## Abstract

In Part I of this study, the atmospheric weather noise for 1951–2000 was inferred from an atmospheric analysis in conjunction with SST-forced AGCM simulations and used to force interactive ensemble coupled GCM simulations of the North Atlantic SST variability. Here, results from those calculations are used in conjunction with a simple stochastically forced coupled model of the decadal time scale North Atlantic tripole SST variability to examine the mechanisms associated with the tripole SST variability. The diagnosed tripole variability is found to be characterized by damped, delayed oscillator dynamics, similar to what has been found by other investigators. However, major differences here, affecting the signs of two of the crucial parameters of the simple model, are that the atmospheric heat flux feedback damps the tripole pattern and that a counterclockwise intergyre gyre-like circulation enhances the tripole pattern. Delayed oscillator dynamics are still obtained because the sign of the dynamically important parameter, proportional to the product of these two parameters, is unchanged. Another difference with regard to the dynamical processes included in the simple model is that the major contribution to the ocean’s dynamical heat flux response to the weather noise wind stress is through a delayed modulation of the mean gyres, rather than from the simultaneous intergyre gyre response. The power spectrum of a revised simple model forced by white noise has a less prominent decadal peak using the parameter values and dynamics diagnosed here than in previous investigations. Decadal time scale retrospective predictions made with this version of the simple model are no better than persistence.

## 1. Introduction

Part I of this study (Fan and Schneider 2011, hereafter Part I) presented results from diagnostic simulations of the observed 1951–2000 North Atlantic tripole variability. The observed tripole pattern has centers of like sign to the east of Newfoundland and in the tropical eastern Atlantic, and a center of opposite sign near the southeastern coast of the United States. The pattern can be represented using the tripole index of Czaja and Marshall (2001, hereafter CM01) Δ*T _{s}*, defined as the SST anomaly averaged over a northern box (40°–55°N, 60°–40°W) minus that in a southern box (25°–35°N, 80°–60°W). Figure 1 in Part I shows the regions, the detrended JFM tripole index for 1951–2000, and the associated tripole pattern found from regression of the SST on the index. Linear regression of the National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research (NCAR) reanalysis (Kalnay et al. 1996) sea level pressure (SLP) on the index produces the North Atlantic Oscillation (NAO, e.g., Barnston and Livezey 1987) pattern.

The simulations were carried out with an interactive ensemble coupled GCM (IE-CGCM; Kirtman and Shukla 2002) based on the Center for Ocean–Land–Atmosphere Studies (COLA) anomaly CGCM (Kirtman et al. 2002). The IE-CGCM reduces the weather noise surface fluxes that are provided to the OGCM but preserves the feedbacks between the ocean and atmosphere. The simulations were forced by the weather noise surface fluxes that occurred in the latter half of the twentieth century, estimated based on the NCEP–NCAR reanalysis. The observed tripole SSTA during this period was reasonably well simulated. These simulations provide the data for the analysis presented here and are described further in section 2.

A simple model for the variability of the tripole index was developed by CM01 (see also Marshall et al. 2001). This model includes forcing of the ocean by the atmospheric heat flux and wind stress weather noise (Hasselmann 1976; Frankignoul and Hasselmann 1977), forcing of the ocean by the atmospheric heat flux and wind stress feedbacks to tripole SST anomalies, and advection of the mean temperature by the anomalous oceanic intergyre gyre circulation. Modulations of the mean gyre were argued not to be relevant. A representation of the Atlantic meridional overturning circulation (AMOC) was included in Marshall et al. (2001) but not in CM01. Parameter values estimated from observations featured a positive atmospheric NAO feedback to the tripole index, implying reduced heat flux damping, and an amplifying effect on tripole anomalies by a clockwise intergyre gyre. For these parameters, the simple model produces a damped oscillatory regime forced by the weather noise.

The basic mechanism in this regime in CM01 involves a positive surface heat flux feedback between the NAO and the SST tripole. A positive NAO weather noise event enhances the westerlies in the northern box, producing negative Δ*T _{s}* (relative cooling in the northern box) through the surface heat flux. The atmospheric feedback to a negative Δ

*T*is enhanced westerlies, producing further cooling. The enhanced westerlies also cause a clockwise intergyre gyre (subtropical gyre extending farther north) after a delay related to the Rossby wave propagation phase speed. The gyre response produces a delayed warming of the northern box, increasing Δ

_{s}*T*and damping the tripole. The delayed damping eventually causes the SST anomaly to change sign, giving oscillatory behavior.

_{s}The simple model has the property that the oscillatory dynamics produces a decadal time scale peak in the power spectrum, related to the delay time, in the response to a white noise heat flux forcing. The intergyre gyre response to the wind stress noise forcing alone can also produce a decadal spectral peak by preferentially damping the very long periods (delay time is small relative to period). It was found in CM01 that the response to the wind stress noise then enhances the sharpness of the peak from the response to the heat flux weather noise.

Weng and Neelin (1998) analyzed a related simplified model of a decadal time scale tripole-like mode of North Atlantic SST variability, including processes similar to those in CM01. Neelin and Weng (1999) and Weng and Neelin (1999) generalized the model and included multiplicative as well as additive noise. The rectangular basin ocean model includes a surface mixed layer SST equation linearized about climatological SST and a state of rest, coupled to an interior represented by a shallow water model linearized about a state of rest. The empirical atmospheric model includes stochastic heat flux and wind stress forcing, and nonlocal heat flux and wind stress feedbacks to the SST anomalies. The CM01 box model is analogous to projecting the equations on a single eigenmode. The gyre response to the wind stress feedbacks leads to an interdecadal peak in the power spectrum of the SST response to white weather noise heat flux and wind stress forcing. The time scale of the peak is related to the time delay for Rossby wave propagation from the wind stress feedback forcing region to the SST anomaly region. Westward propagation of the gyre anomalies produced by the feedback wind stress provides a visual manifestation of the role of Rossby wave dynamics. The ocean dynamics effect on the SST anomalies is through advection of the mean temperature by the geostrophic currents.

Eden and Greatbatch (2003) used an intermediate model consisting of a simple stochastically forced atmosphere, also with a positive NAO feedback to the tripole SSTA, coupled to an OGCM. They found that the intergyre gyre was a rapid rather than delayed response to the wind stress, and moreover that the intergyre gyre contribution to the tripole index tendency was of opposite sign to that taken by Marshall et al. (2001) and CM01. However, delayed oscillator dynamics were still diagnosed, with the necessary delayed damping due to advection by the anomalous currents, and associated with the meridional overturning circulation.

Bellucci et al. (2008) estimated the parameter values for the CM01 model for the roughly 5-yr-period tripole variability simulated by the SINTEX-G CGCM (Gualdi et al. 2008). The parameter regime for the SINTEX-G CGCM was found to be analogous to that of CM01, indicating strongly damped, delayed oscillator dynamics activated by weather noise forcing. A clockwise intergyre gyre produced delayed damping of the tripole SSTA with the delay time related to Rossby wave propagation, and the role of the AMOC variability was found to be secondary.

In the following, the interplay of the mechanisms leading to the tripole variability is analyzed by fitting our IE-CGCM results to the simple model of CM01. Some modifications and extensions to the simple model are also made in order to accommodate the dynamics found in the IE-CGCM simulations.

## 2. Results from the interactive ensemble simulations

Results described in Part I relevant to the application of the CM01 model are briefly summarized here. Three IE-CGCM simulations were carried for 1951–2000 with weather noise surface flux forcing between 15° and 65°N, and no weather noise forcing outside of this region. The specific instance of the weather noise that occurred during this period was estimated by subtracting the ensemble mean surface flux response of the COLA AGCM to the observed evolution of the 1951–2000 SST from the NCEP–NCAR reanalysis surface fluxes (Schneider and Fan 2007). The three simulations were 1) NActl (North Atlantic control), forced with all weather noise components (heat flux, wind stress, and freshwater flux); 2) NAh (North Atlantic heat), forced with weather noise heat flux only; and 3) NAm (North Atlantic momentum), forced with weather noise wind stress only. The tripole index was simulated reasonably well in NActl and NAh, but not in NAm [respective correlations with observed index are 0.74, 0.62, and −0.05 for January–March (JFM) and 0.69, 0.51, and −0.43 for the 7-yr JFM running mean; see Fig. 4 in Part I].

The simulations showed that the observed index is forced primarily by the weather noise heat flux, to the extent that the models used are realistic. The associated ocean dynamical response played a secondary, damping role. An intergyre gyre index was defined as the barotropic streamfunction anomaly averaged over 35°–45°N, 60°–40°W, a region between the north and south tripole boxes. The intergyre gyre index in NActl and NAh lagged the respective tripole index by 3–4 yr, while the NAm intergyre gyre was 180° out of phase with the NAm tripole index (Fig. 8 in Part I). The AMOC index, defined as the maximum of the meridional overturning streamfunction in the North Atlantic, lagged the tripole index by about a year in NActl and NAh (Fig. 8 in Part I). The NAm AMOC was out of phase with the NActl and NAm AMOC. The magnitude of the decadal variability of the intergyre gyre and AMOC indices was about 1 Sv (1 Sv ≡ 10^{6} m^{3} s^{−1}) in each case.

These results were noted to be probably model dependent, since the character of the feedback of the observed SST on the AGCM NAO, important in inferring both the weather noise and the dynamics of the IE-CGCM, is well known to be model dependent. In the particular case of the COLA AGCM, the feedback is negative. Also, the structure of the standard COLA CGCM intrinsic tripole variability is not a good match for the observed structure, and the underlying causes of this bias could then produce misleading results. Therefore, the results both in Part I and here should be viewed as demonstrations of the potential usefulness of our procedures, and as possible rather than definitive explanations of the observed tripole variability.

## 3. Analysis of mechanism for tripole mode variability

First, the heat budget as obtained from the simulations is examined, and then the model of CM01 is applied to diagnose the roles of the weather noise and ocean dynamics in the simulations of the observed tripole variability. Annual-mean rather than winter-mean data are analyzed to avoid the issue of how to include the skipped seasons in the differential equations.

### a. The role of ocean dynamics

The competing processes associated with the tripole variability in the ocean’s heat budget are examined by calculating the difference of the area-averaged heat budgets of the north and south tripole regions. Integrating over the depth of the ocean, the tripole area heat storage tendency is balanced by tendency due to the net surface heat flux and the tendency due to ocean dynamics:

where the vertically varying index Δ*T*(*z*) is defined as the northern minus southern box area-average difference between temperature anomalies at depth *z*, *z _{b}* is the depth of the ocean bottom, Δ

*F*

_{net}is the corresponding net surface heat flux index, is the vertically integrated Δ

*T*, and denotes the tendency of the vertically integrated temperature index due to dynamics. The seawater density

*ρ*and the specific heat

*c*are both taken to be constant. The heat storage (lhs term) and net surface heat flux (first term on rhs) terms in Eq. (1) were evaluated from the ocean model output and detrended, while the ocean dynamics term (second term on rhs) was found as a residual.

The three terms of Eq. (1) are shown in Fig. 1 for NActl, NAh, and NAm. The ocean dynamics term (blue curves in Fig. 1) is of comparable importance to the heat storage and net surface heat flux term in the budgets for both the NActl and NAh tendencies. Since the magnitude of the weather noise heat flux is larger than the feedback heat flux (see Fig. 2 in Part I), the ocean dynamics term plays a primary role in the heat budget. That is, the balance is not equilibrium between weather noise and atmospheric feedback (net flux ≈ 0) or a balance between weather noise and oceanic heat storage. Instead, the surface heat flux contribution is small compared to that of the other two terms in NAm, and the heat content tendency is balanced by advection, similar to the heat budget properties inferred by Kelly and Dong (2004) for an area average over the western North Atlantic. It can be seen that the NAh heat content tendency leads that of NActl, while the NAm heat content tendency lags. The response to the weather noise wind stress, isolated in NAm, then acts to extend the heat content response in time compared to the response to the weather noise heat flux. Additionally, the NAh ocean dynamics response leads the NAh heat content tendency, while the NAh surface heat flux lags.

Regressions of the barotropic streamfunctions for the three North Atlantic simulations against the respective ocean dynamics tendency indices are shown in Fig. 2. In NAm (Fig. 2b), a positive tendency in the tripole heat storage is associated with a structure that resembles a decrease in the magnitude of the mean gyres more than an intergyre gyre. The implied mechanism is that a reduction in the strength of the clockwise subtropical gyre transports less heat poleward, cooling the southern region, and a reduction in the strength of the counterclockwise subpolar gyre transports less cold water equatorward, warming the northern region. The NAm streamfunction regression resembles the response of the barotropic streamfunction lagging the NAO index by 3 yr as found by Eden and Willebrand (2001) in their OGCM simulation forced by the NCEP–NCAR wind stress only, rather than the intergyre gyre-like response that is simultaneous with the NAO wind stress forcing (Eden and Willebrand 2001; Eden and Greatbatch 2003; Fig. 6 in Part I). Our diagnosis then indicates that the primary mode of weather noise wind stress–forced heat transport with respect to the vertically averaged tripole temperature is a delayed response due to modulation of the mean gyres rather than the simultaneous intergyre gyre response.

In NActl (Fig. 2c) and NAh (Fig. 2a), on the other hand, a positive ocean dynamics tendency is associated with a counterclockwise barotropic circulation centered at about 40°N. In NAh, this circulation, although narrower east–west, has some resemblance to the intergyre gyre of Part I (see Fig. 6b therein), extending to near the mean gyre boundary near 50°N between 40° and 50°W. Since the counterclockwise sense of this feature is associated with an increasing tendency of the tripole heat storage, it behaves like the intergyre gyre in Eden and Greatbatch (2003) rather those in Marshall et al. (2001), CM01, and Bellucci et al. (2008), where increasing tendency is associated with clockwise gyres. There are also clockwise barotropic circulation features in Fig. 2 for NActl and NAh in the subpolar gyre region. A similar reduction in the subpolar gyre (consistent with a positive contribution to the tendency for the northern area of the tripole as discussed above) was found in Eden and Willebrand (2001) in response to reanalysis heat flux only forcing, and was interpreted as a gyre response to changes in the AMOC.

Since the ocean dynamics tendencies were found as residuals, we have investigated further to verify that the circulation features shown in Fig. 2 are consistent with the positive ocean dynamics tendencies in NActl and NAh. The direct calculation supports the inference from the residual calculation that the gyres represented by the barotropic streamfunctions in Fig. 2 are responsible for the ocean dynamics tendencies in Fig. 1, and that the heat storage tendency from the counterclockwise intergyre gyre is positive.

The local annual-mean dynamics tendencies associated with the tripole heat storage tendencies due to advection of the mean temperature by the anomalous horizontal currents were calculated from 7-yr running means of annual-mean ocean model output. Tripole tendencies were found by subtracting the area-averaged tendencies of the south from the north region. The horizontal velocity fields for computing the advective tendencies were found by regression against the ocean dynamics tendencies. The temperatures used were the climatological annual means from the respective simulations. The horizontal velocities, although depth dependent, are approximately parallel to the isolines of the barotropic streamfunction regressions from the respective simulations (Fig. 2), in particular in those regions and at those depths contributing significantly to the area averages, located primarily in the northern box. There may be some inaccuracy in these calculations in that the finite difference operations used the Grid Analysis and Display System (GrADS) interpolation and centered differencing functions, rather than the formulations in the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model version 3 (MOM3). The directly calculated advective tendencies are shown in Fig. 3. The vertical integrals of these regressions are positive and on the order of 1 (1.94 for NActl, 1.2 for NAh, and 2.2 for NAm, dimensionless because we are using the regressed streamfunction units), indicating that the advective tendencies are of the correct sign and magnitude to explain the vertically averaged dynamics tendencies for the respective simulations. In each case there is a positive maximum at the surface, a relative minimum at about 100 m, a maximum at an intermediate depth, and a decrease to small amplitude below 1000 m. The largest contribution to the vertically integrated tendency comes from the intermediate layer.

The sign and vertical structure of the layer-by-layer tendencies found using depth-independent velocities calculated from the regressed barotropic streamfunctions are consistent with those calculated from the total horizontal velocities. The contributions from the intergyre gyre and the subpolar gyre feature are both positive and of about the same magnitude. Physically, the climatological tongue of warm water extending northward near the eastern boundary of the northern box is advected into the box by the gyre circulations and is warmer than the water advected out of the box, leading to the positive tendency [similar to the argument given by Eden and Greatbatch (2003) that the mean temperature increases toward the east].

### b. Tripole variability dynamics in the simple model

The model of CM01 is as follows:

Equations (2) and (3) govern the time evolution of the North Atlantic SST anomaly (SSTA) Δ*T _{s}* and the zonal wind stress anomaly index

*τ*defined as the anomalous westerlies in the northern box minus the anomalous trades in the southern box. Because of the structure of the mean surface winds, changes in the turbulent heat flux due to a positive NAO-like structure of enhanced westerlies in the northern box will produce cooling there, while the associated decreased trades in the southern box will produce warming, both effects decreasing the tripole index. The stochastic (weather noise) component of the surface wind stress index is

*N*and the feedback of Δ

*T*on the wind stress index (the atmospheric response to the tripole) is taken to be proportional to . The northward advection of heat by the anomalous gyre across the mean Gulf Stream is represented by

_{s}*gψ*, with

_{g}*ψ*being the intergyre gyre index (positive for a clockwise gyre) and

_{g}*g*the tripole heating rate associated with the intergyre gyre. In CM01, a positive (anticyclonic) intergyre gyre heats the northern box and cools the southern box through advection of the mean temperature by the anomalous flow, leading to a positive tendency for Δ

*T*, so they take

_{s}*g*to be positive. The tripole is damped by the atmospheric heat flux feedback to Δ

*T*with the

_{s}*e*-folding time scale

*λ*

^{−1}. The difference in the weather noise surface heat flux (positive out of the ocean) between the northern and southern boxes is represented by

*αN*, where the relationship between wind stress and turbulent heat flux anomalies is represented by

*α*> 0 for the observed structure of the mean winds. Ekman transport produced by the weather noise wind stress is argued to act to reinforce the weather noise surface heat flux, and can be included as an increase in

*α*. The heat flux associated with the feedback wind stress is proportional to Δ

*T*and is included implicitly in

_{s}*λ*, reducing damping on the tripole index for .

The contribution to *ψ _{g}* from the wind stress feedback to Δ

*T*is estimated from Eq. (3) by integrating time-dependent Sverdrup dynamics along Rossby wave characteristics, and approximating the integral by a midpoint value, giving dimensionally

_{s}where is a constant (Sv K^{−1}), and *a* depends on factors like the geometry of the basin and the position of the forcing. The delay time *t _{d}* is the time for the intergyre gyre to respond to the wind stress, and is related to the phase velocity of the first baroclinic Rossby mode and the location of the feedback wind stress forcing (farther east than the location of the weather noise wind stress in our diagnosis in Fig. 2 of Part I).

Equation (5) has the form of a damped, forced delayed oscillator. If *f* and *g* are positive, then the gyre contribution to the feedback heat flux (second term on rhs) acts as a delayed damping on Δ*T _{s}.* The regime and stability properties of the physical system represented by Eq. (5) are related to the value of the nondimensional parameter

*R*=

*fg*/

*λ*. For

*N*= 0, oscillatory solutions are possible when

*R*> 0, unstable oscillatory growth occurs when

*R*is larger than a critical value

*R*

_{0}depending on

*λ*, and

*R*

_{0}approaches one for large

*λ*(Marshall et al. 2001).

The model of Cessi (2000) produces coupled oscillations without stochastic forcing through the parameterized atmospheric wind stress and transient eddy heat flux feedbacks to the SST. This mechanism is represented in the CM01 model, since the transient eddy heat flux effect can be viewed as being included in the parameter *λ*. The parameterizable wind stress and heat flux feedbacks are included in the interactive ensemble simulations, and are represented in the values of the parameters found from the below regression analysis of the simulations.

#### 1) Estimate of parameters for the gyre model

We estimate the parameters for the CM01 model from the results of the IE-CGCM experiments. Equation (2) can be interpreted as the difference of the area and vertically averaged heat budgets for the tripole regions, north minus south [cf. Eq. (1)] when the temperature anomalies in the vertical have a single time-independent vertical structure and are well correlated to Δ*T _{s}*. The vertically integrated heat content that appears in the vertically averaged tendency is approximated by

*ρcH*Δ

*T*, where the vertically integrated temperature is

_{s}*H*Δ

*T*, and the constant

_{s}*H*is the effective depth of the temperature anomalies. Then Eq. (2) is obtained by dividing the heat budget equation by

*ρcH*.

The parameters *λ*, *f*, and *g* were estimated from the results of experiment NAh. The wind stress anomalies in NAh are due only to the atmospheric feedback response to the SST anomalies, and not to the weather noise, but weather noise surface heat fluxes are included. This configuration corresponds to setting *N* = 0 in Eq. (3), but not in Eq. (2) and therefore satisfies the condition of no wind stress noise used by CM01 to derive Eq. (4). Similarly, there is no Ekman contribution to *α* from the wind stress noise in NAh. The noise forcing term for NAh in Eq. (2) is then given by

where Δ*F*_{noise} is the specified tripole weather noise surface heat flux forcing index, positive into the ocean, defined as the difference of the area-averaged weather noise surface heat flux between the north and south tripole regions. The value for *H* is found to be approximately 500 m by regression of the vertically integrated temperature against the detrended NAh Δ*T _{s}*. This value of

*H*is similar to the winter mixed layer depth in the northern box in the model but larger than the mixed layer depth in the southern box. The vertical structure of northern minus southern region area averaged temperature difference in Fig. 4 shows that the assumption of a time-independent vertical structure of the tripole ocean temperature anomalies can be viewed as only a crude approximation.

We applied the regression procedure described in the appendix to estimate *λ* and *g*. The parameter values found were *λ* = 0.34 yr^{−1} and *g* = −0.054 K Sv^{−1} yr^{−1}. As discussed in section 3a, the negative value of *g* is consistent with the intergyre gyre structure shown in Fig. 2a but is opposite in sign to the value found by CM01.

Finally, the parameters *f* and *t _{d}* were estimated using Eq. (4) by lag regression of

*ψ*on Δ

_{g}*T*. The maximum magnitude was found to be

_{s}*f*= −3 Sv K

^{−1}at . Then

*f*< 0, and the atmospheric feedback on the surface heat flux is negative. The surface winds induced by the tripole SSTA produce a surface turbulent heat flux that damps the tripole. This relationship agrees with what was found in the specified SST ensemble.

Figure 5 shows the regression of the NAh barotropic streamfunction and curl of the wind stress against the 7-yr running mean NAh Δ*T _{s}*, with the wind stress curl lagging by half the delay time (3 yr) and the streamfunction lagging by the 6-yr delay time. The relationships between the fields are consistent with Eqs. (3) and (4) for

*f*< 0, with a positive Δ

*T*preceding a positive “intergyre” wind stress curl 3 yr later, and the wind stress curl associated with an intergyre gyre–like streamfunction anomaly of opposite sign 3 yr later. The opposite signs of the wind stress curl and associated streamfunction anomalies are consistent with the interpretation of the streamfunction anomaly as the response to the wind stress forcing. A lag regression analysis (see Fig. 6.8 in Fan 2008) suggests that the intergyre gyre propagates from northwest to southeast on the

_{s}*t*time scale, not inconsistent with the Rossby wave dynamics underlying Eq. (4). We have not, however, ruled out a significant role for the meridional overturning circulation in producing the intergyre gyre.

_{d}Our estimate for the value of the parameter *R* = *fg/λ* is then *R* = 0.48. Since 0 < *R* < 1, the unforced solutions are damped oscillatory. An estimate for *R* from the results of NActl will be described in the next section.

#### 2) Application to the other simulations: Modified simple model

As shown in section 3a, the gyre configuration that influences the tripole tendencies in simulation NAm is more like a modulation of the mean subtropical and subpolar gyres than an intergyre gyre. To represent this mode of variability, an index was defined as the difference between the area-averaged barotropic streamfunction anomalies, northern minus southern box, called the difference gyre index *ψ _{d}*. The 7-yr running mean of

*ψ*provides a reasonable representation of the NAm ocean dynamics tendency shown in Fig. 1c (correlation = 0.74), whereas

_{d}*ψ*does not fit as well (correlation = −0.66). In the spirit of Eqs. (3) and (4),

_{g}*ψ*is related to the weather noise wind stress forcing, taken to be proportional to Δ

_{d}*F*

_{noise}, with constant of proportionality

*f*and delay time

_{d}*t*:

_{m}A lag regression analysis gives the estimates *t _{m}*/2 = 3 yr and

*f*= 0.081 Sv (W m

_{d}^{−2})

^{−1}.

A modification of the simple model is required to take into account the distinction between the dynamics represented by the difference gyre and the intergyre gyre. The modified heat budget in Eq. (8) includes an additional term representing the heating due to the difference gyre with proportionality constant *g _{d}*:

In application to NAh, Eq. (9) reduces to Eq. (5). For simulation NAm, there is weather noise wind stress, but no weather noise heat flux forcing. This situation corresponds to setting the last term on the rhs of Eq. (9) to zero. Generalizing the regression analysis in the appendix for Eq. (8) and simultaneously estimating the three parameters *λ*, *g*, and *g _{d}* for NActl gives

*λ*= −0.46 yr

^{−1},

*g*= −0.07 K Sv

^{−1}yr

^{−1}, and

*g*= 0.02 K yr

_{d}^{−1}Sv

^{−1}. The value found for

*R*is 0.46, essentially unchanged from that found from the NAh results and in the damped oscillatory regime. The solutions to Eq. (9) using these parameters, corresponding to NAh (second-to-last term on rhs taken to be zero), NAm (last term on rhs taken to be zero), and NActl (all terms included) are compared with the results from the respective simulations in Fig. 6. The simple model produces a reasonable fit to the qualitative properties of the decadal variability in each case.

Other estimates of *λ* can be obtained from the feedback surface heat fluxes, given *H*. The simultaneous regression of the feedback surface heat flux index from the AGCM ensemble used to compute the weather noise surface flux (case 10AGCM in Part I) with the observed annual mean Δ*T _{s}* is −22 W m

^{−2}K

^{−1}, giving

*λ*= 0.33 yr

^{−1}for

*H*= 500 m, close to the estimate obtained from simultaneous regression in NAh. The regressions of the feedback surface heat flux with the simulated annual mean Δ

*T*for the simulations are −26 W m

_{s}^{−2}K

^{−1}for NAh, −27 W m

^{−2}K

^{−1}for NActl, and −11 W m

^{−2}K

^{−1}for NAm. The corresponding alternate estimates of

*λ*from the simulations are then 0.39 yr

^{−1}for NAh, 0.40 yr

^{−1}for NActl, and 0.20 yr

^{−1}for NAm.

Another extension of the simple model that has been considered is the inclusion of the vertical circulation in the heat budget through a tripole vertical velocity index, defined as the difference between the upward velocity at 500 m in the northern minus southern tripole regions. The southern region contribution is small compared to that from the northern region. The inclusion of the vertical circulation in the heat budget parameter estimation gives a minor difference, increasing the damping coefficient slightly and reducing the gyre contribution. The sign of the vertical circulation contribution indicates a positive contribution to the tripole tendency when there is downwelling in the northern region, as expected.

#### 3) Properties of the simple model

We now compare the mechanisms for the tripole SST variability found in our simulations with those found by other investigators. The simple model fit to the weather noise heat flux–forced IE-CGCM simulations of the North Atlantic tripole produces a damping time of about 3 yr, *R* = 0.48, and a delay time of about 6 yr (effective delay time 3 yr). The positive value of *R* places the simple model in an oscillatory regime, and the regime is damped since *R* < 1. These results appear to be similar to CM01, who estimate a damping time of about 1 yr, a delay time of 10 yr, and *R* = 0.4. Certainly both estimates are in the damped oscillatory regime. However, the apparent consistency results from very different atmospheric feedbacks and gyre properties. CM01 and Bellucci et al. (2008) have a positive atmospheric heat flux feedback to the tripole SSTA (*f* > 0) and consequent counterclockwise intergyre gyre (*ψ _{g}* < 0) response to a positive tripole SSTA. In their cases, the heat flux from the counterclockwise intergyre gyre damps the tripole (

*g*> 0). Then

*R*=

*fg/λ*> 0, since

*λ*> 0. Weng and Neelin (1998) and Neelin and Weng (1999) evaluated the wind stress feedback associated with the tripole-like pattern from a canonical correlation analysis (CCA) analysis of simulations made with the ECHAM2 AGCM forced by observed interannually varying SST (Kharin 1995). Their zonal wind stress feedback is essentially opposite in sign in midlatitudes from that found from our simulations; consequently,

*f*> 0 in the Weng and Neelin (1998) simple model. It can be inferred that the sense of their gyre heat transport is such that

*g*> 0, since their simple model produces weakly damped oscillatory eigenmodes (Weng and Neelin 1999), corresponding to

*R*> 0.

In contrast, in our interactive ensemble simulations the atmospheric heat flux feedback to the tripole SSTA is negative (*f* < 0) and therefore produces a clockwise intergyre gyre response to a positive tripole SSTA. The temperature tendency due to the clockwise intergyre gyre damps the tripole (*g* < 0). Thus, the advection of mean temperature by the gyre circulations in both CM01 and in our simulations acts as a delayed damping in the heat budget, despite opposite senses of gyre rotation. Since the signs of the both the atmospheric heat flux feedback (and the associated sign of the atmospheric wind stress response) and the gyre heat flux convergence are reversed, we also obtain *R* > 0.

Another difference is that we find in our simulations that the weather noise wind stress leads to a delayed reinforcement of the tripole through a delayed modulation of the mean subtropical and subpolar gyres. CM01, on the other hand, assume that the instantaneous response of the intergyre gyre to the wind stress noise acts to damp the tripole. The instantaneous response of the intergyre gyre in our simulations is as described in CM01; however, we find no obvious influence of this response on the heat budget. The role found here for the modulation of the mean gyres is similar to that described by Latif and Barnett (1994) for North Pacific decadal SST variability.

Eden and Greatbatch (2003) have a positive heat flux feedback (*f* > 0) as in CM01, and a negative intergyre gyre heat transport effect (*g* < 0), as we have found, but with a short delay time. Therefore, *R* < 0 in their case, and it would be necessary to include AMOC dynamics in the simple model to simulate the oscillatory behavior that they found in their simulations. Their gyre response to the weather noise wind stress increases the tripole tendency, as in our case, but through the intergyre gyre and without the time delay.

The power spectrum of the response to Eq. (9) for stochastic forcing is found by assuming that Δ*T _{s}* =

*T*and Δ

_{ω}e^{iωt}*F*=

_{s}*F*, substituting in Eq. (9), and solving for while taking |

_{ω}e^{iωt}*F*| constant. The result is

_{ω}which is plotted in Fig. 7 for |*F _{ω}*/

*ρcH*| = 1 K yr

^{−1}or |

*F*| ≈ 66 W m

_{ω}^{−2}. This figure should be compared with the corresponding figure in CM01 (their Fig. 13). The no-ocean-dynamics spectrum, found by taking

*f*=

*f*= 0, is similar in both cases, a red noise shape with no peaks. When only the atmospheric feedback to the tripole is included (

_{d}*f*= 0;

_{d}*f*≠ 0), the spectrum is also similar in both cases, with a peak appearing near the delay time

*t*and reduction of the power at very low frequencies compared to the no dynamics spectrum, despite the different signs of

_{d}*f*. Including the ocean dynamical response to the weather noise wind stress but no atmospheric feedback wind stress (

*f*= 0;

*f*≠ 0) leads to major qualitative differences in the spectra, with a spectral peak appearing in CM01 due to a reduction in the power at very low frequencies relative to the no dynamics case because of the damping effect of the wind stress–forced intergyre gyre, but enhancement of the low-frequency power in our case due to the amplification by the gyre modulation. Finally, when all processes are included, a more prominent spectral peak appears in CM01 than with feedback wind stress alone, while the spectral peak is almost eliminated in our case. Of course, the simple model may not be adequate to represent the real climate system with regard to not including or misrepresenting the roles of potentially important processes such as the MOC, frequency dependence of the penetration depth

_{d}*H*, and so on.

In a related case treated by Neelin and Weng (1999), forced by wind stress noise but no heat flux noise, the decadal peak disappears as the coupling to the wind stress feedback is reduced. However, a decadal peak appears with no coupling when both heat flux noise and wind stress noise are included. The power spectrum for the box model associated with the wind stress noise only configuration is given by a modified version of Eq. (10) with the terms [1 + 2*f _{d}g_{d}ρcH* cos(

*ωt*/2)] on the rhs neglected, and produces disappearance of the decadal peak as

_{m}*fg*(positive here) is reduced. The effect of the inclusion of the noise wind stress and heat flux together in producing the spectral peak in the uncoupled case is given by taking

*f*= −

_{d}*f*in the numerator on the rhs of Eq. (10), representing delayed damping by the wind stress noise forced intergyre gyre, and

*fg*= 0 in the denominator, removing the response to the feedback wind stress.

#### 4) Predictability of the tripole index

The predictability of the tripole index has been evaluated using Eq. (9) with the parameters as found in sections 3b(1) and 3b(2). Retrospective forecasts (hindcasts) were made for the annual mean index, using the observed tripole index for 1880 to the present calculated from the Smith et al. (2008) SST for the initial conditions and verification. No detrending was applied. Because of the time delay form of Eq. (9), the forecast initialization requires data for the current year and the three immediately preceding years. The results were not smoothed for verification.

If the noise forcing is taken to be known (1951–2000 period only), the correlation of the forecasts with the observed remains above 0.8 for beyond 10 yr of lead time. Of course, the weather noise is not predictable. A more realistic hindcast verification is then to set the weather noise forcing to zero, equivalent in our linear system to performing an infinite ensemble of predictions with different realizations of stochastic weather noise forcing, and also analogous to making the predictions using the interactive ensemble CGCM with no weather noise forcing and starting from observed initial conditions. A typical forecast anomaly made with this procedure immediately decays, crossing zero after a few years, remaining in a weak phase of opposite sign for about 6 yr, and then effectively disappearing. The forecasts therefore can only be good for decaying events. The anomaly correlations over 1880 to the present are visually indistinguishable from persistence. For 1950 to the present, the hindcast correlations are lower than persistence. Thus, the inclusion of ocean dynamics does not lead to enhancement of the skill as measured by the anomaly correlations. Possibly, the forecast skill could be improved, for example by restricting the forecasts to large anomalies, as these will be more likely to decay than to grow.

## 4. Conclusions

The mechanisms responsible for the 1951–2000 North Atlantic tripole variability were analyzed using results from simulations described in Part I, made with an interactive ensemble configuration of the COLA CGCM forced by weather noise surface fluxes derived from the NCEP–NCAR reanalysis. An analysis of the simulated tripole-area ocean heat budgets indicated that the atmospheric feedback to the tripole SST produced a delayed damping of the tripole heat content anomalies related to an intergyre gyre-like ocean circulation. The response to the wind stress weather noise, more apparent in the heat content than the SST, produced a small delayed amplification effect in the SST related to modulation of the mean gyres.

The diagnosis of the interactive ensemble simulations was quantified in the framework of the simple model of the tripole variability given by CM01. The fitted tripole index variability was found to be primarily the response to the weather noise (NAO) heat flux. The inclusion of the oceanic gyre circulation feedback on the tripole SST led to damped delayed oscillator dynamics, as in earlier studies, but with a third distinct configuration of atmospheric and oceanic dynamics. This configuration produces a decadal peak in the power spectrum of the response to white noise, but it is less prominent than in CM01 or Neelin and Weng (1999). The predictability of the tripole was evaluated with the simple model by making hindcasts of the observed tripole index without weather noise forcing. While the simple model has skill for decaying events, the overall predictability is no better than for persistence forecasts.

Finally, we summarize below the essential similarities and differences of the mechanisms for the decadal tripole SSTA for the three configurations, all of which yield the same end results of delayed oscillator dynamics and a decadal peak in the power spectrum for the tripole SSTA. Further research would be required to test these possibilities in order to determine which configuration, if any, provides an adequate description of the mechanism of the tripole variability in the real climate system.

In CM01, Neelin and Weng (1999), and Bellucci et al. (2008):

Weather noise with the structure of the positive phase of the NAO forces a negative tripole SSTA via the surface heat flux.

The feedback of the tripole SSTA on the atmosphere is positive [enhances the NAO heat flux and wind stress; except primarily wind stress in Neelin and Weng (1999)].

The wind stress feedback to the tripole SSTA forces an intergyre gyre with a time delay of several years.

A clockwise intergyre gyre produces a positive tripole SSTA tendency.

The heat flux due to the intergyre gyre leads to a delayed damping of the tripole SSTA.

The intergyre gyre forced directly by the weather noise wind stress damps the weather noise heat flux–forced tripole SSTA at low frequencies, producing a decadal peak in the power spectrum of the tripole SSTA through uncoupled ocean dynamics.

The effect of the AMOC is minor or not considered.

Weather noise with the structure of the positive phase of the NAO forces a negative tripole SSTA via the surface heat flux.

The feedback of the tripole SSTA on the NAO is positive.

The wind stress feedback to the tripole SSTA forces an intergyre gyre with a short time delay.

A counterclockwise intergyre gyre produces a positive tripole SSTA tendency.

The intergyre gyre response to the wind stress feedback amplifies the tripole SSTA.

The intergyre gyre forced directly by the weather noise wind stress amplifies the tripole SSTA at low frequencies, so that no decadal peak in the power spectrum of the tripole SSTA is produced through uncoupled ocean dynamics.

The AMOC anomalies provide the delayed damping crucial for delayed oscillator dynamics.

In Part I and this study:

Weather noise with the structure of the positive phase of the NAO forces a negative tripole SSTA via the surface heat flux.

The feedback of the tripole SSTA on the NAO is negative.

The wind stress feedback to the tripole SSTA forces an intergyre gyre with a time delay of several years.

A counterclockwise intergyre gyre produces a positive tripole SSTA tendency.

The heat flux due to the intergyre gyre leads to a delayed damping of the tripole SSTA.

Modulation of the mean gyres forced directly by the weather noise wind stress amplifies the tripole SSTA at low frequencies, so that no decadal peak in the power spectrum of the tripole SSTA is produced through uncoupled ocean dynamics.

The AMOC acts to damp the tripole SSTA with a short delay.

## Acknowledgments

We thank Ben Kirtman for the use of his interactive ensemble version of the COLA CGCM and for his assistance in implementing the weather noise forcing in the model, and Mike Fennessy for his help in processing the reanalysis data. Suggestions of relevant publications made by Allessio Bellucci, Yochanan Kushir, Hua Chen, and two anonymous reviewers are greatly appreciated. This research was supported by NSF Grants ATM-0342104, ATM-0653123, and 0830068, NOAA Grant NA09OAR4310058, and NASA Grant NNX09AN50G. The NCAR CISL provided computer resources for the simulations. Data analysis and plotting were done using GrADS.

### APPENDIX

#### Regression Analysis

We performed a multiple regression analysis as in Bellucci et al. (2008) to find the values of the parameters *λ* and *g* that minimize the magnitude of the difference of the temperature tendencies found in NAh and from Eq. (2). However, the simulation of Δ*T _{s}* found by solving Eq. (2) with the parameters found did not produce a good fit. The procedure described below was found to produce good fits.

The procedure to estimate the unknown parameters *λ* and *g* for diagnosis of simulation NAh using Eqs. (2) and (6) was to first integrate the equation

for , an estimate of Δ*T _{s}*, taking Δ

*T*,

_{s}*ψ*, and Δ

_{g}*F*/(

*ρcH*) on the right-hand side as known from the results of NAh. Then after discretization for a time series of

*n*points,

where

and

are accumulated values from the initial time *j* = 1 to time *i*, and Δ*t* is the time step.

The unknown parameters are then found from a least squares fit analysis minimizing the magnitude of the difference between the estimated and true tripole indices: . The minimization leads to a pair of simultaneous equations with unknowns *λ* and *g*:

where *R*(*A*, *B*) is the regression coefficient

and the overbar is the mean value; for example,

In our application, 7-yr running means of annual mean values were used for the tripole, heat flux, and gyre indices, and the time step was 1 yr. The means of Δ*T _{s}*,

*ψ*

_{g}, and Δ

*F*were removed, so that the fit is constrained by , and a linear slope was removed from Δ

*T*so that the target data satisfied the same constraint.

_{s}The results of the curve fitting for NAh are shown in Fig. A1. The weather noise heat flux forcing is the dominant term, balanced by heat flux feedback and gyre terms of approximately equal magnitude.