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Article Type:
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The Impact of Lee Waves on the Southern Ocean Circulation

Luwei Yang aInstitute for Marine and Antarctic Studies, and ARC Centre of Excellence for Climate System Science, University of Tasmania, Hobart, Tasmania, Australia

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Maxim Nikurashin bInstitute for Marine and Antarctic Studies, and ARC Centre of Excellence for Climate Extremes, Australian Antarctic Program Partnership, University of Tasmania, Hobart, Tasmania, Australia

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Andrew McC. Hogg cResearch School of Earth Sciences, and ARC Centre of Excellence for Climate Extremes, Australian National University, Canberra, Australian Capital Territory, Australia

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Bernadette M. Sloyan dOceans and Atmosphere, CSIRO, and Center for Southern Hemisphere Ocean Research, Hobart, Tasmania, Australia

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Online Publication:
02 Sep 2021
Print Publication:
01 Sep 2021
DOI:
https://doi.org/10.1175/JPO-D-20-0263.1
Page(s):
2933–2950
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Abstract

Lee waves play an important role in transferring energy from the geostrophic eddy field to turbulent mixing in the Southern Ocean. As such, lee waves can impact the Southern Ocean circulation and modulate its response to changing climate through their regulation on the eddy field and turbulent mixing. The drag effect of lee waves on the eddy field and the mixing effect of lee waves on the tracer field have been studied separately to show their importance. However, it remains unclear how the drag and mixing effects act together to modify the Southern Ocean circulation. In this study, a lee-wave parameterization that includes both lee-wave drag and its associated lee-wave-driven mixing is developed and implemented in an eddy-resolving idealized model of the Southern Ocean to simulate and quantify the impacts of lee waves on the Southern Ocean circulation. The results show that lee waves enhance the baroclinic transport of the Antarctic Circumpolar Current (ACC) and strengthen the lower overturning circulation. The impact of lee waves on the large-scale circulation are explained by the control of lee-wave drag on isopycnal slopes through their effect on eddies, and by the control of lee-wave-driven mixing on deep stratification and water mass transformation. The results also show that the drag and mixing effects are coupled such that they act to weaken one another. The implication is that the future parameterization of lee waves in global ocean and climate models should take both drag and mixing effects into consideration for a more accurate representation of their impact on the ocean circulation.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Luwei Yang, luweiy@atmos.ucla.edu

Abstract

Lee waves play an important role in transferring energy from the geostrophic eddy field to turbulent mixing in the Southern Ocean. As such, lee waves can impact the Southern Ocean circulation and modulate its response to changing climate through their regulation on the eddy field and turbulent mixing. The drag effect of lee waves on the eddy field and the mixing effect of lee waves on the tracer field have been studied separately to show their importance. However, it remains unclear how the drag and mixing effects act together to modify the Southern Ocean circulation. In this study, a lee-wave parameterization that includes both lee-wave drag and its associated lee-wave-driven mixing is developed and implemented in an eddy-resolving idealized model of the Southern Ocean to simulate and quantify the impacts of lee waves on the Southern Ocean circulation. The results show that lee waves enhance the baroclinic transport of the Antarctic Circumpolar Current (ACC) and strengthen the lower overturning circulation. The impact of lee waves on the large-scale circulation are explained by the control of lee-wave drag on isopycnal slopes through their effect on eddies, and by the control of lee-wave-driven mixing on deep stratification and water mass transformation. The results also show that the drag and mixing effects are coupled such that they act to weaken one another. The implication is that the future parameterization of lee waves in global ocean and climate models should take both drag and mixing effects into consideration for a more accurate representation of their impact on the ocean circulation.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Luwei Yang, luweiy@atmos.ucla.edu
Keywords: Eddies; Internal waves; Ocean circulation; Wind stress

1. Introduction

The Southern Ocean plays an important role in the global ocean and climate by providing an interbasin connection, maintaining the global ocean structure, and regulating the exchange of heat and carbon between the ocean and the atmosphere (e.g., Gnanadesikan 1999; Rintoul et al. 2001; Marshall and Speer 2012; Rintoul and Naveira Garabato 2013; Frölicher et al. 2015; Talley et al. 2016; Rintoul 2018; Gruber et al. 2019). Observations show that the westerly winds over the Southern Ocean have strengthened and shifted poleward for the last few decades (Böning et al. 2008; Jones et al. 2016). It is important to understand the response of the Southern Ocean circulation to the changing wind conditions because the local response in the Southern Ocean will have large-scale impacts on the global ocean and climate (Rintoul 2018, and references therein). The prediction of the Southern Ocean’s response and its associated impacts relies on the understanding and representation of the key dynamic processes that involve surface wind and buoyancy forcing, eddies, mixing, stratification, and topography (Rintoul and Naveira Garabato 2013, and references therein).

The Southern Ocean circulation is regulated by its energetic eddy field and vigorous turbulent mixing (Rintoul and Naveira Garabato 2013, and references therein). The eddy field is responsible for the vertical transfer of momentum and poleward heat fluxes (e.g., Olbers et al. 2004; Hogg et al. 2008; Screen et al. 2009; Thompson and Naveira Garabato 2014; Doddridge et al. 2019) and thereby modulates the volume transport of the Antarctic Circumpolar Current (ACC) and the strength of the meridional overturning circulation (MOC) under the changing climate (Hallberg and Gnanadesikan 2006; Meredith and Hogg 2006; Böning et al. 2008; Hogg et al. 2008; Screen et al. 2009; Farneti et al. 2010; Farneti and Delworth 2010; Abernathey et al. 2011; Dufour et al. 2012; Morrison and Hogg 2013; Munday et al. 2013; Farneti et al. 2015; Gent 2016; Bishop et al. 2016). The interaction between the deep-reaching eddy field and bottom topography has been suggested to play a key role in dissipating eddy energy and contributing to the enhanced turbulent mixing observed at hotspots in the deep Southern Ocean (Rintoul and Naveira Garabato 2013; Marshall et al. 2017). An important process resulting from eddy-topography interactions is the generation, radiation, and breaking of internal lee waves, which is indicated to be a nonnegligible energy sink of the eddy field and a significant energy source of turbulent mixing in the deep Southern Ocean by recent estimates, numerical simulations, and observations (Legg 2020, and references therein). The enhanced turbulent mixing over major topographic features associated with the breaking of internal lee waves leads to water mass transformation in the deep Southern Ocean and hence controls the strength and structure of the MOC (Nikurashin and Ferrari 2013; de Lavergne et al. 2016; Meyer et al. 2015).

The modeling of eddy and mixing fields in the Southern Ocean is challenging because of a lack of physically based representation of eddy energy dissipation processes and the uncertainty in the sources and spatial distribution of turbulent mixing in the ocean interior (e.g., MacKinnon et al. 2017). To date, the removal of eddy energy in eddy-resolving global ocean models depends on the work of bottom frictional drag and subgrid-scale parameterizations that represent all the other unresolved eddy energy dissipation mechanisms (e.g., Kiss et al. 2020). It is important to include the physically based parameterization of individual processes because they not only lead to eddy energy loss but also have other dynamical consequences. One such process is the aforementioned generation, radiation, and breaking of internal lee waves. The generation of lee waves is estimated to result in an energy loss of 0.2–0.5 TW from the geostrophic flow globally (Nikurashin and Ferrari 2011; Scott et al. 2011; Wright et al. 2014), and an energy loss of 0.12 TW from the eddy field in the Southern Ocean (Yang et al. 2018). The subsequent radiation and breaking of lee waves dissipates their energy and drives turbulent mixing, accounting for the elevated turbulent energy dissipation rate and enhanced mixing observed in the ocean (e.g., Polzin and Firing 1997; Heywood et al. 2002; Naveira Garabato et al. 2004; Sloyan 2005; Wu et al. 2011; Liang and Thurnherr 2012; St. Laurent et al. 2012; Sheen et al. 2013, 2014; Waterman et al. 2013; Watson et al. 2013; Meyer et al. 2015). The turbulent mixing is intensified as a result of the eddy energy loss through the energy route provided by the generation, radiation, and breaking of lee waves. Therefore, both the drag effect of lee waves on the eddy field and the mixing effect of lee waves on the tracer field should be considered as the impact of lee waves.

Lee waves have been parameterized in previous studies to represent either their drag effect on the eddy field upon generation or their mixing effect on the tracer field upon breaking (Trossman et al. 2013, 2016; Melet et al. 2014; Broadbridge et al. 2016). Trossman et al. (2016) parameterized a lee-wave drag that represented both the generation of lee waves and topographic blocking in an eddy-resolving global ocean model to study the impact of lee-wave drag on the eddy field. With their drag parameterization, Trossman et al. (2016) found an increase in dissipation and a reduction of eddy kinetic energy (EKE) in the deep ocean as well as its global integral. Melet et al. (2014) used a global estimate of energy flux into lee waves based on Nikurashin and Ferrari (2011) to power lee-wave-driven mixing in an ocean-ice-atmosphere coupled climate model and evaluated its impacts on the Southern Ocean circulation. They found that the mixing parameterization led to a larger lower overturning circulation, stronger stratification, and a higher diffusivity in the Southern Ocean. Broadbridge et al. (2016) implemented a lee-wave-driven mixing parameterization dependent on the resolved flow in an idealized eddy-resolving channel model and assessed the impact of this mixing parameterization on the lower overturning circulation and its sensitivity to changes in wind stress over the Southern Ocean. They found that a higher intensity of lee-wave-driven mixing resulted in a larger lower overturning circulation and a warmer deep ocean. They also showed that the lower overturning circulation strengthened with wind stress in the presence of a lee-wave-driven mixing that was regulated by the flow.

The drag and mixing effects of lee waves have not been parameterized simultaneously and hence their combined impact on the Southern Ocean circulation remains unknown. Compared with the previous attempts to parameterize the impact of lee waves (Trossman et al. 2013, 2016; Melet et al. 2014; Broadbridge et al. 2016), including both drag and mixing effects of lee waves in the lee-wave parameterization introduces an energy link in the models between the eddy field and the lee-wave-driven mixing. As suggested by Stanley and Saenko (2014) and Broadbridge et al. (2016), including this missing energy link might strengthen the lower cell of the Southern Ocean overturning circulation as surface wind stress increases, contrary to the theoretical predictions provided by Ito and Marshall (2008) and Nikurashin and Vallis (2011) that did not take this energy link into account. More recently, Marshall et al. (2017) suggested that the presence of a bottom drag could potentially increase the zonal transport of the ACC through regulating the eddy energy balance. It is unclear how the presence of both drag and mixing effects of lee waves will modify the Southern Ocean circulation and its sensitivity to changes in wind stress.

In this study, we develop an energetically consistent lee-wave parameterization including a lee-wave drag and its associated lee-wave-driven mixing, and implement it into an idealized eddy-resolving ocean model representative of the Southern Ocean to evaluate the impacts of lee waves on the Southern Ocean circulation. We find that the parameterization of lee waves increases the ACC transport in the idealized model by 60 Sv (1 Sv ≡ 106 m3 s−1) (over 40%), which is primarily due to the drag effect with a nonnegligible contribution from the mixing effect. We also find that the parameterization of lee waves results in a net increase of the lower overturning circulation, which is a combination of an increase due to the lee-wave-driven mixing and a decrease due to lee-wave drag. In addition, we show that the drag and mixing effects are coupled, so that the presence of a lee-wave drag decreases the strength of the lee-wave-driven mixing, and vice versa.

The rest of the paper is organized as follows. In section 2, we introduce the formulation for the full lee-wave parameterization (section 2a) and the choice of parameters (section 2b). In section 3, we describe the model configuration used in this study (section 3a) and the experiments conducted using this model configuration (section 3b). In section 4, we present the key results of this study. We first demonstrate that the large-scale flow and bottom fields produced in the reference experiment are representative of those in the Southern Ocean in section 4a. We then describe the characteristics of the lee-wave field in section 4b. The impacts of lee waves on the overturning circulation and the baroclinic transport of the ACC are then shown in sections 4c and 4d, respectively. In section 5, we discuss the mechanisms governing the control of lee waves on the large-scale flow. In section 6, we provide a summary of this study.

2. Lee-wave parameterization

Here, we develop a lee-wave parameterization scheme specifically for eddy-resolving ocean models, assuming that the geostrophic mean and eddy fields are well resolved. To include both the drag and mixing effects in ocean models in an energetically consistent way, we add a lee-wave drag (i.e., the vertical divergence of wave stresses) to the momentum equations and then use the work of lee-wave drag on the resolved flow as an additional energy source for the turbulent mixing (i.e., the lee-wave-driven mixing).

a. Parameterization formulation

1) Lee-wave drag

The lee-wave drag predicted by linear theory (Bell 1975) is an integration of the contribution of radiating lee waves in wavenumber space; this calculation is too computationally expensive for a global ocean model. Thus, a simplification of Bell’s formula is necessary to ensure computational affordability. Depending on how the topography is simplified and assumptions made, the formula of lee-wave drag could be simplified in several ways. When a 2D, horizontally periodic sinusodial topography is assumed, the lee-wave drag is linearly proportional to the velocity (e.g., Pedlosky 2013). For the multichromatic topography representing abyssal hills in the ocean, Nikurashin and Ferrari (2010a) derived a lee-wave drag that was dependent on |U|3/2. Naveira Garabato et al. (2013) proposed to represent depth-integrated lee-wave drag by a quadratic law for mid- and high-latitude oceans under a nonblocking condition; a drag coefficient was introduced while the possibly significant misalignment in direction between the internal wave drag and the near-bottom velocity was neglected. Here, as a first step toward parameterizing lee-wave drag, we use a linear form (Pedlosky 2013).

We decompose the flow u into a resolved component u¯ and an unresolved lee-wave component u′. The momentum equation for u¯ takes the form,
Du¯Dt−fz^×u¯=−∇p¯ρ0+R−1ρ0∂∂z τLW,
where ρ0 is a reference density, f is the Coriolis parameter, z^ is the unit vector in the vertical direction, ⁡(1/ρ0)⁡(∂/∂z)τLW is a body force (referred to as the lee-wave drag hereafter) acting against the resolved flow due to generation, radiation, and breaking of lee waves, given by the vertical divergence of lee-wave stress (in units of N m−2)
τLW=ρ0u′w′¯,
and the term R represents the lateral redistribution of momentum due to viscous stresses.
We construct the vertical profile of lee-wave stress following the methodology of St. Laurent et al. (2002) used for tidal mixing parameterization and assume that it can be represented by the value of lee-wave stress at the bottom τLW|z=−H that arises from the generation of lee waves, and the vertical structure that captures the radiation and breaking of lee waves F(z),
τLW⁡(z)=τLW|z=−HF⁡(z),
where H is the depth of the ocean. The expression of the lee-wave stress at the bottom associated with the generation of lee waves is based on a scaling predicted by linear theory (e.g., Gill 1982; Cushman-Roisin and Beckers 2011; Pedlosky 2013) and takes the form
τLW|z=−H=ρ0u′w′¯|z=−H=−12 ρ0Nbh02ku¯b,
where Nb is the local bottom stratification, h0 is the characteristic amplitude and k the horizontal wavenumber of the topography, u¯b is the horizontal velocity of the resolved flow at the bottom. We use (4) as the form of lee-wave stress in the parameterization because it is more computationally efficient compared with the complete linear wave solution for multiscale topography (Bell 1975) and still captures the leading-order dependence of lee-wave stress on the bottom velocity and stratification. The terms h0 and k are the bulk parameters that represent multiscale abyssal hills in the Southern Ocean.
The vertical structure of lee-wave stress τLW represents the radiation and breaking of lee waves. Since lee waves generated by the geostrophic flows over rough abyssal hills have a vertical scale of 100–1000 m and hence stable to shear instability, other processes, such as wave-wave interactions and wave-mean flow interactions, need to be invoked for lee waves to transfer their energy to smaller-scale internal waves, which can then undergo shear instability and break. The vertical profile is therefore governed by complex nonlinear wave-wave interactions and wave breaking processes, many of which remain poorly understood and parameterized (e.g., MacKinnon et al. 2017; Whalen et al. 2020). Hence, in this study, we use an empirically based vertical structure proposed for tidal mixing parameterization (St. Laurent et al. 2002; Simmons et al. 2004) and also used for lee-wave-driven mixing parameterization (Melet et al. 2014),
F⁡(z)=e−(H+z)/ζ,
where ζ is the vertical decay scale, H is the local depth of the ocean, and z is the vertical coordinate. The vertical structure describes how the momentum (and energy) extracted by lee waves at the bottom is redistributed throughout the water column and satisfies the condition F(−H) = 1. We now obtain the final expression for the lee-wave stress
τLW⁡(z)=−12 ρ0Nbh02ku¯bF⁡(z),
and the lee-wave drag that acts against the resolved flow
FLW=−1ρ0∂∂z τLW=−1ρ0 τLW|z=−H∂∂z F⁡(z)=−Nbh02ku¯b2ζ F⁡(z),
in units of m s−2.
To make an analogy with the linear bottom frictional drag, we introduce a lee-wave drag coefficient (in units of m s−1)
γLW=12 Nbh02k,
so that the expression for the lee-wave stress can be rewritten as
τLW=−ρ0γLWu¯bF⁡(z),
which shows the dependence of the lee-wave stress on its vertical structure and the bottom velocity. The lee-wave drag coefficient γLW is equivalent to Cd|u| commonly used in the expression of the quadratic bottom frictional drag, where Cd is a nondimensional drag coefficient and |u| is the magnitude of the bottom velocity. The lee-wave drag coefficient (8) includes the dependence of the lee-wave stress on the characteristics of the small-scale topography and the local stratification. Substituting Nb = 10−3 s−1, h0 = 50 m, and k = 2π/2000 rad m−1, which are typical values for the Southern Ocean, into (8), we estimate that the characteristic value of the lee-wave drag coefficient is 4.0 × 10−3 m s−1, which is comparable with the linear bottom frictional drag coefficient used in numerical studies (Abernathey et al. 2011; Marshall et al. 2017). However, the key features that distinguish the lee-wave drag from the linear bottom frictional drag is its dependence on its vertical structure, the stratification, and the characteristics of the small-scale topography. These dependencies are the reason why we cannot represent the effects of lee waves in a physical way by increasing the bottom frictional drag coefficient.

In our parameterization, the lee-wave drag acts as a momentum sink on the resolved flow and transfers energy from the resolved flow to lee waves. The response of the resolved flow to the generation of lee waves, represented by the lee-wave drag, was not represented in previous studies (Melet et al. 2014; Broadbridge et al. 2016) that parameterized only the lee-wave-driven mixing.

2) Lee-wave-driven mixing

To make our parameterization energetically consistent, we now derive a diffusivity associated with the breaking of lee waves that will be used in tracer equations. We estimate the work of lee-wave drag and assume that the extracted energy is all converted into lee waves and available for mixing.

We estimate the turbulent energy dissipation rate ϵLW as the work of lee-wave drag against the resolved flow,
ϵLW=|u¯⋅FLW|=|u¯⋅(−1ρ0∂∂z τLW)|,
in units of W kg−1.
The total energy dissipation is estimated by the integral of energy dissipation rate from bottom to 2 km above the bottom
∫−H−H+2000ρ0ϵLWdz=∫−H−H+2000[u¯⋅(−∂∂z τLW)]dz≈∫−H−H+2000∂∂z [u¯⋅(−τLW)] dz=u¯⋅τLW|z=−H−u¯⋅τLW|z=−H+2000≈u¯⋅τLW|z=−H,
which is equivalent to the energy extraction by lee waves at the bottom. Here we assume that the horizontal velocity u¯ is depth independent in the deep ocean (the bottommost 2 km) and that the lee-wave drag decreases to zero within 2 km off the seafloor. Both assumptions are well justified in the ocean (e.g., Sloyan 2005; St. Laurent et al. 2012; Waterman et al. 2013).
The breaking of lee waves enhances turbulent mixing, which can be parameterized as a diffusivity κυLW (in units of m2 s−1), taking the form
κυLW=Γ ϵLWN2,
based on the Osborn relation (Osborn 1980), where Γ is the mixing efficiency and N is the buoyancy frequency. Here we assume that all the energy in lee waves is dissipated locally, i.e., relative to a model grid cell in the horizontal. The diffusivity associated with the lee-wave-driven mixing, κυLW, is then added to the diffusivity used in tracer equations. Note that κυLW is also a function of depth controlled by the vertical profile of the lee-wave stress (5). Compared with the tidal mixing parameterization (Simmons et al. 2004; Jayne 2009), our parameterization of lee-wave-driven mixing relies on an interactive link between momentum and energy, instead of a prescribed map of energy flux into internal waves.

b. Parameter choices

The parameterization of lee waves requires knowledge of the small-scale topography and the vertical structure of lee-wave stress. For our idealized channel configuration (section 3a), we choose the following parameters for the small-scale topography.

The amplitude of the small-scale topography h0 is 50 m. This choice is made based on observations of small-scale roughness (Nikurashin and Ferrari 2010a; Sheen et al. 2013; Meyer et al. 2016) and estimates obtained from analytical datasets of small-scale topography (Goff 2010; Goff and Arbic 2010; Nikurashin and Ferrari 2011). Nikurashin and Ferrari (2010a) fit the free parameters of the model spectrum of abyssal hills (Goff and Jordan 1988) using observations in Drake Passage and estimated the topographic roughness in the radiative wavenumber range to be 60 m. Sheen et al. (2013) found that the roughness of the small-scale topography increased from 30 m in the southeast Pacific to over 100 m in western Drake Passage and the Phoenix Ridge, using multibeam bathymetry data. Meyer et al. (2016) estimated the topographic roughness to be approximately 122.5 m north of Kerguelen Plateau Polar Front Zone where the mixing was the highest. Our estimates of the Southern Ocean-averaged roughness of the small-scale topography using Goff (2010), Goff and Arbic (2010), and Nikurashin and Ferrari (2011) show that a characteristic amplitude of the small-scale topography that contributes significantly to the lee-wave generation is on the order of 10–100 m (Table 1). We choose 50 m as the characteristic amplitude of the small-scale topography in the Southern Ocean. Equation (7) indicates that both the energy extraction from the resolved flow and the lee-wave-driven mixing are sensitive to the choice of h0. This sensitivity has been confirmed in modeling studies (e.g., Nikurashin and Ferrari 2010a; Broadbridge et al. 2016) and shown to modulate the strength of lower overturning circulation in the Southern Ocean (Broadbridge et al. 2016).

Table 1.

Southern Ocean–averaged small-scale roughness (m; first row) calculated using Goff (2010), Goff and Arbic (2010), and Nikurashin and Ferrari (2011). The calculation is repeated by including only the region where the lee-wave generation is larger than 1 m W m−2 (second row), 10 mW m−2 (third row), and 100 mW m−2 (fourth row).

Table 1.

The characteristic horizontal wavenumber of the small-scale topography, k, is 2π/2000 rad m−1, i.e., a wavelength of 2 km is used. This choice is made following Nikurashin and Ferrari (2010a) and Scott et al. (2011). Nikurashin and Ferrari (2010a) found that the characteristic wavenumber of small-scale topography in Drake Passage was 2.3 × 10−4 and 1.3 × 10−4 m−1 along the principal axes of anisotropy. Scott et al. (2011) estimated that the wavenumber of abyssal hills was on the order of 10−4–10−3 rad m−1, accounting for the anisotropy of abyssal hills. In this study, we use one wavenumber to describe the shape of the monochromatic topography as our focus is studying the role of the missing energy link and not the anisotropy of the topography; however, abyssal hills are strongly anisotropic in the ocean and this anisotropy has an impact on the generation of lee waves (Scott et al. 2011; Yang et al. 2018).

The vertical decay scale ζ used in the vertical structure of the lee-wave stress (5) is 500 m. This choice is made following Nikurashin and Ferrari (2013). Nikurashin and Ferrari (2013) showed that the water mass transformation due to lee-wave-driven mixing was sensitive to the choice of vertical decay scale. The comparisons across the sensitivity experiments with the vertical decay scale of 300, 500, 700, and 1000 m in Nikurashin and Ferrari (2013) showed that increased lee-wave-driven water mass transformation occurred when the vertical decay scale was small as more energy dissipation occurred in a given distance above the seafloor. Melet et al. (2014) found that increasing the vertical decay scale from 300 to 900 m strengthened the ACC transport and decreased the lower overturning circulation.

Using the small-scale topographic parameters chosen above, the characteristic steepness parameter is estimated as
s=NHU=10−3×500.1=0.5,
which is smaller than 0.75, the critical steepness parameter commonly used for 2D flow, and indicates that the topography is subcritical and bottom flow falls in the linear regime (Aguilar and Sutherland 2006; Nikurashin and Ferrari 2010b). In this regime, the correction based on steepness parameter is not needed. Based on the characteristic steepness parameter, we expect that most of the lee-wave generation occurs where the bottom flow is linear in our idealized model; the correction is unlikely to affect the results. We therefore choose not to apply the correction to the lee-wave stress in our lee-wave parameterization. However, the correction based on the steepness parameter is important in the parameter space where the steepness parameter is greater than 0.75 and should be applied to develop a more general lee-wave parameterization scheme for a wider parameter space.

3. Model configuration and experiments

a. Model configuration

We use the Modular Ocean Model, version 6 (MOM6; Adcroft et al. 2019). We set up a periodic channel configuration on β-plane, whose domain size is Lx × Ly × Lz = 4000 km × 2500 km × 4 km (Fig. 1), and horizontal resolution is 10 km. There are 72 vertical layers, with vertical resolution of 5 m at the surface increasing to 98 m at the bottom. Key parameters of the configuration are listed in Table 2.

Fig. 1.
Fig. 1.

Model configuration and bathymetry. The gray surface indicates topography used in the model configuration, whose height contours are projected onto the x–y plane. The color at the surface shows a snapshot of surface temperature.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

Table 2.

Key parameters used in the control experiment.

Table 2.

The model is forced by surface wind stress (Fig. 2a), surface temperature restoring (Fig. 2a), and a sponge layer located in the northernmost 100 km of the domain (2400 km ≤ Y ≤ 2500 km, Fig. 2b). In this configuration, the salinity is set as a constant, 35 psu, throughout the domain, and density is only a function of temperature. The surface wind stress takes the form
τwx=τ0 sin⁡(πy/Ly), τwy=0,
where peak wind stress τ0 = 0.2 N m−2. The surface buoyancy flux is determined by the difference between ocean temperature and surface restoring temperature (red line in Fig. 2a) and temperature restoring rate. Surface restoring temperature (red line in Fig. 2a) is zonally uniform. The surface temperature restoring profile follows
Tsurf={12×y/2000,0≤y≤2000 km;12,2000 km≤y≤2500 km.
Fig. 2.
Fig. 2.

Surface forcing, boundary, and initial conditions. (a) Surface forcing consists of wind forcing (black line) and temperature restoring (red line). (b) The vertical diffusivity in the sponge layer, which is increased up to 5 × 10−3 m2 s−1 with respect to the background value of 1 × 10−5 m2 s−1. (c) The vertical profile of the initial temperature field.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

In the northern sponge layer, similar to Hogg (2010), we increase vertical diffusivity (Fig. 2b) to represent downward flux of heat, balancing cold water upwelling, due to the interior mixing occurring in the ocean basins to the north of the Southern Ocean. The maximum diffusivity in the sponge layer is 5 × 10−3 m2 s−1 (Fig. 2b). The adoption of an enhanced diffusivity in the sponge layer allows the stratification to vary and guarantees a physically closed lower overturning cell in an idealized model that has a solid wall bounded in the north. Note that in this case, the upper overturning cell is virtually nonexistent. The purpose of using enhanced diffusivity in the sponge layer is to represent the water mass transformation occurring in the ocean basins north of the Southern Ocean.

A quadratic bottom frictional drag is used in the model with the drag coefficient of 3 × 10−3. The background kinematic viscosity is 10−4 m2 s−1 and the background biharmonic horizontal viscosity is 1010 m4 s−1. The topography consists of a slope at the southern boundary representing the Antarctic slope and two ridges located in the western half of the domain (Fig. 1). To the north of the slope, the topography is a complex combination of ridges and seamounts which gives zonal variability to the simulated circulation.

b. Experiments

In addition to the reference simulation with no lee-wave parameterization described above, we conduct three perturbation experiments to assess the combined drag and mixing effect as well as the contribution of each component: 1) a lee-wave drag only experiment, where only lee-wave drag is parameterized in the momentum equations to reflect the impact of lee-wave generation on the momentum of the resolved flow; 2) a lee-wave-driven mixing only experiment, where only diffusivity is modified to represent enhanced mixing effects due to the breaking of lee waves; and 3) a lee-wave full parameterization experiment, where both lee-wave drag and its associated lee-wave-driven mixing are parameterized. The lee-wave experiments are carried out using the same model configuration as that used in the reference case.

The reference simulation is initialized from a state of rest; the initial temperature field is horizontally uniform and its vertical structure is shown in Fig. 2c. The model is spun up for 200 years until it reaches an equilibrium. When the perturbation experiments are branched from the reference experiment, the model is run from a previously equilibrated state for another 50 years, and the last 5 years of output are used in this study.

4. Results

a. Reference case

The reference case simulation reproduces the large-scale circulation in the Southern Ocean and bottom fields representative of lee-wave generation sites. As the lee-wave parameterization relies on the bottom velocity and stratification fields, we look at the EKE and stratification fields in the bottom-most 500 m from model output.

The zonal baroclinic transport is 144 Sv, which is a good representation of the observed transport at Drake Passage (Donohue et al. 2016). The zonal baroclinic transport is calculated using the density field based on the thermal wind relation, as Eq. (17) in Abernathey and Cessi (2014) with ub = 0. We exclude the latitude band of the Antarctic slope area when calculating the zonal baroclinic transport to focus on the ACC-like region in the domain. Topographic features (Fig. 1) steer the zonal flow and permit a meandering structure of the flow (Fig. 3a). Due to the diffusive northern boundary condition, only the lower overturning cell is produced in the reference case (Fig. 3d). The strength of the lower overturning cell is 9.4 Sv (at Y = 750 km), which is comparable to the basin-scale Antarctic Bottom Water (AABW) transport (Talley 2013). Therefore our channel is a good representation of a sector in the Southern Ocean. The stretched density coordinate (Fig. 3d) shows that the lower cell occupies a small density range (roughly from 36.6 to 36.8 kg m−3), which, in depth space, fills the deep ocean below 2 km (solid lines in Fig. 4).

Fig. 3.
Fig. 3.

Diagnostics for the reference case. (a) Baroclinic streamfunction (Sv). White contours mark streamlines of 50 and 100 Sv. The slope area is excluded in the calculation. (b) Bottom-500-m-averaged EKE, log10 scale (m2 s−2). (c) Bottom-500-m-averaged stratification, log10 scale (s−1). (d) Residual overturning streamfunction (Sv). The dotted and dashed black lines represent the minimum and maximum density, respectively, that occurs at surface. The white block (2400 km ≤ Y ≤ 2500 km) in (b), white dotted line in (c), and gray vertical line in (d) indicate the position of the sponge layer. The white line in (d) marks the density level of 36.6 kg m−3, below which the density coordinate is stretched.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

Fig. 4.
Fig. 4.

Zonal- and time-mean density structure for the reference case. Dashed contours show density levels ranging from 35.4 to 36.4 kg m−3 with an interval of 0.2 kg m−3. Solid contours represent density levels ranging from 36.60 to 36.80 kg m−3 with an interval of 0.05 kg m−3. The gray vertical line denotes the southern boundary of the northern sponge layer.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

Bottom EKE produced in the reference case (Fig. 3b) has a similar magnitude to that in an eddy-rich global ocean model (Fig. 4c in Yang et al. 2018) and shows clear hotspots downstream of topography. The characteristic value of bottom EKE is 10−2 m2 s−2 in hotspots (Fig. 3b), which indicates the bottom eddy flow can be as strong as 0.1 m s−1. Bottom stratification is large over the topography in the shallow ocean and small over the flat area in the abyss (Fig. 3c). The domain-averaged bottom stratification is 3.6 × 10−4 s−1 (Fig. 3c), which is smaller than typical observed value (10−3 s−1) in the abyssal Southern Ocean, especially in lee-wave generation sites (Nikurashin and Ferrari 2011; Scott et al. 2011; Yang et al. 2018); however, similar values are present in the Southern Ocean (Scott et al. 2011) and therefore the bottom stratification in our model is not unrealistic. Weak bottom stratification is typical for the idealized sector models of the Southern Ocean (e.g., Broadbridge et al. 2016). Weaker bottom stratification than that observed in the Southern Ocean implies that the lee-wave generation might be underrepresented.

b. Lee-wave characteristics

To characterize the parameterized lee-wave field in an eddy-resolving configuration of the idealized channel model described in section 3a, we examine lee-wave generation and its subsequent radiation and dissipation from the lee-wave full parameterization experiment. Lee-wave generation is quantified by bottom energy flux into lee waves and lee-wave drag coefficient (γLW) (Figs. 5a,b). Lee-wave dissipation is evaluated using turbulent energy dissipation rate (εLW) and lee-wave-driven mixing (κυLW) (Figs. 5c,d).

Fig. 5.
Fig. 5.

Diagnostics of the lee-wave field from the lee-wave full parameterization experiment. (a) Bottom energy flux into lee waves (mW m−2). The white contour indicates the bottom-500-m-averaged EKE of 30 cm2 s−2, inside which is a bottom EKE hotspot. (b) Lee-wave drag coefficient (m s−1). White contours indicate lee-wave drag coefficient of 0.01 m s−1 and gray contours 0.003 m s−1. In (a) and (b) the Antarctic slope and northern sponge area are not shown. (c) Turbulent energy dissipation rate (log scale) associated with lee-wave generation at Y = 1255 km (W kg−1). The white contour indicates the value of 10−10 W kg−1. (d) Diffusivity associated with lee-wave-driven mixing (log scale) at Y = 1255 km (m2 s−1). The white contour indicates the value of 10−5 m2 s−1. The shallowest depth shown in (c) and (d) is 1 km above the peak of topography. The surface area is masked in (c) and (d) to focus on the lee-wave effects that radiate from the bottom.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

The bottom energy flux into lee waves has an average magnitude of 10 mW m−2 inside the EKE hotspot with a domain maximum value exceeding 100 mW m−2 (Fig. 5a), which is consistent with the offline estimate from an eddy-resolving global ocean model (Yang et al. 2018) where the characteristic lee-wave generation rate in hotspots is on the order of 10–100 mW m−2 (Figs. 8a–c in Yang et al. 2018). The maximum bottom energy flux into lee waves in our idealized model occurs downstream of the two ridges and coincides with the region of largest bottom EKE (Fig. 5a).

The lee-wave drag coefficient spatially varies with bottom stratification (Fig. 5b) as the amplitude and characteristic horizontal wavenumber of the small-scale topography are set to be constant throughout the domain (8). Over the flat abyssal region, the lee-wave drag coefficient is around 6 × 10−4 m s−1; over topography, the lee-wave drag coefficient increases to 0.01 m s−1 due to the strong bottom stratification higher up in the water column (Fig. 5b). The domain averaged lee-wave drag coefficient (0.001 m s−1) is comparable to the linear bottom frictional drag coefficient used in the idealized model of the Southern Ocean in previous studies (e.g., Abernathey et al. 2011; Marshall et al. 2017). The lee-wave drag coefficient in our idealized model is larger than that in Trossman et al. (2013), which is likely due to the simplified representation of small-scale topography.

The turbulent energy dissipation rate is intensified near the bottom and decreases gradually with distance from the bottom within the water column from O(10−8) to O(10−10) W kg−1 at roughly 1–2 km off the bottom (Fig. 5c). This vertical structure is consistent with observations (e.g., Naveira Garabato et al. 2004; St. Laurent et al. 2012; Waterman et al. 2013) and theoretical estimates (Nikurashin et al. 2013) in the Southern Ocean.

Consistent with the turbulent energy dissipation rate, lee-wave-driven mixing is also strongly bottom-intensified, which leads to a diffusivity of up to 10−2 m2 s−1 near the bottom (Fig. 5d). The diffusivity associated with lee-wave-driven mixing decreases away from the bottom reaching 10−5 m2 s−1 at middepth. Overall, the spatial structure and magnitude of the bottom energy flux, lee-wave drag coefficient, turbulent energy dissipation rate, and lee-wave-driven mixing are consistent with previous studies.

c. Lower overturning circulation

We compare the strength of the lower overturning circulation from all three lee-wave parameterization experiments with that from the reference experiment to examine the changes in lower overturning circulation (Fig. 7). We choose the streamfunction of the lower overturning circulation at density level of 36.7 kg m−3 and Y = 750 km for this comparison. The strength of overturning circulation is 9.4 Sv in the reference case (Fig. 6a). It reduces to 6.0 Sv in the lee-wave drag only case (Fig. 6c) and increases to 11.1 and 14.9 Sv, respectively, in the full parameterization (Fig. 6b) and lee-wave-driven mixing only case (Fig. 6d).

Fig. 6.
Fig. 6.

Overturning streamfunction (Sv) in the (a) reference case, (b) full parameterization case, (c) lee-wave-drag-only case, and (d) lee-wave-driven mixing-only case. The values along Y = 750 km marked by orange lines in (a)–(d) are shown in (e). The dotted and dashed black lines represent the minimum and maximum density at the surface, respectively. The gray vertical line indicates the location of the sponge layer. The density range below the white line is stretched. The white dashed dotted line marks the maxima of lower overturning streamfunction along the density axis.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

Consistent with Melet et al. (2014) and Broadbridge et al. (2016), our results show that the parameterization of lee-wave-driven mixing increases the lower overturning circulation. In contrast, our results show that the parameterization of lee-wave drag decreases the lower overturning circulation by 3.4 Sv. Note that the full parameterization leads to an increase of 1.7 Sv in the strength of the lower overturning circulation, which is not as large as that in the lee-wave-driven mixing only experiment (5.5 Sv). The increase in the full parameterization experiment demonstrates that the mixing effect dominates the changes in the strength of lower overturning circulation. The smaller increase in the full parameterization experiment indicates that the effect of lee-wave-driven mixing on the lower overturning circulation is suppressed by the presence of lee-wave drag.

d. Baroclinic transport of the ACC

To examine the impacts of the lee-wave parameterization on the ACC transport, we calculate the baroclinic transport for three sets of experiments with lee-wave parameterization and compare them with that in the reference case. The baroclinic transport increases from 144 Sv in the reference case to 169, 212, and 211 Sv in the lee-wave-driven mixing-only, lee-wave drag only, and lee-wave full parameterization cases, respectively. In the presence of lee waves, the increase in baroclinic transport is as large as 20%–50% of the baroclinic transport in the reference experiment.

In contrast to the opposing effects of lee-wave drag and lee-wave-driven mixing on the lower overturning circulation, both drag and mixing effects lead to an increase of the baroclinic transport. The increase in the baroclinic transport associated with the drag effect is greater than that induced by the mixing effect. When both effects are parameterized, the increase in the baroclinic transport is similar to that in the lee-wave drag only experiment. This increase indicates that the increase due to the full parameterization with both drag and mixing effects is not as large as the linear combination of the increase due to each effect.

5. Discussion

The strength of the lower overturning circulation and the baroclinic transport of the ACC are both regulated by the slope of isopycnals (Nikurashin and Vallis 2011; Marshall et al. 2017). To understand the control of lee-wave drag, lee-wave-driven mixing, and combined drag and mixing effects on the large-scale circulation, we first compare the slope of isopycnals from all lee-wave parameterization experiments with those from the reference case (Figs. 7, 8) and then discuss the regulation of isopycnal slope and stratification by lee-wave drag and lee-wave-driven mixing. We also explore the coupling between lee-wave drag and lee-wave-driven mixing.

Fig. 7.
Fig. 7.

Isopycnals in the experiments without any lee-wave parameterization (solid lines), with only lee-wave-driven mixing (dotted lines), with only lee-wave drag (dashed lines), and full parameterization (dash dotted lines). The colors represent different density levels; from top to bottom, orange, purple, green, red, and black colors represent density levels of 35.6, 36.0, 36.4, 36.6, and 36.65 kg m−3, respectively. The blue solid line denotes the approximate position of the base of the mixed layer. The gray vertical line denotes the southern boundary of the northern sponge layer.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

Fig. 8.
Fig. 8.

(a) Bottom-500-m-averaged stratification [log10 (s−1)] in the reference case. (b)–(d) The difference between the bottom-500-m-averaged stratification in the full parameterization case, lee-wave drag only case, lee-wave-driven mixing only case and that shown in (a), respectively.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

a. Slope of isopycnals and ocean stratification

In the ocean interior, the isopycnals are steepened in all three cases of lee-wave parameterization compared with the reference case (Fig. 7). The steeper isopycnals are also deeper, especially in the northern part of the domain; this is because the boundary conditions in our idealized model strongly constrain the outcrop position of the isopycnals at the surface but allow the northern end of the isopycnals (inside the sponge layer) to adjust in the vertical. Isopycnals are steeper in the lee-wave drag only experiment than in the lee-wave-driven mixing only experiment. However, isopycnals in the lee-wave full parameterization experiments are not distinguishable from those in the lee-wave drag only experiment, suggesting the dominant control of lee-wave drag, via eddies, on the isopycnal slopes. This nonlinear increase in the slope of isopycnals in the full parameterization experiment indicates a complex coupling between the drag effect and the mixing effect on the isopycnal slope.

Changes in the slope of isopycnals imply that the ocean stratification is also modified. Here we examine the bottom-500-m-averaged stratification because it is affected by bottom-intensified lee-wave-driven mixing and it also determines the lee-wave drag coefficient (8). We calculate the bottom-500-m-averaged stratification from the three experiments with lee-wave parameterization and compare them with that from the reference experiment (Fig. 8). The parameterization increases the bottom stratification over the ridges in all three cases (Figs. 8b,c,d). The response of the bottom stratification to the parameterization is more complicated over the flat bottom; the parameterization of lee-wave drag increases the bottom-500-m-averaged stratification, whereas the parameterization of lee-wave-driven mixing decreases the bottom-500-m-averaged stratification (Fig. 8). The combined drag and mixing effect leads to a decrease in the bottom stratification over the flat bottom (Fig. 8b), which indicates a dominant role of lee-wave-driven mixing in controlling the bottom stratification over the flat bottom. The decrease in the bottom stratification in the lee-wave full parameterization experiment is not as significant as that in the lee-wave-driven mixing-only experiment (Figs. 8b,d) due to the superposition and coupling between the drag effect and the mixing effect.

Both lee-wave drag and lee-wave-driven mixing lead to the steepening (and deepening) of isopycnals but modify the ocean stratification in the opposite way because they regulate the slope of isopycnals and ocean stratification under different mechanisms. Lee-wave drag acts against the flow in a way analogous to linear bottom frictional drag; therefore, the inclusion of lee-wave drag yields an effect similar to increasing linear bottom frictional drag coefficient (Marshall et al. 2017). Adding lee-wave drag enhances the energy dissipation of the eddy field (Yang et al. 2018). Consistent with the mechanism proposed by Marshall et al. (2017), establishing a new equilibrium requires steeper isopycnals so that the eddy energy generation rate is increased to compensate for the enhanced eddy energy dissipation rate and balance sources and sinks of EKE. Therefore the isopycnals in the lee-wave drag only experiment are steeper than in the reference case (Fig. 7). As the isopycnals are steepened and displaced deeper into the ocean (Fig. 7), the weak bottom stratfiication in the abyss is replaced by slightly stronger stratification and therefore is enhanced. The increase in the bottom stratification is strongest in the northern part of the domain (Fig. 8c) because this is where the isopycnals experience the largest downward displacement. In contrast, the parameterization of lee-wave-driven mixing decreases the bottom stratification although it also steepens the isopycnal in the interior. The bottom stratification decreases because the parameterization of lee-wave-driven mixing increases the diapycnal diffusivity and the intensified diabatic mixing decreases the vertical temperature gradient (i.e., the stratification in our idealized model). The isopycnals are steeper in the interior because the parameterization of lee-wave-driven mixing results in a larger downward heat flux (the increase in diapycnal diffusivity dominates over the decrease in vertical temperature gradient, not shown) and therefore enables more dense waters being transformed into lighter waters. Therefore, the mixing effect leads to a decrease in density, the deepening of the isopycnals (Fig. 7), and weaker bottom stratification (Fig. 8).

b. Baroclinic transport of the ACC

The regulation of the slope of isopycnals by lee-wave drag and lee-wave-driven mixing explains the changes in the baroclinic transport of the ACC described in section 4d. The increase in the baroclinic transport is consistent with steeper isopycnals shown in all three experiments with lee-wave parameterization (Fig. 7); the steeper the isopycnals are, the larger the baroclinic transport becomes. The increase in the full parameterization case (67 Sv) is not a linear combination of the increase in the lee-wave-drag-only case (68 Sv) and that in the lee-wave-driven only case (25 Sv) due to the coupling between the drag effect and the mixing effect. The increase in the baroclinic transport due to lee-wave drag is also qualitatively consistent with the mechanism proposed by Marshall et al. (2017): the baroclinic transport increases with increased eddy energy dissipation rate; the parameterization of lee-wave drag leads to a larger eddy energy dissipation rate and therefore a larger baroclinic transport.

c. Lower overturning circulation

Lee-wave drag and lee-wave-driven mixing regulate the lower overturning circulation through diapycnal diffusivity and isopycnal slope. The control of the lower overturning circulation by diapycnal diffusivity arises from its diabatic nature. The dependence of the lower overturning circulation on the isopycnal slope can be derived from the advection-diffusion equation for the mean buoyancy (Marshall and Radko 2003; Ito and Marshall 2008; Nikurashin and Vallis 2011):
Jy,z⁡(Ψres,b¯)=∂∂z⁡(κ ∂b¯∂z),
where Ψres is the residual lower overturning streamfunction, b¯ the mean buoyancy, κ the diapycnal diffusivity, and Jy,z⁡(Ψres,b¯)=⁡(Ψres)yb¯z−⁡(Ψres)zb¯y. The physical interpretation of the buoyancy advection-diffusion equation is that, in the steady state, for the equatorward flowing branch of the lower cell, the amount of dense water being advected northward is exactly the amount of water that is being converted by diffusion into lighter waters. Assuming that |b¯z|≫|b¯y|, we can neglect the horizontal advection term on the left hand side of (15) and rewrite the equation as
∂Ψres∂y∂b¯∂z=∂∂z⁡(κ ∂b¯∂z).
Replacing terms in (16) with their characteristic scales, we obtain a scaling relationship for the mixing-driven lower overturning circulation in the Southern Ocean,
Ψres=κh/l=κsρ,
where h and l are the characteristic depth and length scales, respectively, of the isopycnals in the Southern Ocean, and sρ the isopycnal slope.

In the lee-wave drag only experiment, the lower overturning streamfunction decreases by 3.4 Sv compared with the reference case (Figs. 6a,c) and this decrease is attributed to the increase in the isopycnal slope (Fig. 7). In the lee-wave-driven mixing only case, the lower overturning circulation increases by 5.5 Sv compared with that in the reference case (Figs. 6a,d) and this increase is due to the increase in the diapyncal diffusivity. The increase in the lee-wave-driven mixing only case is consistent with the findings in previous studies (e.g., Melet et al. 2014). Despite a slight increase in the isopycnal slope (Fig. 7), which tends to decrease the lower overturning streamfunction, the increase of the lower overturning streamfunction due to diapycnal diffusivity is more significant and therefore leads to an overall increase. In the lee-wave full parameterization case, the lower overturning streamfunction increases by 1.7 Sv compared with the reference case, but not as large as the increase in the lee-wave-driven mixing-only case (Figs. 6a,b,d). This smaller net increase in the full parameterization case again reflects the coupling between lee-wave drag and lee-wave-driven mixing.

The above explanations are based on the comparisons of the ACC and lower overturning circulation in two equilibria without and with the lee-wave parameterization. Here, we discuss how the ACC and the lower overturning circulation react to the lee-wave parameterization during the adjustment period. Introducing a lee-wave drag leads to a larger eddy energy dissipation rate and therefore initially reduces EKE. When the initial response from the eddy field occurs following the inclusion of lee-wave drag, the isopycnal slopes have not yet changed significantly since the adjustment of density layers in the ocean occurs at a much longer time scale. The initial reduction in EKE leads to a smaller eddy diffusivity (Taylor 1922) and weakens the eddy-driven overturning circulation (Gent and McWilliams 1990). The weakening of eddy-driven overturning circulation leads to the steepening of isopycnals as the eddy-driven overturning circulation becomes weaker and, as a result, its flattening effect on the isopycnals can no longer compete with the steepening effect of the wind-driven overturning circulation. Note that the wind-driven overturning circulation is linearly dependent on wind stress and therefore remains the same in the perturbation experiments as in the reference experiment. The steepening of isopycnals is accompanied by an increased vertical shear (through the thermal wind relation), which increases the eddy energy generation rate to match the increased eddy energy dissipation rate. The isopycnals become steeper gradually during the adjustment and remain at a slope that is larger than their original slope in the new equilibrium so that they continue to support the eddy generation at a higher rate to balance the increased eddy energy dissipation rate.

d. The coupling between the drag and mixing effects

The changes in the baroclinic transport are dominated by the drag effect and the changes in the lower overturning circulation are dominated by the mixing effect. However, the coupling between lee-wave drag and lee-wave-driven mixing weakens both drag and mixing effects in the full parameterization case. The coupling arises from the dependence of lee-wave-driven mixing on the bottom kinetic energy (Figs. 9, 10), as well as from the dependence of the lee-wave drag coefficient on the bottom stratification (8), which is in turn regulated by the intensity of mixing. This coupling weakens the mixing effect on the ocean stratification and lower overturning circulation: the presence of lee-wave drag reduces the energy in the deep ocean that can be provided for the lee-wave-driven mixing (Fig. 9). As a result, diffusivity is smaller in the lee-wave full parameterization experiment than in the lee-wave-driven mixing experiment (Fig. 10) and this smaller diffusivity in the lee-wave full parameterization experiment leads to a smaller decrease in the ocean stratification over the flat bottom (Fig. 8) and a smaller increase in the lower overturning streamfunction. This coupling also weakens the drag effect on the ocean stratification (Fig. 11) and baroclinic transport of the ACC: the presence of lee-wave-driven mixing decreases the ocean stratification (Fig. 11), which determines the lee-wave drag coefficient (8). A smaller lee-wave drag coefficient means a smaller increase in the eddy energy dissipation rate, a smaller increase in the eddy energy generation rate as a response, less steepening of the isopycnals, and a smaller increase in the baroclinic transport of the ACC due to the drag effect. Note that the mixing effect on the baroclinic transport is also weaker in the full parameterization experiment than in the lee-wave-driven mixing-only experiment. Even though the combined drag and mixing effects lead to a larger transport in the full parameterization experiment than in the reference experiment, the contribution of each component is reduced due to the coupling effect.

Fig. 9.
Fig. 9.

Bottom-500-m-averaged total kinetic energy is shown in log scale in the (a) full parameterization case and (b) lee-wave-driven mixing only case. (c) The difference between (a) and (b) with a linear scale.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

Fig. 10.
Fig. 10.

Bottom-500-m-averaged diffusivity associated with lee-wave-driven mixing is shown in log scale in the (a) full parameterization case and (b) lee-wave-driven mixing only case. (c) The difference between (a) and (b) with a linear scale.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

Fig. 11.
Fig. 11.

Bottom-500-m-averaged stratification is shown in log scale in (a) the full parameterization case and (b) lee-wave drag only case. (c) The difference between (a) and (b) with a linear scale.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0263.1

6. Summary

Lee-wave drag plays an important role in the energy budget of the eddy field in the Southern Ocean (Yang et al. 2018) and it can potentially modify the ACC and MOC through the EKE balance and isopycnal slope (Ito and Marshall 2008; Nikurashin and Vallis 2011; Marshall et al. 2017). However, to our knowledge, none of the current lee-wave parameterization schemes include both lee-wave drag and its associated lee-wave-driven mixing. Furthermore, it remains unclear how lee-wave drag affects the strength of the ACC and MOC. In this study, we develop and implement a lee-wave parameterization representing both drag and mixing effects of lee waves and use it to investigate the impact of lee waves on the Southern Ocean circulation.

We find that the parameterization of lee waves leads to an increase (over 40%) in the baroclinic transport of the ACC in our idealized model. Lee waves modify the ACC transport mainly through their drag effect on the eddy field. Similar to the frictional drag control of the ACC found in Marshall et al. (2017), the parameterization of lee-wave drag acts to increase the eddy energy dissipation rate and requires a larger eddy energy generation rate to balance. The eddy energy generation rate is primarily governed by baroclinic instability, which is dependent on the vertical shear of the horizontal velocity of the ACC, or equivalently (through the thermal wind relation) the slope of isopycnals. Therefore, the presence of lee-wave drag leads to steeper isopycnals and a larger baroclinic transport. Lee-wave-driven mixing is also found to steepen the isopycnals and enhance the baroclinic transport of the ACC, but to a smaller extent compared with the effect of lee-wave drag. When lee-wave drag and lee-wave-driven mixing are parameterized together, the increase in the baroclinic transport is similar to that in the lee-wave drag only case but larger than that in the lee-wave-driven mixing only case. This result indicates that the ACC transport is underestimated when only the lee-wave drag is parameterized. However, the increase in the baroclinic transport in the lee-wave full parameterization experiment is not a linear combination of that in the lee-wave-drag-only experiment and that in the lee-wave-driven mixing-only experiment. The nonlinear combination is due to the coupling between lee-wave drag and lee-wave-driven mixing.

We also find that the parameterization of lee waves increases the lower overturning circulation. Both lee-wave drag and lee-wave-driven mixing change the strength of the lower overturning circulation. When the diffusivity is constant, we show that the parameterization of lee-wave drag leads to the steepening of isopycnals and therefore a reduction in the lower overturning circulation. However, the weakening effect of lee-wave drag on the lower overturning is much smaller than the strengthening effect of lee-wave-driven mixing. As a result, the lower overturning circulation is increased in the lee-wave full parameterization case than in the reference case. The implication of our result is that the impact of lee waves on the lower overturning circulation might not be as significant as predicted by previous studies that only consider lee-wave-driven mixing.

We find a coupling effect between lee-wave drag and lee-wave-driven mixing. In the full parameterization experiment, the lee-wave drag coefficient is smaller than in the lee-wave drag only experiment, and lee-wave-driven mixing is weaker than that in the lee-wave-driven mixing only experiment. The coupling arises from two aspects; the dependence of lee-wave drag coefficient on the bottom stratification (8), which is regulated by the intensity of mixing; and the dependence of lee-wave-driven mixing on the bottom total kinetic energy, which is affected by lee-wave drag. The coupling leads to a nonlinear combination of the effect of lee-wave drag and that of lee-wave-driven mixing on the ocean stratification, baroclinic transport, and lower overturning circulation.

We use an idealized channel configuration in this study, which is representative of the Southern Ocean circulation albeit a lot simpler. The simplicity allows us to focus on the key governing processes through which lee waves exert a control on the ACC and the lower overturning circulation. This is a key step toward understanding the more complex dynamics that operate in the ocean. However, we acknowledge that more investigation needs to be done in the future because of the limitations of this configuration. Our idealized model does not include or simulate other processes that could modify the lee-wave field, such as internal tides and near-inertial oscillations (Nikurashin and Ferrari 2010b; Shakespeare 2020). The bottom stratification reproduced in our idealized model is smaller than the observed value, which tends to reduce the generation of lee waves. In addition, the salinity is set to be constant in the domain, whereas in the real ocean it can change density.

The lee-wave parameterization we use is novel in that it recovers a missing energy link between the eddy and mixing fields in the model. However, it is a very simple representation of the energy transfer from the eddy flow to lee waves based on linear lee-wave theory (Bell 1975) and is subject to the following limitations. We use a simple representation of small-scale topography with bulk parameters throughout the domain; however, in the real ocean, small-scale topographic features display a variety of shapes and orientations and their spatial distribution also makes a difference to the generation of lee waves (Yang et al. 2018). For a more realistic representation of small-scale topography, the constants chosen in section 2b should be replaced with maps. It is possible that a substantial increase in the thickness of the turbulent bottom boundary layer (e.g., Skyllingstad and Wijesekera 2004), not represented by our parameterization, can suppress lee-wave generation and hence is a source of uncertainty for the results presented here. However, previous lee-wave simulations suggest that these effects lead to quantitative changes and become important for critical and supercritical topography (Nikurashin and Ferrari 2010b), regimes less relevant to lee-wave generation in the Southern Ocean (Nikurashin and Ferrari 2011; Scott et al. 2011). We assume a linear dependence of lee-wave drag on the velocity following linear lee-wave theory (e.g., Gill 1982); however, this dependence can be nonlinear for multiscale topography (Nikurashin and Ferrari 2010a) or for low-level nonpropagating drag (Klymak et al. 2021). Recent high-resolution bottom pressure measurements also show that wave drag can be either linearly (Warner et al. 2013) or quadratically proportional to velocity (Wijesekera et al. 2014). Based on available observations in the Southern Ocean (e.g., Naveira Garabato et al. 2004; St. Laurent et al. 2012; Waterman et al. 2013; Sheen et al. 2013), we choose the exponential vertical profile (5) and the vertical decay scale of 500 m as an ad hoc representation of the wave radiation and breaking processes in the ocean. We assume that all of the energy in lee waves is dissipated locally, i.e., within a grid cell (10 km × 10 km), following Melet et al. (2014). The fraction of local dissipation may be smaller than 1, as suggested by the mismatch between the estimates of lee-wave generation rate and the observations of energy dissipation rate reported in the observational studies (e.g., St. Laurent et al. 2012; Waterman et al. 2013; Sheen et al. 2013; Waterhouse et al. 2014) and investigated in theoretical and modeling studies (e.g., Nikurashin et al. 2014; Kunze and Lien 2019; Zheng and Nikurashin 2019). However, the sensitivity of the large-scale ocean circulation to this parameter choice has been explored for tidal mixing (e.g., Jayne 2009) and lee-wave-driven mixing (e.g., Nikurashin and Ferrari 2013) and shown to lead to mainly quantitative changes. Nevertheless, we believe that improving the lee-wave parameterization will lead to quantitative changes and that our results on the impacts of lee waves on the ACC and MOC and corresponding feedbacks are physically and qualitatively robust. The implication of this study is that the parameterization of lee waves in eddy-resolving global ocean models should represent both lee-wave drag and its associated lee-wave-driven mixing in an energetically consistent way to represent the impact of lee waves on the Southern Ocean circulation.

Acknowledgments

This research was undertaken on the NCI National Facility in Canberra, Australia, which is supported by the Australian Government. LY was supported by the joint CSIRO–UTAS QMS program. MN was supported by the Australian Research Council (ARC) Discovery Project (DP170102162). BMS was jointly funded through CSIRO and the Earth Systems and Climate Change Hub of the Australian Government’s National Environmental Science Program. We thank Aidan Heerdegen, Angus Gibson, and Alistair Adcroft for their help on MOM6-related issues during the implementation of lee-wave parameterization.

Data availability statement

This manuscript is based on output from an eddy-resolving idealized ocean model; processed output and datasets to support the analysis are published with doi: 10.5281/zenodo.5225613. MOM6 source code can be downloaded from https://github.com/NOAA-GFDL/MOM6.

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