## 1. Introduction

The most notable and energetic feature of a subtropical–subpolar gyre system is the strong frontal jet separating the two gyres, which is formed by the confluence of the corresponding western boundary currents (WBCs) after separation from the coast (the so-called WBC extension). The WBC extensions of the North Pacific Ocean and North Atlantic Ocean [the Kuroshio Extension (KE) and the Gulf Stream Extension (GSE), respectively] play a fundamental role in the climate system because they contribute substantially to the global redistribution of heat and affect the ocean–atmosphere system through vigorous air–sea interactions. A fundamental aspect of the KE and GSE that was revealed more clearly in the last decade or so thanks to the mapping of the sea surface height and temperature through remote sensing is an energetic low-frequency variability, which is decadal for the KE and interannual for the GSE (for a recent and comprehensive review, see Kelly et al. 2010).

The KE is particularly interesting, because it yields a clear bimodal behavior connecting a strong zonally elongated stable meandering jet and a weaker and more variable and convoluted jet with a reduced zonal penetration (e.g., Qiu and Chen 2010). The reason why such variability is present is under debate: Pierini and Dijkstra (2009) examine two contrasting views that have recently emerged, for which different processes are believed to control the KE low-frequency variability. In the first view, the variability is basically wind driven: wind stress changes (related to the Pacific decadal oscillation) generate a spatially broad field of baroclinic Rossby waves that is supposed to be capable of determining the observed variations of the KE frontal jet. In the second view, the variability is basically due to intrinsic nonlinear mechanisms all internal to the ocean system: in the context of the so-called double-gyre (DG) approach to the wind-driven ocean circulation (e.g., Dijkstra 2005; Dijkstra and Ghil 2005), an idealized reduced-gravity primitive equation model of the KE forced by steady winds (Pierini 2006) has produced a mean jet and a chaotic bimodal variability in reasonable agreement with altimeter observations. In the different context of the GSE, it is worth noting that Feliks et al. (2011) have recently provided convincing evidence that energetic features of the observed atmospheric low-frequency variability in the North Atlantic region are basically driven by fluctuations of the GSE rather than being the other way around, as the Rossby wave view, extended to the North Atlantic Ocean, would imply. This suggests an important role played by intrinsic mechanisms for the GSE as well.

Because both views are based on convincing experimental and numerical evidence, there should be a way to reconcile them. One possibility is to suppose that the wind variability triggers the KE bimodal cycle, which, being an internal mode of the ocean system, evolves virtually independently of the subsequent wind changes. This hypothesis is supported by both sophisticated general circulation modeling and idealized model results. On the basis of an EOF analysis of an eddy-resolving OGCM for the Earth Simulator (OFES) hindcast, Taguchi et al. (2007) suggested that the wind-driven Rossby wave signal could control the KE bimodal variability through a spatial reorganization of the KE flow, from a meridionally broad to a frontal scale structure, thanks to internal variability of the ocean system [Taguchi et al. (2010) analyzed further the role of eddy forcing in the generation of intrinsic variability]. Using the same model of Pierini (2006), Pierini (2010, hereafter P10) showed that, in a parameter range that precedes the homoclinic bifurcation responsible for the KE relaxation oscillation (RO), the latter needs a red noise wind to be excited in a scenario generally known as coherence resonance (CR): wind changes are in fact necessary to activate a bimodal cycle that, on the other hand, is intrinsic and whose evolution is therefore largely independent on successive wind variations.

CR is a relatively unexplored scenario in physical oceanography, so further modeling studies are needed to better understand the interplay of variable forcing and intrinsic oceanic effects in WBC extension low-frequency variability; moreover, investigating this aspect could contribute to clarifying other phenomena that are also believed to be associated with the occurrence of CR, such as the emergence of Dansgaard–Oeschger (DO) events (Ganopolski and Rahmstorf 2002; Ditlevsen et al. 2007) and the origin of the Atlantic multidecadal oscillation (Frankcombe et al. 2009). After having analyzed this problem through an idealized eddy-permitting primitive equation ocean model (P10), the most obvious next step is to pass to eddy-resolving process studies that include baroclinic instability (e.g., Berloff et al. 2007) or to carry out high-resolution OGCM simulations (e.g., Taguchi et al. 2007).

However, a complementary, opposite approach exists as well: that is, the approach of developing a low-order quasigeostrophic (QG) ocean circulation model through a severe truncation of a Galerkin projection of the original nonlinear partial differential equations. The underlying philosophy is that of investigating the dynamical problem on a more qualitative level (but possibly obtaining also quantitatively correct information) by using an extremely agile mathematical tool if compared with the original full problem (e.g., Lorenz 1982; Olbers 2001). This is an ideal approach when a basic physical mechanism is to be investigated from a theoretical point of view and if its low-order character is to be investigated. To stress the relevance of low-order, conceptual models in the atmospheric sciences, it is enough to quote the work of Lorenz (1963) that gave birth to the theory of chaos. The implementation of the same method for the wind-driven ocean circulation is more complex, because the oceans are embedded in closed basins and intense western boundary layers are present; few but significant low-order models have nonetheless been developed in the past. In a pioneering work, Veronis (1963, hereafter V63) derived a spectral QG ocean model in a square domain retaining only four modes in the Galerkin projection (4D model). More recently, the interest in determining the low-order character of a fundamental process such as the intrinsic midlatitude ocean variability has led Jiang et al. (1995, hereafter J95) to derive a 2D QG model that incorporated an idealized WBC. Virtually the same model was used by Simonnet and Dijkstra (2002) to investigate the physics of the oscillatory gyre mode. Finally, Simonnet et al. (2005, hereafter S05) developed a 4D model with the aim of obtaining a deeper understanding of the occurrence of homoclinic bifurcations in the double-gyre problem.

Following this approach, in this paper a 4D spectral QG model of the wind-driven ocean circulation is derived and used to investigate the low-order character of the intrinsic low-frequency variability of the midlatitude double-gyre ocean circulation and of the corresponding coherence resonance phenomenon. In section 2, the derivation of the model is presented, stressing the improvements over previous low-order QG ocean models; the double-gyre steady and stochastic wind fields used to force the model are then discussed. In section 3, the intrinsic low-frequency variability obtained under steady forcing is presented and discussed also in terms of dynamical systems theory. In section 4a, the coherence resonance phenomenon found acting under stochastic forcing is discussed, whereas in section 4b a method denoted as “phase selection” (PS) is proposed and used to analyze the excitation mechanism. In section 4c, these results are discussed within the general context of climate dynamics. Finally, in section 5, conclusions are drawn.

## 2. The low-order ocean model

### a. Derivation of the 4D truncated ocean model

*β*is the planetary vorticity gradient,

*r*is the Rayleigh friction coefficient,

*ρ*is the mean density,

*H*is the upper-layer thickness, and

*f*is the Coriolis parameter,

*g*′ =

*g*Δ

*ρ*/

*ρ*is the reduced gravity,

*g*is the acceleration of gravity, and Δ

*ρ*/

*ρ*is the relative variation of density across the interface). The interface displacement

*x*and

*y*with two different length scales will be clarified later). With the further definitions

*= [*

**τ***τ*(

*y*,

*t*), 0], one finally gets Eq. (1) in dimensionless form,

*ψ*as

*α*is a real positive constant. This choice follows that of J95, who adopted |

*i*〉,

*i*= 1, 2 as in (7) for their 2D model. The exponential factor allows for a simple but significant representation of the westward intensification, whereas the use of sine functions insures that the free-slip boundary conditions (with

*ψ*= 0) are automatically satisfied in the domain

*x*,

*y*∈ [0,

*π*]. Here, unlike in V63, J95, and S05, the use of two different length scales

*a*and

*b*allows one to consider a rectangular domain with lengths

*L*=

_{x}*a*

*π*and

*L*=

_{y}*b*

*π*, instead of a square domain, which appears to be an undesirable constraint. Here one can, for instance, choose a meridional length

*L*that fits a DG wind system and, independently, a zonal length

_{y}*L*that satisfies conditions related to the structure of the WBC (see the next subsection). On the other hand, the basis (7) differs from the 4D model of S05 in that, in the latter, |3〉 and |4〉 contain the first harmonic in

_{x}*x*and the third and fourth harmonics in

*y*, respectively. The present choice (the same of V63, who, however, did not include the exponential factor) is preferred in order to introduce a further length scale along

*x*, in so achieving a balanced spectral resolution in the two directions.

*i*〉 is orthonormal,

*i*|, one gets

*n*and

_{i}*w*are the nonlinear and wind forcing terms, respectively [the coefficients of the linear terms and

_{i}*w*are reported in (A1); for

_{i}*n*, see below]. Equations (10a) and (10c) and Eqs. (10b) and (10d) can then be combined to get ODEs with a single temporal coefficient derived with respect to time,

_{i}*W*are reported in (A2) and (A3), whereas the relation between

_{i}**n**and

**N**is given in (A5).

List of the main features of the low-order models of the wind-driven ocean circulation.

*N*. The vector components

_{i}*n*in (10) are given by

_{i}*Q*〉(

_{jk}*α*,

*λ*,

*γ*,

**x**) is the symmetric tensor,

*N*appearing in (11),

_{i}*J*(symmetric in

_{ijk}*j*and

*k*) is given by (A6). It is worth noting that the formal expressions (12) and (13) are independent of the choice of the basis and of the scalar product, whereas (14) depends on them.

*ψ*expanded as in (6) and (7) under the scalar product (8) reduces to the set of four coupled nonlinear ODEs [(11a)–(11d); with the nonlinear terms

*N*given by (14); see also (A7)], which can be rewritten in the compact form

_{i}**W**are given by (A6), (A4), and (A3), respectively. Equations (15) have been integrated numerically through the Bulirsch–Stoer method, which makes use of the Richardson extrapolation with a rational function extrapolation routine (Press et al. 1992).

### b. Steady wind forcing and domain of integration

*L*= 3500 km so that a wind stress curl forcing and subtropical and a subpolar gyres with realistic meridional scales can be introduced. The dimensional wind stress curl,

_{y}*θ*= 1 (Fig. 1), is basically the one derived from the European Centre for Medium-Range Weather Forecasts (ECMWF) climatology by Pierini (2006) for the North Pacific. As we will see in section 3, a factor

*θ*≫ 1 is needed to obtain nonlinear effects of reasonable amplitude and consequently a realistic bifurcation scenario in such a severely truncated model. This is an obvious necessity in low-order models of highly nonlinear frontal jets. V63 used exaggerate parameter values to obtain nonlinear effects leading to a rich variety of time-dependent flows; S05 obtained gyre modes and ROs in their full QG model under realistic wind stress curl amplitudes, but a forcing three orders of magnitude more intense was required to obtain an analogous bifurcation scenario in their truncated 4D model. Thus, the tuning of

*θ*must be seen as an empirical renormalization necessary to attain a realistic degree of nonlinearity when only a few degrees of freedom are available.

The zonal dimension *L _{x}* = 3000 km is determined in order to achieve a compromise between an acceptable width of the WBCs,

*L*

_{WBC}, and sufficiently long WBC extensions. From (7), it derives that the narrowest WBC is obtained for

*α*≅ 0.9, so, with that

*L*,

_{x}*L*

_{WBC}≅ 500 km (see Fig. 5). A more realistic width of O(100 km) could only be obtained (with the same

*α*) by choosing an exceedingly small

*L*, which would not allow for sufficiently long WBC extensions. On the other hand, a larger

_{x}*L*would increase

_{x}*L*

_{WBC}even further without really improving the zonal resolution in such a highly truncated model.

### c. Stochastic wind forcing

*dτ*

_{0}(

*y*)/

*dy*is (16) in dimensionless form and

*ε*

_{1}and

*ε*

_{2}are small dimensionless numbers. In the temporal factor

*G*, ζ is a red noise satisfying the Ornstein–Uhlenbeck stochastic differential equation,

*ξ*is a Gaussian white noise with zero mean and unit variance,

*χ*

_{1}and

*χ*

_{2}are positive constants, and

*σ*is the rms of

_{ζ}*ζ*. This is the same stochastic component used by P10 to study CR in his KE model (Pierini 2006); in the applications of sections 4a and 4b, it will be characterized by the corresponding decorrelation time

*T*(see P10 for details). The third term in

_{s}*G*is a periodic signal with period

*T*, where

_{P}*ω*= 2

*π*/

*T*: it will be used in section 4b.

_{p}## 3. Solution under steady forcing: Intrinsic low-frequency variability

Here, we analyze the model behavior by integrating Eqs. (15) under the steady forcing (16) with vanishing initial conditions, for the reference value *r* = 1 × 10^{−8} s^{−1} of the friction coefficient and for varying intensity of the forcing *μ* = *θ*/*θ*_{0}, where *θ*_{0} = 574.85 is the value of *θ* at which a global bifurcation occurs (different values of *r* are used only to derive the bifurcation diagram of Fig. 2b). Other parameters are *β* = 2 × 10^{−11} rad m^{−1} s^{−1}, *ρ* = 1000 kg m^{−3}, *H* = 200 m, and *L _{R}* = 30 km (which, for

*f*= 10

^{−4}rad s

^{−1}, implies Δ

*ρ*/

*ρ*= 4.6 × 10

^{−3}).

Bifurcation diagrams showing the range of variability of Ψ_{1} (scaled with 10^{−5}) (a) vs the normalized wind amplitude *μ*, with *r* = 1 × 10^{−8} s^{−1}, and (b) vs the friction coefficient *r*, with *μ* = 1. The vertical dashed line marks the global bifurcation. In (a), the dot denotes the first Hopf bifurcation, and the thick dashed line denotes the branch of the unstable steady state.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Bifurcation diagrams showing the range of variability of Ψ_{1} (scaled with 10^{−5}) (a) vs the normalized wind amplitude *μ*, with *r* = 1 × 10^{−8} s^{−1}, and (b) vs the friction coefficient *r*, with *μ* = 1. The vertical dashed line marks the global bifurcation. In (a), the dot denotes the first Hopf bifurcation, and the thick dashed line denotes the branch of the unstable steady state.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Bifurcation diagrams showing the range of variability of Ψ_{1} (scaled with 10^{−5}) (a) vs the normalized wind amplitude *μ*, with *r* = 1 × 10^{−8} s^{−1}, and (b) vs the friction coefficient *r*, with *μ* = 1. The vertical dashed line marks the global bifurcation. In (a), the dot denotes the first Hopf bifurcation, and the thick dashed line denotes the branch of the unstable steady state.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

A bifurcation analysis was performed following the empirical continuation method based on a large number of forward integrations for different values of the control parameter (e.g., as done by Pierini et al. 2009). The bifurcation diagram with *μ* chosen as the control parameter is shown in Fig. 2a, where the range of Ψ_{1} after spinup is drawn for *r* = 1 × 10^{−8} s^{−1}. The analogous diagram in which the control parameter is *r*, for *μ* = 1, is shown in Fig. 2b, but, without loss of generality, the following discussion will be referring only to Fig. 2a.

In Fig. 3, 200-yr-long time series of Ψ_{1} are shown for several values of *μ*. The first Hopf bifurcation (at *μ* = 0.348, dot in Fig. 2a) is followed by an extremely small range up to *μ ≈* 0.95, after which the range increases much more rapidly. At *μ* = 0.996, the first period doubling appears (e.g., cf. Figs. 3d,e), followed by the typical period-doubling cascade. At *μ* = 1, a global bifurcation occurs, marked by the appearance of a large-amplitude oscillation bringing the system far away from the unstable fixed point that gave rise to the previous local bifurcations. For increasing values of *μ*, the large-amplitude oscillation appears more frequently but always within a periodic behavior, until *μ* = 1.35, after which homoclinic chaos is found (e.g., Fig. 3l). The duration of each oscillation ranges from 5 to 10 yr before the global bifurcation up to ~15 yr after it.

Time series of Ψ_{1} (scaled with 10^{−5}) obtained for several values of *μ*.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Time series of Ψ_{1} (scaled with 10^{−5}) obtained for several values of *μ*.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Time series of Ψ_{1} (scaled with 10^{−5}) obtained for several values of *μ*.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

The thick dashed line in Fig. 2a represents the unstable steady state around which the system fluctuates and was determined as follows: In a run with a reference value of the wind amplitude chosen as *μ* = 0.87 (*θ* = 500) after the spinup phase (when, therefore, a virtually steady flow is attained), *μ* is gradually increased up to desired value _{1} variability passes from the dashed line to the range given by the two solid lines corresponding to the same value of *μ*). This procedure no longer works if homoclinic chaos is in play.

To obtain a more complete view of the model intrinsic variability, in Fig. 4 the orbits in the Ψ_{1}–Ψ_{3} (black lines) and Ψ_{2}–Ψ_{4} (gray lines) planes are reported for four values of *μ* included in those considered in Fig. 3. Comparing Figs. 4a,b shows the transition from a (virtually) fixed point to a large-amplitude limit cycle; comparing Figs. 4b,c shows the nature of the global bifurcation that leads to the homoclinic orbits for *μ >* 1. Finally, Fig. 4d shows the strange attractor describing the intrinsic variability in the homoclinic chaos regime. The orbits of Figs. 4c,d suggest the occurrence of the Shilnikov scenario (e.g., S05), for which the homoclinic orbit, after having moved through the unstable manifold, spirals back to the unstable fixed point.

Orbits in the Ψ_{1}–Ψ_{3} (black lines) and Ψ_{2}–Ψ_{4} (gray lines) planes for four different values of *μ* (reported in each graph).

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Orbits in the Ψ_{1}–Ψ_{3} (black lines) and Ψ_{2}–Ψ_{4} (gray lines) planes for four different values of *μ* (reported in each graph).

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Orbits in the Ψ_{1}–Ψ_{3} (black lines) and Ψ_{2}–Ψ_{4} (gray lines) planes for four different values of *μ* (reported in each graph).

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

The spatial structure of the circulation patterns is presented in Figs. 5–7 through the streamfunction *ψ*. A first qualitative view of the flow is provided, for *μ* = 1, by the steady state obtained by imposing vanishing nonlinear terms in (15) (

Maps of *ψ* (scaled with 10^{−5}) for *μ* = 1 (a) for the steady state obtained by imposing vanishing nonlinear terms and (b) for the nonlinear unstable steady state.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Maps of *ψ* (scaled with 10^{−5}) for *μ* = 1 (a) for the steady state obtained by imposing vanishing nonlinear terms and (b) for the nonlinear unstable steady state.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Maps of *ψ* (scaled with 10^{−5}) for *μ* = 1 (a) for the steady state obtained by imposing vanishing nonlinear terms and (b) for the nonlinear unstable steady state.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(1)–(8) Sequence of 8 snapshots of *ψ* (scaled with 10^{−5}) for *μ* = 0.991 (time increment of 0.9 yr). The corresponding evolution of Ψ_{1} is shown by the dots in Fig. 3d.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(1)–(8) Sequence of 8 snapshots of *ψ* (scaled with 10^{−5}) for *μ* = 0.991 (time increment of 0.9 yr). The corresponding evolution of Ψ_{1} is shown by the dots in Fig. 3d.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(1)–(8) Sequence of 8 snapshots of *ψ* (scaled with 10^{−5}) for *μ* = 0.991 (time increment of 0.9 yr). The corresponding evolution of Ψ_{1} is shown by the dots in Fig. 3d.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

As in Fig. 6, but for *μ* = 1.043 (time increment of 1 yr). The corresponding evolution of Ψ_{1} is shown by the dots in Fig. 3j.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

As in Fig. 6, but for *μ* = 1.043 (time increment of 1 yr). The corresponding evolution of Ψ_{1} is shown by the dots in Fig. 3j.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

As in Fig. 6, but for *μ* = 1.043 (time increment of 1 yr). The corresponding evolution of Ψ_{1} is shown by the dots in Fig. 3j.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

The character of the intrinsic variability is shown before and after the global bifurcation in Figs. 6 and 7, respectively, through a sequence of eight snapshots of *ψ* describing an entire oscillation (the value of Ψ_{1} corresponding to each snapshot is visualized with a dot in Fig. 3d for Fig. 6 and in Fig. 3j for Fig. 7). For *μ* = 0.991 (Fig. 6; time increment of 0.9 yr), the signal is periodic with a period of 6.9 yr. The system resides for about 3 yr (Fig. 6, panels 3–6) in a DG state similar to the linear solution of Fig. 5; subsequently, the separation point of the WBC extension moves northward until the southern anticyclonic gyre invades the whole western boundary. A rapid relaxation to a virtually zonal WBC extension occurs thereafter (Fig. 6, panels 1–3). For *μ* = 1.043 (Fig. 7; time increment of 1 yr), the signal is again periodic but with a period of 15.1 yr. In this case, the variability is more complex, because now the separation point moves both northward and southward, and the cyclonic and anticyclonic gyres alternatively invade the whole western boundary in a sort of RO with strong temporal asymmetries.

Although it is obvious that a 4D model cannot produce flows with an acceptable degree of realism, the variability of Fig. 7 yields nonetheless interesting features that define a nice low-order prototype of intrinsic DG RO. Indeed, the latter (i) possesses a bimodal character, (ii) presents both zonal and meridional undulations of the WBC extension about a basic DG structure, and (iii) occurs for periods that are basically in agreement with the real oceanic data.

## 4. Solution under stochastic forcing: Coherence resonance and phase selection

### a. Coherence resonance

In this subsection, an analysis will be carried out of the excitation of the RO by stochastic winds in a parameter range that precedes the global bifurcation, following the same procedure applied by P10 in his study of CR in a shallow-water model of the KE.

We start by considering the state with *μ* = 0.957, which, as shown in Fig. 3c (see also Fig. 2a), corresponds to a small-amplitude limit cycle under steady forcing (different values of *μ* will be considered later). The idea of CR is that, if the system before the global bifurcation is excitable (i.e., if it possesses a nonlinear internal mode in the form of a RO that cannot arise autonomously), a stochastic forcing may be able to excite the oscillation, which will then relax, following an evolution virtually independent of the forcing itself. To this respect, the results of six numerical experiments are reported in Fig. 8, where the temporal dependence *G*(*t*) (with *ε*_{1} = 0.2) of the wind stress curl forcing (Figs. 8a–f) is followed by the corresponding model response in terms of Ψ_{1} (Figs. 8a′–f′). Although a white noise wind is unable to excite the RO, red noise winds can induce the excitation, which is most efficient for *T _{s}* = 1 yr (Fig. 8d′). This is in significant agreement with the results in section 4c of P10 found for shallow-water partial differential equations.

Temporal dependence *G*(*t*) of the wind stress curl forcing with *ε*_{1} = 0.2 and *ε*_{2} = 0: (a) for a white noise and a red noise with *T _{s}* = (b) 0.05, (c) 0.1, (d) 1.0, (e) 5.0 , and (f) 10 yr. (a′)–(f′) Corresponding model response of Ψ

_{1}(scaled with 10

^{−5}) with

*μ*= 0.957.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Temporal dependence *G*(*t*) of the wind stress curl forcing with *ε*_{1} = 0.2 and *ε*_{2} = 0: (a) for a white noise and a red noise with *T _{s}* = (b) 0.05, (c) 0.1, (d) 1.0, (e) 5.0 , and (f) 10 yr. (a′)–(f′) Corresponding model response of Ψ

_{1}(scaled with 10

^{−5}) with

*μ*= 0.957.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Temporal dependence *G*(*t*) of the wind stress curl forcing with *ε*_{1} = 0.2 and *ε*_{2} = 0: (a) for a white noise and a red noise with *T _{s}* = (b) 0.05, (c) 0.1, (d) 1.0, (e) 5.0 , and (f) 10 yr. (a′)–(f′) Corresponding model response of Ψ

_{1}(scaled with 10

^{−5}) with

*μ*= 0.957.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

To analyze the dependence of this behavior upon the distance Δ*μ* of *μ* from 1 (Δ*μ* = 1 − *μ*), beyond which the RO arises spontaneously, time series of Ψ_{1} obtained for several values of *μ* under stochastic forcing (with ε_{1} = 0.2, ε_{2} = 0, and *T _{S}* = 1 yr) are shown in the left column of Fig. 9 (the superimposed thick lines show the response to steady forcing ε

_{1}= 0). The results show that, for this value of

*T*, the CR mechanism is very robust, because ROs can sporadically be excited, even starting from a basic state, which is virtually steady, as is the case for

_{S}*μ*= 0.887 (Fig. 9a). In general, the behavior does not appear to be very sensitive to Δ

*μ.*Moreover, to test the typical behavior of CR (Lindner et al. 2004), six more cases are considered: all with

*μ*= 0.957, ε

_{2}= 0, and

*T*= 1 yr but with six different values of ε

_{s}_{1}. The results (Fig. 9, right) show that, in fact, the “activation time” (i.e., the mean temporal distance between two ROs) decreases with increasing noise level, whereas the “excursion time” (i.e., the duration of the RO) is virtually independent of ε

_{1}(see Figs. 9h–k). On the other hand, for exceedingly small noise intensity (e.g., ε

_{1}= 0.025), the wind noise is not able to excite any RO (Fig. 9g), whereas, for exceedingly large noise intensity (e.g., ε

_{1}= 0.8), the ROs themselves are clearly affected by the stochastic nature of the forcing (Fig. 9l). This behavior justifies the use of the term coherence resonance: for an optimal range of noise intensity, the system resonates so as to produce a strong signal characterized by a series of coherent ROs. Finally, it is worth emphasizing that both the weak dependence of CR on Δ

*μ*and the typical dependence of resonance on noise amplitude just discussed are again in substantial agreement with the findings of P10.

(left) Time series of Ψ_{1} obtained for several values of *μ* under stochastic forcing with *ε*_{1} = 0.2, *ε*_{2} = 0, and *T _{s}* = 1 yr (the superimposed thick lines show the response to steady forcing

*ε*

_{1}= 0). (right) Time series of Ψ

_{1}obtained for several values of

*ε*

_{1}under stochastic forcing with

*μ*= 0.957,

*ε*

_{2}= 0, and

*T*= 1 yr. Here, Ψ

_{s}_{1}is scaled with 10

^{−5}.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(left) Time series of Ψ_{1} obtained for several values of *μ* under stochastic forcing with *ε*_{1} = 0.2, *ε*_{2} = 0, and *T _{s}* = 1 yr (the superimposed thick lines show the response to steady forcing

*ε*

_{1}= 0). (right) Time series of Ψ

_{1}obtained for several values of

*ε*

_{1}under stochastic forcing with

*μ*= 0.957,

*ε*

_{2}= 0, and

*T*= 1 yr. Here, Ψ

_{s}_{1}is scaled with 10

^{−5}.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(left) Time series of Ψ_{1} obtained for several values of *μ* under stochastic forcing with *ε*_{1} = 0.2, *ε*_{2} = 0, and *T _{s}* = 1 yr (the superimposed thick lines show the response to steady forcing

*ε*

_{1}= 0). (right) Time series of Ψ

_{1}obtained for several values of

*ε*

_{1}under stochastic forcing with

*μ*= 0.957,

*ε*

_{2}= 0, and

*T*= 1 yr. Here, Ψ

_{s}_{1}is scaled with 10

^{−5}.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Until now, we have judged the substantial invariance of the RO under stochastic forcing by referring only to the temporal evolution of Ψ_{1}. A way to check if such invariance actually holds in the full phase space is to compare the orbits in the Ψ_{1}–Ψ_{3} and Ψ_{2}–Ψ_{4} planes with those obtained with steady forcing past the global bifurcation. Figures 10b–d show the orbits corresponding to the cases of Figs. 9h,j,l, respectively (Fig. 10a shows the response obtained with the same parameters but with *ε*_{1} = 0). If we now compare the orbits of Figs. 10b,c produced by the CR mechanism with, for instance, that of Fig. 4c produced by steady forcing (where ROs arise spontaneously), we get convincing evidence that the character of the RO is in fact virtually the same. In Fig. 10d, on the other hand, the exceedingly large noise intensity produces a significant deformation of the trajectories (as discussed above), although the overall shape of the original RO can still be appreciated.

Orbits in the Ψ–Ψ_{3} (black lines) and Ψ_{2}–Ψ_{4} (gray lines) planes for *μ* = 0.957 (a) under steady forcing and under stochastic forcing with *ε*_{2} = 0; *T _{s}* = 1 yr; and (b)

*ε*

_{1}= 0.05, (c)

*ε*

_{1}= 0.2, and (d)

*ε*

_{1}= 0.8.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Orbits in the Ψ–Ψ_{3} (black lines) and Ψ_{2}–Ψ_{4} (gray lines) planes for *μ* = 0.957 (a) under steady forcing and under stochastic forcing with *ε*_{2} = 0; *T _{s}* = 1 yr; and (b)

*ε*

_{1}= 0.05, (c)

*ε*

_{1}= 0.2, and (d)

*ε*

_{1}= 0.8.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Orbits in the Ψ–Ψ_{3} (black lines) and Ψ_{2}–Ψ_{4} (gray lines) planes for *μ* = 0.957 (a) under steady forcing and under stochastic forcing with *ε*_{2} = 0; *T _{s}* = 1 yr; and (b)

*ε*

_{1}= 0.05, (c)

*ε*

_{1}= 0.2, and (d)

*ε*

_{1}= 0.8.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

### b. Excitation mechanism and phase selection

We have just seen how the occurrence of CR depends in a subtle way both on the decorrelation time and amplitude of the noise forcing. The mechanism that leads to the actual excitation of a single RO is also subtle. Why is, for example, an RO excited in Fig. 9h at *t* ≅ 255 yr but not at a different time? (In the language of dynamical systems theory, the excitation occurs when the system response to the forcing is such that the orbit finds its way out of the basin of attraction of the local attractor following an unstable manifold afterward.) Understanding such a mechanism is of crucial importance, because it would provide valuable information on the functioning and predictability of the system. Unfolding the excitation mechanism requires the identification of the typical features of both the wind forcing and the state of the system just before a RO; such a characterization would then allow one to predict (within a presumably restricted temporal range) the occurrence of the excitation itself. This is an ambitious task that we start to consider here by focusing mainly on the wind forcing.

Characterizing the wind before the excitation, if the time-dependent part of the former is just noise, is problematic because of the variety of temporal scales involved, so an alternative technique is proposed in this section. The system is forced by the usual mean field plus a Fourier component (*ε*_{2} ≠ 0), first without any noise (*ε*_{1} ≠ 0) and then perturbed by a red noise. The advantage of using a fictitious periodic wind lies in the simplicity in characterizing the excitation in terms of the phase of the forcing corresponding to the beginning of the RO, along with its frequency and amplitude. The proposed procedure leads to a phenomenon denoted here as PS, a weak form of phase locking (see section 4c), which will now be defined.

_{1}(thin line) obtained under a mean wind (with

*μ*= 0.957) plus a periodic component (thick line, with

*ε*

_{2}= 0.2 and

*T*= 20 yr) and a stochastic perturbation (with

_{P}*ε*

_{1}= 0.1 and

*T*= 1 yr) is shown. In this 200-yr-long time series, six ROs are excited, and it is clear that their emergence is in some way correlated to the periodic signal. Because, in view of the CR mechanism, some form of synchronization may be expected at the beginning of the oscillation, we will characterize each RO (conventionally defined here as an oscillation for which the relative maximum Ψ

_{S}_{1max}of Ψ

_{1}is

*t*

_{z}at which Ψ

_{1}vanishes just before Ψ

_{1max}. With

*t*(

_{z}*k*) (

*k*= 1, … ,

*N*, with

_{z}*N*= 6) are evidenced by the dots in Fig. 11a. Then, from (17), one can use the phases

_{z}*ϕ*shown by the dots in Fig. 11b yields a clustering around

_{z}*ϕ ≈*200°, suggesting that those ROs are strongly affected by the phase of the periodic signal (with that particular frequency and amplitude): this is what we mean here by PS. Of course, six oscillations are statistically not very significant, so in the following, 1000-yr-long time series will be considered, and the relative frequency

*υ*(

*ϕ*) of occurrence of

*ϕ*will be used to analyze PS.

_{z}(a) Time series of Ψ_{1} (thin line; scaled with 10^{−5}) obtained under a mean wind (with *μ* = 0.957) plus a periodic component (thick line; with *ε*_{2} = 0.2 and *T _{p}* = 20 yr) and a stochastic perturbation (with

*ε*

_{1}= 0.1 and

*T*= 1 yr). The six dots correspond to the times

_{s}*t*(

_{z}*k*),

*k*= 1, … , 6, whereas the threshold

*ϕ*defined by (19).

_{z}Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(a) Time series of Ψ_{1} (thin line; scaled with 10^{−5}) obtained under a mean wind (with *μ* = 0.957) plus a periodic component (thick line; with *ε*_{2} = 0.2 and *T _{p}* = 20 yr) and a stochastic perturbation (with

*ε*

_{1}= 0.1 and

*T*= 1 yr). The six dots correspond to the times

_{s}*t*(

_{z}*k*),

*k*= 1, … , 6, whereas the threshold

*ϕ*defined by (19).

_{z}Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(a) Time series of Ψ_{1} (thin line; scaled with 10^{−5}) obtained under a mean wind (with *μ* = 0.957) plus a periodic component (thick line; with *ε*_{2} = 0.2 and *T _{p}* = 20 yr) and a stochastic perturbation (with

*ε*

_{1}= 0.1 and

*T*= 1 yr). The six dots correspond to the times

_{s}*t*(

_{z}*k*),

*k*= 1, … , 6, whereas the threshold

*ϕ*defined by (19).

_{z}Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

The three simulations shown in Fig. 12 describe the transition to PS for increasing amplitude of the periodic forcing. In Fig. 12a, the temporal dependence *G*(*t*) of the forcing with *ε*_{1} = 0.1, *T _{s}* = 10 yr,

*ε*

_{2}= 0.01, and

*T*= 10 yr is shown by the thin line (the thick line represents the periodic component), whereas the corresponding model response in terms of Ψ

_{p}_{1}with

*μ*= 0.957 is reported in Fig. 12a′. CR is clearly active, but the frequency distribution

*υ*(

*ϕ*) does not yield any significant clustering, as it could be expected in consideration of the weakness of the periodic forcing. If

*ε*

_{2}is increased by a factor of 5 (Figs. 12b,b′), the situation remains basically unchanged, but a further factor of 4 (

*ε*

_{2}= 0.2, Figs. 12c,c′) leads not only to a more effective CR (the total number

*N*of ROs increases from 15 to 33) but also to a clear confinement of

_{z}*υ*(

*ϕ*) centered at

*ϕ ≈*280°, denoting PS.

(a) Temporal dependence *G*(*t*) of the wind stress curl forcing with both red noise and periodic components: *ε*_{1} = 0.1, *T _{s}* = 10 yr,

*ε*

_{2}= 0.01, and

*T*= 10 yr. (b),(c) As in (a), but with (b)

_{p}*ε*

_{2}= 0.05 and (c)

*ε*

_{2}= 0.2. The periodic component and the total signal are shown by the thick and thin lines, respectively. (a′)–(c′) Corresponding model response of Ψ

_{1}(scaled with 10

^{−5}) with

*μ*= 0.957. The three histograms show the relative frequency

*υ*(

*ϕ*) of occurrence of

*ϕ*

_{z}for each experiment. The total number

*N*of

_{z}*t*in each case is reported in the top-left corner.

_{z}Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(a) Temporal dependence *G*(*t*) of the wind stress curl forcing with both red noise and periodic components: *ε*_{1} = 0.1, *T _{s}* = 10 yr,

*ε*

_{2}= 0.01, and

*T*= 10 yr. (b),(c) As in (a), but with (b)

_{p}*ε*

_{2}= 0.05 and (c)

*ε*

_{2}= 0.2. The periodic component and the total signal are shown by the thick and thin lines, respectively. (a′)–(c′) Corresponding model response of Ψ

_{1}(scaled with 10

^{−5}) with

*μ*= 0.957. The three histograms show the relative frequency

*υ*(

*ϕ*) of occurrence of

*ϕ*

_{z}for each experiment. The total number

*N*of

_{z}*t*in each case is reported in the top-left corner.

_{z}Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(a) Temporal dependence *G*(*t*) of the wind stress curl forcing with both red noise and periodic components: *ε*_{1} = 0.1, *T _{s}* = 10 yr,

*ε*

_{2}= 0.01, and

*T*= 10 yr. (b),(c) As in (a), but with (b)

_{p}*ε*

_{2}= 0.05 and (c)

*ε*

_{2}= 0.2. The periodic component and the total signal are shown by the thick and thin lines, respectively. (a′)–(c′) Corresponding model response of Ψ

_{1}(scaled with 10

^{−5}) with

*μ*= 0.957. The three histograms show the relative frequency

*υ*(

*ϕ*) of occurrence of

*ϕ*

_{z}for each experiment. The total number

*N*of

_{z}*t*in each case is reported in the top-left corner.

_{z}Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

The result of a more extensive analysis based on 16 numerical experiments is reported in Figs. 13 and 14, where *υ*(*ϕ*) and Ψ_{1}(t) are shown, respectively. In all cases *μ* = 0.957 (the corresponding Ψ_{1} under steady forcing is the small-amplitude periodic oscillation of Fig. 3c) and *ε*_{2} = 0.2; moreover, each of the four columns corresponds to a different frequency of the periodic component (from left to right, *T _{p}* = 5, 10, 15, and 20 yr). Let us begin by considering a steady forcing plus a purely periodic signal: in such a case in the first row of Fig. 13 (

*ε*

_{1}= 0), a clear phase selection emerges (with vast ranges of

*ϕ*with vanishing

*υ*) that is particularly significant for

*T*= 10 and 15 yr (cases a2 and a3), when

_{p}*N*is so large that the excitation of ROs covers the whole time series (see Fig. 14; the even more pronounced confinement of

_{z}*υ*in cases a1 and a4 corresponds, on the contrary, to a sporadic excitation of the system). The multimodal character of

*υ*, particularly evident in cases a2 and a3, is associated with the emergence of oscillations with different durations (~15 yr and ~25–30 yr), which however correspond to essentially the same RO behavior described in section 3. In simulations (not shown) with

*T*smaller than ≈5 yr or greater than ≈20 yr, the system remains in the unexcited state (

_{p}*N*~ 0). In summary, in the present parameter range a steady forcing plus a purely periodic signal is able to excite the system only if

_{z}*T*is comparable to the duration of the typical RO, with a peak of

_{p}*N*for

_{s}*T*~ 10 yr. Moreover, a PS in the form of a confinement (with multimodal distribution) of

_{p}*υ*(

*ϕ*) always emerges.

(top) Relative frequency *υ*(*ϕ*) of occurrence of *ϕ _{z}* over a 1000-yr-long time series under purely periodic forcing (

*ε*

_{1}= 0) with

*μ*= 0.957,

*ε*

_{2}= 0.2 and

*T*= (a1) 5, (a2) 10, (a3) 15, and (a4) 20 yr; (top middle) As in (top), but also with red noise forcing with

_{p}*ε*

_{1}= 0.1 and

*T*= 1 yr. (bottom middle) As in (top), but also with red noise forcing with

_{s}*ε*

_{1}= 0.2 and

*T*= 1 yr. (bottom) As in (top), but also with red noise forcing with

_{s}*ε*

_{1}= 0.1 and

*T*= 10 yr. The total number

_{s}*N*of

_{z}*t*for each simulation is reported in the top-left corner; the phase in the

_{z}*x*axis is in degrees (in cases a1, b1, c1, and d1 and in cases a2, b2, c2, and d2, the phase is shifted by 160° and 40°, respectively).

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(top) Relative frequency *υ*(*ϕ*) of occurrence of *ϕ _{z}* over a 1000-yr-long time series under purely periodic forcing (

*ε*

_{1}= 0) with

*μ*= 0.957,

*ε*

_{2}= 0.2 and

*T*= (a1) 5, (a2) 10, (a3) 15, and (a4) 20 yr; (top middle) As in (top), but also with red noise forcing with

_{p}*ε*

_{1}= 0.1 and

*T*= 1 yr. (bottom middle) As in (top), but also with red noise forcing with

_{s}*ε*

_{1}= 0.2 and

*T*= 1 yr. (bottom) As in (top), but also with red noise forcing with

_{s}*ε*

_{1}= 0.1 and

*T*= 10 yr. The total number

_{s}*N*of

_{z}*t*for each simulation is reported in the top-left corner; the phase in the

_{z}*x*axis is in degrees (in cases a1, b1, c1, and d1 and in cases a2, b2, c2, and d2, the phase is shifted by 160° and 40°, respectively).

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(top) Relative frequency *υ*(*ϕ*) of occurrence of *ϕ _{z}* over a 1000-yr-long time series under purely periodic forcing (

*ε*

_{1}= 0) with

*μ*= 0.957,

*ε*

_{2}= 0.2 and

*T*= (a1) 5, (a2) 10, (a3) 15, and (a4) 20 yr; (top middle) As in (top), but also with red noise forcing with

_{p}*ε*

_{1}= 0.1 and

*T*= 1 yr. (bottom middle) As in (top), but also with red noise forcing with

_{s}*ε*

_{1}= 0.2 and

*T*= 1 yr. (bottom) As in (top), but also with red noise forcing with

_{s}*ε*

_{1}= 0.1 and

*T*= 10 yr. The total number

_{s}*N*of

_{z}*t*for each simulation is reported in the top-left corner; the phase in the

_{z}*x*axis is in degrees (in cases a1, b1, c1, and d1 and in cases a2, b2, c2, and d2, the phase is shifted by 160° and 40°, respectively).

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Time series of Ψ_{1} (scaled with 10^{−5}) corresponding to the 16 cases reported in Fig. 13.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Time series of Ψ_{1} (scaled with 10^{−5}) corresponding to the 16 cases reported in Fig. 13.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

Time series of Ψ_{1} (scaled with 10^{−5}) corresponding to the 16 cases reported in Fig. 13.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

In the cases presented in the second, third, and fourth rows of Fig. 13, a stochastic perturbation is also present in the forcing. The number *N _{z}* of ROs has its maximum for

*T*= 10 yr, as for the cases without noise, decreasing almost symmetrically for

_{p}*T*= 5 and 15 yr; furthermore,

_{p}*N*is now significantly higher when

_{z}*T*= 5 yr. The clustering of

_{p}*υ*is now less pronounced than in the cases without noise, but it is still evident, yielding similar distributions; however, important differences are worth noticing. With the relatively weak noise amplitude

*ε*

_{1}= 0.1 and with

*T*= 1 yr (second row of Fig. 13),

_{s}*N*is always quite large and PS is significant, with an apparent multimodal character for

_{z}*T*= 5, 10, and 15 yr and with a significant PS for

_{p}*T*= 20 yr (case b4). If

_{p}*T*is increased to

_{s}*T*= 10 yr (fourth row), the distribution is sharper for

_{s}*T*= 10 yr (case d2; the same of Figs. 12c,c′) but is less pronounced in the other three cases (d1, d3, and d4). If, on the other hand,

_{p}*T*is the same but the noise amplitude is increased to

_{s}*ε*

_{1}= 0.2 (third row), then the PS is definitely less pronounced, as could easily be expected. In summary, the PS behavior when noise is added to the forcing is quite complex. In the present parameter range, (i)

*N*experiences a general increase, especially for small and large

_{z}*T*(e.g.,

_{p}*T*= 5 and 20 yr); (ii) PS is particularly effective for relatively small noise amplitude (e.g.,

_{p}*ε*

_{1}= 0.1) and for relatively small noise decorrelation time (e.g.,

*T*= 1 yr); and (iii) PS is less effective for larger noise amplitude (e.g.,

_{s}*ε*

_{1}= 0.2).

### c. Discussion

In section 4a, it was shown that CR can produce ROs in a parameter range in which they do not arise spontaneously (i.e., before the global bifurcation) in a manner very similar, also quantitatively, to what was shown by P10 to occur in a relatively realistic primitive equation ocean model of the KE. We can therefore conclude that such a mechanism, if actually present in the intrinsic low-frequency variability of WBC extensions, is likely to be a low-order phenomenon, being therefore largely independent of the details of the specific oceanographic case. This may well be the case also in other contexts of climate dynamics, where CR is expected to act like, for example, the DO events (Ganopolski and Rahmstorf 2002; Ditlevsen et al. 2007) and the Atlantic multidecadal oscillation (Frankcombe et al. 2009).

A subtle problem deserves to be discussed. We have seen that a RO can arise either spontaneously beyond the global bifurcation or with the aid of noise before it: which of the two alternatives applies? P10 discusses this in the context of the KE and concludes that sufficiently long time series (e.g., of altimeter data; now unavailable) are necessary to provide a definitive answer. Pierini and Dijkstra (2009) stress that a way to justify the observed synchrony (e.g., Qiu and Chen 2010) between the highly nonlinear, spatially sharp decadal bimodal oscillation of the KE (represented here by our RO) and the spatially broad field of Rossby waves generated by wind fluctuations is to suppose that the same fluctuations excite the frontal oscillation, which, being an internal mode of the system, is, in its evolution, virtually independent of the wind itself. This conjecture is obviously in favor of the CR alternative discussed in section 4a.

By using the DO events as an example, Ditlevsen and Johnsen (2010) discuss the same problem by putting it in a general perspective. Let us suppose a system resides in a steady or weakly variable state (like the ones we found in section 3 for *μ* < 1). The transition to a completely different state (the homoclinic regime in our case) can occur either because of variations of some control parameter, with the consequent crossing of a bifurcation, or “tipping” point (here at *μ* = 1), or episodically through the help of stochastic perturbations (the CR scenario). Understanding which of the two alternatives occurs has a profound implication on the predictability of the transition. In fact, Ditlevsen and Johnsen (2010) showed that some form of “early warning” of the transition can be derived in the first case if increased variance (following from the fluctuation–dissipation theorem) and increased autocorrelation (related to critical slow down) are observed, whereas in the noise-induced scenario the system has very limited predictability. In the specific case of the DO events the same authors support the second scenario. The present low-order study of CR can be the base for testing and analyzing in future studies the conclusions drawn by these authors in our context of the wind-driven ocean circulation.

Passing now to the PS mechanism, in section 4b it was shown that the use of fictitious Fourier components in the forcing can induce PS, which can in turn be used to understand the excitation mechanism. Now we discuss a specific case that illustrates how the PS technique can be applied in practice. In Fig. 15a, the meridional profile of the normalized wind stress curl for *ϕ* = 190°, corresponding to the maximum of case b4 of Fig. 13, is shown by the solid line (the two gray arrows indicate the successive profile evolution, whereas the two dashed lines show the maximum and minimum values assumed during a periodic cycle when *ε*_{2} = 0.2). Therefore, this is the periodic wind state for which the RO excitation is most likely to occur. More detailed information can be obtained from Fig. 15b, where two subsequent wind states like the one just discussed (identified by the ovals O_{1} and O_{2}) can be analyzed. Why does the first state induce the excitation (see the RO starting at *t* ~ 910 yr) but the second does not? In O_{1}, the total wind amplitude *G*(*t*) shows a pronounced decrease, in phase with the periodic trend: this in turn produces a state corresponding to an equivalent phase *ϕ* slightly larger than 190° for which a high probability of excitation is expected (see the distribution of case b4 in Fig. 13 to the right of the maximum). On the contrary, in O_{2} the noise produces an equivalent *ϕ* that is much smaller than 190°, corresponding to a smaller *υ*(*ϕ*), which prevents excitation (in Fig. 15c, three maps of *ψ* show the triggering of the RO in the first case and its absence in the second). Obviously, the intrinsic state of the system also plays a role, and the difference of Ψ_{1} corresponding to O_{1} and O_{2} certainly has an influence on the respective evolutions (but, as we have already said, in this preliminary analysis we are only investigating the effect of the wind on the excitation). To end this discussion, it is clear that, without the help of a predominant periodic wind component, the above conclusions would not have been possible.

(a) Meridional profile of the normalized wind stress curl (thick dashed line), maximum and minimum values assumed during a periodic cycle when *ε*_{2} = 0.2 (dashed lines), and profile with *ϕ* = 190° (solid line) corresponding to the maximum of case b4 in Fig. 13 (the two gray arrows indicate the successive profile evolution). (b) As in Fig. 11a (corresponding to case b4 in Fig. 13), but with a zoom over the period 900 yr ≤ *t* ≤ 950 yr. The total temporal dependence *G*(*t*) (thin line) is also reported (the thick line denotes the periodic forcing). For the ovals O_{1} and O_{2}, see the text. (c) (top) Sequence of three snapshots of *ψ* (scaled with 10^{−5}) showing the triggering of the RO and (bottom) same sequence after a complete period (20 yr) showing the absence of the RO.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(a) Meridional profile of the normalized wind stress curl (thick dashed line), maximum and minimum values assumed during a periodic cycle when *ε*_{2} = 0.2 (dashed lines), and profile with *ϕ* = 190° (solid line) corresponding to the maximum of case b4 in Fig. 13 (the two gray arrows indicate the successive profile evolution). (b) As in Fig. 11a (corresponding to case b4 in Fig. 13), but with a zoom over the period 900 yr ≤ *t* ≤ 950 yr. The total temporal dependence *G*(*t*) (thin line) is also reported (the thick line denotes the periodic forcing). For the ovals O_{1} and O_{2}, see the text. (c) (top) Sequence of three snapshots of *ψ* (scaled with 10^{−5}) showing the triggering of the RO and (bottom) same sequence after a complete period (20 yr) showing the absence of the RO.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

(a) Meridional profile of the normalized wind stress curl (thick dashed line), maximum and minimum values assumed during a periodic cycle when *ε*_{2} = 0.2 (dashed lines), and profile with *ϕ* = 190° (solid line) corresponding to the maximum of case b4 in Fig. 13 (the two gray arrows indicate the successive profile evolution). (b) As in Fig. 11a (corresponding to case b4 in Fig. 13), but with a zoom over the period 900 yr ≤ *t* ≤ 950 yr. The total temporal dependence *G*(*t*) (thin line) is also reported (the thick line denotes the periodic forcing). For the ovals O_{1} and O_{2}, see the text. (c) (top) Sequence of three snapshots of *ψ* (scaled with 10^{−5}) showing the triggering of the RO and (bottom) same sequence after a complete period (20 yr) showing the absence of the RO.

Citation: Journal of Physical Oceanography 41, 9; 10.1175/JPO-D-10-05018.1

It is worth noticing that the PS considered here can be seen as a weak form of the nonlinear phase locking (NPL) mechanism invoked to explain—in the context of low-order chaos—the locking of the ENSO events with the seasonal cycle (Tziperman et al. 1994; Tziperman et al. 1995) and, more recently, the Pleistocene 100-kyr ice ages with the Milankovitch forcing (Tziperman et al. 2006): in those cases, a (real) periodic forcing acts as pacemaker of the observed climate cycle. A similar pacemaker role was attributed by Rahmstorf (2003) to a 1470-yr-period forcing (Braun et al. 2005) in determining the timing of Dansgaard–Oeschger events of the last glacial period [but Ditlevsen et al. (2007) are rather in favor of the CR mechanism]. In our case, fictitious periodic forcings introduced to reveal the RO excitation mechanism produce a form of locking, which, however, is less strict than NPL; this may nonetheless suggest the ability of the same system to produce NPL when forced by real periodic winds in different situations.

## 5. Conclusions

In this paper, the low-order character of the intrinsic low-frequency variability of the midlatitude double-gyre ocean circulation and of the corresponding coherence resonance phenomenon has been investigated. A new spectral QG reduced-gravity ocean model retaining only four modes in the Galerkin projection has been derived, thus extending the pioneering work of V63 and the more recent studies of J95, Simonnet and Dijkstra (2002), and S05, in which other low-order models had proved valuable in analyzing fundamental aspects of the wind-driven ocean circulation.

Such a model has been solved under both time-independent and stochastic double-gyre wind forcing following a similar approach adopted by Pierini (2006) and P10 to study the Kuroshio Extension intrinsic low-frequency variability with a primitive equation ocean model. Both the results of the steady and stochastic forcing cases show significant similarities, to some extent also quantitative, with the corresponding primitive equation results. This implies that the transition from a weak gyre mode past a Hopf bifurcation to the vigorous relaxation oscillation of a western boundary current (such as the Kuroshio) extension past a homoclinic bifurcation is an extremely robust feature of midlatitude ocean models. The same can be said of the coherence resonance phenomenon if a parameter range that precedes the homoclinic bifurcation is considered. A technique denoted as phase selection (based on the introduction of fictitious periodic winds and on a statistical description of the observed phase dependence of the relaxation oscillations on the forcing phase) has also been proposed and has proven useful in characterizing the excitation of the oscillations.

In general, although a conceptual model cannot provide direct information about the phenomena under investigation, its use in combination with other more realistic models can be valuable in elucidating specific physical processes. This is particularly true when analyzing abrupt nonlinear transitions of climatic relevance, for which state-of-the-art climate models have not yet proved fully successful. In this context, the results of the present model study may serve as low-order paradigms of important oceanographic phenomena already studied with more realistic models, such as the intrinsic relaxation oscillations of western boundary current extensions and the corresponding coherence resonance mechanism.

Future perspectives include the extension of the model to the multilayer case and its coupling with a low-order atmospheric model to investigate basic ocean–atmosphere feedback mechanisms. Another extension concerns the application, to the results presented here on coherence resonance, of the new methods of random dynamical systems that are currently being used in stochastic climate dynamics (e.g., Ghil et al. 2008). Moreover the phase selection method proposed in this paper will find application to primitive equation model results in order to analyze in detail the mechanism of excitation of the relaxation oscillations that rules the CR phenomenon.

## Acknowledgments

This research was supported by the “Regione Campania” of Italy (L. R. n. 5/2002, Ann. 2005, Mod. 1279).

## APPENDIX

### Model Coefficients

*n*are defined in (12) and (13). The quadratic terms in

_{i}**Ψ**in (A5) are explicitly represented in expression (14), where the rank-3 tensor

*Z*≡ 〈

_{ijk}*i*|

*Q*〉. These coefficients have been evaluated numerically. The structure of the nonlinear terms

_{jk}*N*in the four ODEs of (11) is

_{i}*λ*= 1 (square domain) and Ψ

_{3}= Ψ

_{4}≡ 0, our system coincides with that of J95, but with a difference: the coefficient 2

*J*

_{112}in

*N*

_{1}is equal in amplitude but has the opposite sign of the corresponding coefficient of J95. Checking their analytical derivation shows that the coefficient

*G*

_{1}, as reported in their Eq. (B6c), must in fact have the opposite sign.

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