An empirical model of tropical ocean dynamics

Abstract

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.

Appendix: Construction and validation of the LIM

In any multidimensional statistically stationary system with components x _i , one may define a time lag covariance matrix C(τ) with elements $Cij\left( \tau \right) = \left\langle {x_{i} (t + \tau )x(t)} \right\rangle$, where angle brackets denote a long term average. In linear inverse modeling, one assumes that the system satisfies C(τ) = G(τ)C(0), where importantly G(τ) = exp(Lτ) and L is a constant matrix, which follows from (2). One then uses this relationship to estimate L from observational estimates of C(0) and C(τ ₀ ) at some lag τ_o. For the statistics of this system to be stationary, L must be dissipative, i.e its eigenvalues must have negative real parts. In a forecasting context, G(τ)x(t) represents the “best” forecast (in a least squares sense) of x(t + τ) given x(t). Note that unlike multiple linear regression, determination of G at one lag τ_o identically gives G at all other lags. Also, statistics of the noise forcing, not just the error covariance, are determined by LIM since the positive-definite noise covariance matrix ${\mathbf{Q}} = \left\langle {\xi \xi^{T} } \right\rangle dt$ is determined from a Fluctuation-Dissipation relationship (3), given the observed C(0) and L.

A training lag of τ_ο = 3 months was used to determine L. The EOF truncations (Sect. 2) and training lag were chosen to maximize the LIM’s cross-validated forecast skill for leads up to 18 months, while avoiding the Nyquist problem for L (PS95) that inhibits analysis of interactions amongst the model variables. In no other respect do these choices qualitatively affect the points made in this paper. Estimates of L and of forecast skill were cross-validated by sub-sampling the data record by sequentially removing one five-year period, computing L for the remaining years, and then generating forecasts for the independent years. This procedure was repeated for the entire period. Forecast skill is then determined by comparing the local anomaly correlation between the cross-validated model predictions and gridded untruncated verifications.

Explicitly including both τ_x and Z ₂₀ in the LIM state vector increases skill of 9 (Figs. 12a, b) and 18 (Fig. 12c, d) month T _o forecasts. Skill improvement is mostly due to the inclusion of Z ₂₀ rather than τ_x, confirming that the Z ₂₀ data provides useful information. In addition, the LIM makes Z ₂₀ forecasts whose skill is significantly better than persistence (not shown). The difference in SST skill between the ocean and SST-LIMs generally increases with forecast lead because the ocean LIM captures the slower Z ₂₀ evolution. For example, a second set of “fixed-Z ₂₀” ocean LIM forecasts in which the initial Z ₂₀ anomaly is persisted throughout the forecast period (i.e., Z ₂₀(t) = Z ₂₀(0); not shown) has greatly reduced 18-month forecast skill.

Fig. 12

Cross-validated forecast skill (1959–2000) of T _O for forecast leads of 9 and 18 months, for the LIM (top) and the SST13-LIM (bottom)

We test the validity of the linear approximation with a “tau-test” (PS95). For example, since (2) implies that C(τ) = G(τ) C(0), the LIM should be able to reproduce observed lag-covariance statistics at much longer lags than the 3 month lag on which the LIM was trained (e.g., C(18) = [G(3)]⁶ C(0)). Figure 13 compares the observed and predicted lag-autocovariances of T _O for lags of 9, 18, and 36 months, using both the ocean LIM and the SST13-LIM. The SST13-LIM does a reasonably good job capturing the main aspects of the lag-autocovariance pattern, but for lags less than about a year it tends to overestimate persistence especially along the equator and for longer lags it errs in the amplitude of the negative lag-autocovariance. The ocean LIM improves upon these deficiencies, as well as reproducing observed lag-covariance for Z ₂₀ over the same lags (not shown).

Fig. 13

Observed (top panels), LIM (middle panels) and SST13-LIM (bottom panels) T _O lag-covariance. (left) 9-month lag-covariance; (center) 18-month lag-covariance; (right) 36-month lag-covariance. Note that the observed lag-covariances are based on the full (that is, not truncated in the EOF basis) gridded anomaly fields. Contour interval is 0.04 K²

A complementary test of linearity is to compare observed and LIM-predicted power spectra, by integrating (2) for 42,000 years using the method described in Penland and Matrosova (1994) and Newman et al. (2009) and then collecting statistics. The white noise forcing is determined from the noise covariance matrix Q determined as a residual from (3). Q should be positive-definite but determined this way it is only guaranteed to be symmetric. Ensuring positive-definiteness in the manner of Penland and Matrosova (1994), by rescaling the noise due to one small negative eigenvalue of Q that accounted for less than 0.25% of the trace of Q, resulted in an almost negligible impact on C(0). The resulting model “data” is separated into 1,000 42-year time series. The observed spectra and the ensemble mean of the model spectra for the three leading PCs of T _O and Z ₂₀ are shown in Figs. 14 and 15, respectively. The corresponding EOF pattern for each spectrum is shown in the inset panels. The gray shading shows the 95% confidence intervals of these spectra, estimated using the 1,000 model realizations.

Fig. 14

Power spectra for the three leading SST (T _O ) PCs (red lines), compared to that predicted by the LIM (blue lines). Gray shading represents the 95% confidence interval determined from a 1,000-member ensemble of 42 year LIM model runs (see text for further details). In these log(frequency) versus power times angular frequency (ω) plots, the area under any portion of the curve is equal to the variance within that frequency band. Note that displaying power times frequency slightly shifts the power spectral density peak centered at a period of 4.5 years to a variance peak centered at a period of 3.5 years. Insets in each panel show the corresponding EOF and the variance explained by that pattern

Fig. 15

Same as Fig. 14 but for the three leading 20°C isotherm depth (Z ₂₀) PCs. Note that there is no SST-LIM equivalent

The LIM reproduces the main features of the observed power spectra for the leading PCs of each variable (including τ_x, not shown). Obviously, the mean LIM spectra are much smoother than observed, due to the relatively few degrees of freedom in the truncated EOF space. On the other hand, the irregularity of the observed spectra is at least partly due to sampling, as indicated by the confidence intervals, which show how much variation in the spectra could occur simply from different realizations of noise.

Since C(0) and C(τ) have seasonal dependence, both Johnson et al. 2000 and Xue et al. 2000 suggested that L should also be considered to be seasonally-varying. They constructed “Markov models” in which they determined G(τ_ο) for each season (but not L) so that forecasts are made by an appropriate product of each G. On the other hand, PS95 and Penland (1996) argued that the observed seasonality of ENSO, including its phase locking, can be captured with a fixed L with seasonally varying Q. Newman et al. (2009) found that a tropical atmosphere-SST LIM constructed using weekly data had a poorer representation of coupled ENSO dynamics when segregated by season, although the internal subseasonal atmospheric dynamics were slightly improved. They also found pronounced seasonality of the noise, as did Penland (1996) and Chang et al. (2007). Similar to Penland, we found that the seasonal dependence of the tests above can be generally captured by assuming fixed L but with seasonally varying Q. As in Newman et al. (2009), we also found that seasonal LIMs did poorer in these tests compared to those using a fixed L.