## 1. Introduction

There is a local maximum of kinetic energy on ocean mesoscales (Kamenkovich et al. 1986), which demands from numerical models either to resolve mesoscale features or parameterize them. Although the progress in computers makes global eddy-resolving simulations rather common, they still remain expensive for long enough integrations. In addition to this practical aspect, studies of eddy parameterization contribute to better understanding of ocean dynamics, remaining thereby in the focus of the modern physical oceanography.

Parameterization of eddy fluxes involves both the momentum fluxes (Reynolds stresses) and tracers fluxes (temperature, salinity, density, and other tracers). An important approach in the latter case is the popular scheme by Gent and McWilliams (1990) for eddy-induced thickness (TH) fluxes. This parameterization leads to substantial improvement in a number of oceanic features simulated with coarse models, especially in the Southern Ocean, where it ensures a deep ocean density distribution that is closer to the observed values, a sharper thermocline, eddy-induced cancellation of the Deacon cell, and a better representation of the penetration depth of chemical tracers originating in the atmosphere (see, e.g., Danabasoglu et al. 1994; Hirst and McDougall 1996). This scheme allows one to introduce the so-called bolus velocity and reformulate the basic equations with its assistance (see, e.g., Griffies 2004). The most popular eddy flux parameterization is that of downgradient transport (diffusive parameterization), when the eddy flux of a scalar variable is taken to be proportional to the negative of the gradient of the mean field. However, such a scheme can only be applied to conserved quantities. Momentum is a nonconserved quantity because of the presence of the pressure gradient; parameterizing such quantities presents a challenge. A satisfactory momentum parameterization is absent at present. In most cases, eddy fluxes of momentum are modeled as eddy viscosity (harmonic or biharmonic) that is not uncontroversial, as momentum fluxes can be countergradient and “sharpen” jet flows instead of diffusing them. The popularity of harmonic and biharmonic viscosities lies rather in their contribution to numerical stability than their realism (Harrison 1978; Killworth 1997).

Zonal jets are a common feature of geophysical flows (see, e.g., Rhines 1975; Danilov and Gurarie 2002). They are encountered in many places across the World Ocean. In the midlatitudes the multiple jets have rms zonal velocities estimated as about 6.9 cm s^{−1} and a meridional wavelength of about 300 km (Maximenko et al. 2005). Mesoscale eddies play the major role in creating and supporting such jets through the convergence of lateral eddy momentum fluxes (Dritschel and McIntyre 2008; Eden 2010).

Importantly, the parameterization of potential vorticity (PV) fluxes incorporates the Reynolds stresses, since they are included in the eddy PV fluxes together with fluxes of density [or, with high accuracy, TH]. Note also that there is a better physical base for diffusive parameterization of PV, which is a conserved variable. This explains a growing interest in parameterizations of eddy PV fluxes (Treguier et al. 1997; Wardle and Marshall 2000; Eden and Greatbatch 2008; Eden 2010; Marshall and Adcroft 2010; Ringler and Gent 2011).

The main aim of this study is comparing/contrasting the PV and TH parameterizations in their effect on velocities, energy, and other variables and analyzing their dependence on respective diffusivities. The difference between solutions with TH and PV flux parameterizations comes mainly from the effect of Reynolds stresses (which are not considered, i.e., virtually neglected in the TH scheme) and constraints stemming from the momentum conservation [the parameterized PV fluxes must be compatible with the momentum balance, as stated by the Bretherton theorem (Bretherton 1966), and ensure a positive kinetic energy source in the eddy kinetic energy balance]. Ideally, such a comparison should be done for general 3D primitive equation dynamics, which still presents a challenge. The first step can be made by using strongly simplified quasigeostrophic dynamics in simple channel geometry, which can be treated analytically.

For the quasigeostrophic channel model, the integral constraints have been repeatedly discussed for both momentum (Bretherton 1966; McWilliams et al. 1978; Marshall 1981; Ivchenko 1987; Ivchenko et al. 2008, 2013, 2014) and energy (Ivchenko et al. 2013, 2014). Wood and McIntyre (2010) proved a theorem linking PV mixing and the negative change in angular momentum.

It is hoped that the analysis of analytical solutions here can provide some hints toward understanding more complex dynamics. A caveat of this study is that it cannot be directly applied to channels with varying topography. It is well known that zonal transport in a flat bottom channel can be an order of magnitude higher than in channels with variable topography. The selection of the coefficients for the parameterized eddy fluxes must be done carefully to avoid unphysical and artificially high zonal transport.

However, the use of the simplest possible geometry of flat bottom channel already allows for analytical solutions and better understanding of the eddy flow dynamics.

This paper is organized as follows. We first write down the basic equations for quasigeostrophic dynamics and introduce the TH and PV parameterizations (sections 2–4). In section 5, the difference between solutions is studied for both constant and meridionally varying diffusivities. Section 6 discusses solutions for velocities and parameterizations for coefficients based on the time- and zonal-mean equation for eddy quasigeostrophic enstrophy. Section 7 analyzes the Reynolds stresses diagnosed by comparing solutions for PV and TH parameterizations, and section 8 offers discussion and conclusions.

## 2. Basic quasigeostrophic equation for the zonal channel

### Equations

_{i}is the quasigeostrophic streamfunction, subscripts label the layers of constant mean thicknesses

*H*

_{i}, and

*w*

_{3/2}is the vertical velocity of the interface. The term

*F*

_{i}is the lateral friction that is further omitted in the analytical solution; the Jacobian operator is

*ϵ*is the linear bottom drag; and

**= (**

*τ**τ*

_{x},

*τ*

_{y}) is the wind stress with the components

*τ*

_{y}= 0, where

*τ*

_{0}is the amplitude of wind stress, and

*L*is the width of the channel. The components of fluid velocity

**v**

_{i}= (

*u*

_{i},

*υ*

_{i}) are expressed as

*f*and

*f*

_{0}denote the Coriolis parameter and its reference value, respectively. The interfacial displacement

*h*

_{3/2}is linked to streamfunction as

*g*is the acceleration due to gravity,

*ρ*

_{i}and

*ρ*

_{0}are the density of layer

*i*and the reference density, respectively, and Ψ

_{3/2}is the streamfunction of the interface:

*q*

_{1}and

*q*

_{2}:

*q*

_{i}are given by

## 3. Parameterization of eddy thickness fluxes (TH parameterization)

*K*

_{H}is the coefficient of eddy thickness diffusivity (CT).

Using (21), (22), and (24), after standard manipulations, one can derive the energy balance equations for the mean and eddy mechanical (i.e., sum of kinetic and available potential) energies. Both balances will contain the term *K*_{H} is positive.

The mean mechanical energy is generated by wind, dissipated by the bottom friction, and transferred to the eddy part of the mechanical energy.

*K*

_{H}in expression (29) can take any value. We will use both a constant value of

*K*

_{H}for comparing solutions with those found in the framework of PV parameterization and prescribed meridional profiles that resemble the behavior seen in eddy-resolving experiments (see, e.g., Fig. 12 in McWilliams and Chow 1981):

*A*

_{H}= const is the amplitude and

*F*

_{TH}is the normalized profile,

## 4. Parameterization of eddy potential vorticity fluxes

### a. General expressions

*K*

_{i}≥ 0 is the respective coefficient of QPV diffusivity abbreviated further as CPV. These parameterizations, when substituted into (31) and (32), lead to a closed system that can be solved for any

*K*

_{i}. However, the solutions should satisfy certain integral constraints, which in turn restrict the admissible values for

*K*

_{i}.

Expression for the total eddy mechanical energy based on the parameterization of eddy PV fluxes provides insight into the exchange between the mean and eddy parts of mechanical energy.

*P*

^{ed}are the eddy kinetic energy in the upper and lower layers and the eddy available potential energy, respectively. If (39) is integrated over the channel width, all the “divergent” (with the operator

*K*

_{1}and

*K*

_{2}. Their values can be found from some additional closures based on equations for eddy kinetic energy (Eden and Greatbatch 2008; Eden 2010) or enstrophy (Ivchenko 1984, 1987) or can be prescribed. We will prescribe constant values (to facilitate analytical solutions) or meridional dependencies based on results of fine-resolution numerical experiments.

*K*

_{2}/

*K*

_{1}.

These solutions will be further compared to the solutions obtained for the TH parameterization, in which the Reynolds stresses are neglected. To make the analytical treatment in this case simpler, we resort to further simplification, assuming the QPV diffusivities to be constant.

### b. Constant values of CPV

If the CPV are constant, the solution derived by Ivchenko et al. (1997, 2013) can be used. The derivation relies on the asymptotic expansion in the small parameter *γ* = *L*_{R}/*L*, the ratio of the internal Rossby radius *L*_{R} to the width of the channel. Our equations contain this parameter in the terms with the highest derivatives coming from the relative vorticity. To satisfy the boundary condition of no eddy flux at lateral walls, we set the coefficients *K*_{1} and *K*_{2} linearly to zero in thin layers adjacent to the walls.

*δ*

_{i}=

*H*

_{i}/

*H*,

### c. Variable CPV

*A*

_{PV}= const are the amplitudes and

*F*

_{PV}is the normalized profile,

*y** =

*L*

^{−1}

*y*.

*F*

_{PV}and prescribing

*A*

_{2}, the value of

*A*

_{1}(i.e.,

*K*

_{1}) can easily be found from the Bretherton theorem, which can be written as

## 5. Comparison of solutions for TH parameterization without the contribution of the Reynolds stresses and solutions for PV parameterization

### a. Constraints on the coefficients

#### 1) TH parameterization

There is only a trivial constraint *K*_{H} > 0 that ensures dissipation (transfer to eddies) of the mean available potential energy.

#### 2) PV parameterization

*K*

_{1}and

*K*

_{2}must obey the Bretherton theorem [(38)] and the energy inequality [(42)]. Other constraints may also exist, as hinted by Marshall et al. (2012). They are related to the eddy energy, which is not explicitly modeled here. The Bretherton theorem for the constant (outside the wall boundary layers) CPV can be written in the form

These parameters express the ratio of the stretching term to wind stress in the meridional QPV gradient in the upper layer (*α*_{B}) and of the time scale of zonal baroclinic Rossby wave to that of dissipation by bottom friction (*α*_{U}). It is easy to show that *K*_{2} > *K*_{1} [see (53) and Fig. 2], which is the condition for baroclinic instability (Marshall 1981). The parameters Re and *D* are inversely proportional to *K*_{1} and *K*_{2}, respectively, and therefore (56) provides a link between CPV in upper and lower layers. If *K*_{1} (*K*_{2}) is prescribed, then the value of *K*_{2} (*K*_{1}) must comply with (56).

*G*

_{PV}is the generation of the eddy mechanical energy. This energy inequality (EI) does not allow using the values of

*K*

_{2}in excess of the critical value depending on

*K*

_{1}. In Fig. 2,

*K*

_{2}lies below the red line;

*K*

_{2}is greater than

*K*

_{1}so the points must lie above the black line on the green line corresponding to the Bretherton theorem. In physical terms it implies that for baroclinic instability to develop the vertical shear of the zonally averaged zonal velocity must be higher than the critical value (Marshall 1981; Ivchenko et al. 1997).

### b. Zonal velocity

#### 1) TH parameterization

The zonally averaged zonal velocity [(25)] in the lower layer is proportional to the wind stress and independent of the coefficient *K*_{H} (Fig. 3). The velocity in the upper-layer [(29)] is also proportional to the wind stress [(29)], but retains the dependence on *K*_{H} (Fig. 3).

#### 2) PV parameterization

*K*

_{i}(outside thin wall boundary layers), the meridional profile of zonally averaged zonal velocity in the lower layer comprises two terms. The first one is similar to the expression for the zonally averaged zonal velocity in the case of TH parameterization [proportional to the wind stress and inversely proportional to the

*ϵ H*

_{2}(50)]. The major difference between the TH and PV solutions is the presence of

*δ*

_{2}, and coefficient

*K*

_{1}. It is negative and constant for the prescribed constant coefficients

*K*

_{1}and

*K*

_{2}. Importantly,

These expressions are equal for both parameterizations. It is not surprising because there should be the same balance in the steady state for the same forcing.

The zonally averaged zonal velocity profile in the upper layer is the sum of the lower-layer velocity and two additional terms. The first one is proportional to the wind stress and is equal to the corresponding term for the thickness parameterization up to exchange of *K*_{H} and *K*_{1}. The second term is

*K*

_{1}=

*K*

_{H}(Fig. 3). We call

*K*

_{1}“small” if

*K*

_{1}), which implies accounting for (53):

*K*

_{1}, so that

*K*

_{H}, and therefore the zonally averaged zonal velocity profiles there are close to each other for PV and TH parameterization for small but equal diffusivities

*K*

_{1}=

*K*

_{H}(see Fig. 3a).

In the lower layer, the zonally averaged zonal velocity in the PV parameterization case is determined by the first term in (50) for small CPV because the second term is proportional to *K*_{2}. Since Θ ≈ 1, the velocities for the lower layers for PV and TH parameterization are close to each other for small but equal *K*_{1} = *K*_{H} (see Fig. 3a). Increasing coefficient *K*_{1} (and correspondingly *K*_{H}) from 200 (in Fig. 3a) to 400 and 600 m^{2} s^{−1} increases the difference between the velocity profiles (Figs. 3b,c). Note that we cannot investigate the asymptotic limit of large CPV and CT because *K*_{1} must be less than the critical value set by momentum and energy conservation (Fig. 2).

### c. Production of eddy mechanical energy

#### 1) TH parameterization

*K*

_{H}=

*const*. It is inversely proportional to

*K*

_{H}and strongly increases when

*K*

_{H}decreases (Fig. 4).

#### 2) PV parameterization

The production of eddy energy has already been discussed above.

### d. Zonal transport

#### 1) TH parameterization

*T*can be found by multiplying velocities (25) and (29) with the corresponding layer thicknesses

*H*

_{i}, summing them and integrating over the channel width:

*K*

_{H}is reduced because of the term inversely proportional to

*K*

_{H}(see Fig. 5). For large

*K*

_{H}it approaches the value

#### 2) PV parameterization

*T*can be found by multiplying velocities (51) and (52) with the corresponding layer thicknesses

*H*

_{i}, summing them and integrating over the channel width:

The behavior of zonal transports for PV and TH parameterization appears similar: the transports strongly increase if *K*_{i} and *K*_{H} are reduced, but show no changes for large enough diffusivities. An interesting question is how large should be the differences between the coefficients *K*_{i} and *K*_{H} if we want to get the same zonal transport. If we set *K*_{1} = *K*_{H} (and calculate *K*_{2} from the Bretherton theorem), the transports in PV and TH parameterization will be close to each other (see Fig. 5a). However, if we set *K*_{2} = *K*_{H}, very different values of diffusivities will be required to warrant the same transports (Fig. 5b).

### e. Variable coefficients

The zonal velocities are strongly dependent on the coefficients *K*_{1}, *K*_{2} and *K*_{H}. If diffusivities are small, the velocities in the lower layer show almost identical sinusoidal behavior, but in the upper layer, the PV parameterization results in stronger flows (see Fig. 6a). As *A*_{2} increases, the difference between *K*_{i} and *K*_{H}.

Transport calculated for the PV and TH parameterizations is similar if we set *K*_{2} = *K*_{H} (Fig. 7), however, with higher values for the PV parameterization. It is strongly nonlinear for small values of coefficients.

## 6. Parameterization of eddy PV fluxes based on the eddy enstrophy budget

As we have seen, the zonally averaged zonal velocity profiles strongly depend on the meridional profiles of the CPV. The CPV need to be parameterized either directly in terms of mean fields or linked with the dynamics of other parameterized fields such as eddy kinetic energy or enstrophy. Many studies prefer the eddy kinetic energy (Eden and Greatbatch 2008; Eden 2010). However, in the case of jet flows, the eddy kinetic energy reaches a maximum at the jet centers, while the CPV has a local minimum there (McWilliams and Chow 1981). Reconciling these behaviors requires special hypotheses on the mixing length scale.

*μ*

_{1}is some coefficient (m s

^{2}). We use the term “strong baroclinic regime” to designate the flows with high values of mean meridional QPV gradients. Marshall (1981) used the CPV proportional to the vertical shear of the mean zonal velocities, which provides satisfactory solution for the weak baroclinic regimes, that is, for small meridional gradients of QPV:

*μ*

_{2}(m

^{−1}) to the denominator we get

*K*

_{1}:

*μ*

_{1}:

*K*

_{1}is positive, the first term in (78) must be smaller than the second term, which gives

*πy*/

*L*) and therefore

*μ*

_{1}is most constrained at the channel center

*y*=

*L*/2. For the standard set of parameters, this gives

The coefficient *K*_{1} is plotted in Fig. 8 as a function of the meridional dimensionless coordinate for various values of parameters *μ*_{1} and *μ*_{2}. The respective curves attain either local maxima (red line in Fig. 8) or local minima (other lines) at the center of the channel. The maxima correspond to weak baroclinic regimes with smooth meridional profiles of the zonally mean zonal velocity, and minima correspond to strong baroclinic regimes with the intensification of zonally mean zonal velocity with respect to the sinusoidal profile of wind stress forcing (see Figs. 9a,b). The profiles of zonally mean zonal velocity do not show strong sensitivity to the coefficient *μ*_{2}, which could be called the “coefficient of weak baroclinic regime” (see Fig. 9a). They are more sensitive to the parameter *μ*_{1} (we call it the “coefficient of strong baroclinic regime”) (Fig. 9b), which affects the maximum values at the jet core. It is interesting and important that the total transport in all cases remains almost constant. For the three sets of parameters in the upper and lower panels of Fig. 9, the transports vary around 1700 Sverdrups (Sv; 1 Sv ≡ 10^{6} m^{3} s^{−1}) within 1.5%. The parameterized fluxes redistribute zonal momentum, intensifying the jet, but do not strongly influence the zonal transport.

The zonal channel flows have equivalent-barotropic structure, similar to that of the Antarctic Circumpolar Current (ACC; Killworth 1992; Krupitsky et al. 1996), which allows us to employ similar CPV profiles in both layers with different amplitudes.

## 7. The Reynolds stresses and the TH parameterization

*K*

_{H}? In other words, is it possible by special choice of

*K*

_{H}to introduce in the TH solutions the effect of the Reynolds stress divergence? In section 5, we demonstrate that for constant values of

*K*

_{i}and

*K*

_{H}and for small

*K*

_{1}the solutions for zonal velocities for TH parameterization are close to PV solutions. Similar situations occur for variable coefficients, as well. If choosing

*K*

_{H}=

*K*

_{1}from the PV parameterization, (29) and (25) can be rewritten omitting the Reynolds stresses and introducing

*K*

_{1}instead of

*K*

_{H}:

*K*

_{1}, the second and the third terms in (49) will be much smaller than

*K*

_{1}instead of

*K*

_{H}. Relatedly, using the TH parameterization with the diffusivity coefficient obtained from the PV parameterization allows one to describe the effect of the Reynolds stresses, which gives much sharper zonal velocity profiles.

## 8. Summary and discussion

Any parameterization of eddy fluxes must be based on correct physics. Physical arguments suggest that diffusive parameterizations can be applied only to conserved variables. Momentum is not a conserved variable because of the presence of the pressure gradient, and the parameterization of the Reynolds stresses still presents a great challenge. Commonly used harmonic/biharmonic viscosity operators fail to provide an adequate parameterization of eddy fluxes of momentum. For example, the eddy kinetic energy can in some cases be transferred to the mean kinetic energy in eddying flows, which a harmonic operator can only do if the viscosity was negative.

The interest to parameterizations of eddy fluxes of potential vorticity stems from the fact that they incorporate the redistribution of both momentum and thickness. Adopting them, one does not need to separately parameterize the Reynolds stresses. Their disadvantage is that the mean PV is not a variable of the primitive equation models. In spite of that, recent studies propose the ways of using parameterized PV fluxes in such models (Wardle and Marshall 2000; Eden and Greatbatch 2008; Eden 2010; Marshall and Adcroft 2010; Ringler and Gent 2011; Marshall et al. 2012).

Here, we compared the effects of PV and TH parameterizations on the mean flow driven by wind in a zonally reentrant flat bottom channel. Although the model used by us is a gross oversimplification of realistic situation, it admits analytical treatment or simple analysis, allowing us to learn about the differences caused by the parameterizations. We considered both uniform and variable meridional profiles of diffusivities, prescribing in the latter case the profiles resembling those diagnosed in the eddy-resolving experiments.

For the constant values of the coefficients, the main difference between solutions for the two parameterizations in the lower layer is that mean zonal velocity in the QPV flux parameterization acquires a factor

The major difference between the two approaches relates to the role of constraints. While there are important integral constraints in the PV parameterization case (based on momentum and energy conservation), no such constraints are associated with the TH parameterization. There is only a trivial restriction that *K*_{H} be positive. Using the TH parameterization is thus easier, but the PV parameterization seems to be more complete and allows one to exercise strict control on the conservation of momentum and energy.

The obvious difference between these two parameterizations can be seen when the CPV and CT vary meridionally. According to numerical experiments, the profile of diffusivity in the TH parameterization is almost constant in the center of the jet, whereas the CPV coefficient *K*_{i} has pronounced local minima there (McWilliams and Chow 1981). These minima represent a kind of PV barrier (Dritschel and McIntyre 2008). Therefore, using such a more realistic distribution of coefficients provides a very different distribution of zonal velocities with strong momentum concentration in the jet center for the PV parameterization, but without momentum intensification for the TH parameterization. These results demonstrate the importance of introducing the parameterization scheme with varying diffusivities not only for the Southern Ocean, but for other domains with zonal jets as well. Such closures could be based on eddy enstrophy (Ivchenko 1984; Ivchenko et al. 1997), since the generation term is proportional to the product of the CPV with the square of the mean QPV gradient, that is,

## Acknowledgments

We thank two anonymous reviewers for useful comments and discussion that led to a greatly improved manuscript.

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