## 1. Introduction

In many models of the midlatitude large-scale ocean circulation, driven by steady buoyancy forcing, low-frequency (multidecadal to centennial) variability is found in the meridional overturning circulation. One of the simplest configurations in which this variability occurs spontaneously is that of a thermally driven flow in a single-hemispheric idealized sector basin (Chen and Ghil 1995; Greatbatch and Zhang 1995). In most studies, the steady flow is first determined under restoring conditions and a single meridional overturning cell appears. The heat flux is diagnosed and prescribed in a subsequent integration of the model. By construction, the steady flow is also a solution under the prescribed flux conditions, but it can become unstable and multidecadal oscillations result (Colin de Verdière and Huck 1999; Huck et al. 1999).

From linear stability analyses of the steady flows under prescribed flux conditions, it was shown that the oscillations occur through a so-called Hopf bifurcation when the horizontal eddy diffusivity *K _{H}* becomes small enough (Huck and Vallis 2001; Te Raa and Dijkstra 2002). At this Hopf bifurcation, the growth rate of a particular mode, the so-called interdecadal or multidecadal mode (MM), becomes positive. This mode also exists under restoring conditions and in coupled ocean–atmosphere models (Te Raa and Dijkstra 2003), but in these cases the atmospheric damping is too strong and its growth rate is negative.

The physical mechanism of propagation of the MM was presented in Te Raa and Dijkstra (2002). A slight generalization [relative to that in Te Raa and Dijkstra (2002)] of this mechanism is provided with the help of Fig. 1. A warm anomaly in the north-central part of the basin causes a positive meridional perturbation temperature gradient that induces—via the thermal wind balance—a negative zonal surface flow (Fig. 1a). The anomalous anticyclonic circulation around the warm anomaly causes southward (northward) advection of cold (warm) water to the east (west) of the anomaly, resulting in westward propagation of the warm anomaly. As a result of this westward propagation, the zonal perturbation temperature gradient becomes negative, inducing a negative surface meridional flow (Fig. 1b). The resulting upwelling (downwelling) perturbations along the northern (southern) boundary cause a negative meridional perturbation temperature gradient, inducing a positive zonal surface flow, and the second half of the oscillation starts. The crucial elements in this oscillation mechanism are 1) the phase difference between the zonal and meridional surface flow perturbations and 2) the westward propagation of the temperature anomalies (Te Raa and Dijkstra 2002).

The finite-amplitude flows arising from the growth and nonlinear equilibration of the MM were studied in Te Raa et al. (2004). If continental geometry and wind forcing are taken into account, the propagation of finite-amplitude temperature anomalies can still be attributed to the same processes as those causing the propagation of anomalies in the MM. The spatial patterns of multidecadal variability in the most realistic ocean-only models used in Te Raa et al. (2004) show a SST pattern that has a strong correspondence to those in global climate models (Delworth and Greatbatch 2000; Delworth et al. 2002) and the observed pattern of the Atlantic multidecadal oscillation (AMO) (Kushnir 1994; Kerr 2000; Enfield et al. 2001). It has therefore been proposed that the MM captures the heart of the physics of the AMO (see chapter 6 in Dijkstra 2005).

In the linear stability analyses of thermohaline single-hemispheric flows, also other classes of oscillatory modes, so-called centennial modes (CMs) are found. During one oscillation period, a buoyancy anomaly propagates over the overturning loop of the background flow (Te Raa and Dijkstra 2003). When only thermal forcing is considered, there is only one class of CMs but, when freshwater is also considered, another class of CMs appears. It was shown that the CMs have an essentially two-dimensional character because they also exist in two-dimensional thermohaline flows. This is an essential difference between CMs and MMs as the latter essentially depend on three-dimensional processes (Te Raa and Dijkstra 2003).

Although it is clear that these modes control the linearized dynamics near steady thermohaline flows in a single-hemispheric configuration, the lack of knowledge of their spectral origin has always been quite unsatisfactory. The present study clarifies the nature of these modes and elucidates the underlying dynamical structure of the variability of the single-hemispheric flows. This is accomplished by following the MM and the least stable CM down to the small thermal forcing limit. Here, the results can be connected to analytical results in the zero thermal forcing limit where the modes become pure sea surface temperature (SST) modes. It turns out that there is an elegant structure of SST-mode mergers in the small thermal forcing limit that gives rise to all known classes of multidecadal and centennial modes. The mergers also explain many of the properties of these modes in the realistic forcing regime.

## 2. Ocean flows in a single-hemispheric basin

*θ*= 10°N and

_{s}*θ*= 74°N and longitudes

_{n}*ϕ*= 286° (74°W) and

_{w}*ϕ*= 350° (10°W); the ocean basin has a constant depth

_{e}*D.*The flows in this domain are forced by a heat flux

*Q*(W m

_{H}^{−2}):

*γ*(W m

^{−2}K

^{−1}) is a constant surface heat exchange coefficient. The heat flux

*Q*is proportional to the temperature difference between the sea surface temperature

_{H}*T** and a prescribed atmospheric temperature

*T*, chosen as

_{S}*T*

_{0}= 15°C is a reference temperature and Δ

*T*is the temperature difference between the southern and northern latitude of the domain. The forcing is distributed as a body forcing over the first (upper) layer of the ocean having a depth

*H*.

_{m}*α*is the volumetric expansion coefficient and

_{T}*ρ*

_{0}is a reference density. We neglect inertia in the momentum equations because of the small Rossby number, use the Boussinesq and hydrostatic approximations, and represent horizontal and vertical mixing of momentum and heat by constant eddy coefficients. With

*r*

_{0}and Ω being the radius and angular velocity of the earth, the governing equations for the zonal, meridional, and vertical velocity

*u, υ*, and

*w*, the dynamic pressure

*p*(the hydrostatic part has been subtracted), and the temperature

*T*=

*T** −

*T*

_{0}become

*C*the constant heat capacity, and

_{p}*τ*=

_{T}*ρ*

_{0}

*C*/

_{p}H_{m}*γ*is the surface adjustment time scale of heat. In these equations,

*A*and

_{H}*A*are the horizontal and vertical momentum (eddy) viscosity and

_{V}*K*and

_{H}*K*are the horizontal and vertical (eddy) diffusivity of heat, respectively. In addition,

_{V}The parameters for the standard case are the same as in typical large-scale low-resolution ocean general circulation models and their values are listed in Table 1.

The model setup and governing equations are similar to those used in Te Raa and Dijkstra (2002) and Te Raa and Dijkstra (2003) in which the physics of the multidecadal and centennial modes was studied, respectively. The main difference with the model in those papers is that, because of the B-grid discretization used here (Wubs et al. 2006), we are able to study the flows at a smaller value of lateral friction coefficient *A _{H}*. The value of

*A*in Table 1 is a factor of 100 smaller than in Te Raa and Dijkstra (2002) and thus is similar to values used in earlier studies (Colin de Verdière and Huck 1999; Huck et al. 1999; Huck and Vallis 2001).

_{H}## 3. The structure of the SST modes

*T*= 0) and constant mixing coefficients

*K*and

_{H}*K*, the velocity field is zero [(

_{V}*u*,

*υ*,

*w*) = 0] and the temperature field is determined from the problem

*ν*,

*λ*, and

*μ*are separation constants. The boundary conditions are given by Φ′(

*ϕ*) = Φ′(

_{w}*ϕ*) = 0, Θ′(

_{e}*θ*) = Θ′(

_{s}*θ*) = 0, and

_{n}*Z*′(−

*D*) =

*Z*′(0) = 0.

*θ*=

_{s}*θ*

_{0},

*θ*

_{1}, . . . ,

*θ*

_{M−}_{1},

*θ*=

_{M}*θ*] and solve the corresponding (

_{n}*M*+ 1) × (

*M*+ 1) matrix eigenvalue problem. For each value of

*μ*, we hence find

_{n}*M*+ 1 eigenvalues

*ν*

_{n,m},

*m*= 0, . . . ,

*M*. Since the problem is self-adjoint, all eigenvalues

*ν*

_{n,m}are real. If we define

*α*= −(

*ν*+

*λ*), then the eigenvalues of (8c) are given by

*λ**

_{n,m,l}(s

^{−1}),

This shows that part of the damping of the eigenmodes is determined by the vertical diffusion time scale and part by the horizontal diffusion time scale. The corresponding eigenfunctions *T*_{n,m,l}(*ϕ*, *θ*, *z*) are easily determined from the problem (8) once the eigenfunctions Θ_{n,m} are calculated from (8b).

The discrete version of the eigenvalue problem (8b) was solved numerically by a standard library [Numerical Algorithms Group (NAG)] routine. In Table 2, the damping factors *λ**_{n,m,l} (yr^{−1}) for the least damped eigenmodes are given for two different resolutions *M.* The modes with other indices (*n*, *m*, *l*) are more damped than those in Table 2. Modes with *n* = *m* = 0 are not affected by the resolution because the eigenvalue *ν*_{0,0} = 0. When compared with the results for *M* = 64, the damping factors of the other modes are already quite accurate for *M* = 16. We will use *M* = 16 in the computation of the patterns of the eigenmodes below.

Although the temperature patterns *T*_{n,m,l} are three-dimensional, their vertical structure is so transparent that we will call these modes, in analogy with the nomenclature in the ENSO literature (Jin and Neelin 1993), sea surface temperature modes or SST modes. The essence is that these modes are derived from the time derivative in the temperature equation in contrast to the so-called ocean dynamics modes [e.g., Rossby basin modes: Pedlosky (1987)], which result from the time derivatives of the momentum equations. We can label the SST modes according to the indices (*n*, *m*, *l*) and their spatial structures are easily imagined from these indices. Each index indicates the number of zeros of the eigenfunction in the domain; for example, the eigenfunction of each *n* = 1 mode has precisely one zero in the zonal direction.

Patterns of the four eigenmodes with the least negative damping factors are shown in Fig. 2; note that the amplitude of each pattern is arbitrary. For each mode, the SST pattern is shown as a meridional–depth section of the temperature pattern at *ϕ* = 330°. Apart from the trivial neutral mode (0, 0, 0), the least damped mode is the (0, 0, 1) mode with only a vertical structure (Figs. 2a,b). The next least damped mode is the (0, 1, 0) mode with only a meridional structure (Figs. 2c,d). The third mode is the (1, 0, 0) mode with only structure in the zonal direction (Figs. 2e,f). Figures 2g and 2h show the pattern of the (0, 1, 1) mode, which only has spatial dependence in the meridional and vertical directions.

## 4. The interaction of SST modes

Using the model in section 2, we will now compute steady three-dimensional thermally driven flow solutions at different Δ*T* and determine their linear stability. We will approach the limit of zero forcing (Δ*T* = 0 K) from the realistic forcing case (Δ*T* = 20 K) while tracing the patterns, growth rate, and period of the multidecadal mode and the least damped centennial mode.

### a. Steady states versus ΔT

First the steady governing equations (4) and boundary conditions (5) are discretized on an Arakawa B grid using central spatial differences. We use a horizontal resolution of 4° and a vertical resolution of 250 m. On this 16 × 16 × 16 grid with five unknowns per point (*u*, *υ*, *w*, *p*, and *T*), this leads to a system of 20 480 nonlinear algebraic equations. These are solved with Δ*T* as control parameter using a pseudoarclength continuation method; details on this methodology are provided in Dijkstra (2005).

Starting from the motionless, constant temperature state at Δ*T* = 0 K, we first compute a branch of steady flow solutions of the equations in section 2 under the restoring thermal forcing conditions (2). For each steady flow pattern, the maximum of the meridional overturning streamfunction (*ψ _{M}*) is calculated and is plotted versus Δ

*T*as the dashed–dotted curve in Fig. 3. Since convection, which occurs in the case of an unstable stratification, is not resolved in the low-resolution hydrostatic ocean model, a (convective adjustment) procedure is needed to obtain stably stratified solutions from each computed steady state. In the results below, we use the global adjustment procedure (GAP) as described in Dijkstra et al. (2001). The value of

*ψ*of the stably stratified steady solutions is plotted as the drawn curve in Fig. 3.

_{M}At intermediate Δ*T*, there is quite a difference between the values of *ψ _{M}* of both the nonadjusted and adjusted states, but this difference decreases again for large Δ

*T*. The maximum difference is about 5 Sv (Sv ≡ 10

^{6}m

^{3}s

^{−1}) near Δ

*T*= 7 K. The nonadjusted states have a thermally direct flow and a thermally indirect flow as discussed in Marotzke and Scott (1999), with downwelling at about 50° [for plots, see Dijkstra et al. (2001)]. In our case here, in which

*K*is constant, the overturning of the adjusted states is larger than that of the nonadjusted states. This is different from the results in Marotzke and Scott (1999), who use a model in which

_{V}*K*is only nonzero near the perimeter of the ocean basin. In their model, the strength of the meridional overturning increases with decreasing mixing efficiency associated with convective adjustment. In our case, adjustment acts in addition to the background vertical mixing and as meridional overturning increases with

_{V}*K*, it is larger in the adjusted state.

_{V}The meridional overturning, the near-surface velocity field, the near-surface temperature field, and a meridional slice of the temperature field at *ϕ* = 330° of the stably stratified solution for Δ*T* = 20 K are plotted in Figs. 4a–d, respectively. The maximum of *ψ* occurs at about 50°N and the amplitude is about 26 Sv (Fig. 4a). In the surface velocity field, there is a strong western amplification and the flow is strongly zonal (Fig. 4b). Note that because the value of *A _{H}* is a factor 100 smaller than in Te Raa and Dijkstra (2002), the western boundary thickness is much smaller and is now comparable to that in other model studies (Huck and Vallis 2001). The western intensification of the flow can also be seen in the SST pattern (Fig. 4c) and the flow is nicely stably stratified (Fig. 4d).

In all results that follow below, we only consider the adjusted steady states.

### b. Linear stability of the steady states and mode tracing

*σ*of each perturbation. When this elliptic eigenvalue problem is discretized, a generalized eigenvalue problem is obtained of the form

*u*,

*υ*,

*w*,

*p*, and

*T*), this eigenvalue problem has a dimension of 20 480. We solve for the 12 “most dangerous” modes, that is, those with real part closest to the imaginary axis, using the Jacobi–Davidson

*QZ*method (Sleijpen and Van der Vorst 1996) and order the eigenvalues

*σ*=

*σ*+

_{r}*iσ*according to the magnitude of their real part (the growth factor).

_{i}The growth rate and angular frequency of the modes with largest growth factors are plotted versus Δ*T* in Fig. 5. Each point on the curves labeled MM and CM in Fig. 5a is the growth rate of the multidecadal (MM) and the least damped centennial (CM) mode, respectively. These modes are the ones with the largest growth factors in the linear stability analysis of a steady three-dimensional flow pattern for each value of Δ*T*; the angular frequency *σ _{i}* of the mode is given in Fig. 5b.

Consider first the stability of the stably stratified steady state at the largest value of Δ*T* = 20 K for which steady flow patterns were shown in Fig. 4. In this case, the MM has a positive growth factor: the values of *σ _{r}* and

*σ*are given as the rightmost points on the curves labeled MM in Fig. 5. As the eigenvalues are complex conjugated, there are actually two points representing the MM in Fig. 5b (with ±

_{i}*σ*). The period of this mode for Δ

_{i}*T*= 20 K is about 150 yr and decreases with increasing Δ

*T*(Fig. 5b). The eigenvalues of the second oscillatory mode are plotted as the rightmost point on the curves labeled by CM in Fig. 5. This mode is damped as the growth rate

*σ*is negative (Fig. 5a). For Δ

_{r}*T*= 20 K, the period of this mode is about 350 yr (Fig. 5b) and also slowly decreases with increasing Δ

*T*.

For each eigenvalue *σ* associated with the MM, there is a corresponding eigenvector **x** = **x*** _{r}* +

*i*

**x**

*according to (12). In Fig. 6a, the sea surface temperature field of the real part of the eigenvector (*

_{i}**x**

*) of the MM mode is plotted. A comparison of the pattern in Fig. 6a and the one in Fig. 4d in Te Raa and Dijkstra (2002) demonstrates that the MM here is the multidecadal mode as described in detail in Te Raa and Dijkstra (2002). We confirmed this by looking at two characteristics of the physical mechanism of the multidecadal mode, that is, the phase difference between meridional and zonal flow anomalies (as discussed in the introduction) and the westward propagation.*

_{r}*and*u

*are the steady-state zonal velocity and temperature field. In addition,*T

*η*is a factor set by the vertical structure of the steady-state temperature field. The period is then determined by the travel time of the anomaly over the basin. The reason for the large period here is that the zonal velocities of the basic state for the small value of

*A*used are now much larger than in Te Raa and Dijkstra (2002) while the meridional temperature gradient is about the same. The travel time of the temperature anomalies over the basin is therefore increased and, consequently, so is the period of the mode. By changing the mixing coefficients of heat, the period can be easily tuned in the 50–100-yr range.

_{H}Also for each eigenvalue *σ* associated with the CM, there is a corresponding eigenvector **x** = **x*** _{r}* +

*i*

**x**

*according to (12). In Fig. 7a, the sea surface temperature field of the real part of the eigenvector (*

_{i}**x**

*) of the CM mode is plotted. At Δ*

_{r}*T*= 20 K, the anomaly is of single sign below and above 50°N, which gives it a two-dimensional appearance. The boundary of positive and negative anomalies appears to be set by the sinking region of the basic state (Fig. 4a). The anomaly patterns and period are in correspondence with the patterns of the centennial modes found in Te Raa and Dijkstra (2003). Note that the period is set by the strength of the meridional overturning as it involves the propagation of a temperature anomaly over the background flow.

Next, we trace both the MM and the CM to smaller values of Δ*T*. The path of each mode can be determined by inspection of the patterns of the corresponding eigenfunctions of (12). From Fig. 5a, we see that the growth factor of the MM decreases strongly with decreasing Δ*T* and becomes negative near Δ*T _{c}* ≈ 10 K. For Δ

*T*< Δ

*T*, the steady states are therefore linearly stable. The period of the MM increases with decreasing Δ

_{c}*T*(Fig. 5b) because the steady-state temperature gradient decreases. The growth factor of the CM (Fig. 5a) actually increases with decreasing Δ

*T*, but it never becomes positive. In addition, the period of the CM increases with decreasing Δ

*T*, similar to the MM (Fig. 5b).

*σ.*The time dependence of the oscillatory mode can be represented by the function

**x**

*, we represent the pattern of the mode at*

_{r}*t*= 0. As the eigenvectors

**x**are separately normalized for each value of Δ

*T*, the phases of each mode may not be the same. One could normalize the eigenfunctions such that the phase of the SST pattern of one particular mode is similar for each value of Δ

*T*. However, in the continuation setup the eigenvectors at one value of Δ

*T*are computed from approximations at a previous value of Δ

*T*and hence the changes in the phases turn out to be very small. Hence, we show only the real part of the eigenvector.

In Fig. 6, the sea surface temperature anomaly fields of the MM mode are plotted for six different values of Δ*T*. When Δ*T* is decreased, the pattern becomes more zonal (Figs. 6b–e), but even at Δ*T* = 1.0 one still sees the characteristic northeastward tilt (Fig. 6f) in the boundary between positive and negative temperature anomalies. In the SST patterns of the CM, as shown in Fig. 7 for different values of Δ*T*, the zonal character of the patterns remains the same for the larger values of Δ*T* (Figs. 7b–d) and a slight northeastward tilt appears at the smaller values of Δ*T* (Figs. 7e,f).

### c. SST-mode merging at small ΔT

As the reader may have anticipated, the interesting issue is now how the oscillatory MM and CM modes connect to the stationary SST modes (at Δ*T* = 0 K) as discussed in section 3. From Fig. 5b, one can see that *σ _{i}* for both modes approaches zero and it appears that, for both CM and MM,

*σ*becomes zero in the interval 0 < Δ

_{i}*T*< 0.5. The values of

*σ*and

_{r}*σ*for both modes are plotted (again both: yr

_{i}^{−1}) over this Δ

*T*interval in Fig. 8.

Near Δ*T* = 0.4 K, the angular frequency of the mode CM becomes zero and the complex conjugate pair of eigenvalues splits up into two real eigenvalues. The paths of the different modes can be followed by looking at the SST pattern and the vertical structure of the temperature anomaly. In Figs. 9a and 9b the SST pattern of the two nonoscillatory modes for Δ*T* = 0.34 K is plotted. Both modes derive from the real and imaginary parts of the eigenmode just before merging occurs, and the pattern in Fig. 9a is therefore nearly similar to that in Fig. 7f. Both SST patterns becomes zonally homogeneous for Δ*T* = 0.01 K (Figs. 9c,d). The vertical structure of each of these modes is different (corresponding to *l* = 0 and *l* = 1) and hence the limiting nonoscillatory modes can be identified as the (0, 1, 0) mode and the (0, 1, 1) mode.

Slightly below Δ*T* = 0.3 K, the angular frequency of the MM becomes zero and the complex conjugate pair of eigenvalues splits up into two real eigenvalues. The paths of the different modes can be followed again by looking at the SST pattern and the vertical structure of the temperature anomaly. In Figs. 10a and 10b the SST pattern of the real and imaginary parts at Δ*T* = 0.34 K (where the mode is still oscillatory with a very large period) is plotted. In the real part, the positive anomaly has extended over most of the basin (Fig. 10a), while in the imaginary part a negative temperature anomaly covers the basin (Fig. 10b). Eventually, for small Δ*T*, one of the real modes obtains a structure with hardly any zonal and meridional dependence (Fig. 10c), while for the other mode a clear zonal variation appears (Fig. 10d). Based also on the vertical structure, the limiting nonoscillatory modes can be identified as the (0, 0, 1) mode and the (1, 0, 0) mode.

## 5. Summary and discussion

In this paper, we have demonstrated that both the centennial mode and multidecadal mode have their origin in the interaction of stationary SST modes. This result was obtained by tracing the eigenmodes of the linear stability problem of thermally driven flows through parameter space. Note that this mode-tracing approach has, for example, also been successfully used to determine the origin of the so-called gyre mode in the double-gyre wind-driven circulation (Simonnet and Dijkstra 2002). Another example in which it has been used is that of the interaction of SST and ocean dynamics modes of the coupled ocean–atmosphere system in the equatorial Pacific Ocean (Jin and Neelin 1993).

For the single-hemispheric flows, mergers of nonoscillatory SST modes already occur at small thermal forcing to give rise to oscillatory modes. At Δ*T* = 0.2 K, all eigenvalues are still real and the damping factor of the (0, 0, 1) mode approaches that of the (1, 0, 0) mode. Near Δ*T* = 0.3 K, a merger occurs between the (0, 0, 1) mode and the (1, 0, 0) mode, which leads to the MM. For slightly larger Δ*T* = 0.4 K, the (0, 1, 0) SST mode and the (0, 1, 1) SST mode merge to give rise to the least damped CM. While the growth factor of the CM remains negative for larger Δ*T*, that of the MM eventually becomes positive.

The mergers are induced by the deformation of the eigenmode through the presence of the background flow. This interaction is advective and hence involves the terms *A*_{1} = ** u** ·

**∇**

*T̃*and

*A*

_{2}=

**ũ**·

**∇**

*, where the bar indicates properties of the steady flow and the tilde indicates the perturbations. A zonally homogeneous and meridionally varying SST pattern [such as that of the modes (0, 1, 0) and (0, 1, 1)] induces (through geostrophy) mainly a zonal perturbation flow. As the basic-state temperature field has hardly any zonal dependence at weak forcing, both advective terms*T

*A*

_{1}and

*A*

_{2}will be small. The growth rate is determined by the term 〈

*T̃*· (

*A*

_{1}+

*A*

_{2})〉, where 〈·〉 indicates averaging over the domain (Te Raa and Dijkstra 2002), and there will only be a small effect of the basic state on the growth rate. We see this in the CM, being a merger of the (0, 1, 0) and (0, 1, 1) modes.

A zonally varying temperature pattern such as that of the (1, 0, 0) mode will induce mainly a meridional velocity perturbation. As the basic-state temperature has a strong meridional dependence, the advection term *υ̃* ∂* T*/∂

*θ*will be relatively large. A substantial modification of the SST anomaly pattern will occur because of the presence of the basic state with consequent effect on the growth rate through the term 〈

*T̃υ̃*∂

*/∂*T

*θ*〉. This explains the large change in growth rate of the MM, which finally leads to the destabilization of the basic state.

The actual merger is induced by the effect of the basic flow on the spatial structure of the SST modes, but it is more difficult to understand the details of the physics of this merger. One has to study how advective processes between the background flow and the perturbations lead to patterns of the modes that become more “similar” as Δ*T* increases. As the merger already occurs at small Δ*T*, it may be possible to develop an asymptotic theory for the merger; work on this issue is in progress.

Note, however, that the type of merger explains why the CM is found in two-dimensional models (Dijkstra and Molemaker 1997; Te Raa and Dijkstra 2003) but the MM is not. The CM is a merger between two modes that have no zonal structure [the (0, 1, 0) and (0, 1, 1) modes]. These SST modes will also be present in two-dimensional models and hence a merger can occur. The MM is a merger between one mode that has zonal structure and one that does not, but the first one [the (1, 0, 0) mode] is certainly absent in two-dimensional models. Hence, a merger needed to obtain the MM cannot occur in these models and the MM mode is essentially three-dimensional.

Based on the origin of the oscillatory modes as mergers of the SST modes, one can guess the classes of oscillatory modes present when salinity is included. Note that, in the zero forcing limit, the classes of SST modes and sea surface salinity (SSS) modes (solutions to the diffusion equation for salinity) are totally decoupled and, under similar boundary conditions, the algebraic multiplicity of the damping factors *λ* is 2. When the background flow is only thermally forced and its stability is considered under prescribed flux conditions, mergers between the SST modes give rise to the MM and CM modes as above. However, mergers of SSS modes and between SSS and SST modes are also possible, giving rise to the additional class of CMs found in Te Raa and Dijkstra (2003). Because of the absence of a mean salinity gradient in Te Raa and Dijkstra (2003), this new class of modes remains damped. A mean salinity field can be generated, for example, by forcing the basic flow with mixed boundary conditions. In case the linear stability of this state is also considered under mixed boundary conditions, SST modes under restoring conditions are able to merge with SSS modes under prescribed flux conditions, possibly leading to additional types of oscillatory CMs and MMs.

The modal structure of the midlatitude three-dimensional ocean circulation becomes more and more interesting. In the dynamic limit, such as determined from studies of the pure wind-driven circulation, four classes of modes are known (Sheremet et al. 1997; Nauw and Dijkstra 2001). The Rossby basin modes (Pedlosky 1987) are the oscillatory modes that appear directly from the linear operator of the motionless flow. Other modes, such as the classical baroclinic modes and the gyre modes appear with increasing wind forcing through Hopf bifurcations. Of these, the low-frequency gyre mode arises through a merger of stationary modes (Simonnet and Dijkstra 2002). In thermally driven flows, as considered in this paper, the MM and the CM arise through mergers of SST modes. One can expect that in reality (and in high-resolution ocean models) interesting mergers between gyre modes and multidecadal modes (in so-called two degeneracies) are possible; whether these interactions will occur remains to be investigated.

## Acknowledgments

This work was supported by NSF Grant OCE-0425484.

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Standard values of parameters used in the numerical calculations.

Damping factors *λ**_{n,m,l} (yr^{−1}) of the eight least damped SST modes for *M* = 16 and *M* = 64.