Climate Dynamics

, 37:1823

An empirical model of tropical ocean dynamics


    • CIRES Climate Diagnostics CenterUniversity of Colorado
    • Physical Sciences DivisionNOAA Earth System Research Laboratory
  • Michael A. Alexander
    • Physical Sciences DivisionNOAA Earth System Research Laboratory
  • James D. Scott
    • CIRES Climate Diagnostics CenterUniversity of Colorado
    • Physical Sciences DivisionNOAA Earth System Research Laboratory

DOI: 10.1007/s00382-011-1034-0

Cite this article as:
Newman, M., Alexander, M.A. & Scott, J.D. Clim Dyn (2011) 37: 1823. doi:10.1007/s00382-011-1034-0


To extend the linear stochastically forced paradigm of tropical sea surface temperature (SST) variability to the subsurface ocean, a linear inverse model (LIM) is constructed from the simultaneous and 3-month lag covariances of observed 3-month running mean anomalies of SST, thermocline depth, and zonal wind stress. This LIM is then used to identify the empirically-determined linear dynamics with physical processes to gauge their relative importance to ENSO evolution. Optimal growth of SST anomalies over several months is triggered by both an initial SST anomaly and a central equatorial Pacific thermocline anomaly that propagates slowly eastward while leading the amplifying SST anomaly. The initial SST and thermocline anomalies each produce roughly half the SST amplification. If interactions between the sea surface and the thermocline are removed in the linear dynamical operator, the SST anomaly undergoes less optimal growth but is also more persistent, and its location shifts from the eastern to central Pacific. Optimal growth is also found to be essentially the result of two stable eigenmodes with similar structure but differing 2- and 4-year periods evolving from initial destructive to constructive interference. Variations among ENSO events could then be a consequence not of changing stability characteristics but of random excitation of these two eigenmodes, which represent different balances between surface and subsurface coupled dynamics. As found in previous studies, the impact of the additional variables on LIM SST forecasts is relatively small for short time scales. Over time intervals greater than about 9 months, however, the additional variables both significantly enhance forecast skill and predict lag covariances and associated power spectra whose closer agreement with observations enhances the validation of the linear model. Moreover, a secondary type of optimal growth exists that is not present in a LIM constructed from SST alone, in which initial SST anomalies in the southwest tropical Pacific and Indian ocean play a larger role than on shorter time scales, apparently driving sustained off-equatorial wind stress anomalies in the eastern Pacific that result in a more persistent equatorial thermocline anomaly and a more protracted (and predictable) ENSO event.


Sea-surface temperaturesENSOClimate variabilityOcean-atmosphere interactionCoupled models

1 Introduction

Penland and Sardeshmukh (1995; hereafter PS95) showed that observed tropical SST variability may be viewed as resulting from the linear stochastically-forced system
$$ {\frac{{d{\mathbf{T}}_{o} }}{dt}} = {\mathbf{LT}}_{o} + {\mathbf{F}}_{s} , $$
where To(t) are sea surface temperature (SST) anomalies throughout the tropical domain, L is a stable linear dynamical operator, and Fs is white noise (which can have spatial coherence and seasonal dependence). PS95 derived L empirically by constructing a Linear Inverse Model (LIM); a similar result has also been suggested by several studies using intermediate coupled models (Flügel and Chang 1996; Moore and Kleeman 1997, 1999; Penland et al. 2000; Thompson and Battisti 2001). Equation 1 may be thought of as a multivariate extension to the univariate red noise null hypothesis for SST variability first proposed by Frankignoul and Hasselman (1977), in which the slowly evolving ocean integrates forcing by rapidly evolving weather noise. The “effectively linear” stochastic approximation (1) is similarly valid when nonlinear processes decorrelate much more rapidly than linear processes (e.g., Papanicolaou and Kohle 1974; Hasselmann 1976; see also Penland 1996), so that nonlinear terms may be parameterized as a (second) linear process plus unpredictable white noise. [This is in contrast to “linearization” of the prevailing equations, where it is assumed that the magnitude, rather than the time scale, of the nonlinear terms is small.]

Equation 1 is a good approximation of the observed statistics of ENSO evolution. For example, a LIM constructed from the zero- and one week lagged covariance statistics of weekly atmospheric and SST data more faithfully reproduces the entire power spectrum on seasonal to interannual time scales of monthly ENSO SST variability than do the corresponding spectra from virtually all ensemble members of the “twentieth-century” (20c3m) IPCC AR4 coupled GCMs (Newman et al. 2009). Also, forecasts made using the LIM are not only competitive with operational GCMs such as NCEP’s coupled forecast system (CFS) but, particularly for forecast leads of 6 months and greater, generally have higher skill throughout most of the Pacific Basin (not shown), indicating that (1) is also a good approximation of the case-to-case evolution of SST anomalies.

While (1) has proven highly useful as a dynamical model for SSTs, it is not as satisfying as a dynamical model of the full ocean. Equation 1 assumes that SST is a proxy for the complete coupled atmosphere-ocean state vector, which PS95 notes is a plausible scenario so long as essentially instantaneous (on the seasonal time scale) SST-induced changes in surface winds drive an immediate thermocline response, and/or in the fast-wave limit (Neelin 1991; Neelin and Jin 1993) in which wave dynamics allow the deeper ocean to rapidly adjust to changes in SSTs. In this case, all other physically important ocean variables can be effectively regressed onto SST; that is, X = B To + r, where X is the remaining ocean state vector, B is a regression matrix and r is a white residual. (Note that this regression is generally not local.) Consequently, L in (1) includes all the implicit effects of X upon the evolution of To. Clearly, this is at least partly true, evidenced both by a strong observed relationship between thermocline depth and SST (e.g., Zelle et al. 2004) and by the success of (1) as a model. Also, studies that constructed a LIM by including a variable corresponding to ocean heat content (Johnson et al. 2000; Xue et al. 2000) did not find statistically significant improvement in forecast skill up to a few seasons ahead.

However, for longer time scales (and thus forecast leads) we expect deeper ocean physics is too important for this simple relationship to hold, if for no other reason than r is unlikely to be white. For example, near-equatorial Kelvin and Rossby waves are an integral part of the El Niño phenomena, and within the delayed oscillator paradigm the latter set the time scale for ENSO (Schopf and Suarez 1988; Battisti and Hirst 1989). In the subtropics, thermocline variability clearly propagates west in accordance with Rossby wave theory and appears to influence the upper ocean along the equator via Kelvin waves along the western boundary (Capotondi and Alexander 2001). In the recharge-discharge paradigm for ENSO, zonally-averaged export of heat to and from the equator is the critical process for ENSO dynamics (Jin 1997). Indeed, heat content anomalies at 5°N, important in either paradigm, contribute to ENSO predictability at 6–12 month lead times (Latif and Graham 1992).

Moreover, if the linear stochastically forced paradigm is appropriate for tropical climate dynamics, then the LIM must also appropriately represent the correct balance of coupled air-sea physical processes, both linear and nonlinear. This makes LIM a potentially important diagnostic tool that could be used to understand, for example, how misrepresented physical processes might affect the simulation of ENSO in coupled models, so a LIM based on SST alone is necessarily insufficient and both atmospheric and subsurface information must be included in its state vector. As a first step, in this study we diagnose a LIM constructed from an extended state vector that includes SST, thermocline depth (as represented by the 20°C isotherm depth) and surface zonal wind stress, a combination of variables similar to that earlier employed by Johnson et al. (2000) and Xue et al. (2000) using more limited datasets. The resulting LIM, described in Sect. 2, improves both tropical SST predictions at longer leads and the simulation of power spectra and lag co-variance statistics relative to the SST-only LIM (shown in the “Appendix”), giving confidence in its diagnostic usefulness. Initial analysis of the different terms within the LIM and their effect upon the statistics of SST variability is made in Sect. 3. A discussion of how thermocline depth and surface wind stress variability interacts with the most rapidly amplifying SST anomalies and how they together evolve into ENSO events is then presented in Sect. 4, along with how two key eigenmodes of L that represent different balances of dynamical processes contribute to this evolution. Concluding remarks are made in Sect. 5.

2 Model details and data

LIM may be broadly defined as extracting the dynamical evolution operator L of the system
$$ {\frac{{d{\mathbf{x}}}}{dt}} = {\mathbf{Lx}} + \xi $$
from its observed statistics, as described for example in PS95 (see also Penland 1989, 1996; Winkler et al. 2001; Newman et al. 2003, 2009; Newman 2007; Alexander et al. 2008; Newman and Sardeshmukh 2008; Kallummal and Kirtman 2008). The statistics of the noise forcing \( {\mathbf{Q}} = \left\langle {\xi \xi^{T} } \right\rangle dt \) are then determined by the corresponding Fluctuation–Dissipation relationship
$$ d{\mathbf{C}}\left( 0 \right)/dT = {\mathbf{LC}}\left( 0 \right) + {\mathbf{C}}\left( 0 \right){\mathbf{L}}^{T} + {\mathbf{Q}}, $$
given the observed seasonally-varying (where T is time of year) data covariance matrix \( {\mathbf{C}}\left( 0 \right) = \left\langle {{\mathbf{xx}}^{T} } \right\rangle \). The procedure and its strengths and pitfalls are discussed at length in the above papers, and are summarized in the “Appendix”.
For the LIM presented in this paper, we chose the model state vector x
$$ {\mathbf{x}} = \left[ {\begin{array}{*{20}c} {{\mathbf{T}}_{O} } \\ {{\mathbf{Z}}_{20} } \\ {\varvec{\uptau}_{x} } \\ \end{array} } \right] $$
where TO is anomalous SST, Z20 is anomalous 20°C isotherm depth, and τx is anomalous surface zonal wind stress, determined from 42 years (1959–2000) of 3-month running mean data. Monthly means of TO and Z20 can also be used to construct LIMs, but this time average is problematic when an atmospheric variable is included because of some aliasing of the Madden–Julian oscillation (MJO) signal. SST was obtained from the Hadley Sea Ice and Sea Surface Temperature analysis (HadISST; Rayner et al. 2003), 20°C isotherm depth from the SODA dataset (Carton and Giese 2008), and wind stress from the NCEP Reanalysis (Kalnay et al. 1996). Data were averaged into 2° latitude × 5° longitude gridboxes. Anomalies, determined by removing the long-term monthly mean, were projected onto their leading Empirical Orthogonal Functions (EOFs) determined for the region 25°S–25°N. Prior to computing EOFs, each field was normalized by its domain-averaged climatological root-mean-square amplitude. The leading 13/7/3 EOFs of To/Z20x were retained, explaining about 83/36/32 percent of the variability of their respective fields. Locally, however, variance explained can be considerably higher (or lower), as seen in Fig. 1, which shows the (untruncated) variance of To, Z20, and τx, and the local fraction of variance explained by the truncated EOF bases. The time-varying coefficients of these EOFs, i.e., the principal components (PCs), define the 23-component state vector x. For purposes of comparison, corresponding SST-only LIMs (or “SST-LIMs”) were constructed by representing x with only TO, using either 13 or 23 PCs (SST13-LIM and SST23-LIM). Throughout the remainder of this paper, “ocean LIM” will denote the LIM constructed from the extended To-Z20x state vector.
Fig. 1

Variance (contours) and fraction of local variance explained by EOF truncation (gray shading) for variables used in the LIM. Top SST (TO); contour interval 0.25 K2. 2nd row 20°C isotherm depth (Ζ20); contour interval 100 m2. Bottom zonal wind stress (\( {\varvec{\uptau}}_{x} \)); contour interval 5 × 10−5 N2m−4. Thicker contours indicate larger values, starting at and increasing from 1 K2, 300 m2, and 20 × 10−5 N2m−4, respectively

To use the LIM for diagnosis, we must first validate it by determining whether the effectively linear approximation of (2) holds, through a series of tests (PS95) that include a comparison of the observed evolution of individual events to their prediction by the LIM and a more general comparison of observed anomaly evolution statistics to those linearly predicted by (2). The results of these tests (see “Appendix”) basically agree with both PS95 and subsequent studies that constructed linear empirical models from more limited ocean datasets (Johnson et al. 2000; Xue et al. 2000). The additional variables in the model state vector result in a LIM that passes the above tests with even greater fidelity than the SST-LIMs, although this improvement is not significant until the time scale is longer than a few seasons.

As an example, Fig. 2 shows that the SST23-LIM and the ocean LIM, constructed from the same number of degrees of freedom, have similar Niño 3.4 SST forecast skill for leads up to about 8 months. That is, replacing higher order SST PCs with subsurface information in the LIM does not improve skill for shorter leads, also earlier noted by C. Penland (personal communication) and Johnson et al. (2000). For longer leads, however, the ocean LIM has higher skill than SST-LIMs of any EOF truncation. [In fact, for forecast leads over 1 year, more than 13 EOFs degrades SST-LIM skill.] As a result, if a “good” forecast has anomaly correlation skill greater than or equal to 0.6, then the ocean LIM improves Niño 3.4 forecast skill by about 3–4 months. A similar result also obtains over the Indian Ocean (not shown).
Fig. 2

Cross-validated forecast skill (1959–2000) of Niño3.4, defined as the area averaged SST in the region 5°S–5°N, 170°W–120°W, for forecasts made by (black) the LIM, (blue) the version of the LIM where thermocline interactions are removed (by setting LZT = LTZ = LτΖ = LΖτ = 0 in L), Lnoz, (red) the SST13-LIM, and (orange) the SST23-LIM

3 Processes within the dynamical operator

The dynamics of different tropical oceanic processes may be investigated by rewriting (2) as
$$ \frac{d}{dt}\left[ {\begin{array}{*{20}c} {{\mathbf{T}}_{o} } \\ {{\mathbf{Z}}_{20} } \\ {\varvec{\uptau}_{x} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{L}}_{TT} } & {{\mathbf{L}}_{TZ} } & {{\mathbf{L}}_{T\tau } } \\ {{\mathbf{L}}_{ZT} } & {{\mathbf{L}}_{ZZ} } & {{\mathbf{L}}_{Z\tau } } \\ {{\mathbf{L}}_{\tau T} } & {{\mathbf{L}}_{\tau Z} } & {{\mathbf{L}}_{\tau \tau } } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{T}}_{o} } \\ {{\mathbf{Z}}_{20} } \\ {\varvec{\uptau}_{x} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\varvec{\upxi}_{T} } \\ {\varvec{\upxi}_{Z} } \\ {\varvec{\upxi}_{\tau } } \\ \end{array} } \right]. $$
Note that LTT is distinct from the SST-LIM linear operator, which implicitly includes linear diagnostic relationships between Z20 and \( \varvec{\uptau}_{x} \) and TO. By explicitly separating out the effects of the other two variables on TO and vice versa, we can use (4) to identify LTT with surface ocean processes, LZZ with internal ocean processes, Lττ with surface atmospheric processes, and the off-diagonal submatrices with coupling. Of course, these operators each implicitly retain the influence of variables not included in x, especially considering the more severe EOF truncation of Z20 and \( \varvec{\uptau}_{x} \), and to the extent that the terms are related to the same unspecified variables they may not be entirely independent. Also, processes with time scales much shorter than the 3-month averaging period used to construct x (notably, Kelvin waves) are still represented by a diagnostic relationship plus some noise.
With these important caveats in mind, we “remove” effects of thermocline variability by constructing a new operator Lnoz, where we set LZT = LTZ = LτZ = L = 0 in L. First, to evaluate the impact of thermocline anomalies upon SST forecasts we repeated all forecasts using Lnoz as the forecast operator, finding that while thermocline interaction with SSTs has a small effect on SST skill for forecast leads shorter than about 6 months, it has an increasing impact for longer-lead forecasts, reaching the .05 significance level for leads greater than a year (Fig. 2). How thermocline interactions affect SST variability statistics can likewise be seen by determining a new covariance Cnoz(0) from (3) using Lnoz and the original noise covariance Q, and predicted τ-lag covariance from Cnoz(τ) = exp(Lnozτ)Cnoz(0). The results are compared to the corresponding fields determined from the full LIM in Fig. 3; note that SST variance from the LIM is correct by construction (apart from a difference due to the EOF truncation), and that the 9-month lag-covariance predicted by the LIM is an excellent match to observations (see Fig. 13). Removing the thermocline interactions leads to a small increase of central equatorial Pacific TO variability and simultaneously a pronounced decrease of eastern Pacific variability (cf. top panels of Fig. 3), while central Pacific variability becomes more persistent over a 9-month period (cf. bottom panels of Fig. 3). In essence, ENSO events have characteristics more similar to “Modoki” or Central Pacific type events (e.g., Ashok et al. 2007; Kao and Yu 2009) when thermocline interactions are removed from L. Additionally, overall decadal variability is reduced (not shown).
Fig. 3

TO variance and lag-covariance determined from L (left panels) and from Lnoz, where the thermocline interactions are removed by setting LZT = LTZ = LτΖ = LΖτ = 0 in L (right panels). Top variance; contour interval is 0.16 K2. Bottom 9-month lag-covariance; contour interval is 0.04 K2

The covariance budget of 3-months running mean SST anomalies, averaged over the annual cycle and derived from (3) and (4) as
$$ \mathop {\left[ {{\mathbf{L}}_{TT} {\mathbf{C}}_{TT} + {\mathbf{C}}_{TT} {\mathbf{L}}_{TT}^{T} } \right]}\limits_{\text{SST}} + \mathop {\left[ {{\mathbf{L}}_{TZ} {\mathbf{C}}_{ZT} + {\mathbf{C}}_{TZ} {\mathbf{L}}_{TZ}^{T} } \right]}\limits_{{{\text{Thermocline}}\;{\text{interaction}}}} + \mathop {\left[ {{\mathbf{L}}_{T\tau } {\mathbf{C}}_{\tau T} + {\mathbf{C}}_{T\tau } {\mathbf{L}}_{T\tau }^{T} } \right]}\limits_{{{\text{Wind}}\;{\text{stress}}}} + \mathop {{\mathbf{Q}}_{TT} }\limits_{\text{Stochastic}} = {\mathbf{O}}, \, $$
where \( {\mathbf{C}}_{xy} = \left\langle {{\mathbf{xy}}^{T} } \right\rangle \), is another gauge of anomalous thermocline and wind stress influences on SST variability. The diagonal elements in each bracketed term in (5) may be interpreted as their contribution to the local TO variance tendency. Note that although these terms relate to the maintenance of local variance, they contain non-local effects: LTT and CTT are not diagonal matrices, and thermocline and zonal wind stress variability contribute to SST variability. In principle this “inverse” method of determining the interannual heat budget should agree with a more traditional “forward” approach in which all terms in the surface-layer heat budget are explicitly calculated (e.g., Wang and McPhaden 2000; Kang et al. 2001).
It is interesting to break down the bracketed SST term further into “local” (Fig. 4a) and “non-local” (Fig. 4b) contributions. The local contribution, twice the product of the diagonal elements of LTT with the corresponding diagonal elements of CTT, results in a dissipative term of the form \( \left.- 2 \langle {\left( {T_{\rm O} } \right)^{2} } \rangle \right/\tau_d \) (where τd is a local decay time scale) with stronger damping in the eastern and equatorial central Pacific (τd ~ 6–7 months) than in the western and subtropical central Pacific (τd ~ 10–15 months). This regional variation of damping includes effects both of local SST-related variations of radiative and surface heat fluxes (e.g., Kessler and McPhaden 1995; Wang and McPhaden 2000) and of vertical mixing related to stronger mean upwelling in the east. The non-local term is the difference between the total term (not shown) and the local term, representing how all other values of TO impact local TO variability, such as through advection and/or wave propagation. This term also generally acts to damp equatorial SST variance as might be expected from the mean advection of thermal anomalies (Jin et al. 2006) and meridional transport due to tropical instability waves (Wang and McPhaden 2000; Jochum et al. 2007), except in the central Pacific and around 5–10°N where it acts to generate SST variance, consistent with the downgradient transport of SST variance (cf. Fig. 1) by high frequency eddy fluxes (Wang and McPhaden 2000) and mean equatorial zonal and off-equatorial meridional currents (Kang et al. 2001).
Fig. 4

The local variance budget for SST (see text for description of terms). Note that the panels sum to zero. Contour and shading interval is 0.04 K2 month−1; positive values are red/yellow and negative values are blue, with the zero contour removed for clarity

The primary balance in Fig. 4 is between thermocline interactions (Fig. 4c) and the SST dynamical terms. The thermocline term, which is positive except in the central Pacific, includes the net SST variance generation by local thermocline and upwelling feedbacks, and also non-local zonal advective feedback (Picaut et al. 1996; Jin and An 1999) due to the dependence of anomalous zonal current on the anomalous equatorial thermocline gradient. TO noise effects are very small (Fig. 4e) except in a narrow equatorial band where tropical instability waves are active (Chelton et al. 2000); in fact, atmospheric noise forcing has a greater but indirect effect upon SSTs primarily by maintaining zonal wind stress variability (not shown), which in turn forces thermocline variability. While there is a weaker direct zonal wind stress forcing term (Fig. 4d), it might also implicitly represent other anomalous surface wind effects on latent and sensible heat fluxes, and even radiative fluxes due to cloudiness. Note that advective feedbacks in the east-central Pacific appear to mostly be contained within the thermocline term and do not represent direct effects from the wind stress forcing term as in the “wind stress feedback” (Burgers and van Oldenborgh 2003).

4 Optimal evolution of ENSO

4.1 Maximum amplification of SST anomalies in SST and ocean LIMs

PS95 showed that the “optimal” initial condition for maximum amplification of tropical SST anomalies, obtained via a singular vector decomposition (SVD) of the system propagator G(τ) under the L2 norm (i.e., domain-mean square amplitude) of TO (e.g., Farrell 1988; PS95), is also the most relevant initial condition for ENSO development. The SVD analysis yields a dominant pair u1,v1 of normalized singular vectors and maximum singular value λ1, such that the initial condition v1 leads to the anomaly G v1 = λ1u1 at time t = τ. The maximum possible anomaly growth factor λ12(τ), sometimes called a “maximum amplification” (MA) curve (PS95), is displayed in Fig. 5a for each LIM. For consistent comparison, the L2 norm is defined in the space of the leading 13 EOFs of TO in all three cases.
Fig. 5

a Comparison of maximum amplification (MA) curves, defined as [λ1(τ)]2 determined by the SVD of G(τ) under the L2 norm of TO, for the LIM and the SST-LIMs. b Pattern correlation of v1(9) with v1(τ) for the ocean LIM and the SST-LIMs

The shape and maximum (at τ ~ 8–9 months) of the MA curve for each SST-LIM is consistent with earlier studies including PS95. Although in principle an optimal initial condition for one value of τ does not have to be optimal for another value of τ, in reality PS95 found relatively little difference in the leading optimal structure for a wide range of τ. This result is also true for our SST-LIMs; for example the pattern correlation of v1(9 months) with v1(τ) is greater than 0.9 for τ ranging from about 4 to 14 months (Fig. 5b). That is, for all practical purposes the SST-LIM has only one leading optimal initial structure.

The shape of the MA curve for the ocean LIM is quite different, however, with a secondary maximum at τ ~ 17–18 months and a fairly flat curve between the two maxima. Moreover, these two maxima represent different optimal structures, with an 0.35 pattern correlation between v1(9 months) and v1(18 months) (Fig. 5b). However, although the initial states are different, the end results are similar, with both optimal initial conditions evolving into a large central and eastern Pacific SST anomaly (see below discussion). The increased maximum amplification for the ocean LIM implies increased predictability (Newman et al. 2003) for forecast leads greater than about nine months, consistent with the improved skill in Fig. 2.

4.2 Evolution of 9- and 18-month leading optimal structures

To explore the differences between these two leading optimal structures, we integrate (2) forwards from either initial condition v1(9), the optimal structure for growth over a τ = 9 month interval (Fig. 6a), or from initial condition v1(18), the optimal structure for growth over a τ = 18 month interval (Fig. 7a). For ease of comparison, both initial conditions are chosen with sign leading to a warm event, but in this linear model all signs can be reversed. The evolution from v1(9) [v1(18)] over the next 36 months is displayed in Figs. 6b–d [7b–d]. Forcing by Z20 (\( \varvec{\uptau}_{x} \)) and nonlocal terms in the TO (Z20) tendency [from (4)] are indicated in Figs. 6e, f and 7e, f by shading (contours); these terms are evaluated at the equator but can contain effects driven from other latitudes.
Fig. 6

Evolution of the optimal initial condition for amplification of SST anomalies in the LIM over a 9-month interval. a Initial state, and the evolved states, b 9 and c 18 months later. d Time-longitude cross-section of the LIM evolution averaged between 2°S–2°N. TO is indicated by shading (contour interval 0.175 K), Z20 by contours (contour interval 6 m), and τx by blue vectors (scaled by the reference vector 0.02 N m−2, with values below 0.002 N m−2 removed for clarity). e and f Time-longitude cross-section of selected forcing terms in the tendency equations for TO (shading interval 0.05 K/month) and Z20 (contours interval 0.5 m/month): e Nonlocal terms; fZ20 forcing of TO tendency and τx forcing of Z20 tendency. Note that sign in all panels can be flipped, but that black contours and red shading are of one sign, and green contours and blue shading are the other sign. Amplitudes are also arbitrary, but are scaled by a single factor here to have representative values. Note that TO is doubled in a to be more visible
Fig. 7

Same as Fig. 6 but for the evolution of the optimal initial condition for amplification of SST anomalies in the LIM over an 18-month interval

The initial anomaly for the shorter 9-month optimization time (Fig. 6a) has (1) SST that is broadly similar to the corresponding SST-LIM optimal initial condition (not shown, but see PS95) except in the central equatorial Pacific where it has half the amplitude, (2) a positive basin-wide thermocline anomaly centered in the east central Pacific similar to that often viewed as an important ENSO precursor (e.g., Wyrkti 1985; Jin 1997; Meinen and McPhaden 2000; Clarke and Van Gorder 2003; McPhaden et al. 2006), and (3) a westerly wind stress anomaly at roughly the same longitude with off-equatorial maxima. Optimal SST anomaly amplification over the next 9 months is consistent with Bjerknes (1969) feedback: the thermocline term LTZZ20 (Fig. 6f, shading) forces rapid amplification of anomalously warm SST in the east Pacific, which in turn (along with the initial SST anomaly) drives intensification and westward expansion of anomalous equatorial westerly wind stress (not shown) that drives (Fig. 6f, contours) the Z20 anomaly to intensify and move eastward (Fig. 6d), and so on. Maximum LTZZ20 forcing of SST occurs near the Z20 maximum at about 110°W; a secondary LTZZ20 maximum near 140°W might represent advective feedback since it is located near large zonal gradients of both mean SST and anomalous Z20.

Optimal growth is broadly consistent with the “discharge” (Jin 1997) of anomalous equatorial heat content (as represented by equatorial Z20 integrated in the region 2°S–2°N, 120°E–85°W), which after increasing the first 2 months decreases to zero at the peak SST time of t = 9 months (not shown, but cf. Fig. 6d). The initial increase of equatorial heat content coincides with initial off-equatorial wind stress that weakens after about 2 months, when the initial SST anomaly has evolved into its ENSO pattern. Zonal wind stress forcing of Z20 in the east Pacific and Z20 forcing of SST both peak before the SST anomaly, at t = 7 months, at which time zonal wind stress forcing of equatorial heat content also becomes negative. SST anomaly amplification continues for two more months, until the (negative) total SST term grows to balance the thermocline term (e.g., Jin 1997).

Equatorial heat content decreases during anomalous SST amplification since as the positive Z20 anomaly grows and moves eastward, zonal wind stress forcing \( {\mathbf{L}}_{Z\tau}\varvec{\uptau}_{x} \) (Fig. 6f) generates a negative Z20 anomaly in the west Pacific (Fig. 6d), with minima both north and south of the equator (Fig. 6b) consistent with Ekman pumping from the curl of easterly wind stress anomalies centered at about 15°N and 15°S. The non-local portion of LZZZ20 forcing also enhances the west Pacific negative Z20 anomaly, but is generally much weaker (Fig. 6e) except near the secondary Z20 minimum located around 130°E and 6°N (Fig. 6b). SST anomaly decay accelerates when the negative western Pacific thermocline anomaly finally expands eastwards along the equator (Fig. 6d), reaching South America by about t = 18 months (Fig. 6c) and causing equatorial SSTs to become cool even as subtropical SSTs remain warm. This implies that ocean wave propagation processes may be secondary to direct wind forcing in discharging the heat content during optimal growth, and is consistent with the suggestion that wind forcing in the west Pacific was of primary importance in initiating the termination of the 1997/98 El Nino (McPhaden 1999; Boulanger and Menkes 2001).

Note also that while the growing negative thermocline anomaly drives cooler SSTs via LTZZ20, it is offset in the central equatorial Pacific by the non-local SST term (Fig. 6e, shading). This is consistent with equatorial SST tendencies during the 1997/98 El Niño just east of the dateline, where negative vertical advection was offset by positive advection by the mean zonal current (Boulanger and Menkes 2001).

Repeating this integration by first setting the initial Z20 and τx anomalies to zero results in similar SST evolution but with peak amplitude reduced by almost half (not shown), demonstrating that the initial thermocline and SST anomalies contribute roughly equally to SST amplification. If the initial conditions in Fig. 6a evolve using Lnoz instead (not shown), the peak anomaly is not only weaker but also more localized, centered much closer to the dateline, similar to the results shown in Fig. 3; also, as it grows the SST anomaly propagates westward from its initial location along the South American coast.

In contrast to v1(9), the initial SST portion of v1(18) (Fig. 7a) is relatively much stronger than the initial thermocline anomaly. Notably, the southwest Pacific and Indian Ocean SST anomalies persist for several months (e.g., Fig. 7d) while off-equatorial westerly wind stress anomalies rapidly intensify in both hemispheres of the eastern Pacific (not shown) and deepen the equatorial thermocline there (Fig. 7d, f). [Adding additional τx PCs to x to capture more west Pacific variance does not change this picture.] As the anomaly evolves, it never becomes similar to v1(9), with perhaps the closest correspondence occurring at about t = 6 months primarily for central and eastern Pacific SST (cf. t = 0 in Fig. 6d with t = 6 months in Fig. 7d). Subsequent anomaly evolution (t = 6–18 months) is only broadly similar to that shown for the 9-month optimal and moreover extends over a longer period consistent with more persistent wind stress forcing of Z20 (Fig. 7f).

Most of the evolution from v1(18) is due to the initial SST, in contrast to v1(9); a second integration with the initial Z20 anomaly set to zero reduces peak SST by less than 15% (not shown). However, the evolution of Z20 remains essential, since the thermocline term ultimately drives most of the eastern Pacific SST amplification (Fig. 7f). Alternatively, when v1(18) is evolved with the Lnoz operator, there is very little SST growth.

When considering optimal structures, it is important whether this potential growth is actually observed to occur as predicted by the LIM. In addition, do the optimal structures shown in Figs. 6 and 7 improve upon those from the SST-LIM; that is, are they more relevant to the observed system? To answer this, following PS95 and others we compared the projection of observed anomalies on the initial conditions shown in Figs. 6a and 7a to the projection of observed SST anomalies upon the predicted evolved structure 9 (Fig. 6b) and 18 (Fig. 7c) months later, respectively, as shown in Fig. 8a (τ = 9 months) and Fig. 8c (τ = 18 months). This analysis was repeated using the corresponding SST13-LIM optimal structures (not shown); results are shown in Fig. 8b (τ = 9 months) and d (τ = 18 months). The increased linear correlation and reduced scatter about the least squares lines in the panels on the left relative to those on the right indicate that optimal SST growth is better captured by the ocean LIM. Of course, some scatter about the least squares line is still expected due to the noise Fs. The slopes of the dashed lines should correspond to the amplification factors λ1(τ), and in fact they are quite close. This also serves as additional evidence that L is independent of τ0, since these calculations have all been made for intervals τ ≫ τ0 = 3 months.
Fig. 8

Projection of observations upon the optimal initial condition for amplification of SST anomalies in the LIM over a 9-month interval, versus the optimal evolved SST state 9 months later, in a the ocean LIM and b the SST13-LIM. Projection of observations upon the optimal initial condition for amplification of SST anomalies in the LIM over a 18-month interval, versus the optimal evolved SST state 18 months later, in c the ocean LIM and d the SST13-LIM. The dashed lines in all four panels represent the expected evolution, with slopes λ1(τ). Blue circles represent the 1959–1976 period, and red circles the 1977–2000 period. Red dots represent the 1997–1998 El Niño event

Stratifying into pre- (blue) and post- (red) 1976/1977 periods shows no significant long-term change in the relationship (i.e., change in slope) for the ocean LIM. Observed decadal changes in variability could then be represented by random variation of noise forcing projecting differently upon the optimal initial conditions, as opposed to a decadal change in underlying dynamics. Finally, while the 1997/1998 El Niño event is somewhat better simulated with the ocean LIM than the SST-LIM, it remains a potential outlier and may represent an event for which we cannot exclude a role for predictable nonlinearity.

4.3 Eigenmodes contributing to optimal structures

For the SST-LIM, the leading optimal structure can be well approximated by just three nonnormal eigenmodes of L (PS95; Penland and Matrosova 2006). We find this too for v1(9) and its evolution into u1(9), where again all three are propagating eigenmodes (that is, complex conjugate pairs with complex eigenvalues each representing period and e-folding time) with periods of about 4 years (‘4-year eigenmode’; Fig. 9), 1.9 years (‘2-year eigenmode’; Fig. 10), and 21 years (‘bidecadal eigenmode’; not shown). [The complete set of 23 eigenmodes \( \varvec{\upvarphi }_{j} \) and corresponding complex eigenvalues σj are determined from the eigenanalysis \( {\mathbf{L}}\varvec{\upvarphi }_{j} = \sigma_{j} \varvec{\upvarphi }_{j} \).] While the SST components and periods are similar to the corresponding SST-LIM eigenmodes (not shown, but see Penland and Matrosova 2006), the e-folding times of both the 4-year and bidecadal eigenmodes are about double those of their SST-LIM counterparts; the 2-year eigenmode e-folding time is unchanged.
Fig. 9

The third least damped eigenmode of L, with period of 4.0 years and e-folding time of 2.1 years. a Minimum and b Maximum SST phases, defined from the root-mean-squared domain SST anomaly. c Time-space cross-section of the eigenmode evolution, starting from the phase shown in Fig. 11a, along the transect indicated by d. TO is indicated by shading (contour interval 0.175 K), Z20 by contours (contour interval 6 m), and τx by blue vectors (scaled by the reference vector 0.02 N m−2). Note that sign in all panels is arbitrary, but that black contours and red shading are of one sign, and green contours and blue shading are the other sign. Amplitudes are also arbitrary, but are scaled by a single factor here to have representative values
Fig. 10

The eighth least damped eigenmode of L with period of 1.9 years and e-folding time of 0.65 years. a Minimum and b maximum SST phases, defined from the root-mean-squared domain SST anomaly. c Time-space cross-section of the eigenmode evolution, starting from the phase shown in Fig. 11b, along the transect indicated by d. Plotting conventions as in Fig. 9 except all values in Fig. 10c are scaled by a factor of 3; note that the transect points here are at the same longitudes as the transect points in Fig. 9

The 4- and 2-year eigenmodes are not orthogonal (with a minimum 40° angle between them) as is evident in their similar structures for SST minimum (Figs. 9a, 10a) and SST maximum (Figs. 9b, 10b) phases. Consequently, an initial state quite similar to v1(9) can be simply constructed by taking the 4-year eigenmode in a phase about 9 months prior to its SST maximum (Fig. 11a), and then mostly “covering it up” by adding the 2-year eigenmode with amplitude and phase corresponding to Fig. 11c, resulting in the anomaly shown in Fig. 11e (cf. Fig. 6a). [The statistics of the noise determined by (3) show these two eigenmodes on average are excited together and in opposite phases.] Subsequently, the 2-year eigenmode undergoes relatively more rapid propagation and decay (Fig. 10c), “revealing” the more slowly evolving and decaying 4-year eigenmode (Fig. 9c) and even beginning to reinforce it by t = 9 months (Fig. 11b,d,f). This simple transition from destructive to constructive interference between these two eigenmodes is the basis of optimal growth, with the bidecadal and other eigenmodes playing important but lesser roles.
Fig. 11

Evolution of the 4- and 2-year eigenmode components of v1(9), with the initial state in the left panels and the evolved state in the right panels. a 4-year eigenmode component at t = 0. b 4-year eigenmode component at t = 9. c 2-year eigenmode component at t = 0. d 2-year eigenmode component at t = 9. e 4-year plus 2-year eigenmode components at t = 0. f 4-year plus 2-year eigenmode components at t = 9. Plotting conventions as in Fig. 6

Some features of these empirical eigenmodes appear similar to theoretical eigenmodes determined from various linearized versions of the Zebiak and Cane (ZC; 1987) intermediate complexity coupled atmosphere-ocean model (e.g., Jin and Neelin 1993; Dijkstra and Neelin 1999; Fedorov and Philander 2001; An and Jin 2001; MacMynowski and Tziperman 2008; Bejarano and Jin 2008). In particular, the 4-year eigenmode resembles a damped “recharge-discharge oscillator” (Jin 1997; Burgers et al. 2005), evolving from minimum SST phase when anomalous equatorial heat content is a maximum towards maximum SST phase (Fig. 9b) when anomalous equatorial heat content has decreased to zero, during which time the eastward moving thermocline anomaly forces SST amplification (through the LTZZ20 term, not shown). Subsequent off-equatorial wind stress at and south of the equator, driven by the SST anomaly, then partly “recharges” the equatorial heat content anomaly (through the \( {\mathbf{L}}_{Z\tau}\varvec{\uptau}_{x} \) term, not shown) with opposite sign. This eigenmode also appears to have some features characteristic of a “delayed-oscillator” (Schopf and Suarez 1988; Battisti and Hirst 1989), with a thermocline extremum in the far west equatorial Pacific (at about 5°N, 135°E) driven largely by LZZZ20 that could be consistent with equatorial Rossby waves (e.g., Battisti and Hirst 1989), and a second thermocline anomaly at about 14°N that moves westward (Fig. 9c), curving slightly southward as it crosses the Pacific, consistent with Rossby wave propagation (e.g., Schopf et al. 1981). But these are both fairly weak features, and most of the apparent evolution of the thermocline anomaly along the entire loop shown in Fig. 9c and d can be reproduced without the nonlocal portion of the LZZZ20 (not shown), except in the central subtropical Pacific (transect points 45–55). Additionally, the wind stress term \( {\mathbf{L}}_{Z\tau}\varvec{\uptau}_{x} \) drives amplification of the east–west Z20 gradient on the equator, consistent with many earlier studies dating back to Wyrtki. As a result, the thermocline anomaly evolution does not appear as free Kelvin or Rossby wave propagation.

A different balance of forcing terms related to different τx and SST structure drives the faster evolution of the 2-year eigenmode. Along the equator, τx for the 4-year eigenmode extends from the dateline through the SST maximum at about 120°W, whereas τx reverses sign for the 2-year eigenmode at about 140°W, the location of the eigenmode’s more westward SST maximum. SST and zonal wind stress anomalies also have greater meridional extent, particularly south of the equator, for the slower eigenmode. On the other hand, the two eigenmodes have only minor thermocline differences, primarily of amplitude, and consequently have similar thermocline forcing of SST evolution, including a discharge of anomalous equatorial heat content as SST evolves from minimum to maximum phase. However, while the thermocline forcing term is stronger than the non-local SST term for the 4-year eigenmode, it is weaker for the 2-year eigenmode (not shown). The imbalance between the two terms is also relatively larger, allowing for a more rapid evolution of the 2-year eigenmode in which the SST anomaly moves westward along the equator from the South American coast to near the dateline (Fig. 10c), where the SST anomaly becomes almost exactly out of phase with the thermocline anomaly below and so is weakened (consistent with the overall SST variance sink in Fig. 4c).

The bidecadal eigenmode bears some resemblance to “decadal” ENSO variability (e.g., Zhang et al. 1997) and its contribution to v1(9) is essential for capturing the decadal tail of the TO/PC1 spectrum (Fig. 14). Its much longer time scale is consistent with its more poleward subtropical Z20 anomalies in both hemispheres, which after being induced by anomalies moving from the equator in the east central Pacific propagate more slowly toward the western boundary as might be expected of subtropical Rossby waves (Capotondi and Alexander 2001) at these higher latitudes (e.g., Chelton and Schlax 1996).

For longer optimization periods, the optimal structure projects more strongly upon other eigenmodes besides the three eigenmodes discussed above, although the 4-year eigenmode remains the largest component. In this case, predictable ENSO evolution over periods of a year and more may not be so clearly identified with a few eigenmodes.

5 Concluding remarks

Including thermocline depth and wind stress in the PS95 SST-only LIM has provided a foundation for empirical diagnosis of observed coupled atmosphere-ocean dynamics. We find that the SST pattern leading to optimal ENSO excitation, originally found by PS95, also corresponds to an east-central Pacific thermocline anomaly similar to an ENSO precursor found by many previous studies. Each contributes roughly equally to subsequent ENSO evolution and amplification, consistent with a large ENSO event possibly requiring both surface and subsurface processes as a trigger (e.g., McPhaden 1999; Vimont et al. 2003; Anderson 2007; Chang et al. 2007). Also, since anomalous thermocline evolution is primarily forced by wind stress, free ocean wave dynamics are likely of secondary importance to ENSO evolution, although rapid adjustment of the ocean on a time scale shorter than the 3 month data-averaging period (such as through Kelvin waves) can be a part of the wind stress forcing term.

Our results support the view of ENSO as an episodic and not periodic (e.g., Kessler 2002) effectively linear process. Essentially, a rapidly evolving westward-propagating eigenmode driven predominantly but not exclusively by shallow SST-atmosphere dynamics first destructively and then constructively interferes with a more slowly evolving eastward-propagating eigenmode driven by mixed SST/ocean dynamics. Since the slower mode has a damping time scale about half its period, it continues to predictably evolve for another quarter cycle; that is, an ENSO episode involves anomaly growth, decay, and overshoot, the last stage occurring only along the equator with a weak sign reversal of anomalous SST and zonally-integrated warm water volume. This picture is enriched but not fundamentally altered by the presence of the other eigenmodes. Some previous observational studies have also emphasized the combined contribution of similar tropical Pacific structures with 4- and 2-year periods (e.g., Jiang et al. 1995; White et al. 2003), although no decay time scales were determined. Also, eigenanalysis using the full ZC intermediate coupled model (Bejarano and Jin 2008) results in two leading eigenmodes, again with 4- and 2-year periods, that capture many but not all of the characteristics noted above, including a relative difference in the balance between processes forcing SST anomalies. Theoretical eigenanalyses focusing on the weakly unstable portion of a large parameter space have led to the view that ENSO variations (or “flavors”) are a consequence of changes in the background state altering stability characteristics of the two leading eigenmodes (e.g., Fedorov and Philander 2001; Bejarano and Jin 2008), and/or of nonlinear interaction between them (e.g., Jin et al. 1994). Our empirical results support the alternative view that both (stable) eigenmodes are always relevant since rapid observed ENSO amplification is a consequence of their modal interference. Hence, events differ because noise variations differently excite each eigenmode, although on average these two eigenmodes are preferentially excited with an initial (mostly opposing) relationship to each other. As a consequence, the relative importance of key physical processes changes from event to event, yet background state changes in stability properties may have only minor impact. Moreover, the leading optimal structure is the “best-case scenario”; sub-optimal ENSO events are also possible, including events where anomalies develop less rapidly but in different tropical regions (such as “Modoki” or central Pacific ENSOs), depending on the particular noise realization.

While over short time intervals, the ocean LIM yields minimal SST forecast improvement, over time intervals greater than several months it improves upon the SST-LIM in all measures, including long-lead tropical SST predictions, simulation of power spectra and lag co-variance statistics, and the observed relevance of leading optimal structures (Fig. 8). Much of this improvement may be due to a second optimal structure, allowing for significant SST anomaly growth over time intervals longer than a year, not present in the SST-LIM. Growth still occurs when initial destructive interference with the 4-year mode evolves into constructive interference, although this is now the result of many rather than a few eigenmodes. While the initial anomaly lies mostly within the SST field, subsequent ENSO development cannot be captured by SST alone since it is driven by persistent off-equatorial wind stress in the eastern Pacific that deepens (shallows) the equatorial thermocline to drive warm (cold) SST anomaly amplification (e.g., Wang et al. 2003).

The improved validation of the ocean LIM gives further support to the effectively linear, stochastically forced view of tropical ocean variability. It is particularly encouraging that, despite inherent limitations on ocean data assimilation, Z20 has improved the LIM’s representation of ENSO dynamics. While some remaining deficiencies of this LIM may represent truly predictable (i.e., slowly-varying) nonlinear internal ocean dynamics representable only by a high-dimensional fully nonlinear CGCM (Chen et al. 2004), we might hope for further improvement of the LIM by explicit representation of additional processes within its state vector, such as vertical stratification, currents, the extratropics, and/or atmospheric variability. For example, due to the dependence of current strength on thermocline slope, one clear deficiency of this study is that not explicitly including currents complicates distinguishing advective feedback from other thermocline feedbacks.

One key issue that remains is determining the correct seasonal dependence of both L and Q, given the seasonal dependence of C(0) and C(τ) as well as the tropical base state. We have followed the approach of some previous analyses (e.g., PS95, Penland 1996; Chang et al. 2007; Newman et al. 2009) which found that ENSO seasonality, including its phase locking, can be captured by a LIM where L is fixed but Q is seasonally-varying. As noted in the “Appendix”, we have also investigated potential seasonality of the dynamical operator but obtained results that were actually slightly worse. Whether this is because even 42 years of data is too short to correctly capture the seasonality of L, or because noise effects make the seasonality of the effectively linear operator L weaker than a linearization of the physical equations would lead us to expect, or because seasonality is better determined not piecewise by season (e.g., Johnson et al. 2000; Xue et al. 2000) but instead from the entire year in a cyclo-stationary LIM (e.g., Ortiz-Bevía 1997), remains to be determined.

“Metrics” of important phenomena within coupled GCMs generally do not directly measure dynamical processes of interest but rather their resulting variability. For example, Nino3 power spectra typically used to compare ENSO SST variability (e.g., Guilyardi et al. 2009) do not determine how, only whether, ENSO dynamics differ between models. By constructing even a rudimentary air-SST-subsurface ocean LIM, we go beyond simpler metrics to estimate the relative impact of different coupled feedback terms, and their importance both to the SST variance budget and to the evolution of ENSO events. This can serve as an important baseline for coupled dynamics as simulated by coupled GCMs. In particular, since the 4- and 2-year eigenmodes together provide a fairly compact representation of observed ENSO dynamics, it may prove useful to compare corresponding eigenmodes determined from the output of coupled GCM runs. Moreover, recall that in the extreme case where thermocline interactions were entirely decoupled from the surface in the LIM, SST spectra became more sharply peaked with reduced decadal variability, and SST variability shifted towards the central Pacific. Similarly, Newman et al. (2009) found that removing air-sea coupling in their tropical LIM led to a greatly weakened ENSO whose period was too short and whose maximum amplitude was too far west. These are all common CGCM failings, suggesting that climate LIMs might also be useful for diagnosing feedbacks leading to errors in comprehensive coupled climate models.


The authors thank Antonietta Capotondi, Cécile Penland, Amy Solomon, David Battisti and two anonymous reviewers for helpful comments. This work was partially supported by a grant from NOAA CLIVAR-Pacific.

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