## 1. Introduction

It is a common assumption in physical oceanography that the magnitude and sign of the zonally integrated meridional transport in the ocean [i.e., the meridional overturning circulation (MOC)] is related to the meridional pressure or density gradient. This assumption originates in the discussion of a two-box model by Stommel (1961), in which the exchange flow between the two boxes is parameterized with the density difference between the boxes. The physical basis of this closure is a hypothetical dynamical balance between the pressure difference induced by the different densities of the boxes and friction in a narrow pipe connecting the two parts of the ocean at depth.

*υ*is implemented in the meridional momentum balance, while momentum advection is assumed to be negligible. The last assumption is reasonable for scales larger than the internal Rossby radius. From this regime, a simple diagnostic relation

*γ*depends on the type for frictional parameterization (we will assume Rayleigh friction for simplicity but other forms are possible). Note that the wind stress forcing in Eq. (1) was ignored. It can be included in all closures discussed in the present study.

This relation for *γ*, which, however, depends on latitude. However, their particular choice of this relation is void of any dynamical fundament. Wright et al. (1998) give dynamical arguments to motivate a modified version of the closure, which leads to a relation very similar to Eq. (1) (see appendix A for details).

It is one purpose of this study to demonstrate that the closure from Eq. (1) is physically inconsistent. Although this point was already discussed by Straub (1996) and Greatbatch and Lu (2003), it was apparently not well received by the scientific community: there are currently several coupled earth system models of intermediate complexity with zonally averaged ocean model components relying on the closure given by Eq. (1) (Claussen et al. 2002). Because of their low computational costs, such models are often used for paleoclimate simulations and long-term climate projections—several of them are included in the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC) (Solomon et al. 2007). Ocean-only versions are used for studies discussing the stability of the thermohaline circulation (e.g., Alexander and Monahan 2009). Furthermore, scalings for the global meridional circulation including the Southern Ocean and its impact on the circulation in zonally bounded basins still rely on Eq. (1) (Gnanadesikan 1999; Levermann and Fürst 2010). We would like to point out that the closure by Wright et al. (1995) is an exception; it does not rely on Eq. (1) as we will discuss in appendix A.

It is evident that the closures by Marotzke et al. (1988), Wright and Stocker (1991), and Wright et al. (1998) have in common that they call for a “downgradient” form of the meridional transport similar to what was assumed by Stommel (1961) for the viscous pipe flow in his two-box model, leading to a local relation between

The models by Stommel (1961) and Stommel and Arons (1960) have different conceptual backgrounds and were developed to focus on different aspects of ocean dynamics. Therefore it cannot be a priori expected that both models are consistent with each other. Evidently, both models had success in describing important phenomena of the ocean dynamics. However, applying the strongly simplified assumptions of the Stommel (1961) model to zonally averaged models of Marotzke et al. (1988), Wright and Stocker (1991), and Wright et al. (1998), Straub’s dilemma cannot be ignored anymore because it reveals dynamical inconsistencies of these models.

The central purpose of the present study, however, is to present and validate an alternative closure for zonally averaged models, which generalizes the concept of Wright et al. (1995). Their closure is based on a meridional integration of the vorticity balance in the interior and in the western boundary layer. In their closure, the need emerges for an integration constant that is difficult to determine; but which sets the size and sign of the meridional transports. We also divide the ocean into an interior and a western boundary current, but instead of averaging the vorticity equation over these regions we work with the momentum and buoyancy (layer thickness) equations directly. Although this way we avoid the need to determine an integration constant, we still need parameterizations for the interior pressure difference of the boundary layer and the zonal velocity at the interface between these regions. A detailed comparison of two types of circulation models [a two-layer model (LM) like the one by Stommel and Arons (1960) and a general primitive equation model (PEM) with many levels] with their zonally averaged counterparts demonstrates the feasibility of the closure.

## 2. Straub’s dilemma

### a. Dilemma in a simple layered model

*H*denotes the mean thickness of the lower layer of a two-layer ocean and its perturbation

*h*, with a density difference

*δρ*, between the two layers represented by the reduced gravity

*g*′ =

*gδρ/ρ*

_{0}. The velocities

*u*and

*υ*are the differences between the upper- and lower-layer velocities. A prescribed deepwater source is denoted by

*Q*and the interior upwelling is parameterized by the term −

*λh*in the thickness balance of Eq. (4). The momentum balance in Eqs. (2) and (3) is taken to be linear, and friction induced by subgrid-scale processes is represented by Rayleigh friction with coefficient

*r.*For a detailed derivation of the model equations see, for example, Gill (1982), their section 6.2, or Greatbatch and Lu (2003). There are two equations derived from Eqs. (2)–(4) that we present for later use. The momentum balance yields the vorticity balance

*u*and

*υ*turns this into the familiar form of the quasi-geostrophic vorticity equation

*R*=

*c*/

*|f|*is the baroclinic Rossby radius and

The potential vorticity equation reveals the existence of a western boundary layer of the Stommel type with the familiar width *δ _{W}* =

*r*/

*β*, resulting from the dominant balance between advection of planetary vorticity and the torque by the Rayleigh friction:

*βυ*= −

*r*∂

*or*

_{x}υ*β*∂

*and the upwelling term −*

_{x}h*λh*/

*R*

^{2}dominate. Approaching the northern (or southern) rim of the domain, with

*h*→ const,

*υ*→ 0, and

*Q*≡ 0, upwelling and the Rayleigh friction term

*δh*/

*h*is the relative variation of

*h.*These considerations can be used to construct an approximate analytical solution of Eq. (7). Here, however, a numerical model will be used.

For the experiments with LM we have used the parameter values *r* = 2 × 10^{−6} s^{−1}, *λ* = 1 × 10^{−9} s^{−1}, *g*′ *=* 0.02 m s^{−2}, *β* = 2.3 × 10^{−11} m^{−1} s^{−1}, and *H* = 400 m, which yields *c* = 2.8 m s^{−1}, *R =* 30 km (at *y =* 4000 km), *δ _{W} =* 100 km, and

*δ*

_{NS}

*=*300 km. For the latter,

*δh*/

*h*~ 0.1 is used. The system is integrated on an equatorial

*β*plane and the horizontal resolution is 20 km in the zonal and meridional directions. The zonal and meridional extent of the model domain is 2500 and 10 000 km, respectively. To demonstrate the influence of the transport and pressure field on the location of the deepwater source, we choose three different locations for

*Q.*The results of the three experiments are shown in Fig. 1. The location of the deepwater source is at the northwestern edge of the model domain for Fig. 1a, at the equator at the western boundary for Fig. 1b, and at the southwestern edge of the model domain for Fig. 1c. The lateral scale of

*Q*is

*δ*in both directions. In each experiment two dynamical different regimes exist: a narrow western boundary layer with a strong meridional flow and a weak interior flow whose meridional component is always poleward. The widths of the boundary layers at the western, northern, and southern rims confirm the above considerations.

_{W}*r*/(

*βB*) ≪ 1, where

*B*is the zonal width of the basin. We also need to know that

*h*is related to

*Q*only in an integral sense—that is,

*λh*, which is almost identical for the three cases (i.e., almost uniform and of similar magnitude). The major differences occur, therefore, only in the boundary current of the individual experiments, which has to balance the interior flow and the upwelling in the interior and the different inflows of the localized deepwater source

*Q*.

It is clear that the location and strength of the deepwater source *Q* in the thickness equation determines the total transport in the lower layer—that is, when *Q* is located at the northwestern corner, the total meridional transport is southward in both hemispheres of the domain (Fig. 1g, solid line) and it is anywhere northward for a deepwater source *Q* located at the southwestern corner of the domain (Fig. 1i), while the total transport is poleward in both hemispheres for an equatorial source (Fig. 1h). The source *Q* drives a total transport of about 5 Sv (1 Sv ≡ 10^{6} m^{3} s^{−1}) in the vicinity of the source in each case, which linearly reduces because of the interior upwelling into the upper layer with increasing distance from the source. Although the western boundary layer is much smaller than the total width of the basin, the transport in the western boundary layer is of similar magnitude to the total transport. It also has the same direction as the total transport, except for the region *y < −*2500, *|y|* > 2500, and *y >* 2500 km for the experiment with northern, equatorial, and southern sources, respectively, where it opposes the total transport.

Since the zonal integral of *h* is dominated by the interior, the zonally integrated *h* becomes independent of the location of the deepwater source *Q.* Consequently, *h* and in particular the meridional gradient of *h* become independent of the location of *Q*, and thus the sign and strength of the meridional transport is neither related to the zonal average of *h* nor its meridional gradient. This statement is in contrast to the box model by Stommel (1961) where the flow between the two boxes is parameterized by the meridional density difference between the boxes, and also in contrast to the closures by Marotzke et al. (1988), Wright and Stocker (1991), and Wright et al. (1998), which all depend on Eq. (1). To summarize, the parameterization in the box model by Stommel (1961) and the closures based on Eq. (1) are not consistent with the dynamics of the model by Stommel and Arons (1960). This inconsistency (Straub’s dilemma) was first noted by Straub (1996).

### b. The dilemma in primitive equations

The independence of the meridional gradient of the zonally averaged thickness from the meridional transport is not specific to layered models but is also found in a primitive equation model (Viebahn and Eden 2010; https://wiki.zmaw.de/ifm/TO/cpflame), referred to as PEM, with a similar configuration as the LM. In PEM we have neglected momentum advection (as before), and, for simplicity, the only tracer is temperature. The model domain is identical to the layered model, but there are 20 vertical levels of 50-m thickness, such that the domain is 1000 m deep. PEM is forced by relaxation of temperature in the uppermost grid box toward a target temperature, which is zonally and meridionally uniform except for a small region of meridional width *r*/*β* (equivalent to the western boundary-layer width) at the northern or equatorial region with a 3-K-smaller target temperature. This way, a northern or equatorial deepwater formation region is introduced as in the layered model. A case with a southern source is just a mirror of the one with a northern source and therefore not further discussed. The time scale of relaxation at the surface is 20 days. Convection in the case of unstable stratification is parameterized by setting the vertical diffusivity to very large values. As in LM, there is no wind forcing—that is, we focus here on the thermohaline circulation. Friction is identical to the LM, except that we introduce in addition lateral and vertical friction with viscosities of 3.2 × 10^{4} m^{2} s^{−1} and 1 × 10^{−3} m^{2} s^{−1}, respectively, since otherwise unphysical oscillations on a short time scale develop. We use the Quicker advection scheme (Leonard 1979) for tracers and vertical diffusivity of 1 × 10^{−4} m^{2} s^{−1} in addition.

The steady solution of PEM, shown in Fig. 2, indeed has much resemblance to LM. In the experiment with a northern source, there is a deep temperature minimum at the equator, and isopycnals below about 500-m depth are symmetric with respect to the equator, bending toward the bottom and toward the poles (Fig. 2b). A similar “hill,” symmetric around the equator, can be seen in the experiment with the equatorial source (Fig. 2d), although it is located more to the surface than at depth. Figures 2a,c also show the meridional transports for both experiments with PEM by the meridional streamfunction *ψ* with

(a) The meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) the zonally averaged temperature (contour interval is 0.5°C) of PEM with a deepwater formation region at the northern boundary. (c) The meridional overturning streamfunction (contour interval is 2 Sv) and (d) zonally averaged temperature (contour interval is 0.5°C) for an equatorial deepwater formation region. The results are time averages over the last 100 yr of a 200-yr integration.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a) The meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) the zonally averaged temperature (contour interval is 0.5°C) of PEM with a deepwater formation region at the northern boundary. (c) The meridional overturning streamfunction (contour interval is 2 Sv) and (d) zonally averaged temperature (contour interval is 0.5°C) for an equatorial deepwater formation region. The results are time averages over the last 100 yr of a 200-yr integration.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a) The meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) the zonally averaged temperature (contour interval is 0.5°C) of PEM with a deepwater formation region at the northern boundary. (c) The meridional overturning streamfunction (contour interval is 2 Sv) and (d) zonally averaged temperature (contour interval is 0.5°C) for an equatorial deepwater formation region. The results are time averages over the last 100 yr of a 200-yr integration.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

As for LM, the meridional gradient of the zonally averaged density (i.e., temperature) or pressure (not shown) is similar in both experiments with PEM (cf. Fig. 1d with Fig. 2b and Fig. 1e with Fig. 2d for the case with northern and equatorial deep water sources, respectively) and is of opposite sign in both hemispheres. Their depth dependence differs. The meridional transport, on the other hand, does not show any direct dependency on *p*, which is in turn governed by the frictionless Sverdrup relation in the interior.

Greatbatch and Lu (2003) increased the vertical mixing to unrealistic values in LM and found that the zonally averaged thickness

## 3. A consistent closure

We regard the closure for a zonally averaged model proposed by Wright et al. (1995) as dynamically consistent. It is based on a meridionally integrated, zonally averaged balance of vorticity (see appendix B). However, integrating the vorticity balance requires integration constants to be specified in each hemisphere. Wright et al. (1995) locate these constants at the northern and southern boundaries of the domain and relate them to the interior flow at the respective boundaries. The result is a nonlocal relation between the zonally averaged pressure

### a. Closure for the layer model

*b*or

*i*, respectively. The total meridional transport can be obtained by

*B*,

*B*, and

_{b}*B*denote the total basin width from the western to the eastern boundary, the width of the western boundary layer, and the width of the interior, respectively. Note that the boundary-layer width might be defined as a multiple of

_{i}*r*/

*β.*Obviously,

*B*≪

_{b}*B*

_{i}.#### 1) Motivation and evaluation of the closure

*x*to the offshore edge of the western boundary layer at

_{W}*x*, yields

_{δ}*h*=

_{b}*h*(

*x*=

*x*) −

_{δ}*h*(

*x*=

*x*)

_{W}*= h*In the thickness balance the zonal velocity

_{δ}− h_{W}.*u*=

_{δ}*u*(

*x*=

*x*) at the interface between the interior and the boundary layer appears. Furthermore, it was assumed in Eq. (14) that

_{δ}*u*(

*x*=

*x*) = 0 and the source

_{W}*Q*was located entirely in the western boundary layer. Likewise, the respectively averaged equations for the interior regime, extending from

*x*to

_{δ}*x*, are

_{E}*h*=

_{i}*h*(

*x*=

*x*) −

_{E}*h*(

*x*=

*x*)

_{δ}*= h*−

_{E}*h*Here

_{δ}.*u*(

*x*=

*x*) = 0 is used. To allow for the northern and southern boundary layers, described in section 3a, the friction terms have been retained though they are negligible in the actual interior. The pressure differences over the respective domains, Δ

_{E}*h*and Δ

_{b}*h*, as well as the zonal velocity

_{i}*u*, have to be parameterized. The resulting model will be referred to as the zonally averaged layer model (ZALM).

_{δ}*u*must be close to

_{δ}^{1}

*u*within the western boundary layer would yield

*γ*

_{1}= 2. However,

*u*is not increasing linearly over the western boundary layer (not shown) and we found that the best fit is obtained for

*γ*

_{1}= 1.7 [see also Figs. 3c,f where lhs and rhs of Eq. (18) are shown]. Next we demand that the thickness balance for the interior regime yields the averaged form of the Sverdrup balance from Eq. (8), assuming steady state and vanishing friction. To insert

*βυ*contribution, frictional, and tendency terms remain—that is, to an interior Sverdrup balance analogous to Eq. (8) [assuming that

*βυ*dominates

*h*, the integrated form of this relation reads

_{i}*y*= 0 follows from the steady state zonal balance at the equator:

(a) Thickness difference (m) over the western boundary layer Δ*h _{b}* (solid) and its parameterization from Eq. (23) (dashed) as a function of

*y*for the experiment with the northern source. (b) As in Δ

*h*and its parameterization from Eq. (21). (c) Zonal velocity at the offshore edge of the boundary layer

_{i}*u*(solid) and its parameterization from Eq. (18) (dashed). All tuning coefficients are set to 1. (d)–(f) As in (top), but for experiments with the equatorial source.

_{δ}Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a) Thickness difference (m) over the western boundary layer Δ*h _{b}* (solid) and its parameterization from Eq. (23) (dashed) as a function of

*y*for the experiment with the northern source. (b) As in Δ

*h*and its parameterization from Eq. (21). (c) Zonal velocity at the offshore edge of the boundary layer

_{i}*u*(solid) and its parameterization from Eq. (18) (dashed). All tuning coefficients are set to 1. (d)–(f) As in (top), but for experiments with the equatorial source.

_{δ}Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a) Thickness difference (m) over the western boundary layer Δ*h _{b}* (solid) and its parameterization from Eq. (23) (dashed) as a function of

*y*for the experiment with the northern source. (b) As in Δ

*h*and its parameterization from Eq. (21). (c) Zonal velocity at the offshore edge of the boundary layer

_{i}*u*(solid) and its parameterization from Eq. (18) (dashed). All tuning coefficients are set to 1. (d)–(f) As in (top), but for experiments with the equatorial source.

_{δ}Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

*h*−

_{b}= h_{δ}*h*across the boundary layer. We have experimented with a variety of closures for Δ

_{W}*h*analogous to Eq. (20)—that is, motivated by the potential vorticity budget in the western boundary layer. However, many possible forms for the closure, which often yield an excellent fit to the respective variable in the zonally resolved model, turned out to lead to unstable numerical integrations. The simple ansatz

_{b}*γ*

_{2}of order one, on the other hand, yields stable integrations in all cases, which we have considered, and is also a reasonable fit to Δ

*h*from the zonally resolved model as discussed next [lhs and rhs of Eq. (23) are shown in Figs. 3a,d]. Further, the results of the integrations with the resulting zonally averaged model compare well with the zonally resolved counterparts, as discussed below, giving confidence to the parameterization from Eq. (23).

_{b}Figure 3 shows the western boundary thickness difference Δ*h _{b}* and its parameterization

*h*except for the southernmost (northernmost) part of the domain in the experiment with a northern (southern) source, where Δ

_{b}*h*becomes negative, while

_{b}*h*We also note that the quality of the parameterization depends on the exact definition of the width of the western boundary layer. Here, we have used

_{b}.*B*=

_{b}*r/β*with values for

*r*and

*β*as in the numerical experiments. The middle panels of Fig. 3 display Δ

*h*from the zonally resolved model and its parameterization from Eq. (21) which in fact agree very well. Figure 3 also shows the outflow from the western boundary region,

_{i}*u*, from the zonally resolved model together with its parameterization

_{δ}*u*well with respect to the meridional structure and sign, while the magnitude is underestimated, which might be resolved by tuning the parameter

_{δ}*γ*

_{1}to values greater than one.

#### 2) Performance of the zonally averaged layer model

The complete ZALM consists of Eqs. (12)–(14) for the boundary layer and Eqs. (15)–(17) for the interior domain together with the parameterizations expressed in Eqs. (18), (21), and (23). ZALM was programmed in FORTRAN 90 and the source code together with a detailed documentation of all numerical details can be downloaded from https://wiki.zmaw.de/ifm/TO/zom. The steady state of a numerical integration of ZALM is shown as dashed lines in Fig. 4 and compared with the correspondingly averaged quantities diagnosed from LM. Note that the configuration of ZALM is identical to LM in all respects (except for the zonal extent and the closure). Transports and thickness height are reasonably well reproduced by ZALM for the northern and equatorial sinking case using *γ*_{1} = *γ*_{2} = 1 and a boundary-layer width of *B _{b}* =

*r*/

*β.*However, by changing the tuning parameters to

*γ*

_{1}= 1.7,

*γ*

_{2}= 1.2, and

*B*= 2

_{b}*r*/

*β*, the broad central hill in

(a),(c) Comparison of results of LM and ZALM for the experiment with the northern source and (b),(d) for the experiment with the equatorial source. Shown is (a),(b) the zonally averaged layer thickness *γ*_{1} = *γ*_{2} = 1 and *B _{b}* =

*r*/

*β*, and the dotted lines the results of ZALM with

*γ*

_{1}= 1.7,

*γ*

_{2}= 1.2, and

*B*=

_{b}*2r*/

*β.*In (c) and (d) the thick lines are total transports

*γ*

_{1}=

*γ*

_{2}= 1 and

*B*=

_{b}*r*/

*β*are shown; the results with the tuned parameter set look similar.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a),(c) Comparison of results of LM and ZALM for the experiment with the northern source and (b),(d) for the experiment with the equatorial source. Shown is (a),(b) the zonally averaged layer thickness *γ*_{1} = *γ*_{2} = 1 and *B _{b}* =

*r*/

*β*, and the dotted lines the results of ZALM with

*γ*

_{1}= 1.7,

*γ*

_{2}= 1.2, and

*B*=

_{b}*2r*/

*β.*In (c) and (d) the thick lines are total transports

*γ*

_{1}=

*γ*

_{2}= 1 and

*B*=

_{b}*r*/

*β*are shown; the results with the tuned parameter set look similar.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a),(c) Comparison of results of LM and ZALM for the experiment with the northern source and (b),(d) for the experiment with the equatorial source. Shown is (a),(b) the zonally averaged layer thickness *γ*_{1} = *γ*_{2} = 1 and *B _{b}* =

*r*/

*β*, and the dotted lines the results of ZALM with

*γ*

_{1}= 1.7,

*γ*

_{2}= 1.2, and

*B*=

_{b}*2r*/

*β.*In (c) and (d) the thick lines are total transports

*γ*

_{1}=

*γ*

_{2}= 1 and

*B*=

_{b}*r*/

*β*are shown; the results with the tuned parameter set look similar.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

We next take a closer look at the physical processes that establish the circulation in ZALM. Kawase (1987) showed that the establishment of the deepwater circulation involves basin-wide propagating Kelvin and Rossby waves: a thickness anomaly generated at the northern boundary of the basin propagates along the western boundary southward in the form of a Kelvin wave; at the equator, the Kelvin wave turns into an equatorial Kelvin wave and crosses the basin toward the east where it is again reflected and propagates at the eastern boundary north- and southward; westward propagating long Rossby waves, emanating from the eastern boundary, then transfer the signal into the interior of the ocean. These processes are of course realized in LM and a very similar adjustment process can be found in ZALM.

*u*term in the thickness balances.

_{δ}*τ*=

_{α}*B*/(

_{α}*βR*

^{2}) is the time that a baroclinic Rossby wave needs to cross the respective region

*α = i*,

*b.*The corresponding time scale for the interior is roughly 10 yr in the northern part; it decreases toward the equator to several days (7 days at

*y*= 300 km). For the boundary layer, the time scale is considerably smaller (by the factor

*B*/

_{b}*B*≃ 25). The Rossby wave communication between the two regimes is thus represented by an oscillation of the mean layer thicknesses

_{i}To assess the temporal behavior, Fig. 5 shows *B _{b}.* Increasing (decreasing)

*B*from

_{b}*r*/

*β*—the value used in Fig. 5—to larger (smaller) values, the southward propagation speed decreases (increases) slightly. The reason is the increasing (decreasing) importance of

Establishment of the circulation for (top) LM and (bottom) ZALM for an experiment with a northern deep-water source. Here we show

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Establishment of the circulation for (top) LM and (bottom) ZALM for an experiment with a northern deep-water source. Here we show

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Establishment of the circulation for (top) LM and (bottom) ZALM for an experiment with a northern deep-water source. Here we show

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

The signal is then transported from the boundary layer into the interior (see Figs. 5b,d). This is first achieved by the increasing imbalance of the Kelvin wave dynamics: approaching the equator, the geostrophic relation for

### b. Application to primitive equations

*α = b*,

*i*indicating the boundary or interior part, respectively, and

*ϵ*= 1 and

_{b}*ϵ*= −1. The (scaled) pressure

_{i}*K*in case of unstable stratification.

_{α}*p*=

_{b}*p*(

*x*=

*x*) −

_{δ}*p*(

*x*=

*x*) and Δ

_{W}*p*=

_{i}*p*(

*x*=

*x*) −

_{E}*p*(

*x*=

*x*), respectively, and the zonal velocity

_{δ}*u*at the offshore edge of the western boundary need parameterizations. Analogous to the closure in the layered model, we use

_{δ}*p*(

_{i}*y*= 0), is again set by the steady zonal momentum balance at the equator. The model includes a rigid lid surface boundary condition and a diagnostic relation to find the surface pressure as usual in ocean general circulation models. This will be called the zonally averaged primitive equation model (ZAPEM).

Figure 6 shows Δ*p _{b}*, Δ

*p*, and

_{i}*u*diagnosed in PEM and their parameterizations given by Eq. (31). There is a good agreement concerning sign and structure of the variables and their parameterizations. Only in the southernmost part does the parameterization for Δ

_{δ}*p*not show the correct sign, which is similar to what we have seen for LM (see Fig. 3a). It turns out that an important parameter is the boundary-layer width

_{b}*B*, which we have chosen here as

_{b}*B*= 3.7

_{b}*r*/

*β*since this value seems to match best the boundary-layer width in PEM. Note that the boundary layer is broader than expected from the Rayleigh friction term because we also have included harmonic friction in PEM, which leads to a wider boundary layer.

Comparison of parameterized variables with the model result for PEM in the experiment with a northern deep-water source. (top) Here we show (a) Δ*p _{b}*, (b) Δ

*p*, and (c)

_{i}*u*in the zonally resolved model. (bottom) The respective parameterizations: (d)

_{δ}^{2}s

^{−2}and (c),(f) 0.1 cm s

^{−1}.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Comparison of parameterized variables with the model result for PEM in the experiment with a northern deep-water source. (top) Here we show (a) Δ*p _{b}*, (b) Δ

*p*, and (c)

_{i}*u*in the zonally resolved model. (bottom) The respective parameterizations: (d)

_{δ}^{2}s

^{−2}and (c),(f) 0.1 cm s

^{−1}.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Comparison of parameterized variables with the model result for PEM in the experiment with a northern deep-water source. (top) Here we show (a) Δ*p _{b}*, (b) Δ

*p*, and (c)

_{i}*u*in the zonally resolved model. (bottom) The respective parameterizations: (d)

_{δ}^{2}s

^{−2}and (c),(f) 0.1 cm s

^{−1}.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

ZAPEM was programmed in FORTRAN 90 and the source code as well as a documentation of all important numerical details can be downloaded from https://wiki.zmaw.de/ifm/TO/zom. The results of ZAPEM after 200 yr of integration for two different surface boundary conditions are shown in Fig. 7.

(a) Meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) zonally averaged temperature *ψ* and

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a) Meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) zonally averaged temperature *ψ* and

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

(a) Meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) zonally averaged temperature *ψ* and

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

The configuration and relevant parameters of ZAPEM are identical to the zonally resolved model version (PEM) shown in Fig. 2, except for the zonal extent, the closure, and that we have omitted the harmonic zonal friction terms (meridional friction is kept) in ZAPEM. Two different surface boundary conditions are implemented using the two different target surface temperatures, as described in section 2b. Without parameter optimization (i.e., for *γ*_{1} = *γ*_{2} = 1 and *B _{b}* = 3.7

*r*/

*β*) we found already good agreement between ZAPEM and PEM; we therefore made no further attempt of parameter tuning. However, for the polar sinking case (see Figs. 7a,b) the overturning rate in ZAPEM is slightly too strong while the vertical stratification is slightly too weak. For the equatorial sinking (see Figs. 7c,d) the reverse statement holds.

Figure 8 shows results of an integration using Eq. (32) as closure with *γ*_{WS} = 40. While the overturning circulation is similar to ZAPEM and PEM, the structure of the buoyancy field reveals a major disagreement for the case of a northern deepwater source (Fig. 8b). According to Eq. (1), the sign of the meridional buoyancy (pressure) gradient cannot change if the streamfunction consists of one single overturning cell. This clearly contradicts the results of PEM and ZAPEM. Only for the case of the equatorial deepwater source, the buoyancy distribution (Fig. 8d) conforms better with that of ZAPEM and PEM, although the northern and southern boundary layers as observed in ZAPEM and PEM are not reproduced.

Effect of the inconsistent parameterization analogous to Eq. (1): (a) meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) zonally averaged temperature

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Effect of the inconsistent parameterization analogous to Eq. (1): (a) meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) zonally averaged temperature

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Effect of the inconsistent parameterization analogous to Eq. (1): (a) meridional overturning streamfunction *ψ* (contour interval is 0.5 Sv) and (b) zonally averaged temperature

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Figure 9 illustrates the effect of wind forcing and a Southern Ocean part in simulations with ZAPEM and PEM. It is straightforward to include wind forcing and/or zonally periodic boundary conditions in the zonally averaged models ZALM and ZAPEM: the zonally averaged wind stress is used as upper-boundary condition in the vertical stress divergences contained in *K*_{gm}*f*^{2}/*N*^{2} where *K*_{gm} denotes the isopycnal thickness diffusivity according to the Gent and McWilliams (1990) parameterization. We simply take a constant value of *K*_{gm} = 1000 m^{2} s^{−1}.

Model configuration with Southern Ocean included. Shown are the (a),(c) meridional streamfunction *ψ* (Sv) and (b),(d) zonally averaged temperature

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Model configuration with Southern Ocean included. Shown are the (a),(c) meridional streamfunction *ψ* (Sv) and (b),(d) zonally averaged temperature

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

Model configuration with Southern Ocean included. Shown are the (a),(c) meridional streamfunction *ψ* (Sv) and (b),(d) zonally averaged temperature

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-021.1

For the simulations shown in Fig. 9, we have chosen the same model domain as before, but for the southern quarter of the basin we apply zonally periodic boundary conditions to represent the Southern Ocean. Note that the setup is similar to that used in Viebahn and Eden (2010): the wind stress over the Southern Ocean region is zonally constant and sinusoidal in the meridional coordinate with a maximum of 2 × 10^{−4} m^{2} s^{−2} located at the center of the periodic domain. The wind stress in the zonally bounded part of the domain is set to zero. The surface boundary condition for buoyancy is a relaxation toward a target buoyancy restoring function with a linear increase (with a rate of 10^{−9} s^{−2}) in the Southern Ocean region, a constant value from *y* = −2560 to *y* = 2560 km, and a linear decrease of the target buoyancy at the northern part (with the same rate), which generates an equivalent temperature difference of about 20 K between the equator and the polar boundaries. As before, the zonally averaged model (ZAPEM) with our new closure is compared to a simulation with PEM in an identical (but zonally resolved) setup (Fig. 9). ZAPEM again reproduces well PEM, although ZAPEM again slightly underestimates the overturning and overestimates the stratification in comparison with the corresponding PEM experiment. We further note that temperature and salinity and further passive tracers can be added as variables to the zonally averaged model (not shown). Isopycnal mixing is also implemented as an additional mixing term in the tracer balances. It is also straightforward to include variations in the ocean depth.

## 4. Summary and discussion

The box model by Stommel (1961) and the model by Stommel and Arons (1960) both aim to describe the meridional overturning circulation of the ocean. Much of our knowledge about this important aspect of the ocean’s circulation is based on these models. However, Straub (1996) pointed out an inconsistency between the box model and the Stommel and Arons (1960) model, which proves the assumption in Eq. (1) to be inconsistent, which is an inherent assumption in the box model of Stommel (1961) and also in many zonally averaged ocean models (Claussen et al. 2002). We call this inconsistency Straub’s dilemma, representing the fact that it appears not possible to infer the meridional transport from the meridional gradient of the zonally averaged pressure. This is because the zonally averaged pressure is dominated by the interior pressure, which, on the other hand, is governed by frictionless and linear dynamics expressed by the Sverdrup relation of Eq. (8). Since this vorticity balance is driven only by the interior upwelling, it is unrelated to the sign of the meridional flow.

In this study, we present and evaluate a new and consistent closure for zonally averaged models to replace the inconsistent closure given by Eq. (1), illustrated by numerical integrations with a layered model version and a version based on the full primitive equations. Following Wright et al. (1995), the model domain is divided into an interior part—governed by the Sverdrup relation of Eq. (8)—and a boundary layer part, where friction plays an important role in the vorticity balance. However in contrast to Wright et al. (1995), we do not use the vorticity balances of the interior and the boundary layer directly, but instead use the zonally averaged, interior and boundary layer, momentum and thickness (or buoyancy) budgets. The reason for doing so is that using the meridionally integrated vorticity balances, as suggested by Wright et al. (1995), introduces the need to specify an unknown integration constant. We find the choice of this integration constant to be problematic, since it sets the sign of the meridional transport.

Therefore, we use the vorticity balance only to motivate the parameterization of the zonal pressure difference over the interior, which is needed for the zonally averaged interior zonal momentum balance. The zonal pressure difference across the boundary layer, on the other hand, is parameterized by the difference of the zonally averaged pressure in the interior and the pressure averaged over the boundary layer. The advective exchange between the boundary layer and interior is parameterized using the mean zonal velocity in the boundary layer. The standing-eddy fluxes in the nonlinear buoyancy budgets are simply neglected. Both in the layered model and the primitive equation model we find good agreement with respect to the evaluation of the parameterizations and model results in terms of the mean simulation of the transports and the thickness (buoyancy) and its time changes.

We advocate replacing the inconsistent closure of Eq. (1) with the new closure discussed in this study in zonally averaged ocean models. However, we do not imply that the box model by Stommel (1961) is inconsistent as well. On the one hand, the interpretation of the meridional flow in the ocean as driven by the pressure difference between two boxes and controlled by friction in a hypothetical pipe connecting the two boxes by Stommel (1961) is certainly an incorrect oversimplification of the real dynamics. On the other hand, many results and predictions of the box models can be reproduced by models, including the correct dynamics. This agreement might give some confidence in the box model, although we know that its dynamics are incomplete and only a very rough analog to the real dynamics. We hope that the new zonally averaged model presented here can contribute to further confirm and extend knowledge from the box model about the meridional flow in the ocean.

## Acknowledgments

This work was supported Grant by BMBF-SOPRAN FKZ 3F0611A.

## APPENDIX A

### Some Frequently Used Inconsistent Closures

*γ*= 1/

*r*. Here

*g*′ is the reduced gravity,

*h*the layer thickness,

*υ*the meridional velocity component, and

*r*to connect to the model by Stommel and Arons (1960), while Marotzke et al. (1988) originally used vertical diffusion of momentum. However, the specific choice of the friction does not change the fundamental relation in Eq. (1).

*h*−

_{E}*h*over the basin width

_{W}*B*needs a parameterization. They choose

*ϕ*denotes latitude and γ

_{WS}a constant of order 1. The zonal pressure difference is thus expressed in terms of the local meridional pressure gradient, which is also of the form in Eq. (1), assuming that the meridional flow is in geostrophic balance, with a parameter

*γ*proportional to cos

*ϕ.*This setting is supported by numerical experiments with a three-dimensional (but highly simplified) circulation model, but as argued by Greatbatch and Lu (2003), the support is due to the highly diffusive nature of the ocean model. Note that the closure is not based on any dynamical concepts.

*h*The zonal momentum balance is entirely abandoned. In fact, the zonally averaged meridional momentum balance is written as

_{E}− h_{W}.*f*, the zonally averaged zonal velocity

*B*of the ocean again into a western frictional boundary layer part of width

*B*and an interior part of width

_{b}*B*≫

_{i}= B − B_{b}*B*. They write

_{b}*g*) denotes the geostrophic component of the velocity. The interior flow is largely geostrophic and, thus, Wright et al. (1998) assumed the product

*u*vanishes on the continental side of the layer and should be largely governed by the geostrophic balance on the offshore edge of the western boundary layer. The interior geostrophic component continues only moderately changed into the boundary layer and to the actual boundary. Hence, the magnitudes of

*γ*=

*B*/(

_{b}*rB*). Note that this closure is entirely of geometric nature: it uses the observed structure of a basin-wide circulation with a narrow western boundary current but not any further dynamics.

## APPENDIX B

### The Consistent Closure by Wright et al.

Wright et al. (1995) propose a dynamically consistent closure by splitting the ocean basin into a western boundary layer and an interior and considered the vorticity budgets averaged separately over both regions. Assuming a frictionless interior and using specific parameterizations for friction in the western boundary layer, they derive a nonlocal relation between the meridional transport and zonally averaged pressure. Because the original closure of Wright et al. (1995) needs the specification of integration constants that are difficult to determine, we have presented in section 3a generalization of the concept by Wright et al. (1995), which does not need the specification of integration constants.

*W*,

*E*, and

*δ*denote that the values are taken at the western or eastern boundary or at the interface between interior ocean and boundary layer, respectively. The overbars denote zonal averages over the interior with additional index

*i*or boundary layer with index

*b,*and

*B*and

_{b}*B*denote the width of the boundary layer and the interior, respectively. Only a few approximations now lead to the closure by Wright et al. (1995). First, the friction term in Eq. (B2) will be neglected. Because of the kinematic boundary condition at the eastern boundary,

_{i}*u*= 0, it follows from Eq. (3) that ∂

_{E}*= 0 or*

_{y}h_{E}*h*= const. Second, the thickness perturbation

_{E}*h*along the western boundary in Eq. (B2) is eliminated by the meridional velocity

_{W}*υ*using the steady meridional momentum balance at

_{W}*x*in the form

_{W}*υ*is parameterized by

_{W}*u*= 0 was assumed. Next the friction coefficient

_{W}*r*in Eq. (B3) is replaced by

*βB*using

_{b}*B*=

_{b}*r*/

*β*as the boundary-layer width according to Stommel (1948). The meridional integral of Eq. (B3) with starting point at

*y*

_{0}can be used to eliminate

*h*from Eq. (B1) to end up with

_{W}*y*

_{0}is arbitrary. The meridional velocities

*h*Wright et al. (1995) propose the closure

_{δ}.*h*(

_{W}*y*

_{0}). Note that the frictionless interior balance leads to

*f*and integrated from

*y*

_{0}to

*y*(in the same hemisphere to avoid the singularity at

*y*= 0) to give

Figures 1d–f show indeed that

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^{1}

In this paper we denote by the