# Effects of improved tidal mixing in NEMO one-degree global ocean model

Casimir de Lavergne, Saurabh Rathore, Gurvan Madec, Jean-Baptiste Sallée, Christian Éthé, Antoine Nasser, Bruno Millet, Martin Vancoppenolle

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Casimir de Lavergne, Saurabh Rathore, Gurvan Madec, Jean-Baptiste Sallée, Christian Éthé, et al. Effects of improved tidal mixing in NEMO one-degree global ocean model. 2025. hal-05237363

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Effects of improved tidal mixing in NEMO one-degree global ocean model

Casimir de Lavergne¹, Saurabh Rathore², Gurvan Madec³, Jean-Baptiste Sallée⁴, Christian Ethé⁵, Antoine Nasser⁶, Bruno Millet⁷, and Martin Vancoppenolle⁸

¹Sorbonne Université

²Indian Institute of Technology Delhi

³LOCEAN

⁴LOCEAN-IPSL, CNRS

⁵Institute Pierre Simon Laplace

⁶Université Grenoble Alpes

⁷Laboratoire des Ciences du Climat et de l'Environnement

⁸Sorbonne Université

November 13, 2024

Abstract

Diapycnal mixing in the ocean interior has diverse numerical representations in extant global ocean models. These repre-
sentations affect the simulated transport and storage of oceanic tracers in ways that remain little studied. Here we present
the impacts of three different tidal mixing schemes in thousand-year-long simulations with the NEMO global ocean model at
one-degree resolution. The first scheme includes local bottom-intensified mixing at internal tide generation sites and a constant
background diffusivity. The second explicitly includes both local and remote tidal mixing, with no background diffusivity. The
third scheme is identical to the second but has the added contribution of bottom-trapped (subinertial) internal tides, known to
be important in polar regions. The three simulations show broadly similar circulation and stratification but important regional
differences. Explicit representation of remote tidal mixing strengthens the Atlantic Meridional Overturning Circulation by up
to 1.5 x 106 m3 s-1. Inclusion of bottom-trapped internal tides reduces heat reaching Antarctica by eroding Circumpolar Deep
Water at southern high latitudes, and reduces the mean age of the global deep (> 2 km) ocean by 10%. The results call
for more observational constraints on polar ocean mixing, and point to multi-faceted climatic repercussions of tidal mixing
representations.
Effects of improved tidal mixing in NEMO one-degree global ocean model

Casimir de Lavergne¹, Saurabh Rathore¹,², Gurvan Madec¹,³, Jean-Baptiste Sallée¹, Christian
Ethé¹, Antoine Nasser¹,³, Bruno Millet⁴, Martin Vancoppenolle¹

1 LOCEAN Laboratory, Sorbonne University-CNRS-IRD-MNHN, Paris, France

5 2 Centre for Atmospheric Sciences, Indian Institute of Technology Delhi, India

3 Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France

4 Laboratoire des Sciences du Climat et de l'Environnement, CEA-CNRS-UVSQ, Université Paris-Saclay, Gif-sur-Yvette, France

10 Correspondence: Casimir de Lavergne (casimir.delavergne@locean.ipsl.fr)

Key points:

* Three tidal mixing schemes are compared in long global ocean simulations at 1° resolution.

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* Explicit representation of remote tidal mixing strengthens the Atlantic overturning circulation.

* Subinertial internal tides increase mixing in polar oceans, reducing subsurface heat fringing Antarctica and lowering abyssal ocean age.
**Abstract:** Diapycnal mixing in the ocean interior has diverse numerical representations in extant global ocean models. These representations affect the simulated transport and storage of oceanic tracers in ways that remain little studied. Here we present the impacts of three different tidal mixing schemes in thousand-year-long simulations with the NEMO global ocean model at one-degree resolution. The first scheme includes local bottom-intensified mixing at internal tide generation sites and a constant background diffusivity. The second explicitly includes both local and remote tidal mixing, with no background diffusivity. The third scheme is identical to the second but has the added contribution of bottom-trapped (subinertial) internal tides, known to be important in polar regions. The three simulations show broadly similar circulation and stratification but important regional differences. Explicit representation of remote tidal mixing strengthens the Atlantic Meridional Overturning Circulation by up to $1.5 \times 10^6 \text{ m}^3 \text{ s}^{-1}$. Inclusion of bottom-trapped internal tides reduces heat reaching Antarctica by eroding Circumpolar Deep Water at southern high latitudes, and reduces the mean age of the global deep (> 2 km) ocean by 10%. The results call for more observational constraints on polar ocean mixing, and point to multi-faceted climatic repercussions of tidal mixing representations.

**Plain Language Summary:** Winds generate surface ocean waves that break on beaches. Similarly, at the bottom of the ocean, tides generate internal ocean waves that propagate throughout the ocean and ultimately break. When such internal waves break, they cause turbulence and mixing inside the ocean. This mixing, referred to as tidal mixing, is an important process that redistributes heat, carbon and nutrients across ocean layers. In most computer models of the global ocean circulation, tides are absent but their contribution to mixing is represented by a simplified physical model. In this work, we compare three different representations of tidal mixing in global ocean simulations. We find small differences overall in the simulated ocean currents and density structure. Nonetheless, we show important regional differences and pinpoint the benefits of accounting explicitly for all pathways from tidal wave generation to mixing. The work thus leads the way to better representing tidal mixing in global ocean simulations.
# 1. Introduction

The ocean can be viewed as a stack of density layers that exchange slowly with their neighbors. Cross-layer exchanges of heat and matter occur at ocean boundaries via surface and geothermal fluxes, and in the interior via mixing of ocean tracers at molecular scale (Walin 1982, Iudicone et al. 2008). This irreversible mixing of tracers is catalyzed by small-scale turbulence, which effectively controls the rate at which layers communicate with each other in the ocean interior (Gregg 1987). By transferring mass and tracers across density layers, small-scale turbulence thus shapes the physical ocean state and regulates the transport and storage of climatically important tracers, such as nutrients, carbon and heat (Tuerena et al. 2019, de Lavergne et al. 2022, Melet et al. 2022).

Global ocean models do not resolve small-scale turbulence and must therefore resort to parameterizations of its effects. The common approach is to provide a vertical (or diapycnal) diffusivity that mixes tracers and momentum across the model layers. However, methods to determine the diapycnal diffusivity distribution vary widely across models. In the strongly mixing surface boundary layer, energetic considerations and/or classical turbulence theory are generally invoked to obtain a set of equations relating diffusivity to surface forcing, velocity shear and density stratification (Mellor and Yamada 1982, Gaspar et al. 1990, Large et al. 1994, Reichl and Hallberg 2018). In the deeper ocean, diffusivity is locally enhanced when the resolved shear or stratification is unstable. But this falls short of capturing the effects of unresolved instabilities energized by winds and tides. To account for the latter, most models combine a static background diffusivity (Jochum 2009) with a representation of bottom-enhanced tidal mixing (Simmons et al. 2004). Even so, the choice of background diffusivity and the imposed tidal mixing distribution vary across modelling centers and across model configurations (e.g., Adcroft et al. 2019, Kiss et al. 2020).

Consequences of varying the oceanic background diffusivity or the tidal mixing energy distribution have been recently explored using climate model simulations (Melet et al. 2016, Tatebe et al. 2018, Hieronymus et al. 2019). These studies highlight important changes in the heat balance and overturning circulation of the global ocean, motivating the need to better constrain diapycnal mixing in the ocean interior and to refine its representation in climate models. Studies also emphasized the need to develop energy constrained parameterizations, where diffusivity is not static but instead deduced from a three-
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dimensional distribution of turbulence production (Huang 1998, Eden et al. 2014). This
requires tracking energy routes from forcing to small-scale turbulence production.

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The main such energy route is arguably the generation, propagation and breaking of internal tides—that is, internal waves of tidal frequency sparked by interaction of barotropic tidal currents with sloping topography (Egbert and Ray 2000, Vic et al. 2019, de Lavergne et al. 2019). The traditional parameterization strategy is to estimate the rate of internal tide generation at the bottom, and assume that a third of this power dissipates in the local water column according to an exponential decay from the seafloor upwards (St Laurent et al. 2002, Simmons et al. 2004). However, it has become clear that the remaining two-thirds of power, which correspond to large-scale (or low-mode) internal tides able to propagate far from generation sites, also require explicit parameterization (Eden and Olbers 2014, MacKinnon et al. 2017, de Lavergne et al. 2019). Indeed, low-mode internal tides explain the bulk of observed dissipation patterns in the upper ocean beneath the mixed layer, and cannot be reliably represented by a constant background diffusivity (Pollmann et al. 2017, de Lavergne et al. 2020).

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Two schemes have been developed to track the propagation and dissipation of internal tide energy, including that of low-mode internal tides. The scheme of Eden and Olbers (2014) prescribes two-dimensional maps of internal wave generation, split between a high-mode and a low-mode compartment, and uses internal wave theory to redistribute this power horizontally and vertically within the simulated stratification. An alternative scheme (de Lavergne et al. 2020) prescribes four two-dimensional maps of internal tide dissipation, whose power is then distributed vertically within the simulated stratification according to a process-dependent vertical structure. The latter scheme thus neglects interactivity of the low-mode internal tide energy propagation, but allows for a realistic three-dimensional distribution of mixing energy at low computational cost (de Lavergne et al. 2020).

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Neither scheme includes subinertial internal tides, that is, internal tides whose frequency is lower than the local Coriolis frequency. Such internal tides, which occur poleward of 74.5° for the dominant M₂ semidiurnal tide and poleward of 30° for the K₁ diurnal tide, are trapped at the bottom and ultimately break to cause mixing near the seafloor (Chapman 1989, Tanaka et al. 2013, Falahat and Nycander 2014, Tanaka 2023). They are thought to play an important role in mixing the polar oceans (Rippeth et al. 2015, Fer et al. 2020),

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where they dominate over their freely propagating counterparts (Müller 2013). Yet the potential global-scale impacts of mixing by bottom-trapped internal tides remain unexplored.

Here, we implement the parameterization of local and remote tidal mixing proposed by de Lavergne et al. (2020) in the Nucleus for European Modelling of the Ocean (NEMO), and document its behavior in a long integration of a global ocean configuration with 1° nominal horizontal resolution. We also examine differences with two twin experiments: one which includes the additional – hitherto neglected – contribution of subinertial internal tides, and another which follows the traditional choice (Simmons et al. 2004) of a constant background diffusivity and local bottom-intensified tidal mixing. The configuration and mixing parameterizations are presented in detail in the next section. In section 3, we document the main characteristics and sensitivities of our simulations. We conclude with a summary and discussion of results (section 4).

## 2. Methods

### 2.1 Configuration overview and forcing

Our configuration uses version 4.2 of NEMO (NEMO System Team 2022), which solves the primitive equations in Boussinesq approximation. The ocean model component is coupled to the SI³ dynamic-thermodynamic sea ice model (Rousset et al. 2015, Vancoppenolle et al. 2023). We employ a global Mercator grid of 1° nominal horizontal resolution, referred to as ORCA1. The meridional resolution is refined at high latitudes so that the meridional grid spacing matches the reducing zonal spacing, and refined within 15° of the equator to better resolve equatorial dynamics. As a result, the average area-weighted horizontal resolution is 0.69° or 83 km over the global ocean; in the Southern Ocean south of 40°S, it is 0.57° or 45 km. In the vertical, the model includes 75 layers that follow geopotential surfaces, except for time-dependent stretching or squeezing of layers to accompany changes in sea surface height. The vertical grid spacing varies from about 1 m near the surface to 200 m in the abyssal ocean. Partial cells at the bottom allow for a better representation of the observed bathymetry. The domain covers depths of up to 6,000 m.
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The model is run for 1,000 years with a time step of 45 minutes. Initial hydrographic conditions are Conservative Temperature and preformed salinity fields calculated from the observational climatology of Gourestki (2019) using the Gibbs SeaWater Toolbox (McDougall and Barker 2011). Preformed salinity is used instead of Absolute Salinity because the model, while employing the TEOS-10 equation of state (IOC, SCOR and IAPSO 2010, Roquet et al. 2015), does not represent biological (non conservative) sources of salt included in Absolute Salinity. For brevity, Conservative Temperature and preformed salinity will be referred to as temperature and salinity in all the following. At the simulation start, velocities are zero and sea ice is present where the initial sea surface temperature is within 1.0 °C of the freezing point.

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At the bottom, the model is forced by geothermal heat fluxes mapped by Lucazeau (2019) (Supplementary Figure S1). At the surface, atmospheric conditions are prescribed following version 2 of the Coordinated Ocean-ice Reference Experiments (COREv2) normal year forcing (Large and Yeager 2009), which represents a typical year of the instrumental period (mostly between 1979 and 2006). This forcing includes 6-hourly fields of zonal and meridional wind at 10 m and of air temperature and humidity at 10 m. In addition, it includes daily mean radiation and monthly mean precipitation (rain and snow) reaching the surface. Freshwater forcing also includes river runoff, iceberg melt released in the surface layer following the spatial pattern estimated by Merino et al. (2016), and basal melt from Antarctic ice-shelf cavities released at depth over the mouth of cavities (Mathiot et al. 2017). Basal melt fluxes from each cavity and the total magnitude of Antarctic iceberg melt are prescribed according to estimates of Depoorter et al. (2013); they are not interactive and the cavities are closed.

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Surface fluxes including evaporation are computed using bulk formulae (Large and Yeager 2009), as a function of the simulated sea surface temperature and sea ice cover. To compensate for missing atmospheric feedbacks, restoring to the observed seasonally varying sea surface salinity (as mapped by Gouretski (2019)) is activated, with a timescale of 230 days per 50 m of seawater. This restoring term is expressed as a freshwater flux added to the local evaporation minus precipitation flux. To ensure that the freshwater balance is globally closed and avoid drift in sea level, the global mean freshwater flux leaving the ocean is calculated at each time step and subtracted from the evaporation minus

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precipitation flux of every surface ocean grid cell. This uniform adjustment adds 0.13 m of
sea level every year on average in our simulations, a magnitude that can be compared to the
1.11 m added by precipitation every year. Closure of the freshwater balance in forced-ocean
experiments is a delicate issue: our experience shows that the required adjustment can
grow large in the absence of careful calibration. In addition, this adjustment introduces a
non-physical global interdependence of freshwater fluxes that should be borne in mind
when interpreting temporal variability or sensitivity to forcing and physics.

## 2.2 Representation of advection and diffusion

The vector-invariant form of momentum equations is chosen together with the rotational-divergence form of the stress tensor (Madec et al. 1991). Advection of momentum uses a second-order centered scheme. The relative and planetary vorticity terms are discretized with a scheme that conserves energy and potential enstrophy of non-divergent flow (Burridge and Haseler 1977, Arakawa and Lamb 1981). These choices allow for reduced sensitivity of the simulated circulation to artificial indentation of model isobaths (Nasser et al. 2023). Momentum diffusion is accomplished by a Laplacian operator and a prescribed horizontal eddy diffusivity coefficient of 20,000 m² s⁻¹, reduced to 1,000 m² s⁻¹ around the equator (Delecluse and Madec 1999). The lateral boundary condition is free slip. To aid downslope flow of dense water along the staircase-like topography, the advective scheme of Campin and Goosse (1999) is activated. A quadratic bottom drag is also included, where the drag coefficient is C<sub>d</sub> = 10⁻³ [0.25(u² + v²) + u<sub>t</sub>²]⁰.⁵ with u and v the simulated bottom horizontal velocities and u<sub>t</sub> a constant (tidal) velocity of 0.05 m s⁻¹.

Restratification by baroclinic mesoscale eddies is parameterized by downgradient diffusion of isopycnal layer thickness, following the approach of Gent and McWilliams (1990). The associated thickness diffusivity varies horizontally with the simulated stratification (Supplementary Figure S2): it is proportional to the square of the Rossby radius and to the Eady growth rate (Held and Larichev 1996, Treguier et al. 1997), and tapered equatorward of 20° to account for the increasing sparsity of baroclinic mesoscale eddies there (Chelton et al. 2011). Restratification of the surface mixed layer by submesoscale eddies is parameterized following Fox-Kemper et al. (2008).
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Advection of all tracers is performed with a second-order Flux Corrected Transport scheme. Diffusion of tracers along isopycnals by mesoscale and smaller scale turbulence (Redi 1982, Abernathey et al. 2022) is parameterized with a diffusivity that varies geographically with the local Rossby radius and Eady growth rate (Supplementary Figure S2), like the thickness diffusivity.

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Vertical diffusion of momentum and tracers is parameterized by three components: a Turbulent Kinetic Energy (TKE) closure, a convection scheme, and a tidal mixing scheme. The TKE closure is order 1.5: it includes a prognostic equation for TKE and diagnostic equations relating mixing parameters and TKE (Gaspar et al. 1990, Blanke and Delecluse 1993, NEMO System Team 2022). This scheme produces enhanced vertical mixing when the Richardson number of the resolved flow falls below a critical value of 0.22. Mixing is further enhanced near the base of the mixed layer to account for the effects of unresolved Langmuir cells (Axell 2002) and near-inertial oscillations. Near-inertial oscillations of the mixed layer, generated by high-frequency winds, fuel intense mixing at the base of the mixed layer and relatively weak mixing in the deeper ocean (Plueddemann and Farrar 2006, Alford 2020, von Storch and Lüschow 2023). In the TKE scheme of NEMO their contribution to upper-ocean mixing is represented by adding a vertical profile of TKE that decays exponentially with depth and that peaks at a small fraction of the surface TKE (Rodgers et al. 2014). Here we choose the fraction to be 8%, and the decay scale to vary from 1 m at the equator to 30 m at high latitudes. The advantages of this scheme are its simplicity and interactivity: the induced mixing depends on the surface TKE hence on the instantaneous wind forcing. The drawbacks are the ad hoc vertical profile and the absence of constraint on the power consumed by this mixing.

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The convective scheme simply consists in enhancing the vertical diffusivity to 10 m² s⁻¹ when stratification is statically unstable. No parameterization of double diffusive mixing is included here. Hence, in our experiments, vertical mixing in the ocean interior (away from the surface boundary layer) relies almost exclusively on the tidal mixing parameterization, which is detailed next.

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## 2.3 Tidal mixing parameterization and sensitivity experiments

To investigate the sensitivity of the ocean state to the representation of tidal mixing, we performed three experiments. Experiment OLD includes a constant background diffusivity of $10^{-5} \text{ m}^2 \text{ s}^{-1}$ and local bottom-intensified mixing at internal tide generation sites (Simmons et al. 2004). This bottom-intensified mixing is specified by assuming that: (i) one-third of the power input to internal tides mapped by Nycander (2005) fuels turbulence production in the local water column; and (ii) turbulence production decays exponentially with height above bottom, with a uniform 500 m decay scale (St Laurent et al. 2002). Experiment NEW includes explicit and comprehensive treatment of freely propagating internal tides; that is, it incorporates the tidal mixing scheme of de Lavergne et al. (2020) with the added contribution of shallow internal tides ($E_{sha}$), implemented as described below. Experiment TRA is identical to NEW but includes the additional contribution of subinertial internal tides ($E_{tra}$), also detailed next.

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Tidal mixing representations in both NEW and TRA rest upon the energy-constrained framework of de Lavergne et al. (2020). This framework employs four static maps of depth-integrated internal tide energy dissipation, which have been remapped onto the ORCA1 grid using an energy-conserving interpolation. Each map is associated with a distinct dissipation mechanism and a corresponding vertical structure of turbulence production, which depends on the simulated stratification. Hence, at each time step, the power available in each water column is distributed in the vertical, yielding a global three-dimensional distribution of turbulence production $\epsilon(x,y,z)$. The vertical diffusivity $K_\rho$ is then deduced from turbulence production using a zero-order closure, $K_\rho = R_f \epsilon N^{-2}$, where N is the buoyancy frequency and $R_f$ is the mixing efficiency (Osborn 1980). Mixing efficiency is known to vary depending on local flow and turbulence characteristics (e.g., Mashayek and Peltier 2013, Bouffard and Boegman 2013, Vladoiu et al. 2019). However there is no consensus about, nor comprehensive model of, mixing efficiency variations in the ocean, so that the traditional assumption of a constant $R_f = 1/6$ remains the standard choice (Gregg et al. 2018). For simplicity, we take $R_f = 1/6$ in all three experiments. 500-year simulations performed with a coarser ($2^\circ$) resolution global ocean configuration showed only weak sensitivity of ocean circulation to mixing efficiency variations when parameterized following de Lavergne et al. (2016). An upper bound of $10^{-2} \text{ m}^2 \text{ s}^{-1}$ is imposed on the tidal vertical diffusivity. The lower

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bound is $1.4 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$ for tracers and $1.4 \times 10^{-6} \text{ m}^2 \text{ s}^{-1}$ for momentum, corresponding to molecular heat diffusivity and kinematic viscosity, respectively. No differential mixing between temperature and salinity is implemented.

The mapping of internal tide energy dissipation performed by de Lavergne et al. (2019, 2020) uses analytical estimates of internal tide generation that are valid only at seafloor depths greater than 400 m and equatorward of the turning latitude of each tidal constituent, due to assumptions in the analytical theory (Bell 1975, Falahat et al. 2014). To remedy these two limitations, we turn to an alternative estimate of internal tide generation, obtained with a high-resolution ocean general circulation model (Müller 2013). This model estimate agrees broadly with the analytical calculation of Falahat et al. (2014) within overlapping validity regions, while also including internal tides generated over shallow (< 400 m) topography and poleward of their turning latitude. Müller (2013) provides four two-dimensional power fields, corresponding to the four tidal constituents ($M_2$, $S_2$, $K_1$ and $O_1$) included in the simulation. These fields have positive and realistic global integrals (Table 1) yet contain frequent negative values, like the analytical estimates of Falahat et al. (2014). We correct for these negative values following de Lavergne et al. (2019): we first set negative values to zero, then multiply the resulting fields by a depth-dependent factor that preserves the original depth-profile of internal tide generation. This procedure is ad hoc but yields plausible distributions (de Lavergne et al. 2019; Fig.1). Within each of the four corrected fields, we select regions where the tidal frequency exceeds the Coriolis frequency to obtain the generation rate of *trapped internal tides* ($E_{\text{tra}}$). In the remainder of the ocean, we select grid cells where bathymetry is less than 400 m to obtain the generation rate of *shallow internal tides* ($E_{\text{sha}}$). The total power available from each of the two components and each of the four tidal constituents is given in Table 1.

To implement these two components in the tidal mixing parameterization, we sum over the four tidal constituents, using scaling factors (see Table 1) to account for missing constituents, following de Lavergne et al. (2019). We then remap the total fields, $E_{\text{tra}}$ and $E_{\text{sha}}$, onto our model grid (Fig.1). The two fields correspond to generation rates, not dissipation rates, hence they are not direct estimates of turbulence production integrated over the water column. However, we expect the dissipation to be near local in both cases, because subinertial internal tides are bottom-trapped (Albrecht et al. 2006, Tanaka et al.
2013, Musgrave et al. 2016), and because the decay time of internal tides implied by wave-
wave interactions is very short in shallow seas (de Lavergne et al. 2019). We thus assume
local dissipation at the 1° resolution of our configuration. Subinertial internal tides are
expected and observed to cause mixing intensified near the seafloor (Tanaka et al. 2013,
Rippeth et al. 2015, Musgrave et al. 2017): we choose the same vertical structure as that
attributed to dissipation at critical topography, i.e. an exponential decay from the bottom
upwards with a decay scale determined by local topographic slopes (de Lavergne et al.
2020). Shallow internal tides are expected to dissipate primarily via wave-wave interactions:
we thus distribute their power in the vertical as proportional to the square of the simulated
buoyancy frequency (Gregg 1989, D'Asaro and Lien 2000).

The depth-integrated turbulence production implied by each of the three tidal mixing
representations is mapped in Figure 2. For the OLD experiment, the power implicitly
consumed by the background diffusivity is included, and calculated as the vertical integral of
ρ₀ Kₚ N² R_f⁻¹ with ρ₀ = 1026 kg m⁻³ the reference density of seawater and N the annual mean
buoyancy frequency from the first simulation year. We find that the background diffusivity
of 10⁻⁵ m² s⁻¹ in OLD consumes 0.72 TW of power globally. The global total power amounts
to 1.12 TW in OLD, 1.28 TW in NEW and 1.40 TW in TRA. Although these totals are of the
same order of magnitude, they hide substantial differences in geographical patterns of
mixing (Fig.2). Explicit representation of remote tidal mixing creates a much more
heterogeneous mixing distribution, reflecting the heterogeneous sources and sinks of
internal tide energy. Inclusion of trapped internal tides raises power availability poleward of
30° and most conspicuously in the subarctic Pacific, Arctic and Southern Oceans (Figs. 1a
and 2b,c).

Similarities and differences between the three mixing representations are further illustrated
in Figure 3, which shows turbulence production summed over depth and longitude (Fig.3a)
or along geopotential model layers (Fig.3b). The latitudinal profiles highlight the similarity of
bulk mixing rates across the three simulations. Most important relative differences occur at
high latitudes, where TRA has the highest power input to mixing, except north of 70°N; the
background diffusivity of OLD sustains unrealistically high mixing in the Arctic pycnocline
(Nguyen et al. 2009). The depth profiles highlight the large proportion of power contained in
the top 500 m: contrary to common perception, internal tide-driven mixing is prominent in
the upper ocean, not only in the deep ocean (Kunze et al. 2006, de Lavergne et al. 2020). The tidal mixing parameterization in NEW and TRA generates more mixing than OLD in the ocean's top kilometer, while the opposite is true between 1 and 3 km depth.

It must be emphasized that power consumed by tidal mixing in OLD is not fixed in time: it evolves with stratification changes. The total power effectively consumed in our OLD experiment increases by 1.5% over the full length of the simulation; a relatively small change that reflects the small drift in stratification. However, should stratification increase rapidly, for example under a strong greenhouse forcing scenario, the power consumed by the constant background diffusivity could increase substantially (de Lavergne et al. 2020). The map and curves shown for OLD in Figures 2 and 3 are thus only applicable to the initial state of our experiment: the same diagnostics applied to a different model state would generate different magnitudes and patterns. These considerations underline the inadequacy of static diffusivities: they ignore energy constraints and do not allow for a realistic response of mixing to forcing or stratification changes (Eden et al. 2014, Sigman et al. 2010).

## 2.4 Ideal age and offline simulation

To investigate ventilation timescales, the simulations include an ideal age tracer, which measures the time elapsed since local seawater was last at the surface (England 1995). This tracer is advected and diffused using the same velocities, diffusivities and numerical schemes as those transporting temperature and salinity. Ideal age is restored to zero in the top 10 m of ocean, and increases by 1 year per year deeper than 10 m. It is initialized with uniform null concentrations. After 1,000 years, it is still far from equilibrium in the deep ocean, despite relative stabilization of the circulation and stratification (Fig.4). In order to quantify deep ventilation rates in the near steady state, we extend the simulations of ideal age in offline mode, for an extra 3,000 years. The offline simulation uses physical variables (velocities, diffusivities, layer thicknesses, temperature and salinity) from the last year of the full simulation to evolve the passive tracer distribution. In detail, we output physical variables from year 1,001 of each experiment at five-day frequency, and simulate ideal age as in the online simulation, except that physics are not computed but instead read from this five-day output and time-interpolated at each time step.
# 3. Results

## 3.1 Drift

The global mean ocean temperature increases by 0.29 °C in OLD, 0.34 °C in TRA and 0.35 °C in NEW over 1,000 years (Fig.4a). The drift is not linear and slows progressively. Over the last century of the simulations, global mean temperature increases only by 1.5 x 10⁻³ °C in OLD, 8.6 x 10⁻³ °C in NEW and 4.3 x 10⁻³ °C in TRA; increases in mean deep ocean temperature are similar or smaller. By construction (section 2.1), the salt content and volume of the global ocean are constant over the simulation, hence the mean salinity is also constant. Redistribution between layers can nonetheless occur: over the whole simulation length and across the experiments, the mean salinity in the upper km decreases by 0.022-0.050 g kg⁻¹ whereas the mean salinity in the deeper ocean increases by 0.007-0.016 g kg⁻¹. Overall, the drift toward warmer and saltier deep waters can largely be explained by a warm and salty bias at North Atlantic Deep Water (NADW) formation sites, which is progressively communicated to the global deep ocean (see section 3.3).

The overturning circulation shows relatively rapid stabilization: the Atlantic Meridional Overturning Circulation (AMOC) and the lower cell of the MOC reach a near-steady strength after a few hundred years (Fig.4c-e) in all three experiments. The strength of the lower cell is diagnosed in density coordinate and shown both for the latitude of peak strength (64°S; Fig.4d) and a middle latitude (30°S; Fig.4e). The eastward transport through Drake Passage, which is a key indicator of Southern Ocean circulation and stratification, is relatively stable in all three experiments (Fig.4b; mind the small y-axis range). Specifically, it increases from 163 Sv (1 Sv is 10⁶ m³ s⁻¹) in the initial year to 172 Sv, 174 Sv and 181 Sv at the end of the TRA, NEW and OLD experiments, respectively. The positive drift over the tenth century is largest in TRA but lesser than 0.4 Sv. Observational estimates of the Drake Passage transport range between 134 Sv and 175 Sv (Cunningham et al. 2003, Donohue et al. 2016, Colin de Verdière and Ollitrault 2016); our simulations thus lie near the high end of observations.

To bring out the level of equilibration of the model stratification and circulation in the final state of our experiments, we extended experiment TRA for an additional 500 years. The extended time series show that the circulation of TRA is quasi steady by year 1,000 (Fig.4b-e). Only the global mean ocean temperature displays a persistent, weak drift of 3.6 x 10⁻³ °C
per century (Fig.4a). Given the small degree of variability and drift present in the analyzed time series near the end of our millennial integrations (Fig.4), we focus hereafter on simulation characteristics averaged over years 991-1,000 of each experiment. When discussing anomalies, we will take NEW as the reference experiment and examine differences between OLD and NEW (OLD – NEW) or between TRA and NEW (TRA – NEW).

## 3.2 Surface boundary layer

The surface mixed layer is central to ocean ventilation, being the layer that is effectively in contact with the atmosphere. The depth of this layer, and its spatial and seasonal variations, contribute to shape water-mass characteristics and transports throughout the ocean (Marshall and Nurser 1991, Williams et al. 1995). Figure 5 compares the mixed layer depth (MLD) in experiment NEW to the observational map of Sallée et al. (2021). In both cases MLD is defined as the depth at which potential density exceeds its 10 m value by $0.03 \text{ kg m}^{-3}$ (de Boyer Montégut et al. 2004). The broad geographical patterns are well captured by the model for each season. The main difference is a tendency to overestimate MLD at high latitudes in winter and spring, indicative of overestimated winter deep convection and possibly underestimated spring restratification. Main deep convection sites in the model are the Labrador Sea, Irminger Sea, Norwegian Sea, southern Weddell Sea and a band along the northern edge of the Antarctic Circumpolar Current (ACC).

Overall, the simulated MLDs are weakly sensitive to the representation of tidal mixing. Principle differences between experiments occur in winter. Figure 6 shows the annual maximum MLD of NEW and anomalies in OLD and TRA relative to NEW. In the North Atlantic, OLD has shallower winter deep convection than NEW (Fig.6b), consistent with weaker NADW formation and a weaker AMOC. On the other hand, OLD has deeper winter mixed layers in the polar oceans (Arctic and near-Antarctic), which we attribute to higher levels of mixing within polar haloclines (Figs. 3 and 7) that precondition winter convection (Nguyen et al. 2009, Morawitz et al. 1996). Significant anomalies in subantarctic MLDs are also apparent, with an overall shoaling in the Atlantic and Indian sectors and deepening in the Pacific sector (Fig.6b). These changes are suggestive of slight frontal shifts in the ACC, combined to local preconditioning of deep convection by interior mixing (Sloyan et al. 2010).
Southern Hemisphere MLD anomalies in TRA are qualitatively similar to those in OLD, albeit weaker. This similarity suggests that the MLDs are sensitive to the bulk mixing across the high-latitude permanent pycnocline. Both TRA and OLD have stronger average mixing across the Southern Ocean pycnocline than NEW: OLD has higher mixing throughout most of the upper ocean while TRA has higher mixing along continental slopes (Figs. 7 and 8). In the Northern Hemisphere, differences between TRA and NEW are mostly found in the subpolar North Atlantic. In particular, MLDs deepen in the Irminger Sea and central Labrador Sea, possibly contributing to a slightly stronger AMOC (Fig.4c).

Sea ice coverage exerts a profound influence on deep ocean stratification and ventilation, particularly in the Southern Ocean (Goosse and Fichefet 1999, Haumann et al. 2016, Klocker et al. 2023), and can be sensitive to deep ocean mixing (Tatebe et al. 2018). In our forced ocean experiments, sea ice is insensitive to the chosen tidal mixing scheme in both hemispheres (Fig.9). The seasonal expansion and retreat of the sea ice pack is reasonably well represented, despite Antarctic sea ice dropping too low in summer and retreating too late during the warming season (Fig.9b). Simulated deep mixed layers in southern polar waters are not associated with openings in the sea ice (Fig.5f,h). We will show in the next section that bottom waters are produced on Antarctic shelves rather than in open ocean convection zones.

## 3.3 Water-mass characteristics

The zonal mean temperature simulated by NEW in the Atlantic shows a warm bias at mid-depths (1.5 to 3 km) and a cold bias in the deeper abyssal layer (Fig.10a,b). These biases reflect important biases in NADW, which is too warm and salty (Fig.11), and which does not reach as deep as in the observations. Lack of realistic overflows from the Nordic Seas into the Irminger and Iceland basins may partly explain this enduring bias of ocean general circulation models (Danabasoglu et al. 2014, Heuzé 2021). Lack of the densest varieties of NADW can also stem from the use of climatological atmospheric forcing: interannual variations in atmospheric conditions are key to produce water-mass extremes and to simulate realistic ventilation patterns in the subpolar North Atlantic (Marzocchi et al. 2021). The absence of NADW in the abyss allows Antarctic Bottom Water (AABW) to take over this volume, leading to the cold and fresh bias deeper than 3.5 km. The density of bottom waters in NEW compares well with observations (Fig.11), so that the shallow bias of NADW
cannot be attributed to overly dense AABW. More generally, temperature and salinity biases tend to have opposed effects on density, so that the density distribution matches observations more closely than temperature or salinity (Fig.11).

In the OLD experiment, deep hydrographic biases in the Atlantic are slightly enhanced: NADW is about 0.1 °C warmer and AABW is 0.2 °C colder than in NEW (Fig.10c). Nonetheless, at high southern latitudes, in the Weddell Sea, the colder temperatures of OLD bring the model closer to observations. In the TRA experiment, Atlantic biases are faintly reduced compared to NEW (Fig.10d): NADW reaches a few hundred meters deeper, leading to warmer temperatures below 3 km depth; and the subsurface temperature maximum in the Southern Ocean is reduced, diminishing the warm bias there. The addition of mixing by subinertial internal tides barely affects the main thermocline (Fig.10d), as expected from its geographical distribution (Fig.1a).

In the Pacific Ocean, all three experiments show smaller biases than in the Atlantic (Figs.11 and 12). The main bias is an overly strong temperature stratification in the subarctic Pacific, which is about 0.2 °C too cold deeper than 4 km and 2-3 °C too warm between 200 and 1500 m (Fig.12a,b). Analysis of the emergence of subarctic Pacific biases during the first decades of the simulations shows that the anomalous subsurface warmth comes from the subtropical gyre, pointing to overestimated subsurface heat transport across the subtropical-subpolar gyre boundary in the North Pacific. The inclusion of mixing by bottom-trapped internal tides, which is particularly strong in the Kuril straits and along the Aleutian arc, alleviates subarctic Pacific temperature biases both in the abyss and upper ocean, but is insufficient to eliminate them (Fig.12d). The OLD experiment, on the other hand, simulates slightly colder Pacific bottom waters, as well as a colder thermocline and slightly colder North Pacific intermediate waters. The latter differences can be attributed to a smaller mean diapycnal diffusivity in the main thermocline of the Pacific, due to the absence of explicit mixing by the energetic low-mode internal tides of the western Pacific (Fig.2, de Lavergne et al. 2019).

In the Southern Ocean, the Pacific sector zonal mean shows again that OLD and TRA have colder subsurface waters than NEW (Fig.12c,d). The spatial pattern of this mixing-driven cooling is unveiled in Figure 13, which shows the annual mean maximum temperature of each grid column south of 50°S. Experiment NEW has too warm Circumpolar Deep Water
(CDW) in the high-latitude Southern Ocean (Fig.13a,b). This bias is slightly reduced in OLD (Fig.13c), where stronger background mixing in the pycnocline of subpolar seas (Fig.7b) contributes to vent CDW heat to the atmosphere. Mixing by subinertial internal tides is even more efficient at cooling CDW (Fig.13d): the temperature maximum in TRA is reduced throughout the subpolar seas by a few tenths of a degree or more, the effect reaching over 1 °C in some regions adjacent to Antarctica. Thus, bottom-intensified mixing energized by subinertial internal tides erodes the temperature maximum of CDW. The distribution of CDW cooling suggests that the extra mixing accelerates the transfer of CDW heat to the mixed layer and ultimately to the atmosphere, leaving less heat able to interact with Antarctic ice shelves. This effect of tides – CDW alteration by vertical ocean mixing away from sub-ice-shelf cavities – has the potential to damp other tidal effects that were shown to augment heat exchanges between ocean and ice shelves (Padman et al. 2018, Jourdain et al. 2019).

It is noteworthy that all three experiments have a well-preserved subsurface heat reservoir in the high-latitude Southern Ocean (Figs. 10 and 12). This realistic structure owes much to the absence of large open-ocean polynyas. Such polynyas frequently develop in global ocean models and deplete intermittently or permanently the subsurface heat content of southern subpolar seas, while producing AAPW (Martin et al. 2013, Heuzé et al. 2015). Here, we find that bottom water production occurs on the shelves, near the observed present-day locations (Fig.14; Silvano et al. 2023). Indeed, the bottom neutral density (Jackett and McDougall 1997) does not peak in areas of deep mixed layers (Fig.5f), but instead at four main shelf sites: the southwestern Weddell Sea, the Prydz Bay area, the Adélie coast and the Ross Sea (Fig.14b). The distribution of bottom density further indicates that dense shelf waters lose a (too) large portion of their density excess as they descend the continental slope, but remain the densest bottom waters. A numerical dye release experiment confirms that bottom waters in the model come from the four shelf sites (Fig.14c,d). The passive 'dye' tracer is restored to 1 in the top 10 meters of the ocean and propagated in offline mode using velocities and diffusivities from the last year of NEW. This tracer reaches the bottom in the locations of dense shelf water formation (bottom neutral density > 28.5 kg m⁻³) and then spreads along the seafloor downstream of these locations. We find similarly realistic ventilation pathways in OLD and TRA experiments.
505

## 3.4 Overturning circulation

The meridional overturning circulation is the leading conveyor of tracer transport between the surface boundary layer and the deep ocean. Viewed globally, it is composed of four principal cells (Lumpkin and Speer 2007) that are well captured by the model (Fig.15e,f): (i) an anticlockwise circulation linking the abyss to near-Antarctic surface waters, known as the lower cell; (ii) an overlying clockwise cell connecting both poles and traversing the upper half of the ocean, known as the AMOC; (iii) two subtropical cells confined to the upper few hundred meters and latitudes lower than 40°, circulating clockwise in the Northern Hemisphere and anticlockwise in the Southern Hemisphere. The coherence of each cell is evident when the overturning streamfunction is calculated as a function of density (Fig.15f) rather than depth (Fig.15e). Here we use simulated $σ_2$ (potential density referenced to 2000 dbars) to bin meridional velocities and we reproject the resultant streamfunction onto pseudo-depth (e.g., Zika et al. 2012) (Fig.15b,d,f). All streamfunctions shown include both resolved and parameterized (Gent and McWilliams 1990, Fox-Kemper et al. 2008) components of advection. In the reprojected view, both the AMOC and the lower cell have a magnitude that peaks near the main sinking site and declines monotonously away from this site. In experiment NEW, the lower cell has a peak magnitude of 15 Sv at 64°S, whilst the AMOC has a maximum strength of 21 Sv at 59°N. Subtropical cells reside essentially in the Pacific and Indian basins (Fig.15). When diagnosed in density coordinate, the Indian-Pacific subtropical cells of NEW attain peak strengths of 32 Sv near 10°S and 19 Sv near 5°N.

510

515

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The decomposition between Atlantic and Indian-Pacific basins reveals that sinking in the lower cell occurs in both sectors of the polar Southern Ocean, at comparable rates, yet the bulk of bottom waters is channeled toward the Indian-Pacific (Fig.15b,d). At 30°S, the lower cell of NEW has a strength of 2.9 Sv in the Atlantic, 2.4 Sv in the Indian and 4.2 Sv in the Pacific. These transports are of the same order of magnitude but smaller than recent data-constrained estimates (Holzer et al. 2021). The total northward AABW flow at 30°S is 9.5 Sv in NEW, implying that about 5 Sv of AABW upwell between 64°S and 30°S.

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530

In the North Atlantic, the AMOC has a strength that compares well with observational estimates from the RAPID mooring array (Sinha et al. 2018) which monitors the circulation at 26.5°N (Fig.16). At this latitude, the peak magnitude of the streamfunction in NEW is 17 Sv both in depth and density space. At 30°S, the peak magnitude is 16 Sv, indicating little

535
upwelling of NADW across the low-latitude Atlantic pycnocline (Fig.15d). The bulk of NADW upwells in the high-latitude Southern Ocean (Fig.15f). Nonetheless, a significant fraction is inferred to enter the Indian and Pacific basins at mid-depths and ultimately upwell at lower latitudes of the Southern Hemisphere (Fig.15b). Overlap between NADW densities and the return branch of the lower cell precludes a straightforward quantification of each upwelling pathway.

Only the lower cell is thought to rely strongly on mixing in the ocean interior, the other cells being firstly governed by surface momentum and buoyancy exchanges (Nikurashin and Vallis 2012, de Lavergne et al. 2022). In accord with these principles and the moderate perturbations of bulk mixing imposed (Fig.3, section 2.3), we find only weak differences in overturning across the three simulations (Figs. 4 and 16). For example, the modelled Atlantic overturning streamfunction at 26.5°N has a stable structure, with southward flow between about 1 and 4 km depth, biased shallow compared to RAPID observations (Fig.16). The main difference is a 1 Sv increase in peak AMOC strength from OLD to NEW, and a smaller increase from NEW to TRA (Fig.16). The lower cell also shows a very stable structure across the three experiments (not shown), with small differences in magnitude (Fig.4d,e). Differences are largest in the Pacific: OLD has the strongest Pacific AABW inflow at 30°S (5.6 Sv, compared to 4.2 Sv in NEW and 4.6 Sv in TRA), while TRA has the largest overturning rate in the subarctic Pacific abyss (1.1 Sv at 50°N, compared to 0.4 Sv in NEW and OLD). The latter finding implies that mixing by subinertial internal tides in the North Pacific invigorates the flow of AABW into the subarctic Pacific.

## 3.5 Ventilation rate

Despite small differences in overturning, we find important differences in deep ocean ventilation rates, as measured by ideal age (Fig.17). Indeed, ideal age is not only a function of overturning rate but also of diapycnal and isopycnal mixing rates, which can play a dominant role in ventilating regions where the overturning is weak (Jones and Abernathey 2019, Holzer et al. 2021). The large-scale distribution of ideal age is however closely tied to the overturning structure (Figs. 15b,d and 17a-c): ideal age is low near the sinking branches of the overturning and increases along the circulation cells carrying dense waters away from their sources. The maximum zonal mean age occurs near 3.5-4 km depth in the Atlantic and Indian basins, in the density range that separates NADW and AABW (Fig.17b,c). In the
Pacific, the maximum zonal mean age lies further aloft, near 2 km depth (Fig. 17a). This mid-depth zone of the Pacific is relatively isolated from the surface ocean because it is remote from the NADW source and above the abyssal overturning (de Lavergne et al. 2017, Holzer et al. 2021). In experiment NEW, the maximum zonal mean age reaches about 2400 years in the core of this shadow zone (Fig. 17a), where the overturning is largely absent (Fig. 15b).

As anticipated, the sensitivity of ideal age to the tidal mixing representation peaks in the Pacific shadow zone, where OLD and TRA have ages up to 250 years lower than NEW (Fig. 17d,g), largely due to higher rates of bulk diapycnal mixing within the mid-depth North Pacific (Fig. 2). But OLD and TRA also have lower ages than NEW throughout the realm of the lower cell, below about 2.5 km depth, indicating that AABW is younger from its source (Fig. 17d-i). We attribute this difference to stronger pycnocline mixing in the subpolar Southern Ocean, which lowers the age (and temperature) of CDW entrained into AABW. The effect is larger in TRA than OLD, consistent with greatest erosion of CDW near Antarctic margins in TRA (cf Fig. 13 and section 3.3). Quantitatively, the volume-weighted global mean age below 2 km is 1221 years in TRA, 1237 years in OLD and 1328 years in NEW. The reduction of about 10% of deep-ocean age in TRA relative to NEW is comparable to the percentage difference in tidal mixing power between the two simulations – a rather tight dependence which owes to the strength of subinertial internal tides in the North Pacific and near AABW formation sites (Fig. 1a).

The OLD experiment also shows a slight increase of age across much of the upper half of the ocean (Fig. 17d-f), consistent with a slower AMOC. Additionally, a pronounced aging is simulated between 1 and 2 km depth in the northern Indian Ocean (Fig. 17e). The higher ages can be explained by reduced ventilation by NADW, together with a reduced influence of young and fresh Antarctic intermediate waters, which are more rapidly eroded in OLD than NEW (not shown). The latter effect also occurs in TRA, resulting in similar though weaker and shallower ageing, near 1 km depth in the North Indian Ocean (Fig. 17h). This local age increase in TRA relative to NEW demonstrates that increased diapycnal mixing in the ocean interior can produce reduced ventilation rates on a regional scale.
4. Discussion and conclusion

We examined the impact of a comprehensive representation of mixing by breaking internal tides in a 1° global ocean configuration of NEMO. Simulations were run for 1,000 years under COREv2 normal year forcing (Large and Yeager 2009). Replacement of the constant background diffusivity of 10⁻⁵ m² s⁻¹ (experiment OLD) by an explicit representation of remote tidal mixing (experiment NEW) increases the tidal power input to mixing from 1.12 to 1.28 TW and generates substantial spatial contrasts in diffusivities (Figs. 2, 7 and 8). The improvement causes only moderate changes in the simulated broad-scale hydrography and circulation: the AMOC strengthens by about 1 Sv, the lower overturning cell weakens by about 1 Sv, salinity changes typically by less than 0.1 g kg⁻¹ and temperature by less than 0.5 °C. In general, the simulated hydrographic changes are much smaller than pre-existing model biases. This finding is consistent with the notion that atmospheric forcing and surface boundary layer processes – unchanged across our simulations – play a leading role in setting water-mass characteristics and circulation. Nonetheless, the small sensitivity of hydrography and circulation also owes to the small differences in basin-scale bulk diffusivities between OLD and NEW (Fig. 3).

Similarities between OLD and NEW simulations do not mean that the two representations of tidal mixing are equivalent. The more realistic and energy constrained distribution of tidal diffusivity in NEW increases confidence in model behavior (de Lavergne et al. 2020). For instance, the energy constraint on mixing could modify the response to forcing changes (e.g., Sigman et al. 2010), while spatial variations in upper-ocean tidal mixing rates could affect regional biogeochemical cycles (e.g., Tuerena et al. 2019). Such implications have not been investigated here.

Adding the contribution to mixing of bottom-trapped, subinertial internal tides (experiment TRA, otherwise identical to NEW) augments tidal diffusivities at high latitudes and raises the total tidal power input to 1.40 TW (Figs. 1-3). Again, this modification of tidal mixing causes only modest changes to stratification and circulation at basin scale. However, it induces some important differences in regional hydrography and deep-ocean ventilation rates, and tends to reduce known biases in our configuration. In particular, the extra mixing by subinertial internal tides causes cooling of CDW in southern subpolar seas, reducing the subsurface ocean heat that surrounds Antarctica (Fig. 17), with potential repercussions for
ice-shelf melt (Depoorter et al. 2013, Hutchinson et al. 2023). In addition, mixing by subinertial internal tides reduces deep-ocean age by up to 250 years in the Pacific shadow zone and by over 100 years on average deeper than 2 km (Fig.17). This reduction of deep-ocean age has consequences for carbon storage at depths, and more generally for biogeochemical cycles that depend strongly on deep-ocean ventilation timescales (Ödalen et al. 2018, Eggleston and Galbraith 2018).

Parameterized mixing by bottom-trapped internal tides thus has significant regional impacts on water mass transformation and passive tracer transport. The role of bottom-trapped internal tides in mixing the subsurface Arctic Ocean is well recognized (Rippeth et al. 2015, Fer et al. 2020). Here we find the largest impacts in the subarctic Pacific and high-latitude Southern Ocean. The elevated mixing along Antarctic continental slopes has particular significance because it affects critical water mass transformations occurring in this region, and contributes to shelter Antarctica from subsurface warmth contained in CDW, in synergy with subpolar gyres (Narayanan et al. 2023). This finding calls for new observational and theoretical studies of this mixing process, notably in the subpolar Southern Ocean, to reduce uncertainties in the magnitude and distribution of parameterized mixing. The impacts on mixing of both ‘internal’ and ‘topographic’ modes of conversion of subinertial surface tides need scrutiny (Tanaka 2023). In the present study, a numerical model estimate of barotropic-to-baroclinic tidal conversion (Müller 2013) has been employed to parameterize mixing due to subinertial internal tides, assuming local, bottom-intensified dissipation (section 2.3). The accuracy of the resulting mixing distribution should be assessed in future work.

Our results also depend in part on configuration and calibration choices, and the associated model limitations and biases. For example, experiment NEW simulates relatively high ages in the mid-depth Pacific shadow zone (up to 2400 years compared to peak ages of about 1600 years in data-constrained estimates; Gebbie and Huybers 2012, Holzer et al. 2021). This feature could favor strong sensitivity of the regional ideal age to increased diapycnal mixing. Similarly, biased NADW formation processes and sites could affect the diagnosed sensitivity of the AMOC to the representation of tidal mixing. Perhaps more importantly, the absence of an interactive atmosphere and of interannual variability in forcing may damp or amplify some of the sensitivities documented here. For instance, the restoring of sea surface
salinity toward observations tends to restrain differences in simulated salinity. On the other hand, this forcing protocol helps to achieve relatively realistic hydrography and circulation, and allows to perform millennial simulations at reduced computational cost. Notably, our forcing and calibration choices allow the simulation of AABW production near observed shelf sites (Fig.14), together with realistic and quasi-steady density stratification across the global deep ocean (Fig.11). This makes the present modelling framework suited to further studies of abyssal ocean ventilation and its sensitivity.

**Acknowledgements:** We thank M. Müller for sharing their tidal conversion dataset. We also thank Y. Tanaka and B. Gülk for comments. This work was granted access to the HPC resources of IDRIS under the allocations 2021-A0100107451, 2022-A0120107451 and 2023-A0140107451 made by GENCI. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreements Nr. 821001 and Nr. 101060452. Financial support was also provided by the French National LEFE programme through the ROOF project.

**Data Availability Statement:** NEMO code is available at https://forge.nemo-ocean.eu/nemo/nemo/-/releases/. The employed version is 4.2.0. The configuration files used in this study are available via the data repository at https://doi.org/10.5281/zenodo.14041098. The COREv2 forcing fields from Large and Yeager (2009) can be found at https://data1.gfdl.noaa.gov/nomads/forms/core/COREv2/CNYF_v2.html. The power maps underlying the tidal mixing parameterization of de Lavergne et al. (2020), used in experiments NEW and TRA, are available at https://www.seanoe.org/data/00619/73082/. Tidal conversion output from Müller (2013) is available at https://www.wdc-climate.de/ui/entry?acronym=DKRZ Ita 510. The observational hydrographic climatology of Gouretski (2019) is accessible at https://www.cen.uni-hamburg.de/en/icdc/data/ocean/waghc.html. The sea ice index data employed here is available at https://nsidc.org/data/g02135/versions/3. The observational climatology of
mixed layer depth from Sallée et al. (2021) can be accessed at
https://doi.org/10.5281/zenodo.5776180.

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1085 Zika, J.D., Le Sommer, J., Dufour, C.O., Molines, J.-M., Barnier, B., Brasseur, P., Dussin, R., Penduff, T., Iudicone, D., Lenton, A., Madec, G., Mathiot, P., Orr, J., Shuckburgh, E., Vivier, F., 2012. Vertical Eddy Fluxes in the Southern Ocean. J. Phys. Oceanogr. 43, 941–955. https://doi.org/10.1175/JPO-D-12-0178.1
<table><caption>**Table 1: Total power input to subinertial internal tides (E<sub>tra</sub>) and shallow internal tides (E<sub>sha</sub>).**</caption><thead><tr><th>Tidal constituent</th><th>M<sub>2</sub></th><th>S<sub>2</sub></th><th>K<sub>1</sub></th><th>O<sub>1</sub></th><th>All</th></tr></thead><tbody><tr><td>Power E<sub>tra</sub> (GW)</td><td>1.7</td><td>0.2</td><td>60</td><td>45</td><td>113</td></tr><tr><td>Power E<sub>sha</sub> (GW)</td><td>149</td><td>25</td><td>37</td><td>20</td><td>236</td></tr><tr><td>Scaling factor</td><td>1.050</td><td>1.086</td><td>1.126</td><td>1.046</td><td></td></tr></tbody></table>

These barotropic-to-baroclinic conversion rates, based on Müller (2013), are given for each of four tidal constituents and for the sum of the eight most important constituents (last column). The scaling factors used to obtain the sum for all eight constituents are given in the last row.

<figure><img src="image_3.png" /><figcaption>**Figure 1:** Depth-integrated turbulence production by subinertial internal tides (a) and shallow internal tides (b) remapped onto the model grid. The total power in each field is indicated at the top left of each panel. Note the power is shown on a log<sub>10</sub> scale.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Figure 2:** Depth-integrated turbulence production implied by each tidal mixing scheme. For experiment OLD (**a**), the power shown includes the specified near-field dissipation of internal tides and the contribution of the background diffusivity (calculated as the vertical integral of ρ₀ Kₚ N² Rf⁻¹ with Kₚ = 10⁻⁵ m² s⁻¹, ρ₀ = 1026 kg m⁻³, Rf = 1/6, and N the annual-mean buoyancy frequency from the first simulation year). Experiment NEW (**b**) has no background diffusivity: all tidal mixing derives from the specified internal tide energy dissipation. Experiment TRA (**c**) is identical to NEW except for the addition of subinertial internal tides. Note the power is shown on a log₁₀ scale.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

**Figure 3:** Turbulence production implied by each tidal mixing scheme, summed over depth and longitude (**a**) or over geopotential model layers (**b**). Experiment OLD is shown in black, experiment NEW in light blue and experiment TRA in dark blue. Note the log₁₀ scale of the x-axis of (**b**).
<figure><img src="image_6.png" /><figcaption>**Figure 4:** Evolution of global mean ocean temperature (a), Drake Passage eastward volume transport (b), maximum of the Atlantic meridional overturning streamfunction at 26.5°N (c), and northward Antarctic Bottom Water (AABW) transport at 64°S (d) and 30°S (e). Experiments OLD, NEW and TRA are shown in black, light blue and dark blue, respectively. AABW transport is the maximum northward transport underneath a given $σ_3$ isopycnal (for $σ_3$ > 1041.4 kg m⁻³).</figcaption></figure>
<figure><img src="image_7.png" /></figure>

**Figure 5:** Mixed layer depth in observations and experiment NEW, averaged over January-February-March (**a,b**), April-May-June (**c,d**), July-August-September (**e,f**) and October-November-December (**g,h**). The left column (panels (**a**), (**c**), (**e**), (**g**) corresponds to the observational climatology of Sallée et al. (2021), the right column (panels (**b**), (**d**), (**f**), (**h**) to experiment NEW. In the right column, we also show as red contours the modelled 50% sea ice concentration averaged over the central month of each season.
<figure><img src="image_8.png" /><figcaption>**Figure 6:** Maximum mixed layer depth in experiment NEW (**a**), and anomalies in OLD (**b**) and TRA (**c**) with respect to NEW. The maximum is calculated from monthly-mean output over the last decade of each experiment.</figcaption></figure>
<figure><img src="image_9.png" /></figure>

**Figure 7:** Tidal vertical diffusivity in the permanent pycnocline. Panel (a) shows the diffusivity in experiment NEW; panel (b) shows the ratio of diffusivities in OLD and NEW; and panel (c) shows the ratio of diffusivities in TRA and NEW. Diffusivities are those produced by the tidal mixing parameterization of each experiment, including the background diffusivity in the case of OLD. They are taken at the depth of maximum stratification during the month of maximum mixed layer depth, hence in the local permanent pycnocline. Note that a log₁₀ scale is used in all panels.
<figure><img src="image_10.png" /></figure>

**Figure 8:** Tidal vertical diffusivity at 210°E (or 150°W), in experiments OLD (a), NEW (b) and TRA (c). This diffusivity is that produced by the tidal mixing parameterization in each experiment, including the background diffusivity in the case of OLD. Note the diffusivity is shown on a log₁₀ scale.
<figure><img src="image_11.png" /></figure>

**Figure 9:** Seasonal evolution of Arctic (a) and Antarctic (b) sea ice extents in satellite observations (red) and experiments OLD (black), NEW (light blue) and TRA (dark blue). Observations correspond to the sea ice index of the National Snow and Ice Data Center, based on the NASA Team algorithm (Stroeve and Meier 2018). Observed values shown here are monthly averages over the period 1979-2019; the thick red curve corresponds to the climatological mean while thin red curves denote the observed range for that period.
<figure><img src="image_12.png" /><figcaption>**Figure 10:** Atlantic zonal mean temperature in the observational climatology of Gouretski (2019) (a) and experiment NEW (b). Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown. The same analysis for salinity is shown in Supplementary Figure S3.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 11:** Basin-averaged density (a,d,g), temperature (b,e,h) and salinity (c,f,i) profiles in the observational climatology of Gouretski (2019) (red) and in model experiments OLD (black), NEW (light blue) and TRA (dark blue). The Pacific Ocean (33°S-65°N) is shown in panels (a-c), the Atlantic Ocean (33°S-65°N) in panels (d-f) and the Southern Ocean (80°S-33°S) in panels (g-i). Density is measured as σ₂, the potential density referenced to 2000 dbars.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Figure 12:** Pacific zonal mean temperature in the observational climatology of Gouretski (2019) (a) and experiment NEW (b). Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown. The same analysis for salinity is shown in Supplementary Figure S4.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 13:** Annual mean maximum temperature of the water column in the observational climatology of Gouretski (2019) (a) and experiment NEW (b), shown as a stereographic projection south of 50°S. Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure 14:** Annual maximum neutral density at the seafloor in the observational climatology of Gouretski (2019) (a) and in experiment NEW (b). Panels (c) and (d) show the concentration of an artificial dye tracer in the bottom grid cell of the model (experiment NEW), respectively 10 years and 30 years after initialization. The dye tracer is restored to 1 in the top 10 meters of the model and propagates as a conservative tracer in the deeper ocean. This dye simulation was done in offline mode using five-day output from year 1,001 of experiment NEW. It shows that the bottom ocean fills from the four sites where dense shelf waters (neutral density > 28.5 kg m⁻³) are formed.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Figure 15:** Meridional overturning streamfunction in experiment NEW. The streamfunction is shown for the Indian-Pacific (**a,b**), Atlantic (**c,d**) and global (**e,f**) oceans. It was computed both in depth coordinate (**a,c,e**) and density (**σ**₂) coordinate (**b,d,f**). The density-coordinate streamfunction is reprojected to depth using the pseudo-depth of isopycnal surfaces. The pseudo-depth of an isopycnal surface within a given latitude band is the geopotential which has the same volume beneath it (e.g., Zika et al. 2012). The shown streamfunctions include both resolved and parameterized (Gent and McWilliams 1990, Fox-Kemper et al. 2008) components of advection.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Figure 16:** Meridional overturning streamfunction at 26.5°N in the Atlantic. RAPID observations (Sinha et al. 2018) are shown in red, experiment OLD in black, experiment NEW in light blue and experiment TRA in dark blue.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Figure 17:** Zonal mean ideal age simulated by experiment NEW (a-c). Anomalies in OLD (d-f) and TRA (g-i) with respect to NEW are also shown. The zonal average was taken over the Pacific (a,d,g), Indian (b,e,h) or Atlantic (c,f,i) basin, each extended due south to Antarctica.</figcaption></figure>
Effects of improved tidal mixing in NEMO one-degree global ocean model

Casimir de Lavergne¹, Saurabh Rathore¹,², Gurvan Madec¹,³, Jean-Baptiste Sallée¹, Christian
Ethé¹, Antoine Nasser¹,³, Bruno Millet⁴, Martin Vancoppenolle¹

1 LOCEAN Laboratory, Sorbonne University-CNRS-IRD-MNHN, Paris, France

5 2 Centre for Atmospheric Sciences, Indian Institute of Technology Delhi, India

3 Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France

4 Laboratoire des Sciences du Climat et de l'Environnement, CEA-CNRS-UVSQ, Université Paris-Saclay, Gif-sur-Yvette, France

10 Correspondence: Casimir de Lavergne (casimir.delavergne@locean.ipsl.fr)

Key points:

* Three tidal mixing schemes are compared in long global ocean simulations at 1° resolution.

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* Explicit representation of remote tidal mixing strengthens the Atlantic overturning circulation.

* Subinertial internal tides increase mixing in polar oceans, reducing subsurface heat fringing Antarctica and lowering abyssal ocean age.
**Abstract:** Diapycnal mixing in the ocean interior has diverse numerical representations in extant global ocean models. These representations affect the simulated transport and storage of oceanic tracers in ways that remain little studied. Here we present the impacts of three different tidal mixing schemes in thousand-year-long simulations with the NEMO global ocean model at one-degree resolution. The first scheme includes local bottom-intensified mixing at internal tide generation sites and a constant background diffusivity. The second explicitly includes both local and remote tidal mixing, with no background diffusivity. The third scheme is identical to the second but has the added contribution of bottom-trapped (subinertial) internal tides, known to be important in polar regions. The three simulations show broadly similar circulation and stratification but important regional differences. Explicit representation of remote tidal mixing strengthens the Atlantic Meridional Overturning Circulation by up to $1.5 \times 10^6 \text{ m}^3 \text{ s}^{-1}$. Inclusion of bottom-trapped internal tides reduces heat reaching Antarctica by eroding Circumpolar Deep Water at southern high latitudes, and reduces the mean age of the global deep (> 2 km) ocean by 10%. The results call for more observational constraints on polar ocean mixing, and point to multi-faceted climatic repercussions of tidal mixing representations.

**Plain Language Summary:** Winds generate surface ocean waves that break on beaches. Similarly, at the bottom of the ocean, tides generate internal ocean waves that propagate throughout the ocean and ultimately break. When such internal waves break, they cause turbulence and mixing inside the ocean. This mixing, referred to as tidal mixing, is an important process that redistributes heat, carbon and nutrients across ocean layers. In most computer models of the global ocean circulation, tides are absent but their contribution to mixing is represented by a simplified physical model. In this work, we compare three different representations of tidal mixing in global ocean simulations. We find small differences overall in the simulated ocean currents and density structure. Nonetheless, we show important regional differences and pinpoint the benefits of accounting explicitly for all pathways from tidal wave generation to mixing. The work thus leads the way to better representing tidal mixing in global ocean simulations.
# 1. Introduction

The ocean can be viewed as a stack of density layers that exchange slowly with their neighbors. Cross-layer exchanges of heat and matter occur at ocean boundaries via surface and geothermal fluxes, and in the interior via mixing of ocean tracers at molecular scale (Walin 1982, Iudicone et al. 2008). This irreversible mixing of tracers is catalyzed by small-scale turbulence, which effectively controls the rate at which layers communicate with each other in the ocean interior (Gregg 1987). By transferring mass and tracers across density layers, small-scale turbulence thus shapes the physical ocean state and regulates the transport and storage of climatically important tracers, such as nutrients, carbon and heat (Tuerena et al. 2019, de Lavergne et al. 2022, Melet et al. 2022).

Global ocean models do not resolve small-scale turbulence and must therefore resort to parameterizations of its effects. The common approach is to provide a vertical (or diapycnal) diffusivity that mixes tracers and momentum across the model layers. However, methods to determine the diapycnal diffusivity distribution vary widely across models. In the strongly mixing surface boundary layer, energetic considerations and/or classical turbulence theory are generally invoked to obtain a set of equations relating diffusivity to surface forcing, velocity shear and density stratification (Mellor and Yamada 1982, Gaspar et al. 1990, Large et al. 1994, Reichl and Hallberg 2018). In the deeper ocean, diffusivity is locally enhanced when the resolved shear or stratification is unstable. But this falls short of capturing the effects of unresolved instabilities energized by winds and tides. To account for the latter, most models combine a static background diffusivity (Jochum 2009) with a representation of bottom-enhanced tidal mixing (Simmons et al. 2004). Even so, the choice of background diffusivity and the imposed tidal mixing distribution vary across modelling centers and across model configurations (e.g., Adcroft et al. 2019, Kiss et al. 2020).

Consequences of varying the oceanic background diffusivity or the tidal mixing energy distribution have been recently explored using climate model simulations (Melet et al. 2016, Tatebe et al. 2018, Hieronymus et al. 2019). These studies highlight important changes in the heat balance and overturning circulation of the global ocean, motivating the need to better constrain diapycnal mixing in the ocean interior and to refine its representation in climate models. Studies also emphasized the need to develop energy constrained parameterizations, where diffusivity is not static but instead deduced from a three-
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dimensional distribution of turbulence production (Huang 1998, Eden et al. 2014). This
requires tracking energy routes from forcing to small-scale turbulence production.

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The main such energy route is arguably the generation, propagation and breaking of internal tides—that is, internal waves of tidal frequency sparked by interaction of barotropic tidal currents with sloping topography (Egbert and Ray 2000, Vic et al. 2019, de Lavergne et al. 2019). The traditional parameterization strategy is to estimate the rate of internal tide generation at the bottom, and assume that a third of this power dissipates in the local water column according to an exponential decay from the seafloor upwards (St Laurent et al. 2002, Simmons et al. 2004). However, it has become clear that the remaining two-thirds of power, which correspond to large-scale (or low-mode) internal tides able to propagate far from generation sites, also require explicit parameterization (Eden and Olbers 2014, MacKinnon et al. 2017, de Lavergne et al. 2019). Indeed, low-mode internal tides explain the bulk of observed dissipation patterns in the upper ocean beneath the mixed layer, and cannot be reliably represented by a constant background diffusivity (Pollmann et al. 2017, de Lavergne et al. 2020).

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Two schemes have been developed to track the propagation and dissipation of internal tide energy, including that of low-mode internal tides. The scheme of Eden and Olbers (2014) prescribes two-dimensional maps of internal wave generation, split between a high-mode and a low-mode compartment, and uses internal wave theory to redistribute this power horizontally and vertically within the simulated stratification. An alternative scheme (de Lavergne et al. 2020) prescribes four two-dimensional maps of internal tide dissipation, whose power is then distributed vertically within the simulated stratification according to a process-dependent vertical structure. The latter scheme thus neglects interactivity of the low-mode internal tide energy propagation, but allows for a realistic three-dimensional distribution of mixing energy at low computational cost (de Lavergne et al. 2020).

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Neither scheme includes subinertial internal tides, that is, internal tides whose frequency is lower than the local Coriolis frequency. Such internal tides, which occur poleward of 74.5° for the dominant M₂ semidiurnal tide and poleward of 30° for the K₁ diurnal tide, are trapped at the bottom and ultimately break to cause mixing near the seafloor (Chapman 1989, Tanaka et al. 2013, Falahat and Nycander 2014, Tanaka 2023). They are thought to play an important role in mixing the polar oceans (Rippeth et al. 2015, Fer et al. 2020),

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where they dominate over their freely propagating counterparts (Müller 2013). Yet the potential global-scale impacts of mixing by bottom-trapped internal tides remain unexplored.

Here, we implement the parameterization of local and remote tidal mixing proposed by de Lavergne et al. (2020) in the Nucleus for European Modelling of the Ocean (NEMO), and document its behavior in a long integration of a global ocean configuration with 1° nominal horizontal resolution. We also examine differences with two twin experiments: one which includes the additional – hitherto neglected – contribution of subinertial internal tides, and another which follows the traditional choice (Simmons et al. 2004) of a constant background diffusivity and local bottom-intensified tidal mixing. The configuration and mixing parameterizations are presented in detail in the next section. In section 3, we document the main characteristics and sensitivities of our simulations. We conclude with a summary and discussion of results (section 4).

## 2. Methods

### 2.1 Configuration overview and forcing

Our configuration uses version 4.2 of NEMO (NEMO System Team 2022), which solves the primitive equations in Boussinesq approximation. The ocean model component is coupled to the SI³ dynamic-thermodynamic sea ice model (Rousset et al. 2015, Vancoppenolle et al. 2023). We employ a global Mercator grid of 1° nominal horizontal resolution, referred to as ORCA1. The meridional resolution is refined at high latitudes so that the meridional grid spacing matches the reducing zonal spacing, and refined within 15° of the equator to better resolve equatorial dynamics. As a result, the average area-weighted horizontal resolution is 0.69° or 83 km over the global ocean; in the Southern Ocean south of 40°S, it is 0.57° or 45 km. In the vertical, the model includes 75 layers that follow geopotential surfaces, except for time-dependent stretching or squeezing of layers to accompany changes in sea surface height. The vertical grid spacing varies from about 1 m near the surface to 200 m in the abyssal ocean. Partial cells at the bottom allow for a better representation of the observed bathymetry. The domain covers depths of up to 6,000 m.
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The model is run for 1,000 years with a time step of 45 minutes. Initial hydrographic conditions are Conservative Temperature and preformed salinity fields calculated from the observational climatology of Gourestki (2019) using the Gibbs SeaWater Toolbox (McDougall and Barker 2011). Preformed salinity is used instead of Absolute Salinity because the model, while employing the TEOS-10 equation of state (IOC, SCOR and IAPSO 2010, Roquet et al. 2015), does not represent biological (non conservative) sources of salt included in Absolute Salinity. For brevity, Conservative Temperature and preformed salinity will be referred to as temperature and salinity in all the following. At the simulation start, velocities are zero and sea ice is present where the initial sea surface temperature is within 1.0 °C of the freezing point.

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At the bottom, the model is forced by geothermal heat fluxes mapped by Lucazeau (2019) (Supplementary Figure S1). At the surface, atmospheric conditions are prescribed following version 2 of the Coordinated Ocean-ice Reference Experiments (COREv2) normal year forcing (Large and Yeager 2009), which represents a typical year of the instrumental period (mostly between 1979 and 2006). This forcing includes 6-hourly fields of zonal and meridional wind at 10 m and of air temperature and humidity at 10 m. In addition, it includes daily mean radiation and monthly mean precipitation (rain and snow) reaching the surface. Freshwater forcing also includes river runoff, iceberg melt released in the surface layer following the spatial pattern estimated by Merino et al. (2016), and basal melt from Antarctic ice-shelf cavities released at depth over the mouth of cavities (Mathiot et al. 2017). Basal melt fluxes from each cavity and the total magnitude of Antarctic iceberg melt are prescribed according to estimates of Depoorter et al. (2013); they are not interactive and the cavities are closed.

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Surface fluxes including evaporation are computed using bulk formulae (Large and Yeager 2009), as a function of the simulated sea surface temperature and sea ice cover. To compensate for missing atmospheric feedbacks, restoring to the observed seasonally varying sea surface salinity (as mapped by Gouretski (2019)) is activated, with a timescale of 230 days per 50 m of seawater. This restoring term is expressed as a freshwater flux added to the local evaporation minus precipitation flux. To ensure that the freshwater balance is globally closed and avoid drift in sea level, the global mean freshwater flux leaving the ocean is calculated at each time step and subtracted from the evaporation minus

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precipitation flux of every surface ocean grid cell. This uniform adjustment adds 0.13 m of
sea level every year on average in our simulations, a magnitude that can be compared to the
1.11 m added by precipitation every year. Closure of the freshwater balance in forced-ocean
experiments is a delicate issue: our experience shows that the required adjustment can
grow large in the absence of careful calibration. In addition, this adjustment introduces a
non-physical global interdependence of freshwater fluxes that should be borne in mind
when interpreting temporal variability or sensitivity to forcing and physics.

## 2.2 Representation of advection and diffusion

The vector-invariant form of momentum equations is chosen together with the rotational-divergence form of the stress tensor (Madec et al. 1991). Advection of momentum uses a second-order centered scheme. The relative and planetary vorticity terms are discretized with a scheme that conserves energy and potential enstrophy of non-divergent flow (Burridge and Haseler 1977, Arakawa and Lamb 1981). These choices allow for reduced sensitivity of the simulated circulation to artificial indentation of model isobaths (Nasser et al. 2023). Momentum diffusion is accomplished by a Laplacian operator and a prescribed horizontal eddy diffusivity coefficient of 20,000 m² s⁻¹, reduced to 1,000 m² s⁻¹ around the equator (Delecluse and Madec 1999). The lateral boundary condition is free slip. To aid downslope flow of dense water along the staircase-like topography, the advective scheme of Campin and Goosse (1999) is activated. A quadratic bottom drag is also included, where the drag coefficient is C<sub>d</sub> = 10⁻³ [0.25(u² + v²) + u<sub>t</sub>²]⁰.⁵ with u and v the simulated bottom horizontal velocities and u<sub>t</sub> a constant (tidal) velocity of 0.05 m s⁻¹.

Restratification by baroclinic mesoscale eddies is parameterized by downgradient diffusion of isopycnal layer thickness, following the approach of Gent and McWilliams (1990). The associated thickness diffusivity varies horizontally with the simulated stratification (Supplementary Figure S2): it is proportional to the square of the Rossby radius and to the Eady growth rate (Held and Larichev 1996, Treguier et al. 1997), and tapered equatorward of 20° to account for the increasing sparsity of baroclinic mesoscale eddies there (Chelton et al. 2011). Restratification of the surface mixed layer by submesoscale eddies is parameterized following Fox-Kemper et al. (2008).
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Advection of all tracers is performed with a second-order Flux Corrected Transport scheme. Diffusion of tracers along isopycnals by mesoscale and smaller scale turbulence (Redi 1982, Abernathey et al. 2022) is parameterized with a diffusivity that varies geographically with the local Rossby radius and Eady growth rate (Supplementary Figure S2), like the thickness diffusivity.

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Vertical diffusion of momentum and tracers is parameterized by three components: a Turbulent Kinetic Energy (TKE) closure, a convection scheme, and a tidal mixing scheme. The TKE closure is order 1.5: it includes a prognostic equation for TKE and diagnostic equations relating mixing parameters and TKE (Gaspar et al. 1990, Blanke and Delecluse 1993, NEMO System Team 2022). This scheme produces enhanced vertical mixing when the Richardson number of the resolved flow falls below a critical value of 0.22. Mixing is further enhanced near the base of the mixed layer to account for the effects of unresolved Langmuir cells (Axell 2002) and near-inertial oscillations. Near-inertial oscillations of the mixed layer, generated by high-frequency winds, fuel intense mixing at the base of the mixed layer and relatively weak mixing in the deeper ocean (Plueddemann and Farrar 2006, Alford 2020, von Storch and Lüschow 2023). In the TKE scheme of NEMO their contribution to upper-ocean mixing is represented by adding a vertical profile of TKE that decays exponentially with depth and that peaks at a small fraction of the surface TKE (Rodgers et al. 2014). Here we choose the fraction to be 8%, and the decay scale to vary from 1 m at the equator to 30 m at high latitudes. The advantages of this scheme are its simplicity and interactivity: the induced mixing depends on the surface TKE hence on the instantaneous wind forcing. The drawbacks are the ad hoc vertical profile and the absence of constraint on the power consumed by this mixing.

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The convective scheme simply consists in enhancing the vertical diffusivity to 10 m² s⁻¹ when stratification is statically unstable. No parameterization of double diffusive mixing is included here. Hence, in our experiments, vertical mixing in the ocean interior (away from the surface boundary layer) relies almost exclusively on the tidal mixing parameterization, which is detailed next.

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## 2.3 Tidal mixing parameterization and sensitivity experiments

To investigate the sensitivity of the ocean state to the representation of tidal mixing, we performed three experiments. Experiment OLD includes a constant background diffusivity of $10^{-5} \text{ m}^2 \text{ s}^{-1}$ and local bottom-intensified mixing at internal tide generation sites (Simmons et al. 2004). This bottom-intensified mixing is specified by assuming that: (i) one-third of the power input to internal tides mapped by Nycander (2005) fuels turbulence production in the local water column; and (ii) turbulence production decays exponentially with height above bottom, with a uniform 500 m decay scale (St Laurent et al. 2002). Experiment NEW includes explicit and comprehensive treatment of freely propagating internal tides; that is, it incorporates the tidal mixing scheme of de Lavergne et al. (2020) with the added contribution of shallow internal tides ($E_{sha}$), implemented as described below. Experiment TRA is identical to NEW but includes the additional contribution of subinertial internal tides ($E_{tra}$), also detailed next.

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Tidal mixing representations in both NEW and TRA rest upon the energy-constrained framework of de Lavergne et al. (2020). This framework employs four static maps of depth-integrated internal tide energy dissipation, which have been remapped onto the ORCA1 grid using an energy-conserving interpolation. Each map is associated with a distinct dissipation mechanism and a corresponding vertical structure of turbulence production, which depends on the simulated stratification. Hence, at each time step, the power available in each water column is distributed in the vertical, yielding a global three-dimensional distribution of turbulence production $\epsilon(x,y,z)$. The vertical diffusivity $K_\rho$ is then deduced from turbulence production using a zero-order closure, $K_\rho = R_f \epsilon N^{-2}$, where N is the buoyancy frequency and $R_f$ is the mixing efficiency (Osborn 1980). Mixing efficiency is known to vary depending on local flow and turbulence characteristics (e.g., Mashayek and Peltier 2013, Bouffard and Boegman 2013, Vladoiu et al. 2019). However there is no consensus about, nor comprehensive model of, mixing efficiency variations in the ocean, so that the traditional assumption of a constant $R_f = 1/6$ remains the standard choice (Gregg et al. 2018). For simplicity, we take $R_f = 1/6$ in all three experiments. 500-year simulations performed with a coarser ($2^\circ$) resolution global ocean configuration showed only weak sensitivity of ocean circulation to mixing efficiency variations when parameterized following de Lavergne et al. (2016). An upper bound of $10^{-2} \text{ m}^2 \text{ s}^{-1}$ is imposed on the tidal vertical diffusivity. The lower

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bound is $1.4 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$ for tracers and $1.4 \times 10^{-6} \text{ m}^2 \text{ s}^{-1}$ for momentum, corresponding to molecular heat diffusivity and kinematic viscosity, respectively. No differential mixing between temperature and salinity is implemented.

The mapping of internal tide energy dissipation performed by de Lavergne et al. (2019, 2020) uses analytical estimates of internal tide generation that are valid only at seafloor depths greater than 400 m and equatorward of the turning latitude of each tidal constituent, due to assumptions in the analytical theory (Bell 1975, Falahat et al. 2014). To remedy these two limitations, we turn to an alternative estimate of internal tide generation, obtained with a high-resolution ocean general circulation model (Müller 2013). This model estimate agrees broadly with the analytical calculation of Falahat et al. (2014) within overlapping validity regions, while also including internal tides generated over shallow (< 400 m) topography and poleward of their turning latitude. Müller (2013) provides four two-dimensional power fields, corresponding to the four tidal constituents ($M_2$, $S_2$, $K_1$ and $O_1$) included in the simulation. These fields have positive and realistic global integrals (Table 1) yet contain frequent negative values, like the analytical estimates of Falahat et al. (2014). We correct for these negative values following de Lavergne et al. (2019): we first set negative values to zero, then multiply the resulting fields by a depth-dependent factor that preserves the original depth-profile of internal tide generation. This procedure is ad hoc but yields plausible distributions (de Lavergne et al. 2019; Fig.1). Within each of the four corrected fields, we select regions where the tidal frequency exceeds the Coriolis frequency to obtain the generation rate of trapped internal tides ($E_{\text{tra}}$). In the remainder of the ocean, we select grid cells where bathymetry is less than 400 m to obtain the generation rate of shallow internal tides ($E_{\text{sha}}$). The total power available from each of the two components and each of the four tidal constituents is given in Table 1.

To implement these two components in the tidal mixing parameterization, we sum over the four tidal constituents, using scaling factors (see Table 1) to account for missing constituents, following de Lavergne et al. (2019). We then remap the total fields, $E_{\text{tra}}$ and $E_{\text{sha}}$, onto our model grid (Fig.1). The two fields correspond to generation rates, not dissipation rates, hence they are not direct estimates of turbulence production integrated over the water column. However, we expect the dissipation to be near local in both cases, because subinertial internal tides are bottom-trapped (Albrecht et al. 2006, Tanaka et al.
2013, Musgrave et al. 2016), and because the decay time of internal tides implied by wave-
wave interactions is very short in shallow seas (de Lavergne et al. 2019). We thus assume
local dissipation at the 1° resolution of our configuration. Subinertial internal tides are
expected and observed to cause mixing intensified near the seafloor (Tanaka et al. 2013,
Rippeth et al. 2015, Musgrave et al. 2017): we choose the same vertical structure as that
attributed to dissipation at critical topography, i.e. an exponential decay from the bottom
upwards with a decay scale determined by local topographic slopes (de Lavergne et al.
2020). Shallow internal tides are expected to dissipate primarily via wave-wave interactions:
we thus distribute their power in the vertical as proportional to the square of the simulated
buoyancy frequency (Gregg 1989, D'Asaro and Lien 2000).

The depth-integrated turbulence production implied by each of the three tidal mixing
representations is mapped in Figure 2. For the OLD experiment, the power implicitly
consumed by the background diffusivity is included, and calculated as the vertical integral of
ρ₀ Kₚ N² R_f⁻¹ with ρ₀ = 1026 kg m⁻³ the reference density of seawater and N the annual mean
buoyancy frequency from the first simulation year. We find that the background diffusivity
of 10⁻⁵ m² s⁻¹ in OLD consumes 0.72 TW of power globally. The global total power amounts
to 1.12 TW in OLD, 1.28 TW in NEW and 1.40 TW in TRA. Although these totals are of the
same order of magnitude, they hide substantial differences in geographical patterns of
mixing (Fig.2). Explicit representation of remote tidal mixing creates a much more
heterogeneous mixing distribution, reflecting the heterogeneous sources and sinks of
internal tide energy. Inclusion of trapped internal tides raises power availability poleward of
30° and most conspicuously in the subarctic Pacific, Arctic and Southern Oceans (Figs. 1a
and 2b,c).

Similarities and differences between the three mixing representations are further illustrated
in Figure 3, which shows turbulence production summed over depth and longitude (Fig.3a)
or along geopotential model layers (Fig.3b). The latitudinal profiles highlight the similarity of
bulk mixing rates across the three simulations. Most important relative differences occur at
high latitudes, where TRA has the highest power input to mixing, except north of 70°N; the
background diffusivity of OLD sustains unrealistically high mixing in the Arctic pycnocline
(Nguyen et al. 2009). The depth profiles highlight the large proportion of power contained in
the top 500 m: contrary to common perception, internal tide-driven mixing is prominent in
the upper ocean, not only in the deep ocean (Kunze et al. 2006, de Lavergne et al. 2020). The tidal mixing parameterization in NEW and TRA generates more mixing than OLD in the ocean's top kilometer, while the opposite is true between 1 and 3 km depth.

It must be emphasized that power consumed by tidal mixing in OLD is not fixed in time: it evolves with stratification changes. The total power effectively consumed in our OLD experiment increases by 1.5% over the full length of the simulation; a relatively small change that reflects the small drift in stratification. However, should stratification increase rapidly, for example under a strong greenhouse forcing scenario, the power consumed by the constant background diffusivity could increase substantially (de Lavergne et al. 2020). The map and curves shown for OLD in Figures 2 and 3 are thus only applicable to the initial state of our experiment: the same diagnostics applied to a different model state would generate different magnitudes and patterns. These considerations underline the inadequacy of static diffusivities: they ignore energy constraints and do not allow for a realistic response of mixing to forcing or stratification changes (Eden et al. 2014, Sigman et al. 2010).

## 2.4 Ideal age and offline simulation

To investigate ventilation timescales, the simulations include an ideal age tracer, which measures the time elapsed since local seawater was last at the surface (England 1995). This tracer is advected and diffused using the same velocities, diffusivities and numerical schemes as those transporting temperature and salinity. Ideal age is restored to zero in the top 10 m of ocean, and increases by 1 year per year deeper than 10 m. It is initialized with uniform null concentrations. After 1,000 years, it is still far from equilibrium in the deep ocean, despite relative stabilization of the circulation and stratification (Fig.4). In order to quantify deep ventilation rates in the near steady state, we extend the simulations of ideal age in offline mode, for an extra 3,000 years. The offline simulation uses physical variables (velocities, diffusivities, layer thicknesses, temperature and salinity) from the last year of the full simulation to evolve the passive tracer distribution. In detail, we output physical variables from year 1,001 of each experiment at five-day frequency, and simulate ideal age as in the online simulation, except that physics are not computed but instead read from this five-day output and time-interpolated at each time step.
# 3. Results

## 3.1 Drift

The global mean ocean temperature increases by 0.29 °C in OLD, 0.34 °C in TRA and 0.35 °C in NEW over 1,000 years (Fig.4a). The drift is not linear and slows progressively. Over the last century of the simulations, global mean temperature increases only by 1.5 x 10⁻³ °C in OLD, 8.6 x 10⁻³ °C in NEW and 4.3 x 10⁻³ °C in TRA; increases in mean deep ocean temperature are similar or smaller. By construction (section 2.1), the salt content and volume of the global ocean are constant over the simulation, hence the mean salinity is also constant. Redistribution between layers can nonetheless occur: over the whole simulation length and across the experiments, the mean salinity in the upper km decreases by 0.022-0.050 g kg⁻¹ whereas the mean salinity in the deeper ocean increases by 0.007-0.016 g kg⁻¹. Overall, the drift toward warmer and saltier deep waters can largely be explained by a warm and salty bias at North Atlantic Deep Water (NADW) formation sites, which is progressively communicated to the global deep ocean (see section 3.3).

The overturning circulation shows relatively rapid stabilization: the Atlantic Meridional Overturning Circulation (AMOC) and the lower cell of the MOC reach a near-steady strength after a few hundred years (Fig.4c-e) in all three experiments. The strength of the lower cell is diagnosed in density coordinate and shown both for the latitude of peak strength (64°S; Fig.4d) and a middle latitude (30°S; Fig.4e). The eastward transport through Drake Passage, which is a key indicator of Southern Ocean circulation and stratification, is relatively stable in all three experiments (Fig.4b; mind the small y-axis range). Specifically, it increases from 163 Sv (1 Sv is 10⁶ m³ s⁻¹) in the initial year to 172 Sv, 174 Sv and 181 Sv at the end of the TRA, NEW and OLD experiments, respectively. The positive drift over the tenth century is largest in TRA but lesser than 0.4 Sv. Observational estimates of the Drake Passage transport range between 134 Sv and 175 Sv (Cunningham et al. 2003, Donohue et al. 2016, Colin de Verdière and Ollitrault 2016); our simulations thus lie near the high end of observations.

To bring out the level of equilibration of the model stratification and circulation in the final state of our experiments, we extended experiment TRA for an additional 500 years. The extended time series show that the circulation of TRA is quasi steady by year 1,000 (Fig.4b-e). Only the global mean ocean temperature displays a persistent, weak drift of 3.6 x 10⁻³ °C
per century (Fig.4a). Given the small degree of variability and drift present in the analyzed time series near the end of our millennial integrations (Fig.4), we focus hereafter on simulation characteristics averaged over years 991-1,000 of each experiment. When discussing anomalies, we will take NEW as the reference experiment and examine differences between OLD and NEW (OLD – NEW) or between TRA and NEW (TRA – NEW).

## 3.2 Surface boundary layer

The surface mixed layer is central to ocean ventilation, being the layer that is effectively in contact with the atmosphere. The depth of this layer, and its spatial and seasonal variations, contribute to shape water-mass characteristics and transports throughout the ocean (Marshall and Nurser 1991, Williams et al. 1995). Figure 5 compares the mixed layer depth (MLD) in experiment NEW to the observational map of Sallée et al. (2021). In both cases MLD is defined as the depth at which potential density exceeds its 10 m value by $0.03 \text{ kg m}^{-3}$ (de Boyer Montégut et al. 2004). The broad geographical patterns are well captured by the model for each season. The main difference is a tendency to overestimate MLD at high latitudes in winter and spring, indicative of overestimated winter deep convection and possibly underestimated spring restratification. Main deep convection sites in the model are the Labrador Sea, Irminger Sea, Norwegian Sea, southern Weddell Sea and a band along the northern edge of the Antarctic Circumpolar Current (ACC).

Overall, the simulated MLDs are weakly sensitive to the representation of tidal mixing. Principle differences between experiments occur in winter. Figure 6 shows the annual maximum MLD of NEW and anomalies in OLD and TRA relative to NEW. In the North Atlantic, OLD has shallower winter deep convection than NEW (Fig.6b), consistent with weaker NADW formation and a weaker AMOC. On the other hand, OLD has deeper winter mixed layers in the polar oceans (Arctic and near-Antarctic), which we attribute to higher levels of mixing within polar haloclines (Figs. 3 and 7) that precondition winter convection (Nguyen et al. 2009, Morawitz et al. 1996). Significant anomalies in subantarctic MLDs are also apparent, with an overall shoaling in the Atlantic and Indian sectors and deepening in the Pacific sector (Fig.6b). These changes are suggestive of slight frontal shifts in the ACC, combined to local preconditioning of deep convection by interior mixing (Sloyan et al. 2010).
Southern Hemisphere MLD anomalies in TRA are qualitatively similar to those in OLD, albeit weaker. This similarity suggests that the MLDs are sensitive to the bulk mixing across the high-latitude permanent pycnocline. Both TRA and OLD have stronger average mixing across the Southern Ocean pycnocline than NEW: OLD has higher mixing throughout most of the upper ocean while TRA has higher mixing along continental slopes (Figs. 7 and 8). In the Northern Hemisphere, differences between TRA and NEW are mostly found in the subpolar North Atlantic. In particular, MLDs deepen in the Irminger Sea and central Labrador Sea, possibly contributing to a slightly stronger AMOC (Fig.4c).

Sea ice coverage exerts a profound influence on deep ocean stratification and ventilation, particularly in the Southern Ocean (Goosse and Fichefet 1999, Haumann et al. 2016, Klocker et al. 2023), and can be sensitive to deep ocean mixing (Tatebe et al. 2018). In our forced ocean experiments, sea ice is insensitive to the chosen tidal mixing scheme in both hemispheres (Fig.9). The seasonal expansion and retreat of the sea ice pack is reasonably well represented, despite Antarctic sea ice dropping too low in summer and retreating too late during the warming season (Fig.9b). Simulated deep mixed layers in southern polar waters are not associated with openings in the sea ice (Fig.5f,h). We will show in the next section that bottom waters are produced on Antarctic shelves rather than in open ocean convection zones.

## 3.3 Water-mass characteristics

The zonal mean temperature simulated by NEW in the Atlantic shows a warm bias at mid-depths (1.5 to 3 km) and a cold bias in the deeper abyssal layer (Fig.10a,b). These biases reflect important biases in NADW, which is too warm and salty (Fig.11), and which does not reach as deep as in the observations. Lack of realistic overflows from the Nordic Seas into the Irminger and Iceland basins may partly explain this enduring bias of ocean general circulation models (Danabasoglu et al. 2014, Heuzé 2021). Lack of the densest varieties of NADW can also stem from the use of climatological atmospheric forcing: interannual variations in atmospheric conditions are key to produce water-mass extremes and to simulate realistic ventilation patterns in the subpolar North Atlantic (Marzocchi et al. 2021). The absence of NADW in the abyss allows Antarctic Bottom Water (AABW) to take over this volume, leading to the cold and fresh bias deeper than 3.5 km. The density of bottom waters in NEW compares well with observations (Fig.11), so that the shallow bias of NADW
cannot be attributed to overly dense AABW. More generally, temperature and salinity biases tend to have opposed effects on density, so that the density distribution matches observations more closely than temperature or salinity (Fig.11).

In the OLD experiment, deep hydrographic biases in the Atlantic are slightly enhanced: NADW is about 0.1 °C warmer and AABW is 0.2 °C colder than in NEW (Fig.10c). Nonetheless, at high southern latitudes, in the Weddell Sea, the colder temperatures of OLD bring the model closer to observations. In the TRA experiment, Atlantic biases are faintly reduced compared to NEW (Fig.10d): NADW reaches a few hundred meters deeper, leading to warmer temperatures below 3 km depth; and the subsurface temperature maximum in the Southern Ocean is reduced, diminishing the warm bias there. The addition of mixing by subinertial internal tides barely affects the main thermocline (Fig.10d), as expected from its geographical distribution (Fig.1a).

In the Pacific Ocean, all three experiments show smaller biases than in the Atlantic (Figs.11 and 12). The main bias is an overly strong temperature stratification in the subarctic Pacific, which is about 0.2 °C too cold deeper than 4 km and 2-3 °C too warm between 200 and 1500 m (Fig.12a,b). Analysis of the emergence of subarctic Pacific biases during the first decades of the simulations shows that the anomalous subsurface warmth comes from the subtropical gyre, pointing to overestimated subsurface heat transport across the subtropical-subpolar gyre boundary in the North Pacific. The inclusion of mixing by bottom-trapped internal tides, which is particularly strong in the Kuril straits and along the Aleutian arc, alleviates subarctic Pacific temperature biases both in the abyss and upper ocean, but is insufficient to eliminate them (Fig.12d). The OLD experiment, on the other hand, simulates slightly colder Pacific bottom waters, as well as a colder thermocline and slightly colder North Pacific intermediate waters. The latter differences can be attributed to a smaller mean diapycnal diffusivity in the main thermocline of the Pacific, due to the absence of explicit mixing by the energetic low-mode internal tides of the western Pacific (Fig.2, de Lavergne et al. 2019).

In the Southern Ocean, the Pacific sector zonal mean shows again that OLD and TRA have colder subsurface waters than NEW (Fig.12c,d). The spatial pattern of this mixing-driven cooling is unveiled in Figure 13, which shows the annual mean maximum temperature of each grid column south of 50°S. Experiment NEW has too warm Circumpolar Deep Water
(CDW) in the high-latitude Southern Ocean (Fig.13a,b). This bias is slightly reduced in OLD (Fig.13c), where stronger background mixing in the pycnocline of subpolar seas (Fig.7b) contributes to vent CDW heat to the atmosphere. Mixing by subinertial internal tides is even more efficient at cooling CDW (Fig.13d): the temperature maximum in TRA is reduced throughout the subpolar seas by a few tenths of a degree or more, the effect reaching over 1 °C in some regions adjacent to Antarctica. Thus, bottom-intensified mixing energized by subinertial internal tides erodes the temperature maximum of CDW. The distribution of CDW cooling suggests that the extra mixing accelerates the transfer of CDW heat to the mixed layer and ultimately to the atmosphere, leaving less heat able to interact with Antarctic ice shelves. This effect of tides – CDW alteration by vertical ocean mixing away from sub-ice-shelf cavities – has the potential to damp other tidal effects that were shown to augment heat exchanges between ocean and ice shelves (Padman et al. 2018, Jourdain et al. 2019).

It is noteworthy that all three experiments have a well-preserved subsurface heat reservoir in the high-latitude Southern Ocean (Figs. 10 and 12). This realistic structure owes much to the absence of large open-ocean polynyas. Such polynyas frequently develop in global ocean models and deplete intermittently or permanently the subsurface heat content of southern subpolar seas, while producing AAPW (Martin et al. 2013, Heuzé et al. 2015). Here, we find that bottom water production occurs on the shelves, near the observed present-day locations (Fig.14; Silvano et al. 2023). Indeed, the bottom neutral density (Jackett and McDougall 1997) does not peak in areas of deep mixed layers (Fig.5f), but instead at four main shelf sites: the southwestern Weddell Sea, the Prydz Bay area, the Adélie coast and the Ross Sea (Fig.14b). The distribution of bottom density further indicates that dense shelf waters lose a (too) large portion of their density excess as they descend the continental slope, but remain the densest bottom waters. A numerical dye release experiment confirms that bottom waters in the model come from the four shelf sites (Fig.14c,d). The passive 'dye' tracer is restored to 1 in the top 10 meters of the ocean and propagated in offline mode using velocities and diffusivities from the last year of NEW. This tracer reaches the bottom in the locations of dense shelf water formation (bottom neutral density > 28.5 kg m⁻³) and then spreads along the seafloor downstream of these locations. We find similarly realistic ventilation pathways in OLD and TRA experiments.
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## 3.4 Overturning circulation

The meridional overturning circulation is the leading conveyor of tracer transport between the surface boundary layer and the deep ocean. Viewed globally, it is composed of four principal cells (Lumpkin and Speer 2007) that are well captured by the model (Fig.15e,f): (i) an anticlockwise circulation linking the abyss to near-Antarctic surface waters, known as the lower cell; (ii) an overlying clockwise cell connecting both poles and traversing the upper half of the ocean, known as the AMOC; (iii) two subtropical cells confined to the upper few hundred meters and latitudes lower than 40°, circulating clockwise in the Northern Hemisphere and anticlockwise in the Southern Hemisphere. The coherence of each cell is evident when the overturning streamfunction is calculated as a function of density (Fig.15f) rather than depth (Fig.15e). Here we use simulated $σ_2$ (potential density referenced to 2000 dbars) to bin meridional velocities and we reproject the resultant streamfunction onto pseudo-depth (e.g., Zika et al. 2012) (Fig.15b,d,f). All streamfunctions shown include both resolved and parameterized (Gent and McWilliams 1990, Fox-Kemper et al. 2008) components of advection. In the reprojected view, both the AMOC and the lower cell have a magnitude that peaks near the main sinking site and declines monotonously away from this site. In experiment NEW, the lower cell has a peak magnitude of 15 Sv at 64°S, whilst the AMOC has a maximum strength of 21 Sv at 59°N. Subtropical cells reside essentially in the Pacific and Indian basins (Fig.15). When diagnosed in density coordinate, the Indian-Pacific subtropical cells of NEW attain peak strengths of 32 Sv near 10°S and 19 Sv near 5°N.

510

515

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The decomposition between Atlantic and Indian-Pacific basins reveals that sinking in the lower cell occurs in both sectors of the polar Southern Ocean, at comparable rates, yet the bulk of bottom waters is channeled toward the Indian-Pacific (Fig.15b,d). At 30°S, the lower cell of NEW has a strength of 2.9 Sv in the Atlantic, 2.4 Sv in the Indian and 4.2 Sv in the Pacific. These transports are of the same order of magnitude but smaller than recent data-constrained estimates (Holzer et al. 2021). The total northward AABW flow at 30°S is 9.5 Sv in NEW, implying that about 5 Sv of AABW upwell between 64°S and 30°S.

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530

In the North Atlantic, the AMOC has a strength that compares well with observational estimates from the RAPID mooring array (Sinha et al. 2018) which monitors the circulation at 26.5°N (Fig.16). At this latitude, the peak magnitude of the streamfunction in NEW is 17 Sv both in depth and density space. At 30°S, the peak magnitude is 16 Sv, indicating little

535
upwelling of NADW across the low-latitude Atlantic pycnocline (Fig.15d). The bulk of NADW upwells in the high-latitude Southern Ocean (Fig.15f). Nonetheless, a significant fraction is inferred to enter the Indian and Pacific basins at mid-depths and ultimately upwell at lower latitudes of the Southern Hemisphere (Fig.15b). Overlap between NADW densities and the return branch of the lower cell precludes a straightforward quantification of each upwelling pathway.

Only the lower cell is thought to rely strongly on mixing in the ocean interior, the other cells being firstly governed by surface momentum and buoyancy exchanges (Nikurashin and Vallis 2012, de Lavergne et al. 2022). In accord with these principles and the moderate perturbations of bulk mixing imposed (Fig.3, section 2.3), we find only weak differences in overturning across the three simulations (Figs. 4 and 16). For example, the modelled Atlantic overturning streamfunction at 26.5°N has a stable structure, with southward flow between about 1 and 4 km depth, biased shallow compared to RAPID observations (Fig.16). The main difference is a 1 Sv increase in peak AMOC strength from OLD to NEW, and a smaller increase from NEW to TRA (Fig.16). The lower cell also shows a very stable structure across the three experiments (not shown), with small differences in magnitude (Fig.4d,e). Differences are largest in the Pacific: OLD has the strongest Pacific AABW inflow at 30°S (5.6 Sv, compared to 4.2 Sv in NEW and 4.6 Sv in TRA), while TRA has the largest overturning rate in the subarctic Pacific abyss (1.1 Sv at 50°N, compared to 0.4 Sv in NEW and OLD). The latter finding implies that mixing by subinertial internal tides in the North Pacific invigorates the flow of AABW into the subarctic Pacific.

## 3.5 Ventilation rate

Despite small differences in overturning, we find important differences in deep ocean ventilation rates, as measured by ideal age (Fig.17). Indeed, ideal age is not only a function of overturning rate but also of diapycnal and isopycnal mixing rates, which can play a dominant role in ventilating regions where the overturning is weak (Jones and Abernathey 2019, Holzer et al. 2021). The large-scale distribution of ideal age is however closely tied to the overturning structure (Figs. 15b,d and 17a-c): ideal age is low near the sinking branches of the overturning and increases along the circulation cells carrying dense waters away from their sources. The maximum zonal mean age occurs near 3.5-4 km depth in the Atlantic and Indian basins, in the density range that separates NADW and AABW (Fig.17b,c). In the
Pacific, the maximum zonal mean age lies further aloft, near 2 km depth (Fig. 17a). This mid-depth zone of the Pacific is relatively isolated from the surface ocean because it is remote from the NADW source and above the abyssal overturning (de Lavergne et al. 2017, Holzer et al. 2021). In experiment NEW, the maximum zonal mean age reaches about 2400 years in the core of this shadow zone (Fig. 17a), where the overturning is largely absent (Fig. 15b).

As anticipated, the sensitivity of ideal age to the tidal mixing representation peaks in the Pacific shadow zone, where OLD and TRA have ages up to 250 years lower than NEW (Fig. 17d,g), largely due to higher rates of bulk diapycnal mixing within the mid-depth North Pacific (Fig. 2). But OLD and TRA also have lower ages than NEW throughout the realm of the lower cell, below about 2.5 km depth, indicating that AABW is younger from its source (Fig. 17d-i). We attribute this difference to stronger pycnocline mixing in the subpolar Southern Ocean, which lowers the age (and temperature) of CDW entrained into AABW. The effect is larger in TRA than OLD, consistent with greatest erosion of CDW near Antarctic margins in TRA (cf Fig. 13 and section 3.3). Quantitatively, the volume-weighted global mean age below 2 km is 1221 years in TRA, 1237 years in OLD and 1328 years in NEW. The reduction of about 10% of deep-ocean age in TRA relative to NEW is comparable to the percentage difference in tidal mixing power between the two simulations – a rather tight dependence which owes to the strength of subinertial internal tides in the North Pacific and near AABW formation sites (Fig. 1a).

The OLD experiment also shows a slight increase of age across much of the upper half of the ocean (Fig. 17d-f), consistent with a slower AMOC. Additionally, a pronounced aging is simulated between 1 and 2 km depth in the northern Indian Ocean (Fig. 17e). The higher ages can be explained by reduced ventilation by NADW, together with a reduced influence of young and fresh Antarctic intermediate waters, which are more rapidly eroded in OLD than NEW (not shown). The latter effect also occurs in TRA, resulting in similar though weaker and shallower ageing, near 1 km depth in the North Indian Ocean (Fig. 17h). This local age increase in TRA relative to NEW demonstrates that increased diapycnal mixing in the ocean interior can produce reduced ventilation rates on a regional scale.
4. Discussion and conclusion

We examined the impact of a comprehensive representation of mixing by breaking internal tides in a 1° global ocean configuration of NEMO. Simulations were run for 1,000 years under COREv2 normal year forcing (Large and Yeager 2009). Replacement of the constant background diffusivity of 10⁻⁵ m² s⁻¹ (experiment OLD) by an explicit representation of remote tidal mixing (experiment NEW) increases the tidal power input to mixing from 1.12 to 1.28 TW and generates substantial spatial contrasts in diffusivities (Figs. 2, 7 and 8). The improvement causes only moderate changes in the simulated broad-scale hydrography and circulation: the AMOC strengthens by about 1 Sv, the lower overturning cell weakens by about 1 Sv, salinity changes typically by less than 0.1 g kg⁻¹ and temperature by less than 0.5 °C. In general, the simulated hydrographic changes are much smaller than pre-existing model biases. This finding is consistent with the notion that atmospheric forcing and surface boundary layer processes – unchanged across our simulations – play a leading role in setting water-mass characteristics and circulation. Nonetheless, the small sensitivity of hydrography and circulation also owes to the small differences in basin-scale bulk diffusivities between OLD and NEW (Fig. 3).

Similarities between OLD and NEW simulations do not mean that the two representations of tidal mixing are equivalent. The more realistic and energy constrained distribution of tidal diffusivity in NEW increases confidence in model behavior (de Lavergne et al. 2020). For instance, the energy constraint on mixing could modify the response to forcing changes (e.g., Sigman et al. 2010), while spatial variations in upper-ocean tidal mixing rates could affect regional biogeochemical cycles (e.g., Tuerena et al. 2019). Such implications have not been investigated here.

Adding the contribution to mixing of bottom-trapped, subinertial internal tides (experiment TRA, otherwise identical to NEW) augments tidal diffusivities at high latitudes and raises the total tidal power input to 1.40 TW (Figs. 1-3). Again, this modification of tidal mixing causes only modest changes to stratification and circulation at basin scale. However, it induces some important differences in regional hydrography and deep-ocean ventilation rates, and tends to reduce known biases in our configuration. In particular, the extra mixing by subinertial internal tides causes cooling of CDW in southern subpolar seas, reducing the subsurface ocean heat that surrounds Antarctica (Fig. 17), with potential repercussions for
ice-shelf melt (Depoorter et al. 2013, Hutchinson et al. 2023). In addition, mixing by subinertial internal tides reduces deep-ocean age by up to 250 years in the Pacific shadow zone and by over 100 years on average deeper than 2 km (Fig.17). This reduction of deep-ocean age has consequences for carbon storage at depths, and more generally for biogeochemical cycles that depend strongly on deep-ocean ventilation timescales (Ödalen et al. 2018, Eggleston and Galbraith 2018).

Parameterized mixing by bottom-trapped internal tides thus has significant regional impacts on water mass transformation and passive tracer transport. The role of bottom-trapped internal tides in mixing the subsurface Arctic Ocean is well recognized (Rippeth et al. 2015, Fer et al. 2020). Here we find the largest impacts in the subarctic Pacific and high-latitude Southern Ocean. The elevated mixing along Antarctic continental slopes has particular significance because it affects critical water mass transformations occurring in this region, and contributes to shelter Antarctica from subsurface warmth contained in CDW, in synergy with subpolar gyres (Narayanan et al. 2023). This finding calls for new observational and theoretical studies of this mixing process, notably in the subpolar Southern Ocean, to reduce uncertainties in the magnitude and distribution of parameterized mixing. The impacts on mixing of both ‘internal’ and ‘topographic’ modes of conversion of subinertial surface tides need scrutiny (Tanaka 2023). In the present study, a numerical model estimate of barotropic-to-baroclinic tidal conversion (Müller 2013) has been employed to parameterize mixing due to subinertial internal tides, assuming local, bottom-intensified dissipation (section 2.3). The accuracy of the resulting mixing distribution should be assessed in future work.

Our results also depend in part on configuration and calibration choices, and the associated model limitations and biases. For example, experiment NEW simulates relatively high ages in the mid-depth Pacific shadow zone (up to 2400 years compared to peak ages of about 1600 years in data-constrained estimates; Gebbie and Huybers 2012, Holzer et al. 2021). This feature could favor strong sensitivity of the regional ideal age to increased diapycnal mixing. Similarly, biased NADW formation processes and sites could affect the diagnosed sensitivity of the AMOC to the representation of tidal mixing. Perhaps more importantly, the absence of an interactive atmosphere and of interannual variability in forcing may damp or amplify some of the sensitivities documented here. For instance, the restoring of sea surface
salinity toward observations tends to restrain differences in simulated salinity. On the other hand, this forcing protocol helps to achieve relatively realistic hydrography and circulation, and allows to perform millennial simulations at reduced computational cost. Notably, our forcing and calibration choices allow the simulation of AABW production near observed shelf sites (Fig.14), together with realistic and quasi-steady density stratification across the global deep ocean (Fig.11). This makes the present modelling framework suited to further studies of abyssal ocean ventilation and its sensitivity.

**Acknowledgements:** We thank M. Müller for sharing their tidal conversion dataset. We also thank Y. Tanaka and B. Gülk for comments. This work was granted access to the HPC resources of IDRIS under the allocations 2021-A0100107451, 2022-A0120107451 and 2023-A0140107451 made by GENCI. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreements Nr. 821001 and Nr. 101060452. Financial support was also provided by the French National LEFE programme through the ROOF project.

**Data Availability Statement:** NEMO code is available at https://forge.nemo-ocean.eu/nemo/nemo/-/releases/. The employed version is 4.2.0. The configuration files used in this study are available via the data repository at https://doi.org/10.5281/zenodo.14041098. The COREv2 forcing fields from Large and Yeager (2009) can be found at https://data1.gfdl.noaa.gov/nomads/forms/core/COREv2/CNYF_v2.html. The power maps underlying the tidal mixing parameterization of de Lavergne et al. (2020), used in experiments NEW and TRA, are available at https://www.seanoe.org/data/00619/73082/. Tidal conversion output from Müller (2013) is available at https://www.wdc-climate.de/ui/entry?acronym=DKRZ_Ita_510. The observational hydrographic climatology of Gouretski (2019) is accessible at https://www.cen.uni-hamburg.de/en/icdc/data/ocean/waghc.html. The sea ice index data employed here is available at https://nsidc.org/data/g02135/versions/3. The observational climatology of
mixed layer depth from Sallée et al. (2021) can be accessed at
https://doi.org/10.5281/zenodo.5776180.

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<table><caption>**Table 1:** Total power input to subinertial internal tides (E<sub>tra</sub>) and shallow internal tides (E<sub>sha</sub>). These barotropic-to-baroclinic conversion rates, based on Müller (2013), are given for each of four tidal constituents and for the sum of the eight most important constituents (last column). The scaling factors used to obtain the sum for all eight constituents are given in the last row.</caption><thead><tr><th>Tidal constituent</th><th>M<sub>2</sub></th><th>S<sub>2</sub></th><th>K<sub>1</sub></th><th>O<sub>1</sub></th><th>All</th></tr></thead><tbody><tr><td>Power E<sub>tra</sub> (GW)</td><td>1.7</td><td>0.2</td><td>60</td><td>45</td><td>113</td></tr><tr><td>Power E<sub>sha</sub> (GW)</td><td>149</td><td>25</td><td>37</td><td>20</td><td>236</td></tr><tr><td>Scaling factor</td><td>1.050</td><td>1.086</td><td>1.126</td><td>1.046</td><td></td></tr></tbody></table>

<figure><img src="image_20.png" /><figcaption>**Figure 1:** Depth-integrated turbulence production by subinertial internal tides (a) and shallow internal tides (b) remapped onto the model grid. The total power in each field is indicated at the top left of each panel. Note the power is shown on a log<sub>10</sub> scale.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Figure 2:** Depth-integrated turbulence production implied by each tidal mixing scheme. For experiment OLD (**a**), the power shown includes the specified near-field dissipation of internal tides and the contribution of the background diffusivity (calculated as the vertical integral of ρ₀ Kₚ N² Rf⁻¹ with Kₚ = 10⁻⁵ m² s⁻¹, ρ₀ = 1026 kg m⁻³, Rf = 1/6, and N the annual-mean buoyancy frequency from the first simulation year). Experiment NEW (**b**) has no background diffusivity: all tidal mixing derives from the specified internal tide energy dissipation. Experiment TRA (**c**) is identical to NEW except for the addition of subinertial internal tides. Note the power is shown on a log₁₀ scale.</figcaption></figure>
<figure><img src="image_22.png" /></figure>

**Figure 3:** Turbulence production implied by each tidal mixing scheme, summed over depth and longitude (a) or over geopotential model layers (b). Experiment OLD is shown in black, experiment NEW in light blue and experiment TRA in dark blue. Note the log₁₀ scale of the x-axis of (b).
<figure><img src="image_23.png" /><figcaption>**Figure 4:** Evolution of global mean ocean temperature (a), Drake Passage eastward volume transport (b), maximum of the Atlantic meridional overturning streamfunction at 26.5°N (c), and northward Antarctic Bottom Water (AABW) transport at 64°S (d) and 30°S (e). Experiments OLD, NEW and TRA are shown in black, light blue and dark blue, respectively. AABW transport is the maximum northward transport underneath a given σ₃ isopycnal (for σ₃ > 1041.4 kg m⁻³).</figcaption></figure>
<figure><img src="image_24.png" /></figure>

**Figure 5:** Mixed layer depth in observations and experiment NEW, averaged over January-February-March (**a,b**), April-May-June (**c,d**), July-August-September (**e,f**) and October-November-December (**g,h**). The left column (panels (**a**), (**c**), (**e**), (**g**) corresponds to the observational climatology of Sallée et al. (2021), the right column (panels (**b**), (**d**), (**f**), (**h**) to experiment NEW. In the right column, we also show as red contours the modelled 50% sea ice concentration averaged over the central month of each season.
<figure><img src="image_25.png" /><figcaption>**Figure 6:** Maximum mixed layer depth in experiment NEW (a), and anomalies in OLD (b) and TRA (c) with respect to NEW. The maximum is calculated from monthly-mean output over the last decade of each experiment.</figcaption></figure>
<figure><img src="image_26.png" /></figure>

**Figure 7:** Tidal vertical diffusivity in the permanent pycnocline. Panel (a) shows the diffusivity in experiment NEW; panel (b) shows the ratio of diffusivities in OLD and NEW; and panel (c) shows the ratio of diffusivities in TRA and NEW. Diffusivities are those produced by the tidal mixing parameterization of each experiment, including the background diffusivity in the case of OLD. They are taken at the depth of maximum stratification during the month of maximum mixed layer depth, hence in the local permanent pycnocline. Note that a log₁₀ scale is used in all panels.
<figure><img src="image_27.png" /><figcaption>**Figure 8:** Tidal vertical diffusivity at 210°E (or 150°W), in experiments OLD (a), NEW (b) and TRA (c). This diffusivity is that produced by the tidal mixing parameterization in each experiment, including the background diffusivity in the case of OLD. Note the diffusivity is shown on a log₁₀ scale.</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>**Figure 9:** Seasonal evolution of Arctic (a) and Antarctic (b) sea ice extents in satellite observations (red) and experiments OLD (black), NEW (light blue) and TRA (dark blue). Observations correspond to the sea ice index of the National Snow and Ice Data Center, based on the NASA Team algorithm (Stroeve and Meier 2018). Observed values shown here are monthly averages over the period 1979-2019; the thick red curve corresponds to the climatological mean while thin red curves denote the observed range for that period.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>**Figure 10:** Atlantic zonal mean temperature in the observational climatology of Gouretski (2019) (a) and experiment NEW (b). Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown. The same analysis for salinity is shown in Supplementary Figure S3.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>**Figure 11:** Basin-averaged density (a,d,g), temperature (b,e,h) and salinity (c,f,i) profiles in the observational climatology of Gouretski (2019) (red) and in model experiments OLD (black), NEW (light blue) and TRA (dark blue). The Pacific Ocean (33°S-65°N) is shown in panels (a-c), the Atlantic Ocean (33°S-65°N) in panels (d-f) and the Southern Ocean (80°S-33°S) in panels (g-i). Density is measured as σ₂, the potential density referenced to 2000 dbars.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>**Figure 12:** Pacific zonal mean temperature in the observational climatology of Gouretski (2019) (a) and experiment NEW (b). Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown. The same analysis for salinity is shown in Supplementary Figure S4.</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>**Figure 13:** Annual mean maximum temperature of the water column in the observational climatology of Gouretski (2019) (a) and experiment NEW (b), shown as a stereographic projection south of 50°S. Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>**Figure 14:** Annual maximum neutral density at the seafloor in the observational climatology of Gouretski (2019) (a) and in experiment NEW (b). Panels (c) and (d) show the concentration of an artificial dye tracer in the bottom grid cell of the model (experiment NEW), respectively 10 years and 30 years after initialization. The dye tracer is restored to 1 in the top 10 meters of the model and propagates as a conservative tracer in the deeper ocean. This dye simulation was done in offline mode using five-day output from year 1,001 of experiment NEW. It shows that the bottom ocean fills from the four sites where dense shelf waters (neutral density > 28.5 kg m⁻³) are formed.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>**Figure 15:** Meridional overturning streamfunction in experiment NEW. The streamfunction is shown for the Indian-Pacific (**a,b**), Atlantic (**c,d**) and global (**e,f**) oceans. It was computed both in depth coordinate (**a,c,e**) and density (**σ**₂) coordinate (**b,d,f**). The density-coordinate streamfunction is reprojected to depth using the pseudo-depth of isopycnal surfaces. The pseudo-depth of an isopycnal surface within a given latitude band is the geopotential which has the same volume beneath it (e.g., Zika et al. 2012). The shown streamfunctions include both resolved and parameterized (Gent and McWilliams 1990, Fox-Kemper et al. 2008) components of advection.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>**Figure 16:** Meridional overturning streamfunction at 26.5°N in the Atlantic. RAPID observations (Sinha et al. 2018) are shown in red, experiment OLD in black, experiment NEW in light blue and experiment TRA in dark blue.</figcaption></figure>
<figure><img src="image_36.png" /><figcaption>**Figure 17:** Zonal mean ideal age simulated by experiment NEW (a-c). Anomalies in OLD (d-f) and TRA (g-i) with respect to NEW are also shown. The zonal average was taken over the Pacific (a,d,g), Indian (b,e,h) or Atlantic (c,f,i) basin, each extended due south to Antarctica.</figcaption></figure>
Supplementary Information for

Effects of improved tidal mixing in NEMO one-degree global ocean model

Casimir de Lavergne¹, Saurabh Rathore¹,², Gurvan Madec¹,³, Jean-Baptiste Sallée¹, Christian
Ethé¹, Antoine Nasser¹,³, Bruno Millet⁴, Martin Vancoppenolle¹

1 LOCEAN Laboratory, Sorbonne University-CNRS-IRD-MNHN, Paris, France

2 Centre for Atmospheric Sciences, *Indian Institute of Technology Delhi, India*

3 Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France

4 Laboratoire des Sciences du Climat et de l'Environnement, CEA-CNRS-UVSQ, Université Paris-Saclay, Gif-sur-Yvette, France

This document contains four supplementary figures mentioned in the main text of the article.
<figure><img src="image_37.png" /><figcaption>**Supplementary Figure S1:** Geothermal heat flux through the seafloor from Lucazeau (2019). This flux is included as a bottom thermal forcing in all three experiments.</figcaption></figure>
<figure><img src="image_38.png" /><figcaption>**Supplementary Figure S2:** Map of the isopycnal tracer diffusivity (a) and isopycnal thickness diffusivity (b) averaged over the last decade of experiment TRA. Both diffusivities are independent of depth but vary horizontally with the simulated stratification.</figcaption></figure>
<figure><img src="image_39.png" /><figcaption>**Supplementary Figure S3:** Atlantic zonal mean salinity in the observational climatology of Gouretski (2019) (a) and experiment NEW (b). Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown.</figcaption></figure>
<figure><img src="image_40.png" /><figcaption>**Supplementary Figure S4:** Pacific zonal mean salinity in the observational climatology of Gouretski (2019) (a) and experiment NEW (b). Anomalies in experiments OLD (c) and TRA (d) with respect to NEW are also shown.</figcaption></figure>