General seasonal phase-locking of variance and persistence: application to tropical pacific, north pacific and global ocean

Abstract

A stochastic climate model is used to explain the major features of seasonal phase-locking of climate variability in general. The model is the classical damped persistence model, generalized with seasonal cycles in the growth rate and noise forcing. Our theory predicts distinct phase-locking features for different seasonal forcing. With seasonal growth rate, the forced variance lags the growth rate within a season, with the initial persistence in phase with the variance. With seasonal noise forcing, the variance also lags the noise forcing within a season, but the initial persistence lags the variance by a season. The theory is further applied successfully to the phase-locking of SST variability over the tropical Pacific, North Pacific and the world ocean. Overall, the variance and persistence is forced predominantly by the seasonal growth rate in the tropics with the variance and persistence in phase, but they are forced by the seasonal noise forcing in the mid-latitude with the variance and persistence in quadrature or even out phase. Our theory provides a general framework and a null hypothesis for the understanding of phase locking of climate variability in general.

Appendices

Appendix 1: Phase-locking in a recharge oscillator model

To study the phase-locking of oscillatory climate variability, such as ENSO, we use the recharge oscillator model (Jin et al. 2006; Stein et al. 2010; Levin and; McPhaden 2015):

$$\frac{{dT}}{{dt}} = - bT + \Omega h + \sigma _{N} N\left( t \right),\frac{{dh}}{{dt}} = - \Omega T$$

(14)

where − b is proportional to the growth rate, N(t) is the Gaussian white noise forcing of unit variance, and ${\sigma _N}^{2}$ is the variance of the noise forcing. With all coefficients constant, the eigenvalues can be derived by inserting $T\sim {e^{\lambda t}}~$ into the homogeneous equation of (A1) as

$$\lambda = \frac{{ - b}}{2} \pm i\sqrt {\Omega ^{2} - \left( {\frac{b}{2}} \right)^{2} } .$$

(15)

Therefore, the behaviors of the free modes have two frequency regimes. In the low frequency regime $\Omega < b/2$, the free modes are two purely damped modes and therefore the forced system should still behave similar to an AR1 model. In the high frequency regime $\Omega>b/2$, however, the free modes are damped oscillatory modes with a growthrate − b/2 and frequency $\sqrt {{\Omega^2} - {{\left( {\frac{b}{2}} \right)}^2}}$. The stochastically forced response in (A1) can no longer be described as purely damped persistence model, or AR1 model. The nature of the free modes suggest that the seasonal phase-locking should be similar to our damped persistence model for very low frequency oscillation, but will be different for frequencies oscillations.

Seasonal phase-locking can be studied by imposing a seasonal cycle in the damping rate as in Eq. (2). With the annual frequency $\omega$, we can expect three frequency regimes of phase-locking. First, in the low frequency regime $\Omega<\hbox{min} \left( {b/2,\Omega} \right),$ phase-locking should be similar to an AR1 model, because (1) the free modes are all damped and (2) the long oscillation time scale will not be felt substantially during an annual cycle. Second, in the high frequency regime $\Omega>\hbox{max} \left( {b/2,\Omega} \right)$, phase-locking should differ significantly from AR1 because the free modes exhibit oscillatory behavior and the oscillation is felt strongly within an annual cycle. Third, in the intermediate regime, $\hbox{min} \left( {b/2,\Omega} \right)<\Omega <\hbox{max} \left( {b/2,\Omega} \right)$, a mixture of transitional behavior may occur.

The three regimes of phase-locking can be seen in Fig. 10, which shows 6 examples that are solved numerically from Eq. (14) and Eq. (2), with the parameters as $b=0.1,~~\Omega=2 \pi /12,~~A=1.25.~$ As expected, for primary oscillation in the low frequency regime ($2 \pi /\Omega=400{\text{~months}},$ and 200 months), phase-locking exhibits almost the same behavior as that in the AR1 model studied before, with the phases of the variance and lag-1 persistence almost in phase, peaking about 2.5 months after the maximum growth rate (Fig. 10a, b). When the primary frequency increases to the intermediate regime, $\Omega<\Omega <b/2$, phase-locking exhibits a transition behavior from that almost of AR1 behavior at $2 \pi /\Omega=40{\text{~months}}$ to a very different regime at $2 \pi /\Omega=20{\text{~months}}$ where the peak variance and lag1-1 persistence are shifted earlier by 3 months and 5 months, respectively. The real world ENSO case therefore corresponds to the lower frequency case is similar to the case of $2 \pi /\Omega=40{\text{~months}}$ and therefore can be reasonably represented in the AR1 model. When the frequency further increases to higher than annual $\Omega>\omega$ ($2 \pi /\Omega=10{\text{~months~and~}}5{\text{~months}})$, the phase-locking changes again: while the variance still lags the maximum growthrate by a season, the lag-1 persistence peaks earlier than growth rate and exhibits clear cycle of the primary frequency within a year.

Fig. 10

Seasonal cycle of a variance and b lag-1 autocorrelation in a recharged oscillator model forced by a white noise modulated by the seasonal cycle of growth rate (brown dash). There are 6 cases with the delayed oscillation period ranging from 5 to 400 months, with each variance or correlation normalized by its maximum. The annual mean damping rate normalized by the annual cycle frequency is chosen as ${b_0}/\omega \approx 0.2$

Appendix 2: Variance and persistence in the seasonal stochastic climate model

The general solution of SST variability in the stochastic climate model Eq. (2) as

$$T=\mathop \int \limits_{{ - \infty }}^{t} {e^{ - \mathop \int \limits_{s}^{t} b\left( u \right)du}}N\left( s \right)ds.$$

with seasonal cycles in the growth rate and the noise forcing as in Eq. (3) and (4) ($\beta$ means the phase difference between the growth rate and noise forcing), the lagged covariance function with forecast lead $\tau>0$ can be derived as

$$\begin{aligned} C\left( {t,\tau } \right) & =T\left( t \right),T\left( {t+\tau } \right)~=~\left\langle {\mathop \int \limits_{{ - \infty }}^{t} {e^{ - \mathop \int \limits_{s}^{t} b\left( u \right)du}}N\left( s \right)ds,~\mathop \int \limits_{{ - \infty }}^{{t+\tau }} {e^{ - \mathop \int \limits_{q}^{{t+\tau }} b\left( v \right)dv}}N\left( q \right)dq} \right\rangle \\ & =\mathop \int \limits_{{ - \infty }}^{t} {e^{ - \mathop \int \limits_{s}^{t} b\left( u \right)du}}~\mathop \int \limits_{{ - \infty }}^{{t+\tau }} {e^{ - \mathop \int \limits_{q}^{{t+\tau }} b\left( v \right)dv}}\left\langle {N\left( s \right),N(q)} \right\rangle dsdq \\ & =\sigma _{0}^{2}\mathop \int \limits_{{ - \infty }}^{t} {e^{{b_0}\left( {2s - 2t - \tau } \right) - \frac{{{b_0}A}}{\omega }[2\sin \left( {\omega s} \right) - \sin \left( {\omega t} \right) - \sin \left( {\omega \left( {t+\tau } \right)} \right]}}\left[ {1+Dcos\left( {\omega s - \beta } \right)} \right]ds~ \\ & =\sigma _{0}^{2}{e^{ - B\omega \left( {t+\frac{\tau }{2}} \right)+\frac{{{A_b}}}{2}\left\{ {sin\left( {\omega t} \right)+{\text{sin}}\left[ {\omega \left( {t+\tau } \right)} \right]} \right\}}}\mathop \int \limits_{{ - \infty }}^{t} {e^{B\omega s - {A_b}{\text{sin}}\left( {\omega s} \right)}}\left[ {1+Dcos\left( {\omega s - \beta } \right)} \right]ds~ \\ \end{aligned}$$

(16)

where the nondimensional annualized damping rate is defined as

$$B=\frac{{2{b_0}}}{\omega }.$$

(17)

and the corresponding amplitude of the seasonal cycle of the growth rate is

$${A_b}=AB.$$

(18)

$${D_b}=DB$$

(19)

The SST variance is therefore

$${\sigma ^2}\left( t \right) \equiv C\left( {t,0} \right)=\sigma _{0}^{2}{e^{ - B\omega t+{A_b}\sin \left( {\omega t} \right)}}\mathop \int \limits_{{ - \infty }}^{t} {e^{B\omega s - {A_b}\sin \left( {\omega s} \right)}}\left[ {1+Dcos\left( {\omega s - \beta } \right)} \right]ds$$

(20)

For ${A_b} \ll 1$.

We have the leading order

$$\begin{aligned} {e^{ - {A_b}{\text{sin}}\left( {\omega s} \right)}} & \approx 1 - {A_b}{\text{sin}}\left( {\omega s} \right) \\ {\sigma ^2}\left( t \right) & \approx \sigma _{0}^{2}{e^{ - B\omega t}}(1+{A_b}\sin \left( {\omega t} \right))~\mathop \int \limits_{{ - \infty }}^{t} {e^{B\omega s}}(1 - {A_b}\sin \left( {\omega s} \right))\left[ {1+Dcos\left( {\omega s - \beta } \right)} \right]ds \\ & =\frac{{\sigma _{0}^{2}}}{{2{b_0}}}\left[ {1+\frac{{{A_b}}}{{\sqrt {1+{B^2}} }}\cos \left( {\omega t - \theta } \right)+\frac{{{D_b}}}{{\sqrt {1+{B^2}} }}\cos \left( {\omega t - \theta - \beta } \right)} \right] \\ \end{aligned}$$

where

$$\begin{aligned} ~\cos \theta & =\frac{B}{{\sqrt {1+{B^2}} }},sin\theta =\frac{1}{{\sqrt {1+{B^2}} }} \\ & =\frac{{\sigma _{0}^{2}}}{{2{b_0}}}\left[ {1+\frac{{{A_b}+{D_b}cos\beta }}{{\sqrt {1+{B^2}} }}cos\left( {\omega t - \theta } \right)+\frac{{{D_b}sin\beta }}{{\sqrt {1+{B^2}} }}sin\left( {\omega t - \theta } \right)} \right] \\ & =\frac{{\sigma _{0}^{2}}}{{2{b_0}}}\left[ {1+\sqrt {\frac{{{{({A_b}+{D_b}cos\beta )}^2}+{{({D_b}sin\beta )}^2}}}{{1+{B^2}}}} cos\left( {\omega t - \theta - {\theta ^\prime }} \right)} \right] \\ \end{aligned}$$

(21)

$$\cos {\theta ^\prime }=\frac{{{A_b}+{D_b}cos\beta }}{{\sqrt {{{({A_b}+{D_b}cos\beta )}^2}+{{({D_b}sin\beta )}^2}} }}$$

(22)

$$\sin {\theta ^\prime }=\frac{{{D_b}sin\beta }}{{\sqrt {{{({A_b}+{D_b}cos\beta )}^2}+{{({D_b}sin\beta )}^2}} }}$$

(23)

For the case of noise forcing dominates, that is A = 0, and $\beta =0$, then

$${\sigma ^2}\left( t \right) \approx \frac{{\sigma _{0}^{2}}}{{2{b_0}}}\left[ {1+\frac{{{D_b}}}{{\sqrt {1+{B^2}} }}\cos \left( {\omega t - \theta } \right)} \right]$$

The lagged correlation can be written as

$$\begin{aligned} r(t,\tau ) & =~\frac{{C(t,\tau ~)}}{{\sigma (t)\sigma (t+\tau )}}~=~{e^{ - \frac{B}{2}\omega \tau - \frac{{B{A_b}}}{2}[\sin \left( {\omega t} \right) - \sin \left( {\omega \left( {t+\tau } \right))} \right]}}\frac{{\sigma (t)}}{{\sigma (t+\tau )}} \\ & \approx {e^{ - {b_0}\tau }}\left\{ {1 - \frac{{{b_0}A}}{\omega }\left[ {\sin \left( {\omega t} \right) - \sin (\omega t+\omega \tau )} \right]} \right\}\sqrt {\frac{{1+\frac{{AB}}{{\sqrt {1+{B^2}} }}\cos \left( {\omega t - \theta } \right)+\frac{{DB}}{{\sqrt {1+{B^2}} }}~\cos \left( {\omega t - \theta - \beta } \right)~~~~}}{{1+\frac{{AB}}{{\sqrt {1+{B^2}} }}\cos \left( {\omega (t+\tau ) - \theta } \right)+\frac{{DB}}{{\sqrt {1+{B^2}} }}~\cos \left( {\omega (t+\tau ) - \theta - \beta } \right)~~~~}}} \\ \frac{B}{{\sqrt {1+{B^2}} }} & =\varepsilon ,~\frac{{2{b_0}A}}{\omega }={\varepsilon _0} \\ & \approx {e^{ - {b_0}\tau }}\left\{ {1 - \frac{1}{2}{\varepsilon _0}\left[ {\sin \left( {\omega t} \right) - \sin \left( {\omega t+\omega \tau } \right)} \right]} \right\}\sqrt {\frac{{1+\varepsilon A\cos (\omega t - \theta )+\varepsilon D\cos \left( {\omega t - \theta - \beta } \right)}}{{1+\varepsilon A\cos \left( {\omega \left( {t+\tau } \right) - \theta } \right)+\varepsilon D\cos \left( {\omega (t+\tau ) - \theta - \beta } \right)}}} \\ & \approx {e^{ - {b_0}\tau }}\left\{ {1 - \frac{1}{2}{\varepsilon _0}\left[ {\sin \left( {\omega t} \right) - \sin \left( {\omega t+\omega \tau } \right)} \right]} \right\}\left\{ {1+\frac{\varepsilon }{2}A\left[ {\cos \left( {\omega t - \theta } \right) - \cos (\omega \left( {t+\tau } \right) - \theta )} \right]+\frac{\varepsilon }{2}D\left[ {\cos \left( {\omega t - \theta - \beta } \right) - \cos (\omega \left( {t+\tau } \right) - \theta - \beta )} \right]} \right\} \\ & ={e^{ - {b_0}\tau }}\left\{ {1+{\varepsilon _0}\left[ {\sin \frac{{\omega \tau }}{2}\cos \left( {\omega t+\frac{{\omega \tau }}{2}} \right)} \right]} \right\}\left[ {1+\varepsilon A\sin \frac{{\omega \tau }}{2}sin\left( {\omega \left( {t+\frac{\tau }{2}} \right) - \theta } \right)+\varepsilon D\sin \frac{{\omega \tau }}{2}\sin \left( {\omega \left( {t+\frac{\tau }{2}} \right) - \theta - \beta } \right)} \right] \\ & \approx {e^{ - {b_0}\tau }}\left\{ {1+{\varepsilon _0}\left[ {\sin \frac{{\omega \tau }}{2}\cos \left( {\omega t+\frac{{\omega \tau }}{2}} \right)} \right]+\varepsilon A\sin \frac{{\omega \tau }}{2}\sin \left( {\omega \left( {t+\frac{\tau }{2}} \right) - \theta } \right)+\varepsilon D\sin \frac{{\omega \tau }}{2}\sin \left( {\omega \left( {t+\frac{\tau }{2}} \right) - \theta - \beta } \right)} \right\} \\ \end{aligned}$$

(24)

For the terms of

$$\begin{aligned} & {\varepsilon _0}sin\frac{{\omega \tau }}{2}cos\left( {\omega t+\frac{{\omega \tau }}{2}} \right)+\varepsilon Asin\frac{{\omega \tau }}{2}sin\left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right) \\ & \quad ={\text{AB}}sin\frac{{\omega \tau }}{2}cos\left( {\omega t+\frac{{\omega \tau }}{2}} \right)+\frac{{AB}}{{\sqrt {1+{B^2}} }}sin\frac{{\omega \tau }}{2}sin\left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right) \\ & \quad ={\text{AB}}sin\frac{{\omega \tau }}{2}\left[ {cos\left( {\omega t+\frac{{\omega \tau }}{2}} \right)+\frac{{sin\left( {\omega t+\frac{{\omega \tau }}{2}} \right)cos\theta - cos\left( {\omega t+\frac{{\omega \tau }}{2}} \right)sin\theta }}{{\sqrt {1+{B^2}} }}} \right] \\ & \quad ={\text{AB}}sin\frac{{\omega \tau }}{2}\left[ {\left( {1 - \frac{1}{{1+{B^2}}}} \right)cos\left( {\omega t+\frac{{\omega \tau }}{2}} \right)+\frac{{B \times sin\left( {\omega t+\frac{{\omega \tau }}{2}} \right)}}{{1+{B^2}}}} \right] \\ & \quad =\frac{{A{B^2}}}{{\sqrt {1+{B^2}} }}sin\frac{{\omega \tau }}{2}\left[ {\frac{B}{{\sqrt {1+{B^2}} }}cos\left( {\omega t+\frac{{\omega \tau }}{2}} \right)+\frac{1}{{\sqrt {1+{B^2}} }}sin\left( {\omega t+\frac{{\omega \tau }}{2}} \right)} \right] \\ & \quad =\frac{{A{B^2}}}{{\sqrt {1+{B^2}} }}sin\frac{{\omega \tau }}{2}cos\left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right) \\ \end{aligned}$$

Therefore,

$$\begin{aligned} & r(t,\tau ) \approx {e^{ - {b_0}\tau }}\left[ {1+\frac{{A{B^2}}}{{\sqrt {1+{B^2}} }}\sin \frac{{\omega \tau }}{2}\cos \left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right)+\frac{{BD}}{{\sqrt {1+{B^2}} }}\sin \frac{{\omega \tau }}{2}\sin \left( {\omega t+\frac{{\omega \tau }}{2} - \theta - \beta } \right)} \right] \\ & \quad ={e^{ - {b_0}\tau }}\left\{ {1+\frac{B}{{\sqrt {1+{B^2}} }}\sin \frac{{\omega \tau }}{2}\left[ {AB\cos \left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right)+D\sin \left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right)cos\beta - Dcos\left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right)sin\beta } \right]} \right\} \\ & \quad ={e^{ - {b_0}\tau }}\left\{ {1+\frac{B}{{\sqrt {1+{B^2}} }}\sin \frac{{\omega \tau }}{2}\left[ {(AB - Dsin\beta )\cos \left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right)+Dcos\beta \sin \left( {\omega t+\frac{{\omega \tau }}{2} - \theta } \right)} \right]} \right\} \\ & \quad ={e^{ - {b_0}\tau }}\left\{ {1+\frac{B}{{\sqrt {1+{B^2}} }}\sin \frac{{\omega \tau }}{2}\left[ {\sqrt {{{(AB - Dsin\beta )}^2}+{{\left( {Dcos\beta } \right)}^2}} \cos \left( {\omega t+\frac{{\omega \tau }}{2} - \theta - {\theta ^{\prime \prime }}} \right)} \right]} \right\} \\ \end{aligned}$$

(25)

$$\cos {\theta ^{\prime \prime }}=\frac{{AB - Dsin\beta }}{{\sqrt {{{(AB - Dsin\beta )}^2}+{{\left( {Dcos\beta } \right)}^2}} }}$$

(26)

$$\sin {\theta ^{\prime \prime }}=\frac{{Dcos\beta }}{{\sqrt {{{(AB - Dsin\beta )}^2}+{{\left( {Dcos\beta } \right)}^2}} }}$$

(27)

For the case of growth rate dominates, that is D = 0, according to Eqs. 21 and 25, then

$${\sigma ^2}\left( t \right) \approx \frac{{\sigma _{0}^{2}}}{{2{b_0}}}\left[ {1+\frac{{{A_b}}}{{\sqrt {1+{B^2}} }}\cos \left( {\omega t - \theta } \right)} \right]$$

$$r(t,\tau ) \approx {e^{ - {b_0}\tau }}\left[ {1+\frac{{A{B^2}}}{{\sqrt {1+{B^2}} }}sin\frac{{\omega \tau }}{2}{\text{cos}}(\omega t+\frac{{\omega \tau }}{2} - \theta )} \right]$$

For the case of noise forcing dominates, that is A = 0 and $\beta =0$, according to Eqs. 21 and 25, then

$${\sigma ^2}\left( t \right) \approx \frac{{\sigma _{0}^{2}}}{{2{b_0}}}\left[ {1+\frac{{{D_b}}}{{\sqrt {1+{B^2}} }}\cos \left( {\omega t - \theta } \right)} \right]$$

$$r(t,\tau ) \approx {e^{ - {b_0}\tau }}\left[ {1+\frac{{A{B^2}}}{{\sqrt {1+{B^2}} }}sin\frac{{\omega \tau }}{2}{\text{cos}}(\omega t+\frac{{\omega \tau }}{2} - \frac{\pi }{2} - \theta )} \right].$$