Eddy energy generation and energy fluxes are examined in a realistic eddy-resolving model of the North Atlantic. Over 80% of the wind energy input is found to be released by the generation of eddies through baroclinic instability. The eddy energy generation is located near the surface in the subtropical gyre but deeper down in the subpolar gyre. To reconcile the mismatch between the depth of eddy energy production and the vertical structure of the horizontal dispersion of eddy energy, the vertical eddy energy flux is downward in the subtropical gyre and upward in the subpolar gyre.
The available potential energy built up by large-scale wind-driven Ekman pumping of the main thermocline is believed to be released by the generation of eddies through instabilities of the mean currents (e.g., Gill et al. 1974). This process is parameterized in most coarse-resolution ocean climate models through an eddy-induced transport velocity that adiabatically flattens isopycnals (Gent and McWilliams 1990, hereafter GM). Although this hypothesized energy route has been supported by some idealized model studies (Radko and Marshall 2003, 2004), it is not clear whether results from these idealized studies (e.g., rectangular basin, flat bottom, simplified surface forcing, etc.) are applicable to the ocean, or even to realistic ocean simulations.
Previous attempts to estimate the energy released through baroclinic instability generally have been based on the GM parameterization, which extracts available potential energy from the mean state at a rate (e.g., Gent et al. 1995)
where P is the mean available potential energy (APE), ρ is density, κGM is the thickness diffusion coefficient, g is the gravitational acceleration, and ∇hρ and ρz are the horizontal and vertical density gradients. For example, Huang and Wang (2003) estimate this energy release from hydrographic climatology to be ~1.3 TW (1 TW = 1012 W) using κGM = 1000 m2 s−1, a value commonly adopted for ocean climate models. Wunsch and Ferrari (2004) obtain values of 0.2 and 0.8 TW using the modified eddy closures of Visbeck et al. (1997) and Danabasoglu and McWilliams (1995), respectively. Although these estimates seem to suggest that baroclinic instability is capable of releasing a significant fraction of the wind energy input to the large-scale ocean circulation, they are subject to large error bars owing to large uncertainties associated with κGM. Further studies are required to improve our understanding of where and how much of the wind energy input to the large-scale ocean circulation is released by ocean eddies.
There is also the question of how eddy energy is dispersed in the ocean once generated. Recent eddy parameterization schemes proposed for the interior of the ocean carry the eddy energy as a prognostic variable in the model equations and have desirable features such as fluxing potential vorticity downgradient without generating spurious sources of energy (Eden and Greatbatch 2008; Marshall et al. 2012). However, the new eddy closure has also been found to be sensitive to parameterizations of the dispersion of eddy energy (Marshall and Adcroft 2010). The parameterized eddy energy will be advected by the mean flow and diffused, although it is clear from satellite observations that there is a dominant and ubiquitous westward propagation of eddy energy in the ocean interior (e.g., Chelton et al. 2007, 2011), except in the separated boundary currents.
However, an issue that has received relatively little attention to date is vertical fluxes of eddy energy. In general, the horizontal dispersion of eddy energy is likely to occur at different depths from that of the eddy energy generation. For example, in the subtropical ocean the largest lateral density gradients, and hence the eddy energy generation according to (1), are confined to surface layers; in contrast the horizontal dispersion of eddy energy, even if dominated by the first and higher baroclinic modes, will have a significant component at depth. Hence vertical eddy energy fluxes are required to connect the sources of eddy energy and its horizontal dispersion.
The aim of the present study is threefold:
examining where and how much the wind energy input to the large-scale ocean circulation is released through baroclinic instability using a realistic eddy-resolving model of the North Atlantic;
mapping the vertical eddy energy fluxes in the ocean model;
thus reconciling the mismatch between the depth of eddy energy production and the vertical structure of the horizontal dispersion of eddy energy.
2. A simple eddy energy balance
The momentum and continuity equations for a Boussinesq, hydrostatic ocean are
Here, uh and w are the horizontal and vertical velocities, p is pressure, ρ is density with ρ0 its reference value, f is the Coriolis parameter, k is a vertical unit vector, g is the gravitational acceleration, z is height and ∇h is the horizontal gradient operator.
We now split the flow into time-varying and time-mean components, denoted by primes and overbars respectively. Taking the time-varying component of (3), multiplying by w′, taking a time-average, and using the time-varying components of (2) and (4), we obtain
Equation (5) is a simple way of writing the mechanical energy budget in an incompressible ocean.1 The terms represent (i) contributions to the eddy energy balance from the horizontal momentum equation (terms inside the curly brackets); (ii) the divergence of a vertical eddy energy flux, ; (iii) the source of eddy energy associated with dense fluid sinking and buoyant fluid rising in baroclinic instability, .
The vertical eddy energy flux represents the redistribution of eddy energy in the vertical but is rarely discussed in the literature [although the equivalent vertical energy flux associated with the internal waves/tides has been widely diagnosed in both observations and numerical models; for a notable exception in the context of the mean flow, see Roquet et al. (2011)].2 As will be shown later in section 4, the vertical eddy energy flux is predominantly downward in the subtropical gyre but upward in the subpolar gyre. We propose that these vertical eddy energy fluxes exist due to mismatches between the depths at which eddy energy is generated through baroclinic instability and the horizontal eddy energy fluxes that disperse the eddy energy. Equation (5) describes a nonlocal eddy energy balance analogous to internal wave generation and energy fluxes (e.g., Gill 1982), illustrated schematically in Fig. 1.
Under an assumption of small Rossby number and Ekman number such that the horizontal momentum equation can be approximated by geostrophic balance, the horizontal terms in the eddy energy Eq. (5) reduce to
that is, the zonal divergence of the westward eddy energy flux associated with westward propagation of Rossby waves. Here β is the meridional gradient in the Coriolis parameter and x is the distance in the zonal direction. However, the remaining terms are not negligible in the local eddy energy budget, even if the overall eddy energy flux is westward over much of the ocean (e.g., Chelton and Schlax 1996; Chelton et al. 2007; Zhai et al. 2010). For the purpose of this study, we do not distinguish between the various different contributions to the eddy energy budget from the horizontal momentum equation and, instead, treat the terms within braces in (5) as a single entity.
3. The model
The ocean model used in this study is the Massachusetts Institute of Technology general circulation model (Marshall et al. 1997). The model domain extends from 14°S to 74°N and from 100°W to 20°E, with a horizontal resolution of × . There are 33 geopotential levels whose thickness increases with depth, ranging from 10 m at the surface to 250 m at the bottom. The model topography is generated from the 5′ Gridded Elevations/Bathymetry for the World data. Biharmonic operators are used for horizontal mixing of tracers with background diffusivity of 1 × 1010 m4 s−1, while a highly scale-selective biharmonic friction with a Smagorinsky-like viscosity (Griffies and Hallberg 2000) is adopted for horizontal mixing of momentum. A linear bottom stress is used at the ocean floor with drag coefficient of 1.1 × 10−3.
The model is driven by climatological monthly mean forcing obtained from National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996). Exchange with the Nordic Seas and the rest of the South Atlantic Ocean, which lie outside of the model domain, is crudely taken into account by restoring the model temperature and salinity fields near the boundaries toward the monthly mean climatological values at all depths with a restoring time scale that varies linearly from 3 days to infinity over the 4° wide buffer zones. There is no explicit treatment of sea ice. The model is first spun up for 23 years at the 1/5° resolution, and is then run for another 9 years at 1/10° resolution, during which the model variables are saved every 6 days. Results from the last 7 years are used for this study. The overbars hereafter denote the time-mean quantities averaged over the 7-yr period, and primes pertain to the eddy fields resolved by the model.
Figure 2a shows the annual-mean barotropic transport streamfunction. The large-scale pattern compares well with that from other ocean models with similar horizontal resolutions (e.g., Smith et al. 2000; Eden et al. 2007). For example, the Gulf Stream does separate roughly at Cape Hatteras with a cyclonic recirculation to the north and an anticyclonic recirculation to the South. However, the path of the North Atlantic Current, as well as its eastward turn at the “Northwest Corner” is less well simulated by the model, a problem shared also by other models (e.g., Masumoto et al. 2004; Eden et al. 2007). As a consequence, the eddy field associated with the North Atlantic Current is shifted somewhat eastward (Fig. 3a).
a. Wind energy input
The rate of wind energy input at the sea surface is calculated using Wwind = τ · uo, where τ is the surface wind stress vector and uo is the total ocean surface velocity. The pattern and magnitude of the wind energy input (Fig. 2b) are similar to the previous studies (e.g., Wunsch 1998; Zhai and Greatbatch 2007; Hughes and Wilson 2008; Scott and Xu 2009). The majority of the wind energy input occurs in the tropical, western boundary, and subpolar regions, whereas there is little wind energy input in the interior of the subtropical gyre. The atmospheric winds thus seem to spin the subtropical gyre on its northern and southern edges. Roquet et al. (2011) suggest that the direct wind energy input to the ocean, Wwind, is first transported laterally by the Ekman transport before being pumped into the interior circulation; thus, the energy can be pumped into the ocean interior far from the region of direct surface wind work. Note that this interpretation is degenerate to the extent that any rotational energy flux can be added without modifying the energy budget.
One peculiar feature in Fig. 2b is a hot spot of wind energy input in the Caribbean Sea, north of Venezuela. This energy hot spot seems to be a robust feature, as it also shows up in other studies of wind energy input to the ocean (e.g., Wunsch 1998; Hughes and Wilson 2008; Scott and Xu 2009). The sensitivity of the strength of the subtropical gyre circulation to the wind energy input in the Caribbean Sea is interesting, but left for future study. Integrating over the North Atlantic Ocean, the total wind energy input is estimated to be about 0.14 TW.
In the following subsections we describe how the wind energy input leads to the generation of eddy energy at different depths (Fig. 3) and the subsequent vertical fluxes (Fig. 4), which we explain by relating to the vertical structure of the horizontal dispersion of eddy energy at three different latitudes (Figs. 5–7).
b. Eddy energy generation
Figure 3 shows the rate at which the eddies release the APE stored in the mean stratification at different depths. The eddies are found to release the APE ubiquitously along the whole rim of the subtropical gyre, even though they are strongly western intensified. The vertical structure of eddy energy generation is, however, very different between the subtropical and subpolar gyres. In the subtropical gyre, the eddy energy generation, , is mainly confined to the upper ocean, especially along the eastern and southern gyre boundaries, although it penetrates deeper than 1000 m in the western boundary region. On the other hand, the eddy energy generation in the subpolar gyre has a much deeper structure with its maximum strength at around 2000 m depth on average. This difference in the vertical structure is readily seen in the zonal transects along 24°, 39°, and 59°N plotted in Figs. 5a, 6a, and 7a, representing typical subtropical, western boundary, and subpolar situations, respectively. The source of eddy energy is clearly surface-intensified at both 24° and 39°N and also strongly western intensified at 39°N, while it peaks at great depths and is also more uniformly distributed in the zonal direction at 59°N. The eddy energy generation, , follows the location of the mean lateral density gradients, as one would expect from linear instability theory (e.g., Green 1970; Stone 1972).
Along the continental shelf break the plot of reveals regions of blue as well as red, hinting that the eddies might be fluxing energy back to the mean flow. We interpret this feature along the sloping bottom topography as a manifestation of the “Neptune” effect (e.g., Holloway 1992) or, equivalently, the mixing of potential vorticity along sloping boundaries (e.g., Greatbatch and Li 2000; Adcock and Marshall 2000). Integrating over the whole North Atlantic Ocean, the rate of APE extracted by baroclinic instability amounts to about 0.11 TW, approximately 80% of the wind energy input at the sea surface. Our results thus confirm that the wind energy input to the gyre circulations is, indeed, largely balanced by eddy generation through baroclinic instability in our model, especially in the western boundary current and its extension.
c. Vertical eddy energy flux
Figure 4 shows the vertical eddy energy flux, , at different depths. The most prominent feature near the surface is the downward eddy energy flux in the North Brazil Current and Caribbean Sea. The signal is mixed, although downward on average, in the western boundary region, and weakly upward in the subpolar region. Deeper down in the water column, the vertical eddy energy flux is dominated by the downward flux in the western boundary current and its extension and by the upward flux in the subpolar region, especially in the Labrador Sea. The vertical eddy energy flux along the eastern and southern subtropical gyre boundaries is upward near the surface and downward at depths, but is too weak to be seen in Fig. 4. This difference in the direction of between the subtropical and subpolar gyres is further illustrated in Figs. 5b, 6b, and 7b. There are large downward eddy energy fluxes in the ocean interior underneath regions of strong eddy energy source at 24° and 39°N, for example, the blue patches between 75° and 60°W and at around 20°W in Fig. 5b and between 75° and 50°W in Fig. 6b. This overall pattern of downward eddy energy fluxes is consistent with the fact that sources of eddy energy at these latitudes are located in the upper ocean (Figs. 5a and 6a). Near the sloping bottom topography in the western part of the transect at 39°N, the vertical eddy energy flux appears upward, possibly due to flow topography interactions, analogous to the upward planetary wave energy flux in the atmosphere when planetary waves encounter mountains. On the other hand, the vertical eddy energy flux is mostly upward in the Labrador Basin and Irminger Basin of the subpolar gyre, consistent with the deep eddy energy source there. Some regional features in Figs. 7a and 7b are also interesting: for example, the eddy energy source between 25° and 20°W is located farther up in the water column, and the vertical eddy energy flux is consequently upward in the upper 200 m or so and downward below.
d. Balancing eddy energy sources/sinks
Figures 5c, 6c, and 7c show the balancing eddy energy sources/sinks inside the braces in Eq. (5) along 24°N, 39°N, and 59°N, respectively. These eddy energy sources/sinks are dominated by the divergence/convergence of horizontal eddy energy fluxes (there is a small contribution from the vertical advection of eddy energy; not shown). At 24°N (Fig. 5) the horizontal eddy energy fluxes are surface intensified, as one would expect if the eddies project predominately onto the baroclinic modes, but, nevertheless, with a nonnegligible component at depth consistent with both baroclinic and barotropic modes. This is consistent with the downward eddy energy fluxes found at this latitude. At 39°N (Fig. 6) in the vicinity of the separated Gulf Stream, the balancing eddy energy sources/sinks have substantial magnitude at depth with downward vertical eddy energy fluxes again required to connect the shallow eddy energy sources with the deeper horizontal eddy energy dispersion. In contrast, at 59°N (Fig. 7) in the subpolar gyre a more complicated picture is found, but the balancing eddy energy sources/sinks are generally more surface intensified than the deep energy sources, explaining the upward vertical eddy energy fluxes at this latitude. Thus, our model diagnostics support the simple conceptual picture presented in Fig. 1 for the eddy energy balance in the interior of the subtropical and subpolar gyres.
5. Concluding remarks
The eddy energy balance in the North Atlantic subtropical and subpolar gyres has been investigated using an eddy-resolving ocean model, with a particular focus on the eddy energy generation and vertical eddy energy fluxes. The major results of the present study are as follows.
The majority of the wind energy input to the large-scale ocean circulation is released by the generation of eddies through baroclinic instability.
The eddy energy generation is located near the surface in the subtropical gyre but deeper down in the subpolar gyre.
To reconcile the mismatch between the depth of eddy energy production and the vertical structure of the horizontal dispersion of eddy energy, the vertical eddy energy flux is downward in the subtropical gyre and upward in the subpolar gyre.
Our results suggest that the spreading of eddy energy by both vertical and horizontal eddy energy fluxes needs to be taken into account in eddy closures that carry an explicit eddy energy budget (e.g., Eden and Greatbatch 2008; Marshall et al. 2012). Future work should include analysis of the eddy energy balance in models of higher resolution, with longer simulations, and surface forcing with higher temporal and spatial resolutions.
XZ thanks Dave Munday for many helpful discussions about the MITgcm. Financial support was provided by the U.K. Natural Environment Research Council. The numerical calculations were performed at the Oxford Supercomputing Centre (OSC). We thank two anonymous reviewers for their many constructive suggestions that led to a much improved manuscript.
The eddy energy equation can be written in a more conventional way: where u is the three-dimensional velocity vector, is the eddy kinetic energy, and represents energy exchange with the mean flow through barotropic instability. However, since the focus of the present study is the vertical eddy energy flux , we distinguish from the remaining terms on the left-hand side of the above equation and treat the remaining terms as a single entity as in (5).
Note that the vertical eddy energy flux is different from the vertical eddy heat flux , where cp is specific heat at constant pressure and θ′ is the eddy potential temperature (see Wolfe et al. 2008). The vertical eddy heat flux is closely related to the source of eddy energy, .