lee Wave Generation at Abyssal Hills: Global Calculations

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  • 1 Research School of Earth Sciences, and ARC Centre of Excellence in Climate Extremes, Australian National University, Canberra, Australian Capital Territory, Australia
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Abstract

The generation of internal waves at abyssal hills has been proposed as an important source of bottom-intensified mixing and a sink of geostrophic momentum. Using the theory of Bell, previous authors have calculated either the generation of lee waves by geostrophic flow or the generation of the internal tide by the barotropic tide, but never both together. However, the Bell theory shows that the two are interdependent: that is, the presence of a barotropic tide modifies the generation of lee waves, and the presence of a geostrophic (time mean) flow modifies the generation of the internal tide. Here we extend the theory of Bell to incorporate multiple tidal constituents. Using this extended theory, we recalculate global wave fluxes of energy and momentum using the abyssal-hill spectra, model-derived abyssal ocean stratification and geostrophic flow estimates, and the TPX08 tidal velocities for the eight major constituents. The energy flux into lee waves is suppressed by 13%–19% as a result of the inclusion of tides. The generated wave flux is dominated by the principal lunar semidiurnal tide (M2), and its harmonics and combinations, with the strongest fluxes occurring along midocean ridges. The internal tide generation is strongly asymmetric because of Doppler shifting by the geostrophic abyssal flow, with 55%–63% of the wave energy flux (and stress) directed upstream, against the geostrophic flow. As a consequence, there is a net wave stress associated with generation of the internal tide that reaches magnitudes of 0.01–0.1 N m−2 in the vicinity of midocean ridges.

© 2020 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: Callum J. Shakespeare, callum.shakespeare@anu.edu.au

Abstract

The generation of internal waves at abyssal hills has been proposed as an important source of bottom-intensified mixing and a sink of geostrophic momentum. Using the theory of Bell, previous authors have calculated either the generation of lee waves by geostrophic flow or the generation of the internal tide by the barotropic tide, but never both together. However, the Bell theory shows that the two are interdependent: that is, the presence of a barotropic tide modifies the generation of lee waves, and the presence of a geostrophic (time mean) flow modifies the generation of the internal tide. Here we extend the theory of Bell to incorporate multiple tidal constituents. Using this extended theory, we recalculate global wave fluxes of energy and momentum using the abyssal-hill spectra, model-derived abyssal ocean stratification and geostrophic flow estimates, and the TPX08 tidal velocities for the eight major constituents. The energy flux into lee waves is suppressed by 13%–19% as a result of the inclusion of tides. The generated wave flux is dominated by the principal lunar semidiurnal tide (M2), and its harmonics and combinations, with the strongest fluxes occurring along midocean ridges. The internal tide generation is strongly asymmetric because of Doppler shifting by the geostrophic abyssal flow, with 55%–63% of the wave energy flux (and stress) directed upstream, against the geostrophic flow. As a consequence, there is a net wave stress associated with generation of the internal tide that reaches magnitudes of 0.01–0.1 N m−2 in the vicinity of midocean ridges.

© 2020 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: Callum J. Shakespeare, callum.shakespeare@anu.edu.au

1. Introduction

The generation of internal waves at the sea floor has the potential to significantly impact the circulation and stratification of the ocean. Internal waves are associated with a (pseudo)momentum flux extracted from the solid earth at generation, which forces the ocean flow when and where the wave breaks or otherwise dissipates (Bretherton 1969). The force acts in the direction of wave propagation, and can act to either accelerate or decelerate the local flow, depending on relative orientation. Furthermore, the breaking of the wave can drive diapycnal mixing, modifying the ocean stratification and supporting upwelling of deep water masses, and thus maintaining the global ocean overturning (Munk and Wunsch 1998). While the energetics of waves (mixing) has typically been treated separately to the wave stress (pseudomomentum flux), the two are intrinsically linked for any given wave: the wave stress τ is k/ω times the wave energy flux E, where k is the wavevector and ω is the intrinsic frequency (Bell 1975b).

The flow in the abyssal ocean responsible for the generation of topographic internal waves is composed of two main components: geostrophic flows at multiday–multimonth time scales, dominated by mesoscale eddies (e.g., Yang et al. 2018), and (higher frequency) barotropic tides at the diurnal and semidiurnal frequencies (e.g., Buijsman et al. 2014). Waves generated as a result of the geostrophic flow are called “lee waves”; they are steady (to the extent that the geostrophic flow is steady), are associated with an upstream-directed momentum flux, and yet carry this momentum and energy downstream. Waves generated as a result of the barotropic tide are called “internal tides”; they are unsteady and for smooth topography are generated symmetrically (with zero time-averaged momentum flux) as the tide sloshes back and forth (Shakespeare and Hogg 2019). A key difference between lee waves and internal tides is that lee waves can exist only in the presence of the generating geostrophic flow and will be forced to break above the topography if the geostrophic flow reduces sufficiently (a phenomena known as critical/inertial levels; Booker and Bretherton 1967; Xie and Vanneste 2015). By contrast, internal tides can propagate freely away from their generation site. Bell (1975b) presented a general theory that incorporates both of these generation mechanisms for an infinite-depth ocean [Khatiwala (2003) derived an equivalent finite-depth theory, but only for the cases of time-mean and oscillatory flow in isolation].

From an energy perspective, the largest flux into the ocean wave field is through the interaction of the barotropic tide with large-scale [O(100 km)] bathymetry (St. Laurent et al. 2002). However, smaller-scale [O(10 km)] waves—even if less energetic—are of significant interest because of their greater potential to dissipate locally, near the ocean bottom. The bathymetry at these small scales is dominated by “abyssal hills” that are unresolved in global bathymetric datasets, which are, at best, multikilometer in resolution—fortunately, however, global statistical representations of the abyssal-hill spectra do exist (Goff and Jordan 1988; Goff 1991, 2010; Goff and Arbic 2010). Many recent authors have combined these abyssal-hill spectra with the theory of Bell (1975b) to calculate the wave stress and/or energy flux. Melet et al. (2013) calculated the energy flux associated with M2 internal tide generation at abyssal hills, obtaining a globally integrated value of 0.08 TW (reduced to 0.03 TW when they empirically correct for supercritical topographic slopes). Naveira Garabato et al. (2013), Scott et al. (2011), Nikurashin and Ferrari (2010, 2011), Wright et al. (2014), and Yang et al. (2018) have all calculated the energy flux and/or the wave stress associated with lee wave generation at abyssal hills. These works have suggested that the energy flux into lee waves ranges from 0.2 to 0.85 TW globally, and that lee wave stresses could play a nonnegligible role in the momentum balance of especially energetic regions, such as the Southern Ocean. This latter effect has been demonstrated directly (Trossman et al. 2016) by the implementation of a parameterized lee wave stress in a global model using the Garner (2005) scheme.1

There have also been a significant number of observational studies exploring internal wave generation and dissipation at small scales, with a particular focus on lee wave generation in the Southern Ocean. In many of these studies (Sheen et al. 2013; Waterman et al. 2013, 2014; Cusack et al. 2017), microstructure measurements yield dissipation rates up to an order of magnitude smaller than the energy flux into lee waves predicted by Bell (1975b) theory. Many possible explanations are suggested for these apparent differences between observation and theory, including downstream propagation of lee waves out of the observational domain (e.g., Waterman et al. 2014), nonlinear effects including topographic blocking and supercritical topography (e.g., Nikurashin et al. 2014), and wave-to-mean energy transfers (e.g., Kunze and Lien 2019). Another possible explanation, to be investigated here, is the action of barotropic tides in suppressing the energy flux into lee waves.

Many parameterizations and calculations assume that the generation of the internal tide only occurs equatorward of the latitudes where the tidal frequency Ωt is equal to the inertial frequency f (the so-called critical latitudes: about 30° for the diurnal tides and 70° for the semidiurnal tides), and that generation only occurs where the tidal frequency is less than the deep-ocean buoyancy frequency, Ωt < N. However, these assumptions ignore two important effects. First, higher tidal harmonics can be generated at integer multiples of the primary tidal frequency, permitting generation poleward of the critical latitudes. The generation of these harmonics depends on a large tidal excursion distance (Utt, where Ut is the tidal speed and Ωt is the frequency) relative to the scale of the topography (Bell 1975a; Garrett and Kunze 2007). Harmonic generation is thus potentially less important in the deep ocean regions where tidal amplitudes are small. Nonetheless, it has been argued that tidal harmonics can be important for ocean mixing, at least in certain locations (Iwamae et al. 2009; Hibiya et al. 2017; Robertson et al. 2017). The second effect permitting internal tide generation outside the frequency band f < Ωt < N is Doppler shifting resulting from a geostrophic flow. Geostrophic flows can Doppler upshift the intrinsic frequency of the internal tide such that it is greater than the local inertial frequency or Doppler downshift the frequency such that is less than the local buoyancy frequency. Internal tides generated in these regimes are similar to lee waves in that they exist only by virtue of the geostrophic flow (they are “flow trapped” waves)—and therefore will break and dissipate when and where this flow vanishes.2 Nakamura et al. (2000) consider the role of the Doppler shift in permitting internal tide generation by the K1 barotropic tide in the Kuril Straits (at 46°N, well above the critical latitude), and thereby driving vigorous mixing. Here we will calculate global energy fluxes and wave stresses associated both with tidal harmonics and these flow-trapped internal tides.

The impact of geostrophic flow on internal tide generation has also recently been considered by Shakespeare and Hogg (2019) in an idealized, wave-resolving Southern Ocean model. They showed that the Doppler shift due to the presence of a geostrophic flow causes an asymmetry in the generation of the internal tide and a net upstream momentum flux. As a result, internal tides—in addition to lee waves (e.g., Naveira Garabato et al. 2013)—may provide a net contribution to the momentum budget of the ocean. Here we extend the work of Shakespeare and Hogg (2019) by evaluating this effect over the global ocean.

The paper is arranged as follows. In section 2 we present a summary of the Bell (1975b) theory that elucidates the coupled nature of lee wave and internal tide generation at the scales of abyssal hills. It is shown that 1) barotropic tides suppress lee wave generation and 2) geostrophic flow modifies internal tide generation to occur in three distinct wavenumber bands. While these results follow directly from the theory of Bell (1975b) they have been ignored by subsequent authors in their calculations of global wave fluxes from abyssal hills (e.g., Naveira Garabato et al. 2013; Scott et al. 2011; Nikurashin and Ferrari 2010, 2011; Wright et al. 2014; Yang et al. 2018; Melet et al. 2013). In section 3 we detail the methods used to correctly calculate global energy fluxes and wave stresses at abyssal-hill topography, accounting for the coupled nature of the generation. To make these calculations, it is necessary to extend the Bell (1975b) theory to encompass multiple tidal constituents, as detailed in appendix A. The results of the global calculations are given in section 4. Last, in section 5, we consider the implications of these calculations for wave parameterizations and the ocean circulation.

2. Review of Bell (1975b)

Bell (1975b) presents expressions for the time-averaged stress exerted on topography and the energy flux associated with the generation of internal waves at the ocean bottom, by tides (time oscillating, with amplitude Ut) and geostrophic (time mean, with amplitude U) barotropic flows. The theory is developed for an infinite-depth fluid under the small-amplitude approximation. The background state has stratification N2 with flow u = U + Ut cosΩtt, where Ωt is the tidal forcing frequency. Explicit solutions for wave generation at a two-dimensional ridge, calculated from the Bell (1975b) theory, are shown in Fig. 1 for reference. The pseudomomentum flux of the generated waves, which is equal and opposite to the time-averaged stress exerted on the topography, is given by Bell (1975b) as
τ=ρ04π2k|k|P(k,l)n=(N2ωn2)(ωn2f2)×sgn(ωn)Jn2(|kUt|Ωt)dkdl,
where the Doppler-shifted intrinsic frequency is ωn = nΩtkU, N is the buoyancy frequency, f is the local Coriolis parameter, k = (k, l) is the horizontal wavenumber, P is the power spectrum of the topography, Jn is the nth-order Bessel function of the first kind, and n = …,−2, −1, 0, 1, 2, … is the harmonic number. Previous authors, including Bell (1975b), have considered this equation in two separate limits: 1) internal tide generation in the absence of a mean flow, where the time-mean stress is zero, and 2) (lee) wave generation in the absence of a tide, where the stress is nonzero. (The energy flux is nonzero in both cases.) In reality, there is both nonnegligible tidal and mean flow nearly everywhere in the ocean. The rigorous distinction between internal tides (time varying) and lee waves (steady) breaks down in this general case, since everything is being advected back and forth by the barotropic tide. However, we can use the frequency observed in a frame moving with the barotropic tide (but not the mean flow) to separate the motions: in such a frame, the observed or “co-tidal” frequency is ωn = nΩt. In this frame, classical lee waves are steady motions, identified by the n = 0 harmonic, and internal tides are time-varying motions with frequencies that are integer multiples of the primary tidal frequency, |n| ≥ 1.
Fig. 1.
Fig. 1.

Snapshots of the vertical velocity as predicted by the Bell (1975b) theory for a combined steady and oscillatory flow, u = U + Ut cosΩtt, over a two-dimensional Gaussian ridge, h(x) = 50 exp[−(x/1 km)2]. The tidal flow speed is fixed at Ut = 2 cm s−1, with a varying steady flow speed: (a) U = 0, (b) U = 1 mm s−1, (c) U = 2 cm s−1 and (d) U = 4 cm s−1. (left) The full solution. (right) The solution split into the four wave bands identified herein: upstream beams, downstream beams, classical lee waves, and tidal lee waves. Only harmonics |n| ≤ 1 are shown.

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

Using this generic definition of lee waves, the lee wave stress from (1) is
τlee=ρ04π2k|k|P(k,l)[N2(kU)2][(kU)2f2]×sgn(kU)J02(|kUt|Ωt)dkdl.
An additional Bessel function factor, dependent on the tidal excursion distance parameter ε = |kUt|/Ωt, appears in (2) as compared with the usual expression (e.g., Naveira Garabato et al. 2013; Scott et al. 2011; Nikurashin and Ferrari 2010, 2011; Wright et al. 2014; Yang et al. 2018) used to calculate lee wave fluxes, which assumes |Ut| = 0. The parameter ε quantifies the distance a parcel is advected by the tide over a tidal cycle (~2π|Ut|/Ωt), relative to the scale of the topography (~2π/|k|). If this value is small, then the Bessel function J0(ε) → 1 (see the plot of Bessel functions Fig. A1 in appendix A) and the effect of the tide is negligible. The expression (2) also shows that lee waves are only generated by small-scale topographies, where kU > f. Thus the excursion distance parameter for a situation that generates lee waves satisfies εf|Ut|/(Ωt|U|). Since Ωt ~ f over much of the ocean, the excursion distance parameter essentially depends on the ratio of the tidal to the mean flow speeds. Barotropic tidal speeds in the deep ocean are of order 1 cm s−1 and mean flow speeds are of order 1–10 cm s−1. Therefore, εmin ~ 0.1–1 over most of the ocean—that is, sufficiently large to affect lee wave generation. Now, since J0(ε)2 ≤ 1, it follows that the lee wave flux is reduced (suppressed) by the presence of the tide. This suppression is reinforced by each additional barotropic tidal constituent, as we show by an extension to the Bell (1975b) theory (see appendix A).
The second major result from the Bell (1975b) theory is that mean flow also affects the generation of the internal tide. The internal tide stress from (1) is simply
τtide=ρ02π2k|k|P(k,l)n=1(N2ωn2)(ωn2f2)×sgn(ωn)Jn2(|kUt|Ωt)dkdl,
where we have combined the positive and negative harmonic numbers n into a single summation over all n > 0 (this formulation is possible because switching n → −n in the integrand of (3) is the same as switching k → −k; that is, the integral over all (plus and minus) k is the same for ±n.) If the mean flow is zero, then the intrinsic frequency ωn is independent of k and tidal generation is symmetric (i.e., equal energy and opposite momentum for ±k.). Therefore, there is zero net stress, as noted by Bell (1975b). However, if the mean flow is nonzero, the intrinsic frequency ωn = nΩtkU and thus tidal generation is asymmetric. Therefore, a net momentum flux exists (τtide ≠ 0). Since the asymmetry relies on the difference of the intrinsic frequencies for positive and negative wavenumbers, nΩtkU versus nΩt + kU, it is only significant when the Doppler shift |kU| is nonnegligible, or ε¯=|kU|/Ωt0. By analogy to the tidal excursion parameter above, this “mean excursion distance” parameter ε¯ is the distance the mean flow advects a parcel over one tidal cycle relative to the topographic length scale, and therefore quantifies the extent to which the mean flow modifies tidal generation. For a mean flow of 1 cm s−1, this parameter indicates that the effect of the mean flow on (semidiurnal) tidal generation is important for horizontal wavelengths of ~5 km and smaller (the scales at which ocean bathymetry is dominated by abyssal hills). Shakespeare and Hogg (2019) investigated a special limit of (3) where both the mean and tidal excursion parameters are small (ε ≪ 1 and ε¯1) in which case the net stress scales as the product of the mean flow velocity and the wavenumber to the fourth power [see their (21)].
Internal tides are only generated at frequencies and wavenumbers where the argument inside the square root in (3) is positive. In the absence of a mean flow, this band is simply f < nΩt < N, independent of wavenumber. However, in the presence of a mean flow, internal tides can be generated at both subinertial and superbuoyancy frequencies, but only over specific bands of wavenumber. Three distinct wavenumber bands of internal tides may be identified (Figs. 2b–d) in addition to the permitted wavenumber band for zero-frequency lee waves (Fig. 2a). The explicit solutions to the Bell (1975b) theory shown in Fig. 1 have also been split into these separate wavenumber bands (see the right-hand panel). The slope of lines of constant wave phase from (A12) in appendix A,
|k|m=sgn(nΩtkU)(nΩtkU)2f2N2(nΩtkU)2,
is shown as a function of the scaled wavenumber kU in Fig. 2. Note that here we use a Fourier (plane wave) definition of exp[−i(kx + lymzωt)] throughout the paper. The sign in (4) is m > 0 for waves where ω = nΩtkU > 0 (the left bands in Fig. 2), and m < 0 for waves where ω = nΩtkU < 0 (the right bands in Fig. 2). Since ω and m always have the same sign, the vertical phase speed is negative (cpz=ω/m<0; with the chosen Fourier definition) and the vertical group speed is positive (cgz=mω>0; with the chosen Fourier definition) for all wavenumber bands, as is required for valid wave solutions. Figure 2 also shows the horizontal group speed (cgx=kω), the sign of which indicates whether waves travel upstream (cgx<0) or downstream (cgx>0). First, considering waves with kU < 0, there is only one permitted band of wavenumbers that generate internal waves which we term the “upstream beam”: |f| − nΩt < |kU| < NnΩt. Upstream beam generation always occurs unless the cotidal frequency exceeds the buoyancy frequency, nΩt > N (Fig. 2d). In particular, upstream beam generation occurs when the cotidal frequency is subinertial, nΩt < |f| (Fig. 2b). In the limit of weak mean flow, the upstream beam collapses to a “normal” internal tide beam that is associated with both upstream energy and momentum fluxes (shaded region of Fig. 2c). However, for larger Doppler shifts, while the momentum flux remains directed upstream, the wave energy is swept downstream by the mean flow (cgx<0). Second, considering waves with kU > 0, there are two distinct wavenumber bands for which generation occurs. The first, at lower wavenumber, is the “downstream beam,” nΩtN < kU < nΩt − |f|, which has a downstream-directed momentum flux (it is the only wavenumber band to have this property). The downstream beam always occurs unless the cotidal frequency is subinertial, nΩt < |f| (Fig. 2b). In particular, downstream beam generation occurs when the cotidal frequency is superbuoyancy, nΩt > N (Fig. 2d). In the limit of zero mean flow, the downstream beam collapses to a normal internal tide beam, equal and opposite to the upstream beam. The second kU > 0 wavenumber band, at higher wavenumber, corresponds to “tidal lee waves”: |f| + nΩt < kU < N + nΩt. Unlike the downstream beam, the tidal lee waves’ lines of constant phase slope upward into the oncoming flow (see Figs. 1d and 2e) and they have an upstream-directed wave stress. Hence, these tidal-frequency lee waves are very similar to normal zero-frequency lee waves, except they occur for a wavenumber band shifted up by nΩt/|U| (but of the same width). Thus, while barotropic tides suppress classical lee wave generation, they also generate tidal lee waves at higher wavenumber and (higher) tidal frequencies. The distinction between “upstream beams” and “tidal lee waves” identified here breaks down in the limit of vanishingly small tidal frequency (and hence an almost-steady flow), where both bands collapse to traditional lee waves (i.e., Fig. 2b collapses to Fig. 2a as Ωt → 0).
Fig. 2.
Fig. 2.

The wavenumber spectrum of internal wave generation in the presence of a mean flow and barotropic tide for various parameter regimes: (a) zero frequency (nΩt = 0), (b) subinertial (nΩt < |f| < N), (c) normal internal tide wave band (|f| < nΩt < N), and (d) superbuoyancy (|f| < N < nΩt), where nΩt > 0. For the purposes of the plot, we set f = 1, N = 5, and nΩt = 0.5, 2, and 6 for (b), (c), and (d), respectively. The slope of lines of constant phase is shown in black, and the horizontal group velocity cgx is in blue (left axis), with the (approximate) nondimensionalized wave stress τ in gray (right axis). Generation is only permitted in (up to) four distinct wavenumber bands. These bands are, in order of increasing wavenumber, (i) downstream beams, (ii) upstream beams, (iii) lee waves, and (iv) tidal lee waves. (e) The vertical velocity field associated with each internal tide band generated at an isolated hill (for small excursion length and |f| < nΩt < N). The wavevector (k, −m) (slope of phase lines) is indicated by the black triangle. The direction of the energy flux (group velocity) is indicated by the blue arrow. Only upstream beams can flux energy upstream, against the mean flow, and only for small Doppler shifts [shaded region in (c)].

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

The wavenumber and frequency bands described above may be separated into two categories: flow-trapped waves and free beams. Free beams correspond to the normal upstream and downstream internal tide beams generated at frequencies |f| < nΩt < N. These waves are not reliant on the continued presence of the geostrophic flow for their existence, but can propagate freely away from their generation site (although they may still encounter critical levels because of, for example, their interaction with strong upper-ocean flows; e.g., Shakespeare and Hogg 2019). In contrast, flow-trapped waves exist only by virtue of the Doppler shift associated with a near-bottom geostrophic flow. They include lee waves, tidal lee waves, upstream tidal beams where nΩt < |f|, and downstream tidal beams where nΩt > N. These waves are all similar to lee waves in that they will be prone to dissipation if the geostrophic flow decays with height, and thus are more likely to cause mixing in the abyss near topography as compared with free beams. In the following sections, we will calculate individually the amount of energy flux and wave stresses associated with trapped waves versus free beams.

The magnitude and frequency distribution of the net energy flux

The coupled nature of internal wave generation by combined mean and tidal flows allows that the energy flux in the combined problem is different to that obtained by individually calculating the wave energy flux due to tides and mean flows in isolation. Here we seek to understand the mechanism behind this modification to the energy flux.

The total wave energy flux is, from (1),
E=ρ04π21|k|P(k,l)×n=(N2ωn2)(ωn2f2)|ωn|Jn2(ε)dkdl.
First, suppose we are in a regime where there are no frequency cutoffs for internal wave generation; for example, the hydrostatic and nonrotating regime: f → 0 and N → ∞. In such a regime f ≪ |ωn| ≪ N for all harmonics n, and (5) may be simplified to
Elimit=ρ04π2N|k|P(k,l)n=ωn2Jn2(|kUt|Ωt)dkdl.
The sum over n in (6) may be directly evaluated using known properties of Bessel functions3 to yield
Elimit=ρ04π2N|k|P(k,l)[(kUt)22+(kU)2]dkdl.
Equation (7) implies that the total wave energy flux in this “no-cutoff limit” is independent of whether the source is a tide or a mean flow; the contributions from each are additive and only depend on the kinetic energy in each (the factor of 1/2 in the tidal term is because the average tide kinetic energy is Ut2/2 over a tidal cycle). An immediate corollary is that any change to the net energy flux in the combined mean and tidal wave generation problem, as compared with two considered independently, must arise from the presence of the wave frequency cutoffs—at either the low end (f) or the high (N) end.

Figure 3a shows the net energy flux for the simple case of an isolated Gaussian ridge of varying width, subject to combined mean and tidal flows of equal strength (U = Ut). The solid curves show the energy fluxes obtained by solving the combined problem and the dashed curves those obtained by considering the mean and tidal flows independently. For small excursion parameters (ε < 0.1; wide ridges) there is no difference between these two approaches. For intermediate excursion parameters (0.1 < ε < 1) there is relatively more energy flux in the combined problem. For large excursion parameters (ε > 1; narrow ridges) there is significantly less energy flux in the combined problem. There are two main factors that give rise to these differences.

  1. The first factor is the value of the tidal excursion parameter whose impact is seen in the no-cutoff limit. Figure 3b shows that in this limit, wave energy flux is increasingly contained in higher tidal harmonics with increasing tidal excursion (because a tide-following parcel travels over a hill an increasing number of times in a tidal cycle). Thus, in the no-cutoff limit, an increasing excursion parameter redistributes energy flux from lower to higher frequencies without changing the net amount. However, in reality it is not possible to continuously shift generation to higher harmonics because eventually these frequencies fall outside the allowable frequencies of the internal waves; that is, a point is reached where ω > N. Ignoring the mean flow, this transition occurs at a harmonic number of nNt (shown by a dashed line on Figs. 3b–d). Any waves that would occur beyond this harmonic number in the no-cutoff limit cannot be generated in the physical problem, thus reducing the total energy flux. Figure 3c shows that the energy flux in the full problem tapers to zero for harmonics near this number.

  2. The second important factor in determining the net energy flux is the impact of the mean flow in Doppler shifting internal waves. On average, these Doppler shifts redistribute wave generation from lower to higher intrinsic frequencies [e.g., the energy flux (5) scales with intrinsic frequency |ωn|]. This redistribution is because a higher intrinsic frequency implies a steeper wave slope and thus a larger energy flux.

Fig. 3.
Fig. 3.

The total energy flux radiated by a Gaussian ridge h = h0 exp[−(k0x)2] of varying half-width 1/k0, where U = Ut (as pictured in Fig. 1c), Nt = 5, and ft = 0.7. (a) Energy flux for the combined mean and tidal generation calculation (solid lines) and the calculations made independently (dashed). The distribution of energy flux between harmonics as a function of excursion parameter in the (b) no-cutoff case and (c) combined/total problem. Energy fluxes are expressed relative to the maximum value given by the green line at the very top of (a). (d) The change in the distribution of energy flux (units: ρ0Ut2h02/2) between harmonics for the combined calculation as compared with summing the independent mean and tidal calculations.

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

The difference between the combined and independent calculations is shown in Fig. 3d as a function of harmonic number. At intermediate excursion parameters the net increase in energy in the combined calculation is associated with the primary n = 1 frequency. In this regime, the mean flow is too small to generate lee waves with any significant amplitude, but it is large enough to cause a significant Doppler shift of the internal tide generation, thereby enhancing the energy flux in the internal tide (as per factor ii above). At large excursion parameters, energy flux is naturally redistributed to higher frequencies by the tide (factor i). The additional impact of Doppler shifting by the mean flow (factor ii) further enhances this effect, resulting in more energy flux being shifted above frequency N in the combined calculation, as compared with the independent one. Thus the net energy flux in the combined calculation is smaller at large excursion parameter. In fact, the net energy flux is smaller even than that predicted for the mean flow alone; that is, adding a tidal flow to an existing mean flow reduces the net energy flux from the hill at large excursion parameter.

3. Methods

a. Data sources

Calculation of the wave energy and momentum fluxes requires knowledge of the global near-bottom stratification, near-bottom “mean” flow, barotropic tides and sub-10-km-scale bathymetry. We obtain the stratification and mean flow (Fig. 4) from a normal-year-forced configuration of the 0.1° resolution ACCESS-OM2–01 global ocean–sea ice model (Kiss et al. 2020). The stratification has little temporal variation and here we use the annually averaged value in the second model level from the bottom. The deep-ocean flow, although steady in a statistical sense, varies locally on time scales from weeks to months because of the abyssal component of mesoscale eddies. Consistent with previous work (Naveira Garabato et al. 2013; Scott et al. 2011) here we use 5-day-averaged velocities in the second model level from the bottom. Owing to the computational intensiveness of the calculation, as detailed below, we only present global results for a single 5-day-averaged period (from 1 to 5 January of the normal-year-forced model). However, in appendix B we repeat the calculation for the most energetic wave frequencies for one 5-day period in every month for one year. We find that the magnitude of the fluxes vary by only a few percent over the year, implying that the global results for a single 5-day period are representative of the time-mean values.

Fig. 4.
Fig. 4.

Input model data from the ACCESS-OM2–01 0.1°-resolution global ocean–sea ice model (Kiss et al. 2020): (a) 5-day-averaged near-bottom velocities and (b) year-averaged near-bottom stratification (near bottom is the second model level from the bottom).

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

We consider the eight principal barotropic tidal constituents (four semidiurnal: M2, S2, N2, and K2; four diurnal: K1, O1, P1, and Q1) using global tidal velocities from the TPX08-atlas published by Oregon State University (Egbert and Erofeeva 2002).

The abyssal-hill bathymetry data used here are a spectral formulation from Goff (2010). The topographic power spectrum is given as
P(k,l)=4πνh2kNkS[|k|2kS2cos2(θθ0)+|k|2kN2sin2(θθ0)+1](ν+1),
where k = |k|(cosθ, sinθ) is the wavenumber, h is the root-mean-square height of topography, ν quantifies the high-wavenumber roll-off, θ0 is the strike angle of topography and kS and kN are the transition/cutoff wavenumbers parallel and perpendicular, respectively, to the strike angle. The Goff (2010) dataset provides maps of these five topographic parameters as a function of geographical position with 1/15° resolution.

b. Calculation

The general expression for the wave stress in the presence of multiple tidal constituents c with arbitrary phase—that is, a background flow of
u¯=U+c[Uccos(Ωct),Vccos(Ωct+ϕc)]
—is derived in appendix A as
τ=ncρ04π2kP(k,l)(N2ω2)(ω2f2)×sgn(ω)cJnc2(εc)d|k|dθ,
written here in angular coordinates where (k, l) = |k| (cosθ, sinθ). The summation notation in (10) is shorthand for the sum over all possible harmonic numbers nc for each tidal constituent c:
nc=nM2=nS2=nN2=nK2=nK1=nP1=nO1=nQ1=.
Waves are generated at an infinite number of intrinsic frequencies,
ω=kU+cncΩc,
corresponding to all possible combinations of harmonic numbers nc. The tidal excursion parameter in (10) is
εc=k2Uc2+l2Vc2+2klUcVccosϕcΩc.
The energy flux corresponding to (10) is
E=ncρ04π2P(k,l)(N2ω2)(ω2f2)×|ω|cJnc2(εc)d|k|dθ.
The product of Bessel functions Jnc2(εc) in the above equations [(10) and (14)] causes the suppression of wave fluxes relative to the case without (other) tidal constituents present. For example, for lee waves (corresponding to nc = 0 for all c) the “suppression factor” is cJ02(εc)1. Since the zeroth order Bessel function decreases with increasing excursion parameter εc (for small εc; see the plot of Bessel functions Fig. A1 in appendix A), the amplitude of wave fluxes are suppressed by each additional tidal constituent, if the excursion parameter for those constituents is significantly greater than zero.

The excursion parameter (13) depends on the relative phase of the zonal and meridional components of the tide. For anisotropic topography, this phase dependence means that the value of the integral in (14) further depends on the orientation of the tidal ellipse relative to the steepest slopes of the topography. Figure 5 displays the excursion parameter for each of the 8 major tidal constituents for a wavenumber of k=(kN2+kS2)1/2 where kN are kS are the cutoff wavenumbers for the abyssal-hill topography at a given location. These wavenumbers give as estimate of the expected scales for internal wave generation at abyssal hills, although generation may still occur at smaller scales (and larger excursion parameters). Nonetheless, M2 is clearly the dominant tidal constituent with excursion parameters of 0.1–0.4 over large regions of the ocean. The S2, O1, and K1 constituents are also nonnegligible (~0.1) in many regions.

Fig. 5.
Fig. 5.

Isotropic tidal excursion parameter ε for the eight major tidal constituents; ε=k(Ut2+Vt2+2UtVtcosϕ)1/2/Ωt. The horizontal wavenumber is taken as k=(kN2+kS2)1/2, where kN are kS are the cutoff wavenumbers for the abyssal-hill topography. The actual excursion length may be larger or smaller depending on relative orientation of the tides in comparison with the abyssal hills. Tidal amplitudes are sourced from the TPX08 atlas (Egbert and Erofeeva 2002), and abyssal-hill topography is from Goff (2010).

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

As described above, the sums in the wave energy flux/stress equations [(10) and (14)] are over all possible combinations of harmonics of each tidal constituent. It is evidently impossible to compute all such combinations. Fortunately, since excursion parameters are small (ε ≪ 1) it is straightforward to expand the Bessel functions in (10) and (14) in ε and determine the expected relative magnitudes of various harmonic combinations. As a starting point, suppose there are only two nonnegligible tidal constituents in a given region (e.g., M2 and S2)—in this case the total energy flux from (14) is
E=ρ04π21|k|P(k,l)n1=n2=|ωn1,n2|(N2ωn1,n22)(ωn1,n22f2)Jn12(ε1)Jn22(ε2)dkdl,
where the intrinsic frequency is ωn1,n2 = −kU + n1Ω1 + n2Ω2. For small ε, the Bessel functions in (15) may be expanded as
J0(ε)2=1ε22+3ε432+O(ε6),J1(ε)2=ε24ε416+O(ε6),andJ2(ε)2=ε464+O(ε6).
Let us define the energy flux for a given harmonic combination as
En1,n2=ρ04π21|k|P(k,l)|ωn1,n2|×(N2ωn1,n22)(ωn1,n22f2)Bn1,n2dkdl,
where Bn1,n2 is the amplitude factor resulting from the Bessel functions, and positive and negative frequencies [±(n1, n2)] have been collapsed into a single component. For each harmonic combination, the amplitude factors for small excursion parameter are
B0,0=1ε12+ε222+ε12ε2243(ε14+ε24)32+O(ε6)=1B1,0B0,1B2,0B0,2B1,1B1,1+O(ε6),
B1,0=ε122ε12ε224ε148+O(ε6),
B1,1=B1,1=ε12ε228+O(ε6),and
B2,0=ε1432+O(ε6).
and similarly for B0,1 and B0,2. Higher harmonic combinations are O(ε6) or smaller, as would be harmonics involving additional (three or more) tidal constituents. The existence of finite amplitude factors at higher frequencies leads to an equivalent reduction in the lee wave (0, 0) amplitude factor [e.g., as shown by (18a)]—this is the lee wave suppression effect noted previously. In the no-cutoff limit introduced in the previous section—which equates to ignoring the square root factor in (15)(18) shows that the effect of increasing excursion parameter is merely a transfer between different frequencies without changing the total energy flux. The list (18) gives the fundamental set of harmonics that are expected in a small excursion parameter regime: from largest to smallest amplitude factor these are lee waves (order 1), primary frequencies of each tide (order εi2/2), sums and differences of primary frequencies (order εi2εj2/8), and secondary frequencies of each tide (order εi4/32). Thus, in the most of the ocean we expect wave generation at abyssal hills to be dominated by lee waves, primary frequency internal tides, the secondary frequency M2 tide (since this constituent has an excursion parameter that is ~4 times as large as any other; see Fig. 5), and combinations of the primary M2 frequency with other primary frequencies.

Nonetheless, to ensure that no important frequencies are omitted, here we compute a slightly larger set of frequencies: zero frequency (lee waves), the primary and harmonic (up to n = 8) frequencies of each constituent, sums and differences of any two primary frequencies (e.g., M2 + K1, M2 − S2), and, sums and differences of any primary frequency with any secondary or tertiary harmonic (e.g., 3M2 − K1, K1 + 2O1). In total these choices equate to 233 individual frequencies ranging from 0 at the low end, to the 14-day spring-neap frequency (e.g., M2 − S2), to nearly hourly at the high end (e.g., 8M2). Very low frequencies such as M2 − S2 are problematic for the analysis since the assumption of time scale separation of “fast waves” propagating in a steady geostrophic flow breaks down (i.e., the geostrophic flow will not be steady on a 14-day time scale). However, we find in section 4 that the energy in these low frequencies is negligible and that only a small subset of frequencies identified in our asymptotic analysis above contribute substantial amplitude.

Equations (10) and (14) are integrated over the permitted wavenumber bands using flow, stratification and topography data on a global 0.1° grid to produce maps of the wave stress and energy flux for each frequency. The sums are carried out separately for the lee waves, and each internal tide band (upstream beam, downstream beam, tidal lee waves; see Fig. 2). The numerical summation is done with 300 logarithmically spaced steps in wavenumber |k| from 10−6 to 10−1 m−1, and with 100 evenly spaced steps in angle θ from 0 to 2π.

4. Results

Figure 6 shows a direct comparison between the lee wave energy flux predicted from the full calculations described above, incorporating all tidal constituents and geostrophic flow, and the methodology of previous calculations of considering only the geostrophic flow. Inclusion of the tides results in a 15% suppression of the calculated lee wave energy flux, from a globally integrated value of 69.8 GW to one of 59.2 GW. There is significant spatial variation in the magnitude of the suppression, as shown by the difference (Fig. 6c) and ratio (Fig. 6d) of the suppressed and unsuppressed fluxes. In regions where the M2 tide is large (e.g., along the mid-Atlantic and Indian ridges; see Fig. 5) there is close to 100% suppression of lee waves. The globally integrated energy flux obtained here for the calculation ignoring tides is about one-third of the 200 GW calculated by Nikurashin and Ferrari (2011) for the same problem. As shown in our sensitivity tests in appendix B, differences of this magnitude can be explained by an ~50% difference in the strength of the geostrophic flows in numerical model-derived datasets (which differ between studies). Nonetheless, it is clear that the tides do reduce the lee wave energy flux radiated from abyssal hills, including in the Southern Ocean.

Fig. 6.
Fig. 6.

Lee wave energy flux (mW m−2) radiated from abyssal hills: (a) energy flux at zero frequency (lee waves), ignoring the presence tides, as per previous calculations, (b) energy flux at zero frequency (lee waves), including the impact of the eight major tidal constituents, (c) the difference between (a) and (b), and (d) the lee wave suppression factor—i.e., the ratio of (b) to (a). The globally integrated energy flux in gigawatts is shown above (a) and (b).

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

Of course, in the full calculation, the energy that is removed from lee waves can reappear as additional generation at other frequencies, which we report in detail below.

a. Wave energy flux from abyssal hills

The globally integrated wave energy flux radiated from abyssal hills is shown in Fig. 7 decomposed by frequency, and in Fig. 8 decomposed by wavenumber. The total energy flux summed over all frequencies and wavenumbers is 207 GW, with 80% of this generation occurring at wavelengths between 2 and 40 km, and the peak at 10 km. Figure 7 ranks the contributing frequencies (i.e., the combination of tidal constituents) from largest to smallest energy fluxes. Only eight frequencies contribute in excess of 2 GW and collectively account for 95% of the total flux: these are the primary M2 frequency (M2; 101 GW), zero-frequency lee waves (0; 59.2 GW), the primary S2 frequency (S2; 17.2 GW), the primary K1 frequency (K1; 5.70 GW), the secondary M2 frequency (2M2; 4.98 GW), the primary N2 frequency (N2; 4.03 GW), the sum of M2 and S2 primary frequencies (M2 + S2; 3.20 GW), and the sum of M2 and K1 primary frequencies (M2 + K2; 2.11 GW). The contribution of successive frequencies decays exponentially, with 99% of the total energy coming from the first 20 frequencies. Figure 7 also shows the decomposition of each frequency into the upstream, downstream and lee wavenumber bands. For most nonzero frequencies, the energy flux is dominated by the upstream band (89 GW in total), with a smaller contribution from the downstream (58 GW) and tidal-frequency lee waves (1.1 GW, in addition to the 59.2 GW from zero-frequency lee waves).

Fig. 7.
Fig. 7.

Globally integrated energy flux radiated from abyssal hills for each individual combination of tidal constituents. Frequencies are ordered by radiated flux, with the largest contributions at the bottom. (a) Individual fluxes on a logarithmic scale. (b) Cumulative sum from largest to smallest contribution. The solid gray line corresponds to the sum over all frequencies, and the dashed line is 95% of this value. For both (a) and (b) the fraction of the bar in a given color shows the proportion of the energy flux radiated as upstream (blue), lee (red), or downstream (yellow) waves.

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

Fig. 8.
Fig. 8.

(a) Wavenumber spectra of the energy flux radiated from abyssal hills. Peak generation occurs at a wavelength of 10 km. (b) Cumulative sum of the energy flux from small to large wavenumber; 80% of the generation occurs between wavelengths of 2 and 40 km.

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

The global distribution of the wave energy flux generated at abyssal hills is shown in Fig. 9 for the eight largest contributing frequencies. The fluxes of all nonzero (tidal) frequencies are dominated by generation at the midocean ridges in the Atlantic and Indian Oceans, with generally weaker and patchier generation in the Southern and Pacific oceans. Indeed, the weaker tidal constituents (e.g., N2) and the harmonic combinations (e.g., 2M2, M2 + S2, M2 + K1) only have significant generation along the midocean ridges. As anticipated from our asymptotic analysis in section 3, the M2 + S2 harmonic combination has similar amplitude to the 2M2 harmonic in most regions, illustrating the importance of including multiple tidal constituents in the calculations. In contrast to the tidal frequencies, the lee wave energy flux is large only in regions of intense bottom flow including many parts of the Southern Ocean, and also along the equator (where all topographic scales are able to generate lee waves since f vanishes).

Fig. 9.
Fig. 9.

Wave energy flux (mW m−2) radiated from abyssal hills for the eight largest frequency contributors, accounting for 95% of the total flux, ranked from largest [(a) M2] to eighth largest [(h) M2 + K1]. The K1 critical latitudes are shown by dashed red lines on (d).

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

The inclusion of mean flow in our calculations permits that tidal generation is no longer restricted to regions where the tidal frequency is superinertial—that is, regions poleward of the so-called critical latitudes where the tidal frequency equals f can now generate tidal frequency waves. The only diurnal tide that contributes significant wave energy flux is K1 (Fig. 9d; fourth largest contributor), with the majority of this generation occurring below the K1 critical latitudes of ±30.1°. However, the same patches of the Southern Ocean where lee waves are generated (i.e., mean flows are large) also have nonnegligible wave generation at the K1 primary frequency. The K1 generation is particularly prominent in the vicinity of Drake Passage, a detailed view of which is shown in Fig. 10. These K1-frequency waves exist only by virtue of the deep-ocean geostrophic flow—if this flow was to vanish, for example as the waves propagate upward from the ocean bottom, the waves could no longer exist and must dissipate (similar to zero-frequency lee waves). For this reason, we describe such waves as flow trapped as compared with normal “free” internal tide beams. Table 1 summarizes the globally integrated generation of free and trapped waves in each of the wavenumber bands. Of the 207 GW globally integrated generation from all frequencies, a little over one-third (61.6 GW) is flow trapped, which is dominated by zero-frequency lee waves but also includes trapped tidal-frequency waves (2.4 GW).

Fig. 10.
Fig. 10.

The K1 primary frequency wave energy flux (mW m−2) radiated from abyssal hills in the vicinity of Drake Passage, beyond the critical latitude.

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

Table 1.

Globally summed energy fluxes associated with wave generation at abyssal hills in different wavenumber and frequency bands. Waves that exist purely by virtue of a deep-ocean geostrophic flow are described as flow trapped. Waves that do not require a geostrophic flow to exist (e.g., the normal internal tide) are called free beams. Trapped waves are more likely to break and drive deep-ocean mixing. The frequency used is the cotidal frequency: ω=|cncΩc|.

Table 1.

b. Wave stress

We now consider the pseudomomentum flux (stress) associated with the predicted wave generation at abyssal hills. In the absence of a geostrophic mean flow (e.g., the traditional internal tide calculation) the net wave stress is identically zero, since generation is symmetric with equal and opposite momentum fluxes (Bell 1975b). However, in the presence a geostrophic flow, the momentum flux in the downstream beam is reduced, and the upstream beam increased, yielding a net stress (e.g., Fig. 2) directed opposite the geostrophic flow. (The one exception to this rule is the superbuoyancy regime shown in Fig. 2d, where only the downstream beam is generated, and the net stress is therefore downstream. However, generation in this regime is negligible at 0.05 GW globally—see Table 1.) Fig. 11 shows the modulus of the wave stress in free beams (mostly the M2 primary frequency) and trapped waves (mostly zero-frequency lee waves). The free beam wave stress is largest along the mid-Atlantic ridge and the southwest Indian ridge, consistent with the patterns in M2 energy flux (Fig. 9a). The trapped wave stress is largest in the Southern Ocean and near the equator, consistent with the patterns in lee wave energy flux (Fig. 9b). The magnitude of both wave stresses reaches 0.01–0.1 N m−2, comparable to the input of momentum from winds at the ocean surface in many regions. Thus, in regions of strong wave generation at abyssal hills, the wave stress should play a first-order role in the momentum balance of the ocean. As a specific example, Figs. 11c–j also displays the zonal and meridional components of wave stress in Drake Passage. The free wave stress is largest in the west of the passage where the M2 tide has substantial amplitude (Fig. 11g), while the trapped wave stress is largest farther to the east where eddying flows are stronger (Fig. 11h). Both stresses are directed on average to the west and north, approximately opposite the mean flow of the Antarctic Circumpolar Current. However, the particular impact of the wave stress depends on where in the water column it is applied (i.e., where the waves dissipate). The usual argument is that lee waves (and other flow-trapped waves) must dissipate within the geostrophic flow that generated them (e.g., Trossman et al. 2016). Since their momentum flux is approximately (though not exactly; e.g., Garner 2005) opposite this flow, the dissipation of these waves in the geostrophic flow acts to decelerate this flow, providing a “brake” on the system. In contrast, free tidal beams are not trapped by the generating flow and may propagate to the upper ocean, interacting with the flow there. The wave stress associated with these waves may actually accelerate the upper-ocean flow depending on where precisely the waves dissipate, as shown by Shakespeare and Hogg (2019).

Fig. 11.
Fig. 11.

The modulus of the wave stress (N m−2) generated at abyssal hills decomposed into the (a) free and (b) trapped wave components. Trapped waves are expected to dissipate in the flow that generated them, whereas free waves may propagate to the upper ocean. (c)–(f) The zonal and meridional components of the free and trapped wave stress in Drake Passage [indicated by the black box in (a) and (b)]. (g) The M2 barotropic tide magnitude in Drake Passage. (h) The geostrophic flow in Drake Passage. (i),(j) The trapped (lee) wave stress in the traditional calculation (ignoring tides) minus the trapped wave stress in the full calculation (including tides) for the zonal and meridional components. The trapped wave stress is suppressed by the addition of tides.

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

The lee wave stress—similar to the lee wave energy flux—can also be suppressed by the presence of tides. Figures 11i and 11j shows the difference between the trapped (lee) wave stress in the traditional calculation (ignoring tides) and the trapped wave stress in the full calculation (including tides) in Drake Passage. In the region where the M2 tide is strong (Fig. 11g) the magnitude of the trapped wave stress is significantly reduced, implying a possible reduction in the drag on deep ocean geostrophic flows in this region.

5. Discussion

At the O(10 km) scales of abyssal hills the generation of internal waves by geostrophic flows and each barotropic tide constituent are coupled, and should not be considered in isolation. The coupling gives rises to wave generation at all possible sums and differences of integer multiples of each tidal frequency—a natural extension to the single-tidal-constituent case considered by Bell (1975a) where generation occurs at integer multiples of the (single) tidal frequency. The smallness of the tidal excursion parameter in most of the ocean (ε = kUtt ≪ 1) means that globally 95% of the total wave energy flux from abyssal hills is contained in only eight frequencies. These consist of the zero frequency (lee waves), the primary M2, S2, K1, and N2 frequencies, the 2M2 harmonic, and sums of two primary frequencies (M2 + S2, M2 + K1)—see Fig. 7.

The coupling of geostrophic flow and tides in the generation of waves at abyssal hills also modifies the nature and magnitude of generation at each frequency. In particular, the energy flux in steady (zero frequency) lee waves is suppressed by the presence of tides. The tides act to redistribute energy from the zero-frequency band into higher frequencies, but owing to the dynamical limitation that wave frequencies fall between f and N, the total wave energy flux may either decrease or increase as a result (e.g., Fig. 3). The reduction in lee wave energy fluxes due to tides is in the range 13%–19% averaged over the global ocean, accounting for the uncertainty in inputs to the flux calculation (see appendix B). However, in regions of very strong tides and/or very small-scale topography, the suppression of both lee wave energy fluxes and wave stresses can be close to 100% (e.g., Fig. 6d). These results suggest that proposed parameterizations of lee wave energy and momentum fluxes, and hence lee wave driven mixing and lee wave drag, previously formulated ignoring the impact of tidal suppression (e.g., Yang et al. 2018; Trossman et al. 2016), may need to be revised to include the effect of tides.

A second effect of the tidal-geostrophic coupling is that the geostrophic flow modifies the generation of the internal tide by Doppler shifting the intrinsic frequency by an amount kU, introducing asymmetry between upstream and downstream waves. We calculate that 55%–63% of the global internal tide energy flux from abyssal hills is directed upstream, against the local geostrophic flow (with the range again accounting for uncertainty in inputs; see appendix B). This asymmetry leads to tidal generation being associated with a net wave stress. The dominant sites for generation of these upstream-biased internal tides are the midocean ridges in the Atlantic and western Indian Oceans (Fig. 11). The wave stresses here are of order 0.01 N m−2, comparable to the surface wind stresses, implying that internal tides may play a first-order role in the ocean momentum balance in these locations. Shakespeare and Hogg (2019) found that such waves dissipate predominantly in the upper few hundred meters of the ocean, in regions where they are aligned with the local flow, transferring their momentum to the balanced flow in these locations. Hence, the potential impact of the wave stresses is twofold: (i) the generation of a bulk Ekman-like current orthogonal to the imposed stress, and (ii) an enhancement of the surface eddy field. However, the Shakespeare and Hogg (2019) results come from an idealized channel-model of the Southern Ocean with unrealistic topography—more work is needed to directly demonstrate these effects in a realistically configured simulation.

The tidal-geostrophic coupling at abyssal hills motivates the introduction of new classes of internal waves that move away from the traditional “internal tide” (generated by the barotropic tide) and “lee wave” (generated by the geostrophic flow) denominations. Here we have introduced the nomenclature “flow-trapped wave” as a generalization of the concept of a lee wave; that is, a wave that has an absolute frequency less than f or exceeding N, and therefore only exists by virtue of the Doppler shift resulting from its generating flow. Such waves will encounter critical levels (e.g., Booker and Bretherton 1967) if the geostrophic flow decays with height, and are therefore confined within, and tend to dissipate within, their generating geostrophic flows. Trapped waves include, for example, the K1 primary frequency internal tide beyond the critical latitudes (e.g., Fig. 10) in addition to steady lee waves. Internal waves that are not flow-trapped are termed “free beams” and are a generalization of the normal internal tide; that is, a wave with absolute frequency between f and N that can freely propagate through the ocean depth, but may still be modified (Doppler-shifted) by a geostrophic flow. The presence of barotropic tides leads to a 12% reduction in the trapped wave stress, which is less than than the 13%–19% reduction in lee wave stress noted above because some of the energy flux “lost” from lee waves reappears as flow-trapped internal tides (see Table 1). The net suppression of flow-trapped wave energy flux by tides is likely to be a contributing factor in the smaller dissipation observed in deep-ocean measurements, as compared with theoretical predictions that ignore tides (e.g., Sheen et al. 2013; Waterman et al. 2013, 2014; Cusack et al. 2017).

The tidal-geostrophic flow coupling described above only applies for wave generation at the small scales of abyssal hills. As shown in Fig. 8, 80% of the wave generation occurs for wavelengths between 2 and 40 km. At such small scales both the excursion distance parameter kUtt, quantifying the effect of tidal suppression, and the Doppler shift parameter kUtt, quantifying the extent of upstream bias, are in the range 0.05–1. “Low mode” tidal generation at larger scales (order 100 km), which contains the vast majority of the energy (1–1.5 TW; e.g., Munk and Wunsch 1998; Nycander 2005; Egbert and Ray 2001), will be largely unaffected by either of these effects.

Some caveats apply to our results. First, we have used linear theory and made the small-amplitude approximation in our calculations, consistent with previous work. We have not accounted for the impact of finite amplitudes and associated flow blocking, which would act to reduce fluxes. A rough estimate from a Froude number-based parameterization of flow blocking used by previous authors suggests a reduction of total energy fluxes by up to 20% (see appendix B). We have also not considered the potential for saturation of wave generation in the limit of supercritical topography (i.e., where the topographic slope exceeds the wave slope) which would tend to reduce the overall fluxes further (e.g., Nikurashin and Ferrari 2010, 2011; Scott et al. 2011; Melet et al. 2013). A second limitation is that the precise figures for the energy flux and wave stress obtained in the present study, and the spatial distribution thereof, are entirely dependent on accurate knowledge of deep-ocean stratification and geostrophic flow speeds, which is lacking. Sensitivity estimates (appendix B) show that fluxes may vary by ~160% for only a 50% error in geostrophic flow magnitudes, and ~50% for a 50% error in buoyancy frequency. Such high sensitivity in flux magnitudes is consistent with previous sensitivity estimates, and the broad spread in previous lee wave flux calculations by different authors. These differences illustrate the large variability in model and data-derived estimates of deep-ocean properties. Therefore, significant caution should be exercised in the promulgation of exact figures for wave energy or momentum fluxes. Instead of exact figures, the major result is the demonstration of the dynamical coupling of lee wave and internal tide generation at abyssal hill scales. Both the existence of a net wave stress associated with internal tides, and the suppression of lee wave energy fluxes by barotropic tides, are robust with respect to uncertainties in deep ocean properties (see appendix B).

Acknowledgments

The author thanks David Trossman for providing access to the Goff (2010) abyssal-hill dataset and Yvan Dossmann for valuable comments on an early draft of the paper. Andy Hogg contributed to the paper through many helpful discussions. The work of the COSIMA consortium (http://cosima.org) in producing and making available high-resolution ocean model output is gratefully acknowledged. The author acknowledges support from an ARC Discovery Early Career Researcher Award DE180100087 and an Australian National University Futures Scheme award. Numerical calculations were conducted on the National Computational Infrastructure (NCI) facility in Canberra, Australia.

APPENDIX A

Extension of Bell Theory to Multiple Tidal Constituents

Bell (1975a) calculate the general expression for wave energy and pseudomomentum flux for an oscillatory mean flow u¯=UcosΩttx^ for a two-dimensional, infinite-depth, nonrotating fluid, under the small-amplitude approximation. Bell (1975b) generalizes this result to the three-dimensional rotating case, for a combined oscillatory and mean flow: u¯=U+UtcosΩtt. Here we further generalize the solution for background flow containing multiple oscillatory (tidal) constituents, each of arbitrary phase, as in the ocean:
u¯=U+c[Uccos(Ωct),Vccos(Ωct+ϕc)],
where ϕc is the phase difference between the zonal and meridional components of constituent c. (Note that the relative phase of different tidal constituents is not important to the calculation of the time-averaged fluxes, and is thus ignored here.) The equation for the vertical velocity w is, similar to Bell (1975a),
[(D2+N2)h2+(D2+f2)zz]w=0,
where the material derivative with respect to the background flow is D=t+u¯h. Fourier transforming in x and y yields
[(D^2+N2)|k|2(D^2+f2)zz]w^=0,
where D^=tiku. The linearized bottom boundary condition is given by w(z=0)=u¯hh, where z = h(x, y) is the topographic height. Following Bell (1975a), we now introduce a coordinate moving with the flow X=xu¯dt and set
w^=w˜exp(iku¯dt),
where w˜ is the Fourier transform of w in the flow-following coordinate. The bottom boundary condition in this frame may be written as
w˜(z=0)=h^texp(iku¯dt).
Substituting the mean flow from (A1) into (A5), we have that
w˜(z=0)=h^texp[ikUticAcΩcsin(Ωct+θc)],
where
Ac=k2Uc2+l2Vc2+2klUcVccosϕcandtanθc=lVcsinϕckUc+lVccosϕc.
Following Bell (1975a), we rewrite (A6) using the Jacobi–Anger expansion as
w˜(z=0)=h^t{eikUtc[nc=einc(Ωct+θc)Jnc(AcΩc)]}=h^n1,n2,n3,(iω)eiωtJn1(A1Ω1)Jn2(A2Ω2)Jn3(A3Ω3),
for nc = …, −2, −1, 0, 1, 2, …, where the intrinsic frequency is
ω=kU+n1Ω1+n2Ω2+n3Ω3+
and the Jn are nth-order Bessel functions of the first kind (plotted in Fig. A1 for reference). Thus, the solution will contain all possible sums and differences of integer multiples of the constituent tidal frequencies. On the basis of the form of (A8), following Bell (1975a), it is logical to seek a series solution of the form
w˜(k,l,z,t)=h^n1,n2,W˜(k,l,n1,n2,)ei(ωt+n1θ1+n2θ2+)×Jn1(A1Ω1)Jn2(A2Ω2)Jn3(A3Ω3).
Substituting this form into (A3) and solving subject to the boundary conditions (A8)—noting that we are in the flow-following frame so that D^=t—yields
W˜(k,l,n1,n2,)=(iω)eimz,
where
m=|k|N2ω2ω2f2sgn(ω)
and the sign of the solution is chosen to have upward energy propagation.
Fig. A1.
Fig. A1.

The nth-order Bessel functions of the first kind, Jn(ε), for n = 0, 1, 2, and 3.

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

The wave stress is the negative of the force exerted on the topography, or
τ=phhdxdy=hhpdxdy,
integrating by parts. Expressed in the flow-following coordinate, this becomes
τ=h(X+u¯dt,Y+υ¯dt)(Xp,Yp)dXdY,
or, invoking the Parseval relation,
τ=14π2exp(iku¯dt)h^*(ikp˜)dkdl,
where the asterisk denotes the complex conjugate. The pressure is related to vertical velocity via
p˜=ρ0w˜(N2ω2mω),
and exp(iku¯dt) in (A15) may be expanded as previously. Substituting (A16) into (A15) and averaging in time yields the final expression for wave stress,
τ=ρ04π2k|k|P(k,l)n1,n2,n3,(N2ω2)(ω2f2)×sgn(ω)Jn12(A1Ω1)Jn22(A2Ω2)Jn32(A3Ω3)dkdl.
The wave stress may be recognized as k times the wave action, whereas the energy flux is the wave action multipled by the intrinsic frequency (Bell 1975b)—that is,
E=ρ04π21|k|P(k,l)n1,n2,n3,|ω|(N2ω2)(ω2f2)Jn12(A1Ω1)Jn22(A2Ω2)Jn32(A3Ω3)dkdl.

APPENDIX B

Sensitivity of Flux Calculations

Here we assess the sensitivity of our wave flux calculations to the buoyancy frequency, strength of the geostrophic flow, topographic blocking and temporal variations in the geostrophic flow. We assess the sensitivity based on the magnitude of the globally integrated energy flux for the two largest contributing frequencies: the M2 primary frequency and zero frequency (lee waves). The magnitude of uncertainty in the deep ocean buoyancy frequency and mean flow is hard to quantify given the paucity of observations in the abyssal ocean. However, it is not unreasonable to expect that the model output used in the present work could be in error by ~50%. We will use this figure to estimate the potential error in energy fluxes.

First, we look at buoyancy frequency. A simple estimate of sensitivity is made by uniformly scaling the buoyancy frequency by 0.5 and 1.5 times and repeating the energy flux calculation. The energy flux decreases by 43% for the 50% decreased N and increases by 56% for the 50% increased N for the M2 primary frequency, and the energy flux decreases by 58% for the 50% decreased N and increases by 61% for the 50% increased N for the lee waves—see Tables B1 and B2. This almost one-to-one scaling is consistent with the hydrostatic limit (|ω| ≪ N) of the energy flux equation [(A18)]. The percentage suppression of the lee wave flux by tides varies by only ±1% for the different buoyancy frequencies.

Table B1.

Sensitivity of primary M2-frequency globally integrated wave energy fluxes to buoyancy frequency and mean flow. Global fluxes in gigawatts are reported for each of 0.5 times and 1.5 times the reference buoyancy frequency and reference mean flow. The upstream bias is the percentage of the total energy flux in the upstream wavenumber band.

Table B1.
Table B2.

Sensitivity of the zero-frequency (lee wave) globally integrated energy fluxes to buoyancy frequency and mean flow. Global fluxes in gigawatts are reported for each of 0.5 times and 1.5 times the reference buoyancy frequency and reference mean flow. Values are reported for the both the traditional calculation excluding tides and the full calculation including the effect of tides in suppressing the lee wave generation. The percentage suppression of the lee wave flux by tides is also reported.

Table B2.

Second, we examine geostrophic flow strength. As for the buoyancy frequency, we scale the reference geostrophic flow by 0.5 and 1.5 times and repeat the energy flux calculation. The M2-frequency energy flux decreases by 1% for the 50% decreased U and increases by 4% for the 50% increased U (see Table B1). Increasing mean flow also acts to increase the fraction of the energy flux in the upstream direction. The lee wave energy flux decreases by 82% for the 50% decreased U and increases by 162% for the 50% increased U (see Table B2). However, the suppression of the lee wave flux by tides persists in all cases, varying from 13% suppression for the stronger mean flow to 19% for the weaker mean flow.

Third, we look at temporal variation. The results presented in the paper utilize near-bottom velocities from a single 5-day-averaged period in January of a normal-year-forced model (Kiss et al. 2020). Here we consider the sensitivity of the results to temporal variation of these velocities by repeating our calculations for the first 5-day period of each month over 1 year. The results are shown in Fig. B1. For the primary M2 frequency, the standard deviation in energy flux is less than 1%. For both M2-frequency lee waves and zero-frequency lee waves the standard deviation is 6%, with a peak in March.

Fig. B1.
Fig. B1.

Sensitivity of results to the chosen 5-day analysis period: the globally integrated energy flux for twelve 5-day analysis periods (approximately one per calendar month over a year).

Citation: Journal of Physical Oceanography 50, 3; 10.1175/JPO-D-19-0179.1

Fourth, we consider topographic blocking. Steep hills are known to reduce the generation of waves, as compared with linear theory, by blocking (arresting) the barotropic flow near the boundary. This effect is often quantified in terms of the (mean flow) Froude number Fr = |U|/(Nh), where h is the characteristic (rms) topographic height. For example, to parameterize topographic blocking, Yang et al. (2018) multiplied energy fluxes by
ζ=[min(1,FrcFr)]2,
where Frc is the critical Froude number. Yang et al. (2018) and Nikurashin et al. (2014) used a critical Froude number of 0.4 and Nikurashin and Ferrari (2011) used 0.7. Applying the same parameterization here using a critical Froude number of 0.4 or 0.7 reduces the lee wave energy flux by 46% or 31%, respectively; the M2 primary frequency energy flux by 5% or 2%, respectively; and the total wave energy flux by 17% or 10%, respectively. Applying the same parameterization for the lee wave flux ignoring tides (the traditional calculation) reduces the energy flux by 42% or 27%, respectively.

In summary, the strongest sensitivity of calculated flux magnitudes of all frequencies is to the buoyancy frequency in the deep ocean. Lee waves flux magnitudes in particular are also highly sensitive to the strength of the mean flow (consistent with previous analysis; e.g., Trossman et al. 2013; Wright et al. 2014), and could be in error by as much as 160% for only a 50% error in the mean flow.

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1

However, this scheme also accounts for stresses associated with topographic blocking which was not separated out from the lee wave stress in Trossman et al. (2013, 2016).

2

Kunze and Lien (2019) suggest that only about half of the lee wave energy is dissipated, with the other half being transferred back to the geostrophic flow.

3

The relevant properties are n=n2Jn2(ζ)=ζ2/2,n=nJn2(ζ)=0,andn=Jn2(ζ)=1.

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