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    (a) Global precipitation minus evaporation (PE) (a good proxy for low-level moisture convergence in the tropics). Units are 2 g m−2 s−1; solid contours and light shading denote positive values, dashed contours and dark shading denote negative values, and the zero contour has been omitted. (b) Zonal average PE over the oceans (dark solid line) and approximation (dashed line) used for Kq (Model I) and for ϵM (Model II).

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    (a) Latent heat flux sensitivity to changes in zonal wind, ∂LH/∂u (contour 5 W s m−3). Solid contours and light shading denote a downward latent heat flux anomaly per unit change in zonal wind speed; dashed contours denote the opposite; and the thick line represents the zero contour. (b) Zonal average latent heat flux sensitivity over global oceans (dark solid line) and approximation used for α(y) (dashed line).

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    Eigenspectrum for the coupled Model I (open black circles) shown together with the eigenspectrum of the uncoupled version of that model [the upper-left 3 × 3 matrix in (3); gray dots]. Eigenvalues for the least-damped coupled modes are shown in the offset figure. Horizontal dashed lines indicate the imposed damping rates for the atmosphere (ϵu−1 = 2 days) and for the ocean temperature (ϵT−1 = 120 days).

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    The four least stable eigenvectors for the coupled Model I: the SST (gray contours), lower-level height perturbations (black contours), and lower-level winds (vectors). For SST and lower-level height, contours are shown at increments of 0.3 times the maximum value for the field, solid contours denote positive values and dashed contours denote negative values, and the zero line has been omitted. Units on the x and y axes have been scaled by the zonal wavelength Lx and the appropriate nondimensional length scale Ly = .

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    As in Fig. 3, but for Model II. The horizontal dashed line in the offset figure indicates the imposed damping rates for ocean temperature (ϵT−1 = 120 days).

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    Transient growth σ(τ)2 over time τ (days) for the first three τ optimals of Model I. Open circles, crosses, and triangles represent energy growth for the leading antisymmetric τ optimal, the leading symmetric τ optimal, and the second symmetric τ optimal, respectively.

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    The leading antisymmetric τ optimal (left) shown with the final structure [(right); scaled by amplitude growth σ(τ)] that it evolves into over time τ for Model I, for different τ : dimensional SST (gray contours; contour 0.1°C), lower-level geopotential (black contours; contour 3 m2 s−2), and lower-level winds (vectors). Solid contours denote positive values, dashed contours denote negative values, and the zero line has been omitted. Growth rates σ(τ)2 and τ are labeled to the right and left, respectively, of the figure. Units on the x and y axes have been scaled by the zonal wavelength Lx and the appropriate nondimentional length scale Ly = . Solid black horizontal lines in the left panels are at ±15°, the approximate cutoff for Kq.

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    As in Fig. 6, but for Model II.

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    The leading antisymmetric τ optimal (left) shown with the final structure [(right); scaled by amplitude growth σ(τ)] that it evolves into over time τ for Model II. Shown are the dimensional SST (gray contours; contour 0.1°C), effective geopotential ΦRG = gh − ΓκT (black contours; contour 3 m2 s−2), and winds (vectors). Solid contours denote positive values, dashed contours denote negative values, and the zero line has been omitted. Growth rates [σ(τ)2] and τ are labeled to the right and left, respectively, of the figure. Units on the x and y axes have been scaled by the zonal wavelength Lx and the appropriate nondimentional length scale Ly = . Solid black horizontal lines in the left panels are at ±15°, the approximate cutoff for diagnosing mean moisture convergence.

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    Uncoupled atmospheric response to imposed SST forcing for (a),(c) Model I with constant α and Kq and (b),(d) Model II with constant α and ϵM. The SST structure is given by (16) with Lf = 15° and for (a) and (b) yc = 7.5° or for (c) and (d) yc = 20°. Shown are the dimensional SST (gray contours; contour 0.25°C), lower-level or equivalent geopotential [black contours, contour 50 m2 s−2 in (a),(b), and (d) and 200 m2 s−2 in (c)] and lower-level winds (vectors). Solid contours denote positive values, dashed contours denote negative values, and the zero line has been omitted. Units on the x axes have been scaled by the zonal wavelength Lx.

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    Ratio of SST growth via the WES feedback [first term in (11)] to SST decay via linear damping [second term in (11)] for Models I and II under constant background conditions and for SST forcing centered at progressively higher latitudes. The meridional structure of the SST forcing is given by (16) with width Lf = 15° and maxima/minima centered at yc. Results are shown for Model I (dashed line) and Model II with ϵM = 8 h−1 (thick solid line) and ϵM = 48 h−1 (thin solid line).

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    Ratio of SST growth via the WES feedback [first term in (11)] to SST decay via linear damping [second term in (11)] as a function of zonal wavelength. The meridional structure of the SST forcing is given by (16), with width Lf = 15° and maxima/minima at ±7.5°.

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    Model I (left) optimal steady forcing patterns and (right) steady response for (a),(b) constant Kq and α; (c),(d) constant Kq but standard α(y); and (e),(f) standard parameters. Shown in the left column are SST tendency [gray contours; 5 × 10−3 K (100 day)−1] and lower-level geopotential tendency [black contours; (a),(c) 5 × 10−2 m2 s−2 (100 days)−1, (e) 5 × 10−1 m2 s−2 (100 days)−1]; and in the right column SST (gray contours; 0.1 K) and lower-level geopotential (black contours; 1 m2 s−2). Solid contours denote positive values (or zero, in the case of SST tendency and SST), dashed contours denote negative values, and the zero contour has been omitted for geopotential height only. Thin black lines at ±30° and ±15° in the left panels illustrate the poleward limits of α(y) and Kq, respectively.

  • View in gallery

    Model II (a) optimal steady forcing patterns and (b) steady response for constant α and standard damping. Shown for (a) are SST tendency [gray contours, 1.5 × 10−2 K (100 day)−1] and height-induced boundary layer geopotential anomaly gh [black contours; 0.2 m2 s−2 (100 day)−1]; and in (b), SST (gray contours, 0.1 K) and lower-level equivalent geopotential (black contours, 0.5 m2 s−2). Solid contours denote positive values (or zero, in the case of SST tendency and SST), dashed contours denote negative values, and the zero contour has been omitted for geopotential height only. Thin black lines at ±30° and ±15° in (a) illustrate the poleward limits of α(y) and ϵM, respectively.

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Transient Growth of Thermodynamically Coupled Variations in the Tropics under an Equatorially Symmetric Mean State

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  • 1 Department of Atmospheric and Oceanic Sciences, and Center for Climatic Research, University of Wisconsin—Madison, Madison, Wisconsin
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Abstract

The dynamics of thermodynamically coupled disturbances in the tropics that bear a strong resemblance to observed meridional mode variations are investigated using two simple linear coupled models. Both models involve an ocean equation coupled to the atmosphere via the linearized effect of zonal wind variations on the surface bulk latent heat flux. The two models differ in their atmospheric components, which consist of (i) a Gill–Matsuno style model of the free troposphere in which atmospheric heating is parameterized to be linearly proportional to sea surface temperature and (ii) a reduced-gravity model of the tropical boundary layer in which SST anomalies are associated with hydrostatic pressure perturbations throughout the boundary layer. Both atmospheric models follow the standard shallow-water equations on an equatorial beta plane.

Growth rates and propagation of coupled disturbances are calculated and diagnosed via eigenanalysis of the linear models and singular value decomposition of the Green’s function for each model. It is found that the eigenvectors of either model are all damped, not orthogonal, and not particularly meaningful in understanding observed tropical coupled variability. The nonnormality of the system, however, leads to transient growth over a time period of about 100 days (based on the choice of parameters in this study). The idealized initial and final conditions that experience this transient growth resemble observed tropical meridional mode variations and tend to propagate equatorward and westward in accord with findings from previous theoretical and modeling studies. Instantaneous growth rates and propagation characteristics of idealized transient disturbances are diagnosed via the linearized atmospheric potential vorticity equation and via propagation characteristics of atmospheric equatorial Rossby waves.

Constraints on the poleward extent of initial conditions or imposed steady forcing that can lead to tropical meridional mode variations are identified through analysis of the steady coupled equations. Three constraints limit the poleward extent of forcing that can generate tropical meridional mode variations: (i) a dynamical constraint imposed by the damping rate of the temperature equation as well as the propagation speed of the mode along its wave characteristic; (ii) a constraint imposed by the effectiveness of zonal wind variations in generating surface latent heat flux anomalies; and (iii) the surface moisture convergence, which limits the poleward extent and strength of ocean to atmosphere coupling.

Corresponding author address: Daniel J. Vimont, 1225 W. Dayton St., Madison, WI 53706. Email: dvimont@wisc.edu

Abstract

The dynamics of thermodynamically coupled disturbances in the tropics that bear a strong resemblance to observed meridional mode variations are investigated using two simple linear coupled models. Both models involve an ocean equation coupled to the atmosphere via the linearized effect of zonal wind variations on the surface bulk latent heat flux. The two models differ in their atmospheric components, which consist of (i) a Gill–Matsuno style model of the free troposphere in which atmospheric heating is parameterized to be linearly proportional to sea surface temperature and (ii) a reduced-gravity model of the tropical boundary layer in which SST anomalies are associated with hydrostatic pressure perturbations throughout the boundary layer. Both atmospheric models follow the standard shallow-water equations on an equatorial beta plane.

Growth rates and propagation of coupled disturbances are calculated and diagnosed via eigenanalysis of the linear models and singular value decomposition of the Green’s function for each model. It is found that the eigenvectors of either model are all damped, not orthogonal, and not particularly meaningful in understanding observed tropical coupled variability. The nonnormality of the system, however, leads to transient growth over a time period of about 100 days (based on the choice of parameters in this study). The idealized initial and final conditions that experience this transient growth resemble observed tropical meridional mode variations and tend to propagate equatorward and westward in accord with findings from previous theoretical and modeling studies. Instantaneous growth rates and propagation characteristics of idealized transient disturbances are diagnosed via the linearized atmospheric potential vorticity equation and via propagation characteristics of atmospheric equatorial Rossby waves.

Constraints on the poleward extent of initial conditions or imposed steady forcing that can lead to tropical meridional mode variations are identified through analysis of the steady coupled equations. Three constraints limit the poleward extent of forcing that can generate tropical meridional mode variations: (i) a dynamical constraint imposed by the damping rate of the temperature equation as well as the propagation speed of the mode along its wave characteristic; (ii) a constraint imposed by the effectiveness of zonal wind variations in generating surface latent heat flux anomalies; and (iii) the surface moisture convergence, which limits the poleward extent and strength of ocean to atmosphere coupling.

Corresponding author address: Daniel J. Vimont, 1225 W. Dayton St., Madison, WI 53706. Email: dvimont@wisc.edu

1. Introduction

Coupled ocean and atmosphere variations in the tropics involve variations that can generally (though oversimply) be classified as “zonal modes” [e.g. the El Niño–Southern Oscillation (ENSO) phenomenon, or the so-called Atlantic Niño] or “meridional modes” (Servain et al. 1999; Chiang and Vimont 2004). A major difference between the two classifications is the physical mechanism that acts to destabilize the mode and hence allows the mode to emerge as a dominant pattern of covariability in the ocean and atmosphere. In the case of the zonal modes, the destabilizing effect can be traced to ocean dynamics—in particular, a positive feedback between zonal winds, thermocline depth or zonal currents, and sea surface temperature [see Neelin et al. (1998) for a review of ENSO physics]. In contrast, tropical meridional modes are destabilized owing to thermodynamic coupling between the wind speed, wind-induced evaporation, and SST (WES feedback; Xie and Philander 1994; Chang et al. 1997]. Chiang and Vimont (2004) show that meridional mode variations exist in both the Pacific and Atlantic basin and suggest that the WES feedback plays an important role in their variability. Further results by Wu et al. (2008) have identified meridional modelike behavior in the Indian Ocean that related to concurrent ENSO events.

It is thought that the WES feedback plays a central role in meridional mode variations through the following relationships. The tropical atmospheric response to a meridional SST gradient includes winds that blow toward warmer water and veer to the right (left) in the Northern (Southern) Hemisphere. In the presence of mean tropical easterlies, the veering effect leads to a relaxation (intensification) of winds over warm (cold) water and hence a positive feedback on the original SST gradient through modulation of evaporation. Thus, the WES feedback includes the following three relationships: (i) SST anomalies generate an atmospheric circulation; (ii) the atmospheric circulation includes wind-induced evaporation anomalies that are collocated with the original SST anomalies; and (iii) the evaporation anomalies act to amplify, or at least reduce the damping rate of, the original SST anomalies. We investigate these processes for idealized thermodynamic disturbances in the tropical coupled ocean–atmosphere system.

Theoretical analysis of tropical ocean–atmosphere interactions have identified meridional modes as true dynamical “modes” of the tropical ocean and atmosphere system (Xie 1996, 1997, 1999) that arise via the WES feedback. These models typically couple a Gill–Matsuno style model of the free troposphere (Matsuno 1966; Gill 1980) to an ocean, emphasizing the thermodynamic effects of changes in wind speed on evaporation and surface ocean Ekman advection and, hence, the SST tendency. Results from these analyses identify meridional mode variations as the least stable modes of a dynamical system that can contribute to the observed asymmetric mean state in the tropics (Xie 1996, 1997) and to tropical decadal variations (Xie 1999). In the case of tropical variability, it is generally thought that tropical meridional mode variations are not self-sustaining, instead requiring some sort of external forcing to generate variability (Xie 1999). In the Atlantic, the North Atlantic Oscillation and ENSO have been identified as potential contributors to the external forcing of the Atlantic meridional mode (Xie and Tanimoto 1998; Czaja et al. 2002; Chiang and Vimont 2004; Kossin and Vimont 2007). In the Pacific, Chiang and Vimont (2004) shows that the atmospheric North Pacific Oscillation (NPO) can lead to Pacific meridional mode variations (Chiang and Vimont 2004; Alexander et al. 2008; Vimont et al. 2009), which can eventually affect the evolution of ENSO through the seasonal footprinting mechanism (Vimont et al. 2003a,b; Alexander et al. 2010).

The propagation characteristics of coupled disturbances have also been investigated by other authors (Liu and Xie 1994; Xie 1996, 1999). Xie (1996) identifies the westward propagating nature of equatorially antisymmetric coupled modes and attributes the coupled propagation to the westward phase shift of the atmospheric Rossby wave response relative to the SST portion of the coupled wave. Liu and Xie (1994) and Xie (1999) show that coupled anomalies propagate equatorward and westward. The equatorward propagation is attributed to the existence of westerly (easterly) wind anomalies centered equatorward of warm (cold) SST anomalies. The zonal wind anomalies, in turn, are attributed to geostrophic balance between the temperature-induced meridional pressure gradient and the Coriolis force.

In the present manuscript, we investigate the growth and propagation characteristics of thermodynamically coupled tropical disturbances that bear a strong resemblance to observed meridional mode variations. The analysis herein differs from previous analyses in the following ways: (i) we consider both a Gill–Matsuno-type atmospheric model of the free troposphere and a Battisti et al. (1999, hereafter referred to as BSH)-type model of the boundary layer response to surface temperature variations (differences between the two models largely reflect differences in their equivalent depths); (ii) we investigate transient and instantaneous growth rates for coupled disturbances in addition to calculating the modal growth rates; (iii) we interpret growth and propagation of the coupled disturbances in terms of the quasigeostrophic potential vorticity (PV) and the wave characteristics of atmospheric Rossby waves; and (iv) we identify constraints on the structure of initial conditions and external forcing that can lead to tropical meridional mode variations.

This paper is organized as follows. The linear coupled models and method of analysis are presented in section 2, including the role of the mean state in setting model parameters. In section 3, we present the structure and growth of the dynamical modes and the structures that experience transient growth over a finite time period. In section 4, we investigate the cause of transient growth and propagation characteristics of the coupled signals through analyzing the uncoupled atmospheric response to meridional modelike SST patterns and determine constraints on the structure of initial conditions and external forcing that would be effective at exciting tropical meridional mode variations. Section 5 includes a summary of results and a discussion of the implications of this work.

2. Model formulation

a. Linear coupled model

We investigate modes in simple models of the tropical atmosphere thermodynamically coupled to a constant depth oceanic mixed layer (a slab ocean). The ocean temperature equation is
i1520-0442-23-21-5771-e1
where ρ, co, and Ho represent water density, specific heat capacity of water, and the depth of the oceanic mixed layer (see Table 1 for values). The temperature is linearly and harmonically damped with damping coefficients ϵT and γ, respectively. Harmonic damping is applied to eliminate high meridional wavenumber modes, as in Xie (1999). The ocean is thermodynamically coupled to the atmosphere via changes in latent heat flux induced by zonal wind variations [α(y)u, discussed in section 2b below]. We do not consider the case of an evaporation dependence on meridional wind variations (the effect of mean meridional winds has been investigated in Liu and Xie 1994).
Two formulations of the tropical atmosphere are explored, both of which are versions of the linearized shallow-water equations on an equatorial beta plane. Results do not change appreciably if a more realistic variation of the Coriolis parameter is used. Modal solutions of the form
i1520-0442-23-21-5771-e2
are assumed, where ψ is the model-state vector 〈u, υ, ϕ, T〉, the zonal wavenumber k is prescribed (for nearly all analysis in the present manuscript we use 2πk−1 = 120°, which roughly approximates the length scale of observed meridional modes; Chiang and Vimont 2004), and the meridional structure and angular frequency ω can be determined through eigenanalysis of the linear system. The meridional discretization and method of solution are described in section 2d. The two formulations of the atmospheric models are described below.

1) Model I: The Gill–Matsuno model of the free troposphere

The first model is a Gill–Matsuno style model of the first baroclinic mode of the tropical free troposphere, which is essentially a two-layered model (Matsuno 1966; Gill 1980). The model I equations are
i1520-0442-23-21-5771-e3
where u and υ are the zonal and meridional winds averaged over the lower layer, ϕ is geopotential, and T is the SST. Here ϵu and ϵϕ are linear damping coefficients that represent Rayleigh friction and Newtonian cooling, respectively, and in line with Gill (1980) are set equal to each other, β is the meridional gradient of the Coriolis parameter at the equator, and ca is the reduced-gravity wave speed for the first baroclinic mode in the free troposphere. The atmosphere is coupled to the ocean via atmospheric heating, Kq(y)T, which is parameterized to be linearly proportional to temperature. The heating rate Kq(y) varies in the meridional direction as discussed in section 2b. As the model represents the first baroclinic mode of the free troposphere, we reverse the sign of the atmospheric heating so that atmospheric wind represents the lower-level flow. Values for the various parameters are provided in Table 1.

This model formulation is very similar to that used by Xie (1996) except that we retain atmospheric tendencies, ignore the contribution of ocean advection to SST tendencies, and prescribe a symmetric meridional dependence for the atmospheric heating. Other studies have highlighted the role of wind-induced surface convergence on the atmospheric heating (Zebiak 1993; BSH), which effectively reduces the wave speed in convecting regions (BSH); we ignore this effect except through discussion of results with differing imposed wave speeds.

2) Model II: Reduced-gravity boundary layer

The second model that we consider is the BSH reduced-gravity model of the well-mixed tropical boundary layer. The atmospheric model is similar to that of Lindzen and Nigam (1987) except that pressure perturbations are acted upon by reduced gravity and the time scale for the life cycle of deep convection is more realistic (BSH). The model equations are
i1520-0442-23-21-5771-e4
where u and υ are the mass-weighted zonal and meridional wind in the boundary layer, h is a perturbed boundary layer height from its climatological height Hb, and T is SST. The model assumes that virtual potential temperature is proportional to surface temperature (κT) and is well mixed in the boundary layer. These potential temperature perturbations are responsible for hydrostatic pressure perturbations (g−1κΓRGT). Formally, this approximation is valid under the assumption of a steady atmospheric response to SST variations in the tropics; we will see that for the time scales of interest this steady assumption holds. The resulting hydrostatic pressure perturbations are acted upon by reduced gravity, which is proportional to the mean virtual potential temperature difference across the top of the boundary layer (g′ = gΔΘυ/Θ0). The reduced gravity results in an effective gravity wave speed for Model II that is about ½ that of Model II for the standard parameters in Table 1, though the addition of a convergence feedback to Model I would bring the two wave speeds in line with each other (BSH). The momentum is damped at a rate of ϵu, and mass convergence is assumed to be vented out of the boundary layer with a time scale appropriate for the life cycle of deep convection, ϵM(y)−1. This time scale depends on the presence of deep convection in the mean state, which in turn depends on the mean moisture convergence in the boundary layer. The ventilation rate is discussed in more detail in section 2b. Note that that the full geopotential perturbation in the boundary layer can be written as ΦRG = ghκΓRGT. In this analysis, variations of ΦRG are generally shown to depict coupled variations, while h variations are shown to illustrate initial conditions or sensitivities.

b. Surface convergence, Kq(y) and ϵM(y)

The mean surface convergence determines the meridional structure of parameters Kq(y) (Model I) and ϵM(y) (Model II). For Model I, changes in atmospheric heating are assumed proportional to surface temperature variations [Kq(y)T] through increased moisture availability. However, this will be realized as deep heating only in regions of mean lower-level convergence (e.g., negative heating anomalies are unphysical in regions where mean heating is zero). In Model II, damping in the continuity equation represents ventilation from the boundary layer, and as such ϵM(y) represents a time scale for the development of deep convection. This time scale is determined by the mean lower-level moisture convergence in the tropics, and values for the time scale are consistent with those in BSH. In regions of mean convection (moisture convergence) deep convection is assumed to develop with a time scale of 8 h (ϵM−1ϵC−1 ≈ 8 h); in mean nonconvecting regions (mean moisture divergence) the adjustment times are set equal to ϵu−1 ≈ 2 days (the exact equality to the momentum damping rate is only coincidence). BSH show that the differing time scales for convective adjustment effectively lead to different gravity wave speeds in convecting versus nonconvecting regions:
i1520-0442-23-21-5771-e5
As , the response to surface heating in regions of mean moisture convergence is more horizontally confined than the response in regions of mean moisture divergence. The implications of the differing effective phase speeds are discussed in more detail in section 4.
The spatial structure of Kq(y) and ϵM(y) are determined by the zonally averaged mean convergence over various tropical regions. The vertically integrated equilibrium moisture budget may be written as
i1520-0442-23-21-5771-e6
where the left-hand side represents mean moisture convergence in the column, and the right-hand side is the mean precipitation minus evaporation. As most of the moisture convergence in the tropics occurs in the boundary layer, the mean PE provides a good estimate of mean moisture convergence below the top of the boundary layer. Mean PE from the Climate Prediction Center (CPC) Merged Analysis of Precipitation (standard version) (Xie and Arkin 1997) is plotted over the tropics in Fig. 1, as is the zonal average of PE over the tropical oceans. In this study, we consider an equatorially symmetric mean state with mean convergence within 15° latitude of the equator. This is characteristic of the zonal-mean low-level moisture convergence in the tropics and of conditions in the western equatorial Pacific. In a following study, we consider the case of an equatorially asymmetric mean heating profile. We approximate the meridional structure of convergence using
i1520-0442-23-21-5771-e7
for y0 set to 15°. The functional form of convergence f (y) is plotted in Fig. 1b. The heating in Model I is then Kq = qf (y), and q is a constant (Table 1). The damping rate for the continuity equation in Model II is approximated as ϵM(y) = ϵNC(1 − f (y)) + ϵCf (y), where ϵNC and ϵC are given in Table 1.

c. The WES feedback parameter α(y)

The WES parameter α(y) determines the change in SST per unit change in zonal wind speed and can be diagnosed via the linearized change in latent heat flux per unit change in wind speed (Czaja et al. 2002; Vimont et al. 2009):
i1520-0442-23-21-5771-e8
Using the standard bulk formula in which the latent heat flux is linearly proportional to wind speed w = (u2 + υ2 + w2)1/2, we compute
i1520-0442-23-21-5771-e9
A similar equation results for the sensitivity of latent heat flux to meridional wind variations, though we do not consider the contribution of meridional wind variations to the WES feedback in this analysis. Other studies have incorporated the effect of Ekman advection into the WES parameter (Xie and Philander 1994; Liu and Xie 1994), which can double the amplitude of the WES parameter in some regions; we do not consider Ekman advection here. Equation (9) shows that the WES parameter will be proportional to the strength of the mean zonal wind, divided by the square of the wind speed. For easterly mean winds ∂LH/∂u is negative, implying a reduction in evaporation (positive latent heat flux anomaly) per unit westerly zonal wind anomaly; hence α(y) > 0. Figure 2 shows −∂LH/∂u for the mean state of the tropical oceans, including zonal averages over various regions. The global zonal average of −∂LH/∂u is approximately symmetric about the equator with maxima of ∼15 W s m−3 in the subtropics around ±20°. Locally, maxima in −∂LH/∂u are found around ±10° across the central Pacific. Vimont et al. (2009) show that there is considerable seasonality in the WES parameter, especially in regions strongly affected by monsoonal circulations. In the present study, we consider the equatorially symmetric case and parameterize α(y) as
i1520-0442-23-21-5771-e10
where α̂ is a constant (Table 1), Hn is the nth-order Hermite polynomial, and y1 and y2 are set to 20° and 25°, respectively. The functional form of α(y) is plotted in Fig. 2b.
The WES parameter plays a central role in the stability of the coupled modes for each of the models through its effect on SST. An equation for SST variance can be obtained by multiplying (1) by T and integrating over the domain (Xie 1999):
i1520-0442-23-21-5771-e11
The last two terms on the rhs of (11) represent the effects of linear and harmonic damping, respectively, and act to stabilize the mode. The first term on the right-hand side demonstrates that the WES feedback can reduce this stability if there is a collocation between wind-induced changes in surface heat flux [α(y)u] and SST.

d. Method of solution

The linear set of equations in Eqs. (3) and (4) are coded into a 4N × 4N matrix, where N is the number of points in the meridional direction. All variables are nondimensionalized using their respective equatorial radius of deformation ae = (c/β)−1/2, time scale to = ()−1/2, and reference temperature anomaly T0 = 1K. We choose a nondimensional grid spacing of Δy/ae = 0.1; solutions have been computed for varying meridional resolution to ensure stability. For Model II, the default value of ca is set to 30 m s−1, which implies ae ≈ 1100 km. For Model II, cRG is given by (5), yielding a deformation radius of about 900 km for nonconvecting regions [note that (5) indicates that in convecting regions the effective deformation radius (about 600 km) is less than the deformation radius in nonconvecting regions—this will be useful for understanding instantaneous growth rates in section 4]. For most calculations the meridional domain is bounded by yN/S = ±5ae and a sponge layer is applied to meridional boundaries (damping parameters near the meridional boundaries are ramped up to approximately 10 times their values in the interior). Meridional derivatives are discretized using centered differences. The parameters used in the present study are listed in Table 1 and will be referred to as the “standard” parameters.

Each linear model is coded into a sparse matrix 𝗠(I,II) of rank 4 × (yNyS)/dy, yielding the linear algebraic system
i1520-0442-23-21-5771-e12
where i = , the spatial structure of the modes ξj and the associated eigenvalues ωj are determined through eigenanalysis of each 𝗠, and the subscript (I,II) indicates that the analysis is carried out separately for Model I or II. The convention herein is such that the imaginary and real part of the eigenvalue represent the growth rate and frequency, respectively, of its associated eigenvector. Solutions have been computed for varying meridional resolution to ensure stability.
The nonnormality of each model yields sets of eigenfunctions that are not necessarily orthogonal. In some cases the resulting modes of the system are not well separated from each other, and for the standard parameters we will see that all eigenvectors have negative imaginary parts (all are damped); hence, the eigenvalue has less practical significance in understanding the evolution of the system than the energy growth over a finite time period. This energy growth is identified through singular value decomposition of Green’s function (Penland and Sardeshmukh 1995; Johnson et al. 2000):
i1520-0442-23-21-5771-e13
The right singular vectors (columns of 𝗩) are initial conditions that develop into structures depicted by the left singular vectors (columns of 𝗨) in time τ. The energy growth is given by the square of the associated singular value σi. In the present treatment, we focus on the energy growth over a finite time period rather than on stability of the individual modes, as discussed in section 3. We will refer to the right singular vectors as τ optimals, as they represent structures that produce “optimal” growth over time τ. As (13) is calculated using the nondimensional matrix 𝗠, the growth is somewhat sensitive to the choice of nondimensional parameters, especially to T0. Our choice of T0 = 1 yields an effective norm that tends to emphasize growth of SST anomalies rather than atmospheric variations, which is appropriate for the present analysis. We maintain the standard L2 norm (equal weighting for all nondimensional variables) for calculating transient growth.

3. Modal analysis and transient growth

In this section, we examine the structure and energy growth of solutions for Models I and II for the standard parameters in Table 1. For both models, we show the eigenspectrum, the transient growth rates, and spatial structures of the τ optimals and associated final conditions for specified finite time periods.

a. Eigenanalysis of the coupled models

The eigenspectrum for the fully coupled Model I is shown together with the eigenspectrum of the uncoupled Gill–Matsuno model [the upper-left 3 × 3 matrix in (3)] in Fig. 3. The uncoupled model (gray dots) contains the set of equatorial waves discussed in Matsuno (1966) as well as numerical modes. The uncoupled modes are damped at approximately the atmospheric damping time scale ϵu−1 = 2 days. The coupled model (open black circles) also contains these uncoupled modes (note the collocation of the black circles with the gray dots) as well as an additional set of very low frequency coupled modes with reduced damping rates. It is noteworthy that all modes in the model are damped under the standard parameters. A closer look at the least-damped modes of the system indicates that five of the modes are less damped than the linear damping rate for the temperature equation ϵT−1 = 120 days, indicating that coupling destabilizes these modes (though all modes are still linearly stable). A calculation that eliminates coupling with the ocean by setting α(y) to 0 contains a similar set of SST modes except with zero frequency and with a larger damping rate than ϵT−1 (not shown), highlighting the importance of coupling in the destabilization and propagation of these coupled modes.

The spatial structures of the least stable modes from Model I are shown in Fig. 4. The leading two modes are equatorially symmetric across the equator, and the third and fourth modes are antisymmetric. The leading symmetric mode has an atmospheric structure that resembles the steady response to imposed symmetric heating (Matsuno 1966; Gill 1980). The leading mode propagates westward with a period of 200 days, and is damped with a time scale of 520 day−1, considerably longer than the imposed linear damping time scale of 120 days. The cause of the westward propagation and destabilization can be understood through investigation of the phasing of the zonal wind and SST. Focusing on the half of the wave phase that is characterized by a positive SST perturbation on the equator, the symmetric SST perturbation sets up an atmospheric circulation that includes maximum westerly zonal wind anomalies to the west of the maximum SST. The westward phasing of maximum zonal wind anomalies generates a positive SST tendency through the WES feedback, Eq. (1), to the west of the original maximum SST anomaly, leading to the westward propagation. The destabilization is due to the collocation of zonal wind and SST perturbations, as shown in (11).

The third and fourth least stable modes from Model I (Fig. 4) are equatorially antisymmetric and bear a remote resemblance to the tropical meridional modes in the Pacific and Atlantic, though the theoretical modes have a much finer spatial structure than the broad-scale spatial structure observed in nature (Chiang and Vimont 2004). The third and fourth modes propagate eastward and westward, respectively, and are both damped with a time scale slightly longer than the imposed linear damping. Like the first mode, the propagation and reduced stability is consistent with the phasing of the zonal wind with the SST. The small damping rate of the first few modes indicates that the WES feedback is positive for these modes although not strong enough to overcome the imposed linear damping. Note that a damping rate less than the imposed linear damping is a sufficient, but not necessary, condition to infer a positive WES feedback as the harmonic damping becomes increasingly effective at stabilizing modes with increasing finescale meridional structure.

The eigenspectra for the coupled and uncoupled version of Model II are shown in Fig. 5. Like the eigenspectrum from Model I, a set of coupled and uncoupled modes exist in the system. The overall structure of the eigenspectrum is similar to that of Model I, though the phase speed for the uncoupled modes is slower than that for Model I, in line with the slower gravity wave speed for Model II. The coupled modes are nearly stationary, and four modes have damping time scales that are less than or equal to the imposed linear damping rate for temperature. For the least-damped coupled modes, two branches of modes emerge with eastward and westward propagation. The spatial structure of these leading eigenvectors (not shown) contains finescale features that do not appear particularly physically relevant. The difference between the structures of the eigenvectors between Models I and II likely results from the reduced effective deformation radius in the deep tropics for Model II—when the wave speed is arbitrarily reduced in Model I the eigenvectors also develop finer spatial structure and change order. The change in order is consistent with Xie (1996), who finds that antisymmetric modes are destabilized for longer zonal wavelengths; in this case, zonal wavelength is fixed but the deformation radius is reduced, which has a similar effect. Note especially the lack of an isolated destabilized westward-propagating symmetric mode, as was found in Model I.

b. Transient growth of coupled disturbances

Visual inspection of the modal structures in Fig. 4 show that the set of coupled modes are not orthogonal (this is easily confirmed numerically), a result of the nonnormality that is introduced by the coupling. Thus, eigenanalysis of the linear system matrix, while instructional for understanding the physics of the system, is less useful than an investigation of possible transient growth over a finite time period. As described in section 2d, transient growth can be identified through singular value decomposition of Green’s function (13) for various lags. The square of the resulting singular values indicates the transient energy growth over a finite time period, and is plotted for Model I in Fig. 6. Visual inspection (or actual projections) of the first three τ-optimal patterns (only the leading pattern is shown in Fig. 7) indicates that they represent the leading antisymmetric τ optimal and the leading two symmetric τ optimals (note that the ordering of the leading symmetric and antisymmetric τ optimals is switched for lags less than 45 days). Despite the negative growth rates for all eigenvectors of Model I, Fig. 6 shows that transient growth is possible due to the nonnormality of the system. For very short time periods, maximum growth occurs for a spatial structure that strongly resembles the least damped and fastest propagating eigenvector of the system (Fig. 4a). Beyond 45 days, maximum transient growth occurs for the antisymmetric structure and exceeds a doubling of energy for a time scale of 135 days. Further analysis (not shown) confirms that this long-term growth is the nonnormal growth that is due to the projection of the antisymmetric τ optimal onto several of the nonorthogonal antisymmetric eigenvectors.

The spatial structure of the τ optimals for various lags, and the final conditions into which they evolve, are shown in Fig. 7. Inspection of the final conditions (right column) shows an antisymmetric spatial structure with a strong resemblance to observed meridional mode variations, with a meridional SST gradient, winds that blow toward warmer water and curve to the right (left) in the Northern (Southern) Hemisphere, and an atmospheric height perturbation that is centered poleward and westward of the SST. The τ optimals (left column of Fig. 7) have a similar structure in SST, though the height perturbations are centered poleward and eastward of the SST anomalies; this is a result of the tendency for the optimal growth to emphasize SST growth. Because of the WES feedback maximum, SST growth will occur when zonal wind anomalies have the exact same structure as the SST anomalies; hence, meridional gradients in initial geopotential height tend to be collocated with initial SST anomalies. As τ increases, the spatial structure of the final condition becomes progressively more confined to the equator, and the spatial structure of the initial condition tends to be centered farther poleward and eastward. This indicates that the initial spatial structures tend to evolve equatorward and westward, consistent with the analysis of Liu and Xie (1994). The source of equatorward propagation will be explored in more detail in section 4.

Transient growth for Model II is shown in Fig. 8. In Model II, the leading and third τ-optimal patterns are equatorially antisymmetric, and the second τ optimal is equatorially symmetric. Only the leading antisymmetric τ optimal experiences transient growth, with maximum growth around 75 days. The spatial structures of the fastest growing antisymmetric τ optimal and its evolution are shown in Fig. 9. The structures resemble the leading antisymmetric τ optimals from Model I, except that the structure of the final conditions exhibit much more equatorial confinement in Model II and height anomalies are more collocated with SST anomalies (owing to the reduced wave speed in Model II). As lag increases, the SST maxima in the τ optimals moves farther poleward and eastward and the final conditions become progressively more confined to the equator. As in model I, this indicates equatorward and westward propagation of the SST maxima. The reduced amplitude of effective geopotential perturbations (ΦRG = gh − ΓκT) with lag τ in the final conditions occurs as the actual boundary layer height (h) adjusts to lower-level convergence and counters the temperature-induced pressure perturbation (ΓκT). For maximum growth, the actual height (h) and temperature contributions to the effective geopotential are like signed in the initial conditions—as in Model I, maximum growth occurs when initial zonal wind anomalies overlay like-signed SST anomalies.

4. Growth and propagation mechanisms

In section 3, we found that the fastest growing thermodynamically coupled patterns for Models I and II were equatorially antisymmetric patterns that propagated equatorward and westward. In this section, we examine the instantaneous growth and propagation of tropical disturbances with idealized antisymmetric SST patterns. As the results from section 3 indicate that extratropical forcing is necessary for generation of meridional modelike variations, we focus on constraints to the poleward extent of initial conditions or forcing that can excite the tropical antisymmetric pattern. These constraints will involve both mechanisms that arise via internal dynamics of the mode (determined under a uniform mean state) and mechanisms that are dictated by the meridional variation of the mean state. For the former, we consider the case of constant α and constant Kq (in the case of Model I; we consider the effect of constant ϵM for Model II). For the latter, we examine the constraints that the spatial structure of α(y) and Kq(y) [or ϵM(y)] imposes on the antisymmetric structure.

a. Instantaneous growth and propagation

The effect of the WES feedback on the instantaneous growth rate is inferred through calculating the amplitude of the WES feedback term from the energy Eq. (11), 〈α(y)uT〉. To ensure uniformity in our analysis, we use the steady, uncoupled versions of Models I and II to determine the steady atmospheric response to a uniformly imposed SST distribution. A steady assumption is valid as the time scales of the coupled modes or of transient growth are long compared to the time scale necessary for the atmosphere to equilibrate. The uncoupled models are the upper 3 × 3 matrices of (3) and (4); the fourth column of (3) and (4) is applied as a forcing f:
i1520-0442-23-21-5771-e14
Here, ψ3 is the atmospheric-state vector 〈u, υ, ϕ〉. The steady solution to (14) is simply
i1520-0442-23-21-5771-e15
For uniformity we use an identical SST forcing for both model analyses, with the functional form:
i1520-0442-23-21-5771-e16
This structure consists of opposite-signed ½ wavelength sinusoids on either side of the equator, each with a maximum (minimum) at yc (−yc) and with width Lf . SST is set to zero equatorward and poleward of the two lobes. We use Lf = 15° and a variety of locations for yc, the centers of the anomaly.

In our initial analysis, we examine growth and propagation characteristics imposed by the dynamics of the system, in absence of variations in the mean state (i.e., α and Kq or ϵM constant). The spatial structure of the uncoupled steady atmospheric response to (16) with Lf = 15° and yc = 7.5° is shown for Models I and II under constant mean-state conditions in Figs. 10a and 10b. The structure of the response in Fig. 10a is similar to the response to the 15 day τ optimal in Fig. 7 and shows positive zonal wind anomalies over and equatorward of the center of the positive lobe of the imposed SST forcing. The equatorward and westward propagation of the coupled mode (seen in Fig. 7) is explained by the meridional and zonal phasing of the steady atmospheric circulation associated with the given SST distribution. The equatorward propagation is attributed to positive (negative) zonal wind anomalies centered equatorward of a positive (negative) SST anomaly. This reduces (increases) wind speed and evaporation equatorward of a maximum (minimum) in SST, causing the SST maximum to migrate equatorward. The westward phasing of the atmospheric response relative to the SST causes the westward propagation, under similar arguments.

The zonal and meridional phasing of the zonal wind anomalies relative to the SST plays a central role in the propagation and growth of SST anomalies, and, as such, we examine the causes of the phase shift. The propagation mechanism of similar coupled disturbances has also been discussed elsewhere (Liu and Xie 1994; Xie 1996, 1997, 1999). Because the zonal wind response is dominated by the rotational component of the zonal wind, we can understand the zonal wind variations via either the properties of equatorially antisymmetric Rossby waves that compose the atmospheric response or by the steady, forced quasigeostrophic PV equation for Model I (see, e.g., Gill 1982):
i1520-0442-23-21-5771-e17
where, for model I, ϵu = ϵϕ. Note that a similar expression could be derived for model II via reformulating the continuity equation, as discussed below. The PV equation results from replacing the horizontal convergence (stretching) term from the vorticity equation with geopotential damping (which becomes the βyϕ/ca2 term) and heating [which becomes the (βy/ca2)KqT term] from the continuity equation. In the deep tropics the dominant balance in the continuity equation is between heating and horizontal convergence. Thus, the dominant balance in the potential vorticity equation (17) is between the heating term and damping of relative vorticity, and for the zonal length scales considered here the relative vorticity is dominated by the meridional gradient of zonal wind. In the absence of meridional variations in the Coriolis parameter, the balance between the heating term and damping of relative vorticity would imply an in-phase relationship between heating and relative vorticity and, hence, positive zonal wind anomalies equatorward of positive heating anomalies. This phasing is responsible for the equatorward propagation of anomalies, and is clearly evident in Figs. 10a and 10b. The extension of zonal wind anomalies outside the region of forcing results from the horizontal spreading effect of potential vorticity inversion.

The meridional phasing of u and T also contributes to instantaneous growth through the WES feedback. For the WES feedback to contribute to instability, (11) shows that the integral 〈α(y)uT〉 must be positive. For constant α, if the relative vorticity had the same structure as the heating (e.g., as would occur in the case of a constant Coriolis parameter), then the positive covariance between u and T on the equatorward flank of the SST would be exactly canceled by the negative covariance on the poleward flank. Figures 10a and 10b show that is not the case, which can be traced to the variation in the Coriolis parameter with latitude. The heating term on the rhs of the PV equation (17) is multiplied by the Coriolis parameter; stretching via heating produces relative vorticity in the presence of background planetary vorticity. The increase of planetary vorticity with latitude generates a poleward offset of the stretching term relative to the actual SST anomaly. As the stretching is primarily balanced by damping of relative vorticity, the meridional offset shifts the zonal wind poleward as well, producing positive (negative) zonal wind anomalies over maxima (minima) in the SST. The variation of the Coriolis parameter also contributes to meridional variations in the strength of the zonal component of the geostrophic wind. The geostrophic wind is larger where the Coriolis parameter is smaller, and therefore the zonal component of the geostrophic wind is larger equatorward of a given geopotential distribution. Both of these effects contribute to a net positive WES feedback in Figs. 10a and 10b.

For forcing at higher latitudes, Figs. 10c and 10d show that the atmospheric response is more in phase with the SST, and hence growth via the WES feedback is reduced. At higher latitudes the larger background planetary vorticity reduces the convergent response to heating in the continuity equation, and hence geopotential damping becomes more important. The −βyϕ/ca2 term represents stretching vorticity in (17), which has the same dependence on the Coriolis parameter as the heating (SST) term and hence a more in-phase relationship between SST and the geopotential height field. The reduced wave speed in Model II relative to Model I has the same effect of reducing the convergence response to heating, explaining the difference between Model I and Model II in Fig. 10.

The ratio of growth via the WES feedback [the first term on the rhs of (11)] to the linear damping [the second term on the rhs of (11)] is plotted in Fig. 11 for Models I and II under constant α and Kq (or ϵM) for imposed SST forcing centered at a range of yc. Beyond approximately the equatorial radius of deformation for Model I (about 10°) the strength of the WES feedback gradually decreases. This decrease is due to the increase in background planetary vorticity, as discussed above. This decrease in the WES feedback with latitude places a constraint on the poleward extent of initial conditions that would grow through thermodynamically coupled feedbacks. This is a weak constraint for Model I owing to the relatively large effective wave speed, but is the dominant constraint for Model II that has a smaller effective wave speed, especially in regions of mean moisture convergence. We will discuss additional differences between Models I and II shortly.

Zonal propagation can be explained by the westward phase shift of geopotential anomalies with respect to the SST anomalies seen in Fig. 10. The westward phasing is due to westward propagating atmospheric Rossby signals over the SST forcing (Gill 1980; Xie 1996). We note that this Rossby wave argument differs from the convergence-based argument in Liu and Xie (1994). The Rossby wave argument also suggests that a monopolar SST structure would evolve slightly differently than the imposed harmonic structure, as the eastern edge of a monopolar forcing would not underlie the atmospheric response to the SST of opposite polarity that exists in the case of harmonic forcing. A monopolar SST structure would expand and elongate westward (e.g., as in Xie 1996) rather than propagate westward and would experience a slightly larger instantaneous growth rate.

The Rossby wave argument also predicts that longer zonal wavelengths should experience more growth than shorter wavelengths, as resonant excitation occurs over a longer fetch and the westward phase shift between the atmosphere and SST is reduced (the latter effect is a result of constant atmospheric damping). This is tested via examining the amplitude of SST growth via the WES feedback [the first term on the rhs of (11)] relative to the linear damping rate [the second term on the rhs of (11)]. The ratio of these two terms is calculated over a range of zonal wavelengths for the meridional SST structure in Fig. 10a and is plotted in Fig. 12. Indeed, the WES feedback is strongest for the longest wavelengths, consistent with Xie (1996).

Last, we investigate the differences between Models I and II. The mathematical equivalence of the two models has been noted by BSH and Neelin (1989), who show that under the change in variables ΦRG = ϕκΓT the coupling of the atmosphere to the ocean in Model II is moved from the momentum equations to the continuity equation. The continuity equations for model I and II are then
i1520-0442-23-21-5771-e18
where
i1520-0442-23-21-5771-eq1
Here we have used the fact that ϵϕ for Model I is equal to ϵM for Model II in nonconvecting regions. For a constant background state (δ = 0 and constant Kq), these two equations are mathematically identical, though the parameters are different. In particular, the coupling coefficient from the ocean to the atmosphere is an order of magnitude larger (Kq = 2.5 × 10−3 m2 s−3 K−1) for Model I than for Model II (ϵϕκΓ = 2.9 × 10−4 m2 s−3 K−1). This is compensated by the difference in wave speed for Model I (ca = 30 m s−1) compared to the effective wave speed in Model II [cRG = 17 m s−1 for nonconvecting regions and cRG(1 − δ) ≈ 3 m s−1 for convecting regions]. The reduction in effective wave speed for Model II in convecting regions represents the increased ventilation rate for mass convergence in the boundary layer. Using the nomenclature of Lindzen and Nigam (1987), boundary layer convergence produces a “back pressure” that, in general, counteracts the thermodynamically induced pressure gradient generated by the SST. Enhanced ventilation (increased damping in the original continuity equation) reduces the back pressure, allowing for a stronger convergent response to heating and more growth of the coupled anomalies through the WES feedback in the deep tropics. This explains the difference in growth rates under different ventilation rates for Model II (Fig. 11, solid lines).

b. Response to steady forcing

In this section, we discuss the limitations imposed by the mean state on the poleward extent of forcing that would be able to excite tropical variations through thermodynamically coupled feedbacks. For this analysis, the steady, forced versions of the fully coupled models (3) and (4) are considered. The solution to an imposed steady forcing is given by (15), except for the fully coupled model and state vector (ψ4 = 〈u, υ, ϕ, T〉). Singular value decomposition of yields a set of optimal forcing patterns (right singular vectors) and steady response patterns (left singular vectors). These forcing and response patterns are plotted for Model I for the constant parameter case, the case of constant Kq but standard α(y), and for the full standard parameters in Fig. 13. For the analysis in Fig. 13 we have extended the meridional domain of the model to ±8 deformation radii.

Figures 13a and 13b illustrate the optimal steady forcing and steady response patterns for the case of constant Kq and α. The structure of the forcing and response can be understood in terms of the westward and equatorward wave characteristics for the coupled signals, as discussed earlier and derived in Liu and Xie (1994). The imposed forcing exhibits this westward and equatorward tilt, with maxima occurring around ±40°. The shape of the forcing in Fig. 13a then is such that the coupled waves are resonantly forced as they propagate along their characteristic paths. The response in Fig. 13b is maximized in the deep tropics equatorward of about ±10° and illustrates the same antisymmetric pattern that has emerged in our earlier analysis. If the thermal damping rate is decreased, the forcing maxima shift to higher latitudes (not shown). The thermal damping rate places a weak constraint on the poleward extent of forcing that can excite tropical variability. This is a different constraint than that imposed on initial structures by variation of the Coriolis parameter with latitude; as for the forced case, we are considering equilibrium rather than instantaneous growth. The poleward extent of forcing in Model II is limited by this constraint (Fig. 14), even in the case of standard α(y) (not shown). This is due to the reduced effective atmospheric wave speed for Model II, which implies a more meridionally confined atmospheric response to a given SST distribution and, hence, slower meridional propagation. Finally, we note that in the absence of the WES feedback the imposed forcing would produce a steady SST response of about 0.01 K, two orders of magnitude smaller than the actual response in the deep tropics. This highlights the importance of the WES feedback in the deep tropics.

The mean state imposes two additional constraints on the poleward extent of steady forcing that could excite a response in the deep tropics. Equation (9) shows that the WES parameter α(y) depends on the mean zonal wind. As the zonal wind transitions from mean easterlies in the tropics to mean westerlies in the midlatitudes, our approximation for α(y) (10) (see Fig. 2) approaches zero. Hence, SST forcing beyond this point would generate an atmospheric circulation with zero WES feedback and would be ineffective at generating a coupled response. This limit explains the location of maximum SST forcing in Fig. 13c and its response in Fig. 13d; the SST forcing is located far enough equatorward that the zonal wind anomalies that would be generated along the southern flank of the SST still generate a positive WES feedback. A similar, and more restrictive constraint, is imposed by the mean convergence in the tropics for Model I In regions of mean moisture divergence SST anomalies do not generate deep heating anomalies, so the atmosphere is decoupled from the ocean. This explains the region of maximum forcing centered just poleward of ±15° in Fig. 13e and its response in Fig. 13f (note that Kq is slightly smoothed in the meridional direction, which allows weak forcing slightly poleward of ±15°).

5. Conclusions and discussion

Two linear coupled models are developed for understanding the growth and propagation of thermodynamically coupled structures in the tropics. The first model is based on the Gill–Matsuno model of the free troposphere, and the second is based on the Battisti et al. (1999) reduced-gravity model of the atmospheric boundary layer. The atmospheric models are coupled to the ocean via parameterized deep heating (in the case of the Gill–Matsuno model) or via hydrostatically induced boundary layer pressure perturbations [in the case of the Battisti et al. (1999) model]. The ocean is coupled to the atmosphere through the zonal wind speed, under the assumption that changes in zonal wind speed lead to changes in the surface latent heat flux. The nonnormality of the models yields sets of eigenfunctions that are not orthogonal and have less physical significance than patterns that experience transient growth over a finite time period.

Singular value decomposition of Green’s function yields sets of initial and final structures that may experience growth over a finite time period. For each model, an equatorially antisymmetric pattern is identified as the pattern that experiences the greatest transient growth over a time period of about 100 days. Investigation of the transient growth characteristics over various time periods indicates that antisymmetric initial conditions with maxima at subtropical latitudes propagate equatorward and westward, consistent with the characteristics derived in Liu and Xie (1994). Propagation is explained via the structure of the rotational component of the zonal wind response to the antisymmetric SST patterns. An imposed SST generates relative vorticity anomalies with positive (negative) zonal wind anomalies centered equatorward of positive (negative) SST anomalies. This phasing generates an equatorward propagation through wind-induced evaporation anomalies, under the assumption of mean easterly winds. The westward propagation of the coupled structures is explained via the westward group velocity of the atmospheric Rossby wave response to the coupled SST structure. The resulting westward phase shift of zonal wind anomalies relative to SST anomalies increases as the coupled signal moves equatorward (lower-order Rossby waves have larger group velocities), explaining why the coupled wave characteristics become more zonal as the wave approaches the equator.

The meridional and zonal phasing of the SST and zonal wind also contributes to the instantaneous growth of the coupled signals. Heating (for Model I; the interpretation is the same for Model II) in the presence of background planetary vorticity generates relative vorticity through stretching. The variation of planetary vorticity with latitude implies that this stretching effect is shifted poleward of the surface SST anomalies and, hence, positive (negative) zonal wind anomalies are shifted over the maxima (minima) in SST. The collocation of like-signed zonal wind and SST anomalies leads to a net positive WES feedback and the possibility for instantaneous growth. This effect is most pronounced near the equator where heating is primarily balanced by convergence rather than height perturbations. As such, variance growth is maximized within about an atmospheric equatorial radius of deformation from the equator. Zonal collocation of zonal wind and SST also enhances instantaneous growth (Xie 1996) and is largest for long zonal wavelengths.

Analysis of the steady forced solution identifies equatorially confined and equatorially antisymmetric patterns as optimal steadily forced patterns. The forcing structures associated with these steady forced patterns extend equatorward and westward from off-equatorial maxima, consistent with the wave characteristics derived by Liu and Xie (1994). The structure of the forcing allows the wave structures to be resonantly excited along their propagation characteristics. Three constraints on the poleward extent of forcing that can excite tropical variability through thermodynamically coupled processes are identified. First, the thermal damping rate imposes a constraint on the effectiveness of steady forcing at generating a signal that propagates all the way to the equator. Signals forced at very high latitudes damp away before reaching the deep tropics. This constraint limits the poleward extent of forcing for Model II owing to the small effective atmospheric wave speed and, hence, the slow meridional propagation characteristic of the coupled signals for Model II relative to Model I. A second constraint emerges from the transition of the mean winds from the easterly trades to midlatitude westerlies. Poleward of this point, the imposed WES parameter α(y) approaches zero and forcing cannot excite a coupled response (note that in reality the WES parameter would change sign as mean winds transition to westerly; we do not consider that case here). A third constraint for Model I exists in the presence of mean moisture divergence in the boundary layer. In Model I SST is assumed to generate deep tropospheric heating only in the presence of mean moisture convergence. Without atmospheric heating, SST anomalies are unable to generate an atmospheric response and hence unable to generate a WES feedback. This last constraint limits the poleward extent of forcing that can excite tropical variability in Model I. These constraints generally support the view that the WES feedback is most positive near the equator (Chang et al. 2000), though as the solutions become more confined to the equator, the lack of equatorial ocean dynamics becomes a more serious caveat.

What are the main differences between the results from each model? As the two models can be shown to be mathematically equivalent (Neelin 1989; BSH), the major difference between the two model formulations is the effective atmospheric equatorial radius of deformation, which for the BSH boundary layer model (Model II) is about half that of the Gill–Matsuno free tropospheric model (Model I). As a result, structures in Model II tend to have a more limited meridional extent than their counterparts in Model I. A consequence of this is that the instantaneous growth rates rapidly decrease as SST anomalies move away from the equator (section 4a). Note, however, that the strict constraint on the poleward extent of the atmospheric heating response to SST in Model I is more of a limiting factor to the poleward extent of forcing that can excite tropical meridional mode variations than the dynamical constraint in Model II (cf. Figs. 13e and 14a). This highlights the role of the boundary layer in connecting subtropical climate variations to tropical meridional mode variations (Czaja et al. 2002; Chiang and Vimont 2004).

How do the coupled disturbances discussed herein relate to observed tropical meridional mode variations? First, we note that the simple model structures that closely resemble observed meridional mode variations are not dynamical modes, but rather a collection of antisymmetric coupled modes that constructively interfere to produce transient growth over a finite time period. Given the somewhat accepted use of the phrase “meridional mode,” however, it is suggested that yet another new name would only generate unnecessary confusion. Second, the simple models show that the most effective external forcing for exciting these growing disturbances is located in the tropics or deep subtropics. This indicates that the atmospheric stationary waves that should be most effective at generating observed meridional mode variations would involve variations in the trades that extend far equatorward, like the atmospheric “North Pacific Oscillation” (NPO) or North Atlantic Oscillation (Xie and Tanimoto 1998; Czaja et al. 2002; Chiang and Vimont 2004; Linkin and Nigam 2008). Finally, the equatorward and westward propagation of coupled disturbances matches observed and modeled relationships between atmospheric NPO variations and potential generation of ENSO events through the seasonal footprinting mechanism (Vimont et al. 2001, 2003a,b). As such, we consider the current results broadly consistent with observed and modeled meridional mode variations.

The analysis herein has been limited to a set of very simple coupled anomaly models that exist around a prescribed equatorially symmetric mean state. The models ignore processes such as momentum entrainment across the top of the boundary layer (Deser 1993; Back and Bretherton 2009), convergence feedbacks (Zebiak and Cane 1987), more complete tropical and midlatitude dynamics, and more realistic mean states. These models are generally invalid outside of the tropics, which is more of an issue for Model I than Model II, given the larger effective atmospheric equatorial Rossby radius for Model I. Some results presented herein extend well into the midlatitudes, where the models are clearly invalid. This paper is intended to illustrate some basic concepts related to growth and propagation of meridional modelike disturbances: as such, those results are still useful. Again, the general conclusions about the dynamics of coupled disturbances should remain valid for coupled tropical disturbances. Further analysis will include investigation of asymmetric mean states, which could shed light on influences of the seasonal cycle and basin geometry.

Acknowledgments

This work was supported by NSF Grants ATM-0735030 and ATM-0849689 and NOAA Grant NA04OAR4310139. NCEP Reanalysis data and CPC Precipitation data were provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, available online from their Web site at http://www.cdc.noaa.gov/. The work benefitted greatly from discussions with Eric DeWeaver and from comments from Michael Alexander and two anonymous reviewers.

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Fig. 1.
Fig. 1.

(a) Global precipitation minus evaporation (PE) (a good proxy for low-level moisture convergence in the tropics). Units are 2 g m−2 s−1; solid contours and light shading denote positive values, dashed contours and dark shading denote negative values, and the zero contour has been omitted. (b) Zonal average PE over the oceans (dark solid line) and approximation (dashed line) used for Kq (Model I) and for ϵM (Model II).

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 2.
Fig. 2.

(a) Latent heat flux sensitivity to changes in zonal wind, ∂LH/∂u (contour 5 W s m−3). Solid contours and light shading denote a downward latent heat flux anomaly per unit change in zonal wind speed; dashed contours denote the opposite; and the thick line represents the zero contour. (b) Zonal average latent heat flux sensitivity over global oceans (dark solid line) and approximation used for α(y) (dashed line).

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 3.
Fig. 3.

Eigenspectrum for the coupled Model I (open black circles) shown together with the eigenspectrum of the uncoupled version of that model [the upper-left 3 × 3 matrix in (3); gray dots]. Eigenvalues for the least-damped coupled modes are shown in the offset figure. Horizontal dashed lines indicate the imposed damping rates for the atmosphere (ϵu−1 = 2 days) and for the ocean temperature (ϵT−1 = 120 days).

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 4.
Fig. 4.

The four least stable eigenvectors for the coupled Model I: the SST (gray contours), lower-level height perturbations (black contours), and lower-level winds (vectors). For SST and lower-level height, contours are shown at increments of 0.3 times the maximum value for the field, solid contours denote positive values and dashed contours denote negative values, and the zero line has been omitted. Units on the x and y axes have been scaled by the zonal wavelength Lx and the appropriate nondimensional length scale Ly = .

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for Model II. The horizontal dashed line in the offset figure indicates the imposed damping rates for ocean temperature (ϵT−1 = 120 days).

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 6.
Fig. 6.

Transient growth σ(τ)2 over time τ (days) for the first three τ optimals of Model I. Open circles, crosses, and triangles represent energy growth for the leading antisymmetric τ optimal, the leading symmetric τ optimal, and the second symmetric τ optimal, respectively.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 7.
Fig. 7.

The leading antisymmetric τ optimal (left) shown with the final structure [(right); scaled by amplitude growth σ(τ)] that it evolves into over time τ for Model I, for different τ : dimensional SST (gray contours; contour 0.1°C), lower-level geopotential (black contours; contour 3 m2 s−2), and lower-level winds (vectors). Solid contours denote positive values, dashed contours denote negative values, and the zero line has been omitted. Growth rates σ(τ)2 and τ are labeled to the right and left, respectively, of the figure. Units on the x and y axes have been scaled by the zonal wavelength Lx and the appropriate nondimentional length scale Ly = . Solid black horizontal lines in the left panels are at ±15°, the approximate cutoff for Kq.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 8.
Fig. 8.

As in Fig. 6, but for Model II.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 9.
Fig. 9.

The leading antisymmetric τ optimal (left) shown with the final structure [(right); scaled by amplitude growth σ(τ)] that it evolves into over time τ for Model II. Shown are the dimensional SST (gray contours; contour 0.1°C), effective geopotential ΦRG = gh − ΓκT (black contours; contour 3 m2 s−2), and winds (vectors). Solid contours denote positive values, dashed contours denote negative values, and the zero line has been omitted. Growth rates [σ(τ)2] and τ are labeled to the right and left, respectively, of the figure. Units on the x and y axes have been scaled by the zonal wavelength Lx and the appropriate nondimentional length scale Ly = . Solid black horizontal lines in the left panels are at ±15°, the approximate cutoff for diagnosing mean moisture convergence.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 10.
Fig. 10.

Uncoupled atmospheric response to imposed SST forcing for (a),(c) Model I with constant α and Kq and (b),(d) Model II with constant α and ϵM. The SST structure is given by (16) with Lf = 15° and for (a) and (b) yc = 7.5° or for (c) and (d) yc = 20°. Shown are the dimensional SST (gray contours; contour 0.25°C), lower-level or equivalent geopotential [black contours, contour 50 m2 s−2 in (a),(b), and (d) and 200 m2 s−2 in (c)] and lower-level winds (vectors). Solid contours denote positive values, dashed contours denote negative values, and the zero line has been omitted. Units on the x axes have been scaled by the zonal wavelength Lx.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 11.
Fig. 11.

Ratio of SST growth via the WES feedback [first term in (11)] to SST decay via linear damping [second term in (11)] for Models I and II under constant background conditions and for SST forcing centered at progressively higher latitudes. The meridional structure of the SST forcing is given by (16) with width Lf = 15° and maxima/minima centered at yc. Results are shown for Model I (dashed line) and Model II with ϵM = 8 h−1 (thick solid line) and ϵM = 48 h−1 (thin solid line).

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 12.
Fig. 12.

Ratio of SST growth via the WES feedback [first term in (11)] to SST decay via linear damping [second term in (11)] as a function of zonal wavelength. The meridional structure of the SST forcing is given by (16), with width Lf = 15° and maxima/minima at ±7.5°.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 13.
Fig. 13.

Model I (left) optimal steady forcing patterns and (right) steady response for (a),(b) constant Kq and α; (c),(d) constant Kq but standard α(y); and (e),(f) standard parameters. Shown in the left column are SST tendency [gray contours; 5 × 10−3 K (100 day)−1] and lower-level geopotential tendency [black contours; (a),(c) 5 × 10−2 m2 s−2 (100 days)−1, (e) 5 × 10−1 m2 s−2 (100 days)−1]; and in the right column SST (gray contours; 0.1 K) and lower-level geopotential (black contours; 1 m2 s−2). Solid contours denote positive values (or zero, in the case of SST tendency and SST), dashed contours denote negative values, and the zero contour has been omitted for geopotential height only. Thin black lines at ±30° and ±15° in the left panels illustrate the poleward limits of α(y) and Kq, respectively.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Fig. 14.
Fig. 14.

Model II (a) optimal steady forcing patterns and (b) steady response for constant α and standard damping. Shown for (a) are SST tendency [gray contours, 1.5 × 10−2 K (100 day)−1] and height-induced boundary layer geopotential anomaly gh [black contours; 0.2 m2 s−2 (100 day)−1]; and in (b), SST (gray contours, 0.1 K) and lower-level equivalent geopotential (black contours, 0.5 m2 s−2). Solid contours denote positive values (or zero, in the case of SST tendency and SST), dashed contours denote negative values, and the zero contour has been omitted for geopotential height only. Thin black lines at ±30° and ±15° in (a) illustrate the poleward limits of α(y) and ϵM, respectively.

Citation: Journal of Climate 23, 21; 10.1175/2010JCLI3532.1

Table 1.

Model standard parameters.

Table 1.

* Center for Climatic Research Contribution Number 994.

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