## Abstract

Much of the knowledge about ocean circulation stems from rather simple analytical models. The behavior of the meridional overturning and, more specifically, the thermohaline-induced part of the global ocean circulation, under changing surface conditions, is often judged by the bifurcation structure of box models with very low (low-order) resolution. The present study proposes a new low-order model of the thermohaline-driven circulation, which is constructed by severe truncation of a spectral decomposition of the two-dimensional equations of motion (vorticity and heat/salt balances). The physical ingredients of the new model are superior to box models because it has a continuous lateral and vertical representation of the fields and finite diffusion coefficients for heat and salt. The building of the spectral model involves much mathematical labor because the structure functions must be constructed in accordance with the boundary conditions for conservation of momentum, mass, heat, and salt. Furthermore, a number of complicated coupling coefficients must be evaluated. Like the box models, the spectral model is a dynamical system with mathematical complexity, but in most of the versions that we analyze, it still can be handled by standard analytical procedures. These versions are the spectral counterparts of the classical box models of Stommel, Rooth, and Welander, adjusted to the Atlantic overturning. A detailed comparison of the model types reveals a similar bifurcation pattern of box and spectral low-order configurations under symmetric and asymmetric forcing conditions and slight perturbations thereof (we use mixed boundary conditions for heat and salt and the surface freshwater flux as a continuation parameter). Comparison of the spectral low-order models with models towards a higher resolved range, namely, the two-dimensional overturning models for the meridional plane, reveals a close resemblance as well. A major difference of box and spectral models is the appearance of parameter windows in the latter, where only unstable steady states exist. The spectral models then show limit cycles, as well as chaotic trajectories with time scales of thousands of years.

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## Notes

*a*and*H*are the Earth’s radius and the ocean depth, respectively.*K*_{ v },*K*_{ h }are the vertical and horizontal diffusivities.Density is here a dimensionless perturbation quantity, i.e., \(\rho = (\varrho -\varrho_0)/\varrho_0\), where \(\varrho\) is the in situ density. Pressure is dynamical, i.e., pressure divided by \(\varrho_0\), thus measured in m

^{2}*s*^{ − 2}. The temperature \(\tilde T\) and salinity \(\tilde S\) are deviations from constant reference values.Derivatives of functions of a single independent variable, either

*μ*or*ζ*, will be denoted by a prime, e.g.,*d**P*_{ n }(*μ*)/*dμ*=*P*′_{ n }and*d**g*_{ n }(*ζ*)/*dζ*=*g*′_{ n }. Derivatives with respect to the time*τ*will be denoted as \(\partial/\partial\tau\).The continuation is performed with a MATLAB code bifurk.m, written by Christoph Völker.

Complex models are questionable, the work you do is real, the outcome imaginary (attributed to Henry Stommel).

This looks odd because

*K*seems to depend only on geometric parameters. Note, however, the scaling, given in Table 1: tendency, advection, and surface fluxes are scaled as \(\partial/\partial \tau, q, \delta \sim K_h^{-1}\). Hence,*K*/*C*~*K*_{ h }.

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## Acknowledgements

We appreciate numerous discussions with Leo Maas and Christoph Völker on simple ocean circulation systems and bifurcation theory. Sergey Danilov helped to improve the paper. Martin Losch has kindly provided the data and ocean basin masks for Fig. 2. The use of the MATLAB programs bifurk.m by Christoph Völker is greatly acknowledged.

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Responsible Editor: Richard J. Greatbatch

## Appendix

### Appendix

### 1.1 A Parameters

### 1.2 B Legendre polynomials and other structure functions

The meridional eigenvalue problem Eq. 14 has Legendre polynomials as solutions. We use normalized polynomials *P*
_{
n
}(*μ*) with eigenvalues \(\upsilon_n= \sqrt{n(n+1)}\). The first three are \(P_0=\sqrt{{1}/{2}}, P_1= \sqrt{{3}/{2}} \mu, \text{and } P_2= \sqrt{{5}/{8}} (3\mu^2-1)\). The orthogonality conditions are given by

Notice that all *P*
_{
n
}, *n* > 0 integrate to zero.

The vertical structure of the diffusively forced solutions (*n* = 1,2, ⋯), described in Section 4, is given by

The eigenvalue equations for the function used in the expansion of the deviation fields are \(G''_q + \gamma_q^2 G_q=\) and correspondingly for *H*
_{
q
}, *F*
_{
q
}, but the boundary conditions differ: *κG*′_{
q
} + *θG*
_{
q
} = 0 at *ζ* = 0, *G*′_{
q
} = 0 at *ζ* = − 1, *H*′_{
q
} = 0 at *ζ* = 0, − 1, and *F*
_{
q
} = 0 at *ζ* = 0, − 1. The expansion of the deviation fields is performed using the sinusoidal functions (*q* = 0,1,2, ⋯, without normalization)

where *G*
_{
q
}(*ζ*) is taken with *θ* ≫ *κ*.

### 1.3 C Temperature equations and coefficients

In addition to the haline variables *L*, *M*, *N*, governed by Eqs. 32 to 34, the eight-equations spectral model has active temperature variables *X* = 8 *T*
_{00}/3, *Y* = 8 *T*
_{10}/3, *Z* = 8 *T*
_{20}/3, governed by

The *U* and *V* Eqs. 30 and 31 get *X*, *Y* terms from the temperature gradient in the vorticity balance; hence,

The coupling coefficients are integrals over triple products of Legendre polynomials (or derivatives) and triple products of vertical structure function, e.g.,

For *κ* = 0.25, we have *ξ*
_{1} = 2.8284, *ξ*
_{2} = 4.8990, and the following coefficients

result.

### 1.4 D Box THC models

For comparison with the spectral THC models, this section presents a brief survey on the governing equations and the bifurcations of the most prominent box THC models. The emphasis is on a consistent treatment of lateral diffusion and its influence on the domain of coexisting steady states. Only THC box models with a pole-to-pole extent are considered. For simplicity, all have an inactive temperature, i.e., the temperature differences between the boxes is a prescribed thermal forcing. We consider the models of Stommel (1961), Rooth (1982), and Welander (1986). They have a lateral diffusion with a scaled coefficient^{Footnote 6}
\(K= 2(1-\mu_r^2) /(D_1+D_2)\) (for Stommel’s model) where *μ*
_{
r
} is the bounding latitude between the respective boxes with lateral extent *D*
_{1} and *D*
_{2}.

Stommel’s and Welander’s models (and partly also Rooth’s model) are governed by a unique function of salinity contrast *x*,

which is easily extracted from the governing Eqs. 53 or 59. The form of *f*(*x*) determines the time development of the respective box variables, the zeros determine the steady states, and the derivative determines their stability. Depending on *ε* and *d*, the function *f*(*x*) has three roots, given by

with *ε*
_{±} = *ε*±*k*. Only real values *a*
_{1,2} ≤ *ε* and *a*
_{3} ≥ *ε* are admissible. There is always one solution, and at most, there is a threefold state. Bifurcations deriving from *f*(*x*) are of the saddle-node type. The generic polynomial form behind the modulus function *f*(*x*) is given by the cubic function

which would arise in box models if the factor *C* in the relation Eq. 52 (see below) between the advection *q* and the density difference *ρ*
_{2} − *ρ*
_{1} depended on the density difference, *C*~|*ρ*
_{2} − *ρ*
_{1}|. This type of advection occurs in the spectral models (see Section 4.1) (Fig. 14).

To get the stability, we have calculated the Jacobians of the respective systems. Some typical bifurcations of the above box models are displayed in Figs. 15, 16, and 17. These are used to compare with the respective spectral models.

*Stommel’s model* is depicted in the top panel of Fig. 14 and given by

For the “global” version of the Stommel model, we have *D*_{1} = *D*_{2} = 1 and, hence, *K* = 1. A hydraulic relation *q* = *C* (*ρ*_{2} − *ρ*_{1}) with a constant *C* between *q* and the density differences between the boxes is motivated in the original Stommel model. The transport is assumed to occur in connecting pipes, and the density difference between the boxes causes a pressure drop that accelerates the fluid in the pipes against friction. Hence, *C* must depend on the friction coefficients of the system. In fact, this is exactly what Eq. 8 suggests: for simplicity, we abandon the tendency term (because \(\Pr \gg 1\)) and implement the box mean values of thermal and haline torques by finite differences to get

with a positive *C*. For \({\rm Ra}=\text{2,600}\), we get a *C* of order 10. As *S*_{1} + *S*_{2} = const, the model can be expressed in terms of the (pole to pole in this global version) differences \(\Delta S= S_1-S_2, \Delta T_{\text{pp}}= T_1-T_2\) of the salinities and temperatures. The model is then governed by

For Stommel’s model, we find correspondence with *f*(*x*) taking *d* = *δ*/*C*, *k* = *K*/*C*, and \(\epsilon=\Delta T_{\text{pp}}\). The variable *x* is identified with Δ*S*, and the strength of advection *q*_{
i
} = *C*(*ε* − *a*_{
i
}) is entirely determined by the roots *a*_{
i
}. Hence, there are at most three values of *q* and Δ*S* for fixed *δ*.

*Bifurcation of Stommel’s model* Solutions with \(\Delta T_{\text{pp}} > \Delta S\) are mainly driven by the thermal contrast between the boxes: *q* is positive, and we have the thermal branch (denoted by TH) with sinking in the northern box 2. For negative *δ*, where haline forcing supports the thermal forcing, TH has the strongest size and is the only state. The TH branch continues to exist for positive \(\delta\le C(\Delta T_{\text{pp}} + K/C)^2/4\). The solution with the largest root *a*_{
i
} has the highest salinity difference and a negative *q*: it is the haline-dominated steady state with reverse circulation (denoted by SA). For negative *δ*, it does not occur, and for large positive *δ*, it is the only steady state and has a relatively weak current. It exists for positive \(\delta>K\Delta T_{\text{pp}}\) values with equatorial downwelling and polar upwelling: the current of this SA state is driven predominantly by haline contrast (we have Δ*S* > 0, *q* < 0). Here, the haline forcing is strong enough to overcome the thermal forcing. At intermediate positive values of *δ*, when \((K/C)\Delta T_{\text{pp}} < \delta/C < (\Delta T_{\text{pp}} + K/C)^2/4\), we find three steady states, the larger one with positive *q* is thermal (it is the continuation of TH from the negative *δ* range to positive *δ*) and the one with negative *q* is haline (SA), the intermediate state is thermal as well but unstable. The window of multiple states shrinks to zero for \(K/C = \Delta T_{\text{pp}}\).

*Rooth’s model* is a THC model with no mass conservation between the neighboring boxes (see middle panel of Fig. 14). Mass is conserved only in an overall (global) loop. The model is governed by

for positive *q*. The reversed overturning with *q* < 0 has different equations,

The circulation is assumed to be driven by density contrast between the polar boxes, i.e., \(q =C \left[ T_1 -\right.\)\(\left. T_3 - (S_1 - S_3) \right]\). From the point of view of a forcing pattern by the second Legendre function, the natural box boundaries of the three-box system are at \(\mu= \mu_r=\pm 1/\sqrt{3}\) and, hence, \(D_1=D_3 =1 - 1/\sqrt{3}=D, D_2=2/\sqrt{3}=2(1-D)\), and \(K= 4/(3+ \sqrt{3})=0.85\). This geometric setting also applies to the Welander model introduced below. The total amount of salt 2 *S*_{0} is conserved, which replaces the dynamical balance of box 2 by a simple invariance statement,

The Rooth model is often considered diffusionless, i.e., *K*/*C* = 0. Then, for *q* > 0, the steady box 1 equation in Eq. 54 coincides with the stable and the unstable thermal branches of the Stommel model. Likewise, for *q* < 0, there is correspondence with the haline branch. Without diffusion, the model is approaching a singularity for *q*→0 unless the point coincides with zero forcing *F* = *δ* = 0. The diffusive state of the thermal branches follows from both balances in Eq. 54, which yields a cubic equation

while the corresponding equation

for the haline branch *q* < 0 follows from Eq. 55. They match at *q* = 0. As in the global Stommel model, the temperature difference \(\Delta T_{\text{pp}}=T_1-T_3\) is between the polar boxes. We find the salinities *S*_{
i
}, *i* = 1,2,3, from the steady balances Eq. 54 or Eq. 55 and the salt conservation Eq. 56, which are linear equations with given *q*.

*Bifurcation of Rooth’s model* As mentioned above, the asymmetric case is a replica of the Stommel model. The steady states for symmetric forcing (\(\delta=0, \Delta T_{\text{pp}} =0, F \ne 0\)) are displayed in Fig. 16. Most of the features shown in these solutions can be readily understood from the above cubic Eqs. 57 and 58. The symmetric case has a solution *q* = 0 for all *F* (only for \(K\ne 0\)) and, for positive *F*, a threefold solution with the additional stable branches \(q=-K + \sqrt{CF}\) for *q* > 0 and \(q=K - \sqrt{CF}\) for *q* < 0, while the zero branch becomes unstable. The two parabolas meet on the axis *q* = 0 at *F* = *K*^{2}/*C*, which defines the lower limit multiple steady states. Note that this threefold state ranges to infinite *F*. This contrasts the Rooth model with the Welander model and the spectral THC models.

Figure 16 (right panels) includes the results for slightly perturbing the forcing conditions. While the asymmetric case is only moderately perturbed (not shown), we find, as expected, a breaking of the pitchfork bifurcation, occurring in the symmetric solution, with a nonzero asymmetry *δ*. Of course, this breaking can as well be achieved by a nonzero \(\Delta T_{\text{pp}}\). In passing, we note that the Hopf bifurcation found by Titz et al. (2002) in the highly asymmetric forcing case, for \(F=0, \delta>C \Delta T_{p}^2/2S_0\) with zero lateral diffusion, vanishes above a critical lateral diffusivity.

*Welander’s model* has a deep-water route, which passes through the equatorial box (see bottom panel of Fig. 14), and hence, we have box interfaces of the Stommel type. The model equations are

with \(q_i = C\left[ T_2-T_i - (S_2-S_i)\right],i=1,3\).

In our application of the Welander box system, the equatorial to polar difference \(\Delta T_{\text{ep}} = T_2 - T_1 = T_2 - T_3\) is assumed equal for both hemispheres (implying \(\Delta T_{\text{pp}}=T_1-T_3 =0\)). The steady states follow from the linear system

together with Eq. 56, for \(\epsilon =\Delta T_{\text{ep}}, k=K/C \). In *S*
_{2} − *S*
_{1}, we have to put *d* = *d*
_{ }− = (*F* − *δ*)/*C*, and in *S*
_{2} − *S*
_{3}, we have to put *d* = *d*
_{ + } = (*F* + *δ*)/*C*. These equations define nine (or fewer) solutions. At maximum, four of the steady states are stable.

*Bifurcation of Welander’s model* The bifurcation diagram for symmetric forcing (i.e., *δ* = 0) is displayed in the left seven panels of Fig. 17. In the fluxes *q*_{
i
} and in *S*_{2}, some of the solutions lie on top of each other, but they are distinguished in *S*_{1} and *S*_{3}. In this symmetrically forced experiment, the curves of *S*_{1} and *S*_{3} are identical (likewise *q*_{1} and *q*_{3}), but this does not mean that the individual equilibria have this correspondence: in the range where multiple equilibria exist, solutions may, e.g., have a positive *q*_{1} and a negative *q*_{3}. It is instructive to look at variables that take care of the underlying symmetry occurring in a symmetrically forced experiment: *q*_{1} − *q*_{3} is zero for symmetric overturning (polar up- or downwelling in both hemispheres and equatorial down- or upwelling, a two-cell circulation), whereas *q*_{1} + *q*_{3} would be zero for one-cell overturning with equal strength of the circulation in both hemispheres (note that such a state cannot be realized in the present set-up).

The branches may again be classified in an obvious way as TH or SA states, but more complicated states appear. The four stable solutions in the restricted *F* domain are (1) a two-cell circulation with strong poleward flow in both hemispheres and equatorial upwelling (TH, both *q*
_{
i
} > 0, lying on the upper branch of *q*
_{1} + *q*
_{3} and on the zero branch of *q*
_{1} − *q*
_{3}); (2) a much weaker two-cell circulation with equatorward flow in both hemispheres and equatorial downwelling (SA, both *q*
_{
i
} < 0, lying on the lower branch of *q*
_{1} + *q*
_{3} and on the zero branch of *q*
_{1} − *q*
_{3}); and (3) two single-cell circulations (Atlantic conveyor belts, denoted by ACB in the following) with either strong northward flow in the northern hemisphere and weak northward flow in the southern hemisphere (*q*
_{1} > − *q*
_{3} > 0, lying on the upper branch of *q*
_{1} − *q*
_{3}), or strong southward flow in the southern hemisphere and weak southward in the northern hemisphere (*q*
_{3} > − *q*
_{1} > 0, lying on the lower branch of *q*
_{1} − *q*
_{3}). It has become custom to label the ACB solutions NPP and SPP (pole-to-pole circulations with sinking in the north or the south but no equatorial up- or downwelling). In the (*q*
_{1} + *q*
_{3}) plot of Fig. 17, both branches lie on the middle stable curve.

The equilibria can be deformed considerably through the implementation of lateral diffusion. As for the Stommel model, an increase of *K* shifts the window of multiple states towards larger *F* and shrinks it to a smaller range up to the point where the window closes at a critical *K*, and multiple states cease to exist. On this route, the symmetry of the solutions remains valid. More dramatic is the influence of the asymmetric forcing *δ* because it acts as an imperfection that breaks the symmetry immediately. As demonstrated in the right two panels of Fig. 17, the asymmetric forcing leads to breaking up the bifurcations on the zero line *q*
_{1} = *q*
_{3}, creating an isolated branch that, moreover, shrinks to nothing if *δ* exceeds a critical value.

The ACB solutions can only exist in the limited *F* range where the principle function *f*(*x*) has three roots. The overlapping range depends on the diffusivity *K*/*C* and the asymmetric forcing *δ*. The window can be diminished up to the point where overlapping ceases, and the ACBs have to break down to give way to a two-cell state. Increasing diffusion leads to a smaller range of existence for ACB. Likewise, negative *δ* narrows the overlapping range while a positive *δ* widens the range. This can easily be traced back to the behavior of the roots: ACB/NPP is built by *a*
_{3}(*d*
_{ + }) and *a*
_{2}(*d*
_{ }− ). Exploiting the conditions that the roots have to be real and that *a*
_{2} < *ε*, *a*
_{3} > *ε*, we get

The domain is sketched in Fig. 18. The ACB/NPP exist in the region between the corresponding colored lines.

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Olbers, D., Zhang, J. The global thermohaline circulation in box and spectral low-order models. Part 1: single basin models.
*Ocean Dynamics* **58**, 311–334 (2008). https://doi.org/10.1007/s10236-008-0156-3

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DOI: https://doi.org/10.1007/s10236-008-0156-3