1. Introduction
Ocean mesoscale eddies have horizontal scales of tens to hundreds of kilometers and account for a majority of oceanic kinetic energy (e.g., Ferrari and Wunsch 2009; Storer et al. 2022). These eddies are important for the transport and mixing of momentum, heat, salt, and carbon, as well as other biogeochemical tracers, and impact the large-scale circulation and climate (Wolfe and Cessi 2010; Marshall and Speer 2012; Griffies et al. 2015; Gnanadesikan et al. 2015). These eddies are yet to be fully resolved in climate models due to their relatively small size, and as a result, their effects must be parameterized. Observational and modeling studies have shown that the strength of mesoscale eddies tend to peak at the surface and decay with depth (Wunsch 1997; de La Lama et al. 2016), with the exception for certain mode-water or topographically trapped eddies that intensify in the interior or near the bottom (Zhang et al. 2017; Radko 2023). Parameterizations that properly account for the surface-intensified vertical structure are crucial for simulations of large-scale ocean circulations and density structures (Danabasoglu and Marshall 2007; Eden et al. 2009). This effort requires a better understanding and representation of the eddy vertical structure, which is the topic of this paper.
The vertical structure of eddy motions is often represented using vertical normal modes (Wunsch 1997; Wortham and Wunsch 2014; de La Lama et al. 2016). Traditionally, these modes—including the barotropic and baroclinic modes—are solutions to an eigenvalue problem with flat-bottom and rigid-lid boundary conditions (Gill 1982; Vallis 2017). The surface-intensified structure of ocean eddies is then decomposed as a linear combination of these eigenmodes in the vertical.
Notably, the barotropic and first baroclinic modes together can capture the bulk of midlatitude eddy structure and variability (Wunsch 1997; Zhang et al. 2013). That these two modes are dominant is consistent with geostrophic turbulence theory, which reveals that eddies emerging from baroclinic instability tend to transfer energy from higher baroclinic modes to the lower modes and then to the barotropic mode at scales comparable to the deformation radius (Salmon 1980; Smith and Vallis 2001). Due to surface-intensified stratification and bottom friction, ocean eddies are usually not fully barotropized and the first baroclinic contribution remains considerable (Fu and Flierl 1980; Smith and Vallis 2001). Although the barotropic and first baroclinic modes are useful for diagnosing the vertical structure of eddies, they are inefficient for quantitative predictions since their relative contribution to eddy energy varies substantially in space and time (Wunsch 1997).
Vertical mode structures can change significantly in the presence of bottom topography (Hallberg 1997; LaCasce 2017), leading to recent studies promoting a different set of baroclinic modes subject to zero horizontal velocity at the bottom due to rough bathymetry (LaCasce 2017). The inclusion of the rough bottom tends to decouple the bottom pressure from the interior, yielding modified baroclinic structures that are more surface-intensified (Rhines 1970; Hallberg 1997; Samelson 1992; Tailleux and McWilliams 2001). The first baroclinic mode derived under such rough-bottom setup captures a larger fraction of observed velocity variance than the traditional first baroclinic mode (de La Lama et al. 2016; Ni et al. 2023) and has recently been used for diagnoses and parameterizations of the vertical structure of eddy velocity and mixing (Adcroft et al. 2019; Groeskamp et al. 2020; Stanley et al. 2020; Holmes et al. 2022).
Still, these rough-bottom modes assume a zero buoyancy anomaly at the ocean surface, whereas ocean eddies exhibit strong surface temperature and salinity anomalies (Lapeyre 2009; Hausmann and Czaja 2012; Frenger et al. 2015). Theoretical studies have proposed vertical-mode bases that include surface buoyancy anomalies, associating with surface-trapped responses that decay quasi-exponentially from the surface (Lapeyre and Klein 2006; Smith and Vanneste 2013; Yassin and Griffies 2022a). The surface-trapped mode is governed by the surface quasigeostrophic (SQG) dynamics in the presence of a surface horizontal buoyancy gradient (Blumen 1978; Held et al. 1995; Lapeyre and Klein 2006; Lapeyre 2017). Previous studies noted that the SQG mode can capture the surface intensification of the observed eddies better than the traditional baroclinic structures, pointing to surface buoyancy anomalies being key to establishing the vertical distribution of ocean eddies (Lapeyre and Klein 2006; Lapeyre 2009).
The SQG framework has been adapted to infer the eddy properties in the vertical from surface observations and hydrography in the midlatitude ocean in both modeling (Lapeyre and Klein 2006; Isern-Fontanet et al. 2008; Wang et al. 2013; Ponte and Klein 2013; Liu et al. 2014; Qiu et al. 2016; Fresnay et al. 2018; Qiu et al. 2020; Miracca-Lage et al. 2022) and observational studies (LaCasce and Mahadevan 2006; Rocha et al. 2013; Liu et al. 2017). Such inference, however, depends on the spectra of the surface eddy energy or buoyancy because the vertical structure of the SQG mode depends on the horizontal eddy scale—smaller eddies tend to have shallower structures (Lapeyre 2017; Ajayi et al. 2020). The scale dependence seemingly limits the application of the SQG framework to the parameterization of eddies for ocean models in which mesoscale features are not fully resolved.
Here, we argue that knowing the energy-containing scale is sufficient to determine the dominant SQG mode, supporting a sparse representation of the vertical structure of mesoscale eddies. Assuming the energy-containing scale is given and using this information in lieu of the full surface spectra, we propose an SQG-based, scale-aware parameterization for the vertical structure of ocean eddies. Focusing on the dependence of vertical structure on horizontal scales, offline analyses with output from an idealized adiabatic ocean model demonstrate that the proposed scheme reproduces the desired distributions of unresolved eddies by models with different horizontal resolutions. The only remaining step for a completely closed parameterization is the determination of the horizontal energy-containing scale, which is a topic outside the scope of this paper.
Why does the SQG mode at the energy-containing scale play such an important role in the vertical eddy structure? We propose that geostrophic turbulence theory offers insights to this question. Surface buoyancy anomalies are often represented as surface-confined potential vorticity (PV) anomalies (Bretherton 1966). The horizontal PV gradient tends to be substantial at the ocean surface, yielding surface-trapped instabilities, such as Charney and mixed-layer instabilities (Smith 2007; Fox-Kemper et al. 2008; Tulloch et al. 2011; Roullet et al. 2012; Callies et al. 2016; Schubert et al. 2020; Feng et al. 2021, 2022). The link between baroclinic instability and the vertical structure of mesoscale eddies is investigated here in an idealized adiabatic ocean model. We find that the SQG-like profile of eddy kinetic energy (EKE) is prevalent in purely wind-driven circulations analogous to those in the Atlantic Ocean. The simulated EKE exhibits (i) a surface-intensified profile consistent with the SQG mode at the energy-containing scale and (ii) a dependence of vertical distribution on the horizontal eddy scale. Although our adiabatic model does not have surface buoyancy forcing and a mixed layer, the isopycnal outcropping due to wind forcing gives rise to a strong surface meridional PV gradient. In the upper ocean, the interior PV gradient is weak due to eddy mixing, which tends to homogenize the PV within geostrophic contours (Rhines and Young 1982b). Such a large-scale PV distribution induces surface-trapped baroclinically unstable modes with an SQG-like vertical structure. These surface-trapped eddies supply energy to the energy-containing eddy through an inverse energy cascade, ultimately shaping the EKE profile.
This manuscript is organized as follows. Section 2 covers equations of the baroclinic and surface-trapped modes for theoretical background. The numerical model configuration and Wentzel–Kramers–Brillouin (WKB) solutions for the SQG mode are given in section 3. In section 4, we reveal the key role of SQG modes in setting the vertical structure of EKE, which can be determined by the horizontal energy-containing scale. In section 5, we argue that the dominance of the SQG mode arises from baroclinic instability and the corresponding inverse energy cascade, which are surface-trapped due to the upper-ocean PV structures. Based on these understandings, in section 6, we propose a SQG-based parameterization for the subgrid EKE profile that depends on the numerical model horizontal grid spacing. The results are summarized, and their implications for understanding the eddy vertical structure are discussed in section 7.
2. Theoretical background
In this section, we recap elements from QG theory, focusing on (i) the vertical structure of baroclinic modes and (ii) modifications to take into account a surface buoyancy anomaly.
a. Baroclinic modes
The first four vertical normal modes solved from the eigenvalue problem in Eq. (3) with (a) flat-bottom and rigid lid boundaries and (b) rough-bottom and rigid-lid boundaries. The stratification decays exponentially with depth following an e-folding scale of a third of the water depth in both cases. Each profile of Φm is normalized so that
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
A notable feature of the two Sturm–Liouville problems considered thus far is that the horizontal wavenumbers do not appear in them, except for the bottom topographic waves that are not shown. Therefore, the vertical structure of the normal modes does not depend on the horizontal scales. This property can be attributed to the surface boundary conditions not including buoyancy anomalies (i.e., dΦ/dz = 0 at z = 0)—an assumption shared by both of the flat- and rough-bottom setups.
b. Surface-trapped modes
To incorporate surface buoyancy anomalies, a surface-aware formulation is proposed to include the horizontal buoyancy gradient at the surface (and bottom), as briefly outlined here following Smith and Vanneste (2013) as well as Yassin and Griffies (2022a).
Vertical profiles of the (a) surface-trapped mode (i.e., the SQG mode) and (b) interior modes, determined by solving Eqs. (8) and (9), respectively. The stratification profile is an exponential function of z, as in Fig. 1. The surface mode is calculated for three different horizontal wavenumbers, 1/(πrd), 2/(πrd), and 8/(πrd), where rd is the Rossby deformation radius of the first flat-bottom baroclinic mode (orange line in Fig. 1a). Each surface-trapped mode profile is normalized by its surface value. The lines in (b) indicate the first four interior modes, each of which is normalized in the same way as the modes in Fig. 1.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
Note the similarity between Eq. (9) and the earlier rough-bottom setup (the surface and bottom BCs are switched here). Additionally, the vertical structure of the interior modes does not depend on horizontal scales (see Fig. 2b).
In contrast, Eq. (8) explicitly includes the horizontal wavenumber, yielding an outstanding feature of the surface-trapped mode: Larger eddies (with smaller horizontal wavenumber k) are associated with deeper modes whose vertical response decays slower with depth (see Fig. 2a). Because Φs has a zero interior PV anomaly and is driven by a surface buoyancy anomaly—consistent with the SQG dynamics (Blumen 1978; Held et al. 1995)—it is also referred to as the “SQG mode.” The SQG mode is relevant to the oceanic cases where a horizontal buoyancy gradient intensifies near the surface (Smith 2007; Capet et al. 2016), and this mode is central to this paper.
The decomposition given by Eqs. (8) and (9) is consistent with the surface mode in Smith and Vanneste (2013) [their Eq. (19)] and is different from that proposed by Lapeyre and Klein (2006). Lapeyre and Klein (2006) use a Neumann surface boundary condition (i.e., surface buoyancy anomaly is specified) for the surface-trapped mode, while Φs here has a Dirichlet surface boundary condition (i.e., surface pressure anomaly is specified). Studies have found that the SQG mode using the surface pressure anomaly is better for reproducing the vertical structure of EKE than that using the surface buoyancy anomaly (Isern-Fontanet et al. 2008; Qiu et al. 2016, 2020). More details are provided in appendix A.
Both of the surface-trapped and interior modes described above assume a flat-bottom condition, i.e., vanishing bottom buoyancy anomaly; recall the bottom BCs in Eqs. (8) and (9). These are the limiting cases of Eq. (6) through which more complicated topographic effects can be included (LaCasce 2017; Yassin and Griffies 2022a) and will be explored in future work.
3. Methods
a. Model configuration
The numerical model used in this study is an idealized configuration of the Modular Ocean Model, version 6 (MOM6). This configuration is named “NeverWorld2” by Marques et al. (2022) and has been used to study mesoscale eddy parameterizations (e.g., Loose et al. 2022; Yankovsky et al. 2022). The model domain is a single basin with two hemispheres using spherical coordinates. The domain extends from 70°S to 70°N in the meridional direction, 0°–60°E in the zonal direction, and has a maximum depth of 4000 m. The domain is bounded by a 200-m continental shelf along all side boundaries, except for a zonally reentrant channel located at 60°–40°S, mimicking the Southern Ocean. A ridge topography orienting meridionally spans the entire meridional extent of the domain. The ridge has a width of 40° and maximum height of 2000 m. There is a semicircular ridge with a radius of 10° spanning 0°–10°E longitude and 60°–40°S latitude that mimics the Scotia Arc across the Drake Passage.
The Boussinesq and hydrostatic primitive equations are discretized vertically using isopycnal coordinates, with a total of 15 isopycnal layers with finer spacing near the surface in the initial setup. The horizontal grid spacing is 1/32°. The model is driven solely via mechanical forcing from a zonally uniform zonal wind stress that varies meridionally and is fixed in time. There is no surface buoyancy forcing, no diabatic mixing, nor is there a mixed layer parameterization (i.e., it is a wind-driven adiabatic stacked shallow-water model). Dissipation arises from a background kinematic vertical viscosity (Aυ = 1.0 × 10−4 m2 s−1), a dimensionless bottom drag (Cd = 0.003), and the horizontal friction is given by a biharmonic Smagorinsky viscosity (Griffies and Hallberg 2000). More details about the model setup and spinup are given in Marques et al. (2022).
Figure 3a shows a snapshot of the surface specific kinetic energy (KE). The black lines correspond to the 500-day mean sea surface height (SSH) contours, representing the streamlines of surface geostrophic currents outside the tropics. The circulation pattern is a caricature of the Atlantic Ocean, with subtropical gyres and western boundary currents in both hemispheres. In the Northern Hemisphere, there is a subpolar gyre, while in the Southern Hemisphere, a circumpolar current prevails in the reentrant channel. The KE illustrates the prevalence of mesoscale eddies throughout the domain. The Rossby deformation radius is resolved over most regions of the model, with the exception being the continental shelf and very high latitudes near the boundaries (Yankovsky et al. 2022).
(a) Snapshot of horizontal KE at the surface. Black lines are contours of the 500-day mean SSH. Blue boxes labeled a, b, c, and d indicate the 8.7° × 8.7° regions where the vertical structure of eddies is analyzed in later figures. (b) A meridional section of the EKE per unit mass along the longitude shown by the blue dashed line in (a). Black lines show all the isopycnals in the model. Blue lines indicate the e-folding depth he of the rms eddy velocity (square root of EKE). The green line is the deformation depth h0 of the energy-containing scale, introduced in section 4b.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
b. Solution for the SQG mode
The SQG mode Φs, described by Eq. (8), is diagnosed in the stacked shallow-water model introduced above. In this model, the dynamical impact of surface buoyancy is represented by the surface-layer PV, as described by the layered QG dynamics detailed in appendixes B and D.
Equation (10) reveals the important characteristics of the SQG mode, namely, its exponential decay with zs and its dependence on the horizontal scale of eddies. Smaller eddies with larger wavenumber k decay faster with depth than larger eddies with smaller wavenumber. The scale dependence of the SQG mode provides valuable insights for parameterizing the vertical structure of eddies, as we discuss in section 6.
Note that Eq. (10) differs from the widely used “eSQG” method in ocean studies (Lapeyre and Klein 2006). The eSQG method assumes a constant N with depth, determined empirically to account for the impact of an internal PV gradient (Lapeyre and Klein 2006). In contrast, our formula accounts for the vertical variation of N within the stretched vertical coordinate zs, allowing for a more accurate representation of the SQG mode itself. For example,
4. Vertical structure of EKE
a. Comparison to the vertical modes
Vertical profiles of EKE in the four selected regions are shown in Fig. 4. The EKE decays rapidly with depth in the upper ocean and gradually approaches a constant in the deep ocean. The barotropic component (i.e., the depth-independent flat-bottom mode) accounts for about 78% of the total EKE in the circumpolar current region and about 50% in the other three regions.
Vertical profiles of the horizontally averaged EKE (solid blue lines with dots) in the four labeled regions in Fig. 3. The four regions are located in the (a) circumpolar current, (b) subtropical gyre, (c) western boundary current, and (d) subpolar regions. Dashed orange and green lines are the numerical [Eq. (8)] and WKB [Eq. (15)] solutions of the SQG mode, respectively, summed over horizontal wavenumbers of the surface EKE spectrum. Red dashed line is the WKB solution of the SQG mode at the energy-containing scale [Eq. (17)]. The purple dashed line is the vertical structure of the EBT mode.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
The profile of EEBT is shown by the purple dashed line in Fig. 4. The EBT mode is surface-intensified, but it decays more slowly with depth than the diagnosed EKE. Furthermore, the EBT mode has a zero vertical gradient at the surface due to its boundary condition in Eq. (5), whereas the EKE has a strong vertical gradient near the surface. Another limitation is that the magnitude of the EBT mode is zero at the bottom, but the EKE can be significant at the bottom even with strong gradients in the bottom topography.
b. Relating to the energy-containing scale
(a)–(d) 1D surface KE spectrum (blue lines with dots) in the four regions shown in Fig. 3. The spectrum,
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
The vertical structure given by Eq. (17) is shown by the red dashed line in Fig. 4. It works similarly well as the SQG solution based on the full energy spectrum, meaning that the vertical structure of EKE can be represented by the SQG mode structure at the energy-containing scale.
The e-folding depth he and the deformation depth of the energy-containing scale h0 are compared in the meridional section in Fig. 3b. They compare well to each other in the extratropical regions (poleward of 20°), which is consistent with the good comparison between the EKE vertical structure and the SQG mode in these regions in Fig. 4. In the tropics, h0 decreases to zero toward the equator due to the decreasing magnitude of Coriolis parameter |f| in Eq. (18), while he remains around 1000–2000 m. The large fluctuations of he in the tropics might be due to the spatial variability of strong undercurrent and equatorial waves.
The comparison between he and h0 at other meridional sections is similar to that in Fig. 3b. The e-folding depth he for the eddy velocity is generally shallower than a half of the water depth. The SQG mode captures the vertical structure of EKE well in the extratropical regions, while it is shallower than the EKE profile in the tropics. Figure 3a shows that the kinetic energy pattern is wave-like in the tropics, indicating that EKE is dominated by linear waves there, while in the extratropics, the flow field contains abundant coherent vortices, indicating the dominance of nonlinear eddies. Previous studies have shown that the eddy flux is dominated by linear waves and nonlinear eddies in the tropics and midlatitude ocean, respectively, leading to different scalings of eddy mixing in these regions (Klocker and Abernathey 2014; Zhang and Wolfe 2022). We expect that Eq. (17) is effective in describing the vertical structure of EKE for nonlinear eddies, which are dominant in the extratropical ocean (Chelton et al. 2011).
5. Why is the vertical structure SQG-like?
In section 4, we have seen that the EKE structure tends to be SQG-like. Here, we discuss the underlying physics by showing (i) the EKE profile is consistent with the distributions of the eddy PV and large-scale meridional PV gradient; (ii) the surface meridional PV gradient yields baroclinic instability that produces surface-trapped unstable modes; and (iii) the kinetic energy created by baroclinic instability cascades upscale in the form of a surface-trapped mode, resulting in an SQG-like EKE profile dominated by the energy-containing eddy.
a. PV structure
Meridional and vertical distributions of (a) the rms eddy PV and (b) the meridional PV gradient normalized by β at the longitude indicated by the blue dashed line in Fig. 3a. Black solid lines are the isopycnals in the model. Black dashed lines with characters a, b, c, and d indicate the latitude range of the four regions shown in Fig. 3. Note that a and c are not in the same longitude as this meridional section.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
The distribution of eddy PV is associated with the large-scale meridional PV gradient Qy structure, shown in Fig. 6b. The meridional PV gradient is estimated from Eq. (C3) in appendix C for the layered shallow-water model. The magnitude of meridional PV gradient peaks at the surface due to the isopycnal outcropping. The interior PV gradient is weak compared with the surface PV gradient, especially in the upper 1000 m in the mid- and high latitudes (poleward of 30°). The weak interior PV gradient is likely due to the isopycnal mixing by eddies, which act to homogenize the PV within closed geostrophic contours in the interior (Rhines and Young 1982a,b). The large-scale PV structure shown in Fig. 6b is consistent with observations of upper-ocean circulations (Keffer 1985; Talley 1988; Pedlosky 1996). The presence of such a large-scale PV structure is responsible for the surface-intensified eddy PV distribution in Fig. 6a.
Besides the surface intensification, the magnitude of the PV gradient also intensifies in the near-bottom layers in the circumpolar current, which is consistent with the distribution of eddy PV in this region. In the subtropics, the PV gradient also peaks at about 400 m due to the flattening of isopycnals at the interior pycnocline. This pronounced interior PV gradient is likely the cause of the strong subsurface eddy PV and the deviation of the EKE profile from SQG in the low latitudes. Quantitative analyses of the impact of the PV gradient on the eddy vertical structure are given in the next section.
In addition to the variation of magnitude, the meridional PV gradient also tends to change sign from surface to the interior. In the mid- and high latitudes (except for the circumpolar current), the meridional PV gradient is positive at the surface layer, negative at layers below the surface, and positive again in the deep ocean. In the circumpolar current, the PV gradient is positive at the surface, remains weak in the interior, and becomes negative in the deep ocean. A similar distribution of the meridional PV gradient is found by Tulloch et al. (2011) (their Fig. 2a) based on oceanic reanalysis climatology. Tulloch et al. (2011) also reveals a positive surface PV gradient in the subtropics, a feature that is missing in our simulation. Figure 6b shows that the surface PV gradient is negative in the subtropics due to the upward isopycnal slope toward the equator, while in the real ocean, surface buoyancy forcing can generate a negative surface buoyancy gradient (equivalent to a positive surface PV gradient) by reversing the near-surface isopycnal slope in the subtropics. Since the NeverWorld2 model is forced by wind stress only, this positive surface PV gradient is not captured by the simulation.
b. Linear stability
The meridional PV gradient is often found to play a crucial role in oceanic baroclinic instability, which is an important energy source for mesoscale eddies (Smith 2007; Tulloch et al. 2011; Capet et al. 2016). A necessary condition for baroclinic instability is for the background horizontal PV gradient to change sign in the vertical, which is a typical feature of the meridional PV gradient shown in Fig. 6b.
We analyze baroclinic stability in the four regions discussed in section 4. The vertical profiles of the meridional and zonal PV gradients in the four regions are shown in Fig. 7. Figure 8 shows the distribution of the growth rate σ of unstable modes, which are solved from the eigenvalue problem described by Eq. (C6) in appendix C. Figure 9 shows the vertical structure of the most unstable mode, indicated by the cyan point in Fig. 8. The orange dashed line indicates the SQG mode at the same horizontal wavenumber as the unstable mode. We compare the vertical structure of the most unstable mode with the SQG mode and connect this result to the interpretation of the EKE profile in these four regions.
(a)–(d) Vertical profiles of meridional (blue lines with dots at the layer centers) and zonal (orange lines) PV gradient normalized by β in the four regions shown in Fig. 3.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
(a)–(d) Growth rate of unstable modes as a function of zonal and meridional wavenumbers, kx and ky, respectively, that are normalized by the deformation radius in the same four regions as in Fig. 3. The cyan point indicates the most unstable mode.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
(a)–(d) Vertical structure of the most unstable mode (blue solid line) indicated by the cyan point in Fig. 8. The orange dashed line is the normalized vertical structure of the SQG mode at the same wavenumber as the unstable mode.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
In the circumpolar current, the interior PV gradients are about ten times smaller than the surface and bottom gradients. The surface and bottom meridional PV gradients are opposite to each other, which gives rise to unstable modes at scales larger than the deformation radius. The vertical structure of the most unstable mode intensifies at both surface and bottom, which is a typical feature of the Eady-type instability arising from the interaction between the surface and bottom edge waves (Eady 1949). The Eady unstable mode is also observed in the Southern Ocean (Tulloch et al. 2011; Feng et al. 2021). The unstable mode is similar to the SQG mode in the upper ocean but not in the deep ocean because of the bottom intensification of the unstable mode. Conversely, the vertical structure of EKE only shows a slight intensification near the bottom and appears more like the SQG mode than the unstable mode in this region (Fig. 4a). The difference between the unstable mode and EKE is likely due to the strong damping by friction at the bottom, thereby reducing the bottom intensification of EKE.
In the western boundary current and subpolar regions, both the meridional and zonal PV gradient changes sign near the surface. The meridional PV gradient is generally stronger than the zonal gradient, except near the bottom, where the zonal PV gradient is large due to the ridge topography. The most unstable mode occurs at scales smaller than the deformation radius in these two regions. The vertical structure of the most unstable mode is surface-intensified, indicating a Charney-type instability arising from the interaction between surface edge waves and interior Rossby waves (Charney 1947). These unstable mode structures are consistent with those observed in the Gulf Stream and North Pacific regions (Tulloch et al. 2011; Capet et al. 2016). The vertical structure of the unstable mode resembles the SQG mode, similar to the vertical structure of EKE in the two regions (Figs. 4c,d).
Although the meridional PV gradient also changes sign at 500–1000-m depth, nearly all unstable modes are surface-intensified in the western boundary current and subpolar regions (not shown), indicating that the instability is always associated with surface edge waves. The Rossby wave propagates eastward in the negative PV gradient layers and westward in the positive PV gradient layers below (the zonal mean flow is eastward above 1000 m and becomes weak below), so they cannot couple to each other due to the mismatch between their zonal phase speeds. Consequently, the sign change of meridional PV gradient at 500–1000 m does not lead to the Phillips type instability (Phillips 1954) in the interior. For the same reason, the zonal PV gradient also does not induce Phillips instability in the deep ocean, although the bottom zonal PV gradient is strong due to the topography.
In the subtropics, the meridional PV gradient is negative at the top layer (note that the real ocean has a positive surface PV gradient above the top layer here) and positive in the interior. Such a PV gradient profile induces Charney-type instability—the growth rate of the unstable mode maximizes at scales smaller than the deformation radius, and the vertical structure of the most unstable mode is surface-intensified. This unstable mode structure is similar to that observed in the subtropical Atlantic Ocean (Tulloch et al. 2009), despite the absence of a positive surface PV gradient in the subtropics in our simulation. The unstable mode also exhibits a subsurface maximum, which deviates from the SQG mode. This subsurface intensification is likely attributed to the intensified meridional PV gradient at around 400 m due to the rapid vertical variation of isopycnal slope at the pycnocline in this region (Figs. 6b and 7b). Such subsurface intensification of unstable modes might contribute to the deviation of the vertical structure of EKE from the SQG mode in this region (Fig. 4b).
Lapeyre (2009) also notes that the SQG mode contributes a minor fraction to the velocity variance in the subtropical Atlantic Ocean based on a realistic simulation. This suboptimal performance of the SQG mode is attributed to the negative correlation between interior and surface meridional PV gradients near the surface, a pattern akin to the observation that meridional PV gradient changes sign near the surface in the subtropics in our model (Fig. 7b). In the western boundary current and subpolar regions, the meridional PV gradient also changes sign in the upper ocean, consistent with Lapeyre (2009)’s Fig. 12a where the regression of PV on surface buoyancy is negative at 200–500 m in the Gulf Stream and North Atlantic Current. We attribute the good performance of the SQG mode in these two regions to the significantly weaker negative PV gradient in the upper ocean compared to the positive surface PV gradient (Figs. 7c,d).
The resemblance between the unstable modes and the SQG mode is attributed to the large-scale PV structure, shown in Fig. 6b, manifesting as a much stronger surface meridional PV gradient than an interior PV gradient. Although the large-scale circulations are purely wind-driven in our shallow-water model, a similar PV structure has been observed in the ocean (Talley 1988; Tulloch et al. 2011). As a result, the Charney- and Eady-type instabilities have been found to be prevalent in the midlatitude ocean (Capet et al. 2016; Feng et al. 2021), consistent with the results of our simulation. Phillips-like instability, characterized by subsurface-intensified unstable mode structures, has also been observed in the subtropical and midlatitude ocean, particularly during the summer (Feng et al. 2022). The relatively rare occurrence of such subsurface-intensified instability in our model may be attributed to the absence of surface buoyancy forcing. Such forcing is known to enhance the near surface stratification during the summer, reducing the prevalence of Charney instability (Capet et al. 2016). The cases discussed here are thus more relevant to ocean situations during the winter and spring.
The surface PV gradient also plays an important role in the mixed layer instability, which is not present in our model due to its lack of a mixed layer. Eddies generated by mixed layer instability exhibit characteristics similar to the Eady mode within the mixed layer and resemble SQG modes below the base of the mixed layer (Callies et al. 2016). Mixed layer instability is thus analogous to the Charney instability in its ability to generate surface-trapped unstable modes, with the distinction that the “surface” for the former shifts to the base of the mixed layer.
c. Role of the energy cascade
The advection term Tn quantifies the spectral tendency of KE due to the KE cascade between horizontal wavenumbers. The positive and negative values of Tn(k) indicate that energy cascade deposits and removes KE, respectively, for eddies at the corresponding wavenumber k. The time tendency term on the LHS of Eq. (20) is close to zero when taking a long-time average. In this case, the energy cascade balances the net energy source or sink due to the work by the horizontal pressure gradient [second term on the RHS of Eq. (20)], forcing and dissipation [third term on the RHS of Eq. (20)] at each wavenumber. The work by the horizontal pressure gradient includes the contribution from the divergence of 3D energy transport by pressure and the conversion of potential energy (PE) to KE due to baroclinic instability (Capet et al. 2008).
Figure 10 shows the distribution of Tn at horizontal wavenumber and depth in the four regions. The vertical structure of Tn is generally surface-intensified and decays faster with depth at smaller scales (larger wavenumbers), which is consistent with characteristics of the SQG mode. In the circumpolar current, the energy cascade removes energy at around the deformation radius. This energy sink due to the energy cascade intensifies at both the surface and bottom, which is consistent with the unstable mode of the Eady instability at this region (section 5b), indicating that the energy cascade balances the baroclinic instability at these wavenumbers. At the other three regions, the energy cascade mainly removes energy from scales smaller than the deformation radius. This energy sink concentrates near the surface, which is likely to balance the energy source due to the Charney baroclinic instability in these regions. There is also an energy sink at small wavenumbers at around 1000 m in the subtropical and subpolar regions, though it is about an order of magnitude smaller than the value of Tn at the surface. This sink is likely to balance the baroclinic instability at small wavenumbers in these two regions (see the growth rate distribution in Fig. 8).
Cospectrum of the advection term of KE, Tn, as a function of horizontal wavenumber and depth in the four regions shown in Fig. 3. Blue color indicates that nonlinear advection removes KE from the corresponding wavenumber, which balances the KE source by other terms on the RHS of the KE equation [Eq. (20)], while red color means that nonlinear advection supplies KE to balance the KE sink by the other terms. The black dashed line indicates the inverse of the Rossby deformation radius. The purple dashed line indicates the energy-containing wavenumber k0. Note the different color range for each panel.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
The energy cascade deposits energy at scales larger than the deformation radius in all the four regions (red color in Fig. 10). This energy source due to energy cascade maximizes at the energy-containing scale (purple dashed line in Fig. 10), where it is balanced by energy conversion from KE to PE and friction (not shown). The vertical distribution of the energy source remains surface-trapped at the energy-containing scale, giving rise to the SQG-like EKE profile there.
The energy cycle shown in Fig. 10 is illustrated by the schematic in Fig. 11, which is motivated by a similar schematic in Roullet et al. (2012). The forcing inputs PE to circulation at the large scales. The PE is converted to KE at scales that are close to or smaller than the deformation radius through baroclinic instability. The baroclinic instability is generally dominated by the surface PV gradient, which gives rise to surface-trapped unstable modes. These unstable modes cascade KE from small scales to large scales through their nonlinear interactions until the cascade is halted at the energy-containing scale. Consequently, the vertical structure of EKE is dominated by the mode structure at the energy-containing scale, which is deeper than the original unstable modes but still remains surface-trapped.
Energy schematic modified from Roullet et al. (2012). The forcing supplies available PE (upper horizontal lines). The available PE cascades to smaller scales and is then converted to KE (lower horizontal lines) through baroclinic instability (blue arrows) at a broad range of horizontal scales that are close to or smaller than the deformation radius, 1/kd. The unstable modes arising from the instability are surface-trapped, which decay faster with depth at smaller scales. The KE of these unstable modes cascades from small scales (shallow modes) to large scales (deep modes) until the cascade is halted at the energy-containing scale, 1/k0. Part of the KE is dissipated by the friction, and the remainder is converted back to available PE (orange arrows) at around the energy-containing scale.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
6. A scale-aware parameterization
Section 4 shows that SQG modes play an important role in setting the vertical structure of EKE. How can this finding inform a parameterization of the eddy vertical structure? A crucial property of the SQG mode is the coupling of its vertical and horizontal scales. As shown by the analytical expression (10), smaller horizontal scale eddies, which have a larger horizontal wavenumber, decay more rapidly with depth than larger eddies. In the context of eddy parameterization, the focus is on representing the subgrid EKE that is unresolved by the model. As the model grid spacing is refined, the scale of unresolved eddies becomes smaller, resulting in a shallower vertical structure of subgrid EKE. This result suggests that parameterization of the vertical structure of subgrid EKE should depend explicitly on the model grid spacing.
To examine the scale dependence of the eddy vertical structure, we employ high-pass spatial filters on the eddy velocity fields to compare the vertical structure at different horizontal scales. Figure 12 shows the normalized vertical structures for eddies smaller than 2π/0.25°, 2π/0.5°, and 2π/1°, represented by the solid blue, orange, and green lines, respectively. The vertical structure of EKE becomes shallower as the horizontal scale becomes smaller. This finding is consistent with the prediction by SQG, highlighting the scale dependence of the eddy vertical structure.
The vertical structure of spatially filtered EKE in the four regions shown in Fig. 4. Blue, orange, and green solid lines are the high-pass filtered vertical structures of EKE with cutoff wavenumber as 1/0.25°, 1/0.5°, and 1/1°, respectively. They represent the vertical structures of subgrid EKE that is unresolved by 0.25°, 0.5°, and 1° models, assuming that a minimum of six grid points is required to resolve an eddy. All EKE profiles are normalized by their surface values. Blue, orange, and green dashed lines are the parameterized vertical structure Ep(z, Δ). The Ep depends on the model grid spacing Δ and the energy-containing wavenumber k0. The vertical structure of subgrid EKE is determined by Δ at 0.25° and 0.5° resolutions, where the energy-containing eddy is resolved, and by k0 at 1° resolution. The parameterization captures the scale dependence reasonably well, except for an underestimation in the subtropical region.
Citation: Journal of Physical Oceanography 54, 6; 10.1175/JPO-D-23-0203.1
To evaluate the effectiveness of the parameterization [Eq. (22)], we conduct tests by varying Δ to 0.25°, 0.5°, and 1° and assume c = 1, which means that we assume a minimum of 6 grid points is required to resolve an eddy. Figure 12 shows that the parameterization profiles, estimated by Eq. (22), look similar to the vertical structure of subgrid EKE in 0.25°, 0.5°, and 1° model grid spacings, with the exception for the subtropics where the parameterization tends to decay faster than the subgrid EKE. This result is consistent with the comparison between the SQG mode and the total EKE in Fig. 4.
The parameterization based on Eq. (22) requires a prediction of the energy-containing scale. Studies have estimated the energy-containing scale as the Rossby deformation radius (Stone 1972), width of the baroclinic zone (Visbeck et al. 1997), Eady length scale (Larichev and Held 1995; Jansen et al. 2015), and Rhines scale (Jansen et al. 2015, 2019). Examination of these theories for predicting the eddy scale is beyond the scope of this study and will be pursued in the forthcoming work.
7. Discussion and conclusions
This study addresses the question of why ocean mesoscale eddies are surface-intensified—at least those realized within our adiabatic and wind-driven shallow-water model—with an aim to parameterize the vertical structure of mesoscale eddies. Unlike previous studies that explain the eddy vertical structure from the perspective of baroclinic modes (Wunsch 1997; de La Lama et al. 2016; LaCasce 2017; Brink and Pedlosky 2020; Quan et al. 2023), we attribute the eddy vertical structure to the baroclinic instability and the energy cascade. The vertical structure of EKE is found to be well-represented by a single SQG mode of the energy-containing scale in the extratropics, indicating that the surface PV plays a dominant role in the vertical structure of mesoscale eddies.
a. Baroclinic instability and the surface-trapped mode
The prevalence of the SQG mode is attributed to the surface-trapped baroclinic instability and energy cascade. Although the model used in this study does not have surface buoyancy forcing, the wind forcing induces isopycnal outcropping, leading to pronounced surface meridional PV gradients. The interior PV gradient is much weaker compared with the surface PV gradient due to the interior PV homogenization by eddies. Linear stability analysis shows that the surface PV gradient plays a dominant role in baroclinic instability, giving rise to surface-trapped unstable modes. These unstable modes are often smaller than the Rossby deformation radius, except in the circumpolar current where they are close to the deformation radius. Through nonlinear interactions, unstable modes transfer their kinetic energy upscale to energize mesoscale eddies, which have an energy-containing scale larger than the deformation radius but retain the surface-trapped vertical structure.
Early geostrophic turbulence theory shows that baroclinic eddies tend to transfer their energy to the barotropic (depth independent) mode, and then barotropic eddies cascade their energy upscale (Charney 1971; Salmon 1980). Later studies have found that ocean eddies are not fully barotropic, and as a consequence, the inverse energy cascade can occur in both the barotropic and baroclinic modes (Smith and Vallis 2001; Scott and Arbic 2007). In this study, we propose a different interpretation of the eddy energy cycle. Instead of decomposing the energy into the barotropic and baroclinic modes, we find that the energy cascade can directly occur among surface-trapped modes. The vertical and horizontal scales of these surface-trapped modes are coupled—smaller eddies decay faster with depth. Consequently, as eddies transfer energy to larger horizontal scales, they also grow deeper, in alignment with the barotropization tendency of geostrophic turbulence (Charney 1971; Salmon 1980). Although eddies can become nearly barotropic if their horizontal scales are an order of magnitude larger than the deformation radius, the inverse energy cascade tends to be halted at a scale that is comparable to the deformation radius. As a result, the vertical structure of energy-containing eddies is surface-intensified, similar to the SQG mode.
b. Parameterization of vertical structure based on SQG
The vertical structure of EKE depends on the horizontal scale, with smaller eddies decaying faster with depth. This feature suggests that the parameterization of eddy vertical structure should account for the horizontal scale of unresolved eddies. As the model grid spacing becomes finer, the unresolved eddies will be smaller and have a shallower vertical structure. Based on the WKB solution of the SQG mode, we propose a scale-aware parameterization of the vertical structure of EKE [Eqs. (22) and (23)]. This parameterization requires the specification of an eddy horizontal scale that depends on the model grid spacing. If the energy-containing scale is not resolved, the vertical structure parameterization is determined by the energy-containing scale itself. On the other hand, if the energy-containing scale is resolved—which may be the case in eddy-permitting models now being used for climate studies (Adcroft et al. 2019)—the vertical structure parameterization is determined by the largest unresolved scale that is proportional to the horizontal grid spacing. This parameterization is shown to capture the vertical structure of EKE filtered with different horizontal scales, so that it can be useful for both noneddying and eddy-permitting simulations.
Full closure of the vertical structure parameterization requires a prediction of the horizontal energy-containing scale. The energy-containing scale is associated with the mechanism that halts the inverse energy cascade, for example, bottom friction and the planetary vorticity gradient (Rhines 1975; Larichev and Held 1995). Prediction of such an eddy length scale has been the focus of many studies (Larichev and Held 1995; Held and Larichev 1996; Thompson and Young 2006; Jansen et al. 2015; Kong and Jansen 2017; Chang and Held 2019; Gallet and Ferrari 2020; Chang and Held 2021; Gallet and Ferrari 2021).
The SQG mode does not fully capture the vertical structure of EKE in the lower latitudes and tropics. Previous studies reveal that the scaling for eddy mixing is different between the tropics and midlatitude ocean due to the dominance of linear waves (tropics) versus nonlinear eddies (extratropics) (Klocker and Abernathey 2014; Zhang and Wolfe 2022). Parameterization of the eddy vertical structure in the tropics requires additional work, though eddies there are mostly large enough to be well resolved by models targeted for climate simulations in the near future such as Adcroft et al. (2019).
c. Prospects and future work
Many previous studies consider the importance of bottom topography in causing the surface intensification of ocean eddies (Aoki et al. 2009; de La Lama et al. 2016; LaCasce 2017; Quan et al. 2023). Bottom topography can decouple abyssal eddies from upper-ocean eddies, leading to baroclinic modes with zero horizontal velocity at the bottom (Rhines 1970; Samelson 1992; Hallberg 1997; Bobrovich and Reznik 1999; Tailleux and McWilliams 2001). We find that the first rough-bottom mode (the EBT mode), with zero horizontal velocity at the bottom, does not describe the vertical structure of EKE well in our simulation. This limitation arises from the EBT mode’s ignorance of the surface buoyancy anomaly, which we find to be essential for reconstructing the eddy vertical structure. In fact, both the surface buoyancy and bottom topography can be incorporated in the general boundary conditions, described by Eq. (6), of vertical modes (LaCasce 2017; Yassin and Griffies 2022a). The vertical mode accounting for both effects will be explored in future work.
The model used in this study does not have a mixed layer, which can change the quantitative analyses of the instability and eddy energetics but, we expect, will not qualitatively change the role of surface PV in the vertical structure. Mixed layer baroclinic instability provides another important energy source for submesoscale eddies, which can cascade their energy upscale to energize mesoscale eddies (e.g., Fox-Kemper et al. 2008; Sasaki et al. 2014; Callies et al. 2016; Schubert et al. 2020; Dong et al. 2020; Khatri et al. 2021). Callies et al. (2016) show that the energy generated by mixed layer instability resides in the SQG mode, where the energy cascades upscale to the largest, most energetic eddies. This energy cascade is similar to our finding, with the difference that the SQG mode is mainly excited by the surface Charney instability in our model. The mixed-layer instability thus may play a similar role to the Charney instability in enhancing the surface-trapped mode. In a more realistic numerical setup, the mixed layer and Charney instabilities can coexist, with the relative importance between them depending on the depth of the mixed layer, stratification, and lateral buoyancy gradient in the upper ocean (Capet et al. 2016; Zhang et al. 2023).
This study serves as a step forward to understand and parameterize the eddy vertical structure by emphasizing its dependence on the horizontal scale, resulting from the impact of surface PV. The parameterization, described by Eq. (22), is ripe to be tested in noneddying and eddy-permitting simulations after combining it with a closure of the horizontal energy-containing scale (e.g., Jansen et al. 2015, 2019). We are planning to apply this vertical structure to the kinetic energy backscatter parameterization that has recently been used in both idealized and realistic eddy-permitting simulations (Jansen et al. 2019; Juricke et al. 2019, 2020; Yankovsky et al. 2024, manuscript submitted to J. Adv. Model. Earth Syst.; Chang et al. 2023). Other aspects, including the impacts of bottom topography and mixed layer, will also be incorporated in this scheme to predict the eddy vertical structure in more realistic situations.
As the Surface Water Ocean Topography (SWOT) satellite has become operational, the SQG schemes proposed here [Eqs. (15) or (17)] can be a useful tool for reconstructing the upper-ocean eddy fields using the sea surface height field from SWOT. Since the surface-intensified Charney instability has been found to be prevalent in the midlatitude oceans, particularly during the spring and winter (Capet et al. 2016; Feng et al. 2021, 2022), our schemes are physically justified to interpret and infer the eddy vertical structure in these regions. Such schemes can be evaluated by comparing to observations of the full-depth eddy profile (Ni et al. 2023).
Acknowledgments.
We thank Alistair Adcroft, Laure Zanna, Chiung-Yin Chang, Jacob Steinberg, and Matthew Lobo for comments on drafts of this manuscript, as well as Elizabeth Yankovsky, Malte Jansen, Shafer Smith, and Baylor Fox-Kemper for helpful discussion. We also thank the two anonymous reviewers for their insightful comments that greatly improved the presentation. This work was supported by Award NA18OAR4320123 from the National Oceanic and Atmospheric Administration, U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the author(s) and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration, or the U.S. Department of Commerce. W. Z. and C. L. P. W. were also supported by the NSF (OCE-2048826). Computing and data storage resources, including the Cheyenne supercomputer (https://doi.org/10.5065/D6RX99HX), were provided by the Computational and Information Systems Laboratory at NCAR, under NCAR/CISL Project Number UNYU0004.
Data availability statement.
The model, NeverWorld2, dataset and detailed information on its configuration are available at https://doi.org/10.26024/f130-ev71 (Marques et al. 2022).
APPENDIX A
Comparison of the Surface-Trapped Mode to the Solution in Lapeyre and Klein (2006)
The interior solution ψint is driven by interior PV anomalies and can be projected onto the flat-bottom modes satisfying Eqs. (3) and (4). The surface solution ψsur is driven by surface buoyancy anomalies. Although ψsur and the surface-trapped mode Φs [described by Eq. (8)] share certain properties (i.e., zero interior PV anomaly and horizontal wavenumber dependence), they satisfy different surface boundary conditions—ψsur satisfies a Neumann surface boundary condition (i.e., surface buoyancy anomaly is given), while Φs satisfies a Dirichlet surface boundary condition (i.e., surface pressure anomaly is given). The derivation of Φs [Eq. (8)] indicates that when the mean surface PV gradient becomes much stronger than the interior gradient, the surface-trapped mode Φs will dominate the surface pressure anomaly, while interior modes Φm will have zero surface pressure anomaly. We call Φs as the SQG mode because it has zero interior PV anomaly and is dynamically driven by the surface buoyancy/PV anomaly, while interior modes Φm are driven by both surface and interior PV anomalies.
Although ψsur of Lapeyre and Klein (2006) uses surface buoyancy anomaly as the boundary condition, later studies have modified its surface boundary condition to incorporate surface pressure (or SSH) anomaly and finding such a formulation (i.e., Φs) successfully reconstructs the vertical profile of EKE and vertical velocity in both idealized and realistic ocean simulations (Klein et al. 2009; Isern-Fontanet et al. 2008; Qiu et al. 2016, 2020). Conversely, studies employing ψsur, estimated from surface buoyancy anomaly, reveal its limited contribution in reconstructing the eddy vertical structure (Wang et al. 2013; LaCasce and Wang 2015). Isern-Fontanet et al. (2008) find that ψsur reconstructs the vertical structure of buoyancy and velocity well only in the upper 100–200 m, while Φs reconstructs the buoyancy and velocity well from the base of the mixed layer to at least 1000 m in the North Atlantic Ocean (their Figs. 8 and 9). They attribute the error of ψsur to the deviation of surface buoyancy from its value at the base of the mixed layer due to the presence of mixed layer processes.
APPENDIX B
Surface PV in Layered Quasigeostrophic Models
APPENDIX C
Linear Stability Analysis
APPENDIX D
Analogy to Continuous QG Model
The surface PV and linear stability analysis described in appendixes B and C are for a layered QG framework, which is a natural fit for the shallow-water model used in this study. One may wonder how these equations are relevant to those in a model with continuous stratification. Here, we derive the vertically discretized form of the continuous QG equations and show that they are equivalent to the layered QG equations and the surface buoyancy anomaly can be absorbed into the surface-layer PV.
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