1. Introduction

Eddies are omnipresent in the ocean, with the local maximum of energy on the mesoscale (Kamenkovich et al. 1986). Eddies redistribute momentum, and in large areas of the ocean [notably the Antarctic Circumpolar Current (ACC)] they can increase the kinetic energy of the mean flow (negative viscosity) and play an important role in the downward transport of momentum by inviscid interfacial form stress.

Numerical models either have to resolve eddies or parameterize them. Although eddy-resolving models are becoming increasingly common [e.g., Delworth et al. (2012) employ a coupled ocean–atmosphere model with unprecedented horizontal resolution in the ocean from 8 km at high latitudes to 28 km in the tropics], it is still too costly to run global eddy-resolving models over the large time periods (decades and centuries) required for climatological studies. Therefore, coarse models are used in which eddy effects are parameterized. Correct parameterizations must be based on clear physics and satisfy some basic principles: this is not always the case for some commonly used schemes. It is well known that “…the [eddy] diffusion model does not satisfactorily describe the eddy [heat, momentum or vorticity] terms…” (Harrison 1978) and that harmonic/biharmonic operators of velocity (temperature and salinity) are used for numerical stability rather than for their realism (Killworth 1997).

Green (1970) and Welander (1973) were the first to propose using parameterizations for eddy fluxes of potential vorticity (PV). This idea was implemented in a number of recent studies (Wardle and Marshall 2000; Eden and Greatbatch 2008; Eden 2010; Marshall and Adcroft 2010; Ringler and Gent 2011). However, the coefficients in these parameterizations cannot be selected arbitrarily, as there are integral constraints that have to be satisfied. The importance of integral constraints for the flat bottom case was demonstrated in Marshall (1981), Ivchenko (1984), Ivchenko et al. (1997), Olbers (2005), and Ivchenko et al. (2008). Recognizing the role of constraints, Marshall et al. (2012) propose a framework for parameterizing eddy potential vorticity fluxes that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy flux. The outstanding question of how variable bottom topography modifies the integral constraints and affects the parameterization coefficients has not been fully addressed by previous studies dealing with the parameterization of eddy PV fluxes. The generalized theorem of Bretherton (GTB), expressing the constraint on momentum in a channel with variable bottom topography, was proven by Ivchenko (1987), and Ivchenko et al. (2013) were the first to apply this constraint. However, it is not the only constraint, and parameterizations of PV fluxes have to satisfy additional requirements among which are the other integral constraints—the energy inequality (EI) and the eddy quasigeostrophic (QG) potential enstrophy inequality.¹ The latter is automatically satisfied if the coefficient of quasigeostrophic potential vorticity (QPV) diffusivity (CPV) is not negative (Ivchenko et al. 1997). The EI constraint has not been applied for the case with variable bottom topography. This paper extends previous studies and seeks

• To prove the energy inequality and study its impact on admissible PV diffusivities in a zonal channel with variable bottom topography;

• To extend previous results pertaining to how the integral constraint for momentum (GTB) affects the parameterization in a zonal channel with complex bottom topography when the zonal-mean anomaly of bottom topography differs from zero [in contrast to Ivchenko et al. (2013)];

• To determine how the PV diffusivities constrained by the GTB and EI and the bottom form stress (BFS) respond to the governing parameters; and

• To develop a simple expression linking the mean zonal transport to the governing parameters.

We achieve these aims by considering a simplified time- and zonally averaged ocean, in a channel configuration, with sinusoidal bottom topography. We parameterize the eddy transport of potential vorticity as a diffusive process with unknown diffusion coefficients. This system is simple enough to be solvable analytically. The analytical solution thus gives us the flow field (zonal velocities, etc.) in terms of the unknown diffusion coefficients. We then substitute the analytical solution into mathematical expressions for the integral constraints introduced above (generalized theorem of Bretherton and energy inequality). This process results in some very interesting and useful restrictions on the size of the diffusion coefficients, their variation with depth, and their dependence on external parameters such as the height of the bottom topography.

We consider quasigeostrophic dynamics in a two-layer fluid driven by winds over an uneven bottom. This is a major simplification of the real dynamics and is used here as a conceptual tool, allowing us to make the problem analytically tractable and to illustrate the interaction between the parameterizations and the flow dynamics. We use numerical simulations to demonstrate qualitative agreement of conclusions derived analytically with the full solution.

The analytical solution is facilitated by the smallness of the relative vorticity in zonal channels compared to the planetary vorticity or the “stretching term” in the expression for the QPV. The order of the differential equation is set by the relative vorticity term, so that the equations have a small parameter at the highest derivative leading to thin boundary layers near the walls. The small parameter γ is the ratio of the Rossby radius to the width of the channel. Its presence allows us to use the technique of asymptotic expansion in this parameter.

The paper is organized as follows: We first write down the main equations (fluid flow and integral constraints) and discuss the analytical solution (sections 2 and 3). Although its derivation was presented by Ivchenko et al. (2013), we briefly recapitulate it for convenience, putting details in the appendix. Section 4 discusses how the energy and momentum constraints shape the behavior of the diffusivities. Zonal momentum redistribution by eddies and net zonal transport in the presence of topography is the subject of section 5. Section 6 describes a numerical model used for simulations and presents a comparison with results found with the analytical model. Section 7 consists of a discussion of the results and conclusions.

2. Basic quasigeostrophic equations for the zonal channel and general constraints

a. Equations

We use the quasigeostrophic equations for a two-layer model (McWilliams et al. 1978):

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where q_i and Ψ_i are quasigeostrophic potential vorticity and streamfunction, respectively; subscripts mark the layers of constant mean thicknesses H_i and τ is the wind stress divided by the water reference density. The fluid velocity v_i = (u_i, υ_i) is expressed as and in the zonal and meridional directions, respectively. Lateral friction F_i is biharmonic in the numerical simulations but is omitted in the analytical solution and

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is the Jacobian operator, and ϵ is the linear bottom drag.

The layerwise quasigeostrophic potential vorticities q_i are given by

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where is the reduced gravity; g is the acceleration due to gravity; ρ_i is the density of layer i; ρ₀ is the reference density; f and f₀ denote the Coriolis parameter and its reference value, respectively; and B is the deviation of bottom topography from the constant depth of H = H₁ + H₂.

We derive analytical solutions for the time and zonal mean of Eqs. (1)–(4):

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The overbar here denotes the time and zonal average, and the prime implies the eddy component, that is, the deviation from the time and zonal mean.

To proceed further, we assume a diffusive parameterization for eddy fluxes of QPV in each layer,

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where K_i ≥ 0 is the respective CPV. In doing so we follow numerous studies (Taylor 1915; Green 1970; Rhines 1977; Marshall 1981; Killworth 1997; Treguier et al. 1997; Wardle and Marshall 2000; and others) proposing that the eddy transfer of PV is downgradient. The presence of a rotational component in eddy fluxes and problems with its separation (Fox-Kemper et al. 2003) add complexity to this problem. However, we think that the downgradient assumption of eddy PV flux presents an appropriate starting point. These parameterizations, when substituted into Eqs. (5)–(6), will lead to a closed set of equations that can be solved for any K_i. The point, however, is that the solutions should satisfy certain integral constraints, which in turn restrict the admissible values for K_i. As pointed out by Ivchenko et al. (2013), the GTB presents one such constraint, which immediately imposes a link between the layer diffusivities K₁ and K₂. We shall analyze it in more detail here and also take into account additional constraints as explained below.

b. Generalized theorem of Bretherton

We begin by noting that the integration of the time- and zonal-mean potential vorticity Eqs. (5)–(6) in the meridional direction for a stationary regime, disregarding horizontal friction, yields

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where the constants of integration in Eqs. (10)–(11) are set to zero because the eddy fluxes, wind stress, and zonal-mean velocity are enforced as zero on the solid boundaries in our solutions (Marshall 1981; Ivchenko et al. 2013).

Combining Eqs. (10)–(11) we find

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This expression is valid for arbitrary bottom topography. For a flat bottom channel, the meridional integral of the left-hand side of Eq. (12) becomes zero according to the Bretherton theorem (Taylor 1915; Bretherton 1966; McWilliams et al. 1978; Marshall 1981; Ivchenko et al. 2013):

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Hence, for the flat bottom channel, the main force by wind stress is balanced by the bottom drag:

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It is well known that this balance, valid in the absence of bottom topography, results in an unrealistically high transport of the ACC (Munk and Palmén 1951) if applied in the Southern Ocean.

Substitution of the diffusive parameterization, Eq. (9), in the Bretherton theorem, Eq. (13), yields

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This expression relates to the first instability criterion of Pedlosky: “the potential vorticity gradient must be somewhere positive and somewhere negative for instability to occur” (Pedlosky 1964).

For variable bottom topography the Bretherton theorem is replaced by the GTB (Ivchenko 1987; Ivchenko et al. 2013):

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The term within the integral on the rhs of Eq. (16) is the bottom form stress, since

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where p₂ is the pressure in the lower layer normalized by the reference density.

The bottom form stress is an inviscid mechanism that redistributes momentum. We cannot expect acceleration of the zonal flow by this mechanism, which yields

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This can be used, in addition to the GTB, as an extra constraint the solution has to satisfy.

Again, we can substitute the diffusive parameterization of eddy QPV fluxes given by Eq. (9) into the GTB, Eq. (16), to obtain

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Because of Eq. (18) the rhs of Eq. (19) is positive, and instability (more accurately, states with nonzero eddy kinetic energy) in a channel with variable bottom topography can occur even if the potential vorticity gradient is everywhere positive, in contrast to the flat bottom case.

Eddies redistribute the mean zonal momentum, and following Pedlosky (1964, 1979) and using Eq. (12) this balance can be written

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The first term on the right-hand side of Eq. (20) therefore describes the redistribution of zonal barotropic momentum by eddies.

If we substitute the GTB, Eq. (16), into the meridionally integrated momentum balance Eq. (12), we find that the wind stress is balanced by the bottom friction and the bottom form stress:

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The second term on the left-hand side is commonly much smaller than the wind stress term in the presence of topography, and the balance is between the wind stress and bottom form stress both for a zonal channel with an uneven bottom and the ACC (Munk and Palmén 1951; Ivchenko et al. 1996; Stevens and Ivchenko 1997).

Numerical experiments show that even a small zonal variation in B substantially reduces the zonal transport (McWilliams et al. 1978; Treguier and McWilliams 1990; Sinha and Richards 1999; Wolff et al. 1991). We therefore expect that the link between the diffusivities will be affected by the presence of bottom topography.

c. Energy inequality

By multiplying the instantaneous quasigeostrophic potential vorticity [Eqs. (1)–(2)] by , taking the time and zonal average, summing layer contributions, and integrating in the meridional direction, we obtain the budget of the total eddy mechanical (i.e., kinetic and available potential) energy. The balance is expressed as

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where , , and APE^ed are the eddy kinetic energy for the upper and lower layers and the eddy available potential energy, respectively (Ivchenko 1987; Ivchenko et al. 1997). We assume that the wind is steady and zonally uniform, depending only on the meridional coordinate.

The first term on the right-hand side corresponds to the generation of total eddy mechanical energy by baroclinic and/or barotropic instabilities. The second term on the right-hand side contains dissipation terms due to the horizontal and bottom friction. For steady-state solutions, the eddy mechanical energy is positive only if the generation term is positive:

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Once again, using the diffusive parameterization of eddy QPV fluxes, Eqs. (9) and (23) can be rewritten in the form

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This statement agrees with the second instability condition of Pedlosky: “the product of the zonal velocity and the potential vorticity gradient must in all cases be somewhere positive for instability to occur” (Pedlosky 1964). In the context of this study, it is this constraint on the solutions with parameterized eddy fluxes that leads to further restrictions on the diffusivities. Clearly, all the constraints discussed thus far (the GTB, negativity of the form stress, and the energy inequality) are not independent, yet lead to different limitations.

3. Analytical solution

We set the wind stress to be zonal and to vary as the sine of the meridional coordinate:

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where τ₀ is the amplitude of zonal wind stress. We also prescribe the bottom topography deviation B as

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where B₀ is the amplitude of bottom topography and σ is a nondimensional parameter. If σ ≠ 0, the zonal-mean deviation of bottom topography is not zero, but remains a function of the meridional coordinate. Other than this, the topography selected is zonally periodic and takes zero values at the meridional boundaries (y = 0, L). We substitute Eq. (9) in the time and zonally averaged equations for QPV [Eqs. (5)–(6)], omit the lateral friction, and rewrite them in dimensionless form (Marshall 1981; Ivchenko 1987; Ivchenko et al. 2013):

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where the asterisk marks dimensionless variables y = Ly*, , , and . We also introduce the following parameters , γ = L_RL⁻¹, , ϵ = βLϵ*, and the dimensionless CPVs s_i = K_i(L_Ru_c)⁻¹. The wind amplitude enters through u_s = (πτ₀)(H₁βL)⁻¹, and is the channel velocity scale, chosen so that the stretching term in QPV balances the meridional variation in planetary vorticity (Marshall 1981).

We are interested in obtaining an analytical solution to Eqs. (27)–(28), applying boundary conditions of no flux through the solid boundaries:

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The procedure of solution is similar to that of Ivchenko et al. (2013), except for the addition of the parameter σ to the topography anomaly (with respect to the reference depth H). We assume the coefficients K_i (s_i) to be constant over the channel, except for two thin boundary layers near the walls with thickness Δ ≪ 1, where they are linearly reduced to zero, which allows us to satisfy boundary conditions.

Solving equations and analyzing solutions in nondimensional form offers some advantages, simplifying the use of the asymptotic expansion technique (see below) and introducing nondimensional parameters that reflect the main balances between the terms in the governing equations. However, a caveat is that nondimensional parameters can be dependent on several physical parameters, which should be kept in mind in the analysis. The system [Eqs. (27)–(28)] contains a small parameter γ = L_RL⁻¹. Its presence allows us to apply an asymptotic expansion in this parameter and match solutions in the boundary layers with the solution outside them. The mathematical details of this technique are the same as in Ivchenko et al. (2013). For convenience, they are repeated in the appendix. The solutions for the zonal velocities are given by Eqs. (A15)–(A18), in the appendix. From these we are able to obtain expressions for potential vorticities and streamfunctions in each layer, and from the latter we can obtain the meridional velocity in order to calculate the bottom form stress. After solutions are found, they are substituted in the constraints discussed above, enabling us to analyze the limitations on CPV as a function of model parameters.

4. Results: Integral constraints

In this section we substitute the analytic solution derived in the appendix [Eqs. (A15)–(A18)] into the GTB and energy inequality, given, respectively, by Eqs. (16) and (24) of section 2. Since the analytic solutions are functions of the layerwise CPVs, we thus obtain a set of restrictions on the CPVs that depend on the governing parameters. The impact of these restrictions on the CPVs and the behavior of the BFS are illustrated by Figs. 1–5.

(a) Re (green) and Re_cr (blue) versus B₀, with D = 2. (b) D versus B for Re = 20 (red), 30 (blue), and 40 (green). In both panels σ = 0 (solid), −0.15 (asterisks), and 0.15 (dashed).

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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(a) Re (green) and Re_cr (blue) versus B₀, with D = 2. (b) D versus B for Re = 20 (red), 30 (blue), and 40 (green). In both panels σ = 0 (solid), −0.15 (asterisks), and 0.15 (dashed).

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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(a) Re (green) and Re_cr (blue) versus B₀, with D = 2. (b) D versus B for Re = 20 (red), 30 (blue), and 40 (green). In both panels σ = 0 (solid), −0.15 (asterisks), and 0.15 (dashed).

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Constraints of Re and D by GTB and EI for (a) flat bottom. (b) B₀ = 300 m. Permissible values of Re and D based on the EI lie above the corresponding curves and D must be positive.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Constraints of Re and D by GTB and EI for (a) flat bottom. (b) B₀ = 300 m. Permissible values of Re and D based on the EI lie above the corresponding curves and D must be positive.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Constraints of Re and D by GTB and EI for (a) flat bottom. (b) B₀ = 300 m. Permissible values of Re and D based on the EI lie above the corresponding curves and D must be positive.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of BFS on bottom topography for (a) σ = 0 and (b) D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of BFS on bottom topography for (a) σ = 0 and (b) D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of BFS on bottom topography for (a) σ = 0 and (b) D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of BFS on parameters (a) ϵ and (b) σ at B₀ = 300 m.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of BFS on parameters (a) ϵ and (b) σ at B₀ = 300 m.

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Dependence of BFS on parameters (a) ϵ and (b) σ at B₀ = 300 m.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of BFS on wind stress, where D = 2 for (a) variable B₀ and (b) variable σ.

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Dependence of BFS on wind stress, where D = 2 for (a) variable B₀ and (b) variable σ.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of BFS on wind stress, where D = 2 for (a) variable B₀ and (b) variable σ.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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a. Integral constraint for momentum (GTB)

The GTB, given by Eq. (19), can be written in dimensionless form as

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where S is the scaling of , that is,

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and υ₂ is obtained from Ψ₂ [see appendix Eq. (A6)].

Substituting solutions for the meridional gradients of QPV and bottom form stress into Eq. (30), we obtain

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Equation (32) introduces two nondimensional parameters involving the CPVs. These are Re = u_sLK₁⁻¹, which is an analog of the Reynolds number, and D = ϵu_c(βK₂)⁻¹, which characterizes the relation between the bottom friction and diffusion of QPV in the lower layer.

Four other nondimensional parameters α_B, α_R, α_T, and α_U are now introduced:

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The parameter estimates the relative importance of wind stress and the stretching term in the meridional gradient of QPV for the upper layer. The parameter α_R measures the ratio between the “topographic” (|f₀|B₀)H₂⁻¹ and planetary βL contributions to the QPV in the lower layer. The physical sense of parameter α_T is the relative importance of inviscid bottom form stress against viscous bottom friction. Finally, the parameter α_U is the ratio between the time scale of zonal baroclinic Rossby waves and that of dissipation by bottom friction. Two of these parameters (α_B and α_U) depend on the stratification, amplitude of wind stress, bottom friction, β, Rossby radius, and other parameters, but do not directly involve the characteristics of topography. We will call them the “flow parameters.” The two remaining parameters (α_R and α_T) depend on the amplitude of bottom topography deviation and will be referred to as the “topographic parameters.” We apply our theory for the Southern Hemisphere, where f₀ < 0.

Equation (32) can be rewritten for D as a linear function of Re:

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where

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Although the relationship between D and Re is linear, the relationship between K₂ and K₁ is nonlinear. Since Re > 0 and D > 0, Eq. (37) yields a critical value Re_cr:

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This immediately implies that there is an upper bound on the CPV of the upper layer. Note, that Re_cr > 0 exists even for the flat bottom channel (Ivchenko et al. 1997), which means that if Re < Re_cr the flow is not unstable for baroclinic instability.

The values of Re and Re_cr decrease nonlinearly when the amplitude of B₀ is increased (Fig. 1a). For fixed Re, the value of D increases if the amplitude B₀ is increased for physically meaningful solutions (see Fig. 1b). If B₀ is small and Re insufficiently high, the parameter D could became negative, which is forbidden. For larger prescribed Re, D increases accordingly. The relationship D(B₀) is linear for fixed Re if σ = 0 and weakly nonlinear if σ ≠ 0. For subsequent calculations we use the following set of “standard” values: u_s = 1.5 × 10⁻² m s⁻¹, u_c = 1.4 × 10⁻¹ m s⁻¹, L = 1.5 × 10⁶ m, L_x = 8 × 10⁶ m, H₁ = 10³ m, H₂ = 4 × 10³ m, k = 4, b = −0.16, ϵ = 10⁻⁷ s⁻¹, β = 1.4 × 10⁻¹¹ m⁻¹ s⁻¹, τ₀ = 10⁻⁴ m² s⁻², and L_R = 4 × 10⁴ m (Marshall 1981; Ivchenko et al. 2013).

In a flat bottom channel, Re = 69.6 and Re_cr = π²(2δ₁)⁻¹ = 24.7 for a prescribed value of D = 2 (Fig. 2a). If the bottom topography amplitude B₀ is varied between 0 and 500 m, for D = 2, and parameter σ = ±0.15, the critical value Re_cr is greater than 15. Positive (negative) values of σ yields higher (lower) values of Re_cr (see Fig. 2b). This happens because a negative σ provides a negative contribution to the mean meridional QPV gradient in the lower layer in the central part of the channel, ¼ < y* < ¾, where the wind forcing is strongest [see Eq. (A2)]. Since baroclinic instability plays a dominant role and the onset of instability requires a change in the sign of the mean meridional QPV gradients between the layers, and since this gradient is positive in the upper layer [see Eq. (A1)], the instability occurs for a smaller value of the mean vertical shear in the case of negative σ, that is, smaller Re_cr.

According to Eqs. (37) and (38), the slope M₁ of ∂D/∂Re is always positive and increases if the amplitude of bottom topography is increased, since only the parameter α_T in Eq. (38) is linked to the bottom topography (proportional to B₀). Note that the slope ∂D/∂Re is independent of the parameter α_R and the “geometric” parameter σ. For the flat bottom case, the slope [Eq. (38)] reduces to

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that is, it becomes proportional to the two flow parameters.

b. Bottom form stress: Physical mechanism and dependence on parameters

Since the BFS plays a major role in zonal currents, like the ACC, it is important to understand what affects it and what restrictions it imposes.

The bottom form stress (i.e., the right-hand side of the GTB) can be written in dimensionless form (for the Southern Hemisphere) as

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All the parameters in the rhs before the expression in parentheses are positive, so that the inequality [Eq. (18)] is satisfied if

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It is natural that the rhs of Eq. (43) is independent of the “topographic” parameter α_T (since it includes friction). It depends, however, on the geometric parameter σ. In the center of the channel, the meridional gradient of zonally averaged bottom topography is positive for σ < 0 and negative for σ > 0 that modifies the planetary part of QPV gradient. The respective contribution in the expression for [see Eq. (A2)] is −2πα_Rσ cos(2πy*) and is negative (positive) for negative (positive) σ, which yields decreasing (increasing) total eddy QPV and accordingly BFS.

The expression for the BFS [Eq. (42)] can be rewritten by using the GTB [Eq. (37)]:

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We conclude that

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should be observed if D is small to ensure that BFS > 0.

The BFS depends nonlinearly on the amplitude of the bottom topography (Fig. 3a), especially when it is small, where B₀ < 100 m. The BFS is higher for small values of D, and the parameter σ only weakly affects the BFS for small B₀, but leads to larger effects for higher amplitudes of bottom topography (Fig. 3b) because nondimensional BFS depends on σ only via the last term in the brackets in Eq. (42). The term in brackets, −3σπ³δ₂α_R, is proportional to B₀, since α_R ~ B₀. This leads to a small impact of σ on BFS for small values of bottom topography, which increases with increasing B₀.

To find the dependence of the BFS on Re we use Eq. (32), that is, the GTB, and rewrite the BFS as the sum of fluxes to obtain

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The denominator in Eq. (46) is positive, and the BFS is inversely proportional to Re squared if Re is sufficiently large. The sign of this link depends on the term (1 − 3δ₂πα_Rσ): if (1 − 3δ₂πα_Rσ) > 0, then ∂BFS/∂Re < 0. Note, that this condition is identical to Eq. (45).

The BFS increases with bottom friction ϵ for a high enough amplitude of bottom topography B₀ (see Fig. 4a). However, it is interesting to note that the BFS decreases when ϵ increases for small B₀ (e.g., B₀ = 100 m in our figure). For small values of B₀ the contribution of viscous bottom drag to the mean momentum balance [Eq. (21)] becomes significant, so that the contribution of the BFS gets less important with increasing bottom friction ϵ.

The BFS decreases also if the geometric parameter σ increases (Fig. 4b) or with increasing parameter D.

The value of BFS as a function of wind stress is higher for larger values of B₀ (Fig. 5a). For smaller wind stress this dependence is strongly nonlinear, but for higher wind stress it is independent of wind stress and is a function of B₀. For a high enough wind stress, the dependence of the BFS on σ is insensitive to the stress value (Fig. 5b), but for smaller values (in the range of climatological values along the ACC, i.e., about 10⁻⁴ m² s⁻²) the BFS displays nonlinear dependence on wind stress. This happens because the nondimensional BFS depends on the wind stress via two parameters: and Re ~ u_s ~ τ₀, which leads to the independence of nondimensional BFS on the wind stress for large values of Re; that is, for

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[see Eq. (42)]. This last condition corresponds to high values of wind stress.

c. Integral constraint for energy

The energy inequality [Eq. (24)] can be rewritten in dimensionless form as

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Substituting solutions for velocities [Eqs. (A15)–(A18)] and meridional gradients of QPV into this inequality we find, after some rearrangement,

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In Eq. (49), the parameters Re and D are linked via the GTB [i.e., Eq. (37)]; this will introduce the topographic parameter α_T to the EI.

It is straightforward to find an asymptotic value of Re when D tends to infinity:

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so that Re > 4 for any set of external parameters (including the flat bottom case) (see Figs. 2a,b).

The asymptotic value of Re = 4 corresponds to the averaged value of CPV in the upper layer:

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for the standard set of parameters.

From Eq. (49) and Figs. 2a and 2b, we can see that there are restrictions on the parameter D, but there is no asymptotic form as there is for Re.

Note that the parameter D is proportional to the bottom viscosity ϵ. This means that the limit case of D = 0 cannot provide a physically reasonable solution because in the presence of forcing (wind stress), the system must rely on the bottom drag to reach a stationary regime. The GTB does not exclude the zero value of D.

5. Momentum redistribution by eddies and zonal transport

Eddies redistribute zonal momentum, locally increasing or decreasing it. In this section we examine the zonal redistribution of momentum by the parameterized eddies in our analytic solution and investigate how this redistribution process is modified by the presence of topography compared to the flat bottom case. We also use our analytic solution to investigate the parameters that govern the magnitude of the net zonal transport. The results are illustrated in Figs. 6–11.

QPV eddy flux for (a) upper layer and (b) total flux, with D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV eddy flux for (a) upper layer and (b) total flux, with D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV eddy flux for (a) upper layer and (b) total flux, with D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV total eddy flux for (a) variable bottom friction with B₀ = 300 m, D = 2, and σ = 0 and (b) variable wind stress.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV total eddy flux for (a) variable bottom friction with B₀ = 300 m, D = 2, and σ = 0 and (b) variable wind stress.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV total eddy flux for (a) variable bottom friction with B₀ = 300 m, D = 2, and σ = 0 and (b) variable wind stress.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV total eddy flux with B₀ = 300 m for (a) variable D and σ = 0, and (b) variable σ and D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV total eddy flux with B₀ = 300 m for (a) variable D and σ = 0, and (b) variable σ and D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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QPV total eddy flux with B₀ = 300 m for (a) variable D and σ = 0, and (b) variable σ and D = 2.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of zonal transport on B₀ for (a) variable σ and D = 2, and (b) variable D and σ = 0.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of zonal transport on B₀ for (a) variable σ and D = 2, and (b) variable D and σ = 0.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of zonal transport on B₀ for (a) variable σ and D = 2, and (b) variable D and σ = 0.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of D and K₂ on transport with variable σ and B₀ = 300m for (a) D and (b) K₂.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of D and K₂ on transport with variable σ and B₀ = 300m for (a) D and (b) K₂.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of D and K₂ on transport with variable σ and B₀ = 300m for (a) D and (b) K₂.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of transport, K₁ and K₂ on parameters for (a) variable wind stress, with D = 2, and (b) variable ϵ.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of transport, K₁ and K₂ on parameters for (a) variable wind stress, with D = 2, and (b) variable ϵ.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of transport, K₁ and K₂ on parameters for (a) variable wind stress, with D = 2, and (b) variable ϵ.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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a. Momentum redistribution

We begin by considering the zonal momentum balance, Eq. (20), which can be rewritten in nondimensional form as

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where T₀ is the characteristic time scale of the process; that is, t = T₀t* and T_R = (βL_R)⁻¹ is the time scale associated with a zonal baroclinic Rossby wave. The first term on the right-hand side is the eddy redistribution of momentum, the second term is the external forcing (by wind), and the last term is the bottom friction. According to Eq. (10), the eddy flux of QPV in the upper layer depends on wind stress and H₁ only (see Fig. 6a). Eddies increase the vertically integrated mean momentum when the first term on the right-hand side of Eq. (52) is positive. To find where this happens for a flat bottom channel, we use the solutions for mean zonal velocities of Eqs. (A15)–(A18) and substitute them into the expression for mean meridional gradients of the QPV [Eqs. (A1)–(A2)], which yields

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Therefore from Eqs. (37)–(39) and δ₁α_Bα_URe − D > 0, the region where eddies increase the mean zonal barotropic momentum is (see Fig. 6b, red line, the region where eddy flux is greater than 0)

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The bottom topography strongly affects the eddy redistribution of zonal momentum, especially at small amplitudes, when the flow regime changes from a flat bottom one to one where bottom form stress dominates. Figure 6b, for example, shows that even topography as small as 1–10 m already has a marked impact. In the center of the jet the eddy flux is positive for B₀ = 1 m and at the flanks it is negative. This distribution is similar to the flat bottom case, but the integral of the eddy flux is negative. If B₀ exceeds a value of about 10 m, the fluxes become negative at any point and reach the maximum (by modulus) at the center of the channel. The type of the curve of the eddy flux is changed from concave (which pertains to the flat bottom case) to convex (which pertains to the variable bottom topography channel). Such a change occurs in the range of amplitude of bottom topography between 10 and 100 m. For high enough B₀ increasing D leads to a minor increase of the absolute value of the negative flux at the center of the channel. This result is neither obvious nor trivial because increasing D means increasing the relative importance of bottom drag. However, the increase of D is lifting the value of Re, because of the GTB, which strongly contributes to the total eddy flux of QPV. The meridional distribution of the total eddy flux of QPV is sensitive to the bottom friction (Fig. 7a). If the coefficient of the bottom friction increases between 1.0 and 2.0 × 10⁻⁷ s⁻¹ the type of the curve of the total eddy flux changes from convex to concave (see Fig. 7a). However, the flux does not change its sign and the meridional integral of the flux demonstrates only minor variability for fixed topography. Increasing the bottom friction from 0.3 to 4 × 10⁻⁷ s⁻¹ yields a change (decreasing) of eddy fluxes of only 7%. This occurs because the main balance is between the wind stress and the BFS (which is equal to the total eddy flux of the QPV), whereas the bottom friction contribution plays a smaller role [see Eq. (21)].

Eddy fluxes for small wind stress are almost constant across the channel. However, they tend to a sinusoidal distribution and increase in amplitude when the amplitude of wind stress τ₀ increases. This can be anticipated because the eddy QPV fluxes in the upper layer must be proportional to τ_x by Eq. (10), while in the lower layer they must balance the viscous bottom drag by Eq. (11). As the amplitude of wind stress τ₀ increases, the dimensional bottom form stress increases to balance it, while the bottom drag remains relatively small. The total eddy QPV flux becomes dominated by the upper-layer QPV flux, whose profile resembles that of the wind stress (Fig. 7b).

Eddy fluxes for small D are almost constant across the channel (Fig. 8a). However, increasing D yields redistribution of eddy fluxes, increasing (by modulus) in the center of the channel, but decreasing on the flanks. This happens because the relative importance of the eddy fluxes in the lower and upper layers is proportional to the Re/D, and since Re is a linear function of D from the GTB, their ratio is inversely proportional to D, and this means that the total eddy flux becomes dominated by the upper-layer eddy flux of QPV, whose profile resembles the wind stress [see Eq. (10)].

The geometric parameter σ contributes to the total eddy QPV flux through the term proportional to σ cos(2πy*), which is positive for negative σ at the center of the channel. Because in the total eddy fluxes the upper-layer contribution is dominant and negative, this means reduction in the amplitude in the center of the channel compared to the case σ > 0 (Fig. 8b).

b. Zonal transport

Deriving an expression for the total transport of the Antarctic Circumpolar Current is a challenging task for the dynamics of the Southern Ocean. Its difficulty hinges on the need to properly address the penetration of momentum downward and its balance with the bottom form stress. Both processes are mediated by eddies and using an unsatisfactory parameterization would yield incorrect velocities and unreasonable transport.

Using our results for zonal velocities [Eqs. (A15)–(A18)], we can calculate the zonal transport. We can write the following expression for the transport T in dimensional form:

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According to this expression, the transport appears to depend on both parameters Re and D, but one of them can be eliminated by using the GTB. Substituting Re as a function of D from Eq. (37) yields

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Equation (56) predicts an inversely proportional dependence of the transport on the amplitude of bottom topography B₀ (see Fig. 9a). Increasing the prescribed parameter D results in an increased total transport (see Fig. 9b). This result is not obvious because D is proportional to the bottom friction. The BFS is inversely proportional to D [see Eq. (44)], and increasing D implies a decrease in the major term opposing the wind (i.e., BFS). Also, according to Eq. (56), the transport can be split into three terms: a term linearly increasing with D, a term independent of D, and a term inversely proportional to D. Their sum is increasing with D. Note that the term proportional to D in Eq. (56) is

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It is inversely proportional to the parameter α_U, which is proportional to the bottom viscosity ϵ. The term D/α_U is independent of ϵ, and the term T₁ depends on ϵ only via α_T. Increasing ϵ makes α_T smaller. The total transport quickly decreases with increasing B₀. There is a linear relationship between the transport and parameter σ (not shown). This parameter can strongly affect the transport as B₀ is varied, especially if B₀ is high (see Fig. 9a). Note that the transport depends on the topographic parameter α_R only if the zonal average of the bottom topography deviation is not zero (σ ≠ 0).

The stationary total zonal transport was calculated for fixed D = 2, which corresponds to a value of the CPV in the lower layer K₂ = 500 m² s⁻¹. The coefficient in the upper layer has been calculated by using the GTB. The transport in the flat bottom case reaches an unrealistically high value of 1744 Sverdrups (Sv; 1 Sv ≡ 10⁶ m³ s⁻¹) with the value of K₁ = 322.6 m² s⁻¹. The CPV in the upper layer is smaller than that in the lower layer, which agrees with Marshall (1981). The transport with variable bottom topography drops down drastically, depending on B₀. The CPV in the upper layer strongly increases compared to the flat bottom channel and is higher than the value of K₂. This coefficient decreases linearly when the parameter σ increases.

It is of interest to estimate the range of values of the CPV in the lower layer and the parameter D. The term D increases when transport increases (for fixed B₀) (see Fig. 10a). However, the value of D only increases slightly for small transports, but after about 250 Sv their increase is more substantial.

It is possible to evaluate the CPV for the fixed B₀ and variable transport from Eq. (56). The value K₂ decreases nonlinearly with increasing transport (see Fig. 10b). For B₀ = 300 m and σ = 0, coefficient K₂ decreases from about 970 m² s⁻¹ to about 115 m² s⁻¹. For small transports the values of K₂ vary strongly for different σ, but for transport more than about 350 Sv they are close to each other.

The zonal transport increases linearly with the wind stress (Fig. 11a). For a fixed CPV in the lower layer (K₂ = 500 m² s⁻¹), the coefficient K₁ increases nonlinearly with wind stress (Fig. 11a). There is only weak sensitivity of the total transport to the bottom friction (Fig. 11b), which is not surprising since the main balance for wind stress is provided by inviscid BFS. The increase in Re yields an increase of transport for the same ϵ (Fig. 11b).

6. Numerical model and experiments with the eddy-resolving model

Since our analytical treatment relates to the time- and zonal-mean model with parameterized fluxes, we carry out numerical simulations with a full two-layer model to illustrate the predictions of the analytical model. We do not deal with parameterizations here, but concentrate on eddy fluxes in the presence of topography and show that they behave rather similarly to the predictions of the analytical model, showing also rather similar sensitivity to the governing parameters. We limit ourselves to a qualitative comparison because the analytical results above are obtained for constant K₁ and K₂, while numerical fluxes do not obey Eq. (9) with constant CPV.

The numerical model is implemented on a C grid and uses the Arakawa Jacobian (Arakawa 1966) preserving both energy and enstrophy. The time stepping and implementation of boundary conditions may introduce small nonconservative effects. The boundary conditions for the streamfunctions follow the implementation of McWilliams (1977) and McWilliams et al. (1978). The computational mesh is 800 by 150 points with dx = dy = 10 km. Cyclic boundary conditions are applied in the zonal direction. On solid walls we require that ∂²Ψ_i/∂y² = 0 (free slip) and that ∇⁴Ψ_i = 0 (additional boundary condition required for biharmonic viscosity). Time stepping follows the third-order Adams–Bashforth method. The parameters are selected so as to be in agreement with the analytical model; in particular, the bottom topography is given by Eq. (26). All simulations are run for 20 years, and the results are averaged over the last 10 years. It takes a model about 10 years to reach a quasi-stationary level of energy. Although the next 10-yr period is still insufficient to reach truly stationary statistics, the deviations are already rather moderate for the eddy fluxes. The runs have been performed for three amplitudes of topography: 100, 300, and 500 m. For each of them, we run for three values of parameter σ = −0.15, 0, and 0.15 and three values of bottom drag coefficient corresponding to the inverse of 30, 115, and 360 days (high, standard, and low friction).

Figures 12 and 13 illustrate the behavior of the time- and zonal-mean meridional eddy QPV flux and the form stress for different ϵ, but fixed σ = 0.15 and for different σ and fixed ϵ = 10⁻⁷ s⁻¹ (standard friction), respectively. According to Eq. (10), the zonal-mean meridional PV flux in the upper layer should repeat the wind profile, and numerical simulations for the upper layer show this, as the lateral viscosity contribution is rather small, with some spread due to insufficiently long averaging time. The eddy flux in the lower layer, however, is modified by the presence of topography and becomes progressively more negative and loses its amplitude as the amplitude of the bottom topography B₀ is increased. This modification is compensated by the BFS, and the GTB is maintained with a very high accuracy. Note that the compensation of eddy flux with the BFS is not local, but involves redistribution of flux even in this zonal-mean picture.

Dependence of time- and zonal-mean eddy (a) QPV fluxes and the (b) BFS on the meridional coordinate for various amplitudes of topography [100 m (blue), 300 m (red), and 500 m (green)] and bottom drag ϵ = 1/115 (thick lines), 1/30 (thin lines), and 1/360 (thin dashed lines) day⁻¹ from the numerical eddy-resolving experiment. The flux in the upper layer is defined by wind, so all curves collapse to the same negative sine profile. High friction (thin curves) makes the eddy flux in the lower layer more positive, that is, closer to the flat bottom situation. The case with the smallest friction and topography deviations (dashed blue) shows strong oscillations in the jet position, and the curves for it are based on a longer integration.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of time- and zonal-mean eddy (a) QPV fluxes and the (b) BFS on the meridional coordinate for various amplitudes of topography [100 m (blue), 300 m (red), and 500 m (green)] and bottom drag ϵ = 1/115 (thick lines), 1/30 (thin lines), and 1/360 (thin dashed lines) day⁻¹ from the numerical eddy-resolving experiment. The flux in the upper layer is defined by wind, so all curves collapse to the same negative sine profile. High friction (thin curves) makes the eddy flux in the lower layer more positive, that is, closer to the flat bottom situation. The case with the smallest friction and topography deviations (dashed blue) shows strong oscillations in the jet position, and the curves for it are based on a longer integration.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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Dependence of time- and zonal-mean eddy (a) QPV fluxes and the (b) BFS on the meridional coordinate for various amplitudes of topography [100 m (blue), 300 m (red), and 500 m (green)] and bottom drag ϵ = 1/115 (thick lines), 1/30 (thin lines), and 1/360 (thin dashed lines) day⁻¹ from the numerical eddy-resolving experiment. The flux in the upper layer is defined by wind, so all curves collapse to the same negative sine profile. High friction (thin curves) makes the eddy flux in the lower layer more positive, that is, closer to the flat bottom situation. The case with the smallest friction and topography deviations (dashed blue) shows strong oscillations in the jet position, and the curves for it are based on a longer integration.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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As in Fig. 12, but for the parameter σ = 0.15 (thin lines), 0 (thick lines), and −0.15 (thin dashed lines). Variable σ leads to some spread, but longer averaging is needed to estimate its effect reliably.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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As in Fig. 12, but for the parameter σ = 0.15 (thin lines), 0 (thick lines), and −0.15 (thin dashed lines). Variable σ leads to some spread, but longer averaging is needed to estimate its effect reliably.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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As in Fig. 12, but for the parameter σ = 0.15 (thin lines), 0 (thick lines), and −0.15 (thin dashed lines). Variable σ leads to some spread, but longer averaging is needed to estimate its effect reliably.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0173.1

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According to Fig. 13, varying σ in the selected range leads to relatively small changes. Variations in the bottom drag coefficient modify the balance to a larger extent, as follows from Fig. 12. The low bottom drag corresponds to a more barotropic flow, with larger velocities in the lower layer, and more negative mean meridional QPV flux. In general, an increase in the topographic amplitude and a reduction in bottom drag have similar impact on the fluxes.

This time- and zonal-mean picture is very different from the 2D patterns of time-mean fluxes. The latter are dominated by large rotational contributions associated with the topography. For this reason, we did not attempt to fit the simulated fluxes with Eq. (9).

We note however, that the change in the role of BFS as σ and ϵ are varied is similar to that derived analytically. The contribution from BFS in the GTB increases in absolute value with reduction in ϵ and increase in the topographic amplitude, as could be expected. The sensitivity to friction is the largest for small topographic variations and shows a tendency to saturation as B₀ is increased. Variations in σ lead to small changes in the BFS contribution on the level of several percent. Discussing them would require much longer averaging.

7. Summary and discussion

Both the energy inequality and generalized theorem of Bretherton impose strong restrictions on the eddy PV transfer coefficients. If the chosen CPVs fail to satisfy these constraints, the resulting equations violate the basic laws of energy or momentum conservation.

We explore this issue in an elementary way, taking time- and zonal-mean QG two-layer equations, parameterizing the eddy PV fluxes as downgradient (with constant layer diffusivities), solving the equations, and analyzing the limitations on the diffusivities (CPVs).

We demonstrate that the bottom topography plays an important role in these restrictions, especially for momentum. It explicitly enters the GTB, which should be satisfied by solutions with parameterized QPV fluxes. Any failure to do so leads to violation of the major momentum balance in zonal flows (like the ACC) between the wind stress and the bottom form stress (Munk and Palmén 1951; Ivchenko et al. 1996; Stevens and Ivchenko 1997).

The energy inequality requires that the parameter Re must be greater than the critical value Re_cr = 4, which means that the CPV in the upper layer (main thermocline) must be less than a certain critical value, depending on the amplitude of the wind stress, the layer thickness, and the planetary vorticity gradient β, for any type of bottom topography or flat bottom case. For the standard set of parameters this value is K₁ < 5.6 × 10³ m² s⁻¹.

Because of these restrictions, the parameterization for eddy PV fluxes allows only limited freedom in the choice of coefficients. If we select the value for the coefficient K₁ in the upper layer (which has to comply with the EI), the value of the coefficient in the lower layer K₂ is then prescribed by the GTB. We found a linear relationship between D and Re set by the GTB, which is nonlinear between K₁ and K₂. The link between the coefficients (or parameters Re and D) strongly depends on bottom topography, that is, on its amplitude, zonal, and meridional variability. For flat bottom topography, the GTB requires that the parameter Re > π²(2δ₁)⁻¹. The values of Re_cr for variable bottom topography and the slope of the line D(Re) are strongly dependent on the parameters of bottom topography. For example, changing the mean zonal average value of topography (the parameter σ in the case considered) shifts the value of Re_cr. If D is fixed, the parameter Re has a nonlinear dependence on B₀ (decreasing), but remains higher than Re_cr; for B₀ in the range between 10 and 500 m, the range of Re is within a factor of 3 of its lowest value. The CPV in the upper layer is larger than in the lower layer; the dependence of K₁ on the amplitude of the bottom topography is nonlinear, but it increases for higher values of B₀. For the topography considered here, the value of K₁ (for K₂ fixed) substantially depends on the meridional gradient of zonally averaged bottom topography. The bottom form stress shows a strong nonlinear dependence on the amplitude of the bottom topography for not very high values of B₀, less than about 100 m.

The analytical solution developed here allows us to emphasize the importance of the GTB. For example, it shows that increasing the parameter D leads to an increase of the magnitude of the total eddy flux, which is not immediately apparent (why should a higher bottom friction yield a higher eddy flux?). The relative importance of the eddy fluxes in the lower and upper layers is proportional to Re/D, and since Re is a linear function of D from the GTB, their ratio is inversely proportional to D, and this means that the total eddy flux becomes dominated by the upper-layer eddy flux of QPV.

The solution also allows one to calculate the total zonal transport as a function of external parameters and of diffusivities (or one of Re and D) that should be chosen from observation or numerical simulations. It is rather sensitive to the amplitude of bottom topography, which drastically reduces the transport from unrealistically high values for the flat bottom case to plausible values even for moderately large B₀.

Admittedly, the approach taken here is a simplification and with just two layerwise constant QPV diffusivities the constraint provided by the GTB is much more restrictive than it would be for many layers, leaving alone the limitations of the quasigeostrophic approximation. However, we would like to stress that while the presence of constraints is well recognized, the fact that they are affected by bottom topography is mentioned less frequently. We see the results presented here as posing the question about the implications of these constraints for more realistic, primitive equation models, which would be of great interest to explore.