Orbital modulation of ENSO seasonal phase locking

Abstract

Modern El Niño-Southern Oscillation (ENSO) events are characterized by their phase locking of variability to the seasonal cycle and tend to peak at the end of calendar year. Here, we show that in an idealized NCAR-CCSM3 simulation of the climate of the last 300,000 years, ENSO seasonal phase locking is shifted periodically following the precessional forcing: ENSO tends to peak in boreal winter when perihelion is near vernal equinox, but to peak in boreal summer when perihelion lies in between autumnal equinox and winter solstice. The mechanism for the change of ENSO’s phase locking is proposed to be caused by the change of seasonality of the growth rate, or the intensity of ocean–atmosphere feedbacks, of ENSO. It is found that the December peak of ‘winter ENSO’ is caused by the continuous growth of ENSO anomaly from June to November, while the May–June peak of ‘summer ENSO’ appears to be caused jointly by the seasonal shift of higher growth rate into spring and stronger stochastic noise towards the first half of the year. Furthermore, the change of the seasonal cycle of feedbacks is contributed predominantly by that of the thermodynamic damping. The summer peak of ENSO is proposed to be caused by the following mechanism. A perihelion in the late fall to early winter leads to a cooling of the surface eastern equatorial Pacific (EEP) due to reduced insolation in spring. This cooling, reinforced by an oceanic process, reduces the latent heat flux damping in spring, and therefore favors the growth of the eastern Pacific-like ENSO (as opposed to the central Pacific-like ENSO). This EEP cooling is also likely to generate more effective short wave-cloud-SST feedback and, in turn, increased instability. Ultimately, the weakened thermodynamic damping in spring, combined with relatively intensive stochastic forcing, benefits the subsequent summer peak of ENSO.

Appendices

Appendix 1

1.1 Monthly variance of AR1 model

Following Torrence and Webster (1998), for the autoregressive model ${{\text{X}}_{\text{t}}}={{{{\upalpha}}}_{\text{m}}}{{\text{X}}_{{\text{t}} - 1}}+{{\text{Z}}_{{\text{t}} - 1,{\text{m}}}}$, the monthly variance satisfies:

$$\sigma _{m}^{2}=\sigma _{{Zm}}^{2}+\alpha _{m}^{2}\sigma _{{m - 1}}^{2},$$

(1)

where ${{{\upsigma}}}_{{\text{m}}}^{2}$ is our target—the variance for calendar month m (equals month t − 1), ${{{{\upalpha}}}_{\text{m}}}$ is the AR1 coefficient for calendar month m, and ${{{\upsigma}}}_{{{\text{Zm}}}}^{2}$ is variance of white noise of calendar month m. To eliminate the cross term ${{{\upalpha}}}_{{\text{m}}}^{2}{{{\upsigma}}}_{{{\text{m}} - 1}}^{2}$, we use the basic assumption that the noise is uncorrelated with itself in a different month.

Use recursive substitution, (1) can be rewritten as:

$$\sigma _{m}^{2}=\sigma _{{Zm}}^{2}+\alpha _{m}^{2}\sigma _{{Zm - 1}}^{2}+\alpha _{m}^{2}\alpha _{{m - 1}}^{2}\sigma _{{Zm - 2}}^{2}+ \ldots +\sigma _{{Zm - 11}}^{2}\mathop \prod \limits_{{j=0}}^{{10}} \alpha _{{m - j}}^{2}+{A^2}\sigma _{{m - 12}}^{2},$$

where $A=\mathop \prod \nolimits_{{j=0}}^{{11}} {\alpha _j}$ is a constant equal to the product of all AR coefficients in 12 months. We also have ${{{\upsigma}}}_{{{\text{m}} - 12}}^{2}={{{\upsigma}}}_{{\text{m}}}^{2}$ as the recursion repeats itself. By changing m from 1 to 12, we have 12 such equations to solve the 12 unknowns, when A < 1 (convergence requirement) we have:

$$\sigma _{m}^{2}=\frac{1}{{1 - {A^2}}}\mathop \sum \limits_{{n=0}}^{{11}} \sigma _{{Zm - n}}^{2}\mathop \prod \limits_{{j=0}}^{{n - 1}} \alpha _{{m - j}}^{2},$$

(2)

where $\mathop \prod \nolimits_{{j=0}}^{{ - 1}} {{{\upalpha}}}_{{{\text{m}} - {\text{j}}}}^{2} \equiv 1$. The monthly variance of the model has a form of all the previous noise variances weighted by the AR coefficients, and can be calculated if seasonal cycle of AR coefficients and noises are available.

Appendix 2

2.1 Recharge oscillator model

The recharge oscillator model (Jin 1997) is also used as a statistical approach to study the seasonal phase locking of ENSO variance, and is compared with the AR1 model (Appendix 1). The equations that we performed numerical experiments are:

$${T_t} - {T_{t - 1}}={\alpha _m}{T_{t - 1}}+{\omega _m}{h_{t - 1}}+{{{{\upsigma}}}_{Zm}}{Z_{t - 1,~~m}},$$

(3)

$${h_t} - {h_{t - 1}}= - {R_m}{T_{t - 1}},$$

where T and h are Niño 3.4 SST anomaly and upper ocean heat content anomaly in the tropical western-central Pacific, respectively; ${{{{\upalpha}}}_{\text{m}}}$ is the growth rate of SSTA in calendar month m, ${\omega _m}$ measures the seasonally varying influence of thermocline anomalies on SSTA, and ${Z_m}$ is unit Gaussian white noise with ${{{\upsigma}}}_{{{\text{Zm}}}}^{2}$ controlling its seasonal variance; $- {R_m}$, on the other hand, describes the time scale for slow basin-wide geostrophic adjustment of the mean thermocline depth (or heat content), and forms a coupled system.

Those coefficients in Eq. (2) (${\alpha _m},~{\omega _m}~and~{R_m},~$all with seasonal cycle) can be calculated from 12 multilinear fits, given the time series of T and h. It should be noted that in this study sea surface height (5°S–5°N, 120°E–160°W) is used to approximate heat content. Figure 11a, b show the growth rate (${\alpha _m}$) and noise forcing (${{{{\upsigma}}}_{Zm}}$) for the two composite ENSO cases. The recharge oscillator model indicates that the ‘summer ENSO’ events have similarly higher growth rate during the first half of the year, and are less unstable during the second half of the year; the noise forcing for the two cases are comparable in phase and magnitude, both consistent with results from the AR1 model (Fig. 3a, b).

Fig. 11

Same as Fig. 3a–d, but for the results of the recharge oscillator model

We then perform numerical experiments using the recharge oscillator model, also with two different scenarios of ${\alpha _m}$ and ${{{{\upsigma}}}_{Zm}}$: (i) a seasonally varying growth rate but an annual mean noise level, and (ii) a seasonally varying noise level but an annual mean growth rate. It is similar to the method in AR1 approach (Sect. 4.1 and Fig. 3c, d), in order to quantify the influence of seasonal cycle of growth rate and noise forcing on ENSO variance. For each scenario, 2000 integrations are made, with random initial condition for T and h. The simulated ENSO seasonal phase locking (ensemble mean) is also in line with the AR1 results (Fig. 11c, d; Fig. 3c, d). The phase locking of ‘winter ENSO’ is determined by its growth rate, while the ‘summer ENSO’ is caused jointly by the seasonally varying noise and growth rate. The only noticeable difference in ENSO phase between the two models is that the noise-forced ENSO peaks (Fig. 11c, d, dashed line) are slightly delayed by a month in the recharge oscillator model, and a plausible explanation is because the slow thermocline process is involved.

Finally, we discuss the relationship between AR1 model and recharge oscillator model. The recharge oscillator has two purely damped free modes in the low frequency regime such as ENSO oscillation. In addition, the low frequency oscillation will not be felt substantially in one annual cycle. In other words, ${{{{\upomega}}}_{\text{m}}}$ and $- {{\text{R}}_{\text{m}}}$ in Eq. (2) are sufficiently small. So by nature, a recharge oscillator model is similar to an AR1 model, with respect to ENSO seasonal phase locking. This point is proved explicitly by comparing the analytical solution from AR1 model and numerical solutions from recharge oscillator model by applying different oscillation periods ranging from 5 to 400 months (Jin et al. 2017).

Appendix 3

3.1 Seasonal BJ index

In previous studies, the role of ocean–atmosphere feedback can be quantified in terms of the Bjerknes stability (BJ) index, which indicates the coupled ocean–atmosphere instability (Jin et al. 2006; Kim and Jin 2011a, b; Liu et al. 2014). The BJ index is applied to explain ENSO amplitude, though, regardless of seasonal difference. Inspired by the work of Stein et al. (2010), we here show the derivation the seasonal BJ index.

The BJ index is used to quantitatively evaluates the role of ocean–atmosphere feedbacks and damping effects. The calculation starts from the linearized SST tendency equation in the mixed layer:

$$\frac{{\partial T}}{{\partial t}}=Q - \frac{{\partial (\bar {u}T)}}{{\partial x}} - \frac{{\partial (\bar {v}T)}}{{\partial y}} - \frac{{\partial (\bar {w}T)}}{{\partial z}} - u\frac{{\partial \bar {T}}}{{\partial x}} - v\frac{{\partial \bar {T}}}{{\partial y}} - w\frac{{\partial \bar {T}}}{{\partial z}},$$

(4)

where the overbar denotes the climatology, T is sea temperature anomaly, Q is the total surface heat flux (including shortwave flux, longwave flux, sensible and latent heat fluxes) and (u, v, w) are ocean current velocity. We average Eq. (3) spatially over a rectangular box of the tropical central and eastern Pacific Ocean (5°S–5°N, 180°E–80°W) and integrate above the mixed-layer depth (~ 85 m) following Kim and Jin (2011a, b). Denoting the averaged variables in 〈 〉, it becomes:

$$\frac{{\partial \left\langle T \right\rangle}}{{\partial t}}=\left\langle Q \right\rangle - \left( {\frac{{{{(\bar {u}T)}_{EB}} - {{\left( {\bar {u}T} \right)}_{WB}}}}{{Lx}}+\frac{{{{(\bar {v}T)}_{NB}} - {{\left( {\bar {v}T} \right)}_{SB}}}}{{Ly}}} \right) - \left\langle {\frac{{\partial \bar {T}}}{{\partial x}}} \right\rangle \left\langle u \right\rangle - \left\langle {\frac{{\partial \bar {T}}}{{\partial z}}} \right\rangle \left\langle {H\left( {\bar {w}} \right)w} \right\rangle +\left\langle {\frac{{\bar {w}}}{{{H_1}}}} \right\rangle \left\langle {H(\bar {w}){T_{sub}}} \right\rangle ,$$

(5)

where

$$H(x)=\left\{ {\begin{array}{*{20}{c}} {1,~~\quad x \geq 0} \\ {0,~~\quad x<0} \end{array}} \right.,$$

is the step function. Assuming we have:

$$\frac{{\partial T}}{{\partial t}} \approx RT$$

then the BJ index (R) can be derived as:

$$R=\mathop { - {\alpha _s}}\limits_{{{\text{TD}}}} - \mathop {{\alpha _{MA}}}\limits_{{{\text{MA}}}} +\mathop {{\mu _a}{\beta _u}\left\langle { - {{\bar {T}}_x}} \right\rangle }\limits_{{{\text{ZA}}}} +\mathop {{\mu _a}{\beta _w}\left\langle { - {{\bar {T}}_z}} \right\rangle }\limits_{{{\text{EK}}}} +\mathop {{\mu _a}{\beta _h}\left\langle {\frac{{\bar {w}}}{{{H_1}}}} \right\rangle {a_h}}\limits_{{{\text{TH}}}} ,$$

(6)

while in Eq. (5) the coefficients (without the overbar) can be calculated using least-square regression method (Kim and Jin 2011a, b; see also; Liu et al. 2014 Methods for more details).

The BJ index represents the total feedback strength, which is a sum of five feedback terms, including two negative feedbacks, the surface heat flux feedback, or thermodynamic damping (TD), the mean advection feedback, or dynamic damping (MA), and three positive feedbacks, the zonal advection feedback (ZA), the Ekman local upwelling feedback (EK), and the thermocline feedback (TH). TD consists of shortwave feedback (${\alpha _{SW}}$), longwave feedback (${\alpha _{LW}}$), sensible heat flux feedback (${\alpha _{SHFLX}}$) and latent heat flux feedback (${\alpha _{LHFLX}}$). MA is contributed by zonal and meridional mean advection feedbacks. Each of the three dynamical positive feedbacks is the product of the background state (${\bar {T}_x},{\bar {T}_z}\;{\text{and}}\;\bar {w}$), the atmospheric response sensitivity (or surface wind stress sensitivity) to SST (${{{{\upmu}}}_a}$), and the oceanic response sensitivity to equatorial surface wind stress (${{{{\upbeta}}}_u}$, ${{{{\upbeta}}}_w}$ and ${{{{\upbeta}}}_h}$), reflecting the critical role of each element in the generation of the feedback.

We made two small improvements over the method of feedback calculation of Liu et al. (2014). One is the use of column integrated (within top 85 m) temperature and velocity for surface layer heat budget instead of the surface layer alone. This way, the overestimation of mean advection damping (which is largest at the surface) is reduced. The other is the use of sea surface height (SSH) to estimate thermocline depth instead of heat content which was poorly estimated by column-weighted sea temperature of only three layers (surface, 56 m and 149 m). The positive thermocline feedback is thus more realistic.

By definition the threshold for BJ is 0, and when BJ > 0 the SST anomaly grows. However, in the practical sense when the calculated BJ is > 0, it does not guarantee the ENSO anomaly to grow. This is because the absolute value of BJ index can vary with many factors, such as the choice of region (e.g., eastern Pacific or Niño 3.4), the processing method (e.g. band-pass filter). So in this study we focus more on the relative change rather than the absolute value of BJ index, i.e., a relatively higher/lower BJ index suggests a more unstable/stable state for ENSO anomaly. Nevertheless, we do a transformation to the BJ index definition equation and quantitatively compare it (R in Eq. 5) with AR1 growth rate. The BJ index satisfies

$$({T_{t+\tau }} - {T_t})/\tau \approx R*{T_t},$$

where $\tau$ is 1 month (1/12 year). Rearrange it

$${T_{t+\tau }} \approx (1+R*\tau )*{T_t}~,$$

so $1+R*\tau$ can be directly compared with the growth rate ${\alpha _m}$ in AR1 model.

We next consider the seasonal cycle of each feedback of the BJ index. In particular, the inclusion of seasonal cycle (otherwise it is a constant) of each coefficient will be based on the associated physical process (see below).

First, the heat flux damping effect is represented by the regression coefficient ${\alpha _s}$, which could vary seasonally. In the tropical eastern Pacific, the cloud radiation is closely associated with the seasonal meridional migration of ITCZ position (Timmermann et al. 2007; Stein et al. 2010), and it would be expected to influence the surface heat flux damping process. It is assumed that the surface heat flux anomaly is distributed throughout the mixed layer, so the mixed layer depth in the calculation should also vary seasonally to reflect orbital modulation of ocean stratification (Fig. 6).

Second, the mean advection damping term ${\alpha _{MA}}$ appears to be seasonal dependent. As the mean currents are calculated along the edge of equatorial Pacific rectangular box, intuitively they are controlled by the Pacific equatorial currents and counter-currents which all have seasonal changes (e.g. Philander et al. 1987) that could affect the damping effect.

The first seasonally modulated parameter is ${\mu _a}$, which represents the surface zonal wind response to the SST anomaly. Conceptual models such as Lindzen and Nigam (1987) and Gill (1980) have suggested that the response is sensitive to background climate, e.g. background wind and moisture fields, SST field via the Clausius–Clapeyron relationship and therefore the position of the ITCZ.

The adjustment of the ocean to a wind field disturbance ($\beta$ terms), when related to Ekman pumping (${\beta _w}$), direct forcing (${\beta _u}$, zonal current by zonal wind) or entrainment at thermocline (${a_h}$), is assumed seasonal dependent. In principle, they are associated with seasonal variation of mixed layer depth of the ocean.

The only coefficient that we consider to be associated with the relative slow oceanic process, thereby not tuned by seasonal modulation, is ${\beta _h}$. The assumption is that on interannual timescales, the west-east thermocline tilt in equatorial Pacific is determined largely through Sverdrup balance in response to the zonal wind stress anomaly (Burgers et al. 2005). The temporal scale to reach such quasi-balance is longer than a few months, therefore it is not seasonally dependent.

Additionally, all the background state terms (${\bar {T}_x}$, ${\bar {T}_z}$ and ${{\bar {w}}}$) are seasonally varying as they directly represent the effects of the seasonally varying background state on ENSO dynamics.

Appendix 4

4.1 Estimation of stochastic noises from monthly wind data

As an approximation, we derive the stochastic noise as the daily variance of zonal wind spatially averaged over the tropical western Pacific (TWP, 10°S–10°N, 130°E–170°W). The daily variance of stochastic forcing is calculated from the monthly output of surface zonal wind speed ${{\bar {u}}}$ and squared zonal wind speed $\overline {{{\text{uu}}}}$ (using 500 hPa meridional wind exhibits a similar result, not shown). The monthly mean of ${\text{u}}$ daily variance can be calculated as: $\mathop \sum \nolimits_{{{\text{n}}=1}}^{{\text{N}}} {\text{u}}_{{\text{n}}}^{2}/{\text{N}}=\overline {{{\text{uu}}}} - {\overline {{\text{u}}} ^2}$, where $\overline {{{\text{uu}}}} =\mathop \sum \nolimits_{{{\text{n}}=1}}^{{\text{N}}} {(\overline {{\text{u}}} +{{\text{u}}_{\text{n}}})^2}/{\text{N}}$, and $\overline {{\text{u}}} =\mathop \sum \nolimits_{{{\text{n}}=1}}^{{\text{N}}} (\overline {{\text{u}}} +{{\text{u}}_{\text{n}}})/{\text{N}}$ (we have $\mathop \sum \nolimits_{{{\text{n}}=1}}^{{\text{N}}} {{\text{u}}_{\text{n}}}=0$), where N is the number of days in 1 month and ${{\text{u}}_{\text{n}}}$ is the daily anomaly to month mean of zonal wind speed. It should be pointed out that this estimation is approximate in nature, because there is no unique way for the estimation. The western tropical Pacific selected here is meant to represent the atmospheric processes like MJO or westerly wind bursts (WWBs).

Appendix 5

5.1 The impact of acceleration

The effect of the accelerated boundary conditions (orbital parameters) on the climate evolution here is estimated first using a one-dimensional diffusive ocean model (Timm and Timmerman 2007). The simple diffusive process shows how the temperature profile responds to a surface forcing:

$$\frac{{\partial T(z)}}{{\partial t}}= - \frac{\partial }{{\partial z}}\left( {\kappa \left( z \right)\frac{{\partial T\left( z \right)}}{{\partial z}}} \right)+F(z=0,t).$$

For simplicity, the turbulent background diffusion for the global ocean is prescribed as $\kappa$ =6.2 × 10^− 5 m² s⁻¹ (Timm and Timmerman 2007). The periodic forcing F is prescribed at ocean surface as F(z = 0,t) = Asin($\omega$t), while A is set to 1 °C s⁻¹. The propagation of the forcing is shown in Fig. 12 for 100-fold acceleration $\omega$₁₀₀ = 2$\pi /210$ year⁻¹, 10-fold acceleration $\omega$₁₀ = 2$\pi /2100$ year⁻¹ and non-acceleration $\omega$₁ = 2$\pi /21000$ year⁻¹. The results for the three scenarios of forcing frequencies depict the in-phase temperature anomalies at the surface on a common time axis (stretched by a factor of 100 and 10 for (a) and (b)). The acceleration leads to dampened and delayed response in the deep ocean. The phase lag grows with depth, by 10-fold acceleration is about 600 accelerated years at 200 m depth, while by 100-fold acceleration is about 2000 accelerated years (2 ka). It is obvious the increased factor of acceleration could weaken the surface anomalies that propagate through deep ocean, thus twisting the phase relationship between the deep ocean and the surface forcing.

Fig. 12

Time sequence of the vertical temperature profile in a simple diffusion model under three acceleration scenarios: (upper panel) 100-fold acceleration, (middle panel) tenfold acceleration and (bottom panel) non-acceleration. Y-axis is depth (m) and x-axis is accelerated year, and the contour is the anomalous temperature (°C). Black lines connect the maximum signals in the ocean. The x-axis time scale of upper and middle panels is stretched by a factor of 100 and 10 to illustrate the phase shift in the deep ocean

Furthermore, we make a direct comparison between the last 210-year of the 3000-year accelerated (ORB, representing the last 300 ka) with the 21,000-year unaccelerated (TRACE-ORB, last 21 ka, Liu et al. 2014) orbital single-forcing simulation. Similar to results implied in the simple diffusive model, the intermediate/deep sea temperatures in CCSM3 are biased by the acceleration technique because of their slow response time. We focus on the orbital time scale temporal characteristics of the mean ocean stratification which is closely associated with the Ekman upwelling feedback. Figure 13c show the evolution of vertical temperature gradient in the tropical eastern Pacific (220–280°E, 5°S–5°N), and there are systematic differences between the two simulations. The unaccelerated simulation shows larger forced precessional signals (an amplitude of 0.4 °C vs. 0.2 °C) and leads slightly in phase. The longer adjustment time of subsurface ocean than surface ocean induces the delayed response (Timm and Timmerman 2007) which tend to cancel out the opposite effect below surface water between two successive precessional forcing cases. In summary, the change in ENSO variability and its phase locking feature could be partly influenced from weakened precessional signals in the subsurface ocean which is expected to dampen the response in ENSO feedbacks such as the Ekman upwelling feedback and thermocline feedback.

Fig. 13

a, b orbital parameters obliquity and precession of last 50 ka. c Vertical temperature difference (SST-t@50 m) of the EEP in ORB (red) and TRACE-ORB (grey)