Deep-ocean mixing driven by small-scale internal tides

Turbulent mixing in the ocean is key to regulate the transport of heat, freshwater and biogeochemical tracers, with strong implications for Earth’s climate. In the deep ocean, tides supply much of the mechanical energy required to sustain mixing via the generation of internal waves, known as internal tides, whose fate—the relative importance of their local versus remote breaking into turbulence—remains uncertain. Here, we combine a semi-analytical model of internal tide generation with satellite and in situ measurements to show that from an energetic viewpoint, small-scale internal tides, hitherto overlooked, account for the bulk (>50%) of global internal tide generation, breaking and mixing. Furthermore, we unveil the pronounced geographical variations of their energy proportion, ignored by current parameterisations of mixing in climate-scale models. Based on these results, we propose a physically consistent, observationally supported approach to accurately represent the dissipation of small-scale internal tides and their induced mixing in climate-scale models.


Supplementary Note 1
Supercritical-slope correction for barotropic-to-baroclinic energy conversion.
Although supercritical slopes cover a small fraction of the total seafloor area (∼1%), they accumulate between 500 m and 1500 m (reaching up to 20% of the global seafloor area at a given depth, Supplementary Figure 1a). This is where tidal energy conversion is the strongest (Supplementary Figure 1b), noticeably due to enhanced stratification in the thermocline and steep topographic slopes of continental shelf breaks and isolated seamounts. As such, the modelled energy conversion, which is not formally valid for γ > 1, must be corrected in these areas. We opted for a correctionà la Melet et al. 1 and Falahat et al. 2 The correction is at work at depths shallower than 2000 m (Figs. 1b,c) and overall reduces the total conversion rate below 500 m from 912 GW to 787 GW.
Despite the correction, the calculation gives a very few unrealistic values at shallow depths, mostly in the western Pacific. In these areas, uncertainties in the calculation arise from a lack of hydrographic data, and less reliable tidal velocities, as noticed in 4 . To circumvent this caveat, we use a capping at 1 W m −2 , as commonly done in previous studies 1,2,5 . This capping only affects regions shallower than 700 m (Figs. 1b,c), and further reduces the global conversion rate below 500 m to 737 GW. In the article, energy conversion is systematically corrected and capped, and budgets exclude depths shallower than 700 m to discard most of the calculation suffering from supercritical-slope correction and capping. r 2 = 0.90), i.e., hotspots of barotropic tide dissipation are co-located with hotspots of barotropicto-baroclinic tide energy conversion. We also computed the coefficient of determination r 2 relative to the y = x fit such as: where y i is the observed field (D M 2 ),ȳ is its mean, and f i is the 'prediction' (E 1−∞ M 2 ). We found Supplementary Note 3 Comparison of (∇ · F 1 M 2 ) + and E 1 M 2 .
Supplementary Figure 3 shows that (∇·F 1 M 2 ) + and E 1 M 2 are well correlated (regression coefficient is r 2 = 0.94), i.e., observed and predicted hotspots of mode-1 internal tide generation are colocated. However, we found a negative r 2 y=x , due to the relatively strong deviation of the data from y = x line. In fact, E 1 M 2 seems to overestimate (underestimate) (∇ · F 1 M 2 ) + for weak (strong) generation, i.e., red line above (below) y = x (Supplementary Figure 3).

Supplementary Note 4
Sensitivity of the critical mode number and q on the attenuation length scale.
The critical mode number n crit (and q, subsequently) depends on the attenuation length scale of mode 1, L 1 , which is expected to vary geographically. However, L 1 is impossible to estimate globally since the only in situ observations sample a beam emanating from the Hawaiian Ridge. To test the sensitivity of n crit on L 1 , we doubled or halved our conservative estimate of L 1 (1300 km) and find that the critical mode number only varies by one unit (Supplementary Figure 4). This is due to the sharp decay of L n with mode number (L n ≈ L 1 /n 3 ). Specifically, we find that n crit = 3, 4, 5 for L 1 = 650, 1300, 2600 km, respectively. It is thus reasonable to assume that n crit varies globally within a small range of values, say {3,4,5}.