## 1. Introduction

The role of the ocean in the climate system is a topical and important question. Most climate modelers recognize the importance of the ocean in modulating, increasing, or damping climate variability and include a dynamical ocean in their climate simulations. However, the ocean components of most climate models are necessarily coarse resolution and invariably have, as a consequence, high viscosity, thereby omitting some potentially important aspects of the ocean circulation. For example, the ocean component of the Hadley Centre's Third-Generation Coupled Ocean–Atmosphere GCM (HadCM3; Gordon et al. 2000) uses a 1.25° × 1.25° grid; recent experiments by Roberts et al. (2004) used the same atmosphere as HadCM3, but with a ⅓° × ⅓° ocean, which was designed to permit (but not fully resolve) features of ocean dynamics on the Rossby radius scale (50–100 km). The high resolution allowed better simulation of many ocean features, in particular tropical instability waves. In the midlatitudes, however, the simulated eddy kinetic energy was considerably lower than the values estimated from altimetry measurements (Stammer 1997). It is likely that this is due to marginal resolution of ocean eddies and high viscosity. In short, comprehensive climate models are too computationally expensive for eddy-resolving simulations.

Many studies of midlatitude ocean dynamics reduce the computational expense by using models with simple domains and only a few vertical layers. Despite their simplicity, such models are capable of producing rich behavior provided horizontal resolution is high and viscosity is relatively small. For example, Holland (1978) showed single- and double-gyre models of a wind-driven box ocean in which the spontaneous production of eddies dominated the circulation. Features of this circulation were confirmed by Verron and Le Provost (1991) for a larger ocean basin with 1-, 2-, and 3-layer quasigeostrophic models. McCalpin and Haidvogel (1996) showed that even in a 1½-layer model the tendency for destabilization produced a rich and variable eddy field on shorter time scales, and also resulted in lower-frequency instabilities produced (in their model) by irregular transitions between relatively stable states. Time-dependent models of the circulation have shown that the occurrence of low-frequency variability in such models is commonplace, although the strong eddy field may obscure many features of the circulation, making it difficult to agree upon the mechanisms underpinning this variability (Berloff and McWilliams 1999; Primeau 2002; Nauw et al. 2004). Variability of this type has the potential to alter heat transport in the ocean, and would therefore be important to climate variability (Dewar 2001).

The potential importance of simulated low-frequency variability to climate provides additional motivation for determining its physical causes. In particular, such knowledge may allow the construction of tests on the observational record, as well as improvement of comprehensive climate models. A number of possible mechanisms of low-frequency variability have been proposed. While the details of each proposed mechanism differ, they can be broadly grouped into three categories:

irregular transitions between quasi-stable states (e.g., McCalpin and Haidvogel 1996),

feedback between eddies and the mean flow (e.g., Spall 1996), and

oscillations in the mean flow of low-dimensional dynamical systems (e.g., Simonnet and Dijkstra 2002).

The first mechanism is based on the possibility that the wind-driven flow has multiple stable or unstable solutions that act to steer the flow. Its viability has been demonstrated in several barotropic or 1½-layer models (Cessi and Ierley 1995; McCalpin and Haidvogel 1996; Primeau 1998). In the case of McCalpin and Haidvogel (1996), a time-dependent model is used to show that the model state is more likely to exist close to one of three steady solutions, but that this tendency breaks down in more turbulent parameter regimes.

The eddy feedback mechanism was proposed by Spall (1996) to explain the variability of a three-layer isopcynic model of the North Atlantic. This eddy-resolving simulation showed an oscillation in the position of the Gulf Stream and the deep western boundary current (DWBC) with a period of about 10 yr. The proposed mechanism involved a feedback between baroclinic instability, the position of the DWBC, and the strength of barotropic, eddy-driven inertial recirculating gyres near the Gulf Stream separation point (see Cessi et al. 1987; Cessi 1988) for a description of the development of inertial recirculation gyres). When baroclinic instability was strong, this forced inertial recirculations, which diverted the DWBC from the coast so that it was aligned with the Gulf Stream after separation. The result was a barotropized jet that was more stable than the baroclinic jet: baroclinic instability was reduced, leading to a reduction in the baroclinic eddy field. In the final phase of the oscillation, the weaker eddy field resulted in a smaller driving force on the inertial recirculations, which decreased in intensity. The DWBC returned to its path along the coast, and the jet became more baroclinic and less stable so that increased eddy activity ensued, and the cycle began again. This complex series of events produced low-frequency oscillations because of the time taken to amplify the eddy field via baroclinic instability, the gradual subsequent downward momentum transport by eddies, and the time for the inertial recirculations to decay after eddy activity reduced.

Another similar mechanism was proposed by Qiu and Miao (2000) to explain variability in a ⅙°-resolution model of the North Pacific. Their paper focuses on the Kuroshio Extension region, where there is no DWBC to interact with the inertial recirculations. Nonetheless they find that transitions between a straight and meandering Kuroshio Extension are related to the strength of the southern inertial recirculation gyre, which amplifies slowly by the gradual accumulation of eddy-driven potential vorticity and then diffuses its potential vorticity during episodes of instability. The mechanism they propose is simpler than that of Spall (1996), but there are similarities: the time scale is set by the spinup of the inertial recirculations, and the signature of the variability is a modulation in the strength and position of the ocean jet.

Both Qiu and Miao (2000) and Spall (1996) rely on observations of a single run of a high-resolution primitive equation model. It is far from clear that these models use parameters that are correct for the ocean, and one realization of a model run does not allow testing of the dependence upon these parameters.

The alternative techniques used to attack this problem are derived from nonlinear dynamical systems theory. The general approach here is to solve a simple, idealized model at high viscosity for a steady solution. Then, by gradually decreasing viscosity (or increasing forcing strength, or perhaps asymmetry of forcing) one sees the development of other stable states (Jiang et al. 1995; Cessi and Ierley 1995; Primeau 1998), bifurcations leading to periodic behavior (Nauw and Dijkstra 2001; Nadiga and Luce 2001), quasi-periodic attractors and secondary bifurcations (Berloff and Meacham 1997, 1998, and ultimately chaotic behavior (Jiang et al. 1995; Nadiga and Luce 2001). Such approaches have the advantage that one is able to pinpoint the processes that lead to, for example, a particular Hopf bifurcation that is responsible for periodic behavior in the high viscosity model. For example, Simonnet and Dijkstra (2002) examine the evolution of a particular low-frequency mode of instability, which they call the *gyre* mode. This mode, or a mode with similar spatial and temporal characteristics, appears in several relevant studies, including Dijkstra and Katsman (1997), Berloff and McWilliams (1999), Chang et al. (2001), Nauw and Dijkstra (2001), and Simonnet (2005), and it depends upon the strength of the inertial recirculating gyres for its existence. The advantage of the nonlinear dynamical systems approach is shown by Simonnet and Dijkstra (2002) as they trace the gyre mode from its origins at high viscosity down to the lower viscosities at which it becomes unstable. The gyre mode is demonstrated to be the product of two stationary modes that merge to produce a single oscillatory mode, which ultimately has positive growth rate (see their Fig. 3).

However, the physical understanding obtained by following the dynamical systems approach is offset in part by the high computational cost. As a result, investigations are usually confined to smaller basins with low-resolution barotropic (or 1½ layer) models and relatively high viscosity. In addition, continuation methods cannot be used to find unstable attractors more complex than steady states in a general way; thus, they can never be used for fully turbulent flows. For this reason it is not clear that the physical mechanisms identified are still active at high Reynolds number.

In this paper we show results from time-dependent simulations of a large, high-dimensional, low-viscosity ocean. Our ocean is similar in some respects to the studies of Spall (1996) and Qiu and Miao (2000) except that we use simplified domains and idealized physics while retaining a three-layer structure, which is necessary for realistic simulation of baroclinic instability. These simplifications produce a faster code that allows us to run multiple simulations. The model is employed to investigate the robustness of the proposed mechanisms for low-frequency variability over a wide parameter space. The model is described in section 2. We outline the mean flow and its dependence upon parameters in section 3, and then diagnose the variability of the model as a function of parameters in section 4. These results are analyzed in section 5 with particular reference to the multiple equilibria, eddy–mean flow feedback, and gyre mode mechanisms for low-frequency variability.

## 2. The model

*J*(

**q**,

**p**) =

**p**

_{y}**q**

*−*

_{x}**p**

_{x}**q**

*is the Jacobian, and we use vectors of length three to represent fields in the three ocean layers such as pressure (*

_{y}**p**) and potential vorticity (

**q**), where

*f*

_{0}, the (assumed constant) gradient of the Coriolis parameter with distance northward

*β*, a Laplacian viscosity coefficient

*A*

_{2}, and a biharmonic viscosity coefficient

*A*

_{4}.

*w*

_{ek}is the Ekman pumping calculated from the wind stress curl, Δ

*describes the temperature difference across interface*

_{i}T*i*, and

*δ*

_{ek}is a bottom drag parameter. The matrices 𝗔 and 𝗕 are

*H*are unperturbed layer thicknesses and

_{i}*g*′

_{i}is the reduced gravity for interface

*i*. Note that 𝗕

**e**produces two forcing terms on each layer, representing forcing at the top and bottom of each layer.

*m*) embedded in layer 1. This is included in the coupled model to enable communication of heat and stress between atmosphere and ocean [see Hogg et al. (2003b) for further details]. We retain the mixed layer in these ocean-only simulations for consistency with planned coupled experiments. Mixed layer temperature

*T*evolves according to the equation

_{m}*u*and

_{m}*υ*) are calculated from the wind stress and the geostrophic flow in QG layer 1,

_{m}*K*

_{2}is a diffusion coefficient,

*ρ*is the ocean density,

*C*is the specific heat capacity of the ocean, and

_{p}*H*is the (fixed) mixed layer thickness. The imposed heat forcing at the surface is described by a positive upward heat flux

_{m}*F*and the final term describes the exchange of heat with QG layer 1 (in which it is embedded) due to the diabatic effect of Ekman pumping.

*α*

_{bc}is zero for free slip and infinite for no-slip boundary conditions (although, in practice,

*α*

_{bc}> 2 is a good approximation to no slip), Δ

*x*is the horizontal grid spacing, and subscript

*n*denotes the outward normal derivative. The value of

*f*(

_{k}*t*) is constrained by mass conservation and is the same for all boundaries.

In all experiments shown here, the wind forcing is steady. The wind stress curl field, shown by the Ekman velocity in Fig. 1a is calculated from a coupled simulation using the default parameter set and represents a time mean realistic forcing for the midlatitudes. Heat flux (Fig. 1b) is calculated from the same coupled simulation, has a zero mean and is also steady. Note that the heat flux is largest near the western boundary of the ocean and smaller in the interior. (Simulations over a wide range of parameter space have confirmed that the primary features of the circulation are not strongly dependent upon the form of the heat flux.) Forcing on the QG layers is thus steady, except for the time variation of the diabatic heating term [the second element of the vector **e** defined in (3)] which has a weak feedback through SST. Sensitivity tests indicate that the results shown below do not depend upon this feedback.

We use a geometrically simple rectangular domain, but the large domain size (3840 km × 4800 km), high horizontal resolution (10 km) and low viscosity (*A*_{2} = 0 m^{2} s^{−1} and *A*_{4} = 1 × 10^{10} m^{4} s^{−1}) conspire to yield a dynamically rich circulation. Baroclinic instability, which is much stronger when three or more layers are used than with only two layers, produces ocean eddies that form on length scales comparable with the larger of the Rossby radii (∼50 km in this model) and the low viscosity increases the longevity of the eddies. The simulations therefore have a higher dimensionality than most ocean models. The full parameter set is outlined in Table 1; the first three entries to this table are parameters that are varied in the numerical experiments described in sections 3 and 4.

## 3. The mean circulation

In this paper we will concentrate on the variability of the wind-driven circulation; however, in the simulations presented below, the variability of the circulation is intertwined with properties of the mean flow, and so it is useful to describe the mean circulation first.

### a. Mean flow for the default parameter set

*ψ*

_{1}=

*H*

_{1}

*p*

_{1}

*/f*

_{0}) is then calculated by integrating from the eastern boundary. Figure 2a shows the circulation predicted in the uppermost layer assuming that this linear balance holds everywhere. This prediction differs from the mean uppermost layer flow (averaged over years 21–180 of a 180-yr simulation using default parameters listed in Table 1) shown in Fig. 2b in several respects. First, mass conservation requires a western boundary current to form, which advects potential vorticity along the boundary, thereby altering the flow field. In addition, for high Reynolds number regimes, the flow is inherently unstable; barotropic and baroclinic instability result, and they produce eddies that alter the time-averaged flow. The predicted subtropical and subpolar gyres, the linear response to the forcing, are of the correct order of magnitude; however, the circulation is amplified in the northwestern (southwestern) corner of the subtropical (subpolar) gyre. The increased transports in these regions are known as the inertial recirculations (see Cessi et al. 1987; Cessi 1988) that occur because of the advection of anomalous potential vorticity via the western boundary current, producing a pool of low- (high-) potential-vorticity fluid on the subtropical (subpolar) side of the western boundary current extension. In the case shown here this nonlinear component of the flow is of comparable magnitude to the linear response.

The circulation in layer 2 (Fig. 2c) also includes a direct linear response and a nonlinear component. The forcing on this layer is produced by the diabatic heating term (the second element of **e**), which drives a weak circulation in the opposite direction to the uppermost layer. This weak gyre is swamped by the strong inertial recirculations that extend to full depth in the ocean (layer-3 transport is shown in Fig. 2d). The barotropic nature of the inertial recirculations is noteworthy because, if the flow were steady, then the only forcing on the lowest layer would be the linear bottom drag. It follows that the inertial recirculations, forced in the uppermost layer, can be transferred to the lower layers; this occurs because of the effective diffusion of layer thickness, which homogenizes pressure vertically. We therefore assume that the strong geostrophic turbulence present in this simulation is essential for driving the recirculating gyres in the lower layers.

The extent of turbulence can be seen in Fig. 2e. Here we see a long meandering jet, which is coherent across much of the basin—the variation in position of this jet produces the wider jet seen in the mean flow. In addition the active eddy field, likely to be a product of both barotropic and baroclinic instability, is shown with peak instantaneous transports exceeding 85 Sv (Sv ≡ 10^{6} m^{3} s^{−1}) in the inertial recirculations. The mean SST distribution (Fig. 2f) shows a latitudinal temperature gradient, as expected, with the development of the temperature front near the western boundary current extension.

The forcing terms for this model are asymmetric because the diabatic heating term is stronger in the southern half of the basin. This asymmetry is reflected in both the linear and inertial gyres. In addition, the inertial recirculations act to pull the ocean jet south of the zero wind stress curl line (shown in Fig. 2b). This result is slightly counterintuitive—one might expect that inertia would cause an overshoot of the subtropical gyre so that the jet would sit north of that line. However, it is potential vorticity, not inertia, that controls the dynamics of this flow. This was discussed by Moro (1988) with reference to asymmetric barotropic QG simulations; he found that the position of the stronger recirculating gyre governs the position of the jet. At higher Reynolds number Moro (1988) observed that the jet was pulled toward the stronger recirculating gyre.

### b. Dependence of the mean flow upon parameters

The relative importance of different components of the wind-driven circulation in this model can be investigated by a series of tests in each of which the value of a model parameter is systematically varied. In three specific cases below we present the mean transport in the uppermost and lowest layers. In addition, the variation of eddy and mean kinetic energy (per unit area) over a wider range of parameters is shown for each layer. The first parameter to be varied is the biharmonic viscosity *A*_{4}, which in Figs. 3a and 3b is a factor of 4 larger than the default case (reducing the time scale for the viscous decay of eddies with a length scale comparable to the Rossby radius from 20 to 5 days). Comparison with the mean circulation for the case with default parameters (Figs. 2b,d) shows similarities in the linear circulation in the interior of the uppermost layer gyres, which is virtually unchanged for these two cases. However, there is a marked difference in the size and strength of the inertial recirculations in both layers 1 and 3. There are two possible mechanisms that may account for this reduction in the flow. The first is that eddies, which help to drive the recirculating gyres at depth, are damped by the high viscosity, resulting in a corresponding decrease in the recirculations. The weakening of the eddy field is shown in Fig. 3c where eddy kinetic energy is damped strongly with increasing viscosity. A corresponding decrease in the kinetic energy of the mean circulation (Fig. 3d) is also apparent. But a correlation between the pattern of eddy and mean kinetic energy does not imply causality; a second viable hypothesis is that greater viscosity alters the structure of the western boundary currents, reducing the efficiency of transport of anomalous potential vorticity by the boundary current. Such a mechanism would act to inhibit the generation of the recirculating gyres in the uppermost layer. The information presented in Fig. 3 is not sufficient to distinguish whether it is the weaker eddy field or the structure of the boundary current that is responsible for reduction in the inertial recirculations.

Further insight can be obtained by varying the boundary friction parameter *α*_{bc}. The sensitivity of double gyre circulation to partial slip boundary conditions was described by Haidvogel et al. (1992) who demonstrated that increasing boundary friction results in a state where the subtropical and subpolar western boundary currents separate prematurely to create two distinct eastward jets. We confine our study to the parameter regime close to the transition between single and separated jets. Figure 4 shows the results of this sensitivity test in the same format as Fig. 3.

The mean transport in Figs. 4a and 4b demonstrates that the increasing boundary friction has a similar effect to increasing internal viscosity: the linear circulation is largely unchanged, while the magnitude of the inertial recirculations is reduced markedly. Large boundary friction extracts energy horizontally from each layer, thereby reducing eddy activity over the entire basin (Fig. 4c). The kinetic energy of the mean flow is reduced by a similar proportion to the viscous case. These simulations show that changing boundary friction has a qualitatively similar effect to changing viscosity and does not allow us to distinguish between the relative roles of eddies and the western boundary current in controlling the size of the inertial recirculations.

It is therefore instructive to damp the flow by increasing the bottom drag in the ocean, which acts as a sink of energy on the lowest layer only. Figure 5 shows that sensitivity to bottom drag is different from the previous cases. The mean flow (Figs. 5a,b) where bottom drag has been amplified (by a factor of 4) shows only a slight decrease in the strength of the recirculations. This decrease is significant in the lowest layer, but negligible in the uppermost layer. In addition, the dependence of kinetic energy on the bottom drag (Figs. 5c,d) is surprising. The eddy kinetic energies in layers 2 and 3 are damped by increasing bottom drag; however, the effect on layer 1 is negligible, while the mean kinetic energy in layer 1 increases with bottom drag.

Increasing the bottom drag allows insight into the relative roles of eddies and the western boundary current structure in setting up the inertial recirculations because such tests do not directly alter the upper-layer western boundary current but do damp the eddy field in lower layers. We infer that the crucial element to the formation of strong inertial recirculations in this model is the structure of the western boundary current. In low viscosity (or small boundary friction) cases, the western boundary current can efficiently carry potential vorticity anomalies that drive the inertial recirculations. As viscosity or boundary friction is increased, the effect of the boundary propagates across the western boundary current, thereby decreasing the amount of potential vorticity carried by the current.

The series of experiments presented here demonstrates the way viscosity, boundary friction, and bottom drag affect the mean flow. Despite the high dimensionality of these simulations and the complexity introduced by the turbulent eddies, the primary features of the circulation can be explained qualitatively. The direct (linear) response to the wind stress is to form a pattern of double gyre circulation, which acts to transport potential vorticity anomalies via the western boundary current. The uppermost layer thus develops strong inertial recirculations that are communicated to the lower layers by the ocean eddies. This picture is consistent with previous investigations into this type of circulation (e.g., Chang et al. 2001).

## 4. The variability of the circulation

Simulation of a dynamically rich ocean circulation, containing eddies that exist on a range of spatial scales (see Fig. 2e), results in high temporal variability that occurs on short (monthly) time scales due to the direct effect of eddies and a meandering jet. Of greater interest to climate modeling is the variability of the circulation on time scales longer than a year. To investigate this low-frequency variability we record the pressure at 15-day intervals in each layer over 160 model years; these pressure fields necessarily include the high temporal variability of the eddy field, which is filtered out using a 2-yr low-pass Fourier filter applied at every point in space. The data are further reduced by calculating the Hilbert (or complexified) empirical orthogonal functions (EOFs).

Hilbert EOFs are calculated using a complexified dataset; the imaginary part of this dataset is constructed by taking the Hilbert transform of the real data (von Storch and Zwiers 1999). One then calculates the EOFs in the standard way (eigenvector analysis of the covariance matrix of the data), which decomposes the dataset into complex eigenvectors (spatial patterns of the data that maximize variance) and their principal components (PCs). This allows extraction of statistical modes of variability from the filtered dataset, and in many cases the bulk of the variance can be attributed to only a few spatial modes [see von Storch and Zwiers (1999) for a full treatment of EOFs].

Figure 6 shows the spatial pattern of the first three Hilbert EOFs for the filtered uppermost-layer transport streamfunction in the default case. For each mode we obtain two spatial fields; one of these is the real part of the Hilbert EOF, the other is the imaginary part. These two spatial fields describe the shape of the statistical mode at different phases of the oscillation.

In Fig. 6a the first spatial mode (which represents 47% of the total variance in the filtered dataset) is shown. The real part of the spatial mode forms a dipole focused on the inertial recirculations; the imaginary component comprises a tripole pattern in the same region. The dipole phase of this oscillation describes a strengthening or weakening of the inertial recirculations (and hence the ocean jet that forms in the western boundary current extension), while the tripole pattern represents a shifting of the ocean jet, which is out of phase with the amplitude modulation. The second and third modes show different patterns from the first mode, but represent only 9% and 6% of the variance respectively and therefore are of secondary importance.

At any point in time, the data can be reconstructed by multiplying each spatial mode by its own time-dependent complex coefficient (called a principal component) and summing over all modes. The spectra of the principal components (Fig. 6d) therefore give information about the frequency of any periodic behavior in the modes. At higher frequencies all three modes have similar power, but the first mode has a broad, low-frequency peak centered on periods of 16 yr (indicated by the arrow). Therefore, the first EOF mode for simulations with the default parameter set describes a strong interdecadal oscillation that centers on the inertial recirculation region.

The Hilbert EOF analysis is a useful tool for diagnosing the spatial mode and showing its spectral characteristics, but this statistical method by itself gives no consideration to the physical mechanisms that cause the oscillation. However, by diagnosing the strength and character of the primary EOF mode over the range of simulations shown in section 3, we are able to extract more information about its behavior. The spatial pattern of the first Hilbert EOF modes from three different simulations (with the same parameter sets used for Figs. 3a, 4a and 5a) are shown in Fig. 7. In every case the dominant spatial mode of variability is a tripole–dipole pattern, similar in character to the pattern seen in Fig. 6a. For the high viscosity and boundary friction cases the variance attributed to the first spatial mode is significantly weaker than the default case (23% of the total variance rather than 47%). Both modes are modified by wall-trapped features similar to those observed by Berloff and McWilliams (1999); the tripole–dipole pattern, while reduced in area, is still the first-order signal. When bottom drag is increased (Fig. 7c), the observed mode is only marginally weaker than the default case, and the tripole–dipole pattern centered on the inertial recirculations is again clear.

The spectra of the first principal components are plotted in Fig. 8 for the three sensitivity tests, where in each panel the spectrum drawn with a heavy line is that of the default case. Increasing viscosity (Fig. 8a) yields a reduction in the magnitude of the spectral peak and also acts to reduce the dominant period of the oscillation. In the most viscous case (*A*_{4} = 8 × 10^{10} m^{4} s^{−1}) it is not clear that the peak is statistically significant. Increasing boundary friction (Fig. 8b) also tends to increase the frequency of the mode and reduce the power available to it. These two parameters act to control the size and strength of the inertial recirculations by changing the potential vorticity advection in the western boundary current. The similarity of the spectral signature presented here implies that the characteristics of the inertial recirculation control both the time scale and power of the dominant mode.

The dependence of the spectral characteristics of the first EOF mode upon bottom drag (Fig. 8c) shows that the frequency and magnitude of the peak are relatively insensitive to bottom drag. The exception is the case with smallest bottom friction in which the peak is weaker (but the frequency is unchanged). It is interesting to compare these results with Fig. 5d, which shows the mean kinetic energy as a function of bottom drag. There is a correlation between the mean kinetic energy in layer 1 and the size of the spectral peak of the first principal component, indicating that it is the uppermost-layer mean circulation that controls the oscillation. In contrast, mean kinetic energy in the lower layers and the eddy kinetic energy are not correlated with variability of the first principal component.

The Hilbert EOF analysis is a useful technique for selecting patterns of statistical variability. This technique includes no physical information, but we use it as a diagnostic tool to measure the pattern, strength, and frequency of variability over a wide range of simulations. Thus we are able to gauge that the dominant pattern of variability is robust over the parameter ranges simulated. In addition, the magnitude and frequency of the oscillation appear to be strongly dependent on the size of the inertial recirculations in the uppermost layer.

Hilbert EOF analysis can also provide insight into the spatial variability of SST in the model. This is relevant because oscillations of the ocean circulation may alter heat flux and thereby contribute to climate variability. The first Hilbert EOF for SST in the default case is shown in Fig. 9. The spectrum of the mode is generally red, but a low-frequency (16 yr) peak demonstrates the likelihood that this first SST EOF mode is directly linked to variations in the circulation in layer 1. This link is confirmed by the lagged cross correlation between the PC of this SST mode and the first Hilbert PC of *ψ*_{1} (Fig. 9c), which shows a strong negative correlation at zero lag and a strong positive correlation when the streamfunction leads by 10 years. The quadrature between the first SST and *ψ*_{1} modes indicates that they both participate in the same interdecadal cycle.

The spatial pattern of the SST mode is strongest in the inertial recirculation region where there is a sharp gradient in mean SST (see Fig. 2f). The two phases represent a modulation in the width (real part) and a shifting (imaginary part) of the sharp SST gradient. However, unlike the circulation EOFs, the real part of the spatial SST mode has finite amplitude over a large area of the ocean. This implies that change in the advection of heat between gyres has affected SST across the entire gyre, a consequence of efficient horizontal transport within the gyre (relative to intergyre heat transport). This nonlocal effect contributes to the likelihood that ocean circulation plays a role in midlatitude climate variability. This hypothesis requires investigation with the full coupled model, which is beyond the scope of the current paper.

## 5. Mechanisms driving variability

The data presented above are not the first simulations to suggest that low-frequency variability may arise spontaneously in wind-driven ocean circulation. Previous studies have identified similar trends in a range of different models; however, the cause of this variability is unclear. Three theories were outlined in section 1 that might explain this phenomenon: steering by multiple equilibria, eddy–mean flow feedback, or a nonlinear oscillation in the mean flow, which can be identified at low Reynolds number and persists in the turbulent regime. In this section we examine evidence for and against the relevance of each of these theories to our simulations. To help in this regard, we first refer to Fig. 10 in which two time series (from the run with *δ*_{ek} = 4 m) are plotted. The jet position and jet velocity are calculated from the position and magnitude, respectively, of the maximum eastward velocity and filtered with a 6-month low-pass filter.

These time series show three distinct phases that form the oscillation and are labeled A, B, and C (the latter of these shown by the gray bands) in the diagram. These phases are determined from the jet position. In phase A the jet is in its northernmost position (north of 2000 km, and close to the zero wind stress curl line). Phase B describes a gradual southward shift, while in phase C the jet jumps northward once again. Jet velocity is greatest in phase A and smallest in phase C. The phases marked are approximate but can be used to identify the primary findings of the Hilbert EOF analysis: that the oscillation involves shifting and modulation of the strength of the jet, which are out of phase.

### a. Multiple equilibria

If multiple equilibria exist, low-frequency variability can occur through occasional transitions between these states. Multiple equilibria can be found in low Reynolds number calculations of barotropic and 1½-layer double gyre flows (see, e.g., Cessi and Ierley 1995). It is likely that they also exist in the high Reynolds number cases simulated here, but we are unable to directly calculate these equilibria with a time-stepping model. McCalpin and Haidvogel (1996) faced the same limitations when investigating low viscosity flows in a large (1½ layer) ocean basin; they found high, medium, and low energy states of the system, and energy histograms demonstrated a preference for these states over the intermediate regimes. Primeau (2002) used similar methods but found no evidence of maxima in the energy histograms. We repeat this analysis using Q-GCM, which differs primarily from McCalpin and Haidvogel (1996) in the use of three vertical layers.

Composite means of the nonlinear component of the transport streamfunction [found by subtracting the linear circulation obtained from (10)] in phases A, B, and C are shown for layers 1 and 3 in Fig. 11. Phase A, which was previously identified as being the highest energy state, has the strongest inertial recirculations and the longest jet. The strength of the circulation in phase B is nearly equal; however, the jet is displaced to the south. It is notable that in both of these phases the jet is south of the zero wind stress curl line. Phase C, the transitionary phase, shows a short jet with relatively weak inertial recirculations.

The hypothesis of steering by multiple equilibria implies that the stable states are more likely to exist in a time-dependent model, as shown by McCalpin and Haidvogel (1996) using energy histograms. We have identified three phases in the circulation; however, we find no evidence in our data that the states shown in Fig. 11 are more likely than others. This result does not necessarily imply that multiple equilibria do not exist for the configuration used; it is likely that the strength of the turbulence produced by baroclinic instability prevents steering by multiple equilibria.

### b. Eddy feedback mechanisms

An alternative theory is that eddies rectify the mean flow in a manner that alters stability, generating a feedback loop in which the strength of the eddy field varies. We have already established in section 3 that the eddies act to barotropize the inertial recirculation gyres and that they may play a small role in amplifying these gyres. With this in mind it is noteworthy that in Fig. 11f the recirculating gyre in the lowest layer is almost absent; variation in the strength of lowest layer recirculations was identified by Spall (1996) as being a critical phase in the eddy–mean flow feedback loop. This may be produced by a weakening of the eddy field and may then act to cause the northward transition of the jet that occurs during phase C.

We investigate this possibility using statistics of the eddy field, calculated by tracking individual eddies in the simulation with TRACK, a program originally formulated to track atmospheric features such as low pressure systems and cyclones (see Hodges 1994, 1999. Thus we obtain the position and propagation speed of eddies in our simulation, allowing us to count the number of eddies detected in a region. We count eddies in a default parameter simulation as a function of longitude and time over a 900-km latitude band in the inertial recirculation region in Fig. 12b, alongside a plot of the mean jet velocity against time (which differs from Fig. 10b because of the different value of bottom drag used). It is clear from this data that a strong relationship exists between the eddy count and jet strength. This relationship may imply either that the eddy count depends upon the jet characteristics and responds to changes in the background mean flow or, alternatively, that eddies act to force changes on the background mean flow in a similar manner to that proposed by Spall (1996). The latter option would imply that eddy–mean flow feedback is responsible for the low-frequency variability observed. We aim to distinguish these possibilities by examining phases in the cycle. Spall (1996) showed that there is a delay between eddy forcing and mean flow responses that has a time scale of several years. However, if eddies are responding to the mean flow, there should be a negligible delay.

The data shown in Fig. 12 indicate a large variability in the spatial location of eddies. When the jet is strongest (phase A), the eddy field stretches across the entire basin and, as jet velocity decreases (phases B and C), the eddy field contracts to the western half of the basin. By refining the eddy counting to thinner latitude bands, one can also identify the influence of the meridional shift of the inertial recirculations on the eddy field (data not shown). However there is no indication from these data that the eddy field is leading the dynamics of the mean flow. Instead we find that the eddy counts are consistent with the hypothesis that eddies are reacting to the mean flow.

We have performed a number of additional tests designed to further investigate the possible role of eddies in controlling the low frequency circulation. First, we look at spatial maps of eddy kinetic energy, calculated by subtracting the filtered kinetic energy from the total kinetic energy. An example is shown in Fig. 13, where we average filtered and eddy kinetic energy over a 900-km latitude band and investigate variations over a 60-yr period (using the same simulation as Fig. 10). The oscillation in the path of the ocean jet is clear in the filtered kinetic energy field. The eddy kinetic energy field also follows the jet position to some extent, but eddies distribute their kinetic energy over a wider region. If feedbacks between eddies and the mean flow had occurred in the simulation, then we would expect to see some evidence in the spatial pattern of eddies. It is most likely that such a signal will be clearest in phase C of the oscillation, where the jet shifts northward. However, at all times, this diagnostic shows that eddies are weaker when the low-frequency flow is weaker and stronger when the low-frequency flow is stronger. It is implied (but not proven) by this that eddies simply respond to low-frequency variations in the mean flow.

*q*′ is the contribution from eddies. As such, we can decompose the Jacobian term in (1) as

### c. The gyre mode

The gyre mode is investigated by Simonnet and Dijkstra (2002), with reference to a number of previous studies in which similar modes are observed. Using continuation methods, Simonnet and Dijkstra (2002) extract several key features about their observed mode. This is achieved by solving for steady barotropic flows at low Reynolds number (high viscosity) and finding bifurcation points as viscosity is decreased. The first pitchfork bifurcation produces asymmetry in the flow; once this occurs, the gyre mode can be found but is damped. It becomes unstable at higher Reynolds number. If the gyre mode is traced back to the symmetric flow (Fig. 3 in Simonnet and Dijkstra 2002) it can be shown to derive from two stationary modes: a tripole (jet shifting) mode and a dipole (jet strengthening) mode. These two modes form the real and imaginary components of the gyre mode so that the two components are 90° out of phase with each other. There are numerous other studies (e.g., Jiang et al. 1995; Dijkstra and Katsman 1997; Berloff and McWilliams 1999; Chang et al. 2001) that show spatial modes that resemble the gyre mode, and it would appear to be a robust feature of these wind-driven flows.

The spatial resemblance of the gyre mode to those observed in the Q-GCM experiments is striking. Both modes have real and imaginary components that describe the modulation of the strength and position of the jet. In both cases, the mode only emerges when the inertial recirculating gyres are of sufficient strength and depends upon asymmetry in the ocean flow: in our case asymmetry is caused by the diabatic forcing term, and tests with symmetric forcing (not shown here) reveal that the mode depends on this asymmetry for its survival. (Note that in Simonnet and Dijkstra 2002 the asymmetry arises from a symmetry-breaking pitchfork bifurcation.)

However, there are several difficulties in applying the gyre mode analysis to the flows presented here. First, we need to consider the time scale of the instability. In the barotropic calculations of Simonnet and Dijkstra (2002) the period of oscillation is 1.8 yr, but it is predicted that in baroclinic models this time scale will be longer. The mode presented in the current paper has a period *O*(15 yr). Second, while the spatial pattern of the primary instability is similar across a wide range of models, it might also be argued that the modulation of a strong ocean jet is bound to be the most variable feature in an otherwise steady flow. Therefore we cannot assume, solely from the spatial pattern, that two similar looking modes result from the same physical processes, and so look for further evidence to support the existence of the gyre mode in our high Reynolds number baroclinic simulations.

The ideal strategy would be to extend the continuation methods of Simonnet and Dijkstra (2002) to higher Reynolds number and attempt to track the gyre mode through parameter space. Unfortunately, this is not computationally feasible, and so we follow Chang et al. (2001) using our time-dependent model as a “poor man’s continuation method.” This involves a series of simulations beginning at very low Reynolds number where the gyre mode can be identified. Subsequent runs in the series involve larger Reynolds number, and we examine how the primary mode of variability is altered by the higher Reynolds number. To make these experiments similar to Simonnet and Dijkstra (2002) and computationally efficient, we make several alterations to the model using lower resolution (Δ*x* = 30 km), replacing the biharmonic viscosity with a Laplacian viscosity (coefficient *A*_{2}), and altering layer structure [*H _{i}* = (700, 1000, 2300) m] so that baroclinic instability is reduced. The boundary condition is initially free slip (

*α*

_{bc}= 0) but reverts to partial slip (

*α*

_{bc}= 0.5) at higher Reynolds number.

In Fig. 14 we show the series of simulations. In each case the mean field (gray contours) and the real and imaginary parts of the first Hilbert EOF for the uppermost layer streamfunction are displayed. The field of view is reduced so as to concentrate on the inertial recirculation region. Each panel shows the viscosity in the title; the first three runs are low resolution with Laplacian viscosity, and the next two are the original high-resolution simulations that use biharmonic viscosity.

Figure 14ais the highest viscosity (*A*_{2} = 1500 m^{2} s^{−1}) at which instability occurs. The mean field and Hilbert EOF can be compared with the unstable steady state (Fig. 1) and the modal structure (Fig. 3) of the gyre mode in Simonnet and Dijkstra (2002). The mean field differs from the unstable steady state in some respects (our ocean basin is much larger and inertial recirculations smaller); however, the key characteristics of asymmetrical inertial recirculations and a short-wavelength standing Rossby wave are present in both cases. Comparison of the Hilbert EOFs and modal structure also shows significant similarities: both are dominated by a train of Rossby waves with phase velocities from the northeast, which are amplified in the recirculation region. The time scale of this mode (as seen by the well-defined peak in Fig. 14f) is 0.6 yr, which is shorter than the 1.8 yr found by Simonnet and Dijkstra (2002).

As viscosity decreases (Figs. 14b,c), the key properties of the mode are altered. First, the jet penetrates further into the interior, and the spatial shape of the mode is elongated in the east–west direction. In addition, the spectral peak of the modes broadens and the dominant time scale becomes longer—it increases to 4 yr at *A*_{2} = 300 m^{2} s^{−1} (although the spectrum is almost flat, and the peak may not be statistically significant in this case). One assumes that these characteristics are induced by the onset of turbulence in the domain.

At very high Reynolds number (Figs. 14d,e) there are more significant modifications to the mode. These changes occur against a background mean flow in which the jet penetrates the center of the domain and inertial recirculations are large. Nonetheless, direct comparison of the shape of the mode at *A*_{2} = 300 m^{2} s^{−1} and *A*_{4} = 2 × 10^{10} m^{4} s^{−1} demonstrates the likelihood of a link between the strongest mode in these two simulations. The time scale continues to lengthen with decreasing viscosity.

This analysis shows that there is a link between the gyre mode, which is observed at low Reynolds number, and the primary mode of variability in the circulation at high Reynolds number. However, this link does not imply that exactly the same dynamics are operating. Indeed, the significant modifications to the gyre mode (penetration in the ocean interior, broadband spectral characteristics, and much lower frequency at higher Reynolds number) show how geostrophic turbulence acts to control the oscillation. In particular, one is unable to predict the occurrence or time scale of the mode on analytical grounds. We demonstrate these difficulties by examining the low-order model of the gyre mode proposed by Simonnet and Dijkstra (2002).

*κ*∇

^{2}

_{H}

*h*term to the rhs of the depth tendency equation, where

*κ*is an eddy diffusivity. In the rhs of the vorticity equation [(1)], this becomes a drag term −

*κ*∇

^{2}

_{H}

*p*/

*f*

_{0}

*r*

^{2}

_{d}, where

*r*is the deformation radius. The second modification (since baroclinic eddies cannot occur) is to note that all length scales are much larger than the deformation radius, so that in the time derivative of the vorticity tendency term (∇

_{d}^{2}

_{H}

*p*−

*p*/

*r*

^{2}

_{d})/

*f*

_{0}, the Laplacian term can be considered small compared with the second, vertical stretching, term. The resulting single equation

*κ*of order 10

^{3}m

^{2}s

^{−1}, is typically an order of magnitude larger than the linear drag in the bottom layer of the default model and leads to a Stommel boundary layer width of order 100 km. The model was forced by the wind stress and diabatic terms using a time-mean SST field, giving rise to an effective forcing which is taken to be independent of

*x*but asymmetric in

*y*[and well modeled by a linear combination of terms sin(

*n*

*π*

*y*/

*Y*),

*n*= 1, 2, 3].

For typical default parameters (i.e., the same parameters used in the full model) the system settles into a steady state, with oscillatory behavior requiring distinctly larger forcing amplitudes. The oscillations, when present, were dominated by oscillations in the path (but not the magnitude) of the western boundary current extension and propagation of planetary waves westward through much of the basin. Two Hilbert EOFs were usually adequate to account for 95% of the variance. Thus the behavior differs considerably from that of the full model. The simplified model can be cast onto two or three sinusoids in the north–south direction as in Jiang et al. (1995) or Simonnet and Dijkstra (2002) and a single *x* structure chosen to fit the model solutions. The behavior of the truncated system did not resemble that of either the reduced model or the original 3-layer system. The two-component system (analyzed by both authors cited) is of course insufficient to generate chaotic solutions, and at best only damped oscillations are possible; for realistic parameters the solutions are steady. The three-component system was, predictably, more complicated, but again tended to be steady for realistic parameter ranges.

The failure of these low-order models to reproduce the behavior observed in the full model may be due either to the inadequacy of traditional eddy parameterizations, or else to the role of *x* dependence in the oscillation. The *x* dependence of the gyre mode close to the Hopf bifurcation (Fig. 14a) is very strong, although this decreases with decreasing viscosity. On the other hand, the role of eddies in this suite of models is complicated. The low-order models use a GM parameterization that may not be sufficient; comparison with the eddy-resolving simulations indicate that the eddy forcing is the opposite sign near the boundary and a factor of 10 larger in the inertial recirculation region than the GM parameterization would predict. The low resolution, low Reynolds number experiments use a strong Laplacian viscosity (a traditional replacement for eddies at low resolution, which is employed by many climate models). With this scheme, eddies can be damped out or, in some cases, allowed to persist in a limited (poorly resolved) manner. The fully turbulent simulations allow for an active eddy field by running at high resolution and using biharmonic viscosity. Here we see that the effect of eddies on the mean circulation, and on the low-frequency variability, is substantially different to the proposed parameterizations. In general, eddies act to smooth more actively in the zonal direction, resulting in elongated inertial recirculations and, subsequently, elongated patterns in the modal structure. In addition, the dominant time scale of the oscillation is lengthened by an order of magnitude or more. The physical mechanisms contributing to these effects is not clear from the experiments shown here. Future experiments are required to make accurate measurements of the contribution of eddies to reshaping the low-frequency variability in these wind-driven flows.

## 6. Conclusions

The numerical simulations presented here are the first steps in producing a coupled model that will be used to evaluate the relationship between variability in the wind-driven ocean circulation and climate variability in the midlatitudes. We have chosen an ocean-only configuration where nonlinear ocean dynamics is explicitly calculated in a high-dimensional, low-viscosity simulation. The results are rich and diverse. Variation of controlling parameters and forcing enable us to ascertain that a strong mode of decadal variability in the ocean circulation arises when eddy activity is high, nonlinearity of the circulation is strong, and asymmetry of forcing is present.

The physical characteristics of the simulated mode of variability resemble the gyre mode found in several previous studies. Through a range of experiments we have ascertained that the gyre mode, which becomes active at low Reynolds number, is modified by the turbulence as Reynolds number increases. These modifications produce patterns of variability that occupy a larger part of the domain and have a significantly longer time scale than the low Reynolds number cases.

The dependence of the model characteristics upon Reynolds number suggests either that eddies help to shape the mean flow, which has a natural oscillation frequency, or alternatively that eddies play an active role in the cycle through eddy–mean flow feedback. There was no direct evidence of eddy–mean flow feedback in the results presented here, but we prefer to interpret this result with caution: any such feedback mechanisms may not be amenable to the techniques used here. This aspect of the flow is subject to ongoing enquiries.

The data presented suggest that there are several conditions required for an ocean model to be able to simulate this type of variability. First, the boundary current must be capable of advecting the correct amount of potential vorticity—this depends upon high resolution, the viscosity scheme used, and the boundary conditions. Second, the properties of the eddy field need to be correct to model both the spatial extent and the time scale of the variability. Tests with simple eddy parameterizations revealed that they were inadequate for this task. Third, a source of asymmetry between the two gyres is required; in our model, asymmetry is provided by the forcing. In some symmetric models asymmetry can arise naturally through symmetry-breaking bifurcations (Nadiga and Luce 2001), although this effect may be limited in turbulent regimes. Once these three conditions are met, our model shows that the modified gyre mode is the natural spatial mode of low-frequency variability that arises.

The relationship between the modified gyre mode and observed modes of variability is difficult to quantify. The most relevant comparison is with oscillations in the Kuroshio Extension recorded by satellite altimetry. Qiu (2003) highlights the existence of a 12-yr oscillation (which might be related to the Pacific decadal oscillation) and demonstrates that one viable mechanism is the latitudinal dependence of Rossby wave speeds. Nonlinear mechanisms such as the modified gyre mode described here may equally well explain these observations; however, the short satellite record (11 yr) does not allow a reliable comparison between modeling and observations.

## Acknowledgments

Kevin Hodges provided the eddy tracking routine that was used in section 5b. Andrew Kiss provided feedback on an early draft of the manuscript. This work was supported by NERC, COAPEC Grant NER/T/S/2000/00319. WKD is supported by NSF Grants OCE-020884 and OCE-020700. This is a contribution of the Climate Institute, a Center of Excellence supported by the Research Foundation of the Florida State University. AMH was a visiting fellow at the Research School of Earth Sciences at The Australian National University during the latter stages of this work.

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Default parameter set.