1. Introduction
The ocean is populated by an intense geostrophic eddy field with a dominant energy-containing scale on the order of 100 km at midlatitudes. To obtain plausible turbulent cascades of dynamic and passive tracers, it is necessary for models to resolve spatial scales of at least an order of magnitude higher, with grid spacings of 10 km if not finer (Hecht and Smith 2008). Thus, ocean climate models are unlikely routinely to resolve geostrophic eddies for the foreseeable future and further development and validation of improved parameterizations remains an important task. Moreover, development and validation of improved eddy parameterizations is an excellent strategy for testing and further advancing our knowledge of how geostrophic ocean eddies impact the large-scale circulation.
A popular parameterization of geostrophic eddies is the scheme of Gent and McWilliams (1990). This scheme is motivated by the physical principle that eddies flatten neutral density surfaces adiabatically and do so via extraction of potential energy from the mean state, modeled by the inclusion of an eddy-induced overturning circulation acting to flatten neutral surfaces (Gent et al. 1995), equivalent to a diffusion of the height of neutral surfaces. Despite many notable model improvements offered by the Gent and McWilliams closure (Danabasoglu et al. 1994; Gent 2011), the influence of eddy Reynolds stresses is neglected. This results in a number of processes being poorly modeled by the scheme, including freely decaying turbulence and the establishment of Fofonoff gyres (Cummins 1992; Wang and Vallis 1994), strong recirculations around topographic features (Bretherton and Haidvogel 1976), and the formation of jets as a result of up-gradient momentum fluxes (Starr 1968; McIntyre 1970) (also see Eden 2010, for a recent attempt to parameterize up-gradient momentum fluxes).
However, because potential vorticity is a fundamental dynamical tracer and because the dynamics are constrained by conservation principles, constraints upon κ are implicit in such a down-gradient potential vorticity closure. Specifically, although such a down-gradient closure leads to a consistent eddy enstrophy budget, it does not in general lead to a consistent eddy energy budget or satisfy relevant momentum constraints. Exclusion of these constraints can therefore result in physically inconsistent flows. In the following, we discuss these shortcomings by considering two simple examples, in which the eddy potential vorticity fluxes are strongly constrained by the need to conserve energy and momentum, respectively. For a more general discussion, the reader is referred to Salmon (1998, chapter 6).
a. Conservation of energy
In Adcock and Marshall (2000), an idealized flow over topography is presented, sketched schematically in Fig. 1. The configuration consists of an inverted reduced-gravity model with a moving abyssal layer overlaying a seamount. Complete homogenization of the abyssal potential vorticity requires the layer interface between the abyssal and upper layers to rise completely over the seamount. Such motion is forbidden because of energetic constraints: such a final state has a higher energy than the initial state. Rather, Adcock and Marshall (2000) conclude, based on further numerical integrations that in this case potential vorticity is partially mixed, with the degree of mixing constrained by the initial system energy. Similar behavior was first described in a barotropic model by Bretherton and Haidvogel (1976).
Schematic diagram illustrating a difficulty with eddy closures based on unconstrained potential vorticity mixing (adapted from Adcock and Marshall 2000). Flow is confined to an abyssal layer, underlying an infinitely deep, motionless upper layer. (left) The initial state consists of a set of geostrophically balanced eddies, associated with a deformed layer interface (solid line), above a seamount (solid shading). (right) If the eddies were to completely homogenize the potential vorticity field, this would require the layer interface to rise completely over the seamount and in turn a large anticyclonic circulation around the seamount. However, the energy of this hypothetical end state exceeds that in the initial state, indicating that unconstrained potential vorticity mixing is physically impossible.
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
In Eden and Greatbatch (2008), a modification to the Gent and McWilliams closure is formulated with such an energy constraint integrated into the formulation. The thickness diffusivity is estimated from the eddy energy using mixing length theory, and the eddy energy is carried as a prognostic variable by solving a parameterized eddy kinetic energy budget. In Marshall and Adcroft (2010), it is shown that an energetically constrained down-gradient potential vorticity closure leads to a parameterized analog of Arnold’s first stability theorem (Arnold 1965).
b. Conservation of momentum
Schematic diagram illustrating a further difficulty with eddy closures based on unconstrained potential vorticity mixing. The flow in the zonal channel is such that the meridional potential vorticity gradient is positive definite. Conservation of zonal momentum requires that the meridional eddy potential vorticity flux 
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
c. Aims of this paper
In this paper, we formulate a parameterization framework whereby momentum and energy constraints are satisfied by construction. Specifically, we seek to relate the eddy stress tensor, whose divergence is the eddy potential vorticity flux, to the total (kinetic plus potential) eddy energy. Subject to appropriate boundary conditions on the components of the eddy stress tensor, this satisfies the integral momentum constraint by construction and can satisfy energy conservation subject to solving for a parameterized eddy energy budget. This manuscript represents an initial contribution toward the longer-term goal of developing an eddy parameterization that mixes potential vorticity while conserving energy and momentum.
The paper proceeds as follows. In section 2, we present an overview of the new framework and list the benefits of parameterizing the eddy shape and orientation, rather than the eddy potential vorticity fluxes directly. In section 3, we derive a bound for a norm of the eddy stress tensor expressed in terms of the total eddy energy. In section 4, this bound is used to reexpress the stress tensor in terms of the eddy energy and dimensionless parameters relating to the eddy shape and orientation. In section 5, we show that the functional form of the Eady growth rate is retained by this new framework. We further show that a down-gradient buoyancy closure in the new framework yields a variant of the Gent and McWilliams scheme that is similar in form to that proposed by Visbeck et al. (1997). In section 6, we present results from a three-layer baroclinic quasigeostrophic double-gyre simulation and show that the nondimensional parameters are of tractable order (i.e., that they are not vanishingly small) and have large-scale structure. This confirms that the eddy energy sets the magnitude of the eddy stress tensor. The paper concludes with a summary in section 7, including a discussion of how one might enforce potential vorticity mixing within this framework.
2. The new framework
We seek a framework for parameterizing eddy potential vorticity fluxes in a manner that is consistent with conservation of both energy and momentum. We initially discuss the more general residual-mean equations, where the eddy forcing appears as the divergence of a stress tensor. We then restrict our analysis to quasigeostrophic dynamics, which, although requiring further assumptions for its applicability, provides a tractable framework in which to progress.
a. Residual-mean equations
Our starting point is the equations of motion written in residual-mean form. The key idea, following Greatbatch and Lamb (1990), Wardle and Marshall (2000), and Ferreira and Marshall (2006), is to write the equations in terms of residual-mean variables, such that all of the eddy forcing appears on the right-hand side of the momentum equation through a term involving only the eddy potential vorticity flux.
The quasigeostrophic approximation has a number of obvious disadvantages, including the neglect of finite variations in bottom topography, the importance of which has been emphasized in section 1b. Nevertheless, we hope that the relation between the eddy forcing in the residual-mean primitive equations and in the quasigeostrophic equations presents a clear road map for applying our ideas to the residual-mean primitive equations in future manuscripts.
b. Conservation of energy
Crucially, the first term on the right-hand side of (9) represents the conversion of energy from the mean flow to the eddies; a corresponding term appears in the mean energy equation, such that the total energy (mean plus eddy) is conserved aside from explicit energy sources and sinks. Notwithstanding the challenges of parameterizing the internal eddy energy fluxes and eddy energy dissipation, here we assume that (9) is solved for the total (kinetic plus potential) eddy energy and we use this to constrain the magnitude of the eddy stress tensor, as defined in the following subsection. Note that Eden and Greatbatch (2008), in contrast, solve a prognostic eddy kinetic energy equation and use this to set the value of the eddy diffusivity, but further approximations must be applied in constructing an eddy kinetic energy budget.
c. Eddy stress tensor
The terms involving M and N, as well as the related barotropic “E vector,” are discussed in Hoskins et al. (1983). The lower row (the buoyancy fluxes) contains exactly those terms that are parameterized in Gent and McWilliams (1990) and Greatbatch and Lamb (1990).
Our proposal is to parameterize the four dynamically relevant components of the eddy stress tensor, M, N, R, and S, rather than the eddy potential vorticity flux directly. Note that, although this requires parameterization of four eddy fluxes (M, N, R, and S), this is no more than in the original equations of motion (two eddy Reynolds stresses and two eddy buoyancy fluxes), though two more than in the residual-mean equations (two eddy potential vorticity fluxes).
There are number of potential advantages of this approach:
It follows from (7), (11), and the boundary conditions that appropriate momentum constraints are satisfied. For example, in a zonal channel (2) is satisfied.
It is easily ensured that the parameterized eddy potential vorticity flux preserves the “tensorial properties” and symmetries of the original eddy potential vorticity flux (cf. Popovych and Bihlo 2011, manuscript submitted to J. Math. Phys.).
If we neglect the Reynolds stresses, M and N, then this formulation reduces to parameterizing the eddy form stress, as in Gent and McWilliams (1990) and Greatbatch and Lamb (1990). Thus, the framework is a natural one to use for extending Gent and McWilliams to include contributions from Reynolds stresses.
The two columns of the eddy stress tensor are the Eliassen–Palm vectors, associated with the flux of wave activity, in turn related to the propagation of eddy energy (Eliassen and Palm 1961; Andrews and McIntyre 1976). These may provide useful information for parameterizing the fluxes of eddy energy.
The eddy energy provides a rigorous upper bound on a norm of the eddy stress tensor (detailed in section 3). Note that it is the total eddy energy that is bounded, not the eddy kinetic energy used by Eden and Greatbatch (2008) for which there is no conservation law.
This upper bound allows the eddy stress tensor to be rewritten, with no loss of generality, in terms of the eddy energy, two eddy anisotropy parameters, and trigonometric factors involving three eddy orientations. Assuming that the eddy energy is available from the solution of a parameterized eddy energy equation, all of the remaining unknowns are nondimensional and bounded in magnitude by unity. All dimensional freedom in the parameterization problem is therefore encapsulated in the eddy energy (section 4).
The eddy orientations have a strong connection with classical stability theory, the eddies extracting energy from the mean flow when the eddies lean against the mean shear, and conversely returning energy to the mean flow when the eddies lean with the mean shear (section 4).
As a result of preserving the symmetries of the original eddy potential vorticity flux, the framework is able to retain some results from classical stability theory, such as the functional form of the Eady growth rate (section 5) and, with a further constraint to ensure potential enstrophy is dissipated, Arnold’s first stability theorem (section 7).
Application to the Eady problem suggests a connection between the proposed framework and the Visbeck et al. (1997) form of the Gent and McWilliams eddy closure, in the limit of negligible Reynolds stresses (section 5).
As with any new approach, there are also disadvantages. Here, the most significant disadvantage is the loss of explicit potential vorticity mixing. On the other hand, for the reasons outlined in section 1, we argue that simple-minded down-gradient diffusive closures have fundamental limitations that, despite four decades of research (since Green 1970), remain far from being resolved, if indeed this is possible (see Ringler and Gent 2011, for a related discussion). In section 7, we propose a resolution to this particular issue by solving a prognostic budget for the eddy potential enstrophy, in precisely the same manner as the eddy energy, and using this budget to ensure that potential vorticity is mixed on average through explicit dissipation of eddy potential enstrophy.
3. Bounded norm of the eddy stress tensor
We now proceed to bound the magnitude of the eddy stress tensor (11) by the eddy energy. This will allow us, in section 4, to rewrite the eddy stress tensor in terms of the eddy energy and bounded nondimensional parameters.
4. Eddy flux angles and eddy anisotropies
(a) An elliptic barotropic eddy at horizontal angle ϕm and with major and minor axes of c1 and c2, respectively. It follows that ϕm is the eddy momentum flux angle and that the momentum anisotropy is a measure of the eddy eccentricity, 
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
This geometric interpretation of the eddy fluxes also provides a strong physical connection with linear stability theory (Pedlosky 1987, chapter 7), illustrated schematically in Fig. 4. Eddies extract energy from the mean flow when they lean into the shear, consistent with instability; conversely, eddies return energy to the mean flow when they lean with the shear, consistent with stability. A related discussion of observed and modeled eddy Reynolds stresses decelerating and accelerating the Kuroshio is given in Waterman et al. (2011, particularly their Fig. 24); also see Greatbatch et al. (2010) for an illustration of eddy Reynolds stresses accelerating the separated Gulf Stream.
Schematic diagram illustrating the relation between the orientation of eddies and their stability. The solid arrows represent the background mean flow u, and the dashed contours represent eddy streamfunction contours ψ′: (a) purely barotropic eddies that tilt in the horizontal and (b) purely baroclinic eddies that tilt in the vertical. Eddies that lean against the mean shear are unstable, in the sense that they extract energy from the mean flow and grow; conversely, eddies that lean into the mean shear are stable, in the sense that they surrender energy to the mean flow and decay.
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
5. Application to the Eady model
In the next two sections, we apply the new framework to two limiting cases: (i) baroclinic instability in a channel and (ii) fully developed turbulence in a closed, wind-driven basin.
a. Eady growth rate in the new parameterization framework
Thus, without specifying the form of the eddy closure for the components of the stress tensor, other than that the eddy buoyancy flux is on average down the mean buoyancy gradient, this new approach is guaranteed to preserve the functional form of the Eady growth rate, with an upper bound that is within a factor of 0.62 of the growth rate actually obtained. This is in marked contrast to previous down-gradient eddy closures in which arbitrary, dimensional eddy diffusivities need to be specified.5
b. A down-gradient buoyancy closure in the new parameterization framework
Of the parameters in (30), Ri depends upon the mean flow, whereas α and E are properties of the eddies. Hence a down-gradient buoyancy closure in this general form requires a parameterized eddy energy budget and a parameterization for α. Note, however, that Visbeck et al. (1997) find in numerical experiments that the constant 
6. Application to wind-driven gyres
The eddy stress tensor decomposition (11) is general, but it is not self-evidently applicable as a framework for eddy parameterization. In particular, should the anisotropy parameters be small or should any of the parameters exhibit finescale spatial structure, then the potential vorticity fluxes would be highly sensitive to the details of their parameterization. In such a situation, the problem of parameterizing the geometric parameters would be highly intractable and could not be expected to capture the influence of the eddies on the mean flow.
In this section, we investigate, in an idealized oceanographic context exhibiting fully developed geostrophic turbulence, whether the nondimensional parameters have a magnitude of tractable order for the purposes of parameterization (i.e., that they have an order close to unity). We further investigate whether the parameters of the flux decomposition exhibit large-scale structure with features relating to the mean flow structure. For these purposes, we use a three-layer quasigeostrophic model to simulate a baroclinic double-gyre configuration, as described in Berloff et al. (2007).
a. Model configuration
Model parameters used for the quasigeostrophic three-layer model.
Figure 5 shows the final instantaneous transport streamfunction and potential vorticity. One sees the expected characteristic double-gyre regime, with an intense eastward jet separating two gyres and with transient eddies observed throughout the domain. Partial homogenization of mean potential vorticity in the middle layer and in the central part of the domain is also observed (Holland and Rhines 1980; Rhines and Young 1982a,b).
A three-layer quasigeostrophic baroclinic double-gyre simulation. Snapshots of the transport streamfunction HiΨi in the (a) upper, (b) middle, and (c) lower layers. Snapshots of the potential vorticity q in the (d) upper, (e) middle, and (f) lower layers.
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
b. Eddy energy
The eddy energy and eddy energy partitioning angle λ are shown in Fig. 6. The eddy energy is primarily concentrated in the thin eastward jet, and it decreases in the lower layers, as expected. The eddy energy partitioning indicates that the energy is approximately equipartitioned between potential and kinetic parts, with a slightly larger potential energy part in the middle larger and kinetic part in the lower layer. The normalized histogram of the partitioning angle λ peaks at values of 0.25π, 0.28π, and 0.16π in the upper, middle, and lower layers, respectively. Elevated values of λ, indicating relatively increased potential energy, are observed in the eastward jet and in the western boundary region.
Eddy energy in the three-layer baroclinic double-gyre simulation. The eddy energy E in the (a) upper, (b) middle, and (c) lower layers. The eddy energy partitioning angle λ = arctan(P/K) in the (d) upper, (e) middle, and (f) lower layers.
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
c. Eddy anisotropies
The momentum anisotropy γm and the buoyancy anisotropy γb are shown in Fig. 7. Significant structure is observed in the momentum anisotropy parameter γm, with larger values of γm observed toward the eastern boundary. A thin zonal line of low momentum anisotropy is observed near the northern and southern boundaries. The buoyancy anisotropy γb also exhibits some significant structure. In particular, two clear tracks of increased buoyancy anisotropy are observed on the flanks of the eastward jet in the upper and middle layers. Somewhat decreased values of the buoyancy anisotropy are observed in the recirculation regions. The normalized histograms of γm and γb in each layer are shown in Fig. 8. The histogram of γm exhibits a clear peak in the upper layer at γm ≈ 0.15. The values of γm are observed to increase in the lower layers. The boundary condition for γm on lateral boundaries leads to a second peak at γm = 1 in all layers. The histogram of γb also exhibits a clear peak in the upper layer at γb ≈ 0.15. The values of γb are observed to decrease somewhat in the lower layers.
The eddy anisotropy parameters in the three-layer baroclinic double-gyre simulation. The momentum anisotropy parameter γm in the (a) upper, (b) middle, and (c) lower layers, on a logarithmic color scale. The buoyancy anisotropy parameter γb in the (d) upper, (e) middle, and (f) lower layers, on a logarithmic color scale.
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
Normalized histograms of eddy anisotropy parameters in the three-layer double-gyre simulation. The distribution is formed from a 256-interval histogram (with a linear scale), normalized such that 
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
d. Eddy flux angles
For completeness, the eddy flux angles ϕm and ϕb are shown in Fig. 9. Both ϕm and ϕb show significant variation in the region of the eastward jet and toward the western boundary. We do not consider the detailed structure of ϕm and ϕb here, except to note that the structure in the flux angles is large scale and related to the structure of the eddy anisotropies. The physical processes controlling these structures will be studied in detail in a subsequent manuscript.
The eddy flux angle parameters in the three-layer baroclinic double-gyre simulation. Note that there is a π symmetry in ϕm, and hence ϕm indicates a flux line rather than a unique flux direction. The angles are displayed at every eighth grid point in each direction. The flux angle associated with momentum ϕm in the (a) upper, (b) middle, and (c) lower layers. The buoyancy flux angle ϕb in the (d) upper, (e) middle, and (f) lower layers.
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
e. Implications for parameterization
These simulations represent an initial step toward assessing the utility of the eddy flux decomposition (16). Hence, we have not examined correlations between the parameters, either between each other or with the mean flow. We have also limited attention to the long-term average; one might expect low-frequency variability of the eddy statistics. Crucially, however, the anisotropy parameters are not vanishingly small—rather they are on the order of ~0.1 or larger—and all geometric parameters exhibit large-scale spatial structure with identifiable features that are related to the mean flow. These are crucial requirements for the eddy flux decomposition (16) to serve as a useful framework for eddy parameterization.
7. Concluding remarks
In this manuscript, we have proposed a new framework for parameterizing eddy potential vorticity fluxes in ocean circulation models. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. We have shown that a norm of the eddy stress tensor is bounded by the eddy energy. This allows the elements in the eddy stress tensor to be rewritten in terms of (i) the eddy energy, for which we propose solving prognostic equation for the eddy energy (following Eden and Greatbatch 2008), and (ii) nondimensional parameters that are bounded in magnitude by unity and have geometric interpretation in terms of eddy shapes and orientations. Numerical calculations suggests that the eddy anisotropies are typically on the order of 0.1 in fully developed geostrophic turbulence, indicating that the bound provides a useful constraint on the magnitude of the eddy fluxes.
It is important to emphasize that the new framework does not represent an eddy closure. We have merely moved the need for parameterization from one set of quantities (eddy potential vorticity fluxes) to another (eddy anisotropies and tilts). Nevertheless, assuming that the eddy energy is available at each time level, the fact that the outstanding quantities requiring parameterization are all nondimensional and bounded exerts strong constraints on the resultant eddy potential vorticity fluxes, ensuring that relevant energy and momentum constraints are preserved. We have also demonstrated strong connections between the new framework and classical stability theory.
a. Potential vorticity mixing
Finally, on a more speculative note, in Fig. 10 we show the eddy potential enstrophy and its ratio with the eddy energy. This ratio is equivalent to mean squared wavenumber in eddy energy space. In layer 1, there is close, though not exact, correspondence between the structure of this ratio and that of the eddy anisotropies plotted in Fig. 7, and this ratio, although a similar correspondence is not found in layers 2 and 3, in particular in the pool of nearly homogenous potential vorticity where the potential enstrophy is small. In future work, we will investigate the extent to which the ratio of the eddy potential enstrophy and eddy energy might constrain the eddy anisotropies or otherwise.
Eddy potential enstrophy and its ratio to the eddy energy in the three-layer baroclinic double-gyre simulation. The eddy potential enstrophy Λ in the (a) upper, (b) middle, and (c) lower layers. The ratio of the eddy potential enstrophy to the eddy energy Λ/E in the (d) upper, (e) middle, and (f) lower layers.
Citation: Journal of Physical Oceanography 42, 4; 10.1175/JPO-D-11-048.1
b. Parameterization of eddy tilts
The numerical diagnostics suggest that equipartitioning of eddy energy provides an adequate zero-order parameterization of the vertical eddy tilts. For the horizontal eddy tilts, there are a number of potential approaches.
The first is to use classical stability theory to select eddy angles that maximize the growth of eddy energy, on the assumption that these are the eddies that emerge at finite amplitude. The second is to select eddy angles that maximize mixing of potential vorticity, although preliminary results suggest that such an approach requires calculation of high-order derivatives and is impractical.
Alternatively, to the extent that the eddies can be modeled as linear waves, one might use ray-tracing theory (e.g., Bühler 2009) to determine how the eddies are deformed by the background shear. Intriguingly, the ray-tracing formulation is also closely related to the propagation of wave activity, in turn related to the propagation (and growth) of eddy energy. It is therefore possible that a ray-tracing approach could be used simultaneously to solve for the propagation and growth of eddy energy as well as the deformation of eddies by the mean flow.
c. Divergent eddy potential vorticity fluxes
The new framework relates the total eddy potential vorticity flux to the divergence of the eddy stress tensor. However, it is only the divergent component of the eddy potential vorticity flux that influences the evolution of the mean flow; the rotational flux results in a divergent force on the right-hand side of the momentum Eq. (3) that projects on the pressure gradient and has no influence on the evolution of the flow (in the three-dimensional, nonquasigeostrophic residual-mean equations, the divergent and rotational fluxes must be defined through a three-dimensional decomposition; cf. Marshall and Pillar 2011).
An important issue is therefore the extent to which the new framework provides useful constraints on the magnitude and structure of the divergent, rather than full, eddy potential vorticity fluxes. Previous diagnostics of rotational and divergent eddy potential vorticity fluxes suggest that the latter are typically an order of magnitude smaller (Marshall and Shutts 1981; Roberts and Marshall 2000). The new framework includes an explicit rotational potential vorticity flux through the contribution of the eddy potential energy, but it is likely that the remaining terms also project significantly onto the rotational eddy potential vorticity flux.
d. Application to the primitive equations
Finally, although we have focused on quasigeostrophic eddy potential vorticity fluxes in this manuscript, for reasons of analytical tractability, the fact that eddy fluxes of Rossby–Ertel potential vorticity appear on the right-hand side of the residual-mean Eq. (3) provides a clear road map for applying any parameterization developed using the new framework directly to the primitive equations. Although this will require further assumptions, the use of an eddy stress tensor means that it should be straightforward to enforce any momentum constraints. We also anticipate that the application of the new framework to the primitive equations may be of value for studying the influence of variable bottom topography on the eddy potential vorticity fluxes and their forcing of mean flows.
Acknowledgments
We wish to thank Geoff Vallis and an anonymous reviewer for exceptionally constructive and thorough reviews that led to a much improved manuscript. We also thank Bill Young and Xiaoming Zhai for additional insightful comments. Financial support was provided by the UK Natural Environment Research Council (NE/H020454/1).
APPENDIX
Quasigeostrophic Residual-Mean Equations
REFERENCES
Adcock, S. T., and D. P. Marshall, 2000: Interactions between geostrophic eddies and the mean circulation over large-scale bottom topography. J. Phys. Oceanogr., 30, 3223–3238.
Aiki, H., and K. J. Richards, 2008: Energetics of the global ocean: The role of layer-thickness form drag. J. Phys. Oceanogr., 38, 1845–1869.
Andrews, D. G., 1983: A finite-amplitude Eliassen-Palm theorem in isentropic coordinates. J. Atmos. Sci., 40, 1877–1883.
Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in a horizontal and vertical shear: The generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos. Sci., 33, 2031–2048.
Arnold, V. I., 1965: Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR, 162, 975–978.
Berloff, P., A. M. C. Hogg, and W. Dewar, 2007: The turbulent oscillator: A mechanism of low-frequency variability of the wind-driven ocean gyres. J. Phys. Oceanogr., 37, 2363–2386.
Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325–334.
Bretherton, F. P., and D. B. Haidvogel, 1976: Two-dimensional turbulence above topography. J. Fluid Mech., 78, 129–154.
Bühler, O., 2009: Waves and Mean Flows. Cambridge University Press, 341 pp.
Charney, J. G., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19, 159–172.
Chelton, D. B., M. G. Schlax, R. M. Samelson, and R. A. de Szoeke, 2007: Global observations of large oceanic eddies. Geophys. Res. Lett., 34, L15606, doi:10.1029/2007GL030812.
Cummins, P. F., 1992: Inertial gyres in decaying and forced geostrophic turbulence. J. Mar. Res., 50, 545–566.
Danabasoglu, G., J. C. McWilliams, and P. R. Gent, 1994: The role of mesoscale tracer transports in the global ocean circulation. Science, 264, 1123–1126.
Dewar, W. K., and A. M. Hogg, 2010: Topographic inviscid dissipation of balanced flow. Ocean Modell., 32, 1–13.
Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33–52.
Eden, C., 2010: Parameterising meso-scale eddy momentum fluxes based on potential vorticity mixing and a gauge term. Ocean Modell., 32, 58–71.
Eden, C., and R. J. Greatbatch, 2008: Towards a mesoscale eddy closure. Ocean Modell., 20, 223–239.
Eliassen, A., and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geofys. Publ., 22, 1–23.
Ferreira, D., and J. Marshall, 2006: Formulation and implementation of a “residual-mean” ocean circulation model. Ocean Modell., 13, 86–107.
Gent, P. R., 2011: The Gent-McWilliams parameterization: 20/20 hindsight. Ocean Modell., 39, 2–9.
Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20, 150–155.
Gent, P. R., J. Willebrand, T. J. McDougall, and J. C. McWilliams, 1995: Parameterizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr., 25, 463–474.
Greatbatch, R. J., and K. G. Lamb, 1990: On parameterizing vertical mixing of momentum in non-eddy resolving ocean models. J. Phys. Oceanogr., 20, 1634–1637.
Greatbatch, R. J., X. Zhai, M. Claus, L. Czeschel, and W. Rath, 2010: Transport driven by eddy momentum fluxes in the Gulf Stream Extension region. Geophys. Res. Lett., 37, L24401, doi:10.1029/2010GL045473.
Green, J. S. A., 1970: Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Quart. J. Roy. Meteor. Soc., 96, 157–185.
Hecht, M. W., and R. D. Smith, 2008: Towards a physical understanding of the North Atlantic: A review of model studies in an eddying regime. Ocean Modeling in an Eddying Regime, Geophys. Monogr., Vol. 177, Amer. Geophys. Union, 213–240.
Holland, W. R., and P. B. Rhines, 1980: An example of eddy-induced circulation. J. Phys. Oceanogr., 10, 1010–1031.
Hoskins, B. J., I. N. James, and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci., 40, 1595–1612.
Johnson, G. C., and H. L. Bryden, 1989: On the size of the Antarctic Circumpolar Current. Deep-Sea Res., 36, 39–53.
Killworth, P. D., 1997: On the parameterization of eddy transfer. Part I. Theory. J. Mar. Res., 55, 1171–1197.
Marshall, D. P., and A. C. Naveira Garabato, 2008: A conjecture on the role of bottom-enhanced diapycnal mixing in the parameterization of geostrophic eddies. J. Phys. Oceanogr., 38, 1607–1613.
Marshall, D. P., and A. J. Adcroft, 2010: Parameterization of ocean eddies: Potential vorticity mixing, energetics and Arnold’s first stability theorem. Ocean Modell., 32, 188–204.
Marshall, D. P., and H. R. Pillar, 2011: Momentum balance of the wind-driven and meridional overturning circulation. J. Phys. Oceanogr., 41, 960–978.
Marshall, J. C., 1981: On the parameterization of geostrophic eddies in the ocean. J. Phys. Oceanogr., 11, 257–271.
Marshall, J. C., and G. Shutts, 1981: A note on rotational and divergent eddy fluxes. J. Phys. Oceanogr., 11, 1677–1680.
McIntyre, M. E., 1970: On the non-separable baroclinic parallel flow instability problem. J. Fluid Mech., 40, 273–306.
Molemaker, M. J., J. C. McWilliams, and I. Yavneh, 2005: Baroclinic instability and loss of balance. J. Phys. Oceanogr., 35, 1505–1517.
Munk, W. H., and E. Palmén, 1951: Note on the dynamics of the Antarctic Circumpolar Current. Tellus, 3, 53–55.
Nikurashin, M., and R. Ferrari, 2010: Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory. J. Phys. Oceanogr., 40, 1055–1074.
Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed. Springer-Verlag, 710 pp.
Plumb, R. A., 1986: Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci., 43, 1657–1678.
Polzin, K. L., 2008: Mesoscale eddy–internal wave coupling. Part I: Symmetry, wave capture, and results from the Mid-Ocean Dynamics Experiment. J. Phys. Oceanogr., 38, 2556–2574.
Rhines, P. B., and W. R. Young, 1982a: A theory of wind-driven circulation. I. Mid-ocean gyres. J. Mar. Res., 40 (Suppl.), 559–596.
Rhines, P. B., and W. R. Young, 1982b: Homogenization of potential vorticity in planetary gyres. J. Fluid Mech., 122, 347–367.
Ringler, T., and P. Gent, 2011: An eddy closure for potential vorticity. Ocean Modell., 39, 125–134.
Roberts, M. J., and D. P. Marshall, 2000: On the validity of downgradient eddy closures in ocean models. J. Geophys. Res., 105, 28 613–28 627.
Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.
Sen, A., R. B. Scott, and B. K. Arbic, 2008: Global energy dissipation rate of deep-ocean low-frequency flows by quadratic bottom boundary layer drag: Computations from current-meter data. Geophys. Res. Lett., 35, L09606, doi:10.1029/2008GL033407.
Starr, V. P., 1968: Physics of Negative Viscosity Phenomena. McGraw-Hill, 256 pp.
Stone, P. H., 1972: A simplified radiative-dynamical model for the static stability of rotating atmospheres. J. Atmos. Sci., 29, 405–418.
Treguier, A. M., I. M. Held, and V. D. Larichev, 1997: Evaluating eddy mixing coefficients from eddy resolving models: A case study. J. Phys. Oceanogr., 27, 567–580.
Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 745 pp.
Visbeck, M., J. Marshall, T. Haine, and M. Spall, 1997: Specification of eddy transfer coefficients in coarse-resolution ocean circulation models. J. Phys. Oceanogr., 27, 381–402.
Wang, J., and G. K. Vallis, 1994: Emergence of Fofonoff states in inviscid and viscous ocean circulation models. J. Mar. Res., 52, 83–127.
Wardle, R., and J. Marshall, 2000: Representation of eddies in primitive equation models by a PV flux. J. Phys. Oceanogr., 30, 2481–2503.
Waterman, S., N. G. Hogg, and S. Jayne, 2011: Eddy–mean flow interaction in the Kuroshio extension region. J. Phys. Oceanogr., 41, 1182–1208.
White, A. A., and J. S. A. Green, 1982: A non-linear atmospheric long wave model incorporating parametrizations of transient baroclinic eddies. Quart. J. Roy. Meteor. Soc., 108, 55–85.
Wood, R. B., and M. E. McIntyre, 2009: A general theorem on angular-momentum changes due to potential vorticity mixing and on potential-energy changes due to buoyancy mixing. J. Atmos. Sci., 67, 1261–1274.
Young, W. R., 2012: An exact thickness-weighted average formulation of the Boussinesq equations. J. Phys. Oceanogr., in press.
Zhai, X., H. L. Johnson, and D. P. Marshall, 2010: Significant sink of ocean-eddy energy near western boundaries. Nat. Geosci., 3, 608–612.
One may also consider this constraint as a conservation of angular momentum; conservation of zonal momentum in the rotating frame is equivalent to conservation of angular momentum in an inertial frame.
Young has pointed out that E should be defined as two separate vectors, because E does not, in general, transform as a tensor (W. R. Young 2011, personal communication). However, in the quasigeostrophic limit we consider for the remainder of this paper, E does transform as a tensor under horizontal transformations and might therefore be termed a “quasigeostrophic tensor.”
Note that the signs of the fluxes M and N are defined inconsistently in the literature.
Weighting of vertical components by factors of 