1. Introduction
Roughly 1 TW is input to the balanced ocean circulation and eddy fields by wind work (Wunsch 1998; von Storch et al. 2007; Scott and Xu 2009; Ferrari and Wunsch 2009). But how this power is dissipated to maintain a steady state for balanced flow—as needed to break the perpetual motion cycle of gyre-scale flows destabilizing into Rossby-scale eddies only to inversely cascade energy back to the basin scale—remains uncertain. Suggested sinks include (i) bottom drag, which appears to account for less than 25% (Arbic and Flierl 2004; Sen et al. 2008; Arbic et al. 2009), though form drag associated with topographic blocking may be larger (Klymak 2018); (ii) interior loss of balance to spontaneous and stimulated internal gravity wave generation (Polzin 2010; Plougonven and Zhang 2014; Shakespeare and Taylor 2013, 2014, 2015; Shakespeare 2019), which appears to be an even smaller fraction (Nagai et al. 2015), though this depends on the waves being reabsorbed before dissipating (Shakespeare and Hogg 2017); (iii) surface-layer submesoscale instabilities cascading to dissipation (Molemaker et al. 2010; D’Asaro et al. 2011; Nikurashin et al. 2013); and (iv) wind-work suppression (Ferrari and Wunsch 2009). One of the largest predicted pathways to dissipation at 0.2–0.75 TW is through (v) topographically generated internal lee waves (Scott et al. 2011; Nikurashin and Ferrari 2011; Wright et al. 2014; Melet et al. 2014; Trossman et al. 2016; Yang et al. 2018, 2019). Wright et al. obtained the highest of these estimates by correcting for low abyssal currents in HYCOM, but it had ±0.2-TW uncertainties associated with the buoyancy frequency climatology, as well as unquantified uncertainties associated with their bottom velocity product and theoretical assumptions. Lee waves are expected to be a major energy sink for Antarctic Circumpolar Currents in the Southern Ocean (ACC; Gille 1997), which accounts for 50% of the global wind work.
Internal lee waves are generated by flow U over topography h(x, y) (Bell 1975; Baines 1995). Their intrinsic phase speed is equal and opposite to the flow ωL/k = −U so that they are stationary in an Eulerian frame (ωE = 0), where ωL = |kU| = |k ⋅ V| is the intrinsic or Lagrangian frequency following the water, ωE the Eulerian frequency in a fixed frame, and k the along-stream topographic wavenumber. Free internal lee waves only exist for |f| < |kU| < N because they are internal gravity waves.
However, recent microstructure measurements in the Southern Ocean find turbulent dissipation rates ε as much as an order of magnitude below lee-wave generation predictions (Sheen et al. 2013; Waterman et al. 2013, 2014; Cusack et al. 2017). While the predictions are in doubt because they are based on 1D topography h(x) and neglect nonlinear topographic blocking and splitting by lower-wavenumber topography (Nikurashin et al. 2014; Trossman et al. 2015, 2016; Klymak 2018), observations in Drake Passage and on the north flank of Kerguelen Plateau find elevated internal-wave shear variance and turbulent dissipation rates associated with the deep-reaching Subantarctic, Polar, and South Antarctic Circumpolar Fronts (St. Laurent et al. 2012; Brearley et al. 2013; Waterman et al. 2013; Sheen et al. 2013), qualitatively consistent with interaction of these jets with topography generating upward-propagating internal lee waves which break to produce turbulence in the water column. Quantitative comparisons find that measured dissipation is only a fraction of the estimated upward lee-wave energy-flux (Bell 1975; Nikurashin and Ferrari 2010a; Cusack et al. 2017). Within 500 meters above bottom (mab), Waterman et al. (2013) reported similar dissipation rates inside and outside regions of high predicted lee-wave generation. Elevated dissipation rates were found at 1000–1500 mab in regions of higher predicted lee-wave generation, pointing to transmission of bottom-generated energy aloft by lee waves. Brearley et al. (2013) reported that water column dissipation rates were better-correlated with near-bottom than near-surface flows, suggesting the near-bottom flow field is not barotropic equivalent, that is, not coupled to the surface flow. The dissipation in their mooring measurements near a Scotia Sea undersea hill was due to a few large events and only 20% of the predicted lee-wave radiation, though this may be partially due to advection and radiation away from the mooring site. Cusack et al. (2017) described a lee wave downstream of a Shackleton Fracture Zone ridge with an upward energy-flux of ~1 W m−2. They inferred vertically integrated dissipation rates two orders of magnitude smaller.
Directly measured microstructure dissipation rates also fall below inferences based on a finescale shear-and-strain parameterization (Gregg and Kunze 1991; Polzin et al. 1995; Gregg et al. 2003; Naveira Garabato et al. 2004a,b; Kunze et al. 2006; Wu et al. 2011; Whalen et al. 2012; Ijichi and Hibiya 2017) in Antarctic Circumpolar Currents by factors of 5 ± 0.5 in Drake Passage and on the north flank of Kerguelen Plateau (Waterman et al. 2013, 2014) and factors of 2–3 south of Australia (Takahashi and Hibiya 2019). Brearley et al. (2013) reported that the finescale parameterization was consistent with lee-wave generation predictions. Kunze (2017) inferred integrated dissipations of less than 0.03 TW south of 40°S with a strain-based finescale parameterization applied to CTD casts, a factor of 3–10 below the 0.1–0.3-TW dissipation proposed for lee waves in the Southern Ocean (Scott et al. 2011; Nikurashin and Ferrari 2011; Melet et al. 2014; Wright et al. 2014). Direct microstructure measurements (e.g., St. Laurent et al. 2012; Waterman et al. 2014) find maximum average near-bottom dissipation rates less than 10−8 W kg−1. That microstructure measurements fall below both lee-wave generation and finescale parameterization predictions suggests that lee-wave power is finding another sink.
Waterman et al. (2014) showed that sites of this discrepancy are characterized by (i) bottom-intensified flows U ~ 0.2 m s−1, (ii) moderate topographic roughness hrms ~ O(100) m, (iii) lower-than-typical shear/strain variance ratio
Trossman et al. (2015) revisited this comparison, contrasting the microstructure measurements with the Garner (2005) model, which neglects rotation, horizontal propagation, horizontal advection, boundary reflections, and wave–wave and wave–mean flow interactions. They reported consistency with topographic blocking (inverse topographic Froude number or steepness parameter Nh/U > 0.7) below 1000 mab. Topographic blocking suppresses flow that is a distance ~U/N below ridge crests for 1D topography and forces the flow around, rather than over, 2D obstacles to shed vortices depending on their geometry (Baines 1995). Topographic blocking has thus been argued to induce saturation of lee-wave generation such that the effective topographic height heff does not exceed ~0.7U/N; more generally mheff ~ 0.7 where m is the lee-wave vertical wavenumber. Nikurashin et al. (2014) reported saturation for Nh/U > 0.4 for superinertial 2D topography h(x, y). Trossman et al. reported consistency with observations in the bottom few hundred meters but order-of-magnitude overprediction by the Garner model higher in the water column, consistent with Waterman et al. (2014).
A number of explanations for the suppression of turbulent dissipation at ~1000–1500 mab reported by Waterman et al. (2014) have been put forward:
LADCP and CTD instrument noise at low N (Kunze et al. 2006; Kunze 2017), although Waterman et al. (2014) ruled out instrumental problems on the north flank of Kerguelen Plateau, as well as issues with quantifying turbulent dissipation rates ε from microstructure measurements and the finescale parameterization.
Biased or undersampling of a highly heterogeneous turbulence field (Klymak 2018).
Poor representation of near-bottom flows or small-scale bathymetry (Trossman et al. 2015, 2016). Bathymetry on the scales that generate lee waves in the ocean is not resolved by global datasets such as Smith and Sandwell (1997) (Kunze and Llewellyn Smith 2004; Nikurashin and Ferrari 2010a), so must be treated statistically (Goff and Jordan 1988; Goff 2010; Goff and Arbic 2010). Lee-wave generation is reduced both for 2D topography and by saturation due to topographic blocking for Nh/U > 0.7 in 1D h(x) (Baines 1995; Garner 2005; Nikurashin et al. 2014; Klymak 2018) and >0.4 in 2D h(x, y) (Nikurashin et al. 2014). Treating the bottom boundary condition as linear, with the lee-wave generation band |f/U| < |k| < |N/U| unaffected by blocking and splitting at lower-wavenumber topography, may either under- or overestimate lee-wave generation (Nikurashin et al. 2014; Klymak 2018) depending on details of the bathymetry that are not available from the spectral representations, for example, with blocking, fields of hollows will have lower generation than fields of hills.
The cascade is short-circuited near the bottom, though this would produce higher near-bottom dissipation rates that are not observed (Waterman et al. 2013).
Lee-wave energy is swept downstream to dissipate at a location other than the generation site (Zheng and Nikurashin 2019), though this was not the case in the freely drifting profiling float data of Cusack et al. (2017).
Wave–wave interactions between lee waves and the background wave field drains energy into free waves that can escape the currents (Kunze et al. 1995; M. Claret 2019, personal communication).
Either 1) a narrowband lee-wave signal with ωL = f(k) = g(m), where k is the along-stream topographic horizontal wavenumber and m the vertical wavenumber, or 2) the narrow abyssal frequency bandwidth N/f ~ 3–4 do not lend themselves to as efficient an energy cascade as the broadband canonical internal-wave field. This might explain the finescale parameterization overestimation of turbulent dissipation but not lee-wave power generation overestimation.
Conservation of wave action E/|kU| (Bretherton and Garrett 1968) allows partial reabsorption of higher-frequency lee-wave energy when they are Doppler-shifted to lower frequency (|kU| ↓ |f|) as they propagate upward or cross-stream into diminishing |U|. This is consistent with the bottom-intensified currents characterizing overprediction conditions (Waterman et al. 2014) and will be the focus of this paper treated in detail in section 3.
The “suppression of turbulence” in Antarctic Circumpolar Currents (Waterman et al. 2014) resembles recent numerical simulations of spontaneous generation of internal waves in a meandering Kuroshio Front (Nagai et al. 2015) in which the bulk of the generated waves are reabsorbed elsewhere by the balanced flow rather than being dissipated at small scales. These spontaneously generated waves were characterized by subinertial Eulerian frequencies {ωE ≪ feff = [f(f + ζ)]1/2} and superinertial Lagrangian frequencies (ωL > feff) like internal lee waves. Nagai et al. argued that the waves redistribute rather than dissipate balanced energy. If the same holds true for lee waves, they would not represent the 0.2–0.75-TW sink of balanced circulation energy, nor be as large a source of turbulent diapycnal mixing, as has been proposed (Scott et al. 2011; Nikurashin and Ferrari 2011; Wright et al. 2014; Melet et al. 2014; Trossman et al. 2015). Alternatively, if wave–wave interactions transfer trapped energy to free waves (f < ωE < N) (Kunze et al. 1995) that dissipate elsewhere, that is, point vi above, lee waves would still represent a dominant sink for balanced flows.
Section 2 reviews pertinent lee-wave generation theory. It includes discussion of corrections for topographic blocking (saturation), that is, linearization of the bottom boundary condition, and shear instability, as |kU| ↓ |f. Example topography spans the oceanographically relevant topographic wavenumber and spectral slope parameter range. Since this paper’s focus is on water column process, not generation, lee-wave generation will be presented in terms of a linearized bottom boundary condition and 1D topography h(x) for simplicity, but the impacts of 2D bathymetry h(x, y) are pointed out. Readers familiar with lee-wave theory may wish to skip directly to the new material in section 3 that introduces conservation of wave action E/|kU| (Bretherton and Garrett 1968) and quantifies its relevance for the suppression-of-turbulence problem in the same parameter space. For such readers, the lee-wave radiation cospectrum coS[wp](k) is given by (17) and displayed in Fig. 6. Section 4 provides a summary and conclusions with discussion of assumptions and caveats.
2. Lee-wave generation
This section reviews lee-wave generation theory (e.g., Long 1955; Bell 1975; Baines 1995) in a rotating stratified fluid in order to construct a lee-wave energy-flux cospectrum coS[wp](k). Internal lee waves are generated by flow U over topography h(x, y). Their phase speed is equal and opposite to the flow ωL/k = −U such that their Eulerian frequency ωE is zero and their intrinsic or Lagrangian frequency ωL = −kU = |kU| where k = 2π/λx and λx is the along-stream topographic wavelength, for example, h(x) = h0sin(kx). In the ocean, topographic variability is not confined to a single wavenumber but distributed over a range of lengthscales that can be described by a horizontal wavenumber spectrum for topographic height S[h](k). The flow/topography response radiates upward into the water column as lee waves for |f/U| < |k| < |N/U| but is evanescent (bottom-trapped) outside this topographic wavenumber band (Musgrave et al. 2016). Internal lee-wave properties are determined by flow speed U, buoyancy frequency N, Coriolis frequency f, the topographic height field h(x, y) or its spectral distribution S[h](k, ℓ), and topographic wavevector (k, ℓ), where ℓ is the cross-stream wavenumber.
Although modeling efforts have focused on vertically and horizontally uniform flow (e.g., Nikurashin and Ferrari 2010a,b; Klymak 2018), horizontally and vertically confined bottom currents are typical of the ACC, where lee-wave generation theory and the finescale parameterization overpredict dissipation rates ε (Waterman et al. 2014), as well as deep western boundary currents.1
Lee waves cannot propagate out of a spatially confined current. As |kU| ↓ f in the vertical, they encounter critical layers where their vertical wavenumber |m| → ∞, group speed |Cg| → 0 and amplitudes increase. As |kU| → f + N2k2/(2fm2) in the cross-stream direction, the cross-stream wavenumber ℓ passes through zero and lee waves will deflect inward from turning points.
Lee-wave vertical wavenumber m (2π/λz) as a function of topographic horizontal wavenumber k from (2) for high-latitude Coriolis frequency f = 1.3 × 10−4 rad s−1, abyssal N = 10−3 rad s−1, and bottom flow U = 0.2 m s−1. For topographic wavenumbers k below |f/U| (λh ~10 km) and above N/U (λh ~1 km), vertical wavenumbers m are imaginary (dotted) corresponding to evanescent bottom-trapped motions with no bottom drag in the absence of bottom friction.
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
Sample horizontal wavenumber k spectra S[h](k) (12) for topographic height h going as k−2 (thick solid) and k−3 (thin solid), along with the topographic-blocking or saturation threshold (13) (dotted curve). When the topographic spectrum exceeds the topographic-blocking threshold as k ↓ f/U (k < k*), the flow U will only sense topography at the threshold level 0.7U/N. Thus, the minimum of either the topographic spectrum (solid) or the saturation spectrum (dotted) sets lee-wave radiation for any given k.
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
(left) Horizontal wavenumber k and (right) vertical wavenumber m spectra for horizontal kinetic energy HKE = (u2 + υ2)/2, that is, (8) + (9) in text (red), and available potential energy APE = b2/N2/2, that is, (10) (blue), for k−2 (thick lines) and k−3 (thin lines) topographic height spectra S[h](k). At low k (k < k*) and high m (>N/U), saturated (topographically blocked) spectra are the lower (solid) curves while the full linear responses are the higher (dotted) curves. The peak in the vertical wavenumber m spectrum corresponds to the continuum band where m is approximately constant (Fig. 1). High-latitude Coriolis frequency f, abyssal buoyancy frequency N, and mean flow U are as indicated in the header.
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
Variance-preserving (left) horizontal wavenumber k and (right) vertical wavenumber m spectra for gradient Froude number δN = |Vz|/N (red) and vertical strain ξz = ∂ξ/∂z = mh (blue) for k−2 (thick lines) and k−3 (thin lines) topographic height spectra S[h](k). At low k and high m, saturated (topographically blocked) spectra are the lower (solid) curves in contrast to the full linear topographic response (dotted curves). Note that
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
(top) Turbulent bottom-boundary-layer thickness zbbl = m−1 and (bottom) vertically integrated dissipation rate
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
(left) Variance-preserving horizontal wavenumber k cospectra for lee-wave energy flux ⟨wp⟩ (17) from a k−2 topographic height spectrum S[h](k) (12). Topographic blocking (saturation) is taken into account as |kU| ↓ |f| (k < k*). Wave action E/|kU| conservation (19) has been used to partition the wp cospectra into fractions that will be reabsorbed into the mean (20) (blue) vs dissipated (21) (red) assuming infinitesimal amplitude so that |kU| ↓ |f| before wave breaking; if waves break before reaching the |kU| = |f| critical layer, more energy will be dissipated and less reabsorbed. Numbers inside the left panel correspond to covariances under the spectra. (right) Variance-preserving vertical wavenumber m cospectra reveal that lee-wave radiation peaks at vertical wavenumbers m ~ N/U. Dissipation (red) is skewed to slightly higher m and absorption (blue) to slightly lower m.
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
(left) Horizontal wavenumber k and (right) vertical wavenumber m spectra for vertical energy-flux wp (black), scaled vertical momentum-flux Uuw (blue), and cross-stream buoyancy-flux Ufυb/N2 (green) components of the vertical Eliassen–Palm (EP) flux, and scaled bottom drag Uphx = kUph (red) for a k−2 (12) topographic spectra S[h](k). Variables have been scaled to have the same units as energy-flux to illustrate the equivalence of the different terms as in (18). Topographic blocking (saturation) is taken into account as |kU| ↓ |f| (k < k*). High-latitude Coriolis frequency f, abyssal buoyancy frequency N, and mean flow U are indicated in the header.
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
3. Lee-wave action E/|kU| conservation
For finite-amplitude lee waves, the absorptive fraction will depend on the frequency kU1 at which waves break as (kU0 − kU1)/kU0 = (U0 − U1)/U0, leaving fraction kU1/kU0 = U1/U0 ≥ f/kU0 available for nonlinear transfer to free wave radiation, turbulent dissipation and mixing. Thus, breaking before encountering a critical layer will produce a larger dissipative fraction.
A lee wave’s dissipative fraction (21) is a function of the generating frequency |kU|. The net dissipative fraction from all wavenumbers k depends on the distribution of topographic height h with wavenumber k, that is, the topographic height spectrum S[h](k) ~ kn (12), and the bandwidth N/f. Provided that the spectral slope n is constant, the net dissipative fraction ∫{fS[wp](k)/kU}dk/∫S[wp](k) dk does not depend on U because it can be expressed as f∫[g(ωL)/ωL] dωL/∫g(ωL) dωL with both integrals ranging from f to N {where g(ωL) = [(N2 − ωL2)(ωL2 − f2)]1/2ωLn}. A higher net dissipative fraction is available for (i) lower N/f as found in the abyss and higher latitudes and (ii) redder topographic spectra (more negative topographic spectral slopes n) such that a greater fraction of the lee-wave generation occurs at near-inertial |kU| (Fig. 8). For low-latitude pycnocline stratification (N/f ~100), the net dissipative fraction ranges from 0.2 to 0.5. But it increases for high latitudes and low abyssal stratification. For Coriolis frequency |f| = 1.3 × 10−4 rad s−1, corresponding to the 60° latitude of the Antarctic Circumpolar Currents, N = 10−3 rad s−1 (N/f ~ 7), flow speed U = 0.2 m s−1, rms topographic height hrms = 100 m, and topographic height spectral slopes n = −2.5 ± 0.3 characterizing the Southern Ocean (Nikurashin and Ferrari 2010b; Waterman et al. 2014), lee waves radiate at rate ~420 mW m−2 when adjusted for saturation as |kU| ↓ |f|, peaking in the continuum band (k = 2 × 10−3 rad m−1, m = 5 × 10−3 rad m−1) (Fig. 6). Roughly 2% is lost to bottom-boundary-layer turbulence in the near-inertial band (Fig. 5) where lee waves are shear unstable upon generation (Fig. 4). Of the remainder, roughly half will be lost to the mean as |kU| ↓ |f|, leaving 0.44–0.56 available for the net turbulent dissipative fraction (Fig. 6). Thus, while lee waves are most effective as a dissipative sink for balanced flow in the high-latitude abyss such as the Antarctic Circumpolar Currents in the Southern Ocean, even there they are only ~0.5 effective. This assumes (i) a linearized bottom boundary condition (7) with (ii) the minimum of the topographic spectrum (12) or saturated topographic spectrum (13), which is discussed further in section 4, (iii) |U| decreases with height above bottom consistent with the conditions for suppression of turbulence reported by Waterman et al. (2014), and (iv) waves do not break until |kU| ~ f.
Dependence of the dissipative fraction of lee-wave generation ∫ε/wp (21) on N/f and topographic spectral slope n under the small-amplitude approximation, where S[h](k) ~ kn. For N ↓ f as in the high-latitude and abyssal ocean, the topographic bandwidth f/U < k < N/U becomes narrow and the dissipative fraction becomes ~1, independent of topographic spectral slope. Numbers 0.3–0.6 along the upper and left axes are ∫ε/wp contour labels, replicating the color bar on the right. For the mid and low-latitude pycnocline (N/f ~10–100), the dissipative fraction decreases from about ~0.56 for topographic spectral slopes n = −3 to ~0.35 for topographic spectral slopes n = 0. The black box encloses the parameter space characterizing the Antarctic Circumpolar Currents [n = −2.5 ± 0.3 (Nikurashin and Ferrari 2010b); N/f ~5–10 (Waterman et al. 2014)], where the dissipative fraction is 0.44–0.56.
Citation: Journal of Physical Oceanography 49, 11; 10.1175/JPO-D-19-0052.1
4. Conclusions
This paper explored one of several possible explanations for the suppression of turbulence observed in Southern Ocean where microstructure measurements find dissipation rates that are smaller than predicted by lee-wave generation theory and finescale parameterizations (Sheen et al. 2013; Waterman et al. 2013, 2014; Cusack et al. 2017; Takahashi and Hibiya 2019). Transfer of lee-wave energy back to the balanced flow through wave action E/|kU| conservation (Bretherton and Garrett 1968) can account for a factor of 2 reduction in turbulence production with the remaining portion of lee-wave power reabsorbed by the balanced flow for conditions typical of the Southern Ocean (Waterman et al. 2014). The low N/f of the Southern Ocean abyss favors a lower absorptive fraction (Fig. 8) than lower latitudes and higher stratification. Nevertheless, it cannot fully explain the observed factor of 3–5 suppression (Waterman et al. 2014) so cannot fully account for the reported discrepancy between turbulent dissipation and lee-wave generation. Lee waves also become less effective dissipative sinks for bluer topographic height spectra. This reduction of lee-wave dissipation by wave action conservation to 0.1–0.37 TW makes them comparable to bottom drag (Arbic and Flierl 2004; Sen et al. 2008; Arbic et al. 2009) rather than the dominant sink.
Reabsorption of lee-wave energy back into the balanced flow is a mechanism for redistributing rather than dissipating balanced energy by homogenizing the generating/trapping current. Internal-wave redistribution of balanced flows is a well-established phenomenon in the atmosphere, in particular for (i) mountain lee waves (Alexander 2003; Garner 2005), (ii) driving the quasi-biennial oscillation (Lindzen and Holton 1968), (iii) driving the Madden–Julian oscillation (Biello and Majda 2005), (iv) influencing middle atmosphere winds (Holton and Alexander 2000), and (v) acting as a drag on lower stratospheric and tropospheric winds (McFarlane 1987; Scinocca and McFarlane 2000).
The results found here come with substantial caveats. The model neglects interactions among lee waves as well as with the background internal-wave field, tides, bottom-trapped topographic waves (Rhines 1970), the subinertial (|kU| < |f|) and superbuoyancy (|kU| > N) forced evanescent response (Musgrave et al. 2016; Klymak 2018) including blocking and splitting (Nikurashin et al. 2014), bottom Ekman shear and shed vortices (D’Asaro 1988). A particular concern is the linearized boundary condition (7) which assumes that blocking at lower topographic wavenumbers does not impact lee-wave generation at higher wavenumbers (Fig. 2) following Baines (1995), Nikurashin and Ferrari (2010a,b), Scott et al. (2011), Nikurashin and Ferrari (2011), Wright et al. (2014), and Nikurashin et al. (2014). Since topographic heights at lower wavenumbers are larger than those at higher wavenumbers, this is implausible. Higher topographic wavenumbers cannot be treated in isolation of the influences by lower topographic wavenumbers (that block and split near-bottom flows). For 1D topography h(x), the smaller-scale bottom topography downstream will be shielded from the flow by blocking, quenching lee-wave generation. For 2D topography h(x, y), flow can be steered around topography (for seamounts and saddle points) or blocked (for ridges and depressions) so that smaller-scale topography will feel flow modulated by the larger-scale topography (Klymak 2018). Nikurashin et al. (2014) found reduced lee-wave generation for superinertial 2D topography while wavenumber spectra of temperature from Klymak’s (2018) model show higher levels for full topography than high-passed topography in the internal-lee-wave band at 300 mab, and comparable at 1000 mab (his Fig. 5), indicating that blocking and splitting can diminish or enhance lee-wave generation. Klymak (2018) suggested that amplification of near-bottom flow U by larger-scale topography in his simulations shifted the lee-wave generation band to lower wavenumbers which have higher topographic height variance. Thus, lee-wave generation for k > f/U will be modified from the linear generation model presented in section 2. Lee-wave generation will depend on details of topographic geometry that are not available from the spectral representations (Goff and Jordan 1988; Goff 2010; Goff and Arbic 2010), for example, elevations versus depressions.
Nevertheless, the central point here, that wave action conservation will reduce the fraction of lee-wave generation lost to turbulent dissipation and mixing, still needs to be taken into account in energy budgets for the balanced flow. Moreover, since the net dissipative fraction is independent of U for uniform topographic spectral slope, it will be independent of modulations of the near-bottom flow U by topography at k < ~O(f/U) where blocking and splitting are expected (Fig. 2).
Even if the dissipative fraction is robust, it is unclear that it is responsible for the deficit reported by Waterman et al. (2014). Wave action conservation and other mechanisms for the turbulence deficit are being investigated with ongoing numerical simulations. Other plausible explanations include (i) overestimation of lee-wave generation because of blocking and splitting of near-bottom flows by lower-wavenumber 2D topography h(x, y) (Nikurashin et al. 2014; Trossman et al. 2015, 2016), although Klymak (2018) reported that blocking and hydraulic effects from larger-scale topography (k < f/U) dominated near-bottom dissipation by factors of 2–3; (ii) microstructure sampling biases in a highly heterogeneous turbulence field (Waterman et al. 2014; Klymak 2018); (iii) internal wave–wave interactions transferring energy into waves free to propagate out of the jets (f < ωE < N); and (iv) advection downstream and propagation cross stream (Zheng and Nikurashin 2019). These mechanisms may work in concert to reduce turbulence in the local water column but not all of them reduce lee waves as sinks for balanced flow as does wave action conservation. The puzzle of how the balanced mean and eddy fields dissipate their 1 TW of generation remains (Ferrari and Wunsch 2009; Scott et al. 2011; Nikurashin and Ferrari 2011; Wright et al. 2014; Melet et al. 2014).
Acknowledgments
David Trossman and Jody Klymak provided invaluable guidance. Amala Mahadevan, Amit Tandon, and an anonymous reviewer provided helpful comments for clarifying the manuscript. This work was supported by NSF Grants OCE-1756093, OCE-1829082, and OCE-1829190.
REFERENCES
Alexander, M. J., 2003: Gravity wave fluxes. Encyclopedia of Atmospheric Sciences, Academic Press, 1699–1705, https://doi.org/10.1016/B0-12-227090-8/00309-2.
Arbic, B. K., and G. Flierl, 2004: Baroclinically unstable geostrophic turbulence in the limit of strong and weak bottom Ekman friction: Application to mid-ocean eddies. J. Phys. Oceanogr., 34, 2257–2273, https://doi.org/10.1175/1520-0485(2004)034<2257:BUGTIT>2.0.CO;2.
Arbic, B. K., and Coauthors, 2009: Estimates of bottom flows and bottom-boundary-layer dissipation of the oceanic general circulation from global high-resolution models. J. Geophys. Res., 114, C02024, https://doi.org/10.1029/2008JC005072.
Baines, P. G., 1995: Topographic Effects in Stratified Flows. Cambridge University Press, 482 pp.
Bell, T. H., 1975: Topographically-generated internal waves in the open ocean. J. Geophys. Res., 80, 320–327, https://doi.org/10.1029/JC080i003p00320.
Biello, J., and A. Majda, 2005: A new multiscale model for the Madden–Julian oscillation. J. Atmos. Sci., 62, 1694–1721, https://doi.org/10.1175/JAS3455.1.
Brearley, J. A., K. L. Sheen, A. C. Naveira Garabato, D. A. Smeed, and S. Waterman, 2013: Eddy-induced modulation of turbulent dissipation over rough topography in the Southern Ocean. J. Phys. Oceanogr., 43, 2288–2308, https://doi.org/10.1175/JPO-D-12-0222.1.
Bretherton, F. P., and C. J. R. Garrett, 1968: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. London, 302A, 529–554, https://doi.org/10.1098/rspa.1968.0034.
Cusack, J. M., A. C. Naveira Garabato, D. A. Smeed, and J. B. Girton, 2017: Observation of a large lee wave in the Drake Passage. J. Phys. Oceanogr., 47, 793–810, https://doi.org/10.1175/JPO-D-16-0153.1.
D’Asaro, E. A., 1988: Generation of submesoscale vortices: A new mechanism. J. Geophys. Res., 93, 6685–6693, https://doi.org/10.1029/JC093iC06p06685.
D’Asaro, E. A., C. M. Lee, L. Rainville, R. Harcourt, and L. Thomas, 2011: Enhanced turbulence and energy dissipation at ocean fronts. Science, 332, 318–322, https://doi.org/10.1126/science.1201515.
Eliassen, A., and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geofys. Publ. Oslo, 22, 1–23.
Ferrari, R., and C. Wunsch, 2009: Ocean circulation kinetic energy: Reservoirs, sources and sinks. Annu. Rev. Fluid Mech., 41, 253–282, https://doi.org/10.1146/annurev.fluid.40.111406.102139.
Garner, S. T., 2005: A topographic-drag closure built on an analytical base flux. J. Atmos. Sci., 62, 2302–2315, https://doi.org/10.1175/JAS3496.1.
Gille, S. T., 1997: The Southern Ocean momentum balance: Evidence for topographic effects from numerical model output and altimeter data. J. Phys. Oceanogr., 27, 2219–2232, https://doi.org/10.1175/1520-0485(1997)027<2219:TSOMBE>2.0.CO;2.
Goff, J. A., 2010: Global prediction of abyssal hill root-mean-square heights from small-scale altimetric gravity variability. J. Geophys. Res., 115, B12104, https://doi.org/10.1029/2010JB007867.
Goff, J. A., and T. H. Jordon, 1988: Stochastic modeling of seafloor morphology: Inversion of Sea beam data for second-order statistics. J. Geophys. Res., 93, 13 589–13 608, https://doi.org/10.1029/JB093iB11p13589.
Goff, J. A., and B. K. Arbic, 2010: Global prediction of abyssal hill roughness statistics for use in ocean models from digital maps of paleo-spreading rate, paleo-ridge orientation and sediment thickness. Ocean Modell., 32, 36–43, https://doi.org/10.1016/j.ocemod.2009.10.001.
Gregg, M. C., and E. Kunze, 1991: Shear and strain in Santa Monica Basin. J. Geophys. Res., 96, 16 709–16 719, https://doi.org/10.1029/91JC01385.
Gregg, M. C., T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from the breaking of internal waves in equatorial ocean waters. Nature, 422, 513–515, https://doi.org/10.1038/nature01507.
Holton, J. R., and M. J. Alexander, 2000: The role of waves in the transport circulation of the middle atmosphere. Atmospheric Science Across the Stratopause, Geophys. Monogr., Vol. 123, Amer. Geophys. Union, 21–35.
Ijichi, T., and T. Hibiya, 2017: Eikonal calculations for energy transfer in the deep-ocean internal wave field near mixing hotspots. J. Phys. Oceanogr., 47, 199–210, https://doi.org/10.1175/JPO-D-16-0093.1.
Klymak, J. M., 2018: Nonpropagating form drag and turbulence due to stratified flow over large-scale abyssal-hill topography. J. Phys. Oceanogr., 48, 2383–2395, https://doi.org/10.1175/JPO-D-17-0225.1.
Kunze, E., 2017: Internal-wave-driven mixing: Global geography and budgets. J. Phys. Oceanogr., 47, 1325–1345, https://doi.org/10.1175/JPO-D-16-0141.1.
Kunze, E., and S. G. Llewellyn Smith, 2004: The role of small-scale topography in turbulent mixing of the global ocean. Oceanography, 17, 55–60, https://doi.org/10.5670/oceanog.2004.67.
Kunze, E., R.W. Schmitt and J.M. Toole, 1995: The energy balance in a warm-core ring’s near-inertial critical layer. J. Phys. Oceanogr., 25, 942–957, https://doi.org/10.1175/1520-0485(1995)025<0942:TEBIAW>2.0.CO;2.
Kunze, E., E. Firing, J. M. Hummon, T. K. Chereskin, and A. M. Thurnherr, 2006: Global abyssal mixing inferred from lowered ADCP shear and CTD strain profiles. J. Phys. Oceanogr., 36, 1553–1576, https://doi.org/10.1175/JPO2926.1.
Lindzen, R., and J. R. Holton, 1968: A theory of the quasi-biennial oscillation. J. Atmos. Sci., 25, 1095–1107, https://doi.org/10.1175/1520-0469(1968)025<1095:ATOTQB>2.0.CO;2.
Long, R. R., 1955: Some aspects of the flow of stratified fluids. Part 3: Continuous density gradient. Tellus, 7, 341–357, https://doi.org/10.3402/tellusa.v7i3.8900.
McFarlane, N. A., 1987: The effect of orographically-excited gravity wave drag on the general circulation of the lower stratosphere and troposphere. J. Atmos. Sci., 44, 1775–1800, https://doi.org/10.1175/1520-0469(1987)044<1775:TEOOEG>2.0.CO;2.
Melet, A., R. Hallberg, S. Legg, and M. Nikurashin, 2014: Sensitivity of the ocean state to lee-wave-driven mixing. J. Phys. Oceanogr., 44, 900–921, https://doi.org/10.1175/JPO-D-13-072.1.
Molemaker, M. J., J. C. McWilliams, and X. Capet, 2010: Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech., 654, 35–63, https://doi.org/10.1017/S0022112009993272.
Musgrave, R. C., R. Pinkel, J. A. MacKinnon, and J. R. Mazloff, 2016: Stratified tidal flow over a tall ridge above and below the turning latitude. J. Fluid Mech., 793, 933–957, https://doi.org/10.1017/jfm.2016.150.
Nagai, T., A. Tandon, E. Kunze, and A. Mahadevan, 2015: Spontaneous generation of internal waves by the Kuroshio Front. J. Phys. Oceanogr., 45, 2381–2406, https://doi.org/10.1175/JPO-D-14-0086.1.
Naveira Garabato, A. C., K. L. Polzin, B. A. King, K. J. Heywood, and M. Visbeck, 2004a: Widespread intense turbulent mixing in the Southern Ocean. Science, 303, 210–213, https://doi.org/10.1126/science.1090929.
Naveira Garabato, A. C., K. I. C. Oliver, A. J. Watson, and M. J. Messias, 2004b: Turbulent diapycnal mixing in the Nordic Seas. J. Geophys. Res., 109, https://doi.org/10.1029/2004JC002411.
Nikurashin, M., and R. Ferrari, 2010a: Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory. J. Phys. Oceanogr., 40, 1055–1074, https://doi.org/10.1175/2009JPO4199.1.
Nikurashin, M., and R. Ferrari, 2010b: Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Application to the Southern Ocean. J. Phys. Oceanogr., 40, 2025–2042, https://doi.org/10.1175/2010JPO4315.1.
Nikurashin, M., and R. Ferrari, 2011: Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean. Geophys. Res. Lett., 38, L08610, https://doi.org/10.1029/2011GL046576.
Nikurashin, M., G. K. Vallis, and A. Adcroft, 2013: Routes to energy dissipation for geostrophic flows in the Southern Ocean. Nat. Geosci., 6, 48–51, https://doi.org/10.1038/ngeo1657.
Nikurashin, M., R. Ferrari, N. Grisouard, and K. Polzin, 2014: The impact of finite-amplitude bottom topography on internal-wave generation in the Southern Ocean. J. Phys. Oceanogr., 44, 2938–2950, https://doi.org/10.1175/JPO-D-13-0201.1.
Plougonven, R., and F. Zhang, 2014: Internal gravity waves from atmospheric jets and fronts. Rev. Geophys., 52, 33–76, https://doi.org/10.1002/2012RG000419.
Polzin, K. L., 2010: Mesoscale eddy-internal wave coupling. Part II: Energetics and results from PolyMode. J. Phys. Oceanogr., 40, 789–801, https://doi.org/10.1175/2009JPO4039.1.
Polzin, K. L., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterization of turbulent dissipation. J. Phys. Oceanogr., 25, 306–328, https://doi.org/10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.
Rhines, P., 1970: Edge-, bottom- and Rossby waves in a rotating stratified fluid. Geophys. Astrophys. Fluid Dyn., 1, 273–302, https://doi.org/10.1080/03091927009365776.
Scinocca, J., and N. McFarlane, 2000: The parameterization of drag induced by stratified flow over anisotropic orography. Quart. J. Roy. Meteor. Soc., 126, 2353–2393, https://doi.org/10.1002/qj.49712656802.
Scott, R. B., and Y. Xu, 2009: An update on the wind power input to the surface geostrophic flow of the world ocean. Deep-Sea Res. I, 56, 295–304, https://doi.org/10.1016/j.dsr.2008.09.010.
Scott, R. B., J. A. Goff, A. C. Naveira Garabato, and A. J. G. Nurser, 2011: Global rate and spectral characteristics of internal gravity wave generation by geostrophic flow over topography. J. Geophys. Res., 116, C09029, https://doi.org/10.1029/2011JC007005.
Sen, A., R. B. Scott, and B. K. Arbic, 2008: Global energy dissipation rate of deep-ocean low-frequency flows by quadratic bottom-boundary-layer drag: Computations from current-meter data. Geophys. Res. Lett., 35, L09606, https://doi.org/10.1029/2008GL033407.
Shakespeare, C. J., 2019: Spontaneous generation of internal waves. Phys. Today, 72, 34–39, https://doi.org/10.1063/PT.3.4225.
Shakespeare, C. J., and J. R. Taylor, 2013: A generalized mathematical model of geostrophic adjustment and frontogenesis: Uniform potential vorticity. J. Fluid Mech., 736, 366–413, https://doi.org/10.1017/jfm.2013.526.
Shakespeare, C. J., and J. R. Taylor, 2014: The spontaneous generation of inertia-gravity waves during frontogenesis forced by large strain: Theory. J. Fluid Mech., 757, 817–853, https://doi.org/10.1017/jfm.2014.514.
Shakespeare, C. J., and J. R. Taylor, 2015: The spontaneous generation of inertia-gravity waves during frontogenesis forced by large strain: Numerical solutions. J. Fluid Mech., 772, 508–534, https://doi.org/10.1017/jfm.2015.197.
Shakespeare, C. J., and A. McC. Hogg, 2017: Spontaneous surface generation and interior amplification of internal waves in a regional-scale ocean model. J. Phys. Oceanogr., 47, 811–826, https://doi.org/10.1175/JPO-D-16-0188.1.
Sheen, K. L., and Coauthors, 2013: Rates and mechanisms of turbulent dissipation and mixing in the Southern Ocean: Results from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES). J. Geophys. Res. Oceans, 118, 2774–2792, https://doi.org/10.1002/jgrc.20217.
Smith, W. H. F., and D. T. Sandwell, 1997: Global seafloor topography from satellite altimetry and ship depth soundings. Science, 277, 1956–1962, https://doi.org/10.1126/science.277.5334.1956.
St. Laurent, L. S., A. C. Naveira Garabato, J. R. Ledwell, A. M. Thurnherr, J. M. Toole, and A. J. Watson, 2012: Turbulence and diapycnal mixing in Drake Passage. J. Phys. Oceanogr., 42, 2143–2152, https://doi.org/10.1175/JPO-D-12-027.1.
Takahashi, A., and T. Hibiya, 2019: Assessment of finescale parameterizations of deep-ocean mixing in the presence of geostrophic current shear: Results of microstructure measurements in the Antarctic Circumpolar Current region. J. Geophys. Res. Oceans, 124, 135–153, https://doi.org/10.1029/2018JC014030.
Trossman, D. S., S. Waterman, K. L. Polzin, B. K. Arbic, S. T. Garner, A. C. Naveira Garabato, and K. L. Sheen, 2015: Internal lee-wave closures: Parameter sensitivity and comparison to observations. J. Geophys. Res. Oceans, 120, 7997–8019, https://doi.org/10.1002/2015JC010892.
Trossman, D. S., B. K. Arbic, J. G. Richman, S. T. Garner, S. R. Jayne, and A. J. Wallcraft, 2016: Impact of topographic internal lee-wave drag on an eddying global ocean model. Ocean Modell., 97, 109–128, https://doi.org/10.1016/j.ocemod.2015.10.013.
von Storch, J.-S., H. Sasaki, and J. Marotzke, 2007: Wind-generated power input to the deep ocean: An estimate using a 1/10° general circulation model. J. Phys. Oceanogr., 37, 657–672, https://doi.org/10.1175/JPO3001.1.
Waterman, S., K. L. Polzin, and A. C. Naveira Garabato, 2013: Internal waves and turbulence in the Antarctic Circumpolar Current. J. Phys. Oceanogr., 43, 259–282, https://doi.org/10.1175/JPO-D-11-0194.1.
Waterman, S., K. L. Polzin, A. C. Naveira Garabato, K. L. Sheen, and A. Forryan, 2014: Suppression of internal-wave breaking in the Antarctic Circumpolar Current near topography. J. Phys. Oceanogr., 44, 1466–1492, https://doi.org/10.1175/JPO-D-12-0154.1.
Whalen, C. B., L. D. Talley, and J. A. MacKinnon, 2012: Spatial and temporal variability of global ocean mixing inferred from ARGO profiles. Geophys. Res. Lett., 39, L18612, https://doi.org/10.1029/2012GL053196.
Wright, C. J., R. B. Scott, P. Ailliot, and D. Furnival, 2014: Lee-wave generation rates in the deep ocean. Geophys. Res. Lett., 41, 2434–2440, https://doi.org/10.1002/2013GL059087.
Wu, L., Z. Jing, S. Riser, and M. Visbeck, 2011: Seasonal and spatial variations of Southern Ocean diapycnal mixing from ARGO profiling floats. Nat. Geosci., 4, 363–366, https://doi.org/10.1038/ngeo1156.
Wunsch, C., 1998: The work done by the wind on the oceanic general circulation. J. Phys. Oceanogr., 28, 2332–2339, https://doi.org/10.1175/1520-0485(1998)028<2332:TWDBTW>2.0.CO;2.
Yang, L., M. Nikurashin, A. M. Hogg, and B. M. Sloyan, 2018: Energy loss from transient eddies due to lee-wave generation in the Southern Ocean. J. Phys. Oceanogr., 48, 2867–2885, https://doi.org/10.1175/JPO-D-18-0077.1.
Yang, L., M. Nikurashin, H. Sasaki, H. Sun, and J. Tian, 2019: Dissipation of mesoscale eddies and its contribution to mixing in the northern South China Sea. Sci. Rep., 556, https://doi.org/10.1038/s41598-018-36610-x.
Zheng, K., and M. Nikurashin, 2019: Downstream propagation and remote dissipation of internal waves in the Southern Ocean. J. Phys. Oceanogr., 49, 1873–1887, https://doi.org/10.1175/JPO-D-18-0134.1.
Along-stream confluence and diffluence U(x) will not be considered, though important for hydraulic flows over sills (Baines 1995).