1. Introduction
Mesoscale eddies are a very important element of the global ocean since they usually account for the main peak in the kinetic energy spectrum (Kamenkovich et al. 1986; McWilliams 2008; Wunsch and Stammer 1995). This means that ocean models have to either resolve or parameterize them. To resolve the mesoscale, horizontal grids in models must be much smaller than the internal Rossby radii of deformation. Improvements in computing capability (both memory and speed) allow us to run global models with high resolution. However, weak stratification in the polar regions and the associated small internal Rossby radii (2–3 km) still preclude adequate resolution to explicitly resolve eddies in these areas. Another significant problem is the appearance of strong internal variability with increasing resolution. Small disturbances can result in energetic noise, which can only be removed by averaging over ensembles of numerical experiments; for example, the Met Office routinely runs ensembles of 10 members for decadal predictions and 42 members for seasonal prediction (Smith et al. 2007). Rather than employing an ensemble of high-resolution model simulations to realistically represent eddies and their effects on the mean flow, another approach is to utilize lower-resolution models and include a parameterization of the important effects of the eddies on the large-scale circulation. It is very likely that mesoscale eddy parameterization “will be needed for some decades into the future” (Bachman and Fox-Kemper 2013).
Parameterization of mesoscale eddies is important not only for practical reasons (reduced computational expense) but also for theoretical reasons: a physically correct parameterization allows us to better understand the dynamics of eddy–eddy and eddy–mean flow interactions, that is, fundamental parts of geophysical fluid dynamics. There have been many studies devoted to this problem (e.g., Green 1970; Welander 1973; Marshall 1981; Ivchenko 1984; Gent and McWilliams 1990; Ivchenko et al. 1997; Killworth 1997; Treguier et al. 1997; Olbers et al. 2000; Wardle and Marshall 2000; Olbers 2005; Eden 2010; Marshall and Adcroft 2010; Ringler and Gent 2011; Marshall et al. 2012; Ivchenko et al. 2013, 2014a,b; and many others).
There has been much interest in applying a diffusive parameterization to potential vorticity (PV) (Green 1970; Welander 1973; Marshall 1981). Importantly, if we use a diffusive parameterization of potential vorticity we do not need to separately parameterize eddy momentum and buoyancy fluxes, because they are already included in the eddy flux of potential vorticity. While the parameterization in terms of PV is well suited to approximations such as the quasigeostrophic formulation, primitive equation models widely used today are formulated in terms of the momentum equations and do not lend themselves as easily to a diffusive parameterization of PV.
Using a diffusive closure of eddy PV fluxes requires an integral constraint for the momentum budget known as the theorem of Bretherton to be introduced (Bretherton 1966; McWilliams et al. 1978; Marshall 1981) (see section 3). Some studies (Marshall 1981; Ivchenko 1984; Ivchenko et al. 1997, 2013, 2014a,b; Olbers et al. 2000) satisfy the momentum constraint by a suitable choice of diffusivity coefficient and others by inclusion of a so-called gauge term (Eden 2010).
McWilliams et al. (1978) and McWilliams and Chow (1981) demonstrated sharpening of zonal flow by PV mixing in an eddy-resolving quasigeostrophic zonal channel model. It was further demonstrated that using a diffusive parameterization of quasigeostrophic PV (QGPV) in a zonal channel can result in sharper and stronger currents (Ivchenko 1984; Ivchenko et al. 1997, 2014b), provided a spatially variable positive diffusion coefficient is specified, with local minima in regions occupied by jets. Dritschel and McIntyre (2008) and Wood and McIntyre (2010) also performed theoretical studies of sharpening of zonal flows by PV diffusion.
Application of a diffusive parameterization of PV in a zonal reentrant channel [with application to the Antarctic Circumpolar Current (ACC)] has been studied in many papers both for domains with a flat bottom and domains with bottom topography included, but only for the zonally averaged case (Marshall 1981; Ivchenko 1984; Ivchenko et al. 1997, 2013, 2014a,b). Introduction of bottom topography creates a number of difficult complications (see Constantinou and Young 2017).
There are two major questions associated with application of a diffusive parameterization of PV in the presence of bottom topography:
Is the eddy PV diffusivity coefficient K guaranteed to be positive? The coefficient K varies in space and time. Its local value in some locations probably could occasionally be negative. However, can we be sure that the mean (averaged) value of K is positive? Rhines and Young (1982) suggested that the eddy flux of PV is downgradient (i.e., positive eddy PV diffusivity) in an integral sense. There are not many analytical works that constrain the sign of PV diffusion. Abernathey et al. (2013) made an analysis based on a primitive equation model for a circumpolar channel. However, following Treguier et al. (1997) they calculated certain quasigeostrophic quantities, such as QGPV flux, background QGPV gradient, and corresponding diffusivity, using zonal averaging. The QGPV diffusivity is positive nearly everywhere, except near the surface, where the QG approximation is invalid. Birner et al. (2013), on the other hand, reveal a localized region of significant upgradient eddy PV fluxes on the poleward side of the subtropical free atmospheric jet core during the winter and spring seasons of both hemispheres. However, Birner et al. (2013) have noted that the net PV fluxes are downgradient when averaged over both the equatorward and poleward flanks of the jet.
In this study an analytical solution is provided that supports PV diffusivity being positive (in a domain-averaged sense). The assumption of a spatially constant eddy PV diffusivity is clearly unrealistic; however, it leads to a mathematically tractable problem and the solution provides insights that will remain applicable in the more general case.
How does one deal with the rotational (nondivergent) part of eddy PV flux? Eddy fluxes of PV comprise a rotational component and a divergent component: any vector E can be separated into divergent Ediv and rotational Erot parts (see next section).
The rotational component of the eddy flux of potential vorticity is likely to be substantial for a zonal channel with bottom topography (Sinha 1993). However, the rotational part does not directly influence the flow, because the divergence of the eddy flux appears in the PV equation and so the contribution of the rotational component is zero. The rotational part can, however, influence the flow by influencing the coefficient K via the equation of eddy potential enstrophy (see section 2).
Separation of the eddy PV flux into divergent and rotational components requires a specific boundary condition. Maddison et al. (2015) defined the divergent component of the PV flux by introducing a streamfunction tendency (“force function”). This is equivalent to a zero tangential component boundary condition (zero normal flux) and hence is not completely general. Mak et al. (2016) introduced a new method for diagnosing eddy diffusivity in a gauge-invariant fashion, which is independent of rotational flux components. This was achieved by seeking to match diagnosed and parameterized eddy force functions through an optimization procedure. The method was applied to a multilayer QG ocean gyre experiment, and it was demonstrated that the mean PV diffusivity over the horizontal domain is positive; however, robust locally negative diffusivity takes place even in the absence of rotational fluxes.
An alternative possibility is to determine the sign of the coefficient theoretically. In this study, we derive an analytical solution and construct an expression for kinetic energy, integrated over the whole domain, and use physical constraints on kinetic energy to demonstrate that the sign of K, interpreted as a domain-averaged PV diffusivity, must be positive. This is the first time that an analytical solution using a diffusive parameterization of PV has been derived for a barotropic quasigeostrophic zonal channel flow above zonally varying bottom topography. It is not, however, our intention to compare the relative merits of alternative eddy parameterizations.
The remainder of this paper is organized as follows. In section 2 we present the basic equations for quasigeostrophic barotropic flow and equations for a zonal channel geometry with bottom topography. In section 3 we formulate the generalized theorem of Bretherton. In section 4 we demonstrate an analytical solution for zonal flow, construct an expression for kinetic energy, and present results of our calculations for different types of topography. Section 5 consists of discussion and conclusions.
2. Equations for zonal channel geometry including eddy parameterization
Bottom topography substantially complicates the dynamics. The streamfunction exhibits nonzonal meanders near topographic obstacles (McWilliams et al. 1978), and therefore it is necessary to perform spatial averaging not for the whole zonal length, but for only part of it. The averaged equations depend on both zonal and meridional directions, which creates much more mathematical complexity compared to the fully zonally averaged case, but they remain analytically tractable as we will demonstrate.
Meridional profile of the QGPV diffusion coefficient k normalized by k0.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Meridional profile of the QGPV diffusion coefficient k normalized by k0.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Meridional profile of the QGPV diffusion coefficient k normalized by k0.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
3. Generalized theorem of Bretherton
4. Analytical solution for zonal flow
a. Model setup
The net zonal transport across the channel depends only on U, because 
b. Momentum balance
c. Energy balance
The three terms {E, k}, {E, h}, and {E, β} are proportional to k0 and represent dissipation of energy only if k0 > 0. If k0 < 0 all these terms are physically incorrect.
d. Analytical solution
The zonal flow is perturbed by the presence of topography and diffusion of QGPV. In the case of a flat bottom (i.e., cn = dn = 0) the motion is unperturbed, since 
This analytical solution is possible because only a single meridional component of the bottom topography B is retained. In the case of a more general expression of B it would be much more difficult to obtain an analytical solution because of greatly increased mathematical complexity.
We evaluate the solution for a number of cases with different topography and different k0 using parameter values relevant to the Southern Ocean: channel length Lx = 4 × 106 m and width 106 m; reference depth 5 × 103 m; Coriolis parameter f0 = −10−4 s−1 and β = 1.4 × 10−11 m−1 s−1; and τ0 = 10−4 m2 s−2. We illustrate the streamfunction for three cases: case 1 specifies the topography as c3 = 300 m; case 2 specifies c1 = 300 m and d1 = 300 m; case 3 specifies c2 = 300 m and d5 = 300 m (see Figs. 2–4). Here and later the topographic Fourier coefficients whose values are not explicitly stated are set to zero.
(top) Bottom topography (m) represented by c3 = 300 m (case 1). Here and in subsequent figures the topographic Fourier coefficients whose values are not explicitly stated are set to zero. Streamfunction Ψ, times reference depth H (Sv), with k0 = (second row) 0, (third row) 200, and (fourth row) 400 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
(top) Bottom topography (m) represented by c3 = 300 m (case 1). Here and in subsequent figures the topographic Fourier coefficients whose values are not explicitly stated are set to zero. Streamfunction Ψ, times reference depth H (Sv), with k0 = (second row) 0, (third row) 200, and (fourth row) 400 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
(top) Bottom topography (m) represented by c3 = 300 m (case 1). Here and in subsequent figures the topographic Fourier coefficients whose values are not explicitly stated are set to zero. Streamfunction Ψ, times reference depth H (Sv), with k0 = (second row) 0, (third row) 200, and (fourth row) 400 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 2, but for (top) c1 = 300 m and d1 = 300 m (case 2) and k0 = (second row) 0, (third row) 400, and (fourth row) 800 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 2, but for (top) c1 = 300 m and d1 = 300 m (case 2) and k0 = (second row) 0, (third row) 400, and (fourth row) 800 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 2, but for (top) c1 = 300 m and d1 = 300 m (case 2) and k0 = (second row) 0, (third row) 400, and (fourth row) 800 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 2, but for (top) c2 = 300 m and d5 = 300 m (case 3) and k0 = (second row) 0, (third row) 100, and (fourth row) 200 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 2, but for (top) c2 = 300 m and d5 = 300 m (case 3) and k0 = (second row) 0, (third row) 100, and (fourth row) 200 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 2, but for (top) c2 = 300 m and d5 = 300 m (case 3) and k0 = (second row) 0, (third row) 100, and (fourth row) 200 m2 s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
(top) Zonal transport (Sv) as a function of k0 (m2 s−1). (middle) Components of the domain-averaged energy budget 
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
(top) Zonal transport (Sv) as a function of k0 (m2 s−1). (middle) Components of the domain-averaged energy budget
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
(top) Zonal transport (Sv) as a function of k0 (m2 s−1). (middle) Components of the domain-averaged energy budget
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 5, but for case 2: bottom topography c1 = 300 m and d1 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 5, but for case 2: bottom topography c1 = 300 m and d1 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 5, but for case 2: bottom topography c1 = 300 m and d1 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 5, but for case 3: bottom topography c2 = 300 m and d5 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 5, but for case 3: bottom topography c2 = 300 m and d5 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
As in Fig. 5, but for case 3: bottom topography c2 = 300 m and d5 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Under prescribed topography the maximum transport corresponds to k0 = 0, which varies substantially (depending on topography). The highest transports are 300.5, 445.0, and 115.0 Sv (1 Sv ≡ 106 m3 s−1) in cases 1–3, respectively.
Scatterplot of zonal transport (Sv) vs parameter of topographic roughness D for various realizations of bottom topography. The fitting curve is based on a seventh-order polynomial approximation.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Scatterplot of zonal transport (Sv) vs parameter of topographic roughness D for various realizations of bottom topography. The fitting curve is based on a seventh-order polynomial approximation.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Scatterplot of zonal transport (Sv) vs parameter of topographic roughness D for various realizations of bottom topography. The fitting curve is based on a seventh-order polynomial approximation.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
All the terms on the RHS of Eq. (34) are positive (Figs. 5–7, middle panels) and contribute to balancing the source of kinetic energy 
For small values of k0, the bottom friction dominates the other terms. However, with increasing k0 the terms 
Increasing wind stress leads to increasing zonal transport (see Fig. 9). In case 3 for k0 = 0 a fivefold increase in wind stress amplitude τ0 = 5 × 10−4 m2 s−2 results in a factor-3 increase in transport from 115.0 to 338.3 Sv. Note, however, that the transport does not increase linearly with increasing wind stress: the sensitivity reduces by a factor of 2 from τ0 = 1 × 10−4 m2 s−2 to τ0 = 5 × 10−4 m2 s−2. Note that this reducing sensitivity of the transport for high values of wind stress does not relate to eddy activity (recall we are considering the case k0 = 0). Constantinou and Young (2017) and Constantinou (2018) found barotropic eddy saturation, that is, insensitivity of the transport to wind forcing in QG flow in a barotropic configuration. On the other hand Munday et al. (2013) demonstrated eddy saturation in a three-dimensional baroclinic setting using an ocean-only general circulation model. It would be interesting to verify eddy saturation in our model with parameterized eddies. However, the transport strongly depends on the value of k0 (Fig. 9). It would take additional effort to find the best-fitting coefficient k0 for each wind stress. One approach would be to perform eddy-resolving GCM experiments with given wind stress. Based on values of transport taken from these eddy-resolving experiments, we could use the relationship between transport and k0 (as in Fig. 9) to find the most realistic value of k0 for each wind stress and then verify eddy saturation in the parameterized model.
Zonal transport (Sv) as a function of k0 (m2 s−1) for various wind stress. Case 3: bottom topography c2 = 300 m and d5 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Zonal transport (Sv) as a function of k0 (m2 s−1) for various wind stress. Case 3: bottom topography c2 = 300 m and d5 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Zonal transport (Sv) as a function of k0 (m2 s−1) for various wind stress. Case 3: bottom topography c2 = 300 m and d5 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
5. Discussion and conclusions
Mesoscale eddy parameterization is an important problem of physical oceanography helping to understand the dynamics of interactions of eddies with the mean flow. Moreover, even state-of-the-art high-resolution 1/12° global models do not resolve mesoscale eddies in high latitudes.
There are various approaches to the problem of eddy parameterization. This study focuses on parameterization of eddy QGPV fluxes. PV and QGPV are conserved variables, which allows use of a diffusion type of parameterization, contrary to momentum, which is not conserved, and therefore a diffusive parameterization is unsuitable in this case.
Whether the effective coefficient of potential vorticity diffusion is positive represents the principal question in studies of mesoscale eddy parameterization (Welander 1973; Marshall 1981). If the coefficient is of negative sign a diffusive parameterization cannot be used, since it would be both mathematically and physically incorrect. The sign of this coefficient in a zonal barotropic channel is the topic of the present paper. We have demonstrated that if transient eddies are adequately described as effective PV diffusion, then the mean PV diffusivity over the domain k0 must be positive in eastward flows. This result comes out of the balance of the zonal momentum and kinetic energy: because of the parameterization, a new term appears in these equations with the physical sense of a topographic form stress for unresolved scales. The main zonal momentum balance is between wind stress [the LHS in Eq. (30)], topographic form stress exerted by the mean flow, and topographic form stress exerted by parameterized eddies.
The integral constraint on meridional fluxes of eddy QGPV known as the theorem of Bretherton in the case of a flat-bottom channel is generalized for barotropic zonal flow under variable-bottom relief. This expression allows us to provide a clear physical sense for the βk term, as a topographic form stress exerted by parameterized eddies.
We introduce a new integral measure D of the roughness of the bottom topography, which is the rms of topographic slope. The best-fitting curve representing the relationship between zonal transport and D is of hyperbolic type with a large increase of the transport when D is small and decreasing and a small decrease when D is large and increasing (Fig. 8).
In the kinetic energy balance, the only positive contribution comes from the wind stress 
Only modes represented in the bottom topography contribute to the amplitude of the streamfunction an and bn [see Eqs. (47) and (48)]. Since the modulus of the amplitudes cn and dn of the Fourier topographic modes are finite and 
Constantinou and Young (2017) demonstrated an “eddy saturation” regime, that is, insensitivity of the zonal transport to large changes in the wind stress (provided the wind stress is over a threshold value), in a barotropic configuration. To study eddy saturation in our model we need to choose a value of k0 for each type of topography, since the zonal transport depends strongly on k0. In this context, the appropriate k0 could be estimated using eddy-resolving GCM experiments. For a given wind stress eddy-resolving model experiments can be used to evaluate the associated transport, and the relationship between transport and k0 (similar to Fig. 9) can then be used to obtain an appropriate k0. However, this is beyond the scope of the present paper.
In summary, our study demonstrates conclusively that if QGPV diffusion is a good approximation, then the mean QGPV diffusivity must be positive. Our results will contribute to further understanding and parameterization of the effects of mesoscale eddies in more realistic ocean and climate models in the future.
Acknowledgments
We thank two anonymous reviewers for their substantial efforts in reviewing our paper. Their comments were extremely helpful and led to a much improved manuscript. VOI acknowledges the support of the University of Southampton and the National Oceanography Centre. VBZ was supported by the Russian Science Foundation, Grant 17-77-30001. BS was supported by National Capability funding from the U.K. Natural Environment Research Council.
APPENDIX
Parameters and Further Details of the Solution Method
Equations (47), (48), and (49) together constitute the desired analytic solution. However, because of the mathematical complexity we are unable to obtain explicit solutions for U in terms of the external parameters. Instead we apply an inverse solution method. We seek values of U that are consistent with the specified wind stress τ0 (e.g., 
Relationship between the mean zonal velocity U and the wind stress amplitude τ0. Case 2: bottom topography c1 = 300 m and d1 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Relationship between the mean zonal velocity U and the wind stress amplitude τ0. Case 2: bottom topography c1 = 300 m and d1 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
Relationship between the mean zonal velocity U and the wind stress amplitude τ0. Case 2: bottom topography c1 = 300 m and d1 = 300 m.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0229.1
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