1. Introduction
Meridional modes of climate variability are the dominant source of ocean and atmosphere variability in the tropical Atlantic [the Atlantic meridional mode (AMM); Moura and Shukla 1981; Chang et al. 1997; Servain et al. 1999; Chiang and Vimont 2004] and play a secondary role to El Niño–Southern Oscillation (ENSO) as a source of ocean and atmosphere variability in the Pacific [the Pacific meridional mode (PMM); Chiang and Vimont 2004]. Meridional mode variations have also been studied in the South Pacific (Zhang et al. 2014a), the western Pacific (Wang et al. 2012), and the Indian Ocean (Wu et al. 2008). Meridional modes are characterized by an anomalous meridional SST gradient that drives an anomalous interhemispheric boundary layer flow from cold to warm waters. Associated with this anomalous boundary layer flow is an ITCZ shift in the direction of the anomalously warm hemisphere (Hastenrath and Heller 1977). The ubiquity of meridional mode variations in the tropical climate system motivates further understanding of the role of the mean state in meridional mode structure and variation.
Meridional modes of climate variability emerge as a result of a positive feedback between wind, evaporation, and SST (the WES feedback; Xie and Philander 1994; Chang et al. 1997). The WES feedback has been explained in the following way: An SST anomaly in one hemisphere creates a cross-equatorial, interhemispheric SST and pressure gradient that drives a meridional flow from the cold hemisphere to the warm one. The Coriolis force deflects the meridional flow to the right in the Northern Hemisphere and to the left in the Southern Hemisphere, which would reinforce the mean easterly trades in the cold hemisphere and reduce the mean easterly trades in the warm hemisphere. This change in wind speed modulates the evaporative latent heat flux at the surface, reinforcing the original SST gradient. Vimont (2010) shows that under the quasigeostrophic approximation neither the cross equatorial SST gradient nor the reversal in sign of the Coriolis parameter at the equator is necessary for growth via a WES feedback. Instead, a meridional variation in mean absolute vorticity is sufficient to produce instability and westward propagation of meridional mode structures. Additional feedbacks involving vertical mixing in the ocean (Xie and Philander 1994) and cloud radiative effects (Tanimoto and Xie 2002; Xie and Saito 2001; Evan et al. 2013) have been shown to influence meridional mode variation.
The mean state can influence meridional mode variations through the ITCZ structure, the mean trade wind strength and location, and the variation in stochastic forcing that is likely required for meridional mode variations in nature (Xie 1999; Vimont 2010). The mean ITCZ structure can affect how the atmosphere responds to tropical SST variations via either deep heating (Gill 1980; Zebiak 1986; Battisti et al. 1999) or boundary layer convergence and ventilation (Lindzen and Nigam 1987; Battisti et al. 1999). Mean trade winds influence the strength of the WES feedback via the role of wind speed variations on evaporation (Xie and Philander 1994; Czaja et al. 2002; Vimont et al. 2009; Vimont 2010; Zhang et al. 2014b). The WES feedback has also been implicated in explaining the mean ITCZ state asymmetry in the first place (Xie and Philander 1994; Xie and Saito 2001; Takahashi and Battisti 2007). The study of the ITCZ mean state and variability remains an area of active research (Tomas and Webster 1997; Toma and Webster 2010a,b; Frierson et al. 2013; Schneider et al. 2014; Bischoff and Schneider 2014). External and/or stochastic forcing influences meridional mode variance primarily through geographic variations in trade wind variations and associated surface heat flux to the respective subtropical oceans, especially during winter (Nobre and Shukla 1996). ENSO and the North Atlantic Oscillation (NAO) have both been shown to force the AMM (Xie and Tanimoto 1998; Czaja et al. 2002; Chiang and Vimont 2004; Kossin and Vimont 2007; Penland and Hartten 2014), while in the Pacific the North Pacific Oscillation (NPO) (Rogers 1981; Linkin and Nigam 2008) has been shown as a source of external forcing for the PMM (Chiang and Vimont 2004; Chang et al. 2007; Vimont et al. 2009).
Meridional mode variability over the Atlantic possesses a strong seasonality with peak variance in boreal spring. During this season, meridional mode variability corresponds to the so-called Atlantic SST dipole and consists of an interhemispheric antisymmetric SST pattern, although the anticoherence has been questioned (Houghton and Tourre 1992; Enfield et al. 1999). The boreal spring maximum in meridional mode variability appears to result from enhanced atmospheric forcing during the preceding season (Czaja et al. 2002; Czaja 2004). However, the contribution of the seasonal cycle (ITCZ location and strength) to the seasonality of meridional mode dynamics is not clear. Meridional mode variability has been less extensively studied during boreal summer and fall when, in the Atlantic, the AMM structure consists mostly of a single SST “monopole” (in contrast to the more familiar equatorially antisymmetric dipole) over the Northern Hemisphere subtropics, with little activity over the Southern Hemisphere (Smirnov and Vimont 2011). The relative roles of internal dynamics and external forcing are also not clear during this season, although compared to spring, external atmospheric forcing is not at its peak.
This study investigates the role of mean state asymmetry on meridional mode structure and growth, including both the effect of asymmetry on internal dynamics and on external forcing. This paper extends the scope of Vimont (2010), who investigates transient growth of meridional mode disturbances under an equatorially symmetric mean state. The remainder of this paper is organized as follows. Section 2 will describe the model and methods utilized. Section 3 will present the optimal results under symmetric and asymmetric ITCZ states and some sensitivity tests. Finally section 4 will present the conclusions.
2. Model and analysis
a. Model
Model parameters.
The nondimensionalized (Vimont 2010) version of (1) forms our coupled model, which is spectrally decomposed into a single zonal harmonic (
b. Background states
The background mean state affects the model (1) through the location and intensity of the intertropical convergence zone (ITCZ), which affects the meridional structure of 
1) The Kq and ITCZ mean states
(a) FMA average of 
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
(a) FMA average of
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
(a) FMA average of
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
2) The WES parameter α(y) and surface winds
Zonal mean of the WES parameter (J m−3) over the Atlantic for 1981–2010: FMA average (solid) and ASO average (dashed). Monthly 1981–2010 latent heat and wind data from ERA-Interim (Dee et al. 2011) were used.
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Zonal mean of the WES parameter (J m−3) over the Atlantic for 1981–2010: FMA average (solid) and ASO average (dashed). Monthly 1981–2010 latent heat and wind data from ERA-Interim (Dee et al. 2011) were used.
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Zonal mean of the WES parameter (J m−3) over the Atlantic for 1981–2010: FMA average (solid) and ASO average (dashed). Monthly 1981–2010 latent heat and wind data from ERA-Interim (Dee et al. 2011) were used.
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
c. Transient growth analysis
In the present manuscript, we focus on transient growth of the coupled model rather than growth of individual modes. While Farrell and Ioannou (1996) provide a more complete distinction between the two processes, we provide a brief explanation here. In short, modal growth (growth of a specific eigenvector of the system) is most appropriate for normal systems (which have orthogonal modes), or as 
Equation (6) indicates that growth is norm-dependent. The nondimensionalization of (1) however, results in an arbitrary norm because in our case, the choice of a reference SST for nondimensionalization is arbitrary. In practice, we are interested in coupled structures with time scales that are long compared to the atmospheric damping time scale (2 days). For this reason we can consider only the growth of SST because for time scales of interest the atmosphere will effectively be in equilibrium with the SST. To evaluate our choice of nondimensional parameters and to provide an illustration of the use of a final norm, we reevaluate transient growth under the equatorially symmetric case considered by Vimont (2010). We compare results calculated under a specified final “SST norm” 
Figure 3 left column shows the initial and final 140-day optimals using 
Optimal (a),(c) initial and (b),(d) final structures calculated under the same mean state as used in Vimont (2010) for 140 days under the full (unconstrained) initial norm kernel and SST final norm kernel in (a),(b) and for 100 days under the NH initial norm kernel with full (L2) final norm kernel in (c),(d). SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the field, solid contours denote positive values, and dashed contours negative ones. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Optimal (a),(c) initial and (b),(d) final structures calculated under the same mean state as used in Vimont (2010) for 140 days under the full (unconstrained) initial norm kernel and SST final norm kernel in (a),(b) and for 100 days under the NH initial norm kernel with full (L2) final norm kernel in (c),(d). SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the field, solid contours denote positive values, and dashed contours negative ones. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Optimal (a),(c) initial and (b),(d) final structures calculated under the same mean state as used in Vimont (2010) for 140 days under the full (unconstrained) initial norm kernel and SST final norm kernel in (a),(b) and for 100 days under the NH initial norm kernel with full (L2) final norm kernel in (c),(d). SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the field, solid contours denote positive values, and dashed contours negative ones. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
To illustrate the effect of varying initial norms, we calculate optimal initial and final conditions under a “Northern Hemisphere norm” in which initial conditions are constrained to be north of the equator. This is achieved by setting 
3. Structure and growth under asymmetric mean states
In this section we calculate growth and evolution of optimal structures under various mean state geometries. In section 3a, we analyze growth under “observed” mean states for boreal spring and fall and find that a symmetric (asymmetric) structure produces optimal growth under the spring (fall) conditions. We investigate the role of mean state asymmetry and ITCZ width in section 3b and discuss differences in the structures and growth in section 3c.
a. Optimal structures in boreal spring and fall
In this section we will calculate the optimal structures that maximize transient growth over boreal spring and fall, which are the seasons characterized by extreme mean locations of the ITCZ in the Atlantic. To address the asymmetry between the Northern Hemisphere (NH) and Southern Hemisphere (SH) extratropical forcing on meridional mode variability, we also calculate the optimals and growth rates that arise when we constrain the optimal initial conditions to either the Northern or Southern Hemisphere. The results shown in this section should hold qualitatively well over the eastern Pacific as the mean WES and ITCZ states there are similar to the Atlantic.
Figure 4 shows the 140-day optimal initial and final structures under boreal spring mean states. Figure 4a depicts the unconstrained optimal initial condition, and Fig. 4c shows the case in which the initial condition is constrained to the NH. The unconstrained optimal (Fig. 4a) is nearly symmetric about the equator (it would be perfectly symmetric if both α and 
Optimal (a),(c) initial and (b),(d) final structures calculated under the FMA mean state for 140 days under the full (unconstrained) initial norm kernel in (a),(b) and for 140 days under the NH initial norm kernel in (c),(d). SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the respective initial condition fields, solid contours denote positive values, and dashed contours negative ones. The lengths of the velocity arrows are drawn relative to the maximum wind speed of the respective initial conditions. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Optimal (a),(c) initial and (b),(d) final structures calculated under the FMA mean state for 140 days under the full (unconstrained) initial norm kernel in (a),(b) and for 140 days under the NH initial norm kernel in (c),(d). SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the respective initial condition fields, solid contours denote positive values, and dashed contours negative ones. The lengths of the velocity arrows are drawn relative to the maximum wind speed of the respective initial conditions. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Optimal (a),(c) initial and (b),(d) final structures calculated under the FMA mean state for 140 days under the full (unconstrained) initial norm kernel in (a),(b) and for 140 days under the NH initial norm kernel in (c),(d). SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the respective initial condition fields, solid contours denote positive values, and dashed contours negative ones. The lengths of the velocity arrows are drawn relative to the maximum wind speed of the respective initial conditions. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
When the initial condition is constrained to the Northern Hemisphere (Fig. 4c) the optimal is just the northern half of the unconstrained initial condition (Fig. 4a; see appendix B). Like the unconstrained optimal, the NH-constrained optimal grows into a structure (Figs. 4b,d) that strongly resembles the least stable eigenvector of the system in 140 days. The SH-constrained initial condition (not shown) is very similar in structure to the mirror image of the NH-constrained initial condition, although the NH one grows more than its SH counterpart (Fig. 4e). The WES feedback is stronger in the Northern Hemisphere (Fig. 2) and the ITCZ is centered slightly north of the equator (Fig. 1c) on average over FMA, which results in a stronger effective coupling over the Northern Hemisphere tropics and a larger growth rate for the NH-constrained initial condition. The SH-constrained optimal decays at a slower rate than the 120 day−1 linear oceanic damping rate, implying a positive role of the WES feedback in enhancing SST variance.
The leading optimal does not resemble the antisymmetric AMM structure that dominates variability in the tropical Atlantic during boreal spring. However, we include a discussion of this structure as similar modes do emerge in other coupled models (Noguchi 1998; Xie et al. 1999). We speculate that this optimal could be related to a thermally coupled Walker mode that is observed in more complex models when the atmosphere is coupled to a thermodynamic slab ocean via moisture and heat fluxes (Clement et al. 2011). This optimal also looks similar to one of the unstable modes found in Hirst (1986, their Fig. 9b) when the SST equation includes mean zonal SST advection by an anomalous zonal current but no thermocline feedbacks. In that case, the mean zonal advection produces an effect similar to our WES feedback, leading to similar structure and propagation characteristics.
Over boreal fall the mean state (Figs. 1 and 2) and resulting optimal structures (Fig. 5) are quite different. The unconstrained leading optimal is asymmetric (i.e., a combination of a symmetric and antisymmetric function) about the equator, with most of the amplitude in the NH. The initial condition starts almost exclusively in the NH and evolves westward and equatorward over time, becoming more symmetric in the process (Figs. 5a,b). This type of “nondipole” behavior is in accord with the Atlantic meridional mode variability observed during boreal summer and fall (see Fig. 2 of Smirnov and Vimont 2011). When the initial condition is constrained to the NH, the optimal initial and final conditions (not shown) are virtually identical to the unconstrained optimal, and so is the growth rate (Fig. 5c). Note that for both the unconstrained and constrained optimal, growth rates for the ASO mean state conditions are nearly double that of the FMA unconstrained optimal, despite similar WES parameters and similar width and amplitude of the ITCZ (and hence 
The 110-day (a) initial and (b) final optimal structures for ASO mean states when the initial condition is unconstrained. SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the initial condition fields, solid contours denote positive values, and dashed contours negative ones. The lengths of the velocity arrows are drawn relative to the maximum wind speed of the initial condition. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
The 110-day (a) initial and (b) final optimal structures for ASO mean states when the initial condition is unconstrained. SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the initial condition fields, solid contours denote positive values, and dashed contours negative ones. The lengths of the velocity arrows are drawn relative to the maximum wind speed of the initial condition. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
The 110-day (a) initial and (b) final optimal structures for ASO mean states when the initial condition is unconstrained. SST is represented by gray contours, low-level geopotential height by black contours, and low-level wind fields by the arrows. Contours are shown at increments of 0.3 times the maximum value of the initial condition fields, solid contours denote positive values, and dashed contours negative ones. The lengths of the velocity arrows are drawn relative to the maximum wind speed of the initial condition. Units on the x and y axes have been scaled by the zonal wavelength (
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
During boreal fall the mean ITCZ is completely contained in the NH tropics, implying that convective support for coupled ocean–atmosphere interaction is asymmetric. Any initial condition in the Southern Hemisphere would quickly die off because the system is uncoupled there (in this model) during this season. We note that the SH behavior is a limitation to our study, as these variations may be coupled through boundary layer adjustment (Lindzen and Nigam 1987; Battisti et al. 1999; Chiang et al. 2001) in nature. Optimals constrained to start in the SH are not well separated and show very little large-scale structure (not shown). Any SH-constrained initial optimal decays at a rate close to the linear oceanic damping rate, as can be seen in Fig. 5c for the leading SH optimal.
b. The role of ITCZ location and width
In this section we explore how the mean state affects the growth and symmetry of the optimal response under a symmetric 
(a) Different 
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
(a) Different
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(a) Different
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Growth of the leading symmetric and antisymmetric optimal structures under the different coupling regions (Fig. 6a) are shown in Figs. 6b and 6c, respectively. A general increase of the growth is observed for both symmetric and antisymmetric leading optimals as the coupling region gets wider. The rate of increase of the growth as a function of 
Comparing Fig. 4e to Fig. 5c, it is observed that the leading optimal condition achieves more growth in boreal fall compared to spring. Is this result part of a pattern in which optimal conditions transiently grow more for asymmetric mean states? To investigate this we calculate the transient growth under different 
Growth of the least stable asymmetric mode under mean state given by the shape and magnitude of solid line in Fig. 6a, but centered at y0 = 0° (solid line), 4° (dashed line), 8° (dashed–dotted line), 12° (dotted line), 16° (circle), and 20° (cross).
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Growth of the least stable asymmetric mode under mean state given by the shape and magnitude of solid line in Fig. 6a, but centered at y0 = 0° (solid line), 4° (dashed line), 8° (dashed–dotted line), 12° (dotted line), 16° (circle), and 20° (cross).
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
Growth of the least stable asymmetric mode under mean state given by the shape and magnitude of solid line in Fig. 6a, but centered at y0 = 0° (solid line), 4° (dashed line), 8° (dashed–dotted line), 12° (dotted line), 16° (circle), and 20° (cross).
Citation: Journal of Climate 29, 10; 10.1175/JCLI-D-15-0542.1
c. The mean state as a “mode selector”
The different optimal responses for different symmetric mean states (Fig. 6a) may be understood in terms of the spatial structure of heating of stationary forced atmospheric Rossby and Kelvin waves, which depends on the product 
(a) Norm of 
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(a) Norm of
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(a) Norm of
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The transition from the symmetric mode for narrow ITCZ structures to an antisymmetric mode for broader ITCZ structures is examined as a function of ITCZ width (all equatorially centered) in Fig. 8a. When 
Similar arguments explain why the antisymmetric optimal is preferentially excited for ITCZ structures that are centered off the equator. The norm of 
Comparing the growth and structures of the optimal responses under different symmetric coupling regions is not just a theoretical exercise. As previously mentioned, in this work the ocean is coupled to the atmosphere through SST-induced deep heating anomalies. To a first approximation this Gill-like atmospheric model (Gill 1980) can be transformed into a Lindzen and Nigam (1987) (LN)-type model (Neelin 1989) where SST influences the boundary layer pressure gradients that directly drive the boundary layer winds. The equatorial radius of deformation 
4. Conclusions
The role of the mean state in growth and structure of tropical meridional mode variability was examined using the simple linear coupled model of Vimont (2010). For an equatorial mean state that is largely symmetric (representing boreal spring in the Atlantic) growth is dominated by a zonally propagating eigenmode that does not resemble observed meridional mode variations. Transient growth of meridional-mode-like structures is larger for boreal summer and fall conditions, and the spatial structure of variability is more consistent with observed meridional mode variability.
The role of hemispheric asymmetry in optimal initial conditions that produce transient growth was examined through developing a Northern or Southern Hemisphere initial norm under which to evaluate growth. Leading optimal conditions grow more when they start in the Northern Hemisphere compared to the Southern Hemisphere during boreal spring, and SH-constrained initial conditions do not grow at all during boreal fall. This is consistent with existing research that finds an important role for Northern Hemisphere compared to Southern Hemisphere extratropical forcing in meridional mode variability. Although we caution that these optimals were calculated by assuming that SST influences the atmosphere through its impact on deep convection. A different result may result in a model where SST is coupled to the atmosphere through boundary layer SST-induced pressure variations, an effect that we have not considered in depth in this work.
Optimal structures and growth were examined for different mean states, with a focus on ITCZ width and location as the WES feedback is relatively insensitive to seasonal variations. For an equatorially symmetric ITCZ optimal structures were either purely symmetric or antisymmetric about the equator and evolve conserving their initial symmetry. Internal dynamics do not mix the symmetries for this idealized case. This does not preclude external shocks (like asymmetric external atmospheric forcing) from mixing the symmetry of the optimals. For narrow ITCZ structures equatorially symmetric optimals dominate the growth, while antisymmetric optimal structures dominate growth for wider ITCZ structures. This is traced to the increase in importance of the antisymmetric atmospheric Rossby wave response as the (symmetric) effective coupling region gets broader.
Growth and optimal structures were also examined as a function of ITCZ location, which relates to seasonal variations in the ITCZ. In that case, the optimal response activity was mainly contained in the same hemisphere as the ITCZ, as there is not enough convective support for an atmospheric response that would feed back and grow outside of the ITCZ region. As a consequence of this, optimals under an asymmetric ITCZ are neither symmetric nor antisymmetric. Interestingly, optimal initial conditions experience maximum growth when the ITCZ is located off of the equator, which is counterintuitive considering that maximum meridional mode variance is found in boreal spring (Chiang and Vimont 2004). This could imply that meridional mode variability during boreal summer and fall is more transiently maintained by internal ocean–atmosphere interactions, whereas in spring the thermodynamic coupling might have a secondary role to direct atmospheric forcing. More studies are needed to assess the relative roles of internal thermodynamically coupled ocean–atmosphere interactions and direct atmospheric forcing on meridional mode variability during different seasons.
There are numerous caveats to the findings in this study. First, the atmospheric model is overly simplistic, although the results should be applicable to a Lindzen and Nigam (1987) or Battisti et al. (1999) style model of the boundary layer responding to surface temperature anomalies. Second, the ocean model is clearly too simplistic as lateral Ekman fluxes, upwelling, ocean dynamics, and low-cloud feedbacks will play a role in surface temperature variations in nature. Third, in reality the ITCZ and WES feedback are codetermined by the dynamics of the Hadley and Walker circulations. No attempt was made herein to ensure a physically consistent mean state, aside from deriving the specific mean states for the Atlantic Ocean from observations.
Despite these caveats, the study does provide physical insight into meridional mode behavior under structural variations in the mean state. Specifically, it highlights the importance of the mean state as a “mode selector” of thermodynamically coupled ocean–atmosphere variability; that is, through preferentially enabling ocean–atmosphere coupling for specific atmospheric modes, the mean state can play an important role in determining whether tropical thermodynamically coupled variability is dominated by equatorially symmetric, antisymmetric, or asymmetric structures. Results also identify the convective coupling provided by the ITCZ as a key player in meridional mode dynamics.
Acknowledgments
This work was supported by NSF Climate and Large Scale Dynamics Projects ATM-0849689 and 1463970 and the University of Wisconsin Climate, People, and the Environment Program. We thank Tim Li for handling the manuscript; and S.-P. Xie, A. Subramanian, and one anonymous reviewer for their comments and suggestions.
APPENDIX A
Kq(y)α(y) as the Effective Coupling of the System
The 
This shows that the coupled part of the system depends on the product 
APPENDIX B
Relation between Leading Constrained and Unconstrained Optimals
For example, in this case Fig. 3c corresponds to 
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For the same argument, an SST initial norm does not make physical sense for long lead times, as it would penalize initial SST anomalies and allow unconstrained atmospheric initial anomalies. For long lead times, these atmospheric anomalies are simply the structures that most effectively generate SST anomalies that ultimately are responsible for the long-term growth.
