Supplemental data for this article is available online at

Motivated by a recent active period of Tropical Instability Waves (TIWs) that followed the extreme 2015/2016 El Niño, we developed a stochastically forced linear model for TIWs with its damping rate modulated by the annual cycle and El Niño Southern Oscillation (ENSO). The model’s analytical and numerical solutions capture relatively well the observed Pacific TIWs amplitude variability dominated by annual and ENSO timescales. In particular, our model reproduces the seasonal increase in TIWs variance during summers and falls and the nonlinear relationship with the ENSO phase characterised by a suppression, respectively increase of TIW activity during El Niño, respectively La Niña. A substantial fraction of TIWs amplitude modulation emerges from the deterministic nonlinear interaction between ENSO and the annual cycle. This simple mathematical formulation allows capturing the nonlinear rectifications of TIWs activity onto the annual cycle and ENSO through, for instance, TIWs-induced ocean heat transport. Moreover, our approach serves as a general theoretical framework to quantify the deterministic variability in the covariance of climate transients owing to the combined modulation of the annual cycle and ENSO.

The El Niño–Southern Oscillation (ENSO) is the largest source of interannual climate variability in the world (McPhaden et al.

In addition, a number of studies indicates that higher frequency processes or stochastic atmospheric forcing, such as the Madden-Julian Oscillations (MJO) or Westerly Wind Bursts (WWB) in the Western Pacific, have significant effects on ENSO characteristics (e.g. Kleeman and Moore

On the other side of the Pacific basin, another intraseasonal oceanic process and its connection to ENSO and the annual cycle have been studied but have not received quite as much attention as the MJO/ENSO relationship. Tropical Instability Waves (TIWs) appear generally in the equatorial Eastern Pacific in June and persist until early in the following year as westward-propagating oscillations of the temperature front between the cold upwelled equatorial water and the warmer water to the north (Legeckis

In this study, we propose a simple linear model to study the covariance of the stochastically forced TIWs activity. Our approach differs from a dynamical theory and is rather based on simple parameterizations of the interactions between the dominant time scales. In particular, the inclusion of a time varying damping rate allows capturing the modulation of TIW variability by ENSO and the annual cycle. The rest of the paper will be structured as follows: in

We use the NOAA high-resolution blended analysis of daily SST over the period from January 1^{st} 1997 until October 20^{th} 2016. This product combines observations from different platforms (satellites, ships, buoys) on a regular 1/4° degree global grid (Reynolds et al. _{1}, as the simple equally spaced and weighted (but alternating sign) summation of SST anomalies at 8 fixed points along 0–6°N shown in

(_{1}. See the mathematical definition of _{1} and _{2} indices in Section 2.

It should be pointed out that choosing different numbers and locations of base points (such 4, 6 or 10) and shifting them together in the zonal direction within the TIW active region give similar results. To capture TIWs westward propagation and thus the main TIW period, we define _{2} in the same way as _{1}, except the base points are all shifted by a fixed distance representing a 90° degree zonal phase shift (i.e. roughly 6.25° in longitude).

Thus, the TIWs propagation characteristics (amplitude and phase) are contained in the quantity

This way of quantifying TIWs is original, yet extremely simple compared to prior methods. Historically, complex EOF (Lyman et al.

_{1} index is shown on _{1} shown in ^{th} mode of the Emprirical Orthogonal Functions decomposition, which accounts for TIW variability (_{1} index and the principal component from the EOF decomposition exhibit a significant energy in a broad 20–40 days band as noted by previous studies (_{1} timeseries also reveals a strong annual and interannual modulation of its enveloppe (i.e. TIWs activity/amplitude, cf. _{2} index is nearly a 9-day phase shift of _{1} and has similar spatio-temporal and spectral features (e.g. propagation, amplitude… Not shown).

A stochastically-forced model of climate variability, in which slow changes of climate (i.e. low frequency modulation) are explained as the integral response to continuous white noise forcing, has first been introduced by Hasselmann (_{0}_{A}_{E} = 1100 days, i.e. the ENSO main period (cf. Niño3 spectrum, i.e. red line on _{2} the phase for the interannual damping rate. _{A}_{1}_{0}, _{A}_{N}_{0} of 1/23 days^{−1}.

(_{1} index (1997–2015). Dots denote the 95% statistical significance level. Black filled triangles and circles show the positions of longitudinal nodes (positive for circles, i.e. red lines on _{1}, _{2}) phase diagram. _{1}-_{2} in black and cross correlation between _{1} and _{2} in red.

Several options are available to estimate the model’s parameters. First of all, since our model’s fornulation is similar to a Linear Inverse Model (for a general description, see Penland and Matrosova (_{0}, we choose to estimate _{A}_{N}_{1}-_{2}) and cross correlations between _{1} and _{2} to evaluate the TIWs propagation e-folding times (or growth rates) during different seasons of climatological years (for _{A}_{N}_{A}_{N}

^{nd} order TIW amplitude analytical solution inferred with different reasonable values of γ_{A}/γ_{0} and γ_{N}/γ_{0} (_{12}, black) and ensemble averaged of FFT single-sided amplitude spectra of the 2^{nd} order approximation of the TIW amplitude analytical solution for the different versions of the model (same colour code as ^{st} 1997-Dec 31^{st} 2015.

_{TOT}_{TOT}_{A}+δT_{N}_{A}/γ_{0} _{N}/γ_{0} ^{rd} order polynom, which accounts for the asymmetrical nature of this nonlinearity (i.e. the ENSO asymmetrical feedback on TIW amplitude; An (

Using a 3^{rd}-order polynomial fit, in the least square sense, we estimate _{N}Nino

As long as |_{A}_{0}_{N}_{0}_{N}^{3}_{0}^{2})) to approximate the analytical solution of TIWs amplitude |^{2} (i.e. the low-frequency modulation, diagnosed in the following as the 50-days running mean of the solution ^{2}) and

The 2^{nd} order approximation of our simple model’s solution exhibits similar spectral properties as the observed TIW amplitude (cf. ^{2}

As explained in _{C-MODE}_{AC}_{ENSO}

Additionally, we performed a parameter sensivity analysis presented in _{A}/γ_{0}| and |_{N}/γ_{0}| ratio values. This not only corroborates the parameters estimation presented in the previous section based on a physical analysis of the observed TIWs features (and not from an optimal tuning method through linear regressions), but also illustrates the robustness of our model and its analytical solution within the parameters domain.

Now that the TIW model can reproduce realistically the main spectral and temporal features of the observed TIW amplitude variability, we explore the sensitivity of the analytical and numerical solutions to different forcing terms. To do so, we compare the spectral and temporal properties of the TIW amplitude variability generated by a 50-members ensemble of three different versions of the linear model: the full model forced by both the annually and interannually TIW varying damping rates (noted TIWxTOT, i.e. all terms included, _{N} ≠_{A} ≠_{N}^{3} ≠_{N} =_{N}^{3} =_{A} =^{st} 1997 to August 31^{st} 2016 using a 4^{th} order Runge-Kutta method and the parameters values from

(_{1}-_{2}) autocorrelations (in black) and (_{1},_{2}) cross correlations (in red) logarithm for climatological years (i.e. neither El Niño nor La Niña years) between November and February. The black dotted line represents the best linear fit to these curves local maxima or in other words the estimation of TIWs propagation e-folding time during neutral years. Similarly, the green (magenta) dotted line represents the best linear estimation of TIWs propagation e-folding time for El Niño (La Niña) years. The red (blue) dotted line represents the best linear estimation of TIWs propagation e-folding time for extreme El Niño (La Niña) years.

^{nd} order approximation of the analytical solution (red line). The full model (TIWxTOT i.e. _{N} ≠_{A} ≠_{N}^{3} ≠^{−1} in the spectrum of both the observed and simulated TIW amplitude (as well as in the spectrum of Niño3 index, cf. red line in

The asymmetric feedback between the ENSO state and TIW amplitude first evidenced by An (

As evidenced on

Additionally, _{1} index (real part of Z in the nonlinear model). The richness in TIW spectrum (20–40 days^{−1}) only emerges when both the annually and inter-annually varying TIW damping rates are included in the model and cannot be accounted for without the nonlinear interaction between the Eastern Pacific Annual Cycle and ENSO. Although beyond the scope of this paper, the complex variability of the TIW high-frequency process arises partly from nonlinear dynamics in the Eastern Pacific. TIW amplitude modulation can potentially influence the local SST variability at different timescales, which in turn can alter the typical patterns of the annual and interannual atmospheric variability in this key region where sits the Inter-Tropical Convergence Zone.

In this paper, we developed a simple TIWs stochastic linear model forced by ENSO and the annual cycle to assess the deterministic low frequency modulation of TIWs variance. In particular, we study the effect of the Cold Tongue SST annual cycle and ENSO on the modulation of TIW activity. Both the analytical and numerical solutions of our simple model depict realistically the basic features of TIWs amplitude variability; namely the seasonal increase in TIWs variance during summers and falls and the nonlinear relationship with the ENSO phase characterised by a suppression, respectively increase of TIW activity during El Niño, respectively La Niña. The novelty result evidenced through this innovative framework is that a significant fraction (diagnosed by a significant spectral peak in the C-MODE frequency band as compared to a red noise process) of TIWs amplitude modulation emerges from the deterministic nonlinear interaction between ENSO and the annual cycle and is therefore predictable. This reflects a similar combination mode dynamic as the one described by Stuecker et al. (

Additionally, the observed interannual anomalies of the 50-days running TIW amplitude are significantly correlated with those from the model’s 50-member ensemble average (not shown) and the analytical solution (

^{nd} order analytical solution of TIW amplitude. All time series are normalised and smoothed using a 50-days running mean. ^{nd} order approximation of the analytical solution of TIW amplitude from different sensitivity experiments. Thick black lines for observation, red for the experiments using only a annually-varying TIW damping rate (TIWxAC), green for the experiments using only an interannually-varying TIW damping rate (TIWxNino) and magenta for the experiments using both (TIWxTOT). The dashed blue line indicates the averaged spectrum of 5000 generations of a random red noise process.

(^{nd} order approximation of TIW amplitude (red) using the following set of parameters: γ_{0} = 1/23 days^{−1}_{;} γ_{A}/γ_{0} = 0.3; γ_{N}/γ_{0} = −0.5 and _{N}^{3}^{nd} order approximation of the analytical solution of the interannual TIW amplitude and ENSO.

This formulation not only accounts for the response to the ENSO-Annual Cycle interaction but also for potentially significant higher order nonlinear interactions, i.e. following the notation introduced in

Furthermore, the simple theoretical framework presented here can be generalised to assess the modulation by the annual cycle, ENSO and their combined effect of any so-called state-dependent transients of the tropical climate system. A stochastically forced linear model of high frequency processes with periods much shorter than ENSO, with an annually and interannually varying damping rate, provides a good approximate to infer nonlinear deterministic rectifications through the covariance fields. A follow-up application of this new framework is the study of the deterministic modulation of MJO/WWB activity due to ENSO, the annual cycle and their nonlinear combination. This will be the topic of an oncoming research article.

Supplemental data for this article can be accessed

Approximation of the analytical solution of the low frequency amplitude modulation of TIWs using a Taylor series expansion:

-To stay in a stable damped regime, let

and

^{st}-order solution:

^{nd}-order solution:

…

- From (

- Eqs. (A1)+(A2):

- Let

- From (

- From (

- Eqs. (A4)+(A5)+(A6)+(A7):

- Since

- From (

- Eqs. (A9)+(A10):

- From (A12) and (A8)

- From (

- From (

- Eqs. (A13)+(A14)+(A15)+(A16):

- Since

- From (A18) and (C)

Combining (B), (C), (D) and (E) into (A)